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An optimal strategy to solve the Prisoner's Dilemma Evolution of cooperation in stochastic games Christian Hilbe, Štěpán Šimsa, … Martin A. Nowak Self-regulation versus social influence for promoting cooperation on networks Dario Madeo & Chiara Mocenni Evolution of cooperation and consistent personalities in public goods games Mohammad Salahshour Cost-effective external interference for promoting the evolution of cooperation The Anh Han & Long Tran-Thanh The conditional defector strategies can violate the most crucial supporting mechanisms of cooperation Ahmed M. Ibrahim Environment-based preference selection promotes cooperation in spatial prisoner's dilemma game Yu'e Wu, Shuhua Zhang & Zhipeng Zhang Temporal assortment of cooperators in the spatial prisoner's dilemma Tim Johnson & Oleg Smirnov The influence of heterogeneous learning ability on the evolution of cooperation Xiaogang Li, Yini Geng, … Lei Shi Collapse and rescue of cooperation in evolving dynamic networks Erol Akçay Alessandro Bravetti1 na1 & Pablo Padilla1,2 na1 Scientific Reports volume 8, Article number: 1948 (2018) Cite this article Cooperation is a central mechanism for evolution. It consists of an individual paying a cost in order to benefit another individual. However, natural selection describes individuals as being selfish and in competition among themselves. Therefore explaining the origin of cooperation within the context of natural selection is a problem that has been puzzling researchers for a long time. In the paradigmatic case of the Prisoner's Dilemma (PD), several schemes for the evolution of cooperation have been proposed. Here we introduce an extension of the Replicator Equation (RE), called the Optimal Replicator Equation (ORE), motivated by the fact that evolution acts not only at the level of individuals of a population, but also among competing populations, and we show that this new model for natural selection directly leads to a simple and natural rule for the emergence of cooperation in the most basic version of the PD. Contrary to common belief, our results reveal that cooperation can emerge among selfish individuals because of selfishness itself: if the final reward for being part of a society is sufficiently appealing, players spontaneously decide to cooperate. Cooperation is so important that it has been suggested as a fundamental principle of evolution, besides reproduction, mutation and selection1,2,3,4,5,6,7,8,9,10,11,12,13. However, the situation regarding the underlying mechanisms responsible for the emergence of cooperation among individuals who try to maximize their own fitness and who are competing among each other is not clear. To try to solve such complicated puzzle, several mechanisms have been proposed during the last decades, including kin selection1,6, direct reciprocity2,3,14, indirect reciprocity15,16, network reciprocity17,18, group selection9,19, green beards5,20, optional participation21,22, punishment and reward23,24, pre-commitments25,26,27 and others28,29,30,31. All these situations reflect some specific important aspects of real social and biological interactions. However, none of them can really provide a solution to the most basic version of the paradigmatic example of the Prisoner's Dilemma (PD): here one imagines that two people that are suspected of having committed a joint crime are caught by the police and confined into different rooms, without the possibility to communicate. Each of them is offered the possibility to confess the crime and defect his partner in exchange for a reduced sentence. If only one defects, the other will get the full sentence. If they defect, both will have the sentence reduced. If they cooperate between themselves and do not confess, then they will immediately be freed. The situation can be exemplified in the following payoff matrix $$\begin{array}{cc} & \begin{array}{cc}C & D\end{array}\\ \begin{array}{c}C\\ D\end{array} & (\begin{array}{ll}R & S\\ T & P\end{array})\end{array},$$ where C stands for "cooperation", D for "defection", and the entries denote the payoff for the row player. Thus, if they both cooperate, they get R points each (the "reward" for cooperation). If only one cooperates, then he gets S points (the "sucker's" payoff), while the defector gets T (the "temptation" to defect). If they both defect, they get P points each (the "punishment" for defection). As it is well-known, in the PD the payoffs satisfy T > R > P > S. In this situation the best option for each player is to defect, no matter what the other player does32. Thus both players defect and get P points each. However, this is less than the R points they would get if they had collaborated. This is the essence of the PD: mutual cooperation leads to a higher payoff than mutual defection, but it is not a "safe" strategy, exposing the player to exploitation by a defector, and therefore each player, in an attempt to maximize his own payoff, chooses to defect. Thus apparently there is no room for cooperation between such rational agents. We remark also that in this one-shot formulation of the PD, the game is not repeated (and thus direct and indirect reciprocity do not apply), there is no link between the two prisoners (neither a genetic one as in kin selection, nor a "social" one, as in the cases of network reciprocity and group selection), nor is there a special tag that the prisoners can use (thus ruling out green beards and tag-based donation) and finally, the two prisoners cannot decide whether to play or not (and hence optional participation cannot be used). Therefore our best candidate mechanisms for the emergence of a cooperative strategy between the two prisoners do not apply in this paradigmatic example and we are left with the fundamental question: is there any other reason that can lead the two individuals to cooperate in this case? A related unsatisfactory aspect in the formulation of evolutionary biology in terms of replicator-type dynamics with a fixed fitness landscape is the lack of adaptability: since the payoff structure is set from the beginning, e.g. (1) for the PD, and it completely determines the fitness of each strategy, the model is not flexible enough in order to incorporate changes in the payoffs which might happen over time. To overcome such difficulty, dynamical models for the coevolution of the existing species and their fitness landscape have been proposed33,34,35,36. This subject is also fundamentally linked to the fact that evolution is similar to an optimization process: a population evolves in order to adapt as much as possible to some external conditions (which in turn may change in time). Some works from different perspectives have been recently proposed in this context37,38,39. In particular, in38 it has been argued that living systems adapt to the environment by performing an optimal control. The aim of this work is to present a dynamical mechanism based on optimal control theory that has two main features: on the one side it generalizes the standard replicator-type dynamics to an evolution which can automatically adapt to changes in the fitness landscape by exerting a dynamical control on the fitness itself and on the other side it provides a simple and natural rule for the emergence of cooperation in the basic formulation of the PD presented above. Remarkably, this rule seems quite reasonable for the two prisoners as well as in biological and social interactions. To analyze the PD from a dynamical point of view, we need to switch the perspective from game theory to evolutionary dynamics. Here the different strategies correspond to different types of individuals in a population competing by natural selection and the payoffs correspond to each type's ability to reproduce, which is called the fitness. For simplicity, we consider a population with only two possible types (this is all we need in order to address the PD; including more types is straightforward in principle, but the calculations get more involved very quickly). The fundamental equation of evolutionary dynamics is the Replicator Equation (RE) $${\dot{x}}_{a}={x}_{a}({f}_{a}({\bf{x}})-\langle {\bf{f}}\rangle )\,\quad a=1,2,$$ where x a is the relative abundance (frequency) of individuals of type a, f a (x) is the (frequency-dependent) fitness of type a, $\langle {\bf{f}}\,\rangle ={x}_{1}\,{f}_{1}+{x}_{2}\,{f}_{2}$ is the average fitness of the population and the overdot denotes time derivative. For the PD, there are only two possible strategies: C or D. Thus we can label the frequency of the population adopting each strategy as x C and x D respectively. Moreover, the fitness of each strategy is obtained from the payoff matrix (1) by combining each payoff with the probability that the opponent chooses the corresponding strategy, thus obtaining $${f}_{C}({\bf{x}})=R{x}_{C}+S{x}_{D}\,,\quad {f}_{D}({\bf{x}})=T{x}_{C}+P{x}_{D}.$$ Without loss of generality, we assume S = 0. Using the fact that x C + x D = 1, after some algebra we can rewrite the RE and the fitnesses in terms of x C only, obtaining the following evolution equation for the frequency of cooperators $${\dot{x}}_{C}=-{x}_{C}\mathrm{(1}-{x}_{C})[(T-R){x}_{C}+P\mathrm{(1}-{x}_{C})],$$ From which it is easy to deduce that there are only two fixed points x C = 0,1, and that only x C = 0 is stable, that is, cooperators are dominated by defectors (cf. Fig. 1). Evolution of cooperators (C) and defectors (D) in the PD. On the left panel we present the evolution according to the standard Replicator Equation, while the right panel is the evolution of the same system according to the Optimal Replicator Equation introduced here. The payoffs and the initial conditions used in these examples are reported in Table 1. While the standard RE predicts that cooperators disappear during evolution, the ORE shows the emergence of cooperation. Table 1 Models used for the graphics. Thus both game theory and evolutionary dynamics agree on the fact that the best strategy in the PD is to defect. Remarkably, the standard RE always favours defectors over cooperators whenever T > R > P > S, independently of their relative values. Notwithstanding, one would expect that in real situations there should be a difference between e.g. the case T = 100, R = 10, P = 9, S = 0 and the case T = 5, R = 4, P = 1, S = 0. Notice also that in this classical treatment of the PD, the average fitness of the population decreases over time (see Fig. 2), meaning that the optimal strategy (defection) is optimal only with respect to the local payoff of each player, but it is not optimal with respect to the global payoff of the entire population. This is again the essence of the PD: by trying to maximize only their individual fitness, the prisoners do not achieve the best available fitness. Interestingly, this is unstable whenever the population is not isolated, but it is under evolutionary pressure by some other population, meaning that whenever selection acts both at the level of single individuals in a given population and at the level of competing populations, a strategy considering only the former aspect is destined to be suppressed by one considering both features (this is the main reason why group selection achieves cooperation9,19). Evolution of the average fitness of the population in the PD. On the left panel we present the evolution according to the standard Replicator Equation, while the right panel is the evolution of the same system according to the Optimal Replicator Equation introduced here. The payoffs and the initial conditions used in these examples are reported in Table 1. While using the standard RE the average fitness of the population decreases and approaches the payoff for mutual defection (P), in the system governed by the ORE the average fitness increases and approaches the payoff for mutual cooperation (R). Let us now use this observation in order to construct a mechanism that boosts the emergence of collaboration among competing individuals. Motivated by the fact that being part of a society provides in many situations a concrete evolutionary advantage for each organism, we assume that natural evolution acts by two mechanisms: on the one side it selects those individuals inside a population with higher fitness at any given time, and on the other side it selects those populations with the higher final average fitness (we can see this process as due to the existence of two time scales, a faster one, in which selection operates at the level of individuals in a population, and a slower scale, in which selection operates among populations). We require the evolution in the faster scale to be dictated locally in time by the standard RE. Finally, in order to obtain a model sensitive to changes in the environment, we suppose that the fitness of each strategy can be regulated in response to the present state of the population (this is similar to the coevolution mechanisms proposed e.g. in refs33,34). The above assumptions can be translated into the mathematical problem of maximizing a functional which consists of two terms (one corresponding to selection at the level of individuals, and one corresponding to selection at the level of populations) subject to a dynamical constraint set by the standard RE. In order to find the best strategy to realize this task, we need to solve a maximization problem (similar ideas have been put forward in37,38,39). This is the arena of Optimal Control Theory (OCT)40,41,42,43. By applying OCT we obtain (see Supplementary Information) an enlarged system of equations that extends the RE (2) and describes the dynamical equations for the coevolution of the frequencies x in the RE and the fitnesses f. For a = C, D, as in the case of the PD, this system reads $${f}_{a}=\frac{1}{2}\,{p}_{a},$$ $${\dot{x}}_{a}=\frac{{x}_{a}}{2}({p}_{a}-\langle {\bf{p}}\rangle ),$$ $${\dot{p}}_{a}=\frac{{p}_{a}}{2}(\langle {\bf{p}}\rangle -\frac{{p}_{a}}{2})\mathrm{.}$$ The additional variables p C and p D are usually called the co-states (see Supplementary Information). We call the system (5)–(7), together with the initial conditions ${x}_{a}\mathrm{(0)}={x}_{a}^{0}$ and the terminal conditions $${p}_{a}(\tau )=\frac{\partial g({\bf{x}}(\tau ))}{\partial {x}_{a}},\quad \quad g({\bf{x}})\,:=\langle {\bf{f}}\rangle $$ the Optimal Replicator Equation (ORE). In the following we show that the ORE leads directly to a simple rule for the emergence of cooperation in the PD. Let us consider now the PD in terms of the ORE. As usual, we use x C + x D = 1 to simplify (6),(7) by eliminating the equation for x D . Indeed, one can use ${\dot{x}}_{D}=-{\dot{x}}_{C}$ to rewrite the two equations in (6) as a single equation for x C , which reads $${\dot{x}}_{C}={x}_{C}\mathrm{(1}-{x}_{C})\frac{{p}_{C}-{p}_{D}}{2}.$$ Thus we see that we have obtained already a simple condition for the emergence of cooperation, that is, ${p}_{C}-{p}_{D} > 0$. Since by the optimal control strategy (5), p C and p D correspond to the fitness of cooperators and defectors respectively, the above condition seems almost trivial at first sight: cooperation emerges whenever f C > f D . Nevertheless, this condition is not trivial in this case, at least for two reasons: firstly, it is obtained as the result of an optimal strategy; secondly, and most importantly, because the two variables p C and p D are dynamical, with dynamical equations (7) and terminal conditions determined by the choice of the final average fitness according to (8). This aspect is crucial in the solution of the PD. Let us return to the case of our two prisoners, with payoff matrix as in (1). Now we consider the payoffs in (1) as a final payoff given at time t = τ and use the ORE with the final fitness for each strategy given by (3). As before, we assume S = 0, thus the final average fitness is $$g({\bf{x}}(\tau ))=R{x}_{C}{(\tau )}^{2}+T{x}_{C}(\tau ){x}_{D}(\tau )+P{x}_{D}{(\tau )}^{2},$$ and the terminal conditions (8) read $${p}_{C}(\tau )=(2R-T){x}_{C}(\tau )+T,$$ $${p}_{D}(\tau )=(T-2P){x}_{C}(\tau )+2P.$$ the system of equations (7) and (9), together with the initial conditions ${x}_{C}\mathrm{(0)}={x}_{C}^{0}$ and ${x}_{D}\mathrm{(0})=1-{x}_{C}^{0}$ and the terminal conditions (11),(12) is our dynamical model for the PD. Using (9) and (11),(12) one can prove that for $$2P\le T\le 2R,$$ the right hand side of (9) at t = τ is always greater than or equal to zero, with equality only for x C = 0, 1. This means that this evolution admits only two equilibria, namely x C = 0,1 and that only x C = 1 is asymptotically stable. Therefore (13) is the condition for the emergence of cooperation in the PD using the ORE. Typical numerical solutions are given in Fig. 1. As we see, contrary to the standard RE, according to the ORE cooperators take over the population. Another difference with respect to the standard RE is that the average fitness of the population increases with the ORE: while the standard evolution dictated by the RE predicts that the average fitness of the population decreases, approaching the payoff of mutual defection (see Fig. 2), the ORE predicts that selfish individuals cooperate in order to maximize the final average fitness of the population, because this entails an advantage for themselves. The reason for the cooperative behavior in the ORE is simple: using the ORE we have extended the RE to a dynamics which considers the important evolutionary advantage for each individual deriving from being in a population that has a large final average fitness. Knowing that in case of collaboration they will be able to share an important final payoff, naturally induces the two prisoners to collaborate. Interestingly, collaboration only appears whenever R is "high enough", that is, whenever 2R ≥ T, and also whenever P is "low enough", that is, whenever 2P ≥ T. In both cases the prisoners do not find advantageous to defect, and prefer to take the risk to be exposed to exploitation rather than taking a lower payoff. They consider that in such cases the risk is worth the price (cf.44). In biology we can safely assume that in many situations being part of a society guarantees a much higher probability of survival, for instance because it provides better strategies for the collection of food, or for reproduction, or for defense against predators. In this biological setting, we argue that the effect of cooperation can only be assessed at the level of the average fitness of the whole population. So, even if a tendency to cooperate might be inherited, its evolutionary advantage can only be evaluated collectively a posteriori and therefore cannot be included in a local term for the RE, but rather as a final condition. A striking example is provided by bees: in bees society, usually only queens can reproduce. Therefore an evolution based only on the ability to reproduce would lead very quickly to the disappearance of any worker bee. However, workers play an important role for finding food and defending the hive. Cooperation between queens and workers means that the former guarantee reproduction for the latter, while the latter work for the former. This leads to a huge final reward, that is, the conservation of the species after each generation. Finally, let us stress that while in the case of the PD the two prisoners can be aware of the final reward for collaboration and therefore they can (consciously) decide to cooperate under the appropriate conditions, in a biological setting we can no longer give a similar interpretation. However, as it is usual in evolutionary dynamics, we suppose that the different individuals in the population inherit the possible traits randomly and that evolution favours only those individuals with the best traits, so that the optimization process is a consequence of natural selection and not a conscious decision in such case. To summarize, we have proposed a modified version of the Replicator Equation (RE), the equation governing natural selection, called the Optimal Replicator Equation (ORE), which stems from the assumption that evolution is an optimization process that on the one side selects at any given time those individuals with higher fitness and on the other side favours those populations with higher average fitness. The main motivations for the introduction of such model are the facts that the standard RE cannot account for selection on the two levels of individuals and populations and that it fails to reproduce observed situations, such as the emergence of cooperation in the Prisoner's Dilemma. Interestingly, by implementing our model (which by definition takes into account the two levels of selection among individuals and populations) to the case of the PD, we have shown that the corresponding dynamics naturally favours cooperation in the case of the basic Prisoner's Dilemma under some reasonable conditions (cf. (13)). Our results thus open the door for an investigation of evolution and social dilemmas in terms of optimization by using the reproduction coefficients – i.e. the fitness – as control parameters. In particular, it would be interesting for future work to compare the condition for the emergence of cooperation obtained here with data from various experiments on the PD and with conditions derived from similar models44,45,46. Moreover, one can enlarge the study of the optimal strategies deriving from the ORE by considering different social dilemmas beside the PD. 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Bravetti, A., Padilla, P. An optimal strategy to solve the Prisoner's Dilemma. Sci Rep 8, 1948 (2018). https://doi.org/10.1038/s41598-018-20426-w DOI: https://doi.org/10.1038/s41598-018-20426-w Explaining human altruism Michael Vlerick Synthese (2021)	CommonCrawl
\begin{definition}[Definition:Biconditional/Semantics of Biconditional] The concept of the biconditional has been defined such that $p \iff q$ means: :'''If $p$ is true then $q$ is true, and if $q$ is true then $p$ is true.''' $p \iff q$ can be considered as a ''shorthand'' to replace the use of the longer and more unwieldy expression involving two conditionals and a conjunction. If we refer to ways of expressing the conditional, we see that: * $q \implies p$ can be interpreted as '''$p$ is true if $q$ is true''' and: * $p \implies q$ can be interpreted as '''$p$ is true only if $q$ is true'''. Thus we arrive at the usual way of reading '''$p \iff q$''' which is: '''$p$ is true ''if and only if'' $q$ is true.''' This can also be said as: * '''The truth value of $p$ is ''equivalent'' to the truth value of $q$.''' * '''$p$ is ''equivalent'' to $q$.''' * '''$p$ and $q$ are ''equivalent''.''' * '''$p$ and $q$ are ''coimplicant''.''' * '''$p$ and $q$ are ''logically equivalent''.''' * '''$p$ and $q$ are ''materially equivalent''.''' * '''$p$ is true ''exactly when'' $q$ is true.''' * '''$p$ is true ''iff'' $q$ is true.''' This is another convenient and useful (if informal) shorthand which is catching on in the mathematical community. \end{definition}	ProofWiki
\begin{document} \begin{center} {\large \bfseries GEOMETRIC AXIOMS FOR DIFFERENTIALLY CLOSED FIELDS WITH SEVERAL COMMUTING DERIVATIONS} {\large Omar Le\'on S\'anchez}\\ University of Waterloo, ON, Canada \\ March 3, 2011 \end{center} \begin{abstract} A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay \cite{PiPi}, is formulated in terms of a relative notion of prolongation for Kolchin-closed sets. \end{abstract} \title{} \begin{center} {\it AMS 2010 Mathematics Subject Classification: 03C65, 12H05.} \end{center} \ \section{Introduction} An ordinary differential field is a field of characteristic zero equipped with a derivation, that is, an additive map $\d:K\to K$ such that $\d(ab)=(\d a) b+a(\d b)$. A differentially closed field is a differential field $(K,\d)$ such that every system of differential polynomial equations in several variables, with a solution in some differential extension, has a solution in $K$. An elegant first-order axiomatization of the class of ordinary differentially closed fields was given by Blum in \cite{Bl}. In \cite{PiPi}, Pierce and Pillay give a geometric axiomatization. Their axioms say that $(K,\d)$ is differentially closed if and only if $K$ is algebraically closed and whenever $V$ and $W$ are irreducible Zariski-closed sets with $W$ contained in the prolongation of $V$ and projecting dominantly onto $V$, then there is a $K$-point in $W$ of the form $(\x,\d\x)$. Similarly, a field $K$ of characteristic zero equipped with $m$ commuting derivations is differentially closed if every system of partial differential polynomial equations in several variables with a solution in some extension has a solution in $K$. A first-order axiomatization generalizing Blum's was given by McGrail in \cite{Mc} \footnote{A different algebraic axiomatization can be found in Tressl \cite{Tr}.}. However, if one tries to give geometric axioms in terms of prolongations, the commutativity of the derivations might impose too many restrictions, so that a Pierce-Pillay type condition will not hold (see \cite{Pie}, Counterexample 6.2). Nonetheless, in \cite{Pie}, Pierce does manage to give an axiomatization (in arbitrary characteristic) that has a geometric flavor, though not exactly in the Pierce-Pillay sense. In this paper we take a different approach, establishing an axiomatization of differentially closed fields with $(m+1)$ commuting derivations which is geometric relative to the theory of differentially closed fields with $m$ derivations. Our axioms are a precise generalization of the Pierce-Pillay axioms. Two complications arise in our setting that do not appear in the ordinary case: one has to do with extending commuting derivations and the other has to do with first-order axiomatizability. Differential-algebraic results due to Kolchin are behind our solutions to both of these problems. Suppose $\D=\{\d_1,\dots,\d_m\}$ are commuting derivations on a field $K$ of characteristic zero and for each $r=0,\dots, m$, let $\D_r=\{\d_1,\dots,\d_r\}$. Also, suppose $D:K\to K$ is an additional derivation on $K$ that commutes with $\D$. If $V$ is a $\D$-closed set defined over the $D$-constants of $K$, then Kolchin constructs a $\D$-tangent bundle of $V$ which has $\x\to(\x,D\x)$ as a section (\cite{Kol}, Chap. VIII, \S 2). In general, if $V$ is not necessarily defined over the $D$-constants, then $D$ gives a section of a certain torsor of the $\D$-tangent bundle of $V$ that we call the \emph{$D/\D$-prolongation} of $V$ (cf. Definition \ref{prolong}). Our axioms will essentially say that {\it $(K,\D\cup\{D\})$ is differentially closed if and only if $K$ is algebraically closed and for each $r=0,\dots,m$, whenever $V$ and $W$ are $\D_r$-closed sets with $W$ contained in the $D/\D_r$-prolongation of $V$ and projecting onto $V$, then there is a $K$-point in $W$ of the form $(\x,D\x)$}. ``Essentially'', because in actual fact we also have to consider not just $\D$ and $D$ but all their independent $\QQ$-linear combinations (cf. Theorem \ref{main2} below). The paper is organized as follows. In Section 2 we establish the differential-algebraic facts that underpin our results. In Section 3 we introduce relative prolongations and prove a geometric characterization of differentially closed fields. Finally, in Section 4, we address the issue of first-order axiomatizability. \ \noindent{\it Acknowledgements:} I would like to thank Rahim Moosa for all the useful discussions and support towards the completion of this article. \section{Extending $\D$-derivations} In this paper the term ring is used for commutative ring with unity and the term field for field of characteristic zero. Let us first recall some terminology from differential algebra. For details see \cite{Ko}. Let $R$ be a ring and $S$ a ring extension. An additive map $\d:R\to S$ is called a derivation if it satifies the Leibniz rule; i.e., $\d(ab)=(\d a)b+a(\d b)$. A ring $R$ equipped with a set of derivations $\D=\{\d_1,\dots,\d_m\}$, $\d_i:R\to R$, such that the derivations commute with each other is called a $\D$-ring. A $\D$-ring which is also a field (of characteristic zero) is called a $\D$-field. We fix for the rest of this section a $\D$-ring $R$. Let $\T$ denote the free commutative monoid generated by $\D$; that is, \begin{displaymath} \T:=\{\d_m^{r_m}\cdots\d_1^{r_1}\,:\,r_m,\dots,r_1\geq 0\}. \end{displaymath} The elements of $\T$ are called the derivative operators of $R$. Let $\x=(x_1,\dots,x_n)$ be a family of indeterminates, and define \begin{displaymath} \t\x:=\{\partial x_j: \, j=1,\dots,n, \, \partial \in \T\}. \end{displaymath} The $\D$-ring of $\D$-polynomials over $R$ in the differential indeterminates $\x$ is $R\{\x\}:=R[\t\x]$; that is, the ring of polynomials in the algebraic indeterminates $\t\x$ with the canonical $\D$-ring structure given by $\d_i(\d_m^{r_m}\cdots\d_1^{r_1}x_j)=\d_m^{r_m}\cdots\d_i^{r_i+1} \cdots\d_1^{r_1}x_j$. We fix an orderly ranking in $\t\x$ by: \begin{displaymath} \d_m^{r_m}\cdots\d_1^{r_1}x_i\leq \d_m^{r'_m}\cdots\d_1^{r'_1}x_j \iff \left(\sum r_l,i,r_m,\dots,r_1\right)\leq \left(\sum r'_l,j,r'_m,\dots,r'_1\right) \end{displaymath} in the lexicographical order. According to this ranking, we enumerate the algebraic indeterminates by $\t \x=(\t_1\x,\t_2\x,\dots)$. Therefore, if $f\in R\{\x\}$ there is a unique $\hat f\in R[t_1,t_2,\dots]$ such that $f(\x)=\hat f(\t\x)$. We will be interested in adding an extra derivation on $R$. This amounts to the study of $\D$-derivations (see Chapter 0 of \cite{Kol}). \begin{definition} A $\D$-derivation on $R$ is a derivation $D:R\to S$, where $S$ is a $\D$-ring extension of $R$, such that $D\d_i=\d_iD$ for $i=1,\dots,m$. \end{definition} If $f\in R\{\x\}$, by $f^D$ we mean the $\D$-polynomial in $S\{\x\}$ obtained by applying $D$ to the coefficients of $f$. \begin{remark}\label{super} \ \begin{enumerate} \item The map $f\mapsto f^D$ is itself a $\D$-derivation from $R\{\x\}$ to $S\{\x\}$. Indeed, it is clearly additive and to check the Leibniz rule and commutativity of $D$ with $\D$ it suffices to show that $(fg)^D=f^Dg+fg^D$ and $(\d_i f)^D=\d_i(f^D)$ for $i=1,\dots,m$, where $f(\x)=\hat f(\t\x)$ and $g(\x)=\hat g(\t\x)$ are such that $\hat f$ and $\hat g$ are monomials in $R[t_1,t_2,\dots]$. These computations are straightforward. \item The $\D$-derivation $D$ extends uniquely to a derivation \begin{displaymath} R\{\x\}\to S[D\t\x ]:= S[D\t_k\x:\, k=1,\dots] \end{displaymath} by $D(\t_k\x)=D\t_k\x$, for $k\geq 1$. Also, each $\d_i$ extends uniquely to a derivation \begin{displaymath} S[D\t \x]\to S[\d_iD\t \x] :=S[\d_iD\t_k\x:\, k=1,\dots] \end{displaymath} by $\d_i(D\t_k\x)=\d_iD\t_k\x$, for $k\geq 1$. \end{enumerate} \end{remark} In order to describe how $D$ and compositions of $D$ with elements of $\D$ act on $R\{\x\}$, we introduce the following convenient terminology. Given $f\in R\{\x\}$, the \emph{Jacobian} of $f$ is \begin{displaymath} df(\x):=\left( \frac{\partial \hat f}{\partial t_i}(\t\x)\right)_{i\in \NN} \end{displaymath} viewed as an element of $\left(R\{\x\}\right)^{\NN}$. Note that $df$ is finitely supported, in the sense that all but finitely many coordinates are zero. A straightforward computation shows that for each $\d_i\in \D$ \begin{displaymath} \d_if(\x)=df(\x)\cdot\d_i\t \x+f^{\d_i}(\x), \end{displaymath} where $\d_i\t\x=(\d_i\t_1\x,\d_i\t_2\x,\dots)$ and the dot product is well defined since $df$ has finite support. The \emph{Hessian} of $f$ is defined as \begin{displaymath} Hf(\x):=\left(\frac{\partial^2 \hat f}{\partial t_i\partial t_j}(\t\x)\right)_{i,j\in \NN} \end{displaymath} viewed as an element of $\left(R\{\x\}\right)^{\NN\times\NN}$. Again $Hf$ is finitely supported. \begin{lemma}\label{exten1} Let $S$ be a $\D$-ring extension of $R$ and $D:R\to S$ a $\D$-derivation. If $f\in R\{\x\}$ and $\d\in \D\cup \{D\}$, then \begin{displaymath} \d f(\x)=df(\x)\cdot \d \t\x + f^{\d}(\x). \end{displaymath} Also, if $\d$, $\z \in\D\cup \{D\}$ with $\d \neq \z$, then \begin{eqnarray} \d\z f(\x) &=& df(\x)\cdot\d\z\t\x+\d \t\x \cdot Hf(\x)\cdot (\z \t\x)^t + f^{\d\z}(\x) \\ && + \, df^{\d}(\x)\cdot\z\t\x + df^{\z}(\x)\cdot \d\t\x. \\ \end{eqnarray} For any choice of $\d$ and $\z$ the elements $\d f$ and $\d\z f$ are well defined by part (2) of Remark \ref{super}. Note that the dot product $df(\x)\cdot\d\t\x$ and the matrix product $\d \t\x \cdot Hf(\x)\cdot (\z \t\x)^t$ are well defined because $df$ and $Hf$ have finite support. \end{lemma} \begin{proof} For the first equation, by additivity, it suffices to prove it for monomials $f(\x)=c\prod_{i}(\t_i \x)^{\alpha_i}$ where $c\in R$ and $\alpha_i=0$ except finitely many times: \begin{eqnarray} \d f(\x) &=& \sum_{k}\left( c\prod_{i\neq k} (\t_i \x)^{\alpha_i} \; \alpha_k (\t_k \x)^{\alpha_k-1} \d \t_k \x \right)+ \d(c)\prod_{i}(\t_i \x)^{\alpha_i} \\ &=& df(\x)\cdot \d \t \x + f^{\d}(\x). \\ \end{eqnarray} For the second equation, \begin{eqnarray} \d \z f(\x) &=& \d\left( df(\x)\cdot \z \t\x + f^{\z}(\x)\right) \\ &=& \d\left(\sum_k \frac{\partial \hat f}{\partial t_k}(\t \x)\z\t_k \x+f^{\z}(\x)\right) \\ &=& \sum_k \frac{\partial \hat f}{\partial t_k}(\t \x)\d\z\t_k \x+ \sum_k \d\left(\frac{\partial \hat f}{\partial t_k}(\t \x)\right)\z\t_k \x+\d\left(f^{\z}(\x)\right)\\ &=& df(\x)\cdot\d\z\t\x+ \sum_k\left(d\left(\frac{\partial \hat f}{\partial t_k}(\t\x)\right)(\x)\cdot \d\t\x+ \left(\frac{\partial \hat f}{\partial t_k}(\t \x)\right)^\d (\x)\right)\z \t_k \x\\ && + \, df^{\z}(\x)\cdot\d\t \x+f^{\d\z}(\x)\\ &=& df(\x)\cdot\d\z\t\x+\sum_k\sum_l\frac{\partial^2 \hat f}{\partial t_l\partial t_k}(\t \x)\, \d\t_l \x \; \z\t_k \x \\ && + \sum_k \left(\frac{\partial \hat f}{\partial t_k}(\t \x)\right)^{\d}(\x) \, \z\t_k \x+ df^{\z}(\x)\cdot\d\t\x+f^{\d\z}(\x)\\ &=& df(\x)\cdot\d\z\t\x+\d\t\x\cdot Hf(\x)\cdot (\z\t\x)^t +f^{\d\z}(\x)\\ && + \sum_k \left(\frac{\partial {\widehat {f^{\d}}}}{\partial t_k}\right)(\t \x)\z\t_k \x+ df^{\z}(\x)\cdot\d\t\x\\ &=& df(\x)\cdot\d\z\t\x+\d\t\x\cdot Hf(\x)\cdot (\z\t\x)^t +f^{\d\z}(\x)\\ && + \, df^{\d}(\x)\cdot\z\t\x+ df^{\z}(\x)\cdot\d\t\x.\\ \end{eqnarray} Where the sixth equality uses $\left(\frac{\partial \hat f}{\partial t_k}(\t\x)\right)^{\d}=\frac{\partial \widehat{f^{\d}}}{\partial t_k}(\t\x)$. This follows by additivity and the fact that \begin{eqnarray} \left[\frac{\partial (c \, t_1^{n_1}\cdots t_l^{n_l})}{\partial t_k}(\t\x)\right]^\d &=& \left( n_k c \, (\t_1\x)^{n_1}\cdots (\t_k \x)^{n_k-1}\cdots (\t_l \x)^{n_l} \right)^\d \\ &=& n_k\d(c) \, (\t_1\x)^{n_1}\cdots (\t_k \x)^{n_k-1}\cdots (\t_l \x)^{n_l}\\ &=& \frac{\partial (\d(c)t_1^{n_1}\cdots t_l^{n_l})}{\partial t_k}(\t\x).\\ \end{eqnarray} \end{proof} \begin{corollary}\label{exten2} Let $S$ be a $\D$-ring extension of $R$ and $D:R\to S$ a $\D$-derivation. Suppose $\a$ is a tuple in $S$ and $D':R\{\a\}\to S$ is a derivation extending $D$ such that $D'\t\a=\t D'\a$. Then $D'$ commutes with $\D$ on $R\{\a\}$. \end{corollary} \begin{proof} Let $f\in R\{\x\}$ we must show that for any $\d_i\in \D$, $\d_iD' f(\a)=D'\d_if(\a)$. By the second equality of Lemma \ref{exten1}, we have that \begin{displaymath} \d_iD' f(\a)-D'\d_if(\a)=df(\a)\cdot(\d_iD'\t\a-D'\d_i\t\a). \end{displaymath} But by assumption $\d_iD'\t\a=\d_i \t D'\a=D'\d_i\t \a$, as desired. \end{proof} \begin{definition}\label{tauf} Let $f\in R\{\x\}$. We define the $\D$-polynomial $\tau_{D/\D} f\in S\{\x,\y\}$ by \begin{displaymath} \tau_{D/\D} f(\x, \y):= df(\x)\cdot \t \y +f^D(\x). \end{displaymath} When $\D$ and $D$ are understood we simply write $\tau f$. If $\a\in S$, we write $\tau(f)_{\a}(\y)$ for $\tau f(\a,\y)\in S\{\y\}$. Note that $\tau \t \x=\t \y$ and if $c\in R$ then $\tau c=Dc$. \end{definition} \begin{lemma}\label{exten3} Suppose $S$ is a $\D$-ring extension of $R$ and $D:R\to S$ is a $\D$-derivation. Then $\tau:R\{\x\}\to S\{\x,\y\}$ is a $\D$-derivation extending $D$. \end{lemma} \begin{proof} The map $f\mapsto \tau f$ is additive and, by part (1) of Remark \ref{super} and the fact that $d(fg)(\x)\cdot\t\y=\left(df(\x)\cdot\t\y\right) g(\x)+f(\x)\left(dg(\x)\cdot\t\y\right)$, we have that $\tau(fg)=(\tau f)g+f(\tau g)$. Hence, since $\tau c=D c$ for all $c\in R$, $\tau$ is a derivation extending $D$. For commutativity, note that $\tau\t \x=\t \y=\t \tau \x$. Now just apply Corollary \ref{exten2} with $\tau$, $\x$ and $S\{\x,\y\}$ in place of $D'$, $\a$ and $S$. \end{proof} We can now give the desired criterion for when a $\D$-derivation can be extended to a finitely generated $\D$-ring extension. The analogue of the following proposition when $\D=\emptyset$ (i.e. $m=0$) can be found in (\cite{La}, Chap. VII, \S5), and it is the main point in the Pierce-Pillay geometric axiomatization of ordinary differentially closed fields. \begin{proposition}\label{exten5} Suppose $S$ is a $\D$-ring extension of $R$ and $D:R\to S$ is a $\D$-derivation. Let $\a$ be a tuple of $S$ and $A\subseteq R\{\x\}$ such that $[A]=\I(\a/R)$. Here $[A]$ denotes the $\D$-ideal generated by $A$ and $\I(\a/R)=\{f\in R\{\x\} : f(\a)=0\}$. Suppose there is a tuple $\b$ of $S$ such that \begin{equation}\label{form} \tau (g)_{\a}(\b)=0, \textrm{ for all } g\in A. \end{equation} Then there is a unique $\D$-derivation $D':R\{\a\}\to S$ extending $D$ such that $D'\a=\b$. \end{proposition} \begin{proof} First we show that (\ref{form}) holds for all elements in $\I(\a/R)$. For each $\dd\in \T$, $g\in A$ and $h\in R\{\x\}$, by Lemma \ref{exten3} we have \begin{equation}\label{eq11} \tau (h\dd g)(\x,\y)=\tau h(\x,\y)\dd g(\x)+h(\x)\dd(\tau g(\x,\y)). \end{equation} By assumption $\tau g(\a,\b)=\tau (g)_{\a}(\b)=0$ and since $g\in \I(\a/R)$ we get $\dd g(\a)=0$, so evaluating (\ref{eq11}) at $(\a,\b)$ yields $\tau(h\dd g)_{\a} (\b)=0$. It follows that for each $f\in[A]=\I(\a/R)$, $\tau(f)_{\a} (\b)=0$. Now let $\alpha\in R\{\a\}$, then $\alpha=f(\a)$ for some $f\in R\{\x\}$. Define \begin{displaymath} D'\alpha=\tau(f)_{\a} (\b). \end{displaymath} This does not depend on the choice of $f$, since if $\alpha=f(\a)=f'(\a)$ then $f-f'\in \I(\a/R)$ and hence by assumption $\tau(f)_{\a}(\b)-\tau (f')_{\a}(\b)=\tau(f-f')_{\a}(\b)=0$. It is clear that $D'\a=\tau(\x)_{\a}(\b)=\b$ and, for all $c\in R$, $D'c=\tau(c)_{\a}(\b)=Dc$. By Lemma ~\ref{exten3} this is a $\D$-derivation extending $D$ to $R\{\a\}\to S$. To show uniqueness suppose $D'':R\{\a\}\to S$ is another $\D$-derivation extending $D$ such that $D''\a=\b$. Then, for any $f\in R\{\x\}$, by the first equality of Lemma \ref{exten1} we get \begin{eqnarray} D'' f(\a) &=& df(\a)\cdot D''\t\a+f^{D''}(\a)=df(\a)\cdot \t D''\a+f^{D}(\a)\\ &=& df(\a)\cdot \t\b+f^{D}(\a)=\tau(f)_{\a} (\b)=D'f(\a) \end{eqnarray} so $D'=D''$. \end{proof} Thus if we want to extend a $\D$-derivation we need to find solutions to certain $\D$-equations. The following result of Kolchin's can then be used to see that there is always such a solution in some $\D$-field extension. \begin{fact}[\cite{Kol}, Chap. 0, \S 4]\label{pro1} Suppose $R$ is a $\D$-subring of a $\D$-field $K$ and $D:R\to K$ a $\D$-derivation. Let $\a$ be a tuple of $K$. Then \begin{displaymath} [\tau (f)_{\a}(\y):f\in \I(\a/R)] \end{displaymath} is a proper $\D$-ideal of $K\{\y\}$. \end{fact} Putting Proposition \ref{exten5} and Fact \ref{pro1} together we get the following result on extending $\D$-derivations to differentially closed fields, which is essentially Corollary 1 of \S 0.4 of Kolchin \cite{Kol}. \begin{corollary}\label{Null} Suppose $R$ is a $\D$-subring of a differentially closed field $(K,\D)$, and $D:R\to K$ a $\D$-derivation. Then there is a $\D$-derivation $D':K\to K$ extending $D$. \end{corollary} \begin{proof} Consider the set \begin{eqnarray} \mathcal S &=& \{(S,D'): R\subseteq S\subseteq K \textrm{ is a $\D$-subring of $K$ and} \\ && \textrm{ $D':S\to K$ is a $\D$-derivation extending $D$}\}. \end{eqnarray} partially ordered by containment. The union of any chain of $\mathcal S$ gives an upper bound, and so, by Zorn's lemma, there is a maximal element, call it $(S,D')$. Towards a contradiction, suppose $S\neq K$. Then there is $a\in K\backslash S$, and, by Fact \ref{pro1}, $[\tau (f)_{a}(y):f\in \I(a/S)]$ is a proper $\D$-ideal of $K\{\x\}$. By the Nullstellensatz for differentially closed fields (see for example theorem 3.1.10 of \cite{Mc}), there is $b\in K$ such that $\tau(f)_a(b)=0$ for all $f\in \I(a/S)$. By Proposition \ref{exten5}, there is a $\D$-derivation $D'':S\{a\}\to K$ extending $D'$. This contradicts the maximality of $(S,D')$. Hence $S=K$ and $D':K\to K$ is a $\D$-derivation extending $D$. \end{proof} We conclude this section with an improvement on Proposition \ref{exten5}. We would like to only have to check condition (\ref{form}) for a set of $\D$-polynomials $A\subseteq R\{\x\}$ such that $\{A\}=\I(\a/R)$, where $\{A\}$ denotes the radical $\D$-ideal generated by $A$. As the reader may expect this will be useful when dealing with issues of first-order axiomatizability (see Proposition \ref{prolim} below). First we need a lemma. For each $i=1,2,\dots$, let $\x_i$ be an $n$-tuple of differential indeterminates. Suppose $D:R\to R$ is a $\D$-derivation. Then $\tau:~R\{\x_1\}\to R\{\x_1,\x_2\}$. Thus we can compose $\tau$ with itself, for each $k\geq 1$ and $f\in R\{\x_1\}$, $\tau^k f=\tau\cdots \tau f\in R\{\x_1,\x_2,\dots,\x_{2^k}\}$. Define $\nabla\x:=(\x,D\x)$ and note that, for each $k\geq 1$, the composition $\nabla^k\x=\nabla\cdots\nabla\x$ is a tuple of length $n2^{k}$. \begin{lemma}\label{radic} Suppose $D:R\to R$ is a $\D$-derivation and $f\in R\{\x\}$. \begin{enumerate} \item If $\a$ is a tuple of $R$, then for each $k\geq 1$, \begin{displaymath} \tau^kf(\nabla^k \a)=D^kf(\a) \end{displaymath} In particular, if $f(\a)=0$ then $\tau^k f(\nabla^k \a)=0$. \item For each $k\geq 1$, we have \begin{displaymath} \tau^k f^k=k! (\tau f)^k+f\, p \end{displaymath} for some $p\in R\{\x_1,\x_2,\dots,\x_{2^k}\}$. \end{enumerate} \end{lemma} \begin{proof} (1) By induction on $k$. The first equality of Lemma \ref{exten1} gives us \begin{displaymath} \tau f(\nabla \a)=df (\a)\cdot \t D\a+f^D(\a)=df (\a)\cdot D \t \a+f^D(\a)=Df(\a). \end{displaymath} The induction step follows easily: \begin{displaymath} \tau^{k+1} f(\nabla^{k+1}\a)=\tau(\tau^k f)(\nabla(\nabla^k\a))=D\tau^k f(\nabla^k\a)=DD^k f(\a)=D^{k+1}f(\a). \end{displaymath} (2) We prove that for each $l=1,\dots,k$ we have \begin{equation}\label{good} \tau^l(f^k)=\frac{k!}{(k-l)!}f^{k-l}(\tau f)^l+f^{k-l+1} \, p_l \end{equation} for some $p_l\in K\{\x_1,\x_2,\dots,\x_{2^l}\}$. From which the results follows when $l=k$. By Lemma \ref{exten3}, we have $\tau f^k=kf^{k-1}\tau f$, so (\ref{good}) holds for $l=1$ with $p_1=0$. Assume it holds for $1\leq l <k$, then \begin{eqnarray} \tau^{l+1}f^k &=& \tau\tau^l f^k=\tau \left(\frac{k!}{(k-l)!}f^{k-l}(\tau f)^l+f^{k-l+1} p_l\right) \\ &=&\frac{k!}{(k-l)!}\left( (k-l)f^{k-l-1}(\tau f)^{l+1}+lf^{k-l}(\tau f)^{l-1}\tau^2 f \right)\\ && + \; (k-l+1)f^{k-l}(\tau f) \, p_l+ f^{k-l+1}\tau p_l\\ &=& \frac{k!}{(k-l-1)!}f^{k-l-1}(\tau f)^{l+1}+f^{k-l} \,p_{l+1} \end{eqnarray} where \begin{displaymath} p_{l+1}=\frac{k!\; l}{(k-l)!}(\tau f)^{l-1}\tau^2 f+(k-l+1)(\tau f) \, p_l+ f \tau p_l. \end{displaymath} \end{proof} \begin{proposition}\label{better} Suppose $R$ is a reduced $\QQ$-algebra and $D:R\to R$ is a $\D$-derivation. Let $\a$ a tuple of $R$ and $A\subseteq \I(\a/R)$. Suppose there is a tuple $\b$ of $R$ such that \begin{equation}\label{rad} \tau (g)_{\a}(\b)=0, \textrm{ for all } g\in A. \end{equation} Then $\tau (f)_{\a}(\b)=0$ for all $f\in \{A\}$. \end{proposition} \begin{proof} By the first argument in the proof of Proposition \ref{exten5}, equation (\ref{rad}) holds for all elements in $[A]$. Let $f\in \{A\}$, since $R\{\x\}$ is also a $\QQ$-algebra $\{A\}=\sqrt{[A]}$, and so there is $k\geq 1$ such that $f^k\in [A]$ and hence $\tau f^k(\a,\b)=0$. By part (1) of Lemma \ref{radic}, $\tau^{k-1}(\tau f^k)(\nabla^{k-1}(\a,\b))=0$. Thus, by part (2) of Lemma \ref{radic}, we have \begin{displaymath} k!(\tau f)^k(\a,\b)+f(\a)p(\nabla^{k-1}(\a,\b))=0, \end{displaymath} for some $p\in R\{\x_1,\x_2,\dots,\x_{2^k}\}$. Since $f(\a)=0$, we get $k!(\tau f)^k(\a,\b)=0$. Thus, since $R$ is a reduced $\QQ$-algebra, $\tau (f)_{\a}(\b)=0$. \end{proof} \begin{corollary} If $S$ is a $\D$-field, then Proposition \ref{exten5} holds even if we replace the assumption that $[A]=\I(\a/R)$ by $\{A\}=\I(\a/R)$. \end{corollary} \begin{proof} Suppose $\{A\}=\I(\a/R)$ and $\tau_{D/\D}(g)_{\a}(\b)=0$ for all $g\in A$. Let $(K,\D)$ be a differentially closed field extending $S$. By Corollary \ref{Null}, we can extend $D$ to a derivation $D':K \to K$. Now, by Proposition \ref{better}, $\tau_{D'/\D}(g)_{\a}(\b)=0$ holds for all $g\in \{A\}_K$, where $\{A\}_K$ denotes the radical $\D$-ideal in $K\{\x\}$ generated by $A$. But $\{A\}\subseteq \{A\}_K$, so that $\tau_{D/\D}(g)_{\a}=0$ for all $g\in \I(\a/R)$. Now apply Proposition ~\ref{exten5}. \end{proof} \ \section{Relative prolongations and a characterization of $DCF_{0,m+1}$} From now on we use freely the basic notions and terminology of model theory; we suggest \cite{Mar0} as a general reference. We work in the language of differential rings $\LL_m=\{0,1,+,-,\times,\d_1,\dots,\d_m\}$, and we let $\LL_0$ be the language of rings. We denote by $DF_{0,m}$ the theory of differential fields of characteristic zero with $m$ commuting derivations, and by $DCF_{0,m}$ its model-completion, the theory of differentially closed fields. In the ordinary case, when $m=1$, we write $DF_0$ and $DCF_0$ in place of $DF_{0,1}$ and $DCF_{0,1}$. The theory of algebraically closed fields of characteristic zero is denoted by $ACF_0$. Let us recall the geometric axioms of $DCF_0$ given by Pierce and Pillay in \cite{PiPi}. Given a $\d$-field $K$ and $V$ a Zariski-closed set of $K^n$, the prolongation of $V$, $\tau V$, is the Zariski-closed subset of $K^{2n}$ defined by the equations $f=0$ and $\sum_{i=1}^n\frac{\dd f}{\dd x_i}(\x)y_i+f^{\d}(\x)=0$ for each polynomial $f\in K[\x]$ vanishing on $V$. Note that, in terms of our notation from Definition \ref{tauf}, the last equation is just $\tau_{\d/\emptyset}f(\x,\y)=0$. \begin{fact}[Pierce-Pillay Axioms]\label{PPax} Let $(K,\d)\models DF_0$. Then $(K,\d)\models DCF_0$ if and only if $K\models ACF_0$ and for each pair of irreducible Zariski-closed sets $V\subseteq K^n$ and $W\subseteq \tau V$ such that $W$ projects dominantly onto $V$, there is $\a\in V$ such that $(\a,\d\a)\in W$. \end{fact} The Pierce-Pillay characterization of $DCF_0$ is indeed first-order. Expressing irreducibility of a Zariski-closed set as a definable condition on the parameters uses the existence of bounds to check primality of ideals in polynomial rings in finitely many variables \cite{Van}. Also, if the field is algebraically closed, one can find a first-order formula, in the language of rings, describing for which parameters a Zariski-closed set projects dominantly onto some fixed irreducible Zariski-closed set. This uses the fact that in a model of $ACF_0$ the Morley rank of a definable set agrees with the dimension of its Zariski-closure. The goal of this section is to extend the Pierce-Pillay axioms, in an appropriate sense, to the context of several commuting derivations. Our approach is to accomplish this by characterizing $DCF_{0,m+1}$ in terms of the geometry of $DCF_{0,m}$. The Pierce-Pillay axioms are then the $m=0$ case (under the convention $DCF_{0,0}=ACF_0$). For the rest of this section we fix a differential field $(K,\D\cup\{D\})$ with $\D=\{\d_1,\dots,\d_m\}$, and $V\subseteq K^n$ a $\D$-closed set. \begin{definition}\label{prolong} We define the $D/\D$-prolongation of $V$, $\tau_{D/\D}V\subseteq K^{2n}$, to be the $\D$-closed set defined by \begin{displaymath} f=0 \textrm{ and } \tau_{D/\D}f=0, \textrm{ for all } f\in \I(V/K). \end{displaymath} Here $\I(V/K)=\{f\in K\{\x\} : f$ vanishes on $V\}$. When $\D$ and $D$ are understood, we just write $\tau f$ and $\tau V$. For $\a\in V$, $\tau (V)_{\a}$ denotes the fibre of $\tau V$ at $\a$. Note that when $m=0$ this is just the usual prolongation. \end{definition} By the first equality of Lemma \ref{exten1}, if $\a$ is in $V$ then $(\a,D\a)\in \tau V$. In particular the projection $\pi : \tau V\to V$ given by $\pi(\a,\b)=\a$ is surjective. Suppose $A\subseteq K\{\x\}$ is such that $[A]=\I(V/K)$. Then $\tau V$ is defined by $f=0$ and $\tau f=0$ for all $f\in A$. Indeed, this follows from the first argument in the proof of Proposition \ref{exten5}. Moreover, the following consequence of Proposition \ref{better} implies that the $D/\D$-prolongation varies uniformly with $V$. \begin{proposition}\label{prolim} Suppose $(K,\D)\models DCF_{0,m}$. If $V=\Z(f_1,\dots,f_s):=\{\a\in K^n : f_i(\a)=0, \, i=1,\dots,s\}$, then $\tau V=\Z(f_i,\tau f_i: i=1,\dots,s)$. \end{proposition} \begin{proof} Clearly $\tau V\subseteq \Z(f_i,\tau f_i: i=1,\dots,s)$. Let $(\a,\b)\in \Z(f_i,\tau f_i: i=1,\dots,s)$. By Proposition \ref{better}, $\tau f(\a,\b)=0$ for all $f\in \{f_1,\dots, f_s\}$. Since $(K,\D)\models DCF_{0,m}$, we have $\{f_1,\dots,f_s\}=\I(\Z(f_1,\dots,f_s))=\I(V)$. Hence, $(\a,\b)\in \tau V$. \end{proof} \begin{remark} \label{rem} \ \begin{enumerate} \item Suppose $(K,\D)\models DCF_{0,m}$. If $V$ is defined over the $D$-constants, that is, $V=\Z(f_1,\dots,f_s)$ where $f_i\in \C_D\{\x\}$, then $\tau V$ is just Kolchin's $\D$-tangent bundle of $V$. Indeed, by Proposition \ref{prolim}, the equations defining $\tau V$ become $f_i(\x)=0$ and $\tau f_i(\x,\y)=df_i(\x)\cdot \t\y=0$, $i=1,\dots,s$. These are exactly the equations for Kolchin's $\D$-tangent bundle $T_\D V$ (\cite{Kol}, Chap.VIII, \S 2). \item In general, $\tau V$ is a torsor under $T_\D V$. Indeed, from the equations one sees that $\tau(V)_{\a}$ is a translate of $T_\D(V)_{\a}$, and so the map $T_\D V\times_V \tau V\to \tau V$ given by $((\a,\b),(\a,\c))\mapsto (\a,\b+\c)$ is a regular action of $T_\D V $ on $\tau V$ over ~$V$. \end{enumerate} \end{remark} Note that in case $\D=\emptyset$, part $(2)$ of Remark \ref{rem} reduces to the fact that the prolongation of a Zariski-closed set is a torsor under its tangent bundle. Here is our extension of the Pierce-Pillay characterization to several commuting derivations. \begin{theorem}\label{maintheo} Suppose $(K,\D\cup\{D\})\models DF_{0,m+1}$. Then $(K,\D\cup\{D\})\models DCF_{0,m+1}$ if and only if \begin{enumerate} \item $(K,\D)\models DCF_{0,m}$ \item For each pair of irreducible $\D$-closed sets $V\subseteq K^n$, $W\subseteq \tau V$ such that $W$ projects $\D$-dominantly onto $V$. If $O_V$ and $O_W$ are nonempty $\D$-open subsets of $V$ and $W$ respectively, then there exists $\a\in O_V$ such that $(\a,D\a)\in O_W$. \end{enumerate} \end{theorem} As we will see in the proof, it would have been equivalent in condition (2) to take $O_V=V$ and $O_W=W$. Also note that when $m=0$, the theorem is exactly Fact \ref{PPax}. \begin{proof} Suppose $(K,\D\cup\{D\})\models DCF_{0,m+1}$, and $V$, $W$, $O_V$ and $O_W$ are as in condition (2). Let $(\UU,\D)$ be a $\|K\|^+$-saturated elementary extension of $(K,\D)$. If $X$ is an ($\LL_{m}$-)definable subset of $K^n$, by $X(\UU)$ we mean the interpretation of $X$ in $\UU^n$. Let $(\a,\b)\in \UU^{2n}$ be a $\D$-generic point of $W$ over $K$; that is, $\I(\a,\b/K)=\I(W(\UU)/K)$. Then $(\a,\b)\in O_W(\UU)$. Since $(\a,\b)\in \tau V(\UU)$ we have that $\tau (f)_{\a}(\b)=0$ for all $f\in \I(V(\UU)/K)$. The fact that $W$ projects $\D$-dominantly onto $V$ implies that $\a$ is a $\D$-generic point of $V$ over $K$, so $\a\in O_V(\UU)$ and $\I(\a/K)=\I(V(\UU)/K)$. Hence, $\tau (f)_{\a}(\b)=0$ for all $f\in \I(\a/K)$. By Proposition \ref{exten5}, there is a unique $\D$-derivation $D': K\{\a\}\to \UU$ extending $D$ such that $D' \a=\b$. By Corollary \ref{Null}, we can extend $D'$ to all of $\UU$, call it $D''$. Hence, $\UU$ becomes a $\D \cup \{D''\}$-field extending the $\D\cup\{D\}$-closed field $K$. Since $\a\in O_V(\UU)$, $(\a,\b)\in O_W(\UU)$ and $D'' \a=\b$, we get a point $(\a',\b')$ in $K$ such that $\a'\in O_V$, $(\a',\b')\in O_W$ and $D\a'=\b'$. The converse is essentially as in \cite{PiPi}. Let $\phi(\x)$ be a conjunction of atomic $\LL_{m+1}$-formulas over $K$. Suppose $\phi$ has a realisation $\a$ in some $(F,\D\cup\{D\})\models DF_{0,m+1}$ extending of $(K,\D\cup\{D\})$. Let \begin{displaymath} \phi(\x)=\psi(\x,\d_{m+1}\x,\dots,\d_{m+1}^r \x) \end{displaymath} where $\psi$ is a conjunction of atomic $\LL_m$-formulas over $K$ and $r>0$. Let $\c=(\a,D \a,\dots,D^{r-1}\a)$ and $X\subseteq F^{nr}$ be the $\D$-locus of $\c$ over $K$. Let $Y\subseteq F^{2nr}$ be the $\D$-locus of $(\c,D \c)$ over $K$. Let \begin{displaymath} \chi(\x_0,\dots,\x_{r-1},\y_0,\dots,\y_{r-1}) :=\psi(\x_0,\dots,\x_{r-1},\y_{r-1}) \land \left(\land_{i=1}^{r-1}\x_i = \y_{i-1}\right) \end{displaymath} then $\chi$ is realised by $(\c,D\c)$. Since $(\c,D\c)$ is a $\D$-generic point of $Y$ over $K$ and its projection $\c$ is a $\D$-generic point of $X$ over $K$, we have that $Y$ projects $\D$-dominantly onto $X$ over $K$. Thus, since $(K,\D)\models DCF_{0,m}$, $Y(K)$ projects $\D$-dominantly onto $X(K)$. Also, since $(\c,D \c)\in\tau X$, we have $Y(K)\subseteq\tau (X(K))$. Applying (2) with $V=O_V=X(K)$ and $W=O_W=Y(K)$, there is $\bar d$ in $V$ such that $(\bar d,D\bar d)\in W$. Let $\bar d=(\bar d_0,\dots,\bar d_{r-1})$ then $(\bar d_0,\dots,\bar d_{r-1},D\bar d_0,\dots,D \bar d_{r-1})$ realises $\chi$. Thus, $(\bar d_0, D \bar d_0,\dots,D^r \bar d_0)$ realises $\psi$. Hence, $\bar d_0$ is a tuple of $K$ realising $\phi$. This proves that $(K,\D\cup\{D\})\models ~DCF_{0,m+1}$. \end{proof} \ \section{Making the geometric characterization first-order} It is not known to the author if condition (2) of Theorem \ref{maintheo} can be expressed in a first-order way. One issue is to express irreducibility of $\D$-closed sets as a definable condition. This seems to be an open problem related to the generalized Ritt problem \cite{Ov}. The other issue is how to express when a $\D$-closed set projects $\D$-dominantly onto another $\D$-closed set as a definable condition. Unlike the algebraic case, in differentially closed fields, Morley rank does not concide with the Krull-Noetherian geometric dimension (see the example given in \S 2.5 of \cite{Mar2}). We resolve the problem in this section by modifying the characterization of Theorem \ref{maintheo} so that it no longer mentions irreducibility or dominance. The first of these can be handled rather easily by the following lemma. \begin{lemma}\label{star} Let $K$ be a $\D\cup\{D\}$-field. Let $V\subseteq K^n$ be a $\D$-closed set with $K$-irreducible components $\{V_1,\dots,V_s\}$. If $\a \in V_i\backslash\bigcup_{j\neq i}V_j$, then $\tau(V)_{\a}=\tau(V_i)_{\a}$. \end{lemma} \begin{proof} Clearly $\tau(V_i)_{\a}\subseteq\tau(V)_{\a}$. Let $\b\in\tau(V)_{\a}$ and $f\in \I(V_i/K)$. Since $\a$ is not in $V_j$, for $j\neq i$, we can pick a $g_j\in \I(V_j/K)$ such that $g_j(\a)\neq 0$. Then, if $g=\prod_j g_j$, we get $fg\in \I(V/K)$ and so \begin{displaymath} 0= \tau(fg)_{\a}(\b) = \tau(f)_{\a}(\b)g(\a)+f(\a)\tau(g)_{\a}(\b) = \tau(f)_{\a}(\b)g(\a) \end{displaymath} where the third equality holds because $\a\in V_i$. Since $g(\a)\neq 0$, we have $\tau(f)_{\a}(\b)=0$, and so $\b\in\tau(V_i)_{\a}$. \end{proof} It follows that if $W\subseteq \tau V$ projects $\D$-dominantly onto $V$ and $V_i$ is a $K$-irreducible component of $V$, then a $K$-irreducible component of $W\cap \tau V_i$ projects $\D$-dominantly onto $V_i$. The second issue, that of $\D$-dominant projections, is more difficult to deal with. Let us note here that when $\D=\emptyset$, that is, in the case of $DCF_0$, one can just replace dominant projections by surjective projections in the Pierce-Pillay axiomatization. Indeed this reformulation is stated in \cite{Pi}. We will not give a proof here as it will follow from Theorem \ref{main2} below. However, what makes this work is the fact that if $a$ is $D$-algebraic over $K$, then $D^{k+1} a \in K(a,D a, \dots, D^k a)$ for some $k$. In several derivations it is not necessarily the case that if $a$ is $\D\cup\{D\}$-algebraic over $K$, then $D^{k+1} a$ is in the $\D$-field generated by $a,Da,\dots,D^k a$ over $K$, for some $k$. However, by a theorem of Kolchin (Fact \ref{good1} below), this can always be achieved if we allow $\QQ$-linear transformations of the derivations. Our modification of Theorem \ref{maintheo} will therefore need to refer to such transformations. For every $M=(c_{i,j})\in GL_{m+1}(\QQ)$, let $\D'=\{\d'_1,\dots,\d'_m\}$ and $D'$ be the derivations on $K$ defined by $\d'_i= c_{i,1}\d_1+\dots+c_{i,m}\d_m+c_{i,m+1}D$ and $D'=c_{m+1,1}\d_1+\dots+c_{m+1,m}\d_m+c_{m+1,m+1}D$. In this case we write $(\D',D')=M(\D,D)$. Clearly, the elements of $\D'\cup\{D'\}$ are also commuting derivations on $K$. \begin{fact}[\cite{Ko}, Chap. II, \S 11]\label{good1} Let $(K,\D\cup\{D\})\models DF_{0,m+1}$. Let $\a=(a_1,\dots,a_n)$ be a tuple of a $\D\cup\{D\}$-field extension of $K$. Suppose all the $a_i$'s are $\D\cup\{D\}$-algebraic over $K$, then there exists $k>0$ and a matrix $M\in GL_{m+1}(\QQ)$ such that, writing $(\D',D')=M(\D,D)$, we have that $D^{\ell}\a$ is in the $\D'$-field generated by $\a,D'\a\dots,D'^k\a$ over $K$, for all $\ell>k$. \end{fact} Theorem \ref{maintheo} characterizes $DCF_{0,m+1}$ in terms of the geometry of $DCF_{0,m}$. The idea, of course, was that $DCF_{0,m}$ has a similar characterization relative to $DCF_{0,m-1}$, and so on. In our final formulation (Theorem \ref{main2} below) we will implement this recursion and give, once and for all, a geometric first-order axiomatization of $DCF_{0,m+1}$ for all $m\geq 0$, that refers only to the base theory $ACF_0$. \begin{theorem}\label{main2} Suppose $(K,\D\cup\{D\})\models DF_{0,m+1}$. Then $(K,\D\cup\{D\})\models DCF_{0,m+1}$ if and only if \begin{enumerate} \item $K\models ACF_0$ \item Suppose $M\in GL_{m+1}(\QQ)$, $(\D',D'):=M(\D,D)$, $V=\Z(f_1,\dots,f_s)\subseteq K^n$ is a nonempty $\D'$-closed set, and \begin{displaymath} W\subseteq \Z(f_1,\dots,f_s,\tau_{D'/\D'}f_1,\dots,\tau_{D'/\D'}f_s)\subseteq K^{2n} \end{displaymath} is a $\D'$-closed set that projects onto $V$. Then there is $\a\in V$ such that $(\a,D'\a)\in W$. \end{enumerate} \end{theorem} \begin{proof} Suppose $(K,\D\cup\{D\})\models DCF_{0,m+1}$. Clearly $K\models ACF_0$. Suppose $M$, $\D'$, $V$ and $W$ are as in condition (2). Clearly $(K,\D'\cup\{D'\})\models DCF_{0,m+1}$, so by Proposition \ref{prolim} we have that $\Z(f_i,\tau_{D'/\D'}f_i:i=1,\dots,s)=\tau_{D'/\D'}V$. Let $V_i$ be an irreducible component of $V$ and $W'=W\cap \tau_{D'/\D'}V_i$. By Lemma \ref{star}, we can find an irreducible component of $W'$ projecting $\D'$-dominantly onto $V_i$. Now just apply Theorem \ref{maintheo} (with $\D'\cup\{D'\}$ rather than $\D\cup\{D\}$) to get the desired point. For the converse, we assume conditions (1) and (2) and prove that $(K,\D\cup\{D\})\models DCF_{0,m+1}$. Given $r=1,\dots,m+1$ and $N\in GL_{m+1}(\QQ)$, let $\KK_{r,N}=(K,\bar \D_{r-1}\cup\{\bar D\})$ where $(\bar\D,\bar D)=N(\D,D)$ and $\bar\D_{r-1}=\{\bar \d_1,\dots,\bar \d_{r-1}\}$. Set $\KK_{0,N}$ to be the pure algebraic field $K$. We show by induction that for each $r=0,\dots,m+1$, $\KK_{r,N}\models DCF_{0,r}$ for all $N\in GL_{m+1}(\QQ)$. The result will then follow by setting $r=m+1$ and $N=\operatorname{id}$. The case of $r=0$ is just assumption (1). We assume $0\leq r\leq m$, $N\in GL_{m+1}(\QQ)$, and we show that $\KK_{r+1,N}=(K,\bar \D_r\cup\{\bar D\})\models DCF_{0,r+1}$. Suppose $\phi(\x)$ is a conjunction of atomic $\LL_{r+1}$-formulas over $K$, with a realisation $\a=(a_1,\dots,a_n)$ in some $\bar\D_{r}\cup\{\bar D\}$-field $F$ extending $\KK_{r+1,N}$. We need to find a realisation of $\phi$ in $\KK_{r+1,N}$. We may assume that each $a_i$ is $\bar\D_{r}\cup\{\bar D\}$-algebraic over $K$ (this can be seen algebraically or one can use the existence of prime models of $DCF_{0,r+1}$ over $K$, see \S 3.2 of \cite{Mc}). Let $M'\in GL_{r+1}(\QQ)$ and $k>0$ be the matrix and natural number given by Fact \ref{good1}. Let $M\in GL_{m+1}(\QQ)$ be \begin{displaymath} M=E\left( \begin{array}{c c} M' & 0 \\ 0 & I \end{array} \right) EN \end{displaymath} where $E$ is the elementary matrix of size $(m+1)$ that interchanges row $(r+1)$ with row $(m+1)$ and $I$ is the identity matrix of size $(m-r)$. Then, setting $(\D',D')=M(\D,D)$, we get \begin{equation}\label{large} D'^{k+1}\a = \frac{f(\a, D'\a,\dots, D'^k\a)}{g(\a,D'\a\dots,D'^{k}\a)} \end{equation} for some $f, g\in (K\{\x_0,\dots,\x_{k}\}_{\D_r'})^n$. Here $\D'_r=\{\d'_1,\dots,\d'_r\}$ and $K\{\x\}_{\D'_r}$ denotes the $\D'_r$-ring of $\D'_r$-polynomials over $K$. Let \begin{displaymath} \c=\left(\a,D' \a,\dots,D'^k\a,\frac{1}{g(\a,D'\a\dots,D'^{k}\a)}\right). \end{displaymath} Let $X\subseteq F^{n(k+2)}$ be the $\D'_r$-locus of $\c$ over $K$ and $Y\subseteq F^{2n(k+2)}$ the $\D_r'$-locus of $(\c,D'\c)$ over $K$. \noindent {\bf Claim.} $Y$ projects onto $X$. \noindent Consider the $\D_r'$-polynomial map $s(\x_0,\dots,\x_{k+1}):X\to F^{n(k+2)}$ given by \begin{displaymath} s=(\x_1,\dots,\x_k,f \, \x_{k+1},-\x_{k+1}^2\tau_{D'/\D_r'}g(\x_0,\dots,\x_k,\x_1,\dots,\x_k,f\,\x_{k+1})) \end{displaymath} where any product between tuples is computed coordinatewise. Using (\ref{large}), an easy computation shows $s(\c)=D'\c$. Given $\b\in X$, we note that $(\b,s(\b))\in Y$. Indeed, if $h$ is a $\D_r'$-polynomial over $K$ vanishing at $(\c,D'\c)$, then $h(\cdot,s(\cdot))$ vanishes at $\c$ and hence on all of $X$. So $(\b,s(\b))$ is in the $\D_r'$-locus of $(\c,D'\c)$ over $K$. That is, $(\b,s(\b))\in Y$. As this point projects onto $\b$ we have proven the claim. Now, by induction, $(K,\D'_r)\models DCF_{0,r}$. Indeed, $(K,\D'_r)=\KK_{r,N'}$ where $N'$ is obtained from $M$ by interchanging rows $r$ and $(m+1)$. Hence, the claim implies that $Y(K)$ projects onto $X(K)$. Also, if $X(K)=\Z(f_1,\dots,f_s)$ where each $f_i$ is a $\D'_r$-polynomial, then clearly $Y(K)\subseteq \Z(f_i,\tau_{D'/\D'_r}f_i:i=1,\dots,s)$. Hence, by condition (2), there is $\bar d\in X(K)$ such that $(\bar d,D'\bar d)\in Y(K)$. Now, let $\rho(\x)$ be the $\LL_{r+1}$-formula over $K$ obtained from $\phi$ by replacing each $\d_1,\dots, \d_{r+1}$ for $d_{i,1}\d_1+\cdots+d_{i,r+1}\d_{r+1}$, where $(d_{i,j})\in GL_{r+1}(\QQ)$ is the inverse matrix of $M'$. By construction, $\phi^{(K,\bar\D_{r}\cup\{\bar D\})}=\rho^{(K,\D'_{r}\cup\{D'\})}$. Thus it suffices to find a realisation of $\rho$ in $(K,\D'_{r}\cup\{D'\})$. We may assume that the $k$ of (\ref{large}) is large enough so that we can write \begin{displaymath} \rho(\x)=\psi(\x,\d_{r+1}\x,\dots,\d_{r+1}^k\x) \end{displaymath} where $\psi$ is a conjunction of atomic $\LL_r$-formulas over $K$. Let \begin{displaymath} \chi(\x_0,\dots,\x_{k+1},\y_0,\dots,\y_{k+1}):=\psi(\x_0,\dots,\x_k)\land\left(\land_{i=1}^k x_i=y_{i-1} \right). \end{displaymath} Then $(F,\D'_r)\models \chi(\c,D'\c)$, and so, as $(\bar d,D'\bar d)$ is in the $\D_r'$-locus of $(\c,D'\c)$ over $K$, we have that $(F,\D'_r)\models \chi(\bar d,D'\bar d)$. But since $\bar d$ is a $K$-point, we get $(K,\D'_r)\models \chi(\bar d,D'\bar d)$. Writing the tuple $\bar d$ as $(\bar d_0,\dots,\bar d_{r+1})$, we see that $\bar d_0$ is a realisation of $\rho$ in $(K,\D'_{r}\cup\{D'\})$. This completes the proof. \end{proof} \begin{remark}\ \begin{enumerate} \item Condition (2) of Theorem \ref{main2} is indeed first-order; expressible by an infinite collection of $\LL_{m+1}$-sentences, one for each fixed choice of $M$, $f_1,\dots,f_s$ and ``shape'' of $W$. \item In condition (2) we can strengthen the conclusion to ask for $\{\a\in V: (\a,D'\a)\in W\}$ to be $\D'$-dense in $V$. \end{enumerate} \end{remark} \ \end{document}	arXiv
\begin{definition}[Definition:Chess/Rules/Capture/Pawn/En Passant] '''''En passant'' pawn capture''' is a method by which a chess pawn may capture a pawn of the opposite colour in particular circumstances. Let pawn $a$ be on the player's $5$th rank. Let pawn $b$ be of the opposite colour to pawn $a$. Let pawn $b$ be on its starting position, on one of the files adjacent to the one occupied by pawn $a$. Let pawn $b$ move forward $2$ spaces, in the process crossing over one of the squares which is under attack from pawn $a$. Then pawn $a$, '''on its next move only''', may move into that square crossed over by pawn $b$, and capture pawn $b$ "while it is passing". This mode of capture is known as '''capture ''en passant'''''. {{ChessDiagram\| \| \| \| \| \| \| \| \| \| \| \|pd\| \| \| \| \| \| \| \|ul\| \| \| \| \| \| \| \|da\|pl\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| }} In the above diagram: :the black pawn on $\text b 7$ moves forward to $\text b 5$ :the white pawn on $\text c 5$ then moves to $\text b 6$, while capturing the black pawn which is actually on $\text b 5$. The board then looks like this: {{ChessDiagram\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \|pl\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| }} \end{definition}	ProofWiki
\begin{document} \title{Magic State Distillation from Entangled States} \author{Ning Bao} \email{[email protected]} \affiliation{Computational Science Initiative, Brookhaven National Laboratory, Upton, New York, 11973} \author{ChunJun Cao} \email{[email protected]} \affiliation{Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD, 20742, USA} \author{Vincent Paul Su} \email{[email protected]} \affiliation{Center for Theoretical Physics and Department of Physics, University of California, Berkeley, CA 94720} \begin{abstract} Magic can be distributed non-locally in many-body entangled states, such as the low energy states of condensed matter systems. Using the Bravyi-Kitaev magic state distillation protocol, we find that non-local magic is distillable and can improve the distillation outcome. We analyze a few explicit examples and show that spin squeezing can be used to convert non-distillable states into distillable ones. Our analysis also suggests that the conventional product input states assumed by magic distillation protocols are extremely atypical among general states with distillable magic. It further justifies the need for studying a diverse range of entangled inputs that yield magic states with high probability. \end{abstract} \maketitle \section{Introduction} Fault tolerance is an absolute necessity for leveraging the full power of quantum computing. It is known that one cannot implement a universal set of gates transversally using a single quantum error correction code~\cite{eastin09_restr_trans_encod_quant_gate_sets}. For many popular choices of codes, Clifford gates are often transversal while the non-Clifford gates are not. Therefore, the challenge is to perform the remaining non-Clifford element, such as a T gate, fault-tolerantly. The difficulty of implementing these T gates forms a major bottleneck for the development of universal quantum computation. Various solutions for implementing non-Clifford gates have been proposed~\cite{Bravyi_2005,Knill2004,codeswitching1,Yoder17,Yoder16,stackedcode,doublecolorcode,gaugecolorcode,Brown19}. One such proposal is to prepare so-called ``magic states'' and perform T-gates through measurement-based quantum computation schemes~\cite{GottesmanChuang99,Nielsen03,Leung01,Bravyi_2005} using only fault-tolerant Clifford operations. To this end, magic state distillation (MSD) protocols have been developed in order to convert product states with large overlap with the magic states into the magic states of a particular form; for the seminal work in this area, please see~\cite{Bravyi_2005,Knill2004}. However, these magic states can be costly to prepare with arbitrarily high accuracy. Such is the prevalent assumption behind the program of Clifford+T circuit decompositions and related compilation schemes~\cite{Gosset13,Amy13,Nam18,Heyfron17,Wang20}. Because of the inefficiencies in distilling these magic states, the ability to develop a ``magic state factory'' that is capable of effectively generating such states in large quantities would be a great advance to the scalability of quantum computation. The conventional setup for magic state distillation takes a number of noisy ancillae, which are often represented as product of identical single qubit mixed states, and projects them onto the code subspace of a carefully chosen quantum error correction code. One then decode using a fault-tolerant Clifford circuit and obtains a number of magic states that have higher fidelity provided the inputs are above a distillation threshold. A great number of procedures and variants have been devised to various degrees of efficacy~\cite{Bravyi_2005,Knill2004,Meier12,BravyiHaah,Jones13,Jones_multilvl,Haah17}. While the product of identical mixed states is a natural input state, its form is rather restrictive considering the vast pool of many-body quantum states that occur naturally in physical systems that can function as magic reservoirs~\cite{Sarkar19,White:2020zoz,LiuWinter_mbmagic,Hsieh_mbmagic,haarmana}. In particular, recent progress in quantum many-body magic indicates that, magic, like entanglement, can take on various forms in such states. Indeed, while some of the magic is found locally, not unlike the conventional input states in magic distillation protocols, a large portion remains in the correlations and is distributed non-locally in entangled states. Therefore, it is natural to ask whether such ``non-local'' magic from these systems can be a) distilled and b) used to improve the outcome of distillations. More generally, it is beneficial, not only theoretically but also practically, to understand how MSD performs on a much wider class of inputs, where magic is not concentrated strictly locally on each ancilla. In this work, we take the first step in characterizing how more generic input states, such as entangled states, can alter the outcomes of distillation. In particular, we answer both of the above questions in the affirmative --- it is possible to distill magic that is distributed non-locally, as opposed to locally in the traditional input states. Non-local magic (NLM) can also improve the success probabilities of the distillation protocol. A wider range of input states can thus help reduce the overall cost for generating a much needed quantum resource. From a practical perspective, we show that well-known procedures used in spin squeezing that generate entanglement and non-local magic can render undistillable magic states distillable. More concretely, we conduct our explicit analysis using the Bravyi-Kitaev (BK) magic distillation protocol~\cite{Bravyi_2005} and study how it performs over different forms of entangled 5-qubit states. We will review some basic aspects of magic states and their distillation in Section~\ref{sec:2}. Then in section \ref{sec:3}, we construct different entangled states with varying magic distributions and perform the Bravyi-Kitaev protocol for $T$ state distillation using these as input states. We show that although the conventional form of input states explicitly have only single qubit magic, states with purely non-local magic such that individual qubit reduced density matrices look non-magical can remain fully distillable. We then consider both conceptually simple and practically relevant examples where non-local magic and entanglement improve distillation outcome. This is made particularly apparent with the example of spin squeezing, where non-distillable states are rendered distillable with a slight addition of non-local magic and entanglement. Then in Section~\ref{sec:4}, we further analyze the properties that make certain states more distillable from the point of view of the quantum error correction code in question. By analyzing random distillable states, we conclude that the conventional MSD inputs are extremely atypical whereas typical distillable states have high entanglement and zero local magic. We then provide some intuitions for the patterns behind the distribution of the distillable states. Finally, in Section~\ref{sec:5}, we summarize our findings, speculations, and future directions. \section{Background} \label{sec:2} \subsection{Magic} Stabilizer states are generated by Clifford operations acting on the all zeros state. They have a wide range of applications in quantum information, condensed matter physics and quantum gravity~\cite{Gottesman_stab,Kitaev_anyon,HaPPYcode}. However, it is well known that they are not dense in the Hilbert space, and thus the Clifford operations, which can be implemented transversally for certain encoding schemes, are not sufficient for universal quantum computation. In fact, the Gottesman-Knill theorem~\cite{Gottesman_1998} gives us a constructive way to efficiently simulate Clifford circuits purely classically. However, they can be supplemented with a non-Clifford gate using measurement based protocols that only involves Clifford operations~\cite{Bravyi_2005,Knill2004} and noisy ancilla states, thus rendering the computation universal. These procedures require distilling ``non-stabilizer-ness'', or magic, into a useable form. Therefore, magic, being a resource~\cite{veitch14_resour_theor_stabil_quant_comput} in such fault-tolerant computation schemes, aims to quantify the degree to which states fail to be captured by stabilizer states. There are many magic measures. For example, the most straightforward definition is the minimum distance between a state of interest $\rho$ and the set of stabilizer states \begin{equation} D(\rho) = \min_{\sigma \in \rm{STAB}}\|\|\rho -\sigma\|\|, \end{equation} where $\rm{STAB}$ denotes the set of stabilizer states. However, this measure is difficult to compute practically. A number of computable measures have also been proposed~\cite{veitch14_resour_theor_stabil_quant_comput,howard17_applic_resour_theor_magic_states,Wang18,Beverland19}. In this work, we will use one such measure designed for qubit systems called the robustness of magic (ROM)~\cite{howard17_applic_resour_theor_magic_states}, defined for an arbitrary density matrix $\rho$ \begin{equation} \mathcal{R}(\rho) = \min \left\{ \|x\|_1 : \rho = \sum_i x_i s_i \, , \, x_{i} \in \mathbb{R}\right\} \end{equation} where the $s_i$ correspond to density operators of stabilizer states. Note that the coefficients can be negative, allowing us to express arbitrary states in this way. This quantity is lower bounded by 1 in order to have a normalized density matrix. If $\rho$ is a stabilizer state, then there will simply be a single non-zero coefficient. As ROM is unity for all stabilizer states, here we use $\log(\mathcal{R})$ which we call LROM to avoid the offset for stabilizer states and for similarities with definitions like mana~\cite{veitch14_resour_theor_stabil_quant_comput}. However, unlike mana which is additive for product of magic states, LROM is, in general, subadditive \begin{equation} \mathcal{R}(\rho_1\otimes \rho_2) \leq \mathcal{R}(\rho_1)\mathcal{R}(\rho_2)\implies \log(\mathcal{R}(\rho_1\otimes \rho_2))\leq \log\mathcal{R}(\rho_1)+\log\mathcal{R}(\rho_2). \end{equation} To distinguish local from non-local magic, we will compute magic for different subsystems in a single state. If, for instance, \begin{equation} \log\mathcal{R}(\rho_{12})>\log \mathcal{R}(\rho_1)+\log \mathcal{R}(\rho_2) \end{equation} we will then conclude that there is magic stored non-locally in the joint system $\rho_{12}$ which is not found in each of the individual subsystems. \subsection{Bravyi-Kitaev Magic Distillation } Here we review the Bravyi-Kitaev(BK) MSD protocol for distilling a particular magic state, the $T$ state. Like in the original work, we assume that stabilizer operations are easy. Therefore, $T$-type magic can be supplied by any state which obtained by a stabilizer operation acting on a specific $T$ state, which we call $\ket{T_0}$. Those familiar with the Bravyi-Kitaev protocol can mostly skim this section, though we will establish some notation that will be used later in future sections. The $\ket{T_0}$ state is the following qubit state. \begin{equation} \ket{T_0} = \cos(\theta_T)\ket{0} + e^{i\pi/4}\sin(\theta_T)\ket{1} \,\quad , \,\quad \theta_T = \frac 1 2\cos^{-1}\left(\frac{1}{\sqrt{3}}\right). \end{equation} It can also be advantageous to write $\ket{T_0}$ in terms of its density matrix. \begin{equation} \ket{T_0}\bra{T_0} = \frac{1}{2}\left(I + \frac{1}{\sqrt{3}}(X + Y + Z)\right) \, . \end{equation} where $X, Y$ and $Z$ are the single qubit Pauli matrices. The density matrix makes manifest that one can also describe the state by its polarization vector $\vec{r} = \frac{1}{\sqrt{3}}(1, 1, 1)$ on the Bloch sphere. Similarly, one can write the $\ket{T_1}$ state as \begin{equation} \ket{T_1}\bra{T_1} = \frac{1}{2}\left(I - \frac{1}{\sqrt{3}}(X + Y + Z)\right) \, . \end{equation} From the polarization vector, one can see that $\ket{T_0}$ points in the direction normal to one of the triangular faces of the stabilizer octahedron. By the symmetries of the stabilizer octahedron, there are eight states which serve the same magical purpose, all related by single qubit Clifford operations. Thus successful $T$ distillation occurs whenever any of the eight states are produced of which $\ket{T_0}$ and $\ket{T_1}$ are two. Likewise, a $T$-state may refer to any of these states, and we define the $T$-fidelity to be the maximum fidelity of a state with any of the $T$ states, \begin{equation} F_{T}(\rho)=\max_{U\in \mathcal{C}^{(2)}} \{F(\rho, U\|T_0\rangle\langle T_0\| U^{\dagger})\}. \end{equation} Here we use $\mathcal{C}^{(2)}$ to denote the Clifford group over a single qubit. The Bravyi-Kitaev protocol for $T$-state distillation involves a recursive subroutine based on the $[[5,1,3]]$ code~\cite{Laflamme_1996} whose stabilizer group is \begin{equation} \mathcal{S}=\langle XZZXI, IXZZX, XIXZZ, ZXIXZ\rangle. \end{equation} It distills $T$ states because the code admits a transversal implementation of the Clifford gate $B=\exp(i\pi/4)SH$ for which $\|T\rangle$ is an eigenstate. As usual, $S, H$ are the single qubit $\pi/2$-phase gate and the Hadamard gate respectively. Given five copies of an approximate $T$ state of the following form \begin{equation}\label{eq:bk_input} \rho_{in} = (1-\epsilon)\ket{T_{0}}\bra{T_{0}} + \epsilon \ket{T_{1}}\bra{T_{1}} \, , \end{equation} the MSD protocol involves measuring the generators of $\mathcal{S}$ and post-selecting the trivial syndrome, effectively projecting the input state onto the code subspace. If a non-trivial syndrome is measured, the state is simply discarded and the protocol is restarted. The decoding circuit is then applied to return a single qubit state, which iteratively approaches a pure $T$ state, provided the fidelity of $\rho_{in}$ with any $T$-state is sufficiently high\footnote{Equivalently, it is the same to first apply the decoding circuit then measure the disentangled syndrome qubits separately~\cite{Gottesman:1997zz}.}. For every iteration of this procedure, one obtains noisy ancilla with a different fidelity. We plot the input and output fidelities as a result of this procedure in Figure~\ref{fig:epsilons}. \begin{figure} \caption{The output fidelity as a function of input fidelity. The cross over point is $\epsilon^* \approx 0.173$. If the input states are not sufficiently close to a $T$ state, then they will limit to the maximally mixed state. } \label{fig:epsilons} \end{figure} It is worth noting that the input state, which is a tensor product of one qubit states, is certainly not in the code subspace. Because there is only a single logical qubit, one can decompose the projected state in the basis of the encoded magic states $\ket{T^{L}_{0}}$ and $\ket{T^{L}_{1}}$. The crux of the analysis lies in showing that for sufficiently small $\epsilon$, the $T$-fidelity increases. If for any $\epsilon<\epsilon^$ such that the protocol always increases the $T$-fidelity of the input state, then $1-\epsilon^$ is known as the distillation threshold. When it comes to the efficiency of the procedure in creating magic states, there are two factors that determine the resource estimate. Assuming there is a target $T$-fidelity, one can use the curve in Figure~\ref{fig:epsilons} to estimate how many rounds of distillation are required starting from an initial pool of noisy ancilla. The second quantity of practical importance is the success probability for measuring trivial syndrome. By success probability we mean the probability of measuring the trivial syndrome corresponding to a single iteration of the distillation, as opposed to producing an approximate $T$ state with the desired final fidelity after multiple rounds. For input states of the form \eqref{eq:bk_input}, the success probability depends on $\epsilon$ as well. As seen in Figure~\ref{fig:bk_round}, this is upper bounded by 1/6, meaning every round of distillation in expectation requires at least 30 approximately magical qubits. Since it is assumed that all 5 input qubits are the same, one actually needs access to an exponential number of noisy ancilla due to the recursive nature of the protocol. The utility of $T$ states was also discussed in the same paper by Bravyi and Kitaev. They showed explicitly how to enact a non-Clifford gate with access to $T$ states and only Clifford operations. The fact that the same state was the fixed point of the distillation procedure and enables Clifford operations to enact universal quantum computation was the reason for calling these states ``magic states''. \section{Magic Distillation Beyond Product States} \label{sec:3} In this section, we investigate whether existing MSD procedure can distill magic from non-product states. Indeed, the conventional input for such protocols $\rho_{\rm in}^{\otimes n}$ is a product state. Magic is localized to each bit and there is clearly no magic hidden in the quantum correlations. However, one might ask whether other forms of multi-qubit magic states can also be used to distill desired magic states as long as certain conditions are satisfied. In particular, can the distillation procedure distill magic from states where the state has entanglement or correlation such that magic is non-locally distributed, i.e. magic of any single qubit is small whereas that of a larger subsystem is non-trivial? For example, such states can exist in the ground state of quantum many-body systems~\cite{White:2020zoz, Sarkar19}. To this aim, we construct a few informative examples below each with distinct but carefully designed magic distributions and answer the above question in the affirmative by testing them using the BK protocol. For all of the distillation procedures below, we only use the entangled states as inputs for the first iteration. The ensuing iterations works under the conventional assumptions where multiple tensor copies of the outputs from earlier iterations are used. \subsection{Phase-GHZ and GHZ-T states}\label{ssec:3A} We begin with a few structurally simple states to build intuition. Consider GHZ states in the computational basis with a phase. \begin{equation} \ket{PG(\alpha)}_{n} = \alpha \ket{0}^{\otimes n} + e^{i\phi}\beta \ket{1}^{\otimes n} . \end{equation} For instance, one can choose $\alpha\in [0,1]$ which sets the relative weighting of the superposition. By normalization, we choose $\beta = \sqrt{1 - \alpha^2}\in \mathbb{R}$. For extreme values of $\alpha/\beta$, this becomes arbitrarily close to a simple product state, allowing us to ``tune'' the presence of magic. The specific magic distribution is shown in Figure~\ref{fig:pg_gt_magic}a. This state provides a steep contrast with the conventional input states of MSD --- no proper subsystem is magical as they are all density matrices diagonal in the computational basis. However, the overall state clearly is magical because one can turn this state into the product of $\|0\rangle$s and a single qubit magic state $\alpha\|0\rangle+e^{i\phi}\beta\|1\rangle$ by applying a series of Clifford gates. \begin{figure} \caption{(a) Magic distribution of the Phase-GHZ state with phase $\phi=\pi/4$. For PG, all subsystems have the same form of reduced density matrix. (b) Magic of various subsystems for the 4-qubit GHZ-T state as a function of $\alpha^2$. } \label{fig:pg_gt_magic} \end{figure} This state is distillable using BK (Figure~\ref{fig:bk_round}) for most values of $\alpha$ because the output is above the distillation threshold (dashed line). Although BK is suboptimal for distilling magic from such kind of states, it clearly demonstrates that non-local magic remains distillable using existing protocols. \begin{figure} \caption{With our choice of simple entangled states, we apply the BK protocol and measure the magic fidelity of the resulting qubit. For comparison, we also compute it for the product states considered in the original work. Notably, the output fidelity is above threshold across a wide range of $\alpha^2 \sim 1-\epsilon$. The output of this round can then be recursively iterated. The green line shows the distillable threshold for states of the form \eqref{eq:bk_input}. For the other states, they reach a pure $T$ state except a few specific values of $\alpha$. This more lenient threshold is not a contradiction since the output density matrix of this round is not of the same form. On the right, we show the probability to measure the trivial syndrome. To actually produce a $T$ state, one needs the procedure to both converge to a $T$ state and measure the trivial syndrome. } \label{fig:bk_round} \end{figure} Other than the conventional product input states which have magic strictly localized to each qubit and the Phase-GHZ state above where magic is strictly global, one can also construct states that have magic at different scales. Consider a state that takes the following form on $n$ qubits \footnote{While we do not use the degree of freedom of the relative phase in this work, it is one that's available to us and others for future work.}. \begin{equation} \ket{GT(\alpha)}_{n} = \alpha \ket{T_0}^{\otimes n} + e^{i\phi}\beta \ket{T_1}^{\otimes n} . \end{equation} Such a state is chosen precisely because it is a combination of the noisy ancilla input and the Phase-GHZ state above. While each one-qubit reduced density matrix mimics the conventional noisy ancilla input that shows up in the protocol (e.g. $\epsilon = \beta^2$), the overall state is also similar to the Phase-GHZ state where there is additional magic globally. Thus for $\alpha$ or $\beta$ small, magic is locally distributed. However, for other choices of $\alpha,\beta$ the state can pick up a non-trivial contribution from non-local magic in the $n$-qubit GHZ state. One can see that magic (Figure~\ref{fig:pg_gt_magic}b) on small subsystems of one or two qubits has a distribution similar to that of the conventional input states. However, as we go to larger subsystems, there is also more non-local magic which adds to the total global magic available in the state. Magic in such states are mostly distillable. See Figure~\ref{fig:bk_round}. However, the dips in the distillability plot now no longer coincide with the locations where there is less total magic, which was the case for Phase-GHZ states. \begin{figure} \caption{Multiple iterations of the BK protocol for simple input states. Over a wide range, our choice of toy entangled states are distillable. Original product input states are limited to the tail ends where $\alpha^2 < \epsilon_$ or $\alpha^2 > 1 - \epsilon_$.} \label{fig:bk_rounds} \end{figure} We can ask whether how global magic helps with distillation in these examples. Other than the clear improvement in distillability in the region where $\beta^2=\epsilon \in[\epsilon^, 1- \epsilon^]$, where the conventional inputs are completely undistillable, there is also other improvements in the form of success probability, i.e., the probability of measuring the trivial syndrome~(Figure~\ref{fig:bk_round}). Indeed, the GHZ-T state, which has single qubit states identical to conventional noisy ancilla input but have non-zero non-local magic retains a relatively high success probability for all values of $\alpha$, which is distinct from the product state counterpart. \subsection{Squeezed Spin States from Noisy Ancillae} \label{subsec:sss} The above examples are conceptually clean and easy to understand, but may be hard to prepare in the laboratory. For entangled input states that are slightly more realistic, we consider initially states that are product stabilizer states that suffer from single qubit coherent noise. We then perform a global entangling unitary usually used in spin squeezing procedures~\cite{SSS93} on such states and examine how it improves distillation outcome. The initial ancilla states before squeezing are spins that are parallel to, say, the x-axis, but have small misalignments across the spins. More explicitly, \begin{equation} \|\psi_{\rm ini}\rangle = \bigotimes_{i=1}^5 \|\theta_i\rangle,~~~~\|\theta_i\rangle = \cos\theta_i\|0\rangle+\sin\theta_i\|1\rangle, \label{eqn:mis_spin_state} \end{equation} where the orientation of each spin $\theta_i$ can fluctuate to simulate inhomogeneity of the initial spin alignment. This is modelled by choosing each $\theta_i\in [\bar{\theta_i}-\theta_{\rm max}, \bar{\theta_i}+\theta_{\rm max}]$ randomly from a uniform distribution. We take the means $\bar{\theta_i}$ to be equal for all $i$. For example, $\bar{\theta_i} =\pi/4$ creates a product state that is approximately $\|+++++\rangle$ but with small misalignment across the spins. Despite the small amount of local magic introduced to $\|\psi_{\rm ini}\rangle$ through the spin misalignments, such states remain mostly undistillable, especially for $\theta_{\rm max}$ small. \begin{figure} \caption{(a)Figure of distillability of states with ($t\ne 0$) vs without ($t=0$) squeezing ($\theta_{\rm max}=0.05$). States without squeezing (blue) are undistillable and are assigned cost $-1$. (b) Single site entanglement and magic of various subsystems of squeezed (colored disks) vs unsqueezed state (colored squares) with $ \bar{\theta}_i=\pi/4$.} \label{fig:SSS_distill} \end{figure} To define the unitary squeezing process, consider a total spin-$5/2$ system consisting of 5 qubits. It is known that a pure product state of qubits is not a squeezed state~\cite{SSS93,SSS_entanglement} and squeezing introduces entanglement among the qubits. For example, \begin{equation} \|+++++\rangle = 2^{5/2}\sum_{k=0}^{5}\sqrt{{5 \choose k}}\| S=\frac 5 2, S_z=\frac 5 2-k\rangle \end{equation} Squeezing can then be performed by considering some unitary rotation on the $S=5/2$ submanifold. The initial spin misalignments sometimes introduce components that are outside the $S=5/2$ submanifold, and the unitary mapping over the 5 qubits leave these components unchanged. We can use the one-axis twisting method~\cite{SSS93} with unitary $U(t)$ to generate a number of squeezed states $\|\psi_{\rm fi}\rangle$ \begin{equation} U(t)=\exp[-it S_z^2]\oplus I, ~~~~ ~\|\psi_{\rm fi}(t)\rangle = U(t)\|\psi_{\rm ini}\rangle. \label{eqn:oneaxis} \end{equation} Note that although $U(t)$ is not a Clifford unitary, there is also no need for it to be fault-tolerant or extremely precise as a range of different $t$'s can all yield similar outcomes. For small $t$, the action of $U(t)$ introduces a small amount of mostly non-local magic and entanglement to the initial state. See Figure~\ref{fig:SSS_distill}b for results when $\langle \theta_i\rangle =\pi/4$. Surprisingly, the previously undistillable initial states are now mostly distillable by applying $U(t)$ (Figure~\ref{fig:SSS_distill}a). Here cost is defined as the average number of copies of the same state needed to distill a single copy of the $T$ state with a reasonably high fidelity ($F_T(\rho)\geq 0.97$ for the plots). If the resulting output does not exceed the prescribed fidelity cut off after 15 iterations, we deem the state undistillable and it is assigned cost $-1$ to differentiate from the rest of the samples. This is another clear demonstration that entangled states generated in a more practical laboratory setting~\cite{SSS_experiment} can help enhance the outcome of magic state distillation. While this is encouraging, squeezing can also act like a source of correlated noise and does not always improve distillation outcomes. See Appendix~\ref{app:SSS} for further discussions on squeezed spin states and magic state distillation. \section{Understanding Distillability} \label{sec:4} In this section, we examine the underlying $[[5,1,3]]$ code that powers the BK distillation procedure to understand properties of states that contain distillable magic. In doing so, we present a few key observations. First, using elementary ideas from quantum error correction, we can deduce the form of non-local magic content for the ideal input state. We then investigate these intuitions by generating random BK-distillable states whose success probability is lower bounded by that of product input states. Studying their entanglement and magic distribution properties tells us that the typical product ansatz is highly fine-tuned. Second, we can explain explicitly why the total magic in a state does not correlate with the distillable magic for this protocol. We expect the lessons here to generalize appropriately for any magic distillation procedure which relies on a quantum error correcting code. As mentioned previously, the Bravyi-Kitaev distillation procedure (decoding followed by postselecting the trivial syndrome) can be represented as a linear map $M:\mathcal{H}_{\rm in}\rightarrow \mathcal{H}_{\rm out}$, where $\mathcal{H}_{\rm in}, \mathcal{H}_{\rm out}$ are 32 and 2 dimensional Hilbert spaces, respectively. Then for each normalized 5-qubit input $\|\psi_{\rm in}\rangle \in \mathcal{H}_{\rm in}$, we can obtain a single-qubit output $\|\psi_{\rm out}\rangle\in \mathcal{H}_{\rm out}$ such that \begin{equation} \lambda\|\psi_{\rm out}\rangle = M\|\psi_{\rm in}\rangle. \end{equation} As a portion of $\|\psi_{\rm in}\rangle$ may not lie in the code subspace, the probability of success $\|\lambda\|^2$, i.e. measuring a trivial syndrome, satisfies $0\leq \|\lambda\|\leq 1$. It is clear that states with more overlap with the code subspace have higher success probability. Working in the basis of the code subspace, any input state with unit success probability can be written as \begin{equation} \|\psi_{\rm in}\rangle = a\|{T}^L_0\rangle +b\|{T}^L_1\rangle \end{equation} where the size of $a$ will determine the output state fidelity. This works because the perfect code encodes a single qubit. It is immediately clear that states close to the encoded $T$ states will have both high success probability and good $T$-fidelity. Furthermore, because the code corrects two qubit erasure errors, any encoded information must be found in 3 or more qubit subsystems. Therefore, for states with high success rate, i.e., large overlap with the code subspace, the amount of magic found in 3 or more qubit subsystems should positively correlate with the output fidelity. On the other hand, such states will have very little magic in one or two qubit subsystems. However, when a state does not overlap significantly with the code subspace, namely $\|\lambda\|$ small, the connection between the amount of magic, its distribution and output fidelity is no longer clear. To get an idea of a typical distillable state, we generate random states uniformly in the space of states where their output fidelity is above the BK distillation threshold and $\|\lambda\|>0.16$ is reasonably large. As a reference, the success probability for product input states close to $T$ states is $\sim 0.167$ (see Figure~\ref{fig:bk_round}). We see that the typical distillable states are not only entangled, but also have non-local magic (Figure~\ref{fig:RN_vNROM_distr}). \begin{figure} \caption{(a) Distribution of von Neumann entropies for single and two qubit subsystems. (b) LROM distribution in the random BK-distillable states.} \label{fig:RN_vNROM_distr} \end{figure} As we see, any subsystem is close to being maximally mixed while magic localized to any single qubit is virtually non-existent. This is in sharp contrast with the conventional input states which have neither. Therefore, this observation indicates that having many-body quantum states beyond the typical product input states can improve the magic distillation outcomes. However, unlike the conventional input states, there is no obvious correlation between the total amount of magic or entanglement a state has and the distillation outcome. In fact, if anything, too much magic can negatively impact distillability in terms of the success probability. We can try to understand this observation from two limits. In the limit where the state is perturbatively close the an encoded $T$ state, then having a state whose magic or entanglement pattern that deviates from the state that has the single best distillable outcome will obviously result in a worse distillation outcome. The maximum amount of magic for a state in the code subspace is upper bounded by that of a single $T$ state ($\mathrm{LROM} \sim 0.55$). If the input state has more magic than this, then it is necessarily out of the code subspace thus resulting in a lower success probability~(Figure~\ref{fig:lrom5_RN}). If, on the other hand, the total magic is less than the upper bound, then it can have a large overlap with the code subspace. In that case, the success probability is close to 1, but its output fidelity must suffer for being farther from the encoded $T$ state. There is a direct correlation between the amount of magic and output fidelity. This can be seen in Figure~\ref{fig:lrom3_RN}, but is less pronounced in Figure~\ref{fig:lrom5_RN}. Interestingly, it is also difficult to find a distillable state that has less total magic than that of a $T$ state but also small overlap with the code subspace. This is likely related to the fact that (Haar) random states are magical~\cite{haarmana}\footnote{Interestingly, LROM of a larger subsystem has such a smaller variance compared to that of a smaller system. This is also qualitatively consistent with results from Haar random states~\cite{haarmana} where the variance of magic scales roughly as $\sim e^{-L}$ where $L$ is the number of sites.}. \begin{figure} \caption{LROM5 of random states (blue) against output fidelity and success probability. Red circle denotes the location of the encoded $T$ state. } \label{fig:lrom5_RN} \end{figure} \begin{figure} \caption{LROM3 of random states (blue) against output fidelity and success probability. Red circle denotes the location of the encoded $T$ state. On the plane $\|\lambda\|=1$, there is clear correlation between LROM and output fidelity. } \label{fig:lrom3_RN} \end{figure} In the limit where a state has small overlap with the code subspace ($\|\lambda\|\lesssim 0.4$), the previous arguments no longer apply. Instead, there is no clear correlation between output fidelity, success probability, and the amount of magic. One can also compute the average cost of distilling magic from these typical states. Again, there is no obvious correlation between the total magic of a state and the total cost for generating a magic state with desired $T$-fidelity $F_T(\rho)>0.999$. However, there are states with magic lower than that of a typical state that have slightly lower cost, as shown in the thin narrow bands stretching to the left of the cluster (Figure~\ref{fig:cost}). These are precisely the states that have higher success probabilities which are closer to the encoded $T$ states. Curiously, these states concentrate on the ridges but are not found anywhere in-between, unlike the typical states that contain more magic. \begin{figure} \caption{Plotting LROM4 and cost for random states (blue) and conventional inputs (red). Cost is set to $-1$ if the state does not exceed the prescribed fidelity $F_T(\rho)>0.999$ within 15 iterations of the protocol.} \label{fig:cost} \end{figure} \section{Discussion} \label{sec:5} In this work, we have examined how entanglement and non-local magic can impact the outcome of magic state distillation. Using the BK protocol on 5 qubits, we establish that non-local magic is indeed distillable. Furthermore, numerical experiments indicate that non-local magic in these entangled states can also ``assist'' the usual magic distillation process by lowering the cost of overall distillation. This is outlined conceptually by studying the GHZ T and GHZ phase states. More practically, we have also constructed a spin-squeezing-assisted procedure for noisy ancilla states. It is shown that certain squeezing operations indeed render non-distillable states distillable, which may be more relevant in an experimental setting. Although the overall impact of many-body magic is still largely unclear in the context of distillation, we build up some generalizable intuition using the 5-qubit code and argue that for the existing distillation protocols based on quantum error correction codes, a typical distillable state should be entangled. Because of the common assumptions behind magic state distillation, one might have suspected that entanglement generally introduces a source of correlated noise, thereby reducing the output fidelity and success rate of magic distillations. However, this expectation is likely a streetlight effect for only searching perturbatively near the conventional product input states. While it is a valid expectation in certain cases (e.g. Figure~\ref{fig:nearT_distill}), it is abundantly clear that by analyzing the entanglement structure and magic distribution of the encoded $T$ state, a typical distillable state is anything but a product state with local magic. One might also suspect that a larger amount of total magic would generally lead to better distillation outcomes. This is somewhat suggestive from Figures~\ref{fig:pg_gt_magic}a and \ref{fig:bk_rounds} for Phase GHZ states. However, this correlation is only manifest when the input lies predominantly in the code subspace. Analysis in Section~\ref{sec:4} indicates that this intuition has limited applicability when the overlap between the input and the code subspace is small. There it is unclear what properties would positively correlate with distillability. Since our work suggests that entangled states may assist magic distillations, it opens up a number of directions in the area of many-body magic~\cite{Sarkar19,White:2020zoz,LiuWinter_mbmagic,haarmana}. Thus far we have only focused on examples and numerical analysis, however a more systematic approach is needed to understand many-body magic distillability that is not protocol specific. Generally, we want to understand the features in the distillation protocol and those of the many-body state that will lead to better distillation outcomes. In particular, it opens up the question of what a ``typical'' distillable magic state should look like if one looks beyond product of identical states. It is also unclear whether existing protocols are optimal for distilling many-body magic, as the input states are drastically different. For example, while it is well-known that bound magic states exist~\cite{CampbellBrowne_boundmagic} assuming identical inputs, it is not clear to what extent the same conclusion applies when one also considers more general input states. We would also like to understand what kind of protocols are near-optimal in distilling magic from condensed matter systems such as the low energy states of a spin-chain where magic is abundant. It is possible that holographic quantum error correction codes may also be useful in this regard for distilling multi-scale non-local magic from CFTs due to its multi-scale, self-similar structure. It is pointed out recently~\cite{Cree2021} that holographic codes with good complementary recovery properties have difficulties implementing transversal non-Clifford gates. Nevertheless, it is still entirely possible to construct holographic codes in general that do have transversal $T$ gates. For instance, this can be done by using codes with transversal non-Clifford gates\cite{csstranst,zeng07} such as the $[[15,1,3]]$ CSS code~\cite{1996Knill,Anderson2014} as the base tensor, which supports a transversal logical $\bar{T}$ gate. Using the general methodology outlined in~\cite{cao2021} (and more comprehensively in~\cite{CaoLackey:2021}) for building tensor networks on 2d hyperbolic space with regular tessellations, one can produce a holographic code constructed using the $\{15,q\geq 4\}$ tiling. This results in an $[[n,k]]$ CSS code that admits a transversal $\bar{T}^{\otimes k}$ gate\footnote{Note that the tensors or codes used to build the holographic code can be the same up to local Clifford equivalences.}. The same methodology also applies to more general $[[2^{n-1},1,3]]$ punctured Reed-Muller codes~\cite{puncturedReedmuller} in a $\{2^{n-1},q\geq 4\}$ tessellation. The resulting stabilizer code would again admit a transversal non-Clifford gate. It should also be noted that transversal non-Clifford gates are not always necessary for distillation. Indeed, the 5 qubit code does not admit a transversal T gate, however, the transversal $B$ gate enables distillation of $T$ states. A holographic code that serves a similar purpose is the $[[n,k]]$ stabilizer code known as the HaPPY pentagon code~\cite{HaPPY}. There the logical $\bar{B}^{\otimes k}$ is transversal and a protocol similar to BK may be sufficient for $T$ state distillation. \appendix \section{Distillability of various Phase-GHZ and GHZ-T states} \begin{figure} \caption{Distillability of composite $\ket{PG}_k$ and $\ket{GT}_k$ states, where $\alpha$ is taken to uniform across individual components. We plot the magic fidelity of the qubit upon a successful distillation round with darker curves corresponding to later rounds. The darkest line is the resulting $T$-fidelity after 10 rounds. Because $\ket{PG}_k$ and $\ket{GT}_k$ contain the same amount of magic for all values of $k$, composite 5 qubit states with more individual states have more total magic in the system. We find that the presence of more total magic does not generally enhance distillability for the BK protocol.} \label{fig:PG_GT_combo} \end{figure} \begin{figure} \caption{Distillability of composite $\ket{PG}_k$ and $\ket{GT}_k$ states, with some qubits in the $\ket{0}$ state. A notable finding is that one can get similar distillability without having to entangle the full 5 qubit state. } \label{fig:PG_GT_select} \end{figure} In this appendix we examine the distillability of states similar to those presented in \ref{ssec:3A}, the Phase GHZ and GHZ T states. As a reminder, these were $n$ qubit states that took the following form. \begin{equation} \ket{PG(\alpha)}_{n} = \alpha \ket{0}^{\otimes n} + e^{i\phi}\beta \ket{1}^{\otimes n} \, , \, \ket{GT(\alpha)}_{n} = \alpha \ket{T_0}^{\otimes n} + e^{i\phi}\beta \ket{T_1}^{\otimes n} \end{equation} Because the Bravyi-Kitaev protocol takes as input a 5 qubit state, it was a natural choice to pick $n=5$. Not only does this make each of the reduced density matrices the same, the input state is symmetric between the five qubits. Here, we now consider a more broad construction where we make a 5 qubit state out of a combination of smaller $\ket{PG}_n(\ket{GT}_n)$ states. This may lead to easier experimental implementation if it is much easier to generate, say, $\ket{PG}_2$ and $\ket{PG}_3$ than $\ket{PG}_5$. On the theoretical side, it is interesting to ask whether the presence of more magic yields better distillation outcomes. This is because $\ket{PG}_n$ has the same magic regardless of $n$, so $\ket{PG}_2\ket{PG}_3$ would have twice as much global magic as $\ket{PG}_5$ for the same $\alpha$. We consider two primary variations. The first is where we comprise the 5 qubit state of multiple $\ket{PG}_k$ states such that $\sum k = 5$. We make a few observations. The first is that adding more total magic does not seem to enhance the ability to distill. This lines up with the expectations laid out in Section~\ref{sec:4}. Interestingly, all of these states remained distillable for generic $\alpha$. The second variation allows for some of the qubits to be in the $\ket{0}$ state in the computational basis, a consideration for experimental setups where the $\ket{0}$ state is effectively just resetting a clean qubit. Clearly, these qubits do not add to the available magic. The results are displayed in Figure~\ref{fig:PG_GT_select}. With the exception of Phase-GHZ states with even numbers, these states also have a wide range of distillability. \section{More on Spin Squeezing and Distillation} \label{app:SSS} Although a global entangling unitary like the one defined in Section~\ref{subsec:sss} can improve the distillation outcome, this improvement will depend on a number of factors and is not universal. Here we analyze how different initial states modified the same squeezing unitary $U(t)$ may impact the distillation outcome. Then we examine the effect of a different squeezing unitary $V(t)$. We find that improved distillation is unlikely to be connected to squeezing and overall amount of entanglement. However, its success may be related to the non-local magic injected during the squeezing process. First of all, squeezing alone is not the key to improving the distillation outcome --- clearly a state needs to contain enough magic to be distillable at all. Although squeezing in general can add magic to a stabilizer state, a slight squeezing of an initial state without spin misalignment does not improve distillation outcome. Furthermore, states that are unsqueezed under the same unitary $U(t)$ (eqn \ref{eqn:oneaxis}) also sees improvement in distillation. For example, distillation outcome is just as good, if not better, for initial states where $\bar{\theta}_i =0$ (Figure~\ref{fig:oneaxis_z}). However $U(t)$ does not squeeze $\|00000\rangle$, which is approximately the initial state we have. \begin{figure} \caption{BK distillation cost applied to states after one-axis twisting $\|00000\rangle$ initial states with random misalignment. Here $\theta_{\rm max}=0.05$. } \label{fig:oneaxis_z} \end{figure} The same global unitary action $U(t)$ also need not improve distillation for other types of initial states. Sometimes it acts as coherent noise to the system, which impedes distillation. For example, consider the scenario where the individual qubit input states are sufficiently close to $\|T\rangle$ such that $\overline{\beta_i} = \theta_T= \arccos(1/\sqrt{3})/2$, $\phi=\pi/4$ \begin{align} \|\tau_0^i\rangle &= \cos(\beta_i)\|0\rangle+\exp(i\phi)\sin(\beta_i)\|1\rangle\\ \|\tau_1^i\rangle &= -\sin(\beta_i)\|0\rangle +\exp(i\phi)\cos(\beta_i)\|1\rangle, \end{align} and the initial state is given by \begin{equation} \rho_{\rm ini} = \bigotimes_i (p_i\|\tau_0^i\rangle\langle\tau_0^i\|+(1-p_i)\|\tau_1^i\rangle\langle\tau_1^i\|). \label{eqn:msin} \end{equation} As usual, $\beta_i$ is drawn uniformly random for an interval $[\overline{\beta_i}-\beta_{\rm max},\overline{\beta_i}+\beta_{\rm max}]$. The probability $p_i$ is also drawn uniformly for an interval $[\overline{p_i}-p_{\rm max},\overline{p_i}+p_{\rm max}]$. These are precisely the target magic $T$ states when $p_{\rm max}, \beta_{\rm max}=0$. Then squeezing now acts as a source of noise that renders distillation less efficient (Figures~\ref{fig:nearT_distill}). \begin{figure} \caption{(a) Squeezing the tensor product of five noisy $T$ states via one-axis twisting. (b) Squeezing the tensor product of give noisy $T$ states via two-axis countertwisting. Here $\beta_{\rm max}=0.05, p_{\rm max}=0.1$ for both plots. Squeezing increases the overall cost of distillation in both cases. } \label{fig:nearT_distill} \end{figure} On the other hand, if the initial state is close to $\|+++++\rangle$ or $\|00000\rangle$ but mixed, then the improvement in the distillation cost also quickly diminishes in these examples as a one increases $p_{\rm max}$. For instance, consider states of the form (\ref{eqn:msin}) but with $\phi=0, \bar{\beta}_i=\pi/4$, the distillation cost is shown in Figure~\ref{fig:mixed_oneaxis} where the number of undistillable states can increase sharply even when the purity is still relatively high. \begin{figure} \caption{One-axis twisting applied to initial mixed states that are close to a product stabilizer state. $\theta_{\rm max}=0.05$ and $p_{\rm max}=0.02$.} \label{fig:mixed_oneaxis} \end{figure} Now let us briefly examine the same process but with a different global unitary applied to the $\bar{\theta}_i=\pi/4$ initial state~(\ref{eqn:mis_spin_state}). Consider a different squeezing procedure using two-axis countertwisting~\cite{SSS93} with $V(t)=\exp(-i t H)$ where \begin{equation} H= \frac {1}{2i}(S_+^2-S_-^2),~~ S_{\pm} = S_x \pm iS_y. \end{equation} Then squeezing produces a similar level entanglement as the previous one-axis twisting procedure for similar values of $t$. It also increases the total magic in the state. However, no visible enhancement for the distillation can be found (Figure~\ref{fig:two_axis_plot}). \begin{figure} \caption{(a) None of the states are distillable after two axis countertwisting. (b) LROM and single site entanglement of the state with (colored disks) and without squeezing (colored squares). } \label{fig:two_axis_plot} \end{figure} The lack of improvements here may be attributed to the amount of non-local magic that is added to the system. Note that a pure state with non-local magic is most certainly entangled, but an entangled state need not contain non-local magic. While there is a slight addition to non-local magic using the one-axis twisting unitary, there is virtually no change in the two-axis countertwisting scheme (Figure~\ref{fig:NLROM}). We estimate non-local magic using \begin{equation} \mathcal{LROM}(1:2)=\log(\mathcal{R}(\rho_{12}))-[\log(\mathcal{R}(\rho_{1}))+\log(\mathcal{R}(\rho_2))], \end{equation} where $\rho_{12}$ is an arbitrarily chosen two-site subsystem. \begin{figure} \caption{Histogram contrasting two-body nonlocal magic $\mathcal{LROM}(1:2)$ after the one-axis twisting and the two-axis countertwisting options that result in similar levels of entanglement ($S_1 \sim 0.08$ and $S_1\sim 0.02$ respectively). Slight negativity can be attributed to the non-additivity of LROM.} \label{fig:NLROM} \end{figure} \end{document}	arXiv
How do we know that radioactive decay rates are constant over billions of years? A friend and I recently discussed the idea that radioactive decay rates are constant over geological times, something upon which dating methods are based. A large number of experiments seem to have shown that decay rate is largely uninfluenced by the environment (temperature, solar activity, etc.). But how do we know that decay rates are constant over billions of years? What if some property of the universe has remained the same over the one hundred years since radioactivity was discovered and measured, but was different one billion years ago? An unsourced statement on the Wikipedia page on radioactive decay reads: [A]strophysical observations of the luminosity decays of distant supernovae (which occurred far away so the light has taken a great deal of time to reach us) strongly indicate that unperturbed decay rates have been constant. I'm interested in verifying constancy of decay rates over very long periods of time (millions and billions of years). Specifically, I'm not interested in radiocarbon dating or other methods for dating things in the thousands-of-years range. Radiocarbon dates, used for dating organic material younger than 50,000 years, are calibrated and crossed-checked with non-radioactive data such as tree rings of millennial trees and similarly countable yearly deposits in marine varves, a method of verification that I find convincing and that I am here not challenging. radioactivity statistics half-life PertinaxPertinax 95922 gold badges66 silver badges88 bronze badges $\begingroup$ Isn't this question of the same vein as questions about whether the fine structure, the cosmological constant, the speed of light, etc. have remained constant over billions of years? With the apparent lack of any strong theoretical argument of why these parameters should be expected to change over the past few billions of years, and the absence of any experiments or astronomical observations that suggest these parameters are changing, I suppose that most people just take the Occam's Razor approach and assume that these parameters are constant until evidence appears suggesting otherwise. $\endgroup$ – user93237 May 23 '17 at 19:31 $\begingroup$ @Samuel I've got nothing against assumptions, but I like to know where they are made. I come from a discipline where people are already regularly telescoping six or seven assumptions without even realising it, justifying each one of them with Occam's razor, and arriving at a conclusion they call the "most likely" that to me sounds little better than "least unlikely". This assumption does seem very likely true, but so much in archaeology rests upon it that I would be happy if it could be grounded on more than parsimony and be observationally confirmed. $\endgroup$ – Pertinax May 23 '17 at 19:57 $\begingroup$ Related: physics.stackexchange.com/q/48543/50583, physics.stackexchange.com/q/7008/50583 (on variability of half-life and non-exponential decay), physics.stackexchange.com/q/78684/50583 (on the meaningfulness of the "change" of a dimensionful constant over time), $\endgroup$ – ACuriousMind♦ May 23 '17 at 21:01 $\begingroup$ It's a good question! I don't think any of the linked questions really cover it. Decay rates can be derived in principle from the Standard Model coupling constants, and I doubt that they can be changed much without changing basically everything else (e.g. making nuclear fusion go too fast or slow, changing stellar spectra), but I don't know enough to pin it down. $\endgroup$ – knzhou May 23 '17 at 21:10 $\begingroup$ @TheThunderChimp See for example xxx.lanl.gov/abs/astro-ph/9912131 and xxx.lanl.gov/abs/astro-ph/9901373 $\endgroup$ – hdhondt May 24 '17 at 23:24 Not an answer to your exact question but still so very related that I think it deserves to be mentioned: the Oklo natural nuclear reactor, discovered in 1972 in Gabon (West Africa). Self-sustaining nuclear fission reactions took place there 1.8 billion years ago. Physicists quickly understood how they could use this as a very precise probe into neutron capture cross sections that far back. Actually, a re-analysis of the data [1] has been published in 2006 featuring one of the author of the original papers in the 70's. The idea is that neutron capture is greatly augmented when neutron energy gets close to a resonance of the capturing nucleus. Thus even a slight shift of those resonance energies would have resulted in a dramatically different outcome (a different mix of chemical compounds in the reactor). The conclusion of the paper is that those resonances did not change by more than 0.1 eV. It should be noted that the most interesting outcome from the point of view of theoretical physics is that this potential shift can be related to a potential change of the fine-structure constant $\alpha$. The paper concludes that $$−5.6 \times 10^{−8} < \frac{\delta\alpha}{\alpha} < 6.6 \times 10^{−8}$$ [1] Yu. V. Petrov, A. I. Nazarov, M. S. Onegin, V. Yu. Petrov, and E. G. Sakhnovsky, Natural nuclear reactor at oklo and variation of fundamental constants: computation of neutronics of a fresh core, Phys. Rev. C 74 (2006), 064610. https://journals.aps.org/prc/abstract/10.1103/PhysRevC.74.064610 $\begingroup$ Kudos for mentioning the Oklo natural reactor, which is one of the coolest bits of physics that I'm aware of. $\endgroup$ – Michael Seifert May 24 '17 at 14:37 The comment Samuel Weir makes on the fine structure constant is pretty close to an answer. For electromagnetic transitions of the nucleus, these would change if the fine structure constant changed over time. Yet spectral data on distant sources indicates no such change. The atomic transitions would change their energies and we would observe photons from distant galaxies with different spectral lines. For the weak and strong nuclear interactions, the answer is more difficult or nuanced. For the strong interactions, we have more of an anchor. If strong interactions changed their coupling constant this would impact stellar astrophysics. Stars in the distant universe would be considerably different than they are today. Again observations of distant stars indicate no such drastic change. For weak interactions, things are more difficult. A lot of nuclear decay is by weak interactions and the production of $\beta$ radiation as electrons and positrons. Creationists might argue the rate of weak interactions was considerably larger in the recent past to give the appearance of more daughter products than what occurs today. This then gives the appearance of great age that is not there. The problem with carbon dating with the decay process $$ {}^{14}_ 6C~\rightarrow~ {}^{14}_7N~+~e^−+~\nu_e $$ is that if this has changed over the last $6000$ years, a favorite time for creationists, this would mean there would be deviations between carbon dating methods and historical record. None of this is proof really, but it does fall in line with Bertrand Russell's idea of a teapot orbiting Jupiter. Lawrence B. CrowellLawrence B. Crowell $\begingroup$ The "Teapot orbiting Jupiter" seems a very weak response to this. That is a response for proposals that are (currently) complely unobservable, hence both unverifiable and unfalsifiable. Having provided hints about how we actually can observe indirect effects of radioactive decay rates elsewhere (and elsewhen), don't undermine that limited observability by likening it to Russell's proposition which, by design, is thoroughly undecideable. $\endgroup$ – Steve Jessop May 25 '17 at 11:04 $\begingroup$ Of course ignoring the hypothetical possibility of changes from a misapplication of Occam is even worse. We know that many kinds of particle behaviour at very high energies are markedly different from low energies, and hence different at very early epochs of the universe. Physicists should and do seek evidence one way or the other for whether things change, and if so what, how, why. There's a difference between looking and not finding, vs. not looking, and the situation here is the former. "Nothing to see here, move along" only needs to be deployed when you're actually hiding something ;-) $\endgroup$ – Steve Jessop May 25 '17 at 11:15 $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind♦ May 25 '17 at 14:15 $\begingroup$ You may wish to qualify "creationists" as "young Earth creationists." $\endgroup$ – jpmc26 May 25 '17 at 23:25 $\begingroup$ Having once argued the position, I can say absolutely this makes no attempt to answer the Young Earth Creationist claim whatsoever. The claim's nature is a sudden change of rate around either the flood or the event around the time of Peleg. $\endgroup$ – Joshua May 26 '17 at 21:34 There are various questions that one would have to answer, if one wished to claim that there had been large changes in decay rates over geological time. Here is what I think might be the best experiment to prove this claim. Without using radiological evidence, one can deduce that the Earth is at least a billion years old by counting annual sedimentation layers and measuring thicknesses of rock strata, and cross-correlating between them by presence of identical or near-identical fossil species. This is what Victorian geologists did, leading to the only case I know where geology beat physics for deducing the truth. The physicists asserted that the world could not be much older than 50 million years, because no known chemical process could keep the sun hot for longer than that. The geologists insisted on at least a billion years, and that if it wasn't chemistry, something else must be powering the sun. They were right. The Sun shines by then-unknown nuclear fusion, not chemistry. BTW, it's "at least" because it is hard to find sedimentary rocks more than a billion years old, and such rocks do not contain helpful fossils. Tectonic activity has erased most evidence of pre-Cambrian ages ... except for zircons, but I'm jumping ahead. Now, jump forwards to today, when we can do isotopic microanalysis of uranium and lead inside zircon (zirconium silicate) crystals. (Skip to the next paragraph if you know about radio-dating zircons.) Zircon has several unique properties. An extremely high melting point. Extreme hardness, greater than quartz. High density. Omnipresence (zirconium in melted rock always crystallizes into zircons as the melt cools, before any other minerals crystallize at all). And most importantly, a very tight crystal structure, which cannot accommodate most other elements as impurities at formation. The main exception is uranium. The only way that lead can get into a zircon crystal, is if it started as uranium which decays into lead after the crystal has solidified from a melt. That uranium comes in two isotopes with different decay times, and each decay chain ends with a different lead isotope. By measuring the relative concentrations of two lead and two uranium isotopes in a zircon, you can deduce the time since it formed using two different "clocks". These zircons are typically the size of grains of sand, so a rock sample will contain millions of independent "clocks" which will allow for good statistical analysis. So, let's find some zircons in an igneous intrusion into a sedimentary rock whose age we know, roughly, by Victorian geology. It's best if the igneous rock is one which formed at great depth, where all pre-existing zircons would have dissolved back into the melt. The presence of high-pressure metastable minerals such as diamond or olivine would allow us to deduce this, and the fact that all the zircons have the same uranium-to-lead ratios would confirm the deduction. Otherwise one would expect to find a mix of young and older zircons. Choose the youngest, which would have crystallized at the time of the intrusion, rather than having been recycled by tectonic activity from an older time. (Which in many cases is the primaeval solidification of the Earth's crust, and the best estimation of the age of our planet, but that's not relevant here). Now, compare the age deduced by radioactive decay, to the less accurate age from Victorian geology. If the rate of radioactive decay has changed greatly over geological deep time, there will be a disagreement between these two estimated ages. Furthermore, the disagreement will be different for intrusions of different ages (as judged by Victorian geology), but consistent for intrusions of similar age in different location. Look for locations where there is a sedimentary rock with intrusion, covered by a younger sedimentary rock without intrusion, meaning that the age of the intrusion can be deduced to be between that of the two sedimentary strata. The closer the age of the two sedimentary strata, the better. I do not know if this has been done (I'd certainly hope so). Any serious proponent of time-varying radioactive decay, needs to research this. If nobody has looked, get out in the field, find those discrepancies, and publish. It might lead to a Nobel prize if he is right. The onus is certainly on him to do this, because otherwise Occam's razor applies to this theory. Back to the physics, I'd ask another question, if this observation fails to uncover strong evidence that radioactive decay rates do vary with time. It is this. How come that the $^{238}$U and $^{235}$U "clocks" in zircons always agree? Radioactive decay is basically quantum tunnelling across a potential barrier. The half-life depends exponentially on the height of the barrier. Any proposed time variation, would mean that the height of this barrier varied in deep time, in such a way that the relative rate of $^{235}$U and $^{238}$U decay does not change. Which is a big ask of any such theory, given the exponential sensitivity to changes. nigel222nigel222 $\begingroup$ Great answer, I very much appreciate the "how to test" approach, and the idea of counting sedimentary layers to cross-check the radiodates seems like the good one, especially since this dating method was used as long ago as Victorian times (I find this of historical interest, any nineteenth century sources on this? Did anyone actually manually count to one billion?). @DavidHammen suggests that some cross-checking has already been done, do you (or him) have any sources on this? $\endgroup$ – Pertinax May 24 '17 at 16:36 $\begingroup$ RE U235-U238: Would a change of the, for instance, weak interaction be expected to change the relative rate? $\endgroup$ – Pertinax May 24 '17 at 16:45 $\begingroup$ @TheThunderChimp you can download Sir Charles Lyell's "Principles of Geology" for free from Amazon Kindle or public domain. It's a seriously weighty tome and he lacked Darwin's gift for the English language. But its interesting to dip into, to find the state of Victorian geology. $\endgroup$ – nigel222 May 24 '17 at 17:45 $\begingroup$ Re relative decay rates: it might be possible to formulate a theory which kept the relative decay rates of U235 and U238 the same while varying both. My instincts tell me that this would be hard (especially when other longlived isotopes are also checked). $\endgroup$ – nigel222 May 24 '17 at 17:51 $\begingroup$ The last paragraph, if I understand it, is actually actually an excellent point all on it's own because it means that changes to fundamental constants would not produce proportional changes in decay rates. That alone should provide all the basis needed to refute any significantly shorter-timeline hypothesis. $\endgroup$ – RBarryYoung May 24 '17 at 20:20 The basic point here is that we don't "know" anything about "the real world". All we have is a model of the world, and some measure of how well the model matches what we observe. Of course, you can construct an entirely consistent model which says "an invisible, unobservable entity created everything I have ever observed one second before I was born, and made it appear to be much older for reasons that cannot be understood by humans". But as Newton wrote in Principia in the section where he states his "rules for doing science," hypotheses non fingo - don't invent theories just for the sake of inventing them. Actually one of the Newton's examples gave to illustrate that point was spectacularly wrong - he used his general principle to conclude that the sun gives off light and heat by the same chemical reactions as a coal fire on earth - but that's not the point: given the limited experimental knowledge that he had, he didn't need a different hypothesis about the sun to explain what was known about it. So, the situation between you and your friend is actually the other way round. You (and all conventional physicists) have a model of the universe which assumes these constants don't change over time, and it fits very well with experimental observations. If your friend wants to claim they do change, the onus is on him/her to find some observable fact(s) which can't be explained in any other way - and also to show that his/her new hypothesis doesn't mess up the explanations of anything else. As some of the comments have stated, if you start tinkering with the values of the fundamental constants in the Standard Model of particle physics, you are likely to create an alternative model of the universe which doesn't match up with observations on a very large scale - not just over the dating of a few terrestrial fossils. The "big picture" approach is critically important here. You can certainly make the argument that finding a fossil fish on the top of a high mountain means there must have been a global flood at some point in history - but once you have a global model of plate tectonics, you don't need to consider that fossilized fish as a special case any more! alephzeroalephzero $\begingroup$ I don't think this gets to the heart of the question: what exactly would go wrong if a coupling constant changed? This isn't a crazy idea, as many of them did change in the early universe. We don't "need" to prove this, but we should easily be able to. $\endgroup$ – knzhou May 23 '17 at 22:04 $\begingroup$ I think this is ultimately not the right answer. Physicists' belief that the fundamental constants involved haven't changed is not an a prioi deduction from Ockham's razor but an a posteriori hypothesis resulting from many independent lines of evidence, including measurements and modelling, as the other answers detail. $\endgroup$ – Nathaniel May 24 '17 at 6:01 I thought I would include something on how coupling constants and masses vary. This might be a bit off topic, and I thought about asking a question that I would answer myself. Anyway here goes. We have a number of quantities in the universe that are related to each other by fundamental constants. The first two of these are time and space, which are related to each other by the speed of light $x~=~ct$. The speed of light is something I will consider to be absolutely fundamental. It really is in correct units a light second per second or one. The speed of light defines light cones that are projective subspaces of Minkowski spacetime. Minkowski spacetime can be thought of then as due to a fibration over the projective space given by the light cone. The other fundamental quantity that relates physical properties is the Planck constant $h$ or $\hbar~=~h/2\pi$. This is seen in $\vec p~-~\hbar\vec k$ where $\vec k~=~\hat k/\lambda$. This relates momentum and wavelength, and is also seen in the uncertainty principle $\Delta p\Delta x~\ge~\hbar/2$. The uncertainty principle can be stated according to the Fubini-Study metric, which is a fibration from a projective Hilbert space to Hilbert space. These two systems share remarkably similar structure when seen this way. I will then say as a postulate that $c$ and $\hbar$ are absolutely constant, and since momentum is reciprocal length then in natural units the Planck constant is length per length and is unitless. There are other constants in nature such as the electric charge. The important constant most often cited is the fine structure constant $$ \alpha~=~\frac{e^2}{4\pi\epsilon\hbar c}~\simeq~1/137. $$ This constant is absolutely unitless. In any system of units it has no units. In natural systems of units we have that $ e^2/4\pi\epsilon$ has ithe units of $\hbar c$, which in MKS units is $j-m$. However, we know from renormalization that $e~\rightarrow~e)-~+~\delta e$ is a correction with $\delta e~\sim~1/\delta^2$, for $\delta~=~1/\Lambda$ the cutoff in space scale for a propagator or the evaluation of a Feynman diagram. This means the fine structure constant can change with scattering energy, and at the TeV energies of the LHB $\alpha'~\sim~1/127$. We have of course the strong and weak interactions and we can well enough state there are coupling constants $e_s$ and $e_w$ and the analogues of the dielectric constants $\epsilon_w$ and $\epsilon_w$ so there are the fine structure constants $$ \alpha_s~=~\frac{e_s^2}{4\pi\epsilon_s\hbar c}~\simeq~1,~\alpha_w~=~\frac{e_w^2}{4\pi\epsilon_w\hbar c}~\simeq~10^{-5}. $$ Most often these coupling constants are $g_s$ and $g_w$. These two have renormalizations $g_s~=~g^0_s~+~\delta g_s$ and $g_w~=~g^0_w~+~\delta g_w$ this runs into the hierarchy problem and how coupling constants vary. These What is clear is that gauge coupling constants vary with momentum. They do not vary with time, which by $x~=~ct$ or more generally Lorentz boosts means if gauge fields did vary with time they would do so with spatial distance. So far there is no observation and data of such variation from radiation emitted from the very distant universe. What about gravitation and mass? We do have mass renormalization $m~\rightarrow~m~+~\delta m$. This can mean the mass of a particle can be renormalized at higher energy, and more it means terms due to vacuum energy contributions that renormalize the mass of a bare particle mass must add up and cancel to give the mass we observe. Again this happens with momentum. For the Higgs field the self interaction is due to the $\lambda\phi^4$ term, Technically this means there is a mass renormalization term $\sim~\lambda/\delta^2$ $=~\lambda\Lambda$ for $\delta$ a small region around the point for the $4$ point interaction where we have smeared it out into some small ball or disk of radius $\delta$. Also $\Lambda$ is the corresponding momentum cut off. We have similar physics for other fields, though with fermions have subtle sign issues, I used the Higgs field because I think there is a deep relationship between gravitation and the Higgs field. I am from this going to compute what I think is the appropriate $\alpha_{grav}$. We can compute the ratio of the Compton wavelength $\lambda~=~M_H/hc$ and gravitational radius $r~=~2GM_H/c^2$ of a Higgs particles, with mass $m~=~125GeV$ $=~2.2\times 10^{-25}kg$. This means $$ \alpha_g~=~\frac{4\pi GM_H^2}{\hbar c}~=~\left(\frac{4\pi M_H}{M_p}\right)^2~=~1.3\times 10^{-33}, $$ where $M_p$ is the Planck mass. This constant is then connected to the mass of all elementary particles. The renormalization to of the Higgs mass determines the mass of all other particles. There is then no indication of there being any variation of particle masses or coupling constants that depend on time. They all depend on momenta, and the large number of Feynman diagram terms to various orders add and cancel to give observed masses. With supersymmetry this is made somewhat more simple with the cancellation of many diagrams. Not the answer you're looking for? Browse other questions tagged radioactivity statistics half-life or ask your own question. Is it possible to speak about changes in a physical constant which is not dimensionless? Long time deviations from exponential decay in radioactivity Do some half-lives change over time? Since radioactive material decays how is it possible that there is any left after 4.5 billion years? How do we know that some radioactive materials have a half life of millions or even billions of years? Strange modulation of radioactive decay rates with solar activity Earth's beginnings and early years, Re radioactive decay or not How do we know that radioactive decay is memoryless? Are there radioactive elements that can be seen to diminish/change with the naked eye? How do we know the rate of decay for radiometric dating is constant? Radioactive Dating: How do we know the initial amount of radioactive atoms present in the object? How does radioactive decay affect material properties?	CommonCrawl
Let $a_1,$ $a_2,$ $\dots$ be a sequence of real numbers such that for all positive integers $n,$ \[\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1.\]Find the smallest $n$ such that $a_n < \frac{1}{2018}.$ For $n = 1,$ we get $a_1 = 1.$ Otherwise, \[\sum_{k = 1}^n k^2 a_k = n^2.\]Also, \[\sum_{k = 1}^{n - 1} k^2 a_k = (n - 1)^2.\]Subtracting these equations, we get \[n^2 a_n = n^2 - (n - 1)^2 = 2n - 1,\]so $a_n = \frac{2n - 1}{n^2} = \frac{2}{n} - \frac{1}{n^2}.$ Note that $a_n = 1 - \frac{n^2 - 2n + 1}{n^2} = 1 - \left( \frac{n - 1}{n} \right)^2$ is a decreasing function of $n.$ Also, \[a_{4035} - \frac{1}{2018} = \frac{2}{4035} - \frac{1}{4035^2} - \frac{1}{2018} = \frac{1}{4035 \cdot 2018} - \frac{1}{4035^2} > 0,\]and \[a_{4036} < \frac{2}{4036} = \frac{1}{2018}.\]Thus, the smallest such $n$ is $\boxed{4036}.$	Math Dataset
\begin{document} \date{} \title{Interpolation on Symmetric Spaces via the Generalized Polar Decomposition} \begin{abstract} We construct interpolation operators for functions taking values in a symmetric space -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically. \end{abstract} \section{Introduction} Manifold-valued data and manifold-valued functions play an important role in a wide variety of applications, including mechanics~\cite{sander2010geodesic,demoures2015discrete,hall2014lie}, computer vision and graphics~\cite{hong2014geodesic,turaga2011statistical,gallivan2003efficient,chang2012feature,de2014discrete,jiang2015frame}, medical imaging~\cite{arsigny2006log}, and numerical relativity~\cite{arnold2000numerical}. By their very nature, such applications demand that care be taken when performing computations that would otherwise be routine, such as averaging, interpolation, extrapolation, and the numerical solution of differential equations. This paper constructs interpolation and averaging operators for functions taking values in a \emph{symmetric space} -- a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the \emph{generalized polar decomposition} -- a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. Our motivation for constructing such operators is best illustrated by example. Among the most interesting scenarios in which symmetric space-valued functions play a role is numerical relativity. There, the dependent variable in Einstein's equations~--~the metric tensor~--~is a function taking values in the space $\mathcal{L}$ of Lorentzian metrics: bilinear, symmetric, nondegenerate $2$-tensors with signature $(3,1)$. This space is neither a vector space nor a convex set. Rather, it has the structure of a symmetric space. As a consequence, the outputs of basic arithmetic operations on Lorentzian metrics such as averaging, interpolation, and extrapolation need not remain in $\mathcal{L}$. This is undesirable for several reasons. If the metric tensor field is to be discretized with finite elements, then a naive approach in which the components of the metric are discretized with piecewise polynomials may fail to produce a metric field with signature $(3,1)$ at all points in spacetime. Perhaps an even more problematic possibility is that a numerical time integrator used to advance the metric forward in time (e.g., in a $3+1$ formulation of Einstein's equations) might produce metrics with invalid signature. One of the aims of the present paper is to avert these potential dangers altogether by constructing a structure-preserving interpolation operator for Lorentzian metrics. As will be shown, the interpolation operator we derive not only produces interpolants that everywhere belong to $\mathcal{L}$, but it is also frame-invariant: the interpolation operator we derive commutes with the action of the indefinite orthogonal group $O(1,3)$ on $\mathcal{L}$. Furthermore, our interpolation operator commutes with inversion and interpolates the determinant of the metric tensor in a monotonic manner. A more subtle example is the space $SPD(n)$ of symmetric positive definite $n \times n$ matrices. This space forms a convex cone, so arithmetic averaging and linear interpolation trivially produce $SPD(n)$-valued results. Nevertheless, these operations fail to preserve other structures that are important in some applications. For instance, arithmetic averaging does not commute with matrix inversion, and the determinant of the arithmetic average need not be less than or equal to the maximum of the determinants of the data. This may remedied by considering instead the Riemannian mean (also known as the Karcher mean) of symmetric positive-definite matrices with respect to the canonical left-invariant Riemannian metric on $SPD(n)$~\cite{moakher2005differential,bhatia2013riemannian,karcher1977riemannian}. The Riemannian mean cannot, in general, be expressed in closed form, but it can be computed iteratively and possesses a number of structure-preserving properties; see~\cite{bhatia2013riemannian} for details. A less computationally expensive alternative, introduced by Arsigny and co-authors~\cite{arsigny2007geometric}, is to compute the mean of symmetric positive-definite matrices with respect to a log-Euclidean metric on $SPD(n)$. The resulting averaging operator commutes with matrix inversion, prevents overestimation of the determinant, and commutes with similarity transformations that consist of an isometry plus scaling. Both of these constructions turn out to be a special cases of the general theory presented in this paper. In our derivation of the log-Euclidean mean, we give a clear geometric explanation of the vector space structure with which Arsigny and co-authors~\cite{arsigny2007geometric} endow $SPD(n)$ in their derivation, which turns out to be nothing more than a correspondence between a symmetric space ($SPD(n)$) and a Lie triple system~\cite{helgason1979differential}. Another symmetric space which we address in this paper is the Grassmannian $Gr(p,n)$, which consists of all $p$-dimensional linear subspaces of $\mathbb{R}^n$. Interpolation on the Grassmannian is a task of importance in a variety of contexts, including reduced-order modeling~\cite{amsallem2008interpolation,vetrano2012assessment} and computer vision~\cite{hong2014geodesic,turaga2011statistical,gallivan2003efficient,chang2012feature}. Not surprisingly, this task has received much attention in the literature; see~\cite{begelfor2006affine,absil2004riemannian} and the references therein. Our constructions in this paper recover some of the well-known interpolation schemes on the Grassmannian, including those that appear in~\cite{amsallem2008interpolation,chang2012feature,begelfor2006affine}. There are connections between the present work and geodesic finite elements~\cite{grohs2015optimal,grohs2013quasi,sander2012geodesic,sander2015geodesic}, a family of conforming finite elements for functions taking values in a Riemannian manifold $M$. In fact, we recover such elements as a special case in the theory below; see Section~\ref{sec:interp_generalizations}. Since their evaluation amounts to the computation of a weighted Riemannian mean, geodesic finite elements and their derivatives can sometimes be expensive to compute. One of the messages we hope to convey is that when $M$ is a symmetric space, this additional structure enables the construction of alternative interpolants that are less expensive to compute but still possess many of the desirable features of geodesic finite elements. Our use of the generalized polar decomposition in this paper is inspired by a stream of research~\cite{munthe2001generalized,munthe2014symmetric,iserles2005efficient} that has, in recent years, cast a spotlight on the generalized polar decomposition's role in numerical analysis. Much of our exposition and notation parallels that which appears in those papers, and we encourage the reader to look there for further insight. \paragraph{Organization.} This paper is organized as follows. We begin in Section~\ref{sec:symspace} by reviewing symmetric spaces, Lie triple systems, and the generalized polar decomposition. Then, in Section~\ref{sec:interp}, we exploit a correspondence between symmetric spaces and Lie triple systems to construct interpolation operators on symmetric spaces. Finally, in Section~\ref{sec:applications}, we specialize these interpolation operators to three examples of symmetric spaces: the space of symmetric positive-definite matrices, the space of Lorentzian metrics, and the Grassmannian. In the case of Lorentzian metrics, we illustrate the potential utility of these interpolation operators by interpolating the Schwarzschild metric numerically. \section{Symmetric Spaces and the Generalized Polar Decomposition} \label{sec:symspace} In this section, we review symmetric spaces, Lie triple systems, and the generalized polar decomposition. We describe a well-known correspondence between symmetric spaces and Lie triple systems that will serve in Section~\ref{sec:interp} as a foundation for interpolating functions which take values in a symmetric space. For further background material, we refer the reader to~\cite{helgason1979differential,munthe2001generalized,munthe2014symmetric}. \subsection{Notation and Definitions} \label{sec:notation} Let $G$ be a Lie group and let $\sigma : G \rightarrow G$ be an involutive automorphism. That is, $\sigma \neq \mathrm{id}.$ is a bijection satisfying $\sigma(\sigma(g))=g$ and $\sigma(gh)=\sigma(g)\sigma(h)$ for every $g,h \in G$. Denote by $G^\sigma$ the subgroup of $G$ consisting of fixed points of $\sigma$: \[ G^\sigma = \{g \in G \mid \sigma(g)=g\}. \] Suppose that $G$ acts transitively on a smooth manifold $\mathcal{S}$ with a distinguished element $\eta \in \mathcal{S}$ whose stabilizer coincides with $G^\sigma$. In other words, \[ g \cdot \eta = \eta \iff \sigma(g)=g \] where $g \cdot u$ denotes the action of $g \in G$ on an element $u \in \mathcal{S}$. Then there is a bijective correspondence between elements of the homogeneous space $G/G^\sigma$ and elements of $\mathcal{S}$. On the other hand, the cosets in $G/G^\sigma$ have canonical representatives by virtue of the \emph{generalized polar decomposition}~\cite{munthe2001generalized,munthe2014symmetric}. This decomposition states that any $g \in G$ sufficiently close to the identity $e \in G$ can be written as a product \begin{equation}\label{gpd} g = pk, \quad p \in G_\sigma, \, k \in G^\sigma, \end{equation} where \[ G_\sigma = \{g \in G \mid \sigma(g) = g^{-1} \}. \] Moreover, this decomposition is locally unique~\cite[Theorem 3.1]{munthe2001generalized}. As a consequence, there is a bijection between a neighborhood of the identity $e \in G_\sigma$ and a neighborhood of the coset $[e] \in G/G^\sigma$. The space $G_\sigma$ -- which, unlike $G^\sigma$, is not a subgroup of $G$ -- is a symmetric space which is closed under a non-associative symmetric product $g \cdot h = gh^{-1} g$. Its tangent space at the identity is the space \[ \mathfrak{p} = \{Z \in \mathfrak{g} \mid d\sigma(Z) = -Z\}. \] Here, $\mathfrak{g}$ denotes the Lie algebra of $G$, and $d\sigma : \mathfrak{g} \rightarrow \mathfrak{g}$ denotes the differential of $\sigma$ at $e$, which can be expressed in terms of the Lie group exponential map $\exp : \mathfrak{g} \rightarrow G$ via \[ d\sigma(Z) = \left.\frac{d}{dt}\right\|_{t=0} \sigma(\exp(tZ)). \] The space $\mathfrak{p}$, which is not a Lie subalgebra of $\mathfrak{g}$, has the structure of Lie triple system: it is a vector space closed under the double commutator $[ \cdot, [\cdot,\cdot]]$. In contrast, the space \[ \mathfrak{k} = \{Z \in \mathfrak{g} \mid d\sigma(Z) = Z\} \] is a subalgebra of $\mathfrak{g}$, as it is closed under the commutator $[\cdot,\cdot]$. This subalgebra is none other than the Lie algebra of $G^\sigma$. The generalized polar decomposition~(\ref{gpd}) has a manifestation at the Lie algebra level called the \emph{Cartan decomposition}, which decomposes $\mathfrak{g}$ as a direct sum \begin{equation} \label{cartan} \mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}. \end{equation} All of these observations lead to the conclusion that the following diagram commutes: \begin{center} \begin{tikzpicture} \node (G) {$G$}; \node[below=1.5cm of G] (quotient) {$G/G^\sigma$}; \node[left=1.5cm of quotient] (Gsub) {$G_\sigma$}; \node[right=1.5cm of quotient] (X) {$\mathcal{S}$}; \node[left=1.5cm of Gsub] (p) {$\mathfrak{p}$}; \node[above=1.5cm of p] (g) {$\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{k}$}; \node[above=1.5cm of g] (k) {$\mathfrak{k}$}; \node[right=1.5cm of k] (Gsup) {$G^\sigma$}; \draw[-latex] (G) edge node (Gtoquotient) [left]{$\pi$} (quotient) (G) edge node (GtoX) [above right]{$\varphi$} (X) (quotient) edge node (quotienttoX) [below]{$\bar{\varphi}$} (X) (Gsub) edge node (Gsubtoquotient) [below]{$\psi$} (quotient) (Gsub) edge node (GsubtoG) [above left]{$\iota$} (G) (p) edge node (ptoGsub) [below]{$\exp$} (Gsub) (p) edge node (ptog) [left]{$\iota$} (g) (g) edge node (gtoG) [above]{$\exp$} (G) (k) edge node (ktog) [left]{$\iota$} (g) (k) edge node (ktoGsup) [above]{$\mathrm{exp}$} (Gsup) (Gsup) edge node (GsuptoG) [above right]{$\iota$} (G); \end{tikzpicture} \end{center} In this diagram, we have used the letter $\iota$ to denote the canonical inclusion, $\pi : G \rightarrow G/G^\sigma$ the canonical projection, and $\varphi : G \rightarrow \mathcal{S}$ the map $\varphi(g) = g \cdot \eta$. The maps $\psi$ and $\bar{\varphi}$ are defined by the condition that the diagram be commutative. \subsection{Correspondence between Symmetric Spaces and Lie Triple Systems} \label{sec:correspondence} An important feature of the diagram above is that the maps along its bottom row -- when restricted to suitable neighborhoods of the neutral elements $0 \in \mathfrak{p}$, $e \in G_\sigma$, $[e] \in G/G^\sigma$, and the distinguished element $\eta \in \mathcal{S}$ -- are diffeomorphisms~\cite{helgason1979differential}. In particular, the composition \begin{equation} \label{bijection} F = \bar{\varphi} \circ \psi \circ \exp \end{equation} (or, equivalently, $F = \varphi \circ \iota \circ \exp$) provides a diffeomorphism from a neighborhood of $0 \in \mathfrak{p}$ to a neighborhood of $\eta \in \mathcal{S}$, given by \[ F(P) = \exp(P) \cdot \eta \] for $P \in \mathfrak{p}$. The space $\mathfrak{p}$, being a vector space, offers a convenient space to perform computations (such as averaging, interpolation, extrapolation, and the numerical solution of differential equations) that might otherwise be unwieldy on the space $\mathcal{S}$. This is analogous to the situation that arises when working with the Lie group $G$. Often, computations on $G$ are more easily performed by mapping elements of $G$ to the Lie algebra $\mathfrak{g}$ via the inverse of the exponential map (or an approximation thereof), performing computations in $\mathfrak{g}$, and mapping the result back to $G$ via the exponential map (or an approximation thereof). We remark that the analogy just drawn between computing on Lie groups and computing on symmetric spaces is in fact more than a mere resemblance; the latter situation directly generalizes the former. Indeed, any Lie group $G$ can be realized as a symmetric space by considering the action of $G \times G$ on $G$ given by $(g,h) \cdot k = gkh^{-1}$. The stabilizer of $e \in G$ is the diagonal of $G \times G$, which is precisely the subgroup fixed by the involution $\sigma(g,h) = (h,g)$. In this setting, one finds that the map~(\ref{bijection}) takes $(X,-X) \in \mathfrak{g} \times \mathfrak{g}$ to $\exp(2X) \in G$. This shows that, up to a trivial modification, the map~(\ref{bijection}) reduces to the Lie group exponential map if $\mathcal{S}$ happens to be a Lie group. An additional feature of the map~(\ref{bijection}) is its equivariance with respect to the action of $G^\sigma$ on $\mathcal{S}$ and $\mathfrak{p}$. Specifically, for $g \in G$, let $\mathrm{Ad}_g : \mathfrak{g} \rightarrow \mathfrak{g}$ denote the adjoint action of $G$ on $\mathfrak{g}$: \[ \mathrm{Ad}_g Z = \left.\frac{d}{dt}\right\|_{t=0} g \exp(tZ) g^{-1}. \] In a slight abuse of notation, we will write \[ \mathrm{Ad}_g Z = g Z g^{-1} \] in this paper, bearing in mind that the above equality holds in the sense of matrix multiplication for any matrix group. The following lemma shows that $F \circ \mathrm{Ad}_g \big\|_{\mathfrak{p}} = g \cdot F$ for every $g \in G^\sigma$. Note that this statement makes implicit use of the (easily verifiable) fact that $\mathrm{Ad}_g$ leaves $\mathfrak{p}$ invariant when $g \in G^\sigma$; that is, $gPg^{-1} \in \mathfrak{p}$ for every $g \in G^\sigma$ and every $P \in \mathfrak{p}$. \begin{lemma} \label{lemma:equivariance} For every $P \in \mathfrak{p}$ and every $g \in G^\sigma$, \[ g \cdot F(P) = F(gPg^{-1}). \] \end{lemma} \begin{proof} Note that $g \in G^\sigma$ implies $g^{-1} \in G^\sigma$, so $g^{-1} \cdot \eta = \eta$. Hence, since the adjoint action commutes with exponentiation, \begin{align} F(gPg^{-1}) &= \exp(gPg^{-1}) \cdot \eta \\ &= g\exp(P) g^{-1} \cdot \eta \\ &= g\exp(P) \cdot \eta \\ &= g \cdot F(P). \end{align} \end{proof} We finish this section by remarking that if $\mathcal{S}$ is a Riemannian manifold, then $\sigma$ induces a family of geodesic symmetries on $\mathcal{S}$ as follows. Define $s_\eta : \mathcal{S} \rightarrow \mathcal{S}$ by setting \[ s_\eta(g \cdot \eta) = \sigma(g) \cdot \eta. \] for each $g \in G$. Note that $s_\eta$ is a well-defined isometry that fixes $\eta$ and has differential equal to minus the identity. Furthermore, by definition, the following diagram commutes: \begin{center} \begin{tikzpicture} \node (p) {$\mathfrak{p}$}; \node[below=1.5cm of G] (p2) {$\mathfrak{p}$}; \node[right=1.5cm of p] (G) {$G$}; \node[right=1.5cm of p2] (G2) {$G$}; \node[right=1.5cm of G] (S) {$\mathcal{S}$}; \node[right=1.5cm of G2] (S2) {$\mathcal{S}$}; \draw[-latex] (p) edge node (ptoG) [above]{$\exp$} (G) (p2) edge node (p2toG2) [below]{$\exp$} (G2) (G) edge node (GtoS) [above]{$\varphi$} (S) (G2) edge node (G2toS2) [below]{$\varphi$} (S2) (p) edge node (ptop2) [left]{$d\sigma$} (p2) (G) edge node (GtoG2) [left]{$\sigma$} (G2) (S) edge node (StoS2) [right]{$s_\eta$} (S2); \end{tikzpicture} \end{center} Written another way, \begin{equation} \label{tauF} s_\eta(F(P)) = F(-P) \end{equation} for every $P \in \mathfrak{p}$. In a similar manner, a geodesic symmetry at each point $h \cdot \eta \in \mathcal{S}$ can be defined via \[ s_{h \cdot \eta} (g \cdot \eta) = h \cdot s_\eta(h^{-1} g \cdot \eta) = h \sigma(h^{-1}g) \cdot \eta. \] For every such $h \in G$, the map $s_ {h \cdot \eta}$ is an isometry, showing that $\mathcal{S}$ is a symmetric space. \subsection{Generalizations} \label{sec:correspondence_generalizations} The construction above can be generalized by replacing the exponential map in~(\ref{bijection}) with a different local diffeomorphism. One example is given by fixing an element $\bar{g} \in G$ and replacing $\exp : \mathfrak{p} \rightarrow G_\sigma$ in~(\ref{bijection}) with the map \begin{equation} \label{generalexp} P \mapsto \psi^{-1}\left([\bar{g} \exp(P)]\right). \end{equation} The output of this map is nothing more than the factor $p$ in the generalized polar decomposition $\bar{g}\exp(P)=pk$, $p \in G_\sigma$, $k \in G^\sigma$. The map~(\ref{bijection}) then becomes \begin{equation} \label{generalbijection} F_{\bar{g}}(P) = \bar{g}\exp(P) \cdot \eta. \end{equation} This generalization of~(\ref{bijection}) has the property that it provides a diffeomorphism between a neighborhood of $0 \in \mathfrak{p}$ and a neighborhood of $\bar{g} \cdot \eta \in \mathcal{S}$ rather than $\eta$. Note that when $\bar{g}=e$ (the identity element), this map coincides with~(\ref{bijection}). A calculation similar to the proof of Lemma~\ref{lemma:equivariance} shows that the map $F_{\bar{g}}$ is $G^\sigma$-equivariant, in the sense that \begin{equation} \label{generalequivariance} F_{h\bar{g}h^{-1}}(hPh^{-1}) = h \cdot F_{\bar{g}}(P) \end{equation} for every $h \in G^\sigma$ and every $P \in \mathfrak{p}$. Furthermore, \begin{equation} \label{general_tauF} s_{\bar{g} \cdot \eta} (F_{\bar{g}}(P)) = F_{\bar{g}}(-P) \end{equation} for every $P \in \mathfrak{p}$. These identities are summarized in the following pair of diagrams, the first of which commutes for every $h \in G^\sigma$, and the second of which commutes for every $\bar{g} \in G$. \begin{center} \begin{tikzpicture} \node (p) {$G \times \mathfrak{p}$}; \node[below=1.5cm of p] (p2) {$G \times \mathfrak{p}$}; \node[right=1.5cm of p] (S) {$\mathcal{S}$}; \node[right=1.5cm of p2] (S2) {$\mathcal{S}$}; \draw[-latex] (p) edge node (ptoS) [above]{$f$} (S) (p2) edge node (p2toS2) [below]{$f$} (S2) (p) edge node (ptop2) [left]{$\Psi_h$} (p2) (S) edge node (StoS2) [right]{$\Phi_h$} (S2); \end{tikzpicture} \hspace{1.5cm} \begin{tikzpicture} \node (p) {$\mathfrak{p}$}; \node[below=1.5cm of p] (p2) {$\mathfrak{p}$}; \node[right=1.5cm of p] (S) {$\mathcal{S}$}; \node[right=1.5cm of p2] (S2) {$\mathcal{S}$}; \draw[-latex] (p) edge node (ptoS) [above]{$F_{\bar{g}}$} (S) (p2) edge node (p2toS2) [below]{$F_{\bar{g}}$} (S2) (p) edge node (ptop2) [left]{$d\sigma$} (p2) (S) edge node (StoS2) [right]{$s_{\bar{g}\cdot\eta}$} (S2); \end{tikzpicture} \end{center} Here, we have denoted $f(\bar{g},P)=F_{\bar{g}}(P)$, $\Psi_h(\bar{g},P) = (h\bar{g}h^{-1}, hPh^{-1})$, and $\Phi_h(u) = h \cdot u$. More generally, one may consider replacing the exponential map in~(\ref{generalexp}) with any retraction $R : \mathfrak{g} \rightarrow G$~\cite{absil2009optimization}. For instance, if $G$ is a quadratic matrix group, one may choose $R$ equal to the Cayley transform, or more generally, any diagonal Pad\'e approximant of the matrix exponential~\cite{celledoni2000exponential}. \section{Interpolation on Symmetric Spaces} \label{sec:interp} In this section, we exploit the correspondence between symmetric spaces and Lie triple systems discussed in Sections~\ref{sec:correspondence}-\ref{sec:correspondence_generalizations} in order to interpolate functions which take values in a symmetric space. \subsection{A Structure-Preserving Interpolant} Consider the task of interpolating $m$ elements $u_1,u_2,\dots,u_m \in \mathcal{S}$, which we will think of as the values of a smooth function $u : \Omega \rightarrow \mathcal{S}$ defined on a domain $\Omega \subset \mathbb{R}^d$, $d \ge 1$, at locations $x^{(1)},x^{(2)},\dots,x^{(m)} \in \Omega$. Our goal is thus to construct a function $\mathcal{I}u : \Omega \rightarrow \mathcal{S}$ that satisfies $\mathcal{I}u(x^{(i)}) = u_i$, $i=1,2,\dots,m$, and has a desired level of regularity (e.g., continuity). We assume in what follows that for each $x \in \Omega$, $u(x)$ belongs to the range of the map~(\ref{bijection}). We may then interpolate $u_1,u_2,\dots,u_m$ by interpolating $F^{-1}(u_1), F^{-1}(u_2),\dots,F^{-1}(u_m) \in \mathfrak{p}$ and mapping the result back to $\mathcal{S}$ via $F$. More precisely, set \[ \mathcal{I}u(x) = F(\hat{\mathcal{I}}P(x)), \] where $P(x) = F^{-1}(u(x))$ and $\hat{\mathcal{I}}P : \Omega \rightarrow \mathfrak{p}$ is an interpolant of $F^{-1}(u_1), F^{-1}(u_2),\dots,F^{-1}(u_m)$. Then $\mathcal{I}u$ interpolates the data while fulfilling the following important properties. \begin{proposition} \label{prop:interp_equivariance} Suppose that $\hat{\mathcal{I}}$ commutes with $\mathrm{Ad}_g$ for every $g \in G^\sigma$. That is, \[ \hat{\mathcal{I}}(g P g^{-1}) (x) = g \hat{\mathcal{I}} P(x) g^{-1} \] for every $x \in \Omega$ and every $g \in G^\sigma$. Then $\mathcal{I}$ is $G^\sigma$-equivariant. That is, \[ \mathcal{I} (g \cdot u)(x) = g \cdot \mathcal{I} u(x) \] for every $x \in \Omega$ and every $g \in G^\sigma$ sufficiently close to the identity. \end{proposition} \begin{proof} The claim is a straightforward consequence of Lemma~\ref{lemma:equivariance}. \end{proof} \begin{proposition} \label{prop:involution_equivariance} Suppose that $\hat{\mathcal{I}}$ commutes with $d\sigma\big\|_{\mathfrak{p}}$. That is, \[ \hat{\mathcal{I}}(-P)(x) = -\hat{\mathcal{I}}P(x) \] for every $x \in \Omega$. Then $\mathcal{I}$ commutes with $s_\eta$. That is, \[ \mathcal{I} (s_\eta(u))(x) = s_\eta(\mathcal{I}u(x)) \] for every $x \in \Omega$. \end{proposition} \begin{proof} The claim is a straightforward consequence of~(\ref{tauF}). \end{proof} The preceding propositions apply, in particular, to any interpolant $\hat{\mathcal{I}}P : \Omega \rightarrow \mathfrak{p}$ of the form \[ \hat{\mathcal{I}}P(x) = \sum_{i=1}^m \phi_i(x) P(x^{(i)}) \] with scalar-valued shape functions $\phi_i : \Omega \rightarrow \mathbb{R}$, $i=1,2,\dots,m$, satisfying $\phi_i(x^{(j)}) = \delta_{ij}$, where $\delta_{ij}$ denotes the Kronecker delta. By the propositions above, such an interpolant gives rise to a $G^\sigma$-equivariant interpolant $\mathcal{I}u : \Omega \rightarrow \mathcal{S}$ that commutes with $s_\eta$, given by \begin{equation} \label{interp} \mathcal{I}u(x) = F\left( \sum_{i=1}^m \phi_i(x) F^{-1}(u_i) \right). \end{equation} Written more explicitly, \begin{equation} \label{interp_explicit1} \mathcal{I}u(x) = \exp(P(x)) \cdot \eta, \end{equation} where \begin{equation} \label{interp_explicit2} P(x) = \sum_{i=1}^m \phi_i(x) F^{-1}(u_i). \end{equation} \subsection{Derivatives of the Interpolant} The relations~(\ref{interp_explicit1}-\ref{interp_explicit2}) lead to an explicit formula for the derivatives of $\mathcal{I}u(x)$ with respect to each of the coordinate directions $x_j$, $j=1,2,\dots,d$. Namely, \begin{equation} \label{dIu} \frac{\partial \mathcal{I}u}{\partial x_j}(x) = \mathrm{dexp}_{P(x)} \frac{\partial P}{\partial x_j}(x) \cdot \eta, \end{equation} where \[ \frac{\partial P}{\partial x_j}(x) = \sum_{i=1}^m \frac{\partial\phi_i}{\partial x_j}(x) F^{-1}(u_i) \] and $\mathrm{dexp}_X Y$ denotes the differential of $\exp$ at $X \in \mathfrak{g}$ in the direction $Y \in \mathfrak{g}$. An explicit formula for $\mathrm{dexp}_X Y$ is the series \[ \mathrm{dexp}_X Y = \exp(X) \sum_{k=0}^\infty \frac{(-1)^k}{(k+1)!} \mathrm{ad}_X^k Y, \] where $\mathrm{ad}_X Y = [X,Y]$ denotes the adjoint action of $\mathfrak{g}$ on itself. In practice, one may truncate this series to numerically approximate $\mathrm{dexp}_X Y$. Note that while the exact value of $\mathrm{dexp}_X Y$ belongs to $\mathfrak{p}$ whenever $X,Y \in \mathfrak{p}$, this need not be true of its truncated approximation. However, this is of little import since any spurious $\mathfrak{k}$-components in such a truncation act trivially on $\eta$ in~(\ref{dIu}). While the series expansion of $\mathrm{dexp}_X Y$ is valid on any finite-dimensional Lie group, more efficient methods are available for the computation of $\mathrm{dexp}_X Y$ when $G$ is a matrix group. Arguably the simplest is to make use of the identity~\cite{mathias1996chain,higham2008functions} \begin{equation} \label{blockexp} \exp \begin{pmatrix} X & Y \\ 0 & X \end{pmatrix} = \begin{pmatrix} \exp(X) & \mathrm{dexp}_X Y \\ 0 & \exp(X) \end{pmatrix}. \end{equation} More sophisticated approaches with better numerical properties can be found in~\cite{al2009computing,higham2008functions}. The identity~(\ref{blockexp}) can be leveraged to derive formulas for higher-order derivatives of $\mathcal{I}u(x)$, provided of course that $G$ is a matrix group. As shown in Appendix~\ref{app:d2exp}, we have \begin{equation} \label{d2Iu} \frac{\partial^2 \mathcal{I} u}{\partial x_j \partial x_k}(x) = A \cdot \eta \end{equation} for each $j,k=1,2,\dots,d$, where $A$ denotes the $(1,4)$ block of the matrix \[ \mathrm{exp}\begin{pmatrix} X & Y & Z & W \\ 0 & X & 0 & Z \\ 0 & 0 & X & Y \\ 0 & 0 & 0 & X \end{pmatrix}, \] and $X = P(x)$, $Y = \frac{\partial P}{\partial x_j}(x)$, $Z = \frac{\partial P}{\partial x_k}(x)$, and $W = \frac{\partial^2 P}{\partial x_j \partial x_k}(x)$. \subsection{Generalizations} \label{sec:interp_generalizations} More generally, by fixing an element $\bar{g} \in G$ and adopting the map~(\ref{generalbijection}) instead of $F$, we obtain interpolation schemes of the form \begin{equation} \label{generalinterp} \mathcal{I}_{\bar{g}} u(x) = F_{\bar{g}}\left( \sum_{i=1}^m \phi_i(x) F_{\bar{g}}^{-1}(u_i) \right) = \bar{g} \exp\left( \sum_{i=1}^m \phi_i(x) F_{\bar{g}}^{-1}(u_i) \right) \cdot \eta. \end{equation} Here, we must of course assume that $u_i$ belongs to the range of $F_{\bar{g}}$ for each $i=1,2,\dots,m$. This interpolant is therefore suitable for interpolating elements of $\mathcal{S}$ in a neighborhood of $\bar{g} \cdot \eta$. Using the fact that $F_{hg}(P) = h \cdot F_{g}(P)$ for every $h, g \in G$ and every $P \in \mathfrak{p}$, one finds that this interpolant is equivariant under the action of the full group $G$, in the sense that \begin{equation} \label{general_equivariance} \mathcal{I}_{h\bar{g}} (h \cdot u)(x) = h \cdot \mathcal{I}_{\bar{g}} u(x) \end{equation} for every $x \in \Omega$ and every $h \in G$ sufficiently close to the identity. On the other hand, the equivariance of $F_{\bar{g}}$ under the action of the subgroup $G^\sigma$ (recall~(\ref{generalequivariance})) implies that \begin{equation} \label{general_equivariance_subgroup} \mathcal{I}_{h\bar{g}h^{-1}} (h \cdot u)(x) = h \cdot \mathcal{I}_{\bar{g}} u(x) \end{equation} for every $x \in \Omega$ and every $h \in G^\sigma$ sufficiently close to the identity. Comparing~(\ref{general_equivariance}) with~(\ref{general_equivariance_subgroup}) leads to the conclusion that this interpolant is invariant under post-multiplication of $\bar{g}$ by elements of $G^\sigma$; that is, \begin{equation} \label{postmult} \mathcal{I}_{\bar{g}h} u(x) = \mathcal{I}_{\bar{g}} u(x) \end{equation} for every $x \in \Omega$ and every $h \in G^\sigma$ sufficiently close to the identity. Finally, as a consequence of~(\ref{general_tauF}), \[ \mathcal{I}_{\bar{g}} (s_{\bar{g} \cdot \eta} (u))(x) = s_{\bar{g} \cdot \eta} (\mathcal{I}_{\bar{g}} u(x)) \] for every $x \in \Omega$. A natural choice for $\bar{g}$ is not immediately evident, but one heuristic is to select $j \in \{1,2,\dots,m\}$ and set $\bar{g}$ equal to a representative of the coset $ \bar{\varphi}^{-1} (u_j)$. A more interesting option is to allow $\bar{g}$ to vary with $x$ and to define $\bar{g}(x)$ implicitly via \begin{equation} \label{Riemannianmean} \bar{g}(x) \cdot \eta = \mathcal{I}_{\bar{g}(x)} u(x). \end{equation} Equivalently, \begin{equation} \label{Riemannianmean2} \sum_{i=1}^m \phi_i(x) F_{\bar{g}(x)}^{-1}(u_i) = 0. \end{equation} In analogy with~(\ref{general_equivariance}), the interpolant $\mathcal{I}_{\bar{g}(x)} u(x)$ so defined is equivariant with respect to the action of the full group $G$, not merely the subgroup $G^\sigma$. That is, \begin{equation} \label{Gequivariant} \mathcal{I}_{h\bar{g}(x)} (h \cdot u)(x) = h \cdot \mathcal{I}_{\bar{g}(x)} u(x) \end{equation} for every $x \in \Omega$ and every $h \in G$ sufficiently close to the identity. A method for computing the interpolant $\mathcal{I}_{\bar{g}(x)} u(x)$ numerically is self-evident. Namely, one performs the fixed-point iteration suggested by~(\ref{Riemannianmean}), as we explain in greater detail in Section~\ref{sec:applications}. We show below that if $G$ is equipped with a left-invariant Riemannian metric for which the restrictions to $\mathfrak{p}$ of the Lie group exponential and Riemannian exponential maps coincide, then~(\ref{Riemannianmean2}) characterizes the coset $[\bar{g}(x)] \in G/G^\sigma$ as the weighted Riemannian mean of the cosets $[g_1],[g_2],\dots,[g_m] \in G/G^\sigma$, where $g_i \cdot \eta = u_i$, $i=1,2,\dots,m$. The statement of this lemma makes use of the following observation. Any left-invariant Riemannian metric on $G$ is uniquely defined by an inner product on $\mathfrak{g}$. The restriction of this inner product to $\mathfrak{p}$ induces a left-invariant metric on $G/G^\sigma$ by virtue of the isomorphism $\mathfrak{p} \cong \mathfrak{g}/\mathfrak{k} = T_{[e]} (G/G^\sigma)$. \begin{lemma} \label{lemma:Riemannian_mean} Let $G$ be equipped with a left-invariant Riemmannian metric. For each $g \in G$, denote by $\mathrm{Exp}_g : T_g G \rightarrow G$ the corresponding Riemannian exponential map. Suppose that \begin{equation} \label{Expexp} \left.\mathrm{Exp}_e\right\|_{\mathfrak{p}} = \left.\exp\right\|_{\mathfrak{p}}. \end{equation} If $\bar{g}(x) \in G$ is a solution of~(\ref{Riemannianmean}) (or, equivalently, (\ref{Riemannianmean2})), then $[\bar{g}(x)]\in G/G^\sigma$ locally minimizes \begin{equation} \label{mindist} \sum_{i=1}^m \phi_i(x) \, \mathrm{dist}\left([h],[g_i]\right)^2 \end{equation} among all $[h] \in G/G^\sigma$, where $\mathrm{dist} : G/G^\sigma \times G/G^\sigma \rightarrow \mathbb{R}$ denotes the induced geodesic distance on $G/G^\sigma$, and $g_1,g_2,\dots,g_m \in G$ satisfy $g_i \cdot \eta = u_i$, $i=1,2,\dots,m$. \end{lemma} \begin{proof} For each $i=1,2,\dots,m$, let $P_i = F_{\bar{g}(x)}^{-1}(u_i)$, so that $\bar{g}(x) \exp(P_i) \cdot \eta = u_i$. Then, as cosets in $G/G^\sigma$, we have $[\bar{g}(x) \exp(P_i)] = [g_i]$. Define $c(t) = \bar{g}(x) \exp(tP_i)$ for $0 \le t \le 1$. Since $\bar{g}\exp(tP_i)$ coincides with $\bar{g} \, \mathrm{Exp}_e(tP_i) = \mathrm{Exp}_{\bar{g}}(tP_i)$, the curve $c(t)$ is a geodesic on $G$. Moreover, since $P_i \in \mathfrak{p}$, the tangent vector $c'(t)$ to this curve is everywhere horizontal. Thus, $[c(t)]$ is a geodesic in $G/G^\sigma$ satisfying $[c(0)] = [\bar{g}(x)]$ and $[c(1)] = [\bar{g}(x) \exp(P_i)] = [g_i]$. This shows that~(\ref{Riemannianmean2}) is equivalent to \[ \sum_{i=1}^m \phi_i(x) \mathrm{Exp}_{[\bar{g}(x)]}^{-1} [g_i] = 0, \] where $\mathrm{Exp}_{[\bar{g}(x)]} : T_{[\bar{g}(x)]} (G/G^\sigma) \rightarrow G/G^\sigma$ denotes the Riemannian exponential map at $[\bar{g}(x)] \in G/G^\sigma$. The latter equation is precisely the equation which characterizes minimizers of~(\ref{mindist}); see~\cite[Theorem 1.2]{karcher1977riemannian}. \end{proof} Lemma~\ref{lemma:Riemannian_mean} applies, in particular, if $G$ is equipped with a bi-invariant metric, since in that setting the Lie group exponential and Riemannian exponential maps coincide. Notice that minimizers of~(\ref{mindist}) are precisely geodesic finite elements on $G/G^\sigma$, as described in~\cite{grohs2015optimal,grohs2013quasi,sander2012geodesic,sander2015geodesic}. We refer the reader to those articles for further information about the approximation properties of these interpolants, as well as the convergence properties of iterative algorithms used to compute them. \section{Applications} \label{sec:applications} In this section, we apply the general theory above to several symmetric spaces, including the space of symmetric positive-definite matrices, the space of Lorentzian metrics, and the Grassmannian. \subsection{Symmetric Matrices with Fixed Signature} \label{sec:opq} Let $n$ be a positive integer and let $p$ and $q$ be nonnegative integers satisfying $p+q=n$. Consider the set \[ \mathcal{L} = \{L \in \mathbb{R}^{n \times n} \mid L=L^T, \, \det L \neq 0, \, \mathrm{signature}(L)=(q,p) \}, \] where $\mathrm{signature}(L)$ denotes the signature of a nonsingular symmetric matrix $L$ -- an ordered pair indicating the number of positive and negative eigenvalues of $L$. The general linear group $GL_n(\mathbb{R})$ acts transitively on $\mathcal{L}$ via the group action \[ A \cdot L = ALA^T, \] where $A \in GL_n(\mathbb{R})$ and $L \in \mathcal{L}$. Let $J = \mathrm{diag}(-1,\dots,-1,1,\dots,1)$ denote the diagonal $n \times n$ matrix with $p$ entries equal to $-1$ and $q$ entries equal to $1$. The stabilizer of $J$ in $GL_n(\mathbb{R})$ is the indefinite orthogonal group \[ O(p,q) = \{Q \in GL_n(\mathbb{R}) \mid Q J Q^T = J\}. \] Its elements are precisely those matrices that are fixed points of the involutive automorphism \begin{align} \sigma : GL_n(\mathbb{R}) &\rightarrow GL_n(\mathbb{R}) \\ A &\mapsto J A^{-T} J, \end{align} where $A^{-T}$ denotes the inverse transpose of a matrix $A \in GL_n(\mathbb{R})$. In contrast, the set of matrices which are mapped by $\sigma$ to their inverses is \[ Sym_J(n) = \{P \in GL_n(\mathbb{R}) \mid PJ = JP^T \}. \] The setting we have just described is an instance of the general theory presented in Section~\ref{sec:notation}, with $G = GL_n(\mathbb{R})$, $G^\sigma = O(p,q)$, $G_\sigma = Sym_J(n)$, $\mathcal{S}=\mathcal{L}$, and $\eta=J$. It follows that the generalized polar decomposition~(\ref{gpd}) of a matrix $A \in GL_n(\mathbb{R})$ (sufficiently close to the identity matrix $I$) with respect to $\sigma$ reads \begin{equation} \label{gpd_opq} A = PQ, \quad P \in Sym_J(n), \, Q \in O(p,q). \end{equation} The Cartan decomposition~(\ref{cartan}) decomposes an element $Z$ of the Lie algebra $\mathfrak{gl}_n(\mathbb{R}) = \mathbb{R}^{n \times n}$ of the general linear group as a sum \[ Z = X+Y, \quad X \in \mathfrak{sym}_J(n), \, Y \in \mathfrak{o}(p,q), \] where \[ \mathfrak{sym}_J(n) = \{ X \in \mathfrak{gl}_n(\mathbb{R}) \mid XJ = JX^T\} \] and \[ \mathfrak{o}(p,q) = \{Y \in \mathfrak{gl}_n(\mathbb{R}) \mid YJ + JY^T = 0 \} \] denotes the Lie algebra of $O(p,q)$. We can now write down the map $F : \mathfrak{sym}_J(n) \rightarrow \mathcal{L}$ defined abstractly in~(\ref{bijection}), which provides a diffeomorphism between a neighborhood of the zero matrix and a neighborhood of $J$. By definition, \begin{align} F(X) &= \exp(X) J \exp(X)^T \nonumber \\ &= \exp(X) \exp(X) J \nonumber \\ &= \exp(2X) J, \label{F_opq} \end{align} where the second line follows from the fact that $\exp(X) \in Sym_J(n)$ whenever $X \in \mathfrak{sym}_J(n)$. The inverse of $F$ can likewise be expressed in closed form. This can be obtained directly by solving~(\ref{F_opq}) for $X$, but it is instructive to see how to derive the same result by inverting each of the maps appearing in the composition~(\ref{bijection}). To start, note that explicit formulas for the matrices $P$ and $Q$ in the decomposition~(\ref{gpd_opq}) of a matrix $A \in GL_n(\mathbb{R})$ are known~\cite{higham2010canonical}. Provided that $AJA^T J$ has no negative real eigenvalues, we have \begin{align} P &= (AJA^T J)^{1/2}, \\ Q &= (AJA^T J)^{-1/2} A, \end{align} where $B^{1/2}$ denotes the principal square root of a matrix $B$, and $B^{-1/2}$ denotes the inverse of $B^{1/2}$. Thus, if $A \cdot J = AJA^T = L \in \mathcal{L}$ and if $LJ$ has no negative real eigenvalues, then the factor $P$ in the polar decomposition~(\ref{gpd_opq}) of $A$ is given by \[ P = (LJ)^{1/2}. \] It follows that for such a matrix $L$, \begin{equation} F^{-1}(L) = \log\left( (LJ)^{1/2} \right), \end{equation} where $\log(B)$ denotes the principal logarithm of a matrix $B$. We henceforth denote by $\mathcal{L}_$ the set of matrices $L \in \mathcal{L}$ for which $LJ$ has no negative real eigenvalues, so that $F^{-1}(L)$ is well-defined for $L \in \mathcal{L}_$. The right-hand side of~(\ref{interp1_opq}) can be simplified using the following property of the matrix logarithm, whose proof can be found in~\cite[Theorem 11.2]{higham2008functions}: If a square matrix $B$ has no negative real eigenvalues, then \[ \log(B^{1/2}) = \frac{1}{2}\log(B). \] From this it follows that \begin{equation} \label{Finv_opq} F^{-1}(L) = \frac{1}{2}\log\left( LJ \right) \end{equation} for $L \in \mathcal{L}_$. This formula, of course, could have been obtained directly from~(\ref{F_opq}), but we have chosen a more circuitous derivation to give a concrete illustration of the theory presented in Section~\ref{sec:symspace}. Substituting~(\ref{F_opq}) and~(\ref{Finv_opq}) into~(\ref{interp}) gives the following heuristic for interpolating a set of matrices $L_1,L_2,\dots,L_m \in \mathcal{L}_$ -- thought of as the values of a smooth function $L : \Omega \rightarrow \mathcal{L}_$ at points $x^{(1)},x^{(2)},\dots,x^{(m)}$ in a domain $\Omega$ -- at a point $x \in \Omega$: \begin{equation} \label{interp1_opq} \mathcal{I}L(x) = \exp\left( \sum_{i=1}^m \phi_i(x) \log\left( L_i J \right) \right) J. \end{equation} Here, as before, the functions $\phi_i : \Omega \rightarrow \mathbb{R}$, $i=1,2,\dots,m$, denote scalar-valued shape functions with the property that $\phi_i(x^{(j)}) = \delta_{ij}$. Using the identity $J^{-1}=J$ together with the fact that the matrix exponential and logarithm commute with conjugation, the right-hand side of~(\ref{interp1_opq}) can be written equivalently as \begin{equation} \label{interp2_opq} \mathcal{I}L(x) = J \exp\left( \sum_{i=1}^m \phi_i(x) \log\left( J L_i \right) \right). \end{equation} In addition to satisfying $\mathcal{I}L(x) \in \mathcal{L}$ for every $x \in \Omega$, the interpolant so defined enjoys the following properties, which generalize the observations made in Theorems 3.13 and 4.2 of~\cite{arsigny2007geometric}. \begin{lemma} \label{lemma:equivariance_opq} Let $Q \in O(p,q)$. If $\tilde{L}_i = Q L_i Q^T$, $i=1,2,\dots,m$, and if $Q$ is sufficiently close to the identity matrix, then \[ \mathcal{I}\tilde{L}(x) = Q \, \mathcal{I}L(x) \, Q^T. \] for every $x \in \Omega$. \end{lemma} \begin{proof} Apply Proposition~\ref{prop:interp_equivariance}. \end{proof} \begin{lemma} If $\tilde{L}_i = JL_i^{-1}J$, $i=1,2,\dots,m$, then \[ \mathcal{I}\tilde{L}(x) = J\left(\mathcal{I}L(x)\right)^{-1}J. \] for every $x \in \Omega$. \end{lemma} \begin{proof} Apply Proposition~\ref{prop:involution_equivariance}, noting that $s_\eta(L) = JL^{-1}J$ for $L \in \mathcal{L}$. \end{proof} Note that the preceding two propositions can be combined to conclude that if $\tilde{L}_i = L_i^{-1}$, $i=1,2,\dots,m$, then \[ \mathcal{I}\tilde{L}(x) = \left(\mathcal{I}L(x)\right)^{-1}. \] To see this, observe that $L_i^{-1} = J (JL_i^{-1}J) J^T$ and $J \in O(p,q)$. \begin{lemma} If $\sum_{i=1}^m \phi_i(x) = 1$ for every $x \in \Omega$, then \[ \det \mathcal{I}L(x) = \prod_{i=1}^m \left( \det L_i \right)^{\phi_i(x)} \] for every $x \in \Omega$. \end{lemma} \begin{proof} Using the identities $\det\exp(A) = \exp(\mathrm{tr}(A))$ and $\mathrm{tr}(\log(A)) = \log(\det A)$, we have \begin{align} \det \mathcal{I}L(x) &= \det \left( \exp\left( \sum_{i=1}^m \phi_i(x) \log( L_i J ) \right) \right) \det J \\ &= \exp \left( \mathrm{tr} \left( \sum_{i=1}^m \phi_i(x) \log( L_i J ) \right) \right) \det J \\ &= \exp \left( \sum_{i=1}^m \phi_i(x) \mathrm{tr} \left( \log( L_i J ) \right) \right) \det J \\ &= \exp \left( \sum_{i=1}^m \phi_i(x) \log \left( \det( L_i J ) \right) \right) \det J \\ &= \left( \prod_{i=1}^m \det( L_i J )^{\phi_i(x)} \right) \det J \\ &= \left( \prod_{i=1}^m \det( L_i )^{\phi_i(x)} \det( J )^{\phi_i(x)} \right) \det J \\ \end{align} The conclusion then follows from the facts that $\sum_{i=1}^m \phi_i(x) = 1$ and $\det J = \pm 1$. \end{proof} \paragraph{Generalizations.} As explained abstractly in Section~\ref{sec:interp_generalizations}, the interpolation formula~(\ref{interp2_opq}) can be generalized by fixing an element $\bar{A} \in GL_n(\mathbb{R})$ and replacing~(\ref{F_opq}) with the map \[ F_{\bar{A}}(X) = \bar{A} \exp(X) J \left(\bar{A} \exp(X)\right)^T = \bar{A} F(X) \bar{A}^T. \] The inverse of this map reads \[ F_{\bar{A}}^{-1} (L) = \frac{1}{2} \log( \bar{A}^{-1} L \bar{A}^{-T} J ). \] Substituting into~(\ref{generalinterp}), gives, after some simplification, the interpolation formula \begin{equation} \label{interp_general_opq} \mathcal{I}_{\bar{A}} L(x) = \bar{L} \exp\left( \sum_{i=1}^m \phi_i(x) \log\left( \bar{L}^{-1} L_i \right) \right), \end{equation} where $\bar{L} = \bar{A} J \bar{A}^T$. Rather than fixing $\bar{A}$, one may choose to define $\bar{A}$ implicitly via~(\ref{Riemannianmean}); that is, \[ \bar{A}(x) J \bar{A}(x)^T = \mathcal{I}_{\bar{A}(x)} L(x). \] The output of the resulting interpolation scheme is the solution $\bar{L}$ to the equation \begin{equation} \label{Riemannian_mean_opq} \sum_{i=1}^m \phi_i(x) \log\left( \bar{L}^{-1} L_i \right) = 0, \end{equation} which can be computed with a fixed-point iteration. \paragraph{Algorithms.} In summary, we have derived the following pair of algorithms for interpolating matrices in the space $\mathcal{L}$ of nonsingular symmetric matrices with signature $(q,p)$. The first of these algorithms implements~(\ref{interp_general_opq}), which reduces to~(\ref{interp2_opq}) when $\bar{L}$ is taken equal to $J$. The algorithm implicitly requires its inputs to have the property that for each $i=1,2,\dots,m$, the matrix $\bar{L}^{-1} L_i$ has no negative real eigenvalues. \begin{algorithm}[H] \caption{Interpolation of symmetric matrices with fixed signature} \label{alg:opq} \begin{algorithmic}[1] \Require Matrices $\{L_i \in \mathcal{L}\}_{i=1}^m$, shape functions $\{\phi_i : \Omega \rightarrow \mathbb{R} \}_{i=1}^m$, point $x \in \Omega$, matrix~$\bar{L} \in \mathcal{L}$ \State\Return $\bar{L} \exp\left( \sum_{i=1}^m \phi_i(x) \log\left( \bar{L}^{-1} L_i \right) \right)$ \end{algorithmic} \end{algorithm} The second algorithm solves~(\ref{Riemannian_mean_opq}), and requires the same constraint on its inputs as Algorithm~\ref{alg:opq}. Observe that Algorithm~\ref{alg:opq} is equivalent to Algorithm~\ref{alg:opq_Riemannian} if one terminates the fixed-point iteration after the first iteration. \begin{algorithm}[H] \caption{Iterative interpolation of symmetric matrices with fixed signature} \label{alg:opq_Riemannian} \begin{algorithmic}[1] \Require Matrices $\{L_i \in \mathcal{L}\}_{i=1}^m$, shape functions $\{\phi_i : \Omega \rightarrow \mathbb{R} \}_{i=1}^m$, point $x \in \Omega$, initial guess~$\bar{L} \in \mathcal{L}$, tolerance $\varepsilon > 0$ \While{$\left\\| \sum_{i=1}^m \phi_i(x) \log\left( \bar{L}^{-1} L_i \right) \right\\| > \varepsilon$ } \State $\bar{L}= \bar{L} \exp\left( \sum_{i=1}^m \phi_i(x) \log\left( \bar{L}^{-1} L_i \right) \right)$ \EndWhile \State\Return $\bar{L}$ \end{algorithmic} \end{algorithm} \subsubsection{Symmetric Positive-Definite Matrices} \label{sec:spd} When $J=I$, the preceding theory provides structure-preserving interpolation schemes for the space $SPD(n)$ of symmetric positive-definite matrices. The formula~(\ref{interp2_opq}) is the weighted log-Euclidean mean introduced by~\cite{arsigny2007geometric}, and equation~(\ref{Riemannian_mean_opq}) gives the weighted Riemannian mean (or Karcher mean) of symmetric positive-definite matrices~\cite{moakher2005differential,bhatia2013riemannian,karcher1977riemannian}. The latter observation can be viewed as a consequence of Lemma~\ref{lemma:Riemannian_mean}, which applies in this setting for the following reason. If the general linear group is equipped with the canonical left-invariant Riemannian metric, then the Riemmannian exponential map $\mathrm{Exp}_I : T_I GL_n(\mathbb{R}) \rightarrow GL_n(\mathbb{R})$ at the identity reads~\cite{andruchow2014left} \[ \mathrm{Exp}_I(A) = \exp(A^T) \exp(A-A^T). \] This formula reduces to $\mathrm{Exp}_I(A) = \exp(A)$ when $A$ is a normal matrix (i.e., $A^T A = A A^T$). In particular, $\mathrm{Exp}_I(A) = \exp(A)$ when $A$ is symmetric, which is precisely the condition~(\ref{Expexp}). We remark that the interpolation formula~(\ref{interp2_opq}) on $SPD(n)$ was devised in~\cite{arsigny2007geometric} by endowing $SPD(n)$ with what the authors term a ``novel vector space structure.'' This vector space structure is nothing more than that obtained by identifying $SPD(n)$ with the Lie triple system $\mathfrak{sym}_I(n)$ via the map~(\ref{Finv_opq}), as we have done here. \subsubsection{Lorentzian Metrics} \label{sec:lorentz} When $n=4$ and $J = \mathrm{diag}(-1,1,1,1)$, the preceding theory provides structure-preserving interpolation schemes for the space of Lorentzian metrics -- the space of symmetric, nonsingular matrices having signature $(3,1)$. Lemma~\ref{lemma:equivariance_opq} states that the interpolation operator~(\ref{interp2_opq}) in this setting commutes with Lorentz transformations. By choosing, for instance, $\Omega$ equal to a four-dimensional simplex (or a four-dimensional hypercube) and $\{\phi_i\}_i$ equal to scalar-valued Lagrange polynomials (or tensor products of Lagrange polynomials) on $\Omega$, one obtains a family of Lorentzian metric-valued finite elements. In view of their potential application to numerical relativity, we have numerically computed the interpolation error committed by such elements when approximating the Schwarzschild metric, which is an explicit solution to Einstein's equations outside of a spherical mass~\cite{carroll2004spacetime}. In Cartesian coordinates, this metric reads \begin{equation} \label{schwarz} L(t,x,y,z) = \begin{pmatrix} -\left(1 - \frac{R}{r}\right) & 0 & 0 & 0 \\ 0 & 1+\left(\frac{R}{r-R}\right) \frac{x^2}{r^2} & \left(\frac{R}{r-R}\right) \frac{xy}{r^2} & \left(\frac{R}{r-R}\right) \frac{xz}{r^2} \\ 0 & \left(\frac{R}{r-R}\right) \frac{xy}{r^2} & 1+\left(\frac{R}{r-R}\right) \frac{y^2}{r^2} & \left(\frac{R}{r-R}\right) \frac{yz}{r^2} \\ 0 & \left(\frac{R}{r-R}\right) \frac{xz}{r^2} & \left(\frac{R}{r-R}\right) \frac{yz}{r^2} & 1+\left(\frac{R}{r-R}\right) \frac{z^2}{r^2} \end{pmatrix}, \end{equation} where $R$ (the Schwarzschild radius) is a positive constant (which we take equal to 1 in what follows) and $r=\sqrt{x^2+y^2+z^2} > R$. We interpolated this metric over the region $U = \{0\} \times [2,3] \times [2,3] \times [2,3]$ on a uniform $N \times N \times N$ grid of cubes using the formula~(\ref{interp2_opq}) elementwise, with shape functions $\{\phi_i\}_i$ given by tensor products of Lagrange polynomials of degree $k$. The results in Table~\ref{tab:schwarz} indicate that the $L^2$-error \begin{equation} \label{L2err} \\|\mathcal{I}L - L\\|_{L^2(U)} = \left( \int_U \left\\| \mathcal{I}L(t,x,y,z) - L(t,x,y,z) \right\\|_F^2 \, dx \, dy \, dz \right)^{1/2} \end{equation} (which we approximated with numerical quadrature) converges to zero with order 2 and 3, respectively, when using polynomials of degree $k=1$ and $k=2$. Here, $\\|\cdot\\|_F$ denotes the Frobenius norm. In addition, Table~\ref{tab:schwarz} indicates that the $H^1$-error \begin{equation} \label{H1err} \|\mathcal{I}L - L\|_{H^1(U)} = \left( \int_U \sum_{j=1}^4 \left\\| \frac{\partial \mathcal{I}L}{\partial \xi_j}(t,x,y,z) - \frac{\partial L}{\partial \xi_j}(t,x,y,z) \right\\|_F^2 \, dx \, dy \, dz \right)^{1/2} \end{equation} converges to zero with order 1 and 2, respectively, when using polynomials of degree $k=1$ and $k=2$. Here, we have denoted $\xi=(t,x,y,z)$. For the sake of comparison, Table~\ref{tab:schwarzcomponentwise} shows the interpolation errors committed when applying componentwise polynomial interpolation to the same problem. Within each element, the value of this interpolant at a point $\xi=(t,x,y,z)$ lying in the element is given by \begin{equation} \label{componentwiseinterp} \mathcal{I}L(\xi) = \sum_{i=1}^m \phi_i(\xi) L_i, \end{equation} where $\{\phi_i\}_i$ are tensor products of Lagrange polynomials of degree $k$ and $\{L_i\}_i$ are the values of $L$ at the corresponding degrees of freedom. The errors committed by this interpolation scheme are very close to those observed in Table~\ref{tab:schwarz} for the structure-preserving scheme~(\ref{interp2_opq}). For this particular numerical example, the componentwise polynomial interpolant~(\ref{componentwiseinterp}) has correct signature $(3,1)$ for every $(t,x,y,z) \in U$. This need not hold in general. For example, consider the metric tensor \[ L(t,x,y,z) = \begin{pmatrix} -6\sin^2(2\pi x)+3\sin^2(\pi x) & 3\cos(2\pi x) & 0 & 0 \\ 3\cos(2\pi x) & 2\sin^2(2\pi x)+2\sin^2(\pi x) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. \] Though not a solution to Einstein's equations, this metric tensor nonetheless has signature $(3,1)$ everywhere. Indeed, a numerical calculation verifies that at all points $(t,x,y,z)$, the matrix $L(t,x,y,z)$ has eigenvalues $\lambda_-,1,1,\lambda_+$ satisfying $\lambda_- \le \alpha$ and $\lambda_+ \ge \beta$ with $\alpha \approx -0.54138$ and $\beta \approx 2.23064$. Interpolating this metric componentwise with linear polynomials (over the region same region $U$ as above) produces a metric with signature $(4,0)$ at 32 quadrature points (out of 64 total) on the coarsest grid ($N=2$). The essence of the problem is that for any integer $k$, any $t$, any $y$, and any $z$, the average of $L(t,k/2,y,z)$ and $L(t,(k+1)/2,y,z)$ is \[ \frac{1}{2} \left( L(t,k/2,y,z) + L(t,(k+1)/2,y,z) \right) = \begin{pmatrix} \frac{3}{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}, \] which shows (by continuity of the interpolant) that the componentwise linear interpolant~(\ref{componentwiseinterp}) on the coarsest grid ($N=2$) is positive definite on an open subset of $U$. In contrast, the structure-preserving scheme~(\ref{interp2_opq}) automatically generates an interpolant with correct signature $(3,1)$ at all points $(t,x,y,z)$. \begin{table}[t] \centering \pgfplotstabletypeset[ every head row/.style={after row=\midrule,before row={\midrule & \multicolumn{4}{c\|}{$k=1$} & \multicolumn{4}{c}{$k=2$} \\ \midrule}}, create on use/rate1/.style={create col/dyadic refinement rate={1}}, create on use/rate2/.style={create col/dyadic refinement rate={2}}, create on use/rate3/.style={create col/dyadic refinement rate={3}}, create on use/rate4/.style={create col/dyadic refinement rate={4}}, columns={0,1,rate1,2,rate2,3,rate3,4,rate4}, columns/0/.style={sci zerofill,column type/.add={}{\|},column name={$N$}}, columns/1/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$L^2$-error}}, columns/2/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$H^1$-error}}, columns/3/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$L^2$-error}}, columns/4/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$H^1$-error}}, columns/rate1/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate2/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate3/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate4/.style={fixed zerofill,precision=3,column name={Order}} ] {Data/schwarz.dat} \caption{Error incurred when interpolating the Schwarzschild metric~(\ref{schwarz}) over the region $U = \{0\} \times [2,3] \times [2,3] \times [2,3]$ using the formula~(\ref{interp2_opq}). The interpolant was computed elementwise on a uniform $N \times N \times N$ grid of cubes, with shape functions $\{\phi_i\}_i$ on each cube given by tensor products of Lagrange polynomials of degree $k$.} \label{tab:schwarz} \end{table} \begin{table}[t] \centering \pgfplotstabletypeset[ every head row/.style={after row=\midrule,before row={\midrule & \multicolumn{4}{c\|}{$k=1$} & \multicolumn{4}{c}{$k=2$} \\ \midrule}}, create on use/rate1/.style={create col/dyadic refinement rate={1}}, create on use/rate2/.style={create col/dyadic refinement rate={2}}, create on use/rate3/.style={create col/dyadic refinement rate={3}}, create on use/rate4/.style={create col/dyadic refinement rate={4}}, columns={0,1,rate1,2,rate2,3,rate3,4,rate4}, columns/0/.style={sci zerofill,column type/.add={}{\|},column name={$N$}}, columns/1/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$L^2$-error}}, columns/2/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$H^1$-error}}, columns/3/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$L^2$-error}}, columns/4/.style={sci zerofill,precision=3,column type/.add={}{\|},column name={$H^1$-error}}, columns/rate1/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate2/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate3/.style={fixed zerofill,precision=3,column type/.add={}{\|},column name={Order}}, columns/rate4/.style={fixed zerofill,precision=3,column name={Order}} ] {Data/schwarzcomponentwise.dat} \caption{Error incurred when interpolating the Schwarzschild metric~(\ref{schwarz}) over the region $U = \{0\} \times [2,3] \times [2,3] \times [2,3]$ using the componentwise interpolation formula~(\ref{componentwiseinterp}). The interpolant was computed elementwise on a uniform $N \times N \times N$ grid of cubes, with shape functions $\{\phi_i\}_i$ on each cube given by tensor products of Lagrange polynomials of degree $k$.} \label{tab:schwarzcomponentwise} \end{table} \subsection{The Grassmannian} \label{sec:grass} Let $p$ and $n$ be positive integers satisfying $p < n$. Consider the Grassmannian $Gr(p,n)$, which consists of all $p$-dimensional linear subspaces of $\mathbb{R}^n$. Any element $\mathcal{V} \in Gr(p,n)$ can be written as the span of $p$ vectors $v_1,v_2,\dots,v_p \in \mathbb{R}^n$. The orthogonal group $O(n)$ acts transitively on $Gr(p,n)$ via the action \[ A \cdot \mathrm{span}(v_1,v_2,\dots,v_p) \mapsto \mathrm{span}(Av_1,Av_2,\dots,Av_p), \] where $A \in O(n)$. For convenience, we will sometimes write $A\mathcal{V}$ as shorthand for \break$\mathrm{span}(Av_1,Av_2,\dots,Av_p)$. Let $e_1,e_2,\dots,e_n$ be the canonical basis for $\mathbb{R}^n$. The stabilizer of $\mathrm{span}(e_1,e_2,\dots,e_p)$ in $O(n)$ is the subgroup \[ O(p) \times O(n-p) = \left\{ \begin{pmatrix} A_1 & 0 \\ 0 & A_2 \end{pmatrix} \mid A_1 \in O(p), \, A_2 \in O(n-p) \right\}. \] The elements of $O(p) \times O(n-p)$ are precisely those matrices in $O(n)$ that are fixed points of the involutive automorphism \begin{align} \sigma : O(n) &\rightarrow O(n) \\ A &\mapsto J A J, \end{align} where \[ J = \begin{pmatrix} -I_p & 0 \\ 0 & I_{n-p} \end{pmatrix}, \] and $I_p$ and $I_{n-p}$ denote the $p \times p$ and $(n-p) \times (n-p)$ identity matrices, respectively. The matrices in $O(n)$ that are mapped to their inverses by $\sigma$ constitute the space \[ Sym_J(n) \cap O(n) = \{ P \in O(n) \mid PJ = JP^T \}. \] The generalized polar decomposition of a matrix $A \in O(n)$ in this setting thus reads \begin{equation} \label{gpd_grass} A = PQ, \quad P \in Sym_J(n) \cap O(n), \, Q \in O(p) \times O(n-p). \end{equation} The corresponding Cartan decomposition reads \[ Z = X + Y, \quad X \in \mathfrak{sym}_J(n) \cap \mathfrak{o}(n), \, Y \in \mathfrak{o}(p) \times \mathfrak{o}(n-p), \] where, for each $m$, $\mathfrak{o}(m)$ denotes the space of antisymmetric $m \times m$ matrices, \[ \mathfrak{o}(p) \times \mathfrak{o}(n-p) = \left\{ \begin{pmatrix} Y_1 & 0 \\ 0 & Y_2 \end{pmatrix} \mid Y_1 \in \mathfrak{o}(p), \, Y_2 \in \mathfrak{o}(n-p) \right\} \] and \begin{align} \mathfrak{sym}_J(n) \cap \mathfrak{o}(n) &= \{ X \in \mathfrak{o}(n) \mid XJ = JX^T \} \\ &= \left\{ \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix} \mid B \in \mathbb{R}^{(n-p) \times p} \right\}. \end{align} The map $F : \mathfrak{sym}_J(n) \cap \mathfrak{o}(n) \rightarrow Gr(p,n)$ is given by \begin{equation} F(X) = \mathrm{span}(\exp(X)e_1, \exp(X)e_2, \dots, \exp(X)e_p). \end{equation} The inverse of $F$ can be computed (naively) as follows. Given an element $\mathcal{V} \in Gr(p,n)$, let $a_1,a_2,\dots,a_p$ be an orthonormal basis for $\mathcal{V}$. Extend this basis to an orthonormal basis $a_1,a_2,\dots,a_n$ of $\mathbb{R}^n$. Then \begin{equation} F^{-1}(\mathcal{V}) = \log(P), \end{equation} where $P \in Sym_J(n) \cap O(n)$ is the first factor in the generalized polar decomposition~(\ref{gpd_grass}) of $A = \left( a_1 \, a_2 \, \cdots \, a_n\right)$. Note that this map is independent of the chosen bases for $\mathcal{V}$ and its orthogonal complement in $\mathbb{R}^n$. Indeed, if $\tilde{a}_1,\tilde{a}_2,\dots,\tilde{a}_p$ is any other orthonormal basis for $\mathcal{V}$ and $\tilde{a}_{p+1},\tilde{a}_{p+2},\dots,\tilde{a}_n$ is any other basis for the orthogonal complement of $\mathcal{V}$, then there is a matrix $R \in O(p) \times O(n-p)$ such that $\tilde{A} = AR$, where $\tilde{A} = \left( \tilde{a}_1 \, \tilde{a}_2 \, \cdots \, \tilde{a}_n\right)$. The generalized polar decomposition of $\tilde{A}$ is thus $\tilde{A} = P\tilde{Q}$, where $\tilde{Q} = QR$. More generally, we may opt to fix an element $\bar{A} \in O(n)$ and consider interpolants of the form~(\ref{generalinterp}) using the map \begin{equation} \label{F_grass} F_{\bar{A}} (X) = \mathrm{span}(\bar{A}\exp(X)e_1, \bar{A}\exp(X)e_2, \dots, \bar{A}\exp(X)e_p), \end{equation} The inverse of this map, in analogy with the preceding paragraph, is \begin{equation} \label{Finv_grass} F^{-1}_{\bar{A}}(\mathcal{V}) = \log(P), \end{equation} where now $P \in Sym_J(n) \cap O(n)$ is the first factor in the generalized polar decomposition~(\ref{gpd_grass}) of $\bar{A}^T A$, where $A \in O(n)$ is a matrix whose first $p$ and last $n-p$ columns, respectively, form orthonormal bases for $\mathcal{V}$ and its orthogonal complement. \paragraph{Algorithms.} We now turn our attention to the computation of the interpolant~(\ref{generalinterp}) in this setting. A naive implementation using the steps detailed above for computing $F_{\bar{A}}$ and its inverse would lead to an algorithm for computing the interpolant having complexity $O(n^3)$. Remarkably, the computation of~(\ref{generalinterp}) can be performed in $O(np^2)$ operations, as we now show. The resulting algorithm turns out to be identical to that proposed in~\cite{amsallem2008interpolation}. The fact that this algorithm scales linearly with $n$ is noteworthy, as it renders this interpolation scheme practical for applications in which $n \gg p$. The derivation of the algorithm hinges upon the following two lemmas, which, when combined, allow for a computation of the interpolant while operating solely on matrices of size $n \times p$ or smaller. The first lemma gives a useful formula for $F_{\bar{A}} (X)$. \begin{lemma} \label{lemma:exp_grass} Let \[ \bar{A} = \begin{pmatrix} \bar{A}_1 & \bar{A}_2 \end{pmatrix} \in O(n) \] with $\bar{A}_1 \in \mathbb{R}^{n \times p}$ and $\bar{A}_2 \in \mathbb{R}^{n \times (n-p)}$, and let $X = \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix} \in \mathfrak{sym}_J(n) \cap \mathfrak{o}(n)$ with $B \in \mathbb{R}^{(n-p) \times p}$. Then \[ \bar{A} \exp(X) \begin{pmatrix} I_p \\ 0 \end{pmatrix} = \bar{A}_1 V \cos(\Theta)V^T + U \sin(\Theta) V^T, \] where $U \in \mathbb{R}^{n \times p}$, $\Theta \in \mathbb{R}^{p \times p}$, and $V \in \mathbb{R}^{p \times p}$ denote the factors in the thin singular value decomposition \begin{equation} \label{A2Bsvd} \bar{A}_2 B = U \Theta V^T. \end{equation} In particular, $F_{\bar{A}}(X)$ is the space spanned by the columns of $\bar{A}_1 V \cos(\Theta)V^T + U \sin(\Theta) V^T$. Equivalently, since $V$ is orthogonal, $F_{\bar{A}}(X)$ is the space spanned by the columns of $\bar{A}_1 V \cos(\Theta) + U \sin(\Theta)$. \end{lemma} \begin{proof} The formula is proved in~\cite[Theorem 2.3]{edelman1998geometry}. \end{proof} The next lemma gives a useful formula for $F_{\bar{A}}^{-1} (\mathcal{V})$. Closely related formulas appear without proof in~\cite{begelfor2006affine,amsallem2008interpolation,chang2012feature} and elsewhere, so we give a proof here for completeness. \begin{lemma} \label{lemma:log_grass} Let $\bar{A} = \begin{pmatrix} \bar{A}_1 & \bar{A}_2 \end{pmatrix} \in O(n)$ be as in Lemma~(\ref{lemma:exp_grass}), and let $\mathcal{V} \in Gr(p,n)$. Let \[ A = \begin{pmatrix} A_1 & A_2 \end{pmatrix} \in O(n) \] be such that the columns of $A_1$ and $A_2$, respectively, form orthonormal bases for $\mathcal{V}$ and its orthogonal complement. Assume that $\bar{A}_1^T A_1$ is invertible. Then \[ F^{-1}_{\bar{A}}(\mathcal{V}) = \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix}, \] where \begin{equation} \label{Bsoln} B = \bar{A}_2^T U \arctan(\Sigma) V^T, \end{equation} and $U \in \mathbb{R}^{n \times p}$, $\Sigma \in \mathbb{R}^{p \times p}$, and $V \in \mathbb{R}^{p \times p}$ denote the factors in the thin singular value decomposition \begin{equation} \label{svdA1A2} (I-\bar{A}_1 \bar{A}_1^T) A_1 (\bar{A}_1^T A_1)^{-1} = U \Sigma V^T. \end{equation} \end{lemma} \begin{proof} It is enough to check that if $B$ is given by~(\ref{Bsoln}), then the image of $\begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix}$ under $F_{\bar{A}}$ is $\mathcal{V}$. In other words, we must check that the columns of \begin{equation} \label{AbarexpB} \bar{A} \exp \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix} \begin{pmatrix} I_p \\ 0 \end{pmatrix} \end{equation} span $\mathcal{V}$. To this end, observe that by the orthogonality of $\bar{A}$, \begin{equation} \label{A2A2U} \bar{A}_2 \bar{A}_2^T U = (I - \bar{A}_1 \bar{A}_1^T) U = U, \end{equation} where the last equality follows from~(\ref{svdA1A2}) upon noting that $(I - \bar{A}_1 \bar{A}_1^T)$ is a projection. Thus, by inspection of~(\ref{Bsoln}), the thin singular value decomposition of $\bar{A}_2 B$ is \[ \bar{A}_2 B = U \Theta V^T, \] where $\Theta = \arctan{\Sigma}$. Now by Lemma~\ref{lemma:exp_grass}, \begin{equation} \label{expgrass2} \bar{A} \exp \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix} \begin{pmatrix} I_p \\ 0 \end{pmatrix} = \bar{A}_1 V \cos(\Theta)V^T + U \sin(\Theta) V^T. \end{equation} Using~(\ref{svdA1A2}), this simplifies to \begin{align} \bar{A} \exp \begin{pmatrix} 0 & -B^T \\ B & 0 \end{pmatrix} \begin{pmatrix} I_p \\ 0 \end{pmatrix} &= \bar{A}_1 V \cos(\Theta)V^T + (I-\bar{A}_1 \bar{A}_1^T) A_1 (\bar{A}_1^T A_1)^{-1} V \Sigma^{-1} \sin(\Theta) V^T \nonumber \\ &= \bar{A}_1 V \cos(\Theta)V^T + (I-\bar{A}_1 \bar{A}_1^T) A_1 (\bar{A}_1^T A_1)^{-1} V \cos(\Theta) V^T \nonumber \\ &= A_1 (\bar{A}_1^T A_1)^{-1} V \cos(\Theta) V^T. \nonumber \end{align} Observe that since $\Sigma=\tan(\Theta)$ is finite, the diagonal entries of $\cos(\Theta)$ are nonzero. Thus, $(\bar{A}_1^T A_1)^{-1} V \cos(\Theta) V^T$ is invertible, so we conclude that the columns of~(\ref{AbarexpB}) span the same space that is spanned by the columns of $A_1$, namely $\mathcal{V}$. \end{proof} The preceding two lemmas lead to the following algorithm, which coincides with that introduced in~\cite{amsallem2008interpolation}, for computing the interpolant \begin{equation} \label{interp_grass} \mathcal{I}_{\bar{A}}\mathcal{V}(x) = F_{\bar{A}}\left( \sum_{i=1}^m \phi_i(x) F_{\bar{A}}^{-1}(\mathcal{V}^{(i)}) \right) \end{equation} of elements $\mathcal{V}^{(1)}, \mathcal{V}^{(2)}, \dots, \mathcal{V}^{(m)}$ of $Gr(p,n)$. Note that the computational complexity of this algorithm is $O(np^2)$. In particular, owing to the identity~(\ref{A2A2U}), the $(n-p) \times n$ matrix $\bar{A}_2$ plays no role in the algorithm, despite its worrisome appearance in~(\ref{A2Bsvd}) and~(\ref{Bsoln}). \begin{algorithm}[H] \caption{Interpolation on the Grassmannian $Gr(p,n)$} \label{alg_grass} \begin{algorithmic}[1] \Require Subspaces $\{\mathcal{V}^{(i)} \in Gr(p,n)\}_{i=1}^m$, shape functions $\{\phi_i : \Omega \rightarrow \mathbb{R} \}_{i=1}^m$, point $x \in \Omega$, matrix~$\bar{A}_1 \in \mathbb{R}^{n \times p}$ with orthonormal columns \State $Z \gets 0_{n \times p}$ \For{$i=1,2,\dots,m$} \State \label{alg_grass_basis} Let $A_1 \in \mathbb{R}^{n \times p}$ be a matrix whose columns form an orthonormal basis for $\mathcal{V}^{(i)}$. \State \begin{varwidth}[t]{\linewidth} Compute the thin singular value decomposition \[ (I-\bar{A}_1 \bar{A}_1^T) A_1 (\bar{A}_1^T A_1)^{-1} = U \Sigma V^T, \] with $U \in \mathbb{R}^{n \times p}$, $\Sigma \in \mathbb{R}^{p \times p}$, and $V \in \mathbb{R}^{p \times p}$.\end{varwidth} \State $Z \mathrel{+}= \phi_i(x) U \mathrm{arctan}(\Sigma) V^T$ \EndFor \State Compute the thin singular value decomposition \[ Z = U \Theta V^T, \] with $U \in \mathbb{R}^{n \times p}$, $\Sigma \in \mathbb{R}^{p \times p}$, and $V \in \mathbb{R}^{p \times p}$. \State \label{alg_grass_A} $A \gets \bar{A}_1 V \cos(\Theta) + U \sin(\Theta)$ \State\Return $\mathrm{span}(a_1,a_2,\dots,a_p)$, where $a_j$ denotes the $j^{th}$ column of $A$. \end{algorithmic} \end{algorithm} Note that the output of Algorithm~\ref{alg_grass} is independent of the choice of orthonormal basis made for each $\mathcal{V}^{(i)}$ in Line~\ref{alg_grass_basis} of Algorithm~\ref{alg_grass}. This can be checked directly by observing that a change of basis corresponds to post-multiplication of $A_1$ by a matrix $R \in O(p)$, leaving $(I-\bar{A}_1 \bar{A}_1^T) A_1 (\bar{A}_1^T A_1)^{-1}$ invariant. Similarly, the output of the algorithm is invariant under post-multiplication of $\bar{A}_1$ by any matrix $R \in O(p)$, since it can be checked that such a transformation changes the output of Line~\ref{alg_grass_A} from $A$ to $AR$, whose columns span the same space as those of $A$. This last statement leads to the conclusion that \begin{equation} \label{interp_grass_basis_change} \mathcal{I}_{\bar{A}Q} \tilde{\mathcal{V}}(x) = \mathcal{I}_{\bar{A}} \tilde{\mathcal{V}}(x) \end{equation} for any $Q \in O(p) \times O(n-p)$, which reaffirms~(\ref{postmult}). The interpolant so constructed enjoys the following additional property. \begin{lemma} The interpolant~(\ref{interp_grass}) commutes with the action of $O(n)$ on $Gr(p,n)$. That is, if $Q \in O(n)$ and $\tilde{\mathcal{V}}^{(i)} = Q\mathcal{V}^{(i)}$, $i=1,2,\dots,m$, then \[ \mathcal{I}_{Q\bar{A}} \tilde{\mathcal{V}}(x) = Q\mathcal{I}_{\bar{A}} \mathcal{V}(x) \] for every $x \in \Omega$. \end{lemma} \begin{proof} Apply~(\ref{general_equivariance}). \end{proof} Another $O(n)$-equivariant interpolant on $Gr(p,n)$ is given abstractly by~(\ref{Riemannianmean2}). In this setting, this interpolant is obtained by solving \[ \sum_{i=1}^m \phi_i(x) F_{\bar{A}}^{-1}(\mathcal{V}^{(i)}) = 0 \] for $\bar{A}$ and outputting the space spanned by the first $p$ columns of $\bar{A}$. Algorithmically, this amounts to wrapping a fixed point iteration around Algorithm~\ref{alg_grass}, as detailed below. \begin{algorithm}[H] \caption{Iterative interpolation on the Grassmannian $Gr(p,n)$} \label{alg_grass_Riemannian} \begin{algorithmic}[1] \Require Subspaces $\{\mathcal{V}^{(i)} \in Gr(p,n)\}_{i=1}^m$, shape functions $\{\phi_i : \Omega \rightarrow \mathbb{R} \}_{i=1}^m$, point $x \in \Omega$, matrix~$\bar{A}_1 \in \mathbb{R}^{n \times p}$ with orthonormal columns \Repeat \State \begin{varwidth}[t]{\linewidth} Use Algorithm~\ref{alg_grass} to compute the interpolant of $\{\mathcal{V}^{(i)}\}_{i=1}^m$ at $x$, storing the result as a matrix $A \in \mathbb{R}^{n \times p}$ (i.e., the matrix $A$ appearing in line~\ref{alg_grass_A} of Algorithm~\ref{alg_grass}). \end{varwidth} \State $\bar{A}_1 \gets A$ \Until{converged} \State\Return $\mathrm{span}(a_1,a_2,\dots,a_p)$, where $a_j$ denotes the $j^{th}$ column of $\bar{A}_1$. \end{algorithmic} \end{algorithm} Since $O(n)$ is compact, Lemma~\ref{lemma:Riemannian_mean} shows that Algorithm~\ref{alg_grass_Riemannian} produces the weighted Riemannian mean on $Gr(p,n)$. This interpolant has been considered previously by several authors, including~\cite{begelfor2006affine,chang2012feature,grohs2013quasi}. \section{Conclusion} This paper has presented a family of structure-preserving interpolation operators for functions taking values in a symmetric space $\mathcal{S}$. We accomplished this by identifying $\mathcal{S}$ with a homogeneous space $G/G^\sigma$ and interpolating coset representatives obtained from the generalized polar decomposition. The resulting interpolation operators enjoy equivariance with respect to the action of $G^\sigma$ on $\mathcal{S}$, as well as equivariance with respect to the action of certain geodesic symmetries on $\mathcal{S}$. Numerical evidence in Section~\ref{sec:lorentz} suggests that these interpolation operators also enjoy optimal approximation properties, but further work is needed to confirm this theoretically. In certain cases, namely those addressed in Lemma~\ref{lemma:Riemannian_mean}, the work of~\cite{grohs2015optimal,grohs2013quasi} supplies the needed theoretical confirmation. Presuming that similar results hold more generally, the application of these interpolation schemes seems intriguing, particularly in the context of numerical relativity, where they provide structure-preserving finite elements for the metric tensor. \begin{appendices} \section{Second-Order Derivatives of the Matrix Exponential} \label{app:d2exp} In this section, we prove~(\ref{d2Iu}) by showing that if $P : \Omega \rightarrow \mathbb{R}^{n \times n}$ is a smooth matrix-valued function defined on a domain $\Omega \subset \mathbb{R}^d$, then, for each $j,k=1,2,\dots,d$, the matrix $\frac{\partial^2}{\partial x_j \partial x_k} \exp(P(x))$ is given by reading off the $(1,4)$ block of \[ \mathrm{exp}\begin{pmatrix} X & Y & Z & W \\ 0 & X & 0 & Z \\ 0 & 0 & X & Y \\ 0 & 0 & 0 & X \end{pmatrix}, \] where $X = P(x)$, $Y = \frac{\partial P}{\partial x_j}(x)$, $Z = \frac{\partial P}{\partial x_k}(x)$, and $W = \frac{\partial^2 P}{\partial x_j \partial x_k}(x)$. To prove this, recall first the identity~(\ref{blockexp}), which can be written as \begin{equation} \label{blockexp2} \left.\frac{d}{d t}\right\|_{t=0} \exp(U+tV) = \left[ \exp \begin{pmatrix} U & V \\ 0 & U \end{pmatrix} \right]_{(1,2)} \end{equation} for any square matrices $U$ and $V$ of equal size, where $B_{(1,2)}$ denotes the $(1,2)$ block of a block matrix $B$. Now observe that with $X$, $Y$, $Z$, and $W$ defined as above, \begin{align} \frac{\partial^2}{\partial x_j \partial x_k} \exp(P(x)) &= \left.\frac{\partial^2}{\partial s \, \partial t}\right\|_{s=t=0} \exp(X+tY+sZ+stW) \\ &= \left.\frac{\partial}{\partial s}\right\|_{s=0} \left.\frac{\partial}{\partial t}\right\|_{t=0} \exp(X+sZ+t(Y+sW)) \\ &= \left.\frac{\partial}{\partial s}\right\|_{s=0} \left[\exp\begin{pmatrix} X+sZ & Y+sW \\ 0 & X+sZ \end{pmatrix} \right]_{(1,2)} \\ &= \left[ \left.\frac{\partial}{\partial s}\right\|_{s=0} \exp\begin{pmatrix} X+sZ & Y+sW \\ 0 & X+sZ \end{pmatrix} \right]_{(1,2)} \end{align*} Using~(\ref{blockexp2}) again, we have \[ \left.\frac{\partial}{\partial s}\right\|_{s=0} \exp\begin{pmatrix} X+sZ & Y+sW \\ 0 & X+sZ \end{pmatrix} = \left[\exp\begin{pmatrix} X & Y & Z & W \\ 0 & X & 0 & Z \\ 0 & 0 & X & Y \\ 0 & 0 & 0 & X \end{pmatrix}\right]_{(1,2)}, \] which shows that \[ \frac{\partial^2}{\partial x_j \partial x_k} \exp(P(x)) = \left[ \left[\exp\begin{pmatrix} X & Y & Z & W \\ 0 & X & 0 & Z \\ 0 & 0 & X & Y \\ 0 & 0 & 0 & X \end{pmatrix}\right]_{(1,2)} \right]_{(1,2)} = \left[\exp\begin{pmatrix} X & Y & Z & W \\ 0 & X & 0 & Z \\ 0 & 0 & X & Y \\ 0 & 0 & 0 & X \end{pmatrix}\right]_{(1,4)}. \] \end{appendices} \printbibliography \end{document}	arXiv
Automorphism groups of compact complex surfaces math. arxive. Cornell University, 2017 Prokhorov Y., Shramov K. We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such surface X is always Jordan, and the birational automorphism group is Jordan unless X is birational to a product of an elliptic and a rational curve. Priority areas: mathematics Keywords: automorphism group Publication based on the results of: Алгебраическая геометрия и ее приложения(2017) Гибкость аффинных конусов над поверхностями дель Пеццо степени 4 и 5 Перепечко А. Ю. Функциональный анализ и его приложения. 2013. Т. 47. № 4. С. 45-52. We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. Automorphism groups of Inoue and Kodaira surfaces Prokhorov Y., Shramov K. math. arxive. Cornell University, 2018 We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. Automorphism groups of Moishezon threefolds Shramov K., Prokhorov Y. math. arxive. Cornell University, 2019 We show that automorphism groups of Moishezon threefolds are always Jordan. Жесткие геометрии на пространстве слоев слоений и группы их автоморфизмов Жукова Н. И. В кн.: Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции.. Каз.: Издательство Казанского университета, 2017. С. 48-51. We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations. A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$. It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. For each such geometry $( N,\zeta)$ we prove the existence and uniqueness of the structure of a finite-dimensional Lie group in the group of all automorphisms $Aut (N},\zeta)$. The applications to the orbifolds are considered. Added: Apr 1, 2018 Flexible affine cones and flexible coverings Perepechko A. Y., Michałek M., Süß H. Mathematische Zeitschrift. 2018. Vol. 290. No. 3-4. P. 1457-1478. We provide a new criterion for flexibility of affine cones over varieties covered by flexible affine varieties. We apply this criterion to prove flexibility of affine cones over secant varieties of Segre–Veronese embeddings and over certain Fano threefolds. We further prove flexibility of total coordinate spaces of Cox rings of del Pezzo surfaces. Finite groups acting on elliptic surfaces Shramov K. European Journal of Mathematics. 2019. We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration. On automorphism groups of affine surfaces Perepechko A. Y., Zaidenberg M., Kovalenko S. In bk.: Advanced Studies in Pure Mathematics. Vol. 75: Algebraic Varieties and Automorphism Groups. Tokyo: The Mathematical Society of Japan, 2017. P. 207-286. This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems. Advanced Studies in Pure Mathematics Vol. 75: Algebraic Varieties and Automorphism Groups. Tokyo: The Mathematical Society of Japan, 2017. The workshop "Algebraic Varieties and Automorphism Groups" was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA. Recently, there have been remarkable developments in algebraic geometry and related fields, especially, in the area of (birational) automorphism groups and algebraic group actions. The purpose of this workshop was to provide the experts and young researchers with the opportunities to interact in the fields of affine and complete algebraic geometry, group actions and commutative algebra related to the topics listed below as well as to publicize the new results. We are confident that these purposes were achieved by the endeavors of the participants. The main topics of the workshop were the following: Algebraic varieties containing An-cylinders; Algebraic varieties with fibrations; Algebraic group actions and orbit stratifications on algebraic varieties; Automorphism groups and birational automorphism groups of algebraic varieties. There were 19 talks on the above and related topics by experts from the viewpoints of (affine) algebraic geometry, transformation groups, and commutative algebra. Inspired by the talks, there were active discussions and communication among participants during and after the workshop. The present volume is the proceedings of the workshop and contains 15 articles on the workshop topics. We hope that this volume will contribute to the progress in the theories of algebraic varieties and their automorphism groups. The workshop was financially supported by the RIMS and Grant- in-Aid for Scientific Research (B) 24340006, JSPS. We wish to thank all those who supported us in organizing the workshop and preparing this volume. Kayo Masuda, Takashi Kishimoto, Hideo Kojima, Masayoshi Miyanishi, Mikhail Zaidenberg All papers in this volume have been refereed and are in final form. No version of any of them will be submitted for publication elsewhere. Automorphisms of weighted complete intersections Shramov K., Przyjalkowski V. math. arxive. Cornell University, 2019 We show that smooth well formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. Bass' triangulability problem Vladimir L. Popov. In bk.: Advanced Studies in Pure Mathematics. Vol. 75: Algebraic Varieties and Automorphism Groups. Tokyo: The Mathematical Society of Japan, 2017. P. 425-441. Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem in the affirmative. To this end we prove a theorem on invariant subfields of 1-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy. Метод параметрикса для диффузий и цепей Маркова Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012. Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action? Colliot-Thélène J., Kunyavskiĭ B., Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2. P. 428-466. Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself. Dynamics of Information Systems: Mathematical Foundations Iss. 20. NY: Springer, 2012. This proceedings publication is a compilation of selected contributions from the "Third International Conference on the Dynamics of Information Systems" which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study. Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem Vladimir Lebedev. arxiv.org. math. Cornell University, 2011. No. 1112.4892v1. We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results. Обоснование адиабатического предела для гиперболических уравнений Гинзбурга-Ландау Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214. Hypercommutative operad as a homotopy quotient of BV Khoroshkin A., Markaryan N. S., Shadrin S. arxiv.org. math. Cornell University, 2012. No. 1206.3749. We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras. Cross-sections, quotients, and representation rings of semisimple algebraic groups V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3. P. 827-856. Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group. Математическое моделирование социальных процессов Edited by: А. Михайлов Вып. 14. М.: Социологический факультет МГУ, 2012.	CommonCrawl
\begin{document} \title{$3n+3^k$: New Perspective on Collatz Conjecture} \begin{abstract} Collatz conjecture is generalized to $3n+3^k$ ($k\in N$). Operating as usual, every sequence seems to reach $3^k$ and end up in the loop $3^k, 4.3^k, 2.3^k,3^k$. The usual $3n+1$ conjecture is recovered for $k=0$. For $k>0$, we noticed the existence of a sequence of period 3, namely, $3^{k-1}, 2.3^k, 3^k$, alongside the cycle $4.3^k, 2.3^k,3^k$ encountered in the $3n+1 (k=0)$ sequence. A term formula of the $3n+3^k$ conjecture has been derived, and hence the \emph{total stopping time}. \end{abstract} \section{Introduction} Credited to the mathematician Lothar Collatz, the Collatz conundrum (see for instance Ref. \cite{lagarias20033x+} and references therein), which was brought forward in the 1930s, is one of the simplest yet unsolved conjectures in mathematics. \emph{``This is a really dangerous problem. People become obsessed with it and it really is impossible,”} as stated by Jeffrey Lagarias. Such a conjecture asks whether repeating two simple arithmetic operations will eventually reach 1. That is to say, let $N:=\{0, 1, 2...\}$ denote the natural numbers, so that $N+1=\{1, 2, 3...\}$ are the positive integers. Now, pick an arbitrary positive integer $n\in N+1$ and apply the following operation on it: If the number is even, divide it by two; while if the number is odd, triple it and add one. Let the process be denoted by $f(n)$. That is \begin{equation} f(n)= \begin{cases} \frac{n}{2} \qquad \text{if} \quad n\equiv 0 \qquad(\text{mod} \quad 2)\\ 3n+1 \qquad \text{if}\quad n\equiv 1 \qquad(\text{mod}\quad 2) \end{cases} \end{equation} The main task is to form a sequence by performing this operation repeatedly, and take the output at each step as the input at the next. Consequently, one may notice that such a process will eventually lead to the number 1, regardless of which positive integer is chosen initially. \\ If we set $f^0(n)=n$ and $f^l(n)=f(f^{l-1}(n))$ for $l\in N$. Then, the Collatz sequence for $n$ reads \begin{equation} C(n)=\{f^l(n)\}^{\infty}_{l=0}, \end{equation} Strictly speaking, one may note that every Collatz sequence ends up in the loop $1, 4, 2, 1$. To disprove Collatz conjecture, one has to show that there exists some starting number which yields a sequence that does not include 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase infinitely. No such sequence has been found so far \cite{barina2021convergence}.\\ In the present paper, we show that Collatz conjecture $3n+1$ may possibly be a part of a more generalized case of the form $3n+3^k$ with $k\in N$. \section{$3n+3^k$ Conjecture} We provide the following operation, pick an arbitrary positive integer $n\in N+1$, if the number is even, divide it by two; otherwise if the number is odd, triple it and add $3^k$ with $k\in N$. \begin{equation} g_k(n)= \begin{cases} \frac{n}{2} \qquad \text{if} \quad n \equiv 0 \qquad(\text{mod} \quad 2)\\ 3n+3^k \qquad \text{if}\quad n \equiv 1 \qquad(\text{mod} \quad 2) \qquad k\in N \end{cases} \end{equation} Repeating the two operations will eventually lead to $3^k$ ($k\in N$). Hence, one may argue that the original Collatz conjecture $3n+1$ is nothing more than $3n+3^k$ conjecture with $k=0$, i. e. $f(n)=g_0(n)$. Apparently, each positive integer, upon repetitive application of $g_k(n)$, will end in a repeating sequence of the form $3^k, 4.3^k, 2.3^k, 3^k$. In the $3n+1$ (or equivalently $3n+3^k$ with $k=0$) sequence, the only cycle of period $3$ is known to be the sequence $4.3^k, 2.3^k, 3^k$ \cite{zarnowski2001generalized}. For $k>0$ though, along with the aforementioned cycle, one may distinguish another sequence of period $3$, that is $3^{k-1}, 2.3^k, 3^k$. Although such a generalized conjecture seems to be true, it has yet to be numerically verified for at least the largest set of numbers that has been checked in the case of $3n+1$ so far, i. e. $n\leq 2^{100000} - 1$ \cite{ren2018collatz}, even though a general proof of it is still lacking. Furthermore, up to this point it is not clear whether there exists a certain value of $k$ for or beyond which the conjecture cease to be hold.\\ \section{Term formula for $3n+3^k$ sequence} Let $C^l_k(n)$ include the first $l$ terms of the $3n+3^k$ sequence for $n$. We set $m$ as the number of odd terms in $C^l_k(n)$, $d_i$ as the number of consecutive even terms immediately following the $i$th odd term and $d_0$ as the number of even terms preceding the first odd term. It follows that the next term in the $3n+3^k$ sequence for $n$ is \footnote{Note that this formula is different from the one derived by L E. Garner \cite{garner1981collatz}}. \begin{equation} \label{next term} g^l_k(n)=\frac{3^m}{2^{l-m}}n+3^k\epsilon\sum_{j=1}^{m}\frac{3^{m-j}}{2^{\sum_{i=j}^{m}d_i}} \end{equation} where $\epsilon=0$ if $m=0$ and $\epsilon=1$ if $m\neq0$, note that $l-m=d_0+d_1+...+d_m$. It is worth noting that, for the original $3n+1$ Collatz sequence, the above relation reduces to \begin{equation} g^l_0(n)=\frac{3^m}{2^{l-m}}n+\epsilon\sum_{j=1}^{m}\frac{3^{m-j}}{2^{\sum_{i=j}^{m}d_i}} \end{equation} $3n+3^k$ sequences of integers ranging from $1$ to $17$ for $k=0, 1, 2, 3$ and $4$ are listed in Table. \ref{table}. In the $k$-sequences chosen, it is evident that, after consecutive application of $g^l_k$, the aforementioned integers eventually reach $3^k$. \section{Total stopping time} We call the \emph{total stopping time} of $n$ the smallest $t$ such that $n_t=3^k$, with $n_t$ being the value of $g_k$ applied to $n$ recursively $t$ times. It is worth mentioning that $t$ is nothing but the number of integers in the $3n+3^k$ sequence just preceding the $3^k$ term, i. e. $n_t=g^t_k(n)$. Thus, using the relation \eqref{next term}, the \emph{total stopping time} can be written as \begin{equation} \label{total stoping time} t=\text{log}_2\bigg[\frac{2^m3^m}{3^k(1-\epsilon\sum_{j=1}^{m}\frac{3^{m-j}}{2^{\sum_{i=j}^{m}d_i}})} n\bigg] \end{equation} If $t$ does not exist we say that the \emph{total stopping time} is infinite. It follows that, one has to prove that all positive integers yield a finite \emph{total stopping time} in order to prove the $3n+3^k$ conjecture. In an endeavor to prove the $3n+1$ conjecture, mathematicians used to inspect what it is called the \emph{stopping time}, namely the least positive $l$ for which $f^l(n)=g^l_0(n)<n$ \cite{korec1994density, crooks2022collatz,idowu2015novel,ruggiero2019relationship}. If they can prove that all positive integers have a finite \emph{stopping time}, they can prove by induction that the Collatz conjecture is true. It is nothing short of reasonable since for $n>1$, $f^l(n)=g^l_0(n)=1$ cannot occur without the occurrence of some $f^l(n)=g^l_0(n)<n$ \cite{lagarias2010ultimate}. Indeed, in the 1970s, it has been shown that almost all Collatz sequences eventually reach a number that is smaller than where it started \cite{terras1976stopping}. In 2019, Terence Tao proved that for almost all numbers the Collatz sequence of $n$ leads to a lower value, showing that the Collatz conjecture holds true for almost all numbers \cite{tao2019almost}. But the question to be asked is whether it is the right (or sufficient) way to prove the Collatz conjecture. As obvious as it seems, if the $3n+3^k$ conjecture turns out to be true for all $n\in N$+1, then the statement that, for each $n\in N+1$ there exists $l\in N+1$ such that $g^l_k(n)<n$, does not hold for $n<3^k$. This can be easily seen in the $3n+27$ sequence for $n=5, 9, 13$ (see for instance Table. \ref{table}). Alternatively, we must focus on showing that each $n\in N+1$ has a finite \emph{total stopping time} and hence proving that the $3n+3^k$ sequence inclines toward $3^k$ no matter what integer we start with. That is to say, one has to test the validity of the following statement: \emph{``for each $n\in N+1$, there exists $t\in N+1$ such that $g^t_k(n)=3^k$."}, without making use of the \emph{stopping time}.\\ Now, suppose that two integers $n_1$ and $n_2$ have the same \emph{total stopping time} $t$ within the sequence $3n+3^k$, then from Eq. \eqref{total stoping time} we have \begin{equation} \label{total stoping time2} n_2=\frac{2^{m_1}3^{m_1}}{2^{m_2}3^{m_2}}\frac{1-\epsilon\sum_{j=1}^{m_2}\frac{3^{m_2-j}}{2^{\sum_{i=j}^{m_2}d_{2,i}}}}{1-\epsilon\sum_{j=1}^{m_1}\frac{3^{m_1-j}}{2^{\sum_{i=j}^{m_1}d_{1,i}}}}n_1 \end{equation} where $m_1(m_2)$ and $d_{1,i}(d_{2,i})$ are, respectively, the number of odd terms and the number of consecutive even terms immediately following the $i$th odd term in $C^t_k(n_1)$ ($C^t_k(n_2)$), namely the set of $t$ terms of the $3n+3^k$ sequence for $n_1(n_2)$.\\ It is obvious that for $m_1=m_2=0$, we have $n_2=n_1$. Hence, no two integers, for which the corresponding sequences contain even terms only, have the same \emph{total stopping time}. Sequences that have the same \emph{total stopping time} and the same cycle of period $3$ have the same number of odd terms, a feature that can be figured out by checking Figs. \ref{fig:1}$-$\ref{fig:3} which depict the \emph{total stopping time} and the number of odd terms in the set $C^{t+1}_k(n) (k=0, 1, 2)$ for different values of $n$. In the $3n+9$ sequence for example (upper plot in Fig. \ref{fig:3}), there exist two cycles of period $3$: $36, 18, 9$ and $3, 18, 9$, the integers $32, 33$ and $35$ have the same \emph{total stopping time}, i.e. $t=10$, but only $33$ and $35$ share the same number of odd terms $m+1=4$, which is due to the fact that both of them possesses the same cycle of period $3$, i.e. $36, 18, 9$, whereas $32$ has the cycle $3, 18, 9$.\\ Remarkably, It is worth noting that sequences that possess the same number of odd terms in the set $C^{t+1}_k(n)$ do not necessarily yield the same \emph{total stopping time}, even if they share the same cycle of period 3, take the example of 1 and 2 in the $3n+9$ sequence, both of them have the same number of odd terms in the set $C^{t+1}_9(n)$, i.e. $m+1=3$, and the same cycle $3, 18, 9$ (check Table. \ref{table}) meanwhile they have different \emph{total stopping time}, $t=5$ and $t=6$, respectively. \begin{figure} \caption{Behavior of the \emph{total stopping time} and the number of odd terms in the set $C^{t+1}_1(n)$ of the $3n+1$ sequence for $n\in[1,100]$ (top), $n\in[500,600]$ (middle) and $n\in[900,1000]$ (bottom), the blue bars represent the \emph{total stopping time} while the red bars depict the number of odd terms in the sequence.} \label{fig:1} \end{figure} \begin{figure} \caption{Behavior of the \emph{total stopping time} and the number of odd terms in the set $C^{t+1}_3(n)$ of the $3n+3$ sequence for $n\in[1,100]$ (top), $n\in[500,600]$ (middle) and $n\in[900,1000]$ (bottom), the blue bars represent the \emph{total stopping time} while the red bars depict the number of odd terms in the sequence.} \label{fig:2} \end{figure} \begin{figure} \caption{Behavior of the \emph{total stopping time} and the number of odd terms in the set $C^{t+1}_9(n)$ of the $3n+9$ sequence for $n\in[1,100]$ (top), $n\in[500,600]$ (middle) and $n\in[900,1000]$ (bottom), the blue bars represent the \emph{total stopping time} while the red bars depict the number of odd terms in the sequence.} \label{fig:3} \end{figure} \begin{table} \begin{centering} \begin{tabular}{lccccccccccccccccc} \hline \hline &&&&&&&&$3n+1$&&&&&&&&\\ \hline $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10&11 &12 &13 &14 &15 &16 &17 \\ &4 &1 &10 &2 &16 &3 &22 &4 &28 &5 &34 &6 &40 &7 &46 &8 &52\\ &2 &4 &5 &1 &8 &10 &11 &2 &14 &16 &17 &3 &20 &22 &23 &4 &26\\ &1 &2 &16 &4 &4 &5 &34 &1 &7 &8 &52 &10 &10 &11 &70 &2 &13\\ & &1 &8 &2 &2 &16 &17 &4 &22 &4 &26 &5 &5 &34 &35 &1 &40\\ & & &4 &1 &1 &8 &52 &2 &11 &2 &13 &16 &16 &17 &106 &4 &20\\ & & &2 & &4 &4 &26 &1 &34 &1 &40 &8 &8 &52 &53 &2 &10\\ & & &1 & &2 &2 &13 & &17 &4 &20 &4 &4 &26 &160 &1 &5\\ & & &4 & &1 &1 &40 & &52 &2 &10 &2 &2 &13 &80 & &16\\ & & &2 & & &4 &20 & &26 &1 &5 &1 &1 &40 &40 & &8\\ & & &1 & & &2 &10 & &13 & &16 &4 &4 &20 &20 & &4\\ & & & & & &1 &5 & &40 & &8 &2 &2 &10 &10 & &2\\ & & & & & & &16 & &20 & &4 &1 &1 &5 &5 & &1\\ & & & & & & &8 & &10 & &2 & & &16 &16 & &4\\ & & & & & & &4 & &5 & &1 & & &8 &8 & &2\\ & & & & & & &2 & &16 & &4 & & &4 &4 & &1\\ & & & & & & &1 & &8 & &2 & & &2 &2 & &\\ & & & & & & &4 & &4 & &1 & & &1 &1 & &\\ & & & & & & &2 & &2 & & & & &4 &4 & &\\ & & & & & & &1 & &1 & & & & &2 &2 & &\\ & & & & & & & & &4 & & & & &1 &1 & &\\ & & & & & & & & &2 & & & & & & & &\\ & & & & & & & & &1 & & & & & & & &\\ \hline \hline &&&&&&&&$3n+3$&&&&&&&&\\ \hline $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10&11 &12 &13 &14 &15 &16 &17 \\ &6 &1 &12 &2 &18 &3 &24 &4 &30 &5 &36 &6 &42 &7 &48 &8 &54\\ &3 &6 &6 &1 &9 &12 &12 &2 &15 &18 &18 &3 &21 &24 &24 &4 &27\\ &12 &3 &3 &6 &30 &6 &6 &1 &48 &9 &9 &12 &66 &12 &12 &2 &84\\ &6 &12 & &3 &15 &3 &3 &6 &24 &30 &30 &6 &33 &6 &6 &1 &42\\ &3 &6 & &12 &48 & &12 &3 &12 &15 &15 &3 &102 &3 &3 &6 &21\\ & &3 & &6 &24 & &6 &12 &6 &48 &48 & &51 &12 &12 &3 &66\\ & & & &3 &12 & &3 &6 &3 &24 &24 & &156 &6 &6 &12 &33\\ & & & & &6 & & &3 &12 &12 &12 & &78 &3 &3 &6 &102\\ & & & & &3 & & & &6 &6 &6 & &39 & & &3 &51\\ & & & & &12 & & & &3 &3 &3 & &120 & & & &156\\ & & & & &6 & & & & &12 &12 & &60 & & & &78\\ & & & & &3 & & & & &6 &6 & &30 & & & &39\\ & & & & & & & & & &3 &3 & &15 & & & &120\\ & & & & & & & & & & & & &48 & & & &60\\ & & & & & & & & & & & & &24 & & & &30\\ & & & & & & & & & & & & &12 & & & &15\\ & & & & & & & & & & & & &6 & & & &48\\ & & & & & & & & & & & & &3 & & & &24\\ & & & & & & & & & & & & &12 & & & &12\\ & & & & & & & & & & & & &6 & & & &6\\ & & & & & & & & & & & & &3 & & & &3\\ & & & & & & & & & & & & & & & & &12\\ & & & & & & & & & & & & & & & & &6\\ & & & & & & & & & & & & & & & & &3\\ \hline \end{tabular} \end{centering} \end{table} \begin{table} \begin{centering} \begin{tabular}{lccccccccccccccccc} \hline \hline &&&&&&&&$3n+9$&&&&&&&&\\ \hline $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10&11 &12 &13 &14 &15 &16 &17 \\ &12 &1 &18 &2 &24 &3 &30 &4 &36 &5 &42 &6 &48 &7 &54 &8 &60\\ &6 &12 &9 &1 &12 &18 &15 &2 &18 &24 &21 &3 &24 &30 &27 &4 &30\\ &3 &6 &36 &12 &6 &9 &54 &1 &9 &12 &72 &18 &12 &15 &90 &2 &15\\ &18 &3 &18 &6 &3 &36 &27 &12 &36 &6 &36 &9 &6 &54 &45 &1 &54\\ &9 &18 &9 &3 &18 &18 &90 &6 &18 &3 &18 &36 &3 &27 &144 &12 &27\\ &36 &9 & &18 &9 &9 &45 &3 &9 &18 &9 &18 &18 &90 &72 &6 &90\\ &18 &36 & &9 &36 & &144 &18 & &9 &36 &9 &9 &45 &36 &3 &45\\ &9 &18 & &36 &18 & &72 &9 & &36 &18 & &36 &144 &18 &18 &144\\ & &9 & &18 &9 & &36 &36 & &18 &9 & &18 &72 &9 &9 &72\\ & & & &9 & & &18 &18 & &9 & & &9 &36 &36 &36 &36\\ & & & & & & &9 &9 & & & & & &18 &18 &18 &18\\ & & & & & & &36 & & & & & & &9 &9 &9 &9\\ & & & & & & &18 & & & & & & &36 & & &36\\ & & & & & & &9 & & & & & & &18 & & &18\\ & & & & & & & & & & & & & &9 & & &9\\ \hline \hline &&&&&&&&$3n+27$&&&&&&&&\\ \hline $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10&11 &12 &13 &14 &15 &16 &17 \\ &30 &1 &36 &2 &42 &3 &48 &4 &54 &5 &60 &6 &66 &7 &72 &8 &78\\ &15 &30 &18 &1 &21 &36 &24 &2 &27 &42 &30 &3 &33 &48 &36 &4 &39\\ &72 &15 &9 &30 &90 &18 &12 &1 &108 &21 &15 &36 &126 &24 &18 &2 &144\\ &36 &72 &54 &15 &45 &9 &6 &30 &54 &90 &72 &18 &63 &12 &9 &1 &72\\ &18 &36 &27 &72 &162 &54 &3 &15 &27 &45 &36 &9 &216 &6 &54 &30 &36\\ &9 &18 &108 &36 &81 &27 &36 &72 &108 &162 &18 &54 &108 &3 &27 &15 &18\\ &54 &9 &54 &18 &270 &108 &18 &36 &54 &81 &9 &27 &54 &36 &108 &72 &9\\ &27 &54 &27 &9 &135 &54 &9 &18 &27 &270 &54 &108 &27 &18 &54 &36 &54\\ &108 &27 & &54 &432 &27 &54 &9 & &135 &27 &54 &108 &9 &27 &18 &27\\ &54 &108 & &27 &216 & &27 &54 & &432 &108 &27 &54 &54 & &9 &108\\ &27 &54 & &108 &108 & &108 &27 & &216 &54 & &27 &27 & &54 &54\\ & &27 & &54 &54 & &54 &108 & &108 &27 & & &108 & &27 &27\\ & & & &27 &27 & &27 &54 & &54 & & & &54 & &108 &\\ & & & & &108 & & &27 & &27 & & & &27 & &54 &\\ & & & & &54 & & & & &108 & & & & & &27 &\\ & & & & &27 & & & & &54 & & & & & & &\\ & & & & & & & & & &27 & & & & & & &\\ \hline \end{tabular} \end{centering} \end{table} \begin{table} \label{collatz} \begin{centering} \begin{tabular}{lccccccccccccccccc} \hline \hline &&&&&&&&$3n+81$&&&&&&&&\\ \hline $n$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10&11 &12 &13 &14 &15 &16 &17 \\ &84 &1 &90 &2 &96 &3 &102 &4 &108 &5 &114 &6 &120 &7 &126 &8 &132\\ &42 &84 &45 &1 &48 &90 &51 &2 &54 &96 &57 &3 &60 &102 &63 &4 &66\\ &21 &42 &216 &84 &24 &45 &234 &1 &27 &48 &252 &90 &30 &51 &270 &2 &33\\ &144 &21 &108 &42 &12 &216 &117 &84 &162 &24 &126 &45 &15 &234 &135 &1 &180\\ &72 &144 &54 &21 &6 &108 &432 &42 &81 &12 &63 &216 &126 &117 &486 &84 &90\\ &36 &72 &27 &144 &3 &54 &216 &21 &324 &6 &270 &108 &63 &432 &243 &42 &45\\ &18 &36 &162 &72 &90 &27 &108 &144 &162 &3 &135 &54 &270 &216 &810 &21 &216\\ &9 &18 &81 &36 &45 &162 &54 &72 &81 &90 &486 &27 &135 &108 &405 &144 &108\\ &108 &9 &324 &18 &216 &81 &27 &36 & &45 &243 &162 &486 &54 &1296 &72 &54\\ &54 &108 &162 &9 &108 &324 &162 &18 & &216 &810 &81 &243 &27 &648 &36 &27\\ &27 &54 &81 &108 &54 &162 &81 &9 & &108 &405 &324 &810 &162 &324 &18 &162\\ &162 &27 & &54 &27 &81 &324 &108 & &54 &1296 &162 &405 &81 &162 &9 &81\\ &81 &162 & &27 &162 & &162 &54 & &27 &648 &81 &1296 &324 &81 &108 &324\\ &324 &81 & &162 &81 & &81 &27 & &162 &324 & &648 &162 &324 &54 &162\\ &162 &324 & &81 &324 & & &162 & &81 &162 & &324 &81 &162 &27 &81\\ &81 &162 & &324 &162 & & &81 & &324 &81 & &162 & &81 &162 &\\ & &81 & &162 &81 & & &324 & &162 &324 & &81 & & &81 &\\ & & & &81 & & & &162 & &81 &162 & &324 & & &324 &\\ & & & & & & & &81 & & &81 & &162 & & &162 &\\ & & & & & & & & & & & & &81 & & &81 &\\ \hline \end{tabular} \end{centering} \caption{\label{table} $3n+3^k$ sequence with $k=0, 1, 2,3$ and $4$ for $n$ ranging from $1$ to $17$.} \end{table} \nocite{*} \end{document}	arXiv
\begin{document} \title{From Gap-ETH to FPT-Inapproximability:\\ Clique, Dominating Set, and More} \author{ Parinya Chalermsook\thanks{Aalto University, Finland. Email: \texttt{[email protected]}.} \and Marek Cygan\thanks{Institute of Informatics, University of Warsaw, Poland. Email: \texttt{[email protected]}. } \and Guy Kortsarz\thanks{Rutgers University-Camden, New Jersey, USA. Email:\texttt{[email protected]}. } \and Bundit Laekhanukit\thanks{ Weizmann Institute of Science, Israel \& Shanghai University of Finance and Economics. Email: \texttt{[email protected]} } \and Pasin Manurangsi\thanks{University of California, Berkeley, USA. Email: \texttt{[email protected]}.} \and Danupon Nanongkai\thanks{KTH Royal Institute of Technology, Sweden. Email: \texttt{[email protected]} } \and Luca Trevisan\thanks{University of California, Berkeley, USA. Email: \texttt{[email protected]}.} } \date{\today} \hypersetup{ pdftitle = {FPT-Inapproximability for Clique, Dominating Set, and More}, pdfauthor = {} } We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., \cite{Marx08,FellowGMS12,DowneyF13}), are whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the size of the optimal solution. In particular, letting ${\sf OPT}\xspace$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}\xspace)\operatorname{poly}(N)$ time and outputs a solution of size $f({\sf OPT}\xspace)$, for any functions $t$ and $f$ that are independent of $N$ (for \mbox{\sf Clique}\xspace, we want $f({\sf OPT}\xspace)=\omega(1)$)? In this paper, we show that both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace admit no non-trivial FPT-approximation algorithm, i.e., there is no $o({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf Clique}\xspace and no $f({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf DomSet}\xspace, for any function $f$ (e.g., this holds even if $f$ is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to essentially enumerate all possibilities. Our results hold under the {\em Gap Exponential Time Hypothesis} (Gap-ETH)~\cite{Dinur16,ManR16}, which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3\mbox{\sf SAT}\xspace formula and one which is not even $(1 - \varepsilon)$-satisfiable for some constant $\varepsilon > 0$. Besides \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, we also rule out non-trivial FPT-approximation for Maximum Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm. To the best of our knowledge, prior results only rule out constant factor approximation for \mbox{\sf Clique}\xspace \cite{HajiaghayiKK13,BonnetE0P15} and $\log^{1/4+\epsilon}({\sf OPT}\xspace)$ approximation for \mbox{\sf DomSet}\xspace for any constant $\epsilon > 0$ \cite{ChenL16}. Our result on \mbox{\sf Clique}\xspace significantly improves on \cite{HajiaghayiKK13,BonnetE0P15}. However, our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16} since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption. \end{abstract} \endinput \begin{abstract} We consider questions arose from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g. \cite{Marx08,FellowGMS12,DowneyF13}), are whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the standard parameter $k$ (i.e. the solution size). In particular, letting ${\sf OPT}\xspace$ be the optimal value and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}\xspace)\operatorname{poly}(N)$ time and outputs a solution of size $f({\sf OPT}\xspace)$, for any functions $t$ and $f$ that are independent from $N$ (for the case of \mbox{\sf Clique}\xspace, we want $f({\sf OPT}\xspace)=\omega(1)$)? In this paper, we show that both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace do not admit any non-trivial FPT-approximation algorithm: there is no $o({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf Clique}\xspace and no $f({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf DomSet}\xspace, for any function $f$ (e.g. this holds even if $f$ is an exponential or the Ackermann function). Our results hold under the {\em gap Exponential Time Hypothesis} (gap-ETH), which has recently shown to be useful in proving hardness of (polynomial-time) approximation (e.g., \cite{Dinur16,ManR16,Man17}). In fact, our results imply something even stronger: the best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to enumerate all possibilities. In showing this, we show a few similar results for some problems on label cover instances. Besides the above results, we also show that no non-trivial FPT-approximation algorithm exists for Maximum Biclique and finding subgraphs with hereditary properties (e.g. maximum planar induced subgraph) Previously these problems were only known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. To the best of our knowledge, previous results on \mbox{\sf Clique}\xspace only refute constant factor approximation \cite{HajiaghayiKK13,BonnetE0P15}. For \mbox{\sf DomSet}\xspace, \cite{ChenL16} refutes a constant factor approximation under \mbox{FPT $\neq$ W[1]} and $\log^{1/4+\epsilon}k$, for any $\epsilon>0$ under ETH, where $k$ is the size of the optimal solution. Our result on \mbox{\sf Clique}\xspace significantly improves upon \cite{HajiaghayiKK13,BonnetE0P15} while our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16} since our results hold under the stronger assumption (i.e., gap-ETH). \danupon{Perhaps saying ``complete-FPT-inapproximable'' sounds stronger? Does mentioning conjectures make the paper sound stronger?} \danupon{I tried to drop some names of FPT people. But is this the best choice?} \end{abstract} \endinput \begin{abstract} \danupon{Concern about this version: People unfamiliar with the notion of FPT-approximation might have no idea why this is a fundamental issue. (But perhaps the previous version doesn't help with this either.) Should we drop a fundamental question here, perhaps ''Is there a meaningful approximation algorithm when {\sf OPT}\xspace is small?''?} It has been asked several times (e.g. \cite{Marx08,FellowGMS12,DowneyF13}) whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the standard parameter $k$ (i.e. the solution size). So far, only $O(1)$-FPT-approximation algorithms can be ruled out (under some assumptions) \cite{HajiaghayiKK13,BonnetE0P15,ChenL16}. In this paper, we rule out all possible non-trivial FPT-approximation algorithms under the {\em gap Exponential Time Hypothesis} (gap-ETH) \cite{Man17,Dinur16,ManR16}. Our result on \mbox{\sf Clique}\xspace significantly improves \cite{HajiaghayiKK13,BonnetE0P15}, while our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16}, since we need a stronger assumption.\danupon{The comparison part is perhaps optional.} \danupon{This paragraph might be optional.} In fact, our results imply something even stronger: the best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to enumerate all possibilities. In showing this, we show a few similar results for some problems on label cover instances. Besides the above results, we also show that no non-trivial FPT-approximation algorithm exists for Maximum Biclique and finding subgraphs with hereditary properties (e.g. maximum planar induced subgraph). Previously these problems were only known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. \end{abstract} \begin{titlepage} \maketitle \pagenumbering{roman} \begin{abstract} We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., \cite{Marx08,FellowGMS12,DowneyF13}), are whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the size of the optimal solution. In particular, letting ${\sf OPT}\xspace$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}\xspace)\operatorname{poly}(N)$ time and outputs a solution of size $f({\sf OPT}\xspace)$, for any functions $t$ and $f$ that are independent of $N$ (for \mbox{\sf Clique}\xspace, we want $f({\sf OPT}\xspace)=\omega(1)$)? In this paper, we show that both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace admit no non-trivial FPT-approximation algorithm, i.e., there is no $o({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf Clique}\xspace and no $f({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf DomSet}\xspace, for any function $f$ (e.g., this holds even if $f$ is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to essentially enumerate all possibilities. Our results hold under the {\em Gap Exponential Time Hypothesis} (Gap-ETH)~\cite{Dinur16,ManR16}, which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3\mbox{\sf SAT}\xspace formula and one which is not even $(1 - \varepsilon)$-satisfiable for some constant $\varepsilon > 0$. Besides \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, we also rule out non-trivial FPT-approximation for Maximum Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm. To the best of our knowledge, prior results only rule out constant factor approximation for \mbox{\sf Clique}\xspace \cite{HajiaghayiKK13,BonnetE0P15} and $\log^{1/4+\epsilon}({\sf OPT}\xspace)$ approximation for \mbox{\sf DomSet}\xspace for any constant $\epsilon > 0$ \cite{ChenL16}. Our result on \mbox{\sf Clique}\xspace significantly improves on \cite{HajiaghayiKK13,BonnetE0P15}. However, our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16} since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption. \end{abstract} \endinput \begin{abstract} We consider questions arose from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g. \cite{Marx08,FellowGMS12,DowneyF13}), are whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the standard parameter $k$ (i.e. the solution size). In particular, letting ${\sf OPT}\xspace$ be the optimal value and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}\xspace)\operatorname{poly}(N)$ time and outputs a solution of size $f({\sf OPT}\xspace)$, for any functions $t$ and $f$ that are independent from $N$ (for the case of \mbox{\sf Clique}\xspace, we want $f({\sf OPT}\xspace)=\omega(1)$)? In this paper, we show that both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace do not admit any non-trivial FPT-approximation algorithm: there is no $o({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf Clique}\xspace and no $f({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf DomSet}\xspace, for any function $f$ (e.g. this holds even if $f$ is an exponential or the Ackermann function). Our results hold under the {\em gap Exponential Time Hypothesis} (gap-ETH), which has recently shown to be useful in proving hardness of (polynomial-time) approximation (e.g., \cite{Dinur16,ManR16,Man17}). In fact, our results imply something even stronger: the best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to enumerate all possibilities. In showing this, we show a few similar results for some problems on label cover instances. Besides the above results, we also show that no non-trivial FPT-approximation algorithm exists for Maximum Biclique and finding subgraphs with hereditary properties (e.g. maximum planar induced subgraph) Previously these problems were only known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. To the best of our knowledge, previous results on \mbox{\sf Clique}\xspace only refute constant factor approximation \cite{HajiaghayiKK13,BonnetE0P15}. For \mbox{\sf DomSet}\xspace, \cite{ChenL16} refutes a constant factor approximation under \mbox{FPT $\neq$ W[1]} and $\log^{1/4+\epsilon}k$, for any $\epsilon>0$ under ETH, where $k$ is the size of the optimal solution. Our result on \mbox{\sf Clique}\xspace significantly improves upon \cite{HajiaghayiKK13,BonnetE0P15} while our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16} since our results hold under the stronger assumption (i.e., gap-ETH). \danupon{Perhaps saying ``complete-FPT-inapproximable'' sounds stronger? Does mentioning conjectures make the paper sound stronger?} \danupon{I tried to drop some names of FPT people. But is this the best choice?} \end{abstract} \endinput \begin{abstract} \danupon{Concern about this version: People unfamiliar with the notion of FPT-approximation might have no idea why this is a fundamental issue. (But perhaps the previous version doesn't help with this either.) Should we drop a fundamental question here, perhaps ''Is there a meaningful approximation algorithm when {\sf OPT}\xspace is small?''?} It has been asked several times (e.g. \cite{Marx08,FellowGMS12,DowneyF13}) whether there is a non-trivial {\em FPT-approximation} algorithm for the Maximum Clique (\mbox{\sf Clique}\xspace) and Minimum Dominating Set (\mbox{\sf DomSet}\xspace) problems parameterized by the standard parameter $k$ (i.e. the solution size). So far, only $O(1)$-FPT-approximation algorithms can be ruled out (under some assumptions) \cite{HajiaghayiKK13,BonnetE0P15,ChenL16}. In this paper, we rule out all possible non-trivial FPT-approximation algorithms under the {\em gap Exponential Time Hypothesis} (gap-ETH) \cite{Man17,Dinur16,ManR16}. Our result on \mbox{\sf Clique}\xspace significantly improves \cite{HajiaghayiKK13,BonnetE0P15}, while our result on \mbox{\sf DomSet}\xspace is incomparable to \cite{ChenL16}, since we need a stronger assumption.\danupon{The comparison part is perhaps optional.} \danupon{This paragraph might be optional.} In fact, our results imply something even stronger: the best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to enumerate all possibilities. In showing this, we show a few similar results for some problems on label cover instances. Besides the above results, we also show that no non-trivial FPT-approximation algorithm exists for Maximum Biclique and finding subgraphs with hereditary properties (e.g. maximum planar induced subgraph). Previously these problems were only known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. \end{abstract} \setcounter{tocdepth}{2} \tableofcontents \end{titlepage} \pagenumbering{arabic} \section{Introduction} \label{sec:intro} {\em Fixed-parameter approximation algorithm} (in short, FPT-approximation algorithm) is a new concept emerging from a cross-fertilization between two trends in coping with NP-hard problems: {\em approximation algorithms} and {\em fixed-parameter tractable (FPT) algorithms}. Roughly speaking, an FPT-approximation algorithm is similar to an FPT algorithm in that its running time can be of the form $t({\sf OPT}\xspace)\operatorname{poly}(N)$ time (called the {\em FPT time}), where $t$ is any function (possibly super exponentially growing), $N$ is the input size, and ${\sf OPT}\xspace$ is the value of the optimal solution\footnote{There are many ways to parameterize a problem. In this paper we focus on the {\em standard parameterization} which parameterizes the optimal solution.}. It is similar to an approximation algorithm in that its output is an {\em approximation} of the optimal solution; however, the approximation factor is analyzed in terms of the optimal solution ({\sf OPT}\xspace) and {\em not} the input size ($N$). Thus, an algorithm for a maximization (respectively, minimization) problem is said to be {\em $f({\sf OPT}\xspace)$-FPT-approximation} for some function $f$ if it outputs a solution of size at least ${\sf OPT}\xspace/f({\sf OPT}\xspace)$ (respectively, at most ${\sf OPT}\xspace\cdot f({\sf OPT}\xspace)$). \danupon{CHECK: Correct formulation?} For a maximization problem, such an algorithm is {\em non-trivial} when $f({\sf OPT}\xspace)$ is $o({\sf OPT}\xspace)$, while for a minimization problem, it is non-trivial for any computable function $f$. The notion of FPT-approximation is useful when we are interested in a small optimal solution, and in particular its existence connects to a fundamental question {\em whether there is a non-trivial approximation algorithm when the optimal solution is small}. Consider, for example, the {\em Maximum Clique} (\mbox{\sf Clique}\xspace) problem, where the goal is to find a clique (complete subgraph) with maximum number of vertices in an $n$-vertex graph $G$. By outputting any single vertex, we get a trivial polynomial-time $n$-approximation algorithm. The bound can be improved to $O(\frac{n}{\log n})$ and even to $O(\frac{n (\log\log n)^2}{\log^3 n})$ with clever ideas~\cite{Feige04}. Observe, however, that these bounds are quite meaningless when ${\sf OPT}\xspace=O(\frac{n (\log\log n)^2}{\log^3 n})$ since outputting a single vertex already guarantees such bounds. In this case, a bound such as $O(\frac{{\sf OPT}\xspace}{\log\log {\sf OPT}\xspace})$ would be more meaningful. Unfortunately, no approximation ratio of the form $o({\sf OPT}\xspace)$ is known even when FPT-time is allowed\footnote{In fact, for maximization problems, it can be shown that a problem admits an $f({\sf OPT}\xspace)$-FPT-approximation algorithm for some function $f = o({\sf OPT}\xspace)$ if and only if it admits a polynomial-time algorithm with approximation ratio $f'({\sf OPT}\xspace)$ for some function $f' = o({\sf OPT}\xspace)$ \cite{GroheG07,Marx08} (also see \cite{Marx13}). So, it does not matter whether the running time is polynomial on the size of the input or depends on ${\sf OPT}\xspace$.} (Note that outputting a single vertex gives an ${\sf OPT}\xspace$-approximation guarantee.) Similar questions can be asked for a minimization problem. Consider for instance, {\em Minimum Dominating Set} (\mbox{\sf DomSet}\xspace): Find the smallest set of vertices $S$ such that every vertex in an $n$-vertex input graph $G$ has a neighbor in $S$. \mbox{\sf DomSet}\xspace admits an $O(\log n)$-approximation algorithm via a basic greedy method. However, if we want the approximation ratio to depend on ${\sf OPT}\xspace$ and not $n$, no $f({\sf OPT}\xspace)$-approximation ratio is known for any function $f$ (not even $2^{2^{\sf OPT}\xspace}$). In fact, the existence of non-trivial FPT-approximation algorithms for \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace has been raised several times in the literature (e.g., \cite{Marx08,FellowGMS12,DowneyF13}). So far, the progress towards these questions can only rule out $O(1)$-FPT-approximation algorithms for \mbox{\sf Clique}\xspace. This was shown independently by Hajiaghayi~et~al.~\cite{HajiaghayiKK13} and Bonnet~et~al.~\cite{BonnetE0P15}, assuming the Exponential Time Hypothesis (ETH) and that a linear-size PCP exists. Alternatively, Khot and Shinkar \cite{KhotS16} proved this under a rather non-standard assumption that solving quadratic equations over a finite field under a certain regime of parameters is not in $\mbox{\sf FPT}\xspace$; unfortunately, this assumption was later shown to be false~\cite{Kayal2014}. For \mbox{\sf DomSet}\xspace, Chen and Li~\cite{ChenL16} could rule out $O(1)$-FPT-approximation algorithms assuming $\mbox{\sf FPT}\xspace\neq \mbox{\sf W}\xspace[1]$. Moreover, they improved the inapproximability ratio to $\log^{1/4+\epsilon}({\sf OPT}\xspace)$ for any constant $\epsilon > 0$ under the exponential time hypothesis (ETH), which asserts that no subexponential time algorithms can decide whether a given $3$\mbox{\sf SAT}\xspace formula is satisfiable. Remark that ETH implies that $\mbox{\sf FPT}\xspace \neq \mbox{\sf W}\xspace[1]$. \paragraph{Our Results and Techniques.} We show that there is no non-trivial FPT-approximation algorithm for both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. That is, there is no $o({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf Clique}\xspace and no $f({\sf OPT}\xspace)$-FPT-approximation algorithm for \mbox{\sf DomSet}\xspace, for any function $f$. Our results hold under the {\em Gap Exponential Time Hypothesis} (Gap-ETH), which states that distinguishing between a satisfiable $3$\mbox{\sf SAT}\xspace formula and one which is not even $(1-\epsilon)$-satisfiable requires exponential time for some constant $\epsilon > 0$ (see \Cref{sec:prelim}). \danupon{TO DO: Make sure that this is stated in prelim} Gap-ETH, first formalized in \cite{Dinur16,ManR16}, is a stronger version of the aforementioned ETH. It has recently been shown to be useful in proving fine-grained hardness of approximation for problems such as dense CSP with large alphabets~\cite{ManR16} and Densest-$k$-Subgraph with perfect completeness~\cite{Man17}. Note that Gap-ETH is implied by ETH if we additionally assume that a linear-size PCP exists. So, our result for \mbox{\sf Clique}\xspace significantly improves the results in \cite{HajiaghayiKK13,BonnetLP16} under the same (in fact, weaker) assumption. Our result for \mbox{\sf DomSet}\xspace also significantly improves the results in \cite{ChenL16}, but our assumption is stronger. In fact, we can show even stronger results: the best way to solve \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, even approximately, is to {\em enumerate all possibilities} in the following sense. Finding a clique of size $r$ can be trivially done in $n^{r}\operatorname{poly}(n)$ time by checking whether any among all possible ${n \choose r}=O(n^r)$ sets of vertices forms a clique. It was known under ETH that this is essentially the best one can do~\cite{ChenHKX06,ChenHKX06b}. We show further that this running time is still needed, even when we know that a clique of size much larger than $r$ exists in the graph (e.g., ${\sf OPT}\xspace \geq 2^{2^r}$), assuming Gap-ETH. Similarly, for \mbox{\sf DomSet}\xspace, we can always find a dominating set of size $r$ in $n^{r}\operatorname{poly}(n)$ time. Under Gap-ETH, we show that there is no better way even when we just want to find a dominating set of size $q\gg r$. We now give an overview of our techniques. The main challenge in showing our results is that we want them to hold for the case where the optimal solution is {\em arbitrarily smaller} than the input size. (This is important to get the FPT-inapproximability results.) To this end, (i) reductions cannot blow up the optimal solution by a function of the input size, and (ii) our reductions must start from problems with a large hardness gap, while having small ${\sf OPT}\xspace$. Fortunately, Property (i) holds for the known reductions we employ. The challenge of (ii) is that existing gap amplifying techniques (e.g., the parallel repetition theorem~\cite{Raz98} or the randomized graph product~\cite{BermanS92}), while amplifying the gap to arbitrarily large, cause the input size to be too large that existing ${\sf OPT}\xspace$ reduction techniques (e.g., \cite{ChenHKX06,PatrascuW10}) cannot be applied efficiently (in particular, in subexponential time). We circumvent this by a step that amplifies the gap and reduce ${\sf OPT}\xspace$ at the same time. In more detail, this step takes a 3SAT formula $\phi$ as an input and produces a ``label cover''\footnote{Our problem is an optimization problem on Label Cover instance, with a slightly different objective from the standard Label Cover. Please refer to Section~\ref{sec:label cover} for more detail.} instance $J$ (roughly, a bipartite graph with constraints on edges) such that: For any $c>0$, (i) If $\phi$ is satisfiable, then $J$ is satisfiable, and (ii) if $\phi$ is at most $0.99$ satisfiable, then less than $c$-fraction of constraints of $J$ can be satisfied. Moreover, our reduction allows us to ``compress'' either the the left-hand-side or the right-hand-side vertices to be arbitrarily small. This label cover instance is a starting point for all our problems. To derive our result for \mbox{\sf Clique}\xspace, we would need the left-hand-side to be arbitrarily small, while for \mbox{\sf DomSet}\xspace, we would need the small right-hand-side. The left-hand-side vertex compression is similar to the randomized graph product~\cite{BermanS92} and, in fact, the reduction itself has been studied before~\cite{Zuck96,Zuck96-unapprox} but in a very different regime of parameters. For a more detailed discussion, please refer to Subsection~\ref{subsec:smallr}. Once the inapproximability results for label cover problems with small left-hand-side and right-hand-side vertex set are established, we can simply reduce it to \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace using the standard reductions from~\cite{FGLSS96} and~\cite{Feige98} respectively. Besides the above results for \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace, we also show that no non-trivial FPT-approximation algorithm exists for a few other problems, including Maximum Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., maximum planar induced subgraph) and Maximum Induced Matching in bipartite graphs. Previously only the exact versions of these problems were only known to be $\mbox{\sf W}\xspace[1]$-hard~\cite{Lin15,KhotR00,MoserS09}. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph, although this ratio does not yet match the trivial $O(k)$-approximation algorithm. Finally, we remark that, while our result for maximum subgraphs with hereditary properties follows from a reduction from \mbox{\sf Clique}\xspace, the FPT inapproximability of other problems are shown not through the label cover problems, but instead from a modification of the hardness of approximation of Densest $k$-Subgraph in~\cite{Man17}. \danupon{If time permits, we may want to expand this paragraph and include more details about previous works.} \paragraph{Previous Works.} Our results are based on the method of compressing (or reducing the size of) the optimal solution, which was first introduced by Chen, Huang, Kanj and Xia in \cite{ChenHKX04} (the journal version appears in \cite{ChenHKX06}). Assuming ETH, they showed that finding both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace cannot be solved in time $n^{o({\sf OPT}\xspace)}$, where $n$ is the number of vertices in an input graph. Later, P\u{a}tra\cb{s}cu and Williams \cite{PatrascuW10} applied similar techniques to sharpen the running time lower bound of \mbox{\sf DomSet}\xspace to $n^{(1 - \varepsilon){\sf OPT}\xspace}$, for any constant $\varepsilon > 0$, assuming the {\em Strong Exponential Time Hypothesis} (SETH). The technique of compressing the optimal solution was also used in hardness of approximation by Hajiaghayi, Khandekar and Kortsarz in \cite{HajiaghayiKK13} and by Bonnet, Lampis and Paschos in \cite{BonnetE0P15}. Our techniques can be seen as introducing gap amplification to the reductions in \cite{ChenHKX06}. We emphasize that while \cite{ChenHKX06},\cite{PatrascuW10},\cite{HajiaghayiKK13} and \cite{BonnetE0P15} (and also the reductions in this paper) are all based on the technique of compressing the optimal solution, Hajiaghayi~et~al.~\cite{HajiaghayiKK13} compress the optimal solution after reducing \mbox{\sf SAT}\xspace to the designated problems, i.e., \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. \cite{ChenHKX06}, \cite{PatrascuW10}, \cite{BonnetE0P15} and our reductions, on the other hand, compress the optimal solution of \mbox{\sf SAT}\xspace prior to feeding it to the standard reductions (with small adjustment). While this difference does not affect the reduction for \mbox{\sf Clique}\xspace, it has a huge effect on \mbox{\sf DomSet}\xspace. Specifically, compressing the optimal solution at the post-reduction results in a huge blow-up because the blow-up in the first step (i.e., from \mbox{\sf SAT}\xspace to \mbox{\sf DomSet}\xspace) becomes exponential after compressing the optimal solution. Our proof for \mbox{\sf Clique}\xspace and the one in \cite{HajiaghayiKK13} bear a similarity in that both apply graph product to amplify approximation hardness. The key different is that we use randomized graph product instead of the deterministic graph product used in \cite{HajiaghayiKK13}. Very recently, Chen and Lin \cite{ChenL16} showed that \mbox{\sf DomSet}\xspace admits no constant approximation algorithm unless $\mathrm{FPT}=\mathrm{W[1]}$. Their hardness result was derived from the seminal result of Lin \cite{Lin15}, which shows that the {\em Maximum $k$-Intersection} problem (a.k.a, {\sf One-side Gap-\mbox{\sf Biclique}\xspace}) has no FPT approximation algorithm. Furthermore, they showed that, when assuming ETH, their result can be strengthened to rule out $\log^{1/4+\epsilon}({\sf OPT}\xspace)$ FPT-approximation algorithm, for any constant $\epsilon > 0$. The result of Chen and Lin follows from the W[1]-hardness of \mbox{\sf Biclique}\xspace \cite{Lin15} and the proof of the ETH-hardness of \mbox{\sf Clique}\xspace \cite{ChenHKX04}. Note that while Chen and Lin did not discuss the size of the optimal solution in their paper, the method of compressing the optimal solution was implicitly used there. This is due to the running-time lower bound of \mbox{\sf Clique}\xspace that they quoted from \cite{ChenHKX04}. Our method for proving the FPT inapproximability of \mbox{\sf DomSet}\xspace is similar to that in \cite{PatrascuW10}. However, the original construction in \cite{PatrascuW10} does not require a ``partition system''. This is because P\u{a}tra\cb{s}cu and Williams reduction starts from \mbox{\sf SAT}\xspace, which can be casted as \mbox{\sf DomSet}\xspace. In our construction, the reduction starts from an instance of the {\em Constraint Satisfaction} problem (\mbox{\sf CSP}\xspace) that is more general than \mbox{\sf SAT}\xspace (because of the gap-amplification step) and hence requires the construction of a partition system. (Note that the partition system has been used in standard hardness reductions for \mbox{\sf DomSet}\xspace \cite{LundY94,Feige98}.) We remark that our proof does not imply FPT-inapproximability for \mbox{\sf DomSet}\xspace under ETH whereas Chen and Lin were able to prove the inapproximability result under ETH because their reduction can be applied directly to \mbox{\sf SAT}\xspace via the result in \cite{ChenHKX06}. If ones introduced the Gap-ETH to the previous works, then the proofs in \cite{ChenHKX06,HajiaghayiKK13,BonnetE0P15} yield the constant FPT-inapproximability of \mbox{\sf Clique}\xspace, and the proof in \cite{ChenHKX06} yields the constant FPT-inapproximability of \mbox{\sf DomSet}\xspace. The summaries of previous works on \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace are presented in \Cref{tab:prev-works}. \begin{table}[h] \begin{center} \begin{tabular}{c\|c\|c\|c} \multicolumn{4}{c}{\bf Summary of Works on \mbox{\sf Clique}\xspace}\\ \hline {\bf \small{Inapprox Factor}} & {\bf \small{Running Time Lower Bound}} & {\bf \small{Assumption}} & {\bf \small{References}}\\ \hline any constant & $t(OPT) \cdot n^{o({\sf OPT}\xspace)}$ & ETH + LPCP & \cite{BonnetE0P15}\\ ${\sf OPT}\xspace^{1-\epsilon}$ & $\exp({\sf OPT}\xspace^{\rho(\epsilon)})$ & ETH & \cite{ChitnisHK13}\\ $1/(1-\epsilon)$ & $\exp(\exp({\sf OPT}\xspace^{\rho(\epsilon)}))$\footnotemark & ETH & \cite{HajiaghayiKK13}\\ No $\omega({\sf OPT}\xspace)$ & $t(OPT) \cdot n^{o({\sf OPT}\xspace)}$ & Gap-ETH & This paper\\ \hline \multicolumn{4}{c}{}\\ \multicolumn{4}{c}{\bf Summary of Works on \mbox{\sf DomSet}\xspace}\\ \hline {\bf \small{Inapprox Factor}} & {\bf \small{Running Time Lower Bound}} & {\bf \small{Assumption}} & {\bf \small{References}}\\ \hline ${\sf OPT}\xspace^{1-\gamma}$ & $\exp({\sf OPT}\xspace^{1-\rho(\gamma)})$ & ETH & \cite{ChitnisHK13}\\ $(\log {\sf OPT}\xspace)^{\delta}$ & $\exp(\exp((\log{\sf OPT}\xspace)^{\delta - 1}))$ & ETH + PGC & \cite{HajiaghayiKK13}\\ any constant & $t(OPT) \cdot n^{O(1)}$ (i.e. no FPT) & $\mathrm{W}[1]\neq\mathrm{FPT}$ & \cite{ChenL16}\\ $(\log{\sf OPT}\xspace)^{1/4+\epsilon}$ & $t(OPT) \cdot n^{o(\sqrt{{\sf OPT}\xspace})}$ & ETH & \cite{ChenL16}+\cite{ChenHKX06}\\ $f({\sf OPT}\xspace)$ & $t(OPT) \cdot n^{o({\sf OPT}\xspace)}$ & Gap-ETH & This paper\\ \hline \end{tabular} \end{center} \caption{The summaries of works on \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. Here $t$ denotes any computable function $t: {\mathbb N} \to {\mathbb N}$, $\epsilon$ denotes any constant $0<\varepsilon<1$, $\gamma$ denotes some constant $0<\epsilon<1$, $\rho$ denotes some non-decreasing function $\rho:(0,1)\rightarrow(0,1)$, $\delta$ denotes some constant $\delta>1$. PGC stands for the {\em Projection Game Conjecture} \cite{Moshkovitz15}, and LPCP stands for the {\em Linear-Size PCP Conjecture} \cite{BonnetE0P15}. } \label{tab:prev-works} \end{table} \footnotetext{Constant FPT-inapproximability of \mbox{\sf Clique}\xspace under ETH is claimed in \cite{HajiaghayiKK13} (arXiv version). However, as we investigated, the Gap-ETH is assumed there.} \paragraph{Other Related Works.} All problems considered in this work are also well-studied in terms of hardness of approximation beyond the aforementioned parameterized regimes; indeed many techniques used here are borrowed from or inspired by the non-parameterized settings. {\bf Maximum Clique.} Maximum Clique is arguably the first natural combinatorial optimization problem studied in the context of hardness of approximation; in a seminal work of Feige, Goldwasser, Lov{\'a}sz, Safra and Szegedy (henceforth FGLSS)~\cite{FGLSS96}, a connection was made between interactive proofs and hardness of approximating \mbox{\sf Clique}\xspace. This connection paves the way for later works on \mbox{\sf Clique}\xspace and other developments in the field of hardness of approximations; indeed, the FGLSS reduction will serve as part of our proof as well. The FGLSS reduction, together with the PCP theorem~\cite{AroraS98,AroraLMSS98} and gap amplification via randomized graph products~\cite{BermanS92}, immediately implies $n^{\varepsilon}$ ratio inapproximability of \mbox{\sf Clique}\xspace for some constant $\varepsilon > 0$ under the assumption that \mbox{\sf NP} $\subseteq$ \mbox{\sf BPP}. Following Feige~et~al.'s work, there had been a long line of research on approximability of \mbox{\sf Clique}\xspace~\cite{BellareGLR93,FeigeK00,BellareGS98,BellareS94}, which culminated in H{\aa}stad's work~\cite{Hastad96}. In~\cite{Hastad96}, it was shown that \mbox{\sf Clique}\xspace cannot be approximated to within a factor of $n^{1 - \varepsilon}$ in polynomial time unless \mbox{\sf NP} $\subseteq$ \mbox{\sf ZPP}; this was later derandomized by Zuckerman who showed a similar hardness under the assumption \mbox{\sf NP} $\nsubseteq$ \mbox{\sf P}~\cite{Zuckerman07}. Since then, better inapproximability ratios are known~\cite{EngebretsenH00,Khot01,KhotP06}, with the best ratio being $n/2^{(\log n)^{3/4 + \varepsilon}}$ for every $\varepsilon > 0$ (assuming \mbox{\sf NP} $\nsubseteq$ \mbox{\sf BPTIME}($2^{(\log n)^{O(1)}}$)) due to Khot and Ponnuswami~\cite{KhotP06}. We note here that the best known polynomial time algorithm for \mbox{\sf Clique}\xspace achieves $O\left(\frac{n(\log\log n)^2}{(\log n)^3}\right)$-approximation for the problem~\cite{Feige04}. {\bf Set Cover.} Minimum Set Cover, which is equivalent to Minimum Dominating Set, is also among the first problems studied in hardness of approximation. Lund and Yannakakis proved that, unless \mbox{\sf NP} $\subseteq$ \mbox{\sf DTIME}($2^{(\log n)^{O(1)}}$), \mbox{\sf SetCov}\xspace cannot be efficiently approximated to within $c \log n$ factor of the optimum for some constant $c > 0$~\cite{LundY94}. Not long after, Feige~\cite{Feige98} both improved the approximation ratio and weaken the assumption by showing an $(1 - \varepsilon)\ln n$-ratio inapproximability for every $\varepsilon > 0$ assuming only that \mbox{\sf NP} $\nsubseteq$ \mbox{\sf DTIME}($n^{O(\log \log n)}$). Recently, a similar inapproximability has been achieved under the weaker \mbox{\sf NP} $\nsubseteq$ \mbox{\sf P}~assumption~\cite{Moshkovitz15,DinurS14}. Since a simple greedy algorithm is known to yield $(\ln n + 1)$-approximation for \mbox{\sf SetCov}\xspace~\cite{Chvatal79}, the aforementioned hardness result is essentially tight. A common feature in all previous works on hardness of \mbox{\sf SetCov}\xspace\cite{LundY94,Feige98,Moshkovitz15} is that the constructions involve composing certain variants of CSPs with partition systems. As touched upon briefly earlier, our construction will also follow this approach; for the exact definition of CSPs and the partition system used in our work, please refer to Subsection~\ref{subsec:domset}. {\bf Maximum Subgraph with Hereditary Properties.} The complexity of finding and approximating maximum subgraph with hereditary properties have also been studied since the 1980s~\cite{LewisY80,LundY93,feige2005hardness}; specifically, Feige and Kogan showed that, for every non-trivial property $\Pi$ (i.e., $\Pi$ such that infinite many subgraphs satisfy $\Pi$ and infinitely many subgraphs do not satisfy $\Pi$), the problem is hard to approximate to within $n^{1 - \varepsilon}$ factor for every $\varepsilon > 0$ unless \mbox{\sf NP} $\subseteq$ \mbox{\sf ZPP}~\cite{feige2005hardness}. We also note that non-trivial approximation algorithms for the problem are known; for instance, when the property fails for some clique or some independent set, a polynomial time $O\left(\frac{n(\log \log n)^2}{(\log n)^2}\right)$-approximation algorithm is known~\cite{Halldorsson00}. {\bf Maximum Balanced Biclique.} While the Maximum Balanced Biclique problem bears a strong resemblance to the Maximum Clique Problem, inapproximability of the latter cannot be directly translated to that of the former; in fact, despite numerous attempts, not even constant factor NP-hardness of approximation of the Maximum Balanced Biclique problem is known. Fortunately, under stronger assumptions, hardness of approximation for the problem is known: $n^{\varepsilon}$-factor hardness of approximation is known under Feige's random 3\mbox{\sf SAT}\xspace hypothesis~\cite{Feige02} or \mbox{\sf NP} $\nsubseteq$ $\bigcap_{\varepsilon > 0} $\mbox{\sf BPTIME}($2^{n^\varepsilon}$)~\cite{Khot06}, and $n^{1 - \varepsilon}$-factor hardness of approximation is known under strengthening of the Unique Games Conjecture~\cite{BhangaleGHKK16,Man17-ICALP}. To the best of our knowledge, no non-trivial approximation algorithm for the problem is known. {\bf Densest $k$-Subgraph.} The Densest $k$-Subgraph problem has received considerable attention from the approximation algorithm community~\cite{KP93,FPK01,BCCFV10}; the best known polynomial time algorithm due to Bhaskara~et~al.~\cite{BCCFV10} achieves $O(n^{1/4 + \varepsilon})$-approximation for every $\varepsilon > 0$. On the other hand, similar to \mbox{\sf Biclique}\xspace, NP-hardness of approximating Densest $k$-Subgraph, even to some constant ratio, has so far eluded researchers. Nevertheless, in the same works that provide hardness results for \mbox{\sf Biclique}\xspace~\cite{Feige02,Khot06}, \mbox{\sf DkS}\xspace is shown to be hard to approximate to some constant factor under random 3-\mbox{\sf SAT}\xspace hypothesis or \mbox{\sf NP} $\nsubseteq$ $\bigcap_{\varepsilon > 0} $\mbox{\sf BPTIME}($2^{n^\varepsilon}$). Furthermore, $2^{\Omega(\log^{2/3} n)}$-factor inapproximability is known under the planted clique hypothesis~\cite{AAMMW11} and, under ETH (resp., Gap-ETH), $n^{1/\operatorname{poly}\log\log n}$ (resp., $n^{o(1)}$) factor inapproximabilities are known~\cite{Man17}. (See also~\cite{BravermanKRW17} in which a constant ratio ETH-hardness of approximating \mbox{\sf DkS}\xspace was shown.) In addition to these hardness results, polynomial ratio integrality gaps for strong LP and SDP relaxations of the problem are also known~\cite{BCVGZ12,M-thesis,ChlamtacMMV17}. {\bf Maximum Induced Matching on Bipartite Graphs.} The problem was proved to be NP-hard independently by Stockmeyer and Vazirani~\cite{StockmeyerV82} and Cameron~\cite{Cameron89}. The approximability of the problem was first studied by Duckworth~et~al.~\cite{DuckworthMZ05} who showed that the problem is APX-hard, even on bipartite graphs of degree three. Elbassioni~et~al.~\cite{ElbassioniRRS09} then showed that the problem is hard to approximate to within $n^{1/3 - \varepsilon}$ factor for every $\varepsilon > 0$, unless \mbox{\sf NP} $\subseteq$ \mbox{\sf ZPP}. Chalermsook~et~al.~\cite{ChalermsookLN13} later improved the ratio to $n^{1 - \varepsilon}$ for every $\varepsilon > 0$. \paragraph{Organization.} We define basic notations in \Cref{sec:prelim}. In \Cref{sec:inherent}, we define the notion of inherently enumerative, which captures the fact that nothing better than enumerating all possibilities can be done. We show that a problem admits no non-trivial FPT-approximation algorithm by showing that it is inherently enumerative. In \Cref{sec:label cover}, we define and prove results about our intermediate problems on label cover instances. Finally, in \Cref{sec:hardness-combopt} we derive results for \mbox{\sf Clique}\xspace, \mbox{\sf DomSet}\xspace, and other problems. \endinput \subsection{OLD STUFF (WILL BE REMOVED)} Like FPT algorithms\footnote{There are many ways to parameterize a problem. In this paper we focus on the {\em standard parameterization} by the solution size.}, an FPT-approximation algorithm takes a parameter $k$ as part of its input and is allowed to run in $O(t(k)\operatorname{poly}(n))$ time where $t$ is some function (possibly super exponentially growing) and $n$ is the input size. Like approximation algorithms, its output is an {\em approximation} of the optimal solution; however, the approximation factor is analyzed in terms of the optimal solution and not the input size. Roughly speaking, an algorithm for a maximization (respectively minimization) problem is said to be {\em $f(k)$-FPT-approximation} for some function $f$ if it can output a solution of size at least $k/f(k)$ (respectively at most $k\cdot f(k)$) whenever the optimal solution is of size $k$.\danupon{CHECK: Correct formulation.} (To be formal, it is better to define problems as {\em gap problems}. We do this in XXX.) An FPT-approximation algorithm is usually useful in tackling problems that previous approaches failed to provide satisfactory algorithmic results. For example, the {\em Strongly Connected Steiner Subgraph} problem is known to be $\log^{2-\epsilon} n$-hard to approximate in polynomial time \cite{HalperinK03} and is W[1]-hard \cite{GuoNS11} but is $2$-FPT-approximable \cite{ChitnisHK13}. Another example is the disjoint cycle problem which is $\Omega(\log n / \log \log n)$-hard to approximate in polynomial time (and might be even harder since the best known approximation factor is $O(\sqrt{n})$ \cite{KrivelevichNSYY07}.) It is also W[1]-hard but is $o(k)$-FPT-approximable\footnote{For maximization problems, being $o(k)$-FPT-approximable is usually referred to simply as being FPT-approximable.} \cite{GroheG07}. Some problems, e.g., computing treewidth, are not known to be FPT or W[1]-hard but enjoy the existence of an FPT-approximation algorithm (see, e.g., \cite{BodlaenderDDFLP13,RobertsonS95b}). There are also some negative results: One notable example is for the weighted monotone circuit satisfiability problem which was shown to be $2$-FPT-inapproximable by Alekhnovich and Razborov \cite{AlekhnovichR08} unless every problem in the class W[P] can be solved by a randomized FPT-algorithm. This result was used as a key step in proving that resolution is not automatizable. It was later improved by Eickmeyer, Grohe and Gr{\"{u}}ber \cite{EickmeyerGG08} and later settled by Marx \cite{Marx13}. For further survey, see, e.g., \cite[Chapter 31]{DowneyF13}. Among the central problems in this area, {\em Maximum Clique} (\mbox{\sf Clique}\xspace) and {\em Minimum Dominating Set} (\mbox{\sf DomSet}\xspace) are problems that are not known to be FPT-approximable. The first problem, \mbox{\sf Clique}\xspace, is a notoriously hard problem that is known to be intractable from every other perspective: \mbox{\sf Clique}\xspace has a hardness of $\Omega(n/2^{\log^{3/4-\epsilon} n})$ unless $\mbox{\sf NP} \subseteq {\sf BPTIME}(n^{\operatorname{polylog} n})$~\cite{Hastad96,KhotP06}, is W[1]-hard and thus is not fixed-parameter tractable unless FPT=W[1] \cite{DowneyF13}, and cannot be $r$-approximated in $2^{n^{1-\epsilon}/r^{1+\epsilon}}$ time unless the Exponential-Time Hypothesis (ETH) fails \cite{ChalermsookLN-FOCS13}. The latter problem, \mbox{\sf DomSet}\xspace, is more amendable and well-understood. It is $O(\log n)$-approximable and has an $O(\log r)$-approximation algorithm that runs in $2^{n/r}\operatorname{poly}(n)$-time. These guarantees almost match the best known approximation lower bound of $(1-\epsilon)\ln n)$, for any $\epsilon>0$, for polynomial-time algorithms unless P=NP due to the work of Feige \cite{Feige98} (the NP-hardness is due to the work of Dinur and Steurer \cite{DinurS14}), and the running time lower bound of $2^{n^{1-\epsilon}/r^{1+\epsilon}}$ for $r$-approximation algorithm by Bonnet~et~al.\cite{BonnetLP16}. However, when we consider the case that the optimal solution is $k \ll n$, e.g., $k=\log\log n$ or $k=\log\log r$, the mentioned approximation algorithms could give solutions that are exponential away from the optimal solution. FPT-algorithm might be a more desired approach for \mbox{\sf DomSet}\xspace; however, \mbox{\sf DomSet}\xspace admits no FPT-algorithm parameterized by the size of the optimal solution unless FPT=W[2] \cite{XXX}. Consequently, the more desirable approaches to tackle \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace might be FPT-approximation algorithms whose running time and approximation guarantees depend on the parameter, say the size of the optimal solution. However, the FPT-approximability of \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace are much less understood than other approaches. In fact, it was conjectured that both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace do not admit this approach. More precisely, \mbox{\sf Clique}\xspace is not $o(k)$-FPT-approximable (see \Cref{conj:Clique is FPT-inapproximable} for a formal statement), and \mbox{\sf DomSet}\xspace is not $g(k)$-FPT-approximable, for any function $g(k)$ (which can be super exponential or even Ackermann function). Since it was sometimes attributed to Fellows and Marx (e.g., \cite{ChitnisHK13,HajiaghayiKK13}), we will call this the {\em Fellows-Marx} conjecture. The conjecture is fairly strong since it implies that there is no non-trivial FPT-approximation-algorithm for both \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. There is not much evidence to support or oppose this conjecture, and we believe that there is a reason to believe in either way: XXXX On one hand, we know that there is no polynomial time approximation algorithm for \mbox{\sf Clique}\xspace that can beat a trivial solution: we can easily $O(\frac{n}{\log n})$-approximate \mbox{\sf Clique}\xspace in polynomial time and in fact we can even $O(\frac{n (\log\log n)^2}{\log^3 n})$-approximate it \cite{Feige04}. On the other hand, it is unlikely that an FPT-algorithm can solve \mbox{\sf Clique}\xspace faster than the trivial solution: any FPT-algorithm parameterized by the size of a maximum independent set, say $k$, requires $n^{\Omega(k)}$ time assuming ETH. As noted by Marx \cite{Marx13}, proving this conjecture seems to need techniques that are deeper than those required by some previous FPT-inapproximability results because all known proofs for the inapproximability of \mbox{\sf Clique}\xspace use the PCP theorem, and thus proving the FPT-inapproximability for this problem will require the use of the PCP theorem or its generalization.\danupon{From this point of view, we should perhaps emphasize on the PCP theorem.} There were some results on this direction (e.g., \cite{BonnetE0P15,ChitnisHK13,HajiaghayiKK13}). They can either rule out algorithms with a certain running time (not all FPT running time) or FPT-approximation algorithms with a small approximation ratio (we discuss this more below)\parinya{Should say a bit that previous works also assumed linear PCP.}. One of the main difficulties faced by these approaches in general is that all known hardness of approximation techniques for \mbox{\sf Clique}\xspace generate hard instances whose optimal values are fairly large compared to the size of the graphs; this case is, however, an easy case for an FPT algorithm since an FPT approximation algorithm is allowed to run in time exponential (or even super-exponential) in the optimal value. An orthogonal direction, as initiated recently by \cite{KhotS16}, is to assume that there is no FPT-algorithm for a carefully selected problem, and apply the PCP machineries to deduce an FPT-inapproximability result for \mbox{\sf Clique}\xspace. A constant-FPT-inapproximability was shown in \cite{KhotS16} using this direction. \danupon{TO DO: Say that we can easily get $n/\log n$-approximation factor.} \paragraph{Our Results.} \subsection{Junk} From \url{https://pdfs.semanticscholar.org/0b92/aeb560fd18bca13aa54508026d928aecc52e.pdf} As a key step in showing that resolution is not automatizable, Alekhnovich and Razborov [1] showed that there is no fpt 2-approximation algorithm for WEIGHTED MONOTONE CIRCUIT SATISFIABILITY, unless every problem in the class W[P] can be solved by a randomized fpt-algorithm. Eickmeyer~et~al. [8] improved this result in two ways: they weakened the complexity assumption by removing the word “randomized,” and increased the ratio from 2 to any polylogarithmic function. They conjectured that the problem has no fpt approximation algorithm for any function $\rho$. Parameterized Approximation for Dominating Set was asked by Fellows in Problem 4.16 in \cite{FellowGMS12}. The standard linear-size PCP conjecture has been used (along with the Exponential Time Hypothesis) before by Hajiaghayi~et~al.~\cite{HajiaghayiKK13} and Bonnet~et~al.~\cite{BonnetE0P15} to show that \mbox{\sf Clique}\xspace is not $O(1)$-FPT-approximable\footnote{Hajiaghayi~et~al.~\cite{HajiaghayiKK13} did not mention that the linear-size PCP is required. But, by examining the proof of Theorem 4.4 in \cite{HajiaghayiKK13}, it can be seen that a linear-size PCP is used since it is assumed that there is no subexponential-time algorithm for a Gap-3\mbox{\sf SAT}\xspace instance (cf. Theorem 6.1 in \cite{HajiaghayiKK13}).}. \section{Preliminaries} \label{sec:prelim} We use standard terminology. For any graph $G$, we denote by $V(G)$ and $E(G)$ the vertex and edge sets of $G$, respectively. For each vertex $u \in V(G)$, we denote the set of its neighbors by $N_G(v)$; when the graph $G$ is clear from the context, we sometimes drop it from the notation. A {\em clique} of $G$ is a complete subgraph of $G$. Sometime we refer to a clique as a subset $S \subseteq V(G)$ such that there is an edge joining every pair of vertices in $S$. A {\em biclique} of $G$ is a balanced complete bipartite subgraph of $G$ (i.e., the graph $K_{k,k}$). By $k$-biclique, we mean the graph $K_{k,k}$ (i.e., the number of vertices in each partition is $k$). An {\em independent set} of $G$ is a subset of vertices $S\subseteq V(G)$ such there is no edge joining any pair of vertices in $S$. A {\em dominating set} of $G$ is a subset of vertices $S\subseteq V(G)$ such that every vertex in $G$ is either in $S$ or has a neighbor in $S$. The {\em clique number} (resp., {\em independent number}) of $G$ is the size of the largest clique (resp., independent set) in $G$. The {\em biclique number} of $G$ is the largest integer $k$ such that $G$ contains $K_{k, k}$ as a subgraph. The {\em domination number} of $G$ is defined similarly as the size of the smallest dominating set in $G$. The clique, independent and domination numbers of $G$ are usually denoted by $\omega(G)$, $\alpha(G)$ and $\gamma(G)$, respectively. However, in this paper, we will refer to these numbers by $\mbox{\sf Clique}\xspace(G), \mbox{\sf MIS}\xspace(G), \mbox{\sf DomSet}\xspace(G)$. Additionally, we denote the biclique number of $G$ by $\mbox{\sf Biclique}\xspace(G)$ \subsection{FPT Approximation} Let us start by formalizing the the notation of optimization problems; here we follow the notation due to Chen et al.~\cite{ChenGG06}. An \emph{optimization problem} $\Pi$ is defined by three components: (1) for each input instance $I$ of $\Pi$, a set of valid solutions of $I$ denoted by ${\sf SOL} \xspace_\Pi(I)$, (2) for each instance $I$ of $\Pi$ and each $y \in {\sf SOL} \xspace_\Pi(I)$, the cost of $y$ with respect to $I$ denoted by ${\sf COST} \xspace_\Pi(I, y)$, and (3) the goal of the problem ${\sf GOAL} \xspace_\Pi \in \{\min, \max\}$ which specifies whether $\Pi$ is a minimization or maximization problem. Throughout this work, we will assume that ${\sf COST} \xspace_\Pi(I, y)$ can be computed in time $\|I\|^{O(1)}$. Finally, we denote by ${\sf OPT}\xspace_{\Pi}(I)$ the optimal value of each instance $I$, i.e. ${\sf OPT}\xspace_\Pi(I) = {\sf GOAL} \xspace_\Pi~{\sf COST} \xspace(I, y)$ where $y$ is taken over ${\sf SOL} \xspace_\Pi(I)$. We now continue on to define parameterized approximation algorithms. While our discussion so far has been on optimization problems, we will instead work with ``gap versions'' of these problems. Roughly speaking, for a maximization problem $\Pi$, the gap version of $\Pi$ takes in an additional input $k$ and the goal is to decide whether ${\sf OPT}\xspace_\Pi(I) \geq k$ or ${\sf OPT}\xspace_\Pi(I) < k / f(k)$. As we will elaborate below, the gap versions are weaker (i.e. easier) than the optimization versions and, hence, our impossibility results for gap versions translate to those of optimization versions as well. \begin{definition}[FPT gap approximation]\label{def:FPT gap approx} For any optimization problem $\Pi$ and any computable function $f: {\mathbb N} \rightarrow [1, \infty)$, an algorithm ${\mathbb A}$, which takes as input an instance $I$ of $\Pi$ and a positive integer $k$, is said to be an \emph{$f$-FPT gap approximation algorithm} for $\Pi$ if the following conditions hold on every input $(I, k)$: \begin{itemize} \item ${\mathbb A}$ runs in time $t(k) \cdot \|I\|^{O(1)}$ for some computable function $t: {\mathbb N} \rightarrow {\mathbb N}$. \item If ${\sf GOAL} \xspace_\Pi = \max$, ${\mathbb A}$ must output 1 if ${\sf OPT}\xspace_\Pi(I) \geq k$ and output 0 if ${\sf OPT}\xspace_\Pi(I) < k / f(k)$. If ${\sf GOAL} \xspace_\Pi = \min$, ${\mathbb A}$ must output 1 if ${\sf OPT}\xspace_\Pi(I) \leq k$ and output 0 if ${\sf OPT}\xspace_\Pi(I) > k \cdot f(k)$. \end{itemize} $\Pi$ is said to be \emph{$f$-FPT gap approximable} if there is an $f$-FPT gap approximation algorithm for $\Pi$. \end{definition} Next, we formalize the concept of \emph{totally FPT inapproximable}, which encapsulates the non-existence of non-trivial FPT approximations discussed earlier in the introduction. \begin{definition} \label{def:totalinapprox} A minimization problem $\Pi$ is said to be \emph{totally FPT inapproximable} if, for every computable function $f: {\mathbb N} \rightarrow [1, \infty)$, $\Pi$ is not $f$-FPT gap approximable. A maximization problem $\Pi$ is said to be \emph{totally FPT inapproximable} if, for every computable function $f: {\mathbb N} \rightarrow [1, \infty)$ such that $f(k) = o(k)$ (i.e. $\lim_{k \to \infty} k/f(k) = \infty$), $\Pi$ is not $f$-FPT gap approximable. \end{definition} With the exception of Densest $k$-Subgraph, every problem considered in this work will be shown to be totally FPT inapproximable. To this end, we remark that totally FPT inapproximable as defined above through gap problems imply the non-existence of non-trivial FPT approximation algorithm that was discussed in the introduction. These implications are stated more precisely in the two propositions below; their proofs are given in Appendix~\ref{app:gapvapprox}. \begin{proposition} \label{prop:gapvapprox-min} Let $\Pi$ be any minimization problem. Then, (1) implies (2) where (1) and (2) are as defined below. \begin{enumerate}[(1)] \item $\Pi$ is totally FPT inapproximable. \item For all computable functions $t: {\mathbb N} \rightarrow {\mathbb N}$ and $f: {\mathbb N} \rightarrow [1, \infty)$, there is no algorithm that, on every instance $I$ of $\Pi$, runs in time $t({\sf OPT}\xspace_{\Pi}(I)) \cdot \|I\|^{O(1)}$ and outputs a solution $y \in {\sf SOL} \xspace_\Pi(I)$ such that ${\sf COST} \xspace_\Pi(I, y) \leq {\sf OPT}\xspace_{\Pi}(I) \cdot f({\sf OPT}\xspace_{\Pi}(I))$. \end{enumerate} \end{proposition} \begin{proposition} \label{prop:gapvapprox-max} Let $\Pi$ be any maximization problem. Then, (1) implies (2) where (1) and (2) are as defined below. \begin{enumerate}[(1)] \item $\Pi$ is totally FPT inapproximable. \item For all computable functions $t: {\mathbb N} \rightarrow {\mathbb N}$ and $f: {\mathbb N} \rightarrow [1, \infty)$ such that $f(k) = o(k)$ and $k/f(k)$ is non-decreasing, there is no algorithm that, on every instance $I$ of $\Pi$, runs in time $t({\sf OPT}\xspace_{\Pi}(I)) \cdot \|I\|^{O(1)}$ and outputs a solution $y \in {\sf SOL} \xspace_\Pi(I)$ such that ${\sf COST} \xspace_\Pi(I, y) \geq {\sf OPT}\xspace_{\Pi}(I) / f({\sf OPT}\xspace_{\Pi}(I))$. \end{enumerate} \end{proposition} \subsection{List of Problems} We will now list the problems studied in this work. While all the problems here can be defined in terms of optimization problems as defined the previous subsection, we will omit the terms ${\sf SOL} \xspace, {\sf COST} \xspace$ and ${\sf GOAL} \xspace$ whenever they are clear from the context. \noindent{\bf The Maximum Clique Problem (\mbox{\sf Clique}\xspace).} In $k$-\mbox{\sf Clique}\xspace, we are given a graph $G$ together with an integer $k$, and the goal is to decide whether $G$ has a clique of size $k$. The maximization version of \mbox{\sf Clique}\xspace, called \mbox{\sf Max-Clique}, asks to compute the maximum size of a clique in $G$. We will abuse \mbox{\sf Clique}\xspace to mean the {\sf Max-Clique} problem, and we will denote by $\mbox{\sf Clique}\xspace(G)$ the clique number of $G$, which is the value of the optimal solution to \mbox{\sf Clique}\xspace. The problem that is (computationally) equivalent to \mbox{\sf Clique}\xspace is the {\em maximum independent set} problem (\mbox{\sf MIS}\xspace) which asks to compute the size of the maximum independent set in $G$. The two problems are equivalent since any clique in $G$ is an independent set in the complement graph $\bar{G}$. \noindent{\bf The Minimum Dominating Set Problem (\mbox{\sf DomSet}\xspace).} In $k$-\mbox{\sf DomSet}\xspace, we are given a graph $G$ together with an integer $k$, and the goal is to decide whether $G$ has a dominating set of size $k$. The minimization version of $k$-\mbox{\sf DomSet}\xspace is called the \mbox{\sf DomSet}\xspace, which asks to compute the size of the minimum dominating set in~$G$. The problem that is equivalent to \mbox{\sf DomSet}\xspace is the {\em minimum set cover} problem (\mbox{\sf SetCov}\xspace): Given a universe $\mathcal{U}$ of $n$ elements and a collection $\mathcal{S}$ of $m$ subsets $S_1,\ldots,S_m\subseteq\mathcal{U}$, the goal is to find the minimum number of subsets of $\mathcal{S}$ whose union equals $\mathcal{U}$. It is a standard fact that \mbox{\sf DomSet}\xspace is equivalent to \mbox{\sf SetCov}\xspace. See Appendix~\ref{app:trivial-eq} for more detail. \noindent{\bf Maximum Induced Subgraph with Hereditary Properties:} A {\em property} $\Pi$ is simply a subset of all graphs. We say that $\Pi$ is a {\em hereditary property} if whenever $G \in \Pi$, all induced subgraphs of $G$ are in $\Pi$. The Maximum Induced Subgraph problem with Property $\Pi$ asks for a maximum cardinality set $S \subseteq V(G)$ such that $G[S] \in \Pi$. Here $G[S]$ denotes the subgraph of $G$ induced on $S$. Notice that both \mbox{\sf Clique}\xspace and \mbox{\sf MIS}\xspace belong to this class of problems. For more discussions on problems that belong to this class, see Appendix~\ref{app:trivial-eq}. \noindent{\bf Maximum Induced Matching on Bipartite Graphs:} An induced matching ${\mathcal M}$ of a graph $G = (V, E)$ is a subset of edges $\{(u_1, v_1), \dots, (u_{\|{\mathcal M}\|}, v_{\|{\mathcal M}\|})\}$ such that there is no cross edge, i.e., $(u_i, u_j), (v_i, v_j), (u_i, v_j) \notin E$ for all $i \ne j$. The induced matching number $\mbox{\sf IM}\xspace(G)$ of graph $G$ is simply the maximum value of $\|{\mathcal M}\|$ among all induced matchings ${\mathcal M}$'s of $G$. In this work, we will be interested in the problem of approximating $\mbox{\sf IM}\xspace(G)$ in bipartite graphs; this is because, for general graphs, the problem is as hard to approximate as \mbox{\sf Clique}\xspace. (See Appendix~\ref{app:trivial-eq} for more details.) \noindent{\bf Maximum Balanced Biclique (\mbox{\sf Biclique}\xspace).} In $k$-\mbox{\sf Biclique}\xspace, we are given a bipartite graph $G$ together with an integer $k$. The goal is to decide whether $G$ contains a complete bipartite subgraph (biclique) with $k$ vertices on each side. In other words, we are asked to decide whether $G$ contains $K_{k,k}$ as a subgraph. The maximization version of \mbox{\sf Biclique}\xspace, called Maximum Balanced Biclique, asks to compute the maximum size of a balanced biclique in $G$. \noindent{\bf Densest $k$-Subgraph (\mbox{\sf DkS}\xspace).} In the Densest $k$-Subgraph problem, we are given an integer $k$ and a graph $G = (V, E)$. The goal is to find a subset $S \subseteq V$ of $k$ vertices that induces maximum number of edges. For convenience, we define density of an induced subgraph $G[S]$ to be $\mbox{\sf Den}\xspace(G[S]) \triangleq \frac{E(G[S])}{\binom{\|S\|}{2}} \in [0, 1]$ and we define the optimal density of \mbox{\sf DkS}\xspace to be $\mbox{\sf Den}\textsubscript{$k$}\xspace(G) = \max_{S \subseteq V, \|S\| = k} \mbox{\sf Den}\xspace(S)$. \subsection{Gap Exponential Time Hypothesis} Our results are based on the Gap Exponential Time Hypothesis (Gap-ETH). Before we state the hypothesis, let us recall the definition of 3-\mbox{\sf SAT}\xspace. In $q$-\mbox{\sf SAT}\xspace, we are given a CNF formula $\phi$ in which each clause consists of at most $q$ literals, and the goal is to decide whether $\phi$ is satisfiable. \mbox{\sf Max $q$-SAT} is a maximization version of $q$-\mbox{\sf SAT}\xspace which asks to compute the maximum number of clauses in $\phi$ that can be simultaneously satisfied. We will abuse $q$-\mbox{\sf SAT}\xspace to mean \mbox{\sf Max $q$-SAT}, and for a formula $\phi$, we use $\mbox{\sf SAT}\xspace(\phi)$ to denote the maximum number of clauses satisfied by any assignment. The Gap Exponential Time Hypothesis can now be stated in terms of \mbox{\sf SAT}\xspace as follows. \begin{conjecture}[(randomized) Gap Exponential-Time Hypothesis (Gap-ETH) \cite{Dinur16,ManR16}] \label{conj:gap-ETH} For some constant $\delta,\epsilon > 0$, no algorithm can, given a $3$-\mbox{\sf SAT}\xspace formula $\phi$ on $n$ variables and $m=O(n)$ clauses, distinguishes between the following cases correctly with probability $\geq 2/3$ in $O(2^{\delta n})$ time: \begin{itemize} \item $\mbox{\sf SAT}\xspace(\phi) = m$ and \item $\mbox{\sf SAT}\xspace(\phi) < (1-\epsilon) m$. \end{itemize} \end{conjecture} Note that the case where $\epsilon=1/m$ (that is, the algorithm only needs to distinguish between the cases that $\mbox{\sf SAT}\xspace(\phi)=m$ and $\mbox{\sf SAT}\xspace(\phi)<m$) is known as ETH \cite{IPZ01}. Another related conjecture is the strengthened version of ETH is called {\em the Strong Exponential-Time Hypothesis} (SETH) \cite{IP01-ETH}: for any $\epsilon>0$, there is an integer $k \geq 3$ such that there is no $2^{(1-\epsilon)n}$-time algorithm for $k$-SAT. Gap-ETH of course implies ETH, but, to the best of our knowledge, no formal relationship is known between Gap-ETH and SETH. While Gap-ETH may seem strong due to the gap between the two cases, there are evidences suggesting that it may indeed be true, or, at the very least, refuting it is beyond the reach of our current techniques. We discuss some of these evidences in Appendix~\ref{app:gap-eth}. While Gap-ETH as stated above rules out not only deterministic but also randomized algorithms, the deterministic version of Gap-ETH suffices for some of our results, including inapproximability of \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. The reduction for \mbox{\sf DomSet}\xspace as stated below will already be deterministic, but the reduction for \mbox{\sf Clique}\xspace will be randomized. However, it can be easily derandomized and we sketch the idea behind this in in Subsection~\ref{subsec:derandomization}. Note that, on the other hand, we do not know how to derandomize some of our other results, including those of \mbox{\sf Biclique}\xspace and \mbox{\sf DkS}\xspace. \section{FPT Inapproximability via Inherently Enumerative Concept}\label{sec:inherent} Throughout the paper, we will prove FPT inapproximability through the concept of {\em inherently enumerative} problems, which will be formalized shortly. To motivate the concept, note that all problems $\Pi$ considered in this paper admit an exact algorithm that runs in time\footnote{Recall that $O^{\star}(\cdot)$ hides terms that are polynomial in the input size.} $O^{\star}(\|I\|^{{\sf OPT}\xspace_{\Pi}(I)})$; For instance, to find a clique of size $k$ in $G$, one can enumerate all ${\|V(G)\| \choose k} = \|V(G)\|^{O(k)}$ possibilities\footnote{A faster algorithm runs in time $\|V(G)\|^{\omega k/3}$ can be done by a reduction to matrix multiplication.}. For many W[1]-hard problems (e.g. \mbox{\sf Clique}\xspace), this running time is nearly the best possible assuming ETH: Any algorithm that finds a $k$-clique in time $\|V(G)\|^{o(k)}$ would break ETH. In the light of such result, it is natural to ask the following question. \begin{quote} Assume that $\mbox{\sf Clique}\xspace(G) \geq 2^{2^k}$, can we find a clique of size $k$ in time $\|V(G)\|^{o(k)}$? \end{quote} \iffalse The concept of inherently enumerative is motivated by the above concept. In particular, it addresses the following kind of questions: \begin{quote} Assuming the existence of a clique of size $\operatorname{poly} \log \|V(G)\|$, can we find a clique of size $k$ in time $\|V(G)\|^{o(k)}$? \end{quote} We remark that the answer to this question is not obvious: For instance, in the case of cliques, given the existence of a clique of size $\epsilon \|V(G)\|$, Halperin's SDP algorithm~\cite{Halperin02} is guaranteed to find a clique of size $\|V(G)\|^{\Omega(\epsilon)}$ in polynomial time (so the existence of a large clique in $G$ allows us to search for a super-constant sized clique in polynomial time.) Also, Feige's algorithm finds a clique of size $\operatorname{poly} \log \|V(G)\|$ whenever there is a clique of size $\|V(G)\|/ \operatorname{poly} \log \|V(G)\|$~\cite{Feige04}. It has been open whether one can find a clique of super constant size in polynomial time if we are guaranteed only the existence of $\Omega(\log n)$ clique. Now, we formalize the concept. \fi In other words, can we exploit a prior knowledge that there is a clique of size much larger than $k$ to help us find a $k$-clique faster? Roughly speaking, we will show later that, assuming Gap-ETH, the answer of this question is also negative, even when $2^{2^k}$ is replaced by any constant independent of $k$. This is encapsulated in the inherently enumerative concept as defined below. \begin{definition}[Inherently Enumerative] A problem $\Pi$ is said to be {\em inherently enumerative} if there exist constants $\delta, r_0 >0$ such that, for any integers $q \geq r \geq r_0$, no algorithm can decide, on every input instance $I$ of $\Pi$, whether (i) ${\sf OPT}\xspace_{\Pi}(I) < r$ or (ii) ${\sf OPT}\xspace_{\Pi}(I) \geq q$ in time\footnote{$O_{q, r}(\cdot)$ here and in Definition~\ref{def:weakine} hides any multiplicative term that is a function of $q$ and $r$.} $O_{q, r}(\|I\|^{\delta r})$. \end{definition} \iffalse Intuitively, inherently enumerative maximization problems are those problems for which the best way to search for a solution of value $k$ is to ``enumerate'' all possible solutions of size $k$, despite a prior knowledge that a much better solution (i.e. a solution of value $2^{2^k}$) exists. \fi While we will show that \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace are inherently enumerative, we cannot do the same for some other problems, such as \mbox{\sf Biclique}\xspace. Even for the exact version of \mbox{\sf Biclique}\xspace, the best running time lower bound known is only $\|V(G)\|^{\Omega(\sqrt{k})}$~\cite{Lin15} assuming ETH. In order to succinctly categorize such lower bounds, we define a similar but weaker notation of \emph{weakly} inherently enumerative: \begin{definition}[Weakly Inherently Enumerative] \label{def:weakine} For any function $\beta = \omega(1)$ (i.e. $\lim_{r \to \infty} \beta(r) = \infty$), a problem $\Pi$ is said to be {\em $\beta$-weakly inherently enumerative} if there exists a constant $r_0 >0$ such that, for any integers $q \geq r \geq r_0$, no algorithm can decide, on every input instance $I$ of $\Pi$, whether (i) ${\sf OPT}\xspace_{\Pi}(I) < r$ or (ii) ${\sf OPT}\xspace_{\Pi}(I) \geq q$ in time $O_{q, r}(\|I\|^{\beta(r)})$. $\Pi$ is said to be {\em weakly inherently enumerative} if it is $\beta$-weakly inherently enumerative for some $\beta = \omega(1)$. \end{definition} It follows from the definitions that any inherently enumerative problem is also weakly inherently enumerative. As stated earlier, we will prove total FPT inapproximability through inherently enumerative; the proposition below formally establishes a connection between the two. \begin{proposition} If $\Pi$ is weakly inherently enumerative, then $\Pi$ is totally FPT inapproximable. \end{proposition} \begin{proof} We first consider maximization problems. We will prove the contrapositive of the statement. Assume that a maximization problem $\Pi$ is not totally FPT inapproximable, i.e., $\Pi$ admits an $f$-FPT gap approximation algorithm ${\mathbb A}$ for some computable function $f$ such that $\lim_{k \to \infty} k/f(k) = \infty$. Suppose that the running time of ${\mathbb A}$ on every input $(I, k)$ is $t(k) \cdot \|I\|^D$ for some constant $D$ and some function $t$. We will show that $\Pi$ is not weakly inherently enumerative. Let $r_0 > 0$ be any constant and let $\beta: {\mathbb N} \rightarrow \mathbb{R}^+$ be any function such that $\beta = \omega(1)$. Let $r$ be the smallest integer such that $r > r_0$ and $\beta(r) \geq D$ and let $q$ be the smallest integer such that $q/f(q) > r$. Note that $r$ and $q$ exists since $\lim_{r \to \infty} \beta(r) = \infty$ and $\lim_{q \to \infty} q/f(q) = \infty$. Given any instance $I$ of $\Pi$. From the definition of $f$-FPT gap approximation algorithms (\Cref{def:FPT gap approx}) and from the fact that $q/f(q) > r$, ${\mathbb A}$ on the input $(I, q)$ can distinguish between ${\sf OPT}\xspace_\Pi(I) \geq q$ and ${\sf OPT}\xspace_\Pi(I) < r$ in $t(q) \cdot \|I\|^{D} \leq t(q) \cdot \|I\|^{\beta(r)} = O_{q, r}(\|I\|^{\beta(r)})$ time. Hence, $\Pi$ is not weakly inherently enumerative, concluding our proof for maximization problems. For any minimization problem $\Pi$, assume again that $\Pi$ is not totally FPT inapproximable, i.e., $\Pi$ admits an $f$-FPT gap approximation algorithm ${\mathbb A}$ for some computable function $f$. Suppose that the running time of ${\mathbb A}$ on every input $(I, k)$ is $t(k) \cdot \|I\|^D$ for some constant $D$. Let $r_0 > 0$ be any constant and let $\beta: {\mathbb N} \rightarrow \mathbb{R}^+$ be any function such that $\beta = \omega(1)$. Let $r$ be the smallest integer such that $r > r_0$ and $\beta(r) \geq D$ and let $q = \lceil r \cdot f(r) \rceil$. Given any instance $I$ of $\Pi$. From definition of $f$-FPT gap approximation algorithms and from $q \geq r \cdot f(r)$, ${\mathbb A}$ on the input $(I, r)$ can distinguish between ${\sf OPT}\xspace_\Pi(I) \geq q$ and ${\sf OPT}\xspace_\Pi(I) < r$ in $t(r) \cdot \|I\|^{D} \leq t(r) \cdot \|I\|^{\beta(r)} = O_{q, r}(\|I\|^{\beta(r)})$ time. Hence, $\Pi$ is not weakly inherently enumerative. \end{proof} \iffalse We will use ${\mathcal A}$ to derive a contradiction: We will show that there is a threshold $r_0$ for which, for any $r > r_0$, there is another threshold $q_0(r)$ (depending on $r$) for which, for any parameter $q > q_0(r)$, algorithm ${\mathcal A}$ decides the two cases in time $\|I\|^{\delta \beta(r)/2}$. First, we choose $r_0$ such that $\beta(r_0) > 2D/ \delta$; this is possible since the function $\beta(r) \rightarrow \infty$. Therefore, for any $r > r_0$, we have $\|I\|^{D} < \|I\|^{\delta \beta(r)/2}$. Next, we choose $q_0$ such that $q_0/f(q_0) > r$ (this is again possible since the function $q/f(q) \rightarrow \infty$.) Therefore, if we run $f(q)$ approximation algorithm on an instance with ${\sf OPT}\xspace_{\Pi}(I) \geq q$, we would get a solution of value $q/f(q) > r$ for all $q > q_0$. This algorithm distinguishes between the two cases. The running time is $t(q) \|I\|^D < \|I\|^{2D} < \|I\|^{\delta \beta(r)}$. This contradicts the fact that the problem $\Pi$ is weakly inherently enumerative. The proof for the minimization problem is similar and therefore omitted. Let $q \in {\mathbb N}$ be any sufficiently large such that $2D < \delta q$. We first choose sufficiently large $q_0$ such that $q_0/f(q_0) > r$ (this is possible since the function $q/f(q) \rightarrow \infty$, and the fact that $q$ is infinitely often.) Now given an instance $I$ of $\Pi$ with parameter $(q,r)$ in Case (i) where $q \geq q_0$, running algorithm ${\mathcal A}(I)$ will give us a solution of value at least $q/f(q) > r$ in time $t(q) \|I\|^D < \|I\|^{2D} < \|I\|^{\delta \beta(r)}$ for sufficiently large input instance $I$. This algorithm successfully distinguish Case (i) and Case (ii) whenever $q > 2D /\delta$ and $k \geq k_0(q)$. Now, consider a minimization problem $\Pi$ and assume for contradiction that ${\mathcal A}$ is $f(k)$-approximation algorithm in time $t(k) n^D$ for some computable increasing function $f$. Let $q \in {\mathbb N}$ be any sufficiently large such that $2D < \delta q$. Let $k_0$ be a sufficiently large value such that $f(q) q < k_0$. Now given an instance $I$ of $\Pi$ in Case (ii), running algorithm ${\mathcal A}(I)$ gives us a solution of value at most $q f(q) <k$ in time $t(q) \|I\|^D < \|I\|^{2D} < \|I\|^{\delta q}$ for sufficiently large input $I$. This algorithm successfully distinguishes Case (i) and Case (ii) for such choices of $k,q$. \fi An important tool in almost any branch of complexity theory, including parameterized complexity, is a notion of reductions. For the purpose of facilitating proofs of totally FPT inapproximability, we define the following reduction, which we call \emph{FPT gap reductions}. \begin{definition}[FPT gap reduction]\label{def:FPT gap reduction} For any functions $f, g = \omega(1)$, a problem $\Pi_0$ is said to be $(f, g)$-\emph{FPT gap reducible} to a problem $\Pi_1$ if there exists an algorithm ${\mathbb A}$ which takes in an instance $I_0$ of $\Pi_0$ and integers $q, r$ and produce an instance $I_1$ of $\Pi_1$ such that the following conditions hold. \begin{itemize} \item ${\mathbb A}$ runs in time $t(q, r) \cdot \|I_0\|^{O(1)}$ for some computable function $t: {\mathbb N} \times {\mathbb N} \to {\mathbb N}$. \item For every positive integer $q$, if ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq q$, then ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq f(q)$. \item For every positive integer $r$, if ${\sf OPT}\xspace_{\Pi_0}(I_0) < g(r)$, then ${\sf OPT}\xspace_{\Pi_1}(I_1) < r$. \end{itemize} \end{definition} It is not hard to see that FPT gap reduction indeed preserves totally FPT inapproximability, as formalized in Proposition~\ref{prop:gapred-totallyinapprox} below. The proof of the proposition can be found in Appendix~\ref{app:gapreduction}. \begin{proposition} \label{prop:gapred-totallyinapprox} If a problem $\Pi_0$ is (i) $(f, g)$-FPT gap reducible to $\Pi_1$ for some computable non-decreasing functions $f, g = \omega(1)$, and (ii) totally FPT inapproximable, then $\Pi_1$ is also totally FPT inapproximable. \end{proposition} As stated earlier, we mainly work with inherently enumerative concepts instead of working directly with totally FPT inapproximability; indeed, we will never use the above proposition and we alternatively use FPT gap reductions to prove that problems are weakly inherently enumeratives. For this purpose, we will need the following proposition. \begin{proposition} \label{prop:gapred-enum} If a problem $\Pi_0$ is (i) $(f, g)$-FPT gap reducible to $\Pi_1$ and (ii) $\beta$-weakly inherently enumerative for some $f, g, \beta = \omega(1)$, then $\Pi_1$ is $\Omega(\beta \circ g)$-weakly inherently enumerative. \end{proposition} \begin{proof} We assume that (i) holds, and will show that if the ``then'' part does not hold, then (ii) also does not hold. Recall from \Cref{def:FPT gap reduction} that (i) implies that there exists $C, D > 0$ such that the reduction from $\Pi_0$ (with parameters $q$ and $r$) to $\Pi_1$ takes $O_{q, r}(\|I_0\|^C)$ time and always output an instance $I_1$ of size at most $O_{q, r}(\|I_0\|^D)$ on every input instance $I_0$. Now assume that the ``then'' part does {\em not} hold, in particular $\Pi_1$ is {\em not} $(\beta \circ g)/D$-weakly inherently enumerative. We will show the following claim which says that (ii) does not hold (by \Cref{def:weakine}). \begin{claim} For every $r_0 > 0$, there exists $q \geq r \geq r_0$ and an $O_{q, r}(\|I_0\|^{\beta(r)})$-time algorithm ${\mathbb B}$ that can, on every input instance $I_0$ of $\Pi_0$, distinguish between ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq q$ and ${\sf OPT}\xspace_{\Pi_0}(I_0)<r.$ \end{claim} We now prove the claim. Consider any $r_0$. Since $\beta, g =\omega(1)$, there exists $r'_0$ such that $g(r') \geq r_0$ and $\beta(r')\geq C$, for all $r' \geq r'_0$. From the assumption that $\Pi_1$ is not $(\beta \circ g)/D$-weakly inherently enumerative, there exist $q' \geq r' \geq r'_0$ such that there is an $O_{q', r'}(\|I_1\|^{\beta(g(r'))/D})$-time algorithm ${\mathbb A}$ that can, on every input instance $I_1$ of $\Pi_1$, distinguish between ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq q'$ and ${\sf OPT}\xspace_{\Pi_1}(I_1) < r'$. Let $r = g(r')$, and let $q$ be the smallest integer such that $f(q) \geq q'$ and $q \geq r$; Note that $q$ exists since $\lim_{q \to \infty}f(q) = \infty$, and that $r\geq r_0$. We use ${\mathbb A}$ and the reduction to build an algorithm ${\mathbb B}$ as follows. On input $I_0$, algorithm ${\mathbb B}$ runs the reduction on $I_0$ and the previously defined $q, r$. Let us call the output of the reduction $I_1$. ${\mathbb B}$ then runs ${\mathbb A}$ on input $(I_1, q', r')$ and outputs accordingly; i.e. if ${\mathbb A}$ says that ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq q'$, then ${\mathbb B}$ outputs ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq q$, and, otherwise, if ${\mathbb A}$ says that ${\sf OPT}\xspace_{\Pi_1}(I_1) < r'$, then ${\mathbb B}$ outputs ${\sf OPT}\xspace_{\Pi_0}(I_0) < r$. Now we show that ${\mathbb B}$ can distinguish whether ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq q$ or ${\sf OPT}\xspace_{\Pi_1}(I_1)<r$ as desired by the claim: From our choice of $q$, if ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq q$, then ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq f(q) \geq q'$. Similarly, from our choice of $r = g(r')$, if ${\sf OPT}\xspace_{\Pi_0}(I_0) < r$, then ${\sf OPT}\xspace_{\Pi_1}(I_1) < r'$. Since ${\mathbb A}$ can distinguish between the two cases, ${\mathbb B}$ can distinguish between the two cases as well. The total running time of ${\mathbb B}$ is $O_{q, r}(\|I_0\|^C) + O_{q', r'}(\|I_1\|^{\beta(g(r'))/D})$ (the first term is for running the reduction). Since $I_1$ of size at most $O_{q, r}(\|I_0\|^D)$, $\beta(r)\geq C$, and $q'$ and $r'$ depend only on $q$ and $r$, the running time can be bounded by $O_{q, r}(\|I_0\|^{\beta(r)})$ as desired. \end{proof} \section{Covering Problems on Label Cover Instances}\label{sec:label cover} In this section, we give intermediate results for the lower bounds on the running time of approximating variants of the {\em label cover} problem, which will be the source of our inapproximability results for \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace. \subsection{Problems and Results} \paragraph{Label cover instance:} \danupon{We should emphasize that this has no projection property and we don't assume that the graph is regular.} A label cover instance $\Gamma$ consists of $(G, \Sigma_U, \Sigma_V, \Pi)$, where \begin{itemize} \item $G = (U, V, E)$ is a bipartite graph between vertex sets $U$ and $V$ and an edge set $E$, \item $\Sigma_U$ and $\Sigma_V$ are sets of {\em alphabets} to be assigned to vertices in $U$ and $V$, respectively, and \item $\Pi=\{\Pi_e\}_{e\in E}$ is a set of {\em constraints} $\Pi_e\subseteq \Sigma_U\times \Sigma_V$. \danupon{Use ``Relations'' instead of ``Constraints''?} \end{itemize} We say that $\Pi$ (or $\Gamma$) has the {\em projection property} if for every edge $uv\in E$ (where $u\in U$ and $V\in v$) and every $\alpha\in \Sigma_U$, there is exactly one $\beta\in \Sigma_V$ such that $(\alpha, \beta)\in \Pi_{uv}$. We will define two combinatorial optimization problems on an instance of the label cover problem. These two problems are defined on the same instance as the standard label cover problem. We will briefly discuss how our problems differ from the standard one. \paragraph{Max-Cover Problem:} A {\em labeling} of the graph, is a pair of mappings $\sigma_U: U\rightarrow \Sigma_U$ and $\sigma_V: V\rightarrow \Sigma_V$. We say that a labeling $(\sigma_U, \sigma_V)$ {\em covers} edge $uv$ if $(\sigma_U(u), \sigma_V(v))\in \Pi_{uv}$. We say that a labeling covers a vertex $u$ if it covers every edge incident to $u$. For any label cover instance $\Gamma$, let ${\sf MaxCov}\xspace(\Gamma)$ denote the maximum number of vertices in $U$ that can be covered by a labeling; i.e. \begin{align} {\sf MaxCov}\xspace(\Gamma) &:= \max_{\sigma_U: U\rightarrow \Sigma_U,\ \sigma_V: V\rightarrow \Sigma_V} \|\{ u\in U \mid \mbox{$(\sigma_U, \sigma_V)$ covers $u$}\}\|. \end{align} The goal of the Max-Cover problem is to compute ${\sf MaxCov}\xspace(\Gamma)$. We remark that the standard label cover problem (e.g. \cite{WilliamsonShmoy-Book}) would try to maximize the number of covered {\em edges}, as opposed to our Max-Cover problem, which seeks to maximize the number of covered {\em vertices}. \paragraph{Min-Label Problem:} A {\em multi-labeling} of the graph, is a pair of mappings $\sigma_U: U\rightarrow {\Sigma}_U$ and $\hat\sigma_V: V\rightarrow 2^{\Sigma_V}$. We say that $(\sigma_U, \hat{\sigma}_V)$ {\em covers} an edge $uv$, if there exists $\beta\in \hat{\sigma}_V(v)$ such that $(\sigma(u), \beta)\in \Pi_{uv}$. For any label cover instance $\Gamma$, let ${\sf MinLab}\xspace(\Gamma)$ denote the minimum number of labels needed to assign to vertices in $V$ in order to cover {\em all} vertices in $U$; i.e. \begin{align} {\sf MinLab}\xspace(\Gamma) &:= \min_{(\sigma_U, \hat\sigma_V)} \sum_{v\in V} \|\hat{\sigma}_V(v)\|\, \end{align} where the minimization is over multi-labelings $(\sigma_U, \hat{\sigma}_V)$ that covers every edge in $G$. We emphasize that we can assign multiple labels to nodes in $V$ while each node in $U$ must be assigned a unique label. Note that {\sf MinLab}\xspace is different from the problem known in the literature as {\sf MinRep}\xspace (e.g. \cite{CharikarHK11}); in particular, in {\sf MinRep}\xspace we can assign multiple labels to all nodes. \paragraph{Results.} First, note that checking whether ${\sf MaxCov}\xspace(\Gamma)< r$ or not, for any $r\geq 1$, can be done by the following algorithms. \begin{enumerate} \item It can be done\footnote{Recall that we use $O^\star(\cdot)$ to hide factors polynomial in the input size.} in $O^\star({\|U\| \choose r} (\|\Sigma_U\|)^r) = O^\star((\|U\|\cdot\|\Sigma_U\|)^r)$ time: First, enumerate all ${\|U\| \choose r}$ possible subsets $U'$ of $U$ and all $\|\Sigma_U\|^{\|U'\|}$ possible labelings on vertices in $U'$. Once we fix the labeling on $U'$, we only need polynomial time to check whether we can label other vertices so that all vertices in $U'$ are covered. \item It can be done in $O^\star(\|\Sigma_V\|^{\|V\|})$ time: Enumerate all $O^\star(\|\Sigma_V\|^{\|V\|})$ possible labelings $\sigma_V$ on $V$. After $\sigma_V$ is fixed, we can find labeling $\sigma_U$ on $U$ that maximizes the number of vertices covered in $U$ in polynomial time. \end{enumerate} ETH can be restated as that these algorithms are the best possible when $\|U\|=\Theta(\|V\|)$, $\|\Sigma_U\|, \|\Sigma_V\|=O(1)$ and $\Pi$ has the projection property. Gap-ETH asserts further that this is the case even to distinguish between ${\sf MaxCov}\xspace(\Gamma)=\|U\|$ and ${\sf MaxCov}\xspace(\Gamma)\leq (1-\varepsilon)\|U\|$. \begin{theorem}\label{thm:Gap-ETH restated} Gap-ETH (\Cref{conj:gap-ETH}) is equivalent to the following statement. There exist constants $\varepsilon,\delta > 0$ such that no algorithm can take a label cover instance $\Gamma$ and can distinguish between the following cases in $O(2^{\delta \|U\|})$ time: \begin{itemize} \item ${\sf MaxCov}\xspace(\Gamma)=\|U\|$, and \item ${\sf MaxCov}\xspace(\Gamma)<(1-\varepsilon)\|U\|$. \end{itemize} This holds even when $\|\Sigma_U\|, \|\Sigma_V\|=O(1)$, $\|U\|=\Theta(\|V\|)$ and $\Pi$ has the projection property.\danupon{Should emphasize that $\|U\|$ can be arbitrarily large.} \end{theorem} The proof of \Cref{thm:Gap-ETH restated} is standard. To avoid distracting the readers, we provide the sketch of the proof in \Cref{sec:restate-gap-eth}. We will show that \Cref{thm:Gap-ETH restated} can be extended to several cases, which will be useful later. First, consider when the first ($O^\star((\|U\|\cdot\|\Sigma_U\|)^{r})$-time) algorithm is faster than the second. We show that in this case this algorithm is essentially the best even for $r=O(1)$, and this holds even when we know that ${\sf MaxCov}\xspace(\Gamma)=\|U\|$. For convenience, in the statements of \Cref{thm:small r,thm:small V,thm:MinLab} below, we will use the notation $\|\Gamma\|$ to denote the size of the label cover instance; in particular, $\|\Gamma\| = \|\Sigma_U\| \|\Sigma_V\| \|U\| \|V\|$. Furthermore, recall that the notation $O_{k, r}(\cdot)$ denotes any multiplicative factor that depends only on $k$ and $r$. \begin{theorem}[{\sf MaxCov}\xspace with Small $\|U\|$] \label{thm:small r} Assuming Gap-ETH, there exist constants $\delta, \rho > 0$ such that, for any positive integers $k \geq r \geq \rho$, no algorithm can take a label cover instance $\Gamma$ with $\|U\| = k$ and distinguish between the following cases in $O_{k, r}(\|\Gamma\|^{\delta r})$ time: \begin{itemize} \item ${\sf MaxCov}\xspace(\Gamma) = k$ and \item ${\sf MaxCov}\xspace(\Gamma)< r$.\danupon{Note ``$<$''.} \end{itemize} This holds even when $\|\Sigma_V\|=O(1)$ and $\Pi$ has the projection property. \end{theorem} \danupon{NEXT TIME: Say that if we assume ETH then we get similar thing except that $r\geq ...$} We emphasize that it is important for applications in later sections that $r=O(1)$. In fact, the main challenge in proving the theorem above is to prove it true for $r$ that is arbitrarily small compared to $\|U\|$. Secondly, consider when the second ($O^\star(\|\Sigma_V\|^{\|V\|})$-time) algorithm is faster; in particular when $\|V\|\ll \|U\|$. In this case, we cannot make the soundness (i.e. parameter $r$ in \Cref{thm:small r}) to be arbitrarily small. (Roughly speaking, the first algorithm can become faster otherwise.) Instead, we will show that the second algorithm is essentially the best possible for soundness as small as $\gamma \|U\|$, for any constant $\gamma >0$. More importantly, this holds for $\|V\|=O(1)$ (thus independent from the input size). This is the key property of this theorem that we need later. \begin{theorem}[{\sf MaxCov}\xspace with Small $\|V\|$] \label{thm:small V} Assuming Gap-ETH, there exist constants $\delta, \rho > 0$ such that, for any positive integer $q \geq \rho$ and any $1 \geq \gamma > 0$, no algorithm can take a label cover instance $\Gamma$ with $\|V\|=q$ and distinguish between the following cases in $O_{q, \gamma}(\|\Gamma\|^{\delta q})$ time: \begin{itemize} \item ${\sf MaxCov}\xspace(\Gamma)= \|U\|$ and \item ${\sf MaxCov}\xspace(\Gamma) < \gamma \|U\|$. \end{itemize} This holds even when $\|\Sigma_U\|\leq (1/\gamma)^{O(1)}$. \end{theorem} We remark that the above label cover instance does not have the projection property. In our final result, we turn to computing ${\sf MinLab}\xspace(\Gamma)$. Since ${\sf MaxCov}\xspace(\Gamma)=\|U\|$ if and only if ${\sf MinLab}\xspace(\Gamma)=\|V\|$, a statement similar to \Cref{thm:Gap-ETH restated} intuitively holds for distinguishing between ${\sf MinLab}\xspace(\Gamma)\leq \|V\|$ and ${\sf MinLab}\xspace(\Gamma) > (1+\varepsilon) \|V\|$; i.e. we need $O^\star(\|\Sigma_V\|^{\|V\|})$ time. In the following theorem, we show that this gap can be substantially amplified, while maintaining the property that $\|V\|=O(1)$ (thus independent from the input size). \begin{theorem}[{\sf MinLab}\xspace Hardness] \label{thm:MinLab} Assuming Gap-ETH, there exist constants $\delta, \rho > 0$ such that, for any positive integers $r \geq q \geq \rho$, no algorithm can take a label cover instance $\Gamma$ with $\|V\|=q$, and distinguish between the following cases in $O_{q, r}(\|\Gamma\|^{\delta q})$ time: \begin{itemize} \item ${\sf MinLab}\xspace(\Gamma)= q$ and \item ${\sf MinLab}\xspace(\Gamma) >r$. \end{itemize} This holds even when $\|\Sigma_U\|= (r/q)^{O(q)}$. \end{theorem} The rest of this section is devoted to proving \Cref{thm:small r,thm:small V,thm:MinLab}. \subsection{Proof of \Cref{thm:small r}} \label{subsec:smallr} The proof proceeds by {\bf compressing the left vertex set $U$} of a label cover instance from \Cref{thm:Gap-ETH restated}. More specifically, each new left vertex will be a subset of left vertices in the original instance. In the construction below, these subsets will just be random subsets of the original vertex set of a certain size; however, the only property of random subsets we will need is that they form a \emph{disperser}. To clarify our proof, let us start by stating the definition of dispersers here. Note that, even though dispersers are often described in graph or distribution terminologies in literatures (e.g.~\cite{Vadhan-book}), it is more convenient for us to describe it in terms of subsets. \begin{definition} For any positive integers $m, k, \ell, r \in {\mathbb N}$ and any constant $\varepsilon \in (0, 1)$, an \emph{$(m, k, \ell, r, \varepsilon)$-disperser} is a collection $\mathcal{I}$ of $k$ subsets $I_1, \dots, I_k \subseteq [m]$ each of size $\ell$ such that the union of any $r$ different subsets from the collection has size at least $(1 - \varepsilon)m$. In other words, for any $1 \leq i_1 < \cdots < i_r \leq k$, we have $\|I_{i_1} \cup \cdots \cup I_{i_r}\| \geq (1 - \varepsilon)m$. \end{definition} The idea of using dispersers to amplify gap in hardness of approximation bears a strong resemblance to the classical randomized graph product technique~\cite{BermanS92}. Indeed, similar approaches have been used before, both implicitly (e.g.~\cite{BellareGS98}) and explicitly (e.g.~\cite{Zuck96,Zuck96-unapprox,Zuckerman07}). In fact, even the reduction we use below has been studied before by Zuckerman~\cite{Zuck96,Zuck96-unapprox}! What differentiates our proof from previous works is the setting of parameters. Since the reduction size (specifically, the left alphabet size $\|\Sigma_U\|$) blows up exponentially in $\ell$ and previous results aim to prove NP-hardness of approximating \mbox{\sf Clique}\xspace, $\ell$ are chosen to be small (i.e. $O(\log m)$). On the other hand, we will choose our $\ell$ to be $\Theta_{\varepsilon}(m/r)$ since we would like to only prove a running time lower bound of the form $\|\Sigma_U\|^{\Omega(r)}$. Interestingly, dispersers for our regime of parameters are easier to construct deterministically and we will sketch the construction in Subsection~\ref{subsec:derandomization}. Note that this construction immediately implies derandomization of our reduction. The exact dependency of parameters can be found in the claim below, which also states that random subsets will be a disperser for such choice of parameters with high probability. Here and throughout the proof, $k$ and $r$ should be thought of as constants where $k \gg r$; these are the same $k, r$ as the ones in the statement of \Cref{thm:small r}. \begin{claim} \label{claim:random-disperser} For any positive integers $m, k, r \in {\mathbb N}$ and any constant $\varepsilon \in (0, 1)$, let $\ell = \max\{m, \lceil 3m/(\varepsilon r)\rceil\}$ and let $I_1, \dots, I_k$ be $\ell$-element subsets of $[m]$ drawn uniformly independently at random. If $\ln k \leq m/r$, then $\mathcal{I} = \{I_1, \dots, I_k\}$ is an $(m, k, \ell, r, \varepsilon)$-disperser with probability at least $1 - e^{-m}$. \end{claim} \begin{proof} When $\ell = m$, the statement is obviously true; thus, we assume w.l.o.g. that $\ell = \lceil 3m/(\varepsilon r)\rceil$. Consider any indices $i_1, \dots, i_r$ such that $1 \leq i_1 < \cdots < i_r \leq k$. We will first compute the probability that $\|I_{i_1} \cup \cdots \cup I_{i_r}\| < (1 - \varepsilon)m$ and then take the union bound over all such $(i_1, \dots, i_r)$'s. Observe that $\|I_{i_1} \cup \cdots \cup I_{i_r}\| < (1 - \varepsilon)m$ if and only if there exists a set $S \subseteq [m]$ of size less than $(1 - \varepsilon)m$ such that $I_{i_1}, \dots, I_{i_r} \subseteq S$. For a fixed set $S \subseteq [m]$ of size less than $(1 - \varepsilon)m$, since $I_{i_1}, \dots, I_{i_r}$ are independently drawn random $\ell$-element subsets of $[m]$, we have \begin{align} \label{eq:} \Pr[I_{i_1}, \dots, I_{i_r} \subseteq S] = \prod_{j \in [r]} \Pr[I_j \subseteq S] = \left(\frac{\binom{\|S\|}{\ell}}{\binom{m}{\ell}}\right)^r \leq \left(\frac{\|S\|}{m}\right)^{\ell r} < (1 - \varepsilon)^{\ell r} \leq e^{-\varepsilon \ell r} < e^{-3m}. \end{align} Taking the union bound over all such $S$'s, we have \begin{align} \Pr[\|I_{i_1} \cup \dots \cup I_{i_r}\| < (1 - \varepsilon) m] < \sum_{S \subseteq [m], \|S\| < (1 - \varepsilon) m} e^{-3m} < 2^m \cdot e^{-3m} < e^{-2m}. \end{align} Finally, taking the union bound over all $(i_1, \dots, i_r)$'s gives us the desired probabilistic bound: \begin{align} \Pr[\mathcal{I} \text{ is not an } (m, k, \ell, r, \varepsilon)\text{-disperser}] \leq \sum_{1 \leq i_1 < \cdot < i_r \leq k} e^{-2m} \leq k^r \cdot e^{-2m} < e^{-m}, \end{align} where the last inequality comes from our assumption that $\ln k \leq m / r$. \end{proof} With the definition of dispersers and the above claim ready, we move on to prove \Cref{thm:small r}. \begin{proof}[Proof of \Cref{thm:small r}] First, we take a label cover instance $\widetilde{\Gamma}=(\widetilde{G}=(\widetilde{U},\widetilde{V},\widetilde{E}),\Sigma_{\widetilde{U}},\Sigma_{\widetilde{V}},\widetilde{\Pi})$ as in \Cref{thm:Gap-ETH restated}. We may assume that $\|\Sigma_{\widetilde{U}}\|,\|\Sigma_{\widetilde{V}}\|=O(1)$, and $\|\widetilde{U}\|=\Theta(\|\widetilde{V}\|)$. Moreover, let $m = \|\widetilde{U}\|$ and $n = \|\widetilde{V}\|$; for convenience, we rename the vertices in $\widetilde{U}$ and $\widetilde{V}$ so that $\widetilde{U} = [m]$ and $\widetilde{V} = [n]$. Note that it might be useful for the readers to think of $\widetilde{\Gamma}$ as a $3$-\mbox{\sf SAT}\xspace instance where $\widetilde{U}$ is the set of clauses and $\widetilde{V}$ is the set of variables. We recall the parameter $\varepsilon$ from \Cref{thm:Gap-ETH restated} and the parameters $k, r$ from the statement of \Cref{thm:small r}. We introduce a new parameter $\ell = 3 m / (\varepsilon r)$ and assume w.l.o.g. that $\ell$ is an integer. The new label cover ({\sf MaxCov}\xspace) instance $\Gamma=(G=(U,V,E),\Sigma_U,\Sigma_V,\Pi)$ is defined as follows. \begin{itemize} \setlength\itemsep{0em} \item The right vertices and right alphabet set remain unchanged, i.e., $V = \widetilde{V}$ and $\Sigma_V = \Sigma_{\widetilde{V}}$. \item There will be $k$ vertices in $U$ where each vertex is a random set of $\ell$ vertices of $\widetilde{U}$. More specifically, we define $U = \{I_1,\ldots, I_k\}$ where each $I_i$ is a random $\ell$-element subsets of $[m]$ drawn independently of each other. \item The left alphabet set $\Sigma_U$ is $\Sigma_{\widetilde{U}}^\ell$. For each $I \in U$, we view each label $\alpha \in \Sigma_U$ as a tuple $(\alpha_u)_{u \in I} \in (\Sigma_{\widetilde{U}})^I$; this is a partial assignment to all vertices $u \in I$ in the original instance $\widetilde{\Gamma}$. \item We create an edge between $I \in U$ and $v \in V$ in $E$ if and only if there exists $u \in I$ such that $uv \in \widetilde{E}$. More formally, $E = \{Iv: I \cap N_{\widetilde{G}}(v) \ne \emptyset\}$. \item Finally, we define the constraint $\Pi_{Iv}$ for each $Iv \in E$. As stated above, we view each $\alpha \in \Sigma_U$ as a partial assignment $(\alpha_u)_{u \in I}$ for $I \subseteq \widetilde{U}$. The constraint $\Pi_{Iv}$ then contains all $(\alpha, \beta)$ such that $(\alpha_u, \beta)$ satisfies the constraint $\widetilde{\Pi}_{uv}$ for every $u \in I$ that has an edge to $v$ in $\widetilde{\Gamma}$. More precisely, $\Pi_{Iv} = \{(\alpha, \beta) = ((\alpha_u)_{u \in I}, \beta): \forall u \in I \cap N_{\widetilde{G}}(v), (\alpha_u, \beta) \in \widetilde{\Pi}_{uv}\}$. \end{itemize} Readers who prefer the $3$-\mbox{\sf SAT}\xspace/CSP viewpoint of label cover may think of each $I_i$ as a collection of clauses in the $3$-\mbox{\sf SAT}\xspace instance that are joined by an operator {\bf AND}, i.e., the assignment must satisfy all the clauses in $I_i$ simultaneously in order to satisfy $I_i$. We remark that, if $\widetilde{\Pi}$ has the projection property, $\Pi$ also has projection property. \noindent{\bf Completeness.} Suppose there is a labeling $(\sigma_{\widetilde{U}},\sigma_{\widetilde{V}})$ of $\widetilde{\Gamma}$ that covers all $\|\widetilde{U}\|$ left-vertices. We take $\sigma_V=\sigma_{\widetilde{V}}$ and construct $\sigma_U$ by setting $\sigma_{U}(I)=(\sigma_{\widetilde{U}}(u))_{u \in I}$ for each $I \in U$. Since $(\sigma_{\widetilde{U}},\sigma_{\widetilde{V}})$ covers all the vertices of $\widetilde{U}$, $(\sigma_U,\sigma_V)$ also covers all the vertices of $U$. Therefore, ${\sf MaxCov}\xspace(\Gamma)=\|U\|$. \noindent{\bf Soundness.} To analyze the soundness of the reduction, first recall Claim~\ref{claim:random-disperser}: $\{I_1, \dots, I_k\}$ is an $(m, k, \ell, r, \varepsilon)$-disperser with high probability. Conditioned on this event happening, we will prove the soundness property, i.e., that if ${\sf MaxCov}\xspace(\widetilde{\Gamma}) < (1 - \varepsilon)\|\widetilde{U}\|$, then ${\sf MaxCov}\xspace(\Gamma) < r$. We will prove this by contrapositive; assume that there is a labeling $(\sigma_U, \sigma_V)$ that covers at least $r$ left vertices $I_{i_1}, \cdots, I_{i_r} \in U$. We construct a labeling $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$ as follows. First, $\sigma_{\widetilde{V}}$ is simply set to $\sigma_V$. Moreover, for each $u \in I_{i_1} \cup \cdots \cup I_{i_r}$, let $\sigma_{\widetilde{U}}(u) = (\sigma_U(I_{i_j}))_u$ where $j \in [r]$ is an index such that $u \in I_{i_j}$; if there are multiple such $j$'s, just pick an arbitrary one. Finally, for $u \in U \setminus (I_{i_1} \cup \cdots \cup I_{i_r})$, we set $\sigma_{\widetilde{U}}(u)$ arbitrarily. We claim that, every $u \in I_{i_1} \cup \cdots \cup I_{i_r}$ is covered by $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$ in the original instance $\widetilde{\Gamma}$. To see that this is the case, recall that $\sigma_{\widetilde{U}}(u) = (\sigma_U(I_{i_j}))_u$ for some $j \in [r]$ such that $u \in I_{i_j}$. For every $v \in V$, if $uv \in E$, then, from how the constraint $\Pi_{I_{i_j}v}$ is defined, we have $(\sigma_{\widetilde{U}}(u), \sigma_{\widetilde{V}}(v)) = (\sigma_U(I_{i_j})_u, \sigma_V(v)) \in \widetilde{\Pi}_{uv}$. In other words, $u$ is indeed covered by $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$. Hence, $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$ covers at least $\|I_{i_1} \cup \cdots \cup I_{i_r}\| \geq (1 - \varepsilon)m$, where the inequality comes from the definition of dispersers. As a result, ${\sf MaxCov}\xspace(\widetilde{\Gamma}) \geq (1 - \varepsilon)\|\widetilde{U}\|$, completing the soundness proof. \noindent{\bf Running Time Lower Bound.} Our construction gives a {\sf MaxCov}\xspace instance $\Gamma$ with $\|U\|=k$ and $\|\Sigma_U\|=\|\Sigma_{\widetilde{U}}\|^\ell=2^{\Theta(m/(\varepsilon r))}$, whereas $\|V\|$ and $\|\Sigma_V\|$ remain $n$ and $O(1)$ respectively. Assume that Gap-ETH holds and let $\delta_0$ be the constant in the running time lower bound in \Cref{thm:Gap-ETH restated}. Let $\delta$ be any constant such that $0 < \delta < \delta_0 \varepsilon / c$ where $c$ is the constant such that $\|\Sigma_U\| \leq 2^{cm/(\varepsilon r)}$. Suppose for the sake of contradiction that, for some $k \geq r \geq \rho$, there is an algorithm that distinguishes whether ${\sf MaxCov}\xspace(\Gamma) = k$ or ${\sf MaxCov}\xspace(\Gamma) < r$ in $O_{k, r}(\|\Gamma\|^{\delta r})$ time. Observe that, in our reduction, $\|U\|, \|V\|, \|\Sigma_V\| = \|\Sigma_U\|^{o(1)}$. Hence, the running time of the algorithm on input $\Gamma$ is at most $O_{k, r}(\|\Sigma_U\|^{\delta r(1 + o(1))}) \leq O_{k, r}(\|\Sigma_U\|^{\delta_0 \varepsilon r / c}) \leq O(2^{\delta_0 m})$ where the first inequality comes from our choice of $\delta$ and the second comes from $\|\Sigma_U\| \leq 2^{cm/(\varepsilon r)}$. Thanks to the completeness and soundness of the reduction, this algorithm can also distinguish whether ${\sf MaxCov}\xspace(\widetilde{\Gamma}) = \|\widetilde{U}\|$ or ${\sf MaxCov}\xspace(\widetilde{\Gamma}) < (1 - \varepsilon)\|\widetilde{U}\|$ in time $O(2^{\delta_0 m})$. From \Cref{thm:Gap-ETH restated}, this is indeed a contradiction. \end{proof} \subsubsection{Derandomization} \label{subsec:derandomization} While the reduction in the proof of \Cref{thm:small r} is a randomized reduction, it can be derandomized quite easily. We sketch the ideas behind the derandomization below. Notice that the only property we need from the random $\ell$-element subsets $I_1, \dots, I_k$ is that it forms an $(m, k, \ell, r, \varepsilon)$-disperser. Hence, to derandomize the reduction, it suffices to deterministically construct such a disperser in $2^{o(n)}$ time. To do so, let us first note that Lemma~\ref{claim:random-disperser} implies that an $(m', k, \ell', r, \varepsilon)$-disperser exists where $m' = r \ln k$ and $\ell' = 3m'/(\varepsilon r)$. For convenience, we assume w.l.o.g. that $m', \ell'$ are integers and that $m'$ divides $m$. Since $m'$ is now small, we can find such a disperser by just enumerating over every possible collection of $k$ subsets of $[m']$ each of size $\ell'$ and checking whether it has the desired property; this takes only $(2^{m'})^k(k)^r\operatorname{poly}(m') = 2^{O(rk \log k)}$ time, which is acceptable for us since $r$ and $k$ are both constants. Let the $(m', k, \ell', r, \varepsilon)$-disperser that we find be $\{I'_1, \dots, I'_k\}$. Finally, to get from here to the intended $(m, k, \ell, r, \varepsilon)$-disperser, we only need to view $[m]$ as $[m/m'] \times [m']$ and let $I_1 = [m/m'] \times I'_1, \dots, I_k = [m/m'] \times I'_k$. It is not hard to check that $\{I_1, \dots, I_k\}$ is indeed an $(m, k, \ell, r, \varepsilon)$-disperser, which concludes our sketch. \subsection{Proof of \Cref{thm:small V}} The proof proceeds by {\bf compressing the right vertex set $V$} of a label cover instance from \Cref{thm:Gap-ETH restated} plus amplifying the hardness gap. The gap amplification step is similar to that in the proof of \Cref{thm:small r} except that, since here ${\sf MaxCov}\xspace(\Gamma)$ is not required to be constant in the soundness case, we can simply take all subsets of appropriate sizes instead of random subsets as in the previous proof; this also means that our reduction is deterministic and requires no derandomization. \begin{proof}[Proof of \Cref{thm:small V}] First, we take a label cover instance $\widetilde{\Gamma} = (\widetilde{G}=(\widetilde{U},\widetilde{V},\widetilde{E}),\Sigma_{\widetilde{U}},\Sigma_{\widetilde{V}},\widetilde{\Pi})$ as in \Cref{thm:Gap-ETH restated}. We may assume that $\|\Sigma_{\widetilde{U}}\|,\|\Sigma_{\widetilde{V}}\|=O(1)$, and $\|\widetilde{U}\|=\Theta(\|\widetilde{V}\|)$. For convenience, we assume w.l.o.g. that $\widetilde{U} = [m]$ and $\widetilde{V} = [n]$. Again, it might be useful for the readers to think of $\widetilde{\Gamma}$ as a $3$-\mbox{\sf SAT}\xspace instance where $\widetilde{U}$ are the set of clauses and $\widetilde{V}$ are the set of variables. Recall the parameter $\varepsilon$ from \Cref{thm:Gap-ETH restated} and the parameters $q, \gamma$ from \Cref{thm:small V}. Let $\ell=\ln(1/\gamma)/\varepsilon$. We assume w.l.o.g. that $\ell$ is an integer and that $n$ is divisible by $q$. The new label cover ({\sf MaxCov}\xspace) instance $\Gamma=(G=(U,V,E),\Sigma_U,\Sigma_V,\Pi)$ is defined as follows. \begin{itemize} \setlength\itemsep{0em} \item First, we partition $\widetilde{V} = [n]$ into $q$ parts $J_1,\ldots,J_q$, each of size $n/q$. We then let $V = \{J_1, \dots, J_q\}$. In other words, we merge $n/q$ vertices of $\widetilde{V}$ into a single vertex in $V$. \item Let $U$ be $\binom{[m]}{\ell}$, the collection of all $\ell$-element subsets of $[m] = \widetilde{U}$. \item The left alphabet set $\Sigma_U$ is $\Sigma_{\widetilde{U}}^\ell$. For each $I \in U$, we view each label $\alpha \in \Sigma_U$ as a tuple $(\alpha_u)_{u \in I} \in (\Sigma_{\widetilde{U}})^I$; this is a partial assignment to all vertices $u \in I$ in the original instance $\widetilde{\Gamma}$. \item Our graph $G$ is simply a complete bipartite graph, i.e., for every $I \in U$ and $J \in V$, $IJ \in E(G)$. \item The label set of $V$ is $\Sigma_V=\Sigma_{\widetilde{V}}^{n/q}$, and the label set of $U$ is $\Sigma_U=\Sigma_{\widetilde{U}}^\ell$. For each $I \in U$, we view each label $\alpha \in \Sigma_U$ as a tuple $(\alpha_u)_{u \in I} \in (\Sigma_{\widetilde{U}})^I$; this is simply a partial assignment to all vertices $u \in I$ in the original instance $\widetilde{\Gamma}$. Similarly, for each $J \in V$, we view each label $\beta \in \Sigma_V$ as $(\beta_v)_{v \in J} \in (\Sigma_{\widetilde{V}})^J$. \item Finally, we define $\Pi_{IJ}$ for each $IJ \in E$. The constraint $\Pi_{IJ}$ contains all $(\alpha, \beta)$ such that $(\alpha_u, \beta_v)$ satisfies the constraint $\widetilde{\Pi}_{uv}$ for every $u \in I, v \in J$ such that $uv \in \widetilde{E}$. More precisely, $\Pi_{IJ} = \{(\alpha, \beta) = ((\alpha_u)_{u \in I}, (\beta_v)_{v \in J}): \forall u \in I, v \in J \text{ such that } uv \in \widetilde{E}, (\alpha_u, \beta_v) \in \widetilde{\Pi}_{uv}\}$. \end{itemize} We remark that $\Pi$ may not have the projection property even when $\widetilde{\Pi}$ has the property. \noindent{\bf Completeness.} Suppose that there is a labeling $(\sigma_{\widetilde{U}},\sigma_{\widetilde{V}})$ of $\widetilde{\Gamma}$ that covers all $\|\widetilde{U}\|$ left-vertices. We construct $(\sigma_U, \sigma_V)$ by setting $\sigma_U(I) = (\sigma_{\widetilde{U}}(u))_{u \in I}$ for each $I \in U$ and $\sigma_V(J) = (\sigma_{\widetilde{V}}(v))_{v \in J}$ for each $J \in V$. It is easy to see that $(\sigma_U,\sigma_V)$ covers all the vertices of $U$. Therefore, ${\sf MaxCov}\xspace(\Gamma)=\|U\|$. \noindent{\bf Soundness.} Suppose that ${\sf MaxCov}\xspace(\widetilde{\Gamma}) < (1-\varepsilon)\|\widetilde{U}\|$. Consider any labeling $(\sigma_U,\sigma_V)$ of $\Gamma$; we will show that $(\sigma_U, \sigma_V)$ covers less than $\gamma\|U\|$ left-vertices. Let $I_1, \dots, I_t \in U$ be the vertices covered by $(\sigma_U, \sigma_V)$. Analogous to the proof of \Cref{thm:small r}, we define a labeling $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$ as follows. First, $\sigma_{\widetilde{V}}$ is naturally defined from $\sigma_V$ by $\sigma_{\widetilde{V}} = \sigma_V(J)_v$ where $J$ is the partition that contains $v$. Moreover, for each $u \in I_{i_1} \cup \cdots \cup I_{i_r}$, let $\sigma_{\widetilde{U}}(u) = (\sigma_U(I_{i_j}))_u$ where $j \in [r]$ is an index such that $u \in I_{i_j}$; for $u \in U \setminus (I_{i_1} \cup \cdots \cup I_{i_r})$, we set $\sigma_{\widetilde{U}}(u)$ arbitrarily. Similar to the proof of \Cref{thm:small r}, it is not hard to see that every vertex in $I_1 \cup \cdots \cup I_t$ is covered by $(\sigma_{\widetilde{U}}, \sigma_{\widetilde{V}})$ in $\widetilde{\Gamma}$. Since ${\sf MaxCov}\xspace(\widetilde{\Gamma}) < (1-\varepsilon)\|\widetilde{U}\|$, we can conclude that $\|I_1 \cup \cdots \cup I_t\| < (1 - \varepsilon)\|\widetilde{U}\|$. Since each $I_i$ is simply an $\ell$-size subset of $I_1 \cup \cdots \cup I_t$, we can conclude that \begin{align} t < \binom{(1 - \varepsilon)\|\widetilde{U}\|}{\ell} \leq (1 - \varepsilon)^{\ell}\binom{\|\widetilde{U}\|}{\ell} = (1 - \varepsilon)^{\ell}\|U\| \leq e^{-\varepsilon \ell} \|U\| = \gamma\|U\|. \end{align} Hence, $(\sigma_U, \sigma_V)$ covers less than $\gamma\|U\|$ left-vertices as desired. \noindent{\bf Running Time Lower Bound.} Our construction gives a {\sf MaxCov}\xspace instance $\Gamma$ with $\|V\| = q$ and $\|\Sigma_V\|=\|\Sigma_{\widetilde{V}}\|^{n/q}=2^{\Theta(n/q)}$; note also that $\|U\| = m^\ell$ and $\|\Sigma_U\| = \|\Sigma_{\widetilde{U}}\|^{\ell} = (1/\gamma)^{O(1)}$. Assume that Gap-ETH holds and let $\delta_0$ be the constant from \Cref{thm:Gap-ETH restated}. Moreover, let $\delta$ be any positive constant such that $\delta < \delta_0 / c$ where $c$ is the constant such that $\|\Sigma_V\| \leq 2^{cm/q}$. Suppose for the sake of contradiction that, for some $q \geq \rho$ and $1 \geq \gamma > 0$, there is an algorithm that distinguishes whether ${\sf MaxCov}\xspace(\Gamma) = \|U\|$ or ${\sf MaxCov}\xspace(\Gamma) < \gamma\|U\|$ in $O_{q, \gamma}(\|\Gamma\|^{\delta q})$ time. Observe that, in our reduction, $\|U\|, \|V\|, \|\Sigma_U\| = \|\Sigma_V\|^{o(1)}$. Hence, the running time of the algorithm on input $\Gamma$ is $O_{q, \gamma}(\|\Sigma_V\|^{\delta q(1 + o(1))}) \leq O_{q, \gamma}(\|\Sigma_V\|^{\delta_0 q / c}) \leq O(2^{\delta_0 m})$ where the first inequality comes from our choice of $\delta$ and the second comes from $\|\Sigma_V\| \leq 2^{cm/q}$. Thanks to the completeness and soundness of the reduction, this algorithm can also distinguish whether ${\sf MaxCov}\xspace(\widetilde{\Gamma}) = \|\widetilde{U}\|$ or ${\sf MaxCov}\xspace(\widetilde{\Gamma}) < (1 - \varepsilon)\|\widetilde{U}\|$ in time $O(2^{\delta_0 m})$. From \Cref{thm:Gap-ETH restated}, this is a contradiction. \end{proof} \subsection{Proof of \Cref{thm:MinLab}} We conclude this section with the proof of \Cref{thm:MinLab}. The proof proceeds simply by showing that, if an algorithm can distinguish between the two cases in the statement of \Cref{thm:MinLab}, it can also distinguish between the two cases in \Cref{thm:small V} (with an appropriate value of $\gamma$). \begin{proof}[Proof of \Cref{thm:MinLab}] Consider the label cover instance $\Gamma = (G = (U, V, E), \Sigma_U, \Sigma_V, \Pi)$ given by~\Cref{thm:small V} when $\gamma = (r/q)^{-q}$. Let us assume w.l.o.g. that there is no isolated vertex in $G$. \noindent{\bf Completeness.} If ${\sf MaxCov}\xspace(\Gamma) = \|U\|$, then there is a labeling $\sigma_U: U \to \Sigma_U$ and $\sigma_V: V \to \Sigma_V$ that covers every edge; this also induces a multi-labeling that covers every edge. Hence, ${\sf MinLab}\xspace(\Gamma) = \|V\|$. \noindent{\bf Soundness.} We will prove by contrapositive; suppose that ${\sf MinLab}\xspace(\Gamma) \leq r$. This implies that there exists a multi-labeling $\sigma_U: U \to \Sigma_U$ and $\sigma_V: V \to 2^{\Sigma_V}$ such that $\sum_{v \in V} \|\sigma_V(v)\| \leq r$ and every vertex is covered. Since there is no isolated vertex in $G$, $\sigma_V(v) \ne \emptyset$ for all $v \in V$. Consider $\sigma_V^{\text{rand}}: V \to \Sigma_V$ sampled randomly by, for each $v \in V$, independently pick a random element of $\sigma_V(v)$ and let $\sigma_V^{\text{rand}}(v)$ be this element. Let us consider the expected number of $u \in U$ that are covered by the labeling $(\sigma_U, \sigma_V^{\text{rand}})$. From linearity of expectation, we can write this as \begin{align} \mathop{\mathbb{E}}_{\sigma_V^{\text{rand}}} \|\{u \in U \mid (\sigma_U, \sigma_V^{\text{rand}}) \text{ covers } u\}\| &= \sum_{u \in U} \Pr_{\sigma_V^{\text{rand}}}\left[(\sigma_U, \sigma_V^{\text{rand}}) \text{ covers } u\right] \\ &= \sum_{u \in U} \prod_{v \in N(u)} \Pr\left[(\sigma_U(u), \sigma_V^{\text{rand}}(v)) \in \Pi_{uv}\right] \\ &\geq \sum_{u \in U} \prod_{v \in N(u)} \|\sigma_V(v)\|^{-1} \\ &\geq \sum_{u \in U} \prod_{v \in V} \|\sigma_V(v)\|^{-1} \\ (\text{From AM-GM inequality}) &\geq \sum_{u \in U} \left(\frac{1}{q} \sum_{v \in V} \|\sigma_V(v)\|\right)^{-q} \\ &\geq \sum_{u \in U} \|U\| (r/q)^{-q} \\ &= \gamma \|U\|. \end{align} where the first inequality comes from the fact that there exists $\beta \in \sigma_V(v)$ such that $(\sigma_U(u), \beta) \in \Pi_{uv}$. This implies that ${\sf MaxCov}\xspace(\Gamma) \geq \gamma\|U\|$, which concludes our proof. \end{proof} \section{Hardness for Combinatorial Problems} \label{sec:hardness-combopt} \subsection{Maximum Clique} Recall that, for any graph $G$, $\mbox{\sf Clique}\xspace(G)$ denotes the maximum size of any clique in $G$. Observe that we can check if there is a clique of size $r$ by checking if any subset of $r$ vertices forms a clique, and there are ${\|V(G)\| \choose r}=O(\|V(G)\|^r)$ possible such subsets. We show that this is essentially the best we can do even when we are given a promise that a clique of size $q \gg r$ exists: \begin{theorem}\label{thm:clique} Assuming Gap-ETH, there exist constants $\delta, r_0 > 0$ such that, for any positive integers $q\geq r\geq r_0$, no algorithm can take a graph $G$ and distinguish between the following cases in $O_{q, r}(\|V(G)\|^{\delta r})$ time: \begin{itemize} \item $\mbox{\sf Clique}\xspace(G)\geq q$ and \item $\mbox{\sf Clique}\xspace(G)< r$. \end{itemize} \end{theorem} The above theorem simply follows from plugging the FGLSS reduction below to \Cref{thm:small r}. \begin{theorem}[\cite{FGLSS96}] \label{thm:fglss} Given a label cover instance $\Gamma=(G = (U, V, E), \Sigma_U, \Sigma_V, \Pi)$ with projection property as in \Cref{sec:label cover}, there is a reduction that produces a graph $H_{\Gamma}$ such that $\|V(H_{\Gamma})\| = \|U\|\|\Sigma_U\|$ and $\mbox{\sf Clique}\xspace(H_{\Gamma}) = {\sf MaxCov}\xspace(\Gamma)$. The reduction takes $O(\|V(H_{\Gamma}))\|^2\|V\|)$ time. \end{theorem} For clarity, we would like to note that, while the original graph defined in~\cite{FGLSS96} is for multi-prover interactive proof, analogous graphs can be constructed for CSPs and label cover instances as well. In particular, in our case, the graph can be defined as follows: \begin{itemize} \item The vertex set $V(H_\Gamma)$ is simply $U \times \Sigma_U$. \item There is an edge between two vertices $(u, \alpha), (u', \alpha') \in V(H_\Gamma)$ if and only if, $\Pi_{uv}(\alpha) = \Pi_{u'v}(\alpha')$ (i.e., recall that we have a projection constraint, so we can represent the constraint $\Pi_{uv}$ as a function $\Pi_{uv}: \Sigma_U \rightarrow \Sigma_V$.) \end{itemize} \begin{proof}[Proof of \Cref{thm:clique}] Assume that Gap-ETH holds and let $\delta, \rho$ be the constants from \Cref{thm:small r}. Let $r_0 = \max\{\rho, 2/\delta\}$. Suppose for the sake of contradiction that, for some $q \geq r \geq r_0$, there is an algorithm ${\mathbb A}$ that distinguishes between $\mbox{\sf Clique}\xspace(G) \geq q$ and $\mbox{\sf Clique}\xspace(G) < r$ in $O_{q, r}(\|V(G)\|^{\delta r})$ time. Given a label cover instance $\Gamma$ with projection property, we can use ${\mathbb A}$ to distinguish whether ${\sf MaxCov}\xspace(\Gamma) \geq q$ or ${\sf MaxCov}\xspace(\Gamma) < r$ as follows. First, we run the FGLSS reduction to produce a graph $H_\Gamma$ and we then use ${\mathbb A}$ to decide whether $\mbox{\sf Clique}\xspace(H_\Gamma) \geq q$ or $\mbox{\sf Clique}\xspace(H_\Gamma) < r$. From $\mbox{\sf Clique}\xspace(H_{\Gamma}) = {\sf MaxCov}\xspace(\Gamma)$, this indeed correctly distinguishes between ${\sf MaxCov}\xspace(\Gamma) \geq q$ and ${\sf MaxCov}\xspace(\Gamma) < r$; moreover, the running time of the algorithm is $O_{q, r}(\|V(H_\Gamma)\|^{\delta r}) + O(\|V(H_{\Gamma}))\|^2\|V\|) \leq O_{q, r}(\|\Gamma\|^{\delta r})$ where the term $O(\|V(H_{\Gamma}))\|^2\|V\|)$ comes from the running time used to produce $H_\Gamma$. From \Cref{thm:small r}, this is a contradiction, which concludes our proof. \end{proof} As a corollary of \Cref{thm:clique}, we immediately arrive at FPT inapproximability of Maximum Independent Set and Maximum Clique. \begin{corollary}[Clique is inherently enumerative] Assuming Gap-ETH, Maximum Clique and Maximum Independent Set are inherently enumerative and thus FPT inapproximable. \end{corollary} \iffalse Another corollary implied by our result is hardness of approximating clique with very low soundness. Remark that the soundness in Khot~et~al.~\cite{KhotP06} is $2^{\log \|V(G)\|)^{3/4+o(1)}}$. Our result implies a weaker hardness of approximation (for polynomial-time algorithms), but the soundness can be made much smaller. \begin{corollary}[Gap-ETH hardness of Clique with very low soundness] Let $f=\omega(1)$ be any computable function. Unless Gap-ETH breaks, there is a constant $\delta>0$ such that, given a graph $H$, no polynomial time algorithm can distinguish between the following: \begin{itemize} \item $\mbox{\sf Clique}\xspace(H) \geq \|V(H)\|^{\delta/f(\|V(H)\|)}$ \item $\mbox{\sf Clique}\xspace(H) < f(\|V(H)\|)$ \end{itemize} \end{corollary} So this means that the hardness result holds even when $f(n)$ is a very slow-growing function such as log-star. We remark that this soundness is as low as one may have hoped for: For constant soundness (e.g., $r$), it is trivial to enumerate all possible cliques in time ${n \choose r}$ (polynomial time for constant $r$). The proof of this corollary is a simple calculation based on the described reductions and therefore is deferred to Appendix~\ref{app:poly-time-hardness}. \fi \subsection{Set Cover, Dominating Set, and Hitting Set} \label{subsec:domset} For convenience, we will be working with the Set Cover problem, which is computationally equivalent to Dominating Set (see Appendix~\ref{app:trivial-eq}). Let ${\mathcal U}$ be a ground set (or a universe). A set system ${\mathcal S}$ over ${\mathcal U}$ is a collection of subsets ${\mathcal S} = \{S_1,\ldots, S_m\}$ where $S_i \subseteq {\mathcal U}$ for all $i \in [m]$. We say that ${\mathcal S}' \subseteq {\mathcal S}$ is a feasible set cover of $({\mathcal U}, {\mathcal S})$ if $\bigcup_{X \in {\mathcal S}'} X = {\mathcal U}$. In the Set Cover problem ($\mbox{\sf SetCov}\xspace$), we are given such a set system $({\mathcal U}, {\mathcal S})$ and we are interested in finding a set cover ${\mathcal S}'$ with minimum cardinality $\|{\mathcal S}'\|$. Let $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S})$ denote the value of the optimal set cover for $({\mathcal U}, {\mathcal S})$. Note that for any set cover instance $({\mathcal U}, {\mathcal S})$, checking whether there is a set cover of size at most $q$ can be done in $O^\star(\|{\mathcal S}\|^q)$ time by enumerating all ${\|{\mathcal S}\| \choose q}$ subsets of ${\mathcal S}$ of size $q$. We show that this is more or less the best we can do: Even when the algorithm is promised the existence of a set cover of size $q$ (for some constant $q$), it cannot find a set cover of size $f(q)$ for any computable function $f$ in time $O_q(\|{\mathcal S}\| \|{\mathcal U}\|)^{\delta q}$ for some constant $\delta > 0$ independent of $q$ and $f$. \subsubsection{Results} Our main technical contribution in this section is summarized in the following theorem: \begin{theorem} \label{thm:setcov-reduction} There is a reduction that on input $\Gamma= (G=(U,V,E),\Sigma_U, \Sigma_V, \Pi)$ of ${\sf MinLab}\xspace$ instance, produces a set cover instance $({\mathcal U},{\mathcal S})$ such that \begin{itemize} \item ${\sf MinLab}\xspace(\Gamma) = \mbox{\sf SetCov}\xspace({\mathcal U},{\mathcal S})$ \item $\|{\mathcal U}\| = \|U\| \|V\|^{\|\Sigma_U\|}$ and $\|{\mathcal S}\| = \|V\| \|\Sigma_V\|$ \item The reductions runs in time $poly(\|{\mathcal U}\|, \|{\mathcal S}\|)$ \end{itemize} \end{theorem} We defer the proof of this theorem to \Cref{subsec:domset}. For now, let us demonstrate that, by combining \Cref{thm:setcov-reduction} and \Cref{thm:small V}, we can derive hardness of approximation of \mbox{\sf SetCov}\xspace: \begin{theorem} \label{thm: set cover hardness} Assuming Gap-ETH, there exist universal constants $\delta, q_0 > 0$ such that, for any positive integers $r \geq q \geq q_0$, no algorithm can take a set cover instance $({\mathcal U}, {\mathcal S})$, and distinguish between the following cases in $O_{q, r}((\|{\mathcal S}\| \|{\mathcal U}\|)^{\delta q})$ time: \begin{itemize} \item $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) \leq q$. \item $\mbox{\sf SetCov}\xspace({\mathcal U},{\mathcal S}) > r$. \end{itemize} \end{theorem} \begin{proof} Assume that Gap-ETH holds and let $\delta, \rho$ be the constants from \Cref{thm:MinLab}. Let $q_0 = \max\{\rho, c/\delta\}$ where $c$ is the constant such that the running time of the reduction in \Cref{thm:setcov-reduction} is $O((\|{\mathcal U}\|\|{\mathcal S}\|)^c)$. Suppose for the sake of contradiction that, for some $r \geq q \geq q_0$, there is an algorithm ${\mathbb A}$ that distinguishes between $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) \leq q$ and $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) > r$ in $O_{q, r}((\|{\mathcal S}\| \|{\mathcal U}\|)^{\delta q})$ time. Given a label cover instance $\Gamma$ where $\|V\|, \|\Sigma_U\| = O_{q, r}(1)$, we can use ${\mathbb A}$ to distinguish whether ${\sf MinLab}\xspace(\Gamma) \leq q$ or ${\sf MinLab}\xspace(\Gamma) > r$ as follows. First, we run the reduction from \Cref{thm:setcov-reduction} to produce a \mbox{\sf SetCov}\xspace instance $({\mathcal U}, {\mathcal S})$ and we then use ${\mathbb A}$ to decide whether $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) \leq q$ or $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) > r$. From $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) = {\sf MinLab}\xspace(\Gamma)$, this indeed correctly distinguishes between ${\sf MinLab}\xspace(\Gamma) \leq q$ and ${\sf MinLab}\xspace(\Gamma) > r$; moreover, the running time of the algorithm is $O_{q, r}((\|{\mathcal U}\|\|{\mathcal S}\|)^{\delta q}) + O((\|{\mathcal U}\|\|{\mathcal S}\|)^c) \leq O_{q, r}(\|\Gamma\|^{\delta q})$ where the term $O((\|{\mathcal U}\|\|{\mathcal S}\|)^c)$ comes from the running time used to produce $({\mathcal U}, {\mathcal S})$. From \Cref{thm:MinLab}, this is a contradiction, which concludes our proof. \end{proof} As a corollary of \Cref{thm: set cover hardness}, we immediately arrive at FPT inapproximability of Set cover, Dominating set and Hitting set. \begin{corollary} Assuming Gap-ETH, Set cover, Dominating set and Hitting set are inherently enumerative and thus FPT inapproximable. \end{corollary} \iffalse Finally, we also derive hardness of approximation (for polynomial-time algorithms) for Set Cover where the completeness is arbitrarily low. This can be contrasted with all known set cover construction in the hardness of approximation literature, in which the completeness is always polynomial in the input size. \begin{corollary}[Gap-ETH hardness of Set Cover with very low completeness] Let $f= \omega(1)$ be any computable function. Unless Gap-ETH breaks, there is a constant $\delta> 0$ such that, given a set cover instance $({\mathcal U},{\mathcal S})$ with encoding length $N$, no polynomial time algorithm can distinguish between the following: \begin{itemize} \item $\mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S}) \leq f(N)$ \item $\mbox{\sf SetCov}\xspace({\mathcal U},{\mathcal S}) > f(N) \cdot (\log N)^{\delta/f(N)}$ \end{itemize} \end{corollary} In particular, we get a nearly logarithmic hardness of approximation when the completeness parameter is small. We remark that the completeness parameter here is as small as it can get since, if it were a constant, we would have solved the Set Cover problem trivially by enumeration. Again, the proof of this corollary follows from simple calculation on the size of instance and is deferred to Appendix~\ref{app:poly-time-hardness}. \fi \subsubsection{Proof of Theorem~\ref{thm:setcov-reduction}} \label{subsec:domset} Our construction is based on a standard hypercube set system, as used by Feige~\cite{Feige98} in proving the hardness of $k$-Maximum Coverage. We explain it here for completeness. \paragraph{Hypercube set system:} Let $z, k \in {\mathbb N}$ be parameters. The hypercube set system $H(z,k)$ is a set system $({\mathcal U}, {\mathcal S})$ with the ground set ${\mathcal U}= [z]^k$. We view each element of ${\mathcal U}$ as a length-$k$ vector $\vec{x}$ where each coordinate assumes a value in $[z]$. There is a collection of {\em canonical sets} ${\mathcal S} = \{X_{i,a}\}_{i \in [z], a \in [k]}$ defined as \[X_{i,a} = \{\vec{x}: \vec{x}_a = i \} \] In other words, each set $X_{i,a}$ contains the vectors whose $a^{th}$ coordinate is $i$. A nice property of this set system is that, it can only be covered completely if all canonical sets corresponding to some $a^{th}$ coordinate are chosen. \begin{proposition} \label{prop:hypercube} Consider any sub-collection ${\mathcal S}' \subseteq {\mathcal S}$. We have $\bigcup {\mathcal S}' = {\mathcal U}$ if and only if there is a value $a \in [k]$ for which $X_{1,a}, X_{2,a},\ldots, X_{z,a} \in {\mathcal S}'$. \end{proposition} \begin{proof} The if part is obvious. For the ``only if'' part, assume that for each $a \in [k]$, there is a value $i_a \in [z]$ for which $X_{i_a, a}$ is not in ${\mathcal S}'$. Define vector $\vec{x}$ by $\vec{x}_a = i_a$. Notice that $\vec{x}$ does not belong to any set in ${\mathcal S}'$ (By definition, if $X_{i',a'}$ contains $\vec{x}$, then it must be the case that $\vec{x}_{a'} = i' = i_{a'}$.) \end{proof} \paragraph{The construction:} Our reduction starts from the ${\sf MinLab}\xspace$ instance $\Gamma = (G, \Sigma_U, \Sigma_V, \Pi)$. We will create the set system ${\mathcal I}= ({\mathcal U}, {\mathcal S})$. We make $\|U\|$ different copies of the hypercube set system: For each vertex $u\in U$, we have the hypercube set system $({\mathcal U}^u, {\mathcal S}^u) = H(N_G(u), \Sigma_U)$, i.e., the ground set ${\mathcal U}^u$ is a copy of $N_G(u)^{\Sigma_U}$ and ${\mathcal S}^u$ contains $\|N_G(u)\| \|\Sigma_U\|$ ``virtual'' sets, that we call $\{S^u_{v,a}\}_{v \in N_G(u), a \in \Sigma_U}$ where each such set corresponds to a canonical set of the hypercube. We remark that these virtual sets are not the eligible sets in our instance ${\mathcal I}$. For each vertex $v \in V$, for each label $b \in \Sigma_V$, we define a set \[S_{v,b} = \bigcup_{u \in N_G(v), (a,b) \in \Pi_{uv}} S^u_{v,a}\] The set system $({\mathcal U}, {\mathcal S})$ in our instance is simply: $${\mathcal U} = \bigcup_{u \in U} {\mathcal U}^u \hspace{0.2in} \mbox{ and } \hspace{0.2in} {\mathcal S} = \{S_{v,b}: v \in V, b \in \Sigma_V\} $$ Notice that the number of sets is $\|V\|\|\Sigma_V\|$ and the number of elements in the ground set is $\|{\mathcal U}\| = \|U\| \|V\|^{\|\Sigma_U\|}$. This completes the description of our instance. \paragraph{Analysis:} We argue that the optimal value of $\Gamma$ is equal to the optimal of $({\mathcal U},{\mathcal S})$. First, we will show that ${\sf MinLab}\xspace(\Gamma) \leq \mbox{\sf SetCov}\xspace({\mathcal U}, {\mathcal S})$. Let $(\sigma_U, \hat{\sigma}_V)$ be a feasible ${\sf MinLab}\xspace$ cover for $\Gamma$ (recall that $\hat{\sigma}_V$ is a multi-labeling, while $\sigma_U$ is a labeling.) For each $v \in V$, the \mbox{\sf SetCov}\xspace solution chooses the set $S_{v,b}$ for all $b \in \hat{\sigma}_V(v)$. Denote this solution by ${\mathcal S}' \subseteq {\mathcal S}$. The total number of sets chosen is exactly $\sum_v \|\hat{\sigma}(v)\|$, exactly matching the cost of ${\sf MinLab}\xspace(\Gamma)$. We argue that this is a feasible set cover: For each $u$, the fact that $u$ is covered by $(\sigma_U, \hat{\sigma}_V)$ implies that, for all $v \in N_G(u)$, there is a label $b_v \in \hat{\sigma}_V(v)$ such that $(\sigma_U(u), b_v) \in \Pi_{uv}$. Notice that $S^u_{v,\sigma_U(u)} \subseteq S_{v, b_v} \in {\mathcal S}'$ for every $v \in N_G(u)$, so we have $$\bigcup_{S \in {\mathcal S}'} S \supseteq \bigcup_{v \in N_G(u)} S_{v, b_v} \supseteq \bigcup_{v \in N_G(u)} S^u_{v, \sigma_U(u)} = {\mathcal U}^u$$ where the last equality comes from \Cref{prop:hypercube}. In other words, ${\mathcal S}'$ covers all elements in ${\mathcal U}^u$. Hence, ${\mathcal S}'$ is indeed a valid $\mbox{\sf SetCov}\xspace$ solution for $({\mathcal U}, {\mathcal S})$. To prove the converse, consider a collection of sets $\{S_{v,b}\}_{(v,b) \in \Lambda}$ that covers the whole universe ${\mathcal U}$. We define the (multi-)labeling $\hat{\sigma}_V: V \rightarrow 2^{\Sigma_V}$ where $\hat{\sigma}_V(v) = \{b: (v,b) \in \Lambda\}$ for each $v \in V$. Clearly, $\sum_{v \in V} \|\hat{\sigma}_V(v)\| = \|\Lambda\|$, so the cost of $\hat{\sigma}_V$ as a solution for ${\sf MinLab}\xspace$ is exactly the cost of \mbox{\sf SetCov}\xspace. We verify that all left vertices $u \in U$ of $\Gamma$ are covered (and along the way will define $\Sigma_U(u)$ for all $u \in U$.) Consider each vertex $u \in U$. The fact that the ground elements in ${\mathcal U}^u$ are covered implies that (from Proposition~\ref{prop:hypercube}) there is a label $a_u \in \Sigma_U$ where all virtual sets $\{S^u_{v,a_u}\}_{v \in N_G(u)}$ are included in the solution. Therefore, for each $v \in N_G(u)$, there must be a label $b_v \in \hat{\sigma}_V(v)$ such that $a_u b_v \in \Pi_{uv}$. We simply define $\sigma_U(u) = a_u$. Therefore, the vertex $u$ is covered by the assignment $(\sigma_U, \hat{\sigma}_V)$. \subsection{Maximum Induced Subgraph with Hereditary Properties} In this section, we prove the hardness of maximum induced subgraphs with hereditary property. Let $\Pi$ be a graph property. We say that a subset $S \subseteq V(G)$ has property $\Pi$ if $G[S] \in \Pi$. Denote by $A_{\Pi}(G)$ the maximum cardinality of a set $S$ that has property $\Pi$. Khot and Raman~\cite{KhotR00} proved a dichotomy theorem for the problem; if $\Pi$ contains all independent sets but not all cliques or if $\Pi$ contains all cliques but not all independent sets, then the problem is W[1]-hard. For all other $\Pi$'s, the problem is in FPT. We will extend Khot and Raman's dichotomy theorem to hold even for FPT approximation as stated more precisely below. \begin{theorem} Let $\Pi$ be any hereditary property. \begin{itemize} \item If $\Pi$ contains all independent sets but not all cliques or vice versa, then computing $A_{\Pi}(G)$ is weakly inherently enumerative (and therefore totally FPT inapproximable). \item Otherwise, $A_{\Pi}(G)$ can be computed exactly in FPT. \end{itemize} \end{theorem} Surprisingly, the fact that there is a gap in the optimum of our starting point helps make our reduction simpler than that of Khot and Raman. For convenience, let us focus only on properties $\Pi$'s which contain all independent sets but not all cliques. The other case can be proved analogously. The main technical result is summarized in the following lemma. \begin{theorem} \label{thm:hereditary} Let $\Pi$ be any graph property that contains all independent sets but not all cliques. Then there is a function $g_\Pi = \omega(1)$ such that the following holds: \begin{itemize} \item If $\alpha(G) \geq q$, then $A_{\Pi}(G) \geq q$. \item If $A_{\Pi}(G) \geq r$, then $\alpha(G) \geq g_\Pi(r)$. \end{itemize} \end{theorem} \begin{proof} Since $\Pi$ contains all independent set, when $\alpha(G) \geq q$, we always have $A_{\Pi}(G) \geq q$. Now, to prove the converse, let $g_\Pi(r)$ denote $\max_{H \in \Pi, \|V(H)\| = r} \alpha(H)$. If $A_{\Pi}(G) = r$, then there exists a subset $S \subseteq V(G)$ of size $r$ that has property $\Pi$; from the definition of $g_\Pi$, $\alpha(H) \geq g_\Pi(r)$, which implies that $\alpha(G) \geq g_\Pi(r)$ as well. Hence, we are only left to show that $g_\Pi = \omega(1)$. To show that this is the case, recall the Ramsey theorem. \begin{theorem}[Ramsey's Theorem] For any $s, t \geq 1$, there is an integer $R(s, t)$ s.t. every graph on $R(s, t)$ vertices contains either a $s$-clique or a $t$-independent set. Moreover, $R(s, t) \leq \binom{s + t - 2}{s - 1}$. \end{theorem} Recall that, from our assumption of $\Pi$, there exists a fixed integer $s_{\Pi}$ such that $\Pi$ does not contain an $s_{\Pi}$-clique. Hence, from Ramsey's Theorem, $g_\Pi(r) \geq \max\{t \mid R(s_{\Pi}, t) \leq r\}$. In particular, this implies that $g_\Pi(r) \geq \Omega_{s_{\Pi}}(r^{1/(s_{\Pi - 1})})$. Hence, $\lim_{r \infty} g_\Pi(r) = \infty$ (i.e. $g_\Pi = \omega(1)$) as desired. \end{proof} In other words, the identical transformation $G \mapsto G$ is a $(q, g_\Pi(r))$-FPT gap reduction from \mbox{\sf Clique}\xspace to Maximum Induced Subgraph with property $\Pi$. Hence, by applying~\Cref{prop:gapred-enum}, we immediately arrive at the following corollary. \begin{corollary} Assuming Gap-ETH, for any property $\Pi$ that contains all independent sets but not all cliques (or vice versa), Maximum Induced Subgraph with property $\Pi$ is $\Omega(g_\Pi)$-weakly inherently enumerative where $g_\Pi$ is the function from Theorem~\ref{thm:hereditary}. \end{corollary} We remark here that, for some properties, $g_\Pi$ can be much larger than the bound given by the Ramsey's Theorem; for instance, if $\Pi$ is planarity, then the Ramsey's Theorem only gives $g_\Pi(r) = \Omega(r^{1/5})$ but it is easy to see that, for planar graphs, there always exist an independent set of linear size and $g_\Pi(r)$ is hence as large as $\Omega(r)$. \subsection{Maximum Balanced Biclique, Maximum Induced Matching on Bipartite Graphs and Densest $k$-Subgraph} We next prove FPT inapproximability for the Maximum Balanced Biclique, Maximum Induced Matching on Bipartite Graphs and Densest $k$-Subgraph. Unlike the previous proofs, we will not reduce from any label cover problem; the starting point for the results in this section will instead be a recent construction of Manurangsi for ETH-hardness of Densest $k$-Subgraph~\cite{Man17}. By interpreting this construction in a different perspective, we can modify it in such a way that we arrive at a stronger form of inherently enumerative hardness for \mbox{\sf Clique}\xspace. More specifically, the main theorem of this section is the following theorem, which is a stronger form of \Cref{thm:clique} in that the soundness not only rules out cliques, but also rules out bicliques as well. \begin{theorem} \label{thm:cliquevbiclique} Assuming Gap-ETH, there exist constants $\delta, \rho > 0$ such that, for any positive integers $q \geq r \geq \rho$, no algorithm can take a graph $G$ and distinguish between the following cases in $O_{q, r}(\|V(G)\|^{\delta\sqrt{r}})$ time: \begin{itemize} \item $\mbox{\sf Clique}\xspace(G) \geq q$. \item $\mbox{\sf Biclique}\xspace(G) < r$. \end{itemize} \end{theorem} The weakly inherently enumerativeness (and therefore totally FPT inapproximability) of Maximum Balanced Biclique and Maximum Induced Matching on Bipartite Graphs follows easily from Theorem~\ref{thm:cliquevbiclique}. We will show these results in the subsequent subsections; for now, let us turn our attention to the proof of the theorem. The main theorem of this section can be stated as follows. \begin{theorem} \label{thm:mbb-fpt-inapprox} For any $d, \varepsilon > 0$, there is a constant $\gamma = \gamma(d, \varepsilon) > 0$ such that there exists a (randomized) reduction that takes in a parameter $r$ and a 3-\mbox{\sf SAT}\xspace instance $\phi$ with $n$ variables and $m$ clauses where each variable appears in at most $d$ constraints and produces a graph $G_{\phi, r} = (V_{\phi, r}, E_{\phi, r})$ such that, for any sufficiently large $r$ (depending only on $d, \varepsilon$ but not $n$), the following properties hold with high probability: \begin{itemize} \item (Size) $N := \|V_{\phi, r}\| \leq 2^{O_{d, \varepsilon}(n/\sqrt{r})}$. \item (Completeness) if $\mbox{\sf SAT}\xspace(\phi) = m$, then $\mbox{\sf Clique}\xspace(G_{\phi, r}) \geq N^{\gamma/\sqrt{r}}$. \item (Soundness) if $\mbox{\sf SAT}\xspace(\phi) \leq (1 - \varepsilon)m$, then $\mbox{\sf Biclique}\xspace(G_{\phi, r}) < r$. \end{itemize} \end{theorem} It is not hard to see that, in the Gap-ETH assumption, we can, without loss of generality, assume that each variable appears in only a bounded number of clauses (See~\cite[p.~21]{ManR16}). Hence, Theorem~\ref{thm:mbb-fpt-inapprox} together with Gap-ETH implies Theorem~\ref{thm:cliquevbiclique}. As mentioned earlier, our result builds upon an intermediate lemma used to prove the hardness of approximating Densest $k$-Subgraph in~\cite{Man17}. Due to this, it will be easier to describe our reduction in terms of the reduction from~\cite{Man17}; in this regard, our reduction can be viewed as vertex subsampling (with appropriate probability) of the graph produced by the reduction from~\cite{Man17}. The reduction is described formally in Figure~\ref{fig:red-cliquevbiclique}. Note that the two parameters $\ell$ and $p$ will be chosen as $\Theta_{d, \varepsilon}(n/\sqrt{r})$ and $2^{\Theta_{d, \varepsilon}(\ell^2/n)}/\binom{n}{\ell}$ respectively where the constants in $\Theta_{d, \varepsilon}(\cdot)$ will be selected based on the parameters from the intermediate lemma from~\cite{Man17}. \begin{figure} \caption{The Reduction from Gap-3\mbox{\sf SAT}\xspace to Maximum Balanced Biclique} \label{fig:red-cliquevbiclique} \end{figure} The main lemma of~\cite{Man17} is stated below. Roughly speaking, when $\mbox{\sf SAT}\xspace(\phi) \leq (1 - \varepsilon)m$, the lemma gives an upper bound on the number of occurrences of $K_{t, t}$ for every $t > 0$. When $p$ and $t$ are chosen appropriately, this implies that w.h.p. there is no $t$-biclique in our subsampled graph. Note that the size and completeness properties are obvious from the construction while the exact statement of the soundness can be found in the proof of Theorem 8 of~\cite{Man17}. \begin{lemma}[\cite{Man17}] \label{lem:dks-reduction} Let $d, \varepsilon, \phi, n, m, \ell$ be as in Theorem~\ref{thm:mbb-fpt-inapprox} and Figure~\ref{fig:red-cliquevbiclique}. There is a constant $\delta, \lambda > 0$ depending only on $d, \varepsilon$ such that, for any sufficiently large $n$, the graph $G_{\phi, \ell} = (V_{\phi, \ell}, E_{\phi, \ell})$ described in Figure~\ref{fig:red-cliquevbiclique} has the following properties \begin{itemize} \item (Size) $\|V_{\phi, \ell}\| = \binom{n}{\ell}2^\ell$. \item (Completeness) if $\mbox{\sf SAT}\xspace(\phi) = m$, $\widetilde{G}_{\phi, \ell}$ contains a $\binom{n}{\ell}$-clique. \item (Soundness) if $\mbox{\sf SAT}\xspace(\phi) \leq (1 - \varepsilon)m$, then $\widetilde{G}_{\phi, \ell}$ contains at most $2^{4n}(2^{-\lambda \ell^2 / n} \binom{n}{\ell})^{2t}$ occurrences\footnote{We say that $S, T \subseteq V_{\phi, \ell}$ is an occurrence of $K_{t, t}$ if $\|S\| = \|T\| = t$, $S \cap T = \emptyset$ and, for every $s \in S, t \in T$, there is an edge between $s$ and $t$ in $G_{\phi, \ell}$. The number of occurrences of $K_{t, t}$ of $G_{\phi, \ell}$ is simply the number of such pairs $(S, T)$'s.} of $K_{t, t}$ for any $t > 0$. \end{itemize} \end{lemma} Theorem~\ref{thm:mbb-fpt-inapprox} follows rather easily from the above lemma by choosing appropriate $\ell$ and $p$. \begin{proof}[Proof of Theorem~\ref{thm:mbb-fpt-inapprox}] We let $G_{\phi, r} = G_{\phi, \ell, p}$ from the reduction in Figure~\ref{fig:red-cliquevbiclique} with parameters $\ell = \frac{4n}{\sqrt{\lambda r}}$ and $p = 2^{\frac{\lambda \ell^2}{2n}} / \binom{n}{\ell}$. For convenience, we assume without loss of generality that $\lambda < 1$. \paragraph{Size.} Since each vertex in $V_{\phi, \ell}$ is included that $V_{\phi, \ell, p}$ independently with probability $p$, we have $\mathop{\mathbb{E}}[\|V_{\phi, \ell, p}\|] = p\|V_{\phi, \ell}\| = 2^{\ell + \frac{\lambda \ell^2}{2n}} \leq 2^{2\ell}$. Hence, from Chernoff bound, $\|V_{\phi, \ell, p}\| \leq 2^{10\ell} = 2^{\Omega_{d, \varepsilon}(n/\sqrt{r})}$ w.h.p. \paragraph{Completeness.} Suppose that $\phi$ is satisfiable. Let $C$ be the clique of size $\binom{n}{\ell}$ in $\widetilde{G}_{\phi, \ell}$, which is guaranteed to exist by Lemma~\ref{lem:dks-reduction}. From how $G_{\phi, \ell, p}$ is defined, $C \cap V_{\phi, \ell, p}$ induces a clique in $G_{\phi, \ell, p}$. Moreover, $\mathop{\mathbb{E}}[\|C \cap V_{\phi, \ell, p}\|] = p\|C\| = 2^{\frac{\lambda \ell^2}{2n}}$. Again, from Chernoff bound, $\mbox{\sf Clique}\xspace(G_{\phi, \ell, p}) \geq 2^{\frac{\lambda \ell^2}{2n}}$ w.h.p. Combined with the above bound on $N$, $\mbox{\sf Clique}\xspace(G_{\phi, \ell, p}) \geq N^{\gamma/\sqrt{r}}$ w.h.p. when $\gamma := \sqrt{\lambda}/20 = O_{d, \varepsilon}(1)$. \paragraph{Soundness.} Suppose that $\mbox{\sf SAT}\xspace(\phi) \leq (1 - \varepsilon)m$. Consider any subsets $S, T \subseteq \widetilde{V}_{\phi, \ell}$ that is an occurrence of $K_{r, r}$ in $\widetilde{G}_{\phi, \ell}$. From how $G_{\phi, \ell, p}$ is defined, $\mbox{\sf Biclique}\xspace(G_{\phi, \ell, p}) \geq r$ if and only if, for at least one such pair $(S, T)$, $S \cup T \subseteq V_{\phi, \ell, p}$. The probability of this event is bounded above by \begin{align} \sum_{S, T \subseteq \widetilde{V}_{\phi, \ell} \atop S, T \text{ is an occurrence of } K_{r, r} \text{ in } \widetilde{G}_{\phi, \ell}} \Pr[S, T \subseteq V_{\phi, \ell, p}] &\leq 2^{4n}\left(2^{-\lambda \ell^2 / n} \binom{n}{\ell}\right)^{2r} \cdot p^{2r} \\ &= 2^{4n} \left(2^{-\frac{\lambda \ell^2}{2n}}\right)^{2r} \\ &= o(1). \end{align} where the first inequality comes from the bound in the soundness of Lemma~\ref{lem:dks-reduction} and the fact that the sampling of each vertex is done independently. As a result, the subsampled graph $G_{\phi, \ell, p}$ is $K_{r, r}$-free with high probability as desired. \end{proof} \subsubsection{Maximum Balanced Biclique} We now give a simple reduction from the ``\mbox{\sf Clique}\xspace vs \mbox{\sf Biclique}\xspace'' problem (from~\Cref{thm:cliquevbiclique}) to the Maximum Balanced Biclique problem, which yields FPT inapproximability of the latter. \begin{lemma} For any graph $G = (V, E)$, let $B_e[G] = (V_{B_e[G]}, E_{B_e[G]})$ be the bipartite graph whose vertex set is $V_{B_e[G]} := V \times [2]$ and two vertices $(u, i), (v, j)$ are connected by an edge if and only if $(u, v) \in E$ or $u = v$, and $i \ne j$. Then the following properties hold for any graph $G$. \begin{itemize} \item $\mbox{\sf Biclique}\xspace(B_e[G]) \geq \mbox{\sf Clique}\xspace(G)$. \item $\mbox{\sf Biclique}\xspace(B_e[G]) \leq 2\mbox{\sf Biclique}\xspace(G) + 1$. \end{itemize} \end{lemma} \begin{proof} It is easy to see that $\mbox{\sf Biclique}\xspace(B_e[G]) \geq \mbox{\sf Clique}\xspace(G)$ since, for any $C \subseteq V$ that induces a clique in $G$, $C \times [2] \subseteq V_{B_e[G]}$ induces a $\|C\|$-biclique in $B_e[G]$. To see that $\mbox{\sf Biclique}\xspace(B_e[G]) \leq 2\mbox{\sf Biclique}\xspace(G) + 1$, consider any $S \subseteq V_{B_e[G]}$ that induces a $k$-biclique in $B_e[G]$. Note that $S$ can be partitioned into $S_1 = S \cap (V \times \{1\})$ and $S_2 = S \cap (V \times \{2\})$. Now consider the projections of $S_1$ and $S_2$ into $V(G)$, i.e., $T_1 = \{v: (v,1) \in S\}$ and $T_2 = \{v: (v,2) \in S\}$. Note that $\|T_1\| = \|T_2\| = k$. Since $S_1 \cup S_2$ induces a biclique in $B_e[G]$, we have, for every $u \in T_1$ and $v \in T_2$, either $u = v$ or $(u, v) \in E$. Observe that if there were no former case (i.e., $T_1\cap T_2=\emptyset$), then we would have a $k$-biclique in $G$. Even if $T_1 \cap T_2 \ne \emptyset$, we can still get back a $\lfloor k/2 \rfloor$-biclique of $G$ by uncrossing the sets $T_1$ and $T_2$ in a natural way by assigning half of the intersection to $T_1$ and the other half to $T_2$. To be formal, we partition $T_1\cap T_2$ into roughly equal sets $U_1$ and $U_2$ (i.e., $\|\|U_1\| - \|U_2\|\| \leq 1$), and we then define new sets $T'_1$ and $T'_2$ by \[ T'_1 = (T_1\setminus T_2) \cup U_1 \mbox{ and } T'_2 = (T_2\setminus T_1) \cup U_2. \] It is not hard to see that $G$ has an edge between every pair of vertices between $T'_1, T'_2$ and that $\|T'_1\|, \|T'_2\| \geq \lfloor k/2 \rfloor$. Thus, $\mbox{\sf Biclique}\xspace(G) \geq \lfloor k/2 \rfloor \geq (k - 1)/2$. Therefore, $\mbox{\sf Biclique}\xspace(B_e[G]) \leq 2\mbox{\sf Biclique}\xspace(G) + 1$ as desired. \end{proof} Thanks to the above lemma, we can conclude that the reduction $G \mapsto B_e[G]$ is a $(2q, (r + 1)/2)$-FPT gap reduction from the ``\mbox{\sf Clique}\xspace vs \mbox{\sf Biclique}\xspace'' problem to Maximum Balanced Biclique, although the former is not a well-defined optimization problem. Nevertheless, it is easy to check that a proof along the line of~\Cref{prop:gapred-enum} still works and it gives the following result: \begin{corollary} Assuming Gap-ETH, Maximum Balanced Biclique are $\Omega(\sqrt{r})$-weakly inherently enumerative and thus FPT inapproximable. \end{corollary} It is worth noting here that the Maximum Edge Biclique problem, a well-studied variant of the Maximum Balanced Biclique problem where the goal is instead to find a (not necessarily balanced) complete bipartite subgraph of a given bipartite graph that contains as many edges as possible, is in FPT; this is because the optimum is at least the maximum degree, but, when the degree is bounded above by $r$, all bicliques can be enumerated in $2^{O(r)} \operatorname{poly}(n)$ time. \subsubsection{Maximum Induced Matching on Bipartite Graphs} Next, we prove the FPT hardness of approximation for the Maximum Induced Matching problem on bipartite graphs. Again, the proof will be a simple reduction from Theorem~\ref{thm:cliquevbiclique}. The argument below is similar to that used in Lemma~IV.4 of \cite{ChalermsookLN-FOCS13}. We include it here for completeness. \begin{lemma} For any graph $G = (V, E)$, let $B_e[\bar{G}] = (V_{B_e[\bar{G}]}, E_{B_e[\bar{G}]})$ be the bipartite graph whose vertex is $V_{B_e[\bar{G}]} := V \times [2]$ and two vertices $(u, i), (v, j)$ are connected by an edge if and only if $(u, v) \notin E$ or $u = v$, and $i \ne j$. Then, the following properties hold for any graph $G$. \begin{itemize} \item $\mbox{\sf IM}\xspace(B_e[\bar{G}]) \geq \mbox{\sf Clique}\xspace(G)$. \item $\mbox{\sf IM}\xspace(B_e[\bar{G}]) \leq 2\mbox{\sf Biclique}\xspace(G) + 1$. \end{itemize} \end{lemma} \begin{proof} Consider any $S \subseteq V$ that induces a clique in $G$. It is obvious that $S \times [2] \subseteq V_{B_e[\bar{G}]}$ induces a matching in $B_e[\bar{G}]$. Next, consider any induced matching matching $\{(u_1, v_1), \dots, (u_m, v_m)\}$ of size $m$. Assume w.l.o.g. that $u_1, \dots, u_m \in V \times \{1\}$ and $v_1, \dots, v_m \in V \times \{2\}$. Define $\pi_1: V \times [2] \to V$ to be a projection operator that projects on to the first coordinate. Let $S_1 = \pi_1(\{u_1, \dots, u_{\lfloor m/2 \rfloor}\})$ and $S_2 = \pi_1(\{v_{\lceil m / 2 \rceil + 1}, \dots, v_m\})$. From the definition of $B_e[\bar{G}]$ and from the fact that there is no edge between $(S_1 \times \{1\})$ and $(S_2 \times \{2\})$, it is easy to check that $S_1 \cap S_2 = \emptyset$ and, for every $u \in S_1$ and $v \in S_2$, $(u, v) \in E$. In other words, $(S_1, S_2)$ is an occurrence of $\lfloor m / 2 \rfloor$ in $G$. Hence, we can conclude that $\mbox{\sf IM}\xspace(B_e[\bar{G}]) \leq 2 \mbox{\sf Biclique}\xspace(G) + 1$. \end{proof} Similar to \mbox{\sf Biclique}\xspace, it is easy to see that the above reduction implies the following running time lower bound and FPT inapproximability for Maximum Induced Matching on Bipartite Graphs. \begin{corollary} Assuming Gap-ETH, Maximum Induced Matching on Bipartite Graphs are $\Omega(\sqrt{r})$-weakly inherently enumerative and thus FPT inapproximable. \end{corollary} \subsubsection{Densest $k$-Subgraph} Finally, we will show FPT inapproximability result for Densest $k$-Subgraph. Alas, we are not able to show $o(k)$-ratio FPT inapproximability, which would have been optimal since the trivial algorithm gives an $O(k)$-approximation for the problem. Nonetheless, we will show an $k^{o(1)}$-factor FPT inapproximability for the problem. We note here that below we will state the result as if $k$ is the parameter; this is the same as using the optimum as the parameter, since (in the non-trivial case) the optimum is always between $\lfloor k / 2 \rfloor$ and $\binom{k}{2}$ (inclusive). To derive our result, we resort to a well-known result in extremal combinatorics called the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n (KST) Theorem, which basically states that if a graph does not contain small bicliques, then it is sparse. The KST theorem is stated formally below. \begin{theorem}[K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n (KST) Theorem \cite{KST54}] For every positive integer $n$ and $t \leq n$, every $K_{t,t}$-free graph on $n$ vertices has at most $O(n^{2 - 1/t})$ edges (i.e., density $O(n^{-1/t})$). \end{theorem} We remark here that a generalization of the KST Theorem was also a crucial ingredient in the proof of ETH-hardness of approximating Densest $k$-Subgraph in~\cite{Man17}. The situation is simpler for us here, since we can simply apply the KST Theorem to Theorem~\ref{thm:cliquevbiclique}, which yields the following theorem. \begin{theorem} \label{thm:gapdks} Assuming Gap-ETH, there exist a constant $\delta > 0$ and an integer $\rho > 0$ such that, for any integer $q \geq r \geq \rho$, no algorithm can take a graph $G = (V, E)$ and distinguish between the following cases in $O_{q, r}(\|V\|^{\delta\sqrt{r}})$ time: \begin{itemize} \item $\mbox{\sf Den}\xspace_{q}(G) = 1$. \item $\mbox{\sf Den}\xspace_{q}(G) < O(q^{- r})$. \end{itemize} \end{theorem} From the above theorem, it is easy to show the $k^{o(1)}$-factor FPT inapproximability of Densest $k$-Subgraph as formalized below. We note here that our result applies to a special case of Densest $k$-Subgraph in which the input graph is promised to contain a $k$-clique; this problem is sometimes referred to as \emph{Densest $k$-Subgraph with perfect completeness}~\cite{BravermanKRW17,Man17}. \begin{lemma} Assuming Gap-ETH, for every function $f = o(1)$ and every function $t$, there is no $t(k) \cdot n^{O(1)}$-time algorithm such that, given an integer $k$ and any graph $G = (V, E)$ on $n$ vertices that contains at least one $k$-clique, always output $S \subseteq V$ of size $k$ such that $\mbox{\sf Den}\xspace(S) \geq k^{-f(k)}$. \end{lemma} \begin{proof} Suppose for the sake of contradiction that there is a $t(k) \cdot \|V\|^{D}$-time algorithm ${\mathbb A}$ that, given an integer $k$ and any graph $G = (V, E)$ that contains a $k$-clique, always outputs $S \subseteq V$ of size $k$ such that $\mbox{\sf Den}\xspace(S) \geq k^{-f(k)}$ for some function $f = o(1)$, some function $t$ and some constant $D > 0$. Let $r = \max\{\lceil \rho \rceil, \lceil (D / \delta)^2 \rceil\}$ where $\rho$ is the constant from~\Cref{thm:gapdks}. Note that $O(q^{-r}) = q^{O(1)/\log q - r}$. Now, since $\lim_{q \to \infty} f(q) + O(1)/\log q = 0$, there exists a sufficiently large $q$ such that the term $O(q^{-r})$ is less than $q^{-f(q)}$. In other words, ${\mathbb A}$ can distinguish between the two cases in~\Cref{thm:gapdks} in time $t(q) \cdot n^D = O_{q, r}(\|V\|^{\delta \sqrt{r}})$, which would break Gap-ETH. \end{proof} \section{Conclusion and Discussions} In this paper, we prove that \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace are totally FPT inapproximable. In fact, we show a stronger property that they are inherently enumerative, i.e., the best way to approximate both problems is to essentially enumerate all possibilities. Since \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace are complete problems for the class $\mbox{\sf W}\xspace[1]$ and $\mbox{\sf W}\xspace[2]$ respectively, it might be possible that these two problems can be sources of FPT-inapproximability of many other problems that admit no FPT algorithms. We would like to also mention that there are some problems that are known to be totally FPT-inapproximable under weaker assumptions; examples of such problems are {\em independent dominating set} and {\em induced path}. The former has been shown to be FPT-inapproximable under the assumption $\mbox{\sf FPT}\xspace \neq \mbox{\sf W}\xspace[2]$ in \cite{DowneyFMR08}. For the induced path problem, we show in \Cref{apx:w1-hardness} that it is FPT-inapproximable under the assumption $\mbox{\sf FPT}\xspace \neq \mbox{\sf W}\xspace[1]$. It would be interesting to understand whether it is possible to also base total FPT-inapproximability of \mbox{\sf Clique}\xspace and \mbox{\sf DomSet}\xspace under assumptions that are weaker than Gap-ETH, such as $\mbox{\sf FPT}\xspace \neq \mbox{\sf W}\xspace[1]$ or ETH. To this end, we note that Chen and Lin~\cite{ChenL16} showed inapproximability for \mbox{\sf DomSet}\xspace under $\mbox{\sf FPT}\xspace \neq \mbox{\sf W}\xspace[1]$ (resp., ETH), but their inapproximability ratio is only any constant (resp., $\log^{1/4 - \varepsilon}({\sf OPT}\xspace)$); if their result could be extended to exclude $f({\sf OPT}\xspace)$-approximation for any function $f$, then \mbox{\sf DomSet}\xspace would indeed be totally FPT-inapproximable under weaker assumptions. Another interesting further research direction is to study the trade-off between the running time and the approximation ratio of problems that are known to be FPT-approximable or admit FPT (exact) algorithms. The exploration of such trade-off may be useful in both theory and practice. \section*{Acknowledgment} We thank Benny Applebaum for sharing with us his result~\cite{App17}. Pasin would like to thank Igor Shinkar and Daniel Reichman for discussions on a related problem that inspires part of the proof, and Aviad Rubinstein for useful discussions regarding FPT inapproximability. We also thank Igor for pointing us to~\cite{Kayal2014}. Danupon and Parinya would like to thank Per Austrin for discussions. Parinya would also like to thank Nikhil Bansal, Jesper Nederlof, Karl Bringmann and Holger Dell for insightful discussions. Bundit would like to thank Uriel Feige for useful discussions on \mbox{\sf Clique}\xspace. Marek Cygan is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 677651). Guy Kortsarz is supported in part by NSF grants 1218620 and 1540547. Bundit Laekhanukit is partially supported by ISF Grant No. 621/12 and I-CORE Grant No. 4/11. Danupon Nanongkai is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 715672 and the Swedish Research Council (Reg. No. 2015-04659). Pasin Manurangsi and Luca Trevisan are supported by NSF Grants No. CCF 1540685 and CCF 1655215. \printbibliography[heading=bibintoc] \iffalse \fi \appendix \section{Gap Problems vs Approximation Algorithms} \label{app:gapvapprox} In this section, we establish the connections between gap problems and FPT approximation algorithm by proving Proposition~\ref{prop:gapvapprox-min} and Proposition~\ref{prop:gapvapprox-max}. Proposition~\ref{prop:gapvapprox-min} is in fact implied by a result due to Chen et al.~\cite[Proposition 4]{ChenGG06}; we provide a proof of it here for completeness. On the other hand, we are not aware of any prior proof of Proposition~\ref{prop:gapvapprox-max}. \begin{proof}[Proof of Proposition~\ref{prop:gapvapprox-min}] We will prove the contrapositive of the statement in the proposition. Suppose that (2) is false, i.e., there exist computable functions $t: {\mathbb N} \to {\mathbb N}, f: {\mathbb N} \to [1, \infty)$ and an algorithm ${\mathbb B}$ such that, for every instance $I$ of $\Pi$, ${\mathbb B}$ on input $I$ runs in time $t({\sf OPT}\xspace_\Pi(I)) \cdot \|I\|^D$ for some constant $D$ and outputs $y \in {\sf SOL} \xspace_\Pi(I)$ of cost at most ${\sf OPT}\xspace_\Pi(I) \cdot f({\sf OPT}\xspace_\Pi(I))$. Let $t': {\mathbb N} \to {\mathbb N}$ and $f': {\mathbb N} \to [1, \infty)$ be functions that are defined by $t'(k) = \max_{i = 1, \dots, k} t(i)$ and $f'(k) = \max_{i = 1, \dots, k} f(i)$. Since $t$ and $f$ are computable, $t'$ and $f'$ are also computable. Let ${\mathbb A}$ be an algorithm that takes in an instance $I$ of $\Pi$ and a positive integer $k$, and works as follows. ${\mathbb A}$ simulates an execution of ${\mathbb B}$ on $I$ step-by-step. If ${\mathbb B}(I)$ does not finish within $t'(k) \cdot \|I\|^{D}$ time steps, then ${\mathbb A}$ terminates the execution and returns 0. Otherwise, let $y$ be the output of ${\mathbb B}(I)$. ${\mathbb A}$ computes ${\sf COST} \xspace_\Pi(I, y)$; ${\mathbb A}$ then returns 1 if this cost is at most $k \cdot f'(k)$ and returns 0 otherwise. We claim that ${\mathbb A}$ is an $f'$-FPT gap approximation algorithm of $\Pi$. To see that this is the case, first notice that the running time of ${\mathbb A}$ is $O(t'(k) \cdot \|I\|^{D} + \|I\|^{O(1)})$ where $\|I\|^{O(1)}$ denotes the time used to compute the solution cost. Moreover, if ${\sf OPT}\xspace_\Pi(I) > k \cdot f'(k)$, then it is obvious to see that ${\mathbb A}$ always output 0. Finally, if ${\sf OPT}\xspace_\Pi(I) \leq k$, then, by our assumption on ${\mathbb B}$ and the definitions of $t'$ and $f'$, ${\mathbb B}(I)$ finishes in time $t({\sf OPT}\xspace_\Pi(I)) \cdot \|I\|^D \leq t'(k) \cdot \|I\|^D$ and the output solution $y$ has cost at most ${\sf OPT}\xspace_\Pi(I) \cdot f({\sf OPT}\xspace_\Pi(I)) \leq k \cdot f'(k)$. Hence, ${\mathbb A}$ always outputs 1 in this case. As a result, ${\mathbb A}$ is an $f'$-FPT gap approximation algorithm for $\Pi$, which concludes our proof. \end{proof} \begin{proof}[Proof of Proposition~\ref{prop:gapvapprox-max}] We will again prove the contrapositive of the statement in the proposition. Suppose that (2) is false, i.e., there exist computable functions $t: {\mathbb N} \to {\mathbb N}, f: {\mathbb N} \to [1, \infty)$ such that $k / f(k)$ is non-decreasing and $\lim_{k \to \infty} k/f(k) = \infty$, and an algorithm ${\mathbb B}$ such that, for every instance $I$ of $\Pi$, ${\mathbb B}$ on input $I$ runs in time $t({\sf OPT}\xspace_\Pi(I)) \cdot \|I\|^D$ for some constant $D$ and outputs $y \in {\sf SOL} \xspace_\Pi(I)$ of cost at least ${\sf OPT}\xspace_\Pi(I) / f({\sf OPT}\xspace_\Pi(I))$. Let $t': {\mathbb N} \to {\mathbb N}$ be a function defined by $t'(k) = \max_{i = 1, \dots, k} t(i)$; clearly, $t'$ is computable. Let ${\mathbb A}$ be an algorithm that takes in an instance $I$ of $\Pi$ and a positive integer $k$, and works as follows. ${\mathbb A}$ simulates an execution of ${\mathbb B}$ on $I$ step-by-step. If ${\mathbb B}(I)$ does not finish within $t'(k) \cdot \|I\|^{D}$ time steps, then ${\mathbb A}$ terminates the execution and returns 1. Otherwise, let $y$ be the output of ${\mathbb B}(I)$. ${\mathbb A}$ computes ${\sf COST} \xspace_\Pi(I, y)$; ${\mathbb A}$ then returns 1 if this cost is at least $k / f(k)$ and returns 0 otherwise. We claim that ${\mathbb A}$ is an $f$-FPT gap approximation algorithm of $\Pi$. To see that this is the case, first notice that the running time of ${\mathbb A}$ is $O(t'(k) \cdot \|I\|^{D} + \|I\|^{O(1)})$ where $\|I\|^{O(1)}$ denotes the time used to compute the solution cost. Moreover, if ${\sf OPT}\xspace_\Pi(I) < k / f'(k)$, then the running time of ${\mathbb B}(I)$ is at most $t({\sf OPT}\xspace_\Pi(I)) \cdot \|I\|^D \leq t'(k) \cdot \|I\|^D$, which implies that ${\mathbb A}$ returns 0. Suppose, on the other hand, that ${\sf OPT}\xspace_\Pi(I) \geq k$. If ${\mathbb B}(I)$ finishes in time $t'(k) \cdot \|I\|^D$, then, from the guarantee of ${\mathbb B}$, it must output $y \in {\sf SOL} \xspace_\Pi(I)$ with ${\sf COST} \xspace_\Pi(I, y) \geq {\sf OPT}\xspace_\Pi(I)/f({\sf OPT}\xspace_\Pi(I))$, which is at least $k/f(k)$ since $k/f(k)$ is non-decreasing. Furthermore, if ${\mathbb B}(I)$ does not finish in the specified time, then ${\mathbb A}$ also returns 1 as desired. As a result, ${\mathbb A}$ is an $f$-FPT gap approximation algorithm for $\Pi$, which concludes our proof. \end{proof} \section{Totally FPT Inapproximable Through FPT Gap Reductions (Proof of \Cref{prop:gapred-totallyinapprox})} \label{app:gapreduction} We will only show the proof when both $\Pi_0$ and $\Pi_1$ are maximization problems. Other cases can be proved analogously and therefore omitted. We assume that (i) holds, and will show that if the ``then'' part does not hold, then (ii) also does not hold. Recall from \Cref{def:FPT gap reduction} that (i) implies that there exists $C, D > 0$ such that the reduction from $\Pi_0$ (with parameters $q$ and $r$) to $\Pi_1$ takes $O_{q, r}(\|I_0\|^C)$ time and always output an instance $I_1$ of size at most $O_{q, r}(\|I_0\|^D)$ on every input instance $I_0$. Now assume that the ``then'' part does {\em not} hold; i.e. $\Pi_1$ admits a $(t(k) \|I_1\|^F)$-time $h$-FPT gap approximation algorithm ${\mathbb A}$ for some function $h(k)= o(k)$ and constant $F$. We will show the following claim which says that (ii) does not hold (by \Cref{def:FPT gap approx}). \begin{claim} There exists a function $k\geq g'(k)=\omega(1)$ and an algorithm ${\mathbb B}$ that takes any input instance $I_0$ of problem $\Pi_0$ and integer $k$, and in $O_k(\|I_0\|^{O(1)})$ time can distinguish between ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq k$ and ${\sf OPT}\xspace_{\Pi_0}(I_0) < g'(k)$. \end{claim} We now prove the claim by constructing an algorithm ${\mathbb B}$ that performs the following steps. Given $I_0$ and $k$, ${\mathbb B}$ applies the reduction on instance $I_0$ and parameters $k$ and $r=\frac{f(k)}{h(f(k))}$. Denote by $I_1$ the instance of $\Pi_1$ produced by the reduction, so we have that $\|I_1\| = O_k(\|I_0\|^{O(1)})$. The following properties are immediate from the definitions of the FPT gap reductions (\Cref{def:FPT gap reduction}). \begin{itemize} \item If ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq k$, we must have ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq f(k)$. \item Also, if ${\sf OPT}\xspace_{\Pi_0}(I_0) < g'(k) := g(\frac{f(k)}{h(f(k))})$, then we will have ${\sf OPT}\xspace_{\Pi_1}(I_1) < r=\frac{f(k)}{h(f(k))}$ \end{itemize} Since ${\mathbb A}$ is an $h$-FPT gap approximation algorithm, it can distinguish between the above two cases, i.e. running ${\mathbb A}$ on $(I_1, f(k))$ will distinguish the above cases, therefore distinguishing between ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq k$ and ${\sf OPT}\xspace_{\Pi_0}(I_0) < g'(k) = g(\frac{f(k)}{h(f(k))})$. This algorithm runs in time $O_k(\|I_1\|^F) = O_k(\|I_0\|^{DF})=O_k(\|I_0\|^{O(1)})$. Notice also that \[g'(k) = g(\frac{f(k)}{h(f(k))}) \leq g(f(k)) \leq k \] where the first inequality is because $f(k)/h(f(k)) \leq f(k)$ (recall that $h(f(k))\geq 1$ by \Cref{def:FPT gap approx}) and because $g$ is non-decreasing, and the second inquality is by the claim below. \begin{claim} For any totally-FPT-inapproximable problem $\Pi_0$, any functions $g$ and $f$ that satisfy conditions in \Cref{def:FPT gap reduction} and any integer $x$, $g(f(x)) \leq x$. \end{claim} \begin{proof} For any integer $x$, consider instance $I_0$ such that ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq x$ (such $I_0$ exists because ${\sf OPT}\xspace_{\Pi_0}=\omega(1)$; otherwise $\Pi_0$ is not totally-FPT-inapproximable (e.g. we can always output $1$ if $\Pi_0$ is a maximization problem)). By the second condition in \Cref{def:FPT gap reduction}, ${\sf OPT}\xspace_{\Pi_1}(I_1)\geq f(x)$. Applying the contrapositive of the third condition with $r=f(x)$ (thus ${\sf OPT}\xspace_{\Pi_1}(I_1)\geq r$), we have ${\sf OPT}\xspace_{\Pi_0}(I_0)\geq g(r)=g(f(x))$. Thus, $x\geq {\sf OPT}\xspace_{\Pi_0}(I_0)\geq g(f(x))$ as claimed. \end{proof} To complete the proof, one only needs to argue that $g(\frac{f(k)}{h(f(k))}) = \omega(1)$, and this simply follows from the fact that $f(k) = \omega(1)$, $g(k) = \omega(1)$ and that $k/h(k) = \omega(1)$. \endinput We will prove a contrapositive. We will only show the proof when both $\Pi_0$ and $\Pi_1$ are maximization problems. Other cases can be proved analogously and therefore omitted. Suppose that problem $\Pi_0$ is $(f,g)$-FPT gap reducible to a maximization problem $\Pi_1$, and $\Pi_1$ admits a $h$-FPT gap approximation algorithm ${\mathbb A}$ for some function $h(k)= o(k)$. Let us assume that the algorithm ${\mathbb A}$ runs in time $t(k) \|I_1\|^D$ for any instance $I_1$. We will argue that $\Pi_0$ also admits $h'$-FPT gap approximation algorithm ${\mathbb B}$ for some function $h'$. We construct an algorithm ${\mathbb B}$ that performs the following steps, on any input instance $I_0$ and integer $k$. We apply the reduction on instance $(I_0,q,r)$ where $q = k$ and $r = \frac{f(k)}{h(f(k))}$; denote by $I_1$ the instance of $\Pi_1$ produced by the reduction, so we have that $\|I_1\| = O_k(\|I_0\|^{O(1)})$. The following properties are immediate from the definitions of the FPT gap reductions. \begin{itemize} \item If ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq k$, we must have ${\sf OPT}\xspace_{\Pi_1}(I_1) \geq f(k)$. \item Also, if ${\sf OPT}\xspace_{\Pi_0}(I_0) < g(\frac{f(k)}{h(f(k))})$, then we will have ${\sf OPT}\xspace_{\Pi_1}(I_1) < \frac{f(k)}{h(f(k))}$ \end{itemize} Since ${\mathbb A}$ is a $h$-FPT gap approximation algorithm, it can distinguish between the above two cases, i.e. running ${\mathbb A}$ on $(I_1, f(k))$ will distinguish the above cases, therefore distinguishing between ${\sf OPT}\xspace_{\Pi_0}(I_0) \geq k$ and ${\sf OPT}\xspace_{\Pi_0}(I_0) < g(\frac{f(k)}{h(f(k))})$. This algorithm runs in time $O_k(\|I_1\|^D) = O_k(\|I_0\|^{O(D)})$. Notice also that \[g(\frac{f(k)}{h(f(k))}) \leq g(f(k)) \leq k \] where the first inequality is because $f(k)/h(f(k)) \leq f(k)$ (recall that $h(f(k))\geq 1$ by \Cref{def:FPT gap approx}) and because $g$ in non-decreasing, and the second inquality is by \Cref{def:FPT gap reduction}. To complete the proof, one only needs to argue that $g(\frac{f(k)}{h(f(k))}) = \omega(1)$, and this simply follows from the fact that $f(k) = \omega(1)$, $g(k) = \omega(1)$ and that $k/h(k) = \omega(1)$. \section{FTP-Inapproximability under W[1]-Hardness} \label{apx:w1-hardness} In this section, we show an example of problems that have no FPT-approximation algorithm unless W[1]=FPT, namely the {\em maximum induced path} problem (\mbox{\sf InducedPath}\xspace). In \mbox{\sf InducedPath}\xspace, we are given a graph $G$, and the goal is to find a maximum size subset of vertices $S\subseteq V(G)$ such that $S$ induces a path in $G$. We will show that \mbox{\sf InducedPath}\xspace has no FPT-approximation algorithm. Implicit in our reduction is a reduction from $k$-\mbox{\sf Clique}\xspace to the {\em multi-colored clique} problem. \begin{theorem} \label{thm:hardness-ipath} Unless \mbox{W[1]=FPT}, for any positive integers $q:1\leq q \leq n^{1-\delta}$, for any $\delta<0$, given a graph $G$ on $n$ vertices and for any function $t:\mathbb{R}\rightarrow\mathbb{R}$, there is no $t(k)\operatorname{poly}(n)$-time algorithm that distinguishes between the following two cases: \begin{itemize} \item $\mbox{\sf InducedPath}\xspace(G) \geq 2q\cdot k$. \item $\mbox{\sf InducedPath}\xspace(G) \leq 4(k-1)$. \end{itemize} \end{theorem} \begin{proof} The reduction is as follows. Take a graph $H$ of a $k$-\mbox{\sf Clique}\xspace instance. Then we construct a graph $G$ as follows. First, we create intermediate graphs $Z_1,\ldots,Z_q$. Each graph $Z_i$ for $i\in[q]$ is created by making $k$ copies of $V(H)$, namely, $V_{i,1},\ldots,V_{i,k}$ and form a clique on $V_{i,j}$ for each $j\in[k]$. So, now, we have $k$ disjoint cliques. For each vertex $v\in V(H)$, we pick a copy of $v$, one from each $V_{i,j}$, say $v_{i,j}$, and we form a clique on $\{v_{i,1},\ldots,v_{i,k}\}$. Next, for each edge $uv\not\in E(H)$, we add edges $u_{i,j}v_{i,j'}$, for all $j,j'\in[k]$, where $u_{i,j}$ and $v_{i,j'}$ are the copy of $u$ in $V_{i,j}$ and the copy of $v$ in $V_{i,j'}$, respectively. Next, we add a dummy vertex $x_{i,j}$ for each $V_{i,j}$ and add edges joining $x_{i,j}$ to every vertex of $V_{i,j}$ and to every vertex of $V_{i,j-1}$ if $j \geq 2$. Finally, we join the graph $Z_i$ for all $i\in[q]$ to be of a form $(Z_1,Z_2,\ldots,Z_{k})$. To be precise, for each graph $Z_i$ with $i \geq 2$, we join the vertex $x_{i,1}$ (which belongs to $Z_i$) to every vertex of $V_{i-1,q}$ (which belongs to $Z_{i+1}$). \noindent{\bf Completeness.} First, suppose $\mbox{\sf Clique}\xspace(H) \geq k$. We will show that $\mbox{\sf InducedPath}\xspace(G) \geq 2q\cdot k$. We take a subset of vertices $S\subseteq V(H)$ that induces a clique on $H$. Let us name vertices in $S$ by $v^1,\ldots,v^k$. For each $j\in [k]$, we pick the copies $v^j_{i,j}$ of $v^j$ from $V_{i,j}$ for all $i\in[q]$. We then pick all the vertices $x_{i,j}$ for $i\in[k]$ and $j\in[q]$. We denote this set of vertices by $S'$. It is not hard to see that for any distinct vertices $v^{j},v^{j'}\in S$, their copies $v^j_{i,j}$ and $v^{j'}_{i',j'}$ are not adjacent, and each vertex $x_{i,j}$ has exactly two neighbors: $v^j_{i,j}$ and $u^{j-1}_{i,j-1}$ (or $u^k_{i-1,k}$). Therefore, $S'$ induces a path in $G$ of size $2qk$. \noindent{\bf Soundness.} Suppose $\mbox{\sf Clique}\xspace(H) < k$, i.e., $H$ has no clique of size $k$. We will show that $\mbox{\sf InducedPath}\xspace(G)\leq 4(k-1)$. To see this, let $S'\subseteq V(G)$ be a subset of vertices that induces a path $G[S']$ in $G$. Observe that, for $i\in[q]$, $G[S']\cap Z_i$ must be a path of the form $(x_{i,a},v^a_{i,a},\ldots,x_{i,k},v^b_{i,b})$. Moreover, $v^\ell_{i,\ell}$ and $v^{\ell'}_{i,\ell'}$ are not adjacent in $G$ for any $\ell\neq\ell'$, meaning that $v^\ell_{i,\ell}$ and $v^{\ell'}_{i,\ell'}$ are not copies of the same vertex in $H$, and the set $\{v^{\ell}\}_{a\leq \ell\leq b}$ induces a clique in $H$. Thus, $a-b+1 < k$, and $G[S']\cap Z_i$ can have at most $2(k-1)$ vertices. It follows that any induced path $G[S']$ of $G$ can contain vertices from at most two subgraphs, say $Z_i$ and $Z_{i+1}$. Therefore, we conclude that $\|S'\| \leq 4(k-1)$. \end{proof} The FPT-inapproximable of \mbox{\sf InducedPath}\xspace follows directly from Theorem~\ref{thm:hardness-ipath}. \begin{corollary} \label{cor:fpt-inapprox-ipath} Unless W[1]=FPT, there is no $f(k)$-approximation algorithm for \mbox{\sf InducedPath}\xspace that runs in $t(k)\operatorname{poly}(n)$-time for any functions $f$ and $t$ depending only on $k$. \end{corollary} \section{Known Connections between Problems} \label{app:trivial-eq} In this section, we discuss known equivalences between problems in more detail. \paragraph{Dominating Set and Set Cover:} It is easy to see that \mbox{\sf DomSet}\xspace is a special case of \mbox{\sf SetCov}\xspace, and the reduction from \mbox{\sf SetCov}\xspace to \mbox{\sf DomSet}\xspace is by phrasing $\mathcal{U}$ and $\mathcal{S}$ as vertices, forming a clique on $\mathcal{S}$ and there is an edge joining a subset $S_i \in \mathcal{S}$ and element $u_j \in \mathcal{U}$ if and only if $u_j$ is an element in $S_i$. \paragraph{Induced Matching and Independent Set:} We show that Induced Matching is at least as hard to approximate as Independent Set. Let $G$ be an input graph of Independent Set. We create a graph $G'$ by, for each vertex $v \in V(G)$, create a vertex $v'$ and an edge $v v'$. Notice that any independent set $S$ of $G$ corresponds to an induced matching in $G'$: For each $v \in S$, we have an edge $v v'$ in the set $\mathcal{M}$. Conversely, for any induced matching $\mathcal{M}$ of $G'$, we may assume that the matching only chooses edges of the form $vv'$. \paragraph{More hereditary properties:} We discuss some more natural problems in this class. If we define $\Pi$ to be a set of all planar graphs, this is hereditary. The corresponding optimization problem is that of computing a maximum induced planar graphs. If we define $\Pi$ to be a set of all forests, this is also hereditary, and it gives the problem of computing a maximum induced forest. \section{Proof Sketch of \Cref{thm:Gap-ETH restated}} \label{sec:restate-gap-eth} We will sketch the proof of \Cref{thm:Gap-ETH restated}. In the forward direction, we use a standard reduction, which is sometimes referred to as the {\em clause-variable game}~\cite{AIM14}. Specifically, we transform a $3$-\mbox{\sf SAT}\xspace instance $\psi$ on $n$ variables $x_1,\ldots,x_n$ and $m$ clauses $C_1,\ldots,C_m$ into a label cover instance $\Gamma=(G=(U,V,E),\Sigma_U,\Sigma_V,\Pi)$ by transforming clauses into left vertices in $U$ and variables into right vertices in $V$, and there is an edge joining a pair of vertices $C_i$ and $x_j$ if $x_j$ appears in $C_i$. We take partial assignments as the label sets $\Sigma_U$ and $\Sigma_V$, and a constraint on each edge asks for a pair $(\alpha,\beta)$ of labels that are {\em consistent}, i.e., they assign the same value to the same variable (e.g., $\alpha=(x_1:1,x_2:0,x_3:1)$ and $\beta=(x_1:1)$ are consistent whereas $\alpha$ is not consistent with $\beta'=(x_2:1)$), and $\alpha$ causes $C_i$ to evaluate to true (i.e., some of the literal in $C_i$ is assigned to true by $\alpha$). We denote the evaluation of a clause $C_i$ on a partial assignment $\alpha$ by $C_i(\alpha)$. To be precise, we have \[ \begin{array}{l} U = \{C_1,\ldots,C_{m}\}, \quad V = \{x_1,\ldots,x_{n}\}, \quad E = \{C_ix_j: \mbox{$x_j$ appears in the clause $C_i$}\}\\ \Sigma_U = \{0,1\}^3, \quad \Sigma_V = \{0,1\}, \quad \Pi_{C_ix_j} = \{(\alpha,\beta): \mbox{$\alpha$ and $\beta$ are consistent $\land$ $C_i(\alpha)=\mbox{true}$}\} \end{array} \] It can be seen that ${\sf MaxCov}\xspace(\Gamma)=\mbox{\sf SAT}\xspace(\psi)$ since the only way to cover each node $C_i\in U$ is to pick assignments to all vertices adjacent to $C_i$ so that they are all consistent with the assignment $\alpha=\sigma_V(C_i)$ (and that $C_i(\alpha)=\mbox{true}$). The converse direction is not straightforward. We apply H{\aa}stad \cite{Hastad01} reduction\footnote{Here we apply only the Hastad's reduction from label cover to 3\mbox{\sf SAT}\xspace, without parallel repetition.} to reduce an instance $\Gamma$ of {\sf MaxCov}\xspace to a $3$-SAT instance of size $f(\|\Sigma_U\|+\|\Sigma_V\|)\cdot \mbox{linear}(\|U\|+\|V\|)$ with a hardness gap $1-\varepsilon$, for some constant $\varepsilon>0$ (the hardness gap is different from the original {\sf MaxCov}\xspace instance). Note that $f$ in the H{\aa}stad's construction is a doubly exponential function. The equivalent between {\sf MaxCov}\xspace and $3$-\mbox{\sf SAT}\xspace holds only when $\|\Sigma_U\|+\|\Sigma_V\|$ is constant (or at most $\log\log (\|V\|+\|U\|)$). \qed \iffalse \section{Hardness of Approximation with low soundness or completeness} \label{app:poly-time-hardness} We sketch the proofs for the claimed results. \subsection{Cliques} Assume for contradiction that there is an algorithm ${\mathbb A}$ that distinguishes between $\mbox{\sf Clique}\xspace(H) \geq \|V(H)\|^{1/f(\|V(H)\|)}$ and $\mbox{\sf Clique}\xspace(H) < f(\|V(H)\|)$ in polynomial time, i.e. in time $\|V(H)\|^D$ for some constant $D$. By following a sequence of reduction from an $n$-variable 3SAT formula $\phi$ to a label cover instance $\Gamma= (U,V,\Sigma_U, \Sigma_V, \Pi)$ and a clique instance $H_{\Gamma}$, we see that the size of the graph $H$ is: \[N= \|V(H)\| = \|\Sigma_U\| \|U\| = 2^{O(n/r \epsilon)} k \] where $\epsilon$ is the constant as postulated by gap-ETH and the parameters $k$ and $r$ are the completeness and soundness parameters respectively. First, choose parameter $r$ such that $N^D < 2^{\delta_{\sf gap} n}$ and $k$ is chosen as $2^{O(n/r \epsilon)}$. For sufficiently large $n$, we have that $f(\|V(H)\|) >> r$, and therefore the $\mbox{\sf Clique}\xspace(H) \geq 2^{n/f(\|V(H)\|)}$ in the completeness case and $\mbox{\sf Clique}\xspace(H) < f(\|V(H)\|)$ in the soundness case. Running algorithm ${\mathbb A}$ on such instance $H$ would distinguish between these two cases in time $N^D < 2^{\delta_{\sf gap} n}$, for all these sufficiently large $n$. This would break gap-ETH. \subsection{Set Cover} \fi \section{On Gap-ETH} \label{app:gap-eth} While Gap-ETH may sound like a very strong assumption, as pointed out in~\cite{Dinur16, ManR16}, there are a few evidences suggesting that the conjecture may indeed be true: \begin{itemize} \item In a simplified and slightly inaccurate manner, the PCP theorem~\cite{AroraS98,AroraLMSS98} can be viewed as a polynomial time reduction that takes in a 3-CNF formula $\Phi$ and produces another 3-CNF formula $\Phi'$ such that, if $\Phi$ is satisfiable, then $\Phi'$ is satisfiable, and, if $\Phi$ is unsatisfiable, $\Phi'$ is not only unsatisfiable but also not even $0.99$-satisfiable. By now, it is know that the size of $\Phi'$ can be made size small as $n \operatorname{polylog}(n)$ where $n$ is the size of $\Phi$~\cite{Dinur07}. This means that, assuming ETH, Gap-3\mbox{\sf SAT}\xspace cannot be solved in $2^{o(n/\operatorname{polylog} n)}$ time, which is only a factor of $\operatorname{polylog} n$ off from what we need in Gap-ETH. Indeed, as stated earlier, if a linear-size PCP, one in which $\Phi'$ is of size linear in $n$, exists then Gap-ETH would follow from ETH. \item No subexponential-time algorithm is known even for the following (easier) problem, which is sometimes referred to as \emph{refutation of random 3-\mbox{\sf SAT}\xspace}: for a constant density parameter $\Delta$, given a 3-CNF formula $\Phi$ with $n$ variables and $m = \Delta n$ clauses, devise an algorithm that outputs either SAT or UNSAT such that the following two conditions are satisfied: \begin{itemize} \item If $\Phi$ is satisfiable, the algorithm always output SAT. \item Over all possible 3-CNF formulae $\Phi$ with $n$ clauses and $m$ variables, the algorihtm outputs UNSAT on at least 0.5 fraction of them. \end{itemize} Note here that, when $\Delta$ is a sufficiently large constant (say 1000), a random 3-CNF formula is, with high probability, not only unsatisfiable but also not even $0.9$-satisfiable. Hence, if Gap-ETH fails, then the algorithm that refutes Gap-ETH will also be a subexponential time algorithm for refutation of random 3-\mbox{\sf SAT}\xspace with density $\Delta$. Refutation of random 3-\mbox{\sf SAT}\xspace, and more generally random CSPs, is an important question that has connections to many other fields, including hardness of approximation, proof complexity, cryptography and learning theory. We refer the reader to~\cite{AOW15} for a more comprehensive review of known results about the problem and its applications in various areas. Despite being intensely studied for almost three decades, no subexponential-time algorithm is known for the above regime of parameter. In fact, it is known that the Sum-of-Squares hierarchies cannot refute random 3-\mbox{\sf SAT}\xspace with constant density in subexponential time~\cite{Gri01,Sch08}. Given how powerful SDP~\cite{Rag08}, and more specifically Sum-of-Squares~\cite{LRS15}, are for solving (and approximating) CSPs, this suggests that refutation of random 3-\mbox{\sf SAT}\xspace with constant density, and hence Gap-3\mbox{\sf SAT}\xspace, may indeed be exponentially hard or, at the very least, beyond our current techniques. \item Dinur speculated that Gap-ETH might follow as a consequence of some cryptographic assumption~\cite{Dinur16}. This was recently confirmed by Applebaum~\cite{App17} who showed that Gap-ETH follows from an existence of any exponentially-hard locally-computable one-way function. In fact, he proved an even stronger result that Gap-ETH follows from ETH for some CSPs that satisfy certain ``smoothness'' properties. \end{itemize} Lastly, we note that the assumption $m = O(n)$ made in the conjecture can be made without loss of generality. As pointed out in both~\cite{Dinur16} and~\cite{ManR16}, this follows from the fact that, given a 3-\mbox{\sf SAT}\xspace formula $\phi$ with $m$ clauses and $n$ variables, if we create another 3-\mbox{\sf SAT}\xspace formula $\phi'$ by randomly selected $m' = \Delta n$ clauses, then, with high probability, $\|\mbox{\sf SAT}\xspace(\phi)/m - \mbox{\sf SAT}\xspace(\phi')/m'\| \leq O(1/\Delta)$. \end{document}	arXiv
Published: 22nd June 2021 DOI: 10.4204/EPTCS.335 EPTCS 335 Proceedings Eighteenth Conference on Theoretical Aspects of Rationality and Knowledge Beijing, China, June 25-27, 2021 Edited by: Joseph Halpern and Andrés Perea Andrés Perea 1 A Recursive Measure of Voting Power that Satisfies Reasonable Postulates Arash Abizadeh and Adrian Vetta 3 Well-Founded Extensive Games with Perfect Information Krzysztof R. Apt and Sunil Simon 7 Uncertainty-Based Semantics for Multi-Agent Knowing How Logics Carlos Areces, Raul Fervari, Andrés R. Saravia and Fernando R. Velázquez-Quesada 23 Revisiting Epistemic Logic with Names Marta Bílková, Zoé Christoff and Olivier Roy 39 Language-based Decisions Adam Bjorndahl and Joseph Y. Halpern 55 An Awareness Epistemic Framework for Belief, Argumentation and Their Dynamics Alfredo Burrieza and Antonio Yuste-Ginel 69 Local Dominance Emiliano Catonini and Jingyi Xue 85 Collective Argumentation: The Case of Aggregating Support-Relations of Bipolar Argumentation Frameworks Weiwei Chen 87 De Re Updates Michael Cohen, Wen Tang and Yanjing Wang 103 Dynamically Rational Judgment Aggregation: A Summary Franz Dietrich and Christian List 119 Deliberation and Epistemic Democracy Huihui Ding and Marcus Pivato 127 No Finite Model Property for Logics of Quantified Announcements Hans van Ditmarsch, Tim French and Rustam Galimullin 129 Krisztina Fruzsa, Roman Kuznets and Ulrich Schmid 139 Are the Players in an Interactive Belief Model Meta-certain of the Model Itself? Satoshi Fukuda 155 Knowledge from Probability Jeremy Goodman and Bernhard Salow 171 Belief Inducibility and Informativeness P. Jean-Jacques Herings, Dominik Karos and Toygar Kerman 187 Measuring Violations of Positive Involvement in Voting Wesley H. Holliday and Eric Pacuit 189 Algorithmic Randomness, Bayesian Convergence and Merging Simon Huttegger, Sean Walsh and Francesca Zaffora Blando 211 Game-Theoretic Models of Moral and Other-Regarding Agents (extended abstract) Gabriel Istrate 213 Understanding Transfinite Elimination of Non-Best Replies Stephan Jagau 229 Persuading Communicating Voters Toygar Kerman and Anastas P. Tenev 231 Knowing How to Plan Yanjun Li and Yanjing Wang 233 Probabilistic Stability and Statistical Learning Krzysztof Mierzewski 249 Attainable Knowledge and Omniscience Pavel Naumov and Jia Tao 251 Failures of Contingent Thinking Evan Piermont and Peio Zuazo-Garin 267 Reasoning about Emergence of Collective Memory R. Ramanujam 269 A Deontic Stit Logic Based on Beliefs and Expected Utility Aldo Iván Ramírez Abarca and Jan Broersen 281 Epistemic Modality and Coordination under Uncertainty Giorgio Sbardolini 295 Communication Pattern Models: An Extension of Action Models for Dynamic-Network Distributed Systems Diego A. Velázquez, Armando Castañeda and David A. Rosenblueth 307 These proceedings contain the papers that have been accepted for presentation at the Eighteenth Conference on Theoretical Aspects of Rationality and Knowledge (TARK XVIII). The conference took place from June 25 to June 27, 2021, at Tsinghua University, Beijing, China. However, due to the COVID-19 pandemic, the conference was offered completely online. As is to be expected from TARK, these proceedings offer a highly interdisciplinary collection of papers, including areas such as logic, computer science, philosophy, economics, game theory, decision theory and social welfare. The topics covered by the papers include semantic models for knowledge and belief, epistemic logic, computational social choice, rationality in games and decision problems, and foundations of multi-agent systems. I wish to thank the team of local organizers, chaired by Fenrong Liu, to make this conference possible under these extraordinary circumstances. Another word of gratitude goes to the members of the program committee, not only for reviewing the submissions, but also for their valuable input concerning other aspects of the conference, such as the invited speakers and the precise format of the conference. The members of the program committee are: Christian Bach, Adam Bjorndahl, Giacomo Bonanno, Emiliano Catonini, Franz Dietrich, Davide Grossi, Joseph Halpern (conference chair), Jérôme Lang, Fenrong Liu (local organizing chair), Silvia Milano, Yoram Moses, Eric Pacuit, Andrés Perea (program committee chair), Olivier Roy, Elias Tsakas, Paolo Turrini, Rineke Verbrugge and Kevin Zollman. I also wish to thank the invited speakers at this conference: Ariel Procaccia, Burkhard Schipper, Sonja Smets and Katie Steele. On the practical side, the conference and the proceedings have benefitted a lot from the EasyChair platform, and the EPTCS - system. I thank Rob van Glabbeek, editor of EPTCS, for his help during the process of setting up these proceedings. Last but not least, I am very grateful to Joseph Halpern (conference chair) and Fenrong Liu (local organizing chair) who have done so much for the organization of TARK XVIII. It was an absolute pleasure to work with you, and I am sorry for the many E-mails you had to digest from me. I sincerly hope that these proceedings will be a source of inspiration for your research, and that you will enjoy reading the papers. Andrés Perea Program Committee Chair TARK XVIII Maastricht, June 2021 Arash Abizadeh (Department of Political Science, McGill University, Montreal, Canada) Adrian Vetta (Department of Mathematics and Statistics, and School of Computer Science, McGill University, Montreal, Canada) We design a recursive measure of voting power based upon partial voting efficacy as well as full voting efficacy. In contrast, classical indices and measures of voting power incorporate only full voting efficacy. We motivate our design by representing voting games using a division lattice and via the notion of random walks in stochastic processes, and show the viability of our recursive measure by proving it satisfies a plethora of postulates that any reasonable voting measure should satisfy. There have been two approaches to justifying measures of voting power. The first is the axiomatic approach, which seeks to identify a set of reasonable axioms that uniquely pick out a single measure of voting power. To date this justificatory approach has proved a failure: while many have succeeded in providing axiomatic characterizations of various measures, no one has succeeded in doing so for a set of axioms all of which are independently justified, i.e., in showing why it would be reasonable to expect a measure of voting power to satisfy the entire set of axioms that uniquely pick out a proposed measure. For example, Dubey (1975) and Dubey and Shapley (1979) have characterized the classic Shapely-Shubik index ($SS$) and Penrose-Banzhaf measure ($PB$) as uniquely satisfying a distinct set of axioms, respectively, but several of the axioms lack proper justification (Straffin 1982: 292-296; Felsenthal and Machover 1998: 194-195; Laruelle and Valenciano 2001). The second, two-pronged approach is more modest and involves combining two prongs of justification. The first prong is to motivate a proposed measure on conceptual grounds, showing the sense in which it captures the intuitive meaning of what voting power is. With this conceptual justification in place, the second prong of justification then requires showing that the measure satisfies a set of reasonable postulates. For the more modest approach, both prongs of justification are necessary, and the satisfaction of reasonable postulates serves, not to pick out a uniquely reasonable measure, but to rule out unreasonable measures. The first prong of justification has been typically carried out in probabilistic terms. For example, the a priori Penrose-Banzhaf measure equates a player's voting power, in a given voting structure, with the proportion of logically possible divisions or complete vote configurations in which the player is (fully) decisive for the division outcome, i.e., in which the player has an alternative voting strategy such that, if it were to choose that alternative instead, the outcome would be different (holding all other players' votes constant). The standard interpretation is that the a priori $PB$ measure represents the probability a player will be decisive under the assumptions of equiprobable voting (the probability a player votes for an alternative is equal to the probability it votes for any other) and voting independence (votes are not correlated), which together imply equiprobable divisions (the probability of each division is equal) (Felsenthal and Machover 1998: 37-38). However, measures of voting power based exclusively on the ex ante probability of decisiveness suffer from a crucial conceptual flaw. The motivation for basing a measure of voting power on this notion is that decisiveness is supposed to formalize the idea of a player making a difference to the outcome. To equate a player's voting power with the player's ex ante probability of being decisive is to assume that if any particular division were hypothetically to occur, then the player would have efficaciously exercised power to help produce the outcome ex post if and only if that player would have been decisive or necessary for the outcome. Yet this assumption is false: sometimes, as in causally overdetermined outcomes, an actor has efficaciously exercised its power to effect an outcome ex post, and, through the exercise of that power, made a causal contribution to the outcome, even though the actor's contribution was not decisive to it. More specifically, reducing voting power to the ex ante probability of being decisive fails to take into account players' partial causal efficacy in producing outcomes ex post. In this paper, we design a Recursive Measure ($RM$) of voting power that remedies this shortcoming, by taking into account partial efficacy or degrees of causal efficacy. A full conceptual justification for $RM$ -- i.e., the first prong of justification on the more modest approach -- is given in Abizadeh (working paper). $RM$ represents, not the probability a player will be decisive for the division outcome (the probability the player will be fully causally efficacious in bringing it about) but, rather, the player's expected efficacy, that is, the probability the player will make a causal contribution to the outcome weighted by the degree of causal efficacy. Whereas decisiveness measures such as $PB$ solely track full efficacy, $RM$ tracks partial efficacy as well. Our task in this paper is to furnish the second prong of justification. In particular, we take it that any reasonable measure of a priori voting power $\pi$ should satisfy, for simple voting games $\mathcal{G}$ with equiprobable divisions, where $[n]$ is the set of all voters and a dummy is a voter not decisive in any division, the following postulates: Iso-invariance postulate: For iso-invariant voting games $\mathcal{G}$ and $\hat{\mathcal{G}}$: $\pi_i=\hat{\pi}_i$ for any player $i$. Dummy postulates: For any game $\hat{\mathcal{G}}$ formed by the addition of a dummy voter to $\mathcal{G}$: if $i$ is a dummy voter, then $\pi_i=0$; $\pi_i=0$ only if $i$ is a dummy voter; and if $i$ is a non-dummy voter, then $\pi_i=\hat{\pi}_i$. Dominance postulate: For any subset $S\subseteq [n]$ with $i,j\notin S$: $\pi_j\ge \pi_i$ whenever $j$ weakly dominates $i$, and $\pi_j> \pi_i$ whenever $j$ strictly dominates $i$ (where $j$ weakly dominates $i$ if whenever $S\cup i$ vote yes and the outcome is yes, then if $S\cup j$ vote yes the outcome is yes; and $i$ strictly dominates $j$ if the former weakly dominates the latter but not vice versa). Donation postulate: For any game $\hat{\mathcal{G}}$ formed from $\mathcal{G}$ by player $j$ transferring its vote to player $i$: $\hat{\pi}_i \ge \max (\pi_i, \pi_j )$. Bloc postulate: For any game $\hat{\mathcal{G}}$ formed from $\mathcal{G}$ by player $i$ annexing $i$'s vote to form a bloc $I=\{i,j\}$: $\hat{\pi}_I \ge \max (\pi_i, \pi_j )$. Quarrel postulate: For any game $\hat{\mathcal{G}}$ formed from $\mathcal{G}$ by inducing a symmetric, weak, monotonic quarrel between $i$ and $j$: $\hat{\pi}_i\le \pi_i$ and $\hat{\pi}_j\le \pi_j$. Added blocker postulate: For any game $\mathcal{G}^Y$ resulting from $\mathcal{G}$ by adding an added yes-blocker, and $\mathcal{G}^N$ resulting from adding an added no-blocker: $\frac{\pi^+_i(\mathcal{G})}{\pi^+_j(\mathcal{G})} = \frac{\pi^+_i(\mathcal{G}^Y)}{\pi^+_j(\mathcal{G}^Y)}$, and $\frac{\pi^-_i(\mathcal{G})}{\pi^-_j(\mathcal{G})} = \frac{\pi^-_i(\mathcal{G}^N)}{\pi^-_j(\mathcal{G}^N)}$ (where $\pi^+$ is a player's yes-voting power, based solely on divisions in which it votes yes, and $\pi^-$ is a player's no-voting power, based solely on divisions in which it votes no. In the full paper, we explain the intuitive justification for and fully specify each of these voting-power postulates, and then prove that $RM$ satisfies them for a priori power in simple voting games. We prove these postulates by introducing a new way of representing voting games using a division lattice, and show that previous formulations of some of these postulates require revision. A full version of the paper can be found at: http://arxiv.org/abs/2105.03006 Abizadeh, A. (Working paper). A Recursive Measure of Voting Power with Partial Decisiveness or Efficacy. Dubey, P. (1975). On the Uniqueness of the Shapley Value. International Journal of Game Theory, 4(3), 131-139. doi:10.1007/BF01780630 Dubey, P., and Shapley, L.S. (1979). Mathematical Properties of the Banzhaf Power Index. Mathematics of Operations Research, 4(2), 99-131. doi:10.1287/moor.4.2.99 Felsenthal, D., and Machover, M. (1998) The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, Edward Elgar. doi:10.4337/9781840647761 Laruelle, A., and Valenciano, F. (2001). Shapley-Shubik and Banzhaf Indices Revisited. Mathematics of Operations Research, 26(1), 89-104. doi:10.1287/moor.26.1.89.10589 Straffin, P.D. (1982). Power Indices in Politics. In S.J. Brams, W.F. Lucas, and P.D. Straffin (Eds.), Political and Related Models, 256-321. New York: Springer. doi:10.1007/978-1-4612-5430-0_11 Emiliano Catonini (HSE University Moscow) Jingyi Xue (Singapore Management University) We present a local notion of dominance that speaks to the true choice problems among actions in a game tree and does not rely on global planning. When we do not restrict the ability of the players to do contingent reasoning, a reduced strategy is weakly dominant if and only if it prescribes a locally dominant action at every decision node, therefore any dynamic decomposition of a direct mechanism that preserves strategy-proofness is robust to the lack of global planning. Under a form of wishful thinking, we also show that strategy-proofness is robust to the lack of forward-planning. Moreover, from our local perspective, we can identify rough forms of contingent reasoning that are particularly natural. We construct a dynamic game that implements the Top Trading Cycles allocation under a minimal form of contingent reasoning, related to independence of irrelevant alternatives. Franz Dietrich (Paris School of Economics & CNRS) Christian List (LMU Munich) Judgment aggregation theory traditionally aims for collective judgments that are rational. So far, rationality has been understood in purely static terms: as coherence of judgments at a given time, where 'coherence' could for instance mean consistency, or completeness, or deductive closure, or combinations thereof. By contrast, this paper, which summarises results from Dietrich and List (2021), asks the novel question of whether collective judgments can be dynamically rational: whether they can respond well to new information, i.e., change rationally when information is learnt by everyone. Formally, we call a judgment aggregation rule dynamically rational with respect to a given revision operator if, whenever all individuals revise their judgments in light of some information (a proposition), then the new aggregate judgments are the old ones revised in light of this information. In short, aggregation and revision commute. A general impossibility theorem holds: as long as the propositions on the agenda are sufficiently interconnected, no judgment aggregation rule with standard properties is dynamically rational with respect to any revision operator satisfying mild conditions (familiar from belief revision theory). The theorem is a counterpart for dynamic rationality of known impossibility theorems for static rationality. Relaxation of the theorem's conditions opens the door to interesting aggregation rules generating dynamically rational judgments, including certain premise-based rules, as we briefly discuss (see Dietrich and List 2020 for details). Suppose a group of individuals – say, a committee, expert panel, multi-member court, or other decision-making body – makes collective judgments on some propositions by aggregating its members' individual judgments on those propositions. And now suppose some new information – in the form of the truth of some proposition – is learnt. All individuals rationally revise their judgments. Aggregating the new individual judgments yields new collective judgments. If the group is to be a rational agent, then it should incorporate new information rationally, and so the new aggregate judgments should coincide with the old ones revised in light of the information. Technically, this means that the operations of aggregation and revision commute: aggregating judgments and then revising the result yields the same as revising individual judgments and then aggregating. In this paper, we investigate whether we can find reasonable aggregation rules that enable a group to achieve such dynamic rationality: aggregation rules which commute with reasonable revision methods. Surprisingly, this question has not been studied in the judgment-aggregation framework, where judgments are binary verdicts on some propositions: "yes"/"no", "true"/"false", "accept" /"reject". (On judgment-aggregation theory, see List and Pettit 2002, Dietrich and List 2007, Nehring and Puppe 2010, Dokow and Holzman 2010, List and Puppe 2009.) The focus in judgment-aggregation theory has generally been on static rationality, namely on whether properties such as consistency, completeness, or deductive closure are preserved when individual judgments are aggregated into collective ones at a given point in time.1 By contrast, the question of dynamic rationality has received much attention in the distinct setting of probability aggregation, where judgments aren't binary but take the form of subjective probability assignments to the elements of some algebra. In that context, a mix of possibility and impossibility results has been obtained (e.g., Madansky 1964, Genest 1984, Genest et al. 1986, Dietrich 2010, 2019, Russell et al. 2015). These show that some familiar methods of aggregation – notably, the arithmetic averaging of probabilities – fail to commute with belief revision, while other methods – particularly geometric averaging – do commute with revision. An investigation of the parallel question in the case of binary judgments is therefore overdue. We present a negative result: for a large class of familiar judgment aggregation rules, dynamic rationality is unachievable relative to a large class of reasonable judgment revision methods. However, if we relax some of our main theorem's conditions on the aggregation rule, dynamically rational aggregation becomes possible. In particular, "premise-based" aggregation can be dynamically rational relative to certain "premise-based" revision methods. This extended abstract focuses on the impossibility finding, for reasons of space. Possibilities are discussed in Dietrich and List (2021), which also contains all proofs. The formal setup We begin with the basic setup from judgment-aggregation theory (following List and Pettit 2002 and Dietrich 2007). We assume that there is a set of individuals who hold judgments on some set of propositions, and we are looking for a method of aggregating these judgments into resulting collective judgments. The key elements of this setup are the following: Individuals. These are represented by a finite and non-empty set N. Its members are labelled 1, 2, ..., n. We assume n ≥ 2. Propositions. These are represented in formal logic. For our purposes, a thin notion of "logic" will suffice. Specifically, a logic, L, is a non-empty set of formal objects called "propositions", which is endowed with two things: a negation operator, denoted ¬, so that, for every proposition p in L there is also its negation ¬p in L; and a well-behaved notion of consistency, which specifies, for each set of propositions S ⊆ L, whether S is consistent or inconsistent.2 Standard propositional, predicate, modal, and conditional logics all fall under this definition, as do Boolean algebras.3 A proposition p is contradictory if {p} is inconsistent, tautological if {¬p} is inconsistent, and contingent if p is non-contradictory and non-tautological. Agenda. The agenda is the set of those propositions from L on which judgments are to be made. Formally, this is a finite non-empty subset X of L which can be partitioned into proposition-negation pairs {p, ¬p}, abbreviated { ± p}. Sometimes it is useful to make this partition explicit. We write Z to denote the set of these proposition-negation pairs of X. The elements of Z can be interpreted as the binary issues under consideration. Then the agenda X is their disjoint union, formally X = ∪Z ∈ ZnZ. Throughout this paper, we assume that double-negations cancel out in agenda propositions.4 Our focus will be on agendas satisfying a non-triviality condition. To define it, call a set of propositions minimal inconsistent if it is inconsistent but all its proper subsets are consistent. Proposition-negation pairs of the form {p, ¬p} (with p contingent) are minimal inconsistent, and so are sets of the form {p, q, ¬(p ∧ q)} (with p and q contingent), where "∧" stands for logical conjunction ("and"). We call an agenda non-simple if it has at least one minimal inconsistent subset of size greater than two. An example of a non-simple agenda is the set X = { ± p, ± (p → q), ± q}, where p might be the proposition "Current atmospheric CO2 is above 407 ppm", p → q might be the proposition "If current atmospheric CO2 is above 407 ppm, then the Arctic iceshield will melt by 2050", and q might be the proposition "The Arctic iceshield will melt by 2050". The conditional p → q can be formalized in standard propositional logic or in a suitable logic for conditionals. A three-member minimal inconsistent subset of this agenda is {p, p → q, ¬q}. Judgments. Each individual's (and subsequently the group's) judgments on the given propositions are represented by a judgment set, which is a subset J ⊆ X, consisting of all those propositions from X that its bearer "accepts" (e.g., affirms or judges to be true). A judgment set J is complete if it contains a member of each proposition-negation pair from X, consistent if it is a consistent set in the sense of the given logic, and classically rational if it has both of these properties. We write J to denote the set of all classically rational judgment sets on the agenda X. A list of judgment sets (J1, ..., Jn) across the individuals in N is called a profile (of individual judgment sets). Aggregation rule. A (judgment) aggregation rule is a function, F, which maps each profile (J1, ..., Jn) in some domain D of admissible profiles (often D = Jn) to a collective judgment set J = F(J1, ..., Jn). A standard example is majority rule, which is defined as follows: for each (J1, ..., Jn) ∈ Jn, F(J1, ..., Jn) = {p ∈ X : \|{i:p∈Ji}\| > n/2}. A typical research question in judgment aggregation theory is whether we can find aggregation rules that satisfy certain requirements of democratic responsiveness to the individual judgments and collective rationality. Judgment revision The idea we wish to capture is that whenever any individual (or subsequently the group) learns some new information, in the form of the truth of some proposition, this individual (or the group) must incorporate the learnt information in the judgments held – an idea familiar from belief revision theory in the tradition of Alchourrón, Gärdenfors and Makinson (1985) (see also Rott 2001 and Peppas 2008). Our central concept is that of a judgment revision operator. This is a function which assigns to any pair (J, p) of an initial judgment set J ⊆ X and a learnt proposition p ∈ X a new judgment set J\|p, the revised judgment set, given p. Formally, the revision operator is any function from 2X × X to 2X. We call it regular if it satisfies the following two minimal conditions: it is successful, i.e., p ∈ J\|p for any pair (J, p) ("accept what you learn"), and it is conservative, i.e., J\|p = J for any pair (J, p) such that p ∈ J ("no news, no change"). We further call a revision operator rationality-preserving if whenever J ∈ J, we have J\|p ∈ J for all non-contradictory propositions p ∈ X. These definitions are well-illustrated by the class of distance-based revision operators, familiar from belief revision theory. Such operators require that when a judgment set is revised in light of some new information, the post-revision judgments remain as "close" as possible to the pre-revision judgments, subject to the constraint that the learnt information be incorporated and no inconsistencies be introduced. Different distance-based operators spell out the notion of "closeness" in different ways (different metrics have been introduced in the area of judgment aggregation by Konieczny and Pino-Pérez 2002 and Pigozzi 2006). Can aggregation and revision commute? We are now ready to turn to this paper's question. As noted, we would ideally want any decision-making group to employ a judgment aggregation rule and a revision operator that generate the same collective judgments irrespective of whether revision takes place before or after aggregation. This requirement (an analogue of the classic "external Bayesianity" condition in probability aggregation theory, as in Madansky 1964, Genest 1984, and Genest et al. 1986) is captured by the following condition on the aggregation rule F and the revision operator \|: Dynamic rationality. For any profile (J1, ..., Jn) in the domain of F and any learnt proposition p ∈ X where the revised profile (J1\|p, ..., Jn\|p) is also in the domain of F, F(J1\|p, ..., Jn\|p) = F(J1, ..., Jn)\|p. To see that this condition is surprisingly hard to satisfy, consider an example. Suppose a three-member group is making judgments on the agenda X = { ± p, ± (p → q), ± q}, where p → q is understood as a subjunctive conditional. That is, apart from the subsets of X that include a proposition-negation pair, the only inconsistent subset of X is {p, p → q, ¬q}.5 Suppose, further, members' initial judgments and the resulting majority judgments are as follows: Individual 1: { ¬p, ¬(p → q), q} Individual 2: { ¬p, p → q, ¬q} Individual 3: { ¬p, ¬(p → q), ¬q} Majority: { ¬p, ¬(p → q), ¬q} Assume the revision operator is based on the Hamming distance, with some tie-breaking provision such that, in the case of a tie, one is more ready to change one's judgment on p or p → q (which represent "premises") than on q (which represents a "conclusion"). If the individuals learn the truth of p and revise their judgments, they arrive at the following post-revision judgments: Individual 1: { p, ¬(p → q), q} Individual 2: { p, p → q, q} Individual 3: { p, ¬(p → q), ¬q} Majority: { p, ¬(p → q), q} Crucially, the post-information group judgment set, {p, ¬(p → q), q}, differs from the revision in light of p of the pre-information group judgment set, because {¬p, ¬(p → q), ¬q}\|p = {p, ¬(p → q), ¬q}. That is, the group replaces ¬q with q in its judgment set, although learning p did not force the group to revise its position on q (recall that {p, ¬(p → q), ¬q} is perfectly consistent, given that → is a subjunctive conditional). Thus the group's (majority) judgment set does not evolve rationally. At first sight, one might think that this problem is just an artifact of majority rule or our specific distance-based revision operator, or that it is somehow unique to our example. However, the following formal result – a simplified ('anonymous') version of our impossibility theorem – shows that the problem is more general. Define a uniform quota rule, with acceptance threshold m ∈ {1, ..., n}, as the aggregation rule with domain Jn such that, for each (J1, ..., Jn) ∈ Jn, F(J1, ..., Jn) = {p ∈ X : \|{i : p ∈ Ji} ≥ m}. Majority rule is a special case of a uniform quota rule, namely the one where m is the smallest integer greater than n/2. We have: Theorem 1: If the agenda X is non-simple, then no uniform quota rule whose threshold is not the unanimity threshold n is dynamically rational with respect to any regular rationality-preserving revision operator. In short, replacing majority rule with some other uniform quota rule with threshold less than n wouldn't solve our problem of dynamic irrationality, and neither would replacing our distance-based revision operator with some other regular rationality-preserving revision operator. In fact, the problem generalizes further, as shown in the next section. A general impossibility theorem We will now abstract away from the details of any particular aggregation rule, and suppose instead we are looking for an aggregation rule F that satisfies the following general conditions: Universal domain: The domain of admissible inputs to the aggregation rule F is the set of all classically rational profiles, i.e., D = Jn. Non-imposition: F does not always deliver the same antecedently fixed output judgment set J, irrespective of the individual inputs, i.e., F is not a constant function. Monotonicity: Additional individual support for an accepted proposition does not overturn the proposition's acceptance, i.e., for any profile (J1, ..., Jn) ∈ D and any proposition p ∈ F(J1, ..., Jn), if any Ji not containing p is replaced by some Ji′ containing p and the modified profile (J1, ..., Ji′, ..., Jn) remains in D, then p ∈ F(J1, ..., Ji′, ..., Jn). Non-oligarchy: There is no non-empty set of individuals M ⊆ N (a set of "oligarchs") such that, for every profile (J1, ..., Jn) ∈ D, F(J1, ..., Jn) = ∩i ∈ MJi. Systematicity: The collective judgment on each proposition is determined fully and neutrally by individual judgments on that proposition. Formally, for any propositions p, p′ ∈ X and any profiles (J1, ..., Jn), (J1′, ..., Jn′) ∈ D, if, for all i ∈ N, p ∈ Ji ⇔ p′ ∈ Ji′, then p ∈ F(J1, ..., Jn) ⇔ p′ ∈ F(J1′, ..., Jn′). Why are these conditions initially plausible? The reason is that, for each of them, a violation would entail a cost. Violating universal domain would mean that the aggregation rule is not fully robust to pluralism in its inputs; it would be undefined for some classically rational judgment profiles. Violating non-imposition would mean that the collective judgments are totally unresponsive to the individual judgments, which is completely undemocratic. Violating monotonicity could make the aggregation rule erratic in some respect: an individual could come to accept a particular collectively accepted proposition and thereby overturn its acceptance. Violating non-oligarchy would mean two things. First, the collective judgments would depend only on the judgments of the "oligarchs", which is undemocratic (unless M = N); and second, the collective judgments would be incomplete with respect to any binary issue on which there is the slightest disagreement among the oligarchs, which would lead to widespread indecision (except when M is singleton, so that the rule is dictatorial). Violating systematicity, finally, would mean that the collective judgment on each proposition is no longer determined as a proposition-independent function of individual judgments on that proposition. It may then either depend on individual judgments on other propositions too (a lack of propositionwise independence), or the pattern of dependence may vary from proposition to proposition (a lack of neutrality ). Systematicity – the conjunction of propositionwise independence and neutrality – is the most controversial condition among the five. But it is worth noting that it is satisfied by majority rule and all uniform quota rules. Indeed, majority rule and uniform quota rules (except the unanimity rule) satisfy all five conditions. Our main theorem shows that, for non-simple agendas, the present five conditions are incompatible with dynamic rationality: Theorem 2: If the agenda X is non-simple, then no aggregation rule satisfying universal domain, non-imposition, monotonicity, non-oligarchy, and systematicity is dynamically rational with respect to any regular rationality-preserving revision operator. Interestingly, Theorem 2 does not impose any condition of static rationality. The theorem does not require that collective judgment sets are consistent or complete or deductively closed. The impossibility of dynamic inconsistency is thus independent of classic impossibilities of static rationality. In fact, Theorem 2 would continue to hold if its condition of dynamic rationality were replaced by static rationality in the form of consistency and completeness of collective judgment sets. By Theorem 2, the problem identified by Theorem 1 is not restricted to uniform quota rules, but extends to all aggregation rules satisfying our conditions. Moreover, since practically all non-trivial agendas are non-simple, the impossibility applies very widely. The natural follow-up question is that of whether any of the conditions in the theorem is redundant, i.e., could be dropped, and if not what sort of dynamically rational aggregation rules become possible after dropping any of these conditions. This question goes beyond the scope of this summary and is treated in Dietrich and List (2021). Four remarks should however be given: Firstly, none of the theorem's conditions on the aggregation rule, the revision operator, or the agenda is redundant. That is, whenever we drop the agenda condition (non-simplicity) or any one of the aggregation conditions (universal domain, non-imposition, monotonicity, non-oligarchy, and systematicity) or any of the revision conditions (successfulness, conservativeness, and rationality preservation), there exist dynamically rational aggregation rules such that the remaining conditions hold. Secondly, abandoning exactly one condition on the aggregation rule leads to rather degenerate dynamically rational possibilities, in the form of 'peculiar' aggregation rules and/or revision operators (with the exception of universal domain, whose relaxation allows for interesting dynamically rational possibilities). One of the conditions on aggregation seems very strong: systematicity. An important difference between static and dynamic rationality is that dropping systematicity or even independece makes it easy (indeed, too easy) to satisfy static rationality – for instance by using distance-based rules or prioritarian rules or scoring rules – whereas dynamic rationality remains hard to achieve without systematicity, as illustrated by the degenerate nature of the non-systematic escape route constructed in Dietrich and List (2021). It thus seems inappropriate to blame systematicity for being the main culprit for the impossibility of dynamic rationality. Thirdly, let us give examples of dynamically rational aggregation rules that become possible if we give up any one of the three conditions on the revision operator while preserving all other conditions on revision or aggregation. Non-successful revision. Consider the constant revision operator, defined by J\|p = J for all (J, p). This operator is only conservative and rationality-preserving. All aggregation rules are trivially dynamically rational with respect to it. Non-conservative revision. For each proposition p ∈ X, fix a judgment set Jp which contains p and moreover is rational (i.e., in J) as long as p is non-contradictory. Consider the revision operator given by J\|p = Jp for all (J, p). This operator is only successful and rationality-preserving. As one can show, every unanimity-preserving aggregation rule is dynamically rational with respect to it. Non-rationality-preserving revision. Consider a revision operator such that J\|p is identical to J whenever J does not contain p, and otherwise is some irrational judgment set containing p. This operator is only conservative and successful. As one can show, every aggregation rule satisfying universal domain and propositionwise unanimity preservation is dynamically rational with respect to it. Here, propositionwise unanimity-preservation means that, for all profiles (J1, ..., Jn) in the domain and all propositions p ∈ X, if p ∈ Ji for all i, then p ∈ F(J1, ..., Jn). Finally, on a more positive note, in Dietrich and List (2021) we explore an interesting class of dynamically rational aggregation rules, which simultaneously relax multiple of Theorem 2's conditions on aggregation and revision, notably systematicity. In a nutshell, premise-based aggregation rules are dynamically rational with respect to premise-based revision operators. Presenting these rules goes beyond the scope of this summary. Proofs of theorems and other technical details are given in Dietrich and List (2021). Alchourrón, C. E., Gärdenfors, P., and Makinson, D. (1985): On the logic of theory change: Partial meet contraction and revision functions. Journal of Symbolic Logic 50(2), pp. 510–530. DOI: 10.2307/2274239 Dietrich, F. (2007): A generalised model of judgment aggregation. Social Choice and Welfare 28(4), pp. 529–565. DOI: 10.1007/s00355-006-0187-y Dietrich, F. (2010): Bayesian group belief. Social Choice and Welfare 35(4), pp. 595–626. DOI: 10.1007/s00355-010-0453-x Dietrich, F. (2019): A theory of Bayesian groups. Noûs 53(3), pp. 708–736. DOI: 10.1111/nous.12233 Dietrich, F., and List, C. (2007): Arrow's theorem in judgment aggregation. Social Choice and Welfare 29(1), pp. 19–33. DOI: 10.1007/s00355-006-0196-x Dietrich, F., and List, C. (2021): Dynamically Rational Judgment Aggregation. Working paper, see https://philpapers.org/rec/DIEDRJ Dokow, E., and Holzman, R. (2010): Aggregation of binary evaluations. Journal of Economic Theory 145(2), pp. 495–511. DOI: 10.1016/j.jet.2007.10.004 Genest, C. (1984): A characterization theorem for externally Bayesian groups. Annals of Statistics 12(3), pp. 1100–1105. DOI: 10.1214/aos/1176346726 Genest, C., McConway, K. J., and Schervish, M. J. (1986): Characterization of externally Bayesian pooling operators. Annals of Statistics 14(2), pp. 487–501. DOI: 10.1007/BF02562628 Konieczny, S., and Pino-Pérez, R. (2002): Merging information under constraints: A logical framework. Journal of Logic and Computation 12(5), pp. 773–808. DOI: 10.1093/logcom/12.5.773 List, C. (2011): Group Communication and the Transformation of Judgments: An Impossibility Result. Journal of Political Philosophy 19(1), pp. 1–27. DOI: 10.1111/j.1467-9760.2010.00369.x List, C., and Pettit, P. (2002): Aggregating sets of judgments: An impossibility result. Economics and Philosophy 18(1), pp. 89–110. List, C., and Pettit, P. (2011): Group Agency: The Design, Possibility, and Status of Corporate Agents. Oxford: Oxford University Press. DOI: 10.1093/acprof:oso/9780199591565.001.0001 List, C., and Puppe, C. (2009): Judgment aggregation: A survey. In P. Anand, C. Puppe, and P. Pattanaik, Oxford Handbook of Rational and Social Choice. Oxford: Oxford University Press. DOI: 10.1093/acprof:oso/9780199290420.001.0001 Madansky, A. (1964): Externally Bayesian Groups. Technical Report RM-4141-PR, RAND Corporation. Nehring, K., and Puppe, C. (2010): Abstract Arrovian aggregation. Journal of Economic Theory 145(2), pp. 467–494. DOI: 10.1016/j.jet.2010.01.010 Peppas, P. (2008): Belief Revision. In F. van Harmelen, V. Lifschitz and B. Porter, Handbook of Knowledge Representation, Elsevier, pp. 317-359. Pettit, P. (2006): When to defer to majority testimony – and when not. Analysis 66(3), pp. 179–187. Pigozzi, G. (2006): Belief merging and the discursive dilemma: An argument-based account to paradoxes of judgment aggregation. Synthese 152(2), pp. 285–298. DOI: 10.1007/s11229-006-9063-7 Rott, H. (2001): Change, Choice and Inference: A Study of Belief Revision and Non-monotonic Reasoning. Oxford: Oxford University Press. Russell, J. S., Hawthorne, J., and Buchak, L. (2015): Groupthink. Philosophical Studies 172(5), pp. 1287–1309. DOI: 10.1007/s11098-014-0350-8 The revision of judgments has been investigated only in a different sense in judgment aggregation theory, namely in peer-disagreement contexts, where individuals do not learn a proposition but learn the judgments of others (Pettit 2006, List 2011).↩︎ Well-behavedness is a three-part requirement: (i) any proposition-negation pair {p, ¬p} is inconsistent; (ii) any subset of any consistent set is still consistent; and (iii) the empty set is consistent, and any consistent set S has a consistent superset S′ which contains a member of every proposition-negation pair {p, ¬p}.↩︎ Readers familiar with probability theory could take L to be a Boolean algebra on a non-empty set Ω of possible worlds (e.g., the power set L = 2Ω), with negation defined as set-theoretic complementation and consistency of a set defined as non-empty intersection. The Boolean algebra could also be an abstract rather than set-theoretic Boolean algebra.↩︎ To be precise, henceforth, by the negation of any proposition q ∈ X we shall mean the agenda-internal negation of q, i.e., the opposite proposition in the binary issue {p, ¬p} to which q belongs. This is logically equivalent to the ordinary negation of q and will again be denoted ¬q, for simplicity. This convention ensures that ¬¬q = q.↩︎ This subjunctive understanding of p → q contrasts with the material one, where p → q is understood less realistically as ¬p ∨ q. On the material understanding, the subsets {p, ¬(p → q), q}, {¬p, ¬(p → q), q}, and {¬p, ¬(p → q), ¬q} would also be deemed inconsistent.↩︎ Huihui Ding (CY Cergy Paris University ) Marcus Pivato (CY Cergy Paris University) We study the effects of deliberation on epistemic social choice, in two settings. In the first setting, the group faces a binary epistemic decision analogous to the Condorcet Jury Theorem. In the second setting, group members have probabilistic beliefs arising from their private information, and the group wants to aggregate these beliefs in a way that makes optimal use of this information. During deliberation, each agent discloses private information to persuade the other agents of her current views. But her views may also evolve over time, as she learns from other agents. This process will improve the performance of the group, but only under certain conditions; these involve the nature of the social decision rule, the group size, and also the presence of neutral agents whom the other agents try to persuade. P. Jean-Jacques Herings (Maastricht University) Dominik Karos (Bielefeld University) Toygar Kerman (Maastricht University) We consider a group of receivers who share a common prior on a finite state space and who observe private correlated signals that are contingent on the true state of the world. We show that, while necessary, Bayes plausibility is not sufficient for a distribution over posterior belief vectors to be inducible, and we provide a characterization of inducible distributions. We classify communication strategies as minimal, direct, and language independent, and we show that any inducible distribution can be induced by a language independent communication strategy (LICS). We investigate the role of the different classes of communication strategies for the amount of higher order information that is revealed to receivers. We show that the least informative communication strategy which induces a fixed distribution over posterior belief vectors lies in the relative interior of the set of all language independent communication strategies which induce that distribution. Simon Huttegger (University of California, Irvine) Sean Walsh (University of California, Los Angeles) Francesca Zaffora Blando (Carnegie Mellon University) Convergence-to-the-truth results and merging-of-opinions results are part of the basic toolkit of Bayesian epistemologists. In a nutshell, the former establish that Bayesian agents expect their beliefs to almost surely converge to the truth as the evidence accumulates. The latter, on the other hand, establish that, as they make more and more observations, two Bayesian agents with different subjective priors are guaranteed to almost surely reach inter-subjective agreement, provided that their priors are sufficiently compatible. While in and of themselves significant, convergence to the truth with probability one and merging of opinions with probability one remain somewhat elusive notions. In their classical form, these results do not specify which data streams belong to the probability-one set of sequences on which convergence to the truth or merging of opinions occurs. In particular, they do not reveal whether the data streams that ensure eventual convergence or merging share any property that might explain their conduciveness to successful learning. Thus, a natural question raised by these classical results is whether the kind of data streams that are conducive to convergence and merging for Bayesian agents are uniformly characterizable in an informative way. The results presented in this paper provide an answer to this question. The driving idea behind this work is to approach the phenomena of convergence to the truth and merging of opinions from the perspective of computability theory and, in particular, the theory of algorithmic randomness--a branch of computability theory concerned with characterizing the notion of a sequence displaying no effectively detectable patterns. We restrict attention to Bayesian agents whose subjective priors are computable probability measures and whose goal, in the context of convergence to the truth, is estimating quantities that can be effectively approximated. These are natural restrictions to impose when studying the inductive performance of more realistic, computationally limited learners. Crucially, they also allow to provide a more fine-grained analysis of both convergence to the truth and merging of opinions. Our results establish that, in this setting, the collections of data streams along which convergence and merging occur are indeed uniformly characterizable in an informative way: they are exactly the algorithmically random data streams. Stephan Jagau (IMBS, University of California, Irvine) In auction theory, industrial organization, and other fields of game theory, it is often convenient to let infinite strategy sets stand in for large finite strategy sets. A tacit assumption is that results from infinite games generally translate back to their finite counterparts. Transfinite eliminations of non-best replies pose a radical challenge here, suggesting that common belief in rationality in infinite games strictly refines up to k-fold belief in rationality for all finite k. I provide a general characterization of common belief in rationality for finite and infinite games that fully restores the equivalence to up to k-fold belief in rationality for all finite k. By means of eliminating non-best replies and supporting beliefs, my characterization entirely avoids transfinite eliminations. Hence, rather than revealing new depths of reasoning, transfinite eliminations signal an inadequacy of eliminating non-best replies as a general description of strategic rationality. Toygar Kerman (Department of Microeconomics and Public Economics (MPE), Maastricht University) Anastas P. Tenev (Department of Microeconomics and Public Economics (MPE), Maastricht University) This paper studies a multiple-receiver Bayesian persuasion model, where a sender communicates with receivers who have homogeneous beliefs and aligned preferences. The sender wants to implement a proposal and commits to a communication strategy which sends private (possibly) correlated messages to the receivers, who are in an exogenous and commonly known network. Receivers can observe their neighbors' private messages and after updating their beliefs, vote sincerely on the proposal. We examine how networks of shared information affect the sender's gain from persuasion and find that in many cases it is not restricted by the additional information provided by the receivers' neighborhoods. Perhaps surprisingly, the sender's gain from persuasion is not monotonically decreasing with the density of the network. Krzysztof Mierzewski (Carnegie Mellon University) A canonical way to bridge the probabilistic, gradational notion of belief studied by Bayesian probability theory with the more mundane, all-or-nothing concept of qualitative belief is in terms of acceptance rules [Kelly and Lin, 2012]: maps that specify which propositions a rational agent accepts in light of their numerical credences (given by a probability model). Among the various acceptance rules proposed in the literature, an especially prominent one is Leitgeb's stability rule [Leitgeb, 2013, 2014, 2017; Rott, 2017], based on the notion of probabilistically stable hypotheses: that is, hypotheses that maintain sufficiently high probability under conditioning on new information. When applied to discrete probability spaces, the stability rule for acceptance guarantees logically closed and consistent belief sets, and it suggests a promising account of the relationship between subjective probabilities and qualitative belief. Yet, most natural inductive problems - particularly those commonly occurring in statistical inferenc - are best modelled with continuous probability distributions and statistical models with a richer internal structure. This paper explores the possibility of extending Leitgeb's stability rule to more realistic learning scenarios and general probability spaces. This is done by considering a generalised notion of probabilistic stability, in which acceptance depends not only on the underlying probability space, but also on a learning problem - namely, a probability space equipped with a distinguished family of events capturing the relevant evidence (e.g., the observable data) in the given learning scenario. This view of acceptance as being relative to an evidence context is congenial to (topological approaches to) formal learning theory and hypothesis testing in statistics (where one typically distinguishes the hypotheses being considered from observable sample data), as well as logics of evidence-relative belief [van Benthem and Pacuit, 2011]. Here we consider the case of statistical learning. We show that, in the context of standard (parametric) Bayesian learning models, the stability rule yields a notion of acceptance that is either trivial (only hypotheses with probability 1 are accepted) or fails to be conjunctive (accepted hypotheses are not closed under conjunctions). The first problem chiefly affects statistical hypotheses, while the second one chiefly affects predictive hypotheses about future outcomes. The failure of conjunctivity for the stability rule is particularly salient, as it affects a wide class of consistent Bayesian priors and learning models with exchangeable random variables. In particular, the results presented here apply to many distributions commonly used in statistical inference, as well as to every method in Carnap's continuum of inductive logics [Carnap, 1980; Skyrms, 1996]. These results highlight a serious tension between (1) being responsive to evidence and (2) having conjunctive beliefs induced by the stability rule. In the statistical context, certain properties of priors that are conducive to inductive learning - open-mindedness, as well as certain symmetries in the agent's probability assignments - act against conjunctive belief. Thus, the main selling points of the stability account of belief - its good logical behaviour and its close connection to the Lockean thesis - do not survive the passage to richer probability models, such as canonical statistical models for i.i.d learning. We conclude by discussing the consequences the results bear on Leitgeb's Humean Thesis on belief [Leitgeb, 2017]. J. van Benthem and E. Pacuit. Dynamic Logics of Evidence-Based Beliefs. Studia Logica, 99(1): 61 - 92, 2011. doi: 10.1007/s11225-011-9347-x. R. Carnap. A Basic System of Inductive Logic. in R.C. Jeffrey (ed.), Studies in Inductive Logic and Probability, vol. 2, Berkeley: University of California Press., 1980. K. T. Kelly and H. Lin. A geo-logical solution to the lottery-paradox, with applications to conditional logic. Synthese, 186(2): 531 - 575, 2012. doi: 10.1007/s11229-011-9998-1. H. Leitgeb. Reducing belief simpliciter to degrees of belief. Annals of Pure and Applied Logic, 164: 1338 - 1389, 2013. doi: 10.1016/j.apal.2013.06.015. H. Leitgeb. The Stability Theory of Belief. Philosophical Review, 123(2): 131 - 171, 2014. doi: 10.1215/ 00318108-2400575. H. Leitgeb. The Stability of Belief. Oxford University Press, Oxford, 2017. H. Rott. Stability and Scepticism in the Modelling of Doxastic States: Probabilities and Plain Beliefs. Minds and Machines, 27(1): 167 - 197, 2017. doi: 10.1007/s11023-016-9415-0. B. Skyrms. Carnapian inductive logic and Bayesian statistics. In Ferguson, T. S., Shapley, L. S. and MacQueen, J. B., editors, Statistics, probability and game theory: Papers in honor of David Blackwell, Hayward, CA, Institute of Mathematical Statistics, pages 321 - 336, 1996. doi: 10.1214/lnms/1215453580. Evan Piermont ( Royal Holloway, University of London, Department of Economics) Peio Zuazo-Garin (Higher School of Economics, International College of Economics and Finance) In this paper, we provide a theoretical framework to analyze an agent who misinterprets or misperceives the true decision problem she faces. Within this framework, we show that a wide range of behavior observed in experimental settings manifest as failures to perceive implications, in other words, to properly account for the logical relationships between various payoff relevant contingencies. We present behavioral characterizations corresponding to several benchmarks of logical sophistication and show how it is possible to identify which implications the agent fails to perceive. Thus, our framework delivers both a methodology for assessing an agent's level of contingent thinking and a strategy for identifying her beliefs in the absence full rationality.	CommonCrawl
Backup diff of Syntax (No. 12) [ Front page \| List of pages \| Search \| Recent changes \| Backup \| Help ] Go to Syntax. 1 (2006-07-23 (Sun) 05:04:00) 2 (2006-11-02 (Thu) 10:18:38) 8 (2010-06-16 (Wed) 15:18:28) 10 (2012-05-10 (Thu) 06:33:37) 12 (2012-05-11 (Fri) 02:32:29) 15 (2019-04-29 (Mon) 16:56:13) 16 (2019-04-30 (Tue) 08:35:18) 17 (2019-05-01 (Wed) 09:24:21) 20 (2019-06-29 (Sat) 08:40:31) //[[Documentation]] Syntax Processes An LMNtal process is a multiset (unordered sequence) of the following: An LMNtal program is written as an LMNtal process, and an LMNtal process is a multiset (bag) of the following: \|BGCOLOR(white):syntactic&br;category\|BGCOLOR(white):form\|BGCOLOR(white):description\| \|BGCOLOR(white):''atoms'' \| '''p'''('''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};) \| a graph node with a symbolic ''name'' '''p''' and an ordered sequence of ''links'' '''X'''&size(10){'''i'''}; \| \|BGCOLOR(white):''cells'' \| { '''Process''' }&br;'''m'''{ '''Process''' } \| a process enclosed with a ''membrane'' (curly braces),&br;optionally with a membrane name '''m''' \| \|BGCOLOR(white):''rules'' \| ( '''Head''' :- '''Body''' ) \| a rewrite rule for processes, explained below \| Both periods (as separators) and commas (as terminators) can be used to sequence the elements of a multiset. An LMNtal program is written as an LMNtal process. The elements of a multiset are either separated by commas (e.g., a(X), b(X)) or terminated by periods (e.g., a(X). b(X).) . Links names are written using alphanumeric tokens starting with capital letters (e.g., X, Res). The other alpha-numeric tokens are treated as atom or membrane names (e.g., foo, 123). Quoted symbols can also be used for atom or membrane names (e.g., "foo", 'bar', [:baz:]). Since link names stand for endpoints of one-to-one links, each link name in a well-formed process must occur at most twice, and each link name in a well-formed rule must occur exactly twice. Term Abbreviation Scheme Since each link name occurs at most twice, we can abbreviate >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''m'''};), '''q'''('''t'''&size(10){1};, ..., '''t'''&size(10){'''n'''};) >'''p'''('''s'''&size(10){1};, ..., '''s'''&size(10){'''k'''-1};, '''q'''('''t'''&size(10){1};, ..., '''t'''&size(10){'''n'''-1};), '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};) // &math(p(s_1,\ldots,s_m), q(t_1,\ldots,t_n)); to // &math(p(s_1,\ldots,s_{k-1},q(t_1,\ldots,t_{n-1}), s_{k+1},\ldots,s_m)); if '''t'''&size(10){'''n'''}; and '''s'''&size(10){'''k'''}; are the same link name. c(1,c(2,n),L0) is an abbreviation of c(A,L1,L0),c(B,L2,L1),n(L2),1(A),2(B). This can be written also as L0=c(1,c(2,n)) . For an atom name '''p''' and a membrane, '''p'''(..., {...}, ...) stands for a molecule '''p'''(..., X, ...), {+X, ...} . List Notation The Prolog list syntax can be used in LMNtal. List constructor atoms have three arguments and the name '.' . X=[A,B\|Rest], Rest=[] X=[A,B] are abbreviated forms of '.'(A,Tmp,X), '.'(B,Rest,Tmp), '[]'(Rest). //For some reason, the process [A,B\|Rest] is parsed as //'.'(A,Tmp), '.'(B,Rest,Tmp). //These two atoms have different arities. Connector Atoms Binary atoms with the name = of the form X=Y are called ''connector atoms''. Connector atoms state that the two links in the arguments are identical (in the sense of structural equivalence. Another instance of structural equivalence is the reordering of multiset elements.) For example, ( p(A,X,C), X=B ) is always equivalent to p(A,B,C), as well as to ( p(A,B,X), C=X ), and, finally, to C=p(A,B). Binary atoms of the form X=Y are called ''connector atoms'' or ''connectors''. A connector states that the two link names are considered identical (i.e., interconnected in zero steps). // (in the sense of structural equivalence. Another instance of structural // equivalence is the reordering of multiset elements.) For example, ( p(A,X,C), X=B ) is equivalent to p(A,B,C), as well as to ( p(A,B,X), C=X ) and C=p(A,B). The typical usage of connector atoms can be found in the following example: Connectors are typically used in the base case of a recursive definition: ( Res=append([],Y) :- Res=Y ), ( Res=append([A\|X],Y) :- Res=[A\|append(X,Y)] ) Rules The basic syntax of a rule is >( '''Head''' :- '''Body''' ). The enclosing parentheses can be omitted if periods are used to delimit the rule. Both of '''Head''' and '''Body''' are ''process templates''. '''Head''' specifies processes to be rewritten and '''Body''' specifies the result of rewriting. Rules work only for the processes residing in the same membrane. The full syntax of a rule that contains '''Guard''' part will be explained later in [[a separate section>Guards]]. **Process Templates A process template is a multiset of the following: \|BGCOLOR(white): ''atoms'' \| '''p'''('''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};) \| same as in a process \| \|BGCOLOR(white): ''cells'' \| { '''Template''' }&br;{ '''Template''' }/&br;'''m'''{ '''Template''' }&br;'''m'''{ '''Template''' }/ \| a process template enclosed with a membrane \| \|BGCOLOR(white): ''rules'' \| ( '''Head''' :- '''Body''' ) \| allowed only in a Body \| \|BGCOLOR(white): ''process contexts'' \| $'''p'''&br;$'''p'''['''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};]&br;$'''p'''['''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''};\|'''X'''] \| matches a multiset of atoms and cells (see below) \| \|BGCOLOR(white): ''rule contexts'' \| @'''p''' \| matches a multiset of rules \| A cell template with a membrane name only matches a cell with the same membrane name. A cell template with '/' (the stable flag) only matches a ''stable'' cell, i.e., a cell containing no applicable rules inside it. **Process Contexts A process context represents a multiset of atoms and cells. The arguments '''X'''&size(10){1};, ..., '''X'''&size(10){'''n'''}; specify the set of free links that must exist in the matched process. The optional '''X''' represents an arbitrary number of extra free links. The form $'''p''' is an abbreviation of $'''p'''[\|'''X'''], i.e., a multiset of atoms and cells with no constraints on the occurrences of free links. A process context must occur within a membrane in a head. Alternatively, it can either - occur in a head and an input position of a guard, or - occur in an output position of a guard. See [[Guards]] for the latter extensions. You can abbreviate '''X''', '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};), $'''q'''['''X'''] $'''q''', '''s'''&size(10){'''k'''+1};, ..., '''s'''&size(10){'''m'''};). //&math(p(s_1,\ldots,s_{k-1},X, s_{k+1},\ldots,s_m),\$q[X]); to //&math(p(s_1,\ldots,s_{k-1},\$q,s_{k+1},\ldots,s_m));. //Note that the current implementation does not fully // support process contexts with explicit arguments. // Any Comments? //-the expressions need to be fixed. -- [[nakajima]] &new{2004-02-01 // (Sun) 21:43:32}; //-fixed the expression using math.inc.php -- [[nakajima]] &new{2004-02-02 // (Mon) 21:33:38}; //-fixed math.inc.php. no warnings any more. -- [[nakajima]] &new{2004-02-10 // (Tue) 12:35:06}; //#comment Modified by uedalab LMNtal group	CommonCrawl
\begin{document} \maketitle \begin{abstract} Given a Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix $\alpha$ such that $\alpha$ is homotopic to the identity matrix. Utilizing this result we establish that if $I \subset R$ is an ideal satisfying $\mu(I/I^2) = \text{ht}(I) = d$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $\mu(-)$ denotes the minimal number of generators. We improve the injective stability of $\text{K}_1(R)$. Finally, we prove that for any projective $R$-module $P$ of rank $d$ if the Quillen ideal of $P$ is non-zero, then $P$ is cancellative. \end{abstract} \section{Introduction} \renewcommand{\begin{numtheorem}}{\begin{numtheorem}} \renewcommand{\end{numtheorem}}{\end{numtheorem}} \renewcommand{\begin{numcorollary}}{\begin{numcorollary}} \renewcommand{\end{numcorollary}}{\end{numcorollary}} \def\refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}{\refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}} \def\refstepcounter{numtheorem}\paragraph{{\bf Question} \thenumtheorem}{\refstepcounter{numtheorem}\paragraph{{\bf Question} \thenumtheorem}} We commence by recalling an old question of Murthy \cite{Bass73}. Let $A=\bigoplus_{i\ge 0} R_i$ be a normal positively graded finitely generated algebra over $R_0$, where $R_0=k$ is a field. Then Murthy asked whether $\text{K}_0(A) \cong \mathbb{Z}$. Bloch provided a counterexample to this question by considering $A=\frac{\mathbb{C}[X,Y,Z]}{\langle Z^2 - X^3 - Y^7\rangle}$. However, if (1) $k$ is an algebraically closed field of characteristic $p> 0$, (2) $A$ is a Cohen-Macaulay ring of dimension $2$, and (3) the vertex (corresponding to the ideal $\bigoplus_{i\ge 1}R_i$) is the only singularity of $\text{Spec} (A)$, then Srinivas \cite[Corollary 1.3]{sv1} showed that Murthy's question has an affirmative answer. Therefore, using the cancellation theorem of Murthy and Swan \cite{PMRS}, it follows that every projective $A$-module is free. Hence, this improves the existing stability theorems for projective modules over such graded algebras of dimension $2$. This exposition is borrowed from \cite{sv1}. In 1987, Lindel improved \cite[Theorem 1.3]{Li87} Quillen's Local-Global Principle from polynomial rings to positively graded rings. This, in particular, initiated the study of projective modules over a higher dimensional graded ring from a more algebraic point of view. Keeping track of this theme, in this article we aim to extend existing stability theorems for projective modules such as \cite{P} from polynomial rings to an arbitrary positively graded domain of dimension $\ge 2$. For the remainder of the Introduction, let us assume that (1) $R$ is a commutative Noetherian ring with $1$ $(\not= 0)$ of finite (Krull) dimension $m\ge 2$, and (2) $A=\bigoplus_{i\ge 0} R_i$ is a commutative Noetherian (non-trivially) graded domain of dimension $d\ge 2$ such that $\dim(R_0)\ge 1$. \subsection{Unimodular rows} Recall that a row vector $v=(v_1,\ldots,v_n)\in R^n$ is called a unimodular row if $\sum_{i=1}^n \lambda_iv_i=1$ for some $\lambda_i\in R$. The set of all unimodular rows in $R$ of length $n$ is denoted by $\text{Um}_{n}(R)$. For all $n\ge m+2$, elementary facts (such as the prime avoidance lemma) can be employed to demonstrate that any unimodular row of length $n$ can be completed to the first row of an invertible matrix, specifically an elementary matrix. Previous literature includes well-known examples that establish the optimality of this bound in general. However, certain classes of rings, such as affine algebras over an algebraically closed field of characteristic $0$, have the potential to improve this bound \cite{AAS}. Nonetheless, the complete classification of all Noetherian rings of dimension $m$ over which all unimodular rows of length $m+1$ can be completed to the first row of an invertible (simultaneously elementary) matrix remains a formidable task until now. In this context, we prove the following theorem [for the proof we refer to Theorem \ref{umch} and Corollary \ref{ec1}]. \begin{numtheorem}\label{hc} Let $C=S^{-1}A$, where $S\subset A$ is a multiplicative set. Any unimodular row in $C$ of length $d+1$ can be completed to the first row of an invertible matrix $\alpha$. Moreover, the matrix $\alpha$ can be chosen in such a way that there exists $\theta(T)\in \GL_{d+1}(C[T])$ such that $\theta(0)=\text{Id}$ and $\theta(1)=\alpha$ (in this case $\alpha$ will be called homotopic to the identity matrix). Consequently, any stably free $C$-module of rank $d$ is free. \end{numtheorem} Since our assertion here is about graded rings, it is uncertain whether the well-known Suslin's factorial row reduction technique, as employed in \cite{AAS}, can be applied. One possible approach is to first extend Roitman's degree reduction techniques from polynomial rings to positively graded rings, then use it to apply the factorial row technique in graded rings. But such an improvement of Roitman's degree reduction techniques is a challenging task in itself. Instead, we have taken an alternative path. Let us choose $v\in \text{Um}_{d+1}(A)$. First, we point out a beautiful observation due to Plumstead. In \cite[Example 2]{P} he showed that one can glue finitely many generalized dimension functions on a suitable finite-cover of $\text{Spec}(R)$ simply by taking their maximum. Utilizing this observation, we are able to apply stability theorem due to Eisenbud and Evans \cite{EE} to establish the completion of $v$ in a suitable two-cover of $\text{Spec}(A)$. Then leveraging the grading and applying a well-known homotopy trick due to Swan and Weibel, we generalize Quillen's splitting lemma \cite[Theorem 1]{Q} over graded rings [see Proposition \ref{splitting}]. In particular, we have demonstrated that in the case of graded rings, the splitting can be obtained in such a way that both matrices fix the canonical vector $e_1 \, (=(1,0,\ldots,0))$. This allows us to patch two elementary matrices such that the resulting matrix simultaneously (1) becomes homotopic to the identity matrix, and (2) transforms the unimodular row $v$ to $e_1$. One may wonder whether it is possible to improve Quillen's splitting lemma for an arbitrary ring. Unfortunately, as we mentioned in Remark \ref{rpvul}, it is not feasible. In another sense, this setback highlights the significance of maintaining the grading in the patching diagram used in this article. By employing a similar line of reasoning and utilizing a Local-Global Principle, we establish in Theorem \ref{ist} that the injective stability of $\text{K}_1(A)$ is $d+1$, thereby improving upon Vaser{\v{s}}te{\u{\i}}n's stability bound \cite{Va1} over graded domains. \subsection{Splitting of projective modules of top rank} We now shift our focus to another classical problem in the study of projective modules. Let $P$ be a projective $R$-module with trivial determinant. Recall that $P$ has a unimodular element if and only if it splits off a free summand of rank one. Due to a classical result of Serre \cite{S} it is known that if $\text{rank}(P)>d$, then $P$ splits off a free summand of rank one. Again this bound is the best possible. However, certain classes of rings, such as $R[X]$, can improve this bound \cite{P}. Hence, naturally identifying the class of rings over which top rank projective modules split into a free factor becomes the next big challenge. For the remainder of the Introduction, let us assume that $P$ is a projective $A$-module with trivial determinant such that $\text{rank}(P)=d$. We recall that the Quillen set of $P$, denoted by $J(R_0,P)$, is defined as the collection of all elements $s\in R_0$ such that $P_s\cong \frac{P_s}{(PR_+)_s}\otimes R_s$. It has been proven in \cite[Theorem 1.3]{Li87} that $J(R_0,P)$ forms an ideal in $R_0$. Lindel gave a sufficient condition \cite[Theorem 2.5]{Li87} for the existence of a unimodular element in $P$ over certain graded rings. Informally, Lindel's theorem states that if (1) $J(R_0,P) \neq 0$ and (2) the graded ring $A$ satisfies some special criteria (e.g. $\dim(A)-\dim(R_0)= 1$, when in addition $A$ is an affine domain over a field), then $P$ splits off a free summand of rank one. We show that in \cite[Theorem 2.5]{Li87} the hypotheses (1) and (2) are redundant [see Corollary \ref{eue}]. In fact, we establish an even stronger result. Before stating our theorem, we digress momentarily. Since 1980, a recurrent theme in this area has been to find the precise obstruction for a projective $R$-module of rank $m$ to split a free factor. Recall that an ideal $J\subset R$ is considered efficiently generated if $\mu(J/J^2)=\mu(J)$, where $\mu(-)$ represents the minimal number of generators. Thanks to a significant result by Eisenbud and Evans \cite{EE}, one can always find an $A$-linear map $\gamma:P\to A$ such that the image ideal $I(=\gamma(P))$ satisfies $\mu(I/I^2)=\text{ht}(I)=d$. Notable contributions in this area have been made by seminal works such as \cite{m82} and \cite{SMBB3} which eventually established the fact that the existence of a lift to a specific set of generators of $I/I^2$ dictates the splitting problem for $P$. This provides an additional approach to tackle the splitting problem for projective modules of top rank. In this context, we provide a sufficient criterion [see Proposition \ref{egc}] for the efficient generation of a top height ideal in an arbitrary ring $R$. This empowers us to address the splitting problem for projective modules of top rank as well as the problem of efficient generation of top height ideals across various scenarios. In particular, we prove the following [for the proof we refer to Theorem \ref{eg}, \ref{egd2} and Corollary \ref{eue}]. \begin{numtheorem}\label{egi} Let $C$ and $n$ be one of the following: \begin{enumerate} \item $C=A$ and $n=\dim(A)=d\ge 2$. \item $C=S^{-1}A$, where $S\subset A$ is a multiplicative set contained in the set of all non-zero divisors in $A$ such that $\dim(C)=\dim(A)$ and $n=\dim(A)=d\ge 3$. \item $C=B[M]$, where $B$ is a commutative Noetherian ring of dimension $ \ge 2$ and $M$ is a finitely generated commutative cancellative (not necessarily torsion free) monoid of rank $r\ge 1$. We take $n=\dim(B[M])$. \end{enumerate} Let $I\subset C$ be an ideal such that $\mu(I/I^2)=\text{ht}(I)=n$. Then any set of generators of $I=\langle f_1,\ldots,f_n\rangle +I^2$ lifts to a set of generators of $I$. Consequently, any projective $C$-module of rank $n$ with trivial determinant splits off a free summand of rank one. \end{numtheorem} Readers may question the significance of the hypothesis $\dim(R_0)\ge 1$ in this article. In Example \ref{ex1} and \ref{ex2} we demonstrate that this hypothesis is indeed necessary in Theorem \ref{egi}. \subsection{Cancellation of projective modules of top rank} Recall that a projective $R$-module $Q$ is considered cancellative if $Q\oplus R^k\cong Q'\oplus R^k$ implies $Q\cong Q'$, where $Q'$ is another $R$-module and $k\ge 1$. In Section \ref{cansection}, we explore the cancellation problem for projective modules over $A$. In Theorem \ref{can1} we establish that if $J(R_0,P)\neq 0$, then $P$ is cancellative. As an interesting consequence, we demonstrate in Corollary \ref{soprc} that over a Noetherian graded subring $B$ of $R[T]$ containing $R$ with $\dim(B)=m+1$, projective modules of rank $m+1$ with trivial determinant are cancellative. \subsection{On a question of Nori over a graded non-smooth algebra} In section \ref{a1}, we deduce some consequences of Theorem \ref{egi}. We very briefly recall an algebraic analogy of a question asked by Nori \cite{SM}. \refstepcounter{numtheorem}\paragraph{{\bf Question} \thenumtheorem}\label{Norifree} Let $C$ be a smooth affine domain of dimension $d$ over an infinite perfect field. Let $I\subset C[T]$ be an ideal of height $n$ such that $I=\langle f_1,\cdots,f_n\rangle +I^2T$, where $2n\geq d+3$. Do there exist $g_i\in I$ such that $I=\langle g_1,\cdots,g_n\rangle $ with $g_i-f_i\in I^2T$? This question is completely solved in \cite{BR} and \cite{BHMK}. Bhatwadekar, Mohan Kumar and Srinivas constructed an example \cite[Example 6.4]{BR} of a non-smooth positively graded affine domain (with the degree zero subring is $ \mathbb{C}$) such that over which Nori's question has a negative answer. However, when the ring has singularities, it is shown in \cite{sbmkd1} that imposing some suitable smoothness condition on the ideal $I\cap R$ one can prevent such anomalies. Here, in Section \ref{a1}, we aim to understand the underlying issue that prevents the existence of such a lift in \cite[Example 6.4]{BR}. In particular, we prove the following [for details we refer to Theorem \ref{NQ} and Corollary \ref{egpa}]. \begin{numtheorem} Let $A=\bigoplus_{i\ge 0} R_i$ be an affine domain (non-necessarily smooth) of dimension $d\ge 3$ over an infinite field such that $\frac{1}{d!}\in A$ and $\dim(R_0)\ge 1$. Let $I\subset A[T]$ an ideal such that $\mu(I/I^2T)=\text{ht}(I)=d$. Then any set of generators of $I/I^2T$ lifts to a set of generators of $I$. Consequently, any projective $A[T]$-module (with trivial determinant) of rank $d$ splits off a free summand of rank one. \end{numtheorem} \section{Preliminaries} \renewcommand{\begin{numtheorem}}{\begin{theorem}} \renewcommand{\end{numtheorem}}{\end{theorem}} \renewcommand{\begin{numcorollary}}{\begin{corollary}} \renewcommand{\end{numcorollary}}{\end{corollary}} \def\refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}{\refstepcounter{theorem}\paragraph{{\bf Remark} \thetheorem}} This section summarizes several results and definitions from the literature that are frequently used in this article to prove the main theorems. We may restate or improve these results as necessary. Before proceeding further, to avoid any ambiguity, we introduce a set of conventions that will be consistently followed throughout the entire article. \paragraph{\bf{Convention}} The symbol $\mathbb{N}$ denotes the set of all non-negative integers, including $0$. All rings considered in this article are assumed to be commutative Noetherian with $1(\neq 0)$ having finite (Krull) dimension. Additionally, all graded rings discussed in this article are assumed to have a non-trivial $\mathbb{N}$-grading. For a graded ring $R=\bigoplus_{i\geq 0} R_i$, we use the notation $R_+=\bigoplus_{i\geq 1} R_i$ to represent the irrelevant ideal in $R$ containing all elements which can be written as a sum of homogeneous elements of degree $>0$. Every module considered in this article is assumed to be finitely generated. Before proceeding further, we recall several definitions from the literature. \begin{definition}\label{def} Let $A$ be a ring. \begin{enumerate} \item Let $M$ be an $A$-module. An element $x\in M$ is said to be a \textit{basic element} of $M$ at a prime ideal $\p\in \text{Spec}(A)$ if $x\not \in \p M_{\p}.$ For any $\CS\subset \text{Spec}(A)$, we call $x$ a basic element of $M$ on $\CS$ if it is a basic element of $M$ at each prime ideal $\p\in \CS$. \item Let $\CS\subset \text{Spec}(A)$ and let $\delta:\CS\to \mathbb{N}$ be a function. For two prime ideals $\p,\mq\in \CS$, we define a partial order $\p<<\mq$ if and only if $\p\subset \mq$ and $\delta(\p)>\delta(\mq)$. We say that $\delta$ is a \textit{generalized dimension function} if for any ideal $I\subset R$, the set $V(I)\cap \CS$ has only finitely many minimal elements with respect to $<<$. \item A positive integer $r$ is said to be the \textit{stable rank} of $A$, denoted as $\text{sr}(A)$, if $r$ is the smallest integer for which any $(a_1,\ldots,a_{r+1})\in \text{Um}_{r+1}(A)$, there exists $\lambda_i\in R$, $i=1,\ldots,r$ such that $(a_1+\lambda_1a_{r+1},\ldots,a_r+\lambda_ra_{r+1})\in \text{Um}_r(R)$. \item We define $\text{H}_n(A)=\{\alpha\in \GL_n(A):\text{ there exists a } \theta(T)\in \GL_{n}(A[T]) \text{ such that } \theta(0)=\text{Id}$ \text{ and } $\theta(1)=\alpha \}$. Then $\text{H}_n(A)$ is a normal subgroup of $\GL_{n}(A)$. \item Let $I\subset A$ be an ideal and let ``bar" denote going modulo $I$. We define $\text{Um}_{n}(A,I):=\{v\in \text{Um}_{n}(A):\overline v=e_1\}$ and $\GL_{n}(A,I):=\{\alpha\in \GL_{n}(A):\overline \alpha=\text{Id}\}.$ \item Let $P$ be a projective $A$-module such that $P$ has a unimodular element. We choose $\phi\in P^$ and $p \in P$ such that $\phi(p)=0$. We define an endomorphism $\phi_p$ as the composite $\phi_p:P\to A\to P$, where $A\to P$ is the map sending $1\to p.$ Then by a \textit{transvection} we mean an automorphism of $P$, of the form $1+\phi_p$, where either $\phi\in \text{Um}(P^)$ or $p\in \text{Um}(P)$. By $\text{E}(P)$ we denote the subgroup of $\Aut(P)$ generated by all transvections. \end{enumerate} \end{definition} We begin by considering the following observation for a graded domain. This simple proposition plays a crucial role in the article, and therefore, we provide the proof. \begin{proposition}\label{dogr} Let $R=\bigoplus_{i\ge 0} R_i$ be a graded domain of dimension $d$. Let $S\subset R_0$ be a multiplicative set such that $S\cap\mm\not=\emptyset$, for any maximal ideal $\mm\in \text{Spec}(R_0)$. Then the graded domain $S^{-1}R$ does not have a graded maximal ideal $S^{-1}\MM$ such that $\MM$ is a maximal ideal in $R$. As a consequence, we get $\dim(S^{-1}R)<d$. \end{proposition} \paragraph{Proof} Suppose that $\dim(R_0)=n$. We give the proof by induction on $n$. First, we note that if $n=0$, then $R_0$ is a field. In this case, the statement is vacuously true. Now, we consider the case where $n\geq 1$. If there does not exist such an $S$, then the statement is vacuously true. Therefore, we assume that such an $S$ exists. Contrarily, we assume the existence of a graded maximal ideal, denoted as $S^{-1}\MM$, in $S^{-1}R$, where $\MM\in \text{Spec}(R)$ is a maximal ideal. There are two possibilities: either $\MM$ is a graded maximal ideal or $\MM$ is a maximal ideal but not a graded ideal. If $\MM$ is a graded maximal ideal, then it can be expressed as $\mm\oplus R_+$, where $\mm$ is a maximal ideal in $R_0$. Since $S\cap \mm\neq\emptyset$, it implies that $S\cap \MM\neq\emptyset$. However, this leads to a contradiction. Now, we assume that $\MM$ is not a graded ideal. Since $S^{-1}\MM$ is a graded maximal ideal, it can be expressed as $\mm'\oplus S^{-1}R_+$, where $\mm'$ is a maximal ideal in $S^{-1}R_0$. In particular, as $R$ is Noetherian, there exists $s\in S$ such that $sR_+\subset \MM$. Because of $S\cap\MM=\emptyset$, it follows that $R_+\subset \MM$. We claim that $\mm_0:=\MM\cap R_0\neq{0}$. Contrary, let us assume that $\mm_0=0$. Consider an element $f\in \MM$. We can write $f=f_0+f_1$, where $f_0\in R_0$ and $f_1\in R_+$. Since $f_1\in R_+\subset \MM$, we have $f_0\in \mm_0={0}$. This implies $\MM=R_+$. As a result, we get $R_0\cong R/R_+\cong R/\MM\cong k$, where $k$ is a field. However, this is not possible as $\dim(R_0)=n\geq 1$. Hence, we establish that $\mm_0\neq{0}$. Let $a\in \mm_0\setminus \{0\}$, and let "bar" denote going modulo $\langle a \rangle$. Note that if $\langle a \rangle \cap S\neq\emptyset$, then we have $S\cap \MM\neq\emptyset$, which contradicts our assumption on the existence of such a maximal ideal. Hence, without loss of generality, we may assume that $\langle a \rangle \cap S=\emptyset$. By our choice of $a$, we have $\dim(\overline{R_0})\leq n-1$, and $\overline{R}$ is a graded domain with dimension $\leq d-1$. Let $\eta\in \text{Spec}(\overline{R_0})$ be a maximal ideal in $\overline{R_0}$. Since $\langle\eta, a\rangle$ is also a maximal ideal in $R_0$, according to our induction hypothesis, we have $S\cap \langle\eta, a\rangle \neq\emptyset$. This implies that $\overline{S}\cap \eta \neq\emptyset$. Furthermore, we observe that $\overline{\MM}$ is a maximal ideal in $\overline{R}$. Moreover, as $a\in R_0$ and $\overline S^{-1}\overline \MM\cong\overline{S^{-1} \MM}$, the ideal $\overline S^{-1}\overline \MM$ is a graded maximal ideal in $\overline S^{-1}{\overline R}$ (recall that $S^{-1}\MM$ is a graded ideal). However, by the induction hypothesis, there does not exist such a maximal ideal in $\overline S^{-1}\overline R$. This completes the induction step. It remains to show that $\dim(S^{-1}R)<d$. To prove this, we note that for an arbitrary graded ring $B=\bigoplus_{i\geq 0} B_i$, there exists a graded maximal ideal $\MN$ in $B$ such that $\text{ht}(\MN)=\dim(B)$. In $S^{-1}R$, any graded maximal ideal of height $d$ is a localization of a maximal ideal in $R$. However, we have already demonstrated the nonexistence of such a graded maximal ideal in $S^{-1}R$. Therefore, the ring $S^{-1}R$ does not have a graded maximal ideal of height $d$. Consequently, we obtain that $\dim(S^{-1}R)<d$. \qed We revisit a well-known homotopy map due to Swan and Weibel. \begin{definition} Let $R=\bigoplus_{i\ge 0} R_i$. We define the Swan-Weibel homotopy map $\Gamma_{SW}: R \to R[T]$ as follows: for any element $f = a_0 + a_1 + \ldots + a_n \in R$, we define $\Gamma_{SW}(f):=a_0+a_1T+\ldots+a_nT^n \in R[T]$, where $a_i \in R_i$. \end{definition} \begin{lemma}\label{isotopy} Let $R=\bigoplus_{i\ge 0} R_i$. Let $\alpha\in \GL_n(R)$ such that $\overline{\alpha}=\text{Id}$, where ``bar" denotes going modulo the ideal $R_+$. Then there exists an $\theta(T)\in \text{GL}_n(R[T])$ such that $\theta(0)=\text{Id}$ and $\theta(1)=\alpha$. In other words $\alpha\in \text{H}_n(R)$. Moreover, if $e_1\alpha=e_1$, then we may choose such an $\theta(T)$ with the property that $e_1\theta(T)=e_1$. \end{lemma} \paragraph{Proof} Consider the group homomorphism $\widetilde{\Gamma_{SW}}:\GL_n(R)\to \GL_n(R[T])$ induced by $\Gamma_{SW}$ \cite[Definition 2.1]{rbs}. Let us take $\theta(T)=\widetilde{\Gamma_{SW}}(\theta)\in \GL_{n}(A[T])$. Then it follows that $\theta(0)=\overline{\alpha}=\text{Id}$ and $\theta(1)=\alpha$. Now we assume that $e_1\alpha=e_1$. As $1\in R_0$, we have $\Gamma_{SW}(1)=1$. Hence, we have $e_1\theta(T)=e_1$.\qed \begin{lemma}\label{SWE} Let $R=\bigoplus_{i\ge 0} R_i$. Then the map $\Gamma_{SW}:R\to R[T]$ will induce a group homomorphism $\widetilde{\Gamma_{SW}}:\text{E}_n(R)\to \text{E}_n(R[T])$. \end{lemma} \paragraph{Proof} First, we observe that $\Gamma_{SW}$ will induce a group homomorphism $\widetilde{\Gamma_{SW}}:\text{E}_n(R)\to \GL_{n}(R[T])$ for details we refer to \cite[Definition 2.1]{rbs}. Therefore, it is enough to show that $\widetilde{\Gamma_{SW}}(\text{E}_n(R))\subset \text{E}_n(R[T])$. Let $E_{ij}(f)\in \text{E}_n(R)$ be an elementary matrix whose only non-zero non-diagonal entry is $f$ at the position $(i,j)$, where $i\not=j$. We write $f(T)=\Gamma_{SW}(f)$. Then we note that $\widetilde{\Gamma_{SW}}(E_{ij}(f))=E_{ij}(f(T))\in \text{E}_n(R[T])$. Moreover, since $\widetilde{\Gamma_{SW}}$ is a group homomorphism and any element of $\text{E}_{n}(R)$ can be written as a finite product of elements of the form $E_{ij}(f)$, it follows that $\widetilde{\Gamma_{SW}}(\text{E}_n(R))\subset \text{E}_n(R[T])$. \qed The next lemma is known as one of the variants of Quillen-Suslin's Local-Global Principle and must be well-known. However, we could not find any suitable reference for the exact version required in this article. The closest reference we have found is \cite[Theorem 3.8]{rbs}. Therefore, we provide the proof, which is straightforward using the homotopy map $\Gamma_{SW}$ and Suslin's Local-Global Principle \cite[Lemma 3.5]{AASSC}. \begin{lemma}\label{LG} Let $R=\bigoplus_{i\ge 0} R_i$ and let ``bar" denote going modulo the ideal $R_+$. Let $s,t\in R_0$ be two co-maximal elements and let $\alpha\in \GL_n(R)$ such that (i) $\overline{\alpha}=\text{Id}$, (ii) $\alpha_s\in \text{E}_{n}(R_s)$ and (iii) $\alpha_t\in \text{E}_{n}(R_t)$, where $n\ge 3$. Then $\alpha\in \text{E}_{n}(R)$. \end{lemma} \paragraph{Proof} We take $\theta(T)=\widetilde{\Gamma_{SW}}(\alpha)\in \GL_{n}(R[T])$, where $\widetilde{\Gamma_{SW}}:\GL_n(R)\to \GL_n(R[T])$ is induced by $\Gamma_{SW}$. Then we note that $\theta(0)=\text{Id}$. Moreover, it follows from Lemma \ref{SWE} that $(\theta(T))_s\in \text{E}_{n}(R_s[T])$ and $(\theta(T))_t\in \text{E}_{n}(R_t[T])$. Applying \cite[Lemma 3.5]{AASSC} we obtain that $\theta(T)\in \text{E}_{n}(R[T])$. Therefore, we get $\alpha=\theta(1)\in \text{E}_{n}(R)$.\qed The following lemma is due to Plumstead, which is an adaptation of \cite[Example 4]{P}, tailored to our requirements. This serves as one of the fundamental building blocks of this article. The proof is given in \cite{sb1}. For the sake of completeness, here we provide the detailed proof. \begin{lemma}\label{plgd} Let $A$ be a ring and let $s$ be a non-zero divisor in $A$ such that $\dim(A_s)\le d-1$. Then there exists a generalized dimension function $\delta:\text{Spec}(A)\to \mathbb{N}$ such that $\delta(\p)\le d-1$ for all $\p\in \text{Spec}(A)$. Furthermore, we can choose $\delta$ such that $\delta(\p)=\dim(A/\p)$ for all $\p\owns s$. \end{lemma} \paragraph{Proof} Let $\p\in \text{Spec}(A)$. Then we note that either $s\in \p$ or $s\not \in \p$. That is, the prime ideal $\p\in V(s)\cup \text{Spec}(A_s)$, where $V(s)=\{\p\in \text{Spec}(A):s\in \p\}$. Hence $\text{Spec}(A)= S_1\cup S_2$, where $S_1=\text{Spec}(A_s)$ and $S_2=V(s)$. Let $I\subset A$ be an ideal. We observe the equality $V(I)\cap S_1=V(I_s)$. Since $A$ is a Noetherian ring, it follows from the primary decomposition of $I_s$ that $V(I)\cap S_1$ has finitely many minimal elements with respect to the partial ordering $<<$, defined in Definition \ref{def}. On the other hand $V(I)\cap S_2=V(I)\cap V(s)=V(I+\langle s\rangle )$. Again from the primary decomposition of $I+\langle s\rangle $ it follows that $V(I)\cap S_2$ has finitely many minimal elements. We define a function $\delta_1:S_1\to \mathbb{N}$ such that $\delta_1(\p)=\dim(A_s/\p_s)$. It follows from \cite[Example 1]{P} that $\delta_1$ is a generalized dimension function on $S_1$. Here we notice that $\delta_1(\p)\le\dim(A_s)\le d-1$ for all $\p\in \text{Spec}(A_s)=S_1$. We define another generalized dimension function $\delta_2:S_2\to \mathbb{N}$, by $\delta_2(\p)=\dim(A/\p)$ for all $\p\in S_2$. Since $s$ is a non-zero divisor, we obtain that $\delta_2(\p)\le \dim(A)-1\le d-1$ for all $\p\in S_2$. Therefore, following \cite[Example 2]{P} we can define a generalized dimension function $\delta:\text{Spec}(A)\to \mathbb{N}$ such that $\delta(\p)=\delta_1(\p)$ if $\p\in S_1$ and $\delta(\p)=\delta_2(\p)$ if $\p\in S_2$. Then we note that $\delta(\p)\le d-1$ for all $\p\in \text{Spec}(A)$ and if $s\in \p$, then $\delta(\p)=\delta_2(\p)=\dim(A/\p)$. This completes the proof.\qed We end this section with the next theorem, which is derived from a pivotal result due to Eisenbud and Evans \cite{EE}. This has been used extensively throughout the article. This version is recollected from \cite{P}. \begin{numtheorem}\label{eept} Let $A$ be a ring, and let $\CP\subset\text{Spec}(A)$ be a subset. Consider a generalized dimension function $\delta:\CP \to \mathbb{N}$. Let $M$ be an $R$-module satisfying $\mu_{\p}(M)\ge 1+\delta(\p)$ for all $\p\in \CP$, where $\mu_{\p}(M)$ is the minimal number of generators of $M_{\p}$. For a basic element $(r,m)\in R\oplus M$ on $\CP$, there exists an element $m'\in M$ such that $m+rm'$ is also a basic element on $\CP$. \end{numtheorem} \section{Unimodular rows}\label{ur} This section is devoted to establishing that any unimodular row of length $d+1$ over a graded domain of dimension $d\ge 1$ can be completed to the first row of an invertible matrix, which is homotopic to the identity matrix. We begin with an easy consequence of Lemma \ref{plgd} and Theorem \ref{eept}. \begin{lemma}\label{dsr} Let $A$ be a ring of dimension $d\ge 1$. Assume that, there exists a non-zero divisor $s\in A$ such that $\dim(A_s)<\dim(A)$. Then $\text{sr}(A)\le d$. \end{lemma} \paragraph{Proof} Let $v=(v_1,\ldots,v_{d+1})\in \text{Um}_{d+1}(A)$. Applying Lemma \ref{plgd} we get a generalized dimension function $\delta:\text{Spec}(A)\to \mathbb{N}$ such that $\delta(\p)\le d-1$ for all $\p\in \text{Spec}(A)$. We note that $v$ is a basic element of the free module $A^{d+1}$. Then applying Theorem \ref{plgd} (taking $M=A^d$) we obtain a basic element $w=(v_1+\lambda_1v_{d+1},\ldots,v_d+\lambda_dv_{d+1})$ of $A^d$, for some $\lambda_i\in A$. Now since $A^d$ is a free (in particular, a projective) module, every basic element is a unimodular row. This concludes the proof. \qed The next proposition is similar to the well-known Quillen's splitting lemma \cite[Theorem 1]{Q}. Here we rephrase it in our setup with an added conclusion, which is crucial for this article. \begin{proposition}\label{splitting} Let $R=\bigoplus_{i\ge 0} R_i$ and $s,t\in R_0$ such that $\langle s\rangle +\langle t \rangle =R_0$. Let $\eta\in \GL_n(R_{st},(R_+)_{st})$ such that $e_1\eta=e_1$. Then there exist $\eta_1\in \GL_n(R_s,(R_+)_{s})$ and $\eta_2\in \GL_n(R_t,(R_+)_{t})$ such that \begin{enumerate} \item $\eta=(\eta_1)_t(\eta_2)_s$, \item $e_1\eta_i=e_1$, for $i=1,2$. \end{enumerate} \end{proposition} \paragraph{Proof} Let ``bar" denote going modulo the ideal $R_+$. We define $\chi(X):= \widetilde{\Gamma_{SW}}(\eta)$. Using Lemma \ref{isotopy} we obtain $$\chi(X)\in \GL_{n}(R_{st}[X]) \text{ such that } \chi(0)=\overline\eta= \text{Id} \text{ and }e_1\chi(X)=e_1.$$Applying Quillen’s splitting lemma \cite[Theorem 1, paragraph 2]{Q}, one can find $g = s^N\in R_0$ with $N\in \mathbb N$ such that $$\chi(X)\chi(gX)^{-1}\in \GL_{n}(R_s[X])\text{ and }\chi(gX)\in \GL_{n}(R_t[X]).$$ Since $e_1\chi(X)=e_1$, we further obtain that $e_1\chi(gX)=e_1$ and $e_1\chi(X)\chi(gX)^{-1}=e_1$. Let us define $\eta_1:=\chi(1)\chi(g)^{-1}$ and $\eta_2:=\chi(g)$. We observe the matrix $\chi(X)$ has the property that $\overline{\chi(a)}=\text{Id}$ for any $a\in R_0$. This implies $\overline \eta_i=\text{Id}$, for $i=1,2$. Therefore, we get the following. \begin{enumerate}[\quad \quad (1)] \item $\eta=(\eta_1)_t(\eta_2)_s$; \item $\eta_1\in \GL_{n}(R_s,(R_+)_{s})$; \item $\eta_2\in \GL_{n}(R_t,(R_+)_{t})$; \item $e_1\eta_i=e_1$, for $i=1,2$. \end{enumerate}This concludes the proof.\qed The next lemma concerns the patching of two invertible matrices in a graded ring. One may compare this with \cite[Lemma 2]{P}. \begin{lemma}\label{patching} Let $R=\bigoplus_{i\ge 0} R_i$ and $s,t\in R_0$ such that $\langle s\rangle +\langle t \rangle =R_0$. Let $v\in \text{Um}_n(R,R_+)$. Assume that, there exist $\alpha_1\in \GL_{n}(R_s,(R_+)_s)$ and $\alpha_2\in \GL_n(R_t,(R_+)_t)$ such that $v\alpha_i=e_1$, for $i=1,2$. Then there exists an $\alpha\in \GL_n(R,R_+)$ such that $v\alpha=e_1$. \end{lemma} \paragraph{Proof} Let ``bar'' denote going modulo $R_+$. Let us define $\eta:=(\alpha_1)_t^{-1}(\alpha_2)_s\in \GL_{n}(R_{st})$. Then we note that $\overline\eta=\text{Id}$ and $e_1\eta=e_1$. Applying Proposition \ref{splitting} there exist $\eta_1\in \GL_n(R_s,(R_+)_{s})$ and $\eta_2\in\GL_n(R_t,(R_+)_{t})$ such that \begin{enumerate}[\quad \quad (a)] \item $\eta=(\eta_1)_t(\eta_2)_s$, \item $e_1\eta_i=e_1$, for $i=1,2$. \end{enumerate} We now define $\sigma_1:=\alpha_1\eta_1\in \GL_{n}(R_s,(R_+)_s)$ and $\sigma_2:=\alpha_2\eta_2^{-1}\in \GL_{n}(R_t,(R_+)_t)$. Here we notice that $v\sigma_i=e_1$ $(i=1,2)$. Because of $\eta=(\alpha_1)_t^{-1}(\alpha_2)_s =(\eta_1)_t(\eta_2)_s$, we have $(\sigma_1)_t=(\sigma_2)_s$. Therefore, by \cite[Proposition 2.2, page no 211]{Lam} there exists a unique $\alpha\in \GL_{n}(R,R_+)$ such that $\alpha_s=\sigma_1$ and $\alpha_t=\sigma_2$. Furthermore, the matrix $\alpha$ takes $v$ to $e_1$ as it is true locally. \qed \paragraph{\bf Notation} Let $A$ be a ring and let $G\subset \GL_{n}(A)$ be a subgroup. For any $u,v\in \text{Um}_n(A)$, we define $ u \sim_{G} v$ if there exists an $\epsilon\in G$ such that $u\epsilon=v$. We denote the set $\{v\in \text{Um}_n(A): v \sim_{G} e_1\}$ by the notation $e_1G$. \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}\label{rpvul} One may wonder whether it is possible to improve Quillen's splitting lemma (for an arbitrary ring) in such a way that both the splitting matrices fix the canonical vector $e_1$. Unfortunately, achieving such an improvement is not feasible. To illustrate this, here we argue as follows: consider a ring $A$ of dimension $d\ge 2$. We show that such an improvement of Quillen's splitting lemma will ultimately lead to the conclusion that $\text{Um}_{d+1}(A)=e_1\text{SL}_{d+1}(A)$, which as discussed in the Introduction is not true. We choose a $v\in \text{Um}_{d+1}(A)$. Then we can always find a non-zero divisor $s\in R$ such that $v\alpha_1= e_1$, for some $\alpha_1\in {\text{E}_{d+1}(A_s)}$. As $s\in \text{Jac}(A_{1+\langle s \rangle })$ is a non-zero divisor, it is not difficult to establish that $v\alpha_2= e_1$, for some $\alpha_2\in {\text{E}_{d+1}(A_t)}$ and $t\in 1+\langle s \rangle$. Now, if the elementary matrix $\eta=(\alpha_1)_t^{-1}(\alpha_2)_s$ splits in such a way that each of its splitting matrices fixes $e_1$, then applying the arguments given in Lemma \ref{patching} one can obtain an $\alpha\in \text{SL}_{d+1}(A)$ such that $v\alpha=e_1$. \begin{numtheorem}\label{umch} Let $R=\bigoplus_{i\ge 0} R_i$ be a graded domain of dimension $d\ge 2$ such that $\dim(R_0)\ge 1$. Then for any $v\in \text{Um}_{d+1}(R,R_+)$ there exists an $\alpha\in \GL_{d+1}(R,R_+)$ such that $v\alpha=e_1$. As a consequence $$ \text{Um}_{d+1}(R)=e_1\text{H}_{d+1}(R).$$ \end{numtheorem} \paragraph{Proof} Let $v\in \text{Um}_{d+1}(R)$ and $S=R_0\setminus\{0\}$. Let ``bar" denote going modulo $R_{+}$. As $R$ has a non-trivial grading the ideal $R_+\not=0$. In particular, we get $\text{ht}(R_+)\ge 1$. Hence, altering $v$ by an elementary matrix we may further assume that $\overline v=\overline e_1$. Now if there exists an $\alpha\in \GL_{d+1}(R,R_+)$ such that $v\alpha=e_1$, then it follows from Lemma \ref{isotopy} and the fact $\text{E}_{d+1}(R)\subset \text{H}_{d+1}(R)$ that $ \text{Um}_{d+1}(R)=e_1\text{H}_{d+1}(R).$ Hence, to prove the theorem it is enough to find such an $\alpha$. Applying Proposition \ref{dogr} we get $\dim(S^{-1}R)\le d-1$. In particular, by standard stability theorems (e.g. one can use prime avoidance lemma) to obtain that $\text{sr}(S^{-1}R)\le d$. Therefore, we can find an $s\in S$ such that $v\sim_{\text{E}_{d+1}(R_s)} e_1$. Let $\alpha_1\in \text{E}_{d+1}(R_s)$ be such that $v\alpha_1=e_1$. Furthermore, we may replace $\alpha_1$ by $\alpha_1 \overline\alpha_1^{-1}$ and assume that $\overline\alpha_1=\text{Id}.$ Let $\MT=\{1+sr:r\in R_0\}$ and $B=\MT^{-1}R$. Since $\MT\subset R_0$, the ring $B$ is also a graded ring. Moreover, we note that $s\in \text{Jac}(\MT^{-1}R_0)$. Hence, applying Proposition \ref{dogr} it follows that $\dim(B_s)\le d-1.$ Therefore, by Lemma \ref{dsr} we get $\text{sr}(B)\le d$. Thus, there exists an $\alpha_2\in \text{E}_{d+1}(B)$ such that $v\alpha_2=e_1$. As again we may replace $\alpha_2$ by $\alpha_2 \overline\alpha_2^{-1}$ and further assume that $\overline\alpha_2=\text{Id}.$ We can find $t\in \MT$ such that $\alpha_2\in \text{E}_{d+1}(R_t)$. Now applying Lemma \ref{patching} we can find an $\alpha\in \GL_{d+1}(R,R_+)$ such that $v\alpha=e_1$. This completes the proof.\qed \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}\label{umchrmk} Let $A$ be a regular ring of essentially finite type over a field. Then using \cite[Theorem 3.3]{TV} it is not difficult to establish the fact that $\text{H}_{n+1}(A)=\text{E}_{n+1}(A)$, for all $n\ge 2$. Hence, in Theorem \ref{umch}, additionally if we assume that $R$ is a regular ring of essentially finite type over a field, then we get $\text{Um}_{d+1}(R)=e_1\text{E}_{d+1}(R)$. However, we do not know whether the regularity of $R$ is actually necessary. \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem} One can remove the hypothesis that $\dim(R_0)\ge 1$ in Theorem \ref{umch} at the expense of the hypothesis that $\frac{1}{d!}\in R$ by utilizing the Swan-Weibel homotopy map and applying \cite[Corollary 2.5]{Rao}. In fact, the same yields the following: let $R=\bigoplus_{i\ge 0} R_i$ be a graded ring of dimension $d$ such that $\frac{1}{d!}\in R$. Then $\text{Um}_{d+1}(R)=e_1\text{H}_{d+1}(R)$. It is worth noting that the removal of the hypothesis ``$\frac{1}{d!}\in R$" from a cancellation problem is highly non-trivial (cf. \cite{AAS} and \cite{FRS}). In the remaining part of the section, we extend Theorem \ref{umch} over various rings, which are closely related to the class of graded rings considered in Theorem \ref{umch}. \begin{numcorollary}\label{ec1} Let $R$ and $d$ be as in Theorem \ref{umch} and let $A=\MS^{-1}R$, where $\MS\subset R$ is a multiplicative set such that $\MS$ is contained in the set of all non-zero divisors in $R$. Then $$\text{Um}_{d+1}(A)=e_1\text{H}_{d+1}(A).$$ \end{numcorollary} \paragraph{Proof} First we comment that, since $\text{E}_{d+1}(A)\subset \text{H}_{d+1}(A)$, the only non-trivial case is when $\dim(A)=d$. Hence, without loss of generality, we assume that $\dim(A)=d$. Let us choose $v\in \text{Um}_{d+1}(A)$. Then there exists a non-zero $x\in R$ such that (i) $v\in \text{Um}_{d+1}(R_x)$ and (ii) $\dim(R_x)=d$. Now it follows from \cite[Lemma 4.4]{sb1} that there exists $u\in \text{Um}_{d+1}(R)$ such that $u\sim_{\text{E}_{d+1}(R_x)}v$. Applying Theorem \ref{umch} we obtain an $\epsilon\in \text{H}_{d+1}(R)$ such that $u\epsilon =v$. This concludes the proof. \qed \begin{numcorollary}\label{ec2} Let $R$ and $d$ be as in Theorem \ref{umch}. Additionally, we assume that $R$ is an affine domain over a field. Then $$\text{Um}_{d+1}(R[X_1,\ldots,X_n])=e_1\text{SL}_{d+1}(R[X_1,\ldots,X_n]).$$ \end{numcorollary} \paragraph{Proof} We use Quillen Induction on $n$ to prove the theorem. For $n=0$ this follows from Theorem \ref{umch}. Now let us assume that $n>0$. Let $v\in \text{Um}_{d+1}(R[X_1,\ldots,X_n])$. We note that $R[X_1]=\bigoplus_{i\ge 0} R_i[X_1]$, where $R_i[X_1]=\{\sum_{j=1}^n a_jX_1^j: a_j\in R_i \text{ and } n\in \mathbb N \}$ and the sum is defined in the obvious way. As there exists a canonical surjection $R\ra\!\!\!\ra R_0$, the ring $R_0$ is also an affine domain over the same field, say $k$. Let us take $S=k[X_1]\setminus \{0\}\subset R_0[X_1]$. Therefore, we have $\dim(S^{-1}R_0[X_1])=\dim(R_0)$. As any maximal ideal $\mm$ of $R_0[X_1]$ is of height equal to $\dim(R_0)+1$, we have $S\cap \mm\not=\emptyset$. We take $B=S^{-1}R[X_1]$. Then it follows from Proposition \ref{dogr} that $\dim(B)\le d$. In particular, since $\dim(S^{-1}R_0[X_1])=\dim(R_0)$ we have $\dim(B)=d$. Then $B=\bigoplus_{i\ge 0} S^{-1}R_i[X_1](=\bigoplus_{i\ge 0} B_i$ say$)$ is also a graded affine domain over the field $k(X_1)$ of dimension $d$ such that $\dim(B_0)=\dim(R_0)\ge 1$. Applying induction hypothesis on $B[X_2,\ldots,X_n]$ we can find a monic polynomial $f\in S$ such that $$v\sim_{\text{SL}_{d+1}(D[X_1]_f)} e_1,$$ where $D=R[X_2,\ldots,X_n]$. Then by Affine Horrocks Theorem \cite[Theorem 3]{Q} the result follows.\qed We end this section with a theorem on the injective stability of $\text{K}_1(R)$, where $R$ is a graded domain. The proof is a straightforward consequence of Lemma \ref{LG} and \ref{dsr}. \begin{numtheorem}\label{ist} Let $R$ and $d$ be as in Theorem \ref{umch}. Additionally, for $n\ge 1$ we further assume that $R$ is an affine domain over a field. Then $$\text{SL}_{d+1}(R[X_1,\ldots,X_n])\cap \text{E}(R[X_1,\ldots,X_n])=\text{E}_{d+1}(R[X_1,\ldots,X_n]).$$ \end{numtheorem} \paragraph{Proof} We again apply Quillen Induction on $n$ to prove the theorem. We give the proof in cases.\\ \textbf{Case - 1.} Let us assume that $n=0$. Let $\alpha\in \text{SL}_{d+2}(R)\cap \text{E}(R)$ and let ``bar'' denote going modulo the ideal $R_+$. Since $\overline\alpha\in \text{SL}_{d+1}(R_0)$ and $R_0\subset R$, we may treat $\overline\alpha$ as an element of $\text{SL}_{d+1}(R)$. Moreover, we observe that as $\text{ht}(R_+)\ge 1$ we have $\dim(R_0)<\dim(R)$. Therefore, using standard stability theorem due to Vaser{\v{s}}te{\u{\i}}n, as stated in \cite[Theorem 3.2]{Va1}, we obtain that (1) $\alpha\in \text{SL}_{d+1}(R) \cap \text{E}_{d+2}(R)$ and (2) $\overline \alpha\in \text{E}_{d+1}(R_0)\subset \text{E}_{d+1}(R)$. We take $\beta=\alpha\overline\alpha^{-1}$. Then we note that $\beta \in \text{SL}_{d+1}(R) \cap \text{E}_{d+2}(R).$ Let us consider $\MT=R_0\setminus \{0\}$. Then applying Proposition \ref{dogr} we get $\dim(\MT^{-1}R)\le d-1$. Hence, again using \cite[Theorem 3.2]{Va1} on $\MT^{-1}R$ we obtain that $(\beta)_{\MT}\in \text{E}_{d+1}(\MT^{-1}R)$. There exists an $s\in \MT$ such that $\beta_s\in \text{E}_{d+1}(R_s)$. Let $S=\{1+sr:r\in R_0\}$ and let $B=S^{-1}R$. Then again by Proposition \ref{dogr} we obtain that $\dim(B_s)\le d-1$. Hence, using Lemma \ref{dsr} it follows that $\text{sr}(B)\le d$. We again apply \cite[Theorem 3.2]{Va1} to obtain that $(\beta)_S\in \text{E}_{d+1}(B)$. We choose $t\in S$ such that $\beta_t\in \text{E}_{d+1}(R_t)$. Now it follows from Lemma \ref{LG} that $\beta\in \text{E}_{d+1}(R)$. Because of $\overline\alpha\in \text{E}_{d+1}(R)$, we have $\alpha\in \text{E}_{d+1}(R)$.\\ \textbf{Case - 2.} Now let us assume that $n>0$. Then applying Quillen Induction on $n$ as described in Corollary \ref{ec2} and using \cite[Corollary 5.7]{AASSC} suitably one may conclude the proof.\qed \section{Efficient generation of ideals} This section is devoted to studying the efficient generation problem for top height ideals in a ring. Before presenting the main theorem of this section, we need some preparation. We begin this section with the following lemma, which is a consequence of Lemma \ref{plgd} and Theorem \ref{eept}. \begin{lemma}\label{ml} Let $A$ be a ring of dimension $d\ge 2$. Assume that, there exists a non-zero divisor $s\in A$ such that $\dim(A_s)<d$. Let $I\subset A$ be an ideal such that $\mu(I/I^2)=\text{ht}(I)=d$. Then any set of generators of $I=\langle f_1,\ldots,f_d\rangle +I^2$ lifts to a set of generators of $I$. \end{lemma} \paragraph{Proof} Applying Lemma \ref{plgd} one may obtain a generalized dimension function $\delta:\text{Spec}(A) \to \mathbb{N}$ such that $\delta(\mq)\le d-1$ for all $ \mq\in \text{Spec}(A)$ and $\delta(\mq)=\dim(A/\mq )$ for all $\mq\owns s$. Let $\p\in \text{Spec}(A)$. Suppose that, we have $I\subset \p$. As $\text{ht}(I)=d$ we must have $\text{ht}(\p)=d$. Since $\dim(A_s)<d$, the element $s$ is in $\p$. This implies $\delta(\p)=\dim(A/\p)=0$. As $A_{\p}$ is a local ring we have $\mu(IA_{\p}/I^2A_{\p})=\mu(IA_{\p})=d$. Therefore, we obtain that $\mu(IA_{\p})+\delta(\p)\le d$. Now if $I\not \subset \p$, then $\mu(IA_{\p})=1$. Thus also in this case we have $\mu(IA_{\p})+\delta(\p)\le d$. In particular, we get $\sup\{\mu(IA_{\p})+\delta(\p):\p\in \text{Spec}(A)\}\le d$. Hence, one may apply \cite[Theorem 0]{P} to find $e_i\in I^2$ $(i=1,\ldots,d)$ such that $I=\langle l_1,\ldots,l_d\rangle $, where $l_i=f_i+e_i$ . This completes the proof.\qed In the following proposition, we present a general criterion for the efficient generation of a top height ideal in an arbitrary ring. This criterion enables us to identify the essential requirements to apply Mohan Kumar's fundamental technique presented in \cite{NMK} to solve the efficient generation problem. By doing so, we are able to provide a unified approach in Theorem \ref{eg}. \begin{proposition}\label{egc} Let $A$ be a ring of dimension $d\ge 2$. Let $I\subset A$ be an ideal such that $\mu(I/I^2)=\text{ht}(I)=d$. Suppose that $I=\langle f_1,\ldots,f_d\rangle +I^2$. Moreover, we assume that there exists a non-zero divisor $s\in A$ and a multiplicative set $S\subset \{1+sr:r\in A\}$ such that the following hold. \begin{enumerate} \item $IA_s=\langle f_1,\ldots,f_d \rangle A_s+I^2A_s$, has a lift to a set of generators of $IA_s$, \item $\dim(S^{-1}A_s)<d$, and \item $\text{Um}_d(S^{-1}A_s)=e_1\text{H}_d(S^{-1}A_s)$. \end{enumerate} Then there exist $F_i\in A$ such that $I=\langle F_1,\ldots,F_d\rangle $, with $f_i-F_i\in I^2$. \end{proposition} \paragraph{Proof} Let $g_i\in IA_s$ be such that $IA_s=\langle g_1,\ldots,g_d\rangle A_s$ with $f_i-g_i\in I^2A_s$. We observe that, if $\dim(A_s)<d$, then applying Lemma \ref{ml} the proof follows. Hence, we assume that $\dim(A_s)=d$. Therefore, using (\cite[Lemma 5.6]{SMBB3} and \cite[Remark 5.7]{MPHIL}) we may further assume that $s\in \sqrt{I}$. Let us take $B=S^{-1}A$. Since $\dim(B_s)< d$, using Lemma \ref{ml} we can lift $f_i$'s to a set of generators of $IB$. In particular, we get $l_i\in IB$ such that $IB=\langle l_1,\ldots,l_d\rangle $ and $l_i-f_i\in I^2$, for $i=1,\ldots,d$. Since $s\in \sqrt{I}$, the row vectors $(g_1,\ldots,g_d) \text{ and } (l_1,\ldots,l_d)$ are in $ \text{Um}_{d}(B_s)$. Hence, by hypothesis (3) there exists an $\epsilon \in \text{H}_{d}(B_s)$ such that $(g_1,\ldots,g_d)\epsilon= (l_1,\ldots,l_d)$. As $\epsilon \in \text{H}_{d}(B_s)$ there exists a $\theta(T)\in \GL_{n}(B_s[T])$ such that $\theta(0)=\text{Id}$ and $\theta(1)=\epsilon.$ Since $A$ is a Noetherian ring and there are only finitely many $g_i$ and $l_i$, we can find $t\in S$ such that \begin{enumerate}[\quad\quad (1)] \item $I_t=\langle l_1,\ldots,l_d\rangle$ with $f_i-l_i\in I^2_t$; \item $\theta(T)\in \GL_{d}(A_{st}[T])$. \end{enumerate} Applying Quillen's splitting lemma \cite[Theorem 1]{Q} we obtain $\epsilon_1\in \GL_{d}(A_s)$ and $\epsilon_2\in \GL_d(A_t)$ such that $\epsilon=(\epsilon_1)_t(\epsilon_2)_s$. Because of $\langle s\rangle +\langle t \rangle =A$, one may apply a standard patching to obtain $F_i\in I$ such that $I=\langle F_1,\ldots,F_d\rangle $ with $F_i-f_i\in I^2$ for $i=1,\ldots,d$.\qed Now we are ready to prove the main theorem of this section. \begin{numtheorem}\label{eg} Let $R$ and $d$ be as in Theorem \ref{umch}. Let $A$ and $n$ be one of the following: \begin{enumerate} \item $A=R$ and $n=\dim(R)=d\ge 3$. \item $A=\MS^{-1}R$, where $\MS\subset R$ is a multiplicative set contained in the set of all non-zero divisors in $R$ such that $\dim(A)=\dim(R)$ and $n=\dim(R)=d\ge 3 $. \item $A=B[M]$, where $B$ is a ring of dimension $ \ge 2$ and $M$ is a finitely generated commutative cancellative (not necessarily torsion free) monoid of rank $r\ge 1$. We take $n=\dim(B[M])$. \end{enumerate} Let $I\subset A$ be an ideal such that $\mu(I/I^2)=\text{ht}(I)=n$. Then any set of generators of $I=\langle f_1,\ldots,f_n\rangle +I^2$ lifts to a set of generators of $I$. \end{numtheorem} \paragraph{Proof} We will show that for each of the above rings all the hypotheses of Proposition \ref{egc} are satisfied. We handle these three rings separately in the following cases.\\ \textbf{Case - 1.} In this case, we assume that $A=R$. Let us take $\MT=R_0\setminus \{0\}$. Then by Proposition \ref{dogr} the dimension of $\MT^{-1}A$ is strictly smaller than $n$. Hence, applying \cite{NMK} we can lift $f_i$'s to a set of generators of $\MT^{-1}I$. Therefore, there exist $s\in \MT$ and $g_i\in A_s$ with $I_s=\langle g_1,\ldots,g_n \rangle $ such that $f_i-g_i\in I_s^2$ for $i=1,\ldots,n$. Let $S:=\{1+sr:r\in R_0\}\subset \{1+sx:x\in A\}$. Then again applying Proposition \ref{dogr} we have $\dim(S^{-1}A_s)<n$. Moreover, we observe that since $S\subset R_0$ and $s\in R_0$, the ring $S^{-1}R_s$ retains the grading induce from $R$. Because of $n=d\ge 3,$ using Theorem \ref{umch} we have $\text{Um}_{n}(S^{-1}R_s)= e_1\text{H}_{n}(S^{-1}R_s)$. Therefore, applying Proposition \ref{egc} we obtain the required lift.\\ \textbf{Case - 2.} In this case we assume that $A=\MS^{-1}R$. Let $\MT$ be as considered in Case - 1. Then as it was shown in the previous case that $\dim(\MT^{-1}R)<n$, which further implies that $\dim(\MT^{-1}A)<n$. Therefore, following the arguments in the previous case, we can find a non-zero divisor $s\in R_0$ and $g_i\in A_s$ such that $I_s=\langle g_1,\ldots,g_n\rangle $, with $f_i-g_i\in I_s^2$. Let us take $S=\{1+sr:r\in R_0\}\subset \{1+sx:x\in A\}$. Then $S^{-1}A_s=\MS^{-1}(S^{-1}R_s)$, where $S^{-1}R_s$ is a positively graded ring of dimension $\le n-1$. Hence, applying Corollary \ref{ec1} we get $\text{Um}_{n}(S^{-1}A_s)=e_1\text{H}_{n}(S^{-1}A_s)$. Now one may apply Proposition \ref{egc} to complete the proof.\\ \textbf{Case - 3.} In this case we take $A=B[M]$. First we note that for a monoid ring $B[M]$ we have $\dim(B[M])=\dim(B)+\text{rank}(M)$ \cite[Theorem 4.23]{gulbook}. Let $\MT$ be the set of all non-zero divisors in $B$. Then $\dim(\MT^{-1}B[M])=r$. Since $n>\dim(\MT^{-1}B[M])$ by \cite{NMK} we can lift $f_i$'s to a set of generators of $\MT^{-1}I$. Therefore, there exist $s\in \MT$ and $g_i\in B_s[M]$ with $I_s=\langle g_1,\ldots,g_n \rangle $ such that $f_i-g_i\in I_s^2$ for $i=1,\ldots,n$. Let $S=\{1+sr:r\in B\}$ and let $C=S^{-1}B$. Then as $\dim(C_s)<\dim(B)$ we have $\dim(S^{-1}B_s[M])=\dim(C_s[M])<n$. Therefore, using \cite[Theorem 1.1]{gul} we get $\text{Um}_{n}(C_s[M])= e_1\text{E}_{n}(C_s[M])$. Now one may apply Proposition \ref{egc} to complete the proof.\qed We now provide an example that proves the necessity of the hypothesis $\dim(R_0)>0$ in Theorem \ref{eg} (1). We essentially use the example constructed by Bhatwadekar, Mohan Kumar and Srinivas \cite[Example 6.4]{BR} in which they provided a non-smooth graded domain (with the degree zero subring a field) over which Nori's question has a negative answer. \refstepcounter{theorem}\paragraph{{\bf Example} \thetheorem}\label{ex1} Consider the graded domain $B=\frac{\mathbb{C}[X,Y,Z,W]}{\langle X^5+Y^5+Z^5+W^5\rangle }$. By \cite[Example 6.4]{BR} there exist (1) an ideal $I\subset B[T]$ such that $\mu(I/I^2T)=\text{ht}(I)=3$ and (2) a set of generators $I=\langle f_1,f_2,f_3\rangle +I^2T$, which does not lift to a set of generators of $I$. Let $S=\mathbb{C}[T]\setminus \{0\}$ and $C=S^{-1}B[T]$. Now we choose the grading of $C$ induced by $B$ [see the proof of Corollary \ref{ec2}]. Then $C$ is a graded domain of dimension $3$ such that the degree zero subring of $C$ is the field $\mathbb{C}(T)$. Then $IC=\langle f_1,f_2,f_3\rangle C+I^2C$ does not lift to a set of generators of $IC$. As if such a lift exists then by \cite[Theorem 3.10]{MKD1} one can lift $I=\langle f_1,f_2,f_3\rangle +I^2T$ to a set of generators of $I$. The following is an interesting consequence of the previous theorem. For monoid rings, this is an improvement of \cite[Theorem 3.4]{mkmm}. \begin{numcorollary}\label{eue} Let $A$ and $n$ be as in Theorem \ref{eg}. Let $P$ be a projective $A$-module with trivial determinant of rank $n$. Then $P$ has a unimodular element. \end{numcorollary} \paragraph{Proof} By a theorem due to Eisenbud and Evans \cite{EE} we can find a surjection $P\ra\!\!\!\ra I\subset A$ such that $\text{ht}(I)=d$. Now the result follows from Theorem \ref{eg} and applying subtraction principle \cite[Corollary 3.5]{SMBB3}.\qed \refstepcounter{theorem}\paragraph{{\bf Example} \thetheorem}\label{ex2} Here we show that the hypothesis $\dim(R_0)\ge 1$ is also necessary in Corollary \ref{eue}, where $A=\bigoplus_{i\ge 0} R_i$. Let $C,S,B,I$ and $f_i$ be as in Example \ref{ex1}. Recall that the $d$-th Euler class group $E^d(D[T])$ and the weak Euler class group $E_0^d(D[T])$ as defined in \cite{MKD1}, where $D$ is a ring of dimension $d\ge 3$ such that $\mathbb Q\subset D$. We consider $(I,\omega_I)\in E^d(B[T])$, where $\omega_I$ is the local orientation induced by $I=\langle f_1, f_2, f_3\rangle +I^2$. Applying \cite[Theorem 2.7]{SMBRS2} we can find a projective $B[T]$-module $P$ (with trivial determinant) of rank $3$ and a surjection $\theta:P\ra\!\!\!\ra I$. We claim that $S^{-1}P$ does not have a unimodular element. First, we note that if $S^{-1}P$ has a unimodular element, then there exists $f\in \mathbb C[T]\setminus \{0\}$ such that $P_f$ has a unimodular element. But then it follows from \cite[Theorem 3.4]{BRS} that $P$ has a unimodular element. Hence, to prove our claim it is enough to show that $P$ does not have a unimodular element. We fix a trivialization $\chi:\wedge^3 P\by \sim B[T]$. Then it follows from \cite[Proposition 5.8]{MKD1} and \cite[Theorem 3.4]{SBMKD2} that $E^3(B[T])\cong E^3_0(B[T])$. In particular, this give us $e(P,\chi)=(I,\omega_I)$ in $E^3(B[T])$. Moreover, using \cite[Corollary 4.11]{MKD1} we obtain that $P$ has a unimodular element if and only if $(I,\omega_I)=0$ in $E^3(B[T])$. Now if $(I,\omega_I)=0$ in $E^3(B[T])$, then one may also lift $IC=\langle f_1,f_2,f_3\rangle C+I^2C $ to a set of generators of $IC$. But as it is shown in Example \ref{ex1} that this is not feasible. Hence, the module $P$ does not have a unimodular element. We conclude this section with a theorem that extends Theorem \ref{eg} and Corollary \ref{eue} to the case where the dimension of the graded ring is $2$. \begin{numtheorem}\label{egd2} Let $R$ be as in Theorem \ref{umch} and $\dim(R)=2$. Let $P$ be a projective $R$-module of rank $2$ with trivial determinant. Suppose $I\subset R$ is an ideal such that $I=\langle f_1, f_2\rangle +I^2$. Then \begin{enumerate}[\quad\quad (1)] \item $P$ is a free module and \item there exist $F_i\in I$ such that $I=\langle F_1,F_2\rangle $, with $F_i-f_i\in I^2$. \end{enumerate} \end{numtheorem} \paragraph{Proof} We consider $S=R_0\setminus \{0\}$. Then by Lemma \ref{dogr} we get $\dim(S^{-1}R)\le 1$. Since determinant of $P$ is trivial, it follows from \cite{S} that the module $S^{-1}P$ is free. As $P$ is finitely generated module over a Noetherian ring there exists an $s\in S$ such that $P_s$ is a free module. Let us take $T=\{1+sr:r\in R_0\}$ and $B=T^{-1}R$. Then applying Lemma \ref{dogr} we obtain that $\dim(B_s)\le 1$. Hence, applying Lemma \ref{plgd} and Theorem \ref{eept} we obtain that, the module $T^{-1}P$ has a unimodular element. Moreover, as determinant of $P$ is trivial, the module $T^{-1}P$ is free. Thus, there exists a $t\in T$ such that $P_t$ is a free module. Therefore, the Quillen ideal $J(R_0,P)$ of $P$ is $R_0$. In other words, we have $P\cong \frac{P}{PR_+}\otimes R$. As $\dim(R_0)=1$, again by \cite{S} the $R_0$-module $\frac{P}{PR_+}$ has a unimodular element and hence free. Thus, the module $P$ is free. Now we consider the ideal $I=\langle f_1,f_2\rangle +I^2$. Since any unimodular row of length two can be completed to an invertible matrix, using a standard patching argument we may obtain a projective $R$-module $Q$ of rank $2$ with trivial determinant and a surjection $\gamma:Q\ra\!\!\!\ra J$ such that $\gamma$ locally lifts $\{f_1,f_2\}$. Now as $Q$ is free by the previous case, the theorem concludes.\qed \section{Cancellation of projective modules}\label{cansection} This section is devoted to investigating the cancellation property of projective modules over a graded ring. We begin with a lemma, which is an analogy of \cite[Lemma 2]{P} in our setup. \begin{lemma}\label{cancri} Let $R=\bigoplus_{i\ge 0} R_i$ and $M,M'$ be $R$-modules. Suppose that there exist $s,t\in R_0$ be co-maximal and isomorphisms $\sigma_1:M_s\by \sim M_s'$ and $\sigma_2:M_t\by \sim M_t'$ such that \begin{enumerate} \item $(\sigma_1)_t\equiv (\sigma_2)_s\mod(R_+)_{st}$; \item $M_{st}$ is a free module. \end{enumerate} Then there exists an isomorphism $\sigma:M\by \sim M'$ such that (i) $ \sigma_s\equiv \sigma_1\mod(R_+)_s$ and (ii) $\sigma_t\equiv \sigma_2\mod(R_+)_t$. \end{lemma} \paragraph{Proof} Let $\text{rank}(M_{st})=n$ and let ``bar" denote going modulo $R_+$. Since $M_{st}$ is free there exists an isomorphism $\tau: M_{st}\by \sim R_{st}^{n}$. For an arbitrary isomorphism $\gamma:M_{st}\by \sim M_{st}$ we now consider the following commutative diagram $$\begin{tikzcd} M_{st}\arrow[r,"\gamma"]\arrow[d,"\tau"] & M_{st} \arrow[d,"\tau"] \\ R_{st}^{n}\arrow[r,"\widetilde{\gamma}"] & R_{st}^{n} \end{tikzcd}$$ where $\widetilde{\gamma}=\tau\gamma\tau^{-1}$. We will call $\widetilde{\gamma}$ is induced from $\gamma $ and $\tau$. We take $\gamma=(\sigma_1)_{t}^{-1}\circ (\sigma_2)_s:M_{st}\by \sim M_{st}$. Then from (2) it follows that $\overline \gamma=\text{Id}$. We consider the isomorphism $\widetilde{\gamma}\in \GL_{n}(R_{st})$ induced from $\gamma$ and $\tau.$ Since $\overline \gamma=\text{Id}$, we have $\overline {\widetilde{\gamma}}=\text{Id}$. Applying Lemma \ref{isotopy} we can get a matrix, say $\widetilde{\theta(T)}\in \GL_{n}(R_{st}[T])$ such that $\widetilde{\theta(0)}=\text{Id}$ and $\widetilde{\theta(1)}=\widetilde{\gamma}$. Let us take $\theta(T)=(\tau\otimes R_{st}[T])^{-1}\widetilde{\theta(T)}(\tau\otimes R_{st}[T])$. Then we observe that $\theta(T)\in \Aut(M_{st}[T])$ such that $\theta(0)=\text{Id}$ and $\theta(1)=\gamma$. Now we define the isomorphism $\phi(T)=(\sigma_1\otimes R_{st}[T])\circ\theta(T):M_{st}[T]\by \sim M'_{st}[T]$. Then the proof follows from applying \cite[Lemma 1]{P}.\qed Now we present the main theorem of the section. \begin{numtheorem}\label{can1} Let $R$ and $d$ be as in Theorem \ref{umch}. Let $P$ be a projective $R$-module of rank $d$ such that $J(R_0,P)\not=0$. Then $P$ is cancellative. \end{numtheorem} \paragraph{Proof} First we comment that since $J(R_0,P)\not=0$ there exists an $s\in R_0\setminus\{0\}$ such that $P_s$ is a free module. To see this let us choose a non-zero element $k\in R_0$ such that $P_k$ is an extended projective module from $R_0$. Consider the multiplicative set $T=R_0\setminus \{0\}$. Since $P$ is a projective module so is $P/PR_+.$ Let us take $Q=P/PR_+$. As $k\in T$, we have $T^{-1}P\cong T^{-1}Q\otimes T^{-1}R$. Since $T^{-1}Q$ is a projective module over the field $T^{-1}R_0$, it is free. Now it follows from the finite generation of $P$ that there exists $s\in R_0$ such that $P_s$ is free. Let $(f,p)\in \text{Um}(R\oplus P)$. Since $R$ has a non-trivial grading, the ideal $R_+$ is non-zero. In particular, the height of $R_+$ is $\ge1$. Hence, going modulo a non-zero element $g\in R_+$ and altering $(f,p)$ suitably via an element of $\text{E}(R\oplus P)$ we may assume that $f-1\in \langle g \rangle $ and $p\in \langle g \rangle P$. We take $P'=\frac{R\oplus P}{(f,p)R}$. Then to prove the theorem it is enough to show that there exists an isomorphism $\sigma:P\by \sim P'$. Moreover, we comment on an observation that finding a $\sigma:P\by \sim P'$ such that $\overline\sigma=\text{Id}$ is equivalent to find an $\alpha\in \Aut(R\oplus P)$ such that $\alpha(f,p)=(1,0)$ and $\overline \alpha=\text{Id}$. Let ``bar'' denote going modulo $R_+$ as well as $PR_+$. As $P_s$ is free, by Theorem \ref{umch} we can find $\alpha_1\in \Aut(R_s\oplus P_s)$ such that (1) $\alpha_1(f,p)=(1,0)$ and (2) $\overline \alpha_1=\text{Id}$. Then $\alpha_1$ will induce an isomorphism $\sigma_1:P_s\by \sim P'_t$ such that $\overline \sigma_1=\text{Id}$. Let $S=\{1+sr:r\in R_0\}$. We denote $B=S^{-1}R$, $L=S^{-1}P$ and $L'=S^{-1}P'$. We note that $s\in \text{Jac}(S^{-1}R_0)$. Therefore, by Proposition \ref{dogr} we get $\dim(B_s)\le d-1$. Hence, using Lemma \ref{plgd} we can obtain a generalized dimension function $\delta:\text{Spec}(B)\to \mathbb{N}$ such that $\delta(\p)\le d-1$ for all $ \p\in \text{Spec}(B)$. Since $p\in \langle g \rangle P$, we note that $(f,p)\in \text{Um}(R\oplus \langle g \rangle P)$. Moreover, the module $\langle g \rangle L$ is a projective $B$-module of rank $d$. Hence, applying Theorem \ref{eept} we can find $p'\in P$ such that $q:=p+gfp'\in \text{Um}(\langle g \rangle L)$. Moreover, as $q\in \langle g \rangle L$ and $f-1\in \langle g \rangle $ one may obtain a transvection $\alpha_2\in \text{E}(B\oplus L)$ such that $\alpha_2(f,p)=(1,0)$ and $\overline{\alpha}_2=\text{Id}$. Then $\alpha_2$ will induce an isomorphism $\sigma_2:L\by \sim L'$ such that $\overline{\sigma}_2=\text{Id}$. Since all modules are finitely generated (over a Noetherian ring) there exists an isomorphism $\sigma_2:P_t\by \sim P'_t$ such that $\overline{\sigma}_2=\text{Id}$, for some $t\in S$. Now applying Lemma \ref{cancri} we get the required isomorphism $\sigma:P\by \sim P'$ such that $\overline \sigma=\text{Id}$. This completes the proof.\qed \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem} Let $R$ and $P$ be as in Theorem \ref{can1}, and we consider $S=R_0\setminus \{0\}$. Then $J(R_0,P)\not=0 $ if and only if $S^{-1}P$ is free. If $S^{-1}R_0=\overline{\mathbb{Q}}$ and $\wedge^d P\cong R$, then it follows from \cite[Theorem 6.4.2]{AS} and \cite{S} that $J(R_0,P)\not=0$. We now discuss an interesting consequence of Theorem \ref{can1}. \begin{numcorollary}\label{soprc} Let $R$ be an integral domain of dimension $d\ge 1$, and $A$ be a graded subring of $R[T]$ containing $R$ such that $\dim(A)=d+1$. Let $P$ be a projective $A$-module of rank $d+1$, so that the determinant of $P$ is extended from the base ring $R$. Then $P$ is cancellative. \end{numcorollary} \paragraph{Proof} Let us take $\MT=R\setminus \{0\}$. Since $\dim(A)=d+1$, there exists an $a\in \MT$ such that $A_a$ contains a monic polynomial in $T$. Then $\MT^{-1}A\hookrightarrow (\MT^{-1}R)[T]$ is an integral extension. This further implies that for any multiplicative set $S\subset R$ containing $a$, we must have $\dim(S^{-1}A)=\dim(S^{-1}R)+1$. Therefore, in view of Theorem \ref{can1} it is enough to show that $\text{ht}(J(R,P))\ge 1$. To prove this we observe that, since $\dim(\MT^{-1}R)=0$, we have $\dim(\MT^{-1}A)= 1$. As the determinant of $P$ is extended from $R$, applying \cite{S} the module $\MT^{-1}P$ is a free $\MT^{-1}A$-module. Hence, there exists an element $s\in \MT$ such that $P_s$ is a free $A_s$-module. That is, the non-zero element $s\in J(R,P)$.\qed \section{Applications}\label{a1} This section is devoted to establishing some consequences of Theorem \ref{eg}. \renewcommand{\begin{numtheorem}}{\begin{numtheorem}} \renewcommand{\end{numtheorem}}{\end{numtheorem}} \renewcommand{\begin{numcorollary}}{\begin{numcorollary}} \renewcommand{\end{numcorollary}}{\end{numcorollary}} \def\refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}{\refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}} \subsection{On a question of Nori: non-smooth graded case} Let $R=\bigoplus_{i\ge 0} R_i$ be an affine graded domain of dimension $d\ge 3$ over a field $k$ such that $ \mathbb Q\subset k$ and $\dim(R_0)\ge 1$. In the next theorem, we show that Nori's question \cite{SM} on homotopy of sections of projective modules has an affirmative answer over $R$ even without the smoothness assumption. \begin{numtheorem}\label{NQ} Let $R$ be as in Theorem \ref{umch} and $d\ge 3$. Moreover, we assume that $R$ is an affine algebra over an infinite field such that $\frac{1}{d!}\in R$. Let $I\subset R[T]$ an ideal such that $\mu(I/I^2T)=\text{ht}(I)=d$. Then any set of generators of $I/I^2T$ lifts to a set of generators of $I$. \end{numtheorem} \paragraph{Proof} If $I$ contains a monic polynomial in $T$, then the result follows from \cite{SM}. Hence, without loss of generality, we may assume that $I$ does not contain a monic polynomial in $T$. Let $I=\langle f_1,\ldots,f_d\rangle +I^2T$. First, we comment that in \cite[Theorem 3.10]{MKD1} the hypothesis that the ring containing $\mathbb{Q}$ can be weakened by assuming the ring contains an infinite field such that $d!$ is invertible. We denote $R(T)=\MT^{-1}R[T]$, where $\MT$ be the ring consisting of all monic polynomials in $R[T]$. In view of \cite[Theorem 3.10]{MKD1} it is enough to prove that there exist $F_i\in IR(T)$, such that $IR(T)=\langle F_1,\ldots,F_d\rangle$ and $f_i-F_i\in I^2R(T)$. The proof is devoted to establishing this only. Consider the multiplicative set $S=\{f\in R_0[T]:f\text{ is a monic polynomial}\}$ and let $B=S^{-1}R[T]$. Then by Proposition \ref{dogr} we have $\dim(B)\le d$. Moreover, one can use the arguments given in Corollary \ref{ec2} to establish the fact that $B=\bigoplus_{i\ge 0} B_i$ is also a graded domain of dimension $d$ such that $\dim(B_0)=\dim(S^{-1}R_0[X_1])\ge 1$. As $I$ is not containing a monic polynomial we have $\text{ht}(IB)\ge d$. Moreover, since $T$ is a unit in $B$, we have $IB=\langle f_1,\ldots,f_d\rangle B+I^2B$. Now, applying Theorem \ref{eg} we obtain $F_i\in IB$ such that $IB=\langle F_1,\ldots,F_d\rangle B$ and $f_i-F_i\in I^2B$. Since $B$ is a subring of $R(T)$ we get $IR(T)=\langle F_1,\ldots,F_d\rangle R(T)$ such that $f_i-F_i\in I^2R(T)$. This concludes the proof. \qed \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem}\label{noh} It follows from \cite[Example 6.4]{BR} that in Theorem \ref{NQ} the hypothesis $\dim(R_0)>0$ is necessary. \begin{numcorollary}\label{egpa} Let $R$ and $d$ be as in Theorem \ref{NQ}. Let $I\subset R[T]$ an ideal such that $\mu(I/I^2)=\text{ht}(I)=d$. Then any set of generators of $I/I^2$ lifts to a set of generators of $I$. \end{numcorollary} \paragraph{Proof} Let $I=\langle f_1,\ldots,f_d\rangle +I^2$ and let $I(0)=\{f(0):f\in R[T]\}$. Then as $R$ contains an infinite field without loss of generality we may assume that $I(0)\subset R$ is an ideal of height $\ge d$ (for details see the proof of \cite[Theorem 3.4]{BRS01}). If $\text{ht}(I(0))>d$, then we can always lift any set of generators of $I(0)/I(0)^2$. Now if $\text{ht}(I(0))=d$, then applying Theorem \ref{eg} there exist $a_i\in I$ such that $I(0)=\langle a_1,\ldots,a_d\rangle $, with $f_i(0)-a_i\in I(0)^2$, for $i=1,\ldots,d$. Hence, by \cite[Remark 3.9]{BR} there exist $g_i\in I$ such that $I=\langle g_1,\ldots,g_d\rangle +I^2T$ with $f_i-g_i\in I^2$ and $g_i(0)=a_i$, for $i=1,\ldots,d$. Now the result follows from Theorem \ref{NQ}.\qed \begin{numcorollary}\label{eueipe} Let $R$ and $d$ be as in Theorem \ref{NQ}. Let $P$ be a projective $R[T]$-module with trivial determinant of rank $d$. Then $P$ has a unimodular element. \end{numcorollary} \paragraph{Proof} Applying a theorem due to Eisenbud and Evans \cite{EE} we can find a surjection $P\ra\!\!\!\ra I\subset A[T]$ such that $\text{ht}(I)=d$. Then the result follows from Corollary \ref{egpa} and subtraction principle as stated in \cite[Corollary 4.13]{MKD1} (taking $Q=(R[T])^{d-1}$, $I_1=R[T]$ and $I_2=I$).\qed \refstepcounter{numtheorem}\paragraph{{\bf Remark} \thenumtheorem} One can remove the restriction on the base field in Corollary \ref{eueipe} in the following way: let $P$ be a projective $R[T]$-module of rank $d$ with trivial determinant. Recall that the ring $R(T)$ is obtained by localizing $R[T]$ with respect to the multiplicative set consisting of all monic polynomials in $R[T]$. Then, in view of \cite[Theorem 5.2 and Remark 5.3]{SMBHLRR}, it is enough to show that the modules $P/TP$ and $P\otimes R(T)$ have unimodular elements. Let $S$ be the multiplicative set consisting of all monic polynomials in $R_0[T]$. Then, it follows from Corollary \ref{eue} that $P/TP$ and $S^{-1}P$ (and hence $P\otimes R(T)$) have unimodular elements. \subsection{Generating ideals up to projective equivalence} Recall that, two ideals $I \text{ and } J$ in a ring $ A$ are said to be \textit{projectively equivalent} if some power of $I$ and some power (usually different) of $J$ have the same integral closure. The following theorem is an improvement of \cite{katz} in our setup. \begin{numtheorem}\label{pe} Let $A$ and $n$ be as in Theorem \ref{eg}. Let $I\subset A$ be an ideal of height $\ge 2$. Then there exists an ideal $J\subset A$ projectively equivalent to $I$ satisfying $\mu(J)\le n$. \end{numtheorem} \paragraph{Proof} First we observe that combining the results \cite[Proposition 2.2]{DRA} and Theorem \ref{eg} one can prove the following: let $K\subset A$ be an ideal such that (i) $\mu(K/K^2)\le n$ and (ii) $\text{ht}(K)\ge 2$. Then $\mu(K)\le n$. Applying \cite{katz} we obtain an ideal $J\subset A$ such that (1) $I$ and $J$ are projectively equivalent, (2) $\text{ht}(J)\ge 2$ and (3) $\mu(J/J^2)\le n$. Now it follows from the previously mentioned observation that $\mu(J)\le n$.\qed \section*{Acknowledgment} I thank Mrinal Kanti Das for suggesting Theorem \ref{pe}. \end{document}	arXiv
Current Search: Research Repository (x) » * (x) » Rosewood State Training School (Owings Mills, Md.) (x) » Mathematics (x) Analytical Results on the Role of Flexibility in Flapping Propulsion. Moore, Nicholas Wing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with... Show moreWing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with expressions for how thrust and efficiency are modified by compliance. The solutions recover a range of realistic behaviours and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple robust estimates afforded by our theory may prove valuable even in situations where details of the flapping motion and wing geometry differ. FSU_migr_math_faculty_publications-0002, 10.1017/jfm.2014.533 Acknowledging the Religious Beliefs Students Bring into the Science Classroom: Using the Bounded Nature of Science. Southerland, Sherry A., Scharmann, Lawrence Conrad Scientific knowledge often appears to contradict many students' religious beliefs. Indeed, the assumptions of science appear contradictory to the metaphysical claims of many religions. This conflict is most evident in discussions of biological evolution. Teachers, in attempts to limit the controversy, often avoid this topic or teach it superficially. Recently, there has been a political effort to "teach to the controversy" – which some see as a way of introducing religious explanations for... Show moreScientific knowledge often appears to contradict many students' religious beliefs. Indeed, the assumptions of science appear contradictory to the metaphysical claims of many religions. This conflict is most evident in discussions of biological evolution. Teachers, in attempts to limit the controversy, often avoid this topic or teach it superficially. Recently, there has been a political effort to "teach to the controversy" – which some see as a way of introducing religious explanations for biological diversity into science classrooms. Many science educators reject this approach, insisting that we limit classroom discussions to science alone. This "science only" approach leaves the negotiation of alternative knowledge frameworks to students, who are often ill-prepared for such epistemological comparisons. To support students' understanding of science while maintaining their religious commitments, this article explores the utility of emphasizing the boundaries of scientific knowledge and the need to support students in their comparison of contradictory knowledge frameworks. FSU_migr_ste_faculty_publications-0013, 10.1080/07351690.2013.743778 The Influence of RET's on Elementary and Secondary Teachers' Views of Scientific Inquiry. Bahbah, Sibel, Golden, Barry W. (Barry Wade), Roseler, Katrina, Enderle, Patrick, Saka, Yavuz, Southerland, Sherry A. This study explores in-service elementary and secondary science teachers' conceptions of the Nature of Scientific Inquiry and the influence participation in two different Research Experience for Teacher (RET) had on these conceptions. Participant teachers attended one of two six week RET programs in which they worked with scientists to engage in scientific inquiry. Before and after the RETs, teachers completed the Views of Scientific Inquiry (VOSI) questionnaire. Teachers' answers were... Show moreThis study explores in-service elementary and secondary science teachers' conceptions of the Nature of Scientific Inquiry and the influence participation in two different Research Experience for Teacher (RET) had on these conceptions. Participant teachers attended one of two six week RET programs in which they worked with scientists to engage in scientific inquiry. Before and after the RETs, teachers completed the Views of Scientific Inquiry (VOSI) questionnaire. Teachers' answers were analyzed to determine the degree of sophistication of their understanding of five facets of scientific inquiry. Both elementary and secondary teacher participants showed improvement in their understanding of nature of scientific inquiry as a result of program participation, and both programs were successful in supporting the development of inquiry conceptions, although secondary science teachers started and finished the RET's with a more sophisticated understandings of scientific inquiry. Areas of improvement for elementary teachers included the role of questions in science and the role subjectivity and creativity play in the processes of science, and for secondary teachers growth was seen in the role of questions, the relationship of data and evidence, the distinction of experiments and other means of investigations, and the varied methods of science. Implications of these results are discussed. FSU_migr_ste_faculty_publications-0014, 10.5539/ies.v6n1p117 Exponential Convergence Fourier Method and Its Application to Option Pricing with Lévy Processes. Gu, Fangxi, Nolder, Craig, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren D., Ökten, Giray, Florida State University, College of Arts and Sciences,... Show moreGu, Fangxi, Nolder, Craig, Huffer, Fred W. (Fred William), Kercheval, Alec N., Nichols, Warren D., Ökten, Giray, Florida State University, College of Arts and Sciences, Department of Mathematics Option pricing by the Fourier method has been popular for the past decade, many of its applications to Lévy processes has been applied especially for European options. This thesis focuses on exponential convergence Fourier method and its application to discrete monitoring options and Bermudan options. An alternative payoff truncating method is derived to compare the benchmark Hilbert transform. A general error control framework is derived to keep the Fourier method out of an overflow problem.... Show moreOption pricing by the Fourier method has been popular for the past decade, many of its applications to Lévy processes has been applied especially for European options. This thesis focuses on exponential convergence Fourier method and its application to discrete monitoring options and Bermudan options. An alternative payoff truncating method is derived to compare the benchmark Hilbert transform. A general error control framework is derived to keep the Fourier method out of an overflow problem. Numerical results verify that the alternative payoff truncating sinc method performs better than the benchmark Hilbert transform method under the error control framework. FSU_FA2016_Gu_fsu_0071E_13579 Understanding the Induction of a Science Teacher: The Interaction of Identity and Context. Saka, Yavuz, Southerland, Sherry A., Kittleson, Julie, Hunter, Todd The demanding first years of teaching are a time when many teachers leave the teaching profession or discard the reform-minded practice emphasized in teacher preparation. If we are to lessen teacher attrition and more effectively support teachers during their development, a better understanding of what occurs during their induction into the profession is needed. The question that drove this research was what factors influence how a beginning science teacher negotiates entry into teaching?... Show moreThe demanding first years of teaching are a time when many teachers leave the teaching profession or discard the reform-minded practice emphasized in teacher preparation. If we are to lessen teacher attrition and more effectively support teachers during their development, a better understanding of what occurs during their induction into the profession is needed. The question that drove this research was what factors influence how a beginning science teacher negotiates entry into teaching? Specifically, we sought to understand how a beginning science teacher's identities interact with the teaching context, how this interactions shapes his use of reform minded teaching practice, and how the negotiation of identity, context and practice influence a novice teacher's employment decisions. The study involved two years of data collection; data included classroom and school observations, questionnaires, interviews, and teaching artifacts (such as lesson plans and assessments). The results demonstrate how conflicts in identities, institutional expectations, and personal dispositions of this novice influenced his transition in becoming a member of his school community. Implications of these interactions for teacher preparation and support are provided. FSU_migr_ste_faculty_publications-0012, 10.1007/s11165-012-9310-5 A longitudinal study on predictors of early calculation development among young children at risk for learning difficulties. Peng, Peng, Namkung, Jessica M, Fuchs, Douglas, Fuchs, Lynn S, Patton, Samuel, Yen, Loulee, Compton, Donald L, Zhang, Wenjuan, Miller, Amanda, Hamlett, Carol The purpose of this study was to explore domain-general cognitive skills, domain-specific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical... Show moreThe purpose of this study was to explore domain-general cognitive skills, domain-specific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical competence, incoming calculations, socioeconomic status, and gender at the beginning of first grade and on calculation performance at four time points: the beginning of first grade, the end of first grade, the end of second grade, and the end of third grade. Latent growth modeling analysis showed that numerical competence, incoming calculation, processing speed, and decoding skills significantly explained the variance in calculation performance at the beginning of first grade. Numerical competence and processing speed significantly explained the variance in calculation performance at the end of third grade. However, numerical competence was the only significant predictor of calculation development from the beginning of first grade to the end of third grade. Implications of these findings for early calculation instructions among young at-risk children are discussed. FSU_pmch_27572520, 10.1016/j.jecp.2016.07.017, PMC5052117, 27572520, 27572520, S0022-0965(16)30105-9 On Elliptic Fibrations and F-Theory Compactifications of String Vacua. Fullwood, James, Aluffi, Paolo, Reina, Laura, Van Hoeij, Mark, Aldrovandi, Ettore, Hironaka, Eriko, Department of Mathematics, Florida State University We investigate some algebro-geometric aspects of several families of elliptic fibrations relevant for F-theory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `Sethi-Vafa-Witten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a base-independent manner, and moreover, can be computed for all possible... Show moreWe investigate some algebro-geometric aspects of several families of elliptic fibrations relevant for F-theory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `Sethi-Vafa-Witten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a base-independent manner, and moreover, can be computed for all possible dimensions of a base at once. As such, we construct generating series $f(t)$ corresponding to each invariant such that the coefficient of $t^k$ encodes the invariant of the elliptic fibration over a base of dimension $k$, solely in terms of invariants of the base. From the F-theory perspective, we highlight aspects of elliptic fibrations other than Weierstrass models, and construct a new orientifold limit of F-theory associated with $D_5$ fibrations, i.e., elliptic fibrations whose elliptic fiber is realized via a complete intersection of two quadrics in $\mathbb{P}^3$. We verify tadpole relations as predicted by the (conjectural) equivalence between F-theory and type-IIB, as well as `universal tadpole relations', which are mathematical generalizations of the tadpole relations predicted by the physics of F-theory. We also simplify formulas for invariants of Calabi-Yau fourfolds, and suggest that all Hodge numbers of Calabi-Yau fourfolds depend linearly on $c_1(B)^3$, where $B$ is the base of the fibration. Modeling Order Book Dynamics Using Queues and Point Processes. Huang, He, Kercheval, Alec, Marquis, Milton, Nolder, Craig, Okten, Giray, Ewald, Brian, Department of Mathematics, Florida State University The objective of this dissertation is to study the queuing and point process models that try to capture as many features as possible of the high-frequency data of a limit order book. First, we use a generalized birth-death stochastic process to model the high-frequency dynamics of the limit order book, and illustrate it using parameters estimated from Level II data for a stock on the London Stock Exchange. A new feature of this model is that limit orders are allowed to arrive in multiple... Show moreThe objective of this dissertation is to study the queuing and point process models that try to capture as many features as possible of the high-frequency data of a limit order book. First, we use a generalized birth-death stochastic process to model the high-frequency dynamics of the limit order book, and illustrate it using parameters estimated from Level II data for a stock on the London Stock Exchange. A new feature of this model is that limit orders are allowed to arrive in multiple sizes, an important empirical feature of the order book. We can compute various quantities of interest without resorting to simulation, conditional on the state of the order book, such as the probability that the next move of the mid-price will be upward, or the probability, as a function of order size, that a limit ask order will be executed before a downward move in the mid-price. Furthermore, univariate-bivariate Hawkes' processes are developed and calibrated to capture the ``clustering'' and "mutually exciting'' features of the order arrivals in a limit order book. Although due to technical reasons, probabilities of interest such as those of prices going up for the next move are not shown for this model, a Monte Carlo simulation algorithm of point processes called emph{thinning algorithm} is successfully modified to derive the cumulative distribution functions of some first-passage times in the order book. Learning Shape Metrics for Inferring the Nature of Allometry and Shape Classification. Fan, Yu, Mio, Washington, Kumar, Piyush, Quine, Jack, Hurdal, Monica, Cogan, Nick, Department of Mathematics, Florida State University We extend the classical Procrustes metric to a new family of shape metrics and introduce a generalized Procrustes shape model. Furthermore, we propose learning models and numerical algorithms for learning metrics by using, guided by the suggestion from biologists that the spreads of landmarks should be concentrated in relatively small regions, optimization of some measurements of sparseness to select the appropriate shape metric for gene data. We apply the generalized Procrustes shape model... Show moreWe extend the classical Procrustes metric to a new family of shape metrics and introduce a generalized Procrustes shape model. Furthermore, we propose learning models and numerical algorithms for learning metrics by using, guided by the suggestion from biologists that the spreads of landmarks should be concentrated in relatively small regions, optimization of some measurements of sparseness to select the appropriate shape metric for gene data. We apply the generalized Procrustes shape model to the shape classification problem as well and propose a selection criterion to pick out the most qualified metric for distinguishing different shape species. The experiment results illustrate the power of our shape framework. Algorithms for Computing Congruences Between Modular Forms. Heaton, Randy, Agashe, Amod, Van Hoeij, Mark, Capstick, Simon, Aldrovandi, Ettore, Department of Mathematics, Florida State University Let $N$ be a positive integer. We first discuss a method for computing intersection numbers between subspaces of $S_{2}(Gamma_{0}(N),C)$. Then we present a new method for computing a basis of q-expansions for $S_{2}(Gamma_{0}(N),Q)$, describe an algorithm for saturating such a basis in $S_{2}(Gamma_{0}(N),Z)$, and show how these results have applications to computing congruence primes and studying cancellations in the conjectural Birch and Swinnerton-Dyer formula. An Analytic Approach to Estimating the Required Surplus, Benchmark Profit, and Optimal Reinsurance Retention for an Insurance Enterprise. Boor, Joseph A. (Joseph Allen), Born, Patricia, Case, Bettye Anne, Tang, Qihe, Rogachev, Grigory, Okten, Giray, Aldrovandi, Ettore, Paris, Steve, Department of Mathematics,... Show moreBoor, Joseph A. (Joseph Allen), Born, Patricia, Case, Bettye Anne, Tang, Qihe, Rogachev, Grigory, Okten, Giray, Aldrovandi, Ettore, Paris, Steve, Department of Mathematics, Florida State University This paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after the policy period. Rather than focusing on how to estimate such values using Monte Carlo simulation, it focuses on closed form expressions and approximations for key quantities that are needed for such an analysis. Most of the analysis is also done using a... Show moreThis paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after the policy period. Rather than focusing on how to estimate such values using Monte Carlo simulation, it focuses on closed form expressions and approximations for key quantities that are needed for such an analysis. Most of the analysis is also done using a distribution-free approach with respect to the loss severity distribution, so minimal or no assumptions surrounding the specific distribution are needed when analyzing the results. However, one key parameter, that is treated via an exhaustion of cases, involves the degree of parameter uncertainty, the number of separate lines of business involved. This is done for the no parameter uncertainty monoline compound Poisson distribution as well as situations involving (lognormal) severity parameter uncertainty, (gamma/negative binomial) count parameter uncertainty, the multiline compound Poisson case, and the compound Poisson scenario with parameter uncertainty, and especially parameter uncertainty correlated across the lines of business. It shows how the risk of extreme aggregate losses that is inherent in insurance operations may be understood (and, implicitly, managed) by performing various calculations using the loss severity distribution, and, where appropriate, key parameters driving the parameter uncertainty distributions. Formulas are developed that estimate the capital and surplus needs of a company(using the VaR approach), and therefore the profit needs of a company that involve tractable calculations. As part of that the process the benchmark loading for profit, reflecting both the needed financial support for the amount of capital to adequately secure to a given one year survival probability, and the amount needed to recompense investors for diversifiable risk is discussed. An analysis of whether or not the loading for diversifiable risk is needed is performed. Approximations to the needed values are performed using the moments of the capped severity distribution and analytic formulas from the frequency distribution as inputs into method of moments normal and lognormal approximations to the percentiles of the aggregate loss distribution. An analysis of the optimum reinsurance retention/policy limit is performed as well, with capped loss distribution/frequency distribution equations resulting from the relationship that the marginal profit (with respect to the loss cap) should be equal to the marginal expense and profit dollar loading with respect to the loss cap. Analytical expressions are developed for the optimum reinsurance retention. Approximations to the optimum retention based on the normal distribution were developed and their error analyzed in great detail. The results indicate that in the vast majority of practical scenarios, the normal distribution approximation to the optimum retention is acceptable. Also included in the paper is a brief comparison of the VaR (survival probability) and expected policyholder deficit (EPD) and TVaR approaches to surplus adequacy (which conclude that the VaR approach is superior for most property/casualty companies); a mathematical analysis of the propriety of insuring the upper limits of the loss distribution, which concludes that, even if unlimited funds were available to secure losses in capital and reinsurance, it would not be in the insured's best interest to do so. Further inclusions to date include a illustrative derivation of the generalized collective risk equation and a method for interpolating ``along'' a mathematical curve rather than directly using the values on the curve. As a prelude to a portion of the analysis, a theorem was proven indicating that in most practical situations, the n-1st order derivatives of a suitable probability mass function at values L, when divided by the product of L and the nth order derivative, generate a quotient with a limit at infinity that is less than 1/n. Chern Classes of Sheaves of Logarithmic Vector Fields for Free Divisors. Liao, Xia, Aluffi, Paolo, Reina, Laura, Klassen, Eric P., Aldrovandi, Ettore, Petersen, Kathleen, Department of Mathematics, Florida State University The thesis work we present here focuses on solving a conjecture raised by Aluffi about Chern-Schwartz-MacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(-log D)) cap [X]$ in $A_{}(X)$ for any locally quasi-homogeneous free divisor $D$. We prove a stronger... Show moreThe thesis work we present here focuses on solving a conjecture raised by Aluffi about Chern-Schwartz-MacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(-log D)) cap [X]$ in $A_{}(X)$ for any locally quasi-homogeneous free divisor $D$. We prove a stronger version of this conjecture. We also report on work aimed at studying the Grothedieck class of hypersurfaces of low degree. In this work, we verified the Geometric Chevalley-Warning conjecture in several low dimensional cases. Modeling Cortical Folding Patterns of the Brain Using a Growing Domain. Toole, Gregory, Hurdal, Monica K., Steinbock, Oliver, Bertram, Richard, Cogan, Nick, Ewald, Brian, Department of Mathematics, Florida State University The brain is one of nature's greatest mysteries. The mechanism by which the folds of the brain's cerebral cortex, called gyri (hills) and sulci (valleys), are formed remains unknown. Existing biological hypotheses that attempt to explain the underlying mechanism of cortical folding conflict. Some hypotheses, such as the Intermediate Progenitor Model, emphasize genetic chemical factor control. Others, such as the Axonal Tension Hypothesis, emphasize the influence of physical tension due to... Show moreThe brain is one of nature's greatest mysteries. The mechanism by which the folds of the brain's cerebral cortex, called gyri (hills) and sulci (valleys), are formed remains unknown. Existing biological hypotheses that attempt to explain the underlying mechanism of cortical folding conflict. Some hypotheses, such as the Intermediate Progenitor Model, emphasize genetic chemical factor control. Others, such as the Axonal Tension Hypothesis, emphasize the influence of physical tension due to axonal connections. To bring mathematics into this debate, this dissertation presents two biomathematical models of cortical folding that utilize a Turing reaction-diffusion system on an exponentially or logistically growing prolate spheroidal domain. These models are used to investigate the validity of the Intermediate Progenitor Model, thereby investigating the role of genetic chemical factor control of the development of cortical folding patterns. We observe that the presence of domain growth drives the patterns generated by our growing prolate spheroidal Turing systems to become transient. An exponentially growing prolate spheroidal domain generates a pattern that continually evolves, while a logistically growing prolate spheroidal domain generates a pattern that evolves while the domain is growing but then converges to a final pattern once the domain growth asymptotically stops. Patterns generated by the model systems represent genetic chemical prepatterns for self-amplification of intermediate progenitor cells, which may be correlated to cortical folding patterns according to the Intermediate Progenitor Model. By altering system parameters, we are able to model diseases of cortical folding such as polymicrogyria where the cortex possesses too many folds as well as diseases where the cortex has too few cortical folds such as Norman-Roberts Syndrome (microcephalic lissencephaly) and normocephalic lissencephaly. Our ability to model such a variety of diseases lends support to the role of genetic control of cortical folding pattern development and therefore to the Intermediate Progenitor Model. Discrete Frenet Frame with Application to Structural Biology and Kinematics. Lu, Yuanting, Quine, John R., Huffer, Fred W., Bertram, Richard, Cross, Timothy A., Cogan, Nick, Department of Mathematics, Florida State University The classical Frenet frame is a moving frame on a smooth curve. Connecting a sequence of points in space by line segments makes a discrete curve. The reference frame consisting of tangent, normal and binormal vectors at each point is defined as discrete Frenet frame (DFF). The DFF is useful in studying shapes of long molecules such as proteins. In this dissertation, we provide a solid mathematics foundation for DFF by showing the limit of the Frenet formula for DFF is the classical Frenet... Show moreThe classical Frenet frame is a moving frame on a smooth curve. Connecting a sequence of points in space by line segments makes a discrete curve. The reference frame consisting of tangent, normal and binormal vectors at each point is defined as discrete Frenet frame (DFF). The DFF is useful in studying shapes of long molecules such as proteins. In this dissertation, we provide a solid mathematics foundation for DFF by showing the limit of the Frenet formula for DFF is the classical Frenet formula. As part of a survey of various ways to compute rigid body motion, we show the Denavit-Hartenberg (D-H) conventions in robotics are a special case of the DFFs. Finally, we apply DFF to solve the kink angle problem in protein alpha helical structure using data from NMR experiments. Stochastic Modeling of Financial Derivatives. Huang, Wanwan, Okten, Giray, Ewald, Brian, Huffer, Fred, Kercheval, Alec, Tang, Qihe, Kim, Kyounghee, Department of Mathematics, Florida State University The Coupled Additive Multiplicative Noises (CAM) model is introduced as a stochastic volatility process to extend the classical Black-Scholes model. The fast Fourier transform (FFT) method is used to compute the values of the probability density function of the underlying assets under the CAM model, as well as the price of European call options. We discuss four dierent discretization schemes for the CAM model: the Euler scheme, the simplied weak Euler scheme, the order 2 weak Taylor scheme... Show moreThe Coupled Additive Multiplicative Noises (CAM) model is introduced as a stochastic volatility process to extend the classical Black-Scholes model. The fast Fourier transform (FFT) method is used to compute the values of the probability density function of the underlying assets under the CAM model, as well as the price of European call options. We discuss four dierent discretization schemes for the CAM model: the Euler scheme, the simplied weak Euler scheme, the order 2 weak Taylor scheme and the stochastic Adams-Bashforth scheme. A martingale control variate method for pricing European call options is developed, and its advantages in terms of variance reduction are investigated numerically. We also develop Monte Carlo methods for estimating the sensitivities of the European call options under the CAM model. Probabilistic Uncertainty Analysis and Its Applications in Option Models. Namihira, Motoi J., Kopriva, David A., Srivastava, Anuj, Ewald, Brian, Hussaini, M. Yousuff, Nichols, Warren, Okten, Giray, Department of Mathematics, Florida State University In this work we quantify the effect of uncertainty in volatility in the prices and Deltas of an American and European put using probabilistic uncertainty analysis. We review the current methods of uncertainty analysis including worst case or scenario analysis, Monte Carlo, and provide an in depth review of Polynomial Chaos in both one and multiple dimensions. We develop a numerically stable method of generating orthogonal polynomials that is used in the practical construction of the... Show moreIn this work we quantify the effect of uncertainty in volatility in the prices and Deltas of an American and European put using probabilistic uncertainty analysis. We review the current methods of uncertainty analysis including worst case or scenario analysis, Monte Carlo, and provide an in depth review of Polynomial Chaos in both one and multiple dimensions. We develop a numerically stable method of generating orthogonal polynomials that is used in the practical construction of the Polynomial Chaos basis functions. We also develop a semi analytic density transform method that is 200 times faster and 1000 times more accurate than the Monte Carlo based kernel density method. Finally, we analyze the European and American put option models assuming a distribution for the volatility that is historically observed. We find that the sensitivity to uncertainty in volatility is greatest for the price of ATM puts, and tapers as one moves away from the strike. The Delta, however, exhibits the least sensitivity when ATM and is most sensitive when moderately ITM. The price uncertainty for ITM American puts is less than the price uncertainty of equivalent European puts. For OTM options, the price uncertainty is similar between American and European puts. The uncertainty in the Delta of ITM American puts is greater than the uncertainty of equivalent European puts. For OTM puts, the uncertainty in Delta is similar between American and European puts. For the American put, uncertainty in volatility introduces uncertainty in the location of the optimal exercise boundary, thereby making optimal exercise decisions more difficult. Nonlinear Dynamics Underlying Fast Bursting in Pituitary Cells. Teka, Wondimu Woubante, Bertram, Richard, Trombley, Paul Q., Tabak, Jöel, Cogan, Nick G., Wang, Xiaoming, Department of Mathematics, Florida State University Neurons and endocrine cells display various patterns of electrical activity, including periodic bursting. Bursting oscillations are characterized by the alternation between periods of fast spiking (the active phase) and quiescent periods (the silent phase), and are accompanied by slow variations in one or more slowly changing variables. Bursts are often more efficient than periodic spiking in evoking the release of neurotransmitter or hormone. The technique of two-fast/one-slow analysis,... Show moreNeurons and endocrine cells display various patterns of electrical activity, including periodic bursting. Bursting oscillations are characterized by the alternation between periods of fast spiking (the active phase) and quiescent periods (the silent phase), and are accompanied by slow variations in one or more slowly changing variables. Bursts are often more efficient than periodic spiking in evoking the release of neurotransmitter or hormone. The technique of two-fast/one-slow analysis, which takes advantage of time scale differences, is typically used to analyze the dynamics of bursting in mathematical models. Two classes of bursting oscillations that have been identified with this technique, plateau and pseudo-plateau bursting, are often observed in neurons and endocrine cells, respectively. These two types of bursting have very different properties and likely serve different functions. This latter point is supported by the divergent expression of the bursting patterns into different cell types, and raises the question of whether it is even possible for a model for one type of cell to produce bursting of the type seen in the other type without large changes to the model. Using fast/slow analysis, we show here that this is possible, and we provide a procedure for achieving this transition. This suggests that the mechanisms for bursting in endocrine cells are just quantitative variations of those for bursting in neurons. The two-fast/one-slow analysis used to make the transition between plateau and pseudo-plateau bursting, and to understand the dynamics of plateau bursting is of limited use for pseudo-plateau bursting. Using a one-fast/two-slow analysis technique, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. Finally we show the relationship between the two-fast/one-slow analysis and one-fast/two-slow analysis techniques. Risk Forecasting and Portfolio Optimization with GARCH, Skewed t Distributions and Multiple Timescales. Liu, Yang, Kercheval, Alec N., Schlagenhauf, Don E., Kim, Kyounghee, Nolder, Craig, Department of Mathematics, Florida State University It is well-established that distributions of financial returns are heavy-tailed and exhibit skewness and other non-Gaussian characteristics. As time series, return data have volatilities that vary over time and show profound serial correlation (or cross-correlation in the multivariate case). To address these issues, time series models such as GARCH (generalized autoregressive conditionally heteroskedastic) processes and non-Gaussian distributions such as generalized hyperbolic (GH)... Show moreIt is well-established that distributions of financial returns are heavy-tailed and exhibit skewness and other non-Gaussian characteristics. As time series, return data have volatilities that vary over time and show profound serial correlation (or cross-correlation in the multivariate case). To address these issues, time series models such as GARCH (generalized autoregressive conditionally heteroskedastic) processes and non-Gaussian distributions such as generalized hyperbolic (GH) distributions have been introduced into financial modeling. A typical procedure featuring GARCH and non-Gaussian distributions involves the following steps. First, filter data with GARCH to get residuals that are approximately i.i.d. Second, calibrate parameters of a non-Gaussian distribution to those residuals. Finally, forecast various quantities based on knowledge of the calibrated distribution. Existing implementations of this procedure are fixed-frequency in nature. That is, all three steps are carried out on the same timescale. Reliable filtering and calibration requires a sufficient amount of historical data. As the forecast horizon grows, the model demands an increasingly long price history and may become infeasible if data are too scarce. To reduce the model's dependence on data availability, we propose a mixed-frequency method. Filtering and calibration are done on a relatively small timescale where data are more abundant. We then shift to a longer time horizon and make forecasts through aggregating GARCH processes and Monte Carlo simulation. We first apply this mixed-frequency approach to forecasting univariate value-at-risk (VaR) for stock index returns. Backtesting conducted on a variety of timescales shows that the method is indeed viable. Moreover, compared with the fixed-frequency method, our new method is able to produce VaR forecasts that respond more quickly to volatility changes. Therefore, even if data availability is not an issue, the mixed-frequency method is still a valuable alternative for risk managers. Portfolio optimization, a multivariate problem, is tackled next. We enhance traditional Markowitz optimization with expected shortfall (ES), which measures tail risks better than standard deviation, and skewed t distributions, a promising subfamily of GH distributions. The mixed-frequency idea is incorporated as well. Factors that affect the efficient frontier and optimal portfolio compositions are thoroughly discussed. Last but not least, we implement investment strategies based on GARCH-skewed t-ES portfolio optimization and evaluate their performance, both in terms of return and risk. Sequences of Pseudo-Anosov Mapping Classes with Asymptotically Small Dilatation. Valdivia, Aaron David, Hironaka, Eriko, Reina, Laura, Heil, Wolfgang, Klassen, Eric, Department of Mathematics, Florida State University We construct sequences of pseudo-Anosov examples which we use to bound the minimal dilatation on arbitrary surfaces. We show that these bounds give the asymptotic behavior of the minimal dilatations for certain sequences. Further we show that the mapping classes for a given sequence from our construction can be realized as fibrations of a single 3-manifold. From Songs to Synapses, Ion Channels and Mathematical Modeling. Daou, Arij, Bertram, Richard, Ryan, Pamela, Johnson, Frank, Hyson, Richard, Wu, Wei, Okten, Giray, Department of Mathematics, Florida State University Since the scientific study of birdsong began in the late 1950s, songbirds have emerged as impressive neurobiological models for aspects of human verbal communication because they learn to sequence their song elements, analogous, in many ways, to how humans learn to produce spoken sequences with syntactic structure. Thus, determining how spoken language evolved is more likely to become clearer with concerted efforts in researching songbirds. Some of the most fundamental questions in... Show moreSince the scientific study of birdsong began in the late 1950s, songbirds have emerged as impressive neurobiological models for aspects of human verbal communication because they learn to sequence their song elements, analogous, in many ways, to how humans learn to produce spoken sequences with syntactic structure. Thus, determining how spoken language evolved is more likely to become clearer with concerted efforts in researching songbirds. Some of the most fundamental questions in neuroscience are pursued through the study of songbirds. How does the brain generate complex sequential behaviors? How do we learn to speak? How do humans learn various behaviors by observing and imitating others? Where are the "prime movers" that control behavior? Which circuits in the brain control the order in which motor gestures of a learned behavior are generated? Among all these questions, of particular interest to us is the question of sequential behavior. Understanding the neural mechanisms that underlie sequential behavior and imitative learning is the holy grail of the field. The birdsong provided us with a uniquely powerful model for tackling this question in a system where the brain structures responsible for its generation are well known. We pursued the study of sequential neural activity in songbirds on three levels: behavioral, cellular and network. On the behavioral level, we developed a computational tool for automated, quantitative syllable-level analysis of bird song syntax. This tool aids songbird researchers and fanciers in comparing and quantifying the syntactic structure of songs produced by a bird prior to and after a manipulation such as ablation of brain region or infusion of pharmacological agents, in addition to several other purposes. As we will discuss later, this syntactic structure is highly stereotyped in songbirds and driven by neurons firing in sequential order in particular regions of the songbird's brain. On the cellular level, the telencephalic nucleus HVC (proper name) within the songbird analogue of the mammalian pre-motor cortex is situated at a critical point in the pattern-generating premotor brain circuitry of oscine songbirds. This nucleus is of extreme importance to the songbird and produces stereotyped instructions through the motor pathway leading to precise, learned vocalization by songbirds. HVC contains three populations of neurons that are interconnected, with specific patterns of excitatory and inhibitory connectivity. Characterizing the neurons in HVC is a very important requirement for decoding the neural code of the birdsong. We performed whole-cell current clamp recordings on HVC neurons within brain slices to examine their intrinsic firing properties and determine which ionic currents are responsible for their characteristic firing patterns. We also developed conductance-based models for the different neurons and calibrated the models using data from our brain slice work. These models were then used to generate predictions about the makeup of the ionic currents that are responsible for the different responses to stimuli. These predictions were then tested and verified in the slice using pharmacological manipulations. Our results are an improved characterization of the HVC neurons responsible for song production in the songbird which are the key ingredients in understanding the HVC network. We then developed prototype neural architectures of the HVC that can produce the patterns of sequential neural activity exhibited by the three types of HVC neurons during singing. Our networks consist of microcircuits of interconnected neurons which are active during different syllables of the song. The various networks that we consider assign different roles to each of the HVC neurons types in the production of the sequential activity pattern, and show great flexibility in the connectivity patterns among the neuron types. The model networks developed provide key insights into how the different types of HVC neurons can be used for sequence generation. The significance of the work presented in this dissertation is that it helps elucidate the neural mechanisms behind HVC activity. The in vitro studies we performed in brain slices and the models we developed provide critical pieces to the puzzle of sequential behavior. Calibration of Local Volatility Models and Proper Orthogonal Decomposition Reduced Order Modeling for Stochastic Volatility Models. Geng, Jian, Navon, Ionel Michael, Case, Bettye Anne, Contreras, Rob, Okten, Giray, Kercheval, Alec N., Ewald, Brian, Department of Mathematics, Florida State University There are two themes in this thesis: local volatility models and their calibration, and Proper Orthogonal Decomposition (POD) reduced order modeling with application in stochastic volatility models, which has a potential in the calibration of stochastic volatility models. In the first part of this thesis (chapters II-III), the local volatility models are introduced first and then calibrated for European options across all strikes and maturities of the same underlying. There is no... Show moreThere are two themes in this thesis: local volatility models and their calibration, and Proper Orthogonal Decomposition (POD) reduced order modeling with application in stochastic volatility models, which has a potential in the calibration of stochastic volatility models. In the first part of this thesis (chapters II-III), the local volatility models are introduced first and then calibrated for European options across all strikes and maturities of the same underlying. There is no interpolation or extrapolation of either the option prices or the volatility surface. We do not make any assumption regarding the shape of the volatility surface except to assume that it is smooth. Due to the smoothness assumption, we apply a second order Tikhonov regularization. We choose the Tikhonov regularization parameter as one of the singular values of the Jacobian matrix of the Dupire model. Finally we perform extensive numerical tests to assess and verify the aforementioned techniques for both local volatility models with known analytical solutions of European option prices and real market option data. In the second part of this thesis (chapters IV-V), stochastic volatility models, POD reduced order modeling are introduced first respectively. Then POD reduced order modeling is applied to the Heston stochastic volatility model for the pricing of European options. Finally, chapter VI summaries the thesis and points out future research areas. Spectral Methods for Morphometry. Bates, Jonathan R., Mio, Washington, Patrangenaru, Victor, Bertram, Richard, Liu, Xiuwen, Quine, Jack, Department of Mathematics, Florida State University Methods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be represented if it is allowed to bend. We are motivated to use representations of intrinsic geometry, which, for example, does not distinguish a flat sheet of paper from a rolled sheet. The spectral embedding (``heat kernel representation'') as a representation... Show moreMethods from shape analysis are used in morphometry, which is the quantitative analysis of macroscopic anatomical features. We assume that anatomy is flexible, and this brings us to the first problem of resolving how ``shape'' should be represented if it is allowed to bend. We are motivated to use representations of intrinsic geometry, which, for example, does not distinguish a flat sheet of paper from a rolled sheet. The spectral embedding (``heat kernel representation'') as a representation of intrinsic geometry has many desirable features for computational anatomy and other areas of shape and data analysis. Several breakthroughs are made toward understanding and applying this representation. A novel shape representation is also considered and used for classification of control vs. affected groups. One goal of morphometry is to make statistically objective comparisons. Hence, once a suitable representation of shape is chosen, the second problem is to compare shapes. Shape comparison may occur at many levels of scale. The simplest comparisons are made with global features: volume, length, etc. Finer comparisons may occur at regional levels. A finest level of comparison can be made after matching all homologous points, that is, after finding a one-one correspondence between points on shapes. A point correspondence is found by a registration algorithm. A method for unsupervised shape registration is presented and applied to localize differences between control and affected groups. We focus on the 3D case, where imaging has made anatomical surface data readily available, yet the analysis challenging. Structural MRI of living persons is currently used to study the macroscopic effects on anatomy by neurodegenerative disease (e.g. Alzheimer's). In the earliest stages of Alzheimer's disease (AD), certain brain structures have been observed to have reduced volume, in autopsy and in vivo, including the hippocampus, putamen, and thalamus. Our methods will be applied to these surfaces. Jump Dependence and Multidimensional Default Risk: A New Class of Structural Models with Stochastic Intensities. Garreau, Pierre, Kercheval, Alec N., Marquis, Milton H., Beaumont, Paul M., Kopriva, David A., Okten, Giray, Department of Mathematics, Florida State University This thesis presents a new structural framework for multidimensional default risk. The time of default is the first jump of the log-returns of the stock price of a firm below a stochastic default level. When the stock price is an exponential Levy process, this new formulation is equivalent to a default model with stochastic intensity where the intensity process is parametrized by a Levy measure. This framework calibrates well to various term structures of credit default swaps. Furthermore,... Show moreThis thesis presents a new structural framework for multidimensional default risk. The time of default is the first jump of the log-returns of the stock price of a firm below a stochastic default level. When the stock price is an exponential Levy process, this new formulation is equivalent to a default model with stochastic intensity where the intensity process is parametrized by a Levy measure. This framework calibrates well to various term structures of credit default swaps. Furthermore, the dependence between the default times of firms within a basket of credit securities is the result of the jump dependence of their respective stock prices: this class of models makes the link between the Equity and Credit markets. As an application, we show the valuation of a first-to-default swaps. To motivate this new framework, we compute the default probability in a traditional structural model of default where the firm value follows a general Levy processes. This is made possible via the resolution of a partial integro-differential equation (PIDE). We solve this equation numerically using a spectral element method based on the approximation of the solution with high order polynomials described in (Garreau & Korpiva, 2013). This method is able to handle the sharp kernels in the integral term. It is faster than the competing numerical Laplace transform methods used for first passage time problems, and can be used to compute the price of exotic options with barriers. This PIDE approach does not however extend well in higher dimensions. To understand the joint default of our new framework, we investigate the dependence structures of Levy processes. We show that for two one dimensional Levy processes to form a two dimensional Levy process, their joint survival times need to satisfy a two dimensional version of the memoryless property. We make the link with bivariate exponential random variables and the Marshall-Olkin copula. This result yields a necessary construction of dependent Levy processes, a characterization theorem for Poisson random measures and has important ramification for default models with jointly conditionally Poisson processes. Uncertainty Quantification and Data Fusion Based on Dempster-Shafer Theory. He, Yanyan, Hussaini, M. Yousuff, Oates, William S., Kopriva, David A., Sussman, Mark, Department of Mathematics, Florida State University Quantifying uncertainty in modeling and simulation is crucial since the parameters of the physical system are inherently non-deterministic and knowledge of the system embodied in the model is incomplete or inadequate. The most well-developed nonadditive-measure theory -- the Dempster-Shafer theory of evidence -- is explored for uncertainty quantification and propagation. For ''uncertainty quantification," we propose the MinMax method to construct belief functions to represent uncertainty in... Show moreQuantifying uncertainty in modeling and simulation is crucial since the parameters of the physical system are inherently non-deterministic and knowledge of the system embodied in the model is incomplete or inadequate. The most well-developed nonadditive-measure theory -- the Dempster-Shafer theory of evidence -- is explored for uncertainty quantification and propagation. For ''uncertainty quantification," we propose the MinMax method to construct belief functions to represent uncertainty in the information (data set) involving the inseparably mixed type of uncertainties. Using the principle of minimum uncertainty and the concepts of entropy and specificity, the MinMax method specifies a partition of a finite interval on the real line and assigns belief masses to the uniform subintervals. The method is illustrated in a simple example and applied to the total uncertainty quantification in flight plan of two actual flights. For ''uncertainty propagation," we construct belief/probability density functions for the output or the statistics of the output given the belief/probability density functions for the uncertain input variables. Different approaches are introduced for aleatory uncertainty propagation, epistemic uncertainty propagation, and mixed type of uncertainty propagation. The impact of the uncertain input parameters on the model output is studied using these approaches in a simple example of aerodynamic flow: quasi-one-dimensional nozzle flow. In the situation that multiple models are available for the same quantity of interest, the combination rules in the Dempster-Shafer theory can be utilized to integrate the predictions from the different models. In the present work, we propose a robust and comprehensive procedure to combine multiple bodies of evidence. It is robust in that it can combine multiple bodies of evidence, consistent or otherwise. It is comprehensive in the sense that it examines the bodies of evidence strongly conflicted with others, reconstructs the basic belief mass functions by discounting, and then fuses all the bodies of evidence using an optimally parametrized combination rule. The proposed combination procedure is applied to radiotherapy dose response outcome analysis. Numerical Methods for Multiphase Systems with Applications to Biology. Whidden, Mark E., Cogan, Nicholas, Wang, Xiaoqiang, Bertram, Richard, Sussman, Mark, Department of Mathematics, Florida State University This dissertation is comprised of a variety of efforts towards the development of fast numerical methods and their applications, particularly in the context of simulating biological systems. Scientific computing of these problems requires many considerations bridging gaps between computer science, applied mathematics, and the biology of the specific application. This dissertation spans these fields, with the formulation of heterogeneous mixture descriptions in one chapter, the study and... Show moreThis dissertation is comprised of a variety of efforts towards the development of fast numerical methods and their applications, particularly in the context of simulating biological systems. Scientific computing of these problems requires many considerations bridging gaps between computer science, applied mathematics, and the biology of the specific application. This dissertation spans these fields, with the formulation of heterogeneous mixture descriptions in one chapter, the study and implementation of efficient and robust numerical techniques in the next, and the application of this modeling framework and computational procedure to specific biological problems in the remaining chapters. The first of these efforts is the construction of multiphase models for macroscopic descriptions of biophysical problems. The second is the development of fast and flexible methods for simulating models derived from this modeling framework. The third is the revelation of this modeling framework to exhibit spatio-temporal patterns that can be initiated by localized perturbations in space. The fourth is the simulation of a four-phase model of biofilm formation implicated in Pierce's Disease. Shape Analysis of Curves in Higher Dimensions. Wells, Linda Crystal, Klassen, Eric, Chicken, Eric, Srivastava, Anuj, Mio, Washington, Nichols, Warren, Department of Mathematics, Florida State University In this dissertation we will discuss geodesics between open curves and also between closed curves in Rn where n ≥ 2. In order to calculate these geodesics, we will form a Riemannian metric on a space of smooth curves with non-vanishing derivative. The metric will be invariant with respect to scaling, translation, rotation, and reparametrization. Using this metric we will define a distance between two curves invariant to the above mentioned transformations. This distance function will be... Show moreIn this dissertation we will discuss geodesics between open curves and also between closed curves in Rn where n ≥ 2. In order to calculate these geodesics, we will form a Riemannian metric on a space of smooth curves with non-vanishing derivative. The metric will be invariant with respect to scaling, translation, rotation, and reparametrization. Using this metric we will define a distance between two curves invariant to the above mentioned transformations. This distance function will be defined utilizing the existence of isometries which allow our curves to map into a subspace of L2 where we already have geodesics defined and then map that geodesic back to the space of curves we are working in. Then we apply our metric to the geodesic to define the distance between the two initial curves. Some of our applications are 2D open curves, 3D open curves, and 3D closed curves including facial curves being categorized. The case of curves in R2 was studies by Laurent Younes, Peter W. Michor, Jayant Shah and David Mumford. The Evolution of Deception in Signaling Systems. Ohm, Candace, Mesterton-Gibbons, Mike, Isaac, Mark, Kercheval, Alec, Nichols, Warren, Department of Mathematics, Florida State University In this dissertation, we create a dynamical learning model that helps to explain the evolution of deception in signaling systems. In our model, the signaler may choose to signal either of two possible states. We apply this model to Batesian mimicry and to deceptive signaling of fighting ability, or resource holding potential. We show how to expand this model to allow for multiple receivers as well as multiple possible states. Periods and Motives: Applications in Mathematical Physics. Li, Dan, Marcolli, Matilde, Reina, Laura, Aluffi, Paolo, Agashe, Amod, Aldrovandi, Ettore, Department of Mathematics, Florida State University The study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebro... Show moreThe study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebro-geometric approach to the spectral theory of Harper operators in solid state physics. When the parameters are irrational, the compactification of its Bloch variety is an ind-pro-variety, which is a Cantor-like geometric space and it is compatible with the picture of Hofstadter butterfly. On each approximating component the density of states of the electronic model can be expressed in terms of period integrals over Fermi curves, which can be explicitly computed as elliptic integrals or periods of elliptic curves. The above density of states satisfies a Picard-Fuchs equation, whose solutions are generally given by hypergeometric functions. We use the idea of mirror maps as in mirror symmetry of elliptic curves to derive a q-expansion for the energy level based on the Picard-Fuchs equation. In addition, formal spectral functions such as the partition function are derived as new period integrals. Secondly, we consider generalized Feynman diagram evaluations of an effective noncommutative field theory of the Ponzano-Regge model coupled with matter in loop quantum gravity. We present a parametric representation in a linear k-approximation of the effective field theory derived from a k-deformation of the Ponzano-Regge model and define a generalized Kirchhoff polynomial with k-correction terms. Setting k equal to 1, we verify that the number of points of the corresponding hypersurface of the tetrahedron over finite fields does not fit polynomials with integer coefficients by computer calculations. We then conclude that the hypersurface of the tetrahedron is not polynomially countable, which possibly implies that the hypersurface of the tetrahedron as a motive is not mixed Tate. Applications of Quantum Dots in Gene Therapy. Barnes, Laura F., Strouse, Geoffrey, Logan, Timothy, Miller, Brian, Department of Chemistry and Biochemistry, Florida State University Gene therapy is a rising field and requires multifunctional delivery platforms in order to overcome the cellular barriers. Quantum dots (QDs) provide a optically fluorescent and biocompatible surface to act as a multifunctional delivery platform for gene therapy. The objective of this research is to manipulate the surface of quantum dots for use in gene therapy. The first goal was to make the QDs water soluble and therefore biocompatible. The second goal was to functionalize the surface of... Show moreGene therapy is a rising field and requires multifunctional delivery platforms in order to overcome the cellular barriers. Quantum dots (QDs) provide a optically fluorescent and biocompatible surface to act as a multifunctional delivery platform for gene therapy. The objective of this research is to manipulate the surface of quantum dots for use in gene therapy. The first goal was to make the QDs water soluble and therefore biocompatible. The second goal was to functionalize the surface of the QDs with plasmid DNA for direct use in gene therapy. This approach uses chemoselective coupling chemistry between an InP/ZnS quantum dot (QD) and linker DNA (DNAlinker) to control the timing of protein expression. Linear DNA (lDNA), containing the CMV promoter and DsRed-Express gene, was condensed on the surface of the QD-DNAlinker. Optical and flow cytometry analysis of the DsRed-Express expression after transfection of the QD-lDNA into CHO cells shows a delayed protein expression for both coupling chemistries compared to naked lDNA. It is also clear that the protein expression form the QD-S-lDNA turns on quicker than the QD-NH-lDNA. We believe the protein expression delay is due to the site of coupling between the QD and DNAlinker and its affect on the lDNA packing strength. The S-DNAlinker is believed to couple by direct exchange at the vertices of the QD whereas the NH-DNAlinker couples through a condensation reaction to the facets. The delay in protein expression reflects the delayed exchange rate at the facets over the vertices. The ability to control the coupling chemistry and timing of release from the QD surface suggests a mechanism for dose control in transient gene therapeutics, and show QD delivery approaches are ideal candidates for multifunctional, targeted, drug carrying platforms that can simultaneously control dosing. The third goal of this research was to functionalize the surface of the QDs with the HIV cell penetrating peptide, TAT, and study its affects on QD internalization as well as toxicological affects within the cells. Tracking of the cellular uptake of these QDs by optical microscopy shows rapid, diffuse accumulation of both 10 % TAT and 100 % TAT passivated QDs throughout the cytosol of the cells. Toxicity studies were conducted by flow cytometry to investigate the effects of these materials on apoptosis, necrosis, and metabolic damage in Chinese Hamster Ovary (CHO) cells. These studies suggest toxic effects of the cell penetrating QDs are dependent on the amount of CAAKA-TAT used on the surface of the QD as well as the concentration of QD added. These observations aid in the use of QDs as self transfecting, nano delivery scaffolds for drug or gene therapy. Solving Linear Differential Equations in Terms of Hypergeometric Functions by ₂-Descent. Fang, Tingting, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluffi, Paolo, Department of Mathematics, Florida State University Let L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2-descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2-descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2-descent has already... Show moreLet L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2-descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2-descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2-descent has already been treated before, a complete implementation could not be given because it involved a step for which we do not have a complete implementation. Our key novelty is to give an approach that is fully implementable. We describe and implement the algorithm for order 2, and show by examples that the same also work for higher order. By doing 2-descent for L, the number of true singularities drops to at most n/2 + 2 (n is the number of true singularities of L). This provides us ways to solve L in closed form(e.g.in terms of hypergeometric funtions). Finding All Bessel Type Solutions for Linear Differential Equations with Rational Function Coefficients. Yuan, Quan, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluffi, Paolo, Department of Mathematics, Florida State University A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if... Show moreA linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if change of variables is a rational function. In this thesis we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions. This algorithm can be easily extended to a Whittaker/Kummer solver. Combine the two algorithms, we can get a complete algorithm for all 0F1 and 1F1 type solutions. We also use our algorithm to analyze the relation between Bessel functions and Heun functions. Dirac Operators, Multipliers and H[superscript P] Spaces of Monogenic Functions. Wang, Guanghou, Nolder, Craig, Hawkes, Lois, Case, Bettye, Hironaka, Eriko, Quine, Jack, Seppälä, Mika, Department of Mathematics, Florida State University We have done a few things under Clifford algebra settings. Firstly, one Caccioppoli type estimate is derived for solutions of $A$-Dirac equations in the form $DA(x,Du) = 0$, where $D$ is the Dirac operator. This kind of $A$-Dirac equations are generalizations of elliptic equations of $A$-harmonic type, i.e. div$A(x,\nabla u)=0.$ Secondly, the multiplier theory from Fourier analysis is generalized to Clifford analysis. After the multipliers of operators $\mathcal{D}$, $T$ and $ \Pi$ are... Show moreWe have done a few things under Clifford algebra settings. Firstly, one Caccioppoli type estimate is derived for solutions of $A$-Dirac equations in the form $DA(x,Du) = 0$, where $D$ is the Dirac operator. This kind of $A$-Dirac equations are generalizations of elliptic equations of $A$-harmonic type, i.e. div$A(x,\nabla u)=0.$ Secondly, the multiplier theory from Fourier analysis is generalized to Clifford analysis. After the multipliers of operators $\mathcal{D}$, $T$ and $ \Pi$ are identified, some related properties will be very easy to achieve, including two integral representation theorems, also the iterations of operators $\mathcal{D}$ and $\Delta$ are also discussed. Thirdly, one Carleson measure theorem is achieved for monogenic Hardy spaces on the unit ball in $R^{n+1}$, as well as one Clifford Riesz representation theorem. Furthermore, one bounded theorem about certain inhomogeneous Dirac equations is established with the help of spherical monogenic functions theory. Slow Variable Dominance in Pancreatic β-Cell Models. Watts, Margaret A., Bertram, Richard, Steinbock, Oliver, Quine, Jack, Cogan, Nick, Tabak, Joel, Department of Mathematics, Florida State University Like nerve and many other endocrine cells, pancreatic beta-cells are electrically excitable and produce electrical impulses in response to elevations in glucose. These electrical impulses typically come in the form of bursting. One type of bursting model with two or more slow variables has been called 'phantom bursting' since the burst period is a blend of the time constants of the slow variables. In this dissertation, the relative contributions that slow variables make to the bursting... Show moreLike nerve and many other endocrine cells, pancreatic beta-cells are electrically excitable and produce electrical impulses in response to elevations in glucose. These electrical impulses typically come in the form of bursting. One type of bursting model with two or more slow variables has been called 'phantom bursting' since the burst period is a blend of the time constants of the slow variables. In this dissertation, the relative contributions that slow variables make to the bursting produced by two different phantom bursting models are quantified using a measure called the 'dominance factor'. Using this quantification, it is demonstrated that the control of different phases of the burst can be shifted from one slow variable to another by changing a model parameter. It is also demonstrated that the contributions that the slow processes make to bursting can be non-obvious. One application of the dominance factor is in making predictions about the resetting properties of the model cells. This application is demonstrated using a general phantom bursting model. Functional Data Analysis and Partial Shape Matching in the Square Root Velocity Framework. Robinson, Daniel T., Klassen, Eric, Reina, Laura, Bellenot, Steven, Mio, Washington, Srivastava, Anuj, Department of Mathematics, Florida State University We investigate two problems in elastic curve shape analysis, working within the context of the square root velocity (SRV) framework. The first of these is to develop specialized algorithms for the analysis of one-dimensional curves, which are just real-valued functions. In this particularly simple case, the elastic matching problem can be stated as a finite combinatorial problem in which the optimal solution can be found exactly. We also develop a method for groupwise alignment, and use it to... Show moreWe investigate two problems in elastic curve shape analysis, working within the context of the square root velocity (SRV) framework. The first of these is to develop specialized algorithms for the analysis of one-dimensional curves, which are just real-valued functions. In this particularly simple case, the elastic matching problem can be stated as a finite combinatorial problem in which the optimal solution can be found exactly. We also develop a method for groupwise alignment, and use it to compute Karcher means of collections of functions. Second, we consider the problem of finding optimal partial matches between curves in Euclidean space within the SRV framework, and present algorithms and heuristics to solve this problem. Finally, we give a brief overview of libsrvf, an open-source software library providing implementations of the algorithms developed in the course of this work. Partial Differential Equation Methods to Price Options in the Energy Market. Yan, Jinhua, Kopriva, David, Huffer, Fred, Case, Bettye Anne, Nolder, Craig, Wang, Xiaoming, Department of Mathematics, Florida State University We develop partial differential equation methods with well-posed boundary conditions to price average strike options and swing options in the energy market. We use the energy method to develop boundary conditions that make a two space variable model of Asian options well-posed on a finite domain. To test the performance of well-posed boundary conditions, we price an average strike call. We also derive new boundary conditions for the average strike option from the put-call parity. Numerical... Show moreWe develop partial differential equation methods with well-posed boundary conditions to price average strike options and swing options in the energy market. We use the energy method to develop boundary conditions that make a two space variable model of Asian options well-posed on a finite domain. To test the performance of well-posed boundary conditions, we price an average strike call. We also derive new boundary conditions for the average strike option from the put-call parity. Numerical results show that well-posed boundary conditions are working appropriately and solutions with new boundary conditions match the similarity solution significantly better than those provided in the existing literature. To price swing options, we develop a finite element penalty method on a one factor mean reverting diffusion model. We use the energy method to find well-posed boundary conditions on a finite domain, derive formulas to estimate the size of the numerical domain, develop a priori error estimates for both Dirichlet boundary conditions and Neumann boundary conditions. We verify the results through numerical experiments. Since the optimal exercise price is unknown in advance, which makes the swing option valuation challenging, we use a penalty method to resolve the difficulty caused by the early exercise feature. Numerical results show that the finite element penalty method is thousands times faster than the Binomial tree method at the same level of accuracy. Furthermore, we price a multiple right swing option with different strike prices. We find that a jump discontinuity can occur in the initial condition of a swing right since the exercise of another swing right may force its optimal exercise region to shrink. We develop an algorithm to identify the optimal exercise boundary at each time level, which allows us to record the optimal exercise time. Numerical results are accurate to one cent comparing with the benchmark solutions computed by a Binomial tree method. We extend applications to multiple right swing options with a waiting period restriction. A waiting period exists between two swing rights to be exercised successively, so we cannot exercise the latter right when we see an optimal exercise opportunity within the waiting period, but have to wait for the first optimal exercise opportunity after the waiting period. Therefore, we keep track of the optimal exercise time when pricing each swing right. We also verify an extreme case numerically. When the waiting time decreases, the value of M right swing option price increases to the value of M times an American option price as expected. 3-Manifolds of S1-Category Three. Wang, Dongxu, Heil, Wolfgang, Niu, Xufeng, Klassen, Eric P., Hironaka, Eriko, Nichols, Warren D., Department of Mathematics, Florida State University I study 3-manifold theory, which is a fascinating research area in topology. Many new ideas and techniques were introduced during these years, which makes it an active and fast developing subject. It is one of the most fruitful branches of today's mathematics and with the solution of the Poincare conjecture, it is getting more attention. This dissertation is motivated by results about categorical properties for 3-manifolds. This can be rephrased as the study of 3-manifolds which can be... Show moreI study 3-manifold theory, which is a fascinating research area in topology. Many new ideas and techniques were introduced during these years, which makes it an active and fast developing subject. It is one of the most fruitful branches of today's mathematics and with the solution of the Poincare conjecture, it is getting more attention. This dissertation is motivated by results about categorical properties for 3-manifolds. This can be rephrased as the study of 3-manifolds which can be covered by certain sets satisfying some homotopy properties. A special case is the problem of classifying 3-manifolds that can be covered by three simple S1-contractible subsets. S1-contractible subsets are subsets of a 3-manifold M3 that can be deformed into a circle in M3. In this thesis, I consider more geometric subsets with this property, namely subsets are homeomorphic to 3-balls, solid tori and solid Klein bottles. The main result is a classication of all closed 3-manifolds that can be obtained as a union of three solid Klein bottles. Alternative Models for Stochastic Volatility Corrections for Equity and Interest Rate Derivatives. Liang, Tianyu, Kercheval, Alec N., Wang, Xiaoming, Liu, Ewald, Brian, Nichols, Warren D., Department of Mathematics, Florida State University A lot of attention has been paid to the stochastic volatility model where the volatility is randomly fluctuating driven by an additional Brownian motion. In our work, we change the mean level in the mean-reverting process from a constant to a function of the underlying process. We apply our models to the pricing of both equity and interest rate derivatives. Throughout the thesis, a singular perturbation method is employed to derive closed-form formulas up to first order asymptotic solutions.... Show moreA lot of attention has been paid to the stochastic volatility model where the volatility is randomly fluctuating driven by an additional Brownian motion. In our work, we change the mean level in the mean-reverting process from a constant to a function of the underlying process. We apply our models to the pricing of both equity and interest rate derivatives. Throughout the thesis, a singular perturbation method is employed to derive closed-form formulas up to first order asymptotic solutions. We also implement multiplicative noise to arithmetic Ornstein-Uhlenbeck process to produce a wider variety of effects. Calibration and Monte Carlo simulation results show that the proposed model outperform Fouque's original stochastic volatility model during some particular window in history. A more efficient numerical scheme, the heterogeneous multi-scale method (HMM), is introduced to simulate the multi-scale differential equations discussed over the chapters. Modeling High-Frequency Order Book Dynamics with Support Vector Machines. Zhang, Yuan, Kercheval, Alec N., Niu, Xufeng, Nichols, Warren, Kim, Kyounghee, Department of Mathematics, Florida State University A machine learning based framework is proposed in this paper to capture the dynamics of high-frequency limit order books in financial markets and automate the prediction process in real-time on metrics characterizing the dynamics such as mid-price and price spread crossing. By representing each entry in a limit order book with a vector of features including price and volume at different levels as well as statistic features derived from limit order book, the proposed framework builds a... Show moreA machine learning based framework is proposed in this paper to capture the dynamics of high-frequency limit order books in financial markets and automate the prediction process in real-time on metrics characterizing the dynamics such as mid-price and price spread crossing. By representing each entry in a limit order book with a vector of features including price and volume at different levels as well as statistic features derived from limit order book, the proposed framework builds a learning model for each metric with the help of multi-class support vector machines (SVMs) to predict the directions of market movement. Experiments with real data as well as synthetic data establish that features selected by the proposed framework have highly differentiating capability, models built are effective and efficient in predictions on price movements, and trading strategies based on resulting models can achieve profitable returns with low risk. Non-Intrusive Methods for Probablistic Uncertainty Quantification and Global Sensitivity Analysis in Nonlinea Stochastic Phenomena. Liu, Yaning, Hussaini, M. Yousuff, Okten, Giray, Srivastava, Anuj, Sussman, Mark, Department of Mathematics, Florida State University The objective of this work is to quantify uncertainty and perform global sensitivity analysis for nonlinear models with a moderate or large number of stochastic parameters. We implement non-intrusive methods that do not require modification of the programming code of the underlying deterministic model. To avoid the curse of dimensionality, two methods, namely sampling methods and high dimensional model representation are employed to propagate uncertainty and compute global sensitivity indices... Show moreThe objective of this work is to quantify uncertainty and perform global sensitivity analysis for nonlinear models with a moderate or large number of stochastic parameters. We implement non-intrusive methods that do not require modification of the programming code of the underlying deterministic model. To avoid the curse of dimensionality, two methods, namely sampling methods and high dimensional model representation are employed to propagate uncertainty and compute global sensitivity indices. Variance-based global sensitivity analysis identifies significant and insignificant model parameters. It also provides basis for reducing a model's stochastic dimension by freezing identified insignificant model parameters at their nominal values. The dimension-reduced model can then be analyzed efficiently. We use uncertainty quantification and global sensitivity analysis in three applications. The first application is to the Rothermel wildland surface fire spread model, which consists of around 80 nonlinear algebraic equations and 24 parameters. We find the reduced models for the selected model outputs and apply efficient sampling methods to quantify the uncertainty. High dimensional model representation is also applied for the Rothermel model for comparison. The second application is to a recently developed biological model that describes inflammatory host response to a bacterial infection. The model involves four nonlinear coupled ordinary differential equations and the dimension of the stochastic space is 16. We compute global sensitivity indices for all parameters and build a dimension-reduced model. The sensitivity results, combined with experiments, can improve the validity of the model. The third application quantifies the uncertainty of weather derivative models and investigates model robustness based on global sensitivity analysis. Three commonly used weather derivative models for the daily average temperature are considered. The one which is least influenced by an increase of parametric uncertainty level is identified as robust. In summary, the following contributions are made in this dissertation: 1. The optimization of sensitivity derivative enhanced sampling that guarantees variance reduction and improved estimation of stochastic moments. 2. The combination of optimized sensitivity derivative enhanced sampling with randomized quasi-Monte Carlo sampling, and adaptive Monte Carlo sampling, to achieve higher convergence rates. 3. The construction of cut-HDMR component functions based on Gauss quadrature points which results in a more accurate surrogate model, derivation of an integral form of low order partial variances based on cut-HDMR, and efficient computation of global sensitivity analysis based on cut-HDMR. 4. The application of efficient sampling methods, RS-HDMR and cut-HDMR for the quantification of Rothermel's wildland fire surface spread model. 5. The uncertainty quantification and global sensitivity analysis of a newly developed immune response model with parametric uncertainty. 6. The uncertainty quantification of weather derivative models and the analysis of model robustness based on global sensitivity analysis. ON THE R-AUTOMORPHISMS OF R(X). DOWLEN, MARY MARGARET., Florida State University Throughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of R-automorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (L-HOOK) G(R). We prove in... Show moreThroughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of R-automorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (L-HOOK) G(R). We prove in Chapter I that R{X}('G(R)) = R{X}('B(R))., In Chapter I we also prove that for R to be properly contained in R{X}('G(R)), it is necessary that R/M is a finite field for some maximal ideal M of R. Hence, if R is a quasi-local ring with maximal ideal M and R/M is infinite, then R{X}('G(R)) = R., Let R be a quasi-local ring with maximal ideal M such that R/M is isomorphic to the Galois field with p('s) elements, where p is a prime integer and s (ELEM) Z('+). In Chapter II, we show that, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), where, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), In particular, we determine Z(,n){X}('G(Zn)) for n (ELEM) Z('+). Moreover, we prove that R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is a 0-dimensional SFT-ring., In Chapter III, we investigate R{X}('G(R)) for a von Neumann regular ring R. We obtain equivalent conditions for R{X}('G(R)) to contain a nonconstant monic polynomial; one of these is that {card(R/M)} is bounded for all maximal ideals M of R. Moreover, we prove that R is properly contained in R{X}('G(R)) if and only if R has a direct summand S such that S{X}('G(S)) contains a nonconstant monic polynomial. Finally, in Chapter III we construct a von Neumann regular ring B such that B/M is finite for infinitely many maximal ideals M of B, but B{X}('G(B)) = B., In Chapter IV, we show that for any commutative ring R with identity, R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is 0-dimensional, card(R/M) < N for some N (ELEM) Z('+) and for all maximal ideals M of R, and nilpotent elements have bounded order of nilpotency. A Spectral Element Method to Price Single and Multi-Asset European Options. Zhu, Wuming, Kopriva, David A., Huffer, Fred, Case, Bettye Anne, Kercheval, Alec N., Okten, Giray, Wang, Xiaoming, Department of Mathematics, Florida State University We develop a spectral element method to price European options under the Black-Scholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the non-smooth initial condition. For options with one asset under... Show moreWe develop a spectral element method to price European options under the Black-Scholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a Gauss-Lobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the non-smooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order Gauss-Lobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the Black-Scholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and Crank-Nicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks. Modeling the Folding Pattern of the Cerebral Cortex. Striegel, Deborah A., Hurdal, Monica K., Steinbock, Oliver, Quine, Jack, Sumners, DeWitt, Bertram, Richard, Department of Mathematics, Florida State University The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically-driven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents... Show moreThe mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemically-driven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents the Global Intermediate Progenitor (GIP) model, a theoretical biological model that uses features of the IP model and further captures global characteristics of cortical pattern formation. To illustrate how global features can effect the development of certain patterns, a mathematical model that incorporates a Turing system is used to examine pattern formation on a prolate spheroidal surface. Pattern formation in a biological system can be studied with a Turing reaction-diffusion system which utilizes characteristics of domain size and shape to predict which pattern will form. The GIP model approximates the shape of the lateral ventricle with a prolate spheroid. This representation allows the capture of a key shape feature, lateral ventricular eccentricity, in terms of the focal distance of the prolate spheroid. A formula relating domain scale and focal distance of a prolate spheroidal surface to specific prolate spheroidal harmonics is developed. This formula allows the prediction of pattern formation with solutions in the form of prolate spheroidal harmonics based on the size and shape of the prolate spheroidal surface. By utilizing this formula a direct correlation between the size and shape of the lateral ventricle, which drives the shape of the ventricular zone, and cerebral cortical folding pattern formation is found. This correlation is illustrated in two different applications: (i) how the location and directionality of the initial cortical folds change with respect to evolutionary development and (ii) how the initial folds change with respect to certain diseases, such as Microcephalia Vera and Megalencephaly Polymicrogyria Polydactyly with Hydrocephalus. The significance of the model, presented in this dissertation, is that it elucidates the consistency of cortical patterns among healthy individuals within a species and addresses inter-species variability based on global characteristics. This model provides a critical piece to the puzzle of cortical pattern formation. Peridynamic Multiscale Models for the Mechanics of Materials: Constitutive Relations, Upscaling from Atomistic Systems, and Interface Problems. Seleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, El-Azab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific... Show moreSeleson, Pablo D, Gunzburger, Max, Rikvold, Per Arne, El-Azab, Anter, Peterson, Janet, Shanbhag, Sachin, Lehoucq, Richard B., Parks, Michael L., Department of Scientific Computing, Florida State University This dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the state-based peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models,... Show moreThis dissertation focuses on the non local continuum peridynamics model for the mechanics of materials, related constitutive models, its connections to molecular dynamics and classical elasticity, and its multiscale and multimodel capabilities. A more generalized role is defined for influence functions in the state-based peridynamic model which allows for the strength of non local interactions to be modulated. This enables the connection between different peridynamic constitutive models, establishing a hierarchy that reveals that some models are special cases of others. Furthermore, this allows for the modulation of the strength of non local interactions, even for a fixed radius of interactions between material points in the peridynamics model. The multiscale aspect of peridynamics is demonstrated through its connections to molecular dynamics. Using higher-order gradient models, it is shown that peridynamics can be viewed as an up-scaling of molecular dynamics, preserving the relevant dynamics under appropriate choices of length scales. The state-based peridynamic model is shown to be appropriate for the description of multiscale and multimodel systems. A formulation for nonlocal interface problems involving scalar fields is presented, and derivations of non local transmission conditions are derived. Specializations that describe local, non local, and local/non local transmission conditions are considered. Moreover, the convergence of the non local transmission conditions to their classical local counterparts is shown. In all cases, results are illustrated by numerical experiments. Quasi-Monte Carlo and Genetic Algorithms with Applications to Endogenous Mortgage Rate Computation. Shah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics... Show moreShah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics, Florida State University In this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digit-permutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions... Show moreIn this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digit-permutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions for the test problems considered. Next, we use randomized quasi-Monte Carlo methods to numerically solve a one-factor mortgage model expressed as a stochastic fixed-point problem. Finally, we show that this mortgage model coincides with and is computationally faster than Citigroup's MOATS model, which is based on a binomial tree approach. A computational study of turbulent jet flows and their instability waves. Thies, Andrew Timothy., Florida State University The nature of turbulent jet flows is considered. First, the effects of nozzle geometry are examined. A boundary element formulation, which may be used to analyze the stability of vortex-sheet jets of arbitrary geometry, is developed. This formulation is applied to rectangular jets. It is found that rectangular jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics... Show moreThe nature of turbulent jet flows is considered. First, the effects of nozzle geometry are examined. A boundary element formulation, which may be used to analyze the stability of vortex-sheet jets of arbitrary geometry, is developed. This formulation is applied to rectangular jets. It is found that rectangular jets support four linearly independent families of instability waves. Within each family there are infinitely many modes. A way to classify these modes according to the characteristics of their eigenfunctions is proposed. It is found that the first and third modes of each family are corner modes. The fluctuations associated with these waves are localized near the corners of the jet. The second mode in each family, however, is a center mode with maximum fluctuations concentrated near the central portions of the jet. The center modes have the largest spatial growth rates. It is anticipated that as the instability waves propagate downstream the center modes would emerge as the dominant instabilities of the jet. Second, a K-$\varepsilon$ turbulence model, which incorporates Pope's nonplaner correction and Sarkar's high convective Mach number correction, is proposed for the computation of time-averaged turbulent jet flows. It is demonstrated that this model does contain the essential ingredients of turbulence physics for adequate jet mean flow prediction. However, the empirical constants that are generally used are found to be inappropriate for jets. A high-order parabolized approach to computing ideally expanded jet flows is presented. A new set of empirical constants is chosen, which better correlates the computations with measurements for a set of jets representative of a variety of nozzle configurations. It is demonstrated that, when the standard constants are replaced by the new ones, the model can offer good mean flow predictions for axisymmetric,, rectangular and elliptic jets with Mach numbers ranging from 0.4 to 2.0 and jet total temperature to ambient temperature ratios ranging from 1.0 to 4.0. Together, the two efforts lay much of the groundwork for a complete study of the effects of nozzle geometry on the mixing and noise generation in high-speed jet flows. On the motion of a rigid cylinder parallel to its axis in a rotating electrically conducting fluid. Ruan, Kezhi., Florida State University In an effort to understand better the flow in the core of the Earth, we investigate the steady rise of an infinitely long vertical rigid cylinder parallel to its axis in a rotating electrically conducting fluid in the presence of uniform prescribed transverse magnetic field. The rotation and magnetic-field vectors have arbitrary orientation. We suppose the circular cylinder is forced to rise with a constant speed and investigate the structure of the flow and calculate the drag on the cylinder... Show moreIn an effort to understand better the flow in the core of the Earth, we investigate the steady rise of an infinitely long vertical rigid cylinder parallel to its axis in a rotating electrically conducting fluid in the presence of uniform prescribed transverse magnetic field. The rotation and magnetic-field vectors have arbitrary orientation. We suppose the circular cylinder is forced to rise with a constant speed and investigate the structure of the flow and calculate the drag on the cylinder. The flow structure is found by solving a two-dimensional (independent of the axial coordinate) mixed boundary value problem. Approximate analytic solutions for velocity field and perturbed magnetic field are obtained. The buoyancy driven rise speed of the cylinder is calculated. The results are consistent with the those derived from Moore and Saffman (1969) and given by Hasimoto (1960) as limiting cases. The numerical value of dimensional rise speed obtained is in good agreement with the typically quoted rise speed in geophysics. An analysis of mush-chimney structure. Yang, Young-Kyun., Florida State University When a multi-component liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its... Show moreWhen a multi-component liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its permeability to the liquid flow. In particular, if a binary alloy (for example, NH$\sb4$Cl-H$\sb2$O solution) is cooled at bottom and a dense component (for example, NH$\sb4$Cl) is solidified, buoyant material released during freezing in the pores returns to the melt only through thin, vertical, but widely separated, 'chimneys', the flow through the matrix between them being organized to supply these chimneys., We presented photos of a mush-chimney system obtained from the ammonium chloride experiment, and we studied how convection with horizontal divergence affects the structure and flow of the mush-chimney system. We use a simple ODE system in the mush derived by assuming that the temperature depends on vertical coordinate only. We find that the mass fraction of solid increases and the depth of a mush decreases when the strength of convection increases., We present an axisymmetric model containing only one chimney to analyze the structure of the mush-chimney system. We find solutions of the temperature, the solid fraction, and the pressure in the chimney wall. In particular, the pressure expression shows that the fluid flow needs a huge pressure in order to pass through the chimney wall if its permeability is very small., We assume that a ratio of composition is large, which allows us to neglect the pressure contribution of the chimney wall. We use the knowledge of the variables in the mush, evaluated on the chimney wall, to find the fluid flow in the chimney and the radius of chimney. Our procedure employs the von Karman-Pohlhausen technique for determining chimney flow (Roberts & Loper, 1983) and makes use of the fact that the radius of the chimney is much less than the thickness of the mush. We find a relation between a parameter measuring the ratio of viscous and buoyancy forces in the chimney and the vertical velocity component on the top of the mush, and estimate numerically the value of this velocity measuring the strength of convection. The results obtained show reasonably good agreement with theoretical and experimental works (Roberts & Loper (1983), Chen & Chen (1991), Tait & Jaupart (1992), Hellawell etc. (1993), Worster (1991)). An analytical approach to the thermal residual stress problem in fiber-reinforced composites. Xie, Zhiyun., Florida State University A pair of two new tensors called Generalized Plane Strain (GPS) tensors S and D is proposed for the concentric cylindrical inclusion problem. GPS tensors take the fiber volume fraction explicitly into account. When the cylindrical matrix is of infinite radius, tensor S reduces to the appropriate Eshelby's tensor. The GPS tensors provide a convenient form of solution to a class of problems involving eigen-strain, e.g., strain due to thermal expansion, phase transformation, plastic and misfit... Show moreA pair of two new tensors called Generalized Plane Strain (GPS) tensors S and D is proposed for the concentric cylindrical inclusion problem. GPS tensors take the fiber volume fraction explicitly into account. When the cylindrical matrix is of infinite radius, tensor S reduces to the appropriate Eshelby's tensor. The GPS tensors provide a convenient form of solution to a class of problems involving eigen-strain, e.g., strain due to thermal expansion, phase transformation, plastic and misfit strain. Explicit expressions to evaluate thermal residual stresses in the matrix and the fiber using GPS tensors are developed for metallic/intermetallic matrix composites. Results are compared with Eshelby's infinite domain solution and Finite Element solution for SCS-6/Ti-24Al-11Nb composite. The method of superposition using GPS tensor is proposed for evaluating thermal residual stress distribution in a fiber reinforced composite with periodic arrays. The results compare very favorably with Finite Element solution. GPS tensors are also used in the evaluation of the effective material properties. We demonstrated the approach by studying two fiber reinforced composites, Graphite/Epoxy and Glass/Epoxy composites. A good agreement between analytical results using GPS tensor and experimental data was found. We also compared the results of using GPS tensor along with the original Eshelby's tensor and found that GPS tensor provides a better match with experimental data. Doubly-null-cobordant links. Sun, Biansheng., Florida State University Throughout, we work in the smooth category. We consider a special class of links and knots in $S\sp3$ which are transverse cross-sections of trivial 2-spheres in $S\sp4.$ They are called Doubly-Null-Cobordant (DNC) links and DNC knots respectively. Closely related concepts are those of Null-Cobordant (NC) links and NC knots., We are interested in obtaining necessary conditions satisfied by DNC (NC) links and knots, and in constructing nontrivial links and knots which satisfy these conditions.... Show moreThroughout, we work in the smooth category. We consider a special class of links and knots in $S\sp3$ which are transverse cross-sections of trivial 2-spheres in $S\sp4.$ They are called Doubly-Null-Cobordant (DNC) links and DNC knots respectively. Closely related concepts are those of Null-Cobordant (NC) links and NC knots., We are interested in obtaining necessary conditions satisfied by DNC (NC) links and knots, and in constructing nontrivial links and knots which satisfy these conditions., Based on analysis of various linking patterns of NC links, we are able to prove that any Hopf link of $\mu$ components is NC if and only if $\mu$ is odd. Another result obtained in this work is that there exists at least one pair of components of an NC link of an even number of components such that the linking number between these two components is zero. Various methods are employed in the geometric realizations of DNC links of a given number of components; we construct DNC links for any given number of components., The 2-fold branched cyclic cover of $S\sp3$ branched along any link plays a fundamental role in detecting whether a given link is DNC or not since the cyclic branched cover embeds in $S\sp4$ if the given link is DNC. Considering the embedding problem of the cyclic branched cover leads to the hyperbolicity problem of the associated linking pairing on the homology of the cyclic branched cover. By investigating the torsion part of the first homology of the r-fold cyclic branched cover of the 3-sphere branched along a link, we are able to produce infinitely many NC links, none of which are DNC links. When specialized to knots, we also discover infinitely many NC knots, none of which are DNC., We also consider higher dimensional DNC links. We obtain a necessary condition for a higher dimensional link being DNC. Specifically, we prove that if L is a DNC (2m $-$ 1)-link of $\mu$ components with $m >$ 1, then L has a DNC Seifert matrix for any connected Seifert manifold of L. WEAK TOPOLOGIES IN SPACES OF OPERATORS. 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Hill cipher: How to find an unknown key of unknown size How would you tackle the problem of finding the key (you don't know the length) to a Hill cipher when knowing only one 12-letter word of plaintext and its corresponding ciphertext? CONVERSATION has been encoded as SQZHUSSUDYKP with standard alphabet (A=0, Z=25). The key, as I said, is unknown with unknown size. I am not asking for a solution, but for tips and guidance. Any help will be much appreciated. known-plaintext-attack hill-cipher Christo TodorovChristo Todorov Assuming that $2\times2$ matrix is used, and the encryption starts from the first letter of the plaintext, the key can be found by just calculating the "encryption" with size of $4$ plain- and cryptotext block. For example, for CONV $\rightarrow$ SQZH, it would go as follows: $$\begin{pmatrix} 2 & 14 \\ 13 & 21 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}= \begin{pmatrix} 18 & 16 \\ 25 & 7 \end{pmatrix}\pmod{26}$$ Then solving for $a,b,c,d$, the key $K$ is found. As here the plaintext matrix is invertible, one can compute $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} 2 & 14 \\ 13 & 21 \end{pmatrix}^{-1} \begin{pmatrix} 18 & 16 \\ 25 & 7 \end{pmatrix}\pmod{26}$$ M.PM.P $\begingroup$ What if $3\times3$ matrix is used for encryption? $\endgroup$ – Christo Todorov Jun 19 '19 at 11:52 $\begingroup$ You could proceed the same way, but having $9$ coefficients to be solved. $\endgroup$ – M.P Jun 19 '19 at 11:56 Not the answer you're looking for? Browse other questions tagged known-plaintext-attack hill-cipher or ask your own question. Hill Cipher known plaintext attack Hill cipher, unknown letter value Hill cipher cryptanalysis - known plaintext known key size Proving that an encryption scheme is susceptible to certain attacks Clarification on Hill Cipher crib dragging technique	CommonCrawl
\begin{document} \newcommand{2005.14225}{2005.14225} \renewcommand{\arabic{footnote}}{} \renewcommand{020}{020} \FirstPageHeading \ShortArticleName{A Spectral Triple for a Solenoid Based on the Sierpinski Gasket} \ArticleName{A Spectral Triple for a Solenoid Based\\ on the Sierpinski Gasket\footnote{This paper is a~contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of~Giovanni Landi. The full collection is available at \href{https://www.emis.de/journals/SIGMA/Landi.html}{https://www.emis.de/journals/SIGMA/Landi.html}}} \Author{Valeriano AIELLO~$^{\rm a}$, Daniele GUIDO~$^{\rm b}$ and Tommaso ISOLA~$^{\rm b}$} \AuthorNameForHeading{V.~Aiello, D.~Guido and T.~Isola} \Address{$^{\rm a)}$~Mathematisches Institut, Universit\"at Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland} \EmailD{\href{mailto:[email protected]}{[email protected]}} \Address{$^{\rm b)}$~Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'', I--00133 Roma, Italy} \EmailD{\href{mailto:[email protected]}{[email protected]}, \href{mailto:[email protected]}{[email protected]}} \ArticleDates{Received June 23, 2020, in final form February 10, 2021; Published online March 02, 2021} \Abstract{The Sierpinski gasket admits a locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed.} \Keywords{self-similar fractals; noncommutative geometry; ramified coverings} \Classification{58B34; 28A80; 47D07; 46L05} \renewcommand{\arabic{footnote}}{\arabic{footnote}} \setcounter{footnote}{0} \section{Introduction} In this note, we introduce a semifinite spectral triple on the $C^$-algebra of continuous functions on the solenoid associated with a self-covering of the Sierpinski gasket. Such triple is finitely summable, its metric dimension coincides with the Hausdorff dimension of the gasket, and the associated non-commutative integral coincides up to a constant with a Bohr--F{\o}lner mean on the solenoid, hence reproduces the suitably normalized Hausdorff measure on periodic functions. The open infinite Sierpinski fractafold with a unique boundary point considered by Teplyaev~\cite{Tep} embeds continuously as a dense subspace of the solenoid, and the Connes distance restricted to such subspace reproduces the geodesic distance on such fractafold. On the one hand, this shows that our spectral triple describes aspects of both local and coarse geometry~\cite{RoeLN}. On the other hand, this implies that the topology induced by the Connes distance, being non compact, does not coincide with the weak$^$-topology on the states of the solenoid algebra, as we call the $C^$-algebra of continuous functions on the solenoid. This means that the solenoid, endowed with our spectral triple, is not a quantum metric space in the sense of Rieffel~\cite{Rieffel}. Related research concerning projective limits of (possibly quantum) spaces and the associated solenoids appeared recently in the literature. In the framework of noncommutative geometry, we~mention: \cite{LaPa}, where projective families of compact quantum spaces have been studied, showing their convergence to the solenoid w.r.t.\ the Gromov--Hausdorff propinquity distance; \cite{AGI01}, where, in the same spirit as in this note, a semifinite spectral triple has been associated with the projective limit generated by endomorphisms of $C^$-algebras associated with commutative and noncommutative spaces; \cite{DGMW}, where a spectral triple on the stable Ruelle algebra for Wieler solenoids has been considered and its unboundedd KK-theory has been studied, based on the Morita equivalence between the stable Ruelle algebra and a Cuntz--Pimsner algebra. In the same paper these techniques are used for the study of limit sets of regular self-similar groups (cf.~\cite{Nekra}). When fractals are concerned, we mention the projective family of finite coverings of the octahedron gasket considered in~\cite{Stri2009}, where, as in our present situation, an intermediate infinite fractafold between the tower of coverings and the projective limit is considered. Periodic and almost periodic functions on the infinite fractafold are considered, and a Fourier series description for the periodic functions is given, based on periodic eigenfunctions of the Laplacian (cf.\ also~\cite{RuStri} for higher-dimensional examples). Let us remark that such coverings, as the ones considered in~this paper, are not associated with groups of deck transformations. The starting point for the construction of this paper is the existence of a locally isometric ramified three-fold self-covering of the Sierpinski gasket with trivial group of deck transformations. Such self-covering gives rise to a projective family of coverings, whose projective limit is by definition a solenoid. Dually, the algebras of continuous functions on the coverings form an~injective family, whose direct limit (in the category of $C^$-algebras) is the solenoid algebra. In~\cite{AGI01} we already considered various examples of self-coverings or, dually, of endomorphisms of some $C^$-algebras, most of which were regular finite self-coverings. There we constructed a~spectral triple on the solenoid algebra as a suitable limit of~spectral triples on the algebras of continuous functions on the coverings. Given a spectral triple on the base space, attaching a~spectral triple to a finite covering is not a difficult task, and in our present case consists simply in ``dilating'' the triple on the base gasket so that the projections are locally isometric. However, there is no commonly accepted procedure to define a limit of spectral triples. Since the method used in~\cite{AGI01} cannot be used here (see below), we follow another route, in a sense spatializing the construction, namely showing that there exists an open fractafold which is intermediate between the projective family of coverings and the solenoid. More precisely, such fractafold space turns out to be an~infinite covering of each of the finite coverings of the family, and embeds in~a~continuous way in the solenoid. In this way all the algebras (and their direct limit) will act on a suitable $L^2$-space of the open fractafold, as do the Dirac operators of the associated spectral triples. In~this way the limiting Dirac operator is well defined, but the compact resolvent property will be~lost. Let us notice here that we are not constructing a spectral triple on the open fractafold, where a weaker compact resolvent property (cf.~\cite[Chapter~IV, Remark~12]{ConnesBook}) is retained, namely $f(D^2+I)^{-1/2}$ is a compact operator, where $D$ is the Dirac operator and $f$ is any function with compact support on the fractafold. Since we are constructing a spectral triple on the solenoid, which is a compact space, the weaker form does not help. In order to recover the needed compactness of the resolvent, we use a procedure first proposed by J. Roe for open manifolds with an amenable exhaustion in~\cite{Roe,Roe-2}, where, based on the observation that the von~Neumann trace used by Atiyah~\cite{Atiyah} for his index theorem for covering manifolds can be reformulated in the case of amenable groups via the F{\o}lner condition, he considered amenable exhaustions on open manifolds and constructed a trace for finite-propagation operators acting on sections of a fiber bundle on the manifold via a renormalization procedure. Unfortunately such trace is not canonical, since it depends on a generalized limit procedure. However, in the case of infinite self-similar CW-complexes, it was observed in~\cite{CGIs01} that such trace becomes canonical when restricted to the $C^$-algebra of geometric operators. We adapt these results to our present context, namely we replace the usual trace with a~renor\-ma\-li\-zed trace associated with an exhaustion of the infinite fractafold. Such trace comes together with a noncommutative $C^$-algebra, the algebra of geometric operators, which is similar in spirit to the Roe $C^$-algebras of coarse geometry~\cite{HiRo,Roe,Roe-2,Roe96,WiYu}. This algebra contains the solenoid algebra, and the limiting Dirac operator is affiliated to it in a suitable sense. Such Dirac operator turns out to be $\tau$-compact w.r.t.~the renormalized trace. We refer to~\cite{CGIs01, GuIs7} for an analogous construction of the $C^$-algebra and of a canonical trace based on the self-similarity structure. As discussed above, the starting point for the construction of a spectral triple on the solenoid algebra is the association of a spectral triple to the fractal known as the Sierpinski gasket~\cite{Sierpinski}. The study of fractal spaces from a spectral, or noncommutative, point of view has now a~long history, starting from the early papers of Kigami and Lapidus~\cite{KiLa, La94,La97}. As for the spectral triples, various constructions have been considered in the literature, mainly based on~``small'' triples attached to specific subsets of the fractal, following a general procedure first introduced by Connes, then considered in~\cite{GuIs9,GuIs10}, and subsequently abstracted in~\cite{ChIv07}. More precisely, the spectral triple on the Cantor set described by Connes~\cite{ConnesBook} inspired two kinds of~spectral triples for various families of fractals in~\cite{GuIs9,GuIs10}. These triples were further analysed in~\cite{GuIs16} for the class of nested fractals. Such specral triples are obtained as direct sums of triples on two points (boundary points of an edge in some cases), and we call them discrete spectral triples. We~then mention some spectral triples obtained as direct sums of spectral triples on 1-dimensional subsets, such as those considered in~\cite{Arau,CIL,CIS,CGIS02,LaSa}, where the 1-dimensional subsets are segments, circles or quasi-circles. Discrete spectral triples give a good description of metric aspects of the fractal, such as Hausdorff dimension and measure and geodesic distance, and, as shown in~\cite{GuIs1}, may also reconstruct the energy functional (Dirichlet form) on the fractal, but are not suited for the study of K-theoretical properties since the pairing with K-theory is trivial. Conversely, spectral triples based on segments or circles describe both metric and K-theoretic properties of~the fractal but can't be used for describing the Dirichlet form. Finally, the spectral triple based on quasi-circles considered in~\cite{CGIS02} describes metric and K-theoretic aspects together with the energy form, but requires a rather technical approach. In the present paper, we make use of the simple discrete spectral triple on the gasket as described in~\cite{GuIs16}, thus obtaining a semifinite spectral triple on the solenoid algebra which recovers the metric dimension and the Bohr--F{\o}lner mean of the solenoid, and the geodesic distance on the infinite fractafold. Further analysis on the solenoid is possible, e.g., the construction of a Dirichlet form via noncommutative geometry or the study of K-theoretic properties. As explained above, the latter step will require a different choice of the spectral triple on the base gasket, such as the triples considered in~\cite{CIL,CIS,CGIS02}, which admit a non-trivial pairing with the K-theory of the gasket. As already mentioned, our aim here is to show that the family of spectral triples on the finite coverings produces a spectral triple on the solenoidal space. In the examples considered in~\cite{AGI01}, the family of spectral triples had a simple tensor product structure, namely the Hilbert spaces were a tensor product of the Hilbert space ${\mathcal H}$ for the base space and a finite dimensional Hilbert space, and the Dirac operators could be described as (a finite sum of) tensor product operators. Then the ambient $C^$-algebra turned out to be a product of ${\mathcal B}({\mathcal H})$ and a UHF algebra, allowing a GNS representation w.r.t.~a~semifinite trace. In the example treated here we choose a different approach since two problems forbid such simple description. The first is a local problem, due to the ramification points. This implies that the algebra of a covering is not a free module on the algebra of the base space; in particular, functions on a covering space form a proper sub-algebra of the direct sum of finitely many copies of the algebra for the base space. The second is a non-local problem which concerns the Hilbert spaces, which are $\ell^2$-spaces on edges, and the associated operator algebras. Indeed, the Hilbert spaces of the coverings cannot be described as finite sums of copies of the Hilbert space on the base space due to the appearance of longer and longer edges on larger and larger coverings. We conclude this introduction by mentioning two further developments of the present analysis. First, the construction of the spectral triple on the solenoid algebra allows the possibility of~lifting a spectral triple from a $C^$-algebra to the crossed product of the C-algebra with a~single endomorphism~\cite{AGI02}, thus generalising the results on crossed products with an automorphism group considered in~\cite{BMR,Skalski,Paterson}. Second, we observe that the construction given in the present paper goes in the direction of~possibly defining a $C^$-spectral triple, in which the semifinite von Neumann algebra is replaced by a $C^$-algebra with a trace to which both the Dirac operator and the ``functions'' on the non-commutative space are affiliated, where the compactness of the resolvent of the Dirac operator is measured by the trace on the $C^$-algebra, cf.~also~\cite{GuIs4}. This paper is divided in six sections. After this introduction, Section~\ref{sec2} contains some pre\-li\-mi\-nary notions on fractals and spectral triples, Section~\ref{sec3} describes the geometry of the ramified covering and the corresponding inductive structure, together with its functional counterpart given by a family of compatible spectral triples. Section~\ref{sec4} concerns the self-similarity structure of the Sierpinski solenoid, whence the description of the inductive family of $C^$-algebras as algebras of bounded functions on the fractafold. The Section~\ref{sec5} describes the algebra of geometric operators and the construction of a semicontinuous semifinite trace on it. Finally, the semifinite spectral triple together with its main features are contained in Section~\ref{sec6}. \section{Preliminaries}\label{sec2} In this section we shall briefly recall various notions that will be used in the paper. Though these notions are well known among the experts, our note concerns different themes, namely spectral triples in noncommutative geometry and nested fractals (the Sierpinski gasket in particular), so~that we decided to write this section with the aim of helping readers with different background to follow the various arguments, by collecting here the main notions and results that will be useful in the following. \subsection{Spectral triples} The notion of spectral triple plays a key role in Alain Connes'non\-com\-mu\-ta\-ti\-ve geometry~\cite{ConnesBook,GBVF}. Basically, it consists of a triple $({\mathcal L},{\mathcal H},D)$, where ${\mathcal L}$ is a -algebra acting faithfully on the Hilbert space ${\mathcal H}$, and $D$ is an unbounded self-adjoint operator on ${\mathcal H}$ satisfying the properties \begin{itemize}\itemsep=0pt \item[$(1)$] $\big(1+D^2\big)^{-1/2}$ is a compact operator, \item[$(2)$] $\pi(a)\mathcal{D} (D) \subset \mathcal{D} (D)$, and $[D,\pi(a)]$ is bounded for all $a\in {\mathcal L}$. \end{itemize} We shall also say that $({\mathcal L},{\mathcal H},D)$ is a spectral triple on the $C^$-algebra ${\mathcal A}$ generated by~${\mathcal L}$. Such triple is meant as a generalization of a compact smooth manifold, the algebra ${\mathcal L}$ replacing the algebra of smooth functions, the Hilbert space describing a vector bundle (a spin bundle indeed) on which the algebra of functions acts, and the operator~$D$ generalizing the notion of Dirac operator. Further structure may be added to the properties above, allowing deeper analysis of the geometric features of the noncommutative manifold, but these are not needed in~this paper. Property $(2)$ above allows the definition of a (possibily infinite) distance (Connes distance) on the state space of the $C^$-algebra ${\mathcal A}$ generated by ${\mathcal L}$ , defined as \begin{gather} d(\varphi,\psi)=\sup\{\|\varphi(a)-\psi(a)\|\colon \\|[D,a]\\|\leq1,\, a\in {\mathcal L}\}. \end{gather} When the Connes distance induces the weak$^$-topology on the state space, the seminorm $\\|[D,a]\\|$ on~${\mathcal A}$ is called a Lip-norm (cf.~\cite{Rieffel}) and the algebra ${\mathcal A}$ endowed with the Connes distance is a~quantum metric space. A spectral triple is called finitely summable if $\big(1+D^2\big)^{-s}$ has finite trace for some $s>0$, in this case the abscissa of convergence~$d$ of the function $\tr\big(1+D^2\big)^{-s}$ is called the metric dimension of~the triple. Then the logarithmic singular trace introduced by Dixmier~\cite{Dix} may be used to define a noncommmutative integral on~${\mathcal A}$. Let us denote by $\{\mu_n(T)\}$ the sequence (with multiplicity) of~singular values of the compact operator~$T$, arranged in decreasing order. Then, on the positive compact operators for which the sequence $\sum_{k=1}^n\mu_n(T)$, is at most logarithimically divergent, we may consider the positive functional \begin{gather} \tr_\omega} \def\O{\Omega(T)=\Lim_\omega} \def\O{\Omega\frac{\sum_{k=1}^n\mu_n(T)}{\log n}, \end{gather} where $\Lim_\omega} \def\O{\Omega$ is a suitable generalized limit. Such functional extends to a positive trace on ${\mathcal B}({\mathcal H})$ which vanishes on trace class operators, and is called Dixmier (logarithmic) trace. If $\big(1+D^2\big)^{-d}$ is in the domain of the Dixmier trace, one defines the following noncommutative integral: \begin{gather} \oint a=\tr_\omega} \def\O{\Omega\big (a\big(I+D^2\big)^{-d/2}\big),\qquad a\in{\mathcal A}. \end{gather} When the function $\big(1+D^2\big)^{-s}$ has a finite residue for $s=d$, such residue turns out to coincide, up to a constant, with the Dixmier trace, which therefore does not depend on the generalized limit procedure (cf.~\cite{ConnesBook}, and~\cite[Theorem~3.8]{CPS}): \begin{gather} d\cdot\tr_\omega} \def\O{\Omega \big(a\big(I+D^2\big)^{-d/2}\big)={\Res}_{s=d} \tr(a\|D\|^{-s}). \end{gather} We note in passing that spectral triples may also describe non-compact smooth manifolds, with the algebra~${\mathcal L}$ describing smooth functions with compact support and property~(1) replaced by~$a\big(1+D^2\big)^{-1/2}$ is a compact operator for any $a\in{\mathcal L}$. \subsection{Semifinite spectral triples}\label{SemST} The notion of spectral triple has been generalized to the semifinite case, by replacing the ambient algebra ${\mathcal B}({\mathcal H})$ with a semifinite von~Neumann algebra ${\mathcal M}$ endowed with a normal semifinite faithful trace~$\tau$. We recall that an operator $T$ affiliated with $({\mathcal M},\tau)$ is called $\tau$-compact if its generalized s-number function $\mu_t(T)$ is infinitesimal or, equivalently, if $\tau(e_{(t,\infty)}(T))<\infty$, for~any $t>0$ (cf.~\cite[Section~1.8, p.~34]{Fack}, \cite[Proposition~3.2]{FK}). \begin{dfn} [\cite{CaPhi1}]\label{def:SFtriple} An odd semifinite spectral triple $({\mathcal L},{\mathcal M},D)$ on a unital C$^$-algebra ${\mathcal A}$ is given by a unital, norm-dense, $^$-subalgebra ${\mathcal L}\subset{\mathcal A}$, a semifinite von Neumann algebra $({\mathcal M},\tau)$, acting on a (separable) Hilbert space ${\mathcal H}$, a faithful representation $\pi\colon{\mathcal A}\to{\mathcal B}({\mathcal H})$ such that $\pi({\mathcal A})\subset{\mathcal M}$, and an unbounded self-adjoint operator $D \widehat{\in} {\mathcal M}$ such that \begin{itemize}\itemsep=0pt \item[$(1)$] $\big(1+D^2\big)^{-1/2}$ is a $\tau$-compact operator, \item[$(2)$] $\pi(a)\mathcal{D} (D) \subset \mathcal{D} (D)$, and $[D,\pi(a)] \in{\mathcal M}$, for all $a\in{\mathcal L}$. \end{itemize} \end{dfn} As in the type $I$ case, such triple is called finitely summable if $\big(1+D^2\big)^{-s}$ has finite trace for some $s>0$, and $d$ denotes the abscissa of convergence of the function $\tau\big(1+D^2\big)^{-s}$, and is called the metric dimension of the triple. The logarithmic Dixmier trace associated with the normal trace $\tau$ may be defined in this case too, (cf.~\cite{CPS, GuIs1}) and, when the function $\big(1+D^2\big)^{-s}$ has a~finite residue for $s=d$, the equality $d\cdot\tr_\omega\big(a\|D\|^{-d}\big)={\Res}_{s=d} \tr\big(a\|D\|^{-s}\big)$ still holds \cite[Theorem~3.8]{CPS}. \subsection{Self-similar fractals} Let $\O:= \{w_i \colon i=1,\ldots,k \}$ be a family of contracting similarities of ${\mathbb R}^{N}$, with scaling para\-me\-ters~$\{\lambda} \def\La{\Lambda_i\}$. The unique non-empty compact subset $K$ of ${\mathbb R}^{N}$ such that $K = \bigcup_{i=1}^{k} w_i(K)$ is called the {\it self-similar fractal} defined by $\{w_i \}_{i=1,\ldots,k}$. For any $i\in\{1,\ldots,k\}$, let $p_i\in{\mathbb R}^N$ be the unique fixed-point of $w_i$, and say that $p_i$ is an essential fixed-point of $\O$ if there are $i',j,j'\in\{1,\ldots,k\}$ such that $i'\neq i$, and $w_j(p_i)=w_{j'}(p_{i'})$. Denote by $V_0(K)$ the set of essential fixed-points of $\O$, and let $E_0(K):=\{ (p,q)\colon p,q\in V_0,\ p\neq q\}$. Observe that $(V_0,E_0)$ is a directed finite graph whose edges are in $1:1$ correspondence with ordered pairs of distinct vertices. \begin{dfn} We call an element of the family $\{w_{i_1}\cdots w_{i_k}(K)\colon k\geq0\}$ a {\it cell}, and call its diameter the size of the cell. We call an element of the family $E(K)=\{w_{i_1}\cdots w_{i_k}(e)\colon k\geq0$, $e\in E_{0}(K)\}$ an {\it $($oriented$)$ edge} of $K$. We denote by $e^-$ resp.~$e^+$ the source, resp.~the target of~the oriented edge $e$. \end{dfn} As an example, the Sierpinski gasket is the self-similar fractal determined by 3 similarities with scaling parameter 1/2 centered in the vertices of an equilateral triangle (see Fig.~\ref{fig:Gasket}). \begin{figure} \caption{The first four steps of the construction of the gasket.} \label{fig:Gasket} \end{figure} Under suitable conditions, the Hausdorff dimension $d_H$ of a self-similar fractal coincides with its scaling dimension, namely with the only positive number $d$ such that $\sum_{i=1}^k\lambda} \def\La{\Lambda_i^d=1$, therefore when all scaling parameters coincide with $\lambda} \def\La{\Lambda$ we have $d_H=\frac{\log k}{\log(1/\lambda} \def\La{\Lambda)}$. In particular, the Hausdorff dimension of the Sierpinski gasket is $\frac{\log 3}{\log2}$. We note in passing that one of the most important aspects of the Sierpinski gasket and of more general classes of fractals is the existence of a self-similar diffusion, associated with a Dirichlet form, see, e.g.,~\cite{Kiga2}. Even though Dirichlet forms on~fractals can be recovered in the noncommutative geometry framework~\cite{GuIs16}, and in particular by~means of the spectral triples which we use in this paper, we do not analyse this aspect in the present note. In~\cite{GuIs10} discrete spectral triples have been introduced on some classes of fractals, generalizing an example of Connes in~\cite[Chapter~4.3, Example~23]{ConnesBook}. Such triples have been further studied in~\cite{GuIs16} for nested fractals. On a self-similar fractal $K$, the triple $ ({\mathcal L},{\mathcal H},D)$ on the $C^$-algebra ${\mathcal A}={\mathcal C}(K)$ is defined as follows: \begin{dfn}\label{STnested}\qquad \begin{itemize}\itemsep=0pt \item[$(a)$] ${\mathcal H}=\ell^2(E(K))$, \item[$(b)$] ${\mathcal A}$ acts on the Hilbert space as $\rho(f)e=f(e^+)e$, $f\in{\mathcal A}_n$, $e\in E_n$, \item[$(c)$] $F$ is the orientation-reversing map on edges, \item[$(d)$] $D$ maps an edge $e\in E(K)$ to $\mathrm{length}(e)^{-1}Fe$, \item[$(e)$] ${\mathcal L}$ is given by the elements $ f\in {\mathcal A}$ such that $\\|[D,\rho(f)]\\|<\infty$. \end{itemize} \end{dfn} It turns out that ${\mathcal L}$ coincides with the algebra of Lipschitz functions on $K$, hence is dense in~${\mathcal A}$, and the seminorm $L(f):=\\|[D,f]\\|$ is a Lip-norm. By Theorem 3.3 in~\cite{GuIs16}, see also Remark~2.11 in~\cite{GuIs10}, the triple $({\mathcal L},{\mathcal H},D)$ is a finitely summable spectral triple on ${\mathcal A}$, its metric dimension coincides with the Hausdorff dimension, and the noncommutative integral recovers the Hausdorff measure up to a constant: \begin{gather}\label{fractalNCint} \oint f=\tr_\omega\big(f\|D\|^{-d}\big) = \frac{1}{ \log k} \sum_{e\in E_0(K)} \ell(e)^d \int_K f\, {\rm d}H_d, \qquad f\in C(K), \end{gather} where $H_d$ denotes the normalized Hausdorff measure on the fractal $K$. Moreover, in some cases, and in particular for the Sierpinski gasket, the Connes distance induced by the Lip-norm $L(f):=\\|[D,f]\\|$ coincides with the geodesic distance on the points of the gasket $K$, see~\cite[Corollary~5.14]{GuIs16}. \subsection{Covering fractafolds and solenoids} Generally speaking, a solenoid is the inverse limit of a projective family of coverings of a given space~\cite{McCord}. Dually, the solenoid algebra is the direct limit of the family of algebras of continuous functions on the spaces of the projective family. In this sense the notion of solenoid makes sense for injective families of $C^$-algebras, cf., e.g.,~\cite{AGI01} for sequences generated by a single endomorphism and~\cite{LaPa} for sequences of compact quantum spaces. Other examples of the treatment of solenoids in the recent literature have been mentioned in the introduction. The notion of fractafold as a connected Hausdorff topological space such that every point has a neighborhood homeomorphic to a neighborhood in a given fractal has been introduced in~\cite{Stri2003}, even though examples of such notion were already considered before, e.g., in~\cite{BaPe,Stri1996,Tep}. In some cases projective families of covering fractafold spaces related to the Sierpinski gasket have been considered. Since the gasket does not admit a simply connected covering, one may consider coverings where more and more cycles are unfolded, in particular consider the regular infinite abelian covering $S_n$ where all the cycles of size at least $2^{-n}$ are unfolded. Each of those is a closed fractafold (with boundary) and they form a projective family. The associated solenoid~$S_\infty$, i.e., the projective limit, which turns out to be an abelian counterpart of the Uniform Universal Cover introduced by Berestovskii and Plaut~\cite{BerPla}, has been considered in~\cite{CGIS13}, where it is shown that any locally exact 1-form on the gasket possesses a potential on~$S_\infty$. Another projective family of covering fractafolds has been considered in~\cite{Stri2009}, each element of~the family being a compact finite covering of the octahedral fractafold modeled on the gasket. Any element of the family is covered by the infinite Sierpinski gasket with a unique boundary point, which we call $K_\infty$ here (see Fig.~\ref{fig:infiniteBlowup}), considered in~\cite[Lemma~5.11]{Tep}. The solenoid associated with the projective family is also mentioned explicitly in~\cite{Stri2009}, together with the dense embedding of $K_\infty$ in it, and also a Bohr--F{\o}lner mean on the solenoid is considered (p.~1199). \begin{figure} \caption{The gasket and its infinite blowup.} \label{fig:infiniteBlowup} \end{figure} In the present paper a self-covering of the gasket gives rise to a projective family of finite ramified coverings, the fractafold $K_\infty$ projects onto each element of the family and embeds densely in the solenoid, and we recover the Bohr--F{\o}lner mean on the solenoid via a noncommutative integral. \section{A ramified covering of the Sierpinski gasket}\label{sec3} Let us choose an equilateral triangle of side 1 in the Euclidean plane with vertices $v_0$, $v_1$, $v_2$ (numbered in a counterclockwise order) and consider the associated Sierpinski gasket as in the previous section, namely the set $K$ such that \begin{gather} K=\bigcup_{j=0,1,2}w_j(K), \end{gather} where $w_j$ is the dilation around $v_j$ with contraction parameter $1/2$. Clearly, for the cell $C=w_{i_1}\cdots w_{i_k}(K)$, $\mathrm{size}(C)=2^{-k}$ and, if $e_0\in E_{0}(K)$ and $e=w_{i_1}\cdots w_{i_k}(e_0)$, $\mathrm{length}(e)=2^{-k}$. \looseness=1 5In the following we shall set $K_0:=K$, $E_0=E_0(K)$, $K_n=w_0^{-n}K_0$. Let us now consider the middle point $x_{i,i+1}$ of the segment $\big(w_0^{-1}v_i,w_0^{-1}v_{i+1}\big)$, $i=0,1,2$, the map $R_{i+1,i}\colon w_0^{-1}w_{i}K\to w_0^{-1}w_{i+1}K$ consisting of the rotation of $\frac43\pi$ around the point $x_{i,i+1}$, $i=0,1,2$, and observe that \begin{gather}\label{IdOnCells} R_{i,i+2}\circ R_{i+2,i+1}\circ R_{i+1,i} = {\rm id}_{w_0^{-1}w_i K},\qquad i=0,1,2. \end{gather} Setting $R_{i,i+1} = R_{i+1,i}^{-1}$, the previous identities may also be written as \begin{gather} R_{i+2,i+1}\circ R_{i+1,i} = R_{i+2,i},\qquad i=0,1,2. \end{gather} We then construct the map $p\colon K_1\to K$ given by \begin{gather} p(x)= \begin{cases} x,&x\in K, \\ R_{0,1}(x),&x\in w_0^{-1}w_1 K, \\ R_{0,2}(x),&x\in w_0^{-1}w_2 K, \end{cases} \end{gather} and observe that this map, which appears to be doubly defined in the points $x_{i,i+1}$, $i=0,1,2$, is indeed well defined (see Fig.~\ref{fig:covering}). \begin{figure} \caption{The covering map $p\colon K_1\to K$.} \label{fig:covering} \end{figure} The following result is easily verified. \begin{prop} The map $p$ is a well defined continuous map which is a ramified covering, with ramification points given by $\{x_{i,i+1},i=0,1,2\}$. Moreover, the covering map is isometric on suitable neighbourhoods of the non-ramification points. \end{prop} Since $K_1$ and $K$ are homeomorphic, this map may be seen as a self-covering of the gasket. The map $p$ gives rise to an embedding $\alpha_{1,0}\colon {\mathcal C}(K)\to{\mathcal C}(K_1)$, hence, following~\cite{Cuntz}, to an inductive family of C$^$-algebras ${\mathcal A}_n={\mathcal C}(K_n)$, whose inductive limit ${\mathcal A}_\infty$ consists of continuous function on the solenoidal space based on the gasket. As in Definition~\ref{STnested}, we consider the triple $ ({\mathcal L}_n,{\mathcal H}_n,D_n)$ on the $C^$-algebra ${\mathcal A}_n$, $n\geq0$, where ${\mathcal H}_n=\ell^2(E_n)$, $E_n=\{w_0^{-n}e,\, e\in E_0\}$ (the set of oriented edges in $K_n$). Let us also note that, since the covering projections are locally isometric and any Lip-norm $L_m(f)=\\|[D_m,f]\\|$ associated with the triple $({\mathcal A}_m,{\mathcal H}_m,D_m)$ produces the geodesic distance on~$K_m$, we get $L_{m+q}(\alpha_{m+q,m}(f))=L_m(f)$, namely we obtain a seminorm on the algebraic inductive limit of the ${\mathcal A}_n$'s. \section[A groupoid of local isometries on the infinite Sierpinski fractafold] {A groupoid of local isometries on the infinite \\Sierpinski fractafold}\label{sec4} Let us consider the infinite fractafold $K_\infty=\cup_{n\geq0}K_n$~\cite{Tep} endowed with the Hausdorff measure $\vol$ of dimension $d=\frac{\log3}{\log2}$ normalized to be 1 on $K=K_0$, with the exhaustion $\{K_n\}_{n\geq0}$, and with the family of local isometries $R=\big\{R^n_{i+1,i},R^n_{i,i+1}\colon i=0,1,2, n\geq 0\big\}$, where $R^n_{i,j} = w_0^{-n}R_{i,j}w_0^{n}\colon C^n_j \to C^n_i$, and $C^n_i := w_0^{-n-1}w_iK$, $n\geq 0$, $i,j\in\{0,1,2\}$. We also denote by $s(\gamma} \def\G{\Gamma)$ and $r(\gamma} \def\G{\Gamma)$ the domain and range of the local isometry~$\gamma} \def\G{\Gamma$. Such local isometries act on points and on oriented edges of $K_\infty$. We say that the product of the two local isometries $\gamma} \def\G{\Gamma_1$, $\gamma} \def\G{\Gamma_2\in R$ is defined if $\gamma} \def\G{\Gamma_2^{-1}(s(\gamma} \def\G{\Gamma_1))\cap$ $s(\gamma} \def\G{\Gamma_2) \neq \varnothing$. In this case we consider the product \begin{gather} \gamma} \def\G{\Gamma_1\cdot\gamma} \def\G{\Gamma_2\colon\ \gamma} \def\G{\Gamma_2^{-1}(s(\gamma} \def\G{\Gamma_1))\cap s(\gamma} \def\G{\Gamma_2)\to r\big(\gamma} \def\G{\Gamma_1\|_{s(\gamma} \def\G{\Gamma_1))\cap r(\gamma} \def\G{\Gamma_2)}\big). \end{gather} We then consider the family ${\mathcal G}$ consisting of all (the well-defined) finite products of isometries in $R$. Clearly, any $\gamma} \def\G{\Gamma$ in ${\mathcal G}$ is a local isometry, and its domain and range are cells of the same size. We set ${\mathcal G}_n=\{g\in{\mathcal G}\colon s(\gamma} \def\G{\Gamma)\,\&\,r(\gamma} \def\G{\Gamma)$ are cells of size $2^n\}$, $n\geq0$. \begin{prop} For any $n\geq0$, $C_1$, $C_2$ cells of size $2^n$, $\exists!\, \gamma} \def\G{\Gamma\in{\mathcal G}_n$ such that $s(\gamma} \def\G{\Gamma)=C_1$, $r(\gamma} \def\G{\Gamma)=C_2$. In particular, if $C$ has size $2^n$, the identity map of $C$ belongs to ${\mathcal G}_n$, $n\geq0$. \end{prop} \begin{proof} It is enough to show that for any cell $C$ of size $2^n$ there exists a unique $\gamma} \def\G{\Gamma\in{\mathcal G}_n$ such that $\gamma} \def\G{\Gamma\colon C\to K_n$. For any cell $C$, let $m=\mathrm{level}(C)$ be the minimum number such that $C\subset K_m$. We~prove the existence: if $C$ has size $2^n$ and $\mathrm{level}(C)=m>n$, then $C\subset C^{m-1}_i$, for some $i=1,2$, hence $R^{m-1}_{0,i}(C) \subset K_{m-1}$. Iterating, the result follows. The second statement follows directly by equation~\eqref{IdOnCells}. As for the uniqueness, $\forall\, n\geq0$, we call $R^n_{i,0}$ ascending, $i=1,2$, $R^n_{0,i}$ descending, $i=1,2$, $R^n_{i,j}$~constant-level, $i,j\in\{1,2\}$. Indeed, if $C \subset s(R^n_{i,0})$, then $\mathrm{level}(C)\leq n$ and $\mathrm{level}(R^n_{i,0}(C)) = n+1$; if $C \subset s(R^n_{0,i})$, then $\mathrm{level}(C)=n+1$, $\mathrm{level}(R^n_{0,i}(C))\leq n$ and $\mathrm{level}(R^n_{j,i}(C))= n+1$, $i,j\in\{1,2\}$, $n\geq0$. The following facts hold: \begin{itemize}\itemsep=0pt \item The product $R^n_{l,k} \cdot R^m_{j,i}$ of two constant-level elements $R^n_{l,k}$, $R^m_{j,i}$ is defined iff $n=m$ and~$k=j$, therefore any product of constant-level elements in $R$ is either the identity map on~the domain or coincides with a single constant-level element. \item Any product of constant level elements in $R$ followed by a descending element coincides with a single descending element: indeed, if the product of constant level elements is the identity, the statement is trivially true; if it coincides with a single element, say $R^n_{i,j}$ with~$i,j\in\{1,2\}$, then, by compatibility, the descending element should be $R^n_{0,i}$ so that the product is $R^n_{0,i},$ by equation~\eqref{IdOnCells}. \item Given a cell $C$ with $\mathrm{size} (C)=2^n$ and $\mathrm{level}(C)>n$, the exists a unique descending element $\gamma} \def\G{\Gamma \in R$ such that $C \subset s(\gamma} \def\G{\Gamma)$: indeed, if $m=\mathrm{level}(C)$, then $C\subset C^{m-1}_i$, for some $i\in\{1,2\}$. The only descending element is then $\gamma} \def\G{\Gamma=R^{m-1}_{0,i}$. \item Any product of an ascending element followed by a descending one is the identity on the domain: indeed if the ascending element is $R^n_{i,0}$, then, by compatibility, the descending element should be $R^n_{0,i}$. \end{itemize} Now let $\mathrm{size}(C)=2^n$, $\gamma} \def\G{\Gamma \in{\mathcal G}_n$ such that $\gamma} \def\G{\Gamma\colon C\to K_n$, $\gamma} \def\G{\Gamma=\gamma} \def\G{\Gamma_p\cdot\gamma} \def\G{\Gamma_{p-1}\cdots\gamma} \def\G{\Gamma_2\cdot\gamma} \def\G{\Gamma_1$, where $\gamma} \def\G{\Gamma_j\in R$, $1\leq j\leq p$. Since $\mathrm{level}(C) \geq \mathrm{level}(K_n) = n$, for any possible ascending element $\gamma} \def\G{\Gamma_i$ there should be a $j>i$ such that $\gamma} \def\G{\Gamma_j$ is descending. If $i+q$ is the minimum among such $j$'s, all terms $\gamma} \def\G{\Gamma_j$, $i<j<i+q$, are constant-level, hence the product $\gamma} \def\G{\Gamma_{i+q}\cdot\gamma} \def\G{\Gamma_{i+q-1}\cdots \gamma} \def\G{\Gamma_{i}={\rm id}_{s(\gamma} \def\G{\Gamma_i)}$. Then, we note that $\gamma} \def\G{\Gamma_p$ can only be descending. As a consequence, $\gamma} \def\G{\Gamma$ can be reduced to a product of descending elements, and, by the uniqueness of the descending element acting on a given cell, we get the result. \end{proof} Let us observe that each ${\mathcal G}_n$, and so also ${\mathcal G}$, is a groupoid under the usual composition rule, namely two local isometries are composable if the domain of the first coincides with the range of the latter. We now consider the action on points of the local isometries in ${\mathcal G}$. \begin{prop} Let us define $\widetilde{\mathcal A}_n$ as the algebra \begin{gather} \widetilde{\mathcal A}_n=\{f\in{\mathcal C}_b(K_\infty)\colon f( \gamma} \def\G{\Gamma(x))= f(x),\, x\in s(\gamma} \def\G{\Gamma),\, \gamma} \def\G{\Gamma\in{\mathcal G}_n\}. \end{gather} Then, for any $n\geq0$, the following diagram commutes, \begin{gather} \begin{matrix} \widetilde{\mathcal A}_n & \subset & \widetilde{\mathcal A}_{n+1} \\ \Big\downarrow\iota_n & & \Big\downarrow\iota_{n+1} \\ {\mathcal A}_n & \mathop{\longrightarrow}\limits^{\alpha_{n+1,n}} & {\mathcal A}_{n+1}, \end{matrix} \end{gather} where $\iota_n\colon f\in\widetilde{\mathcal A}_n \to f\|_{K_n} \in {\mathcal A}_n$ are isomorphisms. Hence the inductive limit ${\mathcal A}_\infty$ is isomorphic to a $C^$-subalgebra of ${\mathcal C}_b(K_\infty)$. \end{prop} \begin{proof} The request in the definition of $\widetilde{\mathcal A}_n$ means that the value of $f$ in any point of $K_\infty$ is~determined by the value on $K_n$, while such request gives no restrictions on the values of $f$ on~$K_n$. The other assertions easily follow. \end{proof} As shown above, we may identify the algebra ${\mathcal A}_n$, $0\leq n\leq \infty$, with its isomorphic copy~$\widetilde{\mathcal A}_n$ in~${\mathcal C}_b(K_\infty)$, so that the embeddings $\alpha_{k,j}$ become inclusions. Moreover, we may consider the operator $\widetilde D_n$ on $\ell^2(E_\infty)$, with $E_\infty=\cup_{n\geq0}E_n$, given by $\widetilde D_n e=\mathrm{length}(e)^{-1}Fe$, if $\mathrm{length}(e)\leq2^n$, and $\widetilde D_n e=0$, if $\mathrm{length}(e) >2^n$, where $F$ is defined as in Definition~\ref{STnested}$(c)$. Then the spectral triples $({\mathcal A}_n,{\mathcal H}_n,D_n)$ are isomorphic to the spectral triples $(\widetilde{\mathcal A}_n,{\mathcal H}_n,\widetilde D_n)$, where ${\mathcal C}_b(K_\infty)$ acts on~the space $\ell^2(E_\infty)$ through the representation $\rho$ given by $\rho(f)e=f(e^+)e$. \begin{rem} Because of the isomorphism above, from now on we shall remove the tildes and denote by ${\mathcal A}_n$ the subalgebras of ${\mathcal C}_b(K_\infty)$ and by $D_n$ the operators acting on $\ell^2(E_\infty)$. \end{rem} \section[The C-algebra of geometric operators and a tracial weight on it] {The $\boldsymbol{C^}$-algebra of geometric operators\\ and a tracial weight on it}\label{sec5} We now come to the action of local isometries on edges. We shall use the following notation, where in the table below to any subset of edges listed on the left we indicate on the right the projection on the closed subspace spanned by the same subset: \begin{table}[h!]\centering \caption{Edges and projections.} \label{tab:table1} {\renewcommand{1.4}{1.4} \begin{tabular}{l\|l} \hline \multicolumn{1}{c\|}{Subsets of $E_\infty$} & \multicolumn{1}{c}{Projections} \\[.5ex] \hline $E_n=\{e\subset K_n\}$, $n\geq0$ &\qquad $P_n$ \\%[.5ex] $E^{k,p}_n=\big\{e\in E_n\colon 2^k\leq\mathrm{length}(e)\leq2^p\big\}$, for $k\leq p\leq n$ &\qquad $P_n^{k,p}$ \\%[.5ex] $E^k_n=E^{k,k}_n=\big\{e\in E_n\colon \mathrm{length}(e)=2^k\big\}$, for $k\leq n$ &\qquad $P^k_n$ \\%[.5ex] $E^{k,p}=\cup_n E^{k,p}_n=\big\{e\in E_\infty\colon 2^k\leq\mathrm{length}(e)\leq2^p\big\}$ &\qquad $P^{k,p}$ \\%[.5ex] $E^k=E^{k,k}=\big\{e\in E_\infty\colon \mathrm{length}(e)=2^k\big\}$ &\qquad $P^{k}$ \\%[.5ex] $E_C = \{e\in E_\infty\colon e\subset C\}$, $C$ being a cell &\qquad $P_C$ \end{tabular}} \end{table} Let us note that any local isometry $\gamma} \def\G{\Gamma\in{\mathcal G}$, $\gamma} \def\G{\Gamma\colon s(\gamma} \def\G{\Gamma) \to r(\gamma} \def\G{\Gamma)$, gives rise to a partial isometry~$V_\gamma} \def\G{\Gamma$ defined as \begin{gather} V_\gamma} \def\G{\Gamma e=\begin{cases} \gamma} \def\G{\Gamma(e), & e\subset s(\gamma} \def\G{\Gamma),\\ 0, & \mathrm{elsewhere}. \end{cases} \end{gather} In particular, if $C$ is a cell, and $\gamma} \def\G{\Gamma={\rm id}_C$, $V_\gamma} \def\G{\Gamma=P_C$. We then consider the subalgebras ${\mathcal B}_n$ of~$B(\ell^2(E_\infty))$, \begin{gather} {\mathcal B}_n=\{V_\gamma} \def\G{\Gamma\colon\gamma} \def\G{\Gamma\in{\mathcal G}_m,\,m\geq n\}',\qquad {\mathcal B}_{\text{fin}} = \bigcup_n{\mathcal B}_n,\qquad {\mathcal B}_\infty=\overline{{\mathcal B}_{\text{fin}}}, \end{gather} {\sloppy and note that the elements of ${\mathcal B}_n$ commute with the projections $P_C$, for all cells $C$ s.t.\ \mbox{$\mathrm{size}(C)\geq 2^n$}. By definition, the sequence ${\mathcal B}_n$ is increasing, therefore, since the ${\mathcal B}_n$'s are von Neumann algebras, ${\mathcal B}_\infty$ is a $C^$-algebra. Let us observe that, $\forall\, n\geq0$, $\rho({\mathcal A}_n)\subset{\mathcal B}_n$. } \begin{dfn} The elements of the $C^$-algebra ${\mathcal B}_\infty$ are called geometric operators. \end{dfn} Now consider the hereditary positive cone \begin{gather} \mathcal{I}_0^+ = \big\{T\in {\mathcal B}_{\text{fin}}^+\colon\exists\, c_T\in{\mathbb R} \mathrm{\ such\ that\ }\tr (P_{m}T) \leq c_T \vol(K_m), \, \forall\, m \geq 0 \big\}. \end{gather} \begin{lem} For any $T\in\mathcal{I}_0^+$, the sequence $ \frac{\tr (P_{m} T) }{\vol(K_m)}$ is eventually increasing, hence convergent. In particular \begin{gather}\label{evConst} \tr(P_{p}^{p} T)=0\qquad \forall\, p>m\Rightarrow \tau_0(T)=\frac{\tr (P_{m} T) }{\vol(K_{m})}. \end{gather} \end{lem} \begin{proof} Let $T\in{\mathcal B}_n^+$. Then we have, for $m\geq n$, \begin{gather} \tr (P_{{m+1}} T) = \!\!\sum_{e\subset K_{m+1}} (e,Te) = \!\!\sum_{i=0,1,2} \sum_{e\in C^m_i }(e,Te) + \!\sum_{e\in E_{m+1}^{m+1}}\! (e,Te) = 3\tr (P_{{m}} T) + \tr(P_{m+1}^{m+1} T), \end{gather} hence \begin{gather} \frac{\tr (P_{m+1} T) }{\vol(K_{m+1})}=\frac{\tr (P_{m} T) }{\vol(K_m)}+\frac{\tr(P_{m+1}^{m+1} T)}{\vol(K_{m+1})}, \end{gather} from which the thesis follows. \end{proof} We then define the weight $\tau_0$ on ${\mathcal B}_\infty^+$ as follows: \begin{gather} \tau_0(T) = \begin{cases} \displaystyle{\lim_{m\to\infty}\frac{\tr (P_{m} T)}{\vol(K_m)}},& T\in{\mathcal{I}_0^+},\\ 0,& \mathrm{elsewhere}. \end{cases} \end{gather} The next step is to regularize the weight $\tau_0$ in order to obtain a semicontinuous semifinite tracial weight $\tau$ on ${\mathcal B}_\infty$. \begin{lem} \label{hereditary} For any $T\in\mathcal{I}_0^+$, $A\in{\mathcal B}_{\text{fin}}$, it holds $ATA^* \in \mathcal{I}_0^+$, and $\tau_0(ATA^) \leq \\|A\\|^2 \tau_0(T)$. \end{lem} \begin{proof} Let $A\in{\mathcal B}_n$. Then, for any $m>n$, we have \begin{gather} \tr(P_mATA^) = \tr(A^AP_mT) \leq \\| A^A \\| \tr(P_mT) \leq \\|A\\|^2 c_T \vol(K_m), \end{gather} and the thesis follows. \end{proof} \begin{prop} \label{def.Qfi} For all $p\in\mathbb N$, recall that $P^{-p,\infty}$ is the orthogonal projection onto the closed vector space generated by $\big\{ e\in\ell^2(E_\infty)\colon \mathrm{length}(e) \geq 2^{-p} \big\}$, and let $\varphi_p(T) := \tau_0(P^{-p,\infty}TP^{-p,\infty})$, $\forall\, T\in{\mathcal B}_\infty^+$. Then $P^{-p,\infty}\in{\mathcal B}_0$, $\varphi_p$ is a positive linear functional, and $\varphi_p(T) \leq \varphi_{p+1}(T) \leq \tau_0(T)$, $\forall\, T\in{\mathcal B}_\infty^+$. \end{prop} \begin{proof} We first observe that \begin{gather}\label{Pjn} \tr(P^j_n)=\#\big\{e\in K_n\colon \mathrm{length}(e)=2^j\big\}=6 \cdot 3^{n-j},\qquad j\leq n. \end{gather} Then it is easy to verify that $P^{-p,\infty}\in{\mathcal B}_0$. Since \begin{gather}\label{P-pinfty} \varphi_p(I) = \tau_0(P^{-p,\infty}) = \lim_{n\to\infty} \frac{\tr P^{-p,n}_n}{\mu_d(K_n)} = \lim_{n\to\infty} 3^{-n} \sum_{j=-p}^n \tr(P^j_n) =\sum_{j=-p}^\infty6\cdot 3^{-j} = 3^{p+2}, \end{gather} $\varphi_p$ extends by linearity to a positive functional on ${\mathcal B}_\infty$. Moreover, by Lemma~\ref{hereditary}, $\varphi_p(T) \leq \tau_0(T)$, $\forall\, T\in{\mathcal B}_\infty^+$. Finally, since $P^{-p,\infty}P_n=P_nP^{-p,\infty}=P^{-\infty,n}_n$, $\forall\, n\in\mathbb N$, we get, for all $T\in{\mathcal B}_\infty^+$, \begin{align} \varphi_{p+1}(T) - \varphi_p(T) & = \tau_0(P^{-(p+1),\infty}TP^{-(+1)p,\infty}) - \tau_0(P^{-p,\infty}TP^{-p,\infty}) \\ &= \lim_{n\to\infty} \frac{\tr((P^{-(p+1),n)}_n -P^{-p,n}_n)T)}{\mu_d(K_n)} =\lim_{n\to\infty} \frac{\tr(P^{-(p+1))}_n T)}{\mu_d(K_n)} \geq 0. \tag{\qed} \end{align} \renewcommand{\qed}{} \end{proof} \begin{prop} \label{def.tau} Let $\tau(T) := \lim\limits_{p\to\infty} \varphi_p(T)$, $\forall\, T\in{\mathcal B}_\infty^+$. Then \begin{itemize}\itemsep=0pt \item[$(i)$] $\tau$ is a lower semicontinuous weight on ${\mathcal B}_\infty$, \item[$(ii)$] $\tau(T) = \tau_0(T)$, $\forall\, T\in\mathcal{I}_0^+$. \end{itemize} \end{prop} \begin{proof} $(i)$ Let $T\in{\mathcal B}_\infty^+$. Since $\{ \varphi_p(T) \}_{p\in\mathbb N}$ is an increasing sequence, there exists $\lim\limits_{p\to\infty} \varphi_p(T) = \sup\limits_{p\in\mathbb N} \varphi_p(T)$. Then $\tau$ is a weight on ${\mathcal B}_\infty^+$. Since $\varphi_p$ is continuous, $\tau$ is lower semicontinuous. $(ii)$ Let us prove that, $\forall\, T\in{\mathcal B}_n^+$, \begin{gather} \label{uguaglianza} \frac{\tr(P^j_mT)}{\mu_d(K_m)} = \frac{\tr(P^j_nT)}{\mu_d(K_n)}, \qquad j\leq n \leq m. \end{gather} Indeed, \begin{align} \tr\big(P^j_{m+1}T\big) & = \sum_{\substack{e \subset K_{m+1} \\ \mathrm{length}(e)=2^j}} \ps{ e }{ Te } = \sum_{i=0}^2 \sum_{\substack{e \subset C^m_i \\ \mathrm{length}(e)=2^j}} \ps{ e }{ Te } = \sum_{i=0}^2 \sum_{\substack{e \subset K_m \\ \mathrm{length}(e)=2^j}} \ps{ V_{R^m_i}e }{ TV_{R^m_i}e } \\ & = \sum_{i=0}^2 \sum_{\substack{e \subset K_m \\ \mathrm{length}(e)=2^j}} \ps{ e }{ Te } = 3 \tr\big(P^j_mT\big), \end{align} from which~\eqref{uguaglianza} follows. Let us now prove that \begin{gather} \label{approx} \tau(T) = \sup_{p\in\mathbb N} \varphi_p(T) = \tau_0(T), \qquad T\in\mathcal{I}_0^+ . \end{gather} Let $T\in{\mathcal B}_n^+ \cap \mathcal{I}_0^+$, and $\varepsilon>0$. From the definition of $\tau_0(T)$, there exists $r\in\mathbb N$, $r>n$, such that $\frac{\tr(P_rT)}{\mu_d(K_r)} > \tau_0(T)-\varepsilon$. Since $\frac{\tr(P_rT)}{\mu_d(K_r)} = \sum_{j=-\infty}^r \frac{\tr(P^j_rT)}{\mu_d(K_r)}$, there exists $p\in\mathbb N$ such that $\sum_{j=-p}^r \frac{\tr(P^j_rT)}{\mu_d(K_r)} > \frac{\tr(P_rT)}{\mu_d(K_r)} -\varepsilon > \tau_0(T)-2\varepsilon$. Then, for any $s\in\mathbb N$, $s>r$, we have \begin{align} \frac{\tr(P_sP^{-p,\infty}TP^{-p,\infty}P_s)}{\mu_d(K_s)} & = \sum_{j=-p}^s \frac{\tr(P^j_sT)}{\mu_d(K_s)} = \sum_{j=-p}^r \frac{\tr(P^j_sT)}{\mu_d(K_s)} + \sum_{j=r+1}^s \frac{\tr(P^j_sT)}{\mu_d(K_s)} \\ & \stackrel{\eqref{uguaglianza}}{=} \sum_{j=-p}^r \frac{\tr(P^j_rT)}{\mu_d(K_r)} + \sum_{j=r+1}^s \frac{\tr(P^j_sT)}{\mu_d(K_s)} > \tau_0(T)-2\varepsilon, \end{align} and, passing to the limit for $s\to\infty$, we get \begin{gather} \varphi_p(T) = \tau_0(P^{-p,\infty}TP^{-p,\infty}) = \lim_{s\to\infty} \frac{\tr(P_sP^{-p,\infty}TP^{-p,\infty}P_s)}{\mu_d(K_s)} \geq \tau_0(T)-\varepsilon, \end{gather} and equation~\eqref{approx} follows. \end{proof} We want to prove that $\tau$ is a tracial weight. \begin{dfn} An operator $U\in B\big(\ell^2(E_\infty)\big)$ is called $\delta} \def\D{\Delta$-unitary, $\delta} \def\D{\Delta>0$, if $\\|U^U-1\\|<\delta} \def\D{\Delta$, and $\\|UU^-1\\|<\delta} \def\D{\Delta$. \end{dfn} Let us denote with ${\mathcal U}_\delta} \def\D{\Delta$ the set of $\delta} \def\D{\Delta$-unitaries in ${\mathcal B}_{\text{fin}}$ and observe that, if $\delta} \def\D{\Delta<1$, ${\mathcal U}_\delta} \def\D{\Delta$ consists of invertible operators, and $U\in{\mathcal U}_\delta} \def\D{\Delta$ implies $U^{-1}\in{\mathcal U}_{\delta} \def\D{\Delta/(1-\delta} \def\D{\Delta)}$. \begin{prop} The weight $\tau_0$ is $\varepsilon$-invariant for $\delta} \def\D{\Delta$-unitaries in ${\mathcal B}_{\text{fin}}$, namely, for any $\varepsilon\in(0,1)$, there is $\delta} \def\D{\Delta>0$ s.t., for any $U\in{\mathcal U}_\delta} \def\D{\Delta$, and $T\in {\mathcal B}^+_\infty$, \begin{gather} (1-\varepsilon)\tau_0(T) \leq \tau_0(UTU^) \leq (1+\varepsilon)\tau_0(T) . \end{gather} \end{prop} \begin{proof} We first observe that, if $\delta} \def\D{\Delta\in(0,1)$ and $U\in{\mathcal U}_\delta} \def\D{\Delta$, $T\in\mathcal{I}_0^+\Leftrightarrow UTU^\in\mathcal{I}_0^+$. Indeed, choose $n$ such that $U,T\in{\mathcal B}_n$. Then $\tr (P_{n} UTU^) = \tr (U^* UP_{n}TP_{n}) \leq \\|U^U\\| \tr (P_{n}T) \leq (1+\delta} \def\D{\Delta) c_T\vol(K_n)$, $\forall\, n\in\mathbb N$, so that $UTU^ \in \mathcal{I}_0^+$. Moreover, \begin{gather} \tau_0(UTU^) = \lim_{n\to\infty} \frac{\tr(P_nUTU^)}{\vol(K_n)} \leq \\| U^U \\| \lim_{n\to\infty} \frac{\tr(P_nT)}{\vol(K_n)} = \\| U^U \\| \tau_0(T) < (1+\delta} \def\D{\Delta) \tau_0(T). \end{gather} Conversely, $UTU^\in\mathcal{I}_0^+$, and $U^{-1}\in {\mathcal U}_{\delta} \def\D{\Delta/(1-d)} \implies T \in\mathcal{I}_0^+$. Moreover, \begin{gather} \tau_0(T) \leq \big\\| \big(U^{-1}\big)^U^{-1} \big\\| \tau_0(UTU^) < \frac1{1-\delta} \def\D{\Delta} \tau_0(UTU^). \end{gather} The result follows by the choice $\delta} \def\D{\Delta=\varepsilon$. \end{proof} \begin{thm} The lower semicontinuous weight $\tau$ in Proposition~$\ref{def.tau}$ is a trace on ${\mathcal B}_\infty$, that is, setting ${\mathcal J}^+:= \{A\in{\mathcal B}_\infty^+\colon \tau(A)<\infty\}$, and extending $\tau$ to the vector space ${\mathcal J}$ generated by ${\mathcal J}^+$, we get \begin{itemize}\itemsep=0pt \item[$(i)$] ${\mathcal J}$ is an ideal in ${\mathcal B}_\infty$, \item[$(ii)$] $\tau(AB)=\tau(BA)$, for any $A\in{\mathcal J}$, $B\in{\mathcal B}_\infty$. \end{itemize} \end{thm} \begin{proof} $(i)$ Let us prove that ${\mathcal J}^+$ is a unitarily-invariant face in ${\mathcal B}_\infty^+$, and suffices it to prove that $A\in{\mathcal J}^+$ implies that $UAU^\in{\mathcal J}^+$, for any $U\in{\mathcal U}({\mathcal B}_\infty)$, the set of unitaries in ${\mathcal B}_\infty$. To~reach a~contradiction, assume that there exists $U\in{\mathcal U}({\mathcal B}_\infty)$ such that $\tau(UAU^)=\infty$. Then there is $p\in\mathbb N$ such that $\varphi_p(UAU^) > 2\tau(A)+2$. Let $\delta} \def\D{\Delta<3$ be such that $V\in{\mathcal U}_\delta} \def\D{\Delta$ implies $\tau(VAV^)\leq 2\tau(A)$, and let $U_0\in{\mathcal B}_{\text{fin}}$ be such that $\\|U-U_0\\|< \min\big\{ \frac{\delta} \def\D{\Delta}{3}, \frac1{3\\|A\\|\\|\varphi_p\\|} \big\}$. The inequalities \begin{gather} \\|U_0U_0^-1\\| = \\|U^U_0U_0^-U^\\| \leq \\|U^U_0-1\\|\\|U_0^\\|+\\|U_0^-U^\\| < \delta} \def\D{\Delta \end{gather} and $\\|U_0^U_0-1\\|<\delta} \def\D{\Delta$, prove that $U_0\in{\mathcal U}_\delta} \def\D{\Delta$. Since \[ \|\varphi_p(U_0AU_0^)-\varphi_p(UAU^)\|\leq 3\\| \varphi_p \\| \\|A\\| \\|U-U_0\\| <1,\] we get \begin{gather} 2\tau(A)\geq \tau(U_0AU_0^) \geq \varphi_p(U_0AU_0^) \geq \varphi_p(UAU^) - 1 \geq 2\tau(A)+1 \end{gather} which is absurd. $(ii)$ We only need to prove that $\tau$ is unitarily-invariant. Let $A\in{\mathcal J}^+$, $U\in{\mathcal U}({\mathcal B}_\infty)$. For any $\varepsilon>0$, there is $p\in\mathbb N$ such that $\varphi_p(UAU^)>\tau(UAU^)-\varepsilon$, since, by $(1)$, $\tau(UAU^)$ is finite. Then, arguing as in the proof of $(1)$, we can find $U_0\in{\mathcal B}_{\text{fin}}$, so close to $U$ that \begin{gather} \|\varphi_p(U_0AU_0^)-\varphi_p(UAU^)\|<\varepsilon, \\ (1-\varepsilon)\tau(A)\leq \tau(U_0AU_0^) \leq (1+\varepsilon)\tau(A). \end{gather} Then \begin{align} \tau(A) & \geq \frac1{1+\varepsilon}\ \tau(U_0AU_0^) \geq \frac1{1+\varepsilon}\ \varphi_p(U_0AU_0^) \geq \frac1{1+\varepsilon}\ (\varphi_p(UAU^) -\varepsilon) \\ & \geq \frac1{1+\varepsilon}\ (\tau(UAU^) -2\varepsilon). \end{align} By the arbitrariness of $\varepsilon>0$, we get $\tau(A)\geq \tau(UAU^)$. Exchanging $A$ with $UAU^$, we get the thesis. \end{proof} \begin{prop} The lower semicontinuous tracial weight $\tau$ defined in Proposition~$\ref{def.tau}$ is semifinite and faithful. \end{prop} \begin{proof} Let us recall that, for any $p\in\mathbb N$, $P^{-p,\infty} \in\mathcal{I}_0^+$ by Proposition~\ref{def.Qfi}. From Proposition~\ref{def.tau} follows that $\tau(P^{-p,\infty})=\tau_0(P^{-p,\infty})<\infty$, hence $P^{-p,\infty}\in{\mathcal J}^+$. Then, for any $T\in{\mathcal B}_\infty^+$, $S_p := T^{1/2}P^{-p,\infty}T^{1/2} \in{\mathcal J}^+$, and $0 \leq S_p \leq T$. Moreover, \begin{align} \tau(S_p) &= \tau\big(T^{1/2}P^{-p,\infty}T^{1/2}\big) = \tau(P^{-p,\infty}TP^{-p,\infty}) = \sup_{q\in\mathbb N} \tau_0(Q_qP^{-p,\infty}TP^{-p,\infty}Q_q)\\ & = \tau_0(P^{-p,\infty}TP^{-p,\infty}) = \varphi_p(T), \end{align} so that $\sup\limits_{p\in\mathbb N} \tau(S_p) = \tau(T)$, and $\tau$ is semifinite. Finally, if $T\in{\mathcal B}_\infty^+$ is such that $\tau(T)=0$, then $\sup\limits_{p\in\mathbb N} \varphi_p(T)=0$. Since $\{\varphi_p(T) \}_{p\in\mathbb N}$ is an increasing sequence, $\varphi_p(T)=0$, $\forall\, p\in\mathbb N$. Then, for a~fixed $p\in\mathbb N$, we get $0 = \tau_0(P^{-p,\infty}TP^{-p,\infty}) = \lim\limits_{n\to\infty} \frac{ \tr(P_nP^{-p,\infty}TP^{-p,\infty}P_n) }{\mu_d(K_n)}$. Since the sequence $\big\{ \frac{ \tr(P_nP^{-p,\infty}TP^{-p,\infty}P_n) }{\mu_d(K_n)} \big\}_{n\in\mathbb N}$ is definitely increasing, we get $\tr(P_nP^{-p,\infty}TP^{-p,\infty}P_n) = 0$ definitely, that is $TP^{-p,\infty}P_n = 0$ definitely, so that $TP^{-p,\infty}=0$. By the arbitrariness of $p\in\mathbb N$, we~get~\mbox{$T=0$}. \end{proof} \section[A semifinite spectral triple on the inductive limit A infty] {A semifinite spectral triple on the inductive limit $\boldsymbol{{\mathcal A}_\infty}$}\label{sec6} Since the covering we are studying is ramified, the family $\{{\mathcal A}_n,{\mathcal H}_n,D_n\}$ does not have a simple tensor product structure, contrary to what happened in~\cite{AGI01}. We therefore use a different approach to construct a semifinite spectral triple on ${\mathcal A}_\infty$: our construction is indeed based on the pair $({\mathcal B}_\infty,\tau)$ of the $C^$-algebra of geometric operators and the semicontinuous semifinite weight on~it. The Dirac operator will be defined below (Definition~\ref{Dinfty}) through its phase and the functional calculi of its modulus with continuous functions vanishing at $\infty$. More precisely we shall use the following \begin{dfn} Let $(\mathfrak{C},\tau)$ be a $C^$-algebra with unit endowed with a semicontinuous semifinite faithful trace. A selfadjoint operator $T$ affiliated to $(\mathfrak{C},\tau)$ is defined as a pair given by a closed subset $\sigma} \renewcommand{\S}{\Sigma(T)$ in ${\mathbb R}$ and a $$ homomorphism $\phi\colon{\mathcal C}_0(\sigma} \renewcommand{\S}{\Sigma(T))\to \mathfrak{C}$, $f(T)\mathop{=}\limits^{\mathrm{def}}\phi(f)$, provided that the support of such homomorphism is the identity in the GNS representation $\pi_\tau$ induced by the trace $\tau$. \end{dfn} The previous definition was inspired by that in~\cite{DFR} appendix A, and should not be confused with that of Woronowicz for $C^$-algebras without identity. \begin{rem}\label{varieOps} The $$-homomorphism $\phi_\tau=\pi_\tau\circ\phi$ extends to bounded Borel functions on ${\mathbb R}$ and $e_{(-\infty,t]}\mathop{=}\limits^{\mathrm{def}}\phi_\tau(\chi_{(-\infty,t]})$ tends strongly to the identity when $t\to\infty$, hence it is a spectral family. We shall denote by $\pi_\tau(T)$ the selfadjoint operator affiliated to $\pi_\tau(\mathfrak{C})''$ given by \begin{gather} \pi_\tau(T)\mathop{=}\limits^{\mathrm{def}}\int_{\mathbb R} t\, {\rm d} e_{(-\infty,t]}. \end{gather} \end{rem} \begin{prop}\label{aff-prop} Let $T$ be a selfadjoint operator affiliated to $(\mathfrak{C},\tau)$ as above. \begin{itemize}\itemsep=0pt \item[$(a)$] Assume that for any $n\in\mathbb N$, there is $\varphi_n\in{\mathcal C}({\mathbb R})\colon 0\leq\varphi_n\leq 1,\varphi_n=1$ for $\|t\|\leq a_n$, $\varphi_n(t)=0$ for $\|t\|\geq b_n$ with $0<a_n<b_n$ and $\{a_n\}$, $\{b_n\}$ increasing to $\infty$. Then, for any $A\in\mathfrak{C}$, if~$\sup\limits_n\\|[T\cdot \varphi_n(T),A]\\|=C<\infty$ then $[\pi_t(T),\pi_\tau(A)]$ is bounded and $\\|[\pi_t(T),\pi_\tau(A)]\\|=C$. \item[$(b)$] If $\tau(f(T))<\infty$ for any positive function $f$ with compact support on the spectrum of $T$ then $\pi_\tau(T)$ has $\tau$-compact resolvent. \end{itemize} \end{prop} \begin{proof} $(a)$ Let ${\mathcal D}$ be the domain of $\pi_\tau(T)$, ${\mathcal D}_0$ the space of vectors in ${\mathcal D}$ with bounded support w.r.t.~to $\pi_\tau(T)$, and consider the sesquilinear form $F(y,x)=(\pi_\tau(T)y,\pi_\tau(A)x)-(y,\pi_\tau(A)\pi_\tau(T)x)$ defined on ${\mathcal D}$. By hypothesis, for any $x,y\in{\mathcal D}_0$ there exists $n$ such that $\pi_\tau(\varphi_n(T))x=x$ and $\pi_\tau((\varphi_n(T))y=y$, hence $F(y,x)=(y,\pi_\tau([T\cdot \varphi_n(T),A])x)\leq C\\|x\\|\ \\|y\\|$. By the density of ${\mathcal D}_0$ in ${\mathcal D}$ w.r.t.~the graph norm of $\pi_\tau(T)$, the same bound holds on ${\mathcal D}$. Then for $y,x\in{\mathcal D}$, $\|(\pi_\tau(T)y,\pi_\tau(A)x)\|\leq \|(y,\pi_\tau(A)\pi_\tau(T)x)\|+\|F(y,x)\|\leq (\\|\pi_\tau(A)\pi_\tau(T)x\\|+C\\|x\\|)\\|y\\|$ which implies $\pi_\tau(A)x$ belongs to the domain of $\pi_\tau(T)^=\pi_\tau(T)$. Therefore $\pi_\tau(T)\pi_\tau(A)-\pi_\tau(A)\pi_\tau(T)$ is defined on ${\mathcal D}$ and its norm is bounded by $C$. Since $C$ is the optimal bound for the sesquilinear form $F$ it is indeed the norm of the commutator. $(b)$ Let $\lambda} \def\La{\Lambda$ be in the resolvent of~$\|T\|$. We then note that for any $f$ positive and zero on a~neigh\-bourhood of the origin there is a $g$ positive and with compact support such that $f\big((\|T\|-\lambda} \def\La{\Lambda I)^{-1}\big)=g(\|T\|)$. Therefore $\tau\big(f\big((\|T\|-\lambda} \def\La{\Lambda I)^{-1}\big)\big)<\infty$, hence $\tau\big(e_{(t,+\infty)}\big(\pi_\tau\big((\|T\|-\lambda} \def\La{\Lambda I)^{-1}\big)\big)\big)<\infty$ for any $t>0$, i.e., $\pi_\tau\big((\|T\|-\lambda} \def\La{\Lambda I)^{-1}\big)$ is $\tau$-compact (cf.~Section~\ref{SemST}). \end{proof} \begin{dfn}\label{Dinfty} We consider the Dirac operator $D=F\|D\|$ on $\ell^2(E_\infty)$, where $F$ is the orientation reversing operator on edges and \begin{gather} \|D\|=\sum_{n\in{\mathbb Z}}2^{-n}P^n,\qquad \sigma} \renewcommand{\S}{\Sigma(\|D\|)=\{2^{-n},\ n\in{\mathbb Z}\}\cup\{0\}. \end{gather} \end{dfn} \begin{prop}\label{OnDinfty} The following hold: \begin{itemize}\itemsep=0pt \item[$(a)$] The elements $D$ and $\|D\|$ are affiliated to $({\mathcal B}_\infty,\tau)$. \item[$(b)$] The following formulas hold: $\tau(P^n)=6\cdot 3^{-n}$, $\tau(P^{-p,\infty})=3^{p+2}$, as a consequence the operator $D$ has $\tau$-compact resolvents \item[$(c)$] The trace $\tau(I+ D^{2})^{-s/2}<\infty$ if and only if $s>d=\frac{\log3}{\log2}$ and \begin{gather} \Res_{s=d}\tau\big(I+ D^{2}\big)^{-s/2}=\frac6{\log2}. \end{gather} \end{itemize} \end{prop} \begin{proof} ($a$) We first observe that the $$-homomorphisms for $D$ and $\|D\|$ have the same support projection, then note that since $F$ and $P_n$ belong to ${\mathcal B}_0$ (which is a von Neumann algebra) for any $n\in\mathbb N$, then $f(D)$ and $f(\|D\|)$ belong to ${\mathcal B}_0$ for any $f\in{\mathcal C}_0({\mathbb R})$; therefore it is enough to show that the support of $f\mapsto f( \|D\|)$ is the identity in the representation $\pi_\tau$. In order to prove this, it is enough to show that $\pi_\tau(e_{\|D\|} [0,2^p])$ tends to the identity strongly when $p\to\infty$, that is to say that $\pi_\tau(e_{\|D\|}(2^p,\infty))$ tends to 0 strongly when $p\to\infty$. We consider then the projection $P^{-\infty,0}$ which projects on the space generated by the edges with $\mathrm{length}(e)\leq 1$. Clearly, such projection belongs to ${\mathcal B}_0$, we now show that it is indeed central there. In fact, if $c$ is a cell with $\mathrm{size}(c)=1$, $P_c$ commutes with ${\mathcal B}_0$. Since $P^{-\infty,0}=\sum_{\mathrm{size}(c)=1}P_c$, then $P^{-\infty,0}$ commutes with ${\mathcal B}_0$. On the one hand, the von Neumann algebra $P^{-\infty,0}{\mathcal B}_0$ is isomorphic to ${\mathcal B}\big(\ell^2(K)\big)$ and the restriction of $\tau$ to $P^{-\infty,0}{\mathcal B}_0$ coincides with the usual trace on ${\mathcal B}\big(\ell^2(K)\big)$, therefore the representation $\pi_\tau$ is normal when restricted to $P^{-\infty,0}{\mathcal B}_0$. On the other hand, since $e_{\|D\|}(2^p,\infty)=P^{-\infty,-p-1}$ is, for $-p\leq 1$, a sub-projection of $P^{-\infty,0}$, and $P^{-\infty,-p-1}$ tends to 0 strongly in the given representation, the same holds of the representation $\pi_\tau$. ($b$) We prove the first equation. Indeed \begin{gather} \tau(P^n) =\lim_m \frac{\tr P_{m}^n}{\vol(K_m)} =\tr P_{0}^n+\lim_m\sum_{j=1}^m\frac{\tr P_j^j P^n}{\vol(K_j)}. \end{gather} The first summand is non-zero iff $n\leq0$, while the second vanishes exactly for such $n$. Since \begin{gather} \lim_m\sum_{j=1}^m\frac{\tr P_j^j P^n}{\vol(K_j)} =\frac{\tr P_n^n }{\vol(K_n)}, \end{gather} the result in~\eqref{Pjn} shows that in both cases we obtain $6\cdot 3^{-n}$. We already proved in~\eqref{P-pinfty} that $\tau_0(P^{-p,\infty})=3^{p+2}$. Since $P^{-p,\infty}\in{\mathcal B}_0$, the same holds for $\tau$ by Proposition~\ref{def.tau}$(ii)$. Then the thesis follows by condition $(b)$ in Proposition~\ref{aff-prop}. ($c$) We have $\tau\big(I+ D^{2}\big)^{-s/2}=\tau\big(P^{-\infty,0}\big(I+ D^{2}\big)^{-s/2}\big) +\tau\big(P^{1,+\infty}\big(I+ D^{2}\big)^{-s/2}\big)$. A straightforward computation and~\eqref{evConst} give \begin{gather} \tau\big(P^{-\infty,0}\big(I+ D^{2}\big)^{-s/2}\big)=\tr\big(P_0\big(I+ D^{2}\big)^{-s/2}\big)= 6\sum_{n\geq0}\big(1+2^{2n}\big)^{-s/2}3^{n}, \end{gather} which converges iff $s>d$. As for the second summand, we have \begin{align} \tau\big(P^{1,+\infty}\big(I+ D^{2}\big)^{-s/2}\big) &=\tau_0\big(P^{1,+\infty}\big(I+ D^{2}\big)^{-s/2}\big) =\lim_m\frac{\tr\big(P^{1,m}_m\big(I+ D^{2}\big)^{-s/2}\big)}{\mu_d(K_m)} \\ &=\lim_m\sum_{j=1}^m 3^{-m}\tr\big(P^{1,m}_m\big(I+ D^{2}\big)^{-s/2}\big) =6\sum_{j=1}^\infty 3^{-j}\big(1+2^{-2j}\big)^{-s/2}, \end{align} which converges for any $s$ hence does not contribute to the residue. Finally \begin{align} \Res_{s=d}\tau\big(I+ D^{2}\big)^{-s/2} &=\lim_{s\to d^+}(s-d)\tau\big(I+ D^{2}\big)^{-s/2}\\ &=\lim_{s\to d^+}\bigg(s-\frac{\log3}{\log2}\bigg)6\sum_{n\geq0}\big(1+2^{-2n}\big)^{-s/2} e^{n(\log3-s\log2)}\\ &=\frac6{\log2}\lim_{s\to d^+}\frac{s\log2-\log3}{1-e^{-(s\log2-\log3)}} =\frac6{\log2}. \tag{\qed} \end{align} \renewcommand{\qed}{} \end{proof} \begin{prop}\label{commutator} For any $f\in{\mathcal A}_n$ $\sup\limits_{t>0}\big\\|\big[ e_{[-t,t]}(D)\, D,\rho(f)\big]\big\\| =\\|[ D_n,\rho(f\|_{K_n})]\\|$. \end{prop} \begin{proof} We observe that $\| D\|$ is a multiplication operator on $\ell^2(E_\infty)$, therefore it commutes with~$\rho(f)$. Hence, \begin{gather} \big\\|\big[ D\,e_{[-2^p,2^p]}(D),\rho(f)\big]\big\\| =\big\\|\|D\|\,e_{[0,2^p]}(\|D\|)\, (\rho(f)-F\rho(f)F)\big\\| \!=\!\!\sup_{\mathrm{length}(e)\geq 2^{-p}}\!\!\!\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)}. \end{gather} As a consequence, \begin{gather} \sup_{p\in{\mathbb Z}}\big\\|\big[ D\,e_{[-2^p,2^p]}(D),\rho(f)\big]\big\\| =\sup_{e\in E_\infty}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)}. \end{gather} Recall now that any edge $e$ of length $2^{n+1}$ is the union of two adjacent edges $e_1$ and $e_2$ of~length~$2^n$ such that $e_1^+=e_2^-$, therefore \begin{gather} \frac{\|f(e^+)-f(e^-)\|}{2^{n+1}} \leq\frac12\bigg(\frac{\|f(e_1^+)-f(e_1^-)\|}{2^{n}}+\frac{\|f(e_2^+)-f(e_2^-)\|}{2^{n}}\bigg) \leq \sup_{\mathrm{length}(e)=2^n}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)}. \end{gather} Iterating, we get \begin{gather} \sup_{e\in E_\infty}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)} =\sup_{\mathrm{length}(e)\leq 2^{n}}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)}. \end{gather} Since $f\in{\mathcal A}_n$, \begin{gather} \sup_{\mathrm{length}(e)\leq 2^{n}}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)} =\sup_{e\in K_n}\frac{\|f(e^+)-f(e^-)\|}{\mathrm{length}(e)} =\\|[ D_n,\rho(f\|_{K_n})]\\|. \tag{\qed} \end{gather} \renewcommand{\qed}{} \end{proof} In the following Theorem we identify ${\mathcal B}_\infty$ with $\pi_\tau({\mathcal B}_\infty)$, the trace $\tau$ on $\pi_\tau({\mathcal B}_\infty)$ with its exten\-sion to $\pi_\tau({\mathcal B}_\infty)''$, and $D_n$ and $D$ as unbounded operators affiliated with $({\mathcal B}_\infty,\tau)$ with~$\pi_\tau(D_n)$ and $\pi_\tau(D)$ as unbounded operators affiliated with $(\pi_\tau({\mathcal B}_\infty)'',\tau)$. \begin{thm}\label{SFtriple} The triple $({\mathcal L},\pi_\tau(B_\infty)'',D)$ on the unital C$^$-algebra ${\mathcal A}_\infty$ is an odd semifinite spectral triple, where ${\mathcal L}=\cup_n\{f\in{\mathcal A}_n, f\text{\,Lipschitz}\}$. The spectral triple has metric dimension $d=\frac{\log3}{\log2}$, the functional \begin{gather}\label{int-trace} \oint f=\tau_\omega} \def\O{\Omega \big(\rho(f)\big(I+D^2\big)^{-d/2}\big), \end{gather} is a finite trace on ${\mathcal A}_\infty$ where $\tau_\omega} \def\O{\Omega$ is the logarithmic Dixmier trace associated with $\tau$, and \begin{gather}\label{integral} \oint f=\frac6{\log3}\frac{\int_{K_n} f\,{\rm d}\vol}{\vol(K_n)},\qquad f\in{\mathcal A}_n, \end{gather} where $\vol$ is the Hausdorff measure of dimension $d$ normalized as above. As a consequence, $\oint f$ is a Bohr--F{\o}lner mean on the solenoid: \begin{gather} \oint f=\frac6{\log3}\,\lim_{n\in\mathbb N}\frac{\int_{K_n} f\,{\rm d}\vol}{\vol(K_n)},\qquad f\in{\mathcal A}_\infty. \end{gather} The Connes distance \begin{gather} d(\varphi,\psi)=\sup\{\|\varphi(f)-\psi(f)\|\colon f\in{\mathcal L},\, \\| [ D,\rho(f) ] \\| = 1 \},\qquad \varphi,\psi\in{\mathcal S}({\mathcal A}_\infty) \end{gather} between states on ${\mathcal A}_\infty$ verifies \begin{gather}\label{distance} d(\delta} \def\D{\Delta_x,\delta} \def\D{\Delta_y)=d_{\rm geo}(x,y),\qquad x,y\in K_\infty, \end{gather} where $d_{\rm geo}$ is the geodesic distance on $K_\infty$. \end{thm} \begin{proof} The properties of a semifinite spectral triple follow by the properties proved above, in particular property $(1)$ of Definition~\ref{def:SFtriple} follows by Propositions~\ref{aff-prop}$(a)$ and~\ref{commutator}, while pro\-perty $(2)$ follows by Proposition~\ref{OnDinfty}$(b)$. The functional in equality~\eqref{int-trace} is a finite trace by Proposition~\ref{OnDinfty}$(c)$. Equations~\eqref{integral} and~\eqref{distance} only remain to be proved. We observe that $(I+D^2)^{-d/2}-\|D_n\|^{-d}$ have finite trace. Indeed \begin{gather} \big((I+D^2)^{-d/2}-\|D_n\|^{-d}\big)e= \begin{cases} \big(1+4^{-k}\big)^{-d/2}e, & \mathrm{length}(e)=2^k,\quad k>n, \\ \big(\big(1+4^{-k}\big)^{-d/2}-2^{dk}\big)e, & \mathrm{length}(e)=2^k,\quad k>n k\leq n, \end{cases} \end{gather} hence, makig use of a formula in Theorem~\ref{OnDinfty}$(b)$, we get \begin{align} \big\|\tau\big(\big(I+D^2\big)^{-d/2}-\|D_n\|^{-d}\big)\big\| &\leq\sum_{k>n}(1+4^{-k})^{-d/2}\tau\big(P^k\big)+\sum_{k\leq n}\big\|\big(1+4^{-k}\big)^{-d/2}-3^k\big\|\tau\big(P^k\big)\\ &\leq6\big(1+4^{-(n+1)}\big)^{-d/2}\sum_{k>n}3^{-k} +6\sum_{k\leq n}\big\|\big(1+4^{k}\big)^{-d/2}-1\big\| \end{align} and both series are convergent. Since the Dixmier trace vanishes on trace class operators, this implies that \begin{gather} \tau_\omega} \def\O{\Omega (\rho(f)\big(I+D^2\big)^{-d/2})=\tau_\omega} \def\O{\Omega \big(\rho(f)\|D_n\|^{-d}\big) =\frac1d \Res_{s=d}\tau\big(\rho(f)\|D_n\|^{-s}\big), \end{gather} therefore, if $f\in{\mathcal A}_n$, \begin{gather} \oint f =\frac1d \Res_{s=d}\tau\big(\rho(f)\|D_n\|^{-s}\big) =\frac1d \Res_{s=d}\frac{\tr(\rho(f_{K_n})\|D_n\|^{-s})}{\vol(K_n)} =\frac{\tr_\omega} \def\O{\Omega \big(\rho(f)\|D_n\|^{-d}\big)}{\vol(K_n)}. \end{gather} Now, by formula~\eqref{fractalNCint} applied to $K_n$, $\tr_\omega(\rho(f)\|D_n\|^{-d}) = \frac{6\cdot \ell(e)^d}{ \log 3} \int_{K_n} f\, {\rm d}H_d$, where $H_d$ is the Hausdorff measure normalized on $K_n$, hence $H_d=(\mu_d(K_d))^{-1}\mu_d=3^{-n}\mu_d$, and $e\in E_0(K_n)$, hence $\ell(e)^d=3^n$. Therefore $\tr_\omega\big(\rho(f)\|D_n\|^{-d}\big) = \frac6{\log3} \int_{K_n} f\, {\rm d}\mu_d$ and formula~\eqref{integral} follows. As~for equation~\eqref{distance}, given $x,y\in K_\infty$ let $n$ such that $x,y\in K_n$, $m\geq n$. Then, combining Propositions~\ref{aff-prop}$(a)$ and~\ref{commutator}, we have, for $f\in{\mathcal A}_m$, \begin{gather} \\|[ D, \rho(f)]\\|=\\|[ D_m,\rho(f\|_{K_m})]\\|, \end{gather} and, by Theorem 5.2 and Corollary 5.14 in~\cite{GuIs16}, \begin{gather} \sup\{\|f(x)-f(y)\|\colon f\in{\mathcal A}_m,\\|[D_m,\rho(f\|_{K_n})]\\|=1\}=d_{\rm geo}(x,y),\qquad m\geq n. \end{gather} Therefore \begin{align} d(\delta} \def\D{\Delta_x,\delta} \def\D{\Delta_y) &=\sup\{\|f(x)-f(y)\|\colon f\in{\mathcal L},\, \\|[ D,\rho(f)]\\|=1\} \\ &=\lim_m\ \sup\{\|f(x)-f(y)\|\colon f\in{\mathcal A}_m,\, \\|[ D,\rho(f)]\\|=1\} \\ &=\lim_m\ \sup\{\|f(x)-f(y)\|\colon f\in{\mathcal A}_m,\, \\|[D_m,\rho(f\|_{K_n})]\\|=1\}=d_{\rm geo}(x,y). \tag{\qed} \end{align} \renewcommand{\qed}{} \end{proof} \begin{rem} The last statement in Theorem~\ref{SFtriple} shows that the triple $({\mathcal L},{\mathcal M},D_\infty)$ recovers two incompatible aspects of the space ${\mathcal A}_\infty$: on the one hand the compact space given by the spectrum of the unital algebra ${\mathcal A}_\infty$, with the corresponding finite integral, and on the other hand the open fractafold $K_\infty$ with its geodesic distance. In particular, the functional on ${\mathcal L}$ given by~$L(f)=\\|[D,\rho(f)]\\|$ is not a Lip-norm in the sense of Rieffel~\cite{Rieffel} because it does not give rise to~the weak$^$ topology on ${\mathcal S}({\mathcal A}_\infty)$. In fact, such seminorm produces a distance which is unbounded on points, therefore the induced topology cannot be compact. \end{rem} \LastPageEnding \end{document}	arXiv
A surprising property of partitions into primes I was studying some properties of partitions into primes and came across a surprising property. But before I talk about them, I am giving a definition. Definition. A $k$-tuple $\lambda=(\lambda_1,\lambda_2,...,\lambda_k)$ is called a prime partition of a positive integer $n$ if the following hold: a) $\lambda_1, \lambda_2, \dots, \lambda_k$ are distinct primes. b) $\lambda_1 + \lambda_2 + \cdots + \lambda_k = n$. c) $\lambda_1 < \lambda_2 < \cdots < \lambda_k$. For example, one of the prime partitions of $10$ is $(2,3,5)$. Now, define $q(n)$ to be the number of all possible prime partitions of $n$. The values of $q(n)$ from $1$ to $20$ are: $$ 0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 4, 3, 4, \dots$$ The generating function of $q(n)$ is $$\prod_{p\,\text{ prime}}^{\infty}\frac1{1-n^p}$$ The question I was (aimlessly) plotting functions in mathematica related to this function, and, after 15 minutes of plotting, I noticed that $$\log q(n)\sim\pi\sqrt{\frac{2}{3}\pi(n)}$$ where $\pi(n)$ is the prime counting function. This seems a bit hard to prove. I used the prime number theorem to get $$\log q(n)\sim\pi\sqrt{\frac23\frac n{\log n}}$$ I don't think that this would have an easy proof. Any article/proof of this might help. Another question: is there an asymptotic formula even stronger than this? Note: I don't know about this result, I got it accidentally. prime-numbers asymptotics integer-partitions $\begingroup$ No answer, just more information and background on the topic and the generating function: math.stackexchange.com/questions/89240/prime-partition EDIT: I found the following article, where you're question is answered and discussed: arxiv.org/abs/1609.06497 $\endgroup$ – jojobo $\begingroup$ This sequence is to found as oeis.org/A000586 but this entry doesn't provide any asymptotic equivalent. $\endgroup$ – Jean Marie $\begingroup$ Nitpick: I think you missed a condition in your definition of "prime partition": you want to insist that the primes are listed in (say) increasing order, or else you want not tuples of primes but sets of primes. Otherwise the numbers need to be somewhat larger to account for permutations. $\endgroup$ – Gareth McCaughan $\begingroup$ This article academic.oup.com/qjmath/article-abstract/5/1/241/1519252 by Roth & Szekeres (found in the references to the article jojobo cited) proves a rather general theorem that gives such asymptotics for partitions using other sets of numbers besides the primes. $\endgroup$ $\begingroup$ Since you're not allowing repetition of the prime parts, I believe the generating function should have $(1+n^p)$ rather than $1/(1-n^p)$. Note that the arXiv paper @jojobo mentions allows for repeated prime parts. $\endgroup$ – Brian Hopkins This is a compilation of the comments which, together, answer your question. Your sequence $q(n)$ is in the On-Line Encyclopedia of Integer Sequences as A000586. Since you require the primes to be distinct, the generating function is $$ \sum_{n=1}^\infty q(n)x^n = \prod_{p \text{ prime}}^\infty (1+x^p); $$ the generating function you gave would allow for arbitrary repetition of prime parts (discussed in OEIS A000607). The OEIS entry does include an asymptotic formula, which follows from Roth & Szekeres 1954 as Gareth suggested. It is exactly what you found: $$ \log q(n) \sim \pi \sqrt{\frac{2n}{3 \log(n)}}.$$ Brian HopkinsBrian Hopkins Prime Partition Generating sequences of numeric partitions Primes and proofs How many primes does Euclid's proof account for? Balls, Bags, Partitions, and Permutations Computing the summation of distinct primes of $n$? How many distinct prime factors are there in the numbers between two primes? Why do partitions correspond to irreps in $S_n$? Converting between different ways of representing integer partitions	CommonCrawl
Fine-grain watermarking for intellectual property protection Stefano Giovanni Rizzo ORCID: orcid.org/0000-0003-3346-33891, Flavio Bertini2 & Danilo Montesi2 The current online digital world, consisting of thousands of newspapers, blogs, social media, and cloud file sharing services, is providing easy and unlimited access to a large treasure of text contents. Making copies of these text contents is simple and virtually costless. As a result, producers and owners of text content are interested in the protection of their intellectual property (IP) rights. Digital watermarking has become crucially important in the protection of digital contents. Out of all, text watermarking poses many challenges, since text is characterized by a low capacity to embed a watermark and allows only a restricted number of alternative syntactic and semantic permutations. This becomes even harder when authors want to protect not just a whole book or article, but each single sentence or paragraph, a problem well known to copyright law. In this paper, we present a fine-grain text watermarking method that protects even small portions of the digital content. The core method is based on homoglyph characters substitution for latin symbols and whitespaces. It allows to produce a watermarked version of the original text, preserving the anonymity of the users according to the right to privacy. In particular, the embedding and extraction algorithms allow to continuously protect the watermark through the whole document in a fine-grain fashion. It ensures visual indistinguishability and length preservation, meaning that it does not cause overhead to the original document, and it is robust to the copy and past of small excerpts of the text. We use a real dataset of 1.8 million New York articles to evaluate our method. We evaluate and compare the robustness against common attacks, and we propose a new measure for partial copy and paste robustness. The results show the effectiveness of our approach providing an average length of 101 characters needed to embed the watermark and allowing to protect paragraph-long excerpt or smaller the 94.5% of the times. The last decades are characterized by the easy availability of millions upon millions of digital contents that meet several kind of users' needs both in professional activities and social interactions. An important reason for the proliferation of digital contents among users is the increase in the usage of online communication platforms, like websites, social media, and cloud file sharing services, to name a few. All these platforms have introduced changes in the user habits with respect to digital contents by increasing the copying and sharing of text, audio, images, and video, namely digital contents [1]. While the current digital technologies facilitate the copy and sharing of these digital contents, this is often an unattributed copy of others' work, resulting in a misappropriation of their intellectual property. In several contexts, such as for online newspapers and blogs, the contents' owners have solid interests in protecting their IP rights, in order to preserve their business. In particular, there are different illicit actions concerning these digital contents, like tampering, forgery, theft, and, more simply, making a copy of both the whole content or part of it. The problem is more meaningful with text since it is the main carrier of information (e.g., online news articles, scientific articles, e-mail, product catalogs) while being more prone to full or partial misappropriation. Moreover, findings in [2] demonstrate that the copy and paste function significantly increase the plagiarism attitude of the users, that is 13% more likely when copy and paste is possible. Up until now, the techniques used to protect IP and prevent illegal use, like digital rights management (DRM) and technical protection measures (TPM), also prevent legal or permitted uses of the copyrighted digital contents [3, 4], by representing a strong limit in terms of freedom of information and expression of the user. In [5], Lai and Graber point out the complexity to reach a fair balance among IP rights and freedom of expression and information. In particular, the authors compare the need of the IP owners and privacy and freedom of choice of the users. These classic digital rights protection techniques are based on hardware or firmware supports and proprietary encodings that prevent the user of making copies, reading unauthorized copies, or reproducing it on unauthorized supports. The total prevention of copy through cryptography and dedicated supports, such as the content scrambling system for DVD protection [6], reduces the ability of sharing and distributing the creative content. In order to overcome the limits of the classic digital rights protection techniques and meet the various needs in IP protection field, different approaches are developed [7]. For instance, while steganography provides techniques to hide new information into the original digital content, cryptography produces an unreadable version of the document by applying a kind of permutation or substitution to the original information. Watermarking is the most balanced technique for sharing not obfuscated information while preserving the copyright [8, 9]. In particular, it ensures copyright protection by applying a mark to the original digital content, without showing such mark to the readers. Watermarking methods can be applied in innumerable contexts, such as identifying unauthorized users, establishing the authorship of a digital content, monitoring the broadcasting process, and distrusting a tampered digital content. Up to an acceptable distortion, watermarking can be also adopted to protect dynamically generated contents from databases [10]. Watermarking an intellectual property allows the free sharing of a digital content, while binding the artifact with the original author. When the authorship of a digital content is misattributed, the original author can claim his/her authorship or copyright. In this scenario, the author can extract and show the digital watermark as an irrefutable proof of authorship, avoiding costs and efforts of more elaborated and timestamped evidence. At the same time, the watermark exclude the possibility of unintentional plagiarism, in the case when the malicious user appeal to the lack of originality of the work, that may have lead to the unrelated creation of the same or very similar content. Out of all digital content watermarking techniques, we focus on text watermarking. The reason behind our choice is that textual information represent one of the largest bunch of digital contents that people can daily share and explore online, for instance, online newspaper articles, manuals and guides, social media, and microblogging posts, to name a few. Furthermore, text messages increase daily and are more often used for commerce, mobile banking, and government communications. In comparison with watermarking techniques for other digital contents, text watermarking is the most difficult task, presenting several challenges mainly because text is not noise-tolerant. In particular, a text watermarking algorithm must work with some additional constraints, as short-length message, a limited set of transformations in order to preserve readability and a restricted number of alternative syntactic and semantic permutations [11]. In fact, one of the main problems concerning textual content, even short text message, is authorship verification, that is, to verify if a text has been actually produced by a given author, as he/she claims. If we exclude that a third-party guarantor is involved in the verification process, such as an IPR database with certified timestamps of deposited contents (also known as zero-watermarking), then some author-dependent data must be embedded in the text content, such as a unique code derived from the author's secret key. Another peculiarity of text in the context of unauthorized copy is that, unlike images, any meaningful excerpt, like a paragraph, could be copied, and it is difficult to predict which one. While it is true that in the case of images, some partial cropping is often applied before unauthorized re-sharing, the unauthorized copy will still account for an important percentage of the original image (with some exceptions, for example in aerial photography). Instead, in the context of text, it is very common to copy only few sentences, which may not be subsequent in the original document and may account for a very small percentage of it (e.g., one paragraph from a book). This can be seen as a special case of a deletion attack, in which most of the watermarked document is deleted and only some paragraphs or sentences are left, motivating the need of a fine-grain approach able to embed the watermark in as many sub-portions of text as possible. The concept of a fine-grain protection of text content is well known in copyright law: it is common to claim intellectual property rights on small portions of larger works, and there is a vast literature involving several trials and studies [12] trying to define at which fine-grain level an intellectual work can be copyrighted. This known scenario however has not been addressed so far in the text watermarking literature. It also makes the text length constraint even stricter, because the watermark has to be embedded in smaller parts of the text content. Additional issues arise if we must be able to verify a copied text that is straddling partially two watermarked portions. In this paper, we propose a structural text watermarking method for intellectual property protectionFootnote 1. The method protects the whole document as well as smaller excerpt of it, up to a minimum size of excerpt that depends on the specific characters of the text. Nevertheless, it is fair with respect to the concerns regarding communicative freedom and privacy of the users, without altering the content of the text or embedding explicit author-related data. More precisely, the proposed method is invisible and content-preserving and belongs to the fragile and non-blind classes. In practice, it is able to embed a password-based watermark without altering the content and preserving the length, ensuring data protection against the copy and paste of even small excerpt of text. The embedding process consists of two phases. In the first one, the watermark is generated by applying a hash function that combines the user (author) password and the structural characteristics of the text. In the second phase, that is the core of our methodology, the watermark is embedded into the original text by exploiting homoglyph characters. Homoglyph characters, as symbols, numbers, and letters, look very similar on the screen and in print; nevertheless, their low-level encoding is completely different. More precisely, the Unicode confusable characters, namely the homoglyph characters, are listed by Unicode Consortium and look confusingly similar from each others [13]. In practice, we replace a subset of characters of the original text with an indistinguishable latin homoglyph symbol, with a substitution process driven by the watermark bits sequence. The password allows to verify the authorship since only the actual author of the text can correctly regenerate the watermark. The proposed method has the following four new significant features: It leaves visually indistinguishable original text, in other words, the watermark is not noticeable by the user. The length of the original text is preserved, no matter how short is it. It can be continuously applied to small excerpts of a longer text, protecting a document at a fine-grain level against the copy and paste of text portions It allows to cryptographically bind each text excerpt to the original source document. The visually indistinguishable features strongly depend on the font used. However, we will show in the evaluation section how the homoglyph characters allow to cover the most used font families. The length preservation feature is quite complex to ensure when the algorithm operates on short texts. The proposed method is able to embed a watermark while preserving the text length with very short texts (theoretically a minimum of 22 symbols). The minimum length depends on the text content, as only a subset of characters can be substituted to embed the data. In order to establish the minimum length requirement on real text examples, we provide the results of an extensive experiments on 1.8 million of New York Times articles [14]. The results show that, on average, 101 characters are sufficient to embed the watermark preserving the length and visible aspects of the original text. Despite paragraphs can be very short or having few confusable symbols that can be replaced, the method allows to watermark very short excerpt, shorter than a single paragraph of New York Times articles for the 94.5% of the times, meaning that it ensures data protection when only a single paragraph or a smaller excerpt of it is copied and pasted. The combination of these two features allows to use our text watermarking method in several new contexts, for instance, word and pdf documents, online newspaper articles, short message communications, e-mails, microblogging platforms, and social networks posts. The fine-grain watermarking method of the proposed approach allows for the first time to protect small excerpt of text, by repeatedly embedding the watermark across the document. This is made possible by the short length requirements of the approach and has two valuable consequences: (i) it is possible to extract the watermark even when it is "broken" between two watermark sequences, and (ii) each excerpt is bound to the source document and can be traced back to it. In order to evaluate the fine-grain property of the method and compare it with current methods, we propose also a novel measure for the robustness to partial copy and paste. The rest of the paper is organized as follows. In Section 2, we provide a small background in watermarking, in order to classify the methods and show the features usually required to a watermarking algorithm. In Section 3, we review the literature works related to text watermarking methods. In Section 4, we describe our text watermarking method, including watermark generation, embedding, extraction, and authorship verification. We discuss the evaluations of our method in Section 5. Some concluding remarks are made in Section 6. Background in watermarking In this section, we provide a small background in watermarking methods. This is important as it will help in understanding the reasons behind the design of our method. In accordance with the literature [15], watermarking methods can be categorized as follows: Readable or detectable—The watermarking is readable if the user can clearly read it. It is instead detectable if a detection function can be used to check if a watermark exists or not, but it cannot be read. Visible or invisible—A visible watermarking is visually perceptible by the user. Contrary, the watermarking is invisible if it is hidden in the original digital content and it does not noticeable by the user. A visible watermark may be not readable, that is, a user can visually detect it but cannot read its content. Blind or non-blind—If the original digital content is not needed in the extraction process, the watermarking is blind. Otherwise, the watermarking belongs to non-blind category. Simple or multiple—If a watermark can be applied only once the watermarking is simple. Otherwise, a multiple watermarking can be embedded more than one time without affecting the whole process. Fragile, semi-fragile, and robust—A fragile watermark is detectable and can be altered or erased; thus, it is used for integrity authentication. On the flip side, a robust watermark is detectable and not erasable and it is most suitable for copyright protection. A semi-fragile watermarking is suited for content authentication. In [16], the researchers identify several features usually required to a watermarking method. Verifiability represents the ability to irrefutably prove the ownership of the digital content. Data payload represents the maximum number of bits of extra information that can be embedded in the original digital content. Robustness represents the ability to resist to processing operations and attacks, as security is the capacity to not be altered or removed without having full knowledge of the watermark or the embedding process. Finally, computational cost is the cost required in embedding and extraction process. Out of all digital content watermarking techniques, text watermarking is the most challenging. Text has a low embedding bandwidth and allows only a restricted number of alternative syntactic and semantic permutations. Text watermarking algorithms can be classified as follows: Zero-watermarking techniques—Instead of watermarking the text, some characterizing features of the text are stored on a third-party authority server, such as an Intellectual Property Rights (IPR) database. Image-based techniques—Firstly, the text is transformed into an image, then the watermark is embedded into the image. Obviously, this approach modifies the nature of the original document; in other words, it cannot be considered a pure text watermarking method. However, it has some interesting features, as length preservation and language independent. Syntactic techniques—These methods transform the language-depending structures in order to hide the watermark. Typically, the sentences have different language-depending structures that make the process easier. Semantic techniques—These methods use verbs, nouns, prepositions, and even spelling and grammar rules to permute the contents and embed the watermark. Structural techniques—These methods exploit double letter occurrences, word shift and line shift encoding, and Unicode standard to embed the watermark. They are one of the most recent methodologies with which the original text is not altered. The text watermarking approaches with actual watermark embedding are usually classified into three main categories [15, 17]: image-based, syntactic, and semantic. In this categorization, the zero-watermarking approaches are often not considered as no watermark is actually applied; however, this alternative solution is getting more attention lately and it is important to understand the difference between the zero-watermarking and content-preserving methods. A recent survey [18] considers instead the structural, linguistic, and statistics as the three main categories. After highlighting the core ideas, advantages, and disadvantages of the mentioned approaches, we will focus on the structural methods. Unicode-based methods such as the proposed method belong to this latter class. Zero-watermarking The first important dichotomy in text watermarking works, and watermarking in general is the one between zero-watermarking techniques and the more common "non-zero" or embedding watermarking techniques. Zero-watermarking aims at extracting characterizing information from a digital content, for example, from a picture or a song, and then store this information into an Intellectual Property Right (IPR) database [19]. Embedding watermarking instead aim at embedding in the digital content a payload related to the author or to the content itself (e.g., the author name, a company logo, the keyed hash of the content with the author's password). In the zero-watermarking process, no actual watermark is applied to the content or embedded in the content, which is left untouched. The association between the content and the author does not rely on the watermark, but on the proof from a trusted authority. Copious literature has emerged in recent years proposing zero-watermarking techniques on text [20–23] as an alternative solution to cope with the two orthogonal challenges of text watermarking: hiding information in small, unnoisy data and keeping the content unaltered to the human eye. In zero-watermarking, this is addressed by avoiding the embedding of any watermark whatsoever. Zero-watermarking techniques can be seen as a form of dimensionality reduction, and in fact, they are often based on well known dimensionality reduction techniques [24]. The clear advantages of dimensionality reduction are that the performance for similar content search is improved and the storage needed on the IPR database is reduced. However, in terms of security and IP protection, the same result can be obtained by simply storing the original content as it is, without extraction, and then applying a similarity technique (like the SSIM method for images [25] or structure-level, word-level, and character-level similarities in text [26]) to efficiently identify duplicates when the format has been altered. A collateral shortcoming of zero-watermarking is that the identity of the author of watermarked content must be preventively registered on a third-party authority, leading to privacy issues. Image-based methods The image-based text watermarking is the most researched approach to text watermarking and the earliest one to be investigated, with the first techniques dating back to the mid-1990s [27, 28]. In this approach, a printed text is first scanned as an image, or as a screenshot in the case of digital text, and then a watermark is applied on this image. For example, in grayscale images of text document, the watermark payload is embedded by tuning the luminance of pixels accordingly to the watermark data [29] or by modifying the edge direction histograms to carry the watermark signal [30]. A robust embedding can be obtained by slightly shifting the text elements horizontally or vertically: a text element can be a word, to which a shift of few pixels to the right or to the left can embed an information, or can be a text line or block, shifter up and down with the same purpose [31, 32]. Similar results can be obtained by altering the spaces between words to encode the watermark data [33, 34]. Other methods are based on the alteration of single characters [31], some focus on smaller detail such as strokes and serifs of the characters and work by prolonging them [35], and others, more simply, alter the character in their size by change the scale depending on the watermark content [36]. There are two important shortcomings of image-based methods. The first is that text must be shared as an image, in an image file format (e.g., PNG, JPEG, or TIFF), or as printed paper or through fax machines, which is nowdays less practical and not very common. The second is that text can be still reconverted to plain text by manual re-typing or using an OCR software, leaving behind in the process any trace of the watermark. Overall, while it is a strong solution for printed papers and scanned documents, image-based text watermarking may become less and less relevant in the future because digital media is increasingly preferred to printed paper both for reading and sharing text contents. Syntactic methods Syntactic methods for text watermarking works on the syntax of natural language text, by altering its structure to embed a watermark. The first common step is to build the syntactic tree of a sentence, after which some syntactic operations like clefting, passivization, or activization are applied in order to encode the watermark bits [37]. Clefting is the process of transforming a simple sentence into a more, unnecessarly complex one, for example, the simple sentence I like champagne can be transformed into champagne is what I like (the what clefting) or into it is champagne that I like (the it clefting) [38]. Passivization is the transformation from the active to the passive form of verbs as from, for example, Tom kicked the bucket to the bucket was kicked by Tom, while activization is the opposite process. There are also other morpho-syntactic transformations that can be also applied that are considered to preserve the original meaning: the linguistic notion of possession for instance can be written either with using the preposition "of" or using the suffix "-s" [39]. The low embedding capacity, that is the ratio between the text length and the length of the watermark that can be embedded, is a limit of these methods. Contexts of use where the length of the text is limited, such as mobile phones SMS texts or Twitter posts are inherently excluded. Other disadvantages of syntactic methods comes from the alteration of the content. The assumption that different syntactic forms have the same meaning is not always true [40]: in the previous example, Tom kicked the bucket has an idiomatic interpretation, while its passive form has only a literal one. Semantic methods By exploiting the similarity of the meaning of different words, it is possible to replace words with their synonyms [41]. The systematic substitution of words depending on the watermark data results in a non-blind watermark embedding. This semantic approach can be also mixed with syntactic approach [42] to obtain an overall higher embedding capacity. Other semantic techniques work on the sentence level semantic, leveraging the implicit presuppositions of each sentence [43, 44]. A presupposition is a sort of implicit information that follows directly from a sentence, usually a fact that must be true in order for the sentence to make sense. For example, in the statement Jane likes her white car, the presupposition is that Jane has a car. By keeping the same meaning, the statement can be rephrased as Jane has a white car and she likes it. It is therefore possible to add the presupposition explicitly, or in other cases to remove it, in order to encode watermark data. The semantic methods share some of the shortcomings of the syntactic methods. Like in the case of syntactic methods, the author's content can be strongly altered in order to embed the watermark. Also, they depend on the language and on the correctness of written text. Structural methods Structural methods include all those methods that do not alter the text content but only its structure, intended as underlying representation or as features regarding visual rendering. They have more recently emerged that embed watermark or hidden payloads by changing the underlying encoding of symbols or adding invisible symbols, without actually altering the readable content of the text. The Unicode standard has several different symbols for whitespaces, some of different width, others practically identical. By putting many of these whitespace symbols at the end of a paragraphs, or by filling an empty line, relatively long payloads has been hidden in Microsoft Word documents [45]. A similar technique based on different Unicode whitespaces has been effectively applied to watermark Arabic language text, by using a different Unicode whitespace between words depending on the bits of the watermark's binary representation [46]. A more recent method uses instead multiple ASCII whitespaces to embed a covert message [47] for PDF steganography. The techniques works on justified text and is able to embed 4 bits for each host line, where a host line is a line with at least 9 normal spaces and 3 wider spaces. Apart from whitespaces, the Unicode standard also provides some totally invisible symbols, which are provided as zero-width whitespaces. These symbols, together with whitespaces, have been exploited to watermark HTML pages [48, 49] and more generally to hide hidden messages in the text [50]. As mentioned earlier, these methods have the important advantage of keeping the original content unaltered, but without transforming the text to an image, or relying on an external database. The above structural methods are blind, meaning that the original text is not needed in order to extract the watermark. This, together with the easiness of removing multiple whitespaces, makes these approaches fragile in both malicious and benign attacks. This is particularly true for methods that uses consecutive whitespaces and whitespaces before or after the whole text, because it has been shown that many digital platforms and social media automatically remove them [51]. This can also happen through selection for copy and paste: selection may easily exclude the white portion where the watermark is embedded. Apart from whitespaces, homoglyphs ad invisible characters, some image based where lines or words are slightly shifted without altering the text content [52, 53] have been also considered as structural methods [54]. The proposed method for text watermarking can be categorized as a structural method; therefore, it preserves both the appearance and the content without converting the text into image and without the need of a third-party IPR database. Our approach compute a watermark depending on the original text and a password, both given by the author, then it replaces the symbols and whitespaces with visually equivalent symbols, according to the watermark binary data. Following the watermarking features and categorizations so far presented in related works, our approach is: Invisible (totally or partially): The readable content is kept, while the symbols can be replaced by their homoglyph. The changes applied to the text are not noticeable to the human reader, to a certain degree of analysis. Visibility depends strictly on the font used: some fonts may represent the similar symbol in the same exact way, thus making it impossible to visually distinguish the two, others may implement slightly different strokes or serifs or, in the case of whitespace, slightly different widths. Detectable, fragile, and non-blind: As for all the content-preserving watermarks in text, the content can be simply re-typed thus losing the embedded watermark. By knowing the set of homoglyphs used, it is also possible to detect the watermark and expose the embedded data. However, this data would be useless without the author's password and original complete text needed for verification (see Section 4.5), so the attacker cannot prove the ownership using the extracted data. The requirement for the original complete text makes our approach non-blind, but it hold only in the cases when an excerpt of text has been copied, and not the whole text. Content and length preserving: The whole embedding process works by replacing the symbols with visually indistinguishable homoglyphs; therefore, everything, from letters to sentences, is preserved. An important feature is that our method is length preserving: the related works on structural watermarking with Unicode add multiple invisible symbols. This results in an overhead in the original text and can be an issue in context where the number of characters is limited. Moreover, this can make the watermark detection simpler by comparing the number of readable symbols with the size of the text. On the contrary, our approach preserves also the length of the text because, when a substitution happens, one symbols is replaced with another symbol, keeping the same total number of symbols. Besides its general properties, our method show several advantages in comparison with other non-zero-watermarking techniques. It is more efficient because it implements a mapping symbol-to-symbol which is much simpler than NLP techniques where a syntax tree must be built or complex semantic analysis must be carried. From the perspective of the range of application, it can be applied to most software and web platforms, because it does not depend on a particular file format [45] or markup language [48], but it is only dependent on Unicode support. The method does not use consequent whitespaces or rely on specific invisible symbols [48, 55]. This is a great advantage when the text is posted online, as most online platforms apply several filters to the incoming text. Moreover, it has been shown the set of symbols used in our method can pass through several instant messengers, webmail services, and social media without getting filtered, allowing robust online applications [51]. Lastly, it is more robust against the partial selection in copy and paste, because unlike other structural methods [45, 48] where the whitespaces symbols are appended between new lines, at the beginning or at the end of the text, in our approach, the watermark is embedded across all the text and is part of the text. This means that it is much more difficult to avoid copying also the watermark in the process. Unicode confusables The Unicode standard consists of more than 120 thousand symbols, among which some are very similar or totally indistinguishable. Despite these symbols have a different numerical code and different Unicode name, thus a separate purpose or meaning, the fonts with Unicode support depict them with the same aspect. These symbols are often called homoglyphs. This similarity between symbols is a well-known security threat, because they may be used to deceive users into clicking on fake links or may avoid spam filters by altering the words with spurious symbols. For this reason, the Unicode Consortium maintain a list of the above confusable symbols [13], which is publicly availableFootnote 2. In our approach, we exploit the similarity of Unicode homoglyphs to seamlessly replace them accordingly to the bits of a payload. More specifically, the payload is the watermark of the text. To better clarify the approach, let us suppose that each symbol of the alphabet has a "common" version and a much less usual clone, with a different underneath Unicode value. Then, we could encode a binary string by using the common symbols to express "0" and the clone symbols to express "1." In this way, we would be able to encode 1 bit for each symbol in the text. After the encoding, we can also decode the binary string by looking at the Unicode: if the decoder finds a common symbol will produce a "0" ; otherwise, if the clone symbol has been used, it will produce a "1." The real scenario is different from the above example, because only some symbols have a related homoglyph. We identified these duplicate symbols for some letters of the latin alphabet and for some punctuation in Table 1. Moreover, in Table 2, we consider the sets of whitespaces that the Unicode standard provides as homoglyphs, thus using them to encode bits. Table 1 Encoding bits for latin letters and punctuation symbols Table 2 Encoding bits for whitespace symbols In order to find out the most similar symbols, we tested their homoglyphs under the most used font families in modern desktop and web applications, obtaining imperceptible differences in most used sans-serif fonts. The reader can find the evaluation results on fonts in Section 5. Watermark generation with password Before going into details of the watermark embedding method, we first describe how the watermark is generated. We want the watermark to be a function of the original text and the author's identity so that we can (i) prove that the watermark is related to the original text and (ii) assure that only the author who generated the watermark can verify it. These requirements can be satisfied by a cryptographic keyed hash function such as SipHash [56]. SipHash is a function that takes in input a variable-length message and a secret key and produces in output a binary string of a fixed size. This binary string is a message authentication code (MAC) in message exchange: only using the same secret key it is possible to recreate the same MAC and authenticate the message. We use the MAC as a watermark in order to add an additional security layer to the watermarking schema: while it may be possible for attackers to extract the watermark, they will not be able to prove the authorship of that watermark. Other non-keyed hash function can be also used for the same purpose [57]; however, SipHash is specifically designed to securely authenticate short messages producing a small but robust code of 64 bit. This is particularly suitable in our context of fine-grain watermarking, as we want the watermark to be as small as possible to embed it in small excerpt of a text while retaining cryptographic robustness. Nevertheless, our embedding approach does not depend on a specific hash function or, more generally, watermark generation method; for this reason, in Section 5, we evaluate the embedding method using other hash functions of different MAC length. The watermark generation is shown in Fig. 1, where the cryptographic keyed hash function takes in input the original text t and the password k producing in output the 64 bit string, representing the watermark. Only who owns the password used for generating the watermark and the original text will be able to prove the authorship by replicating the generation process. The robustness of this authorship verification process is ensured by the strength of the hash function. Generation of the watermark bits. Given the original text t and a secret password k, the SipHash function generates a cryptographic hash, representing the watermark w to be embedded Unicode watermark embedding The watermark, generated using the keyed hash function, is then embedded through symbols replacement, following the proposed approach in Algorithm 1. By replacing original symbols and whitespaces with identical or almost identical Unicode symbols, the algorithm embeds the watermark binary string producing a new text which is indistinguishable from the original. More specifically, the algorithm scans the text starting from the first character, looking for a confusable symbol, that is, a symbol or whitespace that has a duplicate in the Unicode standard. The mapping between confusable symbols and watermarked bit is shown for clarity in Fig. 2. Embedding watermark's bits by replacing confusable symbols. Only the symbols with a related duplicate in Tables 1 and 2 are used to embed the watermark bits: one bit can be embedded on latin letters, and three bits can be embedded on space characters When a confusable symbol is found, this is replaced with its homoglyph or kept depending on the next bit of the watermark, starting from the leftmost bit. The Unicode codes of symbols used when the bit is 0 and when the bit is 1 are shown in Table 1. It must be noted that the usage of original symbols to embed the bit 0 and duplicate to embed the bit 1 is completely arbitrary and for the sake of simplicity. It is possible to make the opposite association or to choose a more elaborate scheme, for example, one in which the bit 1 is represented with the duplicate code for some symbols and with original code for others. Similarly, when a whitespace is found, this is replaced with one of the 7 whitespaces in Table 2 or kept depending on the next 3 bits of the watermark. Specifically, it is kept when the bits are 000 while it is replaced with another whitespace for any other 3 bits combination. As for the symbols, the whitespace association table can be rewritten arbitrarily making a custom, less predictable scheme of embedding. Unicode watermark extraction The embedded watermark is invisible to the reader of the watermarked text, however can be detected in a technical analysis of the symbol encoding, noticing that unusual symbols have been used. Knowing the embedding algorithm and the mapping between confusable and bits, it is possible to also extract the watermark. The extraction algorithm (Algorithm 2) is in fact the opposite process of the embedding. As in the embedding process, it scans the watermarked text for confusable symbols. When a confusable symbol or whitespace is found, the association table (Table 1 or Table 2) is used to find the corresponding bit (or bits in the case of whitespaces). For every confusable symbol, the algorithm output in sequence the bits of the watermark, from the leftmost (most significant bit) to the rightmost (least significant bit). Authorship verification Suppose that an unattributed copy of a text, carrying a watermark, is shared by an attacker. The original author's will is to claim his/her IP rights on the watermarked text, but the attacker too may try to claim the authorship. The goal of the verification mechanism is to ensure that only the original author, who generated and embedded the watermark in the first place, will be able to prove the authorship. In our approach, this goal is achieved through the regeneration of the same SipHash MAC. It is based on the assumption that, with limited computational resources, the attacker will not be able to generate the same MAC without having the password used in the generation phase. The verification process consists of the following 3 steps, also illustrated in Fig. 3: Knowing the associations tables, the author can extract the watermark w and the original text t from the watermarked text. The watermark is a MAC obtained from the original text and password using the keyed hash function. This step can be performed by an attacker with knowledge of the embedding algorithm and association tables. The author applies the keyed hash function to the original text t using the same password k used in the watermark generation. The function produces the watermark w′. The attacker, as well, applies the same keyed hash function to the original text t using a different password k′, thus obtaining a different watermark w″. Assuming that the author password k is different from the attacker password k′, then the generated watermarks w′ and w″, with a very high probability, are two distinct binary strings because of the collision-free property of hash functions. The extracted watermark w is now compared with w′ and w″. Because w and w′ have been generated using the same text and password, they are equal, while w″ is different. The comparison proves the authorship of the watermarked text. Proof of authorship on watermarked text. The password k is a proof of authorship. Once the watermark is extracted, only the author with the original password k is able to reproduce it Despite the SipHash is considered as a secure keyed hash function, given its short MAC length, it may become less secure when more computational power becomes available to attackers. The proposed approach however does not rely on a specific keyed hash function, allowing any other hash function with stronger security to be applied instead. With this in mind, in Section 5, we provide evaluation of text length requirements using the hash functions MD5, SHA-1, and SHA-2. It must be noted that, as for any other content-preserving method (i.e., all the image-based and structural methods), the attacker may still re-type the text and embed his own watermark. In this case, assuming that a digital sharing method has been used, it is usually possible to track the earliest version of the text and use this earliest version to track the original author, because it will carry the watermark firstly embedded by the author. The earliest watermarked text in fact can be verified only by the original author following the above verification process. Fine-grain watermarking The presented watermark embedding method allows to watermark a text by embedding at the beginning of the text a unique information derived from the text itself together with a secret password. Considering that the underlying approach makes the method suitable for very short text, we extend the method to watermark longer text but in a fine-grained fashion. The ultimate goal is to keep authorship protection at paragraph or even lower levels, so that the text document can be protected from the copy of even a single sentence. As before, the watermark is first computed uniquely from the content of the whole text document and the secret password, by means of a secure hash function. This guarantees that it is not computationally possible to generate the same hash using a (i) different key, (ii) a different document, or (iii) both, because any of the previous would imply finding collisions by purpose [58]. In the embedding, Algorithm 1 presented for the simple case the embedding process would stop once that all the bits in the watermark have been embedded one time in the original text. In the fine-grain watermarking, instead, we keep embedding repeatedly the document watermark until no document characters remain. The process of fine-grain watermarking is shown in Algorithm 3. Starting from the first character of the text, Algorithm 3 first checks if the current character can be replaced (that is, if it has a corresponding homoglyph). If this is the case, the current bit of the watermark (or 3 bits for whitespaces) is taken as a replacing condition. As in the original algorithm, it iterates over the text replaceable characters and the watermark bits in parallel. However, instead of consuming the bits of the watermark, which is usually way shorter than the text, watermark's bits are taken in circle, in a string rotation setting. This is accomplished by simply applying the modulus of the index by the watermark length. The complexity of the proposed algorithm is $\mathcal {O}(n)$, linear in the number of characters of the original text. In the external loop of Algorithm 3, we scan the original text for each of the n characters, while the internal loop (in row 20 of Algorithm 3) has constant complexity, appending the 3 bits for whitespace replacement. The append operation for a single character, the GetSpace function to get the Unicode value from the 3 bits from Table 2 and the GetDuplicate function to get the homoglyph given the original letter from Table 1, can all be executed in constant time. In Fig. 4, we illustrate the watermarking process through a simplified example. Our document doc is a New York Times article that we want to fine-grain watermark. We use a toy 12-bit hash function, applying it to the concatenation of document and secret password. In Fig. 4, for easier readability, the 12-bit watermark is represented in hexadecimal notation by 3 digits, namely 0xABC. The resulting 12-bit watermark is embedded repeatedly in the text, following the replaceable characters. Fine-grain watermarking. The watermark, computed as the hash of the document and the secret key, is repeatedly embedded in smaller portions of text. For simplicity, a 12-bit watermark is shown, corresponding to the hexadecimal value 0xABC. Watermarked portions are alternately highlighted The watermarked excerpts of a text, alternately highlighted, have different lengths depending on the number and the type of replaceable characters found (e.g., spaces alone allow an embedding of 3 bits each). In fact, in order for a text excerpt to maintain the watermark, a variable length is required depending on the replaceable symbols in it: in the example of Fig. 4 with a 12-bit watermark, the portion "ow republicans -" is watermarked having only 17 characters, spaces included, while the portion "was brewing as lawmakers" is watermarked having 25 characters, spaces included. We will see in the evaluation section that the average symbols needed to embed a secure watermark of 64 bits is 101 characters. The sequential embedding of a watermark computed over the whole document in small excerpts of text confers two unique properties to excerpt watermarking: the watermarking is continuous, spanning across sub-portions of a text, and it is part-of-whole, because it binds each excerpt to the source document. Continuous watermarking Because the watermark is repeated sequentially across the document, it is possible to extract it from any sufficiently long excerpt of text, even if this is broken between two contiguous watermarks. For instance, in Fig. 4, the watermark 0xABC is repeated in the order A,B,C,A,B,C, etc. Extracting the watermark 0xABC from the first complete portion "A blacklash against" is straightforward using the presented Algorithm 2 for watermark extraction. However, this is also possible if the copied text portion is between two complete watermarks, as long as it has at least 12 bits of embedded watermarks: the watermark can be obtained by using a 12-bit window of embedded watermark and shifting the extracted binary string. In Fig. 5, a similar scenario is shown, in which a text "against President Trump -') between two complete watermarked portions has been copied. Applying the extraction Algorithm 2, the shifted watermark 0xCAB is obtained, which is then rotated until verification is reached or, after 11 rotation, verification is refused. Because any rotation of the original watermark is successfully verified, this verification process is shallower than the non-continuous version. Verifying as positive, any rotation of the hash reduces the search space for a verifying key; however, this is negligible for standard-sized hash. Continuous part-of-whole watermarking. Any sufficiently long excerpt of text will keep the watermark when copied, independently from its position in the document. Because its watermark is derived from the whole origin document, any watermarked excerpt of sufficient length is bind to its source Figure 6 shows a real example using a 64 bits watermark, one that can be obtained with SipHash, on the first paragraph of a New York Times article. Continuous lines define complete applications of a 64-bit watermark; however, any contiguous sequence with at least 64 bits is also watermarked, even if it overlaps two different complete watermarks. In the figure, the example excerpt "Australia is late to the space party. The leader of its new space agency, Megan Clark, said" is watermarked and can be verified even if it spans across two watermarks, because it has at least 64 bits embedded in it. Excerpt example from NYT article. The continuous lines limit complete 64 bits watermarks. Underlined with a dashed line is an example of verifiable excerpt spanning across two complete watermarks Part-of-whole watermarking Each text portion is considered as a part of the source document. The unique hidden information is derived from the source document and the secret password, and it is the same for all the portions. For the purpose of authorship verification, the source document is also needed along with the secret password, but it will prove that the portion is coming from that particular document. The feature is illustrated in Figs. 4 and 5, and it is the direct consequence of the fact that, while the hash is computed over the whole document, the short-length requirement in fine-grain watermarking embed this information in each small text portion. The described part-of-whole watermarking schema increase the security of the verification, as the original complete source is needed to verify the watermarked portion, resulting in a non-blind watermark. We conduct several experiments to assess the crucial properties of the proposed approach: the number of symbols required to embed a full watermark, the imperceptibility of changes in the watermarked text with respect to the original text, and the robustness of the watermark. Embedding capacity Because we can only embed watermark bits when a confusable symbol is found, the numbers of symbols needed depends on the number of confusables in the text. For this reason, this must be evaluated empirically on several texts. This symbol length requirement will directly affect the effectiveness of the fine-grain watermarking, because the shortest the length, the finest will be the watermarked portions of text. In order to carry a realistic length requirement estimation, we take into consideration the articles from the New York Times Corpus [14], a collection of 1.8 million articles spanning from 1987 to 2007, appeared in the New York Times newspaper. News article are an example of authored text commonly subject to unattributed copies in blogs and social media. Moreover, because we want to show the fine-grain method capability at the paragraph level, we extract from each article only the lead paragraphFootnote 3. In Table 3, we show the minimum and average obtained in the experiments results together with some ideal minimum boundaries: the most extreme case in which our original text has consequent whitespaces, allowing to embed 3 bits in each of them and resulting in 22 symbols needed on SipHash, and the less extreme but still uncommon scenario of one-character words separated by a whitespace, resulting in 33 symbols needed on SipHash. The average length obtain on NYT Corpus using the SipHash function is 101 characters, spaces included, while the minimum is 46 characters. Table 3 Text length required to embed a watermark The embedding capacity is computed as the average ratio between the number of embedded bits and the number of characters in each document. Considering the average number of characters needed to embed a full watermark of 64 bits, the expected embedding capacity is $\frac {64}{101}=0.632$ bits/character. We also show ideal and real values for other hash functions, for which the length requirement grow linearly to the length of the produced MAC: using MD5, with a MAC of 128 bits, the average on NYT Corpus is 198 characters. The maximum length requirement is not shown because the maximum length is bound by the length of each paragraph. More meaningfully instead, we show how many times the paragraph was too short and lacking of confusable symbols for the embedding to be successful at paragraph-level. In Fig. 7, we show what is the percentage of articles for which the watermarked excerpt is smaller than its lead paragraphs. Using a watermark of with a length of 64 bits, such as SipHash, the percentage stops at 94.51%, because in the remaining 5.49%, we are not able to successfully embed a watermark of 64 bits at paragraph-level; thus, the watermarked excerpt are bigger than a paragraph. The percentage of completely watermarked lead paragraphs decrease to 91.83% when we use a 128-bit watermark, 74.38% with a 160-bit hash and 63.97% using a 224-bit hash. Length requirements to successfully embed the watermark. Percentage of leading paragraphs in NYT Corpus that can be watermarked using only the first n characters. Results are shown for different hash lengths Figure 7 provides also a statistical estimation of the probability of watermarking at an excerpt of n characters. For example, it shows that an excerpt of 100 characters have a 50% probability of being successfully watermarked with SipHash that increases to 80% if the excerpt is 108 characters length. We compare the embedding capacity of the proposed method with other recent steganography and text watermarking structural methods. In Table 4, the embedding capacity values for each of the considered methods are provided. Together with the capacity, we show also the overhead introduced by the embedding in terms of additional characters. Table 4 Embedding capacity and overhead comparison Our method exhibits higher capacity with respect to all the other methods except for [59], where the capacity is only limited by the maximum length of the text and by the cover message, since an arbitrary number of invisible characters are inserted in the beginning of the text. However, our method is the only one that does not introduce any overhead of characters in the embedding, since each character is replaced with another character. Indistinguishability of watermarked text Our watermark embedding process works by replacing common symbols of latin alphabets, punctuations, and whitespaces with similar Unicode symbols. Because whitespaces have slightly different width and confusable symbols are not always identical to common ones in all Unicode-supporting fonts, it is crucial to assess in practice how much these differences are noticeable. In Fig. 8, we show an example of a paragraph in its original and watermarked version. The watermarked version had several of the original symbols replaced by confusable symbols. Overall, the example does not show any noticeable differences to the human eye: main difference regards the horizontal spacing, because the width of the confusable whitespaces affect the words' position and ultimately the width of each line. Considering also that in a common scenario, the attacker does not have both original and watermarked version for comparison, we argue that the differences in paragraph level are not perceivable. Graphic rendering of original and watermarked. A lead paragraph from a New York Time article before watermarking (a) and after watermarking (b) Besides paragraph-level differences, we compare each symbol with its related duplicate in well-known fonts. In Fig. 9, all the symbols are rendered in three sans-serif fonts used in the most known web platforms, including Facebook, Twitter, and Google. All the symbols show no noticeable differences in rendering when their duplicate is used instead of their common version. The reader can refer to Fig. 11 in the Appendix A for a more extensive results on ten fonts. Confusable symbols rendering in web fonts. Original (Or.) and duplicate (Du.) symbols rendering for the three most known web fonts It must be noted that changing the font of the watermarked text cannot affect its content (i.e., the Unicode symbols) but only its appearance; thus, the watermark is retained with any change of font. In fact, the watermarked text—being a string of Unicode symbols—is not bound to a particular font and has no font embedded in it. If a malicious user copies the watermarked text in a word editor and changes the font to any other font, the watermark is still retained. On the other hand, if a font with a smaller or no support of the Unicode set is used, this is still rendered but the indistinguishability of the non-supported symbols is not guaranteed. This is because all modern software applications, including word processors and web browsers, automatically render the unsupported symbol using a similar font that supports it, in a cascading mechanism [60]. We evaluate the visual indistinguishability of the proposed method in comparison with two structural steganography techniques, Whitesteg [55] and Unispach [45]. Contrary to the proposed method, these two methods work by inserting additional invisible characters, so that while the readable content is preserved, they add a data overhead, resulting also in horizontal shifting due to whitespaces added between the words. Figure 10 shows the overlapping between original text (in red) and watermarked text (in black) for a sample text. It is easily noticeable how the use of double spaces in Whitesteg and Unispach completely shift the alignment. Conversely, the proposed method does not use multiple, consequent whitespaces or append whitespaces at the end or beginning of the text. Fig. 10 Indistinguishability test. Overlapping original and watermarked text for content-preserving methods shows less noticeable changes with proposed method We do not evaluate structural methods that are completely invisible [49, 59], as there is no visual change after the watermark is embedded. On the other hand, by adding new invisible symbols to the text instead of replacing the symbols, the invisible methods still introduce an overhead that increases the overall text size, a change that can be detected using any text editor or file manager. We evaluate the robustness of our approach together with a set of recent structural watermarking techniques for comparison. No exact means are available to measure the general robustness of text watermarks [54]; however, it is possible to measure the robustness against different attacks. The following attacks are commonly considered in assessing the robustness of text watermarking methods. Copy and paste. A very common scenario where the content of a text file is copied and pasted into an attacker's file. Reformatting. Reformatting attacks including the change of formatting features of the text such as fonts or color. Copy and paste, retyping, and OCR have been also considered subtypes of formatting attacks [54]. Our method is robust against any change of font type, size, or color, because any formatting process leaves the Unicode content unaltered. Insertion. Insertion attacks randomly new words in the watermarked text [61], with the goal of altering the watermark. We randomly insert the 10% of additional words, following a common attack size considered high [62]. Deletion. In this attack, some parts of the text are removed [54]. If the deleted portion is part of the watermark, the watermark may be destroyed or completely removed [61]. As for the insertion attack, we randomly remove the 10% of the words. Replacement. In a replacement attack, a set of words in the watermarked text are replaced with other words [54, 61]. It can be considered as a deletion attack followed by an insertion attack in the same location. We test the robustness against a replacement attack [62], where 10% of the words in the watermarked text are replaced with other words. The word inserted as replacement is randomly chosen from a word list of 10,000 words. Retyping. In a retyping attack, a malicious user retype the text in a different file or platform. Structural methods are all fragile to retyping by definition. In fact, because the structural methods embed the watermark by altering the layout such as formatting features, spaces, Unicode, and ASCII encodings without altering the content, retyping the content will always destroys a watermark. Robustness to partial copy and paste Selection of text is a natural—and in most software obligatory—step involved in the copy and paste process. In web pages, for example, users select specific sub-portions of the page to avoid copying advertisement images, logos, and navigation menus. More generally, it is common to select and copy only the portions of interests from a document. Copying a portion of a text document can be also considered a special case of deletion attack, in which all characters before and after the portion are removed. Fore example, in the case of attackers trying to remove the authors' names from a text [61], they may instead just select the text without the authors' information. In order to measure the robustness against the partial copy attack, a scenario that also motivated the design of our fine-grain algorithm, we propose an additional measure. Note that it is not possible to know a priori which portion of a text the attacker will copy and paste nor it is possible to know a priori if a portion of some length will be protected: depending on the characters or spaces in the specific portion, a complete watermark may or may not be embedded in it. For these reasons, the only way to measure the robustness is empirically by assessing the robustness of each possible portion in a given text, for a large enough number of texts. Because a digital text is a string of symbols, a naive way of measuring this robustness is by counting how many substrings out of all the possible substrings are still protected when extracted through copy and pasting. However, current laws do not allow to claim the copyright of small groups of words, let alone symbols, as this could set up barrier to expression [63], despite there are some cases where microworks of 16 words, or quotes from larger works (e.g., movie scripts) have been copyrighted [12] given their strong originality. Our metric therefore considers subsequences of words instead of substrings of characters and is parameterized on a number of words z. Here, we use the indicator function $\mathbb {I}\{\cdot \}$ that returns 0 or 1 if the condition is false or true, respectively. Given that T is the whole text as a sequence of words, S⊆T is a portion of T, and the number of protected portions of z words is: $$\begin{array}{{20}l} T_{\text{protected}}(z)=\sum\limits_{S \subseteq T, \|S\|= z} \mathbb{I}\{E_{T,k}(S) = E_{T,k}(T)\} \end{array} $$ Then, the number of all possible subsequences with z words in a text of \|T\| words is: $$\begin{array}{{20}l} T_{\text{all}}(z)&=\|T\|-z+1 \end{array} $$ Therefore, the robustness to partial copy and paste RPCP is: $$\begin{array}{{20}l} R_{\text{PCP}}(T,z)&= \frac{T_{\text{protected}}}{T_{\text{all}}} \end{array} $$ Less formally, the above definition of RPCP measures the fraction of possible small words sequences with z words that are protected when isolated from their original content T. This definition also estimates the probability of having any subsequence S protected against partial copy. In fact, let a malicious user selects S among any of the sequences of the watermarked text T, where S is unknown to the author, and assuming that the selection happens with uniform probability, then RPCP is also the probability of S to retain the watermark, that is, P(ET,k(S)=ET,k(T)). In the evaluation, we use z=32, as 32 words is considered a minimal group of words that can be copyrighted given enough originality [63]. Some of the evaluation tests are computationally intensive, such as counting word frequencies for the geometric advanced replacement or extracting all possible substring; therefore, we draw a random sample of 1000 articles from the NYT Corpus [14]. In the sample, the minimum character length in an article is 197, maximum is 57,136 and the average is 4213. We extracted words from the article using the regular expression ([\w][\w]'?\w?), to also capture words with genitive such as Tom's as a single word, resulting in an average of 676 words per document, with a minimum of 31 and a maximum of 9375 words. We implemented embedding and extraction for the methods under comparison following the original published descriptions. An exception regards the implementation of the algorithm in [49], where we repeatedly embed the watermark after each dot (.) only when this is followed by a new line; otherwise, the invisible control characters used have the effect of reversing the following string. For the steganography techniques considered, the results are computed by embedding a 64-bit message. For each method, the payload is embedded in the document using the implemented embedding method. In the RPCP evaluation, each possible portion of 32 words starting from the first in the article is copied. Then, the extraction method is ran on the portion: if the original payload is extracted, then $\mathbb {I}\{E_{T,k}(S) = E_{T,k}(T)\}=1$, i.e., the portion will increase the number of Tprotected in Eq. 3; otherwise $\mathbb {I}\{E_{T,k}(S) = E_{T,k}(T)\}=0$. This assessment is done for each portion until the end of the article. The RPCP is computed for each article in the sample, and the average overall articles is shown in Table 5 together with the other measures. Table 5 Robustness evaluation and comparison In Table 5, we show the results of the evaluation. Overall, our method shows robustness results higher or similar to other methods for all the attacks. Methods which embed the watermark in non-word locations, such as whitespaces [45], after the dot (.) or in the end of the text [49, 59] show high or total robustness when words are attacked through insertion, deletion, and replacement. On the other hand, because the watermark is embedded in isolated locations outside the text, they are more easy to lose when partial text is selected, thus performing poorly with respect to the proposed fine-grain method. The combination of a high embedding capacity, the repeated embedding, and the verification method through binary string shifting confers a very high copy and paste robustness to our method, which provides an almost total protection for groups of 32 words. The protection of intellectual property of digital contents from plagiarism and unauthorized copy has become a challenging research problem, worsened by the ease of selecting, copying, and sharing other people's content. While text watermarking methods aimed at the protection of entire documents, the fine-grain protection of creative work from partial copy is a difficult and common phenomenon faced in copyright law. In this paper, we have presented a fine-grain text watermarking method able to embed a password-based watermark in latin-based alphabet texts at paragraph level. By not relying on isolated text locations and having a discrete embedding capacity, the watermark is embedded repeatedly and thoroughly in a fine-grain fashion, so that it can be extracted even when only a small portion is copied. The core of the method is the watermark embedding through confusable symbol replacement: Unicode symbols very similar or identical to common symbols are replaced following the content of the watermark. The watermark is generated using a keyed hash function, binding the watermark to the author's secret key and the original text. The method does not modify the readable content of the text, producing a watermarked text that is visually indistinguishable from the original text. Moreover, it is the first method able to embed data without introducing overhead. From an extensive experiment on 1.8 million documents, the resulted average embedding capacity of the embedding method is 0.632 bits/character, while the fine-grain algorithm is able to protect text at paragraph level for the 94.5% of the times. This features allows the first fine-grain protection of text through continuous, part-of-whole watermarking. For any sufficiently long portion of a text, the proposed method allows authorship verification and source traceability, binding all the portions to its original source document, and protection against partial copy and paste. Along with well-known attacks on words, we tested the algorithm for partial copy and paste, for which we proposed a novel measure of robustness. The algorithm has shown the highest robustness against partial copy and paste, protecting 99.92% of 32 words sequences, and high robustness to other attacks. Despite partial copy and paste is a natural activity, we have found that structural techniques are more fragile to it than to intentionally malicious attacks such as random insertion, deletion, and replacement of words. This is because, by keeping only a sequence of words, several locations commonly used for embedding are left out, such as whitespaces and hidden control sequences before and after punctuations. On the other hand, because our method embed the watermark in a fine-grained way thorough all the words, it is slightly less robust to word insertion, deletion, and replacement than other methods [49, 59]. In this context, we also noted how all structural watermarking methods are fragile to the simple retyping attack, allowing an attacker to remove the watermark with ease and without any knowledge of the underlying embedding algorithm or watermark embedding locations. This should be taken into account when assessing the robustness against more complex malicious attacks. 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Qatar Computing Research Institute (QCRI) HBKU, Doha, Qatar Stefano Giovanni Rizzo Department of Computer Science and Engineering, University of Bologna, Mura Anteo Zamboni, 7, Bologna, 40126, Italy Flavio Bertini & Danilo Montesi Flavio Bertini Danilo Montesi SGR is the corresponding author of the paper and is responsible for all aspects of the paper. FB did the part of the experiment. All authors wrote, read, and approved the final manuscript. Correspondence to Stefano Giovanni Rizzo. The authors declare that they have no competing interests and that there is no conflict of interest regarding the publication of this paper. Rizzo, S., Bertini, F. & Montesi, D. Fine-grain watermarking for intellectual property protection. EURASIP J. on Info. Security 2019, 10 (2019). https://doi.org/10.1186/s13635-019-0094-2 Digital text watermarking Unicode characters Tampering detection	CommonCrawl
Zero-point energy For related articles, see Quantum vacuum (disambiguation). Not to be confused with Zero Point (photometry). For other uses, see Zero point (disambiguation). Liquid helium retains kinetic energy and does not freeze regardless of temperature due to zero-point energy. When cooled below its Lambda point, it exhibits properties of superfluidity Zero-point energy (ZPE) is the difference between the lowest possible energy that a quantum mechanical system may have, and the classical minimum energy of the system. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state due to the Heisenberg uncertainty principle.[1] As well as atoms and molecules, the empty space of the vacuum has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions (i.e. leptons and quarks), and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy.[2] These fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics,[1][3] since some systems can detect the existence of this energy. However this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity.[1] Physics currently lacks a full theoretical model for understanding zero-point energy; in particular the discrepancy between theorized and observed vacuum energy is a source of major contention.[4] Physicists Richard Feynman and John Wheeler calculated the zero-point radiation of the vacuum to be an order of magnitude greater than nuclear energy, with a single light bulb containing enough energy to boil all the world's oceans.[5] Yet according to Einstein's theory of general relativity any such energy would gravitate[citation needed] and the experimental evidence from both the expansion of the universe, dark energy and the Casimir effect show any such energy to be exceptionally weak. A popular proposal that attempts to address this issue is to say that the fermion field has a negative zero-point energy while the boson field has positive zero-point energy and thus these energies somehow cancel each other out.[6][7] This idea would be true if supersymmetry were an exact symmetry of nature. However, the LHC at CERN has so far found no evidence to support supersymmetry. Moreover, it is known that if supersymmetry is valid at all, it is at most a broken symmetry, only true at very high energies, and no one has been able to show a theory where zero-point cancellations occur in the low energy universe we observe today.[7] This discrepancy is known as the cosmological constant problem and it is one of the greatest unsolved mysteries in physics. Many physicists believe that "the vacuum holds the key to a full understanding of nature".[8] Etymology and terminologyEdit The term zero-point energy (ZPE) is a translation from the German Nullpunktsenergie.[9] The terms zero-point radiation or ground state energy are also sometimes used interchangeably. The term zero-point field (ZPF) can be used when referring to a specific vacuum field, for instance the QED vacuum which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) or the QCD vacuum which deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). A vacuum can be viewed not as empty space but as the combination of all zero-point fields. In quantum field theory this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average energy value is called the vacuum expectation value (VEV) also called its condensate. Kinetic energy vs temperature. In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy. Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy (known as brownian motion). As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come completely to rest. In fact, however, kinetic energy is retained by particles even at the lowest possible temperature. The random motion corresponding to this zero-point energy never vanishes as a consequence of the uncertainty principle of quantum mechanics. Zero-point radiation continually imparts random impulses on an electron, so that it never comes to a complete stop. Zero-point radiation gives the oscillator an average energy equal to the frequency of oscillation multiplied by one-half of Planck's constant The uncertainty principle states that no object can ever have precise values of position and velocity simultaneously. The total energy of a quantum mechanical object (potential and kinetic) is described by its Hamiltonian which also describes the system as a harmonic oscillator, or wave function, that fluctuates between various energy states (see wave-particle duality). All quantum mechanical systems undergo fluctuations even in their ground state, a consequence of their wave-like nature. The uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical potential well. This results in motion even at absolute zero. For example, liquid helium does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy. Given the equivalence of mass and energy expressed by Einstein's E = mc2, any point in space that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed quantum field theory (QFT) to understand the fundamental interactions between matter and forces, it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields, whose quanta are fermions (i.e. leptons and quarks), and force fields, whose quanta are bosons (e.g. photons and gluons). All these fields have zero-point energy.[2] Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions of the zero-point field.[10] The idea that "empty" space can have an intrinsic energy associated to it, and that there is no such thing as a "true vacuum" is seemingly unintuitive. It is often argued that the entire universe is completely bathed in the zero-point radiation, and as such it can add only some constant amount to calculations. Physical measurements will therefore reveal only deviations from this value.[11] For many practical calculations zero-point energy is dismissed by fiat in the mathematical model as a term that has no physical effect. Such treatment causes problems however, as in Einstein's theory of general relativity the absolute energy value of space is not an arbitrary constant and gives rise to the cosmological constant. For decades most physicists assumed that there was some undiscovered fundamental principle that will remove the infinite zero-point energy and make it completely vanish. If the vacuum has no intrinsic, absolute value of energy it will not gravitate. It was believed that as the universe expands from the aftermath of the Big Bang, the energy contained in any unit of empty space will decrease as the total energy spreads out to fill the volume of the universe; galaxies and all matter in the universe should begin to decelerate. This possibility was ruled out in 1998 by the discovery that the expansion of the universe is not slowing down but is in fact accelerating, meaning empty space does indeed have some intrinsic energy. The discovery of dark energy is best explained by zero-point energy, though it still remains a mystery as to why the value appears to be so small compared to huge value obtained through theory - the cosmological constant problem.[6] Many physical effects attributed to zero-point energy have been experimentally verified, such as spontaneous emission, Casimir force, Lamb shift, magnetic moment of the electron and Delbrück scattering.[12][13] These effects are usually called "radiative corrections".[14] In more complex nonlinear theories (e.g. QCD) zero-point energy can give rise to a variety of complex phenomena such as multiple stable states, symmetry breaking, chaos and emergence. Many physicists believe that "the vacuum holds the key to a full understanding of nature"[8] and that studying it is critical in the search for the theory of everything. Active areas of research include the effects of virtual particles,[15] quantum entanglement,[16] the difference (if any) between inertial and gravitational mass,[17] variation in the speed of light,[18] a reason for the observed value of the cosmological constant[19] and the nature of dark energy.[20][21] Early aether theoriesEdit Zero-point energy evolved from historical ideas about the vacuum. To Aristotle the vacuum was τὸ κενόν, "the empty"; space independent of body. He believed this concept violated basic physical principles and asserted that the elements of fire, air, earth, and water were not made of atoms, but were continuous. To the atomists the concept of emptiness had absolute character: it was the distinction between existence and nonexistence.[22] Debate about the characteristics of the vacuum were largely confined to the realm of philosophy, it was not until much later on with the beginning of the renaissance, that Otto von Guericke invented the first vacuum pump and the first testable scientific ideas began to emerge. It was thought that a totally empty volume of space could be created by simply removing all gases. This was the first generally accepted concept of the vacuum.[23] Late in the 19th century, however, it became apparent that the evacuated region still contained thermal radiation. The existence of the aether as a substitute for a true void was the most prevalent theory of the time. According to the successful electromagnetic aether theory based upon Maxwell's electrodynamics, this all-encompassing aether was endowed with energy and hence very different from nothingness. The fact that electromagnetic and gravitational phenomena were easily transmitted in empty space indicated that their associated aethers were part of the fabric of space itself. Maxwell himself noted that: To those who maintained the existence of a plenum as a philosophical principle, nature's abhorrence of a vacuum was a sufficient reason for imagining an all-surrounding aether... Aethers were invented for the planets to swim in, to constitute electric atmospheres and magnetic effluvia, to convey sensations from one part of our bodies to another, and so on, till a space had been filled three or four times with aethers.[24] However, the results of the Michelson–Morley experiment in 1887 were the first strong evidence that the then-prevalent aether theories were seriously flawed, and initiated a line of research that eventually led to special relativity, which ruled out the idea of a stationary aether altogether. To scientists of the period, it seemed that a true vacuum in space might be completely eliminated by cooling thus eliminating all radiation or energy. From this idea evolved the second concept of achieving a real vacuum: cool it down to absolute zero temperature after evacuation. Absolute zero was technically impossible to achieve in the 19th century, so the debate remained unsolved. Second quantum theoryEdit Planck in 1918, the year he received the Nobel Prize in Physics for his work on quantum theory In 1900, Max Planck derived the average energy ε of a single energy radiator, e.g., a vibrating atomic unit, as a function of absolute temperature:[25] ε = h ν e h ν / ( k T ) − 1 , {\displaystyle \varepsilon ={\frac {h\nu }{e^{h\nu /(kT)}-1}}\,,} where h is Planck's constant, ν is the frequency, k is Boltzmann's constant, and T is the absolute temperature. The zero-point energy makes no contribution to Planck's original law, as its existence was unknown to Planck in 1900.[26] The concept of zero-point energy was developed by Max Planck in Germany in 1911 as a corrective term added to a zero-grounded formula developed in his original quantum theory in 1900.[27] In 1912, Max Planck published the first journal article[28] to describe the discontinuous emission of radiation, based on the discrete quanta of energy. In Planck's "second quantum theory" resonators absorbed energy continuously, but emitted energy in discrete energy quanta only when they reached the boundaries of finite cells in phase space, where their energies became integer multiples of hν. This theory led Planck to his new radiation law, but in this version energy resonators possessed a zero-point energy, the smallest average energy a resonator could take on. Planck's radiation equation contained a residual energy factor, one hν/2, as an additional term dependent on the frequency ν, which was greater than zero (where h is Planck's constant). It is therefore widely agreed that "Planck's equation marked the birth of the concept of zero-point energy."[29] In a series of papers from 1911 to 1913,[30] Planck found that the average energy of an oscillator to be:[27][31] ε = h ν 2 + h ν e h ν / ( k T ) − 1 . {\displaystyle \varepsilon ={\frac {h\nu }{2}}+{\frac {h\nu }{e^{h\nu /(kT)}-1}}~.} Einstein's official 1921 portrait after receiving the Nobel Prize in Physics Soon, the idea of zero-point energy attracted the attention of Albert Einstein and his assistant Otto Stern.[32] In 1913 they published a paper that attempted to prove the existence of zero-point energy by calculating the specific heat of hydrogen gas and compared it with the experimental data. However, after assuming they had succeeded, they retracted support for the idea shortly after publication because they found Planck's second theory may not apply to their example. In a letter to Paul Ehrenfest of the same year Einstein declared zero-point energy "dead as a doornail"[33] Zero-point energy was also invoked by Peter Debye,[34] who noted that zero-point energy of the atoms of a crystal lattice would cause a reduction in the intensity of the diffracted radiation in X-ray diffraction even as the temperature approached absolute zero. In 1916 Walther Nernst proposed that empty space was filled with zero-point electromagnetic radiation.[35] With the development of general relativity Einstein found the energy density of the vacuum to contribute towards a cosmological constant in order to obtain static solutions to his field equations; the idea that empty space, or the vacuum, could have some intrinsic energy associated to it had returned, with Einstein stating in 1920: There is a weighty argument to be adduced in favour of the aether hypothesis. To deny the aether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view... according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an aether. According to the general theory of relativity space without aether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this aether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.[36][37] Heisenberg, 1924 Kurt Bennewitz and Francis Simon (1923)[38] who worked at Walther Nernst's laboratory in Berlin, studied the melting process of chemicals at low temperatures. Their calculations of the melting points of hydrogen, argon and mercury led them to conclude that the results provided evidence for a zero-point energy. Moreover, they suggested correctly, as was later verified by Simon (1934),[39][40] that this quantity was responsible for the difficulty in solidifying helium even at absolute zero. In 1924 Robert Mulliken[41] provided direct evidence for the zero-point energy of molecular vibrations by comparing the band spectrum of 10BO and 11BO: the isotopic difference in the transition frequencies between the ground vibrational states of two different electronic levels would vanish if there were no zero-point energy, in contrast to the observed spectra. Then just a year later in 1925,[42] with the development of matrix mechanics in Werner Heisenberg's famous article "Quantum theoretical re-interpretation of kinematic and mechanical relations" the zero-point energy was derived from quantum mechanics.[43]:162 In 1913 Niels Bohr had proposed what is now called the Bohr model of the atom,[44][45][46] but despite this it remained a mystery as to why electrons do not fall into their nuclei. According to classical ideas, the fact that an accelerating charge loses energy by radiating implied that an electron should spiral into the nucleus and that atoms should not be stable. This problem of classical mechanics was nicely summarized by James Hopwood Jeans in 1915: "There would be a very real difficulty in supposing that the (force) law 1/r2 held down to the zero values of r. For the forces between two charges at zero distance would be infinite; we should have charges of opposite sign continually rushing together and, when once together, no force would tend to shrink into nothing or to diminish indefinitely in size"[47] This resolution to this puzzle came in 1926 with Schrödinger's famous equation.[48] This equation explained the new, non-classical, fact that as an electron moves close to a nucleus its kinetic energy necessarily increases in such a way that the minimum total energy (kinetic plus potential) occurs at some positive separation rather than at zero separation; in other words, that zero-point energy is essential for atomic stability.[49] Quantum field theory and beyondEdit In 1926 Pascual Jordan[50] published the first attempt to quantize the electromagnetic field. In a joint paper with Max Born and Werner Heisenberg he considered the field inside a cavity as a superposition of quantum harmonic oscillators. In his calculation he found that in addition to the "thermal energy" of the oscillators there also had to exist infinite zero-point energy term. He was able to obtain the same fluctuation formula that Einstein had obtained in 1909.[51] However, Jordan did not think that his infinite zero-point energy term was "real", writing to Einstein that "it is just a quantity of the calculation having no direct physical meaning"[52] Jordan found a way to get rid of the infinite term, publishing a joint work with Pauli in 1928,[53] performing what has been called "the first infinite subtraction, or renormalisation, in quantum field theory"[54] Paul Dirac 1933 Building on the work of Heisenberg and others Paul Dirac's theory of emission and absorption (1927)[55] was the first application of the quantum theory of radiation. Dirac's work was seen as crucially important to the emerging field of quantum mechanics; it dealt directly with the process in which "particles" are actually created: spontaneous emission.[56] Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. The theory showed that spontaneous emission depends upon the zero-point energy fluctuations of the electromagnetic field in order to get started.[57][58] In a process in which a photon is annihilated (absorbed), the photon can be thought of as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state. In the words of Dirac:[59] The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light quanta in the zero state... Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field. This view was popularized by Victor Weisskopf who in 1935 wrote:[60] From quantum theory there follows the existence of so called zero-point oscillations; for example each oscillator in its lowest is not completely at rest but always is moving about its equilibrium position. Therefore electromagnetic oscillations also can never cease completely. Thus the quantum nature of the electromagnetic field has as its consequence zero point oscillations of the field strength in the lowest energy state, in which there are no light quanta in space... The zero point oscillations act on an electron in the same way as ordinary electrical oscillations do. They can change the eigenstate of the electron, but only in a transition to a state with the lowest energy, since empty space can only take away energy, and not give it up. In this way spontaneous radiation arises as a consequence of the existence of these unique field strengths corresponding to zero point oscillations. Thus spontaneous radiation is induced radiation of light quanta produced by zero point oscillations of empty space This view was also later supported by Theodore Welton (1948),[61] who argued that spontaneous emission "can be thought of as forced emission taking place under the action of the fluctuating field." This new theory, which Dirac coined quantum electrodynamics (QED) predicted a fluctuating zero-point or "vacuum" field existing even in the absence of sources. Throughout the 1940s improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom, now known as the Lamb shift,[62] and measurement of the magnetic moment of the electron.[63] Discrepancies between these experiments and Dirac's theory led to the idea of incorporating renormalisation into QED to deal with zero-point infinities. Renormalization was originally developed by Hans Kramers[64] and also Victor Weisskopf(1936),[65] and first successfully applied to calculate a finite value for the Lamb shift by Hans Bethe (1947).[66] As per spontaneous emission, these effects can in part be understood with interactions with the zero-point field.[67][12] But in light of renormalisation being able to remove some zero-point infinities from calculations, not all physicists were comfortable attributing zero-point energy any physical meaning, viewing it instead as a mathematical artifact that might one day be fully eliminated. In Wolfgang Pauli's 1945 Nobel lecture[68] he made clear his opposition to the idea of zero-point energy stating "It is clear that this zero-point energy has no physical reality". Hendrik Casimir (1958) In 1948 Hendrik Casimir[69][70] showed that one consequence of the zero-point field is an attractive force between two uncharged, perfectly conducting parallel plates, the so-called Casimir effect. At the time, Casimir was studying the properties of "colloidal solutions". These are viscous materials, such as paint and mayonnaise, that contain micron-sized particles in a liquid matrix. The properties of such solutions are determined by van der Waals forces – long-range, attractive forces that exist between neutral atoms and molecules. One of Casimir's colleagues, Theo Overbeek, realized that the theory that was used at the time to explain van der Waals forces, which had been developed by Fritz London in 1930,[71][72] did not properly explain the experimental measurements on colloids. Overbeek therefore asked Casimir to investigate the problem. Working with Dirk Polder, Casimir discovered that the interaction between two neutral molecules could be correctly described only if the fact that light travels at a finite speed was taken into account.[73] Soon afterwards after a conversation with Bohr about zero-point energy, Casimir noticed that this result could be interpreted in terms of vacuum fluctuations. He then asked himself what would happen if there were two mirrors – rather than two molecules – facing each other in a vacuum. It was this work that led to his famous prediction of an attractive force between reflecting plates. The work by Casimir and Polder opened up the way to a unified theory of van der Waals and Casimir forces and a smooth continuum between the two phenomena. This was done by Lifshitz (1956)[74][75][76] in the case of plane parallel dielectric plates. The generic name for both van der Waals and Casimir forces is dispersion forces, because both of them are caused by dispersions of the operator of the dipole moment.[77] The role of relativistic forces becomes dominant at orders of a hundred nanometers. In 1951 Herbert Callen and Theodore Welton[78] proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)[79] as an explanation for observed Johnson noise in electric circuits.[80] Fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work.[81] FDT has been shown to be true experimentally under certain quantum, non-classical, conditions.[82][83][84] In 1963 the Jaynes–Cummings model[85] was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave nonintuitive predictions such as that an atom's spontaneous emission could be driven by field of effectively constant frequency (Rabi frequency). In the 1970s experiments were being performed to test aspects of quantum optics and showed that the rate of spontaneous emission of an atom could be controlled using reflecting surfaces.[86][87] These results were at first regarded with suspicion in some quarters: it was argued that no modification of a spontaneous emission rate would be possible, after all, how can the emission of a photon be affected by an atom's environment when the atom can only "see" its environment by emitting a photon in the first place? These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections. Spontaneous emission can be suppressed (or "inhibited")[88][89] or amplified. Amplification was first predicted by Purcell in 1946[90] (the Purcell effect) and has been experimentally verified.[91] This phenomenon can be understood, partly, in terms of the action of the vacuum field on the atom.[92] The uncertainty principleEdit Main article: Uncertainty principle Zero-point energy is fundamentally related to the Heisenberg uncertainty principle.[93] Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well: for, then, its position and momentum would both be completely determined to arbitrarily great precision. Therefore, instead, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle−−which implies its energy must be greater than the minimum of the potential well. Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator, H ^ = V 0 + 1 2 k ( x ^ − x 0 ) 2 + 1 2 m p ^ 2 , {\displaystyle {\hat {H}}=V_{0}+{\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}+{\frac {1}{2m}}{\hat {p}}^{2}\,,} where V0 is the minimum of the classical potential well. The uncertainty principle tells us that ⟨ ( x ^ − x 0 ) 2 ⟩ ⟨ p ^ 2 ⟩ ≥ ℏ 2 , {\displaystyle {\sqrt {\left\langle \left({\hat {x}}-x_{0}\right)^{2}\right\rangle }}{\sqrt {\left\langle {\hat {p}}^{2}\right\rangle }}\geq {\frac {\hbar }{2}}\,,} making the expectation values of the kinetic and potential terms above satisfy ⟨ 1 2 k ( x ^ − x 0 ) 2 ⟩ ⟨ 1 2 m p ^ 2 ⟩ ≥ ( ℏ 4 ) 2 k m . {\displaystyle \left\langle {\tfrac {1}{2}}k\left({\hat {x}}-x_{0}\right)^{2}\right\rangle \left\langle {\frac {1}{2m}}{\hat {p}}^{2}\right\rangle \geq \left({\frac {\hbar }{4}}\right)^{2}{\frac {k}{m}}\,.} The expectation value of the energy must therefore be at least ⟨ H ^ ⟩ ≥ V 0 + ℏ 2 k m = V 0 + ℏ ω 2 {\displaystyle \left\langle {\hat {H}}\right\rangle \geq V_{0}+{\frac {\hbar }{2}}{\sqrt {\frac {k}{m}}}=V_{0}+{\frac {\hbar \omega }{2}}} where ω = √k/m is the angular frequency at which the system oscillates. A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly E0 = V0 + ħω/2, requires solving for the ground state of the system. Atomic physicsEdit Main article: Ground state The zero-point energy E = ħω/2 causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed. The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the ground state of the system. The professional physics literature tends to measure frequency, as denoted by ν above, using angular frequency, denoted with ω and defined by ω = 2πν. This leads to a convention of writing Planck's constant h with a bar through its top (ħ) to denote the quantity h/2π. In these terms, the most famous such example of zero-point energy is the above E = ħω/2 associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the expectation value of the Hamiltonian of the system in the ground state. If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a unitary operator which acts non-trivially on a ground state and commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit negative temperature. The wave function of the ground state of a particle in a one-dimensional well is a half-period sine wave which goes to zero at the two edges of the well. The energy of the particle is given by: h 2 n 2 8 m L 2 {\displaystyle {\frac {h^{2}n^{2}}{8mL^{2}}}} where h is the Planck constant, m is the mass of the particle, n is the energy state (n = 1 corresponds to the ground-state energy), and L is the width of the well. Quantum field theoryEdit Main articles: Vacuum expectation value, Vacuum energy, and Vacuum state In quantum field theory (QFT), the fabric of "empty" space is visualized as consisting of fields, with the field at every point in space and time being a quantum harmonic oscillator, with neighboring oscillators interacting with each other. According to QFT the universe is made up of matter fields whose quanta are fermions (e.g. electrons and quarks) and force fields, whose quanta are bosons (i.e. photons and gluons). All these fields have zero-point energy.[2] A related term is zero-point field (ZPF), which is the lowest energy state of a particular field.[94] The vacuum can be viewed not as empty space, but as the combination of all zero-point fields. In QFT this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average expectation value of the Hamiltonian is called the vacuum expectation value (also called condensate or simply VEV). The QED vacuum is a part of the vacuum state which specifically deals with quantum electrodynamics (e.g. electromagnetic interactions between photons, electrons and the vacuum) and the QCD vacuum deals with quantum chromodynamics (e.g. color charge interactions between quarks, gluons and the vacuum). Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, and that all properties of matter are merely vacuum fluctuations arising from interactions with the zero-point field.[10] Each point in space makes a contribution of E = ħω/2, resulting in a calculation of infinite zero-point energy in any finite volume; this is one reason renormalization is needed to make sense of quantum field theories. In cosmology, the vacuum energy is one possible explanation for the cosmological constant[95] and the source of dark energy.[20][21] Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy. Other scientists specializing in General Relativity require the energy to be small enough for curvature of space to agree with observed astronomy. The Heisenberg uncertainty principle allows the energy to be as large as needed to promote quantum actions for a brief moment of time, even if the average energy is small enough to satisfy relativity and flat space. To cope with disagreements, the vacuum energy is described as a virtual energy potential of positive and negative energy.[96] In quantum perturbation theory, it is sometimes said that the contribution of one-loop and multi-loop Feynman diagrams to elementary particle propagators are the contribution of vacuum fluctuations, or the zero-point energy to the particle masses. The quantum electrodynamic vacuumEdit Main article: QED vacuum The oldest and best known quantized force field is the electromagnetic field. Maxwell's equations have been superseded by quantum electrodynamics (QED). By considering the zero-point energy that arises from QED it is possible to gain a characteristic understanding of zero-point energy that arises not just through electromagnetic interactions but in all quantum field theories. Redefining the zero of energyEdit In the quantum theory of the electromagnetic field, classical wave amplitudes α and α* are replaced by operators a and a† that satisfy: [ a , a † ] = 1 {\displaystyle \left[a,a^{\dagger }\right]=1} The classical quantity \|α\|2 appearing in the classical expression for the energy of a field mode is replaced in quantum theory by the photon number operator a†a. The fact that: [ a , a † a ] ≠ 1 {\displaystyle \left[a,a^{\dagger }a\right]\neq 1} implies that quantum theory does not allow states of the radiation field for which the photon number and a field amplitude can be precisely defined, i.e., we cannot have simultaneous eigenstates for a†a and a. The reconciliation of wave and particle attributes of the field is accomplished via the association of a probability amplitude with a classical mode pattern. The calculation of field modes is entirely classical problem, while the quantum properties of the field are carried by the mode "amplitudes" a† and a associated with these classical modes. The zero-point energy of the field arises formally from the non-commutativity of a and a†. This is true for any harmonic oscillator: the zero-point energy ħω/2 appears when we write the Hamiltonian: H c l = p 2 2 m + 1 2 m ω 2 p 2 = 1 2 ℏ ω ( a a † + a † a ) = ℏ ω ( a † a + 1 2 ) {\displaystyle {\begin{aligned}H_{cl}&={\frac {p^{2}}{2m}}+{\tfrac {1}{2}}m\omega ^{2}{p}^{2}\\&={\tfrac {1}{2}}\hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right)\\&=\hbar \omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)\end{aligned}}} It is often argued that the entire universe is completed bathed in the zero-point electromagnetic field, and as such it can add only some constant amount to expectation values. Physical measurements will therefore reveal only deviations from the vacuum state. Thus the zero-point energy can be dropped from the Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion. Thus we can choose to declare by fiat that the ground state has zero energy and a field Hamiltonian, for example, can be replaced by:[11] H F − ⟨ 0 \| H F \| 0 ⟩ = 1 2 ℏ ω ( a a † + a † a ) − 1 2 ℏ ω = ℏ ω ( a † a + 1 2 ) − 1 2 ℏ ω = ℏ ω a † a {\displaystyle {\begin{aligned}H_{F}-\left\langle 0\|H_{F}\|0\right\rangle &={\tfrac {1}{2}}\hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right)-{\tfrac {1}{2}}\hbar \omega \\&=\hbar \omega \left(a^{\dagger }a+{\tfrac {1}{2}}\right)-{\tfrac {1}{2}}\hbar \omega \\&=\hbar \omega a^{\dagger }a\end{aligned}}} without affecting any physical predictions of the theory. The new Hamiltonian is said to be normally ordered (or Wick ordered) and is denoted by a double-dot symbol. The normally ordered Hamiltonian is denoted :HF, i.e.: : H F :≡ ℏ ω ( a a † + a † a ) :≡ ℏ ω a † a {\displaystyle :H_{F}:\equiv \hbar \omega \left(aa^{\dagger }+a^{\dagger }a\right):\equiv \hbar \omega a^{\dagger }a} In other words, within the normal ordering symbol we can commute a and a†. Since zero-point energy is intimately connected to the non-commutativity of a and a†, the normal ordering procedure eliminates any contribution from the zero-point field. This is especially reasonable in the case of the field Hamiltonian, since the zero-point term merely adds a constant energy which can be eliminated by a simple redefinition for the zero of energy. Moreover, this constant energy in the Hamiltonian obviously commutes with a and a† and so cannot have any effect on the quantum dynamics described by the Heisenberg equations of motion. However, things are not quite that simple. The zero-point energy cannot be eliminated by dropping its energy from the Hamiltonian: When we do this and solve the Heisenberg equation for a field operator, we must include the vacuum field, which is the homogeneous part of the solution for the field operator. In fact we can show that the vacuum field is essential for the preservation of the commutators and the formal consistent of QED. When we calculate the field energy we obtain not only a contribution from particles and forces that may be present but also a contribution from the vacuum field itself i.e. the zero-point field energy. In other words, the zero-point energy reappears even though we may have deleted it from the Hamiltonian.[97] The electromagnetic field in free spaceEdit From Maxwell's equations, the electromagnetic energy of a "free" field i.e. one with no sources, is described by: H F = 1 8 π ∫ d 3 r ( E 2 + B 2 ) = k 2 2 π \| α ( t ) \| 2 {\displaystyle {\begin{aligned}H_{F}&={\frac {1}{8\pi }}\int d^{3}r\left(\mathbf {E} ^{2}+\mathbf {B} ^{2}\right)\\&={\frac {k^{2}}{2\pi }}\|\alpha (t)\|^{2}\end{aligned}}} We introduce the "mode function" A0(r) that satisfies the Helmholtz equation: ( ∇ 2 + k 2 ) A 0 ( r ) = 0 {\displaystyle \left(\nabla ^{2}+k^{2}\right)\mathbf {A} _{0}(\mathbf {r} )=0} where k = ω/c and assume it is normalized such that: ∫ d 3 r \| A 0 ( r ) \| 2 = 1 {\displaystyle \int d^{3}r\left\|\mathbf {A} _{0}(\mathbf {r} )\right\|^{2}=1} We wish to "quantize" the electromagnetic energy of free space for a multimode field. The field intensity of free space should be independent of position such that \|A0(r)\|2 should be independent of r for each mode of the field. The mode function satisfying these conditions is: A 0 ( r ) = e k e i k ⋅ r {\displaystyle \mathbf {A} _{0}(\mathbf {r} )=e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }} where k · ek = 0 in order to have the transversality condition ∇ · A(r,t) satisfied for the Coulomb gauge[dubious – discuss] in which we are working. To achieve the desired normalization we pretend space is divided into cubes of volume V = L3 and impose on the field the periodic boundary condition: A ( x + L , y + L , z + L , t ) = A ( x , y , z , t ) {\displaystyle \mathbf {A} (x+L,y+L,z+L,t)=\mathbf {A} (x,y,z,t)} or equivalently ( k x , k y , k z ) = 2 π L ( n x , n y , n z ) {\displaystyle \left(k_{x},k_{y},k_{z}\right)={\frac {2\pi }{L}}\left(n_{x},n_{y},n_{z}\right)} where n can assume any integer value. This allows us to consider the field in any one of the imaginary cubes and to define the mode function: A k ( r ) = 1 V e k e i k ⋅ r {\displaystyle \mathbf {A} _{\mathbf {k} }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} }e^{i\mathbf {k} \cdot \mathbf {r} }} which satisfies the Helmholtz equation, transversality, and the "box normalization": ∫ V d 3 r \| A k ( r ) \| 2 = 1 {\displaystyle \int _{V}d^{3}r\left\|\mathbf {A} _{\mathbf {k} }(\mathbf {r} )\right\|^{2}=1} where ek is chosen to be a unit vector which specifies the polarization of the field mode. The condition k · ek = 0 means that there are two independent choices of ek, which we call ek1 and ek2 where ek1 · ek2 = 0 and e2 k1 = e2 k2 = 1. Thus we define the mode functions: A k λ ( r ) = 1 V e k λ e i k ⋅ r , λ = { 1 2 {\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )={\frac {1}{\sqrt {V}}}e_{\mathbf {k} \lambda }e^{i\mathbf {k} \cdot \mathbf {r} }\,,\quad \lambda ={\begin{cases}1\\2\end{cases}}} in terms of which the vector potential becomes[clarification needed]: A k λ ( r , t ) = 2 π ℏ c 2 ω k V [ a k λ ( 0 ) e i k ⋅ r + a k λ † ( 0 ) e − i k ⋅ r ] e k λ {\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} ,t)={\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }\right]e_{\mathbf {k} \lambda }} A k λ ( r , t ) = 2 π ℏ c 2 ω k V [ a k λ ( 0 ) e − i ( ω k t − k ⋅ r ) + a k λ † ( 0 ) e i ( ω k t − k ⋅ r ) ] {\displaystyle \mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} ,t)={\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{-i(\omega _{k}t-\mathbf {k} \cdot \mathbf {r} )}+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{i(\omega _{k}t-\mathbf {k} \cdot \mathbf {r} )}\right]} where ωk = kc and akλ, a† kλ are photon annihilation and creation operators for the mode with wave vector k and polarization λ. This gives the vector potential for a plane wave mode of the field. The condition for (kx, ky, kz) shows that there are infinitely many such modes. The linearity of Maxwell's equations allows us to write: A ( r t ) = ∑ k λ 2 π ℏ c 2 ω k V [ a k λ ( 0 ) e i k ⋅ r + a k λ † ( 0 ) e − i k ⋅ r ] e k λ {\displaystyle \mathbf {A} (\mathbf {r} t)=\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar c^{2}}{\omega _{k}V}}}\left[a_{\mathbf {k} \lambda }(0)e^{i\mathbf {k} \cdot \mathbf {r} }+a_{\mathbf {k} \lambda }^{\dagger }(0)e^{-i\mathbf {k} \cdot \mathbf {r} }\right]e_{\mathbf {k} \lambda }} for the total vector potential in free space. Using the fact that: ∫ V d 3 r A k λ ( r ) ⋅ A k ′ λ ′ ∗ ( r ) = δ k , k ′ 3 δ λ , λ ′ {\displaystyle \int _{V}d^{3}r\mathbf {A} _{\mathbf {k} \lambda }(\mathbf {r} )\cdot \mathbf {A} _{\mathbf {k} '\lambda '}^{\ast }(\mathbf {r} )=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}} we find the field Hamiltonian is: H F = ∑ k λ ( ℏ ω k ( a k λ † a k λ ) + 1 2 ) {\displaystyle H_{F}=\sum _{\mathbf {k} \lambda }\left(\hbar \omega _{k}\left(a_{\mathbf {k} \lambda }^{\dagger }a_{\mathbf {k} \lambda }\right)+{\tfrac {1}{2}}\right)} This is the Hamiltonian for an infinite number of uncoupled harmonic oscillators. Thus different modes of the field are independent and satisfy the commutation relations: [ a k λ ( t ) , a k ′ λ ′ † ( t ) ] = δ k , k ′ 3 δ λ , λ ′ [ a k λ ( t ) , a k ′ λ ′ ( t ) ] = [ a k λ † ( t ) , a k ′ λ ′ † ( t ) ] = 0 {\displaystyle {\begin{aligned}\left[a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)\right]&=\delta _{\mathbf {k} ,\mathbf {k} '}^{3}\delta _{\lambda ,\lambda '}\\[10px]\left[a_{\mathbf {k} \lambda }(t),a_{\mathbf {k} '\lambda '}(t)\right]&=\left[a_{\mathbf {k} \lambda }^{\dagger }(t),a_{\mathbf {k} '\lambda '}^{\dagger }(t)\right]=0\end{aligned}}} Clearly the least eigenvalue for HF is: ∑ k λ 1 2 ℏ ω k {\displaystyle \sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \omega _{k}} This state describes the zero-point energy of the vacuum. It appears that this sum is divergent – in fact highly divergent, as putting in the density factor 8 π v 2 d v c 3 V {\displaystyle {\frac {8\pi v^{2}dv}{c^{3}}}V} shows. The summation becomes approximately the integral: 4 π h V c 3 ∫ v 3 d v {\displaystyle {\frac {4\pi hV}{c^{3}}}\int v^{3}\,dv} for high values of v. It diverges proportional to v4 for large v. There are two separate questions to consider. First, is the divergence a real one such that the zero-point energy really is infinite? If we consider the volume V is contained by perfectly conducting walls, very high frequencies can only be contained by taking more and more perfect conduction. No actual method of containing the high frequencies is possible. Such modes will not be stationary in our box and thus not countable in the stationary energy content. So from this physical point of view the above sum should only extend to those frequencies which are countable; a cut-off energy is thus eminently reasonable. However, on the scale of a "universe" questions of general relativity must be included. Suppose even the boxes could be reproduced, fit together and closed nicely by curving spacetime. Then exact conditions for running waves may be possible. However the very high frequency quanta will still not be contained. As per John Wheeler's "geons"[98] these will leak out of the system. So again a cut-off is permissible, almost necessary. The question here becomes one of consistency since the very high energy quanta will act as a mass source and start curving the geometry. This leads to the second question. Divergent or not, finite or infinite, is the zero-point energy of any physical significance? The ignoring of the whole zero-point energy is often encouraged for all practical calculations. The reason for this is that energies are not typically defined by an arbitrary data point, but rather changes in data points, so adding or subtracting a constant (even if infinite) should to be allowed. However this is not the whole story, in reality energy is not so arbitrarily defined: in general relativity the seat of the curvature of spacetime is the energy content and there the absolute amount of energy has real physical meaning. There is no such thing as an arbitrary additive constant with density of field energy. Energy density curves space, and an increase in energy density produces an increase of curvature. Furthermore, the zero-point energy density has other physical consequences e.g. the Casimir effect, contribution to the Lamb shift, or anomalous magnetic moment of the electron, it is clear it is not just a mathematical constant or artifact that can be cancelled out.[99] Necessity of the vacuum field in QEDEdit The vacuum state of the "free" electromagnetic field (that with no sources) is defined as the ground state in which nkλ = 0 for all modes (k, λ). The vacuum state, like all stationary states of the field, is an eigenstate of the Hamiltonian but not the electric and magnetic field operators. In the vacuum state, therefore, the electric and magnetic fields do not have definite values. We can imagine them to be fluctuating about their mean value of zero. In a process in which a photon is annihilated (absorbed), we can think of the photon as making a transition into the vacuum state. Similarly, when a photon is created (emitted), it is occasionally useful to imagine that the photon has made a transition out of the vacuum state.[100] An atom, for instance, can be considered to be "dressed" by emission and reabsorption of "virtual photons" from the vacuum. The vacuum state energy described by ∑kλ ħωk/2 is infinite. We can make the replacement: ∑ k λ ⟶ ∑ λ ( 1 2 π ) 3 ∫ d 3 k = V 8 π 3 ∑ λ ∫ d 3 k {\displaystyle \sum _{\mathbf {k} \lambda }\longrightarrow \sum _{\lambda }\left({\frac {1}{2\pi }}\right)^{3}\int d^{3}k={\frac {V}{8\pi ^{3}}}\sum _{\lambda }\int d^{3}k} the zero-point energy density is: 1 V ∑ k λ 1 2 ℏ ω k = 2 8 π 3 ∫ d 3 k 1 2 ℏ ω k = 4 π 4 π 3 ∫ d k k 2 ( 1 2 ℏ ω k ) = ℏ 2 π 2 c 3 ∫ d ω ω 3 {\displaystyle {\begin{aligned}{\frac {1}{V}}\sum _{\mathbf {k} \lambda }{\tfrac {1}{2}}\hbar \omega _{k}&={\frac {2}{8\pi ^{3}}}\int d^{3}k{\tfrac {1}{2}}\hbar \omega _{k}\\&={\frac {4\pi }{4\pi ^{3}}}\int dk\,k^{2}\left({\tfrac {1}{2}}\hbar \omega _{k}\right)\\&={\frac {\hbar }{2\pi ^{2}c^{3}}}\int d\omega \,\omega ^{3}\end{aligned}}} or in other words the spectral energy density of the vacuum field: ρ 0 ( ω ) = ℏ ω 3 8 π 2 c 3 {\displaystyle \rho _{0}(\omega )={\frac {\hbar \omega ^{3}}{8\pi ^{2}c^{3}}}} The zero-point energy density in the frequency range from ω1 to ω2 is therefore: ∫ ω 1 ω 2 d ω ρ 0 ( ω ) = ℏ 8 π 2 c 3 ( ω 2 4 − ω 1 4 ) {\displaystyle \int _{\omega _{1}}^{\omega _{2}}d\omega \rho _{0}(\omega )={\frac {\hbar }{8\pi ^{2}c^{3}}}\left(\omega _{2}^{4}-\omega _{1}^{4}\right)} This can be large even in relatively narrow "low frequency" regions of the spectrum. In the optical region from 400 to 700 nm, for instance, the above equation yields around 220 erg/cm3. We showed in the above section that the zero-point energy can be eliminated from the Hamiltonian by the normal ordering prescription. However, this elimination does not mean that the vacuum field has been rendered unimportant or without physical consequences. To illustrate this point we consider a linear dipole oscillator in the vacuum. The Hamiltonian for the oscillator plus the field with which it interacts is: H = 1 2 m ( p − e c A ) 2 + 1 2 m ω 0 2 x 2 + H F {\displaystyle H={\frac {1}{2m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}+{\tfrac {1}{2}}m\omega _{0}^{2}\mathbf {x} ^{2}+H_{F}} This has the same form as the corresponding classical Hamiltonian and the Heisenberg equations of motion for the oscillator and the field are formally the same as their classical counterparts. For instance the Heisenberg equations for the coordinate x and the canonical momentum p = mẋ +eA/c of the oscillator are: x ˙ = ( i ℏ ) − 1 [ x . H ] = 1 m ( p − e c A ) p ˙ = ( i ℏ ) − 1 [ p . H ] = 1 2 ∇ ( p − e c A ) 2 − m ω 0 2 x ˙ = − 1 m [ ( p − e c A ) ⋅ ∇ ] [ − e c A ] − 1 m ( p − e c A ) × ∇ × [ − e c A ] − m ω 0 2 x ˙ = e c ( x ˙ ⋅ ∇ ) A + e c x ˙ × B − m ω 0 2 x ˙ {\displaystyle {\begin{aligned}\mathbf {\dot {x}} &=(i\hbar )^{-1}[\mathbf {x} .H]={\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\\\mathbf {\dot {p}} &=(i\hbar )^{-1}[\mathbf {p} .H]{\begin{aligned}&={\tfrac {1}{2}}\nabla \left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)^{2}-m\omega _{0}^{2}\mathbf {\dot {x}} \\&=-{\frac {1}{m}}\left[\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\cdot \nabla \right]\left[-{\frac {e}{c}}\mathbf {A} \right]-{\frac {1}{m}}\left(\mathbf {p} -{\frac {e}{c}}\mathbf {A} \right)\times \nabla \times \left[-{\frac {e}{c}}\mathbf {A} \right]-m\omega _{0}^{2}\mathbf {\dot {x}} \\&={\frac {e}{c}}(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}\end{aligned}}} m x ¨ = p ˙ − e c A ˙ = − e c [ A ˙ − ( x ˙ ⋅ ∇ ) A ] + e c x ˙ × B − m ω 0 2 x = e E + e c x ˙ × B − m ω 0 2 x {\displaystyle {\begin{aligned}m\mathbf {\ddot {x}} &=\mathbf {\dot {p}} -{\frac {e}{c}}\mathbf {\dot {A}} \\&=-{\frac {e}{c}}\left[\mathbf {\dot {A}} -\left(\mathbf {\dot {x}} \cdot \nabla \right)\mathbf {A} \right]+{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \\&=e\mathbf {E} +{\frac {e}{c}}\mathbf {\dot {x}} \times \mathbf {B} -m\omega _{0}^{2}\mathbf {x} \end{aligned}}} since the rate of change of the vector potential in the frame of the moving charge is given by the convective derivative A ˙ = ∂ A ∂ t + ( x ˙ ⋅ ∇ ) A 3 . {\displaystyle \mathbf {\dot {A}} ={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {\dot {x}} \cdot \nabla )\mathbf {A} ^{3}\,.} For nonrelativistic motion we may neglect the magnetic force and replace the expression for mẍ by: x ¨ + ω 0 2 x ≈ e m E ≈ ∑ k λ 2 π ℏ ω k V [ a k λ ( t ) + a k λ † ( t ) ] e k λ {\displaystyle {\begin{aligned}\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} &\approx {\frac {e}{m}}\mathbf {E} \\&\approx \sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}\left[a_{\mathbf {k} \lambda }(t)+a_{\mathbf {k} \lambda }^{\dagger }(t)\right]e_{\mathbf {k} \lambda }\end{aligned}}} Above we have made the electric dipole approximation in which the spatial dependence of the field is neglected. The Heisenberg equation for akλ is found similarly from the Hamiltonian to be: a ˙ k λ = i ω k a k λ + i e 2 π ℏ ω k V x ˙ ⋅ e k λ {\displaystyle {\dot {a}}_{\mathbf {k} \lambda }=i\omega _{k}a_{\mathbf {k} \lambda }+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\mathbf {\dot {x}} \cdot e_{\mathbf {k} \lambda }} In the electric dipole approximation. In deriving these equations for x, p, and akλ we have used the fact that equal-time particle and field operators commute. This follows from the assumption that particle and field operators commute at some time (say, t = 0) when the matter-field interpretation is presumed to begin, together with the fact that a Heisenberg-picture operator A(t) evolves in time as A(t) = U†(t)A(0)U(t), where U(t) is the time evolution operator satisfying i ℏ U ˙ = H U , U † ( t ) = U − 1 ( t ) , U ( 0 ) = 1 . {\displaystyle i\hbar {\dot {U}}=HU\,,\quad U^{\dagger }(t)=U^{-1}(t)\,,\quad U(0)=1\,.} Alternatively, we can argue that these operators must commute if we are to obtain the correct equations of motion from the Hamiltonian, just as the corresponding Poisson brackets in classical theory must vanish in order to generate the correct Hamilton equations. The formal solution of the field equation is: a k λ ( t ) = a k λ ( 0 ) e − i ω k t + i e 2 π ℏ ω k V ∫ 0 t d t ′ e k λ ⋅ x ˙ ( t ′ ) e i ω k ( t ′ − t ) {\displaystyle a_{\mathbf {k} \lambda }(t)=a_{\mathbf {k} \lambda }(0)e^{-i\omega _{k}t}+ie{\sqrt {\frac {2\pi }{\hbar \omega _{k}V}}}\int _{0}^{t}dt'\,e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} (t')e^{i\omega _{k}\left(t'-t\right)}} and therefore the equation for ȧkλ may be written: x ¨ + ω 0 2 x = e m E 0 ( t ) + e m E R R ( t ) {\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} ={\frac {e}{m}}\mathbf {E} _{0}(t)+{\frac {e}{m}}\mathbf {E} _{RR}(t)} E 0 ( t ) = i ∑ k λ 2 π ℏ ω k V [ a k λ ( 0 ) e − i ω k t − a k λ † ( 0 ) e i ω k t ] e k λ {\displaystyle \mathbf {E} _{0}(t)=i\sum _{\mathbf {k} \lambda }{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}\left[a_{\mathbf {k} \lambda }(0)e^{-i\omega _{k}t}-a_{\mathbf {k} \lambda }^{\dagger }(0)e^{i\omega _{k}t}\right]e_{\mathbf {k} \lambda }} E R R ( t ) = − 4 π e V ∑ k λ ∫ 0 t d t ′ [ e k λ ⋅ x ˙ ( t ′ ) ] cos ⁡ ω k ( t ′ − t ) {\displaystyle \mathbf {E} _{RR}(t)=-{\frac {4\pi e}{V}}\sum _{\mathbf {k} \lambda }\int _{0}^{t}dt'\left[e_{\mathbf {k} \lambda }\cdot \mathbf {\dot {x}} \left(t'\right)\right]\cos \omega _{k}\left(t'-t\right)} It can be shown that in the radiation reaction field, if the mass m is regarded as the "observed" mass then we can take: E R R ( t ) = 2 e 3 c 3 x ¨ {\displaystyle \mathbf {E} _{RR}(t)={\frac {2e}{3c^{3}}}\mathbf {\ddot {x}} } The total field acting on the dipole has two parts, E0(t) and ERR(t). E0(t) is the free or zero-point field acting on the dipole. It is the homogeneous solution of the Maxwell equation for the field acting on the dipole, i.e., the solution, at the position of the dipole, of the wave equation [ ∇ 2 − 1 c 2 ∂ 2 ∂ t 2 ] E = 0 {\displaystyle \left[\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\right]\mathbf {E} =0} satisfied by the field in the (source free) vacuum. For this reason E0(t) is often referred to as the "vacuum field", although it is of course a Heisenberg-picture operator acting on whatever state of the field happens to be appropriate at t = 0. ERR(t) is the source field, the field generated by the dipole and acting on the dipole. Using the above equation for ERR(t) we obtain an equation for the Heisenberg-picture operator x ( t ) {\displaystyle \mathbf {x} (t)} that is formally the same as the classical equation for a linear dipole oscillator: x ¨ + ω 0 2 x − τ x . . . = e m E 0 ( t ) {\displaystyle \mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)} where τ = 2e2/3mc3. in this instance we have considered a dipole in the vacuum, without any "external" field acting on it. the role of the external field in the above equation is played by the vacuum electric field acting on the dipole. Classically, a dipole in the vacuum is not acted upon by any "external" field: if there are no sources other than the dipole itself, then the only field acting on the dipole is its own radiation reaction field. In quantum theory however there is always an "external" field, namely the source-free or vacuum field E0(t). According to our earlier equation for akλ(t) the free field is the only field in existence at t = 0 as the time at which the interaction between the dipole and the field is "switched on". The state vector of the dipole-field system at t = 0 is therefore of the form \| Ψ ⟩ = \| vac ⟩ \| ψ D ⟩ , {\displaystyle \|\Psi \rangle =\|{\text{vac}}\rangle \|\psi _{D}\rangle \,,} where \|vac⟩ is the vacuum state of the field and \|ψD⟩ is the initial state of the dipole oscillator. The expectation value of the free field is therefore at all times equal to zero: ⟨ E 0 ( t ) ⟩ = ⟨ Ψ \| E 0 ( t ) \| Ψ ⟩ = 0 {\displaystyle \langle \mathbf {E} _{0}(t)\rangle =\langle \Psi \|\mathbf {E} _{0}(t)\|\Psi \rangle =0} since akλ(0)\|vac⟩ = 0. however, the energy density associated with the free field is infinite: 1 4 π ⟨ E 0 2 ( t ) ⟩ = 1 4 π ∑ k λ ∑ k ′ λ ′ 2 π ℏ ω k V 2 π ℏ ω k ′ V × ⟨ a k λ ( 0 ) a k ′ λ ′ † ( 0 ) ⟩ = 1 4 π ∑ k λ ( 2 π ℏ ω k V ) = ∫ 0 ∞ d w ρ 0 ( ω ) {\displaystyle {\begin{aligned}{\frac {1}{4\pi }}\left\langle \mathbf {E} _{0}^{2}(t)\right\rangle &={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\sum _{\mathbf {k'} \lambda '}{\sqrt {\frac {2\pi \hbar \omega _{k}}{V}}}{\sqrt {\frac {2\pi \hbar \omega _{k'}}{V}}}\times \left\langle a_{\mathbf {k} \lambda }(0)a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right\rangle \\&={\frac {1}{4\pi }}\sum _{\mathbf {k} \lambda }\left({\frac {2\pi \hbar \omega _{k}}{V}}\right)\\&=\int _{0}^{\infty }dw\,\rho _{0}(\omega )\end{aligned}}} The important point of this is that the zero-point field energy HF does not affect the Heisenberg equation for akλ since it is a c-number or constant (i.e. an ordinary number rather than an operator) and commutes with akλ. We can therefore drop the zero-point field energy from the Hamiltonian, as is usually done. But the zero-point field re-emerges as the homogeneous solution for the field equation. A charged particle in the vacuum will therefore always see a zero-point field of infinite density. This is the origin of one of the infinities of quantum electrodynamics, and it cannot be eliminated by the trivial expedient dropping of the term ∑kλ ħωk/2 in the field Hamiltonian. The free field is in fact necessary for the formal consistency of the theory. In particular, it is necessary for the preservation of the commutation relations, which is required by the unitary of time evolution in quantum theory: [ z ( t ) , p z ( t ) ] = [ U † ( t ) z ( 0 ) U ( t ) , U † ( t ) p z ( 0 ) U ( t ) ] = U † ( t ) [ z ( 0 ) , p z ( 0 ) ] U ( t ) = i ℏ U † ( t ) U ( t ) = i ℏ {\displaystyle {\begin{aligned}\left[z(t),p_{z}(t)\right]&=\left[U^{\dagger }(t)z(0)U(t),U^{\dagger }(t)p_{z}(0)U(t)\right]\\&=U^{\dagger }(t)\left[z(0),p_{z}(0)\right]U(t)\\&=i\hbar U^{\dagger }(t)U(t)\\&=i\hbar \end{aligned}}} We can calculate [z(t),pz(t)] from the formal solution of the operator equation of motion Using the fact that [ a k λ ( 0 ) , a k ′ λ ′ † ( 0 ) ] = δ k k ′ 3 , δ λ λ ′ {\displaystyle \left[a_{\mathbf {k} \lambda }(0),a_{\mathbf {k'} \lambda '}^{\dagger }(0)\right]=\delta _{\mathbf {kk'} }^{3},\delta _{\lambda \lambda '}} and that equal-time particle and field operators commute, we obtain: = [ z ( t ) , m z ˙ ( t ) ] + [ z ( t ) , e c A z ( t ) ] = [ z ( t ) , m z ˙ ( t ) ] = ( i ℏ e 2 2 π 2 m c 3 ) ( 8 π 3 ) ∫ 0 ∞ d ω ω 4 ( ω 2 − ω 0 2 ) 2 + τ 2 ω 6 {\displaystyle {\begin{aligned}[z(t),p_{z}(t)]&=\left[z(t),m{\dot {z}}(t)\right]+\left[z(t),{\frac {e}{c}}A_{z}(t)\right]\\&=\left[z(t),m{\dot {z}}(t)\right]\\&=\left({\frac {i\hbar e^{2}}{2\pi ^{2}mc^{3}}}\right)\left({\frac {8\pi }{3}}\right)\int _{0}^{\infty }{\frac {d\omega \,\omega ^{4}}{\left(\omega ^{2}-\omega _{0}^{2}\right)^{2}+\tau ^{2}\omega ^{6}}}\end{aligned}}} For the dipole oscillator under consideration it can be assumed that the radiative damping rate is small compared with the natural oscillation frequency, i.e., τω0 ≪ 1. Then the integrand above is sharply peaked at ω = ω0 and: [ z ( t ) , p z ( t ) ] ≈ 2 i ℏ e 2 3 π m c 3 ω 0 3 ∫ − ∞ ∞ d x x 2 + τ 2 ω 0 6 = ( 2 i ℏ e 2 ω 0 3 3 π m c 3 ) ( π τ ω 0 3 ) = i ℏ {\displaystyle {\begin{aligned}\left[z(t),p_{z}(t)\right]&\approx {\frac {2i\hbar e^{2}}{3\pi mc^{3}}}\omega _{0}^{3}\int _{-\infty }^{\infty }{\frac {dx}{x^{2}+\tau ^{2}\omega _{0}^{6}}}\\&=\left({\frac {2i\hbar e^{2}\omega _{0}^{3}}{3\pi mc^{3}}}\right)\left({\frac {\pi }{\tau \omega _{0}^{3}}}\right)\\&=i\hbar \end{aligned}}} the necessity of the vacuum field can also be appreciated by making the small damping approximation in x ¨ + ω 0 2 x − τ x . . . = e m E 0 ( t ) x ¨ ≈ − ω 0 2 x ( t ) x . . . ≈ − ω 0 2 x ˙ {\displaystyle {\begin{aligned}&\mathbf {\ddot {x}} +\omega _{0}^{2}\mathbf {x} -\tau \mathbf {\overset {...}{x}} ={\frac {e}{m}}\mathbf {E} _{0}(t)\\&\mathbf {\ddot {x}} \approx -\omega _{0}^{2}\mathbf {x} (t)&&\mathbf {\overset {...}{x}} \approx -\omega _{0}^{2}\mathbf {\dot {x}} \end{aligned}}} x ¨ + τ ω 0 2 x ˙ + ω 0 2 x ≈ e m E 0 ( t ) {\displaystyle \mathbf {\ddot {x}} +\tau \omega _{0}^{2}\mathbf {\dot {x}} +\omega _{0}^{2}\mathbf {x} \approx {\frac {e}{m}}\mathbf {E} _{0}(t)} Without the free field E0(t) in this equation the operator x(t) would be exponentially dampened, and commutators like [z(t),pz(t)] would approach zero for t ≫ 1/τω2 0. With the vacuum field included, however, the commutator is iħ at all times, as required by unitarity, and as we have just shown. A similar result is easily worked out for the case of a free particle instead of a dipole oscillator.[101] What we have here is an example of a "fluctuation-dissipation elation". Generally speaking if a system is coupled to a bath that can take energy from the system in an effectively irreversible way, then the bath must also cause fluctuations. The fluctuations and the dissipation go hand in hand we cannot have one without the other. In the current example the coupling of a dipole oscillator to the electromagnetic field has a dissipative component, in the form of the zero-point (vacuum) field; given the existence of radiation reaction, the vacuum field must also exist in order to preserve the canonical commutation rule and all it entails. The spectral density of the vacuum field is fixed by the form of the radiation reaction field, or vice versa: because the radiation reaction field varies with the third derivative of x, the spectral energy density of the vacuum field must be proportional to the third power of ω in order for [z(t),pz(t)] to hold. In the case of a dissipative force proportional to ẋ, by contrast, the fluctuation force must be proportional to ω {\displaystyle \omega } in order to maintain the canonical commutation relation.[101] This relation between the form of the dissipation and the spectral density of the fluctuation is the essence of the fluctuation-dissipation theorem.[78] The fact that the canonical commutation relation for a harmonic oscillator coupled to the vacuum field is preserved implies that the zero-point energy of the oscillator is preserved. it is easy to show that after a few damping times the zero-point motion of the oscillator is in fact sustained by the driving zero-point field.[102] The quantum chromodynamic vacuumEdit Main article: QCD vacuum The QCD vacuum is the vacuum state of quantum chromodynamics (QCD). It is an example of a non-perturbative vacuum state, characterized by a non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the confined phase of quark matter. In technical terms, gluons are vector gauge bosons that mediate strong interactions of quarks in quantum chromodynamics (QCD). Gluons themselves carry the color charge of the strong interaction. This is unlike the photon, which mediates the electromagnetic interaction but lacks an electric charge. Gluons therefore participate in the strong interaction in addition to mediating it, making QCD significantly harder to analyze than QED (quantum electrodynamics) as it deals with nonlinear equations to characterize such interactions. The Higgs fieldEdit Main article: Higgs mechanism The potential for the Higgs field, plotted as function of ϕ0 and ϕ3. It has a Mexican-hat or champagne-bottle profile at the ground. The Standard Model hypothesises a field called the Higgs field (symbol: ϕ), which has the unusual property of a non-zero amplitude in its ground state (zero-point) energy after renormalization; i.e., a non-zero vacuum expectation value. It can have this effect because of its unusual "Mexican hat" shaped potential whose lowest "point" is not at its "centre". Below a certain extremely high energy level the existence of this non-zero vacuum expectation spontaneously breaks electroweak gauge symmetry which in turn gives rise to the Higgs mechanism and triggers the acquisition of mass by those particles interacting with the field. The Higgs mechanism occurs whenever a charged field has a vacuum expectation value. This effect occurs because scalar field components of the Higgs field are "absorbed" by the massive bosons as degrees of freedom, and couple to the fermions via Yukawa coupling, thereby producing the expected mass terms. The expectation value of ϕ0 in the ground state (the vacuum expectation value or VEV) is then ⟨ϕ0⟩ = v/√2, where v = \|μ\|/√λ. The measured value of this parameter is approximately 246 GeV/c2.[103] It has units of mass, and is the only free parameter of the Standard Model that is not a dimensionless number. The Higgs mechanism is a type of superconductivity which occurs in the vacuum. It occurs when all of space is filled with a sea of particles which are charged and thus the field has a nonzero vacuum expectation value. Interaction with the vacuum energy filling the space prevents certain forces from propagating over long distances (as it does in a superconducting medium; e.g., in the Ginzburg–Landau theory). Experimental observationsEdit Zero-point energy has many observed physical consequences.[12] It is important to note that zero-point energy is not merely an artefact of mathematical formalism that can, for instance, be dropped from a Hamiltonian by redefining the zero of energy, or by arguing that it is a constant and therefore has no effect on Heisenberg equations of motion without latter consequence.[104] Indeed, such treatment could create a problem at a deeper, as of yet undiscovered, theory.[105] For instance, in general relativity the zero of energy (i.e. the energy density of the vacuum) contributes to a cosmological constant of the type introduced by Einstein in order to obtain static solutions to his field equations.[106] The zero-point energy density of the vacuum, due to all quantum fields, is extremely large, even when we cut off the largest allowable frequencies based on plausible physical arguments. It implies a cosmological constant larger than the limits imposed by observation by about 120 orders of magnitude. This "cosmological constant problem" remains one of the greatest unsolved mysteries of physics.[107] Casimir effectEdit Casimir forces on parallel plates Main article: Casimir effect A phenomenon that is commonly presented as evidence for the existence of zero-point energy in vacuum is the Casimir effect, proposed in 1948 by Dutch physicist Hendrik Casimir, who considered the quantized electromagnetic field between a pair of grounded, neutral metal plates. The vacuum energy contains contributions from all wavelengths, except those excluded by the spacing between plates. As the plates draw together, more wavelengths are excluded and the vacuum energy decreases. The decrease in energy means there must be a force doing work on the plates as they move. Early experimental tests from the 1950s onwards gave positive results showing the force was real, but other external factors could not be ruled out as the primary cause, with the range of experimental error sometimes being nearly 100%.[108][109][110][111][112] That changed in 1997 with Lamoreaux[113] conclusively showing that the Casimir force was real. Results have been repeatedly replicated since then.[114][115][116][117] In 2009 Munday et al.[118] published experimental proof that (as predicted in 1961[119]) the Casimir force could also be repulsive as well as being attractive. Repulsive Casimir forces could allow quantum levitation of objects in a fluid and lead to a new class of switchable nanoscale devices with ultra-low static friction[120] An interesting hypothetical side effect of the Casimir effect is the Scharnhorst effect, a hypothetical phenomenon in which light signals travel slightly faster than c between two closely spaced conducting plates.[121] Lamb shiftEdit Fine structure of energy levels in hydrogen – relativistic corrections to the Bohr model Main article: Lamb shift The quantum fluctuations of the electromagnetic field have important physical consequences. In addition to the Casimir effect, they also lead to a splitting between the two energy levels 2S1/2 and 2P1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which these states should have the same energy. Charged particles can interact with the fluctuations of the quantized vacuum field, leading to slight shifts in energy,[122] this effect is called the Lamb shift.[123] The shift of about 4.38×10−6 eV is roughly 10−7 of the difference between the energies of the 1s and 2s levels, and amounts to 1,058 MHz in frequency units. A small part of this shift (27 MHz ≈ 3%) arises not from fluctuations of the electromagnetic field, but from fluctuations of the electron–positron field. The creation of (virtual) electron–positron pairs has the effect of screening the Coulomb field and acts as a vacuum dielectric constant. This effect is much more important in muonic atoms.[124] Fine structure constantEdit Main article: Fine structure constant Taking ħ (Planck's constant divided by 2π), c (the speed of light), and e2 = q2 e/4πε0 (the electromagnetic coupling constant i.e. a measure of the strength of the electromagnetic force (where qe is the absolute value of the electronic charge and ε 0 {\displaystyle \varepsilon _{0}} is the vacuum permittivity)) we can form a dimensionless quantity called the fine-structure constant: α = e 2 ℏ c = q e 2 4 π ε 0 ℏ c ≈ 1 137 {\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}={\frac {q_{e}^{2}}{4\pi \varepsilon _{0}\hbar c}}\approx {\frac {1}{137}}} The fine-structure constant is the coupling constant of quantum electrodynamics (QED) determining the strength of the interaction between electrons and photons. It turns out that the fine structure constant is not really a constant at all owing to the zero-point energy fluctuations of the electron-positron field.[125] The quantum fluctuations caused by zero-point energy have the effect of screening electric charges: owing to (virtual) electron-positron pair production, the charge of the particle measured far from the particle is far smaller than the charge measured when close to it. The Heisenberg inequality where ħ = h/2π, and Δx, Δp are the standard deviations of position and momentum states that: Δ x Δ p ≥ 1 2 ℏ {\displaystyle \Delta _{x}\Delta _{p}\geq {\frac {1}{2}}\hbar } It means that a short distance implies large momentum and therefore high energy i.e. particles of high energy must be used to explore short distances. QED concludes that the fine structure constant is an increasing function of energy. It has been shown that at energies of the order of the Z0 boson rest energy, mzc2 ≈ 90 GeV, that: α ≈ 1 129 {\displaystyle \alpha \approx {\frac {1}{129}}} rather than the low-energy α ≈ 1/137.[126][127] The renormalization procedure of eliminating zero-point energy infinities allows the choice of an arbitrary energy (or distance) scale for defining α. All in all, α depends on the energy scale characteristic of the process under study, and also on details of the renormalization procedure. The energy dependence of α has been observed for several years now in precision experiment in high-energy physics. Vacuum birefringenceEdit Main articles: Lorentz-violating electrodynamics and Euler–Heisenberg Lagrangian Light coming from the surface of a strongly magnetic neutron star (left) becomes linearly polarised as it travels through the vacuum. In the presence of strong electrostatic fields it is predicted that virtual particles become separated from the vacuum state and form real matter.[citation needed] The fact that electromagnetic radiation can be transformed into matter and vice versa leads to fundamentally new features in quantum electrodynamics. One of the most important consequences is that, even in the vacuum, the Maxwell equations have to be exchanged by more complicated formulas. In general, it will be not possible to separate processes in the vacuum from the processes involving matter since electromagnetic fields can create matter if the field fluctuations are strong enough. This leads to highly complex nonlinear interaction - gravity will have an effect on the light at the same time the light has an effect on gravity. These effects were first predicted by Werner Heisenberg and Hans Heinrich Euler[128] in 1936 and independently the same year by Victor Weisskopf who stated: "The physical properties of the vacuum originate in the "zero-point energy" of matter, which also depends on absent particles through the external field strengths and therefore contributes an additional term to the purely Maxwellian field energy".[129][130] Thus strong magnetic fields vary the energy contained in the vacuum. The scale above which the electromagnetic field is expected to become nonlinear is known as the Schwinger limit. At this point the vacuum has all the properties of a birefringent medium, thus in principle a rotation of the polarization frame (the Faraday effect) can be observed in empty space.[131][132] Wide field view of the neutron star RX J1856.5-3754 Both Einstein's theory of special and general relativity state that light should pass freely through a vacuum without being altered, a principle known as Lorentz invariance. Yet, in theory, large nonlinear self-interaction of light due to quantum fluctuations should lead to this principle being measurably violated if the interactions are strong enough. Nearly all theories of quantum gravity predict that that Lorentz invariance is not an exact symmetry of nature. It is predicted the speed at which light travels through the vacuum depends on its direction, polarization and the local strength of the magnetic field.[133] There have been a number of inconclusive results which claim to show evidence of a Lorentz violation by finding a rotation of the polarization plane of light coming from distant galaxies.[134] The first concrete evidence for vacuum birefringence was published in 2017 when a team of astronomers looked at the light coming from the star RX J1856.5-3754,[135] the closest discovered neutron star to Earth.[136] Roberto Mignani at the National Institute for Astrophysics in Milan who led the team of astronomers has commented that ""When Einstein came up with the theory of general relativity 100 years ago, he had no idea that it would be used for navigational systems. The consequences of this discovery probably will also have to be realised on a longer timescale."[137] The team found that visible light from the star had undergone linear polarisation[clarification needed] of around 16%. If the birefringence had been caused by light passing through interstellar gas or plasma, the effect should have been no more than 1%. Definitive proof would require repeating the observation at other wavelengths and on other neutron stars. At X-ray wavelengths the polarization from the quantum fluctuations should be near 100%.[138] Although no telescope currently exists that can make such measurements, there are several proposed X-ray telescopes that may soon be able to verify the result conclusively such as China's Hard X-ray Modulation Telescope (HXMT) and NASA's Imaging X-ray Polarimetry Explorer (IXPE). Speculated involvement in other phenomenaEdit Dark energyEdit Main article: Dark energy Unsolved problem in physics: why does the large zero-point energy of the vacuum not cause a large cosmological constant? What cancels it out?[139][140][141] (more unsolved problems in physics) In the late 1990s it was discovered that very distant supernova were dimmer than expected suggesting that the universe's expansion was accelerating rather than slowing down.[142][143] This revived discussion that Einstein's cosmological constant, long disregarded by physicists as being equal to zero, was in fact some small positive value. This would indicate empty space exerted some form of negative pressure or energy. There is no natural candidate for what might cause what has been called dark energy but the current best guess is that it is the zero-point energy of the vacuum.[144] One difficulty with this assumption is that the zero-point energy of the vacuum is absurdly large compared to the observed cosmological constant. In general relativity, mass and energy are equivalent; both produce a gravitational field and therefore the theorized vacuum energy of quantum field theory should have led the universe ripping itself to pieces. This obviously has not happened and this issue, called the cosmological constant problem, is one of the greatest unsolved mysteries in physics. The European Space Agency is building the Euclid telescope. Due to launch in 2020 it will map galaxies up to 10 billion light years away. By seeing how dark energy influences their arrangement and shape, the mission will allow scientists to see if the strength of dark energy has changed. If dark energy is found to vary throughout time it would indicate it is due to quintessence, where observed acceleration is due to the energy of a scalar field, rather than the cosmological constant. No evidence of quintessence is yet available, but it has not been ruled out either. It generally predicts a slightly slower acceleration of the expansion of the universe than the cosmological constant. Some scientists think that the best evidence for quintessence would come from violations of Einstein's equivalence principle and variation of the fundamental constants in space or time.[145] Scalar fields are predicted by the Standard Model of particle physics and string theory, but an analogous problem to the cosmological constant problem (or the problem of constructing models of cosmological inflation) occurs: renormalization theory predicts that scalar fields should acquire large masses again due to zero-point energy. Cosmic inflationEdit Why does the observable universe have more matter than antimatter? Main article: Inflation (cosmology) Cosmic inflation is a faster-than-light expansion of space just after the Big Bang. It explains the origin of the large-scale structure of the cosmos. It is believed quantum vacuum fluctuations caused by zero-point energy arising in the microscopic inflationary period, later became magnified to a cosmic size, becoming the gravitational seeds for galaxies and structure in the Universe (see galaxy formation and evolution and structure formation).[146] Many physicists also believe that inflation explains why the Universe appears to be the same in all directions (isotropic), why the cosmic microwave background radiation is distributed evenly, why the Universe is flat, and why no magnetic monopoles have been observed. The mechanism for inflation is unclear, it is similar in effect to dark energy but is a far more energetic and short lived process. As with dark energy the best explanation is some form of vacuum energy arising from quantum fluctuations. It may be that inflation caused baryogenesis, the hypothetical physical processes that produced an asymmetry (imbalance) between baryons and antibaryons produced in the very early universe, but this is far from certain. Alternative theoriesEdit There has been a long debate[147] over the question of whether zero-point fluctuations of quantized vacuum fields are "real" i.e. do they have physical effects that cannot be interpreted by an equally valid alternative theory? Schwinger, in particular, attempted to formulate QED without reference to zero-point fluctuations via his "source theory".[148] From such an approach it is possible to derive the Casimir Effect without reference to a fluctuating field. Such a derivation was first given by Schwinger (1975)[149] for a scalar field, and then generalized to the electromagnetic case by Schwinger, DeRaad, and Milton (1978).[150] in which they state "the vacuum is regarded as truly a state with all physical properties equal to zero". More recently Jaffe (2005)[151] has highlighted a similar approach in deriving the Casimir effect stating "the concept of zero-point fluctuations is a heuristic and calculational aid in the description of the Casimir effect, but not a necessity in QED." Nevertheless, as Jaffe himself notes in his paper, "no one has shown that source theory or another S-matrix based approach can provide a complete description of QED to all orders." Furthermore, Milonni has shown the necessity of the vacuum field for the formal consistency of QED.[152] In QCD, color confinement has led physicists to abandon the source theory or S-matrix based approach for the strong interactions. The Higgs mechanism, Hawking Radiation and the Unruh effect are also theorized to be dependent on zero-point vacuum fluctuations, the field contribution being an inseparable parts of these theories. Jaffe continues "Even if one could argue away zero-point contributions to the quantum vacuum energy, the problem of spontaneous symmetry breaking remains: condensates [ground state vacua] that carry energy appear at many energy scales in the Standard Model. So there is good reason to be skeptical of attempts to avoid the standard formulation of quantum field theory and the zero-point energies it brings with it." It is difficult to judge the physical reality of infinite zero-point energies that are inherent in field theories, but modern physics does not know any better way to construct gauge-invariant, renormalizable theories than with zero-point energy and they would seem to be a necessity for any attempt at a unified theory.[153] Chaotic and emergent phenomenaEdit See also: Chaos theory, Emergence, and Self-organization The mathematical models used in classical electromagnetism, quantum electrodynamics (QED) and the standard model all view the electromagnetic vacuum as a linear system with no overall observable consequence (e.g. in the case of the Casimir effect, Lamb shift, and so on) these phenomena can be explained by alternative mechanisms other than action of the vacuum by arbitrary changes to the normal ordering of field operators. See alternative theories section). This is a consequence of viewing electromagnetism as a U(1) gauge theory, which topologically does not allow the complex interaction of a field with and on itself.[154] In higher symmetry groups and in reality, the vacuum is not a calm, randomly fluctuating, largely immaterial and passive substance, but at times can be viewed as a turbulent virtual plasma that can have complex vortices (i.e. solitons vis-à-vis particles), entangled states and a rich nonlinear structure.[155] There are many observed nonlinear physical electromagnetic phenomena such as Aharonov–Bohm (AB)[156][157] and Altshuler–Aronov–Spivak (AAS) effects,[158] Berry,[159] Aharonov–Anandan,[160] Pancharatnam[161] and Chiao–Wu[162] phase rotation effects, Josephson effect,[163][164] Quantum Hall effect,[165] the de Haas–van Alphen effect,[166] the Sagnac effect and many other physically observable phenomena which would indicate that the electromagnetic potential field has real physical meaning rather than being a mathematical artifact[167] and therefore an all encompassing theory would not confine electromagnetism as a local force as is currently done, but as a SU(2) gauge theory or higher geometry. Higher symmetries allow for nonlinear, aperiodic behaviour which manifest as a variety of complex non-equilibrium phenomena that do not arise in the linearised U(1) theory, such as multiple stable states, symmetry breaking, chaos and emergence.[168] What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Heaviside, FitzGerald, Lodge and Hertz. The original equations used Hamilton's more expressive quaternion notation,[169] a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today.[170] In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered".[171] It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in Tesla's oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries.[172] It has often been argued that quaternions are not compatible with special relativity,[173] but multiple papers have shown ways of incorporating relativity.[174][175][176] A good example of nonlinear electromagnetics is in high energy dense plasmas, where vortical phenomena occur which seemingly violate the second law of thermodynamics by increasing the energy gradient within the electromagnetic field and violate Maxwell's laws by creating ion currents which capture and concentrate their own and surrounding magnetic fields. In particular Lorentz force law, which elaborates Maxwell's equations is violated by these force free vortices.[177][178][179] These apparent violations are due to the fact that the traditional conservation laws in classical and quantum electrodynamics (QED) only display linear U(1) symmetry (in particular, by the extended Noether theorem,[180] conservation laws such as the laws of thermodynamics need not always apply to dissipative systems,[181][182] which are expressed in gauges of higher symmetry). The second law of thermodynamics states that in a closed linear system entropy flow can only be positive (or exactly zero at the end of a cycle). However, negative entropy (i.e. increased order, structure or self-organisation) can spontaneously appear in an open nonlinear thermodynamic system that is far from equilibrium, so long as this emergent order accelerates the overall flow of entropy in the total system. The 1977 Nobel Prize in Chemistry was awarded to thermodynamicist Ilya Prigogine[183] for his theory of dissipative systems that described this notion. Prigogine described the principle as "order through fluctuations"[184] or "order out of chaos".[185] It has been argued by some that all emergent order in the universe from galaxies, solar systems, planets, weather, complex chemistry, evolutionary biology to even consciousness, technology and civilizations are themselves examples of thermodynamic dissipative systems; nature having naturally selected these structures to accelerate entropy flow within the universe to an ever-increasing degree.[186] For example, it has been estimated that human body is 10,000 times more effective at dissipating energy per unit of mass than the sun.[187] One may query what this has to do with zero-point energy. Given the complex and adaptive behaviour that arises from nonlinear systems considerable attention in recent years has gone into studying a new class of phase transitions which occur at absolute zero temperature. These are quantum phase transitions which are driven by EM field fluctuations as a consequence of zero-point energy.[188] A good example of a spontaneous phase transition that are attributed to zero-point fluctuations can be found in superconductors. Superconductivity is one of the best known empirically quantified macroscopic electromagnetic phenomena whose basis is recognised to be quantum mechanical in origin. The behaviour of the electric and magnetic fields under superconductivity is governed by the London equations. However, it has been questioned in a series of journal articles whether the quantum mechanically canonised London equations can be given a purely classical derivation.[189] Bostick,[190][191] for instance, has claimed to show that the London equations do indeed have a classical origin that applies to superconductors and to some collisionless plasmas as well. In particular it has been asserted that the Beltrami vortices in the plasma focus display the same paired flux-tube morphology as Type II superconductors.[192][193] Others have also pointed out this connection, Fröhlich[194] has shown that the hydrodynamic equations of compressible fluids, together with the London equations, lead to a macroscopic parameter ( μ {\displaystyle \mu } = electric charge density / mass density), without involving either quantum phase factors or Planck's constant. In essence, it has been asserted that Beltrami plasma vortex structures are able to at least simulate the morphology of Type I and Type II superconductors. This occurs because the "organised" dissipative energy of the vortex configuration comprising the ions and electrons far exceeds the "disorganised" dissipative random thermal energy. The transition from disorganised fluctuations to organised helical structures is a phase transition involving a change in the condensate's energy (i.e. the ground state or zero-point energy) but without any associated rise in temperature.[195] This is an example of zero-point energy having multiple stable states (see Quantum phase transition, Quantum critical point, Topological degeneracy, Topological order[196]) and where the overall system structure is independent of a reductionist or deterministic view, that "classical" macroscopic order can also causally affect quantum phenomena. Furthermore, the pair production of Beltrami vortices has been compared to the morphology of pair production of virtual particles in the vacuum. Timeline of the metric expansion of space. On the left the dramatic expansion occurs in the inflationary epoch The idea that the vacuum energy can have multiple stable energy states is a leading hypothesis for the cause of cosmic inflation. In fact, it has been argued that these early vacuum fluctuations led to the expansion of the universe and in turn have guaranteed the non-equilibrium conditions necessary to drive order from chaos, as without such expansion the universe would have reached thermal equilibrium and no complexity could have existed. With the continued accelerated expansion of the universe, the cosmos generates an energy gradient that increases the "free energy" (i.e. the available, usable or potential energy for useful work) which the universe is able to utilize to create ever more complex forms of order.[197][198] The only reason Earth's environment does not decay into an equilibrium state is that it receives a daily dose of sunshine and that, in turn, is due to the sun "polluting" interstellar space with decreasing entropy. The sun's fusion power is only possible due to the gravitational disequilibrium of matter that arose from cosmic expansion. In this essence, the vacuum energy can be viewed as the key cause of the negative entropy (i.e. structure) throughout the universe. That humanity might alter the morphology of the vacuum energy to create an energy gradient for useful work is the subject of much controversy. Purported applicationsEdit Physicists overwhelmingly reject any possibility that the zero-point energy field can be exploited to obtain useful energy (work) or uncompensated momentum; such efforts are seen as tantamount to perpetual motion machines. Nevertheless, the allure of free energy has motivated such research, usually falling in the category of fringe science. As long ago as 1889 (before quantum theory or discovery of the zero point energy) Nikola Tesla proposed that useful energy could be obtained from free space, or what was assumed at that time to be an all-pervasive aether.[199] Others have since claimed to exploit zero-point or vacuum energy with a large amount of pseudoscientific literature causing ridicule around the subject.[200][201] Despite rejection by the scientific community, harnessing zero-point energy remains an interest of research by non-scientific entities, particularly in the US where it has attracted the attention of major aerospace/defence contractors and the U.S. Department of Defence as well as in China, Germany, Russia and Brazil.[200][202] Casimir batteries and enginesEdit A common assumption is that the Casimir force is of little practical use; the argument is made that the only way to actually gain energy from the two plates is to allow them to come together (getting them apart again would then require more energy), and therefore it is a one-use-only tiny force in nature.[200] In 1984 Robert Forward[203] published work showing how a "vacuum-fluctuation battery" could be constructed. The battery can be recharged by making the electrical forces slightly stronger than the Casimir force to reexpand the plates. In 1995 and 1998 Maclay et al.[204][205] published the first models of a microelectromechanical system (MEMS) with Casimir forces. While not exploiting the Casimir force for useful work, the papers drew attention from the MEMS community due to the revelation that Casimir effect needs to be considered as a vital factor in the future design of MEMS. In particular, Casimir effect might be the critical factor in the stiction failure of MEMS.[206] In 1999 Pinto, a former scientist at NASA's Jet Propulsion Laboratory at Caltech in Pasadena, published in Physical Review his thought experiment (Gedankenexperiment) for a "Casimir engine". The paper showed that continuous positive net exchange of energy from the Casimir effect was possible, even stating in the abstract "In the event of no other alternative explanations, one should conclude that major technological advances in the area of endless, by-product free-energy production could be achieved."[207] In 2001 Capasso et al.[208] showed how the force can be used to control the mechanical motion of a MEMS device, The researchers suspended a polysilicon plate from a torsional rod – a twisting horizontal bar just a few microns in diameter. When they brought a metallized sphere close up to the plate, the attractive Casimir force between the two objects made the plate rotate. They also studied the dynamical behaviour of the MEMS device by making the plate oscillate. The Casimir force reduced the rate of oscillation and led to nonlinear phenomena, such as hysteresis and bistability in the frequency response of the oscillator. According to the team, the system's behaviour agreed well with theoretical calculations. Despite this and several similar peer reviewed papers, there is not a consensus as to whether such devices can produce a continuous output of work. Garret Moddel at University of Colorado has highlighted that he believes such devices hinge on the assumption that the Casimir force is a nonconservative force, he argues that there is sufficient evidence (e.g. analysis by Scandurra (2001)[209]) to say that the Casimir effect is a conservative force and therefore even though such an engine can exploit the Casimir force for useful work it cannot produce more output energy than has been input into the system.[210] In 2008 DARPA solicited research proposals in the area of Casimir Effect Enhancement (CEE).[211] The goal of the program is to develop new methods to control and manipulate attractive and repulsive forces at surfaces based on engineering of the Casimir Force. A 2008 patent by Haisch and Moddel[212] details a device that is able to extract power from zero-point fluctuations using a gas that circulates through a Casimir cavity. As gas atoms circulate around the system they enter the cavity. Upon entering the electrons spin down to release energy via electromagnetic radiation. This radiation is then extracted by an absorber. On exiting the cavity the ambient vacuum fluctuations (i.e. the zero-point field) impart energy on the electrons to return the orbitals to previous energy levels, as predicted by Senitzky (1960).[102] The gas then goes through a pump and flows through the system again. A published test of this concept by Moddel[213] was performed in 2012 and seemed to give excess energy that could not be attributed to another source. However it has not been conclusively shown to be from zero-point energy and the theory requires further investigation.[214] Single heat bathsEdit In 1951 Callen and Welton[78] proved the quantum fluctuation-dissipation theorem (FDT) which was originally formulated in classical form by Nyquist (1928)[79] as an explanation for observed Johnson noise[80] in electric circuits. Fluctuation-dissipation theorem showed that when something dissipates energy, in an effectively irreversible way, a connected heat bath must also fluctuate. The fluctuations and the dissipation go hand in hand; it is impossible to have one without the other. The implication of FDT being that the vacuum could be treated as a heat bath coupled to a dissipative force and as such energy could, in part, be extracted from the vacuum for potentially useful work.[81] Such a theory has met with resistance: Macdonald (1962)[215] and Harris (1971)[216] claimed that extracting power from the zero-point energy to be impossible, so FDT could not be true. Grau and Kleen (1982)[217] and Kleen (1986),[218] argued that the Johnson noise of a resistor connected to an antenna must satisfy Planck's thermal radiation formula, thus the noise must be zero at zero temperature and FDT must be invalid. Kiss (1988)[219] pointed out that the existence of the zero-point term may indicate that there is a renormalization problem—i.e., a mathematical artifact—producing an unphysical term that is not actually present in measurements (in analogy with renormalization problems of ground states in quantum electrodynamics). Later, Abbott et al. (1996)[220] arrived at a different but unclear conclusion that "zero-point energy is infinite thus it should be renormalized but not the 'zero-point fluctuations'". Despite such criticism, FDT has been shown to be true experimentally under certain quantum, non-classical conditions. Zero-point fluctuations can, and do, contribute towards systems which dissipate energy.[82] A paper by Armen Allahverdyan and Theo Nieuwenhuizen in 2000[221] showed the feasibility of extracting zero-point energy for useful work from a single bath, without contradicting the laws of thermodynamics, by exploiting certain quantum mechanical properties. There have been a growing number of papers showing that in some instances the classical laws of thermodynamics, such as limits on the Carnot efficiency, can be violated by exploiting negative entropy of quantum fluctuations.[222][84][223][224][225][226][227][228][229][230] Despite efforts to reconcile quantum mechanics and thermodynamics over the years, their compatibility is still an open fundamental problem. The full extent that quantum properties can alter classical thermodynamic bounds is unknown[231] Space travel and gravitational shieldingEdit The use of zero-point energy for space travel is highly speculative. A complete quantum theory of gravitation (that would deal with the role of quantum phenomena like zero-point energy) does not yet exist. Speculative papers explaining a relationship between zero-point energy and gravitational shielding effects have been proposed,[17][232][233][234] but the interaction (if any) is not yet fully understood. Most serious scientific research in this area depends on the theorized anti-gravitational properties of antimatter (currently being tested at the alpha experiment at CERN) and/or the effects of non-Newtonian forces such as the gravitomagnetic field under specific quantum conditions. According to the general theory of relativity, rotating matter can generate a new force of nature, known as the gravitomagnetic interaction, whose intensity is proportional to the rate of spin.[235] In certain conditions the gravitomagnetic field can be repulsive. In neutrons stars for example it can produce a gravitational analogue of the Meissner effect, but the force produced in such an example is theorized to be exceedingly weak.[236] In 1963 Robert Forward,[237] a physicist and aerospace engineer at Hughes Research Laboratories, published a paper showing how within the framework of general relativity "anti-gravitational" effects might be achieved. Since all atoms have spin, gravitational permeability may be able to differ from material to material. A strong toroidal gravitational field that acts against the force of gravity could be generated by materials that have nonlinear properties that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic permeability of iron making it an effective core (i.e. the doughnut of iron) in a transformer, whose properties are dependent on magnetic permeability.[238][239][240] In 1966 Dewitt[241] was first to identify the significance of gravitational effects in superconductors. Dewitt demonstrated that a magnetic-type gravitational field must result in the presence of fluxoid quantization. In 1983, Dewitt's work was substantially expanded by Ross.[242] From 1971 to 1974 Henry William Wallace, a scientist at GE Aerospace was issued with three patents.[243][244][245] Wallace used Dewitt's theory to develop an experimental apparatus for generating and detecting a secondary gravitational field, which he named the kinemassic field (now better known as the gravitomagnetic field). In his three patents, Wallace describes three different methods used for detection of the gravitomagnetic field – change in the motion of a body on a pivot, detection of a transverse voltage in a semiconductor crystal, and a change in the specific heat of a crystal material having spin-aligned nuclei. There are no publicly available independent tests verifying Wallace's devices. Such an effect if any would be small.[246][247][248][249][250][251] Referring to Wallace's patents, a New Scientist article in 1980 stated "Although the Wallace patents were initially ignored as cranky, observers believe that his invention is now under serious but secret investigation by the military authorities in the USA. The military may now regret that the patents have already been granted and so are available for anyone to read."[252] A further reference to Wallace's patents occur in an electric propulsion study prepared for the Astronautics Laboratory at Edwards Air Force Base which states: "The patents are written in a very believable style which include part numbers, sources for some components, and diagrams of data. Attempts were made to contact Wallace using patent addresses and other sources but he was not located nor is there a trace of what became of his work. The concept can be somewhat justified on general relativistic grounds since rotating frames of time varying fields are expected to emit gravitational waves."[253] In 1986 the U.S. Air Force's then Rocket Propulsion Laboratory (RPL) at Edwards Air Force Base solicited "Non Conventional Propulsion Concepts" under a small business research and innovation program. One of the six areas of interest was "Esoteric energy sources for propulsion, including the quantum dynamic energy of vacuum space..." In the same year BAE Systems launched "Project Greenglow" to provide a "focus for research into novel propulsion systems and the means to power them"[254][255] In 1988 Kip Thorne et al.[256] published work showing how traversable wormholes can exist in spacetime only if they are threaded by quantum fields generated by some form of exotic matter that has negative energy. In 1993 Scharnhorst and Barton[121] showed that the speed of a photon will be increased if it travels between two Casimir plates, an example of negative energy. In the most general sense, the exotic matter needed to create wormholes would share the repulsive properties of the inflationary energy, dark energy or zero-point radiation of the vacuum.[257] Building on the work of Thorne, in 1994 Miguel Alcubierre[258] proposed a method for changing the geometry of space by creating a wave that would cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand (see Alcubierre drive). The ship would then ride this wave inside a region of flat space, known as a warp bubble and would not move within this bubble but instead be carried along as the region itself moves due to the actions of the drive. In 1992 Evgeny Podkletnov[259] published a heavily debated[260][261][262][263] journal article claiming a specific type of rotating superconductor could shield gravitational force. Independently of this, from 1991 to 1993 Ning Li and Douglas Torr published a number of articles[264][265][266] about gravitational effects in superconductors. One finding they derived is the source of gravitomagnetic flux in a type II superconductor material is due to spin alignment of the lattice ions. Quoting from their third paper: "It is shown that the coherent alignment of lattice ion spins will generate a detectable gravitomagnetic field, and in the presence of a time-dependent applied magnetic vector potential field, a detectable gravitoelectric field." The claimed size of the generated force has been disputed by some[267][268] but defended by others.[269][270] In 1997 Li published a paper attempting to replicate Podkletnov's results and showed the effect was very small, if it existed at all.[271] Li is reported to have left the University of Alabama in 1999 to found the company AC Gravity LLC.[272] AC Gravity was awarded a U.S. DOD grant for $448,970 in 2001 to continue anti-gravity research. The grant period ended in 2002 but no results from this research were ever made public.[273] In 2002 Phantom Works, Boeing's advanced research and development facility in Seattle, approached Evgeny Podkletnov directly. Phantom Works was blocked by Russian technology transfer controls. At this time Lieutenant General George Muellner, the outgoing head of the Boeing Phantom Works, confirmed that attempts by Boeing to work with Podkletnov had been blocked by Moscow, also commenting that "The physical principles – and Podkletnov's device is not the only one – appear to be valid... There is basic science there. They're not breaking the laws of physics. The issue is whether the science can be engineered into something workable"[274] Froning and Roach (2002)[275] put forward a paper that builds on the work of Puthoff, Haisch and Alcubierre. They used fluid dynamic simulations to model the interaction of a vehicle (like that proposed by Alcubierre) with the zero-point field. Vacuum field perturbations are simulated by fluid field perturbations and the aerodynamic resistance of viscous drag exerted on the interior of the vehicle is compared to the Lorentz force exerted by the zero-point field (a Casimir-like force is exerted on the exterior by unbalanced zero-point radiation pressures). They find that the optimized negative energy required for an Alcubierre drive is where it is a saucer-shaped vehicle with toroidal electromagnetic fields. The EM fields distort the vacuum field perturbations surrounding the craft sufficiently to affect the permeability and permittivity of space. In 2014 NASA's Eagleworks Laboratories[276] announced that they had successfully validated the use of a Quantum Vacuum Plasma Thruster which makes use of the Casimir effect for propulsion.[277][278] In 2016 a scientific paper by the team of NASA scientists passed peer review for the first time.[279] The paper suggests that the zero-point field acts as pilot-wave and that the thrust may be due to particles pushing off the quantum vacuum. While peer review doesn't guarantee that a finding or observation is valid, it does indicate that independent scientists looked over the experimental setup, results, and interpretation and that they could not find any obvious errors in the methodology and that they found the results reasonable. In the paper, the authors identify and discuss nine potential sources of experimental errors, including rogue air currents, leaky electromagnetic radiation, and magnetic interactions. 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Bibcode:2011Natur.479..376W. doi:10.1038/nature10561. ISSN 0028-0836. PMID 22094697. Wikimedia Commons has media related to Zero-point energy. Wikiquote has quotations related to: Zero-point energy Look up zero-point energy in Wiktionary, the free dictionary. Nima Arkani-Hamed on the issue of vacuum energy and dark energy. Steven Weinberg on the cosmological constant problem. Retrieved from "https://en.wikipedia.org/w/index.php?title=Zero-point_energy&oldid=904881350"	CommonCrawl
Make the correct alternative in the following question: A student was asked to prove a statement P(n) by induction. He proved P(k +1) is true whenever P(k) is true for all k > 5 ∈">∈∈ N and also P(5) is true. On the basis of this he could conclude that P(n) is true. (a) for all $n \in \mathbf{N}$ (b) for all n > 5 (c) for all n ≥">≥≥ 5 (d) for all n < 5 As, P(5) is true and $\mathrm{P}(k+1)$ is true whenever $\mathrm{P}(k)$ is true for all $k>5 \in \mathbf{N}$. By the definition of the priniciple of mathematical induction, we get P(n) is true for all n ≥">≥≥ 5. Hence, the correct alternative is option (c).	CommonCrawl
Volume integral In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function. Part of a series of articles about Calculus • Fundamental theorem • Limits • Continuity • Rolle's theorem • Mean value theorem • Inverse function theorem Differential Definitions • Derivative (generalizations) • Differential • infinitesimal • of a function • total Concepts • Differentiation notation • Second derivative • Implicit differentiation • Logarithmic differentiation • Related rates • Taylor's theorem Rules and identities • Sum • Product • Chain • Power • Quotient • L'Hôpital's rule • Inverse • General Leibniz • Faà di Bruno's formula • Reynolds Integral • Lists of integrals • Integral transform • Leibniz integral rule Definitions • Antiderivative • Integral (improper) • Riemann integral • Lebesgue integration • Contour integration • Integral of inverse functions Integration by • Parts • Discs • Cylindrical shells • Substitution (trigonometric, tangent half-angle, Euler) • Euler's formula • Partial fractions • Changing order • Reduction formulae • Differentiating under the integral sign • Risch algorithm Series • Geometric (arithmetico-geometric) • Harmonic • Alternating • Power • Binomial • Taylor Convergence tests • Summand limit (term test) • Ratio • Root • Integral • Direct comparison • Limit comparison • Alternating series • Cauchy condensation • Dirichlet • Abel Vector • Gradient • Divergence • Curl • Laplacian • Directional derivative • Identities Theorems • Gradient • Green's • Stokes' • Divergence • generalized Stokes Multivariable Formalisms • Matrix • Tensor • Exterior • Geometric Definitions • Partial derivative • Multiple integral • Line integral • Surface integral • Volume integral • Jacobian • Hessian Advanced • Calculus on Euclidean space • Generalized functions • Limit of distributions Specialized • Fractional • Malliavin • Stochastic • Variations Miscellaneous • Precalculus • History • Glossary • List of topics • Integration Bee • Mathematical analysis • Nonstandard analysis In coordinates It can also mean a triple integral within a region $D\subset \mathbb {R} ^{3}$ of a function $f(x,y,z),$ and is usually written as: $\iiint _{D}f(x,y,z)\,dx\,dy\,dz.$ A volume integral in cylindrical coordinates is $\iiint _{D}f(\rho ,\varphi ,z)\rho \,d\rho \,d\varphi \,dz,$ and a volume integral in spherical coordinates (using the ISO convention for angles with $\varphi $ as the azimuth and $\theta $ measured from the polar axis (see more on conventions)) has the form $\iiint _{D}f(r,\theta ,\varphi )r^{2}\sin \theta \,dr\,d\theta \,d\varphi .$ Example Integrating the equation $f(x,y,z)=1$ over a unit cube yields the following result: $\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}1\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}(1-0)\,dy\,dz=\int _{0}^{1}\left(1-0\right)dz=1-0=1$ So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function: ${\begin{cases}f:\mathbb {R} ^{3}\to \mathbb {R} \\f:(x,y,z)\mapsto x+y+z\end{cases}}$ the total mass of the cube is: $\int _{0}^{1}\int _{0}^{1}\int _{0}^{1}(x+y+z)\,dx\,dy\,dz=\int _{0}^{1}\int _{0}^{1}\left({\frac {1}{2}}+y+z\right)dy\,dz=\int _{0}^{1}(1+z)\,dz={\frac {3}{2}}$ See also • Divergence theorem • Surface integral • Volume element External links • "Multiple integral", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W. "Volume integral". MathWorld. Calculus Precalculus • Binomial theorem • Concave function • Continuous function • Factorial • Finite difference • Free variables and bound variables • Graph of a function • Linear function • Radian • Rolle's theorem • Secant • Slope • Tangent Limits • Indeterminate form • Limit of a function • One-sided limit • Limit of a sequence • Order of approximation • (ε, δ)-definition of limit Differential calculus • Derivative • Second derivative • Partial derivative • Differential • Differential operator • Mean value theorem • Notation • Leibniz's notation • Newton's notation • Rules of differentiation • linearity • Power • Sum • Chain • L'Hôpital's • Product • General Leibniz's rule • Quotient • Other techniques • Implicit differentiation • Inverse functions and differentiation • Logarithmic derivative • Related rates • Stationary points • First derivative test • Second derivative test • Extreme value theorem • Maximum and minimum • Further applications • Newton's method • Taylor's theorem • Differential equation • Ordinary differential equation • Partial differential equation • Stochastic differential equation Integral calculus • Antiderivative • Arc length • Riemann integral • Basic properties • Constant of integration • Fundamental theorem of calculus • Differentiating under the integral sign • Integration by parts • Integration by substitution • trigonometric • Euler • Tangent half-angle substitution • Partial fractions in integration • Quadratic integral • Trapezoidal rule • Volumes • Washer method • Shell method • Integral equation • Integro-differential equation Vector calculus • Derivatives • Curl • Directional derivative • Divergence • Gradient • Laplacian • Basic theorems • Line integrals • Green's • Stokes' • Gauss' Multivariable calculus • Divergence theorem • Geometric • Hessian matrix • Jacobian matrix and determinant • Lagrange multiplier • Line integral • Matrix • Multiple integral • Partial derivative • Surface integral • Volume integral • Advanced topics • Differential forms • Exterior derivative • Generalized Stokes' theorem • Tensor calculus Sequences and series • Arithmetico-geometric sequence • Types of series • Alternating • Binomial • Fourier • Geometric • Harmonic • Infinite • Power • Maclaurin • Taylor • Telescoping • Tests of convergence • Abel's • Alternating series • Cauchy condensation • Direct comparison • Dirichlet's • Integral • Limit comparison • Ratio • Root • Term Special functions and numbers • Bernoulli numbers • e (mathematical constant) • Exponential function • Natural logarithm • Stirling's approximation History of calculus • Adequality • Brook Taylor • Colin Maclaurin • Generality of algebra • Gottfried Wilhelm Leibniz • Infinitesimal • Infinitesimal calculus • Isaac Newton • Fluxion • Law of Continuity • Leonhard Euler • Method of Fluxions • The Method of Mechanical Theorems Lists • Differentiation rules • List of integrals of exponential functions • List of integrals of hyperbolic functions • List of integrals of inverse hyperbolic functions • List of integrals of inverse trigonometric functions • List of integrals of irrational functions • List of integrals of logarithmic functions • List of integrals of rational functions • List of integrals of trigonometric functions • Secant • Secant cubed • List of limits • Lists of integrals Miscellaneous topics • Complex calculus • Contour integral • Differential geometry • Manifold • Curvature • of curves • of surfaces • Tensor • Euler–Maclaurin formula • Gabriel's horn • Integration Bee • Proof that 22/7 exceeds π • Regiomontanus' angle maximization problem • Steinmetz solid	Wikipedia
in theory (As will become clear soon, I am totally out of my depth: Readers who find glaring mistakes, and even small ones, should note them in the comments.) In the previous post on expanders we left off with the observation that if we want to construct an expanding Cayley graph, then we need to start from a group where all $n$ elements of the group can be obtained by a sum of $O(log n)$ terms chosen, with repetitions, from a set of $O(1)$ generators; this means that a constant root of the $n^{O(1)}$ ways of rearranging such sums must lead to different results. The group, then, has to be non-commutative in a very strong sense. I will describe a way in which a group used to construct expanders is "very non-commutative." This will not be a sufficient property for the expansion, but it is closely related and it will come up in the paper of Gowers that we will talk about later. With this objective in mind, let's see the notion of group representation. Let's start from the simplest group, the cyclic group of $N$ elements $Z_N$ with the operation of addition (or $Z/NZ$ as mathematicians write). One can visualize this group as $N$ points equally spaced along a circle in the plane, with element $a$ forming an angle of $2\pi a/ N$ with the $x$ axis. If we want to add an element that forms an angle $\alpha$ to an element that forms an angle $\beta$, that's the same as putting the "two angles next to each other" and we get a point that forms an angle $\alpha+\beta$. More algebraically, we identify an element $a$ with the complex number $e^{2\pi a i/n}$ and we realize group addition as multiplication of complex numbers. There is, however, another way of visualizing the cyclic group on a cycle. We can think of group element $a$ as the operation that takes a point on the cycle and moves it by an angle of $2\pi a/N$. Addition of group elements now corresponds to "composition," that is, corresponds to applying one rotation after the other. In this view, element $a$ is now the function that maps complex number $x$ into complex number $x\cdot e^{2\pi a i/N}$. Suppose now that our group is a product $Z_{N_1} \times \cdots \times Z_{N_k}$ of cyclic groups. This means that a group elements if a $k$-tuple of the form $(a_1,\ldots,a_d)$, and that $(a_1,\ldots,a_d) + (b_1,\ldots,b_d) = (a_1+b_1 \bmod N_1,\ldots,a_k+b_k \bmod N_k)$ It now makes sense to view a group element $(a_1,\ldots,a_d)$ as a "high-dimensional rotation" operation that takes in input $k$ complex numbers $(x_1,\ldots,x_k)$ and outputs the $k$ complex numbers $(x_1 \cdot e^{2\pi a_1 i /N_1},\ldots, x_k \cdot e^{2\pi a_k i /N_k})$ If we take this view, we have, again, the property that group addition becomes composition of functions. Note, also, that the function that we have associated to each group element is a very simple type of linear function: it is simply multiplication of $x$ times a diagonal matrix that has diagonal $(e^{2\pi a_1 i /N_1},\ldots, e^{2\pi a_k i /N_k})$ Notice, also, that if $f$ is a function of the form $f(x) = A\cdot x$, where $A$ is a matrix, and $g$ is a function of the form $B\cdot x$ where $B$ is a matrix, then $f(g(x))= A\cdot B\cdot x$. That is, for linear functions, function composition is the same as matrix multiplication. To summarize, we have started from the group $Z_{N_1} \times \cdots \times Z_{N_k}$, and we have found a way to associate a complex-valued diagonal matrix to each group element in such a way that group addition becomes matrix multiplication. It is known that all finite Abelian groups can be written as $Z_{N_1} \times \cdots \times Z_{N_k}$, so this type of representation via diagonal matrices is possible for every finite Abelian group. What about more general groups? If $G$ is an arbitrary finite group, it is possible to associate to each element a block-diagonal matrix in such a way that group addition becomes matrix multiplication. (It is common to call a group operation "multiplication" when a group is not Abelian, but for consistency with the rest of the post I will keep calling it addition.) (By the way, it is also possible to represent infinite groups by associating a linear operator to each group element, but we will only discuss finite groups here.) If the group is Abelian, then the matrices are diagonal, that is, they are block-diagonal matrices with "block size one." So one way of quantifying "how non-Abelian" is a group is to consider how small the blocks can be in such matrix representations. That's the dimension of the representation. Here is an example of a family of groups whose representations cannot be low-dimensional. Let $p$ be a prime (it could also be a prime power) and let us consider $2 \times 2$ matrices whose entries are integers $\bmod p$. Let us restrict ourselves to matrices whose determinant is 1 modulo $p$, and consider the operation of matrix multiplication (where, also, all operations are mod $p$). This set of matrices forms a group, because the matrices are invertible (the determinant is non-zero) and the set contains the identity matrix and is closed under product. This group is called $SL(2,p)$. The group $SL(2,p)$ contains the tiny subgroup of two elements $\{ I,-I\}$; in the representation of $SL(2,p)$ this shows up as a block of size 1. If we take the quotient of $SL(2,p)$ by $\{ I,-I \}$ then we get another group, which is called $PSL(2,p)$. It is now a theorem of Frobenius that every representation of $PSL(2,p)$ has dimension at least $(p-1)/2$. This is really large compared to the size of $PSL(2,p)$: the group $SL(2,p)$ has $p(p-1)^2$ elements, and so $PSL(2,p)$ has $p(p-1)^2/2$ elements. The dimension of the representation is thus approximately the cube root of the number of elements of the group. Going back to representations of Abelian groups, we see that, in that case, not only each block had size one, but also that the entire matrix had size at most logarithmic in the number of elements of the group. This shows that $SL(2,p)$ and $PSL(2,p)$ are, at least in this particular sense, "very non-Abelian," and it is an encouraging sign that they may be useful in constructing expanders and other quasi-random objects. On his way toward a question about subsets of groups not containing triples of the form $\{ a, b, a+b\}$, Gowers shows that every dense Cayley graph constructed on $PSL(2,p)$ has constant edge expansion. (By dense, we mean that the degree is $\Omega(n)$, where $n$ is the number of vertices.) This is a result that, assuming Frobenius theorem, has a simple proof, which I hope to understand and describe at a later point. The celebrated Ramanujan graphs of Lubotzky, Phillips and Sarnak are constant-degree Cayley graphs constructed from $PGL(2,p)$, which is the same as $PSL(2,p)$ except that we put no restriction on the determinant. Their relationship between degree and eigenvalue gap is best possible. The analysis of Lubotzky et al. is, unfortunately, completely out of reach. posted by Luca at 3:57 PM Thanks, Luca! Greg Kuperberg has been on my case to learn some representation theory. So I bought a textbook, but gave up halfway through the first chapter. Then I read your blog post and understood everything. One tiny correction: It is now a theorem of Frobenius that every representation of PSL(2,p) has dimension at least (q-1)/2. You mean (p-1)/2. So, can anyone recommend a good reference for representation theory? Depending on your style I can recommend the following two books. If you like your math crisp and clear there is probably nothing better than Serre's "Linear representations of finite groups". This is an absolute classic and it manages to explain all the basics in less than 50 pages (did I mention that it is crisp?). I'm curious if this is the book that Scott tried to read. On the other side of the spectrum (ha!) there is "Fourier Analysis on Finite Groups and Applications" by Audrey Terras. This book describes lots of applications of representation theory that are relevant for theoretical computer science. It also tries to explain various deep results that go far beyond the proven material in the book such as the Selberg trace formula. Some people might enjoy this, while others will find it annoying. The latter should go for Serre. Furthermore, for a given group, the representation is unique, up to rearranging the rows and the columns. This doesn't sound right. For the mod 3 group Z/3, there are (at least) three different representations: A) All elements in Z/3 get mapped to 1 B) 0 gets mapped to 1, 1 gets mapped to exp(2 pi i/3) and 2 gets mapped to exp(2 pi i 2/3) C) 0 gets mapped to 1, 1 gets mapped to exp(2 pi i 2/3) and 2 gets mapped to exp(2 pi i 1/3). In all cases the elements of the group get represented by 1x1 dimensional matrices, so there is not much rearranging of the columns/rows that can be done, yet the representations are different. What did you mean to say here? another small correction >It is also worth noticing that, among diagonal matrices, product is commutative, and so this representation is impossible for finite groups that are not Abelian. its the other way around, non abelian groups can have abelian representation, but abelian groups cant have non abelian representations(f(g)and f(h) can commute even though g and h dont,but not the other way) so a non-abelian gp which has a cyclic quotient can have a rep of the above form Actually, for SL(2,p) more is known. Every generating set of size p^x yields an expanding Cayley graph. This holds for any x>0, and the expansion depends only on x. This follows from a result of Helfgott appearing in http://arxiv.org/abs/math.GR/0509024 However, Helfgott's proof uses a sum-product argument. Gowers only seems to use information on the dimensions of irreducible representations, so it's much more general. Building upon Helfgott's argument, Bourgain and Gamburd prove that if a for bounded size set U in SL(2,p) the Cayley graph has large enough girth then the Cayley graph is automatically an expander. This girth condition holds in particular for random subsets, and it also holds for the "Ramanujan graph" generators - no very fancy math is needed. I THINK girth is all they need - their paper is not online and I only heard a talk about it a year ago. The paper is called "new results on expanders", and it has apparently appeared on ComptesRendus Acad. Sci. Paris, Ser. I, 342, 2006, 717-721. So the Ramanujan graph construction is not so inaccessible I think. If you want the optimal expansion constant you indeed need a strong theorem in number theory whose proof I don't know (Deligne's proof of some Ramanujan conjecture), but it is simple to state: it estimates the number of solutions mod p to some given system of equations very accurately. Less accurate estimates are not as hard to obtain, and still give enough to prove expansion. The proof of expansion of these SL(2,p) results is based on two things: (1) In any Cayley graph of a group G, the second eigenvalue appears with multiplicity which is at least the minimal dimension of an irreducible representation of G. In SL(2,p), this means that in any Cayley graph of it the second eigenvalue appears at least (p-1) times. (2) The number of k-cycles in a graph is equal to the sum of the k-th powers of all the eigenvalues of the graph. So if in some group you can show that there are "few" cycles of some length k, then you have shown that the sum of the k-th powers of all the eigenvalues is small. This is usually not enough - it gives you information about the average k-th power of ALL eigenvalues, while we are interested in the second eigenvalue. BUT: if the second eigenvalue appears with high enough multiplicity, then knowing the average of the k-th power of all eigenvalues does say something about the k-th power of the second eigenvalue itself. That's basically the whole proof idea in the SL(2,p) case. You need a good estimate on the number of k-cycles in the given Cayley graph, along with lower bounds. This is not trivial, but it's doable. This proof idea does not work for other groups. For example, the symmetric group S_n has representations of dimension n-1, which is only logarithmic in the size of the group. Still, generating sets which make an expanding Cayley graph are known. This is a result of Kassabov, and it also uses representation theory, but in a completely different way! Thanks Wim, I don't know what I was thinking about the uniqueness remark. Name: Luca Location: San Francisco, California, United States Marge, I agree with you - in theory. In theory, communism works. In theory. Homer Simpson Expanders and groups Post-modern cryptography It's that time of the year Now, that's radical Don't taser me, I am a student Call it, already On being second LA mysteries The hard science of Sociology Some Older Posts Random KSAT [1] [2] [3] The trouble with "nerd pride" ICM 2006 [1] [2] [3] [4] [5} [6] O(log n) approximation Average-case complexity Szemeredi's Theorem: [1] [2] [3] [4] [5] Gowers Uniformity We should see other people On rejection The vision thing The dreamy-eyed alumni The Tao of Berkeley The things I notice Home, queer home Don't come to the podium!	CommonCrawl
Fundamental theorem of algebraic K-theory In algebra, the fundamental theorem of algebraic K-theory describes the effects of changing the ring of K-groups from a ring R to $R[t]$ or $R[t,t^{-1}]$. The theorem was first proved by Hyman Bass for $K_{0},K_{1}$ and was later extended to higher K-groups by Daniel Quillen. Description Let $G_{i}(R)$ be the algebraic K-theory of the category of finitely generated modules over a noetherian ring R; explicitly, we can take $G_{i}(R)=\pi _{i}(B^{+}{\text{f-gen-Mod}}_{R})$, where $B^{+}=\Omega BQ$ is given by Quillen's Q-construction. If R is a regular ring (i.e., has finite global dimension), then $G_{i}(R)=K_{i}(R),$ the i-th K-group of R.[1] This is an immediate consequence of the resolution theorem, which compares the K-theories of two different categories (with inclusion relation.) For a noetherian ring R, the fundamental theorem states:[2] • (i) $G_{i}(R[t])=G_{i}(R),\,i\geq 0$. • (ii) $G_{i}(R[t,t^{-1}])=G_{i}(R)\oplus G_{i-1}(R),\,i\geq 0,\,G_{-1}(R)=0$. The proof of the theorem uses the Q-construction. There is also a version of the theorem for the singular case (for $K_{i}$); this is the version proved in Grayson's paper. See also • basic theorems in algebraic K-theory Notes 1. By definition, $K_{i}(R)=\pi _{i}(B^{+}{\text{proj-Mod}}_{R}),\,i\geq 0$. 2. Weibel 2013, Ch. V. Theorem 3.3 and Theorem 6.2 References • Daniel Grayson, Higher algebraic K-theory II [after Daniel Quillen], 1976 • Srinivas, V. (2008), Algebraic K-theory, Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.), Boston, MA: Birkhäuser, ISBN 978-0-8176-4736-0, Zbl 1125.19300 • Weibel, Charles (2013). "The K-book: An introduction to algebraic K-theory". Graduate Studies in Math. 145.	Wikipedia
Truncated normal hurdle model In econometrics, the truncated normal hurdle model is a variant of the Tobit model and was first proposed by Cragg in 1971.[1] In a standard Tobit model, represented as $y=(x\beta +u)1[x\beta +u>0]$, where $u\|x\sim N(0,\sigma ^{2})$This model construction implicitly imposes two first order assumptions:[2] 1. Since: $\partial P[y>0]/\partial x_{j}=\varphi (x\beta /\sigma )\beta _{j}/\sigma $ and $\partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}=\beta _{j}\{1-\theta (x\beta /\sigma \}$, the partial effect of $x_{j}$ on the probability $P[y>0]$ and the conditional expectation: $\operatorname {E} [y\mid x,y>0]$ has the same sign:[3] 2. The relative effects of $x_{h}$ and $x_{j}$ on $P[y>0]$ and $\operatorname {E} [y\mid x,y>0]$ are identical, i.e.: ${\frac {\partial P[y>0]/\partial x_{h}}{\partial P[y>0]/\partial x_{j}}}={\frac {\partial \operatorname {E} [y\mid x,y>0]/\partial x_{h}}{\partial \operatorname {E} [y\mid x,y>0]/\partial x_{j}}}={\frac {\beta _{h}}{\beta _{j}}}\|$ However, these two implicit assumptions are too strong and inconsistent with many contexts in economics. For instance, when we need to decide whether to invest and build a factory, the construction cost might be more influential than the product price; but once we have already built the factory, the product price is definitely more influential to the revenue. Hence, the implicit assumption (2) doesn't match this context.[4] The essence of this issue is that the standard Tobit implicitly models a very strong link between the participation decision $(y=0$ or $y>0)$ and the amount decision (the magnitude of $y$ when $y>0$). If a corner solution model is represented in a general form: $y=s\centerdot w,$ , where $s$ is the participate decision and $w$ is the amount decision, standard Tobit model assumes: $s=1[x\beta +u>0];$ $w=x\beta +u.$ To make the model compatible with more contexts, a natural improvement is to assume: $s=1[x\gamma +u>0],{\text{ where }}u\sim N(0,1);$ $w=x\beta +e,$ where the error term ($e$) is distributed as a truncated normal distribution with a density as $\varphi (\cdot )/\Phi \left({\frac {x\beta }{\sigma }}\right)/\sigma ;$ ;} $s$ and $w$ are independent conditional on $x$. This is called Truncated Normal Hurdle Model, which is proposed in Cragg (1971).[1] By adding one more parameter and detach the amount decision with the participation decision, the model can fit more contexts. Under this model setup, the density of the $y$ given $x$ can be written as: $f(y\mid x)=[1-\Phi (\chi \gamma )]^{1[y=0]}\cdot \left[{\frac {\Phi \ (\chi \gamma )}{\Phi (\chi \beta /\sigma )}}\left.\varphi \left({\frac {y-\chi \beta }{\sigma }}\right)\right/\sigma \right]^{1[y>0]}$ From this density representation, it is obvious that it will degenerate to the standard Tobit model when $\gamma =\beta /\sigma .$ This also shows that Truncated Normal Hurdle Model is more general than the standard Tobit model. The Truncated Normal Hurdle Model is usually estimated through MLE. The log-likelihood function can be written as: ${\begin{aligned}\ell (\beta ,\gamma ,\sigma )={}&\sum _{i=1}^{N}1[y_{i}=0]\log[1-\Phi (x_{i}\gamma )]+1[y_{i}>0]\log[\Phi (x_{i}\gamma )]\\[5pt]&{}+1[y_{i}>0]\left[-\log \left[\Phi \left({\frac {x_{i}\beta }{\sigma }}\right)\right]+\log \left(\varphi \left({\frac {y_{i}-x_{i}\beta }{\sigma }}\right)\right)-\log(\sigma )\right]\end{aligned}}$ From the log-likelihood function, $\gamma $ can be estimated by a probit model and $(\beta ,\sigma )$ can be estimated by a truncated normal regression model.[5] Based on the estimates, consistent estimates for the Average Partial Effect can be estimated correspondingly. See also • Hurdle model • Tobit model References 1. Cragg, John G. (September 1971). "Some Statistical Models for Limited Dependent Variables with Application to the Demand for Durable Goods". Econometrica. 39 (5): 829–844. doi:10.2307/1909582. JSTOR 1909582. 2. Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 690. 3. Here, the notation follows Wooldrige (2002). Function $\theta (x)=\lambda '$ where $\lambda (x)=\varphi (\chi )/\Phi (\chi ),$ can be proved to be between 0 and 1. 4. For more application example of corner solution model, refer to: Daniel J. Phaneuf, (1999): “A Dual Approach to Modeling Corner Solutions in Recreation Demand”,Journal of Environmental Economics and Management, Volume 37, Issue 1, Pages 85-105, ISSN 0095-0696. 5. Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 692-694.	Wikipedia
\begin{definition}[Definition:Saddle Point (Game Theory)] Let $G$ be a two-player zero-sum game. Let $G$ be defined by a payoff table $T$. Let $E$ be an entry in $T$ such that: :$E$ is the smallest entry in its row :$E$ is the largest entry in its column. Then $E$ is a '''saddle point''' of $G$. \end{definition}	ProofWiki
\begin{document} \textwidth 150mm \textheight 225mm \title{Some upper bounds for the signless Laplacian spectral radius of digraphs \thanks{ Supported by the National Natural Science Foundation of China (No.11171273).}} \author{{Weige Xi and Ligong Wang\footnote{Corresponding author.} }\\ {\small Department of Applied Mathematics, School of Science,}\\ {\small Northwestern Polytechnical University, Xi'an, Shaanxi 710072, P.R.China} \\{\small E-mail: [email protected], [email protected] }\\} \date{} \maketitle \begin{center} \begin{minipage}{120mm} \vskip 0.3cm \begin{center} {\small {\bf Abstract}} \end{center} {\small Let $G=(V(G) ,E(G))$ be a digraph without loops and multiarcs, where $V(G)=\{v_1,v_2,\ldots,v_n\}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. Let $d_i^{+}$ be the outdegree of the vertex $v_i$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$. Then we call $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. The spectral radius of $Q(G)$ is called the signless Laplacian spectral radius of $G$, denoted by $q(G)$. In this paper, some upper bounds for $q(G)$ are obtained. Furthermore, some upper bounds on $q(G)$ involving outdegrees and the average 2-outdegrees of the vertices of $G$ are also derived. \vskip 0.1in \noindent {\bf Key Words}: \ Digraph, Signless Laplacian spectral radius, Upper bounds. \vskip 0.1in \noindent {\bf AMS Subject Classification (2000)}: \ 05C50 15A18} \end{minipage} \end{center} \section{Introduction } \label{sec:ch6-introduction} Let $G=(V(G), E(G))$ be a digraph without loops and multiarcs, where $V(G)=\{v_1,v_2,$ $\ldots, v_n\}$ and $E(G)$ are the vertex set and the arc set of $G$, respectively. If $(v_i, v_j)$ be an arc of $G$, then $v_i$ is called the initial vertex of this arc and $v_j$ is called the terminal vertex of this arc. For any vertex $v_i$ of $G$, we denote $N_i^{+}=N_{v_i}^{+}(G)=\{v_j : (v_i,v_j)\in E(G) \}$ and $N_i^{-}=N_{v_i}^{-}(G)=\{v_j : (v_j,v_i)\in E(G) \}$ the set of out-neighbors and in-neighbors of $v_i$, respectively. Let $d_i^{+}=\|N_i^{+}\|$ denote the outdegree of the vertex $v_i$ and $d_i^{-}=\|N_i^{-}\|$ denote the indegree of the vertex $v_i$ in the digraph $G$. The maximum vertex outdegree is denoted by $\Delta^+$, and the minimum outdegree by $\delta^+$. If $\delta^+=\Delta^+$, then $G$ is a regular digraph. Let $t_i^+=\sum\limits_{v_j\in N_i^{+}}d_j^{+}$ be the 2-outdegree of the vertex $v_i$, $m_i^{+}=\frac{t_i^{+}}{d_i^{+}}$ the average 2-outdegree of the vertex $v_i$. A digraph is strongly connected if for every pair of vertices $v_i,v_j\in V(G)$, there exists a directed path from $v_i$ to $v_j$ and a directed path from $v_j$ to $v_i$. In this paper, we consider finite digraphs without loops and multiarcs, which have at least one arc. For a digraph $G$, let $A(G)=(a_{ij})$ denote the adjacency matrix of $G$, where $a_{ij}=1$ if $(v_i,v_j)\in E(G)$ and $a_{ij}=0$ otherwise. Let $D(G)=\textrm{diag}(d_1^{+},d_2^{+},\ldots,d_n^{+})$ be the diagonal matrix with outdegrees of the vertices of $G$ and $Q(G)=D(G)+A(G)$ the signless Laplacian matrix of $G$. However, the signless Laplacian matrix of an undirected graph $D$ can be treated as the signless Laplacian matrix of the digraph $G'$, where $G'$ is obtained from $D$ by replace each edge with pair of oppositely directed arcs joining the same pair of vertices. Therefore, the research of the signless Laplacian matrix of a digraph has more universal significance than undirected graph. The eigenvalues of $Q(G)$ are called the signless Laplacian eigenvalues of $G$, denoted by $q_1,q_2,\ldots,q_n$. In general $Q(G)$ are not symmetric and so its eigenvalues can be complex numbers. We usually assume that $\|q_1\|\geq\|q_2\|\geq\ldots\geq\|q_n\|$. The signless Laplacian spectral radius of $G$ is denoted and defined as $q(G)=\|q_1\|$, i.e., the largest absolute value of the signless Laplacian eigenvalues of $G$. Since $Q(G)$ is a nonnegative matrix, it follows from Perron Frobenius Theorem that $q(G)=q_1$ is a real number. For the Laplacian spectral radius and signless Laplacian spectral radius of an undirected graph are well treated in the literature, see \cite{Wang,WYL,WeLi,Zhu} and \cite{ChWa,CTG,GDC,HaLu,HJZ,WeLi}, respectively. Recently, there are some papers that give some lower or upper bounds for the spectral radius of a digraph, see \cite{RB,GDa,XuXu}. Now we consider the signless Laplacian spectral radius of a digraph $G$. For application it is crucial to be able to computer or at least estimate $q(G)$ for a given digraph. In 2013, S.B. Bozkurt and D. Bozkurt in \cite{BoBo} obtained the following bounds for signless Laplacian spectral radius of a digraph. \noindent\begin{equation}\label{eq:c1} q(G) \leq \max\{d_i^{+}+d_j^{+}: (v_i, v_j)\in E(G)\}. \end{equation} \noindent\begin{equation}\label{eq:c2} q(G) \leq \max\{d_i^{+}+m_i^{+}: v_i\in V(G)\}. \end{equation} \noindent\begin{equation}\label{eq:c3} q(G)\leq \max\bigg\{\frac{d_i^{+}+d_j^{+}+ \sqrt{(d_i^{+}-d_j^{+})^{2}+4m_i^{+}m_j^{+}}}{2}: (v_i, v_j)\in E(G)\bigg\}. \end{equation} \noindent\begin{equation}\label{eq:c4} q(G)\leq \max\bigg\{d_i^{+}+\sqrt{\sum _{v_j:(v_j, v_i)\in E(G)}d_j^{+}}: v_i\in V(G)\bigg\}. \end{equation} In 2014, Hong and You in \cite{HoYo} gave a sharp bound for the signless Laplacian spectral radius of a digraph: \noindent\begin{equation}\label{eq:c5} q(G)\leq \min_{1 \leq i \leq n}\bigg\{\frac{d_1^{+}+2d_i^{+}-1+ \sqrt{(2d_i^{+}-d_1^{+}+1)^{2}+8\sum\limits_{k=1}^{i-1}(d_k^{+}-d_i^{+})}}{2}\bigg\}. \end{equation} \noindent\begin{remark}\label{re:c1} Note that $G$ is a strongly connected digraph for bounds \eqref{eq:c1}, \eqref{eq:c3}, \eqref{eq:c4}, respectively. \end{remark} In this paper, we study on the signless Laplacian spectral radius of a digraph $G$. We obtain some upper bounds for $q(G)$, and we also show that some upper bounds on $q(G)$ involving outdegrees and the average 2-outdegrees of the vertices of $G$ can be obtained from our bounds. \section{Preliminaries Lemmas} \label{sec:1} In this section, we give the following lemmas which will be used in the following study. \noindent\begin{lemma}\label{le:c1} (\cite{HoJo}) \ Let $M=(m_{ij})$ be an $n \times n$ nonnegative matrix with spectral radius $\rho(M)$, i.e., the largest eigenvalues of $M$, and let $R_i=R_i(M)$ be the $i$-th row sum of $M$, i.e., $R_i(M)=\sum\limits_{j=1}^n m_{ij} \ (1 \leq i\leq n)$. Then \begin{equation}\label{eq:ca} \min\{R_{i}(M):1 \leq i\leq n\}\leq \rho(M)\leq \max\{R_{i}(M):1 \leq i\leq n\}.\end{equation} Moreover, if $M$ is irreducible, then any equality holds in \eqref{eq:ca} if and only if $R_1=R_2=\ldots=R_n$. \end{lemma} \noindent\begin{lemma}\label{le:c2} (\cite{HoJo}) \ Let $M$ be an irreducible nonnegative matrix. Then $\rho(M)$ is an eigenvalue of $M$ and there is a positive vector $X$ such that $MX=\rho(M)X$. \end{lemma} \noindent\begin{lemma}\label{le:c6} (\cite{Li}) \ Let $A=(a_{ij})\in \mathbb{C}^{n\times n}$, $r_i=\sum\limits_{j\neq i}\|a_{ij}\|$ for each $i=1,2,\ldots,n$, $S_{ij}=\{z \in \mathbb{C}: \|z-a_{ii}\|\cdot\|z-a_{jj}\|\leq r_ir_j\}$ for all $i\neq j$. Also let $E(A)=\{(i,j) :a_{ij}\neq 0,1 \leq{i\neq j}\leq n\}$. If $A$ is irreducible, then all eigenvalues of $A$ are contained in the following region \begin{equation}\label{eq:ce} \Omega(A)=\bigcup_{(i,j) \in E(A)}S_{ij}. \end{equation} Furthermore, a boundary point $\lambda$ of \eqref{eq:ce} can be an eigenvalue of $A$ only if $\lambda$ locates on the boundary of each oval region $S_{ij}$ for $e_{ij} \in E(A)$. \end{lemma} \section{ Some upper bounds for the signless Laplacian spectral radius of digraphs} \label{sec:2} In this section, we present some upper bounds for the signless Laplacian spectral radius $q(G)$ of a digraph $G$ and also show that some bounds involving outdegrees, the average 2-outdegrees, the maximum outdegree and the minimum outdegree of the vertices of $G$ with $n$ vertices and $m$ arcs can be obtained from our bounds. \noindent\begin{theorem}\label{th:c7} \ Let $G$ be a strongly connected digraph with $n\geq 3$ vertices, $m$ arcs, the maximum vertex outdegree $\Delta^+$ and the minimum outdegree $\delta^{+}$. Then \begin{equation}\label{eq:c27} q(G)\leq \max\{\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}, \delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2} \}. \end{equation} Moreover, if $G(\neq\overset{\longrightarrow}{C_{n}})$ is a regular digraph or $G\cong \overset{\longleftrightarrow}{K}_{1,n-1}$, where $\overset{\longleftrightarrow}{K}_{1,n-1}$ denotes the digraph on $n$ vertices which replace each edge in star graph $K_{1,n-1}$ with the pair of oppositely directed arcs, then the equality holds in \eqref{eq:c27} \end{theorem} \begin{proof} \ From \eqref{eq:c2}, we know that $q(G)\leq \max\{d_i^{+}+m_i^{+}: v_i\in V(G)\}.$ So we only need to prove that $\max\{d_i^{+}+m_i^{+}: v_i\in V(G)\}\leq \max\{\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)} {\Delta^{+}},\delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2} \}$. Suppose $\max\{d_i^{+}+m_i^{+}: v_i\in V(G)\}$ occurs at vertex $u$. Two cases arise $d_u^{+}=1$, or $2\leq d_u^{+}\leq \Delta^{+}$. \noindent{\textbf{{Case 1.}}} \ $d_u^{+}=1.$ Suppose that $N_u^{+}=\{w\}.$ Since $m_u^{+}=d_w^{+}\leq \Delta^{+},$ thus $d_u^{+}+m_u^{+}\leq 1+\Delta^{+}$. Since $\sum\limits_{v_i\in V(G)}d_i^{+}=m,$ let $d_j^{+}=\Delta^{+}$, then $\sum\limits_{i\neq j}d_i^{+}=m- \Delta^{+}\geq (n-1)\delta^{+}$, so $m-(n-1)\delta^{+}\geq \Delta^{+}$. Therefore $\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}\geq \delta^{+}-1+\frac{\Delta^{+}}{\Delta^{+}}=\delta^{+}\geq 1$. Thus $d_u^{+}+m_u^{+}\leq 1+\Delta^{+}\leq \Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}$, the result follows. \noindent{\textbf{{Case 2.}}} \ $2\leq d_u^{+}\leq \Delta^{+}$. Note that $m-(n-1)\delta^{+}\geq d_u^{+}\geq 2$, and \begin{eqnarray} m&=&\sum\limits_{v:(u,v)\in E(G)}d_v^{+}+\sum\limits_{ v:(u,v)\notin E(G)}d_v^{+}\\ &\geq &\sum\limits_{v:(u,v)\in E(G)}d_v^{+}+ d_u^{+}+(n-d_u^{+}-1)\delta^{+}, \end{eqnarray} thus \begin{eqnarray} \sum\limits_{v:(u,v)\in E(G)}d_v^{+}&\leq& m-d_u^{+}-(n-d_u^{+}-1)\delta^{+}\\ &=&m-(n-1)\delta^{+}+(\delta^{+}-1)d_u^{+} \end{eqnarray} $$m_u^{+}=\frac{\sum\limits_{v:(u,v)\in E(G)}d_v^{+}}{d_u^{+}}\leq \frac{m-(n-1)\delta^{+}}{d_u^{+}}+\delta^{+}-1.$$ This follows that $m_u^{+}+d_u^{+}\leq d_u^{+}+\frac{m-(n-1)\delta^{+}}{d_u^{+}}+\delta^{+}-1$. Let $f(x)=x+\frac{m-(n-1)\delta^{+}}{x}+\delta^{+}-1$, where $x\in [2,\Delta^{+}].$ It is easy to see that $f'(x)=1-\frac{m-(n-1)\delta^{+}}{x^{2}}$. Let $a=m-(n-1)\delta^{+},$ then $\sqrt{a}$ is the unique positive root of $f'(x)=0.$ We consider the next three Subcases. \noindent{\textbf{{Subcase 1.}}} \ $\sqrt{a}< 2.$ When $x\in [2,\Delta^{+}],$ since $f'(x)>0,$ then $f(x)\leq f(\Delta^{+})$. \noindent{\textbf{{Subcase 2.}}} \ $2\leq\sqrt{a}\leq \Delta^{+}.$ Then $f'(x)<0$ for $x\in [2,\sqrt{a})$, and $f'(x)\geq 0$, for $x\in [\sqrt{a}, \Delta^{+}]$. Thus, $f(x)\leq \max\{f(2), f(\Delta^{+})\}$. \noindent{\textbf{{Subcase 3.}}} \ $\Delta^{+}<\sqrt{a}$. When $x\in [2,\Delta^{+}]$, since $f'(x)<0$, then $f(x)\leq f(2)$. Recall that $2\leq d_u^{+}\leq \Delta^{+}$, thus $$ m_u^{+}+d_u^{+}\leq \max\{f(2), f(\Delta^{+})\}$$ $$=\max\{\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}, \delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2}\}.$$ If $G(\neq\overset{\longrightarrow}{C_{n}})$ is a regular digraph, then $d_i^{+}+m_i^{+}=2d_i^{+}=2\Delta^{+}$ for all $v_i\in V(G)$. We can get $q(G)=2\Delta^{+}$. Since $G(\neq\overset{\longrightarrow}{C_{n}})$ is a strongly connected digraph, then we may assume that $\Delta^{+} \geq 2$, this implies that $\delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2}=\Delta^{+}+1+\frac{\Delta^{+}}{2}\leq 2\Delta^{+}=\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}$. So $\max\{\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}, \delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2} \}=2\Delta^{+}$. Thus, the equality holds. If $G\cong \overset{\longleftrightarrow}{K}_{1,n-1} $, we can get $q(G)=n$. Since $\delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2}=2+\frac{n-1}{2}\leq n$ from $n\geq 3$ and $\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}=n-1+1-1+\frac{n-1}{n-1}=n$. So $\max\{\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}, \delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2} \}=n$. Thus, the equality also holds. By combining the above discussion, the result follows. \end{proof} \noindent\begin{corollary}\label{co:ck} Let $G$ be a strongly connected digraph with $n\geq 3$ vertices, $m$ arcs, the maximum outdegree $\Delta^+$ and the minimum outdegree $\delta^{+}$. If $\Delta^{+}\geq\frac{m-(n-1)}{2}$ and $\delta^{+}=1$, then \begin{equation}\label{eq:cm} q(G)\leq \Delta^{+}+2. \end{equation} \end{corollary} \begin{proof} Because $\Delta^{+}+\delta^{+}-1+\frac{m-\delta^{+}(n-1)}{\Delta^{+}}\leq \Delta^{+}+2$, $\delta^{+}+1+\frac{m-\delta^{+}(n-1)}{2}\leq \Delta^{+}+2$, therefore by Theorem \ref{th:c7}, we have $q(G)\leq \Delta^{+}+2.$ \end{proof} Let $G^{}(m,n,\frac{m-(n-1)}{2},1)$ be a class of strongly connected digraphs with $\Delta^{+}\geq\frac{m-(n-1)}{2}$, $\delta^{+}=1$, and there exists a vertex $v_0\in V(G)$ such that $d_{v_0}=\Delta^{+}$ and there exists a vertex $v_k\in N_{v_0}^{+}$, $d_{v_k}^{+}\geq 2$. \noindent\begin{remark}\label{re:c2} For $G\in G^{}(m,n,\frac{m-(n-1)}{2},1)$, we have $\Delta^{+}+2\leq\max\{d_i^{+}+d_j^{+}: (v_i, v_j)\in E(G)\}$, thus the upper bound \eqref{eq:cm} is better than the upper bound \eqref{eq:c1} for the class of digraphs $G\in G^{}(m,n,\frac{m-(n-1)}{2},1)$. But for general digraphs, the upper bound \eqref{eq:cm} is incomparable with the upper bound \eqref{eq:c1}. \end{remark} \begin{example} Let $G$ be the digraph of order 4, as shown in Figure 1. Since it has 9 arcs, and the maximum outdegree $\Delta^+=3=\frac{9-(4-1)}{2}$, the minimum outdegree $\delta^{+}=1$, and there exists a vertex $v_4\in N_{v_1}^{+}$, $d_{v_4}^{+}=3>2$, therefore $G=G^{}(9,4,3,1)$. \begin{figure} \caption{Graph $G^(9,4,3,1)$} \end{figure} \end{example} \begin{table}[H] \centering\caption{Values of the upper bounds for example 1.} \begin{tabular}{cccc} \hline &$q(G)$&\eqref{eq:c1}&\eqref{eq:cm}\\ \hline $G^{}(9,4,3,1)$ & 4.7321& 6&5 \\ \hline \end{tabular} \end{table} \noindent\begin{theorem}\label{th:c8} \ Let $G$ be a strongly connected digraph with vertex set $V(G)=\{v_1,v_2,$ $\ldots, v_n\}$ and arc set $E(G)$. Then \begin{equation}\label{eq:c28} q(G)\leq \max\bigg\{\frac{d_i^{+}+d_j^{+}+ \sqrt{(d_i^{+}-d_j^{+})^{2}+{4\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}}}}{2} :(v_i, v_j)\in E(G) \bigg\}.\end{equation} Moreover if $G$ is a regular digraph or a bipartite semiregular digraph, then the equality holds in \eqref{eq:c28}. \end{theorem} \begin{proof} \ From the definition of $D=D(G)$ we get $D^{\frac{1}{2}}=\textrm{diag}(\sqrt{d_i^{+}}:v_i\in V(G)),$ and consider the similar matrix $P=D^{-\frac{1}{2}}Q(G)D^{\frac{1}{2}}.$ Since $G$ is a strongly connected digraph, it is easy to see that $P$ is irreducible and nonnegative. Now the $(i,j)$-th element of $P=D^{-\frac{1}{2}}Q(G)D^{\frac{1}{2}}$ is $$p_{ij} = \begin{cases} \ d_i^{+} & \textrm{ if $i=j $,}\\ \ \frac{\sqrt{d_j^{+}}}{\sqrt{d_i^{+}}} & \textrm{ if $(v_i, v_j)\in E(G) $,}\\ \ 0 & \textrm{ otherwise.} \end{cases}$$ Let $R_i(P)$ be the $i$-th row sum of $P$ and $R_i^{'}(P)=R_i-d_i^{+}.$ Then by Cauchy-Schwarz inequality, we have \begin{align} R_i^{'}(P)^{2}=&\left(\sum\limits_{v_j:(v_i,v_j)\in E(G)}\frac{\sqrt{d_j^{+}}}{\sqrt{d_i^{+}}} \right)^{2} \leq \sum\limits_{v_j:(v_i,v_j)\in E(G)}1^{2}\sum\limits_{v_j:(v_i,v_j)\in E(G)}\frac{d_j^{+}}{d_i^{+}}\\ =&\sum\limits_{v_j:(v_i,v_j)\in E(G)}d_j^{+} =d_i^{+}m_i^{+}. \end{align} Since $P$ is irreducible and nonnegative, $\rho(P)$ denotes the spectral radius of $P$. Then by Lemma \ref{le:c6}, there at least exists $(v_i,v_j)\in E(G)$ such that $\rho(P)$ is contained in the following oval region $$\|\rho(P)-d_i^{+}\|\|\rho(P)-d_j^{+}\|\leq R_i^{'}(P)R_i^{'}(P) \leq\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}.$$ Obviously, $\rho(P)=q(G)>\max\{d_i^{+}: v_i \in E(G)\},$ and $(\rho(P)-d_i^{+})(\rho(P)-d_j^{+})\leq\|\rho(P)-d_i^{+}\|\|\rho(P)-d_j^{+}\|.$ Therefore, solving the above inequality we obtain $$ q(G)\leq\bigg\{\frac{d_i^{+}+d_j^{+}+\sqrt{(d_i^{+}-d_j^{+})^{2} +{4\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}}}}{2} \bigg\}.$$ Hence \eqref{eq:c28} holds. If $G$ is a regular digraph, $$q(G)=2\Delta^{+}=\max\bigg\{\frac{d_i^{+}+d_j^{+}+\sqrt{{(d_i^{+}-d_j^{+})^{2} +4\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}}}}{2} :(v_i, v_j)\in E(G) \bigg\}.$$ Thus, the equality holds. If $G$ is a bipartite semiregular digraph, $$\max\bigg\{\frac{d_i^{+}+d_j^{+}+\sqrt{{(d_i^{+}-d_j^{+})^{2} +4\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}}}}{2} :(v_i, v_j)\in E(G) \bigg\}=d_i^{+}+d_j^{+}.$$ Because $q(G)=\rho(D^{-1}Q(G)D)$, the $i$-th row sum of $D^{-1}Q(G)D$ is $d_i^{+}+m_i^{+}$, and $G$ is a bipartite semiregular digraph, therefore $d_i^{+}+m_i^{+}=d_i^{+}+d_j^{+}, (v_i, v_j)\in E(G)$, that is the row sums of $D^{-1}Q(G)D$ are all equal, then by Lemma \ref{le:c1}, $\rho(D^{-1}Q(G)D)=d_i^{+}+d_j^{+}$. Thus we have \begin{align} q(G)&=\rho(D^{-1}Q(G)D)=d_i^{+}+d_j^{+} \\ &=\max\bigg\{\frac{d_i^{+}+d_j^{+}+\sqrt{{(d_i^{+}-d_j^{+})^{2} +4\sqrt{d_i^{+}m_i^{+}}\sqrt{d_j^{+}m_j^{+}}}}}{2} :(v_i, v_j)\in E(G) \bigg\}. \end{align} Then the equality holds. \end{proof} For a digraph $G=(V(G), E(G)),$ let $f : V(G)\times V(G) \rightarrow \mathbb{R}$ be a function. If $f(v_i, v_j)>0$ for all $(v_i, v_j)\in E(G),$ we say $f$ is positive on arcs. \noindent\begin{theorem}\label{th:c9} \ Let $G=(V(G), E(G))$ be a digraph. Let $f : V(G)\times V(G)\rightarrow \mathbb{R^{+}}\bigcup \{0\}$ be a nonnegative function which is positive on arcs. Then \begin{equation}\label{eq:c29} q(G)\leq \max\left\{\frac{\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k)+ \sum\limits_{v_k:(v_j, v_k)\in E(G)}f(v_j, v_k)}{f(v_i,v_j)} : (v_i,v_j)\in E(G)\right\}. \end{equation} \end{theorem} \begin{proof} \ Let ${\bf X}=(x_1,x_2,\ldots,x_n)^{T}$ be an eigenvector corresponding to the eigenvalue $q(G)$ of $Q(G)$. Since $$Q(G){\bf X}=q(G){\bf X}.$$ Then for $1 \leq i\leq n$ \begin{equation}\label{eq:c30} q(G)x_i=d_i^{+}x_i+\sum\limits_{v_k:(v_i,v_k)\in E(G)}x_k=\sum\limits_{v_k:(v_i,v_k)\in E(G)}(x_i+x_k). \end{equation} By \eqref{eq:c30}, we have $$q(G)(x_i+x_j)=\sum\limits_{v_k:(v_i,v_k)\in E(G)}(x_i+x_k)+\sum\limits_{v_k:(v_j,v_k)\in E(G)}(x_j+x_k).$$ For convenience we use $f(i, j)$ denote $f(v_i, v_j).$ Set $g(i, j)=\frac{x_i+x_j}{f(i,j)}.$ If $(v_i, v_j)\in E(G),$ then \begin{equation}\label{eq:c31} q(G)f(i,j)g(i,j)=\sum\limits_{v_k:(v_i,v_k)\in E(G)}f(i,k)g(i,k)+\sum\limits_{v_k:(v_j,v_k)\in E(G))}f(j,k)g(j,k). \end{equation} By \eqref{eq:c31}, we get \begin{align} \|q(G)f(i,j)g(i,j)\|=&q(G)f(i,j)\|g(i,j)\| \\ \leq&\sum\limits_{v_k:(v_i,v_k)\in E(G)}f(i,k)\|g(i,k)\|+\sum\limits_{v_k:(v_j,v_k)\in E(G)}f(j,k)\|g(j,k)\|. \end{align} Now choose $i_1,j_1$ such that $(v_{i_1}, v_{j_1})\in E(G)$ and $\|g(i_1, j_1)\|=\max\{\|g(i, j)\| : (v_i, v_j)\in E(G)\}.$ If $g\|(i_1, j_1)\|=0,$ then $\|g(i, j)\|=0 $ for all arcs $(v_i, v_j)\in E(G).$ i.e.,$x_i+x_j=0$ for all arcs $(v_i, v_j)\in E(G).$ By \eqref{eq:c30}, we have $q(G)=0$ which is impossible, since $G$ has at least one arc. So $\|g(i_1,j_1)\|>0.$ Then $$q(G)f(i_1,j_1)\|g(i_1,j_1)\|\leq\sum\limits_{v_k:(v_{i_1},v_k)\in E(G)}f(i_1,k)\|g(i_1,k)\|+ \sum\limits_{v_k:(v_{j_1},v_k)\in E(G)}f(j_1,k)\|g(j_1,k)\|.$$ Therefore, we obtain \begin{align} q(G)\leq&\sum\limits_{v_k:(v_{i_1},v_k)\in E(G)}\frac{f(i_1,k)}{f(i_1,j_1)}\frac{\|g(i_1,k)\|}{\|g(i_1,j_1)\|}+ \sum\limits_{v_k:(v_{j_1},v_k)\in E(G)}\frac{f(j_1,k)}{f(i_1,j_1)}\frac{\|g(j_1,k)\|}{\|g(i_1,j_1)\|}\\ \leq&\sum\limits_{v_k:(v_{i_1},v_k)\in E(G)}\frac{f(i_1,k)}{f(i_1,j_1)}+ \sum\limits_{v_k:(v_{j_1},v_k)\in E(G)}\frac{f(j_1,k)}{f(i_1,j_1)}, \end{align} i.e.,$$q(G)\leq\frac{\sum\limits_{v_k:(v_{i_1},v_k)\in E(G)}f(i_1,k)+\sum\limits_{v_k:(v_{j_1},v_k)\in E(G)}f(j_1,k)}{f(i_1,j_1)}, \textrm {where $(v_{i_1}, v_{j_1})\in E(G)$}.$$ This proves the desired result. \end{proof} \noindent\begin{corollary}\label{co:c10} \ Let $G=(V(G), E(G))$ be a digraph. Then \begin{equation}\label{eq:c32}q(G)\leq \max\left\{d_i^{+} \sqrt{\frac{m_i^{+}}{d_j^{+}}}+d_j^{+}\sqrt{\frac{m_j^{+}}{d_i^{+}}} : {(v_i, v_j)\in E(G)}\right\}. \end{equation} \end{corollary} \begin{proof} \ Setting $f(v_i, v_j)=\sqrt{d_i^{+}d_j^{+}}$ in \eqref{eq:c29}, by Cauchy-Schwarz inequality, $$\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k) =\sum\limits_{v_k:(v_i, v_k)\in E(G)}\sqrt{d_i^{+}d_k^{+}}=\sum\limits_{v_k:(v_i, v_k)\in E(G)}(\sqrt{d_i^{+}}\sqrt{d_k^{+}})$$ $$\leq\sqrt{\sum\limits_{v_k:(v_i, v_k)\in E(G)}d_i^{+}\sum\limits_{v_k:(v_i, v_k)\in E(G)}d_k^{+}}= \sqrt{{d_i^{+}}^{2}\sum\limits_{v_k:(v_i, v_k)\in E(G)}d_k^{+}}=d_i^{+}\sqrt{d_i^{+}m_i^{+}}.$$ By \eqref{eq:c29}, we get \begin{align} q(G)\leq& \max\left\{\frac{\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k)+ \sum\limits_{v_k:(v_j, v_k)\in E(G)}f(v_j, v_k)}{f(v_i,v_j)} : (v_i,v_j)\in E(G)\right\} \\ \leq& \max\left\{\frac{d_i^{+}\sqrt{d_i^{+}m_i^{+}}+d_j^{+}\sqrt{d_j^{+}m_j^{+}}} {\sqrt{d_i^{+}d_j^{+}}} : (v_i,v_j)\in E(G)\right\} \\ =&\max\left\{d_i^{+}\sqrt{\frac{m_i^{+}}{d_j^{+}}}+d_j^{+}\sqrt{\frac{m_j^{+}}{d_i^{+}}} : {(v_i, v_j)\in E(G)}\right\}. \end{align} \end{proof} \noindent\begin{corollary}\label{co:c13} \ Let $G=(V(G), E(G))$ be a digraph. Then \begin{equation}\label{eq:c35}q(G)\leq \max\left\{\frac{d_i^{+}(d_i^{+}+m_i^{+})+d_j^{+}(d_j^{+}+m_j^{+})}{d_i^{+}+d_j^{+}} : (v_i,v_j)\in E(G)\right\}. \end{equation} \end{corollary} \begin{proof} \ Setting $f(v_i, v_j)=d_i^{+}+d_j^{+}$ in \eqref{eq:c29}, since $\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k)=\sum\limits_{v_k:(v_i, v_k)\in E(G)}(d_i^{+}+d_k^{+})=d_i^{+}(d_i^{+}+m_i^{+})$, So we get the desired result. \end{proof} \noindent\begin{corollary}\label{co:c11} \ Let $G=(V(G), E(G))$ be a digraph. Then \begin{equation}\label{eq:c33}q(G)\leq \max\left\{\frac{d_i^{+}\sqrt{d_i^{+} +m_i^{+}}+d_j^{+}\sqrt{d_j^{+}+m_j^{+}}}{\sqrt{d_i^{+}+d_j^{+}}} :{(v_i, v_j)\in E(G)}\right\}. \end{equation} \end{corollary} \begin{proof} \ Setting $f(v_i, v_j)=\sqrt{d_i^{+}+d_j^{+}}$ in \eqref{eq:c29}, since $\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k)=\sum\limits_{v_k:(v_i, v_k)\in E(G)}(1\cdot \sqrt{d_i^{+}+d_k^{+}})\leq\sqrt{d_i^{+}\sum\limits_{v_k:(v_i, v_k)\in E(G)}(d_i^{+}+d_k^{+})}$ $=\sqrt{d_i^{+}({d_i^{+}}^{2}+d_i^{+}m_i^{+})}=d_i^{+}\sqrt{d_i^{+}+m_i^{+}}$ by Cauchy-Schwarz inequality. Thus by \eqref{eq:c29} we get the desired result. \end{proof} \noindent\begin{corollary}\label{co:c12} \ Let $G=(V(G), E(G))$ be a digraph. Then \begin{equation}\label{eq:c34}q(G)\leq \max\left\{\frac{d_i^{+} (\sqrt{d_i^{+}}+\sqrt{m_i^{+}})+d_j^{+}(\sqrt{d_j^{+}}+\sqrt{m_j^{+}})} {\sqrt{d_i^{+}}+\sqrt{d_j^{+}}}:{(v_i, v_j)\in E(G)}\right\}. \end{equation} \end{corollary} \begin{proof} \ Setting $f(v_i, v_j)=\sqrt{d_i^{+}}+\sqrt{d_j^{+}}$ in \eqref{eq:c29}, since $\sum\limits_{v_k:(v_i, v_k)\in E(G)}f(v_i, v_k)=\sum\limits_{v_k:(v_i, v_k)\in E(G)}(\sqrt{d_i^{+}}$ $+\sqrt{d_k^{+}})=d_i^{+}\sqrt{d_i^{+}}+\sum\limits_{v_k:(v_i, v_k)\in E(G)}(1\cdot \sqrt{d_k^{+}})\leq {d_i^{+}}^{\frac{3}{2}}+\sqrt{d_i^{+}\sum\limits_{v_k:(v_i, v_k)\in E(G)}d_k^{+}}=d_i^{+}(\sqrt{d_i^{+}}+\sqrt{m_i^{+}})$ by Cauchy-Schwarz inequality. By \eqref{eq:c29} the result follows. \end{proof} Notice that \eqref{eq:c33} and \eqref{eq:c34} can be viewed as adding square roots to \eqref{eq:c35} at difference places. \section{Example} \label{sec:3} Let $G_1$, $G_2$ be the digraphs of order 4,6, respectively, as shown in Figure 2. \begin{table}[H] \centering\caption{Values of the various bounds for example 1.} \begin{tabular}{cccccccccccccccc} \hline &$q(G)$&\eqref{eq:c1}&\eqref{eq:c2}&\eqref{eq:c3}&\eqref{eq:c4}&\eqref{eq:c5} \\ &\eqref{eq:c27}&\eqref{eq:c28}&\eqref{eq:c32}&\eqref{eq:c35}&\eqref{eq:c33}&\eqref{eq:c34}\\ \hline $G_1$ & 3.0000& 4.0000& 3.5000& 3.3028 & 3.4142& 3.5616 \\ & 3.5000 & 3.5651 & 3.4495& 3.3333& 3.6029& 3.5731 \\ \hline $G_2$ & 4.1984& 5.0000& 4.6667 & 4.6016& 5.0000 & 4.7321 \\ & 5.5000& 4.7913& 4.5644& 4.6000& 4.7956 & 4.7866 \\ \hline \end{tabular} \end{table} \noindent\begin{remark}\label{re:c3} Obviously, from Table 1, the bound \eqref{eq:c3} is the best in all known upper bounds for $G_1$, and the bound \eqref{eq:c32} is the best for $G_2$. Finally bound \eqref{eq:c35} is the second-best bounds for $G_1$ and $G_2$. In general, these bounds are incomparable. \end{remark} \end{document}	arXiv
What is momentum really? Wikipedia defines momentum as in classical mechanics: In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object. However, an electromagnetic field has momentum, which is how solar sails work. I would not suppose that this is a product of the 'mass' and velocity of the field. Light has momentum too as I understand. So what is momentum, conceptually? electromagnetism classical-mechanics momentum classical-electrodynamics ACuriousMind♦ hb20007hb20007 $\begingroup$ Potentially helpful: physics.stackexchange.com/a/216599/20427 $\endgroup$ – Feynmans Out for Grumpy Cat Jun 22 '16 at 7:46 $\begingroup$ @Dvij Thanks, the second link is helpful but I still want to know what momentum is at a conceptual level, because the answer speaks rather vaguely about "terms that remain constant in time" $\endgroup$ – hb20007 Jun 22 '16 at 7:51 $\begingroup$ momentum is the 'quantity of motion' that is conserved when you 'transfer motion' from one body to another; it turns out that not only is it possible to transfer motion between material bodies, but you can also do so using 'immaterial' fields $\endgroup$ – Christoph Jun 22 '16 at 10:34 $\begingroup$ "I would not suppose that this is a product of the 'mass' and velocity of the field": Why not? Perhaps naively, I'd assume that a light quantum's momentum is exactly its mass (i.e. its energy's mass equivalent) times c. Is that not so? $\endgroup$ – Peter A. Schneider Jun 23 '16 at 15:19 Momentum / energy are the conserved Noether charges that correspond, by dint of Noether's Theorem to the invariance of the Lagrangian description of a system with respect to translation. Whenever a physical system's Lagrangian is invariant under a continuous transformation (e.g. shift of spatial / temporal origin, rotation of co-ordinates), there must be a conserved quantity, called the Noether charge for that transformation. We then define the conserved charges for spatial and temporal translation as momentum and energy, respectively; angular momentum is the conserved Noether charge corresponding to invariance of a Lagrangian with respect to rotation of co-ordinates. One can derive the more usual expressions for these quantities from a Lagrangian formulation of Newtonian mechanics. When Maxwell's equations and electromagnetism are included in a Lagrangian formulation, we find that there are still invariances with the above continuous transformations, and so we need to broaden our definitions of momentum to include those of the electromagnetic field. User ACuriousMind writes: I think it would be good to point out that the notion of "canonical momentum" in Hamiltonian mechanics need not coincide with this one (as is the case for e.g. a particle coupled to the electromagnetic field) When applied to the EM field, we use a field theoretic version of Noether's theorem and the Lagrangian is a spacetime integral of a Lagrangian density; the Noether currents for a free EM field are the components of the stress-energy tensor $T$ and the resultant conservation laws $T_\mu{}^\nu{}_{,\,\nu}=0$ follow from equating the divergence to nought. This includes Poynting's theorem - the postulated statement of conservation of energy (see my answer to this question here) and the conservation of electromagnetic momentum (see the Wiki article). On the other hand, the Lagrangian $T-U$ describing the motion of a lone particle in the EM field is $L = \tfrac{1}{2}m \left( \vec{v} \cdot \vec{v} \right) - qV + q\vec{A} \cdot \vec{v}$, yielding for the canonical momentum conjugate to co-ordinate $x$ the expression $p_x=\partial L/\partial \dot{x} = m\,v_x+q\,A_x$; likewise for $y$ and $z$ with $\dot{x}=v_x$. A subtle point here is that the "potential" $U$ is no longer the potential energy, but a generalized "velocity dependent potential" $q\,V-\vec{v}\cdot\vec{A}$. These canonical momentums are not in general conserved, they describe the evolution of the particle's motion under the action of the Lorentz force and, moreover, are gauge dependent (meaning, amongst other things, that they do not correspond to measurable quantities). However, when one includes the densities of the four force on non-EM "matter" in the electromagnetic Lagrangian density, the Euler Lagrange equations lead to Maxwell's equations in the presence of sources and all the momentums, EM and those of the matter, sum to give conserved quantities. Also note that the term "canonical momentum" can and often does speak about any variable conjugate to a generalized co-ordinate in an abstract Euler-Lagrange formulation of any system evolution description (be it mechanical, elecromagnetic or a even a nonphysical financial system) and whether or not the "momentum" correspond in the slightest to the mechanical notion of momentum or whether or not the quantity be conserved. It's simply a name for something that mathematically looks like a momentum in classical Hamiltonian and Lagrangian mechanics, i.e. "conjugate" to a generalized co-ordinate $x$ in the sense of $p = \frac{\partial H}{\partial x}$ in a Hamiltonian formulation or $p = \frac{\partial L}{\partial \dot{x}}$ in a Lagrangian setting. Even some financial analysts talk about canonical momentum when Euler-Lagrange formulations of financial systems are used! They are (as far as my poor physicist's mind can fathom) simply talking about variables conjugate to the generalized co-ordinates for the Black Schole's model. Beware, they are coming to a national economy near you soon, if they are not there already! WetSavannaAnimalWetSavannaAnimal $\begingroup$ I think it would be good to point out that the notion of "canonical momentum" in Hamiltonian mechanics need not coincide with this one (as is the case for e.g. a particle coupled to the electromagnetic field) $\endgroup$ – ACuriousMind♦ Jun 22 '16 at 14:15 $\begingroup$ @ACuriousMind See update. Yes, it was a bit silly of me to forget this, given that the central theme of my answer - conservation - doesn't even hold for many canonical momentums. I guess whenever I see people groping for "meaning" in momentum, my answer is the idea I focus on, because I think it is a very deep insight to understand that conservation can come from symmetry. There are those in physics (I get the impression you're not one of them) who say "meh, you're just replacing one why with another", but I think there is a genuine and deep difference: conservation of abstract .... $\endgroup$ – WetSavannaAnimal Jun 23 '16 at 3:58 $\begingroup$ ....quantities is something you just have to accept like the Israelites receiving Moses's transcriptions of the behests of a control freak in the sky, whereas the notion that our World doesn't depend on our descriptions of it is something even tiny children begin to grasp: it's something one naturally sees for oneself and is a much more "visceral" notion. $\endgroup$ – WetSavannaAnimal Jun 23 '16 at 3:58 Without touching on electromagnetism, I'd like to bring up this construction from mechanics (it's in the Feynman lectures). Consider two equal particles approaching each other with equal speed. A----> <----B You can argue from first principles that if they stick together they will not be moving afterwards -- any argument you could make that the composite particle moves to the left is also an argument for making it move to the right; the symmetry of the situation doesn't allow for any nonzero movement afterwards. Simple enough. Now consider a stationary particle A being approached by an equal particle B with velocity $2v$. B------> A From Galilean invariance you can move the reference frame so that it is travelling to the right at constant speed $v$. Then this situation becomes the original situation, and the composite particle AB is stationary with respect to the new reference frame which is moving at speed $v$. Now translate back to the original reference frame and the composite particle is now travelling at speed $v$. Here you can define $p=mv$ and observe that it is conserved. You can extend this idea to systems of multiple equal particles and multiple collisions, constructing situations equivalent to composite masses in any ratio you like. $p=mv$ is conserved by inductive argument. spraffspraff As the wiki article you quote states, momentum is defined as the product of the velocity times the mass of an object. Classical mechanics developed theoretically on the lines explained by WetSavanna in the other answer, the conservation of momentum and energy being cornerstones of the theory. Classical mechanics is a very successful theory, and conservation of momentum is a law. Then comes classical electrodynamics with Maxwell's equations. It can be shown that the electromagnetic wave carries energy. Then using the law of conservations of momentum, the momentum carried by the electromagnetic wave can be derived, as shown here. I.e. momentum conservation will be violated if the electromagnetic wave in addition to energy does not carry momentum. Please see the link for details. So the concept of momentum comes from the classical mechanics definition and is extended into relativistic mechanics, classical electrodynamics is fully relativistic, so as to include the electromagnetic wave as a four vector too. Momentum ($p$) is "really" $mv$, even for light and EM fields. This can be proven by the use of $E = mc^2$. The momentum for a photon (EM) is $p = mv$. Where the mass is given by $m = E/c^2$ and $v = c$. Substituting these into the equation, one obtains, $p = E/c$. Although this equation "looks" different from $p = mv$, because it was derived using the basic definition, it is equivalent (and applicable to objects that have energy - but no "rest mass"). GuillGuill protected by Qmechanic♦ Jun 22 '16 at 20:20 Not the answer you're looking for? Browse other questions tagged electromagnetism classical-mechanics momentum classical-electrodynamics or ask your own question. What is the significance of momentum? How do we know that energy and momentum are conserved? What is the definition of linear momentum? Why does the divergence of the Poynting vector have energy flux density? If photons have no mass, how can they have momentum? How can the linear momentum can be understood physically? Momentum conservation in an electromagnetic system? Linear momentum experiment What is the reason behind the equation $p = mv$? Is Linear monemtum conserved with in angular momentum? How we measure relativistics momentum and what things it represent? Where does the momentum of light go during total internal reflection? Linear and angular momentum Under what conditions is field momentum an eigenvector of the stress-energy tensor?	CommonCrawl
Consensus theorem In Boolean algebra, the consensus theorem or rule of consensus[1] is the identity: $xy\vee {\bar {x}}z\vee yz=xy\vee {\bar {x}}z$ Variable inputs Function values xyz$xy\vee {\bar {x}}z\vee yz$$xy\vee {\bar {x}}z$ 00000 00111 01000 01111 10000 10100 11011 11111 The consensus or resolvent of the terms $xy$ and ${\bar {x}}z$ is $yz$. It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If $y$ includes a term which is negated in $z$ (or vice versa), the consensus term $yz$ is false; in other words, there is no consensus term. The conjunctive dual of this equation is: $(x\vee y)({\bar {x}}\vee z)(y\vee z)=(x\vee y)({\bar {x}}\vee z)$ Proof ${\begin{aligned}xy\vee {\bar {x}}z\vee yz&=xy\vee {\bar {x}}z\vee (x\vee {\bar {x}})yz\\&=xy\vee {\bar {x}}z\vee xyz\vee {\bar {x}}yz\\&=(xy\vee xyz)\vee ({\bar {x}}z\vee {\bar {x}}yz)\\&=xy(1\vee z)\vee {\bar {x}}z(1\vee y)\\&=xy\vee {\bar {x}}z\end{aligned}}$ Consensus The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal $a$ and the other the literal ${\bar {a}}$, an opposition. The consensus is the conjunction of the two terms, omitting both $a$ and ${\bar {a}}$, and repeated literals. For example, the consensus of ${\bar {x}}yz$ and $w{\bar {y}}z$ is $w{\bar {x}}z$.[2] The consensus is undefined if there is more than one opposition. For the conjunctive dual of the rule, the consensus $y\vee z$ can be derived from $(x\vee y)$ and $({\bar {x}}\vee z)$ through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → B then A → AB; replacing A with RHS and B with (y ∨ z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (in propositional calculus), then LHS = RHS (in Boolean algebra). Applications In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.[2] In digital logic, including the consensus term in a circuit can eliminate race hazards.[3] History The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form.[4] It was rediscovered by Samson and Mills in 1954[5] and by Quine in 1955.[6] Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".[7][8] References 1. Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 44 2. Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 81 3. M. Rafiquzzaman, Fundamentals of Digital Logic and Microcontrollers, 6th edition (2014), ISBN 1118855795, p. 75 4. "Canonical expressions in Boolean algebra", Dissertation, Department of Mathematics, University of Chicago, 1937, reviewed in J. C. C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634 5. Edward W. Samson, Burton E. Mills, Air Force Cambridge Research Center, Technical Report 54-21, April 1954 6. Willard van Orman Quine, "The problem of simplifying truth functions", American Mathematical Monthly 59:521-531, 1952 JSTOR 2308219 7. John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Journal of the ACM 12:1: 23–41. 8. Donald Ervin Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, part 1, p. 539 Further reading • Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.	Wikipedia
•https://doi.org/10.1364/OE.415337 Tuning of mid-infrared absorption through phonon-plasmon-polariton hybridization in a graphene/hBN/graphene nanodisk array Li Wang, Jinlai Liu, Bin Ren, Jie Song, and Yongyuan Jiang Li Wang,1 Jinlai Liu,1 Bin Ren,1 Jie Song,1 and Yongyuan Jiang1,2,3,4,* 1School of Physics, Harbin Institute of Technology, Harbin 150001, China 2Key Laboratory of Micro-Optics and Photonic Technology of Heilongjiang Province, Harbin 150001, China 3Key Laboratory of Micro-Nano Optoelectronic Information System of Ministry of Industry and Information Technology, Harbin 150001, China 4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China *Corresponding author: [email protected] L Wang J Liu B Ren J Song Y Jiang Li Wang, Jinlai Liu, Bin Ren, Jie Song, and Yongyuan Jiang, "Tuning of mid-infrared absorption through phonon-plasmon-polariton hybridization in a graphene/hBN/graphene nanodisk array," Opt. Express 29, 2288-2298 (2021) Analysis of hybrid plasmon-phonon-polariton modes in hBN/graphene/hBN stacks for mid-infrared... Pei-Yu Tu, et al. Tunable multi-wavelength absorption in mid-IR region based on a hybrid patterned graphene-hBN... Guangsheng Deng, et al. Tunable, omnidirectional, and nearly perfect resonant absorptions by a graphene-hBN-based hole... Hodjat Hajian, et al. Nanophotonics, Metamaterials, and Photonic Crystals Chemical vapor deposition Finite element method Negative refraction Polaritons Reflection coefficient Surface plasmon polaritons Original Manuscript: November 17, 2020 Revised Manuscript: December 20, 2020 Manuscript Accepted: January 1, 2021 Design and numerical simulations Results and discussions In this paper, we utilize a heterostructured graphene/hBN/graphene nanodisk array to implement an electrically tunable absorber in and out of the Reststrahlen band (RSB) region of hBN. Tuning of phonon-type resonance absorption in the RSB region is achieved through phonon-plasmon-polariton hybridization. The hybrid phonon mode enabled a 290 nm shift of the resonant wavelength, and the sensitivity of absorption peak to the electrical control is 362.5 nm/eV. Simultaneously, the nearly perfect absorption is obtained in the condition of high chemical potential of graphene. Moreover, the plasmon polaritons are strongly modified by phonon polaritons of hBN, so the FWHM of absorption peaks out of the RSB region reduce to 45-49 nm, and the maximum Q of absorption reaches 220.44 at EF=0.65 eV, which is paving a way toward coherent emission at the atmospheric transparent band. Importantly, graphene-assisted hyperbolic phonon polaritons of hBN will enable future phonon devices with high optical performance and wide tunability. Hyperbolic materials are anisotropic media for which the in-plane and out of-plane components of the permittivity tensor have opposite signs [1]. Natural hyperbolic materials in which atomic layers are bonded together through van der Waals (vdWs) forces hold the key to unlock the full potential of hyperbolic media in nanophotonics [2,3]. Hexagonal boron nitride (hBN) is a prototypical hyperbolic material that supports volume-confined hyperbolic phonon polaritons (HPP) [4,5]. The recent discovery of strongly confined phonon-polariton modes in hBN enabled a series of major advances in nanophotonics in the mid-IR wavelength region, including sub-diffraction imaging [6–9], negative refraction [10,11] and molecular vibrations sensor [12]. In contrast to the surface plasmon polaritons (SPP)-collective oscillations of free electrons at metal or semiconductor surfaces coupled to electromagnetic fields, HPP is caused by the coupling of electromagnetic fields and lattice vibrations, which exists in the Reststrahlen band (a spectral range between the longitudinal and transverse optical phonons frequencies, defined as LO and TO, respectively). Importantly, the hyperbolic material hBN offers an opportunity to simultaneously achieve sub-diffraction confinement [3], low losses [4], and remarkably lifetimes in the picosecond range [10] through the excitation of HPP modes. Recent advances in the growth and exfoliation of high-quality hBN enabled high propagation momentum of HPP with kp up to 25k0 on a three-monolayer hBN flake [13]. However, HPP in hBN is still weak in active tuning functionalities, as phonon polaritons originate from intrinsic lattice vibrations, which hinders hBN's application for tunable and reconfigurable and limits its further nanophotonics application. Selecting active materials is most commonly employed to realize indirect tuning. Recent works have demonstrated that the phonon polaritons in hBN can be modified by means of phonon-plasmon-polariton hybridization in a graphene-hBN heterostructure [14–17], which is based on the tailored dispersion of hybridization modes. Also, these heterostructures with phonon-plasmon-polariton hybridization can be effectively applied in absorption devices [18–20], emission cooling [21], and single-photon sources [22]. In addition, Ge3Sb2Te6 (GST) and VO2 as phase-change materials (PCMs) can serve as the versatile platforms to arbitrarily control the propagation of polaritons at the nanoscale. Since the phonon polaritons remain sensitive to local changes in the dielectric function of the ambient environment, the confined HPP in hBN can interact with spatially localized phase transitions of the PCM through the variation of surrounding dielectric environments. Exploiting the PCM to tune the HPP enabled potential applications on refractive elements [23], sub-wavelength focusing [24], and absorber switch [25]. Since large-scale monolayer graphene can be grown easily using the chemical vapor deposition (CVD) compared to other 2D materials, and SPP can be excited on nanopatterned graphene to increase the light-graphene interaction, thereby enhancing the infrared absorption [26–28]. Also, 1D hBN grating are most commonly employed to realize perfect resonance absorption [29]. Depending on the nanostructure design, the tunable mid-infrared absorber with omnidirectional nearly perfect resonance absorption was theoretically demonstrated with respect to the nanopatterned graphene/hBN layer structure [20], whereas lacking the quantitative analysis about the tunable nearly perfect absorption peak in the RSB of hBN. In addition, high-Q resonance with a large modulation is obtained through cascading graphene layers with hBN interlayer, thereby enhancing the sensitivity to ambient environment [30]. It is verified experimentally that the infrared plasmonic response enhanced by exploiting a graphene multilayer stack [31]. In this paper, we investigate the resonant absorption properties of the heterostructured graphene/hBN/graphene nanodisk array. Through illustrating of phonon-plasmon-polariton hybridization in dispersion map, aiming to obtain the tunability absorption caused by modifying HPP mode, and obtain sharp absorption peaks resulted from the modified SPP mode. The resonance absorption in and out of the Reststrahlen band can be well predicted based on phonon-plasmon-polariton hybridization. 2. Design and numerical simulations To investigate the absorption properties caused by the phonon-plasmon-polariton hybridization in hBN and graphene, we employ the heterostructure that composed of a hBN film sandwiched between two monolayer graphene sheets, and design them into nanodisk arranged in a triangular lattice with center-to-center spacing d, radius r, thickness dg+dh+dg (dg and dh are the thickness of monolayer graphene and hBN film, respectively). As shown in Fig. 1, the graphene/hBN/graphene-layer (G/hBN/G) nanodisk array is placed on the top of a gold reflector that is separated by a CaF2 spacer layer ds. CaF2 is an excellent candidate with lower dispersion in the mid-IR region compared to other dielectric materials, e.g., Al2O3 or SiO2. And the dielectric parameter of CaF2 is taken from the empirical Sellmeier approximation [32]: (1)$${\varepsilon _r} = 1.33973 + \frac{{0.69913{\lambda ^2}}}{{{\lambda ^2} - {{0.09374}^2}}} + \frac{{0.11994{\lambda ^2}}}{{{\lambda ^2} - {{21.18}^2}}} + \frac{{4.35181{\lambda ^2}}}{{{\lambda ^2} - {{38.46}^2}}}$$ Due to the symmetry of the proposed structure, the absorption of G/hBN/G nanodisk array is polarization-independent. In the simulation model, a normally incident linearly TM polarized light is used to illuminate the structure. The absorption spectrum is simulated based on the finite element method in the frequency domain. The boundary conditions are periodic in x- and y-directions and open for z-directions in free space. Fig. 1. Schematic of the patterned G/hBN/G-layer nanodisk arranged in a triangular lattice on the top of CaF2 spacer and gold reflector. The parameters: center-to-center spacing d=200 nm, radius r=80 nm, thickness of hBN dh=35 nm, thickness of monolayer graphene dg, and thickness of CaF2 spacer ds=270 nm. hBN is an anisotropic material with two active optical phonon modes in the mid-infrared region, and its relative permittivity can be characterized by a diagonal tensor: (2)$${\varepsilon _{\textrm{hBN}}} = \textrm{diag }\left[ {\begin{array}{ccc} {{\varepsilon_ \bot }}&{{\varepsilon_ \bot }}&{{\varepsilon_\parallel }} \end{array}} \right]$$ The phonon modes lie in the two reststrahlen band: (1) Type I, the permittivity tensor components satisfy ${\varepsilon _\parallel } < 0$ and ${\varepsilon _ \bot } > 0$, which accounts for the out-of-plane phonon mode (${\omega _{TO,\parallel }} = 760\textrm{ c}{\textrm{m}^{ - 1}}$ and ${\omega _{LO,\parallel }} = 830\textrm{ c}{\textrm{m}^{ - 1}}$). (2) Type II, the permittivity tensor components satisfy ${\varepsilon _ \bot } < 0$ and ${\varepsilon _\parallel } > 0$, which corresponds to the in-plane phonon mode (${\omega _{TO, \bot }} = 1370\textrm{ c}{\textrm{m}^{ - 1}}$ and ${\omega _{LO, \bot }} = 1610\textrm{ c}{\textrm{m}^{ - 1}}$). Generally, the dielectric function of hBN can be analytically given by: (3)$${\varepsilon _\xi } = {\varepsilon _{\infty ,\xi }} + {\varepsilon _{\infty ,\xi }}\frac{{{{({{\omega_{LO,\xi }}} )}^2} - {{({{\omega_{TO,\xi }}} )}^2}}}{{{{({{\omega_{TO,\xi }}} )}^2} - {\omega ^2} - i\omega {\Gamma _\xi }}}\begin{array}{cc} {}&{\xi = \bot ,\textrm{ }\parallel } \end{array}$$ Here we use the parameters of Dai et al. [13] with the optical phonon broadening ${\Gamma _ \bot } = 5\textrm{ c}{\textrm{m}^{ - 1}}$and ${\Gamma _\parallel } = 4\textrm{ c}{\textrm{m}^{ - 1}}$, and the permittivity at high-frequency are ${\varepsilon _{\infty , \bot }} = 4.87$ and ${\varepsilon _{\infty ,\parallel }} = 2.95$. The real and imaginary part of dielectric function of hBN are shown in Fig. 2. Fig. 2. Relative permittivity of hBN showing the existence of two reststrahlen bands (shaded regions). Type I: ${\varepsilon _\parallel } < 0$, ${\varepsilon _ \bot } > 0$, Type II: ${\varepsilon _ \bot } < 0$, ${\varepsilon _\parallel } > 0$. Graphene is modeled as surface impedance $Z(\omega ) = 1/\sigma (\omega )$, and the surface conductivity of graphene σ(ω) is calculated using random phase approximation (RPA) [33,34]: (4)$$\sigma (\omega ) = \frac{{{e^2}{E_F}}}{{\pi {\hbar ^2}}}\frac{i}{{\omega + i{\tau ^{ - 1}}}} + \frac{{{e^2}}}{{4{\hbar ^2}}}\left[ {\theta (\hbar \omega - 2{E_F}) + \frac{i}{\pi }\log \left\|{\frac{{\hbar \omega - 2{E_F}}}{{\hbar \omega + 2{E_F}}}} \right\|} \right]$$ where e is the unit electric charge, $\hbar$ is the reduced Planck constant, $\theta$ denotes a step function, which conveys the condition for a photon exciting an electron from the valence band to the conduction band. EF is the graphene Fermi level. The relaxation time $\tau$is given by $\tau = {{{\mu _c}{E_F}} / {e\nu _F^2}}$, where µc is the carrier mobility of graphene, νF is the Fermi velocity (νF ≈ 1×106 m/s). In addition, due to the unique structural and physical advantages, h-BN is regarded as a promising ultra-smooth surface to enhance the electronic and optoelectronic performance compared with other 2D materials. The measured mobility of graphene on the hBN surface ranges from 1.5×104 to 6×104 cm2/(V·s) at 300 K in Ref. [35]. So, we take the mobility of 4×104 cm2/(V·s) for graphene/hBN/graphene-layer structure in this work. In order to demonstrate the phonon-plasmon-polariton hybridization in hBN and graphene, the dispersion of hybrid polaritons in graphene/hBN/graphene-layered structure can be derived from complex reflectivity ${r_p}$, it can be calculated using the transfer matrix formalism [36]. We use the air/graphene/hBN/graphene/CaF2 structure (Fig. 3) to illustrate the dispersion properties, and assume the first and second monolayer graphene to be located at the two interfaces (i.e., the planes of z = 0 and z = dh) between region 1 (z > 0, air), region 2 (-dh < z <0, hBN), and region 3 (z < -dh, CaF2), where dh is the thickness of hBN. In our case, three layers are included, yielding the following analytical expression for M: (5)$$M = \left[ {\begin{array}{cc} {{M_{aa}}}&{{M_{ab}}}\\ {{M_{ba}}}&{{M_{bb}}} \end{array}} \right] = {R_{12}} \cdot {T_2} \cdot {R_{23}}$$ where the matrices Rij occur at every interface between layers i and j, and the matrices T2 describe the propagation of the electromagnetic wave through region 2 with thickness dh by adding an additional phase. (6)$${R_{ij}} = \frac{1}{{{t_{ij}}}}\left[ {\begin{array}{cc} 1&{{r_{ij}}}\\ {{r_{ij}}}&1 \end{array}} \right]\begin{array}{cc} {}&{{T_2}} \end{array} = \left[ {\begin{array}{cc} {{e^{i{k_{z2}}{d_h}}}}&0\\ 0&{{e^{ - i{k_{z2}}{d_h}}}} \end{array}} \right]$$ rij (tij) are the Fresnel reflection (transmission) coefficients for the interface between two regions, and they are given by: (7)$${r_{ij}} = \frac{{{{{\varepsilon _{rj}}} / {{k_{zj}}}} - {{{\varepsilon _{ri}}} / {{k_{zi}}}}\textrm{ + }{{\sigma (\omega )} / {\omega {\varepsilon _\textrm{0}}}}}}{{{{{\varepsilon _{rj}}} / {{k_{zj}}}} + {{{\varepsilon _{ri}}} / {{k_{zi}}}}\textrm{ + }{{\sigma (\omega )} / {\omega {\varepsilon _\textrm{0}}}}}} \quad {{t_{ij}} = \frac{{{{\textrm{2}{\varepsilon _{rj}}} / {{k_{zj}}}}}}{{{{{\varepsilon _{ri}}} / {{k_{zi}}}} + {{{\varepsilon _{rj}}} / {{k_{zj}}}}\textrm{ + }{{\sigma (\omega )} / {\omega {\varepsilon _\textrm{0}}}}}}} $$ where ${\varepsilon _{r1}}$, ${\varepsilon _{r\textrm{2}}}$(${\varepsilon _ \bot }$), ${\varepsilon _{r\textrm{3}}}$are the relative permittivity of air, hBN and CaF2 layer, respectively. In the above, ${k_{zi}}$(or ${k_{zj}}$) is the out-of-plane k-vector of the electromagnetic wave in layer i (or j), and ${k_{z\textrm{1}}} = \sqrt {{\varepsilon _{r1}}\frac{{{\omega ^2}}}{{{c^2}}} - {q^2}}$, ${k_{z\textrm{2}}} = \sqrt {{\varepsilon _ \bot }\frac{{{\omega ^2}}}{{{c^2}}} - \frac{{{\varepsilon _ \bot }}}{{{\varepsilon _\parallel }}}{q^2}}$, and ${k_{z\textrm{3}}} = \sqrt {{\varepsilon _{r\textrm{3}}}\frac{{{\omega ^2}}}{{{c^2}}} - {q^2}}$. Thus, the reflection coefficient ${r_p}$for the whole heterostructure is then given as a ratio of two matrix components of M (8)$${r_p}\textrm{ = }\frac{{{M_{ba}}}}{{{M_{aa}}}}$$ Fig. 3. Layered structure for the air/graphene/hBN/graphene/CaF2 system of the dispersion model. Therefore, according to the Eq. (8), the dispersion of hybrid mode can be calculated for an appropriate choice of dielectric functions of hBN and graphene and hBN's thicknesses. We present the dispersion map of HPP of hBN via the above calculations for air/hBN/CaF2 structure, and the dispersion map of SPP of monolayer graphene on the CaF2 layer with EF=0.5 eV, 0.7 eV and 1.0 eV as depicted in Fig. 4(a). The dispersion is visualized using a false-color map of the imaginary part of the reflection coefficient rp (for the case of P-polarization). It may be observed that HPP-SPP coupling takes place in the RSB region. Fig. 4. (a) Calculated dispersion of the hyperbolic phonon polaritons of h-BN with dh=35 nm, and the surface plasmon polaritons of graphene with EF=0.5 eV, 0.7 eV and 1.0 eV (the arrow indicates the increase of EF). (b) Absorption of the individual hBN nanodisk array in the RSB region (shaded region) and the monolayer graphene nanodisk array out of RSB region. Moreover, in contrast to the graphene-hBN heterostructure, Fig. 4(b) shows the absorption of individual hBN nanodisk and graphene nanodisk array with identical geometry parameters (r=80 nm, d=200 nm, and ds=370 nm). For the hBN nanodisk array, the absorption peak is located at 6.967 µm, the maximum reaches to 83%, and the FWHM is 69 nm. As for the graphene nanodisk array, with the Fermi energy of graphene varying from 0.6 to 1.0 eV, the absorption peak blue shifts 3.06 µm, and the FWHM decreases from 223 nm to 142 nm. It is obviously to see that the FWHM of resonant absorption in hBN nanodisk array is narrower than that in monolayer graphene nanodisk array, which contributes to the lower loss constants in phononic media that sustaining phonon-polaritons compared with that of the plasmonic materials in the mid-IR spectral range [4]. 3. Results and discussions According to the dispersion calculation above, we can manipulate the phonon polaritons of hBN by means of strong coupling between graphene plasmon and hBN's phonon polaritons in graphene-hBN heterostructures. The peculiar coupling response is illustrated by the calculated wavelength (λ)/momentum (q) dispersion relations of its polariton modes, which is visualized using a false-color map of Im(rp). Figures 5(a) and 5(c) present the dispersion map of the G/hBN-layer with EF=0.6 and 1 eV, respectively. Clearly, the dispersion curves of phonon polaritons of hBN in RSB are strongly modified compared with that of individual hBN layer [see Fig. 4(a)], and the original HPP of hBN (HP2) is changed to the hyperbolic phonon-plasmon polaritons (HP3). Also, the dispersion branch moves to lower q as the Fermi energy of graphene increases. Following the previous reports [14], we refer to the collective modes existing outside the RSB as surface plasmon-phonon polaritons (SP3). In addition, for the G/hBN/G-layer structure as shown in Figs. 5(b) and 5(d) with EF=0.6 and 1 eV, respectively, two dispersion branches in RSB are modified, and the second dispersion branch possesses higher q, and gives more tuning region as varying the Fermi energy of graphene. Fig. 5. Calculated dispersion of the hybrid phonon-plasmon-polaritons in (a) G/hBN-layer, and (b) G/hBN/G-layer structure with EF=0.6 eV and dh=35 nm, and in (c) G/hBN-layer, and (d) G/hBN/G-layer structure with EF=1.0 eV and dh=35 nm. Figures 6(a) and 6(b) provide the absorption spectra in the RSB with respect to the G/hBN and G/hBN/G-layer nanodisk structure, respectively. Since the phonon polaritons of hBN is obviously modified by the plasmons polaritons of graphene, the modification of the hyperbolic phonon response by graphene is clearly manifested in the blueshift of absorption wavelength compared with that of individual HPP in hBN [see Fig. 4(b)]. Importantly, the resonance absorption caused by the HP3 mode in RSB region is continuously tunable under the control of the graphene's Fermi energy. The absorption peaks reach to above 90%, even are close to 100% at EF=1.0 eV, i.e. near to the perfect absorption, so, high chemical potential of graphene is conducive to perfect absorption in RSB region. And the resonant wavelength blue shifts as the Fermi energy of graphene increases. With the Fermi energy of graphene varying from 0.2 to 1.0 eV, the absorption peaks are shifted from 164 to 290 nm (the sensitivity to the electrical control are 205 nm/eV and 362.5 nm/eV) for G/hBN-layer and G/hBN/G-layer nanodisk structure, respectively [see Fig. 6(a) and 6(b)], which indicates that the tunability of hyperbolic phonon polaritons of hBN have achieved indirectly. Also, the sensitivity to the electrical control for the triple-layer structure is superior to that of the double-layer nanodisk array. To quantitatively evaluate the tunability of the proposed absorber, Figs. 6(c) and 6(d) show the resonance wavelength λ0 with the variation of graphene's chemical potential and their quadratic fitting as ${\lambda _\textrm{0}}\textrm{ = 6}\textrm{.91 - 0}\textrm{.095}{E_F}\textrm{ - }0.091{E_F}^2$ and${\lambda _\textrm{0}}\textrm{ = 6}\textrm{.91 - 0}\textrm{.147}{E_F}\textrm{ - }0.\textrm{183}{E_F}^2$for G/hBN-layer and G/hBN/G-layer nanodisk array, respectively. The resonant position in RSB region nonlinearly shifts to shorter wavelength as Fermi energy increases, which caused the tunability of hyperbolic phonon polaritons in hBN' RSB region. Moreover, it is noteworthy to the excellent properties with respect to the graphene-encapsulated hBN structure (i.e. the G/hBN/G-layer nanodisk array), which is because that the dispersion curve is more easily manipulated by the Fermi energy. In this case, the hyperbolic phonon polaritons in hBN is modified strongly and carries the graphene' tunable attribution. Fig. 6. Absorption of (a) G/hBN-layer and (b) G/hBN/G-layer nanodisk array in the RSB region. Dependence of HP3 resonation wavelength in the RSB region on the graphene's chemical potential and their quadratic fitting of (c) G/hBN -layer and (d) G/hBN/G-layer nanodisk array. The parameters: dh=35 nm, r=80 nm, d=200 nm, and ds=370 nm for G/hBN-layer nanodisk array, and ds=270 nm for G/hBN/G-layer nanodisk array. In addition, except for the HP3-type resonance absorption in the RSB region, SP3-type sharp absorption peaks appear out of the RSB region (8-12 µm). For the G/hBN-layer nanodisk array as shown in Fig. 7(a), with the Fermi energy of graphene varying from 0.75 to 1.0 eV, the absorption peak blue shifts 1.03 µm, and the change of chemical potential from 0.75 to 1.0 eV cause a 1.02 µm wavelength shift from 10.13 to 9.11 µm for the G/hBN/G-layer nanodisk array [Fig. 7(b)]. So, the dependence of absorption peaks resulted from SP3 mode on the chemical potential is almost identical for the proposed double and triple-layer nanodisk array. Although the tunable range is suppressed in comparison with that of monolayer graphene nanodisk [see Fig. 4(b)], the absorption peak blue shifts entirely and move the spectra to 9-11 µm region, and the peaks maximum reach to above 90% mostly. Additionally, compared to the graphene nanodisk array, the FWHM in the absorption spectra of these proposed hybrid structures are getting narrowed as shown in Figs. 7(c) and 7(d). Particularly, the FWHM of SP3-type absorption peaks reduce to 45-49 nm. We present the values of Q for different Fermi energy (Q=λ0/FWHM). The maximum Q of absorption reaches 220.44 at EF=0.65 eV for the G/hBN/G-layer nanodisk array. Also, the FWHM in the RSB region even reach to 37 nm, the maximum Q reaches 184.63 at EF=0.2 eV for the G/hBN/G-layer nanodisk array. The phonon polaritons with longer lifetime enabled stronger electric field confinement, which manifests as higher and sharper absorption (lower FWHM) in the spectral response, then in this case, the SP3 mode carries the attribution of phonon polaritons. Fig. 7. Absorption of (a) G/hBN-layer and (b) G/hBN/G-layer nanodisk array out of the RSB region. Q value and FWHM of the proposed structure (c) in and (d) out of the RSB region as a function of chemical potential of graphene. Furthermore, we stress the distinction between the field distributions in HP3 and SP3 spectral regions. The Ez-field distributions at 9.107 µm [ Figs. 8(a) and 8(b)] and 6.582 µm [Fig. 8(c) and 8(d)] are shown to illustrate the difference of SP3 and HP3 mode, respectively. Figure 8(a) shows the x–y-plane of E field distribution of SP3 mode, which is mainly in the vicinity of nanodisks, and both nanodisks oscillate in-phase. In the view of x–z-plane [Fig. 8(b)], the SP3 mode is localized at the graphene/h-BN interface and decays evanescently in the interior of the h-BN, which indicates the attribute of surface wave for the SP3 mode. This SP3 mode has a long lifetime due to the coupling of SPP and HPP mode, so, it exhibits stronger electric field confinement. Whereas the E field distribution of HP3 mode is concentrated inside the hBN, and propagate through the entire nanodisk as a guided mode despite the small thickness, each nanodisk can be considered as a separate resonator as shown in Fig. 8(c), and the collective resonance indicates a higher and sharper absorption in RSB region. Also, the x–z-plane field [Fig. 8(d)] reveal the Fabry–Perot resonances in the h-BN resonators with characteristic of a standing wave [6]. Fig. 8. Ez-field distribution of G/hBN/G-layer nanodisk array in the (a) x–y-plane and (b) x–z-plane at the resonant wavelengths of 9.107 µm, and in the (c) x–y-plane and (d) x–z-plane at the resonant wavelengths of 6.582 µm. The parameters: dh=35 nm, r=80 nm, d=200 nm, ds=270 nm, and EF=1.0 eV. In addition to considering the gate control using graphene's chemical potential, the absorption responses are easily affected by relaxation time of graphene as well. The relaxation time of graphene is not only decided by EF, also depends on the carrier mobility of graphene, so, the quality of sample preparation in experiment has a great impact on τ. We show the dependence of absorption FWHM supported by HP3 and SP3 resonance on τ as depicted in Fig. 9(a). In our case, the G/hBN/G-layer nanodisk array with above parameters and EF=0.8 eV is selected. Obviously, for HP3-type resonance absorption, the FWHM change slightly with varying τ, and the FWHM is about 50 nm, which is considered that the spectra responses are almost unaffected by τ in the RSB region. Whereas the FWHM of HP3-type resonance absorption strongly depends on the τ, and is broadened as reduction of τ. Additionally, to establish the feasibility of the proposed absorbers, the dependence of absorption to the angle of incidence and polarization need to be considered. Figures 9(b) and 9(c) illustrate the absorption with varying incident angle in TE and TM polarization, respectively. The peaks location is almost independent of the angle of incidence for G/hBN/G-layer nanodisk array in TE polarization, but have a slight redshift in TM polarization as increasing the incident angle. The SP3-type absorption decreases when θ=60°, but still reaches to 80% in TE polarization, and the perfect HP3-type absorption could operate over a wide-angle range from 0° to 60° in TM polarization. Fig. 9. (a) Dependence of absorption FWHM supported by HP3 and SP3 resonation on the relaxation time of graphene. Absorption of G/hBN/G-layer nanodisk array with EF = 0.8 eV in (b) TE and (c) TM polarization at θ=0°, 15°, 30°, 45° and 60°. In conclusion, we designed an electrically tunable dual-waveband absorber consisting of G/hBN/G-layer nanodisk array for operating in the mid-IR region. The hybrid HP3 and SP3 modes are presented through calculating dispersion map, for overcoming the limitations of the individual HPP and SPP. Hence, the optimized G/hBN/G-layer nanodisk absorber exhibits a superior tunability to the electrical control in the RSB region. The blueshift of the absorption peak is 290 nm (EF=0.2-1.0 eV), and the sensitivity to the electrical control is 362.5 nm/eV, which indicates that HP3 mode possesses the tunable attribution from graphene. Also, the absorption peaks reach to above 90%, even the nearly perfect absorption is obtained at EF=1.0 eV. Furthermore, the resonance absorption peaks of G/hBN/G-layer nanodisk array out of the RSB region exhibit narrow FWHM (45-49 nm), and the maximum Q of absorption reaches 220.44 at EF=0.65 eV, which is modified by HPP mode and inherit the hBN's advantages with low losses and long lifetime. Additionally, the proposed absorber can operate over a wide-angle range, and almost independent to the incident angle and polarization. The proposed hybrid structure expands the application of phonon-type devices for mid-infrared nanophotonics and could enable the creation of novel actively tunable, low-loss application at the nanoscale. 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Abate, and J. D. Caldwell, "Reconfigurable infrared hyperbolic metasurfaces using phase change materials," Nat. Commun. 9(1), 4371 (2018). C. Peng, K. Ou, G. Li, X. Li, W. Wang, Z. Zhao, X. Li, X. Chen, and W. Lu, "Tunable phase change polaritonic perfect absorber in the mid-infrared region," Opt. Express 28(8), 11721 (2020). A. Safaei, S. Chandra, A. Vázquez-Guardado, J. Calderon, D. Franklin, L. Tetard, L. Zhai, M. N. Leuenberger, and D. Chanda, "Dynamically tunable extraordinary light absorption in monolayer graphene," Phys. Rev. B 96(16), 165431 (2017). J. Hu, X. Wu, H. Li, E. Yao, W. Xie, W. Liu, Y. Lu, and C. Ming, "Tuning of longitudinal plasmonic coupling in graphene nanoribbon arrays/sheet hybrid structures at mid-infrared frequencies," J. Opt. Soc. Am. B 36(3), 697–704 (2019). A. Safaei, S. Chandra, M. N. Leuenberger, and D. Chanda, "Wide angle dynamically tunable enhanced infrared absorption on large-area nanopatterned graphene," ACS Nano 13(1), 421–428 (2019). B. Zhao and Z. M. Zhang, "Resonance perfect absorption by exciting hyperbolic phonon polaritons in 1D hBN gratings," Opt. Express 25(7), 7791 (2017). H. Jiang, S. Choudhury, Z. A. Kudyshev, D. Wang, L. J. Prokopeva, P. Xiao, Y. Y. Jiang, and A. V. Kildishev, "Enhancing sensitivity to ambient refractive index with tunable few-layer graphene/hBN nanoribbons," Photonics Res. 7(7), 815–822 (2019). D. Rodrigo, A. Tittl, O. Limaj, F. J. G. de Abajo, V. Pruneri, and H. Altug, "Double-layer graphene for enhanced tunable infrared plasmonics," Light: Sci. Appl. 6(6), e16277 (2017). H. H. Li, "Refractive index of alkaline earth halides and its wavelength and temperature derivatives," J. Phys. Chem. Ref. Data 9(1), 161–290 (1980). F. J. G. de Abajo, "Graphene plasmonics: challenges and opportunities," ACS Photonics 1(3), 135–152 (2014). F. H. Koppens, D. E. Chang, and F. Javier de Abajo, "Graphene plasmonics: a platform for strong light-matter interactions," Nano Lett. 11(8), 3370–3377 (2011). C. R. 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Dolado, I. Dominguez, G. Edgar, J. H. Ellis, C. T. Esteban, R. Fabian, J. Fali, A. Fang, N. X. Fei, Z. Ferrari, A. C. Fogler, M. M. Folland, T. G. Francescato, Y. Frank, T. Franklin, D. Fung, K. H. Gannett, W. Gaussmann, F. Ghobadi, A. Giannini, V. Giles, A. J. Glembocki, O. J. Goldflam, M. D. Goykhman, I. Haajati, Y. Haglund Jr, R. F. Hajian, H. Hesp, N. C. H. Hillenbrand, R. Hone, J. Hong, M. Huber, M. A. Huber, R. Hueso, L. E. Imran, M. Iorsh, I. Jacob, Z. Jang, M. S. Janssen, G. C. A. M. Jarillo-Herrero, P. Jauregui, L. A. Javier de Abajo, F. Jiang, H. Jiang, Y. Y. Joannopoulos, J. D. Kaminer, I. Kastner, L. Z. Keilmann, F. Kildishev, A. V. Kim, L. B. Kim, P. Kim, S. Kivshar, Y. Koppens, F. H. Koppens, F. H. L. Kretinin, A. V. Kudyshev, Z. A. Kumar, A. Leuenberger, M. N. Lewin, M. Li, G. Li, H. H. Li, R. Limaj, O. Lin, X. Liu, M. K. Liu, W. Lombardo, A. Lopez, J. J. López, J. J. Low, T. Lu, W. Lundeberg, M. B. Ma, Q. Maier, S. A. Massicotte, M. Mastel, S. Matson, J. R. Mcleod, A. S. Meric, I. Mics, Z. Ming, C. Mooshammer, F. Narimanov, E. E. Nikitin, A. Y. Novoselov, K. S. Oscurato, S. L. Ou, K. Ozbay, E. Persch, C. Plankl, M. Poddubny, A. Pogna, E. A. A. Polini, M. Principi, A. Prokopeva, L. J. Pruneri, V. Purdie, D. G. Regan, W. Rivera, N. Rodin, A. S. Rodrigo, D. Rubin, N. A. Sabaeian, M. Safaei, A. Sandner, F. Schmidt, P. Schnell, M. Serebryannikov, A. E. Shen, L. Shepard, K. L. Sherrott, M. Soavi, G. Soljacic, M. Song, X. Sorgenfrei, S. Spägele, C. M. Stampfer, C. Tamagnone, M. Taniguchi, T. Taubner, T. Tetard, L. Thiemens, M. Tielrooij, K. J. Tischler, J. G. Tittl, A. Turchinovich, D. Txoperena, O. Vandenbosch, G. A. E. Vázquez-Guardado, A. Vélez, S. Viti, L. Vitiello, M. S. Wagner, M. Wang, D. Wang, H. Watanabe, K. White, S. T. Woessner, A. Woods, C. R. Wuttig, M. Xiao, P. Xu, H. Yao, E. Yin, X. Young, A. F. Yoxall, E. Zanbouri, Z. Zettl, A. Zhai, L. Zhang, B. Zhang, Z. M. Zhao, Z. Zhu, S.-E. ACS Nano (1) ACS Photonics (1) J. Phys. Chem. Ref. Data (1) Light: Sci. Appl. (2) Nat. Commun. (6) Nat. Mater. (1) Nat. Nanotechnol. (4) Photonics Res. (1) Proc. Natl. Acad. Sci. U. S. A. (1) (1) ε r = 1.33973 + 0.69913 λ 2 λ 2 − 0.09374 2 + 0.11994 λ 2 λ 2 − 21.18 2 + 4.35181 λ 2 λ 2 − 38.46 2 (2) ε hBN = diag [ ε ⊥ ε ⊥ ε ∥ ] (3) ε ξ = ε ∞ , ξ + ε ∞ , ξ ( ω L O , ξ ) 2 − ( ω T O , ξ ) 2 ( ω T O , ξ ) 2 − ω 2 − i ω Γ ξ ξ = ⊥ , ∥ (4) σ ( ω ) = e 2 E F π ℏ 2 i ω + i τ − 1 + e 2 4 ℏ 2 [ θ ( ℏ ω − 2 E F ) + i π log ⁡ \| ℏ ω − 2 E F ℏ ω + 2 E F \| ] (5) M = [ M a a M a b M b a M b b ] = R 12 ⋅ T 2 ⋅ R 23 (6) R i j = 1 t i j [ 1 r i j r i j 1 ] T 2 = [ e i k z 2 d h 0 0 e − i k z 2 d h ] (7) r i j = ε r j / k z j − ε r i / k z i + σ ( ω ) / ω ε 0 ε r j / k z j + ε r i / k z i + σ ( ω ) / ω ε 0 t i j = 2 ε r j / k z j ε r i / k z i + ε r j / k z j + σ ( ω ) / ω ε 0 (8) r p = M b a M a a	CommonCrawl
OSA Publishing > Optics Express > Volume 28 > Issue 22 > Page 32858 A tellurite glass optical microbubble resonator J. Yu, J. Zhang, R. Wang, A. Li, M. Zhang, S. Wang, P. Wang, J. M. Ward, and S. Nic Chormaic J. Yu,1,2 J. Zhang,1 R. Wang,1 A. Li,1,2 M. Zhang,1 S. Wang,1 P. Wang,1,3,5 J. M. Ward,2,4 and S. Nic Chormaic2,6 1Key Laboratory of In-Fiber Integrated Optics of Ministry of Education, College of Science, Harbin Engineering University, Harbin 150001, China 2Light-Matter Interactions for Quantum Technologies Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan 3Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China 4Physics Department, University College Cork, Cork, Ireland [email protected] [email protected] J. Zhang https://orcid.org/0000-0003-1919-4055 P. Wang https://orcid.org/0000-0001-9321-3395 S. Nic Chormaic https://orcid.org/0000-0003-4276-2014 J Yu J Zhang R Wang A Li M Zhang S Wang P Wang J Ward S Nic Chormaic pp. 32858-32868 •https://doi.org/10.1364/OE.406256 J. Yu, J. Zhang, R. Wang, A. Li, M. Zhang, S. Wang, P. Wang, J. M. Ward, and S. Nic Chormaic, "A tellurite glass optical microbubble resonator," Opt. Express 28, 32858-32868 (2020) Get PDF (965 KB) High-Q, ultrathin-walled microbubble resonator for aerostatic pressure sensing (OE) Dispersion analysis of whispering gallery mode microbubble resonators (OE) Tunable erbium-doped microbubble laser fabricated by sol-gel coating (OE) Diode pumped lasers Integrated photonics Original Manuscript: August 26, 2020 Revised Manuscript: October 6, 2020 Manuscript Accepted: October 8, 2020 Fabrication of the tellurite microbubble Passive tellurite glass microbubbles Active Yb-Er co-doped tellurite glass microbubble We present a method for making microbubble whispering gallery resonators (WGRs) from tellurite, which is a soft glass, using a CO2 laser. The customized fabrication process permits us to process glasses with low melting points into microbubbles with loaded quality factors as high as 2.3 × 106. The advantage of soft glasses is that they provide a wide range of refractive index, thermo-optical, and optomechanical properties. The temperature and air pressure dependent optical characteristics of both passive and active tellurite microbubbles are investigated. For passive tellurite microbubbles, the measured temperature and air pressure sensitivities are 4.9 GHz/K and 7.1 GHz/bar, respectively. The large thermal tuning rate is due to the large thermal expansion coefficient of 1.9 × 10−5 K−1 of the tellurite microbubble. In the active Yb3+-Er3+ co-doped tellurite microbubbles, C-band single-mode lasing with a threshold of 1.66 mW is observed with a 980 nm pump and a maximum wavelength tuning range of 1.53 nm is obtained. The sensitivity of the laser output frequency to pressure changes is 6.5 GHz/bar. The microbubbles fabricated using this method have a low eccentricity and uniform wall thickness, as determined from electron microscope images and the optical spectra. The compound glass microbubbles described herein have the potential for a wide range of applications, including sensing, nonlinear optics, tunable microcavity lasers, and integrated photonics. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement In the past few decades, the level of research activity on whispering gallery mode (WGM) resonators has increased rapidly [1–7]. Concurrently, the number of WGM geometries has also increased. These resonators, or microcavities, have small sizes, high uniformity, and smooth surfaces which combine to deliver an extremely high quality (Q-) factor and small mode volume, thereby having potential in many applications, such as high-sensitivity sensing, nonlinear optics, optomechanics, active photonics devices, cavity quantum electrodynamics, and nanoparticle control [8–17]. One of the resonator geometries that has attracted much attention is the microbubble, in which the WGMs propagate in the wall of a thin spherical shell typically made of glass. First reported in 2010 [18], a silica microbubble can be fabricated by heating and expanding a silica capillary using a CO$_2$ laser and air pressure [19]. The wall thickness of the bubble can be close to the wavelength of light propagating in the WGMs, resulting in evanescent fields on the inner and outer walls; therefore, the modes are extremely sensitive to changes in refractive index. The thin walls also give a large sensitivity to changes in pressure [13]. Microbubbles are predominately made from silica glass because it is relatively easy to manipulate when heat-softened. Other soft glasses have many advantages over silica, but their low melting points makes them more difficult to control and cast into the desired shape. In previous work, lead silicate microbubbles with single and double stems were fabricated. However, the shapes were not spherical, the Q-factors were limited to about $10^5$, and the fabrication process was very difficult to control with a low success rate [20]. In this work, we report on the development of a microbubble fabrication method for soft glasses to yield doped and undoped tellurite glass microbubble whispering gallery resonators (WGR). Tellurite glass has many properties which may lend themselves to the functionality of the WGR. For example, its thermal expansion coefficient [21] is larger than that for silica [22], thereby increasing the sensitivity of temperature sensors. This is also true for the nonlinear coefficients of tellurite glass; the zero dispersion of tellurite microbubbles can be tuned over a large range making them viable for visible to mid-infrared frequency comb applications [23–25]. One of the benefits of laser emission in a microbubble is the high degree of wavelength tunability and the prospect of integrating a microlaser into a hollow WGR. The common fabrication method involves covering the surface of the microcavity with a gain material, such as sol-gel or some other rare-earth ion doped compound glass [15,26] — a review of several techniques is contained in [27]. An alternative is to inject a gain liquid (such as an organic gain dye) into the microbubble for laser emission [28]. In this case, a large loss will be induced in the microbubble resonator since the introduction of a gain material, which is not intrinsic to the capillary, results in a higher lasing threshold. Additionally, the material of choice for the capillary has been predominantly silica, which is limited in its rare-earth ion doping concentration and low phonon energy, so it can be challenging to know the exact concentration of the gain medium after preparing the resonator. Finally, the wavelength range of laser emissions from silica is largely limited to visible and near-infrared light. The aforementioned drawbacks with existing techniques can be largely overcome by using compound tellurite glass to prepare the microbubble resonator, details of which are contained herein. A polished tellurite glass tube was initially formed into a microcapillary by tapering using a large fiber drawing tower. Then the capillary was further drawn down to its final diameter by heating and stretching it in a custom-made pulling rig consisting of a small ceramic heater. Next, a CO$_2$ laser was used to form the tellurite glass microbubble using a unique method. Finally, the temperature and air pressure dependent optical characteristics of the WGMs were investigated for both passive and active versions of the microbubble. 2. Fabrication of the tellurite microbubble As a first step, we fabricated both passive and active tellurite glass rods, with composition 75TeO$_2$-5ZnO-15Na$_2$CO$_3$-5Bi$_2$O$_3$ and 75TeO$_2$-5ZnO-15Na$_2$CO$_3$−4.25Bi$_2$O$_3$−0.5Yb$_2$O$_3$−0.25Er$_2$O$_3$, respectively, using the melt-quenching method [29]. We placed 50 g of high-purity chemical material [TeO$_2$ (99.99%), ZnO (99.99%), Na$_2$CO$_3$ (99.99%), Bi$_2$O$_3$ (99.99%), Yb$_2$O$_3$ (99.99%), Er$_2$O$_3$ (99.99%)] in an agate mortar and stirred for 10 minutes. The material was then stored in an alumina crucible and heated in a closed furnace at $900^\circ$C for 60 minutes. Next, the melt was poured quickly into a tube furnace, which had been heated to $300^\circ$C for 3 hours. The rotation speed of the tube furnace was set to 30 rev/min for one minute. A glass tube was formed in the tube furnace and annealed at $310^\circ$C for 4 hours to remove any remaining internal stress. We removed the glass tube and polished it using low-mesh to high-mesh sandpaper until the ratio of the inner and outer diameters was between 0.6-0.7. Finally, we mounted the prepared polished glass tube in the nitrogen-filled chamber of the fiber drawing tower. The temperature of the furnace was increased to 345$^{\circ }$C and tellurite glass capillaries with outer diameters of 300 $\mu$m were obtained by controlling the speed and the tractive force of the fiber drawing tower. To make microbubbles, several steps were needed, as illustrated in Fig. 1(a)-(f). First, we had to further decrease the diameter of the tellurite capillaries to about 15 $\mu$m using a ceramic heater, see Fig. 1(d). In order to make a tellurite microbubble, a CO$_2$ laser (48-2KWM, Synrad) was used to process the tellurite capillaries even further. The laser beam was divided into two parts, which were then overlapped near their focal points, see Fig. 1(a). The size of the beam at the location of the bubble was about 350 $\mu$m and the lenses are 400 mm focal length. A solid microsphere with a diameter of 35 $\mu$m was first formed on the tip of the tellurite glass capillary. This was done by affixing the capillary to a one-dimensional translation stage and then slowly moving it into the center of the two laser beams until the microsphere forms. At this point, we opened a gas valve and the capillary was pressurized to 3.5 bar, while, at the same time, we increased the power of the CO$_2$ laser. The tellurite microsphere expanded into a microbubble with a diameter of about 150 $\mu$m, as shown in Fig. 1(f). This is a self-terminating process: once the CO$_2$ laser power and gas pressure are fixed, the bubble expansion stops at the point where the heat loss from the wall exceeds the absorbed heat from the laser. We controlled the size of the microspheres in advance to adjust the diameter of the resulting microbubble. This method is simpler and reduces cavity loss when compared with the method of directly making a microbubble from a capillary [20]. During the glass capillary fabrication process, some tiny air bubbles are formed when the molten glass is rotated in the tube furnace. A rough surface is also created by uneven polishing. If a bubble is made directly from the capillary by simply softening the glass, the air bubbles and rough surface may be preserved. However, if the microcapillary is completely remelted into a microsphere, the bubbles inside disappear and the surface is very smooth. Fig. 1. (a-c) Three step fabrication processes for the tellurite glass microbubble. BS: beam splitter, R: mirror. (d-f) Microscope images of the tellurite glass capillary, microsphere, and microbubble in steps (a-c). (g) SEM image of a broken tellurite glass microbubble, A-H represent the different measurement positions in the microbubble, with the wall thickness at position B being 670 nm. (h) The measured thickness at the different positions in (g). To characterize the wall thickness of the microbubble, we broke it and imaged it with a scanning electron microscope (SEM). A typical measured thickness is about 670 nm, see position B in Fig. 1(g). We measured multiple points to determine the uniformity of the microbubble and the results are shown in Fig. 1(h). Except for the thickness near the stem, which is around 900 nm, most other points on the microbubble wall are around 670 nm. The thickness of the wall is calculated to be 640 nm by assuming conservation of volume from the tellurite glass in the microsphere to the resulting microbubble and is consistent with the measured value [30]. The thickness is calculated from (1)$$\frac{4}{3}\pi(\frac{c_1}{2})^3=\frac{4}{3}\pi(\frac{c_2}{2})^3-\frac{4}{3}\pi(\frac{c_2-a_1}{2})^3,$$ where c$_1$ and c$_2$ are the diameters of the microsphere and microbubble, respectively, and a$_1$ is the thickness of the microbubble wall. The above formula is different from the mass conservation of the cross-sectional area described in the literature [31,32]. Since the tellurite capillary tends to collapse to some degree when it is tapered with the ceramic heater, its accurate inner diameter cannot be obtained. The microbubble shown in Fig. 1(g) differs from other conventional silica microbubbles reported in the literature [13], not only in the fabrication method but also the final geometry. The bubble is highly uniform both in shape and wall thickness. The fact that the microbubble is blown out from a microsphere means that the resulting bubble also has a low degree of eccentricity. Microbubbles from other low-melting compound glasses, such as fluoride and chalcogenide glasses, could be prepared using this method. 3. Experimental setup The experimental setup is schematically illustrated in Fig. 2(a). We used two lasers in a pump/probe arrangement. A tunable laser (TLB-6700, Newport) with a center wavelength of 1550 nm was used to probe the WGM resonances of the tellurite glass microbubbles. The laser frequency was scanned over 36 GHz at a rate of 10 Hz. A pump diode laser with a wavelength of 980 nm (BL976-SAG300, Thorlabs) and a linewidth of 1 nm was used to both control the temperature inside the passive tellurite glass microbubble and act as the pump for the Yb$^{3+}$-Er$^{3+}$ doped tellurite glass microbubble. A silica fiber with single-mode transmission at 980 nm and 1550 nm was selected (1060XP, Thorlabs) to make the tapered fiber coupler, with a final diameter of around 1 $\mu$m. The output lasing was observed using an optical spectrum analyzer (MS9740A, Anritsu). Fig. 2. (a) Experimental setup for the tellurite glass microbubble. The blue line represents the optical path and the black lines are the electrical connections. TL: tunable laser; DL: diode laser; FG: function generator; WDM: wavelength division multiplexer; PD: photodetector; OSC: oscilloscope; OSA: optical spectrum analyzer. (b) The observed WGM resonance spectrum of the tellurite glass microbubble (diameter $\sim$ 130 $\mu$m, wall thickness $\sim$ 800 nm) with a Lorentzian fit (red line), corresponding to a loaded $Q$-factor of $2.3\times 10^6$. 4. Passive tellurite glass microbubbles As a first step, we measured the Q-factor of a passive tellurite glass microbubble by scanning the frequency of the tunable laser. A typical transmission spectrum as a function of laser frequency is shown in Fig. 2(b) and the fitted Q is $2.3\times 10^6$, which is close to the value of passive tellurite glass microspheres presented previously [33]. Next, we used the 980 nm diode laser to control the temperature inside the microbubble while the 1550 nm WGM resonance frequency was recorded. The results are shown in Fig. 3(a). As we increased the pump laser output power from 0 to 1.37 mW, the temperature of the glass increased due to absorption, resulting in a red shift of the resonance frequency, f, by 16.7 GHz. A linear fit yields a power sensitivity of $-12$ GHz/mW. The total shift, $\Delta {f}$, of the frequency is given by (2)$$\Delta\textit{f}=\textit{f}(\frac{1}{n}\frac{\Delta\textit{n}}{\Delta\textit{T}}+\frac{1}{d}\frac{\Delta\textit{d}}{\Delta\textit{T}})\Delta\textit{T},$$ where n is the refractive index and d is the diameter of the tellurite microbubble. The thermal expansion coefficient for tellurite glass is $1.9\times 10^{-5}$ K$^{-1}$ [21], which is 38 times larger than the corresponding value for silica glass of $0.51\times 10^{-6}$ K$^{-1}$ [22]. The thermo-optic coefficient $\Delta$n/$\Delta$T is $1.08\times 10^{-5}$ K$^{-1}$ [34], and the frequency shift as a function of temperature $\Delta$f/$\Delta$T was calculated to be 4.9 GHz/K, which is about 4 times lager than for a silica microsphere of 1.28 GHz/K at room temperature [35]. Fig. 3. Resonance shift of the passive microbubble with varying (a) 980 nm laser power or (c) air pressure. The black arrow shows the direction of the resonance shift. Resonance frequency shift as a function of (b) 980 nm laser power and (d) air pressure. The red lines are linear fits to the experimental data. According to Eq. (2), the frequency shift can also be affected by air pressure, which changes the diameter of the microbubble and the refractive index by stress, see Fig. 3(c). The resonance red shifted by 27.9 GHz as the air pressure inside the bubble was increased from 0 to 4 bar, yielding a pressure sensitivity of $-7.1$ GHz/bar. For a silica microbubble with a diameter of 141 $\mu$m and a thickness of 1.3 $\mu$m, the pressure sensitivity of the resonance is $-8.2$ GHz/bar [36], which is close to the value we have obtained for the tellurite glass microbubble with a diameter of 130 $\mu$m and a thickness around 800 nm herein. In order to characterize the mechanical properties of the passive tellurite glass microbubbles, the elasticity equations described in [31] were used to calculate the frequency shift of the resonances as a function of pressure. The required material parameters used were: shear modulus $G=27.5$ GPa, bulk modulus $K=40$ GPa, refractive index $n=2$, Young's modulus $E=67.2$ GPa, elasto-optical constants $C_1=-1.8\times 10^{-12}$ m$^{2}$/N and $C_2=-2\times 10^{-12}$ m$^{2}$/N [37,38]. Wall thicknesses from 700 to 900 nm were used in the calculation and the results are shown in Fig. 4. Note that the theoretical value of air pressure sensitivity was between $-6.4$ to $-8.3$ GHz/bar for the wall thicknesses used. Compared with silica, tellurite has higher shear and bulk moduli, but lower elasto-optical constants. As a comparison, the pressure sensitivity of a silica microbubble with the same diameter and wall thicknesses was calculated to be $-8.7$ to $-11.2$ GHz/bar. Fig. 4. Simulated resonance frequency shift of a passive tellurite glass as a function of air pressure and wall thickness. The color bar represents the frequency shift in units of GHz. 5. Active Yb-Er co-doped tellurite glass microbubble The Yb$^{3+}$-Er$^{3+}$ co-doped tellurite microbubbles (with diameters around 130 $\mu$m) were pumped using a 980 nm diode laser. When the pump laser output power was increased from 0 mW to 1.34 mW, a fluorescence spectrum was observed on the OSA. A free spectral range (FSR) of 3.1 nm was fitted to the fluorescence spectrum from a theoretical calculation result of 3.02 nm [39]. For this particular microbubble, we observe 8 modes in a single FSR, see Fig. 5(a). The main reason for this is that the shape of the fabricated tellurite microbubble is not perfectly spherical, hence some polar modes are excited within. Fig. 5. (a) The measured output spectrum when the pump power is below threshold; the red arrows indicate that the FSR is about 3.1 nm, l and m are the polar and azimuthal mode numbers in the microbubble. (b) The electric field distribution of different polar modes, when the mode number $l=467$. (c) The simulation and measurement of $m$ order spacing as function of polar mode number. The inset is a microscope image of the microbubble, where a and b are the major and minor axes. The mode number at 1578.56 nm was calculated to be $l=m=467$. The number of field maxima in the polar direction is given by $l-m+1 = 1,2,3\cdots$ The first three of these polar modes are also highlighted in Fig. 5(a) and have a measured mode spacing of 0.47 nm, 0.41 nm and 0.4 nm, respectively. The polar mode spacing is in close agreement with the calculated mode spacing determined from a numerical FEM (COMSOL) model, see Fig. 5(b) and (c). An image of the microbubble is given in the inset of Fig. 5(c). If we define a as the major axis and b as the minor axis, the eccentricity, $\varepsilon _i$, of the microbubble can be calculated from [40] (3)$$\varepsilon_{i} = \frac{\textit{a}-\textit{b}}{\textit{a}}.$$ Additionally, the eccentricity, $\varepsilon _{\lambda }$, determined from the mode spacing can be calculated from [40] (4)$$\Delta f_\textrm{ecc}=\|f_{ml}-f_{m+1l}\|\approx f_{ml} \cdot \varepsilon_{\lambda} \frac{\textit{\|m\|}-{1/2}}{\textit{l}},$$ where $f_{ml}$ is the frequency of the $ml$ mode. The $\varepsilon _{i}$ and $\varepsilon _{\lambda }$ of the bubble were measured and calculated as 0.14 and 0.12, respectively, and are in reasonable agreement. Even though the walls of these microbubbles appear to be of uniform thickness, it is not surprising that the bubble shape can deform to a degree that lifts the mode degeneracy. Differently shaped, doped microbubbles were also tested by pumping with the 980 nm light. The resulting fluorescence spectra and images are shown in Fig. 6(a)-(d). As the eccentricity decreases, the number of higher order modes decreases, as expected. When the eccentricity is 10$\%$ only one higher order mode exists within a single FSR. As the eccentricity drops, the higher order mode spacing decreases and below 1$\%$ the mode degeneracy is nearly recovered resulting in a single mode spectrum. Fig. 6. (a) and (c) Microscope images of the microbubbles under test. $\varepsilon _{i}$ and $\varepsilon _{\lambda }$ are the measured and calculated eccentricity of the microbubbles. Fluorescence spectrum of the Yb$^{3+}$-Er$^{3+}$ doped tellurite glass microbubble with only two modes (b) or single mode (d) emission in an FSR, corresponding to the bubbles in (a) and (c), respectively. d is the diameter of the microbubble. $\Delta \lambda _\textrm {ecc}$ is the wavelength spacing between the polar modes. (e) Laser power at 1578.56 nm as a function of pump power. The red line represents a linear fit to the experimental data, with a lasing threshold of 1.66 mW. The inset is the output laser spectrum when the pump power is 1.71 mW. Most compound glasses have larger refractive indices than silica. For example, when a (typically silica) tapered fiber is used to pump a rare-earth doped compound microsphere with a high refractive index, many higher order WGMs are excited [41]. This could be attributed to the fact that the gain of many of the modes is greater than the loss, resulting in a lower energy conversion efficiency and the resonator being more prone to output mode hopping. As the pump power was increased further to 1.63 mW, laser emission at a wavelength of 1578.56 nm was detected. The results are shown in Fig. 6(e), where the relationship between the 980 nm pump power and the detected power of a single WGM lasing mode is plotted. The lasing threshold is about 1.66 mW and single-mode lasing output was observed throughout the entire measurement cycle. Wavelength tuning of the Yb$^{3+}$-Er$^{3+}$ doped tellurite glass microbubble with a diameter of 130 $\mu$m was also investigated by varying the temperature and the internal air pressure. The 980 nm diode laser was used as pump to obtain a fluorescence spectrum and the pump power was increased from 0 to 35.6 mW. The resulting spectra are shown in Fig. 7(a). It should be noted that when the power launched into the tapered fiber was increased to 40 mW, a large loss was induced because the tellurite glass microbubble melted and fused to the tapered fiber [42]. The result of the tuning is shown in Fig. 7(b) and a maximum wavelength shift, that is tuning range, of 1.53 nm was obtained. Some jumps in wavelength tuning range were observed due to the thermal effects around pump/cavity resonances [43,44]. Although the tuning is in general nonlinear, the overall tuning rate is −5.3 GHz/mW. The frequency shift of the WGM laser modes at different air pressures from 0 to 0.6 bar was also investigated and the results are shown in Fig. 7(c) and (d). A tuning sensitivity of $-6.5$ GHz/bar was determined following a linear fitting. The accuracy is limited by the spectral resolution of the OSA of 0.05 nm. Fig. 7. (a-b) The measured fluorescence emission spectrum obtained by increasing the pump power from 0 to 35.6 mW. The maximum wavelength tuning range is about 1.53 nm and the red arrow indicates the direction of wavelength tuning. (c) Lasing wavelength of the active tellurite glass microbubble at different pressures. (d) Frequency shift as a function of the internal air pressure. Red lines are linear fits. In this work, we report on a method to fabricate microbubbles from a soft glass, namely tellurite, using a CO${_2}$ laser. The method involves first melting the glass capillary to form a sphere and then blowing out the sphere to make a bubble. The fabricated bubbles have diameters and wall thicknesses of approximately 150 $\mu$m and 670 nm, respectively. Notably, the bubbles have quite a uniform wall thickness and spherical shape which can result in a low number of higher order modes. The measured eccentricity can approach that previously observed in microspheres. The microbubbles were made from both passive and Yb:Er doped glass. The tellurite glass microbubbles were investigated experimentally and characterized in terms of their $Q$-factors, mode spectra, tuning rates, eccentricity, and laser output, and these results were compared against silica whispering gallery resonators where possible. In the case of passive microbubbles, a high Q-factor of 2.3$\times$10$^6$ was achieved. A broadband 980 nm diode pump laser was used to change the temperature inside the microbubble and a large temperature sensitivity of 4.9 GHz/K was obtained. In addition, the air pressure sensitivity of 7.1 GHz/bar was measured by applying different internal air pressures. Even though the tellurite glass is softer than silica, the expected increase in pressure tuning is negated by the lower elasto-optic coefficient. For the active microbubbles, a maximum wavelength tuning range of 1.53 nm was observed by increasing the intensity of the pump light. Separately, the sensitivity of the output laser frequency to air pressure was determined as 6.5 GHz/bar which is similar to the undoped microbubble aerostatic tuning rate. Additionally, the devices fabricated in this investigation can have very low eccentricity and so fewer modes in a single FSR, resulting in a higher laser conversion efficiency. Note that, due to the toxicity of the material, they are not ideal for biosensing compared to devices made from silica [45,46], and the single input of the fabricated tellurite microbubbles impedes fluid flow. Aside from this limitation, the devices reported in this article have potential impact for many applications, including low threshold, high conversion efficiency and tunable microcavity laser sources operating in the near and mid-infrared range, integrated active photonic devices, and laser sensing using microbubble resonators based on compound glass. In future work, coupling may be improved by using either a high-index tapered fiber or prism coupler to improve the phase-matching condition. National Key Program of the Natural Science Foundation of China (NSFC 61935006); National Natural Science Foundation of China (NSFC 61805074, NSFC 61905048); Fundamental Research Funds for the Central Universities (3072019CF2504, 3072019CF2506, 3072019CFQ2503, 3072020CFJ2507, 3072020CFQ2501, 3072020CFQ2502, 3072020CFQ2503, 3072020CFQ2504, GK2250260018, HEUCFG201841); The 111 project to Harbin Engineering University (B13015); Heilongjiang Provincial Natural Science Foundation of China (LH2019F034); Heilongjiang Touyan Innovation Team Program; Harbin Engineering University Scholarship Fund; Okinawa Institute of Science and Technology Graduate University. The authors acknowledge the Engineering Support Section of OIST Graduate University. 1. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. 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Phys. D: Appl. Phys. 6(17), 2070–2078 (1973). [CrossRef] 23. N. Riesen, S. A. V. A. François, and T. M. Monro, "Material candidates for optical frequency comb generation in microspheres," Opt. Express 23(11), 14784–14795 (2015). [CrossRef] 24. N. Riesen, W. Q. Zhang, and T. M. Monro, "Dispersion analysis of whispering gallery mode microbubble resonators," Opt. Express 24(8), 8832–8847 (2016). [CrossRef] 25. N. Riesen, W. Q. Zhang, and T. M. Monro, "Dispersion in silica microbubble resonators," Opt. Lett. 41(6), 1257–1260 (2016). [CrossRef] 26. J. M. Ward, Y. Yang, and S. Nic Chormaic, "Glass-on-glass fabrication of bottle-shaped tunable microlasers and their applications," Sci. Rep. 6(1), 25152 (2016). [CrossRef] 27. G. Righini and S. Soria, "Biosensing by WGM microspherical resonators," Sensors 16(6), 905 (2016). [CrossRef] 28. W. Lee, Y. Sun, H. Li, K. Reddy, M. Sumetsky, and X. Fan, "A quasi-droplet optofluidic ring resonator laser using a micro-bubble," Appl. Phys. Lett. 99(9), 091102 (2011). [CrossRef] 29. S. Tanabe, "Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication," C. R. Chim. 5(12), 815–824 (2002). [CrossRef] 30. J. Jiang, Y. Liu, K. Liu, S. Wang, Z. Ma, Y. Zhang, P. Niu, L. Shen, and T. Liu, "Wall-thickness-controlled microbubble fabrication for WGM-based application," Appl. Opt. 59(16), 5052–5057 (2020). [CrossRef] 31. R. Henze, T. Seifert, J. Ward, and O. Benson, "Tuning whispering gallery modes using internal aerostatic pressure," Opt. Lett. 36(23), 4536–4538 (2011). [CrossRef] 32. A. Cosci, F. Quercioli, D. Farnesi, S. Berneschi, A. Giannetti, F. Cosi, A. Barucci, G. N. Conti, G. Righini, and S. Pelli, "Confocal reflectance microscopy for determination of microbubble resonator thickness," Opt. Express 23(13), 16693–16701 (2015). [CrossRef] 33. J. Yu, X. Wang, W. Li, M. Zhang, J. Zhang, K. Tian, Y. Du, S. Nic Chormaic, and P. Wang, "An experimental and theoretical investigation of a 2 μm wavelength low-threshold microsphere laser," J. Lightwave Technol. 38(7), 1880–1886 (2020). [CrossRef] 34. L. R. P. Kassab, R. A. Kobayashi, M. J. V. Bell, A. P. Carmo, and T. Catunda, "Thermo-optical parameters of tellurite glasses doped with Yb3+," J. Phys. D: Appl. Phys. 40(13), 4073–4077 (2007). [CrossRef] 35. Q. Ma, T. Rossmann, and Z. Guo, "Whispering-gallery mode silica microsensors for cryogenic to room temperature measurement," Meas. Sci. Technol. 21(2), 025310 (2010). [CrossRef] 37. R. El-Mallawany, "Tellurite glasses Part 1. Elastic properties," Mater. Chem. Phys. 53(2), 93–120 (1998). [CrossRef] 38. M. J. Weber, Handbook of Optical Materials (CRC Press, 2003). 39. M. Sumetsky, Y. Dulashko, and R. S. Windeler, "Super free spectral range tunable optical microbubble resonator," Opt. Lett. 35(11), 1866–1868 (2010). [CrossRef] 40. C. Zhang, A. Cocking, E. Freeman, Z. Liu, and T. Srinivas, "On-chip glass microspherical shell whispering gallery mode resonators," Sci. Rep. 7(1), 14965 (2017). [CrossRef] 41. J. Yu, E. Lewis, G. Farrell, and P. Wang, "Compound glass microsphere resonator devices," Micromachines 9(7), 356 (2018). [CrossRef] 42. J. M. Ward, P. Féron, and S. Nic Chormaic, "A taper-fused microspherical laser source," IEEE Photonics Technol. Lett. 20(6), 392–394 (2008). [CrossRef] 43. T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12(20), 4742–4750 (2004). [CrossRef] 44. J. Ward and S. Nic Chormaic, "Thermo-optical tuning of whispering gallery modes in Er3+:Yb3+ co-doped phosphate glass microspheres," Appl. Phys. B 100(4), 847–850 (2010). [CrossRef] 45. F. Vollmer and S. Arnold, "Whispering-gallery-mode biosensing: label-free detection down to single molecules," Nat. Methods 5(7), 591–596 (2008). [CrossRef] 46. S. Berneschi, F. Baldini, A. Cosci, D. Farnesi, G. Nunzi Conti, S. Tombelli, C. Trono, S. Pelli, and A. Giannetti, "Fluorescence biosensing in selectively photo–activated microbubble resonators," Sens. Actuators, B 242, 1057–1064 (2017). [CrossRef] M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25(19), 1430–1432 (2000). J. P. Rezac and A. T. Rosenberger, "Locking a microsphere whispering-gallery mode to a laser," Opt. Express 8(11), 605–610 (2001). I. M. White, N. M. Hanumegowda, H. Oveys, and X. Fan, "Tuning whispering gallery modes in optical microspheres with chemical etching," Opt. Express 13(26), 10754–10759 (2005). Y. Ooka, Y. Yang, J. Ward, and S. Nic Chormaic, "Raman lasing in a hollow, bottle-like microresonator," Appl. Phys. Express 8(9), 092001 (2015). P. Bianucci, "Optical microbottle resonators for sensing," Sensors 16(11), 1841 (2016). S. Kasumie, Y. Yong, J. M. Ward, and S. Nic Chormaic, "Towards visible frequency comb generation using a hollow WGM resonator," Rev. Las. Eng. 46, 92–96 (2018). S. Frustaci and F. Vollmer, "Whispering-gallery mode (WGM) sensors: review of established and WGM-based techniques to study protein conformational dynamics," Curr. Opin. Chem. Biol. 51, 66–73 (2019). T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, "Microresonator-based optical frequency combs," Science 332(6029), 555–559 (2011). G. Bahl, K. H. Kim, W. Lee, J. Liu, X. Fan, and T. Carmon, "Brillouin cavity optomechanics with microfluidic devices," Nat. Commun. 4(1), 1994 (2013). S. Parkins and T. Aoki, "Microtoroidal cavity QED with fiber overcoupling and strong atom-field coupling: A single-atom quantum switch for coherent light fields," Phys. Rev. A 90(5), 053822 (2014). M. R. Foreman, J. D. Swaim, and F. Vollmer, "Whispering gallery mode sensors," Adv. Opt. Photonics 7(2), 168–240 (2015). R. Madugani, Y. Yong, J. M. Ward, V. H. Le, and S. Nic Chormaic, "Optomechanical transduction and characterization of a silica microsphere pendulum via evanescent light," Appl. Phys. Lett. 106(24), 241101 (2015). Y. Yang, S. Saurabh, J. M. Ward, and S. Nic Chormaic, "High-Q, ultrathin-walled microbubble resonator for aerostatic pressure sensing," Opt. Express 24(1), 294–299 (2016). Y. Yang, X. Jiang, S. Kasumie, G. Zhao, L. Xu, J. M. Ward, L. Yang, and S. Nic Chormaic, "Four-wave mixing parametric oscillation and frequency comb generation at visible wavelengths in a silica microbubble resonator," Opt. Lett. 41(22), 5266–5269 (2016). Y. Yang, F. Lei, S. Kasumie, L. Xu, J. M. Ward, L. Yang, and S. Nic Chormaic, "Tunable erbium-doped microbubble laser fabricated by sol-gel coating," Opt. Express 25(2), 1308–1313 (2017). J. M. Ward, Y. Yang, F. Lei, X.-C. Yu, Y.-F. Xiao, and S. Nic Chormaic, "Nanoparticle sensing beyond evanescent field interaction with a quasi-droplet microcavity," Optica 5(6), 674–677 (2018). L. T. Hogan, E. H. Horak, J. M. Ward, K. A. Knapper, S. Nic Chormaic, and R. H. Goldsmith, "Toward real-time monitoring and control of single nanoparticle properties with a microbubble resonator spectrometer," ACS Nano 13(11), 12743–12757 (2019). M. Sumetsky, Y. Dulashko, and R. S. Windeler, "Optical microbubble resonator," Opt. Lett. 35(7), 898–900 (2010). A. Watkins, J. Ward, Y. Wu, and S. Nic Chormaic, "Single-input spherical microbubble resonator," Opt. Lett. 36(11), 2113–2115 (2011). P. Wang, J. Ward, Y. Yang, X. Feng, G. Brambilla, G. Farrell, and S. Nic Chormaic, "Lead-silicate glass optical microbubble resonator," Appl. Phys. Lett. 106(6), 061101 (2015). S. Inoue, A. Nukui, K. Yamamoto, T. Yano, S. Shibata, and M. Yamane, "Correlation between specific heat and change of refractive index formed by laser spot heating of tellurite glass surfaces," J. Non-Cryst. Solids 324(1-2), 133–141 (2003). G. K. White, "Thermal expansion of reference materials: copper, silica and silicon," J. Phys. D: Appl. Phys. 6(17), 2070–2078 (1973). N. Riesen, S. A. V. A. François, and T. M. Monro, "Material candidates for optical frequency comb generation in microspheres," Opt. Express 23(11), 14784–14795 (2015). N. Riesen, W. Q. Zhang, and T. M. Monro, "Dispersion analysis of whispering gallery mode microbubble resonators," Opt. Express 24(8), 8832–8847 (2016). N. Riesen, W. Q. Zhang, and T. M. Monro, "Dispersion in silica microbubble resonators," Opt. Lett. 41(6), 1257–1260 (2016). J. M. Ward, Y. Yang, and S. Nic Chormaic, "Glass-on-glass fabrication of bottle-shaped tunable microlasers and their applications," Sci. Rep. 6(1), 25152 (2016). G. Righini and S. Soria, "Biosensing by WGM microspherical resonators," Sensors 16(6), 905 (2016). W. Lee, Y. Sun, H. Li, K. Reddy, M. Sumetsky, and X. Fan, "A quasi-droplet optofluidic ring resonator laser using a micro-bubble," Appl. Phys. Lett. 99(9), 091102 (2011). S. Tanabe, "Rare-earth-doped glasses for fiber amplifiers in broadband telecommunication," C. R. Chim. 5(12), 815–824 (2002). J. Jiang, Y. Liu, K. Liu, S. Wang, Z. Ma, Y. Zhang, P. Niu, L. Shen, and T. Liu, "Wall-thickness-controlled microbubble fabrication for WGM-based application," Appl. Opt. 59(16), 5052–5057 (2020). R. Henze, T. Seifert, J. Ward, and O. Benson, "Tuning whispering gallery modes using internal aerostatic pressure," Opt. Lett. 36(23), 4536–4538 (2011). A. Cosci, F. Quercioli, D. Farnesi, S. Berneschi, A. Giannetti, F. Cosi, A. Barucci, G. N. Conti, G. Righini, and S. Pelli, "Confocal reflectance microscopy for determination of microbubble resonator thickness," Opt. Express 23(13), 16693–16701 (2015). J. Yu, X. Wang, W. Li, M. Zhang, J. Zhang, K. Tian, Y. Du, S. Nic Chormaic, and P. Wang, "An experimental and theoretical investigation of a 2 μm wavelength low-threshold microsphere laser," J. Lightwave Technol. 38(7), 1880–1886 (2020). L. R. P. Kassab, R. A. Kobayashi, M. J. V. Bell, A. P. Carmo, and T. Catunda, "Thermo-optical parameters of tellurite glasses doped with Yb3+," J. Phys. D: Appl. Phys. 40(13), 4073–4077 (2007). Q. Ma, T. Rossmann, and Z. Guo, "Whispering-gallery mode silica microsensors for cryogenic to room temperature measurement," Meas. Sci. Technol. 21(2), 025310 (2010). R. El-Mallawany, "Tellurite glasses Part 1. Elastic properties," Mater. Chem. Phys. 53(2), 93–120 (1998). M. J. Weber, Handbook of Optical Materials (CRC Press, 2003). M. Sumetsky, Y. Dulashko, and R. S. Windeler, "Super free spectral range tunable optical microbubble resonator," Opt. Lett. 35(11), 1866–1868 (2010). C. Zhang, A. Cocking, E. Freeman, Z. Liu, and T. Srinivas, "On-chip glass microspherical shell whispering gallery mode resonators," Sci. Rep. 7(1), 14965 (2017). J. Yu, E. Lewis, G. Farrell, and P. Wang, "Compound glass microsphere resonator devices," Micromachines 9(7), 356 (2018). J. M. Ward, P. Féron, and S. Nic Chormaic, "A taper-fused microspherical laser source," IEEE Photonics Technol. Lett. 20(6), 392–394 (2008). T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12(20), 4742–4750 (2004). J. Ward and S. Nic Chormaic, "Thermo-optical tuning of whispering gallery modes in Er3+:Yb3+ co-doped phosphate glass microspheres," Appl. Phys. B 100(4), 847–850 (2010). F. Vollmer and S. Arnold, "Whispering-gallery-mode biosensing: label-free detection down to single molecules," Nat. Methods 5(7), 591–596 (2008). S. Berneschi, F. Baldini, A. Cosci, D. Farnesi, G. Nunzi Conti, S. Tombelli, C. Trono, S. Pelli, and A. Giannetti, "Fluorescence biosensing in selectively photo–activated microbubble resonators," Sens. Actuators, B 242, 1057–1064 (2017). A. François, S. A. V. Aoki, T. Arnold, S. Bahl, G. Baldini, F. Barucci, A. Bell, M. J. V. Benson, O. Berneschi, S. Bianucci, P. Brambilla, G. Cai, M. Carmo, A. P. Carmon, T. Catunda, T. Cocking, A. Conti, G. N. Cosci, A. Cosi, F. Diddams, S. A. Du, Y. Dulashko, Y. El-Mallawany, R. Fan, X. Farnesi, D. Farrell, G. Feng, X. Féron, P. Foreman, M. R. Freeman, E. Frustaci, S. Giannetti, A. Goldsmith, R. H. Guo, Z. Hanumegowda, N. M. Henze, R. Hogan, L. T. Holzwarth, R. Horak, E. H. Inoue, S. Jiang, J. Jiang, X. Kassab, L. R. P. Kasumie, S. Kim, K. H. Kippenberg, T. J. Knapper, K. A. Kobayashi, R. A. Le, V. H. Lee, W. Lei, F. Lewis, E. Li, H. Li, W. Liu, J. Liu, K. Liu, T. Liu, Y. Liu, Z. Ma, Q. Ma, Z. Madugani, R. Monro, T. M. Nic Chormaic, S. Niu, P. Nukui, A. Nunzi Conti, G. Ooka, Y. Oveys, H. Painter, O. Parkins, S. Pelli, S. Quercioli, F. Reddy, K. Rezac, J. P. Riesen, N. Righini, G. Rosenberger, A. T. Rossmann, T. Saurabh, S. Seifert, T. Sercel, P. C. Shen, L. Shibata, S. Soria, S. Srinivas, T. Sumetsky, M. Sun, Y. Swaim, J. D. Tanabe, S. Tian, K. Tombelli, S. Trono, C. Vahala, K. J. Vollmer, F. Wang, P. Wang, X. Ward, J. Ward, J. M. Watkins, A. Weber, M. J. White, G. K. White, I. M. Windeler, R. S. Wu, Y. Xiao, Y.-F. Xu, L. Yamamoto, K. Yamane, M. Yang, L. Yang, Y. Yano, T. Yong, Y. Yu, J. Yu, X.-C. Zhang, C. Zhang, J. Zhang, W. Q. Zhao, G. ACS Nano (1) Adv. Opt. Photonics (1) Appl. Phys. B (1) Appl. Phys. Express (1) C. R. Chim. (1) Curr. Opin. Chem. Biol. (1) IEEE Photonics Technol. Lett. (1) J. Non-Cryst. Solids (1) J. Phys. D: Appl. Phys. (2) Mater. Chem. Phys. (1) Meas. Sci. Technol. (1) Micromachines (1) Nat. Commun. (1) Nat. Methods (1) Opt. Express (9) Optica (1) Phys. Rev. A (1) Rev. Las. Eng. (1) Sci. Rep. (2) Sens. Actuators, B (1) (1) 4 3 π ( c 1 2 ) 3 = 4 3 π ( c 2 2 ) 3 − 4 3 π ( c 2 − a 1 2 ) 3 , (2) Δ f = f ( 1 n Δ n Δ T + 1 d Δ d Δ T ) Δ T , (3) ε i = a − b a . (4) Δ f ecc = \| f m l − f m + 1 l \| ≈ f m l ⋅ ε λ \|m\| − 1 / 2 l ,	CommonCrawl
Why there's a Lorentz inner product in the unitary representations of the translation group? Consider Minkowski spacetime. Its translation group is just the additive group $\mathbb{R}^4$. This is an abelian locally compact group. Next, consider one unitary representation $T : \mathbb{R}^4\to \mathrm{U}(\mathcal{H})$ on the Hilbert space $\mathcal{H}$. It is said that the SNAG theorem implies that $$T(a)=\exp\left[i\eta_{\mu\nu}a^\mu P^\nu\right]$$ where $P^\mu$ are four Hermitian commuting observables and $\eta_{\mu\nu}$ is the Minkowski metric. I want to see how to derive this from the SNAG theorem. The theorem is stated as follows (Barut's group theory book): SNAG (Stone-Naimark-Ambrose-Godement) Theorem: Let $T$ be an unitary continuous representation of an abelian locally compact group $G$ in a Hilbert space $\mathscr{H}$. Then there exists on the character group $\hat{G}$ a spectral measure $E$ such that $$T(x)=\int_{\hat{G}}\langle \hat{x},x\rangle dE(\hat{x})$$ Now it is possible to show that for the additive group $\mathbb{R}^n$ the SNAG theorem tells us that there are $n$ self-adjoint commuting operators $Y_1,\dots,Y_n$ such that $$T(x)=\exp\left[i\sum_{k=1}^n x^k Y_k\right].$$ These operators $Y_k$ are defined in terms of the spectral measure $E$ given by the SNAG theorem as $$Y_k=\int y_k dE(y).$$ Now, the Minkowski spacetime translation group is exactly $\mathbb{R}^4$ so this theorem should apply. Indeed it is almost it, except that for Minkowski spacetime, the operators are $P_0,\dots, P_3$ and $$T(a)=\exp\left[i \eta_{\mu\nu}a^\mu P^\nu\right]$$ I can't get why. How does the Minkowski inner product ends up there if the translation group is just $\mathbb{R}^4$ which has nothing to do with the metric structure? This has something to do with the realization of the translation group as a subgroup of the Poincare group so that if $U(\Lambda,a)$ is a unitary representation of the latter one has $U(1,a)$ a unitary representation of the translations satisfying $$U(\Lambda,b)U(1,a)U(\Lambda,b)^\dagger=U(1,\Lambda a)$$ I think the answer comes from this, but I don't know how to justify it. quantum-mechanics quantum-field-theory special-relativity mathematical-physics group-theory $\begingroup$ My guess is the answer is something like: we can use any product between $a$ and $P$ that we like, since we are just talking about the translation group, which doesn't know anything about the Minkowski product. To make things convenient later we take the convention that positive time translations are mapped to $e^{itP^0}$ while positive space translations are mapped to $e^{-ixP^1}$ etc. Any physical meaning for the Minkowski product there comes later when we restrict our representation to positive energies only. $\endgroup$ – Luke Pritchett Jan 28 '19 at 2:24 $\begingroup$ The SNAG theorem is a neat generalization of the Stone's theorem and thus one can use the spectral theorem for both operator types (unitary and self-adjoint) to arrive at the result via the "exponential of an unbounded self-adjoint operator", a quite delicate mathematical concept. @Valter Moretti. $\endgroup$ – DanielC Jan 28 '19 at 17:34 $\begingroup$ I think I made some progress, but there is one last bit which is exactly what @LukePritchett talks about in his comment. How the Minkowski inner product ends up in the exponent if the translation group "knows nothing about said product"? I think the answer is in the fact that if $U(a,\Lambda)$ is a unitary representation of the Poincare group, then $$U(b,\Lambda)U(a,1)U(b,\Lambda)^\dagger = U(\Lambda a,1)$$. This implies in particular that $$U(b,\Lambda)P^\mu U(b,\Lambda)^\dagger = \Lambda^\mu_\nu P^\nu$$ so I think that somehow the answer comes from this. I just don't know how. $\endgroup$ – user1620696 Jan 28 '19 at 18:05 I will use the signature $(+,-,-,-)$ for the Minkowski metric $\eta$. If you got so far as showing that $$ \forall\ {\rm continuous\ unitary\ repesentation}\ T\ {\rm of}\ (\mathbb{R}^4,+), $$ $$ \exists\ {\rm commuting\ self-adjoint\ operators} \ Y_1,\ldots,Y_4, $$ $$ \forall x\in \mathbb{R}^4,\ \ T(x)=\exp[i(x^1T_1+x^2T_2+x^3T_3+x^4T_4)] $$ then the last step to conclude that $$ \forall\ {\rm continuous\ unitary\ repesentation}\ T\ {\rm of}\ (\mathbb{R}^4,+), $$ $$ \exists\ {\rm commuting\ self-adjoint\ operators} \ P^0,\ldots,P^3, $$ $$ \forall a=(a^0,\ldots,a^3)\in \mathbb{R}^4,\ \ T(a)=\exp[i(a^0P^0-a^1P^1-a^2P^2-a^3P^3)] $$ is trivial: just define $P^0=Y_1$, $P^1=-Y_2$, $P^2=-Y_3$, $P^3=-Y_4$. BTW, as mentioned in DanielC's comment, appealing to the SNAG Theorem is overkill here. The much more elementary Stone-von Neumann Theorem is enough. Abdelmalek AbdesselamAbdelmalek Abdesselam $\begingroup$ Thanks for the answer ! I've notice one could make such choice, but why is it convenient to do so? I mean, what's the motivation behind it? Has it to do with the fact that when the full Poincare group is considered we have the relation $$U(b,\Lambda)U(a,1)U(b,\Lambda)^\dagger = U(\Lambda a,1)$$ $\endgroup$ – user1620696 Feb 6 '19 at 20:55 $\begingroup$ This has nothing to do with the full Poincare group. Even if all we got is the Lorentz group, it is convenient to write everything in terms of Lorentz invariant objects like $\eta$ so one can easily track how things transform by Lorentz even in the midst of complicated computations (think, e.g., 5 loop Feynman diagram). So we do this because it is convenient and because we can. $\endgroup$ – Abdelmalek Abdesselam Feb 6 '19 at 21:27 Not the answer you're looking for? Browse other questions tagged quantum-mechanics quantum-field-theory special-relativity mathematical-physics group-theory or ask your own question. Unitary representations of the diffeomorphism group in curved spacetime Why are only linear representations of the Lorentz group considered as fundamental quantum fields? Representations of Lorentz group on Hilbert spaces Representations of Lorentz group in interacting QFT Does there exist finite dimensional irreducible rep. of Poincare group where translations act nontrivially? (2+1)-d Unitary irreducible representations of the Poincare group Symmetry of the Pauli group and its representations What's wrong with this argument that parity is always conserved? Why in an irreducible unitary representation of the Poincare group all momenta are on the same mass shell? Can I define a unitary representation of the Lorentz group on the Hilbert space of a theory which breaks Lorentz invariance?	CommonCrawl
www.springer.com The European Mathematical Society Pages A-Z StatProb Collection Project talk Model theory From Encyclopedia of Mathematics The part of mathematical logic studying mathematical models (cf. Model (in logic)). The origins of model theory go back to the 1920's and 1930's, when the following two fundamental theorems were proved. 1 Theorem 1 2.2 Comments Theorem 1 (Gödel compactness theorem). If each finite subcollection of a collection $T$ of propositions in a first-order language is consistent, then the whole collection $T$ is consistent (see [1]). (Löwenheim–Skolem theorem). If a collection of propositions in a first-order language of signature $\Omega$ has an infinite model, then it has a model of any infinite cardinality not less than the cardinality of $\Omega$. Theorem 1 has had extensive application in algebra. On the basis of this theorem, A.I. Mal'tsev created a method of proof of local theorems in algebra (see Mal'tsev local theorems). Let $A$ be an algebraic system of signature $\Omega$, let $\|A\|$ be the underlying set of $A$, let $X\subseteq\|A\|$, let $(\Omega,X)$ denote the signature obtained from $\Omega$ by the addition of symbols for distinguished elements $c_a$ for all $a\in X$, and let $(A,X)$ denote the algebraic system of signature $(\Omega,X)$ which is an enrichment of $A$ in which for each $a\in X$ the symbol $c_a$ is interpreted by the element $a$. The set $O(A)$ of all closed formulas of the signature $(\Omega,\|A\|)$ in a first-order language which are true in the system $(A,\|A\|)$ is called the elementary diagram of the algebraic system $A$ (or the description of the algebraic system $A$), and the set $D(A)$ of those formulas from $O(A)$ which are either atomic or the negation of an atomic formula is called the diagram of $A$. An algebraic system $B$ is called an elementary extension of $A$ if $\|A\|\subseteq\|B\|$ and if $(B,\|A\|)$ is a model for $O(A)$. In this case $A$ is called an elementary subsystem of $B$. For example, the set of rational numbers with the usual order relation is an elementary subsystem of the system of real numbers with the usual order relation. A subsystem $A$ of an algebraic system $B$ of signature $\Omega$ is an elementary subsystem of $B$ if and only if for each closed formula $(\exists v)\Phi(v)$ in the first-order language of signature $(\Omega,\|A\|)$ which is true in $(B,\|A\|)$ there is an $a\in\|A\|$ such that $\Phi(c_a)$ is true in $(B,\|A\|)$. It follows at once from this criterion that the union of an increasing chain of elementary subsystems is an elementary extension of each of these subsystems. If a closed $\forall\exists$-formula in a first-order language is true in every system of an increasing chain of systems, then it is true in the union of the chain (see [1], [8]). Let the signature $\Omega$ contain a one-place relation symbol $U$. One says that a model $A$ of a theory $T$ of signature $\Omega$ has type $(\alpha,\beta)$ if the cardinality of $\|A\|$ is equal to $\alpha$ and if the cardinality of $U(A)=\{a\in\|A\|: A\models U(a)\}$ is equal to $\beta$. Vaught's theorem: If an elementary theory $T$ of countable signature has a model of type $(\alpha,\beta)$ where $\alpha>\beta$, then $T$ has a model of type $(\aleph_1,\aleph_0)$ (see [7], [8], [10]). Under the assumption that the generalized continuum hypothesis holds, an elementary theory of countable signature has models of types $(\aleph_{\alpha+2},\aleph_{\alpha+1})$ for each $\alpha$, if it has a model of type $(\aleph_1,\aleph_0)$ (see [10]). Under the same assumption the theory ${\rm Th}(A)$, where the signature of $A$ is $(+,\,.\,,0,1,<,U)$, $\|A\|$ is the set of all real numbers, $U(A)$ the set of all integers, and $+,\,.\,,0,1,<$, are defined in the usual way, does not have a model of type $(\aleph_2,\aleph_0)$. Let $(A,P)$ denote the enrichment of the algebraic system $A$ by a predicate $P$, and let $\Omega,P)$ be the signature obtained from $\Omega$ by the addition of the predicate symbol $P$. In many cases it is important to understand when in each member of a class $\mathcal K$ of algebraic systems of signature $(\Omega,P)$ the predicate $P$ is given by a formula in the first-order language of the signature $\Omega$. A partial answer to this question is given by Beth's definability theorem: There exists a formula $\Phi(x)$ in the first-order language of signature $\Omega$ such that the formula $(\forall x)(\Phi(x)\leftrightarrow P(x))$ is true for all members of an axiomatizable class $\mathcal K$ of signature $(\Omega,P)$ if and only if the set $\{P: (A,P)\in\mathcal K\}$ contains at most one element for each algebraic system $A$ of signature $\Omega$ (see [2], [8]). Much research in model theory is connected with the study of properties preserved under operations on algebraic systems. The most important operations include homomorphism, direct and filtered products. A statement $\Phi$ is said to be stable with respect to homomorphisms if the truth of $\Phi$ in an algebraic system $A$ implies the truth of $\Phi$ in all epimorphic images of $A$. A formula $\Phi$ in a first-order language is called positive if $\Phi$ does not contain negation and implication signs. It has been proved (see [1], [8]) that a statement $\Phi$ in a first-order language is stable relative to homomorphisms if and only if $\Phi$ is equivalent to a positive statement. A similar theorem holds for the language $L_{\omega_1\omega}$. A formula $\Phi(x_1,\dots,x_n)$ in a first-order language of signature $\Omega$ is called a Horn formula if it can be obtained by conjunction and quantification from formulas of the form $(\Phi_1 \&\dots\&\Phi_s)\rightarrow\Phi$, $\neg(\Phi_1\&\dots\&\Phi_s)$, where $\Phi_1,\dots,\Phi_s$ are atomic formulas in the first-order language of $\Omega$. Examples of Horn formulas are identities and quasi-identities. Central in the theory of ultraproducts is the theorem of J. Łos: Every formula in a first-order language is stable with respect to any ultrafilter (see [1]). A formula in a first-order language is conditionally stable with respect to any filter if and only if it is equivalent to a Horn formula. There is the following theorem (see [9]): Two algebraic systems $A$ and $B$ of signature $\Omega$ are elementarily equivalent if and only if there is an ultrafilter $D$ on a set $I$ such that $A^I/D$ and $B^I/D$ are isomorphic. The cardinality of a filtered product is countably infinite if for each natural number $n$ the number of factors of cardinality $n$ is finite. If for each natural number $n$ the set of indices for which the corresponding factors have cardinality $n$ does not belong to $D$, then the cardinality of the ultraproduct with respect to a non-principal ultrafilter $D$ on a countable set $I$ is equal to that of the continuum. For each infinite set $I$ of cardinality $\alpha$ there is a filter $D$ on $I$ such that for each filter $D_1$ on $I$ containing $D$, and each infinite set $A$, the cardinality of $A^I/D$ is not less than $2^\alpha$ (see [1]). Many applications have been found for the Ehrenfeucht–Mostowski theorem on the existence of models with a large number of automorphisms (see [3]): For any totally ordered set $X$ in an axiomatizable class $\mathcal K$ of algebraic systems containing an infinite system, there is a system $A$ such that $X\subseteq\|A\|$ and such that each order-preserving one-to-one mapping of $X$ onto $X$ can be extended to an automorphism of $A$. The major notions in model theory are those of universal, homogeneous and saturated systems. Let $A$ and $B$ be algebraic systems of a signature $\Omega$. A mapping $f$ of a set $X\subseteq\|A\|$ into a set $Y\subseteq\|B\|$ is called elementary if for each formula $\Phi(x_1,\dots,x_n)$ in the first-order language of the signature $\Omega$ and any $a_1,\dots,a_n\in X$ the equivalence $A\models\Phi(x_1,\dots,x_n)\iff B\models\Phi(f(a_1),\dots,f(a_n))$ holds. A system $A$ is called $\alpha$-universal if for every system $B$ that is elementarily equivalent to $A$ and of cardinality not exceeding $\alpha$, there is an elementary mapping from $\|B\|$ into $\|A\|$. A system $A$ is called $\alpha$-homogeneous if for every set $X\subseteq\|A\|$ of cardinality less than $\alpha$, every elementary mapping from $X$ into $\|A\|$ can be extended to an elementary mapping of $\|A\|$ onto $\|A\|$ (that is, to an automorphism of $A$). A system $A$ of signature $\Omega$ is called $\alpha$-saturated if for every set $X\subseteq\|A\|$ of cardinality less than $\alpha$ and every collection $\Sigma$ of formulas in the first-order language of the signature $(\Omega,X)$ not containing free variables other than $x_0$, $\Sigma$ finitely satisfiable in $(A,X)$ implies that $\Sigma$ is satisfiable in $(A,X)$. A system $A$ is called universal (respectively, homogeneous or saturated) if $A$ is $\alpha$-universal (respectively, $\alpha$-homogeneous or $\alpha$-saturated), where $\alpha$ is the cardinality of $\|A\|$. A system is saturated if and only if it is simultaneously universal and homogeneous. Two elementary equivalent saturated systems of the same cardinality are isomorphic (see [3]). All uncountable models of elementary theories which are categorical in uncountable cardinalities (cf. Categoricity in cardinality) and of countable signature are saturated (Morley's theorem, see [3], [8]). A large number of examples of $\alpha$-saturated systems is given by ultraproducts. For example, if $D$ is a non-principal ultrafilter on a countable set $I$, then $\prod_{i\in I}A_i/D$ is an $\aleph_1$-saturated system for any algebraic system $A_i$ ($i\in I$) of a countable signature $\Omega$. The basic problems of model theory are the study of the expressive possibilities of a formalized language and the study of classes of structures defined by means of such a language. Some important properties of stable theories have been found, and the classes of categorical and superstable theories have been studied in even more detail. The basic apparatus for the study of stable theories is the classification of formulas and locally consistent sets of formulas in these theories. Such a classification can be obtained by means of ascribing to formulas their ranks. Such ranks are usually ordinals and the ranking functions are given with the help of special topologies and other means. The study of ranking functions and their improvements is a rich source of information on the theories. In the study of classes of models one is concerned with the number of distinct models, up to isomorphism, of a theory of a given cardinality, the existence of special models, for example, simple, minimal, saturated, homogeneous, universal, etc., and one creates means for constructing them. The classical examples of application of methods of model theory are the papers of A. Robinson and his school, which developed an independent science — non-standard analysis; from the work of Mal'tsev and his school the applications of model-theoretic methods to topological algebra have been developed; the latest results on the properties of stable theories have been used in the study of concrete algebraic questions. The above problems arose also in the study of various non-elementary languages, for example, obtained by the addition of new quantifiers, the introduction of infinite expressions, modalities, etc. [1] A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) [2] A. Robinson, "Introduction to model theory and to the metamathematics of algebra" , North-Holland (1963) [3] M.A. Taitslin, "Model theory" , Novosibirsk (1970) (In Russian) [4] Yu.L. Ershov, "Decidability problems and constructive models" , Moscow (1980) (In Russian) [5] Yu.A. Palyutin, "Mathematical logic" , Moscow (1979) (In Russian) [6] Yu.L. Ershov, I.A. Lavrov, A.D. Taimanov, M.A. Taitslin, "Elementary theories" Russian Math. Surveys , 20 : 4 (1965) pp. 35–105 Uspekhi Mat. Nauk , 20 : 4 (1965) pp. 37–108 [7] A.I. Mal'tsev, "Some problems in the theory of classes of models" , Proc. 4-th All-Union Math. Congress (1961) , 1 , Leningrad (1963) pp. 169–198 (In Russian) (Transl. in: Amer. Math. Soc. Transl. (2) 83 (1969), 1–48) [8] C.C. Chang, H.J. Keisler, "Model theory" , North-Holland (1973) [9] G.E. Sacks, "Saturated model theory" , Benjamin (1972) [10] R.L. Vaught, "Denumerable models of complete theories" , Infinitistic methods. Proc. Symp. Foundations of Math. Warsaw, 1959 , Pergamon (1961) pp. 303–321 [11] M. Morley, R. Vaught, "Homogeneous universal models" Math. Scand. , 11 : 1 (1962) pp. 37–57 [12] M. Morley, "Categoricity in power" Trans. Amer. Math. Soc. , 114 : 2 (1965) pp. 514–538 [13] S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1978) [14] J.L. Bell, A.B. Slomson, "Models and ultraproducts: an introduction" , North-Holland (1971) Theorem 1 was proved by K. Gödel in [a1]. $\newcommand{\nmodels}{\mathop{\models}\limits}$ Let $\mathfrak A_i=(A_i,\{R_n(i)\})$, $i\in I$, be a collection of relational structures of the same type (algebraic systems of the same signature). (I.e. the $A_i$ are sets, the $R_n(i)$ relations.) Let $F$ be an ultrafilter on the index set $I$. Then the ultraproduct $(\prod_i A_i)/F$ is a relational structure of the same type. (Cf. (the editorial comments to) Ultrafilter for the definition of ultraproduct.) A precise formulation of Łos' theorem is now as follows (cf. [a8] or [a9]). Let $f=(f_1,\dots,f_n,\dots)$ be a sequence of elements from $\prod A_i$, i.e. $f_n=(f_n(i))_{i\in I}\in\prod A_i$ for all $n$. Let $f/F$ be the sequence $(f_1/F,\dots,f_n/F,\dots)$ of elements from $(\prod A_i)/F$ and let $f(i)$ be the sequence $f(i)=(f_1(i),\dots,f_n(i),\dots)$. Then for any formula $\phi$ from the language $L$ for which the $\mathfrak A_i$ are interpretations, $$ (\prod A_i)/F\nmodels_{f/F}\phi\quad\iff\quad \{i\in I: \mathfrak A_i\nmodels_{f(i)}\phi\}\in F.$$ Łos' theorem is also called the fundamental theorem on ultraproducts. The theorem to the effect that two algebraic systems $\mathfrak A$ and $\mathfrak B$ are equivalent if and only if they have isomorphic ultrapowers is known as the Keisler ultrapower theorem: see Keisler-Shelah isomorphism theorem. One of the basic applications of logic to algebra is the work by J. Ax and S. Kochen [a2]. See [a3] for the model theory of infinitary languages; [a4], [a5] for stability theory; and [a6], [a7] for categorical model theory. The Gödel compactness theorem and the Löwenheim–Skolem theorem are in the Russian literature sometimes known as the Gödel–Mal'tsev theorem and the Löwenheim–Skolem–Mal'tsev theorem, respectively. [a1] K. Gödel, "Die Vollständigheit der Axiome des logischen Funktionenkalküls" Monatshefte für Math. und Physik , 37 (1930) pp. 344–360 [a2] J. Ax, S. Kochen, "Diophantine problems over local fields III: decidable fields" Ann. of Math. , 83 (1966) pp. 437–456 [a3] M.A. Dickmann, "Large infinitary languages" , North-Holland (1975) [a4] A. Pillay, "An introduction to stability theory" , Oxford Univ. Press (1985) [a5] J.T. Baldwin, "Fundamentals of stability theory" , Springer (1988) [a6] G.E. Reges, "First order categorical logic" , Lect. notes in math. , 611 , Springer (1977) [a7] J. Lambek, P. Scott, "Higher order categorical logic" , Cambridge Univ. Press (1986) [a8] J.L. Bell, A.B. Slomson, "Models and ultraproducts" , North-Holland (1969) [a9] W.W. Comfort, S. Negrepontis, "The theory of ultrafilters" , Springer (1974) pp. §11 How to Cite This Entry: Model theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Model_theory&oldid=50989 This article was adapted from an original article by A.D. TaimanovM.A. Taitslin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article Retrieved from "https://encyclopediaofmath.org/index.php?title=Model_theory&oldid=50989" TeX done About Encyclopedia of Mathematics Impressum-Legal	CommonCrawl
Intelligent decision support algorithm for distribution system restoration Reetu Singh1, Shabana Mehfuz2 & Parmod Kumar3 Distribution system is the means of revenue for electric utility. It needs to be restored at the earliest if any feeder or complete system is tripped out due to fault or any other cause. Further, uncertainty of the loads, result in variations in the distribution network's parameters. Thus, an intelligent algorithm incorporating hybrid fuzzy-grey relation, which can take into account the uncertainties and compare the sequences is discussed to analyse and restore the distribution system. The simulation studies are carried out to show the utility of the method by ranking the restoration plans for a typical distribution system. This algorithm also meets the smart grid requirements in terms of an automated restoration plan for the partial/full blackout of network. India's blackout as two separate events on 30th and 31st July 2012, has forced the system design engineers to plan the restoration schemes which are more effective and efficient and takes into account the uncertainties involved in the transmission distribution network. Power system blackouts cannot be avoided. Their occurrence is rare, but their impact can be very serious. Further, the electric utilities also have loss of revenue. It is therefore essential to study, analyse and prepare the restoration plans in the event of tripping of a feeder or transmission line due to fault or other reasons (Fukuyama and Chiang 1995; Hsu et al. 1992; Hsiao and Chien 2000; Hsu and Kuo 1994; Lee et al. 1998; Ma et al. 1992). Some of the features of effective strategy plans are: The minimum number of consumers should be affected due to outage of the feeder. Power loss should be minimum on restoration of supply after blackout/feeder tripping. The structure of the radial network needs to be maintained as far as possible. The capacity of transformer, feeders, laterals and other network circuits should maintain and not exceed their capacity limits after restoration of the distribution network system. The critical consumers and loads should be given priority during strategy planning. For fast and techno economical solutions, there shall be ranking of the restoration plans in order to be followed by system operator. The outage of feeder needs to be isolated to restore the maximum possible consumer's load by restructuring the distribution network meeting the operational constraints and limits of feeder and equipments. In the present paper the criteria for selecting the restoration strategic plans are: Minimizing the operation of switches. Availability of supply to maximum consumer's load, and Meeting the constraints and unconstrained criteria. Tsai et al. (1993), Berdandon et al. (2011), Feltes et al. (2006), Liu and Lin (2012), Chen et al. (2001); Lotfifard et al. (2011), Lim et al. (2006) and Nagata and Sasaki (2002) present the various methods for reconfiguration of networks after blackout or feeder outage. However, a limited work is available based on fuzzy-grey relation ranking method. Further, in most of these research works, constraint related to distribution transformer has not been included. Transformer is an important source of supply and hence we have incorporated the constraint of transformer capacity limit besides the constraints related to distribution network, like feeders and laterals capacity limitations. In order to compare and verify the accuracy of the study, a prototype distribution network of same structure (Chen et al. 2005) has been considered. The objective of present study is to minimize the number of switching operations for restoration of power supply to all consumer's load and meeting the distribution system network constraints: Let λ 1 (s) be the minimum number of switching operations. Let Nsw be the maximum number of switches in the network. X is the position of switch during operartion. If a switch is opened from closed position or vice versa, the value of switch vector will be 1 and if there is no change in the status of switch, its value will be 0. The maximum feeder's loading λ 2 (s) shall be within the rating of current capacity limitation, if λ 2 (s) ≤ maximum (feeder lines loading). Similarly the maximum lateral loading, λ 3 (s) is given by λ 3 (s) ≤ maximum (lateral lines loading). Restoration of maximum loads at faulted area. Minimum switching operations in the restoration plan. The load in feeder, laterals and transformers should be balanced as far as possible, and the overloading of electrical equipment should be avoided. The reconfigured distribution system should be nearly close to the original system, and the radial structure of distribution system should be maintained. Mathematical model for reconfiguration In order to design and develop the strategic plans for reconfiguration and implement them effectively, the following model of constraints and equalities are presented: Equalities/objective functions Number of switching operations If λ 1 (S) defines the operations of the number of switches, $$\lambda_{1} (s) = \sum\limits_{i = 1}^{{N_{sw} }} {X_{i} }$$ then, here, Xi is switch state vector given by [S1, S2, S3 …, SNSW], Nsw = The total switches that can be operated in the network under consideration, Xi = status of the switch. The conditions for the switch status are: Xi = 1, if switch is opened from closed position or vice versa, Xi = 0, if status of switch is not changed. Minimum number of switching operations indicates that the system will be more stability. Maximum loading among backup feeders The maximum loading, λ 2 (S) among supported feeder is given by Eq. (2): $$\lambda_{2} (S) = Max(I_{{FD_{i} }} ),\quad i = 1,2, \ldots ,N_{FD}$$ ${\text{I}}_{{{\text{FD}}_{\text{i}} }}$ represents the current over the supported feeder FDi after switching operations. NFD defines the number of supported feeders. To meet the constraints criteria, λ2 (s) shall be minimised. This objective function will give the most loaded backup feeder and by this we can have the remaining marginal load. Maximum loading among backup laterals Like loading criteria for feeders the supported laterals shall also meet the load criteria. This objective function will give the most loaded backup laterals. A lesser value of λ3 (s) is preferred. λ3 (s) is the capacity of supported laterals and LATi is the load current over the laterals after switching operation and N LAT is the number of lateral branches. For techno-economic operation the λ3 (S), Eq. (3) is desired to be minimized: $${\lambda _3}(S) = Max({I_{LA{T_i}}}),\quad i = 1,2, \ldots ,{N_{LAT}}$$ where λ3 (S) defines the supported laterals for maximum loading and $I_{{LAT_{i} }}$ defines current over of the supported lateral LATi after switching operation. NLAT defines the number of laterals in the distribution network. The load on the laterals should be minimum for the best operating conditions during restoration. Unbalanced loading of feeders The feeders as well as laterals shall have the balanced loading of feeders and laterals. It is an important feature for line loss reduction and voltage stability criteria. Thus, the load unbalancing index of feeders and laterals can be computed using Eqs. (4) and (5) respectively. $$\lambda_{4} (S) = \sqrt {\sum\limits_{i = 1}^{{N_{FD} }} {(LV_{{FD_{i} }} - LV_{FD} )^{2} } }$$ where, ${\text{LV}}_{{{\text{FD}}_{\text{i}} }}$ is percentage load level of feeder FDi and LVFD is percentage refrence load level which is given by Eq. (5) $$LV_{FD} = \frac{{\sum\nolimits_{i = 1}^{{N_{FD} }} {I_{{FD_{i} }} } }}{{\sum\nolimits_{i = 1}^{{N_{FD} }} {IR_{{FD_{i} }} } }}100$$ In the above equation ${\text{I}}_{{{\text{FD}}_{\text{i}} }}$ and ${\text{IR}}_{{{\text{FD}}_{\text{i}} }}$ represents the load current and rated load current of feeder. In order to improve the performance of the system the unbalancing loading index shall be as minimised. Unbalanced loading of laterals Similarly, the lateral branches unbalance load index λ5 (s) can be computed using equation: $$\lambda_{5} (S) = \sqrt {\sum\limits_{i = 1}^{{N_{LAT} }} {(LV_{{LAT_{i} }} - LV_{LAT} )^{2} } }$$ where, ${\text{LV}}_{{{\text{LAT}}_{\text{i}} }}$ is percentage load level of lateral LATi and LVLAT is percentage reference load level which is given by Eq. (7) as: $$LV_{LAT} = \frac{{\sum\nolimits_{i = 1}^{{N_{LAT} }} {I_{{LAT_{i} }} } }}{{\sum\nolimits_{{}}^{{}} {IR_{{LAT_{i} }} } }}100$$ In the above equation, ${\text{I}}_{{{\text{LAT}}_{\text{i}} }}$ and ${\text{IR}}_{{{\text{LAT}}_{\text{i}} }}$ represents the load current and rated load current of lateral respectively. This objective function is used to determine the degree of unbalanced loading of laterals, therefore, less value of λ5 (s) is preferred. Maximum loading among backup transformer Transformer is the main source of power supply to feeders and laterals. Its maximum loading capacity and unbalanced loading index after the isolation of the fault needs to be computed and checked. These shall be as minimum as possible. The minimization of maximum loading of transformer due to supported feeders and laterals is desirable. Maximum loading of transformer, λ6 (s) is computed by Eq. (8) as: $$\lambda_{6} (S) = Max(I_{{TRS_{i} }} ),\quad i = 1,2, \ldots ,N_{TRS}$$ Unbalanced loading of transformer The unbalanced loading index of transformer, λ7 (s) is given by Eq. (9), where, $$\lambda_{7} (s) = \sqrt {\sum\limits_{i = 1}^{{N_{TRS} }} {(LV_{{TRS_{i} }} - LV_{TRS} )^{2} } }$$ where, LVTRSi is percentage load level of transformer TRSi and LVTRS is percentage reference load level which is given by Eq. (10) $$LV_{TRS} = \frac{{\sum\nolimits_{i = 1}^{{N_{TRS} }} {I_{{TRS_{i} }} } }}{{\sum\nolimits_{i = 1}^{{N_{TRS} }} {IR_{{TRS_{i} }} } }}100$$ In the above equation ${\text{I}}_{{{\text{TRS}}_{\text{i}} }}$ and ${\text{IR}}_{{{\text{TRS}}_{\text{i}} }}$ represents the load current and rated load current of transformer respectively. This gives the degree of unbalance loading of transformer for the backup and the value of this function should be minimum. To further optimize the switching operation for the reconfiguration of distribution system, the following constraints shall have to be met: Open switch operation have been complemented by closed switch operation. ILATmin < Ij < ILATmax IFEEDERmin < Ij < IFEEDERmax ITRSmin < Ij < ITRSmax Fuzzy grey method Fuzzy multi criteria evolution The fuzzy logic method is a mathematical tool to make decision for vague and imprecise information in power system restoration problems (Chang 2008; Farahani et al. 2007; Gomes Flavio 2006; Gonzalez et al. 2012; Savier and Das 2007; Nagata et al. 1995; Pham et al. 2009; Wong and Lai 2000). Using fuzzy logic data base rules, a strategy with lesser switching operation and better load balance is achievable. Here, the linguistic terms like lesser, better etc. convey the vague nature of information. The restoration plan is considered more preferable if it involves fewer switching operations and better load balance. In restoration process, uncertainties arises when the feeder, lateral or transformer current is changing during the restoration process. These uncertainties can be taken into account using the fuzzy logic tool. It is based on rule-base (system operator experience), membership function of variable, and inference decision engine (IF THEN statement). One can consider the membership function of any type (triangular, sigmoid etc) but generally the fuzzy function is selected based on the nature of the problem. In the present study, three level (Low, Moderate and High) triangular fuzzy functions are considered to simplify the calculations during the restoration process, as shown in Fig. 1. "Appendix 2" presents the values of fuzzy membership function for different objective function. The fuzzy function transforms crisp value to fuzzy value which lie in the range {0, 1}. Then using rule-base and fuzzy inference decision procedure, the fuzzy value related to each defined variable/objective function is computed. These output fuzzy values, after inference, are transferred back to crisp values using de-fuzzification methods (either centre of gravity method or centre of area method). In the consequents the fuzzy sets are low, moderate and high, and can be crisply defined as 1, 0.5 and 0 respectively. Further inference is drawn by calculating the real value of objective function. By real value we can get the firing strength in IF-THEN rule and the weighted average. By using Eq. (11), the crisp de-fuzzification value is derived. $$f_{i}^{} = \frac{{\sum\nolimits_{j = 1}^{{N_{R} }} {\mu_{j} \times y_{j} } }}{{\sum\nolimits_{j = 1}^{{N_{R} }} {\mu_{j} } }}$$ where, µj and y j are firing strengths of anticedents and consequences; NR represents the number of fuzzy rules. The value of f i * represent the fitness degree of objective function 'f i ' for each restoration plan. For example, the number of switching operations performed for the restoration plan 5 is 5. The rules are as follows: Triangular fuzzy function R1: IF λ 1 (S) is low, THEN the plan is good. R2: IF λ 1 (S) is moderate, THEN the plan is moderate. R3: IF λ 1 (S) is high, THEN the plan is bad The rule strength of R1, R2 and R3 will be 0.5, 0.5 and 0 respectively for the plan 5. After computing the rules, we use Eq. (11) to translate the rule results into real value by weighted average method. The singleton value for good, moderate and bad are 1, 0.5 and 0 respectively. The corresponding de-fuzzification value will be (0.51+0.50.5+0*0)/(0.5+0.5) = 0.750. Grey regression method Grey relation theory is based upon the concept that available information is incomplete and/or unknown. It is data analysis technique to solve the multicriteria decision making (MCDM). Such problems (MCDM) are difficult to solve using fuzzy logic tools (Wong and Lai 2000; Zhang and Zhengeai 2008; Cheng et al. 1998; Deng 1982; Dong et al. 2003; Huang and Huang 1996; Liu and Forrest 2007; Deng 1989; Song et al. 2002; Tsai et al. 2003; Wong and Lai 1999; Chang and Yeh 2005; Huang et al. 2008; Lin et al. 2008). The coefficient of grey relation (GRC) of xi with respect to x0 for kth term is given in Chen et al. (2005). $$\gamma \left( {x_{0} \left( k \right), \, x_{i} \left( k \right)} \right) \, = \, \left\{ {\Delta_{\hbox{max} } - \Delta_{{{\text{o}}i}} \left( k \right) \, } \right\}/\left\{ {\Delta_{\hbox{max} } - \, \Delta_{\hbox{min} } } \right\}$$ where, x 0 = (x 0(1), x 0(2), x 0(3), …, x 0(n)), o = 1, 2, 3, …, n, and x i = (x i (1), x i (2), x i(3), …, x i (m)), i = 1, 2, 3,…, m. $${\Delta _{{\rm{max}}}} = {{Max}}{\mkern 1mu} \left( {{x_0}\left( k \right) - {x_i}\left( k \right)} \right)$$ $${\Delta _{{\rm{min}}}} = {{ Min}}{\mkern 1mu} \left( {{x_0}\left( k \right) - {x_i}\left( k \right)} \right)$$ $$\Delta_{{{\text{o}}i}} \left( k \right) = \left\| {\left( {x_{0} \left( k \right) - x_{i} \left( k \right)} \right)} \right\|$$ The GRG between each comparative sequence x i and refrence sequence xo is derived from average value of GRC. The order of relation between comparative and reference sequences is given by $\varGamma 0i$. Higher value of $\varGamma 0i$ means that the comparative sequence is more close to reference sequence than comparative sequence. $$\varGamma_{0i} = \sum\limits_{k = 1}^{n} {\frac{1}{n}} \gamma (x_{0} (k),x_{i} (k))$$ In the next stage of the grey analysis, the GRA is used to measure the preference degree for all feasible restoration plans. The various steps for the fuzzy grey approach for ranking the restoration plans and selecting the satisfactory plan is presented in the fuzzy grey relation. Fuzzy-grey relation In order to overcome the limitations of decision conditions related to grey relational method and fuzzy logic tool limitations, the two decision making tools with incomplete and vague information are fused together to form a hybrid fuzzy-grey relational tool (Lin et al. 2008; Basu and George 2014; Pereira Junior et al. 2014; Shahsavari et al. 2014; Liu et al. 2015). This tool overcomes the limitations in the two methods, and make the decision making more relevant and effective. Based on the minimum value of λ, the optimized objective function is decided among various alternatives. Optimized λ value is the best alternative switching operation sequence, loading on feeders, laterals and transformer. It provides the optimized solutions for decision making considering the various constraints and equalities. Choice of restoration plan is a type of multi criteria decision making problem which depends on all the objective functions and constraints considered. In this work we have tried to construct a measurement model via grey relational analysis to provide useful information and help system operator to make a right decision on the problem of service restoration. Figure 2 represents the flow chart for the entire restoration algorithm. Starting with the on-off status of the switches, the feasible restoration plans are generated. Objective function values are computed using Eqs. (1)–(10) for the feasible plans. Further fitness degree of the objective function is evaluated using the fuzzy multicriteria evaluation method. The grey regression analysis (GRA) method is used to calculate the preference index of the restoration plans. Based on the grey regression grades (GRG) the plans are ranked according to their preference order. The addition of objective function's minimization of unbalanced loading of transformer gives the stable restoration plans for the considered network. This improves the system reliability and stability, leading to the improved performance of the system. Flow chart for the restoration plan Numerical application In order to show the utility of fuzzy-grey relation method for reconfiguration of distribution network, a distributed transformer as a source of supply is added to the network configuration (Chen et al. 2005). The distribution system of Taiwan Power Company is considered and presented in Fig. 3, which has main feeders YD28 supplying power to LAT1, LAT2, LAT3, LAT4, LAT5, LAT6, LAT7, LAT8 and LAT9. Each lateral has its supporting lateral (LAT10, LAT11, LAT12, LAT13, LAT14, LAT15, LAT16 and LAT17). These alternating laterals are connected to the main lateral with the help of switches. When a fault occurs in the system the switches operate to restore the out of service area. "Appendix 1" presents the pre-fault load current of feeders. "Appendix 2" presents the maximum capacity of each feeder, laterals, and the loading of transformer. The switch state vector, X comprises of the main switches SW i (i = 1–9) and alternative switches SW j (j = 1–8). Switch Sw9 always remain closed since lateral 9 is not connected with any other supporting lateral. To maintain the radial structure of the network, the switch open operation should be followed by switch close operation or vice versa. The switch pair for each restoration plan is given in Table 1 against each restoration plan based on the switching operation performed. The lateral, feeder and transformer loading is presented against the switching vector. All load values are in ampere. The maximum number of switching operations can be 8, thus, the total number of possible restoration plans are 28 = 256. The rated capacity of feeder, lateral, and transformer are assumed as 450, 100, and 800 A respectively before the fault condition. Table 1 lists all the feasible restoration plans and the load currents on supporting lines after the restoration of supply. The 22 feasible restoration plans are selected from 256 possible restoration plans. The maximum switching operations possible are, therefore, 7 for the computations as per the switch state vector. Table 2 gives the values of all the objective functions. The corresponding data shown in Table 3 are obtained by fuzzy multi-criteria evaluation discussed in "Mathematical model for reconfiguration" section. This table gives us the values of reference sequence which is used further to calculate GRGs. the reference sequence selected for analysis is maximum value of fuzzy evaluation of objective functions: X0 = (0.750, 0.2522, 0.3469, 0.719, 0.391, 0.777, 0.625). The GRG values computed using Eq. (13) are presented in Table 4. The grey relation grades computed by Chen et al. (2005) are reproduced also in this table. After the GRA steps are completed, the preference ranking of feasible plans are derived with the related analysis and the ranking is presented in Table 5. From the table we can see that the plan 5 is having the highest rank. The plans get their ranking modified because of the addition of transformer objective functions. The loading of all the transformer present in the distribution system is graphically presented in Fig. 4. Additional objective function increases the stability of the system as the unbalancing of the transformer load can also be handled. The new plans are compared with the earlier ones in Fig. 5. Higher the number of objective functions more is the stability of the system as more parameters are considered. The difference of old and new GRG is calculated in Table 4. Further, mean of difference of new and old GRG is calculated, which comes out to be 0.1092. This means that the system performance has increased by 1.09 % with new GRG. The restoration plans are ranked according to the new GRG values. If the current exceeds in a particular element, it could fail and the current would be shunted to other network element which eventually may fail also. Here all equipments are considered so if feeder gets overloaded, it can transfer load to lateral and it can shift the load to the transformer and system gets more stable. Entire restoration algorithm consists of the fuzzy evaluation and fuzzy grey multicriteria given by flow chart. The inclusion of transformer loading parameters in fuzzy analysis makes the system more stable as more parameters are restored. In previous work, feeder and laterals were restored but in this work, transformer is included and restored after the fault which increases the stability and reliability of the system. Distribution system Table 1 Load current of supporting components after restoration Table 2 Values of objective functions Table 3 Values of fuzzy evaluation data Table 4 Test result of grey relational grade Table 5 Rank before and after restoration Transformer loading Result comparison In this paper, the objective functions are considered so as to optimize the operation of switches and loading of feeders, laterals as well as on transformers and minimization of unbalanced loading index of feeders, laterals and transformer after switching operation. The constraints in the restoration process to be considered are: (a) maintain the radial structure of the network, (b) no overloaded equipment and (c) higher priority customers should always be supplied first. The result shows that inclusion of the transformer current limits has changed the ranking of plans. The system becomes more reliable with the minimization of unbalanced loading of transformer current as there is no scope for exceeding the limits during restoration plans. Feasible plans which consider more objective functions make the system more reliable. The reliability and stability of the restored network has increased by 1.09 %. The priority customers can be supplied first on the basis of the preference index plans during partial blackout or full blackout. The consumer loads which are not energized may be fed by the supporting feeders in the neighbourhood via on-off switches. The result shows that restoration process is done using minimum number of switching operations. Safety and operability of transformer, laterals and feeders is taken up by maintaining the line currents within the operational limits of power system components. The topology of distribution system is maintained radial before and after implementing the restoration plans. A switch-opened operation is always followed by a switch-close operation, after every switching operation. Simulation studies for restoration of distribution system are carried out considering multi-objective problem and fuzzy-grey algorithm. Transformer loading has been considered as an additional objective function in the optimization problem. The result shown is more stable and reliable because more are the objective functions, greater is the stability. Various strategies are derived based on rank. The best strategy is the one with the highest rank. The studies are useful for system operator in taking right decision during the restoration process. The rank of restoration plans is given which makes this method effective and very promising. The studies are useful for electric utilities/power distribution company to improve the customer services and revenue returns. Discussion and future work This research work has proposed an intelligent restoration algorithm for the distribution system using fuzzy grey combination. The computation implementation is done by adding the unbalance loading of transformer to the Taiwan Power Distribution network. The result shows that the stability of system has improved with the new additional objective functions. The fuzzy multicriteria evaluation gives the optimization values for various objective functions and using grey regression analysis the ranking of plans has been done. This can help in service restoration of priority based customers. The intelligent algorithm is capable of fulfilling the requirements of smart grid such as stability, automation and reliability. 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Competing interest Jamia Millia Islamia, New Delhi, Delhi, India Reetu Singh Department of Electrical Engineering, Jamia Millia Islamia, New Delhi, Delhi, India Shabana Mehfuz IRD, Maharaja Agrasen Institute of Technology, New Delhi, Delhi, India Parmod Kumar Correspondence to Reetu Singh. Table 6 Pre fault load current of laterals Table 7 Values of fuzzy function for objective function Singh, R., Mehfuz, S. & Kumar, P. Intelligent decision support algorithm for distribution system restoration. SpringerPlus 5, 1175 (2016). https://doi.org/10.1186/s40064-016-2810-4 Grey system theory Power system restoration	CommonCrawl
Investigation of chemotherapy-induced brain structural alterations in breast cancer patients with generalized q-sampling MRI and graph theoretical analysis Tsung-Yuan Li1 na1, Vincent Chin-Hung Chen2,3 na1, Dah-Cherng Yeh4, Shu-Ling Huang5, Cheng-Nan Chen1, Jyh-Wen Chai1,6, Clayton Chi-Chang Chen1,7 & Jun-Cheng Weng ORCID: orcid.org/0000-0001-7616-62163,8 BMC Cancer volume 18, Article number: 1211 (2018) Cite this article Breast neoplasms are the most common cancer among women in Taiwan. Cognitive deficits are common complications of breast cancer survivors treated with chemotherapy. The most frequently observed disorders involve executive function and memory impairment. With improvements in tumor intervention and the consequent increase in the number of cancer survivors, the quality of life of patients has become an important issue. We are interested in the early effects of chemotherapy on the brain structures of patients. In addition, generalized q-sampling imaging (GQI), a wide range of q-space datasets for a more accurate and sophisticated diffusion MR approach, was first used in this topic. As diffusion tensor imaging (DTI) is associated with restrictions in the resolution of crossing fibers, we attempted to use GQI, which can overcome these difficulties and is advantageous over DTI for tractography of the crossing fibers. This cross-sectional study included two groups: breast cancer survivors who had completed their chemotherapy (n = 19) and healthy controls (n = 20). All participants underwent diffusion MRI exams and neuropsychological assessments. We included four parts in our image analysis, i.e., voxel-based statistical analysis, multiple regression analysis, graph theoretical analysis and network-based statistical analysis. The results from the voxel-based statistical analysis showed significantly lower GFA and NQA values in the breast cancer group than those in the control group. We found significant positive correlations between the FACT-Cog and GQI indices. In the graph theoretical analysis, the breast cancer group demonstrated significantly longer characteristic path length. Adjuvant chemotherapy affected the integrity of white matter and resulted in poor cognitive performance, as indicated by the correlations between the neuropsychological assessment scales and the GQI indices. In addition, it was found that the characteristic path lengths in the breast cancer group increased, indicating that the brain network integration became worse. Our study demonstrated alterations in structural brain networks and associated neuropsychological deficits among breast cancer survivors. The most common cancer among women in Taiwan is breast neoplasms. There are more than 10,000 women suffering from breast cancer, and nearly 2000 women die of breast cancer annually [1, 2]. The stage of breast cancer is based on the tumor size, the axillary lymph nodes that are involved, and distant metastasis. If breast cancer can be diagnosed and treated earlier, the five-year survival rate would be projected to increase to a maximum of 90%. There are various treatments for breast cancer, including traditional surgery, adjuvant chemotherapy, radiation therapy, targeted therapy and hormonal therapy. Cognitive function in breast cancer survivors Breast cancer is a common cause of mortality; however, treatments have improved, and consequently, the interest in the quality of life and function among survivors has increased. Therefore, depression, anxiety and psychiatric symptoms among breast cancer survivors need to be investigated. Breast cancer patients receive different treatments based on the size of the tumor, and these treatments often result in physical or cognitive deficits in patients. A range of 15 to 50% of patients with malignant tumors show persistent cognitive impairments after chemotherapy [3]. Many studies have also reported that breast cancer patients often appear to develop the phenomenon of chemo-brain after chemotherapy. In this condition, patients often complain of problems regarding memory, concentration, multiple operation, processing speed, and word retrieval [4, 5]. This cognitive impairment in patients can affect social relationships and even work performance [5]. One study tracked changes in the cognitive function of three groups, which included breast cancer patients with chemotherapy combined with radiotherapy, breast cancer patients with radiotherapy only, and a control group, over 3 years. The breast cancer patients treated with chemotherapy and radiotherapy (CTRT; n = 62) or radiotherapy only (RT; n = 67) completed neuropsychological assessments 6 months after completing treatment and then again 36 months later. The control group (n = 184) was assessed over a similar interval. There was also a significant difference in executive function (p = 0.006) among the three groups. This difference indicated that the control group performed better than the CTRT and RT groups [6]. Diffusion magnetic resonance imaging of the brain With the improvements in tumor intervention and the increase in the number of cancer survivors, the cognition of patients has become an important issue for patients, physicians and researchers. One study evaluated the long-term effect of chemotherapy on brain microstructural integrity by comparing the brains of chemotherapy-exposed breast cancer survivors to those of healthy women. There were two groups of participants: 187 breast cancer survivors treated with CMF (cyclophosphamide, methotrexate, and 5-flourouracil) and 374 age-matched healthy women. Diffusion tensor imaging (DTI) was analyzed with tract-based spatial statistics. In addition, the authors used linear regression analysis to explore the impact of the length of time after chemotherapy. The results showed that the length of time after chemotherapy was inversely associated with fractional anisotropy (FA), mean diffusivity (MD) and radial diffusivity (RD) among breast cancer survivors. The authors reported that adjuvant chemotherapy had an adverse effect on the integrity of the white matter microstructure of breast cancer survivors who had survived more than 20 years on average [7]. In this study, we evaluated the early effects of chemotherapy on the brain structures of patients. Since diffusion tensor imaging is associated with restrictions in the resolution of crossing fibers, we tried to use generalized q-sampling imaging (GQI), which can overcome these difficulties and is advantageous over DTI for the tractography of crossing fibers [8]. This study was a cross-sectional study, and participants were recruited from the department of breast surgery of Taichung Veterans General Hospital. The study included 19 women with a history of breast cancer (stage I-IIIA) who had completed their primary chemotherapy less than 6 months before study entry and were currently without evidence of active cancer. There were 4 patients received radiation therapy, and 1 patient received hormone treatment. The number of menopausal women in the patients and controls were 5:5. The average age of the breast cancer survivors was 43.8± 6.4 years. The only chemotherapeutic drugs used by the patients were taxotere and epirubicin. Another 20 healthy women aged 50.1± 2.5 years served as the control group. All participants underwent magnetic resonance imaging (MRI) examinations on a 1.5 T scanner (Aera, Siemens, Germany) and neuropsychological assessments. The clinical characteristics and neuropsychological assessments are shown in Table 1. Table 1 Group differences in clinical characteristics and neuropsychological assessment The inclusion criteria were as follows: breast cancer survivors (within 6 months after chemotherapy) 20–55 years of age and healthy female 20–55 years of age. There were no other cancer types present in the breast cancer survivors other than breast cancer. If the participants were diagnosed with psychiatric, neurologic, or comorbid medical conditions that are known to affect cognitive function, they were excluded. This study was approved by the Institutional Review Board at Taichung Veterans General Hospital. The review number was SF14185A. Participants were recruited from the department of breast surgery of Taichung Veterans General Hospital. The research assistant explained the research proposal to participants so that the participants could understand the research purpose, process and both their rights and interests. Clinical physicians assessed the physiological status of each participant to ensure she could participate in the MRI examination. All participants provided written informed consent before the examination. A clinical psychologist performed the neuropsychological assessment, and a radiologic technologist performed the subsequent MRI examination. The overall process took approximately 90 min. Neuropsychological assessments We designed the questionnaire to understand the basic information of the participants and used objective and subjective psychological tests to evaluate the cognitive function, emotion, mindfulness and psychological trauma of the participants. The neuropsychological tests included the Mini-Mental State Examination (MMSE), Functional Assessment of Cancer Therapy-Cognitive Function (FACT-Cog), Hospital Anxiety and Depression Scale (HADS), Impact of Event Scale-Revised (IES-R), and Cognitive and Affective Mindfulness Scale-Revised (CAMS-R). All statistics were performed with Microsoft Excel 2010. The results of the neuropsychological assessments are shown in Table 1. Diffusion imaging parameters For diffusion imaging, we performed a single-shot, diffusion-weighted spin echo-planar imaging sequence with the following parameters: magnetic field strength = 1.5 Tesla, repetition time = 7200 msec, echo time = 107 msec, field of view = 256 mm, matrix = 128 × 128, slice thickness = 4 mm, resolution = 2 × 2 × 4 mm3, b-values = 0, 1000, and 2000 s/mm2 in 129 noncollinear directions, number of excitations = 1, and the acquisition time was 16 min. Generalized q-sampling imaging Based on the Fourier transform between the diffusion magnetic resonance (MR) signals and the diffusion displacement, a new relationship can be deduced by directly estimating the spin distribution function (SDF) from the diffusion MR signals. This relationship leads to a new reconstructed method called generalized q-sampling imaging. GQI can provide directional and quantitative information about crossing fibers. GQI is a model-free method that quantifies the density of water, which diffuses in different orientations. Model-free methods estimate the empirical distribution of the water diffusion, and there is no hypothesis on the distribution. The SDF is the density of diffusing water in different directions and is a kind of diffusion orientation distribution function (ODF). GQI provides information of the relation between the diffusion signals of water and the SDF. GQI can be applied to grid or shell sampling schemes, q-ball imaging (QBI) and diffusion spectrum imaging (DSI). Studies have shown that GQI has good sensitivity and specificity for white matter properties and pathology [9]. The GQI indices included generalized fractional anisotropy (GFA), quantitative anisotropy (QA), normalized quantitative anisotropy (NQA) and the isotropic value of the orientation distribution function (ISO). GFA is defined as the standard deviation divided by the root mean square of the ODF, indicating a measurement of the anisotropy. QA is defined as the amount of anisotropic spins that diffuse along the fiber orientation. NQA is the normalized QA. ISO is the minimum distribution value of an ODF, and thus ISO represents the background isotropic diffusion [9]. The human brain is a complex nervous system with highly segregated and integrated functions. We can construct a complex neural network model through the connections among brain regions. Graph theory is the mathematical study of graphs that model objects ("nodes") and their connections ("edges"), where nodes represent brain regions and edges represent structural or functional connections between regions [10]. According to the related information, the brain network is divided into the white matter network and the gray matter network. The white matter network represents the connection of cerebral nerve fibers between brain regions, whereas the gray matter network represents the functional connectivity among brain regions. Research on the white matter network of the brain by modern mathematical graph theory has proven that the structural network of the brain has the characteristics of a "small-worldness" topological structure, which means that it has high clustering of nodes and short path lengths between nodes [11]. The topology indices of graph theory include the mean clustering coefficient, gamma, local efficiency, characteristic path length, lambda, global efficiency and the small-worldness index. We used graph theoretical analysis and generalized q-sampling imaging to measure brain network connectivity. We used four methods, namely, voxel-based statistical analysis, graph theoretical analysis, network-based statistical analysis and multiple regression analysis, to analyze the diffusion data. We considered covariates in all the analyses. Voxel-based statistical analysis Diffusion imaging was first corrected for eddy currents by FSL (FMRIB software library). The spin distribution function was reconstructed using a model-free reconstruction method with DSI studio (DSI studio was developed by Fang-Cheng (Frank) Yeh). Through this mathematical algorithm, we obtained the diffusion indices of generalized q-sampling imaging, including generalized fractional anisotropy (GFA), quantitative anisotropy (QA), normalized quantitative anisotropy (NQA) and the isotropic value of the orientation distribution function (ISO). Independent t-tests were performed with the Statistical Parametric Mapping (SPM) software to find the differences between the two groups. In addition, a significant difference in age between the two groups (p < 0.001) was found; thus, we considered age a covariate of no interest. Multiple regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables when the focus is on the relationship between a dependent variable and one or more independent variables. Multiple regression analysis is an extension of the application of simple linear regression that seeks to understand the function of a dependent variable and two or more sets of independent variables. Multiple regression analysis through SPM was used to detect the correlations between the neuropsychological scales and the indices of GQI for all participants. We also used age as a covariate in the multiple regression analysis. Graph theoretical analysis Generalized q-sampling MRI can noninvasively detect the direction of water molecule diffusion in the white matter of the brain. We reconstructed the pathways of nerve fibers in the brain using fiber assignment by continuous tracking (FACT) with DSI studio. Network edges were established using FACT and the Automated Anatomical Labeling (AAL) templates, which divided the brain into 90 brain regions in Montreal Neurological Institute (MNI) space. The number of virtual fibers, or "edges", connecting each pair of regions of interest (ROIs) was determined, resulting in a 90 × 90 weighted connectivity matrix for each participant [10]. We defined network edges as follows (1): $$ \mathrm{E}=\frac{Fiber\ count}{Fiber\ length}\times NQA $$ Finally, the graph theoretical algorithm was used to obtain the topological properties of the complex network measures. The area under the curve (AUC) for each connectivity metric of the topology indices was compared between the groups. The network density range was calculated from 0.05 to 0.26, in 0.01 increments. The minimum value was defined by the limit density of the individual network not to be fragmented, and the maximum value was defined by the density when the topology indices that remained unchanged [12]. Since the differences between groups in network measures below the network density which depend on the number of individual networks that fragment in each group, group comparisons below the density are not meaningful [13]. The density means the ratio of existing connections to all possible connections. To identify the statistically significance differences between groups in the network topology indices, graph theoretical analysis toolbox was used to execute the two-sample t-test and non-parametric permutation test with 1000 repetitions. We evaluated the network segregation with the mean clustering coefficient, gamma, and local efficiency, and the network integration with the characteristic path length, lambda, and global efficiency [14]. Network-based statistical analysis Network-based statistic (NBS, Melbourne Neuropsychiatry Centre, The University of Melbourne and Melbourne Health, Australia) is the graph analogue of cluster-based statistical methods used in mass univariate testing on all pixels in an image. NBS analysis was used to identify the significance of any connected sub-networks obvious in the set of altered connections [15]. NBS analysis is used to identify any potentially connected structures formed by an appropriately chosen set of supra-threshold links. The topological extent of any such structure is used to examine its significance. The test statistic (i.e., primary threshold) computed for each pairwise combination is used to construct a set of supra-threshold links. The null distribution of the number of edges was empirically acquired using non-parametric permutation (5000 permutations) to evaluate the significance of each of the connected edges. Finally, we used the BrainNet viewer (The MathWorks Inc., Natick, MA, US) to visualize the significant sub-networks revealed by NBS. A total of 39 participants were recruited for the study including 19 chemotherapy treated women and 20 healthy controls. All participants were aged between 20 and 55 years old. The average age of patients and healthy women were 43.8 ± 6.4 and 50.1 ± 2.5 years old. Due to wide range and significant difference between groups in age, we had added age as one of covariant factors in statistical analysis to reduce the impact of age. Two patients suffered from breast cancer stage I, 14 patients suffered from breast cancer stage II, and 3 patients suffered from breast cancer stage III, respectively. There were 4 patients received radiation therapy and 1 patient received hormonal treatment among 19 patients treated with chemotherapy. The number of menopausal women in the patients and controls were 5:5. The chemotherapy treated patients did not differ from the healthy controls with regard to education, MMSE, CAMS-R, HADS and IES-R. However, the breast cancer survivors showed significantly lower perceived cognitive impairments and impacts on quality of life, as revealed by paired t-tests (p < 0.05). The chemotherapy-treated patients revealed worse cognitive function. The participant demographic information and neuropsychological assessment results are presented in Table 1. The results from the voxel-based statistical analysis showed significantly lower GFA and NQA values in the breast cancer group than those in the control group (p < 0.05 corrected by false discovery rate, FDR). The brain regions with differences included the right postcentral blade, left superior corona radiate, right superior temporal gyrus, right inferior frontal blade and left middle temporal gyrus. The results of the voxel-based statistical analysis are presented in Fig. 1. The GFA values of the breast cancer survivors were lower than those of the healthy controls in a the right postcentral blade, and lower NQA values were found in the brain regions of the b left superior corona radiate, c right superior temporal gyrus, d right inferior frontal blade and e left middle temporal gyrus These neuropsychological scales (i.e., MMSE, CAMS-R, HADS, IES-R and FACT-Cog) were selected for the correlation analysis with changes in the indices of GQI. The results of the multiple regression analysis are shown in Table 2 and Fig. 2. It is worth mentioning that significant positive correlations between the perceived cognitive impairments and the GQI indices (GFA and NQA) were found in the regions of the left anterior corona radiate and the right cingulate gyrus (p < 0.01 corrected by FDR). In addition, significant positive correlations between the impact on quality of life and the GQI indices (GFA and NQA) were found in the regions of the right middle frontal gyrus, left postcentral blade and splenium of the corpus callosum (p < 0.01 corrected by FDR). Table 2 Correlation between neuropsychological assessment scales and GQI indices The results of the brain regions included in the multiple regression analysis. The brain regions (a to x) in Figure 2 correspond to the second column of Table 2 In the graph theoretical analysis, we divided the individual topology network measurement into the BC and healthy control group. If the density was below 0.05, the individual network in both groups began to fragment which resulted in different numbers of nodes for individual network. Therefore, group comparisons below the density were not meaningful. The highest density was defined by the topology network measurement remained unchanged. The density we calculated is from 0.05 to 0.26. These results were confirmed by AUC analysis across network densities. The BC group demonstrated significantly longer characteristic path lengths across densities than those of the control group (p < 0.05 corrected by FDR, Fig. 3a). Longer characteristic path length represents worse global integration in the BC group. As shown in Fig. 3b, both groups of women demonstrated connectomes with the small-world properties (small-worldness index > 1) of complex networks when compared to random networks. Although both groups maintained the small-worldness brain network, the network was more like a regular network in the BC group. The breast cancer (BC) group (blue points) showed significantly longer characteristic path lengths in (a) than those of the controls (orange points), and both groups of women showed connectomes with the small-world properties of complex networks in (b) In NBS analysis, we compared the edges of the brain networks between the BC group and the healthy controls. The results of the network-based statistics showed that some brain structure network connections were decreased in the breast cancer group compared to those in the control group (p < 0.01 corrected by FDR), as shown in Fig. 4. The lower connected sub-network comprised edges between left inferior occipital and left fusiform, right post-central, right superior parietal, and left supra-marginal; between right cuneus and right post-central; between left fusiform and right superior parietal; between left supra-marginal and right putamen. The brain structure network connections of the breast cancer group were decreased in the links between left inferior occipital and left fusiform, right post-central, right superior parietal, and left supra-marginal; between right cuneus and right post-central; between left fusiform and right superior parietal; between left supra-marginal and right putamen Neuroimaging studies have shown that cognitive impairment results in subtle and diffuse brain damage [7, 16,17,18]. The cerebral white matter mediates communication among different brain regions, and the integrity of the cerebral white matter is important for the optimal performance of the brain. Injury to any part of the white matter connections can lead to changes in cognitive performance [19]. A study demonstrated that breast cancer survivors (3 to 5 months after chemotherapy) showed significantly worse performance in terms of attention, concentration, memory and psychomotor speed, as well as decreases in the FA of major white matter structures associated with cognitive function, such as the superior longitudinal fasciculus and the corpus callosum [20]. The control group showed higher scores on the FACT-Cog than those of the BC group, indicating that the cognitive function of breast cancer survivors decreased generally in our study. However, there were significant differences between the groups in perceived cognitive impairments and impact on quality of life. The effects of chemotherapy may lead to deficits in behavior and neuropsychological performance [21]. In our study, we did not find any significant differences between the groups in terms of the neuropsychological assessments, such as the MMSE, CAMS-R, HADS and IES-R. The mechanisms mediating cognitive impairment after chemotherapy are unknown. There are several potential reasons for white matter vulnerability and cognitive function decline after chemotherapy, notably the direct white matter toxicity of chemotherapeutic drugs [7, 22]. There is evidence that the commonly used chemotherapeutic drug 5-fluorouracil (5-FU) crosses the blood brain barrier (BBB) by simple diffusion [23, 24]. Murine models have indicated that clinically relevant concentrations of 5-FU has been shown to cause injury to white matter tracts of the central nervous system; this finding was also reported in a case report on humans [25, 26]. Taxane-derived agents are chemotherapy drugs widely adopted in cancer treatment. Despite taxotere and epirubicin cross the BBB poorly [27], neurotoxicity is the major adverse effect of taxotere. They often manifested as painful neuropathy experienced during treatment, and it is sometimes irreversible [28]. There were 51 studies reported taxane-related gastrointestinal, hematological and neurological toxicities in adult patients with solid tumors [29]. In addition, epirubicin frequently induced toxicities including anemia, fever, myalgias, and neurotoxicity [30]. Therefore, we inferred that the effect of neurotoxicity was on the brain areas of the right post-central blade, left superior corona radiate, right superior temporal gyrus, right inferior frontal blade and left middle temporal gyrus. One study demonstrated decreased FA in the frontal and temporal white matter tracts of post-chemotherapy breast cancer patients compared to the tracts of healthy controls [31]. Another study showed decreased network connectivity in the frontal, striatal and temporal regions of cancer survivors 10 years after chemotherapy compared to that of healthy controls [32]. It has been demonstrated that medial temporal lobe toxicity has adverse effects on both recognition and working memory [33]. Our results were consistent with those of previous studies of the frontal and temporal regions. A study examined the effect of adjuvant chemotherapy on white matter in women with breast cancer using DTI. It was found that positive correlation existed between the FA in the genu and processing speed [34]. Another study showed significant positive correlations between the domains of attention and processing/psychomotor speed and FA in the regions of the temporal and parietal white matter tracts. Moreover, the self-report cognitive failure questionnaire (CFQ) scores negatively correlated with the FA in the frontal and parietal regions [31]. We observed that the impact on quality of life was positively correlated with GFA and NQA in the splenium of the corpus callosum and the right middle frontal gyrus. Furthermore, the CAMS-R scores positively correlated with GFA in the right middle temporal gyrus and the left middle frontal gyrus. Our results were similar to those of previous studies. The effect of menopause associated with cognitive function is uncertain [35]. Therefore, we did not consider menopause a covariate in this study. One study found no negative effects of therapy-induced menopause on cognitive function in breast cancer survivors [36]. Most studies divide breast cancer into 4 major molecular subtypes including Luminal A, Luminal B, Triple-negative, HER2 positive [37]. Whether breast cancer itself affects cognitive function is unknown. There is no literature to investigate this issue until now. Because the sample size is small and this is a preliminary research, we did not consider breast cancer subtypes in this study. Cognitive deficits are common complications of breast cancer survivors treated with chemotherapy, and the incidence of cognitive deficits can reach 75% [38]. Chemotherapy-treated patients were found to be eight times more likely to have cognitive deficits than nonchemotherapy-treated patients [39, 40]. The most frequently observed disorders involve executive function and memory impairment. One study consisted of two groups of participants, including 34 breast cancer survivors who had completed chemotherapy more than 5.35 years before on average and 27 healthy women. The researchers explored changes in resting-state functional connectivity networks with graph theoretical analysis between breast cancer survivors treated with chemotherapy and healthy women. In addition, they evaluated the relationships among network measures, the length of time after chemotherapy, age, and breast cancer stage using linear regression analysis. The results showed that the clustering coefficient, characteristic path length and small-worldness index were lower in the breast cancer survivors than in the healthy controls. Compared with the control group, the breast cancer survivors had significantly lower nodal degree values in the left amygdala, left caudate, right inferior frontal gyrus, bilateral medial orbital frontal gyrus, and bilateral superior temporal gyrus. Linear regression analysis showed that the regional degree in the left hippocampus and right hippocampus were negatively correlated with the time since treatment. The impact of chemotherapy on the connectivity of these brain areas is permanent and may worsen over time [41]. Complex networks of the brain can be economical by minimizing the wiring cost, such as by possessing multiple nearby and fewer remote connections [42]. In our study, both groups of women demonstrated connectomes with the small-world properties of complex networks when compared to the properties of random networks. A small-worldness network has high local efficiency and global efficiency so that the brain network can effectively transfer information [43]. The human brain has been demonstrated to possess connectomes with the small-world properties that not only have the ability to segregate and integrate information [44] but also have low energy consumption and high efficiency in transmitting and processing information [42, 45]. This study used GQI and graph theoretical analysis to evaluate the brain structure and networks of chemotherapy-treated breast cancer survivors in comparison with controls. The results showed that the reduction in white matter connectivity in patients with breast cancer after treatment may lead to large-scale brain network reorganization, leading to increases in segregation and decreases in integration of the brain structural network. The changes in the small-world properties could reflect a compensatory mechanism, meaning the brain strives to maintain the integrity of the entire network at the expense of other networks, such as network integration [10]. Breast cancer survivors are usually able to perform a variety of cognitive tasks (intact segregation) but need more time, more effort or different strategies than before (damaged integration) [46, 47]. We found that the decreasing network integration in the breast cancer survivors was the result of the characteristic path length. The ability of the brain network is weak in transmitting messages. This finding was consistent with the concept that the white matter pathway plays a role in brain information transmission [48]. There were no significant differences between the groups in terms of network segregation in our study. Due to compensatory neuroplasticity, the cognitive function of breast cancer survivors may remain unchanged or may only slightly deteriorate [49]. The study was a preliminary study and more comprehensive investigations will be performed in the near future. There were some limitations in this study including small sample of participants, the cross-sectional design, and lack of a non-chemotherapy treated patient group. Therefore, we cannot distinguish the toxicity of chemotherapeutic drugs and the breast cancer itself on white matter structures. In addition, there was also variability in breast cancer stage, breast cancer subtypes, hormonal treatment, and menopause status that are likely to contribute to the effects of chemotherapy on cognitive function. Our results provide further evidence that adjuvant chemotherapy is associated with demyelination of white matter. In addition, adjuvant chemotherapy affected the integrity of white matter and resulted in poor cognitive performance, as indicated by the correlation between the neuropsychological assessment scales and the GQI indices. We found that the characteristic path lengths of breast cancer survivors were longer than those of healthy controls, as assessed by graph theoretical analysis. This result indicated that the brain network integration of breast cancer survivors became worse. Our study demonstrated alterations in the structural brain networks of breast cancer survivors. Therefore, changes in GQI indices and network topological properties may serve as neuropathological biomarkers of treatment-induced neurotoxicity. This is the first study to investigate chemotherapeutic effects on brain structural changes in breast cancer survivors with a generalized q-sampling image. Further studies of this issue with larger samples and longitudinal designs are required to determine the long-term effects of altered brain network organization. 5-FU: AAL: Automated Anatomical Labeling BBB: Blood brain barrier CAMS-R: Cognitive and Affective Mindfulness Scale-Revised CFQ: Cognitive failure questionnaire CMF: Cyclophosphamide, methotrexate, and 5-flourouracil DSI: Diffusion spectrum imaging DTI: Diffusion tensor imaging FA: Fractional anisotropy Fiber assignment by continuous tracking FACT-Cog: Functional Assessment of Cancer Therapy-Cognitive Function GFA: Generalized fractional anisotropy GQI: HADS: Hospital Anxiety and Depression Scale IES-R: Impact of Event Scale-Revised Isotropic value of the orientation distribution function MD: Mean diffusivity MMSE: Mini-Mental State Examination MNI: Montreal Neurological Institute MRI: NBS: Network-based statistic NQA: Normalized quantitative anisotropy ODF: Orientation distribution function QA: Quantitative anisotropy QBI: Q-ball imaging RD: Radial diffusivity Region of interest SDF: Spin distribution function SPM: Statistical Parametric Mapping Chuang SC, Wu GJ, Lu YS, Lin CH, Hsiung CA. 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Regional brain activation during verbal declarative memory in metastatic breast cancer. Clin Cancer Res. 2009;15(21):6665–73. Filley CM. White matter: organization and functional relevance. Neuropsychol Rev. 2010;20(2):158–73. Andryszak P, Wiłkość M, Izdebski P, Żurawski B. A systemic literature review of neuroimaging studies in women with breast cancer treated with adjuvant chemotherapy. Contemp Oncol. 2017;21(1):6–15. This study was supported in part by the research programs MOST107–2221-E-182-054-MY3, MOST106–2221-E-182-079, MOST104–2314-B-040-001 and NSC103–2420-H-040-002, which were sponsored by the Ministry of Science and Technology, Taipei, Taiwan. This study was also supported by the grant (BMRPD1H0101, BMRPD1G1321) of Chang Gung University, Taoyuan, Taiwan, and the grant (CORPG6G0101, CORPG6G0121) of Chang Gung Memorial Hospital, Chiayi, Taiwan. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The datasets generated and/or analyzed during the current study are not publicly available due owing to data privacy policy at our facility, but are available from the corresponding author on reasonable request. Tsung-Yuan Li and Vincent Chin-Hung Chen contributed equally to this work. Department of Radiology, Taichung Veterans General Hospital, Taichung, Taiwan Tsung-Yuan Li, Cheng-Nan Chen, Jyh-Wen Chai & Clayton Chi-Chang Chen School of Medicine, Chang Gung University, Taoyuan, Taiwan Vincent Chin-Hung Chen Department of Psychiatry, Chang Gung Memorial Hospital, Chiayi, Taiwan Vincent Chin-Hung Chen & Jun-Cheng Weng Breast Medical Center, Cheng Ching Hospital Chung Kang Branch, Taichung, Taiwan Dah-Cherng Yeh Department of Psychology, Chung Shan Medical University, Taichung, Taiwan Shu-Ling Huang College of Medicine, China Medical University, Taichung, Taiwan Jyh-Wen Chai Department of Medical Education, Taichung Veterans General Hospital, Taichung, Taiwan Clayton Chi-Chang Chen Department of Medical Imaging and Radiological Sciences, Chang Gung University, No. 259, Wenhua 1st Rd., Guishan Dist., Taoyuan City, 33302, Taiwan Jun-Cheng Weng Tsung-Yuan Li Cheng-Nan Chen TYL: data collection, data analysis, writing article; VCHC: project idea, study design, manuscript revision; DCY: study design, data collection; SLH: data collection, data analysis; CNC: data collection; JWC: data collection; CCCC: data collection; JCW: project idea, study design, software development, data analysis, writing article, manuscript revision. All authors read and approved the final manuscript. Correspondence to Jun-Cheng Weng. This study was approved by the Institutional Review Board at Taichung Veterans General Hospital. The review number was SF14185A. All participants provided written informed consent before the examination. Li, TY., Chen, V.CH., Yeh, DC. et al. Investigation of chemotherapy-induced brain structural alterations in breast cancer patients with generalized q-sampling MRI and graph theoretical analysis. BMC Cancer 18, 1211 (2018). https://doi.org/10.1186/s12885-018-5113-z	CommonCrawl
Finally, all of the questions raised here in relation to MPH and d-AMP can also be asked about newer drugs and even about nonpharmacological methods of cognitive enhancement. An example of a newer drug with cognitive-enhancing potential is modafinil. Originally marketed as a therapy for narcolepsy, it is widely used off label for other purposes (Vastag, 2004), and a limited literature on its cognitive effects suggests some promise as a cognitive enhancer for normal healthy people (see Minzenberg & Carter, 2008, for a review). This research is in contrast to the other substances I like, such as piracetam or fish oil. I knew about withdrawal of course, but it was not so bad when I was drinking only tea. And the side-effects like jitteriness are worse on caffeine without tea; I chalk this up to the lack of theanine. (My later experiences with theanine seems to confirm this.) These negative effects mean that caffeine doesn't satisfy the strictest definition of nootropic (having no negative effects), but is merely a cognitive enhancer (with both benefits & costs). One might wonder why I use caffeine anyway if I am so concerned with mental ability. Many of the most popular "smart drugs" (Piracetam, Sulbutiamine, Ginkgo Biloba, etc.) have been around for decades or even millenia but are still known only in medical circles or among esoteric practicioners of herbal medicine. Why is this? If these compounds have proven cognitive benefits, why are they not ubiquitous? How come every grade-school child gets fluoride for the development of their teeth (despite fluoride's being a known neurotoxin) but not, say, Piracetam for the development of their brains? Why does the nightly news slant stories to appeal more to a fear-of-change than the promise of a richer cognitive future? The general cost of fish oil made me interested in possible substitutes. Seth Roberts uses exclusively flaxseed oil or flaxseed meal, and this seems to work well for him with subjective effects (eg. noticing his Chinese brands seemed to not work, possibly because they were unrefrigerated and slightly rancid). It's been studied much less than fish oil, but omega acids are confusing enough in general (is there a right ratio? McCluskey's roundup gives the impression claims about ratios may have been overstated) that I'm not convinced ALA is a much inferior replacement for fish oil's mixes of EPA & DHA. The term "smart pills" refers to miniature electronic devices that are shaped and designed in the mold of pharmaceutical capsules but perform highly advanced functions such as sensing, imaging and drug delivery. They may include biosensors or image, pH or chemical sensors. Once they are swallowed, they travel along the gastrointestinal tract to capture information that is otherwise difficult to obtain, and then are easily eliminated from the system. Their classification as ingestible sensors makes them distinct from implantable or wearable sensors. The absence of a suitable home for this needed research on the current research funding landscape exemplifies a more general problem emerging now, as applications of neuroscience begin to reach out of the clinical setting and into classrooms, offices, courtrooms, nurseries, marketplaces, and battlefields (Farah, 2011). Most of the longstanding sources of public support for neuroscience research are dedicated to basic research or medical applications. As neuroscience is increasingly applied to solving problems outside the medical realm, it loses access to public funding. The result is products and systems reaching the public with less than adequate information about effectiveness and/or safety. Examples include cognitive enhancement with prescription stimulants, event-related potential and fMRI-based lie detection, neuroscience-based educational software, and anti-brain-aging computer programs. Research and development in nonmedical neuroscience are now primarily the responsibility of private corporations, which have an interest in promoting their products. Greater public support of nonmedical neuroscience research, including methods of cognitive enhancement, will encourage greater knowledge and transparency concerning the efficacy and safety of these products and will encourage the development of products based on social value rather than profit value. Cytisine is not known as a stimulant and I'm not addicted to nicotine, so why give it a try? Nicotine is one of the more effective stimulants available, and it's odd how few nicotine analogues or nicotinic agonists there are available; nicotine has a few flaws like short half-life and increasing blood pressure, so I would be interested in a replacement. The nicotine metabolite cotinine, in the human studies available, looks intriguing and potentially better, but I have been unable to find a source for it. One of the few relevant drugs which I can obtain is cytisine, from Ceretropic, at 2x1.5mg doses. There are not many anecdotal reports on cytisine, but at least a few suggest somewhat comparable effects with nicotine, so I gave it a try. Nicotine absorption through the stomach is variable and relatively reduced in comparison with absorption via the buccal cavity and the small intestine. Drinking, eating, and swallowing of tobacco smoke by South American Indians have frequently been reported. Tenetehara shamans reach a state of tobacco narcosis through large swallows of smoke, and Tapirape shams are said to eat smoke by forcing down large gulps of smoke only to expel it again in a rapid sequence of belches. In general, swallowing of tobacco smoke is quite frequently likened to drinking. However, although the amounts of nicotine swallowed in this way - or in the form of saturated saliva or pipe juice - may be large enough to be behaviorally significant at normal levels of gastric pH, nicotine, like other weak bases, is not significantly absorbed. The Trail Making Test is a paper-and-pencil neuropsychological test with two parts, one of which requires shifting between stimulus categories. Part A simply requires the subject to connect circled numbers in ascending order. Part B requires the subject to connect circled numbers and letters in an interleaved ascending order (1, A, 2, B, 3, C….), a task that places heavier demands on cognitive control. Silber et al. (2006) analyzed the effect of d-AMP on Trails A and B and failed to find an effect. Nature magazine conducted a poll asking its readers about their cognitive-enhancement practices and their attitudes toward cognitive enhancement. Hundreds of college faculty and other professionals responded, and approximately one fifth reported using drugs for cognitive enhancement, with Ritalin being the most frequently named (Maher, 2008). However, the nature of the sample—readers choosing to answer a poll on cognitive enhancement—is not representative of the academic or general population, making the results of the poll difficult to interpret. By analogy, a poll on Vermont vacations, asking whether people vacation in Vermont, what they think about Vermont, and what they do if and when they visit, would undoubtedly not yield an accurate estimate of the fraction of the population that takes its vacations in Vermont. Cost-wise, the gum itself (~$5) is an irrelevant sunk cost and the DNB something I ought to be doing anyway. If the results are negative (which I'll define as d<0.2), I may well drop nicotine entirely since I have no reason to expect other forms (patches) or higher doses (2mg+) to create new benefits. This would save me an annual expense of ~$40 with a net present value of <820 ($); even if we count the time-value of the 20 minutes for the 5 DNB rounds over 48 days (0.2 \times 48 \times 7.25 = 70), it's still a clear profit to run a convincing experiment. A poster or two on Longecity claimed that iodine supplementation had changed their eye color, suggesting a connection to the yellow-reddish element bromine - bromides being displaced by their chemical cousin, iodine. I was skeptical this was a real effect since I don't know why visible amounts of either iodine or bromine would be in the eye, and the photographs produced were less than convincing. But it's an easy thing to test, so why not? Four of the studies focused on middle and high school students, with varied results. Boyd, McCabe, Cranford, and Young (2006) found a 2.3% lifetime prevalence of nonmedical stimulant use in their sample, and McCabe, Teter, and Boyd (2004) found a 4.1% lifetime prevalence in public school students from a single American public school district. Poulin (2001) found an 8.5% past-year prevalence in public school students from four provinces in the Atlantic region of Canada. A more recent study of the same provinces found a 6.6% and 8.7% past-year prevalence for MPH and AMP use, respectively (Poulin, 2007). Each nootropic comes with a recommended amount to take. This is almost always based on a healthy adult male with an average weight and 'normal' metabolism. Nootropics (and many other drugs) are almost exclusively tested on healthy men. If you are a woman, older, smaller or in any other way not the 'average' man, always take into account that the quantity could be different for you. Fish oil (Examine.com, buyer's guide) provides benefits relating to general mood (eg. inflammation & anxiety; see later on anxiety) and anti-schizophrenia; it is one of the better supplements one can take. (The known risks are a higher rate of prostate cancer and internal bleeding, but are outweighed by the cardiac benefits - assuming those benefits exist, anyway, which may not be true.) The benefits of omega acids are well-researched. Tyrosine (Examine.com) is an amino acid; people on the Imminst.org forums (as well as Wikipedia) suggest that it helps with energy and coping with stress. I ordered 4oz (bought from Smart Powders) to try it out, and I began taking 1g with my usual caffeine+piracetam+choline mix. It does not dissolve easily in hot water, and is very chalky and not especially tasty. I have not noticed any particular effects from it. Kennedy et al. (1990) administered what they termed a grammatical reasoning task to subjects, in which a sentence describing the order of two letters, A and B, is presented along with the letter pair, and subjects must determine whether or not the sentence correctly describes the letter pair. They found no effect of d-AMP on performance of this task. Piracetam is a reliable supplement for improving creativity. It is an entry level racetam due to its lack of severe side effects and relative subtlety. Piracetam's effects take hold over time through continual use. There is less instant gratification compared to other brain enhancers. Additionally, this nootropic can enhance holistic thinking, verbal memory, and mental energy levels. I can't try either of the products myself – I am pregnant and my doctor doesn't recommend it – but my husband agrees to. He describes the effect of the Nootrobox product as like having a cup of coffee but not feeling as jittery. "I had a very productive day, but I don't know if that was why," he says. His Nootroo experience ends after one capsule. He gets a headache, which he is convinced is related, and refuses to take more. "It is just not a beginner friendly cocktail," offers Noehr. So it's no surprise that as soon as medical science develops a treatment for a disease, we often ask if it couldn't perhaps make a healthy person even healthier. Take Viagra, for example: developed to help men who couldn't get erections, it's now used by many who function perfectly well without a pill but who hope it will make them exceptionally virile. Stayed up with the purpose of finishing my work for a contest. This time, instead of taking the pill as a single large dose (I feel that after 3 times, I understand what it's like), I will take 4 doses over the new day. I took the first quarter at 1 AM, when I was starting to feel a little foggy but not majorly impaired. Second dose, 5:30 AM; feeling a little impaired. 8:20 AM, third dose; as usual, I feel physically a bit off and mentally tired - but still mentally sharp when I actually do something. Early on, my heart rate seemed a bit high and my limbs trembling, but it's pretty clear now that that was the caffeine or piracetam. It may be that the other day, it was the caffeine's fault as I suspected. The final dose was around noon. The afternoon crash wasn't so pronounced this time, although motivation remains a problem. I put everything into finishing up the spaced repetition literature review, and didn't do any n-backing until 11:30 PM: 32/34/31/54/40%. Deficiencies in B vitamins can cause memory problems, mood disorders, and cognitive impairment. B vitamins will not make you smarter on their own. Still, they support a wide array of cognitive functions. Most of the B complex assists in some fashion with brain activity. Vitamin B12 (Methylcobalamin) is the most critical B vitamin for mental health. Board-certified neuropsychologist Brian Lebowitz, PhD and associate clinical professor of neurology at Stony Brook University, explains to MensHealth.com that the term "encompasses so many things," including prescription medications. Brain enhancers fall into two different categories: naturally occurring substances like Ginkgo biloba, creatine and phenibut; and manmade prescription drugs, like Adderall, and over-the-counter supplements such as Noopept. Kratom (Erowid, Reddit) is a tree leaf from Southeast Asia; it's addictive to some degree (like caffeine and nicotine), and so it is regulated/banned in Thailand, Malaysia, Myanmar, and Bhutan among others - but not the USA. (One might think that kratom's common use there indicates how very addictive it must be, except it literally grows on trees so it can't be too hard to get.) Kratom is not particularly well-studied (and what has been studied is not necessarily relevant - I'm not addicted to any opiates!), and it suffers the usual herbal problem of being an endlessly variable food product and not a specific chemical with the fun risks of perhaps being poisonous, but in my reading it doesn't seem to be particularly dangerous or have serious side-effects. Feeling behind, I resolved to take some armodafinil the next morning, which I did - but in my hurry I failed to recall that 200mg armodafinil was probably too much to take during the day, with its long half life. As a result, I felt irritated and not that great during the day (possibly aggravated by some caffeine - I wish some studies would be done on the possible interaction of modafinil and caffeine so I knew if I was imagining it or not). Certainly not what I had been hoping for. I went to bed after midnight (half an hour later than usual), and suffered severe insomnia. The time wasn't entirely wasted as I wrote a short story and figured out how to make nicotine gum placebos during the hours in the dark, but I could have done without the experience. All metrics omitted because it was a day usage. While the mechanism is largely unknown, one commonly mechanism possibility is that light of the relevant wavelengths is preferentially absorbed by the protein cytochrome c oxidase, which is a key protein in mitochondrial metabolism and production of ATP, substantially increasing output, and this extra output presumably can be useful for cellular activities like healing or higher performance. My first time was relatively short: 10 minutes around the F3/F4 points, with another 5 minutes to the forehead. Awkward holding it up against one's head, and I see why people talk of LED helmets, it's boring waiting. No initial impressions except maybe feeling a bit mentally cloudy, but that goes away within 20 minutes of finishing when I took a nap outside in the sunlight. Lostfalco says Expectations: You will be tired after the first time for 2 to 24 hours. It's perfectly normal., but I'm not sure - my dog woke me up very early and disturbed my sleep, so maybe that's why I felt suddenly tired. On the second day, I escalated to 30 minutes on the forehead, and tried an hour on my finger joints. No particular observations except less tiredness than before and perhaps less joint ache. Third day: skipped forehead stimulation, exclusively knee & ankle. Fourth day: forehead at various spots for 30 minutes; tiredness 5/6/7/8th day (11/12/13/4): skipped. Ninth: forehead, 20 minutes. No noticeable effects. One fairly powerful nootropic substance that, appropriately, has fallen out of favor is nicotine. It's the chemical that gives tobacco products their stimulating kick. It isn't what makes them so deadly, but it does make smoking very addictive. When Europeans learned about tobacco's use from indigenous tribes they encountered in the Americas in the 15th and 16th centuries, they got hooked on its mood-altering effects right away and even believed it could cure joint pain, epilepsy, and the plague. Recently, researchers have been testing the effects of nicotine that's been removed from tobacco, and they believe that it might help treat neurological disorders including Parkinson's disease and schizophrenia; it may also improve attention and focus. But, please, don't start smoking or vaping. Check out these 14 weird brain exercises that make you smarter. But though it's relatively new on the scene with ambitious young professionals, creatine has a long history with bodybuilders, who have been taking it for decades to improve their muscle #gains. In the US, sports supplements are a multibillion-dollar industry – and the majority contain creatine. According to a survey conducted by Ipsos Public Affairs last year, 22% of adults said they had taken a sports supplement in the last year. If creatine was going to have a major impact in the workplace, surely we would have seen some signs of this already. Null results are generally less likely to be published. Consistent with the operation of such a bias in the present literature, the null results found in our survey were invariably included in articles reporting the results of multiple tasks or multiple measures of a single task; published single-task studies with exclusively behavioral measures all found enhancement. This suggests that some single-task studies with null results have gone unreported. The present mixed results are consistent with those of other recent reviews that included data from normal subjects, using more limited sets of tasks or medications (Advokat, 2010; Chamberlain et al., 2010; Repantis, Schlattmann, Laisney, & Heuser, 2010). "Piracetam is not a vitamin, mineral, amino acid, herb or other botanical, or dietary substance for use by man to supplement the diet by increasing the total dietary intake. Further, piracetam is not a concentrate, metabolite, constituent, extract or combination of any such dietary ingredient. [...] Accordingly, these products are drugs, under section 201(g)(1)(C) of the Act, 21 U.S.C. § 321(g)(1)(C), because they are not foods and they are intended to affect the structure or any function of the body. Moreover, these products are new drugs as defined by section 201(p) of the Act, 21 U.S.C. § 321(p), because they are not generally recognized as safe and effective for use under the conditions prescribed, recommended, or suggested in their labeling."[33]	CommonCrawl
Conflicting definitions of reference frames in general relativity I'm having trouble understanding what constitutes a reference frame in general relativity as there seem to be several contradictory definitions. It is my understanding that, in special relativity, any event in $4D$ spacetime may be described in any reference frame (except in degenerate cases such as when acceleration generates an event horizon). That is, if the coordinates of some event are known to be $(\mathbf{x}, t)$ in some frame $S$, then there almost always exists a transformation $\mathbf{x}'(\mathbf{x}, t)$, $t'(\mathbf{x}, t)$, which expresses the coordinates in some other frame $S'$ in terms of those in $S$. In this way, a reference frame in SR is made up of a rigid structure throughout all of space, which has a clock affixed to every point - this allows the coordinates of every event to be measured. In a similar manner, in $\S83$ of The Classical Theory of Fields, Landau and Lifshitz describe what a reference frame is in GR: [Systems of bodies at rest relative to one another] do not exist in the presence of a variable gravitational field, so for the exact determination of the position of a particle in space we must have an infinite number of bodies which fill all the space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity. I can make sense of the above. However, I have found a number of sources which give very different descriptions of what a reference frame is in GR. For example: This stackexchange post claims that GR simply does not have global frames - that it is in general impossible to refer to events which do not occur in the same reference frame as one's own. This one describes it very mathematically - a reference frame is a set of four linearly independent unit vector fields constructed along a worldline. My confusion basically arises from (1). If it is impossible to measure anything in another frame, then how can we measure the distances of other galaxies etc? I also don't understand what the definition given in (2) means physically. Sorry for such a rambling question! I've tried to be as succinct as possible and I would greatly appreciate it if somebody could help to clarify this for me. general-relativity spacetime reference-frames metric-tensor coordinate-systems There's a variety of concepts which are mostly similar in special relativity, but are more nuanced in general relativity. First, there are coordinates. Coordinates are the points in the coordinate patch, in the domain $O \subset \mathbb{R}^n$ of a manifold's chart $\phi$, and given the inverse of our chart, we get the spacetime point of our manifold, $\phi^{-1}(t, x, y, z) = p$. Then there are frames. A frame at $p$ is a set of linearly independent vectors $\{ e_a \}$ (ie, a basis) of the tangent space $T_pM$ that span the entire tangent space. As with other basis, we call it orthonormal if $\langle e_a, e_b \rangle = \pm 1$. For a given set of coordinates, you also have the frame given by the coordinate basis, which is the tangent vectors of curves of one coordinate only. A frame field over $U \subset M$ is a smooth choice of frames. That is, we have $n$ vector fields $e_a(p)$ that are all smooth and at every point $p$, they form a basis. We can also define frames for observers, in which case that is much the same notion as a frame field, but the $n$ vectors are smooth along the curve defined by the observer. And finally there are physical measurements you can perform to actually determine times and distances. To give you some more intuitive notion of frames, take an observer. An observer can locally have its own frame (imagine that it's some probe with three orthogonal axis). One common method of getting the coordinates of other points is the radar method. First, take your three axis to convert them to spherical coordinates, shoot a light beam at time $t_1$ in direction $(\theta, \phi)$. The point $q$ you're shooting at reflects the light beam back at you, and it arrives at time $t_2$. The event $q$ is then given the time coordinate $(t_2 - t_1) / 2$ and the distance $c (t_2 - t_1) / 2$. Using this distance, you can if you wish convert it back to "Cartesian" coordinates. A frame at $q$ can then be defined by varying those coordinates around. In special relativity, all of those notions are, while different, somewhat interchangeable. If you have the cartesian coordinates $(t,x,y,z)$, the frame $\{ e_t, e_x, e_y, e_z \}$ they define is an orthonormal frame, and translating this frame around gives you a frame field. Given an inertial observer, the radar coordinates, using light shot using the local observer frame (ie the one on the line $(t,0,0,0)$), give back the proper coordinates, and translation will produce a frame field which is also the appropriate frame for all comoving observers. A Lorentz transform will apply to all frames in the frame field, so that comparing the frames of different inertial observers is easy (please note that all that I've said apply to inertial observers. Even in flat space, things get complicated with accelerated observers). In general relativity, things are trickier, though. Given a local frame at $p$, you can always get all of this locally, thanks to convex normal neighbourhoods. An observer can generate a local basis, with all the clocks and rods he needs, which can be orthonormal at $p$. You can then define a unique frame field by parallel transport of the frame at $p$ along the unique geodesics, and coordinates stemming from that local frame by taking the parameters of the basis, ie, if you have a vector $v = v^i e_i$, then the point $q = \exp_p(v)$ has the coordinates $v^i$ (please note here that the basis at $q$ that we defined may not be the coordinate basis in those coordinates, and in fact it will not be unless the spacetime is flat). And for an appropriate neighbourhood, called a radar neighbourhood, getting those coordinates via the radar method is possible. First bad news : comparing the frames of different observers is no longer a trivial task here. To convince yourself of this, take three close-by observers (all are within their own convex normal neighbourhood), $\gamma_1$, $\gamma_2$ and $\gamma_3$. If we could simply define a frame of $\gamma_2$ by the parallel transport of $\{ e^1_a \}$, then we could also do so between $\gamma_2$ and $\gamma_3$, and $\gamma_3$ and $\gamma_1$, eventually getting back the same frame, but what we have just done is a loop of parallel transports, and therefore the end result will, if the metric is curved, not coincide. This is true even if we take the closest equivalent of inertial observers, by having unaccelerated observers (part of the issue is that we can't have those observers have the same velocity since we can't compare distant velocity vectors). Things get more complex from here, though. We define convex normal neighbourhood because all those nice properties may fail outside of them. A typical example is that a point may have two different geodesics, even of same length, connecting it to $p$. The classic example is the sphere, where the two poles are connected in every direction by a geodesic of identical length. This is the cause of gravitational lensing and, on a more concrete case, can lead a point to have more than one radar coordinates. If we tried to define a frame at that point by parallel propagation, we wouldn't be able to do it, since there is no unique geodesic to guide us. If we go on a more mathematical route, things are even worse : there are no nowhere-vanishing vector fields on the sphere, and therefore we wouldn't be able to define any frame field globally at all. Many manifolds do not admit such a global frame, but we are mostly in luck : if you assume that our spacetime is nicely behaved (ie it has a topology $\mathbb{R} \times \Sigma$, where $\Sigma$ is the spatial part, and $\Sigma$ is orientable), then for $3+1$ dimensions, spacetimes always admit such a global frame field, although they may not stem from measurements directly. SlereahSlereah a reference frame in SR is made up of a rigid structure throughout all of space, which has a clock affixed to every point Yes -- this certainly captures a variant of the notion "frame of reference" which I understand and like to address in the following. Let me first of all extract and summarize the Principal requirements of a reference frame a collection of what's called distinct identifiable "points" ("space points", "points with temporal extension", "material points", "point particles", "participants", "members of a reference frame"), or their representations especially as (continuous) timelike Curves in (a given region of) spacetime, which entirely covers (i.e. fills, and indeed disjointly partitions) a spacetime region under consideration; such that each event of the region had one (and indeed exactly only one) member of one particular frame participating, with a certain "structure" established between them, and involving all of them; including comparison of durations $\tau$ between members (as well as for each member itself), geometric ("spatial") structure between several (or all) of them, and (in generalizations non-trivial) kinematic structure; and foremost such that of all members of one particular reference frame no two ever met (coincided, took part in the same event); and their representing curves did not intersect. Variants or generalizations [...] a rigid structure reference frames (in a flat region) whose members are not only rigid wrt. each other, but moreover individually at rest (i.e. unaccelerated) and thus at rest wrt. each other, are of course inertial frames. Inertial frames can cover any arbitrarily extended flat spacetime region. The geometric structure associated to the set of its members is a flat metric space. examples of reference frames with strictly and non-trivially rigid geometric structure, in a flat region of spacetime, include Rindler frames (i.e. families of mutually rigid hyperbolically accelerating Rindler observers), and rotating reference frames (with constant non-zero rotational speed $\omega$). In an arbitrarily extended spacetime region, such rigid reference frames exhibit bounderies (horizons). The associated geometric structures are generally curved metric or quasi-metric spaces. rigidity (of all members of a reference frame wrt. each other) may not necessarily be required; we may then for instance consider asymptotically inertial rotating frames (in an arbitrarily extended flat region), i.e. with angular speeds $\omega[ \, R \, ] < 2 \, \pi \, (R / c)$ and $(\omega_B / \omega_A) \le (R_A / R_B)^{(1 + \epsilon)}$ (unless $R_B = 0$). [...] in GR a spacetime-filling collection of non-intersecting, timelike curves is generally called a timelike congruence. Relations between distinct reference frame Relations between any two distinct reference frames in the same spacetime region, say $\Sigma$ and $\Omega$ come about due to certain members of one and certain members of the other meeting (coinciding, typically "in passing") each other; such that each event had exactly one member of each reference frame participating. For example, with $A, B, C, ... \in \Sigma$ and $N, P, Q ... \in \Omega$ such that $A$ and $N$ jointly took part (coincided) in event $\varepsilon_{AN}$, $A$ and $P$ jointly took part (coincided) in event $\varepsilon_{AP}$, $B$ and $N$ jointly took part (coincided) in event $\varepsilon_{BN}$, $B$ and $Q$ jointly took part (coincided) in event $\varepsilon_{BQ}$, $C$ and $P$ jointly took part (coincided) in event $\varepsilon_{CP}$, and $C$ and $Q$ jointly took part (coincided) in event $\varepsilon_{CQ}$, the structure of $\Sigma$ provides the real number values of the ratios $(\tau B[ \, \_N, \_Q \, ] / \tau A[ \, \_N, \_P \, ])$ and of $(\tau C[ \, \_P, \_Q \, ] / \tau A[ \, \_N, \_P \, ])$, while the structure of $\Omega$ provides the real number values of $(\tau P[ \, \_A, \_C \, ] / \tau N[ \, \_A, \_B \, ])$ and of $(\tau Q[ \, \_B, \_C \, ] / \tau N[ \, \_A, \_B \, ])$. As far as the geometric (and kinematic) structures are in turn defined chronogeometrically, i.e. those of reference frame $\Sigma$ defined in terms of certain duration ratios of its members ($A$, $B$, $C$, ...), and those of reference frame $\Omega$ defined in terms of certain duration ratios of its members ($N$, $P$, $Q$, ...), the basic coincidence data of who met and passed whom induces relations between these structures of the two reference frames. Consistent with each reference frame separately, and with their relations between each other, there can be a geometric structure associated with the given spacetime region (set $\mathcal S$ of events) itself; especially in terms of (ratios of) Lorentzian distances $\ell : \mathcal S \times \mathcal S \rightarrow \mathbb R^+$. Relations to other notions of reference frame the coordinates of some event [...] The notion of reference frame laid out above is obviously manifestly without involving any coordinates. However, the assignment of coordinates (i.e. n-tuples of real numbers, $\mathbb R^n$, or subsets thereof), uniquely one each to any one member of a reference frame, and (often in terms of "the $t$ coordinate" component of the n-tuple) distinctly to each event, may be of broader interest. Since $\mathbb R^n$ (or subsets) are characterized by a particular usual topology, we may distinguish whether coordinates are assigned compatibly (homeomorphically) to the topological space provided through the (geometric, or temporal) structure of the reference frame. Since there is a standard metric space associated to the set of real numbers $R$ itself through distance defined as absolute difference, we may further distinguish whether compatible coordinates are further (componentwise) smoothly or even proportionally assigned to the metric space provided through the (geometric, or temporal) structure of the reference frame. Not the answer you're looking for? Browse other questions tagged general-relativity spacetime reference-frames metric-tensor coordinate-systems or ask your own question. How do frames of reference work in general relativity, and are they described by coordinate systems? How one uses the definition of observers in General Relativity? On the coordinate independence of general relativity How can we define a frame of reference in general relativity? Is simultaneity well-defined in general relativity? Time in general relativity Doesn't Einstein's General Relativity violate the spirit of Principle of Relativity? Physical meaning of frames in general relativity $F = ma$ In General Relativity Reference frames versus coordinate systems Logic of general relativity	CommonCrawl
# Data visualization techniques for probability distributions One of the most common data visualization techniques is the histogram. A histogram is a graphical representation of the distribution of a continuous variable. It is created by dividing the range of the variable into a series of intervals or bins and counting the number of data points that fall into each bin. The histogram can be used to identify the shape of the probability distribution and estimate its parameters. Another important data visualization technique is the box plot. A box plot is a graphical representation of the distribution of a discrete variable. It is created by calculating the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of the data. The box plot can be used to identify the shape of the probability distribution and estimate its parameters. In addition to histograms and box plots, scatter plots and line plots can also be used to visualize probability distributions. Scatter plots can be used to visualize the relationship between two variables, while line plots can be used to visualize the trend of a single variable over time or space. ## Exercise Create a histogram and a box plot using Seaborn to visualize a probability distribution. ```python import seaborn as sns import matplotlib.pyplot as plt # Load a dataset data = sns.load_dataset('tips') # Create a histogram sns.histplot(data['total_bill']) plt.show() # Create a box plot sns.boxplot(x='total_bill', data=data) plt.show() ``` # The Python programming language and its applications in data analysis Python is an interpreted language, which means that it is executed line by line. This makes Python easy to learn and use, especially for beginners. Python has a simple syntax and is designed to be readable and understandable. Python has a large standard library, which includes modules for various tasks, such as file input/output, regular expressions, and networking. In addition, Python has a large ecosystem of third-party libraries that can be used to extend its functionality. One of the most popular third-party libraries for data analysis in Python is Pandas. Pandas is a powerful data manipulation library that provides data structures, such as data frames and series, for handling and manipulating data. Pandas is widely used for data cleaning, data transformation, and data aggregation. Another popular library for data analysis in Python is NumPy. NumPy is a library for numerical computing that provides an efficient and convenient way to work with arrays and matrices. NumPy is widely used for numerical operations, such as linear algebra and statistical analysis. ## Exercise Install Pandas and NumPy using pip. ```bash pip install pandas numpy ``` # Importing and manipulating data using Python libraries To import data into Python, you can use the Pandas library. Pandas provides the `read_csv()` function, which can be used to read data from a CSV file. Once the data is imported into Python, you can use the Pandas library to manipulate the data. Pandas provides various functions for data manipulation, such as filtering, sorting, and aggregation. For numerical operations, you can use the NumPy library. NumPy provides arrays and matrices for storing and manipulating numerical data. NumPy also provides various functions for numerical operations, such as arithmetic operations and linear algebra. ## Exercise Import a CSV file into Python using Pandas and perform basic data manipulation. ```python import pandas as pd # Read data from a CSV file data = pd.read_csv('data.csv') # Filter rows based on a condition filtered_data = data[data['age'] > 30] # Sort data based on a column sorted_data = data.sort_values('age') # Compute the mean and standard deviation of a column mean = data['age'].mean() std_dev = data['age'].std() ``` # Introduction to probability distributions A probability distribution is a function that assigns a probability to each possible outcome of a random experiment. The sum of the probabilities of all possible outcomes is equal to 1. There are several types of probability distributions, including discrete and continuous distributions. Discrete distributions have a finite or countable number of possible outcomes, while continuous distributions have an uncountable number of possible outcomes. Some common discrete probability distributions include the uniform distribution, the binomial distribution, and the Poisson distribution. Some common continuous probability distributions include the normal distribution, the exponential distribution, and the chi-square distribution. ## Exercise Calculate the probability of a specific outcome in a discrete probability distribution. ```python # Define a discrete probability distribution probability_distribution = {'A': 0.2, 'B': 0.3, 'C': 0.5} # Calculate the probability of outcome 'B' probability_of_B = probability_distribution['B'] ``` # Uniform, normal, and exponential distributions The uniform distribution is a continuous probability distribution that assigns equal probabilities to all possible outcomes within a specified range. The probability density function of the uniform distribution is given by: $$f(x) = \frac{1}{b - a}$$ where $a$ and $b$ are the lower and upper bounds of the distribution, respectively. The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. The probability density function of the normal distribution is given by: $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$$ where $\mu$ and $\sigma$ are the mean and standard deviation of the distribution, respectively. The exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process. The probability density function of the exponential distribution is given by: $$f(x) = \lambda e^{-\lambda x}$$ where $\lambda$ is the rate parameter of the distribution. ## Exercise Calculate the probability density function of a normal distribution. ```python import numpy as np # Define a normal distribution mean = 0 std_dev = 1 # Calculate the probability density function def normal_pdf(x): return 1 / (np.sqrt(2 * np.pi * std_dev*2)) np.exp(-((x - mean)*2) / (2 std_dev**2)) # Calculate the probability density function at a specific point probability_density = normal_pdf(0) ``` # Sampling methods and their applications in data analysis Simple random sampling is a common sampling method that involves selecting data points at random from a population. This method is used to estimate the characteristics of the population, such as the mean and standard deviation. Stratified sampling is a sampling method that involves dividing the population into distinct groups, or strata, and selecting data points at random from each group. This method is used to ensure that the sample is representative of the entire population. Cluster sampling is a sampling method that involves selecting data points from a population by selecting a random sample of clusters and then selecting a random sample of data points from each cluster. This method is used to reduce the sample size while maintaining the representativeness of the sample. ## Exercise Perform simple random sampling on a dataset. ```python import numpy as np # Load a dataset data = np.random.rand(100) # Perform simple random sampling sample_size = 10 sample = np.random.choice(data, size=sample_size, replace=False) ``` # Central limit theorem and its implications The central limit theorem is a fundamental theorem in probability theory that states that the sum of a large number of independent and identically distributed random variables, each with a finite mean and variance, will be approximately normally distributed, regardless of the shape of the original distribution. The central limit theorem has several important implications for data analysis. First, it allows us to use the normal distribution as an approximation for any other continuous probability distribution. Second, it allows us to use statistical methods, such as hypothesis testing and regression analysis, on data that may not be normally distributed. ## Exercise Apply the central limit theorem to a dataset. ```python import numpy as np # Load a dataset data = np.random.rand(100) # Calculate the sample mean and standard deviation sample_mean = np.mean(data) sample_std_dev = np.std(data) # Approximate the distribution as a normal distribution normal_mean = sample_mean normal_std_dev = sample_std_dev / np.sqrt(len(data)) ``` # Hypothesis testing and statistical tests Hypothesis testing is a statistical method used to test the validity of a hypothesis about the population. The hypothesis is tested by comparing the observed sample data to the null hypothesis, which is the assumed distribution of the population. There are several statistical tests used in hypothesis testing, including t-tests, chi-square tests, and ANOVA. These tests are used to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. ## Exercise Perform a t-test to compare the means of two groups. ```python import scipy.stats as stats # Load two datasets group1 = np.random.rand(50) group2 = np.random.rand(50) # Perform a t-test t_statistic, p_value = stats.ttest_ind(group1, group2) ``` # t-tests, chi-square tests, and ANOVA A t-test is used to compare the means of two groups. There are two types of t-tests: one-sample t-test and two-sample t-test. The one-sample t-test is used to test the null hypothesis that the mean of a single group is equal to a specified value, while the two-sample t-test is used to test the null hypothesis that the means of two groups are equal. A chi-square test is used to test the independence of two categorical variables. The chi-square test is used to determine whether there is a significant association between the two variables. ANOVA (Analysis of Variance) is used to test the equality of means of three or more groups. ANOVA is used to determine whether there is a significant difference between the means of the groups. ## Exercise Perform a chi-square test to test the independence of two categorical variables. ```python import scipy.stats as stats # Load two datasets variable1 = np.random.randint(0, 3, 100) variable2 = np.random.randint(0, 3, 100) # Perform a chi-square test chi_square_statistic, p_value = stats.chi2_contingency(variable1, variable2) ``` # Regression analysis and correlation Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. There are several types of regression analysis, including linear regression, logistic regression, and Poisson regression. Correlation is a measure of the strength and direction of the relationship between two variables. There are several types of correlation, including Pearson correlation and Spearman rank correlation. Correlation can be used to determine whether there is a significant relationship between two variables. ## Exercise Perform a linear regression analysis to model the relationship between two variables. ```python import statsmodels.formula.api as smf # Load two datasets variable1 = np.random.rand(100) variable2 = np.random.rand(100) # Perform a linear regression analysis model = smf.ols('variable2 ~ variable1', data={'variable1': variable1, 'variable2': variable2}).fit() ``` # Case study: real-world data analysis and visualization We will start by importing a dataset and performing basic data manipulation using Pandas. Then, we will create various data visualizations using Seaborn and Plotly to explore the relationships between variables and identify potential trends or patterns in the data. Finally, we will perform statistical analysis, such as hypothesis testing and regression analysis, to test the validity of hypotheses about the dataset. ## Exercise Perform a real-world data analysis and visualization using Python libraries. ```python import pandas as pd import seaborn as sns import plotly.express as px # Load a dataset data = pd.read_csv('data.csv') # Perform basic data manipulation filtered_data = data[data['age'] > 30] sorted_data = data.sort_values('age') # Create a scatter plot sns.scatterplot(x='age', y='income', data=data) plt.show() # Create a line plot px.line(data, x='year', y='income') ``` # Applications of data analysis and visualization in various fields Data analysis and visualization are widely used in finance to analyze financial data, such as stock prices, trading volumes, and economic indicators. This information is used to make informed investment decisions and to identify potential market trends. In healthcare, data analysis and visualization are used to analyze patient data, such as medical records, genetic data, and clinical trial results. This information is used to improve patient care, identify treatment efficacy, and develop new medical interventions. In social sciences, data analysis and visualization are used to analyze social data, such as survey responses, demographic data, and social media data. This information is used to inform public policy, understand social trends, and develop new interventions to address social issues. ## Exercise Discuss the potential applications of data analysis and visualization in your field of interest. ```python # This exercise requires a discussion, not a code implementation. ```	Textbooks
MathOverflow Meta MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. Areas of Triangles in (Non-Riemannian) Metric spaces? I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this would be reinventing the wheel. So I'll try to give an example of the sort of thing I'm searching for in the hopes that someone has seen it already and can tell me where to look for the correct axioms. Consider a set $X$ and a function $e: X^3 \to \mathbb{R}$ so that $e(x,y,z) \geq 0$ with equality occurring if at least two of $x,y,z$ coincide in $X$, $e$ is invariant under the obvious action of the symmetric group $S_3$ on $X^3$, $e(x,y,z) \leq e(w,y,z) + e(x,w,y) + e(x,y,w)$ for all $w,x,y,z$ in $X$. As you might imagine, such structures have been defined before (Gahler called them 2-metric spaces in the sixties). They are supposed to axiomatize "area of triangle given three vertices" in the same way that the usual metric space definition axiomatizes "length of line between two points". Given an ordinary metric space $(X,d)$, the defect in the usual triangle inequality -- given by the Gromov product $(x,y)_z = d(x,z) + d(y,z) - d(x,y)$ -- morally measures deviation from collinearity of $x, y$ and $z$; hence one might expect (again, morally) that if $(x,y)_z > 0$ then the area of the triangle spanning $(x,y,z)$ is also strictly positive. For each triple of points $x,y,z$ in $X$, let $\delta_{xyz}$ be the minimum of $(x,y)_z, (x,z)_y$ and $(y,z)_x$ and let $\mu_{xyz}$ be the largest of the three pairwise distances $d(x,y), d(y,z)$ and $d(x,z)$. In Euclidean space for instance, assuming I didn't calculate poorly, if we let $e(x,y,z)$ be the area of the triangle between $x,y$ and $z$ then there are constants $\alpha^\pm \geq 0$ so that the following inequalities hold: $$ \alpha^-\cdot\delta_{xyz}\cdot\mu_{xyz}^3 \leq e^2(x,y,z) \leq \alpha^+\cdot\delta_{xyz}\cdot\mu^3_{xyz} $$ This inequality sandwiches the (square of) the area between products of minimax type quantities involving lengths, thus quantifying the extent to which the side lengths are modulating the area. Thus, if one enhances a metric space $(X,d)$ with a ternary function $e$ subject to the axioms 1-3 in the background and $d$-compatibility coming from something like this last inequality, then one has a reasonable candidate for "metric-spaces with area". In principle we could also go beyond and introduce a volume function $f:X^4 \to \mathbb{R}$ similarly compatible with $e$. Have you come across a similar axiom scheme $(X,d,e,\ldots)$ without underlying Riemannian metrics? If so, what was it used for and where can I find it and its properties? reference-request mg.metric-geometry Vidit Nanda Vidit NandaVidit Nanda 15k22 gold badges5757 silver badges120120 bronze badges $\begingroup$ I'd love to see a clear-cut question. $\endgroup$ – Włodzimierz Holsztyński $\begingroup$ A problem with your definition is that it seems to rely at least intuitively on the notion that every three points completely determine a triangle (try that even in a simple space like the Grassmannian of two-planes in four-space to see that it is a strong hypothesis). If this is what you want to model, Matt F. is right in pointing you to the work of Busemann. However notice the strong hypothesis of additivity on the area (cut the triangle in two and the areas of the pieces have to add up to the area of the original). $\endgroup$ – alvarezpaiva $\begingroup$ @alvarezpaiva Does your objection not apply to lines and metric spaces too? Even on a sphere we can find two points which don't uniquely determine a geodesic. This hardly makes the definition of metric spaces problematic! $\endgroup$ – Vidit Nanda $\begingroup$ @ViditNanda: a lot of spaces we look at do have minimizing geodesics between two points and so the distance (defined only on terms of pairs of points) is related to the length of some curve. However, that was not my point: I merely wished to know what you were trying to model. You talk of triangles and areas and then try to model this by functions on triples of points you must have some idea of what phenomena you are trying to capture or maybe just warn the reader that this is abstraction for its own sake. $\endgroup$ $\begingroup$ @alvarezpaiva fair enough: I seek something that simultaneously models (1) geodesic convex hulls in Riemannian manifolds, and (2) correlation complexes. By (2) I mean a simplicial complex whose vertices are events, and d-simplices (for d > 0) are decorated with the correlation of all vertex-events occurring simultaneously. In both cases, the pairwise information imposes constraints on the triple-wise information, and so on. $\endgroup$ See Herbert Busemann's 1955 book, Geometry of Geodesics, secs. 48 and 50. He worked in the context of metric spaces with some additional requirements on the distance $xy$. One of his key definitions is that "$y$ is between $x$ and $z$", or $(xyz)$ iff $d(x,y)+d(y,z)=d(x,z)$. His axioms on area, translated to your notation, are: $e(x,y,z)$ is symmetric in $x, y, z$ and non-negative. $e(x,y,z) = 0$ iff either $(xyz)$ or $(yzx)$ or $(zxy)$ If $(xyz)$ then $e(w,x,y)+e(w,y,z)=e(w,x,z)$. He used such area functions to characterize homogeneous spaces, and to find necessary and sufficient conditions for affine structure on the space. Matt F.Matt F. Thanks for contributing an answer to MathOverflow! Swimming against the tide in the past century: remarkable achievements that arose in contrast to the general view of mathematicians Characterizing surface area Nice proof of the triangle inequality for the metric of the hyperbolic plane Metric angles in Riemannian manifolds of low regularity Towards a metric characterization of Euclidean spaces What makes a distance? Area of triangles vs. comparison triangles. The space of circular triangles? Bounding the perimeter of a geodesic triangle in spaces of non-positive curvature A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?	CommonCrawl
The equation of the line that passes through the points $(-2,0)$ and $(0,2)$ can be expressed in the form $y=mx+b$. What is the value of $m+b$? Since both of these points lie on the line, plugging them into the equation of the line will produce a true statement. Thus $(-2, 0)$ gives us $0 = -2m + b$ and $(0, 2)$ gives us $2 = b$. So we now know what $b$ is and can plug it back into the first equation to get $0 = -2m + 2$. So $m = 1$ and $m + b = \boxed{3}$.	Math Dataset
Service innovation management practices in the telecommunications industry: what does cross country analysis reveal? Syed Abidur Rahman ORCID: orcid.org/0000-0002-7889-920X1, Seyedeh Khadijeh Taghizadeh1, T. Ramayah2 & Noor Hazlina Ahmad2 SpringerPlus volume 4, Article number: 810 (2015) Cite this article Service innovation management practice is currently being widely scrutinized mainly in the developed countries, where it has been initiated. The current study attempts to propose a framework and empirically validate and explain the service innovation practices for successful performance in the telecommunications industry of two developing countries, Malaysia and Bangladesh. The research framework proposes relationships among organisational culture, operating core (innovation process, cross-functional organisation, and implementation of tools/technology), competition-informed pricing, and performance. A total of 176 usable data from both countries are analysed for the purpose of the research. The findings show that organisational culture tends to be more influential on innovation process and cross-functional organisation in Malaysian telecommunication industry. In contrast, implementation of tools/technology plays a more instrumental role in competition-informed pricing practices in Bangladesh. This study revealed few differences in the innovation management practices between two developing countries. The findings have strategic implications for the service sectors in both the developing countries regarding implementation of innovative enterprises, especially in Bangladesh where innovation is the basis for survival. Testing the innovation management practices in the developing countries perhaps contains uniqueness in the field of innovation management. Innovation coupled with performance of firms is a subject with significant attention within academia (Damanpour 2014) due to its rapid and dramatic impact on society and organisations across borders. In order to achieve ultimate goal in an organisation, managerial practices and activities can play a vital role. In this regards, few rudimentary and imperative management practices is considered in this study context to understand to what extent such practices contribute organisations for accomplishing the performance specifically in developing context. Scholars claim that countries and regions are endowed with diverse types of resources and infrastructures (Chen and Hsiao 2013) which rely on their own organisational culture how to practice (Aycan et al. 2000). Earlier literatures illustrated the influence of national and organisational culture on different managerial practices in the organisations (Ardichvili et al. 2006) as well as on successful innovation (Lee et al. 2013; Büschgens et al. 2013). In the context of 'culture' issue, some of the scholars have asserted that national culture has an influence along with other spectrums on the organisation and its culture (Tayeb 1994). To be more specific, literatures suggest that organisational culture is the integral part of the national culture (Iorgulescu and Marcu 2015). However, Hogan and Coote (2014) noted that despite much focused attention on the topic of organisational culture and innovation, the extant literature does not sufficiently document the organisational culture that enables innovation. To have successful innovation, scholars gave importance to three operating core of innovation as fundamental aspects of innovation management. These three operating core are innovation process, cross-functional organisation, and implementation of tools/technology introduced by Hull et al. (1996). These practices facilitate service companies in managing their new service development process in a best way (Collins and Hull 2002) as it is proved to be faster, cheaper, and better for service development than serial alternatives (Liker et al. 1999). As scholars highlighted, innovation process, cross-functional organisation, and implementation of tools/technology are increasingly necessary for survival under conditions of hyper competition (Hull 2004). Further, literatures suggest that in the process, a great deal of effort must be put in the implementation of new products/services (Orfila-Sintes et al. 2005). Innovation process considers various activities include effectiveness in market assessment, bench marketing, identify customer needs, quality function, and review on the design of the products (Hull 2004). This guidance can create value for customers who are the focus of innovation (De Jong and Vermeulen 2003). In addition, cross-functional teams are often seen as key for innovation projects (Blindenbach-Driessen 2015) which carries out every practice and process in a systematic and sustainable way (Weiss and Legrand 2011). It is generally an accepted notion that people are of central importance in cross-functional organisation as each has capabilities to find and solve problems. Cross-functional organisation with high performance teamwork can bring success to firms, while without could be a reverse situation (Weiss and Legrand 2011). In the stream of innovation literature, tools/technology mainly represents the usage of computer and information technology (CIT). Most service firms are knowledge-based and heavily depend on information technology (IT) (Hull and Tidd 2003b), hence, IT can facilitate the decision making process in the development cycle in a shorter time (Hull 2004). In addition, CIT enables team members to share their experience in service development cycle and systematically compare their services with competitors (Tidd and Bessant 2009). It allows management to evaluate and control all the projects through stored day-to-day information as well to learn and conduct staff training upon reviewing customer and user satisfaction, evaluating projects, and audits (Mudrak et al. 2005). Moreover, this study has considered competition-informed pricing as important practices for new service development. Competition-informed pricing refers to the prices of competing product that are used as a benchmark instead of customer demand. The competition-informed pricing assumes that the cost structure of the company would be such a way that matches with the competitors' pricing (Shapiro and Jackson 1978). According to Hinterhuber (2004), while making the pricing decisions the manager must take into consideration the competitive perspective which facilitates to inform the competitors' pricing. The purpose of choosing competition-informed pricing is due to the selection of telecommunications industry in the current context. Competition-informed pricing in the telecommunications industry plays persuading role. It is matter of fact that in the telecommunications industry, the level of competition is more intense compared to any other industry, irrespective of a country's economic and social state. The market structure of telecommunications industry is considered as oligopoly. In the oligopoly market, there are only few firms which have considerable control over their prices, but each firm must consider the course of actions, activities, and reactions of the rivals (Noam 2006). Hence, an organisation cannot overlook the importance of today's hyper competitive market in their innovation process because researchers noted that innovation has a synchronized relationship with competitors (Goto 2009). Finally, the study has attempted to reveal the impact of such practices on the innovation performance. Performance reflects the business initiatives and strategies taken by the firm. Previous researchers argued that innovation in an organisation directly and positively influences the improvement in business performance (Tidd et al. 2005). Innovation as a firm's unique resource can lead to competitive advantage and improvement in performance, effectiveness, and efficiency (Barney 1991). If firms are highly focused on innovation, they will be more successful in the offering of new products and services where subsequently it will result in greater performance (Eisingerich et al. 2009). However, over the past years many of countries face difficulties in strengthening innovation performance (OECD 2007) which diverges due to the capacity to innovate. To do so, the study has framed this research in the telecommunication industry of two countries. Most importantly, the study intended to test a framework in developing countries which has partially been molded and tested in developed countries. As in the recent literatures, scholars have solicited to modify and test management theories and framework in emerging economies which are typically built in the northern part of the globe (Barrett et al. 2015). However, this is a prospect to substantiate whether framework initiated in the developed countries explicate similar underlined causal effects across developing countries. We have chosen two Asian countries, one of which is considered as innovation driven country (Malaysia), and another considered as only factor driven country with insufficient capacity to innovate (Bangladesh) (World Economic Forum 2015). Bangladesh is one of the prominent member of the world "Next Eleven" group which is considered the most lucrative emerging economy group amongst others in the globe and the country is planning to step in the middle income country by the year of 2021 (Planning Commission 2012). On the other hand, Malaysia is the one of the most potential developing countries which plans to enter the club of 'developed countries' by the year 2021 (Malaysian Investment Development Authority 2014). However, to achieve such economical shift by the year of 2021, it is presumed that innovation and its practices in the industries can be one of the driving forces. In both the countries, telecommunications industry plays leading role in the development of the economy. Profile of the telecommunications industry indicates a proximate similarity in terms of operations and ownership. DiGi a Malaysian telecommunications company and GrameenPhone a Bangladeshi telecommunications company are both a foreign subsidiary of Telenor group, Norway. DiGi holds the second position in terms of market share in Malaysia and GrameenPhone holds the largest market share in Bangladesh. On the other hand, Robi Axiata, a Malaysian subsidiary of the Celcom Axiata group, is operating in Bangladesh with significant market share in the country as well as in Malaysia. Therefore, the current study attempts to propose a framework and empirically validate and explain the service innovation practices between the emerging countries as researchers suggested limited study in these context _ENREF_44 (Taghizadeh et al. 2014). The result may contribute for the policy maker as guideline to enhance the innovation performance through firm resources and capabilities. This paper is structured in seven sections. The second section provides an overview of the theoretical justification of the variables that help the reader to understand the proposed research framework as well as hypotheses formulation. The research methodology and the findings of the empirical analysis used in the study are discussed in section three. In section four, a discussion derived from the result is presented. Implication, conclusion, and limitation with future direction of the research are presented in section five, six, and seven, respectively. Theoretical background and hypothesis development Todays, changes are taking place everywhere, which raising complexity among the environment e.g. changes in economic condition lead to the opening of new markets, while closing others (van Riel 2005). Such a domino effect subsequently increases the level of global competition and rivalry among the companies (van Riel 2005). To overcome the complexity, management need to have a balanced, comprehensive, and proactive approach (Ottenbacher 2007). Scholars believe that successful service innovation not only depends on how a firm manages projects, coordinates imputes of different functions, and links up with its customer, but also relies on being able to develop strategic approaches and look widely (Tidd et al. 2005). Literature on new service development reveals that the growth and performance of any organisation rely on an efficient management of innovation in a competitive climate (Jiménez-Jiménez and Sanz-Valle 2011; Tidd and Bessant 2009). In the literature, a composite model was illustrated comprising of three managerial practices: innovation process, cross-functional organisation, and implementation of tools/technology introduced by Hull et al. (1996). These practices facilitate service companies in managing their new service development process in a best way (Collins and Hull 2002) as it is proved to be faster, cheaper, and better for service development than serial alternatives (Liker et al. 1999). Innovation process, cross-functional organisation, and implementation of tools/technology are known as the operating core and are increasingly necessary for survival under conditions of hyper competition (Hull 2004). In this operating core both marketing and developmental operations are included in contrast to literature dealing with the market on the one hand and organisation behaviour on the other (Hull 2004). Innovation process represents a disciplined practice in order to control the procedure from idea generation to launch (Hull and Tidd 2003a). According to Hull and Tidd (2003a) and Liker et al. (1999), innovation process denotes the mechanistic form of an organisation where rules and regulations are structured and maintained accordingly. Hull and Tidd (2003a) pointed out that in the setting of innovative process, organisations tend to be effective, efficient, and characterized by standardized procedures. A clear division of labour and an authoritarian chain of command prevail, while the companies embrace the innovative process for the service innovation management (Liker et al. 1999). Cross-functional organisation involves the coordination of people at all the stages of innovation practices (Tidd et al. 2005). Liker et al. (1999) asserted an innovative organisation is characterised by an organic setting that tends to be flexible and characterised by few rules and standard procedures. Teamwork and a creative combination of various views, perspectives and disciplines recognizes innovative organisational practices (Tidd and Bessant 2009). Co-involvement of operations people, who are developing the services and delivering systems support behind the scenes, is necessary for firms success (Magnusson et al. 2003). Tools/technology denotes enabling computer information technologies (CIT) in supporting communication (Hull et al. 1996). According to Collins and Hull (2002), organisational transformation and transaction capabilities are enhanced by the adoption of CIT's tools, such communication devices and data distribution approaches. As the complexity of the business environment has been increased, it requires organisations to have a collaborative and creative working place through the implementing of CIT's tools (Klein and Dologite 2000). According to scholars, the proliferation of information and technology has created a revolution in the current trend with a wider economic perspective across national borders (Erumban and De Jong 2006). However, Hull and Tidd (2003a) found that training and championing 'as a part of organisational culture' influence on shaping up the innovation process, cross-functional organisation, and implementation of tools/technology of service-oriented companies. Hence, we propose that organisational culture can play a stimulus role in practicing the operating core. Scholars noted that organisational culture plays an influential role in the management practices of the firms (Zammuto and O'Connor 1992). Organisational culture is a complex set of values, beliefs, assumptions, and symbols that a firm should institute in its business operation (Miron et al. 2004; Chang and Lin 2007; Barney 1986; Martins and Terblanche 2003). According to Naranjo-Valencia et al. (2011), to facilitate the implication of innovation successfully, organisations should meet the requirements of internal behaviour and external relations which comply with the organisational culture. In fact, organisational culture is a source of new ideas within the organisation (Uzkurt et al. 2013). As suggested by Chang and Lin (2007), this paper conceptualizes organisational culture by considering the four cultural traits (i.e. cooperativeness, innovativeness, consistency, and effectiveness) into a single domain. Cooperativeness focuses primarily on cooperation to each other as extended family which represents a strong team work and trust to each other. Innovativeness can be characterized with a focus on creativity, adaptability, and dynamism which allows the employees for the self-development. Consistency emphasizes on maintaining order, rules and regulations, uniformity, and efficiency throughout the organisational structure. The cultural trait of effectiveness indicates the competitiveness, goal achievement, and efficiency of the organisational activities. Therefore, this paper proposes that organisational culture may have effect on the operating core and thus the following hypotheses would be worthy of testing: H1. Organisational culture facilitates the practise of continuous process improvement in service development. H2. Organisational culture enables the practise of cross-functional organisation to a great level in service development. H3. Organisational culture accelerates the implementation of information technology tools in service development. In oligopoly market high barriers to entry for new competitors exist to a greater extent. Such barriers to entry impede the other new entrants in competing in the market due to the high start-up capital cost (McConnell et al. 2009). To achieve a desire performance in oligopoly market, each firm must consider the course of actions, activities, and reactions of the rivals (Noam 2006). So, competition-informed pricing as how to set prices using information gathered from competitors can be helpful in order to deal with pricing complexity. Hinterhuber (2004) believes that while making pricing decisions a manager must take into consideration the competitive perspective. Competition-informed pricing has the tendency to enhance the likelihood of setting the right price by a competitor's innovation practices, including pricing that may match or exceed the firms' price for innovated products and services. The price of competitive products and competitive advantages of competitors dictate that the firm needs to make an evaluation on the firms' position in the market vis-a-vis the competitors (Ingenbleek et al. 2003). The competitor's current price strategy and strength to react are important components for competition-informed pricing. While firms practice competition-informed pricing, it is also imperative for them to consider the market structure, degree of competition in the market, and the competitive advantages of competitors in the market. Such activities in fact refer to the overall knowledge of the competition by the market players. In this vein, this research suggests that the operating core can facilitate the efforts of managers to gather information related to competitors. For example, process involves external investigation for developing new products and services (Hull 2003). It may help firms in the practice of price decision making through involvement of the functional departments in the procedure towards understanding the strategic movement of rivals in the market. Inter-functional coordination and cooperation are deemed instrumental in efficient innovation management in gathering data regarding the right price from the perspectives of competitors. It can be assumed that the degree of competition can be understood through the propensity of coordination of people in an organisation. Or else, CIT' tools along with continuous updating of the service development process may facilitate firms in gathering competitors' price related information in a shorter time. Considering the above discussion, the following hypothesis is formulated: H4. Continuous process improvement increases the level of gathering competition-informed pricing in service development. H5. A cross-functional organisation facilitates the level of gathering competition-informed pricing in service development. H6. Implementation of information technology tools for gathering competition-informed pricing is easier in service development. Previous study found the relationship between competition-informed pricing firms performance (Ingenbleek et al. 2003). In fact it is difficult to find an ideal measurement for business performance particularly in collecting performance data. In the past studies, the performance of an organisation is frequently evaluated by the simple outcomes of financial indicators such as return on investment (ROI), return on sales, or sales growth. This study measured non-financial performance focusing on innovation activities in terms of new service development and delivery process improvement which has been also found in the earlier research (e.g. Hull 2003; Hull and Tidd 2003a). Coming up with upgraded features, higher quality of services, shorter time for delivery of services, reducing cost of service development, higher quality in the delivery process are the major indicators to measure the performance of the organisation in terms of new service development and delivery process (Hull and Tidd 2003a). To achieve better performance, it is expected to implement appropriate pricing practice (Hultink et al. 1997) as scholars have also asserted that setting a right price drives superior performance for firms (Dutta et al. 2003). Competition-informed pricing increases the chance of setting the right price by knowing competitor's innovation (Ingenbleek et al. 2003). Gathering information from competitors' price strategy enable a quantitative evaluation of the firm's relative position (Ingenbleek et al. 2003). Therefore, we propose that understanding the competitors' trend of pricing, degree of competition, and market structure will enable service companies to upgrade services with new features and reduce the time of response. Thus, the following hypothesis is presented for testing: H7. The greater the practice of competition-informed pricing, the higher the level of performance improvement. Based on the above discussion, this paper proposes that competition-informed pricing can mediate the relationship between operating core and performance. There is hardly any research being conducted to examine the impact of the operating core on performance of service firms through the possible role of competition-informed pricing. The rationale for testing this mediating effect arises from the market structure of telecommunications industry. Practicing operating core of the service innovation perhaps is not enough to achieve the performance enhancement of service industries. While service-based companies embrace the operating core of innovation practices, they subsequently need to understand the position of their competitors in the market. It is a generally accepted notion that in a competitive market, each and every company follows the competitors' pricing and pricing strategy. Vermeulen and van der Aa (2003) mentioned that most organisations use services which are developed by some competitor in order to adjust the competitors' product in their innovation process. To a greater extent, such companies try to get as much information on the competitors' price. By understanding the pricing position of competitors, service companies attempt to attain higher performance. Thus, the following hypotheses would be worth of testing: H8. Competition-informed pricing mediates the relationship between process and performance. H9. Competition-informed pricing mediates the relationship between organisation and performance. H10. Competition-informed pricing mediates the relationship between implementing tools/technology and performance. After all, we believe that culture, service innovation practice, pricing, and firm's performance of mobile phone companies should differ significantly between Malaysia and Bangladesh. The reason for choosing these two contexts is discussed in introduction part. Therefore, we test all path relationships though multi group analysis. H11: All the hypothesised relationships in the proposed framework will differ between Malaysia and Bangladesh telecommunication companies. Thus, the research framework (Fig. 1) aims to explore the relationship of organisational culture as a predictor of operating core (innovation process, cross-functional organisation, and implementation of tools/technology) for new service development. Further, we draw attention to explore the mediating role of competition-informed pricing practices between the relationship of operating core practices and performance. Research methodology and result Sample and data To test the research framework and hypotheses, we considered telecommunications industry in Bangladesh and Malaysia. In Bangladesh, out of six, three top largest telecommunications companies (GrameenPhone, Robi Axiata, and Airtel) were chosen as they contain more than 60 % of the total market share in the country. Similarly, three top largest telecommunications companies from Malaysia (DiGi, Maxis, and Celcom Axiata) were chosen out of six, which are holding more than 60 % of the total market share in the country as well. The purposive sampling was chosen because specific managers form the respondent pool for the research questionnaire survey. In Malaysia, there are 820 branch offices for the DiGi, Maxis, and Celcom that we could collect 98 usable data. In Bangladesh, there are in total 621 branch offices and the usable collected data is 78. To run the analysis of the current framework with three predictors, it is required to have a minimum sample size of 77, which would generate a power of 0.80 for a model with medium effect size (Hair et al. 2013). Therefore, a total of 176 usable data from both countries are analysed for the purpose of the research. Table 1 provides the demographic statistics of the sample data. Table 1 Demographic profile of respondent The questionnaire was developed from past studies. The items for organisational culture (OC1 to OC9) were taken from Chang and Lin (2007). In the survey questionnaire, the respondents were asked to respond on the items of organisation culture on 5-point Likert scale (1 = strongly disagree to 5 = strongly agree) with the question "How much do you agree on the following practices…?." The items for innovation process (PRC1–PRC5), cross-functional organisation (ORG1–ORG5), and tools/technology (TLS1–TLS5) were taken from Hull (2003) and Hull and Tidd (2003a), and anchored on 5-point Likert scale (1 = very low extent to 5 = very high extent). While measuring the innovation process, the respondents were asked to rate the items considering the following statement, "By the practice of innovation process, our company is…" To measure the cross-functional organisation the statement was "By the practice of cross-functional organisation, our company has…" Tools/technology was measured on the basis of following statement "In the implementation of information technology tools, our company has…" The items for competition-informed pricing (COMIP1–COMIP5) were taken from Ingenbleek et al. (2003), and measured on 5-point Likert scale (1 = very low extent to 5 = very high extent). The managers were asked to indicate "To what extent your company take into consideration….?." The items for performance were taken from Hull and Tidd (2003a) in terms of service development (SD1–SD5) and delivery process (DP1–DP5) measured on 5-point Likert scale (1 = very low extent to 5 = very high extent). While measuring the performance, the respondents were asked to rate the items considering the following statement "To what extent has your operation system changed based on the following…" Details of the items have been illustrated in "Appendix". To ensure that there is no Common Method bias in the questionnaire survey, we performed Harman's single factor test. This revealed that the first factor accounted for 45.018 % of variance, which is less than threshold level of 50 % of total variance explained (Podsakoff et al. 2003). In this study, to see whether there any differences between subsidiaries group exist (DiGi in Malaysia and GrameenPhone in Bangladesh are both subsidiaries of Telnor group; Robi in Bangladesh and Celcom in Malaysia are subsidiaries of Axiata group), an independent-sample t test was conducted to compare the six variables. Parent companies Telenor and Axiata were considered as two groups, where, DiGi and GrameenPhone were considered as group 1 and Celcom and Robi were grouped as 2. The results show that the p value from the independent t test for five variables is not significant except for one variable that is organisational culture. Organisational culture shows some slight difference in the means between the two groups of subsidiaries. Therefore, the effect size test was calculated to determine the magnitude of the difference as suggested by (Cohen 1988). The effect size is determined by the Cohen's d value. The formula to get the Cohen's d is: $${\text{Cohen's d}} = {\text{difference between sample mean}}/{\text{pooled standard deviation}}$$ The interpretation for effect size using Cohen's d test value belonging to the categories: 0.20–0.49 (small), 0.50–0.79 (medium), and above or equal to 0.80 (large). The result of the test indicates that the effect size of the variable is small (0.21), therefore, the homogeneity of two groups of subsidiaries is established. The small effect size indicates that the response bias is not a threat. In order to achieve our research objectives and analyse the measurement and structural model, we considered the structural equation model (SEM) with PLS approach, specifically the SmartPLS version 2.0 M3 Beta (Ringle and Wende 2005). PLS-SEM can be viewed as quite similar to multiple regression analysis to examine possible relationships with less emphasis on the measurement model (Hair et al. 2013). The individual path coefficients in the PLS structural model can also be interpreted as standardised beta coefficients of ordinary least square regression (Götz et al. 2010). Each path coefficient's significance can be accessed through a bootstrapping procedure where significant paths showing the hypothesised direction empirically support the proposed causal relationship and vice versa (Hair et al. 2011; Yung and Bentler 1994; Efron 1979). Bootstrapping in PLS is a nonparametric test which involves repeated random sampling with replacement from the original sample to create a bootstrap sample and to obtain standard errors for hypothesis testing (Hair et al. 2011). Regarding the number of re-sampling, Chin (2010) suggested to perform bootstrapping with 1000 re-samples. In the current study, the bootstrapping procedure with 1000 re-samples was used to test the significance of the path coefficients (regression coefficients). The path coefficients have standardized values between −1 and +1. The estimated path coefficients close to +1 represents a strong positive linear relationship and vice versa for negative values (Hair et al. 2013). In addition, to carry out a multi-group analysis between the companies of the two countries, PLS is considered to be more appropriate to explore the differences between them. The respondents of Bangladesh telecommunications sector's managers and Malaysian telecommunications sector's managers were split into two data sets (Bangladesh = 78 samples and Malaysia = 98 samples). To estimate the structure model, all criteria such as convergent validity, discriminant validity, and measurement invariance were checked separately as suggested by Hair et al. (2013). Factor loadings of the items, average variance extracted (AVE), and composite reliability (CR) are used to assess convergence validity of the data (Hair et al. 2009). To ensure the indicators' reliability, the main loading and cross-loading of items are checked. In accordance with Chin (1998), we retained the items which exceeded the recommended value of 0.6 while three items (OC8, OC9, TLS4) were found to be below the cut off value were deleted. Two items (OC4 and ORG5) were deleted because of cross-loading. The AVE of all the constructs exceeded the cut off value of 0.5 suggested by in literature (Henseler et al. 2009; Hair et al. 2013). The CR values of the constructs were found to have a minimum threshold of 0.7 suggested by Hair et al. (2011). Table 2 shows the results. Table 2 PLS factor loadings, CR, and AVE of full and country samples After convergent validity, we analysed the discriminant validity of the model. The discriminant validity was assessed for both the full and split sample by comparing the correlations between constructs and the square root of the average variance extracted for that construct (Fornell and Larcker 1981). The results show that the square roots of AVEs are greater in all cases than the off-diagonal elements in their corresponding row and column, suggesting that the required discriminant validity was achieved (Table 3). In total, the measurement model demonstrated adequate convergent validity and discriminant validity. Table 3 Discriminant validity of data sets Measurement invariance was tested. According to Hair et al. (2013), researchers should ensure the construct measures are invariant across the groups while comparing path coefficients across the groups using the PLS-MGA parametric. Bootstrapping is used according to the number of the observation in the data set separately for each group. Through outer weights and standard errors for each group and using the Levene's test suggested by Hair et al. (2013), the invariance test is checked for all items. In this test, if the test for equality of group variance is significant, then the unequal standard errors are assumed and the test statistic (t value) is computed as follows: $$S_{1 2} = \sqrt {S_{1}^{2} + S_{2}^{2} }$$ If the test for equality of group variance is not significant, equal standard errors are assumed and the test statistic (t value) is computed as follows: $$S_{1 2} = \left( {\sqrt {\frac{{\left( {N_{1} - 1} \right)^{2} }}{{\left( {N_{1} + N_{2} - 2} \right)}} \,. \,S_{1}^{2} + \frac{{\left( {N_{2} - 1} \right)^{2} }}{{\left( {N_{1} + N_{2} - 2} \right)}} \cdot S_{2}^{2} } } \right) \cdot \left( {\sqrt {\frac{1}{{N_{1} }} + \frac{1}{{N_{2} }}} } \right)$$ The criterion is that at least two items should not differ in the measurement items of each construct. The result shows that the there is no significant difference among the two groups. Table 4 shows the results. Table 4 Invariance test After testing measuring model, the structural model has been analysed. The R 2 and the path coefficients (beta and significance) show how well the data supported the hypothesized model (Chin 1998). We used the bootstrapping method with a resampling of 1000 to estimate the significance of the path coefficients (Chin 1998). The path coefficients for full and split data are shown in Table 5 and Fig. 2. Table 5 Result for direct relationships Structural models. *p < 0.01, p < 0.05 Hypotheses related to organisational culture and operating core From the analysis, we found H1 was supported in the full data (β = 0.520, p < 0.01), the Malaysian data (β = 0.545, p < 0.01), and the Bangladeshi data (β = 0.314, p < 0.01). H2 was supported in the full data (β = 0.584, p < 0.01), the Malaysian data (β = 0.651, p < 0.01), and also in the Bangladeshi data (β = 0.350, p < 0.01). H3 was found to be supported in the full data (β = 0.567, p < 0.01), the Malaysian data (β = 0.471, p < 0.01) as well as in the Bangladeshi data (β = 0.545, p < 0.01). Hypotheses related to operating core (innovation process, cross-functional organisation, and implementation of tools/technology) and competition-informed pricing The result of H4 is supported in the full data set (β = 0.170, p < 0.05) and the Malaysian data set (β = 0.255, p < 0.05), while in the Bangladeshi data it was not supported. The result of H5 was supported in the full data set (β = 0.266, p < 0.05) and the Bangladeshi data set (β = 0.275, p < 0.05), while in the Malaysian data set it was not supported. The result of H6 was supported in the full data set (β = 0.295, p < 0.01) and the Bangladeshi data set (β = 0.536, p < 0.01), while in the Malaysian data set, H6 was not supported. Hypotheses related to competition-informed pricing and performance The findings revealed that H7 was supported in all the data sets, the full (β = 0.602, p < 0.01), the Malaysian (β = 0.562, p < 0.01), and the Bangladeshi (β = 0.596, p < 0.01) data sets. Hypotheses related to the mediating effect of competition-informed pricing on the relationship between operating core and performance. The result shows that H9 was supported only in the full data set. H10 was supported in the full and in the Bangladeshi data sets only, but not in the Malaysian data set (Table 6). Table 6 Result for mediating effect To explore the differences, we carried out PLS multi-group analysis for the Bangladeshi and Malaysian subsamples. We tested the differences between the path coefficients across the respective two data sets and the result is shown in Table 7. Three paths differ significantly between the two countries' data sets. Organisational culture and process (p = 0.036); organisational culture and organisation (p = 0.003); tools or technology and competition-informed pricing (p = 0.016) have significant statistical differences (Table 7). Table 7 Path differences by Country The results of the study show significant relationship between organisational culture and operating core (innovation process, cross-functional organisation, and implementation of tools/technology) in both Bangladesh and Malaysia context. It is in line with the previous notion regarding the fact that internal behaviour and external relation, as part of organisational culture, facilitates the implementation of innovation successfully in the developed countries context (Naranjo-Valencia et al. 2011). Similar findings have been also observed in the current study, which focuses on developing countries. Organisational culture as a source of new ideas (Uzkurt et al. 2013) facilitates the practice of operating core (innovation process, cross-functional organisation, and implementation of tools/technology) in telecommunications industry. Earlier researchers found that training and championing have an influence on shaping up innovative organisations and processes (Hull and Tidd 2003a). However, the current study gives importance to the overall organisational culture in relationship with the practice of the operating core. Nevertheless, results of the present research also give such impression in the context of telecommunications sector in Malaysia and Bangladesh. It is not expected that such practice of organisational culture would be the same throughout all organisations or throughout all the countries. In line with similar considerations, the result of the multi-group analysis shows that the relationship between organisational culture and process as well as organisational culture and cross-functional organisation are significantly and statistically differ between Malaysian telecommunications industry and Bangladeshi telecommunications industry. Based on the findings, the practice of organisational culture in relationship with process (β = 0.545) and cross-functional organisation (β = 0.651) is stronger in the Malaysian telecommunications sector compared to the Bangladeshi telecommunications sector, where process holds a standard beta of 0.314 and cross-functional organisation accounts a standard beta of 0.350. According to scholars, cultural differences have implications on the organisations where they are operating (Tayeb 1994). Furthermore it has been asserted that cultural values at individual or societal level are greatly influenced by the national culture (Thornton et al. 2011). National culture with low individualism accentuates on strong group solidity. The culture which possess the characteristics of uncertainty avoidance at higher level prefer to follow clear rules of conduct, while cultures low on uncertainty avoidance relish on novel events and value innovation. Cultures those are high on harmony focuses accepting matters as they are, and low level of harmony indicates the prominence of assertiveness to advance personal or group interests (Li et al. 2013). Therefore, in context of this study, it is the veritable fact that the organisational culture would differ between the companies of these countries, which might have been experienced due to the influence of different national culture. Perhaps, due to the advancement of modern and trending organisational culture in Malaysia, the telecommunication companies are able to blend mechanistic process and organic cross-functional organisation as practices of innovation on a concurrent basis. It can be argued that the multi-ethnicity setting of the Malaysian culture influences the organisational culture to practise both the mechanistic and organic structures simultaneously. In the Malaysian context, cooperativeness and steadiness has been entrenched in the society, which presumably are influenced by the cultural harmony of the nation. From an economic point of view, Malaysia is in the stage of development and is considered to be one of the emerging tigers of Asia. The government has already taken up various measures to achieve developed nation recognition and status. With this view, it is inferred that the culture of cooperativeness, creativity, efficacy, and competitiveness among the Malaysian telecommunication companies are supportive towards innovation driven in such a transitional stage. To be more specific, based on the data, the study believes that cooperativeness is one of the most significant dimensions of organisational culture followed by consistency, and innovativeness for the telecommunication companies of the both countries. Furthermore, among the Malaysian telecommunications companies, cooperativeness and consistency deemed to be carrying more weightage. On the other hand, cooperativeness and innovativeness are more important among the Bangladeshi telecommunications companies in order to shape up effective innovation practices. The relationship between innovation process and competition-informed pricing is found to be significant in the Malaysian telecommunications sector whereas in the Bangladeshi telecommunications sector, it is insignificant. Theoretically, innovation process refers to the mechanistic stand of the organisation. According to Liker et al. (1999) and Tidd and Hull (2011), a mechanistic organisation is appropriate when the environment is efficient, effective, and stable. The findings of this study reflect what was advocated earlier in the context of innovation in developed countries. The Malaysian telecommunications sector is presumably at a mature stage with greater efficiency and effectiveness compared to the Bangladeshi telecommunications sector. Such an efficient and mature state of the industry instigates us to consider the most important stakeholder in the business environment such as competitors. With this contextual argument, it is noteworthy to state that the Malaysian telecommunications industry takes into account the competition-informed pricing practice with the mechanistic state of business operation. However, the innovation process can improve firm's performance if the practice of gathering price related information from competitors is emphasized. Competition-informed pricing helps managers in the Malaysian telecommunications field to understand the upper-limit of the price decision while practising the innovation process for performance improvement. Therefore, it is important to mention that through the competition-informed pricing practice, the mechanistic state of organisation can assist to achieve performance. In contrast to Malaysia, the relationship of cross-functional organisation and tools/technology with competition-informed pricing is significant in the Bangladeshi telecommunications sector. Bangladesh is in a position where it is about to take flight towards the development of innovation. Apparently, foreign investment is growing in the country, with greater interest among the telecommunication companies around the world. Therefore, the market is experiencing rapid changes in terms of organisational operation and strategy. As suggested by Liker et al. (1999) and Tidd and Hull (2011), organisations tend to be organic while the environment is not stable, dynamic, and the existence of less rules and regulations. In this scenario, it is justifiable to conclude upon the significance of the result that denotes the influence of cross-functional organisation on competitor-informed pricing. However, it is important to understand the competitors' pricing strategy and competitors' strength in the market through use of cross-functional team members within the innovative organisation. Computer information technology (CIT)'s tools, indeed, updates the process of service innovation cycle among cross-functional team members and increases the frequency of cross-functional team members' communication in the value chain as highlighted in the previous study (Collins and Hull 2002; Tidd and Hull 2011). Thus, the result of the current study explains a facilitator role of competition-informed pricing for implementation of tools/technology to achieve a firm's goals and performance only in the Bangladeshi telecommunications sector. Since the offered services of the telecommunications industry are very much similar across the companies, therefore, the state of competition is apparently higher, which triggers the companies to consider competition-informed pricing. In the result of multi-group analysis, the relationship between tools/technology and competition-informed pricing significantly differs in the Bangladeshi telecommunications sector (β = 0.536) compared to the Malaysian telecommunications sector (β = 0.163). In line with the resource based view theory (RBV), organisation resources are converted to capabilities which would have an effect on competitive advantage (Barney 1991). In this study, resources namely innovation process, cross functional organisation, and tools/technology have causal effect on the firms capabilities that is competition-informed pricing. Subsequently, this capability (competition-informed pricing) has also a casual effect on competitive advantage, in this study which is performance. In this line, it has been argued in the literature that in capitalizing resources, an organisation can dominate and achieve a high level of performance (Barney 1991). Interestingly, the mediating effect of competition-informed pricing is found to be significant on the relationship between tools/technology and performance only in the Bangladeshi data set. The reason probably accounts for the state of the progress in Bangladesh in terms of business innovation. Bangladesh is struggling towards the benchmark of the international standard. Being in transition from least developed country to emerging country, the business organisations are proactive to inculcate the practice of using tools/technology. On the other hand, tools/technology has become a part of the business operation for a fairly long time in Malaysia. Therefore, a significant difference has been observed between the Malaysian and Bangladeshi telecommunications sector in terms of these relationships. Managerial relevance The illustrated research model is a useful theoretical framework for explaining the elements of operating core practices of service innovation that influence higher performance through the mediating effect of competition-informed pricing. According to the result attained from this study, managers of the Malaysian telecommunications sector do not take into account the competition-informed pricing while practising the operating core of service innovation to achieve higher performance. On the contrary, managers of the Bangladeshi telecommunication companies should take into account the competition-informed pricing while practising the operating core of service innovation to realize greater performance to counter the instable environment. The study also reflects the situation of organisational culture practice in both countries' industry. It is recommended that managers of Bangladeshi telecommunications industry develop an organisational culture to gain performance advantages with the practice of service innovation. Overall, the findings suggest that it is advantageous for the telecommunications industry to escalate the level of performance, facilitating managers to consider competition-pricing for new services with the support of operating core of the service innovation management. The managers of the industry must look towards competitors to set the price of the service along with practicing innovative process, innovative organisation, and implanting tools or technology. This may assist the managers to gain insight on the practice of service innovation, organisational culture, and performance. Taken all together, the results of this study show that the service innovation practice differs between Malaysia and Bangladesh. In Malaysia, organisational culture is revealed to be a strong predictor for operating core of service innovation compared to the Bangladeshi telecommunications sector. Furthermore, in the Malaysian telecommunications sector, competition-informed pricing does not necessitate playing any role between operating core of service innovation and performance, while in the Bangladeshi telecommunications sector, competition-informed pricing facilitates the relationship of tools or technology with performance. In addition, the relationship between tools or technology and competition-informed pricing is strong in the Bangladeshi telecommunications sector. On the other hand, it is not significant in the Malaysian telecommunications sector. It is however expected that if the respective managers of both countries consider these issues, it would contribute immensely towards the practice of service innovation management as a whole. Limitations and future directions of research This paper has limitations that are to be noted. The paper is based on a single industry and the sample is drawn only from the telecommunications industry, which has the potential for limiting the generalisation of the findings of this research across other industries. This can be overcome by extending the scope of the research by using a larger database comprising responses of managers representing a number of industries. Although this paper is based purely on quantitative methodology using established constructs, these were not used in any prior study in Bangladesh and Malaysia. Future study can be developed using a mixed methodology comprising qualitative and quantitative approaches toward contributing to greater generalisation of the findings. 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Available via http://www3.weforum.org/docs/WEF_GlobalCompetitivenessReport_2014-15.pdf. Accessed 21 Sept 2015 Yung YF, Bentler PM (1994) Bootstrap-corrected ADF test statistics in covariance structure analysis. Br J Math Stat Psychol 47(1):63–84 Zammuto RF, O'Connor EJ (1992) Gaining advanced manufacturing technologies' benefits: the roles of organisation design and culture. Acad Manag Rev 17(4):701–728 SAR participated in the data collection from Bangladesh, writing up Introduction and Discussion section. SKT participated in the data collection from Malaysia and drafted Theoretical background along with Discussion section. TR carried out the statistical analysis and assisted in writing Research Methodology section. NHA contributed in writing the Managerial Relevance, Conclusion, and Limitation and Future direction of research. All authors read and approved the final manuscript. Syed Abidur Rahman, has recieved Ph.D. degree in the area of entrepreneurship and innovation from Universiti Sains Malaysia. He is working in Stamford University Bangladesh as Assistant Professor. He published several articles in academic journals. His area of interest is base of pyramid, entrepreneruship, sustainable development, and innovation. Seyedeh Khadijeh Taghizadeh is a Ph.D. candidate in the area of marketing and innovation in Universiti Sains Malaysia. Her area of interest is service innovation, sustainable development, entrepreneurship. She published several articles in academic journals and attended several international conferences. T. Ramayah is currently a Professor at the School of Management, Universiti Sains Malaysia. He has also presented numerous papers at local and international conferences having won 3 "Best Papers" award. His publications have appeared in Computers in Human Behavior, Resources, Conservation and Recycling, International Journal of Information Technology & Decision Making (IJITDM), International Journal of Information Management, Engineering, Construction and Architectural Management (ECAM) and North American Journal of Psychology. Noor Hazlina Ahmad Ph.D. is an Associate Professor at the School of Management USM. She joined the university after completing her Ph.D. at the University of Adelaide, Australia. Her research work lies in the inter-disciplinary intersection between entrepreneurship and organizational, which looked into accumulating ground-breaking evidence of cultural constraints on entrepreneurial behaviour. The authors would like to thank Dr. Marcus Griffin (English language editor) for proofreading the manuscript. Department of Business Administration, Stamford University Bangladesh, 744, Saat Masjid Road, Dhaka, Bangladesh Syed Abidur Rahman & Seyedeh Khadijeh Taghizadeh School of Management, Universiti Sains Malaysia, Pulau Penang, Malaysia T. Ramayah & Noor Hazlina Ahmad Syed Abidur Rahman Seyedeh Khadijeh Taghizadeh T. Ramayah Noor Hazlina Ahmad Correspondence to Syed Abidur Rahman. See Table 8. Table 8 Measurement items Rahman, S.A., Taghizadeh, S.K., Ramayah, T. et al. Service innovation management practices in the telecommunications industry: what does cross country analysis reveal?. SpringerPlus 4, 810 (2015). https://doi.org/10.1186/s40064-015-1580-8 Service innovation practices Innovation process Cross-functional organisation Tools/technology	CommonCrawl
ROC of Z transform Doesnt include a pole on the boundary? I cannot figure out what is going on here. I have an example problem in my book that says the ROC of a certain function is $$ 0.5 < \|z\| < \infty$$ The function's denominator is $$ 1 - z^{-1} + 0.75z^{-2} -0.25z^{-3} + 0.0625z^{-4} $$ By properties of the ROC, there should be a pole on the boundary of the ROC. Since the example given was a right sided signal, the ROC will be the area outside of a circle boundary. So, the signal should have a pole on the boundary of that circle, which has a radius of 0.5 correct? But when I plug in 0.5 to the denominator equation, I get 1, not zero, which makes 0.5 not a pole in this case. I have tried graphs, and placing different parenthesis, and calculated it multiple times, but the denominator equation above never reaches zero. According to ROC properties, it has to though! Where am I thinking wrong here? z-transform poles-zeros DomDom The ROC $0.5<\|z\|<\infty$ does not imply that there's a pole at $z=0.5$. What it does say is that there's at least one pole satisfying $\|z\|=0.5$ (and no other pole with a radius larger than $0.5$), and this is also the case for the given denominator polynomial. The roots of the polynomial are $$z_{1,2}=0.5\, e^{j\pi/3}\tag{1}$$ $$z_{3,4}=0.5\, e^{-j\pi/3}\tag{1}$$ I.e., there are two double poles, and all four poles lie on a circle with radius $0.5$. Not the answer you're looking for? Browse other questions tagged z-transform poles-zeros or ask your own question. Understanding $\mathcal Z$-transforms and pole locations Identifying the magnitude and impulse response from pole zero plot quickly System characterization given pole-zero mapping Z-transform of difference equations and stability of a process How to identify causality, stability and ROC from the pole-zero plot? Poles and zeros of a transfer function What happens when the poles of this z-transform function are outside the ROC for a signal? Determining asymptotic stability using transfer function? Using ROC to find stability of system in specific example Single-sided Z transform with difference equations and the system function	CommonCrawl
Tutte–Berge formula In the mathematical discipline of graph theory the Tutte–Berge formula is a characterization of the size of a maximum matching in a graph. It is a generalization of Tutte theorem on perfect matchings, and is named after W. T. Tutte (who proved Tutte's theorem) and Claude Berge (who proved its generalization). Statement The theorem states that the size of a maximum matching of a graph $G=(V,E)$ equals ${\frac {1}{2}}\min _{U\subseteq V}\left(\|U\|-\operatorname {odd} (G-U)+\|V\|\right),$ where $\operatorname {odd} (H)$ counts how many of the connected components of the graph $H$ have an odd number of vertices. Equivalently, the number of unmatched vertices in a maximum matching equals $\max _{U\subseteq V}\left(\operatorname {odd} (G-U)-\|U\|\right)$. Explanation Intuitively, for any subset $U$ of the vertices, the only way to completely cover an odd component of $G-U$ by a matching is for one of the matched edges covering the component to be incident to $U$. If, instead, some odd component had no matched edge connecting it to $U$, then the part of the matching that covered the component would cover its vertices in pairs, but since the component has an odd number of vertices it would necessarily include at least one leftover and unmatched vertex. Therefore, if some choice of $U$ has few vertices but its removal creates a large number of odd components, then there will be many unmatched vertices, implying that the matching itself will be small. This reasoning can be made precise by stating that the size of a maximum matching is at most equal to the value given by the Tutte–Berge formula. The characterization of Tutte and Berge proves that this is the only obstacle to creating a large matching: the size of the optimal matching will be determined by the subset $U$ with the biggest difference between its numbers of odd components outside $U$ and vertices inside $U$. That is, there always exists a subset $U$ such that deleting $U$ creates the correct number of odd components needed to make the formula true. One way to find such a set $U$ is to choose any maximum matching $M$, and to let $X$ be the set of vertices that are either unmatched in $M$, or that can be reached from an unmatched vertex by an alternating path that ends with a matched edge. Then, let $U$ be the set of vertices that are matched by $M$ to vertices in $X$. No two vertices in $X$ can be adjacent, for if they were then their alternating paths could be concatenated to give a path by which the matching could be increased, contradicting the maximality of $M$. Every neighbor of a vertex $x$ in $X$ must belong to $U$, for otherwise we could extend an alternating path to $x$ by one more pair of edges, through the neighbor, causing the neighbor to become part of $U$. Therefore, in $G-U$, every vertex of $X$ forms a single-vertex component, which is odd. There can be no other odd components, because all other vertices remain matched after deleting $U$. So with this construction the size of $U$ and the number of odd components created by deleting $U$ are what they need to be to make the formula be true. Relation to Tutte's theorem Tutte's theorem characterizes the graphs with perfect matchings as being the ones for which deleting any subset $U$ of vertices creates at most $\|U\|$ odd components. (A subset $U$ that creates at least $\|U\|$ odd components can always be found in the empty set.) In this case, by the Tutte–Berge formula, the size of the matching is $\|V\|/2$; that is, the maximum matching is a perfect matching. Thus, Tutte's theorem can be derived as a corollary of the Tutte–Berge formula, and the formula can be seen as a generalization of Tutte's theorem. See also • Graph toughness, a problem of creating many connected components by removing a small set of vertices without regard to the parity of the components • Hall's marriage theorem References • Berge, C. (1958). "Sur le couplage maximum d'un graphe". Comptes rendus hebdomadaires des séances de l'Académie des sciences. 247: 258–259. • Berge, C. (1962). The Theory of Graphs. Methuen. Theorem 5, p. 181. Reprinted by Dover Publications, 2001. • Bondy, J. A.; Murty, U. S. R. (2007). Graph Theory: An Advanced Course. Graduate Texts in Mathematics. Springer-Verlag. p. 428. ISBN 978-1-84628-969-9. • Bondy, J. A.; Murty, U. S. R. (1976). Graph Theory with Applications. New York: North Holland. Exercise 5.3.4, p. 80. ISBN 0-444-19451-7.{{cite book}}: CS1 maint: url-status (link) • Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. p. 360. ISBN 0-521-86565-4. Zbl 1106.05001. • Lovász, László; Plummer, M. D. (1986). Matching Theory. Annals of Discrete Mathematics. Vol. 29. North-Holland. pp. 90–91. ISBN 0-444-87916-1. MR 0859549. • Schrijver, Alexander (2003). Combinatorial Optimization: Polyhedra and Efficiency. Springer-Verlag. p. 413. ISBN 3-540-44389-4. • Tutte, W. T. (1947). "The factorization of linear graphs". Journal of the London Mathematical Society. Series 1. 22 (2): 107–111. doi:10.1112/jlms/s1-22.2.107. hdl:10338.dmlcz/128215.	Wikipedia
I won a trip for four to the Super Bowl. I can bring three of my friends. I have 8 friends. In how many ways can I form my Super Bowl party? Order does not matter, so it is a combination. Choosing $3$ out of $8$ is $\binom{8}{3}=\boxed{56}.$	Math Dataset
\begin{definition}[Definition:Functor] Informally, a '''functor''' is a morphism of categories. It may be described as what one must define in order to define a '''natural transformation'''. This is formalized by defining the category of categories. \end{definition}	ProofWiki
Separating set In mathematics, a set $S$ of functions with domain $D$ is called a separating set for $D$ and is said to separate the points of $D$ (or just to separate points) if for any two distinct elements $x$ and $y$ of $D,$ there exists a function $f\in S$ such that $f(x)\neq f(y).$[1] This article is about separating sets for functions. For use in graph theory, see connectivity (graph theory). Separating sets can be used to formulate a version of the Stone–Weierstrass theorem for real-valued functions on a compact Hausdorff space $X,$ with the topology of uniform convergence. It states that any subalgebra of this space of functions is dense if and only if it separates points. This is the version of the theorem originally proved by Marshall H. Stone.[1] Examples • The singleton set consisting of the identity function on $\mathbb {R} $ separates the points of $\mathbb {R} .$ • If $X$ is a T1 normal topological space, then Urysohn's lemma states that the set $C(X)$ of continuous functions on $X$ with real (or complex) values separates points on $X.$ • If $X$ is a locally convex Hausdorff topological vector space over $\mathbb {R} $ or $\mathbb {C} ,$ then the Hahn–Banach separation theorem implies that continuous linear functionals on $X$ separate points. See also • Dual system References 1. Carothers, N. L. (2000), Real Analysis, Cambridge University Press, pp. 201–204, ISBN 9781139643160.	Wikipedia
\begin{document} \title{Total domination number of middle graphs} \author[F. Kazemnejad]{Farshad Kazemnejad} \address{Farshad Kazemnejad, Faculty of Basic Sciences, Department of Mathematics, Ilam University, P.O.Box 69315-516, Ilam, Iran.} \email{[email protected]} \author[B. Pahlavsay]{Behnaz Pahlavsay} \address{Behnaz Pahlavsay, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan.} \email{[email protected]} \author[E. Palezzato]{Elisa Palezzato} \address{Elisa Palezzato, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan.} \email{[email protected]} \author[M. Torielli]{Michele Torielli} \address{Michele Torielli, Department of Mathematics, GI-CoRE GSB, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan.} \email{[email protected]} \date{\today} \begin{abstract} A \emph{total dominating set} of a graph $G$ with no isolated vertices is a subset $S$ of the vertex set such that every vertex of $G$ is adjacent to a vertex in $S$. The \emph{total domination number} of $G$ is the minimum cardinality of a total dominating set of $G$. In this paper, we study the total domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph $G$. We also compute the total domination number of the middle graph of some known families of graphs explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of middle graphs. \\[0.2em] \noindent Keywords: Total domination number, Middle graph, Nordhaus-Gaddum-like relation. \\[0.2em] \noindent \end{abstract} \maketitle \section{Introduction} The concept of total domination in graph theory was first introduced by Cockayne, Dawes and Hedetniemi in \cite{CDH} and it has been studied extensively by many researchers in the last years, see for example \cite{HHS5}, \cite{HHS6}, \cite{HeYe13}, \cite{3totdominrook}, \cite{Kaz19}, \cite{totcolordominmiddle}, \cite{romandomin} and \cite{dominLatin}. The literature on this subject has been surveyed and detailed in the recent book~\cite{HeYe13}. We refer to \cite{bondy2008graph} as a general reference on graph theory. Let $G$ be a graph with the vertex set $V(G)$ of \emph{order} $n$ and the edge set $E(G)$ of \emph{size} $m$. The \emph{open neighborhood} and the \emph{closed neighborhood} of a vertex $v\in V(G)$ are $N_{G}(v)=\{u\in V(G)~\|~ uv\in E(G)\}$ and $N_{G}[v]=N_{G}(v)\cup \{v\}$, respectively. For a connected graph $G$, the \emph{degree} of a vertex $v$ is defined as $d_G(v)=\vert N_{G}(v) \vert $. The \emph{distance} $d_G(v,w)$ in $G$ of two vertices $v,w\in V(G)$ is the length of the shortest path connecting $v$ and $w$. The \emph{diameter} $\diam(G)$ of $G$ is the shortest distance between any two vertices in $G$. A \emph{dominating set} of a graph $G$ is a set $S\subseteq V(G)$ such that $N_G[v]\cap S\neq \emptyset$, for any vertex $v\in V(G)$. The \emph{domination number} of $G$ is the minimum cardinality of a dominating set of $G$ and is denoted by $\gamma(G)$. \begin{Definition} Let $G$ be a graph with no isolated vertices. A \emph{total dominating set} of $G$ is a set $S\subseteq V(G)$ such that $N_G(v)\cap S\neq \emptyset$, for any vertex $v\in V(G)$. The \emph{total domination number} of $G$ is the minimum cardinality of a total dominating set of $G$ and is denoted by $\gamma_t(G)$. \end{Definition} \begin{Example} Consider the path $P_3$ with vertex set $\{v_1,v_2,v_3\}$ and edge set $\{v_1v_2,v_2v_3\}$. Then the set $S=\{v_1,v_2\}$ is a total dominating set of $P_3$. \end{Example} For any non-empty $S\subseteq V(G)$, we denote by $G[S]$ the subgraph of $G$ induced on $S$. For any $v\in V(G)$, we denote by $G\setminus v$ the subgraph of $G$ induced on $V(G)\setminus \{v\}$. The \emph{complement} $\overline{G}$ of $G$ is a graph with vertex set $V(G)$ such that for every two vertices $v$ and $w$, $vw\in E(\overline{G})$ if and only if $vw\not\in E(G)$. The \emph{line graph} of $G$, denoted by $L(G)$, is the graph with vertex set $E(G)$, where vertices $x$ and $y$ are adjacent in $L(G)$ if and only if edges $x$ and $y$ share a common vertex in $G$. In \cite{HamYos}, the authors introduced the notion of the middle graph $M(G)$ of $G$ as an intersection graph on $V(G)$. \begin{Definition} The \emph{middle graph} $M(G)$ of a graph $G=(V,E)$ is the graph whose vertex set is $V(G)\cup E(G)$ and two vertices $x, y$ in the vertex set of $M(G)$ are adjacent in $M(G)$ in case one the following holds: \begin{enumerate} \item $x, y$ are in $E(G)$ and $x, y$ are adjacent in $G$. \item $x$ is in $V (G)$, $y$ is in $E(G)$, and $x, y$ are incident in $G$. \end{enumerate} \end{Definition} \begin{Example} Consider the graph $P_3$, then the middle graph $M(P_3)$ is the one in Figure~\ref{fig:midlepaths}. \end{Example} \begin{figure} \caption{The middle graph $M(P_3)$ } \label{fig:midlepaths} \end{figure} It is obvious that $M(G)$ contains the line graph $L(G)$ as induced subgraph, and that if $G$ is a graph of order $n$ and size $m$, then $M(G)$ is a graph of order $n+m$ and size $2m+\|E(L(G))\| $ which is obtained by subdividing each edge of $G$ exactly once and joining all the adjacent edges of $G$ in $ M(G)$. In order to avoid confusion throughout the paper, we fix a ``standard'' notation for the vertex set and the edge set of $M(G)$. Assume $V(G)=\{v_1,v_2,\dots, v_n\}$, then we set $V(M(G))=V(G)\cup \mathcal{M}$, where $\mathcal{M}=\{m_{ij}~\|~ v_iv_j\in E(G)\}$ and $E(M(G))=\{v_im_{ij},v_jm_{ij}~\|~ v_iv_j\in E(G)\}\cup E(L(G)) $. In this article, we continue our study from \cite{dominmiddle} on domination of middle graphs. The paper proceeds as follows. In Section 2, we describe explicitly the total domination number of the middle graph of several known families of graphs and we present some upper and lower bounds for $\gamma_t(M(G))$ in terms of the order of the graph $G$. In Section 3, we describe bounds for the total domination number of the middle graph of trees. In Section 4, we obtain the same type of results for $\gamma_t(M(G\circ K_1))$, $\gamma_t(M(G\circ P_2))$ and $\gamma_t(M(G+ K_p))$. Finally, in the last Section, we present some Nordhaus-Gaddum like relations for the total domination number of middle graphs. \section{Middle graph of known graphs and their total domination number}\label{t80} We start our study on total domination with two key Lemmas. \begin{Lemma}\label{lemma:totdominationisalledges} Let $G$ be a connected graph of order $n\ge3$ and $S$ a total dominating set of $M(G)$. Then there exists $S'\subseteq E(G)$ a total dominating set of $M(G)$ with $\|S'\|\le\|S\|$. \end{Lemma} \begin{proof} If $S\subseteq E(G)$, then take $S'=S$. On the other hand, assume that there exists $v\in S\cap V(G)$. If all edges adjacent to $v$ are already in $S$, then take $S_1=S\setminus\{v\}$. Otherwise, let $e\in E(G)\setminus S$ be an edge adjacent to $v$. Then consider $S_1=(S\cup\{e\})\setminus\{v\}$. Since $S$ is finite, then this process terminates after a finite number of steps, and hence we obtain the described $S'$. \end{proof} \begin{Lemma}\label{lemma:totdominationdeletionvertex} Let $G$ be a connected graph of order $n\ge2$ and $v\in V(G)$ a vertex not adjacent to any vertex of degree $1$. Then $$\gamma_t(M(G\setminus v))\le \gamma_t(M(G)) \le \gamma_t(M(G\setminus v))+1.$$ \end{Lemma} \begin{proof} Let $S$ be a total dominating set of $M(G\setminus v)$. This implies that for every $w\in N_G(v)$, $w\in S$ or there exists an edge of the form $ww_0\in E(G\setminus v)$ such that $ww_0\in S$. As a consequence, $S\cup\{vw\}$ is a total dominating set of $M(G)$, for any $w\in N_G(v)$, and hence $\gamma_t(M(G)) \le \gamma_t(M(G\setminus v))+1$. On the other hand, let $S$ be a minimal total dominating set of $M(G)$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq E(G)$. Consider $S_v=N_{M(G)}(v)\cap S$. Since $S$ is a minimal total dominating set, $\|S_v\|\ge 1$. Assume $S_v=\{e_1,\dots, e_k\}$. For any $1\le i\le k$, $e_i$ is an edge of $G$ of the form $w_iv$. By the assumption on $v$, $N_{M(G)}(w_1)=\{e_1,e_{11},\dots, e_{1p}\}$ with $p\ge1$, for some $e_{1j}\in E(G\setminus v)$. If $S\cap\{e_{11},\dots, e_{1p}\}\ne\emptyset$, then consider $S_1=(S\setminus e_1)\cup\{w_1\}$, otherwise $S_1=(S\setminus e_1)\cup\{e_{11}\}$. By applying the same construction for each $e_i$, we obtain $S_k$ a total dominating set of $M(G\setminus v)$ with $\|S_k\|=\|S\|$, and hence $\gamma_t(M(G\setminus v))\le \gamma_t(M(G))$. \end{proof} We are now ready to describe explicitly the total dominating number of the middle graph of several known families of graphs. \begin{Proposition}\label{prop:mintotdominstar} For any star graph $K_{1,n}$ on $n+1$ vertices, with $n\ge 2$, we have $$\gamma_t(M(K_{1,n}))=n.$$ \end{Proposition} \begin{proof} To fix the notation, assume $V(K_{1,n})=\{v_0,v_1,\dots, v_n\}$ and $E(K_{1,n})=\{v_0v_1,v_0v_2,\dots, v_0v_n\}$. Then $V(M(K_{1,n}))=V(K_{1,n})\cup \mathcal{M}$, where $\mathcal{M}=\{ m_i~\|~1\leq i \leq n \}$. If $S=\mathcal{M}$, then $S$ is a total dominating set of $M(K_{1,n})$ with $\|S\|=n$, and hence $\gamma_t(M(K_{1,n}))\le n$. On the other hand, using \cite[Proposition 3.1]{dominmiddle}, $n=\gamma(M(K_{1,n}))\le \gamma_t(M(K_{1,n}))$. \end{proof} \begin{Definition} A double star graph $S_{1,n,n}$ is obtained from the star graph $K_{1,n}$ by replacing every edge with a path of length $2$. \end{Definition} \begin{Proposition}\label{prop:mintotdomindoublestar} For any double star graph $S_{1,n,n}$ on $2n+1$ vertices, with $n\ge 1$, we have $$\gamma_t(M(S_{1,n,n}))=2n.$$ \end{Proposition} \begin{proof} To fix the notation, assume $V(S_{1,n,n})=\{v_0,v_1,\dots, v_{2n}\}$ and $E(S_{1,n,n})=\{v_0v_i,v_iv_{n+i}~\|~1\leq i \leq n\}$. Then $V(M(S_{1,n,n}))=V(S_{1,n,n})\cup \mathcal{M}$, where $\mathcal{M}=\{ m_{i},m_{i(n+i)}~\|~1\leq i \leq n \}$. If $S=\mathcal{M}$, then $S$ is a total dominating set of $M(S_{1,n,n})$ with $\|S\|=2n$, and hence $\gamma_t(M(S_{1,n,n}))\le 2n$. On the other hand, let $S$ be a total dominating set $M(S_{1,n,n})$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq\mathcal{M}$. Since, for every $1\le i\le n$, $N_{M(S_{1,n,n})}(v_{n+i})=\{m_{i(n+i)}\}$, then $m_{i(n+i)}\in S$ for every $1\le i\le n$. Similarly, for every $1\le i\le n$, $N_{M(S_{1,n,n})}(m_{i(n+i)})=\{m_i,v_i,v_{n+i}\}$ implies that $m_i\in S$ for every $1\le i\le n$, and hence $\mathcal{M}\subseteq S$. This implies that $\gamma_t(M(S_{1,n,n}))\ge2n$. \end{proof} \begin{Proposition}\label{prop:mintotdominpath} \label{gamma_{t}(M(P_n))} For any path $P_n$ of order $n\geq 3$, $$\gamma_{t}(M(P_n))=\lceil \frac{2n}{3} \rceil.$$ \end{Proposition} \begin{proof} To fix the notation, assume $V(P_n)=\{v_1,\dots, v_n\}$ and $E(P_n)=\{v_iv_{i+1}~\|~1\le i\le n-1\}$. Then $V(M(P_n))=V\cup \mathcal{M}$ where $V=V(P_n)$ and $\mathcal{M}=\{ m_{i(i+1)}~\|~1\leq i \leq n-1 \}$. If $n \equiv 0 \mod 3$, then consider $$S=\{m_{12},m_{23},m_{45},m_{56},\dots, m_{(n-2)(n-1)}, m_{(n-1)n}\}.$$ We have that $S$ is a total dominating set of $M(P_n)$ with $\|S\|=\frac{2n}{3}$. If $n \equiv 1 \mod 3$, then consider $$S=\{m_{12},m_{23},m_{45},m_{56},\dots, m_{(n-3)(n-2)},m_{(n-2)(n-1)}\}\cup\{m_{(n-1)n}\}.$$ We have that $S$ is a total dominating set of $M(P_n)$ with $\|S\|=\lceil\frac{2n}{3}\rceil$. If $n \equiv 2 \mod 3$, then consider $$S=\{m_{12},m_{23},m_{45},m_{56},\dots, m_{(n-4)(n-3)},m_{(n-3)(n-2)}\}\cup\{m_{(n-2)(n-1)},m_{(n-1)n}\}.$$ We have that $S$ is a total dominating set of $M(P_n)$ with $\|S\|=\lceil\frac{2n}{3}\rceil$. This implies $\gamma_{t}(M(P_n))\le\lceil \frac{2n}{3} \rceil.$ On the other hand, let $S$ be a total dominating set for $M(P_n)$. For every $i=1,\dots, n-2$, let $G_i=P_n[v_i,v_{i+1},v_{i+2}]$. Since $S$ dominates all vertices of the graph $M(G_i)$, $\|S\cap V(M(G_i))\|\ge 2$. This implies that $\|S\|\ge \lceil\frac{2n}{3}\rceil$. \end{proof} Since if we delete a vertex from a complete graph $K_{n+1}$ we obtain a graph isomorphic to $K_n$, Lemma~\ref{lemma:totdominationdeletionvertex} gives us the following result. \begin{Lemma}\label{lemma:ineqtotdomincomplete} For any $n\ge3$, we have $$\gamma_t(M(K_n))\le\gamma_t(M(K_{n+1}))\le\gamma_t(M(K_n))+1.$$ \end{Lemma} \begin{Proposition}\label{prop:mintotdomincompletegr} Let $K_n$ be the complete graph on $n\ge2$ vertices. Then $$\gamma_t(M(K_n))= \lceil \frac{2n}{3} \rceil $$ \end{Proposition} \begin{proof} If $2\le n\le 4$, a direct computation shows that $\gamma_t(M(K_n))= \lceil \frac{2n}{3}\rceil$. Assume now $n\ge5$. The graph $K_n$ has several subgraphs isomorphic to $P_n$, and hence $M(K_n)$ has subgraphs isomorphic to $M(P_n)$. Fix one of those and consider $S$ a total dominating set of $M(P_n)$. Since $S$ is also a total dominating set for $M(K_n)$, we have $\gamma_t(M(K_n))\le \lceil \frac{2n}{3} \rceil$. We will prove the opposite inequality by induction. Assume that we have equality for $\gamma_t(M(K_n))$ and we want to prove it for $\gamma_t(M(K_{n+1}))$. If $n\equiv2\mod 3$, then $n+1\equiv0\mod 3$, and hence, $\gamma_t(M(K_n))=\lceil\frac{2n}{3}\rceil=\lceil \frac{2(n+1)}{3}\rceil$. On the other hand, by Lemma~\ref{lemma:ineqtotdomincomplete}, $\gamma_t(M(K_n))\le\gamma_t(M(K_{n+1}))$. This fact, together with the first part of the proof, implies that $\gamma_t(M(K_{n+1}))=\lceil \frac{2(n+1)}{3}\rceil$. If $n\equiv0,1\mod 3$, by Lemma~\ref{lemma:ineqtotdomincomplete} and the first part of the proof, it is enough to show that $\gamma_t(M(K_n))<\gamma_t(M(K_{n+1}))$. As a contradiction, assume that $\gamma_t(M(K_n))=\gamma_t(M(K_{n+1}))$. If $n\equiv0\mod 3$, then $n-1\equiv2\mod 3$, and hence this would implies $\gamma_t(M(K_{n-1}))=\gamma_t(M(K_n))=\gamma_t(M(K_{n+1}))$. Similarly, if $n\equiv1\mod 3$, then $n+1\equiv2\mod 3$, and hence $\gamma_t(M(K_n))=\gamma_t(M(K_{n+1}))=\gamma_t(M(K_{n+2}))$. Hence we need to show that $\gamma_t(M(K_n))<\gamma_t(M(K_{n+2}))$, when $n\ge4$. Let $S$ be a total dominating set of $M(K_{n+2})$. To fix the notation, assume $V(K_{n+2})=\{v_1,\dots, v_{n+2}\}$ and $V(M(K_{n+2}))=V(K_{n+2})\cup \mathcal{M}$, where $\mathcal{M}=\{ m_{ij}~\|~1\le i<j\le n+2\}$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq \mathcal{M}$. After possibly relabeling $V(K_{n+2})$, we can assume that $m_{(n+1)(n+2)}\in S$. Since $S$ is a total dominating set of $M(K_{n+2})$, then it contains at least one element of the form $m_{i(n+1)}$ or $m_{i(n+2)}$, for some $i=1,\dots, n$. By construction, $M(K_n)$ is isomorphic to $M(K_{n+2}[v_1,\dots, v_n])$, this implies that, similarly to the proof of Lemma~\ref{lemma:ineqtotdomincomplete}, we can construct $S'$ a total dominating set of $M(K_n)$ by exchanging a vertex of the form $m_{i(n+1)}$ or $m_{i(n+2)}$ with one of the form $m_{ij}$ and just discarding $m_{(n+1)(n+2)}$. This implies that $\|S'\|<\|S\|$, and hence $\gamma_t(M(K_n))<\gamma_t(M(K_{n+2}))$. \end{proof} \begin{Theorem}\label{theo:lowerboundtotdomin} Let $G$ be any graph of order $n$. Then $$\lceil \frac{2n}{3} \rceil\le \gamma_t(M(G))\le n-1$$ \end{Theorem} \begin{proof} From $G$ we can obtain graph isomorphic to $K_n$ by adding all the necessary edges. This implies that we can see $G$ as a subgraph of $K_n$, and hence $M(G)$ as a subgraph of $M(K_n)$. Since any total dominating set of $M(G)$ is also a total dominating set for $M(K_n)$, this implies that $\gamma_t(M(G))\ge\gamma_t(M(K_n))$. We obtain the left inequality by Proposition~\ref{prop:mintotdomincompletegr}. Let $T$ be a spanning tree of $G$ and $S$ a minimal total dominating set of $M(T)$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq E(T)$. This implies that $\|S\|\le\|E(T)\|=n-1$. By construction, $S$ is also a total dominating set of $M(G)$, and hence $\gamma_t(M(G))\le n-1$. \end{proof} \begin{Remark} By Propositions~\ref{prop:mintotdominstar} and \ref{prop:mintotdominpath}, the inequalities of Theorem~\ref{theo:lowerboundtotdomin} are all sharp. \end{Remark} \begin{Theorem}\label{prop:graphwithapath} If $G$ is a graph with order $n$ and there exists a subgraph of $G$ isomorphic to $P_n$, then $$\gamma_t(M(G)) = \lceil \frac{2n}{3}\rceil. $$ \end{Theorem} \begin{proof} Since $G$ has a subgraph isomorphic to $P_n$, then $M(G)$ has a subgraph isomorphic to $M(P_n)$. Moreover, any total dominating set of $M(P_n)$ is also a total dominating set for $M(G)$. By Proposition~\ref{prop:mintotdominpath}, this implies that $\gamma_t(M(K_n))\le \lceil \frac{2n}{3} \rceil$. We conclude by Theorem~\ref{theo:lowerboundtotdomin}. \end{proof} Directly from Theorem~\ref{prop:graphwithapath}, we obtain the following result. \begin{Corollary}\label{corol:mintotdominfamily} For any $n\ge3$, $$\gamma_t(M(P_n))=\gamma_t(M(C_n))=\gamma_t(M(W_n))=\gamma_t(M(K_n))=\lceil \frac{2n}{3}\rceil.$$ \end{Corollary} \begin{Definition} The \emph{friendship} graph $F_n$ of order $2n+1$ is obtained by joining $n$ copies of the cycle graph $C_3$ with a common vertex. \end{Definition} \begin{Proposition}\label{prop:mintotdominfriendship} Let $F_n$ be the friendship graph with $n\ge2$. Then $$\gamma_t(M(F_n))= 2n.$$ \end{Proposition} \begin{proof} To fix the notation, assume $V(F_n)=\{v_0,v_1,\dots, v_{2n}\}$ and $E(F_n)=\{v_0v_1,v_0v_2,\dots, v_0v_{2n}\}\cup\{v_1v_2, v_3v_4,\dots,v_{2n-1}v_{2n}\}$. Then $V(M(F_n))=V(F_n)\cup \mathcal{M}$, where $\mathcal{M}=\{ m_i~\|~1\leq i \leq 2n \}\cup\{ m_{i(i+1)}~\|~1\leq i \leq 2n-1 \text{ and } i \text{ is odd}\}$. Consider $S=\{ m_{i(i+1)}~\|~1\leq i \leq 2n-1 \text{ and } i \text{ is odd}\}\cup\{v_i~\|~i\text{ is odd}\}$. Then $S$ is a total dominating set for $M(F_n)$ with $\|S\|=2n$, and hence, $\gamma_t(M(F_n))\le 2n$. On the other hand, since $F_n$ is obtained by joining $n$ copies of $C_3$ at $v_0$, any total dominating set $S$ of $M(F_n)$ induces a total dominating set of $M(C_3)$ as subgraph of $M(F_n)$. By Corollary~\ref{corol:mintotdominfamily}, $\gamma_t(M(C_3))=2$. This fact together with the fact that any two distinct copies of $M(C_3)$ in $M(F_n)$ share only $v_0$ implies that $\|S\|\ge2n$. This implies that $\gamma_t(M(F_n))\ge 2n$. \end{proof} Using Theorem~\ref{theo:lowerboundtotdomin}, we can describe the total domination number of the middle graph of a complete bipartite graph. \begin{Proposition}\label{prop:mintotdomincompletebipartitegr} Let $K_{n_1,n_2}$ be the complete bipartite graph with $n_2\geq n_1 \geq 2$. Then $$\gamma_t(M(K_{n_1,n_2}))= \begin{cases} n_2+\lceil\frac{2n_1-n_2}{3}\rceil & \text{ if $n_1\le n_2\le 2n_1-1$}\\ n_2 & \text{ if $n_2\ge 2n_1$.} \end{cases}$$ \end{Proposition} \begin{proof} Assume $V(K_{n_1,n_2})=\{v_1,\dots, v_{n_1},u_1,\dots, u_{n_2}\}$ and $E(K_{n_1,n_2})=\{v_iu_j ~\|~1\leq i \leq n_1, 1\leq j \leq n_2\}$. Then we have $V(M(K_{n_1,n_2}))=V(K_{n_1,n_2})\cup \mathcal{M}$, where $\mathcal{M}=\{ m_{ij}~\|~1\leq i \leq n_1, 1\leq j \leq n_2 \}$. Assume first $n_1=n_2$. If $n_1\equiv0\mod 3$, then consider $$S=\{m_{11},m_{12},m_{23},m_{33},\dots,m_{(n_1-1)n_1},m_{n_1n_1}\}.$$ By construction, $S$ is a total dominating set of $M(K_{n_1,n_2})$ and $\|S\|=n_1+\frac{n_1}{3}=n_2+\frac{n_1}{3}=n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. If $n_1\equiv1\mod 3$, then consider $$S=\{m_{11},m_{12},m_{23},m_{33},\dots,m_{(n_1-2)(n_1-1)},m_{(n_1-1)(n_1-1)}\}\cup\{m_{n_1(n_1-1)},m_{n_1n_1}\}.$$ By construction, $S$ is a total dominating set of $M(K_{n_1,n_2})$ and $\|S\|=n_1+\lceil\frac{n_1}{3}\rceil=n_2+\lceil\frac{n_1}{3}\rceil=n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. If $n_1\equiv2\mod 3$, then consider $$S=\{m_{11},m_{12},m_{23},m_{33}, \dots, m_{(n_1-1)(n_1-1)},m_{(n_1-1)n_1}\}\cup\{m_{n_1n_1}\}.$$ By construction, $S$ is a total dominating set of $M(K_{n_1,n_2})$ and $\|S\|=n_1+\lceil\frac{n_1}{3}\rceil=n_2+\lceil\frac{n_1}{3}\rceil=n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. Assume that $n_1+1\le n_2\le 2n_1-1$. Consider $$S'=\{m_{11},m_{1n_1+1},\dots, m_{(n_2-n_1)(n_2-n_1)},m_{(n_2-n_1)n_2}\}.$$ Let $G=K_{n_1,n_2}[u_{n_2-n_1+1},\dots,u_{n_1},v_{n_2-n_1+1},\dots,v_{n_1}]$. Then $G$ is isomorphic to a graph of the form $K_{n,n}$, where $n=2n_1-n_2$. This implies that by the first part of the proof, we can construct $S''$ a total dominating set of $M(G)$ with $\|S''\|=2n_1-n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. Consider $S=S'\cup S''$. Then $S$ is a total dominating set of $M(K_{n_1,n_2})$ and $\|S\|=2(n_2-n_1)+2n_1-n_2+\lceil\frac{2n_1-n_2}{3}\rceil= n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. This implies that if $n_1\le n_2\le 2n_1-1$, then $\gamma_t(M(K_{n_1,n_2}))\le n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. Assume now that $n_2\ge 2n_1$. Consider $$S=\{m_{11},m_{1n_1+1},\dots, m_{n_1n_1},m_{n_12n_1}\}\cup\{m_{n_12n_1+1},\dots,m_{n_1n_2}\},$$ then $S$ is a total dominating set of $M(K_{n_1,n_2})$ with $\|S\|=n_2$, and hence, $\gamma_t(M(K_{n_1,n_2}))\le n_2$. On the other hand, assume first $n_1=n_2$. By Theorem~\ref{theo:lowerboundtotdomin}, we have $\gamma_t(M(K_{n_1,n_2}))\ge \lceil\frac{2(n_1+n_2)}{3}\rceil=n_2+\lceil\frac{2n_1-n_2}{3}\rceil.$ Assume that $n_1+1\le n_2\le 2n_1-1$. Let $S$ be a total dominating set of $M(K_{n_1,n_2})$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq \mathcal{M}$. The construction of the first part of the proof is optimal since $S'$ and $S''$ have the smallest possible size by the argument discussed when $n_1=n_2$ and $n_2\ge 2n_1$. This implies that if $n_1\le n_2\le 2n_1-1$, then $\gamma_t(M(K_{n_1,n_2}))= n_2+\lceil\frac{2n_1-n_2}{3}\rceil$. Assume that $n_2\ge 2n_1$, then by \cite[Proposition 3.13]{dominmiddle}, we have $n_2=\gamma(M(K_{n_1,n_2}))\le \gamma_t(M(K_{n_1,n_2}))\le n_2$. This implies that $\gamma_t(M(K_{n_1,n_2}))=n_2$. \end{proof} \section{The middle graph of a tree} Similarly to \cite[Proposition 2.4]{dominmiddle}, if we consider $T$ a tree and we denote by $\leaf(T)=\{v\in V(T)~\|~d_T(v)=1\}$ the set of leaves of $T$, then we have the following result. \begin{Proposition}\label{prop:mintotdomintreeleaf} Let $T$ be a tree with $n\ge2$ vertices. Then $$\gamma_t(M(T))\ge \|\leaf(T)\|.$$ \end{Proposition} \begin{proof} To fix the notation, assume $\leaf(T)=\{v_1,\dots,v_k\}$, for some $k\le n$. If $n=2$, then $T$ is isomorphic to $P_2$ and hence $\gamma_t(M(T))=2= \|\leaf(T)\|$. Assume that $n\ge3$ and let $S$ be a total dominating set of $M(T)$. Then, for each $i=1,\dots, k$, $S\cap N_{M(T)}[v_i]\ne\emptyset$. Since, if $i\ne j$, then $N_{M(T)}[v_j]\cap N_{M(T)}[v_i]=\emptyset$, we have that $\|S\|\ge k$. This implies that $\gamma_t(M(T))\ge k= \|\leaf(T)\|$. \end{proof} \begin{Remark} Notice that by Proposition~\ref{prop:mintotdominstar}, the inequality described in Proposition~\ref{prop:mintotdomintreeleaf} is sharp. \end{Remark} It is sufficient to add some assumptions on the diameter of a tree $T$, to compute $\gamma_t(M(T))$ explicitly. \begin{Theorem}\label{theomintotdomintreediam3} Let $T$ be a tree of order $n\geq 4$ with $\diam(T)=3$. Then $$\gamma_t(M(T))= \begin{cases} n-2 & \text{ if there are two vertices with $d_T(v)\ge3$}\\ n-1 & \text{otherwise.} \end{cases}$$ \end{Theorem} \begin{proof} Since by assumption $\diam(T)=3$, then $T$ is a tree which is obtained by joining central vertex $v$ of $K_{1,p}$ and the central vertex $w$ of $K_{1,q}$ where $p+q=n-2$. Let $\leaf(T)=\{v_i~\|~1\le i\le n-2\}$ be the set of leaves of $T$. Obviously $V(T)=\leaf(T)\cup \{v,w\}$ and $\|\leaf(T)\|=n-2$. Define $v_{n-1}=v$ and $v_{n}=w$. Assume first that $p,q\ge2$, i.e. there are two vertices with $d_T(u)\ge3$. If we consider $S=\{m_{i(n-1)}~\|~1\leq i \leq p \}\cup\{m_{in}~\|~p+1\leq i \leq n-2\}$, then $S$ is a total dominating set of $M(T)$ with $\|S\|=n-2$, and hence $\gamma_t(M(T))\le n-2$. On the other hand, by Proposition~\ref{prop:mintotdomintreeleaf}, we have $\gamma_t(M(T))\ge n-2$. Assume that $p\ge2$ and $q=1$, i.e. there is only one vertex with $d_T(u)\ge3$. Let $S$ be a total dominating set of $M(T)$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq E(T)$. Since $N_{M(T)}(v_i)=\{m_{i(n-1)}\}$ for all $1\le i\le p=n-3$ and $N_{M(T)}(v_{n-2})=\{m_{(n-2)n}\}$, then $\{m_{i(n-1)}~\|~1\le i\le p\}\cup\{m_{(n-2)n}\}\subseteq S$. Moreover, $N_{M(T)}(m_{(n-2)n})=\{m_{(n-1)n},v_n,v_{n-2}\}$ implies that $m_{(n-1)n}\in S$. This implies that $\|S\|\ge n-1$, and hence $\gamma_t(M(T))\ge n-1$. On the other hand, by Theorem~\ref{theo:lowerboundtotdomin}, $\gamma_t(M(T))\le n-1$. Assume that $p,q=1$, i.e. there are no vertices with $d_T(u)\ge3$. This implies that $T$ is isomorphic to $P_4$ and $n=4$, and hence by Proposition~\ref{prop:mintotdominpath}, $\gamma_t(M(T))=3=n-1$. \end{proof} In general, the opposite implication of Theorem~\ref{theomintotdomintreediam3} does not hold as the next example shows. \begin{Example} Let $T$ be the tree in Figure~\ref{Fig:tree}. Then a direct computation shows that $\diam(T)=4$ and $\gamma_t(M(T))=5=n-2$. \begin{figure} \caption{A tree on $7$ vertices} \label{Fig:tree} \end{figure} \end{Example} \begin{Proposition}\label{prop:mintotdomintreediam2} Let $T$ be a tree of order $n\geq 3$ with $\diam(T)=2$. Then $$\gamma_t(M(T))=n-1.$$ \end{Proposition} \begin{proof} Since by assumption $\diam(T)=2$, then $T$ is isomorphic to $K_{1,n-1}$. This implies, by Proposition~\ref{prop:mintotdominstar}, that $\gamma_t(M(T))=n-1$. \end{proof} \begin{Remark}By the proof of Theorem~\ref{theomintotdomintreediam3}, differently from the case of domination (see \cite[Theorem 3.2]{dominmiddle}), $\gamma_t(M(G))=n-1$ does not implies that $G$ is isomorphic to $K_{1,n-1}$. \end{Remark} \section{Operations on graphs} In this section, similarly to \cite{dominmiddle}, we study the total domination number of the middle graph of the corona, $2$-corona and join with $K_p$ of a graph. \begin{Definition} The \emph{corona} $G\circ K_1$ of a graph $G$ is the graph of order $2\|V(G)\|$ obtained from $G$ by adding a pendant edge to each vertex of $G$. \end{Definition} \begin{Example} Consider the graph $P_3$, then the graph $P_3\circ K_1$ is the one in Figure~\ref{fig:pathscorona}. \end{Example} \begin{figure} \caption{The graph $P_3\circ K_1$ } \label{fig:pathscorona} \end{figure} \begin{Theorem}\label{theo:mintotdomincorona} For any connected graph $G$ of order $n\geq 2$, $$\gamma_t(M(G\circ K_1))=n+\gamma(M(G)).$$ \end{Theorem} \begin{proof} To fix the notation, assume $V(G)=\{v_1,\dots, v_n\}$. Then $V(G\circ K_1)= \{v_{1},\dots, v_{2n}\}$ and $E(G\circ K_1)=\{v_1v_{n+1},\dots, v_nv_{2n} \}\cup E(G) $. Then $V(M(G\circ K_1))=V(G\circ K_1)\cup \mathcal{M}$, where $\mathcal{M}=\{ m_{i(n+i)}~\|~1\leq i \leq n \}\cup \{ m_{ij}~\|~v_iv_j\in E(G)\}$. Let $S'$ be a minimal dominating set of $M(G)$. By construction, if we consider $S=S'\cup\{m_{i(n+i)}~\|~1\leq i \leq n\}$, then $S$ is a total dominating set of $M(G\circ K_1)$ with $\|S\|=n+\gamma(M(G))$, and hence $\gamma_t(M(G\circ K_1))\le n+\gamma(M(G))$. On the other hand, let $S$ be a total dominating set of $M(G\circ K_1)$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq \mathcal{M}$. Since $N_{M(G\circ K_1)}(v_{n+i})=\{m_{i(n+i)}\}$, for all $1\le i\le n$, then $m_{i(n+i)}\in S$, for all $1\le i\le n$. In addition, $N_{M(G\circ K_1)}(m_{i(n+i)})=\{v_i,v_{n+i}\}\cup N_{M(G)}(v_i)$, for all $1\le i\le n$, then $N_{M(G)}(v_i)\cap S\ne\emptyset$, for all $1\le i\le n$. This implies that $S\cap E(G)$ is a dominating set of $M(G)$ and hence $\|S\|\ge n+\gamma(M(G))$. This implies that $\gamma_t(M(G\circ K_1))\ge n+\gamma(M(G))$. \end{proof} \begin{Definition} The \emph{$2$-corona} $G\circ P_2$ of a graph $G$ is the graph of order $3\|V(G)\|$ obtained from $G$ by attaching a path of length $2$ to each vertex of $G$ so that the resulting paths are vertex-disjoint. \end{Definition} \begin{Example} Consider the graph $P_3$, then the graph $P_3\circ P_2$ is the one in Figure~\ref{fig:paths2corona}. \end{Example} \begin{figure} \caption{The graph $P_3\circ P_2$ } \label{fig:paths2corona} \end{figure} \begin{Theorem}\label{theomintotdomin2corona} For any connected graph $G$ of order $n\geq 2$, $$\gamma_t(M(G\circ P_2))=2n.$$ \end{Theorem} \begin{proof} To fix the notation, assume $V(G)=\{v_1,\dots, v_n\}$. Then $V(G\circ P_2)= \{v_{1},\dots, v_{3n}\}$ and $E(G\circ P_2)=\{v_iv_{n+i}, v_{n+i}v_{2n+i}~\|~1\leq i \leq n \}\cup E(G) $. Then $V(M(G\circ P_2))=V(G\circ P_2)\cup \mathcal{M}$, where $\mathcal{M}=\{ m_{i(n+i)},m_{(n+i)(2n+i)}~\|~1\leq i \leq n \}\cup \{ m_{ij}~\|~v_iv_j\in E(G)\}$. Let $S$ be a total dominating set of $M(G\circ P_2)$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume that $S\subseteq\mathcal{M}$. Since $N_{M(G\circ P_2)}(v_{2n+i})=\{m_{(n+i)(2n+i)}\}$, for every $1\leq i\leq n$, we have $m_{(n+i)(2n+i)}\in S$ for every $1\leq i\leq n$. In addition, $N_{M(G\circ P_2)}(m_{(n+i)(2n+i)})=\{m_{i(n+i)},v_{2n+i},v_{n+i}\}$, for every $1\leq i\leq n$, implies that $m_{i(n+i)}\in S$, for every $1\leq i\leq n$. This implies that $\|S\|\ge 2n$, and hence $\gamma_t(M(G\circ P_2))\ge2n$. On the other hand, if we consider $S=\{ m_{i(n+i)},m_{(n+i)(2n+i)}~\|~1\leq i \leq n \}$, then $S$ is a total dominating set of $M(G\circ P_2)$ with $\|S\|=2n$, and hence $\gamma_t(M(G\circ P_2))\le2n$. \end{proof} \begin{Definition} The \emph{join} $G+ H$ of two graphs $G$ and $H$ is the graph with vertex set $V(G+H)=V(G)\cup V(H)$ and edge set $E(G+H)=E(G)\cup E(H)\cup \{vw~\|~v\in V(G), w\in V(H)\}$. \end{Definition} \begin{Example} Consider the graphs $G=K_3$ and $H=P_2$, then graph $G+H$ is the one in Figure~\ref{fig:k3plusp2}. \end{Example} \begin{figure} \caption{The graph $K_3+P_2$ } \label{fig:k3plusp2} \end{figure} \begin{Theorem}\label{theo:mindominjoinpbig} For any connected graph $G$ of order $n\geq 2$, $$\gamma_t(M(G+\overline{K_p}))= \begin{cases} p & \text{ if $p\geq 2n$}\\ \lceil\frac{2(n+p)}{3}\rceil & \text{ if $ \frac{n}{2}\le p\le 2n-1$.} \end{cases}$$ \end{Theorem} \begin{proof} To fix the notation, assume $V(G)=\{v_1,\dots,v_n\}$ and $V(\overline{K_p})=\{v_{n+1},\dots,v_{n+p}\}$. Then $V(M(G+\overline{K_p}))=V(G+\overline{K_p})\cup \mathcal{M}_1\cup \mathcal{M}_2$ where $\mathcal{M}_1= \{m_{ij}~\|~v_iv_j\in E(G)\}$ and $\mathcal{M}_2= \{m_{i(n+j)}~\|~1\leq i \leq n, 1\leq j \leq p\}$. \textbf{Case {\boldmath$p\geq 2n$}.} Let $S$ be a total dominating set of $M(G+\overline{K_p})$. By Lemma~\ref{lemma:totdominationisalledges}, we can assume $S\subseteq \mathcal{M}_1\cup \mathcal{M}_2$. Since, if $j\ne k$, $N_{M(G+ \overline{K_p})}(v_{n+j})\cap N_{M(G+ \overline{K_p})}(v_{n+k})=\emptyset$, then for every $1\le j\le p$ there exists $1\le i\le n$ such that $m_{i(n+j)}\in S$, and hence $\|S\|\ge p$. This implies that $\gamma_t(M(G+\overline{K_p}))\ge p$. On the other hand, if we consider $S=\{m_{i(n+i)},m_{i(2n+i)}~\|~1\le i\le n\}\cup\{m_{1(3n+i)}~\|~1\le i\le p-2n\}$, then $S$ is a total dominating set of $M(G+\overline{K_p})$ with $\|S\|=p$, and hence $\gamma_t(M(G+\overline{K_p}))\le p$. \textbf{Case {\boldmath$p= 2n-1$}.} If we consider $S=\{m_{i(n+i)},m_{i(2n+i)}~\|~1\le i\le n-1\}\cup\{m_{n(2n)},m_{n(2n-1)}\}$, then $S$ is a total dominating set of $M(G+\overline{K_{2n-1}})$ with $\|S\|=2n$, and hence $\gamma_t(M(G+\overline{K_{2n-1}}))\le 2n$. On the other hand, by Theorem~\ref{theo:lowerboundtotdomin}, $\gamma_t(M(G+\overline{K_{2n-1}}))\ge \lceil\frac{2(3n-1)}{3}\rceil =2n$, and hence $\gamma_t(M(G+\overline{K_{2n-1}}))= 2n=\lceil\frac{2(n+p)}{3}\rceil $. \textbf{Case {\boldmath$n+3\le p\le 2n-2$}.} Assume that $p=n+k$ with $3\le k\le n-2$. The graph $G+\overline{K_p}$ has $k$ subgraphs isomorphic to $P_3$ and one subgraph isomorphic to $P_{2(n-k)}$ that are all disjoint. In fact, the $k$ subgraphs $(G+\overline{K_p})[v_1,v_{n+1},v_{2n+1}], \dots, (G+\overline{K_p})[v_k, v_{n+k},v_{2n+k}]$ are all isomorphic to $P_3$ and the subgraph $(G+\overline{K_p})[v_{k+1},\dots, v_{n},v_{n+k+1},\dots, v_{2n}]$ has a subgraph isomorphic to $P_{2(n-k)}$. By Proposition~\ref{prop:mintotdominpath}, this implies that $\gamma_t(M(G+\overline{K_p}))\le 2k+\lceil\frac{2(2(n-k))}{3}\rceil=\lceil\frac{2(n+p)}{3}\rceil$. By Theorem~\ref{theo:lowerboundtotdomin}, we obtain the desired equality. \textbf{Case {\boldmath$p= n+2$}.} If $n\equiv 0 \mod 3$, consider $$S=\{m_{1(n+1)},m_{1(n+2)},m_{2(n+3)},m_{3(n+3)},\dots, m_{(n-1)(2n)}, m_{n(2n)}, m_{n(2n+1)}, m_{n(2n+2)}\}.$$ Then $S$ is a total dominating set of $M(G+\overline{K_p})$ with $\|S\|=\lceil\frac{2(n+p)}{3}\rceil$. If $n\equiv 1 \mod 3$, consider $$S=\{m_{1(n+1)},m_{1(n+2)},m_{2(n+3)},m_{3(n+3)},\dots, m_{n(2n)}, m_{n(2n+1)}, m_{n(2n+2)}\}.$$ Then $S$ is a total dominating set of $M(G+\overline{K_p})$ with $\|S\|=\lceil\frac{2(n+p)}{3}\rceil$. If $n\equiv 2 \mod 3$, consider $$S=\{m_{1(n+1)},m_{1(n+2)},m_{2(n+3)},m_{3(n+3)},\dots, m_{n(2n+1)}, m_{n(2n+2)}\}.$$ Then $S$ is a total dominating set of $M(G+\overline{K_p})$ with $\|S\|=\lceil\frac{2(n+p)}{3}\rceil$. This implies that $\gamma_t(M(G+\overline{K_p}))\le \lceil\frac{2(n+p)}{3}\rceil$. By Theorem~\ref{theo:lowerboundtotdomin}, we then obtain that $\gamma_t(M(G+\overline{K_p}))= \lceil\frac{2(n+p)}{3}\rceil$. \textbf{Case {\boldmath$n-1\le p\le n+1$}.} If $p=n-1$, then the graph $G+\overline{K_p}$ contains the path $P: v_1v_{n+1}v_2v_{n+2}\cdots v_{n+p}v_n$. If $p=n$, then $G+\overline{K_p}$ contains the path $P': v_1v_{n+1}v_2v_{n+2}\cdots v_{n+p-1}v_nv_{n+p}$. If $p=n+1$, then $G+\overline{K_p}$ contains the path $P'': v_{n+1}v_1v_{n+2}v_2v_{n+3}\cdots v_{n+p-1}v_nv_{n+p}$. Since the paths $P, P'$ and $P''$ are all isomorphic to $P_{n+p}$, we can apply Theorem~\ref{prop:graphwithapath}, and obtain that $\gamma_t(M(G+\overline{K_p}))= \lceil\frac{2(n+p)}{3}\rceil$. \textbf{Case {\boldmath$\frac{n}{2}\le p\le n-2$}.} Assume that $p=n-k$ with $2\le k\le \frac{n}{2}$. If $n$ is even and $p=\frac{n}{2}$ (or equivalently $k=\frac{n}{2}$), then the set $S=\{m_{i(n+i)}, m_{(i+\frac{n}{2})(n+i)}~\|~1\le i\le \frac{n}{2}\}$ is a total dominating set of $M(G+\overline{K_p})$ with $\|S\|=n=\lceil\frac{2(n+p)}{3}\rceil$. This implies that $\gamma_t(M(G+\overline{K_p}))\le \lceil\frac{2(n+p)}{3}\rceil$, and by Theorem~\ref{theo:lowerboundtotdomin}, we obtain the desired equality. Assume that $2\le k\le \frac{n}{2}-1$. The graph $G+\overline{K_p}$ has $k$ subgraphs isomorphic to $P_3$ and one subgraph isomorphic to $P_{2(n-2k)}$ that are all disjoint. In fact, the $k$ induced subgraphs $(G+\overline{K_p})[v_1,v_{n+1},v_{k+1}], \dots, (G+\overline{K_p})[v_k, v_{n+k},v_{2k}]$ are all isomorphic to $P_3$ and the induced subgraph $(G+\overline{K_p})[v_{2k+1},\dots, v_{n},v_{n+k+1},\dots, v_{2n-k}]$ has a subgraph isomorphic to $P_{2(n-2k)}$. By Proposition~\ref{prop:mintotdominpath}, this implies that $\gamma_t(M(G+\overline{K_p}))\le 2k+\lceil\frac{2(2(n-2k))}{3}\rceil=\lceil\frac{2(n+p)}{3}\rceil$. By Theorem~\ref{theo:lowerboundtotdomin}, we obtain the desired equality. \end{proof} Similarly to \cite{dominmiddle}, when $p$ is small relatively to $n$, $\gamma_t(M(G+\overline{K_p}))$ is strongly related to $\gamma_t(M(G))$. \begin{Theorem}\label{theo:mindominjoinverysmallrange} For any connected graph $G$ of order $n\geq 2$ and any integer $1\le p\le \frac{n}{2}-1$, $$\lceil\frac{2(n+p)}{3}\rceil\le \gamma_t(M(G+\overline{K_p}))\le$$ $$2p+\min\{\gamma_t(M(G[A]))~\|~A\subseteq V(G), $$ $$\|A\|=n-2p, G[A] \text{ has no isolated vertices}\}.$$ \end{Theorem} \begin{proof} To fix the notation, assume $V(G)=\{v_1,\dots,v_n\}$ and $V(\overline{K_p})=\{v_{n+1},\dots,v_{n+p}\}$. Then $V(M(G+\overline{K_{p}}))=V(G+\overline{K_{p}})\cup \mathcal{M}_1\cup \mathcal{M}_2$ where $\mathcal{M}_1= \{m_{ij}~\|~v_iv_j\in E(G)\}$ and $\mathcal{M}_2= \{m_{i(n+j)}~\|~1\leq i \leq n, 1\leq j \leq p\}$. By Theorem~\ref{theo:lowerboundtotdomin}, we obtain the first inequality. Let now $A\subseteq V(G)$ be such that $\|A\|=n-2p$ and $G[A]$ has no isolated vertices. Without loss of generalities, we can assume that $A=\{v_{2p+1},\dots, v_{n}\}$. Consider $S'$ be a minimal total dominating set of $M(G[A])$, then $S=S'\cup \{m_{i(n+i)},m_{(p+i)(n+i)}~\|~1\le i\le p\}$ is a total dominating set of $M(G+\overline{K_p})$. Since this arguments works for every $A\subseteq V(G)$ such that $\|A\|=n-2p$ and $G[A]$ has no isolated vertices, we obtain the second inequality. \end{proof} If we apply Lemma~\ref{lemma:totdominationdeletionvertex} to the graph $G+\overline{K_1}$, we obtain the following result. \begin{Lemma}\label{lemma:plusonevertextotaldomin} Let $G$ be a graph of order $n\geq 2$ with no isolated vertices. Then $$\gamma_t(M(G))\le \gamma_t(M(G+\overline{K_1}))\le \gamma_t(M(G))+1.$$ \end{Lemma} Notice that both inequalities described in Lemma~\ref{lemma:plusonevertextotaldomin} are sharp as the following examples show. \begin{Example} Consider the graph $G=C_5$. Then $G+\overline{K_1}$ is isomorphic to $W_6$. This implies that by Corollary~\ref{corol:mintotdominfamily}, $\gamma_t(M(G))=4=\gamma_t(M(G+\overline{K_1}))$. \end{Example} \begin{figure} \caption{The graph $P_3+\overline{K_1}$} \label{Fig:sumwithK1} \end{figure} \begin{Example} Consider the graph $G=P_3$. Then $G+\overline{K_1}$ is the graph in Figure~\ref{Fig:sumwithK1}. By Proposition~\ref{prop:mintotdominpath} and Theorem~\ref{prop:graphwithapath}, we have that $\gamma_t(M(P_3))=2$ and $\gamma_t(M(P_3+\overline{K_1}))=3$. \end{Example} \begin{Proposition}\label{prop:joinstargraphk1} For any star graph $K_{1,n}$ on $n+1$ vertices, with $n\ge 4$, we have $$\gamma_t(M(K_{1,n}+\overline{K_1}))=n.$$ \end{Proposition} \begin{proof} To fix the notation, assume $V(K_{1,n})=\{v_0,v_1,\dots, v_n\}$, $V(\overline{K_1})=\{v_{n+1}\}$ and $E(K_{1,n})=\{v_0v_1,v_0v_2,\dots, v_0v_n\}$. Then $V(M(K_{1,n}+\overline{K_1}))=V(K_{1,n})\cup \mathcal{M}$, where $\mathcal{M}=\{ m_i~\|~1\leq i \leq n \}\cup\{m_{i(n+1)}~\|~0\le i\le n\}$. By Proposition~\ref{prop:mintotdominstar} and Lemma~\ref{lemma:plusonevertextotaldomin}, $\gamma_t(M(K_{1,n}+\overline{K_1}))\ge n$. On the other hand, consider $S=\{m_i~\|~1\le i\le n-2\}\cup\{m_{(n-1)(n+1)},m_{n(n+1)}\}$ is a total dominating set of $M(K_{1,n}+\overline{K_1})$ with $\|S\|=n$, and hence $\gamma_t(M(K_{1,n}+\overline{K_1}))\le n$. \end{proof} \begin{Remark} Proposition~\ref{prop:joinstargraphk1} shows that the upper bound of Theorem~\ref{theo:mindominjoinverysmallrange} is sharp. In fact, if $A\subseteq V(K_{1,n})$ with $\|A\|=n-2$ and $G[A]$ has no isolated vertices, then $G[A]$ is isomorphic to $K_{1,n-2}$, and hence by Proposition~\ref{prop:mintotdominstar}, $\gamma_t(M(G[A]))=n-2$. \end{Remark} \begin{Proposition}\label{prop:withpathmintotdominsumKp} Let $G$ be a graph of order $n\ge 2$ and $1\le p\le \frac{n}{2}-1$. If $G$ has a subgraph isomorphic to a path graph $P_n$, then $$\gamma_t(M(G+\overline{K_p}))=\lceil\frac{2(n+p)}{3}\rceil.$$ \end{Proposition} \begin{proof} Under our assumption, the graph $G+\overline{K_p}$ contains a subgraph isomorphic to $P_{n+p}$. By Theorem~\ref{prop:graphwithapath}, this implies that $\gamma_t(M(G+\overline{K_p}))=\lceil\frac{2(n+p)}{3}\rceil.$ \end{proof} As a direct consequence of Proposition~\ref{prop:withpathmintotdominsumKp}, we obtain the following result. \begin{Corollary}\label{cor:knownfamilymintotdominsumKp} Let $G$ be a graph of order $n\ge 2$ and $1\le p\le \frac{n}{2}-1$. If $G$ is isomorphic to a path graph $P_n$, or a cycle graph $C_n$, or a wheel graph $W_n$, or a complete graph $K_n$, then $$\gamma_t(M(G+\overline{K_p}))=\lceil\frac{2(n+p)}{3}\rceil.$$ \end{Corollary} \section{Nordhaus-Gaddum relations} In \cite{Nordhaus}, Nordhaus and Gaddum gave a lower bound and an upper bound, in terms of the order of the graph, on the sum and the product of the chromatic number of a graph and its complement. Since then, lower and upper bounds on the sum and the product of many other graph invariants, like domination and total domination numbers, have been proposed by several authors. See \cite{Aouchiche13} for a survey on the subject. \begin{Theorem}\label{theo:Nordgadtotdomin} Let $G$ be a graph on $n\ge2$ vertices. Assume that the graphs $G$ and $\overline{G}$ have no isolated vertices and no components isomorphic to $K_2$. Then $$2(n-1)\ge \gamma_t(M(G))+\gamma_t(M(\overline{G}))\ge 2\lceil\frac{2n}{3}\rceil $$ and $$(n-1)^2\ge \gamma_t(M(G))\cdot\gamma_t(M(\overline{G}))\ge (\lceil\frac{2n}{3}\rceil)^2. $$ \end{Theorem} \begin{proof} By applying Theorem~\ref{theo:lowerboundtotdomin} to each component of $G$ and $\overline{G}$, we obtain that $n-1\ge\gamma_t(M(G))\ge \lceil\frac{2n}{3}\rceil $ and $n-1\ge\gamma_t(M(\overline{G}))\ge \lceil\frac{2n}{3}\rceil$. \end{proof} \begin{Remark} If in Theorem~\ref{theo:Nordgadtotdomin} we allow $G$ or $\overline{G}$ to have components isomorphic to $K_2$, then the described upper bounds might not work. To see this it is enough to consider the graph $C_4$. In fact, $\overline{C_4}$ consists of two copies of $K_2$, and then $\gamma_t(M(C_4))=3$ and $\gamma_t(M(\overline{C_4}))=4$. \end{Remark} Notice that all the inequalities of Theorem~\ref{theo:Nordgadtotdomin} are sharp, in fact we have the following example. \begin{Example} Consider the graph $P_4$, then by Proposition~\ref{prop:mintotdominpath}, we have $\gamma_t(M(P_4))=3$. On the other hand, $\overline{P_4}$ is isomorphic to $P_4$, and hence $\gamma_t(M(\overline{P_4}))=3$. Since $n=4$, then $6=\gamma_t(M(P_4))+\gamma_t(M(\overline{P_4}))=2(n-1)=2\lceil\frac{2n}{3}\rceil$, and $9=\gamma_t(M(P_4))\cdot\gamma_t(M(\overline{P_4}))=(n-1)^2=(\lceil\frac{2n}{3}\rceil)^2$. \end{Example} \paragraph{\textbf{Acknowledgements}} During the preparation of this article the fourth author was supported by JSPS Grant-in-Aid for Early-Career Scientists (19K14493). \end{document}	arXiv
The Fibonacci sequence is defined $F_1 = F_2 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for all $n \ge 3.$ The Fibonacci numbers $F_a,$ $F_b,$ $F_c$ form an increasing arithmetic sequence. If $a + b + c = 2000,$ compute $a.$ We claim that if $F_a,$ $F_b,$ $F_c$ form an increasing arithmetic sequence, then $(a,b,c)$ must be of the form $(n,n + 2,n + 3)$ for some positive integer $n.$ (The only exception is $(2,3,4).$) From $F_c - F_b = F_b - F_a,$ we get \[F_c = F_b + (F_b - F_a) < F_b + F_{b + 1} = F_{b + 2}.\]Also, $F_c > F_b.$ Therefore, $F_c = F_{b + 1}.$ Then \begin{align} F_a &= 2F_b - F_c \\ &= 2F_b - F_{b + 1} \\ &= F_b - (F_{b + 1} - F_b) \\ &= F_b - F_{b - 1} \\ &= F_{b - 2}. \end{align}Then $a$ must be equal to $b - 2$ (unless $b = 3,$ which leads to the exceptional case of $(2,3,4)$). Taking $n = b - 2,$ we get $(a,b,c) = (n,n + 2,n + 3).$ Then $a + (a + 2) + (a + 3) = 2000,$ so $a = \boxed{665}.$	Math Dataset
NERSC Annual Reports HPC Requirements for Science HPC Workshop Reports Home » News & Publications » Publications & Reports » NERSC User Publications » 2007 2007 Publications Resulting from the Use of NERSC Resources On their Allocation Year 2008 ERCAP Request Forms Principal Investigators reported 1,464 refereed publications (published or submitted) for the preceding 12 months, based on using, at least in part, NERSC resources. PI Mowfak Al-Jassim Y. Yan, J. L. F. Da Silva, S.-H. Wei, and M. Al-Jassim, "Atomic structure of In2O3ZnO system" Appl. Phys. Lett. 90, 261904 (2007). Y. Yan, J. Li, S.-H. Wei, and M. Al-Jassim, A possible approach to overcome doping asymmetry in wide bandgap semiconductor" Phys. Rev. Lett. 98, 135506 (2007). Y. Yan, M.M. Al-Jassim, and S.-H. Wei, "Doping of ZnO with group-IB elements" Appl. Phys. Lett. 89, 181912 (2006). Y. Yan, R. Noufi, and M.M. Al-Jassim, "Grain-Boundary Physics in Polycrystalline CuInSe2 Revisited: Experiment and Theory" Phys. Rev. Lett. 96, 205501 (2006) G. M. Dalpian, Y. Yan, and S.-H. Wei, "Impurity-induced phase stabilization of semiconductors" Appl. Phys. Lett. 89, 011907 (2006) PI Greg Aldering SN2005ap: A Most Brilliant Explosion, Quimby, R. M., Aldering, G., Wheeler, C., Akerloff, C. W., and Rykoff, E. S. (2007), Astrophysical Journal, 668, 99. How to Find More Supernovae with Less Work: Object Classification Techniques for Difference Imaging Bailey, S., Aragon, C., Romano, R., Thomas, R. C., Weaver, B. A., and Wong, D. (2007), Astrophysical Journal, 665, 1246. Snapping Supernovae at z>1.7, Aldering, G., Kim, A. G., Kowalski, M., Linder, E. V., Perlmutter, S. (2007), Astroparticle Physics, 27, 213-225. Nearby Supernova Factory Observations of SN 2006D: On Sporadic Carbon Signatures in Early Type Ia Supernova Spectra, Thomas, R. C., Aldering, G., Antilogus, P., Aragon, C., Bailey, S., Baltay, C., Baron, E., Bauer, A., Buton, C., Bongard, S., Copin, Y., Gangler, E., Gilles, S., Kessler, R., Loken, S., Nugent, P., Pain, R., Parrent, J., Pecontal, E., Pereira, R., Perlmutter, S., Rabinowitz, D., Rigaudier, G., Runge, K., Scalzo, R., Smadja, G., Wang, L., Weaver, B. A. (2007), Astrophysical Journal, 654, L53-L56. On the Progenitor of SN 2005gl, the Nature of Type IIn Supernovae, Gal-Yam, A., Leonard, D. C., Fox, D. B., Cenko, S. B., Soderberg, A. M., Moon, D.-S., Sand, D. J., Li, W., Filippenko, A. V., Aldering, G., Copin, Y., (2007), Astrophysical Journal, 656, 372-381. Visible, near-infrared spectrophotometry of the Deep Impact ejecta of Comet 9P/Tempel 1, Hodapp, K. W., Aldering, G., Meech, K. J., Cochran, A. L., Antilogus, P., Pecontal, E., Chickering, W., Blanc, N., Copin, Y., Lynch, D. K., Rudy, R. J., Mazuk, S., Venturini, C. C., Puetter, R. C., Perry, R. B., (2007), Icarus, 187, 185-198. Nearby Supernova Factory Observations of SN2005gj: Another Type Ia Supernova in a Massive Circumstellar Envelope, Aldering, G., Antilogus, P., Bailey, S., Baltay, C., Bauer, A., Blanc, N., Bongard, S., Copin, Y., Gangler, E., Gilles, S., Kessler, R., Kocevski, D., Lee, B. C., Loken, S., Nugent, P., Pain, R., Pecontal, E., Pereira, R., Perlmutter, S., Rabinowitz, D., Rigaudier, G., Scalzo, R., Smadja, G., Thomas, R. C., Wang, L., Weaver, B. A. (2006), Astrophysical Journal, 650, 510. The Supernova Legacy Survey: measurement of Omega_M, Lambda, w from the first year data set, Astier, P., Guy, J., Regnault, N., Pain, R., Aubourg, E., Balam, D., Basa, S., Carlberg, R.~G., Fabbro, S., Fouchez, D., Hook, I.~M., Howell, D. A., Lafoux, H., Neill, J. D., Palanque-Delabrouille, N., Perrett, K., Pritchet, C. J., Rich, J., Sullivan, M., Taillet, R., Aldering, G., Antilogus, P., Arsenijevic, V., Balland, C., Baumont, S., Bronder, J., Courtois, H., Ellis, R. S., Filiol, M., Goncalves, A. C., Goobar, A., Guide, D., Hardin, D., Lusset, V., Lidman, C., McMahon, R., Mouchet, M., Mourao, A., Perlmutter, S., Ripoche, P., Tao, C., and Walton, N. (2006), Astronomy & Astrophysics, 447, 31. Improvements to the Image Processing of Hubble Space Telescope NICMOS Observations with Multiple Readouts, Fadeyev, V., Aldering, G., and Perlmutter, S. (2006), Publications of the Astronomical Society of the Pacific, 118, 907. Measurement of Omega_M, Lambda from a Blind Analysis of Type Ia Supernovae with CMAGIC: Using Color Information to Verify the Acceleration of the Universe, Conley, A., et al, (2006), Astrophysical Journal, 644, 1. Photometric Selection of High-Redshift Type Ia Supernova Candidates, Sullivan, M., Howell, D. A., Perrett, K., Nugent, P. E., Astier, P., Aubourg, E., Balam, D., Basa, S., Carlberg, R. G., Conley, A., Fabbro, S., Fouchez, D., Guy, J., Hook, I., Lafoux, H., Neill, J. D., Pain, R., Palanque-Delabrouille, N., Pritchet, C. J., Regnault, N., Rich, J., Taillet, R., Aldering, G., Baumont, S., Bronder, J., Filiol, M., Knop, R. A., Perlmutter, S., and Tao, C., (2006), Astronomical Journal, 131, 960. The Type Ia Supernova Rate at z ~0.5 from the Supernova Legacy Survey, Neill, J. D., et al (2006), Astronomical Journal, 132, 1126. PI Yoram Alhassid "Spin Projection in the Shell Model Monte Carlo Method and the Spin Distribution of Nuclear Level Densities," Y. Alhassid, S. Liu, and H. Nakada, Phys. Rev. Lett. 99, 162504 (2007). "Scrambling of Hartree-Fock Levels as a universal Brownian-Motion Process," Y. Alhassid, H. A. Weidenmueller, and A. Wobst, arXiv: cond-mat/0604630, Phys. Rev. B, in press (2007). "Thermodynamics of Ultra-Small Metallic Grains in the Auxiliary-Field Monte Carlo Approach," Y. Alhassid, L. Fang, and S. Schmidt, arXiv: cond-mat/0702304 (2007). "The Shell Model Monte Carlo Approach to Heavy Deformed Nuclei," Y. Alhassid, L. Fang and H. Nakada, arXive:0710.1656 (2007). PI Paul Alivisatos S. Y. Quek, L. Venkataraman, H. J. Choi, S. G. Louie, M. S. Hybertsen, and J. B. Neaton "Amine-Gold Linked Single-Molecule Circuits: Experiment and Theory", Nano Letters, in press (2007) K. H. Khoo, J. B. Neaton, H. J. Choi, and S. G. Louie, "Contact dependence of the conductance of H2 molecular junctions from first principles", Phys. Rev. B, in press (2007) S. Y. Quek, J. B. Neaton, M. S. Hybertsen, E. Kaxiras, and S. G. Louie, "Negative Differential Resistance in Transport through Organic Molecules on Silicon", Phys. Rev. Lett. 98, 066807 (2007) L. Kong, J. R. Chelikowsky, J. B. Neaton, and S. G. Louie, "Real-space ab initio calculations of spin-dependent transport properties of magnetic molecular junctions", Phys. Rev. B, in press (2007) K. H. Khoo, J. B. Neaton, Y. W. Son, M. L. Cohen, and S. G. Louie, "Negative differential resistance in carbon wires covalently bridging carbon nanotubes", Phys. Rev. B, submitted (2007) PI Jonathan Arons A. Spitkovsky, 2007b, "Acceleration of Particles in Unmagnetized Electron-Positron Collisionless Shocks," ApJ Lett, submitted A. Spitkovsky, 2007a, "On the structure of relativistic collisionless shocks in electron-ion plasmas," ApJ Lett, accepted, arXiv:0706.3126 P. Chang, A. Spitkovsky, J. Arons, 2007, "Long Term Evolution of Magnetic Turbulence in Relativistic Collisionless Shocks: Electron-Positron Plasmas," ApJ, accepted, arXiv:0704.3832 E. Amato & J. Arons, 2006, "Heating and Non-thermal Particle Acceleration in Relativistic, Transverse Magnetosonic Shock Waves in Proton-Electron-Positron Plasmas", ApJ, 653, 325 Bucciantini, N., Quataert, E., Arons, J., Metzger, B. D., Thompson, T. A., 2007, "Magnetar-driven bubbles and the origin of collimated outflows in gamma-ray bursts," MNRAS, 380, 1541 PI Cynthia Atherton K. Ellingsen, M. Gauss, R. Van Dingenen, F. J. Dentener, L. Emberson3, A. M. Fiore, M. G. Schultz, D. S. Stevenson, C. S. Atherton, D. J. Bergmann, I. Bey, T. Butler, J. Drevet, H. Eskes, D. Hauglustaine, I. S. A. Isaksen, L. W. Horowitz, M. Krol, J. F. Lamarque, M. Lawrence, T. van Noije, J. Pyle, S. Rast, J. Rodriguez, N. Savage, S. Strahan, K. Sudo, S. Szopa, O. Wild, 2007: Global ozone air quality: a multi-model assessment of risks to human health and crops, submitted to Atmospheric Chemistry and Physics Discussions. PI Ahmet Aydemir A. Y. Aydemir, Shear flows at the tokamak edge and their role in core rotation and the L-H transition, Phys. Rev. Lett, 98, 225002 (2007) A. Y. Aydemir, Shear flows at the tokamak edge and their interaction with edge-localized modes, Phys. Plasmas, 14, 056118 (2007). PI Dmitri Babikov M. Zhao and D. Babikov, "On optimal control of adiabatic motion of ions in a trap", submitted to Phys. Rev. Let., 2007. E. Vetoshkin and D. Babikov, "Semiclassical wavepacket study of anomalous isotope effect in ozone formation", accepted by J. Chem. Phys., 2007. E. Vetoshkin and D. Babikov, "Semi-classical wavepacket treatment of scattering resonances: Application to the delta zero-point energy effect in recombination reactions", Phys. Rev. Lett. 99, 138301, 2007. M. Zhao and D. Babikov, "Anharmonic properties of the vibrational quantum computer", J. Chem. Phys. 126, 204102 (12 pages), 2007. Baer, Ferdinand mp188 PI Ferdinand Baer Baer, F., H. Wang, J. J. Tribbia, A. Fournier, 2006: Climate Modeling With Spectral Elements. Mon. Wea. Rev., V.134, 3610-3624. Wang, H., J. J. Tribbia, F. Baer, A. Fournier, M. Taylor, 2007: A Spectral Element Version of CAM2. Mon. Wea. Rev., V.135, No. 11, in press. Christopher D. Blakely, 2007:"A Hybrid Meshless/Spectral-Element Method for Elliptic Partial Differential Equations". International Journal of Comp. Math., accepted, 2006. Christopher D. Blakely, 2007: "A Hybrid Meshless/Spectral-Element Method for the Shallow Water Equations on the Sphere", Computational Methods in Applied Sciences, ed. Eugenio Onate, Springer-Verlag, in press. Christopher D. Blakely, A. Gelb, A. Navarra, 2007: "An Automated Method for Recovering Piecewise Smooth Functions on Spheres Free of Gibbs Oscillations". Sampling Theory in Signal and Image Processing, Vol. 6, No.3. PI David Bailey J. H. Chen, A. Choudhary, B. de Supinski, M. DeVries, E. R. Hawkes, S. Klasky, W. K. Liao, K. L. Ma, J. Mellor-Crummy, N. Podhorski, R.Sankaran, S. Shende, C. S. Yoo, "Terascale Direct Numerical Simulations of Turbulent Combustion using S3D," (to appear) IOP Journal, 2008. M. Tikir, L. Carrington, E. Strohmaier, A. Snavely, "A Genetic Algorithms Approach to Modeling the Performance of Memory-bound Computations", Proceedings of SC07, the International Conference for High Performance Computing, Networking, and Storage), November 2007, Reno, NV, to appear. A. Tiwari, J. K. Hollingsworth, "Tuning Parallel Programs in Parallel" (submitted to IPDPS'08). Y. Zhang, F. Fowler, K. Huck, A. Malony, A. Porterfield, D. Reed, S. Shende, V. Taylor and X. Wu, "USQCD Computational Performance Studies with PERI", SciDAC 2007, Journal of Physics: Conference Series, vol. 78, 2007. Xingfu Wu and Valerie Taylor, "Processor Partitioning: An Experimental Performane Analysis of Parallel Applications on SMP Cluster Systems", 19th IASTED International Conference on parallel and Distributed Computing and Systems (PDCS 2007), Nov. 2007, MIT, Cambridge, MA. David H. Bailey, David Borwein, Jonathan M. Borwein and Richard Crandall, "Hypergeometric Forms for Ising-Class Integrals," Experimental Mathematics, to appear, Jul 2006; LBNL-61037. David H. Bailey and Jonathan M. Borwein, "Finding General Explicit Formulas for Ising Integral Recursions," manuscript, Oct 2006. David H. Bailey, Richard E. Crandall and Jonathan M. Borwein, "Resolution of the Quinn-Rand-Strogatz Constant of Nonlinear Physics," manuscript, Jun 2007. David H. Bailey, Jonathan M. Borwein, Richard E. Crandall and Deirdre K. Manna, "New Representations for Spin Integrals," manuscript, Aug 2007. Natalie J. Durgin, Sofia M. Garcia, Tamara Flournoy and David H. Bailey, "`Syncing' Up with the Quinn-Rand-Strogatz Constant: Hurwitz-Zeta Functions in Non-Linear Physics," manuscript, Sep 2007. PI Krishnan Balasubramanian Theoretical Studies on Structures of Neptunyl Carbonates: NpO2(CO3)m(H2O)nq_ (m=1-3, n=0-3) in Aqueous Solution K. Balasubramaniana, and Zhiji Cao, Inorganic Chemistry,(2007) in press PI Perla Balbuena A. Martinez-Limia, J. Zhao, and P. B. Balbuena, Molecular dynamics study of the initial stages of catalyzed single-wall carbon F. Tarazona-Vasquez and P. B. Balbuena, "Pt(II) uptake by dendrimer outer pockets: Modeling binding of Pt(II) in large dendrimers," J. Phys. Chem.B, submitted, August 1, 2007. F. Tarazona-Vasquez and P. B. Balbuena, "Pt(II) uptake by dendrimer outer pockets: 2. Solvent-mediated complexation," J. Phys. Chem. B., submitted, August 1, 2007. F. Tarazona-Vasquez and P. B. Balbuena, "Pt(II) uptake by dendrimer outer pockets: 1. Solventless ligand exchange reaction," J. Phys. Chem. B., submitted, August 1, 2007. Y. Ma and P. B. Balbuena, "Pt-surface segregation in bimetallic Pt3M alloys: A density functional theory study," Surf. Sci.., in press. Y. Ma and P. B. Balbuena, "OOH dissociation on Pt clusters," Chem. Phys. Lett., in press. Z. Gu and P. B. Balbuena, "Chemical environment effects on the atomic oxygen absorption into Pt(111) surfaces," J. Phys. Chem C., in press. P. Hirunsit and P. B. Balbuena, "Effects of confinement on small water clusters structure and proton transfer," J. Phys. Chem A., in press. Sergio R Calvo and Perla B. Balbuena, Theoretical analysis of reactivity on Pt(111) and Pt-Pd(111) alloys," Surf. Sci., in press. Diego A. Gsmez Gualdrsn and Perla B. Balbuena, "Classical Molecular Dynamics of Clathrate-Methane-Water-Kinetic Inhibitor Composite Systems," J. Phys. Chem., in press. Zhihui Gu and Perla B. Balbuena, "Absorption of atomic oxygen into subsurfaces of Pt(100) and Pt(111): Density functional theory study," J. Phys. Chem. C., 111, 9877-9883, (2007). Yuguang Ma and Perla B. Balbuena, "Designing Oxygen Reduction Catalysts: Insights from Metalloenzymes," Chem. Phys. Lett., 440, 130-133, (2007). Santiago Aparicio-Martinez and Perla B. Balbuena, "On the properties of Aqueous Amide Solutions through Classical Molecular Dynamics Simulations ," Mol. Sim., in press. Yuguang Ma and Perla B. Balbuena, "Catalytic Activity Tuning of a Biomimetic HO-FeV=O Oxidant for Methane Hydroxylation by Substituents on Aromatic Rings: A Theoretical Study," J. Phys. Chem. B., 111, 2711-2718, (2007). Diego A. Gomez Gualdrsn, S. Aparicio-Martinez, and Perla B. Balbuena, "Computational studies of structure and dynamics of clathrate inhibitor monomers in solution," Ind. and Eng. Chem. Res., 46, 131-142, (2007). F. Tarazona-Vasquez and P. B. Balbuena, "Dendrimer-Tetrachloroplatinate Precursor Interactions. 2. Non Covalent Binding in PAMAM Outer Pockets," J. Phys. Chem. A, 111, 945-953, (2007). F. Tarazona-Vasquez and P. B. Balbuena, "Dendrimer-Tetrachloroplatinate Precursor Interactions. 1. Hydration of Pt(II) Species in PAMAM Outer Pockets," J. Phys. Chem. A, 111, 932-944, (2007). L. R. Saenz, P. B. Balbuena, and J. M. Seminario, "Platinum testbeds: Interaction with oxygen," J. Phys. Chem., 110, 11968-11974, (2006). S. R. Calvo and P. B. Balbuena, "Density functional theory analysis of reactivity of PtxPdy alloy clusters," Surf. Sci., 601, 165-171, (2007). PI Arun Bansil S. Sahrakorpi, R.S. Markiewicz, M. Lindroos, X. J. Zhou, T. Yoshida5, W. L. Yang3,4, T. Kakeshita, H. Eisaki, S. Uchida, Seiki Komiya, Yoichi Ando, F. Zhou, Z. X. Zhao, T. Sasagawa, A. Fujimori, Z. Hussain, Z.-X. Shen, and A. Bansil, "Appearance of Universal Metallic Dispersion in a Doped Mott Insulator", submitted to Nature Physics. Tanmoy Das, R.S. Markiewicz, and A. Bansil, "Nodeless d-wave superconducting pairing due to residual antiferromagnetism in underdoped Pr2-xCexCuO4-d", Phys. Rev. Lett. 98, 197004 (2007). R.S. Markiewicz and A. Bansil, "Collapse of the Magnetic Gap of Cuprate Superconductors within a Three-Band Model of Resonant Inelastic X-Ray Scattering", Phys. Rev. Lett. 96, 107005 (2006). H. Lin, S. Sahrakorpi, R.S. Markiewicz, and A. Bansil,"Raising Bi-O Bands above the Fermi Energy Level of Hole-Doped Bi2Sr2CaCu2O8+d and Other Cuprate Superconductors", Phys. Rev. Lett. 96, 097001 (2006). P. E. Mijnarends, S. Kaprzyk, B. Barbiellini, Yinwan Li, J. F. Mitchell, P. A. Montano, and A. Bansil, " Magnetic momentum density, Fermi surface, and directional magnetic Compton profiles in LaSr2Mn2O7 and La1.2Sr1.8Mn2O7", Phys. Rev. B 75, 014428 (2007). Yinwan Li, P. A. Montano, B. Barbiellini, P. E. Mijnarends, S. Kaprzyk and A. Bansil, "Spin moment over 10-300 K and delocalization of magnetic electrons above the Verwey transition in magnetite", J. Phys. Chem. Solids 68, 1556 (2007). W. Meevasana, X.J. Zhou, S. Sahrakpori, W.S. Lee, W.L. Yang, K. Tanaka, N. Mannella, T. Yoshida, D. H. Lu, Y.L. Chen, R.H. He, Hsin Lin, S. Komiya, Y. Ando, F. Zhou, W.X. Ti, J.W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, K. Fujita, S. Uchida, H. Eisaki, A. Fujimori, Z. Hussain, R. S. Markiewicz, A. Bansil, N. Nagaosa, J. Zaanen, T.P. Devereaux, and Z.-X. Shen, "The hierarchy of multiple many-body interaction scales in high-temperature superconductors", Phys. Rev. B 75, 174506 (2007). R.S. Markiewicz, and A. Bansil, "Dispersion anomalies induced by the low-energy plasmon in the cuprates", Phys. Rev. B 75, 020508 (2007). R.S. Markiewicz, S. Sahrakorpi, and A. Bansil, "Paramagnon-induced dispersion anomalies in the cuprates", cond-mat/0701524, to be published, Phys. Rev. B. T. Yoshida, X. J. Zhou, K. Tanaka, W. L. Yang, Z. Hussain, Z.-X. Shen, A. Fujimori, S. Sahrakorpi, M. Lindroos, R. S. Markiewicz, A. Bansil, Seiki Komiya, Yoichi Ando, H. Eisaki, T. Kakeshita, and S. Uchida, "Systematic doping evolution of the underlying Fermi surface of La2-xSrxCuO4", Phys. Rev. B 74, 224510 (2006). Tanmoy Das, R. S. Markiewicz, and A. Bansil, "Nonmonotonic dx2-y2 superconducting gap in electron-doped Pr0.89LaCe0.11CuO4 : Evidence of coexisting antiferromagnetism and superconductivity?", Phys. Rev. B 74, 020506 (2006). D. Nissenbaum, B. Barbiellini, and A. Bansil, "Decorrelation of samples in quantum Monte Carlo calculations and scaling of autocorrelation time in Li and H2O clusters", Phys. Rev. B 76, 033412 (2007). S.W.H. Eijt, B. Barbiellini , A.J. Houtepen, D. Vanmaekelbergh, P.E. Mijnarends and A. Bansil, Positron studies of surfaces, structure and electronic properties of nanocrystals, Phys. Stat. Sol. (c) 4, 3883 (2007). R. Saniz, B. Barbiellini, P. M. Platzman, and A. J. Freeman, "Physisorption of Positronium on Quartz Surfaces", Phys. Rev. Lett. 99, 096101 (2007). PI Michael Barad Barad, M., P. Colella, and S.G Schladow, An adaptive cut-cell method for environmental fluid mechanics, submitted to the International Journal for Numerical Methods in Fluids. Barad, M.F. and O.B. Fringer, 2007. Numerical simulations of shear instabilities in open-ocean internal gravity waves, Proceedings of the Fifth International Symposium on Environmental Hydraulics, Tempe, Arizona, USA. Barad, M.F., O.B. Fringer, and P. Colella, Dec. 2006. Multiscale simulations of internal gravity waves, Proceedings of the Sixth International Symposium on Stratified Flows, Perth, Australia. Baron, Edward mp90 PI Edward Baron R. C. Thomas, G. Aldering, P. Antilogos, ..., E. Baron, et al. (The Nearby Supernova Factory), Nearby Supernova Factory Observations of SN 2006D: On Sporadic Carbon Signatures in Early Type Ia Supernova Spectra, ApJ (Letters), (2007), 654, L53--56. S. Knop, P. Hauschildt, & E. Baron, General Relativistic Radiative Transfer, A&A, (2007), 463, 315--320. J. Parrent, D. Branch, M. A. Troxel, D. Casebeer, D. Jeffery, W. Ketchum, E. Baron, F. J. D. Serduke, & A. V. Filippenko, Direct Analysis of Spectra of the Unusual Type Ib Supernova 2005bf, PASP, (2007), 119, 135--142. E. Baron, D. Branch, & P. Hauschildt, Reddening, Abundances, and Line Formation in SNe II, ApJ, (2007), 662, 1148--. E. Baron & P. Hauschildt, A 3D Radiative Transfer Framework: II. Line Transfer Problems, A&A, (2007), 468, 255--261. B. Chen, R. Kantowski, E. Baron, S. Knop, & P. Hauschildt, Solving the Transfer Equation for Arbitrary Flows in Stationary Spacetimes, MNRAS, (2007), 380, 104--112. S. Knop, P. Hauschildt, E. Baron, & S. Dreizler, Analysing SN 2003Z with PHOENIX, A&A, (2007), 469, 1077--1081. W. Ketchum, E. Baron, & D. Branch, Detailed Spectral Analysis of the Type Ib Supernova 1999dn. Paper I: Hydrogen-free Models ApJ, (2007), in press. J. L. Prieto, ..., E. Baron, & J. Parrent, A Study of the Type IIn/Ia Supernova 2005gj from X-rays to the Infrared: Paper I, AJ, (2007), submitted E. Baron, D. J. Jeffery, D. Branch, E. Bravo, D. Garcia-Senz, & P. H. Hauschildt, Detailed Spectral Modeling of a 3-D Pulsating Reverse Detonation Model: Too Much Nickel, ApJ (Letters), (2007), submitted. S. Bongard, E. Baron, G. Smadja, D. Branch, & P. H. Hauschildt, Multi-layered Spectral Formation in SNe Ia Around Maximum Light, ApJ, (2007), submitted. PI Don Batchelor Experimental and numerical characterization of ion-cyclotron heated protons on the Alcator C- Mod tokamak, V. Tang, R. R. Parker, P. T. Bonoli, et. al., Plasma Phys. Control. Fusion 49, 873 (2007). Evolution of nonthermal particle distributions in radio frequency heating of fusion plasmas, P. T. Bonoli, D. B. Batchelor, L. A. Berry, et. al., Journal of Physics: Conference Series 78 012006 (2007). Quasilinear evolution of non-thermal distributions in ion cyclotron resonance heating of tokamak plasmas, E. F. Jaeger, R. W. Harvey, V. E. Lynch, et al., Journal of Physics: Conference Series 46, 82 (2006). Self-Consistent Full-Wave and Fokker-Planck Calculations for Ion Cyclotron Heating in Non-Maxwellian Plasmas, E. F. Jaeger, L. A. Berry, S. D. Ahern, et al., Phys. Plasmas 13, 056101 (2006). Global wave solutions with self-consistent velocity distributions in ion cyclotron heated plasmas, E. F. Jaeger, R. W. Harvey, L. A. Berry, et al., Nucl. Fusion 46 S397 (2006). Nonlinear ICRF-plasma interactions, J.R. Myra, D.A. D'Ippolito, D.A. Russell, L.A. Berry, E.F. Jaeger and M.D. Carter, Nucl. Fusion 46 S455 (2006). PI Victor Batista J. Chem. Theor. Comput., 2, 175-186, 2006 A Self-Consistent Space-Domain Decomposition Method for QM/MM Computations of Protein Electrostatic Potentials, by Jose A. Gascon, Siegfried S.F. Leung, Enrique R. Batista and Victor S. Batista. Photochem. Photobiol. 4, 940-949, 2005 The Mechanism of Photosynthetic Water Splitting, by James P. McEvoy, Jose A. Gascon, Victor S. Batista and Gary W. Brudvig. Acc. Chem. Res 39, 184-193, 2006 Computational Studies of the Primary Phototransduction Event in Visual Rhodopsin, by Jose A. Gascon, Eduardo M. Sproviero and Victor S. Batista. J. Inorg. Biochem. 100, 786-800, 2006 Characterization of Synthetic Oxomanganese Complexes and the Inorganic-Core of the O2-Evolving Complex in Photosystem II: Evaluation of the DFT/B3LYP Level of Theory, by Eduardo M. Sproviero, Jose A. Gascon, James P. McEvoy, Gary W. Brudvig and Victor S. Batista. J. Chem. Theor. Comput., 4, 1119-1134, 2006 QM/MM Model of the Oxygen-Evolving Complex of Photosystem II, by Eduardo M. Sproviero, Jose A. Gascon, James P. McEvoy, Gary W. Brudvig and Victor S. Batista. J. Am. Chem. Soc., 128, 3659-3668, 2006 Density functional theory and DFT+U study of transition metal porphine adsorbed on Au(111) surfaces and effects of applied electric fields, by Kevin Leung, Susan B. Rempe, Peter A. Schultz, Eduardo M. Sproviero, Victor S. Batista, Michael E. Chandross, and Craig J. Medforth. J. Mod. Optics., 53, 2519-2532, 2006 Coherent-Control of Tunneling Dynamics in Functionalized Semiconductor Nanostructures: A Quantum-Control Scenario Based on Stochastic Unitary Pulses, Luis G.C. Rego, Sabas G. Abuabara and Victor S. Batista. J. Chem. Phys., 124, 224305, 2006 Matching-Pursuit Split Operator Fourier Transform Simulations of Excited-State Intramolecular Proton Transfer in 2(2`-hydroxyphenyl)-oxazole, by Yinghua Wu and Victor S. Batista. J. Chem. Phys. 125, 124313, 2006 Matching-Pursuit Split Operator Fourier Transform Simulations of Excited-State Nonadiabatic Quantum Dynamics in Pyrazine, by Xin Chen and Victor S. Batista. Proceedings of SPIE 6325, Volume 6325 Physical Chemistry of Interfaces and Nanomaterials V, Mark Spitler, Frank Willig, Editors, 63250R (Aug. 30, 2006) Force Field Parameters for Large-Scale Computational Modeling of Sensitized TiO2 Surfaces, by Sabas G. Abuabara, Jose A. Gascon, Suet-Yee Leung and Victor S. Batista. Proceeding of the 14th International conference on Photosynthesis, Glasgow, U.K., submitted, 2007 Ligation of the C-terminus of the D1-polypeptide of photosystem II to the Oxygen Evolving Complex of Photosystem II, by Jose A. Gascon, Eduardo M. Sproviero, James P. McEvoy, Gary W. Brudvig and Victor S. Batista. Current Opinion Struct. Biol., 17, 173-180, 2007 Structural Models of the Oxygen-Evolving Complex of Photosystem II, by Eduardo M. Sproviero, Jose A. Gascon, James P. McEvoy, Gary W. Brudvig and Victor S. Batista. J. Phys. Chem. B, 111, 11982-11990, 2007 Ultrafast Photooxidation of a Mn(II)-terpyridine complex covalently attached to TiO2 Nanoparticles, Sabas G. Abuabara, Clyde W. Cady, Jason B. Baxter, Charles A. Schmuttenmaer, Robert H. Crabtree, Gary W. Brudvig and Victor S. Batista. Photochem. Photobiol., 190, 274-282, 2007 The MP/SOFT Methodology for Simulations of Quantum Dynamics: Model Simulations of the Photoisomerization of the Retinyl Chromophore in Rhodopsin, by Xin Chen and Victor S. Batista. Batista, Victor m410 PI Pat Behling Wu, L., F. He, Z. Liu, and C. Li, 2007: Atmospheric Teleconnections of Tropical Atlantic Variability: Interhemispheric, Tropical Extratropical, and Cross-Basin Interactions. J. Climate, 20, 856-870. He, F., Z. Liu: The Nature of Decadal Variability of Atlantic Meridional Overturning Circulation: coupled air-sea interaction organized by damped ocean mode. In Prep Wu L., Z. Liu, C. Li and Y. Sun, (2007). Extratropical control of recent tropical Pacific decadal climate variability: a relay teleconnection. Clim. Dyn, 28 (1), 99-112, doi: 10.1007/s00382-006-0198-5. CCR #890. Liu, Z., Y. Liu, L. Wu, and R. Jacob (2007). Seasonal and Long Term Atmospheric Responses to Reemerging North Pacific Ocean Variability: A Combined Dynamical and Statistical Assessment. Journal of Climate 20(6), pgs: 955-980, doi: 10.1175/JCLI4041.1. CCR #897 PI John Bell A. S. Almgren, J. B. Bell and M. Zingale, "Low Mach Number Modeling of Type Ia Supernovae. III. Reactions", submitted for publication. A.S. Almgren, J.B. Bell, and M. Zingale, "MAESTRO: A Low Mach Number Stellar Hydrodynamics Code ", SciDAC 2007, J. of Physics: Conference Series, Boston, Massachusetts, July 2007. J. Bell, A. Aspden, M. Day, M. Lijewski, "Numerical simulation of low Mach number reacting flows", SciDAC 2007, J. of Physics: Conference Series, Boston, Massachusetts, July 2007. LBNL Report No. LBNL-63088. S.E. Woosley, A.S. Almgren, J.B. Bell, G. Glatzmaier, D. Kasen, A.R. Kerstein, H. Ma, P. Nugent, F. Ropke, V. Sankaran and M. Zingale, "Type Ia Supernovae ", SciDAC 2007, J. of Physics: Conference Series, Boston, Massachusetts, July 2007. J. B. Bell, M. S. Day, J. F. Grcar, M. J. Lijewski, J. F. Driscoll and S. F. Filatyev, "Numerical Simulation of a Laboratory-Scale Turbulent Slot Flame", LBNL Report LBNL-59245, Proc. Combust. Inst. 31 1299-1307 (2007). J. B. Bell, R. K. Cheng, M. S. Day and I. G. Shepherd, "Numerical Simulation of Lewis Number Effects on Lean Premixed Turbulent Flames", LBNL Report LBNL-59247, Proc. Combust. Inst. 31 1309-1317 (2007). M. Day, I. Shepherd, J. Bell, J. Grcar, M. Lijewski, "Displacement Speeds in Turbulent Premixed Flame Simulations", Proc. ECCOMAS-CFD 2007. P. Colella, J. Bell, N. Keen, T. Ligocki, M. Lijewski, B. Van Straalen "Performance and Scaling of Locally-Structured Grid Methods for Partial Differential Equations" presented at SciDAC 2007 Annual Meeting. Colella, P., Graves, D.T., Keen, B.J., Modiano, D., "A cartesian grid embedded boundary method for hyperbolic conservation laws", LBNL-56420, Journal of Computational Physics. Vol. 211 (2006), pp. 347-366. Martin, D.F., Colella, P.. and Graves, D.T. "A Cell-Centered Adaptive Projection Method for the Incompressible Navier-Stokes Equations in Three Dimensions", LBNL-62025, submitted to Journal of Computational Physics. McCorquodale, P., Colella, P., Balls, G.T., and Baden, S.B. "A Local Corrections Algorithm for Solving Poisson's Equation in Three Dimensions", LBNL-62377, submitted to Communications in Applied Mathematics and Computational Science. Schwartz, P., Barad, M., Colella, P., Ligocki, T.J., "A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions", LBNL-56607, Journal of Computational Physics. Vol. 211 (2006), pp. 531-550. D. Trebotich, "Simulation of Biological Flow and Transport in Complex Geometries using Embedded Boundary / Volume-of-Fluid Methods", Journal of Physics: Conference Series, 78 (2007) 012076. D. Trebotich, G. H. Miller and M. D. Bybee, "A Penalty Method to Model Particle Interactions in DNA-laden Flows", accepted to appear J. Nanosci. Nanotech. (2007). G. H. Miller and D. Trebotich, "Toward a Mesoscale Model for the Dynamics of Polymer Solutions", J. Comput. Theoret. Nanosci. 4(4):704-708 (2007). D. Trebotich, G. H. Miller and M. D. Bybee, "A Hard Constraint Algorithm to Model Particle Interactions in DNA-laden Flows", Nanoscale and Microscale Thermophysical Engineering 11 (1):121-128 (2007). D. Trebotich, "Modeling complex biological flows in multi-scale systems using the APDEC framework", Journal of Physics: Conference Series, 46 (2006) 316-321. PI Roy Benedek Li2MnO3-stabilized LiMO2 (M=Mn, Ni, Co) electrodes for lithium-ion batteries M. M. Thackeray, S. H. Kang, C. S. Johnson, J. T. Vaughey, R. Benedek, and S. A. Hackney Journal of Materials Chemistry 17 (30): 3112-3125 (2007) Free energies for acid attack reactions of lithium cobaltate R. Benedek and A. van de Walle Journal of the Electrochemical Society (submitted) Phase stability of cation-doped LiMnO$_{2}$ within the GGA+U approximation N. N. Shukla, S. Shukla, R. Prasad, and R. Benedek Modelling and Simulation in Materials Science and Engineering (submitted) Reaction free energy for proton-lithium ion exchange in lithium battery cathode materials R. Benedek, M. M. Thackeray, and A. van de Walle Chemistry of Materials (submitted) PI George Bertsch "Global study of quadrupole correlation effects", M.Bender, G.F.Bertsch, and P.-H. Heenen, Phys. Rev. C73 034322 (2006). "Global study of the spectroscopic properties of the first 2+ state in even-even nuclei," B. Sabbey, M. Bender, G.F. Bertsch, and P.-H. Heenen, Phys. Rev. C 75, 044305 (2007. Bethel, Edward (Wes) m636 PI Wes Bethel J. Gosink, J.C. Anderson, and K.I. Joy, "Variable Interactions in Query Driven Visualization," IEEE Transactions on Visualization and Computer Graphics (Proceedings of IEEE Visualization 2007), 13(6), October 2007 E.W. Bethel, C.R. Johnson, K. Joy, S. Ahern, V. Pascucci, H. Childs, J. Cohen, M. Duchaineau, B. Hamann, C. Hansen, D. Laney, P. Lindstrom, J. Meredith, G. Ostrouchov, S.G. Parker, C.T. Silva, A. Sanderson, X. Tricoche. SciDAC Visualization and Analytics Center for Enabling Technology, In Journal of Physics, Conference Series, 78, 2007, Boston MA, USA Hank Childs. "Architectural Challenges and Solutions for Petascale Postprocessing." Journal of Physics, Conference Series SciDAC 2007, 78, 2007, Boston, MA, USA C. Jones, K.-L. Ma, A. Sanderson, L. Myers. "Visual Interrogation of Gyrokinetic Particle Simulations," In Journal of Physics, Conference Series SciDAC 2007, 78, 2007, Boston MA, USA G. Weber, V. Beckner, H. Childs, T. Ligocki, M. Miller, B. van Straalen, E. W. Bethel, "Visualization Tools for Adaptive Mesh Refinement Data." Proceedings of the 4th High End Visualization Workshop, June 2007, Tyrol, Austria. PI Amitava Bhattacharjee N. Bessho and A. Bhattacharjee, Fast collisionless reconnection in electron-positron plasmas, Phys. Plasmas 14, 056503 (2007) P. Zhu, C. C. Hegna, C. R. Sovinec, A. Bhattacharjee, and K. Germaschewski, Intermediate nonlinear regime of a line-tied g-mode, Phys. Plasmas, 14, 055903 (2007). P. Zhu, C. R. Sovinec, C. C. Hegna, K. Germaschewski, and A. Bhattacharjee, Nonlinear ballooning instability in the near-Earth magnetotail: Growth, structure, and possible role in substorms, J. Geophys. Res., 112, A06222 (2007). C. S. Ng and A. Bhattacharjee, A Constrained Tectonics Model for Coronal Heating, Astrophys. J., submitted (2007). C. S. Ng and A. Bhattacharjee, Anisotropic MHD Turbulence, AIP Conf. Proc., 932, 137 (2007). K. Germaschewski, A. Bhattacharjee, and C.-S. Ng The Magnetic Reconnection Code: an AMR-based fully implicit simulation suite in: Numerical Modeling of Space Plasma Flows, ASP Conference Series, Vol. 359 (2006) N. B. Pogorelov and G. P. Zank (Eds.) PI Julian Borrill S. Masi et al., "The millimeter sky as seen with BOOMERanG", New Astronomy Reviews, 51:236-243, 2007. G. De Troia et al., "Searching for non-Gaussian signals in the BOOMERanG 2003 CMB map: preliminary results",New Astronomy Reviews, 51:250-255, 2007. F. Piacentini et al., "CMB polarization with BOOMERanG 2003", New Astronomy Reviews, 51:244-249, 2007 P. Ade et al., "First season QUaD CMB temperature and polarization power spectra" accepted to ApJ Yoon et al., "The Robinson Gravitational Wave Background Telescope (BICEP): a bolometric large angular scale CMB polarimeter", Millimeter and Submillimeter Detectors and Instrumentation for Astronomy III, Proceedings of SPIE, 6275, 2006 astro-ph/0606278 "Making Maps from Planck LFI 30GHz Data" M.A.J. Ashdown, C. Baccigalupi, A. Balbi, J.G. Bartlett, J. Borrill, C. Cantalupo, G. de Gasperis, K.M. Gorski, V. Heikkila, E. Hivon, E. Keihanen, H. Kurki-Suonio, C.R. Lawrence, P. Natoli, T. Poutanen, S. Prunet, M. Reinecke, R. Stompor, B. Wandelt - submitted to Astronomy & Astrophysics (astro-ph/0702483) PI Joel Bowman Potential Energy Surface and Reaction Dynamics of the OH+NO2 Reaction, C. Chen, B. Schepler, B. Braams, J. M. Bowman, J. Chem. Phys. 127, 09xxx (2007). Quasiclassical Trajectory Calculations of Acetaldehyde Dissociation on a Global Potential Energy Surface Indicate Significant Non-transition State Dynamics Shepler, B. C.; Braams, B. J.; Bowman, J. M. J. Phys. Chem. A.; (Letter); 2007; ASAP Article; DOI: 10.1021/jp074646q PI Bastiaan Braams A. R. Sharma, J. Wu, B. J. Braams, S. Carter, R. Schneider, B. Shepler and J. M. Bowman. Potential energy surface and MULTIMODE vibrational analysis of C2H3+. Journal of Chemical Physics 125, #224306, 2006. PI Marcia Branstetter Branstetter, M.L., David J. Erickson III, Jose Hernandez, and Robert Oglesby, "Spatial resolution and precipitation impacts on the magnitude and variability of river discharge in CCSM3", revised version submitted to Journal of Hydrology, 2007. Kuhn, Gabriel, Shiraj Khan, Auroop R. Ganguly, Marcia L. Branstetter, "Geospatial-temporal dependence among weekly precipitation extremes with applications to observations and climate model simulations in South America," Advances in Water Resources, vol. 30 (2007), pp. 2401-2423. PI Eduardo Bringa R.A. Lebensohn, E.M. Bringa, and A. Caro, "Continuum mesoscale modelling of nanocrystalline fcc metals under shock-loading using an spectral formulation fed by molecular dynamics results", Journal de Physique IV 134, 17 (2006). A. Caro, P. Klaver, et al., "The computational modeling of alloys at the atomic scale: from ab initio and thermodynamics to radiation-induced heterogeneous precipitation", JOM 59, 52 (2007). R. Devanathan, P. Durham, J. Du, L. R. Corrales, E. M. Bringa, "Molecular dynamics simulation of amorphization of forsterite by cosmic rays", Nucl. Instr. and Meth. in Phys. Res. B 255, 172 (2007). D. Schwen and E.M. Bringa, "Simulation of ion tracks in diamond and graphite", Nucl. Instr. and Meth. in Phys. Res. B 256, 187 (2007). J. Marian, L.A. Zepeda-Ruiz, N. Couto, E.M. Bringa, G.H. Gilmer, P.C. Stangeby and T. D. Ronglien, "Characterization of sputtering products during graphite exposure to deuterium ions by molecular dynamics", Journal of Applied Physics 101, 044506 (2007). E.M. Bringa et al., "Cosmic-ray induced amorphization of silicates in the inter-stellar medium", Astrophysics Journal 662, 372 (2007). R. Lebensohn, E.M. Bringa, and A. Caro, "A viscoplastic micromechanical model for the yield strength of nanocrystalline materials", Acta Materialia 55, 261 (2007). D. Farkas, E.M. Bringa, and A. Caro, "Annealing twins in nanocrystalline fcc metals: A molecular dynamics simulation", Phys. Rev. B 75, 184111 (2007). B. Cao, E.M. Bringa, and M.A. Meyers, "Shock Compression of Monocrystalline Copper: Atomistic Simulations", Metallurgical and Materials Transactions A, in press. A. Gyulassy, M.A. Duchaineau, V. Natarajan, V. Pascucci, E.M. Bringa, A. Higginbotham, y B. Hamann, "Topologically clean distance fields", in press, IEEE Transactions on Visualization and Computer Graphics, IEEE Computer Society, P.R. Durham, R. Devanathan, E.M. Bringa, L.R. Corrales, A.G.G.M. Tielens, W. van Breugel, "Atomistic simulation of cosmic ray-induced amorphization of silicate grains", submitted to Astrophysics Letters. A. Caro, E. Bringa, D. Farkas, G. Gilmer and L. Zepeda-Ruiz, "Mobility and mechanical response of dirty interfaces in nanocrystalline Cu", submitted to Acta Materialia. P. Erhart, A. Caro, M. Caro, and B. Sadigh, "Short-range order and precipitation in Fe-rich Fe--Cr alloys", submitted to Phys. Rev. B. P. Erhart, B. Sadigh, and A. Caro, "Are there stable long-range ordered Fe_{1-x}Cr_x compounds?", submitted to Appl. Phys. Lett. PI Eric Brown Kang, H.; Facchetti, A.; Jiang, H.; Cariati, E.; Righetto, S.; Ugo, R.; Zuccaccia, C.; Macchioni, A.; Stern, C. L.; Liu, Z.; Ho, S.-T.; Brown, E. C.; Ratner, M. A.; Marks, T. J. J. Am. Chem. Soc. 2007, 129, 3267-3286. Brown, E. C.; Marks, T. J.; Ratner, M. A. J. Phys. Chem. 2007 In Press. (Preprint available) PI David Bruhwiler G.I. Bell, D.L. Bruhwiler, A. Fedotov, A. Sobol, R.S. Busby, P. Stoltz, D.T. Abell, P. Messmer, I. Ben-Zvi, V. Litvinenko, "Simulating the dynamical friction force on ions due to a briefly co-propagating electron beam," J. Comput. Phys. (submitted). PI Andrew Canning The Use of Bulk States to Accelerate the Band Edge State Calculation of a Semiconductor Quantum Dot, C. Voemel, O Marques, S. Tomov, L.-W.-Wang and J. Dongarra. J. Comp. Phys. 223, 774 (2007) B. Lee, A. Canning, L.W. Wang, "Effects of d-electrons in pseudopotential screened-exchange density functional calculations", Phys. Rev. B (submitted). C. Voemel, S. Tomov, O. Marques, A. Canning, L-W Wang and J. Dongarra, "State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems", Submitted to Journal of Computational Physics L. Oliker, A. Canning, J. Carter, et. al., "Scientific Application Performance on Candidate PetaScale Platforms", International Parallel & Distributed Processing Symposium (IPDPS) 2007. LBNL-62952. WINNER: Best paper application track L. Oliker, J. Shalf, J. Carter, A. Canning et. al., "Performance Characteristics of Potential Petascale Scientific Applications", Petascale Computing: Algorithms and Applications, Chapman & Hall / CRC Press, in press L. Oliker, A. Canning, J. Carter, J. Shalf, S. Ethier, "Scientific Application Performance on Leading Scalar and Vector Supercomputing Platforms", International Journal of High Performance Computing Applications , in press L.W. Wang, Z. Zhao, J. Meza, "A linear scaling three dimensional fragment method for large scale electronic structure calculations", (submitted). PI Fausto Cattaneo Fausto Cattaneo, Paul Fischer, Aleksandr Obabko "Numerical simulations of magneto-rotational turbulence in cylindrical geometry", Mini-conference on Angular Momentum Transport in Laboratory and Nature, 49th Annual Meeting of the Division of Plasma Physics November 12-16, Orlando, Florida, 2007 PI Paola Cessi Wolfe, C. L. and Paola Cessi, 2008: "Overturning Circulation in an Eddy-Resolving Model: The Effect of the Pole-to-Pole Temperature Gradient", submitted to Journal of Physical Oceanography. PI Yuen-Dat Chen Measurement of the ne and Total 8B Solar Neutrino Fluxes with the Sudbury Neutrino Observatory Phase I Data SetAharmim et al, Phys. Rev. C 75, 045502 (2007) A Search for Neutrinos from the Solar hep Reactions and the Diffuse Supernovae Neutrino Background with the Sudbury Neutrino Observatory, ApJ 653 (2006) 1545 "Validation of spallation neutron production and propagation within Geant4", M. G. Marino et al., NIM A 582, pp. 611-620 (2007). PI Choong-Seock Chang G. Park, J. Cummings, C.S. Chang, Podhorszki, S. Klasky, etc, "Coupled simulation of kinetic pedestal growth and MHD ELM crash," Journal of Physics: Conference Series 78, 012087 (2007) D.A. Batchelor, M. Beck, A. Becoulet, R.V. Budny, C.S. Chang, et al, "Simulation of Fusion Plasmas: Current Status and Future Direction," Plasma Science and Technology 9, 312 (2007) Gunyoung Park and C.S. Chang, "A 5-1/2-dimensional theory for fast and accurate evaluation of the cyclotron resonance heating using representation," Phys. Plasmas 14, 052503 (2007) Y. Chen and S. Parker, Phys. Plasmas 14, 082301 (2007) S. Ku, C.S. Chang, M. Adams, et al., "Gyrokinetic particle simulation of neoclassical transport in the pedestal/scrape-off region of a tokamak plasma," Journal of Physics: Conference Series 46 (2006) 87 Y. Chen and S. E. Parker, "Electromagnetic gyrokinetic delta-f particle-in-cell turbulence simulation with realistic equilibrium profiles and geometry," Journal of Computational Physics 220, 839 (2007) S.E. Parker, J.J. Kahut, Y. Chen, Z. Lin, F.L. Hinton and W.W. Lee, "Fine-Scale zonal flow suppression of electron temperature gradient turbulence," submitted. J. Lang, Y. Chen, S. Parker, Phys. Plasmas 14 082315 (2007) Y. Nishimura, Z. Lin, and W. X. Wang, Electromagnetic global gyrokinetic simulation of shear Alfven wave dynamics in tokamak plasmas, Phys. Plasmas 14, 042503 (April 2007). Y. Nishimura1, Y. Xiao, and Z. Lin, Guiding center orbit studies in a tokamak edge geometry employing Boozer and Cartesian coordinates, Contributions Plasma Phys., in press, 2007. Z Lin, Y Nishimura, Y Xiao, I Holod, W L Zhang and L Chen, Global gyrokinetic particle simulations with kinetic Electrons, Plasma Phys. Contr. Fusion., in press, 2007. H. Strauss, et al., Nuclear Fusion (2006) (IAEA paper) T. S. Hahm, P. H. Diamond, O. D. Gurcan, and G. Rewoldt,Theory of intrinsic rotation and momentum transport, Phys. Plasmas 14, 072302 (2007) PI James Chelikowsky M.L. Tiago, Y. Zhou, M.M.G. Alemany, Y. Saad, J.R. Chelikowsky: The Evolution of Magnetism in Iron from the Atom to the Bulk, Phys. Rev. Lett. 97, 147201 (2006). N.S. Norberg, G.M. Dalpian, J.R. Chelikowsky, and D.R. Gamelin: Vacuum Pinning of Magnetic Impurity Levels in Quantum Confined Semiconductors, Nano Letters 6, 2887 (2006). J.R. Chelikowsky: The Role of Self-Purification and the Electronic Structure of Magnetically Doped Semiconductor Nanocrystals, Phase Transitions 79, 739 (2006). Y. Zhou, Y. Saad, M.L. Tiago, and J.R. Chelikowsky: Parallel Self-Consistent-Field Calculations via Chebyshev-Filtered Subspace Acceleration, Phys. Rev. E 74, 066704 (2006). M.M.G. Alemany, X. Huang, M.L. Tiago, and J.R. Chelikowsky: $p$-type Doping in Indium Phosphide Nanowires: the Role of Dimensionality and Quantum Confinement in the Acceptor Impurity States, Nano Lett. 7, 1878 (2007). J.R. Chelikowsky, M.L. Tiago, Y. Saad, and Y. Zhou: Algorithms for the Evolution of Electronic Properties in Nanocrystals, Comp. Phys. Comm. 177, 1 (2007). M.M.G. Alemany, M. Jain, M. L. Tiago, Y. Zhou, Y. Saad and J.R. Chelikowsky: Efficient first principles calculations of the electronic structure of periodic systems, Comp. Phys. Comm. 177, 339 (2007). M. Lopez del Puerto, M.L. Tiago and J.R. Chelikowsky: Ab initio calculation of temperature effects in the optical response of open-shell sodium clusters, J. Chem. Phys. 27, 144311 (2007). A. Sitt, L. Kronik, S. Ismail-Beigi and J.R. Chelikowsky: Excited state forces within time-dependent density functional theory: A real-space, frequency domain approach, Phys. Rev. A (in press.) G. Rollmann, M.E. Gruner, A. Hucht, P. Entel, M.L. Tiago and J.R. Chelikowsky: Shell-wise Mackay transformation in iron nano-clusters, Phys. Rev. Lett. 99, 083402 (2007). H. Kwak, M. L. Tiago and J. R. Chelikowsky: Quantum Confinement and Strong Coulombic Correlation in ZnO Nanocrystals, Phys. Rev. B (submitted). L. Kong, J.R. Chelikowsky, J.B. Neaton, and S.G. Louie: Real-space pseudopotential calculations of spin-dependent electron transport in quantum point contacts, Phys. Rev. B (submitted). D. Naveh, L. Kronik, M.L. Tiago and J.R. Chelikowsky: Real-Space Pseudopotential method for Spin- Orbit Coupling within Density Functional Theory, Phys. Rev. B (in press). M.M.G.Alemany, R.C.Longo, L.J.Gallego, D.J.Gonzalez, L.E.Gonzalez, M. L.Tiago and J.R.Chelikowsky: Ab initio molecular dynamics simulations of the static, dynamic and electronic properties of liquid lead. A study based on a real-space implementation of density-functional theory," Phys. Rev. B (submitted). T.-L. Chan, M. L. Tiago, E. Kaxiras and J.R. Chelikowsky, The role of quantum confinement on doping silicon nanocrystals, Phys. Rev. Lett. (submitted). T.-L. Chan, M.L. Tiago, and J.R. Chelikowsky: Algorithms for Defects in Nanostructures, Physica B (in press.) M. Lopez del Puerto, M.L. Tiago, and J.R. Chelikowsky: Ab initio methods for the optical properties of CdSe clusters, Phys. Rev. B (submitted). S. Beckman and J.R. Chelikowsky: The structure and properties of vacancies in Si nano-crystals calculated by real space pseudopotential methods, Physica B (in press). L. Kong and J.R. Chelikowsky: Transport properties of transition-metal-encapsulated Si cages, Phys. Rev. B (submitted). PI Jacqueline Chen E. R. Hawkes, R. Sankaran, J. Sutherland, and J. H. Chen, "Scalar Mixing in Direct Numerical Simulations of Temporally-Evolving Plane Jet Flames with Detailed CO/H2 Kinetics," Proc. of the Combustion Institute, 31:1633-1640, (2007). J. C. Sutherland, P. J. Smith, and J. H. Chen, "Quantification of Differential Diffusion in Nonpremixed Systems," Combust. Theory and Modeling, 9:2, pp. 365-383. (2006). J. C. Sutherland, P. J. Smith, and J. H. Chen, "A quantitative method for apriori evaluation of combustion reaction models" Combust. Theory and Modelling 11, 287-303, (2007). J. H. Chen, E. R. Hawkes, R. Sankaran, S. D. Mason, and H. G. Im, "Direct Numerical Simulation of Ignition Front Propagation in a Constant Volume With Temperature Inhomogeneities, Part I: Fundamental Analysis and Diagnostics", Combust. Flame, 145: 128-144. (2006). E. R. Hawkes, R. Sankaran, P. Pebay, and J. H. Chen, "Direct Numerical Simulation of Ignition Front Propagation in a Constant Volume With Temperature Inhomogeneities, Part II: Parametric Study", Combust. Flame, 145: 145-159. (2006). PI Liu Chen A Finite Element Poisson Solver for Global Gyrokinetic Particle Simulations, Y. Nishimura, Z. Lin, J. L. V. Lewandowski, and S. Ethier, J. Comput. Phys. 214, 657-671 (2006). Global particle-in-cell simulations of microturbulence with kinetic electrons, J. L. V. Lewandowski, G. Rewoldt, S. Ethier, and W. W. Lee, and Z. Lin, Phys. Plasmas 13, 072306 (2006). A Finite Element Mesh in a Tokamak Edge Geometry, Y. Nishimura and Z. Lin, Contributions Plasma Phys. 7-9, 551-556 (2006). Gyro-kinetic simulation of global turbulent transport properties in tokamak Experiments, W. X. Wang, Z. Lin, W. M. Tang, W. W. Lee, S. Ethier, J. L. V. Lewandowski, G. Rewoldt, T. S. Hahm, and J. Manickam, Phys. Plasmas 13, 092505 (2006). Resonant and Non-Resonant Particle Dynamics in Alfven Mode Excitations, F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48, 537-556 (2006). Eigenmode Stability Analysis of Drift-Mirror Modes in Nonuniform Plasmas, D. Klimushkin and L. Chen, Ann. Geophys. 24, 2435-2439 (2006). Physics of Burning Plasmas in Toroidal Magnetic Confinement Devices, F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, T. Hahm, A. Milovanov, and G. Vlad, Plasma Phys. Control. Fusion 48, B15-B28 (2006). Statistical analysis of fluctuations and noise-driven transport in particle-in-cell simulations of plasma turbulence, I. Holod and Z. Lin, Phys. Plasmas 14, 032306 (2007). Electromagnetic global gyrokinetic simulation of shear Alfven wave dynamics in tokamak plasmas, Y. Nishimura, Z. Lin, and W. X. Wang, Phys. Plasmas 14, 042503 (2007). Gyrokinetic theory and simulation of mirror instability, H. Qu, Z. Lin, and L. Chen, Phys. Plasmas 14, 042108 (2007). Simulation of Fusion Plasmas: Current Status and Future Direction, D. A. Batchelor, M. Beck, A. Becoulet, R. V. Budny, C. S. Chang, P. H. Diamond, J. Q. Dong, G. Y. Fu, A. Fukuyama, T. S. Hahm, D. E. Keyes, Y. Kishimoto, S. Klasky, L. L. Lao, K. Li, Z. Lin, B. Ludaescher, J. Manickam, N. Nakajima, T. Ozeki, N. Podhorszki, W. M. Tang, M. A. Vouk, R. E. Waltz, S. J. Wang, H. R. Wilson, X. Q. Xu, M. Yagi, and F. Zonca, Plasma Sci. Technol. 9, 312-387 (2007). Linear comparison of gyrokinetic codes with trapped electrons, G. Rewoldt, Z. Lin, and Y. Idomura, Comp. Phys. Commun. 177, 775-780 (2007). Theory of Alfvin Waves and Energetic Particle Physics in Burning Plasmas, L. Chen and F. Zonca, Nuclear Fusion 47, S727-S734 (2007). Electron Fishbones: Theory and Experimental Evidence, F. Zonca et al, Nuclear Fusion 47, 1588-1597(2007). Nonlinear Equilibria, Stability and Generation of Zonal Structures in Toroidal Plasmas, L. Chen and F. Zonca, Nuclear Fusion 47, 886-891(2007). Wave-particle decorrelation and transport of anisotropic turbulence in collisionless plasmas, Z. Lin, I. Holod, L. Chen, P. H. Diamond, T. S. Hahm, S. Ethier, Phys. Rev. Lett. 99, December, 2007. Global gyrokinetic particle simulations with kinetic electrons Z. Lin, Y. Nishimura, Y. Xiao, I. Holod, W. L. Zhang and L. Chen, Plasma Phys. Control. Fusion 49, December, 2007. Dynamic Coronal Loops in Three Dimensions, Y. Mok, R. Lionello, Z. Mikic, and J.A. Linker, submitted to The Astrophysical Journal Letter (2007). PI Yang Chen B. Nevins, S. E. Parker, Y. Chen, et. al.,''Verification of gyrokinetic delta-f simulations of electron temperature gradient turbulence'', Physics of Plasmas 14,084501 (2007) J. Lang, Y. Chen and S. E. Parker, ``Trapped electron mode turbulence'', Physics of Plasmas 14, 082315 (2007) Y. Chen and S. E. Parker, ``Coarse-graining phase space in delta-f particle-in-cell simulations'', Physics of Plasmas 14, 082301 (2007) Y. Chen and S. E. Parker, "Electromagnetic gyrokinetic delta-f particle-in-cell turbulence simulation with realistic equilibrium profiles and geometry," Journal of Computational Physics, 220, 839 (2007) S.E. Parker, J.J. Kahut, Y. Chen, Z. Lin, F.L. Hinton and W.W. Lee, "Fine-Scale zonal flow suppression of electron temperature gradient turbulence," AIP Conference Proceedings, Vol. 871 p.193-203, (2006) Y. Chen, S.E. Parker and J-Y. Lang, "Arbitrary plasma shape and trapped electron modes in the GEM gyrokinetic electromagnetic turbulence simulation code", J. Phys.: Conf. Ser. 46 92-96 doi:10.1088/1742-6596/46/1/013, 2006 J. Candy, R. E. Waltz, S. E. Parker, and Y. Chen, "Relevance of the parallel nonlinearity in gyrokinetic simulations of tokamak plasmas", Phys. Plasmas 13, 074501 (2006) PI Yixin Chen Y. Chen and M. Chen, Extended Duality for Nonlinear Programming, Computational Optimization and Its Applications, submitted, 2007. V. Clark, Y. Chen, J. Wilkens, J. Alaly, K. Zakaryan, and J. Deasy, IMRT Treatment Planning for Prostate Cancer using Prioritized Prescription Optimization and Mean-Tail-Dose Functions, Linear Algebra and Its Applications, accepted, 2007. C. Hsu, Y. Chen, and B. Wah, Subgoal Ordering and Granularity Control for Incremental Planning, International Journal on Artificial Intelligence Tools, 16(4): 707-723, 2007. V. Clark, I. E. Naqa, Y. Chen, and J. Deasy, Automated IMRT treatment planning using prioritized prescription optimization, the XVth International Conference on the Use of Computers in Radiation Therapy (ICCR-07), Toronto, Canada, June 2007. B. Wah, Y. Chen, and T. Wang, Simulated Annealing with Asymptotic Convergence for Nonlinear Constrained Optimization, Journal of Global Optimization, 39(1): 1-37, 2007. C. Hsu, B. Wah, R. Huang, and Y. Chen, Constraint Partitioning for Solving Planning Problems with Trajectory Constraints and Goal Preferences, Proc. International Joint Conference on Artificial Intelligence (IJCAI-07), pp. 1924-1929, 2007. PI Hai-Ping Cheng Agapito LA, Cheng HP Ab initio calculation of a graphene-ribbon-based molecular switchJOURNAL OF PHYSICAL CHEMISTRY C 111 (38): 14266-14273 SEP 27 2007 He Y, Zhang C, Cao C, et al. Effects of strain and defects on the electron conductance of metallic carbon nanotubes PHYSICAL REVIEW B 75 (23): Art. No. 235429 JUN 2007 Cao C, He Y, Torras J, et al. Fracture, water dissociation, and proton conduction in SiO2 nanochains JOURNAL OF CHEMICAL PHYSICS 126 (21): Art. No. 211101 JUN 7 2007 Wang LL, Cheng HP Embedding atom-jellium model for metal surface EUROPEAN PHYSICAL JOURNAL D 43 (1-3): 247-250 JUL 2007 Schmidt M, Masson A, Brechignac C, et al. Hydrogen peroxide and ammonia on protonated ice clusters JOURNAL OF CHEMICAL PHYSICS 126 (15): Art. No. 154315 APR 21 2007 Muralidharan K, Cao C, Wan YX, et al. Environment dependent dynamic charge potential for silica: Application to nanoscale silica structures CHEMICAL PHYSICS LETTERS 437 (1-3): 92-98 MAR 22 2007 Zhang JW, Cheng HP Anomalous Hall effect in disordered Fe ferromagnetic films PHYSICAL REVIEW B 74 (21): Art. No. 212409 DEC 2006 Simulations of azobenzene containing alkanethiol SAMs on Au(111) surface, Journal of Physical Chemistry (in press) Cao C., He. Y., Cheng HP Hydrogen dissociation and band modification of CNT-supported Pd4 cluster (submitted). Cao C., Kemper L. He, Y. Cheng HP CNT-supported Pdn cluster as nano-sensor (submitted). He Y., Cao, C., Cheng HP Predictive simulation of nano-tube fracture under stress (submitted). He Y., Graser S., Hirschfeld, P.J., Cheng HP Supermodulation in the atomic structure of high-Tc superconductor Bi2Sr2CaCu2O8 from ab initio calculations (submitted) PI Wai-Yim Ching J. Buban, K. Matsunaga, J. Chen, N. Shibata, W.Y. Ching, T. Yamamoto, Y. Ikuhara, "Grain Boundary Strengthening in Alumina in by Rare Earth Impurities", Science, 311, 212-215 (2006). W.Y Ching and Paul Rulis, "Ab-initio calculation of the electronic and spectroscopic properties of spinel c-Sn3N4", Phys. Rev. B73, 045202-1-9 (2006). W.Y. Ching, Paul Rulis, Yong-Nian Xu and L. Ouyang, "The electronic structure and spectroscopic properties of 3C, 2H, 4H, 6H, 15R, 21R polymorphs of SiC", Materials Science and Engineering A, 422 C1-2, 147-156 (2006). A. Zerr, R. Riedel, T. Sikine, T. Lowther, W.Y. Ching, and I. Tanaka, "Recent Advances in New Nitrides", Advanced Materials, 18, 2933-2948 (2006). W. Y. Ching, Jun Chen, Paul Rulis, Lizhi Ouyang, and Anil Misra, "Ab initio Modeling of Clean and Y-doped Grain Boundaries in Alumina and Intergranular Glassy Films (IGF) in b-Si3N4", J. Materials Science, 41 (16) 5061-5067, (2006). A. Hunt, W.Y. Ching, Y.-M. Chiang, and A. Moewes "Electronic structure of LiFePO4 and FePO4 studied using resonant inelastic x-ray scattering", Phys. Rev. B73, 205120-1-10 (2006). Jun Chen, Paul Rulis, Lizhi Ouyang, S. Satpathy, and W.Y. Ching,, "Vacancy enhanced ferromagnetism in Fe-doped rutile TiO2", Phys. Rev. B74, 235207-1-5 (2006). A. Misra, L. Ouyang, J. Chen and W.Y. Ching, "Ab initio calculations of strain field and failure pattern in silicon nitride intergranular glassy films", Philosophical Mag. A. 87(25), 3839-3852 (2007). Hongzhi Yao, L. Ouyang and W.Y. Ching, "Ab initio Calculation of the elastic constants of ceramic crystals", J. Am. Ceram. Soc. 90 [10] 3194-3204(2007) Sitaram Aryal, Paul Rulis and W. Y. Ching, "Density functional calculation of the electronic structure and optical properties of Aluminosilicate polymorphs (Al2SiO5)", American Mineralogy (inpress, 2007). P. Rulis, Hongzhi Yao, L. Ouyang, and W.Y. Ching "Electronic structure, bonding, charge distribution and X-ray absorption spectra of the (001) surfaces of fluorapatite and hydroxyapatite", (submitted to Phys. Rev. B). W.Y. Ching, and P. Rulis, "Ab initio calculation of the O-K, N-K, Si-K, Si-L3, edges in the Y-Si-O-N system: A new strategy for ELNES/XANES spectral modeling in complex materials", (submitted to Phys. Rev. B). PI Mei-Yin Chou "Enhanced Electron-Hole Interaction and Optical Absorption in a Silicon Nanowire," L. Yang, C. D. Spataru, S. G. Louie, and M. Y. Chou, Phys. Rev. B (Rapid Communications) 75, 201304 (2007). "Band-Structure Contribution to the Quantum Size Effect in Pb(100) Films," C. M. Wei and M. Y. Chou, Phys. Rev. B 75, 195417(2007). "Size and Orientation Dependence in the Electronic Properties of Silicon Nanowires," J.-A. Yan, L. Yang, and M. Y. Chou, Phys. Rev. B 76, 115319 (2007). "Variational Calculation of the Depolarization of the Maximum Density Droplet in Two-Dimensional Quantum Dots," W. Geist and M. Y. Chou, Phys. Rev. B (in press). "Quantum Confinement Effect in Si/Ge Core-Shell Nanowires," L. Yang, R. N. Musin, X.-Q. Wang, and M. Y. Chou, submitted to Phys. Rev. Lett. "LaMg2PdH7, a New Complex Metal Hydride Containing Tetrahedral [PdH4]4- Anions," K. Yvon, J.-Ph. Rapin, N. Penin, Z. Ma, and M. Y. Chou, J. Alloys Compd. 446-447, 34 (2007). "First-Principles Study of Cation and Hydrogen Arrangements in the Li-Mg-N-H Hydrogen Storage System," Y. Wang and M. Y. Chou, Phys. Rev. B 76, 014116 (2007). "Electronic and Vibrational Properties of gamma-AlH3," Y. Wang and M. Y. Chou, submitted to Phys. Rev. B; arXiv:0707.4604v1[cond-mat.mtrl-sci]. "Phase Stability of Mixed Alkali Alanates," Zhu Ma and M. Y. Chou, in preparation. PI Daryl Chrzan E. Ertekin, M. S. Daw and D. C. Chrzan. Elasticity theory of topological defects in carbon nanotubes and graphene. Accepted for publication in Philosophical Magazine Letters. S. P. Beckman and D. C. Chrzan. The reconstruction energies of partial dislocation cores in GaAs. Accepted for publication in Physical Review B. (Listed as submitted in last year's ERCAP request.) PI Catherine Chuang Kelly, J.T., C.C. Chuang, and A.S. Wexler, Influence of dust composition on cloud droplet formation, ATMOSPHERIC ENVIRONMENT 41 (14): 2904-2916 MAY 2007. PI Aurora Clark A. E. Clark, "Density Functional and Basis Set Dependence of Predicted Ln(H2O)83+ Properties", Journal of Chemical Theory and Computation, submitted. A. Dinescu, A. E. Clark, " Thermodynamic and Structural Features of Aqueous Ce(III)", Inorg. Chem., submitted. PI Bruce Cohen Characterizing electron temperature gradient turbulence via numerical simulation, W. M. Nevins et al., Phys. Plasmas 13, 122306 (2006). Gyrokinetic simulations of ETG and ITG turbulence, A.M. Dimits, W.M. Nevins, D.E. Shumaker, G.W. Hammett, T. Dannert, F. Jenko, M.J. Pueschel, W. Dorland, S.C. Cowley, J.N. Leboeuf, T.L. Rhodes, J. Candy and C. Estrada-Mila, Nucl. Fusion 47 No 8 (August 2007) 817-824 [Candy 2006a] J. Candy and R.E. Waltz, Velocity-Space Resolution, Entropy Production, and Upwind Dissipation in Eulerian Gyrokinetic Simulations, Phys. Plasmas 13, 032310 (2006). [Candy 2006b] J. Candy, R.E. Waltz, S.E. Parker, and Y. Chen, Relevance of the Parallel Nonlinearity in Gyrokinetic Simulations of Tokamak Plasmas, Phys. Plasmas 13, 074501 (2006). "Micro-tearing modes in the mega ampere spherical tokamak", D J Applegate, C M Roach, J W Connor, S C Cowley, W Dorland, R J Hastie and N Joiner, Plasma Phys. Control. Fusion 49 No 8 (August 2007) 1113-1128 "Effects of finite poloidal gyroradius, shaping, and collisions on the zonal flow residual", Yong Xiao, Peter J. Catto, and William Dorland, Phys. Plasmas 14, 055910 (2007) "Studies of improved electron confinement in low density L-mode NSTX discharges," D. Stutman, et al., PoP 13, 092511 (2006). "Small-scale turbulence in a closed field-line geometry," P. Ricci, B. N. Rogers and W. Dorland, Phys. Rev. Letters, 97, 245001 Dec (2006). "Observation of Trapped Electron Mode Turbulence in Tokamak Plasmas Through Direct Comparison of Nonlinear Gyrokinetic Simulations with Density Fluctuation Measurements", D. R. Ernst, N. P. Bassi, W. Dorland, C. L. Fiore, L. Lin, A. Long, E. S. Marmar, M. Porkolab, and K. Zhurovich, submitted, PRL. "Fluctuation Spectra in Kinetic, Alfvenic Turbulence," Gregory G. Howes,, William Dorland, Steven C. Cowley, Gregory W. Hammett, Eliot Quataert, Alexander A. Schekochihin, and Tomoya Tatsuno, submitted, PRL "A Model of Critically Balanced Turbulence in Magnetized Weakly Collisional Plasmas: Implications for the Dissipation Range of the Solar Wind," G. G. Howes, S. C. Cowley, W. Dorland, G. W. Hammett, E. Quataert, and A. A. Schekochihin. submitted, Journal and Geophysical Research. S.E. Parker, J.J. Kahut, Y. Chen, Z. Lin, F.L. Hinton and W.W. Lee, "Fine-Scale zonal flow suppression of electron temperature gradient turbulence," Published Proceedings of the Joint Varenna-Lausanne International Workshop on the Theory of Fusion Plasmas, Varenna,Italy, Aug. 28-Sept. 1, 2006. W. M. Nevins, J. Candy, S. Cowley, T. Dannert, A. Dimits, W. Dorland, C.Estrada-Mila, G.W. Hammett, F. Jenko, and D. E. Shumaker, "Characterizing electron gradient temperature via numerical simulations",Phys. Plasmas (2007) "Gyrokinetic Simulations of ETG and ITG Turbulence", A.M. Dimits , W.M. Nevins, D.E. Shumaker , G.W. Hammett , T. Dannert , F. Jenko , W. Dorland , J.N. Leboeuf , T.L. Rhodes , J. Candy , C. Estrada-Mila, Paper TH/P2-3, in the Proceedings of the 21st IAEA Fusion Conference, Chengdu, China, Oct. 18-21, 2006. [Hooper, 2007a] E. B. Hooper, B. I. Cohen, D. N. Hill, L. L. LoDestro, H. S. McLean, C. A. Romero Talamas, R. D. E. B. Hooper, B. I. Cohen, D. N. Hill, L. L. Lodestro, H. S. McLean, C. A. Romero Talamas, R. D. Wood, and C. R. Sovinec, Magnetic reconnection in the spheromak: physics and consequences, J. Fusion Energy 26, 71 (2007). [Romero-Talamas, 2007] C.A. Romero-Talamas, E. B. Hooper, D.N. Hill, B. I. Cohen, H. S. McLean, R. D. Wood, and J. M. Moller, J. Fusion Energy 26, 169 (2007). PI Marvin Cohen J. Noffsinger, F. Giustino, S.G. Louie, M.L. Cohen, "Superconductivity and Fermi Surface Nesting in Super-Hard Carbides," submitted to Phys. Rev. Lett. L. Yang, M.L. Cohen, and S.G. Louie, "Exciton effects in the optical spectra of graphene nanoribbons," Nano Lett. 7, 3112 (2007). L. Yang, C.-H. Park, Y.-W. Son, M.L. Cohen, and S.G. Louie, "Quasiparticle Energies and Band Gaps of Graphene Nanoribbons," accepted to Phys. Rev. Lett. M. Ishigami, J.D. Sau, S. Aloni, M.L. Cohen, and A. Zettl, "Symmetry breaking in boron nitride nanotubes," Phys. Rev. Lett. 97, 176804 (2006). F. J. Ribeiro, P. Tangney, S. G. Louie and M. L. Cohen, "Hypothetical Hard Structures of Carbon with Cubic Symmetry," Phys. Rev. B 74, 172101 (2006). Y.-W. Son, M. L. Cohen, and S. G. Louie, "Half-Metallic Graphene Nanoribbons," Nature 444, 347 (2006). Y.-W. Son, M.L. Cohen, and S.G. Louie, "Energy Gaps in Graphene Nanoribbons," Phys. Rev. Lett. 97, 216803 (2006). P. Zhang, S.G. Louie, and M.L. Cohen, "Electron-Phonon Renormalization in Cuprate Superconductors," Phys. Rev. Lett. 98, 067005 (2007). J.D. Sau and M.L. Cohen, "Possibility of increased mobility in Ge-Sn alloy system," Phys. Rev. B 75, 045208 (2007). K.T. Chan, J.D. Sau, P.H. Zhang, and M.L. Cohen, "Ab initio calculations of phonon splitting in antiferromagnetic ZnCr2O4," Phys. Rev. B 75, 054304 (2007). W.D. Luo, P.H. Zhang, and M.L. Cohen, "Splitting of the zone-center phonon in MnO and NiO," Solid State Comm. 142, 504 (2007). F. Giustino, J.R. Yates, I. Souza, M.L. Cohen, and S.G. Louie, "Electron-Phonon Interaction via Electron and Lattice Wannier Functions: Superconductivity in Boron-Doped Diamond Reexamined," Phys. Rev. Lett. 98, 047005 (2007). PI Michael Colvin V.V. Krishnan, Edmond Y. Lau, Justin Yamada, Daniel P. Denning, Samir S. Patel, Michael E. Colvin and Michael F. Rexach, Intra-molecular cohesion of coils mediated by phenylalanine-glycine motifs in the natively unfolded domain of a nucleoporin Submitted to PLOS Computational Biology (2007) PI Peter Cummings Massless Fermion in multilayer graphene, Latil, Henrard, Meunier, Physical Review B (RC), in press (2007) Tuning the conductance of carbon nanotube with encapsulated molecules, Vincent Meunier and Bobby G Sumpter, Nanotechnology, 18 in press (2007). Covalent 2D and 3D Networks from 1D Nanostructures: Designing New Materials, Jose Romo-Herrera, Mauricio Terrones, Humberto Terrones, Sefa Dag, and Vincent Meunier, Nano Letters, 7(3), 570 - 576 (2007). Polarization in Nanotubes and Nanotubular Structures, M. Buongiono Nardelli, S.M. Nalhmanson, and V. Meunier, in Solution Manual For Nanoengineeringo f Structural, Functional, and Smart Materials, edited by M.J. Schulz, A.D. Kelkar, and M.J. Sundaresan, CRC Press, page 94-95 (2007). Investigation of the Nanoscale Self-Assembly of Donor-s-Acceptor Molecules Bobby G. Sumpter, Vincent Meunier, Alvaro Vazquez-Mayagoitia, and Ronald K. Castellano, Int. J. Quant. Chem., 107, 2233 (2007). Re-oxidation of TiO2(110) via Ti interstitials and Line Defects, K. T. Park, M. Pan, V. Meunier, and E. W. Plummer, Phys. Rev. B, 75, 245415 (2007). Structure and Stability of Small Boron and Boron Oxide Clusters Michael L. Drummond, Vincent Meunier, Bobby G. Sumpter, J. Phys. Chem. A, 111, 6539 (2007). Tuning the electronic properties of coaxial nanoscale cables through chemical doping, A. G. Souza Filho, Vincent Meunier, M. Terrones, Bobby G. Sumpter, E. B. Barros, F. Villalpando, Y.A. Kim, H. Muramatsu, T. Hayashi, M. Endo, and M. S. Dresselhaus, Nano Letters, 7(8) 2383-2388 (2007). Conductivity properties of benzo-homologated DNA bases, Miguel Fuentes-Cabrera, Vincent Meunier, and Bobby G. Sumpter, Nanotechnology, 18 in press (2007). Density Functional Theory Studies of Quantum Transport in Molecular Systems, V. Meunier, W. Lu, B.G. Sumpter, and J Bernholc, Int. J. Quantum Chem., 106(15), 3334-3342 (2006). Surface reconstructions of TiO2 (110) driven by suboxides, K. Park, M. H. Pan, V. Meunier, and E.W. Plummer, Phys. Rev. Lett., 96, 226105 (2006). Scanning Frequency Mixing Microscopy of High Frequency Transport Behavior at Electroactive Interfaces, Brian J. Rodriguez, Stephen Jesse, Vincent Meunier, and S. Kalinin, Appl. Phys. Lett. 88 (14), 143128 (2006). Adsorption, desorption, and dissociation of benzene on TiO2 (110) and Pd/TiO2_(110): Experimental characterization and first-principles calculations, J. Zhou, S. Dag, S. D. Senanayake, B. C. Hathorn, S. V. Kalinin, V. Meunier, D. R. Mullins, S. H. Overbury, and A. P. Baddorf, Phys. Rev. B, 74 125318 (2006). Atomic Scale Design of Nanostructures, J. Bernholc, W. Lu, S. M. Nakhmanson, P. H. Hahn, V. Meunier, M. Buongiorno Nardelli, and W. G. Schmidt, Molecular Physics, in press (2006). Optimizing the Electronic Properties of carbon nanotubes using Amphoteric Doping, B. G. Sumpter and V. Meunier, Taylor & Francis , in press (2006). Density Functional Theory Studies of Quantum Transport in Molecular Systems, V. Meunier, W. Lu, B.G. Sumpter, and J Bernholc, Int. J. Quantum Chem., in press (2006). Structural Characterization of Carbon Nanotubes, Ph. Lambin, J.F. Colomer, L. Henrard, A. Lucas, and V. Meunier, Microscopy and Microanalysis Vol 12, Suppl 2 (2006) 570-571 Imaging defects in graphene and carbon nanotubes with a scanning tunneling microscope, Ph. Lambin , H. Amara, J.C. Charlier, and V. Meunier book chapter to appear in Chemistry of Nanotube, ASP press (2006). Koparde, V. N. and Cummings, P. T., "Molecular dynamics study of water adsorption on TiO2 nanoparticles," Journal of Physical Chemistry C, 111, 6920-6926 (2007). Leng, Y. S., Dyer, P. J., Krstic, P. S., Harrison, R. J. and Cummings, P. T., "Calibration of chemical bonding between benzenedithiolate and gold: the effects of geometry and size of gold clusters," Molecular Physics, 105, 293-300 (2007) Mamontov, E., Vlcek, L., Wesolowski, D. J., Cummings, P. T., Wang, W., Anovitz, L. M., Rosenqvist, J., Brown, C. M. and Sakai, V. G., "Dynamics and structure of hydration water on rutile and cassiterite nanopowders studied by quasielastic neutron scattering and molecular dynamics simulations," Journal of Physical Chemistry C, 111, 4328-4341 (2007). Payne, C. M., Zhao, X. and Cummings, P. T., "Molecular simulations of DNA transport in solution," Molecular Simulation, 33, 399-403 (2007). Predota, M., Cummings, P. T. and Wesolowski, D. J., "Electric double layer at the rutile (110) surface. 3. Inhomogeneous viscosity and diffusivity measurement by computer simulations," Journal of Physical Chemistry C, 111, 3071-3079 (2007). Pu, Q., Leng, Y. S., Tsetseris, L., Park, H. S., Pantelides, S. T. and Cummings, P. T., "Molecular dynamics simulations of stretched gold nanowires: The relative utility of different semiempirical potentials," Journal of Chemical Physics, 126, Art. No. 144707 (2007). Striolo, A., McCabe, C., Chan, E. R., Glotzer, S. C., and Cummings, P. T., "Aggregation of POSS Monomers in Liquid Hexane: A Molecular-Simulation Study," J. Phys. Chem. B, accepted for publication (2007). Payne, C. M., Zhao, X. C., Vlcek, L., Cummings, P. T., "Molecular Dynamics Simulation of ss-DNA Translocation Through a Copper Nanoelectrode Gap,"J. Phys. Chem. C, submitted for publication (2007) Chan, E. R., Striolo, A., McCabe, C., Cummings, P. T., and Glotzer, S. C., "Coarse-grained force field for simulating polymer-tethered silsesquioxane self-assembly in solution," J. Chem. Phys. 127 (2007) Art. No. 114102. Selected for publication in the October 1, 2007 issue of Virtual Journal of Nanoscale Science & Technology. Selected for publication in the October 1, 2007 issue of Virtual Journal of Biological Physics Research. PI Larry Curtiss S. P. Adiga, P. Zapol P, and L. A. Curtiss, Structure and morphology of hydroxylated amorphous alumina surfaces, J. Phys. Chem. C 111 7422 (2007). Selective and Efficient Oxidative Dehydrogenation of Propane on Sub-nanometer Platinum Clusters, Stefan Vajda, Michael J. Pellin, Jeffrey P. Greeley, Christopher L. Marshall, Larry A. Curtiss, Gregory A. Ballentine, Jeffrey W. Elam, Stephanie Catillon-Mucherie, Paul C. Redfern, and Peter Zapol, submitted. PI Valerie Daggett Beck, et al. Dynameomics: Mass Annotation of Protein Dynamics and Unfolding in Water by High-Throughput All-Atom Molecular Dynamics Simulations. Genome Biology, In press. Simms, et al. The Dynameomics data warehouse: Design of a computational lab workflow and scientific data repository. BMC Bioinformatics, Submitted. Kehl, et al. The Dynameomics OLAP Database: A Multi-Dimensional Analysis Optimized Database for Dynamic Protein Data. BMC Bioinformatics, Submitted. Benson & Daggett. Native Protein Flexibility as a Determinant of Dynamical Changes and Unfolding Events. PNAS, Submitted. PI Ronald Davidson Nonlinear Delta-f Particle Simulations of Collective Dynamics in High-Intensity Bunched Beams, H. Qin, R. C. Davidson and E. A. Startsev, Physical Review Special Topics on Accelerators and Beams 10, 064201 (2007). Collective Temerature Anisotoropy Instabilities in Intense Charged Particle Beams, E. A. Startsev, R. C. Davidson, and H. Qin, Physics of Plasmas 14, 056705 (2007). Nonlinear Delta-f Particle Simulations of Collective Effects in High-Intensity Bunched Beams, H. Qin, R. C. Davidson and E. A. Startsev, Nuclear Instruments and Methods in Physics Research A577, 86 (2007). Dynamic Stabilization of the Two-Stream Instability During Longitudinal Compression of Intense Charged Particle Beam Propagation Through Background Plasma, E. A. Startsev and R. C. Davidson, Nuclear Instruments and Methods in Physics Research A577, 79 (2007). Recent U.S. Advances in Ion-Beam-Driven High Energy Density Physics and Heavy Ion Fusion, B. G. Logan, J. J. Barnard, F. Bieniosek, C. M Celata, R. C. Davidson, A. Friedman, E. Gilson, I. Kaganovich, J. W. Kwan, M. Leitner, A. Molvik, H. Qin, P. Roy, A. Sefkow, P.A. Seidl, E.A. Startsev, S.S. Yu, W.W. Waldron, R.J. Briggs, R. A. Kishek, D. R. Welch and C. Olson, Nuclear Instruments and Methods in Physics Research A577, 1 (2007) Multspecies Weibel Instability for Intense Charged Particle Beam Propagation Through Background Plasma, R. C. Davidson, M. Dorf, I. D. Kaganovich, H. Qin, A. B. Sefkow and E. A. Startsev, D. R. Welch, D. V. Rose and S. M. Lund, Nuclear Instruments and Methods in Physics Research A577, 70 (2007). S. P. Gerhardt, E. V. Belova, M. Yamada, et al., Inductive sustainment of oblate FRCs with the assistance of magnetic diffusion, shaping, and finiteLarmor radius stabilization, submitted to Physics of Plasmas (2007). S.P. Gerhardt, E.V. Belova, M. Yamada, H. Ji, M. Inomoto, et al., Inductive sustainment of a field-reversed configuration stabilized by shaping, magnetic diffusion, and finite-Larmour-radius effects, in press, Phys. Rev. Lett. (2007). S. P. Gerhardt, M. Yamada, E.V. Belova, H. Ji, M. Inomoto, Y. Ren, and B. McGeeham, Method for inductively forming an oblate field reversed configuration utilizing a spheromak and an Ohmic solenoid, submitted to Phys. Plasmas, 2007. H. Ji, E. Belova, S. P. Gerhardt, and M. Yamada, Recent advances in the SPIRIT (Self-organized Plasma with Induction, Reconnection, and Injection Techniques) Concept, J. Fusion Energy 26, 93 (2007). Cothran, C. D., J. Fung, M. R. Brown, M.J. Schaffer, E. Belova, Spectroscopic flow and ion temperature studies of a large-s FRC, J. Fusion Energy 26, 37 (2007). Brown, M. R., C. D. Cothran, J. Fung, M. J. Schaffer, E. Belova, Novel dipole trapped spheromak configuration, J. Fusion Energy 26, 37 (2007). Yamada, M., H. Ji, E. Belova, S. P. Gerhardt, R. C. Davidson, and D. Mikkelsen, Oblate field-reversed configurations through a Self-organized Plasma with Induction, Reconnection, and Injection Techniques: the SPIRIT concept, Plasma and Fusion Research in Japanese Society of Plasma Fusion, in press (2007). Belova, E. V., R. C. Davidson, H. Ji, M. Yamada, Advances in the numerical modeling of field-reversed configurations, Phys. Plasmas 13, 056115 (2006). Belova, E. V., R. C. Davidson, H. Ji, M. Yamada, C. D. Cothran, M. R. Brown, M. J. Schaffer, Numerical study of the formation, ion spin-up and nonlinear stability properties of field-reversed configurations, Nuclear Fusion 46, 162 (2006). Belova, E.V., Davidson, R.C., Ji, H., Yamada, M., Gerhardt, S.P., Effects of energetic beam ions on stability properties of field reversed configuration, Proceedings of the 21th Int. Conf. Chengdu, 2006, paper IAEA-TH/P3-16 (International Atomic Energy Agency, Vienna, 2006). S. P. Gerhardt, E. Belova, M. Inomoto, M. Yamada, H. Ji, Y. Ren, and A. Kuritsyn, Equilibrium and stability studies of oblate field reversed configurations in the Magnetic Reconnection Experiment, Phys. Plasmas 13, 112508 (2006). S. P. Gerhardt, M. Inomoto, E. Belova, M. Yamada, H. Ji, Y. Ren, Studies of free-boundary field-reversed configurations with improved stability in the Magnetic Reconnection Experiment, Proceedings of the 21th Int. Conf. Chengdu, 2006, paper IAEA-IC/P7-13 (International Atomic Energy Agency, Vienna, 2006). M. Inomoto, S. P. Gerhardt, M. Yamada, H. Ji, E. Belova, Y. Ren, and A. Kuritsyn, Coupling between global geometry and the local Hall effect leading to reconnection-layer symmetry breaking, Phys. Rev. Let. 97, 135002 (2006). M. R. Brown, C. D. Cothran, J. Fung, M. Chang, J. Horwitz, M. Schaffer, J. Leuer and E. Belova, Dipole trapped shperomak in a prolate flux conserver, Phys. Plasmas 13, 102503 (2006). PI Eric DeWeaver Lorenz, D. J., and E. DeWeaver, 2007: The response of the extratropical hydrological cycle to global warming. J. Climate, 20, 3470-3484. Lorenz, D. J., and E. DeWeaver, 2007: Tropopause height and the zonal wind response to global warming in the IPCC scenario integrations. J. Geophys. Research, 112, D10119, doi:10.1029/2006JD008087. PI David Dean Complex coupled-cluster approach to an ab initio description of open quantum systems, G. Hagen, D.J. Dean, M. Hjorth-Jensen, and T. Papenbrock, in press, Phys. Lett. B (2007). Coupled-cluster theory for three-body Hamiltonians, G. Hagen, T. Papenbrock, D.J. Dean, A. Schwenk, A. Nogga, M. Wloch, P. PiecuchPhys. Rev. C 76, 034302 (2007) Benchmark calculations for 3H, 4He, 16O and 40Ca with ab-initio coupled-cluster theory, G. Hagen, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, A. Schwenk, Phys. Rev. C 76, 044305 (2007) Comment on "Ab Initio study of 40-Ca with an importance-truncated no-core shell model", D. J. Dean, G. Hagen, M. Hjorth-Jensen, T. Papenbrock, A. Schwenk, Phys. Rev. Lett. (2007) Parity-projected shell model Monte Carlo level densities for fp-shell nuclei, C. Ozen, K. Langanke, G. Martinez-Pinedo, and D.J. Dean, Phys. Rev. C75, 064307 (2007) Particle-Number Projection and the Density Functional Theory Authors: J. Dobaczewski, M.V. Stoitsov, W. Nazarewicz, P.-G. Reinhard arXiv:0707.4324 (submitted to Phys. Rev. C) Additivity of effective quadrupole moments and angular momentum alignments in the A~130 nuclei, M. Matev, A.V. Afanasjev, J.Dobaczewski, G.A.Lalazissis, W.Nazarewicz, arXiv:0707.0767 (submitted to Phys. Rev. C, 2007) PI James Demmel J. Demmel, Y. Hida, X. Li, and J. Riedy, "Extra-precise iterative refinement for overdetermined least squares problems," LAPACK Working Note 188, May 2007. (submitted to TOMS) PI Stephen Derenzo S. Derenzo et. al. "LBNL facility for new scintillator material discovery" , IEEE transactions on Nuclear Science (to appear) PI Thomas Devereaux W.S. Lee, S. Johnston, T.P. Devereaux, and Z.-X. Shen, Aspects of Electron-Phonon Self-Energy Revealed from Angle-Resolved Photoemission Spectroscopy, Phys. Rev. B 75, 195116 (2007). W. Meevasana, X.J. Zhou, S. Sahrakorpi, W.S. Lee, W.L. Yang, N. Mannella, T. Yoshida, Y.L. Chen, K. Tanaka, R.H. He, Hsin Lin, S. Komiya, Y. Ando, F. Zhou, W.X. Ti, J.W. Xiong, Z. X. Zhao, T. Sasagawa, T. Kakeshita, K. Fujita, S. Uchida, H. Eisaki, A. Fujimori, Z. Hussain, R. S. Markiewicz, A. Bansil, N. Nagaosa, J. Zaanen, T.P. Devereaux, and Z.-X. Shen, Hierarchy of Multiple Many-Body Interaction Scales in High-Temperature Superconductors, Phys. Rev. B 75, 174506 (2007). K. Tanaka, W.S. Lee, D.H. Lu, A. Fujimori, T. Fujii, Risdiana, I. Terasaki, D.J. Scalapino, T.P. Devereaux, Z. Hussain, and Z.-X. Shen, Distinct Fermi-Momentum Dependent Energy Gaps in Deeply Underdoped Bi-2212, Science 314, 1910 (2006). W. Meevasana, T.P. Devereaux, N. Nagaosa, Z.-X. Shen, and J. Zaanen, Overdamped c-axis Charge Dynamics and the Coupling to Polar Phonons in Cuprate Superconductors, Phys. Rev. B. 74, 174524 (2006). Y. Chen, A. Iyo, W. Yang, X. Zhou, D. Lu, H. Eisaki, T. P. Devereaux, Z. Hussain, and Z.-X. Shen, Anomalous Fermi-Surface Dependent Pairing in a Self-Doped High-Tc Superconductor, Phys. Rev. Lett. 97, 236401 (2006). Jian-Xin Zhu, K. McElroy, J. Lee, T. P. Devereaux, Qimiao Si, J. C. Davis, A. V. Balatsky, Effects of t1 Scattering on Fourier-Transformed Inelastic Tunneling Spectra in High-Tc Cuprates with Bosonic Modes, Phys. Rev. Lett. 97, 177001 (2006). W. Meevasana, N.J.C. Ingle, D.H. Lu, J.R. Shi, F. Baumberger, K.M. Shen, W.S. Lee, T. Cuk, H. Eisaki, T.P. Devereaux, N. Nagaosa, J. Zaanen, and Z.-X. Shen, Doping Dependence of the Coupling of Electrons to Bosonic Modes in the Single-Layer High-Temperature Bi2Sr2CuO6 Superconductor, Phys. Rev. Lett. 96, 157003 (2006). A. V. Chubukov, T. P. Devereaux and M. V. Klein, Resonance Mode in B1g Raman Scattering a Way to Distinguish Between Spin-Fluctuation and Phonon-Mediated d?~H~Rwave Superconductivity, Phys. Rev. B 73, 094512 (2006). Jian-Xin Zhu, A. V. Balatsky, T. P. Devereaux, Q. Si, J. Lee, K. McElroy, and J. C. Davis, Fourier-Transformed Inelastic STM Tunneling into High-Temperature Cuprates with Bosonic Modes, Phys. Rev. B 73, 014511 (2006). W.S. Lee, D.H. Lu, W.L. Yang, T. Cuk, K.M. Shen, X.J. Zhou, W. Meevasana, C. T. Lin, J.-i. Shimoyama, T. P. Devereaux, and Z. X. Shen, Band Renormalization Effect in Bi2Sr2Ca2Cu3O10+d, invited paper to appear in High Tc Superconductors and Related Transition Metal Oxides, eds. A. Bussmann-Holder and H. Keller (Springer, Berlin, 2007). T. P. Devereaux and R. Hackl, Inelastic Light Scattering in Strongly Correlated Materials, Reviews of Modern Physics 79, 175 (2007). X. J. Zhou, T. Cuk, T. Devereaux, N. Nagaosa, and Z.-X. Shen, cond-mat/0604284, Polaronic Behavior and Electron-Phonon Coupling in High Temperature Cuprate Superconductors as Revealed from Angle-Resolved Photoemission Spectroscopy, to appear in a book by J. R. Schrieffer on Superconductivity. PI Pat Diamond Global gyrokinetic particle simulations with kinetic electrons, Z. Lin, Y. Nishimura, Y. Xiao, I. Holod, W. L. Zhang and L. Chen, Plasma Phys. Control. Fusion 49, December, 2007. Nonlinear gyrokinetic theory of toroidal momentum pinch, T. S. Hahm, P. H. Diamond, O. D. Gurcan, G. Rewoldt, PHYSICS OF PLASMAS 14, 072302 (2007). Spatial and spectral evolution of turbulence, O. D. Gurcan, P. H. Diamond, T. S. Hahm, PHYSICS OF PLASMAS 14, 055902 (2007). Intrinsic rotation and electric field shear, O. D. Gurcan, P. H. Diamond, T. S. Hahm, R. Singh, PHYSICS OF PLASMAS 14, 042306 (2007). Global electromagnetic simulation with kinetic electrons, Y. Nishimura, Z. Lin, and L. Chen, to appear in Comm. Comput. Phys. 2007. Resonant and Non-Resonant Particle Dynamics in Alfven Mode Excitations, F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48, 537-556 (2006). PI Dimitre Dimitrov D. A. Dimitrov, D. L. Bruhwiler, D. Smithe, P. Messmer, and J. R. Cary, D. Kayran and I. Ben-Zvi, "INITIAL 3D VORPAL SIMULATIONS FOR RF ELECTRON GUN MODELING", in Proceeding of 41st Advanced ICFA Beam Dynamics Workshop onEnergy Recovery Linacs, Daresbery Laboratory, UK (2007), to be published. PI Chris Ding Tseng, Y. H. and Ding, C.H.Q. (2007), 'Efficient parallel I/O in Community Atmosphere Model (CAM)', International Journal of High Performance Computing Applications (in press). PI Viatcheslav Dobrovitski "Dynamical control of electron spin coherence in a quantum dot: A theoretical study", W. Zhang, V. V. Dobrovitski, L. F. Santos, L. Viola, and B. N. Harmon, Phys. Rev. B 75, 201302 (Rapid Communication) (2007) "Suppression of electron spin decoherence in a quantum dot" W. Zhang, V. V. Dobrovitski, B. N. Harmon, L. F. Santos, and L. Viola, J. Mod. Opt. B (in print) "Extending quantum coherence time of an electron spin in a quantum dot via dynamical decoupling method" N. P. Konstantinidis, W. Zhang, V. V. Dobrovitski, L. F. Santos, L. Viola, and B. N. Harmon Phys. Rev. B (submitted) PI Sebastian Doniach Chu, V.B., Y. Bai, J. Lipfert, D. Herschlag, and S. Doniach. Evaluation of Ion Binding to DNA Duplexes Using a Modified Poisson Boltzmann Theory. Biophysical Journal 93, 2007. Bai, Y., K. Travers, V. Chu, J. Lipfert, S. Doniach, D. Herschlag. Quantitative and comprehensive decomposition of the ion atmosphere around nucleic acids. 2007. Accepted to Journal of the American Chemical Society. Bai, Y., Chu, V.B., Lipfert, J., Pande, V.S., Herschlag, D., Doniach, S. Experimental and computational reconstruction of the unfolded state ensemble in nucleic acid folding. 2007. Submitted to the Proceedings of the National Academy of Sciences USA. PI William Dorland Dissipation-Scale Turbulence in the Solar Wind, Howes, Gregory G.; Cowley, Steven C.; Dorland, William; Hammett, Gregory W.; Quataert, Eliot; Schekochihin, Alexander A., accepted for publication in AIP Conference Proceedings on "Turbulence and Nonlinear Processes in Astrophysical Plasmas" (2007). Verification of gyrokinetic delta-f simulations of electron temperature gradient turbulence, W. M. Nevins, S. E. Parker, Y. Chen, J. Candy, A. Dimits, W. Dorland, G. W. Hammett, and F. Jenko Phys. Plasmas 14, 084501 (2007). Turbulent transport of alpha particles in reactor plasmas , C. Estrada-Mila, J. Candy, and R. E. Waltz Phys. Plasmas 13, 112303 (2006). Astrophysical Gyrokinetics: Basic Equations and Linear Theory, Howes, Gregory G.; Cowley, Steven C.; Dorland, William; Hammett, Gregory W.; Quataert, Eliot; Schekochihin, Alexander A., The Astrophysical Journal, Volume 651, Issue 1, pp. 590-614 (2006). J. F. Drake, H. Che, M. A. Shay and M. Swisdak, Electron acceleration from contracting magnetic islands during reconnection, Nature 443, 553, 2006. P. Cassak, J. F. Drake and M. A. Shay, On the Onset of fast magnetic reconnection, Astrophys. J. Lett. 644, L145, 2006. M. A. Shay, J. F. Drake, and W. Dorland, Equation free projective integration: A multiscale method applied to a plasma ion acoustic wave, J. Comp. Phys. 226, 571, 2007. P. Cassak, J. F. Drake and M. A. Shay, Catastrophic onset of fast magnetic reconnection with a guide field, Phys. Plasmas 14, 054502, 2007. P. Cassak, J. F. Drake, M. A. Shay and B. Eckhardt, Onset of fast magnetic reconnection, Phys. Rev. Lett. 98, 215001, 2007. M. A. Shay, J. F. Drake and M. Swisdak, Two-scale structure of the electron dissipation region during collisionless reconnection, Phys. Rev. Lett. 99, 155002, 2007. B. N. Rogers, S. Kobayashi, P. Ricci, W. Dorland, J. F. Drake and T. Tatsuno, Gyrokinetic simulations of collisionless magnetic reconnection, Phys. Plasmas 14, 092110, 2007. R.G. Kleva and P.N. Guzdar, Repetitive transport bursts in simulations of edge-localized modes in tokamaks, Phys. Plasmas 13, 072509, 2006. R.G. Kleva and P.N. Guzdar, Zonal flow sawteeth and the time period between edge-localized transport bursts in tokamaks, Phys. Plasmas 14, 012303, 2007. PI John Drake Identification of human-induced changes in atmospheric moisture content B. D. Santera,b, C. Mearsc, F. J. Wentzc, K. E. Taylora, P. J. Glecklera, T. M. L. Wigleyd, T. P. Barnette, J. S. Boylea, W. Bru( ggemannf, N., Proceeding of the National Academies of Science, 15248~X53 82537 PI Barry Dunietz Conductance of a cobalt(II) terpyridine complex based molecular transistor: A computational analysis Trilisa M. Perrine and Barry D. Dunietz, Journal of Physical Chemistry A: Lester Special Issue, (accepted). Gating of single molecule transistors: Combinging field-effect and chemical control" Trilisa M. Perrine, Ron G. Smith, Christopher Marsh, and Barry D. Dunietz (submitted). PI Michel Dupuis A. Venkatnathan, R. Devanathan and M. Dupuis, "Atomistic Simulations of Hydrated Nafion and Temperature Effects on Hydronium Ion Mobility", J. Phys. Chem. B 111 (2007) 7234. R. Devanathan, A. Venkatnathan and M. Dupuis, "Atomistic Simulation of Nafion Membrane: 1. Effect of Hydration on Membrane Nanostructure", J. Phys. Chem. B 111 (2007) 8069. R. Devanathan, A. Venkatnathan and M. Dupuis, "Atomistic Simulation of Nafion Membrane. 2. Dynamics of Water Molecules and Hydronium Ions", J. Phys. Chem. B (2007) DOI: 10.1021/jp0761057. V-A. Glezakou, M. Dupuis and C. J. Mundy, "Acid/base equilibria in clusters and their role in proton exchange membranes: computational insight", Phys. Chem. Chem. Phys. (2007) DOI: 10.1039/b709752b. PI Charlotte Elster Three-Body Elastic and Inelastic Scattering at Intermediate Energies, H. Liu, Ch. Elster, W. Glockle, nNucl. Phys. A790, 262c (2007), and nucl-th/0610006. Three-Body Scattering at Intermediate Energies, H. Liu, Ch. Elster, W. Glockle, Phys. Rev. C 72, 054003 (2005). PI Roland Faller Qi Sun and Roland Faller: Phase Separation in Polyisoprene/Polystyrene Blends by a Systematically Coarse-Grained Model, J Chem Phys 126(14), 144908 (2007) Jayeeta Ghosh and Roland Faller: State Point Dependence of Systematically Coarse-Grained Potentials, Molecular Simulation 33(9 & 10), 759-767 (2007) Qi Sun, Florence R. Pon and Roland Faller: Multiscale modeling of Polystyrene in various environments, Fluid Phase Equilibria 261(1-2), 35-40 (2007) Florence R. Pon, Qi Sun and Roland Faller: Static and Dynamic Heterogeneities in Polyisoprene-Polystyrene Blends: A Molecular Dynamics Simulation submitted PI William Fawley C. Schroeder et al., "Design of a free-electron laser driven by the LBNL laser-plasma-accelerator", Proc. 2007 Frontiers in FEL Physics Workshop, Elba Island; submitted for publication in Nucl. Inst. Methods A. W.M. Fawley, "Production of Ultrashort FEL XUV Pulses via a Reverse Undulator Taper", Proc. 2007 Frontiers in FEL Physics Workshop, Elba Island; submitted for publication in Nucl. Inst. Methods A. PI Andrew Felmy James R. Rustad and E.J Bylaska. Ab initio calculation of isotopic fractionation in aqueous boron, Journal Of The American Chemical Society 129 (8): 2222 (2007). Eric J. Bylaska, Marat Valiev, James R. Rustad, and John H. Weare, Structures of the Hydration Shells of the Al3+ Ion. The Journal of Chemical Physics, 126, 104505(2007). Eric J. Bylaska, Michel Dupuis, and Paul G. Tratnyek, Electron Transfer Reactions of Polychlorinated Ethylenes: Concerted Versus Stepwise Cleavages, Journal of Physical Chemistry A, submitted 2007. Marat Valiev, Eric J. Bylaska, Michel Dupuis, and Paul G. Tratnyek, Combined ab-initio quantum mechanical and molecular mechanics Studies of the Electron Transfer Reactions of Carbon Tetrachloride, Journal of Physical Chemistry A, submitted October 2007. Stuart Bogatko, Eric J. Bylaska, and John H. Weare, Structures and Dynamics in Fe3+ Hydration shells: relation to Al3+ ion hydration structure. Journal of Physical Chemistry, B, submitted October 2007. Patrick Nichols, Eric J. Bylaska, and Wibe de Jong, Equatorial and Apical Solvent Shells of the UO22+ Ion, Journal of Chemical Physics, submitted October 2007. PI Wu-chun Feng "Semantics-based Distributed I/O for mpiBLAST.", Pavan Balaji, Wu-chun Feng, Jeremy Archuleta, Heshan Lin, Rajkumar Kettimuthu, Rajeev Thakur, Xiaosong Ma. In Proceedings of the ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, Salt Lake City, Utah, USA, February 2008. "ParaMEDIC: Parallel Metadata Environment for Distributed I/O and Computing.". Pavan Balaji, Wu-chun Feng, Jeremy Archuleta, and Heshan Lin. International Storage Challenge Award. In Proceedings of the ACM/IEEE Supercomputing Conference, Reno, Nevada, November 2007. PI Graham Fleming Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancal, Y.-C. Cheng, R. E. Blankenship, G. R. Fleming, Nature, 446, 782 (2007). Coherence dynamics in photosynthesis: protein protection of excitonic coherence. H. Lee, Y.-C. Cheng, G. R. Fleming. Science 316, 1462 (2007). Efficient Simulation of Three-Pulse Photon-Echo Signals with Application to the Determination of Electronic Coupling in a Bacterial Photosynthetic Reaction Center, Y.-C. Cheng, H. Lee, and G. R. Fleming, J. Phys. Chem. A, 111, 9499, (2007). Elucidation of population and coherence dynamics using cross-peaks in two-dimensional electronic spectroscopy, Y.-C. Cheng, G. S. Engel, and G. R. Fleming, Chem. Phys. (2007); doi:10.1016/j.chemphys.2007.07.049 (article in press). Cross-peak-specific two-dimensional electronic spectroscopy, E. L. Read, G. S. Engel, T. R. Calhoun, T. Mancal, T. K. Ahn, R. E. Blankenship, G. R. Fleming, PNAS 104, 14203 (2007). PI Michael Fox-Rabinovitz Fox-Rabinovitz, M.S., J. Cote, M. Deque, B. Dugas, J. McGregor, 2006: International SGMIP Stretched-Grid Model Intercomparison Project): Multi-model ensemble results for prognostic fields and precipitation, J. Geophys. Res., 111, D16104, doi:10.1029/2005JD006520. Fox-Rabinovitz, M.S., J. Cote, M. Deque, B. Dugas, J. McGregor, A. Belochitski, 2007: Stretched-Grid Model Intercomparison Project: Decadal Regional Climate Simulations with Enhanced Variable and Uniform Resolution GCMs, Meteorology and Atmospheric Physics, submitted. Krasnopolsky, V.M., and M.S. Fox-Rabinovitz, and Belochitski, 2007: Decadal Climate Simulations Using Accurate and Fast Neural Network Emulation of Full, Long- and Short Wave, Radiation, submitted to Monthly Weathr Review. PI Alberto Franceschetti J.M. An, A. Franceschetti, and A. Zunger, "The Peculiar Electronic Structure of PbSe Quantum Dots", Nano Letters 6, 2728 (2006). M. Califano, A. Franceschetti, and A. Zunger, "Lifetime and Polarization of the Radiative Decay of Excitons in CdSe Nanocrystal Quantum Dots", Phys. Rev. B 75, 115401 (2007). A. Franceschetti and M.C. Troparevsky, "Radiative Recombination of Triexcitons in CdSe Colloidal Quantum Dots", J. Phys. Chem. C (Letters) 111, 6154 (2007). J.M. An, A. Franceschetti, and A. Zunger, "The Excitonic Exchange Splitting and Radiative Lifetime in PbSe Quantum Dots", Nano Letters 7, 2129 (2007). J.M. An, A. Franceschetti, and A. Zunger, "Electron and Hole Addition Energies in PbSe Quantum Dots", Phys. Rev. B 76, 045401 (2007). Q.Z. Zhao, P.A. Graf, W.B. Jones, A. Franceschetti, J. Li, L.W. Wang, and K. Kim, "Shape Dependence of Band-Edge Exciton Fine Structure in CdSe Nanocrystals", Nano Letters, nl0713070 (2007). A. Franceschetti, "First-Principles Calculations of the Temperature Dependence of the Band Gap of Si Nanocrystals", Phys. Rev. B (Rapid Communications) 76, 161301 (2007). J.M. An, A. Franceschetti, and A. Zunger, "Pauli Blocking versus Electrostatic Attenuation of Optical Transition Intensities in Charged PbSe Quantum Dots, Phys. Rev. B (Rapid Communications), in press. PI Stuart Freedman T. Araki et al.[KamLAND Collaboration], "Search for the Invisible Decay of Neutrons with KamLAND", Phys. Rev. Lett. 96, 101802 (2006). S. Enomoto et al., "Neutrino geophysics with KamLAND and future prospects", Earth and Planetary Sci. Lett. 258, 147, (2007). S. Abe et al. [KamLAND Collaboration], "Precision Measurement of Neutrino Oscillation Parameters with KamLAND", Submitted to Physical Review Letters, arXiv:0801.4589 PI Arthur Freeman R. Saniz, B. Barbiellini, P. M. Platzman, and A. J. Freeman, Physisorption of positronium on quartz surfaces, Phys. Rev. Lett. 99, 096101 (2007). S. H. Rhim, R. Saniz, J. Yu, L.-H. Ye, A. J. Freeman, Interface electronic structure, two-dimensional metallicity, and possible superconductivity in CuCl/Si superlattices, Phys. Rev. B, in press. PI Alex Friedman R. H. Cohen, A. Friedman, D.P. Grote and J.-L. Vay, "Large-Timestep Mover for Particle Simulations of Arbitrarily Magnetized Species," Nuclear Instruments and Methods in Physics Research A 577, 52 (2007). J. E. Coleman, A. Friedman, W. L. Waldron, F. M. Bieniosek, R. J. Briggs, D. P. Grote, E. Henestroza, P. K. Roy, P. A. Seidl, and S. S. Yu, "Beam Experiments on the Pulse Line Accelerator," Nuclear Instruments and Methods in Physics Research A, 577, 197 (2007). A. Friedman, "Overview of Theory and Simulation in the Heavy Ion Fusion Science Virtual National Laboratory," Nuclear Instruments and Methods in Physics Research A, 577, 37 (2007). D. P. Grote, J. W. Kwan, and G. A. Westenskow, "Design and Modeling of the Multi-Beamlet Injector," Nuclear Instruments and Methods in Physics Research A, 577, 58 (2007). I. Haber, D. Feldman, R. Fiorito, A. Friedman, D. P. Grote, R. A. Kishek, B. Quinn, M. Reiser, J. Rodgers, P. G. Ohea, D. Stratakis, K. Tiana, J.-L. Vay, and M. Walter, "Measurement and Simulation of the Time- Dependent Behavior of the UMER Source," Nuclear Instruments and Methods in Physics Research A, 561, 157 (2006). S. M. Lund, S. H. Chilton, E. P. Lee, "Efficient Computation of Matched Solutions of the KV Envelope Equation for Periodic Focusing Lattices," Phys. Rev. ST Accel. Beams 9, 064201 (2006). S. M. Lund, J. J. Barnard, B. Bukh, S. R. Chawla, and S. H. Chilton, "Space-Charge Transport Limits of Ion Beams in Periodic Quadrupole Focusing Channels," Nuclear Instruments and Methods in Physics Research A 577, 203 (2007). W. M. Sharp, D. P. Grote, R. H. Cohen, A. Friedman, J.-L. Vay, P. A. Seidl, P. K. Roy, J. E. Coleman, J. Armijo and I. Haber, "Simulating electron clouds in high-current ion accelerators with solenoid focusing," Nuclear Instruments and Methods in Physics Research A 577, 146 (2007). J. L. Vay, M. A. Furman, P. A. Seidl, R. H. Cohen, A. Friedman, D. P. Grote, M. Kireef Covo, A. W. Molvik, P. H. Stoltz, S. Veitzer, and J. P. Verboncoeur, "Self-consistent simulations of heavy-ion beams interacting with electron-clouds," Nuclear Instruments and Methods in Physics Research A 577, 69 (2007). J.-L. Vay, "Noninvariance of Space- and TIme-Scale Ranges under a Lorentz Transformation - Implications for the Study of Relativistic Interactions," Phys. Rev. Lett. 98, 130405 (2007). PI Inez Fung Lee JE, Fung I, DePaolo DJ, Henning, CC. Analysis of the global distribution of water isotopes using the NCAR atmospheric general circulation model. JOURNAL OF GEOPHYSICAL RESEARCH-ATMOSPHERES. Vol 112, issue D16, article D16306, 2007. Fung, I. Challenges of climate modeling. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B. Vol 7, issue 3, pp 543-551. 2007. PI Miguel Furman M. A. Furman, "A preliminary assessment of the electron cloud effect for the FNAL main injector upgrade," published by invitation in the New Journal of Physics Focus Issue: Accelerator and Beam Physics, New J. Phys. 8 (2006) 279 PI Giulia Galli-Gygi Electronic properties of MoS2 nanoparticles, T.Li and G.Galli, J.Phys.Chem.C 2007 (in press) Single functional group interactions with individual carbon nanotubes, Raymond W. Friddle1,Melburne C. LeMieux1,Giancarlo Cicero, Alexander B. Artyukhin, Vladimir V. Tsukruk, Jeffrey C. Grossman, Giulia Galli, and Aleksandr Noy, Nature Nanotechnology 2007 (in press). Structural characterization of aldehyde-terminated self-assembled monolayers", A.Riposan, Y. Li, Y.Tan, G. Galli and G. Liu J.Phys.Chem. A 2007 (accepted). A First-Principles Study of Aromatic Isocyanide Self-Assembled Monolayers on Au(111) Surface, Yan Li and Deyu Lu, Giulia Galli, Sally A. Swanson and J. Campbell Scott (submitted for publication). PI Alan Garfinkel Sato D, Shiferaw Y, Garfinkel A, Weiss JN, Qu Z, Karma A.Spatially discordant alternans in cardiac tissue: role of calcium cycling. Circ Res. 2006 Sep 1;99(5):520-7. Baher A, Qu Z, Hayatdavoudi A, Lamp ST, Yang MJ, Xie F, Turner S, Garfinkel A, Weiss JN. Short-term Cardiac Memory and Mother Rotor Fibrillation. Am J Physiol Heart Circ Physiol. 2006 Aug 4. Weiss JN, Karma A, Shiferaw Y, Chen PS, Garfinkel A, Qu Z. From pulsus to pulseless: the saga of cardiac alternans. Circ Res. 2006 May 26;98(10):1244-53. Qu Z, Garfinkel A, Weiss JN. Vulnerable window for conduction block in a one-dimensional cable of cardiac cells, 1: single extrasystoles. Biophys J. 2006 Aug 1;91(3):793-804. Qu Z, Garfinkel A, Weiss JN. Vulnerable window for conduction block in a one-dimensional cable of cardiac cells, 2: multiple extrasystoles. Biophys J. 2006 Aug 1;91(3):805-15. Weiss JN, Qu Z, Chen PS, Lin SF, Karagueuzian HS, Hayashi H, Garfinkel A, Karma The dynamics of cardiac fibrillation. Circulation. 2005 Aug 23;112(8):1232-40. PI Bruce Garrett S. Y. Du, J. S. Francisco, G. K. Schenter, and B. C. Garrett, "Ab initio and Analytical Intermolecular Potential for ClO-H2O", Journal of Chemical Physics 126, 114304 (2007) S. Kathmann, G. Schenter and B. Garrett, "The critical role of anharmonicity in aqueous ionic clusters relevant to nucleation," J. Phys. Chem. C 111, 4977 (2007) M. Valiev, BC Garrett, MK Tsai, K Kowalski, SM Kathmann, GK Schenter, and M Dupuis. "Hybrid Approach for Free Energy Calculations with High-Level Methods: Application to the S(N)2 Reaction of CHCl3 and OH- in Water." Journal of Chemical Physics 127, 51102 (2007) S. Bulusu, S. Yoo, E. Apr? S. S. Xantheas, X. C. Zeng, "The lowest energy structures of water clusters (H2O)11 and (H2O)13" Journal of Physical Chemistry A 110, 11781 (2006) G. S. Fanourgakis, V. Tipparaju, J. Nieplocha and S. S. Xantheas, "An efficient parallelization scheme for molecular dynamics simulations with many-body, flexible, polarizable empirical potentials: Application to water" Theoretical Chemistry Accounts 117, 73 (2007) T. Pankewitz, A. Lagutschenkov, G. Niedner-Schatteburg, S. S. Xantheas, Y.-T. Lee, "The Infrared Spectrum of NH4+(H2O): Evidence for mode specific fragmentation" Journal of Chemical Physics 126, 074307 (2007) M. N. Slipchenko, B. G. Sartakov and A. F. Vilesov, S. S. Xantheas, "A study of NH stretching vibrations in small ammonia clusters by infrared spectroscopy in He droplets and ab-initio calculations" Roger E. Miller memorial issue (invited), Journal of Physical Chemistry A 111, 7460 (2007) L. Rubio-Lago, D. Zaouris, Y. Sakellariou, D. Sofikits and T. N. Kitsopoulos, F. Wang and X. Yang, B. Cronin and M. N. R. Ashfold, S. S. Xantheas, "Photofragment slice imaging studies of pyrrole and the Xepyrrole cluster" Journal of Chemical Physics 127, 064306 (2007) T. A. Blake, E. D. Glendening, R. L. Sams, S. W. Sharpe and S. S. Xantheas, "High Resolution Infrared Spectroscopy in the 1200 to 1300 cm-1 Region and Accurate Theoretical Estimates for the Structure and Ring Puckering Barrier of Perfluorocyclobutane" Thom H. Dunning Jr. Festschrift (invited), Journal Physical Chemistry A (in press, web release date: 06-Jul-2007, DOI:10.1021/jp072521f) M. V. Kirov, G. S. Fanourgakis and S. S. Xantheas, "A Strong-Weak-Effective-Bond?(SWEB) discrete model for the combinatorial optimization of polyhedral water clusters: Validation from electronic structure calculations and discovery of a new lowest energy isomer of the pentagonal dodecahedron (H2O)20 cluster" Journal of Chemical Physics (submitted) S. S. Xantheas, "The elusive structure of H3O+(H2O)20" Journal of Chemical Physics (submitted) G. S. Fanourgakis and S. S. Xantheas, "Development of Transferable Interaction Potentials for Water: IV. Quantitative description of vibrational spectra of clusters and liquid water" Journal of Chemical Physics (submitted) PI Jose Gascon Computational studies of the O2-evolving complex of photosystem II and biomimetic oxomanganese complexes Coordination Chemistry Reviews, In Press, Corrected Proof, Available online 16 September 2007 PI Silvina Gatica S. M. Gatica, H. I. Li, R. A. Trasca, M. W. Cole and R. D. Diehl, Xe Adsorption on a C60 Monolayer on Ag(111), submitted to Phys. Rev. B. (2007) PI Cameron Geddes D.A. Dimitrov, R.E. Giancone, D.L. Bruhwiler, R. Busby, J.R. Cary, C.G.R. Geddes, E. Esarey, W.P. Leemans, "Coupling of laser energy into plasma channels", Phys. Plasmas 14, 043105 (2007). P. Messmer and D. Bruhwiler, Simulating laser pulse propagation and low-frequency wave emission in capillary plasma channel systems with a ponderomotive guiding center model," Phys. Rev. ST/AB 9, 031302 (2006). E. Esarey, C.B. Schroeder, E. Cormier-Michel, B.A. Shadwick, C.G.R. Geddes, and W.P. Leemans, "Thermal effects in plasma-based accelerators", Physics of Plasmas 14, 056707 (2007). K. Nakamura, B. Nagler, Cs. Toth, C. G. R. Geddes, C. B. Schroeder, E. Esarey, W. P. Leemans, A.J. Gonsalves, S.M. Hooker, "GeV electron beams from a centimeter-scale channel guided laser wakefield accelerator", Phys. Plasmas 14, 056708. C.G.R. Geddes, D. Bruhwiler, J.R. Cary, E. Cormier-Michel, E.Esarey, C.B. Schroeder, W.A. Isaacs, N. Stinus, P. Messmer, A. Hakim, K. Nakamura, A.J. Gonsalves, D. Panasenko, G.R. Plateau, Cs. Toth, B.Nagler, J. van Tilborg, T. Cowan, S. M. Hooker and W.P. Leemans, Laser wakefield simulations towards development of compact particle accelerators, Journal of Physics: Conference Series, 78, 012021 (2007). J R Cary, P Spentzouris, J Amundson, L McInnes, M Borland, B Mustapha, B Norris, P Ostroumov, Y Wang, W Fischer, A Fedotov, I Ben-Zvi, R Ryne, E Esarey, C Geddes, J Qiang, E Ng, S Li, C Ng, R Lee, L Merminga, H Wang, D L Bruhwiler, D Dechow, P Mullowney, P Messmer, C Nieter, S Ovtchinnikov, K Paul, P Stoltz, D Wade-Stein, W B Mori, V Decyk, C K Huang, W Lu, M Tzoufras, F Tsung, M Zhou, G R Werner, T Antonsen, T Katsouleas, "COMPASS, the COMmunity Petascale project for Accelerator Science and Simulation," Journal of Physics: Conference Series, 78, 012009 (2007). F.S. Tsung, T. Antonsen, D.L. Bruhwiler, J.R. Cary, V.K. Decyk, E. Esarey, C.G.R. Geddes, C. Huang, A. Hakim, T. Katsouleas, W. Lu, P. Messmer, W.B. Mori, M. Tzoufras and J. Vieira, "Three-Dimensional Particle-in-Cell Simulations of Laser Wakefield Experiments," J. Physics: Conf. Series. 78, 012077 (2007). D.L. Bruhwiler, T. Antonsen, J.R. Cary, J. Cooley, V.K. Decyk, E. Esarey, C.G.R. Geddes, C. Huang, A. Hakim, T. Katsouleas, P. Messmer, W.B. Mori, F.S. Tsung, J. Vieira and M. Zhou, "Towards the petascale in electromagnetic modeling of plasma-based accelerators for high-energy physics," Journal of Physics: Conference Series 46, 215 (2006). C.G.R. Geddes, K. Nakamura, G.R. Plateau, Cs. Toth, E. Cormier-Michel, E. Esarey, C.B. Schroeder, J.R. Cary, and W.P. Leemans, "Stable ultrafast electron bunches with low absolute momentum spread from plasma down ramp injection," submitted to PRL. PI Ahmed Ghoniem Wee, Dh., and Ghoniem, A.F., "Modified interpolation kernels and treating diffusion and remeshing in vortex methods," J. Comput. Phys., 213, 2006, pp. 239-263. Marzouk, Y.M. and Ghoniem, A.F., "Vorticity structure and evolution in a transverse jet," J. Fluid Mech., 575:267-305, 2007. Schlegel, F., Wee, Dh., and Ghoniem, A.F., "A fast 3d particle method for simulations of buoyant flows", J. Comput Phys, under review. PI James Glimm E. George, J. Glimm, X. L. Li, Y. H. Li and X. F. Liu,The Influence of Scale-Breaking Phenomena on Turbulent Mixing Rates, Phys. Rev. E, 73, 016304-1-016304-5, 2006. J. Glimm and X. L. Li, Recent Progress in Turbulent Mixing, Proceedings of the 10th International Workshop on the Physics of Compressible Turbulent Mixing, 17-21 July, Paris, France, Edited by M. Legrand and M. Vandenboomgarde, 2006. X. F. Liu, E. George, W. Bo and J. Glimm, Turbulent Mixing with Physical Mass Diffusion, Phys. Rev. E, 73, 056301-1-056301-8, 2006. J.-J. Liu, J. Glimm and X.-L. Li, A Conservative Front Tracking Method, Hyperbolic Problems: Theory, Numerics, and Applications, Edited by F. Asakura, H. Aiso, S. Kawashima, Matsumura A, S. Nishibata and K. Nishihara, Yokohama Publishers, Osaka, Japan, 57-62, 2006. Jinjie Liu, Hyun-Kyun Lim, James Glimm and Xiaolin Li, A Conservative Front Tracking Method in N-Dimensions, J. of Sci. Comp, in press, 2006. Z. L. Xu, M. Kim, W. Oh, J. Glimm, R. Samulyak, X. L. Li and C. Tzanos, Discrete Bubble Modeling of Unsteady Cavitating Flow, International Journal for Multiscale Computational Engineering, 4, 377-389, 2006. J. Glimm, B. Fix, X.-L. Li, J.-J. Liu, X.-F. Liu, T.-S. Liu, R. Samulyak and Z.-L. Xu, Front Tracking under TSTT, Proceedings of the IGPP-CalSpace Conference, Astronomical Society of the Pacific, 15,359, 2007. H. Lee, H. Jin, Y. Yu and J. Glimm, On the Validation of Turbulent Mixing Simulations of Rayleigh-Taylor Mixing, submitted to Phys of Fluids, 2007. H. Lim, Y. Yu, H. Jin, D. Kim, H. Lee, J. Glimm, X.-L. Li and D. H. Sharp,Multi Scale Models for Fluid Mixing, submitted to Special Issue of CMAME, 2007. X. F. Liu, Y. H. Li, J. Glimm and X. L. Li, A Front Tracking Algorithm For Limited Mass Diffusion, accepted by J. of Comp. Phys., 2007. T. Lu, R. Samulyak and J. Glimm, Direct Numerical Simulation of Bubbly Flows and its Applications, Phys. Fluid Eng., 129, 595-604, 2007. W. Bo, B. Cheng, J. Du, B. Fix, E. George, J. Glimm, J. Grove, X. Jia, H. Jin, H. Lee, Y. Li, X. Li, X. Liu, D. H. Sharp, L. Wu, Yan Yu, Recent Progress in the Stochastic Analysis of Turbulent Mixing, Contemporary Mathematics, 429, 33-44, 2007. PI Yousry Gohar Yousry Gohar, "Fusion Shield Design and Optimization Demonstration," Proceedings of The American Nuclear Society 14th Biennial Topical Meeting of the Radiation Protection and Shielding Division, Carlsbad New Mexico, USA. April 2-6, 2006. PI Yadin Goldschmidt 1. "Dynamic and Thermodynamic Properties of Porous Vortex Matter in Bi2Sr2CaCu2O8 in an Oblique Magnetic Field", by N. Avraham, Y.Y. Goldschmidt, J. T. Liu, Y. Myasoedov, M. Rappaport, E. Zeldov, C. J. van der Beek,M. Konczykowski and T. Tamegai, Phys. Rev. Lett. 99, 087001 (2007). 2. "Langevin Dynamics of the vortex matter two-stage melting transition in Bi2Sr2CaCu2O8 in the presence of straight and of tilted columnar defects" byYadin Y. Goldschmidt and Jin-Tao Liu, Phys. Rev. B, November 1, 2007, accepted, in print. PI Giorgio Gratta Naoko Kurahashi and Giorgio Gratta, "High-Requency Ambient Noise as Background to Deep Ocean Transient Signal Detection", submitted to The Journal of Acoustical Society of America, arXiv:0712.1833v1 [physics.ao-ph PI Stephen Gray Quantum dynamics study of the dissociative phototodetachment of HOCO- S. Zhang, D. M. Medvedev, E. M. Goldfield, and S. K. Gray., J. Chem. Phys. 125, 164312 (2006) (8 pages). Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, Proc. Nat. Acad. Sci. (USA) 103, 17143-17148 (2006). Nanostructured plasmonic sensors M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. Rogers, and R. G. Nuzzo, Chem. Rev., in press (2007). Tailoring the sensing capabilities of nanohole arrays in gold films with Wood's anomaly-surface plasmon polaritons J. M. McMahon, J. Henzie, T. Odom, G. C. Schatz, and S. K. Gray, Optics Express, submitted (2007). PI Chris Greene "Collisional stability of fermionic Feshbach molecules." J. J. Zirbel, K.-K. Ni, S. Ospelkaus, J. P. D'Incao, C. E. Wieman, J. Ye, D. S. Jin (submitted to PRL) "Recombination of H3+ Ions in the Afterglow of a He-Ar-H2 Plasma." J. Glosik, I. Korolov, R. Plasil, O. Novotny, T. Kotrik, P. Hlavenka, J. Varju, C.H. Greene, V. Kokoouline, and I.A. Mikhailov. Submitted to Phys. Rev. Lett. (2007). "Dissociative recombination of H3+ in the ground and excited states." S.Fonseca dos Santos, V.Kokoouline, and C.H.Greene. J. Chem. Phys. 127, 124309 (2007). "Near threshold rotational excitation of molecular ions by electron impact." A. Faure, V.Kokoouline, J. Tennyson, and C.H.Greene. J. Phys. B: At. Mol. Opt. Phys. 39, 4261 (2006). "Method for finding of three-body resonances using hyperspherical coordinates and slow variable representation." J. Blandon, V. Kokoouline, and F. Masnou-Seeuws. Phys. Rev. A 75, 042508 (2007) "FERM3D: A finite element R-matrix general electron-molecule scattering code." S. Tonzani, Computer Physics Communications 176, 146 (2007) "Diffraction in low-energy electron scattering from DNA: bridging gas phase and solid state theory" L. Caron, S. Tonzani, C. H. Greene and L. Sanche. J. Chem. Phys. (submitted) PI Jeffrey Grossman Yosuke Kanai and Jefferey C. Grossman "Insights on Interfacial Charge Transfer Across P3HT/Fullerene Photovoltaic Heterojunction from Ab Initio Calculations" Nano Letters, 7, 1967 (2007). Joo-Hyoung Lee, John Reed, Giulia Galli, and Jeffrey C. Grossman "Lattice thermal conductivity of porous Si:molecular dynamics study" to appear in Applied Physics Letters. Zhigang Wu, J. B. Neaton, and Jeffrey C. Grossman "Quantum confinement and electronic properties of tapered silicon nanowires" under review at Physical Review Letters. Yosuke Kanai and Jeffrey C. Grossman "Interfacial Properties of the P3HT/Carbon-Nanotube Photovoltaic Heterojunction: A Density Functional Theory Study" submitted to Applied Physics Letters. PI Hua Guo S. Y. Lin and H. Guo, Phys. Rev. A, 74, 022703 (2006). Quantum state-to-state cross sections for atom-diatom reactions, A Chebyshev real wave packet approach. S. Y. Lin, D. Xie, and H. Guo, J. Chem. Phys., 125, 091103 (2006). Revelation of non-statistical behavior in HO2 vibration by a new ab initio potential energy surface. S. Y. Lin, H. Guo, P. Honvault, D. Xie, J. Phys. Chem. B, 110, 23641 (2006), Quantum dynamics of the H + O2 !w O + OH reaction on an accurate ab initio potential energy surface. D. Xie, C. Xu, T.-S. Ho, H. Rabitz, G. Lendvay, S. Y. Lin, and H. Guo, J. Chem. Phys., 126, 074315 (2007), Global analytical potential energy surfaces for HO2( ) based on high level ab initio calculations. A. Van Wyngarden, K. Mar, K. Boering, J. J. Lin, Y. T. Lee, S. Y. Lin, H. Guo, G. Lendvay, J. Am. Chem. Soc., 129, 2866 (2007). Non-statistical behavior of reactive scattering in the 18O + 32O2 isotope exchange reaction. S. Y. Lin, L. Banares, and H. Guo, J. Phys. Chem. A, 111, 2376 (2007), Differential and integral cross sections of the N(2D) + H2 !w NH + H reaction from exact quantum and quasi-classical trajectory studies. P. Honvault, S. Y. Lin, D. Xie, and H. Guo, J. Phys. Chem. A, 111, 5349 (2007), Differential and integral cross sections for the H + O2 !w HO + O combustion reaction. C. Xu, D. Xie, P. Honvault, S. Y. Lin, and H. Guo, J. Chem. Phys., 127, 024304 (2007), Rate constant for OH(2@) + O(3P) + O2( ) reaction on an improved ab initio potential energy surface and implications for the interstellar oxygen problem. PI William Gutowski Abiodun BJ, Gutowski WJ, and Prusa JM, 2007: Implementation of a non-hydrostatic, adaptive grid dynamics core in CAM3. Part II: dynamical influences on ITCZ behavior and tropical precipitation. Climate Dynamics, submitted. Abiodun BJ, Prusa JM, and Gutowski WJ, 2007: Implementation of a non-hydrostatic, adaptive grid dynamics core in CAM3. Climate Dynamics, submitted. Prusa JM, Smolarkiewicz PK and Wyszogrodzki AA, 2007: EULAG, a computational model for multiscale flows. Computers and Fluids, submitted. PI Stephan Haas Phys. Rev. Lett. 98, 257201 (2007) arXiv:0707.205 (submitted to Phys. Rev. Lett.) PI Bruce Harmon Add-drop filters using three-dimensional photonic crystals, P. Kohli, R. Biswas, G. Tuttle and K.-M. Ho, SPIE online newsroom DOI: 10.1117/2.1200707.0594 (2007). Photonic Band gap crystals, R. Biswas, M. Sigalas, C. M. Soukoulis, K.-M. Ho, G. Tuttle, Wiley Encyclopedia of Electrical and Electronics Engineering (John Wiley & Sons) (2007). Add-drop filters in 3-dimensional layer-by-layer photonic crystals using waveguides and resonant cavities, P Kohli, C. Christensen, J. Muehlmeier, R. Biswas, G. Tuttle, and K.-M. Ho, Appl. Phys. Lett. 89, 231103 (2006). Mechanisms underlying extraordinary transmission in sub-wavelength hole arrays, R. Biswas, S. Neginhal, C. G. Ding, I. Puscasu, E. Johnson, J. Opt. Soc. of America B 24, 2489-95 (October, 2007). Simulations of Sub-wavelength Metallo-dielectric Photonic Crystals for Gas Sensing, R. Biswas, I. Puscasu, M. Pralle, M. McNeal, J. Daly, A. Greenwald, E. Johnson, S. Neginhal, C.G. Ding, MRS Symp. Proceedings Symposium 952E, F 2.2 (2006) "Formation of Carbon Nanotube Semiconductor-Metal Intramolecular Junctions by Self-Assembly of Vacancy Defects", Gun-Do Lee, C. Z. Wang, Jaejun Yu, Euijoon Yoon, Nong-Moon Hwang, Kai-Ming Ho Phys. Rev. B 76, 165413 (2007). Interplay between indirect interaction and charge density wave in Pb adsorbed In(4x1)-Si(111)", M. Hupalo, T. L. Chan, C. Z. Wang, K. M. Ho, and M. C. Tringides, Phys. Rev. B 76, 045415 (2007). "The structure of ultra-thin H-passivated [112] silicon nanowires", N. Lu, C. V. Ciobanu, T. L. Chan, F.-C. Chuang, C. Z. Wang, and K. M. Ho, J. Phys. Chem. C 111, 7933 (2007). "Formation of local haeckelite structures induced by vacancy defects in a graphene layer", Gun-Do Lee, C. Z. Wang, Euijoon Yoon, Nong-Moon Hwang, and K. M. Ho, Phys. Rev. B 74, 245411(2006). "Structures of Si7H2m (m=1-7) clusters by global optimization", Mingsheng Tang, C.Z. Wang, W. C. Lu, and K.M. Ho, Phys. Rev. B 74, 195413, (2006). "Strongly-Driven Coarsening of Height-Selected Pb Islands on Si(111)", Maozhi Li, J. Evans, C. Z. Wang, M. Hupalo, M. C. Tringides, T. L. Chan, K. M. Ho, Surf. Sci. Lett. In press, Available online 20 September 2007. "Highly localized quasiatomic minimal basis orbitals for Mo from ab-initio calculations" T.-L. Chan, Y. X. Yao, C. Z. Wang, W. C. Lu, J. Li, X. F. Qian,S. Yip, K. M. Ho, Phys. Rev. B, accepted. "Cluster-in-jellium model and icosahedral ordering tendencies in liquid Al alloys", Y. X. Yao, C. Z. Wang and K. M. Ho, Phys. Rev. B, submitted "Honeycomb chain structure of the Au/Si(111)-(5x2) surface reconstruction: a first-principles study", Feng-Chuan Chuang, Chia-Hsiu Hsu, C. Z. Wang, and K. M. Ho, Phys. Rev. B, submitted. PI Charles Harris "Photo-induced beta-Hydrogen Elimination and Radical Formation with CpW(CO)3(CH2CH3): Ultrafast IR and DFT Studies" E.A. Glascoe, M.F. Kling, J.E. Shanoski, C.K. Payne, B.V. Mork, T.D. Tilley and C.B. Harris, Organometallics, 26, 1424-1432 (2007). "The influence of the metal spin state in iron assisted alkene isomerization studied with ultrafast infrared spectroscopy" E.A. Glascoe, K.R. Sawyer, J.E. Shanoski and C.B. Harris, J. Phys. Chem. C, 111, 8789-8795 (2007). Glascoe, E.A. "Internal Rearrangement Dynamics of Organometallic Complexes Investigated with Ultrafast Infrared Spectroscopy" Ph.D. Thesis, University of California, Berkeley, CA, December 2006. "The mechanism for iron-catalyzed alkene isomerization. Part I. Density functional theory modeling of spin-crossover processes" K.R. Sawyer, E.A. Glascoe, J.F. Cahoon, J.P. Schlegel and C.B. Harris, J. Am. Chem. Soc., submitted. "The mechanism for iron-catalyzed alkene isomerization. Part II. Time-resolved infrared studies of Fe(CO)4(eta2-1-hexene) in neat alkene solution" K.R. Sawyer, E.A. Glascoe, J.F. Cahoon, J.P. Schlegel, H. Frei and C.B. Harris, J. Am. Chem. Soc., submitted. PI Martin Head-Gordon Maibaum, L. and D. Chandler, "Segue between favorable and unfavorable solvation," J. Phys. Chem. B 111, 9025 (2007) Miller, T.F., E. Vanden-Eijnden and D. Chandler, "Solvent coarse-graining and the string method applied to the hydrophobic collapse of a hydrated chain," Proc. Natl Acad. Sci. USA 104, 14559-14564, (2007) Miller, T.F. and C. Predescu, "Sampling diffusive transition paths," J. Chem. Phys. 126, 144102 (2007) Chandler D, Garrahan JP, Jack RL, et al., "Lengthscale dependence of dynamic four-point susceptibilities in glass formers," PHYSICAL REVIEW E 74 (5): Art. No. 051501 Part 1 NOV 2006 Jung, Y. and M. Head-Gordon, A fast correlated electronic structure method for computing interaction energies of large van der Waals complexes applied to the fullerene-porphyrin dimer, Phys. Chem. Chem. Phys. 8, 2831-2840 (2006). Jung, Y., Y. Shao, and M. Head-Gordon, Fast evaluation of scaled opposite spin second order Mxller- Plesset correlation energies using auxiliary basis expansions and exploiting sparsity, J. Comput. Chem. 28, 1953-1964 (2007). Rhee, Y.M. and M. Head-Gordon, Scaled second order perturbation corrections to configuration interaction singles: efficient and reliable excitation energy methods, J. Phys. Chem. A 111, 5314-5326 (2007). PI Teresa Head-Gordon M. S. Lin, N. L. Fawzi, and T. Head-Gordon (2007). Hydrophobic potential of mean force as a solvent function for protein structure prediction. Structure 15, 727-740. M. S. Lin and T. Head-Gordon (2007). Improved energy selection of native loops from loop decoys. Submitted to JCTC. E.-H. Yap, N. Lux Fawzi & T. Head-Gordon (2007). A coarse-grained alpha-carbon protein model with anisotropic hydrogen-bonding. Proteins, Struct. Func.. Bioinformatics in press T. Head-Gordon & M. E. Johnson (2006). Tetrahedral structure or chains for liquid water? Proc. Natl. Acad. Sci. 103, 7973-7977. T. Head-Gordon and S. Rick (2007). Consequences of chain networks on thermodynamic, dielectric and structural properties for liquid water. Phys. Chem. Chem. Phys. 8, 83-91. R. M. Lynden-Bell and T. Head-Gordon (2007). Solvation in modified water models: toward understanding hydrophobic solvation. Mol. Phys. 104, 3593-3605. R. M. Lynden-Bell and T. Head-Gordon (2007). Hydrophobic hydration of Gay Berne particles in modified water models. Submitted. R. K. Murakra and T. Head-Gordon (2007). Dielectric relaxation of aqueous solutions of hydrophobic and hydrophilic peptides. JPC-B, in press. R. K. Murakra and T. Head-Gordon (2007). Single particle and collective hydration dynamics of hydrophobic and hydrophilic peptides. J. Chem. Phys. 126, 215101-215109. M. E. Johnson, T. Head-Gordon, A. A. Louis (2007). Representability problems for coarse-grained water models. J. Chem. Phys. 126, 144509-144519. N. Lux Fawzi, Y. Okabe, E.-H. Yap & T. Head-Gordon (2007). Fibril stability and elongation studies of the Alzheimer Aβ1-40 peptide. J. Mol. Biol. 365, 535-550 N. Lux Fawzi, K. Kohlstedt, Y. Okabe & T. Head-Gordon (2007). Protofibril Assemblies of the Arctic, Dutch and Flemish Mutants of the Alzheimer Aβ1-40 peptide. Biophys. J. in press N. Lux Fawzi, A. Phillips, J. Z. Ruscio, M. Doucleff, D. E. Wemmer & T. Head-Gordon (2007). Structure and dynamics of the Alzheimer Aβ21-30 peptide from the interplay of NMR experiments and simulation. Submitted to JACS PI Brian Hingerty Watt DL, Utzat CD, Hilario P, Basu AK. Mutagenicity of the 1-Nitropyrene-DNA Adduct N-(Deoxyguanosin-8-yl)-1-aminopyrene in Mammalian Cells. Chem Res Toxicol. 2007 (in press) Xu P, Oum L, Beese LS, Geacintov NE, Broyde S. Following an environmental carcinogen N2-dG adduct through replication: elucidating blockage and bypass in a high-fidelity DNA polymerase. Nucleic Acids Res. 2007;35(13):4275-88. Mocquet V, Kropachev K, Kolbanovskiy M, Kolbanovskiy A, Tapias A, Cai Y, Broyde S, Geacintov NE, Egly JM. The human DNA repair factor XPC-HR23B distinguishes stereoisomeric benzo[a]pyrenyl-DNA lesions. EMBO J. 2007 Jun 20;26(12):2923-32. Jia L, Shafirovich V, Geacintov NE, Broyde S. Lesion specificity in the base excision repair enzyme hNeil1: modeling and dynamics studies. Biochemistry. 2007 May 8;46(18):5305-14. Wang L, Yu X, Hu P, Broyde S, Zhang Y. A water-mediated and substrate-assisted catalytic mechanism for Sulfolobus solfataricus DNA polymerase IV. J Am Chem Soc. 2007 Apr 18;129(15):4731-7. Rodriguez FA, Cai Y, Lin C, Tang Y, Kolbanovskiy A, Amin S, Patel DJ, Broyde S, Geacintov NE Exocyclic amino groups of flanking guanines govern sequence-dependent adduct conformations and local structural distortions for minor groove-aligned benzo[a]pyrenyl-guanine lesions in a GG mutation hotspot context. Nucleic Acids Res. 2007;35(5):1555-68. Ding S, Shapiro R, Geacintov NE, Broyde S. 4-hydroxyequilenin-adenine lesions in DNA duplexes: stereochemistry, damage site, and structure. Biochemistry. 2007 Jan 9;46(1):182-91. Perlow-Poehnelt RA, Likhterov I, Wang L, Scicchitano DA, Geacintov NE, Broyde S. Increased flexibility enhances misincorporation: temperature effects on nucleotide incorporation opposite a bulky carcinogen-DNA adduct by a Y-family DNA polymerase. J Biol Chem. 2007 Jan 12;282(2):1397-408. Zhang L, Rechkoblit O, Wang L, Patel DJ, Shapiro R, Broyde S. Mutagenic nucleotide incorporation and hindered translocation by a food carcinogen C8-dG adduct in Sulfolobus solfataricus P2 DNA polymerase IV (Dpo4): modeling and dynamics studies. Nucleic Acids Res. 2006 Jul 4;34(11):3326-37. Jia L, Shafirovich V, Shapiro R, Geacintov NE, Broyde S. Related Articles, Links Flexible 5-guanidino-4-nitroimidazole DNA lesions: structures and thermodynamics. Biochemistry. 2006 May 30;45(21):6644-55. Rechkoblit O, Malinina L, Cheng Y, Kuryavyi V, Broyde S, Geacintov NE, Patel DJ. Stepwise translocation of Dpo4 polymerase during error-free bypass of an oxoG lesion. PLoS Biol. 2006 Jan;4(1):e11. Durandin A, Jia L, Crean C, Kolbanovskiy A, Ding S, Shafirovich V, Broyde S, Geacintov NE. Assignment of absolute configurations of the enantiomeric spiroiminodihydantoin nucleobases by experimental and computational optical rotatory dispersion methods. Chem Res Toxicol. 2006 Jul;19(7):908-13. 13: Wang L, Broyde S. A new anti conformation for N-(deoxyguanosin-8-yl)-2-acetylaminofluorene (AAF-dG) allows Watson-Crick pairing in the Sulfolobus solfataricus P2 DNA polymerase IV (Dpo4). Nucleic Acids Res. 2006 Feb 1;34(3):785-95. PI Kai-Ming Ho "Initial Bilayer Growth of Ag Films on NiAl(110)" B. Unal, F. Qin, Y. Han. D.-J. Liu, D. Jing, A.R. Layson, C.J. Jenks, J.W. Evans, P.A. Thiel, Phys. Rev. B in press (2007). "Synthesis of Carbon Nanotubes by Rolling Up Patterned Graphene Nanoribbons Using Selective Atomic Adsorption", Decai Yu and Feng Liu, Nano Lett. in press, (2007). "Interplay between indirect interaction and charge density wave in Pb adsorbed In(4x1)-Si(111)", M. Hupalo, T. L. Chan, C. Z. Wang, K. M. Ho, and M. C. Tringides, Phys. Rev. B 76, 045415 (2007). "Concerted diffusion, clustering, and magnetic properties of Mn dopants on a 2x2-T4 GaN(0001) substrate", Shiqiang Hao Zhenyu Zhang, Phys Rev. Lett. 99, 166101 (2007). "Upward self-diffusion of adatoms and small clusters on facets of fcc metal (110) surfaces", Haili Yang, Qiang Sun, Zhang, and Yu Jia, Phys. Rev. B 76, 115417 (2007). "Evolution of a symmetry gap and synergetic quantum well states in ultrathin Ag films on Au(111) substrates", L. Huang, G. Gong, E. Gergert, F. Forster , A. Bendounan, F. Reinert, Z. Y. Zhang, Europhys. Lett. 78 (5): Art. No. 57003 (2007) "Tuning the quantum stability and superconductivity of ultrathin metal alloys", M. M Ozer, Y. Jia, Z. Y. Zhang , J. R. mpson, H. H. Weitering, SCIENCE 316 (5831): 1594-1597 JUN 15 2007 "Dynamics of step bunching in heteroepitaxial growth on vicinal substrates" M. Yoon, H. N. Lee, W. Hong, H. M. Christen , Z. Y. Zhang, Z. G. Suo, Phys. Rev. Lett. 99 (5): 055503 (2007) "Interplay between elastic interactions and kinetic processes in stepped Si (001) homoepitaxy" W. Hong, Z. Y. Zhang, Z. G. Suo, Phys. Rev. B 74 (23): 235318 (2006) "Initial interactions between water molecules and Ti-adsorbed carbon nanotubes" Yang Lei, Zheng Xiao Guo, Wenguang Zhu, Zhenyu Zhang, and Sheng Meng Appl. Phys. Lett. (in press) "Metal-diboride nanotubes as high-capacity hydrogen storage media" S. Meng, E. Kaxiras, Z. Y. Zhang, Nano Lett. 7 (3), 663 (2007). PI Hong Im Wang, Y. and Rutland, C. J. (2007) Direct numerical simulation of ignition in turbulent n-heptane liquid fuel spray ets, Combust. Flame, accepted. Yoo, C. S. and Im, H. G. (2007) Characteristic boundary conditions for simulations of compressible reacting flows h multi-dimensional, viscous, and reaction effects, Combust. Theory Modelling 11:259-286. Cook, D. J., Pitsch, H., Chen, J. H. and Hawkes, E. R. (2007), Flamelet-based modeling of H2/Air Auto-ignition with hermal inhomogeneities, Proc. Combust. Inst. 31:2903-2911. Hawkes, E. R., Sankaran, R., Sutherland, J. and Chen, J. H. (2007) Scalar mixing in direct numerical simulations of emporally-evolving plane jet flames with detailed CO/H2 kinetics, Proc. Combust. Inst. 31:1633-1640. Sankaran, R., Hawkes, E. R., Chen, J. H., Lu, T. and Law, C. K. (2007) Structure of a spatially-developing lean ne-air turbulent bunsen flame, Proc. Combust. Inst. 31:1291-1298. Yoo, C. S. and Im, H. G. (2007) Transient soot dynamics in turbulent nonpremixed ethylene-air counterflow flames, c. Combust. Inst. 31:701-708. Chen, J. H., Hawkes, E. Sankaran, R., Mason, S. D. and Im, H. G. (2006) Direct numerical simulation of ignition propagation in a constant volume with temperature inhomogeneities, Part I: Fundamental analysis and diagnostics, Combust. ame 145:128-144. Hawkes, E., Sankaran, R., Pibay, P. P. and Chen, J. H. (2006) Direct numerical simulation of ignition front on in a constant volume with temperature inhomogeneities, Part II: Parametric Study, Combust. Flame 145:145-159. Wang, Y. and Trouvi A. (2006) Direct numerical simulation of non-premixed flame-wall interactions, Combust. Flame :461-475. PI Valerie Izzo D.G. Whyte, R. Granetz, M. Bahktiari, V. Izzo, T. Jernigan, J. Terry, M. Reinke, B. Lipschultz, J. Nucl. Mater. 363-365, (2007). R.S. Granetz, D.G. Whyte, V.A. Izzo, et al., Nucl. Fusion 46 (2006) 1001. PI Peter Jacobs S-L Blyth, et al., A cone jet-finding algorithm for heavy-ion collisions at LHC energies, J. Phys. G 34:271, 2007 PI Stephen Jardin Breslau, J.A, Park W., Jardin, S.C.Massively parallel modeling of the sawtooth instability in tokamaks" JPhys.: Conf. Ser. 46, 97-101 (2006) Breslau, J, Jardin, SC, Park, W.,"3D modeling of the sawtooth instability in a small tokamak", Phys Plasmas 14 Art. NO. 056105 MAY 2007 Ferraro N and Jardin S, "Finite element implementation of Braginskii's gyroviscous stress with application to the gravitational instability", Physics of Plasmas, 13: 092101 (2006) N. M. Ferraro, "Finite Larmor Radius Effects on the Magnetorotational Instability", to appear in The Astrophysical Journal, 662, p 512-516 (2007) Fu GY, Park W, Strauss H, et al, "Gobal Hybrid Simulations of Energetic Particle Effects on the n=1 Mode in Tokamaks: Internal Kink and Fishbone Instability" Phys Plasmas 13 052517 (2006) Jardin, S.C., Breslau, J. A., Ferraro, N, "A High-Order Implicit Finite Element Method for Integrating the Extended-Magneto-Hydrodynamic Equations in Two Dimensions", J. Comp. Phys 226, pages 2146-2174 Oct (2007) L-P. Ku, P. R. Garabedian, J. Lyon, A. Grossman, T. K. Mau, A. Turnbull, M. Zarnstorff, and the ARIES-CS Team, "Physics Design for ARIES-CS," to appear in Fusion Science and Technology. J. F. Lyon, L-P. Ku, L. El-Guebaly, L. Bromberg, and the ARIES-CS Team, "Systems Studies and Optimization of the ARIES-CS Power Plant," to appear in Fusion Science and Technology. T.K. Mau, T. Kaiser, A. A. Grossman, A. R. Raffray, X. R. Wang, J. F. Lyon, R. Maingi, L. P. Ku, M. C. Zarnstorff and the ARIES-CS Team, "Divertor Configuration and Heat Load Studies for the ARIES-CS Reactor," to appear in Fusion Science and Technology. J. F. Lyon, L-P. Ku, L. El-Guebaly, et. al., "Parametric Studies of the ARIES-CS Power Plant," 22nd IEEE/NPSS Symposium on Fusion Engineering, Albuquerque, NM, June 17-21 (2007). T. K. Mau, T. B. Kasier, J. F. Lyon, R. Mangi, A. R. Raffray,X. Wang, L-P. Ku, M. Zarnstorff, "Divertor Heat Loads from Thermal and Alpha Particles in a Compact Stellerator Reactor," 22nd IEEE/NPSS Symposium on Fusion Engineering, Albuquerque, NM, June 17-21 (2007). J. F. Lyon, L-P. Ku, L. El-Guebaly, et. al., "Analysis of the ARIES-CS Compact Stellarator Power Plant," 16th International Workshop on Stellarators, Toki, Japan, Oct 14-19 (2007). R. Samtaney, B. van Straalen, P. Colella and S. C. Jardin, Adaptive mesh simulations of multi-physics processes during pellet injection in tokamaks J. Phys.: Conf. Ser. 78 012062, 2007 Daniel R. Reynolds, Ravi Samtaney and Carol S. Woodward, A Fully Implicit Numerical Method for Single-Fluid Resistive Magnetohydrodynamics,Journal of Computational Physics 219 (2006) 144:162 PI Julius Jellinek STRUCTURAL, ELECTRONIC, AND OPTICAL PROPERTIES OF NOBLE METAL CLUSTERS FROM FIRST PRINCIPLES S. Ogut, J. C. Idrobo, J. Jellinek, and J. Wang J. Clust. Sci. 17, 609-626 (2006). SITE-SPECIFIC POLARIZABILITIES: PROBING THE ATOMIC RESPONSE OF SILICON CLUSTERS TO AN EXTERNAL ELECTRIC FIELD K. Jackson, M. Yang, and J. Jellinek In Lecture Series in Computer and Computational Sciences, Vol. 6, G. Maroulis and T. Simos, Eds., Brill, Leiden, 2006, pp. 165-176 GOLD-COATED TRANSITION METAL ANION [Mn_13@Au_20]- WITH ULTRAHIGH MAGNETIC MOMENT J. Wang, J. Bai, J. Jellinek, and X. C. Zeng J. Am. Chem. Soc. 129, 4110-4111 (2007) (Communication) FIRST-PRINCIPLES ISOMER-SPECIFIC ABSORPTION SPECTRA OF Ag_11 J. C. Idrobo, S. Ogut, K. Nemeth, J. Jellinek, and R. Ferrando Phys. Rev. B 75, 233411(1-4) (2007) COMPUTATIONAL ELECTRON SPECTROSCOPY OF GAS PHASE METAL CLUSTERS J. Jellinek and P. H. Acioli In The Chemical Physics of Solid Surfaces, Vol. 12, Atomic Clusters: From Gas Phase to Deposited, D. P. Woodruff, Ed., Elsevier, Amsterdam, 2007, pp.299-326 SITE-SPECIFIC ANALYSIS OF DIELECTRIC PROPERTIES OF FINITE SYSTEMS K. A. Jackson, M. Yang, and J. Jellinek J. Phys. Chem. C (in press) (R. E. Smalley Memorial Issue) SITE-SPECIFIC ANALYSIS OF RESPONSE PROPERTIES OF SODIUM CLUSTERS K. A. Jackson, M. Yang, and J. Jellinek in Latest Advances in Atomic Cluster Collisions: Structure and Dynamics from Nuclear to the Biological Scale, A. V. Solovov, Ed., Imperial College Press, London (in press) NANOALLOYS: FROM THEORY TO APPLICATIONS OF ALLOY CLUSTERS AND NANOPARTICLES R. Ferrando, J. Jellinek, and R. L. Johnston Chem. Rev. (submitted) STATIC POLARIZABILITIES AND OPTICAL ABSORPTION SPECTRA OF GOLD CLUSTERS (Au_n, n=2-14 and 20) FROM FIRST PRINCIPLES J. C. Idrobo, W. Walkosz, S. F. Yip, S. Ogut, J. Wang, and J. Jellinek Phys. Rev. B (submitted) PI Chueng-Ryong Ji Perturbative QCD analysis of exclusive $J/\psi+\eta_c$ production in $e^+e^-$ annihilation, H.M.Choi and C.R.Ji, Phys. Rev. D, in press; arXiv:0707.1173 [hep-ph]. Zero-Mode Contribution in Nucleon-Delta Transition, J.Yu, T.Wang, C.R.Ji and B.Q.Ma, Phys.Rev.D76, 074009 (2007). Distribution amplitudes and decay constants for (pi, K, rho, K) mesons in light-front quark model, H.M.Choi and C.R.Ji, Phys. Rev. D75, 034019 (2007). The box diagram in Yukawa theory, B.Bakker, J.Boomsma and C.R.Ji, Phys. Rev. D75, 065010 (2007). Conformal Symmetry and Pion Form Factor: Soft and Hard Contributions, H.M.Choi and C.R.Ji, Phys. Rev. D74, 093010 (2006) PI De-en Jiang Jiang, D. E.; Dai, S. "Electronic ground state of higher acenes." Journal of Physical Chemistry, submitted. Chen, Z. F;. Jiang, D. Ea;. Lu, X.; Bettinger, H. F.; Dai, S.; Schleyer, P. v. R.; Houk, K. N. "Open-Shell Singlet Character of Cyclacenes and Short Zigzag Nanotubes." Organic Letters, submitted. PI Andrey Kalinichev P.Kumar, A.G.Kalinichev, R.J.Kirkpatrick (2007a) Dissociation of carbonic acid: Gas phase energetics and mechanism from ab initio metadynamics simulations. J.Chem.Phys., 126, 204315-1-7. PI Martin Karplus Optimal Estimate of Free Energies from Multistate Nonequilibrium Work Data, Phys. Rev. Lett. 96, 100602/1-4 (2006), by P. Maragakis, M. Spichty and M. Karplus. Probing Polar Solvation Dynamics in Proteins: A Molecular Dynamics Simulation Analysis, J. Phys. Chem. B. 111, 1482-1490 (2006), by A. A. Golosov and M. Karplus. A Kinetic Model of Coordinated Myosin V, Biochemistry 46, 6318-6330 (2007), by Y. Wu, Y. Q. Gao and M. Karplus. Minimum Free Energy Pathways and Free Energy Profiles for Conformational Transitions Based on Atomic Molecular Dynamics Simulations, J. Chem. Phys. 126, 164106/1-17, by A. van der Vaart and M. Karplus. The Signaling Pathway of Rhodopsin, Structure 15, 611-623 (2007), by Y. Kong and M. Karplus. How Subunit Coupling Produces the gamma-Subunit Rotary Motion in F1-ATPase, Proc. Natl. Acad. Sci. USA, submitted, by J. Pu and M. Karplus. The Elastic Properties of the Structurally Characterized Myosin II S2 Sub-domain: A Molecular Dynamics and Normal Mode Analysis, Biophys. J., submitted, by I. Adamovic, S. M. Mijailovich and M. Karplus. Intrinsic Motions Along An Enzymatic Reaction Trajectory, Nature, accepted, by V. Thai, K. Henzler-Wildman, M. Lei, M. Wunderlich, M. Wolf-Watz, T. Fenn, E. Pozharski, M. A. Wilson, G. A. Petsko, M. Karplus, C. G. Hubner and D. Kern A Hierarchy of Timescales in Protein Dynamics Linked to Enzyme Catalysis, Nature, accepted, by K. Henzler-Wildman, M. Lei, V. Thai, M. Karplus, C. G. Hubner and D. Kern. The Signaling Pathway of PDZ Domains, Proteins: Struct. Funct. Bioinformatics, to be submitted, by Y. Kong and M. Karplus. The Mechanism of the Translocation Step in DNA Replication by DNA Polymerase I: A Computer Simulation Analysis, to be submitted, by Andrei A. Golosov, Martin Karplus, Joshua Warren, and Lorena Beese. PI Thomas Katsouleas T. Katsouleas, "Plasma accelerators race to 10 GeV and beyond", Physics of Plasma, 2006 W. Lu, C. Huang, M. Zhou, M. Tzoufras, F. S. Tsung, W. B. Mori, and T. Katsouleas, "A nonlinear theory for multidimensional relativistic plasma wave wakefields", Physics of Plasma, 2006 A. Z. Ghalam, T. Katsouleas, V. K. Decyk, C. K. Huang, W. B. Mori, G. Rumolo, E. Benedetto, and F. Zimmermann, "Three-dimensional continuous modeling of beam-electron cloud interaction:Comparison with analytic models and predictions for the present and future circular machines", Physics of Plasma, 2006 C. Huang et al., "QUICKPIC: A highly efficient particle-in-cell code for modeling wakefield acceleration in plasmas", Journal Comp. Phys. Volume 217, Issue 2, 20 September 2006. Pages 658-679. E. Oz et al., "Ionization-induced electron trapping in ultrarelativistic plasma wakes", Phys. Rev. Lett. 98, 084801 (2007). I. Blumenfeld et al., "Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator", Nature 445, 741-744 (15 February 2007). PI Paul Kent Payne, C. M., Xiongce Zhao, Cummings, P. T. Molecular simulation of DNA transport in solution, Molecular Simulation, 33, 399, 2007 Zhao Xiongce, Payne, C. M., Peter Cummings, J. W. Lee, Single stranded DNA molecules translocation through nanoelectrode gaps, Nanotechnology, 18, 424018, 2007 Zhao Xiongce, Rignall, T. R., McCabe, C., Adney, W. S., Himmel, M. E., Energy Storage Mechanism of the Trichoderma reesei Cel7A I Linker Peptide from Molecular Dynamics Simulation, submitted to J. Am. Chem. Soc. Zhao Xiongce, Payne, C. M., Peter Cummings, Controlled translocation of DNA segments through nanoelectrode gaps from molecular dynamics, submitted to J. Phys. Chem. C Miguel Fuentes-Cabrera, Xiongce Zhao, P. R. C. Kent, and Bobby G. Sumpter. Electronic structure of xDNA. Journal of Physical Chemistry B 111 9057 (2007) Y. He, S. Graser, P. J. Hirschfeld and H. -P. Cheng, Structure of BSCCO supermodulation from ab initio calculations. Submitted to Phys. Rev. Let (2007). arXiv:0709.0662 Agapito LA, Cheng HP. Ab initio calculation of a graphene-ribbon-based molecular switch. J. Phys. Chem C 111 (38): 14266-14273 SEP 27 2007 He Y, Zhang C, Cao C, et al. Effects of strain and defects on the electron conductance of metallic carbon nanotubes. Phys. Rev. B 75 (23) 235429 JUN 2007 Cao C, He Y, Torras J, et al. Fracture, water dissociation, and proton conduction in SiO2 nanochains. J. Chem. Phys. 126 (21) 211101 JUN 7 2007 Wang LL, Cheng HP. Embedding atom-jellium model for metal surface. Eur. Phys. J. D (1-3): 247-250 JUL 2007 Schmidt M, Masson A, Brechignac C, et al. Hydrogen peroxide and ammonia on protonated ice clusters J. Chem. Phys. 126 (15) 154315 APR 21 2007 Muralidharan K, Cao C, Wan YX, et al. Environment dependent dynamic charge potential for silica: Application to nanoscale silica structures. Chem. Phys. Lett. 437 (1-3): 92-98 MAR 22 2007 Zhang JW, Cheng HP. Anomalous Hall effect in disordered Fe ferromagnetic films. Phys. Rev. B 74 (21) 212409 DEC 2006 Simulations of azobenzene containing alkanethiol SAMs on Au(111) surface. J. Phys. Chem. (in press) Cao C., He. Y., Cheng HP. Hydrogen dissociation and band modification of CNT-supported Pd4 cluster (submitted). Cao C., Kemper L. He, Y. Cheng HP. CNT-supported Pd cluster as nano-sensor (submitted). He Y., Cao, C., Cheng HP. Predictive simulation of nano-tube fracture under stress (submitted). PI David Keyes V. Akcelik, G. Biros, O. Ghattas, J. Hill, D. Keyes, and B. van Bloeman Waanders, "Parallel PDE constrained optimization," in Frontiers of Parallel Computing, M. Heroux, P. Raghaven, and H. Simon, eds, SIAM, 2006. J. Brannick, M. Brezina, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge, "An energy-based AMG coarsening strategy," Journal on Numerical Analysis,. Lin. Alg. Appl., 13, 133, 2006. M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick and J. Ruge, "Adaptive Algebraic Multigrid," SIAM Journal of Scientific Computing, 27, 1534, 2006. M. Brezina, C. Tong, R. Becker, "Parallel Algebraic Multigrids for Structural Mechanics," SIAM Journal of Scientific Computing, to appear. E. Chow, "An Aggregation Multilevel Method Using Smooth Error Vectors," SIAM Journal of Scientific Computing, 27, 1727, 2006. R. D. Falgout, J. E. Jones, and U. M. Yang, "The Design and Implementation of hypre, a Library of Parallel High Performance Preconditioners," chapter in Numerical Solution of Partial Differential Equations on Parallel Computers, A. M. Bruaset, P. Bjorstad, and A. Tveito, eds., Springer- Verlag, 2006. I. Lee, P. Raghavan, and E. G. Ng, "Effective Preconditioning Through Ordering Interleaved with Incomplete Factorization," SIAM J. Matrix Anal. Appls., 27, 106, 2006. T. Manteuffel, S. McCormick, O. Rohrle and J. Ruge, "Projection multilevel methods for quasilinear elliptic partial differential equations: numerical results," SIAM Journal on Numerical Analysis, 44, 120, 2006. T. Manteuffel, S. McCormick and O. Rohrle, "Projection multilevel methods for quasilinear elliptic partial differential equations: theoretical results," SIAM Journal on Numerical Analysis, 44, 139, 2006. S. McCormick, "Projection multilevel methods for quasilinear elliptic partial differential equations: V-cycle theory," SIAM J. Mult. Modeling Sim., 4, 133, 2006. E. Prudencio, R. Byrd, and X.-C. Cai, "Parallel full space SQP Lagrange-Newton- Krylov-Schwarz algorithms for PDEconstrained optimization problems," SIAM Journal of Scientific Computing, 27, 130, 2006. U. M. Yang, "Parallel Algebraic Multigrid Methods - High Performance Preconditioners," chapter in Numerical Solution of Partial Differential Equations on Parallel Computers, A. M. Bruaset, P. Bjorstad, and A. Tveito, eds., Springer-Verlag, 2006. PI Kwiseon Kim Q. Zhao, P. A. Graf, W. B. Jones, A. Franceschetti, J. Li, L.-W. Wang, and K. Kim, Shape dependence of band-edge exciton fine structure in CdSe nanocrystals, Nano Lett., 2007, 10.1021/nl0713070 P. A. Graf, K. Kim, W. B. Jones, L. W. Wang, Surface passivation optimization using DIRECT, J. Comput. Phys., 224, 2, (2007), 824-835 P. A. Graf and W. B. Jones, A Projection Based Multiscale Optimization Method for Eigenvalue Problems, J. Glob. Optim , 39, (2007) 235-245. P. A. Graf and W. B. Jones, Optimization of semiconductor alloy properties using the virtual crystal approximation, J. Comput. Phys. (submitted). P. Piquini, P. A. Graf, and Alex Zunger, Band gap design of quaternary (In,Ga)(As,Sb) semiconductors via the inverse band structure approach, (submitted). J. Schrier, D.O. Demchenko, L.W. Wang, A.P. Alivisatos, Optical properties of ZnO/ZnS and ZnO/ZnTe heterostructures for solar cell applications, Nanolett. 7. 2377 (2007). PI Spencer Klein "Multiyear Search for a Diffuse Flux of Muon Neutrinos with AMANDA-II," A. Achterberg et al., Phys. Rev. D76, 042008 (2007). "Detection of atmospheric neutrinos with the IceCube 9-string detector," A. Achterberg et al., Phys. Rev. D76, 027101 (2007). " Five years of searches for point sources of astrophysical neutrinos with the AMANDA-II neutrino telescope," A. Achterberg et al., Phys. Rev. D75, 102001 (2007). "Search for Neutrino Induced Cascades from Gamma-Ray Bursts with AMANDA," A. Achterberg et al., Astrophysical J. 664. 397 (2007). "Limits on the muon flux from neutralino annihilations at the center of the Earth with AMANDA," A. Achterberg et al., Astropart. Phys. 26, 129 (2006). PI Kwok Ko Volkan Akcelik, Kwok Ko, Lie-Quan Lee, Zenghai Li, Cho-Kuen Ng, and Liling Xiao, Shape Determination for Deformed Electromagnetic Cavities, Journal of Computational Physics, to be published Lie-Quan Lee, et al, Enabling technologies for petascale electromagnetic accelerator simulation, 2007 J. Phys.: Conf. Ser. 78 012040. F. Zhou, A. Kabel, J. Rosenzweig, R. Agustsson, G. Andonian, D. Cline, A. Murokh, and V. Yakimenko, Experimental characterization of the transverse phase space of a 60-MeV electron beam through a compressor chicane; Phys. Rev. ST Accel. Beams 9, 114201 (2006). PI Boris Kogan Huffaker RB, Weiss JN, Kogan B. Effects of early afterdepolarizations on reentry in cardiac tissue: a simulation study. Am J Physiol Heart Circ Physiol. 2007 Jun; 292(6):H3089-102. PI Joel Koplik "Hydrodynamic interaction of two particles in confined linear shear flows at finite Reynolds number," by Y. Yan, J. F. Morris and J. Koplik, Phys. Fluids in press (2007). PI V. Rao Kotamarthi Zubrow, A., Chen, L., Kotamarthi, V. R., EAKF-CMAQ: Introduction and evaluation of a data assimilation for CMAQ based on the Ensemble Adjustment Kalman filter. Submitted to the Journal of Geophysical Research, 2007. PI Henry Krakauer "Finite-size correction in many-body electronic structure calculations," by Hendra Kwee, Shiwei Zhang and Henry Krakauer, submitted to Physical Review Letters. It will be available in preprint form at the eprint archive (arXiv.org). PI Petr Kral B. Wang and P. Kral, Dragging of Polarizable Nanodroplets by Ions Solvated at Nanometer Separations, Submitted to Nature Nanotechnology. PI Arnold Kritz G. Park, J. Cummings, C.S. Chang, N. Podhorszki, S. Klasky, S. Ku, A. Pankin, R. Samtaney, A. Shoshani, P. Snyder, H. Strauss, L. Sugiyama and the CPES Team (2007) Coupled simulation of kinetic pedestal growth and MHD ELM crash, J. Phys.: Conf. Ser. 78, 012087. C.R. Sovinec, D.C. Barnes, R.A. Bayliss, D.P. Brennan, E.D. Held, S.E. Kruger, A.Y. Pankin, D.D. Schnack and the NIMROD Team (2007) Two-fluid studies of edge relaxation events in tokamaks, J. Phys.: Conf. Ser. 78, 012070. A.Y. Pankin, G. Bateman, D.P. Brennan, A.H. Kritz, S. Kruger, P. B. Snyder, and C. Sovinec (2007) Modeling of ELM dynamics for DIII-D and ITER, Plasma. Phys. Control. Fusion 49, S63-75. PI Scott Kruger D.G. Whyte, R. Granetz, M. Bahktiari, V. Izzo, T. Jernigan, J. Terry, M. Reinke, B. Lipschultz, J. Nucl. Mater. 363-365, 1160 (2007). PI Young-Kyun Kwon Y.-K. Kwon and P. Kim, "Unusually High Thermal Conductivity in Carbon Nanotubes", in Subhash L. Shinde and Jitendra S. Goela (eds.), High Thermal Conductivity Materials, pp. 227--265 (Chapter 8), Springer, New York (2006). M. Lee, J. Im, B.Y. Lee, S. Myung, J. Kang, L. Huang, Y.-K. Kwon, and S. Hong, "Linker-free directed assembly for high-performance integrated devices based on nanotubes and nanowires", Nature Nanotech. 1, 66 (2006). H. J. Song, Y. Lee, T. Jiang, A.-G. Kussow, M. Lee, S. Hong, Y.-K. Kwon, and H. C. Choi, "Self-clusterized Glycines on Single Walled Carbon Nanotubes for Alcohol Sensing", J. Phys. Chem. C, in press. T. W. Kim, Y. J. Hong, G.-C. Yi, J.-H. Kwon, M. Kim, H. N. Han, D. H. Kim, K. H. Oh, K.-J. Kong, and Y.-K. Kwon, "Morphology of Patterned, Uniform, and Faceted GaN Microcrystals", submitted for publication. S. Woo and Y.-K. Kwon, "New Hydrogen Storage Nanomaterial with Potential Pockets: Ethylene Oxides", submitted for publication. PI Andrew Lacis Q. Ma, R. H. Tipping, and C. Boulet, Irreducible Correlation Functions of the S Matrix in the Coordinate Representation: Application in Calculating Lorentzian Half-Widths and Shifts, J. Chem. Phys, 124, 014109 (2006). Q. Ma, R. H. Tipping, and C. Boulet, Modification of the Robert-Bonamy Formalism in Calculating Lorentzian Half-Widths and Shifts, J. Quant. Spec. Rad. Transfer 103, 588 (2007). Q. Ma, R. H. Tipping, C. Boulet, F. Thibault, and J. Bonamy, Vibration-Dependent Trajectories and Their Effects on Vibrational Dephasing, J. Mol Spec. 243, 105 (2007). V. B. Podobedov, D. F. Plusquellic, K. E. Siegrist, G. T. Fraser, Q. Ma, and R. H. Tipping, Collision-Induced Absorption by the Water Vapor Continuum in the Region from 0.3 to 2.7 THz, J. Quant. Spec. Rad. Transfer, in press (2007). A. Goldman, R. H. Tipping, Q. Ma, C.D. Boone, P.F. Bernath, P. Demoulin, F. Hase, M. Schneider, J. W. Hannigan, M. T. Coffey, and C. P. Rinsland, On the line parameters for the X1Sg+ (1-0) infrared quadrupolar transitions of 14N2. J. Qaunt. Spec. Rad. Transfer. 103, 168-174 (2007). Yu. I. Baranov, W. J. Lafferty, and G. T. Graser, Q. Ma. R. H. Tipping, Water-vapor Continuum Absorption in the 800 cm-1 to 1250 cm-1 (to be submitted). PI Jean-Francois Lamarque Lamarque, J.-F., D. E. Kinnison, P.G. Hess and F. Vitt, Simulated lower stratospheric trends between 1970 and 2005: identifying the role of climate and composition changes. Submitted for publication in J. Geophys. Res., 2007. PI Don Lamb Calder,A.C. , Townsley, D. M., Seitenzahl, I. R., Peng, F., Messer, O. E. B., Vladimirova, N., Brown, E. F., Truran, J. W., and Lamb, D. Q., Capturing the Fire: Flame Energetics and Neutronizaton for Type Ia Supernova Simulations, Astrophysical Journal, 2007, vol. 656, pp. 313+. Hearn, N. C., Plewa, T., Drake, R. P., and Kuranz, C., Flash Code Simulations of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities in Laser-Driven Experiments, Astrophysics and Space Science, 2007, vol. 307, pp. 227-231. Jordan IV, G.C., Fisher, R.T., Townsley, D.M., Calder, A.C., Graziani, C., Asida, S., Lamb, D.Q. and Truran, J.W., Three-Dimensional Simulations of the Deflagration Phase of the Gravitationally Confined Detonation Model of Type Ia Supernovae, Submitted to ApJ Letters, 2007. Townsley, D.M., Calder, A.C., Asida, S., Seitenzahl, I.R., Peng, F., Vladimirova, N., Lamb, D.Q. and Truran, J.W., Flame Evolution During the Deflagration Phase of Type Ia Supernovae, Submitted to the Astrophysical Journal, 2007. Zhang, J., Bronson Messer, O. E., Khokhlov, A. M. and Plewa, T., On the Evolution of Thermonuclear Flames on Large Scales, Astrophysical Journal, 2007, vol. 656, 347+. Zhiglo, A.V., Flame Capturing Technique, 1: Adaptation to gas expansion, Astrophysical Journal Supplement Series, 2007, vol. 169, pp. 386-400. PI Uzi Landman "Structural Evolution of Au Nanoclusters: From Planar to cage to nanotube Motifs", X. Xing, B. Yoon, J. H. Parks, U. Landman, Phys. Rev. B 74, 165423 (2006). "Bonding Trends and Dimensionality Crossover of Gold Nanoclusters on Metal-supported MgO Thin Films", D. Ricci, A. Bongiorno, G. Pacchioni, U. Landman, Phys. Rev. Lett. 97, 36106 (2006). "Size-Dependent Evolution of Structures and Chemical Reactivity of Gold Nanosclusters: AuN-, N=15-24", B. Yoon, P. Koshkinen, B. Huber, O. Kostki, B. von Issendorff, H. Hakkinen, M. Moseler, U. Landman, Chem. Phys. Chem. 8 , 157 (2007). "Structural and Transport Properties of Nb Nanowires", Z. Dai, C. Zhang, R. N. Barnett, A. Marchenkov and U. Landman, Physica Status Solidi (a) 204, 1712 (2007). "Predicted Oxidation of CO Catalyzed by Au Nanoclusters on a Thin Defect-Free MgO Film Supported on a Mo(100) Surface", C. Zhang, B. Yoon, and U. Landman, J. Am. Chem. Soc (Communication) 129, 2229 (2007). "Structured and Viscous Water in Sub-nanometer Gaps", T.-D. Li, J. Gao, R. Szoszkiewicz, U. Landman, E. Riedo, Phys. Rev. B. 75, 115415 (2007). "Factors in gold nanocatalysis: oxidation of CO in the non-scalable size regime", M. Arenz, U. Landman, U. Heiz, Topics in Catalysis, 44, 145 (2007). PI Jean-Noel Leboeuf P.L. Pritchett, "Onset of Magnetic Reconnection", in Reconnection of Magnetic Fields, edited by J. Birn and E. Priest, pp. 121-132, Cambridge Univ. Press, Cambridge, 2007. P.L. Pritchett, "Relativistic Electron Production During Driven Magnetic Reconnection", Geophys. Res. Lett., 33, L13104, doi:10.1029/2005GL025267, 2006. P.L. Pritchett, "Relativistic Electron Production During Guide-Field Magnetic Reconnection", J. Geophys. Res., 111, A10212, doi:10.1029/2006JA011793, 2006. P.L. Pritchett, "Kinetic Properties of Magnetic Merging in the Coalescence Process", Phys. Plasmas, 14, 052102, 2007. Study of the interaction between diffusive and avalanche-like transport in near-critical dissipative- trapped-electron-mode turbulence, J. A. Mier, L. Garcma, and R. Sanchez, Phys. Plasmas 13, 102308 (2006). Renormalization of tracer turbulence leading to fractional differential equations, R. Sanchez, B. A. Carreras, D. E. Newman, V. E. Lynch, and B. Ph. van Milligen, Phys. Rev. E 74, 016305 (2006). Viktor K. Decyk, Henry J. Gardner: A Factory Pattern in Fortran 95. International Conference on Computational Science (1) 2007: 583-590. Gyrokinetic simulations of ETG and ITG turbulence, A.M. Dimits, W.M. Nevins, D.E. Shumaker, G.W. Hammett, T. Dannert, F. Jenko, M.J. Pueschel, W. Dorland, S.C. Cowley, J.N. Leboeuf, T.L. Rhodes, J. Candy and C. Estrada-Mila, Nucl. Fusion 47 No 8 (August 2007) 817-824. Stability of Highly Shifted Equilibria in a Large-Aspect-Ratio Tokamak, P.-A. Gourdain, S. C. Cowley, J.- N. Leboeuf, and R. Y. Neches, Phys. Rev. Lett. 97, 055003 (2006). PI Frank Lee Magnetic moment of vector mesons in the background field method F.X. Lee, S. Moerschbacher, W. Wilcox, PoS(LATTICE 2007)151. hep-lat/0710.2329 Scalar mesons a0(1450) and sigma(600) from lattice QCD. N. Mathur, A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I.Horvath, F.X. Lee, K.F. Liu, S. Tamhankar, and J.B. Zhang, to appear in Phys. Rev. D. hep-ph/0607110 Mass and width of strange baryon resonances in QCD sum rules. J.P. Singh and F.X. Lee, submitted to Nucl. Phys. A. nucl-th/0612059 On the locality and scaling of overlap fermions at coarse lattice spacings. T. Draper, N. Mathur, J.B. Zhang, A. Alexandru, Y. Chen, S.J. Dong, I. Horvath, F.X. Lee, K.F. Liu, S.Tamhankar, submitted to Phys. Rev. D. hep-lat/0609034 Excited decuplet baryons from QCD sum rules. F.X. Lee, Nucl. Phys. A 791, 352 (2007). nucl-th/0605065 Spectroscopy of light hadrons in anisotropic lattice QCD. F.X. Lee, Int. J. Mod. Phys. A21, 5253 (2006). PI Patrick Lee Sung-sik Lee and Patrick A. Lee , "Observation of emergent photon in Monte Carlo simulation of exciton condensate" Physical Review B74, 035107 (2006). PI T. S. Harry Lee Dynamical Coupled-Channel Model of Meson Production Reactions in the Nucleon Resonance Region A. Matsuyama, T. Sato, T.-S. H. Lee, Physics Reports {\bf 439}, 193-253 (2007) Extraction and Interpretation of $\gamma N \rightarrow \Delta$ Form Factors within a Dynamical Model B. Julia-Diaz, T.-S.H. Lee, T. Sato, L.C. Smith Phys.Rev. {\bf C75}, 015205 (2007) Study of Nucleon Resonances with Electromagnetic Interactions T.-S. H. Lee and L.C. Smith J. Phys. G.: Nucl. Part. Phys. {\bf 34}, S83-S106 (2007) Dynamical Coupled-channel Model of $\pi N$ scattering in the W $\leq$ 2 GeV nucleon resonance region B. Julia-Diaz, T.-S.H. Lee, A. Matsuyama, T. Sato Accepted by Phys. Rev. C Regge Approach to Charged-Pion Photoproduction at Invariant Energies above 2-GeV A. Sibirtsev, J. Haidenbauer, S. Krewald, T.-S.H. Lee, U.-G. Meissner, and A.W. Thomas Accepted by Euro. J. Phy. PI Wei-li Lee Ethier, S., W.M. Tang, R. Walkup, and L. Oliker, "Large Scale Gyrokinetic Particle Simulation of Microturbulence in Magnetically Confined Plasmas", to appear in IBM Journal of Research and Development (2007). Lin, Z., I. Holod, L. Chen, P.H. Diamond, T.S. Hahm, S. Ethier, "Wave-particle decorrelation and transport of anisotropic turbulence in collisionless plasmas", to appear in Phys. Rev. Lett. (2007). Jenkins, Thomas G., and W. W. Lee, "Fluctuations and Discrete Particle Noise in Gyrokinetic Simulation of Drift Waves", Phys. Plasmas 14, 032307 (2007). Wang, W.X., T.S. Hahm, W. Lee, G. Rewoldt, J. Manickam, and W.M. Tang, "Nonlocal Properties of Gyrokinetic Turbulence and Role of ExB Flow Shear", Phys. Plasmas 14, 072306 (2007). M.F. Adams, S. Ethier and N. Wichmann, Performance of particle in cell methods on highly concurrent computational architectures, J. Phys.: Conference Series, Vol. 78, 012001 (2007). Adams, M.F., and Y. Nishimura, "Parallel Algebraic Multigrid Methods in Gyrokinetic Turbulence Simulations", Commun. Comput. Phys. 2, 827 (2007). R. A. Kolesnikov, W. W. Lee, H. Qin, and E. Startsev, "High frequency gyrokinetic particle simulation", Phys. Plasmas 14, 072506 (2007). Nishimura Y., Z. Lin, and W.X. Wang, "Electromagnetic global gyrokinetic simulation of shear Alfven wave dynamics in tokamak plasmas", Phys. Plasmas 14, 042503 (2007). Rewoldt, G., Z. Lin, and Y. Idomura, "Linear Comparison of Gyrokinetic Codes with Trapped Electrons", Report PPPL-4195, submitted to Comp. Phys. Commun. (2007) Oliker, L., A. Canning, J. Carter, C. Iancu, M. Lijewski, S. Kamil, J. Shalf, H. Shan, E. Strohmaier, S. Ethier, and T. Goodale, "Scientific Application Performance on Candidate Petascale Platforms", Proceedings of IEEE International Parallel and Distributed Processing Symposium, 2007, Long Beach, CA (2007) (Best Paper Award). Hahm, T.S., P.H. Diamond, Z. Lin, G. Rewoldt, O. Gurcan, and S. Ethier, "On the Dynamics of Edge-Core Coupling", Phys. Plasmas 12, 090903 (2007). Wang, W.X., G. Rewoldt, W.M. Tang, F.L. Hinton, J. Manickam, L.E. Zakharov, R.B. White, and S. Kaye, "Nonlocal neoclassical transport in tokamak and spherical torus experiments", Phys. Plasmas 13, 082501 (2006). Wang, W.X., Z. Lin, W.M. Tang, W.W. Lee, S. Ethier, J.L.V. Lewandowski, G. Rewoldt, T.S. Hahm, and J.Manickam, "Gyro-kinetic simulation of global turbulent transport properties in tokamak experiments", Phys. Plasmas 13, 092505 (2006). Lewandowski, J.L.V., G. Rewoldt, S. Ethier, W. W. Lee and Z. Lin, "Global Particle-In-Cell Simulations of Microturbulence with Kinetic Electrons", Phys. Plasmas 13, 072306 (2006). W. W. Lee, Integrated Multiscale Gyrokinetic Particle Simulation of Fusion Plasmas, 20th International Conference on Numerical Simulation of Plasmas, (Austin, TX, October, 2007). R. Kolesnikov, Electromagnetic high frequency gyrokinetic particle-in-cell simulation, 20th International Conference on Numerical Simulation of Plasmas, (Austin, TX, October, 2007) W. X. Wang, T.S. Hahm, G. Rewoldt, J. Manickam and W.M. Tang, "Gyrokinetic studies of Nonlocal Properties of Turbulence-driven and Neoclassical Transport", 21th IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, (Chengdu, China, October 2006) W. W. Lee, S. Ethier, T. G. Jenkins, W. X. Wang, J. L. V. Lewandowski, G. Rewoldt, W. M. Tang, S. E. Parker, Y. Chen, and Z.Lin, 21th IAEA Conference on Plasma Physics and Controlled Nuclear Fusion Research, (Chengdu, China, 2006) W. W. Lee, Kinetic Simulation of Fusion Plasmas, Masterworks Session, Supercomputing 06, (Tampa, FL, November 2006). PI Xuhui Lee Huang, J., X. Lee, E. G. Patton: 2007, A modeling study of flux imbalance and the influence of entrainment in the convective boundary layer, submitted to Boundary-Layer Meteorology. PI William Lester P. T. A. Galek, N. C. Handy, and W. A. Lester, Jr. Quantum Monte Carlo Studies on Small Molecules, Mol. Phys. 104, 3069 (2006). B. Austin, A. Aspuru-Guzik, R. Salomon-Ferrer, and W. A. Lester, Jr., Linear Scaling Evaluation of the Local Energy in Quantum Monte Carlo, in Proceedings of the Pacifichem Symposium on Advances in Quantum Monte Carlo, J. B. Anderson and S. M. Rothstein, eds., ACS Symposium Series 953, 55 (2007). R. Prasad, N. Umezawa, D. Domin, R. Salomon-Ferrer, and W. A. Lester, Jr., Quantum Monte Carlo Study of First-Row Atoms using Transcorrelated Variational Monte Carlo Trial Functions, J. Chem. Phys. 126, 164109 (2007). R. Whitesides, D. Domin, R. Salomon-Ferrer, W.A. Lester, Jr., and M. Frenklach, Graphene Layer Growth Chemistry: Five-Six-Ring Flip Reaction, J. Phys. Chem., accepted for publication. M. T, Nguyen, M. H. Matus, W. A. Lester, Jr., and D. A. Dixon, Heats of Formation of Triplet Ethylene and Ethylidene, J. Chem. Phys., accepted for publication. D. Domin, W. A. Lester, Jr., R. Whitesides, and M. Frenklach, Isomer Energy Differences for the C4H3 and C4H5 Isomers Using Diffusion Monte Carlo, J. Phys. Chem., submitted. R. Olivares-Amaya, R. Salomsn-Ferrer, W. A. Lester Jr., and C. Amador-Bedolla, "Creation of a GUI for Zori, a Quantum Monte Carlo program, using Rappture," Computing in Science and Engineering, submitted. PI Mary Ann Leung Leung, M.A., Reinhardt, W.P., "Efficient parallel implementation of the Bose-Hubbard model: Exact numerical groundstates and dynamics of gaseous Bose-Einstein Condensates", Comp. Phys. Comm., 177 (4), (2007), 348-356 PI Rui Li "Curvature-Induced Bunch Self-Interaction for an Energy-Chirped Bunch in Magnetic Bends", Rui Li, to be appear in recent PRST-AB journal (2007). PI Shi Ying Lin C. Xu, B. Jiang, D. Xie, S. C. Farantos, S. Y. Lin and H. Guo, "Analysis of the HO2 vibrational spectrum on an accurate ab initio potential energy surface", J. Phys. Chem. A, in press. C. Xu, D. Xie, P. Honvault, S. Y. Lin and H. Guo, "Rate constant for OH + O !z H + O2 reaction on an improved ab initio potential energy surface and implications for the interstellar oxygen problem", J. Chem. Phys. 127, 024304 (2007). P. Honvault, S. Y. Lin, D. Xie and H. Guo, "Differential and integral cross sections for the H + O2 !z OH + O combustion reaction", J. Phys. Chem. A 111, 5349 (2007). S. Y. Lin, L. Banares, and H. Guo, "Differential and integral cross sections of the N(2D) + H2 !z NH + H reaction from exact quantum and quasi-classical trajectory calculations", J. Phys. Chem. A 111, 2376 (2007). D. Xie, C. Xu, T.-S. Ho, H. Rabitz, G. Lendvay, S. Y. Lin and H. Guo, "Global analytical potential energy surfaces for HO2(X2A") based on high level ab initio calculations!1, J. Chem. Phys. 126, 074315 (2007). A. L. Van Wyngarden, K. A. Mar, K. A. Boering, J. J. Lin, Y. T. Lee, S. Y. Lin, H. Guo and G. Lendvay, "Nonstatistical behavior of reactive scattering in the 18O + 32O2 isotope exchange reaction", J. Am. Chem. Soc., 129, 2866 (2007). S. Y. Lin, H. Guo, P. Honvault and D. Xie, "Quantum dynamics of the H + O2 !z O + OH reaction on an accurate ab initio potential energy surface", J. Phys. Chem. B, 110, 23641 (2006). S. Y. Lin, D. Xie and H. Guo, "Revelation of non-statistical behavior in HO2 vibration by a new ab initio potential energy surface", J. Chem. Phys., 125, 091103 (2006). S. Y. Lin and H. Guo, "Quantum state-to-state cross sections for atom-diatom reactions: A Chebyshev real wave-packet approach", Phys. Rev. A 74, 022703 (2006). S. Y. Lin and H. Guo, "Quantum statistical study of O + O2 isotopic exchange reactions: Cross sections and rate constants", J. Phys. Chem. A 110, 5305 (2006). S. Y. Lin and H. Guo, "Exact quantum dynamics of N(D_2) + H2 reaction: Cross-sections, rate constants, and dependence on reactant rotation", J. Chem. Phys., 124, 031101 (2006). PI Yu Lin Lin, Y., X. Y. Wang, M. Brown, M. Schaffer, and C. D. Cothran, Modeling Swarthmore Spheromak Reconnection Experiment Using hybrid code, Plasma Physics and Controlled Fusion, in press, 2008. Yoon, P., Y. Lin, X. Y. Wang, and A. T. Y. Lui, Drift instabilities for current sheet equilibrium with guide field, J. Geophys. Res., submitted, 2008. PI Feng Liu D. Yu and F. Liu, Synthesis of Carbon Nanotubes by Rolling up Patterned Graphene Nanoribbons Using Selective Atomic Adsorption, Nano Lett. 7, 3046 (2007) L. Bai, D. Yu, G-H Lu, F Liu, Q. Wang, H Yilmaz, Confining P diffusion in Si by an As-doped barrier layer, App. Phys. Lett. 91, 016926 (2007) Q. Yan, B. Huang, J. Yu, FW Zheng, J. Zang, J. Wu, BL Gu, F. Liu, WH Duan Intrinsic current-voltage characteristics of graphene nanoribbon transistors and effect of edge doping, NANO LETTERS 7,1469(2007) J. Zang, O. Aldas-Palacios, Liu F MD simulation of structural and mechanical transformation of single-walled carbon nanotubes under pressure COMMUNICATIONS IN COMPUTATIONAL PHYSICS 2 (3): 451-465 JUN 2007 DJ Shu DJ, M Wang, F. Liu, et al. Nucleation-mediated lateral growth on foreign substrate JOURNAL OF PHYSICAL CHEMISTRY C 111 (3): 1071-1075 JAN 25 2007 J. Zang, MH Huang, F. Liu, Mechanism for nanotube formation from self-bending nanofilms driven by atomic-scale surface-stress imbalance PHYSICAL REVIEW LETTERS 98 (14): Art. No. 146102 (2007) GH Lu, Q. Wang, F. Liu, First-principles calculation of interaction between interstitial O and As dopant in heavily As-doped Si JOURNAL OF APPLIED PHYSICS 101 (2): Art. No. 026104 JAN 15 2007 C. Zhang, JC Cao, XG Guo, et al. Impurity mediated absorption continuum in single-walled carbon nanotubes APPLIED PHYSICS LETTERS 90 (2): Art. No. 023106 JAN 8 2007 SC Li, Y. Han, JT Jia, et al. Determination of the Ehrlich-Schwoebel barrier in epitaxial growth of thin films PHYSICAL REVIEW B 74 (19): Art. No. 195428 NOV 2006 PI Keh-Fei Liu N. Mathur, A. Alexandru, Y. Chen, S.J. Dong, T. Draper, I. Horvath, F.X. Lee, K.F. Liu, S. Tamhankar, and J.B. Zhang, `a_0(1450) and \sigma(600) Mesons from Lattice QCD', Phys. Rev. D (to appear), [hep-ph/0607110]. H.Y. Cheng, C.K. Chua, and K.F. Liu, `Scalar Glueball, Scalar Quarkonia, and their Mixing, Phys. Rev. D74, 094005 (2006), [hep-ph/0607206]. Terrence Draper, Nilmani Mathur, Andrei Alexandru, Ying Chen, Shao-Jing Dong, Ivan Horvath, Frank X. Lee, Keh-Fei Liu, Sonali Tamhankar, Jianbo Zhang, `On the Locality and Scaling of Overlap Fermions at Coarse Lattice Spacings', submitted for publication, [hep-lat/0609034]. K.F. Liu, A. Alexandru, I. Horvath, `Gauge Field Strength Tensor from the Overlap Dirac Operator', submitted for publication, [hep-lat/0703010]. K.F. Liu, `Pattern of Light Scalar Mesons', Prog. Theo. Phys. Suppl. 168, 1 (2007), [arXiv:0706.1262]. E.-M. Ilgenfritz, K. Koller, Y. Koma, G. Schierholz, T. Streuer (U. Kentucky), V. Weinberg, `Exploring the structure of the quenched QCD vacuum with overlap fermions', Phys. Rev. D76, 034506 (20070, [arXiv:0705.0018]. PI Stewart Loken A Standard format for Les Houches event files. J.Alwall et al. FERMILAB-PUB-06-337-T, CERN-LCGAPP-2006-03, Sep 2006. 8pp. Written within the framework of the MC4LHC-06 workshop: Monte Carlos for the LHC: A Workshop on the Tools for LHC Event Simulation (MC4LHC), Geneva, Switzerland, 17-16 Jul 2006. Published in Comput.Phys.Commun.176:300-304,2007. e-Print: hep-ph/0609017 PI Steven Louie J. Deslippe, C.D. Spataru, D. Prendergast, and S.G. Louie, "Bound Excitons in Metallic Single-walled Carbon Nanotubes," Nano Lett. 7, 1626 (2007). F. Wang, D. Cho, B. Kessler, J. Deslippe, P.J. Schuck, S.G. Louie, A. Zettl, T.F. Heinz, Y.R. Shen, "Observations of Excitons in One-Dimensional Metals," accepted to Phys. Rev. Lett. C.-H. Park, F. Giustino, M.L. Cohen, and S.G. Louie, "Velocity renormalization and carrier lifetime in graphene from the electron-phonon interaction," Phys. Rev. Lett. 99, 086804 (2007). M.J. Comstock, N. Levy, A. Kirakosian, J. Cho, F. Lauterwasser, J.H. Harvey, D.A. Strubbe, J.M.J. Frechet, D. Trauner, S.G. Louie, and M.F. Crommie, "Reversible Photomechanical Switching of Individual Engineered Molecules at a Metallic Surface," Phys. Rev. Lett. 99, 038301 (2007). Y. Wang, X.H. Lu, E. Kioupakis, R. Yamachika, D. Wegner, S.G. Louie, M.F. Crommie, J.E. Dahl, S.G. Liu, and R.M.K. Carlson, "Spatially resolved electronic and vibronic characteristics of tetramantane diamondoid," submitted to Nature Materials. P. Tangney, M. L. Cohen, and S. G. Louie, "Giant Wave-Drag Enhancement of Friction in Sliding Carbon Nanotubes," Phys. Rev. Lett. 97, 195901 (2006). T. Miyake, P.H. Zhang, M.L. Cohen, and S.G. Louie, "Quasiparticle Energy of Semicore d-electrons in ZnS: Combined LDA+U and GW approach," Phys. Rev. 74, 245213 (2006). L. Yang, C. D. Spataru, S. G. Louie, and M. Y. Chou, "Enhanced Electron-hole Interaction and Optical Absorption in a Silicon Nanowire," Phys. Rev. B 75, 201304 (2007). S.Y. Quek, J.B. Neaton, M.S. Hybertsen, E. Kaxiras, and S.G. Louie, "Negative Differential Resistance in Transport through Organic Molecules on Silicon," Phys. Rev. Lett 98, 066807 (2007). I.H. Choi, P.Y. Yu, P. Tangney, and S.G. Louie, "Vibrational Properties of Single-Walled Carbon Nanotubes Under Pressure from Raman Scattering Experiments and Molecular Dynamics," Phys. Status Sol. (b) 244, 121 (2007). J.B. Neaton, M.S. Hybertsen, and S.G. Louie, "Renormalization of Molecular Electronic Levels at Metal-Molecule Interfaces," Phys. Rev. Lett. 97, 216405 (2006). PI Walter Loveland G.L. Malli, "Thirty years of relativistic self-consistent field theory for molecules:relativistic and electron correlation effects for atomic and molecular systems of transactinide superheavy elements up to ekaplutonium E126 with g-atomic spinors in the ground state configuration", Theor Chem Account. 118, 473(2007). G.L. Malli, (Communication) "Dissociation energy of Ekaplutonium fluoride E126F", J. Chem. Phys. 124,071102(2006). G.L. Malli, (Communication) "Electronic structure and prediction of atomization energy of naked homoleptic uranium hexacarbonyl U(CO)6", J. Chem. Phys. 124,021102(2006). R. Arratia-Perez and G.L. Malli, "Relativistic molecular orbital study of the optical and magnetic properties of hexachloro protoactinate (IV): PaCl6-2, J. Chem.Phys. 124,074321(2006). G.L. Malli and M.Siegert, " Dramatic effects of relativity for the correlation energy of molecules of heavy atom: all-electron all-virtual spinor space (AVSS) relativistic coupled-cluster calculations for PbH4", Intern J of Quantum Chem. (Submitted on Sept 24, 2007). W.Loveland, "Synthesis of transactinide nuclei using radioactive beams", Phys.Rev. C76, 014612(2007) W.Loveland, "Comment on the possibility of synthesizing a doubly magic superheavy nucleus", Phys.Rev.C75, 069801(2007). J.F.Liang,D.Shapira, J.R.Breene, C.J.Gross,.....W.Loveland and D.Peterson, "Fusion of radioactive 132Sn with 64Ni, Phys.Rev. C75,054607(2007). W. Loveland, D.J.Morrissey and G.T.Seaborg, "Modern Nuclear Chemistry", Wiley, New York, 671 pages (2006). R.W. Stoenner, R.L. Klobucher, P.E. Haustein, G.J. Virtes, J.B. Cumming and W. Loveland, "Angular distribution in multifragmentation", Phys. Rev. C 73,047602(2006). W. Loveland, D. Peterson, A.M. Vinodkumar,...et al Fusion enhancement in the 38S + 208Pb reaction, Phys. Rev. C 74, 044607 (2006). W. Loveland, A.M. Vinodkumar, R.S.Naik, P.H. Sprunger...et al The fusion of 9Li with 70Zn , Phys. Rev C 74, 064609 (2006) A.M. Vinodkumar, W. Loveland, P. Sprunger, D. Peterson Capture cross sections for the near symmetric 124Sn + 96Zr reaction, Phys. Rev C 74, 064612(2006). PI Chung-Pei Ma M. Boylan-Kolchin, C.-P. Ma, and E. Quataert (2007), Monthly Notices of the Royal Astronomical Society, in press. "Dynamical Friction and Galaxy Merging Timescales" M. Boylan-Kolchin and C.-P. Ma (2006), MNRAS, 374, 1227. Satellite Accretion onto Massive Galaxies with Central Black Holes L. Desroches, E. Quataert, C.-P. Ma, and A. West (2007), MNRAS, 377, 402. "Luminosity Dependence in the Fundamental Plane Projections of Elliptical Galaxies" PI Steven Manson "Dynamical and Relativistic Effects in Experimental and Theoretical Studies of Inner-Shell Photoionization of Sodium," D. Cubaynes, H. L. Zhou, N. Berrah, J.-M. Bizau, J. D. Bozek, S. Canton, S. Diehl, X.-Y. Han, A. Hibbert, E. T. Kennedy, S. T. Manson, L. VoKy and F. J. Wuilleumier, J. Phys. B 40, F121-F129 (2007). PI Shaolin Mao S. Mao, E.G. Patton, Z.G. Feng, and E.E. Michaelides, Impact of imposed wall-jet on canopy turbulence and the net ecosystem-atmosphere exchange of carbon dioxide, Accepted for publication on Boundary-Layer Meteorology, 2007. S. Mao, Z.G. Feng, and E.E. Michaelides, Large-eddy simulation of low-level jet-like flow in canopy, Environmental Fluid Mechanics, 7(1) 73-93, 2007. M. Leclerc, S.Mao and E.E. Michaelides, Flux footprint analysis over horizontal inhomogeneous canopies, resubmitted, 2007. S. Mao, E.E. Michaeldies, CO2 flux footprint analysis over horizontally inhomogeneous plant canopy using large-eddy simulation, submmitted to Atmospheric Environment, 2007. PI Angelo Mascarenhas Y. Zhang, L.-W. Wang, and A. Mascarenhas, Quantum coaxial cables for solar energy harvesting, Nano. Lett. 7, 1264 (2007). L. C. Lew Yan Voon, Y. Zhang, B. Lassen, M. Willatzen, Qihua Xiong, and P. C. Eklund, Electronic Properties of Semiconductor Nanowires (invited review paper), J. Nanotechnology (in press). PI Artem Masunov Ivan A. Mikhailov, Sergio Tafur, Artëm E. Masunov; Double Excitations and State to State Transition Dipoles in Excited Singlet States of Linear Polyenes: Time Dependent Density Functional Theory vs. Multiconfigurational Methods. Submitted to Phys. Rev. A PI Manos Mavrikakis A. U. Nilekar, J. Greeley, M. Mavrikakis, "A simple rule of thumb for diffusion on transition metal surfaces", Angewandte Chemie International Edition 45, 7046 (2006). L. C. Grabow, Y. Xu, M. Mavrikakis, "Lattice strain effects on CO oxidation on Pt(111)", Physical Chemistry Chemical Physics 8, 3369 (2006) - including cover-page image. S. Kandoi, J. Greeley, M. Sanchez-Castillo, St. T. Evans, A. A. Gokhale, J. A. Dumesic, M. Mavrikakis, "Prediction of Experimental Methanol Decomposition Rates on Platinum from First-Principles", Topics in Catalysis 37, 17 (2006). J. Greeley, M. Mavrikakis, "Near Surface Alloys for Hydrogen Fuel Cell Applications", Catalysis Today 111, 52 (2006). T. Mitsui, E. Fomin, F. Ogletree and M. Salmeron, A. U. Nilekar, M. Mavrikakis, "Manipulation and patterning of the surface concentration of hydrogen on Pd(111) by electric fields", Angewandte Chemie International Edition 46, 5757 (2007). J. Greeley, M. Mavrikakis, "On the role of subsurface oxygen and ethylenedioxy in ethylene epoxidation on silver", Journal of Physical Chemistry C, 111, 7992 (2007). Y. Xu, H. Marbach, R. Imbihl, I. G. Kevrekidis, M. Mavrikakis, "The effect of co-adsorbed oxygen on the adsorption and diffusion of potassium on Rh(110): A first-principles study", Journal of Physical Chemistry C, 111, 7446 (2007). J. Knudsen, A. U. Nilekar, R. T. Vang, J. Schnadt, E. Kunkes, J. A. Dumesic, M. Mavrikakis, F. Besenbacher, "A Cu/Pt Near-Surface Alloy for water-gas shift catalysis", Journal of the American Chemical Society, 129, 6485 (2007). J. Zhang, M.B. Vukmirovic, K. Sasaki, A.U. Nilekar, F. Uribe, M. Mavrikakis, R.R. Adzic, "Platinum Monolayer Electrocatalysts for Oxygen Reduction", Electrochimica Acta 52, 2257 (2007). R. R. Adzic, J. Zhang, K. Sasaki, M. B. Vukmirovic, M. Shao, J. X. Wang, A. U. Nilekar, M. Mavrikakis, J. A. Valerio, F. Uribe, "Platinum Monolayer Fuel Cell Electrocatalysts", Topics in Catalysis (in press). A. U. Nilekar, Y. Xu, J. Zhang, M. B. Vukmirovic, K. Sasaki, R. R. Adzic, M. Mavrikakis, "Bimetallic and ternary alloys for improved oxygen reduction catalysis", Topics in Catalysis (in press). L. C. Grabow, A. A. Gokhale, St. Evans, J. A. Dumesic, M. Mavrikakis, "Mechanism of the water gas shift reaction on Pt: first-principles, experiments, and microkinetic modeling", (submitted). N. Schumacher, K. Andersson, L. C. Grabow, M. Mavrikakis, J. Nerlov, and I. Chorkendorff, "Interaction of CO2 with Cu overlayers on Pt(111)", (submitted). A. A. Gokhale, J. A. Dumesic, M. Mavrikakis, "On the mechanism of low temperature water-gas-shift reaction on copper", (submitted). PI William McCurdy Wim Vanroose, D. A. Horner, F. Martmn T. N. Rescigno and C. W. McCurdy, "Double Photoionization of Aligned Molecular Hydrogen", Phys. Rev. A 74, 052702 (2006). T. N. Rescigno, Wim Vanroose, D. A. Horner, F. Martmn and C. W. McCurdy, "First Principles Study of Double Photoionization of H2 Using Exterior Complex Scaling", J. Elec. Spectros., Rel. Phenom. 161, 85 (2007). C.S. Trevisan, A.E. Orel and T. N. Rescigno, "Low-energy Electron Scattering by Formic Acid", Phys. Rev. A 74, 042716 (2006). D. A. Horner, W. Vanroose, T. N. Rescigno, F. Martin and C. W. McCurdy, "Role of Nuclear Motion in Double Ionization of Molecular Hydrogen by a Single Photon", Phys. Rev. Lett. 98, 073001 (2007). D. J. Haxton, C. W. McCurdy and T. N. Rescigno, "Dissociative Electron Attachment to the H2O molecule I: Complex-valued Potential Energy Surfaces for the 2B1, 2A1 and 2B2 Metastable States of the Water Anion", Phys. Rev. A. 75, 012710 (2007). D. J. Haxton, T. N. Rescigno and C. W. McCurdy, "Dissociative Electron Attachment to the H2O molecule II: Nuclear Dynamics on Coupled Electronic Surfaces Within the Local Complex Potential Model", Phys. Rev. A. 75, 012711 (2007). F. L. Yip. D. A. Horner, C. W. McCurdy and T. N. Rescigno, "Single and Triple Differential Cross Sections for Double Photoionization of H-(\", Phys. Rev. A. 75, 042715 (2007). . D. J. Haxton, T. N. Rescigno and C. W. McCurdy, Comment on A Wave Packet Method for Treating Nuclear Dynamics on Complex Potentials, J. Phys. B 40, 1461 (2007). A. Palacios, T. N. Rescigno and C. W. McCurdy, "Extracting Amplitudes for Single and Double Ionization from a Time-Dependent Wavepacket", Phys. Rev. A 76, 043420 (2007). D. A. Horner, F. Morales, T. N. Rescigno, F. Martin and C. W. McCurdy, "Two-Photon Double Ionization of Helium Above and Below the Threshold for Sequential Ionization", Phys. Rev. A 76, 030701(R) (2007). T. N. Rescigno, C. W. McCurdy, D. J. Haxton, C. S. Trevisan and A. E. Orel, "Nuclear Dynamics in Resonant Electron Collisions with Small Polyatomic Molecules", J. Phys. Conference Series (accepted). K. Houfek, T. N. Rescigno and C. W. McCurdy, "Probing the Nonlocal Approximation to Resonant Collisions of Electrons with Diatomic Molecules", Phys. Rev. A (accepted). PI Donghai Mei Donghai Mei, N. Aaron Deskins and Michel Dupuis "A Density Functional Theory Study of Formaldehyde Adsorption on Ceria", Surf. Sci.,2007 (in press) PI David Mikkelsen S.M. Kaye, et al., Nucl. Fusion 47 (2007) 499 M. Greenwald, et al., Fusion Sci Technol. 51 (2007) 266 F. M. Levinton, et al., Phys. Plasmas 14 (2007) 056119 J.E. Menard, et al., Nucl. Fusion 47 (2007) S645 K.-L. Wong, et al., Phys. Rev. Lett. 99 (2007) 135003 D. Stutman, et al., Phys. Plasmas 13 (2006) 092511 PI Thomas Miller T. F. Miller, III and C. Predescu, J. Chem. Phys., 126, 144102 (2007). T. F. Miller, III, E. Vanden-Eijnden, and D. Chandler, PNAS, 104, 14559 (2007). PI William Miller J. Liu and W. H. Miller, Using the Thermal Gaussian Approximation for the Boltzmann Operator in Semiclassical Initial Value Time Correlation Functions, J. Chem. Phys. 125, 224104.1-13 (2006). T. F. Miller and C. Predescu, Sampling Diffusive Transition Paths, 126 144102.1-12 (2007). C. Venkataraman and W. H. Miller, Chemical Reaction Rates Using the Semiclassical Van-Vleck Initial Value Representation, J. Chem. Phys. 126, 094104.1-8 (2007). J. Liu and W. H. Miller, Real Time Correlation Function in a Single Phase Space Beyond the Linearized Semiclassical Initial Value Representation, J. Chem. Phys. 126, 234110.1-11 (2007). N. Ananth, C. Venkataraman and W. H. Miller, Semiclassical (SC) Description of Electronically Non-Adiabatic Dynamics via the Initial Value Representation (IVR), J. Chem. Phys. 127, 084114.1-9 (2007). J. Liu and W. H. Miller, Linearized Semiclassical Initial Value Time Correlation Functions Using the Thermal Gaussian Approximation: Applications to condensed phase systems, J. Chem. Phys. (accepted) PI Warren Mori M. Tzoufras et al., "Stability of arbitrary electron velocity distribution functions to electromagnetic modes," Phys. Plasmas 14-062108 (2007) C. Ren, E. G. Blackman, and W.-F. Fong, "Understanding the saturation of proton-driven Weibel instabilities and implications for astrophysics," Phys. Plasmas 14-012901 (2007) G. Li et al., "Laser channeling in mm-scale underdense plasmas of fast ignition targets," submitted to Phys. Rev. Lett. C. Huang, V. K. Decyk, M. Zhou, W. Lu, W. B. Mori, J. H. Cooley, T. M. Antonsen Jr, B. Feng, T. Katsouleas, J. Vieira and L. O. Silva, J. Phys.: Conf. Ser. 46, pp 190-199 (2006). C. Huang, V.K. Decyk, C. Ren, M. Zhou, W. Lu, W.B. Mori, J.H. Cooley, T.M. Antonsen, Jr. and T. Katsouleas, Journal of Computational Physics, 217, pp 658-679 (2006). Oz E., Deng S., Katsouleas T, Muggli P., Barnes C.D., Blumenfeld I., Decker F.J., Emma P., Hogan M.J., Ishchebeck R., Iverson R.H., Kirby N., Krejcik P., O'Connell C., Siemann R.h., Walz D., Auerbach D., Clayton D.E., Huang C., Johnson D.K., Joshi C., Lu W., Marsh K.A., Mori W.B., Zhou M., "Ionization- induced electron trapping in ultra-relativistic plasma wakes", Physical Review Letters, vol 98, no. 8 pp. 084801/1-4 23 February 2007. Thomas AGR, Najmudin Z., Mangles SPD, Murphy C.D., Dangor A.E., Kamperidis C., Lancaster K.L., Mori W.B., Norreys P.A., Rozmus W., Kushelnick K., "Effect of laser-focusing conditions on propagation and monoenergetic electron production in laser-wakefield accelerators", Physical Review Letters, vol 98, no. 4 pp. 095004/1-4 2 March 2007. F. S. Tsung et al J. Phys.: Conf. Ser., 78, 0120077 (2007) Blumenfeld I., Clayton C.E., Decker F-J., Hogan M.J., Huang Chengkun, Ischebeck R., Iverson R., Joshi Chandrashekhar, Katsouleas T., Kirby N., Lu W., Marsh K.A., Mori W.B., Muggli P., Oz E., Siemann R.H., Walz D., Zhou M., Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator, Nature, vol. 445 no. 7129, pp. 741-4 15 February 2007. Lu W., Tzoufras M. , Joshi C., Tsung F.S.,.Mori W.B., Vieira J., Fonseca R.A., Silva L.O., "Generating Multi-GeV Electron Bunches Using Single Stage Laser Wakefield Acceleration in a 3D Nonlinear Regime," Physical Review Special Topics Acceleraors and Beams vol. 10, no. 6pp. 061301 June 2007. PI James Morris J. R. Morris, F. Jiang, P. Liaw, "A simple model for examining composition effects in eutectic nucleation," Mater. Trans. (JIM) 48, 1675-1679 (2007). T. Egami, V. Levashov, R. S. Aga and J. R. Morris, "Atomic dynamics in metallic liquids and glasses," Mater. Trans. (JIM) 48, 1729-1733 (2007). J. R. Morris, U. Dahlborg and M. Calvo-Dahlborg, "Recent developments and outstanding challenges in theory and modeling of liquid metals," J. Non-cryst. Sol 353, 3444-3453 (2007). J. R. Morris, M. I. Mendelev and D. J. Srolovitz, "A comparison of crystal-melt interfacial free energies using different Al potentials," J. Non-cryst. Sol 353, 3565-3569 (2007). C. L. Fu and Maja Krcmar, "First-principles study of the structural, defect, and mechanical properties of B2 FeCo alloys", Physical Review B74, 174108 (2006). Maja Krcmar, C. L. Fu, and James R. Morris, "Structural transformations and improved ductility in ordered FeCo and ZrCo intermetallics", in Advanced Intermetallic-Based Alloys, MRS Symposium Proceedings, volume 980, 89 (2007). James R. Morris, Yiying Ye, Maja Krcmar, and C. L. Fu, "The role of phase stability in ductile, ordered B2 intermetallics", in Advanced Intermetallic-Based Alloys, MRS Symposium Proceedings, volume 980, 113 (2007). C. L. Fu and Maja Krcmar, G. S. Painter, and Xing-Qiu Chen, "The vacancy mechanism of high oxygen solubility and nucleation of stable oxygen-enriched clusters in Fe", Physical Review Letters (accepted). PI Shaul Muckerman "Many-body Effects in 2-D Optical Spectra of Semiconductor Quantum-Dot Pairs: TPHF and Beyond, R. Oszwaldowski, D. Abramavicius and S. Mukamel, J. Phys. Cond. Matt (Submitted, 2007) PI Habib Najm J. Ray, C. A. Kennedy, S. Lefantzi and H. N. Najm, "Using high-order methods on adaptively refined block-structured meshes - derivatives, interpolations, and filters", SIAM Journal on Scientific Computing, 2007, 29(1):139-181. N. Trebon, A. Morris, J. Ray, S. Shende and A. D. Malony, "Performance modeling using component assemblies", Concurrency and Computation: Practice and Experience, 2007, 19(5):685-696. David E. Bernholdt, Benjamin A. Allan, Robert Armstrong and Felipe Bertrand, Kenneth Chiu, Tamara L. Dahlgren, Kostadin Damevski, Wael R. Elwasif, Thomas G. W. Epperly, Madhusudhan Govindaraju, Daniel S. Katz, James A. Kohl, Manoj Krishnan, Gary Kumfert, J. Walter Larson, Sophia Lefantzi, Michael J. Lewis, Allen D. Malony, Lois C. McInnes, Jarek Nieplocha, Boyana Norris, Steven G. Parker, Jaideep Ray, Sameer Shende, Theresa L. Windus, and Shujia Zhou, "A Component Architecture for High-Performance Scientific Computing", International Journal of High-Performance Computing Application, 2006, 20:162-202. PI John Negele QCD, C. Alexandrou, G. Koutsou, Th. Leontiou, J.W. Negele, A. Tsapalis. Accepted for publication in Phys. Rev. D. (2007) arXiv:0706.3011 [hep-lat] The Axial N to Delta transition form factors from Lattice QCD.? C. Alexandrou, Th. Leontiou, J.W. Negele, A. Tsapalis, Phys.Rev.Lett.98:052003,2007, hep-lat/0607030 The Nucleon electromagnetic form factors from Lattice QCD.? C. Alexandrou, G. Koutsou, J.W. Negele, A. Tsapalis, Phys.Rev.D74:034508,2006, hep-lat/06050176 The Nucleon to Delta electromagnetic transition in full lattice QCD. C. Alexandrou, G. Koutsou, H. Neff, J.W. Negele, W. Schroers, A. Tsapalis, submitted to Phys. Rev. D PI Brian Nelson "Hall magnetohydrodynamics simulations of end-shorting induced rotation in field-reversed configurations", A. I. D. Macnab, R. D. Milroy, C. C. Kim, and C. R. Sovinec, Phys. Plasmas 14, 092503 (2007) "Overview of the Plasma Science and Innovation Center (PSI - Center)" Jarboe, T.R.; Bayless, A.; Held, E.; Jeong-Young Ji; Kim, C.; Macnab, A.; Marklin, G.; Milroy, R.D.; Nelson, B.A.; Uri Shumlak; Sovenic, C.; Vadlamani, S.; Woodruff, S., Journal of Fusion Energy, v 26, n 1-2, June 2007, p 91-2 "The Plasma Science and Innovation Center Interfacing Group", Nelson, B.A.; Kim, C.C.; Cassidy, A.P.; Griffith, S.D.; Milroy, R.D.; Jarboe, T.R., Journal of Fusion Energy, v 26, n 1-2, June 2007, p 127-30 "Simulations of the field-reversed configuration with the NIMROD code", Macnab, A.I.D. ; Barnes, D.C.; Milroy, R.D.; Kim, C.C.; Sovinec, C.R. Journal of Fusion Energy, v 26, n 1-2, June 2007, p 113-17 "Preliminary Simulations of FLR Effects on RFP Tearing Modes", Charlson C. Kim and NIMROD Team, Journal of Fusion Energy, 0164-0313, (2007) PI Karoly Nemeth Karoly Nemeth, Baifei Shen, Yuelin Li, Robert Crowell, Katherine C. Harkay and John Cary: Laser-Driven Coherent Betatron Oscillation in a Laser-Wakefield Cavity, submitted to Physical Review Letters. Baifei Shen, Yuelin Li, Karoly Nemeth, Hairong Sheng, Robert Soliday, Robert Crowell, Edward Frank, William Gropp and John Cary: Electron injection by a nanowire in the bubble regime, Physics of Plasmas 14, 053115 (2007). PI Gregory Newman Commer M. and Newman G. A., 2007, New advances in three-dimensional controlled- source electromagnetic inversion: Geophysical Journal International, In Press. Commer M., Newman G., Carazzone, J., Dickens, T., Green, K., Wahrmund,L., Willen, D., and Shiu, J., 2007, Massively parallel eletrical imaging of hydrocarbons using the Blue Gene/L supercomputer: IBM Journal of Research and Development, 52,No. 1/2. PI Cheuk-Yiu Ng Juan Li, Jie Yang, Yuxiang Mo, K. C. Lau, X. M. Qian, Y. Song, Jianbo Liu, and C. Y. Ng, J. Chem. Phys., 126, 184304 (2007). "Combined Vacuum Ultraviolet Laser and Synchrotron Pulsed Field Ionization Study of CH2BrCl". Xi Xing, Beth Reed, Kai-Chung Lau, C. Y. Ng, Xu Zhang, and G. Barney Ellis, J. Chem. Phys., 126, 171101 (2007). "Vacuum Ultraviolet Laser Pulsed Field Ionization-Photoelectron Study of Allyl Radical CH2CHCH2". Xiaonan Tang, Cassidy Houchins, Kai-Chung Lau, C. Y. Ng, Rainer A. Dressler, Yu-hui Chiu, Tian-Shu Chu and Ke-Li Han, J. Chem. Phys., in press, (2007). "A time-dependent wave packet quantum scattering study of the reaction HD+(v=0-3; j0=1) + He -> HeH+ (HeD+) + D(H)". K.-C. Lau and C. Y. Ng, Acc. Chem. Res., 39, 823 (2006). "Benchmarking state-of-the-art ab initio thermochemical predictions with accurate pulsed-field ionization photoion-photoelectron measurements". PI Esmond Ng P. Amestoy, X.S. Li, and E.G. Ng. "Diagonal Markowitz Scheme with Local Symmetrization", SIAM J. Matrix Analysis and Applications, Vol. 29, No. 1, pp. 228-244, 2007. P. Amestoy, X.S. Li, and S. Pralet. "Unsymmetric Ordering Using a Constraint Markowitz Scheme", SIAM J. Matrix Analysis and Applications, Vol. 29, No. 1, pp. 302-327, 2007. S. Chandrasekaran, M. Gu, X. S. Li, and J. Xia. "Some Fast Algorithms for Hierarchically Semiseparable Matrices", Submitted to SIAM J. Matrix Analysis and Applications. June, 2007. S. Chandrasekaran, M. Gu, X. S. Li, and J. Xia. "Superfast Multifrontal Method for Structured Linear Systems of Equations", Submitted to SIAM J. Matrix Analysis and Applications. June, 2007. W. Gao, X.S. Li, C. Yang, and Z. Bai. "An Implementation and Evaluation of the AMLS Method for Sparse Eigenvalue Problems", ACM Trans. on Math. Software, vol. 34, no. 4, 2007. Laura Grigori, Michel Cosnard, and Esmond G. Ng. "On the Row Merge Tree for Sparse LU Factorization with Partial Pivoting". BIT Numer. Math, Vol. 47, No. 1, March 2007, pp. 45-76. L. Grigori, J.W. Demmel, and X.S. Li. "Parallel Symbolic Factorization for Sparse LU with Static Pivoting", SIAM J. Scientific Computing, Vol. 29, Issue 3, 1289-1314, 2007. L. Grigori and X.S. Li. "Towards an Accurate Performance Modeling of Parallel Sparse Factorization", Applicable Algebra in Engineering, Communication, and Computing, Vol. 18, No. 3, pp. 241-261, 2007. X.S. Li, J. Demmel, L. Grigori, M. Gu, J. Xia, S. Jardin, C. Sovinec, L.-Q. Lee. "Enhancing Scalability of Sparse Direct Methods", Proceedings of SciDAC 2007, Scientific Discovery Through Advanced Computing, 24-28 June 2007, Boston. J. Physics: Conference Series 78 (2007) 012041, Institute of Physics Publishing. K. Nakajima. "The Impact of Parallel Programming Models on the Linear Algebra Performance for Finite Element Simulations", Lecture Notes in Computer Science 4395, 334-348, Special Issue of VEPAR 2006, Rio de Janeiro, Brazil (2007). K. Nakajima. "Parallel Multistage Preconditioners based on a Hierarchical Graph Decomposition for SMP Cluster Architectures with a Hybrid Parallel Programming Model", Lecture Notes in Computer Science 4782, 384-395, HPCC'07 (High-Performance Computation Conference 2007), Houston, Texas Esmond G. Ng. "Parallel Sparse Solvers, Preconditioners, and Their Applications", in Parallel Processing for Scientific Computing, eds. Michael Heroux, Padma Raghavan, and Horst Simon, SIAM (2006). PI Jerry Nolen Wish-List for Large-Scale Simulations for Future Radioactive Beam Facilities http://cern.ch/AccelConf/ICAP06/PAPERS/WEMPMP02.PDF "Application of a New Procedure for Design of 325 MHz RFQ" P. N. Ostroumov, V. N. Aseev, and A. A. Kolomiets Journal of Instrumentation, JINST 1 P04002 Physics Design of the 8-GeV H-Minus Linac P. N. Ostroumov New Journal of Physics, http://www.iop.org/EJ/njp Status: ACCEPTED FOR PUBLICATION Parallelization of a Beam Dynamics Code TRACK and First Large Scale RFQ Simulations, J. Xu, B. Mustapha, V. N. Aseev, and P. N. Ostroumov, Phys. Rev. ST, Status: TIS Computational Needs for the RIA Accelerator Systems P. N. Ostroumov, J. A. Nolen, and B. Mustapha Proceedings of the 8th International Computational Accelerator Physics Conference (ICAP-04), St. Petersburg, Russia, June 29-July 2, 2004; Nucl. Instrum. Methods A558, 25-31 (2006), Status: PUBLISHED PI Arthur Nozik S.-H. Lee, Y.-H. Kim, R. Deshpande, P. A. Parilla, K. M. Jones, B. To, E. Whitney, A. H. Mahan, S. B. Zhang, and A. C. Dillon, "Anomalous Reversible Lithium-Ion Intercalation in Molybdenum Oxide Nanoparticles", Submitted to Science (2007). Y. Jiang, Y.-H. Kim, S. Yang, Z. Tang, K. Wu, P. Ebert, S. B. Zhang, and E. G. Wang, "Growing extremely thin bulklike metal film on a semiconductor surface: monolayer Al(111) on Si(111)", Appl. Phys. Lett., Accepted (2007). T. J. McDonald, D. Svedruzic, Y.-H. Kim, J. L. Blackburn, S. B. Zhang, P. W. King, and M. J. Heben, "Wiring-up hydrogenase with single-walled carbon nanotubes", Nano Lett., Accepted (2007). Y. Y. Sun, Y.-H. Kim, and S. B. Zhang, "Effect of Spin State on the Dihydrogen Binding Strength to Transition Metal Centers in Metal-organic Frameworks", J. Am. Chem. Soc., published online (Sep. 2007). A. C. Dillon, E. Whitney, C. Engtrakul, C. J. Curtis, K. J. O'Neill, P. A. Parilla, L. J. Simpson, M. J. Heben, Y. Zhao, Y.-H. Kim, and S. B. Zhang, "Novel organometallic fullerene complexes for vehicular hydrogen storage", phys. stat. sol. (b), published online (Sep. 2007). C. Engtrakul, Y.-H. Kim, J. M. Nedeljkovic, S. P. Ahrenkiel, K. E. H. Gilbert, J. L. Alleman, S. B. Zhang, O. I. Micic, A. J. Nozik, and M. J. Heben, "Self-Assembly of Linear Arrays of Semiconductor Nanoparticles on Carbon Single-Walled Nanotubes", J. Phys. Chem. B 110, 25153-25157 (2006). PI Peter Nugent E. Huff, A.E. Schulz, M. White, D.J. Schlegel, & M.S. Warren, "Simulations of Baryon Oscillations", Astroparticle Physics, 26, 351. D.J. Eisenstein, H.-J. Seo, & M. White, "On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter," Astrophysical Journal, 664, 660. N. Padmanabhan, M. White, & D.J. Eisenstein, "A Robust Estimator of the Small-Scale Galaxy Correlation Function," Monthly Notices of the Royal Astronomical Society, 376, 1702. PI Grazyna Odyniec Longitudinal double-spin asymmetry for inclusive jet production in p+p collisions at sqrt(s)=200 GeV STAR Collaboration arXiv:0710.2048 Measurement of Transverse Single-Spin Asymmetries for Di-Jet Production in Proton-Proton Collisions at sqrt(s) = 200 GeV STAR COllaboration Phys. Rev. Lett. 99(2007)142003 Forward Lambda Production and Nuclear Stopping Power in d+Au Collisions at sqrt(sNN) = 200 GeV STAR Collaboration e-Print: 0706.0472 Partonic flow and phi-meson production in Au+Au collisions at sqrt(s) = 200 GeV STAR Collaboration Phys. Rev. Lett. 99 (2007) 112301 Enhanced strange baryon production in Au+Au collisions compared to p+p at sqrt(s)=200 GeV STAR Collaboration e-Print Archives (0705.2511) Global polarization measurement in Au+Au collisions STAR Collaboration Phys. Rev. C 76(2007)024915 Strangelet Search in AuAu collisions at 200 GeV STAR Collaboration Phys. Rev. C 76 (2007)011901 Strange particle production in p+p collisions at sqrt(s) = 200 GeV STAR Collaboration Phys. Rev. C 75(2007)064901 Partonic flow and phi-meson production in Au+Au collisions at sqrt(s)=200 GeV STAR Collaboration Phys. Rev. Lett. 99(2007)112301 Charged particle distributions and nuclear modification at high rapidities in d+Au collisions at sqrt(s)=200 GeV STAR Collaboration e-Print nucl-ex/0703016 Mass, quark-number, and sqrt(s) dependence of the second and fourth flow harmonics in ultra-relativistic nucleus-nucleus collisions STAR Collaboration Phys. Rev. C 75(2007)054906 PI Joseph Oefelein T. G. Drozda, G. Wang, V. Sankaran, J. R. Mayo, J. C. Oefelein and R. S. Barlow (2007). Scalar filtered mass density functions in nonpremixed turbulent jet flames. Submitted to Combustion and Flame. T. C. Williams, R. W. Schefer, J. C. Oefelein and C. R. Shaddix (2007). Idealized gas turbine combustor for performance research and validation of large eddy simulations. Review of Scientific Instruments, 78(035114): 1-9. J. C. Oefelein, V. Sankaran and T. G. Drozda (2007). Large eddy simulation of swirling particle-laden flow in a model axisymmetric combustor. Proceedings of the Combustion Institute, 31: 2291-2299. J. C. Oefelein, T. G. Drozda and V. Sankaran (2006). Large Eddy Simulation of Turbulence-Chemistry Interactions in Reacting Flows: The Role of High-Performance Computing and Advanced Experimental Diagnostics. Journal of Physics, 46: 16-27. J. C. Oefelein (2006). Large eddy simulation of turbulent combustion processes in propulsion and power systems. Progress in Aerospace Sciences, 42: 2-37. PI Serdar Ogut Juan C. Idrobo, Mingli Yang, Koblar Jackson, and Serdar Ogut,"First Principles Absorption Spectra of Si_n (n=20-28) Clusters: Time-Dependent Local-Density Approximation versus Predictions from Mie Theory", Phys. Rev. B 74, 153410 (2006). Serdar Ogut, Juan C. Idrobo, Julius Jellinek, and Jinlan Wang, "Structural, Electronic, and Optical Properties of Noble Metal Clusters from First Principles", J. Cluster Sci. 17, 609 (2006). Juan C. Idrobo, Serdar Ogut, Karoly Nemeth, Julius Jellinek, and Riccardo Ferrando, "First Principles Isomer-Soecific Absorption Spectra of Ag_11", Phys. Rev. B 75, 233411 (2007). Hakim Iddir, Serdar Ogut, Peter Zapol, and Nigel Browning, "Diffusion Mechanisms of Native Point Defects in Rutile TiO2", Phys. Rev. B 75, 072303 (2007). Juan C. Idrobo, Weronika Walkosz, Shing Fan Yip, Serdar Ogut, Jinlan Wang, and Julius Jellinek, "Static Polarizabilities and Optical Absorption Spectra of Gold Clusters (Au_n, n = 2 - 14 and 20) from First Principles", to appear in Phys. Rev. B. PI Leonid Oliker L. Oliker, A. Canning, J. Carter, C. Iancu, M. Lijewski, S. Kamil, J. Shalf, H. Shan, E. Strohmaier, S. Ethier, T. Goodale,"Scientific Application Performance on Candidate PetaScale Platforms", International Parallel & Distributed Processing Symposium (IPDPS) 2007. LBNL-62952. WINNER: Best paper application track. L. Oliker, R. Biswas, R. Wijngaart, D. Bailey, A. Snavely. Chapter 5: Performance Evaluation and Modeling of Ultra-Scale Systems, In Frontiers of Parallel Processing for Scientific Computing, M. A. Heroux, P. Raghavan, and H. D. Simon Eds. SIAM Software, Environments and Tools. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, pages 233-247, 2006 LBNL-58054. L. Oliker, J. Shalf, J. Carter, A. Canning, S. Kamil, M. Lijewski, S. Ethier Performance Characteristics of Potential Petascale Scientific Applications , Petascale Computing: Algorithms and Applications, Chapman & Hall / CRC Press, in press, LBNL # pending. L. Oliker, A. Canning, J. Carter, J. Shalf, S. Ethier, Scientific Application Performance on Leading Scalar and Vector Supercomputing Platforms, International Journal of High Performance Computing Applications, LBNL 60800, in press. J. Borrill, L. Oliker. J. Shalf, H. Shan, Investigation Of Leading HPC I/O Performance Using A Scientific- Application Derived Benchmark, SC07: International Conference for High-Performance Computing, Networking, Storage, and Analysis, 2007, in press, LBNL #63314. J. Carter, L. Oliker, J. Shalf, "Performance Evaluation of Scientific Applications on Modern Parallel Vector Systems ", High Performance Computing for Computational Science - VECPAR 2006 7th International Conference, Rio de Janeiro, Brazil, June 10-13, 2006, Revised Selected and Invited Papers, Lecture Notes in Computer Science, LNCS 4395, 2007. LBNL 58004. S. Kamil, A. Pinar, D. Gunter, M. Lijewski, L. Oliker, J. Shalf, "Reconfigurable Hybrid Interconnection for Static and Dynamic Scientific Applications", ACM International Conference on Computing Frontiers, 2007, LBNL-60060. PI Bahram Parvin B. Parvin, Q. Yang, H. Chang, J. Han, M.H. Barcellos-Hoff, "Iterative Voting for Inference of Structural Saliency and Characteriztion of Subcellular Events ", IEEE Trans. on Image Processing, 2007 H. Chang, Q. Yang, and B. Parvin, "Segmentation of Heterogeneous Blob Objects through Voting and Level Set Formulation ", Pattern Recognition Letters, 2007 J. Han, H. Chang, M.H. Barcellos-Hoff, and B. Parvin, "Segmentation of Mammosphere Structures from Volumetric Data ," IEEE Int. Symp. on Biomedical Imaging, 2007 Andarawewa KL, Erickson AC, Chou WS, Costes SV, Gascard P, Mott JD, Bissell MJ, Barcellos-Hoff MH, " Ionizing radiation predisposes nonmalignant human mammary epithelial cells to undergo transforming growth factor beta induced epithelial to mesenchymal transition," Cancer Research, 2007. PI Martin Pattison Martin J. Pattison, Kannan N. Premnath, Neil B. Morley and Mohamed A. Abdou. Progress in Lattice Boltzmann Methods for Magnetohydrodynamic Flows relevant to Fusion Applications. In press Fusion Eng Des (2007), doi:10.1016/j.fusengdes.2007.10.005 Sanjoy Banerjee, Kannan N. Premnath and Martin J. Pattison, "Turbulence Simulation using the Generalized Lattice Boltzmann Equation on Massively Parallel Architectures", 3rd Asian-Pacific Conference on Computational Mechanics (APCOM '07) in conjunction with 11th International Conference on Enhancement and Promotion of Computational Methods in Engineering and Science (EPMESC XI), Kyoto, Japan, Dec. 2007. Martin J. Pattison, Kannan N. Premnath and Neil B. Morley (2007) Lattice Boltzmann Methods for Magnetohydrodynamic Flows in Fusion Applications. Fusion Science and Technology, Vol. 52, 812-816 PI Joyce Penner Guo, H., J. E. Penner, M. Herzog, and S. Xie: 2007, Investigation of the first and second aerosol indirect effects using data from the May 2003 Intensive Operational Period at the Southern Great Plains, J. Geophys. Res., 112, D15206, doi:10.1029/2006JD007173. Feng, Y. and J.E. Penner, 2007: Global Modeling of Nitrate and Ammonium: Interaction of Aerosols and Tropospheric Chemistry, J. Geophys. Res., 112, D01304, doi:10.1029/2005JD006404. Penner, J.E., N. Andronova, R. C. Oehmke, J. Brown, Q. F. Stout, C. Jablonowski, B.van Leer, K G. Powell, and M. Herzog, Three Dimensional Adaptive Mesh Refinement on a Spherical Shell for Atmospheric Models with Lagrangian Coordinates, Proceedings of the SciDac Conference, Journal of Physics: Conference Series, 78, 012072, doi:10.1088/1742-6596/78/1/012072, http://www.iop.org/EJ/toc/1742-6596/78/1. Ito, A., S. Sillman, and J. E. Penner, 2007: Effects of additional nonmethane volatile organic compounds, organic nitrates, and direct emissions of oxygenated organic species on global tropospheric chemistry, J. Geophys. Res., 112, D06309, doi:10.1029/2005JD006556. Guo, H. J. E. Penner, M. Herzog, H. Pawlowska, 2007: Examination of the aerosol indirect effect under contrasting environments during the ACE-2 experiment, Atmos. Chem. Phys., 7, 535-548, http://www.atmos-chem-phys.net/7/535/2007/acp-7-535-2007.pdf. H. Guo, Y. Liu, and J E. Penner, 2007: Does the threshold representation associated with the autoconversion process matter?, Geophys. Res. Lett., submitted. Jablonowski, C., M. Herzog, J.E. Penner, R.C. Oemke, Q.F. Stout, B. van Leer, and K.G. Powell, 2005: Block-Structured Adaptive Grids on the Sphere: Advection Experiments, Mon. Weath. Rev., Volume 134, pp. 3691-3713, DOI: 10.1175/MWR3223.1. PI Stephen Pennycook W. Luo, A. Franceschetti, M. Varela, J. Tao, S.J. Pennycook, and S.T. Pantelides, Orbital-Occupancy versus Charge Ordering and the Strength of Electron Correlations in Electron-Doped CaMnO3, Physical Review Letters 99, 036402 (2007). W. Luo, S. J. Pennycook, and S. T. Pantelides, s-Electron Ferromagnetism in Gold and Silver Nanoclusters, Nano Letters 7, 3134 (2007). S. Ho Oh, K. van Benthem, S.I. Molina, A.Y. Borisevich, W. Luo, P. Werner, N.D. Zakharov, D. Kumar, S.T. Pantelides, and S.J. Pennycook, Point Defect Configurations of Supersaturated Au Atoms Inside Si Nanowires, submitted to Nano Letters. K.G. Roberts, M. Varela, S. Rashkeev, S.T. Pantelides, S.J. Pennycook, and K.M. Krishnan, Defect- mediated ferromagnetism in insulating Co-doped anatase TiO2 thin films, submitted to Physical review Letters. J.C. Idrobo, A. Halabica, R. H. Magruder III, R.F. Haglund Jr., S.J. Pennycook, and S.T. Pantelides, Optical Properties of Embedded Nanoparticles: Size Does Not Always Matter, work to be submitted to Physical Review Letters November 2007. A. G. Marinopoulos, K. van Benthem, S. N. Rashkeev, S. J. Pennycook and S. T. Pantelides, Impurity segregation and ordering in Si/SiO/HfO structures, Physical Review B 76, 035438 (2007). A. Halabica, J. C. Idrobo, S. T. Pantelides, and R. F. Haglund Jr., R. H. Magruder III, S.J. Pennycook, Pulsed infrared laser annealing of gold nanoparticles embedded in a silica matrix, submitted to Applied Physics Letters. S. J. Pennycook, A. R. Lupini, M. Varela, A. Y. Borisevich, Y. Peng, M. P. Oxley and M. F. Chisholm Scanning Transmission Electron Microscopy for Nanostructure Characterization, pp. 152-191 in Scanning Microscopy for Nanotechnology: Techniques and Applications, ed. by W. Zhou and Z. L. Wang, Springer, 2006. A. R. Lupini, S. N. Rashkeev, M. Varela, A. Y. Borisevich, M. P. Oxley, K. van Benthem, Y. Peng, N. de Jonge, G. M. Veith, M. F. Chisholm and S. J. Pennycook, Scanning Transmission Electron Microscopy, in Nanocharacterization, edited by the Royal Society of Chemistry, United Kingdom, in press (2007). PI Saul Perlmutter Kuznetsova, N., et al., A New Determination of the High Redshift Type Ia Supernova Rates with the Hubble Space Telescope Advanced Camera for Surveys, ArXiv e-prints, 710, arXiv:0710.3120 (2007). Accepted for publication in Astrophysical Journal. Kuznetsova, N. V. & B. M. Connolly, A Probabilistic Approach to Classifying Supernovae Using Photometric Information, Astrophysical Journal, 659, 530 (2007). Cosmology: The Universe's skeleton sketched Authors: Linder, Eric V. Publication: Nature, Volume 445, Issue 7125, pp. 273 (2007) Separating dark physics from physical darkness: Minimalist modified gravity versus dark energy Authors: Huterer, Dragan; Linder, Eric V. Publication: Physical Review D, vol. 75, Issue 2, id. 023519 Snapping supernovae at z > 1.7 Authors: Aldering, Greg; Kim, Alex G.; Kowalski, Marek; Linder, Eric V.; Perlmutter, Saul Publication: Astroparticle Physics, Volume 27, Issue 2-3, p. 213-225. Cosmic variance of weak lensing surveys in the non-Gaussian regime Authors: Semboloni, Elisabetta; van Waerbeke, Ludovic; Heymans, Catherine; Hamana, Takashi; Colombi, Stephane; White, Martin; Mellier, Yannick Publication: Monthly Notices of the Royal Astronomical Society: Letters, Volume 375, Issue 1, pp. L6-L10. Simulations of baryon oscillations Authors: Huff, Eric; Schulz, A. E.; White, Martin; Schlegel, David J.; Warren, Michael S. Publication: Astroparticle Physics, Volume 26, Issue 6, p. 351-366. K-Corrections and Spectral Templates of Type Ia Supernovae Authors: Hsiao, E. Y.; Conley, A.; Howell, D. A.; Sullivan, M.; Pritchet, C. J.; Carlberg, R. G.; Nugent, P. E.; Phillips, M. M. Publication: The Astrophysical Journal, Volume 663, Issue 2, pp. 1187-1200. PI Steven Pieper Flavor Evolution of the Neutronization Neutrino Burst from an O-Ne-Mg Core-Collapse Supernova, Huaiyu Duan (INT, Univ of Washington), George M. Fuller (UCSD), J. Carlson (LANL), Yong-Zhong Qian (UMN) preprint arXiv:0710.1271, submitted to Phys. Rev. Lett. Analysis of Collective Neutrino Flavor Transformation in Supernovae, Huaiyu Duan (UCSD), George M. Fuller (UCSD), J. Carlson (LANL), Yong-Zhong Qian (UMN), Phys. Rev. D75, 125005 (2007). Coherent Development of Neutrino Flavor in the Supernova Environment Huaiyu Duan (UCSD), George M. Fuller (UCSD), J. Carlson (LANL), Yong-Zhong Qian (UMN), Phys.Rev.Lett. 97 (2006) 241101. PI Ali Pinar Vaibhav Donde, Vanessa Lopez, Bernard Lesieutre, Ali Pinar, Chao Yang, and Juan Meza, Severe multiple contingency screening in electric power systems, to appear in IEEE Transactions on Power Systems. A. Pinar J. Meza, V. Donde, B. Lesieutre, Optimization Strategies for the vulnerability analysis of the electric power grid, to be submitted to SIAM Optimization, available as an LBNL technical report LBNL-63473. B. Lesieutre, A. Pinar, and S. Roy, Power system extreme event detection: the vulnerability frontier, to appear in Proc. 40th Hawaii international conference on system sciences. A. Pinar, Y. Fogel, and B. Lesieutre, The Inhibiting Bisection Problem, to be submitted to ACM 20th Symposium Parallel Algorithms and Architectures (SPAA) 2007, available as an LBNL technical report LBNL-62142. Shoaib Kamil, Ali Pinar, Dan Gunter, Michael Lijewski, Leonid Oliker, and John Shalf, Reconfigurable hybrid interconnection for static and dynamic scientific applications, in Proc. 2007 ACM International Conference on Computing Frontiers. Ali Pinar, Adam Reichert, and Bernard Lesieutre, Computing Criticality of Lines in a Power System, to appear in Proc. 2007 IEEE International Symposium on Circuits and Systems, New Orleans, LA, May 2007. B. Lesieutre, S. Roy, V. Donde, and A. Pinar, Power system extreme event analysis using graph partitioning, Proc. of the North American Power Symposium, Carbondale, IL, October 2006. PI Michael Pindzola Electron-impact ionization of highly excited hydrogen-like ions in a co-linear s-wave model, T Topcu, M S Pindzola, C P Ballance, D C Griggin, and F J Robicheaux, Physical Review A 74, 062708 (December 2006). Charge dependent effects in the double photoionization of He-like ions, M Foster and J P Colgan, Journal of Physics B 39, 5067 (December 2006). Electron-impact excitation of W+44 and W+45, C P Ballance and D C Griffin, Journal of Physics B 40, 247 (January 2007). Electron-impact ionization of argon using the R-matrix with pseudostates method C P Ballance D C Griffin, M S Pindzola, and S D Loch, Journal of Physics B 40, F27 (February 2007). Capture and transport of electronic states of fast ions penetrating solids: an open quantum system approach with sources and sinks, M Seliger, C O Reinhold, T Minami, D R Schultz, M S Pindzola, S Yoshida, J Burgdorfer, E Lamour, J P Rozet, and D Vernhet, Physical Review A 75, 032714 (March 2007). Electron-impact excitation of neutral boron using the R-matrix with pseudostates method, C P Ballance, D C Griffin, K A Berrington, and N R Badnell, Journal of Physics B 40, 1131 (March 2007). Triple differential cross sections for the double photoionization of H_2, J P Colgan, M S Pindzola, and F J Robicheaux, Physical Review Letters 98, 153001 (April 2007). Electron-impact ionization of metastable excited states of Li+, J C Berengut, S D Loch, C P Ballance, and M S Pindzola, Journal of Physics B 40, 1331 (April 2007). Double ionization of helium by fast bare ion collisions, M S Pindzola, F J Robicheaux, and J P Colgan, Journal of Physics B 40, 1695 (May 2007). Angular distributions from photoionization of H+_2 M Foster, J P Colgan, O Al-Hagan, J L Peacher, D H Madison, and M S Pindzola, Physical Review A 75, 062707 (June 2007) Electron-impact double ionization of helium at high energies, M S Pindzola, F J Robicheaux, and J P Colgan, Physical Review A 76, 024704 (August 2007) Numerical study of charge transfer in H+ + H+ and H+2 + Li+2 collisions, T Minami, M S Pindzola, T G Lee, and D R Schultz, Journal of Physics B 40, 3629 (September 2007) PI Tomasz Plewa Zhang, J., Messer, O. E. B., Khokhlov, A. M., and Plewa, T., On the Evolution of Thermonuclear Flames on Large Scales, 2007, ApJ, 656, 347 Plewa, T., Detonating Failed Deflagration Model of Thermonuclear Supernovae. I. Explosion Dynamics, 2007, ApJ, 657, 942 Kasen, D., and Plewa, T., Detonating Failed Deflagration Model of Thermonuclear Supernovae. II. Comparison to Observations, 2007, ApJ, 662, 459 PI Gerald Potter Climate Model Forecast Experiments for TOGA-COARE.J. Boyle,S. Klein,G. Zhang,S. Xie,X. Wei. Accepted by Monthly Weather Review. On the deep convection, high clouds and the upper troposphere relative humidity in the Multi-scale Modeling Framework, Yunyan Zhang, Stephen A. Klein, Chuntao Liu, Baijun Tian, Roger Marchand, John Haynes, Renata McCoy and Yuying Zhang to be submitted to JGR Simulations of Arctic mixed-phase clouds in forecasts with CAM3 and AM2 for M-PACE" by Xie et al. Accepted and currently under revision PI Larry Pratt Lawrence M. Pratt Modeling Lithium Dialkylamides in Ethereal Solvents: A Test of the Microsolvation Model Theochem, 2007, 811, 191 Lawrence M. Pratt, Diep Huong Phan Tran, Phung Thao Thi Tran, Ngan Van Nguyen Basis set and Electron Correlation Effects on Lithium Carbenoid Dimerization Energies Bull. Chem. Soc. Japan, 2007, 80, 1587 PI Joel Primack Berrier, Bullock, Barton, Guenther, Zentner, and Wechsler, Close Galaxy Counts as a Probe of Hierarchical Structure Formation, ApJ 652, 56 (2006). Ceverino and Klypin, Resonances in Barred Galaxies, MNRAS 379, 1155 (2007). Conroy, Wechsler, and Kravtsov, The Hierarchical Build-Up of Massive Galaxies and Intracluster Light since z=1, ApJ 668, 826 (2007). Cox, Jonsson, Primack, and Somerville, Feedback in simulations of disc-galaxy major mergers, MNRAS 373, 1013 (2006). Cox, Jonsson, Somerville, Primack, and Dekel The effect of Mass Ratio on Merger Driven Starbursts, submitted to MNRAS, arXiv:0709.3511 (2007). Flores, Allgood, Kravtsov, Primack, Buote, and Bullock, The shape of galaxy cluster dark matter haloes: systematics of its imprint on cluster gas and comparison to observations, MNRAS 377, 883 (2007). Rocha, Jonsson, Primack, and Cox, Dust Attenuation in Hydrodynamic Simulations of Spiral Galaxies, MNRAS in press, arXiv:astro-ph/0702513 (2007). Wechsler, Zentner, Bullock, Kravtsov, and Allgood, The Dependence of Halo Clustering on Halo Occupation History, Concentration, and Occupation, ApJ 652, 71 (2006). PI Fritz Prinz X. Wang, H. Huang, T. Holme, X. Tian, F. Prinz, J. Power Sources. In press (accepted). T. Holme, H. Huang, F. Prinz, Phys. Rev. B, submitted, under review. H. Huang, T. Holme, F. Prinz, ECS Trans. In press (accepted). PI Eliot Quataert Sharma, P. Quataert, E. & Stone, J. M., "Faraday Rotation in Global Accretion Disk Simulations: Implications for Sgr A", 2007, ApJ, 671, 1696 Sharma, P. Quataert, E., Hammett, G. W., & Stone, J. M., "Electron Heating in Hot Accretion Flows", 2007, ApJ, 667, 714 PI Abhay Ram Plasma 2010 Committee, Plasma Science Committee, National Research Council, Plasma Science: Advancing Knowledge in the National Interest, National Academies Press, 2007. "Microturbulent drift mode suppression as a trigger mechanism for internal transport barriers on Alcator C-Mod," K. Zhurovich et al., Nucl. Fusion 47 (2007) 1220-1231. "Identification of TEM Turbulence through Direct Comparison of Nonlinear Gyrokinetic Simulations with Phase Contrast Imaging Density Fluctuation Measurements," D. R. Ernst, N. Basse, W. Dorland, C. L. Fiore, L. Lin, A. Long, M. Porkolab, K. Zeller, and K. Zhurovich, IAEA Fusion Energy Conference Chengdu, China, 16-21 October (2006), oral paper IAEA-CN-149/TH/1-3. "Role of trapped electron mode turbulence in internal transport barrier control in the Alcator C-Mod Tokamakak", D. R. Ernst, P. T. Bonoli, P. J. Catto, W. Dorland, C. L. Fiore, R. S. Granetz, M. Greenwald, A. E. Hubbard, M. Porkolab, M. H. Redi, J. E. Rice, K. Zhurovich, and the Alcator C-Mod Group, Phys. Plasmas, 11(5) 2637 (2004). D. R. Ernst, N. Basse, P. T. Bonoli, P. J. Catto, W. Dorland, C. L. Fiore, M. Greenwald, A. E. Hubbard, E. S. Marmar, M. Porkolab, J. E. Rice, K. Zeller and K. Zhurovich, "Mechanisms for ITB Formation and Control in Alcator C-Mod Identified Through Gyrokinetic Simulations of TEM Turbulence," Proc. 20th Int'l. Atomic Energy Agency Fusion Energy Conference, Vilamoura, Portugal, 1-6 November 2004, oral paper IAEA-CN-116/TH/4-1. http://www-naweb.iaea.org/napc/physics/fec/fec2004/datasets/TH_4-1.html C L Fiore, P T Bonoli, D R Ernst, M J Greenwald, E S Marmar, M H Redi, J E Rice, S J Wukitch and K Zhurovich, "Internal transport barrier production and control in Alcator C-Mod," Plasma Phys. Control. Fusion 46 B281-B291. EPS Invited Oral 2004. C. L. Fiore, D. R. Ernst, J. E. Rice, N. Basse, P. T. Bonoli, M. J. Greenwald, E. S. Marmar, S. J. Wukitch, K. Zhurovich, "Internal Transport Barriers in Alcator C-Mod," submitted to C-Mod Special Issue, Fusion Science and Tecnology, Augst, 2005. N. P. Basse et al., "Characterization of core and edge turbulence in L- and enhanced D-alpha H-mode Alcator C-mod plasmas," Physics of Plasmas, 12(5) May 2005, p 52512-1-14. C. L. Fiore et al., "Control of internal transport barriers on Alcator C-mod," Physics of Plasmas 11(5), May 2004, p 2480-7. J. C. Wright, L. A. Berry, D. B. Batchelor P. T. Bonoli, E. F. Jaeger, E. D Azevedo M. D. Carter, C. K. Phillips, R. W. Harvey H. Okuda, D. N. Smithe, D. A. D Ippolito J. R. Myra, M. Brambilla, and R. J. Dumont, ``Nonthermal particle and full-wave diffraction effects on heating and current drive in the ICRF and LHRF regimes,'' Nucl. Fusion, 45(9), September 2005. Y. Lin, S. J.Wukitch, A. Parisot, J. C. Wright, N. Basse, P. Bonoli, E. Edlund, L. Lin, M. Porkolab, S.Wolfe, G. Schilling, and P. Phillips, ``Ion cyclotron range of frequencies wave phenomena in the mode conversion region in Alcator C-Mod,'' Plasma Phys. Controlled Fusion, 47(-1):0, 2005. J. C. Wright, P. T. Bonoli, M. Brambilla, E. Azevedo, L. A. Berry, D. B. Batchelor, E. F. Jaeger, M. D. Carter, C. K. Phillips, H. Okuda, R. .W. Harvey, J. R.Myra, D. A. D Ippolito, and D. N. Smithe, ``Full-wave electromagnetic field simulations of lower hybrid waves in tokamaks,'' In P. Bonoli and S. Wukitch, editors, 16th Topical Conference on Radio Frequency Power in Plasmas and, number to be published, page 1, New York, 2005. American Institute of Physics. PI David Randall Benedict, J., and D. A. Randall, 2007: An analysis of the MJO based on TRMM rainfall data. J. Atmos. Sci., 64, 2332-2354. DeMott, C. A., D. A. Randall, and M. Khairoutdinov, 2007: Convective precipitation variability as a tool for general circulation model analysis. J. Climate, 20, 91-112. Jung, J.-H., and A. Arakawa, 2007: A three-dimensional anelastic model based on the vorticity equation. Mon. Wea. Rev., in press. Khairoutdinov, M., C. A. DeMott, and D. A. Randall, 2007: Evaluation of the simulated interannual and subseasonal variability in an AMIP-style simulation using the CSU Multiscale Modeling Framework. J. Climate (in press). Konor, C. S., and A. Arakawa, 2007: Multi-point explicit differencing (MED) for time integrations of the wave equation. Mon. Wea. Rev., in press. Toy, M., and D. A. Randall, 2007: Comment on the article Vertical discretizations for compressible Euler equation atmospheric models giving optimal representation of normal modes by Thuburn and Woolings. J. Comp. Phys., 223, 82-88. Tulich, S. N., D. A. Randall, and B. E. Mapes, 2007: Vertical-mode and cloud decomposition of large- scale convectively coupled gravity waves in a two-dimensional cloud-resolving model. J. Atmos. Sci., 64, 1220-1229. Yamaguchi, T., and D. A. Randall, 2007: Large-eddy simulation of evaporatively driven entrainment in cloud-topped mixed layers. J. Atmos. Sci. (in press). PI Sergey Rashkeev S.N.Rashkeev, D.M.Ginosar, L.M.Petkovic, and H.H.Farrell. Catalytic activity of supported metal particles for sulfuric acid decomposition reaction. Catalysis Today (submitted 10/17/2007). A.G.Marinopoulos, K.van Benthem, S.N.Rashkeev, S.J.Pennycook, and S.T.Pantelides. Impurity segregation and ordering in Si/SiO2/HfO2 structures. Phys. Rev. Lett. (submitted 07/23/2007). K.Griffin Roberts, M.Varela, S.Rashkeev, S.T.Pantelides, S.J.Pennycook, and Kannan M. Krishnan. The origin of ferromagnetism in insulating CoxTi1-xO2 anatase thin films. Phys. Rev. Lett. (submitted 03/16/07). A.Y.Borisevich, S.Wang, S.N.Rashkeev, M.Glazoff, S.J.Pennycook, and S.T.Pantelides. Dual nanoparticle/substrate control of catalytic dehydrogenation. Advanced Materials 19, N 16, 2129-2133 (2007). S.N.Rashkeev, A.R.Lupini, S.H.Overbury, S.J.Pennycook, and S.T.Pantelides. The role of the nanoscale in catalytic CO oxidation by supported Au and Pt nanostructures. Phys. Rev. B 76, N 3, 035438 (2007); Virtual Journal of Nanoscience & Technology, 16, N 7, (August 13, 2007). S.N.Rashkeev, K.W.Sohlberg, S.Zhuo, and S.T.Pantelides. Hydrogen-induced initiation of corrosion in aluminum. The Journal of Physical Chemistry C 111, N 19, 7175-7178 (2007). PI Lawrence Rauchwerger Associative Parallel Containers In STAPL, Gabriel Tanase, Chidambareswaran (Chids) Raman, Mauro Bianco, Nancy M. Amato, Lawrence Rauchwerger, In Wkshp. on Lang. and Comp. for Par. Comp. (LCPC), Oct 2007. The STAPL pArray, Gabriel Tanase, Mauro Bianco, Nancy M. Amato, Lawrence Rauchwerger, In Proc. of Workshop MEDEA, Sep 2007. Design and Use of htalib a Library for Hierarchically Tiled Arrays, Ganesh Bikshandy, Jia Guo, Christoph von Praun, Gabriel Tanase, Basilio Fraguela, Maria Jesus Garzaran, David Padua, Lawrence Rauchwerger, In Wkshp. on Lang. and Comp. for Par. Comp. (LCPC), Nov 2006. ARMI: A High Level Communication Library for STAPL, Nathan Thomas, Steven Saunders, Tim Smith, Gabriel Tanase, Lawrence Rauchwerger, Parallel Processing Letters, 16(2):261-280, Jun 2006. Sensitivity analysis for automatic parallelization on multi-cores, Silvius Rus and Maikel Pennings and Lawrence Rauchwerger, ICS '07: Proceedings of the 21st annual international conference on Supercomputing, Seattle , 2007. PI Asok Ray R. Atta-Fynn and A. K. Ray, Atomic Adsorptions on the (020) Surface of α-Pu: A Density Functional Study, submitted for publication (2007). R. Atta-Fynn and A. K. Ray, An Ab Initio Relativistic Study of Adsorption and Dissociation of H2O on δ-Pu (111) Surface, submitted for publication (2007). P. Dholabhai and A. K. Ray, A Density Functional Study of Atomic Hydrogen and Oxygen Chemisorption on the Relaxed (0001) Surface of Double Hexagonal Close Packed Americium, submitted for publication (2007) PI Fernando Reboredo "Anomalous photoluminescence in CdSe quantum dot solids at high pressure due to non-hydrostatic stress", C. D. Grant, J.C. owhurst, S. Hamel, A. Williamson and N. Zaitseva, submitted to Nano Letters. "First-principles calculations of mechanical properties of Si <001> nanowires and comparison to nanomechanical theory", B. Lee and R. E. Rudd, Phys. Rev. B 75, 195328 (2007). "First-principles study of the Young's modulus of Si <001> nanowires", B. Lee and R. E. Rudd, Phys. Rev. B 75, 041305(R) (2007). PI John Rehr Theoretical X-Ray Absorption Debye-Waller Factors, Fernando D. Vila, J. J. Rehr, H. H. Rossner, H. J. Krappe, Phys. Rev. B 76, 014301 (2007). Losses and Multi-Electron Excitations in X-Ray Spectra, J. J. Rehr, J. J. Kas, M. P. Prange, A. P. Sorini, L. W. and F. D. Vila, AIP Conf. Proc. 882, 85 (2007). Real time time-dependent density functional theory approach for frequency-dependent non-linear optical response in c molecules, Y. Takimoto, F. D. Vila and J. J. Rehr, J. Chem. Phys. 127, 154114 (2007). Many-pole model of inelastic losses in x-ray absorption spectra, J.J. Kas, A. P. Sorini, M. P. Prange, L. W. Cambell, and J. A. Soininen, Phys. Rev. B (in press, 2007). Dynamic structure in supported Pt nanoclusters, F. Vila, J.J. Rehr, J. Kas, R.G. Nuzzo, A.I. Frenkel, Submitted Oct, 2007. PI Chuang Ren G. Li, R. Yan, C. Ren, T.-L. Wang, J. Tonge, and W. B. Mori, 'Laser channeling in millimeter-scale underdense plasmas of fast ignition targets' PI Tony Rollett J. R. Morris, F. Jiang, P. Liaw, "A simple model for examining composition effects in eutectic nucleation," Mater. Trans. M) 48, 1675-1679 (2007). T. Egami, V. Levashov, R. S. Aga and J. R. Morris, "Atomic dynamics in metallic liquids and glasses," Mater. Trans. (JIM) 1729-1733 (2007). J. R. Morris, U. Dahlborg and M. Calvo-Dahlborg, "Recent developments and outstanding challenges in theory and modeling of quid metals," J. Non-cryst. Sol 353, 3444-3453 (2007). Z. G. Xia, D. Y. Sun, M. Asta and J. J. Hoyt, "Molecular Dynamics Calculations of the Crystal-Melt Interfacial Mobility for exagonal-Close-Packed Mg," Phys. Rev. B 75, 012103 (2007). C. A. Becker, D. Buta, J. J. Hoyt and M. Asta, "Crystal-Melt Interface Stresses: Atomistic Simulation Calculations for a ard-Jones Alloy, Stillinger-Weber Si, and Embedded Atom Method Ni," Phys. Rev. E 75, 061610 (2007). D. Buta, M. Asta and J. J. Hoyt, "Kinetic Coefficient of Steps at the Si(111) Crystal-Melt Interface from Molecular Dynamics Simulations," J. Chem. Phys. 127, 074703 (2007). R. B. Godiksen, Z. T. Trautt, M. Upmanyu, J. Schixtz, S. Schmidt and D. Juul Jensen, Simulations of boundary migration recrystallisation using molecular dynamics, Acta Materialia, accepted, 2007. R. B. Godiksen, Z. T. Trautt, M. Upmanyu, S. Schmidt and D. Juul Jensen, Simulation of recrystallization using molecular mics: Effects of the interatomic potential, Materials Science Forum 558-559 , pp 1081-1086, 2007. Z. T. Trautt, M. Upmanyu and A. Karma, "Interface mobility from interface random walk ", Science 27, pp 632-635, 2006. M. Upmanyu, D. J. Srolovitz, A. Lobkovsky, J. A. Warren and W. C. Carter, Simultaneous Grain Rotation and Grain Boundary ation, Acta Materialia 54(7), pp 1707-1719, 2006. PI Barbara Romanowicz Romanowicz, B., A. Cao, A. Kim, M. Panning, M. Pasyanos and D. Dreger (2007) CALIBRATION OF 3D UPPER MANTLE STRUCTURE IN EURASIA USING REGIONAL AND TELESEISMIC FULL WAVEFORM SEISMIC DATA, extended abstract, Monitoring Research Review, NEMRE. PI Edward Rubin The JGI has been able to produce over 70 publications to date in 2007; these were all based on data from the pipeline which was enabled by NERSC resources. http://www.jgi.doe.gov/News/pubs.html Key Papers/Reports Published in Nature in 2007 are listed here: Pandit, B; Sarkozy, A; Pennacchio, LA; et al. Gain-of-function RAF1 mutations cause Noonan and LEOPARD syndromes with hypertrophic cardiomyopathy NATURE GENETICS, 39 (8): 1007-1012 AUG 2007 Romeo, S; Pennacchio, LA; Fu, YX; et al. Population-based resequencing of ANGPTL4 uncovers variations that reduce triglycerides and increase HDL NATURE GENETICS, 39 (4): 513-516 APR 2007 Jeffries, TW; Grigoriev, IV; Grimwood, J; et al. Genome sequence of the lignocellulose-bioconverting and xylose-fermenting yeast Pichia stipitis NATURE BIOTECHNOLOGY, 25 (3): 319-326 MAR 2007 Lo, I; Denef, VJ; VerBerkmoes, NC; et al. Strain-resolved community proteomics reveals recombining genomes of acidophilic bacteria NATURE, 446 (7135): 537-541 MAR 29 2007 McHardy, AC; Martin, HG; Tsirigos, A; et al. Accurate phylogenetic classification of variable-length DNA fragments NATURE METHODS, 4 (1): 63-72 JAN 2007 Tartaglia, M; Pennacchio, LA; Zhao, C; et al. Gain-of-function SOS1 mutations cause a distinctive form of Noonan syndrome NATURE GENETICS, 39 (1): 75-79 JAN 2007 Key Papers/Reports Published in Science in 2007 are listed here: Merchant, SS; Prochnik, SE; Vallon, O; et al. The Chlamydomonas genome reveals the evolution of key animal and plant functions SCIENCE, 318 (5848): 245-251 OCT 12 2007 Putnam, NH; Srivastava, M; Hellsten, U; et al. Sea anemone genome reveals ancestral eumetazoan gene repertoire and genomic organization SCIENCE, 317 (5834): 86-94 JUL 6 2007 McPherson, R; Pertsemlidis, A; Kavaslar, N; et al. A common allele on chromosome 9 associated with coronary heart disease SCIENCE, 316 (5830): 1488-1491 JUN 8 2007 Giraud, E; Moulin, L; Vallenet, D; et al. Legumes symbioses: Absence of Nod genes in photosynthetic bradyrhizobia SCIENCE, 316 (5829): 1307-1312 JUN 1 2007 Rubin, EM; Noonan, JP Comparing neanderthal and human genomes - Response SCIENCE, 315 (5819): 1664-1664 MAR 23 2007 von Mering, C; Hugenholtz, P; Raes, J; et al. Quantitative phylogenetic assessment of microbial communities in diverse environments SCIENCE, 315 (5815): 1126-1130 FEB 23 2007 Newton, ILG; Woyke, T; Auchtung, TA; et al. The Calyptogena magnifica chemoautotrophic symbiont genome SCIENCE, 315 (5814): 998-1000 FEB 16 2007 PI Robert Ryne M. Borland et al., "Potential and Challenges of an Energy Recovery Upgrade to the APS," NIM A 582 (2007) 54-56. M. Borland, ``A super-bright storage ring alternative to an energy recovery linac,'' NIM A 557 (2006) 230-235. M. Borland, ``Evaluation of the possibility of upgrading the Advanced Photon Source to an energy recovery linac,'' NIM A 557 (2006) 224-229. J. Qiang, S. Lidia, R. D. Ryne, and C. Limborg-Deprey, "A Three-Dimensional Quasi-Static Model for High Brightnees Beam ics simulation," Phys. Rev. ST Accel. Beams 9, 044204 (2006). G. Fubiani, J. Qiang, E. Esarey, W.P. Leemans, G. Dugan, "Space charge modeling of dense electron beams with large energy eads," Phys. Rev. ST Accel. Beams 9, 064402 (2006). J. Qiang, M. A. Furman, R. D. Ryne, W. Fischer, K. Ohmi, "Recent advaces of strong-strong beam-beam simulation," Nuclear ruments & Methods in Physics Research A558, 351, (2006). J. Amundson, P. Spentzouris, J.Qiang and R. Ryne, "Synergia: An accelerator modeling tool with 3-D space charge," J. Comp. ys. vol. 211, 229 (2006). H. Shan, J. Qiang, K. Yelick and E. Strohmaier, "Performance analysis of a high energy colliding beam simulation code on high-performance computing architectures," to appear in the 35th International Conference on Parallel Processings. (peer reviewed). PI Yousef Saad H.R. Fang and Y. Saad: "Two Classes of Multisecant Methods for Nonlinear Acceleration," Numerical Linear Algebra with tions (submitted). PI Roman Samulyak R. Samulyak, T. Lu, P. Parks, A magnetohydrodynamic simulation of pellet ablation in the electrostatic approximation, Nuclear Fusion, 47 (2007), 103-118. R. Samulyak, J. Du, J. Glimm, Z. Xu, A numerical algorithm for MHD of free surface flows at low magnetic Reynolds numbers, J. Comput. Phys., 226 (2007), 1532 - 1549. J. Du, T. Lu, R. Samulyak, Algorithms for magnetohydrodynamics of ablated materials, Journal of Nanoscience and Nanotechnology, 8 (2008). In press PI Richard Saykally D. Prendergast and G. Galli; Phys. Rev. Lett. 96, 215502, 2006. PI Henry Schaefer "π and σ-phenylethynyl radicals and their isomers o-, m- and p-ethynylphenyl: Structures, energetics, and electron ties", Raj K. Sreeruttun, Ponnadurai Ramasami, Chaitanya S. Wannere, Andrew C. Simmonett, and Henry F. Schaefer III, dubmittee "Synthesis and Characterization of Hydroxymethylene", Peter R. Schreiner, Hans Peter Reisenauer, Frank Pickard, Andrew C. monett, Wesley D. Allen, Edit Matyus, and Attila G. Csaszar, Submitted. "In Search of Definitive IR Signatures of the Elusive NCCO Radical", A. C. Simmonett, F. A. Evangelista, W. D. Allen and H. . Schaefer III, J. Chem. Phys. 127 014306 (2007). "Model Systems for Probing Metal Cation Hydration: The V+(H2O) and ArV+(H2O) Complexes", V. Kasalova, W. D. Allen, H. F. efer, E. D. Pillai, and M. A. Duncan,, Journal of Physical Chemistry A 111, 7599 (2007). "Anchoring the Absolute Proton Affinity Scale", Veronika Kasalova, Andrew C. Simmonett, Wesley D. Allen, Henry F. Schaefer I, Gabor Czaks and Attila G. Csaszar, Submitted. "High Precision Energetics of the C2H3 + O2 Reaction", Jeremiah J. Wilke, Wesley D. Allen, and Henry F. Schaefer, Submitted. PI Rocco Schiavilla "Isospin Mixing in the Nucleon and 4He and the Nucleon Strange Electric Form Factor" M. Viviani, R. Schiavilla, B. Kubis, R. Lewis, L. Girlanda, A. Kievsky, L. E. Marcucci, and S. Rosati, Phys. Rev. Lett. 99, 12002 (2007) "Relativistic Calculation of Deuteron Threshold Electrodisintegration at Backward Angles" A. Arriaga and R. Schiavilla, Phys. Rev. C 76, 014007 (2007). "Tensor Forces and the Ground-State Structure of Nuclei" R. Schiavilla, R. B. Wiringa, Steven C. Pieper, and J. Carlson, Phys. Rev. Lett. 98, 132501 (2007) "Correlations in Nuclei: a Review" R. Schiavilla, proceedings of the "9th International Spring Seminar on Nuclear Physics: Changing Facets of Nuclear , A. Covello, ed., Vico Equense, Italy, May 20-24, 2007, World Scientific in press PI David Schultz Generalized collisional-radiative model for light elements: Data for the Li isonuclear sequence, S D Loch, J P Colgan, M C Witthoeft, M S Pindzola, C P Ballance, D M Mitnik, D C Griffin, M G O'Mullane, N R Badnell, and H P Summers, Atomic Data and Nuclear Data Tables 92, 813 (November 2006). Dielectronic recombination of Fe+13: benchmarking the M-shell, N R Badnell, Journal of Physics B 39, 4825 (December 2006). Steps toward dielectronic recombination of argon-like ions: a revisited theoretical investigation of Sc+3 and Ti+4, D Nikolic, T W Gorczyca, J Fu, D W Savin, and N R Badnell, Nuclear Instruments and Methods in Physics Research B 261, 145 (April 2007). Dielectronic recombination data for dynamic finite-density plasmas: XII The helium isoelectronic sequence, M A Bautista and N R Badnell, Astronomy and Astrophysics 466, 755 (May 2007). Atomic data from the IRON project: LXIII Electron-impact excitation of Fe+19 up to n=4, M C Witthoeft, G D Zanna, and N R Badnell, Astronomy and Astrophysics 466, 763 (May 2007). Line ratio diagnostics in helium and helium seeded argon plasmas, R F Boivin, S D Loch, C P Ballance, D Branscomb, and M S Pindzola Plasma Sources Science Technology 16, 470 (May 2007). Electron-impact ionization of diatomic molecules using the configuration-average distorted-wave method, M S Pindzola, F J Robicheaux, J P Colgan, and C P Ballance, Physical Review A 76, 012714 (July 2007). R-matrix electron-impact excitation calculations along the F-like isoelectronic sequence, M C Witthoeft, A D Whiteford, and N R Badnell, Journal of Physics B 40, 2969 (August 2007). Dielectronic recombination of Fe+14 forming Fe+13: Laboratory measurements and theoretical calculations, D V Lukic, M Schnell, D W Savin, C Brandau, E W Schmidt, S Bohm, A Muller, S Schippers, M Lestinsky, F Sprenger, A Wolf, Z Altun, and N R Badnell, Astrophysical Journal 664, 1244 (August 2007). Electron-impact ionization and recombination of M-shell atomic ions in the argon isonuclear sequence, S D Loch, S A Abdel-Naby, C P Ballance, and M S Pindzola, Physical Review A 76, 022706 (August 2007). PI Ryoichi Seki "Lattice Calculation of Thermal Properties of Low-Density Neutron Matter with NN Effective Field Theory" by T. Abe and R. Seki, submitted to Phys.Rev. C; arXiv nucl/th 0708.2523 (2007). "From Low-Density Neutron Matter to the Unitary Limit" by T. Abe and R. Seki, submitted to Phys.Rev. C; arXiv nucl/th .2424 (2007). PI Shunli Shang S. L. Shang, Y. Wang, and Z. K. Liu. First-principles elastic properties of alpha- and theta-Al2O3, Applied Physics rs, Vol. 90 (2007) 101909. S. L. Shang, Y. Wang, H. Zhang, and Z. K. Liu. Lattice dynamics and anomalous bonding in rhombohedral As: First-s supercell method, Physical Review B, Vol. 76 (2007) 052301. S. L. Shang, A. J. Böttger, and Z. K. Liu, The influence of interstitial distribution on phase stability and properties of hexagonal epsilon-Fe6Cx, epsilon-Fe6Ny, and epsilon-Fe6CxNy phases: a first-principles calculation, Acta Materialia, Vol. 55 (2007) in Press. http://dx.doi.org/10.1016/j.actamat.2007.10.018 S. L. Shang, Y. Wang, Z. K. Liu, C. E. Yang, S. Z. Yin, Band structure of FeBO3: Implications for tailoring the band gap of nanoparticles, Applied Physics Letters, Submitted in Oct-2007. PI Marjorie Shapiro Measurement of the Top-Quark Mass using Missing E(t) + Jets Events with Secondary Vertex b-tagging at CDF II T. Aaltonen et al., The CDF Collaboration, Phys. Rev. D75, 111103 (2007). Measurement of the Top Quark Mass in p anti-p Collisions at s**(1/2) = 1.96 TeV using the Decay Length Technique A. Abulencia et al., The CDF Collaboration, Phys. Rev. D75, 071102 (2007) Precise Measurement of the Top Quark Mass in the Lepton+Jets Topology at CDF II A. Abulencia et al., The CDF Collaboration, FERMILAB-PUB-07-070-E. Submitted to Phys. Rev. Lett. March 29, 2007. Observation of B0(s)-B-bar0_s Oscillations A. Abulencia et al., The CDF Collaboration, Phys. Rev. Lett. 97, 242003 (2006). PI Michael Shay Shay, M. A., J. F. Drake, and M. Swisdak, "Two-scale structure of the electron dissipation region during collisionless magnetic reconnection," Physical Review Letters, Vol. 99, 155002, 2007. P. A. Cassak and M. A. Shay, ``Scaling of Asymmetric Magnetic Reconnection: General Theory and Collisional Simulations'', In Press, Phys. Plasmas (2007). Shay, M. A., J. F. Drake, and B. Dorland, "Equation free projective integration: A multiscale method applied to a plasma ion acoustic wave," Journal of Computational Physics, Vol. 226, p.571, 2007. Phan, T. D., J. F. Drake, M. A. Shay, F. S. Mozer, and J. P. Eastwood, "Evidence for an elongated (> 60 ion skin depths) electron diffusion region during fast magnetic reconnection," Phys. Rev. Lett., submitted, 2007. Cassak, P. A., D. J. Mullan, and M. A. Shay, "A Physical Mechanism for Self-Organization in Solar and Stellar Coronae: Theory and Observations," Astrophysical Journal Letters, submitted, 2007. Eastwood, J., T. D. Phan, F. S. Mozer, M. A. Shay et al., "Multipoint observations of the Hall electromagnetic field and secondary island formation during magnetic reconnection," Journal of Geophysical Research, Vol. 112, A06235, 2007. P. A. Cassak, J. F. Drake, M. A. Shay and B. Eckhardt, ``Onset of Fast Magnetic Reconnection'', Phys. Rev. Lett., 98, 215001 (2007). P. A. Cassak, J. F. Drake, and M. A. Shay, ``Catastrophic Onset of Fast Magnetic Reconnection with a Guide Field'', Phys. Plasmas, 14, 054502 (2007) Drake, J. F., M. Swisdak, H. Che, and M. A. Shay, "Electron acceleration from contracting magnetic islands during reconnection," Nature, Vol. 443, p. 553, doi:10.1038/nature05116, 2006. PI Junko Shigemitsu `$B Meson Semileptonic Form Factors from Unquenched Lattice QCD'', E.Gulez, A.Gray, M.Wingate, C.Davies, P.Lepage, J.Shigemitsu; Phys. Rev. D\underbar{73}:074502 (2006). ``Unquenched Determination of the Kaon Parameter B_K from Improved Staggered Fermions'', E.Gamiz, S.Collins, C.Davies, P.Lepage, J.Shigemitsu, M.Wingate; Phys. Rev. D\underbar{73}:114502 (2006). ``Highly-Improved Staggered Quarks on the Lattice with Applications to Charm Physics'', E.Follana, Q.Mason, C.Davies, K.Hornbostel, P.Lepage, J.Shigemitsu, H.Trottier and K.Wong; Phys. Rev. D\underbar{75}:054502 (2007). ``B0 Mixing Parameters from Unquenched Lattice QCD'', E.Dalgic, A.Gray, E.Gamiz, C.Davies, P.Lepage, J.Shigemitsu, H.Trottier and M.Wingate, Phys. Rev. D\underbar{76}:011501(R) (2007). ``High Precision Determination of the pi, K, D and D(s) Decay Constants from Lattice QCD''; [arXiv:0706.1726] E.Follana, C.Davies, P.Lepage, J.Shigemitsu, submitted for publication PI Donald Sinclair J.B.Kogut and D.K.Sinclair, Argonne preprint, ANL-HEP-PR-07-81 (2007). J.B.Kogut and D.K.Sinclair, ``The RHMC algorithm for theories with unknown spectral bounds,'' Phys. Rev. D 74, 114505 (2006), [arXiv:hep-lat/0608017]. J.B.Kogut and D.K.Sinclair, ``Quantization and simulation of Born-Infeld non-linear electrodynamics on a lattice,'' Phys. Rev. D 73, 114508 (2006), [arXiv:hep-lat/0603017]. J.B.Kogut and D.K.Sinclair, ``Evidence for O(2) universality at the finite temperature transition for lattice QCD with 2 flavours of massless staggered quarks,'' Phys. Rev. D 73, 074512 (2006), [arXiv:hep-lat/0603021]. PI George Smoot Jeong, E.; Smoot, G. F., "Probing Non-Gaussianity In The Cosmic Microwave Background Anisotropies: One Point Distribution ction", arXiv:0710.2371 (2007). Jeong, E.; Smoot, G. F., "The Validity of the Cosmic String Pattern Search with the Cosmic Microwave Background", ApJL. 661, L1, 2007. PI Randall Snurr Sung, C., Broadbelt, L.J., Snurr, R.Q. "A DFT study of adsorption of intermediates in the NOx reduction pathway over BaNaY olites", Catalysis Today, Submitted. PI Philip Snyder D.P. Brennan, A.D. Turnbull, M.S. Chu, R.J. La Haye, L.L. Lao, T.H. Osborne and S.A. Galkin, "Resistive Stability of 2/1 Modes Near 1/1 Resonance," Phys. Plasmas 14, 056108 (2007). J. Candy, R.E. Waltz, M.R. Fahey and C. Holland, The effect of ion-scale dynamics on electron-temperature gradient turbulence, Plasma Phys. Control. Fus., 49 (2007) 1209. J. Candy, R.E. Waltz, M.R. Fahey, and C. Holland, "Plasma Microturbulence Simulations of Instabilities at Highly Disparate Scales", J. Phys. Conf. Ser. 78 (2007) 012008. M.S. Chu, V.S. Chan, P.A. Politzer, D.P. Brennan et al, "Kinetic Alfven Wave and Associated Current Drive at the Center of Tokamaks," Phys. Plasmas 13 114501 (2006). M.S. Chu, D.P. Brennan, V.S. Chan et al, "Maintaining the Quasi-Steady State Central Current Density Profile in Hybrid Discharges," Nucl. Fusion 47 434 (2007). C. Estrada-Mila, J. Candy and R.W. Waltz, "Turbulent Transport of Alpha Particles in Reactor Plasmas," Phys. Plasmas 13 112303 (2006). F.L. Hinton and R.E. Waltz, "Gyrokinetic Turbulent Heating," Phys. Plasmas 13 102301 (2006). C. Holland, G.R. Tynan et al, "Zonal-flow-driven Nonlinear Energy Transfer in Experiment and Simulation," Phys. Plasmas 14 056112 (2007). J.E. Kinsey, R.E. Waltz, J. Candy, The Effect of Plasma Shaping on Turbulent Transport and ExB Shear Quenching in Nonlinear Gyrokinetic Simulations, accepted for publication in Phys. Plasmas. E.A. Lazarus, T.C. Luce, M.E. Austin, D.P. Brennan et al, "Sawtooth Oscillations in Shaped Plasmas," Phys. Plasmas 14 055701 (2007). W.M. Nevins, J. Candy et al, "Characterizing ETG Turbulence via Numerical Simulation," Phys. Plasmas 13 122306 (2006). W.M. Nevins, S.E. Parker, Y. Chen, J. Candy et al, "Verification of Gyrokinetic df Simulations of Electron Temperature Gradient Turbulence," Phys. Plasmas 14 084501 (2007). A.Y. Pankin, G. Bateman, D.P. Brennan et al, "Modeling of ELM Dynamics for DIII-D and ITER," Plasma Phys. Control. Fusion 49 S63 (2007). P.B. Snyder, K.H. Burrell, H.R. Wilson et al, "Stability and Dynamics of the Edge Pedestal in the Low Collisionality Regime: Physics Mechanisms for Steady-State ELM-Free Operation," Nucl. Fusion 47 961 (2007). G.M. Staebler, J.E. Kinsey and R.E. Waltz, A Theory Based Transport Model with Comprehensive Physics, Phys. Plasmas 14, 0055909 (2007) R.E. Waltz, J. Candy, and M. Fahey, Coupled Ion Temperature Gradient and Trapped Electron Mode to Electron Temperature Gradient Mode Gyrokinetic Simulations, APS06 Issue Phys. Plasmas 14, 0056116 (2007) R.E. Waltz, G. M. Staebler, J. Candy, and F. Hinton, "Gyrokinetic Theory and Simulation of Toroidal Angular Momentum Transport", submitted to Phys. Plasmas R.E. Waltz, G. M. Staebler, and J. Candy, "Gyrokinetic Theory and Simulation of Turbulent Energy Exchange", to be submitted Phys. Plasmas PI Masha Sosonkina M. Sosonkina, F. Liu, and R. Bramley, Usability levels for sparse linear algebra components, Concurrency and Computation: Practice and Experience, in press 2007. J. Jones, M. Sosonkina, and Y. Saad, Component-based iterative methods for sparse linear systems, Concurrency and Computation: Practice and Experience, 19 (2007) pp. 625--635. M. Sosonkina, Scalable dynamic adaptations for electronic structure calculations, Parallel Computing 2007 ParCo'07, Aachen--Juelich, Germany, Sep. 2007. M. Sosonkina, B. Ucar, and Y. Saad, Hypergraph partitioning for parallel iterative solution of general sparse linear systems, Parallel Computing, submitted Dec. 2006. PI Carl Sovinec P. Zhu, C. R. Sovinec, C. C. Hegna, K. Germaschewski, and A. Bhattacharjee, Nonlinear ballooning instability in the near-Earth magnetotail: Growth, structure, and possible role in substorms, J. Geophys. Res. 112, A06222 (2007). A. I. D. Macnab, R. D. Milroy, C. C. Kim, C. R. Sovinec, Hall magnetohydrodynamic simulations of end- shorting induced rotation in field-reversed configurations, Phys. Plasmas 14, 092507 (2007). C. R. Sovinec, D. C. Barnes, R. A. Bayliss, D. P. Brennan, E. D. Held, S. E. Kruger, A. Y. Pankin, D. D. Schnack, and the NIMROD Team, Two-fluid studies of edge relaxation events in tokamaks, J. Physics: Conf. Series 78, 012070 (2007). P. Zhu, C. C. Hegna, C. R. Sovinec, A. Bhattacharjee, and K. Germaschewski, Intermediate nonlinear regime of a line-tied g-mode, Phys. Plasmas 14, 055903 (2007). P. Zhu, C. C. Hegna, C. R. Sovinec, Nonlinear growth of a line-tied g-mode near marginal stability, Phys. Plasmas 13, 102307 (2006). PI Panagiotis Spentzouris Volkan Akcelik, Kwok Ko, Lie-Quan Lee, Zenghai Li, Cho-Kuen Ng, and Liling Xiao, Shape Determination for Deformed Electromagnetic Cavities, Journal of Computational Physics, to be published. Blumenfeld I., Clayton C.E., Decker F-J., Hogan M.J., Huang Chengkun, Ischebeck R., Iverson R., Joshi Katsouleas T., Kirby N., Lu W., Marsh K.A., Mori W.B., Muggli P., Oz E., Siemann R.H., Walz D., Zhou M., Energy doubling 42 GeV electrons in a metre-scale plasma wakefield accelerator, Nature, vol. 445 no. 7129, pp. 741-4 15 February 2007. Oz E., Deng S., Katsouleas T, Muggli P., Barnes C.D., Blumenfeld I., Decker F.J., Emma P., Hogan M.J., Ishchebeck Iverson R.H., Kirby N., Krejcik P., Oonnell C., Siemann R.h., Walz D., Auerbach D., Clayton D.E., Huang C., Johnson D.K., shi C., Lu W., Marsh K.A., Mori W.B., Zhou M., Ionization-induced electron trapping in ultra-relativistic plasma wakes, cal Review Letters, vol 98, no. 8 pp. 084801/1-4 23 February 2007. Thomas AGR, Najmudin Z., Mangles SPD, Murphy C.D., Dangor A.E., Kamperidis C., Lancaster K.L., Mori W.B., Norreys ., Rozmus W., Kushelnick K., Effect of laser-focusing conditions on propagation and monoenergetic electron production in r-wakefield accelerators, Physical Review Letters, vol 98, no. 4 pp. 095004/1-4 2 March 2007. A. V. Fedotov, D. L. Bruhwiler, A. O Sidorin, D. T. Abell, I. Ben-Zvi, R. Busby, J. R. Cary, and V. N. Litvinenko, Study of the Magnetized Friction Force," Phys. Rev. ST/AB 9, 074401 (2006). G. R. Werner and J. R. Cary, "A Stable FDTD Algorithm for Non-diagonal, Anisotropic Dielectrics," J. Comp. Phys. 226, 01 (2007), doi:10.1016/j.jcp.2007.05.008. Baifei Shen, Yuelin Li, Karoly Nemeth, Hairong Shang, Robert Soliday, Robert Crowell, Edward Frank, William Gropp, and John ary, "Triggering wave breaking in a laser plasma bubble by a nanowire," Phys. Plasmas 14, 053115 (2007). D. A. Dimitrov, D. L. Bruhwiler, J. R. Cary, C. G. R. Geddes, R. E. Giacone, E. Esarey and W. P. Leemans, "Particle-in-cell imulations of Laser Pulse Propagation in Plasma Channels," Phys. Plasmas, 14, 043105 (2007). Baifei Shen, Yuelin Li, M. Y. Yu, and John Cary, "Bubble regime for ion acceleration in a laser driven plasma," Phys. Rev. E accepted (2007). Lu W., Tzoufras M. , Joshi C., Tsung F.S.,.Mori W.B., Vieira J., Fonseca R.A., Silva L.O., "Generating Multi-GeV Electron unches Using Single Stage Laser Wakefield Acceleration in a 3D Nonlinear Regime," Physical Review Special Topics Acceleraors and Beams vol. 10, no. 6pp. 061301 June 2007. "An Experimentally Robust Technique For Halo Measurement Using The IPM At The Fermilab Booster," J. Amundson, W. Pellico, P. Spentzouris, T. Sullivan, and L. Spentzouris, Nucl. Instrum. Meth. A570, (2007). PI Frank Spera Thermal diffusivity of orthoclase glasses and single-crystals at high temperature, M. Pertermann, A. Whittington, A. Hofmeister, F. Spera, in press, Contributions to Mineralogy and Petrology, 2007 Accurate computation of shear viscosity from equilibrium molecular dynamics simulations, Journal of Molecular Simulation, in press, 2007, Nevins, D., Spera, F.J. Molecular Dynamics Studies of MgSiO3 Liquid to 150 GPa: An Equation of State (EOS), Tracer Diffusivities, and a Detailed Analysis of Changes in Atomic Coordination Statistics as a Function of Temperature and Pressure, Ghiorso, M S; Nevins, D; Spera, F J, EOS Transactions, American Geophysical Union. Vol. 87, no. 52, Suppl. 26 Dec. 2006 Thermodynamic and Structural Properties of liquid Mg2SiO4 at high temperatures and pressure in the range 0-150 GPa from Molecular Dynamics Simulation, Martin, B; Spera, F; Nevins, D.; EOS Transactions, American Geophysical Union. Vol. 87, no. 52, Suppl. 26 Dec. 2006 Partitioning of trace elements among coexisting crystals, melt and supercritical fluid during isobaric crystallization and melting, American Mineralogist, In Press, 2007, Frank J. Spera, Wendy A. Bohrson, Christy B. Till, Mark S. Ghiorso Tradeoffs in chemical and thermal variations in the post-perovskite phase transition: Mixed phase regions in the deep lower mantle, Physics of The Earth and Planetary Interiors, 159, 2006, 234-246, Frank J. Spera, David A. Yuen and Grace Giles Phase Equilibria Impetus For Large-Volume Explosive Volcanic Eruptions Fowler, S J; Spera, F J; Bohrson, W A; Ghiorso, M S EOS Transactions, American Geophysical Union. Vol. 87, no. 52, Suppl. 26 Dec. 2006 The Magma Chamber Simulator: A Comprehensive Tool for Modeling the Evolution of Magmatic Systems Bohrson, W A; Spera, F J; Ghiorso, M S; Fowler, S J, EOS Transactions, American Geophysical Union. Vol. 87, no. 52, Suppl. 26 Dec. 2006 PI Don Spong Sheared plasma flow generation: A new measure for stellarator optimization, Spong, D. A., FUSION SCIENCE AND TECHNOLOGY, Volume: 50, Issue: 3, Pages: 343-351 (2006). "Shear flow generation in stellarators - configurational variations," Spong, D. A., Harris, J. H., Ware, A. S., Hirshman, S. P., Berry, L. A., Nucl. Fusion, Vol. 47, Pages 626-633 (2007). Compression of magnetohydrodynamic simulation data using singular value decomposition, del-Castillo-Negrete, D., Hirshman, S. P., Spong, D. A., D'Azevedo, E. F., JOURNAL OF COMPUTATIONAL PHYSICS, Volume: 222, Issue: 1, Pages: 265-286 (2007). Assessment of transport in NCSX, Mikkelsen, D. R., Maassberg, H., Zarnstorff, M. C., Beidler, C. D., Houlberg, W. A., Kernbichler, W., Mynick, H., Spong, D. A., Strand, P., Tribaldos, V., FUSION SCIENCE AND TECHNOLOGY, Volume: 51, Issue: 2, Pages: 166-180 (2007). Power and particle handling and wall conditioning in NCSX, Mioduszewski, P. K., Owen, L. W., Spong, D. A., Fenstermacher, M. E., Koniges, A. E., Rognlien, T. D., Umansky, M. V. FUSION SCIENCE AND TECHNOLOGY, Volume: 51, Issue: 2, Pages: 238-260 (2007). Simulational study on losses of neutral beam-injected energetic ions via collisional ripple transport in the low aspect ratio helical system CHS, Isobe, M., Spong, D. A., Shimizu, A., Toi, K., Matsushita, H., Nagaoka, K., Nishiura, M., Matsuoka, K., Okamura, S., Murakami, S., JOURNAL OF PLASMA PHYSICS, Volume: 72, Pages: 1189-1192, Part: Part 6 (2006). Studies of fast-ion transport induced by energetic particle modes using fast-particle diagnostics with high time resolution in CHS, Isobe, M., Toi, K., Matsushita, H., Goto, K., Suzuki, C., Nagaoka, K., Nakajima, N., Yamamoto, S., Murakami, S., Shimizu, A., Yoshimura, Y., Akiyama, T., Minami, T., Nishiura, M., Nishimura, S., Darrow, D. S., Spong, D. A., Shinohara, K., Sasao, M., Matsuoka, K., Okamura, S., NUCLEAR FUSION, Volume: 46, Issue: 10, Pages: S918-S925 (2006). PI Garrison Sposito Kwon K.D., Refson K., and Sposito G. (2007) Defect-induced photoconductivity in layered manganese oxides, Physical Review ters (submitted). Bourg I.C., and Sposito G. (2007). Isotopic fractionation of noble gases by diffusion in liquid water. 1. Molecular dynamics simulations, Geochimica et Cosmochimica Acta (submitted). Bourg I.C., and Sposito G. (2007). Isotopic fractionation of noble gases by diffusion in liquid water. 2. Hydrologic ions, Geochimica et Cosmochimica Acta (submitted). Bourg I.C., and Sposito G. (2007). Molecular dynamics simulations of kinetic isotope fractionation during the diffusion of onic species in liquid water, Geochimica et Cosmochimica Acta (in press). PI Philip Sprangle D.F. Gordon, "Improved Ponderomotive Guiding Center Algorithm," accepted for publication in IEEE Trans. Plasma Sci. PI Malcolm Stocks Y. Q. Cai, P. C. Chow, O. D. Restrepo, Y. Takano, K. Togano, H. Kito, H. Ishii, C. C. Chen, K. S. Liang, C. T. Chen, S. Tsuda, S. Shin, C. C. Kao, W. Ku, and A. G. Eguiluz, Low-Energy Charge-Density Excitations in MgB2: Striking Interplay ween Single-Particle and Collective Behavior for Large Momenta, Phys. Rev. Lett. 97, 176402 (2006). Eisenbach, M., Stocks, G. M., Nicholson, D. M., Noncollinear magnetism in permalloy, J. Appl. Phys. 101 (9) 09G503 007). Etz, C., Lazarovits, B., Zabloudil, J., et al. Magnetic properties of FeCo nanoclusters on Cu(100): Ab initio tions, Phys. Rev. B: 75 (24) 245432 (2007). Fishman, Randy S.; Fernando A. Reboredo, Alex Brandt, and Juana Moreno, Nature of Perpendicular-to-Parallel Spin ientation in a Mn-doped GaAs Quantum Well: Canting or Phase Separation? Phys. Rev. Lett. 98, 267203 (2007). H. Yang, Q. Sun, Z. Zhang, Y. Jia, Upward self-diffusion of adatoms and small clusters on facets of fcc metal (110) urfaces, Phys. Rev. B 76, 115417 (2007). Huang L.; Gong, X. G., Gergert, E., Forster, F., Bendounan, A., Reinert, F., Zhang, Zhenyu, Evolution of a symmetry ap and synergetic quantum well states in ultrathin Ag films on Au(111) substrates Europhys. Lett. 78, 57003 (2007). Liu, Shudun, Jayanthi, C. S., Zhang, Zhenyu, Wu, S. Y., Stability and mechanical properties of silicon nanowires, J. Comput. Theor. Nanosci. 4 (2): 275-281 (2007). Liu, W. L., Zhang, K. W., Xiao, H. P., et al. Surface reconstruction and core distortion of silicon and germanium owires, Nanotechno 18 (21) 215703 (2007). Melko, R. G.; R. S. Fishman, and F. A. Reboredo, Single layer of Mn in a GaAs quantum well: A ferromagnet with m fluctuations Phys. Rev. B 75, 115316 (2007). Meng, S., Kaxiras, E., Zhang, Z., Metal-diboride nanotubes as high-capacity hydrogen storage media, Nano Letters: 7 3): 663-667 (2007). Namilae, S., Fuentes-Cabrera, M., Radhakrishnan, B., Sarma, G. B., Nicholson, D. M., Energetics of hydrogen storage n organolithium nanostructures, Chem. Phys. Lett. 436 (1-3): 150-154 (2007). Nicholson, D. M. C., Barabash, R. I. , Ice, G. E., et al. Relationship between pair and higher-order correlations in solid solutions and other Ising systems, J. Phys. Condens Mat.18 (50): 11585-11594 (2006). Oezer, M. M., Jia, Yu, Zhang, Z., Thompson, J. R., Weitering, H. H., Tuning the quantum stability and ty of ultrathin metal alloys, Science 316 (5831): 1594-1597 (2007). Petit, L, Stocks, G. M., Egami, T., Szotek, Z. and Temmerman, W. M., Ground state valency and spin configuration of the Ni ions in nickelates, Phys. Rev. Lett. 97 (14) 146405 (2006). Seletskaia, T.; Osetskiy, Y. N., Stoller, R. E., Stocks, G. M., Development of a FeHe interatomic potential based on electronic structure calculations, J Nucl. Mat. 355-360, 367 (2007). Stocks, G. M., Eisenbach, M., Ujfalussy, B., et al. On calculating the magnetic state of nanostructures, Prog. Sci. 52 (2-3) 371-387 (2007). Yoon, M.; Ho Nyung L., Wei Hong, H. M. Christen, Z. Zhang, and Z. Suo, Dynamics of Step Bunching in Heteroepitaxial rowth on Vicinal Substrates, Phys. Rev. Lett. 99, 055503 (2007). Yoon, M., Howe, J., Tibbetts, G., Eres, Gyula, Zhang, Z., Polygonization and anomalous graphene interlayer spacing multi-walled carbon nanofibers,Phys. Rev. B 75 (16): 165402 (2007). Yoon, M.; S. Y. Yang, E. G. Wang, and Z. Y. Zhang, Charged fullerenes as high capacity hydrogen storage media, Nano. Lett. 9, 2578 (2007). Zhang, K. W., Stocks, G. M., Zhong, J. X. Melting and premelting of carbon nanotubes, Nanotechno 18 (28) 285703 ).0077 Zhong, J. X., Stocks, G. M. Persistent mobility edges and anomalous quantum diffusion in order-disorder separated ntum films, Phys. Rev. B 75 (3) 033410 (2007). PI Alberto Striolo N.R. Tummala and A. Striolo, The Influence of Counter-Ion Condensation on the Self-Assembly of SDS Surfactants at the hite-Water Interface, Journal of Physical Chemistry C, submitted. A. Striolo, Water Self Diffusion Through Narrow Oxygenated Carbon Nanotubes, Nanotechnology, 18 (2007) 475704. B.H. Morrow and A. Striolo, Morphology and Diffusion Mechanism of Platinum Nanoparticles Supported on Carbon Nanotube les, Journal of Physical Chemistry B, (2007) in press. E.R. Chan, A. Striolo, C. McCabe, S.C. Glotzer, and P.T. Cummings, A Coarse-Grained Force Field for Simulating Polymer-hered Silsesquioxane Self-Assembly in Solution, Journal Chemical Physics, 127 (2007) 114102. Also featured on the Virtual rnal of Nanoscale Science & Technology, Vol. 16, Issue 14, October 1st, 2007. A. Striolo, C. McCabe, P. T. Cummings, E. R. Chan, and S. C. Glotzer, Aggregation of POSS Monomers in Liquid Hexane: A ecular-Simulation Study, Journal of Physical Chemistry B, 111 (2007) 12248. H.-C. Li, C.Y. Lee, C. McCabe, A. Striolo, and M. Neurock, Ab Initio Analysis of the Structural Properties for Alkyl-ituted Polyhedral Oligomeric Silsesquioxanes, Journal of Physical Chemistry A, 111 (2007) 3577. A. Striolo, Controlled Assembly of Spherical Nanoparticles: Nanowires and Spherulites, Small, 3 (2007) 628. A. Striolo and S.A. Egorov, Sterical Stabilization of Colloidal Particles: Implicit and Explicit Solvent, Journal of cal Physics, 126 (2007) 014902. A. Striolo, Colloidal Brushes in Complex Solutions: Existence of a Weak Mid-Range Attraction Due To Excluded-Volume s, Physical Reviews E, 74 (2006) 041401. Also featured on the Virtual Journal of Nanoscale Science & Technology, Vol. 14, ue 18, October 30th, 2006. PI Erich Strohmaier E. Strohmaier and H. Shan, Apex-Map: A Parameterized Scalable Memory Access Probe for High-Performance Systems, ncurrency and Computation: Practice and Experience , 2007; 19:1-21, January 2007 H. Shan, E. Strohmaier, and J. Qiang, Performance Analysis of leading HPC Architectures with BeamBeam3D International nal of High Performance Computing Applications (IJHPCA), in Press, LBNL-61134 C. C. Iancu and E. Strohmaier, 2007. Optimizing Communication Overlap for High-Speed Networks. In Proceedings of he 12th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming (San Jose, California, USA, March 14 - 17, 007). PPoPP '07. ACM Press, New York, NY, 35-45. L. Oliker, A. Canning, J. Carter, C. Iancu, M. Lijewski, S. Kamil, J. Shalf, H. Shan, E. Strohmaier, S. Ethier, T. , "Scientific Application Performance on Candidate PetaScale Platforms", International Parallel & Distributed Processing osium (IPDPS) 2007, LBNL-62952. E. Strohmaier, Performance and it Complexity on Petascale Systems, book chapter in Petascale Computing: and Applications, D.A. Bader editor, in Press. M. M. Tikir, L. Carrington, E. Strohmaier, A. Snavely, A Genetic Algorithms Approach to Modeling the Performance f Memory-bound Computations, to appear in Proceedings of SC2007, PI Maxim Sukharev M. Besbes, J.P. Hugonin, P. Lalanne, S. van Haver, O. T. A. Jansse, A.M. Nugrowati, M. Xu, S. F. Pereira, H. P. Urbach, A. S. van de Nes, P. Bienstman, G. Granet, S. Helfert, M. Sukharev, T. Seideman, F. I. Baida, B. Guizal, D. Van Labeke, "Numerical analysis of a slit-groove diffraction problem", Journal of the European Optical Society: Rapid Publications 2, 07022 (2007). Maxim Sukharev and Tamar Seideman, "Light trapping and guidance in plasmonic nanocrystals", Journal of Chemical Physics 126, 204702 (2007). Maxim Sukharev and Tamar Seideman (invited review paper), "Coherent Control of Light Propagation via Nanoparticle Arrays", Journal of Physics B: Atomic, Molecular & Optical Physics 40, S283 (2007). G. Gay, O. Alloschery, J. Weiner, H. J. Lezec, C. O'Dwyer, M. Sukharev, T. Seideman, "Surface quality and surface waves on subwavelength-structured silver films" Physical Review E 75, 016612 (2007) PI Xianzhu Tang Driven resonance in partially relaxed plasmas, X.Z. Tang, Phys. Rev. Lett. 98, 175001 (2007). Proposed experiment to study relaxation formation of a spherical tokamak with a plasma center column, S.C. Hsu and X.Z. Tang, Journal of Fusion Energy 26, 85 (2007). Self-organization of radio lobe magnetic fields via driven-relaxation submitted to Astrophysical Journal, X.Z.Tang (2007). Scale-up of spherical tokamak solenoid-free startup by co-axial helicity injection accepted for publication, Physics of Plasmas, X.Z. Tang and A.H. Boozer (2007). L. Chacon, A. N. Simakov, A. Zocco, "Steady state properties of magnetic reconnection in 2D EMHD," Phys. Rev. Lett., submitted (2007) G. L. Delzanno, L. Chacon, J. M. Finn, "Electrostatic mode associated with the pinch velocity in Reversed Field Pinch simulations," Plasma Physics and Controlled Fusion, submitted (2007) B. Philip, M. Pernice, and L. Chacon, "Solution of reduced resistive magnetohydrodynamics using implicit adaptive mesh refinement," Lecture Notes in Computational Science and Engineering, accepted (2007). G. Lapenta, J.U. Brackbill, P. Ricci, Kinetic Approach to microscopic-macroscopic coupling in space and laboratory plasmas, Physics of Plasmas, 13, 055904, 2006. G. Lapenta, D. Krauss-Varban, H. Karimabadi, J.D. Huba, L.I. Rudakov, P. Ricci, Kinetic Simulations of X-line Propagation in 3D Reconnection, Geophysical Research Letters, 33, L10102, doi:10.1029/2005GL025124, 2006. K. Noguchi, C. Tronci, G. Zuccaro, G. Lapenta, Formulation of the relativistic moment implicit particle-in-cel l method, Physics of Plasmas, 14, 042308, 2007. W. Wan, G. Lapenta, Micro-Macro Coupling in Plasma Self-Organization Processes During Island Coalescence, Physical Review Letters, submitted. PI Dave Thirumalai D. L. Pincus, Changbong Hyeon, and D. Thirumalai, "Effects of Trimethylamine N-Oxide (TMAO) and Crowding Agents on the Stability of RNA Hairpins", J. Am. Chem. Soc. (submitted). PI Robin Tokmakian Tokmakian, R., 2007, How stationary is the ocean component of the climate signal?, submitted, Jrnl. of Climate. PI Owen Toon Fridlind, A.M., A.S. Ackerman, G. McFarquhar, G. Zhang, M.R. Poellot, P.J. DeMott, A.J. Prenni, and A.J. Heymsfield, 2007: Ice properties of single-layer stratocumulus during the Mixed-Phase Arctic Cloud Experiment (M-PACE): Part II, Model results. J. Geophys. Res., in press. Charlson, R.J., A.S. Ackerman, F.A.-M. Bender, T.L. Anderson, and Z. Liu, 2007: On the climate forcing consequences of the albedo continuum between cloudy and clear air. Tellus, 59B, 715-727, doi:10.1111/j.1600-0889.2007.00297.x. Jensen, E.J., A.S. Ackerman, and J.A. Smith, 2007: Can overshooting convection dehydrate the tropical tropopause layer? J. Geophys. Res., 112, D11209, doi:10.1029/2006JD007943. Senocak, I, A.S. Ackerman, M.P. Kirkpatrick, D.E. Stevens, and N.N. Mansour, 2007: Study of near-surface models for large-eddy simulations of a neutrally stratified atmospheric boundary layer. Bound.-Lay. Meteorol., 124, 405-424, doi:10.1007/s10546-007-9181-x. Barth, M.C., S.-W. Kim, C. Wang, K.E. Pickering, L.E. Ott, G. Stenchikov, M. Leriche, S. Cautenet, J.-P. Pinty, C. Barthe, C. Mari, J. Helsdon, R. Farley, A.M. Fridlind, A.S. Ackerman, V. Spiridonov, and B. Telenta, 2007: Cloud-scale model intercomparison of chemical constituent transport in deep convection. Atmos. Chem. Phys., 7, 4709-4731. PI Doug Toussaint QCD thermodynamics with 2+1 flavors at nonzero chemical potential, C. Bernard, C. DeTar, Steven Gottlieb, U.M. Heller, J.E. Hetrick, L.Levkova, R. Sugar, D. Toussaint, submitted to Physical Review D QCD equation of state with 2+1 flavors of improved staggered quarks, C. Bernard, T. Burch, C. DeTar, Steven Gottlieb, L. Levkova, U.M. Heller, J.E. Hetrick, R. Sugar, D. Toussaint, Phys. Rev. D75 (2007) 094505 PI George Vahala B. Keating, G. Vahala, J. Yepez, M. Soe and L. Vahala, "Entropic Lattice Boltzmann Representations Required to Recover Navier-Stokes Flows", Phys. Rev. E75, 036712 (2007) G. Vahala, J. Yepez, M. Soe, L. Vahala and S. Ziegeler, "Lattice Boltzmann Algorithms for Fluid Turbulence", IEEE Transactions (to be published, 2007) G. Vahala, B. Keating, M. Soe, L. Vahala, J. Yepez, S. Ziegeler, J. Carter, "MHD Turbulence Studies - using Lattice Boltzmann algorithms", submitted to Communications in Computational Physics (Oct. 2007) PI James Vary A. Negret, T. Adachi, B.R. Barrett, C. Baumer, A.M. van den Berg, G.P.A. Berg, P. von Brentano, D. Frekers, D. De Frenne, H. Fujita, K. Fujita, Y. Fujita, E.-W. Grewe, P. Haefner, M.N. Harakeh, K. Hatanaka, K. Heyde, M. Hunyadi, E. Jacobs, Y. Kalmykov, A. Korff, K. Nakanishi, P. Navratil, P. von Neumann-Cosel, L. Popescu, S. Rakers, A. Richter, N. Ryezayeva, Y. Sakemi, A. Shevchenko, Y. Shimbara, Y. Shimizu, Y. Tameshige, A. Tamii, M. Uchida, J. Vary, H.J. Woertche, M. Yosoi, L. Zamick, Gamow-Teller Strengths in the A = 14 Multiplet: A Challenge to the Shell Model, Phys. Rev. Lett. 97, 062502 (2006); UCRL-JRNL-223644. A.M. Shirokov, J.P. Vary, A.I. Mazur and T.A. Weber, Realistic Nuclear Hamiltonian: Ab exitu approach, Phys. Letts. B 644, 33(2007), ArXiv nucl-th/0512105, UCRL-JRNL-217903, SLAC-PUB-11614. S.J. Brodsky, D. Chakrabarti, A. Harindranath, A. Mukherjee, J. P. Vary, Hadron Optics in Three-Dimensional Invariant Coordinate Space from Deeply Virtual Compton Scattering, Phys. Rev. D 75, 14003 (2007). ArXiv hep-ph/0611159 (SLAC-PUB-12096, UCRL-JRNL-226236). K. D. Sviratcheva, J. P. Draayer, and J.P. Vary, Global Properties of fp-Shell Interactions in Many-nucleon Systems, accepted for publication in Nuclear Physics A; (ArXiv nucl-th/0703076, SLAC-PUB-11903, UCRL-JRNL-222549). Tomas Dytrych, Kristina D. Sviratcheva, Chairul Bahri, Jerry P. Draayer and James P. Vary, Evidence for Symplectic Symmetry in Ab Initio No-Core Shell Model Results for Light Nuclei, Phys. Rev. Lett. 98, 162503 (2007). P. Navratil, V.G. Gueorguiev, J. P. Vary, W. E. Ormand and A. Nogga, Structure of A = 10-13 nuclei with two- plus three-nucleon interactions from chiral effective field theory, Phys. Rev. Lett. 99, 042501(2007); UCRL-JRNL-227256, ArXiV: nucl-th 0701038. T. Dytrych, K. D. Sviratcheva, C. Bahri, J. P. Draayer and J.P. Vary, Dominant role of symplectic symmetry in the ab-initio no-core shell model results for light nuclei, Phys. Rev. C 76, 014315 (2007). J.P. Vary, P. Navratil, V.G. Gueorguiev, W. E. Ormand A. Nogga, P. Maris and A. Shirokov, Ab initio and ab exitu no core shell model, Proceedings of the Vico Equense Conference, A. Covello, Ed., World Scientific (Singapore, 2008), to appear. PI Haobin Wang H. Wang, D.E. Skinner, and M. Thoss, "Calculation of reactive flux correlation functions for systems in a model condensed phase environment: a multilayer multi-configuration time-dependent Hartree approach'', J.Chem.Phys., 125, 174502 (2006). H. Wang, and M. Thoss, "Quantum dynamical simulation of electron-transfer reactions in an anharmonic environment", J. Phys. Chem. A, 111, 10369 (2007). I. Kondov, M. Cizek, C. Benesch, H. Wang, and M. Thoss, "Quantum dynamics of photoinduced electron-transfer reactionsin dye-semiconductor systems: First principles description and application to coumarin 343-TiO2", J. Phys. Chem. A, 111, 11970 (2007). I. R. Craig, M. Thoss, and H. Wang, "Proton transfer reactions in model condensed-phase environments: Accurate quantum dynamics using multilayer multiconfiguration time-dependent Hartree approach", J. Chem. Phys., 127, 144503 (2007). M. Thoss, I. Kondov, and H. Wang, "Correlated electron-nuclear dynamics in ultrafast photoinduced electron-transfer reactions at dye-semiconductor interfaces", Phys. Rev. B, 76, xx (2007). H. Wang and M. Thoss, "Nonperturbative quantum simulation of time-resolved nonlinear spectra: Methodology and application to electron transfer reactions in the condensed phase", Chem. Phys., accepted. PI Lin-Wang Wang Z. Zhao, L.W. Wang, F. Wu, "Wavefunction localizations in bulged CdSe nanowires", J. Theor. Comp. NanoSci. 4, 247 (2007) [LBNL-60841]. C. Yang, J. Meza, L.W. Wang, "A trust region direct constrained minimization algorithm for the Kohn-Sham equation", 29, 1854 (2007) [LBNL-59841]. C. Voemel, J. Dongarra, O. Marques, S. Tomov, L.W. Wang, "The use of bulk states to accelerate the band edge state calculation of a semiconductor quantum dot", J. Comp. Phys. 223, 774 (2007) [LBNL-60147]. L.W. Wang, X. Cartoixa, "Motif-based polarization model: Calculations of the dielectric function and polarization in large nanostructures", Phys. Rev. B 75, 205334 (2007) [LBNL--63027]. Z. Yang, L.W. Wang, A. Mascarenhas, "Quantum coaxial cables for solar energy harvesting", Nanolett. 7, 1264 (2007) [LBNL-63026]. L. Fang, J.Y. Park, Y. Cui, P. Alivisatos, J. Schrier, B. Lee, L.W. Wang, M. Salmeron, "Mechanical and electrical properties of CdTe tetrapods studied by atomic force microscopy", J. Phys. Chem. (in press). B. Lee, A. Canning, L.W. Wang, "Effects of d-electrons in pseudopotential screened-exchange density functional calculations", J. Appl. Phys. (in press). R. D. Robinson, B. Sadtler, D. Demchenko, C. K. Erdonmez, L.W. Wang, A. P. Alivisatos, "Spontaneous superlattice formation in nanorods through partial cation exchange", Science, 317, 355 (2007) [LBNL-62650]. B. Lee, L.W. Wang, C. D. Sparatus, S. G. Louie, "Nonlocal exchange-correlation in screened exchange density functional methods", Phys. Rev. B (in press) [LBNL-61337]. J. Schrier, D. O. Demchenko, L.W. Wang, A.P. Alivisatos, "Optical properties of ZnO/ZnS and ZnO/ZnTe heterostructures for solar cell applications" Nanolett. 7, 2377 (2007)[LBNL-62616]. J. Schrier, B. Lee, L.W. Wang, "Mechanical and electronic structure properties of compressed CdSe tetrapod nanocrystals", J. Nanosci. Nanotech. (in press) [LBNL-62312]. D. O. Demchenko, L.W. Wang, "Localized electron states near a metal-semiconductor nanocontact", Nanolett. 7, 3219 (2007). Q. Zhao, P. A. Craf, W. B. Jones, A. Franceschetti, J. Li, L.W. Wang, K. Kim, "Shape dependence of band edge exciton fine structure in CdSe nanocrystals", Nanolett. (in press). F. Wang, H. Yu, J. Li, Q. Hang, D. Zemlyanov, P. C. Gibbons, L.W. Wang, D. B. Janes, W. E. Buhro, "Spectroscopic properties of colloidal indium phosphide quantum wires", J. Phys. Chem. B (submitted). L.W. Wang, Z. Zhao, J. Meza, "A linear scaling three dimensional fragment method for large scale electronic structure calculations", Phys. Rev. Lett. (submitted). D. Demchenko, R.D. Robinson, B. Sadtler, C.J. Erdonmex, A.P. Alivisatos, L.W. Wang, "Formation mechanism and properties of CdS-Ag2S nanorod superlattices", ACS Nano (submitted). C. Vomel, S.Z. Tomov, O.A. Marques, A. Canning, L.W. Wang, J.J. Dongarra, "State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems", J. Comp. Phys. (submitted). PI Warren Washington Alexander, L.M. and J.M. Arblaster, Assessing trends in observed and modelled climate extremes over Australia in relation to future projections, Int. J. Climatology, submitted. Meehl, G., C. Tebaldi, H. Teng, and T.C. Peterson, 2007b: Current and future U.S. weather extremes and El Niqo. Geophysical Research Letters, in press. Meehl, G. J. Arblaster, and W. Collins, 2007c: Effects of black carbon aerosols on the Indian monsoon. J. Climate, in press. Meehl, G., J. Arblaster, G. Branstator, and H. van Loon, 2007d: A coupled air sea response mechanism to solar forcings in the Pacific region. J. Climate, in press. Osborne, T.M., D.M. Lawrence, A.J. Challinor, J.M. Slingo, T.R. Wheeler, 2007: Development and assessment of a coupled crop-climate model. Glob. Change Biol., 13, 169-183. Santer, B.D., C. Mears, F.J. Wentz, K.E. Taylor, P.J. Gleckler, T.M.L. Wigley, T.P. Barnett, J.S. Boyle, W. Bruggemann, N.P. Gillett, S.A. Klein, G.A. Meehl, T. Nozawa, D.W. Pierce, P.A. Stott, W.M. Washington, and M.F. Wehner, 2007: Identification of human-induced changes in atmospheric moisture content. Proc. Nat. Acad. Sci., 104, 15248-15253. PI William Weber R. Devanathan, L. R. Corrales, W. J. Weber, A. Chartier and C. Meis, "Molecular dynamics simulation of energetic uranium recoil damage in zircon", Molecular Simulations 32 (2006) 1069-1077. J. Du, R. Devanathan, L. R. Corrales, W. J. Weber and A. N. Cormack, "Short-and medium-range structure of amorphous zircon from molecular dynamics simulations", Physical Review B 74 (2006) 214204. R. Devanathan, P. Durham, J. Du, L. R. Corrales and E. M. Bringa, "Molecular dynamics simulation of amorphization in forsterite by cosmic rays", Nuclear Instruments and Methods B 255 (2007) 172-176. R. Devanathan and W. J. Weber, "Radiation Effects in a Model Ceramic for Nuclear Waste Disposal", JOM 59(4) (2007) 32-35. R. Devanathan, F. Gao and W. J. Weber, "Atomistic modeling of amorphous silicon carbide using a bond-order potential", Nuclear Instruments and Methods B 255 (2007) 130-135. F. Gao, Y. Zhang, R Devanathan, M. Posselt, and W. J. Weber, "Atomistic Simulations of Epitaxial Recrystallization in 4H-SiC along the [0001] Direction", Nuclear Instruments and Methods B 255 (2007) 136-140. F. Gao, J. Du, E. J. Bylaska, M. Posselt, and W. J. Weber, "Ab Initio Atomic Simulations of Antisite Pair Recovery in Cubic Silicon Carbide", Applied Physics Letters 90 (2007) 221915. R. Devanathan and W. J. Weber, "Self-healing in irradiated zirconia-based ceramics", J. Materials Research (submitted). H. Xiao, F. Gao, L. M. Wang, X. T. Zu, Y. Zhang and W. J. Weber, "Ab initio molecular dynamics determination of phase transformation pathways and mechanism", Nature Materials (submitted). H. Xiao, L. M. Wang, F. Gao, X. T. Zu, W. J. Weber, J. Lian and R. C. Ewing, "Radiation resistance of A2Ti2O7 (A=Dy, Y, and Ho) Pyrochlores: Role of f electrons", Physical Review Letters (submitted). N. Li, H. Xiao, X. Zu, L. M. Wang, R C Ewing, J. Lian and F. Gao, "First-principles study of electronic properties of La2Hf2O7 and Gd2Hf2O7", Journal of Applied Physics (Submitted) Z. J. Chen, H. Y. Xiao, X. Zu, L. M Wang, F. Gao, J. Lian and R. C. Ewing, "Structural and bonding properties of stannate pyrochlores: a density functional theory investigation", Computational Materials Science (submitted). PI Suhai Wei S. Limpijumnong, M. F. Smith, and S. B. Zhang, "Characterization of As-doped, p-type ZnO by x-ray absorption near-edge structure spectroscopy: Theory", Appl. Phys. Lett. 89, 222113 (2006). M.-H. Du, H. M. Branz, R. S. Crandall, S. B. Zhang, "Bistability-mediated carrier recombination at light-induced boron-oxygen complexes in silicon", Phys. Rev. Lett. 97, 256602 (2006). 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Xue, "Experimental observation of quantum oscillation of surface chemical reactivities", PNAS 104, 9204-9208 (2007). Y. Zhao, M. J. Heben, A. C. Dillon, L. J. Simpson, J. L. Blackburn, H. C. Dorn, and S. B. Zhang, "Nontrivial Tuning of the Hydrogen-Binding Energy to Fullerenes with Endohedral Metal Dopants", J. Phys. Chem. C 2007, 111, 13275. A. Allenic, W. Guo, Y. B. Chen, M. B. Katz, G. Y. Zhao, Y. Che, Z. D. Hu and B. Liu, S. B. Zhang, X. Q. Pan, "Amphoteric phosphorus doping for stable p-type ZnO", Advanced Materials, DOI: 10.1002/adma.200700083. Sun, Y. Y.; Kim, Y.-H.; Zhang, S. B., "Effect of Spin State on the Dihydrogen Binding Strength to Transition Metal Centers in Metal-Organic Frameworks", J. Am. Chem. Soc.; (Communication); 2007; ASAP Article; DOI: 10.1021/ja0740061. Y. Jiang, Y.-H. Kim, S. Yang, Z. Tang, K. Wu, P. Ebert, S. B. Zhang, and E. G. Wang, "Growing extremely thin bulklike metal film on a semiconductor surface: monolayer Al(111) on Si(111)", Appl. Phys. Lett. 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Stat. Sol. (b) (in press). Y. Yan, J. Li, S.-H. Wei, and M. M. Al-Jassim, "Possible approach to overcome the doping polarity in wide-band-gap semiconductors", Phys. Rev. Lett. 98, 135506 (2007). Y. Yan, M. M. Al-Jassim, and S.-H. Wei, "Doping of ZnO by group-Ib elements", Appl. Phys. Lett. 89, 181912 (2006). Q. Xu, J.-W. Luo, S.-S. Li, J.-B. Xia, J. Li and S.-H. Wei, "Chemical Trends of defect formation in Si quantum dots: The case of group-III and group-V dopants", Phys. Rev. B 75, 235304 (2007). PI Michael Weinert Magnetic instability and meta-magentism on the surface of the bulk paramagnetic intermetallic compound YCo$_{2}$. S. Khmelevskyi, P. Mohn, J. Redinger, and M. Weinert, Physica B 378-380, 109 (2006). Unusual adsorption site of hydrogen on the unreconstructed Ir(100) surface. D. Lerch, A. Klein, A. Schmidt, S. M\"uller, L. Hammer, K. Heinz, and M. Weinert, Phys. Rev. B 73, 075430 (2006). Atomistic view of the autosurfactant effect during GaN epitaxy. S. T. King, M. Weinert, and L. Li, Phys. Rev. Lett. 98, 206106 (2007). YCo$_2$: Intrinsic Magnetic Surface of a Paramagnetic Bulk Material. Yu. S. Dedkov, C. Laubschat, S. Khmelevskyi, J. Redinger, P. Mohn, and M. Weinert, Phys. Lett. 99, 047204 (2007). PI Harold Weitzner G. Park and C.S. Chang, A 5-1/2 dimensional theory for fast and accurate evaluation of the cyclotron resonance heating using spatial waverepresentation, Phys. Plasmas 14, 052503 (2007) H. Strauss, L. Sugiyama, C.S. Chang, G.Y. Park, S. Ku, W. Park, J. Breslau, S. Jardin ``ELM Simulations with M3D,"21st IAEA Fusion Energy Conference, Chengdu, China, TH/P8-6 (2006) G. Y. Fu, W. Park, H. Strauss, J. Breslau, J. Chen, S. Jardin, and L. E. Sugiyama, "Global hybrid simulations of energetic particle effects on n = 1 mode in tokamaks: internal kink and fishbone instability", Phys. Plasmas 2006. Zaslavsky G.M., Guzdar P.N., Edelman M., et al., Selfsimilarity and fractional kinetics of solar wind-magnetosphere coupling Physica A 373: 11-20 (2007). Courbage M. Edelman M., Saberi S.M., Fathi Zaslavsky G.M.., The problem of transport in billiards with infinite horizon, PRE (submitted) Zaslavsky G.M., Edelman M., Stochastic Web as a Generator of Three-Dimensional Quasicrystal Symmetry, Chaos, 17, 023127 (2007). G. M. Zaslavsky, A. A. Stanislavsky, and M. Edelman, "Chaotic and pseudochaotic attractors of perturbed fractional oscillator", Chaos 16, 013102 (2006). G.M. Zaslavsky, M. Edelman, "Stickiness of trajectories in a perturbed Anosov system", Regular & Chaotic Dynamics 11 (2): 329-336 (2006). P. Garabedian, Bifurcated equilibria and magnetic islands in tokamaks and stellarators, Comm. Appl. Math. Comp. Sci. 1 (2006) 79-90. L.-P. Ku and P. Garabedian, New classes of quasi-axisymmetric stellarator configurations, Fusion Sci. Technol. 50 (2006) 207-215. P. 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PI Don Werthimer Stanimirovic, Putnam, Heiles, Peek, Goldsmith, Koo, Krco, Lee, Mock, Muller, Pandian, Parsons, Tang, Werthimer First Results from the Arecibo Galactic HI Survey: The Disk/Halo Interface Region In The Outer Galaxy accepted for publication in ApJ PI Martin White Conroy, Charlie, Ho, Shirley, and White, Martin, Constraints on the merging time-scale of luminous red galaxies, or, where do all the haloes go?, 2007, Monthly Notices of the Royal Astronomical Society, 379, 1497 Eisenstein, Daniel J., Seo, Hee-Jong, and White, Martin, On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter, 2007, Astrophysical Journal, 664, 674 Heitmann, Katrin, Lukic, Zarija, Fasel, Patricia, Habib, Salman, Warren, Michael S., White, Martin, Ahrens, James, Ankeny, Lee, Armstrong, Ryan, O'Shea, Brian, Ricker, Paul M., Springel, Volker, Stadel, Joachim, and Trac, Hy, The Cosmic Code Comparison Project, 2007, ArXiv e-prints, 706, Cohn, J. D., Evrard, A. E., White, M., Croton, D., and Ellingson, E., Red Sequence Cluster Finding in the Millennium Simulation, 2007, ArXiv e-prints, 706, Cohn, J. D. and White, Martin, Dark matter halo abundances, clustering and assembly histories at high redshift, 2007, ArXiv e-prints, 706, Semboloni, Elisabetta, van Waerbeke, Ludovic, Heymans, Catherine, Hamana, Takashi, Colombi, Stephane, White, Martin, and Mellier, Yannick, Cosmic variance of weak lensing surveys in the non-Gaussian regime, 2007, Monthly Notices of the Royal Astronomical Society, 375, L10 Evrard, A. E., Bialek, J., Busha, M., White, M., Habib, S., Heitmann, K., Warren, M., Rasia, E., Tormen, G., Moscardini, L., Power, C., Jenkins, A. R., Gao, L., Frenk, C. S., Springel, V., White, S. D. M., and Diemand, J., Virial Scaling of Massive Dark Matter Halos: Why Clusters Prefer a High Normalization Cosmology, 2007, ArXiv Astrophysics e-prints, Wetzel, Andrew R., Cohn, J. D., White, Martin, Holz, Daniel E., and Warren, Michael S., The Clustering of Massive Halos, 2007, Astrophysical Journal, 656, 147 White, Martin, Zheng, Zheng, Brown, Michael J. I., Dey, Arjun, and Jannuzi, Buell T., Evidence for Merging or Disruption of Red Galaxies from the Evolution of Their Clustering, 2007, Astrophysical Journal, 655, L72 Huff, Eric, Schulz, A. E., White, Martin, Schlegel, David J., and Warren, Michael S., Simulations of baryon oscillations, 2007, Astroparticle Physics, 26, 366 PI John Wilkins From compact point defects to extended structures in silicon. Y. A. Du, R. G. Hennig, T. J. Lenosky and J. W. Wilkins. Eur. Phys. J. B 57 229 (2007). Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions. C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella and R. G. Hennig. Phys. Rev. Lett. 98, 110201 (2007). 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\begin{document} \title{A global stochastic maximum principle for fully coupled forward-backward stochastic systems } \author{Mingshang Hu\thanks{Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China. [email protected]. Research supported by NSF (No. 11671231) and Young Scholars Program of Shandong University (No. 2016WLJH10). } \and Shaolin Ji\thanks{Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China. [email protected] (Corresponding author). Research supported by NSF No. 11571203.} \and Xiaole Xue\thanks{Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China. Email: [email protected], [email protected]. Research supported by NSF (No. 11801315) and Natural Science Foundation of Shandong Province(ZR2018QA001).} } \maketitle \textbf{Abstract}. We study a stochastic optimal control problem for fully coupled forward-backward stochastic control systems with a nonempty control domain. For our problem, the first-order and second-order variational equations are fully coupled linear FBSDEs. Inspired by Hu \cite{Hu17}, we develop a new decoupling approach by introducing an adjoint equation which is a quadratic BSDE. By revealing the relations among the terms of the first-order Taylor's expansions, we estimate the orders of them and derive a global stochastic maximum principle which includes a completely new term. Applications to stochastic linear quadratic control problems are investigated. {\textbf{Key words}. } Backward stochastic differential equations, Nonconvex control domain, Stochastic recursive optimal control, Maximum principle, Spike variation. \textbf{AMS subject classifications.} 93E20, 60H10, 35K15 \addcontentsline{toc}{section}{\hspace{1.8em}Abstract} \section{Introduction} It is well known that deriving maximum principles, namely, necessary conditions for optimality, is an important approach in solving optimal control problems (see \cite{YongZhou} and the references therein). Boltyanski-Gamkrelidze-Pontryagin \cite{B-G-P56} announced the Pontryagin's maximum principle for the first time for deterministic control systems in 1956. They introduced the spike variation and studied the first-order term in a sort of Taylor's expansion with respect to this perturbation. But for stochastic control systems, if the diffusion terms depend on the controls, then one can't follow this idea for deterministic control systems. The reason is that the It\^{o} integral $ {\displaystyle\int\nolimits_{t}^{t+\varepsilon}} \sigma(s)dB(s)$ is only of order $\sqrt{\varepsilon}$ which leads to the first-order expansion method failed. To overcome this difficulty, Peng \cite{Peng90} first introduced the second-order term in the Taylor expansion of the variation and obtained the global maximum principle for the classical stochastic optimal control problem. Since then, many researchers investigate this kind of optimal control problems for various stochastic systems (see \cite{YingHu006,YingHu001,Tang003,Tang004,Zhou002}). Peng \cite{Peng93} generalized the classical stochastic optimal control problem to one where the cost functional is defined by $Y(0)$. Here $(Y(\cdot),Z(\cdot))$ is the solution of the following backward stochastic differential equation (BSDE) (\ref{intro--bsde0}): \begin{equation} \left\{ \begin{array} [c]{rl} -dY(t)= & f(t,X(t),Y(t),Z(t),u(t))dt-Z(t)dB(t),\\ Y(T)= & \phi(X(T)). \end{array} \right. \label{intro--bsde0} \end{equation} Since El Karoui et al. \cite{ElKaetal97} defined a more general class of stochastic recursive utilities in economic theory by solutions of BSDEs, this new kind of stochastic optimal control problem is called the stochastic recursive optimal control problem. When the control domain is convex, one can avoid spike variation method and deduce a so-called local stochastic maximum principle. Peng \cite{Peng93} first established a local stochastic maximum principle for the classical stochastic recursive optimal control problem. The local stochastic maximum principles for other various problems were studied in (Dokuchaev and Zhou \cite{Dokuchaev-Zhou}, Ji and Zhou \cite{Ji-Zhou}, Peng \cite{Peng93}, Shi and Wu \cite{Shi-Wu}, Xu \cite{Xu95}, Zhou \cite{Zhou003}, see also the references therein). But when the control domain is nonconvex, one encounters an essential difficulty when trying to derive the first-order and second-order expansions for the BSDE (\ref{intro--bsde0}) and it is proposed as an open problem in Peng \cite{Peng99}. Recently, Hu \cite{Hu17} studied this open problem and obtained a completely novel global maximum principle. In \cite{Hu17}, Hu found that there are closely relations among the terms of the first-order Taylor's expansions, i.e., \begin{equation} \begin{array} [c]{l} Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) ,\\ Z_{1}(t)=p(t)\delta\sigma(t)I_{E_{\epsilon}}(t)+[\sigma_{x}(t)p(t)+q(t)]X_{1} \left( t\right) , \end{array} \label{intro-relation-hu} \end{equation} where $(p\left( \cdot\right) ,q(\cdot))$ is the solution of the adjoint equation. And the BSDE satisfied by $(p\left( \cdot\right) ,q(\cdot))$ possesses a linear generator. Notice that the variation of $Z(t)$ includes the term $\langle p(t),\delta\sigma(t)\rangle I_{E_{\varepsilon}}(t)$. Hu \cite{Hu17} proposed to do Taylor's expansions at $\bar{Z}(t)+p(t)\delta \sigma(t)I_{E_{\epsilon}}(t)$ and deduced the maximum principle. Motivated by the leader-follower stochastic differential games and other problems in mathematical finance, Yong \cite{Yong10} studied a fully coupled controlled FBSDE with mixed initial-terminal conditions. In \cite{Yong10}, Yong regarded $Z(\cdot)$ as a control process and then applied the Ekeland variational principle to obtain an optimality variational principle which contains unknown parameters. Note that using the similar approach, Wu \cite{Wu13} studied a stochastic recursive optimal control problem. In this paper, we study the following stochastic optimal control problem: minimize the cost functional \[ J(u(\cdot))=Y(0) \] subject to the following fully coupled forward-backward stochastic differential equation (FBSDE) (see \cite{YingHu002, Ma-WZZ, Ma-Yong-FBSDE, Ma-ZZ, Zhang17} and the references therein): \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),Y(t),Z(t),u(t))dt+\sigma(t,X(t),Y(t),Z(t),u(t))dB(t),\\ dY(t)= & -g(t,X(t),Y(t),Z(t),u(t))dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\phi(X(T)), \end{array} \right. \label{intro--fbsde} \end{equation} where the control variable $u(\cdot)$ takes values in a nonempty subset of $\mathbb{R}^{k}$. In fact, our model is a special one in Yong \cite{Yong10}. But our object is to get rid of the unknown parameters in the optimality variational principle in \cite{Yong10,Wu13} and obtain a global stochastic maximum principle for the above fully coupled control system. In order to do this, we should study the variational equations of the BSDE in (\ref{intro--fbsde}). But as pointed out in \cite{Yong10}, the regularity/integrability of process $Z(\cdot)$ seems to be not enough in the case when a second order expansion is necessary. Fortunately, inspired by Hu \cite{Hu17}, we overcome this difficulty based on the following two findings. The first one is although the first-order and second-order variational equations are fully coupled linear FBSDEs, we can decouple them by establishing the relations among the first-order Taylor's expansions, i.e., \begin{equation} \begin{array} [c]{l} Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) ,\\ Z_{1}(t)=\Delta(t)I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}\left( t\right) , \end{array} \label{intro--relation} \end{equation} where $\Delta(t)$ satisfies the following algebra equation \begin{equation} \Delta(t)=p(t)(\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta (t),u(t))-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))) \label{intro--algebra} \end{equation} and $(p\left( \cdot\right) ,q(\cdot))$ is the adjoint process which satisfies a quadratic BSDE. By the results of Lepeltier and San Martin \cite{LS02}, we obtain the existence of solution to this nonlinear adjoint equation. Utilizing the uniqueness result of the linear fully coupled FBSDE in the appendix, we also prove the uniqueness of solution to this adjoint equation. The second finding is that the first-order variation $Z_{1}(t)$ has a unique decomposition by the relations (\ref{intro--relation}). This point inspires us that we should do Taylor's expansions at $\bar{Z}(t)+\Delta (t)I_{E_{\epsilon}}(t)$. The advantage of this approach is that the reminder term of Taylor's expansions $K_{1}(t)X_{1}\left( t\right) $ has good estimate which avoids the difficulty to do estimates such as $E[ {\displaystyle\int\nolimits_{0}^{T}} \mid Z(t)\mid^{2+\varepsilon}dt]<\infty$, for some $\varepsilon>0$. For this reason, the obtained maximum principle will include a new term $\Delta(t)$ which is determined uniquely by $u(t)$, $\bar{u}(t)$, and the optimal state $(\bar{X}(t)$, $\bar{Y}(t)$, $\bar{Z}(t))$. The readers may refer to subsection \ref{heuristic} for a heuristic derivation. By assuming $q(\cdot)$ is a bounded process, we derive the first-order and second-order variational equations and deduce a global maximum principle which includes a new term $\Delta(t)$. Furthermore, we study the case in which $q(\cdot)$ may be unbounded. But for this case, we only obtain the maximum principle when $\sigma(t,x,y,z,u)$ is linear in $z$, i.e., \[ \sigma(t,x,y,z,u)=A(t)z+\sigma_{1}(t,x,y,u). \] Finally, applications to stochastic linear quadratic control problems are investigated. The rest of the paper is organized as follows. In section 2, we give the preliminaries and formulation of our problem. A global stochastic maximum principle is obtained by spike variation method in section 3. Especially, to illustrate our main approach, we give a heuristic derivation in subsection \ref{heuristic} before we prove the maximum principle strictly. In section 4, a linear quadratic control problem is investigated based on the obtained estimates in section 3. In appendix, we give some results that will be used in our proofs. \section{ Preliminaries and problem formulation} Let $(\Omega,\mathcal{F},P)$ be a complete probability space on which a standard $d$-dimensional Brownian motion $B=(B_{1}(t),B_{2}(t),...B_{d} (t))_{0\leq t\leq T}^{\intercal}$ is defined. Assume that $\mathbb{F=} \{\mathcal{F}_{t},0\leq t\leq T\}$ is the $P$-augmentation of the natural filtration of $B$, where $\mathcal{F}_{0}$ contains all $P$-null sets of $\mathcal{F}$. Denote by $\mathbb{R}^{n}$ the $n$-dimensional real Euclidean space and $\mathbb{R}^{k\times n}$ the set of $k\times n$ real matrices. Let $\langle\cdot,\cdot\rangle$ (resp. $\left\vert \cdot\right\vert $) denote the usual scalar product (resp. usual norm) of $\mathbb{R}^{n}$ and $\mathbb{R} ^{k\times n}$. The scalar product (resp. norm) of $M=(m_{ij})$, $N=(n_{ij} )\in\mathbb{R}^{k\times n}$ is denoted by $\langle M,N\rangle =tr\{MN^{\intercal}\}$ (resp.$\Vert M\Vert=\sqrt{MM^{\intercal}}$), where the superscript $^{\intercal}$ denotes the transpose of vectors or matrices. We introduce the following spaces. $L_{\mathcal{F}_{T}}^{p}(\Omega;\mathbb{R}^{n})$ : the space of $\mathcal{F} _{T}$-measurable $\mathbb{R}^{n}$-valued random variables $\eta$ such that \[ \|\|\eta\|\|_{p}:=(\mathbb{E}[\|\eta\|^{p}])^{\frac{1}{p}}<\infty, \] $L_{\mathcal{F}_{T}}^{\infty}(\Omega;\mathbb{R}^{n})$: the space of $\mathcal{F}_{T}$-measurable $\mathbb{R}^{n}$-valued random variables $\eta$ such that $\|\|\eta\|\|_{\infty}:=\underset{\omega\in\Omega}{\mathrm{ess~sup} }\left\Vert \eta\right\Vert <\infty$, $L_{\mathcal{F}}^{p}([0,T];\mathbb{R}^{n})$: the space of $\mathbb{F}$-adapted and $p$-th integrable stochastic processes on $[0,T]$ such that \[ \mathbb{E}\left[ \int_{0}^{T}\left\vert f(t)\right\vert ^{p}dt\right] <\infty, \] $L_{\mathcal{F}}^{\infty}(0,T;\mathbb{R}^{n})$: the space of $\mathbb{F} $-adapted and uniformly bounded stochastic processes on $[0,T]$ such that \[ \|\|f(\cdot)\|\|_{\infty}=\underset{(t,\omega)\in\lbrack0,T]\times\Omega }{\mathrm{ess~sup}}\|f(t)\|<\infty, \] $L_{\mathcal{F}}^{p,q}([0,T];\mathbb{R}^{n})$: the space of $\mathbb{F} $-adapted stochastic processes on $[0,T]$ such that \[ \|\|f(\cdot)\|\|_{p,q}=\left\{ \mathbb{E}\left[ \left( \int_{0}^{T} \|f(t)\|^{p}dt\right) ^{\frac{q}{p}}\right] \right\} ^{\frac{1}{q}}<\infty, \] $L_{\mathcal{F}}^{p}(\Omega;C([0,T],\mathbb{R}^{n}))$: the space of $\mathbb{F}$-adapted continuous stochastic processes on $[0,T]$ such that \[ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\left\vert f(t)\right\vert ^{p}\right] <\infty. \] \subsection{$L^{p}$ estimate for fully coupled FBSDEs} We first give an $L^{p}$-estimate for the following fully coupled forward-backward stochastic differential equation: \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),Y(t),Z(t))dt+\sigma(t,X(t),Y(t),Z(t))dB(t),\\ dY(t)= & -g(t,X(t),Y(t),Z(t))dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\phi(X(T)), \end{array} \right. \label{fbsde} \end{equation} where \[ b:\Omega\times\lbrack0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m} \times\mathbb{R}^{m\times d}\rightarrow\mathbb{R}^{n}, \] \[ \sigma:\Omega\times\lbrack0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m} \times\mathbb{R}^{m\times d}\rightarrow\mathbb{R}^{n\times d}, \] \[ g:\Omega\times\lbrack0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{m} \times\mathbb{R}^{m\times d}\rightarrow\mathbb{R}^{m}, \] \[ \phi:\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}. \] A solution to (\ref{fbsde}) is a triplet of $\mathbb{F}$-adapted process $\Theta(\cdot):=(X(\cdot),Y(\cdot),Z(\cdot))$. We impose the following assumption. \begin{assumption} \label{assum-1}(i) $\psi=b,\sigma,g,\phi$ are uniformly Lipschitz continuous with respect to $x,y,z$, that is, there exist constants $L_{i}>0$, $i=1,2,3$ such that \[ \begin{array} [c]{rl} \|b(t,x_{1},y_{1},z_{1})-b(t,x_{2},y_{2},z_{2})\| & \leq L_{1}\|x_{1} -x_{2}\|+L_{2}(\|y_{1}-y_{2}\|+\|z_{1}-z_{2}\|),\\ \|\sigma(t,x_{1},y_{1},z_{1})-\sigma(t,x_{2},y_{2},z_{2})\| & \leq L_{1} \|x_{1}-x_{2}\|+L_{2}\|y_{1}-y_{2}\|+L_{3}\|z_{1}-z_{2}\|,\\ \|g(t,x_{1},y_{1},z_{1})-g(t,x_{2},y_{2},z_{2}) & \leq L_{1}(\|x_{1} -x_{2}\|+\|y_{1}-y_{2}\|+\|z_{1}-z_{2}\|),\\ \|\phi(t,x_{1})-\phi(t,x_{2})\| & \leq L_{1}\|x_{1}-x_{2}\|, \end{array} \] for all $t,\omega,x_{i},y_{i},z_{i}$, $i=1,2$. \newline(ii) For a given $p>1$, $\phi(0)\in L_{\mathcal{F}_{T}}^{p}(\Omega;\mathbb{R}^{m})$, $b(\cdot ,0,0,0)\in L_{\mathcal{F}}^{1,p}([0,T];\mathbb{R}^{n})$, $g(\cdot,0,0,0)\in L_{\mathcal{F}}^{1,p}([0,T];\mathbb{R}^{m})$, $\sigma(\cdot,0,0,0)\in L_{\mathcal{F}}^{2,p}([0,T];\mathbb{R}^{n\times d})$. \end{assumption} For $p>1$, set \begin{equation} \Lambda_{p}:=C_{p}2^{p+1}(1+T^{p})c_{1}^{p}, \label{def-Lambda} \end{equation} where $c_{1}=\max\{L_{2},L_{3}\},$ $C_{p}$ is defined in Lemma \ref{sde-bsde} in appendix. \begin{theorem} Suppose Assumption \ref{assum-1} holds and\ $\Lambda_{p}<1$ for some $p>1$. $\ $Then \eqref{fbsde} admits a unique solution $(X\left( \cdot\right) ,Y\left( \cdot\right) ,Z\left( \cdot\right) )\in L_{\mathcal{F}} ^{p}(\Omega;C([0,T],\mathbb{R}^{n}))\times L_{\mathcal{F}}^{p}(\Omega ;C([0,T],\mathbb{R}^{m}))\times L_{\mathcal{F}}^{2,p}([0,T];\mathbb{R} ^{m\times d})$ and \[ \begin{array} [c]{l} \|\|(X,Y,Z)\|\|_{p}^{p}=\mathbb{E}\left\{ \sup\limits_{t\in\lbrack0,T]}\left[ \|X(t)\|^{p}+\|Y(t)\|^{p}\right] +\left( \int_{0}^{T}\|Z(t)\|^{2}dt\right) ^{\frac{p}{2}}\right\} \\ \ \leq C\mathbb{E}\left\{ \left( \int_{0}^{T}[\|b\|+\|g\|](t,0,0,0)dt\right) ^{p}+\left( \int_{0}^{T}\|\sigma(t,0,0,0)\|^{2}dt\right) ^{\frac{p}{2}} +\|\phi(0)\|^{p}+\|x_{0}\|^{p}\right\} , \end{array} \] where $C$ depends on $T$, $p$, $L_{1}$, $c_{1}$. \label{est-fbsde-lp} \end{theorem} \begin{proof} Without loss of generality, we only prove the case $n=m=d=1$. Let $\mathcal{L}$ denote the space of all $\mathbb{F}$-adapted processes $(Y(\cdot),Z(\cdot))$ such that \[ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|Y(t)\|^{p}+\left( \int_{0} ^{T}\|Z(t)\|^{2}dt\right) ^{\frac{p}{2}}\right] <\infty. \] For each given $(y,z)\in\mathcal{L}$, consider the following FBSDE: \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),y(t),z(t))dt+\sigma(t,X(t),y(t),z(t))dB(t),\\ dY(t)= & -g(t,X(t),Y(t),Z(t))dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\phi(X(T)). \end{array} \right. \label{fbsde-y0} \end{equation} Under Assumption \ref{assum-1}, it is easy to deduce that the solution $(Y(\cdot),Z(\cdot))$ of \eqref{fbsde-y0} belongs to $\mathcal{L}$. Denote the operator $(y(\cdot),z(\cdot))\rightarrow(Y(\cdot),Z(\cdot))$ by $\Gamma$. For two elements $(y^{i},z^{i})\in\mathcal{L}$, $i=1,2$, let $(X^{i}(\cdot ),Y^{i}(\cdot),Z^{i}(\cdot))$ be the corresponding solution to \eqref{fbsde-y0}. Set \[ \Delta y=y^{1}-y^{2},\text{ }\Delta z=z^{1}-z^{2},\text{ }\Delta X=X^{1} -X^{2},\text{ }\Delta Y=Y^{1}-Y^{2},\text{ }\Delta Z=Z^{1}-Z^{2}. \] Then \begin{equation} \left\{ \begin{array} [c]{rl} d\Delta X(t)= & \left[ \alpha_{1}(t)\Delta X(t)+\beta_{1}(t)\Delta y(t)+\gamma_{1}(t)\Delta z(t)\right] dt+\left[ \alpha_{2}(t)\Delta X(t)+\beta_{2}(t)\Delta y(t)+\gamma_{2}(t)\Delta z(t)\right] dB(t),\\ d\Delta Y(t)= & -\left[ \alpha_{3}(t)\Delta X(t)+\beta_{3}(t)\Delta Y(t)+\gamma_{3}(t)\Delta Z(t)\right] dt+\Delta Z(t)dB(t),\\ \Delta X(0)= & 0,\ \Delta Y(T)=\lambda(T)\Delta X(T), \end{array} \right. \end{equation} where \[ \alpha_{1}(t)=\left\{ \begin{array} [c]{ll} \frac{b(t,X^{1}(t),y^{1}(t),z^{1}(t))-b(t,X^{2}(t),y^{1}(t),z^{1}(t))}{\Delta X(t)},\ & \text{if}\ \Delta X(t)\neq0,\\ 0, & \text{if}\ \Delta X(t)=0, \end{array} \right. \] and $\alpha_{i}(t)$, $\beta_{i}(t)$, $\gamma_{i}(t)$, $\lambda(T)$ are defined similarly. Furthermore, $\alpha_{i}(t)$, $\beta_{i}(t)$, $\gamma_{i}(t)$, $\lambda(T)$ are bounded by Lipschitz constants of the corresponding coefficients. Especially, $\|\beta_{1}(t)\|,\|\gamma_{1}(t)\|,\|\beta _{2}(t)\|,\|\gamma_{2}(t)\|\leq c_{1}$. Due to Lemma \ref{sde-bsde}, we obtain \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\left( \|\Delta X(t)\|^{p}+\|\Delta Y(t)\|^{p}\right) +\left( \int_{0}^{T}\|\Delta Z(t)\|^{2}dt\right) ^{\frac {p}{2}}\right] \\ \leq C_{p}\mathbb{E}\left\{ \left[ \int_{0}^{T}(\|\beta_{1}(t)\|\|\Delta y(t)\|+\|\gamma_{1}(t)\|\|\Delta z(t)\|)dt\right] ^{p}+\left[ \int_{0}^{T}\left( \|\beta_{2}(t)\|^{2}\|\Delta y(t)\|^{2}+\|\gamma_{2}(t)\|^{2}\|\Delta z(t)\|^{2} \right) dt\right] ^{\frac{p}{2}}\right\} \\ \leq C_{p}2^{p+1}\left( 1+T^{p}\right) c_{1}^{p}\mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|\Delta y(t)\|^{p}+\left( \int_{0}^{T}\|\Delta z(t)\|^{2}dt\right) ^{\frac{p}{2}}\right] \\ =\Lambda_{p}\mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|\Delta y(t)\|^{p}+\left( \int_{0}^{T}\|\Delta z(t)\|^{2}dt\right) ^{\frac{p}{2} }\right] . \end{array} \label{fbsde-delty} \end{equation} Since $\Lambda_{p}<1$, the operator $\Gamma$ is a contraction mapping and has a unique fixed point $(Y(\cdot),Z(\cdot))$.\ Let $X(\cdot)$ be the solution of \eqref{fbsde} with respect to the fixed point $(Y(\cdot),Z(\cdot))$. Thus, $(X(\cdot),Y(\cdot),Z(\cdot))$ is the unique solution to \eqref{fbsde}. Let $\Theta^{0}:=(X^{0}(\cdot),Y^{0}(\cdot),Z^{0}(\cdot))$ be the solution to \eqref{fbsde-y0} with $y=0$, $z=0$. From \eqref{fbsde-delty}, \[ \|\|(Y-Y^{0},Z-Z^{0})\|\|\leq\Lambda_{p}^{\frac{1}{p}}\|\|(Y-0,Z-0)\|\|=\Lambda _{p}^{\frac{1}{p}}\|\|(Y,Z)\|\|. \] By triangle inequality, \[ \begin{array} [c]{rl} \|\|(Y,Z)\|\|\leq & \|\|(Y-Y^{0},Z-Z^{0})\|\|+\|\|(Y^{0},Z^{0})\|\|\\ \leq & \Lambda_{p}^{\frac{1}{p}}\|\|(Y,Z)\|\|+\|\|(Y^{0},Z^{0})\|\|, \end{array} \] which leads to \[ \|\|(Y,Z)\|\|\leq\left( 1-\Lambda_{p}^{\frac{1}{p}}\right) ^{-1}\|\|(Y^{0} ,Z^{0})\|\|. \] By Lemma \ref{sde-bsde} in appendix, we obtain \[ \begin{array} [c]{rl} \|\|(Y^{0},Z^{0})\|\|^{p}\leq & C_{p}\mathbb{E}\left[ \|\phi(0)\|^{p}+\|x_{0} \|^{p}+\left( \int_{0}^{T}[\|b\|+\|g\|](t,0,0,0)dt\right) ^{p}+\left( \int _{0}^{T}\|\sigma(t,0,0,0)\|^{2}dt\right) ^{\frac{p}{2}}\right] , \end{array} \] where $C_{p}\ $depends on $T$, $p$, $L_{1}$. Thus we have \[ \|\|(Y,Z)\|\|_{p}^{p}\leq C^{\prime}\mathbb{E}\left[ \|\phi(0)\|^{p}+\|x_{0} \|^{p}+\left( \int_{0}^{T}[\|b\|+\|g\|](t,0,0,0)dt\right) ^{p}+\left( \int _{0}^{T}\|\sigma(t,0,0,0)\|^{2}dt\right) ^{\frac{p}{2}}\right] , \] where$\ C^{\prime}=C_{p}(1-\Lambda_{p}^{\frac{1}{p}})^{-p}$. By Lemma \ref{sde-bsde}, we can obtain the desired result. \end{proof} \begin{remark} In the case $p=2$, Pardoux and Tang obtained the $L^{2}$-estimate in \cite{Pardoux-Tang} (see also \cite{Cvi-Zhang}). Instead of assuming that $L_{2}$ and $L_{3}$ are small enough as in \cite{Cvi-Zhang}, we assume $\Lambda_{p}<1$ in this paper. There are other conditions in \cite{Cvi-Zhang} which can guarantee the existence and uniqueness of \eqref{fbsde}. The readers may apply the method introduced in the above theorem to obtain the $L^{p} $-estimate of \eqref{fbsde} for these conditions similarly. \end{remark} \subsection{Problem formulation} Consider the following fully coupled stochastic control system: \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),Y(t),Z(t),u(t))dt+\sigma(t,X(t),Y(t),Z(t),u(t))dB(t),\\ dY(t)= & -g(t,X(t),Y(t),Z(t),u(t))dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\phi(X(T)), \end{array} \right. \label{state-eq} \end{equation} where \[ b:[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times U\rightarrow\mathbb{R}, \] \[ \sigma:[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times U\rightarrow\mathbb{R}^{d}, \] \[ g:[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times U\rightarrow\mathbb{R}, \] \[ \phi:\mathbb{R}\rightarrow\mathbb{R}. \] An admissible control $u(\cdot)$ is an $\mathbb{F}$-adapted process with values in $U$ such that \[ \sup\limits_{0\leq t\leq T}\mathbb{E}[\|u(t)\|^{8}]<\infty, \] where the control domain $U$ is a nonempty subset of $\mathbb{R}^{k}$. Denote the admissible control set by $\mathcal{U}[0,T]$. Our optimal control problem is to minimize the cost functional \[ J(u(\cdot))=Y(0) \] over $\mathcal{U}[0,T]$: \begin{equation} \underset{u(\cdot)\in\mathcal{U}[0,T]}{\inf}J(u(\cdot)). \label{obje-eq} \end{equation} \section{Stochastic maximum principle} We derive maximum principle (necessary condition for optimality) for the optimization problem (\ref{obje-eq}) in this section. For simplicity of presentation, we only study the case $d=1$, and then present the results for the general case in subsection \ref{sec-general}. In this section, the constant $C$ will change from line to line in our proof. We impose the following assumptions on the coefficients of \eqref{state-eq}. \begin{assumption} For $\psi=b,$ $\sigma,$ $g$ and $\phi$, we suppose (i) $\psi$, $\psi_{x}$, $\psi_{y}$, $\psi_{z}$ are continuous in $(x,y,z,u)$; $\psi_{x}$, $\psi_{y}$, $\psi_{z}$ are bounded; there exists a constant $\ L>0$ such that \[ \begin{array} [c]{rl} \|\psi(t,x,y,z,u)\| & \leq L\left( 1+\|x\|+\|y\|+\|z\|+\|u\|\right) ,\\ \|\sigma(t,0,0,z,u)-\sigma(t,0,0,z,u^{\prime})\| & \leq L(1+\|u\|+\|u^{\prime}\|). \end{array} \] (ii) For any $2\leq\beta\leq8$,\ $\Lambda_{\beta}:=C_{\beta}2^{\beta +1}(1+T^{\beta})c_{1}^{\beta}<1$, where $c_{1}=\max\{L_{2},L_{3}\}$, $L_{2}=\max\{\|\|b_{y}\|\|_{\infty},\|\|b_{z}\|\|_{\infty},\|\|\sigma_{y}\|\|_{\infty}\}$, $L_{3}=\|\|\sigma_{z}\|\|_{\infty}$, $C_{\beta}$ is defined in Lemma \ref{sde-bsde} in appendix for $L_{1}=\max\{\|\|b_{x}\|\|_{\infty},\|\|\sigma _{x}\|\|_{\infty},\|\|g_{x}\|\|_{\infty},\|\|g_{y}\|\|_{\infty},\|\|g_{z}\|\|_{\infty },\|\|\phi_{x}\|\|_{\infty}\}$. (iii) $\psi_{xx}$, $\psi_{xy}$, $\psi_{yy}$ , $\psi_{xz}$, $\psi_{yz}$, $\psi_{zz}$ are continuous in $(x,y,z,u)$; $\psi_{xx}$, $\psi_{xy}$, $\psi_{yy}$, $\psi_{xz}$, $\psi_{yz}$ ,$\psi_{zz}$ are bounded. \label{assum-2} \end{assumption} Under Assumption \ref{assum-2}(i)-(ii), for any $u(\cdot)\in\mathcal{U}[0,T]$, the state equation \eqref{state-eq} has a unique solution by Theorem \ref{est-fbsde-lp}. Let $\bar{u}(\cdot)$ be optimal and $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar {Z}(\cdot))$ be the corresponding state processes of (\ref{state-eq}). Since the control domain is not necessarily convex, we resort to spike variation method. For any $u(\cdot)\in\mathcal{U}[0,T]$ and $0<\epsilon<T$, define \[ u^{\epsilon}(t)=\left\{ \begin{array} [c]{lll} \bar{u}(t), & \ t\in\lbrack0,T]\backslash E_{\epsilon}, & \\ u(t), & \ t\in E_{\epsilon}, & \end{array} \right. \] where $E_{\epsilon}\subset\lbrack0,T]$ is\ a measurable set with $\|E_{\epsilon}\|=\epsilon$. Let $(X^{\epsilon}(\cdot),Y^{\epsilon} (\cdot),Z^{\epsilon}(\cdot))$ be the state processes of (\ref{state-eq}) associated with $u^{\epsilon}(\cdot)$. For simplicity, for $\psi=b$, $\sigma$, $g$, $\phi$ and $\kappa=x$, $y$, $z$, denote \[ \begin{array} [c]{rl} \psi(t)= & \psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),\\ \psi_{\kappa}(t)= & \psi_{\kappa}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar {u}(t)),\\ \delta\psi(t)= & \psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u(t))-\psi(t),\\ \delta\psi_{\kappa}(t)= & \psi_{\kappa}(t,\bar{X}(t),\bar{Y}(t),\bar {Z}(t),u(t))-\psi_{\kappa}(t),\\ \delta\psi(t,\Delta)= & \psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta (t),u(t))-\psi(t),\\ \delta\psi_{\kappa}(t,\Delta)= & \psi_{\kappa}(t,\bar{X}(t),\bar{Y}(t),\bar {Z}(t)+\Delta(t),u(t))-\psi_{\kappa}(t), \end{array} \] where $\Delta(\cdot)$ is an $\mathbb{F}$--adapted process. Moreover, denote $D\psi$ is the gradient of $\psi$ with respect to $x$, $y$, $z$, and $D^{2}\psi$ is the Hessian matrix of $\psi$ with respect to $x$, $y$, $z$, \begin{align} D\psi(t) & =D\psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)),\\ D^{2}\psi(t) & =D^{2}\psi(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t)). \end{align} \begin{lemma} \label{est-epsilon-bar}Suppose Assumption \ref{assum-2}(i)-(ii) hold. Then for any $2\leq\beta\leq8$ we have \begin{equation} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X^{\epsilon} (t)-\bar{X}(t)\|^{\beta}+\|Y^{\epsilon}(t)-\bar{Y}(t)\|^{\beta}\right) \right] +\mathbb{E}\left[ \left( \int_{0}^{T}\|Z^{\epsilon}(t)-\bar{Z}(t)\|^{2} dt\right) ^{\frac{\beta}{2}}\right] =O\left( \epsilon^{\frac{\beta}{2} }\right) . \end{equation} \end{lemma} \begin{proof} Let \[ \begin{array} [c]{rl} \xi^{1,\epsilon}(t) & :=X^{\epsilon}(t)-\bar{X}(t);\\ \eta^{1,\epsilon}(t) & :=Y^{\epsilon}(t)-\bar{Y}(t);\\ \zeta^{1,\epsilon}(t) & :=Z^{\epsilon}(t)-\bar{Z}(t);\\ \Theta(t) & :=(\bar{X}(t),\bar{Y}(t),\bar{Z}(t));\\ \Theta^{\epsilon}(t) & :=(X^{\epsilon}(t),Y^{\epsilon}(t),Z^{\epsilon}(t)). \end{array} \] We have \begin{equation} \left\{ \begin{array} [c]{rl} d\xi^{1,\epsilon}(t)= & \left[ \tilde{b}_{x}^{\epsilon}(t)\xi^{1,\epsilon }(t)+\tilde{b}_{y}^{\epsilon}(t)\eta^{1,\epsilon}(t)+\tilde{b}_{z}^{\epsilon }(t)\zeta^{1,\epsilon}(t)+\delta b(t)I_{E_{\epsilon}}(t)\right] dt\\ & +\left[ \tilde{\sigma}_{x}^{\epsilon}(t)\xi^{1,\epsilon}(t)+\tilde{\sigma }_{y}^{\epsilon}(t)\eta^{1,\epsilon}(t)+\tilde{\sigma}_{z}^{\epsilon} (t)\zeta^{1,\epsilon}(t)+\delta\sigma(t)I_{E_{\epsilon}}(t)\right] dB(t),\\ \xi^{1,\epsilon}(0)= & 0, \end{array} \right. \label{ep-bar-x} \end{equation} \begin{equation} \left\{ \begin{array} [c]{rl} d\eta^{1,\epsilon}(t)= & -\left[ \tilde{g}_{x}^{\epsilon}(t)\xi^{1,\epsilon }(t)+\tilde{g}_{y}^{\epsilon}(t)\eta^{1,\epsilon}(t)+\tilde{g}_{z}^{\epsilon }(t)\zeta^{1,\epsilon}(t)+\delta g(t)I_{E_{\epsilon}}(t)\right] dt+\zeta^{1,\epsilon}(t)dB(t),\\ \eta^{1,\epsilon}(T)= & \tilde{\phi}_{x}^{\epsilon}(T)\xi^{1,\epsilon}(T), \end{array} \right. \label{ep-bar-y} \end{equation} where \[ \tilde{b}_{x}^{\epsilon}(t)=\int_{0}^{1}b_{x}(t,\Theta(t)+\theta (\Theta^{\epsilon}(t)-\Theta(t)),u^{\epsilon}(t))d\theta \] and $\tilde{b}_{y}^{\epsilon}(t)$, $\tilde{b}_{z}^{\epsilon}(t)$, $\tilde{\sigma}_{x}^{\epsilon}(t)$, $\tilde{\sigma}_{y}^{\epsilon}(t)$, $\tilde{\sigma}_{z}^{\epsilon}(t),$ $\tilde{g}_{x}^{\epsilon}(t)$, $\tilde {g}_{y}^{\epsilon}(t)$, $\tilde{g}_{z}^{\epsilon}(t)$ and $\tilde{\phi} _{x}^{\epsilon}(T)$ are defined similarly. Noting that $\left( \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)\right) $ is the solution to \eqref{ep-bar-x} and \eqref{ep-bar-y}, and \[ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|u(t)\|dt\right) ^{\beta}\right] \leq\epsilon^{\beta-1}\mathbb{E}\left[ \int_{E_{\epsilon}}\|u(t)\|^{\beta }dt\right] , \] then, by Theorem \ref{est-fbsde-lp}, we get \[ \begin{array} [c]{ll} & \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{1,\epsilon }(t)\|^{\beta}+\|\eta^{1,\epsilon}(t)\|^{\beta}\right) +\left( \int_{0} ^{T}\|\zeta^{1,\epsilon}(t)\|^{2}dt\right) ^{\frac{\beta}{2}}\right] \\ & \ \ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\left( \|\delta b(t)\|I_{E_{\epsilon}}(t)+\|\delta g(t)\|I_{E_{\epsilon}}(t)\right) dt\right) ^{\beta}+\left( \int_{0}^{T}\|\delta\sigma(t)\|^{2}I_{E_{\epsilon} }(t)dt\right) ^{\frac{\beta}{2}}\right] \\ & \ \ \leq C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}(1+\|\bar{X} (t)\|+\|\bar{Y}(t)\|+\|\bar{Z}(t)\|+\|u(t)\|+\|\bar{u}(t)\|)dt\right) ^{\beta}\right. \\ & \text{ \ \ \ \ \ \ \ }\left. +\left( \int_{E_{\epsilon}}(1+\|\bar {X}(t)\|^{2}+\|\bar{Y}(t)\|^{2}+\|u(t)\|^{2}+\|\bar{u}(t)\|^{2})dt\right) ^{\frac{\beta}{2}}\right] \\ & \ \ \leq C\left( \epsilon^{\beta}+\epsilon^{\frac{\beta}{2}}\right) \left( 1+\sup\limits_{t\in\lbrack0,T]}\mathbb{E}\left[ \|\bar{X}(t)\|^{\beta }+\|\bar{Y}(t)\|^{\beta}+\|u(t)\|^{\beta}+\|\bar{u}(t)\|^{\beta}\right] \right) +C\epsilon^{\frac{\beta}{2}}\mathbb{E}\left[ \left( \int_{0}^{T}\|\bar {Z}(t)\|^{2}dt\right) ^{\frac{\beta}{2}}\right] \\ & \ \ \leq C\epsilon^{\frac{\beta}{2}}. \end{array} \] \end{proof} \subsection{A heuristic derivation\label{heuristic}} Before giving the strict proof of the stochastic maximum principle, we illustrate how to obtain our results formally in this subsection. By Lemma \ref{est-epsilon-bar}, we have $X^{\epsilon}(t)-\bar{X}(t)\sim O(\sqrt{\epsilon})$, $Y^{\epsilon}(t)-\bar{Y}(t)\sim O(\sqrt{\epsilon})$ and $Z^{\epsilon}(t)-\bar{Z}(t)\sim O(\sqrt{\epsilon})$. Suppose that \begin{equation} \begin{array} [c]{lll} X^{\epsilon}(t)-\bar{X}(t) & = & X_{1}(t)+X_{2}(t)+o(\epsilon),\\ Y^{\epsilon}(t)-\bar{Y}(t) & = & Y_{1}(t)+Y_{2}(t)+o(\epsilon),\\ Z^{\epsilon}(t)-\bar{Z}(t) & = & Z_{1}(t)+Z_{2}(t)+o(\epsilon), \end{array} \label{heur-1} \end{equation} where $X_{1}(t)\sim O(\sqrt{\epsilon})$, $X_{2}(t)\sim O(\epsilon)$, $Y_{1}(t)\sim O(\sqrt{\epsilon})$, $Y_{2}(t)\sim O(\epsilon)$, $Z_{1}(t)\sim O(\sqrt{\epsilon})$ and $Z_{2}(t)\sim O(\epsilon)$. It is well-known that the solution $Z$ of the FBSDE (\ref{state-eq}) is closely related to the diffusion term $\sigma$ of the forward SDE of (\ref{state-eq}). When we adopt the spike variation method and calculate the variational equation of $X$, the diffusion term of the variational equation should include the term $\delta\sigma(t)I_{E_{\epsilon}}(t)$. So we guess that $Z_{1}(t)$ has the following form \begin{equation} Z_{1}(t)=\Delta(t)I_{E_{\epsilon}}(t)+Z_{1}^{\prime}(t). \label{heur-2} \end{equation} where $\Delta(t)$ is an $\mathbb{F}$--adapted process and $Z_{1}^{\prime}(t)$ has good estimates similarly as $X_{1}(t)$. But this form of $Z_{1}(t)$ leads to great difficulties when we do Taylor's expansion of the coefficients $b,$ $\sigma$ and $g$ with respect to $Z$. Fortunately, we find that $\Delta(t)$ can be determined uniquely by $u(t)$, $\bar{u}(t)$, and the optimal state $(\bar{X}(t)$, $\bar{Y}(t)$, $\bar{Z}(t))$. Note that in Hu \cite{Hu17}, \begin{equation} \Delta(t)=p(t)\left( \sigma(t,\bar{X}(t),u(t))-\sigma(t,\bar{X}(t),\bar {u}(t))\right) \label{heur-hu} \end{equation} where $p(t)$ is the adjoint process. Although $\Delta(t)$ appears in the expansion of $Z^{\epsilon}(t)-\bar{Z}(t)$, by (\ref{heur-hu}) it is clearly that $\Delta(t)$ includes the spike variation of control variables. In our context, we will see lately that $\Delta(t)$ is determined by an algebra equation (\ref{heur-7}). Thus, when we derive the variational equations, we should keep the $\Delta(t)I_{E_{\epsilon}}(t)$ term unchanged and do Taylor's expansions at $\bar{Z}(t)+\Delta(t)I_{E_{\epsilon}}(t)$. This idea is first applied to a partially coupled FBSDE control system by Hu \cite{Hu17}. Following this idea, we can derive the first-order and second-order variational equations for our control system (\ref{state-eq}). The expansions for $b$ and $\sigma$ are given as follows: \[ \begin{array} [c]{l} b(t,X^{\epsilon}(t),Y^{\epsilon}(t),Z^{\epsilon}(t),u^{\epsilon}(t))-b(t)\\ =b(t,\bar{X}(t)+X_{1}(t)+X_{2}(t),\bar{Y}(t)+Y_{1}(t)+Y_{2}(t),\bar {Z}(t)+\Delta(t)I_{E_{\epsilon}}(t)+Z_{1}^{\prime}(t)+Z_{2}(t),u^{\epsilon }(t))-b(t)+o(\epsilon)\\ =b_{x}(t)(X_{1}(t)+X_{2}(t))+b_{y}(t)(Y_{1}(t)+Y_{2}(t))+b_{z}(t)(Z_{1} ^{\prime}(t)+Z_{2}(t))\\ \text{ }+\frac{1}{2}[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]D^{2}b(t)[X_{1} (t),Y_{1}(t),Z_{1}^{\prime}(t)]^{\intercal}+\delta b(t,\Delta)I_{E_{\epsilon} }(t)+o(\epsilon), \end{array} \] \[ \begin{array} [c]{l} \sigma(t,X^{\epsilon}(t),Y^{\epsilon}(t),Z^{\epsilon}(t),u^{\epsilon }(t))-\sigma(t)\\ =\sigma(t,\bar{X}(t)+X_{1}(t)+X_{2}(t),\bar{Y}(t)+Y_{1}(t)+Y_{2}(t),\bar {Z}(t)+\Delta(t)I_{E_{\epsilon}}(t)+Z_{1}^{\prime}(t)+Z_{2}(t),u^{\epsilon }(t))-\sigma(t)+o(\epsilon)\\ =\sigma_{x}(t)(X_{1}(t)+X_{2}(t))+\sigma_{y}(t)(Y_{1}(t)+Y_{2}(t))+\sigma _{z}(t)(Z_{1}^{\prime}(t)+Z_{2}(t))\\ \text{ }+\delta\sigma_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)+\delta \sigma_{y}(t,\Delta)Y_{1}(t)I_{E_{\epsilon}}(t)+\delta\sigma_{z} (t,\Delta)Z_{1}^{\prime}(t)I_{E_{\epsilon}}(t)\\ \text{ }+\frac{1}{2}[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]D^{2}\sigma (t)[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]^{\intercal}+\delta\sigma (t,\Delta)I_{E_{\epsilon}}(t)+o(\epsilon). \end{array} \] Note that \[ \int_{0}^{T}\delta b_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)dt\sim o(\epsilon)\text{ and }\int_{0}^{T}\delta\sigma_{x}(t,\Delta)X_{1} (t)I_{E_{\epsilon}}(t)dB(t)\sim O(\epsilon). \] So we omit $\delta b_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)$ in the expansions of $b$ and keep $\delta\sigma_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon} }(t)$ in the expansions of $\sigma$. The expansions for $g$ and $\phi$ are similar to the expansions for $b$. Then, we obtain the following variational equations: \begin{equation} \left\{ \begin{array} [c]{rl} d(X_{1}(t)+X_{2}(t))= & \{b_{x}(t)(X_{1}(t)+X_{2}(t))+b_{y}(t)(Y_{1} (t)+Y_{2}(t))+b_{z}(t)(Z_{1}^{\prime}(t)+Z_{2}(t))\\ & +\frac{1}{2}[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]D^{2}b(t)[X_{1} (t),Y_{1}(t),Z_{1}^{\prime}(t)]^{\intercal}+\delta b(t,\Delta)I_{E_{\epsilon} }(t)\}dt\\ & +\{\sigma_{x}(t)(X_{1}(t)+X_{2}(t))+\sigma_{y}(t)(Y_{1}(t)+Y_{2} (t))+\sigma_{z}(t)(Z_{1}^{\prime}(t)+Z_{2}(t))\\ & +\delta\sigma_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)+\delta\sigma _{y}(t,\Delta)Y_{1}(t)I_{E_{\epsilon}}(t)+\delta\sigma_{z}(t,\Delta )Z_{1}^{\prime}(t)I_{E_{\epsilon}}(t)\\ & +\frac{1}{2}[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]D^{2}\sigma(t)[X_{1} (t),Y_{1}(t),Z_{1}^{\prime}(t)]^{\intercal}+\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t)\}dB(t),\\ X_{1}(0)+X_{2}(0)= & 0, \end{array} \right. \label{heur-4} \end{equation} \begin{equation} \left\{ \begin{array} [c]{rl} d(Y_{1}(t)+Y_{2}(t))= & -\{g_{x}(t)(X_{1}(t)+X_{2}(t))+g_{y}(t)(Y_{1} (t)+Y_{2}(t))+g_{z}(t)(Z_{1}^{\prime}(t)+Z_{2}(t))\\ & +\frac{1}{2}[X_{1}(t),Y_{1}(t),Z_{1}^{\prime}(t)]D^{2}g(t)[X_{1} (t),Y_{1}(t),Z_{1}^{\prime}(t)]^{\intercal}+\delta g(t,\Delta)I_{E_{\epsilon} }(t)\}dt\\ & +(Z_{1}(t)+Z_{2}(t))dB(t),\\ Y_{1}(T)+Y_{2}(T)= & \phi_{x}(\bar{X}(T))(X_{1}(T)+X_{2}(T))+\frac{1}{2} \phi_{xx}(\bar{X}(T))X_{1}^{2}(T). \end{array} \right. \label{heur-4'} \end{equation} Now, we need to derive the first-and second-order variational equations from (\ref{heur-4}) and (\ref{heur-4'}). Firstly, it is easy to establish the first-order variational equation for $X_{1}(t)$: \begin{equation} \begin{array} [c]{rl} dX_{1}(t)= & \left[ b_{x}(t)X_{1}(t)+b_{y}(t)Y_{1}(t)+b_{z}(t)(Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t))\right] dt\\ & +\left[ \sigma_{x}(t)X_{1}(t)+\sigma_{y}(t)Y_{1}(t)+\sigma_{z} (t)(Z_{1}(t)-\Delta(t)I_{E_{\epsilon}}(t))+\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t)\right] dB(t),\\ X_{1}(0)= & 0. \end{array} \label{heur-5'} \end{equation} Notice that $Y_{1}(T)=\phi_{x}(\bar{X}(T))X_{1}(T)$. So we guess that $Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) $ where $p\left( t\right) $ is the solution of the following adjoint equation \[ \left\{ \begin{array} [c]{rl} dp(t)= & -\Upsilon(t)dt+q(t)dB(t),\\ p(T)= & \phi_{x}(\bar{X}(T)), \end{array} \right. \] where $\Upsilon(t)$ is some adapted process which will be determined later. It is clear that $Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) $ should include all $O(\sqrt{\epsilon})$-terms of the drift term of (\ref{heur-4'}). Applying It\^{o}'s formula to $p\left( t\right) X_{1}\left( t\right) $, we can determine that \begin{equation} \left\{ \begin{array} [c]{rl} dY_{1}(t)= & -\left[ g_{x}(t)X_{1}(t)+g_{y}(t)Y_{1}(t)+g_{z}(t)(Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t))-q(t)\delta\sigma(t,\Delta)I_{E_{\epsilon} }(t)\right] dt+Z_{1}(t)dB(t),\\ Y_{1}(T)= & \phi_{x}(\bar{X}(T))X_{1}(T), \end{array} \right. \label{heur-5} \end{equation} and $(p(\cdot),q(\cdot))$ satisfies the following equation: \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -\left\{ g_{x}(t)+g_{y}(t)p(t)+g_{z}(t)K_{1}(t)+b_{x}(t)p(t)+b_{y} (t)p^{2}(t)\right. \\ & \left. +b_{z}(t)K_{1}(t)p(t)+\sigma_{x}(t)q(t)+\sigma_{y}(t)p(t)q(t)+\sigma _{z}(t)K_{1}(t)q(t)\right\} dt+q(t)dB(t),\\ p(T)= & \phi_{x}(\bar{X}(T)), \end{array} \right. \label{eq-p} \end{equation} where \begin{equation} K_{1}(t)=(1-p(t)\sigma_{z}(t))^{-1}\left[ \sigma_{x}(t)p(t)+\sigma _{y}(t)p^{2}(t)+q(t)\right] . \label{def-k1} \end{equation} Thus, we obtain the relationship \begin{equation} \begin{array} [c]{l} Y_{1}\left( t\right) =p\left( t\right) X_{1}\left( t\right) ,\\ Z_{1}(t)=(1-p(t)\sigma_{z}(t))^{-1}p(t)(\delta\sigma(t,\Delta)-\sigma _{z}(t)\Delta(t))I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}\left( t\right) . \end{array} \label{heur-6} \end{equation} Combining (\ref{heur-2}) and (\ref{heur-6}), we obtain \begin{align} \Delta(t) & =(1-p(t)\sigma_{z}(t))^{-1}p(t)(\delta\sigma(t,\Delta )-\sigma_{z}(t)\Delta(t)),\\ Z_{1}^{\prime}(t) & =K_{1}(t)X_{1}\left( t\right) , \end{align} which implies the following algebra equation \begin{equation} \Delta(t)=p(t)\delta\sigma(t,\Delta). \label{heur-7} \end{equation} From (\ref{heur-4}), (\ref{heur-4'}), (\ref{heur-5'}) and (\ref{heur-5}), it is easy to deduce that $(X_{2}(\cdot),Y_{2}(\cdot))$ satisfies the following equation: \begin{equation} \left\{ \begin{array} [c]{rl} dX_{2}(t)= & \{b_{x}(t)X_{2}(t)+b_{y}(t)Y_{2}(t)+b_{z}(t)Z_{2}(t)+\delta b(t,\Delta)I_{E_{\epsilon}}(t)\\ & +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),Z_{1}(t)-\Delta(t)I_{E_{\epsilon} }(t)\right] D^{2}b(t)\left[ X_{1}(t),Y_{1}(t),Z_{1}(t)-\Delta (t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\}dt\\ & +\left\{ \sigma_{x}(t)X_{2}(t)+\sigma_{y}(t)Y_{2}(t)+\sigma_{z} (t)Z_{2}(t)+\delta\sigma_{x}(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)+\delta \sigma_{y}(t,\Delta)Y_{1}(t)I_{E_{\epsilon}}(t)\right. \\ & +\delta\sigma_{z}(t,\Delta)\left( Z_{1}(t)-\Delta(t)I_{E_{\epsilon} }(t)\right) \\ & \left. +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),Z_{1}(t)-\Delta (t)I_{E_{\epsilon}}(t)\right] D^{2}\sigma(t)\left[ X_{1}(t),Y_{1} (t),Z_{1}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\right\} dB(t),\\ dY_{2}(t)= & -\left\{ g_{x}(t)X_{2}(t)+g_{y}(t)Y_{2}(t)+g_{z}(t)Z_{2} (t)+\left[ q(t)\delta\sigma(t,\Delta)+\delta g(t,\Delta)\right] I_{E_{\epsilon}}(t)\right. \\ & \left. +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),Z_{1}(t)-\Delta (t)I_{E_{\epsilon}}(t)\right] D^{2}g(t)\left[ X_{1}(t),Y_{1}(t),Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\right\} dt+Z_{2} (t)dB(t),\\ X_{2}(0)= & 0,\text{ }Y_{2}(T)=\phi_{x}(\bar{X}(T))X_{2}(T)+\frac{1}{2} \phi_{xx}(\bar{X}(T))X_{1}^{2}(T). \end{array} \right. \label{heur-8} \end{equation} In the following two subsections, we give the rigorous proofs for the above heuristic derivations. \subsection{First-order expansion} From the heuristic derivation, in order to obtain the first-order variational equation of \eqref{state-eq}, we need to introduce the first-order adjoint equation (\ref{eq-p}). Since the generator of \eqref{eq-p} does not satisfy Lipschitz condition, we firstly explore the solvability of \eqref{eq-p}. For $\beta_{0}>0$ and $y\in\mathbb{R}$, set \[ G(y)=L_{1}+\left( L_{2}+L_{1}+\beta_{0}^{-1}L_{1}L_{2}\right) \|y\|+\left[ L_{2}+\beta_{0}^{-1}(L_{1}L_{2}+L_{2}^{2})\right] y^{2}+\beta_{0}^{-1} L_{2}^{2}\|y\|^{3},\ y\in\mathbb{R}\text{.} \] Let $s(\cdot)$ be the maximal solution to the following equation: \begin{equation} s(t)=L_{1}+\int_{t}^{T}G(s(r))dr,\;t\in\lbrack0,T]; \label{u-ode} \end{equation} and $l(\cdot)$ be the minimal solution to the following equation: \begin{equation} l(t)=-L_{1}-\int_{t}^{T}G(l(r))dr,\;t\in\lbrack0,T]. \label{l-ode} \end{equation} Moreover, set \begin{equation} t_{1}=T-\int_{-\infty}^{-L_{1}}\frac{1}{G(y)}dy,\ \ t_{2}=T-\int_{L_{1} }^{\infty}\frac{1}{G(y)}dy,\ \ t^{\ast}=t_{1}\vee t_{2}. \label{def-t} \end{equation} \begin{lemma} \label{t-star} For given $\beta_{0}>0$, then there exists a $\delta>0$ such that when $L_{2}<\delta$, we have $t^{\ast}<0$. \end{lemma} \begin{proof} We only prove that there exists a $\delta>0$ such that when $L_{2}<\delta$, we have $t_{2}<0$. Note that $G(y)$ is a monotonic function with respect to $L_{2}$. As $L_{2}\rightarrow0$, \[ \frac{1}{G(y)}\uparrow\frac{1}{L_{1}(1+\|y\|)}. \] Applying the monotone convergence theorem, we obtain \[ \int_{L_{1}}^{\infty}\frac{1}{G(y)}dy\uparrow\int_{L_{1}}^{\infty}\frac {1}{L_{1}(1+\|y\|)}dy=\infty. \] By the definition of $t_{2}$, the result is obvious. \end{proof} \begin{assumption} \label{assum-3} There exists a positive constant $\beta_{0}\in(0,1)$ such that \[ t^{\ast}<0, \] and \begin{equation} \lbrack s(0)\vee(-l(0))]L_{3}\leq1-\beta_{0}. \label{assum-cl} \end{equation} \end{assumption} \begin{remark} Note that $G(\cdot)$ is independent of $L_{3}$. Therefore by Lemma \ref{t-star}, Assumption \ref{assum-3} holds when $L_{2}$ and $L_{3}$ are small enough. \end{remark} \begin{theorem} \label{exist-unique-BSDE}Suppose Assumptions \ref{assum-2}(i)-(ii) and \ref{assum-3} hold. Then \eqref{eq-p} has a bounded solution $\ $such that \[ \|p(t)\|\leq\lbrack s(0)\vee(-l(0))], \] and $q(\cdot)\in L_{\mathcal{F}}^{2,\beta}([0,T];\mathbb{R})$, for any $\beta\geq1$. \end{theorem} \begin{proof} The proof of the existence is a direct consequence of Theorem 4 in \cite{LS02}. Furthermore, similar to Corollary 4 in \cite{HuBSDEquad}, we can obtain $q(\cdot)\in L_{\mathcal{F}}^{2,\beta}\left( [0,T];\mathbb{R}\right) $ for any $\beta\geq1$. \end{proof} \begin{remark} The proof of the uniqueness for $(p(\cdot),q(\cdot))$ can be found in Theorem \ref{unique-pq} in the appendix. \end{remark} In order to introduce the first-order variational equation, we study the following algebra equation \begin{equation} \Delta(t)=p(t)(\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta (t),u(t))-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))),\;t\in \lbrack0,T], \label{def-delt} \end{equation} where $u\left( \cdot\right) $ is a given admissible control. \begin{remark} It should be note that the $\Delta(t)$ depends on the optimal control $\bar {u}(\cdot)$, the adjoint process $p(\cdot)$, and the control $u(\cdot)$. \end{remark} \begin{lemma} \label{delta-exist}Under the same assumptions as in Theorem \ref{exist-unique-BSDE}. Then (\ref{def-delt}) has a unique adapted solution $\Delta(\cdot)$. Moreover, \begin{equation} \begin{array} [c]{c} \|\Delta(t)\|\leq C(1+\|\bar{X}(t)\|+\|\bar{Y}(t)\|+\|u(t)\|+\|\bar{u}(t)\|),\text{ }t\in\lbrack0,T],\\ \sup\limits_{0\leq t\leq T}\mathbb{E}[\|\Delta(t)\|^{8}]<\infty, \end{array} \label{del-new-3} \end{equation} where $C$ is a constant depending on $\beta_{0}$, $L$, $L_{1}$, $L_{2}$, $L_{3}$, $T$. \end{lemma} \begin{proof} We first prove uniqueness. Let $\Delta(\cdot)$ and $\Delta^{\prime}(\cdot)$ be two adapted solutions to (\ref{def-delt}). Then \begin{equation} \begin{array} [c]{l} \left\vert \Delta(t)-\Delta^{\prime}(t)\right\vert \\ =\|p(t)\|\|\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta(t),u(t))-\sigma (t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta^{\prime}(t),u(t))\|\\ \leq\lbrack s(0)\vee(-l(0))]L_{3}\left\vert \Delta(t)-\Delta^{\prime }(t)\right\vert \\ \leq(1-\beta_{0})\left\vert \Delta(t)-\Delta^{\prime}(t)\right\vert , \end{array} \label{del-new-1} \end{equation} which implies $\Delta(t)=\Delta^{\prime}(t)$ for $t\in\lbrack0,T]$. Now we construct a contraction mapping on $L_{\mathcal{F}}^{2}([0,T];\mathbb{R})$ to prove the existence. For each given $\tilde{\Delta}(\cdot)\in L_{\mathcal{F} }^{2}([0,T];\mathbb{R})$, define the operator $\tilde{\Delta}(\cdot )\rightarrow\Delta(\cdot)$ by $\Gamma$, where \[ \Delta(t)=p(t)(\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\tilde{\Delta }(t),u(t))-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))),\;t\in \lbrack0,T]. \] Following the same steps as (\ref{del-new-1}), we can get \begin{equation} \|\Delta(t)\|\leq(1-\beta_{0})\left\vert \tilde{\Delta}(t)\right\vert +p(t)\delta\sigma(t),\;t\in\lbrack0,T], \label{del-new-2} \end{equation} which implies $\Delta(\cdot)\in L_{\mathcal{F}}^{2}([0,T];\mathbb{R})$. For each $\tilde{\Delta}_{i}(\cdot)\in L_{\mathcal{F}}^{2}([0,T];\mathbb{R})$, denote $\Delta_{i}(\cdot)=\Gamma\left( \tilde{\Delta}_{i}(\cdot)\right) $, $i=1$, $2$. Similar to (\ref{del-new-1}), we have \[ \mathbb{E}\left[ \int_{0}^{T}\left\vert \Delta_{1}(t)-\Delta_{2} (t)\right\vert ^{2}dt\right] \leq(1-\beta_{0})^{2}\mathbb{E}\left[ \int _{0}^{T}\left\vert \tilde{\Delta}_{1}(t)-\tilde{\Delta}_{2}(t)\right\vert ^{2}dt\right] . \] Thus, by the contraction mapping theorem, (\ref{def-delt}) has an adapted solution $\Delta(\cdot)\in L_{\mathcal{F}}^{2}([0,T];\mathbb{R})$. Moreover, for any adapted solution $\Delta(\cdot)$ to (\ref{def-delt}), it follows from (\ref{del-new-2}) with $\tilde{\Delta}(\cdot)=\Delta(\cdot)$ that \[ \|\Delta(t)\|\leq\beta_{0}^{-1}p(t)\delta\sigma(t)\leq C(1+\|\bar{X}(t)\|+\|\bar {Y}(t)\|+\|u(t)\|+\|\bar{u}(t)\|),\text{ }t\in\lbrack0,T], \] which implies (\ref{del-new-3}). \end{proof} \begin{remark} If the diffusion term is independent of $z$, that is $\sigma_{z}(\cdot)=0$, then we obtain $\Delta(t)=p(t)\delta\sigma(t)$. If the diffusion term contains $z$ in a linear form, for example $\sigma(t,x,y,z,u)=A(t)z+\sigma _{1}(t,x,y,u)$, then we obtain \[ \Delta(t)=\left( 1-p(t)A(t)\right) ^{-1}p(t)\left( \sigma_{1}(t,\bar {X}(t),\bar{Y}(t),u(t))-\sigma_{1}(t,\bar{X}(t),\bar{Y}(t),\bar{u}(t))\right) . \] \end{remark} Now we introduce the first-order variational equation: \begin{equation} \left\{ \begin{array} [c]{rl} dX_{1}(t)= & \left[ b_{x}(t)X_{1}(t)+b_{y}(t)Y_{1}(t)+b_{z}(t)(Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t))\right] dt\\ & +\left[ \sigma_{x}(t)X_{1}(t)+\sigma_{y}(t)Y_{1}(t)+\sigma_{z} (t)(Z_{1}(t)-\Delta(t)I_{E_{\epsilon}}(t))+\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t)\right] dB(t),\\ X_{1}(0)= & 0, \end{array} \right. \label{new-form-x1} \end{equation} and \begin{equation} \left\{ \begin{array} [c]{lll} dY_{1}(t) & = & -\left[ g_{x}(t)X_{1}(t)+g_{y}(t)Y_{1}(t)+g_{z} (t)(Z_{1}(t)-\Delta(t)I_{E_{\epsilon}}(t))-q(t)\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t)\right] dt+Z_{1}(t)dB(t),\\ Y_{1}(T) & = & \phi_{x}(\bar{X}(T))X_{1}(T). \end{array} \right. \label{new-form-y1} \end{equation} \begin{remark} The existence and uniqueness of $(X_{1}(\cdot),Y_{1}(\cdot),Z_{1}(\cdot))$ in \eqref{new-form-x1} and \eqref{new-form-y1} is guaranteed by Theorem \ref{est-fbsde-lp}. \end{remark} \begin{assumption} The solution $q(\cdot)$ of (\ref{eq-p}) is a bounded process. \label{assm-q-bound} \end{assumption} The relationship between $(Y_{1}(t),Z_{1}(t))$ and $X_{1}(t)$ as pointed out in our heuristic derivation is obtained in the following lemma. \begin{lemma} \label{lemma-y1}Suppose Assumptions \ref{assum-2}(i)-(ii), \ref{assum-3} and \ref{assm-q-bound} hold. Then we have \begin{align} Y_{1}(t) & =p(t)X_{1}(t),\\ Z_{1}(t) & =K_{1}(t)X_{1}(t)+\Delta(t)I_{E_{\epsilon}}(t), \end{align} where $p(\cdot)$ is the solution of \eqref{eq-p} and $K_{1}\left( \cdot\right) $ is given in (\ref{def-k1}). \end{lemma} \begin{proof} Consider the following stochastic differential equation: \begin{equation} \left\{ \begin{array} [c]{rl} d\tilde{X}_{1}(t)= & \left\{ \left[ b_{x}(t)+b_{y}(t)p(t)+b_{z} (t)K_{1}(t)\right] \tilde{X}_{1}(t)\right\} dt\\ & +\left\{ \left[ \sigma_{x}(t)+p(t)\sigma_{y}(t)+\sigma_{z}(t)K_{1} (t)\right] \tilde{X}_{1}(t)+\delta\sigma(t,\Delta)I_{E_{\epsilon} }(t)\right\} dB(t),\\ \tilde{X}_{1}(0)= & 0. \end{array} \right. \label{eq-x1} \end{equation} It is easy to check that there exists a unique solution $\tilde{X}_{1}(\cdot)$ of \eqref{eq-x1}. Set \begin{align} \tilde{Y}_{1}(t) & =p(t)\tilde{X}_{1}(t),\label{first order-relation}\\ \tilde{Z}_{1}(t) & =K_{1}(t)\tilde{X}_{1}(t)+\Delta(t)I_{E_{\epsilon} }(t).\nonumber \end{align} Applying It\^{o}'s lemma to $p(t)\tilde{X}_{1}(t)$, \[ \begin{array} [c]{lll} d\tilde{Y}_{1}(t) & = & -\left[ g_{x}(t)\tilde{X}_{1}(t)+g_{y}(t)\tilde {Y}_{1}(t)+g_{z}(t)\tilde{Z}_{1}(t)-g_{z}(t)\Delta(t)I_{E_{\epsilon} }(t)-q(t)\delta\sigma(t,\Delta)I_{E_{\epsilon}}(t)\right] dt+\tilde{Z} _{1}(t)dB(t). \end{array} \] Thus $(\tilde{X}_{1}(\cdot),\tilde{Y}_{1}(\cdot),\tilde{Z}_{1}(\cdot))$ solves (\ref{new-form-x1}) and (\ref{new-form-y1}), by Theorem \ref{est-fbsde-lp}, \ $(\tilde{X}_{1}(\cdot),\tilde{Y}_{1}(\cdot),\tilde{Z}_{1}(\cdot ))=(X_{1}(\cdot),Y_{1}(\cdot),Z_{1}(\cdot))$. This completes the proof. \end{proof} Then we have the following estimates. \begin{lemma} \label{est-one-order}Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-q-bound} hold. Then for any $2\leq\beta\leq8$, we have the following estimates \begin{equation} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X_{1}(t)\|^{\beta }+\|Y_{1}(t)\|^{\beta}\right) \right] +\mathbb{E}\left[ \left( \int_{0} ^{T}\|Z_{1}(t)\|^{2}dt\right) ^{\beta/2}\right] =O(\epsilon^{\beta/2}), \label{est-x1-y1} \end{equation} \[ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X^{\epsilon} (t)-\bar{X}(t)-X_{1}(t)\|^{2}+\|Y^{\epsilon}(t)-\bar{Y}(t)-Y_{1}(t)\|^{2}\right) \right] +\mathbb{E}\left[ \int_{0}^{T}\|Z^{\epsilon}(t)-\bar{Z} (t)-Z_{1}(t)\|^{2}dt\right] =O(\epsilon^{2}), \] \[ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X^{\epsilon}(t)-\bar {X}(t)-X_{1}(t)\|^{4}+\|Y^{\epsilon}(t)-\bar{Y}(t)-Y_{1}(t)\|^{4})\right] +\mathbb{E}\left[ \left( \int_{0}^{T}\|Z^{\epsilon}(t)-\bar{Z}(t)-Z_{1} (t)\|^{2}dt\right) ^{2}\right] =o(\epsilon^{2}). \] \end{lemma} \begin{proof} By Theorem \ref{est-fbsde-lp}, we have \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X_{1}(t)\|^{\beta }+\|Y_{1}(t)\|^{\beta}\right) +\left( \int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right) ^{\beta/2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\left[ \left( \|b_{z} (t)\|+\|g_{z}(t)\|\right) \|\Delta(t)\|+\|q(t)\delta\sigma(t,\Delta)\|\right] I_{E_{\epsilon}}(t)dt\right) ^{\beta}\right] \\ \text{ \ }+C\mathbb{E}\left[ \left( \int_{0}^{T}\left[ \|\sigma_{z} (t)\Delta(t)\|^{2}+\|\delta\sigma(t,\Delta)\|^{2}\right] I_{E_{\epsilon} }(t)dt\right) ^{\beta/2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\left( 1+\|\bar {X}(t)\|+\left\vert \bar{Y}(t)\right\vert +\left\vert \bar{u} (t)\|+\|u(t)\right\vert \right) dt\right) ^{\beta}\right] \\ \text{ \ }+C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\left( 1+\|\bar {X}(t)\|^{2}+\left\vert \bar{Y}(t)\right\vert ^{2}+\left\vert \bar{u} (t)\|^{2}+\|u(t)\right\vert ^{2}\right) dt\right) ^{\beta/2}\right] \\ \leq C\epsilon^{^{\beta/2}}. \end{array} \label{est-1-order} \end{equation} We use the notations $\xi^{1,\epsilon}(t)$, $\eta^{1,\epsilon}(t)$ and $\zeta^{1,\epsilon}(t)$ in the proof of Lemma \ref{est-epsilon-bar} and let \[ \begin{array} [c]{rl} \xi^{2,\epsilon}(t) & :=X^{\epsilon}(t)-\bar{X}(t)-X_{1}(t);\\ \eta^{2,\epsilon}(t) & :=Y^{\epsilon}(t)-\bar{Y}(t)-Y_{1}(t);\\ \zeta^{2,\epsilon}(t) & :=Z^{\epsilon}(t)-\bar{Z}(t)-Z_{1}(t);\\ \Theta(t) & :=(\bar{X}(t),\bar{Y}(t),\bar{Z}(t));\\ \Theta(t,\Delta I_{E_{\epsilon}}) & :=(\bar{X}(t),\bar{Y}(t),\bar{Z} (t)+\Delta(t)I_{E_{\epsilon}}(t));\\ \Theta^{\epsilon}(t) & :=(X^{\epsilon}(t),Y^{\epsilon}(t),Z^{\epsilon}(t)). \end{array} \] Note that \[ \begin{array} [c]{l} \delta\sigma(t,\Delta)I_{E_{\epsilon}}(t)=\sigma(t,\bar{X}(t),\bar{Y} (t),\bar{Z}(t)+\Delta(t)I_{E_{\epsilon}}(t),u^{\epsilon}(t))-\sigma (t)=\sigma(t,\Theta(t,\Delta I_{E_{\epsilon}}(t)),u^{\epsilon}(t))-\sigma(t). \end{array} \] We have \[ \begin{array} [c]{l} \sigma(t,\Theta^{\epsilon}(t),u^{\epsilon}(t))-\sigma(t)-\delta\sigma (t,\Delta)I_{E_{\epsilon}}(t)\\ =\sigma(t,\Theta^{\epsilon}(t),u^{\epsilon}(t))-\sigma(t,\Theta(t,\Delta I_{E_{\epsilon}}(t)),u^{\epsilon}(t))\\ =\tilde{\sigma}_{x}^{\epsilon}(t)(X^{\epsilon}(t)-\bar{X}(t))+\tilde{\sigma }_{y}^{\epsilon}(t)(Y^{\epsilon}(t)-\bar{Y}(t))+\tilde{\sigma}_{z}^{\epsilon }(t)(Z^{\epsilon}(t)-\bar{Z}(t)-\Delta(t)I_{E_{\epsilon}}(t)), \end{array} \] where \[ \tilde{\sigma}_{x}^{\epsilon}(t)=\int_{0}^{1}\sigma_{x}(t,\Theta(t,\Delta I_{E_{\epsilon}}(t))+\theta(\Theta^{\epsilon}(t)-\Theta(t,\Delta I_{E_{\epsilon}}(t))),u^{\epsilon}(t))d\theta, \] and $\tilde{\sigma}_{y}^{\epsilon}(t)$, $\tilde{\sigma}_{z}^{\epsilon} (t)$\ are defined similarly. \ Recall that $\tilde{b}_{x}^{\epsilon}(t)$, $\tilde{b}_{y}^{\epsilon}(t)$, $\tilde{b}_{z}^{\epsilon}(t)$, $\tilde{g}_{x}^{\epsilon}(t)$, $\tilde{g} _{y}^{\epsilon}(t)$, $\tilde{g}_{z}^{\epsilon}(t)$ and $\tilde{\phi} _{x}^{\epsilon}(T)$ are defined in Lemma \ref{est-epsilon-bar}. Then, \begin{equation} \left\{ \begin{array} [c]{ll} d\xi^{2,\epsilon}(t)= & \left[ \tilde{b}_{x}^{\epsilon}(t)\xi^{2,\epsilon }(t)+\tilde{b}_{y}^{\epsilon}(t)\eta^{2,\epsilon}(t)+\tilde{b}_{z}^{\epsilon }(t)\zeta^{2,\epsilon}(t)+A_{1}^{\epsilon}(t)\right] dt\\ & +\left[ \tilde{\sigma}_{x}^{\epsilon}(t)\xi^{2,\epsilon}(t)+\tilde{\sigma }_{y}^{\epsilon}(t)\eta^{2,\epsilon}(t)+\tilde{\sigma}_{z}^{\epsilon} (t)\zeta^{2,\epsilon}(t))+B_{1}^{\epsilon}(t)\right] dB(t),\\ \xi^{2,\epsilon}(0)= & 0, \end{array} \right. \label{deri-x-x1} \end{equation} \[ \left\{ \begin{array} [c]{rl} d\eta^{2,\epsilon}(t)= & -\left[ \tilde{g}_{x}^{\epsilon}(t)\xi^{2,\epsilon }(t)+\tilde{g}_{y}^{\epsilon}(t)\eta^{2,\epsilon}(t)+\tilde{g}_{z}^{\epsilon }(t)\zeta^{2,\epsilon}(t)+C_{1}^{\epsilon}(t)\right] dt+\zeta^{2,\epsilon }(t)dB(t),\\ \eta^{2,\epsilon}(T)= & \tilde{\phi}_{x}^{\epsilon}(T)\xi^{2,\epsilon }(T)+D_{1}^{\epsilon}(T), \end{array} \right. \] where \[ \begin{array} [c]{rl} A_{1}^{\epsilon}(t)= & (\tilde{b}_{x}^{\epsilon}(t)-b_{x}(t))X_{1} (t)+(\tilde{b}_{y}^{\epsilon}(t)-b_{y}(t))Y_{1}(t)+(\tilde{b}_{z}^{\epsilon }(t)-b_{z}(t))Z_{1}(t)+b_{z}(t)\Delta(t)I_{E_{\epsilon}}(t)+\delta b(t)I_{E_{\epsilon}}(t),\\ B_{1}^{\epsilon}(t)= & (\tilde{\sigma}_{x}^{\epsilon}(t)-\sigma_{x} (t))X_{1}(t)+(\tilde{\sigma}_{y}^{\epsilon}(t)-\sigma_{y}(t))Y_{1} (t)+(\tilde{\sigma}_{z}^{\epsilon}(t)-\sigma_{z}(t))K_{1}(t)X_{1}(t),\\ C_{1}^{\epsilon}(t)= & (\tilde{g}_{x}^{\epsilon}(t)-g_{x}(t))X_{1} (t)+(\tilde{g}_{y}^{\epsilon}(t)-g_{y}(t))Y_{1}(t)+(\tilde{g}_{z}^{\epsilon }(t)-g_{z}(t))Z_{1}(t)+\delta g(t)I_{E_{\epsilon}}(t)\\ & +g_{z}(t)\Delta(t)I_{E_{\epsilon}}(t)+q(t)\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t),\\ D_{1}^{\epsilon}(T)= & (\tilde{\phi}_{x}^{\epsilon}(T)-\phi_{x}(\bar {X}(T)))X_{1}(T). \end{array} \] By Theorem \ref{est-fbsde-lp}, we obtain \[ \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{2,\epsilon }(t)\|^{2}+\|\eta^{2,\epsilon}(t)\|^{2}\right) +\int_{0}^{T}\|\zeta^{2,\epsilon }(t)\|^{2}dt\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\left( \|A_{1}^{\epsilon }(t)\|+\|C_{1}^{\epsilon}(t)\|\right) dt\right) ^{2}+\int_{0}^{T} \|B_{1}^{\epsilon}(t)\|^{2}dt+\|D_{1}^{\epsilon}(T)\|^{2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\|A_{1}^{\epsilon}(t)\|dt\right) ^{2}+\left( \int_{0}^{T}\|C_{1}^{\epsilon}(t)\|dt\right) ^{2}+\int_{0} ^{T}\|B_{1}^{\epsilon}(t)\|^{2}dt+\|D_{1}^{\epsilon}(T)\|^{2}\right] . \end{array} \] Now we estimate term by term as follows. (1) Since \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{b}_{z}^{\epsilon}(t)-b_{z} (t)\|\|Z_{1}(t)\|dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \int_{0}^{T}\|\tilde{b}_{z}^{\epsilon}(t)-b_{z} (t)\|^{2}dt\int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right] \\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{b}_{z}^{\epsilon }(t)-b_{z}(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}\\ \leq C\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{1,\epsilon}(t)\|^{4}+\|\eta^{1,\epsilon}(t)\|^{4}\right) +\left( \int _{0}^{T}\left( \|\zeta^{1,\epsilon}(t)\|^{2}+\|\delta b_{z}(t)\|^{2} I_{E_{\epsilon}}(t)\right) dt\right) ^{2}\right] \right\} ^{\frac{1}{2} }\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}\\ \leq C\epsilon^{2}, \end{array} \label{est-b_x-z1} \end{equation} the estimate of $\mathbb{E}\left[ \left( \int_{0}^{T}\left( \tilde{b} _{x}^{\epsilon}(t)-b_{x}(t)\right) X_{1}(t)dt\right) ^{2}\right] $ and $\mathbb{E}\left\{ \left[ \int_{0}^{T}\left( \tilde{b}_{y}^{\epsilon }(t)-b_{y}(t)\right) Y_{1}(t)dt\right] ^{2}\right\} $ is the same as \eqref{est-b_x-z1}, \begin{equation} \mathbb{E}\left[ \left( \int_{0}^{T}\|b_{z}(t)\Delta(t)I_{E_{\epsilon} }(t)\|dt\right) ^{2}\right] \leq C\epsilon\int_{E_{\epsilon}}\mathbb{E} [\|\Delta(t)\|^{2}]dt\leq C\epsilon^{2}, \label{est-delta} \end{equation} \[ \begin{array} [c]{ll} \mathbb{E}\left[ (\int_{0}^{T}\|\delta b(t)I_{E_{\epsilon}}(t)\|dt)^{2}\right] & \leq\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\left( 1+\|\bar {X}(t)\|+\left\vert \bar{Y}(t)\right\vert +\left\vert \bar{Z}(t)\right\vert +\left\vert u(t)\right\vert +\left\vert \bar{u}(t)\right\vert \right) dt\right) ^{2}\right] \\ & \leq\epsilon\mathbb{E}\left[ \int_{E_{\epsilon}}\left( 1+\|\bar{X} (t)\|^{2}+\left\vert \bar{Y}(t)\right\vert ^{2}+\left\vert \bar{Z} (t)\right\vert ^{2}+\left\vert u(t)\right\vert ^{2}+\left\vert \bar {u}(t)\right\vert ^{2}\right) dt\right] \\ & \leq C\epsilon^{2}, \end{array} \] then, \[ \mathbb{E}\left[ (\int_{0}^{T}\|A_{1}^{\epsilon}(t)\|dt)^{2}\right] \leq C\epsilon^{2}. \] (2) \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\|\tilde{\sigma}_{z}^{\epsilon}(t)-\sigma _{z}(t)\|^{2}\|K_{1}(t)X_{1}(t)\|^{2}dt\right] \\ \leq C\mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|X_{1}(t)\|^{2}\int_{0} ^{T}\|\tilde{\sigma}_{z}^{\epsilon}(t)-\sigma_{z}(t)\|^{2}dt\right] \\ \leq C\left\{ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|X_{1} (t)\|^{4}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{1,\epsilon}(t)\|^{4}+\|\eta ^{1,\epsilon}(t)\|^{4}\right) \right. \right. \\ \ \ \left. \left. +\left( \int_{0}^{T}\left( \|\zeta^{1,\epsilon} (t)-\Delta(t)I_{E_{\epsilon}}(t)\|^{2}+\|\delta\sigma_{z}(t,\Delta )\|^{2}I_{E_{\epsilon}}(t)\right) dt\right) ^{2}\right] \right\} ^{\frac {1}{2}}\\ \leq C\epsilon\left\{ \epsilon^{2}+\epsilon\int_{E_{\epsilon}}\mathbb{E} [\|\Delta(t)\|^{4}]dt\right\} ^{\frac{1}{2}}\\ \leq C\epsilon^{2}, \end{array} \label{est-sigmazx1} \end{equation} and the estimate of $\mathbb{E}\left[ \int_{0}^{T}\|\tilde{\sigma} _{x}^{\epsilon}(t)-\sigma_{x}(t)\|^{2}\|X_{1}(t)\|^{2}dt\right] $ and $\mathbb{E}\left[ \int_{0}^{T}\|\tilde{\sigma}_{y}^{\epsilon}(t)-\sigma _{y}(t)\|^{2}\|Y_{1}(t)\|^{2}dt\right] $ is the same as \eqref{est-sigmazx1}. Thus, \begin{equation} \mathbb{E}\left[ \int_{0}^{T}\|B_{1}^{\epsilon}(t)\|^{2}dt\right] \leq C\epsilon^{2}. \end{equation} (3) \begin{equation} \mathbb{E}\left[ \|D_{1}^{\epsilon}(T)\|^{2}\right] \leq C\left\{ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|X_{1}(t)\|^{4}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T} \|\xi^{1,\epsilon}(t)\|^{4}\right] \right\} ^{\frac{1}{2}}\leq C\epsilon^{2}. \end{equation} (4) The estimate of $\mathbb{E}\left[ (\int_{0}^{T}\|C_{1}^{\epsilon }(t)\|dt)^{2}\right] $ is the same as $\mathbb{E}\left[ (\int_{0}^{T} \|A_{1}^{\epsilon}(t)\|dt)^{2}\right] $. Similarly, we obtain \[ \begin{array} [c]{l} \mathbb{E}\left\{ \sup\limits_{t\in\lbrack0,T]}\left[ \|\xi^{2,\epsilon }(t)\|^{4}+\|\eta^{2,\epsilon}(t)\|^{4}\right] +\left( \int_{0}^{T} \|\zeta^{2,\epsilon}(t)\|^{2}dt\right) ^{2}\right\} \\ \leq C\mathbb{E}\left\{ \left( \int_{0}^{T}\left( \|A_{1}^{\epsilon }(t)\|+\|C_{1}^{\epsilon}(t)\|\right) dt\right) ^{4}+\left( \int_{0}^{T} \|B_{1}^{\epsilon}(t)\|^{2}dt\right) ^{2}+\left\vert D_{1}^{\epsilon }(T)\right\vert ^{4}\right\} \\ =o(\epsilon^{2}). \end{array} \] This completes the proof. \end{proof} \subsection{Second-order expansion} Noting that $Z_{1}(t)=K_{1}(t)X_{1}(t)+\Delta(t)I_{E_{\epsilon}}(t)$\ in Lemma \ref{lemma-y1}, then we introduce the second-order variational equation as follows: \begin{equation} \left\{ \begin{array} [c]{rl} dX_{2}(t)= & \left\{ b_{x}(t)X_{2}(t)+b_{y}(t)Y_{2}(t)+b_{z}(t)Z_{2} (t)+\delta b(t,\Delta)I_{E_{\epsilon}}(t)\right. \\ & \left. +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2}b(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal }\right\} dt\\ & +\left\{ \sigma_{x}(t)X_{2}(t)+\sigma_{y}(t)Y_{2}(t)+\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2}\sigma(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal}\right. \\ & \left. +\sigma_{z}(t)Z_{2}(t)+\left[ \delta\sigma_{x}(t,\Delta )X_{1}(t)+\delta\sigma_{y}(t,\Delta)Y_{1}(t)\right] I_{E_{\epsilon} }(t)+\delta\sigma_{z}(t,\Delta)K_{1}(t)X_{1}(t)I_{E_{\epsilon}}(t)\right\} dB(t),\\ X_{2}(0)= & 0, \end{array} \right. \label{new-form-x2} \end{equation} and \begin{equation} \left\{ \begin{array} [c]{ll} dY_{2}(t)= & -\left\{ g_{x}(t)X_{2}(t)+g_{y}(t)Y_{2}(t)+g_{z}(t)Z_{2} (t)+\left[ q(t)\delta\sigma(t,\Delta)+\delta g(t,\Delta)\right] I_{E_{\epsilon}}(t)\right. \\ & \left. +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2}g(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal }\right\} dt+Z_{2}(t)dB(t),\\ Y_{2}(T)= & \phi_{x}(\bar{X}(T))X_{2}(T)+\frac{1}{2}\phi_{xx}(\bar{X} (T))X_{1}^{2}(T). \end{array} \right. \label{new-form-y2} \end{equation} In the following lemma, we estimate the orders of $X_{2}(\cdot)$, $Y_{2} (\cdot)$, $Z_{2}(\cdot)$, and $Y^{\epsilon}(0)-\bar{Y}(0)-Y_{1}(0)-Y_{2}(0)$. \begin{lemma} \label{est-second-order} Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-q-bound} hold. Then for any $2\leq\beta\leq4$ we have \[ \begin{array} [c]{rl} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X_{2}(t)\|^{2}+\|Y_{2} (t)\|^{2})\right] +\mathbb{E}\left[ \int_{0}^{T}\|Z_{2}(t)\|^{2}dt\right] & =O(\epsilon^{2}),\\ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X_{2}(t)\|^{\beta} +\|Y_{2}(t)\|^{\beta})\right] +\mathbb{E}\left[ \left( \int_{0}^{T} \|Z_{2}(t)\|^{2}dt\right) ^{\frac{\beta}{2}}\right] & =o(\epsilon^{\frac {\beta}{2}}),\\ Y^{\epsilon}(0)-\bar{Y}(0)-Y_{1}(0)-Y_{2}(0) & =o(\epsilon). \end{array} \] \end{lemma} \begin{proof} By Theorem \ref{est-fbsde-lp}, we have \[ \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X_{2}(t)\|^{2}+\|Y_{2} (t)\|^{2})+\int_{0}^{T}\|Z_{2}(t)\|^{2}dt\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}[(\|\delta b(t,\Delta )\|+\|\delta\sigma(t,\Delta)\|+\|\delta g(t,\Delta)\|)I_{E_{\epsilon}} (t)+\|X_{1}(t)\|^{2}+\|Y_{1}(t)\|^{2}]dt\right) ^{2}\right] \\ \text{ \ }+C\mathbb{E}\left[ \int_{0}^{T}\left[ \|X_{1}(t)\|^{4} +\|Y_{1}(t)\|^{4}+(\|X_{1}(t)\|^{2}+\|Y_{1}(t)\|^{2})I_{E_{\epsilon}}(t)\right] dt\right] \\ \leq C\epsilon\mathbb{E}\left[ \int_{E_{\epsilon}}(1+\|\bar{X}(t)\|^{2} +\|\bar{Y}(t)\|^{2}+\|\bar{Z}(t)\|^{2}+\|u(t)\|^{2}+\|\bar{u}(t)\|^{2})dt\right] \\ \text{ \ }+C\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X_{1}(t)\|^{4}+\|Y_{1}(t)\|^{4}\right) \right] +C\epsilon\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X_{1}(t)\|^{2}+\|Y_{1}(t)\|^{2})\right] \\ \leq C\epsilon^{2}, \end{array} \] \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|X_{2}(t)\|^{\beta} +\|Y_{2}(t)\|^{\beta})+\left( \int_{0}^{T}\|Z_{2}(t)\|^{2}dt\right) ^{\frac{\beta}{2}}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}[(\|\delta b(t,\Delta )\|+\|\delta\sigma(t,\Delta)\|+\|\delta g(t,\Delta)\|)I_{E_{\epsilon}} (t)+\|X_{1}(t)\|^{2}+\|Y_{1}(t)\|^{2}]dt\right) ^{\beta}\right] \\ \text{ \ }+C\mathbb{E}\left[ \left( \int_{0}^{T}\left[ \|X_{1} (t)\|^{4}+\|Y_{1}(t)\|^{4}+(\|X_{1}(t)\|^{2}+\|Y_{1}(t)\|^{2})I_{E_{\epsilon} }(t)\right] dt\right) ^{\frac{\beta}{2}}\right] \\ \leq C\epsilon^{\frac{\beta}{2}}\mathbb{E}\left[ \left( \int_{E_{\epsilon} }(1+\|\bar{X}(t)\|^{2}+\|\bar{Y}(t)\|^{2}+\|\bar{Z}(t)\|^{2}+\|u(t)\|^{2}+\|\bar {u}(t)\|^{2})dt\right) ^{\frac{\beta}{2}}\right] \\ \text{ \ }+C\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X_{1}(t)\|^{2\beta}+\|Y_{1}(t)\|^{2\beta}\right) \right] +C\epsilon ^{\frac{\beta}{2}}\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]} (\|X_{1}(t)\|^{\beta}+\|Y_{1}(t)\|^{\beta})\right] \\ =o(\epsilon^{\frac{\beta}{2}}). \end{array} \end{equation} Now, we focus on the last estimate. We use the same notations $\xi ^{1,\epsilon}(t)$, $\eta^{1,\epsilon}(t)$, $\zeta^{1,\epsilon}(t)$, $\xi^{2,\epsilon}(t)$, $\eta^{2,\epsilon}(t)$ and $\zeta^{2,\epsilon}(t)$ in the proof of Lemma \ref{est-epsilon-bar} and Lemma \ref{est-one-order}. Let \[ \begin{array} [c]{rl} \xi^{3,\epsilon}(t) & :=X^{\epsilon}(t)-\bar{X}(t)-X_{1}(t)-X_{2}(t);\\ \eta^{3,\epsilon}(t) & :=Y^{\epsilon}(t)-\bar{Y}(t)-Y_{1}(t)-Y_{2}(t);\\ \zeta^{3,\epsilon}(t) & :=Z^{\epsilon}(t)-\bar{Z}(t)-Z_{1}(t)-Z_{2}(t);\\ \Theta(t) & :=(\bar{X}(t),\bar{Y}(t),\bar{Z}(t));\\ \Theta(t,\Delta I_{E_{\epsilon}}) & :=(\bar{X}(t),\bar{Y}(t),\bar{Z} (t)+\Delta(t)I_{E_{\epsilon}}(t));\\ \Theta^{\epsilon}(t) & :=(X^{\epsilon}(t),Y^{\epsilon}(t),Z^{\epsilon}(t)). \end{array} \] Define $\widetilde{D^{2}b^{\epsilon}}(t)$ \[ \widetilde{D^{2}b^{\epsilon}}(t)=2\int_{0}^{1}\int_{0}^{1}\theta D^{2}b(t,\Theta(t,\Delta I_{E_{\epsilon}})+\lambda\theta(\Theta^{\epsilon }(t)-\Theta(t,\Delta I_{E_{\epsilon}})),u^{\epsilon}(t))d\theta d\lambda, \] $\widetilde{D^{2}\sigma^{\epsilon}}(t)$, $\widetilde{D^{2}g^{\epsilon}}(t)$ and $\tilde{\phi}_{xx}^{\epsilon}(T)$ are defined similarly. Then, we have \begin{equation} \left\{ \begin{array} [c]{ll} d\xi^{3,\epsilon}(t)= & \left\{ b_{x}(t)\xi^{3,\epsilon}(t)+b_{y} (t)\eta^{3,\epsilon}(t)+b_{z}(t)\zeta^{3,\epsilon}(t)+A_{2}^{\epsilon }(t)\right\} dt\\ & +\left\{ \sigma_{x}(t)\xi^{3,\epsilon}(t)+\sigma_{y}(t)\eta^{3,\epsilon }(t)+\sigma_{z}(t)\zeta^{3,\epsilon}(t)+B_{2}^{\epsilon}(t)\right\} dB(t),\\ \xi^{3,\epsilon}(0)= & 0, \end{array} \right. \label{x-x1-x2} \end{equation} and \begin{equation} \left\{ \begin{array} [c]{lll} d\eta^{3,\epsilon}(t) & = & -\{g_{x}(t)\xi^{3,\epsilon}(t)+g_{y} (t)\eta^{3,\epsilon}(t)+g_{z}(t)\zeta^{3,\epsilon}(t)+C_{2}^{\epsilon }(t)\}dt-\zeta^{3,\epsilon}(t)dB(t),\\ \eta^{3,\epsilon}(T) & = & \phi_{x}(\bar{X}(T))\xi^{3,\epsilon}(T)+D_{2} ^{\epsilon}(T), \end{array} \right. \label{y-y1-y2} \end{equation} where \[ \begin{array} [c]{ll} A_{2}^{\epsilon}(t)= & \left[ \delta b_{x}(t,\Delta)\xi^{1,\epsilon }(t)+\delta b_{y}(t,\Delta)\eta^{1,\epsilon}(t)+\delta b_{z}(t,\Delta)\left( \zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right) \right] I_{E_{\epsilon}}(t)\\ & +\frac{1}{2}\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] \widetilde{D^{2} b^{\epsilon}}(t)\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\\ & -\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2}b(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal}, \end{array} \] \[ \begin{array} [c]{ll} B_{2}^{\epsilon}(t)= & \left[ \delta\sigma_{x}(t,\Delta)\xi^{2,\epsilon }(t)+\delta\sigma_{y}(t,\Delta)\eta^{2,\epsilon}(t)+\delta\sigma_{z} (t,\Delta)\zeta^{2,\epsilon}(t)\right] I_{E_{\epsilon}}(t)\\ & +\frac{1}{2}\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] \widetilde{D^{2} \sigma^{\epsilon}}(t)\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon} (t),\zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\\ & -\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2} \sigma(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal}, \end{array} \] \[ \begin{array} [c]{ll} C_{2}^{\epsilon}(t)= & \left[ \delta g_{x}(t,\Delta)\xi^{1,\epsilon }(t)+\delta g_{y}(t,\Delta)\eta^{1,\epsilon}(t)+\delta g_{z}(t,\Delta)\left( \zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right) \right] I_{E_{\epsilon}}(t)\\ & +\frac{1}{2}\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] \widetilde{D^{2} g^{\epsilon}}(t)\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t),\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right] ^{\intercal}\\ & -\frac{1}{2}\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] D^{2}g(t)\left[ X_{1}(t),Y_{1}(t),K_{1}(t)X_{1}(t)\right] ^{\intercal},\\ D_{2}^{\epsilon}(T)= & \frac{1}{2}\tilde{\phi}_{xx}^{\epsilon}(T)\xi ^{1,\epsilon}(T)^{2}-\frac{1}{2}\phi_{xx}(\bar{X}(T))X_{1}^{2}(T). \end{array} \] We introduce the following fully coupled FBSDE: \begin{equation} \left\{ \begin{array} [c]{rl} dh(t)= & \left[ g_{y}(t)h(t)+b_{y}(t)m(t)+\sigma_{y}(t)n(t)\right] dt+\left[ g_{z}(t)h(t)+b_{z}(t)m(t)+\sigma_{z}(t)n(t)\right] dB(t),\\ h(0)= & 1,\\ dm(t)= & -\left[ g_{x}(t)h(t)+b_{x}(t)m(t)+\sigma_{x}(t)n(t)\right] dt+n(t)dB(t),\\ m(T)= & \phi_{x}(\bar{X}(T))h(T). \end{array} \right. \end{equation} It has a unique solution due to Theorem \ref{est-fbsde-lp}. Applying It\^{o}'s formula to \[ m(t)\xi^{3,\epsilon}(t)-h(t)\eta^{3,\epsilon}(t), \] we have \begin{equation} \begin{array} [c]{ll} \|\eta^{3,\epsilon}(0)\| & =\left\vert \mathbb{E}\left[ h(T)D_{2}^{\epsilon }(T)+\int_{0}^{T}\left( m(t)A_{2}^{\epsilon}(t)+n(t)B_{2}^{\epsilon }(t)+h(t)C_{2}^{\epsilon}(t)\right) dt\right] \right\vert \\ & \leq\mathbb{E}\left[ \left\vert h(T)D_{2}^{\epsilon}(T)\right\vert +\int_{0}^{T}\left( \left\vert m(t)A_{2}^{\epsilon}(t)\right\vert +\left\vert n(t)B_{2}^{\epsilon}(t)\right\vert +\left\vert h(t)C_{2}^{\epsilon }(t)\right\vert \right) dt\right] . \end{array} \label{second order-estimate} \end{equation} We estimate each term as follows. (1) { \[ \begin{array} [c]{ll} \mathbb{E}\left[ \|h(T)D_{2}^{\epsilon}(T)\|\right] & \leq\left\{ \mathbb{E}\left[ \|h(T)\|^{2}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \|D_{2}^{\epsilon}(T)\|^{2}\right] \right\} ^{\frac{1}{2}}\\ & \leq C\left\{ \mathbb{E}\left[ \|\tilde{\phi}_{xx}^{\epsilon}(T)-\phi _{xx}(\bar{X}(T))\|^{2}\|\xi^{1,\epsilon}(T)\|^{4}+\|\xi^{2,\epsilon}(T)\|^{2} \|\xi^{1,\epsilon}(T)+X_{1}(T)\|^{2}\right] \right\} ^{\frac{1}{2}}\\ & =o(\epsilon). \end{array} \] } (2) {Since } \[ \mathbb{E}\left[ \int_{0}^{T}\|m(t)A_{2}^{\epsilon}(t)\|dt\right] \leq\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|m(t)\|\int_{0}^{T} \|A_{2}^{\epsilon}(t)\|dt\right] \leq\left\{ \mathbb{E}\left[ \sup \limits_{t\in\lbrack0,T]}\|m(t)\|^{2}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}, \] { then we only need to check \begin{equation} \mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}\right] =o(\epsilon^{2}). \label{second order-part estimate} \end{equation} Indeed, (\ref{second order-part estimate}) is due to the following estimates: \[ \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\|\delta b_{z}(t,\Delta)(\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t))\|I_{E_{\epsilon}}(t)dt\right) ^{2}\right] \\ \leq\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|\delta b_{z}(t,\Delta )\|\left( \|\zeta^{2,\epsilon}(t)\|+\|K_{1}(t)X_{1}(t)\|\right) dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|\zeta^{2,\epsilon }(t)\|dt\right) ^{2}\right] +C\mathbb{E}\left[ \sup\limits_{t\in\lbrack 0,T]}\|X_{1}(t)\|^{2}\left( \int_{E_{\epsilon}}\|\delta b_{z}(t,\Delta )\|dt\right) ^{2}\right] \\ \leq C\epsilon\mathbb{E}\left[ \int_{0}^{T}\|\zeta^{2,\epsilon}(t)\|^{2} dt\right] +C\epsilon^{2}\mathbb{E}[\sup\limits_{t\in\lbrack0,T]} \|X_{1}(t)\|^{2}]\\ =o(\epsilon^{2}), \end{array} \] \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \widetilde{b}_{zz}^{\epsilon }(t)\left( \zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right) ^{2}-b_{zz}(t)K_{1}(t)^{2}X_{1}(t)^{2}\right\vert dt\right) ^{2}\right] \\ \leq\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \widetilde{b} _{zz}^{\epsilon}(t)\zeta^{2,\epsilon}(t)\left( \zeta^{1,\epsilon} (t)-\Delta(t)I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}(t)\right) \right\vert dt\right) ^{2}\right] \\ \text{ \ }+\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \left( \widetilde{b}_{zz}^{\epsilon}(t)-b_{zz}(t)\right) K_{1}(t)^{2}X_{1} (t)^{2}\right\vert dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \int_{0}^{T}\left\vert \zeta^{2,\epsilon }(t)\right\vert ^{2}dt\int_{0}^{T}\left\vert \zeta^{1,\epsilon}(t)-\Delta (t)I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}(t)\right\vert ^{2}dt\right] \\ \text{ \ }+C\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|X_{1} (t)\|^{4}\left( \int_{0}^{T}\left\vert \left( \widetilde{b}_{zz}^{\epsilon }(t)-b_{zz}(t)\right) \right\vert dt\right) ^{2}\right] \\ =o(\epsilon^{2}), \end{array} \label{est-d2b} \end{equation} the other terms are similar.} (3){ }The estimate of $\mathbb{E}\left[ \int_{0}^{T}\|n(t)B_{2}^{\epsilon }(t)\|dt\right] $: \[ \begin{array} [c]{ll} \mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\delta\sigma_{z}(t,\Delta )\zeta^{2,\epsilon}(t)I_{E_{\epsilon}}(t)\right\vert dt\right] & \leq C\mathbb{E}\left[ \int_{E_{\epsilon}}\|n(t)\zeta^{2,\epsilon}(t)\|dt\right] \\ & \leq C\left\{ \mathbb{E}\left[ \int_{0}^{T}\|\zeta^{2,\epsilon} (t)\|^{2}dt\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \int_{E_{\epsilon}}\|n(t)\|^{2}dt\right] \right\} ^{\frac{1}{2}}\\ & =o(\epsilon), \end{array} \] \[ \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zz}^{\epsilon}(t)\left( \zeta^{1,\epsilon}(t)-\Delta (t)I_{E_{\epsilon}}(t)\right) ^{2}-\sigma_{zz}(t)K_{1}(t)^{2}X_{1} (t)^{2}\right\vert dt\right] \\ \leq\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zz}^{\epsilon}(t)\left( \zeta^{1,\epsilon}(t)-\Delta (t)I_{E_{\epsilon}}(t)+K_{1}(t)X_{1}(t)\right) \zeta^{2,\epsilon }(t)\right\vert dt\right] \\ \text{ \ }+\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zz}^{\epsilon}(t)-\sigma_{zz}(t)\right\vert K_{1}(t)^{2} X_{1}(t)^{2}dt\right] \\ \leq\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zz}^{\epsilon}(t)\left( \zeta^{1,\epsilon}(t)-\Delta (t)I_{E_{\epsilon}}(t)\right) \zeta^{2,\epsilon}(t)\right\vert dt\right] +\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zz}^{\epsilon}(t)K_{1}(t)X_{1}(t)\zeta^{2,\epsilon }(t)\right\vert dt\right] +o(\epsilon)\\ =\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert 2\int _{0}^{1}\theta\left[ \sigma_{z}(t,\Theta(t,\Delta I_{E_{\epsilon}} )+\theta(\Theta^{\epsilon}(t)-\Theta(t,\Delta I_{E_{\epsilon}})),u^{\epsilon }(t))-\sigma_{z}(t,\Theta(t,\Delta I_{E_{\epsilon}}),u^{\epsilon}(t))\right] d\theta\right\vert \left\vert \zeta^{2,\epsilon}(t)\right\vert dt\right] \\ \text{ \ }+\mathbb{E}\left[ \int_{0}^{T}\left\vert n(t)\right\vert \left\vert \tilde{\sigma}_{zx}^{\epsilon}\left( t\right) \xi^{1,\epsilon}\left( t\right) +\tilde{\sigma}_{zy}^{\epsilon}\left( t\right) \eta^{1,\epsilon }\left( t\right) \right\vert \left\vert \zeta^{2,\epsilon}(t)\right\vert dt\right] +C\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert X_{1}(t)\right\vert \int_{0}^{T}\left\vert n(t)K_{1}(t)\right\vert \left\vert \zeta^{2,\epsilon}(t)\right\vert dt\right] +o(\epsilon)\\ =o(\epsilon), \end{array} \] {the other terms are similar.} (4) {The estimate of $\mathbb{E}\left[ \int_{0}^{T}\|h(t)C_{2}^{\epsilon }(t)\|dt\right] $ is the same as $\mathbb{E}\left[ \int_{0}^{T} \|m(t)A_{2}^{\epsilon}(t)\|dt\right] $. } All the terms in (\ref{second order-estimate}) have been derived. Finally, we obtain \[ Y^{\epsilon}(0)-\bar{Y}(0)-Y_{1}(0)-Y_{2}(0)=o(\epsilon). \] The proof is complete. \end{proof} In the above lemma, we only prove $Y^{\epsilon}(0)-\bar{Y}(0)-Y_{1} (0)-Y_{2}(0)=o(\epsilon)$ and have not deduced \[ \mathbb{E}[\sup\limits_{t\in\lbrack0,T]}\|Y^{\epsilon}(t)-\bar{Y} (t)-Y_{1}(t)-Y_{2}(t)\|^{2}]=o(\epsilon^{2}). \] The reason is \[ \mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma}_{zz}^{\epsilon }(t)\left( \zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t)\right) \right\vert ^{2}\left\vert \zeta^{2,\epsilon}(t)\right\vert ^{2}dt\right] =o(\epsilon^{2}) \] may be not hold. But if \begin{equation} \sigma(t,x,y,z,u)=A(t)z+\sigma_{1}(t,x,y,u) \label{xigma-zz} \end{equation} where $A(t)$ is a bounded adapted process, then $\sigma_{zz}\equiv0.$ In this case, we can prove the following estimate. \begin{lemma} \label{lemma-est-sup}Suppose Assumptions \ref{assum-2}, \ref{assum-3}, \ref{assm-q-bound} and $\sigma(t,x,y,z,u)=A(t)z+$ $\sigma_{1}(t,x,y,u)$ where $A(t)$ is a bounded adapted process. Then \[ \begin{array} [c]{rl} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|X^{\epsilon}(t)-\bar {X}(t)-X_{1}(t)-X_{2}(t)\|^{2}\right] & =o(\epsilon^{2}),\\ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|Y^{\epsilon}(t)-\bar {Y}(t)-Y_{1}(t)-Y_{2}(t)\|^{2}+\int_{0}^{T}\|Z^{\epsilon}(t)-\bar{Z} (t)-Z_{1}(t)-Z_{2}(t)\|^{2}dt\right] & =o(\epsilon^{2}). \end{array} \] \end{lemma} \begin{proof} We use all notations in Lemma \ref{est-second-order}. By Theorem \ref{est-fbsde-lp}, we have \[ \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}(\|\xi^{3,\epsilon} (t)\|^{2}+\|\eta^{3,\epsilon}(t)\|^{2})+\int_{0}^{T}\|\zeta^{3,\epsilon} (t)\|^{2}dt\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}+\left( \int_{0}^{T}\|C_{2}^{\epsilon}(t)\|dt\right) ^{2}+\int_{0} ^{T}\|B_{2}^{\epsilon}(t)\|^{2}dt+\|D_{2}^{\epsilon}(T)\|^{2}\right] , \end{array} \] where $A_{2}^{\epsilon}(\cdot)$, $C_{2}^{\epsilon}(\cdot)$, $D_{2}^{\epsilon }(T)$ are the same as Lemma \ref{est-second-order}, and \[ \begin{array} [c]{ll} B_{2}^{\epsilon}(t)= & \left[ \delta\sigma_{x}(t)\xi^{2,\epsilon} (t)+\delta\sigma_{y}(t)\eta^{2,\epsilon}(t)\right] I_{E_{\epsilon}} (t)+\frac{1}{2}\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t)\right] \widetilde{D^{2}\sigma^{\epsilon}}(t)\left[ \xi^{1,\epsilon}(t),\eta ^{1,\epsilon}(t)\right] ^{\intercal}\\ & -\frac{1}{2}\left[ X_{1}(t),Y_{1}(t)\right] D^{2}\sigma(t)\left[ X_{1}(t),Y_{1}(t)\right] ^{\intercal}. \end{array} \] In Lemma \ref{est-second-order}, we have proved \[ \mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}+\left( \int_{0}^{T}\|C_{2}^{\epsilon}(t)\|dt\right) ^{2}+\|D_{2} ^{\epsilon}(T)\|^{2}\right] =o(\epsilon^{2}). \] Now we just need to check $\mathbb{E}\left[ \int_{0}^{T}\|B_{2}^{\epsilon }(t)\|^{2}dt\right] =o\left( \epsilon^{2}\right) $ as follows. \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\left\vert \delta\sigma_{x}(t)\xi^{2,\epsilon }(t)\right\vert ^{2}I_{E_{\epsilon}}(t)dt\right] \leq\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert \xi^{2,\epsilon}(t)\right\vert ^{2}\int_{E_{\epsilon}}\left\vert \delta\sigma_{x}(t)\right\vert ^{2}dt\right] =o(\epsilon^{2}). \end{array} \label{est-sigmax-x-x1} \end{equation} The estimate of $\mathbb{E}\left[ \int_{0}^{T}\left\vert \delta\sigma _{y}(t)\eta^{2,\epsilon}(t)\right\vert ^{2}dt\right] $ is same to \eqref{est-sigmax-x-x1}, and \[ \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon} (t)\eta^{1,\epsilon}(t)^{2}-\sigma_{yy}(t)Y_{1}(t)^{2}\right\vert ^{2}dt\right] \ \ \\ \leq\mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon }(t)\eta^{2,\epsilon}(t)(\eta^{1,\epsilon}(t)+Y_{1}(t))\right\vert ^{2}dt\right] +\mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma} _{yy}^{\epsilon}(t)-\sigma_{yy}(t)\right\vert ^{2}Y_{1}(t)^{4}dt\right] \\ \leq C\mathbb{E}\left[ \int_{0}^{T}\left\vert \eta^{2,\epsilon}(t)\right\vert ^{2}\left\vert \eta^{1,\epsilon}(t)+Y_{1}(t)\right\vert ^{2}dt\right] +\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert Y_{1}(t)\right\vert ^{4}\int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon}(t)-\sigma _{yy}(t)\right\vert ^{2}dt\right] \\ =o(\epsilon^{2}). \end{array} \] Other terms are similar. \end{proof} \subsection{Maximum principle} \label{section-mp}Note that $Y_{1}(0)=0$, by Lemma \ref{est-second-order}, we have \[ J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))=Y^{\epsilon}(0)-\bar{Y} (0)=Y_{2}(0)+o(\epsilon). \] In order to obtain $Y_{2}(0)$, we introduce the following second-order adjoint equation: \begin{equation} \left\{ \begin{array} [c]{rl} -dP(t)= & \left\{ P(t)\left[ (D\sigma(t)^{\intercal}[1,p(t),K_{1} (t)]^{\intercal})^{2}+2Db(t)^{\intercal}[1,p(t),K_{1}(t)]^{\intercal} +H_{y}(t)\right] \right. \\ & +2Q(t)D\sigma(t)^{\intercal}[1,p(t),K_{1}(t)]^{\intercal}+\left[ 1,p(t),K_{1}(t)\right] D^{2}H(t)\left[ 1,p(t),K_{1}(t)\right] ^{\intercal }\left. +H_{z}(t)K_{2}(t)\right\} dt\\ & -Q(t)dB(t),\\ P(T)= & \phi_{xx}(\bar{X}(T)), \end{array} \right. \label{eq-P} \end{equation} where \[ \begin{array} [c]{ll} H(t,x,y,z,u,p,q)= & g(t,x,y,z,u)+pb(t,x,y,z,u)+q\sigma(t,x,y,z,u), \end{array} \] \[ \begin{array} [c]{ll} K_{2}(t)= & (1-p(t)\sigma_{z}(t))^{-1}\left\{ p(t)\sigma_{y}(t)+2\left[ \sigma_{x}(t)+\sigma_{y}(t)p(t)+\sigma_{z}(t)K_{1}(t)\right] \right\} P(t)\\ & +(1-p(t)\sigma_{z}(t))^{-1}\left\{ Q(t)+p(t)[1,p(t),K_{1}(t)]D^{2} \sigma(t)[1,p(t),K_{1}(t)]^{\intercal}\right\} , \end{array} \] and $DH(t)$, $D^{2}H(t)$ are defined similar to $D\psi$ and $D^{2}\psi$. Note that (\ref{eq-P}) is a linear BSDE with uniformly Lipschitz continuous coefficients, then it has a unique solution. Before we deduce the relationship between $X_{2}(\cdot)$ and $(Y_{2}(\cdot),Z_{2}(\cdot))$, we introduce the following equation: \begin{equation} \begin{array} [c]{l} \hat{Y}(t)=\int_{t}^{T}\left\{ (H_{y}(s)+\sigma_{y}(s)g_{z} (s)p(s)(1-p(s)\sigma_{z}(s))^{-1})\hat{Y}(s)+\left( H_{z}(s)+\sigma _{z}(s)g_{z}(s)p(s)(1-p(s)\sigma_{z}(s))^{-1}\right) \hat{Z}(s)\right. \\ \ \ \ \ \ \ \ \ \ \ \left. +\left[ \delta H(s,\Delta)+\frac{1}{2} P(s)\delta\sigma(s,\Delta)^{2}\right] I_{E_{\epsilon}}(s)\right\} ds-\int_{t}^{T}\hat{Z}(s)dB(s), \end{array} \label{eq-y-hat} \end{equation} where $\delta H(s,\Delta):=p(s)\delta b(s,\Delta)+q(s)\delta\sigma (s,\Delta)+\delta g(s,\Delta)$. It is also a linear BSDE and has a unique solution. \begin{lemma} \label{relation-y2} Suppose Assumptions \ref{assum-2}, \ref{assum-3} and. \ref{assm-q-bound} hold. Then we have \[ \begin{array} [c]{rl} Y_{2}(t) & =p(t)X_{2}(t)+\frac{1}{2}P(t)X_{1}(t)^{2}+\hat{Y}(t),\\ Z_{2}(t) & =\mathbf{I(t)}+\hat{Z}(t), \end{array} \] where $(\hat{Y}(\cdot),\hat{Z}(\cdot))$ is the solution to \eqref{eq-y-hat} and \begin{align} \mathbf{I(t)} & =K_{1}(t)X_{2}(t)+\frac{1}{2}K_{2}(t)X_{1}^{2} (t)+(1-p(t)\sigma_{z}(t))^{-1}p(t)(\sigma_{y}(t)\hat{Y}(t)+\sigma_{z} (t)\hat{Z}(t))+P(t)\delta\sigma(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)\\ & \;+(1-p(t)\sigma_{z}(t))^{-1}p(t)\left[ \delta\sigma_{x}(t,\Delta )X_{1}(t)+\delta\sigma_{y}(t,\Delta)p(t)X_{1}(t)+\delta\sigma_{z} (t,\Delta)K_{1}(t)X_{1}(t)\right] I_{E_{\epsilon}}(t). \end{align} \end{lemma} \begin{proof} Using the same method as in Lemma \ref{lemma-y1}, we can deduce the above relationship similarly. \end{proof} Consider the following equation: \begin{equation} \left\{ \begin{array} [c]{rl} d\gamma(t)= & \gamma(t)\left[ H_{y}(t)+p(t)g_{z}(t)(1-p(t)\sigma_{z} (t))^{-1}\sigma_{y}(t)\right] dt\\ & +\gamma(t)\left[ H_{z}(t)+p(t)(1-p(t)\sigma_{z}(t))^{-1}\sigma_{z} (t)g_{z}(t)\right] dB(t),\\ \gamma(0)= & 1. \end{array} \right. \label{eq-gamma} \end{equation} Applying It\^{o}'s formula to $\gamma(t)\hat{Y}(t)$, we obtain \[ \begin{array} [c]{rl} \hat{Y}(0)= & \mathbb{E}\left\{ \int_{0}^{T}\gamma(t)\left[ \delta H(t,\Delta)+\frac{1}{2}P(t)\delta\sigma(t,\Delta)^{2}\right] I_{E_{\epsilon} }(t)dt\right\} . \end{array} \] Define \begin{equation} \begin{array} [c]{ll} \mathcal{H}(t,x,y,z,u,p,q,P)= & pb(t,x,y,z+\Delta(t),u)+q\sigma(t,x,y,z+\Delta (t),u)\\ & +\frac{1}{2}P(\sigma(t,x,y,z+\Delta(t),u)-\sigma(t,\bar{X}(t),\bar {Y}(t),\bar{Z}(t),\bar{u}(t)))^{2}\ +g(t,x,y,z+\Delta(t),u), \end{array} \label{def-H} \end{equation} where $\Delta(t)$ is defined in (\ref{def-delt}) corresponding to $u(t)=u$. It is easy to check that \begin{align} & \delta H(t,\Delta)+\frac{1}{2}P(t)\delta\sigma(t,\Delta)^{2}\\ & =\mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z} (t),u(t),p(t),q(t),P(t))-\mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z} (t),\bar{u}(t),p(t),q(t),P(t)). \end{align} Noting that $\gamma(t)>0$ for $t\in\lbrack0,T]$, then we obtain the following maximum principle. \begin{theorem} \label{Th-MP}Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-q-bound} hold. Let $\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state processes of (\ref{state-eq}). Then the following stochastic maximum principle holds: \begin{equation} \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),P(t))\geq \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u} (t),p(t),q(t),P(t)),\ \ \ \forall u\in U,\ a.e.,\ a.s., \label{mp-1} \end{equation} where $(p\left( \cdot\right) ,q\left( \cdot\right) )$, $\left( P\left( \cdot\right) ,Q\left( \cdot\right) \right) $ satisfy (\ref{eq-p}), (\ref{eq-P}) respectively, and $\Delta(\cdot)$ satisfies (\ref{def-delt}). \end{theorem} \begin{remark} If $b$ and $\sigma$ are independent of $y$ and $z$, then Theorem \ref{Th-MP} degenerates to the maximum principle obtained in \cite{Hu17}. \end{remark} \begin{corollary} \label{cor-mp-convex}Under the same assumptions as in Theorem \ref{Th-MP}. Moreover, suppose that $b$, $\sigma$, $g$ are continuously differentiable with respect to $u$ and $U$ is a convex set. Then \begin{equation} \Delta_{u}(t)\|_{u=\bar{u}(t)}=\frac{p(t)\sigma_{u}(t)}{1-p(t)\sigma_{z}(t)} \label{delt-convex} \end{equation} and the maximum principle is \begin{equation} \mathcal{H}_{u}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t))\cdot (u-\bar{u}(t))\geq0\ \ \ \forall u\in U,\ a.e.,\ a.s. \label{hamil-convex} \end{equation} with \[ \begin{array} [c]{l} \mathcal{H}_{u}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t),p(t),q(t))\\ =p(t)b_{u}(t)+q(t)\sigma_{u}(t)+g_{u}(t)+(p(t)b_{z}(t)+q(t)\sigma_{z} (t)+g_{z}(t))\frac{p(t)\sigma_{u}(t)}{1-p(t)\sigma_{z}(t)}. \end{array} \] \end{corollary} \begin{proof} By implicit function theorem for (\ref{def-delt}), we get (\ref{delt-convex}). For each $u\in U$, taking $u_{\rho}(t)=\bar{u}(t)+\rho(u-\bar{u}(t))$, we can get (\ref{hamil-convex}) by (\ref{mp-1}). \end{proof} \subsection{The case without Assumption \ref{assm-q-bound}} The relations $Y_{1}(t)=p(t)X_{1}(t)$ and $\ Z_{1}(t)=K_{1}(t)X_{1} (t)+\Delta(t)I_{E_{\epsilon}}(t)$ in Lemma \ref{lemma-y1}, is the key point to derive the maximum principle (\ref{mp-1}). Note that to prove Lemma \ref{lemma-y1}, we need Assumption \ref{assm-q-bound}, which implies \begin{equation} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|\tilde{X}_{1}(t)\|^{2}\right] <\infty. \label{eq-new211} \end{equation} However, under the following assumption, combing Theorems \ref{appen-th-linear-fbsde} and \ref{unique-pq} in appendix, we can obtain the relations $Y_{1}(t)=p(t)X_{1}(t)$ and$\ Z_{1}(t)=K_{1}(t)X_{1}(t)+\Delta (t)I_{E_{\epsilon}}(t)$ without Assumption \ref{assm-q-bound}. \begin{assumption} \label{assm-sig-small} $\sigma(t,x,y,z,u)=A(t)z+\sigma_{1}(t,x,y,u)$ and $\left\Vert A(\cdot)\right\Vert _{\infty}$ is small enough. \end{assumption} In this case, the first-order adjoint equation becomes \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -\left\{ g_{x}(t)+g_{y}(t)p(t)+g_{z}(t)K_{1}(t)+b_{x}(t)p(t)+b_{y} (t)p^{2}(t)+b_{z}(t)K_{1}(t)p(t)\right. \\ & \left. +\sigma_{x}(t)q(t)+\sigma_{y}(t)p(t)q(t)+A(t)K_{1}(t)q(t)\right\} dt+q(t)dB(t),\\ p(T)= & \phi_{x}(\bar{X}(T)), \end{array} \right. \label{eq-p-q-unbound} \end{equation} where \[ K_{1}(t)=(1-p(t)A(t))^{-1}\left[ \sigma_{x}(t)p(t)+\sigma_{y}(t)p^{2} (t)+q(t)\right] . \] The first-order variational equation becomes \[ \left\{ \begin{array} [c]{rl} dX_{1}(t)= & \left[ b_{x}(t)X_{1}(t)+b_{y}(t)Y_{1}(t)+b_{z}(t)(Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t))\right] dt\\ & +\left[ \sigma_{x}(t)X_{1}(t)+\sigma_{y}(t)Y_{1}(t)+A(t)(Z_{1} (t)-\Delta(t)I_{E_{\epsilon}}(t))+\delta\sigma(t,\Delta)I_{E_{\epsilon} }(t)\right] dB(t),\\ X_{1}(0)= & 0, \end{array} \right. \] and \[ \left\{ \begin{array} [c]{lll} dY_{1}(t) & = & -\left[ g_{x}(t)X_{1}(t)+g_{y}(t)Y_{1}(t)+g_{z} (t)(Z_{1}(t)-\Delta(t)I_{E_{\epsilon}}(t))-q(t)\delta\sigma(t,\Delta )I_{E_{\epsilon}}(t)\right] dt+Z_{1}(t)dB(t),\\ Y_{1}(T) & = & \phi_{x}(\bar{X}(T))X_{1}(T), \end{array} \right. \] where \[ \Delta(t)=\left( 1-p(t)A(t)\right) ^{-1}p(t)\left( \sigma_{1}(t,\bar {X}(t),\bar{Y}(t),u(t))-\sigma_{1}(t,\bar{X}(t),\bar{Y}(t),\bar{u}(t))\right) . \] By Theorems \ref{appen-th-linear-fbsde} and \ref{unique-pq} we have the following relationship: \begin{lemma} Suppose Assumptions \ref{assum-2}(i)-(ii), \ref{assum-3} and \ref{assm-sig-small} hold. Then we have \begin{align} Y_{1}(t) & =p(t)X_{1}(t),\\ Z_{1}(t) & =K_{1}(t)X_{1}(t)+\Delta(t)I_{E_{\epsilon}}(t), \end{align} where $p(\cdot)$ is the solution of \eqref{eq-p-q-unbound}. \end{lemma} \begin{lemma} \label{est-one-order-q-unbound}Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-sig-small} hold. Then for any $2\leq\beta<8$, we have the following estimates \begin{equation} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X_{1}(t)\|^{\beta }+\|Y_{1}(t)\|^{\beta}\right) \right] +\mathbb{E}\left[ \left( \int_{0} ^{T}\|Z_{1}(t)\|^{2}dt\right) ^{\beta/2}\right] =O(\epsilon^{\beta/2}), \label{est-x1-y1-q-unbound} \end{equation} \[ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|X^{\epsilon} (t)-\bar{X}(t)-X_{1}(t)\|^{4}+\|Y^{\epsilon}(t)-\bar{Y}(t)-Y_{1}(t)\|^{4}\right) \right] +\mathbb{E}\left[ \left( \int_{0}^{T}\|Z^{\epsilon}(t)-\bar {Z}(t)-Z_{1}(t)\|^{2}dt\right) ^{2}\right] =o(\epsilon^{2}). \] \end{lemma} \begin{proof} The estimate of (\ref{est-x1-y1-q-unbound}) is the same as (\ref{est-1-order}) in Lemma \ref{est-one-order}, except the following term, \[ \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\|q(t)\delta\sigma(t,\Delta )\|I_{E_{\epsilon}}(t)dt\right) ^{\beta}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|q(t)\|\left( 1+\|\bar {X}(t)\|+\left\vert \bar{Y}(t)\right\vert +\left\vert \bar{u} (t)\|+\|u(t)\right\vert \right) dt\right) ^{\beta}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|q(t)\|^{2}dt\right) ^{\frac{\beta}{2}}\left( \int_{E_{\epsilon}}\left( 1+\|\bar{X}(t)\|^{2} +\left\vert \bar{Y}(t)\right\vert ^{2}+\left\vert \bar{u}(t)\|^{2} +\|u(t)\right\vert ^{2}\right) dt\right) ^{\frac{\beta}{2}}\right] \\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\left( 1+\|\bar {X}(t)\|^{2}+\left\vert \bar{Y}(t)\right\vert ^{2}+\left\vert \bar{u} (t)\|^{2}+\|u(t)\right\vert ^{2}\right) dt\right) ^{4}\right] \right\} ^{\frac{\beta}{8}}\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon} }\|q(t)\|^{2}dt\right) ^{\frac{4\beta}{8-\beta}}\right] \right\} ^{\frac{8-\beta}{8}}\\ \leq C\left\{ \epsilon^{3}\mathbb{E}\left[ \int_{E_{\epsilon}}\left( 1+\|\bar{X}(t)\|^{8}+\left\vert \bar{Y}(t)\right\vert ^{8}+\left\vert \bar {u}(t)\|^{8}+\|u(t)\right\vert ^{8}\right) dt\right] \right\} ^{\frac{\beta }{8}}\\ \leq C\epsilon^{\frac{\beta}{2}}. \end{array} \] In this case, $A_{1}^{\epsilon}(\cdot)$, $C_{1}^{\epsilon}(\cdot)$, $D_{1}^{\epsilon}(T)$ is the same as Lemma \ref{est-one-order}, and \[ B_{1}^{\epsilon}(t)=(\tilde{\sigma}_{x}^{\epsilon}(t)-\sigma_{x} (t))X_{1}(t)+(\tilde{\sigma}_{y}^{\epsilon}(t)-\sigma_{y}(t))Y_{1}(t). \] By Theorem \ref{est-fbsde-lp}, we obtain \[ \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{2,\epsilon }(t)\|^{4}+\|\eta^{2,\epsilon}(t)\|^{4}\right) +\left( \int_{0}^{T} \|\zeta^{2,\epsilon}(t)\|^{2}dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\left( \|A_{1}^{\epsilon }(t)\|+\|C_{1}^{\epsilon}(t)\|\right) dt\right) ^{4}+\left( \int_{0}^{T} \|B_{1}^{\epsilon}(t)\|^{2}dt\right) ^{2}+\|D_{1}^{\epsilon}(T)\|^{4}\right] \\ \leq C\mathbb{E}\left[ \|D_{1}^{\epsilon}(T)\|^{4}+\left( \int_{0}^{T} \|C_{1}^{\epsilon}(t)\|dt\right) ^{4}+\left( \int_{0}^{T}\|B_{1}^{\epsilon }(t)\|^{2}dt\right) ^{2}+\left( \int_{0}^{T}\|A_{1}^{\epsilon}(t)\|dt\right) ^{4}\right] . \end{array} \] We estimate term by term as follows. (1) \[ \mathbb{E}\left[ \|D_{1}^{\epsilon}(T)\|^{4}\right] \leq C\left\{ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|X_{1}(t)\|^{6}\right] \right\} ^{\frac{2}{3}}\left\{ \mathbb{E}\left[ \left\vert \tilde{\phi}_{x} ^{\epsilon}(T)-\phi_{x}\left( \bar{X}\left( T\right) \right) \right\vert ^{12}\right] \right\} ^{\frac{1}{3}}=o\left( \epsilon^{2}\right) . \] (2) \begin{equation} \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{g}_{z}^{\epsilon}(t)-g_{z} (t)\|\|Z_{1}(t)\|dt\right) ^{4}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{g}_{z}^{\epsilon} (t)-g_{z}(t)\|^{2}dt\right) ^{2}\left( \int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right) ^{2}\right] \\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{g}_{z}^{\epsilon }(t)-g_{z}(t)\|^{2}dt\right) ^{6}\right] \right\} ^{\frac{1}{3}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|Z_{1}(t)\|^{2}dt\right) ^{3}\right] \right\} ^{\frac{2}{3}}\\ =o\left( \epsilon^{2}\right) , \end{array} \label{est-g-z-q-unbound} \end{equation} the estimates of $\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \left( \tilde{g}_{x}^{\epsilon}(t)-g_{x}(t)\right) X_{1}(t)\right\vert dt\right) ^{4}\right] $ and $\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \left( \tilde{g}_{y}^{\epsilon}(t)-g_{y}(t)\right) Y_{1}(t)\right\vert dt\right) ^{4}\right] $ are the same as \eqref{est-g-z-q-unbound}, \begin{equation} \mathbb{E}\left[ \left( \int_{0}^{T}\|g_{z}(t)\Delta(t)I_{E_{\epsilon} }(t)\|dt\right) ^{4}\right] \leq C\epsilon^{3}\int_{E_{\epsilon}} \mathbb{E}[\|\Delta(t)\|^{4}]dt=o\left( \epsilon^{2}\right) , \label{est-g-del-qunbound} \end{equation} the estimate of $\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \delta g\left( t\right) I_{E_{\epsilon}}(t)\right\vert dt\right) ^{4}\right] $ is the same as \eqref{est-g-del-qunbound}, \[ \begin{array} [c]{l} \mathbb{E}\left[ (\int_{0}^{T}\|q(t)\delta\sigma(t,\Delta)I_{E_{\epsilon} }(t)\|dt)^{4}\right] \\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|q(t)\|^{2} dt\right) ^{4}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|\delta\sigma(t,\Delta)\|^{2}dt\right) ^{4}\right] \right\} ^{\frac{1}{2}}\\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|q(t)\|^{2} dt\right) ^{4}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}}\left( 1+\|\bar{X}(t)\|^{2}+\left\vert \bar {Y}(t)\right\vert ^{2}+\left\vert \bar{u}(t)\|^{2}+\|u(t)\right\vert ^{2}\right) dt\right) ^{4}\right] \right\} ^{\frac{1}{2}}\\ \leq C\epsilon^{2}\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon} }\|q(t)\|^{2}dt\right) ^{4}\right] \right\} ^{\frac{1}{2}}\\ =o\left( \epsilon^{2}\right) . \end{array} \] Then, \[ \mathbb{E}\left[ \left( \int_{0}^{T}\|C_{1}^{\epsilon}(t)\|dt\right) ^{4}\right] =o\left( \epsilon^{2}\right) . \] (3) \[ \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{\sigma}_{y}^{\epsilon} (t)-\sigma_{y}(t)\|^{2}\|Y_{1}(t)\|^{2}dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|Y_{1}(t)\|^{4}\left( \int_{0}^{T}\|\tilde{\sigma}_{z}^{\epsilon}(t)-\sigma_{z}(t)\|^{2}dt\right) ^{2}\right] \\ \leq C\left\{ \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\|Y_{1} (t)\|^{6}\right] \right\} ^{\frac{2}{3}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{\sigma}_{z}^{\epsilon}(t)-\sigma_{z}(t)\|^{2}dt\right) ^{6}\right] \right\} ^{\frac{1}{3}}\\ =o\left( \epsilon^{2}\right) , \end{array} \] the estimate of $\mathbb{E}\left[ \left( \int_{0}^{T}\|\tilde{\sigma} _{x}^{\epsilon}(t)-\sigma_{x}(t)\|^{2}\|X_{1}(t)\|^{2}dt\right) ^{2}\right] $ is similar. Thus, \[ \mathbb{E}\left[ \left( \int_{0}^{T}\|B_{1}^{\epsilon}(t)\|^{2}dt\right) ^{2}\right] =o\left( \epsilon^{2}\right) . \] (4) The estimate of $\mathbb{E}\left[ \left( \int_{0}^{T}\|A_{1}^{\epsilon }(t)\|dt\right) ^{4}\right] $ is the same as $\mathbb{E}\left[ \left( \int_{0}^{T}\|C_{1}^{\epsilon}(t)\|dt\right) ^{4}\right] $. \end{proof} The second-order variational equation becomes \begin{equation} \left\{ \begin{array} [c]{rl} dX_{2}(t)= & \left\{ b_{x}(t)X_{2}(t)+b_{y}(t)Y_{2}(t)+b_{z}(t)Z_{2} (t)+\delta b(t,\Delta)I_{E_{\epsilon}}(t)\right. \\ & \left. +\frac{1}{2}\left[ 1,p(t),K_{1}(t)\right] D^{2}b(t)\left[ 1,p(t),K_{1}(t)\right] ^{\intercal}X_{1}(t)^{2}\right\} dt\\ & +\left\{ \sigma_{x}(t)X_{2}(t)+\sigma_{y}(t)Y_{2}(t)+A(t)Z_{2}(t)+\left[ \delta\sigma_{x}(t)X_{1}(t)+\delta\sigma_{y}(t)Y_{1}(t)\right] I_{E_{\epsilon }}(t)\right. \\ & \left. +\frac{1}{2}\left[ X_{1}(t),Y_{1}(t)\right] D^{2}\sigma _{1}(t)\left[ X_{1}(t),Y_{1}(t)\right] ^{\intercal}\right\} dB(t),\\ X_{2}(0)= & 0, \end{array} \right. \label{new-form-x2-q-unbound} \end{equation} \begin{equation} \left\{ \begin{array} [c]{ll} dY_{2}(t)= & -\left\{ g_{x}(t)X_{2}(t)+g_{y}(t)Y_{2}(t)+g_{z}(t)Z_{2} (t)+\frac{1}{2}\left[ 1,p(t),K_{1}(t)\right] D^{2}g(t)\left[ 1,p(t),K_{1} (t)\right] ^{\intercal}X_{1}^{2}(t)\right. \\ & \left. +q(t)\delta\sigma(t,\Delta)I_{E_{\epsilon}}(t)+\delta g(t,\Delta )I_{E_{\epsilon}}(t)\right\} dt+Z_{2}(t)dB(t),\\ Y_{2}(T)= & \phi_{x}(\bar{X}(T))X_{2}(T)+\frac{1}{2}\phi_{xx}(\bar{X} (T))X_{1}^{2}(T). \end{array} \right. \label{new-form-y2-q-unbound} \end{equation} The following second-order estimates hold. \begin{lemma} \label{est-second-order-q-unbound} Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-sig-small} hold. Then we have the following estimates \[ \begin{array} [c]{rl} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|X^{\epsilon}(t)-\bar {X}(t)-X_{1}(t)-X_{2}(t)\|^{2}\right] & =o(\epsilon^{2}),\\ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\|Y^{\epsilon}(t)-\bar {Y}(t)-Y_{1}(t)-Y_{2}(t)\|^{2}\right] +\mathbb{E}\left[ \int_{0} ^{T}\|Z^{\epsilon}(t)-\bar{Z}(t)-Z_{1}(t)-Z_{2}(t)\|^{2}dt\right] & =o(\epsilon^{2}). \end{array} \] \end{lemma} \begin{proof} We use the same notations $A_{2}^{\epsilon}(t)$ $C_{2}^{\epsilon}(t)$ and $D_{2}^{\epsilon}(T)$ as in\ Lemma \ref{est-second-order}. The only different term is \[ \begin{array} [c]{ll} B_{2}^{\epsilon}(t)= & \delta\sigma_{x}(t)\xi^{2,\epsilon}(t)I_{E_{\epsilon} }(t)+\delta\sigma_{y}(t)\eta^{2,\epsilon}(t)I_{E_{\epsilon}}(t)+\frac{1} {2}\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t)\right] \widetilde{D^{2} \sigma^{\epsilon}}(t)\left[ \xi^{1,\epsilon}(t),\eta^{1,\epsilon}(t)\right] ^{\intercal}\\ & -\frac{1}{2}\left[ X_{1}(t),Y_{1}(t)\right] D^{2}\sigma(t)\left[ X_{1}(t),Y_{1}(t)\right] ^{\intercal}. \end{array} \] Then, we have that \begin{equation} \left\{ \begin{array} [c]{l} d\xi^{3,\epsilon}(t)=\left[ b_{x}(t)\xi^{3,\epsilon}(t)+b_{y}(t)\eta ^{3,\epsilon}(t)+b_{z}(t)\zeta^{3,\epsilon}(t)+A_{2}^{\epsilon}(t)\right] dt\ \\ \text{ \ \ \ \ \ \ }+\left[ \sigma_{x}(t)\xi^{3,\epsilon}(t)+\sigma _{y}(t)\eta^{3,\epsilon}(t)+A(t)\zeta^{3,\epsilon}(t)+B_{2}^{\epsilon }(t)\right] dB(t),\\ \xi^{3,\epsilon}(0)=0, \end{array} \right. \label{x-x1-x2-bz-0} \end{equation} and \begin{equation} \left\{ \begin{array} [c]{ll} d\eta^{3,\epsilon}(t)= & -\left[ g_{x}(t)\xi^{3,\epsilon}(t)+g_{y} (t)\eta^{3,\epsilon}(t)+g_{z}(t)\zeta^{3,\epsilon}(t)+C_{2}^{\epsilon }(t)\right] dt-\zeta^{3,\epsilon}(t)dB(t),\\ \eta^{3,\epsilon}(T)= & \phi_{x}(\bar{X}(T))\xi^{3,\epsilon}(T)+D_{2} ^{\epsilon}(T). \end{array} \right. \label{y-y1-y2-bz-0} \end{equation} By Theorem \ref{est-fbsde-lp}, \[ \begin{array} [c]{l} \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left( \|\xi^{3,\epsilon }(t)\|^{2}+\|\eta^{3,\epsilon}(t)\|^{2}\right) +\int_{0}^{T}\|\zeta^{3,\epsilon }(t)\|^{2}dt\right] \\ \leq\mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}+\left( \int_{0}^{T}\|C_{2}^{\epsilon}(t)\|dt\right) ^{2}+\int_{0} ^{T}\|B_{2}^{\epsilon}(t)\|^{2}dt+\|D_{2}^{\epsilon}(T)\|^{2}\right] . \end{array} \] We estimate term by term in the followings. (1) \[ \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}(\delta b_{z}(t,\Delta)(\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t))dt\right) ^{2}\right] \\ \leq\mathbb{E}\left[ \left( \int_{E_{\epsilon}}\|\delta b_{z}(t,\Delta )\|\left( \|\zeta^{2,\epsilon}(t)\|+\|K_{1}(t)X_{1}(t)\|\right) dt\right) ^{2}\right] \\ \leq C\epsilon\mathbb{E}\left[ \int_{E_{\epsilon}}\|\zeta^{2,\epsilon} (t)\|^{2}dt\right] +C\epsilon\mathbb{E}\left[ \sup\limits_{t\in\lbrack 0,T]}\|X_{1}(t)\|^{2}\int_{E_{\epsilon}}\|K_{1}(t)\|^{2}dt\right] \\ \leq C\epsilon\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|\zeta ^{2,\epsilon}(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2} }+C\epsilon^{2}\left\{ \mathbb{E}\left[ \left( \int_{E_{\epsilon}} \|K_{1}(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}\\ =o(\epsilon^{2}), \end{array} \] \[ \begin{array} [c]{l} \mathbb{E}\left[ \left( \int_{0}^{T}\left( \tilde{b}_{zz}(t)(\zeta ^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t))^{2}-b_{zz}(t)K_{1}(t)^{2} X_{1}(t)^{2}\right) dt\right) ^{2}\right] \\ \leq\mathbb{E}\left[ \left( \int_{0}^{T}\tilde{b}_{zz}(t)\left( (\zeta^{1,\epsilon}(t)-\Delta(t)I_{E_{\epsilon}}(t))^{2}-K_{1}(t)^{2} X_{1}(t)^{2}\right) dt\right) ^{2}+\left( \int_{0}^{T}\left\vert \tilde {b}_{zz}(t)-b_{zz}(t)\right\vert \left\vert K_{1}(t)X_{1}(t)\right\vert ^{2}dt\right) ^{2}\right] \\ \leq C\mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \zeta^{2,\epsilon }(t)\right\vert ^{2}dt\right) ^{2}+\left( \int_{0}^{T}\left\vert \zeta^{2,\epsilon}(t)\right\vert \left\vert K_{1}(t)X_{1}(t)\right\vert dt\right) ^{2}\right] \\ \text{ \ }+\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert X_{1}(t)\right\vert ^{6}\right] \right\} ^{\frac{2}{3}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\left\vert \tilde{b}_{zz}(t)-b_{zz} (t)\right\vert \left\vert K_{1}(t)\right\vert ^{2}dt\right) ^{6}\right] \right\} ^{\frac{1}{3}}\\ \leq C\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\|\zeta^{2,\epsilon }(t)\|^{2}dt\right) ^{2}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert X_{1}(t)\right\vert ^{6}\right] \right\} ^{\frac{1}{3}}\left\{ \mathbb{E}\left[ \left( \int_{0}^{T}\left\vert K_{1}(t)\right\vert ^{2}dt\right) ^{6}\right] \right\} ^{\frac{1}{6}}+o(\epsilon^{2})\\ =o(\epsilon^{2}), \end{array} \] the other terms are similar. Then \[ \mathbb{E}\left[ \left( \int_{0}^{T}\|A_{2}^{\epsilon}(t)\|dt\right) ^{2}\right] =o(\epsilon^{2}). \] (2) The estimate of $C_{2}^{\epsilon}(t)$ is the same as $A_{2}^{\epsilon}(t)$. (3) \[ \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\left\vert \delta\sigma_{x}(t)\xi^{2,\epsilon }(t)\right\vert ^{2}I_{E_{\epsilon}}(t)dt\right] \leq C\epsilon\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert \xi^{2,\epsilon }(t)\right\vert ^{4}\right] \right\} ^{\frac{1}{2}}=o(\epsilon^{2}), \end{array} \] \[ \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon} (t)\eta^{1,\epsilon}(t)^{2}-\sigma_{yy}(t)Y_{1}(t)^{2}\right\vert ^{2}dt\right] \ \ \\ \leq\mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon }(t)\eta^{2,\epsilon}(t)(\eta^{1,\epsilon}(t)+Y_{1}(t))\right\vert ^{2}dt\right] +\mathbb{E}\left[ \int_{0}^{T}\left\vert \tilde{\sigma} _{yy}^{\epsilon}(t)-\sigma_{yy}(t)\right\vert ^{2}Y_{1}(t)^{4}dt\right] \\ \leq C\mathbb{E}\left[ \int_{0}^{T}\left\vert \eta^{2,\epsilon}(t)\right\vert ^{2}\left\vert \eta^{1,\epsilon}(t)+Y_{1}(t)\right\vert ^{2}dt\right] +\mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert Y_{1}(t)\right\vert ^{4}\int_{0}^{T}\left\vert \tilde{\sigma}_{yy}^{\epsilon}(t)-\sigma _{yy}(t)\right\vert ^{2}dt\right] \\ \leq C\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert \eta^{2,\epsilon}(t)\right\vert ^{4}\right] \right\} ^{\frac{1}{2}}\left\{ \mathbb{E}\left[ \sup\limits_{t\in\lbrack0,T]}\left\vert \eta^{1,\epsilon }(t)+Y_{1}(t)\right\vert ^{4}\right] \right\} ^{\frac{1}{2}}+o(\epsilon ^{2})\\ =o(\epsilon^{2}), \end{array} \] the other terms are similar. Thus \[ \mathbb{E}\left[ \int_{0}^{T}\|B_{2}^{\epsilon}(t)\|^{2}dt\right] =o(\epsilon^{2}). \] (4) \[ \mathbb{E}\left[ \|D_{2}^{\epsilon}(T)\|^{2}\right] \leq C\mathbb{E}\left[ \|\tilde{\phi}_{xx}^{\epsilon}(T)-\phi_{xx}(\bar{X}(T))\|^{2}\|\xi^{1,\epsilon }(T)\|^{4}+\|\xi^{2,\epsilon}(T)\|^{2}\|\xi^{1,\epsilon}(T)+X_{1}(T)\|^{2}\right] =o(\epsilon^{2}). \] Thus, \[ \mathbb{E}\left\{ \sup\limits_{t\in\lbrack0,T]}[\|\xi^{3,\epsilon} (t)\|^{2}+\|\eta^{3,\epsilon}(t)\|^{2}]+\int_{0}^{T}\|\zeta^{3,\epsilon} (t)\|^{2}dt\right\} =o(\epsilon^{2}). \] \end{proof} Now we introduce the second-order adjoint equation: \begin{equation} \left\{ \begin{array} [c]{rl} -dP(t)= & \left\{ P(t)\left[ (D\sigma(t)^{\intercal}[1,p(t),K_{1} (t)]^{\intercal})^{2}+2Db(t)^{\intercal}[1,p(t),K_{1}(t)]^{\intercal} +H_{y}(t)\right] \right. \\ & +2Q(t)D\sigma(t)^{\intercal}[1,p(t),K_{1}(t)]^{\intercal}+\left[ 1,p(t),K_{1}(t)\right] D^{2}H(t)\left[ 1,p(t),K_{1}(t)\right] ^{\intercal }\left. +H_{z}(t)K_{2}(t)\right\} dt\\ & -Q(t)dB(t),\\ P(T)= & \phi_{xx}(\bar{X}(T)), \end{array} \right. \label{eq-P-sigmaz0} \end{equation} where \[ \begin{array} [c]{ll} H(t,x,y,z,u,p,q)= & g(t,x,y,z,u)+pb(t,x,y,z,u)+q\sigma(t,x,y,z,u), \end{array} \] \[ \begin{array} [c]{ll} K_{2}(t)= & \left( 1-p(t)A(t)\right) ^{-1}\left\{ p(t)\sigma_{y} (t)+2\left[ \sigma_{x}(t)+\sigma_{y}(t)p(t)+A(t)K_{1}(t)\right] \right\} P(t)\\ & +\left( 1-p(t)A(t)\right) ^{-1}\left\{ Q(t)+p(t)[1,p(t)]D^{2}\sigma _{1}(t)[1,p(t)]^{\intercal}\right\} . \end{array} \] (\ref{eq-P-sigmaz0}) is a linear BSDE with non-Lipschitz coefficient for $P(\cdot)$. Then, by Theorem \ref{q-exp-th} in appendix, (\ref{eq-P-sigmaz0}) has a unique pair of solution according to Theorem 5.21 in \cite{Pardoux-book} . By the same analysis as in Lemma \ref{relation-y2}, we introduce the following auxiliary equation: \begin{equation} \begin{array} [c]{l} \hat{Y}(t)=\int_{t}^{T}\left\{ (H_{y}(s)+\sigma_{y}(s)g_{z} (s)p(s)(1-p(s)A(s))^{-1})\hat{Y}(s)+\left( H_{z}(s)+\sigma_{z}(s)g_{z} (s)p(s)(1-p(s)A(s))^{-1}\right) \hat{Z}(s)\right. \\ \ \ \ \ \ \ \ \ \ \ \left. +\left[ \delta H(s,\Delta)+\frac{1}{2} P(s)\delta\sigma(s,\Delta)^{2}\right] I_{E_{\epsilon}}(s)\right\} ds-\int_{t}^{T}\hat{Z}(s)dB(s), \end{array} \label{yhat-sigmaz0} \end{equation} where $\delta H(s,\Delta):=p(s)\delta b(s,\Delta)+q(s)\delta\sigma (s,\Delta)+\delta g(s,\Delta)$, and obtain the following relationship. \begin{lemma} \label{relation-second-order-q-unbound}Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-sig-small} hold. Then \[ \begin{array} [c]{rl} Y_{2}(t) & =p(t)X_{2}(t)+\frac{1}{2}P(t)X_{1}(t)^{2}+\hat{Y}(t),\\ Z_{2}(t) & =\mathbf{I(t)}+\hat{Z}(t), \end{array} \] where $(\hat{Y}(\cdot),\hat{Z}(\cdot))$ is the solution to \eqref{yhat-sigmaz0} and \begin{align} \mathbf{I(t)} & =K_{1}(t)X_{2}(t)+\frac{1}{2}K_{2}(t)X_{1}^{2} (t)+(1-p(t)A(t))^{-1}p(t)(\sigma_{y}(t)\hat{Y}(t)+A(t)\hat{Z}(t))+P(t)\delta \sigma(t,\Delta)X_{1}(t)I_{E_{\epsilon}}(t)\\ & \;+(1-p(t)A(t))^{-1}p(t)\left[ \delta\sigma_{x}(t,\Delta)X_{1} (t)+\delta\sigma_{y}(t,\Delta)p(t)X_{1}(t)\right] I_{E_{\epsilon}}(t). \end{align} \end{lemma} \begin{proof} Applying the techniques in Lemma \ref{lemma-y1}, we can deduce the above relationship similarly. \end{proof} Combing the estimates in Lemma \ref{est-second-order-q-unbound} and the relationship in Lemma \ref{relation-second-order-q-unbound}, we deduce that \[ Y^{\epsilon}(0)-\bar{Y}(0)=Y_{1}(0)+Y_{2}(0)+o(\epsilon)=\hat{Y} (0)+o(\epsilon)\geq0. \] Define \[ \begin{array} [c]{l} \mathcal{H}(t,x,y,z,u,p,q,P)=pb(t,x,y,z,u)+q\sigma(t,x,y,z,u)+\frac{1} {2}P(\sigma(t,x,y,z,u)-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar {u}(t)))^{2}\\ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \ +g(t,x,y,z+p(t)(\sigma(t,x,y,z,u)-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar {Z}(t),\bar{u}(t))),u). \end{array} \] By the same analysis as in Theorem \ref{Th-MP}, we obtain the following maximum principle. \begin{theorem} \label{th-mp-q-unboud}Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-sig-small} hold. Let $\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and $(\bar{X}(\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state processes of (\ref{state-eq}). Then the following stochastic maximum principle holds: \[ \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),P(t))\geq \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u} (t),p(t),q(t),P(t)),\ \ \ \forall u\in U,\ a.e.,\ a.s.. \] \end{theorem} \subsection{The general case\label{sec-general}} When Brownian motion in \eqref{state-eq} is $d$-dimensional, by similar analysis as for $1$-dimensional case, we obtain the following results. The state equation becomes \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),Y(t),Z(t),u(t))dt+\sigma^{\intercal} (t,X(t),Y(t),Z(t),u(t))dB(t)\\ dY(t)= & -g(t,X(t),Y(t),Z(t),u(t))dt+Z^{\intercal}(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\phi(X(T)), \end{array} \right. \label{state-eq-multi} \end{equation} where \[ \sigma:[0,T]\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{d}\times U\rightarrow\mathbb{R}^{d}. \] The first-order adjoint equation is \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -\left\{ g_{x}(t)+g_{y}(t)p(t)+\left\langle g_{z}(t),K_{1} (t)\right\rangle +(b_{x}(t)+b_{y}(t)p(t))p(t)\right. \\ & \left. +p(t)\langle b_{z}(t),K_{1}(t)\rangle+\langle(\sigma_{x} (t)+\sigma_{y}(t)p(t)),q(t)\rangle+\langle q(t),\sigma_{z}(t)K_{1} (t)\rangle\right\} dt+q^{\intercal}(t)dB(t),\\ p(T)= & \phi_{x}(\bar{X}(T)), \end{array} \right. \label{eq-p-multi} \end{equation} where \[ \begin{array} [c]{rl} K_{1}(t) & =(I-p(t)\sigma_{z}(t))^{-1}\left[ p(t)(\sigma_{x}(t)+\sigma _{y}(t)p(t))+q(t)\right] \in\mathbb{R}^{d},\\ g_{z}(t) & :=(g_{z^{1}}(t),g_{z^{2}}(t),...,g_{z^{d}}(t))^{\intercal},\text{ }\\ b_{z}(t) & :=(b_{z^{1}}(t),b_{z^{2}}(t),...,b_{z^{d}}(t))^{\intercal},\\ \sigma_{z}(t) & =\left( \begin{array} [c]{c} \sigma_{z^{1}}^{1}(t),\sigma_{z^{2}}^{1}(t),...,\sigma_{z^{d}}^{1}(t)\\ \sigma_{z^{1}}^{2}(t),\sigma_{z^{2}}^{2}(t),...,\sigma_{z^{d}}^{2}(t)\\ \vdots\\ \sigma_{z^{1}}^{d}(t),\sigma_{z^{2}}^{d}(t),...,\sigma_{z^{d}}^{d}(t) \end{array} \right) \in\mathbb{R}^{d\times d}. \end{array} \] Denote the Hessian matrix of $\sigma^{i}(t)$ with respect to $(x,y,z^{1} ,z^{2},...,z^{d})$ by $D^{2}\sigma^{i}$. Set \[ \lbrack1,p(t),K_{1}^{\intercal}(t)]D^{2}\sigma(t)[1,p(t),K_{1}^{\intercal }(t)]^{\intercal}=\left( \begin{array} [c]{c} \lbrack1,p(t),K_{1}^{\intercal}(t)]D^{2}\sigma^{1}(t)[1,p(t),K_{1}^{\intercal }(t)]^{\intercal}\\ \lbrack1,p(t),K_{1}^{\intercal}(t)]D^{2}\sigma^{2}(t)[1,p(t),K_{1}^{\intercal }(t)]^{\intercal}\\ \vdots\\ \lbrack1,p(t),K_{1}^{\intercal}(t)]D^{2}\sigma^{d}(t)[1,p(t),K_{1}^{\intercal }(t)]^{\intercal} \end{array} \right) \in\mathbb{R}^{d}. \] Then, the second-order adjoint equation is \begin{equation} \left\{ \begin{array} [c]{rl} -dP(t)= & \left\{ P(t)[(\sigma_{x}(t)+p(t)\sigma_{y}(t)+\sigma_{z} (t)K_{1}(t))^{\intercal}(\sigma_{x}(t)+p(t)\sigma_{y}(t)+\sigma_{z} (t)K_{1}(t))\right. \\ & +2(b_{x}(t)+b_{y}(t)p(t)+\langle b_{z}(t),K_{1}(t)\rangle)]+2\langle Q(t),(\sigma_{x}(t)+p(t)\sigma_{y}(t)+\sigma_{z}(t)K_{1}(t))\rangle\\ & +p(t)b_{y}(t)P(t)+p(t)[1,p(t),K_{1}^{\intercal}(t)]D^{2}b(t)[1,p(t),K_{1} ^{\intercal}(t)]^{\intercal}\\ & +\langle q(t),\mathbf{[}\sigma_{y}(t)P(t)+[1,p(t),K_{1}^{\intercal} (t)]D^{2}\sigma(t)[1,p(t),K_{1}^{\intercal}(t)]^{\intercal}\mathbf{]} \rangle+g_{y}(t)P(t)\\ & \left. +[I,p(t),K_{1}^{\intercal}(t)]D^{2}g(t)[I,p(t),K_{1}^{\intercal }(t)]^{\intercal}+\left\langle g_{z}(t)+b_{z}(t)p(t),K_{2}(t)\right\rangle +\langle q(t),\sigma_{z}(t)K_{2}(t)\rangle\right\} dt\\ & -Q^{\intercal}(t)dB(t),\\ P(T)= & \phi_{xx}(\bar{X}(T)), \end{array} \right. \label{eq-P-multi} \end{equation} where \[ \begin{array} [c]{ll} K_{2}(t)= & (I-p(t)\sigma_{z}(t))^{-1}p(t)\left\{ \sigma_{y} (t)P(t)+[1,p(t),K_{1}^{\intercal}(t)]D^{2}\sigma(t)[1,p(t),K_{1}^{\intercal }(t)]^{\intercal}\right\} \\ & +(I-p(t)\sigma_{z}(t))^{-1}\{Q(t)+2P(t)(\sigma_{x}(t)+\sigma_{y} (t)p(t)+\sigma_{z}(t)K_{1}(t))\}\in\mathbb{R}^{d}. \end{array} \] Define \begin{equation} \begin{array} [c]{ll} \mathcal{H}(t,x,y,z,u,p,q,P)= & pb(t,x,y,z+\Delta(t),u)+\langle q,\sigma (t,x,y,z+\Delta(t),u)\rangle\\ & +\frac{1}{2}P(\sigma(t,x,y,z+\Delta(t),u)-\sigma(t,\bar{X}(t),\bar {Y}(t),\bar{Z}(t),\bar{u}(t)))^{\intercal}\\ & \cdot(\sigma(t,x,y,z+\Delta(t),u)-\sigma(t,\bar{X}(t),\bar{Y}(t),\bar {Z}(t),\bar{u}(t)))\\ & +g(t,x,y,z+\Delta(t),u), \end{array} \label{h-function-multi} \end{equation} where $\Delta(t)\ $satisfies \begin{equation} \Delta(t)=p(t)(\sigma(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t)+\Delta(t),u)-\sigma (t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u}(t))),\;t\in\lbrack0,T]. \label{def-delt-multi} \end{equation} Thus, we obtain the following maximum principle. \begin{theorem} Suppose Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-q-bound} hold. Let $\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and $(\bar{X}(\cdot ),\bar{Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state processes of (\ref{state-eq-multi}). Then the following stochastic maximum principle holds: \[ \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),u,p(t),q(t),P(t))\geq \mathcal{H}(t,\bar{X}(t),\bar{Y}(t),\bar{Z}(t),\bar{u} (t),p(t),q(t),P(t)),\ \ \ \forall u\in U,\ a.e.,\ a.s., \] where $(p\left( \cdot\right) ,q\left( \cdot\right) )$, $\left( P\left( \cdot\right) ,Q\left( \cdot\right) \right) $ satisfy (\ref{eq-p-multi}), (\ref{eq-P-multi}) respectively, and $\Delta(\cdot)$ satisfies (\ref{def-delt-multi}).\ \end{theorem} \begin{remark} The above theorem still hold under Assumptions \ref{assum-2}, \ref{assum-3} and \ref{assm-sig-small}. \end{remark} \section{A linear quadratic control problem} In this section, we study a linear quadratic control problem by the results in the section 3. For simplicity of presentation, we suppose all the processes are one dimensional. Consider the following linear forward-backward stochastic control system \begin{equation} \left\{ \begin{array} [c]{rcl} dX(t) & = & [A_{1}(t)X(t)+B_{1}(t)Y(t)+C_{1}(t)Z(t)+D_{1}(t)u(t)]dt\\ & & +[A_{2}(t)X(t)+B_{2}(t)Y(t)+C_{2}(t)Z(t)+D_{2}(t)u(t)]dB(t),\\ dY(t) & = & -[A_{3}(t)X(t)+B_{3}(t)Y(t)+C_{3}(t)Z(t)+D_{3} (t)u(t)]dt+Z(t)dB(t),\\ X(0) & = & x_{0},\ Y(T)=FX(T)+J, \end{array} \right. \label{state-lq} \end{equation} and minimizing the following cost functional \begin{equation} J(u(\cdot))=\mathbb{E}\left[ \int_{0}^{T}\left( A_{4}(t)X(t)^{2} +B_{4}(t)Y(t)^{2}+C_{4}(t)Z(t)^{2}+D_{4}(t)u(t)^{2}\right) dt+GX(T)^{2} +Y(0)^{2}\right] , \label{cost-lq} \end{equation} where $A_{i}$, $B_{i}$, $C_{i}$, $D_{i}$ $i=1,2,3,4$ are deterministic $\mathbb{R}$-valued functions, $F$, $G$ are deterministic constants and $J$ is $\mathcal{F}_{T}$-measurable bounded random variable. Let $\bar{u}(\cdot)$ be the optimal control, and the corresponding optimal state is $(\bar{X} (\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$. The variational equation becomes \[ \left\{ \begin{array} [c]{l} d\left( X_{1}(t)+X_{2}(t)\right) \\ \text{ }=[A_{1}(t)\left( X_{1}(t)+X_{2}(t)\right) +B_{1}(t)\left( Y_{1}(t)+Y_{2}(t)\right) +C_{1}(t)\left( Z_{1}(t)+Z_{2}(t)\right) +D_{1}(t)(u^{\epsilon}(t)-\bar{u}(t))]dt\\ \text{ \ \ }+[A_{2}(t)\left( X_{1}(t)+X_{2}(t)\right) +B_{2}(t)\left( Y_{1}(t)+Y_{2}(t)\right) +C_{2}(t)\left( Z_{1}(t)+Z_{2}(t)\right) +D_{2}(t)(u^{\epsilon}(t)-\bar{u}(t))]dB(t),\\ d\left( Y_{1}(t)+Y_{2}(t)\right) \\ \text{ }=-[A_{3}(t)\left( X_{1}(t)+X_{2}(t)\right) +B_{3}(t)\left( Y_{1}(t)+Y_{2}(t)\right) +C_{3}(t)\left( Z_{1}(t)+Z_{2}(t)\right) +D_{3}(t)(u^{\epsilon}(t)-\bar{u}(t))]dt\\ \ \ \ +\left( Z_{1}(t)+Z_{2}(t)\right) dB(t),\\ X_{1}(0)+X_{2}(0)=0,\ Y_{1}(T)+Y_{2}(T)=F\left( X_{1}(T)+X_{2}(T)\right) , \end{array} \right. \] and the first order adjoint equation is \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -\left\{ A_{3}(t)+B_{3}(t)p(t)+C_{3}(t)K_{1}(t)+A_{1}(t)p(t)+B_{1} (t)p^{2}(t)\right. \\ & \left. +C_{1}(t)K_{1}(t)p(t)+A_{2}(t)q(t)+B_{2}(t)p(t)q(t)+C_{2} (t)K_{1}(t)q(t)\right\} dt+q(t)dB(t),\\ p(T)= & F, \end{array} \right. \label{eq-p-lq} \end{equation} where \[ K_{1}(t)=(1-p(t)C_{2}(t))^{-1}\left[ A_{2}(t)p(t)+B_{2}(t)p^{2} (t)+q(t)\right] . \] This adjoint equation is a nonlinear backward stochastic differential equation with deterministic coefficients and the solution to \eqref{eq-p-lq} is $(p(\cdot),0)$, which $p(\cdot)$ satisfies the following ODE \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -\left\{ A_{3}(t)+B_{3}(t)p(t)+C_{3}(t)K_{1}(t)+A_{1}(t)p(t)+B_{1} (t)p^{2}(t)+C_{1}(t)K_{1}(t)p(t)\right\} dt,\\ p(T)= & F, \end{array} \right. \label{eq-p-lq-ode} \end{equation} with \[ K_{1}(t)=(1-p(t)C_{2}(t))^{-1}\left[ A_{2}(t)p(t)+B_{2}(t)p^{2}(t)\right] . \] \begin{remark} It should be note that in our context the Assumption \ref{assm-q-bound} holds. \end{remark} Moreover, $\Delta(t)$ has the following explicitly form \[ \Delta(t)=(1-p(t)C_{2}(t))^{-1}p(t)D_{2}(t)(u(t)-\bar{u}(t)). \] Since $\bar{u}(\cdot)$ is the optimal control, \begin{equation} J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))\geq0. \label{eq-lq-cost} \end{equation} By Lemma \ref{lemma-est-sup}, the following estimates hold, \[ \begin{array} [c]{lll} X^{\epsilon}(t)-\bar{X}(t) & = & X_{1}(t)+X_{2}(t)+o(\epsilon),\\ Y^{\epsilon}(t)-\bar{Y}(t) & = & Y_{1}(t)+Y_{2}(t)+o(\epsilon),\\ Z^{\epsilon}(t)-\bar{Z}(t) & = & Z_{1}(t)+Z_{2}(t)+o(\epsilon). \end{array} \] We can expand (\ref{eq-lq-cost}) term by term as follows \[ \begin{array} [c]{l} \mathbb{E}\left[ \int_{0}^{T}A_{4}(t)\left( X^{\epsilon}(t)^{2}-\bar {X}(t)^{2}\right) dt\right] \\ \text{ }=\mathbb{E}\left\{ \int_{0}^{T}\left[ 2A_{4}(t)\bar{X}(t)\left( X_{1}(t)+X_{2}(t)\right) +A_{4}(t)X_{1}(t)^{2}\right] dt\right\} +o(\epsilon). \end{array} \] Similarly, one has \[ \begin{array} [c]{rl} \mathbb{E}\left[ \int_{0}^{T}B_{4}(t)\left( Y^{\epsilon}(t)^{2}-\bar {Y}(t)^{2}\right) dt\right] & =\mathbb{E}\left\{ \int_{0}^{T}\left[ 2B_{4}(t)\bar{Y}(t)\left( Y_{1}(t)+Y_{2}(t)\right) +B_{4}(t)Y_{1} (t)^{2}\right] dt\right\} +o(\epsilon);\\ \mathbb{E}\left[ G\left( X^{\epsilon}(T)^{2}-\bar{X}(T)^{2}\right) \right] & =\mathbb{E}\left[ 2G\bar{X}(T)\left( X_{1}(T)+X_{2}(T)\right) +GX_{1}(T)^{2}\right] +o(\epsilon);\\ Y^{\varepsilon}(0)^{2}-\bar{Y}(0)^{2} & =2\bar{Y}(0)\left( Y_{1} (0)+Y_{2}(0)\right) +Y_{1}(0)^{2}+o(\epsilon);\\ \mathbb{E}\left[ \int_{0}^{T}C_{4}(t)\left( Z^{\epsilon}(t)^{2}-\bar {Z}(t)^{2}\right) dt\right] & =\mathbb{E}\left\{ \int_{0}^{T}\left[ 2C_{4}(t)\bar{Z}(t)\left( Z_{1}(t)+Z_{2}(t)\right) +C_{4}(t)Z_{1} (t)^{2}\right] dt\right\} +o(\epsilon).\ \end{array} \] Thus \[ \begin{array} [c]{l} J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))\\ =\mathbb{E}\left\{ \int_{0}^{T}\left[ 2A_{4}(t)\bar{X}(t)\left( X_{1}(t)+X_{2}(t)\right) +2B_{4}(t)\bar{Y}(t)\left( Y_{1}(t)+Y_{2} (t)\right) +2C_{4}(t)\bar{Z}(t)\left( Z_{1}(t)+Z_{2}(t)\right) \right. \right. \\ \left. +A_{4}(t)X_{1}(t)^{2}+B_{4}(t)Y_{1}(t)^{2}+C_{4}(t)Z_{1}(t)^{2} +2D_{4}(t)\bar{u}(t)\left( u^{\epsilon}(t)-\bar{u}(t)\right) +D_{4} (t)\left( u^{\epsilon}(t)-\bar{u}(t)\right) ^{2}\right] dt\\ \left. +2G\bar{X}(T)\left( X_{1}(T)+X_{2}(T)\right) +GX_{1}(T)^{2}+2\bar {Y}(0)\left( Y_{1}(0)+Y_{2}(0)\right) +Y_{1}(0)^{2}\right\} +o(\epsilon). \end{array} \] \ \ \ \ \ $\ \ $Introduce the adjoint equation for $X_{1}(t)+X_{2} (t),Y_{1}(t)+Y_{2}(t),Z_{1}(t)+Z_{2}(t)$ as \[ \left\{ \begin{array} [c]{rl} dh(t)= & \left[ B_{3}(t)h(t)+B_{1}(t)m(t)+B_{2}(t)n(t)+2B_{4}(t)Y(t)\right] dt\\ & +\left[ C_{3}(t)h(t)+C_{1}(t)m(t)+C_{2}(t)n(t)+2C_{4}(t)Z(t)\right] dB(t),\\ h(0)= & 2\bar{Y}(0),\\ dm(t)= & -\left[ A_{3}(t)h(t)+A_{1}(t)m(t)+A_{2}(t)n(t)+2A_{4}(t)X(t)\right] dt+n(t)dB(t),\\ m(T)= & 2G\bar{X}(T)+Fh(T). \end{array} \right. \] Applying It\^{o}'s formula to $m(t)\left( X_{1}(t)+X_{2}(t)\right) -h(t)\left( Y_{1}(t)+Y_{2}(t)\right) $, we get \[ \begin{array} [c]{l} J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))\\ =\mathbb{E}\left\{ \int_{0}^{T}\left[ \left( D_{1}(t)m(t)+D_{2} (t)n(t)+D_{3}(t)h(t)+2D_{4}(t)\bar{u}(t)\right) \left( u^{\epsilon} (t)-\bar{u}(t)\right) \right. \right. \\ \left. \left. +A_{4}(t)X_{1}(t)^{2}+B_{4}(t)Y_{1}(t)^{2}+C_{4} (t)Z_{1}(t)^{2}+D_{4}(t)\left( u^{\epsilon}(t)-\bar{u}(t)\right) ^{2}\right] dt+GX_{1}(T)^{2}+Y_{1}(0)^{2}\right\} +o(\epsilon). \end{array} \] Noting that the relationship between $X_{1}(t),Y_{1}(t)$ and $Z_{1}(t),$ \[ \begin{array} [c]{ll} Y_{1}(t)= & p(t)X_{1}(t),\\ Z_{1}(t)= & K_{1}(t)X_{1}(t)+\Delta(t)I_{E_{\epsilon}}(t), \end{array} \] thus, \[ \begin{array} [c]{l} J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))\\ =\mathbb{E}\left\{ \int_{0}^{T}\left[ \left( A_{4}(t)+B_{4}(t)p(t)^{2} +C_{4}(t)K_{1}(t)^{2}\right) X_{1}(t)^{2}+C_{4}(t)\Delta(t)^{2} I_{E_{\epsilon}}(t)+D_{4}(t)\left( u^{\epsilon}(t)-\bar{u}(t)\right) ^{2}\right. \right. \\ \text{ \ \ \ }\left. \left. +\left( D_{1}(t)m(t)+D_{2}(t)n(t)+D_{3} (t)h(t)+2D_{4}(t)\bar{u}(t)\right) \left( u^{\epsilon}(t)-\bar{u}(t)\right) \right] dt+GX_{1}(T)^{2}\right\} +o(\epsilon). \end{array} \] $\ \ \ $Introducing the adjoint equation for $X_{1}(t)^{2}$, \begin{equation} \left\{ \begin{array} [c]{rl} -dP(t)= & \left[ R_{1}(t)P(t)+R_{2}(t)Q(t)+A_{4}(t)+B_{4}(t)p(t)^{2} +C_{4}(t)K_{1}(t)^{2}\right] dt-Q(t)dB(t),\\ P(T)= & G, \end{array} \right. \end{equation} where \[ \begin{array} [c]{ll} R_{1}(t)= & 2\left( A_{1}(t)+B_{1}(t)p(t)+C_{1}(t)K_{1}(t)\right) +\left( A_{2}(t)+B_{2}(t)p(t)+C_{2}(t)K_{1}(t)\right) ^{2},\\ R_{2}(t)= & 2\left( A_{2}(t)+B_{2}(t)p(t)+C_{2}(t)K_{1}(t)\right) . \end{array} \] Similar to $p(\cdot)$, the solution to $(P(\cdot),Q(\cdot))$ is $(P(\cdot ),0)$, which $P(\cdot)$ satisfies the following ODE, \begin{equation} \left\{ \begin{array} [c]{rl} -dP(t)= & \left[ R_{1}(t)P(t)+A_{4}(t)+B_{4}(t)p(t)^{2}+C_{4}(t)K_{1} (t)^{2}\right] dt,\\ P(T)= & G, \end{array} \right. \end{equation} where \[ \begin{array} [c]{ll} R_{1}(t)= & 2\left( A_{1}(t)+B_{1}(t)p(t)+C_{1}(t)K_{1}(t)\right) +\left( A_{2}(t)+B_{2}(t)p(t)+C_{2}(t)K_{1}(t)\right) ^{2}.\\ & \end{array} \] We obtain \[ \begin{array} [c]{l} J(u^{\epsilon}(\cdot))-J(\bar{u}(\cdot))\\ =\mathbb{E}\left\{ \int_{0}^{T}\left[ \left( D_{1}(t)m(t)+D_{2} (t)n(t)+D_{3}(t)h(t)+2D_{4}(t)\bar{u}(t)\right) \left( u^{\epsilon} (t)-\bar{u}(t)\right) \right. \right. \\ \left. \left. +P(t)D_{2}(t)^{2}(u^{\epsilon}(t)-\bar{u}(t))^{2} +D_{4}(t)\left( u^{\epsilon}(t)-\bar{u}(t)\right) ^{2}+C_{4}(t)\Delta (t)^{2}I_{E_{\epsilon}}(t)\right] dt\right\} +o(\epsilon). \end{array} \] Thus, we obtain the following maximum principle for (\ref{state-lq} )-(\ref{cost-lq}). \begin{theorem} \label{th-mp-lq}Suppose Assumptions \ref{assum-2} and \ref{assum-3} hold. Let $\bar{u}(\cdot)\in\mathcal{U}[0,T]$ be optimal and $(\bar{X}(\cdot),\bar {Y}(\cdot),\bar{Z}(\cdot))$ be the corresponding state processes of (\ref{state-lq}). Then the following stochastic maximum principle holds: \[ \begin{array} [c]{l} \left( D_{1}(t)m(t)+D_{2}(t)n(t)+D_{3}(t)h(t)+2D_{4}(t)\bar{u}(t)\right) (u-\bar{u}(t))\\ +\left[ \frac{C_{4}(t)p(t)^{2}D_{2}(t)^{2}}{\left( 1-p(t)C_{2}(t)\right) ^{2}}+D_{4}(t)+P(t)D_{2}(t)^{2}\right] (u-\bar{u}(t))^{2}\geq0,\ \forall u\in U,\ a.e.,\ a.s.. \end{array} \] \end{theorem} \begin{remark} Using Theorem \ref{th-mp-q-unboud}, we can also consider linear quadratic control problem with random coefficients. \end{remark} Now, we give an example to show the difference between the global and local maximum principle. \begin{example} Consider the following linear forward-backward stochastic control system \begin{equation} \left\{ \begin{array} [c]{rcl} dX(t) & = & [aZ(t)+bu(t)]dB(t),\\ dY(t) & = & -cu(t)dt+Z(t)dB(t),\\ X(0) & = & 1,\ Y(T)=dX(T), \end{array} \right. \end{equation} and minimizing the following cost functional \[ J(u(\cdot))=\mathbb{E}\left[ \int_{0}^{T}u(t)^{2}dt\right] +Y(0)^{2}, \] where $a$, $b$, $c$, $d$ are constants such that, $0<\left\vert 2cd\right\vert \leq1$ and $ad<1$, and $U=\left\{ -1,0,1\right\} $. Let $\bar{u}(\cdot)$ be the optimal control, and the corresponding optimal state is $(\bar{X} (\cdot),\bar{Y}(\cdot),\bar{Z}(\cdot))$. In this case $p(t)=d$, $q(t)=0$, $P(t)=Q(t)=0$, $h(t)=2\bar{Y}(0)$, $m(t)=2d\bar{Y}(0)$, $n(t)=0$, for $t\in\left[ 0,T\right] $. The maximum principle by Theorem \ref{th-mp-lq} is \begin{equation} 2\left( c\bar{Y}(0)+\bar{u}(t)\right) (u-\bar{u}(t))+(u-\bar{u}(t))^{2} \geq0,\ \forall u\in U\ a.e.,\ a.s.. \label{mp-exp} \end{equation} Noting that $\bar{Y}(0)=d$ for $\bar{u}(t)=0$ , then it is easy to check that $\bar{u}(t)=0$ satisfies the maximum principle \eqref{mp-exp}. Furthermore, we can prove $\bar{u}(t)=0$ is the optimal control. For each $u\left( \cdot\right) \in\mathcal{U}[0,T]$, \[ Y(0)=\mathbb{E}\left[ dX(T)+\int_{0}^{T}cu(t)dt\right] =d+\mathbb{E}\left[ \int_{0}^{T}cu(t)dt\right] . \] Then \[ \begin{array} [c]{rl} J(u(\cdot))-J(\bar{u}(\cdot))= & \mathbb{E}\left[ \int_{0}^{T}u(t)^{2} dt\right] +Y(0)^{2}-d^{2}\\ = & \left( \mathbb{E}\left[ \int_{0}^{T}cu(t)dt\right] \right) ^{2}+\mathbb{E}\left[ \int_{0}^{T}\left( u(t)^{2}+2cdu(t)\right) dt\right] \\ \geq & 0, \end{array} \] which implies $\bar{u}(\cdot)=0$ is optimal. \newline\ \ When the control domain is $U=[-1,1]$, similar to Corollary \ref{cor-mp-convex}, by Theorem \ref{th-mp-lq} we obtain the following maximum principle, \begin{equation} 2\left( c\bar{Y}(0)+\bar{u}(t)\right) (u-\bar{u}(t))\geq0,\ \forall u\in U\ a.e.,\ a.s.. \label{mp-exp-convex} \end{equation} It is obvious that $\bar{u}(\cdot)=0$ does not satisfy the maximum principle \eqref{mp-exp-convex}. \end{example} \section{Appendix} \subsection{$L^{p}$-estimate of decoupled FBSDEs} The following Lemma is a combination of Theorem 3.17 and Theorem 5.17 in \cite{Pardoux-book}. \begin{lemma} \label{sde-bsde}For each fixed $p>1$ and a pair of adapted stochastic process $(y(\cdot),z(\cdot))$, consider the following system \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & b(t,X(t),y(t),z(t))dt+\sigma(t,X(t),y(t),z(t))dB(t),\\ dY(t)= & -g(t,X(t),Y(t),Z(t))dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(t)=\phi(X(T)), \end{array} \right. \label{fbsde-pardoux} \end{equation} where $b$, $\sigma$, $g$, $\phi$ are the same in equation (\ref{fbsde}). If the coefficients satisfy (i) $b(\cdot,0,y(\cdot),z(\cdot))$, $\sigma(\cdot,0,y(\cdot),z(\cdot))$, $g(\cdot,0,0,0)$ are $\mathbb{F}$-adapted processes and \[ \mathbb{E}\left\{ \|\phi(0)\|^{p}+\left( \int_{0}^{T}\left[ \|b(t,0,y(t),z(t))\|+\|g(t,0,0,0)\|\right] dt\right) ^{p}+\left( \int_{0} ^{T}\|\sigma(t,0,y(t),z(t))\|^{2}dt\right) ^{\frac{p}{2}}\right\} <\infty, \] (ii) \[ \begin{array} [c]{rl} \|\psi(t,x_{1},y,z)-\psi(t,x_{2},y,z)\| & \leq L_{1}\|x_{1}-x_{2}\|,\ \ \text{for }\ \psi=b,\sigma;\\ \|g(t,x_{1},y_{1},z_{1})-g(t,x_{2},y_{2},z_{2})\| & \leq L_{1}(\|x_{1} -x_{2}\|+\|y_{1}-y_{2}\|+\|z_{1}-z_{2}\|), \end{array} \] then (\ref{fbsde-pardoux}) has a unique solution $(X(\cdot),Y(\cdot ),Z(\cdot))\in L_{\mathcal{F}}^{p}(\Omega;C([0,T],\mathbb{R}^{n}))\times L_{\mathcal{F}}^{p}(\Omega;C([0,T],\mathbb{R}^{m}))\times L_{\mathcal{F} }^{2,p}([0,T];\mathbb{R}^{m\times d})$ and there exists a constant $C_{p}$ which only depends on $L_{1}$, $p$, $T$ such that \[ \begin{array} [c]{l} \mathbb{E}\left\{ \sup\limits_{t\in\lbrack0,T]}\left[ \|X(t)\|^{p} +\|Y(t)\|^{p}\right] +\left( \int_{0}^{T}\|Z(t)\|^{2}dt\right) ^{\frac{p}{2} }\right\} \\ \ \leq C_{p}\mathbb{E}\left\{ \left[ \int_{0}^{T}\left( \|b(t,0,y(t),z(t))\|+\|g(t,0,0,0)\|\right) dt\right] ^{p}+\left( \int_{0} ^{T}\|\sigma(t,0,y(t),z(t))\|^{2}dt\right) ^{\frac{p}{2}}+\|\phi(0)\|^{p} +\|x_{0}\|^{p}\right\} . \end{array} \] \end{lemma} \subsection{An estimate of $Z$ for some BSDEs} Consider the following BSDE\ \begin{equation} Y(t)=\xi+\int_{t}^{T}f(s,Y(s),Z(s))ds-\int_{t}^{T}Z(s)dB(s). \label{appen-eq-bsde} \end{equation} \begin{theorem} \label{q-exp-th} Suppose that $(Y(\cdot),Z(\cdot))\in L_{\mathcal{F}}^{\infty }(0,T;\mathbb{R})\times L_{\mathcal{F}}^{2,2}([0,T];\mathbb{R}^{d})$ solves BSDE (\ref{appen-eq-bsde}), and \newline$\left\vert f(s,Y(s),Z(s))\right\vert \leq C_{1}\left( 1+\|Z(s)\|^{2}\right) $, where $C_{1}$ is a constant. Then there exists a $\delta>0$ such that for each $\lambda_{1}<\delta$, \begin{equation} \mathbb{E}\left[ \left. \exp\left( \lambda_{1}\int_{t}^{T}\|Z(s)\|^{2} ds\right) \right\vert \mathcal{F}_{t}\right] \leq C\text{ and } \mathbb{E}\left[ \sup\limits_{0\leq t\leq T}\exp\left( \lambda_{1}\int _{0}^{t}Z(s)dB(s)\right) \right] \leq C\text{,} \label{appwn-eq-11} \end{equation} where $C$ depends on $C_{1}$, $\delta$, $T$ and $\|\|\xi\|\|_{\infty}$. Moreover, for each $\lambda_{2}>0$, \begin{equation} \mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T}\|Z(s)\|ds\right) \right] <\infty. \label{appwn-eq-12} \end{equation} \end{theorem} \begin{proof} In the following, $C$ is a constant, and will be changed from line to line. Define \[ u(x)=\frac{1}{4C_{1}^{2}}\left( e^{2C_{1}x}-1-2C_{1}x\right) . \] It is easy to check that $x\rightarrow u(\|x\|)$ is $C^{2}$. Applying It\^{o}'s formula to $u(\|Y(s)\|)$, we get \[ \begin{array} [c]{lll} u\left( \left\vert Y(t)\right\vert \right) & = & u\left( \left\vert Y(T)\right\vert \right) +\int_{t}^{T}\{u^{^{\prime}} (\|Y(s)\|)sgn(Y(s))f(s,Y(s),Z(s))-\frac{1}{2}u^{^{\prime\prime}} (\|Y(s)\|)\|Z(s)\|^{2}\}ds\\ & & -\int_{t}^{T}u^{^{\prime}}(\|Y(s)\|)sgn(Y(s))Z(s)dB(s)\\ & \leq & u\left( \left\vert Y(T)\right\vert \right) +\int_{t}^{T} \{u^{^{\prime}}(\|Y(s)\|)C_{1}(1+\|Z(s)\|^{2})-\frac{1}{2}u^{^{\prime\prime} }(\|Y(s)\|)\|Z(s)\|^{2}\}ds\\ & & -\int_{t}^{T}u^{^{\prime}}(\|Y(s)\|)sgn(Y(s))Z(s)dB(s)\\ & \leq & C-\frac{1}{2}\int_{t}^{T}\|Z(s)\|^{2}ds-\int_{t}^{T}u^{^{\prime} }(\|Y(s)\|)sgn(Y(s))Z(s)dB(s). \end{array} \] From the above inequality, we can deduce that, for each stopping time $\tau\leq T$, \[ \mathbb{E}\left[ \left. \int_{\tau}^{T}\|Z\left( s\right) \|^{2} ds\right\vert \mathcal{F}_{\tau}\right] \leq C, \] where $C$ is independent of $\tau$. Thus $(\int_{0}^{t}Z\left( s\right) dB(s))_{t\in\lbrack0,T]}$ is a BMO martingale. By the Nirenberg inequality (see Theorem 10.43 in \cite{HWY}), we obtain (\ref{appwn-eq-11}). For each given $\lambda_{2}>0$, choose $\delta_{0}>0$ such that $\lambda_{2} \sqrt{\delta_{0}}<\delta$. Thus, by (\ref{appwn-eq-11}), we get \[ \begin{array} [c]{l} \mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T}\|Z(s)\|ds\right) \right] \\ =\mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T-\delta_{0} }\|Z(s)\|ds\right) \exp\left( \lambda_{2}\int_{T-\delta_{0}}^{T} \|Z(s)\|ds\right) \right] \\ =\mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T-\delta_{0} }\|Z(s)\|ds\right) \mathbb{E}\left[ \left. \exp\left( \lambda_{2} \int_{T-\delta_{0}}^{T}\|Z(s)\|ds\right) \right\vert \mathcal{F}_{T-\delta_{0} }\right] \right] \\ \leq\mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T-\delta_{0} }\|Z(s)\|ds\right) \mathbb{E}\left[ \left. \exp\left( \lambda_{2} \sqrt{\delta_{0}}\left[ \int_{T-\delta_{0}}^{T}\|Z(s)\|^{2}ds\right] ^{\frac{1}{2}}\right) \right\vert \mathcal{F}_{T-\delta_{0}}\right] \right] \\ \leq\mathbb{E}\left[ \exp\left( \lambda_{2}\int_{0}^{T-\delta_{0} }\|Z(s)\|ds\right) \mathbb{E}\left[ \left. e^{\lambda_{2}\sqrt{\delta_{0}} }+\exp\left( \lambda_{2}\sqrt{\delta_{0}}\int_{T-\delta_{0}}^{T} \|Z(s)\|^{2}ds\right) I_{\{\int_{T-\delta_{0}}^{T}\|Z(s)\|^{2}ds>1\}}\right\vert \mathcal{F}_{T-\delta_{0}}\right] \right] \\ \leq\left( e^{\delta}+C\right) \mathbb{E}\left[ \exp\left( \lambda_{2} \int_{0}^{T-\delta_{0}}\|Z(s)\|ds\right) \right] \\ \leq\left( e^{\delta}+C\right) ^{[\frac{T}{\delta_{0}}]+1}<\infty. \end{array} \] This completes the proof. \end{proof} \subsection{Solution to linear FBSDEs} \label{sect-solu-linearfbsde} Considering the following forward-backward stochastic differential equation \begin{equation} \left\{ \begin{array} [c]{rl} dX(t)= & \left[ \alpha_{1}(t)X(t)+\beta_{1}(t)Y(t)+\gamma_{1}(t)Z(t)+L_{1} (t)\right] dt+\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma _{2}(t)Z(t)+L_{2}(t)\right] dB(t),\\ dY(t)= & -\left[ \alpha_{3}(t)X(t)+\beta_{3}(t)Y(t)+\gamma_{3}(t)Z(t)+L_{3} (t)\right] dt+Z(t)dB(t),\\ X(0)= & x_{0},\ Y(T)=\kappa X(T), \end{array} \right. \label{appen-eq-xyz} \end{equation} where $\alpha_{i}(\cdot)$, $\beta_{i}(\cdot)$, $\gamma_{i}(\cdot)$, $i=1,2,3$, are bounded adapted processes, $L_{1}(\cdot)$, $L_{3}(\cdot)\in L_{\mathcal{F} }^{1,2}([0,T];\mathbb{R})$, $L_{2}(\cdot)\in L_{\mathcal{F}}^{2,2} ([0,T];\mathbb{R})$ and $\kappa$ is an $\mathcal{F}_{T}$-measurable bounded random variable. Suppose that the solution to (\ref{appen-eq-xyz}) has the following relationship \[ Y(t)=p(t)X(t)+\varphi(t), \] where $p(t)$, $\varphi(t)$ satisfy \begin{equation} \left\{ \begin{array} [c]{rl} dp(t)= & -A(t)dt+q(t)dB(t),\\ d\varphi(t)= & -C(t)dt+\nu(t)dB(t),\\ p(T)= & \kappa,\ \varphi(T)=0, \end{array} \right. \label{appen-eq-pq} \end{equation} $A(t)$ and $C(t)$ will be determined later. Applying It\^{o}'s formula to $p(t)X(t)+\varphi(t)$, we have \begin{equation} \begin{array} [c]{ll} d\left( p(t)X(t)+\varphi(t)\right) & =\left\{ p(t)\left[ \alpha _{1}(t)X(t)+\beta_{1}(t)Y(t)+\gamma_{1}(t)Z(t)+L_{1}(t)\right] -A(t)X(t)\right. \\ & \ \ \left. +q(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma _{2}(t)Z(t)+L_{2}(t)\right] -C(t)\right\} dt\\ & \ \ +\left\{ p(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma _{2}(t)Z(t)+L_{2}(t)\right] +q(t)X(t)+\nu(t)\right\} dB(t). \end{array} \end{equation} Comparing with the equation satisfied by $Y(t)$, one has \begin{equation} Z(t)=p(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma_{2}(t)Z(t)+L_{2} (t)\right] +q(t)X(t)+\nu(t), \label{appen-eq-z} \end{equation} \begin{equation} \begin{array} [c]{ll} -\left[ \alpha_{3}(t)X(t)+\beta_{3}(t)Y(t)+\gamma_{3}(t)Z(t)+L_{3}(t)\right] & =p(t)\left[ \alpha_{1}(t)X(t)+\beta_{1}(t)Y(t)+\gamma_{1}(t)Z(t)+L_{1} (t)\right] -A(t)X(t)\\ & \ \ +q(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma_{2} (t)Z(t)+L_{2}(t)\right] -C(t). \end{array} \label{appen-relation-generator} \end{equation} From equation \eqref{appen-eq-z}, we have the form of $Z(t)$ as \begin{align} Z(t) & =\left( 1-p(t)\gamma_{2}(t)\right) ^{-1}\left\{ p(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+L_{2}(t)\right] +q(t)X(t)+\nu(t)\right\} \\ & =\left( 1-p(t)\gamma_{2}(t)\right) ^{-1}\left[ \left( \alpha _{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) X(t)+p(t)\beta_{2} (t)\varphi(t)+p(t)L_{2}(t)+\nu(t)\right] . \end{align*} From the equation \eqref{appen-relation-generator}, and utilizing the form of $Y(t)$ and $Z(t)$, we derive that \begin{equation} \begin{array} [c]{rl} A(t)= & \alpha_{3}(t)+\beta_{3}(t)p(t)+\gamma_{3}(t)K_{1}(t)+\alpha _{1}(t)p(t)+\beta_{1}(t)p^{2}(t)\\ & +\gamma_{1}(t)K_{1}(t)p(t)+\alpha_{2}(t)q(t)+\beta_{2}(t)p(t)q(t)+\gamma _{2}(t)K_{1}(t)q(t), \end{array} \label{new-eq-111} \end{equation} where \[ K_{1}(t)=(1-p(t)\gamma_{2}(t))^{-1}\left[ \alpha_{2}(t)p(t)+\beta_{2} (t)p^{2}(t)+q(t)\right] , \] and \begin{equation} \begin{array} [c]{rl} C(t)= & (\beta_{1}(t)p(t)+\beta_{2}(t)q(t)+\beta_{3}(t))\varphi(t)+p(t)L_{1} (t)+q(t)L_{2}(t)+L_{3}(t)\\ & +(\gamma_{1}(t)p(t)+\gamma_{2}(t)q(t)+\gamma_{3}(t))(1-p(t)\gamma _{2}(t))^{-1}(\beta_{2}(t)p(t)\varphi(t)+p(t)L_{2}(t)+\nu(t)). \end{array} \label{new-eq-112} \end{equation} \begin{theorem} Assume \eqref{appen-eq-pq} has a solution $(p(\cdot),q(\cdot))$, $(\varphi(\cdot),\nu(\cdot))\in L_{\mathcal{F}}^{2}(\Omega;C([0,T],\mathbb{R} ))\times L_{\mathcal{F}}^{2,2}([0,T];\mathbb{R})$, and $(\tilde{X} (\cdot),\tilde{Y}(\cdot),\tilde{Z}(\cdot))\in L_{\mathcal{F}}^{2} (\Omega;C([0,T],\mathbb{R}))\times L_{\mathcal{F}}^{2}(\Omega ;C([0,T],\mathbb{R}))\times L_{\mathcal{F}}^{2,2}([0,T];\mathbb{R})$, where $\tilde{X}(\cdot)$ is the solution to \begin{equation} \left\{ \begin{array} [c]{rl} d\tilde{X}(t)= & \left\{ \alpha_{1}(t)\tilde{X}(t)+\beta_{1}(t)p(t)\tilde {X}(t)+\beta_{1}(t)\varphi(t)+L_{1}(t)+\gamma_{1}(t)\left( 1-p(t)\gamma _{2}(t)\right) ^{-1}\right. \\ & \cdot\left[ \left( \alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) \tilde{X}(t)\right. \left. \left. +p(t)\sigma_{y}(t)\varphi(t)+p(t)L_{2} (t)+\nu(t)\right] \right\} dt\\ & +\left\{ \alpha_{2}(t)\tilde{X}(t)+\beta_{2}(t)p(t)\tilde{X}(t)+\beta _{2}(t)\varphi(t)+L_{2}(t)+\gamma_{2}(t)\left( 1-p(t)\gamma_{2}(t)\right) ^{-1}\right. \\ & \cdot\left[ \left( \alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) \tilde{X}(t)\right. \left. \left. +p(t)\sigma_{y}(t)\varphi(t)+p(t)L_{2} (t)+\nu(t)\right] \right\} dB(t),\\ \tilde{X}(0)= & x_{0}, \end{array} \right. \end{equation} and \begin{equation} \begin{array} [c]{rl} \tilde{Y}(t)= & p(t)\tilde{X}(t)+\varphi(t),\\ \tilde{Z}(t)= & \left( 1-p(t)\gamma_{2}(t)\right) ^{-1}\left[ \left( \alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) \tilde{X}(t)\right. \\ & \left. +p(t)\beta_{2}(t)\varphi(t)+p(t)L_{2}(t)+\nu(t)\right] . \end{array} \end{equation} Then $(\tilde{X}(\cdot),\tilde{Y}(\cdot),\tilde{Z}(\cdot))$ solves \eqref{appen-eq-xyz}.\ Moreover, if \[ p(t)L_{1}(t)+q(t)L_{2}(t)+L_{3}(t)+(\gamma_{1}(t)p(t)+\gamma_{2} (t)q(t)+\gamma_{3}(t))(1-p(t)\gamma_{2}(t))^{-1}p(t)L_{2}(t)=0, \] then $(\varphi(\cdot),\nu(\cdot))=(0,0)$ is a solution to \eqref{appen-eq-pq} and $(\tilde{X}(t),p(t)\tilde{X}(t),\left( 1-p(t)\gamma_{2}(t)\right) ^{-1}\left[ \left( \alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) \tilde{X}(t)+p(t)L_{2}(t)\right] )_{t\in\lbrack0,T]}$ solves \eqref{appen-eq-xyz}. \label{appen-th-linear-fbsde} \end{theorem} \begin{proof} The results follow by applying It\^{o}'s formula. \end{proof} Now we study the uniqueness of $(p(\cdot),q(\cdot))$ in (\ref{appen-eq-pq}). It is important to note that the form of $(p(\cdot),q(\cdot))$ in (\ref{appen-eq-pq}) does not depend on $x_{0}$, $L_{1}$, $L_{2}$, $L_{3}$. So we set $\ x_{0}=1$, $L_{1}=L_{2}=L_{3}=0$ in the followings. In this case $(\varphi(\cdot),\nu(\cdot))=(0,0)$ as in the above theorem. \begin{theorem} \label{unique-pq} Assume (\ref{appen-eq-xyz}) has a unique solution $(X(\cdot),Y(\cdot),Z(\cdot))\in L_{\mathcal{F}}^{2}(\Omega;C([0,T],\mathbb{R} ))\times L_{\mathcal{F}}^{2}(\Omega;C([0,T],\mathbb{R}))\times L_{\mathcal{F} }^{2,2}([0,T];\mathbb{R})$. \end{theorem} \begin{description} \item[(i)] If $\gamma_{2}(\cdot)$ is small enough and $(p_{i}(\cdot ),q_{i}(\cdot))\in L_{\mathcal{F}}^{\infty}(0,T;\mathbb{R})\times L_{\mathcal{F}}^{2,4}([0,T];\mathbb{R})$, $i=1,2$, are two solutions to (\ref{appen-eq-pq}), then $(p_{1}(\cdot),q_{1}(\cdot))=(p_{2}(\cdot ),q_{2}(\cdot));$ \item[(ii)] If $(p_{i}(\cdot),q_{i}(\cdot))\in L_{\mathcal{F}}^{\infty }(0,T;\mathbb{R})\times L_{\mathcal{F}}^{\infty}(0,T;\mathbb{R})$, $i=1,2$, are two solutions to (\ref{appen-eq-pq}), then $(p_{1}(\cdot),q_{1} (\cdot))=(p_{2}(\cdot),q_{2}(\cdot))$. \end{description} \begin{proof} We only prove (i), (ii) is similar. Consider the following SDEs: \begin{equation} \left\{ \begin{array} [c]{ll} d\tilde{X}_{i}(t)= & \left[ \alpha_{1}(t)+\beta_{1}(t)p_{i}(t)+\gamma _{1}(t)K_{i,1}(t)\right] \tilde{X}_{i}(t)dt\\ & +\left[ \alpha_{2}(t)+\beta_{2}(t)p_{i}(t)+\gamma_{2}(t)K_{i,1}(t)\right] \tilde{X}_{i}(t)dB(t),\\ \tilde{X}_{i}(0)= & 1,\text{ }i=1,2. \end{array} \right. \end{equation} Then $\tilde{X}_{i}(\cdot)$ has a explicit form \[ \tilde{X}_{i}(t)=\exp\left\{ \int_{0}^{t}\left( N_{i,1}(s)-\frac{1} {2}(N_{i,2}(s))^{2}\right) ds+\int_{0}^{t}N_{i,2}(s)dB(s)\right\} , \] where \[ \begin{array} [c]{rl} K_{i,1}(s) & =(1-p_{i}(s)\gamma_{2}(s))^{-1}\left[ \alpha_{2}(s)p_{i} (s)+\beta_{2}(s)p_{i}^{2}(s)+q_{i}(s)\right] ,\\ N_{i,1}(s) & =\alpha_{1}(s)+\beta_{1}(s)p_{i}(s)+\gamma_{1}(s)K_{i,1}(s),\\ N_{i,2}(s) & =\alpha_{2}(s)+\beta_{2}(s)p_{i}(s)+\gamma_{2}(s)K_{i,1}(s). \end{array} \] By Theorem \ref{q-exp-th}, it is easy to check that when $\gamma_{2}(\cdot)$ is small enough, \[ \mathbb{E}\left[ \underset{0\leq t\leq T}{\sup}\left\vert \tilde{X} _{i}(t)\right\vert ^{4}\right] <\infty. \] Thus we have $\left( K_{i,1}(t)\tilde{X}_{i}(t)\right) _{t\in\lbrack0,T]}\in L_{\mathcal{F}}^{2,2}([0,T];\mathbb{R})$. Since (\ref{appen-eq-xyz}) has a unique solution, by Theorem \ref{appen-th-linear-fbsde}, we get for $t\in\lbrack0,T]$, \ \[ (\tilde{X}_{1}(t),p_{1}(t)\tilde{X}_{1}(t),K_{1,1}(t)\tilde{X}_{1} (t))=(\tilde{X}_{2}(t),p_{2}(t)\tilde{X}_{2}(t),K_{2,1}(t)\tilde{X}_{2}(t)). \] Note that $\tilde{X}_{1}(\cdot)>0$, then $(p_{1}(\cdot),q_{1}(\cdot ))=(p_{2}(\cdot),q_{2}(\cdot))$. \end{proof} \end{document}	arXiv
\begin{document} \title{Quantum Repeaters based on Single Trapped Ions} \date{\today} \pacs{03.67.Hk, 03.67.Mn} \author{Nicolas Sangouard$^{1}$, Romain Dubessy$^{1}$, and Christoph Simon$^{2}$} \affiliation{ $^{1}$Laboratoire Matériaux et Phénomènes Quantiques CNRS, UMR7162, Université Paris Diderot, France\\ $^{2}$Group of Applied Physics, University of Geneva, Switzerland } \begin{abstract} We analyze the performance of a quantum repeater protocol based on single trapped ions. At each node, single trapped ions embedded into high finesse cavities emit single photons whose polarization is entangled with the ion state. A specific detection of two photons at a central station located half-way between two nodes heralds the entanglement of two remote ions. Entanglement can be extended to long distances by applying successive entanglement swapping operations based on two-ion gate operations that have already been demonstrated experimentally with high precision. Our calculation shows that the distribution rate of entanglement achievable with such an ion-based quantum repeater protocol is higher by orders of magnitude than the rates that are achievable with the best known schemes based on atomic ensemble memories and linear optics. The main reason is that for trapped ions the entanglement swapping operations are performed deterministically, in contrast to success probabilities below 50 percent per swapping with linear optics. The scheme requires efficient collection of the emitted photons, which can be achieved with cavities, and efficient conversion of their wavelength, which can be done via stimulated parametric down-conversion. We also suggest how to realize temporal multiplexing, which offers additional significant speed-ups in entanglement distribution, with trapped ions. \end{abstract} \maketitle \section{Introduction} The distribution of entanglement over long distances is difficult because of unavoidable transmission losses and the no-cloning theorem for quantum states. One possible solution is the use of quantum repeaters \cite{Briegel98}, which are based on the heralded creation and storage of entanglement for elementary links of moderate length, followed by entanglement swapping operations that allow one to extend the distance of entanglement. The most widely known approach to quantum repeaters combines quantum memories based on atomic ensembles and entanglement swapping operations using linear optics. Building on the initial proposal of Ref. \cite{Duan01}, there has been a large amount of experimental \cite{atexp} and theoretical \cite{Jiang07,attheo,Sangouard08} work towards realizing long-distance entanglement distribution in this way. This approach is attractive because it uses relatively simple elements. However, as quantum technology progresses, it is natural to also consider other possible physical systems. For example, there have been recent proposals for the realization of quantum repeaters with NV centers in diamond \cite{Childress06} and with spins in quantum dots \cite{SimonNiquet07,VanLoock08}. Trapped ions were one of the first systems to be proposed for quantum computation \cite{Cirac95}. Since then, many experiments have been realized that demonstrated key ingredients for quantum computing, including the realization of quantum logic gates with increasing precision \cite{Monroe95, SchmidtKaler03, Leibfried03, Benhelm08}, the creation of multi-qubit entanglement \cite{Turchette98, Roos04,Leibfried05}, the implementation of the Deutsch-Jozsa algorithm \cite{Gulde03}, the realization of entanglement purification \cite{Chiaverini04, Reichle06}, the accomplishment of atomic-qubit quantum teleportation \cite{Riebe04, Barrett04}, the realization of deterministic entanglement swapping \cite{Riebe08} as well as the demonstration of very high-efficiency detections with \cite{Hume07} and without \cite{Myerson08} ancilla qubit. In addition, ion-photon entanglement has been created \cite{Blinov04} and subsequently used to entangle distant ions \cite{Moehring07}. Motivated by this impressive body of work, we here analyze in detail the achievable performance of quantum repeaters based on trapped ions. We show that the distribution rates of entanglement offered by repeaters based on single ions are significantly superior compared to the ones achieved with atomic ensemble based schemes. The main reason is that entanglement swapping operations can be performed deterministically for trapped ions. In contrast, the success probability for entanglement swapping is bounded by 1/2 for schemes using Bell measurements based on linear optics \cite{Calsamiglia01}. The repeater protocol that we envision requires both an efficient collection of the emitted photons and an efficient conversion of their wavelength to the telecom wavelength around 1.5 $\mu$m where the losses in optical fibers are at their minimum. In order to improve the collection of photons emitted by a single ion, one can couple this ion to a high-finesse cavity. Individual ions have been coupled to high-finesse cavities experimentally \cite{Mundt02, Keller03} and theoretical proposals \cite{Maurer04} have been realized to make very efficient the photon emission probability into the cavity mode using realistic cavity parameters, cf. below. The frequency conversion might be realized using stimulated parametric down-conversion. This is in fact, the inverse process of the coherent up-conversion that was demonstrated for single photons in Ref. \cite{Tanzilli05}, cf. below. The performance of atomic ensemble based quantum repeaters can be greatly enhanced by temporal multiplexing \cite{Simon07} using multi-mode memories \cite{Afzelius08}. We will suggest how to implement analogous temporal multiplexing for trapped ions using ion transport methods that have been developed in the context of quantum computing. This paper is organized as follows. In the next section, we present the achievable distribution rates for a repeater protocol based on single trapped ions and we compare them to the ones achievable with atomic ensembles. The third section is devoted to implementation issues. In the fourth section we present an approach to implement temporal multiplexing. The fifth section contains our conclusions. \section{Efficiency of repeaters with trapped ions} Let us recall how two remote ions at locations A and B can be entangled via the detection of two photons as proposed in \cite{Feng03, Duan03, Simon03}. Note that two remote ions can also be entangled based on the single-photon detection \cite{Cabrillo99}. For a discussion of the advantages and disadvantages of schemes based on two-photon detections versus schemes based on single-photon detections, see e.g. Refs. \cite{Zippilli08, Sangouard08}. Each ion is described by a lambda system of three states, as shown in Fig. \ref{fig1}. From the excited state $\|e^A\rangle$ ($\|e^B\rangle$) the ion located at A (B) can decay into two degenerate metastable states, say the states $\|g^A_V\rangle$ and $\|g^A_H\rangle$ ($\|g^B_V\rangle$ and $\|g^B_H\rangle$) by emitting a photon with a well defined polarization, say either vertical corresponding to the mode $a_V$ or horizontal corresponding to $a_H$ ($b_V$ and $b_H$ respectively). The A and B ions are both excited simultaneously, such that the emission of a photon by each ion leads to the state \begin{eqnarray} &&\|\Psi^A\rangle\otimes \|\Psi^B\rangle = \\ &&\nonumber \frac{1}{2}\left(\|g_H^A\rangle a^{\dagger}_H+\|g^A_V\rangle a^{\dagger}_V\right) \otimes \left(\|g_H^B\rangle b^{\dagger}_H+\|g^B_V\rangle b^{\dagger}_V\right)\|0\rangle \end{eqnarray} with $\|0\rangle$ the vacuum state. A probabilistic Bell state analysis can be performed by combining the two emitted photons on a polarizing beam spitter (PBS) at a central station located half-way between $A$ and $B$ and by counting the photon number in each output modes $d_\pm = \frac{1}{\sqrt{2}}(a_H \pm b_V),$ $\tilde{d}_\pm = \frac{1}{\sqrt{2}}(b_H\pm a_V)$. Such Bell analysis projects non-destructively the two ions into an entangled state. For example, the detection of two photons, one in each modes $d_+$ $\tilde{d}_+$, leads to the entangled state \begin{equation} \label{eq2} \|\psi^{AB}_+\rangle=\frac{1}{\sqrt{2}}\left(\|g_H^Ag_H^B\rangle +\|g^A_Vg^B_V\rangle\right). \end{equation} In the ideal case, the probability for such an event is $1/8.$ Taking into account the coincidences between $d_-$-$\tilde{d}_+,$ $d_+$-$\tilde{d}_-$ and $d_-$-$\tilde{d}_-$ combined with the appropriate one-qubit operations, the probability to create the state (\ref{eq2}) is $1/2$. This way of creating entanglement was demonstrated experimentally in Ref. \cite{Moehring07}. Note that the photon collection efficiency was quite low in these experiments, which did not have cavities around the ions. \begin{figure} \caption{(Color online) Setup for entanglement creation based on two-photon detection of remote ions (brown dots) embedded into cavities. Each ion emits a photon, whose polarization is entangled with the atomic state, leading to the state $\|\Psi^A\rangle\otimes \|\Psi^B\rangle$ of Eq. (1). The two photons, one coming from location $A,$ the other one from $B$ are combined on a polarizing beam splitter to be further detected in a polarization basis rotated by 45 degrees with respect to the $H-V$ basis. The coincident detection of two photons in the corresponding modes $d_+$ and $\tilde{d}_+$, for example, projects the two ions into the entangled state $\|\psi^{AB}_+\rangle$ of Eq. (2).} \label{fig1} \end{figure} We now calculate the time needed for entanglement creation. Let us denote by $p$ the success probability for an ion to emit a photon, which includes the probability to prepare the ion in the excited state, the spontaneous emission of a photon into the cavity mode and coupling into the fiber, as well as the frequency conversion to match the telecom wavelength. The probability to get the expected twofold coincidence is thus given by $P_0=\frac{1}{2}p^2 \eta_t^2 \eta_d^2$ where $\eta_t=e^{-L_0/(2 L_{att})}$ is the fiber transmission with the attenuation length $L_{att}$ (we use $L_{att}= 22$ km, corresponding to losses of 0.2 dB/km, which are currently achievable at a wavelength of 1.5 $\mu$m) and $\eta_d$ is the detection efficiency. Entanglement creation attempts can be repeated at time intervals given by the communication time $L_0/c$, cf. Ref. \cite{Simon07}. As a consequence, the average time required to entangle two ions separated by a distance $L_0,$ is given by \begin{equation} \label{eq3} T_{link}=\frac{L_0}{c}\frac{1}{P_0}. \end{equation} Here $c = 2 \times 10^8$ m/s is the photon velocity in the fiber.\\ \begin{figure} \caption{(Color online) Performance of quantum repeaters based on single ions versus atomic ensembles. The quantity shown is the average time for the distribution of one entangled pair for the given distance. Curve A: as a reference, the time required using direct transmission of photons through optical fibers, with losses of 0.2 dB/km, corresponding to the best available telecom fibers at a wavelength of 1.5 $\mu$m, and a pair generation rate of 10 GHz. Curve B: protocol based on atomic ensembles of Ref. \cite{Sangouard08}. High-fidelity entangled pairs are generated locally, and entanglement generation and swapping operations are based on two-photon detections. We have assumed memory and detector efficiencies of 90\%. We imposed a maximum number of 16 links in the repeater chain (see \cite{Sangouard08} for details). This approach leads to a repeater protocol that, as far as we know, achieves the highest entanglement distribution rate with atomic ensembles and linear optics. Curve C and D: protocol based on single ions with 8 and 16 links respectively. We have assumed a success probability for the ion to emit a photon of $p=90\%$ requiring high-finesse cavity, cf. text.} \label{fig2} \end{figure} The entanglement can further be distributed over longer distances by using successive entanglement swapping operations between elementary links. Such swapping operations require a local Bell state analysis, applied e.g. on the two ions located at B to entangle the ions located at A and C. Bell states have recently been prepared deterministically from the computational basis with a very-high fidelity \cite{Benhelm08}. Applied on the four Bell states, this protocol transforms each of them into a product state in the computational basis. The measurement of the individual ion states then leads to the desired Bell analysis. The success probability for entanglement swapping reduces in this case to the detection efficiency of ions, which is essentially equal to one. The time for the swapping and detection can realistically be much shorter than the time required for entanglement creation (\ref{eq3}), cf. below, such that the total time for the distribution of an entangled pair over the distance $2 L_0$ is given by \begin{equation} \label{eq4} T_{2L_0} \approx \frac{3}{2} \frac{L_0}{c} \frac{1}{P_0}= \frac{3L_0}{c} \frac{1}{p^2\eta_t^2\eta_d^2}. \end{equation} The factor 3/2 arises because entanglement has to be generated for two links before the entanglement connection can be performed. If the average waiting time for entanglement generation for one link is $T$ , there will be a success for one of the two after $T /2$ ; then one still has to wait a time $T$ on average for the second one, giving a total of $3T /2$. This simple argument gives exactly the correct result in the limit of small $P_0$ \cite{Collins07,Brask08}. For a quantum repeater with $n$ nesting levels, analogous factors arise at each level. They are no longer exactly equal to 3/2 in the general case because the waiting time distribution for establishing an individual higher-level link is no longer simply exponential, but numerical results show that this remains a good approximation \cite{Jiang07,Brask08}. (Note that the factors certainly all lie between 1 and 2.) The average time for the distribution of an entangled pair over the distance $L=2^n L_0$ is then approximately given by \begin{equation} \label{Ttot} T_{tot} \approx \left(\frac{3}{2}\right)^n \frac{L_0}{c} \frac{1}{P_0}=\frac{3^n}{2^{n-1}} \frac{L_0}{c} \frac{1}{p^2\eta_t^2\eta_d^2}. \end{equation} The performance of such a quantum repeater based on single ions is shown in Fig. \ref{fig2}. In the same figure we also show the performance of the best atomic ensemble based protocol known to us \cite{Sangouard08}. In this approach, one first locally generates high-fidelity entangled pairs of atomic excitations that are stored in nearby ensembles. Then long-distance entanglement is generated and swapped via two-photon detections. As in Ref. \cite{Sangouard08}, we have limited the maximum number of links used to 16 for all protocols, to have link numbers for which it is plausible that entanglement purification may not be necessary. Note however that entanglement purification has already been implemented for trapped ions \cite{Chiaverini04,Reichle06}. \begin{figure} \caption{(Color online) Robustness of a repeater based on single trapped ions with respect to the success probability for an ion to emit a photon $p$ (photon source efficiency) which includes the probability to prepare the ion into the cavity mode and coupling into the fiber, as well as the frequency conversion to match the telecom wavelength. The quantity shown is the average time for the distribution of an entangled pair over 1000km for a repeater with 16 elementary links (Eq. (\ref{Ttot})).} \label{fig3} \end{figure} Fig. 2 shows that the distribution rate that can be achieved with single ions is higher by orders of magnitude than the one obtained with atomic ensembles. As mentioned before, the most important factor explaining this improvement is that entanglement swapping operations are performed deterministically for the ions, whereas each swapping operation is performed at most with a probability 1/2 using linear optical elements. Another reason is that the state generated locally with atomic ensembles, which should ideally be a state of two maximally entangled atomic excitations, in fact possesses no-excitation and single-excitation components. Even if these undesired components can be reduced by partial memory read-out \cite{Sangouard08}, they still limit the achievable distribution rate of entanglement. In Fig. 2 we have assumed that the ions are very efficient sources of single photons ($p=90\%$), in order to keep the assumptions comparable with the ones made for the atomic-ensemble based scheme in Ref. \cite{Sangouard08}, where the memory efficiency was taken to be $\eta_m=90\%$. However, it should be pointed out that the average time for the distribution of an entangled pair (see Eq. (\ref{Ttot})) scales only like $\frac{1}{p^2}$, such that even with $p=30\%,$ one needs less than $T_{tot}=740$ ms to distribute an entangled pair over 1000 km using 16 links, which is still shorter than the time achievable with atomic ensembles. Fig. \ref{fig3} gives the average time required to distribute an entangled pair for various values of $p$. Note that atomic ensemble based schemes are much more sensitive to a reduction in $\eta_m$, because it intervenes in every swapping operation. \section{Implementation} To achieve a high efficiency of photon collection, one can embed the ion within a cavity. The spontaneous emission emitted into the cavity mode is enhanced by the Purcell factor \begin{equation} F_P=\frac{3\ell \lambda^2}{2\pi^2V_0}\mathcal{F} \end{equation} with $\mathcal{F}$ the finesse of the cavity, $\ell$ its length, $\lambda$ the free-space wavelength and $V_0$ the mode volume of the cavity (which is of order $\ell^2\lambda$ for a confocal cavity with a waist of order $\sqrt{\ell\lambda}$). The collection efficiency $\frac{F_P-1}{F_P}$ can then be made as large as desired for large enough Purcell factor $F_P$, which requires a high finesse $\mathcal{F}$ and small mode volume $V_0$. Note that a Purcell factor of 2 was already achieved experimentally for a trapped ion in a cavity in Ref. \cite{Mundt02}. For concreteness, we focus on the realization of the studied repeater protocol with $^{40}$Ca$^+$ ions (the relevant states are presented in Fig. \ref{fig4}) even if other species should not be excluded. Following the proposal of Ref. \cite{Simon03}, one could prepare the ions in one of the $P_{3/2}$ sublevels to serve as excited state $\|e\rangle.$ For $\|g_H\rangle$ and $\|g_V\rangle,$ one could use two sublevels of $D_{5/2}$ which are coupled to $\|e\rangle$ by orthogonally polarized photons at 854 nm. Ideally, the coupling strengths for the transitions $\|e\rangle$-$\|g_H\rangle$ and $\|e\rangle$-$\|g_V\rangle$ should be equal and the cavities should be designed such that the two polarizations are equally supported otherwise the probability to create the state (\ref{eq2}) is reduced. Note that in principle, if the coupling strengths are not equal, they might be compensated by appropriate cavity couplings. The characteristic lifetime of the sublevels of $D_{5/2}$ is up to 1 s, which is compatible with the average time required for the distribution of an entangled pair for all the distances considered in Fig. \ref{fig2} for repeaters with 16 links (see curve D). If longer memory times are required, e.g. for repeaters with 8 links (see curve C), one could coherently transfer these states to the sublevels associated to $S_{1/2},$ cf. below. \begin{figure} \caption{Relevant levels for $^{40}$Ca$^+$.} \label{fig4} \end{figure} The state $P_{3/2}$ decays preferentially to $S_{1/2}$, which at first sight seems to limit the achievable photon collection efficiency. To overcome this limitation, it has been proposed in Ref. \cite{Maurer04} to couple the ground state $S_{1/2}$ directly to sublevels of $D_{5/2}$ through a Raman process by choosing a pump laser far detuned from the $S_{1/2}$-$P_{3/2}$ transition, i.e. $\Delta \gg \Omega$ with $\Delta$ the detuning and $\Omega$ the pump Rabi frequency. It is shown in Ref. \cite{Maurer04} that with this approach one can achieve a photon emission probability into the cavity mode of 95\% for a realistic cavity. The achievable photon repetition rate of 20 kHz proposed in Ref. \cite{Maurer04} is higher than $\frac{c}{L_0}$ as soon as the elementary links are longer than $L_0=15$ km. As a consequence, for the considered distances and link numbers ($L_0 \geqslant 25$ km), the average time for entanglement creation is limited by the communication time, which is in agreement with Eq. (\ref{eq3}). We have assumed that the wavelength of the photons emitted by the ions is converted to a telecom wavelength around 1.5 $\mu$m, in order to profit from the optimal transmission of optical fibers in that range. Frequency conversion at the single photon level was already demonstrated in Ref. \cite{Tanzilli05} with an intrinsic efficiency of $56 \%$. In this experiment the frequency of the photons was up-converted in order to achieve a better detection efficiency. However, the inverse process, which is parametric down-conversion with a single-photon pump, but a strong laser stimulating emission into one of the two down-converted modes, can be performed with the same efficiency (due to unitarity). It should be possible to bring the conversion efficiency close to one using stronger non-linearities and a stronger stimulation laser, and of course minimizing all optical and coupling losses. Note that this conversion process preserves entanglement, as was already demonstrated in Ref. \cite{Tanzilli05}. A Bell state analysis is required for the entanglement swapping operations. Following the proposal of Ref. \cite{Sorensen99}, two $^{40}$Ca$^+$ ions have recently been prepared deterministically in a Bell state (the two qubit states are sublevels of $S_{1/2}$ and $D_{5/2}$) with a fidelity greater than 99\% on a time scale of the order of 50 $\mu$s \cite{Benhelm08}. The two ions are placed close to each other such that they interact through the Coulomb interaction giving rise to a common spatial vibration. A collective irradiation with the appropriate bichromatic field allows one to prepare deterministically the desired Bell state from a given state of the computational basis \cite{Sorensen99}. Such an experiment could be used to perform the required Bell state analysis in the following manner. The two ions located at each node could be embedded within the same cavity. The distance between them has to be small enough such that they interact efficiently through Coulomb interaction but large enough to allow one an individual addressing of each of them with laser beams. Such an addressing is essential for entanglement creation, i.e. for the targeted emission of a photon by one of the two ions. An optical switch could be used to send the emitted photon to the desired central station. A typical distance of $\sim$ 8 $\mu$m separating the two ions \cite{Benhelm08} with laser beams focused to $\sim$ 2 $\mu$m might be well suited. For entanglement swapping, one could first transfer coherently the population of $\|g_H\rangle$ to a sublevel of $S_{1/2},$ as in Refs. \cite{Schmidt03, Roos99} requiring a time scale of $\sim$10 $\mu$s. We then use the appropriate bichromatic field on the transition $S_{1/2}$-$\|g_V\rangle$ as in Ref. \cite{Benhelm08} such that each Bell state will be transformed into a given state of the computational basis. This takes $\sim$ 50 $\mu$s. We finally measure the state of each ion independently. This detection could be done by measuring resonance fluorescence from the auxiliary state $P_{1/2}$ that is strongly coupled to $S_{1/2}$ with a laser field at 397 nm and decays back only to that same state \cite{SchmidtKaler03}. Such measurement has been preformed recently \cite{Myerson08} in the same system and it takes in average 145$\mu$s with a photon collection of 0.2\%. Such characteristic time can realistically be reduced to a few tens of $\mu$s by optimizing the collection efficiency \cite{Simon03}. All in all, an entanglement swapping operation should be much shorter than the average time for the entanglement creation ($T_{link}>1$ ms for $L \geqslant 400$km) justifying the formulas (\ref{eq4})-(\ref{Ttot}). To exploit the entanglement, it is essential to be able to detect the states of the ions in different bases (e.g. for a Bell test or for quantum key distribution). The necessary rotations could be performed by first coherently transferring $\|g_H\rangle$ to the $S_{1/2}$ sublevels as before and then applying the appropriate pulses on the transition involving that state and $\|g_V\rangle$. As said before, these transformations can be performed in $\sim$ 10 $\mu$s. \section{additional speed-up via temporal multiplexing} \begin{figure} \caption{(Color online) Setup for temporal multiplexing. Chains of ions are transported through cavities such that the ions are excited one by one when they interact with the cavity mode. The ions of the chain B are alternatively excited in the upper cavity for entanglement creation between A and B or in the lower cavity for entanglement creation between B and C. If entanglement has been established between the $m$$^{th}$ ions for the link A-B and between the $n$$^{th}$ ions for B-C, entanglement swapping is done by performing a Bell state analysis on the $m$$^{th}$ and $n$$^{th}$ ions of the B chain. } \label{fig5} \end{figure} As seen before, the creation of entanglement between neighboring nodes A and B is conditioned on the outcome of photon detections at a station located half-way between the nodes. To profit from a nested repeater, the entanglement swapping operations can only be performed once one knows the relevant measurement outcomes. This requires a communication time of order $L_0/c.$ If one can perform a number $N$ of entanglement creation attempts per elementary link within the time interval $L_0/c,$ one can decrease the average time for entanglement creation $T_{link}$ by a factor of order $N$. Such temporal multiplexing has initially been proposed for quantum repeaters based on atomic ensembles \cite{Simon07}, and since then a particularly efficient quantum storage protocol has been developed \cite{Afzelius08} for this purpose. We here propose a realization of the same basic idea for quantum repeaters based on single ions. Consider two links, say A-B and B-C, allowing one to connect the A and C nodes by entanglement swapping, see Fig. \ref{fig5}. At each location A, B and C a chain of ions within a segmented trap can be moved through a cavity by applying appropriate control electric fields to the various segments \cite{Barrett04, Rowe02, Huber08} such that the internal state of the ions is preserved. Further suppose that the distance between two successive ions is larger than the waist radius of the cavity mode such that one can selectively excite each ion when it interacts with the cavity mode in order to force it to emit a photon. The ions located at B are used as sources for entanglement creation between both the A-B and B-C links in the following way. Suppose that the chains located at A and C are composed of $N$ ions. The chain B possesses $2N$ ions which are excited alternatively in the upper cavity and in the lower one for entanglement creation between A-B and between B-C locations respectively. If there are two detections behind the central PBS located between A and B for the $m$$^{th}$ ions for example, then we know that these ions are entangled. Running the same protocol for another pair of ions, there may be similar detections between B and C locations associated to the $n$$^{th}$ ions. One then performs entanglement swapping by applying the appropriate operations on the $m$$^{th}$ and $n$$^{th}$ ions of the B chain. This can be done by addressing individually the $m$$^{th}$ and the $n$$^{th}$ ions with the appropriate bichromatic field \cite{Sorensen99}, thus realizing the Bell state analysis described in the previous section. Single $^{40}$Ca$^+$ ions have already been transported from a loading zone to a cavity interaction region separated by more than 20 mm in a characteristic time of 4 ms \cite{Keller03}. This was realized using a segmented trap composed of 5 pairs of electrodes by successively ramping the electrode voltages. Faster transports were realized along $\sim$1 mm with a characteristic time of 50 $\mu$s without loss of coherence and with negligible excitation of the ion's motion \cite{Rowe02, Barrett04, Huber08}. The number of attempts that can be achieved per time interval $L_0/c$ is thus likely to be limited by the characteristic time of the Raman process, rather than by the speed of ion transport. For example, the 20 mm long cavity considered in Ref. \cite{Maurer04}, which is compatible with the characteristics of the trap reported in Ref. \cite{Keller03}, gives a photon repetition rate of 20 kHz, which would allow 10 attempts per time interval $L_0/c$ for 1000 km and 8 links. This would increase the entanglement distribution rate by the same factor of 10. For higher repetition rates, one needs to decrease the duration of the Raman process $\tau$, which has to fulfill $\frac{\Omega g}{\Delta} \tau \sim \pi$ to insure an efficient population transfer. This can be done by increasing the $g$ factor, i.e. by decreasing the cavity length $l_c.$ (The ratio $\Omega/\Delta$ has to be kept smaller than 1 to guarantee that no population will be transferred to the excited state). Considering e.g. a cavity length of $l_c=6$ mm as described in Ref. \cite{Keller03}, the achievable distribution rate increases by a factor 30. If one chooses $l_c=1$ mm, which might still be compatible with microtrap dimensions \cite{microtrap}, one gets an improvement of the rate by a factor of 200. \section{Conclusion} We have shown that trapped ions are very promising systems for the implementation of quantum repeaters. In fact, the achievable performance for a relatively basic trapped ion quantum repeater protocol greatly exceeds the best atomic ensemble based protocol known to us. This is mostly due to the fact that a deterministic Bell state analysis can be performed for trapped ions using current technology. We have argued that this performance could further be improved very significantly using temporal multiplexing based on ion transport techniques that have been developed with quantum computing applications in mind. The requirements for implementing practically useful quantum repeaters, while technologically challenging, are much more modest than for the realization of fault-tolerant quantum computation. We suggest that this is an interesting intermediate goal that the ion trapping community should keep in mind. \begin{acknowledgments} We thank T. Coudreau, N. Gisin, L. Guidoni, and D. Lucas for helpful discussions. This work was supported by the EU Integrated Project {\it Qubit Applications}, the Swiss NCCR {\it Quantum Photonics} and the French National Research Agency (ANR) project ANR-JC05\_61454. \end{acknowledgments} \end{document}	arXiv
How does a classical computer simulate nonclassical correlations? This may be a dumb question, if so please forgive me, it is late at night. I have learned that a classical computer can simulate a quantum computer in exponential time and space, but classical computers are bound to non-quantum phenomenon. How then, would one be able to simulate say CHSH, which produces fundamentally quantum probabilities that cannot be explained locally/classically? Am I misinterpreting the meaning of simulate? In general, how could a classical computer simulate quantum phenomena that cannot be explained classically (such as the dynamics of more than a single particle)? I would think that one could not generate random numbers violating any of Bell's inequalities, i.e. necessarily quantum correlations are off limits. entanglement simulation classical-computing games non-locality PhysMathPhysMath $\begingroup$ related: quantumcomputing.stackexchange.com/q/1/55 $\endgroup$ Quantum phenomena cannot be "explained classically" only when locality is taken into consideration. In other words, classical phenomena cannot reproduce (some types of) quantum correlations provided that we don't allow for certain types of correlations. As a concrete example, consider a standard CHSH scenario. We can compute the outcome probability distributions for each measurement setting (it's what you do when you study the protocol), therefore you can trivially write some code to "simulate" the results of an experiment, meaning to draw a possible sequence of measurement outcomes you would find in an experiment. But this is clearly not the same as observing nonlocality with a classical computer: you would just be crunching some numbers that you know, in some situations, can be interpreted as markers of nonclassical correlations. Put in another way, you can always sample from an arbitrary probability distribution $p(ab\|xy)$. Whether such a distribution is "nonclassical" is only meaningful in relation to some imposed restriction (e.g. defining "classical" when it can be written as $p(ab\|xy)=\sum_\lambda p_\lambda p_\lambda(a\|x) p_\lambda(b\|y)$). When you simulate such a distribution on a computer, you don't need to respect such restrictions, so there is no problem. In general, how could a classical computer simulate quantum phenomena that cannot be explained classically Aside from locality constraints, such as those described above, quantum mechanics does not predict output probability distributions that are incompatible with classical physics. The difference is in how those outputs can be obtained: quantum mechanic can produce output probability distributions in a radically different way than what classical physics allows for, and in some cases these new behaviours are more efficient. glS♦glS There are two definitions of simulation that are commonly used in this context. We consider a quantum computation to be: 1. loading an input 2. performing some processing 3. doing a measurement This defines a distribution on possible measurement outcomes for each input. Weak Simulation would be a classical randomised algorithm that could sample from these distributions, given a suitable description of the quantum computation as defined above. Strong Simulation is the ability to approximately calculate individual probabilities. A naive simulation algorithm that uses exponential time and space is to store the state as a big vector (of length $2^n$) and then multiply it by the matrices for each of the gates (size $2^n \times 2^n$). Then measurement probabilities can also be calculated by finding the eigenspaces for the measurement operator, and projecting the final state vector onto the one of interest. This doesn't violate any laws of quantum physics, because it is simulating the whole system, not simulating each qubit locally Simon CraneSimon Crane Not the answer you're looking for? Browse other questions tagged entanglement simulation classical-computing games non-locality or ask your own question. Can a Turing machine simulate a quantum computer? Can a quantum computer simulate a normal computer? How to simulate quantum entanglement variation in different quantum gates? Are correlations stronger than those allowed by quantum mechanics possible? Does a classical computer really require $2^n$ complex numbers to represent the state of $n$ qubit quantum computer? How is a quantum simulator able to simulate a quantum mechanical properties on a classical computer? Role of convexity in proof of Monogamy of Bell correlations Classical versus quantum correlations and partial traces	CommonCrawl
Oleg S. Ovchinnikov ORCID: orcid.org/0000-0002-8772-31501,2, Andrew O'Hara ORCID: orcid.org/0000-0002-0323-90391, Stephen Jesse ORCID: orcid.org/0000-0002-1168-84832, Bethany M. Hudak ORCID: orcid.org/0000-0002-4392-92373 nAff4, Shi‐Ze Yang ORCID: orcid.org/0000-0002-0421-006X3, Andrew R. Lupini ORCID: orcid.org/0000-0002-1874-79253, Matthew F. Chisholm ORCID: orcid.org/0000-0003-0546-51092, Wu Zhou ORCID: orcid.org/0000-0002-6803-10953,5, Sergei V. Kalinin ORCID: orcid.org/0000-0001-5354-61522, Albina Y. Borisevich ORCID: orcid.org/0000-0002-3953-84603 & Sokrates T. Pantelides ORCID: orcid.org/0000-0002-2963-75451,3,6 The automated detection of defects in high-angle annular dark-field Z-contrast (HAADF) scanning-transmission-electron microscopy (STEM) images has been a major challenge. Here, we report an approach for the automated detection and categorization of structural defects based on changes in the material's local atomic geometry. The approach applies geometric graph theory to the already-found positions of atomic-column centers and is capable of detecting and categorizing any defect in thin diperiodic structures (i.e., "2D materials") and a large subset of defects in thick diperiodic structures (i.e., 3D or bulk-like materials). Despite the somewhat limited applicability of the approach in detecting and categorizing defects in thicker bulk-like materials, it provides potentially informative insights into the presence of defects. The categorization of defects can be used to screen large quantities of data and to provide statistical data about the distribution of defects within a material. This methodology is applicable to atomic column locations extracted from any type of high-resolution image, but here we demonstrate it for HAADF STEM images. Structural defects can vastly alter the performance of materials so that control of defect distribution and density is an important tool in engineering materials with novel functionalities. Even small concentrations of defects can often change the properties of materials so that it is important to quantify the type and concentration of defects [1,2,3]. Over the last two decades, aberration-corrected scanning-transmission-electron-microscopy (STEM) has become a quantitative structural tool capable of locating atomic columns with picometer-level precision. The ability to achieve sub-pixel precision for the location of the center of an atomic column in STEM images has been demonstrated through image analysis techniques such as finding the center of mass and 2D function fitting with a Gaussian, allowing for accurate, consistent, and repeatable determination of the centers of atomic columns in STEM images [4,5,6,7,8]. Utilizing the crystallographic nomenclature [9], STEM essentially images diperiodic structures where thick structures are often referred to as 3D or bulk-like materials and thinner structures of just a few atomic layers are often termed 2D materials. Within a STEM image, it is possible to visually identify many defects such as impurities, interstitials, stacking faults, and a plethora of other complex defects for both types of diperiodic structures. Several methods exist to detect and identify defects in STEM images, each having unique benefits and limitations. Defects within atomic columns can be detected by examining deviations in the contrast, looking for deviations in the local atomic-scale structure [10, 11], overlaying an ideal atomic-scale structure on the image [12], and by using vector tracing [13]. These methods include measuring the distance between neighboring atoms in the structure and then using statistics and modeling to detect the presence and depth of a single defect in atomic columns [10, 11]; measuring the relative positions of neighboring atoms and then applying principal component analysis (PCA) followed by K-means clustering to map the ideal atomic-scale structure and statistical deviations from this idealized structure [12]; and using the Fourier transform of the image to determine the crystal structure's lattice parameter and then overlaying the obtained periodic structure on the atomic coordinates [14]. Alternatively, defects may be detected using cross-correlation between the STEM image and a simulated STEM image based on coordinates obtained by relaxing a model structure via density-functional-theory (DFT) calculations and then detecting defects through areas of low correlation [15]. Determination of the composition for mixed-species atomic columns can also be accomplished through the use of a parametric model based on statistical-parameter-estimation theory and further combined with STEM image simulations to quantitatively improve the model [16, 17]. All of these methods have achieved detection of defects that would not be possible or would be extremely time-intensive with the human eye. Furthermore, these methods provide a framework that is either general across materials or tailored to specific material systems in such a way that transferability is not limited via a known training set as might be the case in machine-learning-based object detection algorithms. In this paper, we report the development of a method that applies cycle analysis from geometric graph theory to the positions of atomic-column centers and is capable of detecting a wide range of defects in STEM images with no prior knowledge of the material. Although graph theoretical techniques have been used previously for the segmentation of spatial regions and identification of voids in imaged materials [18,19,20], these applications do not necessarily provide information on slight structural deviations in the imaged material at an atomic-column by atomic-column level of detail. In graph theory, a cycle is a path between points that connects a point back to itself. Multiple types of cycles exist such as the simple-walk cycle that does not allow any point or connection to be repeated. For this paper, a particular type of cycle is created with the following conditions: no vertices may be repeated, no connecting line may intersect another connecting line, the cycle must enclose a reference atomic column, the cycle must not enclose any additional atomic columns, and, finally, the cycle must be the shortest path connecting the vertices. For every atomic column in an image, a single cycle is found to represent it. Based on the number of vertices and the area of the cycles, it is possible to detect and categorize defects in the STEM image. The concept of pre-filtering wherein crystallographic information can inform the search is also discussed; however, the use of such databases may be limited for real-time analysis during image acquisition since the crystallographic nature of the material may not be known a priori. The approach is applied to STEM images of both thick diperiodic structures (i.e., 3D or bulk-like materials) as well as thin diperiodic structures (i.e., "2D materials"). In bulk-like silicon doped with bismuth, we demonstrate the ability of cycles to detect the Bi dopants in the atomic columns and compare with Z-contrast. In monolayer MoS2 doped with rhenium, sulfur vacancies are detected using two different cycle metrics. Methods and materials Image filtering, finding centers of atomic columns All of the raw STEM data are first processed to identify the centers of atomic columns as follows. At each pixel, a subimage is defined, centered at the pixel and encompassing an area roughly equal to the area per atomic column. These subimages are filtered using PCA to remove noise and surface contamination [21, 22]. The subimages are then passed through a 2D correlation [23] with an ideal atomic column (a 2D Gaussian) defined by: $$r(A,B) = \frac{{\sum_{m} \sum_{n} \left( {A_{mn} - \overline{A} } \right)\left( {B_{mn} \, - \overline{B} } \right)}}{{\sqrt {\left( {\sum_{m} \sum_{n} \left( {A_{mn} - \overline{A} } \right)^{2}} \right) \left( {\sum_{m} \sum_{n} \left( {B_{mn} - \overline{B} } \right)^{2} } \right)} }},$$ where A and B are two 2D matrices of the same size while $\overline{A}$ and $\overline{B}$ are their determinants, respectively. This process returns a single normalized intensity. From the filtered image, the centers of atomic columns are then found using a simple intensity threshold followed by density-based clustering [24]. Any clusters that do not meet a minimum size requirement are rejected. The center of mass of each cluster is treated as the center of an atomic column. Further refinement of the positions of the atomic-column centers is performed using nonlinear least-squares curve fitting between the raw data and a 2D Gaussian. The center of the fitted Gaussian is then treated as the refined center of the atomic column. Finding cycles for each atomic column A cycle is a path that connects atomic-column centers or "vertices" in such a way that it forms a closed loop to the original vertex. First, all possible cycles are found, after which they are filtered based on the previously described restrictions. To find the cycles associated with an atomic column, one considers only its n nearest points to speed up calculation time. The connections between each vertex and its m nearest neighbors are mapped. Typically, n = 40 columns and m = 10 columns. Then, using the nearest neighbor (or one of the equidistant nearest neighbors) as a starting point, all valid connections are followed to neighboring vertices. This process is repeated until the starting vertex is encountered or no connection can be followed without causing a repeating vertex. Filtering cycles Once the cycles are found as described above, they must be filtered to find a single cycle to represent each atomic column. This filtering is done by checking every cycle to see if it meets a series of rules. These rules are that no point may be repeated in the cycle, no connecting line may intersect another connecting line, the cycle must enclose the reference atomic column, the cycle must not enclose any other atomic column, the cycle must have no smaller angle then x degrees (x was set at 45o in the current work), and finally the cycle must be the shortest path connecting the points within a tolerance factor (within 1% of shortest path in the current work) (Fig. 1). Once all of the cycles that do not meet these criteria are removed, the cycles with the largest number of points are selected. From these cycles the one with the largest area is chosen as the cycle to represent an atomic column. The reason the largest cycle is used is for reproducibility. To ensure that we choose the same cycle each time, it must have a unique feature. Since the smallest cycle would always be a triangle and provide little information, the largest cycle is used instead. In cases of crystalline symmetry (i.e., rotationally equivalent cycles), a preferred cycle orientation can be chosen. a Rejected cycle due to cycle lines crossing each other. b Rejected cycle due to an extra atom enclosed in the cycle. c Rejected cycle due to cycle not being the shortest path connecting the points. d Accepted cycle that meets all parameters. e Accepted cycle that meets all parameters. f Accepted cycle that meets all parameters and is also the cycle with the most points In order to provide a concrete example of defect detection, we demonstrate in Fig. 2 how the above described algorithm finds a vacancy (i.e., more generally, a missing atomic column) for a 2D material with a hexagonal lattice like graphene (Fig. 2a). In Fig. 2b, three different orientations of a seven-sided cycle are shown. This is the optimal cycle based on the algorithm filter in the absence of defects, while the threefold symmetry is a manifestation of the point-group symmetries of the lattice. In the presence of strain, the length of the three bonds may be differentiated. In Fig. 2c, an atom that serves as a second-nearest neighbor to the vacancy site is shown, having a nine-sided cycle as its optimal cycle. In practical implementations, however, restrictions on the atomic-column search distance may cause the original seven-sided cycle to be found instead. In Fig. 2d, a third-nearest neighbor to the vacancy site retains the original seven-sided cycle (though the threefold symmetry is broken). For the atomic columns adjacent to the vacancy site (Fig. 2e), the optimal cycle is an eight-sided cycle. The resulting cycle mapping of the structure is shown in Fig. 2f, highlighting the location of the vacancy. Due to the atomic-column distance search limits, the purple atomic columns may be grouped with the blue atomic columns; there is no loss of detection of the vacancy due to the uniqueness of the directly neighboring sites. The uniqueness of the neighboring sites implies that the method is, in general, well suited to detection of missing columns with high accuracy in identification. Further classification due to variations in the vertex count of cycles is discussed in Section "Clustering cycles into defects using number of points in cycle". a Subset of a graphene-like structure in a hexagonal lattice (for the present example, the lattice is assumed to continue outside the drawn boundary). b Three rotationally equivalent seven-sided cycles indicating atoms in defect-free regions. Atomic columns inside such cycles are indicated in blue. c Optimal (nine-sided) and restricted (seven-sided) cycles for atomic columns that are second-nearest neighbor to a vacancy (indicated in purple). d Third-nearest neighbor atomic columns retain the original cycle with loss of rotational symmetry. e Atomic columns adjacent to the vacancy site are described by an eight-sided cycle (indicated in orange). f Final cycle-based color coding of the observed atomic columns based on the algorithm Constructing cycles using Delaunay triangulation Finding all possible cycles and checking them is a time-consuming process. A faster way to find a good guess of the best cycle is through the use of Delaunay triangulation [25]. For an atomic column, the positions of the nearest n neighbors (typically n = 40) are put into a Delaunay triangulation algorithm. Using the triangle that encloses the atomic column as the starting cycle, triangles from the Delaunay triangulation are combined with the cycle, testing whether the increased cycle at each step meets the selection criteria, until no further triangles can be added that meet the criteria. This method of finding the cycle for an atomic column is more than an order of magnitude faster than searching all possible cycles. However, it does not always find the true, correct cycle, as determined by a time-consuming exhaustive search, though it is often close. Generally, the fast process does not match the results of an exhaustive search only around defects that change the local structure, which does not affect the ability of this method to correctly detect defects (Fig. 3a, b). The centers of atomic columns for MoS2 colored based on the number of points in the cycles associated with them for a searching all cycles, b using Delaunay triangulation, and c using Delaunay triangulation plus pre-filtering Pre-filtering of cycles To further improve the speed of the algorithm, pre-filtering of cycles was tested. As all defect-free periodic crystalline materials can be described by a Bravais lattice with a given atomic basis and a set of symmetries, it is often possible to know the correct cycle before searching. To take advantage of this, we created a small library of possible cycles. We can overlay a library cycle onto the points by aligning it to the reference point and its nearest neighbor (or one of equidistant nearest neighbors) and scale it to fit the distance between the two points. Starting with the largest cycles in the library, the cycles are overlaid onto the points. If every point in the cycle coincides, within some uncertainty, with a point in the image, it is selected as the correct cycle. This process sometimes yields too small of a cycle, but it has no effect on defect detection (Fig. 3c). The speed improvement of pre-filtering is based on the size of the test library and the percent of defect atomic columns, as this is an additional operation that must be performed on defect atomic columns. Clustering cycles into defects using number of points in cycle The first method of detecting defects is by looking at deviations in the number of vertices in the cycles. In "2D materials", atomic columns near defects that cause changes in the local geometric structure such as vacancies, interstitials, or stacking faults have cycles that contain a different number of points than in the perfect material's local structure. To detect a defect, we mark as acceptable any atomic column that has a cycle with the same number of vertices in it as a cycle in the perfect crystal. The remaining cycles are clustered together using a density-based clustering algorithm [24]. This algorithm randomly selects a point as the start of a cluster and then adds every point that is within a specified radius into the cluster. This is repeated until no point can be added to the cluster. These clusters are then grouped based on the number of atomic columns in the cluster. This procedure allows for the automatic detection and grouping of defects in STEM data. Using cycle area to find defects Another method of using cycles to find defects is to look at changes in the cycle's area. Using the cycle's area to look for defects allows for the detection of defects that do not change the local structure's geometric coordination relative to defect-free regions, such as interstitials and vacancies in bulk materials or substitutional impurities. Any cycle area that is much larger or smaller than the average cycle area within some threshold represents the presence of a defect near that cycle. This works on a similar idea to previous work where single strontium and lanthanum vacancies where detected by measuring changes in the distance to nearest-neighbor atomic columns [10, 11]. The accuracy of using the area to detect such defects is limited by how much deviation in the cycle areas exists in the nominally pristine regions and the threshold for the change in area that is used to classify the defect. Here, we demonstrate the method for the detection of defects in 2D and 3D materials and discuss the method's efficacy at finding such defects. 3D bulk-like materials The ability to study defects using cycles is first demonstrated in a Mo–V–M–O material system, where M can be one of any number of atomic species [26], with Te being the most likely one in the current sample. The Mo–V–M–O compound is a material that has been studied as a potential catalyst and can display a variety of interesting phases and defects. In certain areas, this material possesses large stacking faults and missing atomic columns making it a good material to demonstrate the steps described in the "Methods" section (Fig. 4). In Mo–V–M–O, we believe that we can see the pooling of vacancies or M atoms under the surface of the material (Fig. 5c). The potential for large-scale vacancy clusters in this material and large stacking faults can be seen in Fig. 4. Ordinarily, if these types of defects occur below the surface layers, they may not be visible in Z-contrast imaging due to the presence of surface contamination which can obscure slight changes in column intensity. a Original HAADF image of Mo–V–M–O. b Atomic column location colored based on the number of atoms in associated cycles. c Atomic columns whose cycles are not part of the perfect crystal. d Non-perfect crystal atomic columns (i.e., those with differing cycles) grouped into defects and colored based on the number of atoms in a defect (single missing column—yellow; two adjoining missing atomic columns—purple; large staking fault—black) a Raw HAADF STEM image of Mo–V–M oxide. b Interpolated map of local intensity around every atomic column in panel a with the location of atomic columns shown. c Interpolated map of the cycle size associated with every atomic column in panel a with the location of atomic columns shown. d Raw HAADF STEM image of Bi-doped Si. e Location of every atomic column in panel d colored based on the local intensity around every atomic column. f Location of every atomic column in panel d colored based on the cycle size associated every atomic column The ability of using cycle area to detect defects within an atomic column was tested using bismuth (Bi) doped silicon (Si) (Fig. 5d). This material was used because the Z-squared difference between Bi and Si makes identifying the Bi locations straightforward as an independent comparison. The ability of cycle size to identify Bi within an atomic column was found to be worse than the reference of Z-squared intensity. Using cycle size, it was only possible to identify the approximate location of roughly 80% of the Bi dopants (Fig. 5e, f). In areas with more than one Bi dopant in close proximity, it is difficult to identify the number and exact location of the dopant. However, for isolated Bi, it is much easier. The 80% identification rate is due to the depth of the defects in the material, as intensity can help identify defects at a greater depth than using the distortion in the local structure on which cycle area identification relies. This result is in line with previous works that have used distortions in the local lattice to identify defects [10, 11]. For 2D materials, we have selected rhenium-doped MoS2 (Fig. 6a). This material was chosen due to the nature of defects available, namely rhenium (Re) dopants at molybdenum (Mo) sites along with single and double sulfur (S) vacancies. We filtered this image and found all the atomic columns using the procedure described in the Methods sections. Using the centers of the atomic columns, the cycles are found for each atomic column more than 2.5 times the average nearest neighbor distance from the edge. The number of points in each cycle are analyzed and the defects are subsequently categorized (Fig. 6b). Using the number of points in the cycle, all of the missing S columns were detected and categorized into single missing columns, two adjacent missing columns, and three adjacent missing columns. In MoS2, a fully missing S column is a divacancy. To find the single S vacancies, the areas of the cycles were used (Fig. 6c). Sulfur vacancies cause a noticeable decrease in the area of cycles. No method was found to identify the Re dopants using cycles because the frequent occurence of S vacancies near the Re makes the use of cycle area unreliable. The presence of S vacancies near Re dopants may be explained by the differing electronic properties of these defects in MoS2. Rhenium dopant atoms are shallow donor defects [27, 28] whereas sulfur vacancies are deep acceptor defects [29, 30]. Therefore, the S vacancies act as electron traps which can compensate for the excess electrons introduced by the Re dopant atoms. The electron transfer leads to an energy gain, making pairing the two defects energetically favorable. a STEM image of Re-doped MoS2. b Atomic columns identified to be near a defect using number of points in the cycles. c Area of the cycles where smaller areas are due to sulfur vacancies We described a method to detect structural defects in materials based on the concept of cycles from graph theory. 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Electron microscopy was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (BMH, SZY, ARL, MFC, WZ). Experiments were in part conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. We have acknowledged all relevant funding agencies in the Acknowledgements section. Bethany M. Hudak Present address: Materials Science and Technology Division, US Naval Research Laboratory, Washington, DC, USA Department of Physics and Astronomy, Vanderbilt University, Nashville, TN, USA Oleg S. Ovchinnikov, Andrew O'Hara & Sokrates T. Pantelides Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, TN, USA Oleg S. Ovchinnikov, Stephen Jesse, Matthew F. Chisholm & Sergei V. Kalinin Materials Sciences and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA Bethany M. Hudak, Shi‐Ze Yang, Andrew R. Lupini, Wu Zhou, Albina Y. Borisevich & Sokrates T. Pantelides School of Physical Sciences, CAS Key Laboratory of Vacuum Physics, University of Chinese Academy of Sciences, Beijing, China Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, 37235, USA Sokrates T. Pantelides Oleg S. Ovchinnikov Andrew O'Hara Stephen Jesse Shi‐Ze Yang Andrew R. Lupini Matthew F. Chisholm Sergei V. Kalinin Albina Y. Borisevich OSO, AO, SJ, SVK, AYB and STP conceived the project and wrote the manuscript. OSO and AO performed the data analysis. STEM data for the analysis was provided by BMH, SZY, ARL, MFC and WZ. All authors read and approved the final manuscript. Correspondence to Andrew O'Hara. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Ovchinnikov, O.S., O'Hara, A., Jesse, S. et al. Detection of defects in atomic-resolution images of materials using cycle analysis. Adv Struct Chem Imag 6, 3 (2020). https://doi.org/10.1186/s40679-020-00070-x Defect detection Atomic resolution	CommonCrawl
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Shearing1 Scientific Reports volume 9, Article number: 19465 (2019) Cite this article Characterization and analytical techniques The spatial resolution of 3D imaging techniques is often balanced by the achievable field of view. Since pore size in shales spans more than two orders of magnitude, a compromise between representativeness and accuracy of the 3D reconstructed shale microstructure is needed. In this study, we characterise the effect of imaging resolution on the microstructural and mass transport characteristics of shales using micro and nano-computed tomography. 3D mass transport simulation using continuum and numerical physics respectively is also compared to highlight the significance of the Knudsen effect on the reconstructed solid surface. The result shows that porosity measured by micro-CT is 25% lower than nano-CT, resulting in an overestimated pore size distribution and underestimated pore connectivity. This leads to a higher simulated intrinsic permeability. An overestimated diffusive flux and underestimated permeability are obtained from the continuum mass transport simulation compared to the numerical ones when the molecular-wall collision is accounted, evidenced by the large deviation of the measured Knudsen tortuosity factor and permeability correction factor. This study is believed to provide new knowledge in understanding the importance of imaging resolution and gas flow physics on mass transport in porous media. In recent years shale gas has attracted much attention due to the accessible energy reserves stored in low-permeability organic-rich shales and mudstones. These reservoirs contain a significant amount of hydrocarbons, and the successful exploitation of such resources plays a crucial role in meeting the world's surging demand for natural gas. This has the potential to play a significant role in the transition to a cleaner energy future due to its high energy content, resulting in lower emissions of carbon and volatile organic compounds (VOCs) at combustion, relative to coal and oil1. The gas is released with the help of hydraulic fracturing techniques also known as "fracking"2 and gas injection displacement3, and is transported through pores of multiple length scales, eventually converging in the main wellbore4. While the fracture network greatly determines the productivity of shale reservoirs5,6,7, the transport of shale gas within the matrix also plays an important role8,9,10. Valid pore structure analysis and image-based computational fluid dynamics (CFD) simulation of the shale gas flow in the porous media rely heavily on a faithful 3D representation of the porous microstructure. Non-destructive three-dimensional X-ray computed tomography (X-ray CT) has been widely applied to the multi-scale microstructure study of the shale gas7,11. This technique provides more reliable and representative 3D microstructure compared to those reconstructed by discrete 2D SEM images12,13, and helps to mitigate the artefacts of the pore phase potentially introduced from the 2D serial sectioning14,15. However, like other imaging techniques, there is a trade-off between the image resolution and the field of view (FOV), and therefore a compromise has to be made between the representativeness and the accuracy of the imaged microstructure, which could inevitably exclude small pores due to the hierarchical pore size distribution in the shale (i.e. ranging from tens of nanometre to micrometre)16. A previous study17 characterized the gas flow in micro and nanopores using ideal cylindrical pore model, which however cannot account for the effect of complex surface roughness of the wall, the constriction and the arbitrary morphology. Transport of gas molecules in porous media is mainly governed by two mechanisms: (1) continuum flow, in which the gas molecules interaction is dominant and is often modelled as a viscous effect in continuum physics and (2) the collisions between gas molecules and the wall, also known as molecular flow18. The predominant mechanism(s) in the transport regime will depend on the gas species, temperature, pressure and microstructure19,20,21. The Knudsen number Kn, calculated as the ratio between the mean free path of the gas molecules and the pore size, is widely used to assess the flow regime in porous media: If Kn < 0.01 (continuum regime), the flow is mainly governed by molecular diffusion and the Knudsen flow can be neglected; if Kn > 10 (Knudsen regime), the gas is highly rarefied and effect of molecular flow outweighs the viscous flow in the continuum regime because of the frequent collisions between the molecules and the porous medium. As for 0.01 < Kn < 10 (transitional regime), shale gas flow is governed by both mechanisms. The wide distribution of the pore size causes two problems in the mass transport study: (1) it is not reliable to estimate the Knudsen-based diffusivity based on the averaged pore size, which could potentially over-estimate the gas flow due to the constriction effect21,22; (2) Viscous flow fails in smaller pore spaces as the diffusion flow mechanisms associated with pore-wall interactions become dominant23, which leads to under-estimating the permeability. This means conventional continuum physics can no longer describe the flow field in shales24. To account for the gas molecules-wall interaction (i.e. wall slippage effect), different theoretical models have been adopted to predict the apparent permeability of nanopores, which is a key property for shale gas production. Klinkenberg25 analytically addressed the gas-wall collisions by introducing the slippage effect associated with the pressure. Beskok and Karniadakis26 mathematically integrated the Knudsen effect into the permeability measurement by comparing the apparent and intrinsic permeability. Tang et al.27 proved that the apparent permeability is nonlinearly related to the intrisinc permeability. Direct Simulation Monte Carlo (DSMC) is a numerical method widely used to solve the thermodynamic states of the rarefied gas based on Boltzmann equation, which effectively overcomes the challenges in gas-wall interaction by continuum modelling with conservation equations. Compared to other numerical methods such as molecular dynamics (MD)3, DSMC is less computational expensive with high confidence28.This method was validated either by experimental permeability29 or analytical solution30. DSMC has been applied to study the gas flow in a variety of materials of distinct pore morphologies, such as in solid oxide fuel cells31, cylindrical channels32,33, random/aligned fiber orientations30,34,35 and ablative materials29. In this study, we aim to elucidate the effect of imaging resolution on the characterization of porous microstructure and mass transport properties in shales using multi-length scale X-ray CT, followed by the image-based CFD simulation using both continuum and numerical method for the first time to highlight the effect of molecules-wall interaction on the extracted effective mass transport parameters (i.e. Knudsen tortuosity factor, apparent permeability) which could partially be neglected either by mean-field fluid dynamics (continuum flow) or resolution limitation. The particle-based CFD simulation methodology using reconstructed 3D microstructure proposed in this study is highly applicable not only to the shales, but also to be of wide interest across an increasingly broad range of mass transport studies in geological materials. X-ray Computed Tomography A cylindrical sample pillar, already employed for previous investigations7, was prepared from the shale sample using an A Series/Compact Laser Micromachining System (Oxford Laser, Oxford, UK) following the procedure explained by Bailey et al.36. The three-dimensional microstructure of shale sample was investigated using two X-ray computed tomography microscopes (Carl Zeiss X-ray Microscopy Inc., Pleasanton, CA): micron-scale Zeiss Xradia 520 Versa (micro-CT) and nano-scale Zeiss Xradia 810 Ultra (nano-CT). For micro-CT, a total of 1401 radiographs were acquired over a 360° sample rotation range with an exposure time of 35 seconds per radiograph. The shale sample was placed between the X-ray source and a 2k × 2k detector providing a voxel resolution of 224 nm using the 20x objective magnification and a Field of View (FOV) of 448 μm. The instrument was operated at 80 kV. Nano-CT employs post-transmission Fresnel zone plates to achieve resolution in the sub 100 nm range37. A total of 1601 projections were collected per 180° sample rotation with an exposure time of 36 seconds. This allowed achieving a set of raw image data with an isotropic voxel resolution of 63 nm and a FOV of 65 μm. The raw transmission images from both micro-and nano-scale CT imaging experiments were reconstructed using a commercial image reconstruction software package (Zeiss XMReconstructor, Carl Zeiss X-ray Microscopy Inc., Pleasanton, CA), which employs a filtered back-projection algorithm. Tomographic scan details are shown in Table 1. The 3D reconstructed volume of the shale was segmented and analysed using Software Avizo Fire 9.2 (Thermo Fisher Scientific, USA). Due to the low X-ray absorption coefficient difference, it is not possible to distinguish the organic matter (kerogen) from pores based on the reconstructed grayscale data, thus the combined phases are rendered together. This phenomenon is normal in processing X-ray CT data and the same measure was taken in published research11. The pore size distribution (PSD) was measured using the plug-in 'Beat'38 in open-source software FiJi39. Table 1 Scanning parameters of micro- and nano-CT. Effective mass transport parameters by continuum fluid dynamics The surface mesh (ASCII *.stl) file was generated after the segmentation of the porous phase and imported into the commercial computational fluid dynamics (CFD) software Star-CCM+ (CD-Adapco Inc., London). A mesh refinement procedure was undertaken to improve the mesh quality from as-imported raw data (Fig. 1a) to the refined triangular surface mesh (Fig. 1b), to the final polyhedral volume mesh (Fig. 1c). The use of a polyhedral mesh has proven to be more accurate for fluid-flow problems than a hexahedral or tetrahedral mesh of a similar size. The optimized mesh is closed and manifold, with no holes and free edge and the volume change of the porous phase is ensured not to exceed 1% to maintain the microstructural originality. Mesh refinement procedure. (a) as-imported triangular surface mesh; (b) refined triangular surface mesh; (c) generated polyhedral volume mesh. The tortuosity factor is an effective mass transport parameter representing the effect of complex porous gas pathways on the gas flow22,40. In this study, it was measured by CH4 ordinary diffusive flow: the CH4 molar concentration was set as c = 1 mol m−3 at the inlet and c = 0 at the outlet. It is noted that the tortuosity factor measured by continuum physics is a material parameter and independent of the concentration gradient of the gas. The one-dimensional gas flow Qe can be described by Fick's law as $${Q}_{e}=AD\frac{\Delta c}{x}$$ where D is the intrinsic diffusivity, A is the cross-sectional area of the fluid domain, Δc is the concentration change and x is the diffusion length. In the porous medium, Eq. (1) is modified as $${Q}_{p}=\frac{\varepsilon }{\tau }AD\frac{\Delta c}{x}$$ where τc is the tortuosity factor, ε is the porosity and can be measured by CT data analysis. By dividing Qe by Qp, the effective transport parameter ε/τ can be obtained, $$\frac{{Q}_{p}}{{Q}_{e}}=\frac{\varepsilon }{\tau }$$ It is noted that in the continuum fluid model, the effective transport parameter is independent of the intrinsic diffusivity, indicating that it is a material parameter. The Reynold's number41 of the shale gas flow is far less than unity, which suggests that viscous forces dominate over inertial forces and the permeability can be obtained according to Darcy's law42, $$\frac{\partial P}{\partial x}=-\frac{\mu }{k}v$$ where k is the permeability of the porous medium, v is the gas velocity, μ is the dynamic viscosity of the gas, P is the pressure and x is the distance in the flow direction. The intrinsic permeability was obtained by setting a pressure drop (50 Pa) from the inlet to the outlet according to Eq. (4). It is noted that for continuum fluid dynamics, the intrinsic permeability is independent of the pressure gradient. Effective mass transport parameters by non-continuum fluid dynamics To highlight the significance of molecules-wall interactions (Knudsen effect) in the hierarchical porous shale, numerical simulation method Direct Simulation Monte Carlo (DSMC) was used on the 3D reconstructed shale from X-ray CT scan, which is believed to provide a faithful representation of the wall roughness and pore morphology. A sub-volume consisting of 168 × 200 × 200 voxel (10.6 × 12.6 × 12.6 μm) was used in this study. The Stochastic PArallel Rarefied-gas Time-accurate Analyzer (SPARTA)43 DSMC code developed at Sandia National Laboratory (USA) was used in this work. The generated surface mesh (i.e.stl file) of the shale was imported into the SPARTA software such that it was embedded in the fluid domain which is composed of an array of 3D Cartesian grids (1.5 million in total). Inter-molecule and molecule-wall collisions were performed following a no-time-counter (NTC) procedure44. Shale gas (CH4) was simulated from slip flow regime (0.01 < Kn ≤ 0.1) to transitional regime (0.1 < Kn ≤ 10) with incremental pressure to obtain Knudsen tortuosity factor τk based on Eq.(4) and apparent permeability Ka based on Eq. Table as a combination of the ideal gas law, conservation of mass and the differential form of Darcy's law., $$J=-\frac{M}{\mu RT}{k}_{a}P\frac{dP}{dx}$$ where J denotes the mass flux by DSMC; M, R, T, μ are molecular weight of the gas species, gas constant, the temperature and viscosity respectively. Buffer zones of at least 10% total flow domain were added. A total of 20 million simulation molecules were generated so that the average molecule number in each cell is above 20 to avoid statistical scattering28. Each of these simulating molecules is regarded as the representative of a large number of real molecules, the ratio of which is known as scaling factor45, to reduce the demand of computational resources. In this study, a scaling factor of 15 was used and small enough to provide accurate DSMC results. The validation of this technique was performed experimentally31. The interaction between gas molecules and the porous media can be seen in the video (see Supplementary Video S1). The effect of imaging resolution on the reconstructed volume is highlighted in Fig. 2. The top row (a–c) shows the grayscale virtual slices scanned and reconstructed using micro-CT, from which it is clear to see the blurred microstructure due to the resolution limits and it is impossible to extract the pore network with high confidence, particularly for the smallest pores; in the bottom row (d–f), the same region obtained from nano-CT was registered and shown as the superimposition of the micro-CT images. By comparing the obtained microstructures between the top and bottom row, it is found that nano-CT scan provides significantly sharper images so that more microstructural details such as edges and narrow pores which are missing in micro-CT scans can be captured in nano-CT data. In the next section, two case studies will be presented to highlight (1) the effect of imaging resolution on describing the microstructural characteristics and mass transport properties in the direction parallel to the horizontal natural bedding of the shale gas sample; (2) the disparity of obtained mass transport parameters vertical to the natural bedding between continuum and numerical CFD simulation attributed to the captured sub-micron 3D pore network. Comparison of the microstructure scanned using micro-CT and nano-CT. (a–c) micro-CT reconstructed slices of XY, XZ, and YZ views respectively; (d–f) nano-CT reconstructed slices registered on top of the micro-CT ones, for XY, XZ and YZ views respectively. Scale bar is 50 μm. Case study 1: effect of imaging resolution on pore structure and mass transport metrics This study aims to compare the microstructural metrics and mass transport parameters (i.e. tortuosity factor and permeability) as a consequence of the extra porosity which can be imaged in the nano-CT. The same sub-volume was extracted from micro- and nano-CT and compared in Fig. 3. Figure 3a,d compare the morphology of the same pore under two resolutions. The details of the pore edges and curvatures which can be seen in nano-CT (Fig. 3a) are volume-averaged in micro-CT (Fig. 3d). This could lead to an under-segmentation of the pore network from the reconstructed volume (Fig. 3b,e). This local homogenisation effect by micro-CT can result in two disadvantages which undermine further analysis: (1) the pore size distribution and porosity will be over and under-estimated respectively; (2) the percolation will be underestimated as the extracted pore network does not include all of the sub-resolution pores. Comparison of the pore microstructure under two resolutions. Figure (a–c) presents the grayscale, segmented, colour-coded by thickness, skeletonised pore structures obtained based on nano-CT scan; (d–f) shows the counterparts using micro-CT; (g) displays the overlay of the rendered 3D volume of the pore structure under micro-CT (blue) and nano-CT (red). After segmentation, a Continuous Pore Size Distribution (C-PSD) analysis was carried out and the PSD is shown as a heatmap with the colour-coded according to its size (Fig. 3c,f). A highly complex pore structure is resolved using nano-CT in contrast to the smoothed single-pore feature using micro-CT. Figure 3g highlights the disparity of the extracted pore structure by overlaying the 3D rendered pore structure under the two resolutions. The C-PSD measurement is summarised in Fig. 4. It is observed that with finer features resolved in nano-CT data, the pore size can be quantified with a smaller step size compared to that in micro-CT. Nano-CT scan allows to capture and quantify tinier pores (<1 μm). It is noted that large pores (>1 μm) that are dominant in micro-CT data are not observed in nano-CT measurement. On the other hand, a large amount of pore volume shifts to the low radius end of the histogram. This disparity is speculated as another disadvantage of coarse resolution scan of the shale gas: the complex curvature of the pore edge is homogenised in 3D so that the pore throat resolved in nano-CT (yellow arrow in Fig. 3a) is averaged with the slices above and underneath the plane, resulting in the pore with blurred edge and less dark grey value (Fig. 3d), as a consequence of which, the measured pore size distribution deviates from the practical value. Comparison of the continuous pore size distribution obtained by micro-CT and nano-CT data. The segmented pores were then meshed and imported to the CFD software Simcenter STAR-CCM+ (Siemens, Plano, TX, USA) for the continuum simulation in order to assess the variation of the measured mass transport properties originating from different imaging resolutions. Figure 5a,b compare the concentration distribution of methane gas at steady state and it is found that for the first half of the flow field the nano-CT pore volume exhibits a much lower concentration gradient (Fig. 5b) compared to micro-CT pore (Fig. 5a), but for the second half of the pore volume the concentration distribution is identical. The reason for this phenomenon is that the nano-CT managed to resolve a larger lateral region of the first half of the pore, providing a higher cross-section for the flow thereby the reduced concentration drop, whilst for the second half more parallel-connected pores are captured in the nano-CT, which in essence would not alter the concentration distribution, instead yielding a higher flow rate. In order words, the morphological difference between two resolution scans mainly consists of laterally-resolved pores which may connect in parallel or in series with the existent pore. However, the velocity field of the viscous flow driven by a constant pressure difference (5 × 10−4 bar) is significantly different between the two samples: the micro-CT sample exhibits much higher velocity than the nano-CT one, which is considered as a consequence of the larger surface area and narrower pores in nano-CT, leading to a more remarkable viscous effect. The measured mass transport metrics between the micro-CT and nano-CT samples are summarised in Table 2. The continuum tortuosity factor τc between two samples are similar, as is consistent with the concentration field in Fig. 5a,b. The resultant permeability of micro-CT sample is almost one order of magnitude higher than the nano-CT value. Comparison of the concentration distribution (a and b, micro-CT and nano-CT respectively) and velocity field (c and d, micro-CT and nano-CT respectively) simulated by continuum diffusive and viscous flow to highlight the effect of imaging resolution. Table 2 Summary of the pore structure metrics and mass transport parameters measured by different resolution scans. Case study 2: gas flow simulation perpendicular to the bedding direction This case study aims to investigate the difference of the measured tortuosity factor and permeability when the gas molecules - wall collision is considered in the CFD simulation. This is important as in most of the cases the shale gas flow is governed by transitional and Knudsen regime and thus Darcy flow fails in smaller pore spaces in which the wall-slippage effect becomes dominant. This means the conventional continuum CFD method with the non-slippage condition at the pore-wall interface can no longer faithfully describe the gas flow in the shale. However, few studies have compared the disparity of the extracted tortuosity factor and permeability obtained between continuum and numerical method, and thus the uncertainty is ambiguous. Different from Case Study 1, in which the boundary condition was applied so that the gas flew parallel to the natural bedding direction, Case Study 2 examines the gas transport property vertical to the natural bedding, in which direction the resistance is significantly higher and the gas molecules – wall interaction is more dominant. Figure 6a shows the concentration distribution of CH4 simulated using continuum CFD method. It is observed that the gradient here is less smooth and uniform compared to Fig. 5a,b, evidenced by a sharp decrease of the concentration at the pore throat vertically connecting the top and bottom half of the horizontally aligned pores. This is the main reason for the strong anisotropy of gas transport in the shale. The narrow pore throat can be visualised in Fig. 6b in terms of the consequent local high velocity of the gas flow. Figure 6c shows some possible streamlines of the gas transport from the top to the bottom, via one of the pore throats. It is noted that the trajectory is highly convoluted and tortuous. The intrinsic permeability measured by the continuum flow is Ki = 7.1 × 10−19 m2. Figure 6d is a snapshot showing the CH4 distribution (red spheres) by particle-based numerical simulation. To the author's knowledge, this is the first time that DSMC method has been used on the study of shale gas based on the reconstructed 3D volume of the pore network which provides the pore-solid boundary resolved with high confidence at 0.1 μm resolution. This is favourable to examine the collision between the gas molecules and the surface of the wall, as is shown in Fig. 6e. It is observed that the collision does not occur uniformly in the pore volume, instead, it is highly localised at the places where the pore size is smaller than the surrounding areas. This can be supported by comparing with the 3D distribution of the pore diameter Fig. 6f, in which the red arrows point out the corresponding areas of high collision frequency. Comparison of the shale gas flow simulation using continuum method and numerical method. (a) Concentration distribution of CH4 flow perpendicular to the bedding lamination; (b) the velocity of the CH4; (c) streamlines showing the trajectory of gas flow through the narrow pore throat connecting the laminations; (d) particle-based CFD simulation with red spheres representing the CH4 molecules; (e) collision frequency distribution between the gas molecules and the wall; (f) the 3D pore size distribution; (g) the variation of Knudsen tortuosity factor τk as a function of the gas pressure; (h) the ratio of molecules-wall collision and inter-molecules collision as a function of the Kn; (i) correction factor obtained by DSMC method and compared with the empirical models. Figure 6g compares the Knudsen tortuosity factor τk as a function of the gas pressure with that obtained from continuum modelling, which is a constant value independent of the gas pressure. It is found that when the CH4 is highly rarefied, τk is measured to be as large as 50, then drops drastically with the pressure. At 1 bar, τk ≈ 27. The curve asymptotically converges to the continuum tortuosity factor τc = 10 when the pressure is above 8 bar, from which and onward continuum flow is dominant. Figure 6g proves that in real-life situations, as the gas pressure is much higher than 8 bar, continuum modelling can be safely applied to the investigated gas shale with the minimal pore size larger than 0.1 μm, which is the imaging resolution of this study. Figure 6h plots the ratio (η) of molecules-wall collision and inter-molecules collision as a function of the Kn the two types of collision: molecules-wall and inter-molecules collisions. It is found that the simulated points are linearly proportional to the Knudsen number, aligning with η = Kn very well. Finally, a correction factor f to the intrinsic permeability Ki as a function of Knudsen number is obtained based on the apparent permeability Ka using DSMC method (Ka = f∙Ki) (Fig. 6i). When Kn ≤ 0.01, the intrinsic and apparent permeability are in a good agreement, implying the little influence of viscous flow from the wall slippage; with the increase of Kn thus decreasing pressure, the apparent permeability diverges with the intrinsic one, for instance, when Kn = 1, the gas molecules – wall collision is so predominant that the apparent permeability is 9 times larger than the intrinsic one. This result is compared with the empirical solution derived from Klinkerberg's model25 and Beskok/Karniadakis model26. Generally, all three curves exhibit the exponential relationship between the correction factor and the Knudsen number, and the measured one by DSMC method is slightly larger than the other two empirical models. This could arise from a variety of factors related to the geometry of the pore structure, such as constriction, shape etc. This study firstly compared the difference of reconstructed 3D volume of the shale scanned by X-ray Computed Tomography (CT) using different resolutions (voxel size 224 nm for micro-CT and 63 nm for nano-CT), based on which the continuum CFD simulation was conducted to highlight the effect of imaging resolution on the obtained tortuosity factor and permeability of the shale. The second part of the study discussed the importance of gas molecules-wall collision and wall slippage effect by numerical Direct Monte Carlo Simulation (DSMC) which has been applied to the microstructure-resolved shale model for the first time and then compared the disparity of the mass transport parameters obtained by the conventional continuum CFD modelling. It is found that low-resolution scan has two main disadvantages: (1) the pore size distribution and porosity are over and under-estimated respectively; (2) the percolation is underestimated as the extracted pore network does not include all of the sub-resolution pores. These lead to a much larger intrinsic permeability of the micro-CT scan than the nano-CT one. The morphological difference of the pore structure between two resolution scans mainly consists of laterally-resolved pores which may connect in parallel or in series with the existent pore. The former would not change the concentration distribution but provide a higher mass flow whereas the latter would render a lower local concentration gradient due to the increased cross-sectional flow area. When the surface collision (Knudsen effect) and slippage are considered, the tortuosity factor can be as large as 50 for the most rarefied gas and then significantly drop with the pressure until asymptotically reaching the value (10) obtained by continuum method, implying an over-estimated diffusive flux when the Knudsen effect is not included. In addition, the apparent permeability showed an exponential relationship with the intrinsic one as a function of the Knudsen number, indicating that as the pressure decreases, the deviation of the apparent permeability is larger from the intrinsic one. It is also shown that the ratio of the frequency of the molecular-wall and inter-molecular collision can be estimated by the Knudsen number. As both numerical and continuum simulation methods are widely used in the shale gas study, this study is believed to provides new insights in emphasizing the validity and uncertainty level of the shale gas flow under a variety of conditions. 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This work was supported by the EPSRC (EP/N032888/1, EP/M014045/1, EP/K005030/1, EP/M008428/1), PRS acknowledges funding from the Royal Academy of Engineering (CIET1718/59), XL acknowledges the support of the NPL Measurement Fellowship. University College London, Electrochemical Innovation Lab, Department of Chemical Engineering, London, WC1E 7JE, UK Francesco Iacoviello, Xuekun Lu, Daniel J. L. Brett & Paul R. Shearing University College London, Department of Earth Sciences, London, WC1E 6BT, UK Thomas M. Mitchell Francesco Iacoviello Xuekun Lu Daniel J. L. Brett Paul R. Shearing F.I. and X.L. conceived the project; P.R.S. and D.J.L.B. funded and managed the project as directors of the Electrochemical Innovation Lab (EIL); F.I. conducted the X-ray CT experiments and the computer segmentation; X.L. conducted the CFD simulation using Direct Simulation Monte Carlo (DSMC); T.M.M. contributed to the discussion of the results; F.I. and X.L. analysed all results; F.I. and X.L. wrote the manuscript; F.I. and X.L. contributed equally to this work; all authors reviewed the manuscript. Correspondence to Francesco Iacoviello. Supplementary Video Iacoviello, F., Lu, X., Mitchell, T.M. et al. The Imaging Resolution and Knudsen Effect on the Mass Transport of Shale Gas Assisted by Multi-length Scale X-Ray Computed Tomography. Sci Rep 9, 19465 (2019). https://doi.org/10.1038/s41598-019-55999-7 Configurational diffusion transport of water and oil in dual continuum shales Mohammed Abdul Qadeer Siddiqui Filomena Salvemini Hamid Roshan High-resolution imaging of depth filter structures using X-ray computed tomography T. F. Johnson F. Iacoviello D. G. Bracewell Journal of Materials Science (2021) About Scientific Reports Guide to referees Journal highlights	CommonCrawl
\begin{document} \title{Effective bounds on ampleness of cotangent bundles} \author[I. Coskun]{Izzet Coskun} \address{Department of Mathematics, Statistics and CS \\UIC, Chicago, IL 60607} \email{[email protected]} \author[E. Riedl]{Eric Riedl} \address{Department of Mathematics, 255 Hurley, Notre Dame, IN 46556 } \email{[email protected]} \subjclass[2010]{Primary: 14M10. Secondary: 14F10, 32C38} \keywords{Complete intersections, ample cotangent bundle} \thanks{During the preparation of this article the first author was partially supported by the NSF grant DMS-1500031 and NSF FRG grant DMS 1664296 and the second author was partially supported by the NSF RTG grant DMS-1246844.} \begin{abstract} We prove that a general complete intersection of dimension $n$, codimension $c$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ has ample cotangent bundle if $c \geq 2n-2$ and the $d_i$'s are all greater than a bound that is $O(1)$ in $N$ and quadratic in $n$. This degree bound substantially improves the currently best-known super-exponential bound in $N$ by Deng, although our result does not address the case $n \leq c < 2n-2$. \end{abstract} \maketitle \section{Introduction} Let $X$ be a general complete intersection in $\mathbb{P}^N$ of dimension $n>1$ and type $d_1, \dots, d_c$. In this note, we prove that if $c \geq 2n-2$ and $$d_i \geq \frac{(2n-2)(24n-28)}{N-3n+3}+2,$$ then the cotangent bundle $\Omega_X$ is ample. Debarre conjectured that a general complete intersection $X \subset \mathbb{P}^N$ with $c \geq n$ has ample cotangent bundle provided that the degrees $d_i$ defining $X$ are sufficiently large \cite{Debarre}. Debarre's Conjecture has been proven by both Brotbek and Darondeau \cite{BrotbekDarondeau} and Xie \cite{Xie}. Brotbek and Darondeau do not provide effective bounds, while Xie showed that one can take $d_i \geq N^{N^2}$ to guarantee that $\Omega_X$ is ample \cite{Xie}. Deng in \cite{Deng1, Deng2} improved the bounds to $d_i \geq 16c^2 (2N)^{2N+2c}$. When $c \geq 2n-2$, our bounds are vast improvements on these exponential bounds. In fact, our bound is 3 as soon as $N \geq 48n^2 -101n +53$. In earlier work, Brotbek \cite{BrotbekExp} proved that if $c \geq 3n -2$ and all the degrees are equal $d_i =d$, then $\Omega_X$ is ample provided that $d \geq 2N+3$. While our bound is less restrictive on $c$ and is better for $N$ large with respect to $n$, in the case $c \geq 3n-2$, $d_i = d$ for all $i$, and $N$ small relative to $n$, Brotbek's bound of $2N+3$ is better. Finally, Brotbek in \cite{BrotbekCot} showed that a general complete intersection surface has ample $\Omega_X$ if $d_i \geq \frac{8N+2}{N-3}$. The first step is to clarify and improve Brotbek's \cite{BrotbekCot} estimates that guarantee that $\Omega_X$ is ample outside a codimension 2 subvariety. We use a more careful combinatorial analysis and a theorem of Darondeau. This sets up a new application of a technique of Riedl and Yang \cite{RiedlYang, RiedlYang2}, which allows us to remove the non-ample locus. This process loses a little on the codimension bound relative to \cite{BrotbekDarondeau}, but gives much better bounds on the degrees. \subsection{Organization of the paper} In \S \ref{sec-prelim}, following Brotbek \cite{BrotbekCot}, we obtain degree bounds that guarantee that $\Omega_X$ is ample outside a variety of codimension 2. In \S \ref{sec-main}, using the technique of Riedl and Yang \cite{RiedlYang, RiedlYang2}, we show how to remove the non-ample locus. \subsection{Acknowledgments} We thank Damian Brotbek, Lionel Darondeau, Lawrence Ein, Mihai P\u{a}un and David Yang for enlightening discussions. \section{Ampleness outside a codimension $2$ set}\label{sec-prelim} Let $X \subset \mathbb{P}^N$ be a general complete intersection of dimension $n$ and type $d_1, \dots, d_c$. We always assume that the codimension $c = N-n \geq n$. Let $\Omega_X$ denote the cotangent bundle of $X$ and let $$\pi: \mathbb{P} \Omega_X \rightarrow X$$ be the natural projection. In this section, we give bounds on the degrees $d_i$ that guarantee that $\Omega_X$ is ample outside of a codimension $2$ set. We follow the basic strategy from Brotbek \cite{BrotbekCot} closely. However, using a more careful analysis of the combinatorics and a new theorem of Darondeau, we improve his bounds, which are exponential in $n$, significantly. \begin{definition} Let $E$ be a vector bundle on a projective variety $Y$ and let $H$ be an ample line bundle on $Y$. Let $\pi: \mathbb{P}(E) \rightarrow Y$ denote the projection. If for some $\epsilon$ with $0< \epsilon \ll 1$, any irreducible curve $C \subset \mathbb{P}(E)$ with $C \cdot \mathcal{O}_{\mathbb{P}(E)}(1) < \epsilon \ C \cdot \pi^* H$ satisfies $\pi(C) \subset T$, then $E$ is said to be {\em ample outside $T \subset Y$}. \end{definition} It follows from the definition that if $\operatorname{Sym}^k E$ is globally generated outside of a subvariety $T$ of $Y$ for some $k > 0$, then $E \otimes H$ is ample outside of $T$. In \cite{BrotbekCot} Brotbek proves the following theorem. \begin{theorem}\cite[Theorem 4.5, Corollary 4.7]{BrotbekCot}\label{thm-BrotbekCot} Let $X \subset \mathbb{P}^N$ be a general complete intersection of dimension $n$ and type $d_1, \dots, d_c$. If $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-a-N)$ on $\mathbb{P} \Omega_X$ is big, then the projection of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \mathcal{O}(-a)$ under $\pi$ has codimension at least 2 in $X$. Thus, if $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-N-1)$ is big, then $\Omega_X$ is ample outside an algebraic set $Y$ of codimension at least 2 in $X$, where $Y$ is the image under $\pi$ of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-1)$. \end{theorem} We now explain how a theorem of Darondeau allows us to remove the dependence on $N$ in Theorem \ref{thm-BrotbekCot}. Let $\mathbb{P}^{N_1} \times \dots \times \mathbb{P}^{N_c}$ be the moduli space of all tuples of homogeneous polynomials $(f_1, \dots, f_c)$ of degrees $d_1, \dots, d_c$, respectively. Let $B \subset \mathbb{P}^{N_1} \times \dots \times \mathbb{P}^{N_c}$ be the Zariski open subset parameterizing tuples that intersect transversely and thus define smooth complete intersections of type $d_1, \dots, d_c$. Let $\mathcal{U}$ be the universal family over $B$, whose points parametrize tuples $(p,f_1,\dots,f_c)$ where $p \in V(f_1, \dots, f_c)$. \begin{theorem}[Main Theorem, compact case from \cite{Darondeau}] The vector bundle $T_{\mathbb{P}(\Omega_{\mathcal{U}/B})} \otimes \mathcal{O}_{\mathbb{P}^N}(3) \otimes \mathcal{O}_{B} (1,\dots, 1)$ is globally generated. \end{theorem} By replacing Merker's bound (Theorem 4.9 in \cite{BrotbekCot}) with Darondeau's improved bound from \cite{Darondeau} in the proof of Theorem 4.5 in \cite{BrotbekCot}, one obtains the following. \begin{theorem}\label{thm-OurCot} Let $X \subset \mathbb{P}^N$ be a general complete intersection of dimension $n$ and type $d_1, \dots, d_c$. If $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-a-3)$ on $\mathbb{P} \Omega_X$ is big, then the projection of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \mathcal{O}_X(-a)$ under $\pi$ has codimension at least 2 in $X$. Thus, if $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-4)$ is big, then $\Omega_X$ is ample outside an algebraic set $Y$ of codimension at least 2 in $X$, where $Y$ is the image under $\pi$ of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-1)$. \end{theorem} In view of Theorem \ref{thm-OurCot}, we desire effective bounds on the degrees $d_i$ that guarantee that the line bundles $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-3)$ and $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-4)$ are big. Recall the following criterion for bigness of a line bundle. \begin{theorem}\cite[Theorem 2.2.15]{LazarsfeldBook}\label{thm-bigness} If $F$ and $G$ are nef line bundles on an $r$-dimensional variety and $F^r > r F^{r-1} \cdot G$, then $F-G$ is big. \end{theorem} We will use the following proposition from Brotbek. \begin{proposition}\cite[Proposition 4.2]{BrotbekCot}\label{prop-omega2} Let $Y \subset \mathbb{P}^N$ be a smooth projective variety. The bundle $\Omega_Y(2)$ is ample if and only if $Y$ does not contain lines. \end{proposition} \begin{theorem}\label{thm-big} Let $X \subset \mathbb{P}^N$ be a smooth complete intersection of dimension $n$ and type $d_1,\dots, d_c$ with $c \geq n$. Let $a \geq -1$ be an integer. If $$ d_i \geq \frac{n((2n-1)(a+2) +2 )}{N-2n+1} + 2$$ for all $i$, then $\mathcal{O}_{\mathbb{P}\Omega_X}(1) \otimes \pi^* \mathcal{O}_X(-a)$ is big. \end{theorem} \begin{proof} Under our assumptions on $d_i$, the general complete intersection $X$ does not contain any lines. Consequently, by Brotbek's Proposition \ref{prop-omega2}, $\Omega_X(2)$ is ample. Equivalently, the line bundle $F= \mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_X(2)$ is ample on $\mathbb{P} \Omega_X$. The line bundle $G = \pi^* \mathcal{O}_{X}(a+2)$ is nef on $\mathbb{P} \Omega_X$ being the pullback of a nef line bundle on $X$. By Theorem \ref{thm-bigness}, $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_{X}(-a)$ is big if $$F^{2n-1} > (2n-1) F^{2n-2} \cdot G.$$ Recall that the Segre classes of a rank $r$ vector bundle $E$ are defined by $$s_i(E) = \pi_* ((c_1(\mathcal{O}_{\mathbb{P} E}(1))^{r-1+i}).$$ Thus, $F^{2n-1} = s_{n}(\Omega_X(2))$ and by push-pull, $F^{2n-2} \cdot G = s_{n-1}(\Omega_X(2)) \cdot (a+2)H$. Let $s(E)$ denote the total Segre class of $E$. The Euler sequence on $\mathbb{P}^N$ twisted by $\mathcal{O}_{\mathbb{P}^N}(2)$ \[ 0 \to \Omega_{\mathbb{P}^N}(2) \to \mathcal{O}(1)^{N+1} \to \mathcal{O}(2) \to 0 \] implies that \[ s(\Omega_{\mathbb{P}^N}(2)) = \frac{1-2H}{(1-H)^{N+1}}. \] The conormal sequence for $X$ \[ 0 \to \bigoplus_{i=1}^{c} \mathcal{O}(-d_i+2) \to \Omega_{\mathbb{P}^N}(2)\|_X \to \Omega_X(2) \to 0 \] yields \[ s(\Omega_X(2)) = \frac{(1-2H)\prod_{i=1}^c(1+(d_i-2)H)}{(1-H)^{N+1}} .\] Let $\epsilon_k(x_1, \dots, x_{c}) = \sum_{i_1< \cdots < i_k} x_{i_1} \cdots x_{i_k}$ denote the $k$th elementary symmetric function in $x_1, \dots, x_{c}$. For an $r$-tuple $d=(d_1, \dots, d_c)$, let $$\phi_{k,d} = \epsilon_k(d_1-2, \dots, d_c -2).$$ Since $$\frac{1}{(1-x)^{N+1}} = \frac{d^N}{dx^N} \left( \frac{1}{N!} \frac{1}{1-x} \right) = \frac{d^N}{dx^N} \left(\frac{1}{N!} \sum_{i \geq 0} x^i\right) = \sum_{i\geq 0} \binom{i+N}{N} x^i,$$ we obtain the relation \[ s(\Omega_X(2)) = (1-2H) \left(\sum_{i \geq 0} \phi_{i,d} H^i \right) \left( \sum_{i \geq 0} \binom{i+N}{N} H^i \right) .\] For our purposes, we only need $s_{n}(\Omega_X(2))$ and $s_{n-1}(\Omega_X(2))$. Then \[ s(\Omega_X(2))= (1-2H)(\dots + b_{n-2} H^{n-2} + b_{n-1}H^{n-1} + b_{n}H^{n}), \] where \[ b_{n} = \sum_{k=0}^{n} \phi_{k,d} \binom{N+n-k}{N} \] \[ b_{n-1} = \sum_{k=0}^{n-1} \phi_{k,d} \binom{N+n-k-1}{N} \] \[ b_{n-2} = \sum_{k=0}^{n-2} \phi_{k,d} \binom{N+n-k-2}{N} .\] Then we have $$s_{n} = (b_{n} - 2b_{n-1})H^{n} \quad \mbox{and} \quad s_{n-1} = (b_{n-1} - 2b_{n-2})H^{n-1}.$$ We would like to determine when $s_{n} - (2n-1)(a+2)s_{n-1}$ is positive. This quantity equals \[b_{n} - ((2n-1)(a+2) +2 )b_{n-1} + 2(2n-1)(a+2)b_{n-2} . \] Expanding out this expression using the convention that $\phi_{k,d}=0$ for $k < 0$, we obtain \begin{equation}\label{eq1} \sum_{k=0}^n \binom{N+n-k}{N} \left( \phi_{k,d} - ((2n-1)(a+2)+2) \phi_{k-1,d} + 2(2n-1)(a+2) \phi_{k-2, d} \right) \end{equation} This quantity is positive if $$\frac{\phi_{k,d}}{\phi_{k-1,d}} \geq ((2n-1)(a+2)+2)$$ for all $1 \leq k \leq n$. Lemma \ref{lem-symmetricPolys} shows that $$\frac{\phi_{k,d}}{\phi_{k-1,d}} \geq \frac{c-k+1}{k} \min_i\{d_i -2\}. $$ Hence, the quantity (\ref{eq1}) is positive if $$\frac{c-k+1}{k} \min_i\{d_i -2\} \geq ((2n-1)(a+2)+2)$$ for all $1 \leq k \leq n$. Recalling that $c= N-n$, this inequality is satisfied for $1 \leq k \leq n$ when $$d_i \geq \frac{n}{N-2n+1} ((2n-1)(a+2) +2 ) + 2. $$ This concludes the proof of the theorem modulo the proof of Lemma \ref{lem-symmetricPolys}. \end{proof} \begin{lemma} \label{lem-symmetricPolys} Let $k < r$ and let $x_i$ be positive real numbers. Then the following inequality holds \[ \frac{\epsilon_k(x_1,\dots,x_r)}{\epsilon_{k-1}(x_1, \dots, x_r)} \geq \frac{r-k+1}{k} \min_i\{ x_i \} . \] \end{lemma} \begin{proof} First, we show that the quotient $\epsilon_k / \epsilon_{k-1}$ is an increasing function in $x_i$. This allows us to replace all of the $x_i$ with $\min \{x_i\}$. Recall that $$\frac{\partial}{\partial x_i} \epsilon_k(x_1, \dots, x_r) = \epsilon_{k-1}(x_1, \dots, \hat{x}_i, \dots, x_r).$$ For simplicity, denote $\epsilon_k(x_1, \dots, x_r)$ by $\epsilon_k$ and $\epsilon_k(x_1, \dots, \hat{x}_i, \dots, x_r)$ by $\hat{\epsilon}_{k, i}$. Hence, \[ \frac{\partial}{\partial x_i} \frac{\epsilon_k(x_1,\dots,x_r)}{\epsilon_{k-1}(x_1, \dots, x_r)} = \frac{\epsilon_{k-1}\hat{\epsilon}_{k-1,i} - \epsilon_k \hat{\epsilon}_{k-2, i}}{\epsilon_{k-1}^2} .\] We would like to show this quantity is positive. It suffices to show the numerator is positive. We compute the coefficient of $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_{k-1}\hat{\epsilon}_{k-1,i}$ and $\epsilon_k \hat{\epsilon}_{k-2, i}$. First, both coefficients are zero unless $$0 \leq a_j \leq 2 \ \mbox{ for all} \ j \not= i,\ 0 \leq a_i \leq 1, \ \mbox{ and} \ \sum_{j=1}^{r} a_j = 2k-2.$$ Let $S$ be the set of $j$ such that $a_j =2$ and let $\|S\|=m$. Let $I \subset \{1, \dots, r\}$ be the set of $j$ such that $a_j = 1$. If $i \in I$, then the coefficient of $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_{k-1}\hat{\epsilon}_{k-1,i}$ is given by $\binom{2k-3 - 2m}{k-2-m}$. This is the number of ways of writing $\prod_{j=1}^r x_j^{a_j}$ as a product of two monomials $m_1 m_2$ of length $k_1$ such that the terms in $m_1$ and $m_2$ are all distinct and $x_i \mid m_1$. Since the terms in $m_1$ and $m_2$ are distinct, $x_j \| m_1$ and $x_j \| m_2$ for $j \in S$. Hence, the coefficient is given by the number of ways of choosing $k-2-m$ elements in $I \backslash \{i\}$. Similarly, if $i \in I$, the coefficient of $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_k \hat{\epsilon}_{k-2, i}$ is given by $\binom{2k-3 - 2m}{k-1-m}$. This corresponds to choosing $k-1-m$ elements out of $I \backslash \{i\}$. Hence, when $i \in I$, the coefficients $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_{k-1}\hat{\epsilon}_{k-1,i}$ and $\epsilon_k \hat{\epsilon}_{k-2, i}$ are equal. By similar reasoning, if $i \not\in I$, then the coefficient of $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_{k-1}\hat{\epsilon}_{k-1,i}$ is given by $\binom{2k-2 - 2m}{k-1-m}$, with the convention that $\binom{0}{0}=1$. The coefficient of $\prod_{j=1}^r x_j^{a_j}$ in $\epsilon_k \hat{\epsilon}_{k-2, i}$ is given by $\binom{2k-2 - 2m}{k-m}$. Since $\binom{2k-2 - 2m}{k-1-m}> \binom{2k-2 - 2m}{k-m},$ we conclude that the numerator is positive. Hence, the quotient $\frac{\epsilon_k(x_1,\dots,x_r)}{\epsilon_{k-1}(x_1, \dots, x_r)}$ increases as $x_i$ increases. Let $x = \min\{x_i\}$. Hence, we get a lower bound for the quotient by setting each of the $x_i=x$. We obtain $\epsilon_k(x,\dots, x) = \binom{r}{k} x^k$. This gives \[ \frac{\epsilon_k(x_1,\dots,x_r)}{\epsilon_{k-1}(x_1, \dots, x_r)} \geq \frac{\binom{r}{k}x^k}{\binom{r}{k-1}x^{k-1}} = \frac{r-k+1}{k} x \] This concludes the proof of the lemma. \end{proof} Combining Theorem \ref{thm-OurCot} and Theorem \ref{thm-big}, we obtain the following corollary. \begin{corollary}\label{cor-mainbig} Let $X \subset \mathbb{P}^N$ be a general complete intersection of dimension $n$ and type $d_1,\dots, d_c$ with $c \geq n$. If $$d_i \geq \frac{(2n^2-n)(a+5)+2n}{N-2n+1}+2$$ for all $i$, then the projection of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \mathcal{O}(-a)$ has codimension at least 2 in $X$. In particular, if $$d_i \geq \frac{12n^2-4n}{N-2n+1}+2$$ for all $i$, then the projection of the stable base locus of $\mathcal{O}_{\mathbb{P} \Omega_X}(1) \otimes \pi^* \mathcal{O}_X(-1)$ has codimension at least 2 in $X$, which implies $\Omega_X$ is ample outside a variety of codimension at least 2 in $X$. \end{corollary} \section{Ampleness everywhere}\label{sec-main} In this section, using a technique of Riedl and Yang introduced in \cite{RiedlYang} and further developed in \cite{RiedlYang2}, we remove the base locus at the expense of slightly worse bounds. For simplicity, let $d= (d_1, \dots, d_c)$. Let $\mathcal{U}_{N,d}$ denote an open subvariety of the universal complete intersection parameterizing pairs $(p,X)$, where $X$ is a complete intersection in $\mathbb{P}^N$ of dimension $n$ and type $d_1, \dots, d_c$ and $p$ is a point of $X$. The main tool is the following theorem of Riedl and Yang. \begin{theorem}\cite[Theorem 2.3]{RiedlYang2} \label{thm-Grassmann} Let $M$ and $t$ be positive integers. Suppose that for every $N$, we have a countable union of locally closed subvarieties $Z_{N,d} \subset \mathcal{U}_{N,d}$ satisfying the following two conditions: \begin{enumerate} \item The codimension of $Z_{M,d}$ in $\mathcal{U}_{M,d}$ is at least $t$. \item If $(p, X_0) \in Z_{N-1,d}$ is a linear section of $(p,X) \in \mathcal{U}_{N,d}$, then $(p,X) \in Z_{N,d}$. \end{enumerate} Then for any $u \geq 0$, $Z_{M-u,d} \subset \mathcal{U}_{M-u,d}$ has codimension at least $u+t$. \end{theorem} Applying Theorem \ref{thm-Grassmann} to Corollary \ref{cor-mainbig}, we can obtain the main result of this note. \begin{theorem} Let $X \subset \mathbb{P}^N$ be a general complete intersection of dimension $n$ and type $d_1, \dots, d_c$, and suppose $n > 1$. \begin{enumerate} \item If $c \geq 2n-1$, $a \geq -1$ and \[ d_i \geq \frac{(8n^2-10n+3)a+ 40n^2 -46n+13}{N-3n+2} + 2 \] for all $i$, then the stable base locus of $\mathcal{O}_{\mathbb{P}\Omega_X}(1) \otimes \pi^\mathcal{O}_X(-a)$ is empty and some multiple is globally generated. \item If $c \geq 2n-2$ and $$d_i \geq \frac{(2n-2)(24n-28)}{N-3n+3}+2$$ for all $i$, then $\Omega_X$ is ample. \end{enumerate} \end{theorem} If $X$ is a curve, $\Omega_X$ is a line bundle of degree $-N-1+\sum_i d_i$, which is globally generated if $\sum_i d_i \geq N+1$ and ample if $\sum_i d_i > N+1$. \begin{proof} By Corollary \ref{cor-mainbig}, if $M \geq 2m$ and $$d_i \geq \frac{(2m^2 -m)(a+5) + 2m}{M-2m+1}+ 2$$ then for some $k \gg 0$ we have $\operatorname{Sym}^k(\Omega_X)(-ka)$ is globally generated outside a subvariety of codimension at least $2$ for a general complete intersection $X \subset \mathbb{P}^M$ of dimensinon $m$. Let $\mathcal{U}_{N,d}$ be the subvariety of the universal complete intersection of type $d_1, \dots, d_c$ in $\mathbb{P}^N$ consisting of pairs $(p,X)$ such that all sections of $\operatorname{Sym}^k(\Omega_X)(-ka)$ extend to the general complete intersection. Let $Z_{N,d}$ be the locus of points $(p,X)$ where $\operatorname{Sym}^k(\Omega_X)(-ka)$ is not globally generated. When $N=M$, $Z_{M,d}$ has codimension at least 2 in $\mathcal{U}_{M,d}$, so satisfies (1) in Theorem \ref{thm-Grassmann} with $t=2$. Combining the restriction sequence $$0 \rightarrow \Omega_X(-1) \rightarrow \Omega_X \rightarrow \Omega_X\|_{X \cap H} \rightarrow 0$$ and the conormal sequence $$0 \rightarrow \mathcal{O}_{X\cap H} (-1) \rightarrow \Omega_X\|_{X\cap H} \rightarrow \Omega_{X \cap H} \rightarrow 0,$$ we see that there is a surjective map $\Omega_X \rightarrow \Omega_{X \cap H} \rightarrow 0$. Consequently, we obtain a surjective map $\operatorname{Sym}^k(\Omega_X)(-ka) \rightarrow \operatorname{Sym}^k(\Omega_{X \cap H})(-ka)$. Hence, if the latter is not globally generated at $p$, the former is certainly not globally generated at $p$ either. Hence, $Z_{N,d}$ satisfies (2) in Theorem \ref{thm-Grassmann}. We conclude that $Z_{M - u, d}$ has codimension at least $u+2$ in $\mathcal{U}_{M-u, d}$. If $u+2 > m-u$, then the projection of $Z_{M - u, d}$ to the space of complete intersections cannot be dominant. Letting $N= M-u$, $n= m-u$, $u=n-1$ and substituting into the degree bounds for $d_i$, we obtain the first statement. Similarly, by Corollary \ref{cor-mainbig}, if $$d_i \geq \frac{12m^2 -4m}{M-2m+1} + 2,$$ then $\Omega_X$ is ample outside a subvariety of codimension at least $2$. Let $Z_{N,d}$ be the locus of points $(p,X)$ where $\Omega_X$ fails to be ample. Then for $N=M$ this locus has codimension at least 2, so satisfies (1) in Theorem \ref{thm-Grassmann} with $t=2$. The surjection $\Omega_X \rightarrow \Omega_{X \cap H} \rightarrow 0$ induces a map $\mathbb{P} \Omega_{X \cap H} \rightarrow \mathbb{P} \Omega_X$ such that the restriction of $\mathcal{O}_{\mathbb{P} \Omega_X}(1)$ to the image coincides with $\mathcal{O}_{\mathbb{P} \Omega_{X\cap H}}(1)$. Consequently, given a curve $C \in X\cap H$ passing through $p$ satisfying $\mathcal{O}_{\mathbb{P} \Omega_{X\cap H}}(1) \cdot C < \epsilon \ \pi^ H \cdot C$, the same curve satisfies $\mathcal{O}_{\mathbb{P} \Omega_{X}}(1) \cdot C < \epsilon \ \pi^* H \cdot C$. Hence, $Z_{N,d}$ satisfies (2) in Theorem \ref{thm-Grassmann}. We conclude that $Z_{M - u, d}$ has codimension at least $u+2$ in $\mathcal{U}_{M-u, d}$. If $u+2 \geq m-u$, then the projection of $Z_{M - u, d}$ to the space of complete intersections cannot be dominant. If it were dominant, then the fibers would be finite. However, if the fibers are nonempty, then they have to be at least 1 dimensional since they contain curves. Letting $N= M-u$, $n=m-u$ and $u= n-2$ and substituting into the degree bounds for $d_i$, we obtain the second statement. Note that taking $u=n-2$ in this last step requires $n > 1$. \end{proof} \begin{corollary}\label{cor-fantastic} Assume that $N \geq 48n^2 -101n +53$. Then the general complete intersection of dimension $n$ in $\mathbb{P}^N$ of type $d_1, \dots, d_{N-n}$ has ample cotangent bundle if $d_i \geq 3$. \end{corollary} \begin{remark} Inspired by the case of curves, one could speculate that a complete intersection of dimension $n$ and type $d_1, \dots, d_c$ in $\mathbb{P}^N$ will have ample cotangent bundle if $d_i \geq 2$ provided that $c \gg n$. \end{remark} \end{document}	arXiv
# Understanding vectors and their properties A vector can be represented in Cartesian coordinates as an ordered list of its components. For example, a 2-dimensional vector can be represented as (x, y), where x and y are the coordinates of the vector in the Cartesian plane. Vectors can be added and subtracted component-wise. For example, if we have two vectors A = (1, 2) and B = (3, 4), their sum is C = A + B = (1 + 3, 2 + 4) = (4, 6). Similarly, their difference is D = A - B = (1 - 3, 2 - 4) = (-2, -2). The dot product of two vectors A and B is defined as the sum of the products of their corresponding components. Mathematically, the dot product is denoted as A · B and is calculated as: $$A \cdot B = \sum_{i=1}^{n} A_i B_i$$ where A_i and B_i are the i-th components of vectors A and B, respectively. The magnitude (or length) of a vector A is defined as the square root of the dot product of A with itself. Mathematically, the magnitude of A is denoted as \|\|A\|\| and is calculated as: $$\|\|A\|\| = \sqrt{A \cdot A}$$ The angle between two vectors A and B is defined as the angle between their corresponding unit vectors. The angle can be calculated using the dot product formula: $$\cos(\theta) = \frac{A \cdot B}{\|\|A\|\| \|\|B\|\|}$$ where \|\|A\|\| and \|\|B\|\| are the magnitudes of vectors A and B, respectively. ## Exercise Calculate the dot product and magnitude of the vector A = (1, 2, 3). # Linear algebra concepts and their impact on machine learning Linear algebra is a branch of mathematics that deals with linear equations and linear transformations. It provides a foundation for many machine learning algorithms, including linear regression, principal component analysis, and support vector machines. A linear transformation is a function that maps vectors from one vector space to another while preserving the operations of addition and scalar multiplication. Linear transformations can be represented as matrices. The inverse of a linear transformation is a transformation that reverses the effect of the original transformation. If a linear transformation can be represented as a matrix A, its inverse is represented as the matrix A^(-1). Eigenvectors and eigenvalues are important concepts in linear algebra. An eigenvector of a linear transformation is a non-zero vector that is left unchanged by the transformation when scaled by a scalar multiple. The corresponding eigenvalue is the scalar multiple. The singular value decomposition (SVD) is a factorization of a matrix A into three matrices U, Σ, and V^T, where U and V^T are orthogonal matrices and Σ is a diagonal matrix. SVD is used in machine learning algorithms like principal component analysis and is closely related to the concept of singular vectors. ## Exercise Calculate the SVD of the matrix A = [[1, 2], [3, 4]]. # Distance metrics and their importance in machine learning Distance metrics are mathematical functions that measure the similarity or dissimilarity between two data points. They are used to compare data points and to define clusters or class boundaries in machine learning algorithms. The Euclidean distance between two points in n-dimensional space is defined as the square root of the sum of the squared differences of their coordinates. Mathematically, the Euclidean distance between points A and B is denoted as d(A, B) and is calculated as: $$d(A, B) = \sqrt{\sum_{i=1}^{n} (A_i - B_i)^2}$$ where A_i and B_i are the i-th coordinates of points A and B, respectively. The Manhattan distance between two points in n-dimensional space is defined as the sum of the absolute differences of their coordinates. Mathematically, the Manhattan distance between points A and B is denoted as d(A, B) and is calculated as: $$d(A, B) = \sum_{i=1}^{n} \|A_i - B_i\|$$ The cosine similarity between two points in n-dimensional space is defined as the cosine of the angle between their corresponding unit vectors. Mathematically, the cosine similarity between points A and B is denoted as s(A, B) and is calculated as: $$s(A, B) = \frac{A \cdot B}{\|\|A\|\| \|\|B\|\|}$$ where \|\|A\|\| and \|\|B\|\| are the magnitudes of points A and B, respectively. ## Exercise Calculate the Euclidean, Manhattan, and cosine similarities between points A = (1, 2, 3) and B = (4, 6, 9). # Kernel functions and their role in machine learning Kernel functions are mathematical functions that transform input data into a higher-dimensional space, making it easier to separate data points in machine learning algorithms. The most common kernel function used in machine learning is the Gaussian radial basis function (RBF) kernel. It is defined as the product of the Gaussian functions evaluated at the pairwise distances between data points. Mathematically, the RBF kernel is denoted as K(A, B) and is calculated as: $$K(A, B) = \exp(-\frac{(\|\|A - B\|\|)^2}{2 \sigma^2})$$ where \|\|A - B\|\| is the Euclidean distance between points A and B, and σ is a positive constant called the bandwidth. The polynomial kernel is another commonly used kernel function. It is defined as the sum of the product of the coordinates of data points raised to a positive integer power. Mathematically, the polynomial kernel is denoted as K(A, B) and is calculated as: $$K(A, B) = (A \cdot B + c)^d$$ where A · B is the dot product of points A and B, c is a constant, and d is a positive integer. ## Exercise Calculate the RBF kernel between points A = (1, 2, 3) and B = (4, 6, 9) with bandwidth σ = 1. # Classification algorithms and their use of Euclidean space Classification algorithms are machine learning algorithms that learn to classify data points into different classes based on their features. They use Euclidean space concepts like distance metrics and kernel functions to define similarity and dissimilarity between data points. K-nearest neighbors (KNN) is a popular classification algorithm that classifies a data point based on the majority class of its k nearest neighbors. The distance between data points is used to determine the k nearest neighbors. Support vector machines (SVM) are another classification algorithm that uses Euclidean space concepts to find the optimal hyperplane that separates data points into different classes. The kernel functions are used to transform the data into a higher-dimensional space, making it easier to find the optimal hyperplane. ## Exercise Train a KNN classifier and an SVM classifier on a given dataset and compare their performance. # Higher dimensions and their impact on machine learning Higher dimensions refer to data points with more than two or three coordinates. In some cases, data points in higher-dimensional spaces can be more easily separated by a hyperplane or a kernel function. Principal component analysis (PCA) is a dimensionality reduction technique that transforms the coordinates of data points into a lower-dimensional space while preserving the maximum amount of variance. PCA is often used to reduce the dimensionality of data before applying machine learning algorithms. t-distributed stochastic neighbor embedding (t-SNE) is a dimensionality reduction technique that aims to preserve the pairwise distances between data points in the lower-dimensional space. It is often used to visualize high-dimensional data points in a 2D or 3D space. ## Exercise Apply PCA and t-SNE to a given dataset and compare the results. # Handling missing data and outliers in Euclidean space Handling missing data and outliers is an important step in machine learning, as they can have a significant impact on the performance of algorithms. One common approach to handling missing data is to impute the missing values based on the available data. This can be done using mean imputation, median imputation, or more advanced techniques like k-nearest neighbors imputation. Outliers can be detected using statistical methods like the IQR method or machine learning algorithms like isolation forests. Once detected, outliers can be handled using techniques like removing, capping, or transforming the data. ## Exercise Impute missing values in a given dataset using mean imputation and detect outliers using the IQR method. # Applying Euclidean space concepts to real-world examples Euclidean space concepts can be applied to real-world examples to solve various problems. Here are a few examples: - In recommendation systems, Euclidean space concepts can be used to measure the similarity between users or items based on their features. - In image recognition, Euclidean space concepts can be used to measure the similarity between images based on their pixel values. - In natural language processing, Euclidean space concepts can be used to measure the similarity between words or documents based on their embeddings. ## Exercise Apply Euclidean space concepts to a real-world example, such as movie recommendation or image similarity. # Evaluating and improving machine learning models using Euclidean space Evaluating and improving machine learning models often involves using Euclidean space concepts to assess the performance of models, select feature subsets, and optimize hyperparameters. Model evaluation can be done using metrics like accuracy, precision, recall, and F1 score, which are based on the Euclidean distance between data points. Feature selection can be done using techniques like filter methods, wrapper methods, or embedded methods, which use Euclidean space concepts to identify the most important features for a given task. Hyperparameter optimization can be done using techniques like grid search, random search, or Bayesian optimization, which use Euclidean space concepts to search for the best combination of hyperparameters. ## Exercise Evaluate and improve a machine learning model using Euclidean space concepts. # Future directions and advancements in Euclidean space and machine learning The integration of Euclidean space concepts with machine learning is an active area of research. Some potential future directions and advancements include: - Developing more efficient and scalable algorithms for high-dimensional data. - Investigating the relationship between Euclidean space concepts and other geometric concepts like Riemannian geometry or topology. - Exploring the use of Euclidean space concepts in deep learning and reinforcement learning. - Developing new kernel functions and distance metrics that better capture the structure of high-dimensional data. ## Exercise Discuss potential future directions and advancements in Euclidean space and machine learning. ## Exercise Calculate the Euclidean, Manhattan, and cosine similarities between points A = (1, 2, 3) and B = (4, 6, 9). The Euclidean distance between points A and B is 5.196, the Manhattan distance is 9, and the cosine similarity is 0.999. ## Exercise Calculate the RBF kernel between points A = (1, 2, 3) and B = (4, 6, 9) with bandwidth σ = 1. The RBF kernel between points A and B is 0.003. ## Exercise Impute missing values in a given dataset using mean imputation and detect outliers using the IQR method. For the given dataset, the missing values are imputed using mean imputation, and the outliers are detected using the IQR method. ## Exercise Evaluate and improve a machine learning model using Euclidean space concepts. For the given machine learning model, the accuracy, precision, recall, and F1 score are calculated using Euclidean space concepts. The model is improved by selecting the most important features and optimizing the hyperparameters. ## Exercise Discuss potential future directions and advancements in Euclidean space and machine learning. Some potential future directions and advancements include developing more efficient and scalable algorithms for high-dimensional data, investigating the relationship between Euclidean space concepts and other geometric concepts, exploring the use of Euclidean space concepts in deep learning and reinforcement learning, and developing new kernel functions and distance metrics that better capture the structure of high-dimensional data.	Textbooks
Showing posts tagged "graph theory". Show all posts. So you've calculated how much you should expect the World Cup sticker book to cost and recorded how much it actually cost. You might be wondering what else you can do with your sticker book. If so, look no further: this post contains 5 mathematical things involvolving your sticker book and stickers. Test the birthday paradox Stickers 354 and 369: Alisson and Roberto Firmino In a group of 23 people, there is a more than 50% chance that two of them will share a birthday. This is often called the birthday paradox, as the number 23 is surprisingly small. Back in 2014 when Alex Bellos suggested testing the birthday paradox on World Cup squads, as there are 23 players in a World Cup squad. I recently discovered that even further back in 2012, James Grime made a video about the birthday paradox in football games, using the players on both teams plus the referee to make 23 people. In this year's sticker book, each player's date of birth is given above their name, so you can use your sticker book to test it out yourself. Sticker 022: Kaliningrad One of the cities in which games are taking place in this World Cup is Kaliningrad. Before 1945, Kaliningrad was called Königsberg. In Königsburg, there were seven bridges connecting four islands. The arrangement of these bridges is shown below. The people of Königsburg would try to walk around the city in a route that crossed each bridge exactly one. If you've not tried this puzzle before, try to find such a route now before reading on... In 1736, mathematician Leonhard Euler proved that it is in fact impossible to find such a route. He realised that for such a route to exist, you need to be able to pair up the bridges on each island so that you can enter the island on one of each pair and leave on the other. The islands in Königsburg all have an odd number of bridges, so there cannot be a route crossing each bridge only once. In Kaliningrad, however, there are eight bridges: two of the original bridges were destroyed during World War II, and three more have been built. Because of this, it's now possible to walk around the city crossing each bridge exactly once. A route around Kaliningrad crossing each bridge exactly once. I wrote more about this puzzle, and using similar ideas to find the shortest possible route to complete a level of Pac-Man, in this blog post. If you didn't convince many of your friends to join you in collecting stickers, you'll have lots of swaps. You can use these to practice performing your favourite sorting algorithms. Bubble sort In the bubble sort, you work from left to right comparing pairs of stickers. If the stickers are in the wrong order, you swap them. After a few passes along the line of stickers, they will be in order. Insertion sort In the insertion sort, you take the next sticker in the line and insert it into its correct position in the list. In the quick sort, you pick the middle sticker of the group and put the other stickers on the correct side of it. You then repeat the process with the smaller groups of stickers you have just formed. Sticker 007: The official ball Sticker 007 shows the official tournament ball. If you look closely (click to enlarge), you can see that the ball is made of a mixture of pentagons and hexagons. The ball is not made of only hexagons, as road signs in the UK show. Stand up mathematician Matt Parker started a petition to get the symbol on the signs changed, but the idea was rejected. If you have a swap of sticker 007, why not stick it to a letter to your MP about the incorrect signs as an example of what an actual football looks like. Psychic pets Speaking of Matt Parker, during this World Cup, he's looking for psychic pets that are able to predict World Cup results. Why not use your swaps to label two pieces of food that your pet can choose between to predict the results of the remaining matches? Timber using my swaps to wrongly predict the first match Tags: football, world cup, sport, stickers, news, probability, graph theory, matt parker, sorting, geometry @Matthew: Thank you for the calculations. Good job I ordered the stickers I wanted #IRN. 2453 stickers - that's more than the number you bought (1781) to collect all stickers! @Milad: Here is how I calculated it: You want a specific set of 20 stickers. Imagine you have already $n$ of these. The probability that the next sticker you buy is one that you want is $$\frac{20-n}{682}.$$ The probability that the second sticker you buy is the next new sticker is $$\mathbb{P}(\text{next sticker is not wanted})\times\mathbb{P}(\text{sticker after next is wanted})$$ $$=\frac{662+n}{682}\times\frac{20-n}{682}.$$ Following the same method, we can see that the probability that the $i$th sticker you buy is the next wanted sticker is $$\left(\frac{662+n}{682}\right)^{i-1}\times\frac{20-n}{682}.$$ Using this, we can calculate the expected number of stickers you will need to buy until you find the next wanted one: $$\sum_{i=1}^{\infty}i \left(\frac{20-n}{682}\right) \left(\frac{662+n}{682}\right)^{i-1} = \frac{682}{20-n}$$ Therefore, to get all 682 stickers, you should expect to buy $$\sum_{n=0}^{19}\frac{682}{20-n} = 2453 \text{ stickers}.$$ @Matthew: Yes, I would like to know how you work it out please. I believe I have left my email address in my comment. It seems like a lot of stickers if you are just interested in one team. @Milad: Following a similiar method to this blog post, I reckon you'd expect to buy 2453 stickers (491 packs) to get a fixed set of 20 stickers. Drop me an email if you want me to explain how I worked this out. Thanks for the maths, but I have one probability question. How many packs would I have to buy, on average, to obtain a fixed set of 20 stickers? To prove you are not a spam bot, please type "rotcaf" backwards in the box below (case sensitive): This is an article which I wrote for the first issue of Chalkdust. I highly recommend reading the rest of the magazine (and trying to solve the crossnumber I wrote for the issue). In the classic arcade game Pac-Man, the player moves the title character through a maze. The aim of the game is to eat all of the pac-dots that are spread throughout the maze while avoiding the ghosts that prowl it. While playing Pac-Man recently, my concentration drifted from the pac-dots and I began to think about the best route I could take to complete the level. Seven bridges of Königsberg In the 1700s, Swiss mathematician Leonhard Euler studied a related problem. The city of Königsberg had seven bridges, which the residents would try to cross while walking around the town. However, they were unable to find a route crossing every bridge without repeating one of them. Diagram showing the bridges in Königsberg. If you have not seen this puzzle before, you may like to try to find a route crossing them all exactly once before reading on. In fact, the city dwellers could not find such a route because it is impossible to do so, as Euler proved in 1735. He first simplified the map of the city, by making the islands into vertices (or nodes) and the bridges into edges. A graph of the seven bridges problem. This type of diagram has (slightly confusingly) become known as a graph, the study of which is called graph theory. Euler represented Königsberg in this way as he realised that the shape of the islands is irrelevant to the problem: representing the problem as a graph gets rid of this useless information while keeping the important details of how the islands are connected. Euler next noticed that if a route crossing all the bridges exactly once was possible then whenever the walker took a bridge onto an island, they must take another bridge off the island. In this way, the ends of the bridges at each island can be paired off. The only bridge ends that do not need a pair are those at the start and end of the circuit. This means that all of the vertices of the graph except two (the first and last in the route) must have an even number of edges connected to them; otherwise there is no route around the graph travelling along each edge exactly once. In Königsberg, each island is connected to an odd number of bridges. Therefore the route that the residents were looking for did not exist (a route now exists due to two of the bridges being destroyed during World War II). This same idea can be applied to Pac-Man. By ignoring the parts of the maze without pac-dots the pac-graph can be created, with the paths and the junctions forming the edges and vertices respectively. Once this is done there will be twenty-four vertices, twenty of which will be connected to an odd number of edges, and so it is impossible to eat all of the pac-dots without repeating some edges or travelling along parts of the maze with no pac-dots. The Pac-graph. The odd nodes are shown in red. This is a start, but it does not give us the shortest route we can take to eat all of the pac-dots: in order to do this, we are going to have to look at the odd vertices in more detail. The Chinese postman problem The task of finding the shortest route covering all the edges of a graph has become known as the Chinese postman problem as it is faced by postmen—they need to walk along each street to post letters and want to minimise the time spent walking along roads twice—and it was first studied by Chinese mathematician Kwan Mei-Ko. As the seven bridges of Königsberg problem demonstrated, when trying to find a route, Pac-Man will get stuck at the odd vertices. To prevent this from happening, all the vertices can be made into even vertices by adding edges to the graph. Adding an edge to the graph corresponds to choosing an edge, or sequence of edges, for Pac-Man to repeat or including a part of the maze without pac-dots. In order to complete the level with the shortest distance travelled, Pac-Man wants to add the shortest total length of edges to the graph. Therefore, in order to find the best route, Pac-Man must look at different ways to pair off the odd vertices and choose the pairing which will add the least total distance to the graph. The Chinese postman problem and the Pac-Man problem are slightly different: it is usually assumed that the postman wants to finish where he started so he can return home. Pac-Man however can finish the level wherever he likes but his starting point is fixed. Pac-Man may therefore leave one odd node unpaired but must add an edge to make the starting node odd. One way to find the required route is to look at all possible ways to pair up the odd vertices. With a low number of odd vertices this method works fine, but as the number of odd vertices increases, the method quickly becomes slower. With four odd vertices, there are three possible pairings. For the Pac-Man problem there will be over 13 billion ($1.37\times 10^{10}$) pairings to check. These pairings can be checked by a laptop running overnight, but for not too many more vertices this method quickly becomes unfeasible. With 46 odd nodes there will be more than one pairing per atom in the human body ($2.53\times 10^{28}$). By 110 odd vertices there will be more pairings ($3.47\times 10^{88}$) than there are estimated to be atoms in the universe. Even the greatest supercomputer will be unable to work its way through all these combinations. Better algorithms are known for this problem that reduce the amount of work on larger graphs. The number of pairings to check in the method above increases like the factorial of the number of vertices. Algorithms are known for which the amount of work to be done increases like a polynomial in the number of vertices. These algorithms will become unfeasible at a much slower rate but will still be unable to deal with very large graphs. Solution of the Pac-Man problem For the Pac-Man problem, the shortest pairing of the odd vertices requires the edges marked in red to be repeated. Any route which repeats these edges will be optimal. For example, the route in green will be optimal. One important element of the Pac-Man gameplay that I have neglected are the ghosts (Blinky, Pinky, Inky and Clyde), which Pac-Man must avoid. There is a high chance that the ghosts will at some point block the route shown above and ruin Pac-Man's optimality. However, any route repeating the red edges will be optimal: at many junctions Pac-Man will have a choice of edges he could continue along. It may be possible for a quick thinking player to utilise this freedom to avoid the ghosts and complete an optimal game. Additionally, the skilled player may choose when to take the edges that include the power pellets, which allow Pac-Man to reverse the roles and eat the ghosts. Again cleverly timing these may allow the player to complete an optimal route. Unfortunately, as soon as the optimal route is completed, Pac-Man moves to the next level and the player has to do it all over again ad infinitum. Since writing this piece, I have been playing Pac-Man using MAME (Multiple Arcade Machine Emulator). Here is one game I played along with the optimal edges to repeat for reference: This video is not displaying because you are a bot. Tags: pac-man, graph theory, video games, games, chalkdust magazine @William: You're right. In a number of places I could've turned round a few pixels earlier. There seems to be no world record for just one Pac-Man level (and I don't have time to get good enough to speed run all 255 levels before it crashes!) This vid was billed as an "optimal" run but around 40 seconds in you eat one "pill" that you don't need to eat. Why don't you just speedrun the first level? This must have been done before. Can you beat the world record? To prove you are not a spam bot, please type "t" then "h" then "e" then "o" then "r" then "e" then "m" in the box below (case sensitive): chess php mathslogicbot bodmas martin gardner raspberry pi bubble bobble big internet math-off national lottery platonic solids latex propositional calculus folding paper christmas curvature folding tube maps rugby noughts and crosses flexagons cross stitch rhombicuboctahedron golden ratio polynomials game of life hats interpolation tennis london underground chebyshev hexapawn pac-man stickers captain scarlet logic chalkdust magazine pythagoras pizza cutting oeis trigonometry manchester royal baby statistics news graph theory puzzles christmas card programming plastic ratio the aperiodical reuleaux polygons sound harriss spiral python weather station mathsjam a gamut of games dataset radio 4 countdown manchester science festival video games asteroids approximation sorting nine men's morris braiding machine learning matt parker menace accuracy data electromagnetic field game show probability javascript dragon curves people maths london triangles final fantasy probability frobel dates games misleading statistics mathsteroids reddit gerry anderson arithmetic draughts fractals palindromes coins light go error bars craft golden spiral football wool world cup sport ternary binary speed realhats books european cup map projections geometry inline code estimation twitter wool misleading statistics bubble bobble puzzles world cup folding paper geometry final fantasy sound javascript bodmas approximation countdown asteroids chebyshev golden spiral coins statistics noughts and crosses weather station books flexagons sport speed pizza cutting oeis reddit arithmetic gerry anderson probability christmas realhats game of life propositional calculus video games logic curvature craft pac-man programming chess reuleaux polygons european cup electromagnetic field matt parker london data cross stitch python polynomials royal baby braiding error bars hats accuracy captain scarlet dataset palindromes go light stickers fractals plastic ratio php mathsjam dragon curves martin gardner manchester science festival people maths national lottery estimation mathsteroids machine learning big internet math-off draughts triangles a gamut of games trigonometry christmas card nine men's morris golden ratio ternary graph theory chalkdust magazine twitter sorting binary folding tube maps rugby harriss spiral radio 4 football the aperiodical raspberry pi menace news mathslogicbot map projections platonic solids manchester interpolation inline code rhombicuboctahedron hexapawn dates game show probability latex tennis games london underground frobel pythagoras	CommonCrawl
GridPP5 Brunel University London Staff Grant Lead Research Organisation: Brunel University Department Name: Electronic and Electrical Engineering This proposal, submitted in response to the 2014 invitation from STFC, aims to provide and operate a computing Grid for the exploitation of LHC data in the UK. The success of the current GridPP Collaboration will be built upon, and the UK's response to production of LHC data in the period April 2016 to March 2020 will be to ensure that there is a sustainable infrastructure providing "Distributed Computing for Particle Physics" We propose to operate a distributed high throughput computational service as the main mechanism for delivering very large-scale computational resources to the UK particle physics community. This foundation will underpin the success and increase the discovery potential of UK physicists. We will operate a production-quality service, delivering robustness, scale and functionality. The proposal is fully integrated with international projects and we must exploit the opportunity to capitalise on the UK leadership already established in several areas. The Particle Physics distributed computing service will increasingly be integrated with national and international initiatives. The project will be managed across various domains and will deliver the UK's commitment to the Worldwide LHC Computing Grid (WLCG) and ensure that worldwide activities directly benefit the UK. By 2015, the UK Grid infrastructure will have expanded in size to 50,000 cores, with more than 35 PetaBytes of storage. This will enable the UK to exploit, in an internationally competitive way, the unique physics potential of the LHC. GridPP's knowledge exchange activities fall into two main areas: firstly, those aimed at other academic disciplines, and secondly, business and industry. GridPP has a strong outreach programme to a public and academic audience, and intends to continue this in GridPP5. The Dissemination Officer will organise GridPP's presence at conferences and events. This includes booking and manning booths, arranging backdrops, material, posters, screens, and rotas where appropriate. Examples of events that we have attended include The British Science Festival, The Royal Society Summer Exhibition, the British Science Association Science Communication Conference and Meet The Scientist at the Museum of Science and Industry in Manchester. GridPP has developed an extensive website that is central to project communications. The Dissemination Officer will be responsible for producing news items for the website and drafting GridPP press releases. We have had broad coverage from these in the past, including many national newspapers and online publications. Additional activities will include producing GridPP material, such as leaflets, posters, t-shirts, bags and magic cubes. We have found these very valuable in raising GridPP's and LHC's profile at minimal cost. The Dissemination Officer will also promote outreach training for members of the collaboration, will identify GridPP staff who have specific expertise in this area and will arrange occasional GridPP events, such as the Tier-1 open day. On KE, our initial work has proved that GridPP's technology can be of use across a range of disciplines and sectors, and we plan to continue this work during GridPP5. The objectives of this program will be to improve awareness of the technologies developed by GridPP and its partners in academia and industry, and hence facilitate the increase in use of these technologies within new areas. Apr 16 - Sep 20 ST/N001273/1 Peter Robert Hobson Paul Kyberd Particle physics - experiment (50%) Particle physics - theory (50%) Beyond the Standard Model (50%) The Standard Model (50%) Brunel University, United Kingdom (Lead Research Organisation) Rutherford Appleton Laboratory, Oxford (Collaboration) University of Bristol, United Kingdom (Collaboration) Imperial College London, United Kingdom (Collaboration) European Organization for Nuclear Research (CERN) (Collaboration) Peter Robert Hobson (Principal Investigator) http://orcid.org/0000-0002-5645-5253 Paul Kyberd (Principal Investigator) ascending (press to sort descending) \|< < 1 2 3 4 5 6 > >\| Sirunyan AM (2018) Measurement of the Z / ? * ? t t cross section in pp collisions at s = 13 TeV and validation of t lepton analysis techniques. in The European physical journal. C, Particles and fields Sirunyan A (2018) Search for beyond the standard model Higgs bosons decaying into a b b ¯ $$ \mathrm{b}\overline{\mathrm{b}} $$ pair in pp collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan AM (2018) Evidence for the Associated Production of a Single Top Quark and a Photon in Proton-Proton Collisions at sqrt[s]=13 TeV. in Physical review letters Sirunyan A (2018) Search for resonances in the mass spectrum of muon pairs produced in association with b quark jets in proton-proton collisions at s = 8 $$ \sqrt{s}=8 $$ and 13 TeV in Journal of High Energy Physics Sirunyan A (2018) Event shape variables measured using multijet final states in proton-proton collisions at s=13$$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan AM (2018) Observation of the ?_{b1}(3P) and ?_{b2}(3P) and Measurement of their Masses. in Physical review letters Sirunyan A (2018) Constraints on models of scalar and vector leptoquarks decaying to a quark and a neutrino at s = 13 TeV in Physical Review D Sirunyan A (2018) Search for the decay of a Higgs boson in the ll? channel in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for an exotic decay of the Higgs boson to a pair of light pseudoscalars in the final state of two muons and two t leptons in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for supersymmetry in events with a t lepton pair and missing transverse momentum in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for decays of stopped exotic long-lived particles produced in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for physics beyond the standard model in high-mass diphoton events from proton-proton collisions at s = 13 TeV in Physical Review D Sirunyan A (2018) Search for a singly produced third-generation scalar leptoquark decaying to a t lepton and a bottom quark in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan AM (2018) Measurement of b hadron lifetimes in pp collisions at s = 8 TeV. in The European physical journal. C, Particles and fields Sirunyan AM (2018) Measurement of Prompt D^{0} Meson Azimuthal Anisotropy in Pb-Pb Collisions at sqrt[s_{NN}]=5.02 TeV. in Physical review letters Sirunyan A (2018) Search for an exotic decay of the Higgs boson to a pair of light pseudoscalars in the final state with two b quarks and two t leptons in proton-proton collisions at s = 13 TeV in Physics Letters B Sirunyan A (2018) Search for new physics in final states with an energetic jet or a hadronically decaying W or Z boson and transverse momentum imbalance at s = 13 TeV in Physical Review D Sirunyan A (2018) Search for long-lived particles with displaced vertices in multijet events in proton-proton collisions at s = 13 TeV in Physical Review D Sirunyan AM (2018) Observation of Medium-Induced Modifications of Jet Fragmentation in Pb-Pb Collisions at sqrt[s_{NN}]=5.02 TeV Using Isolated Photon-Tagged Jets. in Physical review letters Sirunyan A (2018) Search for new long-lived particles at s = 13 TeV in Physics Letters B Sirunyan AM (2018) Search for third-generation scalar leptoquarks decaying to a top quark and a t lepton at s = 13 Te . in The European physical journal. C, Particles and fields Sirunyan AM (2018) Studies of B s 2 * ( 5840 ) 0 and B s 1 ( 5830 ) 0 mesons including the observation of the B s 2 * ( 5840 ) 0 ? B 0 K S 0 decay in proton-proton collisions at s = 8 TeV. in The European physical journal. C, Particles and fields Sirunyan A (2018) Measurement of the groomed jet mass in PbPb and pp collisions at s N N = 5.02 $$ \sqrt{s_{\mathrm{NN}}}=5.02 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for high-mass resonances in final states with a lepton and missing transverse momentum at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan AM (2018) Search for Leptoquarks Coupled to Third-Generation Quarks in Proton-Proton Collisions at sqrt[s]=13 TeV. in Physical review letters Sirunyan A (2018) Search for pair-produced resonances decaying to quark pairs in proton-proton collisions at s = 13 TeV in Physical Review D Sirunyan A (2018) Search for heavy resonances decaying into a vector boson and a Higgs boson in final states with charged leptons, neutrinos and b quarks at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for new physics in events with two soft oppositely charged leptons and missing transverse momentum in proton-proton collisions at s = 13 TeV in Physics Letters B Sirunyan A (2018) Search for vector-like T and B quark pairs in final states with leptons at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan AM (2018) Search for Heavy Neutral Leptons in Events with Three Charged Leptons in Proton-Proton Collisions at sqrt[s]=13 TeV. in Physical review letters Sirunyan A (2018) Measurements of Higgs boson properties in the diphoton decay channel in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for Z? resonances using leptonic and hadronic final states in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Performance of the CMS muon detector and muon reconstruction with proton-proton collisions at v s =13 TeV in Journal of Instrumentation Sirunyan A (2018) Measurement of quarkonium production cross sections in pp collisions at s = 13 TeV in Physics Letters B Sirunyan A (2018) Measurement of the cross section for top quark pair production in association with a W or Z boson in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Measurement of the production cross section for single top quarks in association with W bosons in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Pseudorapidity and transverse momentum dependence of flow harmonics in p Pb and PbPb collisions in Physical Review C Sirunyan A (2018) Search for a heavy resonance decaying into a Z boson and a vector boson in the ? ? ¯ q q ¯ $$ \nu \overline{\nu}\mathrm{q}\overline{\mathrm{q}} $$ final state in Journal of High Energy Physics Sirunyan A (2018) Search for a charged Higgs boson decaying to charm and bottom quarks in proton-proton collisions at s = 8 $$ \sqrt{s}=8 $$ TeV in Journal of High Energy Physics Sirunyan A (2018) Search for excited quarks of light and heavy flavor in ? + jet final states in proton-proton collisions at s = 13 TeV in Physics Letters B Sirunyan A (2018) Azimuthal correlations for inclusive 2-jet, 3-jet, and 4-jet events in pp collisions at $$\sqrt{s}= 13~\hbox {TeV}$$ s = 13 TeV in The European Physical Journal C Khachatryan V (2017) Measurements of differential production cross sections for a Z boson in association with jets in pp collisions at s = 8 $$ \sqrt{s}=8 $$ TeV in Journal of High Energy Physics Sirunyan A (2017) Search for electroweak production of a vector-like quark decaying to a top quark and a Higgs boson using boosted topologies in fully hadronic final states in Journal of High Energy Physics Sirunyan AM (2017) Relative Modification of Prompt ?(2S) and J/? Yields from pp to PbPb Collisions at sqrt[s_{NN}]=5.02 TeV. in Physical review letters Khachatryan V (2017) Measurement of the production cross section of a W boson in association with two b jets in pp collisions at [Formula: see text]. in The European physical journal. C, Particles and fields Khachatryan V (2017) Search for electroweak production of charginos in final states with two t leptons in pp collisions at s = 8 $$ \sqrt{s}=8 $$ TeV in Journal of High Energy Physics Khachatryan V (2017) Search for high-mass diphoton resonances in proton-proton collisions at 13 TeV and combination with 8 TeV search in Physics Letters B Khachatryan V (2017) Search for CP violation in t t ¯ $$ t\overline{t} $$ production and decay in proton-proton collisions at s = 8 $$ \sqrt{s}=8 $$ TeV in Journal of High Energy Physics Sirunyan A (2017) Mechanical stability of the CMS strip tracker measured with a laser alignment system in Journal of Instrumentation Khachatryan V (2017) Measurement of the transverse momentum spectra of weak vector bosons produced in proton-proton collisions at s = 8 $$ \sqrt{s}=8 $$ TeV in Journal of High Energy Physics Description GridPP6 Brunel Staff Grant Amount £112,482 (GBP) Funding ID ST/T001291/1 Description CMS Organisation European Organization for Nuclear Research (CERN) Department Compact Muon Solenoid (CMS) PI Contribution Construction, comissioning and operation of the CMS experiment. Data analysis in top-quark physics studies. Provision (via GridPP London Tier-2) of computing resources. Collaborator Contribution Data acquistion, computing resources (Tier 0), co-authorship of publications, access to data, scientific leadership and support Impact Over 200 refereed journal publications in experimental particle physics. Along with LHC data analysed by the ATLAS collaboration CMS determined the existence of the Higgs boson which was the subject of the 2013 Nobel Prize in Physics. Several STFC funded doctoral students have been trained in data analysis, computer programming and large-scale distributed Grid computing techniques. Organisation Imperial College London Department Department of Physics Organisation Rutherford Appleton Laboratory Department Particle Physics Department Organisation University of Bristol Department School of Physics	CommonCrawl
ETD Guide/Print version The latest reviewed version was checked on 6 June 2016. There are template/file changes awaiting review. This is the print version of ETD Guide You won't see this message or any elements not part of the book's content when you print or preview this page. ETD Guide The current, editable version of this book is available in Wikibooks, the open-content textbooks collection, at https://en.wikibooks.org/wiki/ETD_Guide Permission is granted to copy, distribute, and/or modify this document under the terms of the Creative Commons Attribution-ShareAlike 3.0 License. 2 Introduction/What are ETDs 3 Introduction/ETDs as new genre of documents 4 Introduction/Why ETDs? 5 Introduction/Minimize duplication of effort 6 Introduction/Improve visibility 7 Introduction/Accelerate workflow: graduate more quickly, make ETDs available faster to outside audience 8 Introduction/Costs and benefits 9 Introduction/Purpose, goals, objectives of ETD activities 10 Introduction/Helping students be better prepared as knowledge workers 11 Introduction/Helping students be original 12 Introduction/Helping students network professionally 13 Introduction/Improving graduate education, and quality/expressiveness of ETDs 13.1 Graduate Education 13.2 ETDs in Graduate Education 13.3 Aids in Communication 13.4 Readership and Quality 14 Introduction/Helping faculty 15 Introduction/Increasing readership of ETDs, communicating research results 16 Introduction/Helping universities develop digital library services & infrastructure 17 Introduction/Increasing sharing and collaboration among universities and students 18 Introduction/Enhancing access to university research 19 Introduction/Searching 20 Introduction/Browsing: Classification systems, classification schemes used in different disciplines 21 Introduction/Well known sites and resources for ETDs 22 Introduction/Brief history of ETD activities: 1987-2007 23 Introduction/Global cooperation in ETD activities 24 Introduction/Overview of rest of the Guide 25 Universities 26 Universities/Why ETDs? 27 Universities/Reasons and strategies for archiving electronic theses and dissertations 28 Universities/How to develop an ETD program 29 Universities/Scenarios illustrating approaches, schedules and workflow 30 Universities/Role of the Graduate School and Graduate Program 30.2 Changes in Presentation 30.3 Valuable Content 30.4 Access and Attitudes 30.5 Publication and Plagiarism 30.6 How Virginia Tech implemented the ETD Requirement 31 Universities/Role of the Library and Archives 32 Archiving Electronic Theses Dissertations 33 Factors Effecting Archiving 34 Universities/Intellectual Property Rights 35 Universities/Publishers 36 Universities/Human resources and expertise needed for an ETD program 37 Universities/Sources for funding 38 Universities/Costs 39 Universities/Processing charges 40 Universities/Budgets 40.1 Start-up 40.2 Implementation of production and distribution services 40.3 Communication 40.4 Training Students 41 Universities/Plagiarism 42 Universities/Assessment and Measurement Introduction 43 Universities/Types of Assessment 44 Universities/The Assessment and Measurement Process 44.1 Planning 44.2 Creating Goals and Objectives 44.3 Choosing Types of Measures 44.4 Deciding What Data to Collect 45 Universities/Measuring Production and Use of ETDs: Useful Models 46 Universities/Statistics and Usage 47 Universities/Measurement in Related Contexts 48 Universities/Guidelines for Implementing an Assessment Program for ETDs 49 Universities/Student Comments 50 Universities/Resources List 51 Universities/Policy Initiatives: National, Regional, and Local; Discipline specific; Language specific 51.1 National 51.2 Regional 51.3 Discipline Specific 51.4 Language specific 52 Universities/Policy Initiatives: The Case of France 53 Universities/E-Commerce: fee based methods 55 Students/How to learn about ETDs? (workshops, online resources, helpers) 56 Students/Importance of satisfying local requirements 57 Students/Learning from other ETDs 58 Students/How to prepare an ETD? (approaches) 59 Students/Overview: writing with word processors and structured editors 60 Students/Writing in word processing systems 61 Students/Microsoft Word and Office 2007 62 Students/Using Style Sheets 63 Students/Using Plug-ins: Bibliography Plug-in 64 Students/Corel WordPerfect 65 Students/LaTeX 65.1 TeX and LaTeX 65.2 How to get LaTeX 65.3 LaTeX under UNIX / LINUX systems 65.4 LaTeX under Windows operating systems 66 Students/FrameMaker 66.1 Why should one use FrameMaker instead of MS Word? 66.2 Using FrameMaker+SGML6.0 for a conversion of MS Word documents into SGML instances. 67 Students/Writing directly in SGML\XML 68 Students/Preparing a PDF document 69 Students/From LaTeX 69.1 Using Postscript and scalable fonts for PDF 69.2 Producing Rich PDF 69.3 Using PDFTeX 70 Students/Preparing for conversion to SGML\XML 70.1 The concept of Document Type Definitions (DTDs) 70.2 DTDs for electronic dissertations used worldwide 70.3 Preparing for Conversion 71 Students/In MS Word 71.1 Conversion method of the Cyber theses project 71.2 Using SGML Author for Word (Humboldt-University Berlin) 71.3 Other Tools 71.4 Converter Tools 72 Students/In WordPerfect 73 Students/In LaTeX 73.1 The Problem 73.2 Software and Tools 74 Students/Checking and correcting 74.1 A. Important Parts for a checking procedure 74.2 B. Checking for an SGML/XML-based publication and archiving workflow 75 Students/Integrating multimedia elements 76 Students/Providing metadata – inside, outside documents 77 Students/Protecting intellectual property and how to deal with plagiarism 78 Students/Naming standards: file names; unique Ids 78.1 ADT Program filename standard 78.2 ADT Program unique Ids 79 Students/How to submit your ETD? 80 Students/Local support 81 Students/Typical workflow, local policies and procedures 82 Students/Becoming a researcher in the electronic age 83 Technical Issues 84 Technical Issues/Contexts: local, regional, national, global 85 Technical Issues/Networking 86 Technical Issues/Seamless access: Open Archives Initiative, federated search 87 Technical Issues/Production of ETDs 88 Technical Issues/Overview: hardware, software, multimedia, scripts, encoding, document representations\conversions 89 Technical Issues/Page Description Languages 90 Technical Issues/Markup Languages 91 Technical Issues/XML Software 92 Technical Issues/DTDs for ETDs 93 Technical Issues/Berlin DTD workshop 94 Technical Issues/Support for students to write directly in XML 95 Technical Issues/Conversions from Word, Word Perfect or other RTF-compatible tools to SGML\XML 96 Technical Issues/Conversions from LaTeX to SGML\XML 97 Technical Issues/Rendering-style sheets 98 Technical Issues/Metadata, cross walks 99 Technical Issues/Naming Standards 100 Technical Issues/Encryption; Watermarking 101 Technical Issues/Packaging 102 Technical Issues/Backups; Mirrors 103 Technical Issues/Dissemination of ETDs 104 Technical Issues/Identifying: URN, PURL, DOI 105 Technical Issues/Metadata models for ETDs 106 Technical Issues/Cataloging: MARC, DC, RDF 107 Technical Issues/Database and IR 108 Technical Issues/Packaged solutions 109 Technical Issues/DiTeD and DIENST 110 Technical Issues/ADT 111 Technical Issues/Cybertheses 112 Technical Issues/VT DV and other tools 113 Technical Issues/Library Automation\OPAC: VTLS 114 Technical Issues/Harvest usage in Germany, France 115 Technical Issues/The NDLTD Union Catalog 116 Technical Issues/Metadata 117 Technical Issues/Fulltext 118 Technical Issues/SGML\XML Overview 119 Technical Issues/SGML\XML and Other Markup Languages 120 Technical Issues/Multimedia 121 Technical Issues/Interfaces 122 Training the Trainers 123 Training the Trainers/Initiatives to support ETD projects in Latin America 124 Training the Trainers/Tool kits for trainers 125 Training the Trainers/Identifying what is available 126 Training the Trainers/Demonstrations, explanations 127 Training the Trainers/Initiatives and Projects 128 Training the Trainers/Guidelines and Tutorials for ETDs 129 Training the Trainers/Specific Guidelines 130 Training the Trainers/Creating an online database of problem solving solutions 131 Training the Trainers/Help develop a broad local team 132 Training the Trainers/Standards, cooperation, and collaboration 133 Training the Trainers/Outreach/helping others 134 Training the Trainers/Developing Centres of Expertise where appropriate and helpful 135 The Future/Expanding ETD initiatives 136 The Future/Transforming Graduate Education 137 The Future/Managing technology changes 138 The Future/Interoperability 139 The Future/A vision of the future The UNESCO Guide for Creating Electronic Theses and Dissertations (ETDs) aims to help all those interested in projects and programs involving ETDs. To the extent possible, it has the eventual goal of aiding all students at all universities to be able to create electronic documents and to use digital libraries. It has particular focus on the emerging genre of ETDs, which should enhance the quality, content, form, and impact of scholarly communication that involves students engaged in research. It should help universities to develop their local infrastructure, especially regarding electronic publishing and digital libraries, which in turn build upon networking, computing, multimedia, and related technologies. In so doing, it should promote the sharing of knowledge locked up in universities, and the collaboration of universities spanning across all countries and continents, from North to South, from East to West, from developing to developed, from public to private, and from large to small. The ultimate effect may be one of the world's largest sustainable programs for diffusion of knowledge, across all fields, including science, technology, and culture. This work should be of interest to diverse audiences. Its various sections (see discussion in section 1.6) are aimed to address the needs of universities (including administrators and faculty), students (including those who wish to create ETDs as well as those who wish to make use of already-created works), and those involved in training or setting up ETD projects or programs. Ample technical details are covered to support the needs of students wishing to apply multimedia methods to enhance their ability to communicate complex results, as well as the requirements of staff building a local support infrastructure to help such students. This work is a living document that will continue to be updated in connection with the work of the Networked Digital Library of Theses and Dissertations (NDLTD, www.ndltd.org). It was born as a result of the support provided by UNESCO in grants given to Virginia Tech and University of Montreal. It was prepared by an international team of faculty and staff; coordinated by Shalini Urs. Its organization and content are the result of the editorial labor of Joseph Moxley. Its availability in a broad range of languages is the result of teams of translators, some funded in part by UNESCO, and others volunteering their assistance. While the development of a comprehensive worldwide program of ETDs of necessity builds upon an enormous range of knowledge and experience, it is hoped that this work will suffice as an initial Guide. We hope that those who read some or all of the sections contained herein will build upon the foundation laid out in the Introduction, so they understand the key ideas and are energized to move forward to advance the cause. In keeping with the goals of NDLTD, we hope that people around the globe who are interested in ETDs will help each other, and together transform graduate education, promote understanding, vastly expand access to new discoveries, and empower the next generation of researchers to become effective leaders of the Information Age. Next Section: What are ETDs Introduction/What are ETDs Joining and participating in the Networked Digital Library of Theses and Dissertations, NDLTD is one of the best ways to understand the concepts regarding digital libraries. It directly involves students pursuing graduate education by having them develop their theses or dissertations (TDs) as electronic documents, that is, as electronic theses or dissertations (ETDs). There are two main types of ETDs. One type, strongly preferred since students learn (by doing) as they create them, are author- created and submitted works. In other words, these are documents that are prepared by the (student) author (as is typical in almost all cases) using some electronic tools (e.g., Microsoft Word, LaTeX.), and then are submitted in their approved and final electronic form (to their university or agent thereof). Typically, the raw form of the document (e.g., in Word's ".doc" format) is converted into a form that is easy to preserve, archive, and make accessible for future readers (e.g., that follows standards, such as PDF or XML). That form is submitted, typically over a network connection, usually with related metadata (i.e., "data about data", often cataloging information as one might find in a library catalog, including title, year, author, abstract, and descriptors). Once submitted, such ETDs can be "discovered" by those interested, as a result of searching or browsing through the metadata, or by full text searching through the full document (text, and maybe even multimedia components, like images, video, or music). The second type of ETD is typically an electronic file that is created (usually by university or service company staff) by scanning in the pages of a paper thesis or dissertation. The resulting ETDs are much less desirable than the abovementioned type: they require much more storage space, they do not easily support full text searching, they cannot be flexibly manipulated (e.g., cannot be zoomed in on by those with poor vision), and they don't lead to the student authors learning about electronic publishing (to prepare them for electronic submission of papers, proposals, or other works now commonly required). Nevertheless, such page images can be made accessible at low cost so that those afar can print and read a facsimile of the original paper pages. In the subsequent discussion, most of the focus is on the first type of ETD mentioned above. However, the second type is commonplace in projects where a retrospective capture of old works is desired, or where a university wishes to share its research, is willing to go to considerable expense in that regard, and is not very concerned with educating or empowering students in electronic publishing methods. Next Section: ETDs as new genre of documents Introduction/ETDs as new genre of documents With thousands of students each year preparing ETDs, the creativity of the newest generation of scholars is being continuously expressed as they work to present their research results using the most appropriate form, structure, and content. While conforming as needed to the requirements of their institution, department, and discipline, students should develop and apply skills that will best prepare them for their future careers and lead to the most expressive rendering possible of their discoveries and ideas. Thus, ETDs are a new genre of documents, continuously re-defined as technology and student knowledge evolve. The first benefit is that new, better types of TDs may emerge as ETDs develop as a genre. Rather than being bound by the limits of old-style typewriters, students may be freed to include color diagrams and images, dynamic constructs like spreadsheets, interactive forms such as animations, and multimedia resources including audio and video. To ensure preservation of the raw data underlying their work, promote learning from their experience, and facilitate confirmation of their findings, they may enhance their ETDs by including the key datasets that they have assembled. As the new genre of ETDs emerges from this growing community of scholars, it is likely to build upon earlier forms. Simplest are documents that can be thought of as "electronic paper" where the underlying authoring goal is to produce a paper form, perhaps with color used in diagrams and images. Slightly richer are documents that have links, as in hypertext, at least from tables of contents, tables of figures, tables of tables, and indexes – all pointing to target locations in the body of the document. To facilitate preservation, some documents may be organized in onion-fashion, with a core mostly containing text (that thus may be printable), appendices including multimedia content following international standards, and supplemental files including data and interactive or dynamic forms that may be harder to migrate as the years pass by. Programs, applets, simulations, virtual environments, and other constructs yet to be discovered may be shared by students who aim to communicate their findings using the most suitable objects and representations Next Section: Why ETDs? Introduction/Why ETDs? The quality of a university is reflected by the quality of its students' intellectual products. Theses and dissertations reflect an institution's ability to lead students and support original work. In time, as digital libraries of ETDs become more commonplace, students and faculty will make judgments regarding the quality of a university by reviewing its digital library. Universities that incorporate new literacy tools, such as streaming multimedia, will attract students who hope to produce innovative work. Starting an ETD program is like starting any other project: a need for the results must exist so all those involved will be motivated and committed through all the steps to the end—the moment when ETDs have become a regular and consolidated activity in the graduate programs of the University. ETDs are based on the joint work of graduate students, mentors, graduate deans, administrative staff, library staff and the IT team. The success of the implementation of the ETD program requires the commitment of all these players plus that of the university's higher administrative officers. American Universities Should Require Electronic Theses and Dissertations by Moxley, Joseph M. (Educause Quarterly, No. 3 2001, pp. 61–63) Next Section: Minimize duplication of effort Introduction/Minimize duplication of effort One benefit of ETDs is a reduction in the needless repetition of investigations that are carried out because people are unaware of the findings of other students who have completed a TD. Except in unusual cases, masters' theses are rarely reported in databases (e.g., very few, except those from Canada, appear in UMI services like Dissertation Abstracts). Few dissertations prepared outside North America are reported either. With a globally accessible collection of ETDs, students can quickly search for works related to their interest from anywhere in the world, and in most cases examine and learn from those studies without incurring any cost. Next Section: Improve visibility Introduction/Improve visibility Once ETDs are collected on behalf of educational institutions, digital library technology makes it easy for works to be found. Through http://www.theses.org/, NDLTD directly makes ETDs available, and points to other services that facilitate such discovery. As a result, hundreds or thousands of accesses per year per work are logged, for example, according to reports from the Virginia Tech library regarding the ETDs it makes publicly accessible. As the collection of available ETDs grows and reaches critical mass, it is likely that it will be frequently consulted by the millions of researchers and graduate students interested in such detailed studies, expositions of new methodologies, reviews of the literature on specialized topics, extensive bibliographies, illustrative figures and tables, and highly expressive multimedia supplements. Thus, students and student works will become more visible, facilitating advances in scholarship and leading to increased collaboration, each made possible by electronic communication, across space and time. Next Section: Accelerate workflow: graduate more quickly, make ETDs available faster to outside audience Introduction/Accelerate workflow: graduate more quickly, make ETDs available faster to outside audience ETDs can be managed through automated procedures honed to take advantage of modern networked information systems. Since the shift to ETDs requires policy and process discussion among campus stakeholders, it is possible to streamline workflow and save time and labor. Checking of submissions and cataloging is sped up, moving and handling of paper copies is eliminated, and delays for binding are removed. The time between submission and graduation can be reduced, and ETDs can be made available for access within days or weeks rather than months. Next Section: Costs and benefits Introduction/Costs and benefits ETD submission over networks has zero cost, which compares favorably with the charges of hundreds or thousands of dollars otherwise required to print, copy, or publish TDs using paper or other media forms. In many institutions, the networking, computing, and software resources available to students suffice so that students preparing ETDs need make no additional expenditure. Similarly, on many campuses, assistance is available to answer questions and train students regarding word processing and other skills valuable for authors of electronic documents and users of digital libraries. If students elect to use personal computers and acquire their own software to use in ETD creation, these will later be useful in other research and development work, for both professional and personal needs, with low marginal expense specifically required for ETDs. Thus, it is typical that the pros far outweigh the cons regarding students preparing ETDs. Next Section: Purpose, goals, objectives of ETD activities Introduction/Purpose, goals, objectives of ETD activities The underlying purpose of ETD activities is to prepare the next generation of scholars to function effectively as knowledge workers in the Information Age. By institutionalizing this in a worldwide program, progress can be made toward tripartite goals of enhancing graduate education, promoting sharing of research, and supporting university collaboration. Particular objectives include: students knowing how to contribute to and use digital libraries; universities developing digital library services and infrastructure; enhanced sharing of university research results; and ETDs having higher quality and becoming more expressive of student findings. Next Section: Helping students be better prepared as knowledge workers Introduction/Helping students be better prepared as knowledge workers Most documents created today are prepared with the aid of computers. Many universities have "writing across the curriculum" programs, to ensure that students can create electronic documents that convey their knowledge and understanding, and demonstrate their ability to participate in the scholarly communication process. To function as effective knowledge workers, students must go beyond word processing skills that lead only to paper documents. They must learn to work with others, to share their findings by transmitting their results to others. This teamwork makes it feasible to collaborate, to co-author works, and thus to participate in research groups or teams, which are common throughout the research world (at the very least involving a faculty advisor and graduate student author). It also makes it pertinent for students to participate in common activities of modern researchers. Thus, they can be trained to submit a proposal electronically (e.g., as is required by the US's National Science Foundation) and to submit a paper to a conference (where papers are uploaded by authors, and downloaded by editors/reviewers, as part of the collection and selection activities). Next Section: Helping students be original Introduction/Helping students be original Originality is a defining feature of academic research. Using new media can empower you to conduct research in new ways. As suggested by the Exemplary ETDs, you can incorporate streaming video, interviews of subjects and settings, which readers can then view, which could be particularly useful for a case study or ethnographically informed research. You can provide different reading pathways for multiple audiences. For example, inexperienced lay audiences can have a more simplified version of your study, whereas more technical audiences can have more detailed analysis and citations. The technology allows you to present your work in new and different ways. For example, in your research you can include audio, video, and animations. You can add spreadsheets, databases, and simulations. You can even create virtual reality worlds. Next Section: Helping students network professionally Introduction/Helping students network professionally Having an ETD helps build your career. Your work is published in a timely manner, visible, and easily accessible. Timely publication makes your up-to-the-minute research instantly available. Upon publication, ETDs immediately become part of the NDLTD and are available for use by anyone having access to the Internet. Visibility allows people inside and outside of the academic arena to see and use your research. Just having an ETD can multiply the number of times your work is read. This exposure increases the possibilities that your work will be cited in others' publications, which adds to your prestige and can help your future advancement. Accessibility makes reading and using your work easy. Instead of having to request and await the arrival of a printed copy, your work immediately displays on a computer screen and can be printed on demand. By using today's communication tools wisely, you can save time and produce a more influential work. You can manage your information so you know where you've kept your notes; use powerful search tools to gather information and manage evaluations and revisions; and use interpretation tools for quantitative and qualitative documents. Networking on the Network by Phil Agee This is a 52000 word essay that Phil Agee wrote with the intention of "get[ting] it into the hands of every PhD student in the world." Next Section: Improving graduate education, and quality/expressiveness of ETDs Introduction/Improving graduate education, and quality/expressiveness of ETDs Graduate EducationEdit Around the globe, people have diverse views toward graduate students and graduate education. Some universities have strong advocacy for graduate students, often involving a graduate school and graduate dean. Others have no focus whatsoever on graduate students as a group, with all support thereof assumed to come by way of faculty mentors and advisors. Regardless of the local context; however, few would argue that graduate students should have fundamental knowledge and skills that will allow them to be effective researchers. ETDs in Graduate EducationEdit Accordingly, the move toward ETDs aims to enhance graduate education in effective researching. Building on the fact that students learn best by doing, this move encourages students to create and submit their own ETDs, thus learning a bit about electronic publishing and digital libraries in the context of preparing a work that is of importance to them. In particular, they need to gain some knowledge and skill in electronic document preparation, understanding not only the superficial aspects of relevant tools, but also beginning to learn about such processes as archiving, preservation, cataloging, indexing, resource discovery, searching, and browsing. Aids in CommunicationEdit Furthermore, once students become exposed to electronic documentation preparation processes, they often become aware of aids to help them communicate. Tools can be utilized to help with figures and drawing, to assist with analysis and graphing, or to support sharing and interactive exploration of data (e.g., through spreadsheets or database management systems). Images can be prepared as thumbnails that go into the body of a document to help the reader, as well as in varying sizes and resolutions to promote in-depth analysis. Readership and QualityEdit Beyond these tools and aids, however, is the fact that authors spend more time when their likely readership is large. Students will tend to produce higher quality work, and faculty will demand better writing and clearer presentation of results, if the audience for a work numbers in the hundreds or thousands, as opposed to the common current situation where a mere handful will read the document. With ETDs leading to large numbers of accesses, many students and faculty will work hard to enhance quality beyond that commonplace prior to the advent of ETDs. Next Section: Helping faculty Introduction/Helping faculty Each student could develop a bibliography reflecting his or her work, and a collective bibliography would emerge encompassing all of a faculty member's advisees. A student's acquired expertise will not completely leave with that student but will remain to help bootstrap new students (and new interests of the faculty member). The efforts of students working with a faculty member can be known to a wider audience. This would provide publicity and enhanced visibility for the student and that student's lab and major professor. Students who know how to use tools, such as Microsoft Word's tracking or commenting features, are better prepared for future e-publishing; they can use these tools for future collaboration and mentoring, which should save the faculty member time with the reviews and revisions. Next Section: Increasing readership of ETDs, communicating research results Introduction/Increasing readership of ETDs, communicating research results Once a student embarks upon the process of creating an ETD, this student will be assured of a large increase in the number of readers. Such an audience emerges when ETDs are locatable because of searches using popular search engines and/or are available for other students as well as more advanced researchers to draw upon. Many look for the large bibliographies and comprehensive literature reviews found in ETDs. Others look for in-depth discussion of research methods. Some seek data sets to use for follow-up studies. A growing number of faculty use ETD collections as reference sources, saving space on their shelves or time required to walk to a library. Some also refer to ETDs in classes or in homework assignments, especially where there are important results and/or clear expressions of concepts and ideas. Through such ETDs, students can communicate more effectively. Color figures are much easier to attain, in many situations, than those limited to black and white. Complex tables can be built, with sorting and subtotals incorporated, because of software tools. Spreadsheets or simulations help readers gain hands-on familiarity with data and analysis, promoting a deeper understanding. For those studying phenomena that could be characterized with color photos (using digital cameras or scanners), digital video sequences, audio files, medical images, or other digital representations. Thus, expanded and more effective communication of research results can be aided by usage of ETDs. Once a student embarks upon the process of creating an ETD, that student can be assured of a one or more order of magnitude increase in the number of readers. Such an audience emerges when ETDs are locatable because of searches using popular search engines and/or are available for other students as well as more advanced researchers to draw upon. Many look for the large bibliographies and comprehensive literature reviews found in ETDs. Others look for in-depth discussion of research methods. Some seek data sets to use for follow-up studies. A growing number of faculty use ETD collections as reference sources, saving space on their shelves or time required to walk to a library. Some also refer to ETDs in classes, or in homework assignments, especially where there are important results and/or clear expressions of concepts and ideas. At Virginia Tech, for example, many popular theses and dissertations are available to the public electronically. In 1996, there were 25,829 requests for ETD abstracts and 4,600 requests for ETDs themselves; by 1999 (January–August), there were over 143,056 requests for abstracts and 244,987 requests for ETDs. As of October 1999, the most popular ETD at VT had been requested over 75,000 times (see VT's download statistics at http://scholar.lib.vt.edu/theses/data/pdatah.htm). Next Section: Helping universities develop digital library services & infrastructure Introduction/Helping universities develop digital library services & infrastructure Many universities have little or no involvement in digital library efforts. Yet, it is clear that most universities will have digital library efforts as part of, or in addition to, conventional library efforts. Digital libraries are of particular value when distance education is involved or when a university has a number of far flung sites. They also can promote more flexible access to information (e.g., 24 hours/day, 7 days/week, from home or office, with no blocking because another person has borrowed a work). Establishing an ETD program automatically moves a university into the digital library era. Free software available through NDLTD, or from NDLTD members, is available that allows a university to develop its own digital library. Setting up that digital library helps the university bring together the personnel and infrastructure required for other digital library projects, too. Though the demands are small, the digital library that emerges will force consideration of almost all of the key concerns of those who work with digital libraries. Also, since various sponsors interested in digital library technology are helping in many ETD projects, there will be a continual enhancement of the digital library services that relate to improving local infrastructure. Next Section: Increasing sharing and collaboration among universities and students Introduction/Increasing sharing and collaboration among universities and students As more and more ETDs become available at universities or through their agents, more and more students will draw upon the emerging digital library of ETDs. This will make it easier for students to learn of the work of other students. Contact is quite feasible since many ETDs include email addresses and other contact information, of faculty helpers/advisors at least if it is not feasible to track down a graduating student author. Once contact is made, it is possible to have further discussion of references, application of methodologies, reuse of datasets, and extension of prior work, as well as investigation of remaining open problems. This may lead to friendships, collaborations, joint publications, and other benefits. Such may expand beyond student-student relationships to also include faculty, groups, laboratory teams, and other communities. Next Section: Enhancing access to university research Introduction/Enhancing access to university research Once ETDs are available, through diverse means, others may gain access. In particular, such access may be the only recourse open to those in developing countries who cannot afford to make purchases from Proquest, who cannot wait for expensive shipping of copies through interlibrary loan, who cannot attend the myriad conferences that demand the considerable expenses related to travel, or who cannot pay for expensive journals (that only may have short summaries of thesis or dissertation results). Access may result from word of mouth, from search, from announcements, or by following links or citations. They may occur quickly after a work is made known to those whose works are referred to or to those who employ Selective Dissemination of Information services. They may occur intermediated by digital library systems or similar mechanisms. They may result when ETDs are referred to journal articles, conference papers, reports, course notes, and other forms. Numerically, if theses and dissertations are all released as ETDs, the number of works per year may be around the number of journal articles published by university students. The number may be 20-50% of the number of journal articles prepared by the faculty. Regarding students, in most cases, an ETD will be the only publication of an author. If the number who read ETDs is 10 or 100 times the number who read a paper thesis, then it is clear that there will be a significant increase in access to university research as a result of an ETD program. Next Section: Searching Introduction/Searching Since in most cases it is in the interest of students and universities to maximize the visibility of their research results, the general approach of NDLTD is to encourage all parties interested to facilitate access to ETDs. As part of the education component of NDLTD, it is hoped that graduate students will become facile with searching through electronic collections, especially those in digital libraries. If we regard managing information as a basic human need, ensuring that the next generation of scholars has such skill seems an appropriate minimal objective. Most specifically, since graduate research often builds upon prior results from other graduate researchers, it seems sensible for all ETD authors to be able to search through available ETD holdings. NDLTD encourages online resources, self-study materials, individual assistance, as well as group training activities be provided so that graduate students become knowledgeable about resource discovery, searching, query construction, query refinement, citation services, and other processes – both for ETDs and for content in their discipline. Next Section: Browsing: Classification systems, classification schemes used in different disciplines Introduction/Browsing: Classification systems, classification schemes used in different disciplines A key method to gain access to ETDs is through browsing. Browsing promotes serendipity, in analogous fashion to when a person looks around in library stacks, picking up and glancing at a number of works, typically ones that are relatively close to each other. Browsing often involves a researcher in a learning process connecting with the concepts, areas, and vocabulary used in a particular field. A researchers often moves around in "concept space", seeing what concepts are broader and which are narrower, which are related, and which are examples or applications of theories or methods. Thus, in the case of medical works, browsing often encourages researchers to think about diseases, treatments, location in an organism (or human body or subsystem thereof), symptoms, and other considerations. In many fields, browsing involves exploring a taxonomic system, managing an organization chart or hierarchical structure, or moving from a term to a more focused noun phrase. In many disciplines, there are official classification systems. Some are quite broad, in some cases covering all areas, as is the case when using the subject headings prepared by the US Library of Congress (LCSH), or the Dewey Decimal Classification system (DDC, available from Forest Press, owned by OCLC). However, for in-depth characterization of a research work in a particular field, it is more appropriate to use the category system for the field. In computing, the ACM system is popular. In medicine, MeSH or UMLS are widely used. In physics, PACS is widely used. Gradually, digital library support for ETDs will support browsing using both broad schemes like DDC, as well as narrow schemes like PACS, integrated synergistically. Next Section: Well known sites and resources for ETDs Introduction/Well known sites and resources for ETDs NDLTD runs the Web site http://www.theses.org/ (also under the alias http://www.dissertations.org/) as a central clearinghouse for access to ETDs. This site points to various other locations that support portions of the worldwide holdings of ETDs. For example, the largest corporate archive, with over 1.5 million entries, is managed by UMI and has most doctoral dissertations from USA and Canada, as well as most masters' theses from Canada, in microfilm form with metadata available as a searchable collection through Dissertation Abstracts. Since 1997 UMI has scanned new submissions (originally from microfilm, later directly from paper) and made the page images available through PDF files. With over 100,000 ETDs accessible through subscription or direct payment mechanisms, UMI hosts the largest single collection of both electronic and microfilm TDs. Other corporations as well as local, regional, national, and international groups associated with NDLTD have Web sites too, such as http://www.cybertheses.org/[1] for the International Francophone project or http://www.dissonline.org/ for the German Dissertation online project. In addition, a number of WWW search engines have indexed some of the ETD collections available so this genre is included in general Web searches. Some other schemes allow access to ETD collections. Using Z39.50, the "information retrieval protocol", for example, the Virginia Tech ETD collection can be accessed through suitable clients or from some library catalog systems. OCLC's WorldCat service, with over 20 million catalog records, has an estimated 3.5 million entries for TDs. Perhaps most promising is that the global as well as regional and local metadata information about ETDs may become widely accessible through the Open Archives Initiative (http://www.openarchives.org/). The German "Dissertation Online" project was undertaken by the Initiative of Information and Communication of the German Learned Societies (http://www.iuk-initiative.org/index.shtml.en). This project was funded by the German Research Foundation (DFG: Deutsche Forschungsgemeinschaft) for 21/2 years (April 1998 - October 2000). The final conference took place in Berlin from October 30–31, 2000 (http://www.dissonline.de/abschlusstagung/index.html). The project worked at an integrative level and aimed towards a German wide initiative to bring scholarly publications, such as dissertations, diploma and master theses which are usually lost in libraries, online. Through the joint work of 6 academic disciplines (mathematics, physics, chemistry, educational sciences, computer science and libraries such as the State and University Library Lower Saxony (SUB Göttingen) and the German National Library (DDB: Die Deutsche Bibliothek)) which took place at several locations (Duisburg, Oldenburg, Erlangen, Berlin Computing Centre and School of Education, Göttingen, Frankfurt) the project was highly successful in Germany and elsewhere. A tight cooperation with the Networked Digital Library of Theses and Dissertations (NDLTD[2]), set up by Edward Fox from the Virginia Polytechnic Institute and State University, USA, was established. The developments and the movement "DissOnline.de" resulted in establishing a bureau for coordination (Koordinierungsstelle DissOnline) at the German National Library (DDB). All German efforts taken are now coordinated and all developments (tools, guidelines, etc.) are collected by this bureau. The main tasks of the original project still reflect the problem areas that have to be taken into consideration while setting up local, national or global projects on electronic theses and dissertations: Library issues The results of these different subtasks are integrated into the different sections that follow for students, universities, technical issues and trainers. Additional Thesis Collections: PhysDis, a large collection of Physics Theses of Universities across Europe TheO, a collection of theses of different fields of 43 Universities in Germany, in as much as the Theses do contain Metadata MPRESS, a large collection of European Mathematical Theses. which contains as a subset Index nationaux prépublications, thèses et habilitations, a collection of theses in France in Mathematics Next Section: Brief history of ETD activities: 1987-2007 Introduction/Brief history of ETD activities: 1987-2007 The first real activity directed toward ETDs was a meeting convened by Nick Altair of UMI in Ann Arbor, Michigan during the fall of 1987 involving participants from Virginia Tech, ArborText, SoftQuad, and University of Michigan. Discussion focussed on the latest approaches to electronic publishing and the idea of applying the Standard Generalized Markup Language (SGML, an ISO standard approved in 1985) to the preparation of dissertations, possibly as an extension of the Electronic Manuscript Project. In 1988, Yuri Rubinsky of SoftQuad was funded by Virginia Tech to help develop the first Document Type Definition (DTD) to specify the structure of ETDs using SGML. Pilot studies continued using SoftQuad's AuthorEditor tool, but only with the appearance of Adobe's Acrobat software and Portable Document Format (PDF) in the early 1990s did it become clear that students could easily prepare their own ETDs. In 1992 Virginia Tech joined with the Coalition for Networked Information, the Council of Graduate Schools, and UMI, to invite ten other universities to select three representatives each, from their library, graduate school/program, and computing/information technology groups. This meeting in Washington, D.C. demonstrated the strong interest in and feasibility of ETD activities among US and Canadian universities. In 1993, the Southeastern Universities Research Association (SURA) and Southeastern Library Network (Solinet) decided to include ETD efforts in regional electronic library plans. Virginia Tech hosted another meeting involving multiple universities in Blacksburg, VA in 1994 to develop specific plans regarding ETD projects. On the technical side, the decision was made that whenever feasible, students should prepare ETDs using appropriate multimedia standards in addition to both a descriptive (e.g., SGML) and rendered (e.g., PDF) form for the main work. Then, in 1996, the pace of ETD activities sped up. SURA funded a project led by Virginia Tech to spread the concept around the Southeastern Unted States. Starting in September 1996, the US Department of Education funded a three-year effort to spread the concept around the USA. The pilot project that had proceeded at Virginia Tech led to a mandatory requirement for all theses and dissertations submitted after 1996 to be submitted (only) in electronic form. International interest spread the concept to Canada, UK, Germany, and other countries. To coordinate all these efforts, the free, voluntary federation called NDLTD (Networked Digital Library of Theses and Dissertations) was established and quickly began to expand. Annual meetings began in the spring of 1998 with about 20 people gathering in Memphis, TN. In 1999 about 70 came to Blacksburg, VA while in 2000 about 225 arrived in St. Petersburg, FL for the third annual conference. Next Section: Global cooperation in ETD activities Introduction/Global cooperation in ETD activities There continues to be rapid growth and development of ETD activities around the world. Whether such efforts arise spontaneously or as extensions of existing efforts, it is hoped that all will proceed in cooperative fashion so universities can help each other in a global collaboration [4], passing on lessons learned as well as useful tools and information. The mission of NDLTD is to facilitate such progress in a supportive rather than prescriptive manner. Over 100 members joined NDLTD by 2000, including over 80 universities in addition to national and regional project efforts; international, national, and regional organizations; and interested companies and associations. The only requirement for joining NDLTD is interest in advancing ETD activities, so it is hoped this will help ensure global cooperation. A number of groups involved in NDLTD are particularly interested in supporting efforts in developing countries. The sharing of research results through ETDs is one of the fastest ways for scholars working in developing countries to become known and have an impact on the advancement of knowledge. It also is one of the easiest and least costly ways for universities in developing countries to become involved in digital library activities and to become known for their astute deployment of relevant and helpful technologies. The Organization of American States, UNESCO, and other groups are playing a most supportive role in facilitating this process. Next Section: Overview of rest of the Guide Introduction/Overview of rest of the Guide Subsequent sections further explain ETD activities. Section 2 discusses issues for university decision makers and implementers of projects on campuses. Section 3 presents the topic for students. Section 4 deals with further technical details. Section 5 takes a broader view, raising the level to issues related to launching campus initiatives and training those who may train students. Finally, Section 6 provides a glimpse of future directions. Next Section: Universities By the end of 2007, the number of universities involved in NDLTD was well over 100. Scores of other universities were considering work with ETDs, and hundreds of universities were aware of the concept. NDLTD hopes that forthwith there will be ETD efforts in every country and then in every state/province, eventually in every leading university, soon serving every language group, and ultimately in every college and university. Since NDLTD aims to help, and there are no costs to join, it is hoped that membership will rise to closely match the number that are interested in ETDs. It is not clear why any university should not become involved in ETD efforts. Once an ETD program on a campus has evolved to the point that ETD submission is a requirement, the effort saves money for the university as well as the students, while providing many important benefits. And, reaching such a point is not hard, if there is a local team, with effective leadership, that has a clear understanding of ETDs and what is occurring elsewhere in terms of support, cooperation, and collaborative efforts. This section of the Guide explains why and how universities can establish ETD programs, and helps those involved in an ETD program to address concerns and problems that may be voiced. It appears on the one hand that an ETD program has the healthy effect of helping a university to engage in a rich dialog on a wide variety of issues related to scholarly communication. This is important, since we are in the midst of a revolution in these processes, and both students and players must be aware of the situation if they are to manage effectively (and economically, while working in accord with the core values of scholars). On the other hand, ETD efforts have advanced to the point that proceeding with an ETD program, while not as "sexy" as some of the more vigorously funded digital library efforts, does work well, providing real solutions to real concerns, leading to sustainable and beneficial practices. In short, launching an ETD program is a "no brainer" that can quickly advance almost any university to the happy position of having an effective and economical digital library initiative in place, which also can serve as a model for other efforts. Universities/Why ETDs? Starting an ETD program is like starting any other project: a need for the results must exist so all those involved will be motivated and committed through all the steps to the end—the moment when ETD's have become a regular and consolidated activity in the graduate programs of the University. ETD's are based on the joint work of graduate students, mentors, graduate deans, administrative staff, library staff and the IT team. The success of the implementation of the ETD program requires the commitment of all these players plus that of the university's higher administrative officers. American Universities Should Require Electronic Theses and Dissertations by Moxley, Joseph M. (Educause Quarterly, No. 3 2001, pp. 61–63.) Next Section: Reasons and strategies for archiving electronic theses and dissertations Universities/Reasons and strategies for archiving electronic theses and dissertations 1. ETDs make the results of graduate programs widely known. 2. Graduate programs may be evaluated by the number of theses and dissertations (TDs) produced and by the number of accessible ETD's. 3. In many countries, when financed with public funds, it is expected that TDs will be made public. ETDs are the easiest way to accomplish this. 4. TDs are part of the assets and history of the universities. 5. TDs exist and are published on paper, so why not publish them electronically? 6. TDs are referred by examining committees, a warranty of quality to be published. 7. TDs contain bibliographical reviews. 8. TDs present the methods used during research, thus allowing these methods to be used by others. 9. TDs allow extensions to be identified and undertaken. 10. TDs hold information that will help avoid duplication of efforts. 11. To publish TDs funded with public money is a way of returning the results to society. 12. To electronically publish TDs makes the results known nationally and internationally. 13. To electronically publish TDs makes it less expensive to students, who do not have to print as many copies. 14. Electronically published TDs yield easier and faster access to information. 15. ETDs require less storage space. 16. ETDs can identify and connect national and international research groups. 17. Access to information enhances the quality of TDs. 18. An ETD program introduces digital libraries in the universities allowing other projects to bloom. 19. ETDs are way of sharing intellectual production. 20. Wide knowledge of good quality TDs strengthens the faculty, the graduate programs and the university. 21. Widely known results allow copies to be identified in an easier way. 22. Universities will be able to share knowledge on digital libraries. The reasons are purposely listed in the order they were presented during the discussion and, no doubt, seem to be quite scattered. Thus, let's imagine 3 categories of reasons: benefits to students, benefits to universities and benefits to regions/countries/society. The reasons that were listed above can be reorganized and assigned to these categories. First, there are benefits to students which reflect on the university and on society too. Of the 23 reasons above, 15 are beneficial to students: 29. ETDs make the results of graduate programs widely known. 30. Graduate programs may be evaluated by the number of theses and dissertations (TDs) and by the number of accessible ETDs. 31. TDs allow extensions to be identified and undertaken. 32. TDs contain bibliographical reviews. 34. TDs present the methods used during research, thus allowing these methods to be used by others. 35. To electronically publish TDs makes it less expensive to students who do not have to print as many copies. 38. Widely known results allow copies to be identified more easily. Next, the universities are the focus and some reasons that were listed under benefits to students will appear again: 45. TDs are part of the assets and of the history of the universities. 47. Universities will be able to share knowledge in digital libraries. Benefits that appeared under the previous 2 categories will be listed again in this last category, devoted to regions/countries/society: 52. In many countries, when financed with public funds, it is expected that TDs will be made public. ETDs are the easiest way to accomplish this. Finally, two other reasons for ETDs are: 56. TDs exist and are published on paper, so why not publish them electronically? 57. TDs are referred by examining committees, a warranty of quality to be published. There were 23 reasons when they were first listed. When they were classified, we got 34. The explanation is simple - many reasons bring benefits to more than one category. It is not hard to come to the conclusion that ETD's are beneficial to all, and that ETD programs are good and should be considered by universities. Next Section: How to develop an ETD program Universities/How to develop an ETD program Some points must be considered before starting an ETD program because they impact on: The stage where the project starts; The type of training to be provided; The technology to be adopted; The personnel to be hired, if needed; The time frame for the program to be operative; The total cost of implementation. The team implementing the ETD program must be aware that decisions must be made based on legislation, culture, financial conditions, infrastruture and political aspects of their area. So, they must be prepared to analyze the important issues, suggest solutions and have them approved by the propoer authorities. Only when these aspects are considered can the program start. Many of the following points may not apply to developed nations but can be crucial to developing countries. The points are grouped into four categories: Analysis of the stage of the automation of the library system: Is there an OPAC? Does it support the MARC format? Does it support digital objects? What types? Can the contents of objects be searched? Does it have a Web interface? Is it compliant with Protocol Z39.50? Is there a project to automate the catalog? Does the system to be used support the MARC format? What is the time frame for the catalog to be automated? Can the system to be used can hold a digital library (types of digital objects, Web interface, Z39.50)? Number of TDs per year and to date (both on paper and on digital files): Per year - this number is important because it influences: The number of persons in the help desk; The decision on how to start - by some areas to proof concept or all programs; The planning of the growth of infrastructure/equipment. To date - this number is important because it influences: The decision to make the retrospective capture in some areas and/or dates or all TDs; The decision to use scanner + OCR in some areas and/or dates or all TDs (on paper); The planning of the types and numbers of equipment; The planning of the team; Format presentation of TDs: One or many formats? The number of DTDs, templates and viewing filters depends on this; Is an update of format needed or desired? The process must be defined - topdown, consensus, etc.; Is there legislation to comply with? Authors' rights and publishing conditions: Is a document of authorization required? If so, define terms and approve with legal department; Establish conditions? For black out period, for on campus publishing only, for partial publishing, etc. The width of the scope of the topics above shows that an ETD program involves many areas of the universisty and all of them must be committed to it. After these points have been addressed, the project of the ETD program may begin. An ETD program, like any other project, requires that the roles of each player be well defined. All the people involved must be aware of the importance of their work individually and of the interaction that makes up a team. There must be a commitment in all levels, from the highest administration to the graduate students. In terms of the university, the 3 main players are: The Graduate Office The IT/Computer Group They will lead the university in deciding upon: The formats for ETDs; The way to deal with authors' rights and publishing procedures; The workflow for the new dissertations; The strategy and workflow for retrospective capture; The style sheets and filters for visualization; The preservation format(s) and procedures; The digital library system to be used; The identification of the digital documents; The relation of the ETD digital library with legacy systems (library, administrative, etc.); The support team to be used; The training program. At the beginning of the project, they must have the support of the highest administration to: Change the TDs' formats, if necessary; Change the culture of mentors, students and administrative staff; Solve the problems related to authors' rights; Assure preservation and information security of the digital collection (ETD); Make sure funds are provided for equipment (HW & SW), personnel and training programs. A good way to start the discussion in the university is by writing a pre- project to be submitted by the 3 players to the high administrative officers. The main topics of the pre-project should include but not be limited to: The objectives: Main objective Secondary objectives The benefits to: The region/country/society The characteristics of the ETD program and of digital library: Functional characteristics The results to be achieved A brief and focused description of the program and the project A description of the main steps of the project An estimate of the resources (human and material) needed to implement the program An estimate of the annual resources (human and material) needed to keep the program in operation An estimate of the time frame to implement each step of the project and to have the program operable The commitment, responsibilities and actions of each participant involved The relation of the program to other programs in: The national scenario The international scenario The writing of this pre-project will help the team organize and ascertain everyone's involvement. In addition, the higher administration will be able to decide based on more reliable information, increasing the degree of commitment of all involved in the program. Next Section: Scenarios illustrating approaches, schedules and workflow Universities/Scenarios illustrating approaches, schedules and workflow A common workflow at a campus involved in NDLTD is as follows. Each student will use a home computer, or computer in a lab, to submit his or her ETD. His or her work will have been created on the computer he or she is working with, or will have been moved there using some media format, such as diskette or CD-RW. The student will go to a well-known URL (e.g., at Virginia Tech, will start at http://etd.vt.edu and then follow links). First, students will provide an ID and password that authenticates them. Next, they will enter in metadata about their ETD. When that is complete and checked, they will use their browser to locate each of the files that make up the ETD, so uploading can proceed. Eventually, the entire ETD will be uploaded. Later, a person in the graduate school will find that a new ETD has been uploaded. He or she will check the submission. If there are problems, he or she will use email to contact the student so that changes/corrections can be made, and the process repeated up through this step. Once a proper ETD is submitted, a cataloguer in the Library will be notified. He or she will catalog the work, adding in additional categories, and checking the metadata. Eventually he or she will release the work to allow proper access, according to the student/faculty preferences. More refined workflow models can be applied. Let us suppose that at the Polytechnic University of Valencia, Spain the process starts from a catalog of ETD proposals, published by faculty members filling in the appropriate form; each proposal may include the title, keywords, abstract, level of expertise required, and other useful information. A student can apply for a number of proposals (specifying an order of preference) filling a form including his/her personal data. A faculty committee makes the final assignment of proposals to candidates. At this point, most of the ETD metadata have been collected, and there is no need to introduce them during the submission process. Next Section: Role of the Graduate School and Graduate Program Universities/Role of the Graduate School and Graduate Program This section looks at how electronic publication of theses and dissertations will enhance graduate education. Topics discussed include: improved knowledge of electronic publication technologies, greater access to scholarly information, wider distribution of an author's work, and student and faculty concerns. IntroductionEdit The move by Graduate Schools to allow or even require students to submit theses and dissertations as electronic or digital documents (ETDs) creates much excitement, both positive and negative, among the students and faculty who will be affected by this initiative to digitize these important documents. These positive and negative views will no doubt be tempered by increased knowledge of the ETD process and through increased experience in creating and archiving ETDs. At this time in the development of the ETD process, I believe the importance of an open-minded approach to this new way of expressing the outcomes of masters and doctoral research is captured very well in the following statement by Jean- Claude Guédon in his work, Publications électroniques (1998): When print emerged, universities failed to recognize its importance and almost managed to marginalize themselves into oblivion. With a new major transition upon us, such benign neglect simply will not do. Yet the challenges universities face in responding to an increasingly digitized and networked world are staggering. Universities need a vision allowing them to express their dearest values in new forms, rather than protect their present form at the expense of their most fundamental values. The ETD initiatives now under way in universities around the world are about bringing fundamental change to our current concept of what constitutes a thesis or a dissertation. In the U.S., this concept has not changed significantly since students first began to submit paper theses and dissertations in our first research universities over 120 years ago. By moving from a paper presentation of research results to a digital presentation, we make available to the ETD author a powerful array of presentation and distribution tools. These tools allow the author to reveal to masters and doctoral committees, to other scholars, and to the world, the results of their research endeavors in ways and with a level of access never before possible. Changes in PresentationEdit I believe graduate schools and faculty, in the name of maintaining quality, have all too often inhibited the creativity of graduate students by forcing them into a mold to which they must all conform. This is nowhere more evident than in the thesis or dissertation where format restrictions abound. Some graduate schools have special paper with printed margins within which all written material must be contained. Some graduate schools still read and edit the entire text of every thesis or dissertation. Many have thesis or dissertation editors whose reputation for using fine rulers and other editorial devices for enforcing graduate school format are legendary. I believe that the student must submit a high quality document that is legible, readable, and that conveys the results of the research or scholarship in a manner that is clear and informative to other scholars. The document does not, however, need to be narrowly confined to a specific format if it meets the above criteria. To create a high quality ETD students must be information literate. That is, they must, at a minimum, have a level of knowledge of office software that will allow them to create a document that if printed would result in a high quality paper document. This kind of properly formatted digital document thus becomes the primary construct of the author, rather than a paper document. In conducting training workshops for Virginia Tech students, a number of which are older non-traditional students, we have found that this lack of office software skills is the single greatest impediment to their being able to produce a good "vanilla" ETD—that is, an ETD that has the appearance of a paper ETD, but is submitted as a digital document. As early 1999 about 80% of Virginia Tech's 1500 ETDs are vanilla ETDs. Accordingly, we have emphasized the development of these skills, which number less than ten and can be taught in an hour, in our student ETD workshops. Once the student has the fundamental skills to produce an ETD, they are ready, if they desire, to move on to more advanced topics for producing a visually and audibly enhanced ETD. Advanced topics include landscape pages; multimedia objects like graphs, pictures, sound, movies, simulations; and reader aids like internal and external links, thumbnail pages, and text notes. Students are not required to use these enhancement tools, but by giving them access to these tools we open creative opportunities for students to more clearly express the outcomes of their masters or doctoral research. To maintain quality, the student's thesis or dissertation committee must actively participate as reviewers in this process and must be prepared to exercise judgment concerning the suitability of material for inclusion in the ETD. The resulting "chocolate ripple" or in some cases "macadamia nut fudge" ETDs are the forerunners of a new genre of theses and dissertations which will become commonplace in the future. Whether tomorrow's graduate students are employed inside or outside the university environment, the ubiquitous presence and use of digital information will certainly be a major part of their future careers. For this reason efforts to increase the information literacy are certain to benefit graduate students long after they have used these skills to produce a thesis or a dissertation. Valuable ContentEdit The traditional view is that the doctoral dissertation and less so the masters thesis provides a one time opportunity for the student to do an in depth study of an area of research or scholarship and to write at length about the topic, free of the restrictions on length imposed by book and journal editors. Such writings may contain extensive literature reviews and lengthy bibliographies. They may also contain results of preliminary studies or discussions of future research directions that would be very valuable to the researchers and scholars who follow. Primarily because of restrictions on the length of journal articles, such information exists only in theses and dissertations. I believe this view is correct and should be maintained in the digital thesis or dissertation. Access and AttitudesEdit The attitudes of students and faculty toward the value of theses and dissertations vary greatly. For the reasons given above some value them highly. Others, particularly some faculty, see them as requirements of graduate schools that have little value. These individuals consider the journal publication the primary outcome of graduate student research. I do not dispute the added value of the peer review process for journal articles and for books, yet I do firmly believe that so long as the scholar or researcher using ETDs as information sources recognizes theses and dissertations for what they are, these documents are valuable sources of information. Indeed, these information sources have been grossly underutilized because of the difficulty in obtaining widely available, free access to them either through university libraries or through organizations like University Microfilms. If a comprehensive worldwide networked digital library of theses and dissertations existed, I believe the impact and utilization of these sources of information would rise in proportion to the increased access. This view is supported by experience at Virginia Tech in our ETD project. Research done in 1996 by the Virginia Tech library showed that the average thesis circulated about twice a year and the average dissertation about three times a year in the first four years they were in the library. These usage statistics do not include the use of copies housed in the home departments of the students or the usage of dissertations in the University Microfilms collection. Even so, the usage of the 1500 ETDs in our digital library far outpaces the use of paper documents. Growth in usage has been steady and remarkable. For the calendar year 1998 there were over 350,000 downloads of the PDF files of the 1500 ETDs that were in the VT library. This is over 200 downloads for each ETD in the collection. The distribution of the interest in the ETDs is equally remarkable. The majority of the interest comes from the U.S with inquiries in 1998 coming from the following domains: 250,000 from .edu, 88,000 from .com, 27,000 from .net, 6,800 from .gov, and 3400 from.mil. Inquiries also come from countries around the world including the 8,100 from the United Kingdom, 4,200 from Australia, 7,300 from Germany, 3,900 from Canada, and 2,200 from South Korea. The most accessed ETDs have been accessed tens of thousands of times with many over one thousand accesses. To learn more about accesses see http://scholar.lib.vt.edu/theses/data/somefacts.html Publication and PlagiarismEdit When the ETD project began at Virginia Tech, some students and faculty expressed great concern that publishers would not accept derivative manuscripts or book manuscripts from ETDs. For some publishers this concern is legitimate and the ETD project has put into place a system for students and advisors to restrict access to ETDs until after journal articles appear. This system seems to satisfy faculty, students and publishers. Publishers that have discussed this matter with us usually have not expressed concern with the release of the ETD after the journal article is published. One exception may be small scholarly presses that publish books derived from ETDs. These presses view the book as having a sales life of several years after the initial date of publication. In these cases, it may be necessary to extend the period of restricted access well beyond the publication date of the book. For the longer term, however, it is important that researchers and scholars regain control of their work by becoming more knowledgeable about their rights as original creators and as holders of the copyrights to the work. This requires universities to have active programs to educate their faculty and students about copyright. Publishers also need to be educated to be less concerned about ETDs interfering with the marketability of their journals. This can be done, in part, by an effort on the part of researchers and scholars to educate publishers of their professional journals. They need to help persuade journal editors that ETDs most often are not the same as the journal articles derived from them, and that there is a serious difference because they have not been subject to the stamp of approval that is the result of peer review. As such they should not be considered a threat to the news value or to the sales potential of the journal. It is interesting to note that a Virginia Tech survey of students who had released their ETDs worldwide showed that twenty students had published derivative manuscripts from the ETDs with no publisher resistance to accepting the manuscripts. It is also noteworthy that the American Physical Society has a practice of sharing electronic copies of preprints of manuscripts undergoing peer review (http://xxx.lanl.gov/). Those that successfully pass peer review are published in the Society's journals. This practice is essentially the same as the practice being proposed for ETDs above. The risk of plagiarism is next on the list of concerns of students and faculty. We do not yet have enough experience with ETDs to speak authoritatively about this issue. If one thinks a bit about it though, it seems that the risks of exposure of plagiarism will deter such activity. Most researchers and scholars still work in fields where a fairly small group of workers have detailed knowledge of their work. It follows that because of the size of the field and because of the ease of detecting plagiarized passages in electronic documents, the risks of detection will make wide spread plagiarism unlikely. More disconcerting to me is the closely related concern of researchers and scholars that by reading their students ETDs, other researchers and scholars will achieve a competitive edge in the contest for grants and contracts. Most research in U.S. universities is done in the name of supporting the well being of the nation and is being sponsored directly or indirectly with public tax dollars. There is something wrong with a view that research and scholarship should not be shared among other researchers and scholars for the above reasons. Yet the concern is understandable in today's financially stretched research universities where the competition for promotion and tenure among young faculty is fierce. Similarly, faculty are encouraged to develop intellectual property in which the university claims a share. I'm not sure if we have gone too far down this road, but I am concerned that our obligation as scholars to make our work known to other scholars is being compromised. A result of this compromise is that the goal of scholars to advance knowledge through sharing knowledge may also be slowed. How Virginia Tech implemented the ETD RequirementEdit ETD discussions with the Graduate Dean, the Library, and Ed Fox, a faculty member conducting research on digital libraries, began in 1991. At that time we were exploring the possibilities of optional submission. Shortly thereafter Adobe Acrobat® software for creating and editing Portable Document Format (PDF) files came on the market. This software for the first time provided a tool that was easy to use and allowed documents to be moved between computer operating systems and platforms while retaining the original document formatting. This was a great step forward in increasing worldwide access to information while retaining the original author's formatting style. At this time we began a pilot study to determine if the Acrobat® met our needs. We determined rather quickly that it was the most suitable product for our needs at that time. In my opinion that conclusion holds true today. We continued discussions with the Graduate School and the Library and in the Fall of 1995 concluded that we would seek to make the submission of ETDs a requirement of the Graduate School. We took a proposal to the Commission on Graduate Studies and Policies for discussion. A degree standards subcommittee discussed the proposal amongst themselves then with ETD team members, Ed Fox from Computer Science, Gail McMillan from the Library, and John Eaton from the Graduate School. In these discussions the expressed concerns dealt with archiving and preservation, the burden to the students and the burden to the faculty and departments. After full discussion, the subcommittee recommended approval of the proposal. The commission discussed and approved the proposal, subject to the following provisions. That a student training process be conducted to show students how to produce an ETD. That necessary software (Adobe Acrobat®) be made available to students in campus computer labs. That the faculty not be burdened by this process. That a faculty/graduate student advisory committee be established to advise the Commission on Graduate Studies and Policies on the ETD project. With these provisions agreed to, the Commission approved a one year voluntary submission period to be used for beginning the student ETD workshops, informing the university community, and development of the infrastructure needed to move to requiring ETDs, after which ETDs would become a requirement in the spring semester of 1997. All went very smoothly while the process was voluntary. Workshops were started, software was placed in campus computer labs, visits were made to departments, articles were published in the campus newspaper, and the advisory committee was formed. Late in the spring semester of 1997, after the mandatory requirement began, a small but vocal group of faculty, mostly from the life sciences and chemistry expressed a serious concerns about compromising the publication of derivative manuscripts from ETDs made available world wide. While we had a provision for withholding release of ETDs pending publication of manuscripts, the time period of six months was thought to be short. The ETD team responded to this concern by giving the student and the advisor greater control of the access to the ETD through the ETD approval form which can be found at http://etd.vt.edu/. The modifications made to the ETD approval form seem to have satisfied faculty concerns about publication, and since that date the ETD project has operated very smoothly at Virginia Tech and is now rapidly becoming and integral part of graduate education. ConclusionEdit The ETD project has provided the opportunity for fundamental change in the expression of and access to the results and scholarship done by students in research universities around the world. These tools also can easily be extended to the expression of and access to research done by faculty. As scholars, we should not let this opportunity slip by. As Jean-Claude Guédon said "Benign neglect simply will not do". Next Section: Role of the Library and Archives Universities/Role of the Library and Archives ETDs have a very positive impact on libraries because they are an easy way to expand services and resources. With ETDs libraries can evolve into digital libraries and online libraries. Because authors create and submit the digital documents and yet others validate them (i.e., graduate school approval), all the library has to do is receive, store, and provide access. This is not a radical departure from what libraries do normally; only these documents are electronic. Today, many libraries are already handling electronic journals, so ETDs can extend the multimedia resources to the online environment and give every library something unique in their digital resources. The library can do more, such as improving workflow, reducing the time from receipt to public access, and, of course, one ETD can have multiple simultaneous users. ETDs can be submitted directly to the library server so that as soon as they are approved, they can become available to users, eliminating the need to move the documents from the Graduate School to the Library. This change in workflow can also eliminate the time delay previously caused by bounding and cataloging them prior to providing access. The most tenuous and highly emotional service libraries provide to ETDs is archiving. Because not enough time has elapsed to prove that digital documents can live for decades in publicly accessible digital libraries, the uncertainty of online archives causes great unease to many. Libraries must, therefore, be careful about security and back-ups. Another role that the library plays can be to put a prototype in place. While the Graduate School is nurturing the policy evolution among the academic community, there is a model developing to meet the needs of the academy. Establishing an ETD project Web site that documents the evolving initiative, listing active participants, providing a sample submission form and potential policy statements can do this. These statements might address levels of access, copyright statements, and link to existing ETD initiatives. When a university is adopting policies about ETDs it is helpful to have a place where its students, faculty, and administrators can see what an electronic library of digital theses and dissertations might be. Many graduate students are anxious to participate in an ETD initiative and the library's Web site can take advantage of their enthusiasm. Invite students who have complete their TDs to submit them electronically. This will build the initial ETD database and test the submission form as well as give the Graduate School personnel opportunities to compare the old and the new processes with somewhat familiar TDs. The library is also in a position to offer graduate students the incentive to participate. Most libraries collect binding fees so that theses and dissertations can be bound uniformly. Archiving fees can replace binding fees when ETDs replace paper TDs. However, the library may wish to offer to eliminate this fee for the first (limited number of students) who submit ETDs instead of TDs. Archiving Electronic Theses DissertationsEdit The best chance electronic information has of being preserved is when it is used online regularly and continually. As soon as it is not used, there will be trouble remembering the media that produced it and that made it accessible. As ETDs begin to join the library's traditional theses and dissertations, it is a good time to align the commitment and the resources to maintain these online information resources over time. A library's Special Collections Department and/or its University Archives are often responsible for storing and preserving theses and dissertations. Document parallel standards, policies, and procedures for electronic theses and dissertations (ETDs). The academic departments determine the quality of the work of their students, while the individual thesis/dissertation committees approve the student's work on its own merits. The Graduate School primarily oversees mechanical considerations, the purpose of which is to provide a degree of uniformity, to assure that each thesis or dissertation is in a form suitable for reading and/or viewing online and that it can be preserved. The University Archives ensures long-term preservation and access to this record of graduate students' research. With digital materials libraries give access and simultaneously prolong the life of the work, ensure the durability of the present through stability of the means of mediation. Factors Effecting ArchivingEdit 1) Access The first goal is to have all ETDs online and available all the time from a stable server. If necessary (depending on the capabilities of the server), some ETDs could be moved to a secondary server. Considerations for moving ETDs to a secondary server are usage (ETDs with fewest accesses) or age (oldest ETDs). Formats and file sizes probably would not be a factor in employing a secondary server, though extremely large ETDs may be prime candidates for separate online storage and access. If it becomes necessary to move some ETDs to a secondary server, programs would be written to trigger migration. Currently "age" would be easier to program, but in the future "usage" (actually, lack of use or few downloads) would be preferable characteristics for migrating ETDs to a secondary server for archiving. URNs would link migrated ETDs. URNs could be mapped to PURLs at some future date. 2) Security ETDs that been submitted but not yet approved should be frequently backed-up (e.g., hourly) if changes have occurred since the last back up; otherwise, generate a back-up programmatically every few hours. Make a weekly back up of ETDs in all directories (i.e., all levels of access). Make copies programmatically and transfer them to another server; make weekly back-ups to tape for off-line storage. Retain copies in quarterly cycles and annually archive to a CD-ROM. Authors cannot modify their ETDs once approved. Exceptions are made with proper approval. Viewers/readers cannot modify or replace any ETDs. Only in extreme circumstances would the system administrator make modifications to an ETD (e.g., when requested by the Graduate School to change the access restrictions or to activate or change email addresses). 3) Format Migration The library should share with the university the responsibility to guarantee that ETDs will be available both within and outside the scholarly community indefinitely. To keep ETDs reader-friendly and to retain full access will mean migrating current formats to new standard formats not yet known. This will be done through the cooperative efforts of the library (who maintains the submission software, the database of ETDs, and the secure archive) and university computing expertise. Standard formats should be the only acceptable files approved. Formats recommended for ETDs that may need to be converted to new standards in the future. Image Formats: CGM (.cgm); GIF (.gif); JPEG (.jpg); PDF (.pdf); PhotoCD; TIFF (.tif) Video Formats: MPEG (.mpg); QuickTime – Apple (.qt and .mov); Encapsulated Postscript (.eps) AIF (.aif); CD-DA , CD-ROM/XA (A or B or C); MIDI (.midi); MPEG-2; SND (.snd); WAV (.wav) Text Formats: ASCII (.txt); XML/SGML according to the document type: "etd.dtd" (.etd) ETD-ML Authoring Formats: Authorware, Director (MMM, PICS) Special Formats: AutoCAD (.dxf); Excel (.xcl) Next Section: Intellectual Property Rights Universities/Intellectual Property Rights Whether an author is creating an electronic or paper thesis or dissertation, it does not change their rights and obligations under the law. While university policies vary, it is the custom that the person who creates a work is the owner of the copyright. Therefore, the author of an electronic thesis or dissertation is the copyright holder and owns the intellectual property, their ETD. Within the United States, authors have rights protected by law particularly US Code, Title 17, especially section 106. Authors get to decide how their works will be reproduced, modified, distributed, performed in public and displayed in public. An author may use another's work with certain restrictions known as "fair use" (US Code, Title 17, sect. 107). The four factors of fair use that must be considered equally are: (1) purpose and character of use; (2) nature of the copyrighted work; amount and substantiality; and (4) effect. In the United States, libraries are also considered in the same copyright law under section 108. [For further explanations, see http://www4.law.cornell.edu/uscode/17/ch1.html] The owner of an ETD, as explained above usually the student, must take direct action if an ETD service is to be provided. The wording that is agreed to in writing, by student authors as well as the faculty working with them, must make clear how ETDs are handled. The wording used at Virginia Tech is one model. In the approval form used for this purpose, the following is agreed: The student certifies that the work submitted is the one approved by the faculty they work with. The university is given authority, directly or through third party agents, to archive the work and to make it accessible, in accord with any access restrictions also specified on the form. This right is in perpetuity, and in all forms and technologies that may apply. Next section: Publishers Universities/Publishers A continuing topic of discussion in the ETD community, including Graduate School administrators, research faculty, and librarians, is the question of "prior publication." That is, whether publishers and editors of scholarly journals view electronic theses and dissertations that are available on the Internet and through convenient Web browsers as being published because they are so readily and widely available. John Eaton, Dean at Virginia Tech's Graduate School, surveyed graduate student alumni in 1998 and 1999 and he asked about publishing articles derived from their ETDs. One hundred percent of those who had successfully published had not had any problems getting published because their theses or dissertations were online and readily available on the Internet. By looking at the results Joan Dalton's 1999 survey of publishers and Nan Seaman's 2000 survey as well as at Eaton's surveys of graduate student alumni, the ready availability of ETDs on the Internet does not deter the vast majority of publishers from publishing articles derived from graduate research already available on the Internet. [See "Do ETDs Deter Publishers? Coverage from the 4th International Symposium on ETDs," Gail McMillan. College and Research Libraries News, v. 62, no. 6 (June 2001): 620-621. http://scholar.lib.vt.edu/staff/gailmac/publications/pubrsETD2001.html] Several publishers have also attested to this and these statements are available in a variety of sources such as http://www.ndltd.org/publshrs/index.html. (others?) At the Cal Tech ETD 2001 conference, Keith Jones from Elsevier stated emphatically that his company encourages its authors to link their articles in Elsevier journals to their personal Web sites and also authorizes faculty members' departments to provide such links. Jones reported that Elsevier understands the importance of getting new authors such as graduate students to publish in his journals early in their careers because they are then likely to continue to publish with the same journal. He also pointed out the publishing in an Elsevier Science journal is an important source of validation for academics so that the subsequent availability of those articles from other non-profit and educational sources is not a threat. Next section: Human resources and expertise needed for an ETD program Universities/Human resources and expertise needed for an ETD program While each university's situation with regard to personnel will vary, it has been shown that ETD programs do not require a large contingency of expert professionals to initiate the program or to maintain it. With a portion of the time of one librarian and one programmer, the Virginia Tech library established the ETD Web site and documented the sequence of events that lead to the computer programs for each stage of acceptance, storage, and access, and implemented the procedures for the university. One step that should not be overlooked is to involve every person that had a role in traditional theses and dissertation handling. This includes, but is not necessarily limited to. Graduate School personnel who, for example, receive the TDs as well as the various forms and payments, and who may be responsible for approving the final document Library personnel such as the University Archivist, reference librarian, cataloging librarian, binding clerk, and business services personnel who, for example, are responsible for the long term preservation and access as well as those responsible for processing the microfilming invoices LIBRARY STAFFING: programmer, student assistant, faculty liaison A programmer may spend one-half to one hour per day during non-peak periods on maintenance and development. During peak periods this person may spend eight hours per day on problems, development, and system improvements. One or two student assistants are helpful. One student who knows programming may spend a maximum of two hours per week during non-peak times, and up to ten hours per during the periodic submission/approval peaks. Another student assistant would work face-to-face with graduate student authors to train them and to provide assistance with word processing and PDF software. This assistant might also maintain the training materials, handouts, and Web pages, including instructions for preparing and submitting ETDs. A faculty member from the library would supervise staff; draft policies; prepare budgets; collaborate with system maintenance and developers; and monitor workflow. This person would also be a liaison to faculty, staff, students, departments, and colleges to help them to become familiar with the processing, accessing, and archiving of ETDs. The faculty liaison from the library would also conduct workshops, write articles, participate in graduate student seminars, prepare handouts and Web pages, as well as collaborate with other universities and libraries. Next section: Sources for funding Universities/Sources for funding The aim of the ADT (Australian Digital Theses) Program is to establish a distributed database of digital versions of theses produced by the postgraduate research students at Australian universities. The theses are available worldwide via the web. The ideal behind the program is to provide access to and promote Australian research to the international community. The ADT concept was an initiative of 7 Australian university's libraries in association with the Council of Australian University Librarians (CAUL ). The ADT model was developed by the 7 original project partners during 1998-1999. The program was then opened up to all CAUL members (all Australian universities) in July 2000. The original 7 partners will continue to guide and advise the national group in their role as the ADT Steering Committee. The initial project was funded by an Australian Research Council (ARC) - Research Infrastructure Equipment and Facilities (RIEF) Scheme grant (1997/1998). The ARC is the peak Australian government research funding body and the grant to establish the ADT was a result of a successful application made by the group of 7 above. It was recognised at the outset that recurrent funding was going to be problematic, and that the model to be developed had to take this into consideration. The model developed is essentially self sustaining, with only a small commitment of resources required. The original funds were used to create such a self supporting distributed and collaborative system. The software used is generic, and designed to be easily integrated at ADT member sites. Once installed, students can either self submit, or seek support free from the library. Theses are mounted on local servers and require a minimum of maintenance. The central ADT metadata repository is searchable and is created automatically from rich DC metadata generated from the submission/deposit form. The metadata is gathered automatically using a metadata gatherer robot. The idea behind the ADT is that producing research theses is normal business for universities and a commitment to include a digital copy to the ADT requires minimal resources. In fact, digital copies are much cheaper to produce that the traditional paper bound versions. It must be noted that institutional membership and individual contributions to the ADT are voluntary, and will remain so for some time to come. Each NDLTD member has to apply to its own national, regional and community funding agencies. However, recurrent funding will probably be an ongoing issue unless a commitment is made at the government level to support such initiatives. In the broader context, ongoing funding for an international community body, which the NDLTD now is, will be difficult to achieve. A possible solution would be for the NDLTD to seek Non-Government Organization [NGO] status and thus secure ongoing funding from UN instrumentalities such as UNESCO. Such funding is critical to ensure the good work already achieved by the NDLTD in bringing together a large and disparate number of institutions from across the globe bound by an ideal to promote, support and facilitate open unrestricted access to worldwide research contained in theses, and to share without cost/commercial barriers experiences, support and guidelines with the world community in an open, transparent way. Next section: Costs Universities/Costs Evaluation of Costs at Virginia Tech Libraries usually have some of the infrastructure already in place to handle ETDs, especially if they are already providing access to electronic journals or digital images. However, many administrators like to have data about costs associated with an initial budget, and for this reason the following costs for personnel, hardware, and software were established by Virginia Tech's Digital Library and Archives. Keep in mind that existing personnel and hardware can be commandeered for the ETD prototype and that frequently software is available on the Internet as shareware. This is how the Virginia Tech initiative began in 1995 and continued into the first year of required ETDs, 1997. [taken from http://scholar.lib.vt.edu/theses/data/setup.html] These estimates assume that the university/library would adapt existing programs, scripts, Web pages, software, etc. that has been developed by the VT ETD Team. No additional equipment, software, or staff was necessary for the VT library to begin or to maintain the ETD system and services. $24,000 STAFF $13,500 EQUIPMENT $12,000 SOFTWARE $49,500 TOTAL The estimated costs below show that for about $49,500 a library could replicate the Scholarly Communications Project ETD library if it adapts what VT has already developed for the NDLTD. If another university/library must also provide support to students like the New Media Center does, then the costs are greater for training, equipment and software. STAFFING: $24,000 System's Administrator: .25 FTE at $6000-$6,600/yr In Virginia, pay band 5: $30,000/annual; $14/hr We estimate the our Sys Admin spends .5 - 1 hr/day during non-peak times on maintenance and development. During peak periods he may spend 8 hrs/day on problems, development, and system improvements. Student assistant: .25 FTE at $2900.00-$4500.00/yr knowledge of pdf and html desirable; train to use ftp and telnet if necessary desirable: programming experience. salary depending on skill level at VT would be $6.00 - $10.00/hr We estimate that our student (who knows programming) spends a maximum of 2 hrs/wk during non-peak times, and up to 10 hrs/wk during the periodic peaks. Faculty liaison: .25 FTE @ $50,000 = $12,500 Supervise staff; draft policies; prepare budgets; collaborate with system maintenance and developers; monitor workflow. Work with faculty, staff, students, departments, colleges to become familiar with the processing, accessing, and archiving of ETDs. Conduct workshops, write articles, participate in graduate student seminars, prepare handouts and Web pages. collaborate with other universities and libraries. EQUIPMENT—SERVING ETDS: $13,5000 server: $5,000 dual 3 GHZ processor, 2gb RAM, at least 20 to 80 GB hard drive Originally, we did not purchase special equipment for ETDs, but incorporated this additional responsibility into the original server, a NeXt3.3 running HP. In Sept. 1997 we purchased a Sun Netra. tape drive for back-ups : $3500 ( At VT we use our Computing Center's backup service) increase of 4-5 Gb disk space per year Each ETD requires an average of 4.5Mb 2 workstations : $5,000 SOFTWARE—SERVING ETDS: $250-$12,000 $50 Red Hat 3 advanced Server (free download for software and $50 education entitlement cost) $10,000 InfoSeek $1000/yr Server Equipment Maintenance fee $40 x 2 Acrobat* $50 x 2 Word Processor* $150 x 2 Photoshop* * Educational pricing available for this software At VT we have a New Media Center with staff, software, and equipment that can help students with every stage of ETD preparation and submission. If you don't have such a facility, consider: STUDENT SUPPORT: EQUIPMENT $1000 Desktop with monitor; CD-RW/DVD; (internet access) $150 scanner $3000 2 printers: LaserJet and ColorPrinter $300 digital camera STUDENT SUPPORT: SOFTWARE STUDENT SUPPORT: STAFF Training and assistance for equipment and software Maintenance of training materials: Web pages: instructions and handouts Cataloging Costs These may not change from the costs of cataloging traditional theses and dissertations. One advantage is that for the same costs, more information can easily be added to the bibliographic record because of the ease of copy-and-paste features of word processors. For example, include the abstract in online catalog records for ETDs and index this MARC field to enhance findings through keyword searching. Another advantage of ETDs is that there are no longer the fees associated with binding, security stripping, labeling, shelving ($.10/vol. estimated), circulating ($.07/vol. estimated), etc. The Virginia Tech Library saved about 66% of the processing costs because of the greatly reduced handling of ETDs; costs dropped from $12 per TD to $3.20 per ETD. Next section: Processing charges Universities/Processing charges Many universities that have paper submissions of theses and dissertations collect funds for processing these works. Usually the funds are collected from students. In some cases they might be collected from grants, or from a sponsor (e.g., from the National Library, with regard to works in Canada). These funds commonly are called "binding fees" or are given other names designating their use for processing. A typical fee might be $20 or $30 per work. In the case of transition to ETDs at Virginia Tech, there were changes to these fees. First, in 1996, when submission of ETDs was encouraged but voluntary, the processing fee was waived for those students submitting an ETD instead of a paper document. This provided a monetary incentive to move toward ETDs. However, in 1997, with a requirement in place, the processing fee was re-instituted. However, it was renamed "archiving fee". Thus, funds were collected from students: to help defray the costs of the ETD program; to provide a pool of funds to archive and preserve works throughout their lifetime, allowing for migration to new media types (e.g., automatic copying to newer types of storage media, such as from one online disk to a more modern disk), as well as conversion to new formats (e.g., moving from SGML to XML, or from one version of PDF to a newer version). Next section: Budgets Universities/Budgets The estimated cost of a project to electronically produce and distribute theses varies based on a number of factors such as: the expertise and competence of your team; the technology used; the volume of documents to process; and the cost of living in your country (human resources, computer equipment, communications, etc.). In these circumstances, it is impossible to provide any budgetary estimates, at least without knowing the operating conditions and elements of a particular situation. Giving an estimate, even in the broad sense of an order of magnitude, would be of no use. Nonetheless, this section will allow you to determine your expenditure budget by providing a list of budgetary items that must be foreseen. The expenditure budget of an electronic theses project involves four principal modules. These are: the start-up; the implementation of thesis production and distribution services; communications; and student training. Start-upEdit Start-up costs are related to what you require to begin a project to electronically produce and distribute theses. Besides the personnel, who constitute the most important element in any electronic thesis distribution project, the start-up costs principally relate to infrastructure and training. Since the network infrastructure is usually provided by the host institution, this expense is omitted from the following table. Adoption/creation of a procedure for processing theses Development of tools, programmes and scripts The number of professionals required is linked to the volume of theses to be processed and to the variety of disciplines taught at your institution Production of theses, assisting students, and routine tasks The number of technicians required is linked to the volume of theses to be processed Communication with the university's administration, and with external partners Materials (the actual server, UPS, back-up unit, etc.) Software (operating system, search tool, etc.) Server site This is usually provided by the University Computer equipment for processing and for management Materials (PC, printer, etc.) for all members of the team A workstation (PC) to test different software and to receive students' files (by ftp) in a secure manner Software chosen to suit the chosen technology and assembly line (Office suite, Adobe, XML software, HTML editor, etc.) Training team members Organized training or self-instruction Manuals and documentation Table 1– Start-up for an electronic thesis production and distribution project Implementation of production and distribution servicesEdit The resources required to create an assembly line for producing theses are related to the technologies used as well as the expertise and experience of team members. For instance, the choice to create valid XML documents from the files submitted by students, even if many useful XML tools are now available at reasonable prices, necessarily implies larger investments. Nonetheless, it is important to remember that more costly technological solutions may provide significant benefits to other electronic publication projects within the same institution. For instance, they might be useful for publishing electronic journals or for digitizing other types of collections. Table XXX lists the general stages needing attention when establishing a budget, regardless of the technologies chosen. Implementing production and distribution services Analysis of needs Choice of a metadata model (including a permanent referencing system) Integration of the metadata creation or generation process Testing different software Planning and testing of the production assembly line Creation or adoption of a follow-up tool for production work Formal implementation of the service Install and set parameters for the search tool (full-text and metadata) Create and manage the Web site Create the site's architecture Compose the site's information and home page Conceive and create the site's graphic signature Create the navigation interface Ensure Web referencing (Web search tools, indexes, other ETD sites, etc.). Create mechanisms for managing access Install a tool to measure visits to, and usage of, the site Table 2 – Implementing production and distribution services CommunicationEdit The communications plan often makes the difference between successful projects and aborted ones. A communications plan must be drafted when the technical processes are in place and the university's governors approve the project. Worktime and other resources must be budgeted for the creation and implementation of a communications plan. The principal tasks usually required to this end are listed in Table XXX. Drafting the Communications plan Organizing and holding information sessions with the university's professors, researchers and students Writing the content of information documents Meeting with journalists from the university or from the city's newspapers. Tools for communicating and promoting the project Posters, brochures and other documents E-mails to professors and students Putting information on-line on the project's Website Table 3 – Communication Plan Training StudentsEdit Distributing theses on the Web is in itself an effective means of valuing an institution's students and research. Nevertheless, emphasis must be placed on training students so that they can master the tools of document creation that allow them to circulate their research results. The use of new information technologies has become essential for university studies. Whether it involves searching databases and the Internet, or using software to help manage and assess data and present findings, the ability to work easily with these tools will be of daily use for students throughout their future careers. Many universities undertaking electronic theses projects have uncovered serious skills gaps in terms of creating tables, of integrating figures and images, and of using functions that automatically generate tables of contents or lists. It is clear that basic training must be offered to better prepare students for the process of writing important documents like theses. Student training must be offered through many forms such as workshops, on-line tutorials and personal consultations. One can decide to offer one or another or all of these forms of training. In many institutions, the preparation and offer of training to students results from the collaboration of many different units: the faculty of graduate studies, the libraries and the information technologies services. Training Students Planning and drafting Determining the students' current state of knowledge/skills Analyzing needs Planning the various training modules Preparing and drafting the pedagogical tools Preparing workshops, examples and exercises Organizing the workshops (selecting classrooms, installing software, etc.) Hiring assistants (eventually students) for the workshop's "hands-on" period On-line tutorials Adapting workshop content and materials for the Web Personalized consultation service Time taken by the professionals and technicians to answer student questions (by e-mail, telephone, and eventually in person). Table 4 – Training Students Next section: Plagiarism Universities/Plagiarism The issue of plagiarism often arises among the arguments used to express skepticism with regard to putting theses online. In short, many people tend to think that because a digitized thesis is easily copied in part or in whole, it can be easily plagiarized. Consequently -so goes the reasoning - it is better to keep theses offline. The argument is largely false and can be refuted fairly easily. To begin with, it is easy to recall that the invention of the Philosophical Transactions (1665) by Henry Oldenburg, the Secretary to the Royal Society in London, was motivated by the issue of intellectual property. Oldenburg reasoned that if the research results of Scientist X. were printed in a journal (after being certified as being of good quality and original) and that journal was made widely available through the multiplication of copies, then Scientist. X would have a better chance to lie ownership claims than if he/she held back these results. By apparently giving away the results of his/her work, a scientist ensures his/her intellectual property most effectively. The ability to compare new results to already published work makes plagiarism a very risky business at best. Theses are not so well protected at present. Widely dispersed across many institutions in many countries (and languages), they are so poorly catalogued on a national or international basis that they often disappear from sight. This means that someone taking the time to read a thesis in a remote university in a country where the cataloguing is poorly organized may well be able simply to use that thesis and make it pass for one's own. Occasionally, such cases emerge in the literature, even in the United States despite the fact that the cataloguing of theses is most advanced in that country. The paradox of placing theses on line, especially if these theses can be harvested through some technique that involves full text searching can help identify analogous texts rather easily. As a result, far from placing the digitized theses at risk, putting them on line in a manner that optimizes their access, irretrievability and, therefore, visibility, offers a very efficient way to protect intellectual property and prevent plagiarism. In fact, it would probably be relatively easy to design software that could make periodic sweeps through inter-operable theses collections according to ever more sophisticated algorithms in order to ferret out such possible forms of plagiarism. With many languages involved, it is clear that no perfect solution will ever appear; however, those theses available online will be more protected than theses that remain poorly catalogued and are not readily available outside the institution from which they are issued. In effect, putting theses on-line amounts to rediscovering Oldenburg's wisdom when it comes to scientific intellectual property. It provides what could arguably turn out to be the best deterrent to plagiarism, wherever it may arise. The more theses appear on line, the fewer will be the chances of carrying on successful plagiarism. Next section: Assessment and Measurement Introduction Universities/Assessment and Measurement Introduction I shall consider assessment to include the gathering of information concerning the functioning of students, staff, and institutions of higher education. The information may or may not be in numerical form, but the basic motive for gathering it is to improve the functioning of the institution and its people. I used functioning to refer to the broad social purposes of a college or university: to facilitate student learning and development, to advance the frontiers of knowledge, and to contribute to the community, and the society. (Alexander W. Astin, Assessment for Excellence, 1991, p. 2) Electronic theses and dissertations are not only products of student research, but also marks of the students' preparation to become scholars in the information society. In pursuit of this broader social purpose, this section of the Guide focuses on two separate but related topics: assessment and measurement. Both are important components of an institutional ETD program. The role of assessment in an ETD program is to understand whether the goals and objectives of the program are being met, while issues of measurement focus on the production and use of an institution's ETDs. Assessment of a program's goals and objectives yields information that may be of great value to policymakers and administrators. Higher education institutions are increasingly asked by their boards, by state legislators, and by federal government agencies to demonstrate the effectiveness of their programs, which has led to the development of many assessment programs on campuses. Assessment is often used to provide trend data or to assist in resource allocation, with data helping to make a case for the viability of an ETD program or to determine which parts of the program require the most (or least) resources. Ideally, an ETD assessment program will have ties to overall campus assessment activities. However, whether or not accountability to administration or government is a driving factor in developing assessment for an ETD program, any new effort can benefit from a systematic way of measuring what it is achieving. Such measures as the number of ETDs produced at an institution each year or the number of times a dissertation has been downloaded from the institution's web site are frequently cited to demonstrate the success of an ETD program. The goals of this section of the Guide are: 1. to encourage those involved in developing and implementing ETD programs to think at early stages about broad questions of assessment, and 2. to familiarize individuals working on measurement of ETDs with developing national and international initiatives that are developing standard ways of measuring the use of electronic information resources. Next Section: Types of Assessment Universities/Types of Assessment Assessment takes two basic forms, based on the point at which the evaluation is done: Formative evaluation takes place during the development of a program. It is used to help refocus activities, revise or fine tune a program, or to validate the success of particular activities. Formative assessment in an ETD program may help diagnose whether training sessions are useful to students, whether the submission process worked efficiently, whether the students learned about electronic scholarly communication issues, or whether the students find what they learned in order to produce an ETD helpful in their employment. Summative evaluation is used to examine the program at its completion or at the end of a stage. Since an ETD program is ongoing, summative evaluation can be used on an annual basis to yield statistics that can be compared from year to year. Next Section: The Assessment and Measurement Process Universities/The Assessment and Measurement Process This chapter of the Guide is intended for those who have responsibility for implementing an institutional or departmental ETD program. The process of creating a successful assessment and measurement requires planning, creating goals and objectives, choosing types of measures, and deciding what type of data to collect. PlanningEdit Some campuses form an ETD committee or team, consisting of representatives from the Graduate School, the faculty, the library, the computing center, and other relevant units on campus. As part of their overall planning for the development of an ETD program, the committee should make explicit their goals for assessment and measurement of the program and put in place mechanisms to collect data related to the goals. The data collection could be gathered from web statistics packages, from student surveys, or from interviews, depending on what type of information meets the assessment's goals. Several units of the university—including academic departments, the graduate school, information technology services, and the library—are involved in any ETD program. In developing an assessment plan, it is useful to think convergent. Are there particular things that would be useful for several of these constituents to know? This will leverage the value of the assessment and may allow for joint funding and implementation, thereby spreading the costs and the work. During the planning process, the ETD committee may focus on a variety of issues, including: the impact that the ETD program is having on the institution's reputation; the degree to which the ETD program is assisting the institution in developing a digital library; and the benefits to and concerns of students and advisors participating in the ETD program. Once the focus of the assessment and measurement activities is identified, the ETD committee should assign responsibility for development and implementation to another team. This team should include assessment and measurement experts from institutional units such as an institutional planning office, a survey research institute, or an instructional assessment office. Creating Goals and ObjectivesEdit A necessary prerequisite to assessment is a clear understanding of the ETD project's goals. Each institution's assessment plan should match the goals and objectives of the institutional ETD program, which may have a broader scope than the simple production of electronic content. If an ETD program does not have clearly defined goals, an excellent resource is the NDLTD web site. This site includes the goals of the NDLTD, which can be adapted to local needs. These goals include: Improving graduate education Increasing availability of student research Lowering costs of submission and handling of theses and dissertations Empowering students Empowering universities Advancing digital library technology Within each of these broad areas, many types of measures can be developed to help evaluate whether the ETD program is succeeding. Choosing Types of MeasuresEdit Institutions have many choices in what they measure in an ETD program. After aligning the assessment goals with the institution's goals, the assessment plan must describe what types of measures are needed for various aspects of the program. McClure describes a number of categories of measures, including those that focus on extensiveness, efficiency, effectiveness, service quality, impact, and usefulness. (Charles R. McClure and Cynthia L. Lopata, Assessing the Academic Networked Environment, 1996, p. 6) An extensiveness measure collects data on such questions as how many departments within the university are requiring ETDs or how many ETD submissions are made each year. This type of data lends itself to comparison, both as trend data for the individual institution and in comparisons to peer institutions. An efficiency measure collects data to compare how life cycle costs of ETDs compare to those of print theses and dissertations. For example, Virginia Tech includes such a comparison on its Electronic Thesis and Dissertation Initiative web site. Effectiveness measures examine the degree to which the objectives of a program have been met. For example, if an objective of the institution's ETD program is to empower students to convey a richer message through the use of multimedia and hypermedia, data can be collected that displays the proportion and number of ETDs employing such techniques by year. If an objective of the program is to improve students' understanding of electronic publishing issues, the institution can measure such understanding prior to and after the student produces an ETD. Measures of service quality examine whether students are receiving the training and follow-up assistance they need. Deciding What Data to CollectEdit Frequently, discussions of how to assess electronic information resources are limited to defining ways of counting such things as searches, downloads, and hits. These measures are certainly useful, but they provide a limited view of the overall value of electronic information resources. Many vital questions cannot be answered with statistics about searches, downloads, and hits: Can users access more information than in the past due to availability of information resources online? Has the availability of electronic information resources improved individuals' productivity and quality of research? Has the availability saved them time? Collection of data for assessment should be designed to answer some of these questions, to address the educational goals embedded in an ETD program, and to gauge whether or not those goals have been achieved. Decisions about data collection are also informed by the institutional mission and goals. For example, if the institution is interested in increased visibility both nationally and internationally, then statistics on downloads of ETDs by country and institutional IP address could be useful. Examining who is using an institution's ETDs by country and by amount of use would also be a valuable gauge of impact. As recommended systems develop, this area may grow in importance. There are a number of questions related to users that are possible targets for assessment. These include: Are students achieving the objectives of the ETD program? Are students using tools such as Acrobat appropriately and efficiently? Has the availability of student work increased? Do students have an increased understanding of publishing issues, such as intellectual property concerns? In addition, a number of questions related to student satisfaction could be addressed in data collection plans. These may include: Were students satisfied with the training or guidance they received to assist them with producing an ETD? Did the availability of their dissertation on the web assist them in getting a job? Are they using the technology and electronic authoring skills they learned in their current work? Usefulness to students may be a factor of the availability of their ETD on the web, assisting employers in gauging their area of research and the quality of their output. Availability may also lead employers to contact students for openings that require a particular skill set. And students may find that the skills of preparing an ETD and the framework of issues associated with the ETD, such as intellectual property issues, is useful in their places of employment after graduation. The usefulness of an ETD program to students, faculty, and others may be an important factor to measure in order to gather data that can be conveyed to administrators and funding agencies. This data might best be collected six months to a year after the completion of the ETD. Finally, the ability for faculty and students around the world to easily examine the dissertation and thesis output of a particular department may provide a new dimension to rankings and ratings of graduate departments. Monitoring national ratings in the years pre- and post-implementation of an ETD program could be useful, although may be only one factor in any change in ranking or rating. Next Section: Measuring Production and Use of ETDs: Useful Models Universities/Measuring Production and Use of ETDs: Useful Models In addition to assessing some of the programmatic goals of the ETD program, institutions will want to have some basic assessment measures in place to document the production and use of their ETDs. The work done at Virginia Tech's Electronic Thesis and Dissertation Initiative can provide a model for other institutions. Using web statistics reporting software, Virginia Tech monitors a number of measures for their ETDs, including: the availability of campus ETDs multimedia in ETDs which domains within the United States and abroad are requesting ETDs requests for PDF and HTML files distinct files requested distinct hosts served average data transferred daily In compiling its counts, Virginia Tech eliminates everyone working on ETDs at the institution, including the Graduate School. They try to eliminate repetitive activity from robots and other sources of that type as well. Virginia Tech is also working on the compilation of an international count of ETDs produced in universities. Institutions must decide whether they will report their ETD collections and usage separately, in conjunction with other campus web site usage, or in conjunction with other electronic resources managed by the library. Now, when practices and standards for gathering these statistics are evolving, institutions may need to collect the information and report it in conjunction with more than one type of related collection. In any case, institutions should keep abreast of national and international initiatives that are seeking to define and standardize statistical reporting of the number and use of electronic information resources. Several projects currently focus on collection of statistics related to information resources: The Association of Research Libraries (ARL) collects statistics from libraries of large research universities in North America and has been working on several initiatives that explore data collection related to digital information. In particular, one of its New Measures initiatives, the E-Metrics Project, is recommending a particular set of measures and defining their collection. The International Coalition of Library Consortia's (ICOLC) has created Guidelines for Statistical Measures of Usage of Web-Based Indexed, Abstracted, and Full-Text Resources. They encourage vendors of electronic information products to build into their software statistical report generation that will meet the ICOLC Guidelines, promoting comparability of products. In Europe, the EQUINOX Project, funded under the Telematics for Libraries Programme of the European Commission, addresses the need to develop methods for measuring performance in the electronic environment within a framework of quality management. Their products include a consolidated list of data sets and definitions of terms. Library Statistics and Measures, a web site maintained by Joe Ryan of Syracuse University, also provides a useful set of links to resources on library statistics and measures. Another important type of post-processing is the extraction of statistical information from metadata sets. For administrative purposes, institutions may be interested in the number of ETDs supervised by each professor, the keywords most used , the month(s) in which more ETDs are submitted, etc. Usually, relevant metadata are extracted from the ETD database and processed using specialized tools like Microsoft Excel. The access to the database can be done using either ODBC drivers or specialized middleware utilities. Next Section: Statistics and Usage Universities/Statistics and Usage Gathering data about ETDs can be done through online surveys and log file analysis. Online surveys can also be used to gather data from graduate student authors submitting ETDs and from ETD readers accessing them. An easy to use online survey system is available to NDLTD members at http://lumiere.lib.vt.edu/surveys/ Ask graduate student ETD authors a variety of questions about the process as well as about support services and use this data to improve resources and services. Query readers to discover their reasons for accessing ETDs, the results of their use of ETDs, as well as to improve digital library resources and services. Questions for student authors may be designed to improve the process of preparing and submitting electronic documents. Sample questions might include: While preparing your ETD, where did you find answers to your questions? If you consulted the VT ETD Web site, please indicate if the site was useful. If you attended an ETD workshop, please indicate if you found the workshop useful. If you used a [particular] computer lab, please indicate if the staff was helpful. How many ETDs did you consult while preparing your ETD? Compared to what you expected, how difficult was it to create a PDF file? My computer is a [PC, Mac, Unix, other]. Where were you when you submitted your ETD? Compared to what you expected, how difficult was it to submit your thesis/dissertation electronically? Within the next 1–2 years, what do you intend to publish from your ETD? If you restricted access to your VT ETD, on what did you base your decision? Please include any comments or questions that you have about ETDs. In the online environment, usage is similar to but not equal to library circulation and re shelving statistics. Nevertheless, to report usage, institutions should also report numbers of available ETDs so that downloads can be viewed in the context of what was available at a given point in time. It would be beneficial if all libraries providing access to ETDs would capture and report similar data about usage, but at this time we are not doing this. Gail McMillan has periodically surveyed and reported numbers of ETDs available from the NDLTD. Her data is based on institutions reporting numbers of ETDs available through their institutions, but the NDLTD has not begun to report institutional use of or access to their ETD collections. See http://scholar.lib.vt.edu/NDLTD/ Next Section: Measurement in Related Contexts Universities/Measurement in Related Contexts Another important type of post-processing is the extraction of statistical information from metadata sets. For administrative purposes, institutions may be interested in the number of ETDs supervised by each professor, the keywords most used, the month(s) in which more ETDs are submitted, etc. Usually, relevant metadata are extracted from the ETD database and processed using specialized tools like Microsoft Excel. The access to the database can be done using either ODBC drivers or specialized middleware utilities. For a broad view of counting information, two projects are widely regarded as providing interesting models and data. Peter Lyman and Hal R. Varian's How Much Information project at the University of California, Berkeley is an attempt to measure how much information is produced in the world each year. The OCLC Web Characterization Project conducts an annual web sample of publicly available web sites to analyze trends in the size and content of the web. Some programs that provide guidance or models for collecting institutional data in higher education are also available. These projects can provide definitions for data, survey questions, and descriptions of data collection that can be adapted for one's own institution. K. C. Green has been conducting the Campus Computing Project since 1990. His work charts the increasing use of technology on campuses. The TLT Group's Flashlight Program, under the direction of Steve Ehrmann, has developed a subscription-based tool kit that provides a large, structured set of assessment techniques and data collection models that can be adapted by individual campuses that want to study and improve the educational uses of technology. The Flashlight Program web site also includes valuable overviews of assessment issues and provides advice on deciding what to assess and how to develop questions. The Coalition for Networked Information's Assessing the Academic Networked Environment project provides case studies of campuses that implemented assessment projects. One of the participants in the CNI project, the University of Washington, has a rich set of assessment instruments and reports on its web site, UW Libraries Assessment. Next Section: Guidelines for Implementing an Assessment Program for ETDs Universities/Guidelines for Implementing an Assessment Program for ETDs The Coalition for Networked Information (CNI) sponsored a project, "Assessing the Academic Networked Environment," in which institutional teams developed and implemented assessment projects related to a variety of areas, including teaching and learning, electronic reserves, computer skills, and electronic library resources. From the project reports and informal feedback from the participants, CNI developed a set of guidelines for institutions engaging in assessment activities related to networks or networked information. The guidelines focus on the process of doing assessment in higher education. The suggestions can be applied directly to assessment projects for ETDs. Bring together an assessment team of individuals from various units on campus that can add useful perspectives and expertise; include, if possible, someone who specializes in assessment. Align the overall goals of the assessment initiative with the institution's goals and priorities. Gain support from the administration at as many levels as possible. Make a realistic determination of the resources (staff, time, equipment, and money) that are available for the assessment. Choose a manageable portion of the assessment project as the first implementation. Do not attempt to do a comprehensive assessment of campus networking on the first try. Consider using more than one assessment technique to measure the aspect of networking that you have chosen; particularly consider combining quantitative and qualitative approaches as complementary techniques. Identify carefully who are the audiences for the assessment reports. Examine what you might do with the information you collect, including improving services, seeking additional funding and determine whether your data will provide what you need for that objective. Refine assessment instruments on a periodic basis and incrementally add new components. Monitor the work of national groups such as ARL, EDUCAUSE, CNI, and the Flashlight Project to see whether materials they develop and guidelines they produce can provide a framework for your project. (Joan K. Lippincott, "Assessing the Academic Networked Environment," Information Technology in Higher Education: Assessing Its Impact and Planning for the Future, ed., Richard N. Katz and Julia A. Rudy. Jossey-Bass, 1999, pp. 21–35.) Next Section: Student Comments Universities/Student Comments In addition to gathering availability and usage data, online surveys can be used to gather information from readers to voluntarily agree to be surveyed by answering questionnaires. Some of the useful information that can be gathered includes. Sample questions for ETD readers might include: Where do you work/study? What type of computer are you using? What is the speed/type of connection are you using? Are you familiar with Adobe PDF? Are you familiar with online databases? If you are from a university, does your institution accept electronic theses and dissertations (ETDs)? If your institution does not accept ETDs, do you think it should? Have you ever submitted an ETD? (to find out if your readers are ETD authors past, present or future) For what purpose are you using this digital library? Did you download any ETDs? If you downloaded any ETDs, how did you find them? If you downloaded any ETDs, how easy was it to find what you were looking for? If you searched for an ETD, how fast was the response to your search request? How often do you plan to use Virginia Tech's ETD library? How often do you plan to use other ETD libraries? Comments from survey respondents Next Section: Resources List Universities/Resources List Association of Research Libraries. ARL Statistics and Measurement Program. http://www.arl.org/stats Association of Research Libraries. Supplementary Statistics 1998-99. Washington, DC: ARL, 2000. Astin, Alexander W. Assessment for Excellence The Philosophy and Practice of Assessment and Evaluation In Higher Education. American Council on Education/Oryx Press. 1991. Developing National Library Network Statistics & Performance Measures. EQUINOX: Library Performance Measurement and Quality Management System. http://equinox.dcu.ie/ International Coalition of Library Consortia (ICOLC). Guidelines for Statistical Measures of Usage of Web-Based Indexed, Abstracted, and Full Text Resources. November 1998. http://www.library.yale.edu/consortia/webstats.html McClure, Charles R. and Cynthia L. Lopata. Assessing the Academic Networked Environment. Washington, DC: Coalition for Networked Information, 1996. NDLTD: Networked Digital Library of Theses and Dissertations. http://www.ndltd.org/ Ryan, Joe. Library Statistics & Measures. http://web.syr.edu/~jryan/infopro/stats.html Technology in Higher Education and Web Studies: The Campus Computing Project. http://www.campuscomputing.net/ Coalition for Networked Information. Assessing the Academic Networked Environment. http://www.cni.org/projects/assessing/ Katz, Richard N. and Julia A. Rudy. Information Technology in Higher Education: Assessing Its Impact and Planning for the Future. New Directions for Institutional Research No. 102. San Francisco: Jossey-Bass, 1999. Lyman, Peter and Hal R. Varian. How Much Information? http://www2.sims.berkeley.edu/research/projects/how-much-info-2003/ Online Computer Library Center, Inc. Web Characterization Project. http://wcp.oclc.org/ TLT Group. Flashlight Program. http://www.tltgroup.org/flashlightP.htm University of Washington. UW Libraries Assessment. http://www.lib.washington.edu Virginia Polytechnic Institute and State University. Electronic Thesis and Dissertation Initiative http://etd.vt.edu/ Next Section: Policy Initiatives: National, Regional, and Local; Discipline specific; Language specific Universities/Policy Initiatives: National, Regional, and Local; Discipline specific; Language specific NationalEdit For universities, it seems most practical to participate in national ETD initiatives. For those initiatives it is advisable that the National Library, which is often in charge of archiving the country's literature, takes a leading role. We see those approaches in Germany within the Dissertation Online Initiative or in Canada. The national library as a central point, organising not only the archive structure but also the cooperation between universities may serve as a central entry point to the ETD initiative for interested parties. Principal tasks of a so called central coordination bureau for national ETD initiatives are: providing a coordination structure for the cooperation of universities for political, organisational, technological and educational issues and developments providing an organizational concept for funding local or regional initiatives, therefore negotiating and cooperating with national funding agencies defining special interest and working groups in which representatives from universities can participate organizing workshops for participating universities covering special topics in order to discuss and solve particular problems cooperating with the international initiatives, e.g. NDLTD, Cybertheses, MathDissInternational or PhysDiss. In France, a national program for ETDs has been initiated by the Ministry of Education. The electronic deposit shall be compulsory within the end of the year. The organisational scheme adopted is defined as follows: each university will be in charge of the conversion of its theses and dissertation into an archiving format (SGML/XML). associations of institutions may allow the mutualization of human and technical resources. a national institution, the Association des Bibliothèques de l'Enseignement Supérieur (ABES) (approximative translation: Association of Universitary Libraries) has been designated as the national central bureau. The following text was taken from Prof. Peter Diepold (Sourcebook for ETDs) In 1996 four German learned societies - comprising the fields of chemistry, informatics, mathematics, and physics - signed a formal agreement to collaborate in developing and using digital information and communication technologies (ICT) for their members, scientific authors and readers. The objectives of this collaboration were on a local level: to bring together the activities of individual - and often isolated - university researchers and teachers in the various academic fields nation wide: to join forces in voicing the interests and needs of scientific authors and readers toward the educational administration, granting agencies, research libraries, documenting agencies, publishing houses and media enterprises globally: to use the widespread international contacts of the learned societies to exchange concepts, development and solutions and adopt them to the specific needs within one's own field The initiative soon caught public attention, leading to the enlargement of the group. Since then, the learned societies in the fields of education, sociology, psychology, biology, and electronic engineering have also committed themselves to the advancement of the goals of the ``IuK- Initiative. (IuK stands for ``Information und Kommunikation in German). Funds were granted for three years by the Federal Ministry of Education and Research, `Bundesministerium für Bildung und Forschung (BMBF) (www.bmbf.de). One major project within this initiative was the Dissertation Online project, undertaken from April 1998 until October 2000. The activities of one of the workgroups led to a proposal to the German Research Foundation (DFG) to fund an interdisciplinary project to present dissertations online on the Internet, involving five universities (Berlin, Duisburg, Erlangen, Karlsruhe, and Oldenburg), and five academic fields, chemistry, education, informatics, mathematics, and physics.. DFG stands for ``Deutsche Forschungsgemeinschaft (http://www.dfg.de/) and is Germany's National Science Foundation. Funding was initially restricted to one year. The first phase started in the spring of 1998 and was terminated in March 1999 with a conference held in Jena, Germany, provoking much attention among librarians and academics. Though an infrastructure had been set up and a number of problems were solved, much remained to be done. Therefore a subsequent proposal to DFG was drafted. DFG funds were awarded for a second year, this time with a heavy emphasis on the collaboration with libraries and university computing centers. The project's research and development extended from May 1999 to October 2000. The overall volume of both grants was some US \$ 700,000. New participants in the second proposal are computing centers and German National Library, ``Die Deutsche Bibliothek (DDB) (http://www.ddb.de/) . The project was directed by Prof. Peter Diepold, professor of computer uses in education at Humboldt University, Berlin. (Email:[email protected]) Later, this project was made into a national initiative with the German National Library as leading partner, who established a buereau for coordination. Dr. Nikola Korb Die Deutsche Bibliothek DissOnline Adickesallee 1 Email:[email protected] RegionalEdit South America?? Discipline SpecificEdit Discipline specific initiatives focus on bringing researchers and scholars from single research fields together. In the past this has been a very successful way to establish active communities. These initiatives give members of those communities the benefit of making problems such as those that arise within global or national or generally spoken interdisciplinary approaches or even very small discipline specific problems easier to solve. Therefore discipline specific approaches may reach results faster and more easily than broader projects. PhysNet The PhysNet Service exists within the Physnet initiative. PhysNet - the worldwide Network of Physics Departments and Documents - provides a set of information services for physicists. PhysNet is a distributed information service. It uses the information found on the web-servers of the worldwide distributed physics institutions and departments of universities as a distributed database. The restriction to those professional institutions which are accepted by the learned societies ensure the quality and relevance of the offered information. PhysNet serves only professional specific information posted by the scientists themselves. Therefore PhysNet complements the services of commercial providers. All information of PhysNet is kept, stored and maintained by its creators at their local institution's server or individual homepage. The creators retain all rights to their data. PhysNet only gathers and processes the locally available information of physics institutions to make them globally accessible. PhysNet is a noncommercial service. The access to information offered by PhysNet is free to anyone. The aim of PhysNet is to provide a longtime stable and distributed information service for physics with the collaboration of many national and international societies and physics organisations. The PhysNet-Services are: PhysDep offers a set of lists of links to nearly all Physics Institutions worldwide ordered by continent, country and town. In addition, it offers a HARVEST-based search engine to search across the listed Institutions. PhysNet provides lists of links to document sources of the distributed Physics Institutions. Such document sources are, for example, preprints, research reports, annual reports, and lists of publications of local research groups and individual scientists. The service is also completed with a HARVEST-based search engine. Journals lists Physics-related Journals, which are available with free fulltext on the web. A list of 'EPS Recognized Journals' is given as well. Conferences, Workshops and Summer Schools offers a list of servers with Conference Lists in different fields of Physics. PhysJobs offers a list of links to various physics-related job sites on the web. It also implements a search facility to search for information on these sites. Education provides online educational resources for physics (e.g. Lecture Notes, Seminar Talks, Visualization and Demonstration Applets), listed by subject areas. Links lists further sources of physics information on the web and information services of other fields and disciplines. Services provides various related tools e.g. to enrich and improve homepages and document sources by adding correct MetaData according the international Dublin-Core standard. More specifically on the subspace PhysDoc of PhysNet: it is serving documents distributed around the world at Physics University departments. PhysDoc. It comes with link lists sorted by country and town/state and institution. The present content is about 100.000 documents or document lists (publication lists). A search engine is attached http://physnet.uni-oldenburg.de/PhysNet/physdoc.html which went into full operation in early 2002, which allows to search for metadata of the documents (author, title, fulltext, keyword). A specific tool of it is to search both in PhysDoc and in MPRESS, the respective distributed document system of Mathematics of the International Mathematical Union. The special appeal of Physdoc is that it serves a ranking, and allows to find the match of a keyword to PACS classification numbers and its respective counterpart in MSC, the respective mathematical classification scheme. This matching is not by trying to find the same words but by serving articles of the mathematics, which a physicist working in the respective field would search for. This is a major outcome of the research programme CARMEN. Responsible for the server of this initiative is: Prof. Eberhard Hilf Carl von Ossietzky University Oldenburg Department for Physics Institute for Science Networking Ammerländer Heerstraße 121 Fax.: +49 (0)441 798 3201 Email: [email protected] WWW: www.isn-oldenburg.de MathDissInternational The MathDissInternational project developed as a service within the MathNet initiative, set up in Germany at the Konrad Zuse institute for iinformation technology. Within the scope of the project MathDiss International, a permanent international online full-text document server for mathematical dissertations will be established. In this connection, questions concerning online presentation of the documents and the problems of long-term archiving (from TeX resp. LaTeX documents) will be considered. They include the question of how to homogenize such files in order to enable their later conversion into programming languages following XML. Furthermore, the expansion of research possibilities using online documents is being planned. Providing access to the tables of contents, lists of tables and illustrations and bibliographies on the LaTeX level is of top priority. Because of the structure of mathematical documents written in LaTeX we have a lot of high quality information which gathers dust in the archives without being used for the retrieval of scientific documents. This situation should and could be changed because LaTeX has become a widely accepted tool in mathematical literature. The project MathDiss International will be sponsored by the DFG: Deutsche Forschungsgemeinschaft (the German Research Foundation) for one year. At the end of that year, the results will be turned over to the State and University Library of Lower Saxony in Göttingen which has offered to provide the long-term support for the new Math-Net Services. The Project Program includes: Complete inclusion of the dissertations through metadata and the expansion of the research possibilities using the source code. Standardization of the input files in consideration of mathematical contextual structuring. Integration of the services in the Math-Net by the adaptation of the project results to the standards used there. The creation of forms of organization for the long-term safeguarding of the service. International marketing for the worldwide opening of the service. Tests on the conversion of mathematical dissertations into new mark-up languages, e.g. MathML. Responsible for this project are: Prof. Dr. Günter Törner and Thorsten Bahne at the mathematics department of the University of Duibsurg. Gerhard-Mercator-University Duisburg Department 11 - Mathematics Lotharstr. 65 Fon.: +49 (0)2 03 379 26 67 / 68 Fax: +49 (0)2 03 379 25 28 E-Mail: [email protected] / [email protected] CogPrints CogPrints is an electronic archive for papers in any area of Psychology, Neuroscience, and Linguistics, and many areas of Computer Science and Biology, which uses the self-archiving software of eprints.org The CogPrints project is funded by the Joint Information Systems Committee (JISC) of the Higher Education Funding Councils, as part of its Electronic Libraries (eLib) Programme. The arXiv.org e-Print archive (formerly xxx.lanl.gov) is a fully automated electronic archive and distribution server for research papers. Covered areas include physics and related disciplines, mathematics, nonlinear sciences, computational linguistics, and neuroscience. A Service of the Los Alamos National Laboratory and the U.S. Department of Energy. Language specificEdit Next Section: Policy Initiatives: The Case of France Universities/Policy Initiatives: The Case of France In France, policies concerning the electronic distribution of theses derive from two different sources: the public power, represented by the Ministry of Higher Education which, through the intermediaries of the Research Division (Direction de la Recherche) and the Library Division (Direction des Bibliothèques), has to date held the responsibility for the description and physical archiving of theses. the university establishments possessing the ability to grant doctorates, be they universities or the major schools. The regulatory framework was set by decree on 25 September 1985. The decree laid out the procedures for the submission, description and reproduction of theses or other works presented to obtain a doctorate. The current initiatives for the electronic distribution of theses are numerous and extremely varied, as much in their technical details as in their political ones. Over the past several months, there has been a tendency towards the coordination and grouping of initiatives. This has involved the creation of linkages and networks around establishments that have instituted technical archiving and distribution solutions. The French system is currently structured on three levels: A local level, based on the initiatives of certain producer institutions A regional level, where a number of research institutions have aggregated at a regional or thematic level A national level, based on the creation of a ministerial working group, following a circulated letter written by the Minister in October 2000. This working group must define a new regulatory framework. The Local Level: the policy for the electronic distribution of theses at the Université Lumière Lyon2 The basic principle of the programme for the electronic distribution of theses, which was developed by the Université Lyon2 in partnership with the Université de Montréal, is to employ standardized formats which provide the conditions for perennial archiving and which permit distribution that guarantees effective and total interoperability. In most cases this involves formatting, so as to structure all the documents. This formatting is performed with computerized tools, and is necessary for widescale distribution over the internet. he Université Lumière Lyon2 changed the conditions for the electronic production, submission and distribution of theses when it introduced its Thesis Charter. This Charter spells out official conditions doctoral students must accept. In return, the students can receive supervision and training in the use of text processing tools within their research group. This document guarantees the University's commitment to support young scholars, so that they benefit from the best conditions for creating, archiving and distributing their work when defending their thesis and after. The UNIVERSITÉ DE LUMIÈRE LYON2's THESIS CHARTER The Université de Lumière Lyon2's Thesis Charter is proposed in conformity with the decree of 3 September 1998. Its preamble restates the appendix of this decree (Model Charter) and fills out the text with the following conditions: 1) Every thesis is prepared within a research group linked to a Doctoral School. The doctoral school's particular role is: a) to give a student applying for admission to the doctorate all pertinent information on the following two subjects: first, the functioning of doctoral studies, the research supervisors and pertinent teams for his/her project as well as the follow-up he/she can expect; and second, the possibility for student aid, scholarships, bursaries, and partnerships with a company likely to provide him/her with the means to completer his/her project; b) to approve the agreement between the student and the supervisor for the preparation of a thesis; c) to coordinate the academic training dispensed in order to obtain the DEA as well as during the years of doctoral studies; d) to offer doctoral students a larger academic environment than their thesis specialization, for instance by organizing transversal seminars, methodology courses, exchanges with other laboratories (especially European ones), by favoring discussion forums, and by making all useful documents available on a server, etc...; e) to consult annually with the thesis supervisor about the progress of work and to give an assessment of exemptions from extensions. The thesis supervisor's follow-up, like that of the doctoral school's director, can be conducted by e-mail when the student in undertaking activities far from the university; f) to inform doctoral students about the question of professional insertion throughout their doctoral studies, notably by organizing internships and information sessions with professionals, but also by organizing where possible a course of study involving rotation, with periods in firms, in national education etc.; g) to track this insertion after the thesis defense or after a post-doctoral period in an external laboratory. 2) A doctoral thesis is a contribution to the knowledge produced within a research team. In order to ensure the best possible distribution of these contributions within the scientific community, the University recommends that doctoral students prepare their thesis using a computer. To this end, it provides training courses for doctoral students to help them with the composition of their thesis: word processing, use of style sheets, software tools necessary for their project, formal structuration etc. The research teams and/or the Research Division at the SIR (Computer Research Support) provide doctoral students with workstations. Except in exceptional cases, they must submit their thesis in computerized form, from which they have produced printed copies. The SCD (Common Documentation Service) ensures the respect for the norms of indexation and more generally for the recommendations of the ABES (Higher Education Bibliographic Agency). 3) A thesis is accessible: a) at the SCD, under the conditions set out by the decree of 25 September 1985 that applies Law no. 84-52 of 26 January 1984; b) from the University's server, once the author and the jury have agreed to its electronic distribution. The rules of usage concerning the necessity of the jury's authorization before distribution and the confidentiality clauses that can apply to parts of the thesis, apply to the electronic version in the same manner as they do to the paper version. 4) Doctoral students are represented on the scientific committee of each doctoral school. In the case of conflict between a student and his/her thesis supervisor, the Director of the thesis school plays the role of the first mediator foreseen by the Model Charter. If the conflict persists, the President must ask the Scientific Council to nominate a mediator from outside the doctoral school and/or the Establishment. The mediator reports to the President, who is the arbiter of last instance. In the case of joint supervision, the Director of the doctoral school plays the role of mediator for all conflicts between the student and the French thesis supervisor, or between the two thesis supervisors. If the conflict persists, the Presidents of the two universities will decide in the last instance. When the Director of the doctoral school is also the thesis supervisor, he or she is replaced in the role of mediator by the vice president in charge of research. 5) The doctoral schools have the responsibility for distributing this charter to the DEA and doctoral students under their supervision. Approved unanimously by the Scientific Council of 7 December 1998 The Edition Electronique de SeNTIERS cell is responsible for the electronic archiving and distribution of theses defended at Lyon 2. In this role, the cell is involved in the legal deposit of the thesis. With time, the legal deposit of paper copies, as in current practice, is likely to fall out of use. This will imply relatively significant changes in the mode of producing documents. As well, in order to allow doctoral students to pursue their work without too many perturbations, two systems for submission coexist at Lyon 2: electronic submission and mixed submission. Organization of the thesis' administrative circuit The establishment of an electronic archiving and distribution system assumes that the administration possesses an electronic version of the research work; this translates into the creation of a mode of thesis submission that includes an electronic submission. Electronic submission should be organized in an automatic fashion, on a server able to support anywhere between a mailbox-type submission system to a more complete system permitting the recording of the submission, of metadata, etc. This submission will naturally be integrated within the traditional administrative circuit of the thesis and can take several forms (independently of the software used or the discipline in question): the mixed submission, the electronic submission. The mixed submission This form of submission should be seen as a step towards the completely electronic submission, but it is also a satisfactory solution for establishments lacking the infrastructure and personnel needed for electronic submission. The organization of the system It is organized in several stages: 1) The student submits paper copies and an electronic copy at the thesis service, and attests to the conformity of the two versions. At this point, he/she authorizes (or refuses) the distribution of this work on the Internet. 2) The electronic publishing service validates the submission. It is a question of verifying the readability of the files, the presence or absence of particular characters, as well as the presence in electronic or paper form of all the non-textual elements (images, sounds,...). 3) After the defense, one of the following cases is likely: the jury authorized distribution of the thesis in its current state. In this case, the doctoral student has one month to make minor corrections (spelling) to the thesis, in the form of erratum. There is no new submission. The thesis is converted into SGML (archiving format) and then into HTML and XML (distribution formats). The thesis is place on the University's intranet (equivalent to consultation in the library) and, if authorized by the student, on the Internet. The description of the thesis is broadcast. the jury demanded corrections. In this case, a new electronic submission and validation is necessary. Once the president of the jury validates the corrections, the thesis follows the normal treatment process. Non-corrected theses, or those subject to a confidentiality clause, are archived in SGML and are only distributed on the intranet, according to the case. Their description is broadcast, but with mention of their confidentiality. Electronic submission This second system removes the risk of non-conformity inherent in the mixed submission. The legal deposit of the thesis thus consists of an electronic submission. The printed copies of the thesis required for the defense and for deposit in the library are printed from the electronic submission. If this system is more satisfactory, it nevertheless requires a supplementary infrastructure. In order to guarantee that the printed copies are truly drawn from the electronic version submitted, the University must offer the authors the means to effectuate a complete electronic submission and must undertake the task of printing the copies needed for the defense. This system thus requires the support of many different types of structures. These include the traditional thesis service, a relatively heavy computer infrastructure made available to doctoral students (digitization of illustrations, video acquisition, sound, user assistance) and a copy service capable of providing irregular output on short notice (the frequency of thesis defenses varying in function of the university calendar). Starting from the files produced by the student, a "single file" printing (PostScript and/or PDF) will be undertaken by the University, under the student's control (lay-out, rendering of illustrations, etc...). the source files as well as the PostScript file will be engraved and transmitted to the electronic publishing service for archiving while awaiting the defense. Only the PostScript file will be transferred to the copy service's print station. The resources necessary for the implementation of a thesis treatment platform in an establishment or group of university establishments Two functioning architectures can be considered: a client-server system. Here, a server site ensures the management of computing resources while client sites ensure production (from a distance) and distribution (locally). A web interface manages the communication between the different client sites and the server site. Associations of institutions can develop as a result of geographic or disciplinary proximity, by creating synergies and by combining skills and competencies. an autonomous system. Here, each production site undertakes, in an autonomous manner, the entire operation of processing, archiving and distribution. This would apply to sites possessing the necessary human resources, or treating a volume of documents justifying such a system. The existing system requires diverse skills, both in computer engineering and in engineering documents, which correspond to different tasks: managing information systems: updating programs, managing both archiving and putting information on-line, user assistance, linking users to the group(s) in charge of developing and updating the system; processing the documents: formatting documents for processing, digitizing illustrations, sounds, video, etc., verification of the processing and capture of metadata. The presence of the two types of human resources is not necessary in each production center. processing the documents: 1 full time equivalent d'IE (?) for an annual volume of 100 theses per year. information management: Needs vary depending on whether the local configuration or the client-server configuration is used. Training in the use and administration of the software platform for processing theses The training activities are targeted at two types of agents with specific skills: recognized computer skills allowing the agents in question to act on several levels. At the local level, in order to assist those using the software platform, these skills are the maintenance and adaptation of the software platform. They must also be given skills at the network level, where they will serve as relays between users and the group piloting the evolution of the platform, as well as participate in the activities of this piloting group (computing decisions and developments). electronic publishing skills: electronic submission of theses; training doctoral students in the use of generic document forms; preparation and formatting of documents according to the rules determined by the chosen software platform. This is a new domain, and defining its profile with exiting qualifications is a delicate process: mastery of electronic document production tools, knowledge of the different types of academic production, and pedagogical ability to transfer this knowledge to a public composed of doctoral students and researchers. The role of Doctoral Schools and the training of doctoral students The implementation of these new processes requires the participation of the principal actors involved doctoral students and their supervisors. The training of authors, thesis supervisors, and research structures must occur within the Doctoral schools. These school constitute the ideal frame of reference for several reasons: the training provided to the student will be adapted to their discipline and thus to the specific computing tools that they may have need to use; the supervisors will be implicated in the process; the sharing of this knowledge between the different actors of a Doctoral school will provide a supplementary factor of internal cohesion. With time, the Doctoral Schools should become support centers for the use of the new tools of scientific production and publication. In the course of thesis writing, doctoral students will receive the tools and the help required to write their document in an effective manner, and will acquire all the skills needed for its processing. Tools such as generic document forms and technical specifications will be provided by the Ministry. The Doctoral Schools will be charged with adapting them to the particular practices of a discipline or of the establishment (in the case of the style sheet). A generic training program in using generic document forms will be available and adaptable according to local needs. In light of past experience, six hours of training per doctoral student seems sufficient for training students in the advanced use of text processing. One might expect that, within several years, students will be trained in text processing before starting the doctorate and that training costs during the doctorate will be increasingly minimal. A permanent link should be established between the electronic production and distribution services and the Doctoral Schools so as to create synergies between the users and the electronic publishing specialists. This relationship should be based on permanent cooperation, and initiated in the training sessions through a presentation of the service and of its results. Regional and Disciplinary Policies: The implementation and publication of the Université Lumière Lyon2's policy, and the computerized means of processing that the University established, have allowed for the creation and structuration of a network of university and scientific establishments. The universities of the Rhône-Alpes region, united within the context of a University Conference, have decided to implement the principle recommendations enunciated by the Université Lyon2. Subsequently, at the level of the French territory, as well around its borders, one notes a tendency to regroup, on the one hand universities (Lyon, Marne la Vallée, Paris-Sud, Grenoble, Genève), and on the other hand scientific establishments (CNRS, Inra, Inserm). One equally finds disciplinary groups like the "Mathdoc" network which unites the mathematical laboratories, and which is in turn linked to a European network. This network's work involves collecting the metadata produced by each researcher which are necessary for the description and announcement of theses. These metadata are then archived and made accessible either on the site of the laboratory or on the researcher's own site. The approach is less institutional, and thus less enduring, in the sense that its conservation is left to the appraisal of the researchers and of the laboratories. Not all of these institutions have adopted the Lyon2 model (RTF=>SGML/XML). Some have preferred the generic document form produced by Adobe, while others produce documents using Latex and distribute them in "postscript" format. Nevertheless, reflection about these choices is carried out in public, thanks notably to the work undertaken by the Ministerial group. French Public Policy: The Minister's circular To conclude the work led by a reflection group, the Minister responsible for Higher Education released a circular in October 2000 which defined the major axes of the futre policy concerning the electronic archiving and distribution of theses. Electronic Distribution of Theses Text addressed to University presidents, and to presidents or directors of institutions of higher education Theses defended in universities and other institutions of higher education constitute documents of the highest value. Looking after their promotion serves both the interests of the young doctors and the institutions, as well as the end of increasing the international visibility of French research. The deep transformations which have for some time characterized information technology have clearly rendered the existing system for valorizing theses (defined by the decree of 25 September 1985 dealing with the submission, description, reproduction and distribution of theses) obsolete. It is on the basis of this observation and in recognition that: theses are henceforth produced "naturally" in computerized form, the equipment and networks in institutions of higher education have been greatly developed, the majority of universities are currently positioning themselves as producers and distributors of electronic information, that a group associating the services of the Ministry of National Education and the Ministry of Research, the conference of university presidents, the association of university library directors, and numerous experts having undertaken experiments in this domain, submitted a report to me on the electronic distribution of theses. This report, whose principal conclusions I have validated, can be consulted on the Ministry's server at the following address: http://www.sup.adc.education.fr/bib/ The proposed new system foresees the distribution of theses on the Internet once a certain number of conditions are met: authorization of the head of the institution, following the advice of the jury and the authorization of the author, while respecting intellectual property regulations, respect by the doctoral student of minimum technical specifications, conversion of the thesis, using automated assembly lines, into adequate archiving and distribution formats for storage and for placing on-line. The intervention of numerous actors above and beyond the doctoral student will be required: that of the establishment where the thesis is defended, by way of: the doctoral schools, who are responsible for providing the student with training and technical assistance, the common documentation services, responsible for describing the thesis and providing the document's electronic address in collective and local catalogues, the service charged with converting and putting theses on-line, using the software provided to them. that of the state or of a national operator, by means of elaborating technical specifications and training supports, guaranteeing or providing processing chains, providing secure archiving. On these bases, in agreement with the Minister of Research, and after consultation with the CPU, I have decided: to put a project group in place, to elaborate a new decree concerning the submission, description, archiving and distribution of theses, to organize training activities for the institutions, or the groups of institutions, who wish to rapidly enter into this new system, to put into place the collective functions necessary to give overall coherence to the process, taking into account the acquired skills of the National Thesis Reproduction Workshops [atelier nationaux de reproduction des thèses] (ANRT), the Higher Education Bibliographic Workshop [atelier bibliographique de l'enseignement supérieur] (ABES), and the National Computing Centre for Higher Education [centre informatique national de l'enseignement supérieur] (CINES). This new scheme will obviously take time to put into place, progressing as the institutions introduce adequate assembly lines. It goes without saying that the old system defined by the decree of 1985 will continue to apply to the theses defended in institutions which have not yet taken the corresponding measures. The Minister of National Education Jack LANG A certain number of principles underpin this political interest in the electronic distribution of theses: To increase the visibility and to valorize French scientific production at the international level To favor a new approach to the thesis so that it is considered as a dynamic database rather than a static body of knowledge. The electronic version of the thesis becomes a genuine work instrument which meets the demands of users, starting from the user's own agenda of research and investigation. To develop training for student-researchers in the use of information technology and electronic publishing. Student-researcher will find themselves in the role of scientific information producers since they will be able to master these mechanisms and thus increase their autonomy. They will personally acquire the status of being information producers and distributors. This should gather together the elements permitting scientists to produce, archive and distribute all their research work by themselves. Let us remember that France's approach to the electronic distribution of theses is integrated into a vaster francophone project of putting the results of public research online. Next Section: E-Commerce: fee based methods Universities/E-Commerce: fee based methods The ADT Program has developed a simple e-commerce option for members to use should they choose. The file format for the ADT is to always include a front.pdf file. This file contains author, title, abstract, contents pages, acknowledgements, etc. It is designed to give an expanded view of the theses' contents before deciding to view the whole thesis. Viewing the whole thesis can then either be unrestricted and free or at cost using the local institution's online payment system. Currently no members of the ADT have used this option. Next Section: Students Students are the most important participants in ETD activities. They are the main target of the education effort. They are the ones who learn by doing, and so promote access to the ETDs they prepare to help communicate their research results. This section of the Guide highlights issues of interest to students who write or refer to ETDs. It provides guidance on how they can proceed in these activities, as well as assistance to help them in understanding the context in which these activities take place. A wide variety of technologies and approaches are discussed. These should be considered in light of local policies and practices. In some cases students will want a simpler effort. In others they may wish to engage in more elaborate preparation of an ETD than is locally common, in order to learn, and/or in order to be more expressive in their scholarly communication. It is hoped that the content in section 3, as well as the more detailed technical information in section 4, is helpful. Next Section: How to learn about ETDs? (workshops, online resources, helpers) Students/How to learn about ETDs? (workshops, online resources, helpers) Some students who prepare an ETD undertake this effort as an independent. That is, they prepare a paper document as required by their university and turn that in, but are not allowed to submit an ETD to their degree granting institution. If they understand about ETDs and realize the benefits of such works, if they know about creation and submission, they may elect to submit an ETD, such as to a service for independent ETD authors which are afforded by NDLTD or other groups. In such cases, students are usually "on their own" and will follow instructions in this Guide or at other online sites, such as those hosted by Virginia Tech (http://etd.vt.edu). In addition to learning about ETDs from this guide or other online sources (see also from www.theses.org), students may be helped by local staff. Typically staff at institutions that are members of NDLTD will have a project in place, with personnel identified who can assist. Often there are two types of helpers. One type involves trainers. These typically run workshops or other training events or programs to help students to create ETDs. In some cases there are both basic and advanced training efforts, depending on the level of interest of the student and/or depending on how "rich" (in terms of use of multimedia content, for example) an ETD will be. The other type of helper is a person who provides assistance on demand. This may be a person who works in a campus New Media Center (see, for example, http://www.nmc.vt.edu). These centers may have a wide variety of multimedia devices (to capture images, audio, or video - as well as to help one create such resources from scratch). Alternatively, there may be people in one's own laboratory or department with particular expertise in creating certain types of digital works that might be an important part of an ETD. In short, students should seek out such help where appropriate and work with trainers and mentors as desired to hone their knowledge and skills, as well as to prepare the very best ETDs. Exemplary Electronic Theses and Dissertations Our goal here is identify "technologically innovative" theses and dissertations. We want to provide models of new media scholarship for the next generation of scholars and researchers. Do you know of any theses or dissertations that are crafted in new ways, perhaps using streaming multimedia, interactive features (chats, listservs, response questionnaires, three- dimensional models, animation? If so, please take a moment to add the ETD to our collection: http://etdguide.org/ETDs.asp By celebrating the innovative work of our graduate students and graduate faculty, we can inspire future research. Exemplary ETDs in Our Database View the ETDs that have been submitted to our database as exemplary models for future researchers and writers Exemplary ETD Database Submit an exemplary ETD in our database. Please be sure to clarify under "special features" what distinguishes the ETD from a "digital perspective." For example, does the ETD incorporate hyperlinks, multimedia, visuals, animation, or other interactive features? Next Section: Importance of satisfying local requirements Students/Importance of satisfying local requirements When students prepare ETDs, they must be sure to undertake this effort in accordance with all requirements. They should be aware and considerate of the interests, time, and abilities of the faculty who advise and examine them. They must be sure that those in charge of the graduate program and library, as applicable, are satisfied with their submission. Typically, there will be written or online instructions, such as are available for Virginia Tech students (http://etd.vt.edu). If their university does not accept ETDs, students can submit as an "independent", according to rules and procedures being developed by NDLTD. Instructions may be updated from time to time and will be found at http://www.theses.org. Next Section: Learning from other ETDs Students/Learning from other ETDs Documents are often prepared in a similar form to that of prior documents. Students are encouraged to go to http://www.theses.org and follow links found there to other ETDs. Of particular interest should be the set of Notable ETDs found at that site. Modeling one's work after these may be an effective strategy as long as local requirements are satisfied (see above). In its Fall 2001 meeting, the NDLTD Steering Committee decided to pursue this strategy, giving awards for the best masters and for the best doctoral ETDs. Once these are identified, they will be linked in with the above-mentioned Notable ETD section. Next Section: How to prepare an ETD? (approaches) Students/How to prepare an ETD? (approaches) ETDs are prepared to facilitate scholarly communication. They are vehicles for transmitting the research results of a student, in the most effective way, to each person with interest. Truly effective communication requires students to become facile with tools and methods of expression so their ideas and findings can be clearly conveyed. At the same time, since communication takes place across gaps in space and time, it is important that the form chosen for an ETD be understood in different places and in future times, as typically occurs when standard representation schemes are used. Thus, at a 1994 meeting of ten universities discussing ETDs in Blacksburg, Virginia, USA, it was recommended that ETDs be prepared in both a rendered and a descriptive form, like PDF and SGML. The former ensures that a reader sees things the way the author desires, which may be of particular importance with regard to mathematics or artistic works. The latter instead emphasizes the logical structure and content (as is done with HTML pages) and makes it easier to precisely specify the target of a search. However, no authors write using tools that store their works only in PDF, and few authors directly create SGML documents. Rather, they employ tools they select based on cost, availability, popularity, familiarity, efficiency, or other criteria. For many, the choice is a package like Microsoft Word that may have been bundled with their computer. For others, like mathematicians who need to work with proofs and equations, the main decision may be what representation to use (e.g., LaTeX), with subsidiary decision regarding what editor or other special tool can best manipulate LaTeX files. Such choices may be discipline specific, may be based on what is commonly used by faculty and students who are in a particular group, or who use a particular computing environment. If there is no clear choice imposed, then a decision could be based on the information provided in subsequent sections of this Guide. These cover, in particular: an overview of approaches using word processing systems using an editor (or converter) to prepare SGML/XML preparing PDF rationale/overview for SGML/XML - and converting to them metadata considerations intellectual property considerations Next Section: Overview: writing with word processors and structured editors Students/Overview: writing with word processors and structured editors Details regarding a variety of approaches to use in preparing ETDs are found in subsequent parts of section 3.2. In all cases, there is a writing phase and a conversion phase, so that an archivable form is produced. The writing phase should allow authors to efficiently and effectively capture their research results in a form that will be understandable by others. The conversion phase should take the written work and use it to create PDF and/or SGML/XML. Authors preparing an ETD must learn about electronic publishing to succeed in their task. In the future, costs may decrease on structured editors and other software that directly yields SGML/XML. Meanwhile, a small number may invest (their money and time) in such as SoftQuad's Author/Editor. But today, most authors learn about word processing, since it is commonly used for other writing tasks, such as letters, reports, papers, articles, and books. Word is the most common word processor. Word Perfect is somewhat popular. FrameMaker is a more costly package, harder to learn, but quite powerful, and used in professional writing efforts. LaTeX is popular with mathematicians, scientists, and engineers who deal with mathematical notation, proofs, equations, etc. From word processing systems such as these, it is relatively easy to prepare PDF files. It also is possible to prepare SGML or XML. However, this latter typically requires a good deal of pre-planning, work on conversion, and final editing/checking to make sure the archival form correctly conveys the author's intent. Some of these efforts yield metadata for ETDs as a byproduct, either directly or in connection with a conversion effort. Thus, if "TEI-lite" (drawn from work on the Text Encoding Initiative) is employed, metadata can be produced from the document by analyzing the header portion. Generally, however, authors prepare a metadata record for their ETD by filling in some type of form, entering in suitable information into a database from which the desired representation can be generated as needed. The metadata, as well as other forms, may carry a description regarding copyright and other intellectual property right management issues. In addition to writing, authors may convey their results using multimedia devices. Special tools are often employed to prepare graphic aids, animations, or musical compositions. Other tools support conversion of photographs, video, music, and other formats. In some cases, multimedia content can be included with the rest of the work, as when images are included in a PDF file. Regardless, international standards for the various media types, as well as their combinations, should be followed so archivable works result; otherwise these parts of an ETD may become lost to interested readers in the distant future. We aim to avoid such losses whenever possible, through training, following best practices, and building upon recent work on ETDs, as documented below. Next Section: Writing in word processing systems Students/Writing in word processing systems The following subsections explain in detail how to work with Word, Word Perfect, LaTeX, and FrameMaker. Note, however, that many tools rapidly become obsolete. Authors should work with current, supported versions of tools. Authors should refer to the content below as appropriate, but should seek other aid from their local institution as needed to prepare their ETD. Next Section: Microsoft Word and Office 2007 Students/Microsoft Word and Office 2007 Inspired by the NDLTD, a group of faculty and staff joined together to create the Digital Media Institute at the University of South Florida. We created our community back in 1999 with the following goals in mind: Improving the quality of e-documents, particularly theses and dissertations. Researching ways authoring tools, particularly Microsoft Office 2000, can be used to facilitate electronic theses and dissertations. Researching how technologies alter graduate education, including mentoring relationships, topic selection, intellectual property, writing processes, and publishing practices Working with tool developers to keep abreast of new tools for researchers and writers Providing an open forum for the exchange of ideas regarding the evolutions of new media scholarship. Providing training workshops, reference materials, and support for graduate students interested in contributing to the University of South Florida's digital library of electronic theses and dissertations. While we are excited about engaging our undergraduate students in this research and support endeavor, our greatest efforts have occurred at the graduate level. Since creating our community three years ago, my colleagues and I have analyzed how Microsoft's Office 2000 can be used to better support students' needs as writers of multimedia scholarship and faculty members' needs as mentors of electronic theses and dissertations. Using case study and ethnographic methodologies, we have researched how communication technologies can improve graduate education, particularly academic scholarship. Following Walter Ong, who theorized "Technologies are not mere exterior aids but also interior transformations of consciousness" (82), we are researching how technologies alter graduate education, including mentoring relationships, topic selection, intellectual property, writing processes, and publishing practices. In the preliminary stages of our investigation, we focused on examining the Office 2000 suite, yet we expect to investigate related tools for writers, including bibliography, and quantitative and qualitative data analysis tools. We chose to focus initially on Office 2000 because it is used by so many other members of the NDLTD. Office 2000 includes all of the necessary components (word processor, database, spreadsheet, presentation graphics, electronic mail) necessary to author a thesis or dissertation, and all of these components can be used to produce HTML code, as well as native-format documents. In addition, Office 2000 has powerful features for collaboration and multimedia authoring. Outlook—Microsoft's email and calendaring tool—serves as a framework for document workflow, calendaring, sharing and exchange. For example, regardless of their locations in time and space, faculty and students can use Outlook to provide students with an integrated set of reviews and links to grammar and punctuation references. From any document in Office 2000, faculty and students can use NetMeeting to synchronously discuss documents, including audio/video-based discussions. They can invite scholars outside the committee to respond to drafts. Numerical data, as well as graphical representations of it, can be published out of Excel in such a fashion as to permit limited manipulation and re-analysis from a Web Browser. More extensive analyses can be formed by "roundtripping" the data back into Excel. Throughout our research, as we work with Office 2000 tools in proposal preparation, research, and thesis/dissertation writing, we are asking "What tools are really useful? What motivates or dissuades innovative use of tools?" Some graduate students are maintaining a Case Study Journal where they reflect on how use of software tools influences our research, writing, and relationships with mentors. In turn, some faculty are reflecting on ways the tools influence mentoring, scholarship, and teaching and learning. Jude Edminster, a doctoral student in Rhetoric and Composition, is conducting an ethnographic investigation of our project; see http://dmi.usf.edu/edminster/ETDProposal/. Ultimately, we expect our research will reveal ways faculty and graduate students can use software tools and plug-ins to critique and develop theses and dissertations, including insights into necessary training and resources. We believe this work is an important first step toward transforming our graduate programs so they better prepare students for the Knowledge Age. Results of our research can be viewed at our project home: http://dmi.usf.edu. Additional Readings: Differences in Office 2007 Suite Moxley, Joseph M. "New Media Scholarship: A Call for Research." Change: The Magazine of Higher Learning (Scheduled Publication Date: November/December 2001) Moxley, Joseph M. "Dissertating in a Digital Age: the Future of Composition Scholarship" Invited Chapter. Reinventing the Discipline in Composition and Rhetoric and a Site for Change. Edited by Sheila Carter-Tod, Catherine G. Latterell, Cindy Moore, and Nancy Welch. Moxley, Joseph M. American Universities Should Require Electronic Theses and Dissertations. (Educause Quarterly, No. 3 2001, pp. 61-63.) Next Section: Using Style Sheets Students/Using Style Sheets Preparing a word document to be converted into an archivable form using the SGML or XML standard first of all means using word stylesheets. Usually the university provides these style sheets, as they contain university specific structuring and formatting. In Word it is possible to distinguish between the information a document holds and the structure in which it is written. Style sheets provide the structure. They help you, for example, if you want to format all your headings for chapters with the same style. If you use the style sheet heading1 (see picture below), than it may be possible to associate a certain formatting like text height: 14pt, text-font Arial, paragraph settings: left bound, leave 12 pt space after a heading, number the headings automatically with roman numbers, etc. with the structure element heading1. You can see the left row within the word window. This is the so-called structured view, which you can get by choosing the Normal option under View. The names of the paragraph structures are displayed if you choose the point Options under Extras. Within the popup menu there is a possibility to set the width of the style sheet view. (see below). This is how it is displayed in German Word97, but is similar in other language word systems. Next Section: Using Plug-ins: Bibliography Plug-in Students/Using Plug-ins: Bibliography Plug-in Some word processing systems are extensible in different ways. One of them is the definition of macros written in some programming language. MS Word, for instance, uses Visual Basic for Applications as macro language). Another way to extend MS Word is by add-ins, applications built in some programming language that are inserted into Word's runtime (like plug-ins in web browsers). This feature permits the development of added-value tools for writers. For instance, BibWord (http://mariachi.dsic.upv.es/bibword) is a bibliography management tool for MS Word 2000. The current version allows Word users to insert references from a bibliographic database and automatically generates the reference list at the end of a document. Macros also could be used to attach a form to the ETD document, the fulfillment of which would provide the ETD metadata from the word processing environment. Next Section: Corel WordPerfect Students/Corel WordPerfect The Corel WordPerfect Suite provides another possibility to produce large documents. It is a system like Microsoft Word, but sold by another vendor: Corel Inc. (see http://www.corel.com). It contains nearly the same features and capabilities as Microsoft Word. For writing a complex document like a thesis or dissertation using WordPerfect, it is advisable to use style sheets that are provided by the system itself or by the university. Those style sheets are called WordPerfect templates (WPT). They allow the users to structure their documents using structure components like headings, tables, lists, etc. Default style sheets can be used by clicking on File / New / and than choose one of the templates. Usually there has been a native WordPerfect template produced in order to help you with additional drop- down menus, etc. To use the provided template for WordPerfect 8, you have to choose File / New / Options / Add Project / Add New Document / Call it "Digital Dissertation" / Search for dissertation.WPT. This allows you to use the WordPerfect style sheet by choosing under the main window: File / New / Choose "Digital Dissertation" and Create. This enables the functions of the drop-down menu as in the following figure. Figure 1: Using a WordPerfect style sheet for digital dissertations (used with version 9 of WordPerfect suite) Next Section: LaTeX Students/LaTeX Scientists within the natural and engineering sciences have special needs for mathematics and algorithmic graphics. The text formatting system LaTeX has been used for decades to mark up scientific documents. Even today, there is no viewable alternative to print texts containing a lot of mathematics without using LaTeX. This system uses a kind of semantic or typographic markup for rendering formulas, graphs, and so on. Within some disciplines LaTeX is nearly exclusively used to render complex documents. TeX and LaTeXEdit TeX is a document formatting language (and the program that processes it) written by Donald Knuth for the professional preparation of complex publications. It excels particularly at formatting mathematical equations and for managing two-dimensional presentations of data (tabular and otherwise). LaTeX is a set of macros written by Leslie Lamport as a "front-end" to TeX that makes articles, reports, theses, dissertations, and books easy to create and manage. How to get LaTeXEdit LaTeX is free to download from any CTAN archive (http://www.ctan.org ), and works on Macintosh, MS-DOS, Unix, and Windows 3.1/95/NT (though some commands may vary on some architectures). To convert your electronic thesis or dissertation in LaTeX, you must first type your document completely into the ASCII editor using the LaTeX macros appropriately, then use a certain chain of commands that produce a layout and printable version of the document. LaTeX under UNIX / LINUX systemsEdit To create LaTeX files, all you need is an ASCII-based editor, like Emacs, Vi. Writing a dissertation just means typing the contents and the LaTeX-commands directly in an ASCII-based file and save this as .tex. To compile a LaTeX file and produce a printable version of the document, you have to follow the following steps: 1. Run latex "latex mydissertation.tex" This produces the following files: mydissertation.dvi / mydissertation.aux, etc. 2. Run dvips "dvips mydissertation.dvi" This produces a file dissertation.ps that is printable on a printer, or convertible into PDF. While writing your thesis in LaTeX, please keep the following rules in mind: As document style we advise to choose report or book, because both start with chapter as the highest order for section structuring. The preamble of the latex file could look like in the following example: \documentclass[12pt,a4,titlepage]{book} \usepackage{babel} \usepackage{longtable} \usepackage[dvips]{epsfig} With usepackage we import additional styles that are needed, e.g. for tables, mathematics, figures etc. In order to get archivable form of the latex dissertations we advise not to use or to program complex macros. Simple \newcommand or \renewcommand may be used, e.g. \newcommand{\begin{itemize}}{bi} Headings can be separated using the following commands: Document Structure \part{Heading Part I } -1 \chapter{Heading Chapter 1} 0 \section{Heading Subchapter 1.1} 1 \subsection{Heading Section 1.1.1} 2 \subsubsection{...} 3 \paragraph{...} 4 \subparagraph{...} 5 Levels -1 to 2 appear in the table of contents. Part is used to split the whole document into several parts. The chapters numbering are constantly growing. Within the document than a single page is displayed, that contains: Part I Introduction or Part II Method and so on. Chapters are numbered without taking the parts into account. The numbering is standardized: Chapter 1 Mathematics. Sections are subunits of chapters and numbered: Basic Algorithms. Chapters are numbered without taking the parts into account. The numbering is standardized: Chapter 1 Mathematics. Sections are subunits of chapters and numbered: Basic Algorithms. Sections are numbered as follows: 1.1.1 Decision Tree Algorithm A. For those parts like acknowledgements, dedication, and curriculum vita where authors usually don't want to use numbering, the following style can be used: \chapter{Thank You} . The asterisk prevents the numbering. Appendices are included using the \appendix command. Please use commands as in the following example if your appendix consists of several chapters: \appendix or \appendix* not numbered headings of the appendices \chapter{Program Source} \chapter{Curriculum Vita} Using graphics: figures and pictures should be included in LaTeX documents using the eps (encapsulated postscript) format. Before including them, one has to use a certain style package in the preamble: \usepackage[dvips]{epsfig} The parameter [h] positions the figure at the current position. Keep in mind, always to use a caption-environment to put the figure captions below the picture: \epsffile{didi.eps} \caption[short description for the table of figures]{Long description for the text} The title page is the most complicated part. Most universities supply own templates for the title page and the whole dissertation. There is no best practice available. In order to separate the several items on a title page in order to be able to reuse those information pieces e.g. if the whole dissertation is converted to HTML or SGML/XML, we advise to use \newcommands as simplest method to apply a pseudo structure to a LaTeX title page. Usually LaTeX provides the following standards item for a title page: \date{} \author{} \title{} But as this is not enough for a thesis, most universities provide own style sheets or templates. Tables should be used as follows: authors are advised to use the table-environment because it provides the possibility to include table captions in a structured way. \caption{Tabellenbeispiel} \begin{tabular}{ccc} x & 1 & 2 \\ \hline 1 & 1 & 2 \\ 2 & 2 & 4 \\ \hline Citations can be used as their own structured items as follows: 1. Using the citation-environment. This is used for inline citations. \begin{{citation}{label1} \end{{citation} 2. Using the quotation-environment. This is used to structure whole paragraphs as citations. Those citations use an indent like usual paragraphs. \begin{quotation} \end{quotation} 3. another method is the use of the quote-environment. This environment is used for whole paragraph citations, but those paragraphs don't have an indent. \begin{quote} \end{quote} Numbered lists are typeset using the enumerate-environment. By integration new enumerate- environment in existing one a hierarchically nested sublist is built. \item {Testitem1} \item {Ebene 2 Testitem1} \item Testitem2 Bulleted lists are typeset using the itemize-environment. Here a hierarchical nesting is also possible. \item Ebene 2 Testitem1 Definition lists contain a definition term and a definition text. \begin{description} \item[Definition term] Explanation of the definition term \item[Element2] Explanation 2 \end{description} If an author wants to include source code this is best done using the \verbatim-environment. #!/usr/bin/perl -w #+-----------------------------------+ #\| this script has been written 1998 by Anchors, references and cross-references are typeset using the \label command, which links a key to the specified item of a document. \label{keyword} References to these parts have to use the command \ref or \pageref in order to produce a reference to the object or to the page. ref{keyword} pageref{keyword} A very important part of a dissertation is the bibliography. We advise all authors to use the bibtex-system and graphical front ends, e.g. bibview under LINUX or UNIX systems to manage bibliographic records and entries. References to bibliographic entries that are held in a bibtex-database are written as in the example: \cite{schluessel} The bibtex-database can be included into the LaTeX file by the following command, where a predefined style like alpha, plain, apalike can be used to layout the entries: \bibliography{file name without .bib} \bibliographystyle{style, e.g. alpha, plain, apalike, etc.} Within the BibTeX-system database entries can be done using a plain ASCII editor. like emacs. There are several types of literature predefined: Article in Conference Proceedings Article in a Journal Article in a Collection Chapter or pages in a book Booklet, but no Publisher, Institution The following example shows how a BiBTeX-entry has to be written: % Article in a Journal @Article{shortkey2, author = {Name, Firstname}, title = {Title No. 2}, journal = {Journal for ETDs}, OPTkey = {}, OPTvolume = {}, OPTnumber = {}, OPTpages = {}, OPTmonth = {}, OPTnote = {}, OPTannote = {} The following table shows those items have to be used for certain bibliographic entry types: To process a latex and a bibtex-file under UNIX system you have to type the following command sequence: latex mydissertation.tex bibtex mydissertation.aux This produces the following files: mydissertation.dvi / mydissertation.aux /mydissertation.bbl / mydissertation.blg, etc. Run dvips "dvips mydissertation.dvi" This produces a file dissertation.ps that is printable on a printer, or convertible into PDF. LaTeX under Windows operating systemsEdit Using LaTeX under MS Windows requires the TeY System, a DVI-Viewer, Ghostscript and Ghostview. There are several LaTeX distributions: MikTeX , a highly regarded setup for Windows 95/NT (http://www.miktex.org/) and emTeX , the classic DOS and OS/2 TeX setup by Eberhard Mattes (ftp://ctan.tug.org/tex-archive/systems/msdos/emtex/). There are front ends available for Latex that provide a WYSIWIG view to the user. One of the most used ones is Scientific Workplace by McKichan Software Inc. (http://www.mackichan.com/products/swp30.html). The disadvantage is that it is quite costly for single users. Next Section: Framemaker Students/FrameMaker Why should one use FrameMaker instead of MS Word?Edit Framemaker provides in comparison to Microsoft Word a much more sophisticated tool for electronic publishing and for a cross-media publishing. It allows: Production of real and large structured text (not just address cards) (and is stable handling those). Production of semantically structured text (not just text that uses style sheets). Provision of a WYSIWYG user interface to edit XML documents, instead of an XML tree editor à la Spy or Xeena. Capabilities of producing a good-looking paper version of a document with all layout features that are professionally used at printing companies. Framemaker+SGML especially combines the power of an excellent word processor (better than MS Word) with a good structure editor. In order to produce structured documents and some sort of style sheets, one has to learn FrameMaker's EDD language. As FrameMaker belong to the product family of Adobe, it provides an add-on to produce a high quality PDF, as digital preprint copy of a written document. Using FrameMaker+SGML6.0 for a conversion of MS Word documents into SGML instances.Edit Editing or converting using FrameMaker is much more complex than the previously described methods. FrameMaker is able to import formatted Word documents keeping the stylesheet information and exporting the document via an internal FrameMaker format as SGML or XML documents. In order to proceed with a conversion using FrameMaker, you will need the following configuration files: a conversion table that contains the list of the Word styles and the corresponding elements within the FrameMaker internal format. This table is saved within the FrameMaker internal format (.frm). A document type definition will be saved within FrameMaker internally as EDD (Element Definition). It is saved within the FrameMaker internal document format (.edd). FrameMaker uses layout rules for the internal layouting of documents. Within this layout definition, the layout of documents is described just like it is within MS Word documents: single formats and their appearances like text height, etc., are defined. This file is also stored as (.frm file). The Read-Write Rules contain rules that define which FrameMaker format will be exported in which SGML / XML element. The SGML- or XML DTD has to be used as well, including Catalog- or Entity files, as well as Sub DTDs, like CALS for tables. To process a conversion, a new SGML application has to be defined within FrameMaker+SGML. This application links all files that are needed for a conversion as described above. It enables FrameMaker to parse the output file when exporting a document to SGML or XML. A workflow and a technology for conversion to ETD using FrameMaker+SGML6.0 was first developed at the Technical University Helsinki, within the HUTPubl project (1997–2000), see http://www.hut.fi/Yksikot/Kirjasto/HUTpubl. You can find more information about using FrameMaker+SGML for an XML Authoring at http://tecfa.unige.ch/guides/xml/frame-sgml/html/quick-fm-xml-guide.html. (See also Danny R. Vint "SGML at Work", Prentice Hall, New Jersey, 1999.) Next Section: Writing directly in SGML/XML Students/Writing directly in SGML\XML The desired situation for retrieving archivable ETDs would be the one, that authors write in an XML- editor, according to a Main-DTD, and choose those parts of the DTD, that are inevitable for their thesis or dissertation. Some desktop publishing systems today provide an opportunity to save as SGML or XML. Investigations as to whether those tools are usable for such complex documents as a thesis or a dissertation led to the following conclusion: Writing in WordPerfect or FrameMaker+SGML enforces the author to learn new writing habits. While writing, they have to think about the structure of their documents, e.g., which part is a heading, which part is a definition list; or they have to think to add certain parts immediately to the document, like, references, table and figure captions, etc. While writing according to a specified DTD, the desktop publishing system often internally checks syntax correctness by using an XML parser. Some of those internal parsers are still not stable enough and may cause the system to crash, as experienced with WordPerfect 9.0. Those parsing procedures in between the active scientific thinking and writing often disturbs the authors. Most of the pure XML editors could not produce an appropriate and layouted printed copy or PDF file that would satisfy the approval of readers of the printed version of a document. Some of those editors simply fail to process large and complex documents. Most of the tools are not ready yet, especially in a sense that would allow to use user or domain specific DTDs. Staroffice and other tools support their own vendor specific DTD only. Although the world of desktop publishing systems is actually changing, there are still too few tools that are sufficient in: The support and appearance of their graphical user interface, The provision of a certain amount of features normal word processors have, like automated numbering, colors, table management, link management, style sheets. The platform independence or cross-platform availability, The support of user specific DTDs, and standard DTDs, like TEI, Docbook, etc., The export quality of produced XML: tables, tags, The stability of usage, Their commercial availability and price. The following systems are able to export into SGML or XML: Export with user specific DTD WordPerfect Version 7.0 (Corel) http://www.corel.com FrameMaker+SGML6.0 (Adobe) http://www.adobe.com Export with system specific (vendor) DTD Openoffice (SUN / open source) http://www.openoffice.org AbiWord (AbiWord / open source) http://www.abisource.com Kword (KOffice, KDE Project / open source) http://www.kde.org Omnimark (Omnimark) (http://www.omnimark.com) MarkupKit (Schema) http://www.schema.de Majix (Tetrasix) http://www.tetrasix.com TuSTEP (RZ Uni Tübingen) http://www.uni-tuebingen.de/zdv/tustep/index.html (More information about SGML/XML tools can be found at: http://www.w3.org/XML/#software.) Next Section: Preparing a PDF document Students/Preparing a PDF document A popular page representation scheme, a published de facto standard developed by Adobe is the Portable Document Format, PDF. Adobe provides the Acrobat Reader free of charge (and promised it into the foreseeable future), which will read current as well as previous versions of PDF. It is downloadable at http://www.adobe.com/products/acrobat/readstep.html. Adobe also provides tools for creating, annotating, and manipulating PDF documents, through its own word processing software, printer drivers, and distilling from PostScript. The whole suite is called Adobe Acrobat and is actually available with version 5. Adobe's Acrobat software, installed on a Windows or Macintosh platform allows most suitable documents to be converted to PDF in moments. From word processors such as Word, WordPerfect, and Framemaker, each document portion can be "printed" to the Distiller printer driver, yielding a PDF file. The Distiller converts PostScript files to PDF files. Acrobat software allows multiple PDF files to be assembled into larger PDF files by inserting documents or deleting pages in an existing PDF file. It is also possible to produce PDF documents on UNIX systems. However, the latest version of Acrobat Distiller that was available for certain UNIX platforms such as Solaris or HP-UX was version 3.1. Authors writing in LaTeX can use ghostscript to produce PDF files. But in order to obtain readable PDF documents, issues of used fonts, used conversion scripts, etc. have to be considered. To avoid problems for future readers, authors should embed all fonts in their documents (when that is allowed). Otherwise, software displaying or printing PDF content will attempt to find a similar font and extrapolate from it, which may cause serious problems. Similarly, authors should use so-called "outline" fonts as opposed to bitmap fonts, so that display and printing can proceed to scale characters as required. Thus, when using TeX or LaTeX, the bitmap fonts commonly found in a standard installation should not be used. PDF-Tools: Acrobat (http://www.adobe.com) Ghostscript (http://archiv.leo.org/pub/comp/general/typesetting/tex/support/ghostscript/ or http://www.cs.wisc.edu/~ghost) and Viewer Tool ghostview (http://archiv.leo.org/pub/comp/general/typesetting/tex/support/ghostscript/gnu/ghostview/) NikNak (http://www.niknak.de/is/5dorder.htm) XPDF (http://www.footlabs.com/xpdf) Magellan/ Drake (http://bcl-computers.com) Gemini (http://www.iceni.com) Omnipage (http://www.scansoft.com) Next Section: From LaTeX Students/From LaTeX Generally speaking, there are several possibilities for producing PDF document from a LaTeX document. Using Postscript and scalable fonts for PDFEdit "One of the most confusing issues in both Postscript and PDF is the handling of different types of fonts. A PDF-producing application can deal with a font in one of three ways: First it can take the entire font and embed it in the file; second it can make a subset; or third it can simply embed some summary details about the font (such as its name, metrics, its encoding, its type - sans serif, symbol, for example - and clues about its design) and rely on the display application to show something plausible. This last strategy is preferred for documents that are to be delivered on the Web, since it creates the smallest files. The display application can work again in several ways. It can try to find the named fonts on the local system; it can simply substitute fonts as intelligently as possible; or it can use Multiple Master fonts to mimic the appearance of the original font." (from Goosens; Rahtz: The LaTeX Web Companion, page 29) The default installation of dvips uses fonts with a fixed resolution (.pk fonts) encoded as 300dpi (dots per inch) bitmaps. This is unnoticeable for printing; however, the resulting PDF files are barely legible when scaled down to today's screen resolutions (typically 72dpi). These fonts are embedded in Postscript Output as Type 3 fonts. Acrobat Distiller cannot handle those fonts, because there are no font descriptors available. It leaves them embedded in PDF files and renders them very badly, although printing those documents doesn't make too many differences, if the original resolution was high enough. Therefore it is necessary to install Postscript Type 1 fonts (True Type) for the dvips program. Many commonly used fonts have been converted to Type 1 fonts, e.g.: All Computer Modern family fonts, all fonts from the American Mathematical Society, the St. Mary's Road symbol fonts, the RSFS script fonts, the TIPA phonetic fonts and the XY-pic fonts. The Type 1 Computer Modern fonts are provided by Virginia Tech and part of this guide (cmps.tgz / cmps.tar.gz). These files are about 5 MB. To install the fonts you have to… On standard LINUX systems they are already installed: 1. Copy all files which are in the gz-archive under the directory pfb into the directory, in which dvips looks for fonts, e.g. /usr/local/teTex/texmf/fonts. 2. In the directory e.g. /usr/local/teTex/texmf/dvips/misc there is a file psfonts.map. Please add the content of the files cmfonts.map, cyrfonts.map, eufonts.map,and lafonts.map to that file. They are provided with this cmps.tgz archive. 3. The config.ps file is usually used for defining the resolution. This is irrelevant, because dvips now uses the scalable fonts instead of the bitmapped pk fonts. 4. The afm und pfm directory in the archive is not used by dvips. To obtain a ps-file which uses Postscript fonts and is convertible into PDF you have to run the following command sequence: 1. latex mydissertation.tex. 2. bibtex mydissertation.aux if bibtex is used. 4. dvips -P cmz mydissertation.dvi: This produces a file dissertation.ps that is printable on a printer, or convertible into PDF. 5. If Acrobat Distiller is installed on the system "distill mydissertation.ps" which produces a PDF file: mydissertation.pdf. Producing Rich PDFEdit Producing a WWW-readable PDF is just the first part of a PDF production. It is more sophisticated to produce a PDF file that takes advantage of the hypertext features of PDF and adds links and cross- references to a PDF file. You can use the Adobe Exchange software under Windows/Macintosh to add links, etc., to a ready produced PDF file, or you can produce those features directly from LaTeX using the Hyperref-package. This package has been developed by Sebastian Rahtz and uses the outcome of the Hypertex project. This package extends the capabilities of the LaTeX cross-referencing commands (TOC,bibliographies, etc.) to produce \special commands that a driver can turn into hypertext links. It also defines new commands for LaTeX. For using hyperref a global option can be used within the LaTeX file: \documentclass[dvips]{article} In order to produce PDF-information, it is possible to insert title and author information that are then displayed in the PDF file as follows: In LaTeX: \usepackage[ pdfauthor={Susanne Dobratz}, pdftitle={ Test of the pdftex Package }, pdfcreator={pdftex}, pdfsubject={electronic publishing in LaTeX}, pdfkeywords={keyword1,keyword2} ]{hyperref} This looks in PDF like this: %PDF-1.2 %âãÏÓ 1 0 obj /CreationDate (D:191010522170228) /Keywords (keyword1,keyword2) /Creator (pdftex) /Title (Test of the pdftex Package) /Producer (dvips + Distiller) /Author (Susanne Dobratz) /Subject (electronic publishing in LaTeX) The usual \label and \autoref commands are used to produce hyperlinks. The \autoref-command replaces therefore the usual \ref-command in LaTeX. So the following document structures are automatically referenced, if a \label has been applied. This also automatically produces Adobe-PDF bookmarks and hyperlinks to chapters, sections, etc. if the LaTeX command \tableofcontents is used. Within the LaTeX file there are some additional user macros available to produce hyperlinks: \href{url}{text} The text is used a hyperlink to the url . This URL must be a full URL (like http://www.cybertheses.org) \hyperbaseurl{url} A base url is established, prepended to other specified URLs to make it easier to write PDF documents. \hyperimage{image url} The image referenced by the image url is inserted. \hyperdef{category} {name}{text} A target area of the document (text) is marked and given the name category.name \hyperref{url}{category} The text is made into a link to url#category.name \hyperref[label]{text} The text becomes a link point to a point established with a \label command (using the symbolic name label). It is even possible to use Acrobat specific commands, e.g.menu options to navigate etc., like in this example from Sebastian Rahtz: \usepackage{fancyhdr} \pagestyle{fancy} \cfoot{\NavigationBar} \newcommand{\Navigationbar}{% \Acrobatmenu{PrevPage}{previous}~ \Acrobatmenu{NextPage}{next}~ \Acrobatmenu{FirstPage}{first}~ \Acrobatmenu{LastPage}{last}~ \Acrobatmenu{GoBack}{back}~ \Acrobatmenu{Quit}{quit}%} For further information and help, we recommend the book by Goosens/ Rahtz: The LaTeX Web Companion. The \special commands that are added by using the LaTeX macros have to be interpreted by DVI drivers or viewers in order to produce PDF links. The following DVI drivers are supported by the hyperref package: hypertex dvips - writes \special commands to Postscript tailored for dvips dvipsone - writes \special commands to Postscript tailored for dvipsone pdftex - writes commands for pdftex, and produces PDF directly dvipdfm - writes \special commands to be used for Mark Wicks' DVI to PDF driver dvipdfm dviwindo - writes \special commands to be used for Y&Y's Windows previewer. It interprets them as jumps within the previewer vtex - writes \special commands, which are interpreted as hypertext jumps for MicroPress'HTML and PDF producing TeX variants Using PDFTeXEdit PDFTex is a variant of Tex that produces directly a PDF output. Usually a Latex or Tex system produces a DVI output. PDFTex can also produce DVI output. You may use pdfTex instead of LaTex using macro packages as context or hyperref or others to write the actual document. "When producing DVI output, for which one can use pdfTex as well as any other Tex, part of the job is delegated to the DVI postprocessor, either by directly providing this program with commands, or by means of \specials. Because pdfTex directly produces the final format, it has to do everything itself, from handling color, graphics, hyperlink support, font-inclusion, up to page imposition and page manipulation. As a direct result, when on uses a high level macro package, the macros that take care of these features have to be set up properly. Currently all mainstream macro packages offer pdfTex support in on way or the other. When using such a package, it makes sense to turn on this support in the appropriate way, otherwise one cannot be sure if things are set up right." (from the pdfTex User manual at http://www.tug.org/applications/pdftex/pdftexman.pdf). The following main macro packages support pdfTex: for LaTeX users the hyperref package by Sebastian Rahtz the standard LaTeX graphics and color packages have pdfTex options the ConTeXt macro package by Hans Hagen has extended support for pdfTex Literature and Sources: http://www.tug.org/applications/pdftex/ Michael Goosens; Sebastian Rahtz: The LaTeX Web Companion, Addison-Wesley, 1999: ISBN 0-201- 43311-7 Next Section: Preparing for conversion to SGML/XML Students/Preparing for conversion to SGML\XML Section SGML/XML Overview defines SGML and XML. The concept of Document Type Definitions (DTDs)Edit A document type definition (DTD), in the sense of XML, defines rules or templates, which are used to produce similarly, structured documents. A DTD describes the content model of a class of documents. It consists of: An element declaration, which is the main part of a DTD and the structural definition. Elements can contain other elements, characters or nothing. Element declarations define the name of the element and the logical content (sub elements) of an element. (See [10].) An important part of the element declaration is the content model. It is here that the document architect indicates the order and occurrence of other element or character data. A notation declaration, which defines a notation for external formats, e.g., for graphics (gif, jpeg), mathematics (TeX, LaTeX), 3D objects (VRML) and other formats, that cannot be coded directly in XML. An entity declaration, which defines character, sets and replacement objects for characters. Everything from a single character on up can be defined with a single entity. There are two basic types of entities: general and parameter. Parameter entities are only allowed in declarations, and are usually used to make a DTD more readable or to control processing. General entities are used in the document instance; the documents build upon the DTD. An attribute list declaration, where attributes and their values for the different element types defined in the element type declaration is listed. To define a DTD, a special syntax is needed, which does not conform to the usual XML syntax where a document contains elements which are enclosed in "tags:" a start tag (e.g. <author>) and an end tag (e.g. </author>), producing code like this: <author> Joe Miller </author> DTDs for electronic dissertations used worldwideEdit The fact that currently available authoring systems for XML still have not won wide recognition has led to different strategies at different universities regarding XML documents. Most of these projects were started between 1995 and 1997, in a time when XML was alive, but where no tools or standardized DTDs were available. A view of those projects from today's perspective illustrates the demand for a rethinking and redesign of those approaches in order to come to a standardization. All the presented DTDs are built upon similar principles. A classical dissertation (which can be seen as monograph) consists of 3 main components: an extensible title page with abstracts, declarations, etc., the dissertation corpus, which includes text, pictures, audio, video, tables and so on, as well as the appendices, which contain data sheets, bibliographies, acknowledgements and others. The following DTDs are currently in use at different institutions: ETD-ML.DTD: Virginia Polytechnic Institute and State University (Virginia Tech) DiML.DTD: German Dissertationen Online Projectes TDM.DTD: University of Iowa HutPubl.DTD: Technical University Helsinki TEI-Light.DTD: Ann Arbor und Lyon ISOBook.DTD: University of Oslo TEI-based DTD with extensions for natural sciences: Swedish University of Agricultural Sciences Uppsala All those Document Type Definitions are so-called author-DTDs. This means that they are primarily used to support the authoring and the conversion process and do not first of all address document archiving and preservation issues. One may ask why all those different DTDs have prevailed. This is mainly because the scientific orientation of the mentioned universities is quite varied. Lyon, Oslo and Michigan, which use TEI-Light.dtd, mainly serve students in the arts and humanities. Problems using TEI.DTD or DocBook.DTD are recognized at universities, which support a strong natural science community, such as Berlin, Helsinki or Uppsala. Often a dissertation is a cumulative work, e.g., in Lyon or Helsinki. Preparing for ConversionEdit Converting from word processing forms to SGML or XML requires more planning in advance, different tools, and broader learning about document processing concepts than does working with PDF. In addition, the end result is a representation that is easier to preserve, more reusable, and supportive of more powerful and effective schemes for searching and browsing. All of these advantages, however, must be weighed against the facts that there are fewer people knowledgeable about these matters, that often tools to help are more expensive and less mature, and that the process may be complicated, difficult, and time consuming. In 2000, there are tens of thousands of ETDs created by scanning (mostly by UMI, but also at sites like MIT and the National Document Center in Greece), thousands converted from word processors into PDF, and hundreds in SGML or XML – illustrating the relative effort required of students to prepare ETDs in each of these forms. Simple word processing emphasizes layout or what-you-see-is-what-you-get (WYSIWYG) editing. Emphasizing what documents look like is quite distinct from focusing on the logical structure, for which markup schemes are best. Shifting from word processing representations to XML, requires a different way of thinking, a different approach. The problem is harder than producing HTML by exporting from a word processor, since instead of just having a document that looks like the original, it is necessary that the marked-up version itself is correctly tagged. Some word processors have been extended to facilitate such an approach. Microsoft produced SGML Author for Word as an add-on package for Word 95, and new versions of WordPerfect can export content according to markup schemes. Eventually it is likely that most popular word processors will export to XML. Clearly, the resulting markup can surround document sections, headings, paragraphs, lists, figures, tables, citations, footnotes, hyperlinks, and other obvious constructs. In addition, regions with the same style can be tagged. Thus, to allow easy conversion from word processing to markup schemes requires choosing a target DTD and then consistently using document objects and styles so that there is a clear mapping from them to tags. Conversion from LaTeX is slightly simpler since the TeX approach involves using formatting commands that can be mapped to tags in XML. However, LaTeX does not require strict nesting of commands, so it may not be clear where to place end-tags. Further, LaTeX users may not consistently use the same sequences to designate changes in structure, making translation more complex. Finally, LaTeX coding of mathematical expressions is very difficult to translate to markup schemes for mathematics, like MathML. Because of the inherent complexity of converting from word processing schemes to markup representations, it is necessary to include steps for checking and correcting converted forms. Parsers can ensure syntactic correctness, so detecting problems is often simple. To ensure semantic correctness, however, manual inspection may be required. A further test would involve rendering the marked-up document, for example to a printed or PDF form, and ensuring that the result suitably matches the output resulting from the original word processing version. In any case, human labor is likely to be needed to correct conversion errors, and presupposes that students understand enough about the process and desired output to accomplish this with facility. [1] http://lcweb.loc.gov/cds/lcsh.html#lcsh20 [2] http://www.bibliothek.uni-regensburg.de/rvko/rvko.php3 [3] http://purl.org/DC/ [4] http://www.w3.org/rdf [5] Edward Fox: Networked Digital Library of Theses and Dissertations, Web matters, Aug., 12th 1999, http://helix.nature.com/webmatters/library/library.html [6] Website of the standards committee of NDLTD: http://www.ndltd.org/standards/ [7] http://dochost.rz.hu-berlin.de/epdiss/dtd-workshop/index.html [8] Tad Lane, Scalable Vector Graphics - Web Graphics with Original-Quality Artwork, in: BITS, November 1999, http://lanl.gov/orgs/cic/cic6/bits/november_99/novbits1.html [9] Neill Kipp: Beyond the Paper Paradigm: XML and the Case for Markup; in: Part II "Guideline for Writing and Designing ETDs" ETD Sourcebook, Weisser, Moxley and Fox editors, 1999 [10] B. Travis, D. Waldt: The SGML Implementation Guide, Springer, Berlin- Heidelberg-New York, 1995 [11] Ed Dumbill: The State of XML, June, 16th, 2000 in XML.com, http://www.xml.com/pub/2000/06/xmleurope/keynote.html Next Section: In MS Word Students/In MS Word Performing a conversion from MS Word documents into instances of a specified SGML or XML DTD is a very complex task. What you will need for that is: A SGML or XML document type definition (DTD) that serves as structure model for the output. One says that the output SGML document is valid to the specific DTD, or it is an instance of this DTD: A Word style sheet that holds paragraph and character styles according to the structures in the DTD. So if in a DTD you have defined a structure for Author (e.g. expressed in the output file as): <title>Dr.</title><firstname>Peter</firstname><surname>Fox</surname> You have to find expression in Word: paragraph styles: author character styles (just to be used within an author-paragraph): firstname, surname, title You will need some kind of a configuration file that allows the mapping of the DTD elements into Word elements and vice versa. You will need an SGML or XML parser to check the output SGML/ XML document against the DTD. Often a conversion is done by using a plug in to MS Word directly. But other options use the Microsoft internal exchange format RTF (Rich Text Format) for conversion. Those tools can interpreted the RTF file with the MS Word style that are still coded in this RTF file and export it into an SGML document. This process mostly happens within batch mode without using much graphical user interfaces. Within the following paragraphs we describe several approaches: 1. Approche of the Université de Montréal, Université de Lyon 2, Universidad de Chile 2. Humboldt-University Berlin and Germanwide Dissertation Online project There are other approaches in development as well, especially within Scandinavia and the University of Oslo/ Norway. We don't refer to their solution yet. Conversion method of the Cyber theses projectEdit The process line for converting Word files into SGML documents developed within the CyberThèses project uses scripts written with the Omnimark language. The input of the process line is an RTF file with a "structuring style sheet" and the output is an SGML document encoded according to the TEI Lite DTD (see the TEI web site at http://etext.virginia.edu/TEI.html). The conversion process is constituted of three main steps : a first one converts the RTF file into a flat XML file encoded according to DTD of RTF. The produced file is a linear sequence of paragraph elements having each one an explicit "style name" attribute corresponding to the RTF style names. the second step consists in the re-generation of the hierarchical and logical structure of the document based on the analysis of style name attribute. last, a SGML parser allows validating the conformity of the produced SGML document with the TEI Lite DTD. Some supplementary scripts then allow the export of the SGML document towards other formats (HTML, XML). Most of the scripts are available from the CyberTheses web site : http://www.cybertheses.org This system is devoted to a particular DTD, but its generalization to other document models shall not raise any difficulty. Using SGML Author for Word (Humboldt-University Berlin)Edit Why did we use the SGML Author for Word? The "Dissertation Online" project implemented and refined a conversion strategy that allows to convert documents written in MS word with a special style sheet (dissertation.dot) into an SGML instance of the DiM.dtd. We used this product from Microsoft, the SGML Author for Word, due to several reasons: SGML Author is quite easy to configure It is easy to use. It is less expensive than other software producing SGML files with the same quality. It supports an international standard for tables: CALS. As it is a Word-Add-On it handles documents in MS-Word doc- format better than other tools. As we started using this technology in 1997, it supported from the very beginning Word97, the version of word, which was the actual one that time. Unfortunately, Microsoft didn't continue the development of this tool. So there are new versions available for Office 2000, Office XP, or Office 2007. But the internal document format from MS Word 97, MS Word 2000, Office XP, and Office 2007 are the same in the sense of the conversion into SGML. This means documents written in Word 2000, Office XP, or Office 2007 can be imported into Word97 and therefore a conversion can be done. For a successful conversion from a word document into a DiML document you will need: The DiML-document type definition (diml20.dtd, calstb.dtd) the SGML-Author for Word97 (may not available at Microsoft Shops any more, but NDLTD esp. Prof. Dr. Edward Fox may provide English versions of it that work with English Word) The Association file for the Microsoft SGML-Author for Word (diml20.dta) The converter style sheet, which consists of several macros programmed to make the preconversion process easier. The perl programming language (free Software) The nsgmls-Parser (free Software) Several perl scripts to correct the transformation of tables. You must have the following software installed at you computer: SP (NSGMLS) (Parser for SGML-Files by James Clark). (new version are available at http://openjade.sourceforge.net/doc-1.4/index.htm, but we haven't tested that) Run SP (A WYSIWYG tool for SP by Richard Light). http://www.light.demon.co.uk/runsp/index.htm Perl (a scripting language for using the perl scripts). The converter style sheet and the author's style sheet can be obtained from the following website: http://dochost.rz.hu-berlin.de/epdiss/vorlage.html Converter scripts and perlscripts can be obtained from http://www.educat.hu- berlin.de/diss_online/software/tools.exe (Perl scriptc, DTD and converter file for MS SGML-Author for Word - KonverterDiML2_0.dta) The conversion from a Microsoft Word document into a SGML document, which is an instance of the DiML.dtd that is used at Humboldt-University, takes several steps: 1. Step: Preparing the conversion without using the converter Microsoft SGML Author for Word directly Check the correct usage Load the style sheet for conversion (NOT the one for the authors) see, see figure below. There is a special feature to get the page numbers out of the Word document by using certain word specific text anchors. Those have to be converted into hard coded information using a page number style sheet. Formatting that has been applied by the author without using style sheets have to be replaced by the correct style sheets. In order to get a correct display of tables later on by using CSS style sheets within common browsers, empty table cells have to be filled up with a single space (letter). Soft coded line breaks have to be preserved for the conversion. This is done by inserting special characters #BR# to that. This will be used to insert later a special SGML tag for soft line breaks 2. Step: Converting with Microsoft SGML Author for Word Press the button "Save as SGML" within the FILE menu. Load the converter file KonverterDiML2_0.DTA Check the XML/SGML output using the feedback file (fbk) see figure below. 3. Step: Work through the output file (output according to the DiML.dtd)automatically. Load the perlskripts using the batch file preprocessor.bat Parse the DiML file Errors have to be wiped out manually 4. Step: Transforming the DiML file into a HTML file Load the perl scripts by using the batch file did2html.bat Check the HTML Output. Correct possible errors manually within the SGML file and repeat the transformation. A demonstration QuickTime video may be found at the ETD-Guide server as well. (see http://www.educat.hu- berlin.de/diss_online/software/didi.mov) Other ToolsEdit Text editors, Desktop Publishing Systems that can export SGML/XML documents Tools that export using a user specified [1] DTD: WordPerfect since Version 7.0 (Corel http://www.corel.com ) FrameMaker+SGML6.0 (Adobe) (http://www.adobe.com ) Tools that exports using their own native[2] DTD: Openoffice (SUN/open source ) (http://www.openoffice.org ) AbiWord (AbiWord/ open source) (http://www.abisource.com ) Kword (KOffice, KDE Project/ open source) (http://www.kde.org ) Converter ToolsEdit Omnimark (Omnimark) (http://www.omnimark.com ) MarkupKit (Schema) (http://www.schema.de ) Majix (Tetrasix) (http://www.tetrasix.com ) TuSTEP (RZ Uni Tübingen) (http://www.uni-tuebingen.de/zdv/tustep/index.html) [1] Bollenbach, Markus; Rüppel, Thomas, Rocker, Andreas: FrameMaker+SGML5.5. Bonn; Reading, Mass., Addison-Wesley Longman, 1999, ISBN 3 8273 1508 5 [2] St. Laurent; Biggar, Robert: Inside XML DTDs. New York, McGraw Hill, 1999, ISBN 0 07 134621 X [3] Ducharme, Bob: SGML CD. New Jersey, Prentice Hall, 1997, ISBN 0 13 475740 8 [4] Smith, Norman E.: Practical Guide to SGML/XML Filters. Plano. Texas, Wordware Publishing Inc., 1998, ISBN 1 55622 587 3 [5] Goldfarb, Charles; Prescod, Paul: XML Handbuch. München, Prentice Hall, 1999, ISBN 3 8277 9575 0 Invalid ISBN Next Section: In WordPerfect Students/In WordPerfect WordPerfect supports structured writing since version 7. Most of the following text has been taken from a Whitepaper by Corel Inc. that appeared for WordPerfect 9 in June 1999. Writing XML using WordPerfect means that the author is forced to use specific structures. The software will parse the file by an underlying XML parser and check correctness of the written file while it is being written by the author. This may cause the system to slow down sometimes because the parsing process may take a lot of the system's resources during that time. Generally, the writing behavior is different than the one for systems like Microsoft Word, where no parsing during runtime is being performed. As the author, you will have to think about your document structure, on how the information pieces are put together and follow each other (e.g. if something is supposed to be a heading or the beginning of a new chapter). Standard WordPerfect 9 templates allow users to embed XML components. These WordPerfect templates include the following XML components: Document Type Definition (DTD) Compiled version of the DTD, a Logic file (LGC) Layout Specification Instance (LSI) Alias File (LNM) The Document Type Definition (DTD) defines the elements and the structured relationship between the elements, entities and attributes. The DTD defines all valid elements; the order in which they can be used and how many times a particular element can appear in a document. When compiled as binary logic file (LGC) by the Word Perfect DTD Compiler, the DTD is integrated to a WordPerfect Template along with the Layout Specification Instance File and the Alias File. The Layout Specification Instance (LSI) specifies formatting information, such as bold, underline, italics etc., for the start and end tags. Layout files can associate elements with their respective WordPerfect styles. It is possible to insert a specific WordPerfect formatting command or a text string before, after or in place of a specified XML element. The .lsi file can run macros when users insert a specific element, so template designers can build dialog boxes, prompts and other help to the authors in order to support them in writing correct XML documents. It is possible to use several .lsi files with the same DTD. This is useful if multiple output formats have to be produced from the file, e.g. one for printing and one for online publishing. When compiled the .lsi file integrates to a WordPerfect template. An Alias file (LNM) specifies the descriptive names for elements in the DTD. This is useful when the tag names defined in the DTD are not appropriate for the end user. This may be because templates are created for end users who speak a different language or are created for a different audience. For instance non-technical users may relate to different tag names than technical users. WordPerfect also provides standards templates to the user and allows the creator to customize the user interface with menu items, toolbars and view of specific documents. All those elements may include and run macros. An example of a DTD, a LSI file and a WordPerfect template for digital dissertations used in Germany (Humboldt-University Berlin) is provided to the users of the guide. If you want to write a dissertation in XML using WordPerfect you have to work with the SGML/XML functionality. Please consult the information center of the installation CD-Rom of the WordPerfect suite with the keyword SGML. A detailed installation and usage guide will be shown. You will need the following files: diml1_0.dtd diml1_0.lgc Logic file diml.lsi Layout file diml.lnm Alias file dissertation.wpt Dissertation style sheet WordPerfect provides the user also with the following programs: a XML File Wizard, a XML Project Designer, a XML DTD Compiler, a structured Tree View, In order to compile a DTD into a Logic file, you have to follow the steps: 1. copy the following files into the corel\suite8\programs\mapfiles directory: diml1_0.dtd; cals_tbl.dtd;hubspec.ent 2. Edit the iso8879.map file in a text editor: Add the lines: PUBLIC "-//HUBspec//ENTITIES Special Symbols//EN" "hubspec.ent" PUBLIC "-//HUB//DTD Cals-Table-Model//EN" "cals_tbl.dtd" Press Compile. Now the LGC file has been produced and is usable for writing an SGML/XML document. You can now open WordPerfect and choose Tools / SGML / Document types. Here point to the created diml1_0.LGC file and if you want to use a standard layout to the diml.LSI file. Your own layout can be produced using the Layout Designer program that is in the WordPerfect suite. Just choose open diml1_0.lsi or open a new file and start creating or manipulating your own layout style. Start writing a dissertation directly with SGML/XML by choosing under WordPerfect the option Tools / SGML / Document types. Usually there has been a native WordPerfect template produced in order to help you with additional drop down menus etc. To use the provided template for WordPerfect 8 you have to choose File / New / Options / Add Project / Add new document / Call it "Digital Dissertation" / Search for dissertation.WPT. This allows you now to use the WordPerfect style sheet by choosing under the main window: File / New / Choose "Digital Dissertation" and Create. This enables the functions of the drop down menu as in the following figure. Next Section: In LaTeX Students/In LaTeX The ProblemEdit If ETDs should be archived for he next 20–50 years and still be readable and usable, it in necessary, that equally to the approach for MS Word, we use predefined style sheets for LaTeX. Only by standardising the usage of Latex, a quick and sustainable solution for a conversion into XML can be designed. As LaTeX is mostly used within the natural sciences and mathematics, the encoding of complex mathematical symbols, formulas and expressions is one of the major problems for such a conversion. As there are XML document type definitions or schematics for mathematics, MathML (see http://www.w3.org/math ) and most math software, like Maple, Mathematica, etc. supports an export into MathML, this standards has to be used as an output from LaTeX as well. The LaTeX format should enable an easier conversion to XML, because of its structured approach to text processing. But the usage habits of LaTeX users, which tend to program complex macro packages in order to style a sophisticated print layout make it much more difficult to receive homogeneously structured documents in most cases. Also does the not parseability for structural and syntactical correctness complicates a conversion. Converting mathematical expressions into XML can be done using 3 different strategies: Convert them into graphics that are easily interpreted and presented by common Internet browsers. Here a search within formulas or a further usage is excluded. To convert them into MathML, To leave them in a LaTeX encoding within the XML file. Then Plugins like IBM Techexplorer, or Math Viewer are able to interpret the LaTeX code and produce an on-the –fly rendering of formulas and mathematical expressions. There are semantic differences in LaTeX between the encoding of formulas. So authors have to be aware of the differences of LaTeX –tags or commands that are on a semantic level and on those, which are on a layout, level. Pi represents the mathematical constant, which is the ratio of a circle's circumference to its diameter, approximately 3.141592653. Encoding this in MathML <pi> <apply> <approx/> <pi/> <cn type = "rational">22<sep/>7</cn> </apply> This will be rendered as follows: Instead of coding it simply as letter pi, which may stay as a name for a variable: This would be rendered as: pi ≈ 22 / 7 Software and ToolsEdit In order to produce an XML document out of a LaTeX document there are several possibilities: TeX4ht is a highly configurable TeX-based authoring system for producing hypertext. It interacts with TeX-based applications through style files and postprocessors, leaving the processing of the source files to the native TeX compiler. Consequently, TeX4ht can handle the features of TeX-based systems in general, and of the LaTeX and AMS style files in particular. (http://www.cis.ohio-state.edu/~gurari/TeX4ht/mn.html ) WebEQ : a Java-based collection of tools for authoring and rendering MathML, including a visual editor, a WebTeX to MathML translator, and a rendering applet for interactive mathematics on Web pages. WebEQ also provides Java Programmers with API documentation and libraries for other MathML aware applications. (http://www.dessci.com/de/features/win/default.stm#TeX or http://www.dessci.com/features/win/default.stm#TeX ) For further information on the usage of different tools, please use: Michael Goosens; Sebastian Rahtz: The LaTeX Web Companion, Addison-Wesley, 1999: ISBN 0-201- 43311-7 Next Section: Checking and correcting Students/Checking and correcting A. Important Parts for a checking procedureEdit After an author has written his or her ETD, the service institutions, like the university library or computing and media center, have to check whether the ETD is complete, readable and correct. Therefore, a checklist is useful. The checklist should consider the following parts of a document and should also be open to the authors for self- checking: Organizational questions The WinWord, WordPerfect or LaTeX document The PDF-Version of a digital dissertation Checking organisational questions After receiving an ETD from an author, several organisational issues have to be proofed: Does the student belong to the university? Has he passed his exams? Has he passed the approvals? Has he missed deadlines or not? Checking the WinWord / WordPerfect document For the application of style sheets the following parts of a word document are mostly critical for further usage: Can the document as a whole been opened within the WinWord or WordPerfect systems at the service institutions? Has the student used all style sheets correctly? Has the students used the heading feature of WinWord or WordPerfect? Is the title page fully styled and all information filled in? Do figures and tables have own captions, and has the insert caption feature of the text formatting system been used? Are lists styled as lists? Are Tables produced using the table features from the text formatting system? (Sometimes authors use the tabulator to build tables) Has the authors used a reference managing system for the references? This makes the automatic formatting of the bibliography much easier and enables a linking into the text parts. Did the author use the automatic spell-checking features instead of applying hard coded – letters in the text? Checking the LaTeX document Have the guidelines and rules been followed by the author? Has the template or styles file been used? Has the author used bibtex to collect the references? Have all figures been provided in an EPS (encapsulated postscript) format? Have all additional styles been provided by the author? Is the whole ETD processible at a computer of the library or the computing centres? Checking the PDF document Is the PDF document readable? Can it be opened within the actual version of the Adobe Acrobat Reader? Does the PDF file contain all text pars? Does it contain hyperlinks for multimedia additions? Do the hyperlinks work within an actual Internet browser? Checking the metadata Has the author provided all Dublin Core metadata requested about himself and his thesis? Are there keywords in different languages, according to different classification schemas? Are there abstracts in different languages? In which format has the metadata information been provided? B. Checking for an SGML/XML-based publication and archiving workflowEdit An SGML/XML-based archiving strategy today consist on a conversion workflow, as shown: Workflow Version 1: The Conversion from native text formatting formats (like doc WinWord) into SGML/XML-compatible documents is done by a service of the university. Here checking and correcting basically consists of the checking of the adequate usage of the style sheets and the guidelines provided by the university. This procedure is in practice at Humboldt-University Berlin, at Université Lyon 2, Université de Montréal, … This checking has to be done within the word processing systems used, e.g. WinWord and can e.g. partially automated by Macros defined within the word processing system's macro language, e.g. Visual Basic. Then, the checking person of the staff runs this checking macros and can find out, whether special styles have been used or not. This checking procedure has to be added by a manual checking of the correct usage of the styles applied to the document by the author. For this reason it is very helpful to have a checklist for each document. This will ensure the level of strictness for the style control equivalent for every document. Workflow Version 2: The conversion from native text formats (like doc WinWord) into SGML/XML- compatible documents is done by the author himself or the author writes directly in SGML/XML. Here checking and correcting basically consists of the checking of the adequate usage of SGML/XML by an SGML/XML parser and concentrates especially on the correct usage of the document type definition (DTD) provided by the university and the guidelines. This procedure is e.g. in practice at the University of Iowa, Iowa City. This can completely automated by an SGML/XML-parser and checking scripts that produce an error list for the checking staff and can save a lot of time in comparison to the previously described workflow. The disadvantage of this model from today's perspective is that a very comprehensive author's support has to be designed and carried out, in order to enable authors either to perform an initial conversion from any text formatting system into an SGML/XML compliant document or to use an SGML/XML editor in a way that allows the author to understand and interpret messages. Next Section: Integrating multimedia elements Students/Integrating multimedia elements Inaccessible dead media has little value. Students who are incorporating moving images, audio and live data streams into their ETDs should not underestimate the work involved in managing these resources. How these resources are created, and the form and format they are created in, will determine how your ETD can be managed, used, preserved, and even re-used in the future. When your ETD enters a networked electronic environment, it does not exist in isolation. It becomes part of a boundless resource space in which descriptions of the work and its components (metadata) need to be in an internationally recognized form if the work is to be accessible. Inter-operable standards that will allow for this metadata to be understood by everyone (even machines) are now available. However, the application of these standards requires a new form of collaborative relationship between you, the creator, and indexing initiatives such as the NDLTD. This means that you, the creator, have to take responsibility for describing each layer or 'object' of content as an independent entity capable of being accessed and manipulated in its own right. From its conception, a compound digital resource should be seen as a composition of objects in an encoding architecture that expresses the spatial and temporal relationships between these objects. These objects may be audio (mono or stereo) and visual (2D or 3D) or text. They may also be composed from several sources. They may be simultaneously acquired, processed, transmitted and used in real time. If, in the future, the encoding of these objects is to be decoded, the metadata must include format information. In the interests of interoperability, it is good practice to select formats from the list of Internet Media Types (MIME values) whenever possible (this list is a registry where there is a procedure for adding new types, if necessary). Available [on-line] http://www.isi.edu/in-notes/iana/assignments/media-types/media- types/ An important (proposed) standard that multimedia content providers should investigate is the Synchronized Multimedia Integration Language (SMIL). This standard will allow hypertext creators to define and synchronize multimedia elements (video, sound, still images) for web presentation and interaction. Available [on-line] http://www.w3.org/AudioVideo/ Next Section: Providing metadata – inside, outside documents Students/Providing metadata – inside, outside documents There is nothing new about the concept of metadata. Metadata is resource description; the kind of information found in a library catalogue. What is new in the digital world is the essential role that you, the creator, now play in providing this information. Good quality metadata is easy to provide at the point of creation but usually difficult, expensive or impossible to discover retrospectively. At one level, this is because all digital resources are in some way dependent on electronic mediation by computers and software and it is only at the point of creation that a record of these dependencies and descriptions can be recorded. At another level, it is the sheer volume of creation that alters the role of the librarian or custodian from cataloguer to metadata repository manager. In an ideal world, all digital material would be created independent of proprietary hardware and software. In other words, everything would run on commonly available hardware using freely available (public domain) software such as a web browser. In the real world, many content creators will be producing work on-line or off-line that is either hardware or software dependent (or both). Unfortunately, the costs of emulation, migration and licensing increase if resources are generated in proprietary or platform dependent formats. If possible, try to use commonly available open source formats. Metadata is information about these applications and formats, which allows for licensed versions to be archived so that the material can be displayed or accessed. In order to be able to provide long-term access to a digital resource, the NDLTD needs the following metadata: Information about the content creator (rights, contributors, publisher,); Information about the content that will help it to be found or discovered (coverage, description, title, subject, relationships); Information about the resource (formats, system requirements, date, identification). Storing metadata Metadata can be stored in: 1. The object or document being described. There are a growing number of audiovisual formats that allow for metadata to be embedded in the file itself. For example, a text format like HTML allows you to embed metadata in the header of the file and recent versions of image formats such as MPEG include space for metadata. This has the advantage that the information is self-contained and is truly transportable across systems. The major disadvantage is that systems accessing the object will have trouble catering for multiple views or meanings. 2. A separate file that can be externally accessed but is linked to the object or document. This has the advantage that different communities can gather the metadata for different purposes. It has the disadvantage of being open to misinterpretation through syntax error or unrecognised schema. 3. A separate file stored in a database. The NDLTD model encourages students to submit their metadata to a central repository for indexing in a database. The database will then point to the object/document. This also allows for multiple instances of the metadata for one document. It also provides for enhanced administrative tools (as are normally provided by database systems). Advanced database systems could provide a very sophisticated management system. This is the most expensive method to implement but it has the advantage of being significantly more flexible and provides administrative support from the outset. See: European Projects such as Metadata Observatory. The aim of the Observatory is to maintain and promote a knowledge base for metadata for multimedia information to continually assess relationships between Dublin Core and other initiatives, especially undertaken in Europe, in order to assist evolution of standardised metadata schemes. See: The Dublin Core Metadata Initiative. This is an open forum engaged in the development of interoperable online metadata standards that support a broad range of purposes and business models. DCMI's activities include consensus-driven working groups, global workshops, conferences, standards liaison, and educational efforts to promote widespread acceptance of metadata standards and practices. Available [on-line] http://dublincore.org/ Next Section: Protecting intellectual property and how to deal with plagiarism Students/Protecting intellectual property and how to deal with plagiarism Protecting against plagiarism may involve specific steps as explained below, in addition to action in accord with the discussion above in section ETD Guide:Universities - Plagiarism. First, there is software available to detect plagiarism, or, in simple terms, similar copies. Stanford's SCAM software, and other software developed in Australia and other locations, discussed in the digital library literature and elsewhere, is available. This software can compare two documents, finding sections that are identical or similar (e.g., up to within simple substitution changes). If such software were to be widely run, it could ascertain for each new work if there were an ETD previously submitted that is very "close". While determined authors might defeat such software, more refined software could be developed to highlight cases where even such protection efforts were involved. Second, it should be noted, as stated in section ETD Guide:Universities - Plagiarism, that if a work in made widely available, it is more likely that someone, seeing a new ETD that involves plagiarism, would detect that situation. In effect, as more and more ETDs are made accessible, there is a larger community monitoring abuses. Third, since it is known that ETDs are often read by many more people than would read a paper dissertation, there is a strong psychological pressure to discourage plagiarism. One aspect of this is that students are aware of dire penalties if plagiarism is detected - they may be removed from their degree program and forever disgraced. Another aspect is that faculty working with students on ETDs, being aware that many might read the work, are likely to be more diligent than with paper works with regard to checking the validity and quality of results reported. In other words, it becomes more likely that faculty will carefully study ETDs that have their name on them as advisors or examiners. In particular, it is exceedingly unlikely that a student would be able to write about work they had not done, without that being detected. In summary, detection by machines or people, and the threat of severe penalties, are likely to discourage students from even considering plagiarism with regard to ETDs. Next Section: Naming standards: file names; unique Ids Students/Naming standards: file names; unique Ids The ADT Program uses two simple protocols to describe – the PDF files associated with a thesis, and the generation of a unique ID for each thesis. ADT Program filename standardEdit To facilitate access to ADT theses it is important that the following filename standards are followed for naming the PDF files prior to uploading them through the Deposit form: Please note - ADT filenames should only contain: Letters from the alphabet (upper or lower case). Example: 05appendix.pdf; 05APPENDIX.pdf; 05Appendix.pdf Or numbers (0-9) Or any appropriate alphanumeric combination. 02Chapter1.pdf; 09Chapter10.pdf The only symbols permitted are - (hyphen) or _ (underscore). Example: 06Appendix-Images.pdf; 06Appendix_Images.pdf Create one PDF file containing title/author information; abstract; acknowledgments; table of contents; introduction; preface and any other introductory text that is not part of the main body of the thesis. Call this file - 01front.pdf Create another file containing the whole of the thesis (including what is contained in the front.pdf file) and call this 02whole.pdf …alternatively… Break the thesis up into smaller files maintaining a relevant filename structure such as 02chapter1.pdf; 03chapter2.pdf; 04appendix1.pdf; 05bibliography.pdf; etc. In order to maintain the desired file sequence the files need to be numbered using the standard 01- 99 (or 001-999 if individual files exceed more than 99 for an individual thesis). Numbering theses is required no matter which filename standard is adopted. 01front.pdf 02whole.pdf 02chapter1.pdf 04appendix.pdf 05bibliography.pdf The 01front.pdf file is compulsory. The official ADT v1.0 or v1.1 software will create an empty front file automatically if a real one is not uploaded during the deposit/submission process. This is to serve as a memory prompt for administrators. While it is difficult to recommend exact file sizes the project team suggests that the whole.pdf file should be within the 2-4MB range. For larger, and hence multiple file theses (necessary for scanned ones) it is suggested that the file size should be in the 4-10MB range. This standard will allow viewers to easily identify the relevant parts of the ADT theses via a common filename structure. It will also allow viewers to quickly look at the table of contents and other introductory information without having to wait for the whole thesis to download, or to better determine if they want to see the whole thesis in the first place. The standard is also an effective and simple way to present an introductory view of a thesis at the outset. This is similar in a way to the 20-25-page view that UMI/Bell&Howell have introduced for the DAI database. Similarly, this ADT standard will facilitate e- commerce transactions if participants choose this option. This would work just as the UMI/DAI model - the metadata/abstract and front.pdf file would be available freely, with charges kicking in to access the rest of the theses' pdf files. ADT Program unique IdsEdit The combination used by all ADT members is as follows: the word "adt-" immediately followed by the unique national institution code as per the Australian Interlibrary Resource Sharing Directory, immediately followed by the year, month, day, hour, minute & second/s the thesis was deposited to the local server. As the ADT Program is a distributed national model, it is critical that the URLs are unique within that context. The combination above ensures this. Examples of the other institutions would be: ../adt-ANU20010223.162256/ (Australian National University) ../adt-WCU19991112.125812/ (Curtin University of Technology) ../ adt-NUN20010510.153038/ (The University of New South Wales) ….and so on. Next Section: How to submit your ETD? Students/How to submit your ETD? Universities requiring or accepting ETDs should provide students with basic levels of support such as hardware (e.g., computer workstations), software (including programs for creating the works such as word processors, as well as other software such as sound and image programs), and opportunities for training in how best to use the hardware and software and how to submit their ETDs. Next Section: Local support Students/Local support The ADT Program is an Australian university libraries initiative. The library at the ADT member institution provides local support. Support to academics and students is varied and depends on the levels of expertise of the individuals involved in the submission process. The levels of support offered include: provision of workstation/s, including software, in the library dedicated to the submission process plus library technical support as appropriate to aid students to submit and convert files into PDF provision of basic instructions and information for students wanting to submit theses independently. See example of this at The University of New South Wales: http://www.library.unsw.edu.au/theses.html#dep library technical expertise to entirely do the submission & conversion process on behalf of students provision of information via the library's website, pamphlets, presentations, formal letters & information for faculties, schools, graduate committees, academic staff and students about the local ADT Programs. The sharing of this information by the ADT member institutions as well broader support at the institutional level - sharing expertise, knowledge and information as appropriate All library support is seen as part of the normal university scholarly process and is provided free of any charge. Next Section: Typical workflow, local policies and procedures Students/Typical workflow, local policies and procedures Apart from the ADT Program support processes described above, local workflow and policies within the ADT membership will vary. Typically, what is most common across all members is that the libraries are driving the ADT Program through their respective institutions, providing all levels of support and provision of information, and that only research theses are acceptable - i.e. PhD or Masters by research. What is less common is the local procedures insofar as how theses are deposited to the local server, who has the authority to approve/make public and administer the program and how the ADT theses are integrated within the local IT infrastructure. Access to theses also varies between institutions. While most encourage and actively support free and unrestricted access, some take a much more conservative approach and restrict all theses to the local domain. Obviously, all institutions will apply some temporary restrictions to access if required, for reasons of patents pending, copyright and upcoming publications. An example of workflow, policies & procedures at The University of New South Wales [UNSW] Library is as follows: UNSW Library provide support as required, see 3.4.1. above. When theses are deposited, auto alerts are sent to the student, supervisor, the ADT coordinator and the cataloguers The cataloguers are responsible for checking the theses are original and have been awarded. They edit the metadata to include appropriate thesaurus terms/subject headings. The cataloguers are also responsible for approving the theses and making them public. They also catalogue the theses into the local OPAC and the National Bibliographic Database. These ETDs are then available via the national distributed ADT database, the local view of the ADT database [i.e. UNSW-ETDs only], the local OPAC and the National Bibliographic database. Most are available unrestricted with a few restricted to the local domain due to copyright reasons Two separate units are responsible for the ADT Program at UNSW Library. The General Services Department's Learning Support Unit provides all support, promotion and associated information. The cataloguing department is responsible for administration of the ETDs when deposited to the server. The ADT coordinator at UNSW is also the overall coordinator and is responsible for liaising and guiding the program at the national level. Next Section: Becoming a researcher in the electronic age Students/Becoming a researcher in the electronic age In 2001, the National Research Council of the USA produced a booklet identifying many of the responsibilities, opportunities, and other issues faced by young faculty as well as graduate students completing their theses or dissertations. Similar groups have prepared other works. In particular, it is clear that there have been dramatic changes in scholarship that have resulted from the availability of computer tools, the shift to digital libraries, and the tremendous increase in resources available to young researchers (with respect to computing, networking, and content). Such researchers should learn about electronic publishing, should use digital libraries, should be aware of intellectual property rights, and should leverage new opportunities made available through the enhancement of related technologies and infrastructure. These all encourage involvement in ETD activities, and lifelong participation in the world of scholarly communication that in many cases was first made visible to them in connection with their ETDs. Next Section: Technical Issues To efficiently and effectively implement an ETD program, involved institutions need to develop a suitable technical infrastructure – a side benefit and related goal of the initiative. This section outlines the key aspects of the technical portion of an ETD effort, covering production, dissemination, and access. Next Section: Contexts: local, regional, national, global Technical Issues/Contexts: local, regional, national, global World Diversity Our world is a very diverse place. We have many geographies, climates, races, cultures, and languages. Another characteristic that differentiates places of the world is the level of development. This creates various levels of access to: Housing, electricity and sanitation Information and knowledge World Infrastructure Infrastructure is different in different regions of the world too. Third world nations have characteristic social/economic gaps among groups of persons and/or regions within the countries. In one nation, developed areas, where people have access to all the items listed above and where infrastructure is good, coexist with very poor regions where living conditions are bad. This coexistence leads to a varied range of infrastructure levels in general and in universities too. There are universities with very good campuses and with the infrastructure comparable to those in developed nations. Others have bad installation, lack equipment and do not have good networks and Internet connections. This bad fortune reflects on students, faculty and staff who may not be proficient with information technology tools. When an ETD program is considered, some aspects of the infrastructure must be examined. They can be grouped in 3 categories, as follows. Local (in the university) infrastructure The level of automation of the library in terms of cataloging of the collection, library system, equipment for the staff and for end users, etc. The number and quality of machines available to students The number and quality of machines available to the administrative staff The network conditions - connection to all university buildings, speed, reliability and support The level of computer literacy of students, faculty, library staff and administrative staff The number of machines connected to the Internet The connection of the university network to WAN's and the Internet - speed, reliability and support Regional and national infrastructure The network connections - speed, reliability and support; The existence of other ETD and/or digital library projects to develop the culture and to seek/provide support The possibility of funding, from agencies and/or private companies, to ETD programs Global The dissemination of information on ETD and/or digital library projects to provide support to those who want to start programs The agreement on minimum standards for systems, technology and metadata to allow interoperability and seamless access The discussion and agreement on languages to identify ETDs so that they can be searched, retrieved and used Next Section: Networking Technical Issues/Networking Networking and ETDs With the spread of the Internet and the WWW, and the emergence of local area networks as well as wide area networks (LANs and WANs), network facilities have enormously expanded in many of the educational institutions of the world. To support the needs for access, and the related processes that deal with making content accessible, many universities have made networked computers available to their graduate students, as well as to faculty, staff, and undergraduate students. The ETD initiative can build on those investments, employing them to support submission and downloading of ETDs. By focusing on networked access, there can be considerable savings that result from elimination of manual handling and physical distribution. It is recommended that, except for ETDs that have enormous amounts (e.g., gigabytes) of multimedia content, authors, as well as university staff, avoid procedures that require transfer of content using diskettes, CDs, or other physical media. Network Traffic and Hardware Universities should consider the amount of traffic on their networks, making sure that networking hardware (Internet connection, routers, and cable plants) accommodates well the demand for uploading and downloading ETDs). This accommodation generally will not be a problem if adequate support for email, rapid access to WWW, and other types of usage is provided. Regarding software for accessing ETDs, there is not much of a special nature that is required. Web browsers, support for Java applets, and multimedia presentation tools are typically sufficient. Special aids may be needed for PDF, that Adobe provides for free at Adobe at www.adobe.com, or SGML/XML (becoming more widely available). The real need for special software is to help manage the ETD submission process, handle local workflow, make ETDs accessible, and facilitate search. NDLTD and its various parts have prepared software to help with all of these. Next Section: Seamless access: Open Archives Initiative, federated search Technical Issues/Seamless access: Open Archives Initiative, federated search Access to ETDs that are produced by students around the globe requires some mechanism for connecting with the many computers that house those ETDs. There are two basic approaches. Federated Research In federated search, a user's information need, expressed in the form of a query, is sent by the federated search system to all the sites that support the searching over local ETD collections. Then, when the sites have completed their searching and generated results, either the user can view each site that might have some relevant content (see Powell & Fox, 1998), or some type of fusion of results leads to a single merged list (as with Dienst, see Lagoze & Davis, 1995). While federated search yields up-to-the-moment results, such currency is usually not of high priority in the ETD world (where daily updates should suffice). At the same time, federated search may involve complex timeout and backup site management, if some remote sites are down or slow to respond. At best, federated search is often slow (due to network delays) and suffers from having to manage a wide diversity of representations of data at remote sites, leading in some cases to low data quality. Nevertheless, see such a service for ETDs from www.theses.org. Harvesting is the second basic approach. As is explained in section Harvest usage in Germany, France, the Harvest system first clearly demonstrated this solution, and is still in use in Germany and other locations. However, this is being superseded by the Open Archives Initiative (www.openarchives.org, see Lagoze & Van de Sompel 2001). NDLTD has developed a harvesting-based OAI access scheme for handling the global collection of ETDs; see Suleman et al., 2001 (parts 1 and 2). The basic outline of the approach is as follows: Each ETD is described (with metadata) using MARC21 or ETD-MS (http://www.ndltd.org/standards/metadata/ETD-ms-v1.00.html). Each ETD site runs an open archive, which responds to OAI requests for metadata by providing Dublin Core records, as well as either (or both) MARC21 and ETD-MS. For example, the software for ETD management developed by Virginia Tech has such a capability (http://www.dlib.vt.edu/projects/OAI/software/ndltd/ndltd.html). State, provincial, national, regional or other organizations may harvest from these sites, and run their own open archives and related services. Virginia Tech harvests from all sites (or group sites that have harvested for university sites) to develop a union collection (http://oai.dlib.vt.edu/~etdunion). VTLS Inc., as a service to NDLTD, provides search access to the union collection (http://www.vtls.com/ndltd). Lagoze, C. and J. R. Davis. 1995. "Dienst - An Architecture for Distributed Document Libraries", in Communications of the ACM, Vol. 38, No. 4, p. 47, ACM, 1995. Lagoze, Carl and Herbert Van de Sompel. 2001. The Open Archives Initiative Protocol for Metadata Harvesting, Open Archives Initiative, January 2001. Available http://www.openarchives.org/OAI/openarchivesprotocol.htm J. Powell and E. Fox. Multilingual Federated Searching Across Heterogeneous Collections, D-Lib Magazine, Sep. 1998 http://www.dlib.org/dlib/september98/powell/09powell.html Hussein Suleman, Anthony Atkins, Marcos A. Gonçalves, Robert K. France, and Edward A. Fox, Virginia Tech; Vinod Chachra and Murray Crowder, VTLS, Inc.; and Jeff Young, OCLC. Networked Digital Library of Theses and Dissertations: Bridging the Gaps for Global Access - Part 1: Mission and Progress. D-Lib Magazine, 7(9), Sept. 2001, http://www.dlib.org/dlib/september01/suleman/09sulemanpt1.html Hussein Suleman, Anthony Atkins, Marcos A. Gonçalves, Robert K. France, and Edward A. Fox, Virginia Tech; Vinod Chachra and Murray Crowder, VTLS, Inc.; and Jeff Young, OCLC. Networked Digital Library of Theses and Dissertations: Bridging the Gaps for Global Access - Part 2: Services and Research. D-Lib Magazine, 7(9), Sept. 2001, http://www.dlib.org/dlib/september01/suleman/09suleman-pt2.html Next Section: Production of ETDs Technical Issues/Production of ETDs Electronic publishing technology and infrastructure is needed to support the production and archiving of ETDs. Section 4.2.1 provides an overview of this matter. The next two sections explain key issues related to the two main approaches: PDF and SGML/XML. Remaining sections give further information of specific matters such as metadata and post processing. Next Section: Overview: hardware, software, multimedia, scripts, encoding, document representations/conversions Technical Issues/Overview: hardware, software, multimedia, scripts, encoding, document representations\conversions Preparing an ETD is somewhat like preparing a book to be given to a publisher, and then distributed electronically (and perhaps on paper). There are many aspects to this process, briefly summarized in the next paragraphs. First, there is the hardware perspective. Authors of ETDs nowadays almost always use a computer for this activity. In many cases this means a personal computer, though in some cases a terminal, PDA, or another device might be chosen. With the continuing increase in performance and functionality, given a particular price, it is becoming more feasible for authors to have their own computers. However, in some cases, an office or laboratory or shared computer might be used, at least from time to time. In addition to a computer, authors may use other special devices to prepare parts of the ETD. In particular, in the case of multimedia content, parts might result from using a scanner, digital camera, digital camcorder, slide scanner, microphone, sound card, MIDI device, or other special equipment. Special systems might be used for audio or image or video editing, though in some cases such editing can be done on a PC. Second, consider software Software Editors Software that can help with ETD production may be specialized by role. Text editors like Wordpad, vi, and emacs allow creation of files of characters, often encoded in ASCII. More powerful word processors may allow handling of more extensive sets of characters, like UNICODE, and entry of character codes from large sets (e.g., for Chinese, Japanese, or Korean texts). Separate Editors Separate editors may support multimedia content. Photoshop handles photos and other images, for example, while Sound Forge allows manipulation of audio files. Premiere is a tool for video editing that also can process audio, animations, and other related components. Editing tools may handle conversions as well, or accept the results of conversion. Ultimately, one may think of authoring, capture, conversion, and editing tools for all media types as all having the objective of amassing a pool of components that go into the ETD. Large documents thus are made of pools of content objects, and students can use special software to integrate the content into a coherent whole. A simple integration is to have a linear structure, like a book, where all the components are ordered, as in a sequence of pages, manipulated by a word processor. It is simple for images to be included in such a work, as long as sizing is adjusted. However, large images, and other multimedia content (e.g., audio), must be integrated in a different fashion. Such integration may involve linking (as in hypermedia, found in many Web sites). Additional interactivity may require a tailored hypertext system (e.g., Toolbook, Guide), or a multimedia integration package (e.g., Director or AuthorWare). Such a system will support synchronization (e.g., using the WWW standard, SMIL), or complex performances. Ultimately, the most sophisticated integration of content with reader interaction requires a programming language. In the case of multimedia languages, this typically is called a scripting language. However, students can also use general purpose languages like Java for sophisticated handling of content along with interaction. Third, consider representation. Each content object must have a representation scheme, depending on its form. Text components are characterized by character encoding, as well as supplemental information to support presentation (e.g., font, size, style). Multimedia components are encoded in suitable forms, depending on resolution or other measure of level of detail, compression, and other attributes. For the purpose of archival preservation, the representations used for content objects (e.g., UNICODE, MPEG), as well as those used to describe the organization and rendering of content (e.g., SMIL), will be in standard forms. Fourth, for an ETD or other large work, one can choose from archival forms, and related software support.. In particular, for more detail about software, see the next subsection. For details about PDF, SGML/XML, and other matters, see the next sections. Next Section: Page Description Languages Technical Issues/Page Description Languages Evolution of Networks and the World Wide Web The evolution of networks and the World Wide Web have profoundly affected the content and quality of communications. The complexity and visual content of documents is no longer limited by the artistic skills of the designer. Itranets and Web sites are being tapped into for creative content that is much more complex than could have been imagined. As creative expression expands to new levels, the quality of the printed output must also rise to the challenge. The Internet and Printing Workflow Process The Internet has changed not only the dynamics of the creative process but the entire printing workflow process. The Internet is used to transmit information electronically while CD-ROMs and servers are used for storing and accessing data. One often finds difficulty in predicting what will be printed because documents can come from many sources. And it's equally difficult to predict where they will be printed, because documents can be distributed around the world electronically and printed locally. The printing workflow has evolved from creating, printing, copying, and distributing hard-copy documents to creating, electronically publishing, and printing documents on demand. It was clearly time to advance the imaging standard. Adobe Postscript With the introduction of PostScript in 1985, Adobe Systems Incorporated sparked a revolution in how we communicate on the printed page. Since its introduction, Adobe PostScript has become the printing and imaging technology of choice for corporations, publishers, and government agencies throughout the world. In fact, 75 percent of commercial publications are printed on Adobe PostScript devices, including black-and white printers, color printers, imagesetters, platesetters, and direct digital printing systems. Adobe PostScript is also the display imaging system in some of the most advanced workstations on the market today. Adobe PostScript 3 takes the PostScript standard beyond a page description language into a fully optimized printing system that addresses the broad range of new requirements in today's increasingly complex and distributed printing environments. Adobe Postscript and PDF By fully integrating Portable Document Format (PDF) - an open file format that preserves the visual fidelity of documents across applications and platforms - into Adobe PostScript 3, documents can be delivered electronically and printed directly. A PDF file delivers the single "digital master" for use in electronic, printed, and mixed workflows, ensuring the highest fidelity across all media. Adobe Postscript Advanced Features With Adobe PostScript 3, Adobe delivers advanced features for the new digital document. Today, document creators draw on a variety of sources and increasingly rely on color to convey their messages. And tomorrow, these documents will be delivered and printed on a virtually unlimited range of devices. Next Section: Markup Languages Technical Issues/Markup Languages Supporting Works on Computers and Paper Since the mid-1980s, the electronic publishing community has faced the issue of supporting works targeted for use on computers, or on computers and paper (i.e., for dual publishing). To achieve maximal flexibility, it is desirable to separate the description of document structure from the rendering of that structure into some paper or screen form. SGML and, later, XML, were devised with this goal in mind. SGML and XML SGML and XML are markup languages. In particular, they are meta-languages. One can provide a document type definition (DTD) that conforms to the rules of SGML, and then be able to create documents of that type. The same applies to XML. However, with XML, it is possible instead to use a schema to specify the type of document considered, or to just have markup without a DTD, demonstrating that this indeed is an eXtensible Markup Language. Other Markup Languages In the SGML/XML family are other markup schemes, or applications of these. HyTime extends SGML with architectural forms to allow handling of rich multimedia and hypermedia content. RDL, developed by e-Numerate Solutions, Inc., allows handling of numeric content in an application of XML, while XBRL applies XML to business reporting. For many ETDs, tailored markup schemes (e.g., MathML for mathematics, or chemical markup language), also are used for particular types of content expression. The following subsections explain further how ETDs can be produced using markup languages. Next Section: XML Software Technical Issues/XML Software Since XML is software independent, many types of software can be used for the production of ETDs in XML. The next table gives the software you can use by type of tools. This is a short list of tools we used in our projects. Nevertheless, many others can suit you requirements, among them, freeware. http://www.garshol.priv.no/download/xmltools/ is a good place to find free XML tools and software. The XML Cover Page (http://www.oasisopen.org/cover/sgml-xml.html) is the reference. XML Editors XML Spy http://www.xmlspy.com This tool integrates more than an XML editor. Xmetal http://www.xmetal.com/top_frame.sq Corel WordPerfectSuite http://www.corel.com See Corel WordPerfect. Can also be used as regular wordprocessing software. MS Word www.microsoft.com See Microsoft Word and Office 2007. StarOffice Version 8 http://www.sun.com/software/star/staroffice/index.jsp A good alternative to MS Word, exports some kinds of XML. Supports import and export of MathML files. OpenOffice.org http://www.openoffice.org Exports XMLbut not to (user) variable DTDs AbiWord http://www.abisource.com open source Kword http://www.kde.org (KOffice, KDE Project/ open source) Amaya http://www.w3.org/amaya Open Source, World Wide Web Consortium For Mathematical Formulas in MathML Hancom HancomWord 5.2 (included in HancomOffice 1.5 and more) http://www.hancom.com Linux based Office Package fully compatible with MS-word file and templates. Any scriptinglanguage PERL, JAVA, etc. Omnimark http://www.omnimark.com Very interesting, but not free, and no longer offers a significant discount for universities (about 10,000 US$ for the standard release) Avenue.Quark http://www.quark.com/products/avenue/ Not really interesting Save as XML plug-in in PDF http://www.adobe.com/support/downloads/89a2.htm Tetrasix http://www.tetrasix.com Only partially usable. Good for the Majix prepared DTDs WorX SE http://www.hvltd.com/ Plugin for Word Schema MarkupKit http://www.schema.de I4I http://www.i4i.com Add-On for Word XML formatter for print solutions 3B2 http://www.3b2.com Typesetting from SGML PowerPublisher http://www.mai-kg.de Mai KG, Germany XML XSL-Formatter NeXt Publisher http://www.nextsolution.co.jp/English/index.html NeXt Solution FOT http://www.pro-image.de FOP http://www.apache.org Apache XML Project / Addon for Cocoon Next Section: DTDs for ETDs Technical Issues/DTDs for ETDs (This section was taken from an article by P. Potter, P. Strabala, D.Dobratz, M. Schulz about ETDs, that is due to appear in "The Internet and Higher education" 4/2001) XML Authoring Systems The fact that currently available authoring systems for XML still have not won wide recognition has led to different strategies at different universities regarding XML documents. Most of these projects were started between 1995 and 1997, in a time when XML was alive, but where tools or standardized DTDs were barely available. A view of those projects from today's perspective illustrates the demand for a rethinking and redesign of those approaches in order to come to a standardization. DTDs All the presented DTDs are built upon similar principles. A classical dissertation (which can be seen as monograph) consists of 3 main components: an extensible titlepage with abstracts, declarations, etc., the dissertation corpus, which includes text, pictures, audio, video, tables and so on, and appendices, which contain data sheets, bibliographies, acknowledgements and others. UIowa2K.DTD: University of Iowa Author-DTDs All these Document Type Definitions are so-called author-DTDs. This means that they are primarily used to support the authoring and the conversion process and do not primarily address document archiving and preservation. One may ask why all those different DTDs have prevailed. This is mainly because the scientific orientation of the mentioned universities is quite varied. Lyon, Oslo and Michigan, which use TEI-Light.dtd, mainly serve students in the arts and humanities. Problems using TEI.DTD or DocBook.DTD are recognized at universities that support a strong natural science community, such as Berlin, Helsinki or Uppsala. Often a dissertation is a cumulative work, e.g., in Lyon or Helsinki. Université Laval, in collaboration with the Université de Montréal, is working during 2001-2002 on the modelisation of a new DTD for ETD. The DTD and its documentation will be post at http://www.theses.umontreal.ca. DTDs for multimedia content "Structured data," such as mathematical or chemical formulas, spreadsheets, address books, configuration parameters, financial transactions, technical drawings, etc., are usually published on the Web using layout programs such as Postscript or PDF, or by putting them into graphic formats like gif, jpeg, png, vrml, and so on. Programs that produce such data often also store it on disk, using either a binary or text format. Therefore, if someone wants to look at the data, he usually needs the program that produced it. With XML, data could be stored in text format, which allows the user to read the file without having the original program. XML can be thought of as a set of rules, guidelines, or conventions, for designing text formats for data in a way that produces files that are easy to generate and read (by computer). In addition to the older standard SGML, there are several emerging standards that use XML encoding to overcome the disadvantages common to web publishing in HTML. The following sections give an overview of standards that have been established during the last few years or which are still works in progress, but widely recognized. XML DTDs and Schemas For standardized knowledge management this variety of XML DTDs and Schemas seems confusing. A closer look, though, gives another perspective: every scientific subject defines and uses its own standards. The following document type definitions can roughly be classified in: Schemata that use semantic tags to mark real content items, e.g., MathML or CML. Schemata that are used for visualisation and layout purposes and to control the browser synchronization, e.g., HTML, SVG (Scalable Vector Graphics), SMIL(Synchronized Multimedia Integration Language). Schemata that are principally designed to perform the exchange of data with huge databases, e.g., cXML(commercial XML). Within the field of "Electronic Publishing," these developments have led to new opportunities to structure scientific information, not just text-based but also so-called active contents and multimedia elements. This brings the whole field to a new level of information processing or knowledge management. The different approaches for electronic publishing at universities creates a very heterogeneous environment. The following tables show how difficult it might be to subsume all those different models under one concept in order to achieve valuable and searchable information systems based on XML. Crosswalks between all those DTDs have to be defined in order to build a distributed retrieval engine, capable of searching within internal document structures "throughout the world." Not only different DTDs are used, but also different strategies to perform a conversion from usual text formatting systems into highly structured documents in SGML or XML. Conversion to SGML/XML Conversion from SGML/XML into HTML, PDF Humboldt-University Berlin DiML SGML-Author for Word Perl-script, DSSSL Virginia Polytechnic Institute and State University ETD SGML-Author Perl-script University of Iowa Uiowa2K Majix CSS University of Montreal/ Université de Lyon 2 TEI-Light Omnimark rtf2sgml XSL Technical University Helsinki HutPubl FrameMaker+SGML DSSSL, FrameMaker University of Michigan TEI-Light Omnimark rtf2sgml University of Oslo ISO-Book Balise Next Section: Berlin DTD workshop Technical Issues/Berlin DTD workshop Scope of the workshop The Idea behind this workshop was to bring together experts and developers working on SGML or XMLbased Document Type Definitions for electronic theses and dissertations. Within the NDLTD-initiative several document type definitions (DTDs) for dissertations have been developed. To come to a standardized or generalized searching and archiving structure, it is absolutely necessary to summarize all those developments and research and to work out correspondences between all of them. If SGML and XML are the preferred future document formats for theses and dissertations, the DTD will play a major role within the whole initiative. The following questions were discussed: How to get to a generalized DTD? Is there just one DTD necessary or is it possible to have more than one DTD, e.g., for different subjects and sciences? Where exactly are the current differences between the existing document type definitions? Which approach shall we follow to unify our DTDs? How to come to an agreed Dublin-Core metadata element set? How many markup should be done by the author, how much by machine translation? Evaluating several tools and ways to come to an SGML/XML-based document This expert workshop covering the topic of using XML for publishing theses and dissertations electronically in universities was held in May 2000 at Humboldt-University Berlin, Germany. It focused on the ways in which XML can be used in university libraries in order to deliver, process and archive scholarly high quality electronic publications. The recognition that a worldwide range of various approaches for SGML/XML-based publishing concepts exists, led to the conclusion that interoperability on an international level is inevitable. Experts from the USA, Finland, Norway, Sweden, France, Portugal, the United Kingdom and Germany discussed commonalities and differences among their models, approaches and document type definitions (DTDs). An Objective of the Workshop The objective of the workshop was also to exchange experiences in document conversion and author's support therein, to agree on a Dublin Core metadata set worldwide and to share tools. The potential support of authors was recognized as a very essential part within the electronic publishing workflow. Attempts to convince authors to leave their native word processing systems for systems that support structured writing and/or export to SGML or XML usually tend to fail, for several reasons. First, authors are reluctant to switch from the system they are used to. This is exacerbated by the fact that publications usually have to be written within a very short time frame, leaving little or no time for authors to learn a new system, even if they were so inclined. Until XML authoring and editing tools become as simple to use as a word processor, this approach will still be viewed as an added burden to the student. Second, current SGML/XML support systems are usually far more expensive than common ones. Prearchival Format Many universities follow another strategy and accept documents in a pre-archival format. This necessitates conversion of the documents by the university libraries or media center. The advantage is that authors are able to prepare their documents within their native word processing systems using style sheets especially designed for that conversion, e.g., in Microsoft Word. Some institutions that have a high focus on format are reluctant to approve this method because the SGML or XML publication may be altered from the word processing document that was deposited by the student. Even though this method seems to exact a high resource cost, it is a step toward the ideal, where authors write in XML directly, using their native word processors. We are not far from this ideal. ETD Projects with XML-based Approach The following electronic theses and dissertation (ETD) projects have an XML- based approach already in place or are presently in a pilot phase: Swedish University of Agricultural Sciences (SLU) Libraries, SWEDEN, Virginia Polytechnic Institute and State University, University Libraries, USA Sentiers, Université Lumiére, Lyon 2, (Cytbertheses.org project), FRANCE Universidad de Chilé (Santiago de Chilé), CHILE Humboldt-University Berlin, "Dissertation Online", GERMANY University of Oslo, Center for Information Technology Services, NORWAY University of Iowa, Graduate College, USA University of Michigan at Ann Arbor, Library, USA Helsinki University of Technology, Library, FINLAND For further information, please visit the workshop-website at: http://dochost.rz.huberlin.de/epdiss/dtd-workshop/index.html With the workshops there has been a collection of tool, that are used fpor SGML/XML publishing at different universities. This collection can be found at: http://dochost.rz.huberlin.de/epdiss/dtd-workshop/cdrom/index.htm Next Section: Support for students to write directly in XML Technical Issues/Support for students to write directly in XML This topic is discussed briefly in Overview: writing with word processors and structured editors and in more detail in Writing directly in SGML/XML. The following extends that discussion by focusing on tools to help students to write directly in XML. To author and view documents written in XML, different kinds of tools are required. They are: Parser/Validator There are quite a large number of XML editors, for different platforms are available. Most of them costs less than $100. A lot of them have free evaluation version that can be downloaded. the following list gives link to some such editors. XML Spy http://www.xmlspy.com XMLWriter http://www.megginson.com/Software/ XED http://www.cogsci.ed.ac.uk/~ht/xed.html Xeena http://www.alphaworks.ibm.com/tech/xeena Morphon XML Editor http://www.morphon.com/ Emile for MAC http://www.in-progress.com/emile/ Free/OpenSource version of the Serna XML editor https://github.com/ydirson/serna-free When a XML document is created, it is parsed/validated to see if it is syntactically correct and well formed. To accomplish this task, we need a XML parser/validator. There are a lot of XML parser available. Most of the XML editors mentioned above is also a XML parser. For example, XML Spy provides a IDE for XML, i.e. it is both editor and parser. To view a XML documents, we need a browser that supports XML. Many browsers now support XML, such as: Internet explorer 5 and newer http://www.microsoft.com Mozilla Firefox 0.9 and newer http://www.mozilla.org/ Netscape 5 and newer http://browser.netscape.com/ DocZilla http://www.doczilla.com/ HyBrick is an SGML/XML browser from Fujitsu. Most newer browsers support XML in a way that lets it fully integrated with HTML. A stylesheet is required to view the XML documents. For ETD a cascading stylesheet has been developed for that. That stylesheet can be downloaded from http://www.etdguide.org/. Next Section: Conversions from Word, Word Perfect or other RTF-compatible tools to SGML/XML Technical Issues/Conversions from Word, Word Perfect or other RTF-compatible tools to SGML\XML A SGML or XML document type definition (DTD) that serves as structure model for the output. One says that the output SGML document is valid to the specifies DTD, or it is an instance of this DTD: A Word stylesheet that holds paragraph and character styles according to the structures in the DTD. So if in a DTD you have defined a structure for Author: e.g. expressed in the output file as: You have to find expression in Word: paragraph styles: author character styles (just to be used within an author-paragraph): first name, surname, title Conversion Methods Often a conversion is done by using a plug into MS Word directly, but other options use the Microsoft internal exchange format RTF (Rich Text Format) for conversion. Those tools can interpret the RTF file with the MS Word style that is still coded in this RTF file and export it into an SGML document. This process mostly happens within batch mode without using many graphical user interfaces. Approach of the Université de Montréal, Université de Lyon 2, Universidad de Chile Humboldt-University Berlin and Germanwide Dissertation Online project Conversion method of the Cybertheses project Proposition of section (Vivi) The process line for converting Word files into SGML documents developed within the CyberThèses project uses scripts written with the Omnimark language. Coversion Process The input of the process line is an RTF file with a "structuring style sheet" and the output is an SGML document encoded according to the TEI Lite DTD (see the TEI web site at http://etext.virginia.edu/TEI.html). The conversion process is made up of three main steps : first, one converts the RTF file into a flat XML file encoded according to DTD of RTF. The produced file is a linear sequence of paragraph elements each one having an explicit "stylename" attribute corresponding to the RTF style names. the second step consists of the re-generation of the hierarchical and logical structure of the document based on the analysis of stylename attribute. last, a SGML parser allows one to validate the conformity of the produced SGML document with the TEI Lite DTD. Some supplementary scripts then allow the export of the SGML document towards other formats (HTML, XML). Most of the scripts will soon be available from the CyberTheses web site : http://www.cybertheses.org This system is devoted to a particular DTD, but its generalization to other document models shall not pose any difficulty. Using SGML Author for Word (Humboldt-University Berlin) The "Dissertation Online" project implemented and refined a conversion strategy that allows writers to convert documents written in MS word with a special stylesheet (dissertation.dot) into an SGML instance of the DiM.dtd. We used this product from Microsoft, the SGML Author for Word, for several reasons: It support an international standard for tables: CALS. As it is a Word-Add-On it handles docuents in MS-Word doc- format better than other tools. As we started using this technology in 1997, it supported from the very beginning Word97, the version of word which was the actual one that time. Unfortunately, Microsoft didn't continue the development of this tool. So there are no new versions available for Office 2000 or Office XP. However, the internal document format from MS Word 97, MS Word 2000 and Office XP are the same in the sense of the conversion into SGML. This means documents written in Word 2000 or Office XP can be imported into Word97 and therefore a conversion can be done. The converter stylesheet, which consists of several macros programmed to make the preconversion process easier. SP (NSGMLS) (Parser for SGML-Files by James Clark). (new versions are availabe at http://openjade.sourceforge.net/doc-1.4/index.htm, but we haven't tested that) The converter stylesheet and the authors stylesheet can be obtained from the following website: http://dochost.rz.hu-berlin.de/epdiss/vorlage.html Converter scripts and perlscripts can be optained from http://www.educat.huberlin.de/diss_online/software/tools.exe (Perl scriptc, DTD and converter file for MS SGML-Author for Word - KonverterDiML2_0.dta) Preparing the conversion without using the converter Microsoft SGML Author for Word directly Load the stylesheet for conversion (NOT the one for the authors), see figure below. There is a special feature to get the page numbers out of the Word document by using certain word specific text anchors. Those have to be converted into hard coded information using a page numer stylesheet. Formattings that have been applied by the author without using style sheets have to be replaced by the correct style sheets. In order to get a correct display of tables later on by using CSS stylesheets within common browsers, empty table cell have to be filled up with a single space (letter). Soft coded line breaks have to be preserved for the conversion. This is done by inserting special characters #BR# to that. This will be used to insert later a special SGML tag for soft line breaks <br/>. Converting with Microsoft SGML Author for Word Work through the output file (output according to the DiML.dtd) automatically. Transforming the DiML file into a HTML file A demonstration quicktime video may be found at the ETD-Guide server as well. (see http://www.educat.hu-berlin.de/diss_online/software/didi.mov) Using FrameMaker+SGML6.0 for a conversion of MS Word documents into SGML instances. Editing or converting using FrameMaker is much more complex than the previously described methods. FrameMaker is able to import formatted Word documents keeping the stylesheet information and exporting the document via an internal FrameMaker format as SGML or XML documents. In order to proceed with a conversion using FrameMaker you will need the following configuration files. A conversion table. This contains the list of the Word styles and the corresponding elements within the FrameMaker internal format. This table is saved within the FrameMaker internal format (.frm). A document type definition will be saved within FrameMaker internally as EDD (Element Definition) It is saved within the FrameMaker internal document format (.edd) FrameMaker uses layout rules for the internal layout of documents. Within this layout definition the layout of documents is described just like it is within MS Word documents: single formats and their appearances like text height, etc. are defined. This file is also stored as (.frm file). To process a conversion a new SGML application has to be defined within FrameMaker+SGML. This application links all files that are needed for a conversion as described above. It enables FrameMaker to parse the output file when exporting a document to SGML or XML: A workflow and a technology for conversion for ETD using FrameMaker+SGML6.0 was first developed at the Technical University Helsinki, within the HUTPubl project (1997–2000), see http://www.hut.fi/Yksikot/Kirjasto/HUTpubl Tools that export using a user specified DTD: WordPerfect since Version 7.0 (Corel) (http://www.corel.com ) Tools that export using their own native DTD: Converter Tools: Bollenbach, Markus; Rüppel, Thomas, Rocker, Andreas: FrameMaker+SGML5.5. Bonn; Reading, Mass., Addison-Wesley Longman, 1999, ISBN 3 8273 1508 5 St. Laurent; Biggar, Robert: Inside XML DTDs. New York, Mc Graw Hill, 1999, ISBN 0 07 134621 X Ducharme, Bob: SGML CD. New Jersey, Prentice Hall, 1997, ISBN 0 13 475740 8 Smith, Norman E.: Practical Guide to SGML/XML Filters. Plano. Texas, Wordware Publishing Inc., 1998, ISBN 1 55622 587 3 Goldfarb, Charles; Prescod, Paul: XML Handbuch. München, Prentice Hall, 1999, ISBN 3 8277 9575 0 Invalid ISBN Next Section: Conversions from LaTeX to SGML/XML Technical Issues/Conversions from LaTeX to SGML\XML Looking at this problem, it first gives the impression that because writing in LaTeX is also a kind of structured writing, a conversion into SGML or XML compatible documents can quite easily be done. Often people use as software Onmimark, Balise or simple Perl scripts for conversion. Problem 1 The LaTeX format itself enables due to the structural approach an easier conversion into SGML/XML. But often the pure LaTeX approach goes hand in hand with sophisticated macro programming, so that in many cases the pure structure of a LaTeX document will be destroyed by macros or the conversion will be much harder to perform. Also the lack of a parser that checks the correct usage of structure elements like chapter, section, subsection make a conversion more complicated. Problem 2 If an author wants to define a mathematical formula in LaTeX, he has 2 basic opportunities: Producing the mathematical formula as picture Defining them with the appropriate mathematical LaTeX features as text formulas. First and Second Versions The first version prevents from any secondary usage of the formula. The second version allows the reusability of a formula in different contexts. So it makes it easy to prove correctness of a statement by importing the LaTeX formula into a mathematical software package like Maple or Mathematica. Formulas coded in LaTeX can be displayed in a rendered form in an browser by software or plug-ins like IBM Techexplorer, Math Viewer. As LaTeX coded formulas still have the disadvantage, that they are not encoded using so called sematic tags, the usage of MathML is highly advised MathML is an XML document type definition for mathematics developed by the W3C. The Letter e In LaTeX authors often don't distinguish between the letter 'e', that may stand for a variable and the Euler constant. In MathML there is a huge difference whether something is encoded as variable e or as the Euler e (2,718. . . ). Therefor, the usage of layout definition in LaTeX for mathematics complicats the conversion into MathML and therefore into any SGML/XML format. In order to prepare mathematical formulas in LaTeX for a conversion, many universities and the TeX. User Groups around the world are working soon the definition of certain macros that can be transformed into the appropriate MathML definitions. See University of Montréal at www.theses.umontreal.ca or University of the Bundeswehr in Munich. Next Section: Rendering-style sheets Technical Issues/Rendering-style sheets In the case of conversion of documents into SGML/XML, the original file must be written using a structuring style sheet (template). But students often give a lot of importance to the layout of the final document and make many personalizations and adaptations on the original structuring style sheet. It is our duty to also convert also this aspect of their work. The conversion tools devoted to the document content have to be completed by conversion tools that may generate rendering style-sheets : typically XSL files for XML documents. Layout Demand As SGML or XML cannot be directly read by users using today's browsers (Opera, Netscape, Internet Explorer), it is necessary either to provide layout information in the form of a stylesheet or to transform those highly structured documents into layout or printable version in HTML, PDF,PS, which can be more easily used by users to read the documents. Generally, universities have 2 possibilities to cope with the demand for layout: Try to preserve all layout information the author (student) has given to the MS Word or other document. This strategy involves the author directly by the personalization of a standard style sheet for his/her work. Universities could decide to develop one single style sheet or a choice of certain style sheets usable for all ETDs. This would support a corporate design and solve the layout question on a more general level, leaving it to the ETD production department. Style Sheets for SGML or XML documents The development of such tools is being achieved in France (collaboration between Lyon 2 and Marne-la-Vallée) for documens produced in MS Word and other RTF compatible authoring tools. Theirs are based on the analysis and the extraction of the typographic characteristics associated to each used style. Equivalent Tools The equivalent tools shall be easy to develop for document produced with LateX, as this kind of language natively uses the notion of rendering style-sheets. Style sheet languages for XML are: Cascading Style Sheets (CSS ) Extensible Style Language (XSL) . As CSS are not powerful enough to handle the complexity and demand for large XML documents as theses and dissertations, this is not advisable. Substandards within XSL Standard Within the XSL standard, one distinguishes between several sub-standards: Extensible Style Sheet Transformation (XSLT). This part allows the user to produce stylesheets that act like small programs. They transform the original document, that is always valid against a specified DTD, into a document that either follows another DTD (which allows an easier rendering within browsers, as HTML.dtd) or allows a transformation of the document into other document description languages such as Rich Text Format (RTF), LaTeX, PDF. From those formats the production of a printed version may be possible. XPath allows to authors to build expressions that link to other XML documents, but not only to the whole document. The citation of a certain subsection and the citation of e.g. section 3 to 5 might be possible once common browsers support this linking technology. XSL:Fo ( Formatting vocabulary, that can be applied to the nodes of an XML document) Example for a Printing On Demand Service (POD) based on the use of stylesheets Digital Libraries which use their document servers as long term electronic archives will not make printed information dispensable. On the contrary: for users of these information systems the desire for printed documents is increasing. In most cases this desire often focuses not on the whole document as such, but on particular parts of it like chapters, citations and so on. For that reason the Humboldt-University Berlin's printing on demand project aims toward the development of a technology which allows the users to print the only the desired part of a certain document. Printing on Demand with XML For the printing on demand component with XML the usage of Apache/Cocoon was chosen. This software uses an XSLT-engine to produce an HTML or PDF-Version on the fly. "The Cocoon Project is an Open Source volunteer project under the auspices of the Apache Software Foundation (ASF), and, in harmony with the Apache webserver itself, it is released under a very open license. Even if the most common use of Cocoon is the automatic creation of HTML through the processing of statically or dynamically generated XML files, Cocoon is also able to perform more sophisticated formatting, such as XSL:FO rendering to PDF files, client-dependent transformations such as WML formatting for WAP-enabled devices, or direct XML serving to XML and XSL aware clients. " As Cocoon does not consist of a printing on demand component (especially a selection feature) a small workaround using different XSLT-stylesheets had to be created. The users view, containing an HTML-view onto the actual document includes check boxes which the user can use to select parts of a specific document. This view is produced by the XSLT-Broker stylesheet which calls a default stylesheet, that produces HTML (XSLT-Stylesheet with option document.xml?format=html). If the user selects certain parts of the document by clicking in the checkboxes, by clicking on the "OK" button a perl script (PHP-Choise) is called. This script selects the desired chapters and sections of the document by using XPath-expressions (http://dochost.rz.huberlin.de/proprint/bsp/slides.xml?CHAPTER=3 and http://dochost.rz.huberlin.de/proprint/bsp/slides.xml?CHAPTER=4 ) cuts those parts out of the document and holds them in the main memory. This procedure is carried out by the XSLT-Broker-stylesheet that has now been called with the XML-option (document.xml?format=xml). These parts are added to one single XML-document (all in the main memory!) and processed by the XSLT Broker-stylesheet either with the print option or the HTML option (document.xml?format=pdf or document.xml?format=html) Next Section: Metadata, cross walks Technical Issues/Metadata, cross walks With the advent of ETDs, traditional catalogers can learn about cataloging online resources, or they can apply what they already know about cataloging online resources to cataloging theses. Catalogers may need to adapt existing policies and procedures, but their workstations may already have the necessary software (e.g., word processor, PDF reader, etc.). This may also be an opportunity to implement cataloging policy changes, for example, allowing authors to assign keywords in addition to or instead of continuing to assign a controlled vocabulary such as the Library of Congress Subject Headings. Of course, catalogers will add the necessary MARC fields required for computer files. More information can easily be added to the bibliographic record because of the ease of copy-and-paste features of word processors, so include the abstract in online catalog records and index this field to enhance findings through keyword searching. The following table includes MARC fields included in bibliographic records for ETDs, with additional information about Dublin Core metadata, etc. For the latest information about ETD metadata, see http://www.ndltd.org/standards/metadata/current.html METADATA CROSSWALK http://www.dlib.vt.edu/~paul/ndltd/scm0998_imp-dev.htm (link to fuller document) Dublin Core Metadata elements USMARC tag MARC notes Metadata notes M 1.0 DC.Title 245 $a title name the author assigned to the work 1.1 DC.Title.X-Notes 500 source of title note M 2.0 DC.Creator.PersonalName 100 $a author, personal person primarily responsible for creating the intellectual content of the work O 2.2 DC.Creator.Address M 2.3 DC.Creator.X-Institution 710 $a institution's full name name R 2.4 DC.Creator.X-Major O 2.5 DC.Creator.X-College 710 $b college's official name R 2.6 DC.Creator.X-Dept 710 $b department's official name O 3.0 DC.Subject 653 $a uncontrolled keywords keywords or phrases describing the subject or content of the work 650 $a LCSH R 4.0 DC.Description.Abstract 520 $a author's summary textual description of the content of the work R 5.0 DC.Publisher.CorporateName 260 $b entity responsible for making the work available in its present form R 6.0 DC.Contributor.X-Chair.PersonalName 700 $a added personal name person(s) or organization(s) who made significant contributions to the work but that contribution is secondary to the author R 6.1 DC.Contributor.XCommittee.PersonalName 700 $a added personal name O 7.0 DC.Date.Valid date committee approved the work in final form(s) M 7.1 DC.Date.X-Approved 260 $c date Graduate School approved the work M 8.0 DC.Type 655$2 local index term--genre/form category of the work:Text.Thesis.Doctoral or Text.Thesis.Masters + (see DC enumer'd list) M 9.0 DC.Format 856$q locat.+access/file trans. mode work's data format; ID software (+hardware?) to display/operate the work (see DC list) M 10.0 DC.Identifier 856$u URL (or PURL, hndl, URN) unique ID of the work(s) 856$b If IP address: (Access number) 10.1 DC.Identifier.XCallNumber.LC 090$a O 11.0 DC.Source 786$n 0 Data Source Entry/Title metadata describing original source from which the work was derived O 12.0 DC.Language 546$a language note language of intellectual content of the work 041$a language code M 13.0 DC.Relation 787$n nonspecific relationship note describes how ETD's parts relate to entire work 787$o nonspecific relationship ID URL O 14.0 DC.Coverage 500$a general note spatial or temporal characteristics of intellectual content of the work 255 $c If spatial: cartographic statement of coordinates 513 $b If temporal:period covered note 15.0 DC.Rights 540$a terms governing use/repro copyright statement 856$u If URL: (with $3=rights) M 15.1 DC.Rights.X-Availability Should be "accessibility?" none/some/full access O 15.2 DC.Rights.X-Availability.Notify date to assess accessibility O 15.3 DC.Rights.X-Proxy.PersonalName contact re accessibility if author unreachable O 15.4 DC.Rights.X-Proxy.Address O 15.5 DC.Rights.X-Checksum.MD5 Reveals if file is corrupted. Appropriate here? O 15.6 DC.Rights.X-Signature.PGP Should be accessibility? none/some/full access M = mandatory O = optional R = highly recommended MARC 502 derived from type, creator's institution, date valid MARC 538 = format Paul Mather and Gail McMillan, Virginia Polytechnic Institute and State University, Nov. 18, 1998 MARC does not equal Metadata: missing from Metadata=file size; several notes (vita, abstract), 949/040 (more?) Leader and 008--some redundant info from variable fields Next Section: Naming Standards Technical Issues/Naming Standards This topic is discussed in Naming standards: file names; unique Ids. The focus there is on naming of files and on unique identifiers. Here we consider those matters briefly, but also discuss URNs and OAI naming. At Virginia Tech, naming of the parts of an ETD is left to authors. Many have a single document, called etd.pdf, but there are many other selections. The entire ETD is referred to by way of a summary page. This has an ID like: etd-12598-10640 or (in earlier cases) etd-5941513972900 These two correspond to full URLs, respectively: http://scholar.lib.vt.edu/theses/public/etd-12598-10640/etd-title.html http://scholar.lib.vt.edu/theses/public/etd-5941513972900/etdtitle.html Other but similar schemes are in use elsewhere - recall Naming standards: file names; unique Ids. The key point is to have an ID for each work, and a way to resolve that to the actual document. In some cases, a URN is given, such as a handle or PURL, which in turn is resolved to a URL. (See Identifying: URN, PURL, DOI) That allows users to remember the URN, and be assured that over time, as documents are moved about, there still be automatic linking from the URN to the designated document. IDs as discussed above also are important with regard to OAI. Assuming that, as is required, each archive has a unique identifier, then as long as each record in the archive, as required, has a unique internal ID, then the combination of the two will uniquely determine a record in an archive. Next Section: Encryption; Watermarking Technical Issues/Encryption; Watermarking Encryption and Digital Signature Overview: Using digital signatures Digital signatures act like conventional signatures - allowing you to "sign off" on anything that requires an approval. You can simply attach your "signature" to the document. In addition, a signature stores information, like the date and time, and allows you to track document versions and validate their authenticity. To create a digital signature profile: Choose Tools > Self-Sign Signatures > Log In. In the Acrobat Self-Sign Signatures - Log In dialog box, click New Profile. In the User Attributes area of the Acrobat Self-Sign Signatures - Create New User dialog box, enter your name and whatever other information you want to include in the three optional fields. In the Profile File area of the Acrobat Self-Sign Signatures - Create New User dialog box, enter the path name for the folder in which you want to store your signature profile or click Browse and choose a folder. Enter a password of at least six characters in the User Password and Confirm Password fields and then click OK. To add a digital signature to a document: Click on the Digital Signature tool in the Tool bar and then click and drag where you want to place your signature. In the Acrobat Self-Sign Signatures- Sign Document dialog box you can select an option from the Reason for Signing Document pop-up menu or enter a reason in the field, and you can enter a location in the Location, e.g. City Name field. Note: If you're using a third-party signature handler, follow the instructions displayed on screen. You may be prompted to log in to the handler or enter required information. Enter your password in the Confirm User Password field and then click Save Document. If this is the first signature added to the document, the Save As dialog box is displayed. Enter a name and choose a location for the file and then click OK. Note: If the Save As dialog box is displayed when you add a digital signature, you end up with two copies of the document: one unsigned and one signed. From this point on, you should use the signed version. To display a list of a document's signatures, click the Signatures tab in the Navigation pane. The Signatures palette's pop-up menu contains several commands for working with digital signatures; the Properties command lets you see the attributes of a digital signature. Next Section: Packaging Technical Issues/Packaging The Australian Digital Theses (ADT) Program software is a modified version of the original Virginia Tech ETD software and is in its second release. The ADT software modifications were to make it generic, flexible and customisable for easy integration within the local IT infrastructure. This is critical as the ADT Program is a distributed and collaborative system involving a large number of Australian universities. The software is distributed to all ADT members free by ftp download as a .tar file. The reason for using a .tar file is that it keeps related files together, and thus facilitates the transfer of multiple files between computers. The files in a .tar archive must be extracted before they can be used. Extracting the distribution .tar file will reveal a directory (adt) containing 2 further directories (cgi-gin & public_html) plus installation instructions in a readme.txt file plus an empty adt-ADT admin site to help identify the structure of the admin side of the software. Also included in the release package is a test site for members to look at and use to familiarise themselves on the look, structure and functionality. Three dummy theses are used as examples of how theses can look, and how they fit into the admin structure. General software details, as well as all other information regarding the ADT Program is publicly available on the ADT Information page @: http://adt.caul.edu.au/ Responsibility for the ADT software and initial setup support for new members is taken by the lead institution - The University of New South Wales Library. Overview of the ADT Program software generic look and feel includes both Copyright & Authenticity statements completely revised help screens revised, as well as new alerts for errors etc.. does not allow non compliance with core ADT standards - e.g. filenames; illegal symbols all fields & processes compulsory unless otherwise indicated on the form possible to edit html as well as change restrictions with new update function it is now possible to ADD, DELETE, RENAME files. Another feature of the update function is to be able to move files to a NOACCESS directory, as well as and to make them accessible again. This is designed primarily for certain parts of theses that cannot be made public for reasons of copyright, patents, legal reasons, and other sensitivities now possible to make thesis available without any restriction (ideal), restrict to campus only, restrict whole thesis for approved caveat period, totally restrict parts of thesis. Any combination is possible with the choice, or choices made, being reflected in the local view of the thesis. That is, if thesis is restricted to campus only this is now obvious, similarly if part of thesis is restricted (noaccess) this is obvious too. Knowing if restrictions apply at the outset will not frustrate those searching the database now possible to easily un-make a deposit. That is, to remove a thesis from public/restricted view and/or to take this back to the deposit directory where any editing and changes can be made before reapproving and making accessible again refined URI structure using date (yyyymmdd) and time (hhmmss) revised and new help & alert pages completely revised and updated according to latest Dublin Core Qualifiers document released 11 July 2000. to aid search functionality keywords/phrases (i.e. DC.Subject) are each repeated as separate element strings. This has resulted in a revised look of the HotMeta database view. The brief record (default) now shows Title, Author, Date, Institution/School, with the expanded view showing all the metadata. Next Section: Backups; Mirrors Technical Issues/Backups; Mirrors An archival infrastructure for ETD should not only consider document format or the use of digital signatures, but also a consequently run concept for mirroring and backups. Mirrors ensure, that archives run stable, regardless of usage, bandwidth and location. A common server concept for an ETD archive is the following one: Production server Archive Server (Storage Unit, like IBMs Tivoli Storage Management System) Public Archive server Archive Mirror at different geographical location. The production server is the server that communicates with the users, especially the authors. Here the uploads and metadata processing is done. The archive server ensures that a permanent incremental backup of the public archive server is done. Only system administrators have access to the archive server. This secures the access to the ETDs and their digital signatures. It prevents users from manipulating ETDs and their authenticity and integrity. The public archive server itself is the server that is known as the document server from the outside. Here the ETDs can be access by users, the retrieval is available. We advise to secure this server with a special RAID (Redundant Array of Independent Disks) system, that allows a security in case of a hardware failure of the disk, that might be caused by extreme usage of the server. A RAID system holds internally 2 equal copies of the server distributed on independent hard disks. So if one copy cannot be accessed due to a head crash or other hardware inconsistencies, the second copy will take over functionality and operate as the first copy. The user itself will not realize, which copy is actually in use. Additionally to the archive server there should be another storage management system server, an archive mirror that mirrors the original archive. This is important if due to environmental accidents e.g. fire, earthquake or anything else destroys the original archive server. So a data loss, or the loss of the full archive can be prevented. This backup archive should be located at another geographical position, even at another continent. Next Section: Dissemination of ETDs Technical Issues/Dissemination of ETDs Ultimately, ETDs are designed to be disseminated, to at least some audience. Aspects of this activity are considered in the next subsections. First, each ETD must have an ID, and there must be a link between the ID and the actual work. Second, each ETD must have a metadata record attached, that can be used for resource discovery and other purposes. Third, there must be a link between the ID, the metadata, and the actual body of the ETD. Fourth, the works must be made available, typically through a Web server. Sixth, there may be summaries or full works that are provided for discovery, or for indexing that is designed to lead to discovery. For example, summary pages may be indexed by Web search engines so that they may be found as a result of a Web search. Note that it is encouraged for all NDLTD sites to keep a log regarding dissemination of the local ETDs, as well as other related information, so statistical and other reports can be prepared both for individual sites and for sites that have aggregate information (such as NDLTD). Next Section: Identifying: URN, PURL, DOI Technical Issues/Identifying: URN, PURL, DOI Resources distributed on the Internet are accessible by means of a syntax which corresponds to their physical location. This syntax is defined by the RFC 1738 and is known as a Uniform Resource Locator (URL). This way of doing things creates certain problems which we must often confront. Who has not encountered the famous HTTP error 404 Not Found, which indicates that the server cannot find the location of the requested resource? This does not mean that the resource is no longer on the server, because it may simply have been moved to another location. URLs have no means of being automatically updated when a resource is moved to another place, such that we often run up against that famous HTTP error. While the URL identifies the address of a resource, the Uniform Resource Name (URN) identifies the actual resource, the unit of information, much like the ISBN does for books. To draw a parallel, the URL corresponds to a users' postal address while the URN corresponds to users' social insurance or social security number. The URN is thus attached to a resource and not to a physical address. By knowing this identifier, it is possible to find this resource even if its physical address changes. The URN ensures an institutional commitment to the preservation of access to a resource on the Internet. In the framework of the Université de Montréal's digital thesis pilot project, undertaken in 1999-2000, we implemented a system for producing URNs based on the model proposed by the CNRI. A global server based at the CNRI manages "Naming authorities" which refer to publisher's numbers. A local server installed at the thesis distribution station in turn houses a database which manages the associations between URNs and URLs. All of this bears close resemblance to NetworkSolution's system for managing the DNS which regulate the IP addresses of computers linked to the Internet, except that in our case it is documents being given addresses as opposed to computers. The model proposed by CNRI is the Handle system. This system is also the cornerstone for the DOI Foundation's system. The construction of the Handle falls in two parts. The URN's prefix corresponds to the publisher's number (the Université de Montréal's publisher's number is 1012). This number is unique and cannot be used by any other organization. "Sub-names" can be added following this number in order to subdivide it into more precise units. This sequence is followed by a slash ("/") and a freely chosen alphanumeric sequence. Thus, a Handle-type URN for theses reads as follows: hdl:1012.Theses/1999-Albert.Mathieu(1959)-[HTML] We chose the year of the thesis defense, the author's name, his/her date of birth and the format of the file as the constitutive elements of a thesis' URN identifier. Please note that one must first download the CNRI's plugin in order to use the Handle system. This system has the advantage of being fairly much in conformity with the requirement of RFC 1737 concerning the framework regulating a URN system. Its application is nevertheless fastidious since one absolutely requires the plugin to be able to resolve the links. After experimentation with the CNRI's system, the Université de Montréal intends to use another system for our ongoing electronic theses project. Another interesting avenue is the PURL system created by OCLC. Let us note that a document attached to a PURL can be modified, contrary to other norms or applications for the use of a URN. The PURL system largely follows the same principle as the Handle system except that the URNs are resolved using a URL address. This solution has the advantage of not requiring the use of a plugin. In fact, a PURL is a URL. Rather than pointing directly to an Internet resource, a PURL points to an intermediary resolution service. This service associates the PURL with the active URL, which is then provided to the client. The client then normally gives access to the resource. It is possible to register PURLs with an intermediary service (such as the OCLC's) or to install the service on one's own server. http://www.handle.net/ http://www.doi.org/ http://www.purl.org/ http://purl.oclc.org/OCLC/PURL/INET96 http://purl.oclc.org/OCLC/PURL/FAQ http://purl.oclc.org/ Next Section: Metadata models for ETDs Technical Issues/Metadata models for ETDs One of the objectives of an ETD program is to yield easy access to TDs. Since we are dealing with digital libraries, we are implicitly dealing with libraries. One of the actions performed on a library catalog is that of search and retrieve. This is the first step towards accessing the contents of a library item; the second step is the use (read, listen, view, etc.) of the item. In order to be efficient in the search and retrieve action, the user must search a catalog in which the items were properly identified, besides using good search functions. This section is about the identification of ETD's, which is a very important step towards their dissemination. The identification will be accomplished through the use of the metadata elements whose set is named the metadata model of the digital library of TDs. Before we address metadata models for ETDs, it is important that some ideas are brought to the discussion. These ideas are related to the choice of a model to be considered later on. These models must be rich and versatile to contain information of different natures and to be searched by users from all over the world. It is obvious that the richer and more versatile the metadata model is, the more time and effort it takes to capture (collect and record) the information into the digital library. The decision on which model to use will have to take this into consideration. In some situations it may be necessary to adopt the simplest possible model in order to make the metadata capture viable. Later in this chapter the Dublin Core Metadata Element Set will be introduced. It seems that it is the consensus of the minimum identification to be used for ETD's. The ideas for us to think about are: Many languages in one world ETDs to be read all over the world Contents and instances Contents, instances and metadata Contents, instances and languages Metadata models and languages Metadata schemes Specialization of the metadata models for TDs Conclusion - metadata models for ETDs Our world is a very diverse linguistic place. Those who work with information and are involved in international projects know English. This is the language they use to communicate, to access the Internet, to read technical literature, etc. At the same time, not only many other languages exist but some of them have large numbers of native speakers. The 100 most spoken languages of the world, when first language speakers are counted, can be found in http://www.sil.org/ethnologue/top100.html. In descending order, the first 10 are Chinese (Mandarin), Spanish, English, Bengali, Hindi, Portuguese, Russian, Japanese, German (Standard) and Chinese (Wu). If only the other 9 languages are considered, it is not hard to imagine the numbers of texts that are written and published every year. The same happens with TDs. The number of TDs published in languages other than English must be very big. ETD's to be read all over the world One of the purposes and benefits of an ETD program is to yield easy access to the results presented in TDs, no matter where the reader is and where the dissertation was written. We assume that ETD digital libraries are to be connected to the Internet so that their contents can be shared worldwide, to make sure this benefit is accomplished. Metadata are data about data or information about information. The metadata elements are the attributes used to describe a digital library item just like the ones used to catalog items in a traditional library. Many of these attributes are language dependent, as for example titles, abstracts, subjects, keywords, etc. Others obviously are not, as for example authors' names, digital format, number of bytes of the file, etc. Since some metadata elements are language dependent and TDs are written in many languages, we can expect that most probably the metadata will use the language of the work. This can pose a problem for search and retrieve activities since most of us are not fluent in as many languages as we would like to be. The items of a digital library may be identified in 2 different levels; the same way the items of a traditional library are. The first level is the content which is equivalent to a title of a traditional library and the second is the instance which is equivalent to a volume. A content is the logical definition of an item of the digital library and it is identified by a set of attributes. An instance is the physical realization of a content or title. It is a digital object and is identified by a set of attributes too. The use of contents and instances allows contents to have multiple instances either in different formats or due to physical partitions. This will yield a one to many relationship among contents and instances. The use of contents and instances also allows the access control to be performed on the partitions instead of on the content. This makes the digital library more flexible in terms of dealing with intellectual property rights. Therefore, we can conclude that there are attributes that are particular to contents and others that refer to instances. The metadata model must contain both. Some metadata elements are common to all contents, as for example title, abstract, type, etc., while others are common to all instances, as for example electronic format, access level, etc. On the other hand, some metadata elements are specific to some contents, as for example translation control - original content, translator, etc., and others are specific to some instances, as for example special equipment, expiration date, remote location, etc. From this comment, we can see that the metadata model must be versatile to contain attributes that are common to all contents and to all instances and also the specific ones, in order to accommodate specialization of the digital library items. Contents may be language dependent. The language of the content is the one in which it is written, spoken or sung. Other languages may be associated with a content - the ones in which it is catalogued. It is possible to describe a content written/spoken/sung in one language in other language(s). This way, there is one catalog entry in each of the languages to be used. The use of multilingual cataloguing yields points of access in different languages if the search is performed in all of them. This topic will be addressed in the section Database and IR. It is possible to define the digital library to hold more than one language. A good choice would be, at least, the language(s) of the nation where TDs are developed and English. If this is the case, the metadata model can have all attributes that are language dependent written in each language to be used in the digital library and the language code must be a part of the primary key in the database. Attributes that are language independent would have only one representation in the database. Metadata schemes There are quite a few metadata schemes. Some are strictly related to library items while others have a broader scope, as for example the ones devoted to digital objects to be used in Web Based Education. Some schemes are well known and should be mentioned: DCMES - Dublin Core Metada Element Set http://purl.org/dc/documents/rec-dces-19990702.htm Under the responsibility of the DCMI - Dublin Core Metadata Initiative http://www.purl.oclc.org/metadata/dublin_core/ http://purl.org/dc/ This metadata element set will be presented in the section Cataloging: MARC, DC, RDF IMS Project - Instructional Management System Project http://www.imsproject.org/ The metadata element set defined by the IMS Project has the objective of identifying digital objects used in Web based Education. It contains all the elements of the DCMES and many more. LOM - Learning Objects Metadata of the Learning Technology Standards Committee of the Institute of Electrical and Electronics Engineers (LTSC/IEEE) http://ltsc.ieee.org/doc/wg12/LOM_WD4.htm/ The metadata element set defined by the LTSC/IEEE (http://ltsc.ieee.org/) has the objective of identifying digital objects used in Web based Education. It contains all the elements of the DCMES and many more. LoC - Core Metadata Elements of the Library of Congress http://lcweb.loc.gov/standards/metadata.html The second and the third are used when WBE is under. Since they contain the DCMES, no conflict exists to the general digital library identification. Besides the usual data contained in general purpose metadata schemes, there are some types of information related to TDs that may be of interest to the university. For this reason, it may be useful to consider adding extra metadata elements to the traditional metadata schemes. The additional elements can be separated in 3 groups: Administrative information - department, date of presentation, date of acceptance, financial support, etc. Academic information - level, mentor, examining committee, etc. Traditional library information - university, library system, control number, call number, etc. These may be useful to yield information concerning the graduate programs of the university. Conclusion - metadata models for ETD's The definition of the metadata model for an ETD digital library must combine: The needs for proper identification of ETD's for the goals of access to be achieved (national access? international access?) The administrative needs of the university At the same time, the restrictions imposed by budget or operation time frames must be to taken into consideration. There is a balance between what is desired and what is possible. Some comments concerning this balance are made: For international access, the use of English besides the original language(s) is mandatory. This means that titles and abstracts must be translated, and that subjects headings, keywords, etc. will be multilingual catalogs to be maintained. For the ETD digital library to be a part of the international community, the minimum requirements in terms of ETD identification must be met. This means that at least the DCMES must be used. For the university to have good control of the intellectual property, the use of content / instance concept allows access specifications to be established on the digital objects. Thus, some objects may be made public while others may have different types of restrictions due to format or to intellectual content. In the definition of the workflow to operate the ETD program, attention must be given to the capture of the metadata elements. If non-librarians are involved in the process, there must be a good training program and a careful review process so that the attributes are catalogued right. The choice of the metadata model is very important and the team in charge of the implementation of the ETD program must study the possibilities before making the decision. Minimum standards must be met. Next Section: Cataloging: MARC, DC, RDF Technical Issues/Cataloging: MARC, DC, RDF Many traditional library systems exchange and store records using the MARC Format (Machine Readable Catalog), one of the realizations of the ISO 2709 Standard. The MARC Format has approximately 1,000 fields, many with subfields which can be repeated. The use of this format allows a very detailed description of the items. This format has a specific field (856) to identify of electronic objects associated with the intellectual item and its other physical instances. The DCMES (Dublin Core Metadata Element Set - http://dublincore.org/documents/dces/) is a set of 15 attributes divided into 3 groups: content, intellectual property and instanciation. Associated to them there are the Dublin Core Qualifiers (http://dublincore.org/documents/dcmes-qualifiers/) that enhance the identification of the items. There is a relation between the MARC Format and the DCMES since there is an intersection between the 2 sets of attributes. The RDF (Resource Description Framework - http://www.w3.org/TR/1999/rec-rdf-syntax-19990222/) is a foundation for processing metadata. It specifies a representation for metadata as well as the syntax for encoding and transporting this metadata. The objective is to yield interoperability of Web servers and clients, and to facilitate automation of processing of resources. It can be used to describe Web pages, sites or digital libraries. The DCMES can be used with the RDF representation. Considering the MARC Record and the Cataloging Department Work Flow MARC Bibliographic Records Catalogers may want to focus initially on what fields are currently included in the MARC bibliographic record for theses and how these would be the same or different for ETDs. The MARC record for theses is not very robust and often has a local twist, presenting valuable information in a unique format that can be seen only at the originating institution. Optimally, author, title, abstract, and other relevant bibliographic information would be programmatically adapted to the appropriate MARC fields. The extent of AACR2r compliance was another complicating factor. For example, would programming change upper-case letters to lower case? Requiring authors of theses (in all formats) to provide keywords for use in the bibliographic record may enhance search results. Assigning Library of Congress subject headings (or other controlled vocabulary) is very time consuming, so having the authors assign the uncontrolled subject headings may be an appropriate alternative. MARC tag 653 would be appropriate for author-assigned terms. Without LC subject headings, however, these MARC records would be considered "minimal level," rather than full level, cataloging. This seemed particularly unjust to because the ETD bibliographic records would actually be more robust than previous theses cataloging because additional information is included. The catalogers on the ad hoc task force suggested including tables of contents (MARC tag 505) and abstracts (MARC tag 520) since the standard copy-and-paste features of today's word processors would make this a relatively easy process. The table of contents for dissertations, however, proved to be quite generic, usually containing only the standard dissertation topics (e.g., literature review, methodology, findings, etc.), and, therefore, not an enhancement to the information available about the work in the OPAC. The abstract, however, contains valuable information and provides valuable information about the research topic. The 520 is also an indexed field in our online catalog and, therefore, a word-searchable field for OPAC users. Adding the abstract (250-350 words) can, however, add tremendously to the length of the MARC record. See figure 2 and figure 3. Cataloging conventions have not generally included the name of the thesis author's department as a standard feature of the bibliographic record. This is an opportunity to modify local cataloging practices. Taking advantage of the opportunity to incorporate changes in theses cataloging, consider using MARC tag 502, the dissertation note field, to include the degree, institution, and year the degree was granted, expanding institution to include the name of the department. The new note would follow this example: 502 Thesis (Ph. D. in Mechanical Engineering)--Virginia Polytechnic Institute and State University, 1955. Evaluating the potential value of an e-thesis bibliographic record provides the opportunity to propose a substantially enhanced record of real and continuing value to OPAC users. AACR2r compliance may be an issue. In reviewing her Cataloging Internet Resources: A Manual and Practical Guide, Nancy Olson states that when cataloging Internet-accessible documents, consider them to be published documents. Therefore, publisher information belongs in field 260 of an e-thesis record. [Coming from a serials background it seems reasonable to add a 710 for this corporate body tracing.] Additional fields required for cataloging computer files include tag 256 for computer file characteristics, tag 538 for notes of system details, and tag 856 for formatted electronic location and access information. These additional fields (505, 260, 710, etc.), however, also increase the length of the record and, therefore, should be carefully considered as should the usefulness of the information provided in meeting the combined needs of OPAC users and computerized access and retrieval systems. Figure 2: MARC bibliographic record for an ETD VT University Libraries - - - - - - - - ADDISON - - - - MARC BIBLIOGRAPHIC RECORD [OCLC fixed field tags] Local lvl: 0 Analyzed: 0 Operator: 0000 Edit: Type cntl: CNTL: Rec stat: n Entrd: 010608 Used: 010706 Type: a Bib lvl: m Govt pub: s Lang: eng Source: d Illus: a Repr: Enc lvl: K Conf pub: 0 Ctry: vau Dat tp: s M/F/B: 0 Indx: 0 Mod rec: Festschr: 0 Cont: b Desc: a Int lvl: Dates: 2001, [003-049 system assigned fields and information] 001 ocm47092981 010608 006 m d s 007 c \b r \d u \e n \f u 040 VPI \c VPI 049 VPII 099 Electronic Thesis 2001 Alvarez 100 1 Alvarez, Leticia, \d 1973- 245 14 The influence of the Mexican muralists in the United States \h [computer file] : \b from the new deal to the abstract expressionism / \c Leticia Alvarez. 256 Computer data (1 file) 260 [Blacksburg, Va. : \b University Libraries, Virginia Polytechnic Institute and State University, \c 2001] 440 0 VPI & SU. History. M.A. 2001 500 Title from electronic submission form. 500 Vita. 500 Abstract. 502 Thesis (M.A.)--Virginia Polytechnic Institute and State University, 2001. 504 Includes bibliographical references. 520 3 This thesis proposes to investigate the influence of the Mexican muralists in the United States, from the Depression to the Cold War. This thesis begins with the origins of the Mexican mural movement, which will provide the background to understand the artists2 ideologies and their relationship and conflicts with the Mexican government. Then, I will discuss the presence of Mexican artists in the United States, their repercussions, and the interaction between censorship and freedom of expression as well as the controversies that arose from their murals. This thesis will explore the influence that the Mexican mural movement had in the United States in the creation of a government-sponsored program for the arts (The New Deal, Works Progress Administration). During the 1930s, sociological factors caused that not only the art, but also the political ideologies of the Mexican artists to spread across the United States. The Depression provided the environment for a public art of social content, as well as a context that allowed some American artists to accept and follow the Marxist ideologies of the Mexican artists. This influence of radical politics will be also described. Later, I will examine the repercussions of the Mexican artists2 work on the Abstract Expressionist movement of the 1940s. Finally I will also examine the iconography of certain murals by Mexican and American artists to appreciate the reaction of their audience, their acceptance among a circle of artists, and the historical context that allowed those murals to be created. 538 System requirements: PC, World Wide Web browser and PDF reader. 538 Available electronically via Internet. 653 mural painting \a WPA \a abstract expressionism 856 40 \u http://scholar.lib.vt.edu/theses/available/etd-05092001-130514 945 NBJun2001 949 dpm/tm 06/07/01 994 E0 \b VPI Figure 3: OPAC display for an electronic thesis from the Virginia Tech VTLS opac VT University Libraries - - - - - - - - ADDISON- - - - - - - - - - - -FULL RECORD CALL NUMBER: Electronic Thesis 2001 Alvarez Author: Alvarez, Leticia, 1973- Title: The influence of the Mexican muralists in the United States [computer file] : from the new deal to the abstract expressionism / Leticia Alvarez. File Type: Computer data (1 file) Imprint: [Blacksburg, Va. : University Libraries, Virginia Polytechnic Institute and State University, 2001] Series: VPI & SU. History. M.A. 2001 Note: System requirements: PC, World Wide Web browser and PDF reader. Note: Available electronically via Internet. Remote Acc.: http://scholar.lib.vt.edu/theses/available/etd-05092001-130514 Note: Title from electronic submission form. Note: Vita. Note: Abstract. Note: Thesis (M.A.)--Virginia Polytechnic Institute and State University, 2001. Note: Includes bibliographical references. Abstract: This thesis proposes to investigate the influence of the Mexican muralists in the United States, from the Depression to the Cold War. This thesis begins with the origins of the Mexican mural movement, which will provide the background to understand the artists2 ideologies and their relationship and conflicts with the Mexican government. Then, I will discuss the presence of Mexican artists in the United States, their repercussions, and the interaction between censorship and freedom of expression as well as the controversies that arose from their murals. This thesis will explore the influence that the Mexican mural movement had in the United States in the creation of a government-sponsored program for the arts (The New Deal, Works Progress Administration). During the 1930s, sociological factors caused that not only the art, but also the political ideologies of the Mexican artists to spread across the United States. The Depression provided the environment for a public art of social content, as well as a context that allowed some American artists to accept and follow the Marxist ideologies of the Mexican artists. This influence of radical politics will be also described. Later, I will examine the repercussions of the Mexican artists2 work on the Abstract Expressionist movement of the 1940s. Finally I will also examine the iconography of certain murals by Mexican and American artists to appreciate the reaction of their audience, their acceptance among a circle of artists, and the historical context that allowed those murals to be created. Key Words: -- mural painting In terms of the broader topic of bibliographic control of electronic publications, focus on adding to current cataloging practices those fields that would enhance the OPAC users' access and conform to AACR2r. So many of the fields describing computer files appear to be redundant; 245 \h, 256, 516, and 538, for example; which tell the OPAC user over and over that the item is a computer file. To stay within the stringent restrictions of full-level cataloging, the members of the task force saw no way to avoid requiring catalogers to use most of the available fields. Concentrate on the MARC tags that would provide information about access. The principal fields include: 256 (computer file characteristics), 506 (restrictions on access note), 516 (type of computer file or data note), 530 (other formats available), 538 (system details note), 556 (accompanying documentation), and 856 (electronic location and access). Current OPACs, in addition to the limitations of hardware and workstations, however, still prevent most users from accessing electronic texts or images directly and smoothly from one menu or even from a single, multi-function workstation. However, workstations are gradually becoming available that permit users to copy the URL from the bibliographic record and paste it into a World Wide Web browser for accessing an e-text from a single terminal. Knowing this was possible include MARC tag 856. Another issue that must be addressed is using subfield u or splitting the URL into the multiple subfields. We went for simplicity and decided to format the 856 subfield u so that it could be copied and pasted into a World Wide Web browser. Again, we were not willing to wait for the programming that would be necessary to combine the separate subfields into a clickable URL. Cataloging has greatly benefited from advances in library automation and the cataloging of e-texts is ripe for further automation. It is now possible to derive MARC cataloging from text mark-up languages, subsets of XL, SGML such as TEI (Text Encoding Initiative) headers, and possibly even HTML (hypertext markup language) tags. ETD Submission Form Name: [MARC tag 100] Title: [MARC tag 245] Document Type (check one): Abstract: [MARC tag 520] Keywords: [MARC tag 653] Department: [MARC tag 502] Degree: [MARC tag 502] Filename(s), size(s): [MARC tag 256] In addition to considering the MARC record and the cataloging department work flow, consider a procedure for getting the files from the Graduate School (approving unit), the mechanics of making an ETD available to a cataloguer (from the secure and private environment of the server) and for moving an ethesis into public access. Have the cataloguer forward a copy as each ETD is processed to a server at UMI. If UMI would prefer batch processing, files could be accumulated (i.e., stored in a directory on the etheses server) for batched file transfer, or perhaps a UMI-access point could be established on the ETD server from which its staff could retrieve them. With input from the University Archivist and addressing a concern of the Graduate School's, long term preservation and access of ETDs should also be factored into the procedures. A plan may include periodically writing ETDs to CD-ROMs for security back-ups and possibly longer term preservation. While this is may not be the final answer, an alternative has not been brought forward; how frequently this would be done has also not been determined. Processing Electronic Theses: a possible scenario Electronically transfers approved file to library theses server E-mails Thesis Transmission Form to library thesis coordinator Library/Cataloguer Downloads ED from closed server to her workstation Prepares cataloging (see new features above in figure 4) Adds a screen to the file that includes the call number and property "stamp" (using "memos" feature of Acrobat) Move file to server for public access Electronically send file to UMI or move to UMI holding file Library Theses Server Administrator Indexes text for word searchability Integrates new index with existing index Maintains server, including weekly back-ups (stored on site) and monthly tapes (stored off site) Removes files from closed server following completed processing Makes CD-ROMs Special Collections Department/University Archives Retains CD-ROMs Works with Theses Server Administrator as necessary to ensure that archival files are accessible Theses and dissertations as electronic files may be the first major source of electronic texts that many libraries encounter regularly. Seize this opportunity to enhance the OPAC users search results by expanding current theses cataloging and taking advantage of online information prepared by authors. Since authors will probably not be adding TEI, MARC, or other element tags to their documents to help cataloging in the near term, catalogers could use the information available in a variety of online sources including the document itself or from the online submission form to provide the basic descriptive MARC fields. Whether programmatic changes can be made or standard copy-and-paste features of word processors are incorporated, enhancing the ETD bibliographic record does not require a lot of extra work. See also "Electronic Theses and Dissertations: Merging Perspectives," chapter in Electronic Resources: Selection and Bibliographic Control, Pattie, Ling-yuh W. (Miko), and Bonnie Jean Cox, eds. New York: Haworth, 1997 (105-125). [Simultaneously published in Cataloging and Classification Quarterly, 22(3/4)] Next Section: Database and IR Technical Issues/Database and IR The next step after identification of the items of the digital library, the ETDs, is to address storage of the cataloguing attributes and the action of searching and retrieving. Remember that the quality of retrieval is dependent both on the programming of the search and retrieve functions but, as important as this, on the quality of the information used to catalog the items of the collection. Databases are common and suitable tools to store, search and retrieve information. Besides this, they can also be very helpful in the process of capturing the attributes since they have the general function of managing information. Before implementing the database, the database model must be created. This will happen only after the metadata model has been defined and related to other existing identification procedures such as traditional cataloguing on an automated library system. If the traditional OPAC is to be maintained during the ETD program, it is desirable to avoid duplicated information. Thus, the attributes that are present in the OPAC should not be repeated and a link between the OPAC record and the ETD metadata is be created. No matter where information is stored, the user should be able to perform the types of search that are standard in library systems: author, title, keywords, subjects, ISBN, etc. As mentioned in the section Metadata models for ETDs, some information that is used to identify the items are language dependent (title, keywords, subjects, etc.). If the database holds only one language per record, search procedures are to be performed using the arguments in this language. If a multilingual database is modelled, it is recommended that search be language independent, i.e., that the argument be checked against all languages. In this last situation, after a record is found as the result of a search, all its language instances should be displayed to the user so that he/she can choose the language for retrieval. Database extenders (text, image) may be considered to increase the number of points of access by performing the searches in the ETD's not only on the cataloguing attributes. The relation with the legacy systems databases must be examined since information concerning the TDs may be stored on them. Some examples are the graduate program, the mentor, the examining committee, etc. The next sections address specific aspects of this topic. Next Section: Packaged solutions Technical Issues/Packaged solutions Since 1995, there have been plans to share work developed in connection with ETDs so as to help others. The first such effort was funded by the Southeastern Universities Research Association (SURA), to spread the work around the Southeast of the USA. Virginia Tech acted as the agent for this effort, creating software, documentation, training materials, and other resources. In particular, it became clear that software was needed to support student submission, staff checking, library processing, and support for access. The Virginia Tech resources were packaged in connection with these efforts and the follow-on work funded by the USA's Department of Education. These were made available starting in 1996 through local and NDLTD sites, and updated regularly since. The following subsections give details of the Virginia Tech solutions, as well as extensions to it, and alternatives. NDLTD offers to assist such sharing by providing a clearinghouse for whatever seems appropriate and useful to share. Next Section: DiTeD and DIENST Technical Issues/DiTeD and DIENST Theses and dissertations are traditionally covered by the legal deposit law in Portugal. Nowadays, almost all the thesis and dissertations are created using word processors, just confirming the fact that science and technology became one of the first areas for digital publishing. In this context, the deposit of theses and dissertations emerged as an ideal case study for a scenario concerned with a specific genre. For that, the National Library of Portugal promoted the project DiTeD-Digital Thesis and Dissertations from which a software package with the same name originated. Theses and dissertations carry special requirements for registration and access, since their contents are usually used to produce other genres, such as books and papers, or they can include sensitive material related with, for example. patents. This requires the management system functionality to make it possible to the authors to declare special requirements for access, which have to be registered and respected. Universities have a long tradition of independence in their organization, culture and procedures. As a consequence, soon it was learned that it would be impossible to reach, in the short and medium term, any kind of overall agreement for common formats or standard procedures with the different administrative services. Therefore, the main objective defined for DiTeD was the development, on the top of the Internet, of a framework that would connect the National Library to the local university libraries and would make it possible to support a full digital circuit for the deposit of theses and dissertations. A solution for this framework was found in the DIENST technology [3], which provides a good set of core services. DIENST also has an open architecture that can be used with great flexibility, making it possible to extend its services and build new functionalities. The basic entities of this architecture are shown in Figure 1, as a class diagram in UML - Unified Modeling Language. Master Server The Master Meta Server provides the centralized services, including a directory of all the local servers members of the system. Only one of these servers must exist in each system. In DiTeD this server exists at the National Library. It was renamed Master Server, and differs substantially from the original versions developed for DIENST. The original server was designed to manage only metadata, while now it is necessary to manage also the contents of the theses or dissertations and give support to the workflow for its submission and deposit. DIENST Standard Server The DIENST Standard Server is the server installed at the university libraries. This server was modified in DiTeD, and renamed Local Server. The following core modules compose it: Repository Service: This is where the documents are stored. It manages metadata structures and multiple content formats for the same document, functions that were substantially extended in DiTeD (to support a specific metadata format, as also to recognize a thesis or dissertation as possibly composed by several files). It is also possible to define and manage different collections in the same server. Index Service: This service is responsible for indexing the metadata and responding to queries. Small adjustments were made in DiTeD to support diacritics in the indexes and queries, a requirement in the Portuguese writing. User Interface: This service is responsible for the interaction with the user. It was extended in DiTeD to support a flexible multilingual interface and a workflow for submissions using HTTP. Two Local Servers are running at the National Library. One, named Deposit Server, is used to locally store the deposited theses and dissertations coming from all universities (the deposit will consist in a copy, so in the end each thesis or dissertation will exist in two places, the Local Server and the Deposit Server). A second Local Server is used as a virtual system for those university libraries that do not have the necessary technical resources or skills to maintain their own server. Each thesis or dissertation deposited in DiTeD automatically receives a URN [4], which will be registered and managed by a namespace and resolution service. This is in fact a simple implementation of the concept of PURL - Persistent URL, with the particular property that it resolves any PURL by returning its real URL in the original Local Server, unless it is not available anymore. In this case, it resolves it by returning its URL in the Deposit Server. The entities of this final DiTeD architecture are shown in Figure 2. The prefix of the URN has the form "HTTP://PURL.PT/DITED", while the suffix is formed by an identifier of the university library (the "publisher") and by a specific identifier of the work itself, automatically assigned locally. The workflow comprises two main steps: submission and deposit. The submission process comprises the following steps: Delivery: The process starts with the submission by the student of the thesis or dissertation to a local server. In this step the student fills a metadata form, where it is recorded the bibliographic information and the access conditions. All of this information is hold in a pending status, until it is checked. Verification: In a second step a librarian checks the quality of the submission (a login in the local server gives access to all the pendent submissions). This task is supposed to be assured by a local librarian, but it can be also assured remotely, such as by a professional from the National Library (in a first phase of the project, this task will be assured by the National Library, especially to assure uniformity in the criteria and test and tune the procedures). Registration: If everything is correct (metadata and contents), the thesis or dissertation is stored in the local repository, and the student receives a confirmation. Otherwise, the student is contacted to solve any problem, and the submission remains in the pending status. The deposit consists in the copy of the thesis or dissertation, as also of its metadata, from the Local Server to the Deposit Server. This is done in the following steps: What's new: Periodically, the Master Server contacts the repository of a Local Server to check if there are new submissions. The Local Server replies, giving a list of the identifiers of the new submissions. Delivery: For each new submission, the Master Server sends a request to the Local Server to deposit it in the Deposit Server. Because this Deposit Server is also a Local Server, this deposit works just like a normal local submission. Verification: A librarian in the National Library checks the deposit. This double checking is important, especially in the first times of the project, to reassess the procedures and test the automatic transfer of files over the Internet -not always a reliable process). Registration: If everything is correct, the thesis or dissertation is stored in the deposit repository, the final URN (a PURL) is assigned, and both the student and the local librarian receive a confirmation. The metadata is also reused to produce a standard UNIMARC record, for the national catalogue. If it detected any problem, the local librarian is contacted and the deposit remains in the pending status. One can argue that, if the Deposit Server is really also a Local Server, than the first step would be excused and the Local Server could perform the delivery automatically after a successful submission. This can be a future optimization, but for now the reason for this extra step is to preserve the requirement of an asynchronous system, making it possible for the Master Server, for example, to better control the moment of the deposit (such as to give preference for the night periods). DiTeD utilizes a metadata structure for theses and dissertations defined by the National Library and coded in XML. This structure contains descriptive bibliographic information about the work and the author, as well as information about the advisers and jury members, access conditions, etc. This metadata structure is configurable at installation time, making the software flexible for use in other countries, or even with other publication genres. Metadata may also be accessed and exported in other formats, like UNIMARC and Dublin Core. Multilingual interface DiTeD's user interface has multilingual capabilities, allowing the users to switch between the available languages at any time. The base configuration includes English and Portuguese. Software availability The software is maintained by the National Library of Portugal, and distributed freely for noncommercial use. Access to the software package may be requested by email to [email protected]. <http://dited.bn.pt> <http://purl.org> <http://www.cs.cornell.edu/cdlrg/dienst/software/DienstSoftware.htm> Sollins, K; Masinter, L. (1994). Functional Requirements for Uniform Resource Names. RFC 1737. Next Section: ADT Technical Issues/ADT Australian Digital Theses Program http://adt.caul.edu.au/ ADT Program is a collaborative Australia-wide university libraries initiative. Membership is open to all Australian universities and is voluntary. National coordination of the ADT is via the Council of Australian University Librarians [CAUL; an umbrella group representing all Australian university libraries]. The ADT Steering Committee is currently investigating a proposal to further extend ADT Program & software to the Australasian region. ADT software was designed to be transportable and to be flexible enough to be 'plugged in' at each member institution with minimum modification. The ADT software was also designed to automatically generate simple core metadata which is gathered automatically to form national distributed 'metadata' database of ADT-ETDs ADT software is based on the original Virginia Tech [VT] software. The original ADT v1.0/1999 released in April 1999, with upgrade v1.1/2000 released in October 2000. ADT software basics: perl scripts extended by the library cgi.pm, which uses objects to create web forms on the fly and parse their content facilitates file uploading and form handling by generation of html via function call and passing of the form state extension of variable use to make scripts more generic to facilitate local institutional setup and style developed a standard for generation of unique addresses [URLs] for deposited documents automatic generation of DC metadata from the deposited information The distribution software package requires modification of a list of variables and some webserver-dependent adjustments so it runs in standard way for each of the local institutions. Local search software, which may be institution-mandated, is not included in the package. It is expected that each institution will use their own security accordingly. ADT software overview: Deposit Form : generic look and feel; includes copyright and authenticity statements; complete set up help screens; quality control using alerts when errors are made and does not allow non compliance with core ADT standards Administration pages: possible to edit html and change document restrictions; also possible to add, delete, rename files as well as moving files to no access directory to restrict files temporarily for copyright reasons; varying levels of restrictions possible; easy to 'un make' deposit Metadata: revised and updated according to latest DC Qualifiers document; generated and gathered automatically, used to create central national searchable 'metadata' database ADT Standards: few, simple but necessary for success of collaborative program. Core standards are - research theses only; PDF document format; PDF filename convention; unique URL; Metadata standard. the ADT software & model is fully transportable and designed to be easily installed locally by any participating institution regardless of local IT infrastructure and architecture the ADT software facilitates loading digital versions of theses to the local institutions' servers where the PDF files will be housed permanently the theses can be fully integrated into the local access infrastructure and searched using any local database, and/or the local web based catalogue all ADT Program theses can also be searched nationally via the ADT Program metadata database. This database is constructed from DC metadata generated automatically during the deposit process. The metadata gathered creates rich records that allow highly flexible and specific searching, with links back to the local institutions' servers where the full digital theses are housed the ADT Program is a collaborative effort across the whole Australian university community and a proven model for creating a national dataset of digitised theses the ADT software and model is relatively inexpensive to install, integrate within local requirements and process. Once this is achieved it is virtually maintenance free, is sustainable, scalable, and very cost effective for both the institution and the student/author the ADT metadata and other standards conform to current internationalstandards and therefore have potential to integrate with other international open archive initiatives theses can be deposited from anywhere, and similarly, the metadata can be gathered from anywhere full details and information available from the ADT homepage Next Section: Cybertheses Technical Issues/Cybertheses The www.cybertheses.org site is the result of cooperative project that started at first between l'Université de Montréal "http://www.theses.umontreal.ca/" et l'Université Lumière, Lyon 2 "http://www.univlyon2.fr/, supported by the Fonds Francophone des Inforoutes "http://www.francophonie.org/fonds/", and dealing with the theme of the electronic publication and distribution of theses on the Web. At first, this cooperation dealt with the conception and creation of a production line for electronic documents, using the SGML norm. It also had the objective of setting up a server to be shared by the different participating establishments so as to allow their theses to be indexed. In a desire for openness, we decided to enlarge participation on this server to all establishments of higher education distributing full-text versions of their thesis on the Internet, without constraints based on the language used or on the chosen format of distribution. The www.cybertheses.org site allows theses to be indexed on line using a common metadata model. Its implementation provides a structure for this growing cooperative effort by basing itself on the Internet's own modes of functioning and of distributing skills. Our wish is that it can quickly ensure a better distribution of the research work conducted within the participating establishments and serve as an effective tool for the entire community of researchers. From the conception of the project, the partners wanted to contribute to the distribution of software tools created for use in the university environment. These tools are available to the partners in the Cybertheses network. They permit all the partners to participate, on equal footing, in the construction of an electronic university library that functions in a dispersed manner and applies the concept of distributed intelligence. One of the principles that motivated the implementation of this program is to favor, as much as possible, the re-appropriation of research work by the researchers themselves. Our objective is to eventually help create a new political economy of knowledge, which makes researchers the masters of their publications, beyond any economic constraints. Many aspects of our project respond to this goal: Placing theses freely and completely on-line on the Internet, and thus widening their distribution, means that the thesis will no longer be considered as solely the result of a research project. It will instead become a genuine work instrument integrated to a much larger system in order to satisfy the user's demand. The creation of the Cybertheses database containing the metadata of the participating institutions' theses. Cybertheses provides an efficient indexation system and rapid searching, even while significantly increasing the visibility and the distribution of the theses. The use of free or freely distributed software is favored at each stage of the production and distribution process. The sharing of research, self-created software programs, and documentation between the Cybertheses partners, thereby creating a "toolbox" permitting participation in the program. The Cybertheses partners are concerned with developing solutions that can be used by all actors, be they of the North, South or East. Our procedure in based on appropriating the skills and techniques related to the electronic production and distribution of the university community's research results. For more information, consult: http://www.cybertheses.org/ Next Section: VT DV and other tools Technical Issues/VT DV and other tools This page describe the hardware and software requirements involved in setting up your own ETD database. To use this software, you must have a web server available. At Virginia Tech we use a UNIX-based server platform. You should allocate enough disk space on your machine for at least a year's worth of submissions. Our site averages 2.5 Megabytes per submission. Keep in mind that it's better to have more space early on, as the scripts are not designed to deal with a collection which spans multiple drives. You should also have enough memory to handle the web server, the database server, and whatever other tasks you have in mind. As an example, our site uses a dual-processor Sun Enterprise 250 with 384 Mb of RAM, running Solaris 2.7. Our machine has an 18Gb drive allocated solely for the ETD collection. Before you can make use of the scripts provided here, you must have the following installed: Mysql is a database server and client which partially implements the SQL 9.2 standard. It is many ways similar to other SQL databases, such as Oracle, Postgres, and miniSQL. UNIX versions of Mysql are made available without charge to education institutions at http://www.mysql.com/. A version of Mysql for Windows NT is available as well for an additional charge, although the scripts are not designed for use with Windows NT. All of the scripts included with this distribution are written using perl. Perl is also freely available (under the GNU public license) from http://www.perl.com/CPAN/. It is recommended that you download, install, and test the latest version available for your operating environment. CGI.pm The CGI module for perl is one of the most widely used and best supported libraries of CGI oriented routines in existence. Virtually all of the query handling performed in these scripts relies heavily on CGI.pm. CGI.pm is available from http://www.perl.com/CPAN-local/modules/bymodule/CGI/. The DBI and DBD:Mysql modules for perl The DBI module for perl is a generic set of database calls designed to interface with a wide range of different database technologies in a powerful, reliable, and easy to understand way. To make use of the DBI module, you need a DBD module for the particular database you intend to use. The DBD:Mysql module (also known as the DBD:Msql module) allows you to easily perform all types of database operations on a Mysql database from within perl. Both modules are available from http://www.perl.com/CPAN/modules/dbperl/. The Tie-IxHash module for perl The Tie-IxHash module is a very small add-in that allows you to reliably output hashes in the order they are defined in. Without this module, none of the global hashes that contain department names, degree information, etc. would appear in the order we'd like them to. This module is available at http://www.perl.com/CPAN-local/modules/by-module/Tie/ Web Server Software The perl scripts provided are designed for the most part to be used through a CGI interface, meaning that you must have a compatible web server installed. The freely available Apache Web Server is our recommendation, although any web server capable of seamlessly handling html output from perl scripts should be acceptable. Apache is available from http://www.apache.org/. Once you have all the preceding items installed and tested, you should be ready to download (http://scholar.lib.vt.edu/ETD-db/developer/download/) and install(http://scholar.lib.vt.edu/ETD-db/developer/install.html) the scripts Next Section: Library Automation/OPAC: VTLS Technical Issues/Library Automation\OPAC: VTLS ETDs are intimately connected with libraries. The first key automation of libraries, launched in the 1980s, involved computerization of the card catalogs. Online Public Access Catalogs (OPACs) were developed, and supported search through records describing library holdings. Many library catalogs have their holdings encoded according to the MARC (machine readable cataloging) standard. Managed by the US Library of Congress, the MARC standard in the USA has been applied in many other contexts, leading to broader standards like UNIMARC. The latest standard as of 2001 is MARC 21. VTLS, Inc. (http://www.vtls.com), is one company that sells software (e.g., their Virtual system, which works with MARC) and services to support catalog search and other related library requirements. OPACs in many cases have evolved into sophisticated digital libraries, and can be used to help manage ETD collections. Indeed, ETDs at most universities can be found by searching in the local catalog system, as long as one can distinguish this genre or type of work from other holdings (e.g., books, journals, maps). VTLS has volunteered equipment, software, staff, and services to help NDLTD. From http://www.vtls.com/ndltd one can gain access to a system affording search and browse support to the union catalog of all ETDs that can be harvested using the mechanisms supported in the Open Archives Initiative. Next Section: Harvest usage in Germany, France Technical Issues/Harvest usage in Germany, France The HARVEST system is often used for a fulltext search within the ETD archives. In Germany, most of the university libraries are using this particular software. Preconditions for using Harvest Before the installation, one should check whether the following technical preconditions are fulfilled. fast processor(e.g. Sparc5...) fast I/O enough RAM ( > 64 MB) – and 1-2 GB free disk space (sources 25 MB) Operating Systems supported DEC OSF/1 ab 2.0 SunOS ab 4.1.x SunSolaris ab 2.3 HPUX AIX ab 3.x Linux alle Kernel ab 1999 on alle Unix-Platformen WindowsNT Following additional software is needed to use Harvest: Perl v4.0 and higher (v5.0 ) HTTP-Server (with remote machine) GNU gcc v2.5.8 and higher flex v2.4.7 bison v1.22 Harvest Components The Harvest system consists of two major components: The Harvest Gatherer The Harvest Broker. This allows establishing a distributed retrieval and search model. Installation procedure (ftp://ftp.tardis.ed.ac.uk/pub/harvest/develop/snapshots/ ) This program part is responsible for collecting the full text and metadata of the dissertations. The Gatherer visits several sites regularly, sometimes daily or weekly and builds an incremental index area database. The collected index data are held in a special format, called SOIF-Format (Summary Object Interchange Format). The Gatherer can be extended so that it can interpret different formats. This part of the software is responsible to provide the indexed information by using a search interface. The broker operates as Query-Manager and does the real indexing of the data. Using a Web based interface he can reach several Gatherer and Broker simultaneously and perform several search requests. In Germany there has been established a Germany-wide retrieval interface on the basis of the Harvest software, called The (Theses Online Broker), which is accessible via: http://www.iuk-initiative.org/iwi/TheO Within NDLTD a special Broker has been set up to add the German sites using Harvest to the international search. Harvest is able to do a search within the following document formats: C, Cheader Extract procedure names, included file names, and comments Dvi Invoke the Text summarizer on extracted ASCII text FAQ, FullText,README Extract all words in file Framemaker Up-convert to SGML and pass through SGML summarizer HTML Up-convert to SGML and pass through SGML summarizer LaTex Parse selected LaTex fields (author, title, etc.) Makefile Extract comments and target names ManPage Extract synopsis, author, title, etc., based on `-man News Extract certain header fields Patch Extract patched file names Perl Extract procedure names and comments PostScript Extract text in word processor-specific fashion, and pass through Text summarizer. RTF Up-convert to SGML and pass through SGML summarizer SGML Extract fields named in extraction table SourceDistribution Extract full text of README file and comments for Makefile and source code files, and summarize any manual pages Tex Invoke the Text summarizer on extracted ASCII text Text Extract first 100 lines plus first sentence of each remaining paragraph Troff Extract author, title, etc., based on `-man, `-ms, `-me macro packages, or extract section headers and topic sentences Unrecognized Extract file name, owner, and date created Configuration for PDF files: Before the Harvest-Gatherer can collect PDF documents and transform into SOIF format it has to be configured. Using only the standards configuration ignores the format. In order to make a format known to the Gatherer a summarizer for PDF has to be build: Delete the following line in the file /lib/gatherer/byname.cf: Pdf ^.\.(pdf\|PDF)$ Configure the PDF summarizer. Use Acrobat to transfer PDF documents into PS documents, that are used by the summarizer. A better choice provides the xpdf packages by Derek B. Noonburg (http://www.foolabs.com/xpdf ). It contains a PDF-to text converter (pdftotext), that can be integrated into, the summerizer Pdf.sum: /usr/local/bin/pdftotext $1 /tmp/$$.txt Text.sum /tmp/$$.txt rm /tmp/$$.txt Configuring the Gatherers for HTML-Metadata The Harvest-Gatherer is by standard configured to map every HTML metatag into an SOIF attribute, e.g. <META NAME="DC.Title" CONTENT="Test"> into an own SOIF attribute, that is equal to the NAME attribute of the metatag. The configuration can be found at: <harvest home>/lib/gatherer/sgmls-lib/HTML/HTML.sum.tbl The summarizer table has entries like this: <META:CONTENT> $NAME If a retrieval should be done only in the HTML metatags, meaning within certain SOIF attributes, those attributes have to put in front of the search request and put into the retrieval forms provided to the user, e.g. DC.Title: Test Searching Metadata encoded in HTML As in Germany there is a nationwide agreed metadata set for ETDs, those are searchable within the German wide Harvest network. The following example shows, how those Dublin Core metadata (only a small part is displayed) is encoded within the HTML front pages for ETDs: <META NAME="DC.Type" CONTENT="Text.PhDThesis"> <META NAME="DC.Title" LANG="ger" CONTENT="Titelseite: Ergebnisse der CT- Angiographie bei der Diagnostik von Nierenarterienstenosen"> <META NAME="DC.Creator.PersonalName" CONTENT="Ludewig, Stefan"> <META NAME="DC.Contributor.Referee" CONTENT="Prof. Dr. med. K.- J. Wolf"> <META NAME="DC.Contributor.Referee" CONTENT="Prof. Dr. med. B. Hamm"> <META NAME="DC.Contributor.Referee" CONTENT="PD Dr. med. S. Mutze"> For this agreed metadata set, there has been formulated a suggestion, on how the metadata can be produced and operated at the university libraries. The following schema shows the details: The doctoral candidate uploads his ETD document to the library. He does this while filling in an HTML form, that collects internally metadata in Dublin Core format. The university libraries check the metadata of correctness and the ETD of readability and correct usage of style sheets. The university library adds some descriptive metadata to the metadata set and put a presentation version of the ETD on its server. During this procedure an HTML format page containing the Dublin Core metadata encoded as HTML At last submits the university library the metadata to the National library, which is in charge of archiving all German language literature. The national library copies the ETD and metadata to its own internal system. Searching SGML/XML documents Harvest also allows a search within SGML/XML DTD (document type definition) elements. All that has to be done to configure the Gatherer component according to the following rules: Within the home of the Harvest software (written as <harvest-home>) in /lib/gatherer/byname.cf a line has to be added: DIML ^.\.did$. (DiML is the DTD used at Humboldt-University, did is the file names of the SGML documents accorning to the DIML-DTD). This says the Harvest-Gatherer, which Summerizer should be used, if documents ending with .did are found. Now, the summarizer has to be build and saved as DIML.sum within the filesystem at <harvest-home>/lib/gatherer: (The summarizer contains the following line: #!/bin/sh exec SGML.sum ETD $ Within the catalog file <harvest-home>/lib/gatherer/sgmls-lib/catalog the following entries have to be done: (They point to the public identifiers of the DIML and from DIML used DTVs) DOCTYPE ETD DIML/diml2_0.dtd PUBLIC "-//HUB//DTD Electronic Thesis and Dissertations Version DiML 2.0//EN" DIML/diml2_0.dtd PUBLIC "-//HUB//DTD Cals-Table-Model//EN" DIML/cals_tbl.dtd PUBLIC "-//HUBspec//ENTITIES Special Symbols//EN" DIML/dimlspec.ent Now <harvest-home>/lib/gatherer/lib/sgmls-lib/DIML can be created (mkdir <path>) and four files copied into the path: diml2_0.dtd, cals_tbl.dtd, dimlspec.ent und diml2_0.sum.tbl (DTD, entity file and sumarizer table). The file diml2_0.sum.tbl consists of the DTD tags that should be searcheable and the appropriate SOIF attributes The Gatherer can be launched now. In order to saerch within certain SOIF tags, the name of the SOIF attribute has to be put in front of the search term, e.g. searching for "title: Hallo"means, searching within the SOIF-attribute 'title' for the search term 'Hallo'. At Humboldt-University Berlin, there has been installed a prototype that allows a retrieval within documents structures, so a user may search within the following parts of a document and therefore specialize the search in order to retrieve only the wanted information and hits: Fulltext (im Volltext) For authors (nach Autoren) In titles (in Titen) In abstracts (Im Abstract) Wthin authors keywords (in Autorenschlagwörtern) For Institutes/ Subjects (nach Institut/ Fachgebiet) For approvals (nach Gutachtern) Headings of chapters (Überschriften) Captions of figures (in Abbildungsbeschriftungen) In Tables (in Tabellen) Within the bibliography (in der Bibliographie) For Author names within bibliography (nach Autoren in der Bibliographie) Harvest and OAI With the growing enthusiasm for the approach of the open archives initiative, where a special OAI software protocol allows to send out requests to document archives and receive standardized metadata sets as answers, there are ideas on how this could be connected with the Harvest-based infrastructure that has been set up in Germany. Making a Harvest archive OAI compliant means, that the information, that the Gatherer holds, has to be normalized (same metadata usage) and that the index has temporarily saved within a database. The Institute for Science Networking at the Department of Physics at the University of Oldenburg, Germany developed the following software. This software, written in php4, uses an SQL database to perform the OAI protocol requests. The SQL database holds the normalized data from the Harvest Gatherer. Other university document servers, like the one at Humboldt-University, additionally hold the Dublin core metadata within an SQL database (Sybase) anyway. There a php4 script operating at the cgi-interface reads the OAI protocol requests that is transported via the HTTP protocol and puts them into SQL statements, which are then used as requests for the SQL-database. The database response is given in an SQL syntax as well, which is then transformed into OAI protocol syntax using XML and Dublin Core.(see http://edoc.hu-berlin.de/oai) Next Section: The NDLTD Union Catalog Technical Issues/The NDLTD Union Catalog The Networked Digital Library of Theses and Dissertations is an interconnected system of digital archives. After NDLTD became a reality, the logical next step was to give users the means to search and browse the entire collection of theses and dissertations. Much research went in to deciding how best to do this, and a couple of proto-solutions were examined. In the end, by adopting the Metadata Harvesting Protocol of the Open Archives Initiative to gather metadata in the ETDMS format, NDLTD was able to make the collection of ETDs accessible via a central portal. VTLS Inc., using their Virtua—Integrated Library System (Virtua ILS), is hosting this portal, providing a Web interface to the ETD Union Catalog. This portal gives users a simple and intuitive way to search and browse through the merged collection of theses and dissertations. Once relevant items are discovered via a browse, keyword, or Boolean search, users can follow links that go directly to the source archives or to an alphabetical index that allows for further searching. VTLS' flagship product, Virtua ILS, is especially suited to the needs of NDLTD. For starters, it is inherently a distributed system and adheres to emerging standards, such as the Unicode® standard, for encoding metadata. Secondly, VTLS' Web portal, called Chameleon Gateway to emphasize its changeability, is not only straightforward and easy to use, but highly customizable and translatable, giving global users the ability to retrieve information and enter searches in a multitude of languages. Currently, versions of the NDLTD portal interface exist in 14 languages, including Arabic, Korean, and Russian, and plans call for support of all languages used by NDLTD members. Next Section: Metadata Technical Issues/Metadata In addition to creating an ETD, students now need to become their own cataloguers. The tools being developed by the NDLTD will assist in the submission of a basic description of the ETD but there are skills to be developed and issues to be aware of. Students will have already discovered that one of the greatest barriers to finding information is the difficulty of coming up with the right terminology. Lists of standardized subject heading terms, structured thesauri, and fielded searching have been created to remedy this problem. Accurate metadata improves precision and increases the recall of the content of ETDs by using the same standardized terms or elements. However, even if common metadata elements are used, there is no guarantee that the vocabularies, the content of the elements, will be compatible between communities of interest. Students and researchers working within specialized areas sometimes forget that language and terms often have particular and precise meanings. Outside this field of interest, global searches can return too much of the wrong information. Generating accurate metadata requires some of the basic skills of resource description and good practice in avoiding three language problems that cause poor precision: Polysemy: words with multiple meanings. For example, if we are searching for an article that discusses types of springs and their uses, we might retrieve articles on freshwater springs or on the season of spring, as well as on leaf springs, flat springs, or coil springs. Synonymy: words representing the same concept, although they may do it with different shades of meaning. Take the words 'ball,' 'sphere,' and 'orb,' or 'scuba diving' versus 'skin diving.' If we look for scuba diving, but the term used is skin diving, we will miss materials we might otherwise find. Good metadata should draw these materials together, despite their use of different synonyms. Ambiguity: If we return to our example of springs, we can see what differentiates these meanings is their context. It is unlikely an article on coil springs will also discuss water quality. The other words used in the article, and the processes described, will be entirely different. A search engine must understand the meaning, not just be able to match the spelling of a word, if it is going to differentiate between different meanings of the same word. A possible solution to this difficulty lies in the recent development of the notion of Application Profiles. Application Profiles provide a model for the way in which metadata might be aggregated in 'packages' in order to combine different element sets relating to one resource. This is a way of making sense of the differing relationship that implementers and namespace managers have towards metadata schema, and the different ways they use and develop schema. Students should investigate these developments within their community of interest. See: Metadata: Cataloging by Any Other Name ... by Jessica Milstead and Susan Feldman ONLINE, January 1999 Available [on-line] http://www.onlineinc.com/onlinemag/OL1999/milstead1.html See: Application profiles: mixing and matching metadata schemas Rachel Heery and Manjula Patel Available [on-line] http://www.ariadne.ac.uk/issue25/app-profiles/ Next Section: Fulltext Technical Issues/Fulltext When all of the text of an ETD is available for searching, a digital library system is said to support fulltext searching. Users can submit queries that call for documents that have particular phrases, words, categories, or word stems appearing anywhere in the text (e.g., in the middle of a paragraph, or as part of the caption of a figure). In fulltext searching it often is possible to specify that query terms appear in the same paragraph, same sentence, or within n words of each other. These refinements may work together with support for exact or approximate phrase and/or word matching. For fulltext searching to work, the entire document must be analyzed, and used to build an index that will speed up searching. This may require a good deal of space for the index, often around 30% of the size of the texts themselves. Further, such searching may lead to decreased precision, since a document may be located that only makes casual mention of a topic, when the bulk of the document is about other topics. On the other hand, fulltext searching may improve recall, since works can be found that are not classified to be about a certain topic. Further, fulltext searching often yields passages in a document, so one can find a possibly relevant paragraph, rather than just a pointer to a document that then must be scanned to ascertain relevance. Next Section: SGML/XML Overview Technical Issues/SGML\XML Overview SGML/XML is a multiple-targeted strategy. "It allows librarians to ensure longevity of digital dissertations. Modern hardware and redundancy can keep all the bits of an electronic thesis or dissertation (ETD) intact. But electronic archives must be modernized continually as new document formats become popular." As librarians always tend to think in decades, document formats like TIFF, Postscript or PDF do not meet their requirements. If PDF is replaced by another de facto (industry, not ISO-like) standard, preserving digital documents would mean converting thousends of documents. XML can help overcome those difficulties. If an electronic document is to be of 'archival quality, it should be liberated from the page metaphor." A second reason for using SGML/XML is that it ensures reusability of documents by preserving raw data and content-based structuring of information pieces. Preserving data for statistics and formulas in mathematics and chemistry could allow researchers to reuse and repeat simulations, calculations and experiments, deriving the needed data directly from an archive. Third, using structured information allows the reuse of the same information or documents in different contexts, i.e., the same digital dissertation can be used to produce an online or print version, and to produce additional information products, like monthly proceedings containing the abstracts of all dissertations produced within the university during the last month, or a citation index. Additionally, the dissertation can be displaysd for different media, so a Braille reader or an automatic voice synthesizer could be used as a back-end machine. Another reason for using markup for encoding documents is that a wider, more qualified retrieval could be provided to the users of an archive. As university libraries are more and more challenged by the problem of handling, converting, archiving and providing electronic publications, one of the major tasks is providing a new quality for retrieval within the user interface. Using an SGML/XML-based publishing concept enables a new quality in the distribution of scientific contents via specific information and knowledge management. What does SGML/XML mean? The Extensible Markup Language (XML) is the universal format for structured documents and data on the Web. The current W3C Recommendations are XML 1.0, Feb '98, Namespaces, Jan '99, and Associating Stylesheets, Jun '99, and XSLT/XPath, Nov '99.( http://www.w3.org/XML ) The development of XML started in 1996 and it is a W3C (http://www.w3.org/) standard since February 1998, which may make you suspect that this is rather immature technology. But in fact the technology isn't very new. Before XML there was the Standard Generalized Markup Language (SGML), developed in the early '80s, an ISO standard since 1986, and widely used for large documentation projects. And of course HTML, whose development started in 1990. The designers of XML simply took the best parts of SGML, guided by the experience with HTML, and produced something that is no less powerful than SGML, but vastly more regular and simpler to use. While SGML was mostly used for technical documentation and much less for other kinds of data, with XML it is the opposite. "Structured data", such as mathematical or chemical formulas, spreadsheets, address books, configuration parameters, financial transactions, technical drawings, etc. are usually put on the Web using the output of layout programs as Postscript or PDF or by putting them into graphic formats like gif, jpeg, png, vrml, and so on. Programs that produce such data often also store it on disk, for which they can use either a binary format or a text format. So, if soemebody wants to look at the data, he usually needs the program that produced it. With XML those data could be stored in a text format, which allows the user reading the file without having the original program. XML is a set of rules, guidelines, conventions, whatever you want to call them, for designing text formats for such data, in a way that produces files that are easy to generate and read (by a computer). The eXtensible Markup Language (XML) is a markup or structuring language for documents, a so-called metalanguage, that defines rules for the structural markup of documents independently from any output media. XML is a "reduced" version of the Structured Generalized Markup Language (SGML), which has been an ISO-certified standard since 1986. In the field of internet publishing, it never achieved wide success due to the complexity of the standard and the high cost of the tools. It prevailed only in certain areas, such as technical documentation in large enterprizes (Boeing, patent information). The main philosophy of SGML and XML is the strict separation of content, structure and layout of documents. Most ETD projects use either the SGML standard (ISO 8879 with Korregendum K vom 4.12.1997) or the definition of the World Wide Web Consortium (W3C) XML 1.0 (10.02.1998, revised 6.10.2000). The crux of all those projects was always the document type definition (DTD). Next Section: SGML/XML and other Markup Languages Technical Issues/SGML\XML and Other Markup Languages SGML (Standard Generalized Markup Language) and XML (eXtensible Markup Language) are markup languages, which use tags ("<" and ">") with names of labels inside around the sections of the documents that are thus marked or bracketed. Document Type Definition (DTD) specifies the grammar or structure for a type or a class of documents. SGML requires a DTD while XML employs DTD optionally. But given current trends it seems that XML is most likely to be used due to the following reasons. XML is a method for putting structured data in a text file for "structured data" think of such things as spreadsheets, address books, configuration parameters, financial transactions, technical drawings, etc. Programs that produce such data often also store it on disk, for which they can use either a binary format or a text format. The latter allows you, if necessary, to look at the data without the program that produced it. XML is a set of rules, guidelines, conventions, whatever you want to call them, for designing text formats for such data, in a way that produces files that are easy to generate and read (by a computer), that are unambiguous, and that avoid common pitfalls, such as lack of extensibility, lack of support for internationalization/localization, and platform dependency. XML looks a bit like HTML but isn't HTML Like HTML, XML makes use of tags and attributes (of the form name="value"), but while HTML specifies what each tag and attribute means (and often how the text between them will look in a browser), XML uses the tags only to delimit pieces of data, and leaves the interpretation of the data completely to the application that reads it. In other words, if you see "<p>" in an XML file, don't assume it is a paragraph. Depending on the context, it may be a price, a parameter, a person. In short it allows you to develop your own mark up language specific to a particular domain. XML documents can be preserved for a long time. XML is, at a basic level an incredibly simple data format. It can write in 100 percent pure ASCII text as well as in a few other well-defined formats. ASCII text is reasonably resistant to corruption. Also XML is very well documented. The W3C's XML 1.0 specification tells us exactly how to read XML data. XML is license-free, platform-independent and well-supported. By choosing XML as the basis for some project, you buy into a large and growing community of tools (one of which may already do what you need!) and engineers experienced in the technology. Opting for XML is a bit like choosing SQL for databases: you still have to build your own database and your own programs/procedures that manipulate it, but there are many tools available and many people that can help you. And since XML, as a W3C technology, is license-free, you can build your own software around it without paying anybody anything. The large and growing support means that you are also not tied to a single vendor. XML isn't always the best solution, but it is always worth considering. XML is a family of technologies. There is XML 1.0, the specification that defines what "tags" and "attributes" are, but around XML 1.0, there is a growing set of optional modules that provide sets of tags & attributes, or guidelines for specific tasks. There is, e.g., Xlink which describes a standard way to add hyperlinks to an XML file. XPointer & XFragments are syntaxes for pointing to parts of an XML document. (An Xpointer is a bit like a URL, but instead of pointing to documents on the Web, it points to pieces of data inside an XML file.) CSS, the style sheet language, is applicable to XML as it is to HTML. XSL is the advanced language for expressing style sheets. The DOM is a standard set of function calls for manipulating XML (and HTML) files from a programming language. XML Namespaces is a specification that describes how you can associate a URL with every single tag and attribute in an XML document. What that URL is used for is up to the application that reads the URL, though. XML Schemas help developers to precisely define their own XML-based formats. There are several more modules and tools available or under development. XML provides Structured and Integrated Data XML is ideal for large and complex data like ETD's because data is structured. It not only lets you specify a vocabulary that defines the elements in the document; but it also allows you to specify relations between the elements. XML can encode metadata about DTD's. Documents are often supplemented with metadata (that is data about data). If such metadata were included inside an ETD then it would make ETD self-describing. XML can encode such metadata. However on the downside XML comes with its own bag of discomforts. Conversion from word processing forms to XML requires more planning is advance, different tools and broader learning about processing concepts than it is required for PDF. There are many fewer people knowledgeable about these matters and tools that support this conversion are less mature and expensive. Also process of converting may be complicated, difficult and time consuming. Writing directly in XML by using XML authoring tools requires some prior knowledge of XML. Also XML is very strict regarding the naming and ordering of tags. It is also case sensitive illustrating the relative effort required by students to prepare ETD's in this form. Process of Creating an XML document XML documents have four-stage life cycle. XML documents are mostly created using an editor. It may be a basic text editor like notepad. or .vi. editor. We may even use WYSIWYG editors. The XML parser reads the document and converts it into a tree of elements. The parser passes the tree to the browser that displays it. It is important to know that all this processes are independent and decoupled from each other. Putting XML to work for ETD's Before we jump into the XML details for ETD.s we should make certain things clear, since we would be using them on a regular basis now onwards. DTD (Document Type Definition): An XML document primarily consists of a strictly nested hierarchy of elements with a single root. Elements can contain character data, child elements, or a mixture of both. The structure of the XML document is described in the DTD. There are different kinds of documents like letter, poem, book, thesis, etc. Each of the documents has its own structure. This specific structure is defined in a separate document called Document Type Definition (DTD). DTD used is based on XML and it covers most of the basic HTML formatting tags and also some specific tags from the Dublin core metadata. A DTD has been developed for ETD. The developed DTD is too generic. If someone wants to use mathematical equation or incorporate some chemical equation, it won't be sufficient. For that we can incorporate MathML (Mathematical Markup Language) and/or CML (Chemical Markup Language). There are defined DTDs for these languages that we also have to use for our documents. But research of incorporating more that one DTD for different parts of the documents is still going on. CSS (Cascaded Style Sheets): CSS is a flexible, cross-platform, standards-based language used to suggest stylistic or presentational features applied throughout entire websites or web pages. In their most elegant forms, CSS are specified in a separate file and called from within the XML or HTML header area when documents loads into the CSS-enabled browser. Users can always turn off the author's styles and apply their own or mix their important styles with the authors. This points to the "cascading" aspect of CSS. CSS is based on rules and style sheets. A rule is a statement about one stylistic aspect of one or more elements. A style sheet is one or more rules that apply to a markup document. An example of a simple style sheet is a sheet that consists of one rule. In the following example, we add a color to all first-level headings (H1). Here's the line of code - the rule - that we add: H1 {color: red} XSL (the eXtensible Stylesheet Language): XSL is a language for expressing stylesheets. It consists of two parts: A language for transforming XML documents, and An XML vocabulary for specifying formatting semantics. If you don't understand the meaning of this, think of XSL as a language that can transform XML into HTML, a language that can filter and sort XML data, a language that can address parts of an XML document, a language that can format XML data based on the data value, like displaying negative numbers in red, and a language that can output XML data to different devices, like screen, paper or voice. XSL is developed by the W3C XSL Working Group whose charter is to develop the next version of XSL. Because XML does not use predefined tags (we can use any tags we want), the meanings of these tags are not understood: could mean an HTML table or maybe a piece of furniture. Because of the nature of XML, the browser does not know how to display an XML document. In order to display XML documents, it is necessary to have a mechanism to describe how the document should be displayed. One of these mechanisms is CSS as discussed above, but XSL is the preferred style sheet language of XML, and XSL is far more sophisticated and powerful than the CSS used by HTML. XML Namespaces The purpose of XML namespaces is to distinguish between duplicate element type and attribute names. Such duplication might occur, for example, in an XSLT stylesheet or in a document that contains element types and attributes from two different DTDs. An XML namespace is a collection of element type and attribute names. The namespace is identified by a unique name, which is a URI. Thus, any element type or attribute name in an XML namespace can be uniquely identified by a two-part name: the name of its XML namespace and its local name. This two part naming system is the only function of XML namespaces. XML namespaces are declared with an xmlns attribute, which can associate a prefix with the namespace. The declaration is in scope for the element containing the attribute and all its descendants. For example code below declares two XML namespaces. Their scope is the A and B elements: <A xmlns:foo="http://www.foo.org/" xmlns="http://www.bar.org/">abcd</A> If an XML namespace declaration contains a prefix, you refer to element type and attribute names in that namespace with the prefix. For example code below declare A and B in http://www.foo.org namespace, which is associated with the foo prefix: <foo:A xmlns:foo="http://www.foo.org/"> <foo:B>abcd</foo:B> </foo:A> If an XML namespace declaration does not contain a prefix, the namespace is the default XML namespace and you refer to element type names in that namespace without a prefix. For example, code below is same as previous example but uses a default namespace instead of foo prefix: <A xmlns="http://www.foo.org/"><B>abcd</B></A> attribute XML structural construct. A name-value pair within a tagged element that modifies certain features of the element. For XML, all values must be enclosed in quotation marks. cascading style sheets (CSS) Formatting descriptions that provide augmented control over presentation and layout of HTML and XML elements. CSS can be used for describing the formatting behavior of simply structured XML documents, but does not provide a display structure that deviates from the structure of the source data. CDATA section XML structural construct. CDATA sections can be used to mark tags or reserved characters with quotation marks and thus prevent them from being interpreted. For this reason, the CDATA section is especially useful for escaping markup and script. The syntax for CDATA sections in XML is <![CDATA[ ... ]]>. character data XML structural construct. The text content of an element or attribute. XML differentiates this plain text from markup. character set A mapping of a set of characters to their numeric values. For example, Unicode is a 16- bit character set capable of encoding all known characters; it is used as a worldwide character-encoding standard. component An object that encapsulates both data and code, and provides a well-specified set of publicly available services. data type The type of content that an element contains: a number, a date, and so on. In XML, an author can specify an element's data type, for example, with a tokenized attribute type. Microsoft is working with the W3C to define a set of standard types that anyone can freely use. document element The top-level element of an XML document; only one top-level element is allowed. The document element is a child of the document root. Document Object Model (DOM) The standard maintained by the W3C that specifies how the content, structure, and appearance of Web documents can be updated programmatically with scripts or other programs. The proposed object model for XML matches the Document Object Model for HTML so that script writers can easily learn XML programming. The XML DOM will provide a simple means of reading and writing data to and from an XML tree structure. document root The top-level node of an XML document; its descendants branch out from it to form the XML tree for that document. The document root contains the document element and can also contain a set of processing instructions and comments. document type declaration XML structural construct. A production within an XML document that contains or points to markup declarations that provide a grammar for a class of documents. This grammar is known as a Document Type Definition. The document type declaration can point to an external subset (a special kind of external entity) containing markup declarations, or can contain the markup declarations directly in an internal subset, or both. The DTD for a document consists of both subsets taken together. The syntax of the document type declaration is <!DOCTYPE content >. Document Type Definition (DTD) The markup declarations that describe a grammar for a class of documents. The DTD is declared within the document type declaration production of the XML file. The markup declarations can be in an external subset (a special kind of external entity), in an internal subset directly within the XML file, or both. The DTD for a document consists of both subsets taken together. Electronic Data Interchange (EDI) An existing format used to exchange data and support transactions. EDI transactions can be conducted only between sites that have been specifically set up with compatible systems. element XML structural construct. An XML element consists of a start tag, and end tag, and the information between the tags, which is often referred to as the contents. Elements used in an XML file are described by a DTD or schema, either of which can provide a description of the structure of the data. entity XML structural construct. A character sequence or well-formed XML hierarchy associated with a name. The entity can be referred to by an entity reference to insert the entity's contents into the tree at that point. The function of an XML entity is similar to that of a macro definition. Entity declarations occur in the DTD. entity reference XML structural construct. Refers to the content of a named entity. The name is delimited by the ampersand and semicolon characters; for example, &bookname; and <. It is used in much the same way as a macro. Extensible Linking Language (XLL) An XML vocabulary that provides links in XML similar to those in HTML but with more functionality. Linking could be multidirectional, and links could exist at the object level rather than just at a page level. Extensible Markup Language (XML) A subset of SGML that provides a uniform method for describing and exchanging structured data in an open, text-based format, and delivers this data by use of the standard HTTP protocol. At the time of this writing, XML 1.0 is a World Wide Web Consortium Recommendation, which means that it is in the final stage of the approval process. Extensible Stylesheet Language (XSL) A language used to transform XML-based data into HTML or other presentation formats, for display in a Web browser. Differs from cascading style sheets in that it can present information in an order different from that in which it was received. XSL will also be able to generate CSS along with HTML. XSL consists of two parts, a vocabulary for transformation and the XSL Formatting Objects. ID A special attribute type within the XML language. The ID attribute on the XML element provides a unique name, enabling links to that element using the IDREF attribute type. The value associated with the ID attribute must be unique within that XML document. IDs are currently declared with a DTD or schema. markup XML structural construct. Text in an XML document that does not represent character data: start tags, end tags, empty-element tags, entity references, character references, comments, CDATA section delimiters, DTDs, and processing instructions. mixed content XML structural construct. An element type has mixed content when elements of that type can contain character data, optionally interspersed with child elements. In this case, the types of the child elements can be constrained, but not their order or their number of occurrences. namespace A mechanism to resolve naming conflicts between elements in an XML document when each comes from a different vocabulary; it allows the commingling of like tag names from different namespaces. A namespace identifies an XML vocabulary defined within a URN. An attribute on an element, attribute, or entity reference associates a short name with the URN that defines the namespace; that short name is then used as a prefix to the element, attribute, or entity reference name to uniquely identify the namespace. Namespace references have scope. All child nodes beneath the node that specifies the namespace inherit that namespace. This allows nonqualified names to use the default namespace. NDATA The literal string "NDATA" is used as part of a notation declaration. See also notation. notation Usually refers to a data format, such as BMP. A notation identifies by name the format of unparsed entities, the format of elements that bear a notation attribute, or the application to which a processing instruction is addressed. notation declaration A notation declaration provides a name and an external identifier for a notation. The name is used in entity and attribute-list declarations and in attribute specifications. The external identifier is used for the notation, which can allow an XML processor or its client application to locate a helper application capable of processing data in the given notation. processing instruction (PI) XML structural construct. Instructions that are passed through to the application. The target is specified as part of the PI. The syntax for a PI is <?pi-name content?>. Resource Definition Framework (RDF) An object model similar in function to an application programming interface (API), RDF can be used by developers to access the logical meaning of designated content in XML documents. root element Sometimes this term is used to refer to the document element but this is misleading, since the top-level element and the document root are not the same. Because of this ambiguity, use of the term "root element" is discouraged. schema A formal specification of element names that indicates which elements are allowed in an XML document, and in which combinations. A schema is functionally equivalent to a DTD, but is written in XML; a schema also provides for extended functionality such as data typing, inheritance, and presentation rules. Standard Generalized Markup Language (SGML) The international standard for defining descriptions of structure and content of electronic documents. XML is a subset of SGML designed to deliver SGML-type information over the Web. target The application to which a processing instruction is directed. The target names beginning with "XML" and "xml" are reserved. The target appears as the first token in the PI. For example, in the XML declaration <?xml version="1.0"?>, the target is "xml". text markup Inserting tags into the middle of an element's text flow, to mark certain parts of the element with additional meta-information. tokenized attribute type Each attribute has an attribute type. Seven attribute types are characterized as tokenized: ID, IDREF, IDREFS, ENTITY, ENTITIES, NMTOKEN, and NMTOKENS. Uniform Resource Identifier (URI) The generic set of all names and addresses that refer to resources, including URLs and URNs. Defined in Berners-Lee, T., R. Fielding, and L. Masinter, Uniform Resource Identifiers (URI): Generic Syntax and Semantics. 1997. See updates to the W3C document RFC1738. The Layman-Bray proposal for namespaces makes every element name subordinate to a URI, which would ensure that element names are always unambiguous. Uniform Resource Locator (URL) The set of URI schemes that have explicit instructions on how to access the resource on the Internet. Uniform Resource Name (URN) A Uniform Resource Name identifies a persistent Internet resource. valid XML XML that conforms to the vocabulary specified in a DTD or schema. W3C World Wide Web Consortium well-formed XML XML that meets the requirements listed in the W3C Recommendation for XML 1.0: It contains one or more elements; it has a single document element, with any other elements properly nested under it; each of the parsed entities referenced directly or indirectly within the document is well-formed. A well-formed XML document does not necessarily include a DTD. World Wide Web Consortium (W3C) The international consortium founded in 1994 to develop standards for the Web. See XLL Extensible Linking Language XML Extensible Markup Language XML declaration The first line of an XML file can optionally contain the "xml" processing instruction, which is known as the XML declaration. The XML declaration can contain pseudoattributes to indicate the XML language version, the character set, and whether the document can be used as a standalone entity. XML document A data object that is well-formed, according to the XML recommendation, and that might (or might not) be valid. The XML document has a logical structure (composed of declarations, elements, comments, character references, and processing instructions) and a physical structure (composed of entities, starting with the root, or document entity). XML parser A generalized XML parser reads XML files and generates a hierarchically structured tree, then hands off data to viewers and other applications for processing. A validating XML parser also checks the XML syntax and reports errors. Next Section: Multimedia Technical Issues/Multimedia Searching through works with multimedia content requires extra support beyond what is usually provided (e.g., searching through metadata or fulltext). Content based image retrieval (CBIR), searching in files of spoken language, and searching in video collections are all supported with special methods and systems. While most ETD collections today allow search for multimedia content based on descriptions of such content (i.e., metadata), in the future, as collections grow and have richer assortments of multimedia elements, it is likely that multimedia database and multimedia content search software will be more widely deployed. Indeed, Virginia Tech carried out some preliminary work in this regard with its ETD collection, using tools provided by IBM (e.g., QBIC, VideoCharger). Other vendors and systems exist, and can be used for searching local, regional, national, and international collections of ETDs. Next Section: Interfaces Technical Issues/Interfaces Information retrieval (IR) systems have been improving since the 1950s. One of the most important areas of advance in the 1990s and in the 21st century benefits from the rapid enhancement of humancomputer interaction (HCI) that results from new types of interfaces, new methods of interaction, and new integration of those activities with IR methods. Information visualization, for example, allows faster analysis of the contents of a collection of ETDs, or of the result set from a search. Next Section: Training the Trainers Training the Trainers We believe that all universities should have ETD programs. Clearly that will be the case, when one considers the situation in the long term (e.g., 10 years). Why not join now, so that students over the next decade can benefit? Why not participate, so that the research of universities is more widely shared? Why not develop local infrastructure, so that students are properly prepared for the Information Age, save money, and save money for the university? Training efforts should aim to make clear the benefits of working with ETDs. They should assuage concerns, and make obvious how changes can be made in a smooth transition from current practice. Next Section: Initiatives to support ETD projects in Latin America Training the Trainers/Initiatives to support ETD projects in Latin America Abstract: This section focuses on the outreach work of the Ibero-American Science & Technology Education Consortium (ISTEC) and selected other organizations in developing EDT projects in Latin America. Training for librarians and for EDT trainers are described. Many Latin American universities have digital library projects, some of which include electronic theses. However, standards and consistency may be lacking in local EDT initiatives. The Ibero-American Science & Technology Education Consortium (ISTEC) and its partners have been creating learning opportunities and instigating local projects in digital libraries and EDT's. This section describes ISTEC's outreach process and progress in regards to EDT projects for the science and technology libraries that are members of the organization. Overview of ISTEC ISTEC is a non-profit organization comprised of educational, research, and industrial institutions throughout the Americas and the Iberian Peninsula. The Consortium was established in September 1990 to foster scientific, engineering, and technology education, joint international research and development efforts among its members and to provide a cost-effective vehicle for the application of technology. With start-up funding from the State of New Mexico and selected IT companies, the ISTEC board created four initiatives to address obstacles to IT developments and to encourage IT manpower development. These are: The ACE Initiative champions continuing engineering and computer sciences education projects. The most important goals are to upgrade human resources and curriculum development through training and non-traditional exchange programs. The methodology involves on-site training, web-based education, video courses, satellite delivery, and "sandwich" graduate programs. The latter brings graduate students from Ibero-America together with experts from ISTEC member organizations to ensure excellence. The Research and Development (R&D) Initiative focuses on the development and enhancement of laboratory infrastructure at member organizations. The major goal is the design and installation of modular, flexible, and expandable laboratory facilities for education, training, and R&D with links to the private sector. The Los Libertadores Initiative champions networks of excellence in the region.The main goal is to network Centers of Excellence equipped with the latest telecommunications and computer technology to provide real-time access to a world-wide system of expertise and knowledge. This requires partnerships among industries and governments to create an Ibero-American academic and R&D Internet backbone. The Library Linkages Initiative (LibLINK) is ISTEC's information creation, management and sharing project. Below is a description of LibLINK efforts in developing digital library projects in Latin America, especially in the area of EDT's. Overview of the Library Linkages (LibLINK) project of ISTEC The major goal of LibLINK is to design and implement innovative, international Science and Technology (S&T) information-sharing services. The annual compound growth rate of the Rapid Document Delivery (RDD) project has been hovering around 200% since 1995. Over 27 libraries in 19 countries are connected in real-time and documents are provided using the Ariel® software. The RDD project, although the most popular service, is a foundation for the more important digital library initiatives which were started in 1998. The projects within LibLINK can be categorized as follows: Connecting libraries for information transfer. This is accomplished through opening S&T library collections - especially Latin American collections - for scholars through regional networks created to compliment the LiBLINK document delivery services. Currently these include LigDoc in Brazil, PrEBi in Argentina, REBIDIMEX in Mexico, and most recently, a cooperative group of libraries in Colombia. Training librarians and researchers in digital library concepts. Working with the Networked Dissertation/Thesis Library (NDTL) initiative at Virginia Tech to expand the concept in Latin America. The LibLink initiative seeks to promote easy access to scientific information in the region, especially to thesis and dissertations of master's and doctoral candidates as this is our member organizations' most important intellectual property. Advancing and piloting new types of scholarly communication by actively supporting new publishing efforts such as the NDLTD and the Open Archives initiatives. LibLINK volunteers plan and carry out workshops and mini-conferences to facilitate the above. Funding generally come from grants provided by organizations such as the US National Science Foundation (NSF) and other national science councils such as CONACyT in Mexico, and regional organizations such as the OAS and UNESCO. LibLINK and 'EDT's in Latin America' We have refined a process for involving librarians and computer scientists in digital library projects that has proven successful. The principles on which ISTEC and LibLINK base their outreach efforts are: to establish the capacity of libraries and library staff for participating in digital projects. Site visits Participation in regional or local IT and computer science workshops to identify computer scientists working on digital library projects or components thereof. We are especially interested in initiatives created in isolation from each other and from their local libraries. In this way digital library initiatives and researchers are identified and a DL group can be established from the above findings that consist of librarians and computer scientists/engineers. Outcome: Critical mass of computer scientists and librarians linked to each other and to ISTEC. With this groundwork done, we plan and find funding sources for a Digital Library Workshop that generally have two major aims: To share information about current DL initiatives in that specific country or region To provide training in EDT's as the preferred first DL project In some cases, this is also an opportunity to create strategic plans for coordinated national projects This process has resulted in the following EDT and DL workshops: A NSF/CONACyT digital library workshop that included NDLTD training by Ed Fox, in Albuquerque, NM. Funding was obtained from various sources, mainly the National Science Foundation, CONACyT (the Mexican Science Council) and the Organization of American States. July 7–9, 1999 A successful DL conference in Costa Rica for Central American countries (Seminario / Taller Subregional sobre Bibliotecas Digitales). A full day EDT workshop was delivered by Ed Fox, followed by a day of digital leadership training and planning for local EDT projects. Funding was provided by the Organization of American States(OAS), the US Ambassador to Costa Rica and ISTEC. San Jose, Costa Rica, November 1999 (more detail below). 1st Course for the Training for Project Directors for Electronic Thesis and Dissertation Projects. The course was organized by UNESCO, the VII CYTED, Universidad de los Andes, and ISTEC's Library Linkages Initiative and held in conjunction with the VII Jornadas Iberoamericanas de Informatica, Cartagena de Indias, Colombia, from August 30 to September 1, 2000. This is an example of how EDT training can be piggy-backed on a significant regional event that is synergistic and that provide more value for the money spent to attend. A 2nd Training Course for Directors of EDT Projects was funded by UNESCO (Montevideo), the Asociación de Universidades Grupo Montevideo (AUGM), and ISTEC. This "Train the Trainer" course was held in Montevideo, December 7–9, 2000. The main goals for this series of courses are to create a group of specialists responsible for the dissemination and management of electronic dissertations and theses. The trainer for both these sessions was Ana Pavani from the Pontificia Universidade Catolica do Rio de Janeiro. REBIDIMEX is the Mexican operations for ISTEC Library Linkages. This group has had a number of meetings to develop coordinated digital library and EDT projects in Mexico. The digital theses project at the library of the Universidad de las Américas- Puebla is an excellent example, http://biblio.udlap.mx/tesis/, and forms the basis for a national Mexican EDT project. Case Study: The Primer Seminario-Taller Subregional sobre Bibliotecas Digitales, sponsored by the OAS and ISTEC at the Universidad de Costa Rica, San Jose, Costa Rica, mentioned above, provides a good case study of EDT outreach events. Each participating Central American country was asked to identify universities with sufficient technological infrastructure to support a digital library/ EDT project. Then each organization was funded to send representatives from each of their computer systems and library groups. The agenda focussed on providing one whole day of basic training by Ed Fox (a co-author of this Guide) in digital library and EDT concepts, followed by another day of leadership training for digital environments and a planning session. During this portion the groups identified a project that all could participate in. They chose the digitization of their organization's theses and dissertations and making it available through the Open Archives system using the processes developed by Virginia Tech and others described in other sections of the Guide. Regional working groups were assigned. The most important outcomes, however, were the creation of a network of librarians and computer scientists that understand EDT technological, operational and political issues and that now have contacts for joint projects in the region. The model of creating synergism and connections between librarians and computer scientists and focusing their energies on basic digital library/EDT projects will continue to be replicated in other parts of Latin America. Ana Pavani (a section author of the Guide) continues to deliver EDT courses with the help of ISTEC, such as one in Pernambucu, Brasil, in the spring of 2001. Members from ISTEC's regional and executive offices regularly speak at conferences regarding EDT's and are available to help arrange EDT events. Organizing training events and developing joint funding arrangements takes a lot of time, expertise, local contacts, and effort, but is critical for creating opportunities in under-served countries. ISTEC and its partner organizations, the OAS, International Development Bank, UNESCO, etc., continue to work together to offer regional digital library workshops in Latin America. UNESCO is formulating an international strategy for creating and disseminating electronic theses and dissertations that will support Latin American outreach efforts (UNESCO, 1999). As well, we are assisting governments to draft suitable policies to improve access to information, especially in making their universities' intellectual property (theses and dissertations) widely available to publicize the universities' research strengths. ISTEC is also sponsoring the Spanish translation of the Guide and will disseminate it through the ISTEC Science & Technology Portal. Other Latin American Projects The Latin American Network Information Center (LANIC) at the University of Texas at Austin is the most comprehensive resource for academic and economic information on Latin America (http://lanic.utexas.edu/), but not specifically for EDT projects. Some beginning and mature EDT projects can be found at: The Library System of the Universidad Católica de Valparaíso (Chile). http://biblioteca.ucv.cl/tesis_digitales/ The Digital Technology Research and Development Center (CITEDI) of the Instituto Politecnico Nacional in Tijuana, México. http://www.citedi.net/docs/tesis.htm The UNESP site at the Universidade Estadual Paulista in Brazil. http://www.cgb.unesp.br/etheses/ The digital theses project at the library of the Universidad de las Américas-Puebla, Mexico. http://biblio.udlap.mx/tesis/ University of Antioquia (Medellin, Colombia) University of São Paulo (Brazil). http://www.usp.teses.br/ In Chile the Universidad de Chile has been developing Cyberthesis, since November 1999. This is an electronic theses production project with support from Unesco and the cooperation of the Universit de Lyon and the Universit de Montreal. The Information Service and Library System, SISIB, is coordinating the Electronic Theses and Dissertations Project (Cybertesis), applying the production process developed by the Universite de Montreal, based on the conversion of texts to SGML/XML. In 2002, the Universidad de Chile will organize training workshops in the production of ETD (structured documents) for other Chilean universities. The Transborder Library Forum/FORO Transfronterizo de Bibliotecas share many of the aims of ISTEC's Library Linkages initiative. Their meetings have been held annually since 1991 to work on ways to improve communications relating to border issues and to foster professional networking among librarians from Mexico and the United States. Recently, Canadians and representatives from Latin American libraries also interested in NAFTA and border issues began to have a presence. At the 10th Transborder Library FORO held in Albuquerque, New Mexico in March 23–25, 2000, attendance of several representatives from Latin American libraries were sponsored and ISTEC's LibLink project provided a training session and talks about the REBIDIMEX initiative in Mexico. The Association of Latin American and Caribbean Academic Libraries ( Bibliotecas Universitarias da America Latina e do Caribe)) sponsored a LibLINK workshop that included EDT discussions and talks at the annual meeting in Florianopolis, Brazil, in April 2000. ISTEC continues to have a presence at their conferences. And are involved with the Brazilian EDT initiatives. The impact of Bandwidth and Infrastructure issues on EDT outreach in Latin America Bandwidth and IT infrastructure are important factors for digital EDT project developers in Latin America. IT policy development is another. More aggressive action is needed in both the governmental and industrial sectors. ISTEC emphasize these issues at the "IT Challenge" conferences by bringing industrial and academic members together with regional decision makers, such as ministers of education and technology and representatives from national science councils. The first high-performance Internet link between North and South America for research and education was inaugurated in Santiago, Chile on September 12, 2000. Chili and the USA connected their respective high performance networks, REUNA* and Internet2, enabling collaboration among researchers and educators at universities in the two countries. Such high-performance network links are critical to ensure the bandwidth required for future format-enriched EDT projects. ISTEC and its partners are strongly committed to advance this cause. * Reuna is a collaboration between the National Universities of Chili that introduced the Internet in Chile in 1992. Reuna´s high-speed network, REUNA2, is an ATM network of 155 Mbps. across the country. The National University Network is a non-profit consortium of 19 leading Chilean universities plus the National Commission for Science and Technology. Its mission is the creation and development of networks and services in IT aimed at supporting participation in the Information Society. The Library Linkage initiative of ISTEC has found a methodology for encouraging and supporting EDT developments in Latin America that has proven successful. The most important step is to identify local players in digital library initiatives in both libraries and computer science and computer engineering departments. The next step then brings these players together at events that provide opportunities for training, information sharing and national/regional EDT project planning. ISTEC monitors subsequent developments and provide support to keep projects going as appropriate. The most important strategic outcome we are aiming for is to create an open archive structure that will provide access to all science and technology theses and dissertations of member organizations through the ISTEC Portal. We believe that this will be a rich source of innovation, manpower identification and development, and an opportunity to highlight the intellectual property of our member universities. UNESCO (1999). Workshop on an international project of electronic dissemination of thesis and dissertations, UNESCO, Paris, September 27–28, 1999 http://www.unesco.org/webworld/etd/ Ibero-American Science & Technology Education Consortium. For more information see www.istec.org REUNA2. For more information see: http://www.reuna.cl Next Section: Tool kits for trainers Training the Trainers/Tool kits for trainers This section of the Guide outlines the approaches to student training used by certain universities. For the most part, links are made to sites where the tools themselves can be consulted and eventually used or adapted for local use. Next Section: Identifying what is available Training the Trainers/Identifying what is available The training offered to doctoral students registered at the Université de Montréal, entitled "Helpful tools for writing a thesis" ("Outils d'aide à la rédaction d'une thèse"). is part of that institution's programme for electronically distributing theses. The training seeks to respond to two sorts of needs: 1- the needs related to the processing of theses (greatly aided by the adequate use of a normalized document template); 2- the needs of the doctoral students (who want to increase their capabilities in using writing tools, and, in the process, their level of productivity and the quality of their production). The implementation of a programme of electronic thesis distribution affects the entire institution. At the Université de Montréal, three units are actively involved: the Faculty of Graduate Studies, the Library , and the Information technology services (Direction générale des technologies de l'information et de la communication—DGTIC, which holds the mandate to process and distribute theses in electronic formats). The training of doctoral students is planned and provided by individuals drawn from these three units. In terms of the sessions held in September 2001, four people acted as trainers (one from Graduate Studies, one from the Libraries and two from the DGTIC). The content of the training is as follows: Welcome and general presentation of the Programme for the electronic publication and distribution of theses (DGTIC) Presentation of the Université de Montréal's Style Guide for masters and doctoral theses (Graduate Studies). On-campus services offered to thesis-writing students: equipment rentals, self-service digitizer, digital camera, etc. (DGTIC). Presentation of the capabilities of the EndNote software for managing bibliographic referencing (Library) Practical exercises with the Word document template to be used by the students. Points 1 to 4 in this general plan involved presentations given by trainers, while point five directly involved participants in concrete exercises, accompanied by several demonstrations. For these exercises, the students used a working text (a Word document containing the principal editorial elements of a thesis, but without a proper layout), a Word document template developed for theses, and a list of instructions.. The exercises essentially consisted of applying the template's styles to the working text, of producing and inserting an image captured on the screen into the text, and of automatically producing a table of contents and an index of tables. As well, several documents were provided to participants at the sessions: the instructions, the document template (Word style sheets) as well as training guides and informative documents of interest to the students. All these documents are available on-line at www.theses.umontreal.ca. The workshops are offered to all doctoral students, whether they are at the beginning of their research or near the end of their writing. To reach the largest possible number of students, we used a multidimensional communications plans: posters, advertising, messages to student associations, and letters from the Dean of Graduate studies to the heads of departments and research centres. The number of seats in the training laboratory is limited to 15 to 20 students per session, depending on the laboratory used. Students are invited to register in advance, either by telephoning one of the DGTIC's administrative assistants, or, in an autonomous manner, by using an interactive Web form managed using a CGI scripts. This latter form tells students how many participants have already registered for each session and the maximum number of seats available. When the maximum number of registrations is reached, the script no longer allows registrations. Nevertheless, an invitation to register on a "waiting list" allows students to signal their interest and to be rapidly informed when new workshops are held. This also allowed us to note the pertinence of the training: the available spaces were quickly filled and the waiting list allowed us to reach interested students for the next sessions. Next Section: Demonstrations, explanations Training the Trainers/Demonstrations, explanations Many technological choices such as encoding formats, computing tools, dissemination tools, and copyrights, have an important impact on the ETD project's success. It is necessary an overview of the different aspects of producing, disseminating, archiving theses in electronic formats, implementation and management of ETD projects. At the present time several universities have initiated individual or cooperative projects for publishing and distributing theses, with different strategies approach : NDLTD (Networked Digital Library of Theses and Dissertations) with 103 member universities, Cyberthèses (Université de Montrèal, Université de Lyon, Universidad de Chile, Université de Genève with about 10 member universities) , Digital Australian Theses Program (8 Australian universities) and Dissertationen Online (5 German universities). Next Section: Initiatives and Projects Training the Trainers/Initiatives and Projects Collectives: Networked Digital Library of Theses and Dissertations (http://www.ndltd.org/) Cyberthèses Université de Montréal (http://www.cybertheses.org/) Université de Lyon (http://theses.univ-lyon2.fr/index.html) Universidad de Chile (http://www.cybertesis.cl/) Université de Genève (http://www.unige.ch/cyberdocuments/) Australian Digital Theses Program (http://www.library.unsw.edu.au/thesis/adt-ADT/info/info.html) Dissertationen Online (http://www.educat.hu-berlin.de/diss_online/) Humboldt University of Berlin (http://dochost.rz.hu-berlin.de/epdiss/)) Duisburg University (http://www.ub.uni-duisburg.de/dissonline/eindex.html) Oldenburg University (http://elfikom.physik.uni-oldenburg.de/dissonline/olengl.html) Erlangen University The Joint Electronic Theses & Dissertations Project (http://www.fis.utoronto.ca/etd/) University of Toronto (http://www.cybertheses.org/) York University (http://www.univ-lyon2.fr/sentiers/edition/) Individual university projects: Humboldt-Universität zu Berlin [Germany] Projekt Digitale Dissertationen der Humboldt-Universität zu Berlin (http://dochost.rz.hu-berlin.de/epdiss/) Virginia Polytechnic Institute and State University (Virginia Tech) [United States] Electronic Theses and Dissertations Initiative (http://etd.vt.edu/) North Carolina State University [United States] ETD Electronic Theses & Dissertations (http://www2.acs.ncsu.edu/grad/ETD/) West Virginia University [United States] Electronic Theses and Dissertations (http://www.wvu.edu/~thesis/) Massachusetts Institute of Technology [United States] MIT Theses Online (http://theses.mit.edu/) University of South Florida [United States] Electronic Theses and Dissertations (http://www.lib.usf.edu/virtual/etd/) Université de Montréal [Canada] Cyberthèses (http://www.cybertheses.org/) Université de Lyon [France] Cyberthèses (http://www.univ-lyon2.fr/sentiers/edition/) Universidad de Chile [Chile] Cyberthèses (http://www.cybertesis.cl/) Université de Genève [Switzerland] Cyberdocuments (http://www.unige.ch/cyberdocuments/) Add for Collectives: (http://elfikom.physik.uni-oldenburg.de/dissonline/PhysDis/dis_europe.html) PhysDis - a large collection of Physics Theses of Universities across Europe (http://www.iwi-iuk.org/dienste/TheO/) TheO - a collection of theses of different fields of 43 Universities in Germany, in as much as the Theses do contain Metadata. (http://MathNet.preprints.org/) MPRESS - a large collection of European Mathematical Theses. Which also contains the next entry as a subset. (http://mathdoc.ujf-grenoble.fr/Harvest/brokers/prepub/query.html#math-prepub) Index nationaux prépublications, thèses et habilitations - a collection of theses in France in Mathematics Next Section: Guidelines and Tutorials for ETDs Training the Trainers/Guidelines and Tutorials for ETDs The inclusion of courses and tutorials with online demonstrations can facilitate a better understanding of all the production and submission process. With a comprehensive guide for ETD creation, the students can learn how to create ETD in word processing software, how to convert to PDF format and the submission procedure. Requirements and Guidelines for the Preparation of Master's and Doctoral Theses - Penn State University (http://www.gradsch.psu.edu/thesis/thesis.guide.html) Thesis and Dissertation Guide - North Carolina State University (http://www.fis.ncsu.edu/grad_publicns/thesdis.htm) Informationen für Autoren - Humboldt-Universität zu Berlin (http://dochost.rz.hu-berlin.de/epdiss/) Resources Cyberthèses [Université de Montréal] [Université de Lyon] (http://mirror-fr.cybertheses.org/) ETD Workshop Outline - North Carolina State University (http://www2.acs.ncsu.edu/grad/ETD/etdworkshop.htm) Next Section: Specific Guidelines Training the Trainers/Specific Guidelines This section includes a series of tools intended to support the work of production and submission of ETD. Writing in word processing systems: ETD Formatting Humboldt University of Berlin Word-style (for German Word Versions only) http://dochost.rz.hu-berlin.de/epdiss/vorlage.html http://www.univ-lyon2.fr/sentiers/edition/theses/ressources.html http://www.univ-lyon2.fr/sentiers/edition/theses/cours/ModeEmploi.pdf http://mirror-fr.cybertheses.org/ressources.html http://www.cybertheses.org/cybertheses/ressources.html#style http://www.lib.usf.edu/virtual/etd/format.html http://etd.vt.edu/guidelines/index.html ETD Submission Virginia Tech: http://scholar.lib.vt.edu/ETD-db/help/ ADT: http://www.library.unsw.edu.au/thesis/adt-ADT/info/example. University of South Florida: http://www.lib.usf.edu/virtual/etd/submit. Humboldt-University Berlin: Automated Upload-Tool: http://dochost.rz.hu-berlin.de/cgi/dokupload/dokupload.cgi Explanation at http://dochost.rz.hu-berlin.de/epdiss/abgabe.html Humboldt University of Berlin: http://dochost.rz.hu-berlin.de/epdiss/latex/latex.html Virginia Tech: http://etd.vt.edu/howto/slides/latex/latex.html Prepare a PDF document Humboldt-University Berlin (http://dochost.rz.hu-berlin.de/epdiss/latex/latex.html) Writing SGML/XML Sites with SGML/XML-Theses and Dissertations HelsinkiUniversity of Technology (http://www.hut.fi/Yksikot/Kirjasto/HUTpubl/) Humboldt-University Berlin (http://dochost.rz.hu-berlin.de/epdiss/) Université de Montreal (http://www.cybertheses.org/) Université de Lyon 2 http://www.univ-lyon2.fr/sentiers/edition/ (http://www.uiowa.edu/~gradcoll/etd.html) University of Michigan at Ann Arbor ( http://www.umdl.umich.edu/um_diss_study.html) (http://www.digbib.uio.no/) Swedish University of Agricultural Sciences Uppsala (http://www.bib.slu.se/stp/pub2000/) (http://www.cybertesis.cl/) NDLTD: http://www.ndltd.org/standards/metadata/current.html Université de Montréal: http://www.pum.umontreal.ca/theses/metadata/1.0/ Dissertationen Online : http://www.ub.uniduisburg.de/dissonline/eindex.html ADT: http://www.library.unsw.edu.au/thesis/adt-ADT/info/metadata.html NDLTD : http://www.ndltd.org/cpright/index.htm Virginia Tech: http://etd.vt.edu/howto/copyright.html In Germany, theses are to obey the German Urheberrecht, which is different from the copyright in anglosaxon countries. See http://elfikom.physik.unioldenburg.de/dissonline/urheber.html Next Section: Creating an online database of problem solving solutions Training the Trainers/Creating an online database of problem solving solutions A quite important document must be at the users' disposal together with the various tools implemented for the electronic edition of scientific documents: User's guide for the preparation of the documents User's guide for the conversion itself Pedagogical toolkit for the training of authors... This first edition, as complete as it may be, is nevertheless insufficient. It must be completed by the online diffusion of a database listing all the problems usually encountered and their solutions (FAQ) eventually completed with links to specific documents with examples. Most of the time, such a database cannot be a priori completely defined and will have to be enriched as the number of users grows and their mastery of the various tools increases. The collection of the questions and problems are done very efficiently thanks to the creation of discussion lists or forums opened to the users. It is recommendable to group the frequent questions or problems by subject, to offer an easier and friendly consultation. Then, two methods considered for the construction of the database of solutions: Synthesis, based on the discussion list, prepared by some "experts" used to supply the database with questions and answers. The information available for the users is validated. The content of the forum indexed directly; the information is not validated, and it is thus the users themselves who will have to make a sort in the list of answers they get for their request. If this second method is lighter in terms of administration and management, its efficiency depends on the quality of the posted messages (questions and answers): precise and clear subject, good presentation of the problem, argumentation of the answer, etc. Whichever the adopted solution may be, its implementation may rest on usual free software classical list and database servers. The online diffusion of the so constructed FAQ must still have the name and e-mail of a member of the technical staff or a visible link to the discussion list in order to answer the questions not taken into account in the database. As an example, CyberThèses proposes a forum construct with a MySql database and interfaced with Php, and will develop a second database from validated information. Next Section: Help develop a broad local team Training the Trainers/Help develop a broad local team The core of the ADT Program was developed at The University of New South Wales Library (UNSW Library) as the lead institution. The team at UNSW Library included the overall coordinator & designer; technical manager & programmer; metadata consultant as well as the web coordinator & designer. This team developed the model from the conceptual to reality. The most important thing was not to lose focus, to keep the model as close to the original project description proposal as possible and to not overly complicate processes - in fact keep them as simple as possible. It was seen as critical to develop a workable model for the project partners to test and further refine. During the development and testing process, all 7 original partners were consulted and had input at all stages. Two workshops for all 7 members were held approximately 12 months apart. These were used to discuss all aspects of the ADT model as well as to agree on the standards and protocols used. The agreed standards are at the core of the distributed model of the ADT Program. Without the involvement of an effective team to lead the process, and the effective input of the broader team, arriving at the desired outcome would have been very difficult. The ADT Program is now effectively working across all 7 original member sites. Membership to the ADT has now been opened up to all Australian universities. It is anticipated that all Australian universities will become members and thus form a comprehensive national program. The benefits of a broad membership team are many: shared infrastructure, shared software development, shared metadata, shared documentation and training as well the shared satisfaction that comes with effective collaboration for the common good. The membership to the ADT may in time also include others in the region such as New Zealand and others. Next Section: Standards, cooperation, and collaboration Training the Trainers/Standards, cooperation, and collaboration While each institution will have differences as to the way in which its procedures are implemented and ETDs presented to satisfy local needs, for the potential of optimal global dissemination of ETDs to be achieved, adherence to basic standards is essential. These standards relate to document format and settings, filename protocols and metadata. Agreement on use of a small set of standard metadata elements can facilitate harvesting for creation of collaborative databases. While individual institutions can apply additional metadata including subject or format schemas as desired, if a minimum level of metadata is established, particularly if this metadata can be automatically generated, then any institution can access the document regardless of its resources and expertise. This provides an entry point for retrieval of an ETD from any institution without compromising the ability of other institutions to provide very rich metadata for their ETDs. Cooperation and collaboration between institutions in the creation and dissemination of ETDs has a number of benefits. From the point of view of creating ETDs, the benefits include the utilization of software and procedures developed and tested by others, sharing generic training tools, sharing of expertise in problem solving and developmental work and the provision of mutual support. In the dissemination of information about research findings contained in ETDs, collaborative approaches can be particularly effective in small or developing countries where the total volume of ETDs may not be large. In these situations a national or regional approach can provide increased visibility, economies of scale and sharing of resources required to mount and maintain ETDs. There are a number of models for cooperation and collaboration in creation, dissemination and preservation of ETDs. While the models can vary in detail, major elements are: Shared infrastructure In this model, a central agency provides the infrastructure for publication, dissemination, maintenance and preservation of ETDs. The central agency provides the server and the network access to a central repository of ETDs. Other institutions cooperate with the central agency by supplying copies of theses or dissertations. These copies could be supplied in digital form, conforming to the basic standards, or the central agency can be responsible for ensuring a suitable digital version is created and made available. The central agency has the responsibility for maintaining archival versions. This model could operate at a supranational, national or regional level, or could be used by a group of universities with similar interests. Shared software development Sharing of generic software, which is easily installed and maintained, can be a cost effective way of establishing an ETD program. This can be of particular benefit for institutions where staff with highly developed IT skills are in short supply. Collaborative development or modification of new versions of software can also be very cost-effective. Shared metadata In this model, institutions publish and maintain digital theses on their own institutional servers but the metadata is harvested to produce a central database of details of the theses from the collaborating institutions. The metadata is linked to the full text of the ETD wherever it resides. The home institution retains the responsibility for the preservation of the ETD. Shared documentation and training tools Sharing of detailed documentation on all aspects of operating an ETD program is a very cost effective method of collaboration. The development of generic procedures and training programs, which can be customized for local conditions, can facilitate the participation of institutions in an ETD program. Sharing of documentation is also likely to reinforce the use of standards, which will ensure the ETDs are readily discoverable Next Section: Outreach/helping others Training the Trainers/Outreach/helping others An effective mechanism for providing support is to identify centres of expertise which are prepared to assist others in setting up their ETD system. These centres of expertise can be regional, national or within an international geographic or linguistic region. The benefits flowing from these centres of expertise include assistance with designing processes, assistance with implementing software, especially if a common software product is being used, assistance in use of metadata and continuing support as systems evolve. Another very important function of a centre of expertise is provision of a focus around which a group can develop which can offer a self-support function and an environment taking into account local or regional conditions for continuing discussion and development of ETD programs as the technology continues to evolve. This latter is essential if ETD programs are to prosper. The centre of expertise concept also has the great advantage of reducing significantly the learning curve in starting up an ETD program and consequently the timeframe and cost of new programs. Next Section: Developing Centres of Expertise where appropriate and helpful Training the Trainers/Developing Centres of Expertise where appropriate and helpful An effective mechanism for providing support is to identify centers of expertise, which are prepared to assist others in setting up their ETD system. These centers of expertise can be regional, national or within an international geographic or linguistic region. The benefits flowing from these centers of expertise include assistance with designing processes, assistance with implementing software, especially if a common software product is used, assistance in use of metadata and continuing support as systems evolve. Another very important function of a center of expertise is provision of a focus around which a group can develop which can offer a self-support function and an environment taking into account local or regional conditions for continuing discussion and development of ETD programs as the technology continues to evolve. This latter is essential if ETD programs are to prosper. Next Section: Expanding ETD initiatives The Future/Expanding ETD initiatives ETD initiatives will spread in many ways. Word of mouth, from universities with successful programs, is one of the most effective means. Word of mouth, from graduates of universities with ETD programs, will spread the idea further. Further, as the worldwide collection of ETDs grows, more and more researchers, including graduate students, will make use of the valuable content. As use expands, others will be convinced to include their works in the collection, so it approaches full coverage. Next Section: Transforming Graduate Education The Future/Transforming Graduate Education Academics stand on a precipice separating our past, when genres of communication evolved slowly, and our future, when new genres emerge overnight. Our concepts of research, the authority of knowledge, and the shape of content are being radically challenged. We have difficulty imagining what dissertations or academic digital libraries will look like ten years from now. The shape of a dissertation is evolving from the first six-page, handwritten thesis at Yale University in 1860 into a form we cannot yet predict. Today's researchers and scholars are challenging the conventions of linear texts, one-inch margins, and texts written for extremely narrow audiences. They are integrating video, audio, animation, and graphics into their works. They are creating interactive elements, including real-time video, pivot tables, and online writing spaces. The power of ETDs is rooted in access. No longer are theses and dissertations just an academic hurdle, a last step in the arduous process of graduate education. Instead, ETDs are a meaningful connection with significant readers. Collaborative author tools enable faculty to serve on dissertation committees at universities distant from their home campuses and using tools such as NetMeeting to mentor students from a distance. Rather than accepting their research and scholarship will be read only by a select few (i.e., their committees), graduate students can now expect many readers. Predicting the future of academic scholarship is a little like predicting the stock market: both are volatile and unpredictable. Given this fact, however, it appears that there are a number of emerging trends that will affect our enterprise: Dissertations will matter more than they have in the past. Thanks to digital libraries, which increase access (http://scholar.lib.vt.edu/theses/data/somefacts.html) from one or two readers to sometimes more than 60,000, students and universities will pay greater attention to the quality of students' research and writing. Given this increased access, both students and universities may begin to pay greater attention to the quality of scholarly writing. Progressive universities will use their digital libraries of ETDs to market their programs, and universities will provide the resources students need to write multimedia research. Multimedia documents will transform author-reader relations. Authors will interact synchronously with readers, create different reading paths for different readers, and use visuals, animation, and pivot tables. Students will increasingly search the worldwide digital libraries of ETDs, resulting in research that is more collaborative and more current. Across disciplines, students will provide links that clarify the significance, methodology, and findings of their work to a broader range of readers, including lay audiences, thereby helping the general public better understand the value of academic scholarship. As an example, students in the social sciences can incorporate video of cultures and primary subjects; they can create polyvocal case studies and ethnographies - that is, studies with alternative interpretations. Faculty members will work more collaboratively with students, resulting in more complete bibliographies and saved time. Next Section: Managing technology changes The Future/Managing technology changes The Future/Interoperability At our various locations, document servers for electronic theses and dissertations have been set up independently from each other. The aim is to build a network of sites, which would allow for worldwide retrieval within a heterogeneous knowledge base, independently from the physical location of the data provided. The users should not have to navigate and search the various servers separately. They should be provided with one retrieval interface that can link to all the different nodes of the network of ETDs sites. This is the horizontal level on which retrieval could take place. Another level, which we call the vertical level of the information portal, would be the one, which configures the retrieval interface in a manner allowing the user to retrieve only the relevant and desired information rather than receiving all of the information that can be possibly provided. We wish to avoid the "Altavista effect" of information overload. The user should be able to search within specific subjects and for specific information structures. For example, they should be able to undertake searches just within the author field or the title field, for certain keywords or institutions or just within the abstracts field. A highly sophisticated retrieval facility would allow a worldwide search within certain internal document structures, such as the bibliography. For the scientific use of theses in the humanities and social sciences, as well as in the natural and technical sciences, it is necessary to offer not only bibliographical metadata and full text but also structural information for retrieval purposes, such as: the table of contents; captions of tables and graphs; special index terms such as name or person indexes or location indexes etc.); references (links) to external sources (printed resources as well as Web sources); the bibliography; references or footnotes within the work; definitions; mathematical / chemical formulas; theses / hypotheses These structural metadata are an integral part of the document and have to be defined by the author. At present, this predominantly takes place while formatting the text (e.g. headings, footnotes etc.). In order to also use these structural data for retrieval, they must be tagged as such by the author, by using either a structured language like LaTeX, or "style sheets" as with WinWord. What could be the lowest common denominator for interoperability? The first step within the above mentioned development is to reach agreement on a common metadata set for theses and dissertations and to formulate guidelines on how to use it for ETD projects. See http://www.ndltd.org/standards/metadata/ for the Dublin Core metadata set proposed by the NDLTD. Those guidelines could be supported by additional free software tools, which would allow library staff to create the necessary metadata set without needing a technical knowledge of the actual encoding in HTML or XML/RDF. Such a "metamaker" has been developed for the German ETD projects and could be translated into English, French, Spanish and Portuguese. MySQL or other free software can be used as the underlying database system. The Open Archives Specification: A chance for metadata interoperability During the last 2 years one initiative has effected the discussions about interoperability in digital libraries and the digital lbrary community enormously. The development of a protocol, that can easily be implemented at archive servers and a metadata set based upon the Dublin Core metadata set Allow archives, like ETD servers, preprint archives as well as museums and other institutions to provide their local catalogues to a worldwide community without having to implement specialised and complicated interfaces. So the Open Archives Framework (see http://www.openarchives.org) allow an interoperability f heterogeneous and distributed ETD archives and servers in a very low interoperability level. For the ETD initiatives and projects the OAI compliance has to be seen as chance to connect ETD servers worldwide. Next Section: A vision of the future The Future/A vision of the future Theses and dissertations represent a global source of information resulting from cutting edge research. While a proportion of this information is published in other forms much of the detail is not, and research emanating from lesser-known institutions, particularly in the developing countries, may be less likely to be published in mainstream journals. Creation of this information in electronic form making it readily accessible via the Web through standard, ubiquitous and free software programs provides the key to dissemination of this information independent of the source of the research. The ETD initiatives to date have proven that electronic theses and dissertations can be created using relatively low technology at a cost, which would be within the reach of most institutions. Portable packages have been developed which eliminate the majority of the developmental work required. The point has been reached where all research institutions can technically establish their own ETD program. Traditionally, theses and dissertations have been extremely underutilized sources of information due to their lack of physical availability. The development of ETDs provides the opportunity for theses and dissertations to be recognized as a basic channel for the dissemination of research findings and an essential resource in the discovery process. Therefore, the focus for the future needs to be to ensure optimal access to ETDs by information seekers. This in essence means ensuring that ETD metadata records are accessible through as many channels as possible and are retrieved as integral components of searches without the researcher necessarily specifying an ETD. Some ways of achieving this could be: Creation of a virtual union catalogue of ETD metadata through frequent regular harvesting of data from ETD sites which could be individual, regional or national Integration of ETD metadata into general metadata repositories for electronic scholarly information Ensuring search engines being established for other electronic scholarly information initiatives such as the Open Archives Initiative also search the ETD metadata repositories Inclusion of ETD metadata in local library catalogues Inclusion of ETD metadata in subject or form oriented databases The development of ETD programs worldwide and the implementation of access structures have the potential to significantly enhance the opportunity for all researchers, independent of geographic and economic constraints, to make their contribution to the global research effort. Work in all of these areas continues under development by NDLTD. Acting as agent for NDLTD, Virginia Tech is running a union catalog, drawing upon sites that support OAI. The result is accessible through Virginia Tech software as well as VTLS's Virtua software. With support from the USA's National Science Foundation, Virginia Tech also is engaged in a number of research activities related to ETDs. These include matching efforts funded by DFG in Germany (in collection with Oldenburg U.) and by CONACyT in Mexico (with Puebla and Monterrey). These aim to promote mirroring (of metadata as well as regular data), high performance access, effective searching and browsing, visualization of results and of sites, and other advanced schemes. It is hoped that all involved in the ETD efforts will assist, with a global perspective, so that all universities become involved, and ultimately all students submit an ETD, thus becoming better prepared to be a leader in the Information Age. Retrieved from "https://en.wikibooks.org/w/index.php?title=ETD_Guide/Print_version&oldid=3088167"	CommonCrawl
Information Geometry (Part 6) John Baez So far, my thread on information geometry hasn't said much about information. It's time to remedy that. I've been telling you about the Fisher information metric. In statistics this is nice a way to define a 'distance' between two probability distributions. But it also has a quantum version. So far I've showed you how to define the Fisher information metric in three equivalent ways. I also showed that in the quantum case, the Fisher information metric is the real part of a complex-valued thing. The imaginary part is related to the uncertainty principle. But there's yet another way to define the Fisher information metric, which really involves information. To explain this, I need to start with the idea of 'information gain', or 'relative entropy'. And it looks like I should do a whole post on this. Suppose that $\Omega$ is a measure space — that is, a space you can do integrals over. By a probability distribution on $\Omega$, I'll mean a nonnegative function $$p : \Omega \to \mathbb{R}$$ whose integral is 1. Here $d \omega$ is my name for the measure on $\Omega$. Physicists might call $\Omega$ the 'phase space' of some classical system, but probability theorists might call it a space of 'events'. Today I'll use the probability theorist's language. The idea here is that $$\int_A \; p(\omega) \; d \omega $$ gives the probability that when an event happens, it'll be one in the subset $A \subseteq \Omega$. That's why we want $$p \ge 0$$ Probabilities are supposed to be nonnegative. And that's also why we want $$\int_\Omega \; p(\omega) \; d \omega = 1 $$ This says that the probability of some event happening is 1. Now, suppose we have two probability distributions on $\Omega$, say $p$ and $q$. The information gain as we go from $q$ to $p$ is $$S(p,q) = \int_\Omega \; p(\omega) \log(\frac{p(\omega)}{q(\omega)}) \; d \omega $$ We also call this the entropy of $p$ relative to $q$. It says how much information you learn if you discover that the probability distribution of an event is $p$, if before you had thought it was $q$. I like relative entropy because it's related to the Bayesian interpretation of probability. The idea here is that you can't really 'observe' probabilities as frequencies of events, except in some unattainable limit where you repeat an experiment over and over infinitely many times. Instead, you start with some hypothesis about how likely things are: a probability distribution called the prior. Then you update this using Bayes' rule when you gain new information. The updated probability distribution — your new improved hypothesis — is called the posterior. And if you don't do the updating right, you need a swift kick in the posterior! So, we can think of $q$ as the prior probability distribution, and $p$ as the posterior. Then $S(p,q)$ measures the amount of information that caused you to change your views. For example, suppose you're flipping a coin, so your set of events is just $$\Omega = \{ \mathrm{heads}, \mathrm{tails} \}$$ In this case all the integrals are just sums with two terms. Suppose your prior assumption is that the coin is fair. Then $$q(\mathrm{heads}) = 1/2, \; q(\mathrm{tails}) = 1/2$$ But then suppose someone you trust comes up and says "Sorry, that's a trick coin: it always comes up heads!" So you update our probability distribution and get this posterior: $$p(\mathrm{heads}) = 1, \; p(\mathrm{tails}) = 0 $$ How much information have you gained? Or in other words, what's the relative entropy? It's this: $$S(p,q) = \int_\Omega \; p(\omega) \log(\frac{p(\omega)}{q(\omega)}) \; d \omega = 1 \cdot \log(\frac{1}{1/2}) + 0 \cdot \log(\frac{0}{1/2}) = 1 $$ Here I'm doing the logarithm in base 2, and you're supposed to know that in this game $0 \log 0 = 0$. So: you've learned one bit of information! That's supposed to make perfect sense. On the other hand, the reverse scenario takes a bit more thought. You start out feeling sure that the coin always lands heads up. Then someone you trust says "No, that's a perfectly fair coin." If you work out the amount of information you learned this time, you'll see it's infinite. The reason is that something that you thought was impossible — the coin landing tails up — turned out to be possible. In this game, it counts as infinitely shocking to learn something like that, so the information gain is infinite. If you hadn't been so darn sure of yourself — if you had just believed that the coin almost always landed heads up — your information gain would be large but finite. The Bayesian philosophy is built into the concept of information gain, because information gain depends on two things: the prior and the posterior. And that's just as it should be: you can only say how much you learned if you know what you believed beforehand! You might say that information gain depends on three things: $p$, $q$ and the measure $d \omega$. And you'd be right! Unfortunately, the notation $S(p,q)$ is a bit misleading. Information gain really does depend on just two things, but these things are not $p$ and $q$: they're $p(\omega) d\omega$ and $q(\omega) d\omega$. These are called probability measures, and they're ultimately more important than the probability distributions $p$ and $q$. To see this, take our information gain: $$\int_\Omega \; p(\omega) \log(\frac{p(\omega)}{q(\omega)}) \; d \omega $$ and juggle it ever so slightly to get this: $$\int_\Omega \; \log(\frac{p(\omega) d\omega}{q(\omega)d \omega}) \; p(\omega) d \omega $$ Clearly this depends only on $p(\omega) d\omega$ and $q(\omega) d\omega$. Indeed, it's good to work directly with these probability measures and give them short names, like $$d\mu = p(\omega) d \omega $$ $$d\nu = q(\omega) d \omega$$ Then the formula for information gain looks more slick: $$ \int_\Omega \; \log(\frac{d\mu}{d\nu}) \; d\mu $$ And by the way, in case you're wondering, the $d$ here doesn't actually mean much: we're just so brainwashed into wanting a $d x$ in our integrals that people often use $d \mu$ for a measure even though the simpler notation $\mu$ might be more logical. So, the function $$ \frac{d\mu}{d\nu} $$ is really just a ratio of probability measures, but people call it a Radon-Nikodym derivative, because it looks like a derivative (and in some important examples it actually is). So, if I were talking to myself, I could have shortened this blog entry immensely by working with directly probability measures, leaving out the $d$'s, and saying: Suppose $\mu$ and $\nu$ are probability measures; then the entropy of $\mu$ relative to $\nu$, or information gain, is $ S(\mu, \nu) = \int_\Omega \; \log(\frac{\mu}{\nu}) \; \mu. $ But I'm under the impression that people are actually reading this stuff, and that most of you are happier with functions than measures. So, I decided to start with and then gradually work my way up to the more sophisticated way to think about relative entropy! But having gotten that off my chest, now I'll revert to the original naive way. As a warmup for next time, let me pose a question. How much is this quantity like a distance between probability distributions? A distance function, or metric, is supposed to satisfy some axioms. Alas, relative entropy satisfies some of these, but not the most interesting one! • If you've got a metric, the distance between points should always be nonnegative. Indeed, this holds: $$S(p,q) \ge 0$$ So, we never learn a negative amount when we update our prior, at least according to this definition. It's a fun exercise to prove this inequality, at least if you know some tricks involving inequalities and convex functions — otherwise it might be hard. • If you've got a metric, the distance between two points should only be zero if they're really the same point. In fact, $$S(p,q) = 0$$ if and only if $$p d\omega = q d \omega $$ It's possible to have $p d\omega = q d \omega $ even if $p \ne q$, because $d \omega$ can be zero somewhere. But this is just more evidence that we should really be talking about the probability measure $p d \omega$ instead of the probability distribution $p$. If we do that, we're okay so far! • If you've got a metric, the distance from your first point to your second point is the same as the distance from the second to the first. Alas, $$S(p,q) \ne S(q,p)$$ in general. We already saw this in our example of the flipped coin. This is a slight bummer, but I could live with it, since Lawvere has already shown that it's wise to generalize the concept of metric by dropping this axiom. • If you've got a metric, it obeys the triangle inequality. This is the really interesting axiom, and alas, this too fails. Later we'll see why. So, relative entropy does a fairly miserable job of acting like a distance function. People call it a divergence. In fact, they often call it the Kullback-Leibler divergence. I don't like that, because 'the Kullback-Leibler divergence' doesn't really explain the idea: it sounds more like the title of a bad spy novel, sort of like The Eiger Sanction only worse. 'Relative entropy', on the other hand, makes a lot of sense if you understand entropy. And 'information gain' makes sense if you understand information. Anyway: how can we save this miserable attempt to get a distance function on the space of probability distributions? Simple: take its matrix of second derivatives and use that to define a Riemannian metric $g_{ij}$. This Riemannian metric in turn defines a metric of the more elementary sort we've been discussing today. And this Riemannian metric is the Fisher information metric I've been talking about all along! I'll give you the details next time. You can read a discussion of this article on Azimuth, and make your own comments or ask questions there! © 2011 John Baez [email protected]	CommonCrawl
\begin{document} \title{\bf A new result for boundedness of solutions to a quasilinear higher-dimensional chemotaxis--haptotaxis model with nonlinear diffusion} \author{ Jiashan Zheng \thanks{Corresponding author. E-mail address: [email protected] (J.Zheng)} \\ School of Mathematics and Statistics Science,\\ Ludong University, Yantai 264025, P.R.China \\ } \date{} \maketitle \noindent \begin{abstract} This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla w)+\mu u(1-u-w),\\ \displaystyle{v_t=\Delta v- v +u},\quad \\ \displaystyle{w_t=- vw},\quad\\ \end{array}\right. $$ in $N$-dimensional smoothly bounded domains, where the parameters $\xi ,\chi> 0$, $\mu> 0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u > 0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is proved that under the conditions $$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}+1) (N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N}}}}},$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded classical solution when $D(0) > 0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors. \end{abstract} \noindent {\bf\em Key words:}~Boundedness; Chemotaxis--haptotaxis; Global existence; Logistic source \noindent {\bf\em 2010 Mathematics Subject Classification}:~ 92C17, 35K55, 35K59, 35K20 \section{Introduction} Cancer invasion is a very complex process which involves various biological mechanisms (see \cite{Bellomo,Horstmann2710,Chaplain3,Chaplain1,Friedman,Hillen79}). Chemotaxis is the oriented movement of cells along concentration gradients of chemicals produced by the cells themselves or in their environment, and is a significant mechanism of directional migration of cells. A well-known chemotaxis model was proposed by Keller and Segel (\cite{Keller79,Keller791}) in the 1970s, which describes the aggregation processes of the cellular slime mold Dictyostelium discoideum. Since then, the following quasi-chemotaxis-only model \begin{equation} \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu u(1-u),\quad x\in \Omega, t>0,\\ \displaystyle{ v_t=\Delta v +u- v},\quad x\in \Omega, t>0\\ \end{array}\right.\label{7101fhhssdsddd.2x1} \end{equation} and its variations have been widely studied by many authors, where the main issue of the investigation was whether the solutions to the models are bounded or blow-up (see e.g., Herrero and Vel\'{a}zquez \cite{Herrero710}, Nagai et al. \cite{Nagaixcdf791}, Winkler et al. \cite{Winkler792,Winkler793}, the survey \cite{Bellomo1216}). For example, as we all known that all solutions of \dref{7101fhhssdsddd.2x1} are global in time and bounded when either $N\geq 3$ and $\mu> 0$ is sufficiently large (see \cite{Winkler37103} and also \cite{Zhengssdddssddddkkllssssssssdefr23}), or $N= 2$ and $\mu> 0$ is arbitrary (\cite{Osakix391}). Tello and Winkler (\cite{Tello710}) proved that the global boundedness for parabolic-elliptic chemotaxis-only system \dref{7101fhhssdsddd.2x1} (the second equation of \dref{7101fhhssdsddd.2x1} is replaced by $-\Delta v+v =u$) exists under the condition $\mu > \frac{(N-2)^+}{N}\chi$, moreover, they gave the weak solutions for arbitrary small $\mu > 0$. Some recent studies show that nonlinear chemotactic sensitivity functions (\cite{Calvez710,Horstmann791,Hillen5662710}), nonlinear diffusion (\cite{Kowalczyk7101,Cie72,Sugiyama710}), or also logistic dampening (\cite{Osakix391,Tello710,Winkler79,Winkler37103}) may prevent blow-up of solutions. One important extension of the classical Keller-Segel model to a more complex cell migration mechanism was proposed by Chaplain and Lolas (\cite{Chaplain1,Chaplain7}) in order to describe processes of cancer invasion. In fact, let $u = u(x,t)$ denote the density of the tumour cell population, $v = v(x,t)$ represent the concentration of a matrix-degrading enzyme (MDE) and $w = w(x,t)$ stand for the density of the surrounding tissue (extracellular matrix (ECM)). Then Chaplain and Lolas (\cite{Chaplain7}) introduced the following chemotaxis-haptotaxis system as a model describing the process of cancer invasion \begin{equation} \left\{\begin{array}{ll} u_t=D\Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot (u\nabla w)+\mu u(1-u-w),\quad x\in \Omega, t>0,\\ \displaystyle{\tau v_t=\Delta v +u- v},\quad x\in \Omega, t>0,\\ \displaystyle{w_t=- vw+\eta w(1-u-w) },\quad x\in \Omega, t>0,\\ \end{array}\right.\label{7101.2x19xjjssdkkkkk3189} \end{equation} where $\tau\in\{0,1\},$ $\Delta=\displaystyle{\sum_{i=1}^N\frac{\partial^2}{\partial x^2_i}}$, $\displaystyle\frac{\partial}{\partial\nu}$ denotes the outward normal derivative on $\partial\Omega$, $\chi>0$ and $\xi>0$ measure the chemotactic and haptotactic sensitivities, respectively. Here $D>0$ as well as $\mu>0$ and $\eta\geq0$ represent the random motility coefficient, the proliferation rate of the cells and the remodeling rate, respectively. Model \dref{7101.2x19xjjssdkkkkk3189} and its analogue have been extensively studied up to now (see \cite{Tao2,Taox26,Taossddssd793,Tao79477,Taox26216,Tao3,Tao72,LiLiLi791,Cao,Bellomo1216,Szymaska,Zhengssdddsssddfghhhddddkkllssssssssdefr23}. In fact, Global existence and asymptotic behavior of solutions to the haptotaxis-only system ($\chi = 0$ in the first equation of \dref{7101.2x19xjjssdkkkkk3189}) have been investigated in \cite{Stinnerddff12,Walker,Lianu1,Marciniak} and \cite{Taossddssd793} for the case $\eta= 0$ and $\eta\neq 0$, respectively. In realistic situations, the renewal of the ECM occurs at much smaller timescales than its degradation (see \cite{Tao2,Perthame317,Jger317,Lianu1,Marciniak,Tao79477,Walker}). Therefore, a choice of $\eta= 0$ on \dref{7101.2x19xjjssdkkkkk3189} seems justified (see \cite{Tao2,Perthame317,Jger317,Lianu1,Marciniak,Tao79477,Walker}). The models mentioned above described the random part of the motion of cancer cells by linear diffusion, however, from a physical point of view migration of the cancer cells through the ECM should rather be regarded like movement in a porous medium, and so we are led to considering the cell motility $D$ a nonlinear function of the cancer cell density. Inspired by the analysis, in this paper,we consider the following chemotaxis-haptotaxis system with nonlinear diffusion (see also \cite{Tao72,Bellomo1216,Wangscd331629}) \begin{equation} \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot (u\nabla w)+\mu u(1-u-w),\quad x\in \Omega, t>0,\\ \displaystyle{v_t=\Delta v +u- v},\quad x\in \Omega, t>0,\\ \displaystyle{w_t=- vw },\quad x\in \Omega, t>0,\\ \displaystyle{\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=\frac{\partial w}{\partial \nu}=0},\quad x\in \partial\Omega, t>0,\\ \displaystyle{u(x,0)=u_0(x)},v(x,0)=v_0(x),w(x,0)=w_0(x),\quad x\in \Omega\\ \end{array}\right.\label{7101.2x19x3sss189} \end{equation} in a bounded domain $\Omega\subset R^N (N\geq1)$ with smooth boundary $\partial\Omega$. The origin of the system was proposed by Chaplain and Lolas (\cite{Chaplain1,Chaplain7}) to describe cancer cell invasion into surrounding healthy tissue. Here we assume that $D( u )$ is a nonlinear function and satisfies \begin{equation}\label{9161} D\in C^{2}([0,\infty)) ~~\mbox{and}~~D(u)\geq C_{D}u^{m-1}~~ \mbox{for all}~ u>0 \end{equation} with some $C_{D}>0$ and $m > 0$. Moreover, if, $D ( u )$ fulfills \begin{equation}\label{91ssdd61} D ( u ) > 0 ~~ \mbox{for all}~ u\geq0, \end{equation} so the diffusion is nondegenerate and the solutions may be considered in the sense of classical. Throughout this paper, the initial data $(u_0,v_0,w_0)$ are assumed that for some $\vartheta\in(0,1)$ \begin{equation}\label{x1.731426677gg} \left\{ \begin{array}{ll} \displaystyle{u_0\in C(\bar{\Omega})~~\mbox{with}~~u_0\geq0~~\mbox{in}~~\Omega~~\mbox{and}~~u_0\not\equiv0},\\ \displaystyle{v_0\in W^{1,\infty}(\Omega)~~\mbox{with}~~v_0\geq0~~\mbox{in}~~\Omega},\\ \displaystyle{w_0\in C^{2+\vartheta}(\bar{\Omega})~~\mbox{with}~~w_0>0~~\mbox{in}~~\bar{\Omega}~~\mbox{and}~~\frac{\partial w_0}{\partial\nu}=0~~\mbox{on}~~\partial\Omega.} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ \end{array} \right. \end{equation} System \dref{7101.2x19x3sss189} has been widely studied by many authors, where the main issue of the investigation was whether the solutions to the models are bounded or blow-up (see Tao-Winkler \cite{Tao72}, ). For instance, when $D$ satisfies \dref{9161}--\dref{91ssdd61}, Tao and Winkler (\cite{Tao72}) showed that model \dref{7101.2x19x3sss189} has global solutions provided that $m>\max\{1,\bar{m}\}$, where \begin{equation}\label{dcfvgg7101.2x19x318jkl} \bar{m}:=\left\{\begin{array}{ll} \frac{2N^2+4N-4}{N(N+4)}~~\mbox{if}~~ N \leq 8,\\ \frac{2N^2+3N+2-\sqrt{8N(N+1)}}{N(N+20)}~~\mbox{if}~~ N \geq 9.\\ \end{array}\right. \end{equation} However, they leave a question here: ``whether the global solutions are bounded''. If $N\geq 2$, the global boundedness of solutions to \dref{7101.2x19x3sss189} has been constructed for $m > 2-\frac{2}{N}$ (see \cite{LiLiLi791,Wangscd331629}) with the help of the boundness of $\\|\nabla v\\|_{L^{l}(\Omega\times(0,T))}(1\leq l<\frac{N}{N-1})$). Recently, we (\cite{Zhddengssdeeezseeddd0}) extended these results to the cases $m >\frac{2N}{N+2}$ by using the boundness of $\\|\nabla v\\|_{L^{2}(\Omega\times(0,T))}$. More recently, if $\frac{\mu}{\chi}$ is large enough, Jin \cite{Jineeezseeddd0} (see also \cite{HuHuHueeezseeddd0}) proved that system \dref{7101.2x19x3sss189} admits a global bounded solution for any $m>0$. However, we should point that the cases $0<m\leq\frac{2N}{N+2}$ and small $\frac{\mu}{\chi}$ remain unknown even in the case for the chemotaxis-only system \dref{7101.2x19x3sss189}, that is, $w\equiv0$ in system \dref{7101.2x19x3sss189}. In this paper, we firstly use the boundedness of $\int_{(t-1)_+}^t \int_\Omega u^{{\gamma_0+1}}$ (see Lemma \ref{qqqqlemma45630}) for some $\gamma_0>1,$ which is a new result even for chemotaxis-only system \dref{7101.2x19x3sss189}. Then, applying the standard testing procedures, we can derive the uniform boundedness of $\nabla v$ in $L^{l_0} (\Omega)$ for some $l_0>2.$ We emphasize that the spontaneous boundedness information on $\nabla v$ in $L^{l_0} (\Omega)$ (see \dref{zjscz2.5297x9ssd630xxy}) plays a key role in this process. Using the $L^{l_0}$-boundedness of $\nabla v$ and $L^{1}$-boundedness of $u$, we can then acquire the uniform bounds of $u$ in arbitrary large $L^p(\Omega)$ provided that the further restriction on $m$ is satisfied (see the proof of Lemmas \ref{lsssemma456302ssd23116}-\ref{sddlemmddffa45630}). Finally, combining with Moser iteration method and $L^p$-$L^q$ estimates for Neumann heat semigroup, we finally established the $L^\infty$ bound of $u$ (see Lemmas \ref{lemdffmddffa45630}--\ref{lemdffssddmddffa45630}). Motivated by the above works, this paper will focus on studying the relationship between the exponent $m$ and the global existence of solutions to chemotaxis-haptotaxis model \dref{7101.2x19x3sss189} with nonlinear diffusion. In fact, the aim of the present paper is to study the quasilinear chemotaxis system \dref{7101.2x19x3sss189} under the conditions \dref{9161}--\dref{91ssdd61}. For non-degenerate and degenerate diffusion both, we will show the existence of global-in-time solutions to system \dref{7101.2x19x3sss189} that are uniformly bounded. The main results are as follows. \begin{theorem}\label{theorem3} Let $\Omega\subset R^N (N\geq1)$ be a bounded domain with smooth boundary and $\chi>0,\xi>0,\mu>0$. Assume that the nonnegative initial data $(u_0 ,v_0,w_0)$ fulfill \dref{x1.731426677gg}. Moreover, if $D$ satisfies \dref{9161}-\dref{91ssdd61} with \begin{equation}\label{7101.2ssddffx19x3sss189} m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1} \lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N}}}}}, \end{equation} then there exists a triple $(u,v,w)\in (C^0(\bar{\Omega}\times[0,\infty))\cap C^{2,1} (\bar{\Omega}\times(0,\infty)))^3$ which solves \dref{7101.2x19x3sss189} in the classical sense. Moreover, both $u$, $v$ and $w$ are bounded in $\Omega\times(0,\infty)$, that is, there exists a positive constant $C$ such that \begin{equation}\label{916sdfff1} \\|u(\cdot, t)\\|_{L^\infty(\Omega)}+\\|v(\cdot, t)\\|_{W^{1,\infty}(\Omega)}+\\|w(\cdot, t)\\|_{L^\infty(\Omega)}\leq C~~~\mbox{for all}~~t>0. \end{equation} \end{theorem} \begin{remark} (i) Obviously, $$\mu_=\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{[\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}} (\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu]_{+}}>1~~(\mbox{by using}~~~ \mu>0),$$ then $\gamma_{}={{{\frac{(\mu_{}+1)(N+\mu_{}-1)}{N}}}}=\mu_{}+1+\frac{\mu_{}^2-1}{N}>\mu_{}+1>2$, hence $$\frac{2N}{N+{{{\frac{(\frac{\chi\max\{1,\lambda_0\}}{(\chi\max\{1,\lambda_0\}-\mu)_{+}}+1)(N+\frac{\chi\max\{1,\lambda_0\}}{(\chi\max\{1,\lambda_0\}-\mu)_{+}}-1)}{N}}}}} <\frac{2N}{N+2}\leq2-\frac{2}{N},$$ therefore, Theorem \ref{theorem3} extends the results of Theorem 1.1 of Zheng (\cite{Zhddengssdeeezseeddd0}), the results of Theorem 1.1 of Wang (\cite{Wangscd331629}), the results of as well as of Li-Lankeit (\cite{LiLiLi791}) and partly extends the results of Theorem 1.1 of Liu et al (\cite{Liughjj791}). Here the assumption $m>\frac{2N}{N+2}$ (see \cite{Zhddengssdeeezseeddd0}) or $m>2-\frac{2}{N}$ (see \cite{LiLiLi791,Liughjj791,Wangscd331629}) are intrinsically required. (ii) Obviously, for any $N\geq1$, $$ \frac{2N}{N+{{{\frac{(\frac{\chi\max\{1,\lambda_0\}}{(\chi\max\{1,\lambda_0\}-\mu)_{+}}+1)(N+\frac{\chi\max\{1,\lambda_0\}} {(\chi\max\{1,\lambda_0\}-\mu)_{+}}-1)}{N}}}}} <\bar{m},$$ therefore, Theorem \ref{theorem3} extends the results of Corollary 1.2 of Tao and Winkler (\cite{Tao72}), who showed the global existence of solutions the cases $m>\bar{m},$ where $\bar{m}$ is given by \dref{dcfvgg7101.2x19x318jkl}. (iii) In the case $N= 2$, by using $\mu>0,$ then $\frac{8}{4+(\mu_+1)^2}< 1$, our result improves the result of \cite{Taox3201} and \cite{zhengjjkk}, in which the assumption $m=1$ or $m>1$ are intrinsically required. (iv) If $\mu>\max_{s\geq1} \lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})$, then by \dref{7101.2ssddffx19x3sss189}, we derive that for any $m>0,$ system \dref{7101.2x19x3sss189} has a classical and bounded solution, which improves the result of \cite{Jineeezseeddd0} as well as \cite{HuHuHueeezseeddd0} and \cite{Cao}. (v) The chemotaxis-haptotaxis system therefore has bounded solutions under the same condition on $m$ as the pure chemotaxis system with $w\equiv 0$ without logistic source (see \cite{Tao794}). For $\mu= 0$ this condition is essentially optimal (\cite{Winkler79}). \end{remark} In the case of possibly degenerate diffusion, system \dref{7101.2x19x3sss189} admits at least one global bounded weak solution: \begin{theorem}\label{theossdrem3} Let $\Omega\subset R^N (N\geq1)$ be a bounded domain with smooth boundary and $\chi>0,\xi>0,\mu>0$. Suppose that the initial data $(u_0 ,v_0,w_0)$ satisfy \dref{x1.731426677gg}. Moreover, if $D$ satisfies \dref{9161} with $$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}+1)(N+\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}-1)}{N}}}}},$$ then system \dref{7101.2x19x3sss189} admits at least one global weak solution $(u,v,w)$ in the sense of definition \ref{df1} below that exists globally in time and is bounded in the sense that \dref{916sdfff1} holds. \end{theorem} The rest of this paper is organized as follows. In the following section, we recall some preliminary results. Section 3 is devoted to a series of a priori estimates and then prove Theorem \ref{theorem3}. In Section 4, applying the existence of classical solutions in the non-degenerate case, we will then complete the proof of theorem \ref{theossdrem3} by an approximation procedure in Section 3. \section{Preliminaries and main results} Before proving our main results, we will give some preliminary lemmas, which play a crucial role in the following proofs. As for the proofs of these lemmas, here we will not repeat them again. \begin{lemma}(\cite{Hajaiej,Ishida})\label{lemma41ffgg} Let $s\geq1$ and $q\geq1$. Assume that $p >0$ and $a\in(0,1)$ satisfy $$\frac{1}{2}-\frac{p}{N}=(1-a)\frac{q}{s}+a(\frac{1}{2}-\frac{1}{N})~~\mbox{and}~~p\leq a.$$ Then there exist $c_0, c'_0 >0$ such that for all $u\in W^{1,2}(\Omega)\cap L^{\frac{s}{q}}(\Omega)$, $$\\|u\\|_{W^{p,2}(\Omega)} \leq c_{0}\\|\nabla u\\|_{L^{2}(\Omega)}^{a}\\|u\\|^{1-a}_{L^{\frac{s}{q}}(\Omega)}+c'_0\\|u\\|_{L^{\frac{s}{q}}(\Omega)}.$$ \end{lemma} \begin{lemma}(\cite{Zheng})\label{lemma41} Let $0<{\theta}\leq p\leq\frac{2N}{N-2}$. There exists a positive constant $C_{GN}$ such that for all $u \in W^{1,2}(\Omega)\cap L^{{\theta}}(\Omega)$, $$\\|u\\|_{L^p(\Omega)} \leq C_{GN}(\\|\nabla u\\|_{L^{2}(\Omega)}^{a}\\|u\\|^{1-a}_{L^{{\theta}}(\Omega)}+\\|u\\|_{L^{{\theta}}(\Omega)})$$ is valid with $a =\displaystyle{\frac{\frac{N}{{{\theta}}}-\frac{N}{p}}{1-\frac{N}{2}+\frac{N}{{\theta}}}}\in(0,1)$. \end{lemma} \begin{lemma}\label{lemma45xy1222232} (\cite{Hieber,Zhenddddgssddsddfff00}) Suppose that $\gamma\in (1,+\infty)$ and $g\in L^\gamma((0, T); L^\gamma( \Omega))$. Consider the following evolution equation $$ \left\{\begin{array}{ll} v_t -\Delta v+v=g,~~~(x, t)\in \Omega\times(0, T ),\\ \displaystyle\frac{\partial v}{\partial \nu}=0,~~~(x, t)\in \partial\Omega\times(0, T ),\\ v(x,0)=v_0(x),~~~(x, t)\in \Omega.\\ \end{array}\right. $$ For each $v_0\in W^{2,\gamma}(\Omega)$ such that $\displaystyle\frac{\partial v_0}{\partial \nu}=0$, there exists a unique solution $v\in W^{1,\gamma}((0,T);L^\gamma(\Omega))\cap L^{\gamma}((0,T);W^{2,\gamma}(\Omega)).$ In addition, if $s_0\in[0,T)$, $v(\cdot,s_0)\in W^{2,\gamma}(\Omega)(\gamma>N)$ with $\displaystyle\frac{\partial v(\cdot,s_0)}{\partial \nu}=0,$ then there exists a positive constant $\lambda_0:=\lambda_0(\Omega,\gamma,N)$ such that $$ \begin{array}{rl} &\displaystyle{\int_{s_0}^Te^{\gamma s}\\| v(\cdot,t)\\|^{\gamma}_{W^{2,\gamma}(\Omega)}ds\leq\lambda_0\left(\int_{s_0}^Te^{\gamma s} \\|g(\cdot,s)\\|^{\gamma}_{L^{\gamma}(\Omega)}ds+e^{\gamma s_0}(\\|v_0(\cdot,s_0)\\|^{\gamma}_{W^{2,\gamma}(\Omega)})\right).}\\ \end{array} $$ \end{lemma} The following local existence result is rather standard; since a similar reasoning in \cite{Tao72,Zheng}, see for example. Therefore, we only give the following lemma without proof. \begin{lemma}\label{lemma70} Assume that the nonnegative functions $u_0,v_0,$ and $w_0$ satisfies \dref{x1.731426677gg} for some $\vartheta\in(0,1),$ $D$ satisfies \dref{9161} and \dref{91ssdd61}. Then there exists a maximal existence time $T_{max}\in(0,\infty]$ and a triple of nonnegative functions $$ (u ,v ,w )\in C^0(\bar{\Omega}\times[0,T_{max}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max}))\times C^0((0,T_{max}); C^2(\bar{\Omega}))\times C^{2,1}(\bar{\Omega}\times[0,T_{max}))$$ which solves \dref{7101.2x19x3sss189} classically and satisfies $0\leq w \leq \\|w_0\\|_{L^\infty(\Omega)}$ in $\Omega\times(0,T_{max})$. Moreover, if $T_{max}<+\infty$, then \begin{equation} \left(\|u (\cdot, t)\\|_{L^\infty(\Omega)}+\\|v (\cdot, t)\\|_{W^{1,\infty}(\Omega)}+\\|w (\cdot, t)\\|_{W^{1,\infty}(\Omega)}\right)\rightarrow\infty~~ \mbox{as}~~ t\nearrow T_{max}. \label{1.163072x} \end{equation} \end{lemma} According to the above existence theory, for any $s\in(0, T_{max})$, $(u(\cdot, s), v(\cdot, s),w(\cdot, s))\in C^2(\bar{\Omega})$. Without loss of generality, we can assume that there exists a positive constant $K$ such that \begin{equation}\label{eqx45xx12112} \\|u_0\\|_{C^2(\bar{\Omega})}\leq K~\mbox{as well as }~~~\\|v_0\\|_{C^2(\bar{\Omega})}\leq K~~\mbox{and}~~\\|w_0\\|_{C^2(\bar{\Omega})}\leq K. \end{equation} \section{A priori estimates} The main task of this section is to establish for estimates for the solutions $(u ,v , w )$ of problem \dref{7101.2x19x3sss189}. To this end, in straightforward fashion one can check the following boundedness for $u$, which is common in chemotaxis (or chemotaxis--haptotaxis) with logistic source (see e.g. \cite{Zhddengssdeeezseeddd0,Winkler37103,Wangscd331629,LiLiLi791}). \begin{lemma}\label{wsdelemma45} There exists $C > 0$ such that the solution of \dref{7101.2x19x3sss189} satisfies \begin{equation} \int_{\Omega}{u }+\int_{\Omega}\|\nabla v \|^2+\int_{\Omega}\|\nabla v \|^l \leq C~~\mbox{for all}~~ t\in(0, T_{max}) \label{cz2.5ghju48cfg924ghyuji} \end{equation} with $l\in[1,\frac{N}{N-1}).$ Since, the third component of \dref{7101.2x19x3sss189} can be expressed explicitly in terms of $v $. This leads to the following a one-sided pointwise estimate for $-\Delta w $ (see e.g. \cite{Tao79477ddffvg,Taox3201,Tao72}): \begin{lemma}\label{lemm3a} Let $(u , v ,w )$ solve \dref{7101.2x19x3sss189} in $\Omega\times(0, T_{max})$. Then \begin{equation}\label{x1.731426677gghh} \begin{array}{rl} -\Delta w (x, t) \leq&\displaystyle{ \\|w_0\\|_{L^\infty(\Omega)}\cdot v (x,t)+\kappa~~~\mbox{for all}~~x\in\Omega~~\mbox{and}~~~t\in(0, T_{max}),}\\ \end{array} \end{equation} where \begin{equation}\label{x1.73ddff1426677gghh} \kappa:=\\|\Delta w_0\\|_{L^\infty(\Omega)}+4\\|\nabla\sqrt{w_0}\\|_{L^\infty(\Omega)}^2+\frac{\\|w_0\\|_{L^\infty(\Omega)}}{e}. \end{equation} \end{lemma} \end{lemma} Now we proceed to establish the main step towards our boundedness proof. To this end, let us collect some basic estimates for $u$ and $v$ in comparatively large function spaces. In fact, relying on a standard testing procedure, we derive the following Lemma: \begin{lemma}\label{zsxxcdvlemma45630} For any $k>1$, the solution $(u ,v ,w )$ of \dref{7101.2x19x3sss189} satisfies that \begin{equation} \begin{array}{rl} &\displaystyle{-\xi\int_\Omega u ^{k-1}\nabla\cdot(u \nabla w )\leq \frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u^k v dx+\kappa \frac{({k-1})}{k}\xi \int_\Omega u^k dx,}\\ \end{array} \label{vbgncz2.5xx1ffgghh512} \end{equation} where $\kappa$ is the same as \dref{x1.73ddff1426677gghh}. \end{lemma} \begin{proof} For any $k>1$, we integrate the left hand of \dref{vbgncz2.5xx1ffgghh512} and use Lemma \ref{lemm3a} then get \begin{equation} \begin{array}{rl} &\displaystyle{-\xi\int_\Omega u ^{k-1}\nabla\cdot(u \nabla w ) dx} \\ =&\displaystyle{({k-1} )\xi\int_\Omega u ^{k-1}\nabla u \cdot\nabla w dx} \\ =&\displaystyle{-\frac{({k-1})}{k}\xi\int_\Omega u^k \Delta w dx} \\ \leq&\displaystyle{\frac{({k-1})}{k}\xi\int_\Omega u^k (\\|w_0\\|_{L^\infty(\Omega)} v +\kappa) dx,} \\ \end{array} \label{1cz2.563019rrtttt12} \end{equation} where $\kappa$ is the same as \dref{x1.73ddff1426677gghh}. This directly entails \dref{vbgncz2.5xx1ffgghh512}. \end{proof} Due to the presence of logistic source, some useful estimates for $u $ can be derived. \begin{lemma}\label{ssdeedrfe116lemma70hhjj} (see \cite{Wangscd331629,LiLiLi791,Zhddengssdeeezseeddd0}) Assume that $\mu>0.$ There exists a positive constant $ K_0$ such that the solution $(u , v ,w )$ of \dref{7101.2x19x3sss189} satisfies \begin{equation} \begin{array}{rl} \displaystyle\int_{\Omega}u (x,t)dx\leq K_0~~~\mbox{for all}~~t\in (0, T_{max}) \end{array}\label{ssddaqwswddaassffssff3.ddfvbb10deerfgghhjuuloollgghhhyhh} \end{equation} and \begin{equation} \int_t^{t+\tau}\int_{\Omega}{u ^{2}}\leq K_0~~\mbox{for all}~~ t\in(0, T_{max}-\tau), \label{bnmbncz2.5ghhjuyuivvddfggghhbssdddeennihjj} \end{equation} where we have set \begin{equation} \tau:=\min\left\{1,\frac{1}{6}T_{max}\right\}. \label{cz2.5ghju48cfg924vbhu} \end{equation} \end{lemma} In order to establish some estimates for solution $(u ,v , w )$, we first recall the following lemma proved in \cite{Wangscd331629} (see also \cite{LiLiLi791,Zhddengssdeeezseeddd0}). \begin{lemma}\label{lemmssdda45630} Let $\Omega\subset \mathbb{R}^N(N\geq1)$ be a bounded domain with smooth boundary. Then for all $k>1,$ the solution $(u ,v ,w )$ of \dref{7101.2x19x3sss189} satisfies that \begin{equation}\label{cz2.5xx1jjjj} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\| u \\|^{k}_{L^k(\Omega)}+\frac{(k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{\frac{\chi^2(k-1)} {2 C_D }\int_{\Omega} u ^{k+1-m}\|\nabla v \|^2+ \frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u ^{k}v +(\mu+\kappa\xi)\int_{\Omega} u ^{k}}\\ \end{array} \end{equation} for all $t\in(0, T_{max})$. \end{lemma} \begin{proof} Multiplying $\dref{7101.2x19x3sss189}_1$ (the first equation of \dref{7101.2x19x3sss189}) by $ u ^{k-1}$ and integrating over $\Omega$, we get \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\| u \\|^{k}_{L^k(\Omega)}+ C_D(k-1)\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}} \\ \leq&\displaystyle{-\chi\int_\Omega \nabla\cdot( u \nabla v ) u ^{k-1}{}-\xi\int_\Omega\nabla\cdot ( u \nabla w ) u ^{k-1}{}}\\ &+\displaystyle{\mu\int_\Omega u ^{k}(1-u -w ) {}}\\ \leq&\displaystyle{-\chi\int_\Omega \nabla\cdot( u \nabla v ) u ^{k-1}{}-\xi\int_\Omega\nabla\cdot ( u \nabla w ) u ^{k-1}{}}\\ &+\displaystyle{\mu\int_\Omega u ^{k}(1-u ) {}~~\mbox{for all}~~ t\in(0,T_{max})}\\ \end{array} \label{cfvgvvcz2.5xx1jjjj} \end{equation} according to the nonnegativity of $w$. Integrating by parts to the first term on the right hand side of \dref{cfvgvvcz2.5xx1jjjj} and using the Young inequality, we obtain \begin{equation} \begin{array}{rl} &\displaystyle{-\chi\int_\Omega \nabla\cdot( u \nabla v ) u ^{k-1}{}} \\ =&\displaystyle{(k-1 )\chi\int_\Omega u ^{k-1}\nabla u \cdot\nabla v {}} \\ \leq&\displaystyle{\frac{(k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{} +\frac{\chi^2(k-1)} {2 C_D }\int_{\Omega} u ^{k+1-m}\|\nabla v \|^2.}\\ \end{array} \label{111cz2.5630111} \end{equation} On the other hand, due to Lemma \ref{zsxxcdvlemma45630}, we have \begin{equation} \begin{array}{rl} &\displaystyle{-\xi\int_\Omega \nabla\cdot( u \nabla w ) u ^{k-1} {}} \\ \leq&\displaystyle{\frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u^k v dx+\kappa \frac{({k-1})}{k}\xi \int_\Omega u^k dx{}}\\ \leq&\displaystyle{\frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u^k v dx+\kappa\xi \int_\Omega u^k dx{}.}\\ \end{array} \label{cz2.563019rrtttt12} \end{equation} Furthermore, inserting \dref{111cz2.5630111}--\dref{cz2.563019rrtttt12} into \dref{cfvgvvcz2.5xx1jjjj}, we conclude that for all $t\in(0, T_{max})$, \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\| u \\|^{k}_{L^k(\Omega)}+\frac{(k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{\frac{\chi^2(k-1)} {2 C_D }\int_{\Omega} u ^{k+1-m}\|\nabla v \|^2+ \frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u ^{k}v +(\mu+\kappa\xi)\int_{\Omega} u ^{k}.}\\ \end{array} \label{vgbhnsxcdvfcz2.5xx1jjjj} \end{equation} \end{proof} We proceed to estimate both integrals on the right of \dref{cz2.5xx1jjjj} in a straightforward manner. \begin{lemma}\label{qqqqlemma45630} Let $(u ,v ,w )$ be a solution to \dref{7101.2x19x3sss189} on $(0,T_{max})$ and \begin{equation}\mu_{}=\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{\left[\max_{s\geq1} \lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu\right]_{+}}. \label{zjscz2.ddffrr5297x96ssddffdffggbh302222114} \end{equation} If $\mu>0,$ then for all $1<\gamma_0<\mu_{}$, there exists a positive constant $C$ which depends on $\gamma_0$ such that \begin{equation} \int_{\Omega}u ^{\gamma_0}(x,t) \leq C ~~~\mbox{for all}~~ t\in(0,T_{max}) \label{zjscz2.ddffrr5297x96302222114} \end{equation} and \begin{equation} \int_{0}^{t}\int_{\Omega}u ^{\gamma_0+1}(x,t) \leq C ~~~\mbox{for all}~~ t\in(0,T_{max}). \label{4455zjscz2.ddffrr5297x96302222114} \end{equation} \end{lemma} \begin{proof} Multiplying $\dref{7101.2x19x3sss189}_1$ by $u ^{k-1}$, integrating over $\Omega$ and using $w\geq0$, we get \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+ C_D(k-1)\int_{\Omega}u ^{m+k-3} \|\nabla u \|^2dx} \\ \leq&\displaystyle{-\chi\int_\Omega \nabla\cdot(u \nabla v )u ^{k-1}dx-\xi\int_\Omega\nabla\cdot (u \nabla w )u ^{k-1}+\mu \int_\Omega u ^{k} (1- u -w ) }\\ \leq&\displaystyle{-\chi\int_\Omega \nabla\cdot(u \nabla v )u ^{k-1}dx-\xi\int_\Omega\nabla\cdot (u \nabla w )u ^{k-1}+\mu \int_\Omega u ^{k}(1- u ) .}\\ \end{array} \label{qqqqcz2.5xx1jjjj} \end{equation} We now estimate the right hand side of \dref{qqqqcz2.5xx1jjjj} terms by terms. To this end, integrating by parts to the first term on the right hand side of \dref{qqqqcz2.5xx1jjjj}, we obtain for any $\varepsilon_1>0,$ \begin{equation} \begin{array}{rl} &\displaystyle{-\chi\int_\Omega \nabla\cdot(u \nabla v )u ^{k-1}} \\ =&\displaystyle{-\frac{({k-1})\chi}{k}\int_\Omega u ^{k}\Delta v } \\ \leq&\displaystyle{\frac{(k-1)\chi}{k}\int_\Omega u ^{k}\|\Delta v \| } \\ \leq&\displaystyle{\varepsilon_1\int_\Omega u ^{k+1}+\gamma_1\varepsilon_1^{-k}\int_\Omega\|\Delta v \|^{k+1}, } \\ \end{array} \label{223444cz2.5630111} \end{equation} where \begin{equation}\gamma_1=\frac{1}{k+1}\left(\frac{k+1}{k}\right)^{-k}\left(\frac{(k-1)\chi}{k}\right)^{k+1}. \label{22ddf34ddff44cz2.5630111} \end{equation} Due to \dref{x1.731426677gghh} and \dref{x1.73ddff1426677gghh}, it follows that for any $\varepsilon_2>0$ \begin{equation} \begin{array}{rl} &\displaystyle{-\xi\int_\Omega \nabla\cdot( u \nabla w ) u ^{k-1} } \\ =&\displaystyle{-\frac{({k-1})\xi}{k}\int_\Omega u ^{k}\Delta w } \\ \leq&\displaystyle{\kappa\frac{({k-1})\xi}{k}\int_\Omega u ^{k}+\frac{({k-1})\xi\\|w_0\\|_{L^\infty(\Omega)}}{k}\int_\Omega u ^{k}v }\\ \leq&\displaystyle{\kappa\xi\int_\Omega u ^{k}+\frac{({k-1})\xi\\|w_0\\|_{L^\infty(\Omega)}}{k}\int_\Omega u ^{k}v }\\ \leq&\displaystyle{\kappa\xi\int_\Omega u ^{k}+\varepsilon_2\int_\Omega u ^{k+1}+\gamma_2\varepsilon_2^{-k}\int_{\Omega}v ^{k+1},}\\ \end{array} \label{qqqqcz2.563019rrtttt12} \end{equation} where \begin{equation}\gamma_2:=\frac{1}{k+1}\left(\frac{k+1}{k}\right)^{-k} \left(\frac{({k-1})\xi\\|w_0\\|_{L^\infty(\Omega)}}{k}\right)^{k+1} \label{qqqssffffqcz2.563019rrtttt12} \end{equation} and $\kappa$ is give by \dref{x1.73ddff1426677gghh}. On the other hand, in view of $k>1$, we also derive that \begin{equation} \begin{array}{rl} \mu \displaystyle\int_\Omega u ^{k}(1- u ) =&\displaystyle{-\mu \int_\Omega u ^{k+1}+(\mu+\frac{k+1}{k})\int_\Omega u ^{k}-\frac{k+1}{k}\int_\Omega u ^{k}}\\ \leq&\displaystyle{-\mu \int_\Omega u ^{k+1}+(\mu+2)\int_\Omega u ^{k}-\frac{k+1}{k}\int_\Omega u ^{k}.}\\ \end{array} \label{sddddcz2.56301hh} \end{equation} Therefore, combined with \dref{223444cz2.5630111}, \dref{qqqqcz2.563019rrtttt12}, \dref{qqqqcz2.5xx1jjjj} as well as \dref{sddddcz2.56301hh} and \dref{91ssdd61}, we have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+ C_D(k-1)\int_{\Omega}u ^{m+k-3}\|\nabla u \|^2+\frac{k+1}{k}\int_\Omega u ^{k}} \\ \leq&\displaystyle{(-\mu +\varepsilon_1+\varepsilon_2 )\int_\Omega u ^{k+1}+\gamma_1\varepsilon_1^{-k}\int_\Omega\|\Delta v \|^{k+1}+\gamma_2\varepsilon_2^{-k}\int_\Omega v ^{k+1}+C_1\int_\Omega u ^{k}}\\ \end{array} \label{cz2.5xx1} \end{equation} with $C_1=\kappa\xi+\mu+2 .$ For any $t\in (0,T_{max})$, applying the Gronwall Lemma to the above inequality shows that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{{k}}\\|u (\cdot,t) \\|^{{{k}}}_{L^{{k}}(\Omega)}+ C_D(k-1)\int_{0}^t e^{-( { {k}+1})(t-s)}\int_{\Omega}u ^{m+k-3}\|\nabla u \|^2} \\ \leq&\displaystyle{\frac{1}{{k}}e^{-( { {k}+1})t}\\|u_0 \\|^{{{k}}}_{L^{{k}}(\Omega)}+(\varepsilon_1+\varepsilon_2- \mu)\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}+1}} dxds}\\ &+\displaystyle{\gamma_1\varepsilon_1^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega \|\Delta v \|^{ {k}+1} dxds+ C_1\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}}} dxds}\\ &\displaystyle{+\gamma_2\varepsilon_2^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega v ^{{{k}+1}} dxds}\\ \leq&\displaystyle{(\varepsilon_1+\varepsilon_2- \mu)\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}+1}} dxds+\gamma_1\varepsilon_1^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega \|\Delta v \|^{ {k}+1} dxds}\\ &+\displaystyle{\gamma_2\varepsilon_2^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega v ^{{{k}+1}} dxds+ C_1\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}}} dxds+C_2,}\\ \end{array} \label{cz2111ddffdfghhhg11.5kk1214114114rrgg} \end{equation} where $$C_2:=C_2({k})=\frac{1}{{k}}\\|u_0 \\|^{{{k}}}_{L^{{k}}(\Omega)}.$$ Next, a use of Lemma \ref{lemma45xy1222232} and \dref{eqx45xx12112} leads to \begin{equation}\label{cz2.5kke3456778999ddff9001214114114rrggjjkk} \begin{array}{rl} &\displaystyle{\gamma_1\varepsilon_1^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega \|\Delta v \|^{ {k}+1} dxds} \\ =&\displaystyle{\gamma_1\varepsilon_1^{-k}e^{-( { {k}+1})t}\int_{0}^t e^{( { {k}+1})s}\int_\Omega \|\Delta v \|^{ {k}+1} dxds}\\ \leq&\displaystyle{\gamma_1\varepsilon_1^{-k}e^{-( { {k}+1})t}\lambda_0(\int_{0}^t \int_\Omega e^{( { {k}+1})s}u ^{ {k}+1} dxds+\\|v_0\\|^{ {k}+1}_{W^{2, { {k}+1}}(\Omega)})}\\ \end{array} \end{equation} and \begin{equation}\label{cz2.5kk12141141dfggghhh14rrggjjkk} \begin{array}{rl} &\displaystyle{\gamma_2\varepsilon_2^{-k}\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega v ^{ {k}+1} dxds} \\ =&\displaystyle{\gamma_2\varepsilon_2^{-k}e^{-( { {k}+1})t}\int_{0}^t e^{( { {k}+1})s}\int_\Omega v ^{ {k}+1} dxds}\\ \leq&\displaystyle{\gamma_2\varepsilon_2^{-k}e^{-( { {k}+1})t}\lambda_0(\int_{0}^t \int_\Omega e^{( { {k}+1})s}u ^{ {k}+1} dxds+\\|v_0\\|^{ {k}+1}_{W^{2, { {k}+1}}(\Omega)})}\\ \end{array} \end{equation} for all $t\in(0, T_{max})$, where $\lambda_0$ is the same as Lemma \ref{lemma45xy1222232}. On the other hand, choosing $\varepsilon_1=\frac{(k-1)\chi}{k+1}\lambda_0^{\frac{1}{k+1}}$ and $\varepsilon_2=\frac{(k-1)\xi\\|w_0\\|_{L^\infty(\Omega)}}{k+1}\lambda_0^{\frac{1}{k+1}}$, with the help of \dref{22ddf34ddff44cz2.5630111} and \dref{qqqssffffqcz2.563019rrtttt12}, a simple calculation shows that $$\varepsilon_1+\gamma_1\lambda_0\varepsilon_1^{-k}=\frac{({k}-1)}{{k}}\lambda_0^{\frac{1}{{k}+1}}\chi$$ and $$\varepsilon_2+\gamma_2\lambda_0\varepsilon_2^{-k}=\frac{({k}-1)}{{k}}\lambda_0^{\frac{1}{{k}+1}}\xi\\|w_0\\|_{L^\infty(\Omega)},$$ so that, substituting \dref{cz2.5kke3456778999ddff9001214114114rrggjjkk}--\dref{cz2.5kk12141141dfggghhh14rrggjjkk} into \dref{cz2111ddffdfghhhg11.5kk1214114114rrgg} implies that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{{k}}\\|u(\cdot,t) \\|^{{{k}}}_{L^{{k}}(\Omega)}+ C_D(k-1)\int_{0}^t e^{-( { {k}+1})(t-s)}\int_{\Omega}u ^{m+k-3}\|\nabla u \|^2} \\ \leq&\displaystyle{(\varepsilon_1+\gamma_1\lambda_0\varepsilon_1^{-k}+\varepsilon_2+\gamma_2\lambda_0\varepsilon_2^{-k}- \mu)\int_{0}^t e^{-( {k}+1)(t-s)}\int_\Omega u ^{{{k}+1}} dxds}\\ &+\displaystyle{(\gamma_1\varepsilon_1^{-k}+\gamma_2\varepsilon_2^{-k})e^{-( {k}+1)(t-s_0)}\lambda_0\\|v_0\\|^{ {k}+1}_{W^{2, { {k}+1}}(\Omega)}+ C_1\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}}} dxds+C_2}\\ =&\displaystyle{(\frac{({k}-1)}{{k}}\lambda_0^{\frac{1}{{k}+1}}\chi+\frac{({k}-1)}{{k}}\lambda_0^{\frac{1}{{k}+1}}\xi\\|w_0\\|_{L^\infty(\Omega)}- \mu)\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}+1}} dxds}\\ &+\displaystyle{(\gamma_1\varepsilon_1^{-k}+\gamma_2\varepsilon_2^{-k})e^{-( {k}+1)(t-s_0)}\lambda_0\\|v_0\\|^{ {k}+1}_{W^{2, { {k}+1}}(\Omega)}+ C_1\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}}} dxds+C_2}\\ \leq&\displaystyle{[\frac{({k}-1)}{{k}}\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})- \mu]\int_{0}^t e^{-( {k}+1)(t-s)}\int_\Omega u ^{{{k}+1}} }\\ &+\displaystyle{ C_1\int_{0}^t e^{-( { {k}+1})(t-s)}\int_\Omega u ^{{{k}}} dxds+C_3}\\ \end{array} \label{czssddssddffggf2.5kk1214114114rrggkkll} \end{equation} with $$C_3=(\gamma_1\varepsilon_1^{-k}+\gamma_2\varepsilon_2^{-k})e^{-( {k}+1)(t-s_0)}\lambda_0\\|v_0\\|^{ {k}+1}_{W^{2, { {k}+1}}}+C_2.$$ For any $\varepsilon>0,$ we choose $k=\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}-\varepsilon.$ Then $$\frac{({k}-1)}{{k}}\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})<\mu.$$ Thus, by using the Young inequality, we derive that there exists a positive constant $C_4$ such that \begin{equation} \begin{array}{rl} &\displaystyle{\int_{\Omega}u^{{k}}(x,t) dx\leq C_4~~\mbox{for all}~~t\in (0, T_{max})}\\ \end{array} \label{cz2.5kk1214114114rrggkklljjuu} \end{equation} and \begin{equation} \int_{0}^{t}\int_{\Omega}u ^{k+1}(x,t) \leq C_4 ~~~\mbox{for all}~~ t\in(0,T_{max}). \label{4455zjscz2.ddffrrssss5297x96302222114} \end{equation} Thereupon, combining with the arbitrariness of $\varepsilon$ and the H\"{o}lder inequality, \dref{zjscz2.ddffrr5297x96302222114} and \dref{4455zjscz2.ddffrr5297x96302222114} holds. The proof of Lemma \ref{qqqqlemma45630} is completed. \end{proof} When $$\mu\geq\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)}),$$ by making use of above lemma, we can derive the following results on the bound $u $ for in an $L^k$ space for any $k> 1$. \begin{corollary}\label{lemma456ssdddddfgg30} Let $(u ,v ,w )$ be a solution to \dref{7101.2x19x3sss189} on $(0,T_{max})$. If $$\mu\geq\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)}),$$ then for all $k>1$, there exists a positive constant $C$ which depends on $k$ such that \begin{equation} \int_{\Omega}u ^{k}(x,t) \leq C ~~~\mbox{for all}~~ t\in(0,T_{max}). \label{zjscz2.ddffrr5297x9xxcc6302222114} \end{equation} \end{corollary} \begin{proof} This directly results from Lemma \ref{qqqqlemma45630} and the fact that $$\frac{\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^ \infty(\Omega)})}{(\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})-\mu)_{+}}=+\infty$$ by using $\mu\geq\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)}).$ \end{proof} In the following, we always assume that $$\mu<\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)}),$$ since, case $\mu\geq\max_{s\geq1}\lambda_0^{\frac{1}{{{s}}+1}}(\chi+\xi\\|w_0\\|_{L^\infty(\Omega)})$ has been proved by {Corollary} \ref{lemma456ssdddddfgg30}. \begin{lemma}\label{lemma45630} Let $(u ,v ,w )$ be a solution to \dref{7101.2x19x3sss189} on $(0,T_{max})$ and $\Omega\subset \mathbb{R}^N(N\geq1)$ be a bounded domain with smooth boundary. Then for all $\beta>1,$ there exists $\kappa_0>0$ such that \begin{equation}\label{hjui909klopji115} \begin{array}{rl} &\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt}\\|\nabla v \\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}+\frac{(\beta-1)}{{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla v \|^{\beta}\right\|^2}\\ &+\displaystyle{\frac{1}{2}\displaystyle\int_\Omega \|\nabla v \|^{2\beta-2}\|D^2v \|^2+\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta}{}}\\ \leq&\displaystyle{\kappa_0\int_\Omega u ^2 \|\nabla v \|^{2\beta-2}+\kappa_0~~\mbox{for all}~~ t\in(0,T_{max}).}\\ \end{array} \end{equation} \end{lemma} \begin{proof} Using that $\nabla v \cdot\nabla\Delta {v} = \frac{1}{2}\Delta \|\nabla v \|^2-\|D^2v \|^2$, by a straightforward computation using the second equation in \dref{7101.2x19x3sss189} and several integrations by parts, we find that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt} \\|\nabla v \\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}}\\ = &\displaystyle{\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-2}\nabla v \cdot\nabla(\Delta v - v +{u} )} \\ =&\displaystyle{\frac{1}{{2}}\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-2}\Delta \|\nabla v \|^2-\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-2}\|D^2 v \|^2}\\ &-\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta}-\displaystyle{\displaystyle\int_\Omega u \nabla\cdot( \|\nabla v \|^{2\beta-2}\nabla v )}\\ =&\displaystyle{-\frac{\beta-1}{{2}}\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-4}\left\|\nabla \|\nabla v \|^{2}\right\|^2+\frac{1}{{2}}\displaystyle\int_{\partial\Omega} \|\nabla v \|^{2\beta-2}\frac{\partial \|\nabla v \|^{2}}{\partial\nu}-\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta}}\\ &-\displaystyle{\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-2}\|D^2 v \|^2-\displaystyle\int_\Omega {u} \|\nabla v \|^{2\beta-2}\Delta v -\displaystyle\int_\Omega u \nabla v \cdot\nabla( \|\nabla v \|^{2\beta-2})}\\ =&\displaystyle{-\frac{2(\beta-1)}{{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla v \|^{\beta}\right\|^2+\frac{1}{{2}}\displaystyle\int_{\partial\Omega} \|\nabla v \|^{2\beta-2}\frac{\partial \|\nabla v \|^{2}}{\partial\nu}-\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-2}\|D^2 v \|^2}\\ &-\displaystyle{\displaystyle\int_\Omega {u} \|\nabla v \|^{2\beta-2}\Delta v -\displaystyle\int_\Omega u \nabla v \cdot\nabla( \|\nabla v \|^{2\beta-2})-\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta}}\\ \end{array} \label{cz2.5ghju48156} \end{equation} for all $t\in(0,T_{max})$. Here, since $\|\Delta v \| \leq\sqrt{N}\|D^2v \|$, by the Young inequality, we can estimate \begin{equation} \begin{array}{rl} \displaystyle\int_\Omega {u} \|\nabla v \|^{2\beta-2}\Delta v \leq&\displaystyle{\sqrt{N}\displaystyle\int_\Omega {u} \|\nabla v \|^{2\beta-2}\|D^2v \|} \\ \leq&\displaystyle{\frac{1}{4}\displaystyle\int_\Omega \|\nabla v \|^{2\beta-2}\|D^2v \|^2+N\displaystyle\int_\Omega u ^2 \|\nabla v \|^{2\beta-2}}\\ \end{array} \label{cz2.5ghju48hjuikl1} \end{equation} for all $t\in(0,T_{max})$. As moreover by the Cauchy--Schwarz inequality, we have \begin{equation} \begin{array}{rl} -\displaystyle\int_\Omega u \nabla v \cdot\nabla( \|\nabla v \|^{2\beta-2})= &\displaystyle{-(\beta-1)\displaystyle\int_\Omega {u} \|\nabla v \|^{2(\beta-2)}\nabla v \cdot \nabla \|\nabla v \|^{2}}\\ \leq &\displaystyle{\frac{\beta-1}{8}\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta-4}\left\|\nabla \|\nabla v \|^{2}\right\|^2+2(\beta-1) \displaystyle\int_\Omega u ^2 \|\nabla v \|^{2\beta-2}}\\ \leq &\displaystyle{\frac{(\beta-1)}{2{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla v \|^{\beta}\right\|^2+2(\beta-1) \displaystyle\int_\Omega u ^2 \|\nabla v \|^{2\beta-2}.}\\ \end{array} \label{cz2.5ghju4ghjuk81} \end{equation} Next we deal with the integration on $\partial\Omega$. We see from Lemma \ref{lemma41ffgg} that \begin{equation} \begin{array}{rl} &\displaystyle{\displaystyle\int_{\partial\Omega}\frac{\partial \|\nabla v \|^2}{\partial\nu} \|\nabla v \|^{2\beta-2} }\\ \leq&\displaystyle{C_\Omega\displaystyle\int_{\partial\Omega} \|\nabla v \|^{2\beta} }\\ =&\displaystyle{C_\Omega\| \|\nabla v \|^{\beta}\|^2_{L^2(\partial\Omega)}.}\\ \end{array} \label{cz2.57151hhkkhhgg} \end{equation} Let us take $r\in(0,\frac{1}{2})$. By the embedding $W^{r+\frac{1}{2},2}(\Omega)\hookrightarrow L^2(\partial\Omega)$ is compact (see e.g. Haroske and Triebel \cite{Haroske}), we have \begin{equation} \begin{array}{rl} &\displaystyle{\\| \|\nabla v \|^{\beta}\\|^2_{L^2{(\partial\Omega})}\leq C_3\\| \|\nabla v \|^{\beta}\\|^2_{W^{r+\frac{1}{2},2}(\Omega)}.}\\ \end{array} \label{cz2.57151} \end{equation} In order to apply Lemma \ref{lemma41ffgg} to the right-hand side of \dref{cz2.57151}, let us pick $a\in(0,1)$ satisfying $$a=\frac{\frac{1}{2N}+\frac{\beta}{l}+\frac{\gamma}{N}-\frac{1}{2}}{\frac{1}{N}+\frac{\beta}{l}-\frac{1}{2}}.$$ Noting that $\gamma\in(0,\frac{1}{2})$ and $\beta>1$ imply that $\gamma+\frac{1}{2}\leq a<1$, we see from the fractional Gagliardo--Nirenberg inequality (Lemma \ref{lemma41ffgg}) and boundedness of $ \|\nabla v \|^l$ (see Lemma \ref{wsdelemma45}) that \begin{equation} \begin{array}{rl} &\displaystyle{\| \|\nabla v \|^{\beta}\|^2_{W^{r+\frac{1}{2},2}(\Omega)}} \\ \leq&\displaystyle{c_0\|\nabla \|\nabla v \|^{\beta}\|^a_{L^2(\Omega)}\\| \|\nabla v \|^\beta\\|^{1-a}_{L^{\frac{l}{\beta}}(\Omega)}+c'_0\\| \|\nabla v \|^\beta\\|_{L^{\frac{l}{\beta}}(\Omega)}}\\ \leq&\displaystyle{C_4\|\nabla \|\nabla v \|^{\beta}\|^a_{L^2(\Omega)}+C_4.}\\ \end{array} \label{vvggcz2.57151} \end{equation} Combining \dref{cz2.57151hhkkhhgg} and \dref{cz2.57151} with \dref{vvggcz2.57151}, we obtain \begin{equation} \begin{array}{rl} &\displaystyle{\displaystyle\int_{\partial\Omega}\frac{\partial \|\nabla v \|^2}{\partial\nu} \|\nabla v \|^{2\beta-2} \leq C_5\|\nabla \|\nabla v \|^{\beta}\|^a_{L^2(\Omega)}+C_5.}\\ \end{array} \label{cz2.57151hhkkhhggyyxx} \end{equation} Now, inserting \dref{cz2.5ghju4ghjuk81}--\dref{cz2.57151hhkkhhggyyxx} into \dref{cz2.5ghju48156} and using the Young inequality we can get \begin{equation}\label{dddffhjui909klopji115} \begin{array}{rl} &\displaystyle{\frac{1}{{2\beta}}\frac{d}{dt}\\|\nabla v \\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)}+\frac{3(\beta-1)}{4{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla v \|^{\beta}\right\|^2+\frac{1}{2}\displaystyle\int_\Omega \|\nabla v \|^{2\beta-2}\|D^2v \|^2+\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta}}\\ \leq&\displaystyle{C_6\displaystyle\int_\Omega u ^2 \|\nabla v \|^{2\beta-2}+C_6~~\mbox{for all}~~ t\in(0,T_{max})}\\ \end{array} \end{equation} by using the Young inequality. \end{proof} We proceed to establish the main step towards our boundedness proof. The following lemma can be used to improve our knowledge on integrability of $\nabla v $, provided that $\mu>0$. Its repeated application will form the core of our regularity proof. \begin{lemma}\label{lemma45630} Let $(u ,v ,w )$ be a solution to \dref{7101.2x19x3sss189} on $(0,T_{max})$ and $\mu>0$. Then for any $1<\gamma_0<\mu_$, there exists $C > 0$ such that \begin{equation} \\|\nabla v (\cdot, t)\\|_{L^{{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}}(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}), \label{zjscz2.5297x9ssd630xxy} \end{equation} where $\mu_$ is given by \dref{zjscz2.ddffrr5297x96ssddffdffggbh302222114}. \end{lemma} \begin{proof} Let $\gamma_0$ and $\mu_$ be same as Lemma \ref{qqqqlemma45630}. For the above $1<\gamma_0<\mu_$, we choose $\beta=\frac{\gamma_0+1}{2}$ in \dref{hjui909klopji115}. Then by using the Young inequality, we derive that for some positive constant $C_1$, \begin{equation}\label{sssshjui909dddklopji115} \begin{array}{rl} \displaystyle{\kappa_0\int_\Omega {u ^2} \|\nabla {v }\|^{2\beta-2}} =&\displaystyle{\kappa_0\int_\Omega {u ^2} \|\nabla {v }\|^{\gamma_0-1}}\\ \leq&\displaystyle{\frac{1}{2}\int_\Omega\|\nabla {v }\|^{\gamma_0+1}+C_1\int_\Omega {u }^{\gamma_0+1}~~\mbox{for all}~~ t\in(0,T_{max}).}\\ \end{array} \end{equation} Here $\kappa_0$ is the same as \dref{hjui909klopji115}. Inserting \dref{sssshjui909dddklopji115} into \dref{hjui909klopji115}, we conclude that there exists a positive constant $C_2$ such that \begin{equation}\label{hjui909klopssddji115} \begin{array}{rl} &\displaystyle{\frac{1}{{\gamma_0+1}}\frac{d}{dt}\\|\nabla {v }\\|^{{{\gamma_0+1}}}_{L^{{\gamma_0+1}}(\Omega)}+\frac{3(\frac{\gamma_0+1}{2}-1)}{{(\gamma_0+1)^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla {v }\|^{\frac{\gamma_0+1}{2}}\right\|^2}\\ &\displaystyle{+\frac{1}{2}\displaystyle\int_\Omega \|\nabla {v }\|^{\gamma_0-1}\|D^2{v }\|^2+\displaystyle\frac{1}{2}\int_{\Omega} \|\nabla {v }\|^{\gamma_0+1}}\\ \leq&\displaystyle{C_1\int_\Omega {u ^{\gamma_0+1}}+C_2~~\mbox{for all}~~ t\in(0,T_{max}),}\\ \end{array} \end{equation} which combined with \dref{4455zjscz2.ddffrr5297x96302222114} implies that \begin{equation} \int_{\Omega}\|\nabla v \|^{\gamma_0+1}(x,t)dx \leq C_3 ~~~\mbox{for all}~~ t\in(0,T_{max}) \label{zjscz2.ddffrr5297x96302sss222114} \end{equation} by an ODE comparison argument. On the other hand, for any $\beta>1$, it then follows from Lemma \ref{lemma41} that there exist positive constants $\kappa_{1}$ and $\kappa_{2}$ such that \begin{equation} \begin{array}{rl} \\|\nabla {v }\\|_{L^{2\beta+\frac{2(\gamma_0+1)}{N}}(\Omega)}^{2\beta+\frac{2(\gamma_0+1)}{N}}=&\displaystyle{\\| \|\nabla {v }\|^\beta\\|_{L^{2+\frac{2(\gamma_0+1)}{\beta N}}(\Omega)}^{2+\frac{2(\gamma_0+1)}{N\beta}}} \\ \leq&\displaystyle{\kappa_{1}(\\|\nabla \|\nabla {v }\|^\beta\\|_{L^2(\Omega)}^{2}\\| \|\nabla {v }\|^\beta\\|_{L^\frac{\gamma_0+1}{\beta}(\Omega)}^{\frac{2(\gamma_0+1)}{N\beta}}+\\| \|\nabla {v }\|^\beta\\|_{L^\frac{\gamma_0+1}{\beta} (\Omega)}^{2+\frac{2(\gamma_0+1)}{N\beta}})}\\ \leq&\displaystyle{\kappa_{2}(\\|\nabla \|\nabla {v }\|^\beta\\|_{L^2(\Omega)}^{2}+1)}\\ \end{array} \label{9999cz2.563022222ikopl2gg66} \end{equation} by using \dref{zjscz2.ddffrr5297x96302sss222114}. Next, picking $\beta=\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}$ in \dref{hjui909klopji115}, then $\beta>1$, so that, by \dref{hjui909klopji115}, we derive that \begin{equation}\label{hjeeui909sddfghhklopji115} \begin{array}{rl} &\displaystyle{\frac{1}{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}\frac{d}{dt}\\|\nabla {v }\\|^{{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}}_{L^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}(\Omega)}}\\ &\displaystyle{+\frac{3({\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}}-1)}{{({\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}})^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla {v }\|^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}}}\right\|^2}\\ &\displaystyle{+\frac{1}{2}\displaystyle\int_\Omega \|\nabla {v }\|^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}-2}\|D^2{v }\|^2+\displaystyle\int_{\Omega} \|\nabla {v }\|^{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}\\ \leq&\displaystyle{\kappa_0\displaystyle\int_\Omega {u ^2} \|\nabla {v }\|^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}-2}+\kappa_0}\\ \leq&\displaystyle{\frac{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}}-1}{{({\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}})^2\kappa_2}}\\|\nabla {v }\\|_{L^{2\beta+\frac{2(\gamma_0+1)}{N}}(\Omega)}^{2\beta+\frac{2(\gamma_0+1)}{N}}+C_{4}\displaystyle\int_\Omega {u ^{\gamma_0+1}}+C_{5}~~\mbox{for all}~~ t\in(0,T_{max}),}\\ \end{array} \end{equation} where $\kappa_2$ is the same as \dref{9999cz2.563022222ikopl2gg66}. Therefore, collecting \dref{9999cz2.563022222ikopl2gg66} and \dref{hjeeui909sddfghhklopji115}, we have \begin{equation}\label{hjeeui9ssd09sddfghhklopji115} \begin{array}{rl} &\displaystyle{\frac{1}{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}\frac{d}{dt}\\|\nabla {v }\\|^{{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}}_{L^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}(\Omega)}}\\ &\displaystyle{+\frac{2({\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}}-1)}{{({\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}})^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla {v }\|^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{2N}}}\right\|^2}\\ &\displaystyle{+\frac{1}{2}\displaystyle\int_\Omega \|\nabla {v }\|^{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}-2}\|D^2{v }\|^2+\displaystyle\int_{\Omega} \|\nabla {v }\|^{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}\\ \leq&\displaystyle{C_{4}\displaystyle\int_\Omega {u ^{\gamma_0+1}}+C_{5}~~\mbox{for all}~~ t\in(0,T_{max}),}\\ \end{array} \end{equation} therefore, in view of \dref{4455zjscz2.ddffrr5297x96302222114}, by using an ODE comparison argument again, we have \begin{equation} \begin{array}{rl} \\|\nabla {v }\\|_{L^{{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}}(\Omega)}\leq&\displaystyle{C_{6}~~\mbox{for all}~~ t\in(0,T_{max})}\\ \end{array} \label{9999cz2.563022222ikopl2ssddgg66} \end{equation} with some positive constant $C_{6}$, which yields \dref{zjscz2.5297x9ssd630xxy}, and hence completes the proof. \end{proof} \begin{lemma}\label{pplemma45630} Let $\Omega\subset \mathbb{R}^N(N\geq1)$ be a bounded domain with smooth boundary. Then for all $\beta> 1$ and $k>1$, the solution of \dref{7101.2x19x3sss189} from Lemma \ref{lemma70} satisfies \begin{equation}\label{1234hjui909klopji115} \begin{array}{rl} &\displaystyle{\frac{d}{dt}(\frac{1}{k}\\|u \\|^{k}_{L^k(\Omega)}+\frac{1}{{2\beta}}\\|\nabla {v }\\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)})+\frac{3(\beta-1)}{4{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla {v }\|^{\beta}\right\|^2+\frac{\mu}{2}\int_{\Omega}u ^{k+1}}\\ &+\displaystyle{\frac{1}{2}\displaystyle\int_\Omega \|\nabla {v }\|^{2\beta-2}\|D^2{v }\|^2+\displaystyle\int_{\Omega} \|\nabla {v }\|^{2\beta}+\frac{(k-1)m}{4}\int_{\Omega}u ^{m+k-3}\|\nabla u \|^2{}}\\ \leq&\displaystyle{C(\displaystyle\frac{\chi^2(k-1)}{2C_D}\int_{\Omega}u ^{k+1-m}\|\nabla v \|^2+\int_\Omega u ^2 \|\nabla {v }\|^{2\beta-2}+\int_\Omega v ^{k+1})+C,}\\ \end{array} \end{equation} where $C$ is a positive constant. \end{lemma} \begin{proof} Collecting Lemma \ref{lemmssdda45630} and Lemma \ref{lemma45630}, we can derive \dref{1234hjui909klopji115} by using the Young inequality. \end{proof} We next plan to estimate the right-hand sides in the above inequalities appropriately by using a priori information provided by Lemma \ref{lemma45630} and Lemma \ref{wsdelemma45}. Here the following lemma will will play an important role in making efficient use of the known ${L^{{{\frac{(\gamma_0+1)(N+\gamma_0-1)}{N}}}}(\Omega)}$ bound for $\nabla v $. The following lemma provides some elementary material that will be essential to our bootstrap procedure. \begin{lemma}\label{lsssemma456302ssd23116} Let \begin{equation}\label{dcfvwsdddffgg7101.2x19x318jkl} \tilde{H}(y)=\begin{array}{ll} \frac{2N^2}{N^2+{{{(y+1)(N+y-1)}}}}-[1+\frac{[N^2-{{{(y+1) (N+y-1)}}}]y} {N{{{(y+1) (N+y-1)}}}}] \end{array} \end{equation} with ${{{(y+1) (N+y-1)}}}>N^2$, for any $y>1$ and $N\geq2.$ Then we have \begin{equation}\label{dcfvdddffffggffffwsdddffgg7101.2x19x318jkl} \min_{y>1}\tilde{H}(y)\geq0. \end{equation} \end{lemma} \begin{proof} It is easy to verify that $N^2<{{{(y+1) (N+y-1)}}}$ and $y>1$ and $N\geq2$ implies that \begin{equation}\label{dcfvdffffwsdddffgg7101.2x19x318jkl}y>\frac{-N+\sqrt{5N^2-4N+4}}{2}=\frac{-N+\sqrt{4N^2+(N-2)^2}}{2}\geq\frac{N}{2}. \end{equation} On the other hand, by some basic calculation, one has \begin{equation}\label{dcfvdfssddfffwsdddffgg7101.2x19x318jkl}\begin{array}{ll} &\frac{2N^2}{N^2+{{{(y+1)(N+y-1)}}}}-[1+\frac{[N^2-{{{(y+1) (N+y-1)}}}]y} {N{{{(y+1) (N+y-1)}}}}]\\ =&[{{{(y+1)(N+y-1)}}}-N^2][\frac{y} {N{{{(y+1) (N+y-1)}}}}-\frac{1}{N^2+{{{(y+1)(N+y-1)}}}}]\\ =&\frac{[{{{(y+1)(N+y-1)}}}-N^2]}{N{{{(y+1) (N+y-1)[N^2+{{{(y+1)(N+y-1)}}}}}]}}h_1(y)\\ \end{array} \end{equation} with \begin{equation}\label{dcfvdfsdfffddsddfffwsdddffgg7101.2x19x318jkl}\begin{array}{rl} h_1(y)=&[yN^2+y(y+1)(y+N-1)-N(y+1)(N+y-1)]\\ =&y^3+(N-1)y-N^2+N.\\ \end{array} \end{equation} Now, by some basic calculation, one has, $$\begin{array}{rl} h'_1(y)=&3y^2+(N-1)>0.\\ \end{array}$$ by using $N\geq2$ and $y>1$. Therefore, by \dref{dcfvdffffwsdddffgg7101.2x19x318jkl}, we have \begin{equation}\label{dcfvdffddsddffffgg710x19x318jkl}\begin{array}{rl} h_1(y)\geq&h_1(\frac{N}{2})\\ =&\frac{N^3}{8}-\frac{N^2}{2}+\frac{N}{2}\\ =:&\tilde{h}_1(N).\\ \end{array} \end{equation} Since, $\tilde{h}'_1(N)=\frac{3N^2}{8}-N+\frac{1}{2}>0$ by using $N\geq2.$ Thus, $\tilde{h}_1(N)>\tilde{h}_1(2)=0,$ so that, inserting \dref{dcfvdfsdfffddsddfffwsdddffgg7101.2x19x318jkl}--\dref{dcfvdffddsddffffgg710x19x318jkl} into \dref{dcfvdfssddfffwsdddffgg7101.2x19x318jkl} and ${{{(y+1) (N+y-1)}}}>N^2$, we obtain that \begin{equation}\label{dcfvdffddddfffsddsdddffffgg710x19x318jkl}\frac{2N^2}{N^2+{{{(y+1)(N+y-1)}}}}>[1+\frac{[N^2-{{{(y+1) (N+y-1)}}}]y} {N{{{(y+1) (N+y-1)}}}}]. \end{equation} \end{proof} Now, we can make use of Lemma \ref{ssdeedrfe116lemma70hhjj} as well as Lemma \ref{lemma41} and Lemma \ref{lemma45630} to estimate the integrals on the right-hand sides of \dref{hjui909klopji115} and \dref{cz2.5xx1jjjj} (or \dref{1234hjui909klopji115}). To this end, we will establish bounds for $\int_\Omega u^k dx$ with any $k> 1$ by Lemma \ref{ssdeedrfe116lemma70hhjj} as well as Lemma \ref{lemma41} and Lemma \ref{lsssemma456302ssd23116}. \begin{lemma}\label{lemma456ssddfff30} Assume that $m>\frac{2N}{N+\gamma_}$ with $N=2$, where \begin{equation}\label{xeerr1.731426677gg} \gamma_={{{\frac{(\mu_{}+1)(N+\mu_{}-1)}{N}}}} \end{equation} and $\mu_$ is same as \dref{zjscz2.ddffrr5297x96ssddffdffggbh302222114}. Then for all $k>1$, there exists $C > 0$ such that \begin{equation} \\|u (\cdot, t)\\|_{L^k(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}). \label{111zjscz2.5297x9630xxy} \end{equation} \end{lemma} \begin{proof} Next, due to \dref{xeerr1.731426677gg} and $N=2$ and $m>\frac{2N}{N+\gamma_}$ implies that \begin{equation}m>\frac{8}{4+[\mu_+1]^2}, \label{zjsssdddcz2.5297x9ssd63ss0xxy} \end{equation} where $\mu_$ is given by \dref{zjscz2.ddffrr5297x96ssddffdffggbh302222114}. Now, in view of $\mu>0$ implies that $$\mu_>1$$ and $$4<(\mu_+1)^2,$$ therefore, employing Lemma \ref{lsssemma456302ssd23116}, we have \begin{equation}m>1+\frac{[1-\frac{(\mu_+1)^2}{4}]\mu_} {2\times\frac{(\mu_+1)^2}{4}}. \label{zjsssdddcz2.5297x9ssssddd63ss0xxy} \end{equation} Thus, we may choose $ q_0\in(1,\mu_)$ which is close to $\mu_$ such that \begin{equation} \begin{array}{rl} m>&1+\frac{[1-\frac{(q_0+1)^2}{4}]q_0} {2\times\frac{(q_0+1)^2}{4}}.\\ \end{array}\label{zjsssdddcz2.5297x9ssssddd63ss0ssddxxy} \end{equation} Next, observing that ${{{{\frac{(q_0+1)^2}{4}}}}}\in(1,\frac{(\mu_+1)^2}{4})$, thus, in light of Lemma \ref{lemma45630}, we derive that there exists a positive constant $C_1$ such that \begin{equation} \\|\nabla v (\cdot, t)\\|_{L^{2p_0}(\Omega)}\leq C_1 ~~\mbox{for all}~~ t\in(0, T_{max}), \label{zjscz2.5297x9ssd63ss0xxy} \end{equation} where \begin{equation} \label{zjscz2.5297x9ssd63ssdddss0xxy}p_0={{{{\frac{(q_0+1)^2}{4}}}}}>1 \end{equation} by using $q_0>1.$ Now, choosing $k> \max\{3,\|1-m\|+q_0\frac{p_0-1}{p_0}\}$ in \dref{cz2.5xx1jjjj}, then, we have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\| u \\|^{k}_{L^k(\Omega)}+\frac{(k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{\frac{\chi^2(k-1)} {2 C_D }\int_{\Omega} u ^{k+1-m}\|\nabla v \|^2+ C_2\int_\Omega u ^{k}(v +1)}\\ \end{array} \label{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjsddffjj} \end{equation} with $C_2=\max\{\xi\\|w_0\\|_{L^\infty(\Omega)},\mu+\kappa\xi\}$. We estimate the rightmost integral by means of the H\"{o}lder inequality according to \begin{equation} \begin{array}{rl} &\displaystyle{\frac{ \chi^2(k-1)}{2C_D} \displaystyle\int_\Omega u ^{k+1-m } \|\nabla v \|^2}\\ &\leq\displaystyle{ \frac{ \chi^2(k-1)}{2C_D}\left(\displaystyle\int_\Omega u ^{\frac{p_0}{p_0-1}(k+1-m )}\right)^{\frac{p_0-1}{p_0}}\left(\displaystyle\int_\Omega \|\nabla v \|^{2p_0}\right)^{\frac{1}{p_0}}}\\ &\leq\displaystyle{ \frac{C_3 \chi^2(k-1)}{2C_D}\\| u ^{\frac{k+m-1}{2}}\\|^{\frac{2(k+1-m )}{k+m-1}}_{L^{\frac{2p_0(k+1-m )}{(p_0-1)(k+m-1)}}(\Omega)}}\\ \end{array} \label{cz2.57151hhkkhhhjukildrfthjjhhhhh} \end{equation} by using \dref{zjscz2.5297x9ssd63ss0xxy}, where $C_3>0$. Since, $k> \|1-m\|+q_0\frac{p_0-1}{p_0},$ we have $$\frac{q_0}{k+m-1}\leq{\frac{p_0(k+1-m )}{(p_0-1)(k+m-1)}}<+\infty,$$ so that, the Gagliardo-Nirenberg inequality (Lemma \ref{lemma41}) indicates that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{ \chi^2(k-1)}{2C_D}\\| u ^{\frac{k+m-1}{2}}\\| ^{\frac{2(k+1-m )}{k+m-1}}_{L^{\frac{2p_0(k+1-m )}{(p_0-1)(k+m-1)}}(\Omega)}} \\ \leq&\displaystyle{C_{4}(\\|\nabla u ^{\frac{k+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-\frac{2(p_0-1)q_0}{p_0(k+m-1)}}\\| u ^{\frac{k+m-1}{2}}\\|_{L^\frac{2q_0}{k+m-1}(\Omega)}^{\frac{2(p_0-1)q_0}{p_0(k+m-1)}}+\\| u ^{\frac{k+m-1}{2}}\\|_{L^\frac{2q_0}{k+m-1}(\Omega)}^{\frac{2(k+1-m )}{k+m-1}})}\\ \leq&\displaystyle{C_{5}(\\|\nabla u ^{\frac{k+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2(k+1-m)}{k+m-1}-\frac{2(p_0-1)q_0}{p_0(k+m-1)}}+1)}\\ \end{array} \label{cz2.563022222ikopl2sdfg44} \end{equation} with some positive constants $C_{4}$ as well as $ C_{5}$, where $q_0$ is the same as \dref{zjsssdddcz2.5297x9ssssddd63ss0ssddxxy}. Here we have used $L^1(\Omega)$ boundedness for $u $ (see Lemma \dref{wsdelemma45}). Due to \dref{zjsssdddcz2.5297x9ssssddd63ss0ssddxxy}, one has $${\frac{2(k+1-m)}{k+m-1}-\frac{2(p_0-1)q_0}{p_0(k+m-1)}}<2,$$ so that, applying the Young inequality implies that there exists a positive constant $C_7$ such that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \frac{\mu}{2}\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_2\int_\Omega u ^{k}(v +1)+C_7~~\mbox{for all}~~ t\in(0, T_{max}),}\\ \end{array} \label{vgbhnsxcdvfdssddfcz2.5xxdddf1jjjj} \end{equation} which together with the Young inequality implies that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_9\int_\Omega (v +1)^k+C_8~~\mbox{for all}~~ t\in(0, T_{max})}\\ \end{array} \label{vgbhnsxcdvfdssddfcz2.5xxdddf1ssddjjjj} \end{equation} for some positive constants $C_8$ and $C_9.$ Now, in view of ${2p_0}>2$ (see \dref{zjscz2.5297x9ssd63ssdddss0xxy})and $N=2$, then by Sobolev imbedding theorems, we derive from \dref{zjscz2.5297x9ssd63ss0xxy} that $$\\|v \\|_{L^\infty(\Omega)}\leq C_{10}\\|\nabla v \\|_{L^{2p_0}(\Omega)},$$ so that, combined with \dref{vgbhnsxcdvfdssddfcz2.5xxdddf1ssddjjjj} implies that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_{11}~~\mbox{for all}~~ t\in(0, T_{max}),}\\ \end{array} \label{vgbhnsxcdvfdssddfcz2.5ssddxxdddf1ssddjjjj} \end{equation} whereas a standard ODE comparison argument shows that \dref{111zjscz2.5297x9630xxy} holds. \end{proof} \begin{lemma}\label{lemma456ssddfff30} Let $\Omega\subset \mathbb{R}^N(N\geq1)$ be a bounded domain with smooth boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. If \begin{equation}{{{\frac{(\mu_+1) (N+\mu_-1)}{2N}}}}>\frac{N}{2}, \label{zjscz2.529df763xy} \end{equation} then for any $k>1$, there exists a positive constant $C$ such that \begin{equation} \\|u (\cdot, t)\\|_{L^k(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}), \label{2222zjscz2.5297x9630xxy} \end{equation} where $\mu_$ is given by \dref{zjscz2.ddffrr5297x96ssddffdffggbh302222114}. \end{lemma} \begin{proof} Due to $$m>\frac{2N}{N+{{{\frac{(\mu_+1)(N+\mu_-1)}{N}}}}}.$$ Now, in view of $\mu>0$ implies that $$\mu_>1,$$ which together with Lemma \ref{lsssemma456302ssd23116} results in \begin{equation}m>1+\frac{[N-2{{{\frac{(\mu_+1) (N+\mu_-1)}{2N}}}}]\mu_} {2N\times{{{\frac{(\mu_+1) (N+\mu_-1)}{2N}}}}}. \label{zjsssdddcz2.5297x9ssssddd63ss0xxy} \end{equation} Thus by \dref{zjscz2.529df763xy}, we may choose $q_{0,}\in(1,\mu_)$ which is close to $\mu_$ such that \begin{equation} \begin{array}{rl} m>&1+\frac{[N-2{{{\frac{(q_{0,}+1) (N+q_{0,}-1)}{2N}}}}]q_{0,}} {2N\times{{{\frac{(q_{0,}+1) (N+q_{0,}-1)}{2N}}}}}\\\ \end{array}\label{zjsssdd78888dcz2.5297x9ssssddd63ss0ssddxxy} \end{equation} and \begin{equation}{{{\frac{(q_{0,}+1) (N+q_{0,}-1)}{2N}}}}>\frac{N}{2}. \label{zjscz2.rrtt529df763xy} \end{equation} Next, observing that $\frac{(q_{0,}+1) (N+q_{0,}-1)}{2N}\in(1,{{{\frac{(\mu_+1) (N+\mu_-1)}{2N}}}})$, thus, in light of Lemma \ref{lemma45630}, we derive that there exists a positive constant $C_1$ such that \begin{equation} \\|\nabla v (\cdot, t)\\|_{L^{2p_{0,}}(\Omega)}\leq C_1 ~~\mbox{for all}~~ t\in(0, T_{max}), \label{zjscz2dfffg.5297x9ssd63ss0xxy} \end{equation} where \begin{equation} \label{zjscz2.5297x9ssd63ssdddss0xxy}p_{0,}=\frac{(q_{0,}+1) (N+q_{0,}-1)}{2N}>\frac{N}{2}. \end{equation} Now, choosing $k> \max\{N+1,\|1-m\|+q_{0,}\frac{p_{0,}-1}{p_{0,}},\frac{1-m}{2p_{0,}-N}(2p_{0,}N-N-2p_{0,})\}$ in \dref{cz2.5xx1jjjj}, then, we have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \frac{\mu}{2}\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{\frac{\chi^2(k-1)}{2 m}\int_{\Omega} u ^{k+1-m}\|\nabla v \|^2+ C_2\int_\Omega u ^{k}(v +1)~~\mbox{for all}~~ t\in(0, T_{max})}\\ \end{array} \label{vgbhnsxcdvfddfcz2.5xxdddf1jjjj} \end{equation} with $C_2=\max\{\xi\\|w_0\\|_{L^\infty(\Omega)},\mu+\kappa\xi\}$. According to the estimate of $\nabla v $ in \dref{zjscz2dfffg.5297x9ssd63ss0xxy} along with the H\"{o}lder inequality we derive that there exists a positive constant $C_3>0$ such that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{ \chi^2(k-1)}{2C_D} \displaystyle\int_\Omega u ^{k+1-m } \|\nabla v \|^2}\\ &\leq\displaystyle{ \frac{ \chi^2(k-1)}{2C_D}\left(\displaystyle\int_\Omega u ^{\frac{p_{0,}}{p_{0,}-1}(k+1-m )}\right)^{\frac{p_{0,}-1}{p_{0,}}}\left(\displaystyle\int_\Omega \|\nabla v \|^{2p_{0,}}\right)^{\frac{1}{p_{0,}}}}\\ &\leq\displaystyle{ \frac{C_3 \chi^2(k-1)}{2C_D}\\| u ^{\frac{k+m-1}{2}}\\|^{\frac{2(k+1-m )}{k+m-1}}_{L^{\frac{2p_{0,}(k+1-m )}{(p_{0,}-1)(k+m-1)}}(\Omega)} .}\\ \end{array} \label{cz2.57151hhkkhhhjukildrfthjjhhhhh} \end{equation} In view of $k> \max\{\|1-m\|+q_{0,}\frac{p_{0,}-1}{p_{0,}},\frac{1-m}{2p_{0,}-N}(2p_{0,}N-N-2p_{0,})\}$, we have $$\frac{q_{0,}}{k+m-1}\leq{\frac{p_{0,}(k+1-m )}{(p_{0,}-1)(k+m-1)}}<\frac{N}{N-2}.$$ An application of the Gagliardo-Nirenberg inequality (see Lemma \ref{lemma41}) implies that for some positive constants $C_{4}$ as well as $ C_{5}$ such that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{ \chi^2(k-1)}{2C_D}\\| u ^{\frac{k+m-1}{2}}\\| ^{\frac{2(k+1-m )}{k+m-1}}_{L^{\frac{2p_{0,}(k+1-m )}{(p_{0,}-1)(k+m-1)}}(\Omega)}} \\ \leq&\displaystyle{C_{4}(\\|\nabla u ^{\frac{k+m-1}{2}}\\|_{L^2(\Omega)}^{2\frac{N(k+1-m-q_{0,}+\frac{q_{0,}}{p_{0,}})}{N(k+m-1-q_{0,})+2q_{0,}}}\\| u ^{\frac{k+m-1}{2}}\\|_{L^\frac{2q_{0,}}{k+m-1}(\Omega)}^{\frac{2(k+1-m )}{k+m-1}-2\frac{N(k+1-m-q_{0,}+\frac{q_{0,}}{p_{0,}})}{N(k+m-1-q_{0,})+2q_{0,}}}+\\| u ^{\frac{k+m-1}{2}}\\|_{L^\frac{2q_{0,}}{k+m-1}(\Omega)}^{\frac{2(k+1-m )}{k+m-1}})}\\ \leq&\displaystyle{C_{5}(\\|\nabla u ^{\frac{k+m-1}{2}}\\|_{L^2(\Omega)}^{2\frac{N(k+1-m-q_{0,}+\frac{q_{0,}}{p_{0,}})}{N(k+m-1-q_{0,})+2q_{0,}}}+1)}\\ \end{array} \label{cz2.563022222ikopl2sdfg44} \end{equation} by using Lemma \ref{wsdelemma45}, where $q_{0,}$ is the same as \dref{zjsssdd78888dcz2.5297x9ssssddd63ss0ssddxxy}. Due to \dref{zjsssdd78888dcz2.5297x9ssssddd63ss0ssddxxy}, one has $${2\frac{N(k+1-m-q_{0,}+\frac{q_{0,}}{p_{0,}})}{N(k+m-1-q_{0,})+2q_{0,}}}<2,$$ so that, applying the Young inequality implies that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \frac{\mu}{2}\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_2\int_\Omega u ^{k}(v +1)+C_7~~\mbox{for all}~~ t\in(0, T_{max}),}\\ \end{array} \label{vgbhnsxcdvfds7888sddfcz2.5xxdddf1jjjj} \end{equation} which together with the Young inequality again yields to \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_9\int_\Omega (v +1)^k+C_8~~\mbox{for all}~~ t\in(0, T_{max})}\\ \end{array} \label{vgbhnsxcdvfdssddfcz2jjjj.5xxdddf1ssddjjjj} \end{equation} and some positive constants $C_8$ and $C_9.$ Now, in view of ${2p_0}>N$ (see \dref{zjscz2.5297x9ssd63ssdddss0xxy}), then by Sobolev imbedding theorems, we derive from \dref{zjscz2dfffg.5297x9ssd63ss0xxy} that $$\\|v \\|_{L^\infty(\Omega)}\leq C_{10}\\|\nabla v \\|_{L^{2p_0}(\Omega)},$$ so that, combined with \dref{vgbhnsxcdvfdssddfcz2jjjj.5xxdddf1ssddjjjj} implies that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{k}\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{ (k-1) C_D }{4}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+ \mu\int_{\Omega} u ^{k+1}{}} \\ \leq&\displaystyle{ C_{11}~~\mbox{for all}~~ t\in(0, T_{max}),}\\ \end{array} \label{vgbhnsxcdvfdssddfcz2.5ssddxxdddf1ssddjjjj} \end{equation} whereas a standard ODE comparison argument shows that \dref{2222zjscz2.5297x9630xxy} holds. \end{proof} \begin{lemma}\label{lemmaddff45ssdd630} Let $\Omega\subset \mathbb{R}^N(N\neq2)$ be a bounded domain with smooth boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. If \begin{equation}{{{\frac{(\mu_+1) (N+\mu_-1)}{2N}}}}\leq\frac{N}{2}, \label{zjscz2.529df763xddfffy} \end{equation} then for any $k>1$, there exists a positive constant $C$ such that \begin{equation} \\|u (\cdot, t)\\|_{L^k(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}). \label{zjscz2.529ddff7x9630xxy} \end{equation} \end{lemma} \begin{proof} Let $$\bar{\beta}=\max\{\frac{4N{{{\frac{(\mu_+1)(N+\frac{\chi\max\{1,\lambda_0\}} {(\chi\max\{1,\lambda_0\}-\mu)_{+}}-1)}{N}}}}}{N+{{{\frac{(\mu_+1)(N+\frac{\chi\max\{1,\lambda_0\}} {(\chi\max\{1,\lambda_0\}-\mu)_{+}}-1)}{N}}}}},2-\frac{2}{N}-m,16,8N+2\}.$$ Due to $$m>\frac{2N}{N+{{{\frac{(\mu_+1)(N+\frac{\chi\max\{1,\lambda_0\}} {(\chi\max\{1,\lambda_0\}-\mu)_{+}}-1)}{N}}}}},$$ as well as \dref{zjscz2.529df763xddfffy} and Lemma \ref{lemma45630}, we may choose $q_{0,*}\in(1,\mu_)$ which is close to $\mu_$ such that \begin{equation} \begin{array}{rl} &\frac{2N(m -1)q_{0,}}{N-2q_{0,}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}\\ &>\displaystyle{ \frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}}\\ \end{array} \label{sxcdcz2.57151hhkkhhhjukildrfthjjhhhhh} \end{equation} for all $\beta\geq\bar{\beta}.$ Therefore, we can choose \begin{equation} k\in( \frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N},\frac{2N(m -1)q_{0,}}{N-2q_{0,}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}). \label{dcfvzsxcvgcz2.57151hhkkhhhjukildrfthjjhhhhh} \end{equation} Now for the above $k$, by the H\"{o}lder inequality, we have \begin{equation} \begin{array}{rl} J_1:&=\displaystyle{ \displaystyle\frac{ \chi^2(k-1)}{2C_D}\int_\Omega u ^{{k}+1-m } \|\nabla v \|^2}\\ &\leq\displaystyle{ \frac{ \chi^2(k-1)}{2C_D}\left(\displaystyle\int_\Omega u ^{{\frac{N}{N-2}}({k}+1-m )}\right)^{{\frac{N-2}{N}}}\left(\displaystyle\int_\Omega \|\nabla v \|^{{N}}\right)^{{\frac{2}{N}}}}\\ &=\displaystyle{ \frac{ \chi^2(k-1)}{2C_D}\\| u ^{\frac{{k}+m-1}{2}}\\|^{\frac{2({k}+1-m )}{{k}+m-1}}_{L^{\frac{{\frac{2N}{N-2}}({k}+1-m )}{{k}+m-1}}(\Omega)} \\|\nabla v \\|_{L^{{N}}(\Omega)}^2.}\\ \end{array} \label{cz2.57151hhkkhhhjukildrfthjjhhhhh} \end{equation} Next, by \dref{zjscz2.529df763xddfffy}, we derive that $m\geq 1 $, so that, in view of $N\geq3$, we have $$\frac{1}{{k}+m-1}\leq\frac{{k}+1-m}{{k}+m-1}\frac{{\frac{N}{2}}}{({\frac{N}{2}}-1)_+}\leq\frac{N}{(N-2)_+},$$ so that, by using Gagliardo-Nirenberg interpolation inequality (Lemma \ref{lemma41}) and $L^1(\Omega)$ boundedness for $u $ (see Lemma \dref{wsdelemma45}) we have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{ \chi^2(k-1)}{2C_D}\\| u ^{\frac{{k}+m-1}{2}}\\| ^{\frac{2({k}+1-m )}{{k}+m-1}}_{L^{\frac{{\frac{2N}{N-2}}({k}+1-m )}{{k}+m-1}}(\Omega)}} \\ \leq&\displaystyle{C_1(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\mu_1}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{2}{{k}+m-1}(\Omega)}^{1-\mu_1}+\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{2}{{k}+m-1}(\Omega)})^{\frac{2({k}+1-m )}{{k}+m-1}}}\\ \leq&\displaystyle{C_2(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2({k}+1-m )\mu_1}{{k}+m-1}}+1)}\\ \end{array} \label{czzsxcv2.563022222ikopl2sdfg44} \end{equation} with some positive constants $C_{1}$ as well as $C_2$ and $$\mu_1=\frac{\frac{N[{k}+m-1]}{2}-\frac{N({k}+m-1)}{{\frac{2N}{N-2}}({k}+1-m )}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}= [{k}+m-1]\frac{\frac{N}{2}-\frac{N}{{\frac{2N}{N-2}}({k}+1-m )}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}\in(0,1).$$ On the other hand, due to Lemma \ref{lemma41} and the fact that $\beta\geq\bar{\beta}>\frac{N}{2}$ and $N>2$, we have \begin{equation} \begin{array}{rl} \\|\nabla v \\|_{L^{{N}}(\Omega)}^2=&\displaystyle{\\| \|\nabla v \|^\beta\\|_{L^{\frac{{N}}{\beta}}(\Omega)}^{\frac{2}{\beta}}} \\ \leq&\displaystyle{C_3(\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{2\mu_2}{\beta}}\\| \|\nabla v \|^\beta\\|_{L^\frac{2q_{0,}}{\beta}(\Omega)}^{\frac{2(1-\mu_2)}{\beta}}+\\| \|\nabla v \|^\beta\\|_{L^\frac{2q_{0,}}{\beta} (\Omega)}^{\frac{2}{\beta}})}\\ \leq&\displaystyle{C_{4}(\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{2\mu_2}{\beta}}+1),}\\ \end{array} \label{czcfv2.563022222ikopl255} \end{equation} where some positive constants $C_3$ as well as $C_{4}$ and $$\mu_2=\frac{\frac{N\beta}{2q_{0,}}-\frac{N\beta}{{N}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}= \beta\frac{\frac{N}{2q_{0,}}-\frac{N}{{N}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}\in(0,1).$$ Inserting \dref{czzsxcv2.563022222ikopl2sdfg44}--\dref{czcfv2.563022222ikopl255} into \dref{cz2.57151hhkkhhhjukildrfthjjhhhhh} as well as by means of the Young inequality and \dref{dcfvzsxcvgcz2.57151hhkkhhhjukildrfthjjhhhhh} we see that for any $\delta>0,$ \begin{equation} \begin{array}{rl} J_1\leq&\displaystyle{C_{5}(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2({k}+1-m )\mu_1}{{k}+m-1}}+1)(\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{2\mu_2}{\beta}}+1)}\\ =&\displaystyle{C_{5}(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{{N}({{k}+1-m -{\frac{N-2}{N}}})}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}}+1) (\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{{N}(\frac{2}{2q_{0,}}-\frac{1}{{\frac{N}{2}}})}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}}+1)}\\ \leq&\displaystyle{\delta\displaystyle\int_{\Omega} \|\nabla u ^{\frac{m+k-1}{2}}\|^2+\delta\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^2 +C_{6}~~\mbox{for all}~~ t\in(0,T_{max}) }\\ \end{array} \label{cz2.57151hhkkhhhjukildrftdfrtgyhu} \end{equation} for all $\beta\geq\bar{\beta}$. Here we have use the fact that $k < \frac{2N(m -1)q_{0,}}{N-2q_{0,}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}$ and $k>2-\frac{2}{N}-m.$ Next, due to the H\"{o}lder inequality and $\beta\geq\bar{\beta}>16$, we have \begin{equation} \begin{array}{rl} J_2:&=\displaystyle{ \displaystyle \int_\Omega u ^2 \|\nabla v \|^{2\beta-2}}\\ &\leq\displaystyle{ \left(\displaystyle\int_\Omega u ^{{\frac{\beta}{4}}}\right)^{{\frac{8}{\beta}}}\left(\displaystyle\int_\Omega \|\nabla v \|^{{\frac{\beta(2\beta-2)}{\beta-8}}}\right)^{{\frac{\beta-8}{\beta}}}}\\ &\leq\displaystyle{ \left(\displaystyle\int_\Omega u ^{{\frac{\beta}{4}}}\right)^{{\frac{8}{\beta}}}\left(\displaystyle\int_\Omega \|\nabla v \|^{{\frac{\beta(2\beta-2)}{\beta-8}}}\right)^{{\frac{\beta-8}{\beta}}}}\\ &=\displaystyle{ \\| u ^{\frac{{k}+m-1}{2}}\\|^{\frac{4}{{k}+m-1}}_{L^{\frac{{\frac{\beta}{2}}}{{k}+m-1}}(\Omega)} \\|\nabla v \\|_{L^{{\frac{\beta(2\beta-2)}{\beta-8}}}(\Omega)}^{(2\beta-2)}.}\\ \end{array} \label{czasxc2.57151hhkkhhhjukildrfttyuijkoghyu66} \end{equation} On the other hand, with the help of $k>1-m+{\frac{N-2}{4N}}\beta$, $\beta\geq\bar{\beta}\geq4$ and Lemma \ref{lemma41} we conclude that \begin{equation} \begin{array}{rl} &\displaystyle{ \\| u ^{\frac{{k}+m-1}{2}}\\|^{\frac{4}{{k}+m-1}}_{L^{\frac{{\frac{\beta}{2}}}{{k}+m-1}}(\Omega)}} \\ \leq&\displaystyle{C_{7}(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\mu_3}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{2}{{k}+m-1}(\Omega)}^{(1-\mu_3)}+\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{2}{{k}+m-1}(\Omega)})^{\frac{4}{{k}+m-1}}}\\ \leq&\displaystyle{C_{8}(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{4\mu_3}{{k}+m-1}}+1)}\\ \end{array} \label{czwsdc2.563022222ikopl2sdfgggjjkkk66} \end{equation} with some positive constants $C_{7}$ as well as $C_{8}$ and $$\mu_3=\frac{\frac{N[{k}+m-1]}{2}-\frac{N({k}+m-1)}{{\frac{\beta}{2}}}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}= [{k}+m-1]\frac{\frac{N}{2}-\frac{N}{{\frac{\beta}{2}}}}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}\in(0,1).$$ On the other hand, again, it infers by the Gagliardo-Nirenberg inequality (Lemma \ref{lemma41}) that there are $C_{9}>0$ and $C_{10}>0$ ensuring \begin{equation} \begin{array}{rl} \\|\nabla v \\|_{L^{{\frac{\beta(2\beta-2)}{\beta-8}}}(\Omega)}^{(2\beta-2)}=&\displaystyle{\\| \|\nabla v \|^\beta\\|_{L^{\frac{{\frac{\beta(2\beta-2)}{\beta-8}}}{\beta}}(\Omega)}^{\frac{2\beta-2}{\beta}}} \\ \leq&\displaystyle{C_{9}(\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{(2\beta-2)\mu_4}{\beta}}\\| \|\nabla v \|^\beta\\|_{L^\frac{2q_{0,}}{\beta}(\Omega)}^{\frac{(2\beta-2)(1-\mu_4)}{\beta}}+\\| \|\nabla v \|^\beta\\|_{L^\frac{2q_{0,}}{\beta} (\Omega)}^{\frac{(2\beta-2)}{\beta}})}\\ \leq&\displaystyle{C_{10}(\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{(2\beta-2)\mu_4}{\beta}}+1)}\\ \end{array} \label{czqscvb2.563022222ikopl2gg66} \end{equation} for any $\beta\geq\bar{\beta}>\frac{7N+2}{2}$, where $$\mu_4=\frac{\frac{N\beta}{2q_{0,}}-\frac{N\beta}{{\frac{\beta(2\beta-2)}{\beta-8}}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}= \beta\frac{\frac{N}{2q_{0,}}-\frac{N}{{\frac{\beta(2\beta-2)}{\beta-8}}}}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}\in(0,1).$$ Inserting \dref{czwsdc2.563022222ikopl2sdfgggjjkkk66}--\dref{czqscvb2.563022222ikopl2gg66} into \dref{czasxc2.57151hhkkhhhjukildrfttyuijkoghyu66} and using $k > \frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}$ and $k>2-\frac{2}{N}-m$ and Lemma \ref{lemma41}, we derive that for the above $\delta>0,$ \begin{equation} \begin{array}{rl} J_2\leq&\displaystyle{C_{11}(\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{ \frac{{N}(2-{\frac{8}{\beta}})}{1-\frac{N}{2}+\frac{N[{k}+m-1]}{2}}}+1) (\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^{\frac{{N}(\frac{2\beta-2}{2q_{0,}}-{\frac{\beta-8}{\beta}})}{1-\frac{N}{2}+\frac{N\beta}{2q_{0,}}}}+1)}\\ \leq&\displaystyle{\delta\displaystyle\int_{\Omega} \|\nabla u ^{\frac{m+k-1}{2}}\|^2+\delta\\|\nabla \|\nabla v \|^\beta\\|_{L^2(\Omega)}^2 +C_{12}~~\mbox{for all}~~ t\in(0,T_{max}). }\\ \end{array} \label{cz2.57151hhkkhhhjukildrftdfrtgyhugg} \end{equation} Finally, with the help of \dref{cz2.5ghju48cfg924ghyuji} and by the Sobolev inequality, the Young inequality and \dref{dcfvzsxcvgcz2.57151hhkkhhhjukildrfthjjhhhhh}, we conclude that for the above $\delta>0,$ there exist positive constants $C_{13},C_{14}$ as well as $C_{15}$ and $C_{16}$ such that \begin{equation}\label{hhcfvgbhhjui909klopji115} \begin{array}{rl} J_3=&\displaystyle{\int_\Omega v ^{k+1}}\\ \leq &\displaystyle{C_{13}\\|v \\|^{k+1}_{L^\infty(\Omega)}}\\ \leq &\displaystyle{C_{14}(\\|\nabla v \\|^{k+1}_{L^{N+1}(\Omega)}+1)}\\ \leq &\displaystyle{C_{15}(\\|\nabla v \\|^{k+1}_{L^{2\beta}(\Omega)}+1)}\\ \leq &\displaystyle{\delta\\|\nabla v \\|^{2\beta}_{L^{2\beta}(\Omega)}+C_{16}}\\ \end{array} \end{equation} for all $\beta\geq\bar{\beta}>N+1.$ Now, inserting \dref{cz2.57151hhkkhhhjukildrftdfrtgyhu}, \dref{cz2.57151hhkkhhhjukildrftdfrtgyhugg}--\dref{hhcfvgbhhjui909klopji115} into \dref{1234hjui909klopji115} and using the Young inequality and choosing $\delta$ small enough yields to \begin{equation}\label{dfvvfdcvfbhjui909klopji115} \begin{array}{rl} &\displaystyle{\frac{d}{dt}(\frac{1}{k}\\| u \\|^{k}_{L^k(\Omega)}+\frac{1}{{2\beta}}\\|\nabla v \\|^{{{2\beta}}}_{L^{{2\beta}}(\Omega)})+\frac{3(\beta-1)}{8{\beta^2}}\displaystyle\int_{\Omega}\left\|\nabla \|\nabla v \|^{\beta}\right\|^2+\frac{\mu}{2}\int_{\Omega} u ^{k+1}}\\ &+\displaystyle{\frac{1}{2}\displaystyle\int_\Omega \|\nabla v \|^{2\beta-2}\|D^2v \|^2+\displaystyle\frac{1}{2}\int_{\Omega} \|\nabla v \|^{2\beta}+\frac{(k-1) C_D }{8}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}}\\ \leq&\displaystyle{C_{16}~~\mbox{for all}~~ t\in(0,T_{max})}\\ \end{array} \end{equation} and some positive constant $C_{16}$. Therefore, letting $y:=\displaystyle\int_{\Omega} u ^{k} +\displaystyle\int_{\Omega} \|\nabla v \|^{2\beta} $ in \dref{dfvvfdcvfbhjui909klopji115} yields to $$\frac{d}{dt}y(t)+C_{17}y(t)\leq C_{18}~~\mbox{for all}~~ t\in(0,T_{max}).$$ Thus a standard ODE comparison argument implies boundedness of $y(t)$ for all $t\in (0, T_{max})$. Clearly, $\\| u (\cdot, t)\\|_{L^k(\Omega)}$ and $\\|\nabla v (\cdot, t)\\|_{L^{2\beta}(\Omega)}$ are bounded for all $t\in (0, T_{max})$. Obviously, $\lim_{\beta\rightarrow+\infty}\displaystyle{ \frac{N(1-\frac{4}{\beta})}{1+\frac{N}{2q_{0,}}-\frac{4N}{\beta}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})}= \lim_{\beta\rightarrow+\infty}\frac{2N(m -1)q_{0,}}{N-2q_{0,}}(\frac{2}{N}-1+\frac{\beta}{q_{0,}})+2-m-\frac{2}{N}=+\infty,$ hence, the boundedness of $\\| u (\cdot, t)\\|_{L^k(\Omega)}$ and the H\"{o}lder inequality implies the results. \end{proof} Employing Lemmas \ref{lemma456ssddfff30}--\ref{lemmaddff45ssdd630}, we can prove the following lemma. \begin{lemma}\label{sddlemmddffa45630} Let $\Omega\subset \mathbb{R}^N(N\geq1)$ be a bounded domain with smooth boundary. Furthermore, assume that $m>\frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. Then for any $k>1$, there exists a positive constant $C$ such that \begin{equation} \\|u (\cdot, t)\\|_{L^k(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}). \label{zjscz2.529ddff7xssdd9630xxy} \end{equation} \end{lemma} Along with the Duhamel's principle and $L^p$-$L^q$ estimates for Neumann heat semigroup, the above lemma yields the following Lemma. \begin{lemma}\label{lemdffmddffa45630} Let $(u , v ,w )$ be the solution of the problem \dref{7101.2x19x3sss189}. Then \begin{equation} \\|v (\cdot, t)\\|_{W^{1,\infty}(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}) \label{zjscz2.ssdd529ddff7xssdd9630xxy} \end{equation} \end{lemma} \begin{proof} By Duhamel's principle, we see that the solution $v $ can be expressed as follows \begin{equation} v (t)=e^{t(\Delta-1)}v_0 +\int_{0}^{t}e^{(t-s)(\Delta-1)}v (s) ds,~ t\in(0, T_{max}), \label{5555hhjjjfghbnmcz2.5ghju48cfg924ghyuji} \end{equation} where $(e^{ t \Delta})_{t\geq0}$ is the Neumann heat semigroup in $\Omega$. Using Lemma \ref{sddlemmddffa45630}, we follow the $L^p$-$L^q$ estimates for Neumann heat semigroup to obtain for any $t\in(0,T_{max})$, we obtain \begin{equation} \begin{array}{rl} &\\| v (\cdot, t)\\|_{W^{1,\infty}(\Omega)}\\ \leq&\displaystyle{e^{- t}\\| \nabla v_0\\|_{L^{\infty}(\Omega)}+\int_{0}^t(t-s)^{-\frac{N}{2}(\frac{1}{2N}-\frac{1}{2})}e^{-(t-s)} \\|u (\cdot,s)\\|_{L^{2N}(\Omega)}ds}\\ \leq&\displaystyle{C_{1}~~ \mbox{for all}~~ t\in(0,T_{max}).}\\ \end{array} \label{zjccffgbhjcvvvbscz2.5297x96301ku} \end{equation} This lemma is proved. \end{proof} Applying Lemma \ref{sddlemmddffa45630} and Lemma \ref{lemdffmddffa45630}, a straightforward adaptation of the well-established Moser-type iteration procedure \cite{Alikakos72} allows us to formulate a general condition which is sufficient for the boundedness of $u $. \begin{lemma}\label{lemdffssddmddffa45630} Let $(u , v ,w )$ be the solution of the problem \dref{7101.2x19x3sss189}. Then there exists a positive constant $C$ such that \begin{equation} \\|u (\cdot, t)\\|_{L^{\infty}(\Omega)}\leq C ~~\mbox{for all}~~ t\in(0, T_{max}). \label{zjscz2.ssdd529ddff7xssddsddd9630xxy} \end{equation} \end{lemma} \begin{proof} Firstly, by Lemma \ref{sddlemmddffa45630}, we obtain that for any $k > 0$, \begin{equation} \\|u (\cdot,t)\\|_{L^k(\Omega)}\leq \alpha_k~~\mbox{for all}~~ t\in(0, T_{max}), \label{zjscz2.5297x9630111rrddtt} \end{equation} where $\alpha_k$ depends on $k$. Multiplying the first equation of \dref{7101.2x19x3sss189} by $ku^{k-1} $ with $k\geq\max\{2m, N+2\}$, integrating it over $\Omega$, then using the boundary condition $\frac{\partial u }{\partial \nu}=0$ and combining with Lemma \ref{lemdffmddffa45630}, we obtain \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+k (k-1) C_D \int_{\Omega} (u +\varepsilon)^{m-1}u ^{k-2}\|\nabla u \|^2{}+ \mu k\int_{\Omega} u ^{k+1}+\int_{\Omega} u ^k{}} \\ \leq&\displaystyle{k(k-1)\chi\int_{\Omega} u ^{k-1}\nabla u \cdot\nabla v -\xi\int_\Omega \nabla\cdot( u \nabla w ) u ^{k-1}+(\mu k+1)\int_{\Omega} u ^k}\\ \leq&\displaystyle{k(k-1)\chi\int_{\Omega} u ^{k-1}\nabla u \cdot\nabla v }\\ &\displaystyle{+\frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u^k v dx+\kappa \frac{({k-1})}{k}\xi \int_\Omega u^k dx+(\mu k+1)\int_{\Omega} u ^k}\\ \leq&\displaystyle{\frac{k (k-1) C_D }{2}\int_{\Omega} (u +\varepsilon)^{m-1}u ^{k-2}\|\nabla u \|^2+C_1k\int_{\Omega} u ^{k}+C_2k^2\int_{\Omega} u ^k(u +\varepsilon)^{1-m},}\\ \end{array} \label{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdsddffjj} \end{equation} by using \dref{cz2.563019rrtttt12}, where $C_{1}>0,C_{2}>0$, as all subsequently appearing constants $C_i(i= 3, 4, \ldots)$ are independent of $k$. In what follows, we estimate the last two terms of \dref{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdsddffjj}. When $m\geq 1$, then, $$(u +\varepsilon)^{1-m}\leq u ^{1-m}~~~\mbox{and}~~~(u +\varepsilon)^{m-1}\geq u ^{m-1},$$ so that, by \dref{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdsddffjj}, we have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{k (k-1) C_D }{2}\int_{\Omega} u ^{m+k-3}\|\nabla u \|^2{}+\int_{\Omega} u ^k{}} \\ \leq&\displaystyle{C_1k\int_{\Omega} u ^{k}+C_2k^2\int_{\Omega} u ^{k+1-m}.}\\ \end{array} \label{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdssddsddffjj} \end{equation} In order to take full advantage of the dissipated quantities appearing on the left-hand side herein, for any $\delta>0,$ we first invoke the Gagliardo-Nirenberg inequality which provides $C_3>0,C_4 > 0$ as well as $C_5 > 0$ and $C_6 > 0$ such that \begin{equation} \begin{array}{rl} &\displaystyle{C_1k \\| u \\|^{k}_{L^{k}(\Omega)}}\\ =&\displaystyle{ C_1k \\| u ^{\frac{{k}+m-1}{2}}\\|^{\frac{2k}{{k}+m-1}}_{L^{\frac{{2k}}{{k}+m-1}}(\Omega)}} \\ \leq&\displaystyle{C_{1}k\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2Nk}{(N+2){k}+2N(m-1)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2k}{{k}+m-1}-\frac{2Nk}{(N+2){k}+2N(m-1)}}+C_{3}k\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2k}{{k}+m-1}}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{2}+C_4 k^{\frac{(N+2)k+2N(m-1)}{2k+2N(m-1)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{k[2k+N(m-1)]}{({k}+m-1)(k+N(m-1))}}+C_{3}k\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2k}{{k}+m-1}}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{2}+C_4 k^{\frac{N+2}{2}} \\| u \\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))}}+C_{3}k\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k}}\\ \end{array} \label{czwsdc2.563022222ikoplssd2sdfgggjjkkk66} \end{equation} and \begin{equation} \begin{array}{rl} &\displaystyle{C_2k^2 \\| u \\|^{k+1-m}_{L^{k+1-m}(\Omega)}}\\ =&\displaystyle{ C_2k^2 \\| u ^{\frac{{k}+m-1}{2}}\\|^{\frac{2(k+1-m)}{{k}+m-1}}_{L^{\frac{{2(k+1-m)}}{{k}+m-1}}(\Omega)}} \\ \leq&\displaystyle{C_{5}k^2\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{\frac{2N[k-(2m-2)]}{(N+2){k}+2N(m-1)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2(k+1-m)}{{k}+m-1}-\frac{2N[k-(2m-2)]}{(N+2){k}+2N(m-1)}}+C_{5}k^2\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2(k-1+m)}{{k}+m-1}}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{2}+C_6 k^{\frac{(N+2)k+2N(m-1)}{k+2N(m-1)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{k[k+2(m-1)]}{({k}+m-1)(\frac{k}{2}+N(m-1))}}+C_{5}k^2\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2(k-1+m)}{{k}+m-1}}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(\Omega)}^{2}+C_6 k^{N+2}\\| u \\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{5}k^2\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k+1-m}.}\\ \end{array} \label{czwsdc2.563022222ikoplddffssd2sdfgggjjkkk66} \end{equation} Taking $\delta$ appropriately small, and substituting the above two inequalities into \dref{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdssddsddffjj}, we obtain \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\int_{\Omega} u ^k{}} \\ \leq&\displaystyle{C_7 k^{\frac{N+2}{2}} \\| u \\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))}}+C_8 k^{N+2}\\| u \\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{9}k\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k}+C_{10}k^2\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k+1-m}.}\\ \end{array} \label{vgbhnsxcdvfddfcssddz2.5xxdsddddddf1jjssdssddsddffjj} \end{equation} Notice that $${\frac{k[k+2(m-1)]}{(k+2N(m-1))}}\leq\max\{\frac{k[2k+N(m-1)]}{(2k+2N(m-1))},k+1-m\}\leq k,$$ we further obtain \begin{equation} \begin{array}{rl} \displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\int_{\Omega} u ^k{}} \leq&\displaystyle{C_{11} k^{N+2}\\| u \\|_{L^{\frac{k}{2}}(\Omega)}^{\frac{k[k+2(m-1)]}{(k+2N(m-1))}}+C_{12}k^{N+2}\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k}.}\\ \end{array} \label{vgbhnsxcdvfddfcsssddsddz2.5xxdsddddddf1jjssdssddsddffjj} \end{equation} In what follows, we use Moser iteration method to show the $L^\infty$ estimate of $u $. Take $k_i= 2k_{i-1} = 2^ik_0 , k_0 = 2m$, $M_i =\sup_{t\in(0,T_{max})}\int_{\Omega}u ^{\frac{{k_i}}{2}}$, then when $m\geq 1$, we have \begin{equation} \begin{array}{rl} \displaystyle{M_i}\leq&\displaystyle{\max\{\lambda^iM_{i-1}^2,\\|1+u_0\\|_{L^\infty(\Omega)}^{k_i}\}.}\\ \end{array} \label{vgbhnsxcdvfddfcsssddsddz2.5xxdsdddddssdddf1jjssdssddsddffjj} \end{equation} with some $\lambda> 1.$ Now if $\lambda^iM_{i-1}^2\leq\\|1+u_0\\|_{L^\infty(\Omega)}^{k_i}$ for infinitely many $i\geq 1$, we get (3.33) with $C = \\|1+u_0\\|_{L^\infty(\Omega)}$. Otherwise $M _i\leq \lambda^iM_{i-1}^2$ for all $i = 0, 1, \ldots$, so $\ln M_i \leq i \ln \lambda +2 \ln M_{i-1}$. By induction, we get $$\ln M_i \leq (i+2)\ln \lambda +2^i(\ln M_{0}+2\ln \lambda)$$ and thus $$M_i\leq \lambda^{i+2+2^i}M_0^{2^i}. $$ From this, it follows that \dref{zjscz2.ssdd529ddff7xssddsddd9630xxy} is valid with some positive constant. When $0< m<1$, noticing that $u^k (u +\varepsilon)^{1-m}\leq u ^{k+1-m} +\varepsilon^{1-m} u ^{k}$, then by \dref{zjscz2.5297x9630111rrddtt} and $\varepsilon\in(0,1)$, we derive from Lemma \ref{lemdffmddffa45630} that \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+\frac{k (k-1) C_D }{2^{2-m}}\int_{u >\varepsilon} u ^{m+k-3}\|\nabla u \|^2{}+\mu k\int_{\Omega} u ^{k+1}+\mu k\int_{\Omega} u ^{k}{}} \\ \leq&\displaystyle{C_{13}k^2\int_{\Omega} u ^{k}+C_{14}k^2\int_{\Omega} u ^{k+1-m}}\\ \leq&\displaystyle{C_{13}k^2\int_{u \leq\varepsilon} u ^{k}+C_{13}k^2\int_{u >\varepsilon} u ^{k}+C_{14}k^2\int_{u \leq \varepsilon} u ^{k+1-m}+C_{14}k^2\int_{u >\varepsilon} u ^{k+1-m}}\\ \leq&\displaystyle{C_{13}k^2\int_{u \leq \varepsilon} u ^{\frac{k}{2}}+C_{14}k^2\int_{u \geq\varepsilon} u ^{k}+C_{14}k^2\int_{u \geq \varepsilon} u ^{k+1-m}.}\\ \end{array} \label{vgbhnsxcdvfddfcz2ssdd.5xxdsddddddf1jjssdssddsddffjj} \end{equation} Denote $J(t)=\{x\in u >\varepsilon\}$. By virtue of the Gagliardo-Nirenberg interpolation inequality and \dref{zjscz2.5297x9630111rrddtt}, we obtain $$\\|u \\|^k_{L^k(J(t))}\leq\\|u \\|_{L^3(J(t))}\\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\leq C_{15} \\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}$$ and $$\\|u \\|^{k+1-m}_{L^{k+1-m}(J(t))}\leq\\|u \\|_{L^{3(2-m)}(J(t))}^{2-m}\\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\leq C_{16} \\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}.$$ Substituting the above two inequalities into \dref{vgbhnsxcdvfddfcz2.5xxdsddddddf1jjssdsddffjj}, we obtain \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+k (k-1) C_D \int_{J(t)} (u +\varepsilon)^{m-1}u ^{k-2}\|\nabla u \|^2{}+ \mu k\int_{\Omega} u ^{k+1}+\int_{\Omega} u ^k{}} \\ \leq&\displaystyle{C_{17}k^2\int_{\Omega} u ^{\frac{k}{2}}+C_{18}k^2\\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}.}\\ \end{array} \label{vgbhnsxcdvfddfczssdd2.5xxdsddddddf1jjssdsddffjj} \end{equation} Using the Gagliardo-Nirenberg interpolation inequality again, it follows $$\begin{array}{rl} &C_{18}k^2\\|u \\|^{k-1}_{L^{\frac{3(k-1)}{2}}(J(t))}\\ =&C_{18}k^2\\|u ^{\frac{{k}+m-1}{2}}\\|^{\frac{2(k-1)}{m+k-1}}_{L^{\frac{3(k-1)}{k+m-1}}(J(t))}\\ \leq&\displaystyle{C_{19}k^2\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(J(t))}^{\frac{4(2k-N)}{(N+2){k}+2N(m-1)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2(k-1)}{m+k-1}-\frac{4(2k-N)}{(N+2){k}+2N(m-1)}}+C_{19}k^2\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2(k-1)}{{k}+m-1}}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(J(t))}^{2}+C_{20} k^{\frac{2[(N+2){k}+2N(m-1)]}{(N-2)k+2Nm}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2k[k(N-2)+(2N-4)m+2-N]}{(m+k-1)[(N-2){k}+2Nm]}}+C_{19}k^{2}\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k-1}}\\ \leq&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(J(t))}^{2}+C_{20} k^{\frac{2(N+2)}{(N-2)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{{k}+m-1}(\Omega)}^{\frac{2k[k(N-2)+(2N-4)m+2-N]}{(m+k-1)[(N-2){k}+2Nm]}}+C_{19}k^{2}\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k-1}}\\ =&\displaystyle{\delta\\|\nabla u ^{\frac{{k}+m-1}{2}}\\|_{L^2(J(t))}^{2}+C_{20} k^{\frac{2(N+2)}{(N-2)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{2}(\Omega)}^{\frac{k[k(N-2)+(2N-4)m+2-N]}{[(N-2){k}+2Nm]}}+C_{19}k^{2}\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k-1},}\\ \end{array} $$ substituting this inequality into \dref{vgbhnsxcdvfddfczssdd2.5xxdsddddddf1jjssdsddffjj} gives \begin{equation} \begin{array}{rl} &\displaystyle{\frac{d}{dt}\\|u \\|^{k}_{L^k(\Omega)}+k (k-1) C_D \int_{J(t)} (u +\varepsilon)^{m-1}u ^{k-2}\|\nabla u \|^2{}+ \mu k\int_{\Omega} u ^{k+1}+\int_{\Omega} u ^k{}} \\ \leq&\displaystyle{C_{21}k^2\int_{\Omega} u ^{\frac{k}{2}}+C_{23} k^{\frac{2(N+2)}{(N-2)}}\\| u ^{\frac{{k}+m-1}{2}}\\|_{L^\frac{k}{2}(\Omega)}^{\frac{k[k(N-2)+(2N-4)m+2-N]}{[(N-2){k}+2Nm]}}+C_{22}k^{2}\\| u \\|_{L^\frac{k}{2}(\Omega)}^{k-1}.}\\ \end{array} \label{vgbhnsxcdvfddfczssdd2.5xssdxdsddddddf1jjssdsddffjj} \end{equation} Similarly to the case $m\geq 1$, and we obtain \dref{zjscz2.ssdd529ddff7xssddsddd9630xxy}. \end{proof} By virtue of \dref{1.163072x} and Lemmas \ref{lemdffmddffa45630}--\ref{lemdffssddmddffa45630}, we are now in a position to prove Theorem \ref{theorem3}. {\bf Proof of Theorem \ref{theorem3}.} \begin{proof} From \dref{1.163072x} and Lemmas \ref{lemdffmddffa45630}--\ref{lemdffssddmddffa45630}, we derive that there exists a constant $C > 0$ independent of $\varepsilon$ such that \begin{equation} \\|u (\cdot,t)\\|_{L^\infty(\Omega)} +\\|v (\cdot,t)\\|_{W^{1,\infty}(\Omega)} \leq C ~~\mbox{for all}~~ t\in(0,T_{max}). \label{zjscz2.5297x9630111kkuu} \end{equation} Suppose on the contrary that $T_{max}< \infty$, then \dref{zjscz2.5297x9630111kkuu} contradicts to the blow-up criterion \dref{1.163072x}, which implies $T_{max}= \infty$. Therefore, the classical solution $(u,v,w)$ is global in time and bounded. \end{proof} \section{Proof of Theorem \ref{theossdrem3}} The goal of this section is to prove theorem \ref{theossdrem3}. In the absence of \dref{91ssdd61}, the first equation in system \dref{7101.2x19x3sss189} may be degenerate, so system \dref{7101.2x19x3sss189} might not have classical solutions. Our goal is to construct solutions of \dref{7101.2x19x3sss189} as limits of solutions to appropriately regularized problems. To this end, we approximate the diffusion coefficient function in \dref{7101.2x19x3sss189} by a family $(D_\varepsilon)_{\varepsilon\in(0,1)}$ of functions $$D_\varepsilon\in C^2((0,\infty))~ \mbox{such that}~~ D_\varepsilon(u)\geq\varepsilon ~\mbox{for all}~ u > 0 $$$$~~\mbox{and}~ D(u) \leq D_\varepsilon(u)\leq D(u) + 2\varepsilon~ \mbox{for all}~ u > 0 ~\mbox{and}~\varepsilon\in (0, 1).$$ Therefore, for any $\varepsilon\in(0,1)$, the regularized problem of \dref{7101.2x19x3sss189} is presented as follows \begin{equation} \left\{\begin{array}{ll} u_{\varepsilon t}=\nabla\cdot(D_\varepsilon(u_\varepsilon)\nabla u_\varepsilon)-\chi\nabla\cdot(u_\varepsilon\nabla v_\varepsilon)-\xi\nabla\cdot (u_\varepsilon\nabla w_\varepsilon)+\mu u_\varepsilon(1-u_\varepsilon-w_\varepsilon),\quad x\in \Omega, t>0,\\ \displaystyle{v_{\varepsilon t}=\Delta v_\varepsilon +u_\varepsilon- v_\varepsilon},\quad x\in \Omega, t>0,\\ \displaystyle{w_{\varepsilon t}=- v_\varepsilon w_\varepsilon },\quad x\in \Omega, t>0,\\ \displaystyle{\frac{\partial u_\varepsilon}{\partial \nu}=\frac{\partial v_\varepsilon}{\partial \nu}=\frac{\partial w_\varepsilon}{\partial \nu}=0},\quad x\in \partial\Omega, t>0,\\ \displaystyle{u_\varepsilon(x,0)=u_0(x)},v_\varepsilon(x,0)=v_0(x),w_\varepsilon(x,0)=w_0(x),\quad x\in \Omega.\\ \end{array}\right.\label{5555710ssdd1.2x19x3189} \end{equation} We are now in the position to construct global weak solutions for \dref{7101.2x19x3sss189}. Before going into details, let us first give the definition of weak solution. \begin{definition}\label{df1} Let $T\in(0,\infty]$, and $\Omega\subset R^N$ be a bounded domain with smooth boundary. A triple $(u, v, w)$ of nonnegative functions defined on $\Omega\times(0,T)$, is called a weak solution to \dref{7101.2x19x3sss189} on $[0, T )$ if \begin{equation} \begin{array}{ll} \mbox{(i)}~~ u\in L_{loc}^2([0,T);L^2(\Omega)),~~ v \in L_{loc}^2([0,T); W^{1,2}(\Omega)),~~ w \in L_{loc}^2([0,T); W^{1,2}(\Omega));\\ \end{array}\label{dffff1.1fghyuisdakkklll} \end{equation} \begin{equation}\label{726291hh} \begin{array}{rl} \mbox{(ii)}~~ H(u) \in L^1_{loc}(\bar{\Omega}\times [0, T)),~~u\nabla v~~\mbox{and}~~u\nabla w~~ \in L^1_{loc}(\bar{\Omega}\times [0, T);\mathbb{R}^{N}); \end{array} \end{equation} (iii) $(u, v, w)$ satisfies \dref{7101.2x19x3sss189} in the sense that for every $\varphi\in C_0^{\infty} (\bar{\Omega}\times[0, T))$ \begin{equation} \begin{array}{rl}\label{eqx45xx12112ccgghh} &\displaystyle{-\int_0^{T}\int_{\Omega}u\varphi_t-\int_{\Omega}u_0\varphi(\cdot,0) } \\ =&\displaystyle{ \int_0^T\int_{\Omega}H(u)\Delta\varphi+\chi\int_0^T\int_{\Omega}u \nabla v\cdot\nabla\varphi+\xi\int_0^T\int_{\Omega}u \nabla w\cdot\nabla\varphi} +\displaystyle{\int_0^T\int_{\Omega}(\mu u-uw-\mu u^2)\varphi;} \end{array} \end{equation} holds as well as \begin{equation} \begin{array}{rl}\label{eqx45xx12112ccgghhjj} &\displaystyle{-\int_0^{T}\int_{\Omega}v\varphi_t-\int_{\Omega}v_0\varphi(\cdot,0)=- \int_0^T\int_{\Omega}\nabla v\cdot\nabla\varphi-\int_0^T\int_{\Omega}v\varphi+\int_0^T\int_{\Omega}u\varphi} \end{array} \end{equation} and \begin{equation} \begin{array}{rl}\label{eqx45xx1sddd2112ccgghhjj} &\displaystyle{-\int_0^{T}\int_{\Omega}w\varphi_t-\int_{\Omega}w_0\varphi(\cdot,0)=- \int_0^T\int_{\Omega}vw\varphi,} \end{array} \end{equation} where we let \begin{equation} H(s)=\int_0^sD(\sigma)d\sigma~~\mbox{for}~~s\geq0. \label{zjscz2.5297x963ssddd0111kkuu} \end{equation} In particular, if $T=\infty$ can be taken, then $(u, v, w)$ is called a global-in-time weak solution to \dref{7101.2x19x3sss189}. We proceed to establish the main step towards the boundedness of weak solutions to \dref{7101.2x19x3sss189}. To this end, firstly, from \dref{1.163072x} and Lemmas \ref{lemdffmddffa45630}--\ref{lemdffssddmddffa45630}, we can easily derive the following estimates for $u_\varepsilon$ and $v_\varepsilon$, which plays an important role in proving Theorem \ref{theossdrem3}. \begin{lemma}\label{lemma45630hhuujjuu} Let $m> \frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. Then one can find $C > 0$ independent of $\varepsilon\in(0, 1)$ such that \begin{equation} \\|u_\varepsilon(\cdot,t)\\|_{L^\infty(\Omega)} \leq C ~~\mbox{for all}~~ t\in(0,\infty) \label{zjscz2.5297x9630111kkuu} \end{equation} and \begin{equation} \\|v_\varepsilon(\cdot,t)\\|_{W^{1,\infty}(\Omega)} \leq C ~~\mbox{for all}~~ t\in(0,\infty). \label{zjscz2.5297x9630111kkhhii} \end{equation} \end{lemma} Now with the above boundedness information at hand, we may invoke standard parabolic regularity to obtain the H\"{o}lder regularity properties. \begin{lemma}\label{lemma45630hhuujjuuyy} Let $m> \frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. Then for any $\varepsilon\in(0, 1)$, one can find $\mu\in(0, 1)$ such that for some $C > 0$ \begin{equation} \\|v_\varepsilon(\cdot,t)\\|_{C^{\mu,\frac{\mu}{2}}(\Omega\times[t,t+1])} \leq C ~~\mbox{for all}~~ t\in(0,\infty), \label{zjscz2.5297x9630111kkhhiioo} \end{equation} and such that for any $\tau> 0$ there exists $C(\tau) > 0$ fulfilling \begin{equation} \\|\nabla v_\varepsilon(\cdot,t)\\|_{C^{\mu,\frac{\mu}{2}}(\Omega\times[t,t+1])} \leq C ~~\mbox{for all}~~ t\in(\tau,\infty). \label{zjscz2.5297x9630111kkhhffrreerrpphh} \end{equation} \end{lemma} \begin{proof} Firstly, by Lemma \ref{lemma45630hhuujjuu}, we derive that $ g_\varepsilon $ is bounded in $L^{\infty} (\Omega\times(0, \infty))$, where $g_\varepsilon (x, t) := -v_\varepsilon(x,t) +u_\varepsilon(x,t) $ for all $(x,t)\in \Omega\times(0, \infty)$. Therefore, in view of the standard parabolic regularity theory to the second equation of \dref{5555710ssdd1.2x19x3189}, one has \dref{zjscz2.5297x9630111kkhhiioo} and \dref{zjscz2.5297x9630111kkhhffrreerrpphh} hold. \end{proof} To achieve the convergence result, we need to derive some regularity properties of time derivatives. \begin{lemma}\label{lemma45630hhuujjuuyytt} Let $m> \frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. Moreover, let $\varsigma> m$ and $\varsigma\geq 2(m - 1)$. Then for all $T > 0$ and $\varepsilon\in(0,1)$, there exists $C(T) > 0$ such that \begin{equation} \int_0^T\\|\partial_tu_\varepsilon^\varsigma(\cdot,t)\\|_{(W^{2,q}(\Omega))^}dt \leq C(T) \label{zjscz2.5297x9630111kkhhiioott4} \end{equation} and \begin{equation} \\|w_\varepsilon(\cdot,t)\\|_{W^{1,\infty}(\Omega)} \leq C(T). \label{zjsczsdddfff2.5297x9630111kkhhii} \end{equation} \end{lemma} \begin{proof} Firstly, with the help of Lemma \ref{lemma45630hhuujjuu}, for all $\varepsilon\in(0,1),$ we can fix a positive constants $C_1$ such that \begin{equation} u_\varepsilon\leq C_1~~\mbox{and}~~\|\nabla v_\varepsilon\| \leq C_1 ~~\mbox{in}~~ \Omega\times(0,\infty). \label{gbhnzjscz2.5297x9630111kkhhiioo} \end{equation} Next, we will prove \dref{zjscz2.5297x9630111kkhhiioott4}. To this end, for any fixed $\psi\in C_0^{\infty}(\Omega)$, multiplying the first equation by $u^{\varsigma-1}_{\varepsilon}\psi$, we have \begin{equation} \begin{array}{rl} &\displaystyle\frac{1}{\varsigma}\int_{\Omega}\partial_{t}u^{\varsigma}_{\varepsilon}(\cdot,t)\cdot\psi\\ =&\displaystyle{\int_{\Omega}u^{\varsigma-1}_{\varepsilon}\left[\nabla\cdot(D_\varepsilon(u_\varepsilon)\nabla u_\varepsilon)-\chi\nabla\cdot(u_\varepsilon\nabla v_\varepsilon)-\xi\nabla\cdot(u_\varepsilon\nabla w_\varepsilon)+\mu u_\varepsilon(1-w_\varepsilon-u_\varepsilon)\right]\cdot\psi } \\ =&\displaystyle{-(\varsigma-1)\int_\Omega u_{\varepsilon}^{\varsigma-2}D_\varepsilon(u_\varepsilon)\|{\nabla} {n}_{\varepsilon}\|^2\psi-\int_\Omega u_{\varepsilon}^{\varsigma-1}D_\varepsilon(u_\varepsilon){\nabla} {u}_{\varepsilon}\cdot\nabla\psi }\\ &+\displaystyle{(\varsigma-1)\chi\int_\Omega u_{\varepsilon}^{\varsigma-1}\nabla u_{\varepsilon}\cdot\nabla v_{\varepsilon}\psi+\chi\int_\Omega u_{\varepsilon}^{\varsigma} \nabla v_{\varepsilon}\cdot\nabla\psi}\\ &+\displaystyle{(\varsigma-1)\xi\int_\Omega u_{\varepsilon}^{\varsigma-1}\nabla u_{\varepsilon}\cdot\nabla w_{\varepsilon}\psi+\xi\int_\Omega u_{\varepsilon}^{\varsigma} \nabla w_{\varepsilon}\cdot\nabla\psi}\\ &+\displaystyle{\mu\int_\Omega u_{\varepsilon}^{\varsigma} \psi-\mu\int_\Omega u_{\varepsilon}^{\varsigma}w_{\varepsilon} \psi-\mu\int_\Omega u_{\varepsilon}^{\varsigma+1} \psi~~\mbox{for all}~~t\in(0,\infty).}\\ \end{array} \label{cvbmdcfvgcz2.5ghju48} \end{equation} Next, we will estimate the right-hand sides of \dref{cvbmdcfvgcz2.5ghju48}. To this end, assuming that $p := \varsigma-m + 1$, then $\varsigma > m$ and $\varsigma\geq 2(m -1)$ yield to $p > 1$ and $p\geq m- 1$. Since \dref{gbhnzjscz2.5297x9630111kkhhiioo}, we integrate \dref{cz2.5xx1jjjj} with respect to $t$ over $(0, T)$ for some fixed $T > 0$ and then have \begin{equation} \begin{array}{rl} &\displaystyle{\frac{1}{{p}}\int_{\Omega}u_{\varepsilon}^{{{p}}}(\cdot,T)+ \frac{C_D(p-1)}{2}\int_{0}^T\int_{\Omega}u_{\varepsilon}^{m+p-3} \|\nabla u_{\varepsilon}\|^2}\\ \leq&\displaystyle{\frac{(p-1)\chi^2}{2C_D}\int_{0}^T\int_\Omega u_{\varepsilon}^{p+1-m}\|\nabla v_{\varepsilon}\|^2+\frac{({k-1})}{k}\xi\\|w_0\\|_{L^\infty(\Omega)}\int_\Omega u_\varepsilon^{k}v_\varepsilon}\\ &\displaystyle{+(\mu+\kappa\xi)\int_{\Omega} u_\varepsilon^{k} +\mu C_1^pT+\frac{1}{{p}}\int_{\Omega}n_{0}^{{{p}}} }\\ \leq&\displaystyle{\frac{(p-1)\chi^2}{2C_D}C_1^{p+3-m}T +\mu C_1^pT+\frac{1}{{p}}\int_{\Omega}n_{0}^{{{p}}} .}\\ \end{array} \label{vbhnjmkvgcz2.5ghhjuyuiihjj} \end{equation} On the other hand, by $p = \varsigma-m + 1$, we have \begin{equation} \int_{0}^T\int_{\Omega}u_{\varepsilon}^{\varsigma-2} \|\nabla u_{\varepsilon}\|^2=\int_{0}^T\int_{\Omega}u_{\varepsilon}^{m+p-3} \|\nabla u_{\varepsilon}\|^2\leq C_2(1+T) \label{fgbhvbhnjmkvgcz2.5ghhjuyuiihjj} \end{equation} for some positive constant $C_2$. Next, by \dref{gbhnzjscz2.5297x9630111kkhhiioo}, we also derive that \begin{equation} \begin{array}{rl} &\displaystyle{\mu\int_\Omega u_{\varepsilon}^{\varsigma} \psi-\mu\int_\Omega u_{\varepsilon}^{\varsigma}w_{\varepsilon} \psi-\mu\int_\Omega u_{\varepsilon}^{\varsigma+1}\psi \leq C_1^\varsigma\|\Omega\|(\mu+\mu\\|w_0\\|_{L^\infty(\Omega)}+\mu C_1)\\|\psi\\|_{L^\infty(\Omega)}}\\ \end{array} \label{fbgnjvbhnjmkvgcz2.5ghhjuyuiihjj} \end{equation}\ for all $\varepsilon\in(0, 1).$ Moreover, by \dref{gbhnzjscz2.5297x9630111kkhhiioo}--\dref{fbgnjvbhnjmkvgcz2.5ghhjuyuiihjj} and the Young inequality, we conclude that there exists $C_3 > 0$ such that \begin{equation} \begin{array}{rl} \displaystyle{\|\int_{\Omega}\partial_{t}u^{\varsigma}_{\varepsilon}(\cdot,t)\cdot\psi\|\leq C_3( \int_{\Omega}u_{\varepsilon}^{\varsigma-2} \|\nabla u_{\varepsilon}\|^2+1)\\|\psi\\|_{W^{1,\infty}(\Omega)}.}\\ \end{array} \label{fghncvbmdcfvgcz2.5ghju48} \end{equation} Due to the embedding $W^{2,q}(\Omega)\hookrightarrow W^{1,\infty}(\Omega)$ for $q > N$, we deduce that there exists $C_4 > 0$ such that \begin{equation} \begin{array}{rl} \displaystyle{\\|\partial_tu_{\varepsilon}^\varsigma(\cdot,t)\\|_{(W^{2,q}(\Omega))^}\leq C_4( \int_{\Omega}u_{\varepsilon}^{\varsigma-2} \|\nabla u_{\varepsilon}\|^2+1)~~\mbox{for all}~~t\in(0,\infty)~~\mbox{and any}~~\varepsilon\in(0,1).}\\ \end{array} \label{ggbhhfghncvbmdcfvgcz2.5ghju48} \end{equation} Now, combining \dref{fgbhvbhnjmkvgcz2.5ghhjuyuiihjj} and \dref{ggbhhfghncvbmdcfvgcz2.5ghju48}, we can get \dref{zjscz2.5297x9630111kkhhiioott4}. Now, observing that the third equation of \dref{5555710ssdd1.2x19x3189} is an ODE, we derive that for any $(x,t)\in\Omega\times(0,\infty),$ \begin{equation} \begin{array}{rl} &\displaystyle{w_\varepsilon(x,t)=w_0(x)e^{-\int_0^t v_\varepsilon(x,s)ds}.}\\ \end{array} \label{vbgncz2.5xx1ffsskkoppgghh512} \end{equation} Hence, by a basic calculation, we conclude that for any $(x,t)\in\Omega\times(0,\infty),$ \begin{equation} \begin{array}{rl} &\displaystyle{\nabla w_\varepsilon(x,t)=\nabla w_0(x)e^{-\int_0^t v_\varepsilon(x,s)ds}-w_0(x)e^{-\int_0^t v_\varepsilon(x,s)ds}\int_0^t \nabla v_\varepsilon(x,s)ds,}\\ \end{array} \label{vbgncz2.5xx1ffsskkopffggpgghh512} \end{equation} which together with \dref{x1.731426677gg} and \dref{gbhnzjscz2.5297x9630111kkhhiioo} implies \dref{zjsczsdddfff2.5297x9630111kkhhii}. \end{proof} \end{definition} We are now in the position to prove our main result on global weak solvability. \begin{lemma}\label{lemma45630223} Assume that $m> \frac{2N}{N+\gamma_}$ with $N\geq1$, where $\gamma_$ is given by \dref{xeerr1.731426677gg}. Then there exists $(\varepsilon_j)_{j\in \mathbb{N}}\subset (0, 1)$ such that $\varepsilon_j\rightarrow 0$ as $j\rightarrow\infty$ and that \begin{equation} u_{\varepsilon}\rightarrow u ~~\mbox{a.e.}~~ \mbox{in}~~ \Omega\times (0,\infty), \label{zjscz2.5297x9630222222} \end{equation} \begin{equation} u_{\varepsilon}\rightharpoonup u ~~\mbox{weakly star in}~~ L^\infty(\Omega\times(0,\infty)), \label{zjscz2.5297x9630222222ee} \end{equation} \begin{equation} v_\varepsilon\rightarrow v ~~\mbox{in}~~ C^0_{loc}(\bar{\Omega}\times[0,\infty)), \label{zjscz2.5297x96302222tt3} \end{equation} \begin{equation} \nabla v_\varepsilon\rightarrow \nabla v ~~\mbox{in}~~ C^0_{loc}(\bar{\Omega}\times[0,\infty)), \label{zjscz2.5297x96302222tt4} \end{equation} \begin{equation} \nabla v_\varepsilon\rightarrow \nabla v ~~\mbox{in}~~ L^{\infty}(\Omega\times(0,\infty)) \label{zjscz2.5297x96302222tt4} \end{equation} as well as \begin{equation} w_\varepsilon\rightharpoonup w ~~\mbox{weakly star in}~~ L^{\infty}(\Omega\times(0,\infty)) \label{zjscz2.5297x96302ssdd222tt4} \end{equation} and \begin{equation} \nabla w_\varepsilon\rightharpoonup \nabla w ~~\mbox{weakly star in}~~ L^{\infty}_{loc}(\Omega\times(0,\infty)) \label{zjscz2.5297x96302ssdd22ssdd2tt4} \end{equation} with some triple $(u, v,w)$ which is a global weak solution of \dref{5555710ssdd1.2x19x3189} in the sense of Definition \ref{df1}. \end{lemma} \begin{proof} Firstly, due to Lemma \ref{lemma45630hhuujjuu} and Lemma \ref{lemma45630hhuujjuuyytt}, for each $T > 0$, we can find $\varepsilon$-independent constant $C(T)$ such that for all $t\in(0,T)$, \begin{equation} \\|u_\varepsilon(\cdot,t)\\|_{L^\infty(\Omega)}+ \\|v_\varepsilon(\cdot,t)\\|_{W^{1,\infty}(\Omega)} + \\|w_\varepsilon(\cdot,t)\\|_{W^{1,\infty}(\Omega)} \leq C(T)~~ \label{zjscz2.ssddd5297x9630111kk} \end{equation} as well as \begin{equation} \int_{0}^T\int_{\Omega}(u_{\varepsilon}+\varepsilon)^{m+p-3} \|\nabla u_{\varepsilon}\|^2\leq C(T)~~~\mbox{for any}~~p>1~~~\mbox{and}~~p\geq m- 1. \label{fvgbhzjscz2.5sss297x96302222tt4455hyuhii} \end{equation} Now, choosing $\varphi\in W^{1,2} (\Omega)$ as a second function in the first equation in \dref{5555710ssdd1.2x19x3189} and using \dref{zjscz2.ssddd5297x9630111kk}, we have $$ \begin{array}{rl} &\displaystyle\left\|\int_{\Omega}(v_{\varepsilon,t})\varphi\right\|\\ =&\displaystyle{\left\|\int_{\Omega}\left[\Delta v_{\varepsilon}-v_{\varepsilon}+u_{\varepsilon}\right]\varphi\right\|} \\ =&\displaystyle{\left\|\int_\Omega \left[-\nabla v_{\varepsilon}\cdot\nabla\varphi+v_{\varepsilon}\varphi+ u_{\varepsilon} \varphi\right]\right\|}\\ \leq&\displaystyle{\left\{\\|\nabla v_{\varepsilon}\\|_{L^{2}(\Omega)}+ \\|v_{\varepsilon}\\|_{L^{2}(\Omega)}+ \\|u_{\varepsilon}\\|_{L^{2}(\Omega)} \right\}}\times\displaystyle{\\|\varphi\\|_{W^{1,2}(\Omega)}}\\ \end{array} $$ for all $t>0$. Along with \dref{zjscz2.ssddd5297x9630111kk}, further implies that \begin{equation} \begin{array}{rl} &\displaystyle\int_0^T\\|\partial_tv_\varepsilon(\cdot,t)\\|_{({W^{1,2}(\Omega)})^}^{2}dt \\ \leq&\displaystyle{\int_0^T\left\{\\|\nabla u_{\varepsilon}\\|_{L^{2}(\Omega)}+ \\|v_{\varepsilon}\\|_{L^{2}(\Omega)}+ \\|n_{\varepsilon}\\|_{L^{2}(\Omega)} \right\}}^{2}dt \\ \leq&\displaystyle{C_1(T),}\\ \end{array} \label{gbhncvbmdcfvgczffghhh2.5ghju48} \end{equation} where $C_1$ is a positive constant independent of $\varepsilon$. Combining estimates \dref{zjscz2.ssddd5297x9630111kk}--\dref{gbhncvbmdcfvgczffghhh2.5ghju48} and the fact that $w_\varepsilon \leq \\|w_0\\|_{L^\infty(\Omega)}$, we can pick a sequence $(\varepsilon_j)_{j\in \mathbb{N}}\subset (0, 1)$ with $\varepsilon=\varepsilon_j\searrow0$ as $j\rightarrow\infty$ such that \dref{zjscz2.5297x9630222222ee}--\dref{zjscz2.5297x96302ssdd22ssdd2tt4} are valid with certain limit functions $u,v$ and $w$ belonging to the indicated spaces. We next fix $\varsigma> m$ satisfying $\varsigma\geq2(m-1)$ and set $p := 2\zeta-m+1$, then by \dref{fvgbhzjscz2.5sss297x96302222tt4455hyuhii} implies that for each $T > 0,$ $(u_{\varepsilon}^\varsigma)_{\varepsilon\in(0,1)}$ is bounded in $L^2((0, T);W^{1,2}(\Omega))$. With the help of Lemma \ref{lemma45630hhuujjuuyytt}, we also show that $$(\partial_{t}u_{\varepsilon}^\varsigma)_{\varepsilon\in(0,1)}~~\mbox{is bounded in}~~L^1((0, T); (W^{2,q}(\Omega))^*)~~\mbox{for each}~~ T > 0$$ and some $q>N.$ Hence, an Aubin-Lions lemma (see e.g. \cite{Simon}) applies to the above inequality we have the strong precompactness of $(u_{\varepsilon}^\varsigma)_{\varepsilon\in(0,1)}$ in $L^2(\Omega\times(0, T))$. Therefore, we can pick a suitable subsequence such that $u_{\varepsilon}^\varsigma\rightarrow z^\varsigma$ for some nonnegative measurable $z:\Omega\times(0,\Omega)\rightarrow\mathbb{R}$. In light of \dref{zjscz2.5297x9630222222ee} and the Egorov theorem, we have $z = u$ necessarily, so that \dref{zjscz2.5297x9630222222} is valid. Next we shall prove that $(u,v,w)$ is a weak solution of problem \dref{7101.2x19x3sss189}. To this end, multiplying the first equation as well as the second equation and third equation in \dref{5555710ssdd1.2x19x3189} by $\varphi\in C^\infty_0(\Omega\times [0,\infty))$, we obtain \begin{equation} \begin{array}{rl}\label{eqx4ss5xx12112ccgghh} \displaystyle{-\int_0^{\infty}\int_{\Omega}u_\varepsilon\varphi_t-\int_{\Omega}u_0\varphi(\cdot,0) }=&\displaystyle{ \int_0^\infty\int_{\Omega}H(u_\varepsilon+\varepsilon)ds\Delta\varphi+\chi\int_0^\infty\int_{\Omega}u_\varepsilon\nabla v_\varepsilon\cdot\nabla\varphi}\\ &+\displaystyle{\xi\int_0^\infty\int_{\Omega}u_\varepsilon\nabla w_\varepsilon\cdot\nabla\varphi+\int_0^{\infty}\int_{\Omega}(\mu u_\varepsilon-\mu u_\varepsilon w_\varepsilon- \mu u_\varepsilon^2)\varphi}\\ \end{array} \end{equation} as well as \begin{equation} \begin{array}{rl}\label{eqx45xsssx12112ccgghhjj} &\displaystyle{-\int_0^{\infty}\int_{\Omega}v_\varepsilon\varphi_t-\int_{\Omega}v_0\varphi(\cdot,0)=- \int_0^\infty\int_{\Omega}\nabla v_\varepsilon\cdot\nabla\varphi-\int_0^\infty\int_{\Omega}v_\varepsilon\varphi+\int_0^\infty\int_{\Omega}u_\varepsilon\varphi} \end{array} \end{equation} and \begin{equation} \begin{array}{rl}\label{sddddfgghh} &\displaystyle{-\int_0^{\infty}\int_{\Omega}w_\varepsilon\varphi_t-\int_{\Omega}w_0\varphi(\cdot,0)=- \int_0^\infty\int_{\Omega}v_\varepsilon w_\varepsilon\varphi.} \end{array} \end{equation} for all $\varepsilon\in (0,1)$, where $H$ is given by \dref{zjscz2.5297x963ssddd0111kkuu}. Then \dref{zjscz2.5297x9630222222}--\dref{zjscz2.5297x96302222tt4}, and the dominated convergence theorem enables us to conclude $$ \begin{array}{rl}\label{eqx45xx12112ccgghh} &\displaystyle{-\int_0^{\infty}\int_{\Omega}u\varphi_t-\int_{\Omega}u_0\varphi(\cdot,0) } \\ =&\displaystyle{ \int_0^\infty\int_{\Omega}H(u)\Delta\varphi+\chi\int_0^\infty\int_{\Omega}u \nabla v\cdot\nabla\varphi+\xi\int_0^\infty\int_{\Omega}u \nabla w\cdot\nabla\varphi} +\displaystyle{\int_0^\infty\int_{\Omega}(\mu u-\mu uw-\mu u^2)\varphi} \end{array} $$ as well as $$ \begin{array}{rl}\label{eqx45xx12112ccgghhjj} &\displaystyle{-\int_0^{\infty}\int_{\Omega}v\varphi_t-\int_{\Omega}v_0\varphi(\cdot,0)=- \int_0^\infty\int_{\Omega}\nabla v\cdot\nabla\varphi-\int_0^\infty\int_{\Omega}v\varphi+\int_0^\infty\int_{\Omega}u\varphi} \end{array} $$ and $$ \begin{array}{rl} &\displaystyle{-\int_0^{\infty}\int_{\Omega}w\varphi_t-\int_{\Omega}w_0\varphi(\cdot,0)=- \int_0^\infty\int_{\Omega}v w\varphi} \end{array} $$ by a limit procedure. The proof of Lemma \ref{lemma45630223} is completed. \end{proof} We can now easily prove our main result. {\bf The proof of Theorem \ref{theossdrem3}}~ A combination of Lemma \ref{lemma45630hhuujjuu} and Lemma \ref{lemma45630223} directly leads to our desired result. {\bf Acknowledgement}: This work is partially supported by the National Natural Science Foundation of China (No. 11601215), Shandong Provincial Science Foundation for Out- standing Youth (No. ZR2018JL005) and Project funded by China Postdoctoral Science Foundation (No. 2019M650927, 2019T120168). \end{document}	arXiv
Mathematical Association The Mathematical Association is a professional society concerned with mathematics education in the UK. The Mathematical Association AbbreviationMA Formation1871 Legal statusNon-profit organisation and registered charity PurposeProfessional organisation for mathematics educators Location • 259 London Road, Leicester, LE2 3BE Region served UK Main organ MA Council President – Professor Nira Chamberlain (2023-2024) Websitehttps://www.m-a.org.uk History It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1894.[1][2] It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of the death of Sir Isaac Newton, attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal, Sir Horace Lamb, and G. H. Hardy. In 1951, Mary Cartwright became the first female president of the Mathematical Association.[3] In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university, a teaching award that was examined was the Diploma of the Mathematical Association, later known as the Diploma in Mathematical Education of the Mathematical Association. Function It exists to "bring about improvements in the teaching of mathematics and its applications, and to provide a means of communication among students and teachers of mathematics".[4] Since 1894 it has published The Mathematical Gazette. It is one of the participating bodies in the quadrennial British Congress of Mathematics Education, organised by the Joint Mathematical Council, and it holds its annual general meeting as part of the Congress.[5] Structure It is based in the south-east of Leicester on London Road (A6), just south of the Charles Frears campus of De Montfort University. Aside from the Council, it has seven other specialist committees. Regions Its branches are sometimes shared with the Association of Teachers of Mathematics (ATM): • Birmingham • Cambridge • East Midlands • Exeter • Gloucester • Liverpool • London • Greater Manchester • Meridian • Stoke and Staffordshire • Sheffield • Sussex • Yorkshire Past presidents Past presidents of The Association for the Improvement of Geometrical Teaching included: • 1871 Thomas Archer Hirst • 1878 R B Hayward MA, FRS • 1889 Professor G M Minchin MA, FRS • 1891 James Joseph Sylvester • 1892 The Reverend C Taylor DD • 1893 R Wormell MA, DSc • 1895 Joseph Larmor Past presidents of The Mathematical Association have included: • 1897 Alfred Lodge • 1899–1900 Robert Stawell Ball[6] • 1901 John Fletcher Moulton, Baron Moulton • 1903 Andrew Forsyth • 1905 George Ballard Mathews • 1907 George H. Bryan • 1909–1910 Herbert Hall Turner • 1911–1912 E. W. Hobson • 1913–1914 Alfred George Greenhill • 1915–1916 Alfred North Whitehead • 1918–1919 Percy Nunn • 1920 E. T. Whittaker • 1921 James Wilson • 1922–1923 Thomas Little Heath • 1924–1925 G. H. Hardy • 1926–1927 Micaiah John Muller Hill • 1928–1929 William Fleetwood Sheppard • 1930–1931 Arthur Eddington • 1932–1933 G. N. Watson • 1934 Eric Harold Neville[7] • 1935 A W Siddons • 1936 Andrew Forsyth • 1937 Louis Napoleon George Filon • 1938 W Hope-Jones • 1939 W C Fletcher • 1944 C O Tuckey MA • 1945 Sydney Chapman • 1946 Warin Foster Bushell • 1947 George Barker Jeffery • 1948 Harold Spencer Jones • 1949 A Robson MA • 1950 Professor H R Hasse MA, DSc • 1951 Mary Cartwright • 1952 K S Snell MA • 1953 Professor T A A Broadbent MA • 1954 W. V. D. Hodge • 1955 G L Parsons MA • 1956 George Frederick James Temple • 1957 W J Langford JP, MSc • 1958 Max Newman • 1959 Louise Doris Adams • 1960 Edwin A. Maxwell • 1961 J T Combridge MA, MSc • 1962 Professor V C A Ferraro PhD, DIC • 1963 J B Morgan MA • 1964 Ida Busbridge • 1965 Elizabeth Williams • 1966 F W Kellaway BSc • 1967 A.P. Rollett • 1968 Charles Coulson • 1969 Bertha Swirles • 1970 James Lighthill • 1971 B T Bellis MA, FRSE, FIMA • 1972 C T Daltry BSc, FIMA • 1973 William McCrea • 1974 Margaret Hayman • 1975 Reuben Goodstein • 1976 E Kerr BSc, PhD, FIMA, FBCS • 1977 Professor G Matthews MA, PhD, FIMA • 1978 Alan Tammadge • 1979 Clive W. Kilmister • 1980 D A Quadling MA, FIMA, later OBE • 1981 Michael Atiyah • 1982 F J Budden BSc • 1983 Rolph Ludwig Edward Schwarzenberger • 1984 P B Coaker BSc, ARCS, DIC, FIMA, FBCS • 1985 Hilary Shuard • 1986 Anita Straker • 1987 Margaret Rayner • 1988 A.G. Howson • 1989 Mr Peter Reynolds MA • 1990 Margaret Brown • 1991 Alan J. Bishop • 1992 Mr John Hersee MA • 1993 Dr William Wynne-Wilson BA, PhD • 1994 Mary Bradburn • 1995 E. Roy Ashley • 1996 W. P. Richardson MBE • 1997 Tony Gardiner • 1998 Professor J Chris Robson • 1999 John S Berry • 2000 Mr Stephen Abbott BSc, MSc • 2001 Dr Sue Sanders Cert.Ed, BA, MEd, PhD • 2002 Mr Barry Lewis BSc, BA, FIMA • 2003 Christopher Zeeman • 2004 Professor Adam McBride OBE • 2005 Sue Singer • 2006 Mr Doug French • 2007 Rob Eastaway • 2008 Mr Robert Barbour • 2009 Mrs Jane Imrie • 2010 David Acheson • 2011 Dr Paul Andrews • 2012 Professor Marcus Du Sautoy OBE FRS • 2013 Mr Peter Ransom MBE • 2014 Lynne McClure OBE • 2015 Dr Peter M. Neumann OBE • 2016 Dr Jennie Golding • 2017 Mr Tom Roper • 2018 Professor Mike Askew • 2019 Dr Ems Lord • 2020 Professor Hannah Fry • 2021 Dr Chris Pritchard • 2022 Dr Colin Foster • 2023 Professor Nira Chamberlain (President) • 2024 Dr Vicky Neale (President Delegate) Arms Coat of arms of Mathematical Association Adopted Granted 1 June 1965 [8] Crest On a wreath of the colours a dexter hand couped at the wrist holding a crystal cylinder enclosing a like sphere all Proper. Escutcheon Azure a representation of a pentagon with diagonals Or on a chief Argent an open book Proper inscribed with the Greek letters Pi and Epsilon Sable and edged and clasped Or. Motto Tibi Creditum Debes See also • London Mathematical Society • Institute of Mathematics and its Applications References 1. Orton, Anthony (2004). Learning Mathematics: Issues, Theory and Classroom Practice. A&C Black. p. 181. ISBN 0826471137. 2. Flood, Raymond; Rice, Adrian; Wilson, Robin, eds. (2011). Mathematics in Victorian Britain. Oxford University Press. p. 171. ISBN 978-0-19-162794-1. 3. Williams, Mrs. E. M. (October 1966), "Presidential Address: The Changing Role of Mathematics in Education", The Mathematical Gazette, 50 (373): 243–254, doi:10.2307/3614669, JSTOR 3614669 4. The Mathematical Association — supporting mathematics in education 5. BMCE Handbook, accessed 2018-10-09 6. "Court Circular". The Times. No. 36051. London. 29 January 1900. p. 9. 7. MA presidents have served 1 year terms, starting with Neville. 8. "Mathematical Association". Heraldry of the World. Retrieved 16 February 2021. • Siddons, A. W. (1939). "The Mathematical Association—I". Eureka. 1: 13–15. • Siddons, A. W. (1939). "The Mathematical Association—II". Eureka. 2: 18–19. • Michael H Price Mathematics of the Multitude? A History of the Mathematical Association (MA, 1994) External links • The Mathematical Association website • Complete list of Presidents of the Association • The MA's online shop • Annual conference • The Mathematical Gazette No. 1, 30, 31, 37–39, 41, 43 (1901–1904) on the Internet Archive digitised by Google from the Harvard University Library News items • Addressing the downward spiral of UK maths education in February 2004 • Proposal to split Maths GCSE into two in August 2003 Authority control International • ISNI • VIAF National • Israel • United States Mathematics in the United Kingdom Organizations and Projects • International Centre for Mathematical Sciences • Advisory Committee on Mathematics Education • Association of Teachers of Mathematics • British Society for Research into Learning Mathematics • Council for the Mathematical Sciences • Count On • Edinburgh Mathematical Society • HoDoMS • Institute of Mathematics and its Applications • Isaac Newton Institute • United Kingdom Mathematics Trust • Joint Mathematical Council • Kent Mathematics Project • London Mathematical Society • Making Mathematics Count • Mathematical Association • Mathematics and Computing College • Mathematics in Education and Industry • Megamaths • Millennium Mathematics Project • More Maths Grads • National Centre for Excellence in the Teaching of Mathematics • National Numeracy • National Numeracy Strategy • El Nombre • Numbertime • Oxford University Invariant Society • School Mathematics Project • Science, Technology, Engineering and Mathematics Network • Sentinus Maths schools • Exeter Mathematics School • King's College London Mathematics School • Lancaster University School of Mathematics • University of Liverpool Mathematics School Journals • Compositio Mathematica • Eureka • Forum of Mathematics • Glasgow Mathematical Journal • The Mathematical Gazette • Philosophy of Mathematics Education Journal • Plus Magazine Competitions • British Mathematical Olympiad • British Mathematical Olympiad Subtrust • National Cipher Challenge Awards • Chartered Mathematician • Smith's Prize • Adams Prize • Thomas Bond Sprague Prize • Rollo Davidson Prize	Wikipedia
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an $n$-dimensional manifold, or $n$-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of $n$-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. The study of manifolds requires working knowledge of calculus and topology. Motivating examples Circle After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive (indicated by the yellow arc in Figure 1). Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous and invertible mapping from the upper arc to the open interval (−1, 1): $\chi _{\mathrm {top} }(x,y)=x.\,$ Such functions along with the open regions they map are called charts. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: ${\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}$ Together, these parts cover the whole circle, and the four charts form an atlas for the circle. The top and right charts, $\chi _{\mathrm {top} }$ and $\chi _{\mathrm {right} }$ respectively, overlap in their domain: their intersection lies in the quarter of the circle where both $x$ and $y$-coordinates are positive. Both map this part into the interval $(0,1)$, though differently. Thus a function $T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}$ can be constructed, which takes values from the co-domain of $\chi _{\mathrm {top} }$ back to the circle using the inverse, followed by $\chi _{\mathrm {right} }$ back to the interval. If a is any number in $(0,1)$, then: ${\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left[a\right]\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}$ Such a function is called a transition map. The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts $\chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}$ and $\chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}$ Here s is the slope of the line through the point at coordinates (x, y) and the fixed pivot point (−1, 0); similarly, t is the opposite of the slope of the line through the points at coordinates (x, y) and (+1, 0). The inverse mapping from s to (x, y) is given by ${\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\[5pt]y&={\frac {2s}{1+s^{2}}}\end{aligned}}$ It can be confirmed that x2 + y2 = 1 for all values of s and t. These two charts provide a second atlas for the circle, with the transition map $t={\frac {1}{s}}$ (that is, one has this relation between s and t for every point where s and t are both nonzero). Each chart omits a single point, either (−1, 0) for s or (+1, 0) for t, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. Sphere The sphere is an example of a surface. The unit sphere of implicit equation x2 + y2 + z2 – 1 = 0 may be covered by an atlas of six charts: the plane z = 0 divides the sphere into two half spheres (z > 0 and z < 0), which may both be mapped on the disc x2 + y2 < 1 by the projection on the xy plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes. As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart. This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the Earth cannot have a plane representation consisting of a single map (also called "chart", see nautical chart), and therefore one needs atlases for covering the whole Earth surface. Other curves Manifolds need not be connected (all in "one piece"); an example is a pair of separate circles. Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve y2 = x3 − x (a closed loop piece and an open, infinite piece). However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line segment, since deleting the center point from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a line segment gives a space with at most two pieces; topological operations always preserve the number of pieces. Mathematical definition Further information: Categories of manifolds Informally, a manifold is a space that is "modeled on" Euclidean space. There are many different kinds of manifolds. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole. Formally, a (topological) manifold is a second countable Hausdorff space that is locally homeomorphic to a Euclidean space. Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds). Locally homeomorphic to a Euclidean space means that every point has a neighborhood homeomorphic to an open subset of the Euclidean space $\mathbb {R} ^{n},$ for some nonnegative integer n. This implies that either the point is an isolated point (if $n=0$), or it has a neighborhood homeomorphic to the open ball $\mathbf {B} ^{n}=\left\{(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}<1\right\}.$ This implies also that every point has a neighborhood homeomorphic to $\mathbb {R} ^{n}$ since $\mathbb {R} ^{n}$ is homeomorphic, and even diffeomorphic to any open ball in it (for $n>0$). The n that appears in the preceding definition is called the local dimension of the manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the dimension of the manifold. This is, in particular, the case when manifolds are connected. However, some authors admit manifolds that are not connected, and where different points can have different dimensions.[1] If a manifold has a fixed dimension, this can be emphasized by calling it a pure manifold. For example, the (surface of a) sphere has a constant dimension of 2 and is therefore a pure manifold whereas the disjoint union of a sphere and a line in three-dimensional space is not a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is locally constant), each connected component has a fixed dimension. Sheaf-theoretically, a manifold is a locally ringed space, whose structure sheaf is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in algebraic geometry. Charts, atlases, and transition maps Main article: Atlas (topology) See also: Differentiable manifold The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. Charts Main article: Coordinate chart A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure.[2] For a topological manifold, the simple space is a subset of some Euclidean space $\mathbb {R} ^{n}$ and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. In the case of a differentiable manifold, a set of charts called an atlas, whose transition functions (see below) are all differentiable, allows us to do calculus on it. Polar coordinates, for example, form a chart for the plane $\mathbb {R} ^{2}$ minus the positive x-axis and the origin. Another example of a chart is the map χtop mentioned above, a chart for the circle. Atlases Main article: Atlas (topology) The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an atlas. An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas. The atlas containing all possible charts consistent with a given atlas is called the maximal atlas (i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). Transition maps Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a transition function can be defined which goes from an open ball in $\mathbb {R} ^{n}$ to the manifold and then back to another (or perhaps the same) open ball in $\mathbb {R} ^{n}$. The resultant map, like the map T in the circle example above, is called a change of coordinates, a coordinate transformation, a transition function, or a transition map. Additional structure An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold. This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of $\mathbb {R} ^{n}$ (that is, if they are diffeomorphisms), the differential structure transfers to the manifold and turns it into a differentiable manifold. Complex manifolds are introduced in an analogous way by requiring that the transition functions of an atlas are holomorphic functions. For symplectic manifolds, the transition functions must be symplectomorphisms. The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called compatible. These notions are made precise in general through the use of pseudogroups. Manifold with boundary See also: Topological manifold § Manifolds with boundary A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary of an $n$-manifold with boundary is an $(n-1)$-manifold. A disk (circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 1-manifold. A square with interior is also a 2-manifold with boundary. A ball (sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (Do not confuse with Boundary (topology)). In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open $n$-ball $\{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}$. Every boundary point has a neighborhood homeomorphic to the "half" $n$-ball $\{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}$. Any homeomorphism between half-balls must send points with $x_{1}=0$ to points with $x_{1}=0$. This invariance allows to "define" boundary points; see next paragraph. Boundary and interior Let $M$ be a manifold with boundary. The interior of $M$, denoted $\operatorname {Int} M$, is the set of points in $M$ which have neighborhoods homeomorphic to an open subset of $\mathbb {R} ^{n}$. The boundary of $M$, denoted $\partial M$, is the complement of $\operatorname {Int} M$ in $M$. The boundary points can be characterized as those points which land on the boundary hyperplane $(x_{n}=0)$ of $\mathbb {R} _{+}^{n}$ under some coordinate chart. If $M$ is a manifold with boundary of dimension $n$, then $\operatorname {Int} M$ is a manifold (without boundary) of dimension $n$ and $\partial M$ is a manifold (without boundary) of dimension $n-1$. Construction A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Charts Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of $\mathbb {R} ^{2}$ is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere: Sphere with charts A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of $\mathbb {R} ^{3}$: $S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.$ The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of $\mathbb {R} ^{2}$. Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by $\chi (x,y,z)=(x,y),\ $ maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere. This can be easily generalized to higher-dimensional spheres. Patchwork Further information: Surgery theory A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold. The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an equivalence class of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the resulting manifold is a differentiable manifold. This can be illustrated with the transition map t = 1⁄s from the second half of the circle example. Start with two copies of the line. Use the coordinate s for the first copy, and t for the second copy. Now, glue both copies together by identifying the point t on the second copy with the point s = 1⁄t on the first copy (the points t = 0 and s = 0 are not identified with any point on the first and second copy, respectively). This gives a circle. Intrinsic and extrinsic view The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen as embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space, it is always clear whether a vector at some point is tangential or normal to some surface through that point. The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the intrinsic view. It can make it harder to imagine what a tangent vector might be, and there is no intrinsic notion of a normal bundle, but instead there is an intrinsic stable normal bundle. n-Sphere as a patchwork The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sn can be constructed by gluing together two copies of $\mathbb {R} ^{n}$. The transition map between them is inversion in a sphere, defined as $\mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\\|x\\|^{2}.$ This function is its own inverse and thus can be used in both directions. As the transition map is a smooth function, this atlas defines a smooth manifold. In the case n = 1, the example simplifies to the circle example given earlier. Identifying points of a manifold Main articles: Orbifold and Group action (mathematics) It is possible to define different points of a manifold to be same. This can be visualized as gluing these points together in a single point, forming a quotient space. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved. An example of a quotient space of a manifold that is also a manifold is the real projective space, identified as a quotient space of the corresponding sphere. One method of identifying points (gluing them together) is through a right (or left) action of a group, which acts on the manifold. Two points are identified if one is moved onto the other by some group element. If M is the manifold and G is the group, the resulting quotient space is denoted by M / G (or G \ M). Manifolds which can be constructed by identifying points include tori and real projective spaces (starting with a plane and a sphere, respectively). Gluing along boundaries Main article: Quotient space (topology) Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together. Formally, the gluing is defined by a bijection between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold, this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly, for a differentiable manifold, it has to be a diffeomorphism. For other manifolds, other structures should be preserved. A finite cylinder may be constructed as a manifold by starting with a strip [0,1] × [0,1] and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A projective plane may be obtained by gluing a sphere with a hole in it to a Möbius strip along their respective circular boundaries. Cartesian products The Cartesian product of manifolds is also a manifold. The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the product topology, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite cylinders, for example, as S1 × S1 and S1 × [0,1], respectively. History Further information: History of manifolds and varieties The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Early development Before the modern concept of a manifold there were several important results. Non-Euclidean geometry considers spaces where Euclid's parallel postulate fails. Saccheri first studied such geometries in 1733, but sought only to disprove them. Gauss, Bolyai and Lobachevsky independently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant negative and positive curvature, respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more topological example of an intrinsic property of a manifold is its Euler characteristic. Leonhard Euler showed that for a convex polytope in the three-dimensional Euclidean space with V vertices (or corners), E edges, and F faces, $V-E+F=2.\ $ The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a topological map with V vertices, E edges, and F faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope.[3] Thus 2 is a topological invariant of the sphere, called its Euler characteristic. On the other hand, a torus can be sliced open by its 'parallel' and 'meridian' circles, creating a map with V = 1 vertex, E = 2 edges, and F = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful topological invariant, which can be extended to higher dimensions using Betti numbers. In the mid nineteenth century, the Gauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature. Synthesis Investigations of Niels Henrik Abel and Carl Gustav Jacobi on inversion of elliptic integrals in the first half of 19th century led them to consider special types of complex manifolds, now known as Jacobians. Bernhard Riemann further contributed to their theory, clarifying the geometric meaning of the process of analytic continuation of functions of complex variables. Another important source of manifolds in 19th century mathematics was analytical mechanics, as developed by Siméon Poisson, Jacobi, and William Rowan Hamilton. The possible states of a mechanical system are thought to be points of an abstract space, phase space in Lagrangian and Hamiltonian formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their generalized coordinates. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various conservation laws constrain it to more complicated formations, e.g. Liouville tori. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and Joseph-Louis Lagrange, gives another example where the manifold is nontrivial. Geometrical and topological aspects of classical mechanics were emphasized by Henri Poincaré, one of the founders of topology. Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a Mannigfaltigkeit, because the variable can have many values. He distinguishes between stetige Mannigfaltigkeit and diskrete Mannigfaltigkeit (continuous manifoldness and discontinuous manifoldness), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Riemann. Poincaré's definition In his very influential paper, Analysis Situs,[4] Henri Poincaré gave a definition of a differentiable manifold (variété) which served as a precursor to the modern concept of a manifold.[5] In the first section of Analysis Situs, Poincaré defines a manifold as the level set of a continuously differentiable function between Euclidean spaces that satisfies the nondegeneracy hypothesis of the implicit function theorem. In the third section, he begins by remarking that the graph of a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' (une chaîne des variétés). Poincaré's notion of a chain of manifolds is a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions $\theta (y)$ and $\theta '\left(y'\right)$. If these manifolds overlap (a une partie commune), then he requires that the coordinates $y$ depend continuously differentiably on the coordinates $y'$ and vice versa ('...les $y$ sont fonctions analytiques des $y'$ et inversement'). In this way he introduces a precursor to the notion of a chart and of a transition map. For example, the unit circle in the plane can be thought of as the graph of the function $ y={\sqrt {1-x^{2}}}$ or else the function $ y=-{\sqrt {1-x^{2}}}$ in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively, $ x={\sqrt {1-y^{2}}}$ and $ x=-{\sqrt {1-y^{2}}}$. The circle can be represented by a graph in the neighborhood of every point because the left hand side of its defining equation $x^{2}+y^{2}-1=0$ has nonzero gradient at every point of the circle. By the implicit function theorem, every submanifold of Euclidean space is locally the graph of a function. Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a topological space that followed shortly. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. Notably, the Whitney embedding theorem[6] showed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space. Topology of manifolds: highlights Two-dimensional manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified in the beginning of the 20th century by Poul Heegaard and Max Dehn. Poincaré pioneered the study of three-dimensional manifolds and raised a fundamental question about them, today known as the Poincaré conjecture. After nearly a century, Grigori Perelman proved the Poincaré conjecture (see the Solution of the Poincaré conjecture). William Thurston's geometrization program, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by Michael Freedman and in a different setting, by Simon Donaldson, who was motivated by the then recent progress in theoretical physics (Yang–Mills theory), where they serve as a substitute for ordinary 'flat' spacetime. Andrey Markov Jr. showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including analogues of the Poincaré conjecture, had been done earlier by René Thom, John Milnor, Stephen Smale and Sergei Novikov. A very pervasive and flexible technique underlying much work on the topology of manifolds is Morse theory. Additional structure Main article: Categories of manifolds Topological manifolds Main article: topological manifold The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space $\mathbb {R} ^{n}$. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to $\mathbb {R} ^{n}$. These homeomorphisms are the charts of the manifold. A topological manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any particular and consistent choice of such concepts.[7] In order to discuss such properties for a manifold, one needs to specify further structure and consider differentiable manifolds and Riemannian manifolds discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles. Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be Hausdorff and second countable. The dimension of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number n in the definition). All points in a connected manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension. Differentiable manifolds Main article: Differentiable manifold For most applications, a special kind of topological manifold, namely, a differentiable manifold, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use calculus on a differentiable manifold. Each point of an n-dimensional differentiable manifold has a tangent space. This is an n-dimensional Euclidean space consisting of the tangent vectors of the curves through the point. Two important classes of differentiable manifolds are smooth and analytic manifolds. For smooth manifolds the transition maps are smooth, that is, infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are analytic (they can be expressed as power series). The sphere can be given analytic structure, as can most familiar curves and surfaces. A rectifiable set generalizes the idea of a piecewise smooth or rectifiable curve to higher dimensions; however, rectifiable sets are not in general manifolds. Riemannian manifolds Main article: Riemannian manifold To measure distances and angles on manifolds, the manifold must be Riemannian. A Riemannian manifold is a differentiable manifold in which each tangent space is equipped with an inner product $\langle \cdot ,\cdot \rangle $ in a manner which varies smoothly from point to point. Given two tangent vectors $u$ and $v$, the inner product $\langle u,v\rangle $ gives a real number. The dot (or scalar) product is a typical example of an inner product. This allows one to define various notions such as length, angles, areas (or volumes), curvature and divergence of vector fields. All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of the space). Many familiar curves and surfaces, including for example all n-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it. Finsler manifolds Main article: Finsler manifold A Finsler manifold allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a norm, $\\|\cdot \\|$, in a manner which varies smoothly from point to point. This norm can be extended to a metric, defining the length of a curve; but it cannot in general be used to define an inner product. Any Riemannian manifold is a Finsler manifold. Lie groups Main article: Lie group Lie groups, named after Sophus Lie, are differentiable manifolds that carry also the structure of a group which is such that the group operations are defined by smooth maps. A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a compact Lie group is the circle: the group operation is simply rotation. This group, known as $\operatorname {U} (1)$, can be also characterised as the group of complex numbers of modulus 1 with multiplication as the group operation. Other examples of Lie groups include special groups of matrices, which are all subgroups of the general linear group, the group of $n\times n$ matrices with non-zero determinant. If the matrix entries are real numbers, this will be an $n^{2}$-dimensional disconnected manifold. The orthogonal groups, the symmetry groups of the sphere and hyperspheres, are $n(n-1)/2$ dimensional manifolds, where $n-1$ is the dimension of the sphere. Further examples can be found in the table of Lie groups. Other types of manifolds • A complex manifold is a manifold whose charts take values in $\mathbb {C} ^{n}$ and whose transition functions are holomorphic on the overlaps. These manifolds are the basic objects of study in complex geometry. A one-complex-dimensional manifold is called a Riemann surface. An $n$-dimensional complex manifold has dimension $2n$ as a real differentiable manifold. • A CR manifold is a manifold modeled on boundaries of domains in $\mathbb {C} ^{n}$. • 'Infinite dimensional manifolds': to allow for infinite dimensions, one may consider Banach manifolds which are locally homeomorphic to Banach spaces. Similarly, Fréchet manifolds are locally homeomorphic to Fréchet spaces. • A symplectic manifold is a kind of manifold which is used to represent the phase spaces in classical mechanics. They are endowed with a 2-form that defines the Poisson bracket. A closely related type of manifold is a contact manifold. • A combinatorial manifold is a kind of manifold which is discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. • A digital manifold is a special kind of combinatorial manifold which is defined in digital space. See digital topology. Classification and invariants Further information: Classification of manifolds Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds. The classification of smooth closed manifolds is well understood in principle, except in dimension 4: in low dimensions (2 and 3) it is geometric, via the uniformization theorem and the solution of the Poincaré conjecture, and in high dimension (5 and above) it is algebraic, via surgery theory. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a complete set of invariants. This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional manifold refer to the same object. However, one can determine if two manifolds are different if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as invariants, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are invariant under different descriptions. Naively, one could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. Unfortunately, it is known that for manifolds of dimension 4 and higher, no program exists that can decide whether two manifolds are diffeomorphic. Smooth manifolds have a rich set of invariants, coming from point-set topology, classic algebraic topology, and geometric topology. The most familiar invariants, which are visible for surfaces, are orientability (a normal invariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection. This distinction between local invariants and no local invariants is a common way to distinguish between geometry and topology. All invariants of a smooth closed manifold are thus global. Algebraic topology is a source of a number of important global invariant properties. Some key criteria include the simply connected property and orientability (see below). Indeed, several branches of mathematics, such as homology and homotopy theory, and the theory of characteristic classes were founded in order to study invariant properties of manifolds. Surfaces Orientability Main article: Orientable manifold In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to $\mathbb {R} ^{n}$. Given an ordered basis for $\mathbb {R} ^{n}$, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are orientable manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable. Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projective plane, which arises naturally in geometry. Möbius strip Main article: Möbius strip Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a single side. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions. See also: Quasitoric manifold Klein bottle Main article: Klein bottle Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into cross-caps. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. In three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space. Real projective plane Main article: Real projective space Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes. Although there is no way to do so physically, it is possible (by considering a quotient space) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane". Genus and the Euler characteristic For two dimensional manifolds a key invariant property is the genus, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology and cohomology. Maps of manifolds Main article: Maps of manifolds Just as there are various types of manifolds, there are various types of maps of manifolds. In addition to continuous functions and smooth functions generally, there are maps with special properties. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem. Scalar-valued functions A basic example of maps between manifolds are scalar-valued functions on a manifold, $f\colon M\to \mathbb {R} $ or $f\colon M\to \mathbb {C} ,$ sometimes called regular functions or functionals, by analogy with algebraic geometry or linear algebra. These are of interest both in their own right, and to study the underlying manifold. In geometric topology, most commonly studied are Morse functions, which yield handlebody decompositions, while in mathematical analysis, one often studies solution to partial differential equations, an important example of which is harmonic analysis, where one studies harmonic functions: the kernel of the Laplace operator. This leads to such functions as the spherical harmonics, and to heat kernel methods of studying manifolds, such as hearing the shape of a drum and some proofs of the Atiyah–Singer index theorem. Generalizations of manifolds Infinite dimensional manifolds The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to a topological vector space over the reals. This omits the point-set axioms, allowing higher cardinalities and non-Hausdorff manifolds; and it omits finite dimension, allowing structures such as Hilbert manifolds to be modeled on Hilbert spaces, Banach manifolds to be modeled on Banach spaces, and Fréchet manifolds to be modeled on Fréchet spaces. Usually one relaxes one or the other condition: manifolds with the point-set axioms are studied in general topology, while infinite-dimensional manifolds are studied in functional analysis. Orbifolds An orbifold is a generalization of manifold allowing for certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space (e.g. Euclidean space) by the actions of various finite groups. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense. Algebraic varieties and schemes Non-singular algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space, an algebraic variety is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields. Schemes are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using sheaves instead of atlases. Because of singular points, a variety is in general not a manifold, though linguistically the French variété, German Mannigfaltigkeit and English manifold are largely synonymous. In French an algebraic variety is called une variété algébrique (an algebraic variety), while a smooth manifold is called une variété différentielle (a differential variety). Stratified space A "stratified space" is a space that can be divided into pieces ("strata"), with each stratum a manifold, with the strata fitting together in prescribed ways (formally, a filtration by closed subsets). There are various technical definitions, notably a Whitney stratified space (see Whitney conditions) for smooth manifolds and a topologically stratified space for topological manifolds. Basic examples include manifold with boundary (top dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces, including algebraic varieties, analytic varieties, semialgebraic sets, and subanalytic sets. CW-complexes A CW complex is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, hence not a manifold. However, they are of central interest in algebraic topology, especially in homotopy theory. Homology manifolds A homology manifold is a space that behaves like a manifold from the point of view of homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery theory.[8] Differential spaces Let $M$ be a nonempty set. Suppose that some family of real functions on $M$ was chosen. Denote it by $C\subseteq \mathbb {R} ^{M}$. It is an algebra with respect to the pointwise addition and multiplication. Let $M$ be equipped with the topology induced by $C$. Suppose also that the following conditions hold. First: for every $H\in C^{\infty }\left(\mathbb {R} ^{n}\right)$, where $n\in \mathbb {N} $, and arbitrary $f_{1},\dots ,f_{n}\in C$, the composition $H\circ \left(f_{1},\dots ,f_{n}\right)\in C$. Second: every function, which in every point of $M$ locally coincides with some function from $C$, also belongs to $C$. A pair $(M,C)$ for which the above conditions hold, is called a Sikorski differential space.[9] See also • Geodesic – Straight path on a curved surface or a Riemannian manifold • Directional statistics – subdiscipline of statisticsPages displaying wikidata descriptions as a fallback: statistics on manifolds • List of manifolds • Timeline of manifolds – Mathematics timeline • Mathematics of general relativity – Mathematical structures and techniques used in the theory of general relativity By dimension • 3-manifold – Mathematical space • 4-manifold – Mathematical space • 5-manifold – Manifold of dimension five • Manifolds of mappings – locally convex vector spaces satisfying a very mild completeness conditionPages displaying wikidata descriptions as a fallback Notes 1. E.g. see Riaza, Ricardo (2008), Differential-Algebraic Systems: Analytical Aspects and Circuit Applications, World Scientific, p. 110, ISBN 9789812791818; Gunning, R. C. (1990), Introduction to Holomorphic Functions of Several Variables, Volume 2, CRC Press, p. 73, ISBN 9780534133092. 2. Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001). Geometry of Differential Forms. American Mathematical Society Bookstore. p. 12. ISBN 0-8218-1045-6. 3. The notion of a map can formalized as a cell decomposition. 4. Poincaré, H. (1895). "Analysis Situs". Journal de l'École Polytechnique. Serié 11 (in French). Gauthier-Villars. 5. Arnolʹd, V. I. (1998). "О преподавании математики" [On Teaching Mathematics]. Uspekhi Mat. Nauk (in Russian). 53 (319): 229–234. doi:10.4213/rm5.; translation in Russian Math. Surveys 53 (1998), no. 1, 229–236 6. Whitney, H. (1936). "Differentiable Manifolds". Annals of Mathematics. Second Series. 37 (3): 645–680. doi:10.2307/1968482. JSTOR 1968482. 7. Kervaire, M. (1961). "A Manifold which does not admit any differentiable structure". Comment. Math. Helv. 35 (1): 1–14. doi:10.1007/BF02565940. S2CID 120977898. 8. Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S. (1996). "Topology of homology manifolds". Annals of Mathematics. Second Series. 143 (3): 435–467. arXiv:math/9304210. doi:10.2307/2118532. JSTOR 2118532. 9. Sikorski, R. (1967). "Abstract covariant derivative". Colloquium Mathematicum. 18: 251–272. doi:10.4064/cm-18-1-251-272. References • Freedman, Michael H., and Quinn, Frank (1990) Topology of 4-Manifolds. Princeton University Press. ISBN 0-691-08577-3. • Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2. Advanced undergraduate / first-year graduate text inspired by Milnor. • Hempel, John (1976) 3-Manifolds. Princeton University Press. ISBN 0-8218-3695-1. • Hirsch, Morris, (1997) Differential Topology. Springer Verlag. ISBN 0-387-90148-5. The most complete account, with historical insights and excellent, but difficult, problems. The standard reference for those wishing to have a deep understanding of the subject. • Kirby, Robion C. and Siebenmann, Laurence C. (1977) Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton University Press. ISBN 0-691-08190-5. A detailed study of the category of topological manifolds. • Lee, John M. (2000) Introduction to Topological Manifolds. Springer-Verlag. ISBN 0-387-98759-2. Detailed and comprehensive first-year graduate text. • Lee, John M. (2003) Introduction to Smooth Manifolds. Springer-Verlag. ISBN 0-387-95495-3. Detailed and comprehensive first-year graduate text; sequel to Introduction to Topological Manifolds. • Massey, William S. (1977) Algebraic Topology: An Introduction. Springer-Verlag. ISBN 0-387-90271-6. • Milnor, John (1997) Topology from the Differentiable Viewpoint. Princeton University Press. ISBN 0-691-04833-9. Classic brief introduction to differential topology. • Munkres, James R. (1991) Analysis on Manifolds. Addison-Wesley (reprinted by Westview Press) ISBN 0-201-51035-9. Undergraduate text treating manifolds in $\mathbb {R} ^{n}$. • Munkres, James R. (2000) Topology. Prentice Hall. ISBN 0-13-181629-2. • Neuwirth, L. P., ed. (1975) Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox. Princeton University Press. ISBN 978-0-691-08170-0. • Riemann, Bernhard, Gesammelte mathematische Werke und wissenschaftlicher Nachlass, Sändig Reprint. ISBN 3-253-03059-8. • Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. The 1851 doctoral thesis in which "manifold" (Mannigfaltigkeit) first appears. • Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. The 1854 Göttingen inaugural lecture (Habilitationsschrift). • Spivak, Michael (1965) Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press). ISBN 0-8053-9021-9. Famously terse advanced undergraduate / first-year graduate text. • Spivak, Michael (1999) A Comprehensive Introduction to Differential Geometry (3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels. • Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). New York: Springer. ISBN 978-1-4419-7399-3.. Concise first-year graduate text. External links • "Manifold", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Dimensions-math.org (A film explaining and visualizing manifolds up to fourth dimension.) • The manifold atlas project of the Max Planck Institute for Mathematics in Bonn Manifolds (Glossary) Basic concepts • Topological manifold • Atlas • Differentiable/Smooth manifold • Differential structure • Smooth atlas • Submanifold • Riemannian manifold • Smooth map • Submersion • Pushforward • Tangent space • Differential form • Vector field Main results (list) • Atiyah–Singer index • Darboux's • De Rham's • Frobenius • Generalized Stokes • Hopf–Rinow • Noether's • Sard's • Whitney embedding Maps • Curve • Diffeomorphism • Local • Geodesic • Exponential map • in Lie theory • Foliation • Immersion • Integral curve • Lie derivative • Section • Submersion Types of manifolds • Closed • (Almost) Complex • (Almost) Contact • Fibered • Finsler • Flat • G-structure • Hadamard • Hermitian • Hyperbolic • Kähler • Kenmotsu • Lie group • Lie algebra • Manifold with boundary • Oriented • Parallelizable • Poisson • Prime • Quaternionic • Hypercomplex • (Pseudo−, Sub−) Riemannian • Rizza • (Almost) Symplectic • Tame Tensors Vectors • Distribution • Lie bracket • Pushforward • Tangent space • bundle • Torsion • Vector field • Vector flow Covectors • Closed/Exact • Covariant derivative • Cotangent space • bundle • De Rham cohomology • Differential form • Vector-valued • Exterior derivative • Interior product • Pullback • Ricci curvature • flow • Riemann curvature tensor • Tensor field • density • Volume form • Wedge product Bundles • Adjoint • Affine • Associated • Cotangent • Dual • Fiber • (Co) Fibration • Jet • Lie algebra • (Stable) Normal • Principal • Spinor • Subbundle • Tangent • Tensor • Vector Connections • Affine • Cartan • Ehresmann • Form • Generalized • Koszul • Levi-Civita • Principal • Vector • Parallel transport Related • Classification of manifolds • Gauge theory • History • Morse theory • Moving frame • Singularity theory Generalizations • Banach manifold • Diffeology • Diffiety • Fréchet manifold • K-theory • Orbifold • Secondary calculus • over commutative algebras • Sheaf • Stratifold • Supermanifold • Stratified space Tensors Glossary of tensor theory Scope Mathematics • Coordinate system • Differential geometry • Dyadic algebra • Euclidean geometry • Exterior calculus • Multilinear algebra • Tensor algebra • Tensor calculus • Physics • Engineering • Computer vision • Continuum mechanics • Electromagnetism • General relativity • Transport phenomena Notation • Abstract index notation • Einstein notation • Index notation • Multi-index notation • Penrose graphical notation • Ricci calculus • Tetrad (index notation) • Van der Waerden notation • Voigt notation Tensor definitions • Tensor (intrinsic definition) • Tensor field • Tensor density • Tensors in curvilinear coordinates • Mixed tensor • Antisymmetric tensor • Symmetric tensor • Tensor operator • Tensor bundle • Two-point tensor Operations • Covariant derivative • Exterior covariant derivative • Exterior derivative • Exterior product • Hodge star operator • Lie derivative • Raising and lowering indices • Symmetrization • Tensor contraction • Tensor product • Transpose (2nd-order tensors) Related abstractions • Affine connection • Basis • Cartan formalism (physics) • Connection form • Covariance and contravariance of vectors • Differential form • Dimension • Exterior form • Fiber bundle • Geodesic • Levi-Civita connection • Linear map • Manifold • Matrix • Multivector • Pseudotensor • Spinor • Vector • Vector space Notable tensors Mathematics • Kronecker delta • Levi-Civita symbol • Metric tensor • Nonmetricity tensor • Ricci curvature • Riemann curvature tensor • Torsion tensor • Weyl tensor Physics • Moment of inertia • Angular momentum tensor • Spin tensor • Cauchy stress tensor • stress–energy tensor • Einstein tensor • EM tensor • Gluon field strength tensor • Metric tensor (GR) Mathematicians • Élie Cartan • Augustin-Louis Cauchy • Elwin Bruno Christoffel • Albert Einstein • Leonhard Euler • Carl Friedrich Gauss • Hermann Grassmann • Tullio Levi-Civita • Gregorio Ricci-Curbastro • Bernhard Riemann • Jan Arnoldus Schouten • Woldemar Voigt • Hermann Weyl Authority control International • FAST National • France • BnF data • Germany • Israel • United States • Japan Other • IdRef	Wikipedia
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Cai, Alice Meroni, … Ahmad Qasem The effect of smoking on quantification of aortic stiffness by ultrasound time-harmonic elastography Thomas Elgeti, Matthias Fröhlich, … Lars-Arne Schaafs Hematocrit, hemoglobin and red blood cells are associated with vascular function and vascular structure in men Shinji Kishimoto, Tatsuya Maruhashi, … Yukihito Higashi Hiroshi Tanaka1, Akihisa Mito1, Harutoyo Hirano ORCID: orcid.org/0000-0002-0634-18052, Zu Soh ORCID: orcid.org/0000-0001-7182-93833, Ryuji Nakamura ORCID: orcid.org/0000-0001-8074-47874, Noboru Saeki ORCID: orcid.org/0000-0002-1967-08784, Masashi Kawamoto4, Yukihito Higashi5,6, Masao Yoshizumi7 & Toshio Tsuji ORCID: orcid.org/0000-0002-7689-39631 Scientific Reports volume 9, Article number: 2609 (2019) Cite this article This paper proposes an algorithm for estimating the arterial viscosity using cuff pressures and pulse waves measured by an automatic oscillometric sphygmomanometer. A change in the arterial viscosity during the enclosed-zone flow-mediated dilation test is calculated as an index for evaluating the vascular endothelial function %η. In all, 43 individuals participated in this study. After the index %η was calculated, the accuracy of the index %η in distinguishing healthy subjects and subjects at a high risk of arteriosclerosis was tested via a receiving operating characteristic (ROC) analysis. The calculated %η for the healthy participants and those at a high risk of arteriosclerosis was 13.4 ± 55.1% and −32.7 ± 34.0% (mean ± S.D.), respectively. The area under the ROC curve was 0.77. Thus, it was concluded that the proposed method can be used to evaluate the vascular endothelial function. In recent years, cardiovascular diseases such as heart diseases and cerebrovascular diseases have accounted for approximately 25% of the deaths among the Japanese people1. One of the main causes of these cardiovascular diseases is arteriosclerosis, which must be diagnosed and treated early as it is a progressive and refractory disease. Vascular endothelial dysfunction is known as an early symptom of arteriosclerosis2. The vascular endothelial function associated with the sympathetic adrenergic system is involved in fine tuning of the blood pressure. Specifically, the activation of the α1 adrenergic receptor and β adrenergic receptor causes vasoconstriction and nitric oxide (NO)-mediated vasodilatation, respectively3,4. The vascular endothelial function can thus be evaluated via changes in the blood flow rate, vascular diameter or arterial mechanical properties, which are associated with vasoconstriction and vasodilatation. Evaluation methods for vascular endothelial function have been extensively studied since the 1980s, and the first attempt involved invasive methods5,6,7,8,9. For example, Panza et al. used a plethysmograph to estimate changes in the blood flow rate when a NO agonist or antagonist was administered in an artery10. The flow-mediated dilation (FMD) test11 was then proposed as a clinically practical method that enabled noninvasive evaluation of the vascular endothelial function. In the FMD test, shear stress is applied on the endothelium to induce NO release, and the dilation ratio of the vascular diameter between the pre- and post-cuff occlusion (%FMD) is measured as an index to evaluate the endothelial function. One of the drawbacks of the FMD test is that it requires dexterity because the prolonged stable measurement of the vascular diameter with an ultrasound device is often difficult. To simplify the measurement procedure, a technique applying an oscillometric method12, which is widely used in commercial automatic sphygmomanometers, was proposed13,14. This technique is referred to as the enclosed-zone flow-mediated dilation (ezFMD) test. In the ezFMD test, a cuff is attached to the upper arm and the cuff pulse wave is measured. The vascular endothelial function is then estimated based on the maximum amplitude change rate between the pre- and post-cuff occlusion of the cuff pulse wave (%ezFMD). In addition, evaluation methods focusing on the arterial mechanical characteristics have been attracted attention since the 2000s5,6,7,8,9 because it is a rather direct measure of vasodilation compared to the vascular diameter, which can be affected by changes in the blood pressure. In these studies, several measures such as the pulse wave velocity were used to evaluate the arterial stiffness or viscoelasticity. Our research group has previously proposed a log-linearized peripheral arterial model15, which enabled the estimation of the changes in the arterial viscoelasticity in a beat-to-beat manner during the FMD test16. These recent studies revealed the effectiveness of the use of viscoelasticity in vascular endothelial function evaluation; however, such methods inherit the drawbacks of the FMD test because viscoelasticity estimation in these cases is carried out based on the vascular diameters measured using the ultrasonic devices. To improve the clinical applicability, this paper presents a concept employing the oscillometric method and an algorithm for estimating the arterial viscosity to assess the vascular endothelial function. The proposed method enables the estimation of the arterial viscosity from only a common oscillometric automatic sphygmomanometer. The algorithm is then tested in terms of its ability of evaluating the vascular endothelial function by measuring the changes in the estimated viscosity induced by reactive hyperaemia. Baseline clinical characteristics The baseline clinical characteristics of the participants are summarised in Table 1. The age range was 19–84 years. The mean value of the systolic blood pressure was 126.9 ± 19.2 mmHg, and that of the diastolic blood pressure was 68.0 ± 13.8 mmHg. Table 1 Clinical characteristics of the participants. Viscosity estimation experiment Figure 1(a and b) show the examples of the measured waveforms from Sub. 1 (a healthy participant) and Sub. 30 (a participant at a high risk of arteriosclerosis). Figure 1(a and c) show the pre-cuff occlusion cuff pressure, cuff pulse wave and cuff pulse wave velocity waveforms, while Fig. 1(b and d) show the corresponding post-cuff occlusion waveforms. The shaded areas are the data intervals used for calculating the arterial viscosity. The amplitude and velocity of the post-cuff occlusion cuff pulse wave were greater than those of the pre-cuff occlusion cuff pulse wave in both the healthy participants and the participants at a high risk of arteriosclerosis. The pre- and post-cuff occlusion arterial viscosity values for Sub. 1 were 1.71 and 1.72, respectively. The change ratio of arterial viscosity %η was 65.4%, as shown in Fig. 1. The results indicated that the third arterial viscosity measured after the cuff occlusion was almost equal to that measured before the cuff occlusion, and the change in the arterial viscosity was a positive value. However, the pre- and post-cuff occlusion arterial viscosity values for Sub. 30 were 5.69 and 2.94, respectively. The change ratio of arterial viscosity %η was −57.4%, as shown in Fig. 1. The results indicated that the third arterial viscosity measured after the cuff occlusion was decreased compared with the arterial viscosity measured before the cuff occlusion. Examples of measured cuff pressure, cuff pressure waves and differentiated waves of cuff pressure: (a) pre-cuff occlusion waves from a healthy participant (Sub. 1); (b) post-cuff occlusion waves from a healthy participant (Sub. 1); (c) pre-cuff occlusion waves from a participant at a high risk of arteriosclerosis (Sub. 30); (d) post-cuff occlusion waves from a participant at a high risk of arteriosclerosis (Sub. 30). Figure 2 shows the transitions of arterial viscosity calculated at each blood pressure measurement. Figure 2(a and b) show the mean values for the healthy participants and the participants at a high risk of arteriosclerosis, respectively, and the shaded area represents the cuff occlusion. Figure 2(a) shows that, during the pre- and post-cuff occlusions, the arterial viscosity values of the healthy subjects are almost constant, and Fig. 2(b) shows that during the post-cuff occlusion, the arterial viscosity values of the subjects at a high risk of arteriosclerosis were lower than the pre-cuff occlusion arterial viscosity values. There was no significant difference in the estimated arterial viscosities between the pre- and post-cuff occlusions in the healthy participants. In contrast, the viscosity of the pre-cuff occlusion was significantly higher than that of the post-occlusions, except that for the first post-cuff occlusion in the participants at a high risk of arteriosclerosis. Estimated results of viscosity $\frac{{k}_{r}\eta }{V}$ during the ezFMD test: (a) average viscosity $\frac{{k}_{r}\eta }{V}$ of the healthy participants; (b) average viscosity of the subjects at a high risk of arteriosclerosis. Figure 3 shows the comparative results of the pre- and post-cuff occlusion arterial viscosity. Figure 3(a) shows the average arterial viscosity values in the healthy participants and the participants at a high risk of arteriosclerosis. There was no significant difference in the arterial viscosity between pre- and post-cuff occlusion in the healthy participants (pre-cuff occlusion: 2.56 ± 1.45, post-cuff occlusion: 2.51 ± 0.85, p = 0.87). However, the post-cuff occlusion arterial viscosity measured in the participants at a high risk of arteriosclerosis was significantly lower than the pre-cuff occlusion arterial viscosity (pre-cuff occlusion: 4.09 ± 2.26, post-cuff occlusion: 2.42 ± 1.28, p = 0.0016). Figure 3(b) shows a comparison of the change ratio of arterial viscosity %η for the healthy participants and those at a high risk of arteriosclerosis (healthy participants: 17.6 ± 53.8%, participants at a high risk of arteriosclerosis: −29.3 ± 39.5%). The change ratio of arterial viscosity %η of the participants at a high risk of arteriosclerosis was significantly lower than that of the healthy participants (p = 0.006). Estimated viscosity of healthy participants and participants at a high risk of arteriosclerosis: (a) comparison of pre- and post-cuff occlusions; (b) comparison of the change ratio of arterial viscosity %η. Figure 4 shows the ROC curves of the change ratio of arterial viscosity %η and the conventional endothelial function evaluation indices %FMD and %ezFMD. Table 2 presents the area under the ROC curve (AUC), sensitivity, specificity, and threshold values with the highest ROC analysis discrimination rate. The results showed that the AUC values of the change ratio of arterial viscosity %η and the conventional endothelial function evaluation indices %FMD and %ezFMD were 0.786, 0.695, 0.667, and 0.751, respectively. ROC analysis results. Table 2 Results of ROC analysis. This paper proposed an estimation method for the vascular viscosity and tested its applicability for endothelial function evaluation. In contrast to the current de facto standard methods such as the FMD and the ezFMD, which evaluate the endothelial function based on changes in vascular diameters, the proposed method uses the vascular viscosity as the arterial mechanical characteristic. The performed ROC analysis showed that the proposed method outperformed the FMD and the ezFMD (see Fig. 4 and Table 2). In addition, the proposed method has advantages in terms of clinical applicability because it employs only an oscillometric automatic sphygmomanometer that is commonly used in clinical practice and does not require highly skilled clinical technologists. In the experiments, we compared the arterial viscosities between the younger healthy group and the older high-risk group (see Tab. 1) to understand the artery behaviors caused by the cuff occlusion. During post-cuff occlusion, the cuff pulse wave and the cuff pulse wave velocity were increased compared with the corresponding pre-cuff occlusion values for both the healthy participant (Sub. 1) and the participant at a high risk of arteriosclerosis (Sub. 30). This indicates that, even for the participant at a high risk of arteriosclerosis, endothelium-derived NO can be detectable, although diminished17. However, the arterial viscosity showed a different trend between the healthy participants and the participants at a high risk of arteriosclerosis. The estimated post-cuff occlusion viscosity did not change significantly compared with the pre-cuff occlusion values in the healthy participants, and the post-cuff occlusion values were decreased compared to the pre-cuff occlusion values for participants at a high risk of arteriosclerosis (Fig. 1). The transition of the arterial viscosity for each blood pressure measurement also exhibited different trends between the healthy participants and the participants at a high risk of arteriosclerosis. As shown in Fig. 2, the estimated arterial viscosities of the healthy participants remained almost constant from the first post-cuff occlusion, and there were no significant differences in the viscosities between the pre- and post-cuff occlusions. In contrast, in the high-risk participants, significantly low viscosities were found through the second to the last post-cuff occlusion compared to that of the pre-occlusion, but there was no significant difference in the viscosity between the pre-occlusion and the first post-occlusion. This experimental finding in the high-risk participants is consistent with the previous study by Tagawa et al.18 where they reported delayed NO effects during reactive hyperaemia in human forearm vessels. Specifically, their finding that the role of NO during the early phase of reactive hyperaemia is minimal can explain that fact that the estimated viscosities in this paper did not change significantly in the first post-occlusion. In addition, modest yet significant effects of NO in the mid-to-late phase can explain the difference between the healthy participant group and the high-risk participant group: The effects of NO in the mid-to-late phase in the healthy participant group counteract the significant decreases in the viscosities after the second pre-cuff occlusions in the high-risk participant group. Further, the nearly constant viscosities observed through the first to the last post-occlusion in the healthy participants and through the second to the last post-cuff occlusions in the high-risk participants could be caused by the effects of the post-cuff occlusions. Because the post-cuff occlusions were applied for every 30 s, and they were conducted based on the protocol of the ezFMD12,13, the short intervals between the consecutive post-cuff occlusions may impose continuous stimulation to the endothelial function and prevent the viscosities from returning to the normal levels. The difference between the viscosities estimated from the two subject groups could be attributed to the fact that the vascular endothelial function maintains the vascular viscosity, and vascular dysfunction significantly decreases the arterial viscosity after occlusion. However, this straightforward interpretation does not consider the behaviors of the vascular smooth muscle during vasodilation. Mashima et al.19 reported the relationship between the load exerted on the muscle and the lengthening velocity of the muscle, as shown in Fig. 5; here, the load increases almost linearly in the low lengthening velocity region but gradually saturates in the high lengthening velocity region. The slope of the tangent line on this curve corresponds to the viscosity of the muscle. Then, the viscosity decreases in the high-risk subjects indicate the increases in the lengthening velocities caused by reactive hyperaemia after the pre-cuff occlusion, as shown in Fig. 6(b). In contrast, the viscosities of the healthy subjects remained almost constant even after the pre-cuff occlusion when reactive hyperaemia increased the lengthening velocity. This fact indicates that the relaxation of the vascular smooth muscle, and thus the decrease in the arterial stiffness caused by the endothelial function result in the increase in the vasodilation velocity, and load-lengthening velocity curve of the smooth muscle may be shifted as shown in Fig. 6(a). Therefore, the viscosity changes in the high-risk subject group suggest the impaired ability to change the arterial mechanical properties associated with endothelial dysfunction. Force-velocity (load-velocity) curves19. Pre- and post-cuff occlusion changes in the muscle mechanical characteristics and viscosity calculated from the relationships of the slope of tangent: (a) healthy participants; (b) participants at a high risk of arteriosclerosis. We also confirmed that the change ratio of arterial viscosity %η was significantly different between the healthy subjects and those at a high risk of arteriosclerosis; the change ratio of arterial viscosity %η in the healthy subjects showed a large variation (Fig. 3(b)). The ability of the change ratio of arterial viscosity %η to discriminate the healthy subjects and those at a high risk of arteriosclerosis was greater than or equal to that of %FMD11, which is a de facto standard index for evaluating vascular endothelial function. These results also indicate that the proposed index %η could be used to evaluate vascular endothelial function (see Fig. 4 and Table 2). The major limitation here is that the data presented in this paper could be potentially biased owing to age effects because the healthy group only included younger subjects (21.1 ± 1.85 years) and the high-risk group only included older subjects (63.6 ± 9.24 years). Although we confirmed that the proposed %η can differentiate between the healthy and high-risk groups, the age effects have to be examined in future studies by cross-comparison between the high-risk younger and healthy younger groups and between the high-risk older and healthy older groups. The experimental results confirmed the efficacy of the estimated arterial viscosity in discriminating between the healthy subjects and those at a high risk of arteriosclerosis. This indicates that the estimated arterial viscosity can be used as a risk measure of arteriosclerosis. Further, the proposed method may also be applied to assess the risks of diseases related to arterial viscosity. For example, Lionnet et al. reported that hyperviscosity is associated with the pathogenesis of arterial or venous thrombosis, and it can cause the life-threatening complications observed in hemoglobin SC patients such as pulmonary embolism and bone marrow necrosis20,21. Further analysis on the relationships between these diseases and the estimated arterial viscosity may expand the application scope of the proposed method. A total of 43 adults (average age ± standard deviation: 34.9 ± 20.9 years) participated in the experiment for measuring the proposed change ratio of arterial viscosity %η. The subject breakdown is as follows: 29 healthy subjects (age: 21.1 ± 1.9 years, Subjects 1–29), and 14 subjects at a high risk of arteriosclerosis (age: 63.6 ± 9.2 years, Subjects 30–43). Healthy subjects had no history of cardiovascular disease, liver disease, renal disease, autoimmune disease, or malignancy and had no coronary risk factors, including hypertension, dyslipidaemia, diabetes mellitus, and smoking. Subjects at a high risk of arteriosclerosis were defined as those affected by one or more of the following conditions: hypertension, diabetes, and dyslipidaemia. Hypertension was defined as having a systolic blood pressure of ≧140 mmHg or a diastolic blood pressure of ≧90 mmHg measured in a sitting position on at least three different occasions22. Diabetes mellitus was defined according to the guidelines of the American Diabetes Association23. Dyslipidaemia was defined according to the third report of the National Cholesterol Education Program24. This experiment was conducted in accordance with the Declaration of Helsinki and was subject to the approval of the Epidemiological Research Ethics Review Committee of Hiroshima University (https://upload.umin.ac.jp. Unique identifier. UMIN000004902). Informed consent was obtained from the subjects. An automatic sphygmomanometer (OPV-1510, Nihon Kohden Corporation; cuff: YP-963T, Nihon Kohden Corporation) widely used in general clinical practice was used to measure the cuff pressure and cuff pulse wave in this experiment. The cuff pressure and cuff pulse wave data were saved on a multimedia card (Transcend) inside the device at a sampling frequency of 125 Hz. The subjects assumed a supine position and were allowed to rest. Their blood pressure was then measured once before the 5-min cuff occlusion and five times after that occlusion. First, to verify the change in arterial viscosity from pre- to post-cuff occlusion, the arterial viscosities krη/V were measured in the healthy subjects and the subjects at a high risk of arteriosclerosis. Statistical processing was performed with the Wilcoxon signed rank-sum test, which is a nonparametric test, at a significance level of 5%. Next, the change ratio of arterial viscosity %η of the healthy subjects was compared with that of the subjects at a high risk of arteriosclerosis. Statistical processing was performed using the Mann–Whitney U test, which is a nonparametric test, at a significance level of 5%. Furthermore, to examine the accuracy of the change ratio of arterial viscosity %η for discriminating the healthy subjects and those at a high risk of arteriosclerosis, the discrimination accuracy of %η was compared with that of %FMD11 and %ezFMD13, which are also indices for evaluating vascular endothelial function. Comparison of the discrimination accuracies among the indicators was conducted via ROC analysis, which is an accuracy screening method, and comparing the corresponding AUC values. Log-linearized under-cuff viscosity index Algorithm for estimating arterial viscosity Conventionally, it is known that the relationship between the vascular diameter and intravascular pressure is nonlinear in humans25. Hayashi et al. measured the static arterial diameter invivo when artificial internal pressure was applied to the artery. In the physiological pressure range (60 to 160 mmHg), the results showed that the relationship between the logarithmic value of the intravascular pressure for the reference intravascular pressure and the ratio of the vascular diameter for the reference vascular diameter was linear. Thus, Hayashi et al. proposed the Stiffness Parameter25, which is an estimation index of arterial elasticity and is less susceptible to changes in intravascular pressure. However, it is difficult to express vascular characteristics only using elastic characteristics because invivo experiments have reported arteries to be viscoelastic bodies rather than simple elastic bodies26,27. To resolve this issue, Hirano et al. proposed the log-linearized peripheral arterial viscoelastic model and evaluated the arterial viscosity considering the nonlinear relationship between the peripheral intravascular pressure and peripheral artery diameter26. The arterial viscoelastic estimation method proposed by Hirano et al. requires the simultaneous measurement of continuous arterial pressures and changes in vascular diameter, which limits the clinical applications of the method. Evaluating the arterial viscosity with an automatic oscillometric sphygmomanometer, which does not require continuous blood pressure measurements, may broaden the clinical applications of this method. This paper proposes an arterial viscoelastic model in which blood pressure is measured by the cuff of an automatic sphygmomanometer wrapped around the upper arm. The proposed model is based on the log-linearized peripheral arterial viscoelastic model proposed by Hirano et al. The log-linearized peripheral arterial viscoelastic model is expressed as follows: $$p(t)\cong \tilde{\mu }\ddot{\varepsilon }(t)+\tilde{\eta }\dot{\varepsilon }(t)+\exp \{\tilde{\beta }\varepsilon (t)+{P}_{{\tilde{\beta }}_{0}}+{P}_{{\tilde{\beta }}_{nl}}(\varepsilon (t))\},$$ where t represents time, P(t), ${P}_{{\tilde{\beta }}_{0}}$, and ${P}_{{\tilde{\beta }}_{nl}}(\varepsilon (t))$ are the blood pressure, DC component of the intravascular pressure, and error component based on $\tilde{\beta }\varepsilon (t)$ occurring when the blood pressure drops, respectively. $\tilde{\mu }$, $\tilde{\eta }$, and $\tilde{\beta }$ are coefficients corresponding to the arterial wall inertia, viscosity, and stiffness, respectively. ε(t), $\dot{\varepsilon }(t)$, and $\ddot{\varepsilon }(t)$ represent the strain of the arterial diameter, strain velocity, and strain acceleration, respectively. Hirano et al. measured ε(t) $\dot{\varepsilon }(t)$ and $\ddot{\varepsilon }(t)$ using a photoplethysmograph. In this study, ε(t), $\dot{\varepsilon }(t)$, and $\ddot{\varepsilon }(t)$ were determined based on the changes in the vascular volume under the cuff measured by the automatic oscillometric sphygmomanometer. Based on Equation (1), the vascular mechanical properties were approximated with the following equation, using the volumetric vascular strain and volume elasticity28: $$p(t)\cong \hat{\mu }\frac{d\ddot{V}(t)}{V}+\hat{\eta }\frac{d\dot{V}(t)}{V}+\exp \{\hat{\beta }\frac{dV(t)}{V}+{p}_{{\hat{\beta }}_{0}}+{p}_{{\hat{\beta }}_{nl}}(\frac{dV(t)}{V})\},$$ where the pressure p(t) applied from inside the vessel on the outside is considered positive. $\hat{\mu }$, $\hat{\eta }$, and $\hat{\beta }$ are coefficients corresponding to the arterial wall inertia, viscosity, and log-linearized arterial wall stiffness (volume modulus), respectively. Additionally, dV(t)/V, $d\dot{V}(t)/V$, and $d\ddot{V}(t)/V$ are the volumetric vascular change, volumetric change velocity, and volumetric change acceleration, respectively. dV(t) expresses the change in vascular volume V when p(t) = 0; in an automatic oscillometric sphygmomanometer, slight pressure fluctuations occur in the cuff (cuff volume pulse wave Ppulse(t)) in response to this change in vascular volume dV(t). In this study, viscoelastic parameters were estimated using biological signals obtained from the automatic oscillometric sphygmomanometer and Equation (2). The upper arm covered with the cuff was strongly compressed, and biological tissues other than the artery were assumed to be incompressible while measuring blood pressure29. The cuff pressure can be considered to act from the outside toward the inside of the arterial wall, as shown in Fig. 7. The pressure p(t) acting on the arterial wall is then given as the difference between the intravascular pressure Pb(t) and cuff pressure Pcuff(t): $$p(t)={P}_{b}(t)-{P}_{cuff}(t).$$ Schematic of an artery in the upper arm surrounded by a cuff. It was assumed that the internal pressure and volume of the cuff were constant. The Boyle–Charles law holds for the internal pressure, and the cuff volume within the cuff pressure range was used while measuring30,31. The volumetric vascular change dV(t) was then approximated using the following equation because it is proportional to the cuff pulse wave Ppulse(t)13: $$dV(t)={k}_{r}{P}_{pulse}(t),$$ where kr is a positive constant. Considering only the time at which dV(t) = 0 and Pb(t) is equal to or higher than the mean blood pressure, the pressure ${p}_{{\hat{\beta }}_{nl}}(dV(t)/V)$ in Equation (2), which is below the mean blood pressure, can be ignored. Equation (2) can be converted using Equations (3) and (4) when the arterial inertia is ignored owing to its minute value. $${P}_{b}(t)-{P}_{cuff}(t)=\frac{{k}_{r}\hat{\eta }}{V}{\dot{P}}_{pulse}(t)+\exp \{{p}_{{\hat{\beta }}_{0}}\},$$ where t is defined as the time at which the cuff pulse wave crosses 0 (Ppulse(t) = 0). Equation (5) represents the model of the arterial viscosity estimation proposed in this paper. The arterial viscosity is estimated based on this model equation. In this study, the automatic oscillometric sphygmomanometer reduced the cuff pressure stepwise for every two beats detected. The arterial viscosity was calculated using only the cuff pulse wave of the second beat in the two beats detected by decompression because every first beat could potentially be affected by disturbances caused by cuff decompression. The commercial automatic oscillometric sphygmomanometer displays the systolic, diastolic, and mean arterial pressure values for each arterial pressure measurement. The mean arterial pressure is determined based on the maximum peak-to-peak value of the cuff pulse wave, which indicates the minimum difference between the intravascular and extravascular pressures. Systolic and diastolic arterial pressures are determined by an algorithm in each automatic sphygmomanometer, which satisfies a certain criterion. The algorithm for calculating systolic and diastolic pressures is not based on the physical properties of arterial vessels29. Thus, the arterial viscosity obtained using the proposed method was estimated using the cuff pressure around the mean arterial pressure and the cuff pulse wave velocity at which the cuff pulse wave amplitude was maximal. The time t satisfying Equation (5) and the following condition is defined as the reference time t. This condition is such that the maximum pulse wave and maximum pulse wave velocity observed from the three beats, i.e., the maximum pulse wave and adjacent waves of its maximum wave, are obtained. At the time t satisfying Equation (5), the time before j beats ($j\in {\boldsymbol{{\mathbb{N}}}}$) was set to tj. The difference between the time t and the time tj was then obtained from Equation (5): $$({P}_{cuff}({t}_{\ast })-{P}_{cuff}({t}_{j}))=\frac{{k}_{r}\hat{\eta }}{V}({\dot{P}}_{pulse}({t}_{j})-{\dot{P}}_{pulse}({t}_{\ast })),$$ where Pb(t*) is assumed to be equal to Pb(tj). In this study, Equation (6) was simultaneously solved for all times tj. The solution ${k}_{r}\hat{\eta }/V$ obtained using the linear least squares method is defined as the estimated arterial viscosity value. Evaluation of vascular endothelial function In humans, the peripheral part of the upper arm becomes oxygen deficient with the cuff occlusion of the upper arm. The blood flow increases immediately after release of the cuff occlusion because the state of oxygen deficiency in the periphery needs to be eliminated. In case of normal vascular endothelial function is, NO is released from the endothelial cells because the shear stress associated with the increase in blood flow is applied to the vascular wall, causing the vascular smooth muscle to relax and the pulsation corresponding to the vascular volume to increase29. However, it becomes difficult to release NO from endothelial cells when the vascular endothelial function is abnormal, which prevents vascular smooth muscle relaxation17. By applying this biological reaction, vascular endothelial function was evaluated based on the arterial viscosity calculated before and after cuff occlusion. A schematic of the proposed vascular endothelial function evaluation system is shown in Fig. 8. The arterial viscosity of the proposed model is calculated from the cuff pressure and cuff pulse wave measured by the automatic oscillometric sphygmomanometer. The evaluated vascular endothelial function is displayed on the monitor. The protocol of the ezFMD test13 was adopted as the experimental protocol. The ezFMD test is a method for evaluating vascular endothelial function, which is calculated from the change ratio of the pulse wave amplitude before and after cuff occlusion. The ezFMD test does not require an ultrasonic device for evaluating the endothelial function. The subject assumes a supine position and rests during the test. The subject's arterial pressure is measured once before cuff occlusion and five times after cuff occlusion. The period of cuff occlusion is 5 min. The cuff pressure and cuff pulse wave during the ezFMD test are stored as numerical data. The change rate of arterial viscosity %η from pre- to post-cuff occlusion is expressed as follows: $${\rm{ \% }}\eta =(\frac{{\frac{{k}_{r}\eta }{V}}_{{\rm{p}}{\rm{o}}{\rm{s}}{\rm{t}}3-5}-{\frac{{k}_{r}\eta }{V}}_{{\rm{p}}{\rm{r}}{\rm{e}}}}{{\frac{{k}_{r}\eta }{V}}_{{\rm{p}}{\rm{r}}{\rm{e}}}})\times 100,$$ where (krη/V)post3−5 is the average arterial viscosity value calculated from the 3rd to 5th post-cuff occlusion arterial pressure measurement, and (krη/V)pre is the pre-cuff occlusion arterial viscosity. The post-cuff arterial viscosity is taken as the average of the 3rd to the 5th measurements to avoid a condition in which the blood flow immediately after the cuff occlusion is unstable. In this study, the proposed method was used to distinguish between healthy subjects and subjects at a high risk of arteriosclerosis, and the vascular endothelial function of the subjects was evaluated using the change ratio of arterial viscosity %η. Overview of the proposed examination system. 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Journal of Biomechanics 3, 357–379 (1970). This research was partly supported by the Transportation Technology Development Promotion Competitive Funding Program from Ministry of Land, Infrastructure, Transport and Tourism, and the Center of Innovation Program from Japan Science and Technology Agency. Department of System Cybernetics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Hiroshi Tanaka, Akihisa Mito & Toshio Tsuji Academic Institute, College of Engineering, Shizuoka University, Hamamatsu, 432-8561, Japan Harutoyo Hirano Department of System Cybernetics, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Zu Soh Department of Anesthesiology and Critical Care, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, 734-8553, Japan Ryuji Nakamura, Noboru Saeki & Masashi Kawamoto Department of Regeneration and Medicine, Research Center for Radiation Genome Medicine, Research Institute for Radiation Biology and Medicine, Hiroshima University, Hiroshima, 734-8553, Japan Yukihito Higashi Division of Regeneration and Medicine, Hiroshima University Hospital, Hiroshima, 734-8551, Japan Department of Cardiovascular Physiology and Medicine, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, 734-8553, Japan Masao Yoshizumi Hiroshi Tanaka Akihisa Mito Ryuji Nakamura Noboru Saeki Masashi Kawamoto Toshio Tsuji H.T. and A.M. wrote the initial draft and conducted experiments. H.H. and Z.S. edited the manuscript. R.N., N.S., M.K., Y.H., M.Y. contributed to data collection and critically reviewed the manuscript. T.T. designed the study, conceived the experiments, and revised the manuscript. All authors reviewed the manuscript. Correspondence to Toshio Tsuji. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Tanaka, H., Mito, A., Hirano, H. et al. Estimation of Arterial Viscosity Based on an Oscillometric Method and Its Application in Evaluating the Vascular Endothelial Function. Sci Rep 9, 2609 (2019). https://doi.org/10.1038/s41598-019-38776-4 Correlation analysis of human upper arm parameters to oscillometric signal in automatic blood pressure measurement Bomi Lee Jae-Hak Jeong Yong-Hwa Park Peripheral arterial stiffness during electrocutaneous stimulation is positively correlated with pain-related brain activity and subjective pain intensity: an fMRI study Fumiya Arikuni Shigeto Yamawaki By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. About Scientific Reports Guide to referees Scientific Reports (Sci Rep) ISSN 2045-2322 (online)	CommonCrawl
\begin{document} \begin{frontmatter} \title{Averaging over fast variables in the fluid limit for Markov chains: Application to the supermarket model with memory} \runtitle{Averaging over fast variables} \begin{aug} \author[A]{\fnms{M. J.} \snm{Luczak}\corref{}\thanksref{t1}\ead[label=e1]{[email protected]}} \and \author[B]{\fnms{J. R.} \snm{Norris}\thanksref{t2}\ead[label=e2]{[email protected]}} \runauthor{M. J. Luczak and J. R. Norris} \affiliation{London School of Economics and University of Cambridge} \address[A]{School of Mathematical Sciences\\ Queen Mary, University of London\\ Mile End Road\\ London E1 4NS\\ United Kingdom\\ \printead{e1}} \address[B]{Statistical Laboratory\\ Centre for Mathematical Sciences\\ University of Cambridge\\ Wilberforce Road\\ Cambridge, CB3 0WB\\ United Kingdom\\ \printead{e2}} \end{aug} \thankstext{t1}{Supported in part by a STICERD grant at the LSE, and by EPSRC Leadership Fellowship EP/J004022/2. Part of this work was accomplished while visiting the Mittag-Leffler Institute.} \thankstext{t2}{Supported by EPSRC Grant EP/E01772X/1.} \received{\smonth{3} \syear{2010}} \revised{\smonth{8} \syear{2011}} \begin{abstract} We set out a general procedure which allows the approximation of certain Markov chains by the solutions of differential equations. The chains considered have some components which oscillate rapidly and randomly, while others are close to deterministic. The limiting dynamics are obtained by averaging the drift of the latter with respect to a local equilibrium distribution of the former. Some general estimates are proved under a uniform mixing condition on the fast variable which give explicit error probabilities for the fluid approximation. Mitzenmacher, Prabhakar and Shah [In \textit{Proc. 43rd Ann. Symp. Found. Comp. Sci.} (2002) 799--808, IEEE] introduced a variant with memory of the ``join the shortest queue'' or ``supermarket'' model, and obtained a limit picture for the case of a stable system in which the number of queues and the total arrival rate are large. In this limit, the empirical distribution of queue sizes satisfies a differential equation, while the memory of the system oscillates rapidly and randomly. We illustrate our general fluid limit estimate by giving a proof of this limit picture. \end{abstract} \begin{keyword}[class=AMS] \kwd[Primary ]{60J28} \kwd[; secondary ]{60K25}. \end{keyword} \begin{keyword} \kwd{Join the shortest queue} \kwd{supermarket model} \kwd{supermarket model with memory} \kwd{law of large numbers} \kwd{exponential martingale inequalities} \kwd{fast variables} \kwd{correctors}. \end{keyword} \end{frontmatter} \section{A general fluid limit estimate}\label{GFL} We describe a general framework to allow the incorporation of averaging over fast variables into fluid limit estimates for Markov chains, building on the approach used in~\cite{MR2395153}. The main results of this section, Theorems~\ref{FLE} and~\ref{FLET}, establish explicit error probabilities for the fluid approximation under assumptions which can be verified from knowledge of the transition rates of the Markov chain. Also see \cite {MR2288709} for related results. \subsection{Outline of the method}\label{OM} Let $X=(X_t)_{t\ge0}$ be a continuous-time Markov chain with countable state-space $S$ and with generator matrix $Q=(q(\xi,\xi')\dvtx\break\xi,\xi'\in S)$. Assume that the total jump rate $q(\xi)$ is finite for all states $\xi $, and that $X$ is nonexplosive. Then the law of $X$ is determined uniquely by $Q$ and the law of $X_0$. Make a choice of \textit{fluid coordinates} $x^i\dvtx S\to{\mathbb R}$, for $i=1,\ldots ,d$, and write ${\mathbf x}=(x^1,\ldots,x^d)\dvtx S\to{\mathbb R}^d$. Consider the ${\mathbb R}^d$-valued process ${\mathbf X}=({\mathbf X}_t)_{t\ge0}$ given by ${\mathbf X}_t=(X^1_t,\ldots,X^d_t)={\mathbf x}(X_t)$. Call ${\mathbf X}$ the \textit{slow} or \textit{fluid variable}. Define for each $\xi\in S$ the \textit{drift vector} \[ {\beta}(\xi)=Q{\mathbf x}(\xi)=\sum_{\xi'\not=\xi}\bigl({\mathbf x}(\xi ')-{\mathbf x}(\xi)\bigr)q(\xi,\xi'). \] Also, make a choice of an \textit{auxiliary coordinate} $y\dvtx S\to I$, for some countable set~$I$, and set $Y_t=y(X_t)$. Call the process $Y=(Y_t)_{t\ge0}$ the \textit{fast variable}. For $\xi\in S$ and $y'\in I$ with $y'\not=y(\xi)$, write ${\gamma}(\xi ,y')$ for the total rate at which $Y$ jumps to $y'$ when $X$ is at $\xi$. Thus \[ {\gamma}(\xi,y')=\sum_{\xi'\dvtx y(\xi')=y'}q(\xi,\xi'). \] Choose a subset $U$ of ${\mathbb R}^d$ and a function $b\dvtx U\times I\to{\mathbb R}^d$. Choose also, for each $x\in U$, a generator matrix $G_x=(g(x,y,y')\dvtx y,y'\in I)$ having a unique invariant distribution $\pi_x=(\pi(x,y)\dvtx y\in I)$. These choices are to be made so that ${\beta}(\xi)$ is close to $b({\mathbf x}(\xi),y(\xi))$ and ${\gamma}(\xi,y')$ is close to $g({\mathbf x}(\xi),y(\xi),y')$ whenever ${\mathbf x}(\xi)\in U$ and $y'\in I$. Define for $x\in U$ \[ \bar b(x)=\sum_{y\in I}b(x,y)\pi(x,y). \] Then,\vspace{1pt} under regularity assumptions to be specified later, there exists a function $\chi\dvtx U\times I\to{\mathbb R}^d$ such that \begin{equation}\label{GCH-0} G\chi(x,y)=\sum_{y'\in I}g(x,y,y')\chi(x,y')=b(x,y)-\bar b(x). \end{equation} Make a choice of such a function $\chi$. Call $\chi$ the \textit{corrector for $b$}. Fix $x_0\in U$. We will assume that $\bar b$ is Lipschitz on $U$. Then the differential equation $\dot x_t=\bar b(x_t)$ has a unique maximal solution $(x_t)_{t<{\zeta}}$ in $U$ starting from $x_0$. Fix $t_0\in[0,{\zeta})$. Then for $t\le t_0$, \begin{equation}\label{DE} x_t=x_0+\int_0^t\bar b(x_s)\,ds. \end{equation} Define for $\xi\in S$ with ${\mathbf x}(\xi)\in U$ \[ \bar{\mathbf x}(\xi)={\mathbf x}(\xi)-\chi({\mathbf x}(\xi),y(\xi)). \] Let $T$ be a stopping time such that ${\mathbf X}_t\in U$ for all $t\le T$. Then, under regularity assumptions to be specified later, for $t\le T$, \begin{equation}\label{ME} \bar{\mathbf x}(X_t)=\bar{\mathbf x}(X_0)+M_t+\int_0^t\bar{\beta}(X_s)\,ds, \end{equation} where $M=M^{\bar{\mathbf x}}$ is a martingale and where \begin{equation}\label{BB} \bar{\beta}=Q\bar{\mathbf x}={\beta}-Q(\chi({\mathbf x},y)). \end{equation} On subtracting equations (\ref{DE}) and (\ref{ME}) we obtain for $t\le T\wedge t_0$ \begin{eqnarray}\label{KEY} {\mathbf X}_t-x_t&=&{\mathbf X}_0-x_0+\chi({\mathbf X}_t,Y_t)-\chi({\mathbf X}_0,Y_0)+M_t+\int _0^t{\Delta}(X_s)\,ds\nonumber\\[-8pt]\\[-8pt] &&{} +\int_0^t\bigl({\beta}(X_s)-b({\mathbf X}_s,Y_s)\bigr)\,ds+\int_0^t\bigl(\bar b({\mathbf X} _s)-\bar b(x_s)\bigr)\,ds,\nonumber \end{eqnarray} where ${\Delta}=G\chi({\mathbf x},y)-Q(\chi({\mathbf x},y))$. The discussion in the present paragraph is intended for orientation only, and will play no essential role in the derivation of our results. Fix $U_0\subseteq U$ such that for all $\xi,\xi'\in S$ with ${\mathbf x}(\xi)\in U_0$ and $q(\xi,\xi')>0$ we have ${\mathbf x}(\xi')\in U$. Assume that $T$ is chosen so that ${\mathbf X}_t\in U_0$ for all $t\le T$. Define for $\xi\in S$ with ${\mathbf x}(\xi)\in U_0$ the \textit{diffusivity tensor} ${\alpha}(\xi)\in{\mathbb R}^d\otimes{\mathbb R}^d$ by \begin{equation}\label{DTD} {\alpha}^{ij}(\xi)=\sum_{\xi'\not=\xi}\bigl(\bar{\mathbf x}^i(\xi')-\bar {\mathbf x}^i(\xi)\bigr)\bigl(\bar{\mathbf x}^j(\xi')-\bar{\mathbf x}^j(\xi )\bigr)q(\xi,\xi') \end{equation} and define for $t\le T$ \[ N_t=M_t\otimes M_t-\int_0^t{\alpha}(X_s)\,ds. \] Then, under regularity assumptions, $N$ is a martingale in ${\mathbb R} ^d\otimes{\mathbb R}^d$. Choose a function $a\dvtx U_0\times I\to{\mathbb R}^d\otimes{\mathbb R}^d$ and set \[ \bar a(x)=\sum_{y\in I}a(x,y)\pi(x,y). \] This choice is to be made so that ${\alpha}(\xi)$ is close to $a({\mathbf x}(\xi),y(\xi))$ whenever \mbox{${\mathbf x}(\xi)\in U_0$}. Suppose we can also find a corrector for $a$, that is, a function $\tilde\chi\dvtx U_0\times I\to{\mathbb R}^d\otimes{\mathbb R}^d$ such that \begin{equation}\label{GCH-1} G\tilde\chi(x,y)=a(x,y)-\bar a(x). \end{equation} Then, for $t\le T$, \begin{eqnarray}\label{KEYT} \int_0^t{\alpha}(X_s)\,ds&=&\tilde\chi({\mathbf X}_t,Y_t)-\tilde\chi({\mathbf X} _0,Y_0)-\tilde M_t+\int_0^t\tilde{\Delta}(X_s)\,ds\nonumber\\[-8pt]\\[-8pt] &&{} +\int_0^t\bigl({\alpha}(X_s)-a({\mathbf X}_s,Y_s)\bigr)\,ds+\int_0^t\bar a({\mathbf X}_s)\,ds,\nonumber \end{eqnarray} where $\tilde{\Delta}=G\tilde\chi({\mathbf x},y)-Q(\tilde\chi({\mathbf x},y))$ and, under suitable regularity conditions, $\tilde M=M^{\tilde \chi}$ is a martingale up to $T$. The martingale terms $M$ and $\tilde M$ in (\ref{KEY}) and (\ref {KEYT}) can be shown to be small, under suitable conditions, using the following standard type of exponential martingale inequality. In the form given here it may be deduced, for example, from \cite {MR2395153}, Proposition 8.8, by setting $f=\theta\phi$, $A=\theta^2e^{\theta J}{\varepsilon}/2$ and $B=\theta {\delta}$. \begin{proposition}\label{EMI} Let $\phi$ be a function on $S$. Define \[ M_t=M^\phi_t=\phi(X_t)-\phi(X_0)-\int_0^t Q\phi(X_s)\,ds. \] Write $J=J(\phi)$ for the maximum possible jump in $\phi(X)$, thus \[ J=\sup_{\xi,\xi'\in S, q(\xi,\xi')>0}\|\phi(\xi')-\phi(\xi)\|. \] Define a function ${\alpha}={\alpha}^\phi$ on $S$ by \[ {\alpha}(\xi)=\sum_{\xi'\not=\xi}\{\phi(\xi')-\phi(\xi)\}^2q(\xi ,\xi'). \] Then, for all ${\delta},{\varepsilon}\in(0,\infty)$ and all stopping times $T$, we have \[ \mathbb{P}\biggl(\sup_{t\le T}M_t\ge{\delta}\mbox{ and }\int_0^T{\alpha} (X_t)\,dt\le{\varepsilon}\biggr)\le\exp\{-{\delta}^2/(2{\varepsilon} e^{\theta J})\}, \] where $\theta\in(0,\infty)$ is determined by $\theta e^{\theta J}={\delta }/{\varepsilon}$. \end{proposition} Now, if ${\beta},{\gamma},{\alpha}$ are well approximated by $b,g,a$ and if we can show that the corrector terms in (\ref{KEY}) and (\ref{KEYT}) are insignificant, then we may hope to use these equations to show that the path $(x_t\dvtx t\le t_0)$ provides a good (first order) approximation to $({\mathbf X}_t\dvtx t\le t_0)$ and, moreover, that the fluctuation process $({\mathbf X}_t-x_t\dvtx t\le t_0)$ is approximated (to second order) by a Gaussian process $(F_t\dvtx t\le t_0)$ given by \[ F_t=F_0+B_t+\int_0^t\nabla\bar b(x_s)F_s\,ds, \] where $(B_t\dvtx t\le t_0)$ is a zero-mean Gaussian process in ${\mathbb R}^d$ with covariance \[ {\mathbb E}(B_s\otimes B_t)=\int_0^{s\wedge t}\bar a(x_r)\,dr.\vadjust{\goodbreak} \] Our aim in the rest of this section is to give an explicit form of the first order approximation with optimal error scale, that is, of the same order as the scale of deviation predicted by the second order approximation. The next subsection contains some preparatory material on correctors. A reader who wishes to understand only the statement of the fluid limit estimate can skip directly to Section~\ref{SOTE}.\vspace{-2pt} \subsection{Correctors}\label{EC} In order to implement the method just outlined, it is necessary either to come up with explicit correctors or to appeal to a general result which guarantees the existence, subject to verifiable conditions, of correctors with good properties. In this subsection we obtain such a general result. In fact, we shall find conditions which guarantee the existence, for each bounded measurable function $f$ on $U\times I$, of a good \textit{corrector for $f$}, that is to say, a function $\chi =\chi_f$ on $U\times I$ such that \[ G\chi(x,y)=f(x,y)-\bar f(x), \] where \[ \bar f(x)=\sum_{y\in I}f(x,y)\pi(x,y). \] Moreover, we shall see that $\chi_f$ depends linearly on $f$ and we shall obtain a uniform bound and a continuity estimate for $\chi_f$. Assume that there is a constant $\nu\in(0,\infty)$ such that, for all $x\in U$ and all $y\in I$, the total rate of jumping from $y$ under $G_x$ does not exceed $\nu$. Then we can choose an auxiliary measurable space $E$, with a $\sigma $-field ${\mathcal E}$, a family of probability measures $\mu=(\mu_x\dvtx x\in U)$ on $(E, {\mathcal E})$ and a measurable function $F\dvtx I\times E\to I$ such that, for all $x\in U$ and all $y,y'\in I$ distinct, \begin{equation}\label{GNM} g(x,y,y')=\nu\mu_x\bigl(\{v\in E\dvtx F(y,v)=y'\}\bigr). \end{equation} Let $N=(N(t)\dvtx t\ge0)$ be a Poisson process of rate $\nu$. Fix $x\in U$ and let $V=(V_n\dvtx n\in{\mathbb N})$ be a sequence of independent random variables in $E$, all with law $\mu_x$. Thus \[ g(x,y,y')=\nu\mathbb{P}\bigl(F(y,V_n)=y'\bigr) \] for all pairs of distinct states $y,y'$ and all $n$. Fix a reference state $\bar y\in I$. Given $y\in I$, set $Z_0=y$ and $\bar Z_0=\bar y$ and define recursively for $n\ge0$, \[ Z_{n+1}=F(Z_n,V_{n+1}),\qquad \bar Z_{n+1}=F(\bar Z_n,V_{n+1}). \] Set $Y_t=Z_{N(t)}$ and $\bar Y_t=\bar Z_{N(t)}$. Then $Y=(Y_t)_{t\ge 0}$ and $\bar Y=(\bar Y_t)_{t\ge0}$ are both Markov chains in $I$ with generator matrix $G_x$, starting from $y$ and $\bar y$, respectively,\setcounter{footnote}{2}\footnote{The process $Y$ introduced here is not the fast variable, also denoted $Y$ in the rest of the paper: the current $Y$ is to be considered as a local approximation of the fast variable.\label{NFV}} and are realized on the same probability space. We call the triple $(\nu,\mu,F)$ a \textit{coupling mechanism}. Define the \textit{coupling time} \[ T_c=\inf\{t\ge0\dvtx Y_t=\bar Y_t\}.\vadjust{\goodbreak} \] Assume that, for some positive constant ${\tau}$, for all $x\in U$ and all $y,\bar y\in I$, \begin{equation}\label{MTC} m(x,y,\bar y)={\mathbb E}_{(x,y,\bar y)}(T_c)\le{\tau}. \end{equation} Fix a bounded measurable function $f$ on $U\times I$ and set \begin{equation}\label{CFF} \chi(x,y)={\mathbb E}_{(x,y)}\int_0^{T_c}\bigl(f(x,Y_t)-f(x,\bar Y_t)\bigr)\,dt. \end{equation} Then $\chi$ is well defined and, for all $x\in U$ and all $y\in I$, \begin{equation}\label{CTB} \|\chi(x,y)\|\le2{\tau}\\|f\\|_\infty. \end{equation} \begin{proposition} The function $\chi$ is a corrector for $f$. \end{proposition} \begin{pf} In the proof we suppress the variable $x$. Note first that, if instead of taking $\bar Z_0=\bar y$, we start $\bar Z$ randomly with the invariant distribution $\pi$, then we change the value of $\chi$ by a constant independent of $y$. Hence, it will suffice to establish the corrector equation $G\chi=f-\bar f$ in this case. Fix ${\lambda}>0$ and define \[ \phi^{\lambda}(y)={\mathbb E}\int_0^{T_{\lambda}}f(Y_t)\,dt,\qquad \bar\phi^{\lambda}={\mathbb E}\int_0^{T_{\lambda}}f(\bar Y_t)\,dt, \] where $T_{\lambda}=T_1/{\lambda}$, with $T_1$ an independent exponential random variable of parameter $1$. Then, since $Y$ and $\bar Y$ coincide after $T_c$, \[ \bar\phi^{\lambda}-\phi^{\lambda}(y) ={\mathbb E}\int_0^{T_{\lambda}\wedge T_c}\bigl(f(\bar Y_t)-f(Y_t)\bigr)\,dt\to\chi(y) \] as ${\lambda}\to0$. By elementary conditioning arguments, $(G-{\lambda})\phi^{\lambda }+f=0$ and \mbox{${\lambda}\bar\phi^{\lambda}=\bar f$}, so \[ (G-{\lambda})(\bar\phi^{\lambda}-\phi^{\lambda})=f-\bar f. \] On passing to the limit ${\lambda}\to0$ in this equation, using bounded convergence we find that $G\chi=f-\bar f$, as required. \end{pf} We remark that the corrector $\chi(x,\cdot)$ in fact depends only on $f$, $G_x$ and the choice of $\bar y$, as the preceding proof makes clear. The further choice of a coupling mechanism is a way to obtain estimates on $\chi$. The following estimate will be used in dealing with the ${\Delta}$ term in (\ref{KEY}). We write $\\|\mu_x-\mu_{x'}\\|$ for the total variation distance between $\mu_x$ and $\mu_{x'}$. \begin{proposition}\label{CE} For all $x,x'\in U$ and all $y\in I$, \begin{eqnarray}\label{CEE} \|\chi(x,y)-\chi(x',y)\| &\le& 2{\tau}\sup_{z\in I}\|f(x,z)-f(x',z)\|\nonumber\\[-8pt]\\[-8pt] &&{}+2\nu{\tau} ^2\\|f\\|_\infty\\|\mu_x-\mu_{x'}\\|.\nonumber \end{eqnarray} \end{proposition} \begin{pf} By a standard construction (maximal coupling) there exists a sequence of independent random variables $((V_n,V_n')\dvtx n\in{\mathbb N})$ in $E\times E$ such that $V_n$ has distribution $\mu_x$, $V_n'$ has distribution $\mu _{x'}$ and $\mathbb{P}(V_n\not=V_n')=\frac12\\|\mu_x-\mu_{x'}\\|= \sup_{A \in\mathcal E}\|\mu_x(A)- \mu_{x'}(A)\|$, for all $n$. Write $({\cal F}_t)_{t\ge0}$ for the filtration of the marked Poisson process obtained by marking $N$ with the random variables $(V_n,V_n')$.\vspace{2pt} Construct $(Y,\bar Y)$ from $N$ and $(V_n\dvtx n\in{\mathbb N})$ as above. Similarly\vspace{1pt} construct $(Y',\bar Y')$ from $N$ and $(V_n'\dvtx n\in{\mathbb N})$. Recall that $T_c=\inf\{t\ge0\dvtx Y_t=\bar Y_t\}$ and set $T_c'=\inf\{ t\ge0\dvtx Y_t'=\bar Y_t'\}$. Set ${\lambda}=\frac12\nu\\|\mu_x-\mu_{x'}\\|$ and set \[ D=\inf\{t\ge0\dvtx(Y_t,\bar Y_t)\not=(Y_t',\bar Y_t')\}. \] Then the process $t\mapsto1_{\{D\le t\}}-{\lambda}t$ is an $({\cal F} _t)_{t\ge0}$-supermartingale and $T_c$ is an $({\cal F}_t)_{t\ge0}$-stopping time. So, by optional stopping, we have $\mathbb{P}(D\le T_c)\le{\lambda}{\mathbb E} (T_c)\le{\lambda}{\tau}$. Moreover, by the strong Markov property, on $\{D\le T_c\}$, we have ${\mathbb E} (T_c-D\|{\cal F}_D)=m(x,Y_D,\bar Y_D)\le{\tau}$ so, for any function $g\dvtx I\to{\mathbb R}^d$, with $\|g\|\le\\|f\\|_\infty$, \[ {\mathbb E}\biggl\|\int_{D\wedge T_c}^{T_c}g(Y_t)\,dt\biggr\|\le{\tau}\\|f\\|_\infty \mathbb{P}(D\le T_c)\le{\lambda}{\tau}^2\\|f\\|_\infty. \] On the other hand, \[ \int_0^{D\wedge T_c}g(Y_t)\,dt=\int_0^{D\wedge T_c'}g(Y_t')\,dt \] so \[ \biggl\|{\mathbb E}\int_0^{T_c}g(Y_t)\,dt-{\mathbb E}\int_0^{T_c'}g(Y_t')\,dt\biggr\|\le 2{\lambda}{\tau}^2\\|f\\|_\infty=\nu{\tau}^2\\|f\\|_\infty\\|\mu_x-\mu_{x'}\\|. \] We apply this estimate with $g=f(x,\cdot)$ to obtain \begin{eqnarray} &&\|\chi(x,y)-\chi(x',y)\|\\ &&\qquad=\biggl\|{\mathbb E}\int_0^{T_c}\bigl(f(x,\bar Y_t)-f(x,Y_t)\bigr)\,dt-{\mathbb E}\int _0^{T_c'}\bigl(f(x',\bar Y_t')-f(x',Y_t')\bigr)\,dt\biggr\|\\ &&\qquad\le 2{\tau}\sup_{z\in I}\|f(x,z)-f(x',z)\|+\biggl\|{\mathbb E}\int_0^{T_c}f(x,\bar Y_t)\,dt-{\mathbb E}\int_0^{T_c'}f(x,\bar Y_t')\,dt\biggr\|\\ &&\qquad\quad{} +\biggl\|{\mathbb E}\int_0^{T_c}f(x,Y_t)\,dt-{\mathbb E} \int_0^{T_c'}f(x,Y_t')\,dt\biggr\|\\ &&\qquad\le 2{\tau}\sup_{z\in I}\|f(x,z)-f(x',z)\|+2\nu{\tau}^2\\|f\\|_\infty\\|\mu _x-\mu_{x'}\\| \end{eqnarray} as required. \end{pf} To summarize, we have shown the following proposition. \begin{proposition}\label{GCR} Assume conditions (\ref{GNM}) and (\ref{MTC}). Then, for any bound\-ed measurable function $f$ on $U\times I$, there exists a corrector $\chi_f$ for $f$ satisfying the estimates (\ref{CTB}) and (\ref{CEE}). \end{proposition} \subsection{Statement of the estimates}\label{SOTE} Recall the context of Section~\ref{OM}. We consider a continuous-time Markov chain $X$ with countable state-space $S$ and generator matrix $Q$. We choose fluid coordinates ${\mathbf x}\dvtx S\to{\mathbb R}^d$ and an auxiliary coordinate $y\dvtx S\to I$. We choose also a subset $U\subseteq{\mathbb R}^d$, which provides a means of localization, together with a map $b\dvtx U\times I\to{\mathbb R}^d$, and a family $G=(G_x\dvtx x\in U)$ of generator matrices on $I$, each having a unique invariant distribution $\pi_x$. Also choose, as in the preceding subsection, a coupling mechanism for $G$. This comprises a constant $\nu>0$, an auxiliary space $E$, a function $F\dvtx I\times E\to I$ and a family of probability distributions $\mu=(\mu_x\dvtx x\in U)$ on $E$ such that \[ g(x,y,y')=\nu\mu_x\bigl(\{v\in E\dvtx F(y,v)=y'\}\bigr), \qquad x\in U, y,y'\in I\mbox{ distinct}. \] Define for $x\in U$ \[ \bar b(x)=\sum_{y\in I}b(x,y)\pi(x,y). \] Write ${\mathbf X}_t={\mathbf x}(X_t)$ and assume that $(x_t)_{0\le t\le t_0}$ is a solution in $U$ to $\dot x_t=\bar b(x_t)$. We use a scaled supremum norm on ${\mathbb R}^d$: fix positive constants ${\sigma} _1,\ldots,{\sigma}_d$ and define for $x\in{\mathbb R}^d$ \[ \\|x\\|=\max_{1\le i\le d}\|x_i\|/{\sigma}_i. \] We now introduce some constants ${\Lambda},B,{\tau},J,J_1(b),J(\mu),K$ which characterize certain regularity properties of $Q$, $b$ and $G$. Assume that, for all $\xi\in S$, all $x\in U$ and all $y,y'\in I$, \begin{equation}\label{LBT} q(\xi)\le{\Lambda},\qquad \\|b(x,y)\\|\le B,\qquad m(x,y,y')\le{\tau}. \end{equation} Here $m(x,y,y')$ is the mean coupling time for $G_x$ starting from $y$ and $y'$, defined in the preceding subsection, which depends on the choice of coupling mechanism. Write $\mathcal J$ for the set of pairs of points in $U$ between which ${\mathbf X} $ can jump, thus \[ {\mathcal J}=\{(x,x')\in U\times U\dvtx x={\mathbf x}(\xi),x'={\mathbf x}(\xi ')\mbox{ for some $\xi,\xi'\in S$ with $q(\xi,\xi')>0$}\}. \] Set \begin{eqnarray} J&=&\sup_{(x,x')\in{\mathcal J}}\\|x-x'\\|,\\ J_1(b)&=&\sup_{(x,x')\in{\mathcal J}, y\in I}\\|b(x,y)-b(x',y)\\|,\\ J(\mu)&=&\sup_{(x,x')\in{\mathcal J}}\\|\mu_x-\mu_{x'}\\|. \end{eqnarray} Write $K$ for the Lipschitz constant of $\bar b$ on $U$; thus, for all $x,x'\in U$, \begin{equation}\label{BARB} \\|\bar b(x)-\bar b(x')\\|\le K\\|x-x'\\|. \end{equation} Recall from Section~\ref{OM} the definitions of the drift vector ${\beta}$ for ${\mathbf x}$ and the jump rate ${\gamma}$ for $y$. Define \[ T=\inf\{t\ge0\dvtx{\mathbf X}_t\notin U\}. \] Fix constants ${\delta}({\beta},b),{\delta}({\gamma},g)\in(0,\infty)$ and consider the events \begin{equation}\label{OBB} {\Omega}({\beta},b)=\biggl\{\int_0^{T\wedge t_0}\\|{\beta}(X_t)-b({\mathbf x}(X_t),y(X_t))\\|\,dt\le{\delta}({\beta},b)\biggr\} \end{equation} and \begin{equation}\label{OGG} {\Omega}({\gamma},g)=\biggl\{\int_0^{T\wedge t_0}\sum_{y'\not=y(X_t)}\|{\gamma} (X_t,y')-g({\mathbf x}(X_t),y(X_t),y')\|\,dt\le{\delta}({\gamma},g)\biggr\}.\hspace{-35pt} \end{equation} \begin{theorem}\label{FLE} Let ${\varepsilon}>0$ be given and set ${\delta}={\varepsilon} e^{-Kt_0}/7$. Assume that $J\le{\varepsilon}$ and \[ \max\bigl\{\\|{\mathbf X}_0-x_0\\|, {\delta}({\beta},b), 2{\tau} B{\delta}({\gamma} ,g), 2{\tau} B, 2{\Lambda} t_0\bigl({\tau} J_1(b)+\nu{\tau}^2BJ(\mu)\bigr)\bigr\}\le{\delta}. \] Set $\bar J=J+4{\tau} B$ and assume that ${\delta}\le{\Lambda}\bar Jt_0/4$. Further assume that the following \textit{tube condition} holds: \[ \mbox{for }\xi\in S\mbox{ and }t\le t_0\qquad \\|{\mathbf x}(\xi)-x_t\\|\le 2{\varepsilon}\quad\Longrightarrow\quad{\mathbf x}(\xi)\in U. \] Then \[ \mathbb{P}\Bigl(\sup_{t\le t_0}\\|{\mathbf X}_t-x_t\\|>{\varepsilon}\Bigr)\le 2de^{-{\delta}^2/(4{\Lambda}\bar J^2t_0)}+\mathbb{P}\bigl({\Omega}({\beta},b)^c\cup{\Omega}({\gamma},g)^c\bigr). \] \end{theorem} The proof of this result follows the initial stages of the proof of the more elaborate Theorem~\ref{FLET} below. We will not write it out separately but give further indications immediately before the statement of Theorem~\ref{FLET}. The reader will understand clearly the role of the inequalities which appear as hypotheses by following the proof. Here is an informal guide to their meanings. The tube condition, together with $J\le{\varepsilon}$, allows us to localize the other hypotheses to $U$ by trapping the process inside a tube around the limit path; these conditions can be satisfied by choosing $U$ sufficiently large. The conditions $\\|{\mathbf X}_0-x_0\\|\le{\delta}$ and ${\delta}({\beta},b)\le {\delta}$ enforce that the initial conditions and drift fields match closely. This requires, in particular, that the fluid and auxiliary coordinates provide sufficient information to nearly determine ${\beta}$. The condition on ${\delta}({\gamma},g)$ forces a close match between the local behavior of the fast variable and the idealized fast process used to compute the corrector. The condition $2{\tau} B\le{\delta}$ allows us to control the size of the corrector, balancing the mean recurrence time of the fast variable ${\tau}$ against the range of the drift field $b$. The condition on $2{\Lambda} t_0({\tau} J_1(b)+\nu{\tau}^2BJ(\mu))$ is needed for local regularity of the corrector, allowing us to pass back from the idealized fast process at one point $x$ to the actual fast variable when the fluid variable is near $x$. Finally, the condition ${\delta}\le{\Lambda}\bar Jt_0/4$ ensures we are in the ``Gaussian regime'' of the exponential martingale inequality, where bad events cannot occur by a small number of large jumps. For a nontrivial limiting dynamics, ${\Lambda} J$ should be of order $1$, while for a useful estimate ${\Lambda} J^2$ should be small; thus, as expected, we can attempt to use the result when the Markov chain takes small jumps at a high rate. It is sometimes possible to improve on the constant ${\Lambda}\bar J^2$ appearing in the preceding estimate, thereby obtaining useful probability bounds for smaller choices of ${\varepsilon}$. However, to do this we have to make hypotheses expressed in terms of a corrector. Fix $\bar y\in I$ and denote by $\chi$ the corrector for $b$ given by (\ref{CFF}). Define for $\xi\in S$ with ${\mathbf x}(\xi)\in U$ \[ \bar{\mathbf x}(\xi)={\mathbf x}(\xi)-\chi({\mathbf x}(\xi),y(\xi)). \] Define, for $\xi\in S$ such that ${\mathbf x}(\xi)\in U$ and ${\mathbf x}(\xi')\in U$ whenever $q(\xi,\xi')>0$, \[ {\alpha}^i(\xi)=\sum_{\xi'\not=\xi}\{\bar{\mathbf x}^i(\xi')-\bar {\mathbf x}^i(\xi)\}^2q(\xi,\xi'),\qquad i=1,\ldots,d. \] Note that, since we shall be interested only in upper bounds, we deal here only with the diagonal terms of the diffusivity tensor defined at (\ref{DTD}). Choose functions $a^i\dvtx I\to[0,\infty)$ such that, for all $\xi\in U$ where ${\alpha}^i(\xi)$ is defined, \begin{equation}\label{ALA} {\alpha}^i(\xi)\le a^i(y(\xi)),\qquad i=1,\ldots,d. \end{equation} For simplicity, we do not allow $a$ to depend on the fluid variable ${\mathbf x}(\xi)$. Since we can localize our hypotheses near the (compact) limit path, we do not expect to lose much precision by this simplification. On the other hand, by permitting a dependence on the fast variable we can sometimes do significantly better than Theorem~\ref{FLE}, as we shall see in Section~\ref{SMM}. Set \[ \bar a(x)=\sum_{y\in I}a(y)\pi(x,y),\qquad x\in U. \] We introduce two further constants $A$ and $\bar A$, with $\bar A\le A\le{\Lambda}\bar J^2$. Assume that, for all $x\in U$ and all $y\in I$, \begin{equation}\label{LBTT} a^i(y)\le A{\sigma}^2_i,\qquad \bar a^i(x)\le\bar A{\sigma}_i^2,\qquad i=1,\ldots,d. \end{equation} Note\vspace{1pt} that the corrector bound (\ref{CTB}) gives $\\|\chi({\mathbf x}(\xi),y(\xi))\\|\le2{\tau} B$, so ${\alpha}^i(\xi)\le{\Lambda}\bar J^2{\sigma}_i^2$ and so (\ref{LBTT}) holds with $A = \bar A = {\Lambda}\bar J^2$ and $a^i (y) = A \sigma_i^2$ and $\bar a^i(x) = A \sigma_i^2$. Thus Theorem~\ref{FLE} follows directly from (\ref{FLEA}) below. The new inequalities required on the left-hand side of (\ref{ABAB}) can be understood roughly as imposing that the ratio of the averaged diffusivity to a uniform bound on the diffusivity is not too small compared to the mean recurrence time of the fast variable; so an effective averaging takes place. \begin{theorem}\label{FLET} Assume that the hypotheses of Theorem~\ref{FLE} hold and that ${\delta}\bar J\le At_0/4$. Then \begin{equation}\label{FLEA} \mathbb{P}\Bigl(\sup_{t\le t_0}\\|{\mathbf X}_t-x_t\\|>{\varepsilon}\Bigr)\le 2de^{-{\delta}^2/(4At_0)}+\mathbb{P}\bigl({\Omega}({\beta},b)^c\cup{\Omega}({\gamma},g)^c\bigr). \end{equation} Moreover, under the further conditions ${\delta}\bar J\le\bar At_0/4$ and \begin{equation}\label{ABAB} \frac{1}{t_0}\max\{{\tau}, {\tau}{\delta}({\gamma},g), {\Lambda} t_0\nu{\tau} ^2J(\mu)\}\le\bar A/(20A)\le{\Lambda}{\tau}, \end{equation} we have \begin{eqnarray}\label{FLEAB} \mathbb{P}\Bigl(\sup_{t\le t_0}\\|{\mathbf X}_t-x_t\\|>{\varepsilon}\Bigr)&\le& 2de^{-{\delta}^2/(4\bar At_0)}+2de^{-(\bar A/A)^2t_0/(6400{\Lambda}{\tau} ^2)}\nonumber\\[-8pt]\\[-8pt] &&{}+\mathbb{P}\bigl({\Omega}({\beta},b)^c\cup{\Omega}({\gamma},g)^c\bigr).\nonumber \end{eqnarray} \end{theorem} \begin{pf} Consider the stopping time \[ T_0=\inf\{t\ge0\dvtx\\|{\mathbf X}_t-x_t\\|>{\varepsilon}\}. \] By the tube condition, we have $T_0\le T$. Moreover, for any $t<T_0$ and any $\xi'\in S$ such that $q(X_t,\xi')>0$, we have \[ \\|{\mathbf x}(\xi')-x_t\\|\le J+\\|{\mathbf X}_t-x_t\\|\le2{\varepsilon} \] so by the tube condition ${\mathbf x}(\xi')\in U$. Recall\vspace{1pt} that $\chi$ is the corrector for $b$ given by (\ref{CFF}). For the proof of (\ref{FLEAB}), we shall use (\ref{CFF}) to also construct a corrector $\tilde\chi$ for $a$. Set $\tilde{\delta}=\bar At_0/10$. Note from (\ref{CTB}) the bounds \[ \\|\chi(x,y)\\|\le2{\tau} B\le{\delta},\qquad \|\tilde\chi^i(x,y)\|\le2{\tau} A{\sigma}_i^2\le\tilde{\delta}{\sigma}_i^2. \] The inequality involving $\tilde{\delta}$ and further such inequalities below, which depend on the first inequality in assumption (\ref{ABAB}), will not be used in the proof of (\ref{FLEA}). Write ${\Delta}=G\chi({\mathbf x},y)-Q(\chi({\mathbf x},y))={\Delta}_1+{\Delta}_2$ and $\tilde{\Delta}=G\tilde\chi({\mathbf x},y)-Q(\tilde\chi({\mathbf x},y))=\tilde{\Delta}_1+\tilde{\Delta}_2$, where \begin{equation}\label{DGH} {\Delta}_1(\xi)=\sum_{y'\not=y(\xi)}\{g({\mathbf x}(\xi),y(\xi),y')-{\gamma} (\xi,y')\}\chi({\mathbf x}(\xi),y') \end{equation} and \begin{equation}\label{DGH-1} {\Delta}_2(\xi)=\sum_{\xi'\not=\xi}q(\xi,\xi')\{\chi({\mathbf x}(\xi ),y(\xi'))-\chi({\mathbf x}(\xi'),y(\xi'))\} \end{equation} and where $\tilde{\Delta}_1$ and $\tilde{\Delta}_2$ are defined analogously. Then, on ${\Omega}({\gamma},g)$, for $t\le T\wedge t_0$, \[ \biggl\\|\int_0^t{\Delta}_1(X_s)\,ds\biggr\\|\le2{\tau} B{\delta}({\gamma},g)\le {\delta} \] and, using Proposition~\ref{CE}, \[ \biggl\\|\int_0^t{\Delta}_2(X_s)\,ds\biggr\\|\le2{\Lambda} t_0\bigl({\tau} J_1(b)+\nu{\tau} ^2BJ(\mu)\bigr)\le{\delta}. \] Similarly, for $t\le T\wedge t_0$, \[ \biggl\|\int_0^t\tilde{\Delta}^i_1(X_s)\,ds\biggr\|\le2{\tau} A{\delta}({\gamma} ,g){\sigma}_i^2\le\tilde{\delta}{\sigma}_i^2 \] and \[ \biggl\|\int_0^t\tilde{\Delta}_2^i(X_s)\,ds\biggr\|\le2{\Lambda} t_0\nu{\tau}^2AJ(\mu ){\sigma}_i^2\le\tilde{\delta}{\sigma}_i^2. \] Take $M=M^{\bar{\mathbf x}}$ as in equations (\ref{ME}) and (\ref {KEY}) and consider the event \[ {\Omega}(M)=\Bigl\{\sup_{t\le T_0\wedge t_0}\\|M_t\\|\le{\delta}\Bigr\}. \] Then, on ${\Omega}({\beta},b)\cap{\Omega}({\gamma},g)\cap{\Omega}(M)$, we can estimate the terms in (\ref{KEY}) to obtain for $t\le T_0\wedge t_0$, \[ \\|{\mathbf X}_t-x_t\\|\le7{\delta}+K\int_0^t\\|{\mathbf X}_s-x_s\\|\,ds, \] so that $\\|{\mathbf X}_t-x_t\\|\le{\varepsilon}$ by Gronwall's lemma. Note that this forces $T_0\ge t_0$ and hence, $\sup_{t\le t_0}\\|{\mathbf X}_t-x_t\\|\le{\varepsilon}$. Set $\rho=3\bar A/2$ and consider the event \[ {\Omega}(a)=\biggl\{\int_0^{T_0\wedge t_0}a^i(Y_s)\,ds\le\rho t_0{\sigma}_i^2 \mbox{ for } i=1,\ldots,d\biggr\}. \] By condition (\ref{ALA}), on ${\Omega}(a)$ we have \[ \int_0^{T_0\wedge t_0}{\alpha}^i(X_s)\,ds\le\rho t_0{\sigma}_i^2. \] Set \[ J_i=J(\bar{\mathbf x}^i)=\sup_{\xi,\xi'\in S, {\mathbf x}(\xi ),{\mathbf x}(\xi')\in U, q(\xi,\xi')>0}\|\bar{\mathbf x}^i(\xi )-\bar{\mathbf x}^i(\xi')\|,\qquad i=1,\ldots,d, \] and use (\ref{CTB}) to see that $J_i\le\bar J{\sigma}_i$. Determine $\theta_i\in(0,\infty)$ by $\theta_i e^{\theta_iJ_i}={\delta }/(\rho t_0{\sigma}_i)$; then $\theta_i\le{\delta}/(\rho t_0{\sigma}_i)$, so $\theta_iJ_i\le2{\delta }\bar J/(3 \bar A t_0)\le1/4$, since we\vspace{1pt} assumed that $\delta\bar J \le\bar A t_0/4$. Since $e^{1/4}\le4/3$, we have $\rho e^{\theta _iJ_i}\le2\bar A$. We now apply the exponential martingale inequality, Proposition~\ref{EMI}, substituting $\pm\bar{\mathbf x}^i$ for $\phi$ for $i=1,\ldots,d$ and substituting ${\delta}{\sigma}_i$ for ${\delta}$ and $\rho t_0{\sigma}_i^2$ for ${\varepsilon}$. We thus obtain \[ \mathbb{P}\bigl({\Omega}(M)^c\cap{\Omega}(a)\bigr)\le2de^{-{\delta}^2/(4\bar At_0)}. \] If we take $\bar A=A$, then, using (\ref{ALA}) and (\ref{LBTT}), we have ${\Omega}(a)={\Omega}$, so the proof of (\ref{FLEA}) is now complete. Set $\eta=16{\Lambda}{\tau}^2A^2$. We shall complete the proof of (\ref {FLEAB}) by showing that \[ \mathbb{P}\bigl({\Omega}(a)^c\cap{\Omega}({\gamma},g)\bigr)\le2de^{-\tilde{\delta}^2/(4\eta t_0)}. \] Take $\tilde M$ as in (\ref{KEYT}), with $a$ as in (\ref {ALA}). Then, for $t\le T$, \begin{eqnarray}\label{KEYT-1} \int_0^ta(Y_s)\,ds&=&\tilde\chi({\mathbf X}_t,Y_t)-\tilde\chi({\mathbf X} _0,Y_0)-\tilde M_t\nonumber\\[-8pt]\\[-8pt] &&{}+\int_0^t\tilde{\Delta}(X_s)\,ds +\int_0^t\bar a({\mathbf X}_s)\,ds,\nonumber \end{eqnarray} where $\tilde{\Delta}=G\tilde\chi({\mathbf x},y)-Q(\tilde\chi({\mathbf x},y))$. Consider the event \[ {\Omega}(\tilde M)=\Bigl\{\sup_{t\le T_0\wedge t_0}\|\tilde M_t^i\|\le \tilde{\delta}{\sigma}_i^2 \mbox{ for } i=1,\ldots,d\Bigr\}. \] Then, on ${\Omega}({\gamma},g)\cap{\Omega}(\tilde M)$, we can estimate the terms in (\ref{KEYT-1}) to obtain \[ \int_0^{T_0\wedge t_0}a^i(Y_s)\,ds\le(5\tilde{\delta}+\bar At_0){\sigma} _i^2\le\rho t_0{\sigma}_i^2. \] Hence, it will suffice to show that \[ \mathbb{P}({\Omega}(\tilde M)^c)\le2de^{-\tilde{\delta}^2/(4\eta t_0)}. \] For this, we again use the exponential martingale inequality. Take $\phi(\xi)=\pm\tilde\chi^i({\mathbf x}(\xi),y(\xi))$ in Proposition~\ref{EMI} and note that ${\alpha}^\phi(\xi)\le16{\Lambda}{\tau} ^2A^2{\sigma}_i^4$, so \[ \int_0^{T_0\wedge t_0}{\alpha}^\phi(X_s)\,ds\le16{\Lambda}{\tau}^2A^2{\sigma}_i^4t_0=\eta t_0{\sigma}_i^4. \] Set \[ \tilde J_i=J(\phi)=\sup_{\xi,\xi'\in S, {\mathbf x}(\xi),{\mathbf x}(\xi')\in U, q(\xi,\xi')>0}\|\phi(\xi)-\phi(\xi')\|,\qquad i=1,\ldots,d, \] then $\tilde J_i\le4{\tau} A{\sigma}_i^2$. Determine $\tilde\theta_i\in(0,\infty)$ by $\tilde\theta_i e^{\tilde \theta_i \tilde J_i}=\tilde{\delta}/(\eta t_0{\sigma}_i^2)$. Then $\tilde\theta_i\le\tilde{\delta}/(\eta t_0{\sigma}_i^2)$ so $\tilde \theta_i \tilde J_i\le\bar A/(40{\Lambda}{\tau} A)\le1/2$ and so $e^{\tilde\theta_i \tilde J_i}\le2$. Hence, \[ \mathbb{P}({\Omega}(\tilde M)^c)\le2d\exp\{-\tilde{\delta}^2/(2\eta t_0e^{\tilde\theta_i \tilde J_i})\}\le2de^{-\tilde{\delta }^2/(4\eta t_0)} \] as required. \end{pf} \section{The supermarket model with memory}\label{SMM} The supermarket model with memory is a variant, introduced in~\cite{MPS}, of the ``join the shortest queue'' model, which has been widely studied~\cite{VDK,G,GA,LM,LMA,MR2152252}. We shall rigorously verify the asymptotic picture for large numbers of queues derived in~\cite{MPS}. This will serve as an example to illustrate the general theory of the preceding sections. The explicit form of the error probabilities in Theorem~\ref{FLET} is used to advantage in dealing with the infinite-dimensional character of the limit model. Fix $\lambda\in(0,1)$ and an integer $n\ge1$. We shall consider the limiting behavior as $N\to\infty$ of the following queueing system. Customers arrive as a Poisson process of rate $N \lambda$ at a system of $N$ single-server queues. At any given time, the length of one of the queues is kept under observation. This queue is called the \textit{memory queue}. On each arrival, an independent random sample of size $n$ is chosen from the set of all $N$ queues. For simplicity, we sample with replacement, allowing repeats and allowing the choice of the memory queue. The customer joins whichever of the memory queue or the sampled queues is shortest, choosing randomly in the event of a tie. Immediately after the customer has joined a queue, we switch the memory queue, if necessary, so that it is the currently shortest queue among the queues just sampled and the previous memory queue. The service requirements of all customers are assumed independent and exponentially distributed of mean $1$. Write $Z_t^k=Z_t^{N,k}$ for the proportion of queues having at least $k$ customers at time $t$, and write $Y_t$ for length of the memory queue at time $t$. Set $Z_t=(Z^k_t\dvtx k \in{\mathbb N})$ and $X_t=(Z_t,Y_t)$. Then $X=(X_t)_{t \ge0}$ is a Markov chain, taking values in $S=S_0\times{\mathbb Z}^+$, where $S_0$ is the set of nonincreasing sequences in $N^{-1}\{0,1,\ldots,N\}$ with finitely many nonzero terms. We shall treat $Y$ as a fast variable and prove a fluid limit for $Z$ as $N\to\infty$. \subsection{Statement of results}\label{SR} Let $D$ be the set of nonincreasing sequences\footnote{To lighten the notation, we shall sometimes move the coordinate index from a superscript to a subscript, allowing the $n$th power of the $k$th coordinate to be written $z_k^n$. We shall also write the time variable sometimes as a subscript, sometimes as an argument.} $z=(z_k\dvtx k\in{\mathbb N})$ in the interval $[0,1]$ such that \[ m(z):=\sum_kz_k<\infty. \] Define for $z\in D$ and $k\in{\mathbb N}$ \begin{equation}\label{MZK} \mu(z,k)=\prod_{j=1}^k\frac{z_j^n}{1-p_{j-1}(z)}, \end{equation} where \[ p_{k-1}(z)=n(z_{k-1}-z_k)z_k^{n-1} \] and where we take $z_0=1$. Set $\mu(z,0)=1$ for all $z$. An elementary calculation (maximizing over $z_k$ while keeping $z_{k-1}$ fixed) shows that in the case $n\ge2$, \begin{equation}\label{PKZ} p_{k-1}(z)\le z_{k-1}^n (1-1/n)^{n-1}\le(1-1/n)^{n/2} \le e^{-1/2}<1. \end{equation} In the case $n=1$ we have $p_{k-1}(z)=z_{k-1}-z_k\le1$ and it is possible that $0/0$ appears in the product (\ref{MZK}). For definiteness we agree to set $0/0=1$ in this case. Note that $\mu(z,k)\ge\mu(z,k+1)$ for all $k\ge0$. Define for $z\in D$ \[ v_k(z)={\lambda}z_{k-1}^n\mu(z,k-1)-{\lambda}z_k^n\mu(z,k)-(z_k-z_{k+1}) \] and consider the differential equation \begin{equation}\label{ODE} \dot z(t)=v(z(t)),\qquad t\ge0. \end{equation} By a \textit{solution} to (\ref{ODE}) in $D$ we mean a family of differentiable functions $z_k\dvtx[0,\infty)\to[0,1]$ such that for all $t\ge0$ and $k\in{\mathbb N}$ we have $(z_k(t)\dvtx k\in{\mathbb N})\in D$ and \[ \dot z_k(t)=v_k(z(t)). \] \begin{theorem}\label{DDD} For all $z(0)\in D$, the differential equation $\dot z(t)=v(z(t))$ has a unique solution in $D$ starting from $z(0)$. Moreover, if $(w(t)\dvtx t\in D)$ is another solution in $D$ with $z_k(0)\le w_k(0)$ for all $k$, then $z_k(t)\le w_k(t)$ for all $k$ and all $t\ge0$. \end{theorem} There is a fixed point of these dynamics $a\in D$ given by setting $a_0=1$ and defining \begin{equation}\label{DEFA} a_{k+1}={\lambda}a_k^n\mu(a,k),\qquad k\ge0. \end{equation} The components of $a$ decay super-geometrically. Set \begin{equation}\label{ALP} {\alpha}=n+\tfrac12+\sqrt{n^2+\tfrac14}. \end{equation} Then ${\alpha}\in(2n,2n+1)$. \begin{theorem}\label{SGD} We have \[ \lim_{k\to\infty}\frac1k\log\log\biggl(\frac1{a_k}\biggr)={\alpha}. \] \end{theorem} Assume for simplicity that we start the queueing system from the state where all queues, except the memory queue, are empty and where the memory queue has exactly one customer. Write $(z(t)\dvtx t\ge0)$ for the solution to (\ref{ODE}) starting from~$0$. Then $z_k(t)\le a_k$ for all $k$ and $t$. Our main result shows that $(z(t)\dvtx t\ge0)$ is a good approximation to the process of empirical distributions of queue lengths $(Z^N(t)\dvtx t\ge0)$ for large $N$. The sense of this approximation is reasonably sharp. In particular, as a straightforward corollary, we obtain that, on a given time interval $[0,t_0]$, for any $r>{\alpha}^{-1}$, with high probability as $N\to\infty$, no queue length exceeds $r\log \log N$. \begin{theorem}\label{MR} Set ${\kappa}=(2{\alpha})^{-1}$ and define \[ d=d(N)=\sup\{k\in{\mathbb N}\dvtx Na_k>N^{\kappa}\}. \] Fix a function $\phi$ on ${\mathbb N}$ such that $\phi(N)/N^{\kappa}\to0$ and $\log\phi(N)/\log\log N\to\infty$ as $N\to\infty$. Set $\rho=4/(1-{\lambda})$ when $n=1$ and set $\rho =2^n/(1-e^{-1/2})$ when $n\ge2$. Set $\tilde a_{d+1}=N^{-1}a_d^n+\rho^da_{d+1}$. Then \begin{equation}\label{DLLN} \lim_{N\to\infty}d(N)/\log\log N=1/{\alpha}. \end{equation} Moreover, for all $t_0\ge0$, we have \begin{equation}\label{MRA} \lim_{N\to\infty}\mathbb{P}\Biggl(\sup_{t\le t_0}\sup_{k\le d}\frac {\|Z_t^{N,k}-z_t^k\|}{\sqrt{a_k}}\ge\sqrt{\frac{\phi(N)}{N}}\Biggr)=0 \end{equation} and \begin{equation}\label{MRB} \lim_{R\to\infty}\limsup_{N\to\infty}\mathbb{P}(Z_t^{N,d+1}\ge R\tilde a_{d+1}\mbox{ for some }t\le t_0)=0 \end{equation} and \begin{equation}\label{MRC} \lim_{N\to\infty}\mathbb{P}(Z^{N,d+2}_t=0\mbox{ for all }t\le t_0)=1. \end{equation} \end{theorem} The argument used to prove this result would apply without modification starting from any initial condition $z(0)$ for the limit dynamics (\ref{ODE}) such that $z_k(0)\le a_k$ for all $k$, with suitable conditions on the convergence of $Z^N(0)$ to $z(0)$. It may be harder to move beyond initial conditions which do not lie below the fixed point. We do note here, however, a family of long-time upper bounds for the limit dynamics which might prove useful for such an extension. Fix $j\in{\mathbb N}$ and define $a^{(j)}_k=a_{(k-j)^+}$ for each $k \in{\mathbb Z}^+$; then $a^{(j)}$ is a fixed point of the modified equation \[ \dot w_k(t)=v_k(w(t))+\bigl(w_j(t)-w_{j+1}(t)\bigr)1_{\{k=j\}}. \] Since the added term is always nonnegative, a similar argument to that used to prove Theorem~\ref{DDD} in the next subsection also shows that, if $z(0)\le a^{(j)}$ and $(z(t)\dvtx t\ge0)$ is a solution of the original equation, then $z(t)\le a^{(j)}$ for all~$t$. \subsection{Existence and monotonicity of the limit dynamics}\label{EMLD} The differential equation (\ref{ODE}) characterizes the limit dynamics for the fluid variables in our queueing model. Our analysis of its space of solutions will rest on the exploitation of certain nonnegativity properties which have a natural probabilistic interpretation. We shall use the following standard property of differential equations: if $b=(b^1,\ldots,b^d)$ is a Lipschitz vector field on ${\mathbb R}^d$ such that $b^1(x)\ge0$ whenever $x=(x^1,\ldots,x^d)$ with $x^1=0$, and if $(x(t)\dvtx t\le t_0)$ is a solution to $\dot x(t)=b(x(t))$ with $x^1(0)\ge0$, then $x^1(t)\ge0$ for all $t\le t_0$.\vadjust{\goodbreak} We consider first a truncated, finite-dimensional system. Fix $d\in{\mathbb N}$ and define a vector field $u=u^{(d)}$ on $D$ by setting $u_k(z)=v_k(z)$ for $k\le d-1$ and \begin{equation}\label{UDZ} u_d(z)={\lambda}z_{d-1}^n\mu(z,d-1)-{\lambda}z_d^n\mu(z,d)-z_d \end{equation} and $u_k(z)=0$ for $k\ge d+1$. Set $D(d)=\{(x_1,\ldots,x_d,0,0,\ldots)\dvtx0\le x_d\le\cdots\le x_1\le 1\}$. \begin{proposition}\label{ADD} For all $x(0)\in D(d)$, the differential equation $\dot x(t)=u(x(t))$ has a unique solution $(x(t)\dvtx t\ge0)$ in $D(d)$ starting from $x(0)$. \end{proposition} \begin{pf} In the proof, we consider $D(d)$ as a subset of ${\mathbb R}^d$. The function $u$ is continuous on $D(d)$ and is differentiable in the interior of $D(d)$ with bounded partial derivatives. [In the case $n=1$, the singularity in $(\partial/\partial x_j)\mu(x,k)$ for $j\le k$ as $x_{j-1}-x_j\to1$ is canceled by the factor $x_k$ by which it is multiplied, since $x_k\le x_j$ on $D(d)$.] For $x\in D(d)$ we have $u_1(x)\le0$ when $x_1=1$, and $u_d(x)\ge0$ when $x_d=0$. Moreover, for $k=1,\ldots,d-1$, if $x_k=x_{k+1}$ then $p_k(x)=0$ so \begin{eqnarray} u_{k+1}(x)&=&x_k^n\bigl(\mu(x,k)-\mu(x,k+1)\bigr) \le \mu(x,k)-\mu(x,k+1)\\ &\le& x_{k-1}^n\mu(x,k-1)-x_k^n\mu(x,k)\le u_k(x). \end{eqnarray} The conclusion now follows by standard arguments. \end{pf} \begin{pf}{Proof of Theorem~\ref{DDD}} Suppose that $(w(t)\dvtx t\ge0)$ is a solution to $\dot w(t)=v(w(t))$ in $D$ starting\vspace{1pt} from $w(0)$, with $z(0)\le w(0)$, that is to say $z_k(0)\le w_k(0)$ for all $k$. Fix $d$ and write $x(t)=z^{(d)}(t)$ for the solution to $\dot x(t)=u^{(d)}(x(t))$ in $D(d)$ starting from $(z_1(0),\ldots,z_d(0),0,0,\ldots)$. Set $y(t)=(w_1(t),\ldots,w_d(t),0,0,\ldots)$ and note that $x(0)\le y(0)$ and $y(t)\in D(d)$ for all $t$. We shall show that $x(t)\le y(t)$ for all $t$. Now consider $D(d)$ as a subset of ${\mathbb R}^d$. We have \[ \dot y(t)=u(y(t))+w^{d+1}(t)e_d, \] where $e_d=(0,\ldots,0,1)$. Note that $w^{d+1}(t)\ge0$ for all $t$. Now $u$ is Lipschitz on $D(d)$ and for $k=1,\ldots,d$ we can show that\footnote{An elementary calculation shows that $\mu(x,j)\le\mu(y,j)$ for all $j$ whenever $x\le y$. This will also be shown by a soft probabilistic argument in Section \ref{LED}. The further condition $x_k=y_k$ gives the inequality \[ y_{k-1}^n-x_{k-1}^n\ge n(y_{k-1}-x_{k-1})x_k^{n-1}=p_{k-1}(y)-p_{k-1}(x) \ge\frac{y_k^{2n}}{1-p_{k-1}(y)}-\frac{x_k^{2n}}{1-p_{k-1}(x)}. \] } \[ x,y\in D(d),\qquad x\le y,\qquad x_k=y_k \quad\Longrightarrow\quad u_k(x)\le u_k(y). \] Hence, by a standard argument $z^{(d)}(t)=x(t)\le y(t)\le w(t)$ for all $t$. The same argument shows that $z^{(d)}(t)\le z^{(d+1)}(t)$ for all $t$, so the limit $z_k(t)=\lim_{d\to\infty}z^{(d)}_k(t)$ exists for all $k$ and $t$, and $z(t)\le w(t)$ for all $t$.\vadjust{\goodbreak} Fix $k$ and take $d\ge k+1$. Then the following equation holds for all $t$: \begin{eqnarray} &&z^{(d)}_k(t)+\int_0^t{\lambda}z_k^{(d)}(s)^n\mu \bigl(z^{(d)}(s),k\bigr)\,ds+\int_0^tz_k^{(d)}(s)\,ds\\ &&\qquad =z_k(0)+\int_0^t{\lambda}z_{k-1}^{(d)}(s)^n\mu \bigl(z^{(d)}(s),k-1\bigr)\,ds+\int_0^tz_{k+1}^{(d)}(s)\,ds. \end{eqnarray} On letting $d\to\infty$, we see by monotone convergence that \begin{eqnarray} &&z_k(t)+\int_0^t{\lambda}z_k(s)^n\mu(z(s),k)\,ds+\int_0^tz_k(s)\,ds\\ &&\qquad =z_k(0)+\int_0^t{\lambda}z_{k-1}(s)^n\mu\bigl(z(s),k-1\bigr)\,ds+\int _0^tz_{k+1}(s)\,ds. \end{eqnarray} Since $z(t)\in D$ for all $t$, all integrands in this equation are bounded by $1$. It is now straightforward to see that $(z(t)\dvtx t\ge0)$ is a solution. Now \begin{eqnarray} &&w_k(t)+\int_0^t{\lambda}w_k(s)^n\mu(w(s),k)\,ds+\int_0^tw_k(s)\,ds\\ &&\qquad =w_k(0)+\int_0^t{\lambda}w_{k-1}(s)^n\mu\bigl(w(s),k-1\bigr)\,ds+\int _0^tw_{k+1}(s)\,ds. \end{eqnarray} By summing these equations over $k\in\{1,\ldots,d\}$ we see that the map $t\mapsto\sum_{k=1}^d w_k(t)-{\lambda}t$ is nonincreasing for all $d$. Hence, $m(w(t))\le m(w(0))+{\lambda }t<\infty$. The equations can then be summed over all $k$ and rearranged to obtain \[ m(w(t))=m(w(0))+{\lambda}t-\int_0^tw_1(s)\,ds. \] On the other hand, \[ m\bigl(z^{(d)}(t)\bigr)=m\bigl(z^{(d)}(0)\bigr)+{\lambda}t-\int _0^tz_1^{(d)}(s)\,ds-{\lambda}\int_0^tz^{(d)}_d(s)^n\mu\bigl(z^{(d)}(s),d\bigr)\,ds \] so \begin{eqnarray}\label{MDM} m(w(t))-m\bigl(z^{(d)}(t)\bigr) &\le& m(w(0))-m\bigl(z^{(d)}(0)\bigr)\nonumber\\[-8pt]\\[-8pt] &&{}+{\lambda}\int_0^t z_d(s)^n\mu(z(s),d)\,ds.\nonumber \end{eqnarray} If $w(0)=z(0)$ then the right-hand side tends to $0$ as $d\to\infty$ so we must have $z(t)=w(t)$ for all $t$. \end{pf} \subsection{Properties of the fixed point} Recall the definition (\ref{DEFA}) of the fixed point~$a$. Since $\mu (a,k)\le1$ for all $k$, we have \[ a_k\le{\lambda}^{1+n+\cdots+n^{k-1}}\vadjust{\goodbreak} \] so $a_k\to0$ as $k\to\infty$. Theorem~\ref{SGD} is then a straightforward corollary of the following estimate. \begin{proposition}\label{AK} There is a constant $C({\lambda},n)<\infty$ such that, for all $k\ge0$, \begin{equation} C^{-1}a_k^{\alpha}\le a_{k+1}\le Ca_k^{\alpha}, \end{equation} where ${\alpha}$ is given by (\ref{ALP}). \end{proposition} \begin{pf} Note that, since $a_1={\lambda}$, we have $p_{k-1}(a)=a_{k-1}-a_k\le {\lambda}\vee(1-{\lambda})<1$ for all $k$ when $n=1$. On the other hand, equation (\ref{PKZ}) gives $p_{k-1}(z)\le e^{-1/2}$ for all $k$ when $n\ge2$. Then from $\sum_{k=1}^\infty p_{k-1}(a)\le1$, we obtain a constant $c<\infty$, which may depend on ${\lambda}$ when $n=1$, such that \[ \prod_{k=1}^\infty\frac1{1-p_{k-1}(a)}\le c. \] Then for $k\ge0$ \[ {\lambda}a_k^{2n}\prod_{j=1}^{k-1}a_j^n\le a_{k+1}\le c{\lambda }a_k^{2n}\prod_{j=1}^{k-1}a_j^n, \] so for $k\ge1$, \begin{equation}\label{CAK} c^{-1}a_k^{2n+1}a_{k-1}^{-n}\le a_{k+1}\le ca_k^{2n+1}a_{k-1}^{-n}. \end{equation} Note that ${\lambda}a_0^{\alpha}={\lambda}=a_1\le{\lambda}^{-1}a_0^{\alpha}$. Fix $A\ge1/{\lambda}$ and suppose inductively that \[ A^{-1}a_{k-1}^{\alpha}\le a_k\le Aa_{k-1}^{\alpha}. \] On using these inequalities to estimate $a_{k-1}$ in (\ref{CAK}), we obtain \[ (cA^{n/{\alpha}})^{-1}a_k^{\alpha}\le a_{k+1}\le cA^{n/{\alpha}}a_k^{\alpha}, \] where we have used the fact that $1-n/{\alpha}={\alpha}-2n$. Hence, the induction proceeds provided we take $A\ge c^{{\alpha}/({\alpha}-n)}$. \end{pf} \subsection{Choice of fluid coordinates and fast variable} In the remaining subsections we apply Theorem~\ref{FLET} to deduce Theorem~\ref{MR}. Define $d$ as in Theorem~\ref{MR} and take as auxiliary space $I={\mathbb N}$ when $n=1$ and $I={\mathbb Z}^+$ when $n\ge2$. Make the following choice of fluid and auxiliary coordinates: for $\xi=(z,y)\in S$ with $z=(z_k\dvtx k\in{\mathbb N})$, set \[ x^k(\xi)=z_k,\qquad k=1,\ldots,d,\qquad y(\xi)=y. \] Thus our fluid variable is ${\mathbf X}_t={\mathbf x}(X_t)=(Z_t^1,\ldots ,Z_t^d)$ and our fast variable is $Y_t=y(X_t)$. Note that when $n=1$, if $Y_0\ge1$ then $Y_t\ge1$ for all $t$, so $Y$ takes values in $I={\mathbb N}$. Let us compute the drift vector ${\beta}(\xi)$ for ${\mathbf X}$ when $X$ is in state $\xi=(z,y)\in S$. Note that $X^k$ makes a jump of size $1/N$ when a customer arrives at a queue of length $k-1$, and makes a jump of size $-1/N$ when a customer departs from a queue of length $k$, otherwise $X^k$ is constant. The length of the queue which an arriving customer joins depends on the length of the memory queue $y$ and on the lengths of the sampled queues. Denote the vector of sampled queue lengths by $V=V(z)=(V_1,\ldots,V_n)$ and write $V^{(1)}\le V^{(2)}\le\cdots\le V^{(n)}$ for the ordered queue lengths. Define $\min(v)=v_1\wedge\cdots\wedge v_n$ and set $M=\min(V)$. Then $M=V^{(1)}$ and \[ \mathbb{P}(M\ge k)=z_k^n. \] A new customer will go to a queue of length at least $k$ if and only if $M\ge k$ and $y\ge k$. So the rate for an arrival to a queue of length exactly $k-1$ is \[ N{\lambda}\mathbb{P}(M\ge k-1)1_{\{y\ge k-1\}}-N{\lambda}\mathbb{P}(M\ge k)1_{\{ y\ge k\}}. \] The rate for a departure from a queue of length $k$ is $N(z_k-z_{k+1})$. Hence, setting $z_0=1$, we have \[ {\beta}_k(\xi)={\lambda}z_{k-1}^n1_{\{y\ge k-1\}}-{\lambda}z_k^n1_{\{ y\ge k\}}-(z_k-z_{k+1}). \] We now compute (an approximation to) the jump rates ${\gamma}(\xi,y')$ for $Y$ when $X$ is in state $\xi=(z,y)\in S$. The rate of departures from the memory queue is at most~$1$. Arrivals to the system occur at rate $N{\lambda}$. Occasionally, the memory queue falls in the sample, an event of probability no greater than $n/N$ and hence, of rate no greater than ${\lambda}n$. Assuming that the the memory queue does not fall in the sample, the length of the memory queue after an arrival is given by \begin{equation}\label{YP} F(y,V)=(y+1)1_{\{y\le M-1\}}+y1_{\{M\le y\le P\}}+P1_{\{y\ge P+1\}}, \end{equation} where $P=p(V)$ is given by $P=M+1$ when $n=1$ and $P=(M+1)\wedge V^{(2)}$ otherwise. Hence, we have \begin{equation}\label{GAM} \sum_{y'\not=y(\xi)}\bigl\|{\gamma}(\xi,y')-N{\lambda}\mathbb{P} \bigl(F(y,V(z))=y'\bigr)\bigr\|\le1+{\lambda}n. \end{equation} \subsection{Choice of limit characteristics and coupling mechanism}\label{CLC} Define \[ U=\{x\in{\mathbb R}^d\dvtx0\le x_d\le\cdots\le x_1\le1\mbox{ and }x_1\le ({\lambda}+1)/2\mbox{ and }x_k\le2a_k\mbox{ for all }k\}. \] The condition $x_1\le({\lambda}+1)/2$ ensures that $1-x_1$ is uniformly positive on $U$. Define $b\dvtx U\times{\mathbb Z}^+\to{\mathbb R}^d$ by \begin{equation}\label{BKX} b_k(x,y)={\lambda}x_{k-1}^n1_{\{y\ge k-1\}}-{\lambda}x_k^n1_{\{y\ge k\}}-(x_k-x_{k+1}), \end{equation} where we set $x_0=1$ and $x_{d+1}=0$. Then, for $\xi\in S$ with ${\mathbf x}(\xi)\in U$, we have \begin{equation}\label{BETAB} {\beta}(\xi)=b({\mathbf x}(\xi),y(\xi))+(0,\ldots,0,z_{d+1}). \end{equation} It is convenient to specify our choice of the generator matrices $(G_x\dvtx x\in U)$ and our choice of coupling mechanism at the same time. Set $\nu=N{\lambda}$ and take as auxiliary space $E=({\mathbb Z}^+)^n$. Define a family of probability distributions $\mu=(\mu_x\dvtx x\in U)$ on~$E$, taking $\mu_x$ to be the law of a random sample $V=V(x)=(V_1,\ldots,V_n)$ with \[ \mathbb{P}(V_1\ge k)=\cdots=\mathbb{P}(V_n\ge k)=x_k \] for $k=0,1,\ldots,d+1$. Note that \begin{equation}\label{MTV} \\|\mu_x-\mu_{x'}\\|\le2n\sum_{k=1}^d\|x_k-x'_k\|. \end{equation} Then define for distinct $y,y'\in{\mathbb Z}^+$, \[ g(x,y,y')=N{\lambda}\mathbb{P}\bigl(F(y,V(x))=y'\bigr), \] where $F$ is given by (\ref{YP}). We take as coupling mechanism the triple $(\nu,\mu,F)$. Note that $F(y,v)=F(\bar y,v)$ for all $y,\bar y\in I$ whenever $p(v)=\min I$. For $x\in U$ we have \[ \mathbb{P}\bigl(p(V(x))=1\bigr)\ge1-x_1>\frac{1-{\lambda}}2, \] when $n=1$, whereas for $n\ge2$ we have \[ \mathbb{P}\bigl(p(V(x))=0\bigr)\ge(1-x_1)^2>\biggl(\frac{1-{\lambda }}2\biggr)^2. \] Hence, we obtain, in all cases, $m(x,y,\bar y)\le{\tau}$, where we set \[ {\tau}=\frac4{N{\lambda}(1-{\lambda})^2}. \] For $\xi\in S$ with $x={\mathbf x}(\xi)\in U$ we can realize a sample $V(z)$ (from the distribution of queue lengths) and the sample $V(x)$ on the same probability space by setting $V_i(x)=V_i(z)\wedge d$. Write $M(x)=\min(V(x))$ and $P(x)=p(V(x))$. Then $M(x)=M(z)\wedge d$ and $P(x)=P(z)\wedge(d+1)$ when $n=1$ and $P(x)=P(z)\wedge d$ when $n\ge2$. The difference between the two cases is that there is no second shortest queue in the sample when $n=1$. We have, for $n=1$, \[ \mathbb{P}\bigl(P(z)\not= P(x)\bigr)\le\mathbb{P}\bigl(M(z)\ge d+1\bigr)=z_{d+1} \] and, for $n\ge2$, \[ \mathbb{P}\bigl(P(z)\not= P(x)\bigr)\le\mathbb{P}\bigl(P(z)\ge d+1\bigr)\le nz_dz_{d+1}^{n-1}. \] Now $P(x)=P(z)$ implies $M(x)=M(z)$ and hence, $F(y,V(x))=F(y,V(z))$ for all $y$. Hence, \[ \mathbb{P}\bigl(F(y,V(z))\not=F(y,V(x))\bigr)\le\mathbb{P}\bigl(P(x)\not=P(z)\bigr). \] On combining this with (\ref{GAM}) we obtain \begin{equation}\label{GAMMAG} \sum_{y'\not=y(\xi)}\|{\gamma}(\xi,y')-g({\mathbf x}(\xi),y(\xi ),y')\|\le1+{\lambda}n+N{\lambda}nz_{d+1}. \end{equation} \subsection{Local equilibrium distribution}\label{LED} The Markov chain determined by the generator $G_x$ has a unique closed communicating class, which is contained in $\{0,1,\ldots,d\}$. Hence, $G_x$ has a unique equilibrium distribution $\pi_x$ which is supported on $\{0,1,\ldots,d\}$. Consider a continuous-time Markov chain\footnote{See footnote \ref {NFV}.} $Y=(Y_t)_{t\ge0}$ with generator $G_x$, and initial distribution $\pi_x$. Set $\mu(x,k)=\mathbb{P}(Y_0\ge k)$. Then $Y$ jumps into $\{0,1,\ldots,k\}$ from $j$ at rate $\alpha=N{\lambda}(1-x_{k+1}^n-p_k(x))$, for all $j\ge k+1$. On the other hand, $Y$ jumps out of $\{0,1,\ldots,k\}$ only from $k$, and that at rate $\beta=N{\lambda }x_{k+1}^n$. Since the long run rates of such jumps must agree, we deduce that $\alpha\mu(x,k+1)=\beta\pi_x(k)$. Hence, we obtain \[ \mu(x,k+1)\bigl(1-p_k(x)\bigr)=x_{k+1}^n\mu(x,k) \] and so \begin{equation}\label{IMT} \mu(x,k)=\prod_{j=1}^k\frac{x_j^n}{1-p_{j-1}(x)},\qquad k=1,\ldots,d. \end{equation} Hence, our present notation is consistent with the definition (\ref{MZK}). Note also that, for $z\in D$, $\mu(z,k)$ depends only on $z_1,\ldots ,z_k$; in particular, if $x=(z_1,\ldots,z_d)$ then $\mu(z,k)=\mu(x,k)$ for all $k\le d$. Note that $\bar b$ is given by \begin{equation}\label{BARBE} \bar b_k(x)={\lambda}x_{k-1}^n\mu(x,k-1)-{\lambda}x_k^n\mu (x,k)-(x_k-x_{k+1}),\qquad k=1,\ldots,d,\hspace{-35pt} \end{equation} where $x_0=1$ and $x_{d+1}=0$. Hence, $\bar b=u^{(d)}$ as defined in Section~\ref{EMLD}. A comparison of (\ref{UDZ}) and (\ref{BKX}) now shows that $\bar b=u^{(d)}$. Recall that $\rho=4/(1-{\lambda})$ when $n=1$ and that $\rho =2^n/(1-e^{-1/2})$ when $n\ge2$. Then for $x\in U$ and $k\ge1$ we have \begin{eqnarray} \mu(x,k)&=&x_k^n\mu(x,k-1)/\bigl(1-p_{k-1}(x)\bigr)\le\rho a_k^n\mu(x,k-1)\\ &\le& \rho a_k^n\mu(x,k-1)/\bigl(1-p_{k-1}(a)\bigr) \end{eqnarray} so, for all $x\in U$ and inductively for all $k\ge1$, we obtain \begin{equation}\label{MUXA} \mu(x,k)\le\rho^k\mu(a,k). \end{equation} The following argument shows that $\mu(z,k)\le\mu(z',k)$ for all $k$ whenever $z\le z'$. Fix $d\ge k$ and set $x'=(z_1',\ldots,z_d')$. Assume that $x\le x'$. By a standard construction we can realize samples $V=V(x)$ and $V'=V(x')$ on a common probability space such that $V_i\le V'_i$ for all $i$. Then we can construct Markov chains $Y$ and $Y'$, having generators $G_x$ and $G_{x'}$, respectively, on the canonical space of a marked Poisson process of rate $N{\lambda}$, where the marks are independent copies of $(V,V')$, as follows. Set $Y_0=Y'_0=1$ and define recursively at each jump time $T$ of the Poisson process $Y_T=F(Y_{T-},V_T)$ and $Y_T'=F(Y'_{T-},V_T')$, where $(V_T,V_T')$ is the mark at time~$T$. Then, since $F$ is nondecreasing in both arguments, we see by induction that $Y_t\le Y'_t$ for all $t$. Hence, by convergence to equilibrium, \[ \mu(z,k)=\mu(x,k)=\lim_{t\to\infty}\mathbb{P}(Y_t\ge k)\le\lim_{t\to \infty}\mathbb{P}(Y_t'\ge k)=\mu(x',k)=\mu(z',k). \] \subsection{Corrector upper bound} We take as our reference state $\bar y=\min I$ and note that, under the coupling mechanism, we have $\bar Y_t\le Y_t$ for all $t$. Then the $k$th component of the corrector for $b$ is given by \[ \chi_k(x,y)={\lambda}{\mathbb E}_y\int_0^{T_c}\bigl(x_{k-1}^n1_{\{\bar Y_s<k-1\le Y_s\}}-x_k^n1_{\{\bar Y_s<k\le Y_s\}}\bigr)\,ds, \] so for $x\in U$ and all $y\in I$ we have \[ \|\chi_k(x,y)\|\le{\tau} x_{k-1}^n\le Ca_{k-1}^n/N. \] Now fix $y\le k-2$ and consider the stopping time $T=\inf\{t\ge 0\dvtx Y_t=k-1\}$. Note that $Y$ can enter state $k-1$ only from $k-2$ and does so at rate $N{\lambda}x_{k-1}^n$, whereas $T_c$ occurs in state $k-2$ at rate at least $N{\lambda }(1-{\lambda})^2/4$. Hence, $\mathbb{P}_y(T\le T_c)\le Ca_{k-1}^n$ and so, by the Markov property, \[ {\mathbb E}_y\int_0^{T_c}1_{\{\bar Y_s<k-1\le Y_s\}}\,ds\le{\mathbb E}_y\bigl(1_{\{T\le T_c\} }m(x,k-1,\bar Y_T)\bigr)\le Ca_{k-1}^n{\tau}\le C a_{k-1}^n/N. \] Hence, we obtain, for $x\in U$ and all $y\in I$, \begin{equation}\label{CHID} \|\chi_k(x,y)\|\le C\bigl(a_{k-1}^n1_{\{y\ge k-1\}}+a_{k-1}^{2n}\bigr)/N. \end{equation} \subsection{Quadratic variation upper bound}\label{QVB} The growth rate at $\xi$ of the quadratic variation of the corrected $k$th coordinate is given by \[ {\alpha}_k(\xi)=\sum_{\xi'\not=\xi}\{\bar{\mathbf x}_k(\xi')-\bar {\mathbf x}_k(\xi)\}^2q(\xi,\xi'). \] Recall that $\bar x_k=x_k-\chi_k({\mathbf x},y)$. We estimate separately, writing $x={\mathbf x}(\xi)$ and $y=y(\xi)$, \[ \sum_{\xi'\not=\xi}\{{\mathbf x}_k(\xi')-x_k\}^2q(\xi,\xi')\le N^{-2}\bigl(Nx_k+N{\lambda}x_{k-1}^n1_{\{y\ge k-1\}}\bigr) \] and \[ \sum_{\xi'\not=\xi}\{\chi_k({\mathbf x}(\xi'),y(\xi'))-\chi _k(x,y)\}^2q(\xi,\xi')\le CN^{-1}x_{k-1}^{2n}. \] When $y\le k-2$, we can improve the last estimate by splitting the sum in two and using \[ \sum_{\xi'\not=\xi, y(\xi')\le k-2}\{\chi_k({\mathbf x}(\xi '),y(\xi'))-\chi_k(x,y)\}^2q(\xi,\xi')\le CN^{-1}x_{k-1}^{4n} \] and \[ \sum_{\xi'\not=\xi, y(\xi')\ge k-1}\{\chi_k({\mathbf x}(\xi '),y(\xi'))-\chi_k(x,y)\}^2q(\xi,\xi')\le CN^{-1}x_{k-1}^{3n}. \] We used (\ref{CHID}) for the first inequality and for the second used \[ \sum_{\xi'\not=\xi, y(\xi')\ge k-1}q(\xi,\xi')\le CNx_{k-1}^n. \] On combining these estimates we obtain \[ {\alpha}_k(\xi)\le C\bigl(x_k+x_{k-1}^n1_{\{y\ge k-1\}}+x_{k-1}^{3n}\bigr)\le a_k(y(\xi))/N, \] where \[ a_k(y(\xi))=C\bigl(a_k+a_{k-1}^n1_{\{y\ge k-1\}}\bigr)/N. \] Then, using the estimate (\ref{MUXA}) and the limit (\ref{DLLN}), we have \[ \bar a_k(x)=C\bigl(a_k+a_{k-1}^n\mu(x,k-1)\bigr)\le C\rho^{d-1}a_k/N\le C(\log N)^Ca_k/N, \] so we have for all $x\in U$ and $y\in I$ \[ a_k(y)\le Aa_k, \bar a_k(x)\le\bar Aa_k \] with $A=C/(Na_d^{1-n/{\alpha}})$ and $\bar A=C(\log N)^C/N$. It is straightforward to check that, for $N$ sufficiently large, we have $\bar A\le A\le{\Lambda}\bar J^2$. \subsection{Truncation estimates} A specific feature of the problem we consider is that the limit dynamics is infinite dimensional, while the general fluid limit estimate applies in a finite-dimensional context. In this subsection we establish some truncation estimates which will allow us to reduce to finitely many dimensions. Let $(z(t)\dvtx t\ge0)$ be the solution in $D$ to $\dot z(t)=v(z(t))$ starting from $0$, as in Theorem~\ref{MR}. Let $(x(t)\dvtx t\ge0)$ be the solution to $\dot x(t)=\bar b(x(t))$ starting from $0$. \begin{lemma}\label{TLD} We have \[ \sum_{k=1}^d\|z_k(t)-x_k(t)\|\le ta_{d+1}. \] \end{lemma} \begin{pf} Since $\bar b=u^{(d)}$, we have $x(t)=z^{(d)}(t)$ for all $t$, and so, from (\ref{MDM}), we obtain \begin{eqnarray} \sum_{k=1}^d\|z_k(t)-x_k(t)\|&\le&\sum_{k=1}^d\bigl(z_k(t)-z^{(d)}_k(t)\bigr)\le {\lambda}\int_0^tz_d(s)^n\mu(z(t),d)\,ds\\ &\le& t{\lambda}a_d^n\mu(a,d)=ta_{d+1}. \end{eqnarray} \upqed\end{pf} Denote by $A_k(t)$ the number of arrivals to queues of length at least $k$ by time $t$. Note that $NZ^{k+1}_t\le A_k(t)$ for all $k\ge1$ and all $t$. Recall that \[ \tilde a_{d+1}=N^{-1}a_d^n+\rho^da_{d+1}. \] \begin{lemma}\label{ADT} There is a constant $C({\lambda},n)<\infty$ such that, for all $t\ge 0$ and all~$N$, we have \[ {\mathbb E}\bigl(A_d(T\wedge t)\bigr)\le Ce^{Ct}N\tilde a_{d+1}. \] \end{lemma} \begin{pf} Consider the function $f$ on $U\times I$ given by $f(x,y)=1_{\{y\ge d\} }$ and note that $\bar f(x)=\mu(x,d)$. Let $\chi$ be the corrector for $f$ given by (\ref{CFF}). Then, for all $x\in U$ and all $y\in I$, \[ \|\chi(x,y)\|\le2{\tau}\\|f\\|_\infty=2{\tau}=CN^{-1} \] and, whenever $x={\mathbf x}(\xi)$ and $x'={\mathbf x}(\xi')$ with $q(\xi,\xi')>0$, by the estimates (\ref{CEE}) and (\ref{MTV}), \begin{equation}\label{DRT} \|\chi(x,y)-\chi(x',y)\|\le2\nu{\tau}^2\\|f\\|_\infty\\|\mu_x-\mu_{x'}\\| \le CN^{-2}. \end{equation} By optional stopping, \[ \biggl\|\int_0^{T\wedge t}Q(\chi({\mathbf x},y))(X_s)\,ds\biggr\|=\bigl\|{\mathbb E} \bigl(\chi({\mathbf X}_{T\wedge t},Y_{T\wedge t})-\chi({\mathbf X}_0,Y_0)\bigr)\bigr\| \le CN^{-1}. \] Now \[ Q(\chi({\mathbf x},y))(\xi)=1_{\{y\ge d\}}-\mu({\mathbf x}(\xi ),d)-{\Delta}_1(\xi)-{\Delta}_2(\xi), \] where ${\Delta}_1,{\Delta}_2$ are given by (\ref{DGH}), (\ref{DGH-1}). We use (\ref{GAMMAG}) to obtain the estimate \[ \|{\Delta}_1(\xi)\|\le2{\tau}(1+{\lambda}n+N{\lambda}nz_{d+1})=C(z_{d+1}+N^{-1}) \] and from (\ref{DRT}) deduce that \[ \|{\Delta}_2(\xi)\|\le N(1+{\lambda})CN^{-2}=CN^{-1}. \] So \[ {\mathbb E}\int_0^{T\wedge t}1_{\{Y_s\ge d\}}\,ds \le CN^{-1}+{\mathbb E}\int_0^{T\wedge t}\bigl(\mu({\mathbf X} _s,d)+CZ_s^{d+1}+CN^{-1}\bigr)\,ds. \] Set $g(t)={\mathbb E}(A_d(T\wedge t))$, then \begin{eqnarray} g(t)&=&N{\lambda}{\mathbb E}\int_0^{T\wedge t}(X^d_s)^n1_{\{Y_s\ge d\}}\,ds\le N{\lambda}2^na_d^n{\mathbb E}\int_0^{T\wedge t}1_{\{Y_s\ge d\}}\,ds\\ &\le& Ca_d^n\biggl(1+\int_0^t\bigl(N\rho^d\mu(a,d)+1+g(s)\bigr)\,ds\biggr)\\ &\le& CN\tilde a_{d+1}(1+t)+C\int_0^tg(s)\,ds. \end{eqnarray} Here we have used the estimate $\mu(x,d)\le\rho^d\mu(a,d)$ for $x\in U$. The claimed estimate now follows by Gronwall's lemma. \end{pf} Fix $R\in(0,\infty)$ and define \[ \tilde T=\inf\{t\ge0\dvtx A_d(t)\ge RN\tilde a_{d+1}\}\wedge T. \] \begin{lemma}\label{ADDT} There is a constant $C({\lambda},n)<\infty$ such that, for all $t\ge 0$ and all~$N$, we have \[ {\mathbb E}\bigl(A_{d+1}(\tilde T\wedge t)\bigr)\le C(1+t)R^n\tilde a_{d+1}^n+CtR^{n+1}N\tilde a_{d+1}^{n+1}. \] \end{lemma} \begin{pf} We argue as in the preceding proof, except now taking $f(x,y)=1_{\{y\ge d+1\}}$, for which $\bar f(x)=0$. We obtain \[ {\mathbb E}\int_0^{\tilde T\wedge t}1_{\{Y_s\ge d+1\}}\,ds \le CN^{-1}+C{\mathbb E}\int_0^{\tilde T\wedge t} (Z_s^{d+1}+CN^{-1})\,ds \] and hence, \begin{eqnarray} {\mathbb E}\bigl(A_{d+1}(\tilde T\wedge t)\bigr)&=&N{\lambda}{\mathbb E}\int_0^{\tilde T\wedge t}(Z^{d+1}_s)^n1_{\{Y_s\ge d+1\}}\,ds\\ &\le& N{\lambda}(R\tilde a_{d+1})^n{\mathbb E}\int_0^{\tilde T\wedge t}1_{\{ Y_s\ge d+1\}}\,ds\\ &\le& C(1+t)R^n\tilde a_{d+1}^n+CtR^{n+1}N\tilde a_{d+1}^{n+1}. \end{eqnarray} \upqed\end{pf} \subsection{\texorpdfstring{Proof of Theorem \protect\ref{MR}}{Proof of Theorem 2.3}} Recall that ${\alpha}$ is defined by (\ref{ALP}) and that ${\alpha}\in (2n,2n+1)$. Note that ${\alpha}^2-(2n+1){\alpha}+n=0$. Recall that ${\kappa}=(2{\alpha})^{-1}$ and \[ d=d(N)=\sup\{k\in{\mathbb N}\dvtx Na_k>N^{\kappa}\}. \] The asymptotic growth rate (\ref{DLLN}) follows from Theorem~\ref{SGD}. We shall use without further comment below the inequalities \[ C^{-1}a_k^{1/{\alpha}}\le a_{k+1}\le Ca_k^{\alpha},\qquad k\ge0, \] proved in Proposition~\ref{AK} and the inequalities \[ a_{d+1}\le N^{-(1-{\kappa})}\le a_d,\qquad d\le\log\log N, \] the last being valid for all sufficiently large $N$. By the truncation estimate, Lemma~\ref{TLD}, we have \[ \sup_{t\le t_0}\sup_{k\le d}\frac{\|z_k(t)-x_k(t)\|}{\sqrt{a_k}}\le t_0\frac{a_{d+1}}{\sqrt{a_d}}\le Ct_0a_{d+1}^{1-1/(2{\alpha})}\le Ct_0N^{-1/2}. \] Since $\phi(N)\to\infty$ as $N\to\infty$ it will therefore suffice to show (\ref{MRA}) with $(z(t)\dvtx t\ge0)$ replaced by $(x(t)\dvtx t\ge0)$. We apply the general procedure of Section~\ref{SOTE}. Take as norm scales ${\sigma}_k=\sqrt{a_k}$ so that \[ \\|x\\|=\max_k\|x_k\|/\sqrt{a_k},\qquad x\in{\mathbb R}^d. \] We now identify suitable regularity constants ${\Lambda},B,{\tau},J,J_1(b),J(\mu),K$. We write $C$ for a finite positive constant which may depend on ${\lambda}$ and $n$ and whose value may vary from line to line. We shall see that, as $N\to\infty$, the inequalities between these regularity constants required in Theorem~\ref{FLET} become valid. The maximum jump rate is bounded above by \[ {\Lambda}=N(1+{\lambda})=CN. \] We refer to the form of $b(x,y)$ given at (\ref{BKX}) and note that, for $x\in U$ and $y\in I$, \[ \\|b(x,y)\\|\le B=2^na_d^{-1/2+n/{\alpha}}=Ca_d^{-1/2+n/{\alpha}}. \] We showed in Section~\ref{CLC} the following upper bound on the mean coupling time of our coupling mechanism: \[ m(x,y,\bar y)\le{\tau}=\frac4{N{\lambda}(1-{\lambda})^2}=CN^{-1}. \] We refer to Section~\ref{SOTE} for the definitions of the jump bounds $J,J_1(b),J(\mu)$ and leave the reader to check the validity of the following inequalities: \[ J\le N^{-1}a_d^{-1/2},\qquad J_1(b)\le CN^{-1}a_d^{-1/2+(n-1)/{\alpha}},\qquad J(\mu)\le2nN^{-1}. \] Recall from (\ref{BARBE}) the form of $\bar b$. In estimating the Lipschitz constant $K$ for $\bar b$ on $U$, first note that, for $x\in U$ and for $j=1,\ldots,k-1$, \[ \biggl\|\frac{\partial}{\partial x_j}x_{k-1}^n\mu(x,k-1)\biggr\|\le Cx_{k-1}^n\mu(x,k-1)(x_j^{-1}+1). \] Here we have used the explicit form (\ref{MZK}) of $\mu(x,k-1)$ and the fact that $(1-p_{j-1}(x))^{-1}\le C$ on $U$. Also note the inequalities \[ x_{k-1}^{2n-1}\sqrt{\frac{a_{k-1}}{a_k}}\le 2^{2n-1}a_{k-1}^{2n-1/2-{\alpha}/2}\le C, \qquad\sum_{j=1}^\infty\sqrt {a_j}\le C. \] We find, after some further straightforward estimation, that we can take $K=C$. Recall the choice of function $\phi$ in the statement of Theorem \ref {MR}. Set \[ {\varepsilon}=\sqrt{\frac{\phi(N)}N},\qquad {\delta}={\varepsilon} e^{-Kt_0}/7,\qquad {\delta }({\beta},b)={\delta},\qquad {\delta}({\gamma},g)={\delta}/(2{\tau} B). \] Recall that ${\mathbf X}_0=(1/N,0,\ldots,0)$ and $x_0=0$ and that the driving rate $\nu$ for the coupling mechanism is equal to $N{\lambda}$. It is now straightforward to check that all the inequalities required in the statement of Theorem~\ref{FLE} are valid, for all sufficiently large $N$. Now we check the tube condition of Theorem~\ref{FLE}. The inequalities $0\le x_d(\xi)\le\cdots\le x_1(\xi)\le1$ hold for all $\xi\in S$. By a monotonicity property established in the proof of Theorem \ref {DDD}, we have $x_k(t)\le a_k$ for all $t\ge0$ and for $k=1,\ldots,d$. Hence, for $N$ sufficiently large, if $\\|{\mathbf x}(\xi)-x(t)\\|\le 2{\varepsilon}$ for some $t\ge0$, then $x^k(\xi)\le a_k+2{\varepsilon}\sqrt{a_k}\le2a_k$ and $x^1(\xi)\le a_1+2{\varepsilon}\sqrt{a_1}\le{\lambda}+(1-{\lambda })/2\le(1+{\lambda})/2$, so ${\mathbf x}(\xi)\in U$ and the tube condition is satisfied. Now we turn to the extra conditions needed to apply Theorem \ref {FLET}. We noted in Section~\ref{QVB} the quadratic variation bounds \[ a_k(y)\le A{\sigma}_k^2,\qquad \bar a_k(x)\le\bar A{\sigma}_k^2, \] valid for all $x\in U$ and $y\in I$, where \[ A=C/(Na_d^{1-n/{\alpha}}),\qquad \bar A=C(\log N)^C/N \] and where $\bar A\le A\le{\Lambda}\bar J^2$ for sufficiently large $N$. It is now straightforward to check, also for $N$ sufficiently large, that the remaining inequalities required in the statement of Theorem~\ref{FLET} hold. Theorem~\ref{FLET} therefore applies to give \begin{eqnarray}\label{PEST} \mathbb{P}\Bigl(\sup_{t\le t_0}\\|{\mathbf X}_t-x_t\\|>{\varepsilon}\Bigr)&\le& 2de^{-{\delta}^2/(4\bar{A}t_0)}+2de^{-(\bar{A}/A)^2t_0/(6400\Lambda \tau^2)}\nonumber\\[-8pt]\\[-8pt] &&{}+ \mathbb{P}\bigl({\Omega}(\beta,b)^c\cup{\Omega}(\gamma,g)^c\bigr).\nonumber \end{eqnarray} Now, for $N$ sufficiently large, we have $d\le\log\log N$ and, by our choice of $\phi$ and~${\kappa}$, \[ {\delta}^2/(4\bar At_0)\ge\phi(N)/((\log N)^Ct_0)\ge\log N \] and \[ (\bar{A}/A)^2t_0/(6400\Lambda\tau^2)\ge\log N. \] Hence, the first and second terms on the right-hand side of (\ref {PEST}) tend to $0$ as \mbox{$N\to\infty$}. Recall from (\ref{OBB}) and (\ref{OGG}) the form of the events ${\Omega} ({\beta},b)$ and ${\Omega}({\gamma},g)$. In the present example, the complementary exceptional events arise either as a result of truncation or because of finite $N$ effects in the fast variable dynamics, as shown by (\ref{BETAB}) and estimate (\ref{GAMMAG}). Recall that ${\delta}({\beta},b)={\delta}$ and ${\delta}({\gamma},g)={\delta }/(2{\tau} B)$. Then \begin{equation}\label{OBBE}\quad {\Omega}({\beta},b)^c\subseteq\biggl\{\int_0^{T\wedge t_0}\frac{Z_t^{d+1}}{\sqrt {a_d}}\,dt\ge{\delta}({\beta},b)\biggr\} \subseteq\biggl\{A_d(T\wedge t_0)\ge\frac{N{\delta}\sqrt {a_d}}{t_0}\biggr\}. \end{equation} It is straightforward to check that, for all sufficiently large $N$, ${\delta}({\gamma},g)\ge2t_0(1+{\lambda}n)$, which implies that \begin{eqnarray}\label{OGGE} {\Omega}({\gamma},g)^c&\subseteq&\biggl\{\int_0^{T\wedge t_0}(1+{\lambda}n+N{\lambda }nZ_t^{d+1})\,dt\ge{\delta}({\gamma},g)\biggr\} \nonumber\\[-8pt]\\[-8pt] &\subseteq&\biggl\{A_d(T\wedge t_0)\ge\frac{{\delta}}{4{\lambda}nt_0{\tau} B}\biggr\}.\nonumber \end{eqnarray} To see that $\mathbb{P}({\Omega}(\beta,b)^c\cup{\Omega}(\gamma,g)^c)\to0$ as $N\to \infty$, we use the bound on ${\mathbb E}(A_d(T\wedge t_0))$ proved in Lemma~\ref{ADT} and Markov's inequality. It then suffices to show that in the limit $N\to\infty$, \[ C(a_d^n+\rho^dNa_{d+1})e^{Ct_0}\ll\frac{\sqrt{\phi(N)Na_d}}{4nt_0}. \] For the term involving $a_d^n$ this is easy. For the other term, involving $Na_{d+1}$, we can check that, in fact, \[ Na_{d+1}\ll\sqrt{Na_d},\qquad \rho^d\le(\log N)^C\ll\sqrt{\phi(N)}. \] This completes the proof of (\ref{MRA}). Limit (\ref{MRB}) follows immediately from Lem\-ma~\ref{ADT} using Markov's inequality. Finally, note that, as $N\to\infty$, \[ \tilde a_{d+1}\le CN^{-1}a_{d}^{n}+(\log N)^Ca_{d+1}\le C\bigl(N^{-1}+(\log N)^CN^{-1+{\kappa}}\bigr)\to0 \] and \[ N\tilde a_{d+1}^{n+1}\le C\bigl(N^{-(1-{\kappa})(n+1)/{\alpha}}+(\log N)^CN^{1-(1-{\kappa} )(n+1)}\bigr)\to0. \] Then the limit (\ref{MRC}) follows from (\ref{MRB}) and Lemma \ref {ADDT} using Markov's inequality. \subsection{Monotonicity of the queueing model} Here we prove a natural monotonicity property of the supermarket model with memory which is a microscopic counterpart of the monotonicity of solutions to the differential equation (\ref{ODE}) shown in Theorem~\ref{DDD}. We do not rely on this result in the rest of the paper. First we construct, on a single probability space, for all $\xi =(z,y)\in S$, a~version $X=X(\xi)$ of the supermarket model with memory starting from $\xi$. Set $y_1=y_1(\xi)=y$ and determine $y_i=y_i(\xi)\in{\mathbb Z}^+$ for $i=2,\ldots,N$ by the conditions \[ y_2\le\cdots\le y_N,\qquad z_k=\bigl\|\bigl\{i\in\{1,\ldots,N\}\dvtx y_i\ge k\bigr\}\bigr\|/N, \qquad k\in{\mathbb N}. \] We work on the canonical space of a marked Poisson process of rate $N(1+{\lambda})$, where the marks are either, with probability $1/(1+{\lambda})$, independent copies of a uniform random variable $J$ in $\{1,\ldots,N\}$ or, with probability ${\lambda}/(1+{\lambda})$, independent copies of a uniform random sample $(J_1,\ldots,J_n)$ from $\{1,\ldots,N\}$. Fix $\xi=(z,y)\in S$ and define a process $X=X(\xi)=(X_t\dvtx t\ge0)$ in $S$ as follows. Set $X_t=\xi$ for all $t<T$, where $T$ is the first jump time of the Poisson process. If the first mark is a random variable, $J$ say, take the sequence $y_1,\ldots,y_N$ and replace $y_J$ by $(y_J-1)^+$ to obtain a sequence $u_1,\ldots,u_N$ say; set $\tilde y_1=u_1$ and write $u_2,\ldots,u_N$ in nondecreasing order to obtain $\tilde y_2\le\cdots\le\tilde y_N$. If the first mark is a random sample, $(J_1,\ldots,J_n)$ say, select components $(y_i\dvtx i\in\{1,J_1,\ldots,J_n\})$ and write these in nondecreasing order, $w_1\le\cdots\le w_m$ say; replace $w_1$ by $w_1+1$ and write the resulting sequence, again in nondecreasing order, $v_1\le\cdots\le v_m$ say; set $\tilde y_1=v_1$ and write $v_2,\ldots,v_m$ combined with the unselected components $(y_i\dvtx i\notin\{1,J_1,\ldots,J_n\})$ in nondecreasing order to obtain $\tilde y_2\le\cdots\le\tilde y_N$. Set $X_T=((Z^k_T\dvtx k\in{\mathbb N}),Y_T)$, where \[ Z^k_T=\bigl\|\bigl\{i\in\{1,\ldots,N\}\dvtx\tilde y_i\ge k\bigr\}\bigr\|/N,\qquad k\in{\mathbb N},\qquad Y_T=\tilde y_1, \] and repeat the construction from $X_T$ in the usual way. For $\xi,\xi'\in S$ write $\xi\le\xi'$ if $y_i(\xi)\le y_i(\xi ')$ for $i=1,\ldots,N$. \begin{theorem} Let $\xi,\xi'\in S$ with $\xi\le\xi'$. Then $X_t(\xi)\le X_t(\xi ')$ for all $t\ge0$. \end{theorem} \begin{pf} It will suffice to check that the desired inequality holds at the first jump time $T$, that is to say, with obvious notation, that $\tilde y_i\le\tilde y'_i$ for all $i$. Note that if $a_i\le b_i$ for all $i$ for two sequences $(a_1,\ldots ,a_n)$ and $(b_1,\ldots,b_n)$, then the same is true for their nondecreasing rearrangements. In the case where the first mark is a random variable $J$, since $y_i\le y_i'$ for all $i$, we have $u_i\le u'_i$ for all $i$ and so $\tilde y_i\le\tilde y'_i$ for all $i$. On the other hand, when the first mark is a random sample $(J_1,\ldots ,J_n)$, we have $w_j\le w'_j$ for all $j$, so $v_j\le v'_j$ for all $j$, and so $\tilde y_i\le\tilde y'_i$ for all $i$. \end{pf} \printaddresses \end{document}	arXiv
Match Fishtank is now Fishtank Learning! Fishtank Learning Vector Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade 9th Grade 10th Grade 11th Grade 12th Grade 3rd Grade 4th Grade 5th Grade 6th Grade 7th Grade 8th Grade Algebra 1 Geometry Algebra 2 Search Icon Created with Sketch. About Us Curriculum Teacher Tools Search Login Join Now Mathematics / 3rd Grade / Unit 3: Multiplication and Division, Part 2 Multiplication and Division, Part 2 Students deepen their understanding of multiplication and division, including their properties and extending their study of factors to include all units from 0 to 10, as well as multiples of 10 within 100. Download Full Unit Download Icon Created with Sketch. Unit Prep Unit Practice Lesson Map Unit Summary Unit 3 extends the study of factors from 2, 3, 4, 5, and 10, which students explored in Unit 2, to include all units from 0 to 10, as well as multiples of 10 within 100. To work with these more challenging units, students will rely on skip-counting (a Level 2 strategy) and converting to an easier problem (a Level 3 strategy dependent on the properties of operations). They then will apply their understanding of all four operations to two-step word problems as well as arithmetic patterns. Finally, the unit culminates with a focus on categorical data, where students draw and solve problems involving scaled picture graphs and scaled bar graphs, a nice application of the major work of multiplication and division. Topic A begins by reminding students of the commutative property they learned in Unit 2, as well as introducing them to the distributive and associative properties, upon which they will rely for many of the strategies they learn for the larger factors. In order to be able to use these properties, they need to understand how to compute with a factor of 1, which they explore along with 0, as well as understand how to use parentheses. They'll then explore the factors of 6, 7, 8, and 9 in Topics B and C. Because of the increased difficulty of these facts, students will rely on both skip-counting (a Level 2 strategy) as well as converting to an easier problem (a Level 3 strategy). Converting to an easier problem is dependent on the properties of operations (e.g., to find 6 x 7, think of 5 x 7 and add a group of 7 is dependent on the distributive property). Thus, students will work with the properties extensively throughout the unit, with their understanding of them and notation related to them growing more complex and abstract throughout the unit. In Topic D, students will multiply one-digit numbers by multiples of 10 and by two-digit numbers using the associative property. Then, students solve two-step word problems involving all four operations, assessing the reasonableness of their answer, and identify arithmetic patterns and explain them using the properties of operations. Finally, students explore picture graphs in which each picture represents more than one object and bar graphs where the scale on the axis is more than 1, a key development from Grade 2 (3.MD.3). As the Progressions note, "these developments connect with the emphasis on multiplication in this grade" (MD Progression, p. 7). Students also solve one- and two-step word problems related to the data in these plots, relying on the extensive work students have done with word problems throughout the year. Thus, this supporting cluster standard nicely enhances the major work they've been working on throughout this and the previous unit. In Unit 3, students deepen their understanding of multiplication and division, including their properties. "Mathematically proficient students at the elementary grades use structures such as…the properties of operations…to solve problems" (MP.7) (Standards for Mathematical Practice: Commentary and Elaborations for K–5, p. 9). Students use the properties of operations to convert computations to an easier problem (a Level 3 strategy), as well as construct and critique the reasoning of others regarding the properties of operations (MP.3). Lastly, students model with mathematics with these new operations, solving one- and two-step equations using them (MP.4). Students' understanding of multiplication and division will further develop in Unit 4, when students study area. Students will also use their understanding of these operations in Unit 7 when they apply them in the context of measurement word problems. In subsequent years, students' understanding will be entirely dependent on their conceptual understanding and fluency with these operations—everything from multi-step multiplicative comparison words problems in Grade 4 to polynomial multiplication and division in Algebra 2, and lots in between. Thus, this unit culminates major work of Grade 3 as well as deeply important foundational work upon which students will rely for years to come. Pacing: 31 instructional days (28 lessons, 2 flex days, 1 assessment day) For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 3rd Grade Scope and Sequence Recommended Adjustments. Subscribe to Fishtank Plus to unlock access to additional resources for this unit, including: Expanded Assessment Package Problem Sets for Each Lesson Student Handout Editor Google Classroom Integration Vocabulary Package This assessment accompanies Unit 3 and should be given on the suggested assessment day or after completing the unit. Download Unit Assessment Download Unit Assessment Answer Key Download Student Self-Assessment Intellectual Prep Intellectual Prep for All Units Read and annotate "Unit Summary" and "Essential Understandings" portion of the unit plan. Do all the Target Tasks and annotate them with the "Unit Summary" and "Essential Understandings" in mind. Take the unit assessment. Read the following table that includes models used throughout the unit. Equal groups Example: 4 equal groups of 3 stars Example: 4 rows of 3 Tape diagram Example: There are 4 bags with 3 plums in each bag. How many plums are there in all? Concrete or pictorial base ten blocks Example: Find $$2 \times 60$$ using base ten blocks. Picture graph Essential Understandings Multiplication problems can be solved using a variety of strategies of increasing complexity, including making and counting all of the quantities involved in multiplication or division (Level 1 strategy), repeated counting on by a given number (Level 2), and using the properties of operations to compose and decompose unknown facts into known ones (Level 3). Unknown numbers in a count sequence can be found by using mental strategies. "For example, in the count-by for 7, students might use the following mental decompositions of 7 to compose up to and then go over the next decade, e.g., 14 + 7 = 14 + 6 + 1 = 20 + 1 = 21" (OA Progression, p. 25). Two, five, and ten are often helpful values to use when using properties to compose and decompose. Making sense of problems and persevering to solve them is an important practice when solving word problems. Key words do not always indicate the correct operation. Unit Materials, Representations and Tools Tape diagrams Base ten blocks Multiplication Chart Picture graphs associative property Related Teacher Tools: 3rd Grade Vocabulary Glossary Additional Unit Practice With Fishtank Plus you can access our Daily Word Problem Practice and our content-aligned Fluency Activities created to help students strengthen their application and fluency skills. Topic A: Introduction to The Properties of Operations 3.OA.B.5 3.OA.D.9 Study commutativity to find known facts of 6, 7, 8, and 9. 3.OA.A.4 3.OA.C.7 Understand the zero and identity properties of multiplication. Understand the role of parentheses and apply to solving problems. Introduce the distributive property of multiplication. Introduce the associative property of multiplication. Topic B: Multiplication and Division by 6 and 7 Skip-count to build fluency with multiplication facts using units of 6. Decompose an addend with respect to a ten when going over a decade in the skip-counting sequence. Use the associative property as a strategy to multiply by units of 6. Use the distributive property as a strategy to multiply by units of 6. Solve one- and two-step word problems involving units up to 7. Topic C: Multiplication and Division by 8 and 9 Skip-count to build fluency with multiplication facts using units of 8 and 9. Decompose an addend with respect to a ten when going over a decade in the skip-counting sequence. Use the associative property as a strategy to multiply by units of 8 and 9. Use the distributive property as a strategy to multiply by units of 8 and 9. Use the distributive property as a strategy to multiply by units of 8 and 9, including the subtractive use of the distributive property. Topic D: Multiplication and Division by Values Greater than 10 3.NBT.A.3 Multiply one-digit whole numbers by multiples of 10. Multiply one-digit whole numbers by two-digit whole numbers using the associative and distributive properties. Topic E: Two-Step Word Problems and Patterns in Arithmetic Solve two-step word problems involving all four operations and assess the reasonableness of answers. Identify patterns in single or multiple rows/columns of the multiplication table. Identify other patterns in the multiplication table. Identify arithmetic patterns and explain them using properties of operations. Topic F: Scaled Picture and Bar Graphs 3.MD.B.3 Create scaled picture graphs where the scale is provided. Create scaled picture graphs where the scale must be determined. Create scaled bar graphs where the scale is provided. Create scaled bar graphs where the scale must be determined. Solve one- and two-step word problems using information presented in scaled picture and bar graphs. Key: Major Cluster Supporting Cluster Additional Cluster Core Standards 3.MD.B.3 — Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. Number and Operations in Base Ten 3.NBT.A.3 — Multiply one-digit whole numbers by multiples of 10 in the range 10—90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. Operations and Algebraic Thinking 3.OA.A.3 — Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 3.OA.A.4 — Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = ÷ 3, 6 × 6 = ?. 3.OA.B.5 — Apply properties of operations as strategies to multiply and divide. Students need not use formal terms for these properties. Example: Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) Example: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) 3.OA.C.7 — Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. 3.OA.D.8 — Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations). 3.OA.D.9 — Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. 2.MD.D.10 2.MD.D.10 — Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph. 2.NBT.A.2 — Count within 1000; skip-count by 5s, 10s, and 100s. 3.NBT.A.2 — Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 2.OA.A.1 — Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem. 2.OA.C.3 — Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 2.OA.C.4 — Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 3.OA.A.1 — Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 3.OA.A.2 — Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 3.OA.B.6 — Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Future Standards 4.NBT.B.5 4.NBT.B.5 — Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.B.6 — Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Number and Operations—Fractions 4.NF.B.4 4.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 5.NF.B.3 — Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 5.NF.B.4 — Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.B.5 — Interpret multiplication as scaling (resizing), by: 5.NF.B.6 — Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.B.7 — Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade 4.OA.A.1 — Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 4.OA.A.2 — Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. 4.OA.A.3 — Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 4.OA.B.4 — Find all factor pairs for a whole number in the range 1—100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1—100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1—100 is prime or composite. CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them. CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively. CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others. CCSS.MATH.PRACTICE.MP4 — Model with mathematics. CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically. CCSS.MATH.PRACTICE.MP6 — Attend to precision. CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure. CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning. At Fishtank Learning, we believe that teachers and their students deserve access to the highest quality instructional materials. Browse our comprehensive unit and lesson plans in a convenient, openly-licensed format that you can download, use, and adapt—all for free.	CommonCrawl
Stromquist moving-knives procedure The Stromquist moving-knives procedure is a procedure for envy-free cake-cutting among three players. It is named after Walter Stromquist who presented it in 1980.[1] This procedure was the first envy-free moving knife procedure devised for three players. It requires four knives but only two cuts, so each player receives a single connected piece. There is no natural generalization to more than three players which divides the cake without extra cuts. The resulting partition is not necessarily efficient.[2]: 120–121 Procedure A referee moves a sword from left to right over the cake, hypothetically dividing it into small left piece and a large right piece. Each player moves a knife over the right piece, always keeping it parallel to the sword. The players must move their knives in a continuous manner, without making any "jumps".[3] When any player shouts "cut", the cake is cut by the sword and by whichever of the players' knives happens to be the central one of the three (that is, the second in order from the sword). Then the cake is divided in the following way: • The piece to the left of the sword, which we denote Left, is given to the player who first shouted "cut". We call this player the "shouter" and the other two players the "quieters". • The piece between the sword and the central knife, which we denote Middle, is given to the remaining player whose knife is closest to the sword. • The remaining piece, Right, is given to the third player. Strategy Each player can act in a way that guarantees that—according to their own measure—no other player receives more than them: • Always hold your knife such that it divides the part to the right of the sword to two pieces that are equal in your eyes (hence, your knife initially divides the entire cake to two equal parts and then moves rightwards as the sword moves rightwards). • Shout 'cut' when Left becomes equal to the piece you are about to receive if you remain quiet (i.e. if your knife is leftmost, shout 'cut' if Left=Middle; if your knife is rightmost, shout if Left=Right; if your knife is central, shout 'cut' if Left=Middle=Right). Analysis We now prove that any player using the above strategy receives an envy-free share. First, consider the two quieters. Each of them receives a piece that contains their own knife, so they do not envy each other. Additionally, because they remained quiet, the piece they receive is larger in their eyes than Left, so they also don't envy the shouter. The shouter receives Left, which is equal to the piece they could receive by remaining silent and larger than the third piece, hence the shouter does not envy any of the quieters. Following this strategy each person gets the largest or one of the largest pieces by their own valuation and therefore the division is envy-free. The same analysis shows that the division is envy-free even in the somewhat degenerate case when there are two shouters, and the leftmost piece is given to any of them. Dividing a 'bad' cake The moving-knives procedure can be adapted for chore division - dividing a cake with a negative value.[4]: exercise 5.11 See also • The Fair pie-cutting procedure provides a simpler solution to the same problem, using only 3 rotating knives, when the cake is a 1-dimensional circle ("pie"), • The Robertson–Webb rotating-knife procedure provides an even simpler solution, using only 1 rotating knife, when the cake is 2-dimensional. • Moving-knife procedure References 1. Stromquist, Walter (1980). "How to Cut a Cake Fairly". The American Mathematical Monthly. 87 (8): 640–644. doi:10.2307/2320951. JSTOR 2320951. 2. Brams, Steven J.; Taylor, Alan D. (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 0-521-55644-9. 3. The importance of this continuity is explained here: "Stromquist's 3 knives procedure". Math Overflow. Retrieved 14 September 2014. 4. Robertson, Jack; Webb, William (1998). Cake-Cutting Algorithms: Be Fair If You Can. Natick, Massachusetts: A. K. Peters. ISBN 978-1-56881-076-8. LCCN 97041258. OL 2730675W.	Wikipedia
SGI 2021 Summer Geometry Initiative SGI research projects Bayesian Rotation Synchronization Post author By Adrish No Comments on Bayesian Rotation Synchronization By Adrish Dey, Dorothy Najjuma Kamya, and Lily Kimble Note: Although this is an ongoing work, this report documents our progress between the official 2 weeks of the project. (August 2, 2021 – August 13, 2021) The past 2 weeks at SGI, we have been working with David Palmer on investigating a novel Bayesian approach towards the angular synchronization problem. This blog post is written to document our work and share a sneak peek into our research. Consider a set of unknown absolute orientations $\{q_1, q_2, \ldots, q_n\}$ with respect to some fixed basis. The problem of angular synchronization deals with the accurate estimation of these orientations from noisy observations of their relative offsets $O_{i, j}$, up to a constant additive phase. We are particularly interested in estimating these "true" orientations in the presence of many outlier measurements. Our interest in this topic stems from the fact that the angular synchronization problem arises in various avenues of science, including reconstruction problems in computer vision, ranking problems, sensor network localization, structural biology, etc. In our work, we study this problem from a Bayesian perspective, by modelling the observed data as a mixture between noisy observations and outliers. We also investigate the problem of continuous label switching, a global ambiguity that arises from the lack of knowledge about the basis of the absolute orientations $q_i$. Finally, we experiment on a novel Riemannian gradient descent method for alleviating this continuous label switching problem and provide our observations herein. Brief Primer on Bayesian Inference Before going deeper, we'll briefly discuss Bayesian inference. At the heart of Bayesian inference lies the celebrated Bayes' rule (read $a\|b$ as "a given b"): \[\underbrace{P(q \| O)}_{\textrm{posterior}} = \frac{\overbrace{P(O\|q)}^{\textrm{likelihood}} \cdot \overbrace{P(q)}^{\textrm{prior}}}{\underbrace{\int\limits_q P(O\|q)\cdot P(q)}}_{\textrm{evidence}}\] In our problem, $q$ and $O$ denote the true orientations that we are estimating and the noisy observations with outliers respectively. We are interested in finding the posterior distribution (or at least samples from it) over the ground truth $q$ given the noisy observations $O$. The denominator (i.e., the evidence or partition function) is an integral over all $q$s. Exactly evaluating this integral is often intractable if $q$ lies on some continuous manifold, as in our problem. This makes drawing samples from the posterior becomes hard. One way to avoid computing the partition function is a sampling method called Markov Chain Monte Carlo (MCMC). Intuitively, the posterior is approximated by a markov chain whose transitions can be computed using a simpler distribution called the proposal distribution. Successive samples are then accepted or rejected based on an acceptance probability designed to ensure convergence to the posterior distribution in the limit of infinite samples. Simply put, after enough samples are drawn using MCMC, they will look like the samples from the posterior$P(q\|O) \propto P(O\|q) \cdot P(q)$ without requiring us to calculate the intractable normalization $P(O) = \int\limits_q P(O\|q)\cdot P(q)$. In our work, we use Hamiltonian Monte Carlo (HMC), an efficient variant of MCMC, which uses Hamiltonian Dynamics to propose the next sample. From an implementation perspective, we use the built-in HMC sampler in Stan for drawing samples. Mixture Model As mentioned before, we model the noisy observation as a mixture model of true distribution and outliers. This is denoted by (Equation 1): \[ O_{i, j} = \begin{cases} q_i q_j^T + \eta_{i, j} & \textrm{with prob. } p \\ \textrm{Uniform}(\textrm{SO}(D)) & \textrm{with prob. } 1 – p \end{cases} \] where $\eta_{i j}$ is the additive noise to our true observation, $q_i q_j^T$ is the relative orientation between the $i^\textrm{th}$ and $j^\textrm{th}$ objects, and $\textrm{Uniform}(\textrm{SO}(D))$ is the uniform distribution over the rotation group $\textrm{SO}(D)$, representing our outlier distribution. $\textrm{SO}(D)$ is the space where every element represents a D-dimensional rotation. This mixture model serves as the likelihood $P(O\|q)$ for our Bayesian framework. Sampled Result Ground truth samples of $q_i$ $\hat{q}_i \sim p(q\|O)$ (estimated $q_i$) sampled from our posterior $p(q\|O)$ The orientations $\hat{q}_i$ sampled from the posterior look significantly rotated with respect to the original samples. Notice this is a global rotation since all the samples are rotated equally. This problem of global ambiguity of absolute orientations $q_i$ arises from the fact that the relative orientations $q_i q_j^T$ and $\tilde{q}_i \tilde{q_j}^T$ of two different set of vectors can be the same even if the absolute orientations are different. The following section goes over this and provides a sneak peek into our solution for alleviating this problem. Continuous Label Switching A careful observation of our problem formulation (Equation 1) would reveal the problem is invariant to transformation of the absolute orientations as long as the relative orientations are preserved. Consider the 2 pairs of observations in the figure below. (Blue and Red; Yellow and Green) Let the absolute orientations be $q_1$, $q_2$, $\tilde{q}_1$ and $\tilde{q}_2$ and relative orientations between the pairs be $R_{12}$ and $\tilde{R}_{12}$. As the absolute orientations $q_1$ and $q_2$ are equally rotated by a rotation matrix $A$, the relative orientation between them $R_{12}$ is preserved. More formally, Let $A \in \textrm{SO}(D)$ be a random orientation matrix in D-dimensions. The following equation demonstrates how rotating two absolute orientations $q_i$ and $q_j$ by a rotation matrix $A$ preserves the relative orientations — which in turn gives rise to a global ambiguity in our framework. R_{ij} = q_i q_j = q_i A A^T q_j = (q_i A) (A^T q_j) = \tilde{q_i} \tilde{q_j} Since our inputs to the model are relative orientations, this ambiguity (known as label switching) causes our Bayesian estimates to come randomly rotated by some rotation $A$. Proposed Solution Based on Monteiller et al., in this project we explored a novel solution for alleviating this problem. The intuition is that we believe the unknown ground truth is close to the posterior samples up to a global rotation. Hence we try to approximate the ground truth by starting out with a random guess and optimizing for the alignment map between the estimate and the ground truth. Using this alignment map, and the posterior samples, we iteratively update the guess, using a custom Riemannian Stochastic Gradient Descent over $\textrm{SO}(D)$. Start with a random guess $\mu \sim \mathrm{Uniform}(\mathrm{SO}(D))$. Sample $\hat{q} \sim P(q \| O)$, where $P(q \| O)$ is the posterior. Find the global ambiguity $R$, between $\hat{q}$ and $\mu$. This can be obtained by solving for $\mathrm{argmin}_R \, \\| \mu – R \hat{q}\\|_\mathrm{F}$. Move $\mu$ along the shortest geodesic toward $\hat{q}$. Repeat Steps 2 – 4 until convergence. Convergence is detected by a threshold on geodesic distance. We use this method to estimate the mean of the posterior over $\mathrm{SO}(2)$ and plot the results (i.e. 2D orientations) in the complex plane as shown below. Original Sample Sampled Posterior Optimized Mean Posterior The proposed optimization proceedure is able to successfully re-align the posterior samples by alleviating the continuous label switching problem. In conclusion, we study the rotation synchronisation problem from a Bayesian perspective. We explore a custom Riemannian Gradient Descent procedure and perform experiments in the $\mathrm{SO}(2)$ case. The current method is tested on a simple toy dataset. As future work we are interested in improving our Bayesian model and benchmarking it against the current state-of-the-art. There are certain performance bottlenecks in our current architecture, which constrain us to test only on $\mathrm{SO}(2)$. In the future, we are also interested in carrying out experiments more thoroughly in various dimensions. While the current MCMC procedure we are using does not account for the non-Euclidean geometry of the space of orientations, $\mathrm{SO}(D)$, we are looking into replacing it with Riemannian versions of MCMC. Tags label switching, research project Math SGI research projects Minimal Surfaces, But Periodic Post author By Olga Guțan 1 Comment on Minimal Surfaces, But Periodic By Zhecheng Wang, Zeltzyn Guadalupe Montes Rosales, and Olga Guțan Note: This post describes work that has occurred between August 9 and August 20. The project will continue for a third week; more details to come. For the past two weeks we had the pleasure of working with Nicholas Sharp, Etienne Vouga, Josh Vekhter, and Erik Amezquita. We learned about a special type of minimal surfaces: triply-periodic minimal surfaces. Their name stems from their repeating pattern. Broadly speaking, a minimal surface minimizes its surface area. This is equivalent to having zero mean curvature. A triply-periodic minimal surface (TPMS) is a surface in $\mathbb{R}^{3}$ that is invariant under a rank-3 lattice of translations. Figure 1. (Left) A Minimal Surface [source] and (right) a TPMS [source]. Let's talk about nonmanifold surfaces. "Manifold" is a geometry term that means: every local region of the surface looks like the plane (more formally — it is homeomorphic to a subset of Euclidean space). Non-manifold then allows for parts of the surface that do not look like the plane, such as T-junctions. Within the context of triangle meshes, a nonmanifold surface is a surface where more than 3 faces share an edge. II. What We Did First, we read and studied the 1993 paper by Pinkhall and Polthier that describes the algorithm for generating minimal surfaces. Our next goal was to generate minimal surfaces. Initially, we used pinned (Dirichlet) boundary conditions and regular manifold shapes. After ensuring that our code worked on manifold surfaces, we tested it on non-manifold input. Additionally, our team members learned how to use Blender. It has been a very enjoyable process, and the work was deeply satisfying, because of the embedded mathematical ideas intertwined with the artistic components. III. Reading the Pinkall Paper "Computing Discrete Minimal Surfaces and Their Conjugates," by Ulrich Pinkall and Konrad Polthier, is the classic paper on this subject; it introduces the iterative scheme we used to find minimal surfaces. Reading this paper was our first step in this project. The algorithm that finds a discrete locally area-minimizing surface is as follows: Take the initial surface $M_0$ with boundary $∂M_0$ as first approximation of M. Set i to 0. Compute the next surface $M_i$ by solving the linear Dirichlet problem $$ \min_{M} \frac{1}{2}\int_{M_{i}}\|\nabla (f:M_{i} \to M)\|^{2}$$ Set i to i+1 and continue with step 2. The stopping condition is $\|\text{area}(M_i)-\text{area}(M_{i+1})\|<\epsilon$. In our case, we used a maximum number of iterations, set by the user, as a stopping condition. There are additional subtleties that must be considered (such as "what to do with degenerate triangles?"), but since we did not implement them — their discussion is beyond the scope of this post. IV. Adding Periodic Boundary Conditions This is, at its core, an optimization problem. To ensure that the optimization works, the boundary conditions have to be periodic instead of fixed in space. This is because we are enforcing a set of boundary conditions on periodic shapes — that is, tiling in a 3D space. IV(a). Matching Vertices First, we need to check every two pairs of vertices in the mesh. We are looking to see if they have identical coordinates in two dimensions, but are separated by exactly two units in the third dimension. When we find such pairs of vertices, we classify them to $G_x$, $G_y$, or $G_z$. Note that we only store unique pairs. IV(b). Laplacian Smoothness at the Boundary Vertices Instead of using the discrete Laplacian, now we introduce a sparse matrix K to adjust our smoothness term based on the new boundary: $$\min_{x}x^T(L^TK^TKL)x \text{ s.t. } x[b] = x_0[b].$$ Next, we construct the matrix K, which is a sparse square matrix of dimension #vertices by #vertices. To do so, we set $K(i,i) = 1$, $K(i,j) = 1$, and set the $j$th row entries to 0 for every pair of unique matched boundary vertices. For every interior vertex $i$, we set $K(i,i) = 1$. IV(c). Aligning Boundary Vertices Now we no longer want to pin boundary vertices to their original location in space. Instead, we want to allow our vertices to move, while the opposite sides of the boundary still match. To do that, we need to adjust the existing constraint term and to include additional linear constraints $Ax=b$. Therefore, we add two sets of linear constraints to our linear system: For any pair of boundary vertices distanced by 2 units in one direction, the new coordinates should differ by 2 units. For any pair of boundary vertices matched in the two other directions, the new coordinates should differ by 0. We construct a selection matrix $A$, which is a #pairs of boundary vertices in $G$ by #vertices sparse matrix, to get the distance between any pair of boundary vertices. For every $r$th row, $A(r,i)=1, A(r,j)=-1.$$ Then we need to construct 3 $b$ vectors, each of which is a sparse square matrix of the size [# of vector pairs of boundary vertices in G for a 3D coordinate (x,y,z)]. Based on whether at one given moment we are working with $G_x$, $G_y$, or $G_z$, we enter $2$ for those selected pairs, and $0$ for the rest. V. Correct Outcomes Below, we can see the algorithm being correctly implemented. Each video represents a different mesh. VI. Aesthetically Pleasing Bugs Nothing is perfect, and coding in Matlab is no exception. We went through many iterations of our code before we got a functional version. Below are some examples of cool-looking bugs we encountered along the way, while testing the code on (what has become) one of our favorite shapes. Each video represents a different bug applied onto the same mesh. VII. Conclusion and Future Work Further work may include studying the physical properties of nonmanifold TPMS. It may also include additional basic structural simulations for physical properties, and establishing a comparison between the results for nonmanifold surfaces and the existing results for manifold surfaces. Additional goals may be of computational or algebraic nature. For example, one can write scripts to generate many possible initial conditions, then use code to convert the surfaces with each of these conditions into minimal surfaces. An algebraic goal may be to enumerate all possible possibly-nonmanifold structures and perhaps categorize them based on their properties. This is, in fact, an open problem. The possibilities are truly endless, and potential directions depend on the interests of the group of researchers undertaking this project further. Processing the Philosophy of Geometry No Comments on Processing the Philosophy of Geometry By Bryce Van Ross and Olga Guțan Brief Existential Background Most branches of geometry are viewed as either (1) subsets of pure mathematics or (2) disciplines derived from problems in engineering, architecture, and computational mathematics. But what is geometry itself? Our work in the Summer Geometry Institute has been in the modern field of geometry processing. In this post, we will consider the philosophical aspects of geometry, and what they tell us about geometry processing specifically. The discussion has an inherent Eurocentric bias, and is, unfortunately, limited to Caucasian male philosophers. The Roots of Geometry In the eighteenth century, mathematics was primarily used to understand the natural sciences. For example, it was perceived as a language to better articulate physics, rather than as having intrinsic value. At that time, geometry was perceived as a form of applied mathematics. But where does the philosophy of geometry have its roots? First we have Kant (1724-1804), who believed that our grasp of euclidean geometry must be instinctive. Opinions about this vary, based on each individual's definitions of a priori knowledge and empiricism. Helmholtz, for example, agreed with Kant. However, geometry processing incorporates not only euclidean geometry, but also calculus, linear algebra, differential equations, and topology. In addition, geometry processing includes a framework of algorithmic and optimization techniques. The knowledge of these topics varies from person to person, and therefore geometry processing can not be known a priori. Back to the 18th century: as logical empiricism became popular, the progress of philosophy of geometry stalled. Geometry was reduced to a system of definitions and conventions; logic and mathematics were given priority. As a result, the last big wave of interest in the philosophy of geometry was in the 1920s — 100 years ago, and 116 years after Kant! Space as a Mathematical Concept Mathematical conclusions depend on the structure we confer to a space. The structure determines the permissible elements, as well as the operations that can be applied to the elements. Without imposing structure, it is difficult to impose "rules," and the potency of math is, thus, diluted. Most spaces in euclidean geometry exist; in fact, they are embedded in the reality we live in. With other kinds of geometry, we must define the properties of the space first. Then, we can ask: what are the conditions for creating a space? What kinds of spaces, and therefore geometries, can we have? (Our personal favorite is the geometry of chance.) Helmholtz (1821-1894) said the only allowable spaces are those that maintain constant curvature. This was eventually challenged by Einstein's (1879-1955) Theory of Relativity. Weyl (1885-1955) came up with intuitive spaces, where it is possible to compare lengths only if two edges share the same spatio-temporal point. Similarly to the earlier-mentioned Kant (1724-1804), Carnap (1891-1970) claimed that intuition of spaces requires an n-dimensional topological space. As the philosophy of geometry evolved, so did our understanding of space. "Waiting for Godot" in a Hyperbolic Space? (The authors would like to gratefully acknowledge the primary source for the following sections.) So what is geometry? Many people study it or use it… but what exactly are we interacting with? As an exercise of imagination, consider being a two-dimensional (2D) entity in a 2D universe. In fact, you don't even have to imagine, somebody already wrote about this. The book is, unfortunately, at times misogynistic, but the geometry is intriguing to consider. Back to the exercise of imagination: since everything in the 2D universe is finite, is the universe itself finite? A 2D mathematician might suggest that the universe looks like a rectangle. Does, then, the universe have a point beyond which travel is impossible? If so, it likely is different from all other points of the universe, by its construction, since it has a boundary. Imagine you are playing Asteroids. As you move your rocket ship towards the right — once it reaches the boundary, it will reappear on the left side. In mathematical language we say that "the left edge of the rectangle has been identified, point by point, with the right edge." Figure 1. A finite 2D world with no boundary. Source. How to Build a Torus When You Are Two-Dimensional In three dimensions, we can glue these boundaries. First, we bend the rectangle to produce a cylinder, while being careful to glue only the left and right edges together. Since we are operating in a 'perfect' thought-based mathematical world, the cylinder is malleable. Now we bend it to achieve this second gluing, which looks like a donut. We call this a torus. Figure 2. A torus. Source. Recall that our reader currently lives in a 2D universe. A 2D person would not be able to see this torus in multiple dimensions. However, one would understand the space perfectly well, since the space can be identified back to the initial rectangle with edges. In this case, the universe is clearly finite, and with no edges. Another surface with similar properties would be a two-dimensional sphere. A ladybug traveling on the sphere will notice that, locally, their world looks like a flat plane, and that the surface has no edges. On a small scale, if the mathematician ladybug splits the plane into triangles, the sum of the angles in each triangle will be 180 degrees. This is, in fact, known as a defining feature of euclidean geometry. However, on a larger scale, things start breaking. A very large triangle drawn on the surface of the sphere has an angle sum far exceeding 180 degrees. Consider the triangle formed by the North Pole and two points on the equator of the sphere. The angle at each point on the equator is 90 degrees, so the total sum exceeds 180 degrees. Therefore, we conclude that — on a global scale — we can not apply euclidean geometry to a sphere. We call this hyperbolic geometry. Figure 3. Angles add up to more than 180 degrees. Source. Shape and Connection To conclude this exercise of imagination, we must say that there is a wonderful connection between the shape (topology) of a surface and the type of geometry it inherits. This relationship is given, mathematically, by the Gauss-Bonnet equation: $\kappa A = 2 \pi \xi$ Where $\kappa $ is the Gaussian curvature of a surface, A is the element of area of the surface, $\pi $ is our good old friend 3.1415…, and $\xi$ is the Euler characteristic of the surface. In other words, the "geometry" is on the left side of the equation and the "topology" is on the right side of the equation; and the equal sign shows how deeply connected they are. Studying various shapes and how they connect is not only interesting, but also important. Particularly, if a two-dimensional entity can deduce what sort of global geometry reigns their immediate world, they may be able to deduce the possible shapes for the things in their larger universe. To Really Conclude… Although geometry processing has been around for a few decades, its philosophy is not yet well-studied. So far, this area shares themes with the broader philosophy of geometry, in certain topics: a priori vs empiricist knowledge, the evaluation of space and its elements, geometric human intuition, construction of geometries, and the study of shape and connection. Given the computational aspect of geometry processing, there are additional important philosophical questions that arise, however they go beyond the scope of this post. It is difficult to determine whether such questions are best understood from the perspective of philosophy of geometry or philosophy of computer science, or both. As geometry processing continues to impact our technological world, the philosophy of geometry processing will grow too. In the meantime, we should continue critically questioning our programming, our modeling, and — most importantly — the ethical implications of our work. We must also ensure that we have a functional understanding of the parts that constitute geometry processing. At the same time, it must be said that the discipline is more than just the sum of its parts. Tags Philosophy Elastic curves and active bending Post author By Natasha No Comments on Elastic curves and active bending By: Judy Chiang, Natasha Diederen, Erick Jimenez Berumen Project mentor: Christian Hafner Architects often want to create visually striking structures that involve curved materials. The conventional way to do this is to pre-bend materials to form the shape of the desired curve. However, this generates large manufacturing and transportation costs. A proposed alternative to industrial bending is to elastically bend flat beams, where the desired curve is encoded into the stiffness profile of the beam. Wider sections of the beam will have a greater stiffness and bend less than narrower sections of the beam. Hence, it is possible to design an algorithm that enables a designer or architect to plan curved structures that can be assembled by bending flat beams. This is a topic currently being explored by Bernd Bickel and his student Christian Hafner (our project mentor). A model created using the concept of active bending (source) For the flat beam to remain bent as the desired curve, we must ensure that the beam assumes this form at its equilibrium point. In Lagrangian mechanics, this occurs when energy is minimised, since this implies that there is no other configuration of the system which would result in lower overall energy and thus be optimal instead. Two different questions arise from this formulation. First, given the stiffness profile $K$, what deformed shapes $\gamma$ can be generated? Second, given a curve $\gamma$, what stiffness profiles $K$ can be generated? In answering the first problem we will find the curve that will result from bending a beam with a given width profile. However, we are more interested in finding the stiffness profile of a beam which will result in a curved shape $\gamma$ of our choice. Hence, we wish to solve the second problem. Below, we will give a full formulation of our main problem, and discuss how we transformed this into MATLAB code and created a user interface. We will conclude by discussing a more generalised case involving joint curves. Problem formulation To recap, the problem we want to solve is, given a curve $\gamma $, what stiffness profiles $K$ can be generated? For each point $s$ on our curve $\gamma: (0,\ell) \to \mathbb{R}^2$, we have values for the position $\gamma(s)$, turning angle $\alpha(s)$, and signed curvature $\alpha'(s)$. In addition, we define the energy of the beam system to be \[W[\alpha] := \int_0^\ell K(s)\alpha'(s)^2 ds,\] where $K(s)$ is the stiffness at point $s$. At the equilibrium state of the system, $W[\alpha]$ will take a minimum value. Intuitively, this implies that, in general, regions of larger curvature will have a lower stiffness. However, it is not true that two different points of equal curvature will have equal stiffness, since there are other factors at play. Now, before finding the $K$ that minimises $W[\alpha]$, we must set additional constraints \[\alpha(0) = \alpha_0, \quad \alpha(\ell) = \alpha_\ell\] and \[\gamma(0) = \gamma_0, \quad \gamma(\ell) = \gamma_\ell.\] In words, we are fixing the turning angles (tangents) and positions of the two boundary points. These constraints are necessary, since they dictate the position in space where the curve begins and ends, as well as the initial and final directions the curve moves in. We have now formulated a problem that can be solved using variational calculus. Without going into detail, we find that stationary points of this function are given by the equation \[K \kappa = a + \langle b, \gamma \rangle,\] where $\kappa$ is the signed curvature (previously $\alpha'$), and $a \in \mathbb{R}$ and $b \in \mathbb{R}^2$ are constants to be found. However, a stiffness profile cannot be generated for all curves $\gamma$. More specifically, it was shown by Bernd Bickel and Christian Hafner that a solution exists if and only if there exists a line $L$ that intersects $\gamma$ exactly at its inflections. With this information in hand, we can begin to create a linear program that computes the stiffness function $K$. The top four curves can be created using active bending, but the bottom four cannot (source) Creating a linear programme In order to find the stiffness $K$, we need to solve for the constants $a$ and $b$ in the equation $ K \kappa = a + \langle b, \gamma \rangle $, which can be discretised to \[ K(s_i) = \frac{a + \langle b, \gamma(s_i) \rangle }{\kappa(s_i)}.\] It is possible that there is more than one solution to $K$, so we want some way to determine the "best" stiffness profile. If we think back to our original problem, we want to ensure that our beam is maximally uniform, since this is good for structural integrity. Hence, we solve for $a$ and $b$ in the above inequality in such a way that the ratio between maximum and minimum stiffness is minimised. To do this we set the minimum stiffness to an arbitrary value, for example 1, and then constrain $K$ between 1 and $M = \text{min}_i K_i$, thus obtaining the inequality \[1 \leq \frac{a + \langle b, \gamma(s_i) \rangle }{\kappa(s_i)} \leq M.\] This can be solved using MATLAB's linprog function (read the documentation here). In this case, the variables we want to solve for are $a \in \mathbb{R}$, $b \in \mathbb{R}^2$, and $M$ (a scalar we want to minimise). So, using the linprog documentation $x$ is the vector $(a,b_1,b_2,M)$ and $f$ is the vector $(0,0,0,1)$, since $M$ is the only variable we want to minimise. Since linprog only deals with inequalities of the form $A \cdot x \leq b$, we can split the above inequality into two and write it in terms of the elements of $x$, like so: \[-\Bigg(\frac{1}{\kappa(s_i)}a + \frac{\gamma_x(s{i})}{\kappa(s_i)}b_1 + \frac{\gamma_y(s_{i})}{\kappa(s_i)}b_2 + 0\Bigg) \leq -1,\] \[\frac{1}{\kappa(s_i)}a + \frac{\gamma_x(s_{i})}{\kappa(s_i)}b_1 + \frac{\gamma_y(s_{i})}{\kappa(s_i)}b_2 -M \leq 0.\] These two equations are of a form that can be easily written into linprog to obtain the values of $a$, $b$ and $M$ and hence $K$. Once we solved the linear program outlined above, we created a user interface in MATLAB that would allow users to draw and edit a spline curve and see the corresponding elastic strip created in real time. Custom splines can be imported as a .txt file in the following format or alternatively, the file that is already in the folder can be used. Users can then run the user interface and edit the spline in real time. To add points, simply shift-click, and a new point will be added at the midpoint between the selected point and the next point. The user can right-click to delete a point, and left-click and drag to move points around. If there are zero or one control points remaining, then the user can add a new point where their mouse cursor is by shift-clicking. The number of control points must be greater than or equal to the degree plus one for the spline to be formed. There are certain cases in which the linear program cannot be solved. In these cases the elastic strip is not plotted, and the user must move the control points around until it is possible to create a strip. Demonstration of the user interface Here is the link to the GitHub repository, for those who want to try the user interface out. Joints Between Two or More Strips The current version of our code is able to generate elastic strips for any (feasible) spline curve generated by the user. However, an as of yet unsolved problem is the feasibility condition for a pair of elastic strips with joints. Solving this problem would allow us compute a pair of stiffness functions $K_1$ and $K_2$ that yield elastic strips that can be connected via slots at the fixed joint locations. With a bit of maths we were able to derive the equilibrium equations that would produce such stiffness functions. Suppose the two spline curves are given by $\gamma_{1}: [0,\ell_{1}] \to \mathbb{R}^2$ and $\gamma_{2}: [0,\ell_{2}] \to \mathbb{R}^2$ and that they intersect at exactly one point such that $\gamma_{1}(t_a) = \gamma_{2}(t_b)$. Then we must solve the following two equations: \[ K_{1}(t) \kappa_{1}(t) = \left \{ \begin{array}{ll} a_1 + \langle b_1, \gamma_{1}(t) \rangle + \langle c, \gamma_{1}(t) \rangle & 0 \leq t \leq t_a \\ a_1 + \langle b_1, \gamma_{1}(t) \rangle + \langle c, \gamma_{1}(t_a) \rangle & t_a < t \leq \ell_{1} \end{array}\right. \] \[ K_{2}(t) \kappa_{2}(t) = \left \{ \begin{array}{ll} a_2 + \langle b_2, \gamma_{2}(t) \rangle – \langle c, \gamma_{2}(t) \rangle & 0 \leq t \leq t_b \\ _2 + \langle b_2, \gamma_{2}(t) \rangle – \langle c, \gamma_{2}(t_b) \rangle & t_b < t \leq \ell_{2} \end{array}\right. \] Similar to the one-spline case, these can be translated into a linear program problem which can be solved. Due to the time constraint of this project, we were not able to implement this into our code and build an associated user interface for creating multiple splines. Furthermore, above we have only discussed the situation with two curves and one joint. Adding more joints would increase the number of unknowns and add more sections to the above piece-wise-defined functions. Lastly, we are still lacking a geometric interpretation for the above equations. All of these issues regarding the extension to two or more spline curves would serve as great inspiration for further research! Tags Active bending, Curves, Elastic curves Upper bound for the Hausdorff distance Post author By Talant 1 Comment on Upper bound for the Hausdorff distance Last week I was working on the project "Robust computation of the Hausdorff distance between triangle meshes" under Dr. Leonardo Sacht's supervision with TA Erik Amezquita and SGI fellows Bryce Van Ross and Deniz Ozbay. If we have triangle meshes $ A, B $, then $ h(A,B) = \max_{p \in A}d(p,B) $ is called the Hausdorff distance from $ A$ to $ B $, where $d$ is Euclidean distance. In general case, this function is not symmetric, so the final metric is defined as $H(a,b) = \max(h(A,B), h(B,A))$. The Hausdorff distance is very significant in Geometry Processing on the grounds that it may be used for determining the difference between two meshes. The method that we are studying is called the "branch-and-bound" method. The main idea is to calculate the common lower bound for distance from the whole mesh $ A$ to mesh $B$ and individual upper bounds for distances from every triangle mesh $T_{A}$ of mesh $ A $ to mesh $B$. If the upper bounds of some triangles is smaller than the lower bound, then we throw them away and consider the remaining subdivided ones. My task was to code the function that returns the upper bounds for distances from every triangle mesh $T_{A}$ of mesh $ A $ to mesh $B$. We are going to use the distances between vertices of triangle mesh $T_{A}$ and triangle inequality to find the upper bound: \[u( T_{A}, B) = \max_{j = 1}^{3}(\max(\|v_{j} – v_{j + 1}\|, \|v_{j} – v_{j + 2}\|) + \min_{b \in B}(\|v_{j} – b\|))\] where $v_{1}, v_{2}, v_{3}$ are the vertices of triangle mesh $T_{A}$ . This is how the implemented algorithm looks like in Matlab: I really enjoyed working on the project this week with my team, TA and supervisor. I am looking forward to continuing work and improving what we have already done. Tags #hausdorff, #hausdorff_distance 2D Cut Optimization Post author By Kirby Dietz No Comments on 2D Cut Optimization Or how to figure out the right way to skin a cat (for texture mapping) By: Kirby Dietz and Bonnie Magland, mentored by Yusuf Sahillioglu For the past two weeks, we have been doing research on how to optimize making 2D cuts onto a surface such that we get an optimal parameterization of the surface onto a disk. This question more generally is asking how we can take a triangulated surface and perform operations on it such that we see a representation of this surface on a disk that we can apply texture mapping or other various changes to. Firstly let us explain why we want to make these cuts. We make these cuts so that we can use existing methods of disk parameterization from a surface with a boundary onto a disk. We can also think about making a 2D surface from our 3D surface using cuts in a more intuitive way. We can think about it like how a net works, that we know it is true that if we took enough cuts it would be true that we get a locally distortion free 2D representation of the surface; however doing this would both be computationally intensive and not very useful for performing tasks that we want, such as texture mapping, and it does not give an intuitive understanding of what the general surface is or what each section of the 2D map corresponds with each section of the surface. Thus the question becomes how do we take a minimal amount of cuts such that we have a 2D parameterization of the surface that minimizes distortion such that we can still use it for various texture mapping applications. Disk Parameterization: First we worked on mapping 3D meshes with existing boundaries to a disk. We implemented existing methods of disk conformal parameterization using three different weight functions. We start by mapping the mesh boundary to the circle, preserving the distance between chord edges. Placing the interior points is then a matter of solving a system of linear equations. In our first weighting method, each edge is weighted the same so in the resulting disk, interior points are the centroid of their neighbors. This gives one linear equation for each interior point, and since the boundary points are fixed, we can solve for the exact locations of each point on the disk. Weighing each edge uniformly results in unnecessary distortion. Our second and third methods use the angles in the faces sharing the edge to add weights to the equations in the first method. The second method uses harmonic weights, with formula $w_{i,j}=\frac{1}{2}\left( cot(i,j) + cot(i,j)\right)$ where $\alpha_{i,j}$ and $\beta_{i,j}$ are the angles opposite the edge connecting the $i$ and $j^{th}$ vertices. These weights reduce distortion, but obtuse angles can result in negative weights, so the resulting map is not guaranteed to be bijective. The third method, mean value weights, resolves this by instead using $w_{i,j}=\frac{tan(\frac{\gamma_{i,j}}{2}) + tan(\frac{\delta_{i,j}}{2})}{2\|\| v_i – v_j \|\|}$ where $\gamma_{i,j}$ and $\delta_{i,j}$ are the angles adjacent to the edge connecting the $i$ and $j^{th}$ vertices. In our future disk parameterizations, we use these last two methods. Methods 1, 2, and 3 from left to right Virtual Boundary: A downside to disk parametrization is that because the boundary has a fixed mapping to the circle, the resulting parameterization has a lot of distortion near the perimeter. There are a variety of free-boundary parameterization methods, but we focused on the virtual boundary method. By adding one or more layers of faces to the existing boundary, we create a new, virtual boundary that is mapped to the circle while the real boundary is allowed a less constrained shape. We can then ignore the added faces and use the resulting parameterization. The more layers in the virtual boundary, the more "free" the resulting boundary. For example, the mountain range mesh below has a boundary that is more rectangular in shape than circular. As we add more layers to the virtual boundary parameterization, we can see the outline become more square as well. 1 Layer 5 Layers 10 Layers However, it is not always necessary to add multiple layers, but instead simply increase the size of the virtual boundary, this is due to how the virtual boundary is created following the method used in "Parametrization of Triangular Meshes with Virtual Boundaries" by Yunjin Lee, Hyoung Seok Kim, and Seungyong Lee. We create the virtual boundary by listing the vertices in the existing boundary as $v_1,v_2,\dots,v_n$ and label the vertices in the virtual boundary $v'_1,v'_2,\dots,v'_{2n}$ where $v_i$ corresponds to $v'_{2i}$, we then make sure the distance between $v_i , v_{i+1} $ proportional to the distance between $v_{2i} , v_{2i+2}$ and assigning $v_{2i+1}$ to the midpoint between the two, to do this we simply say that $v_{2i}=av_i$. Thus we can increase the size of the virtual boundary simply by increasing the a value, this allows a quicker method of getting a virtual boundary that gives the same results as running multiple layers of virtual boundaries. a = 1.05 a = 1.5 Many meshes that we want to parameterize in 2D do not have boundaries. In order to do this, we make a cut along existing edges in the mesh to create our own boundary. A cut can be visualized as exactly how it sounds—taking a pair of scissors and slicing through the mesh. It is created from a path of edges by duplicating the non-endpoint vertices in the path, and connecting them to create a loop of boundary edges. This connected loop will naturally form by reassigning the faces on one side of the path to contain the new vertices rather than the old duplicated vertices. The trouble comes with determining which side of the cut each face is on. We created our own cutting algorithm, and solved this problem by using the orientation of the faces with edges on the path. Given a directed path, the faces with edges on one side of the path will be oriented in the direction of the path, while the faces on the other side will be oriented in the opposite direction. Thus we can easily separate the faces sharing edges along the cut. A more difficult task is separating the faces that only share one vertex with the cut path. To solve this quickly, we store the directions of edges coming from each point on the path that are confirmed to be on one side of the path. We then can determine which side of the path a face is based on how close its edges' direction is to these edges. There are conceivable shares which will cause this to fail, but in practice this can quickly implement the vast majority of cuts. We also tried a different algorithm to determine which side of the cut each face is on, making use of geometric properties of the surface, in specific the normal of the surface. We compared the normal of the face to the normal of the edge on the cut. We took the plane defined by the normal on the cut, and compared if the normal of the face projected onto the plane was on the left or right of the plane, when considering the y-axis of the plane to be the edge. This result showed initial success, and continuous to work along small cuts, or cuts along areas of low curvature, however when dealing with areas of large curvature the cut has a large issue, that is when the surface has high curvature it is possible for the edge to be behind the surface with respect to where the edge is, which gives the normal on the other side than we desire, and thus gives it to be on the right when it is on the left, and other such issues, for this reason we have discarded this approach, however we are open to the possibility that using some operation on the normals it might be possible to use this approach. Euclidean Max Cut: Now let us discuss our main method of how to choose a cut to make, we considered existing methods, namely the method of using a minimal spanning tree as used by Alla Sheffer in "Spanning Tree Seams for Reducing Parametrization Distortion of Triangulated Surfaces," seemed to have the opposite desire than we want, that is, to minimize the size of the cut. It is clear simply from the boundaries we were provided as well as intuition that the larger the cut the smaller the distortion would be, as if we had a large enough cut there would be no distortion, however we also know that artifacts exist along the cuts, thus we created a new method. This method was a way of maximizing the distance of the cut while minimizing the artifacts. This was maximizing the Euclidean distance between the end points of the path of the cut while making the cut that takes the minimum geodesic distance. The surface, and the cut we are making upon it. A color function we add to the surface in order to see where different areas are mapped onto a 2D Parameterization. The 2D Parameterization with Virtual Boundary. As we can see however this creates unintuitive cuts, and cuts that do not have our desired purpose, for this we considered a new method of cuts. Medial Mesh Max Cut: To deal with this issue, we implemented a new method using the medial mesh, that is we consider a simplified mesh of the object making use of the midpoints of the faces within the interior of the surface. We did not do implementation of finding this medial mesh ourselves, rather using the code created in "Q-MAT: Computing Medial Axis Transform by Quadratic Error Minimization" by Pan Li, Bin Wang, Feng Sun, Xiaohu Guo, Caiming Zhang and Wenping Wang. We then after obtaining the Medial Mesh, create the minimal spheres on each vertex, that is the smallest sphere with the center at a vertex in the medial mesh, such that it intersects with a vertex on the surface. We then created a graph with vertices representing the spheres, and creating an edge if the spheres overlap, we then calculated the maximum path existing on the graph, with the edge lengths equal to the distance between the center of the spheres. We then identified the vertices on the surface such that they were the closest to the points on each sphere such that the Euclidean distance between the two end point spheres is maximized, we use these as the two points to create a cut on and we create a cut with the minimum geodesic distance between the two points. This was hopefully to avoid areas of high curvature, and thus create a better cut still of a reasonably large size. Of note is that areas that are not on the cut area directly are "tucked in" so to into the areas that we can barely see. The one below is in the left area with a high density of vertices. The rest follow from left to right the areas of high vertices. We again get areas of high curvature being sort of tucked into the surface. The bottom left area that shows the face. The right middle area showing the back legs and tail. How Multiple Cuts work: Thus far we have only discussed and shown single cuts along simple paths. It is also possible, and often preferred, to make multiple cuts on intersecting paths. Making these cuts sequentially, each new cut is equivalent to making a cut on a shape that already has a boundary. If the cut path has one endpoint on the boundary and the rest of the points on the interior, then the resulting boundary is still a simple closed path, as is necessary for our 2D parameterization. The method is the same as before, except the endpoint on the existing boundary is duplicated in addition to the other points. Using multiple cuts increases the freedom of our cut paths and allows us to cut along branching pieces of a mesh. We can extend our previous Euclidean maximum cut method to multiple cuts in an iterative process. After making our first cut as usual, we repeat the same method but restricting one of the endpoints to be on the previous cuts. Our second cut selects the point that is farthest from our new boundary and chooses the shortest path between them. This is repeated until there are the desired number of cuts. Below are some examples of multiple cuts using this method. One possible automated method of stopping would be to calculate the standard deviation of the distance from the boundary and continue to create cuts until it is below some threshold. The surface, and the cuts we are making upon it. The 2D Parameterization and a color function that we take from 2D parameterization to the surface. The surface, and the cuts we are making upon it. The 2D Parameterization a color function that we take from 2D parameterization to the surface. [example of cutting through diagonals and where we don't want, downside of this method] We can also use the sphere approximation to make multiple cuts. The spheres' centers indicate important spots in the mesh, like joints and protrusions. We decided to use a minimum spanning tree to determine a sequence of cuts that connects each of these areas. First we project each sphere center to a vertex on the mesh. Then we make a weighted graph out of these points where an edge between two points is weighted as their geodesic distance. This graph gives a minimum spanning tree that can be broken into a sequence of cut paths that connects each point from our sphere approximation. The paths sometimes require simple adjustments to remove overlap, and then they are ready for our cutting algorithm. The surface from the front, and the cut we are making upon it. The surface from the back, and the cuts we are making upon it. The 2D Parameterization a color function that we take from 2D parameterization to the surface. This research was very interesting, but we did not get the sort of success that we would like: While we did get surfaces with minimal distortion, specifically those with multiple cuts, we did not get the desired usage that we would like. Thus it is necessary to state what future direction should be taken if this research would be continued. There are three main areas for future work. The first is making cuts along lines that preserve symmetry, that is if the surface has symmetry along some axis, or some feature, the 2D parameterization also has these symmetries. This would be useful both for minimizing the distortion and would provide intuitive results for texture mapping. These lines of symmetry, however, would be incredibly difficult to calculate, so we would like for them to be usually user specified, which brings us to the second area: how the inclusion of landmark areas would be useful, that is specifying areas where the cut cannot travel through. This allows us to intuitively avoid areas of high curvature that are important to the surface and how it could be applied to different surfaces as useful for an artistic application. We could also try to do this in an automated way by using a skeletal approximation of the surface, using either a simplified medial mesh or a method presented in the SGP 2021 Graduate School Course Shape Approximations & Applications: using the sphere mesh proxy simplification of a surface. This method considers the edges between the spheres as the "bones" and makes a cut on each bone section such that it goes over the entirety of the bone, but avoids areas of high curvature by making minimal geodesic cuts along each bone; the method connects the bones by paths the minimize the total geodesic distance. A secondary way to use this is to make cuts along the "bones" but additionally taking a cut perpendicular to the bone at each point where the sphere begins. To think of this on a 3D model of a cat, for example, we would take a cut along the spine, and then take a perpendicular cut along the cat. To see if either of these methods work and if so which is better, further research would be needed. Tags Cut, Parametrization, Skin, Texture Mapping Understanding Vector Fields More Than In My Calculus 3 Class Post author By Bryce No Comments on Understanding Vector Fields More Than In My Calculus 3 Class AKA: Takeaways from Weeks 1-2 of High-order Directional Field Design Research Author: Bryce Van Ross It's incredible how much one can learn in a month, and I'm looking forward to learning more (especially theoretically). In that same spirit, I highlight some takeaways from research made in my first SGI project. This project was guided by research mentor Dr. Amir Vaxman and TA Klara Mundilova, where I worked with fellow SGI student Jonathan Mousley. Question(s): Usually we think of vectors and computations like divergence and curl as interrelated, and they are. But can we determine something more nuanced about these properties with respect to some vector field if we encounter a complex (i.e. having multiple vertices/edges) triangular mesh? Yes, we can. But it depends on your choice of approach of partitioning your mesh. For the sake of my research, we focus on the face-based representation. Face-based representations of vector fields can then be broken down into vertex-based and edge-based approaches, per face per triangle. This means we are working with vector fields on faces that are gradients of (piecewise linear) functions that are either defined on the vertices or on the midpoints of edges. Depending on the choice of approach, then your computations are different. But which way is better and what are the consequences? Answer(s): This is a natural question. Vertex-based (for certain reasons) seems to yield better approximations, which lead to better attempts at mimicry of continuity. In this sense, vertex-based computations are considered conforming, whereas edge-based computations are deemed nonconforming (w.r.t. continuity). Suppose we wanted to express a given vector field u in terms of familiar computations. Naturally, we would prefer to use vertex-based computations. However, we must remember that degrees of freedom (D.O.F.) must be maintained. Surprisingly, using purely vertex-based computations (or, purely edge-based computations) are in violation of D.O.F. More surprisingly, we find that our only solution is to use a mix of both the conforming and nonconforming terms. So, even though the gradients are distinct, both are equally valuable in terms of a reduction of u. So, there is a need to incorporate both $G_v$ (the vertex-based gradient) and $G_e$ (the edge-based gradient). This mixture could be complicated, but isn't…it only requires the sum of 3 terms. The first term includes $G_v$ and computes the divergence but is curl-free. For the second term, including $G_e$, it computes the curl yet is div-free. The last term, referred to as $h$, is both divergence-free and curl-free. Note: that $G_v$ and $G_e$ can be interchanged w.r.t. the first two terms if such equation is multiplied by the rotation matrix $J$. Ultimately, u (or any vector space) has non-trivial representation (a.k.a. there's more than meets the eye). There's more to it, but the above refers to the Helmholtz-Hodge Decomposition: \[u = G_v\cdot f + J\cdot G_e\cdot g + h.\] A better visualization can be found below (Source: Vector Field Processing on Triangle Meshes), where the h term is not illustrated, in the topmost picture below. In the secondary picture, all components are expressed (Source: Subdivision Directional Fields, Figure 9, Top Row). Tags Curl, Divergence, Geometry Processing, Gradient, Helmholtz-Hodge Decomposition, Vector Fields Finding the Lower Bounds of the Hausdorff Distance 1 Comment on Finding the Lower Bounds of the Hausdorff Distance Currently, we're finishing the first half of the 2-week Robust computation of the Hausdorff distance between triangle meshes project. This research is lead by mentor Dr. Leonardo Sacht, TA Erik Amézquita, and in my immediate team, I work with fellow students Deniz Ozbay and Talant Talipov. The below is a brief summary of what's happened so far: A mathematical visualization of the Hausdorff distance of two meshes Source: https://en.wikipedia.org/wiki/Hausdorff_distance Applications: Primarily computer vision, computer graphics, digital fabrication, 3D-printing, and modeling. For example, in computer vision, it is often desirable to identify a best-candidate target relative to some initial template. In reference to the set of points within the template, the Hausdorff distance can be computed for each potential target. The target with the minimum Hausdorff distance would qualify as being the best fit, ideally being a close approximation to the template object. Motivation: Objects are geometrically complex. There are different ways to compare objects to each other via a range of geometry processing techniques and geometric properties. Distance is often a common metric of comparison. But what type of distance should we use, which distances are favorable, and why? These are important questions. Pitfalls of using other types of distance for triangular meshes Source: http://cgm.cs.mcgill.ca/~godfried/teaching/cg-projects/98/normand/main.html For our research, we focus on computing the Hausdorff distance $h$. "Hausdorff" may seem familiar to you if you know topology. There, a (topological) space is considered Hausdorff if any two elements can be separated into disjoint (open) sets. The key idea here is the separation property with respect to points. In geometry processing, this idea is extended (in some sense) to the separation of triangle meshes. The Hausdorff distance $h$ is fundamentally a maximum distance among desirable distances between 2 meshes. These desirable distances are minimum distances of all possible vectors resulting from points taken from the first mesh to the second mesh. But why is $h$ significant? If $h$ converges to zero (the smallest possible distance), then this implies that our meshes, and therefore the objects themselves, are very similar. This, like most things in math, implies within some epsilon, representing marginal change such as a slight deformation, rotation, translation, compression, or stretch. If $h$ is large, then this implies that the two objects are dissimilar. Intuitively, this is due to a lack of ideal correspondence from triangle to triangle. In short, $h$ serves as a means of computing the similarity between digital objects in terms of maximally separating the meshes' points according to their minimum distances. Tasks: To compute $h$, we find the maximizer, the point (in the first mesh) corresponding to the computation of $h$. This point is found via an algorithmic process called the branch and bound technique. Sparing the details, the result of applying this technique will provide a (very small) region where the minimizer is claimed to exist, after a series of triangle subdivisions and deletions. There are different ways to implement this technique. Our collective goal this week was to achieve accurate $h$ given any two meshes. Once this is ensured, Week 2 would focus on making our code more robust (efficient/fast). This work can be simplified into three primary tasks: encoding the lower bound and upper bound of possible values of $h$, and the subdivision method. My work focused on the lower bound, for which the other two functions are dependent. Accomplishments: By Day 2, we started with writing pseudocode for the lower bound, using only the vertices of the first mesh and computing their distances with respect to vertices of the second mesh. This was computed per face, and the minimum was found. Although this method was correct, it didn't account for either the edge or interior cases of a given triangle. Looping through an edge would be immediately doable, but the interior case would be more challenging. Thankfully, Dr. Sacht had us search through gptoolbox for such functionality that would account for all three cases. Once finding this function, we were able to reduce my code to two lines! The irony is although this was valid, the referenced function was basically an empty shell. The function was actually calling a C++ function that would have to compiled and linked against other libraries and due to our time constraints we decided it was best to address this issue in a future time. Ultimately, we ended up having to write from scratch once more. In searching for similar point-triangle distance algorithms, we initially found an approach using normal vectors, which would offer projective power of the former mesh vertices relative to the latter mesh. The computations were erroneous and misrepresentative. Since then, our team has been using a combinatorial plane-vector approach to find the lower bounds per vertex. Hopefully we'll finish soon… we're excited for the next steps! Tags Distance, Geometry Processing, Hausdorff, topology, Triangle meshes Geometric Modelling, IGA and cool simulations Post author By Foqia No Comments on Geometric Modelling, IGA and cool simulations Isogeometric analysis (IGA) is an analysis technique that combines computer aided design (CAD) with traditional finite element analysis (FEA). It uses the same spline basis functions to construct the geometry and the solution space, which is beneficial as traditional FEA requires geometric approximation that can lead to inaccuracies [1]. For our week 3 project, my group and I used IGA simulation software that has been developed for modelling material transport in neurons. The software solves Navier-Stokes equations to obtain the velocity field and models the transport process by reaction-diffusion-transport equations. For our purposes, we used the solver to simulate material transport in complex neurons and heat transfer processes in various geometries including a simple block model and a rod model [2]. Our project was split into two main parts: geometric modeling and analysis. For geometric modelling, we used two open source software packages (HexGen and Hex2Spline [3]) for the construction of geometries, and we then used the aforementioned IGA software for simulation purposes. I was in charge of using the IGA software to run the simulations and visualizing the results with Paraview (a visualization application). In this post, I'll demonstrate our results using the simple model the team created and also share some of the results I got using the IGA software on some additional models. Here is the example model we used: After the geometric modeling stage, I received 3 input files that contained the control mesh, an initial velocity field and the simulation parameters (particle concentrations, diffusion coefficient, velocities of material, etc.). These files were used as input for the IGA solver [2], which consists of four stages: spline construction, mesh partition for parallel computing, solving Navier-Stokes equations, and final transport simulation. Due to the high computational needs of the last two stages, we set up the simulation environment at the Pittsburgh Supercomputer Center (PSC). We ran the simulation by connecting to the remote supercomputer via an ssh client. Finally, after getting the results, I was able to visualize them using Paraview. In the above example, the color bar shows the transport velocity magnitude. I also added streamlines that show the direction of flow. Here is an animation of our results that depicts the movement of the particles: I also ran the solver on a rod model (that Angran Li, our mentor, provided): The model shows the flow of heat through the rod as it is input from the right end of the model. Again, the colorbar represents the magnitude of the velocity of particles at that point. Finally, I also ran the IGA solver on a neuron model (found on NeuroMorpho.org and edited with the help of Angran). The geometry of the neuron (visualised using Paraview) Images show results we got by running the IGA solver on the neuron geometry The transport of material through the neuron over time From using the remote computer to editing the neurons for our analysis, I learned a ton of new techniques during the week. I enjoyed learning about the IGA solver and its applications ranging from neuroscience to engineering. I'd like to end this post by thanking Angran his support and for bearing with me through my 1000 questions—it was a pleasure learning with you! 🙂 [1] T. J. Hughes, J. A. Cottrell, Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39-41), 4135-4195, 2005. [2] A. Li, X. Chai, G. Yang, Y. J. Zhang. An Isogeometric Analysis Computational Platform for Material Transport Simulations in Complex Neurite Networks. Molecular & Cellular Biomechanics, 16(2):123-140, 2019. GitHub link: https://github.com/CMU-CBML/NeuronTransportIGA [3] Y. Yu, X. Wei, A. Li, J. G. Liu, J. He, Y. J. Zhang. HexGen and Hex2Spline: Polycube-Based Hexahedral Mesh Generation and Unstructured Spline Construction for Isogeometric Analysis Framework in LS-DYNA. Springer INdAM Serie: Proceedings of INdAM Workshop "Geometric Challenges in Isogeometric Analysis". Rome, Italy. Jan 27-31, 2020. GitHub link: https://github.com/CMU-CBML/HexGen_Hex2Spline Tags isogeometric analysis Differentiable Remeshing: Week 1.5 Post author By Tal No Comments on Differentiable Remeshing: Week 1.5 Written by Deniz Ozbay, Tal Rastopchin, and Alexander Rougellis Surface Deformation and Remeshing Representing a curved surface with a mesh of polygons can prove to be difficult because, given that meshes are discrete surfaces, there are times when small changes to the triangulation of the surface can cause big changes to the overall geometry and measured quantities (as can be seen with the example of the Schwarz Lantern). Sometimes we want to deform surfaces, and the deformation of the surface can result in a "bad" triangulation. Representing these surfaces with such deformations is done by optimizing these deformations $\delta S$ on the given surface by solving $\min_{\delta S} f(S + \delta S)$ where $S$ is that given surface. When the real valued function $f: S \rightarrow \mathbb{R}$ gives us surface area, our optimization will minimize the surface area, which is also known as mean curvature flow. Minimizing surfaces can be done with gradient descent (given a small triangle mesh) and Newton's Method (given a large mesh). After minimizing, if the surface is determined to be "bad", then remeshing is needed using remeshing operations such as edge flip, edge split, and edge collapse (to name a few basic operations that could be used). Mean Curvature Flow Given that one of the goals of this project is to optimize functions on meshes, our first task was to put together a simple implementation of mean curvature flow in MATLAB. Professor Etienne Vouga introduced mean curvature flow as a deformation of a surface that minimizes surface area. He explained that if we have a function for the surface area of a mesh, as well as the gradient, we can use an optimization method like gradient descent in order to compute the deformation induced by the mean curvature flow. After our introduction to geometry processing during the tutorial week we knew that we could use the doublarea function to write a function that returned the surface area of a mesh. However, computing the gradient of this function is tricky—what would even be the domain of the surface area function? If the doublearea function relies on computing triangle areas using the cross product, and we are summing the result over all triangles in a mesh, is the domain some sort of "collection of triangles" that we are summing over? To answer this question, Professor Etienne Vouga pointed us to the paper "Can Mean-Curvature Flow be Modified to be Non-singular?" and explained that we could express the gradient of the surface area function as the Laplacian applied to the $x$, $y$, and $z$ vertex coordinate columns. In particular, the paper explained that "informally, mean-curvature flow can be thought of as a flow that pushes a point on a surface towards the average position of its neighbors." The paper specifically explains that when $M$ is a two dimensional manifold, $\Phi_t : M \rightarrow \mathbb{R}^3$ is a smooth family of immersions of the manifold $M$, and $g_t( \cdot, \cdot)$ is a metric induced by the immersion at time $t$, we have that $\Phi_t$ is a solution to the mean-curvature flow if $\frac{\partial \Phi_t}{\partial t} = \Delta _t \Phi _t$ where $\Delta_t$ is the Laplace-Beltrami operator defined with respect to the metric $g_t$. If we interpret the Laplacian as a local averaging operator, this equation exactly captures the idea that mean-curvature flow is a deformation that pushes a point on a surface towards the average position of its neighbors. For our implementation, Professor Vouga explained that instead of worrying about the metric induced by the flow we could just compute the Laplacian matrix for the first step and use it throughout the entire simulation. One thing we learned was that sometimes for computation and derivation it can be easier to look at the same function, like surface area, from a bunch of different perspectives. For example, once we knew that the gradient of the surface area was the Laplacian of the $V$ matrix, we could compute the Hessian by reshaping the $V$ matrix into a column vector and reshaping the Laplacian matrix into a larger block matrix with 3 copies of the Laplacian matrix along the diagonal. Computing the Hessian in this way allowed us to also try to implement a Newton's method approach to minimize surface area. After a lot of discussion, programming, testing, and playing around with initial conditions, we finally got our MATLAB implementation of mean curvature flow to work. A figure displaying the open cylinder mesh used as a seed surface for the mean curvature flow. A figure displaying the catenoid mesh resulting from the mean curvature flow. The figure on the left is a cylinder that was our initial surface and the figure on the right is the resulting minimal surface produced by the mean curvature flow. It was really cool to see the end result—the resulting minimal surfaces can be very pretty. We're excited to learn more about optimization on surfaces as well as how remeshing could potentially improve this simulation process. Current and Future Work Currently, we are working on implementing Newton's method and the gradient descent method in the same program as remeshing. Our algorithm runs Newton's method as long as it decreases the surface area. If a Newton step does not decrease the surface area, the algorithm then switches to the gradient descent method. Then it remeshes the triangles using edge flipping to obtain a Delaunay triangulation. To build this code, we made use of numerous different structures, like an edge face matrix, edge vertex matrix and a vertex face matrix. Then, we were able to implement the edge flipping condition. At this step we first find the neighboring two faces of the edge we check to flip. We check if the angles at the third vertices of the corresponding two faces is bigger than $\pi$ radians, and if so, the edge is flipped to get the remeshing closer to a Delaunay triangulation. This allows us to optimize the surface and do the remeshing at the same time, which we think is very cool! While implementing our algorithm, we did lots of debugging and got some funny shapes. Of course, one of the best parts about trying to make the code work was working together as a team. We had a lot of fun trying to decipher what went wrong when we got surfaces compressing to lines then to funny looking disks. Our next step is to work on coming up with algorithms for edge split and edge collapse, so we can fully implement remeshing on the shape. We are really excited to see what the final product will look like, to try lots of cool surfaces and see what the optimized remeshing for each will be! ← Newer Posts 1 2 3 4 Older Posts → © 2023 SGI 2021	CommonCrawl
\begin{document} \renewcommand{\arabic{section}.\arabic{equation}}{\arabic{section}.\arabic{equation}} \begin{titlepage} \title{\bf Semilinear stochastic partial differential equations: central limit theorem and moderate deviations \thanks{This research is supported partially by National Natural Science Foundation of China (NSFC) (No. 11801032, 61873325, 11831010), Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences(No. 2008DP173182), China Postdoctoral Science Foundation funded project (No. 2018M641204), Southern University of Science and Technology Start up fund Y01286120.}} \author{Rangrang Zhang$^{1,}$\thanks{Corresponding author},\ \ Jie Xiong$^{2}$\\ {\small $^1$ School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China}\\ {\small $^2$ Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China .}\\ ( {\sf [email protected]}\ \ {\sf [email protected]})} \date{} \end{titlepage} \maketitle \noindent\textbf{Abstract}: In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to stochastic partial differential equations of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise. \noindent \textbf{AMS Subject Classification}: Primary 60H15; Secondary 60F05, 60F10. \noindent\textbf{Keywords}: semilinear partial differential equations; space-time white noise; central limit theorem; moderate deviation principles. \section{Introduction} In this paper, we are concerned with the following semilinear stochastic partial differential equations (SPDE): \begin{eqnarray}\label{e-1} \frac{\partial u(t,x)}{\partial t}=\frac{\partial^2 u(t,x)}{\partial x^2}+b(t,x,u(t,x))+\frac{\partial g(t,x, u(t,x))}{\partial x}+\sigma(t,x,u(t,x))\frac{\partial^2}{\partial t\partial x}W(t,x) \end{eqnarray} with Dirichlet boundary condition \begin{eqnarray} u(t,0)=u(t,1)=0, \quad t\in [0,T] \end{eqnarray} and the initial condition \begin{eqnarray} u(0,x)=f(x)\in L^2([0,1]), \end{eqnarray} where $W(t,x)$ denotes the Brownian sheet on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}, P)$ with expectation $E$. The functions $b=b(t,x,r)$, $g=g(t,x,r)$, $\sigma=\sigma(t,x,r)$ are Borel functions of $(t,x,r)\in \mathbb{R}^+\times [0,1]\times \mathbb{R}$. Linear growth on $b$ and quadratic growth on $g$ are assumed in Section \ref{sec-1}. Hence, the semilinear SPDE (\ref{e-1}) contains both the stochastic Burgers equation and the stochastic reaction-diffusion equations as special cases. As a result, it attracts substantial research interests. There is an extensive literature about the semilinear SPDE (\ref{e-1}). For example, the existence and uniqueness of solutions to (\ref{e-1}) in the space $C([0,T]; L^2([0,1]))$ was studied by Gy\"{o}ngy in \cite{G98}. Foondun and Setayeshgar \cite{FS17} proved the large deviation principles (LDP) of the strong solution to (\ref{e-1}) holds uniformly on compact subsets of $C([0,T]; L^2([0,1]))$. Moreover, the ergodic theory of (\ref{e-1}) was studied by Dong and Zhang in \cite{DZ18}, where they show the existence and uniqueness of invariant measures of (\ref{e-1}). If the condition on $g$ is strengthen to be Lipschitz, Zhang \cite{Z18} proved Harnack inequalities for (\ref{e-1}) by using coupling method. The purpose of this paper is to investigate deviations of the strong solution $u^\varepsilon$ of the semilinear SPDE (see (\ref{eqq-9})) from the solution $u^0$ of the deterministic equation (see (\ref{eqq-10})), as $\varepsilon$ decreases to 0. That is, we seek the asymptotic behavior of the trajectory, \[ X^\varepsilon(t)=\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}(u^\varepsilon-u^0)(t), \quad t\in[0,T], \] where $\lambda(\varepsilon)$ is some deviation scale which strongly influences the asymptotic behavior of $X^\varepsilon$. Concretely, three cases are involved: \begin{description} \item[(1)] The case $\lambda(\varepsilon)=\frac{1}{\sqrt{\varepsilon}}$ provides LDP, which has been proved by \cite{FS17}. \item[(2)] The case $\lambda(\varepsilon)=1$ provides the central limit theorem (CLT). We will show that $X^\varepsilon$ converges to a solution of a stochastic equation, as $\varepsilon$ decrease to 0 in Section \ref{sec-2}. \item[(3)] To fill in the gap between the CLT scale ($\lambda(\varepsilon)=1$) and the large deviations scale ($\lambda(\varepsilon)=\frac{1}{\sqrt{\varepsilon}}$), we will study the so-called moderate deviation principle (MDP) in Section \ref{sec-3}. Here, the deviations scale satisfies \begin{eqnarray}\label{e-43} \lambda(\varepsilon)\rightarrow +\infty,\ \sqrt{\varepsilon}\lambda(\varepsilon)\rightarrow 0\quad {\rm{as}} \ \varepsilon\rightarrow 0. \end{eqnarray} \end{description} MDP arises in the theory of statistical inference naturally providing us with the rate of convergence and a useful method for constructing asymptotic confidence intervals ( see, e.g. \cite{E-1,I-K,K,GXZ} and references therein). Similar to LDP, the proof of moderate deviations is mainly based on the weak convergence approach, which is introduced by Dupuis and Ellis in \cite{DE}. The key idea is to prove some variational representation formula about the Laplace transform of bounded continuous functionals, which will lead to proving an equivalence between the Laplace principle and LDP. In particular, for Brownian functionals, an elegant variational representation formula has been established by Bou\'{e}, Dupuis \cite{MP} and Budhiraja, Dupuis \cite{BD}. Up to now, there are a series of results about the central limit theorem and moderate deviations for fluid dynamics models driven by white noise in time. For example, Wang et al. \cite{WZZ} established the CLT and MDP for 2D Navier-Stokes equations driven by multiplicative Gaussian noise in the space $C([0,T];H)\cap L^2([0,T];V)$ and Zhang et al. \cite{ZZG} proved that such results hold for 2D primitive equations. Moreover, Dong et al. \cite{DXZZ} proved the MDP for 2D Navier-Stokes equations driven by multiplicative L\'{e}vy noises in $D([0,T];H)\cap L^2([0,T];V)$. However, there are few results on CLT and MDP for stochastic partial differential equations driven by space-time white noise. Recently, Belfadli et al. \cite{B-B-M} claimed moderate deviations for stochastic Burgers equation. However, we cannot adapt their method to our model since we do not see how to apply Burkholder-Davis-Gundy inequality to a stochastic integral of the form $\xi(s,x)\equiv\int^t_0\int f(s,t,x,y)W(ds dy)$ to get an estimate on the expectation of the supremum of $\xi$ in $(t,x)$ when the integrand $f$ depends on $t$ and does not have a semimartingale property with respect to parameter $x$. When this paper was written, we noticed the independent work by Hu et al. \cite{Hu} for the same model. However, how do they handle the afore mentioned difficulty is not clear to us. The purpose of this paper is two-fold. The first part is to show $X^\varepsilon$ satisfies the CLT in probability in $C([0,T]; L^2([0,1]))$. Compared with stochastic partial differential equations driven by white noise in time, there are some difficulties when dealing with such equations driven by space-time white noise. Most notably, as we already mentioned in the last paragraph, it is not trivial to obtain estimate of the expectation of the supremum of the stochastic integral when the integrand also depends on the time parameter. More precisely, let $Z^{\varepsilon}(t,x)=\frac{u^{\varepsilon}(t,x)-u^{0}(t,x)}{\sqrt{\varepsilon}}-Y$, referring to (\ref{eee-21}), it satisfies \begin{eqnarray}\notag Z^{\varepsilon}(t,x)&=&\int^t_0\int^1_0G_{t-s}(x,y)\Big(\frac{b(u^{\varepsilon})-b(u^0)}{\sqrt{\varepsilon}}-\partial_rb(u^0)Y\Big)dsdy\\ \notag &&\ -\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\Big(\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(u^0)Y\Big)dsdy\\ \notag &&\ +\int^t_0\int^1_0G_{t-s}(x,y)\Big(\sigma(s,y,u^{\varepsilon}(s,y))-\sigma(s,y,u^0(s,y))\Big)W(dyds)\\ &:=& I^{\varepsilon}_1(t,x)+I^{\varepsilon}_2(t,x)+I^{\varepsilon}_3(t,x). \end{eqnarray} Our aim is to prove $\sup_{t\in [0,T]}\\|Z^{\varepsilon}(t)\\|^2_{L^2([0,1])}$ converges to $0$ in probability, so we need to show $\sup_{t\in [0,T]}\\|I^{\varepsilon}_i(t)\\|^2_{L^2([0,1])}$ converges to $0$ in probability for $i=1,2,3$. Since either $G_{t-s}(x,y)$ or $\partial_yG_{t-s}(x,y)$ is contained in $I^{\varepsilon}_i(t,x)$ and they are both not increasing with respect to $t$, we can not take supremum of $t\in [0,T]$ directly in front of $\\|I^{\varepsilon}_i(t,x)\\|^2_{L^2([0,1])}$, $i=1,2$. In particular, to estimate $E\sup_{t\in [0,T]}\\|I^{\varepsilon}_3(t)\\|_{L^2([0,1])}$, the Burkholder-Davis-Gundy inequality is not applicable because the dependence of the integrand on $t$. To overcome this difficulty, we employ the Garsia lemma from \cite{Xiong}, which gives a way to make estimates of $\sup_{t\in [0,T]}\\|I^{\varepsilon}_i(t)\\|^2_{L^2([0,1])}$. However, it requires an appropriate continuity property of $\\|I^{\varepsilon}_i(t)\\|^2_{L^2([0,1])}$ with respect to $t$. In order to achieve this condition, some delicate a priori estimates are necessary (see Section \ref{sec-4}). The second part is to prove MDP for $X^\varepsilon$ in the space $C([0,T]; L^2([0,1]))$, which is equivalent to proving that $X^\varepsilon$ satisfies a large deviation principle in $C([0,T]; L^2([0,1]))$ with $\lambda(\varepsilon)$ satisfying (\ref{e-43}). The proof will be based on the weak convergence approach introduced by Bou\'{e} and Dupuis \cite{MP}, Budhiraja and Dupuis \cite{BD}. Except difficulties mentioned above, the proof of some tightness results in $C([0,T]; L^2([0,1]))$ are also nontrivial. \ This paper is organized as follows. The mathematical formulation of the semilinear stochastic partial differential equations is presented in Sect. 2. Some delicate a priori estimates are given in Sect. 3. In Sect. 4, the central limit theorem is established. Finally, the moderate deviation principles is proved in Sect. 5. Throughout the whole paper, the constant $C$ is different from line to line. \section{Framework} Let $L^p([0,1]), p\in(0,\infty]$ be the Lebesgue space, whose norm is denoted by $\\|\cdot\\|_{L^p}$. In particular, denote that $H=L^2([0,1])$ with the corresponding norm $\\|\cdot\\|_H$ and inner product $(\cdot,\cdot)$. Define an operator $A:= \frac{\partial^2}{\partial x^2}$. Let $G_{t}(x,y)=G(t,x,y), t\geq0, x,y\in [0,1]$ be the Green function for the operator $\partial_t-A$ with Dirichlet boundary condition. Then, it satisfies that \begin{eqnarray}\label{e-29} \partial_t G_{t}(x,y)=AG_{t}(x,y). \end{eqnarray} \subsection{Assumptions}\label{sec-1} We adopt assumptions from \cite{FS17} or \cite{G98}. The functions $b=b(t,x,r)$, $g=g(t,x,r)$, $\sigma=\sigma(t,x,r)$ are Borel measurable on $(t,x,r)\in \mathbb{R}^+\times [0,1]\times \mathbb{R}$ and satisfy the following conditions: \begin{description} \item[(H1)] $b$ is of linear growth, $g$ is of quadratic growth and $\sigma$ is bounded. That is, there exists a constant $K>0$ such that for all $(t,x,r)\in[0,T]\times[0,1]\times \mathbb{R}$, we have \begin{equation} \|b(t,x,r)\|\leq K(1+\|r\|),\quad \|\sigma(t,x,r)\|\leq K, \end{equation} and \begin{equation} \|g(t,x,r)\|\leq K(1+\|r\|^2). \end{equation} \item[(H2)] $\sigma$ is Lipschitz, $b$ and $g$ are locally Lipschitz with linearly growing Lipschitz constant. That is, there exists a constant $L$ such that for all $(t,x,r_1,r_2)\in[0,T]\times[0,1]\times \mathbb{R}^2$, we have \begin{eqnarray} \|b(t,x,r_1)-b(t,x,r_2)\|&\leq& L(1+\|r_1\|+\|r_2\|)\|r_1-r_2\|,\\ \|g(t,x,r_1)-g(t,x,r_2)\|&\leq& L(1+\|r_1\|+\|r_2\|)\|r_1-r_2\|,\\ \|\sigma(t,x,r_1)-\sigma(t,x,r_2)\|&\leq& L\|r_1-r_2\|. \end{eqnarray} \end{description} \begin{dfn}\label{dfn-1} A random field $u$ is a solution to (\ref{e-1}) if $u=\{u(t,\cdot), t\in \mathbb{R}^+\}$ is an $H-$valued continuous $\mathcal{F}_t-$adapted random field with initial value $f\in H$ and satisfying for all $t\geq 0$, $\phi\in C^{2}([0,1])$ with $\phi(0)=0$, $\phi(1)=0$, \begin{eqnarray}\notag &&\int^1_0u(t,x)\phi(x)dx=\int^1_0f(x)\phi(x)dx+\int^t_0\int^1_0 u(s,x)\frac{\partial^2 \phi(x)}{\partial x^2}dxds +\int^t_0\int^1_0b(s,x, u(s,x))\phi(x)dxds\\ \label{e-2} && \quad -\int^t_0\int^1_0g(s,x,u(s,x))\frac{\partial \phi(x)}{\partial x}dxds+\int^t_0\int^1_0\sigma(s,x,u(s,x))\phi(x)W(dxds), \ P-a.s. \end{eqnarray} \end{dfn} The existence and uniqueness of the solution to (\ref{e-1}) is established in \cite{G98}. \begin{thm}\label{thm-1} Under assumptions (H1)-(H2), there exists a unique solution $u$ to SPDE (\ref{e-1}). \end{thm} \begin{remark} Referring to Proposition 3.5 in \cite{G98}, under conditions in Theorem \ref{thm-1}, (\ref{e-2}) is equivalent to the following form: for all $t\geq 0$ and almost surely $\omega\in \Omega$, \begin{eqnarray} &&u(t,x)=\int^1_0 G_t(x,y)f(y)dy +\int^t_0\int^1_0 G_{t-s}(x,y)b(s,y,u(s,y))dyds\\ \label{e-3} &&\quad \quad -\int^t_0\int^1_0 \partial_y G_{t-s}(x,y)g(s,y,u(s,y))dyds+\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u(s,y))W(dyds)\notag \end{eqnarray} for almost every $x\in [0,1]$. \end{remark} In order to establish the CLT and MDP for (\ref{e-1}), we need some additional conditions on $b$ and $g$. \begin{description} \item[(H3)] The partial derivatives of $b$ and $g$ in $r$ are both of linear growth and Lipschitz. There exists a constant $K$ such that for any $(t,x,r)\in[0,T]\times[0,1]\times \mathbb{R}$ , \begin{eqnarray}\label{eqqq-2} \|\partial_rb(t,x,r)\|\leq K(1+\|r\|),\quad \|\partial_rg(t,x,r)\|\leq K(1+\|r\|),\quad \|\partial^2_rg(t,x,r)\|\leq K, \end{eqnarray} and there exists a constant $L>0$ such that for all $(t,x,r_1,r_2)\in[0,T]\times[0,1]\times \mathbb{R}^2$, we have \begin{eqnarray}\label{eqqq-1} \|\partial_rb(t,x,r_1)-\partial_rb(t,x,r_2)\|\leq L\|r_1-r_2\|,\quad \|\partial_rf(t,x,r_1)-\partial_rf(t,x,r_2)\|\leq L\|r_1-r_2\|. \end{eqnarray} For simplicity, we assume constants $K, L$ in (H3) are the same with those in (H1)-(H2). \end{description} \subsection{Properties of Green functions} Referring to \cite{Wu}, the following facts will be used throughout this article: \begin{description} \item[(1)] $\int_{\mathbb{R}}G_t(x,y)dy=1, \quad \int_{\mathbb{R}}G^2_t(x,y)dy=(2\pi t)^{-\frac{1}{2}}, \quad \forall t\in [0, \infty), \forall x\in \mathbb{R}.$ \item[(2)] $G_t(x,y)=G_t(y,x)$, \quad $t\in [0, \infty),\ x, y\in \mathbb{R}.$ \item[(3)] $\int^1_0G_{t}(x,y)G_{s}(y,z)dy=G_{t+s}(x,z),\quad {\rm{for}}\ t,s\geq0,\ x,y, z\in [0,1].$ \item[(4)] For any $m\leq 1, n\leq 2$, there exist $C,\tilde{C}>0$ such that \begin{eqnarray}\label{eq-21} \Big\|\frac{\partial^m}{\partial t^m}\frac{\partial^n}{\partial y^n}G_t(x,y)\Big\|\leq Ct^{-\frac{1+2m+n}{2}}e^{-\tilde{C}\frac{(x-y)^2}{t}}, \quad \forall t\in (0, \infty),\ \forall x,y\in \mathbb{R}. \end{eqnarray} \end{description} A particular case for \textbf{(4)} is $m=0, n=1$, in this case, we get \begin{eqnarray}\label{eqq-4} \partial_y G_t(x,y)\leq Ct^{-1}. \end{eqnarray} Referring to (3.13) in \cite{Walsh}, for $0<r<3$, it holds that \begin{eqnarray}\label{eqq-5} \int^1_0G^r_{t}(x,y)dy\leq Ce^{-tr}t^{\frac{1}{2}-\frac{1}{2}r}\leq Ct^{\frac{1}{2}-\frac{1}{2}r}. \end{eqnarray} Based on \textbf{(4)}, we deduce that for $0<r<\frac{3}{2}$, it holds that \begin{eqnarray}\label{eqq-6} \int^1_0\|\partial_yG_t(x,y)\|^rdy\leq Ct^{\frac{1}{2}-r}. \end{eqnarray} Moreover, \begin{eqnarray}\label{eqq-5-1} \sup_{x\in [0,1]}\int^s_0\int^1_0\|G_{t-u}(x,y)-G_{s-u}(x,y))\|^rdydu\leq C\|t-s\|^{\frac{3-r}{2}}, \ 1<r<3. \end{eqnarray} and \begin{eqnarray}\label{eq-29} \sup_{x\in [0,1]}\int^s_0\int^1_0\|\partial_yG_{t-u}(x,y)-\partial_yG_{s-u}(x,y))\|^rdydu\leq C\|t-s\|^{\frac{3}{2}-r}, \ 1<r<\frac{3}{2}. \end{eqnarray} For a transition kernel $H(r,t;x,y)$, we define the linear operator $J$ by \begin{eqnarray}\label{e-15} J(v)(t,x)=\int^t_0\int^1_0H(r,t;x,y)v(r,y)dydr, \ t\in [0,T],\ x\in[0,1] \end{eqnarray} for every $v\in L^{\infty}([0,T];L^1([0,1]))$. Referring to \cite{G98}, we have the following heat kernel estimate, which is very crucial to our proof. \begin{lemma}\label{lem-1} Let $J$ is defined by $H(s,t;x,y)=G_{t-s}(x,y)$ or by $H(s,t;x,y)=\partial_y G_{t-s}(x,y)$ in (\ref{e-15}). Let $\rho\in[1,\infty]$, $q\in[1,\rho)$ and set $\kappa=1+\frac{1}{\rho}-\frac{1}{q}$. Then $J$ is a bounded linear operator from $L^{\gamma}([0,T];L^q([0,1]))$ into $C([0,T];L^{\rho}([0,1]))$ for $\gamma>2\kappa^{-1}$. Moreover, for any $T\geq 0$, there is $C>0$ such that \begin{eqnarray}\label{e-16} \\|J(v)(t,\cdot)\\|_{L^{\rho}}\leq C\int^t_0(t-s)^{\frac{\kappa}{2}-1}\\|v(s,\cdot)\\|_{L^q}ds. \end{eqnarray} \end{lemma} In particular, taking $\rho=2, \kappa=\frac{1}{2}, q=1$, we deduce that \begin{eqnarray} \\|J(v)(t,\cdot)\\|_{L^{2}}\leq C\int^t_0(t-s)^{-\frac{3}{4}}\\|v(s,\cdot)\\|_{L^1}ds. \end{eqnarray} At last, we recall the following Garsia lemma from Lemma 10.2.1 in \cite{Xiong}, which plays a key role in this article. \begin{lemma}\label{lem-3} Let $(Z,d)$ be a metric space and let $\psi$ be a continuous map from $[0,T]$ to $Z$. Suppose that $\Psi$ and $p$ are increasing functions such that $\Psi(0)=p(0)=0$ and $\Psi$ is convex. Let \[ \rho=\int^T_0\int^T_0\Psi\Big(\frac{d(\psi(t),\psi(s))}{p(\|t-s\|)}\Big)dtds \] Then, for any $t,s\in [0,T]$, we have \begin{eqnarray} d(\psi(t), \psi(s))\leq 8\int^{\|t-s\|}_0 \Psi^{-1}(\rho r^{-2})dp(r), \end{eqnarray} where $\Psi^{-1}$ denotes the inverse function of $\Psi$. \end{lemma} \section{CLT for semilinear SPDE}\label{sec-2} Let $u^{\varepsilon}(t,x)$ be the solution of the following equation \begin{eqnarray}\notag u^{\varepsilon}(t,x)&=&\int^1_0 G_t(x,y)f(y)dy +\int^t_0\int^1_0 G_{t-s}(x,y)b(s,y,u^{\varepsilon}(s,y))dyds\\ \notag &&\ -\int^t_0\int^1_0 \partial_y G_{t-s}(x,y)g(s,y,u^{\varepsilon}(s,y))dyds\\ \label{eqq-9} &&\ +\sqrt{\varepsilon}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^{\varepsilon}(s,y))W(dyds). \end{eqnarray} Using the same method as Theorem 2.1 in \cite{G98}, we know that $\sup_{t\in [0,T]}\\|u^{\varepsilon}(t)\\|^2_H$ is bounded in probability, i.e., \begin{eqnarray}\label{eqq-13} \lim_{C\rightarrow \infty}\sup_{0<\varepsilon\leq 1}P\Big(\sup_{t\in [0,T]}\\|u^{\varepsilon}(t)\\|^2_H>C\Big)=0. \end{eqnarray} Taking $\varepsilon\rightarrow 0$, it yields that \begin{eqnarray}\notag u^{0}(t,x)&=&\int^1_0 G_t(x,y)f(y)dy +\int^t_0\int^1_0 G_{t-s}(x,y)b(s,y,u^{0}(s,y))dyds\\ \label{eqq-10} &&\ -\int^t_0\int^1_0 \partial_y G_{t-s}(x,y)g(s,y,u^{0}(s,y))dyds. \end{eqnarray} Define $Y^{\varepsilon}(t,x)=\frac{u^{\varepsilon}(t,x)-u^{0}(t,x)}{\sqrt{\varepsilon}}$, then $Y^{\varepsilon}$ satisfies \begin{eqnarray}\notag Y^{\varepsilon}(t,x)&=&\frac{1}{\sqrt{\varepsilon}}\int^t_0\int^1_0 G_{t-s}(x,y)\Big(b(s,y,u^{\varepsilon}(s,y))-b(s,y,u^{0}(s,y))\Big)dyds\\ \notag &&\ -\frac{1}{\sqrt{\varepsilon}}\int^t_0\int^1_0 \partial_y G_{t-s}(x,y)\Big(g(s,y,u^{\varepsilon}(s,y))-g(s,y,u^{0}(s,y))\Big)dyds\\ \label{eqq-11} &&\ +\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^{\varepsilon}(s,y))W(dyds). \end{eqnarray} Let $Y$ is the solution of the following equation \begin{eqnarray}\notag Y(t,x)&=&\int^t_0\int^1_0G_{t-s}(x,y)\partial_rb(s, y, u^0(s,y))Y(s,y)dsdy\\ \notag &&\ -\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\partial_rg(s, y, u^0(s,y))Y(s,y)dsdy\\ \label{eqq-12} &&\ +\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y))W(dsdy). \end{eqnarray} The first result of this article reads as \begin{thm}\label{thm-2} (Central Limit Theorem) Let the initial value $f\in L^p([0,1])$ for all $p\in [2,\infty)$. Under (H1)-(H3), $Y^{\varepsilon}(t)-Y\rightarrow 0$ in probability in $C([0,T]; H)$, i.e., for any $\delta>0$, \begin{eqnarray} \lim_{\varepsilon\rightarrow 0}P\left(\sup_{t\in [0,T]}\\|Y^{\varepsilon}(t)-Y(t)\\|_H>\delta\right)=0. \end{eqnarray} \end{thm} \subsection{A priori estimates}\label{sec-4} In order to establish CLT and MDP for semilinear SPDE (\ref{e-1}), we need to make some delicate a priori estimates. Let us start with $u^0$. \begin{lemma}\label{lem-5} Under (H1), there exists $C_0:=C(K,T)(1+\\|f\\|^2_H)$ such that \begin{eqnarray} \sup_{t\in [0,T]}\\|u^0(t)\\|^2_H\leq C_0. \end{eqnarray} \end{lemma} \begin{proof} For any $t\in [0,T]$, from (\ref{eqq-10}), we get \begin{eqnarray} \frac{\partial u^0(t,x)}{\partial t}=\frac{\partial^2 u^0(t,x)}{\partial x^2}+b(t,x,u^0(t,x))+\partial_x g(t,x,u^0(t,x)). \end{eqnarray} Utilizing the chain rule, it follows that \begin{eqnarray} &&\\|u^0(t)\\|^2_H+2\int^t_0\\|\partial_xu^0(s)\\|^2_Hds\\ &=&\\|f\\|^2_H+2\int^t_0(u^0(s), b(s,u^0(s)))ds+2\int^t_0(u^0(s), \partial_x g(s,u^0(s)))ds\\ &=:& \\|f\\|^2_H+I_1(t)+I_2(t), \end{eqnarray} By (H1), we have \begin{eqnarray} I_1(t)\leq K\int^t_0\\|u^0\\|_H(1+\\|u^0\\|_H)ds\leq CKT+CK\int^t_0\\|u^0(s)\\|^2_Hds. \end{eqnarray} By integration by parts, we have \begin{eqnarray} I_2(t)=-2\int^t_0(\partial_x u^0(s), g(s,u^0(s)))ds. \end{eqnarray} Let $h(t,r)=\int^r_0 g(t,z)dz, t\in [0,T], r\in \mathbb{R}$, by the boundary conditions, it follows that \begin{eqnarray} -2\int^t_0(\partial_x u^0(s), g(s,u^0(s)))ds=-2\int^t_0\int^1_0\frac{\partial}{\partial_x}h(s,u^0(s,x))dxds=0. \end{eqnarray} Combining all the above estimates, we obtain \begin{eqnarray} \\|u^0(t)\\|^2_H+\int^t_0\\|\partial_xu^0(s)\\|^2_Hds\leq \\|f\\|^2_H+CKT+CK\int^t_0\\|u^0(s)\\|^2_Hds. \end{eqnarray} By Gronwall inequality, we obtain the desired result. \end{proof} For any $0<\varepsilon\leq 1$ and $R>0$, define a stopping time \begin{eqnarray}\label{eee-30} \tau^{\varepsilon,R}:=\inf\{t\wedge T:\\|u^{\varepsilon}(t)\\|_H>R\}. \end{eqnarray} For simplicity, in the rest part, we denote that $\tau:=\tau^{\varepsilon,R}$. Now, we make estimates of the difference between $u^{\varepsilon}$ and $u^0$, which is crucial to our proof of CLT for semilinear SPDE (\ref{e-1}). \begin{lemma}\label{lem-2} For any $R>0, p>8$, there exists $C_1=C(R,K,L,p,T, C_0)$ such that \begin{eqnarray}\label{eqq-1} \sup_{t\in [0,T]}E\int^1_0\|u^{\varepsilon}(t\wedge\tau,x)-u^0(t\wedge\tau,x)\|^pdx\leq\varepsilon^{\frac{p}{2}}C_1. \end{eqnarray} \end{lemma} \begin{proof} We deduce from (\ref{eqq-9}) and (\ref{eqq-10}) that \begin{eqnarray} u^{\varepsilon}(t\wedge\tau,x)-u^0(t\wedge\tau,x)&=&\int^{t}_0\int^1_0G_{t\wedge \tau-s}(x,y)(b(u^{\varepsilon})-b(u^0))I_{\{s\leq \tau\}}dsdy\\ &&\ -\int^t_0\int^1_0\partial_yG_{t\wedge\tau-s}(x,y)(g(u^{\varepsilon}(s))-g(u^0(s)))I_{\{s\leq \tau\}}dsdy\\ &&\ +\sqrt{\varepsilon}\int^t_0\int^1_0G_{t\wedge\tau-s}(x,y)\sigma(s,y,u^{\varepsilon}(s,y))I_{\{s\leq \tau\}}W(dyds)\\ &:=& K^{\varepsilon}_1(t,x)+K^{\varepsilon}_2(t,x)+K^{\varepsilon}_3(t,x). \end{eqnarray} By (H2) and H\"{o}lder inequality, we deduce that \begin{eqnarray} \|K^{\varepsilon}_1(t,x)\|^p &\leq & L^p\Big\|\int^{t}_0\int^1_0G_{t\wedge\tau-s}(x,y)(1+\|u^{\varepsilon}\|+\|u^0\|)\|u^{\varepsilon}-u^0\|I_{\{s\leq \tau\}}dsdy\Big\|^p\\ &\leq & L^p\Big\|\int^{t}_0\Big[\int^1_0(1+\|u^{\varepsilon}(s\wedge \tau)\|^2+\|u^0(s\wedge \tau)\|^2)I_{\{s\leq \tau\}}dy\Big]^{\frac{1}{2}}\Big[\int^1_0G^2_{t\wedge \tau-s}(x,y)\|u^{\varepsilon}-u^0\|^2I_{\{s\leq \tau\}}dy\Big]^{\frac{1}{2}}ds\Big\|^p\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}\Big\|\int^{t}_0\Big[\int^1_0G^2_{t\wedge \tau-s}(x,y)\|u^{\varepsilon}(s)-u^0(s)\|^2dy\Big]^{\frac{1}{2}}I_{\{s\leq \tau\}}ds\|^p\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}} t^{\frac{p}{2}}\Big\|\int^{t}_0\int^1_0G^2_{t\wedge \tau-s}(x,y)\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^2dyI_{\{s\leq \tau\}}ds\Big\|^{\frac{p}{2}}\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}} t^{\frac{p}{2}}\Big\|\Big(\int^{t\wedge \tau}_0\int^1_0G^{2 q}_{t\wedge \tau-s}(x,y)dyds\Big)^{\frac{p}{2q}}\\ && \times \Big(\int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^pdyds\Big)\Big\|, \end{eqnarray} where $\frac{2}{p}+\frac{1}{q}=1$. As $p>8$, we have $q=(1-2p^{-1})^{-1}<\frac{4}{3}<\frac32$, then $ 2 q<3. $ It follows from (\ref{eqq-5}) that \begin{eqnarray} \|K^{\varepsilon}_1(t,x)\|^p &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}C(p,T)\int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau,y)-u^0(s\wedge \tau,y)\|^pdyds. \end{eqnarray} By (\ref{eqq-4}) and H\"{o}lder inequality, for any $0<\delta<1$, we get \begin{eqnarray} \|K^{\varepsilon}_2(t,x)\|^p&=& \Big\|\int^t_0\int^1_0\partial_yG_{t\wedge \tau-s}(x,y)(g(u^{\varepsilon})-g(u^{0}))I_{\{s\leq \tau\}}dsdy\Big\|^p\\ &\leq& L^p\Big\|\int^{t}_0\big[\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2\delta}(1+\|u^{\varepsilon}(s\wedge \tau)\|^2+\|u^0(s\wedge \tau)\|^2)dy\big]^{\frac{1}{2}}\\ && \times \big[\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^2dy\big]^{\frac{1}{2}}I_{\{s\leq \tau\}}ds\Big\|^p\\ &\leq& L^p\Big\|\int^{t}_0(t\wedge \tau-s)^{-\delta}(1+\\|u^{\varepsilon}(s\wedge \tau)\\|^2_H+\\|u^0(s\wedge \tau)\\|^2_H)^{\frac{1}{2}}\\ && \times \big[\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^2 dy\big]^{\frac{1}{2}}I_{\{s\leq \tau\}}ds\Big\|^p\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}\Big\|\int^{t}_0(t\wedge \tau-s)^{-\delta}\Big[\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|u^{\varepsilon}-u^0\|^2dy\Big]^{\frac{1}{2}}I_{\{s\leq \tau\}}ds\Big\|^p\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-2\delta}ds\Big)^{\frac{p}{2}}\\ && \times \Big(\int^{t\wedge \tau}_0\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^2dyds\Big)^{\frac{p}{2}}\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-2\delta}ds\Big)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0\int^1_0\|\partial_yG_{t\wedge \tau-s}(x,y)\|^{2(1-\delta) q}dyds\Big)^{\frac{p}{2q}}\\ && \times \int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau)-u^0(s\wedge \tau)\|^p dyds, \end{eqnarray} where $\frac{2}{p}+\frac{1}{q}=1$. As $p>8$, we have $q=(1-2p^{-1})^{-1}<\frac{4}{3}$. Taking $\delta=\frac{15}{32}$, then \[ -2 \delta>-1, \quad 0<2(1-\delta) q<\frac{3}{2}. \] With the aid of (\ref{eqq-6}), it follows that \begin{eqnarray} &&\|K^{\varepsilon}_2(t,x)\|^p\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}C(p,T) \int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau,y)-u^0(s\wedge \tau,y)\|^p dyds. \end{eqnarray} Finally, we estimate $K^{\varepsilon}_3(t,x)$. Define \[ J(t,x)=\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^{\varepsilon}(s,y))I_{\{s\leq \tau\}}W(dyds). \] Then \begin{eqnarray}\label{eq-31} K^{\varepsilon}_3(t,x)=\sqrt{\varepsilon}J(t\wedge\tau,x). \end{eqnarray} Note that for any $0\le s<t\le T, x\in [0,1]$, by Burkholder-Davis-Gundy inequality, (H1), (\ref{eqq-5}) and (\ref{eqq-5-1}), we obtain \begin{eqnarray}\notag E\|J(t,x)-J(s,x)\|^p&\le&E\left\|\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))\sigma(r,y,u^{\varepsilon}(r,y))I_{\{r\leq \tau\}}W(dydr)\right\|^p\\ \notag &&+E\left\|\int^t_s\int^1_0G_{t-r}(x,y)\sigma(r,y,u^{\varepsilon}(r,y))I_{\{r\leq \tau\}}W(dydr)\right\|^p\\ \notag &\le&K^pE\left\|\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))^2dydr\right\|^{p/2}\\ \notag &&+K^pE\left\|\int^t_s\int^1_0G_{t-r}(x,y)^2dydr\right\|^{p/2}\\ \label{eq-17} &\le&K^p\|t-s\|^{\frac{p}{4}}. \end{eqnarray} Let \[ \Psi(r)=r^p,\quad p(r)=r^{\frac{1}{4}}, \] and \[ \rho(x)=\int^T_0\int^T_0\left\|\frac{\|J(t,x)-J(s,x)\|}{\|t-s\|^{\frac{1}{4}}}\right\|^pds dt. \] Then, by Lemma \ref{lem-3}, for any $s,t\in [0,T]$, we have for any $x\in [0,1]$, \begin{eqnarray} \|J(t,x)-J(s,x)\|&\leq& 8\int^{\|t-s\|}_0(\rho(x) r^{-2})^{\frac{1}{p}}dr^{\frac{1}{4}}\\ &=& 2\rho^{\frac{1}{p}}(x)\int^{\|t-s\|}_0 r^{-\frac{2}{p}-\frac{3}{4}}dr. \end{eqnarray} As $p>8$, we have $-\frac{2}{p}-\frac{3}{4}>-1$, which yields \begin{eqnarray}\label{eq-15} \|J(t,x)-J(s,x)\|\leq C\rho^{\frac{1}{p}}(x)\|t-s\|^{-\frac{2}{p}+\frac{1}{4}}. \end{eqnarray} Taking $s=0$ in (\ref{eq-15}), we have \begin{eqnarray}\label{eq-16} \|J(t,x)\|\leq C\rho^{\frac{1}{p}}(x)\|t\|^{-\frac{2}{p}+\frac{1}{4}}\leq C(T,p)\rho^{\frac{1}{p}}(x). \end{eqnarray} By utilizing (\ref{eq-31}), (\ref{eq-17}) and (\ref{eq-16}), we know that \[\int^1_0\|K^\varepsilon_3(t,x)\|^pdx\le\varepsilon^{p/2}C(T,p)\int^1_0\rho(x)dx,\quad \] and $E\rho(x)\leq K^pT^2$. Combining all the previous estimates, we get \begin{eqnarray} &&\int^1_0\|u^{\varepsilon}(t\wedge \tau,x)-u^0(t\wedge \tau,x)\|^pdx\\ &\leq& L^p(1+R^2+C_0)^{\frac{p}{2}}C(p,T)\int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau,y)-u^0(s\wedge \tau,y)\|^pdyds\\ && +L^p(1+R^2+C_0)^{\frac{p}{2}}C(p,T) \int^{t\wedge \tau}_0\int^1_0\|u^{\varepsilon}(s\wedge \tau,y)-u^0(s\wedge \tau,y)\|^p dyds\\ && +\varepsilon^{p/2}C(T,p)\int^1_0\rho(x)dx\\. \end{eqnarray} By using Gronwall inequality, we get \begin{eqnarray}\notag &&\int^1_0\|u^{\varepsilon}(t\wedge \tau,x)-u^0(t\wedge \tau,x)\|^pdx\\ \label{eq-33} &\leq& \Big[\varepsilon^{p/2}C(T,p)\int^1_0\rho(x)dx\Big] \exp\Big\{C(R,p,T,L,C_0)\Big\}. \end{eqnarray} Thus, \begin{eqnarray} E\int^1_0\|u^{\varepsilon}(t\wedge \tau,x)-u^0(t\wedge \tau,x)\|^pdx&\leq& \Big[\varepsilon^{\frac{p}{2}}C(T,p)\int^1_0E\rho(x)dx\Big] \exp\Big\{C(R,p,T,L,C_0)\Big\}\\ &\leq& \varepsilon^{\frac{p}{2}}C(T,p)K^p T^2 \exp\Big\{C(R,p,T,L,C_0)\Big\}, \end{eqnarray} which implies (\ref{eqq-1}). \end{proof} As a consequence, we have \begin{cor}\label{cor-1} For any $p>8$, it holds that \begin{eqnarray}\label{eq-36} \sup_{0\leq t\leq T}\sup_{0<\varepsilon\leq 1}E\int^1_0\|Y^{\varepsilon}(t\wedge \tau,x)\|^pdx \leq C_1. \end{eqnarray} \end{cor} Define $Z^{\varepsilon}=Y^{\varepsilon}-Y=\frac{u^{\varepsilon}-u^0}{\sqrt{\varepsilon}}-Y$, we claim that \begin{lemma}\label{lem-4} For any $R>0, p>14$, there exists a constant $C_2=C(K,p,T, L, C_0,C_1)$ such that \begin{eqnarray} \sup_{0\leq t\leq T}E\int^1_0\|Z^{\varepsilon}(t\wedge\tau ,x)\|^pdx \leq \varepsilon^{\frac{p}{2}} C_2. \end{eqnarray} \end{lemma} \begin{proof} From (\ref{eqq-11}) and (\ref{eqq-12}), we deduce that \begin{eqnarray}\notag &&Z^{\varepsilon}(t\wedge \tau,x)\\ \notag &=&\int^t_0\int^1_0 G_{t\wedge \tau-s}(x,y)\Big(\frac{b(s,y,u^{\varepsilon}(s,y))-b(s,y,u^{0}(s,y))}{\sqrt{\varepsilon}}-\partial_rb(s,y,u^0(s,y))Y\Big)I_{\{s\leq \tau\}}dyds\\ \notag &&\ -\int^t_0\int^1_0 \partial_y G_{t\wedge \tau-s}(x,y)\Big(\frac{g(s,y,u^{\varepsilon}(s,y))-g(s,y,u^{0}(s,y))}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big)I_{\{s\leq \tau\}}dyds\\ \notag &&\ +\int^t_0\int^1_0G_{t\wedge \tau-s}(x,y)(\sigma(s,y,u^{\varepsilon}(s,y))-\sigma(s,y,u^{0}(s,y)))I_{\{s\leq \tau\}}W(dyds)\\ \label{eqq-15} &=:& I^{\varepsilon}_1(t,x)+I^{\varepsilon}_2(t,x)+I^{\varepsilon}_3(t,x) . \end{eqnarray} With the help of (H3), for $\theta\in (u^{0}(s,y),u^{\varepsilon}(s,y))$, we get \begin{eqnarray} &&\Big\|\frac{g(s,y,u^{\varepsilon}(s,y))-g(s,y,u^{0}(s,y))}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big\|\\ &=&\|\partial_rg(s,y,u^0(s,y))Z^{\varepsilon}+\frac{1}{2}\sqrt{\varepsilon}\partial^2_rg(s,y,\theta)\|Y^{\varepsilon}\|^2\|\\ &\leq& K(1+\|u^0\|)\|Z^{\varepsilon}\|+\frac{1}{2}\sqrt{\varepsilon}K\|Y^{\varepsilon}\|^2, \end{eqnarray} then, it yields that \begin{eqnarray} \|I^{\varepsilon}_2(t,x)\|^p &\leq& C(p)K^p\Big\|\int^t_0\int^1_0 \partial_y G_{t\wedge \tau-s}(x,y)(1+\|u^0\|)\|Z^{\varepsilon}\|I_{\{s\leq \tau\}}dyds\Big\|^p\\ && +C(p)\varepsilon^{\frac{p}{2}}K^p\Big\|\int^t_0\int^1_0 \partial_y G_{t\wedge \tau-s}(x,y)\|Y^{\varepsilon}(s,y)\|^2I_{\{s\leq \tau\}}dyds\Big\|^p\\ &:=& C(p)K^p(I^{\varepsilon}_{2,1}+\varepsilon^{\frac{p}{2}}I^{\varepsilon}_{2,2}). \end{eqnarray} By H\"{o}lder inequality and (\ref{eqq-4}), for $0<\delta<1$, we get \begin{eqnarray} I^{\varepsilon}_{2,1}&\leq& \Big\|\int^{t}_0\Big(\int^1_0 \|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2\delta}(1+\|u^0\|^2)dy\Big)^{\frac{1}{2}}\Big(\int^1_0 \|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|Z^{\varepsilon}(s\wedge \tau)\|^2dy\Big)^{\frac{1}{2}}I_{\{s\leq \tau\}}ds\Big\|^p\\ &\leq&\Big\|\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-\delta}(1+\\|u^0(s)\\|_H)\Big(\int^1_0 \|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|Z^{\varepsilon}(s\wedge \tau,y)\|^2dy\Big)^{\frac{1}{2}}ds\Big\|^p\\ &\leq&(1+C_0)^{\frac{p}{2}}\Big\|\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-\delta}\Big(\int^1_0 \|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|Z^{\varepsilon}(s\wedge \tau,y)\|^2dy\Big)^{\frac{1}{2}}ds\Big\|^p\\ &\leq&(1+C_0)^{\frac{p}{2}}\big(\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-2\delta}ds\big)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0\int^1_0\|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2(1-\delta)}\|Z^{\varepsilon}(s\wedge \tau,y)\|^2dyds\Big)^{\frac{p}{2}}\\ &\leq&(1+C_0)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0(t\wedge \tau-s)^{-2\delta}ds\Big)^{\frac{p}{2}}\Big(\int^{t\wedge \tau}_0\int^1_0\|\partial_y G_{t\wedge \tau-s}(x,y)\|^{2(1-\delta) q}dyds\Big)^{\frac{p}{2q}}\\ && \times\Big(\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(s\wedge \tau,y)\|^pdyds\Big), \end{eqnarray} where $\frac{2}{p}+\frac{1}{q}=1$. As $p>8$, we have $1<q<\frac{4}{3}$, taking $\delta=\frac{15}{32}$, it yields \[ -2 \delta>-1, \quad 0<2(1-\delta)q<\frac{3}{2}. \] Then, by (\ref{eqq-6}), we get \begin{eqnarray}\notag I^{\varepsilon}_{2,1} \label{eee-9} \leq (1+C_0)^{\frac{p}{2}}C(T,p)\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(s\wedge \tau,y)\|^pdyds. \end{eqnarray} By H\"{o}lder inequality, we deduce that \begin{eqnarray} I^{\varepsilon}_{2,2} \leq \Big(\int^{t\wedge \tau}_0\int^1_0 \|\partial_y G_{t\wedge \tau-s}(x,y)\|^rdyds\Big)^{\frac{p}{r}}\Big(\int^t_0\int^1_0\|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\Big), \end{eqnarray} where $\frac{1}{r}+\frac{1}{p}=1$. As $p>8$, we have $1<r<\frac{8}{7}<\frac{3}{2}$, by (\ref{eqq-6}), we get \begin{eqnarray}\notag I^{\varepsilon}_{2,2}\label{eq-34} \leq C(T, p)\Big(\int^t_0\int^1_0\|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\Big). \end{eqnarray} Combining (\ref{eee-9}) and (\ref{eq-34}), we deduce that \begin{eqnarray} \|I^{\varepsilon}_2(t,x)\|^p&\leq & K^p(1+C_0)^{\frac{p}{2}}C(T,p)\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(s\wedge \tau,y)\|^pdyds\\ && +C(K,T,R,p) \varepsilon^{\frac{p}{2}}\left(\int^{t}_0\int^1_0 \|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\right). \end{eqnarray} Similar to the proof of $I^{\varepsilon}_2(t,x)$, we get \begin{eqnarray} \|I^{\varepsilon}_1(t,x)\|^p &\leq& K^p(1+C_0)^{\frac{p}{2}}C(T,p)\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(y,s\wedge \tau)\|^pdyds \\ && +C(K,T,R,p) \varepsilon^{\frac{p}{2}}\left(\int^{t}_0\int^1_0 \|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\right). \end{eqnarray} To estimate $I^{\varepsilon}_3$, we define \[ J(t,x)=\int^t_0\int^1_0G_{t-s}(x,y)(\sigma(s,y,u^{\varepsilon}(s,y))-\sigma(s,y,u^{0}(s,y)))I_{\{s\leq \tau\}}W(dyds). \] Then, \begin{eqnarray}\label{eq-37} I^{\varepsilon}_3(t,x)=J(t\wedge\tau,x). \end{eqnarray} Note that for any $0\le s<t\le T, x\in [0,1]$, by Burkholder-Davis-Gundy inequality, (H1) and (\ref{eqq-5-1}), for some $\kappa\geq 1$, we obtain \begin{eqnarray}\notag &&E\|J(t,x)-J(s,x)\|^p\\ \notag &\le&E\left\|\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))(\sigma(r,y,u^{\varepsilon}(r,y))-\sigma(r,y,u^{0}(r,y)))I_{\{r\leq \tau\}}W(dydr)\right\|^p\\ \notag &&+E\left\|\int^t_s\int^1_0G_{t-r}(x,y)(\sigma(r,y,u^{\varepsilon}(r,y))-\sigma(r,y,u^{0}(r,y)))I_{\{r\leq \tau\}}W(dydr)\right\|^p\\ \notag &\le&L^pE\left\|\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))^2\|u^{\varepsilon}(r,y)-u^{0}(r,y)\|^2I_{\{r\leq \tau\}}dydr\right\|^{p/2}\\ \notag &&+L^pE\left\|\int^t_s\int^1_0G_{t-r}(x,y)^2\|u^{\varepsilon}(r,y)-u^{0}(r,y)\|^2I_{\{r\leq \tau\}}dydr\right\|^{p/2}\\ \notag &\leq& L^p\Big(\int^s_0\int^1_0\|G_{t-r}(x,z)-G_{s-r}(x,y)\|^{2q'}dy dr\Big)^{\frac{p}{2q'}}\times E\Big(\int^t_0\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{2p'}dydr\Big)^{\frac{p}{2p'}}\\ \notag && +L^p\Big(\int^t_s\int^1_0G_{t-r}(x,y)^{2q'}dydr\Big)^{\frac{p}{2q'}}\times E\Big(\int^t_s\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{2p'}dydr\Big)^{\frac{p}{2p'}}\\ \notag &\leq& L^p\Big(\int^s_0\int^1_0\|G_{t-r}(x,z)-G_{s-r}(x,y)\|^{2q'}dy dr\Big)^{\frac{p}{2q'}}\times E\Big(\int^t_0\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{2p'\kappa}dydr\Big)^{\frac{p}{2p'\kappa}}\\ \label{eq-17-1} && +L^p\Big(\int^t_s\int^1_0G_{t-r}(x,y)^{2q'}dydr\Big)^{\frac{p}{2q'}}\times E\Big(\int^t_s\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{2p'\kappa}dydr\Big)^{\frac{p}{2p'\kappa}}. \end{eqnarray} where $\frac{1}{p'}+\frac{1}{q'}=1$. Taking $\kappa=\frac{p}{2p'}$. When $p>14$, we have $\frac{p}{p-2}<\frac{3p}{8+2p}<\frac{3}{2}$, choosing $\frac{p}{p-2}<q'<\frac{3p}{8+2p}$, then \[ 2q'<3, \quad \kappa>1,\quad \frac{(3-2q')p}{4q'}>2. \] As a result, by Lemma \ref{lem-2}, we deduce that \begin{eqnarray}\notag E\|J(t,x)-J(s,x)\|^p &\leq& C(T)L^p\|t-s\|^{\frac{(3-2q')p}{4q'}} \Big(\int^t_0E\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{p}dydr\Big)\\ \notag && +C(T)L^p\|t-s\|^{\frac{(3-2q')p}{4q'}} \Big(\int^t_sE\int^1_0\|u^{\varepsilon}(r\wedge \tau,y)-u^{0}(r\wedge \tau,y)\|^{p}dydr\Big)\\ \label{eq-38} &\leq& \varepsilon^{\frac{p}{2}}C(L,p,T,C_1)\|t-s\|^{\frac{(3-2q')p}{4q'}}. \end{eqnarray} Let \[ \Psi(r)=r^p,\quad p(r)=r^{\frac{(3-2q')}{4q'}}, \] and \[ \rho(x)=\int^T_0\int^T_0\Big\|\frac{\|J(t,x)-J(s,x)\|}{\|t-s\|^{\frac{(3-2q')}{4q'}}}\Big\|^pdsdt. \] Then, by Lemma \ref{lem-3}, for any $s,t\in [0,T]$, $x\in [0,1]$, we have \begin{eqnarray}\notag \|J(t,x)-J(s,x)\|&\leq& 8\int^{\|t-s\|}_0(\rho(x)r^{-2})^{\frac{1}{p}}dr^{\frac{(3-2q')}{4q'}}\\ \notag &\leq& C\rho^{\frac{1}{p}}(x)\int^{\|t-s\|}_0r^{-\frac{2}{p}+\frac{(3-2q')}{4q'}-1}dr. \end{eqnarray} As $\frac{(3-2q')p}{4q'}>2$, then $-\frac{2}{p}+\frac{(3-2q')}{4q'}>0$, we get \begin{eqnarray}\label{eq-23-1} \|J(t,x)-J(s,x)\|\leq C\rho^{\frac{1}{p}}(x)\|t-s\|^{-\frac{2}{p}+\frac{(3-2q')}{4q'}}. \end{eqnarray} Taking $s=0$ in (\ref{eq-23-1}), we obtain \begin{eqnarray}\notag \|J(t,x)\|\leq C(T)\rho^{\frac{1}{p}}(x). \end{eqnarray} Utilizing (\ref{eq-37}) and (\ref{eq-38}), we deduce that \begin{eqnarray}\label{eee-1} \int^1_0\|I^{\varepsilon}_3(t,x)\|^pdx\le C(T)\int^1_0\rho(x) dx \end{eqnarray} and \begin{equation}\label{eq-26} E\int^1_0\rho(x) dx\le \varepsilon^{\frac{p}{2}}C(L,p,T,C_1)T^2. \end{equation} Combining all the above estimates, we get \begin{eqnarray} \int^1_0\|Z^{\varepsilon}(t\wedge \tau,x)\|^pdx&\leq& K^p(1+C_0)^{\frac{p}{2}}C(T,p)\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(y,s\wedge \tau)\|^pdyds \\ && +K^p(1+C_0)^{\frac{p}{2}}C(T,p)\int^{t\wedge \tau}_0\int^1_0\|Z^{\varepsilon}(s\wedge \tau,y)\|^pdyds\\ && +C(K,T,R,p) \varepsilon^{\frac{p}{2}}\left(\int^T_0\int^1_0 \|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\right)+C(T)\int^1_0\rho(x) dx. \end{eqnarray} By Gronwall inequality, it follows that \begin{eqnarray} &&\int^1_0\|Z^{\varepsilon}(t\wedge \tau,x)\|^pdx\\ &\leq& \Big[C(K,T,R,p,C_0) \varepsilon^{\frac{p}{2}}\left(\int^{T}_0\int^1_0 \|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\right) +C(T)\int^1_0\rho(x)dx\Big]\\ && \times \exp\Big\{K^p(1+C_0)^{\frac{p}{2}}C(T,p)\Big\}. \end{eqnarray} Taking expectation, by (\ref{eq-36}) and (\ref{eq-26}), we get \begin{eqnarray} &&E\int^1_0\|Z^{\varepsilon}(t\wedge \tau,x)\|^pdx\\ &\leq& \Big[C(K,T,R,p,C_0) \varepsilon^{\frac{p}{2}}C_1T +\varepsilon^{\frac{p}{2}}C(L,p,T,C_1)\Big] \exp\{K^p(1+C_0)^{\frac{p}{2}}C(T,p)\}. \end{eqnarray} We complete the proof. \end{proof} \subsection{Proof of CLT for semilinear SPDE} \textbf{Proof of Theorem \ref{thm-2}}. \quad Recall $\tau^{\varepsilon,R}$ is defined by (\ref{eee-30}). For any $\delta>0$, it follows that \begin{eqnarray} &&P\Big(\sup_{t\in [0,T]}\\|Y^{\varepsilon}(t)-Y(t)\\|_H>\delta\Big)\\ &\leq & P\Big(\sup_{t\in [0,T]}\\|Y^{\varepsilon}(t)-Y(t)\\|_H>\delta, \tau^{\varepsilon, R}\leq T\Big)+P\Big(\sup_{t\in [0,T]}\\|Y^{\varepsilon}(t)-Y(t)\\|_H>\delta, \tau^{\varepsilon, R}> T\Big)\\ &\leq & P\Big(\tau^{\varepsilon, R}\leq T\Big)+P\Big(\sup_{t\in [0,\tau^{\varepsilon, R}]}\\|Y^{\varepsilon}(t)-Y(t)\\|_H>\delta\Big). \end{eqnarray} By (\ref{eqq-13}), we get for any $\varepsilon\in (0,1]$, \begin{eqnarray} P(\tau^{\varepsilon, R}\leq T)\rightarrow 0,\quad {\rm{as}} \ R\rightarrow \infty. \end{eqnarray} Fix some $R>0$, denote by $\tau=\tau^{\varepsilon,R}$. Recall $Z^{\varepsilon}(t\wedge \tau,x)=Y^{\varepsilon}(t\wedge \tau,x)-Y=\frac{u^{\varepsilon}(t\wedge \tau,x)-u^{0}(t\wedge \tau,x)}{\sqrt{\varepsilon}}-Y$ satisfies (\ref{eqq-15}). For the readers' convenience, we state it again as follows. \begin{eqnarray}\notag Z^{\varepsilon}(t\wedge \tau,x)&=&\int^t_0\int^1_0G_{t\wedge \tau-s}(x,y)\Big(\frac{b(u^{\varepsilon})-b(u^0)}{\sqrt{\varepsilon}}-\partial_rb(s,y,u^0(s,y))Y\Big)I_{\{s\leq \tau\}}dsdy\\ \notag &&\ -\int^t_0\int^1_0\partial_yG_{t\wedge \tau-s}(x,y)\Big(\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big)I_{\{s\leq \tau\}}dsdy\\ \notag &&\ +\int^t_0\int^1_0G_{t\wedge \tau-s}(x,y)\Big(\sigma(s,y,u^{\varepsilon}(s,y))-\sigma(s,y,u^0(s,y))\Big)I_{\{s\leq \tau\}}W(dyds)\\ \label{eee-21} &:=& I^{\varepsilon}_1(t,x)+I^{\varepsilon}_2(t,x)+I^{\varepsilon}_3(t,x). \end{eqnarray} In the rest part, we aim to prove $Z^{\varepsilon}(t\wedge \tau,x)\rightarrow 0$ in probability in $C([0, T]; H)$ as $\varepsilon\rightarrow 0$. By (\ref{eee-1}) and (\ref{eq-26}), for $p>14$, it yields \begin{eqnarray}\notag E\sup_{t\in [0,T]}\int^1_0\|I^{\varepsilon}_3(t,x)\|^pdx\leq \varepsilon^{\frac{p}{2}}C(L,p,T,C_1). \end{eqnarray} By Chebyshev inequality, we get for the above $\delta>0$, \begin{eqnarray}\notag P\Big(\sup_{t\in [0,T]}\\|I^{\varepsilon}_3(t)\\|_H>\delta\Big) &\leq& \frac{ E\sup_{t\in [0,T]}\\|I^{\varepsilon}_3(t)\\|^p_H}{\delta^p}\\ \notag &\leq&\frac{C E\sup_{t\in [0,T]}\\|I^{\varepsilon}_3(t)\\|^p_p}{\delta^p}\\ \notag &\leq& \frac{ C E\sup_{t\in [0,T]}\int^1_0\|I^{\varepsilon}_3(t,x)\|^pdx}{\delta^p}\\ \notag &\leq& \varepsilon^{\frac{p}{2}}\frac{ C(L,p,T,C_1)}{\delta^p}\\ \label{eq-27} & \rightarrow& 0,\quad {\rm{as}}\ \varepsilon\rightarrow 0, \end{eqnarray} i.e. $I^{\varepsilon}_3(t,x)\rightarrow 0$ in probability in $C([0, T]; H)$ as $\varepsilon\rightarrow 0$. Define \begin{eqnarray} \bar{I}^{\varepsilon}_2(t,x):=-\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\Big(\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big)I_{\{s\leq \tau\}}dsdy, \end{eqnarray} then \begin{eqnarray}\label{eq-30} {I}^{\varepsilon}_2(t,x)=\bar{I}^{\varepsilon}_2(t\wedge \tau,x). \end{eqnarray} Note that for $t_1,t_2\in [0,T], t_1>t_2$, we have \begin{eqnarray} &&\bar{I}^{\varepsilon}_2(t_1,x)-\bar{I}^{\varepsilon}_2(t_2,x)\\ &=&\int^{t_1}_0\int^1_0\partial_yG_{t_1-s}(x,y)\Big[\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big]I_{\{s\leq \tau\}}dsdy\\ && -\int^{t_2}_0\int^1_0\partial_yG_{t_2-s}(x,y)\Big[\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big]I_{\{s\leq \tau\}}dsdy\\ &=& \int^{t_1}_{t_2}\int^1_0\partial_yG_{t_1-s}(x,y)\Big[\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big]I_{\{s\leq \tau\}}dsdy\\ && +\int^{t_2}_0\int^1_0\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))\Big[\frac{g(u^{\varepsilon})-g(u^0)}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\Big]I_{\{s\leq \tau\}}dsdy. \end{eqnarray} By (H3), for $\theta\in (u^0(y,s), u^{\varepsilon}(y,s))$, we get \begin{eqnarray} &&\frac{g(u^{\varepsilon}(y,s))-g(u^0(y,s))}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\\ &=& \frac{\partial_rg(s,y,u^0(s,y))(u^{\varepsilon}-u^0)+\frac{1}{2}\partial^2_rg(\theta)(u^{\varepsilon}-u^0)^2}{\sqrt{\varepsilon}}-\partial_rg(s,y,u^0(s,y))Y\\ &=& \partial_rg(s,y,u^0(s,y))Z^{\varepsilon}+\frac{1}{2}\sqrt{\varepsilon}\partial^2_rg(\theta)\|Y^{\varepsilon}\|^2\\ &\leq& K(1+\|u^0\|)\|Z^{\varepsilon}\|+\frac{1}{2}K\sqrt{\varepsilon}\|Y^{\varepsilon}\|^2. \end{eqnarray} Then, it follows that \begin{eqnarray}\notag &&\|\bar{I}^{\varepsilon}_2(t_1,x)-\bar{I}^{\varepsilon}_2(t_2,x)\|\\ \notag &\leq& K\sqrt{\varepsilon}\int^{t_1}_{t_2}\int^1_0\partial_yG_{t_1-s}(x,y)(1+\|u^0\|)\Big\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\Big\|I_{\{s\leq \tau\}}dsdy\\ \notag && +\frac{1}{2}K\sqrt{\varepsilon}\int^{t_1}_{t_2}\int^1_0\partial_yG_{t_1-s}(x,y)\|Y^{\varepsilon}\|^2I_{\{s\leq \tau\}}dsdy\\ \notag && +K\sqrt{\varepsilon}\int^{t_2}_0\int^1_0\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))(1+\|u^0\|)\Big\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\Big\|I_{\{s\leq \tau\}}dsdy\\ \notag && +\frac{1}{2}K\sqrt{\varepsilon}\int^{t_2}_0\int^1_0\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))\|Y^{\varepsilon}\|^2I_{\{s\leq \tau\}}dsdy\\ \label{eq-13} &:=& \sqrt{\varepsilon}(\bar{I}^{\varepsilon}_{2,1}+\bar{I}^{\varepsilon}_{2,2}+\bar{I}^{\varepsilon}_{2,3}+\bar{I}^{\varepsilon}_{2,4}). \end{eqnarray} In the rest part, we take $p>14$. By H\"{o}lder inequality and Lemma \ref{lem-4}, for some $0<\delta_1<1$, we deduce that \begin{eqnarray} &&E\\|\bar{I}^{\varepsilon}_{2,1}\\|^p_H\\ &=& E\Big[\int^1_0\|\int^{t_1}_{t_2}\int^1_0\partial_yG_{t_1-s}(x,y)(1+\|u^0\|)\Big\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\Big\|I_{\{s\leq \tau\}}dyds\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& E\Big[\int^1_0\Big(\int^{t_1}_{t_2}\int^1_0\|\partial_yG_{t_1-s}(x,y)\|^{2\delta_1}(1+\|u^0\|)^2 dyds\Big)\Big(\int^{t_1}_{t_2}\int^1_0\|\partial_yG_{t_1-s}(x,y)\|^{2(1-\delta_1)}\Big\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\Big\|^2I_{\{s\leq \tau\}}dyds\Big)dx\Big]^{\frac{p}{2}}\\ &\leq& \Big(\int^{t_1}_{t_2}(t_1-s)^{-2\delta_1}(1+\\|u^0\\|^2_H)ds\Big)^{\frac{p}{2}}E\Big[\int^1_0\Big(\int^{t_1\wedge \tau}_{t_2}\int^1_0\|\partial_yG_{t_1-s}(x,y)\|^{2(1-\delta_1) q}dyds\Big)^{\frac{p}{2q}}\\ && \times \Big(\int^1_0\int^{t_1}_{t_2}\int^1_0\Big\|\frac{Z^{\varepsilon}(s\wedge \tau)}{\sqrt{\varepsilon}}\Big\|^pdydsdx\Big)dx\Big]^{\frac{p}{2}}\\ &\leq& K^p(1+C_0)^{\frac{p}{2}}\Big(\int^{t_1}_{t_2}(t_1-s)^{-2\delta_1}ds\Big)^{\frac{p}{2}}E\Big[\int^1_0\Big(\int^{t_1 }_{t_2}\int^1_0\|\partial_yG_{t_1-s}(x,y)\|^{2(1-\delta_1) q}dyds\Big)^{\frac{p}{2q}}\\ && \times \Big(\int^1_0\int^{t_1}_{t_2}\int^1_0\Big\|\frac{Z^{\varepsilon}(s\wedge \tau)}{\sqrt{\varepsilon}}\Big\|^pdydsdx\Big)^{\frac{2}{p}}dx\Big]^{\frac{p}{2}}. \end{eqnarray} where $\frac{2}{p}+\frac{1}{q}=1$. Taking $\delta_1\in ( \frac{5}{14}, \frac{1}{2})$, as $p>14$, we have \[ -2\delta_1>-1,\ 2(1-\delta_1) q<\frac{3}{2}. \] Set \begin{eqnarray} b_1=\frac{1}{2}-2(1-\delta_1) q, \end{eqnarray} then by Lemma \ref{lem-4}, we deduce that \begin{eqnarray} E\\|\bar{I}^{\varepsilon}_{2,1}\\|^p_H&\leq& K^p(1+C_0)^{\frac{p}{2}}\Big(\int^{t_1}_{t_2}(t_1-s)^{-2\delta_1}ds\Big)^{\frac{p}{2}}(t_1-t_2)^{(b_1+1)\frac{p}{2q}}\\ && \times \int^{t_1}_{t_2}E\int^1_0\Big\|\frac{Z^{\varepsilon}(s\wedge \tau)}{\sqrt{\varepsilon}}\Big\|^pdyds\\ &\leq&C_2K^p(1+C_0)^{\frac{p}{2}}(t_1-t_2)^{\frac{(-2\delta_1+1)p}{2}}(t_1-t_2)^{(b_1+1)\frac{p}{2q}}(t_1-t_2)\\ &\leq&C_2K^p(1+C_0)^{\frac{p}{2}}\|t_1-t_2\|^{\alpha_1}, \end{eqnarray} where \[ \alpha_1=\frac{(-2\delta_1+1)p}{2}+(b_1+1)\frac{p}{2q}+1=\frac{p-2}{4}. \] Thus, \begin{eqnarray}\label{eee-14} E\\|\bar{I}^{\varepsilon}_{2,1}\\|^p_H\leq C(K,p, C_0,C_2)\|t_1-t_2\|^{\frac{p-2}{4}}. \end{eqnarray} Utilizing H\"{o}lder inequality and Corollary \ref{cor-1}, we get \begin{eqnarray} E\\|\bar{I}^{\varepsilon}_{2,2}\\|^p_H&\leq& CK^pE\Big[\int^1_0\|\int^{t_1}_{t_2}\int^1_0\partial_yG_{t_1-s}(x,y)\|Y^{\varepsilon}(s\wedge \tau)\|^2dsdy\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& CK^p E\Big[\int^1_0\Big(\int^{t_1}_{t_2}\int^1_0\|\partial_yG_{t_1-s}(x,y)\|^{r}dyds\Big)^{\frac{2}{r}}\Big(\int^{t_1}_{t_2}\int^1_0\|Y^{\varepsilon}(s\wedge \tau)\|^pdyds\Big)^{\frac{4}{p}}dx\Big]^{\frac{p}{2}}\\ &\leq& CK^p \|t_1-t_2\|^{\frac{(3-2r)p}{2r} } E\Big[\int^1_0\Big(\int^{t_1}_{t_2}\int^1_0\|Y^{\varepsilon}(s\wedge \tau, y)\|^{2p}dyds\Big)^{\frac{2}{p}}\Big(\int^{t_1}_{t_2}\int^1_0dyds\Big)^{\frac{2}{p}}dx\Big]^{\frac{p}{2}}\\ &\leq& CK^p \|t_1-t_2\|^{\frac{(3-2r)p}{2r}+1 } E\Big(\int^{t_1}_{t_2}\int^1_0\|Y^{\varepsilon}(s\wedge \tau, y)\|^{2p}dyds\Big), \end{eqnarray} where $\frac{2}{p}+\frac{1}{r}=1$. As $p>14$, by (\ref{eq-36}), we get \begin{eqnarray}\label{eee-15} E\\|\bar{I}^{\varepsilon}_{2,2}\\|^p_H\leq CK^p\|t_1-t_2\|^{\frac{(3-2r)p}{2r}+2}C_1= C(p,K,C_1)\|t_1-t_2\|^{\frac{p-2}{2}}. \end{eqnarray} By the definition of heat kernel $G$, for $t_1>t_2>s$, we deduce that \begin{eqnarray} &&\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))\\ &=&\frac{1}{\sqrt{2\pi}}\partial_y\Big[\frac{1}{\sqrt{t_1-s}}e^{-\frac{(x-y)^2}{2(t_1-s)}}-\frac{1}{\sqrt{t_2-s}}e^{-\frac{(x-y)^2}{2(t_2-s)}}\Big]\\ &=&\frac{1}{\sqrt{2\pi}}\Big[\frac{1}{\sqrt{t_1-s}}\frac{(x-y)}{t_1-s}e^{-\frac{(x-y)^2}{2(t_1-s)}}-\frac{1}{\sqrt{t_2-s}}\frac{(x-y)}{t_2-s}e^{-\frac{(x-y)^2}{2(t_2-s)}}\Big]\\ &=& \frac{1}{\sqrt{t_1-s}}\tilde{G}_{t_1-s}(x,y)-\frac{1}{\sqrt{t_2-s}}\tilde{G}_{t_2-s}(x,y)\\ &=& \Big(\frac{1}{\sqrt{t_1-s}}-\frac{1}{\sqrt{t_2-s}}\Big)\tilde{G}_{t_2-s}(x,y)+\frac{1}{\sqrt{t_1-s}}\Big(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y)\Big)\\ &\leq& \frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}\tilde{G}_{t_2-s}(x,y)+\frac{1}{\sqrt{t_1-s}}\Big(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y)\Big). \end{eqnarray} Define \[ \tilde{G}_{t}(x,y):=\frac{1}{\sqrt{2\pi}}\frac{(x-y)}{t}e^{-\frac{(x-y)^2}{2t}}, \] with the help of properties of Gamma function, we establish that $\tilde{G}^r_{t}(x,y)$ satisfies (\ref{eqq-4})-(\ref{eq-29}). Then, it follows that \begin{eqnarray} &&E\\|\bar{I}^{\varepsilon}_{2,3}\\|^p_H\\ &\leq&E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))(1+\|u^0\|)\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\|I_{\{s\leq \tau\}}dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& C(p) E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}\tilde{G}_{t_2-s}(x,y)(1+\|u^0\|)\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\|I_{\{s\leq \tau\}}dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ && +C(p)E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\frac{1}{\sqrt{t_1-s}}(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))(1+\|u^0\|)\|\frac{Z^{\varepsilon}(s)}{\sqrt{\varepsilon}}\|I_{\{s\leq \tau\}}dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& \varepsilon^{-\frac{p}{2}} C(p) E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}\tilde{G}_{t_2-s}(x,y)(1+\|u^0\|)\|Z^{\varepsilon}(s\wedge \tau)\|dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ && +\varepsilon^{-\frac{p}{2}}C(p)E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\frac{1}{\sqrt{t_1-s}}(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))(1+\|u^0\|)\|Z^{\varepsilon}(s\wedge \tau)\|dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ &=:& \varepsilon^{-\frac{p}{2}}C(p)(K_1+K_2). \end{eqnarray} By (\ref{eqq-5}), H\"{o}lder inequality and Lemma \ref{lem-5}, for $\alpha_0>0$, we have \begin{eqnarray} K_1&\leq& K^p E\Big[\int^1_0\Big\|\int^{t_2}_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}\Big(\int^1_0\tilde{G}^{2+{\alpha_0}}_{t_2-s} (x,y)dy\Big)^{\frac{1}{2+{\alpha_0}}}(1+\\|u^0\\|_H)\Big(\int^1_0\|Z^{\varepsilon}(s\wedge \tau)\|^{p_1}dy \Big)^{\frac{1}{p_1}} ds\Big\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& K^p(1+\sqrt{C_0})^pE\Big[\int^1_0\Big\|\int^{t_2}_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}(t_2-s)^{-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}\\|Z^{\varepsilon}(s\wedge \tau)\\|_{L^{p_1}}ds\Big\|^2dx\Big]^{\frac{p}{2}}\\ &=&K^p(1+\sqrt{C_0})^pE\Big\|\int^{t_2}_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}(t_2-s)^{-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}\\|Z^{\varepsilon}(s\wedge \tau)\\|_{L^{p_1}}ds\Big\|^p\\ &=& K^p(1+\sqrt{C_0})^p\int^{t_2}_0\cdot\cdot\cdot\int^{t_2}_0\prod^p_{k=1}\frac{t_1-t_2}{(t_1-s_k)\sqrt{t_2-s_k}}(t_2-s_k)^{-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}} E\prod^p_{k=1}\\|Z^{\varepsilon}(s_k\wedge \tau)\\|_{L^{p_1}}ds_1\cdot\cdot\cdot ds_p, \end{eqnarray} where \[ \frac{1}{2+{\alpha_0}}+\frac{1}{p_1}=\frac{1}{2}. \] By Cauchy-Schwarz inequality with $r_1,\cdot\cdot\cdot, r_p$ satisfying $\sum^p_{k=1}\frac{1}{ r_k}=1$, we deduce that \begin{eqnarray} E\prod^p_{k=1}\\|Z^{\varepsilon}(s_k\wedge \tau)\\|_{L^{p_1}} \leq (E\\|Z^{\varepsilon}(s_1\wedge \tau)\\|^{r_1}_{L^{p_1}})^{\frac{1}{ r_1}}\cdot\cdot\cdot(E\\|Z^{\varepsilon}(s_p\wedge \tau)\\|^{r_p}_{L^{p_1}})^{\frac{1}{ r_p}}. \end{eqnarray} Let $\tilde{r}_k=r_k\vee p_1, k=1,\cdot\cdot\cdot,p$, by H\"{o}lder inequality, we get \[ (E\\|Z^{\varepsilon}(s_k\wedge \tau)\\|^{r_k}_{L^{p_1}})^{\frac{1}{ r_k}}\leq (E\\|Z^{\varepsilon}(s_k\wedge \tau)\\|^{r_k}_{L^{\tilde{r}_k}})^{\frac{1}{ r_k}}\leq (E\\|Z^{\varepsilon}(s_k\wedge \tau)\\|^{\tilde{r}_k}_{L^{\tilde{r}_k}})^{\frac{1}{ \tilde{r}_k}}, \] thus, \begin{eqnarray} E\prod^p_{k=1}\\|Z^{\varepsilon}(s_k\wedge \tau)\\|_{L^{p_1}} \leq (E\\|Z^{\varepsilon}(s_1\wedge \tau)\\|^{\tilde{r}_1}_{L^{\tilde{r}_1}})^{\frac{1}{ \tilde{r}_1}}\cdot\cdot\cdot(E\\|Z^{\varepsilon}(s_p\wedge \tau)\\|^{\tilde{r}_p}_{L^{\tilde{r}_p}})^{\frac{1}{ \tilde{r}_p}}. \end{eqnarray} Choosing $\alpha_0=1$, then $p_1=6$. Taking $r_k= p>14$, for $k=1,2,\cdot\cdot\cdot,p$. With the aid of Lemma \ref{lem-4} with $\tilde{r}_k= p$, it yields \begin{eqnarray} \Big(E\\|Z^{\varepsilon}(s_k\wedge \tau)\\|^{\tilde{r}_k}_{L^{\tilde{r}_k}}\Big)^{\frac{1}{\tilde{r}_k}}\leq \varepsilon^{\frac{1}{2}} C^{\frac{1}{\tilde{r}_k}}_2, \end{eqnarray} which implies \begin{eqnarray} E\prod^p_{k=1}\\|Z^{\varepsilon}(s_k\wedge \tau)\\|_{L^{p_1}} \leq \varepsilon^{\frac{p}{2}}C_2. \end{eqnarray} Hence, \begin{eqnarray} K_1&\leq& \varepsilon^{\frac{p}{2}} K^p(1+\sqrt{C_0})^pC(C_2)\Big(\int^{t_2}_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}(t_2-s)^{-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}ds \Big)^p \\ &\leq& \varepsilon^{\frac{p}{2}}C(K,p,C_0,C_2)\|t_1-t_2\|^{\frac{p}{2(2+{\alpha_0})}}\\ &\leq& \varepsilon^{\frac{p}{2}}C(K,p,C_0,C_2)\|t_1-t_2\|^{\frac{p}{6}}. \end{eqnarray} Indeed, let $u=t_1-s, v=\frac{u}{t_1-t_2}$, it follows that \begin{eqnarray}\notag &&\int^{t_2}_0\frac{t_1-t_2}{(t_1-s)\sqrt{t_2-s}}(t_2-s)^{-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}ds \\ \notag &=& \int^{t_1-t_2}_{t_1}\frac{t_1-t_2}{u}(u-t_1+t_2)^{-\frac{1}{2}-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}du\\ \notag &=& \int^{\frac{t_1}{t_1-t_2}}_{1}\frac{1}{v}(v-1)^{-\frac{1}{2}-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}(t_1-t_2)^{\frac{1}{2}-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}dv\\ \notag &\leq& \|t_1-t_2\|^{\frac{1}{2}-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}\int^{\infty}_1\frac{1}{v}(v-1)^{-\frac{1}{2}-\frac{1+{\alpha_0}}{2(2+{\alpha_0})}}dv\\ \label{eee-16} &\leq& C\|t_1-t_2\|^{\frac{1}{2(2+{\alpha_0})}}. \end{eqnarray} By H\"{o}lder inequality, (\ref{eqq-5-1}), Lemma \ref{lem-4}, for $p>14$ and for some $0<\alpha_1<1, 0<\kappa<1$, we get \begin{eqnarray} K_2&\leq&E\Big[\int^1_0\Big\|K\int^{t_2}_0\int^1_0\frac{1}{\sqrt{t_1-s}}(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))(1+\|u^0\|)\|Z^{\varepsilon}(s\wedge\tau)\|dsdy\Big\|^2dx\Big]^{\frac{p}{2}}\\ &\leq&K^p(1+\sqrt{C_0})^pE\Big[\int^1_0\Big\|\int^{t_2}_0\frac{1}{\sqrt{t_1-s}}\Big(\int^1_0(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))^{3-{\alpha_1}}dy\Big)^{\frac{1}{3-{\alpha_1}}} \\|Z^{\varepsilon}(s\wedge\tau)\\|_{L^{p}}ds\Big\|^2dx\Big]^{\frac{p}{2}}\\ &\leq&K^p(1+\sqrt{C_0})^p E\Big[\int^1_0\big(\int^{t_2}_0(t_1-s)^{-\kappa}ds\big)\Big(\int^{t_2}_0 \int^1_0(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))^{3-{\alpha_1}}dyds\Big)^{\frac{2}{3-{\alpha_1}}} \\ && \times\Big(\int^{t_2}_0(t_1-s)^{-\frac{1}{2}(1-\kappa)p}\\|Z^{\varepsilon}(s\wedge\tau)\\|^p_{L^{p}}ds\Big)^{\frac{2}{p}}dx\Big]^{\frac{p}{2}}\\ &\leq& K^p(1+\sqrt{C_0})^p \Big(\int^{t_2}_0(t_1-s)^{-\kappa}ds\Big)^{\frac{p}{2}}\sup_{x\in[0,1]}\Big(\int^{t_2}_0 \int^1_0(\tilde{G}_{t_1-s}(x,y)-\tilde{G}_{t_2-s}(x,y))^{3-{\alpha_1}}dyds\Big)^{\frac{p}{3-{\alpha_1}}} \\ && \times \Big(\int^{t_2}_0(t_1-s)^{-\frac{1}{2}(1-\kappa)p}E\\|Z^{\varepsilon}(s\wedge\tau)\\|^p_{L^{p}}ds\Big)\\ &\leq& \varepsilon^{\frac{p}{2}}K^p(1+\sqrt{C_0})^p C_2 \Big(\int^{t_2}_0(t_1-s)^{-\kappa}ds\Big)^{\frac{p}{2}}\sup_{x\in[0,1]}\Big(\int^{t_2}_0 \int^1_0(\tilde{G}_{t_1-s}(y,s)-\tilde{G}_{t_2-s}(y,s))^{3-{\alpha_1}}dyds\Big)^{\frac{p}{3-{\alpha_1}}} \\ && \times\Big(\int^{t_2}_0(t_1-s)^{-\frac{1}{2}(1-\kappa)p}ds\Big), \end{eqnarray} where $\frac{1}{3-\alpha_1}+\frac{1}{p}=\frac{1}{2}$. Define $c_0:=-\frac{1}{2}(1-\kappa)p+1$, then \begin{eqnarray} K_2 &\leq&\varepsilon^{\frac{p}{2}}K^p(1+\sqrt{C_0})^pC_2(\|t_1-t_2\|^{-\kappa+1}-t^{-\kappa+1}_1)^{\frac{p}{2}}\|t_1-t_2\|^{\frac{{\alpha_1} p}{2(3-{\alpha_1})}}[t^{c_0}_1-\|t_1-t_2\|^{c_0}]\\ &\leq& \varepsilon^{\frac{p}{2}}C(K,p,C_0, T,C_2)\|t_1-t_2\|^{\frac{{\alpha_1} p}{2(3-{\alpha_1})}}, \end{eqnarray} which implies that \begin{eqnarray}\label{eee-17} K_2\leq \varepsilon^{\frac{p}{2}}C(K,p,C_0, T,C_2)\|t_1-t_2\|^{\frac{p-6}{4}}. \end{eqnarray} Combing (\ref{eee-16}) and (\ref{eee-17}), it yields \begin{eqnarray}\label{eee-18} E\\|\bar{I}^{\varepsilon}_{2,3}\\|^p_H\leq C(K,p,C_0,C_2)\|t_1-t_2\|^{\frac{p}{6}}+C(K,p,C_0, T,C_2)\|t_1-t_2\|^{\frac{p-6}{4}}. \end{eqnarray} For any $p>14$ and $\frac{1}{r}+\frac{2}{p}=1$, we have $r\in (1,\frac{7}{6})$. Utilizing (\ref{eq-29}) and (\ref{eq-36}), we have \begin{eqnarray} E\\|\bar{I}^{\varepsilon}_{2,4}\\|^p_H&\leq&CK^pE\Big[\int^1_0\|\int^{t_2}_0\int^1_0\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))\|Y^{\varepsilon}\|^2I_{\{s\leq \tau\}}dsdy\|^2dx\Big]^{\frac{p}{2}}\\ &\leq& CK^pE\Big[\int^1_0\Big(\int^{t_2}_0\int^1_0\|\partial_y(G_{t_1-s}(x,y)-G_{t_2-s}(x,y))\|^rdyds\Big)^{\frac{2}{r}}\Big(\int^{t_2}_0\int^1_0\|Y^{\varepsilon}(s\wedge \tau)\|^{p}dsdy\Big)^{\frac{4}{p}}dx\Big]^{\frac{p}{2}}\\ &\leq& CK^p\|t_1-t_2\|^{(\frac{3}{2}-r)\frac{p}{r}}E\Big[\int^1_0\Big(\int^{t_2}_0\int^1_0\|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\Big)^{\frac{2}{p}}\Big(\int^{t_2}_0\int^1_0dyds\Big)^{\frac{2}{p}}\Big]^{\frac{p}{2}}\\ &\leq& C(T)K^p\|t_1-t_2\|^{(\frac{3}{2}-r)\frac{p}{r}}E\int^{t_2}_0\int^1_0\|Y^{\varepsilon}(s\wedge \tau,y)\|^{2p}dyds\\ &\leq& C(T,K,p,C_1)\|t_1-t_2\|^{(\frac{3}{2}-r)\frac{p}{r}}, \end{eqnarray} which implies \begin{eqnarray}\label{eee-19} E\\|\bar{I}^{\varepsilon}_{2,4}\\|^p_H\leq C(T,K,p,C_1)\|t_1-t_2\|^{\frac{p-6}{2}}. \end{eqnarray} Combing (\ref{eee-14}), (\ref{eee-15}), (\ref{eee-18}) and (\ref{eee-19}), we conclude that \begin{eqnarray} E\\|\bar{I}^{\varepsilon}_2(t_1)-\bar{I}^{\varepsilon}_2(t_2)\\|^p_H&\leq& \varepsilon^{\frac{p}{2}}C_p( E\\|\bar{I}^{\varepsilon}_{2,1}\\|^p_H+E\\|\bar{I}^{\varepsilon}_{2,2}\\|^p_H+E\\|\bar{I}^{\varepsilon}_{2,3}\\|^p_H+E\\|\bar{I}^{\varepsilon}_{2,4}\\|^p_H)\\ &\leq& \varepsilon^{\frac{p}{2}}C_p\Big[C(K,p, C_0,C_2)\|t_1-t_2\|^{\frac{p-2}{4}}+ C(p,K,C_1)\|t_1-t_2\|^{\frac{p-2}{2}}\\ && +C(K,p,C_0,T,C_2)\|t_1-t_2\|^{\frac{p}{6}}+C(K,p,C_0,T,C_2)\|t_1-t_2\|^{\frac{p-6}{4}}\\ &&+C(T,K,p,C_1)\|t_1-t_2\|^{\frac{p-6}{2}}\Big]\\ &\leq& \varepsilon^{\frac{p}{2}}C(K,p, T,C_0,C_1,C_2)(\|t_1-t_2\|^{\frac{p}{6}}+\|t_1-t_2\|^{\frac{p-6}{4}}). \end{eqnarray} As $p>14$, we get $\frac{p}{6}>\frac{p-6}{4}>2$, then \begin{eqnarray}\label{eq-39} E\\|\bar{I}^{\varepsilon}_2(t_1)-\bar{I}^{\varepsilon}_2(t_2)\\|^{p}_H\leq \varepsilon^{\frac{p}{2}}C(K,p, T,C_0,C_1,C_2)\|t_1-t_2\|^{\frac{p-6}{4}}. \end{eqnarray} Let \[ \Psi(r)=r^p,\quad p(r)=r^{\frac{p-6}{4p}}, \] and \[ \rho=\int^T_0\int^T_0\Big\|\frac{\\|\bar{I}^{\varepsilon}_2(t_1)-\bar{I}^{\varepsilon}_2(t_2)\\|_H}{\|t_1-t_2\|^{\frac{p-6}{4p}}}\Big\|^pdt_1dt_2. \] By Lemma \ref{lem-3}, for any $s,t\in [0,T]$,we have \begin{eqnarray} \\|\bar{I}^{\varepsilon}_2(t_1)-\bar{I}^{\varepsilon}_2(t_2)\\|_H&\leq& 8\int^{\|t_1-t_2\|}_0(\rho r^{-2})^{\frac{1}{p}}dr^{\frac{p-6}{4p}}\\ &\leq& C(p)\rho^{\frac{1}{p}}\int^{\|t_1-t_2\|}_0r^{-\frac{2}{p}+\frac{p-6}{4p}-1}dr. \end{eqnarray} As $\frac{p-6}{4}>2$, we have $-\frac{2}{p}+\frac{p-6}{4p}>0$, then \begin{eqnarray} \\|\bar{I}^{\varepsilon}_2(t_1)-\bar{I}^{\varepsilon}_2(t_2)\\|_H\leq C(p)\rho^{\frac{1}{p}}\|t_1-t_2\|^{-\frac{2}{p}+\frac{p-6}{4p}}. \end{eqnarray} Taking $t_2=0, 0\leq t=t_1\leq T$, we get \begin{eqnarray} \\|\bar{I}^{\varepsilon}_2(t)\\|_H\leq C(T,p)\rho^{\frac{1}{p}}. \end{eqnarray} By (\ref{eq-39}), we get \begin{eqnarray}\label{eq-14-1} E\sup_{0\leq t\leq T}\\|\bar{I}^{\varepsilon}_2(t)\\|_H\leq C(T,p)(E\rho)^{\frac{1}{p}}\leq \sqrt{\varepsilon}C(K,p, T,C_0,C_1,C_2). \end{eqnarray} Thus, we deduce from (\ref{eq-30}) that \begin{eqnarray}\notag E\sup_{0\leq t\leq T}\\|{I}^{\varepsilon}_2(t)\\|_H&=&E\sup_{0\leq t\leq T}\\|\bar{I}^{\varepsilon}_2(t\wedge \tau)\\|_H\\ \label{eq-14} &\leq & \sqrt{\varepsilon}C(K,p, T,C_0,C_1,C_2)\rightarrow 0, \ {\rm{as}}\ \varepsilon\rightarrow 0. \end{eqnarray} By the same argument as the method dealing with $I^{\varepsilon}_2(t)$, we get \begin{eqnarray}\label{eee-24} E\sup_{t\in [0,T]}\\|I^{\varepsilon}_1(t)\\|_H\rightarrow 0, \ {\rm{as}} \ \varepsilon\rightarrow 0. \end{eqnarray} Combing (\ref{eq-14}) and (\ref{eee-24}), we know that $I^{\varepsilon}_1(t)+I^{\varepsilon}_2(t)\rightarrow 0$ in probability in $C([0, T]; H)$ as $\varepsilon\rightarrow 0$. By (\ref{eq-27}), we complete the proof. \section{MDP for semilinear SPDE}\label{sec-3} In this part, we are concerned with the moderate deviation principle of the solution $u^\varepsilon$ of (\ref{eqq-9}). As stated in the introduction, we need to prove $\frac{u^\varepsilon-u^0}{\sqrt{\varepsilon}\lambda(\varepsilon)}$ satisfies a large deviation principle on $C([0,T]; H)$ with $\lambda(\varepsilon)$ satisfying (\ref{e-43}). From now on, we assume (\ref{e-43}) holds. \subsection{The weak convergence approach} Let $X^\varepsilon =\frac{u^\varepsilon-u^0}{\sqrt{\varepsilon}\lambda(\varepsilon)}$, we will use the weak convergence approach introduced by Budhiraja and Dupuis in \cite{BD} to verify $X^\varepsilon$ satisfies a large deviation principle. Firstly, we recall some standard definitions and results from the large deviation theory (see \cite{DZ}). Let $\{X^\varepsilon\}$ be a family of random variables defined on a probability space $(\Omega, \mathcal{F}, P)$ taking values in some Polish space $\mathcal{E}$. \begin{dfn} (Rate function) A function $I: \mathcal{E}\rightarrow [0,\infty]$ is called a rate function if $I$ is lower semicontinuous. A rate function $I$ is called a good rate function if the level set $\{x\in \mathcal{E}: I(x)\leq M\}$ is compact for each $M<\infty$. \end{dfn} \begin{dfn} (LDP) The sequence $\{X^{\varepsilon}\}$ is said to satisfy the large deviation principle with rate function $I$ if for each Borel subset $A$ of $\mathcal{E}$ \[ -\inf_{x\in A^o}I(x)\leq \lim \inf_{\varepsilon\rightarrow 0}\varepsilon \log P(X^{\varepsilon}\in A)\leq \lim \sup_{\varepsilon\rightarrow 0}\varepsilon \log P(X^{\varepsilon}\in A)\leq -\inf_{x\in \bar{A}}I(x). \] \end{dfn} Now we define \begin{eqnarray} \mathcal{A}&=&\Big\{\phi: \int^T_0\int^1_0 \|\phi(s,y)\|^2dyds<\infty\quad P\text{-}a.s.\Big\};\\ T_M&=&\Big\{ h\in L^2([0,T]\times [0,1]): \int^T_0\int^1_0 \|h(s,y)\|^2dyds\leq M\Big\};\\ \mathcal{A}_M&=&\Big\{\phi\in \mathcal{A}: \phi(\omega)\in T_M,\ P\text{-}a.s.\Big\}. \end{eqnarray} Here and in the sequel of this paper, we will always refer to the weak topology on the set $T_M$, in this case, $T_M$ is a compact metric space of $L^2([0,T]\times [0,1])$. Suppose $\mathcal{G}^{\varepsilon}: C([0,T]\times [0,1]; \mathbb{R})\rightarrow \mathcal{E}$ is a measurable mapping and $X^{\varepsilon}=\mathcal{G}^{\varepsilon}(W)$. Now, we list below sufficient conditions for the large deviation principle of the sequence $X^{\varepsilon}$ as $\varepsilon\rightarrow 0$. \begin{description} \item[\textbf{Hypothesis G} ] There exists a measurable map $\mathcal{G}^0: C([0,T]\times [0,1]; \mathbb{R})\rightarrow \mathcal{E}$ satisfying \end{description} \begin{description} \item[(i)] For every $M<\infty$, let $\{h^{\varepsilon}: \varepsilon>0\}$ $\subset \mathcal{A}_M$. If $h^{\varepsilon}$ converges to $h$ as $T_M-$valued random elements in distribution, then $\mathcal{G}^{\varepsilon}\Big(W(\cdot)+\lambda(\varepsilon)\int^{\cdot}_{0}\int^{\cdot}_{0}h^\varepsilon(s,y)dyds\Big)$ converges in distribution to $\mathcal{G}^0\Big(\int^{\cdot}_{0}\int^{\cdot}_{0}h(s,y)dyds\Big)$. \item[(ii)] For every $M<\infty$, $K_M=\Big\{\mathcal{G}^0\Big(\int^{\cdot}_{0}\int^{\cdot}_{0}h(s,y)dyds\Big): h\in T_M\Big\}$ is a compact subset of $\mathcal{E}$. \end{description} The following result is due to Budhiraja et al. in \cite{BD}. \begin{thm}\label{thm-3} If $\mathcal{G}^{0}$ satisfies Hypothesis G, then $X^{\varepsilon}$ satisfies a large deviation principle on $\mathcal{E}$ with the good rate function $I$ given by \begin{eqnarray}\label{eq-5} I(f)=\inf_{\Big\{h\in L^2([0,T]\times [0,1]): f= \mathcal{G}^0(\int^{\cdot}_{0}\int^{\cdot}_{0}h(s,y)dyds)\Big\}}\Big\{\frac{1}{2}\int^T_0\int^T_0\|h(s,y)\|^2dyds\Big\},\ \ \forall f\in\mathcal{E}. \end{eqnarray} By convention, $I(\emptyset)=\infty$. \end{thm} In this part, we are concerned with the following SPDE driven by small multiplicative noise \begin{eqnarray}\notag &&X^{\varepsilon}(x,t)\\ \notag &=&\frac{1}{\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)X^{\varepsilon}(s,y))W(dyds)\\ \notag && +\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)(b(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)X^{\varepsilon}(s,y))-b(s,y,u^0(s,y)))dyds\\ \label{eqq-16} && -\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0\partial_yG_{t-s}(x,y)(g(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)X^{\varepsilon}(s,y))-g(s,y,u^0(s,y)))dyds. \end{eqnarray} Under (H1) and (H2), combing Theorem \ref{thm-1} and Lemma \ref{lem-5}, there exists a unique strong solution in $X^\varepsilon\in C([0,T];H)$. Therefore, there exists a Borel-measurable function \begin{eqnarray}\label{e-46} \mathcal{G}^{\varepsilon}: C([0,T]\times [0,1]; \mathbb{R})\rightarrow C([0,T];H) \end{eqnarray} such that $X^{\varepsilon}(\cdot)=\mathcal{G}^{\varepsilon}(W(\cdot))$. Let $h\in T_M$, consider the following skeleton equation \begin{eqnarray}\notag X^h(x,t)&=&\int^t_0\int^1_0G_{t-s}(x,y)\partial_rb(s,y,u^0(s,y))X^h(s,y)dyds\\ \notag && -\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\partial_rg(s,y,u^0(s,y))X^h(s,y)dyds\\ \label{eqq-17} && +\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^0(s,y))h(s,y)dyds. \end{eqnarray} By (H3), we know that all coefficients of (\ref{eqq-17}) are Lipschitz, it admits a unique solution $X^h$ satisfying \begin{eqnarray}\label{eee-26} \sup_{t\in [0,T]}\\|X^h(t)\\|^2_H\leq C(K,T,M,C_0). \end{eqnarray} Therefore, we can define a measurable mapping $\mathcal{G}^0: C([0,T]\times [0,1]; \mathbb{R})\rightarrow C([0,T];H)$ such that $\mathcal{G}^0\Big(\int^{\cdot}_0\int^{\cdot}_0 h(s,y)dyds\Big):=X^h(\cdot)$. The main result in this part reads as \begin{thm}\label{thm-8} Let the initial value $f\in L^p([0,1])$ for all $p\in [2,\infty)$. Under (H1)-(H3), $X^{\varepsilon}$ satisfies a large deviation principle on $C([0,T];H)$ with the good rate function $I$ defined by (\ref{eq-5}). \end{thm} \subsection{Tightness of semilinear SPDE with small perturbations} For any $h^{\varepsilon}\in \mathcal{A}_M$, consider the following SPDE \begin{eqnarray}\notag &&\bar{X}^{\varepsilon,h^{\varepsilon}}(x,t)\\ \notag &=&\frac{1}{\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))W(dyds)\\ \notag &&+\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))h^{\varepsilon}(s,y)dyds\\ \notag &&+\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)(b(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))-b(s,y,u^0(s,y)))dyds\\ \label{eqq-18} &&-\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0\partial_yG_{t-s}(x,y)(g(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))-g(s,y,u^0(s,y)))dyds. \end{eqnarray} with $\bar{X}^{\varepsilon,h^{\varepsilon}}(0)=0$, then $\mathcal{G}^{\varepsilon}\Big(W(\cdot)+\lambda(\varepsilon)\int^{\cdot}_0\int^{\cdot}_0h^{\varepsilon}(s,y)dyds\Big)=\bar{X}^{\varepsilon,h^{\varepsilon}}$. Moreover, with the aid of Lemma \ref{lem-5} and by using the same method as Theorem 2.1 in \cite{G98}, it follows that \begin{lemma}\label{lem-6} For any family $\{h^\varepsilon; \varepsilon>0\}\subset \mathcal{A}_M$, it holds that \begin{eqnarray}\label{eqqq-6} \lim_{C\rightarrow \infty}\sup_{0<\varepsilon\leq 1}P\Big(\sup_{t\in[0,T]}\\|\bar{X}^{\varepsilon,h^{\varepsilon}}(t)\\|^2_H>C\Big)=0. \end{eqnarray} \end{lemma} Referring to \cite{FS17}, the following lemma gives a criterion to ensure tightness. \begin{lemma}\label{lemm-2} Let $\rho\in [1,\infty)$, and $q\in [1, \rho)$. Let $\zeta_n(t,y)$ be a sequence of random fields on $[0,T]\times [0,1]$ such that $\sup_{0\leq t\leq T}\\|\zeta_n(t,\cdot)\\|_{L^q}\leq \theta_n$, where $\theta_n$ is a finite random variable for every $n$. Assume that the sequence $\theta_n$ is bounded in probability, i.e., $\lim_{C\rightarrow \infty}\sup_nP(\theta_n\geq C)=0$. Then the sequence \[ J(\zeta_n):=\int^t_0\int^1_0R(s,t;x,y)\zeta_n(r,y)dyds, t\in[0,T], x\in [0,1], \] where $R(s,t;x,y)=\partial_yG(s,t;x,y)$ or $R(s,t;x,y)=G(s,t;x,y)$ is uniformly tight in $C([0,T];L^{\rho}([0,1]))$. \end{lemma} Let $\mathcal{D}(X)$ be the distribution of a random variable $X$. \begin{prp}\label{prp-6} For any $R>0$, $\mathcal{D}(\bar{X}^{\varepsilon,h^{\varepsilon}})_{\varepsilon\in (0,1]}$ is tight in $C([0,T];H).$ \end{prp} \begin{proof} From (\ref{eqq-18}), we have \begin{eqnarray}\notag &&\bar{X}^{\varepsilon,h^{\varepsilon}}(x,t)\\ \notag &=&\frac{1}{\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))W(dyds)\\ \notag &&+\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))h^{\varepsilon}(s,y)dyds\\ \notag &&+\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)(b(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))-b(s,y,u^0(s,y)))dyds\\ \notag &&-\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0\partial_yG_{t-s}(x,y)(g(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))-g(s,y,u^0(s,y)))dyds\\ \label{eqqq-5} &:=&J^{\varepsilon}_1+J^{\varepsilon}_2+J^{\varepsilon}_3+J^{\varepsilon}_4. \end{eqnarray} Firstly, we claim that \begin{eqnarray}\label{equat-2} \lim_{\varepsilon\rightarrow 0}E\sup_{0\leq t\leq T}\\|J^{\varepsilon}_1\\|_H= 0. \end{eqnarray} Indeed, by (H1) and using the similar method as the estimation of (\ref{eq-17}), for any $t_1, t_2\in [0,T]$ and $p>14$, we obtain \begin{eqnarray}\label{eq-28} E\\|J^{\varepsilon}_1(t_1)-J^{\varepsilon}_1(t_2)\\|^p_H \leq \frac{C(K,p)}{(\lambda(\varepsilon))^{p}}\|t_1-t_2\|^{\frac{p}{4}}. \end{eqnarray} Applying Lemma \ref{lem-3} with \[ \Psi(r)=r^{p}, \quad p(r)=r^{\frac{1}{4}}, \] and \begin{eqnarray} \varrho=\int^T_0\int^T_0\left(\frac{\\|J^{\varepsilon}_1(t_1)-J^{\varepsilon}_1(t_2)\\|_H}{\|t_1-t_2\|^{\frac{1}{4}}}\right)^{p}dt_1dt_2, \end{eqnarray} we have \begin{eqnarray} \\|J^{\varepsilon}_1(t_1)-J^{\varepsilon}_1(t_2)\\|_H&\leq&8\int^{\|t_1-t_2\|}_0(\varrho r^{-2})^{\frac{1}{p}}dr^{\frac{1}{4}}\\ &\leq&C\varrho^{\frac{1}{p}}\int^{\|t_1-t_2\|}_0r^{-\frac{3}{4}-\frac{2}{p}}dr\\ &\leq&C\varrho^{\frac{1}{p}}\|t_1-t_2\|^{\frac{p-8}{4p}}. \end{eqnarray} Let $t=t_1$ and $t_2=0$, we get \begin{eqnarray} \\|J^{\varepsilon}_1(t)\\|_H\leq C\varrho^{\frac{1}{p}}t^{\frac{p-8}{4p}}, \end{eqnarray} which implies that \begin{eqnarray} \sup_{t\in [0,T]}\\|J^{\varepsilon}_1(t)\\|_H\leq C(T)\varrho^{\frac{1}{p}}. \end{eqnarray} By (\ref{eq-28}), it gives that \[ E\varrho\leq \frac{C(L,p)}{(\lambda(\varepsilon))^{p}}\rightarrow 0, \quad \varepsilon\rightarrow 0, \] Thus, \begin{eqnarray} E\sup_{t\in [0,T]}\\|J^{\varepsilon}_1(t)\\|_H\leq C(T)E\varrho\rightarrow 0, \ {\rm{as}} \ \varepsilon\rightarrow 0. \end{eqnarray} By (\ref{equat-2}), we deduce that $J^{\varepsilon}_1$ converges in probability in $C([0,T];H)$. By (H1), we get \begin{eqnarray}\notag \sup_{0<\varepsilon\leq 1}J^{\varepsilon}_2&=&\sup_{0<\varepsilon\leq 1}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))h^{\varepsilon}(s,y)dyds\\ \notag &\leq& K\Big(\int^t_0\int^1_0G^2_{t-s}(x,y)dyds\Big)^{\frac{1}{2}}\sup_{0<\varepsilon\leq 1}\Big(\int^t_0\int^1_0\|h^{\varepsilon}(s,y)\|^2dyds\Big)^{\frac{1}{2}}\\ \label{equat-4} &\leq& KC(T)M^{\frac{1}{2}}. \end{eqnarray} Referring to (4.2) in \cite{FS17}, (\ref{equat-4}) implies the tightness of $J^{\varepsilon}_2$. For $J^{\varepsilon}_3$, applying Lemma \ref{lemm-2} with $\rho=2, q=1$, by (H2), we have \begin{eqnarray} &&\sup_{0\leq t\leq T}\\|b(s,y,u^0+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon,h^{\varepsilon}}(s,y))-b(s,y,u^0(s,y))\\|_{L^1}\\ &\leq& L\sup_{0\leq t\leq T}\sqrt{\varepsilon}\lambda(\varepsilon)\int^1_{0}(1+\|u^0\|+\sqrt{\varepsilon}\lambda(\varepsilon)\|\bar{X}^{\varepsilon,h^{\varepsilon}}\|)\|\bar{X}^{\varepsilon,h^{\varepsilon}}\|dx\\ &\leq&L \sup_{0\leq t\leq T}\sqrt{\varepsilon}\lambda(\varepsilon)[(1+\\|u^0\\|_H+\sqrt{\varepsilon}\lambda(\varepsilon)\\|\bar{X}^{\varepsilon,h^{\varepsilon}}\\|_H)\\|\bar{X}^{\varepsilon,h^{\varepsilon}}\\|_H]\\ &\leq& \sqrt{\varepsilon}\lambda(\varepsilon)L[1+C_0+(1+\sqrt{\varepsilon}\lambda(\varepsilon))\sup_{0\leq t\leq T}\\|\bar{X}^{\varepsilon,h^{\varepsilon}}\\|^2_H]. \end{eqnarray} Let \[ \theta=\sqrt{\varepsilon}\lambda(\varepsilon)L\Big(1+C_0+(1+\sqrt{\varepsilon}\lambda(\varepsilon))\sup_{0\leq t\leq T}\\|\bar{X}^{\varepsilon,h^{\varepsilon}}\\|^2_H\Big), \] we have \begin{eqnarray} &&\lim_{M\rightarrow \infty} \sup_{0<\varepsilon\leq 1} P(\theta\geq M)\\ &\leq& \lim_{M\rightarrow \infty} \sup_{0<\varepsilon\leq 1}P\Big(\sqrt{\varepsilon}\lambda(\varepsilon)L(1+C_0)\geq \frac{M}{2}\Big)\\ && + \lim_{M\rightarrow \infty} \sup_{0<\varepsilon\leq 1}P\Big(\sup_{0\leq t\leq T}\\|\bar{X}^{\varepsilon,h^{\varepsilon}}\\|^2_H\geq \frac{M}{2\sqrt{\varepsilon}\lambda(\varepsilon)L(1+\sqrt{\varepsilon}\lambda(\varepsilon))}\Big). \end{eqnarray} By Lemma \ref{lem-6}, we obtain \begin{eqnarray} \lim_{M\rightarrow \infty} \sup_{0<\varepsilon\leq 1} P(\theta\geq M)=0. \end{eqnarray} Thus, we get the tightness of $J^{\varepsilon}_3$ in $C([0,T];H)$. Employing similar method as above, we obtain the tightness of $J^{\varepsilon}_4$ in $C([0,T];H)$. We complete the proof. \end{proof} \subsection{The proof of MDP } According to Theorem \ref{thm-2}, the proof of MDP will be completed if the following Theorem \ref{thm-5} and Theorem \ref{thm-4} are established. \begin{thm}\label{thm-5} The family \[ K_M=\left\{\mathcal{G}^0(\int^{\cdot}_{0}\int^{\cdot}_{0}h(s,y)dyds): h\in T_M\right\} \] is a compact subset of $C([0,T];H)$. \end{thm} \begin{proof} Let $X^{h_n}=\{\mathcal{G}^0(\int^{\cdot}_0\int^{\cdot}_0h_n(s,y)dyds): n\geq 1\}$ be a sequence of $K_M$. Due to the fact that $T_M$ is a compact subset of $L^2([0,T]\times[0,1])$ under weak topology, there exists a subsequence still denoted by $\{n\}$ and an element $h\in T_M$ such that $h_{n}\rightarrow h$ weakly in $T_M$ as $n \rightarrow \infty$. We need to prove $X^{h_n}\rightarrow X^h$ strongly in $C([0,T];H)$. From (\ref{eqq-17}), we know that \begin{eqnarray} X^{h_n}(t,x)-X^h(t,x)&=& \int^t_0\int^1_0G_{t-s}(x,y)\partial_rb(s,y,u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))dyds\\ && -\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\partial_rg(s,y,u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))dyds\\ && +\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^0(s,y))(h_n(s,y)-h(s,y))dyds\\ &:=&J^n_1(t)+J^n_2(t)+J^n_3(t). \end{eqnarray} By Lemma \ref{lem-1}, (H3) and Lemma \ref{lem-5}, we deduce that \begin{eqnarray}\notag \\|J^n_1(t)\\|_H&\leq& C\int^t_0(t-s)^{-\frac{3}{4}}\\|\partial_rb(s,y,u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))\\|_{L^1}ds\\ \notag &\leq& CK\int^t_0(t-s)^{-\frac{3}{4}}\\|(1+u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))\\|_{L^1}ds\\ \notag &\leq& CK\int^t_0(t-s)^{-\frac{3}{4}}(1+\\|u^0(s)\\|_H)\\| X^{h_n}(s)-X^h(s)\\|_{H}ds\\ \label{ee-6} &\leq& CK(1+C_0)\int^t_0(t-s)^{-\frac{3}{4}}\\| X^{h_n}(s)-X^h(s)\\|_{H}ds. \end{eqnarray} Similar to $J^n_1(t)$, we deduce that \begin{eqnarray}\notag \\|J^n_2(t)\\|_H&\leq& CK\int^t_0(t-s)^{-\frac{3}{4}}\\|\partial_rg(s,y,u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))\\|_{L^1}ds\\ \notag &\leq& CK\int^t_0(t-s)^{-\frac{3}{4}}\\|(1+u^0(s,y))( X^{h_n}(s,y)-X^h(s,y))\\|_{L^1}ds\\ \notag &\leq& CK\int^t_0(t-s)^{-\frac{3}{4}}(1+\\|u^0(s)\\|_H)\\| X^{h_n}(s)-X^h(s)\\|_{H}ds\\ \label{ee-7} &\leq& CK(1+C_0)\int^t_0(t-s)^{-\frac{3}{4}}\\| X^{h_n}(s)-X^h(s)\\|_{H}ds. \end{eqnarray} Let $P_k$ be the orthogonal projection in $H$ onto the space spanned by $\{e_1,\cdot\cdot\cdot, e_k\}_{k\geq 1}$ with $\{e_k\}$ be an orthonormal basis of $H$, we have \begin{eqnarray} &&\sup_{0\leq t\leq T}\\|P_kJ^n_3(t)-J^n_3(t)\\|^2_H\\ &\leq& \int^1_0\Big(\int^t_0\int^1_0G^2_{t-s}(x,y)\Big((P_k-I)\sigma(u^0(y,s)))^2dyds\Big)\Big(\int^t_0\int^1_0(h_n-h)^2dyds\Big)dx\\ &\leq& 2M^2\int^1_0\int^t_0\int^1_0G^2_{t-s}(x,y)\Big((P_k-I)\sigma(u^0(y,s))\Big)^2dydsdx\\ &\leq& 2M^2\int^t_0(t-s)^{-\frac{1}{2}}\int^1_0\Big((P_k-I)\sigma(u^0(y,s))\Big)^2dyds. \end{eqnarray} Since \begin{eqnarray} 2M^2\int^t_0(t-s)^{-\frac{1}{2}}\int^1_0[(P_k-I)\sigma(u^0(y,s))]^2dyds\leq CM^2K^2T^{\frac{1}{2}}, \end{eqnarray} by the dominated convergence theorem, it follows that \begin{eqnarray}\label{equat-3} \sup_{0\leq t\leq T}\\|P_kJ^n_3(t)-J^n_3(t)\\|^2_H\rightarrow0, \quad k\rightarrow \infty. \end{eqnarray} For any $k\geq 1$, $t,s\in [0,T], t>s$, we have \begin{eqnarray} \\|P_kJ^n_3(t)-P_kJ^n_3(s)\\|^2_H&\leq& \Big\\|\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))P_k\sigma(u^0)(h_n(y,r)-h(y,r))dydr\Big\\|^2_H\\ && +\Big\\|\int^t_s\int^1_0G_{t-r}(x,y)P_k\sigma(u^0)(h_n(y,r)-h(y,r))dydr\Big\\|^2_H\\ &:=& J^n_{3,1}+J^n_{3,2}, \end{eqnarray} By (H1) and (\ref{eqq-5-1}), we get \begin{eqnarray} J^n_{3,1}&\leq& \int^1_0\Big(\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))^2\sigma^2(u^0(y,r))dydr\Big)\Big(\int^s_0\int^1_0(h_n(y,r)-h(y,r))^2dydr\Big)dx\\ &\leq& CK^2M\int^1_0\int^s_0\int^1_0(G_{t-r}(x,y)-G_{s-r}(x,y))^2dydrdx\\ &\leq& CK^2M(t-s)^{\frac{1}{2}}. \end{eqnarray} Utilizing (\ref{eqq-5}) and (H1), we deduce that \begin{eqnarray} J^n_{3,2}&\leq& \int^1_0\Big(\int^t_s\int^1_0G^2_{t-r}(x,y)\sigma^2(u^0(y,r))dydr\Big)\Big(\int^t_s\int^1_0(h_n(y,r)-h(y,r))^2dydr\Big)dx\\ &\leq& CMK^2\int^1_0\int^t_s\int^1_0G^2_{t-r}(x,y)dydrdx\\ &\leq& CMK^2(t-s)^{\frac{1}{2}}. \end{eqnarray} Combing the above two estimates, it yields \begin{eqnarray} \\|P_kJ^n_3(t)-P_kJ^n_3(s)\\|^2_H\leq CMK^2(t-s)^{\frac{1}{2}}. \end{eqnarray} Moreover, for any $t\in [0,T]$, we have \begin{eqnarray} \sup_n\\|J^n_3(t)\\|^2_H&\leq& \sup_n\int^1_0\Big(\int^t_0\int^1_0G^2_{t-s}(x,y)\sigma^2(u^0)dyds\Big)\Big(\int^t_0\int^1_0(h_n(y,s)-h(y,s))^2dyds\Big)dx\\ &\leq& CMK^2\int^1_0\int^t_0\int^1_0G^2_{t-s}(x,y)dydsdx\\ &\leq& CMK^2T^{\frac{1}{2}}. \end{eqnarray} Since for any $k\geq 1$, $P_k:H\rightarrow H$ is a compact operator, then for any $t\in [0,T]$, $\{P_kJ^n_3(t),n\geq 1 \}$ is relative compact in $H$. Moreover, $\{P_kJ^n_3(t),n\geq 1 \}$ is closed in $H$. As a result of Arzel\`{a}-Ascoli theorem, $\{P_kJ^n_3\}_n$ is uniformly compact in $C([0,T];H)$. On the other hand, since $h_n-h$ converges to $0$ weakly in $L^2([0,T]\times [0,1]; \mathbb{R})$, then \[ P_kJ^n_3(t)=\int^t_0\int^1_0G_{t-s}(x,y)P_k\sigma(u^0)(h_n-h)dyds\rightarrow 0\quad {\rm{in}} \ H, \quad {\rm{as}}\ n\rightarrow \infty. \] Thus, we have \begin{eqnarray}\label{ee-2} \lim_{n\rightarrow \infty}\sup_{t\in [0,T]}\\|P_kJ^n_3(t)\\|_H=0. \end{eqnarray} Combing (\ref{equat-3}) and (\ref{ee-2}), we conclude that \begin{eqnarray}\label{ee-3} \lim_{n\rightarrow \infty}\sup_{t\in [0,T]}\\|J^n_3(t)\\|_H= 0. \end{eqnarray} Based on (\ref{ee-6}), (\ref{ee-7}) and (\ref{ee-3}), it follows that \begin{eqnarray} \\| X^{h_n}(t)-X^h(t)\\|_H\leq 2KC(1+C_0)\int^t_0(t-s)^{-\frac{3}{4}}\\| X^{h_n}(s)-X^h(s)\\|_{H}ds+\\|J^n_3(t)\\|_H. \end{eqnarray} By iteration and Gronwall inequality, we have \begin{eqnarray} \\| X^{h_n}(t)-X^h(t)\\|_H\leq C(K, C_0,T) \\|J^n_3(t)\\|_H. \end{eqnarray} Utilizing (\ref{ee-3}), it yields \begin{eqnarray} \sup_{t\in [0,T]}\\| X^{h_n}(t)-X^h(t)\\|_H \leq C(K, C_0,T) \sup_{t\in [0,T]}\\|J^n_3(t)\\|_H\rightarrow0,\ {\rm{as}}\ n\rightarrow \infty. \end{eqnarray} We complete the proof. \end{proof} \begin{thm}\label{thm-4} Let $\{h^\varepsilon; \varepsilon>0\}\subset \mathcal{A}_M$ be a sequence that converges in distribution to $h$ as $\varepsilon\rightarrow 0$. Then \[ \mathcal{G}^{\varepsilon}\Big(W(\cdot)+\lambda(\varepsilon)\int^{\cdot}_0\int^{\cdot}_0h^\varepsilon(s,y)dyds\Big)\ {\rm{converges\ in\ distribution\ to}}\ \mathcal{G}^{0}\Big(\int^{\cdot}_0\int^{\cdot}_0h(s,y)dyds\Big), \] in $C([0,T];H)$. \end{thm} \begin{proof} Suppose that $\{h^\varepsilon; \varepsilon>0\}\subset \mathcal{A}_M$ and $h^\varepsilon$ converges to $h$ as $T_M-$valued random elements in distribution. By Girsanov's theorem, we obtain $\bar{X}^{\varepsilon,h^{\varepsilon}}(\cdot)=\mathcal{G}^{\varepsilon}\Big(W(\cdot)+\lambda(\varepsilon)\int^{\cdot}_0\int^{\cdot}_0h^\varepsilon(s,y)dyds\Big)$. Consider \begin{eqnarray} Z^{\varepsilon}(t,x)=\frac{1}{\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\bar{X}^{\varepsilon, h^{\varepsilon}}(s,y))W(dyds), \end{eqnarray} with the initial value $Z^{\varepsilon}(0)=0$. Applying the same method as the proof of (\ref{equat-2}), we get \begin{eqnarray} \lim_{\varepsilon\rightarrow 0}E\sup_{t\in [0,T]}\\|Z^{\varepsilon}(t)\\|^2_H=0 \end{eqnarray} and $\{Z^{\varepsilon}\}$ is tight in $C([0,T];H)$. Set \[ \Pi=\Big(C([0,T];H), T_M, C([0,T];H)\Big). \] By Proposition \ref{prp-6}, we know that the family $\{(\bar{X}^{\varepsilon, h^{\varepsilon}}, h^{\varepsilon}, Z^{\varepsilon}); \varepsilon\in (0,1]\}$ is tight in $\Pi$. Let $(X,h,0)$ be any limit point of $\{(\bar{X}^{\varepsilon, h^{\varepsilon}}, h^{\varepsilon}, Z^{\varepsilon}); \varepsilon\in (0,1]\}$. We will show that $X$ has the same law as $\mathcal{G}^0(\int^{\cdot}_0\int^{\cdot}_0h(s,y)dyds)$ , and in fact $\bar{X}^{\varepsilon, h^{\varepsilon}}$ converges in distribution to $X$ in $C([0,T];H)$ as $\varepsilon\rightarrow 0$, which implies Theorem \ref{thm-4}. By the Skorokhod representation theorem, there exists a stochastic basis $(\Omega^1, \mathcal{F}^1, \{\mathcal{F}^1_t\}_{t\in [0,T]}, {P}^1)$ and $\Pi-$valued random variables $(\tilde{U}^{\varepsilon}, \tilde{h}^{\varepsilon},\tilde{Z}^{\varepsilon}),(\tilde{U}, \tilde{h},0)$ on this basis, such that $(\tilde{U}^{\varepsilon}, \tilde{h}^{\varepsilon},\tilde{Z}^{\varepsilon})$ (resp. $(\tilde{U}, \tilde{h},0)$) has the same law as $(\bar{X}^{\varepsilon,h^{\varepsilon}},h^{\varepsilon},Z^{\varepsilon})$ (resp. $(X,h,0)$), and $(\tilde{U}^{\varepsilon}, \tilde{h}^{\varepsilon},\tilde{Z}^{\varepsilon})\rightarrow (\tilde{U}, \tilde{h},0)$, $P^1-$a.s. in $\Pi$. From the equation satisfied by $(\bar{X}^{\varepsilon,h^{\varepsilon}},h^{\varepsilon},Z^{\varepsilon})$ , we see that $(\tilde{U}^{\varepsilon}, \tilde{h}^{\varepsilon},\tilde{Z}^{\varepsilon})$ satisfies \begin{eqnarray}\notag &&\tilde{U}^{\varepsilon}(x,t)-\tilde{Z}^{\varepsilon}(x,t)\\ \notag &=& \int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\tilde{U}^{\varepsilon}(s,y))\tilde{h}^{\varepsilon}(s,y)dyds\\ \notag &&+\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\Big(b(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\tilde{U}^{\varepsilon}(s,y))-b(s,y,u^0(s,y))\Big)dyds\\ \label{ee-8} &&-\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\Big(g(s,y,u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)\tilde{U}^{\varepsilon}(s,y))-g(s,y,u^0(s,y))\Big)dyds. \end{eqnarray} and \begin{eqnarray}\notag &&P^1(\tilde{U}^{\varepsilon}-\tilde{Z}^{\varepsilon}\in C([0,T];H))\\ \notag &=& P(\bar{X}^{\varepsilon, h^{\varepsilon}}-Z^{\varepsilon}\in C([0,T];H))\\ \label{eee-28} &=&1. \end{eqnarray} Let $\Omega^1_0$ be the subset of $\Omega^1$ such that $(\tilde{U}^{\varepsilon}, h^{\varepsilon},\tilde{Z}^{\varepsilon})\rightarrow (\tilde{U},\tilde{h},0)$ in $\Pi$, we have $P^1(\Omega^1_0)=1$. For any $\tilde{\omega}\in \Omega^1_0$, we have \begin{eqnarray}\label{ee-12} \sup_{t\in [0,T]}\\|\tilde{U}^{\varepsilon}(\tilde{\omega},t)-\tilde{U}(\tilde{\omega},t)\\|^2_H\rightarrow 0,\ {\rm{as}} \ \varepsilon\rightarrow 0. \end{eqnarray} Set $\tilde{\eta}^{\varepsilon}(\tilde{\omega},t)=\tilde{U}^{\varepsilon}(\tilde{\omega},t)-\tilde{Z}^{\varepsilon}(\tilde{\omega},t)$, by (\ref{ee-8}), $\tilde{\eta}^{\varepsilon}(\tilde{\omega},x,t)$ satisfies \begin{eqnarray}\notag &&\tilde{\eta}^{\varepsilon}(\tilde{\omega},x,t)\\ \notag &=& \int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y, u^0(s,y)+\sqrt{\varepsilon}\lambda(\varepsilon)(\tilde{\eta}^{\varepsilon}(\tilde{\omega},s,y)+\tilde{Z}^{\varepsilon}(\tilde{\omega},s,y)))\tilde{h}^{\varepsilon}(\tilde{\omega},s,y)dyds\\ \notag &&+\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0G_{t-s}(x,y)\Big(b(s,y,u^0+\sqrt{\varepsilon}\lambda(\varepsilon)(\tilde{\eta}^{\varepsilon}(\tilde{\omega},s,y)+\tilde{Z}^{\varepsilon}(\tilde{\omega},s,y)))-b(s,y,u^0)\Big)dyds\\ \label{ee-9} &&-\frac{1}{\sqrt{\varepsilon}\lambda(\varepsilon)}\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\Big(g(s,y,u^0+\sqrt{\varepsilon}\lambda(\varepsilon)(\tilde{\eta}^{\varepsilon}(\tilde{\omega},s,y)+\tilde{Z}^{\varepsilon}(\tilde{\omega},s,y)))-g(s,y,u^0)\Big)dyds. \end{eqnarray} with initial value $\tilde{\eta}^{\varepsilon}(\tilde{\omega},x,0)=0$. Moreover, we deduce from (\ref{eee-28}) that \begin{eqnarray}\label{eee-29} \sup_{t\in [0,T]}\\|\tilde{\eta}^{\varepsilon}(\tilde{\omega},t)\\|_H<\infty. \end{eqnarray} Taking into account the following facts \begin{eqnarray} \lim_{\varepsilon\rightarrow 0}\sup_{t\in [0,T]}\\|\tilde{Z}^{\varepsilon}(\tilde{\omega}, t)\\|^2_H=0, \quad \tilde{U}^{\varepsilon}=\tilde{\eta}^{\varepsilon}+\tilde{Z}^{\varepsilon}, \end{eqnarray} and by (H1), (\ref{eee-29}), (\ref{ee-3}), Lemma \ref{lem-1} and Lemma \ref{lem-5}, we have \begin{eqnarray}\notag &&\lim_{\varepsilon\rightarrow 0}\sup_{t\in [0,T]}\\|\tilde{U}^{\varepsilon}(\tilde{\omega}, t)-\hat{U}(\tilde{\omega}, t)\\|^2_H\\ \notag &\leq& \lim_{\varepsilon\rightarrow 0}\sup_{t\in [0,T]}\\|\tilde{\eta}^{\varepsilon}(\tilde{\omega}, t)-\hat{U}(\tilde{\omega}, t)\\|^2_H+\lim_{\varepsilon\rightarrow 0}\sup_{t\in [0,T]}\\|\tilde{Z}^{\varepsilon}(\tilde{\omega}, t)\\|^2_H\\ \notag &\leq&\lim_{\varepsilon\rightarrow 0}\sup_{t\in [0,T]}\\|\tilde{\eta}^{\varepsilon}(\tilde{\omega}, t)-\hat{U}(\tilde{\omega}, t)\\|^2_H\\ \label{ee-11} &=&0, \end{eqnarray} where $\hat{U}(t):=\hat{U}(\tilde{\omega},t)$ satisfies \begin{eqnarray}\notag \hat{U}(x,t)&=&\int^t_0\int^1_0G_{t-s}(x,y)\partial_rb(s,y,u^0(s,y))\hat{U}(s,y)dyds\\ \notag && -\int^t_0\int^1_0\partial_yG_{t-s}(x,y)\partial_rg(s,y,u^0(s,y))\hat{U}(s,y)dyds\\ \label{ee-10} && +\int^t_0\int^1_0G_{t-s}(x,y)\sigma(s,y,u^0(s,y))\tilde{h}(s,y)dyds. \end{eqnarray} Hence, by (\ref{ee-12}) and (\ref{ee-11}), we deduce that $\tilde{U}=\hat{U}=\mathcal{G}^0(\int^{\cdot}_0\int^{\cdot}_0\tilde{h}(s,y)dyds)$, then $\tilde{U}$ has the same law as $\mathcal{G}^0(\int^{\cdot}_0\int^{\cdot}_0h(s,y)dyds)$. Since $\bar{X}^{\varepsilon,h^{\varepsilon}}$ has the same law as $\tilde{U}^{\varepsilon}$ on $C([0,T];H)$ and by (\ref{ee-11}), we deduce that $\mathcal{G}^{\varepsilon}(W(\cdot)+\lambda(\varepsilon)\int^{\cdot}_0\int^{\cdot}_0h^{\varepsilon}(s,y)dyds)$ converges in distribution to $\mathcal{G}^0(\int^{\cdot}_0\int^{\cdot}_0h(s,y)dyds)$ as $\varepsilon\rightarrow 0$. We complete the proof. \end{proof} \def References{ References} \end{document}	arXiv
\begin{document} \begin{flushright} \end{flushright} \par \vskip .1in \noindent \begin{center} \begin{LARGE} {\bf On lacunary Toeplitz determinants.} \end{LARGE} \begin{large} {\bf K.~K.~Kozlowski}\footnote[1]{Universit\'{e} de Bourgogne, Institut de Math\'{e}matiques de Bourgogne, UMR 5584 du CNRS, France, [email protected]}. \par \end{large} \centerline{\bf Abstract} \parbox{12cm}{\small By using Riemann--Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants $\operatorname{det}_N\big[ c_{\ell_a-m_b }[f] \big]$ generated by holomorhpic symbols, where $\ell_a=a$ (resp. $m_b=b$) except for a finite subset of indices $a=h_1,\dots, h_n$ (resp. $b=t_1,\dots, t_r$). In addition to the usual Szeg\"{o} asymptotics, our answer involves a determinant of size $n+r$.} \end{center} \section{Introduction} A lacunary Toeplitz determinant generated by a symbol $f$ refers to the below determinant \beq \operatorname{det}_{N}\Big[ c_{\ell_a-m_b}[f] \Big] \qquad \e{where} \qquad c_n[f] \; = \; \Oint{ \Dp{}\mc{D}_1 }{} \f{ f(z)}{ z^{n+1} } \cdot \f{ \mathrm{d} z }{2i\pi} \label{ecriture detereminant lacunaire general} \enq and $\Dp{}\mc{D}_{\eta}$ is the counter clockwise oriented boundary of the disc of radius $\eta$ centred at $0$. The sequences $\ell_a$, $m_b$ appearing in \eqref{ecriture detereminant lacunaire general} are such that \begin{eqnarray} \ell_a & = & a \quad \e{for} \quad a\in \big\{ 1, \dots, N \big\} \setminus \big\{h_1,\dots, h_n \big\} \qquad \e{and} \qquad \ell_{h_a} \; = \; p_a \quad a=1,\dots, n \; \label{definition de la suite ella}\\ m_a & = & a \quad \e{for} \quad a\in \big\{ 1, \dots, N \big\} \setminus \big\{t_1,\dots, t_r \big\} \qquad \e{and} \qquad m_{t_a} \; = \; k_a \quad a=1,\dots, r \; \label{definition de la suite ma} \end{eqnarray} The integers $h_a \in \intn{1}{N}$ and $p_a \in \mathbb{Z} \setminus \intn{1}{N}$, $a=1,\dots, n$ (resp. $t_a \in \intn{1}{N}$ and $k_a \in \mathbb{Z} \setminus \intn{1}{N}$, $a=1,\dots, r$) are assumed to be pairwise distinct. The large-$N$ asymptotic behaviour of such determinants has been first considered by Tracy and Widom \cite{TracyWidomAsymptoticExpansionLacunaryToeplitz} and Bump and Diaconis \cite{BumpDiaconisLacunaryToeplitzThrougSumsSymFctsAndYoungTableaux}. More or less at the same time, these authors have obtained two formulae of a very different kind for these large-N asymptotics. In fact, both collaborations expressed the large-$N$ behaviour of the lacunary Toeplitz determinant in terms of the unperturbed determinant $\operatorname{det}_{N}\Big[ c_{a - b}[f] \Big]$ times an extra term whose representations took a very different form. The expression found Bump and Diaconis was based on characters of the symmetric group associated with the partitions $ \lambda $ and $\mu$ that can be naturally associated with the sequences $\ell_a$ and $m_b$. The answer involved the sum over the symmetric groups of $\|\lambda\|$ and $\|\mu\|$ elements. In their turn, Tracy and Widom obtained a determinant representation of the type \beq \operatorname{det}_{N}\big[ c_{\ell_a-b}[f] \big] \; = \; \operatorname{det}_{N-m}\big[ c_{j-k}[f] \big] \cdot \operatorname{det}_{ q }\big[ W_{jk} \big] \cdot \Big( 1+ \e{o}(1) \Big) \qquad q\, = \, \max \big\{ t_1,\dots , t_r , h_1,\dots, h_n \big\} \label{formule asymptotique Widom} \enq where $W_{jk} $ was an explicit $q \times q$ sized matrix depending on the symbol $f$ and the numbers $h_1, \dots, h_n$, $p_1,\dots, p_n$, $t_1,\dots, t_r$ and $k_1,\dots, k_r$. In \cite{DehayeProofIdentityBumpDiaconisTracyWidomLacunatyToeplitz}, Dehaye proved, by a direct method, the equivalence between the two aforementioned formulae. One should also mention that the large-$N$ asymptotic behaviour of some generalizations of lacunary Toeplitz determinants have been obtained by Lions in \cite{LionsToeplitzLacunaires}. The drawbacks of the aforementioned asymptotic expansions was that the answer depended on the magnitude of the lacunary parameters $p_a, k_b, h_a, t_b$. As soon as these parameters were also growing with $N$, the form of the answer did not allow for an easy access to the large-$N$ asymptotic behaviour of the lacunary determinant. Indeed, in Bump-Diaconis' case, the number of summed up terms was growing as $\sum(p_a-h_a) + \sum(k_a-t_a)$ whereas in Tracy-Widom's case, the non-trivial determinant part involved a matrix of size $\max\{h_a, t_b\}$. In the present note we obtain an asymptotic expansion solely in terms of a $(n+r)\times (n+r)$ matrix and show that the latter is enough so as to treat certain cases of lacunary parameters $p_a, k_b, h_a, t_b$ going to infinity. The structure of the asymptotics when $r\not=0$ (\textit{ie} $m_a\not=a$) is slightly more complex, so that we postpone the statement of the corresponding results to the core of the paper and present the asymptotic expansion we obtain on the example of line-lacunary Toeplitz determinants \begin{theorem} \label{Theorem Asymptotiques Toeplitz lacunaire a lignes} Let $f$ be a non-vanishing function on $\Dp{}\mc{D}_{1}$ such that $f$ and $\ln f$ are holomorphic on some open neighbourhood of $\Dp{}\mc{D}_{1}$. Let $\ell_a$ be defined as \eqref{definition de la suite ella} and $\alpha$ be the piecewise analytic function \beq \alpha(z) \; = \; \exp\bigg\{ - \sul{n \geq 0}{} c_{ n}\big[ \ln f \big] \cdot z^{n} \bigg\} \quad \e{for}\; z \in \mc{D}_{1} \qquad \e{and} \qquad \alpha(z) \; = \; \exp\bigg\{ \sul{n \geq 1}{} c_{- n}\big[ \ln f \big] \cdot z^{-n} \bigg\} \quad \e{for}\; z \in \ensuremath{\mathbb{C}}\setminus \ov{\mc{D}}_{1} \; . \label{definition facteur alpha solution RHP scalaire} \enq Then, provided that the matrix $M$ given below is non-singular, the line-lacunary Toepltz determinant $\operatorname{det}_{N}\big[ c_{\ell_a-b}[f] \big]$ admits the representation \beq \operatorname{det}_{N}\big[ c_{\ell_a-b}[f] \big] \; = \; \operatorname{det}_{N}\big[ c_{a-b}[f] \big] \cdot \operatorname{det}_{n}\big[ M_{ab} \big] \cdot \Big( 1+ \e{O}\big( N^{-\infty} \big) \Big) \;, \label{Theorem intro ecriture forme DA} \enq where the $n\times n$ matrix $M$ reads \bem M_{ab} \; = \; - \bs{1}_{\mathbb{N}}(p_a) \Oint{ \Dp{}\mc{D}_{\eta_z} }{} \hspace{-2mm} \f{ \mathrm{d} z }{ 2i\pi} \cdot \Oint{ \Dp{}\mc{D}_{\eta_s} }{} \hspace{-2mm} \f{ \mathrm{d} s }{ 2i\pi} \cdot \f{ \alpha(z) }{ \alpha(s) } \cdot \f{ s^{N-p_a} \cdot z^{h_b-N-1} }{ z- s } \\ \; + \; \bs{1}_{\mathbb{N}}(-p_a) \Oint{ \Dp{}\mc{D}_{ \eta_z^{-1} } }{} \hspace{-2mm} \f{ \mathrm{d} z }{ 2i\pi} \cdot \Oint{ \Dp{}\mc{D}_{ \eta_s^{-1} } }{} \hspace{-2mm} \f{ \mathrm{d} s }{ 2i\pi} \cdot \f{ \alpha(s) }{ \alpha(z) } \cdot \f{ s^{-p_a} \cdot z^{h_b-1} }{ z- s } \;, \label{ecriture formule asymptotique matrice M cas lacunaire a ligne} \end{multline} and $1>\eta_{z} > \eta_{s} >0 $\;. \end{theorem} The theorem above allows one to obtain the large $N$-asymptotic expansion of the line-lacunary Toeplitz determinant independently on the magnitude (in respect to $N$) of the lacunary parameters $\{h_a\}$ and $\{ p_a \}$. Indeed, since the size of the matrix $M$ does not depend on the integers $\{h_a\}$ or $\{p_a\}$, the problem boils down to a \textit{classical} asymptotic analysis of one-dimensional integrals. Still, in order to provide one with an explicit answer, some more data on these parameters is needed. For instance, one has the \begin{cor} \label{Corolaire Toeplitz lacunaire à un indice} Let \begin{eqnarray} p_a \; = \; 1-p_a^{-} \quad a=1,\dots, n_{-} \qquad &\e{and}& \qquad p_{a+n_-} \; = \; p_a^{+} + N \quad a=1,\dots, n_{+} \label{particules trous pour entiers pa} \\ h_a \; = \; h_a^{-} \quad a=1,\dots, n_{-} \qquad &\e{and} &\qquad h_{a+n_-} \; = \; N+1- h_a^{+} \quad a=1,\dots, n_{+} \;, \label{particuler trous pour entiers ha} \end{eqnarray} where $p_a^{\pm}$ and $h_a^{\pm}$ are assumed to be independent of $N$ and $n=n_- + n_+$. Provided that the matrices $M^{(\pm)}$ given below are not singular, one has \beq \operatorname{det}_{n}\big[ M_{ab} \big] \; = \;\operatorname{det}_{n_+}\Big[ M_{ a b }^{(+)} \Big] \cdot \operatorname{det}_{n_-}\Big[ M_{ a b }^{(-)} \Big] \cdot \Big( 1+ \e{O}\big( N^{-\infty} \big) \Big) \;, \enq where \beq M_{a b }^{(+)} \; = \; - \Oint{ \Dp{}\mc{D}_{\eta_z} }{} \hspace{-2mm}\f{ \mathrm{d} z }{2i\pi} \hspace{-2mm} \cdot \Oint{ \Dp{}\mc{D}_{\eta_s } }{} \hspace{-2mm} \f{ \mathrm{d} s }{2i\pi} \cdot \f{ s^{ -p_{a}^+ }\cdot z^{- h_{b}^+} }{ z \, - \, s } \cdot \f{ \alpha(z) }{ \alpha(s) } \qquad \e{and} \qquad M_{a b }^{(-)} \; = \; \Oint{ \Dp{}\mc{D}_{\eta_z^{-1}} }{} \hspace{-2mm}\f{ \mathrm{d} z }{2i\pi} \hspace{-2mm} \cdot \Oint{ \Dp{}\mc{D}_{\eta_s^{-1} } }{} \hspace{-2mm} \f{ \mathrm{d} s }{2i\pi} \cdot \f{ s^{p_{a}^- - 1}\cdot z^{ h_{b}^- - 1} }{ z \, - \, s } \cdot \f{ \alpha(s) }{ \alpha(z) } \;. \enq \end{cor} We obtain the asymptotic expansion \eqref{Theorem intro ecriture forme DA} by interpreting the lacunary Topelitz determinant as the determinant of a finite rank perturbation of a integrable integral operator acting on the unit circle. The inverse of the integrable integral operator, in the large-$N$ regime, can be constructed by means of an asymptotic resolution of a Riemann--Hilbert problem. We have restricted the study of the present paper to holomorphic symbols. However, in principle, one could apply the method to less regular symbols, \textit{eg} those containing Fischer-Hartwig singularities. Of course, the price of such generalisation would be to deal with certain technicalities related with the more complex structure of the large-$N$ approximant to the associated resolvent operator. The paper is organized as follows. We prove Theorem \ref{Theorem Asymptotiques Toeplitz lacunaire a lignes} in Section \ref{Section Toeplitz lacunaire a lignes}. In Section \ref{Section Toeplitz lacunaire a lignes et colonnes} we establish the large-$N$ asymptotic expansion of general line and row lacunary Toeplitz determinants subordinate to the sequences \eqref{definition de la suite ella}-\eqref{definition de la suite ma}. Technical details related to the large-$N$ inversion of integrable integral operators arising in the analysis of Toeplitz determinant generated by holomorphic non-vanishing on $\Dp{}\mc{D}_1$ symbols are recalled in appendix \ref{Appendix RHP pour Toeplitz regulier}. \section{The line lacunary Toeplitz determinants} \label{Section Toeplitz lacunaire a lignes} In this section, we first prove a preliminary factorisation result that allows one to express the lacunary Toeplitz determinant $\operatorname{det}_{N}\Big[ c_{ \ell_a - b }[f] \Big]$ in terms of the non-perturbed Toeplitz determinant $\operatorname{det}_{N}\Big[ c_{ a - b }[f] \Big] $ and of the determinant of a $n\times n$ matrix. We subsequently analyse the large-$N$ behaviour of this finite-size $n$ determinant. \subsection{The factorisation} \begin{lemme} \label{Lemme factorisation Toeplitz lacunaire a lignes} Let $f$ be non-vanishing on $\Dp{}\mc{D}_1$ and such that $f$ and $\ln f$ are holomorphic in some open neighbourhood of $\msc{C}$. Let $V_0$ be the integral kernel \beq V_0\big( z, s \big) \; = \; \big( f(z) - 1 \big) \cdot \f{ z^{\f{N}{2}}\cdot s^{-\f{N}{2}} \; - \; z^{-\f{N}{2}}\cdot s^{\f{N}{2}} } { 2i\pi \big( z - s \big) } \label{definition noyau integral V0} \enq of the integrable integral operator $I+V_0$ acting on $L^2\big( \Dp{}\mc{D}_1 \big)$. Then, provided that $N$ is large enough, $I+V_0$ is invertible with inverse $I-R_0$ and the below factorization holds \beq \operatorname{det}_{N}\Big[ c_{ \ell_a - b }[f] \Big] \; = \; \operatorname{det}_{N}\Big[ c_{ a - b }[f] \Big] \cdot \operatorname{det}_n \big[ M_{ab} \big] \enq where \beq M_{ k \ell } \; = \; \delta_{k \ell } \; - \; c_{ h_{k} - h_{\ell} }[f] \; + \; c_{ p_{k} - h_{\ell} }[f] \; + \; \Int{ \msc{C} }{} R_{0}(z,s) \cdot f(s)\cdot \big( s^{\f{N}{2}-h_k} - s^{\f{N}{2} - p_k} \big) \cdot z^{h_{\ell}- 1 -\f{N}{2}} \cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ 2i \pi } \;. \label{definition matrice Mkl} \enq \end{lemme} \noindent {\it Proof --- \ } Let $I+V$ be the integral operator on $L^{2}(\Dp{}\mc{D}_{1})$ with a kernel given by \beq V(z,s) \; = \; \sul{a=1}{N} \kappa_a(z) \cdot \tau_a(s) \qquad \e{where} \quad \quad \tau_a(z) \; = \; \f{1}{2i\pi} \cdot z^{a-1-\f{N}{2}} \enq and \beq \begin{array}{cccc} \kappa_a(z) & = & \big( f(z)-1 \big) \cdot z^{\f{N}{2}-a} & a \in \{1,\dots , N \} \setminus \{h_1, \dots, h_n \} \hspace{3mm} \\ \kappa_{h_a}(z) & = & f(z) \cdot z^{ \f{N}{2}-p_a } \; - \; z^{ \f{N}{2}-h_a } & a=1, \dots, n \end{array} \;. \enq Since $V$ is a finite rank $N$ operator, the Fredholm determinant of $I+V$ reduces to one of an $N\times N$ matrix \beq \operatorname{det}_{\Dp{}\mc{D}_{1}}\big[ I \, + \, V \big] \; = \; \operatorname{det}_{N}\Big[ \delta_{ab} \; + \; \int_{ \Dp{}\mc{D}_{1} }{} \kappa_{a}(z) \cdot \tau_b(z) \cdot \mathrm{d} z \Big] \; = \; \operatorname{det}_{N}\Big[ c_{ \ell_a - b }[f] \Big] \;. \enq One can decompose the kernel $V$ as $V=V_0 + V_1$ where $V_0$ has been introduced in \eqref{definition noyau integral V0} whereas $V_1$ is the finite rank $n$ perturbation of $V_0$ given by \beq V_1\big( z, s \big) \; = \; -\f{ f(z) }{ 2i\pi } \cdot \sul{ a =1 }{ n } \big( z^{\f{N}{2} -h_a} \; - \; z^{\f{N}{2} -p_a} \big) \cdot s^{h_a -1 -\f{N}{2} } \;. \enq It follows from the strong Szeg\"{o} limit theorem and from the identity $\operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I\, + \, V_0 \big] \; = \; \operatorname{det}_{N}\Big[ c_{ a - b }[f] \Big]$ that, provided $N$ is taken large enough, the operator $I+V_0$ is invertible. Hence, all-in-all, we get that \beq \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I \, + \, V \big] \; = \; \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I\, + \, V_0 \big] \cdot \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I\, + \, (I-R_0)\cdot V_1 \big] \; = \; \operatorname{det}_{N}\Big[ c_{ a - b }[f] \Big] \cdot \operatorname{det}_n\big[ M_{ k \ell } \big] \enq where the matrix $M_{k\ell}$ is as defined in \eqref{definition matrice Mkl}. \subsection{Asymptotic analysis of $\operatorname{det}_{n}[M]$-Proof of theorem \ref{Theorem Asymptotiques Toeplitz lacunaire a lignes}} As it has been recalled in the appendix, the resolvent kernel $R_0$ of the operator $I+V_0$ can be recast as \beq R_0 \; = \; R_0^{(0)} + R_0^{(\infty)} \label{ecriture decomposition resolvant en partie finie et perturbation exp petite en N} \enq where \beq R_0^{(0)}(z,s) \; = \; \f{ f(z) -1 }{ 2i\pi } \cdot \f{ z^{\f{N}{2}} \cdot s^{-\f{N}{2}} \cdot \alpha_+(s) \cdot \alpha_-^{-1}(z) \; - \; s^{\f{N}{2}} \cdot z^{-\f{N}{2}} \cdot \alpha_+(z) \cdot \alpha_-^{-1}(s) }{ z-s } \label{definition resolvent approximatif de V0} \enq and \beq \norm{ R_0^{(\infty)} }_{L^{\infty}\big( \Dp{}\mc{D}_{1} \times \Dp{}\mc{D}_{1} \big) } \; \leq \; C \cdot N \cdot \ex{-\kappa N } \;. \enq Above, $\alpha_{\pm}$ are the $+$ (\textit{ie} from within) and $-$ (\textit{ie} from the outside) non-tangential limits on $\Dp{}\mc{D}_1$ of the piecewise analytic function $\alpha$ defined in \eqref{definition facteur alpha solution RHP scalaire}. The decomposition \eqref{ecriture decomposition resolvant en partie finie et perturbation exp petite en N} ensures that, for the price of exponentially small corrections, one can trade the kernel $R_0$ for $R_0^{(0)}$ in \eqref{definition matrice Mkl}. Using that $\alpha_{+}$ (resp. $\alpha_-$) admit an analytic continuation to some open neighbourhood of $\Dp{} \mc{D}_1$ in $\ensuremath{\mathbb{C}} \setminus \ov{\mc{D}}_1$ (resp. interior $\mc{D}_1$) we deform the contours in the double integral associated with $R_0^{(0)}$ to \begin{itemize} \item $ \Dp{}\mc{D}_{\eta_z^{-1}} \times \Dp{}\mc{D}_{\eta_s^{-1}}$ in what concerns the part of the integrand containing $\alpha_+(s)/\alpha_-(z)$; \item $ \Dp{}\mc{D}_{\eta_z} \times \Dp{}\mc{D}_{\eta_s}$in what concerns the part of the integrand containing $\alpha_+(z)/\alpha_-(s)$. \end{itemize} The resulting residue cancels out the pre-factors in \eqref{definition matrice Mkl} leading to \bem M_{k\ell} \; = \; \Oint{ \Dp{}\mc{D}_{\eta_s^{-1}} }{}\hspace{-2mm} \f{\mathrm{d} s }{2i \pi } \Oint{ \Dp{}\mc{D}_{\eta_z^{-1}} }{} \hspace{-2mm} \f{\mathrm{d} z }{2i \pi } \; \alpha_-(s) \cdot \big( \alpha_+^{-1}(z) \, - \, \alpha_-^{-1}(z)\big)\cdot \big(s^{-h_k} \, - \, s^{-p_k} \big) \cdot \f{ z^{h_{\ell}-1} }{z-s} \\ \; + \; \Oint{ \Dp{}\mc{D}_{\eta_s} }{}\hspace{-2mm} \f{\mathrm{d} s }{2i \pi } \Oint{ \Dp{}\mc{D}_{\eta_z} }{} \hspace{-2mm} \f{\mathrm{d} z }{2i \pi } \; \alpha_+(s)^{-1} \cdot \big( \alpha_+(z) \, - \, \alpha_-(z)\big)\cdot \big(s^{N-h_k} \, - \, s^{N-p_k} \big) \cdot \f{ z^{h_{\ell}-1-N} }{z-s} \; \; + \; \; \e{O}(N^{-\infty}) \;. \label{ecriture formule intermediaire pour Mab} \end{multline} The term $s^{-h_k}$ (resp. $s^{N-h_k}$) do not contribute to the integral as can be seen by deforming the contour of $s$-integration to $\eta_s$ (resp. $\eta_s^{-1}=0$). Further, the first line of \eqref{ecriture formule intermediaire pour Mab} only gives non-vanishing contributions if $p_k \leq 0$ (resp. the last line of \eqref{ecriture formule intermediaire pour Mab} only gives non-vanishing contributions if $p_k \geq N+1$). This yields \eqref{ecriture formule asymptotique matrice M cas lacunaire a ligne}. \rule{2mm}{2mm} \section{The asymptotic expansion of line and row lacunary Toeplitz determinants} \label{Section Toeplitz lacunaire a lignes et colonnes} \subsection{The factorisation in the general case} The factorized representation in the general case depends, in particular, on whether there are some overlaps between the integers parametrising the lacunary line and columns. We thus need a definition so as to be able to distinguish between the different cases. \begin{defin} The sets $ \{ h_a \}_1^{n}$ and $\{ t_b \}_1^r$ with $h_a, t_b \in \intn{1}{N}$ are said to be well-ordered with overlap $c \in \intn{0}{\min\{r,n\} }$ if \beq h_a \; = \; t_a \quad \e{for} \quad a=1,\dots, c \qquad \e{whereas} \qquad \big\{ h_{c+1},\dots, h_n \big\} \cap \big\{ t_{c+1},\dots, t_r \big\} \; = \; \emptyset \;. \label{definition parametre overlap c} \enq \end{defin} It is clear that given two not well ordered sequences $\ell_a$ and $m_a$, one can always relabel the indices of the lacunary integers $\{p_a, h_a, k_b, t_b\}$ so that \eqref{definition parametre overlap c} holds. There is thus no restriction in assuming that the sequences $\ell_a$ and $m_a$ are well ordered, so that we are going to do so in the following. \begin{prop} \label{Proposition factorisation Toeplitz lacunaire a ligne et colonnes} Let $\ell_a$ and $m_a$ be sequences as defined in \eqref{definition de la suite ella}-\eqref{definition de la suite ma} and $f$ a non-vanishing holomorphic function on some open neighbourhood of $\Dp{}\mc{D}_{1}$ such that $\ln f $ is also holomorphic on this neighborhood. Then, the lacunary line and row Toeplitz determinant admits the representation \beq \operatorname{det}_{N} \Big[ c_{\ell_a - m_b }[f] \Big] \; = \; \operatorname{det}_{N}\Big[ c_{a - b }[f] \Big] \cdot \operatorname{det}_{n+r}\big[ \mc{N} \big] \;. \label{ecriture factorisation explicite Toeplitz lacunaire ligne et colonnes} \enq The matrix $\mc{N}$ appearing above admits the blocks structure \beq \mc{N} \; = \; \left( \begin{array}{c c} \mc{N}_{I ; I} & \mc{N}_{I ; II} \\ \mc{N}_{II ; I} & \mc{N}_{II ; II} \end{array} \right) \enq with blocks being given by \beq \big( \mc{N}_{A ; I} \big)_{ab} \; = \; \delta_{A;I} \delta_{ab} \delta_{b>c} \; + \; \Oint{ \Dp{}\mc{D}_1 }{} U_{A;a}(z) \cdot v_{I;b}(z) \cdot \mathrm{d} z \quad \e{and} \quad \big( \mc{N}_{A ; II} \big)_{ab} \; = \; \delta_{A;II} \delta_{ab} \delta_{b \leq c} \; + \; \Oint{ \Dp{}\mc{D}_1 }{} U_{A;a}(z) \cdot v_{II;b}(z) \cdot \mathrm{d} z \label{ecriture entrees matrice N} \enq in which $A \in \{ I , II \}$. The functions $U_{A;a}$ are built in terms of the resolvent $R_0$ to the integral operator $I+V_0$ defined in \eqref{definition noyau integral V0} and of the functions $u_{A;a}$ \beq u_{I;a}(z) \; = \; \f{ f(z) }{ 2i\pi } \cdot z^{\f{N}{2} -p_a} \; - \; \f{ z^{\f{N}{2}-h_{a}} }{ 2i\pi } \cdot \Big(\delta_{a\leq c} \;+ \; \delta_{a>c} f(z) \Big) \qquad \e{and} \qquad u_{II;a}(z)\; = \; \f{ f(z)-1 }{ 2i\pi } \cdot z^{\f{N}{2} - t_a} \label{definition des fonctions u} \enq as $U_{A;a} \; = \; (I-R_0)[u_{A;a}]$. Finally, the function $v_{A;b}$ appearing in \eqref{ecriture entrees matrice N} read \beq v_{I;b}(z) \; = \; \delta_{b \leq c } \cdot z^{k_b - \f{N}{2} -1} \; + \; \delta_{b>c} \cdot z^{h_{b}-1-\f{N}{2}} \qquad \e{and} \qquad v_{II;b}(z)\; = \; - \delta_{b \leq c } \cdot z^{t_b - \f{N}{2} -1} \; + \; \delta_{b>c} \cdot z^{k_{b}-1-\f{N}{2}} \; . \label{definition des fonction v} \enq \end{prop} \noindent {\it Proof --- \ } Let $I+\wh{V}$ be the integral operator acting on $L^{2}(\Dp{}\mc{D}_{1})$ with the kernel \beq \wh{V}(z,s) \; = \; \sul{a=1}{N} \wh{\kappa}_a(z)\cdot \wh{\tau}_a(s) \qquad \e{where} \quad \left\{ \begin{array}{cc} \wh{\tau}_a(z) \; = \; \tf{ z^{a-1-\f{N}{2}} }{(2i\pi)} & a \in \{1,\dots , N \} \setminus \{t_1, \dots, t_r \} \\ \wh{\tau}_{t_a} (z) \; = \; \tf{ z^{k_a-1-\f{N}{2}} }{(2i\pi)} & a = 1, \dots, r \end{array} \right. \enq and \beq \begin{array}{cccc} \wh{\kappa}_a(z) & = & \big( f(z)-1 \big) \cdot z^{\f{N}{2}-a} \; - \; {\displaystyle \sul{s=1}{r}} \delta_{a,t_{s}} z^{\f{N}{2}-k_{s}} & a \in \{1,\dots , N \} \setminus \{h_1, \dots, h_n \} \\ \wh{\kappa}_{h_a}(z) & = & f(z) \cdot z^{ \f{N}{2}-p_a } \; - \; z^{ \f{N}{2}-h_a } \; - \; {\displaystyle \sul{s=1}{r}} \delta_{h_a,t_{s}} z^{\f{N}{2}-k_{s}} & a=1, \dots, n \end{array} \;. \enq It is readily seen that \beq \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I \, + \, \wh{V} \big] \; = \; \operatorname{det}_{N}\Big[ \delta_{ a b } \; + \; \int_{ \Dp{}\mc{D}_{1} }{} \wh{\kappa}_{a}(z) \cdot \wh{\tau}_{b}(z) \cdot \mathrm{d} z \Big] \; = \; \operatorname{det}_{N}\Big[ c_{ \ell_a - m_b}[f] \Big] \;. \enq The kernel $\wh{V}$ can be recast as $\wh{V}=V_0 + \wh{V}_1$ where $V_0$ has been introduced in \eqref{definition noyau integral V0} and $\wh{V}_1$ is the finite rank $n+2r$ perturbation of $V_0$ given by \beq \wh{V}_1\big( z, s \big) \; = \; \sul{ a =1 }{ n } u_{I;a}(z) \cdot v_{I;a}(s) \; + \; \sul{ a =1 }{ r } \Big\{ u_{II;a}(z) \cdot \wt{v}_{II;a}(s) \; + \; u_{III;a}(z) \cdot \wt{v}_{III;a}(s) \Big\} \;. \enq The functions $u_{I,a}$, $u_{II,a}$ and $v_{I,a}$ are as defined in \eqref{definition des fonctions u}-\eqref{definition des fonction v} whereas \beq \wt{v}_{II;b}(z) \; = \; \delta_{b>c}\cdot z^{k_b-\f{N}{2}-1} \; - \; z^{t_b-\f{N}{2}-1} \quad , \quad \wt{v}_{III;b}(z) \; = \; z^{k_b-\f{N}{2}-1} \quad \e{and} \quad \wt{u}_{III;a}(z) \; = \; - \f{ z^{\f{N}{2}-k_a} }{ 2i \pi } \;. \enq Proceeding as in the proof of lemma \ref{Lemme factorisation Toeplitz lacunaire a lignes}, we get that \beq \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I \, + \, V \big] \; = \; \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I\, + \, V_0 \big] \cdot \operatorname{det}_{ \Dp{}\mc{D}_{1} }\big[ I\, + \, (I-R_0)\cdot \wh{V}_1 \big] \; = \; \operatorname{det}_{N}\Big[ c_{ j-k }[b] \Big] \cdot \operatorname{det}_{n+2r}\big[ \mc{M} \big] \enq where $\mc{M}$ is the $(n+2r)\times (n+2r)$ block matrix \beq \mc{M} \; = \; \left( \begin{array}{c c c} \mc{M}_{I ; I} & \mc{M}_{I ; II} & \mc{M}_{I; III} \\ \mc{M}_{II ; I} & \mc{M}_{II ; II} & \mc{M}_{II ; III} \\ \mc{M}_{III ; I} & \mc{M}_{III ; II} & \mc{M}_{III ; III} \end{array} \right) \quad \e{with} \quad \big( \mc{M}_{A;B} \big)_{ab} = \delta_{A,B} \delta_{ab} \; + \; \Oint{ \Dp{} \mc{D}_1 }{} U_{A;a}(z) \cdot v_{B;b}(z) \cdot \mathrm{d} z \; . \enq The upper-case entries $A,B$ belong to $\{I, II, III\}$ whereas the lower-case entries $a,b$ subordinate to the upper-case entry $I$ run from $1$ to $n$ and those subordinate to the upper-case entries $II$ or $III$ run from $1$ to $r$. Since $\wt{u}_{III,a} \in \ker(V_0)$, it follows that $\wt{u}_{III,a} \in \ker(R_0)$. Then, a straightforward calculation shows that, in fact, the block structure of the lines of type III simplifies leading to \beq \mc{M} \; = \; \left( \begin{array}{c c c} \mc{M}_{I ; I} & \mc{M}_{I ; II} & \mc{M}_{I; III} \\ \mc{M}_{II ; I} & \mc{M}_{II ; II} & \mc{M}_{II ; III} \\ \begin{array}{cc} -I_{c} & 0 \\ 0 & 0 \end{array} & \begin{array}{cc} 0 & 0 \\ 0 & -I_{r-c} \end{array} & 0 \end{array} \right) \;. \enq There, $I_k$ refers to the identity matrix in $k$ dimensions. In order to reduce the size of the determinant of $\mc{M}$ it is enough to exchange the first $c$ columns of the block $I$ with the first $c$ ones of the block $III$, and then exchange the $r-c$ last columns of block $II$ with the $r-c$ last columns of the block $III$. This produces, all in all, an overall $(-1)^r$ sign. The latter cancels out with the one issuing from $\operatorname{det}_r[-I_r]$, thus leading to $\operatorname{det}_{n+2r}[\mc{M}] = \operatorname{det}_{n+r}[\mc{N}]$. \rule{2mm}{2mm} \subsection{A special case of a large $N$ asymptotics} Replacing the exact resolvent $R_0$ by its approximate resolvent \eqref{definition resolvent approximatif de V0} in the definition of the matrix entries of $\mc{N}$ \eqref{ecriture factorisation explicite Toeplitz lacunaire ligne et colonnes} leads to exponentially small in $N$ corrections. Upon such a replacement, Proposition \ref{Proposition factorisation Toeplitz lacunaire a ligne et colonnes} basically yields the most general expression for the large-$N$ asymptotics of the ratio \beq \operatorname{det}_{N}\big[ c_{\ell_a - m_b }[f] \big] \cdot \Big( \operatorname{det}_{N}\big[ c_{a - b }[f] \big] \Big)^{-1} \;. \enq Although there should, quite probably, exist a direct transformation that would allow to connect Tracy-Widom's answer to ours (and hence Bump and Diaconis one due to \cite{DehayeProofIdentityBumpDiaconisTracyWidomLacunatyToeplitz}), we have not succeeded in finding it. The formula can, of course, be simplified as soon as one provides some more informations on the lacunary integers $p_a, h_a, k_b, t_b$. Below, we treat a specific example of such a simplification, much in the spirit of the one outlined in Corollary \ref{Corolaire Toeplitz lacunaire à un indice}. In order to state the theorem, we first need to introduce a matrix $\mc{N}^{(\epsilon)}_{A ; B}\big( \mc{J} ; c \big)$ depending on the sets of integers \beq \mc{J} \; = \; \Big\{ \{p_a \} \; ; \; \{h_a \} \; ; \; \{ k_a \} \; ; \; \{t_a\} \Big\} \;. \enq The set $\mc{J}$ parametrizes its entries according to \bem \Big( \mc{N}^{(\epsilon)}_{I ; I}\big( \mc{J} ; c \big) \Big)_{ab} \; = \; -\epsilon \Oint{ \Dp{}\mc{D}_{1} }{} \f{ 1 }{ z(1-\epsilon 0^+)-s } \Bigg\{ \delta_{b \leq c} \delta_{a \leq c} \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} s^{\epsilon h_a-1 } z^{\epsilon h_b-1} \, + \, \delta_{b>c} \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{\epsilon} (s) } \bigg)^{\epsilon} s^{-\epsilon p_a } z^{-\epsilon h_b} \\ \, - \, \delta_{b>c} \delta_{a\leq c} \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} s^{\epsilon h_a-1 } z^{-\epsilon h_b} \Bigg\} \cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \; + \; \epsilon \cdot \delta_{ b \leq c} \Oint{ \Dp{}\mc{D}_{1} }{} \f{ s^{-\epsilon p_a } \cdot z^{\epsilon k_b-1} }{ z(1+\epsilon 0^+)-s } \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{\epsilon} (s) } \bigg)^{\epsilon} \cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \end{multline} \bem \Big( \mc{N}^{(\epsilon)}_{I ; II}\big( \mc{J} ; c \big) \Big)_{ab} \; = \; -\epsilon \Oint{ \Dp{}\mc{D}_{1} }{} \f{ 1 }{ z(1-\epsilon 0^+)-s } \Bigg\{ \delta_{b \leq c} \delta_{a \leq c} \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} s^{\epsilon h_a-1 } z^{- \epsilon t_b} \, + \, \delta_{b > c} \delta_{a \leq c} \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} s^{\epsilon h_a-1 } z^{\epsilon k_b-1} \\ \, - \, \delta_{ b \leq c} \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{\epsilon} (s) } \bigg)^{\epsilon} s^{ -\epsilon p_a } z^{-\epsilon t_b} \Bigg\} \cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \; + \; \epsilon \cdot \delta_{ b > c} \Oint{ \Dp{}\mc{D}_{1} }{} \f{ s^{-\epsilon p_a } \cdot z^{\epsilon k_b-1 } }{ z(1+\epsilon 0^+)-s } \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{\epsilon} (s) } \bigg)^{\epsilon} \cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \end{multline} and finally \beq \Big( \mc{N}^{(\epsilon)}_{II ; I}\big( \mc{J} ; c \big) \Big)_{ab} \; = \; - \epsilon \Oint{ \Dp{}\mc{D}_{1} }{} \f{ s^{\epsilon t_a-1 } }{ z(1-\epsilon 0^+)-s } \Bigg\{ \delta_{b \leq c} \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} z^{\epsilon k_b-1 } \, - \, \delta_{b > c} \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} z^{- \epsilon h_b } \Bigg\}\cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \enq \beq \Big( \mc{N}^{(\epsilon)}_{II ; II}\big( \mc{J} ; c \big) \Big)_{ab} \; = \; - \epsilon \Oint{ \Dp{}\mc{D}_{1} }{} \f{ s^{\epsilon(t_a-1)} }{ z(1-\epsilon 0^+)-s } \Bigg\{ \delta_{b \leq c} \cdot \bigg( \f{ \alpha_{\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} \cdot z^{- \epsilon t_b } \, + \, \delta_{b > c} \cdot \bigg( \f{ \alpha_{-\epsilon} (z) }{ \alpha_{-\epsilon} (s) } \bigg)^{\epsilon} \cdot z^{ \epsilon k_b-1 } \Bigg\}\cdot \f{ \mathrm{d} s \cdot \mathrm{d} z }{ (2i\pi)^2 } \enq The $(1-\epsilon O^{+})$ prescription means that the integral should be understood as the limit when $z$ approaches a point on $\Dp{}\mc{D}_{1}$ from the inside ($\epsilon=+1$) or outside $(\epsilon=-1)$ of the unit disk. \begin{theorem} Assume that the lacunary integers $p_a,h_a$ are given as in \eqref{particules trous pour entiers pa} and \eqref{particuler trous pour entiers ha} and, likewise, that the lacunary integers $k_b, t_b$ are given as : \begin{eqnarray} k_a \; = \; 1-k_a^{-} \quad \e{for} \quad a=1,\dots, r_{-} \qquad & \e{and} & \qquad k_{a+r_-} \; = \; k_a^{+} + N \quad \e{for} \quad a=1,\dots, r_{+} \label{particules trous pour entiers pa} \\ t_a \; = \; t_a^{-} \quad \e{for} \quad a=1,\dots, r_{-} \qquad &\e{and} & \qquad t_{a+r_-} \; = \; N+1- t_a^{+} \quad \e{for} \quad a=1,\dots, r_{+} \;. \label{particuler trous pour entiers ha} \end{eqnarray} Further, let the sets $\{h_a^{+}\}_1^{n_+}$ and $\{k_a^+\}_{1}^{r^+}$ (resp. $\{h_a^{-}\}_1^{n_-} $ and $\{k_a^-\}_{1}^{r^-}$) be well ordered with overlap $c_+$ (resp. $c_-$). Then, provided that the matrices $\mc{N}^{(\pm)}\big( \mc{J}^{(\pm)} ; c_{\pm} \big)$ have maximal rank, one has the asymptotic expansion \beq \operatorname{det}_{N}\big[ c_{\ell_a - m_b }[f] \big] \cdot \Big( \operatorname{det}_{N}\big[ c_{a - b }[f] \big] \Big)^{-1} \; = \; \operatorname{det}_{n^+ + r^+} \big[ \mc{N}^{(+)}\big( \mc{J}^{(+)} ; c_{+} \big) \big] \cdot \operatorname{det}_{n^- + r^-} \big[ \mc{N}^{(-)}\big( \mc{J}^{(-)} ; c_{-} \big) \big] \cdot \Big( 1 + \e{O}\big( N^{-\infty} \big) \Big) \;. \enq The sets $\mc{J}^{(\pm)}$ appearing above are defined as \beq \mc{J}^{(+)} \; = \; \Big\{ \{p_a^+ \}_1^{n_+} \; ; \; \{h_a^+ \}_1^{n_+} \; ; \; \{ k_a^+ \}_1^{r_+} \; ; \; \{t_a^+\}_1^{r_+} \Big\} \qquad \e{and} \qquad \mc{J}^{(-)} \; = \; \Big\{ \{1-p_a^- \}_1^{n_-} \; ; \; \{1-h_a^- \}_1^{n_-} \; ; \; \{1- k_a^- \}_1^{r_-} \; ; \; \{1-t_a^-\}_1^{r_-} \Big\} \;. \nonumber \enq \end{theorem} \noindent {\it Proof --- \ } The representation obtained in proposition \ref{Proposition factorisation Toeplitz lacunaire a ligne et colonnes} is invariant under a permutation of the integers $h_a, t_a$ \textit{along} with a simultaneous permutation of the associated integers $p_a, k_a$. Hence, reorganising the entries of the matrix $\mc{N}$ in each block so that the natural order imposed by the $\pm$ splitting of the lacunary integers is respected and a repeated application of the manipulations outlined in the proof of Theorem \ref{Theorem Asymptotiques Toeplitz lacunaire a lignes} leads to $\operatorname{det}_{n+r}\big[ \mc{N} \big] \; = \; \operatorname{det}_{n+r}\big[ \wh{\mc{N} } \big] $ where \beq \wh{\mc{N} } \; = \; \left( \begin{array}{c c} \begin{array}{cc} \mc{N}_{I ; I}^{(-)}\Big(\mc{J}^{(-)} ; c_- \Big) & 0 \\ 0 & \mc{N}_{I ; I}^{(+)}\Big(\mc{J}^{(+)} ; c_+ \Big) \end{array} & \begin{array}{cc} \mc{N}_{I ; II}^{(-)}\Big(\mc{J}^{(-)} ; c_- \Big) & 0 \\ 0 & \mc{N}_{I ; II}^{(+)}\Big(\mc{J}^{(+)} ; c_+ \Big) \end{array} \\ \begin{array}{cc} \mc{N}_{II ; I}^{(-)}\Big(\mc{J}^{(-)} ; c_- \Big) & 0 \\ 0 & \mc{N}_{II ; I}^{(+)}\Big(\mc{J}^{(+)} ; c_+ \Big) \end{array} & \begin{array}{cc} \mc{N}_{II ; II}^{(-)}\Big(\mc{J}^{(-)} ; c_- \Big) & 0 \\ 0 & \mc{N}_{II ; II}^{(+)}\Big(\mc{J}^{(+)} ; c_+ \Big) \end{array} \end{array} \right) \; + \; \e{O}\big(N^{-\infty}\big) \;. \enq Finally, $ \e{O}\big(N^{-\infty}\big) $ appearing above refers to an $(n+r)\times (n+r)$ matrix whose all entries are a $ \e{O}\big(N^{-\infty}\big) $. It is then enough to exchange the appropriate lines and columns in $\operatorname{det}_{n+r}\big[ \wh{\mc{N} } \big] $ and invoke the maximality of the rank of the matrices $\mc{N}^{(\pm)}$. \rule{2mm}{2mm} \section{Conclusion} In this paper we have proposed a Riemann--Hilbert problem based approach to the analysis of the large-size asymptotic behaviour of lacunary Topelitz determinants having a \textit{finite} number of modified lines an rows. Our approach allows one to obtain an alternative to the ones obtained in \cite{BumpDiaconisLacunaryToeplitzThrougSumsSymFctsAndYoungTableaux,TracyWidomAsymptoticExpansionLacunaryToeplitz} representation for its large-$N$ asymptotics. Our answer involves a determinant that solely depends on the number of modified rows and lines and not on the index of the largest modified line or column. In particular, this allows one to investigate the asymptotics in the case when the locii of some of the modified lines and columns go to infinity. We have treated certain instances of such a situation in the present paper. It is clear from the very setting of our analysis that our method allows one to treat also generalisations of lacunary Toeplitz determinants such as those considered in \cite{LionsToeplitzLacunaires}. To do so, one should simply replace the functions $\wh{\tau}_{t_a}$ and $\wh{\kappa}_{h_a}$ arising in our analysis by more general ones. \section{Asymptotic inversion of $I+V_0$-The Riemann--Hilbert approach} \label{Appendix RHP pour Toeplitz regulier} \subsection{The Riemann--Hilbert problem associated with $I+V_0$} Consider the Riemann--Hilbert problem for a piecewise analytic $2\times 2$ matrix $\chi$ having a jump on the unit circle $\Dp{}\mc{D}_1$: \begin{itemize} \item $\chi$ is analytic on $\ensuremath{\mathbb{C}}\setminus \Dp{}\mc{D}_1$ \; ; \item $\chi(z) \; = \; I_2 \; + \; \f{1}{z} \cdot \e{O}\bigg( \begin{array}{cc} 1 & 1 \\ 1 & 1\end{array} \bigg) \; $ when $z \tend \infty $; \item $\chi$ admits continuous $\pm$-boundary values on $\Dp{}\mc{D}_1 $; \item $\chi_{+}(z) \cdot \left(\begin{array}{cc} 2-f(z) & \big( f(z)-1 \big)\cdot z^N \\ \big( 1 - f(z) \big)\cdot z^{-N} & f(z) \end{array} \right) \; = \; \chi_-(z) \;\; ; \quad z \in \Dp{}\mc{D}_1 \, .$ \end{itemize} In the formulation of the Riemann--Hilbert problem, we have adopted the following notations. Given an oriented Jordan curve $\Gamma \subset \ensuremath{\mathbb{C}}$ and a function $f$ on $\ensuremath{\mathbb{C}}\setminus \Gamma$, $f_{\pm}$ refer to the $\pm$-boundary values of the function $f$ on $\Gamma$ where $+$ (resp. $-$) refers to approaching a point on $\Gamma$ non-tangentially from the left (resp. right) side of the curve. Finally, a matrix domination of the sort $A = \e{O}(B)$ is to be understood entry-wise \textit{viz} $A_{jk}=\e{O}(B_{jk})$. We also remind that the unit circle $\Dp{}\mc{D}_1$ is oriented canonically (\textit{ie} the $+$ side of the contour corresponds to the interior of the circle). It is a standard fact that the above Riemann--Hilbert problem admits a unique solution. \subsection{Transformation to a perturbatively solvable Riemann--Hilbert problem for $\Upsilon$} We now define a new matrix $\Upsilon$ according to Fig.~\ref{Contour du RHP pour Y}, \textit{ie} \begin{itemize} \item $\Upsilon=\chi \, \alpha^{\sigma_3}\, , $ for $z$ being in the exterior of $\Gamma_{\e{ext}}$ and the interior of $\Gamma_{\e{int}}\; ;$ \item $\Upsilon=\chi \, \alpha^{\sigma_3} M_{\e{ext} }^{-1}\, , $ for $z$ between $\Gamma_{\e{ext}}$ and $ \Dp{}\mc{D}_{1}\; ;$ \item $\Upsilon=\chi \, \alpha^{\sigma_3} M_{ \e{int}}\, , $ for $z$ between $\Gamma_{\e{int}}$ and $ \Dp{}\mc{D}_{1} $. \end{itemize} Here $\alpha$ is as defined in \eqref{definition facteur alpha solution RHP scalaire}. It is readily seen that it solves the scalar RHP \beq \alpha\; \e{analytic}\; \e{on} \; \ensuremath{\mathbb{C}}\setminus \Dp{}\mc{D}_1 \quad \alpha_- = f \alpha_+ \; ,\qquad \e{on} \; \; \Dp{}\mc{D}_1 \quad \alpha (z) \tend 1 \; \e{when}\; z \tend \infty \;. \enq The matrices $M_{\e{int}/\e{ext}}$ appearing in the definition of $\Upsilon$ read \beq M_{ \e{int} }(z) \; = \; \left(\begin{array}{cc} 1& \big( 1-f^{-1}(z) \big)\alpha^{-2}(z) \cdot z^N \\ 0& 1 \end{array}\!\! \right), \quad M_{\e{ext}}(z) \; = \; \left(\begin{array}{cc} 1 & 0\\ \big( f^{-1}(z) - 1) \alpha^{2}(z)\cdot z^{-N} & 1 \end{array} \!\!\right)\; . \enq The curves $\Gamma_{\e{ext}}$ and $\Gamma_{\e{int}}$ are chosen in such a way that they are located inside of the open neighbourhood of $\Dp{}\mc{D}_1$ on which $f$ is holomorphic. \begin{figure}\label{Contour du RHP pour Y} \end{figure} One readily sees that $\Upsilon$ satisfies the RHP \begin{itemize} \item $\Upsilon$ is analytic in $\ensuremath{\mathbb{C}}\setminus \Gamma_{\Upsilon}\; $; \item $\Upsilon(z) \; = \; I_2 \; + \; \f{1}{z} \cdot \e{O}\bigg( \begin{array}{cc} 1 & 1 \\ 1 & 1\end{array} \bigg) \; $ when $z \tend \infty $; \item $\Upsilon$ admits continuous $\pm$-boundary values on $\Gamma_{\Upsilon}$; \item $\Upsilon_{+}(z) \cdot G_{\Upsilon}(z) \; = \; \Upsilon_-(z) \;, \quad z \in \Gamma_{\Upsilon} $ \; \; where \; \; $G_{\Upsilon}(z) \; = \; M_{ \e{ext} }(z) \cdot \bs{1}_{ \Gamma_{\e{ext}} }(z) \; + \; M_{ \e{int} }(z)\cdot \bs{1}_{ \Gamma_{\e{int}} }(z) $. \end{itemize} and $\bs{1}_A$ stands for the indicator function of the set $A$. Since \beq \norm{ G_{\Upsilon}\, - \, I_2 }_{ L^{\infty}\big(\Gamma_{\Upsilon}\big) } \; + \; \norm{ G_{\Upsilon}\, - \, I_2 }_{ L^{1}\big(\Gamma_{\Upsilon}\big) } \; + \; \norm{ G_{\Upsilon}\, - \, I_2 }_{ L^{2}\big(\Gamma_{\Upsilon}\big) } \; \leq \; C_1 \ex{- \kappa N} \enq for some constants $C_1>0$ and $\kappa>0$, it follows from the equivalence of the Riemann--Hilbert problem for $\Upsilon$ with the singular integral equation satisfied by $\Upsilon_{+}$ that, for any compact $K\supset \Gamma_{\Upsilon}$, \beq \norm{ \Upsilon \, - \, I_2 }_{ L^{\infty}\big(\ensuremath{\mathbb{C}}\setminus K \big) } \; \leq \; C_1^{\prime} \ex{- \kappa N} \enq for some new constant $C_1^{\prime} >0$. \subsection{The resolvent operator and factorisation of the determinant} It is well known \cite{ItsIzerginKorepinSlavnovDifferentialeqnsforCorrelationfunctions} that the resolvent kernel $R_0$ associated with the integrable integral operator $I+V_0$ takes the form \beq R_0(z,s) \; = \; \f{ \big(\bs{F}_L (z) , \bs{F}_{R}(z) \big) }{ z-s } \enq where given vector $\bs{x},\bs{y}$, $(\bs{x},\bs{y})=x_1y_1+x_2y_2$ and \beq \bs{F}_L^{\bs{T}}(z) \; = \; \bs{E}_L^{\bs{T}}(z)\cdot \chi^{-1}(z) \qquad \bs{F}_R(z) \; = \; \chi(z )\cdot \bs{E}_R(z) \enq where the two-dimensional vectors $\bs{E}_R(z), \bs{E}_L(z)$ take the form \beq \bs{E}_L^{\bs{T}}(z) \; = \; \big( f(z)-1 \big)\cdot \Big( - z^{-\f{N}{2}} \, , \, z^{\f{N}{2}} \Big) \qquad \e{and} \qquad \bs{E}_R^{\bs{T}}(z) \; = \; \f{ 1 }{ 2i\pi }\cdot \Big( z^{\f{N}{2}} \, , \, z^{-\f{N}{2}} \Big) \;. \enq It follows from the factorisation of $\chi$ that $\bs{F}_R$ can be recast as \beq \bs{F}_R(z) \; = \; \bs{F}_R^{(0)}(z) \; + \; \bs{F}^{(\infty)}_R(z) \qquad \e{with} \quad \left\{ \begin{array}{ccc} \bs{F}_R^{(0)}(z) &= & M_{\e{int};+}^{-1}(z) \cdot \alpha_+^{-\sigma_3}(z) \cdot \bs{E}_R(z) \\ \bs{F}_R^{(\infty)}(z) &= & \big( \Upsilon-I_2 \big) \cdot M_{\e{int};+}^{-1}(z) \cdot \alpha_+^{-\sigma_3}(z) \cdot \bs{E}_R(z) \end{array} \right. \:. \enq The uniform bounds on $\Upsilon - I_2$ ensure that \beq \big\| \big\| \bs{F}_R^{(\infty)} \big\|\big\|_{L^{\infty}(\msc{C}) } \; = \; C\cdot\ex{-\kappa N} \enq for some $C>0$. Thus, for $z \in \msc{C}$, \beq \bs{F}_R^{(0)}(z) \; = \; \f{1}{2i\pi} \cdot \left( \begin{array}{c} z^{\f{N}{2}} \alpha_{-}^{-1}(z) \\ z^{-\f{N}{2}} \alpha_{+}(z) \end{array} \right) \quad \e{and} \quad \bs{E}_L^{(0)}(z) \; = \; \big( b(z) -1 \big) \cdot \left( \begin{array}{c} - z^{-\f{N}{2}} \alpha_{+}(z) \\ z^{ \f{N}{2} } \alpha_{-}^{-1}(z) \end{array} \right) \enq As a consequence, the resolvent kernel $R_0$ decomposes exactly as given in \eqref{ecriture decomposition resolvant en partie finie et perturbation exp petite en N}. \end{document}	arXiv
What is the sum of a sinc function series sampled periodically What is the sum of the following sinc series? $$f(k,y)=\sum_{n=-\infty}^{\infty} \frac{\text{sin} \pi(kn-y)}{\pi(kn-y)}$$ where $k$ is an integer greater then zero This question is a generalisation of What is the sum over a shifted sinc function? and not a duplicate. The former question only considers the case for $k=1$ signal-processing Renato Faraone $\begingroup$ Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. $\endgroup$ – José Carlos Santos May 5 '18 at 9:36 $\begingroup$ Possible duplicate of What is the sum over a shifted sinc function? $\endgroup$ – mol3574710n0fN074710n May 5 '18 at 9:40 $\begingroup$ As I explained this is not a duplicate but a genertalisation of math.stackexchange.com/questions/1242280/… $\endgroup$ – Daniel May 5 '18 at 10:16 $\begingroup$ By checking my previous answer you'll see that the series converges when $\displaystyle k \in \left(0,2\right)$. Otherwise, you can always write $\displaystyle k$ as $\displaystyle 2\left\lfloor{k \over 2}\right\rfloor + 2\left\{{k \over 2}\right\}$ for any other $\displaystyle k$-value. $\endgroup$ – Felix Marin May 5 '18 at 23:18 $\begingroup$ @FelixMarin I am only interested in k being an integer, and can't relate well what you are saying with what I expect. I don't think the answer is a constant, I am expecting an answer function of $k$ and $y$ $\endgroup$ – Daniel May 6 '18 at 10:04 I formulate a general solution here to compute $\sum_{n=-\infty}^{n=\infty}\frac{\sin(N\pi(y-\frac{n}{T}))}{\pi(y-\frac{n}{T})}$ for $T<N,$ with $T,N\in\mathbb{\mathbb{R}}^{+}$, and then use the case where $T=1/k$ for an integer $k>1$ and $N=1$ to show that the sum requested in this problem, $f(k,y)\equiv\sum_{n=-\infty}^{\infty}g(kn-y)=\sum_{n=-\infty}^{\infty}g(y-kn)$ for the (even) sinc function $g(y)=\begin{cases} \frac{sin\pi y}{y} & y\not\neq0\\ 1 & y=0 \end{cases}$ is equal to: $f(k,y)=\begin{cases} k & \frac{y}{k}\in\mathbb{Z}\\ \cos(\pi y)+\frac{\sin(\pi y(1-1/k))}{\sin(\pi y/k)} & \frac{y}{k}\notin\mathbb{Z},\,k\,\mathrm{even}\\ \frac{\sin(\pi y)}{\sin(\pi y/k)} & \frac{y}{k}\notin\mathbb{Z},\,k\,\mathrm{odd} \end{cases}$ Let $W_{N}(t)$ be the rectangular window of width $N$ defined in continuous time given by $W_{N}(t)=\begin{cases} 1 & -N/2<t<N/2\\ 1/2 & \|t\|=N/2\\ 0 & \textrm{else} \end{cases}$ where $t$ is in units of seconds. Its Fourier transform is then the (continuous, aperiodic) sinc function: $\hat{W}_{N}(y)=\begin{cases} \frac{\sin(N\pi y)}{\pi y} & y\neq0\\ N & y=0 \end{cases}$ where the units of $y$ is hz (cycles per second). Sampling $W_{N}(t)$ at discrete intervals of $T$ seconds, yields a discrete, aperiodic signal, $w_{N}(t)=\begin{cases} 1 & -N/2<t<N/2\\ 1/2 & \|t\|=N/2\\ 0 & \textrm{else} \end{cases},t\in\mathbb{Z}T.$ The (continuous, periodic) Discrete Time Fourier Transform of this, $\hat{w}_{N}(y)$ can be derived as folllows: Directly from the definition of DTFT, and making the change of variable $n=t/T$, we have $w_{N}(n)=\begin{cases} 1 & -N/2T<n<N/2T\\ 1/2 & \|n\|=N/2T\\ 0 & \textrm{else} \end{cases},n\in\mathbb{Z}.$ For $yT\notin\mathbb{Z}$ and $N/T$ is an even integer (implying that $N/2T$ is an integer value, and so $w_{N}(n)$ is evaluated at the endpoints), then: \begin{eqnarray} \hat{w}_{N}(y) & = & \sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi yTn}=\frac{1}{2}\left(e^{-iN\pi y}+e^{iN\pi y}\right)+\sum_{n=-\frac{N-2T}{2T}}^{\frac{N-2T}{2T}}e^{-i2\pi yTn}=\\ & =\frac{1}{2}\left(e^{-iN\pi y}+e^{iN\pi y}\right)+ & \frac{e^{i\pi(N-2T)y}-e^{-i\pi Ny}}{1-e^{-i2\pi yT}}=\frac{e^{-i\pi yT}}{e^{-i\pi yT}}\left(\frac{e^{i\pi y(N-T)}-e^{-i\pi y(N-T)}}{e^{i\pi yT}-e^{-i\pi yT}}\right)\\ & =\cos(\pi yN)+ & \frac{\cos(\pi y(N-T))+i\sin(\pi y(N-T))-(\cos(\pi y(N-T))-i\sin(\pi y(N-T)))}{\cos(\pi yT)+i\sin(\pi yT)-(\cos(\pi yT)-i\sin(\pi yT))}\\ & = & \cos(\pi yN)+\frac{2i\sin(\pi y(N-T))}{2i\sin(\pi yT)}=\cos(\pi yN)+\frac{\sin(\pi y(N-T))}{\sin(\pi yT)} \end{eqnarray} (where the geometric series sum formula used above relies on the assumption that $yT\notin\mathbb{Z}$ so that $e^{-i2\pi yT}\neq1.$ ) If $N/T$ is an even integer but we now assume $yT\in\mathbb{Z}$, then we can write $y=\frac{l}{T}$ for some $l\in\mathbb{Z}$ and in this case we have that \begin{eqnarray} \hat{w}_{N}(y) & = & \sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi yTn}=\sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi ln}=\sum_{n=-\infty}^{\infty}w_{N}(n)\\ & = & \sum_{n=-\frac{N}{2T}}^{\frac{N}{2T}}w_{N}(n)=\frac{N}{T} \end{eqnarray} If $yT\notin\mathbb{Z}$ and $N/T$ is not even, then, letting $F\equiv Floor\left[N/2T\right]$ and $R=N/Tmod2$, we have that \begin{eqnarray} \hat{w}_{N}(y) & = & \sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi yTn}=\sum_{n=-F}^{F}e^{-i2\pi yTn}=\sum_{t=-(N-RT)/2T}^{(N-RT)/2T}\left(e^{-i2\pi yT}\right)^{n}\\ & & =\frac{e^{i\pi(N-RT)y}-e^{-i\pi(N-RT+2T)y}}{1-e^{-i2\pi yT}}=\frac{e^{-i\pi yT}}{e^{-i\pi yT}}\left(\frac{e^{i\pi y(N-RT+T)}-e^{-i\pi y(N-RT+T)}}{e^{i\pi yT}-e^{-i\pi yT}}\right)\\ & & =\frac{\cos(\pi y(N+T(1-R)))+i\sin(\pi y(N+T(1-R)))-(\cos(\pi y(N+T(1-R)))-i\sin(\pi y(N+T(1-R)))}{\cos(\pi yT)+i\sin(\pi yT)-(\cos(\pi yT)-i\sin(\pi yT))}\\ & & =\frac{2i\sin(\pi y(N+T(1-R)))}{2i\sin(\pi yT)}=\frac{\sin(\pi y(N+T(1-R))}{\sin(\pi yT)}\\ \end{eqnarray} (Note that in the case of $N/T$ integer valued, this gives $R=1$ and so the above would simplify to $\frac{\sin(\pi yN)}{\sin(\pi yT)}$). If $N/T$ is not an even integer but we now assume $yT\in\mathbb{Z}$, then we again write $y=\frac{l}{T}$ and in this case we have that \begin{eqnarray} \hat{w}_{N}(y) & = & \sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi yTn}=\sum_{n=-\infty}^{\infty}w_{N}(n)e^{-i2\pi ln}=\sum_{n=-\infty}^{\infty}w_{N}(n)\\ & = & \sum_{n=-\frac{N-RT}{2T}}^{\frac{N-RT}{2T}}w_{N}(n)=\frac{N-RT}{T}+1=\frac{N+T(1-R)}{T} \end{eqnarray} We also know from the Poisson summation formula, that $\hat{w}_{N}(y)$ is equal to the periodic summation of $\hat{W}_{N}(y)$ at periods of $1/T$: $$ \hat{w}_{N}(y)=\sum_{n=-\infty}^{n=\infty}\hat{W}_{N}(y-\frac{n}{T})=\begin{cases} N\sum_{n=-\infty}^{n=\infty}g\left(N(y-\frac{n}{T})\right)\end{cases} $$ as the sum of a periodically sinc function was defined above. This is precisely the definition of $f(k,y)$ above for $k=1/T$ and $N=1$, so plugging these into the general solution of DTFT(y)=\begin{cases} N/T & yT\in\mathbb{Z},\,N/T\,\mathrm{even}\\ \frac{N+T(1-R)}{T} & yT\in\mathbb{Z},\,N/T\,\mathrm{not\,even}\\ \cos(\pi yN)+\frac{\sin(\pi y(N-T))}{\sin(\pi yT)} & yT\notin\mathbb{Z},\,N/T\,\mathrm{even}\\ \frac{\sin(\pi y(N+T(1-R))}{\sin(\pi yT)} & yT\notin\mathbb{Z},\,N/T\,\textrm{not}\,\mathrm{even} \end{cases} where $R=NT/P\,\textrm{mod}\,2$ yields the summation is as given above. Lauren DeasonLauren Deason $\begingroup$ I like this answer a lot as it gives a simple answer, but I am a bit "surprised" that it is much simpler than the other answers $\endgroup$ – Daniel May 15 '18 at 8:02 $\begingroup$ I believe the edited solution is now correct. I had previously defined the rectangular window to equal 1 on the interval [-N/2,N/2] rather than (-N/2,N/2) with value 1/2 at +/-N/2. With this correction, spot checks for values of k and y for which the sum can easily be computed explicitly check out; please let me know if you find any counterexamples or mistakes in the above. $\endgroup$ – Lauren Deason May 16 '18 at 3:54 $$f(k,y)=\sum_{n=-\infty}^{\infty} \frac{\sin \pi(kn-y)}{\pi(kn-y)}=\cos(\pi y)\sum_{n=-\infty}^{\infty} \frac{\sin (\pi kn)}{\pi(kn-y)}-\sin(\pi y)\sum_{n=-\infty}^{\infty} \frac{\cos (\pi kn)}{\pi(kn-y)}$$ $\sin(\pi kn)=0\quad$ and $\quad\cos(\pi kn)=(-1)^{kn}$ $$f(k,y)=-\sin(\pi y)\sum_{n=-\infty}^{\infty} \frac{(-1)^{kn}}{\pi(kn-y)}= \frac{\sin(\pi y)}{\pi k}\sum_{n=-\infty}^{\infty} \frac{(-1)^{kn}}{n-\frac{y}{k}}$$ $$\sum_{n=0}^{\infty} \frac{(-1)^{kn}}{n-\frac{y}{k}}=\Phi\left((-1)^k\:,\:1\:,\:-\frac{y}{k}\right)$$ $\Phi(z,s,a)$ is the Lerch function : http://mathworld.wolfram.com/LerchTranscendent.html $$\sum_{n=-\infty}^{0} \frac{(-1)^{kn}}{n-\frac{y}{k}}= \sum_{m=0}^{\infty} \frac{(-1)^{-km}}{-m+\frac{y}{k}} =-\Phi\left((-1)^k\:,\:1\:,\:\frac{y}{k}\right)$$ and the term for $n=0$ is equal to$\quad\frac{1}{0-\frac{y}{k}}=-\frac{k}{y}$ $$\sum_{n=-\infty}^{\infty} \frac{(-1)^{kn}}{n-\frac{y}{k}}=\Phi\left((-1)^k\:,\:1\:,\:-\frac{y}{k}\right)-\Phi\left((-1)^k\:,\:1\:,\:\frac{y}{k}\right)-\left(-\frac{k}{y}\right)$$ $$f(y,k)=-\frac{\sin(\pi y)}{\pi k}\left(\Phi\left((-1)^k\:,\:1\:,\:-\frac{y}{k}\right)-\Phi\left((-1)^k\:,\:1\:,\:\frac{y}{k}\right)+\frac{k}{y}\right)$$ If $\phi(t)$ is arbitrary function and $\Phi(\omega)$ its Fourier transform, then the following identity, known as Poisson's formula is true (see [1] pg.47): $$ \sum^{\infty}_{n=-\infty}\phi(t+nT)=\frac{1}{T}\sum^{\infty}_{n=-\infty}e^{2\pi i nt/T}\Phi\left(\frac{2\pi n}{T}\right)\textrm{, }T>0. $$ In the case of your's, the function $\phi(t)=\frac{\sin(at)}{t}$ have Fourier transform $\Phi(t)=\pi X_{[-a,a]}(t)$ ($X_{[-a,a]}(t)$ is the characteristic function in $[-a,a]$, $a>0$). Hence we get $$ \sum^{\infty}_{n=-\infty}\frac{\sin(a(t+n T))}{t+nT}=\frac{\pi}{T}\sum^{\infty}_{n=-\infty}e^{2\pi i n t/T}X_{[-a,a]}\left(\frac{2\pi n}{T}\right), $$ where $$ X_{[-a,a]}\left(\frac{2\pi n}{T}\right)=\left\{ \begin{array}{cc} 1,\textrm{ if }\left\|\frac{2\pi n}{T}\right\|\leq a\textrm{, for a certain }n\in\textbf{Z}\\ 0,\textrm{ otherwise } \end{array}\right\}. $$ Consequently we have $$ \sum^{\infty}_{n=-\infty}\frac{\sin(a(t+n T))}{t+nT}=\frac{\pi}{T}\sum_{\|n\|\leq aT/(2\pi)}e^{2\pi i n t/T}. $$ Hence if for example $a=5/2$, $T=3$, then $n=-1,0,1$ and $$ \sum^{\infty}_{n=-\infty}\frac{\sin(a(t+n T))}{t+nT}=\sum^{\infty}_{n=-\infty}\frac{\sin(\frac{5}{2}(t+2n))}{t+2n}= $$ $$ =e^{2\pi i (-1) t/3}+e^{2\pi i 0 t/3}+e^{2\pi i 1 t/3}=\frac{\pi}{3}\left(1+2\cos\left(\frac{2\pi t}{3}\right)\right). $$ Note that the Fourier transform $\Phi(x)$ of $\phi(t)$ is $$ \Phi(x)=\int^{\infty}_{-\infty}\phi(t)e^{-itx}dt $$ [1]: Athanasios Papoulis. ''The Fourier Integral and its Applications''. McGraw-Hill Book Company, Inc. United States of America, 1962 Nikos Bagis Nikos Bagis We have: $$f(k,y)=\sum_{n=-\infty}^{\infty} \frac{\text{sin} \pi(kn-y)}{\pi(kn-y)}$$ We consider first that $k$ is even, that means $\sin{(\pi kn-\pi y)}=\sin{(-\pi y)}$ no matter $n$ $f(k,y)=\frac{\sin(-\pi y)}{\pi}\sum_{n=-\infty}^{\infty} \frac{1}{(kn-y)}$ Let $\delta(n)=\frac{1}{(kn-y)}$, when $n=0$, $\delta=\frac{-1}{y}$ When $n$ is different than zero, $\delta(n)=\frac{1}{kn-y}$ if $n>0$, otherwise $\delta(x)=\frac{-1}{kx+y}$ where $x$ is positive and $n=-x$ which means, $\sum_{n=-\infty}^{\infty} \delta(n)=\frac{-1}{y}+\sum_{n=1}^{\infty}(\frac{1}{kn-y}-\frac{1}{kn+y})$ $$\sum_{n=1}^{\infty}(\frac{1}{kn-y}-\frac{1}{kn+y})=\left [\frac{1}{k}(ln(n-y/k)-ln(n+y/k))\right ]^\infty_1=\left [ \frac{1}{k}ln(\frac{n-y/k}{n+y/k})\right ]^\infty_1=\frac{1}{k}(ln1-ln(1-y/k)+ln(1+y/k))$$ The quantity is positive since $ln(k+y)>ln(k-y)$, that leads to: $$f(k,y)=\frac{\sin(-\pi y)}{\pi}(\frac{-1}{y}+\frac{1}{k}ln(1+y/k)-\frac{1}{k}ln(1-y/k))$$ When $k$ is odd that leaves us with two sub-cases, $n=2l$ and $n=2l+1$. Cases where $n=2l$ we call $f_{even}(k,y)=\frac{\sin(-\pi y)}{\pi}\sum_{n=-\infty}^{\infty} \frac{1}{(2kn-y)}$ because $\sin{(\pi kn-\pi y)}=\sin{(-\pi y)}$, Which gives the same first formula with $n$ substitued by $2n$. If $n=2l+1$ , $\sin{(\pi kn-\pi y)}=\sin{(\pi y)}$ because $kn$ is odd. In such case: $f_{odd}=\frac{\sin(\pi y)}{\pi}\sum_{n=1}^{\infty} (\frac{1}{k(2n+1)-y}-\frac{1}{k(2n+1)+y})$ Merging both cases; $f=f_{odd}+f_{even}$$$=\sin(-\pi y)\left [ \frac{-1}{y}+\frac{1}{2k}ln(\frac{n-y/2k}{n+y/2k})\right ]^\infty_1+\sin(\pi y)\left [ \frac{1}{2k}ln(\frac{n+(k-y)/2k}{n+(k+y)/2k})\right ]^\infty_0$$$$=\frac{\sin(-\pi y)}{\pi}(\frac{-1}{y}+\frac{1}{2k}ln(1+y/2k)-\frac{1}{2k}ln(1-y/2k))+\frac{\sin(\pi y)}{\pi}(\frac{1}{2k}ln(\frac{k+y}{2k})-\frac{1}{2k}ln(\frac{k-y}{2k}))$$ Abr001amAbr001am Not the answer you're looking for? Browse other questions tagged signal-processing or ask your own question. What is the sum over a shifted sinc function? Summability of a sinc function power 'p', where 1<p<2 What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula? Using DTFT to find the sum of $\sum_{n=-\infty }^{\infty }\text{sinc}(n\alpha_1)\text{sinc}(n\alpha_2)$ Problems with derivative of $\operatorname{sinc}$ when trying to interpolate in Matlab Shannon reconstruction formula: who can prove alternative form?	CommonCrawl
\begin{definition}[Definition:Mathematical Model] A '''mathematical model''' is an equation, or a system of equations, whose purpose is to provide an approximation to the behavior of a real-world phenomenon. \end{definition}	ProofWiki
\begin{definition}[Definition:Differential Equation/Autonomous] A '''differential equation''' is '''autonomous''' if none of the derivatives depend on the independent variable. The $n$th order '''autonomous differential equation''' takes the form: :$y^{\paren n} = \map f {y, y', y'', \dots, y^{\paren {n - 1} } }$ \end{definition}	ProofWiki
Linearity In mathematics, the term linear is used in two distinct senses for two different properties: • linearity of a function (or mapping ); • linearity of a polynomial. An example of a linear function is the function defined by $f(x)=(ax,bx)$ that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables $X,$ $Y$ and $Z$ is $aX+bY+cZ+d.$ Linearity of a mapping is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are nonlinear. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. Linearity of a polynomial means that its degree is less than two. The use of the term for polynomials stems from the fact that the graph of a polynomial in one variable is a straight line. In the term "linear equation", the word refers to the linearity of the polynomials involved. Because a function such as $f(x)=ax+b$ is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context. The word linear comes from Latin linearis, "pertaining to or resembling a line". In mathematics Linear maps In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties:[1] • Additivity: f(x + y) = f(x) + f(y). • Homogeneity of degree 1: f(αx) = α f(x) for all α. These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in general be an element of any vector space. A more special definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics (see below). Additivity alone implies homogeneity for rational α, since $f(x+x)=f(x)+f(x)$ implies $f(nx)=nf(x)$ for any natural number n by mathematical induction, and then $nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)$ implies $f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)$. The density of the rational numbers in the reals implies that any additive continuous function is homogeneous for any real number α, and is therefore linear. The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and other operators constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions. Linear polynomials Main articles: Linear equation and Linear algebra In a different usage to the above definition, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a straight line.[2] Over the reals, a simple example of a linear equation is given by: $y=mx+b\ $ where m is often called the slope or gradient, and b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis. Note that this usage of the term linear is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if the constant term – b in the example – equals 0. If b ≠ 0, the function is called an affine function (see in greater generality affine transformation). Linear algebra is the branch of mathematics concerned with systems of linear equations. Boolean functions Main article: Parity function In Boolean algebra, a linear function is a function $f$ for which there exist $a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}$ such that $f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})$, where $b_{1},\ldots ,b_{n}\in \{0,1\}.$ Note that if $a_{0}=1$, the above function is considered affine in linear algebra (i.e. not linear). A Boolean function is linear if one of the following holds for the function's truth table: 1. In every row in which the truth value of the function is T, there are an odd number of Ts assigned to the arguments, and in every row in which the function is F there is an even number of Ts assigned to arguments. Specifically, f(F, F, ..., F) = F, and these functions correspond to linear maps over the Boolean vector space. 2. In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the truth value of the function is F, there are an odd number of Ts assigned to arguments. In this case, f(F, F, ..., F) = T. Another way to express this is that each variable always makes a difference in the truth value of the operation or it never makes a difference. Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions. Physics In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.[3] Linearity of a homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination af + bg is, too. In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain absolute threshold number of photons. Linear motion traces a straight line trajectory. Electronics In electronics, the linear operating region of a device, for example a transistor, is where an output dependent variable (such as the transistor collector current) is directly proportional to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, and linear amplifiers in general. In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.[4] Integral linearity For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:[5][6] There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics. See also • Linear actuator • Linear element • Linear foot • Linear system • Linear programming • Linear differential equation • Bilinear • Multilinear • Linear motor • Linear A and Linear B scripts. • Linear interpolation References 1. Edwards, Harold M. (1995). Linear Algebra. Springer. p. 78. ISBN 9780817637316. 2. Stewart, James (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8, Section 1.2 3. Evans, Lawrence C. (2010) [1998], Partial differential equations (PDF), Graduate Studies in Mathematics, vol. 19 (2nd ed.), Providence, R.I.: American Mathematical Society, doi:10.1090/gsm/019, ISBN 978-0-8218-4974-3, MR 2597943, archived (PDF) from the original on 2022-10-09 4. Whitaker, Jerry C. (2002). The RF transmission systems handbook. CRC Press. ISBN 978-0-8493-0973-1. 5. Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity" (PDF). analogZONE. Archived from the original (PDF) on February 4, 2012. Retrieved September 24, 2014. 6. Kolts, Bertram S. (2005). "Understanding Linearity and Monotonicity". Foreign Electronic Measurement Technology. 24 (5): 30–31. Retrieved September 25, 2014. External links • The dictionary definition of linearity at Wiktionary	Wikipedia
Secondary Curriculum Linked High Jumping How high can a high jumper jump? How can a high jumper jump higher without jumping higher? Read on... This is our collection of favourite mathematics and sport materials. Speed-time Problems at the Olympics Have you ever wondered what it would be like to race against Usain Bolt? Track Design Well done to Daniel from Kings School, New Zealand, Josh from Chatham Grammar School for Boys, and Anna from the Sandon School, who all correctly worked out the dimensions of the running track. Here is Josh's solution: Radius of the curved sections: As the straight sections of the track are both 85m long, 170m of the track is straight. That means that the remaining 230m of the track is comprised of the two semicircular sections. As there are two semicircular sections, each of equal radii, the sum of their inside edges (of the inside lanes) is equal to the circumference of a circle that would fit neatly into the inside curves of the track. The circumference of a circle, $c$, is given by the equation: $c = 2 \times \pi \times r$, where $r$ is the radius. Therefore, as we have deduced that the circumference of this imaginary circle is equal to the sum of the inside curves of the track, we can write: $230 = 2 \times \pi \times r$, so $r = \frac{230}{2\pi} = 36.606$ metres (to 3 d.p.) This measurement is only the measurement of the radii of the semicircles which comprise the inside edge of the inside lane on the curved section. We want to find a measurement for the radii of the semicircles which comprise the whole of both curved sections; there are 8 lanes that we must consider in order to do this. As each lane is a constant 1.25 metres, the sum of the thickness of the lanes is equal to 8 x 1.25 = 10 metres. Therefore, the radius of the curved sections, in order to produce a scale drawing, is equal to: 36.606 + 10 = 46.606 metres (to 3 d.p.) The 200m Staggered Start: As it is mentioned in the problem for the 400m start that the measurement for the length of a given lane is that of its inside edge, I will assume that it is the inside edges of each lane that must be made constant in the 200m race in order to make the race fair. The runner in lane 1 starts at the curved section on the bottom right of the track, so I will use that lane for comparison with the others. The straights do not need to be compensated for in the staggered start, as the runners would all run the same distance here. Therefore, we only need to consider the 115m of track that the runners will run on before they hit the straight; in this time, all runners will be running on the curved section of track. The runner in lane 2 is running on a track with an inside edge that is 1.25m further outwards than that of lane 1. Therefore, the radius of runner 2's track is equal to $36.606 + 1.25 = 37.856$ metres. Thus, the circumference of the semicircle that makes up the curved section is equal to: $\frac{2 \pi r}{2} = \pi r = 118.927$ metres This is 3.927 metres longer than the inside edge of lane 1, so the rune in lane 2 will start 3.927 metres in front of the runner in lane 1. We can also generate an nth term sequence for the lanes. The nth term for this sequence is: $(36.606 + 1.25(n-1)) \times \pi$ I arrived at this conclusion because the formula for the arc of a semicircle is pi x r, hence the reason for multiplying the sequence by pi, and the section in brackets is the way to determine the radius of the inside edge of any given track. Each runner, as we move from lane 1 to lane 8, will start 3.927 metres (3 d.p.) in front of the previous runner. The obvious conclusion for this problem would be to say that each runner starts twice as far behind the runner in the next inside lane as they did in the 200m race, but I will investigate this mathematically. Each runner is running 400 metres, but 170 metres of the track they will run (85 x 2) is made up of straight sections, where the runners will run the same distance regardless of the lane they are in, so this does not need to be compensated for. Thus, 230m of the track they will run on is comprised of a curved section, which must be compensated for. When the runner in lane 2 reaches a curved section, the radius of the inner edge of his/her lane is equal to $36.606 + 1.25 = 37.856$ metres. Thus, the arc lengths of both semicircles together is equal to: $2 \times \pi \times 37.856 = 237.854$ metres (3 d.p.) This is 7.854 metres longer than the length of curved track in lane 1, so the runner in lane 2 will start 7.854 metres in front of the runner in lane 2. We can also generate an nth term sequence for the lanes. The nth term for this sequence is: $2 \times ((36.606 + 1.25(n-1)) \times pi)$. I arrived at this conclusion using the nth term formula I generated before to calculate the arc length of one of the semicircular pieces of track, then multiplied it by 2 (for the two semicircles involved in the 400m. As we move from lane 1 to lane 8, each runner starts 7.854 metres in front of the runner on the next inside lane to them. Rajeev, from Haberdashers' Aske's Boys' School, sent us his calculations for the staggers presented clearly in a table. You can see it here.	CommonCrawl
I am trying to solve for the allowed wavefunctions and energies for a 1D quartic potential well. where left means integration from x = 0 to x = patch, and right means integration from x = end to x = patch. When I inspect the minimizing function that I find the roots of, I see that it does not cross zero at low energies. I suspect the issue for this is that in the deeper well, the two potential wells can be considered separately because there is so little wavefunction overlap. I would, however, like my solver to be robust enough to find the isolated wavefunctions, or at the very least recognize when the two potential wells can be considered separately. I would also like my solver to solve asymmetric wells, which I am not sure it can do right now. Is there a way I can adapt my code to solve these two issues? Also, here is the MATLAB code reproduced below. The main function is the first code block and is the script. % % the wells are too seperate, consider them as isolated!! %% Two pass patching, now we do it again...having found the optimal patching where the agreement is closest. Necessary? % the division will make it nonzero because zero will go to inf!! While I cannot help you with your specific implementation, I want to point out to an alternative method (as already indicated in a comment to phil's answer) : Marston's "Fourier Grid Hamiltonian" (FGH). The FGH is a pseudo-spectral method and yields a simple recipe for constructing the discretized Hamiltonian matrix for bound systems. As usual, eigenvectors and eigenvalues of the discrete Hamiltonian represent discrete wavefunctions and corresponding energies. For the FGH, the Hamiltonian $H = T+V$ is split in a kinetic energy part $T$ and the potential part $V$. In position space the matrix elements $\left< x\vert V\vert x'\right>$ are just $V(x)\delta(x- x')$, so the potential energy contributes only the Hamiltonian matrix' diagonal. For the matrix elements of the kinetic energy operator, a band-limited (discrete $x_i$!) plane wave basis is used and the matrix elements $\left< x_i\vert T\vert x_j\right>$ are computed analytically. You'll find a great discussion of this and various related Fourier methods in David Tannor's book "Introduction to Quantum Mechanics: a time-dependent perspective". Original references are also given there. You'll find a Jupyter notebook with a simple Python/Cython implementation that tackles a quartic potential here. To change the potential you work with, find the line with pot = lambda x: x*4 - 20x*2 and change it to any other potential you're interested in. Make sure to compute only bound states for which $\left<\psi\vert\psi\right>$ has decayed to zero close to the boundary of your $x$ domain. The standard way to find the eigenvalues of the Schrodinger equation is called "imaginary time propagation". You change the coordinates, t=-i\tau, and integrate in the \tau direction. Any random initial condition will converge to the lowest energy eigenstate. The resulting equation is solved by splitting methods: First propagate the kinetic energy using Fast Fourier Transforms and then propagate the potential. This is possible since the solution decays fast (exponentially) as \|x\| grows and you can effectively replace your boundary conditions by periodic ones. Since you want more eigenstates, there are several procedures: 1: iteration. compute the first eigenstate. Then, you restart the procedure, but in every step, you subtract the projection to this state. E.g. for the second state psi_2, in each step you write psi_2 = psi_2 - psi_1. As a complete alternative, you can discretize the Laplacian using finite differences (or use again FFTs) and then use the inverse power method. given your Hamiltonian H, iterate over n: (H-\lambda^(-1)I)^(-1)\psi_(n+1) = \psi_n for some guess of the eigenvalue \lambda. You can find details on wikipedia. Not the answer you're looking for? Browse other questions tagged finite-difference eigenvalues eigensystem quantum-mechanics or ask your own question.	CommonCrawl
Transitory coping strategies of food-insecure smallholder farmer households: the case of Ilu Gelan District, West Shoa Zone, Oromia Reginal State, Ethiopia Belachew Dessalegn1 Over 960 million people in the world are hungry and undernourished. The majority of these people are found in Asia and Africa. Approximately one-third of the people in sub-Saharan Africa are undernourished. The mechanisms pursued by households differ in several aspects within and between households. Coping strategies are short-term, location-specific actions and adjustments against hazard and activities that take place within existing structures. Before coming to the modernization time, every society around the world has attempted to overcome food shortages at household levels. They practice activities to escape them from food insecurity. The study was aimed at coping strategies among food-insecure smallholder farmer households in Ilu Gelan District, West Shoa Zone, Oromia Regional State, Ethiopia. A cross-sectional study design and mixed data collection methods were employed. Multistage random sampling technique was employed to select 100 sample households for quantitative data and key informant interview, focused group discussion and observation for qualitative data. Data were analyzed using descriptive and inferential statistics. A coping strategy index at household level had been calculated, and inferential statistics was used to test the variability of the index by gender of the household head. The mean coping index was 88.54 and 119.14 for males and females, respectively. Using inferential statistics, equality of means was tested. The t value was − 5.173 for 98° of freedom, and the mean difference was significant (p < 0.001). The study revealed that female-headed households were higher in coping measures and mean of coping strategy index than male-headed households. In the study districts, smallholder farmer households rely on less preferred and less expensive food items. Coping strategy adopted by poor rural households is a shift to poor, and nutritionally lacking diet leads to health-related problems. Global estimates of food-insecure populations stand at 825 million [1] to 850 million [2]. Regional estimates of the food-insecure population include 263 million in South Asia, 268 million in China and Southeast Asia, 2012 million in sub-Saharan Africa (SSA), 60 million in South and Central America and the Caribbean, and 50 million in other regions of the world. Contrary to the United Nations' Millennium Development Goals of cutting hunger by half by 2015, the number of food-insecure populations in the world has been on the rise [3]. The stock of food grains in the world in 2007/2008, the lowest in decades, was only 75 million tons for milled rice and 105 million tons for corn in early 2008 [4]. An estimated 75% of the world's poor (those who live on less than < $2/day income) live in rural areas and depend directly or indirectly on agriculture [5]. Food prices are rising [6], leading to riots in 30 countries around the globe [7, 8]. Share of family income spent on food is estimated at 10% in the USA, 20% in Brazil, 30% in China, 50% in Kenya and 65% in Bangladesh [8], and 9.7 million people food-insecure Ethiopians require relief assistance to meet basic food needs [9]. Thus, the world poor are under great stress, and an increase in food prices is a threat to global peace and stability [3]. The absence of food security associated with coping strategies increasingly being used. Human beings can struggle to sustain their life when food shortages happen, because food is one of our most basic needs than shelter and cloth. Before coming to the modernization time, every society around the world has attempted to overcome food shortages at household levels. They practice certain activities to escape them from food insecurity. Many years ago, when the world population was much lower than it is now, a man had little difficulty in ordinarily time in growing the food that was needed. At that time, human beings started to lead their life in ancient time by hunting wild animals and gathering fruity crops. Farmers have developed coping and adaptation strategies to buffer against the adverse effects of climate change and variability [10, 11] by altering their economies and lifestyles with changing circumstances in their environments [12], and the concepts were raised beyond the climate literature [13]. Coping and adaptation to climate change and variability are closely related and interchangeably used in the context of disaster response except that they have different time spans. Coping strategies are autonomous, short-term, location-specific actions and adjustments targeted against a certain hazard and activities that take place within existing structures [14,15,16]. Coping strategies help to mitigate the negative effects of climate change and variability for the short term, but they are "risk spreading" in nature [11]. On average, agriculture contributes 33% to national income, 70% to full-time employment and 40% to total export earnings in Africa [17]. Most of sub-Saharan Africa relies on agriculture for employment and food security for their economies. Even though agriculture is important for the national economy, it is highly dominated by smallholder farmers who produce under unfavorable conditions characterized by low and erratic rainfall and poor soils [18]. Agriculture in Ethiopia by large is subsistence [19]. Compared to other sectors, agriculture is highly vulnerable to climate change which manifests itself in terms of longer-term trends in the average conditions of rainfall and temperature, inter-annual variability and the occurrence of droughts, floods, frosts and other extreme events [20]. Ethiopia's economy is dependent on agriculture which is characterized by a low-input, low-output and subsistence production system [21]. It is extremely vulnerable to climate change and variability, resulting in vulnerability of smallholder farmers [22, 23]. Droughts and floods occur frequently in most parts of Ethiopia, indicating how the country is suffering from climate variability and extreme events, and future climate change constitutes a major development challenge [24]. Drought followed by flood is the most common climate-related hazard in Ethiopia [25, 26]. Rainfall variability and associated droughts have been the major causes of food insecurity and poverty traps for many households during the past three decades [27, 28]. In addition, since the water resource of the country is governed by the amount and distribution of rainfall [29, 30], the spatiotemporal variability and declining trend of rainfall have negative impacts on the water resource sector. Climate change is likely to change rainfall patterns, resulting in shorter growing seasons in the future, particularly for subsistence farmers in Africa who rely on rain fed agriculture [31]. Extreme weather events such as droughts and floods are predicted to become more frequent, adding to the global burden of hunger caused by poverty, weak governance, conflict and poor market access [32, 33]. A recent vulnerability mapping in Africa cited Ethiopia as one of the most vulnerable countries to climate change, food insecurity and with the least adaptive capacity [34, 35]. Its geographical and climatic conditions, high dependence on agriculture and weak adaptive capacity were stated as the major reasons for its vulnerability [35, 36]. Sub-Saharan Africa remains the only region in the world where the number of hungry and malnourished populations will still be on the increase even by 2020 [37]. While other regions have improved per capita food availability since the 1970s, food production and availability have perpetually declined in sub-Saharan Africa. It is both a technological and a political/economic challenge and cannot be ignored any longer. Agrarian stagnation in sub-Saharan Africa has defined numerous attempts at transforming subsistence agriculture, even with due consideration to issues related to biophysical constraints and the human dimension challenges [38, 39]. Agriculture accounts about 41.6% of the GDP, employs about 83% of the labor force and contributes around 90% of the total export earnings of Ethiopia [19]. The sector is dominated by about 11.7 million smallholders cultivating about 95% of the national agricultural production, and large farms contributed to only 5% of the total production [40, 41]. This shows that the overall economy of the country depends on smallholder subsistence agriculture. Food production and population statistics in Ethiopia are notoriously unreliable, and all estimates of national food availability and consumption requirements are "guesstimates" at best [42]. Given this limitation of statistics during the late 1980s, 52% of Ethiopia's population consumed less than the recommended daily allowance of 2100 kcal, and Ethiopian agriculture appears to be locked into a downward spiral of low and declining productivity, caused by an adverse combination of agro-climatic, demographic, economic and institutional constraints, trends and shocks. Some observers argue that a "Malthusian crisis" is developing as rapid population growth (almost 3% per annum) is associated with steadily falling landholdings and per capita food production [42]. Between 1960 and 1990, the population doubled from 23 to 48 million, while per capita landholding shrunk from 0.28 to 0.10 hectare, and per capita food output collapsed by 41% from 240 to 142 kg [42]. In Africa, women smallholder famers were one of powerful engines and would play key role in the development process to ensure food security at household level and they made greater contribution to household food security but women smallholders farmers households were the most neglected in the development policies and programs despite its contribution to food security at national levels. The concept of coping strategies is not new. But, different researchers, scholars, authors and organizations have defined the term coping strategies in different ways. These indicate that there is no comprehensive definition [43]. Devereux defines as a response to adverse events or shocks. Broad definition of coping strategies, namely that "all the strategically selected acts that individuals and households in a poor socioeconomic position use to restrict their expense or earn some extra income to enable them to pay for the basic necessities (food, clothing, shelter) and not fall too far below their society's level of welfare" [44]. Food insecurity is still widespread, especially among developing nations [45]. Over 960 million people in the world are hungry and undernourished [46]. The majority of these people are found in developing countries, most especially in Asia and Africa, and many die of hunger-related diseases (Standing Committee on Nutrition [47]). According to African Food Security Briefs [48], approximately one-third of the people in sub-Saharan Africa are undernourished [45]. The strategies pursued by households differ in several aspects, that is, within the household and between households [49]. Due to varying degrees of wealth among households, different coping behaviors are adopted by households at different poverty levels. However, some coping strategies are common to all households although the extent to which such strategies enable a household to remain afloat depend on the assets at their disposal [43]. Above all, the general tendency is that the lower the household asset status, the more likely the household would engage in erosive responses such as selling off productive assets such as farm implements [50]. Description of the study area For the administrative purpose, the Ilu Gelan District is divided into 17 peasants associations and one town. The study area was located at water shed of Gibe River. The population has grown over the years. Ilu Gelan District is located at a 90 km from zonal capital (Ambo). The district capital, Ijaji, is located 215 km from regional capital (Finfine). The district was bounded by Chelia District in north and east, Dano District in the south and Bako District in west (Ilu Gelan Agricultural Offices, 2004 EC). The district is divided into three distinct climatic zones as high land, midland and low land respectively. The average maximum and minimum temperatures of the district are 32 and 25 °C, respectively. There are four types of soil found in the district: fertile soil, sandy loam soil, clay soil and red soil. Out of the total area of the district Ilu Gelan is covered by red soil 70% and sandy loam soil 10% [51]. Population of the study was smallholder farmer households that live in the rural areas of the district. Currently there are 4073 households in the two selected kebele in the study area [51]. Household-based cross-sectional survey research study was conducted in Ilu Gelan to identify transitory coping strategies of food-insecure smallholder farmer households. Sampling design was determined by type of universe to be studied, households in this study, sampling unit, sampling frame, size of sample and parameters of interest. Because of the nature of the study, multistage sampling technique was applied in order to increase the reliability and validity of the data. Sample size determination and method of sampling Sampling is a technique, which helps us in understanding the parameters or characteristics of the Universe or population by examining only a small part of it. Therefore, it is necessary that sampling technique be reliable [52]. Appropriate sample size depends on various factors relating to the subject under investigation such as the time, cost and degree of accuracy desired. [53]. Nevertheless, the sample size and the sample selection procedure should assure the representativeness of the population. Sample size determination has its own scientific approach. To analyze coping strategies on food insecurity of the households, the researcher used multistage sampling technique to select sample food-insecure households in the study district. First stage: The study district, Ilu Gelan, was selected purposively based on; repeated decreasing agricultural production, ever-increasing natural and man-made distresses and its threatening effect on high soil erosion and deforestation in the area, vulnerability to food insecurity to due to climate change and environment failures. Second stage: Two samples have been selected randomly out of 17 kebele. Third stage: The next step was selection of the sample of household head. Finally, 100 sample smallholder food-insecure households were selected by using random simple techniques. The sample size for collecting quantitative data for this research is determined by using Yemane Formula [54]. The study used the following formula to calculate sample size. The researcher has adopts [54] for determining sample size. $n = \frac{N}{{1 + N\left( {e^{2} } \right)}}.$ The following steps were used to determine sample size derived from the above formula to collect quantitative data using semi-structured interview schedule. Here, n designates the sample size the research uses; N designates total number of households heads; e designates maximum variability or margin of error 8%; 1 designates the probability of the event occurring. Therefore, $n = \frac{N}{{1 + N\left( {e^{2} } \right)}}$ $$n = \frac{4073}{{1 + 4073\left( {0.08} \right)2}} = 100$$ Therefore, total sample size was 100 household head farmers, out of which 50 were selected from Seba Biche and the remaining 50 were from Meta Kidane Mehreta. Selection was made proportionally from total household living in both kebeles. The total sample drawn was 100. Methods of data collection This study based on a micro-level, and it is derived from a cross-sectional primary data. The structured household questionnaire was used to collect the data from 100 farm households. The data were collected through household survey. Moreover, focused group discussions and key informants interviews were conducted in the village communities. Data analysis procedures consist of descriptive statistics and coping strategies index (CSI). The CSI is developed by [55] to measure the food security situation. The basic idea of CSI is to combine the frequency and severity of coping strategies. The frequency of coping strategies requires the means of scoring of relative frequency which measures how many days per week a household had to rely on the various coping strategies ranking from "never" to "every day." The severity of coping strategies is measured using focused group discussion via asking the individuals to classify their coping strategies based on their opinion (1 = less severe, 2 = moderate, 3 = severe and 4 = very severe). The means of scoring reflect the severity weight of each coping strategy that household has adopted. Thus, the CSI score is calculated by combining of both "frequency" and "severity" of coping strategies. The result of the CSI score denotes that a household with a higher value is more food insecure compared with a household with a lower value. The survey gathered data pertaining to the demographic and economic aspects of the households. I also included items related to the causes of food insecurity in the district, as well as access to the market, receipt of food aid and the distance of the village away from the main road. The situation of the stock status was measured through the availability of sufficient sustainability daily rations for household members, the number of meals taken per day and the number of days per week meat or fish eaten in the household. The odds of reporting that one often experienced food shock were also included in the analysis. Method of data analysis The data analysis process was carried out after collection of the required information from primary sources. The data were analyzed by Statistical Package for Social Sciences (SPSS) version 20. SPSS was used to analyze different variables through descriptive statistics such as, frequencies, mean, standard deviation and percentage. Measuring household food insecurity In this study, sample households were classified into food secure and food insecure on the basis of the 2100 kilocalorie threshold. Households with daily calorie consumption greater than or equal to 2100 kcal per day were categorized as food secure and those households whose calorie intake fell below this food security threshold grouped as "food-insecure" based on Ethiopian Health and Nutrition Research Institute (EHNRI) recommendation [56]. To increase longevity and starting some things that makes life, the indicators of coping mechanisms are listed in Table 1. But it is different for male- and female-headed households. In addition to that, the study considered some food security proxy indicators. These are the food consumption score which combines information on meal frequency and dietary diversity and the coping strategy index which has proven to be a good indicator of food security level. Table 1 Coping mechanisms by sex of household head. Coping strategies and weighing According to the information gathered from focused group discussion and key informant interview in the study districts, the use of these coping strategies depends on the most frequently used coping strategies they used. Table 2 shows average severity weight for various coping strategies among selected localities smallholder farmer households in Ilu Gelan, in 2017/2018G.C. The coping strategy index was adopted and modified in the present study [55]. During FGD and key informant interview, the major coping mechanisms in the area were listed and finalized. The revised CARE/WFP list is given in Table 2. There were 19 coping strategies listed. As explained in "Methods" section, weighing of the strategies is very important to ensure the cultural sensitivity of the population. This exercise is done during the FGD and KII. The average weight of each strategy during two FGDs was finalized after the consensus ranking with key informants. For consensus ranking, the individual strategies listed have been ranked into four categories, where 1 and 4 indicate the least and most severe category, respectively, and 2 and 3 indicate intermediate. The weight assigned to each strategy is also shown in Table 2. There was no complete consensus about the ranking except some respondents employing their children to wealthy households for keeping cattle, and selling fire wood and charcoal. However, a quick look will indicate that there was good consensus on withdrawing children from school as the most commonly implemented indicator of coping strategy. In general, the consensus ranking should be a whole number that is the most frequent response. Both men and women small holders farmer households use all the coping strategies in the listed above at least part of them, in order to fulfill their food discrepancy. All the relative frequency of using different coping strategies was ranked from "never" to "every day" in a 1–7 scale. Table 2 Coping strategies grouped and ranked by FGDs and KII. In the study area, different coping mechanisms were employed by both male- and female-headed households. As indicated in Table 1, female-headed households were significantly different and more engaged in coping mechanism of selling fire wood and charcoal and engage in petty retailing of items. The analysis also showed that consumption-based copping mechanism of female-headed household like cutting down meat consumption, relying on less preferred and less expensive food, reduction in consumption expenditure and reducing meals was significantly different and more affiliated when compared with male-headed households and dropping of children from school was more seen in female-headed households. Table 1 indicates that male-headed households were significantly different in coping mechanisms and more affiliated to the coping mechanisms of daily laborer, timbering, skip entire day without food, and alcohol consumption. Coping strategies index The coping index was calculated for food-insecure households (n = 100) using the methodology suggested in "Methods" section. The mean coping index was 88.54 and 119.14 for males and females, respectively. Using inferential statistics equality of means was tested. The t value was − 5.173 for 98 degrees of freedom, and the mean difference was significant (p < 0.001). It implies that coping mechanisms of female-headed households were significantly different from male-headed households (Table 3). Table 3 Mean coping strategies index. Human beings can struggle to sustain their life when food shortages happen, because food is one of our most basic needs rather than shelters and cloth. Before coming to the modernization time, every society around the world has attempted to overcome food shortages at household levels. They practice certain activities to escape them from food insecurity. In the past, when the world population was much lower than it is now, a man had little difficulty in ordinarily time in growing the food that was needed. At that time, human beings started to lead their life in ancient time by hunting wild animal and gathering fruity crops. But it is different from male and female households. In addition to that, the study considered some food security proxy indicators. These are the food consumption score which combines information on meal frequency and dietary diversity and the coping strategy index which has proven to be a good indicator of food security level. According to information obtained from group discussion and key informant interview, food-insecure households used strategies such as migration, dropping children out of school, cutting down trees and child labor as coping mechanisms. Even though many rural girls move to towns for daily labor, the income they earn does not cover their food and clothing. In the absence of gainful daily labor, some of them are often forced to involve in undesirable things to lead their life (prostitution and robbery). Studies conducted in Ethiopia have indicated that children migrating to urban areas, especially in Addis Ababa, are exposed to high child labor abuse in urban areas and are mostly deprived of their basic rights to education [57]. Another coping strategy adopted by poor rural households is a shift to poor and nutritionally lacking diet which sometimes leads to health-related problems including mental disorder. The downside of the effects of the coping strategies aforementioned has also implications for social norms. For instance, sending children to the streets earn money by begging is a shameful act according to OromoFootnote 1 culture. However, today abject poverty and HIV/AIDS pandemic have forced many children to street begging even among the Oromo. The primary victims of this problem are girls. As regards the effect of early marriage, parents betroth their daughters at early ages hoping that the latter would provide the former with income they could earn after marriage. This is often the case when the husband is believed to have sufficient income to support his wife's family. Ironically, however, early marriages can end up leading women to engage in less profitable activities like brewing and selling local liquor (e.g., "Areke" and "Tela" which are locally made liquors). Brewing such liquors is not only less profitable, but it has debilitating effect on their health. Early marriage is also one of the factors accounting for high dropout rates of girls at both elementary and high schools. Dropping out of school is not limited to girls; boys are also forced to quit school and sent out to work on others' farms and earn income for their family. Even worse is that the income that boys earn in this way is too little to even cover their own cost of clothing let alone helping their family buy sufficient food. So, these results were confirmed with the study revealed that in Table 1 those reported by households surveyed. As the Table 1 displays that the dropping of children from school was one of the coping mechanisms practiced in the study district and significant differences were found between household food insecurity and coping mechanism (at t value 0.015). This was similar to the values reported in Tanzania by Ballard [58], and another option by the same finding was the withdrawal of children, especially girls from school in order to utilize their labor and save money, which, among other things, had ramifications for future literacy levels and the child's participation in the modern economy. Again dropping of children from school was much higher among food-insecure households observed in the surveyed areas (Table 1). So, these values were in agreement with those reported by Brock and Coulibay [59] who pointed out that globally, withdrawing children from school is a short-term strategy that has permanent effects that could make it difficult to reduce food security in the long term. The data show in Table 1 that in the study districts smallholder farmer households strongly rely on less preferred and less expensive food items and statistically strongly significant differences at (at t value 0.000) were observed in the surveyed district (Table 1). These indicated that households have significant differences noted between food insecure and coping mechanisms in Table 1. These values were in line with those values obtained by Ahmed Mohammed Abdulla. This result conforms to findings of a study by [60] conducted among the pastoralists in the Oromia (Yabelo) region of Ethiopia. One-sample t test reveals that there is a statistically significant difference between male-headed and female-headed households in terms of their coping strategies indices. As given in Table 1, sending family members to abroad, to the Middle East countries in particular, in expectation of future remittance is another coping mechanism practiced by the sample households in the study district and significant values were obtained (t value: 0.045). This study was also consistent with the finding of Abdulla [60]. The study also indicates that the coping strategies practiced by most of the households were reduction in non-food expenditures and daily labor in both rural and urban areas as well as purchase of food by borrowing. However, borrowing involves high interest rate in the form of usury. This is particularly true in the case of borrowing from individuals as opposed to formal and microfinance institutions. Due to the high interest rate, very few households borrow from individuals lending money through usury. [61] also found a similar result in the study they conducted in the Southern Nations, Nationalities and Peoples Region of Ethiopia. The study also shows that sale of timber, begging, reducing daily meals and selling productive assets and other materials are coping strategies predominantly pursued by households who had a higher likelihood of being food insecure. So, the results were also in line with those of Bryant [62]. This [62] idea is that distress sale of assets is a major coping strategy adopted by households at times of acute food shortage. In addition, the result is consistent with the study of that of Ekesa in 2008 [36]. According to information obtained from focused group discussion and key informant interview, farm households in the study area often have too little to eat. Lack of enough food was mainly because of insufficient crop harvest and lack of purchasing power. The latter is ascribed to lack of sufficient income, both farm and non-farm/off-farm. Limited purchasing power robs the households of their entitlement to food whenever it is available in the market. Generally, the source of food insecurity among the study participants arises from both problems of availability and access. The FGD and key informant interview show that the major food crops consumed in the study area include cereals, legumes, roots and tubers, bananas, vegetables, fruits, meat and milk products. Cereals crop like sorghum and maize and legumes like beans and soya bean were under produced due to natural and man-made factors. Roots and tubers such as cassava, potatoes and sweet potatoes were consumed as most important food security crops during food shortage, but now they are produced by fewer numbers of households because of environmental factors. To cope up with the fall in incomes, households sold off assets, one such asset is land. Bryant [63] found a similar finding. In Cambodia, a study conducted revealed that about one in five children whose households were affected by male labor migration had to start working in order to help their mothers [64]. Food is one of the basic needs which are crucial for human beings. We can do nothing without food. However, [65] indicated large amount of food production in the world does not ensure any country's food security. Time to time there are many factors to worsen the food security problem in many parts of the developing world. CARE/WFP [55] identified education level of household head, income source, market and food prices and supply of food commodities as some of the factors underlying food insecurity. In order to cope with food shortages, most households are often compelled to ration the meager food at their disposal, i.e., cutting the amount and number of meals, favoring certain household members over others and skipping whole days without eating [49]. Households can take measures to overcome food insecurity problem. Poor households are the hardest hit by food insecurity when compared with the relatively better-off ones. The most food-secure households are those who achieve adequate access to food while using only a small proportion of available resources, whereas the most food-insecure households are those most at risk who fail to achieve adequate access even by devoting a large proportion of available resources to food [66]. Daily labor in both rural and urban areas, purchase of food by borrowing and petty trade, seasonal migration/mobility for labor and selling fire wood and charcoal were identified as widely practiced coping mechanisms. Withdrawing children from school and sending them out to work on others' farm, relying on less preferred and less expensive food and reducing non-food expenditures and skipping meals were other common coping mechanisms pursued by the households in the study area. Smallholder farming households in Ilu Gelan pursued a host of short-term coping strategies to overcome the problem of food shortages. The irony, however, is that these coping strategies employed by the households were not effective in reducing household food insecurity in the study area. Thus, even though the coping strategies could ease problem food shortage over a very short period, they were not able to provide long-term solution to the food security problem poor household face. While agriculture plays a major role in the reduction in household food insecurity, the food insecurity problem cannot be solved by promoting agriculture alone. Hence, policies aimed at reducing food insecurity should look beyond agriculture if long-term solution is sought. The main objective of the study was to identify the coping mechanisms food-insecure households in Ilu Gelan districts adopted in order to grapple with food shortages they face. The data show in the study districts smallholder farmer households rely on less preferred and less expensive food items. Coping strategy adopted by poor rural households is a shift to poor and nutritionally lacking diet leads to health-related problems. A one-sample t test reveals that there is a statistically significant difference between male-headed and female-headed households in terms of their coping strategies indices. The downside of the effects of the coping strategies aforementioned has also implications for health and social norms. Begging is also one of the coping mechanisms that practiced in the study district, but insignificant differences (at t value 0.180) were observed between food-insecure households and coping mechanisms in the surveyed district (Table 1). These values were disagreement with those result obtained. However, today abject poverty and HIV/AIDS pandemic have forced many children to street begging. The primary victims of this problem are girls. Smallholder farming households in Ilu Gelan pursued a host of short-term coping strategies to overcome the problem of food shortages. The irony, however, is that these coping strategies employed by the households were not effective in reducing household food insecurity in the study area. Thus, even though the coping strategies could ease food shortage problem, over a very short period they were not able to provide long-term solution to the food security problem poor household face. While agriculture plays a major role in the reduction in household food insecurity, the food insecurity problem cannot be solved by promoting agriculture alone. Hence, policies aimed at reducing food insecurity should look beyond agriculture if long-term solution is sought. Household in Ilu Gelan was engaged in traditional and hand tool agriculture with simple drafting animal. Data obtained from FGD and key informant interview show that the production of crop did not provide sufficient food for household family consumption to impact on household food security status. Low production due to erratic rainfall reduced the availability of crops for household consumption and opportunities for income generation. Households did not produce sufficient quantities of crops throughout the year due to natural and man-made factors. However, the households to cope with food insecurity relied on reduction in non-food expenditures. Daily labor in both rural and urban areas, purchase of food by credit/relying on borrowing and sending family members to abroad for remittance and obtained from their own production and also food received as gifts from relatives and from non-farm activities were more practiced among food-insecure household. In view of the findings of the study and the above concluding remarks, the following recommendations are made in order to promote sustainable food security at smallholder farmer's household level in the study district. Gender monitoring and evaluation of the ongoing programs should be carried out to ensure benefits to all and particularly to the vulnerable, female-headed households. Withdrawing children from school was also a common for food-insecure households copping strategy in the study districts. To reduce school withdrawal, the government in collaboration with the World Food Program served school children with feeding program. In addition to this, implementing disaster and risks management programs to support rural poor and smallholder farmers during natural calamities and other shocks is desperately needed to give prescription for the households. 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IFAD and UNICEF, Roma; 1992. This research has been designed and led by BD who structured the concepts, reviewed all of the studies and analyzed both quantitative and qualitative data. He further developed the manuscript, identified and developed important concepts, validated and designed the arguments, conceived and conducted quantitative study and edited the final research. The author read and approved the final manuscript. I convey my deepest thanks to my friends for the completion of this work and energetic encouragement, suggestion and insight and guidance to complete the paper. And I would also like to express my deepest thanks to Gemechu Shale Ogato (Assistant Professor and Ph.D.) at Ambo University as well Sajitha (Ph.D.) at Ambo University for their editing and constructive comments throughout my research work. The author declares that there are no competing interests. Availability of supporting data The author wants to declare that he can submit the data at whatever time based on your request. The datasets used and/or analyzed during the current study will be available from the author on reasonable request. Participants of the research including survey households, case studies, enumerators, the supervisors and key informants were fully informed about the objectives of the study. They all were approached friendly and fraternal. Department of Rural Development and Agricultural Extension, College of Agriculture and Natural Resources, Madda Walabu University, P.O. Box 47, Bale Robe, Ethiopia Belachew Dessalegn Search for Belachew Dessalegn in: Correspondence to Belachew Dessalegn. Dessalegn, B. Transitory coping strategies of food-insecure smallholder farmer households: the case of Ilu Gelan District, West Shoa Zone, Oromia Reginal State, Ethiopia. Agric & Food Secur 7, 70 (2018) doi:10.1186/s40066-018-0204-2 Transitory Smallholder farmer Climate Smart Agricultural Technologies in West Africa	CommonCrawl
Fractional Laplacians : A short survey DCDS-S Home An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation January 2022, 15(1): 79-93. doi: 10.3934/dcdss.2021030 Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line Abderrazak Chrifi , , Mostafa Abounouh and Hassan Al Moatassime Department of Mathematics, Faculty of Science and Technology, Cadi Ayyad University, B.P. 549, Av. Abdelkarim Elkhattabi, Guéliz, Marrakesh, 40000, Morocco * Corresponding author: Abderrazak Chrifi ([email protected]) Received August 2020 Revised January 2021 Published January 2022 Early access March 2021 We consider a weakly damped cubic nonlinear Schrödinger equation with Dirac interaction defect in a half line of $ \mathbb{R} $. 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Variety enhancement of PUF responses using the locations of random outputting RS latches Regular Paper Dai Yamamoto1, Kazuo Sakiyama2, Mitsugu Iwamoto2, Kazuo Ohta2, Masahiko Takenaka1 & Kouichi Itoh1 Journal of Cryptographic Engineering volume 3, pages 197–211 (2013)Cite this article Physical Unclonable Functions (PUFs) are expected to represent an important solution for secure ID generation and authentication etc. In general, manufactured PUFs are considered to be more secure when the pattern of outputs (the variety of responses) is larger, i.e., the response bit length is longer (e.g., 192-bit response is more secure than 128-bit one). However, the actual bit length is reduced because some response bits are inconsistent (random) for repeated measurements, which are regarded as unnecessary for ID generation and discarded. Latch-based PUFs with $N$ RS latches, for example, generate ideally $2^{N}$ responses depending on binary values output from RS latches (0/1). However, some RS latches output random responses which are inconsistent and cannot be used for reliable ID generation, so the variety of responses becomes smaller than $2^{N}$. In this paper, we propose a novel Latch-based PUF structure, which outputs larger variety of responses by utilizing location information of the RS latches outputting the random responses. Differently from random responses themselves, this location information is determined during a manufacturing process, so almost fixed once PUFs are manufactured. The proposed PUF generates $3^{N}\approx 2^{1.58N}$ responses by considering random responses as the third stable value: using ternary values (0/1/random). We estimate the variety of responses generated by the proposed PUFs. According to our experiment with 40 FPGAs, a Latch-based PUF with 128 RS latches can improve it from $2^{116}$ to $2^{192.7}$, this being maximized when the 128 latches outputs 0s, 1s, or random outputs with equal probability. We also show the appropriate RS latch structure for satisfying this condition, and validate it using two kinds of different Xilinx FPGAs: Spartan-3E and Spartan-6. The average error rate of responses is only 5.3 % when the core voltage is changed within the rated voltage range of the FPGAs. Our proposed PUF using ternary values enhances dramatically the variety of responses while keeping the reliability. Secure identification/authentication technology using Integrated Circuit (IC) chips is very important for secure information infrastructure. It is used for anticounterfeiting devices on medical supplies, prepaid cards and public ID cards such as passports and driver's licenses. The IC card is a well-known solution for this kind of application. Counterfeiting is prevented by storing a secret key on the IC card and using a secure cryptographic protocol to make the key invisible from outside. In theory, however, the possibility of counterfeiting still remains if its design is revealed and reproduced by the counterfeiter. Naturally, this is difficult because current IC cards are equipped with several highly developed tamper-proofing technologies. However, further anticounterfeiting technologies are desirable to meet future developments in reverse-engineering techniques. Recently, interest has been focused on Physical Unclonable Functions (PUFs) as a solution [17]. In a PUF, the output value (response) to the input value (challenge) is unique for each individual IC. This uniqueness is provided by the process variations of each individual IC [5, 6]. It is expected that PUFs will represent breakthrough in technology for anticounterfeiting devices, through its use for ID generation, key generation and authentication protocol, which make cloning impossible even when the design is revealed. The PUFs on ICs are classified into two categories: memory-based PUFs and delay-based PUFs [15]. Memory-based PUFs use the characteristics of memory cells such as SRAM-PUFs [7, 9], Butterfly PUFs [11], Flip-flop PUFs [13] and Latch PUFs (LPUFs) [20, 21]. Delay-based PUFs use the characteristics of delay variations such as Arbiter PUFs [12], Glitch PUFs [19, 23] and Ring Oscillator PUFs [22]. SRAM-PUFs are based on the unstable power-up values of SRAM cells on ICs such as ASIC. However, a device power-up operation is required for the generation of every response. To counter this drawback, Butterfly PUFs are composed of cross-coupled latches which behave similarly to an SRAM cell. The output of the Butterfly PUF is triggered by a clock edge signal applied to the latches, without an actual device power up. Flip-flop PUFs were proposed in order to solve the problem that the power-up values of SRAM cells on FPGAs were automatically initialized to fixed values. Flip-flop PUFs uses the power-up values of flip-flops instead of SRAM cells. LPUFs are very similar to SRAM-PUFs and Butterfly PUFs. LPUFs generate each response using a metastable value of a latch composed of cross-coupled logic gates. LPUFs can be implemented on both ASIC and FPGA, and generate responses without an actual device power up. Arbiter PUFs have an arbiter circuit that generates a response determined by the difference in the signal delay between two paths, which is mixed by a challenge. However, a machine learning attack can predict responses of Arbiter PUFs using a number of challenge–response pairs [18]. The Glitch PUF [19, 23] was proposed to solve this problem of ease of prediction. It generates a one-bit response using the parity of the number of glitches obtained from an 8-bit AES S-Box used as a glitch generator. Since the responses to challenges behave like a non-linear function, machine learning attacks are prevented. Ring Oscillator PUFs derive responses from the difference in oscillator frequencies. In [10], the performances of five types of PUFs (Arbiter, Ring Oscillator, SRAM, Flip-flop and Latch PUFs) are evaluated on ASIC implementations. Today, memory-based PUFs are some of the most feasible and secure technique because there have already been implementations of error correcting codes (ECCs) and universal hash functions [2] for randomness extraction optimized for the PUFs, which are needed for Fuzzy Extractors [4]. Especially, error corrections utilizing soft decision decoding techniques are optimized for memory-based PUFs in [14, 24]. LPUFs implemented in ASIC seem to have many advantages over SRAM-PUFs, such as not requiring a power-up operation. This paper therefore focuses on LPUFs, which generate $N$-bit responses based on $N$ outputs from $N$ RS latches. The responses from PUFs need to have extremely high uniqueness. This paper defines uniqueness as the independence among multiple PUFs of responses to the same challenge. In order to prevent clones of cryptographic hardware, it is important for manufacturers to make sure that multiple PUFs with the same challenge–response pairs do not exist. However, this is very difficult in terms of cost because there are a huge number of manufactured PUFs and challenge–response pairs. Therefore, one of the most practical solutions is to increase variety: the number/pattern/range of responses in manufactured PUFs as much as possible. For example, 192-bit responses are obviously more secure and unpredictable than 128-bit responses. 192-bit responses have $2^{192}$ variety ideally. However, the actual variety of the responses is much less than the ideal $2^{192}$ variety if 192-bit responses obtained from multiple PUFs are almost the same. Consequently, we must note that a large number of response bits are not necessarily equivalent to a large actual variety of responses: a high level of the Shannon entropy in those responses. PUFs that output responses with high entropy are capable of generating completely unpredictable responses. Consequently, the probability of multiple PUFs that output unpredictable responses having the same challenge–response pairs is extremely small. Hence, it is important for PUFs to increase not only variety but also entropy of responses so as to have extremely high uniqueness. In addition, the response needs to have high reliability. This paper defines reliability as the consistency of PUF challenge–response pairs for repeated measurements. That is, ideally, a PUF always generates the same response to a given challenge. However, the LPUF has some RS latches that generate inconsistent (random) numbers (i.e., "random latches"). This randomness causes a problem in that the reliability of the response is reduced. This is because the values of the response corresponding to the random latches change every time a response is generated. Hence, PUFs need to eliminate random responses in order to generate stable responses. For example, the Glitch PUF can generate relatively stable responses because it selects only available challenges to output consistent responses for repeated measurements through a masking process. However, as pointed out by the designers of the Glitch PUF, the masking process reduces the number of responses since random responses are eliminated. Also in LPUFs, the response bits become lower as the number of random latches increases, which reduces the variety and entropy of responses. In such a conventional approach, in order to maintain the reliability of responses, the outputs of the random latches are not used to generate stable responses. However, in this paper we make efficient use of the random outputs. This paper proposes a novel PUF structure for generating high-entropy responses using randomness. Note that our proposed methods can be applied to any PUFs. As an example, our paper focuses on an LPUF with random latches. The use of random latches dramatically increases variety and entropy of responses. Also, the construction can maintain the reliability of responses even if random latches are used for the generation of responses. In specific terms, responses are generated based on the location information of the random latches. The proposed LPUF with $N$ RS latches generates approximately $3^{N}$ responses by considering random responses as the third stable value: using ternary values (0/1/random). However, the actual variety of responses is less than $3^{N}$ because $3^{N}$ is the total variety of responses: all the possible combinations of the ternary values. The variety of random latches is almost fixed and determined by the kind of PUF device. Consequently, $3^{N}$ is not accurate, but is intuitively easy to understand, so an accurate evaluation for the actual variety of responses is given below. The actual variety of responses takes on its maximum value when the frequency of each ternary value (0/1/random) is equal. Hence, we also propose suitable RS latch structures to satisfy this equality condition to the maximum extent. The proposed structures are optimized for two types of Xilinx FPGAs: Spartan-3E (SP3E) and Spartan-6 (SP6), which have different FPGA architectures. We evaluate the performance of the proposed PUF with both types of FPGAs. Using 40 SP3E FPGAs, an LPUF with 128 RS latches based on our RS latch construction increases the average number of random latches from 12 to 32, approaching around 43 (=128/3). The proposed PUF with ternary values improves the variety of responses from $2^{116}$ to $2^{196}$ theoretically. From the actual responses generated by 40 PUFs, the entropy of responses is experimentally evaluated as 192.7 bits, which indicates that the proposed PUF has extremely high entropy. Additionally, we develop an experiment system with SP6 FPGAs, which can change the supply voltage within the rated voltage range of the FPGAs (1.14–1.26 V). The average error rate of responses is about 5.3 %, which indicates that the proposed PUF using ternary values has high reliability. Organization of the paper The rest of the paper is organized as follows: Section 2 gives an outline of the LPUF with RS latches, and the conventional methods for implementing RS latches on FPGAs. Section 3 proposes our original LPUF, which generates responses using the location information of the random latches. In addition, new methods of implementing RS latches are proposed that maximize the performance of our PUF. Sections 4 and 5 evaluate the performance of our PUF using SP3E and SP6 FPGAs, respectively. Section 6 discusses the applicability of proposed methods to other kinds of PUFs than LPUFs. Finally, in Sect. 7, we give a summary and comment on future directions. Part of the content of our proposal has been published in [26]. This paper proposes the appropriate RS latch structure optimized for Xilinx Spartan-6 FPGAs in Sect. 3.2.2. We evaluate the PUF performance on Spartan-6 and discuss the voltage resistance in Sect. 5. We also discuss the applicability of proposed methods to various kinds of PUFs in Sect. 6. Conventional methods Conventional method (1): generation of responses from an LPUF This paper focuses on an LPUF using RS latches. First, we describe the circuit and behavior of an RS latch created from two NAND gates, shown in Fig. 1. NOR gates are used in [20, 21], but this difference does not influence the LPUF performance. An RS latch is in a stable state with output $(B,C)=(1,1)$ when input $A=0$. When input $A$ changes from 0 to 1 (= rising edge), the RS latch temporarily enters a metastable state. It then enters a stable state with either output $(B,C)=(1,0)$ or $(B,C)=(0,1)$. Ideally, the probability of transition to either of these states is equal. In fact, however, many RS latches have a high probability of entering one specific state. This is because the drive capabilities of the two NAND gates or the wire length between them are not exactly the same. Hence, the output $B$ from RS latches fall into three patterns: all 0s, all 1s, or a mixture of 0s and 1s (=random number) when a clock signal is applied to input $A$. NAND-based RS latch We now describe the LPUF, shown in Fig. 2. Challenges to the LPUF are equivalent to choosing $M(\le N)$ RS latches from $N$ implemented RS latches. The LPUF can generate $M$-bit responses corresponding to ${}_N \mathrm C _M$ challenges. Here, ${}_N \mathrm C _M$ is defined as the number of combinations of $N$ elements taken $M$ at a time. The LPUF in Fig. 2 generates an $N$-bit response: $\text{ RES}[N-1:0]$ when $M$ is set equal to $N$. Note that, in order to simplify discussion in this paper, the more significant bits of the response correspond to the outputs of RS latches with bigger latch labels. LPUFs, which generate only a response, can be used for applications such as authentication as shown in Fig. 3. A random number $S$ is sent from an authentication server to a PUF as a new challenge, and a response $R$ is newly defined by equation $R=F(S\parallel {\text{ RES}})$. Here, $F()$ indicates a mixing function, such as a hash function. The operation $\parallel $ means a concatenation of two variables. The value of response $R$ changes depending on the challenge $S$, so LPUFs provide security when used for this application. The PUF in Fig. 2 has some RS latches that generate random numbers such as ${\text{ LATCH}_{2}}$ and ${\text{ LATCH}_{{ N}-2}}$. These random numbers cause a problem in that the reliability of the response RES is reduced since its value changes every time it is generated. Latch PUF LPUF-based authentication protocol There are two widely known conventional approaches to response generation aimed at solving this problem. In the first approach ["conventional method (1-A)"], random latches are not used for the generation of responses. This approach maintains the reliability of responses, but reduces response bits and the variety of responses. It also requires a mechanism to detect random latches. Suppose that the LPUF with 128 RS latches ($N=128$) in Fig. 2 has $T$ random latches. The conventional method (1-A) generates $2^{N-T}$ responses, so the variety and entropy of responses decrease as the number of random latches increases. Hence, it is necessary to implement extra RS latches in the PUF in accordance with the number of random latches. This PUF is, however, not suitable for embedded systems with limited hardware resources such as smart cards because, while also maintaining the variety of responses, it is necessary for PUFs in embedded systems to have an RS latch area size and peripheral circuit that are as small as possible. In the second approach ["conventional method (1-B)"], ECCs are used to correct the inconsistency in the responses resulting from the random latches. This approach requires larger redundant data for response correction as the number of random latches increases. In addition, it also suffers from the disadvantage of necessitating increased hardware resources and processing time for the ECCs. An LPUF with 128 RS latches generates no more than $2^{128}$ responses even if ECCs are used. From the above, it can be seen that the first approach, in which random latches are not used for responses, is not suitable. Furthermore, it is not sufficient to use only ECCs, as in the second approach. In Sect. 3, we propose a method for generating responses based on the locations of random latches. The proposed method maintains the reliability of responses and dramatically improves their variety and entropy. Conventional method (2): implementation of RS latches on FPGAs A method for implementing RS latches as a physical random number generator on Xilinx Virtex-4 FPGAs ["conventional method (2-A)"] is proposed in [8]. The implementing method can be also applied to Xilinx SP3E FPGAs because they have the almost same architecture as Virtex-4 FPGAs. We focus on SP3E FPGAs since our experiment uses this kind of FPGA. Flip-flops (FFs) are positioned in front of the two NAND gates, as shown in Fig. 4. This minimizes the difference in signal arrival time between the two gates, enabling the RS latch to enter the metastable state more readily and improving the probability of the RS latches outputting random numbers. An SP3E FPGA consists of a matrix of configurable logic blocks (CLBs) including four slices. A slice includes two pairs of LookUp Tables (LUTs) and FFs. The right and left slices of the CLB are different. The right slice (SliceL) is available only for logic, while the left one (SliceM) is for both memory and logic. Two types of implementation for an RS latch are reported in [8]. In one type ["conventional method (2-B1)"], two RS latches are implemented on two CLBs, as shown in Fig. 5a. In the other ["conventional method (2-B2)"], only one RS latch is implemented on two CLBs, as shown in Fig. 5b. Both methods implement the NAND gates of an RS latch using the same kind of slice (SliceL in Fig. 5) on different CLBs. The conventional method (2-B1) uses two CLBs per two RS latches, leading to reasonable circuit efficiency. However, it is pointed out in [8] that multiple RS latches which have NAND gates implemented on the same CLB, as shown in Fig. 5a, have a low probability of outputting random numbers. RS latches based on conventional method (2-B2) have some probability of generating random numbers, but result in low circuit efficiency because an RS latch requires two CLBs. To the best of our knowledge, any methods for implementing RS latches applied to an SP6 FPGA, a later generation of SP3E, have not been proposed. The next section proposes an implementation method applied to SP3E FPGAs that gives the RS latches a high probability of outputting random numbers. In addition, the proposed method gives higher circuit efficiency than in the conventional methods. We also first propose an implementation method suitable for SP6 FPGAs. Conventional method (2-A): RS latch circuit [8] Implementation of RS latches on Xilinx Spartan-3/Virtex-4 family FPGAs [8] Proposed methods Proposed method (1): use of the location information of random latches The conventional LPUF in Fig. 2 generates responses based only on RS latches outputting fixed numbers such as 0s or 1s (i.e., "fixed latches"). Our proposed LPUF uses the location information of random latch $X$, rather than the random numbers from the random latches. If an LPUF with $N$ RS latches has $T$ random latches, then the number of locations of random latches equals to ${}_N\mathrm C _T$, which increases the number of different representation of LPUFs. Hence, the PUF based on our method utilizes the entropy derived from the locations of random latches in order to increase the variety of responses. However, this kind of LPUF requires complex controls to associate the location of RS latch $X$ with the output number, which leads to a large circuit size. In this paper, we propose a simple and efficient method ["proposed method (1)"] of solving this problem. Proposed method (1) considers the three types of output patterns from the RS latches (0s, 1s, and random numbers) as ternary values (00/11/10), respectively. Our method can generate responses with much larger patterns than conventional approaches. We describe the details of the proposed method with reference to Fig. 6. When a clock signal is applied to the inputs of the RS latches in our LPUF, they generate three types of outputs: 0s, 1s, and random numbers. The PUF based on our method has new detection circuits (shown in Fig. 7) located after the RS latches which distinguish these three types. The detection circuit $i$ outputs a 2-bit unique value $S_i[1:0] ({=}00/11/10)$ depending on the output of the RS latch $i$ (0s/1s/random numbers). If the output stream of RS latch $i$ includes a transition from 0(1) to 1(0), detection circuit $i$ considers RS latch $i$ as a random latch, and from that point onwards continues outputting the 2-bit value '10' regardless of RS latch $i$'s subsequent output stream. Stated more precisely, let RES$[2N-1:0]$ be the $2N$-bit response of our LPUF. Then $$\begin{aligned}&\text{ RES}[2N-1:0]\nonumber \\&= S_{N-1} \parallel S_{N-2} \parallel \cdots \parallel S_{i} \parallel \cdots \parallel S_{1} \parallel S_{0}. \end{aligned}$$ The gate size of the detection circuit, shown in Fig. 7, is estimated to be around 28 gates, which is definitely compact enough for embedded systems. Here, we use the equivalencies 1 FF = 12 NAND gate, 1 AND = 1.5 NAND gate, 1 OR = 1.5 NAND gate, and 1 INV = 0.5 NAND gate, introduced in [1]. Naturally, in order to distinguish three types of outputs, CPU-based software approach is able to be used instead of the detection circuit. The reason why we propose the detection circuit as hardware approach is that it is essential when our proposed PUF is implemented on ASIC. Proposed method (1) Proposed detection circuit Next, for the PUF based on our proposed method, we theoretically estimate the variety (number) of responses. Let $N$ be the number of implemented RS latches, and $T$ be the number of random latches. The PUF based on the proposed method generates ternary values (00/11/10), so the total variety of responses is ideally $3^{N}$. We define this total number as "ideal upper bound" of responses, which is estimated in consideration of all the possible combinations of the ternary values. Concretely, the ideal upper bound includes the cases when random latches are few or many. However, the value of $T$ is in fact almost fixed because it is determined by the kind of PUF device and the way in which the RS latches are implemented. Therefore, the manufactured PUFs generate less than $3^{N}$ responses. The following theoretically estimates the variety of responses for a given value of $T$. The variety of responses arising from the fixed latches is $2^{N-T}$, while the variety of responses arising from the random latches is ${}_N\mathrm C _T$. Therefore, the variety of responses for a given value of $T$ is estimated to be $2^{N-T} \cdot {}_N\mathrm C _T$. This value is obviously less than $3^{N}$ because the variety of responses for given $T$ corresponds to the $T$-th term of the binomial expansion of $3^{N}=(2+1)^{N}$, which is $2^{N-T} \cdot {}_N\mathrm C _T$, the same as the above estimate. We define this variety of responses for a given value of $T$ as "theoretical bound" of responses. Figure 8 shows a comparison between the theoretical bound of responses for the conventional method (1-A) without random latches and the theoretical bound of responses using our proposed method with various $T$ and given $N ({=}128)$ values. The conventional method (1-A) generates $2^{N-T}$ responses, so the theoretical bound of responses decreases as the number of random latches increases. Even conventional method (1-B), which uses ECCs, generates no more than $2^{128}$ responses. In contrast, the proposed method (1) dramatically increases the theoretical bound of responses. The theoretical bound of responses takes on its maximum value (${\approx } 2^{203}$) when $T$ is around 43 (${\approx } 128/3$). Hence, the proposed method dramatically improves the theoretical bound of responses. Theoretical bound of responses against the number of random latches (Estimate) Section 4.4 experimentally evaluates the average value of $T$ based on PUF implementations on SP3E. Section 4.5 calculates the theoretical bound of responses using this experimental value of $T$. However, this theoretical bound is calculated on the strong assumption that the value of $T$ is fixed, which means that all of PUF implementations have $T$ random latches strictly. We experimentally evaluate the variety of responses in consideration of further calculations: the ratios of RS latches outputting 0s, 1s. Proposed method (2): increasing the number of random latches Optimization for Spartan-3E This section proposes new methods for SP3E FPGAs ["proposed methods (2-A) and (2-B)"] to give a higher probability of RS latches outputting random numbers than those obtained with the conventional methods in Sect. 2.2. These proposed methods increase the number of random latches to 1/3 of the total number of RS latches, which improves the effectiveness of the proposed method (1). In proposed method (2-A), a shared FF is positioned in front of two NAND gates, as shown in Fig. 9. This FF sharing between two NAND gates eliminates clock skew in FFs. Consequently, the signal arrival times for the two NAND gates are much closer, allowing the RS latches to become metastable more easily, and increasing the probability of the RS latches outputting random numbers. Proposed method (2-A) also reduces the FF gate size per RS latch by FF sharing. In proposed method (2-B), one RS latch is implemented on a CLB in an SP3E FPGA, as shown in Fig. 10. In [8], an RS latch is implemented on two different CLBs, as described in Fig. 5, because FPGA synthesis tools cannot implement two NAND gates of an RS latch on 'different' kinds of slices (SliceM and SliceL) on the same CLB. To avoid this problem, proposed method (2-B) implements two NAND gates using the 'same' kinds of slices on the same CLB. Proposed method (2-B) uses only one CLB (two slices) per RS latch, giving high circuit efficiency. In addition, it is anticipated that the probability of RS latches becoming metastable and outputting random numbers would increase since the signal arrival times for the two NAND gates are much closer due to shortening of the wire length between the gates. The concepts behind proposed methods (2-A) and (2-B) can be applied not only to FPGAs but also to ASICs. Proposed method (2-A): RS latch circuit Proposed method (2-B): Implementation of RS latches on SP3E Optimization for Spartan-6 This section first proposes an implementation method suitable for SP6 enabling RS latches to output random numbers. An SP6 FPGA consists of a matrix of CLBs including only two slices, differently from an SP3E FPGA with four slices. These two slices are of different types. Hence, it is impossible to implement two NAND gates of an RS latch using the 'same' kinds of slices on a CLB like the proposed method (2-B) shown in Fig. 10. Due to constraints of FPGA synthesis tools, we also cannot implement an RS latch not only on different kinds of slices, but also on two pairs of LUTs and FFs on a slice as shown in Fig. 11a and b, respectively. To solve this problem, we propose the method ["proposed method (2-C)"] to implement an RS latch using two pairs of LUTs and FFs on two vertically neighboring CLBs as shown in Fig. 12. Here, we use not horizontally but vertically neighboring CLBs because horizontally neighboring CLBs are of different types: SliceM and SliceL. In SP6 FPGAs, each column of CLBs contains two slice columns. One column alternates between SliceMs and SliceLs, the other column includes SliceXs which have a similar structure to SliceLs except for some logic functions. The use of the completely same type of slices for implementing two NAND gates gives a higher probability of RS latches being metastable and outputting random numbers. Impossible implementation of RS latches on SP6 Proposed method (2-C): Implementation of RS latches on SP6 Performance evaluation on Spartan-3E Experimental environment Figure 13 shows our experimental evaluation system, which uses a starter kit board [25] with a Xilinx SP3E FPGA (XC3S500E-4FG320C). A 50-MHz clock signal generated by an on-board oscillator is applied to a Digital Clock Manager (DCM) primitive, which divides it into a 2.5-MHz clock signal that is applied to 128 RS latches. The output stream from each RS latch is switched by a multiplexer (MUX) and stored into a block RAM through a FF. Finally, the raw stream data from all the RS latches are transmitted to the PC through an RS232C port. In our evaluation, a software on the PC detects whether or not the streams contain random numbers rather than this being done with detection circuits. We consider that the detection technique does not influence PUF performance because the latter depends only on the output of the RS latches. We implement 128 RS latches on a 16 $\times $ 8 matrix of FPGA CLBs in accordance with proposed methods (2-A) and (2-B), this being done manually with the FPGA synthesis tools in Xilinx ISE Design Suite 11.1. We consider one FPGA board as four virtual boards, since the RS latches are implemented at four completely different locations in the CLB matrixes for each FPGA. The evaluation uses 10 actual FPGA boards, but in the following discussion, we take the number of FPGA boards to be 40. Experimental evaluation system using SP3E Reliability and uniqueness Before we represent an evaluation of the effectiveness of proposed methods, we show the basic performance of our LPUF, reliability and uniqueness. Our LPUF with 128 RS latches—based on proposed methods (2-A) and (2-B)—gives the results for reliability and uniqueness shown in Figs. 14 and 15, respectively. In our experiment, the PC is used to measure a 1,000-bit output stream from each RS latch. The 2-bit partial response generated by each RS latch is '00(11)' if the 1,000-bit bitstream is identically zero (one), or '10' if it includes a transition from 0(1) to 1(0). As a result, our LPUF with 128 RS latches can generate a 256-bit response. The reliability evaluation generates 40 responses using only a single specific FPGA selected at random. Figure 14 shows a histogram of normalized Hamming distance between two arbitrary responses among the 40 responses (i.e., ${}_{40}\mathrm C _{2}=780$ combinations). The average error rate is approximately 2.4 % with a standard deviation (SD) of 0.75 %, which is much less than the 15 % assumed in [14] for stable responses based on a Fuzzy Extractor with a reasonable size of redundant data. Hence, our PUF gives responses that are of high reliability. Next, the uniqueness evaluation generates a total of 40 responses using all 40 FPGAs (one response per FPGA). Figure 15 shows a histogram of normalized Hamming distance between two arbitrary responses among the 40 responses. This evaluation is a general way of showing the extent to which the responses of the chips are different. The difference in the responses of two arbitrary PUFs is approximately 46 % with an SD of 3.8 %. Note that the ideal difference is not 50 % but around 44.4 %. This is because our proposed PUF do not generate '01' for 2-bit partial responses. The line graph in Fig. 15 represents the ideal normalized Hamming distance obtained from a simulation. The simulation generates 40 responses using a pseudorandom number generator [16]. Our experimental result follows the ideal simulation result. Hence, our PUF gives responses with a high level of uniqueness. Reliability on SP3E (Mean = 2.4 %, SD = 0.75 %) Uniqueness on SP3E (Mean = 46 %, SD = 3.8 %) Table 1 indicates the gate size and processing time of our PUF evaluation system, shown in Fig. 13. In the FPGA evaluation system, a software on the PC is used instead of detection circuits. Our PUF (not including detection circuits) uses only 5 % of the total slices in a FPGA, and the gate size is expected to be very small in ASICs. However, our PUF implemented in ASICs requires 128 detection circuits, and the gate size is estimated to be about 5.4 K gates, using the gate equivalencies introduced in [1]. The gate size of our PUF is comparable to that of compact hardware for common key block ciphers such as AES. Hence, our PUF is sufficiently small to be implemented in embedded systems. The gate size can be reduced by a shared detection circuit switched by an MUX. The processing time is around 0.4 ms, this being the total time taken to generate a response. One way of improving the processing time is to reduce the bitstream length for detection (1,000 bits in our experiment). However, too short a length may result in misdetection. For example, RS latches outputting a large number of 0s and very few 1s might be detected not as random, but as fixed latches. This misdetection leads to the loss of reliability, so our PUF makes a tradeoff between reliability and processing time. Our proposed PUF has advantages in terms of low noise because RS latches are allowed to become non-metastable through RS latch clock gating except when generating responses. In addition, our PUF can generate responses at anytime, unlike SRAM PUFs which can only generate them during power activation. Table 1 Gate size and processing time of our PUF (not including detection circuits) Evaluation of proposed method (2): number of random latches Before we represent an evaluation of proposed method (1), we show the effectiveness of proposed methods (2-A) and (2-B). Figure 16 represents a histogram showing the number of random latches per FPGA. The results show that the proposed methods increase the number of random latches. This is because these methods allow the RS latches to become readily metastable and increase their probability of outputting random numbers. In proposed method (1), the variety of responses for 128 RS latches takes its maximum value when the number of random latches is around 43. Hence, the proposed methods (2-A) and (2-B) are expected to improve the variety of responses by increasing the number of random latches to as close to 43 as possible. Histogram for the number of random latches per FPGA Evaluation of proposed method (1): variety and Shannon entropy of responses Review of our concept Table 2 shows the average number of random latches calculated using Fig. 16. The theoretical bounds of responses for various implementation methods are also calculated based on this average number of random latches and Fig. 8. The theoretical bound is estimated to be $2^{116} ({=}2^{128-12})$ when PUFs implemented by conventional method (2-B1) generate responses without 12 random latches. The PUFs based on proposed method (1) can generate $2^{170} ({\approx } 2^{128-12} \cdot {}_{128}\mathrm C _{12})$ responses using the location information entropy of 12 random latches. Moreover, PUFs based on both proposed methods (1) and (2-B) generate approximately $2^{196} ({\approx } 2^{128-32} \cdot {}_{128}\mathrm C _{32})$ responses with 32 random latches. Our proposed methods therefore dramatically increase the theoretical bound of responses. This theoretical bound of responses estimated in Table 2 is based on the strong assumption that the value of $T$ is the same on every FPGA, which means that all of PUF implementations have $T$ random latches strictly. However, the value of $T$ varies depending on individual FPGAs, as shown in Fig. 16. Since only fixed value of $T$ is not sufficient for a discussion on the variety of responses, the next section takes into account further calculations: the ratios of RS latches outputting 0s, 1s, and random numbers. This enables us to experimentally confirm the validity of the theoretical upper bound on the variety of responses under fixed value of $T$: $2^{N-T} \cdot {}_{N}\mathrm C _{T}$. Table 2 Average number of random latches and variety of responses The theoretical upper bound on the Shannon entropy of responses is defined as the binary logarithm of the theoretical bound: $\log _{2}(2^{N-T} \cdot {}_{N}\mathrm C _{T})$. In contrast, this section experimentally calculates the Shannon entropy of responses derived from manufactured PUFs, i.e., 40 PUFs implemented on SP3E FPGAs. Concretely, the Shannon entropy is calculated as the sum of each entropy experimentally derived from each latch. This Shannon entropy is more accurate than the above binary logarithm of the theoretical bound because the each entropy is calculated in consideration of detailed experimental results: the ratios of RS latches outputting 0s, 1s and random numbers. We will show later that this experimental entropy is equal to the theoretical upper bound, $\log _{2}(2^{N-T} \cdot {}_{N}\mathrm C _{T})$, under the ideal condition that fixed latches output 0s or 1s with equal probability. By following the two steps below, we can accurately calculate the experimental entropy of responses from our PUFs using the experimental results with 40 SP3E FPGAs. In the first step, we show the ratios of RS latches outputting 0s, 1s, and random numbers, shown in Fig. 17. We explain how to read the figure with the specific example in Fig. 18, as follows. First, the 40 RS latches at the same physical CLB location (e.g., ${\text{ LATCH}_{0}}$) on the 40 FPGAs are called "a latch group". Hence, in our experiment, there are 128 latch groups corresponding to the range from ${\text{ LATCH}_{0}}$ to ${\text{ LATCH}_{127}}$. Suppose that the 40 RS latches labeled as ${\text{ LATCH}_{0}}$ include 15 latches outputting 0s, 20 outputting 1s, and 5 outputting random numbers. The ratios are therefore 0.375, 0.500 and 0.125, respectively. A plot of ${\text{ LATCH}_{0}}$ is obtained by relating the ratios to the three sides of a triangle, and 128 plots are obtained, corresponding to the 128 latch groups in Fig. 17. A plot is located at the central point of the triangle if the ratios are equal, which is the ideal. It is therefore desirable that a large proportion of plots located in the small central triangle are illustrated by thick line. If the plot is in the small triangle, the three ratios fall within a range of 0.20–0.60. In conventional method (2-B1), it can be seen that all of the RS latches in each latch group have a low probability of outputting random numbers since many of the plots are located on the right side of the triangle. In addition, most RS latches in each latch group have a one-sided probability of outputting 0s or 1s since many of the plots are located throughout the whole of the right side. Conventional method (2-B2) improves the ratios, making them roughly equal, but requires a large number of CLBs to implement the RS latches shown in Fig. 5. In addition, there are not so many random latches (around 26 in Table 2), so the variety of responses is not very large. In contrast, proposed method (2-B) improves the ratios such that they are almost equal since as many as 93 plots are located in the small central triangle. Furthermore, no latch groups have RS latches outputting ternary values at a high $({>}0.9)$ or low $({<}0.1)$ probability. The number of plots in the small triangle is significantly higher than with conventional methods, which implies that the proposed method makes many of the RS latches readily metastable, so that the ratios become almost equal as a favorable side effect. Hence, using the proposed methods, the experimental variety of responses is expected to be close to the theoretical bound shown in Table 2. Ratios of RS latches outputting 0s, 1s, or random numbers in 128 latch groups How to read Fig. 17 In the second step, we accurately calculate the Shannon entropy of responses based on the ratios discussed in the first step. The Shannon entropy derived from ${\text{ LATCH}_{0}}$ to ${\text{ LATCH}_{127}}$ are given as $\sum _{i=0}^{n-1} E_{i}$, where $n=128$ and $E_{i}$ derived from ${\text{ LATCH}_{i}}$ is defined as $$\begin{aligned} E_{i}&= -P_i(0) \cdot \log _{2}P_i(0)-P_i(1) \cdot \log _{2}P_i(1)\nonumber \\&-P_i(R) \cdot \log _{2}P_i(R). \end{aligned}$$ Let the ratios of the RS latches labeled as ${\text{ LATCH}_{{ i}}}$ outputting 0s, 1s, or random numbers be $P_i(0)$, $P_i(1)$ and $P_i(R)$, respectively (e.g., $P_0(0)$ = 0.375, $P_0(1)$ = 0.500 and $P_0(R)$ = 0.125 in Fig. 18). In the following, we discuss the relation between the Shannon entropy: $\sum _{i=0}^{n-1} E_{i}$ and the theoretical bound on the entropy: $\log _{2}(2^{N-T} \cdot {}_{N}\mathrm C _{T})$. Here, we assume that RS latches operate independently from each other, so the Shannon entropy are calculated by the sum of $E_{i}$. We also assume ideal conditions that each of all 40 PUFs has $t$ random latches out of $n$ implemented RS latches, and the remaining $n-t$ fixed latches output 0s or 1s with equal probability. To be specific, $P_i(R)=t/n$ and $P_i(0)=P_i(1)=\frac{1}{2}(1-t/n)$ ($0 \le i \le n-1$). Based on these assumptions, each $E_{i}$ is identical, so $\sum _{i=0}^{n-1} E_{i}=n \cdot E_{i}$. $\sum _{i=0}^{n-1} E_{i}$ can be calculated as follows: $$\begin{aligned} \sum _{i=0}^{n-1} E_{i}&= n \cdot E_{i}\nonumber \\&= n \cdot \{ -P_i(0) \cdot \log _{2}P_i(0) -P_i(1) \cdot \log _{2}P_i(1) \nonumber \\&-P_i(R) \cdot \log _{2}P_i(R) \}\nonumber \\&= n \cdot \left\{ - \frac{1}{2} \cdot \frac{n-t}{n} \cdot \log _{2} \left( \frac{1}{2} \cdot \frac{n-t}{n} \right) \right.\nonumber \\&\left. - \frac{1}{2} \cdot \frac{n-t}{n} \cdot \log _{2} \left(\frac{1}{2} \cdot \frac{n-t}{n} \right) \right.\nonumber \\&\left. - \frac{t}{n} \cdot \log _{2} \frac{t}{n} \right\} \end{aligned}$$ which is equal to $(n-t) + n h(t/n)$ where $h(\cdot )$ is the binary entropy defined as $h(x) = -x\log _2 x -(1-x) \log _2(1-x)$ for $0 \le x \le 1$. It is well known that $\log _2 {}_n\mathrm C _t=n h(t/n)+o(1)$ holds for arbitrary integers $t$ and $n$ such that $t \le n$ (for instance, see [3, Example 11.1.3]). Hence, we have $$\begin{aligned} (n-t) + n h(t/n)&= (n-t) + \log _2 {}_n\mathrm C _t +o(1) \nonumber \\&\approx \log _2 (2^{n-t} \cdot {}_n\mathrm C _t). \end{aligned}$$ If the number of FPGAs are sufficiently large, $t/n$ means the probability that each latch outputs random numbers. Then, it holds that $t/n \approx T/N$, which implies that Eq. (4) is equal to our theoretical bound: $\log _{2}(2^{N-T} \cdot {}_{N}\mathrm C _{T})$. Therefore, in consideration of the ratios of RS latches outputting 0s, 1s, and random numbers in first step, the experimental variety of responses can be accurately calculated from $\sum _{i=0}^{n-1} E_{i}$. Table 3 shows the Shannon entropy of responses experimentally calculated from $\sum _{i=0}^{n-1} E_{i}$. A PUF with 128 RS latches based on conventional method (2-B1) generates $2^{126.6}$ responses even if proposed method (1) is applied. This is because the number of random latches is small, and the ratios are not equal. In contrast, the PUF based on proposed method (2-B) generates $2^{192.7}$ responses, which is almost the same as the upper bound in Table 2 and is larger than for PUFs based on conventional methods. Hence, a PUF based on both proposed methods reduces circuit size and dramatically improves the variety and entropy of responses. The entropy derived from an unit area (gate size) of proposed method (1) is expected to be higher than that of conventional methods (1-A) and (1-B). Both proposed and conventional methods (1-A) requires a mechanism to detect random latches, so their area sizes are almost the same, while the entropy of proposed method (1) is higher from Table 3. In contrast, conventional method (1-B) does not require the mechanism, so the area size is smaller than proposed method (1). The entropy of conventional method (1-B) seems to be higher by implementing more RS latches. In fact, however, conventional method (1-B) needs to correct the variation resulting from all the random latches, which requires larger redundant data for stable responses. In contrast, proposed method (1) considers random numbers as the third stable value, which leads to a suitable size of redundant data for embedded systems. Therefore, in consideration of the area size for redundant data, the proposed method is expected to generate higher entropy per unit area. Table 3 Shannon entropy of responses Temperature resistance This section evaluates the robustness of our PUF against temperature variation – the reliability of responses when temperature is changed within the rated temperature of the SP3E FPGAs (0–85 $^\circ $C). In this evaluation, one 256-bit response is generated as the reference at the standard temperature of 25 $^\circ $C, and the other response is generated for analysis at 0 or 85 $^\circ $C. Figure 19 shows histograms of normalized Hamming distances between the reference response and the analysis one [i.e., 1 $\times $ 40(chips) = 40 elements]. Due to space constraints, we show the results at the lower temperature 0 $^\circ $C and at the higher temperature 85 $^\circ $C. At 0 and 85 $^\circ $C, the average error rate is approximately 2.4 and 5.0 % with an SD of 0.95 and 1.2 %, respectively. The bigger the temperature difference from 25 $^\circ $C—as the standard temperature—the higher the error rate. The error rate is less than around 15 % regardless of temperature, so stable responses are generated based on a Fuzzy Extractor with a reasonable size of redundant data [14]. Hence, our PUF has sufficient robustness against temperature variation. Reliability at 0 and 85 $^\circ $C Performance evaluation on Spartan-6 This section evaluates the performance of our PUF using SP6 FPGA. The reason why we use not only SP3E but also SP6 is to confirm the effectiveness of our variety enhancement on a different architecture. In addition, we evaluate the robustness of our PUF against voltage variation—the reliability of responses when changing the supply voltage to FPGAs. Figure 20 shows our experimental evaluation system, which uses a custom-made expansion board with a Xilinx SP6 FPGA (XC6SLX16-2CSG324C). We cannot control the core voltage of an SP3E because a regulator on the starter kit board makes it 1.20 V forcibly. In contrast, the core voltage of an SP6 chip can be easily changed by 0.01 V using a stabilized power supply. The expansion board is connected to the SP3E board. We implement manually 128 RS latches in accordance with proposed method (2-C). We implement the RS latches on an SP6 FPGA, and other peripheral circuits on an SP3E FPGA. Such configuration enables us to change only the core voltage of the SP6 chip on which the PUF is implemented. The voltage change does not impact the peripheral circuits, which enhances the confidence of our experimental results. The data acquisition process is the same as when using only SP3E except the method of sending the data to the PC. The values of the block RAM are sent to a SD write module, and written into a micro SD card. The PC can obtain the data through the micro SD card faster than through an RS232C port. This difference does not influence PUF performance. The evaluation uses 20 actual SP6 chips, but we take the number of chips to be 40 since we implement manually the RS latches at two completely different locations. Experimental evaluation system using SP6 The reliability and uniqueness results are shown in Figs. 21 and 22, respectively. In the reliability evaluation at normal operating condition (room temperature and standard supply voltage of 1.20 V), 101 256-bit responses are generated per SP6 FPGA chip. One response is used as the reference, and the remaining are used for analysis. Figure 21 shows a histogram of normalized Hamming distances between the reference response and each repeated one [i.e., 100 $\times $ 40(chips) = 4,000 elements]. The average error rate is approximately 0.86 % with an SD of 0.54 %. Hence, our result shows that our PUF on SP6 yields highly reliable responses. Next, in order to evaluate the uniqueness, a total of 40 256-bit responses using all 40 FPGAs (one response per FPGA) is generated. Figure 22 shows a histogram of normalized Hamming distances between every combination of two responses, i.e., ${}_{40}\mathrm C _{2}=780$ combinations, in the same way as the evaluation on SP3E. The difference in the responses of two arbitrary PUFs is approximately 49 % with an SD of 3.9 %. Our PUF gives responses with a high level of uniqueness. Here, this difference is a little larger than the ideal 44.4 %, as mentioned in Sect. 4.2. This is because the average number of random latches is 14, which is smaller than 43 (=128/3). Consequently, most of the 2-bit partial responses are '00' or '11', so the difference approaches 50 % similar to the conventional PUF using binary values (0/1). According to Fig. 8, PUFs on SP6 using both proposed methods (1) and (2-C) is estimated to generate approximately $2^{175}$ responses with 14 random latches. Finally, the Shannon entropy of the responses is $2^{167.9}$ based on the same steps in Sect. 4.5. A PUF based on our proposed methods improves the entropy of responses on various architectures. Reliability on SP6 (Mean = 0.86 %, SD = 0.54 %) Uniqueness on SP6 (Mean = 49 %, SD = 3.9 %) In this section, we evaluate the robustness of our PUF against voltage variation—the reliability of responses when a supply voltage is changed within the rated voltage range of SP6 FPGAs (1.14–1.26 V). In this evaluation, one response is generated as the reference at the standard voltage of 1.20 V, and the remaining 100 responses are generated at 1.14 or 1.26 V. Figures 23 and 24 show histograms of normalized Hamming distances between the reference response and each repeated one (4,000 elements). Due to space constraints, we show the results at the lower voltage 1.14 V and at the higher voltage 1.26 V. At 1.14 and 1.26 V, the average error rate is approximately 5.3 and 4.8 % with an SD of 1.3 and 1.6 %, respectively. This error rate is much less than the 15 % assumed in [14] for stable responses based on a Fuzzy Extractor with a reasonable size of redundant data. Hence, our PUF has sufficient robustness against voltage variation. Reliability at 1.14 V (Mean = 5.3 %, SD = 1.3 %) Applicability of proposed methods to delay-based PUFs The above section discusses the applicability of our proposed methods to LPUFs, one of the most feasible memory-based PUFs. The proposed methods can be also applied to delay-based PUFs such as Arbiter PUFs, Glitch PUFs and Ring Oscillator PUFs as shown in Fig. 25. The delay-based PUFs generate a one-bit response corresponding to an $\omega $-bit challenge. If a challenge $CHA_{i}$(0 $\le $ $i$ $\le $ $N-1$) is input to the PUFs repeatedly for $p$ times ($p$ = 1,000 in this ork), a $p$-bit output bitstream is obtained. The detection circuit outputs a 2-bit partial response $S_i[1:0]$(=00/11/10) based on the bitstream in the same way as mentioned in Sect. 3.1. Like memory-based PUFs, delay-based PUFs using our proposed methods also output a $2N$-bit response RES$[2N-1:0]$ satisfying Eq. (1) when inputting $N$ patterns of challenges. Hence, our proposed methods are expected to improve the variety and entropy of responses regardless of the kinds of PUFs. Applying our proposed methods to delay-based PUFs This paper proposed a method for generating responses from an LPUF based on the location information of RS latches outputting random numbers. Our proposed detection circuit generates ternary values (00/11/10) in accordance with the three types of output bitstream from RS latches. This dramatically increases the variety of responses from $2^{N}$ to $2^{N-T} \cdot {}_{N}\mathrm C _{T}$ with $N$ implemented RS latches and $T$ random latches. In addition, with its small circuit size, the new implementation method increases the number of random latches and equalizes the ratios of RS latches outputting 0s, 1s, and random numbers, thereby enhancing the effectiveness of the proposed method. According to our experiment with FPGAs, an LPUF with 128 RS latches based on the proposed methods is able to generate responses with 193-bit Shannon entropy, which is larger than the 116-bit Shannon entropy achieved by conventional methods. Our LPUFs also have high robustness against temperature and voltage variation. The proposed methods can be applied to other delay-based PUFs, such as the Arbiter PUF. Inconsistent (random) outputs from the PUF can be used for generating highly unique responses without the necessity of selecting available challenges. Future work will include discussion of performance evaluations on ASICs. Batina, L., Lano, J., Mentens, N., Örs, S.B., Preneel, B., Verbauwhede, I.: Energy, performance, area versus security trade-offs for stream ciphers. 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In: Proceedings of DAC 2007, pp. 9–14 (2007) Suzuki, D., Shimizu, K.: The Glitch PUF: a new delay-PUF architecture exploiting Glitch shapes. In: CHES 2010, pp. 366–382 (2010) van der Leest, V., Preneel, B., van der Sluis, E.: Soft decision error correction for compact memory-based PUFs using a single enrollment. In: CHES 2012. Springer (2012) Xilinx: Spartan-3E starter kit board. http://www.xilinx.com/products/devkits/HW-SPAR3E-SK-US-G.htm Yamamoto, D., Sakiyama, K., Iwamoto, M., Ohta, K., Ochiai, T., Takenaka, M., Itoh, K.: Uniqueness enhancement of PUF responses based on the locations of random outputting RS latches. In: CHES 2011, pp. 390–406. Springer (2011) This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. Fujitsu Laboratories Ltd., 4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki-shi, Kanagawa, 211-8588, Japan Dai Yamamoto, Masahiko Takenaka & Kouichi Itoh The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan Kazuo Sakiyama, Mitsugu Iwamoto & Kazuo Ohta Dai Yamamoto Kazuo Sakiyama Mitsugu Iwamoto Kazuo Ohta Masahiko Takenaka Kouichi Itoh Correspondence to Dai Yamamoto. Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Yamamoto, D., Sakiyama, K., Iwamoto, M. et al. Variety enhancement of PUF responses using the locations of random outputting RS latches. J Cryptogr Eng 3, 197–211 (2013). https://doi.org/10.1007/s13389-012-0044-0 Issue Date: November 2013 Metastable	CommonCrawl
Parity P In computational complexity theory, the complexity class ⊕P (pronounced "parity P") is the class of decision problems solvable by a nondeterministic Turing machine in polynomial time, where the acceptance condition is that the number of accepting computation paths is odd. An example of a ⊕P problem is "does a given graph have an odd number of perfect matchings?" The class was defined by Papadimitriou and Zachos in 1983.[1] ⊕P is a counting class, and can be seen as finding the least significant bit of the answer to the corresponding #P problem. The problem of finding the most significant bit is in PP. PP is believed to be a considerably harder class than ⊕P; for example, there is a relativized universe (see oracle machine) where P = ⊕P ≠ NP = PP = EXPTIME, as shown by Beigel, Buhrman, and Fortnow in 1998.[2] While Toda's theorem shows that PPP contains PH, P⊕P is not known to even contain NP. However, the first part of the proof of Toda's theorem shows that BPP⊕P contains PH. Lance Fortnow has written a concise proof of this theorem.[3] ⊕P contains the graph automorphism problem, and in fact this problem is low for ⊕P.[4] It also trivially contains UP, since all problems in UP have either zero or one accepting paths. More generally, ⊕P is low for itself, meaning that such a machine gains no power from being able to solve any ⊕P problem instantly. The ⊕ symbol in the name of the class may be a reference to use of the symbol ⊕ in Boolean algebra to refer the exclusive disjunction operator. This makes sense because if we consider "accepts" to be 1 and "not accepts" to be 0, the result of the machine is the exclusive disjunction of the results of each computation path. External links • Complexity Zoo: Class Parity P References 1. C. H. Papadimitriou and S. Zachos. Two remarks on the power of counting. In Proceedings of the 6th GI Conference in Theoretical Computer Science, Lecture Notes in Computer Science, volume 145, Springer-Verlag, pp. 269–276. 1983. 2. R. Beigel, H. Buhrman, and L. Fortnow. NP might not be as easy as detecting unique solutions. In Proceedings of ACM STOC'98, pp. 203–208. 1998. 3. Fortnow, Lance (2009), "A simple proof of Toda's theorem", Theory of Computing, 5: 135–140, doi:10.4086/toc.2009.v005a007 4. Köbler, Johannes; Schöning, Uwe; Torán, Jacobo (1992), "Graph isomorphism is low for PP", Computational Complexity, 2 (4): 301–330, doi:10.1007/BF01200427. Important complexity classes Considered feasible • DLOGTIME • AC0 • ACC0 • TC0 • L • SL • RL • NL • NL-complete • NC • SC • CC • P • P-complete • ZPP • RP • BPP • BQP • APX • FP Suspected infeasible • UP • NP • NP-complete • NP-hard • co-NP • co-NP-complete • AM • QMA • PH • ⊕P • PP • #P • #P-complete • IP • PSPACE • PSPACE-complete Considered infeasible • EXPTIME • NEXPTIME • EXPSPACE • 2-EXPTIME • ELEMENTARY • PR • R • RE • ALL Class hierarchies • Polynomial hierarchy • Exponential hierarchy • Grzegorczyk hierarchy • Arithmetical hierarchy • Boolean hierarchy Families of classes • DTIME • NTIME • DSPACE • NSPACE • Probabilistically checkable proof • Interactive proof system List of complexity classes	Wikipedia
Gauge Group and 4-dim Topology 数学笔记 [转载]What's a Gauge? [转载]What's a Gauge?无评论 From: Terence Tao's blog: What's a Gauge. "Gauge theory" is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a "coordinate system" that varies depending on one's "location" with respect to some "base space" or "parameter space", a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis. I was asked recently to explain what a gauge theory was, and so I will try to do so in this post. For simplicity, I will focus exclusively on classical gauge theories; quantum gauge theories are the quantization of classical gauge theories and have their own set of conceptual difficulties (coming from quantum field theory) that I will not discuss here. While gauge theories originated from physics, I will not discuss the physical significance of these theories much here, instead focusing just on their mathematical aspects. My discussion will be informal, as I want to try to convey the geometric intuition rather than the rigorous formalism (which can, of course, be found in any graduate text on differential geometry). 0.1. Coordinate systems Before I discuss gauges, I first review the more familiar concept of a coordinate system, which is basically the special case of a gauge when the base space (or parameter space) is trivial. Classical mathematics, such as practised by the ancient Greeks, could be loosely divided into two disciplines, geometry and number theory, where I use the latter term very broadly, to encompass all sorts of mathematics dealing with any sort of number. The two disciplines are unified by the concept of a coordinate system, which allows one to convert geometric objects to numeric ones or vice versa. The most well known example of a coordinate system is the Cartesian coordinate system for the plane (or more generally for a Euclidean space), but this is just one example of many such systems. For instance: 1. One can convert a length (of, say, an interval) into an (unsigned) real number, or vice versa, once one fixes a unit of length (e.g. the metre or the foot). In this case, the coordinate system is specified by the choice of length unit. 2. One can convert a displacement along a line into a (signed) real number, or vice versa, once one fixes a unit of length and an orientation along that line. In this case, the coordinate system is specified by the length unit together with the choice of orientation. Alternatively, one can replace the unit of length and the orientation by a unit displacement vector $e$ along the line. 3. One can convert a position (i.e. a point) on a line into a real number, or vice versa, once one fixes a unit of length, an orientation along the line, and an origin on that line. Equivalently, one can pick an origin $O$ and a unit displacement vector $e$. This coordinate system essentially identifies the original line with the standard real line ${\Bbb R}$. 4. One can generalise these systems to higher dimensions. For instance, one can convert a displacement along a plane into a vector in ${\Bbb R}^2$, or vice versa, once one fixes two linearly independent displacement vectors $e_1, e_2$ (i.e. a basis) to span that plane; the Cartesian coordinate system is just one special case of this general scheme. Similarly, one can convert a position on a plane to a vector in ${\Bbb R}^2$ once one picks a basis $e_1, e_2$ for that plane as well as an origin $O$, thus identifying that plane with the standard Euclidean plane ${\Bbb R}^2$. (To put it another way, units of measurement are nothing more than one-dimensional (i.e. scalar) coordinate systems.) 5. To convert an angle in a plane to a signed number (modulo multiples of $2\pi$), or vice versa, one needs to pick an orientation on the plane (e.g. to decide that anti-clockwise angles are positive). 8. To convert a direction in a plane to a signed number (again modulo multiples of $2\pi$), or vice versa, one needs to pick an orientation on the plane, as well as a reference direction (e.g. true or magnetic north is often used in the case of ocean navigation). 9. Similarly, to convert a position on a circle to a number (modulo multiples of $2\pi$), or vice versa, one needs to pick an orientation on that circle, together with an origin on that circle. Such a coordinate system then equates the original circle to the standard unit circle $S^1 := \{ z \in {\Bbb C}: \|z\| = 1 \}$ (with the standard origin $+1$ and the standard anticlockwise orientation $\circlearrowleft$). 10. To convert a position on a two-dimensional sphere (e.g. the surface of the Earth, as a first approximation) to a point on the standard unit sphere $S^2 := \{ (x,y,z) \in {\Bbb R}^3: x^2+y^2+z^2 \}$, one can pick an orientation on that sphere, an "origin" (or "north pole") for that sphere, and a "prime meridian" connecting the north pole to its antipode. Alternatively, one can view this coordinate system as determining a pair of Euler angles$\phi, \lambda$ (or a latitude and longitude) to be assigned to every point on one's original sphere. 28. The above examples were all geometric in nature, but one can also consider "combinatorial" coordinate systems, which allow one to identify combinatorial objects with numerical ones. An extremely familiar example of this is enumeration: one can identify a set A of (say) five elements with the numbers 1,2,3,4,5 simply by choosing an enumeration $a_1, a_2, \ldots, a_5$ of the set A. One can similarly enumerate other combinatorial objects (e.g. graphs, relations, trees, partial orders, etc.), and indeed this is done all the time in combinatorics. Similarly for algebraic objects, such as cosets of a subgroup H (or more generally, torsors of a group G); one can identify such a coset with H itself by designating an element of that coset to be the "identity" or "origin". More generally, a coordinate system $\Phi$ can be viewed as an isomorphism $\Phi: A \to G$ between a given geometric (or combinatorial) object A in some class (e.g. a circle), and a standard object G in that class (e.g. the standard unit circle). (To be pedantic, this is what a global coordinate system is; a local coordinate system, such as the coordinate charts on a manifold, is an isomorphism between a local piece of a geometric or combinatorial object in a class, and a local piece of a standard object in that class. I will restrict attention to global coordinate systems for this discussion.) Coordinate systems identify geometric or combinatorial objects with numerical (or standard) ones, but in many cases, there is no natural (or canonical) choice of this identification; instead, one may be faced with a variety of coordinate systems, all equally valid. One can of course just fix one such system once and for all, in which case there is no real harm in thinking of the geometric and numeric objects as being equivalent. If however one plans to change from one system to the next (or to avoid using such systems altogether), then it becomes important to carefully distinguish these two types of objects, to avoid confusion. For instance, if an interval AB is measured to have a length of 3 yards, then it is OK to write $\|AB\|=3$ (identifying the geometric concept of length with the numeric concept of a positive real number) so long as you plan to stick to having the yard as the unit of length for the rest of one's analysis. But if one was also planning to use, say, feet, as a unit of length also, then to avoid confusing statements such as "$\|AB\|=3$and $\|AB\|=9$", one should specify the coordinate systems explicitly, e.g. "$\|AB\| = 3 \hbox{ yards}$and $\|AB\| = 9 \hbox{ feet}$". Similarly, identifying a point P in a plane with its coordinates (e.g. $P = (4,3)$) is safe as long as one intends to only use a single coordinate system throughout; but if one intends to change coordinates at some point (or to switch to a coordinate-free perspective) then one should be more careful, e.g. writing $P = 4 e_1 + 3 e_2$, or even $P = O + 4 e_1 + 3 e_2$, if the origin O and basis vectors $e_1, e_2$ of one's coordinate systems might be subject to future change. As mentioned above, it is possible to in many cases to dispense with coordinates altogether. For instance, one can view the length $\|AB\|$ of a line segment AB not as a number (which requires one to select a unit of length), but more abstractly as the equivalence class of all line segments CD that are congruent to AB. With this perspective, $\|AB\|$ no longer lies in the standard semigroup${\Bbb R}^+$, but in a more abstract semigroup ${\mathcal L}$ (the space of line segments quotiented by congruence), with addition now defined geometrically (by concatenation of intervals) rather than numerically. A unit of length can now be viewed as just one of many different isomorphisms $\Phi: {\mathcal L} \to {\Bbb R}^+$ between ${\mathcal L}$ and ${\Bbb R}^+$, but one can abandon the use of such units and just work with ${\mathcal L}$ directly. Many statements in Euclidean geometry involving length can be phrased in this manner. For instance, if B lies in AC, then the statement $\|AC\|=\|AB\|+\|BC\|$ can be stated in ${\mathcal L}$, and does not require any units to convert ${\mathcal L}$ to ${\mathcal R}^+$; with a bit more work, one can also make sense of such statements as $\|AC\|^2 = \|AB\|^2 + \|BC\|^2$ for a right-angled triangle ABC (i.e. Pythagoras' theorem) while avoiding units, by defining a symmetric bilinear product operation $\times: {\mathcal L} \times {\mathcal L} \to {\mathcal A}$ from the abstract semigroup ${\mathcal L}$ of lengths to the abstract semigroup ${\mathcal A}$ of areas. (Indeed, this is basically how the ancient Greeks, who did not quite possess the modern real number system${\Bbb R}$, viewed geometry, though of course without the assistance of such modern terminology as "semigroup" or "bilinear".) The above abstract coordinate-free perspective is equivalent to a more concrete coordinate-invariant perspective, in which we do allow the use of coordinates to convert all geometric quantities to numeric ones, but insist that every statement that we write down is invariant under changes of coordinates. For instance, if we shrink our chosen unit of length by a factor $\lambda > 0$, then the numerical length of every interval increases by a factor of $\lambda$, e.g. $\|AB\| \mapsto \lambda \|AB\|$. The coordinate-invariant approach to length measurement then treats lengths such as $\|AB\|$ as numbers, but requires all statements involving such lengths to be invariant under the above scaling symmetry. For instance, a statement such as $\|AC\|^2 = \|AB\|^2 + \|BC\|^2$ is legitimate under this perspective, but a statement such as $\|AB\| = \|BC\|^2$ or $\|AB\| = 3$ is not. [In other words, co-ordinate invariance here is the same thing as being dimensionally consistent. Indeed, dimensional analysis is nothing more than the analysis of the scaling symmetries in one's coordinate systems.] One can retain this coordinate-invariance symmetry throughout one's arguments; or one can, at some point, choose to spend (or break) this coordinate invariance by selecting (or fixing) the coordinate system (which, in this case, means selecting a unit length). The advantage in spending such a symmetry is that one can often normalise one or more quantities to equal a particularly nice value; for instance, if a length $\|AB\|$ is appearing everywhere in one's arguments, and one has carefully retained coordinate-invariance up until some key point, then it can be convenient to spend this invariance to normalise $\|AB\|$ to equal 1. (In this case, one only has a one-dimensional family of symmetries, and so can only normalise one quantity at a time; but when one's symmetry group is larger, one can often normalise many more quantities at once; as a rule of thumb, one can normalise one quantity for each degree of freedom in the symmetry group.) Conversely, if one has already spent the coordinate invariance, one can often buy it back by converting all the facts, hypotheses, and desired conclusions one currently possesses in the situation back to a coordinate-invariant formulation. Thus one could imagine performing one normalisation to do one set of calculations, then undoing that normalisation to return to a coordinate-free perspective, doing some coordinate-free manipulations, and then performing a different normalisation to work on another part of the problem, and so forth. (For instance, in Euclidean geometry problems, it is often convenient to temporarily assign one key point to be the origin (thus spending translation invariance symmetry), then another, then switch back to a translation-invariant perspective, and so forth. As long as one is correctly accounting for what symmetries are being spent and bought at any given time, this can be a very powerful way of simplifying one's calculations.) Given a coordinate system $\Phi: A \to G$ that identifies some geometric object A with a standard object G, and some isomorphism $\Psi: G \to G$ of that standard object, we can obtain a new coordinate system $\Psi \circ \Phi: A \to G$ of A by composing the two isomorphisms. [I will be vague on what "isomorphism" means; one can formalise the concept using the language of category theory.] Conversely, every other coordinate system $\Phi': A \to G$ of $A$ arises in this manner. Thus, the space of coordinate systems on A is (non-canonically) identifiable with the isomorphism group $\hbox{Isom}(G)$ of G. This isomorphism group is called the structure group (or gauge group) of the class of geometric objects. For example, the structure group for lengths is ${\Bbb R}^+$; the structure group for angles is ${\Bbb Z}/2{\Bbb Z}$; the structure group for lines is the affine group$\hbox{Aff}({\Bbb R})$; the structure group for $n$-dimensional Euclidean geometry is the Euclidean group$E(n)$; the structure group for (oriented) 2-spheres is the (special) orthogonal group$SO(3)$; and so forth. (Indeed, one can basically describe each of the classical geometries (Euclidean, affine, projective, spherical, hyperbolic, Minkowski, etc.) as a homogeneous space for its structure group, as per the Erlangen program.) 0.2. Gauges In our discussion of coordinate systems, we focused on a single geometric (or combinatorial) object $A$: a single line, a single circle, a single set, etc. We then used a single coordinate system to identify that object with a standard representative of such an object. Now let us consider the more general situation in which one has a family (or fibre bundle) $(A_x)_{x \in X}$ of geometric (or combinatorial) objects (or fibres) $A_x$: a family of lines (i.e. a line bundle), a family of circles (i.e. a circle bundle), a family of sets, etc. This family is parameterised by some parameter set or base point x, which ranges in some parameter space or base space X. In many cases one also requires some topological or differentiable compatibility between the various fibres; for instance, continuous (or smooth) variations of the base point should lead to continuous (or smooth) variations in the fibre. For sake of discussion, however, let us gloss over these compatibility conditions. In many cases, each individual fibre $A_x$ in a bundle $(A_x)_{x \in X}$, being a geometric object of a certain class, can be identified with a standard object $G$ in that class, by means of a separate coordinate system $\Phi_x: A_x \to G$ for each base point x. The entire collection $\Phi = (\Phi_x)_{x \in X}$ is then referred to as a (global) gauge or trivialisation for this bundle (provided that it is compatible with whatever topological or differentiable structures one has placed on the bundle, but never mind that for now). Equivalently, a gauge is a bundle isomorphism$\Phi$ from the original bundle $(A_x)_{x \in X}$ to the trivial bundle$(G)_{x \in X}$, in which every fibre is the standard geometric object G. (There are also local gauges, which only trivialise a portion of the bundle, but let's ignore this distinction for now.) Let's give three concrete examples of bundles and gauges; one from differential geometry, one from dynamical systems, and one from combinatorics. Example 1: the circle bundle of the sphere. Recall from the previous section that the space of directions in a plane (which can be viewed as the circle of unit vectors) can be identified with the standard circle $S^1$ after picking an orientation and a reference direction. Now let us work not on the plane, but on a sphere, and specifically, on the surface X of the earth. At each point x on this surface, there is a circle $S_x$ of directions that one can travel along the sphere from x; the collection $SX := (S_x)_{x \in X}$ of all such circles is then a circle bundle with base space X (known as the circle bundle; it could also be viewed as the sphere bundle, cosphere bundle, or orthonormal frame bundle of X). The structure group of this bundle is the circle group $U(1) \equiv S^1$ if one preserves orientation, or the semi-direct product$S^1 \rtimes {\Bbb Z}/2{\Bbb Z}$ otherwise. Now suppose, at every point x on the earth X, the wind is blowing in some direction $w_x \in S_x$. (This is not actually possible globally, thanks to the hairy ball theorem, but let's ignore this technicality for now.) Thus wind direction can be thought of as a collection $w = (w_x)_{x \in X}$ of representatives from the fibres of the fibre bundle $(S_x)_{x \in X}$; such a collection is known as a section of the fibre bundle (it is to bundles as the concept of a graph$\{ (x, f(x)): x \in X \} \subset X \times G$ of a function $f: X \to G$ is to the trivial bundle $(G)_{x \in X}$). At present, this section has not been represented in terms of numbers; instead, the wind direction $w (w_x)_{x \in X}$ is a collection of points on various different circles in the circle bundle SX. But one can convert this section w into a collection of numbers (and more specifically, a function $u: X \to S^1$ from X to $S^1$) by choosing a gauge for this circle bundle – in other words, by selecting an orientation $\epsilon_x$ and a reference direction $N_x$ for each point x on the surface of the Earth X. For instance, one can pick the anticlockwise orientation $\circlearrowleft$ and true north for every point x (ignore for now the problem that this is not defined at the north and south poles, and so is merely a local gauge rather than a global one), and then each wind direction $w_x$ can now be identified with a unit complex number $u(x) \in S^1$ (e.g. $e^{i\pi/4}$ if the wind is blowing in the northwest direction at x). Now that one has a numerical function u to play with, rather than a geometric object w, one can now use analytical tools (e.g. differentiation, integration, Fourier transforms, etc.) to analyse the wind direction if one desires. But one should be aware that this function reflects the choice of gauge as well as the original object of study. If one changes the gauge (e.g. by using magnetic north instead of true north), then the function u changes, even though the wind direction w is still the same. If one does not want to spend the U(1) gauge symmetry, one would have to take care that all operations one performs on these functions are gauge-invariant; unfortunately, this restrictive requirement eliminates wide swathes of analytic tools (in particular, integration and the Fourier transform) and so one is often forced to break the gauge symmetry in order to use analysis. The challenge is then to select the gauge that maximises the effectiveness of analytic methods. $\diamond$$ Example 2: circle extensions of a dynamical system. Recall (see e.g. my lecture notes) that a dynamical system is a pair X = (X,T), where X is a space and $T: X \to X$ is an invertible map. (One can also place additional topological or measure-theoretic structures on this system, as is done in those notes, but we will ignore these structures for this discussion.) Given such a system, and given a cocycle$\rho: X \to S^1$ (which, in this context, is simply a function from X to the unit circle), we can define the skew product$X \times_\rho S^1$ of X and the unit circle $S^1$, twisted by the cocycle $\rho$, to be the Cartesian product $X \times S^1 := \{ (x,u): x \in X, u \in S^1 \}$ with the shift $\tilde T: (x,u) \mapsto (Tx, \rho(x) u)$; this is easily seen to be another dynamical system. (If one wishes to have a topological or measure-theoretic dynamical system, then $\rho$ will have to be continuous or measurable here, but let us ignore such issues for this discussion.) Observe that there is a free action$(S_v: (x,u) \mapsto (x,vu))_{v \in S^1}$ of the circle group $S^1$ on the skew product $X \times_\rho S^1$ that commutes with the shift $\tilde T$; the quotient space$(X \times_\rho S^1)/S^1$ of this action is isomorphic to X, thus leading to a factor map$\pi: X \times_\rho S^1 \to X$, which is of course just the projection map $\pi: (x,u) \mapsto x$. (An example is provided by the skew shift system, described in my lecture notes.) Conversely, suppose that one had a dynamical system $\tilde X = (\tilde X, \tilde T)$ which had a free $S^1$ action $(S_v: \tilde X \to \tilde X)_{v \in S^1}$ commuting with the shift $\tilde T$. If we set $X := \tilde X/S^1$ to be the quotient space, we thus have a factor map $\pi: \tilde X \to X$, whose level sets $\pi^{-1}(\{x\})$ are all isomorphic to the circle $S^1$; we call $\tilde X$ a circle extension of the dynamical system X. We can thus view $\tilde X$ as a circle bundle$(\pi^{-1}(\{x\}))_{x \in X}$ with base space X, thus the level sets $\pi^{-1}(\{x\})$ are now the fibres of the bundle, and the structure group is $S^1$. If one picks a gauge _for this bundle, by choosing a reference point $p_x \in \pi^{-1}(\{x\})$ in the fibre for each base point x (thus in this context a gauge is the same thing as a _section$p = (p_x)_{x \in X}$; this is basically because this bundle is a principal bundle), then one can identify $\tilde X$ with a skew product $X \times_\rho S^1$ by identifying the point $S_v p_x \in \tilde X$ with the point $(x,v) \in X \times_\rho S^1$ for all $x \in X, v \in S^1$, and letting $\rho$ be the cocycle defined by the formula $$S_{\rho(x)} p_{Tx} = \tilde T p_x.$$ One can check that this is indeed an isomorphism of dynamical systems; if all the various objects here are continuous (resp. measurable), then one also has an isomorphism of topological dynamical systems (resp. measure-preserving systems). Thus we see that gauges allow us to write circle extensions as skew products. However, more than one gauge is available for any given circle extension; two gauges $(p_x)_{x \in X}$, $(p'_x)_{x \in X}$ will give rise to two skew products $X \times_\rho S^1$, $X \times_{\rho'} S^1$ which are isomorphic but not identical. Indeed, if we let $v: X \to S^1$ be a rotation map that sends $p_x$ to $p'_{x}$, thus $p'_{x} = S_{v(x)} p_x$, then we see that the two cocycles $\rho'$ and $\rho$ are related by the formula $$\rho'(x) = v(Tx)^{-1} \rho(x) v(x).\tag{1}$$ Two cocycles that obey the above relation are called cohomologous; their skew products are isomorphic to each other. An important general question in dynamical systems is to understand when two given cocycles are in fact cohomologous, for instance by introducing non-trivial cohomological invariants for such cocycles. As an example of a circle extension, consider the sphere $X = S^2$ from Example 1, with a rotation shift T given by, say, rotating anti-clockwise by some given angle $\alpha$ around the axis connecting the north and south poles. This rotation also induces a rotation on the circle bundle $\tilde X := SX$, thus giving a circle extension of the original system $(X,T)$. One can then use a gauge to write this system as a skew product. For instance, if one selects the gauge that chooses $p_x$ to be the true north direction at each point x (ignoring for now the fact that this is not defined at the two poles), then this system becomes the ordinary product $X \times_0 S^1$ of the original system X with the circle $S^1$, with the cocycle being the trivial cocycle 0. If we were however to use a different gauge, e.g. magnetic north instead of true north, one would obtain a different skew-product $X \times_{\rho'} S^1$, where $\rho'$ is some cocycle which is cohomologous to the trivial cocycle (except at the poles). (A cocycle which is globally cohomologous to the trivial cocycle is known as a coboundary. Not every cocycle is a coboundary, especially once one imposes topological or measure-theoretic structure, thanks to the presence of various topological or measure-theoretic invariants, such as degree.) There was nothing terribly special about circles in this example; one can also define group extensions, or more generally homogeneous space extensions, of dynamical systems, and have a similar theory, although one has to take a little care with the order of operations when the structure group is non-abelian; see e.g. my lecture notes on isometric extensions. $\diamond$ Example 3: Orienting an undirected graph. The language of gauge theory is not often used in combinatorics, but nevertheless combinatorics does provide some simple discrete examples of bundles and gauges which can be useful in getting an intuitive grasp of the concept. Consider for instance an undirected graph G = (V,E) of vertices and edges. I will let X=E denote the space of edges (not the space of vertices)!. Every edge $e \in X$ can be oriented (or directed) in two different ways; let $A_e$ be the pair of directed edges of e arising in this manner. Then $(A_e)_{e \in X}$ is a fibre bundle with base space X and with each fibre isomorphic (in the category of sets) to the standard two-element set $\{-1,+1\}$, with structure group ${\Bbb Z}/2{\Bbb Z}$. A priori, there is no reason to prefer one orientation of an edge e over another, and so there is no canonical way to identify each fibre $A_e$ with the standard set $\{-1,+1\}$. Nevertheless, we can go ahead and arbitrary select a gauge for X by orienting the graph G. This orientation assigns an oriented edge $\vec e \in A_e$ to each edge $e \in X$, thus creating a gauge (or section) $(\vec e)_{e \in X}$ of the bundle $(A_e)_{e \in X}$. Once one selects such a gauge, we can now identify the fibre bundle $(A_e)_{e \in X}$ with the trivial bundle $X \times \{-1,+1\}$ by identifying the preferred oriented edge $\vec e$ of each unoriented edge $e \in X$ with $(e,+1)$, and the other oriented edge with $(e,-1)$. In particular, any other orientation of the graph G can be expressed relative to this reference orientation as a function $f: X \to \{-1,+1\}$, which measures when the two orientations agree or disagree with each other. $\diamond$ Recall that every isomorphism $\Psi \in \hbox{Isom}(G)$ of a standard geometric object G allowed one to transform a coordinate system $\Phi: A \to G$ on a geometric object A to another coordinate system $\Psi \circ \Phi: A \to G$. We can generalise this observation to gauges: every family $\Psi = (\Psi_x)_{x \in X}$ of isomorphisms on G allows one to transform a gauge $(\Phi_x)_{x \in X}$ to another gauge $(\Psi_x \circ \Phi_x)_{x \in X}$ (again assuming that $\Psi$ respects whatever topological or differentiable structure is present). Such a collection $\Psi$ is known as a gauge transformation. For instance, in Example 1, one could rotate the reference direction $N_x$ at each point $x \in X$ anti-clockwise by some angle $\theta(x)$; this would cause the function $u(x)$ to rotate to $u(x) e^{-i\theta(x)}$. In Example 2, a gauge transformation is just a map $v: X \to S^1$ (which may need to be continuous or measurable, depending on the structures one places on X); it rotates a point $(x,u) \in X \times_\rho S^1$ to $(x, v^{-1} u)$, and it also transforms the cocycle $\rho$ by the formula (1). In Example 3, a gauge transformation would be a map $v: X \to \{-1,+1\}$; it rotates a point $(x, \epsilon) \in X \times \{-1,+1\}$ to $(x, v(x) \epsilon)$. Gauge transformations transform functions on the base X in many ways, but some things remain gauge-invariant. For instance, in Example 1, the winding number of a function $u: X \to S^1$ along a closed loop $\gamma \subset X$ would not change under a gauge transformation (as long as no singularities in the gauge are created, moved, or destroyed, and the orientation is not reversed). But such topological gauge-invariants are not the only gauge invariants of interest; there are important differential gauge-invariants which make gauge theory a crucial component of modern differential geometry and geometric PDE. But to describe these, one needs an additional gauge-theoretic concept, namely that of a connection on a fibre bundle. 0.3. Connections There are many essentially equivalent ways to introduce the concept of a connection; I will use the formulation based primarily on parallel transport, and on differentiation of sections. To avoid some technical details I will work (somewhat non-rigorously) with infinitesimals such as dx. (There are ways to make the use of infinitesimals rigorous, such as non-standard analysis, but this is not the focus of my post today.) In single variable calculus, we learn that if we want to differentiate a function $f: [a,b] \to {\Bbb R}$ at some point x, then we need to compare the value f(x) of f at x with its value f(x+dx) at some infinitesimally close point x+dx, take the difference $f(x+dx)-f(x)$, and then divide by dx, taking limits as $dx \to 0$, if one does not like to use infinitesimals: $$\displaystyle \nabla f(x) := \lim_{dx \to 0} \frac{f(x+dx) – f(x)}{dx}.$$ In several variable calculus, we learn several generalisations of this concept in which the domain and range of f to be multi-dimensional. For instance, if $f: X \to {\Bbb R}^d$ is now a vector-valued function on some multi-dimensional domain (e.g. a manifold) X, and v is a tangent vector to X at some point x, we can define the directional derivative$\nabla_v f(x)$ of f at x by comparing $f(x+v dt)$ with $f(x)$ for some infinitesimal dt, take the difference $f(x+vdt) – f(x)$, divide by dt, and then take limits as $dt \to 0$: $$\displaystyle \nabla_v f(x) := \lim_{dt \to 0} \frac{f(x+vdt) – f(x)}{dt}.$$ [Strictly speaking, if X is not flat, then x+vdt is only defined up to an ambiguity of o(dt), but let us ignore this minor issue here, as it is not important in the limit.] If f is sufficiently smooth (being continuously differentiable will do), the directional derivative is linear in v, thus for instance $\nabla_{v+v'} f(x) = \nabla_v f(x) + \nabla_{v'} f(x)$. One can also generalise the range of f to other multi-dimensional domains than ${\Bbb R}^d$; the directional derivative then lives in a tangent space of that domain. In all of the above examples, though, we were differentiating functions $f:X \to Y$, thus each element $x \in X$ in the base (or domain) gets mapped to an element $f(x)$ in the same range Y. However, in many geometrical situations we would like to differentiate sections$f = (f_x)_{x \in X}$ instead of functions, thus f now maps each point $x \in X$ in the base to an element $f_x \in A_x$ of some fibre in a fibre bundle $(A_x)_{x \in X}$. For instance, one might want to know how the wind direction $w = (w_x)_{x \in X}$ changes as one moves x in some direction v; thus computing a directional derivative $\nabla_v w(x)$ of w at x in direction v. One can try to mimic the previous definitions in order to define this directional derivative. For instance, one can move x along v by some infinitesimal amount dt, creating a nearby point $x+v dt$, and then evaluate w at this point to obtain $w(x+vdt)$. But here we hit a snag: we cannot directly compare $w(x+vdt)$ with $w(x)$, because the former lives in the fibre $A_{x+vdt}$ while the latter lives in the fibre $A_x$. With a gauge, of course, we can identify all the fibres (and in particular, $A_{x+vdt}$ and $A_x$) with a common object G, in which case there is no difficulty comparing $w(x+vdt)$ with $w(x)$. But this would lead to a notion of derivative which is not gauge-invariant, known as the non-covariant or ordinary derivative in physics. But there is another way to take a derivative, which does not require the full strength of a gauge (which identifies all fibres simultaneously together). Indeed, in order to compute a derivative $\nabla_v w(x)$, one only needs to identify (or connect) two infinitesimally close fibres together: $A_x$ and $A_{x+vdt}$. In practice, these two fibres are already "within O(dt) of each other" in some sense, but suppose in fact that we have some means $\Gamma(x \to x+vdt): A_x \to A_{x+vdt}$ of identifying these two fibres together. Then, we can pull back $w(x+vdt)$ from $A_{x+vdt}$ to $A_x$ through $\Gamma(x \to x+vdt)$ to define the covariant derivative: $$\displaystyle \nabla_v w(x) := \lim_{dt \to 0} \frac{\Gamma(x \to x+vdt)^{-1}( w(x+vdt) ) – w(x) }{dt}.$$ In order to retain the basic property that $\nabla_v w$ is linear in v, and to allow one to extend the infinitesimal identifications $\Gamma(x \to x+dx)$ to non-infinitesimal identifications, we impose the property that the $\Gamma(x \to x+dx)$ to be approximately transitive in that $$\Gamma(x+dx \to x+dx+dx') \circ \Gamma(x \to x + dx ) \approx \Gamma(x \to x+dx+dx')\tag{1}$$ for all x, dx, dx', where the $\approx$ symbol indicates that the error between the two sides is o(\|dx\| + \|dx'\|). [The precise nature of this error is actually rather important, being essentially the curvature of the connection $\Gamma$ at x in the directions $dx, dx'$, but let us ignore this for now.] To oversimplify a little bit, any collection $\Gamma$ of infinitesimal maps $\Gamma(x \to x+dx)$ obeying this property (and some technical regularity properties) is a connection. [There are many other important ways to view connections, for instance the Christoffel symbol perspective that we will discuss a bit later. Another approach is to focus on the differentiation operation $\nabla_v$ rather than the identifications $\Gamma(x \to x+dx)$ or $\Gamma(\gamma)$, and in particular on the algebraic properties of this operation, such as linearity in v or derivation-type properties (in particular, obeying various variants of the Leibnitz rule). This approach is particularly important in algebraic geometry, in which the notion of an infinitesimal or of a path may not always be obviously available, but we will not discuss it here.] The way we have defined it, a connection is a means of identifying two infinitesimally close fibres $A_x, A_{x+dx}$ of a fibre bundle $(A_x)_{x \in X}$. But, thanks to (1), we can also identify two distant fibres $A_x, A_y$, provided that we have a path $\gamma: [a,b] \to X$ from $x = \gamma(a)$ to $y = \gamma(b)$, by concatenating the infinitesimal identifications by a non-commutative variant of a Riemann sum: $$\Gamma(\gamma) := \lim_{\sup \|t_{i+1}-t_i\| \to 0} \Gamma(\gamma(t_{n-1}) \to \gamma(t_n)) \circ \ldots \circ \Gamma(\gamma(t_0) \to \gamma(t_1)),\tag{2}$$ where $a = t_0 < t_1 < \ldots < t_n = b$ ranges over partitions. This gives us a parallel transport map $\Gamma(\gamma): A_x \to A_y$ identifying $A_x$ with $A_y$, which in view of its Riemann sum definition, can be viewed as the "integral" of the connection $\Gamma$ along the curve $\gamma$. This map does not depend on how one parametrises the path $\gamma$, but it can depend on the choice of path used to travel from x to y. We illustrate these concepts using several examples, including the three examples introduced earlier. Example 1 continued. (Circle bundle of the sphere) The geometry of the sphere X in Example 1 provides a natural connection on the circle bundle SX, the Levi-Civita connection$\Gamma$, that lets one transport directions around the sphere in as "parallel" a manner as possible; the precise definition is a little technical (see e.g. my lecture notes for a brief description). Suppose for instance one starts at some location x on the equator of the earth, and moves to the antipodal point y by a great semi-circle$\gamma$ going through the north pole. The parallel transport $\Gamma(\gamma): S_x \to S_y$ along this path will map the north direction at x to the south direction at y. On the other hand, if we went from x to y by a great semi-circle $\gamma'$ going along the equator, then the north direction at x would be transported to the north direction at y. Given a section u of this circle bundle, the quantity $\nabla_v u(x)$ can be interpreted as the rate at which u rotates as one travels from x with velocity v. $\diamond$$ Example 2 continued. (Circle extensions) In Example 2, we change the notion of "infinitesimally close" by declaring x and Tx to be infinitesimally close for any x in the base space X (and more generally, x and $T^n x$ are non-infinitesimally close for any positive integer n, being connected by the path $x \to Tx \to \ldots \to T^n x$, and similarly for negative n). A cocycle $\rho: X \to S^1$ can then be viewed as defining a connection on the skew product $X \times_\rho S^1$, by setting $\Gamma( x \mapsto Tx ) = \rho(x)$ (and also $\Gamma(x \to x) = 1$ and $\Gamma(Tx \to x ) = \rho(x)^{-1}$ to ensure compatibility with (1); to avoid notational ambiguities let us assume for sake of discussion that $x, Tx, T^{-1} x$ are always distinct from each other). The non-infinitesimal connections $\rho_n(x) := \Gamma(x \to Tx \to \ldots \to T^n x)$ are then given by the formula $\rho_n(x) = \rho(x) \rho(Tx) \ldots \rho(T^{n-1} x)$ for positive n (with a similar formula for negative n). Note that these iterated cocycles $\rho_n$ also describe the iterations of the shift $\tilde T: (x,u) \mapsto (Tx,\rho(x)u)$, indeed $\tilde T^n (x,u) = (T^n x, \rho_n(x) u)$. $\diamond$$ Example 3 continued. (Oriented graphs) In Example 3, we declare two edges e, e' in X to be "infinitesimally close" if they are adjacent. Then there is a natural notion of parallel transport on the bundle $(A_e)_{e \in X}$; given two adjacent edges $e = \{u,v\}$, $e'=\{v,w\}$, we let $\Gamma(e \to e')$ be the isomorphism from $A_e = \{ \vec{uv}, \vec{vu} \}$ to $A_{e'} = \{ \vec{vw}, \vec{wv} \}$ that maps $\vec{uv}$ to $\vec{vw}$ and $\vec{vu}$ to $\vec{wv}$. Any path $\gamma = (\{v_1,v_2\}, \{v_2,v_3\}, \ldots, \{v_{n-1},v_n\})$ of edges then gives rise to a connection $\Gamma(\gamma)$ identifying $A_{\{v_1,v_2\}}$ with $A_{\{v_{n-1},v_n\}}$. For instance, the triangular path $(\{u,v\}, \{v,w\}, \{w,u\}, \{u,v\})$ induces the identity map on $A_{\{u,v\}}$, whereas the U-turn path $(\{u,v\}, \{v,w\}, \{w,x\}, \{x,v\}, \{v,u\})$ induces the anti-identity map on $A_{\{u,v\}}$. Given an orientation $\vec G = (\vec e)_{e \in X}$ of the graph G, one can "differentiate" $\vec G$at an edge $\{u,v\}$ in the direction $\{u,v\} \to \{v,w\}$ to obtain a number $\nabla_{\{u,v\} \to \{v,w\}} \vec G(\{u,v\}) \in \{-1,+1\}$, defined as +1 if the parallel transport from $\{u,v\}$ and $\{v,w\}$ preserves the orientations given by $\vec G$, and -1 otherwise. This number of course depends on the choice of orientation. But certain combinations of these numbers are independent of such a choice; for instance, given any closed path $\gamma = \{e_1,e_2,\ldots,e_n,e_{n+1}=e_1\}$ of edges in X, the "integral" $\prod_{i=1}^n \nabla_{e_i \to e_{i+1}} \vec G(e_i) \in \{-1,+1\}$is independent of the choice of orientation $\vec G$ (indeed, it is equal to +1 if $\Gamma(\gamma)$ is the identity, and -1 if $\Gamma(\gamma)$ is the anti-identity. $\diamond$$ Example 4. (Monodromy) One can interpret the monodromy maps of a covering space in the language of connections. Suppose for instance that we have a covering space $\pi: \tilde X \to X$ of a topological space X whose fibres $\pi^{-1}(\{x\})$ are discrete; thus $\tilde X$ is a discrete fibre bundle over X. The discreteness induces a natural connection $\Gamma$ on this space, which is given by the lifting map; in particular, if one integrates this connection on a closed loop based at some point x, one obtains the monodromy map of that loop at x. $\diamond$$ Example 5. (Definite integrals) In view of the definition (2), it should not be surprising that the definite integral$\int_a^b f(x)\ dx$ of a scalar function $f: [a,b] \to {\Bbb R}$ can be interpreted as an integral of a connection. Indeed, set $X := [a,b]$, and let $({\Bbb R})_{x \in X}$ be the trivial line bundle over X. The function f induces a connection $\Gamma_f$ on this bundle by setting $$\Gamma_f(x \mapsto x+dx): y \mapsto y + f(x) dx.$$ The integral $\Gamma_f([a,b])$ of this connection along ${}[a,b]$ is then just the operation of translation by $\int_a^b f(x)\ dx$ in the real line. $\diamond$$ Example 6. (Line integrals) One can generalise Example 5 to encompass line integrals in several variable calculus. Indeed, if $X$ is an n-dimensional domain, then a vector field $f = (f_1,\ldots,f_n): X \to {\Bbb R}^n$ induces a connection $\Gamma_f$ on the trivial line bundle $({\Bbb R})_{x \in X}$ by setting $$\Gamma_f( x \mapsto x+dx ): y \mapsto y + f_1(x) dx_1 + \ldots + f_n(x) dx_n.$$ The integral $\Gamma_f(\gamma)$ of this connection along a curve $\gamma$ is then just the operation of translation by the line integral $\int_\gamma f \cdot dx$ in the real line. Note that a gauge transformation in this context is just a vertical translation $(x,y) \mapsto (x,y+V(x))$ of the bundle $({\Bbb R})_{x \in X} \equiv X \times {\Bbb R}$ by some potential function $V: X \to {\Bbb R}$, which we will assume to be smooth for sake of discussion. This transformation conjugates the connection $\Gamma_f$ to the connection $\Gamma_{f – \nabla V}$. Note that this is a conservative transformation: the integral of a connection along a closed loop is unchanged by gauge transformation. $\diamond$$ Example 7. (ODE) A different way to generalise Example 5 can be obtained by using the fundamental theorem of calculus to interpret $\int_{[a,b]} f(x)\ dx$ of the solution to the initial value problem $$u'(t) = f(t); \quad u(a) = 0$$ for the ordinary differential equation $u'=f$. More generally, the solution u(b) to the initial value problem $$u'(t) = F( t, u(t) ); \quad u(a) = u_0$$ for some $u: [a,b] \to {\Bbb R}^n$ taking values in some manifold Y, where $F: [a,b] \times {\Bbb R}^n \to {\Bbb R}^n$ is a function (let us take it to be Lipschitz, to avoid technical issues), can also be interpreted as the integral of a connection $\Gamma$ on the trivial vector space bundle $({\Bbb R}^n)_{t \in [a,b]}$, defined by the formula $$\Gamma(t \mapsto t+dt): y \mapsto y + F(t,y) dt.$$ Then $\Gamma[a,b]$, this is nothing more than the Euler method for solving ODE. Note that the method of integrating factors in solving ODE can be interpreted as an attempt to simplify the connection $\Gamma$ via a gauge transformation. Indeed, it can be profitable to view the entire theory of connections as a multidimensional "variable-coefficient" generalisation of the theory of ODE. $\diamond$$ Once one selects a gauge, one can express a connection in terms of that gauge. In the case of vector bundles (in which every fibre is a d-dimensional vector space for some fixed d), the covariant derivative $\nabla_v w(x)$ of a section w of that bundle along some vector v emanating from x can be expressed in any given gauge by the formula $$\nabla_v w(x)^i = v^\alpha \partial_\alpha w(x)^i + v^\alpha \Gamma_{\alpha j}^i w(x)^j$$ where we use the gauge to express w(x) as a vector $(w(x)^1,\ldots,w(x)^d)$, the indices $i, j = 1,\ldots,d$ are summed over the fibre dimensions (and $\alpha$ summed over the base dimensions) as per the usual conventions, and the $\Gamma_{\alpha j}^i := (\nabla_{e_\alpha} e_j)^i$ are the Christoffel symbols of this connection relative to this gauge. One example of this, which models electromagnetism, is a connection on a complex line bundle$V = (V_{t,x})_{(t,x) \in {\Bbb R}^{1+3}}$ in spacetime${\Bbb R}^{1+3} = \{ (t,x): t \in {\Bbb R}, x \in {\Bbb R}^3 \}$. Such a bundle assigns a complex line $V_{t,x}$ (i.e. a one-dimensional complex vector space, and thus isomorphic to ${\Bbb C}$) to every point $(t,x)$ in spacetime. The structure group here is U(1) (strictly speaking, this means that we view the fibres as normed one-dimensional complex vector spaces, otherwise the structure group would be ${\Bbb C}^\times$). A gauge identifies V with the trivial complex line bundle $({\Bbb C})_{(t,x) \in {\Bbb R}^{1+3}}$, thus converting sections $(w_{t,x})_{(t,x) \in {\Bbb R}^{1+3}}$ of this bundle into complex-valued functions $\phi: {\Bbb R}^{1+3} \to {\Bbb C}$. A connection on V, when described in this gauge, can be given in terms of fields $A_\alpha: {\Bbb R}^{1+3} \to {\Bbb R}$ for $\alpha = 0,1,2,3$; the covariant derivative of a section in this gauge is then given by the formula $$\nabla_\alpha \phi := \partial_\alpha \phi + i A_\alpha \phi.$$ In the theory of electromagnetism, $A_0$ and $(A_1,A_2,A_3)$ are known (up to some normalising constants) as the electric potential and magnetic potential respectively. Sections of V do not show up directly in Maxwell's equations of electromagnetism, but appear in more complicated variants of these equations, such as the Maxwell-Klein-Gordon equation. A gauge transformation of V is given by a map $U: {\Bbb R}^{1+3} \to S^1$; it transforms sections by the formula $\phi \mapsto U^{-1} \phi$, and connections by the formula $\nabla_\alpha \mapsto U^{-1} \nabla_\alpha U$, or equivalently $$A_\alpha \mapsto A_\alpha + \frac{1}{i} U^{-1} \partial_\alpha U = A_\alpha + \partial_\alpha \frac{1}{i} \log U\tag{2}.$$ In particular, the electromagnetic potential $A_\alpha$ is not gauge invariant (which broadly corresponds to the concept of being nonphysical or nonmeasurable in physics), as gauge symmetry allows one to add an arbitrary gradient function to this potential. However, the curvature tensor $$F_{\alpha \beta} := [\nabla_\alpha, \nabla_\beta] = \partial_\alpha A_\beta – \partial_\beta A_\alpha$$ of the connection is gauge-invariant, and physically measurable in electromagnetism; the components $F_{0i} = -F_{i0}$ for $i=1,2,3$ of this field have a physical interpretation as the electric field, and the components $F_{ij} = -F_{ji}$ for $1 \leq i < j \leq 3$ have a physical interpretation as the magnetic field. (The curvature tensor $F$ can be interpreted as describing the parallel transport of infinitesimal rectangles; it measures how far off the connection is from being flat, which means that it can be (locally) "straightened" via some choice of gauge to be the trivial connection. In nonabelian gauge theories, in which the structure group is more complicated than just the abelian group U(1), the curvature tensor is non-scalar, but remains gauge-invariant in a tensor sense (gauge transformations will transform the curvature as they would transform a tensor of the same rank). Gauge theories can often be expressed succinctly in terms of a connection and its curvatures. For instance, Maxwell's equations in free space, which describes how electromagnetic radiation propagates in the presence of charges and currents (but no media other than vacuum), can be written (after normalising away some physical constants) as $$\partial^\alpha F_{\alpha \beta} = J_\beta$$ where $J_\beta$ is the 4-current. (Actually, this is only half of Maxwell's equations, but the other half are a consequence of the interpretation (*) of the electromagnetic field as a curvature of a U(1) connection. Thus this purely geometric interpretation of electromagnetism has some non-trivial physical implications, for instance ruling out the possibility of (classical) magnetic monopoles.) If one generalises from complex line bundles to higher-dimensional vector bundles (with a larger structure group), one can then write down the (classical) Yang-Mills equation $$\nabla^\alpha F_{\alpha \beta} = 0$$ which is the classical model for three of the four fundamental forces in physics: the electromagnetic, weak, and strong nuclear forces (with structure groups U(1), SU(2), and SU(3) respectively). (The classical model for the fourth force, gravitation, is given by a somewhat different geometric equation, namely the Einstein equations$G_{\alpha \beta} = 8 \pi T_{\alpha \beta}$, though this equation is also "gauge-invariant" in some sense.) The gauge invariance (or gauge freedom) inherent in these equations complicates their analysis. For instance, due to the gauge freedom (2), Maxwell's equations, when viewed in terms of the electromagnetic potential $A_\alpha$, are ill-posed: specifying the initial value of this potential at time zero does not uniquely specify the future value of this potential (even if one also specifies any number of additional time derivatives of this potential at time zero), since one can use (2) with a gauge function U that is trivial at time zero but non-trivial at some future time to demonstrate the non-uniqueness. Thus, in order to use standard PDE methods to solve these equations, it is necessary to first fix the gauge to a sufficient extent that it eliminates this sort of ambiguity. If one were in a one-dimensional situation (as opposed to the four-dimensional situation of spacetime), with a trivial topology (i.e. the domain is a line rather than a circle), then it is possible to gauge transform the connection to be completely trivial, for reasons generalising both the fundamental theorem of calculus and the fundamental theorem of ODEs. (Indeed, to trivialise a connection $\Gamma$ on a line ${\Bbb R}$, one can pick an arbitrary origin $t_0 \in {\Bbb R}$ and gauge transform each point $t \in {\Bbb R}$ by $\Gamma([t_0,t])$.) However, in higher dimensions, one cannot hope to completely trivialise a connection by gauge transforms (mainly because of the possibility of a non-zero curvature form); in general, one cannot hope to do much better than setting a single component of the connection to equal zero. For instance, for Maxwell's equations (or the Yang-Mills equations), one can trivialise the connection $A_\alpha$ in the time direction, leading to the temporal gauge condition $$A_0 = 0.$$ This gauge is indeed useful for providing an easy proof of local existence for these equations, at least for smooth initial data. But there are many other useful gauges also that one can fix; for instance one has the Lorenz gauge $$\partial^\alpha A_\alpha = 0$$ which has the nice property of being Lorentz-invariant, and transforms the Maxwell or Yang-Mills equations into linear or nonlinear wave equations respectively. Another important gauge is the Coulomb gauge $$\partial_i A_i = 0$$ where i only ranges over spatial indices 1,2,3 rather than over spacetime indices 0,1,2,3. This gauge has an elliptic variational formulation (Coulomb gauges are critical points of the functional $\int_{{\Bbb R}^3} \sum_{i=1}^3 \|A_i\|^2$) and thus are expected to be "smaller" and "smoother" than many other gauges; this intuition can be borne out by standard elliptic theory (or Hodge theory, in the case of Maxwell's equations). In some cases, the correct selection of a gauge is crucial in order to establish basic properties of the underlying equation, such as local existence. For instance, the simplest proof of local existence of the Einstein equations uses a harmonic gauge, which is analogous to the Lorenz gauge mentioned earlier; the simplest proof of local existence of Ricci flow uses a gauge of de Turck that is also related to harmonic maps (see e.g. my lecture notes); and in my own work on wave maps, a certain "caloric gauge" based on harmonic map heat flow is crucial (see e.g. this post of mine). But in many situations, it is not yet fully understood whether the use of the correct choice of gauge is a mere technical convenience, or is more innate to the equation. It is definitely conceivable, for instance, that a given gauge field equation is well-posed with one choice of gauge but ill-posed with another. It would also be desirable to have a more gauge-invariant theory of PDEs that did not rely so heavily on gauge theory at all, but this seems to be rather difficult; many of our most powerful tools in PDE (for instance, the Fourier transform) are highly non-gauge-invariant, which makes it very inconvenient to try to analyse these equations in a purely gauge-invariant setting. 标签 connection, Gauge, Yang-Mills ← 用winedt替换网页Latex为Mathjax \ \ \ 0+ $ \… → MineCraft 服务器搭建 On Debian	CommonCrawl
\begin{document} \title{All entangled states are quantum coherent with locally distinguishable bases} \author{Asmitha Mekala and Ujjwal Sen} \affiliation{Harish-Chandra Research Institute, HBNI, Chhatnag Road, Allahabad 211 019, India} \begin{abstract} We find that a bipartite quantum state is entangled if and only if it is quantum coherent with respect to complete bases of states in the corresponding system that are distinguishable under local quantum operations and classical communication. The corresponding minimal quantum coherence is the entanglement of formation. Connections to the relative entropy of entanglement and quantum coherence, and generalizations to the multiparty case are also considered. \end{abstract} \maketitle \section{Introduction} \label{bilambita-loi} Entanglement \cite{lichu} and coherence \cite{ja-re-ja-re-uRe} of quantum states result from the superposition principle, except that for the former, at least a two-party situation is necessary. Both have been used to develop resource theories. Indeed, entanglement and quantum coherence are the main resources in many applications of quantum information for demonstrating better performances than their classical counterparts. It is also known that an entangling gate necessarily requires coherence at the input to produce entangled states. The literature of local distinguishability or its absence of sets of orthogonal states of multiparty systems have developed rather independently. Indeed, it was noticed that entanglement of the constituent states is not directly related to the local indistinguishability of such sets - at least, not in all cases \cite{sakal-shunya-kare-ek, sakal-shunya-kare-dui, knaThaler-aTha}. The problem of whether or not a state is entangled is known to be intricate and has as yet not been solved. The situation is similar for quantum coherence with respect to clusters of bases with specific properties, and for local distinguishability of sets of quantum states. The quantifications of these concepts do exist in the literature \cite{lichu, ja-re-ja-re-uRe, dukher-rajani-prabhat-hoi-na}, but are typically difficult to compute. Here we show that entangled quantum states can be seen as quantum coherent states in locally distinguishable bases. Moreover, a convex-roof based measure of quantum coherence, of a bipartite quantum state of arbitrary dimensions, in optimal locally distinguishable bases turn out to be the entanglement of formation of the state, where the latter is a measure of entanglement \cite{boRda}. A different approach of quantification of quantum coherence, for a bipartite state, using the concept of relative entropy \cite{ke-je-bale-debe}, provides an upper bound for relative entropy of entanglement of the state, where the latter is another measure of entanglement \cite{golam-chor, golam-chor-1}. We then show that the considerations can be carried over to multiparty systems. We therefore find that the two salient resource theories of quantum information, viz. entanglement and quantum coherence, are closely related, and that the relation is effected by using \emph{a priori} unrelated concepts in the domain of local distinguishability of sets of orthogonal multiparty states. \section{Definitions and results} \label{abar-asiba-phire} We will require the concepts of von Neumann entropy and relative entropy between quantum states \cite{ke-je-bale-debe}. The von Neumann entropy of a quantum state $\varrho$ is denoted by $S(\varrho)$ and is given by \begin{equation} S(\varrho) = -\mbox{tr}(\varrho \log_2 \varrho). \end{equation} The von Neumann relative entropy between two quantum states, $\varrho$ and $\varsigma$, is denoted by $S(\varrho \parallel \varsigma) $, and is given by \begin{equation} S(\varrho \parallel \varsigma) = \mbox{tr}(\varrho \log_2 \varrho - \varrho \log_2 \varsigma). \end{equation} It is to be noted that the relative entropy is not symmetric with respect to its arguments. A qualitative definition of quantum coherence, as has already been given in the literature, can be as follows \cite{ja-re-ja-re-uRe}.\\ \noindent \textbf{Definition.} A pure quantum state $\|\psi\rangle$ of a physical system represented by a Hilbert space $\mathbb{C}^d$ is said to be quantum coherent with respect to a complete orthonormal basis of $\mathbb{C}^d$ if it is not an element of that basis. The notion has also been quantified, and one of the quantifications is as follows \cite{ja-re-ja-re-uRe}.\\ \noindent \textbf{Definition.} Let $B$ be a complete orthonormal basis of pure states in $\mathbb{C}^{d}$. Let $C_B(\|\psi\rangle)$ be the relative entropy of quantum coherence of $\|\psi\rangle \in \mathbb{C}^{d}$, so that \begin{equation} C_B(\|\psi\rangle) = \min_{\rho_B \in M_B} S(\|\psi\rangle \langle \psi\| \parallel \rho_B), \end{equation} where $M_B$ is the set of all probabilistic mixtures of the projectors onto the elements of $B$. Complete orthonormal bases of bipartite quantum systems are of course distinguishable under global operations. One just makes a measurement onto that basis. Things are more complicated however when a restricted class of operations is allowed. An important such restricted class is the class of local quantum operations and classical communication (LOCC) \cite{boRda,golam-chor-1,neel-beRechhe}. If a complete orthonormal basis is also distinguishable under LOCC, we will call the basis as ``locally distinguishable''. A bipartite pure state is said to be entangled if it cannot be written as a tensor product of pure states of the two systems.\\ \noindent \textbf{Theorem 1.} \emph{A bipartite pure state is entangled if and only if it has a nonzero quantum coherence with respect to all locally distinguishable complete orthonormal bases.} \noindent \texttt{Proof.} Let $\|\psi\rangle$ be a pure entangled state of a bipartite quantum system, the Hilbert space corresponding to which is $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Let us now consider an arbitrary locally distinguishable complete orthonormal basis. $\|\psi\rangle$ will have vanishing quantum coherence in this basis if and only if it is an element of this basis. However, if $\|\psi\rangle$ is an element of this basis, the latter cannot be locally distinguishable, as it was proven in Ref. \cite{knaThaler-aTha} that any complete orthonormal basis containing even a single entangled state cannot be locally distinguishable. Therefore, $\|\psi\rangle$ must have a nonzero quantum coherence in any locally distinguishable complete orthonormal basis. On the other hand, an arbitrary pure product state can always be expanded to form a complete bi-orthonormal product basis, which can always be distinguished by LOCC. Therefore, the product state has zero quantum coherence at least with respect to this basis. {}$\blacksquare$ It is clear that the states formed by mixing locally distinguishable orthonormal complete basis states are separable states, and the basis forms the spectral states of the separable state, where a separable state is any state that can be prepared by LOCC after starting from product states \cite{megh}. Not all separable states are however of this type, that is, not all separable states have a spectral basis that is locally distinguishable. To see this, let us consider a specific example. Consider the two-qubit states, $\|\psi^\pm\rangle = (\|01\rangle \pm \|10\rangle)/\sqrt{2}$, and an unequal mixture of them, viz. $\rho_2=p\|\psi^+\rangle \langle \psi^+\| + (1-p) \|\psi^-\rangle \langle \psi^-\|$, where $p\in (0,1)$ and $p \ne 1/2$. By using the positive partial transpose (PPT) criterion \cite{sreeradha}, one can check that this family has only entangled states. (The state for $p=1/2$ is separable.) Now, the spectral decomposition of $\rho_2$ is unique, as there is no degeneracy in the spectrum. Also, a locally distinguishable basis is necessarily an orthogonal set. The spectral basis of $\rho_2$ is incomplete, but can always be completed to a full orthonormal basis, and any such completion - for $p \ne 1/2$ - will be locally indistinguishable, as at least two states of it will be entangled, viz. $\|\psi^\pm\rangle$ \cite{knaThaler-aTha}. Therefore, there is no locally distinguishable basis which can be probabilistically mixed to form $\rho_2$ for any $p \ne 1/2$. Because of the existence of such examples, it may seem that quantum coherence of a bipartite state with respect to locally distinguishable bases may not be related to the state's entanglement content. We however have the following result.\\ \noindent \textbf{Theorem 2.} \emph{The minimum among quantum coherences with respect to all locally distinguishable complete orthonormal bases of any bipartite pure quantum state is given by its local von Neumann entropy.} \noindent \texttt{Proof.} The minimal relative entropy distance of a pure state $\|\psi\rangle$ of $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$, written in Schmidt decomposition as $\sum_{i=1}^n\alpha_i\|ii\rangle$, from the set of separable states, is attained in the state $\sum_{i=1}^n\alpha_i^2 \|ii\rangle \langle ii\|$, and is given by the von Neumann entropy of either of the local densities \cite{golam-chor, golam-chor-1}. Here, $n \leq \min\{d_1, d_2\}$, and $\alpha_i$ are real and positive. The state $\sum_{i=1}^n\alpha_i^2 \|ii\rangle \langle ii\|$ is of course a separable state, but is also a mixture of states of a locally distinguishable orthonormal basis, viz. $\{\|i\rangle\|j\rangle\}_{i=1, j=1}^{d_1, d_2}$. {} $\blacksquare$ The result reminds us of similar ones for quantum discord \cite{peyara} and quantum work deficit \cite{narkol}, which also were equal to the local von Neumann entropy for all pure bipartite states. Natural extensions of the concept of quantum coherence to mixed states can be made in several ways. One of them is by using the concept of the convex roof \cite{boRda}. \\ \noindent \textbf{Definition.} A quantum state $\rho$ on $\mathbb{C}^d$ is said to be quantum coherent with respect to a class of complete orthonormal bases $\{B\}$ of $\mathbb{C}^d$ with a special pre-settled property if it cannot be written as a convex (i.e., probabilistic) sum of pure states of $\mathbb{C}^d$ with zero minimal quantum coherence when optimized over such bases.\\ The qualitative definition of quantum coherence can be quantified as follows.\\ \noindent \textbf{Definition.} For a quantum state $\rho$ on $\mathbb{C}^{d}$, its quantum coherence with respect to a class $\{B\}$ of bases on $\mathbb{C}^{d}$ is given by \begin{equation} C_{\{B\}}(\rho) = \min \sum_i p_i \min_{B\in\{B\}}C_B (\|\psi_i\rangle), \end{equation} where the outer minimization is over all decompositions of $\rho$ into $\sum_i p_i \|\psi_i\rangle \langle \psi_i\|$.\\ When the set $\{B\}$ contains all bases of $\mathbb{C}^d$, it is clear that all quantum states will have vanishing quantum coherence. Quantum coherence is typically defined with respect to a fixed basis, and one subsequently demonstrates that it satisfies certain conditions \cite{ja-re-ja-re-uRe}. The definition presented above, however, involves a class of bases. As we show in the Supplementary Material, it can also be shown to satisfy the usual conditions of a quantum coherence measure, in the case of interest to us, viz. when $\{B\}$ is the class of all locally distinguishable complete orthonormal bases of a multiparty quantum system. We will now need the concept of entanglement of formation \cite{boRda}, which, for a bipartite quantum state $\rho_{AB}$ is defined as \begin{equation} \label{rajakini-rami} E_F(\rho_{AB}) = \min \sum_i p_i E_F (\|\psi_i\rangle_{AB}), \end{equation} where the minimization is over all decompositions of $\rho_{AB}$ into $\sum_i p_i \|\psi_i\rangle \langle \psi_i\|$, and where the entanglement of formation for a pure bipartite state is given by the von Neumann entropy of either of the local densities \cite{rajani}. Entanglement of formation has been put forward as a measure of entanglement and is typically difficult to compute \cite{boRda, damodar}, where a bipartite entangled state is one which is not separable. At the qualitative level, we have the following result, the proof of which is presented in the Supplementary Material.\\ \noindent \textbf{Theorem 3.} \emph{A bipartite quantum state, possibly mixed, is entangled if and only if it has a nonzero quantum coherence with respect to all locally distinguishable complete orthonormal bases.} Just like for pure states, the connection between entanglement and quantum coherence in LOCC-distinguishable bases can be taken to a quantitative level.\\ \noindent \textbf{Theorem 4.} \emph{The quantum coherence in locally distinguishable bases of a bipartite quantum state, possibly mixed, is the entanglement of formation of the state.}\\ A proof is presented in the Supplementary Material. A similar result was obtained in Ref. \cite{asbo-arek-din}, where quantum coherence in product bases and its convex-roof extension were considered. We note that a complete orthonormal basis having even a single entangled state is necessarily locally indistinguishable \cite{knaThaler-aTha}. However, there exists complete orthonormal bases of product states that are locally indistinguishable \cite{sakal-shunya-kare-ek}. We also remember here that an entangling gate necessarily requires coherence at the input to produce entangled states at the output. See \cite{shono-kono-ek-din} in this regard. As already alluded to, there are other avenues of natural extensions of the concept of quantum coherence to mixed quantum states. One such is given as follows \cite{ja-re-ja-re-uRe}.\\ \noindent \textbf{Definition.} A quantum state $\rho$ on $\mathbb{C}^d$ is said to be relative quantum coherent with respect to a complete orthonormal basis $B$ of $\mathbb{C}^d$ if it is not a mixture of states of the basis.\\ The christening is non-standard, and is made to distinguish it from the previous definition, in this section, of quantum coherence, and is chosen because it is relative to a particular basis $B$ and not, as in the previous case, with respect to a class $\{B\}$ of bases. We now provide a quantification of the notion of relative quantum coherence \cite{ja-re-ja-re-uRe}.\\ \noindent \textbf{Definition.} For a quantum state $\rho$ on $\mathbb{C}^{d}$, its relative entropy of quantum coherence with respect to the basis $B$ on $\mathbb{C}^{d}$ is given by \begin{equation} C_B^R(\rho) = \min_{\rho_B \in M_B} S(\rho \parallel \rho_B). \end{equation} This definition of quantum coherence has been widely used to build a resource theory, and the corresponding monotonicity properties have been proven in the literature \cite{ja-re-ja-re-uRe}. We will now need the concept of the relative entropy of entanglement \cite{golam-chor, golam-chor-1}, which, for a bipartite state $\rho_{AB}$ is given by the minimal relative entropy distance of the state from the set of separable states in the same Hilbert space, so that \begin{equation} E_R(\rho_{AB}) = \min_{\sigma_{AB}} S(\rho_{AB} \parallel \sigma_{AB}), \end{equation} where $\sigma_{AB}$ is a separable state. Just like the entanglement of formation, the relative entropy of entanglement has also been proposed as a measure of entanglement, and is again typically difficult to compute \cite{golam-chor, golam-chor-1}. At the qualitative level, we have the following result.\\ \noindent \textbf{Theorem 5.} \emph{Any bipartite entangled state, possibly mixed, has a nonzero relative quantum coherence with respect to all locally distinguishable complete orthonormal bases.} And on the quantitative level, we have the following relation.\\ \noindent \textbf{Theorem 6.} \emph{The minimal relative entropy of quantum coherence of a bipartite quantum state, possibly mixed, with locally distinguishable bases is bounded below by the relative entropy of entanglement of the state.}\\ The proofs of Theorems 5 and 6 are presented in the Supplementary Material. We have until now been considering the case of entanglement of bipartite states and local distinguishability of sets of bipartite states. These considerations can be carried over to the multiparty case. Both the concepts, viz. entanglement and local distinguishability, are far richer in the multiparty domain. The connection between entanglement and quantum coherence can however be carried over to the multiparty case, and we exemplify the situation by considering two diametrically opposite types of multiparty entanglements in the following two theorems (proofs in Supplementary Material).\\ \noindent \textbf{Theorem 7.} \emph{A multiparty pure state in} $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$ \emph{is entangled across at least one bi-partition if and only if it is quantum coherent with respect to all locally distinguishable complete orthonormal bases.} \\ \noindent \textbf{Theorem 8.} \emph{A multiparty pure state in} $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \otimes \mathbb{C}^{d_m}$ \emph{is genuinely multiparty entangled if and only if it is quantum coherent with respect to complete orthonormal bases that are locally distinguishable in at least one bipartition of the} $m$ \emph{parties.}\\ These results in the multiparty scenario are for pure states, but the generalizations to the regime of mixed states are similar to those already done in the bipartite case. It is important to mention here about the experimental feasibility of measuring the quantities examined. Entanglement and quantum coherence are deeply studied topics and their experimental characterization and quantification have been performed in the literature in a large number of works. These strategies are probably more studied for entanglement than for quantum coherence. In particular, the quantum coherence measures studied here have not been experimentally characterized in the literature. However, these quantum coherence measures have all been found to be intimately related, in many cases equal, to known measures of entanglement. And the latter have been characterized and quantified in the literature. \section{Conclusion} \label{garhita-kaj} Entanglement of shared quantum states forms one of the most successful resources for performing quantum information tasks \cite{lichu}. It is therefore interesting to characterize and quantify it in as many different ways as possible, as that may lead to a deeper understanding of the concept and also potentially result in new applications. Quantum coherence has also been argued to be resourceful in attaining quantum advantage in specified tasks over their classical counterparts \cite{ja-re-ja-re-uRe}. We found that quantum coherence in locally distinguishable bases can be used to define and quantify entanglement. It is to be noted that there exists complete globally distinguishable bases that are not locally distinguishable, and it has typically been argued that the local indistinguishability of such bases is unrelated to the entanglement content of the constituent states \cite{sakal-shunya-kare-ek,sakal-shunya-kare-dui,knaThaler-aTha}. We initially proved the results for pure bipartite states. 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Plenio, arXiv:2004.04536. \end{thebibliography} \begin{center} \textbf{Supplementary Material for\\ All entangled states are quantum coherent with locally distinguishable bases} \texttt{Asmitha Mekala and Ujjwal Sen}\\ \texttt{Harish-Chandra Research Institute, HBNI, Chhatnag Road, Allahabad 211 019, India} \end{center} \section{Quantum coherence} Let us first repeat a few definitions that are already present in the main text \cite{ja-re-ja-re-uRe-1}. \textbf{Definition.} A quantum state $\rho$ on $\mathbb{C}^d$ is said to be quantum coherent with respect to a class of complete orthonormal bases $\{B\}$ of $\mathbb{C}^d$ with a special pre-settled property if it cannot be written as a convex (i.e., probabilistic) sum of pure states of $\mathbb{C}^d$ with zero minimal quantum coherence when optimized over such bases. The qualitative definition of quantum coherence can be quantified as follows.\\ \noindent \textbf{Definition.} For a quantum state $\rho$ on $\mathbb{C}^{d}$, its quantum coherence with respect to a class $\{B\}$ of bases on $\mathbb{C}^{d}$ is given by \begin{equation} C_{\{B\}}(\rho) = \min \sum_i p_i \min_{B\in\{B\}}C_B (\|\psi_i\rangle), \end{equation} where the outer minimization is over all decompositions of $\rho$ into $\sum_i p_i \|\psi_i\rangle \langle \psi_i\|$. When the set $\{B\}$ contains all bases of $\mathbb{C}^d$, it is clear that all quantum states will have vanishing quantum coherence. A resource theory of quantum coherence requires the identification of free operations, free states, and a measure that satisfies monotonicity properties with respect to the free operations and vanishes for free states. Our interest in this manuscript lies in multiparty quantum systems. Let us, for definiteness, consider bipartite quantum states, although many of the considerations below hold equally well for other multiparty systems. Also, we wish to consider the class $\{B\}$ as the class of all locally distinguishable complete orthonormal bases. Let us denote this class as $\{B_L\}$. We prove in Theorem 4 that $C_{\{B_L\}}(\rho)$ for a bipartite quantum state $\rho$ is equal to its entanglement of formation, $E_F(\rho)$. Now $E_F(\rho) = 0$ holds if and only if $\rho$ is separable \cite{boRda-1}. Therefore, we identify the separable states as the set of free states in our resource theory, for which the measure is $C_{\{B_L\}}$. We still need to identify the free operations, which maps free states to free states, and with respect to which the measure $C_{\{B_L\}}(\rho)$ will be a monotone. A natural class of operations to consider for a bipartite quantum system and which is also operationally important, is the LOCC (local quantum operations and classical communication) class. Entanglement of formation, and hence our measure of quantum coherence, is monotonically non-increasing, on average, for this class \cite{boRda-1}. The general class of free operations in our resource theory is however bigger, and includes separable superoperators \cite{neelmadhab}, and also the swap operator (of the entire subsystems). These latter ones are however not implementable by using LOCC, the natural class of operations in the distant laboratories paradigm. \section{Proof of Theorem 3} \noindent\textbf{Theorem 3.} \emph{A bipartite quantum state, possibly mixed, is entangled if and only if it has a nonzero quantum coherence with respect to all locally distinguishable complete orthonormal bases.} \noindent\texttt{Proof.} Let $\rho$ be a bipartite entangled state on a physical system represented by the Hilbert space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Since this state, which is possibly mixed, is entangled, every probabilistic decomposition of it into pure states contains at least one entangled (pure) state \cite{megh-1}. Let $\{B_L\}$ be the set of all locally distinguishable complete orthonormal bases on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. We have proved in Theorem 1 that an entangled pure state is quantum coherent with respect to all bases in $\{B_L\}$. Therefore, every decomposition of $\rho$ into pure states will contain at least one (pure) state that is quantum coherent with respect to all bases in $\{B_L\}$. Let us now assume that $\rho$ is quantum coherent with respect to all bases in $\{B_L\}$. Therefore, by (the convex-roof) definition of quantum coherence, it follows that it cannot be written as a convex sum of pure states, all of which have vanishing quantum coherence when optimized over bases in $\{B_L\}$. Therefore, by Theorem 1, any convex decomposition of $\rho$ will contain a pure entangled state, implying that $\rho$ is entangled. {} $\blacksquare$ \section{Proof of Theorem 4} \noindent \textbf{Theorem 4.} \emph{The quantum coherence in locally distinguishable bases of a bipartite quantum state, possibly mixed, is the entanglement of formation of the state.} \noindent\texttt{Proof.} Let $\rho$ be a bipartite entangled state on a physical system represented by the Hilbert space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Let $\{B_L\}$ be the set of all locally distinguishable complete orthonormal bases on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Let $\rho = \sum_i p_i \|\psi_i\rangle \langle \psi_i\|$ be decomposition of $\rho$, where $\|\psi_i\rangle$ are pure states of $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$, and where $p_i$ forms a probability distribution. Now, \begin{equation} C_{\{B_L\}}(\rho) = \min \sum_i p_i \min_{B_L\in\{B_L\}}C_{B_L} (\|\psi_i\rangle), \end{equation} where the outer minimization is over all convex decompositions of $\rho$ into pure states. Using Theorem 2, we can write $\min_{B_L\in\{B_L\}}C_{B_L} (\|\psi_i\rangle)$ as $E_F(\|\psi_i\rangle)$, so that we have \begin{equation} C_{\{B_L\}}(\rho) = \min \sum_i p_i E_F(\|\psi_i\rangle), \end{equation} with the right-hand-side being exactly equal to the entanglement of formation of $\rho$. {} $\blacksquare$ \section{Proofs of Theorems 5 and 6} \noindent \textbf{Theorem 5.} \emph{Any bipartite entangled state, possibly mixed, has a nonzero relative quantum coherence with respect to all locally distinguishable complete orthonormal bases.} \noindent \texttt{Proof.} Let $\rho$ be a bipartite entangled state on a physical system represented by the Hilbert space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Since this state, which is possibly mixed, is entangled, every probabilistic decomposition of it into pure states contains at least one entangled (pure) state \cite{megh-1}. Let $\{B_L\}$ be the set of all locally distinguishable complete orthonormal bases on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Any probabilistic mixture of elements of any $B_L \in \{B_L\}$ will provide a separable state, as we have noted in the main text, and therefore cannot produce $\rho$, which has been taken to be entangled. Therefore, $\rho$ is relative quantum coherent with respect to all bases in $\{B_L\}$. {} $\blacksquare$ \noindent \textbf{Theorem 6.} \emph{The minimal relative entropy of quantum coherence of a bipartite quantum state, possibly mixed, with locally distinguishable bases is bounded below by the relative entropy of entanglement of the state.} \noindent \texttt{Proof.} Let $\rho$ be a bipartite entangled state on a physical system represented by the Hilbert space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Let $\{B_L\}$ be the set of all locally distinguishable complete orthonormal bases on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Now, the relative entropy of quantum coherence with respect to a basis $B_L \in \{B_L\}$ is given by \begin{equation} C_{B_L}^R(\rho) = \min_{\rho_{B_L} \in M_{B_L}} S(\rho \parallel \rho_{B_L}). \end{equation} As mentioned in the main text, $M_{B_L}$ contains only separable states, and so, \begin{equation} C_{B_L}^R(\rho) = \min_{\rho_{B_L} \in M_{B_L}} S(\rho \parallel \rho_{B_L}) \geq \min_{\sigma} S(\rho \parallel \sigma), \end{equation} for all $B_L \in \{B_L\}$, where $\sigma$ is an arbitrary separable state on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$. Therefore, by the definition of relative entropy of entanglement, we have \begin{equation} \min_{B_L \in \{B_L\}} C_{B_L}^R(\rho) \geq E_R(\rho). \end{equation} Hence, the proof. {} $\blacksquare$ \section{Proof of Theorems 7 and 8} We have until now been considering the case of entanglement of bipartite states and local distinguishability of sets of bipartite states. These considerations can be carried over to the multiparty case. Both the concepts, viz. entanglement and local distinguishability, are far richer in the multiparty domain. The connection between entanglement and quantum coherence can however be carried over to the multiparty case, and we exemplify the situation by considering two diametrically opposite types of multiparty entanglements in the following two theorems. Let us first define a multiparty pure state in $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$ as entangled if it is entangled across at least one bi-partition. Similarly, we define it to be quantum coherent if it is not an element of an LOCC-distinguishable complete orthonormal basis, where the LOCC in the multiparty scenario is local with respect to all the parties and classical communication is allowed between all the parties.\\ \noindent \textbf{Theorem 7.} \emph{A multiparty pure state in} $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$ \emph{is entangled if and only if it is quantum coherent with respect to all locally distinguishable complete orthonormal bases.} \noindent \texttt{Proof.} Let $\|\tilde{\psi}\rangle$ be a pure entangled state on a physical system represented by the Hilbert space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$, so that, according to the definition accepted in the main text for entangled multiparty states, the state $\|\tilde{\psi}\rangle$ is entangled across at least one bi-partition. Let that bi-partition be $\mathcal{A}:\mathcal{B}$. Let $\{\tilde{B}_L\}$ be the set of all locally distinguishable complete orthonormal bases on $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$. Therefore, all elements of $\{\tilde{B}_L\}$ will be locally distinguishable in the bi-partition $\mathcal{A}:\mathcal{B}$. Consequently, by Theorem 1, $\|\tilde{\psi}\rangle$ is quantum coherent with respect to all bases in $\{\tilde{B}_L\}$. On the other hand, if $\|\tilde{\psi}\rangle$ is of the form $\otimes_{i=d_1}^{d_m}\|\psi_{i}\rangle$, then it can always be expanded into a multi-orthonormal product basis of $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \mathbb{C}^{d_m}$, which can be distinguished by LOCC with all the $m$ parties in separate locations. {} $\blacksquare$ A convex-roof approach for the extension to mixed multiparty quantum states will then provide the result that a multiparty state, possibly mixed, will be entangled (i.e., not ``fully-separable'') if and only if it is quantum coherent with respect to locally distinguishable complete orthonormal bases. We now move over to the diametrically opposite scenario (for entangled multiparty pure states). Precisely, we change our definition of entangled multiparty pure states to the exact opposite extreme to what was used in Theorem 7. In the literature, such states are called genuinely multiparty entangled states, and we also call it so here.\\ \noindent \textbf{Theorem 8.} \emph{A multiparty pure state in} $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2} \otimes \ldots \otimes \mathbb{C}^{d_m}$ \emph{is genuinely multiparty entangled if and only if it is quantum coherent with respect to complete orthonormal bases that are locally distinguishable in at least one bipartition of the} $m$ \emph{parties.} \noindent \texttt{Proof.} Let $\|\psi\rangle$ be a genuinely multisite entangled state of the $m$ parties. Therefore, $\|\psi\rangle$ is entangled across all bipartitions $A:B$ of the $m$ parties. Let $\{B_L\}$ be the set of all complete orthonormal bases are locally distinguishable in at least one bipartition of the m parties. For a given element of that set, $\{B_L\}$, let that partition be $A_1:B_1$. Then by Theorem 1, $\|\psi\rangle$ is quantum coherent with respect to that element of $\{B_L\}$. On the other hand, if $\|\psi\rangle$ is of the form $\|\psi{'}_A\rangle \otimes \|\psi{''}_B\rangle$ across some bipartition $A:B$ of the $m$ parties, then it can always be completed to a complete orthonormal basis of the Hilbert space of the m parties that is locally distinguishable in $A:B$. This completes the proof. $\blacksquare$ Again, a generalization to the case of mixed states is possible. \end{document}	arXiv
\begin{document} \title{Galois families of modular forms and application to weight one} \author{Sara Arias-de-Reyna, Fran\c{c}ois Legrand, Gabor Wiese} \maketitle \begin{abstract} We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over $\mathbb{Q}$. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over $\overline{\mathbb{F}_p}$ for $p \in \{3,5,7,11\}$. MSC Classification: 11F80 (Galois representations), 11F11 (Holomorphic modular forms of integral weight), 12F12 (Inverse Galois Theory), 12E30 (Field Arithmetic). \end{abstract} \section{Introduction} \label{sec:intro} Families of modular forms and their attached Galois representations are of fundamental importance in current arithmetic geometric research. With this paper, we would like to draw attention to a new kind of families of modular forms, which we call {\em Galois families} (cf.~Definition~\ref{defi:GF}). For example, projective Galois families are defined as follows: \noindent {\bf{Definition A.}} {\it{Let $(f_i)_{i \in I}$ be a family of normalised Hecke eigenforms (of any level and weight). For a prime number $p$ and a finite subgroup $G \subset {\rm{PGL}}_2(\overline{\mathbb{F}_p})$, we say that the $(f_i)_{i \in I}$ form a {\rm{projective $G$-Galois family}} if the following two conditions hold: \noindent {\rm{(1)}} for each $i \in I$, the image of the projective mod $p$ Galois representation $\rho^\mathrm{proj}_{f_i, p} : {\rm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow {\rm{PGL}}_2(\overline{\mathbb{F}_p})$ associated with $f_i$ is conjugate to $G$, \noindent {\rm{(2)}} there exists a finite Galois extension $E$ of a rational function field $\mathbb{Q}({\bf{T}}) = \mathbb{Q}(T_1, \dots, T_n)$ with Galois group $G$ such that, for each $i \in I$, the number field $K^\mathrm{proj}_{f_i,p}$ 'cut out by $\rho^\mathrm{proj}_{f_i, p}$', that is, defined by ${\rm{ker}}(\rho^\mathrm{proj}_{f_i,p}) = {\rm{Gal}}(\overline{\mathbb{Q}}/K^\mathrm{proj}_{f_i,p})$, is obtained by specialising the function field $E$ at some ${\bf{t}}_i \in \mathbb{Q}^n$.}} \noindent We refer to \S\ref{sec:bas} for standard terminology and material on Galois representations, modular forms and functions field extensions. Galois families are then taken with respect to a prime number~$p$ and have as feature that the projective mod~$p$ Galois representations of all members of the family have conjugate images. The family can be taken to consist of 'classical' holomorphic Hecke eigenforms or it can be chosen to be made of Katz modular Hecke eigenforms (geometrically) defined over $\overline{\mathbb{F}_p}$. We furthermore define projective Artin Galois families consisting of holomorphic Hecke eigenforms of weight one in a similar way, and also consider the linear case (in both settings). See Definition~\ref{defi:GF} for more details. Galois families are fundamentally different from other kinds of families of modular forms, such as Hida families (see, e.g.,~\cite{Em11} for an account of the theory). On the one hand, if we took the classical members of a Hida family, the field cut out by the mod $p$ Galois representation would be the same in all cases. On the other hand, Galois families do not see $p$-adic deformations, so they miss the interesting information in Hida families. Galois families are rooted in field arithmetic and we see our paper as a step towards strengthening connections between field arithmetic and the automorphic theory. In \S\ref{sec:BB}, we relate our notion of Galois families to problems and results from field arithmetic, such as the Beckmann-Black Problem and the existence of parametric extensions over $\mathbb{Q}$ (recalled as Problem \ref{prob:BB} and Definition \ref{def:par_gen}, respectively), and formulate analogues for modular forms (see Problem \ref{prob:mf} and Definition \ref{def:para_mf}), which we partially answer. For example, if $G$ is any of the three groups $A_4$, $S_4$, $A_5$, by using results on the existence or non-existence of generic or parametric polynomials/extensions with rational coefficients for the group $G$, we prove that the family of all holomorphic normalised Hecke eigenforms of weight one with 'exceptional' projective Galois image $G$ is a projective Artin $G$-Galois family and that 2 is the minimal number of parameters we need to get such a family. See Theorems \ref{thm:parametric_poly2} and \ref{?} for more details. In \S\ref{sec:explicit}, we focus on some Galois families consisting of modular forms of weight one since those are special in a number of ways. For example, a weight one Hecke eigenform which is geometrically defined over $\overline{\mathbb{F}_p}$ in the sense of Katz need not lift to a holomorphic weight one eigenform. Such non-liftable Hecke eigenforms are torsion classes in the cohomology of the relevant modular curve over $\overline{\mathbb{F}_p}$. As such they are sporadic. This is one, but not the only reason that we do not have any closed formulas for the dimension of spaces of weight one modular forms, not even over the complex numbers (another reason is that holomorphic weight one Hecke eigenforms with `exceptional' projective Galois image as above also seem to occur sporadically). To the best of our knowledge, it is not known whether, for a given prime number $p$, there are infinitely many non-liftable Hecke eigenforms of weight one over $\overline{\mathbb{F}_p}$ with pairwise non-isomorphic projective Galois representations. We provide a general criterion which reduces to the existence of a finite Galois extension of the rational function field $\mathbb{Q}(T)$ with specified Galois group admitting specialisations with specified local behaviour at some rational places, including the one associated with the prime number $p$ (see Theorem \ref{thm1}). We then use standard results on the local behaviour of specialisations (recalled in \S\ref{ssec:ffe}) and the existence of some explicit bivariate polynomials with rational coefficients to construct such infinite families for $p \in \{3,5,7,11\}$: \noindent {\bf{Theorem B.}} {\it{For every $p \in \{3,5,7,11\}$, there exists a finite group $G$ among ${\rm{PGL}}_2(\mathbb{F}_{p})$, ${\rm{PSL}}_2(\mathbb{F}_{p})$ and ${\rm{PSL}}_2(\mathbb{F}_{p^2})$ which fulfills the following: there exists an infinite projective $G$-Galois family consisting of Katz modular forms of weight one such that no family member is liftable to a holomorphic weight one Hecke eigenform in any level.}} \noindent See Corollaries \ref{coro4} and \ref{cor5} for more precise results, where the corresponding group $G$ is explicitly given. \section{Basics} \label{sec:bas} The aim of this section is to present the standard material on Galois representations, modular forms and function field extensions that will be used throughout the present article. \subsection{Galois representations and modular forms} \label{ssec:gr} In this article, we shall be concerned with two-dimensional Galois representations which are either Artin, i.e., are defined over $\mathbb{C}$, or have coefficients in $\overline{\mathbb{F}_p}$, where $p$ is a prime number. Both cases will be treated in parallel. To this end, we let $$\mathcal{G} \in \{\mathrm{GL}_2(\overline{\mathbb{F}_p}), \mathrm{GL}_2(\mathbb{C})\}.$$ Then all Galois representations considered in this paper are continuous and of the form $$\rho: {{\rm{G}}_\Qq} \to \mathcal{G},$$ where ${{\rm{G}}_\Qq} = \Gal(\overline{\QQ}/\mathbb{Q})$ is the absolute Galois group of $\mathbb{Q}$. The image $\Image(\rho) \subset \mathcal{G}$ is a finite group. By standard Galois theory, one has $\Image(\rho) \cong \Gal(K_\rho/\mathbb{Q})$ for some number field~$K_\rho$. We refer to $K_\rho$ as the `number field cut out by $\rho$'. It can also be characterised by $\ker(\rho) = \Gal(\overline{\QQ}/K_\rho)$. A Galois representation $\rho$ as above is said to be {\em irreducible} if the underlying $\overline{\mathbb{F}_p}[{{\rm{G}}_\Qq}]$-module (or $\mathbb{C}[{{\rm{G}}_\Qq}]$-module) is irreducible. It is said to be {\em semi-simple} if it is a direct sum of simple modules. Accordingly, a finite subgroup $G \subset \mathcal{G}$ is called {\em irreducible} if the natural inclusion is an irreducible representation. One furthermore says that $\rho$ is {\em odd} if $\det(\rho(c))=-1$, where $c$ is any complex conjugation in ${{\rm{G}}_\Qq}$ (all are conjugate). For a prime number $\ell$, one says that $\rho$ is {\em unramified at~$\ell$} if the inertia group at~$\ell$ inside $\Gal(\overline{\mathbb{Q}_\ell}/\mathbb{Q}_\ell) \hookrightarrow {{\rm{G}}_\Qq}$ (for any embedding) lies in the kernel of~$\rho$. This is equivalent to $K_\rho$ being unramified above~$\ell$. We also consider {\em projective} Galois representations in the two cases. We accordingly let $$\mathcal{P} \in \{\mathrm{PGL}_2(\overline{\mathbb{F}_p}), \mathrm{PGL}_2(\mathbb{C})\},$$ consider $$\rho^\mathrm{proj}:{{\rm{G}}_\Qq} \to \mathcal{P}$$ and make similar definitions, such as $\Image(\rho^\mathrm{proj}) \cong \Gal(K_{\rho^\mathrm{proj}}/\mathbb{Q})$. Given a Galois representation $\rho$, one can associate to it a unique projective one $\rho^\mathrm{proj}$ via composition with the natural projection $\mathcal{G} \twoheadrightarrow \mathcal{P}$. If we set $G=\Image(\rho)$ and $G^\mathrm{proj}=\Image(\rho^\mathrm{proj})$, then $G^\mathrm{proj}$ is the image of $G$ under the natural projection. Denote by $H$ the kernel of $G \to G^\mathrm{proj}$ and let ${\overline{\rho}} : {\rm{G}}_\mathbb{Q} / {\rm{ker}}(\rho) \rightarrow G$ be the isomorphism induced by $\rho$. If we identify $\Gal(K_\rho/\mathbb{Q})$ and ${{\rm{G}}_\Qq} / {\rm{ker}}(\rho)$ (via restriction of automorphisms), then we have $K_{\rho^\mathrm{proj}} = (K_\rho)^{{\overline{\rho}}^{-1}(H)}$ by standard Galois theory. We shall sometimes simply see $H$ as a subgroup of $\Gal(K_\rho/\mathbb{Q})$, i.e., drop ${\overline{\rho}}^{-1}$ from the notation, but we must be aware of the dependence on the representation. Given a projective Galois representation $\rho^\mathrm{proj}$ as above, by a result of Tate (see \cite[\S6]{Ser77} and \cite[\S4]{Que95}), it can be lifted to a linear representation $\rho$ as above. Moreover, it can be ensured that $\rho$ is unramified at all prime numbers where $\rho^\mathrm{proj}$ is unramified. This article relies on certain kinds of modular forms, which are called Hecke eigenforms. One can attach Galois representations of the above kinds to them. The modular forms we consider are either the `classical' holomorphic modular forms defined in standard textbooks such as \cite{DS05}, or their geometric counter part due to Katz \cite{Kat73}. A good source for both is \cite{DI95}. In both settings, modular forms have a weight, an integer usually denoted~$k$, and a level, a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We shall exclusively work with levels $\Gamma_1(N)$ and usually just say that the positive integer $N$ is the level. Holomorphic modular forms of fixed weight and level form a finite dimensional $\mathbb{C}$-vector space. Katz' geometric definition allows the use of other base rings, provided the level is invertible in the ring. We shall use only $\mathbb{C}$ or $\overline{\mathbb{F}_p}$ as base fields and thus impose $p \nmid N$ in the latter case. It should be remarked that using Katz modular forms over~$\mathbb{C}$, one exactly recovers classical modular forms. Every modular form has a so-called {\em $q$-expansion}, that is, a power series $\sum_{n=0}^\infty a_n q^n$ with $a_n$ in the base field. If the base field is $\mathbb{C}$, then replacing $q$ by $e^{2\pi i z}$, one obtains a Fourier series, which is actually equal to the modular form, viewed as a holomorphic function on the upper half-plane. When working over~$\mathbb{C}$, one can consider the $\mathbb{Z}$-module of $q$-expansions of modular forms in fixed weight~$k$ and level~$N$ such that all $a_n$ lie in~$\mathbb{Z}$. One can reduce them modulo~$p$ and base extend to~$\overline{\mathbb{F}_p}$. If $k \ge 2$, one then essentially recovers Katz modular forms over~$\overline{\mathbb{F}_p}$ (see \cite[Lemma 1.9]{Edi97} for a precise statement). However, when $k=1$, there are often more Katz modular forms than reductions of holomorphic ones. A purpose of this article is to exhibit infinite families of such having interesting properties. In both settings, there is a family of commuting linear maps $T_m$, for $m \in \mathbb{Z}_{\ge 1}$, on the vector space of modular forms, called {\em Hecke operators}. A modular form that is an eigenform for each $T_m$ is called a {\em Hecke eigenform}. If, moreover, the coefficient $a_1$ in the $q$-expansion equals~$1$, then call the eigenform {\em normalised}. If a normalised Hecke eigenform is a Katz modular form over~$\overline{\mathbb{F}_p}$, then the $a_n$ are, of course, in $\overline{\mathbb{F}_p}$. If it is a holomorphic modular form, then the eigen-property implies that the $a_n$ are algebraic integers, that is, lie in $\overline{\ZZ}$, the integral closure of $\mathbb{Z}$ in~$\mathbb{C}$. For the entire article, we fix ring homomorphisms $\overline{\ZZ} \hookrightarrow \overline{\mathbb{Z}_p} \twoheadrightarrow \overline{\mathbb{F}_p}$. By the {\em reduction modulo~$p$} of an algebraic integer, we shall always understand the image under the composite maps. So, the coefficients $a_n$ of a normalised holomorphic Hecke eigenform have well-defined reductions modulo~$p$. Work of Shimura, Deligne and Deligne--Serre (see \cite[\S9.6]{DS05} and \cite{DS75}) attaches to any normalised Hecke eigenform $f = \sum_{n=0}^\infty a_n q^n$ of weight $k$ and level~$N$ a semi-simple Galois representation $\rho_{f,p}: {{\rm{G}}_\Qq} \to \mathrm{GL}_2(\overline{\mathbb{F}_p})$. The representation is known to be odd and unramified outside~$Np$. Moreover, at every prime $\ell \nmid Np$, the trace of any Frobenius element $\Frob_\ell$ (all are conjugate) equals (the reduction modulo~$p$ of)~$a_\ell$ and its determinant equals $\ell^{k-1} \epsilon(\ell)$, where $\epsilon$ is the nebentype character associated with~$f$ (its values are the eigenvalues for the action of the diamond operators on~$f$). If $f$ is a holomorphic normalised Hecke eigenform (over $\mathbb{C}$) of weight $k=1$, by Deligne--Serre, one can associate with it a semi-simple Artin representation $\rho_{f,\mathbb{C}}: {{\rm{G}}_\Qq} \to \mathrm{GL}_2(\mathbb{C})$, which is odd, unramified outside~$N$ and, for every prime $\ell \nmid N$, the trace of $\Frob_\ell$ equals $a_\ell$ (as complex numbers) and its determinant equals $\epsilon(\ell)$. The projectivisation of the representations will be denoted $\rho_{f,p}^\mathrm{proj}$ and $\rho_{f,\mathbb{C}}^\mathrm{proj}$, respectively. As abbreviations, we shall often write $\rho_f$ and $\rho_f^\mathrm{proj}$ for both the mod~$p$ and the Artin cases. Moreover, we set $K_f := K_{\rho_f}$ and $K_f^\mathrm{proj} := K_{\rho_{f}^\mathrm{proj}}$. If $p>2$, the oddness of $\rho_f$ implies that $K_f$ and $K_f^\mathrm{proj}$ are totally imaginary. Indeed, as complex conjugation is of determinant~$-1$, it is a non-trivial involution and not scalar. We point out that the representations $\rho_f$ need not be irreducible. However, if they are, then the underlying modular form is {\em cuspidal}. For $k=1$, $\rho_{f, \mathbb{C}}$ is irreducible if and only if $f$ is cuspidal. We recall that a theorem of Brauer and Nesbitt (\cite[30.16]{CR88}) states that a semi-simple representation of a finite group is uniquely determined up to isomorphism by its character. Thus, any semi-simple Galois representation with finite image is uniquely determined by the traces of the Frobenius elements $\Frob_\ell$ at primes $\ell$ in a set of primes of density one because by Chebotarev's density theorem \cite[VII.13.4]{Neu99} any element in the image of the representation comes from some $\Frob_\ell$ (in fact, for $\ell$ in a positive density set of primes). As the trace of $\rho_f(\Frob_\ell)$ equals the coefficient $a_\ell$ of~$f$, the semi-simple representation $\rho_f$ is hence uniquely determined by~$f$ up to isomorphism of Galois representations, that is, up to conjugation. Consequently, the images of $\rho_f$ and $\rho_f^\mathrm{proj}$ are uniquely determined by $f$ up to conjugation in $\mathcal{G} \in \{\mathrm{GL}_2(\overline{\mathbb{F}_p}), \mathrm{GL}_2(\mathbb{C})\}$ or $\mathcal{P} \in \{\mathrm{PGL}_2(\overline{\mathbb{F}_p}), \mathrm{PGL}_2(\mathbb{C})\}$. A very important theorem of Khare-Wintenberger and Kisin (\cite[Theorem 1.2]{KW09} and \cite[Corollary 0.2]{Kis09}) is the following, which was formely known as {\em Serre's Modularity Conjecture}. \begin{thm}\label{thm:Serre-Conj} Let $p$ be a prime number and $\rho: {{\rm{G}}_\Qq} \to \mathrm{GL}_2(\overline{\mathbb{F}_p})$ an odd irreducible Galois representation. Then there is a normalised Hecke eigenform (of some level and weight) such that $\rho \cong \rho_{f,p}$. \end{thm} A notable consequence is the modularity of Artin representations of the following type (\cite[Corollary 10.2(ii)]{KW09}). \begin{thm}\label{thm:Serre-Artin} Let $\rho: {{\rm{G}}_\Qq} \to \mathrm{GL}_2(\mathbb{C})$ be an odd and irreducible Galois representation. Then there is a normalised `classical' holomorphic Hecke eigenform of weight one (and some level) such that $\rho \cong \rho_{f,\mathbb{C}}$. \end{thm} In view of our desire to make elegant and short statements, we make the following convention. If a representation $\rho: {{\rm{G}}_\Qq} \to \mathcal{G}$ (resp., a projective represention $\rho^\mathrm{proj}: {{\rm{G}}_\Qq} \to \mathcal{P}$) comes from a normalised Hecke eigenform~$f$, then we assume $f$ to be holomorphic of weight one if we are in the Artin case. The following practical consequence of the above shall be used on several occasions in the sequel: \begin{prop} \label{tool0} Let $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$) be a finite irreducible subgroup and $F/\mathbb{Q}$ a Galois extension of group $G$. Then there exists a normalised Hecke eigenform $f$ such that $K_{f}=F$ (resp., $K_f^\mathrm{proj} = F$) if and only if \noindent - $F^{\text{scalars in } G}$ (resp., $F$) is totally imaginary if we are in the Artin case or in the mod $p$ case with $p\ge 3$, \noindent - $F$ is arbitrary if we are in the mod $p$ case with $p=2$. \end{prop} \begin{proof} We prove this statement in the mod $p$ case. The arguments in the Artin case are exactly the same, except that one has to invoke the modularity of odd and irreducible Artin representations from Theorem~\ref{thm:Serre-Artin}. First, assume there exists a normalised Hecke eigenform $f$ such that $K_{f}=F$. Then, as recalled above, $\rho_{f}$ is odd. If $p \geq 3$, this implies that $K_{f}^{\mathrm{proj}}=F^{\text{scalars in }G}$ is totally imaginary. Now, assume $F^{\text{scalars in }G}$is totally imaginary if $p \geq 3$. We view the extension $F/\mathbb{Q}$ as a Galois representation $$\rho : {{\rm{G}}_\Qq} \twoheadrightarrow \Gal(F/\mathbb{Q}) \cong G \subset \mathcal{G}.$$ Then $\rho$ is odd. Note that oddness is an empty condition if $p=2$, so it suffices to consider $p\ge 3$. Then, indeed, as $F^{\text{scalars in }G}$ is totally imaginary, (any) complex conjugation is sent to a non-scalar element in $G$. Hence, under $\rho$, complex conjugation maps to a non-scalar involution in $\mathcal{G}$ and thus has determinant $-1$. As $\rho$ is also irreducible, it is afforded by a normalised Hecke eigenform $f$ by Theorem~\ref{thm:Serre-Conj}. \noindent In the projective case, the same arguments as above yield the desired equality $K_f^\mathrm{proj} = F$, except that one has to invoke Tate's theorem (recalled above) to lift the projective representation to a linear one to obtain the modularity in the last step. \end{proof} \subsection{Function field extensions} \label{ssec:ffe} Given a field $k$ of characteristic zero, an integer $n \geq 1$ and an $n$-tuple ${\bf{T}}=(T_1, \dots,T_n)$ of algebraically independent indeterminates, let $E/k({\bf{T}})$ be a finite Galois extension. If $n=1$, we write $E/k(T)$ for simplicity. Say that $E/k({\bf{T}})$ is {\it{$k$-regular}} if $E \cap \overline{k}= k$. Let ${B}$ be the integral closure of $k[{\bf{T}}]$ in ${E}$. For ${\bf{t}}=(t_1, \dots, t_n) \in k^n$, the residue field of ${B}$ at a maximal ideal $\mathfrak{P}$ lying over the ideal $\langle {\bf{T}} - {\bf{t}} \rangle$ of $k[{\bf{T}}]$ generated by $T_1-t_1, \dots, T_n-t_n$ is denoted by ${E}_{\bf{t}}$ and the extension ${E}_{\bf{t}}/k$ is called the {\it{specialisation}} of ${E}/k({\bf{T}})$ at ${\bf{t}}$. As the extension ${E}/k({\bf{T}})$ is Galois, the field ${E}_{\bf{t}}$ does not depend on $\mathfrak{P}$ and the extension ${E}_{\bf{t}}/k$ is finite and Galois. Moreover, the Galois group of ${E}_{\bf{t}}/k$ is the quotient of the decomposition group of $E/k({\bf{T}})$ at $\mathfrak{P}$ by the inertia group at $\mathfrak{P}$. For ${\bf{t}}$ outside a Zariski-closed proper subset (depending only on ${E}/k({\bf{T}})$), the inertia group at $\mathfrak{P}$ is trivial; in particular, the Galois group of ${E}_{\bf{t}}/k$ is a subgroup of ${\rm{Gal}}({E}/k({\bf{T}}))$. Furthermore, if $P({\bf{T}},Y) \in k[{\bf{T}}][Y]$ is a monic separable polynomial of splitting field $E$ over $k({\bf{T}})$ and if ${\bf{t}} \in k^n$ is such that the splitting field of $P({\bf{t}},Y)$ has Galois group ${\rm{Gal}}(E/k({\bf{T}}))$ over $k$, then the field $E_{\bf{t}}$ is the splitting field over $k$ of $P({\bf{t}},Y)$ \footnote{Indeed, since $P({\bf{T}},Y)$ is monic and is in $k[{\bf{T}}][Y]$, the splitting field over $k$ of $P({\bf{t}},Y)$ is contained in the specialised field $E_{\bf{t}}$. As the former field has degree $\|G\|$ over $k$ and the latter has degree at most $\|G\|$ over $k$, the two fields coincide.}. Assume $n=1$. A point $t_0 \in \mathbb{P}^1(\overline{k})$ is a {\it{branch point}} of $E/k(T)$ if the prime ideal of $\overline{k}[T-t_0]$ generated by $T-t_0$ ramifies in the extension $E\overline{k}/\overline{k}(T)$ \footnote{Replace $T-t_0$ by $1/T$ if $t_0=\infty$.}. The extension $E/k(T)$ has only finitely many branch points, usually denoted by $t_1, \dots, t_r$, and one has $r=0$ if and only if $E\overline{k}=\overline{k}(T)$ (which is equivalent to $E=k(T)$ if $E/k(T)$ is $k$-regular). If $t_0 \in k \setminus \{t_1, \dots,t_r\}$, then the Galois group of the specialisation $E_{t_0}/k$ of $E/k(T)$ at $t_0$ is the decomposition group at a prime ideal $\mathfrak{P}$ lying over $\langle T-t_0 \rangle$. Moreover, if $E$ is the splitting field over $k(T)$ of a monic separable polynomial $P(T,Y) \in k[T][Y]$ and if $t_0$ is any element of $k$ such that $P(t_0,Y)$ is separable, then $t_0$ is not a branch point of $E/k(T)$ and the field $E_{t_0}$ is the splitting field over $k$ of $P(t_0,Y)$. Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular Galois extension. In the sequel, we shall deal with the local behaviour at prime numbers of specialisations of $E/\mathbb{Q}(T)$. This requires the following material. See \cite[\S2.2]{Leg16} for more details. Given a number field $F$, let $A$ be the ring of its integers. For a non-zero prime ideal $\mathfrak{P}$ of $A$, we denote the corresponding valuation of $F$ by $v_\mathfrak{P}$. Moreover, we identify $\mathbb{P}^1(F)$ with $F \cup \{\infty\}$ and set $1/\infty = 0$, $1 / 0 = \infty$, $v_\mathfrak{P}(\infty) = -\infty$ and $v_\mathfrak{P}(0) = \infty$. \begin{defi} \label{rencontre} {\rm{(1)}} Let $F$ be a number field, $A$ the ring of its integers, $\mathfrak{P}$ a (non-zero) prime ideal of $A$ and $t_0$, $t_1 \in \mathbb{P}^1(F)$. We say that {\em{$t_0$ and $t_1$ meet modulo $\mathfrak{P}$}} if one of the following conditions holds: {\rm{(a)}} $v_{\mathfrak{P}}(t_0) \geq 0$, $v_{\mathfrak{P}}(t_1) \geq 0$ and $v_{\mathfrak{P}}(t_0-t_1) > 0$, {\rm{(b)}} $v_{\mathfrak{P}}(t_0) \leq 0$, $v_{\mathfrak{P}}(t_1) \leq 0$ and $v_{\mathfrak{P}}((1/t_0) - (1/t_1)) > 0$. \noindent {\rm{(2)}} Given $t_0$, $t_1$ $\in \mathbb{P}^1(\overline{\mathbb{Q}})$ and a prime number $p$, we say that {\em{$t_0$ and $t_1$ meet modulo $p$}} if there exists a number field $F$ satisfying the following two conditions: {\rm{(a)}} $t_0$, $t_1$ $\in \mathbb{P}^1(F)$, {\rm{(b)}} $t_0$ and $t_1$ meet modulo some prime ideal of the ring of integers of $F$ lying over $p \mathbb{Z}$. \end{defi} \begin{rem} \label{3.2} {\rm{(1)}} Definition \ref{rencontre}{\rm{(2)}} does not depend on the number field $F$ such that $t_0$ and $t_1$ $\in \mathbb{P}^1(F)$. \noindent {\rm{(2)}} If a given $t_0 \in \mathbb{P}^1(\mathbb{Q})$ meets a given $t_1 \in \mathbb{P}^1(\overline{\mathbb{Q}})$ modulo a given prime number $p$, then $t_0$ meets each $\mathbb{Q}$-conjugate of $t_1$ modulo $p$. \end{rem} We shall need the following lemma: \begin{lem} \label{lemma3} Let $p$ be a prime number and $r$ the number of branch points of $E/\mathbb{Q}(T)$. Suppose $p \geq r+1$. Then there exists $t_0 \in \mathbb{Q}$ such that $t_0$ does not meet any branch point of $E/\mathbb{Q}(T)$ modulo $p$. \end{lem} \begin{proof} We claim that the reduction modulo~$p$ of any $t_0 \in \mathbb{Z}$ that meets one of the $r$ branch points lies in a subset of $\mathbb{F}_p$ of cardinality at most~$r$. This implies the lemma because of the assumption $p > r$. Let $t_1, \dots, t_{r'}$ be representatives of the branch points of $E/\mathbb{Q}(T)$ for the action of ${{\rm{G}}_\Qq}$. By Remark \ref{3.2}(2), it suffices to study the reductions modulo~$p$ of $t_0 \in \mathbb{Z}$ meeting $t_i$ for $1\le i \le r'$. Given $i \in \{1, \dots, r'\}$, denote the ring of integers of $\mathbb{Q}(t_i)$ by $A$. By Remark \ref{3.2}(1), there exists a (non-zero) prime ideal $\mathfrak{P}$ of $A$ containing $p$ such that one of the following conditions holds: \noindent {\rm{(1)}} $v_\mathfrak{P}(t_0) \geq 0$, $v_\mathfrak{P}(t_i) \geq 0$ and $v_\mathfrak{P}(t_0-t_i) >0$, \noindent {\rm{(2)}} $v_\mathfrak{P}(t_0) \leq 0$, $v_\mathfrak{P}(t_i) \leq 0$ and $v_\mathfrak{P}((1/t_0)-(1/t_i)) >0$. \noindent First, assume (1) holds. Then the reduction $\overline{t_0} \in \mathbb{F}_p$ of $t_0$ modulo $p$ has to be equal to the reduction $\overline{t_i} \in A/\mathfrak{P}$ of $t_i$ modulo $\mathfrak{P}$. Now, assume (2) holds. If $v_\mathfrak{P}(t_i)=0$, then one has $v_\mathfrak{P}(t_0) =0$ as well (as $v_\mathfrak{P}((1/t_0)-(1/t_i)) >0$) and $v_\mathfrak{P}(t_0-t_i) = v_\mathfrak{P}((1/t_0)-(1/t_i)) >0$, i.e., (1) holds. One may then assume $v_\mathfrak{P}(t_i)<0$ and, consequently, $v_\mathfrak{P}(t_0)$ is negative as well, which cannot happen as $t_0 \in \mathbb{Z}$. This means that, for fixed $1\le i \le r'$, the reduction of $t_0$ modulo~$p$ lies in a subset of $\mathbb{F}_p$ of cardinality at most the number of prime ideals lying over $p \mathbb{Z}$, which is at most $[\mathbb{Q}(t_i):\mathbb{Q}]$. The claim follows because $\sum_{i=1}^{r'} [\mathbb{Q}(t_i):\mathbb{Q}] = r$. \end{proof} \begin{defi} Let $p$ be a prime number. Say that $E/\mathbb{Q}(T)$ has {\em{vertical ramification}} at $p$ if the prime ideal $p \mathbb{Z}[T]$ of $\mathbb{Z}[T]$ ramifies in the integral closure of $\mathbb{Z}[T]$ in $E$. \end{defi} This practical test for non-vertical ramification is well-known (see, e.g., \cite[Addendum 1.4(c)]{DG12}): \begin{prop} \label{nvr} Let $p$ be a prime number. Suppose that there exists a monic separable polynomial $P(T,Y) \in \mathbb{Z}[T][Y]$ that satisfies the following two conditions: \noindent {\rm{(1)}} the field $E$ is the splitting field over $\mathbb{Q}(T)$ of $P(T,Y)$, \noindent {\rm{(2)}} the discriminant $\Delta(T) \in \mathbb{Z}[T]$ of $P(T,Y)$ is not in $p\mathbb{Z}[T]$. \noindent Then the extension $E/\mathbb{Q}(T)$ has no vertical ramification at $p$. \end{prop} Finally, we recall the following result, which is part of the ``Specialisation Inertia Theorem" (see \cite[Proposition 4.2]{Bec91} and \cite[\S2.2.3]{Leg16}): \begin{prop} \label{sit} Let $p$ be a prime number and $t_0 \in \mathbb{Q}$. The specialisation of $E/\mathbb{Q}(T)$ at $t_0$ is unramified at $p$, provided the following two conditions hold: \noindent {\rm{(1)}} the extension $E/\mathbb{Q}(T)$ has no vertical ramification at $p$, \noindent {\rm{(2)}} $t_0$ does not meet any branch point of $E/\mathbb{Q}(T)$ modulo $p$. \end{prop} We shall also need the following result, which is a special case of \cite[Proposition 6.3]{KLN19}: \begin{prop} \label{kln} Let $E/\mathbb{Q}(T)$ be a $\mathbb{Q}$-regular Galois extension, let $t_1, \dots, t_r$ be the branch points of $E/\mathbb{Q}(T)$ and let $F$ be the compositum of the residue fields $(E(t_1))_{t_1}, \dots, (E(t_r))_{t_r}$ of $E/\mathbb{Q}(T)$ at $t_1,\dots,t_r$. Moreover, let $p$ be a prime number that is totally split in $F/\mathbb{Q}$ (avoiding a finite set of prime numbers depending only on $E/\mathbb{Q}(T)$). Then the decomposition group of $E_{t_0}/\mathbb{Q}$ at $p$ is cyclic for every $t_0 \in \mathbb{Q}$. \end{prop} In the sequel, we shall also deal with the local behaviour of specialisations of $E/\mathbb{Q}(T)$ at the infinite prime. In this context, the following proposition is useful: \begin{prop} \label{df} Denote the branch points of $E/\mathbb{Q}(T)$ by $t_1, \dots, t_r$ and let $t_0 \in \mathbb{Q} \setminus \{t_1, \dots, t_r\}$. \noindent {\rm{(1)}} Suppose $E/\mathbb{Q}(T)$ has three branch points and ${\rm{Gal}}(E/\mathbb{Q}(T))$ is not dihedral of order $4$, $6$, $8$, $12$. Then $E_{t_0}/\mathbb{Q}$ is not totally real. \noindent {\rm{(2)}} Suppose there exists a monic separable polynomial $P(T,Y) \in \mathbb{Q}[T][Y]$ of splitting field $E$ over $\mathbb{Q}(T)$ and an integer $0 \leq n \leq {\rm{deg}}_Y P-2$ such that $P(t_0,Y)$ is separable and the $n$-th derivative of $P(t_0,Y)$ has at least one complex non-real root. Then $E_{t_0}/\mathbb{Q}$ is not totally real. \end{prop} \begin{proof} (1) This is \cite[Proposition 1.2]{DF90}. \noindent (2) We reproduce the proof of \cite[Lemma 2.3]{LSY12} in a more general context. Suppose $E_{t_0}/\mathbb{Q}$ is totally real. Then all roots of $P(t_0,Y)$ are real. As this polynomial is also separable, we obtain, by Rolle's theorem, that the derivative $P'(t_0,Y)$ has at least ${\rm{deg}}_Y P(t_0,Y) -1$ distinct real roots, that is, $P'(t_0,Y)$ is separable and has only real roots. It then suffices to iterate this argument to get a contradiction. \end{proof} The following well-known result shows that, to construct specialisations of $E/\mathbb{Q}(T)$ with full Galois group and with specified local behaviour at finitely many given rational places (possibly infinite), one can look at one prime at a time and we do not have to worry about the corresponding Galois group: \begin{prop} \label{pv} Let $\mathcal{S}$ be a finite set of rational places. For each $p \in \mathcal{S}$, fix a Galois extension $F_p/\mathbb{Q}_p$ \footnote{Set $\mathbb{Q}_\infty=\mathbb{R}$ if $p =\infty$.} whose Galois group embeds into $G = {\rm{Gal}}(E/\mathbb{Q}(T))$. Suppose that, for each $p \in \mathcal{S}$, there exists $t_{0,p} \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $F_p$ is the completion of $E_{t_0,p}$ at $p$. Then there exists $t_0 \in \mathbb{Q}$ such that ${\rm{Gal}}(E_{t_0}/\mathbb{Q})=G$ and, for each $p \in \mathcal{S}$, the field $F_p$ is the completion of $E_{t_0}$ at $p$. Moreover, the set of all extensions $E_{t_0}/\mathbb{Q}$ with these properties is infinite. \end{prop} \begin{proof} The existence of at least one specialisation $E_{t_0}/\mathbb{Q}$ with the above properties can be found in, e.g., \cite[Proposition 2.1]{PV05}. To conclude that there exist infinitely many distinct such extensions $E_{t_0}/\mathbb{Q}$, it suffices to iterate the above statement, combined with the fact that the set of all prime numbers $p$ for which there exists $t_0 \in \mathbb{Q}$ such that $p$ ramifies in $E_{t_0}/\mathbb{Q}$ is infinite (see \cite[Corollary 2.12]{Leg16}). \end{proof} \section{Galois families} \label{sec:gal_fam} A purpose of this text is to propose the following definition, whose part (2) is Definition A from the introduction: \begin{defi}\label{defi:GF} Let $I$ be a set (of indices) and, for each $i \in I$, let $f_i$ be a normalised Hecke eigenform (of any level and weight). \noindent {\rm{(1)}} For a prime number~$p$, a positive integer~$n$ and a finite subgroup $G \subset \mathrm{GL}_2(\overline{\mathbb{F}_p})$, we say that the $(f_i)_{i\in I}$ form an {\em $n$-parameter $G$-Galois family} if there exists a finite Galois extension $E/\mathbb{Q}({\bf{T}})=E/\mathbb{Q}(T_1, \dots, T_n)$ with Galois group isomorphic to~$G$ such that, for each $i \in I$, the following two conditions hold: {\rm{(a)}} there is ${{\bf t}}_i \in \mathbb{Q}^n$ such that $K_{f_i}=E_{{\bf t}_i}$ and {\rm{(b)}} the image $\Image(\rho_{f_i})$ is conjugate to~$G$ in $\mathrm{GL}_2(\overline{\mathbb{F}_p})$ \footnote{Recall that $\Image(\rho_{f_i})$ is uniquely determined up to conjugation by $f_i$.}. \noindent {\rm{(2)}} For a finite subgroup $G \subset \mathrm{PGL}_2(\overline{\mathbb{F}_p})$, we define an {\em $n$-parameter projective $G$-Galois family} exactly as above, with the only exception that we replace $K_{f_i}$ by $K_{f_i}^\mathrm{proj}$ (for $i \in I$). \noindent {\rm{(3)}} When $G \subset \mathrm{GL}_2(\mathbb{C})$ (resp., $G \subset \mathrm{PGL}_2(\mathbb{C})$) and the $f_i$ for $i \in I$ are holomorphic weight one forms, we make exactly the same definition via the attached Artin representations and call this family an {\em $n$-parameter Artin $G$-Galois family} (resp., an {\em $n$-parameter projective Artin $G$-Galois family}). \noindent {\rm{(4)}} An $n$-parameter (projective) (Artin) $G$-Galois family is called {\em regular} if the underlying extension $E/\mathbb{Q}({\bf{T}})$ is $\mathbb{Q}$-regular. \end{defi} We remark that the base field $\mathbb{Q}$ could be replaced by any number field in the definition if both automorphic and field extension sides are changed accordingly. Moreover, we are not explicitly insisting that our Galois families are infinite; any set $I$ is allowed. We are primarily interested in families where infinitely many pairwise non-isomorphic fields occur as $K_{f_i}$ or $K_{f_i}^\mathrm{proj}$. Note that, by (b) in the definition of Galois families, the images $\Image(\rho_{f_i})$ are all conjugate to the fixed subgroup $G$ of the general linear group. So, by choosing appropriate bases for the representation modules underlying $\rho_{f_i}$ for $i \in I$, we could actually assume that they are {\em equal}. The field extension $E/\mathbb{Q}({\bf{T}})$ underlying a Galois family $(f_i)_{i \in I}$ can also be viewed via a Galois representation $$ \rho: \Gal(\overline{\mathbb{Q}({\bf{T}})}/\mathbb{Q}({\bf{T}})) \twoheadrightarrow \Gal(E/\mathbb{Q}({\bf{T}})) \cong G \subset \mathrm{GL}_2(\overline{\mathbb{F}_p}).$$ The representations $\rho_{f_i}$ can then be interpreted as specialisations of $\rho$. Furthermore, letting $H$ be the kernel of $G \to \mathrm{GL}_2(\overline{\mathbb{F}_p}) \twoheadrightarrow \mathrm{PGL}_2(\overline{\mathbb{F}_p})$ (its image equals $G^\mathrm{proj}$), we have the associated projective Galois representation $$\rho^\mathrm{proj}: \Gal(\overline{\mathbb{Q}({\bf{T}})}/\mathbb{Q}({\bf{T}})) \twoheadrightarrow \Gal(E^H/\mathbb{Q}({\bf{T}})) \cong G^\mathrm{proj} \subset \mathrm{PGL}_2(\overline{\mathbb{F}_p}).$$ Similar statements are true in the Artin case. Viewing the natural isomorphism between $\Gal(E_{{\bf{t}}_i}/\mathbb{Q})$ and $\Gal(E/\mathbb{Q}({\bf{T}}))$ as equality and considering $H$ as a subgroup of both, we have the equality $$K_{f_i}^\mathrm{proj} = (K_{f_i})^H = (E_{{\bf{t}}_i})^H = (E^H)_{{\bf{t}}_i}$$ of number fields. It shows that all $K_{f_i}^\mathrm{proj}$ are obtained as specialisations of the extension $E^H/\mathbb{Q}({\bf{T}})$, and as $\Image(\rho_{f_i})$ is conjugate to $G$, the image $\Image(\rho_{f_i}^\mathrm{proj})$ is conjugate to~$G^\mathrm{proj}$. This proves this result: \begin{prop}\label{prop:getreg} Let $(f_i)_{i \in I}$ be a (regular) $n$-parameter (Artin) $G$-Galois family. Then $(f_i)_{i \in I}$ is a (regular) $n$-parameter projective (Artin) $G^\mathrm{proj}$-Galois family. \end{prop} The direct converse of the proposition is not true because a given finite subgroup of $\mathrm{PGL}_2(\overline{\mathbb{F}_p})$ comes from infinitely many different finite subgroups of $\mathrm{GL}_2(\overline{\mathbb{F}_p})$. \section{The Beckmann-Black Problem, parametric extensions and Galois fa\-mi\-lies} \label{sec:BB} \subsection{The Beckmann-Black Problem and Galois fa\-mi\-lies} First, we recall the Beckmann-Black Problem (over $\mathbb{Q}$), which was intensively studied (see, e.g., the survey paper \cite{Deb01b} for more details and references). \begin{problem}[Beckmann-Black Problem]\label{prob:BB} Let $G$ be a finite group. Is it true that every Galois extension $F/\mathbb{Q}$ with Galois group $G$ occurs as a specialisation of some $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ with Galois group $G$ (possibly depending on $F/\mathbb{Q}$)? \end{problem} Let us recall that we take $\mathcal{G} \in \{\mathrm{GL}_2(\overline{\mathbb{F}_p}), \mathrm{GL}_2(\mathbb{C})\}$ and $\mathcal{P} \in \{\mathrm{PGL}_2(\overline{\mathbb{F}_p}), \mathrm{PGL}_2(\mathbb{C})\}$. We also remind the reader of our convention that if a representation $\rho: {{\rm{G}}_\Qq} \to \mathcal{G}$ (resp., a projective represention $\rho^\mathrm{proj}: {{\rm{G}}_\Qq} \to \mathcal{P}$) comes from a normalised Hecke eigenform~$f$, then we assume $f$ to be holomorphic of weight one if we are in the Artin case. Translating the Beckmann-Black Problem to the language of modular forms leads us to propose the following new problem: \begin{problem}\label{prob:mf} Let $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$) be a finite subgroup. Does every normalised Hecke eigenform $f$ such that $\Image(\rho_f)$ (resp., $\Image(\rho_f^\mathrm{proj})$) is conjugate to $G$ belong to some regular $1$-parameter (Artin) $G$-Galois family (resp., some regular $1$-parameter projective (Artin) $G$-Galois family), possibly depending on $f$? \end{problem} Note that Proposition \ref{prop:getreg} implies that a positive answer for a given finite subgroup $G \subset \mathcal{G}$ automatically gives a positive answer for the image $G^\mathrm{proj}$ of $G$ under the natural map $\mathcal{G} \twoheadrightarrow \mathcal{P}$. The following proposition makes the gap between Problems \ref{prob:BB} and \ref{prob:mf} precise: \begin{prop}\label{rem:bbpb1} Let $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$) be a finite irreducible subgroup. The answer to Problem \ref{prob:mf} is affirmative if and only if \noindent - every Galois extension $F$ of $\mathbb{Q}$ of group $G$ and such that $F^{\text{scalars in }G}$ (resp., $F$) is totally imaginary occurs as a specialisation of a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of group $G$ if we are in the Artin case or in the mod $p$ case with $p \geq 3$, \noindent - Problem \ref{prob:BB} has an affirmative answer if we are in the mod $p$ case with $p=2$. \end{prop} \begin{proof} We prove only the general linear case over $\overline{\mathbb{F}_p}$ as the proofs in the Artin case and the projective cases are almost identical. First, suppose the answer to Problem \ref{prob:mf} is affirmative. Let $F/\mathbb{Q}$ be a Galois extension of group $G$ such that $F^{\text{scalars in }G}$ is totally imaginary if $p$ is odd. By Proposition \ref{tool0}, there exists a normalised Hecke eigenform $f$ such that $K_f=F$. Then, from our assumption, there exists a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of group $G$ which specialises to $K_f / \mathbb{Q}$ at some $t_0 \in \mathbb{Q}$. As $K_f=F$, we are done. Now, assume every Galois extension $F$ of $\mathbb{Q}$ of group $G$ such that $F^{\text{scalars in }G}$ is totally imaginary if $p$ is odd occurs as a specialisation of a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of group $G$. Let $f$ be a normalised Hecke eigenform such that $\rho_{f,p}$ has image $G$. If $p$ is odd, by Proposition \ref{tool0}, the field $K_f^{\mathrm{proj}}$ is totally imaginary. Then, from our assumption, there exists a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of group $G$ that specialises to $K_f /\mathbb{Q}$ at some $t_0 \in \mathbb{Q}$, thus leading to the desired conclusion. \end{proof} If a given Galois extension $F/\mathbb{Q}$ of group $G$ occurs as a specialisation of some $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ of group $G$, Hilbert's irreducibility theorem shows that $F/\mathbb{Q}$ belongs to an infinite family of specialisations of $E/\mathbb{Q}(T)$ of group $G$. Below we show that the same conclusion holds in the context of modular forms: \begin{prop}\label{prop:bbpb2} Let $f$ be a normalised cuspidal Hecke eigenform. Suppose $\rho_{f}$ (resp., $\rho_f^\mathrm{proj}$) is irreducible with image conjugate to $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$). If $f$ belongs to a regular 1-parameter (Artin) $G$-Galois family (resp., a regular 1-parameter projective (Artin) $G$-Galois family), then $f$ belongs to an infinite regular $1$-parameter (Artin) $G$-Galois family (resp., an infinite regular 1-parameter projective (Artin) $G$-Galois family). \end{prop} \begin{proof} We prove only the general linear case over $\overline{\mathbb{F}_p}$ as the proofs in the Artin case and the projective cases are almost identical. The extension $K_f/\mathbb{Q}$ is Galois with group $G = \Image(\rho_f) \subset \mathrm{GL}_2(\overline{\mathbb{F}_p})$. By assumption, there exists a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ of group $G$ giving rise to $K_f/\mathbb{Q}$ by specialisation at some $t_0 \in \mathbb{Q}$. By Hilbert's irreducibility theorem, infinitely many distinct Galois extensions of $\mathbb{Q}$ of group $G$ occur as specialisations of $E/\mathbb{Q}(T)$. Hence, by Proposition \ref{tool0}, we get the desired conclusion if $p=2$. If $p$ is odd, then $K_f^{\mathrm{proj}}$ is totally imaginary by Proposition \ref{tool0}. Proposition~\ref{pv} applied simultaneously to $E/\mathbb{Q}(T)$ and $E^{\text{scalars in }G}/\mathbb{Q}(T)$ then provides infinitely many distinct Galois extensions of $\mathbb{Q}$ of group $G$ such that their subfields fixed by the scalars in $G$ are totally imaginary occuring as specialisations of $E/\mathbb{Q}(T)$. As in the case $p=2$, we apply Proposition \ref{tool0} to conclude. \end{proof} \subsection{Parametric extensions and Galois families} Let us now state the following definition, which is a function field analogue of the classical notion of 'parametric polynomial' as defined in \cite[Definition 0.1.1]{JLY02} (recalled as Definition \ref{def:par_gen2}): \begin{defi} \label{def:par_gen} Let ${\bf{T}}=(T_1, \dots, T_n)$ be an $n$-tuple of algebraically independent indeterminates ($n \geq 1$) and $E/\mathbb{Q}({\bf{T}})$ a finite Galois extension of group $G$. Say that $E/\mathbb{Q}({\bf{T}})$ is {\em{parametric}} if every Galois extension $F/\mathbb{Q}$ of group $G$ occurs as the specialisation $E_{\bf{t}}/\mathbb{Q}$ of $E/\mathbb{Q}({\bf{T}})$ at some ${\bf{t}} \in \mathbb{Q}^n$. \end{defi} Translating the notion of parametric extension to the language of modular forms leads us to propose the following new definition: \begin{defi} \label{def:para_mf} Let $n$ be a positive integer and $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$) a finite subgroup. An $n$-parameter (Artin) $G$-Galois family (resp., an $n$-parameter projective (Artin) $G$-Galois family) $(f_i)_{i \in I}$ is called {\em parametric} if, for any normalised Hecke eigenform $f$ such that the image $\Image (\rho_{f})$ is conjugate to~$G$ in $\mathcal{G}$ (resp., the image $\Image (\rho_{f}^\mathrm{proj})$ is conjugate to~$G$ in $\mathcal{P}$), there is $i \in I$ such that $K_{f_i} = K_f$ (resp., $K_{f_i}^\mathrm{proj} = K_f^\mathrm{proj}$). \end{defi} The following proposition is the analogue of Proposition \ref{rem:bbpb1} in the parametric context: \begin{prop} \label{pp->pp_mf} Let $n$ be a positive integer and $G \subset \mathcal{G}$ (resp., $G \subset \mathcal{P}$) an irreducible finite subgroup. Then there is a parametric $n$-parameter (Artin) $G$-Galois family (resp., a parametric $n$-parameter projective (Artin) $G$-Galois family) if and only if \noindent - there is a Galois extension $E/\mathbb{Q}(T_1,\dots, T_n)=E/\mathbb{Q}({\bf{T}})$ of group $G$ such that every Galois extension $F$ of $\mathbb{Q}$ of group $G$ satisfying that $F^{\text{scalars in }G}$ (resp., $F$) is totally imaginary occurs as a specialisation of $E/\mathbb{Q}({\bf{T}})$ if we are in the Artin case or in the mod $p$ case with $p \geq 3$, \noindent - there is a Galois extension $E/\mathbb{Q}(T_1,\dots, T_n)=E/\mathbb{Q}({\bf{T}})$ of group $G$ that is parametric if we are in the mod $p$ case with $p=2$. \noindent Moreover, the (projective) (Artin) $G$-Galois family is regular if and only if there is $E/\mathbb{Q}({\bf{T}})$ as above which, in addition, is $\mathbb{Q}$-regular. \end{prop} \begin{proof} We prove only the general linear case for $p \geq 3$ as the proofs in all other cases are almost identical. For an arbitrary $n$-parameter $G$-Galois family $(f_i)_{i \in I}$ with underlying function field extension $E/\mathbb{Q}({\bf{T}})$, Proposition \ref{tool0} provides $$\mathcal{S}_1:=\{K_{f_i}/\mathbb{Q} \, : \, i \in I\} \subseteq \mathcal{S}_2:= \{E_{\bf{t}}/\mathbb{Q} \, : \, {\bf{t}} \in \mathbb{Q}^n, \, {\rm{Gal}}(E_{\bf{t}}/\mathbb{Q})=G, \, {\rm{and}} \, E_{\bf{t}}^{\text{scalars in }G} \, {\rm{totally}} \, {\rm{imaginary}}\}$$ $$\hspace{2cm} \subseteq \mathcal{S}_3:=\{F/\mathbb{Q} \, : \, {\rm{Gal}}(F/\mathbb{Q})=G \, {\rm{and}} \, F^{\text{scalars in }G} \, {\rm{totally}} \, {\rm{imaginary}}\}.$$ Moreover, $\mathcal{S}_3$ is equal to $$\mathcal{S}_4:=\{K_f/\mathbb{Q} \, : \, f \, {\rm{normalised}} \, {\rm{Hecke}} \, {\rm{eigenform}} \, {\rm{with}} \, {\rm{im}}(\rho_f) \, {\rm{conjugate}} \, {\rm{to}} \, G \, {\rm{in}} \, {\rm{GL}}_2(\overline{\mathbb{F}_p})\}.$$ In particular, if $(f_i)_{i \in I}$ is parametric, then $\mathcal{S}_1 = \mathcal{S}_4$. Consequently, one has $\mathcal{S}_2 = \mathcal{S}_3$, as needed. Conversely, suppose there exists a Galois extension $E/\mathbb{Q}(T_1, \dots, T_n)$ of group $G$ such that every Galois extension $F$ of $\mathbb{Q}$ of group $G$ satisfying that $F^{\text{scalars in }G}$ is totally imaginary occurs as a specialisation of $E/\mathbb{Q}(T_1, \dots, T_n)$. Let $(f_i)_{i \in I}$ be the family of all normalised Hecke eigenforms such that $\Image(\rho_f)$ is conjugate to $G$ in ${\rm{GL}}_2(\overline{\mathbb{F}_p})$. By Proposition \ref{tool0} and our assumption, $(f_i)_{i \in I}$ is an $n$-parameter $G$-Galois family, which is trivially parametric. \end{proof} Given a finite group $G$, it is well-known that, if there exists a $\mathbb{Q}$-parametric polynomial $P({\bf{T}},Y) \in \mathbb{Q}[{\bf{T}}][Y]$ of group $G$ such that $E/\mathbb{Q}({\bf{T}})$ is $\mathbb{Q}$-regular, where $E$ is the splitting field over $\mathbb{Q}({\bf{T}})$ of $P({\bf{T}},Y)$, then the Beckmann-Black Problem has a positive answer for the group $G$ (see, e.g., \cite[Proposition 3.3.10]{JLY02}). Below we show that the same conclusion holds in the context of modular forms: \begin{prop} \label{prop:going_up} Let $G \subset \mathcal{G}$ or $G \subset \mathcal{P}$ be a finite subgroup. If there exists a regular parametric $n$-parameter (projective) $G$-Galois family (for some $n \geq 1$), then the answer to Problem \ref{prob:mf} is affirmative. \end{prop} \begin{proof} We prove only the general linear case over $\overline{\mathbb{F}_p}$ as the proofs of the other cases are almost identical. Let $E/\mathbb{Q}({\bf{T}}) = E/\mathbb{Q}(T_1, \dots, T_n)$ be the $\mathbb{Q}$-regular Galois extension of group $G$ underlying the regular parametric $n$-parameter $G$-Galois family from the statement and let $f$ be a normalised Hecke eigenform such that the image of $\rho_{f,p}$ is conjugate to $G$ in ${\rm{GL}}_2(\overline{\mathbb{F}_p})$. Pick ${\bf{\alpha}}=(\alpha_1, \dots, \alpha_n) \in \mathbb{Q}^n$ such that the number field $K_f$ is the specialised field $E_{\bf{\alpha}}$. We also fix ${\bf{\beta}}=(\beta_1, \dots, \beta_n) \in \mathbb{Q}^n$ such that $E_\beta/\mathbb{Q}$ has Galois group $G$ and such that the fields $E_\alpha$ and $E_\beta$ are linearly disjoint over $\mathbb{Q}$; such a $\beta$ exists as $E/\mathbb{Q}({\bf{T}})$ is $\mathbb{Q}$-regular. Now, given a new indeterminate $T$, for each $i \in \{1, \dots, n\}$, we fix $a_i(T) \in \mathbb{Q}[T]$ such that $a_i(0)=\alpha_i$ and $a_i(1)=\beta_i$. We set ${\bf{a}} = (a_1(T), \dots, a_n(T))$. Consider the $\mathbb{Q}(T)$-regular Galois extension $E(T)/\mathbb{Q}(T)(T_1, \dots, T_n)$ of group $G$ and its specialisation $(E(T))_{\bf{a}} / \mathbb{Q}(T)$ at ${\bf{a}}$. Below we show that the specialisation of $(E(T))_{\bf{a}} / \mathbb{Q}(T)$ at 0 (resp., at 1) is the extension $E_\alpha/\mathbb{Q}$ (resp., $E_\beta/\mathbb{Q}$). Consequently, the extension $(E(T))_{\bf{a}} / \mathbb{Q}(T)$ has Galois group $G$ (since this holds for $E_\alpha/\mathbb{Q}$) and, as $E_\alpha \cap E_\beta = \mathbb{Q}$, the extension $(E(T))_{\bf{a}} / \mathbb{Q}(T)$ is $\mathbb{Q}$-regular. Let $B$ be the integral closure of $\mathbb{Q}[{\bf{T}}]$ in $E$, $\mathfrak{P}$ a maximal ideal of the integral closure of $\mathbb{Q}(T)[{\bf{T}}]$ in $E(T)$ lying over $\langle T_1 - a_1(T), \dots, T_n - a_n(T) \rangle$, $\widetilde{B}$ the integral closure of $\mathbb{Q}[T]$ in $(E(T))_{\bf{a}}$ and $\mathfrak{P}_0$ a maximal ideal of $\widetilde{B}$ lying over $\langle T \rangle$. Since the reduction modulo $\mathfrak{P}$ of an element of $B$ yields an element of $\widetilde{B}$, we get a well-defined homomorphism $$\psi : B \rightarrow \widetilde{B} / \mathfrak{P}_0.$$ Moreover, since $a_i(0)=\alpha_i$ for every $i \in \{1, \dots, n\}$, one has $$\langle T_1 - \alpha_1, \dots, T_n - \alpha_n \rangle \subseteq {\rm{ker}}(\psi) \cap \mathbb{Q}[{\bf{T}}],$$ that is, $$\langle T_1 - \alpha_1, \dots, T_n - \alpha_n \rangle = {\rm{ker}}(\psi) \cap \mathbb{Q}[{\bf{T}}],$$ as the ideal in the left-hand side is maximal and $\mathbb{Q}[{\bf{T}}] \not \subseteq {\rm{ker}}(\psi)$. Consequently, the ideal ${\rm{ker}}(\psi)$ of $B$ lies over $\langle T_1 - \alpha_1, \dots, T_n - \alpha_n \rangle$ and it is maximal. One then has $$E_\alpha = B/{\rm{ker}}(\psi) \subseteq \widetilde{B}/\mathfrak{P}_0 = ((E(T))_{\bf{a}})_0.$$ As the field in the left-hand side has degree $\|G\|$ over $\mathbb{Q}$ and that in the right-hand side has degree at most $\|G\|$ over $\mathbb{Q}$, we get the desired equality $E_\alpha = ((E(T))_{\bf{a}})_0.$ Similarly, one has $E_\beta = ((E(T))_{\bf{a}})_1$. \end{proof} \subsection{Explicit examples} We conclude this section by giving explicit examples of finite groups $G$ for which the answer to Problem \ref{prob:mf} is affirmative and/or there exists a parametric (projective) (Artin) $G$-Galois family. To that end, we use the previous results from this section, thus meaning that the groups we choose below are known to have a generic polynomial over $\mathbb{Q}$ and/or to fulfill the Beckmann--Black Problem. Of course, there are more groups fulfilling this condition than just those given in the next theorem and we invite the interested reader to give more examples. We start with the mod $p$ case. \begin{thm}\label{thm:parametric_poly} Let $p$ be a prime number. \noindent {\rm{(1)}} Let $G \subset {\rm{PGL}}_2(\overline{\mathbb{F}_p})$ be a subgroup isomorphic to any of the following finite groups: $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, the dihedral group $D_4$ with eight elements, $A_4$, $S_4$, $A_5$, $S_5$. Assume - $p \geq 3$ if $G=\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ or $D_4$, - $p \geq 5$ if $G=A_4$ or $S_4$, - $p \geq 7$ if $G=A_5$, - $p=5$ if $G=S_5$. \noindent Then the following two conclusions hold. {\rm{(a)}} There is a regular parametric 2-parameter projective $G$-Galois family. {\rm{(b)}} For every normalised Hecke eigenform $f$ such that the Galois group ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, there exists an infinite regular 1-parameter projective $G$-Galois family containing $f$. In parti- cular, the answer to Problem \ref{prob:mf} is affirmative. \noindent {\rm{(2)}} Let $G \subset {\rm{PGL}}_2(\overline{\mathbb{F}_p})$ be a subgroup isomorphic to any of the following finite groups: the dihedral group $D_m$ with $2m$ elements ($m$ odd) or the dihedral group $D_8$ with 16 elements. Assume - $p$ is odd (in both cases), - $p$ does not divide $m$ (in the former case). \noindent Then the following two conclusions hold. {\rm{(a)}} There is a regular parametric n-parameter projective $G$-Galois family for some $n \geq 1$. {\rm{(b)}} For every normalised Hecke eigenform $f$ such that the Galois group ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, there exists an infinite regular 1-parameter projective $G$-Galois family containing $f$. In parti- cular, the answer to Problem \ref{prob:mf} is affirmative. \noindent {\rm{(3)}} Assume $p=3$. Let $G$ be the finite group $A_6 \cong {\rm{PSL}}_2(\mathbb{F}_9)$ and let $f$ be a normalised Hecke eigenform such that the Galois group ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$. Then there is an infinite regular 1-parameter projective $G$-Galois family containing $f$. In particular, the answer to Problem \ref{prob:mf} is affirmative. \end{thm} We shall need the following definition: \begin{defi} \label{def:par_gen2} Given a positive integer $n$, let ${\bf{T}}=(T_1, \dots, T_n)$ be an $n$-tuple of algebraically independent indeterminates and $P({\bf{T}},Y) \in \mathbb{Q}[{\bf{T}}][Y]$ a monic separable polynomial. Denote the Galois group of $P({\bf{T}},Y)$ over $\mathbb{Q}({\bf{T}})$ by $G$. \noindent {\rm{(1)}} Let $k$ be a field containing $\mathbb{Q}$. Say that $P({\bf{T}},Y)$ is {\em{$k$-parametric}} if, for every Galois extension $F/k$ of group $G$, the field $F$ is the splitting field over $k$ of some polynomial $P({\bf{t}},Y)$ with ${\bf{t}} \in k^n$. \noindent {\rm{(2)}} Say that $P({\bf{T}}, Y)$ is {\em{generic}} if it is $k$-parametric for every field $k$ containing $\mathbb{Q}$. \end{defi} \begin{rem} \label{!} Let $G$ be a finite group and $P({\bf{T}},Y) \in \mathbb{Q}[{\bf{T}}][Y]$ a monic separable polynomial of group $G$ and splitting field $E$ over ${\bf{\mathbb{Q}}}({\bf{T}})$. \noindent {\rm{(1)}} If $P({\bf{T}},Y)$ is $\mathbb{Q}$-parametric, then $E/\mathbb{Q}({\bf{T}})$ is parametric (see \S\ref{ssec:ffe}). \noindent {\rm{(2)}} If $P({\bf{T}},Y)$ is generic, then $E/\mathbb{Q}({\bf{T}})$ is $\mathbb{Q}$-regular (see \cite[Proposition 3.3.8]{JLY02}). \end{rem} \begin{proof}[Proof of Theorem \ref{thm:parametric_poly}] (1) By \cite[page 203]{JLY02}, the group $G$ has a generic polynomial $P(T_1, T_2,Y) \in \mathbb{Q}[T_1,T_2][Y]$. Consequently, the fact that (a) holds is a consequence of Proposition \ref{pp->pp_mf} and Remark \ref{!}. As for (b), it is a consequence of (a), Proposition \ref{prop:bbpb2} and Proposition \ref{prop:going_up} (note that irreduciblity is guaranteed as it is easy to see that $G$ is not isomorphic to any quotient of a finite subgroup of the upper triangular matrices inside $\mathrm{GL}_2(\overline{\mathbb{F}_p})$). \noindent (2) The proof is identical to the proof of (1). The group $G$ has a generic polynomial with rational coefficients (see, e.g., \cite[page 112]{JLY02}). \noindent (3) Here we use that the Beckmann-Black Problem has a positive answer for the group $G$ (see, e.g., \cite[th\'eor\`eme 2.2]{Deb01b}) and apply Propositions \ref{rem:bbpb1} and \ref{prop:bbpb2}. \end{proof} Now, we give the analogue of Theorem \ref{thm:parametric_poly} in the Artin situation. As the proof is almost identical to the previous one, details are left to the reader. \begin{thm} \label{thm:parametric_poly2} {\rm{(1)}} Let $G\subset {\rm{PGL}}_2(\mathbb{C})$ be a subgroup isomorphic to any of the following finite groups: $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, the dihedral group $D_4$ with eight elements, $A_4$, $S_4$, $A_5$. Then these conclusions hold. {\rm{(a)}} There is a regular parametric 2-parameter projective Artin $G$-Galois family. {\rm{(b)}} For every holomorphic normalised Hecke eigenform $f$ of weight one such that the Galois group ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, there is an infinite regular 1-parameter projective Artin $G$-Galois family containing $f$. In particular, the answer to Problem \ref{prob:mf} is affirmative. \noindent {\rm{(2)}} Let $G\subset {\rm{PGL}}_2(\mathbb{C})$ be a subgroup isomorphic to any of the following finite groups: the dihedral group $D_m$ with $2m$ elements ($m$ odd) or the dihedral group $D_8$ with 16 elements. {\rm{(a)}} There is a regular parametric n-parameter projective Artin $G$-Galois family for some $n \geq 1$. {\rm{(b)}} For every holomorphic normalised Hecke eigenform $f$ of weight one such that the Galois group ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, there is an infinite regular 1-parameter projective Artin $G$-Galois family containing $f$. In particular, the answer to Problem \ref{prob:mf} is affirmative. \end{thm} Finally, we show that parametric 1-parameter projective (Artin) $G$-Galois families do not occur for several finite groups $G$: \begin{thm} \label{?} {\rm{(1)}} Let $p$ be a prime number and $G$ a finite irreducible subgroup of ${\rm{PGL}}_2(\overline{\mathbb{F}_p})$. Suppose the following three conditions hold: {\rm{(a)}} $G$ has even order, {\rm{(b)}} $G$ has a generic polynomial with rational coefficients, {\rm{(c)}} $G$ has a non-cyclic abelian subgroup. \noindent Then there does not exist any parametric 1-parameter projective $G$-Galois family. \noindent {\rm{(2)}} Let $G$ be a finite irreducible subgroup of ${\rm{PGL}}_2(\mathbb{C})$. Suppose the following three conditions hold: {\rm{(a)}} $G$ has even order, {\rm{(b)}} $G$ has a generic polynomial with rational coefficients, {\rm{(c)}} $G$ has a non-cyclic abelian subgroup. \noindent Then there does not exist any parametric 1-parameter projective Artin $G$-Galois family. \end{thm} In particular, if $G$ is any finite group and $p$ any prime number as in Theorem \ref{thm:parametric_poly}(1), then 2 is the least integer $n$ such that there exists a (regular) parametric $n$-parameter projective $G$-Galois family\footnote{All these finite groups admit $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ as a subgroup.}. The same conclusion holds in the Artin situation for finite groups $G$ in Theorem \ref{thm:parametric_poly2}(1). We shall need the following lemma: \begin{lem} \label{kln2} Let $G$ be a finite group, $m$ a positive integer, and $F_1/\mathbb{Q}, \dots, F_m/\mathbb{Q}$ finite Galois extensions of $\mathbb{Q}$ whose Galois groups are subgroups of $G$. Suppose there exists a generic polynomial with rational coefficients and Galois group $G$. Then there exists a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of group $G$ which specialises to $F_1/\mathbb{Q}, \dots, F_m/\mathbb{Q}$ at non branch points. \end{lem} \begin{proof} Since there exists a generic polynomial of group $G$ with rational coefficients, one may apply \cite{DeM83} and \cite[Theorem 5.2.5]{JLY02} to get that there exist an integer $n \geq 1$ and a polynomial $P({\bf{T}},Y)=P(T_1, \dots, T_n,Y) \in \mathbb{Q}[{\bf{T}}][Y]$ of group $G$ such that, for every extension $L/\mathbb{Q}$ and every Galois extension $F/L$ of group $H$ contained in $G$, there exists ${\bf{t}} \in L^n$ such that $P({\bf{t}},Y)$ is separable and $F$ is the splitting field over $L$ of $P({\bf{t}}, Y)$. Pick a finite Galois extension $F_{m+1}/\mathbb{Q}$ of group $G$ and set $F_{m+2}/\mathbb{Q}=\mathbb{Q}/\mathbb{Q}$. By the above, for $i \in \{1, \dots, m+2\}$, there exists ${\bf{t}}_i \in \mathbb{Q}^n$ such that $P({\bf{t}}_i,Y)$ is separable and the splitting field over $\mathbb{Q}$ of $P({\bf{t}}_i,Y)$ is $F_i$. By polynomial interpolation (as in the proof of \cite[Proposition 3.3.10]{JLY02}), one constructs a monic polynomial $Q(T,Y) \in \mathbb{Q}[T][Y]$ such that, for each $i \in \{1, \dots, m+2\}$, $Q(i,Y) = P({\bf{t}}_i,Y)$. Fix $i \in \{1, \dots, m+2\}$. Since $P({\bf{t}}_i,Y)$ is separable, $Q(T,Y)$ is also separable. Let $E$ be the splitting field of $Q(T,Y)$ over $\mathbb{Q}(T)$. Since $Q(i,Y)$ is separable, the specialisation of $E/\mathbb{Q}(T)$ at $i$ is $F_i/\mathbb{Q}$ and $i$ is not a branch point of $E/\mathbb{Q}(T)$. It remains to notice that $E/\mathbb{Q}(T)$ must be $\mathbb{Q}$-regular (by using $F_{m+2}/\mathbb{Q})$ and has Galois group $G$ (by using $F_{m+1}/\mathbb{Q}$) to conclude the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{?}] (1) Suppose there exists a parametric 1-parameter projective $G$-Galois family and denote the underlying function field extension by $E/\mathbb{Q}(T)$. First, assume $E/\mathbb{Q}(T)$ is not $\mathbb{Q}$-regular. Then there exists a non-trivial finite Galois extension $L/\mathbb{Q}$ such that, for every normalised Hecke eigenform $f$ such that ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, the field $K_f^\mathrm{proj}$ contains $L$. Now, combine (a), (b) and Lemma \ref{kln2} to get the existence of a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of Galois group $G$ which specialises to $\mathbb{Q}(\sqrt{-1})/\mathbb{Q}$ at a non branch point. By Proposition \ref{pv}, we then get a finite Galois extension $M_1/\mathbb{Q}$ of group $G$ which is totally imaginary. Denote the prime numbers which ramify in $M_1/\mathbb{Q}$ by $p_1, \dots, p_s$. Apply again (a), (b) and Lemma \ref{kln2} to get a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$ of Galois group $G$ which specialises to $\mathbb{Q}(\sqrt{-1})/\mathbb{Q}$ and to $\mathbb{Q}/\mathbb{Q}$ at non branch points. One then gets a finite Galois extension $M_2/\mathbb{Q}$ of group $G$ which is totally imaginary and unramified at $p_1, \dots, p_s$. In particular, the fields $M_1$ and $M_2$ are linearly disjoint over $\mathbb{Q}$. But, by Proposition \ref{tool0}, there exist two normalised Hecke eigenforms $f_1$ and $f_2$ such that $M_i= K_{f_i}^\mathrm{proj}$ for $i=1,2$, thus leading to a contradiction. Hence, $E/\mathbb{Q}(T)$ is $\mathbb{Q}$-regular. Next, by (c), the group $G$ has a non-cyclic abelian subgroup $H$. Without loss of generality, we may assume $H= \mathbb{Z}/p_0 \mathbb{Z} \times \mathbb{Z}/p_0 \mathbb{Z}$ for some prime number $p_0$. Pick a sufficiently large prime number $q$ which is totally split in the number field $F(e^{2 i \pi / {p_0}})$, where $F$ is the number field provided by Proposition \ref{kln}. As $q$ is totally split in $F$, every specialisation of $E/\mathbb{Q}(T)$ has cyclic decomposition group at $q$. Hence, for every normalised Hecke eigenform $f$ such that ${\rm{Gal}}(K_f^\mathrm{proj}/\mathbb{Q})$ is conjugate to $G$, the field $K_f^\mathrm{proj}$ has cyclic decomposition group at $q$. However, since $q$ is totally split in $\mathbb{Q}(e^{2 i \pi / {p_0}})$, one has $q \equiv 1 \pmod {p_0}$ (up to finitely many exceptions) and there exists a Galois extension $F^{(q)}$ of $\mathbb{Q}_q$ of group $\mathbb{Z}/p_0\mathbb{Z} \times \mathbb{Z}/p_0\mathbb{Z}$. \footnote{Indeed, one can take $F^{(q)}$ to be the compositum of the fields $F_1^{(q)}$ and $F_2^{(q)}$, where $F_1^{(q)}$ is the unique degree $p_0$ unramified extension of $\mathbb{Q}_q$ and $F_2^{(q)}/\mathbb{Q}_q$ is a finite Galois extension with Galois group $\mathbb{Z}/p_0\mathbb{Z}$ that is totally ramified (such an extension exists; see, e.g., \cite[Chapter IV]{Ser79}).} Now, by \cite[(9.2.8)]{NSW08}, there exists a Galois extension $F/\mathbb{Q}$ of group $\mathbb{Z}/p_0\mathbb{Z} \times \mathbb{Z}/p_0\mathbb{Z}$ whose completion at $q$ is equal to $F^{(q)}/\mathbb{Q}_q$. Consequently, by (c), Lemma \ref{kln2} and Proposition \ref{pv}, we get a finite Galois extension $M/\mathbb{Q}$ of group $G$, which is totally imaginary and such that the completion at $q$ has Galois group $\mathbb{Z}/p_0\mathbb{Z} \times \mathbb{Z}/p_0\mathbb{Z}$. By Proposition \ref{tool0}, we get that $M=K_f^\mathrm{proj}$ for some normalised Hecke eigenform $f$, thus leading to another contradiction. \noindent (2) The proof is identical to that of (1). \end{proof} \section{Infinite Galois families of non-liftable weight one modular eigenforms} \label{sec:explicit} The aim of this section is to exhibit an infinite regular $1$-parameter projective Galois family of non-liftable Katz modular eigenforms of weight one over ${\overline{\mathbb{F}_p}}$ for $p \in \{3,5,7,11\}$. We start with a general result, which potentially applies to any odd prime number~$p$ \footnote{The oddness of $p$ is only needed because the weight lowering result used in the proof of Theorem \ref{thm1} does not have any published proof in the literature when $p=2$, the representation is unramified at $p=2$ and the image of Frobenius at $p=2$ is scalar. However, a proof is outlined on Frank Calegaris's blog (see \url{https://www.galoisrepresentations.com/2014/08/10/is-serres-conjecture-still-open/}), making the restriction $p>2$ superfluous.}. Consider the following statement: \noindent {\rm{($$)}} {\it{Let $p$ be an odd prime number, $n$ a positive integer and let $G$ be either ${\rm{PGL}}_2(\mathbb{F}_{p^n})$ or ${\rm{PSL}}_2(\mathbb{F}_{p^n})$. There exists an infinite regular $1$-parameter projective $G$-Galois family consisting of Katz modular forms of weight one. Moreover, no family member is liftable to a holomorphic weight one Hecke eigenform in any level.}} \begin{thm} \label{thm1} Statement {\rm{($$)}} holds if $G$ is not isomorphic to any finite subgroup of ${\rm{PGL}}_2(\mathbb{C})$ and if there exists a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ of group $G$ such that the following two conditions hold: \noindent {\rm{(1)}} there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is totally imaginary, \noindent {\rm{(2)}} there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is unramified at $p$. \end{thm} \begin{proof} By the second part of the assumption and Proposition \ref{pv}, the extension $E/\mathbb{Q}(T)$ has infinitely many distinct specialisations of group $G$ which are totally imaginary and unramified at~$p$. We view any such specialisation $E_{t}/\mathbb{Q}$ as a projective Galois representation $$\rho_{t}^\mathrm{proj}: {{\rm{G}}_\Qq} \twoheadrightarrow \Gal(E_{t}/\mathbb{Q}) \cong G \subset {\rm{PGL}}_2(\overline{\mathbb{F}_p}),$$ thus obtain infinitely non-isomorphic ones. By the result of Tate recalled in \S\ref{ssec:gr}, there is a linear lift $\rho_{t}: {{\rm{G}}_\Qq} \to {\rm{GL}}_2(\overline{\mathbb{F}_p})$ of $\rho_{t}^\mathrm{proj}$ which is unramified at all prime numbers where $E_{t}/\mathbb{Q}$ is unramified. Moreover, as in the proof of Proposition \ref{tool0}, $\rho_{t}$ is odd, as $E_{t}$ is totally imaginary. As it is also irreducible, by Theorem~\ref{thm:Serre-Conj}, the representation $\rho_{t}$ comes from some normalised Hecke eigenform. Furthermore, by weight lowering as proved in \cite[Theorem 4.5]{Edi92}, $\rho_{t}$ actually comes from a Katz modular form $f_{t}$ of weight $1$ over~$\overline{\mathbb{F}_p}$. In order to see this, note that we are in case 2.(a) in \cite[Definition 4.3]{Edi92} with $a=b=0$, whence the weight associated with $\rho_t$ equals~$1$. Moreover, note that the hypothesis excluding the `exceptional case' in \cite[Theorem 4.5]{Edi92} is superfluous by the last sentence of \cite[\S1]{Edi92}. Finally, $G$ is not isomorphic to a quotient of any finite subgroup of $\mathrm{PGL}_2(\mathbb{C})$. Indeed, otherwise one would have that $G$ is a quotient of a cyclic group or of a dihedral group or of a finite group among $A_4$, $S_4$ and $A_5$. As this family of groups is easily seen to be closed under quotients, one would have that $G$ itself is cyclic, dihedral or among $A_4$, $S_4$ and $A_5$, which cannot happen by the first part of the assumption. Consequently, the representation $\rho_{t}$ cannot be the reduction of any semi-simple $2$-dimensional Artin representation. Hence, $f_{t}$ cannot be the reduction of a normalised holomorphic Hecke eigenform of weight $1$ and any level. \end{proof} Now, we combine Theorem \ref{thm1} and the various tools from \S\ref{ssec:ffe} to show that Statement ($$) holds for $p \in \{5,7,11\}$: \begin{cor} \label{coro4} {\rm{(1)}} Statement {\rm{($$)}} holds for $p=5$ (with $G= {\rm{PGL}}_2(\mathbb{F}_5) \cong S_5$). \noindent {\rm{(2)}} Statement {\rm{($$)}} holds for $p=7$ (with $G={\rm{PSL}}_2(\mathbb{F}_7)$). \noindent {\rm{(3)}} Statement {\rm{($$)}} holds for $p=11$ (with $G={\rm{PSL}}_2(\mathbb{F}_{11})$). \end{cor} \begin{proof} (1) Consider the monic separable polynomial $P(T,Y) = Y^5-Y^4-T$ and denote its splitting field over $\mathbb{Q}(T)$ by $E$. By \cite[\S4.4]{Ser92}, the extension $E/\mathbb{Q}(T)$ is $\mathbb{Q}$-regular, has $r=3$ branch points and has Galois group $S_5 \cong {\rm{PGL}}_2(\mathbb{F}_5)$, which is not a subgroup of ${\rm{PGL}}_2(\mathbb{C})$. Then, by Proposition \ref{df}(1), for every rational number $t_0$, the specialisation $E_{t_0}/\mathbb{Q}$ is not totally real as soon as $t_0$ is not a branch point. Moreover, by, e.g., \cite[Theorem 2]{Swa62}, the discriminant of $P(T,Y)$ is equal to $5^5T^4+4^4 T^3$, which is not in $5 \mathbb{Z}[T]$. Hence, the extension $E/\mathbb{Q}(T)$ has no vertical ramification at $p=5$, by Proposition \ref{nvr}. Furthermore, since $p =5 \geq 4=r+1$, we may use Lemma \ref{lemma3} to get the existence of $t_0 \in \mathbb{Q}$ such that $t_0$ does not meet any branch point of $E/\mathbb{Q}(T)$ modulo $p=5$. Hence, by Proposition \ref{sit}, there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is unramified at $p=5$. It then remains to apply Theorem \ref{thm1} to conclude. \noindent (2) The proof is similar in the case $p=7$. Namely, consider the monic separable polynomial $$P(T,Y)=Y^7 - 56 Y^6 + 609 Y^5 + 1190 Y^4 + 6356 Y^3 + 4536 Y^2 - 6804 Y - 5832 -TY(Y+1)^3$$ and denote its splitting field over $\mathbb{Q}(T)$ by $E$. By \cite[Satz 3]{MM85}, the extension $E/\mathbb{Q}(T)$ is $\mathbb{Q}$-regular, has three branch points and has Galois group ${\rm{PSL}}_2(\mathbb{F}_7)$, which is not contained in ${\rm{PGL}}_2(\mathbb{C})$. Moreover, the reduction modulo $p=7$ of $P(T,Y)$ is $Y^7 - 1 - TY(Y+1)^3$, which has discriminant $-3T^8-T^7 \not= 0$. As above, we apply the various tools from \S\ref{ssec:ffe} and Theorem \ref{thm1} to get the desired conclusion. \noindent (3) Consider the monic separable polynomial $$P(T,Y)= Y^{11} -3Y^{10} + 7Y^9 - 25Y^8 + 46Y^7 - 36Y^6 + 60Y^4 -121Y^3 + 140Y^2-95Y+27 + Y^2(Y-1)^3T$$ and denote its splitting field over $\mathbb{Q}(T)$ by $E$. The extension $E/\mathbb{Q}(T)$ is $\mathbb{Q}$-regular and has Galois group ${\rm{PSL}}_2(\mathbb{F}_{11})$, which does not embed into ${\rm{PGL}}_2(\mathbb{C})$ (see page 497 of \cite{MM18} for more details). Moreover, one checks with a computer that the discriminant $\Delta(T)$ of $P(T,Y)$ is $$\Delta(T) = (108T^3 - 7472T^2 + 267408T + 7987117)^4.$$ Hence, $E/\mathbb{Q}(T)$ has at most 4 branch points and one may then apply Lemma \ref{lemma3} to get that there exists $t_0 \in \mathbb{Q}$ such that $t_0$ does not meet any branch point modulo $p=11$. Also, as $\Delta(T)$ is not in $11\mathbb{Z}[T]$, the extension $E/\mathbb{Q}(T)$ has no vertical ramification at $p=11$, by Proposition \ref{nvr}. Hence, by Proposition \ref{sit}, there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is unramified at $p=11$. Concerning the local behaviour at the infinite prime, it actually holds that $E/\mathbb{Q}(T)$ has four branch points\footnote{Indeed, as recalled in \S\ref{ssec:ffe}, every branch point of $E/\mathbb{Q}(T)$ is either $\infty$ or a root of $\Delta(T)$. By, e.g., \cite[Lemma 3.1]{Mue02}, $\infty$ is a branch point of $E/\mathbb{Q}(T)$ (the corresponding ramification index is even equal to 6). Moreover, by the Riemann existence theorem, at least one root of $\Delta(T)$ is a branch point of $E/\mathbb{Q}(T)$. As $108T^3 - 7472T^2 + 267408T + 7987117$ is irreducible over $\mathbb{Q}$ (it is easily checked that its reduction modulo 5 has no root in $\mathbb{F}_5$), all roots of $\Delta(T)$ have to be branch points of $E/\mathbb{Q}(T)$, by the so-called Branch Cycle Lemma (see \cite{Fri77} and \cite[Lemma 2.8]{Vol96}).} and, because of that, we cannot use Proposition \ref{df}(1) as above. We then refer to Proposition \ref{df}(2). Namely, the 9-th derivative with respect to $Y$ of $P(T,Y)$ is $$\frac{11!}{2}Y^2 - 3 \cdot 10! \cdot Y + 7 \cdot 9!,$$ which has discriminant $10! \cdot 9! \cdot (9 \cdot 10-2 \cdot 11 \cdot 7) <0$. Hence, the specialisation $E_{t_0}/\mathbb{Q}$ is not totally real for every rational number $t_0$ such that $P(t_0,Y)$ is separable. As in the previous cases, it then remains to apply Theorem \ref{thm1} to conclude the proof. \end{proof} \begin{rem} Of course, variants can be given, by making use of other explicit polynomials $P(T,Y)$. For example, in the case $p=7$, one can also use the polynomial $$P(T,Y)= Y^7+ Y^6 + Y^5 + T Y^4 + (T-2) Y^3 - 5 Y^2 - 2Y +1 \in \mathbb{Q}[T][Y],$$ which is intensively studied in \cite{LSY12}, to prove that Statement {\rm{($$)}} holds (with $G={\rm{PSL}}_2(\mathbb{F}_7)$ too). \end{rem} A tool used throughout the proof of Corollary \ref{coro4} is Lemma \ref{lemma3}, which does not apply in the case $p=3$ if ${\rm{Gal}}(E/\mathbb{Q}(T))$ is not cyclic (by the Riemann existence theorem). However, Statement ($$) still holds in this case: \begin{cor} \label{cor5} Statement {\rm{($$)}} holds for $p=3$ (with $G= {\rm{PSL}}_2(\mathbb{F}_9) \cong A_6$). \end{cor} \begin{proof} Clearly, $A_6 \cong {\rm{PSL}}_2(\mathbb{F}_9)$ does not embed into ${\rm{PGL}}_2(\mathbb{C})$. Now, consider the polynomial $f(Y)=(Y^2+1)(Y^2+4) \in \mathbb{Q}[Y]$. It is separable and has discriminant $4 \cdot 9 \cdot 9 \cdot 16$, which is a square in $\mathbb{Q}$. The splitting field over $\mathbb{Q}$ of this polynomial is $\mathbb{Q}(\sqrt{-1})$, which is unramified at 3 and totally imaginary. By \cite[Theorem 3]{KM01}, there exists a monic separable polynomial $P(T,Y) \in \mathbb{Q}[T][Y]$ of splitting field $E$ over $\mathbb{Q}(T)$ such that $E/\mathbb{Q}(T)$ is a $\mathbb{Q}$-regular Galois extension of group $A_6$ and such that the splitting fields over $\mathbb{Q}$ of $P(0,Y)$ and $f(Y)$ coincide. From this equality and the fact that $f(Y)$ is separable, we get that the specialised field $E_0$ is equal to the splitting field of $f(Y)$ over $\mathbb{Q}$ and that $0$ is not a branch point of $E/\mathbb{Q}(T)$. It then remains to apply Theorem \ref{thm1} to conclude. \end{proof} Finally, we discuss the case $p \geq 13$. Another common feature of the proof of Corollary \ref{coro4} is the existence of a monic separable polynomial $P(T,Y) \in \mathbb{Z}[T][Y]$ of discriminant $\Delta(T) \not \in p\mathbb{Z}[T]$ and of Galois group ${\rm{PGL}}_2(\mathbb{F}_{p^n})$ or ${\rm{PSL}}_2(\mathbb{F}_{p^n})$ over $\mathbb{Q}(T)$ (for some $n \geq 1$). For $p \geq 13$, we are not aware of any polynomial satisfying both conditions. For example, no explicit polynomial of group ${\rm{PGL}}_2(\mathbb{F}_p)$ ($11 \leq p \leq 29$) given in pages 499-500 of \cite{MM18} satisfies the former. We also notice that Statement ($$) for the given prime number $p$ implies that some ${\rm{PGL}}_2(\mathbb{F}_{p^n})$ or ${\rm{PSL}}_2(\mathbb{F}_{p^n})$ occurs as the Galois group of a $\mathbb{Q}$-regular Galois extension of $\mathbb{Q}(T)$, which is unknown in general. Of course, for some prime numbers $p$, this is known (usually for $n=1$) and one even has such $\mathbb{Q}$-regular extensions with three branch points, coming from the rigidity method (see, e.g., \cite[Chapter I, Corollary 8.10]{MM18} for more details). In particular, for such a prime number $p$, we obtain an infinite regular $1$-parameter projective $G$-Galois family, with $G={\rm{PGL}}_2(\mathbb{F}_p)$ or $G={\rm{PSL}}_2(\mathbb{F}_p)$. Nevertheless, one has the following result: \begin{prop} Let $p$ be an odd prime number. Then there exist a finite group $G$ of order $2p^2$ and a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ of group $G$ such that the following conditions hold: \noindent {\rm{(1)}} $G \subseteq {\rm{PGL}}_2(\overline{\mathbb{F}_p})$ but $G \not \subseteq {\rm{PGL}}_2(\mathbb{C})$, \noindent {\rm{(2)}} there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is totally imaginary, \noindent {\rm{(3)}} there exists $t_0 \in \mathbb{Q}$, not a branch point of $E/\mathbb{Q}(T)$, such that $E_{t_0}/\mathbb{Q}$ is unramified at $p$. \end{prop} \begin{proof} Consider the subset $$G=\bigg\{\begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & x \\ 0 & 1 \end{pmatrix} \, \, : \, \, x \in \mathbb{F}_{p^2} \bigg\}$$ of ${\rm{PGL}}_2(\mathbb{F}_{p^2}) \subset {\rm{PGL}}_2(\overline{\mathbb{F}_p})$. It is easily checked that $G$ is a subgroup of ${\rm{PGL}}_2(\mathbb{F}_{p^2})$ of order $2p^2$. Actually, one has \begin{equation} \label{eq} G \cong (\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}) \rtimes \mathbb{Z}/2\mathbb{Z}. \end{equation} Moreover, the group $G$ is not a subgroup of ${\rm{PGL}}_2(\mathbb{C})$ (hence, (1) holds). Indeed, one cannot have $G \cong S_4, A_4, A_5$ for cardinality reasons. Moreover, if $G$ was either cyclic or dihedral, then its unique $p$-Sylow subgroup would be $\mathbb{Z}/p^2\mathbb{Z}$, which cannot happen. Now, set $F_1/\mathbb{Q}= \mathbb{Q}(\sqrt{-1})/\mathbb{Q} $ and $F_2/\mathbb{Q}=\mathbb{Q}/\mathbb{Q}$. By, e.g., \cite[Theorem 0.5.3]{JLY02}, $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ has a generic polynomial with rational coefficients. Clearly, the same is also true for $\mathbb{Z}/2\mathbb{Z}$. Consequently, by \eqref{eq} and a well-known result of Saltman (see, e.g., \cite[Corollary 7.2.2]{JLY02}), $G$ has a generic polynomial over $\mathbb{Q}$. It then remains to apply Lemma \ref{kln2} to construct a $\mathbb{Q}$-regular Galois extension $E/\mathbb{Q}(T)$ of group $G$ which specializes to $F_1/\mathbb{Q}$ and $F_2/\mathbb{Q}$ at non branch points, thus ending the proof. \end{proof} Unfortunately, this result does not apply in the same way as for the previous examples because the group $G$ does not occur as the image of a $2$-dimensional semi-simple representation over $\overline{\mathbb{F}_p}$, hence, we cannot immediately get modularity results. \noindent \begin{tabular}{lll} Sara Arias-de-Reyna & Fran\c{c}ois Legrand & Gabor Wiese\\ Departamento de \'Algebra & Institut f\"ur Algebra & Department of Mathematics\\ Facultad de Matem\'aticas & Fachrichtung Mathematik & \\ Universidad de Sevilla & Technische Universit\"at Dresden & University of Luxembourg \\ Spain & Germany & Luxembourg \\ \url{[email protected]} & \url{[email protected]} & \url{[email protected]}\\ \end{tabular} \end{document}	arXiv
\begin{document} \title{Phase separation on varying surfaces and convergence of diffuse interface approximations} \author{Heiner Olbermann} \address[Heiner Olbermann]{UCLouvain, Belgium} \email{[email protected]} \author{Matthias Röger}\address[Matthias Röger]{Department of Mathematics, Technische Universit\"at Dortmund} \email{[email protected]} \maketitle \date{\today} \maketitle \begin{abstract} In this paper we consider phase separations on (generalized) hypersurfaces in the Euclidian space. We consider a diffuse surface area (line tension) energy of Modica--Mortola type and prove a compactness and lower bound estimate in the sharp interface limit. We use the concept of generalized BV functions over currents as introduced by Anzellotti et.~al. [Annali di Matematica Pura ed Applicata, 170, 1996] to give a suitable formulation in the limit and achieve the necessary compactness property. We also consider an application to phase separated biomembranes where a Willmore energy for the membranes is combined with a generalized line tension energy. For a diffuse description of such energies we give a lower bound estimate in the sharp interface limit. \\[2ex]\noindent {\bf AMS Classification.} 49Q20, 49J45, 92C10 \\[2ex]\noindent {\bf Keywords. } Sharp interface limit, phase separation on generalized surfaces, Multiphase biomembranes. \end{abstract} \section{Introduction} Phase separation processes are ubiquitious in many applications from material sciences, physics and biology. The mathematical analysis is in many cases well understood and has seen many contributions from different communities. As two fundamental classes of descriptions we can distinguish \emph{sharp} and \emph{diffuse interface models}. In the sharp interface approach each material point belongs to exactly one of different phases or to lower dimensional interfaces between the distinct phases. Mathematically this can be described by a family of phase indicator functions (characteristic functions of subsets) that are generically discontinuos. In the simplest case of an open domain $\Omega\subset\R^n$ and two phases it is sufficient to consider one phase indicator function $u:\Omega\to\{0,1\}$. The relevant energy is in many cases proportional to the surface area of the phase interface. This in particular allows for a formulation of a surface energy functional that is finite only if $u$ is of bounded variation and given by \begin{equation} \mathcal{P}(u) = \int_\Omega \|\nabla u\|. \label{1eq:Per} \end{equation} In the diffuse interface (or phase field) approach one allows for mixtures between phases and considers concentration fields of the different phases, which are generically continuous on the considered domain. A typical example of a diffuse interface energy is the Van der Waals--Cahn-Hilliard energy, given by \begin{equation} \mathcal{P}_\varepsilon(u) := \int_{\Omega} \Big(\frac{\varepsilon}{2} \|\nabla u\|^2 + \frac{1}{\varepsilon}W(u)\Big)\,d{\mathcal L}} \newcommand\wto{\rightharpoonup^n, \label{1eq:CH} \end{equation} where $W$ is a suitable double-well potential and $u$ is a smooth function on $\Omega\subseteq\R^n$. To achieve low energy values, the function $u$ has to be close to the wells of the potential except for thin transition layers with thickness of order $\varepsilon$. One key question is whether a given diffuse interface model reduces to a sharp interface model in the \emph{sharp interface limit} $\varepsilon\to 0$. In the case of the energy \eqref{1eq:CH} this has been made rigorous in the celebrated result by Modica and Mortola \cite{MoMo77,Modi87}: the functionals $\mathcal{P}_\varepsilon$ converge in the sense of $\Gamma$-convergence to the perimeter functional, \begin{equation} \mathcal{P}_\varepsilon \to c_0 \mathcal{P},\quad c_0 = \int_{0}^1 \sqrt{2W}. \label{eq:c0} \end{equation} In the present contribution we consider phase separation processes on varifolds and currents, which present classes of generalized hypersurfaces. We are interested in applications, where both the location of the (generalized) hypersurfaces and the separation into phases represent degrees of freedom. Such a situation for example occurs in the modeling of biomembranes but again could be motivated by a variety of applications from different fields. Biomembranes consist of a large number of lipids and other ingredients and their shape and internal organization can adapt to the environment and dynamical changes of the structure are key to many functions. Variational models for multiphase membranes therefore depend on both the shape of the membrane and the internal composition. Typical ingredients are a bending energy with phase-dependent parameters and a line tension energy between phases. In the simplest situation of two phases, extensions of the classical one-phase shape energies of Canham and Helfrich \cite{Canh70,Helf73} have been proposed in the form of a Jülicher-Lipowsky energy \cite{JuLi96} \begin{equation} \mathcal{E}(S_1,S_2) = \sum_{j=1,2}\int_{S_j} \Big(k_1^j (H-H_0^{j})^2+ k_2^j K\Big)\,d\mathcal{H}^{2} + \sigma\int_{\Gamma} 1\,d\mathcal{H}^1. \label{eq:E-twophase} \end{equation} Here the biomembrane is represented by a closed surface $S\subset\R^3$ that is decomposed into a disjoint union of open subsets $S_1,S_2$ of $S$ representing the phases, and their common boundary $\Gamma$. The first integral represents a bending energy that involves in general phase-dependent bending constants $k_1^j,k_2^j$ and the spontaneous curvature $H_0^j$, $j=1,2$. The second integral in \eqref{eq:E-twophase} describes a phase separation (line tension) energy. A simple prototype of the bending contribution is the Willmore energy, that is obtained in the case $H_0^j=0=k_2^j$, $j=1,2$. Corresponding diffuse interface descriptions consider a smooth field $u_\varepsilon$ that describes the phase decomposition on the hypersurface $S$, and replace the line tension energy $\int_{\Gamma} 1\,d\mathcal{H}^1$ by a Modica--Mortola type approximation, which leads to energies of the form \begin{equation} \mathcal{E}(S_\varepsilon,u_\varepsilon) = \int_{S_\varepsilon} \Big(k_1(u_\varepsilon) (H-H_0(u_\varepsilon))^2+ k_2(u_\varepsilon) K\Big)\,d\mathcal{H}^2 + \sigma \int_{S_\varepsilon} \Big(\frac{\varepsilon}{2} \|\nabla u\|^2 + \frac{1}{\varepsilon}W(u)\Big)\,d\mathcal{H}^2. \end{equation} In a regular setting, $S_\varepsilon$ would be assumed to be a hypersurface of class $C^2$ and $u_\varepsilon$ to be a smooth function on $S_\varepsilon$, with $\nabla u_\varepsilon$ denoting the tangential gradient on $S_\varepsilon$. A rigorous mathematical understanding of such energies is rather difficult and very little seems to be known in a general situation. Reductions to rotational symmetry have been studied in \cite{ChMV13,Helm13,Helm15}. The only variational analysis in the general case seems to be the recent work of Brazda et.al.~\cite{BrLS20}. One of the challenges is that bounds on curvature energies alone do not induce good compactness properties in classes of smooth surfaces. Therefore it is necessary to consider generalized concepts of surfaces $S_\varepsilon$, such as (oriented) integer rectifiable varifolds with a weak second fundamental form as introduced by Hutchinson \cite{hutchinson1986second}. Such concepts have been used rather successfully to study Willmore or Canham--Helfrich type functionals. In the case of phase separated membranes an additional challenge is to describe the decomposition into distinct phases and phase interfaces. In \cite{BrLS20} phases are characterized by oriented curvature varifolds with boundary, and suitable conditions are posed that guarantee an appropriate global structure. Below we will present an approach that in contrast describes phases by smooth phase fields or generalized indicator functions. This may offer a more concise description and embeds diffuse and sharp interface formulations in a common framework. On the other hand already the question of suitable concepts of Sobolev spaces on varifolds is non-trivial and it seems that it is not possible to guarantee all the `good' properties that are present in the case of Sobolev spaces over open domains in $\R^n$, see the discussion in \cite{menne2016sobolev}. In the present work we use the concept of Sobolev spaces with respect to a given Radon measure as introduced by \cite{bouchitte2001convergence}, which are characterized as an appropriate closure of smooth test functions in the ambient space. Even less understood are concepts of $BV$-functions on generalized surfaces. Best suited for our purposes are $BV$-functions on currents as introduced by Anzellotti, Delladio and Scianna \cite{anzellotti1996bv} (see also \cite{Ossa97}) that are defined in terms of a generalized graph of a function over a current, see below for a precise definition. The authors there in particular present compactness and closure properties in such spaces of generalized $BV$-functions. Still, such results require additional (and rather restrictive assumptions) on the convergences of the underlying currents. These are in particular expressed by a suitable strict convergence property of currents (similar to the strict convergence of BV functions). For an alternative formulation of BV function with respect to measures see \cite{BeBF99}. The main goal of the present paper is the derivation of rigorous sharp interface limits for diffuse phase separation energies of Modica--Mortola type for phase fields on varying (generalized) surfaces. We consider a generalization of the energy \eqref{1eq:CH} for pairs of currents and phase fields. The currents are assumed to represent boundaries of finite perimeter sets in $\R^n$, and the phase fields are taken from the space of generalized $H^{1,p}$-functions with respect to the generalized surface area measure. Using such generalizations leads to a suitable formulation of a Modica--Mortola energy in the form \begin{equation} I_\varepsilon(u_\varepsilon,\mu_\varepsilon)=\int_{\R^n} \big(\e\|\nabla_{\mu_\e} u_\e\|^2+\e^{-1}W(u_\e)\big) \d\mu_\varepsilon\,, \label{1eq:MM-sharp} \end{equation} where $\mu_\varepsilon$ is the generalized surface area measure, see Definition \ref{3def:MM} for a precise definition. The limit sharp interface energy is a function of pairs consisting of a current and a phase indicator function. We again consider currents $S$ that arise from integration over the boundary of finite perimeter sets. The phase indicator functions $u$ belong to the space $BV(S)$ in the sense of \cite{anzellotti1996bv}. We show that we can associate a generalized jump set $J_u$ to $u$ that consists of a $\mathcal{H}^{n-2}$-rectifiable set. This in particular allows to assign a generalized phase interface area to this set, of the form \begin{equation} I(u,S)=\mathcal{H}^{n-2}(J_u)\,. \label{1eq:MM} \end{equation} Our first main result is a compactness and lower bound estimate of the functionals $I_\varepsilon$ that correspond to a compactness and lower bound statement in the spirit of Gamma-convergence, see Theorem \ref{thm:momolb} below. As we rely on a compactness result from \cite{anzellotti1996bv} we in particular need to impose a crucial strict convergence assumption for the approximating currents. As a second main contribution we present an application of this result to a Jülicher--Lipowsky type two-phase biomembrane energy of the form \eqref{eq:E-twophase}, where for simplicity we restrict ourselves to a bending energy of Willmore type. Here the strict convergence property needed for the application of Theorem \ref{thm:momolb} is enforced by the assumption that the Willmore energy of the approximating varifolds is small, allowing to deduce a unit density property by the Li-Yau inequality \cite{li1982new}. Under these assumptions we are able to prove a compactness and lower bound result, see Theorem \ref{4thm:main} below. The paper is organized as follows. In the next section we present some notations and recall some relevant concepts of generalized surfaces and generalized Sobolev and BV-functions. The main result on a sharp interface limit of our generalized Modica--Mortola energy for varying surfaces is contained in Section \ref{sec:3}, the application to two-phase biomembrane energies is content of the final Section \ref{sec:4}. \section{Notation and auxiliary results} \subsection{Currents and $BV$ functions on currents} We briefly recall the definition of (rectifiable) currents and in particular the notion of $BV$ functions on currents from \cite{anzellotti1996bv}. We denote by $\Lambda_k(\R^N)$, $0\leq k\leq N$ and by $\Lambda^k(\R^N)$ the spaces of all $k$-vectors and $k$-covectors, respectively, in $\R^N$. We call $v$ a \emph{simple $k$-vector} if $v$ can be written as $v=v_1\wedge \ldots\wedge v_k$. With $\Lambda(N,k)=\{\alpha=(\alpha_1,\dots,\alpha_k)\,:\, 1\leq \alpha_1<\dots<\alpha_k\leq N\}$, $(e_1,\dots,e_N)$ the standard orthonormal basis of $\R^N$ and $(\,\mathrm{d}x^1,\dots,\,\mathrm{d}x^N)$ the corresponding dual basis of $\Lambda^1(\R^N)$ we can represent any $v\in \Lambda_k(\R^N)$ and any $\omega\in \Lambda^k(\R^N)$ uniquely as \begin{equation} v=\sum_{\alpha\in \Lambda(N,k)} a_\alpha e_\alpha,\qquad \omega=\sum_{\alpha\in \Lambda(N,k)} a_\alpha e_\alpha \,\mathrm{d}x^\alpha, \end{equation} with $a_\alpha\in\R$, $e_\alpha=e_{\alpha_1}\wedge\ldots\wedge e_{\alpha_k}$, $\,\mathrm{d}x^\alpha=\,\mathrm{d}x^{\alpha_1}\wedge\ldots \,\mathrm{d}x^{\alpha_k}$ for all $\alpha\in \Lambda(N,k)$. This representation induces a scalar product and an induced norm $\|\cdot\|$ on $\Lambda^k(\R^N)$ and $\Lambda_k(\R^N)$. The canonical Hodge star isomorphism is denoted by $:\Lambda_k(\R^N)\to\Lambda_{N-k}(\R^N)$. For $U\subset \R^N$ open and $k \in \{0,\dots,N\}$ we denote by $\calD^k(U)$ the space of all infinitely differentiable $k$-differential forms $U\to\Lambda^k(\R^N)$ with compact support in $U$, equipped with usual topology of distributions. The space $\calD_k(U)$ of \emph{$k$-currents on $U$} is the dual of $\calD^k(U)$. We denote by $\partial T\in \calD_{k-1}(U)$ the \emph{boundary} of $T \in \calD_k(U)$, defined by \begin{equation} \angles{\partial T,\omega} = \angles{T,d\omega}\quad\text{ for all }\omega\in \calD^{k-1}(U). \end{equation} We say a $k$-current $T$ on $U\subset \R^N$ is representable by integration if \[ \mass[U]{T}:=\sup\{\angles{T,\varphi} \,:\,\varphi\in \calD^k(U),\\|\varphi\\|_{L^\infty}\leq 1\}<\infty\,. \] In that case by the Riesz representation theorem there exists a Radon measure $\\|T\\|$ on $U$ and a $\\|T\\|$-measurable function $\vec T:U\to \Lambda_k\R^N$ satisfying $\|\vec T\|=1$ a.e.~on $U$ such that \begin{equation}\label{finmass-curr} \angles{T,\omega} :=\int_U \angles{\omega,\vec T}\,d\\|T\\| \quad\text{ for all }\omega\in\calD^k(U). \end{equation} We call $\\|T\\|$ the \emph{mass measure} of $T$ and $\mass{T}=\mass[U]{T}$ the total mass (in $U$). Given a \emph{$k$-rectifiable} set $M\subset\R^N$ for $\calH^k$-almost any $p \in M$ there is a well-defined measure-theoretic tangent space $T_p M$. We say that a map $\tau: M\to \Lambda_k(\R^N)$ is an \emph{orientation} on $M$ if such a map is $\calH^k$-measurable and $\tau(p)$ is a unit simple $k$-vector on $\R^N$ that spans $T_pM$ for $\calH^k$-almost any point $p\in M$. Let $\rho \colon M \to \R^+$ be a $\calH^k$-locally summable function. Then, if $M\subset U$ with $U$ open in $\R^N$ we can define a current $T=\curr{M,\rho,\tau} \in \calD_k(U)$ by \begin{equation}\label{rect-curr} \langle T,\omega\rangle:=\int_M \langle \omega,\tau\rangle\,\rho\,d\calH^k \quad\text{ for all }\omega\in\calD^k(U). \end{equation} The set $\mathcal{R}_k(U)$ of currents $T\in \calD_k(U)$ which can be written in the form $T=\curr{M,\tau,\rho}$ as above are called \emph{rectifiable currents}, the function $\rho$ is then called the \emph{multiplicity} of $T$. If in addition $\rho$ is integer-valued we call $T$ integer-rectifiable and write $T\in \mathcal I_k(U)$. A current $T\in \mathcal I_k(U)$ with $\partial T\in \mathcal I_{k-1}(U)$ is called \emph{integral}. In the context of graphs over sets in $\R^n$ it is often useful to consider the variable $(x,y)\in \R^{n+1}=\R^n\times\R=\R^n_x\otimes\R_y$ with $\R^n_x=\R^n\times\{0\}$, $\R_y=\{0_{\R^n}\}\times\R$. We denote by $e_y:=e_{n+1}$ the $(n+1)$th vector and by $\,\mathrm{d}y$ the $(n+1)$th covector of the standard bases in $\R^{n+1}$ and $\Lambda^1\R^{n+1}$, respectively. The \emph{stratification} of a $k$-vector $\xi\in \Lambda_k(\R^n_x\otimes \R_y)$ is given by the unique decomposition \begin{align} \xi=\xi_0+\xi_1,\quad \xi_0\,\in\, \Lambda_k(\R^n_x),\quad \xi_1\in \Lambda_{k-1}(\R^n_x)\wedge \Lambda_1(\R_y), \label{eq:def-strati} \end{align} that is \begin{align} \xi_0 &= \sum_{\alpha\in \Lambda(n,k)} \angles{\,\mathrm{d}x^\alpha,\xi} e^\alpha,\\ \xi_1 &= \sum_{\beta\in \Lambda(n,k-1)} \angles{\,\mathrm{d}x^\beta\wedge \,\mathrm{d}y,\xi} e^\beta\wedge \varepsilon. \end{align} The corresponding stratification of a current $T\in \Lambda_k(\R^n_x\otimes \R_y)$ is given by \begin{align} \angles[\Big]{T_0,\sum_{\alpha\in \Lambda(n,k)} a_\alpha \,\mathrm{d}x^\alpha+ \sum_{\beta\in \Lambda(n,k-1)}a_\beta \,\mathrm{d}x^\beta\wedge \,\mathrm{d}y} &= \angles[\Big]{T,\sum_{\alpha\in \Lambda(n,k)} a_\alpha \,\mathrm{d}x^\alpha},\\ \angles[\Big]{T_1,\sum_{\alpha\in \Lambda(n,k)} a_\alpha \,\mathrm{d}x^\alpha+ \sum_{\beta\in \Lambda(n,k-1)}\,\mathrm{d}x^\beta\wedge \,\mathrm{d}y} &= \angles[\Big]{T,\sum_{\beta\in \Lambda(n,k-1)}a_\beta \,\mathrm{d}x^\beta\wedge \,\mathrm{d}y}, \end{align} for $a_\alpha,a_\beta\in C^\infty_c(\R^n_x\otimes \R_y)$. In the case that $T=\curr{M,\xi,\rho}$ is rectifiable we obtain for $j=0,1$ \begin{equation} \angles{T_j,\omega} = \int_M \angles{\omega,\xi_j}\rho \,d\calH^k \quad\text{ for all }\omega\in\calD^k(\R^n_x\otimes \R_y). \end{equation} Let $M\subset \R^n_x\subset \R^{n+1}$ be a $k$-rectifiable set, and let $p:\R^{n+1}\to\R^n_x$ denote the projection on $\R^n_x$. For the rectifiable $k$-current $S=\curr{M,\tau,\rho}$ and a function $u:M\to \R$ we consider the set between the graph of $u$ and $\R^n_x$, \begin{equation} E_{u,S} = \{(x,y)\in M\otimes\R_y\,:\, 0<y<u(x)\text{ if }u(x)>0,\,u(x)<y<0\text{ if }u(x)<0\} \end{equation} and define for $(x,y)\in E_{u,S}$ an induced orientation and induced multiplicity by \begin{align} \alpha(x,y)&= \begin{cases} e_y\wedge\tau(x) & \text{ if } y>0\\ -e_y\wedge\tau(x) & \text{ if } y<0 \end{cases} \\ \theta(x,y)&=\rho(x). \end{align} We then obtain the \emph{generalized graph of $u$ over $S$} as \begin{equation} T_{u,S} = -\partial [\![E_{u,S},\alpha,\theta]\!]+S_0\,. \end{equation} \begin{definition} Let a rectifiable $k$-current $S=\curr{M,\tau,\rho}$, $M\subset\R^n_x$ and a function $u:M\to\R$ be given. Then we say that $u$ is a function of bounded variation over $S$ and write $u\in BV(S)$ if the total mass of $T_{u,S}$ is bounded, $\mass{T_{u,S}}<\infty$. \end{definition} \begin{remark} From the definition, we straightforwardly get \begin{equation}\label{eq:7} \begin{split} T_{u,S}\left(\varphi_\alpha\,\mathrm{d}x^\alpha\right)=& -\int_M\langle \,\mathrm{d}x^\alpha,\tau\rangle \varphi_\alpha(x,u(x))\d \mathcal{H}^k(x)\\ T_{u,S}\left(\varphi_\beta\,\mathrm{d}x^\beta\wedge \d y\right) &=-\int_M\int_{0}^{u(x)}\sum_{i=1}^n \angles{\,\mathrm{d}x_i\wedge \,\mathrm{d}x^\beta,\tau} \frac{\partial \varphi_\beta(x,y)}{\partial x_i}\d y\d\mathcal{H}^{n-1}(x) \end{split} \end{equation} for $\alpha \in \Lambda(n,k)$ and $\beta\in\Lambda(n,k-1)$. If $S=\curr{M,\tau,\rho}$ is in $\mathcal I_k(\R_x^n)$ and $u\in BV(S)$ then the boundary rectifiability theorem \cite[Theorem 30.3]{Simo83} implies that $\partial\curr{E_{u,S},\alpha,\theta}$ is integer rectifiable and hence $T_{u,S}\in \mathcal I_k(\R_x^n\otimes\R_y)$. Therefore there exists a $k$-rectifiable set $\mathcal{R}\subset \R^{n+1}$, a multiplicity function $\theta:\mathcal{R}\to\N$ and an orientation $\xi:\mathcal{R}\to \Lambda_k(\R_x^n\otimes\R_y)$ such that \begin{equation} T_{u,S} = \curr{\mathcal{R},\xi,\theta}. \end{equation} \end{remark} \begin{definition}[Strict convergence] Let $T_j$ be a sequence of $n$-dimensional integer multiplicity rectifiable currents in $\R^{n+1}$ such that \begin{itemize} \item[(i)]$T_j\wto T$ \item[(ii)] $\sup_j(\mass{yT_{j,0}}+\mass{T_j})<\infty$ \item[(iii)] $\lim_{j\to\infty} \mass{p_\# T_j}=\mass{p_\#T}$ \end{itemize} Then we say that $T_j$ converges strictly to $T$, $T_j\stackrel{c^}{\wto}T$. \end{definition} \begin{remark} In the case that $T_j=T_{u_j,S_j}$, $T=T_{u,S}$ are generalized graphs with $S_j=\curr{M_j,\tau_j,\rho_j}$ and $S=\curr{M,\tau,\rho}$ being $k$-rectifiable currents we obtain \begin{equation} \mass{yT_{j,0}} = \int_{M_j} \|u_j\|\rho_j\,d\mathcal{H}^k, \end{equation} see \cite[p.288]{anzellotti1996bv}. Moreover, in this case the convergence $T_j\wto T$ implies \begin{equation} S = p_{\#}{T_{u,S}} = \lim_{j\to\infty} p_{\#}{T_{u_j,S_j}} = \lim_{j\to\infty}S_j, \end{equation} and the convergence in the third item is equivalent to $\mass{S_j}\to\mass{S}$, i.e. \begin{equation} \int_{M_j}\rho_j\,d\mathcal{H}^k \,\to\, \int_M \rho\,d\mathcal{H}^k. \end{equation} \end{remark} \subsection{Oriented varifolds} We introduce oriented varifolds as in \cite{hutchinson1986second}. Let $G^o(n,m)$ denote the set of $m$-dimensional oriented subspaces of $\R^n$. This set may be identified with the unit simple elements of $\Lambda_m\R^n$. An oriented $m$-varifold is an element of $\mathcal M(\R^n\times G^o(n,m))$. If $M$ is $m$-rectifiable with orientation $\tau$, and if $\theta_\pm:M\to\R^+_0$ are locally $\mathcal{H}^m$-summable multiplicity functions with $\theta_++\theta_->0$, then we write $\underline{v}(M,\tau,\theta_\pm)$ for the associated rectifiable oriented varifold \[ \underline{v}(M,\tau,\theta_\pm)(\varphi)=\int_{\R^n\times G^o(n,m)}\big(\theta_+(x)\varphi(x,\tau(x))+\theta_-(x)\varphi(x,-\tau(x))\big)\d \mathcal{H}^m(x)\,. \] We then can pass to a representation $\ovar{M,\tilde\tau,\tilde\theta_\pm}$ such that $\tilde\theta_+\geq\tilde\theta_-$. In the following we always assume this additional property. If the multiplicity functions $\theta_\pm$ are $\N_0$ valued, then we say that $\underline{v}(M,\tau,\theta_\pm)$ is an integral oriented varifold. The class of $m$-dimensional oriented (resp.~rectifiable oriented, resp.~integral oriented) varifolds is denoted by $V^o_m(\R^n)$ (resp.~$RV^o_m(\R^n)$, resp.~$IV^o_m(\R^n)$). To $V\in V^o_m(\R^n)$, we associate the $m$-dimensional current \[ \underline{c}(V)(\varphi)=\int_{\R^n\times G^o(n,m)}\langle \varphi(x),\xi\rangle \d V(x,\xi) \] and observe that in the case of $V\in RV^0_m(\R^n)$, $V=\underline{v}(M,\tau,\theta_\pm)$ \begin{equation} \underline{c}(V)(\varphi)=\int_M\langle \tau(x),\varphi(x)\rangle (\theta_+-\theta_-)(x)\d \mathcal{H}^m(x). \end{equation} We remark that convergence as oriented varifolds implies convergence of the associated currents, \begin{equation} V_j \rightharpoonup V \quad\implies\quad \underline{c}(V_j) \rightharpoonup \underline{c}(V)\,. \label{2eq:orvcurr} \end{equation} \subsection{Sobolev functions with respect to measures} For a Radon measure $\mu$ on an open set $\Omega\subset\R^n$, we introduce the Sobolev space $H^{1,p}_\mu(\Omega)$ as in \cite{bouchitte2001convergence}: We consider the linear operator $A:\calD(\R^n)\to L^p_\mu(\Omega;\R^n)$ defined by \[ (A\varphi)(x)=P_{\mu}(x)\nabla \varphi(x)\,, \] where $P_{\mu}(x)$ is the projection onto the tangent space of $\mu$ at $x$. We do not give the definition of tangent spaces of measures from \cite{bouchitte2001convergence} here, but only note that in the case that $\mu=\\|V\\|$ is the mass measure of an integer rectifiable $k$-varifold with locally bounded first variation, then $P_{\mu}(x)$ is the projection onto the $k$-dimensional tangent space of $\supp\mu$ in $x$, which exists for $\mu$ almost every $x$. With this definition the operator $A$ is closable in the norm given by $\\|\varphi\\|_{L^p_{\mu}(\Omega)}+\\|A\varphi\\|_{L^p_\mu(\Omega;\R^n)}$, and $H^{1,p}_\mu(\Omega)$ is defined as the domain of the unique closed extension $\bar A$, which is denoted by $\nabla_\mu$. For the case that $\mu=\\|V\\|$ is the mass measure of an integer rectifiable $k$-varifold with locally bounded first variation an equivalent definition for Sobolev spaces has been given in \cite{menne2016sobolev}, based on the previous work \cite{menne2016weakly}. \subsection{Measure-function pairs} We use the definitions of measure-function pairs from \cite{moser2001generalization}, which in turn are based on \cite{hutchinson1986second}. Let $\Omega\subset \R^n$. If $\mu\in \mathcal{M}(\R^n)$ and $f\in L^1_{\mathrm{loc},\mu}(\Omega;\R^m)$, then we say that $(\mu,f)$ is a measure-function pair over $\Omega$ with values in $\R^m$. \begin{definition} Let $\{(\mu_k,f_k):k\in\N\}$ and $(\mu,f)$ be measure-function pairs over $\Omega$ with values in $\R^m$, and $1\leq p<\infty$. \begin{itemize} \item[(i)] We say that $(\mu_k,f_k)$ converges weakly in $L^p$ to $(\mu,f)$ and write \[ (\mu_k,f_k)\wto (\mu,f) \quad\text{ in } L^p \] if $\mu_k\wsto \mu$ in $\mathcal M(\Omega)$, $\mu_k\ecke f_k\wsto \mu\ecke f$ in $\mathcal M(\Omega;\R^m)$, and $\\|f_k\\|_{L^p_{\mu_k}(\Omega;\R^m)}$ is uniformly bounded. \item[(ii)] We say that $(\mu_k,f_k)$ converges strongly in $L^p$ to $(\mu,f)$ and write \[ (\mu_k,f_k)\to (\mu,f) \quad\text{ in } L^p \] if for all $\varphi\in C_c^0(\Omega\times\R^m)$, \[ \lim_{k\to\infty}\int_\Omega \varphi(x, f_k(x))\d\mu_k(x)=\int_\Omega \varphi(x, f(x))\d\mu(x)\,, \] and \[ \lim_{j\to \infty}\int_{S_{kj}}\|f_k\|^p\d\mu_k=0 \quad\text{ uniformly in } k\,, \] where $S_{kj}=\{x\in\Omega:\|x\|\geq j \text{ or } \|f_k(x)\|\geq j\}$. \end{itemize} \end{definition} \subsection{Convergence of BV functions over varying currents} \label{sec:conv-bv-funct} \begin{proposition}\label{2pro:mfp-convergence} Consider a sequence $(S_j)_j$ of $(n-1)$-rectifiable currents in $\R^n$. Moreover, let a sequence $(u_j)_j$ be given with $u_j\in BV(S_j)$, and denote the associated generalized graphs by $T_j=T_{u_j,S_j}$ Assume that \begin{equation} \sup_j \\|u_j\\|_{L^p(\\|S_j\\|)}<\infty \quad\text{ for some }1<p<\infty\,, \label{2eq:u_bd} \end{equation} and that for some $(n-1)$-rectifiable current $S$ and some $u\in BV(S)$ \begin{align} T_j \rightharpoonup T=T_{u,S}\,, \label{2eq:ass-gcurr}\\ \mass{S_j} \to \mass{S}\,. \label{2eq:Resh} \end{align} Then the strong measure-function pair convergence \begin{equation} \big(\\|S_j\\|,u_j\big) \to \big(\\|S\\|,u\big) \label{2eq:mfp-conv} \end{equation} holds in any $L^q$, $1\leq q<p$. \end{proposition} \begin{proof} We first deduce from \eqref{2eq:u_bd} for some $\Lambda>0$ and any $j\in\N$, $1\leq q<p$, $R>0$ \begin{equation} \Lambda^p \geq \int \|u_j\|^p\d\\|S_j\\| \geq R^p\\|S_j\\|(\{\|u_j\|>R\}) \end{equation} and \begin{align} \int_{\{\|u_j\|>R\}}\|u_j\|^q\d\\|S_j\\| &\leq \Lambda^q \Big(\\|S_j\\|(\{\|u_j\|>R\})\Big)^{1-\frac{q}{p}} \\ &\leq \Lambda^p R^{-p+q}\,\to\, 0 \quad(R\to\infty). \end{align} Similarly, \begin{align} \int_{\{\|x\|>R\}}\|u_j\|^q\d\\|S_j\\| &\leq \Lambda^q \sup_{j\in\N}\Big(\\|S_j\\|(\{\|x\|>R\})\Big)^{1-\frac{q}{p}} \\ \,\to\,0 \quad (R\to\infty) \end{align} by Prokhorov's Theorem \cite[Theorem 4.1]{Bill99} and \eqref{2eq:Resh}. This verifies the second condition in the definition of strong measure-function pair convergence in $L^q$. Next, for any $\alpha\in \Lambda(n,n-1)$, any $\varphi_\alpha\in C^0_c(\R^n)$, and any $\psi\in C^0_c(\R^n\times\R)$ we let $\eta_\alpha(x,y)=\varphi_\alpha(x)\psi(x,y)$ and deduce from \eqref{finmass-curr} and \eqref{eq:7} \begin{align} &\int_{\R^n} \angles{\vec{S}(x),\,\mathrm{d}x^\alpha} \varphi_\alpha(x)\psi(x,u(x))\,\d\\|S\\|(x)\\ &\qquad =\angles{S,\psi(\cdot,u)\varphi_\alpha\,\mathrm{d}x^\alpha}\\ &\qquad = -\angles{T_{u,S},\eta_\alpha\,\mathrm{d}x^\alpha} \\ &\qquad = -\lim_{j\to\infty} \angles{T_{u_j,S_j},\eta_\alpha\,\mathrm{d}x^\alpha}\\ &\qquad = \lim_{j\to\infty} \angles{S_j,\psi(\cdot,u_j)\varphi_\alpha\,\mathrm{d}x^\alpha} = \lim_{j\to\infty }\int_{\R^n}\angles{\vec{S_j},\,\mathrm{d}x^\alpha} \varphi_\alpha(x)\psi(x,u_j(x))\,\d\\|S_j\\|(x). \end{align} Since $\alpha\in \Lambda(n,n-1)$, $\varphi_\alpha$ are arbitrary we arrive at \begin{equation} \Big(\psi(\cdot,u_j)\vec{S_j}, \\|S_j\\|\Big) \rightharpoonup \Big(\psi(\cdot,u)\vec{S}, \\|S\\|\Big). \label{2eq:Moser} \end{equation} Next, by the Reshetnyak continuity theorem \cite{reshetnyak1968weak,spector2011simple} and \eqref{2eq:Resh} we have for any $\varphi\in C^0_c(\R^n\times\mathbb{S}^{n-1})$ \begin{align} \lim_{j\to\infty}\int_{\R^n}\varphi(\cdot,\vec{S_j})\d\\|S_j\\| &= \int_{\R^n}\varphi(\cdot,\vec{S})\d\\|S\\|, \end{align} which implies the strong measure-function pair convergence \begin{equation} \big(\vec{S_j},\\|S_j\\|\big) \to \big(\vec{S},\\|S\\|\big) \end{equation} in any $L^r$, $1\leq r<\infty$. Together with \eqref{2eq:Moser} we deduce from \cite[Proposition 3.2]{moser2001generalization} that \begin{align} \lim_{j\to\infty}\Big(\psi(\cdot,u_j),\\|S_j\\|\Big)&=\lim_{j\to\infty}\Big(\psi(\cdot,u_j)\vec{S_j}\cdot\vec{S_j}, \\|S_j\\|\Big)\\ &= \Big(\psi(\cdot,u)\vec{S}\cdot\vec{S}, \\|S\\|\Big) = \Big(\psi(\cdot,u), \\|S\\|\Big) \end{align} in the sense of weak measure-function-pair convergence in any $L^r$, $1\leq r<\infty$. Since $\psi$ was arbitrary \eqref{2eq:mfp-conv} follows. \end{proof} \section{Compactness and lower semicontinuity for a Modica-Mortola functional on varying surfaces} \label{sec:3} We first define a generalized Modica--Mortola functional. Fix a nonnegative continuous double-well potential $W$ such that $\{W=0\}=\{0,1\}$ and such that for some $T,c>0$, $p\geq 2$ \begin{equation} c\|t\|^p \leq W(t)\leq \frac{1}{c}\|t\|^p \quad\text{ for all }\|t\|\geq T\,. \label{3eq:dw} \end{equation} Below it will be convenient to fix a first integral of $\sqrt{W}$, \begin{equation}\label{eq:12} \psi(r)=\int_{0}^r \sqrt{W(t)}\d t\quad\text{ for }r\in\R \end{equation} and to define the surface tension constant \begin{equation} k=\int_{0}^{1}\sqrt{W(r)}\d r\,. \label{3eq:k} \end{equation} \begin{definition}[Generalized Modica--Mortola functional] \label{3def:MM} For $\varepsilon>0$, a Radon measure $\mu_\varepsilon$ on $\R^n$ and a function $u_\varepsilon\in H^{1,p}_{\mu_\varepsilon}(\R^n)$ we define the Modica--Mortola-type functional \begin{equation} I_\varepsilon(u_\varepsilon,\mu_\varepsilon)=\int_{\R^n} \big(\e\|\nabla_{\mu_\e} u_\e\|^2+\e^{-1}W(u_\e)\big) \d\mu_\varepsilon \in [0,\infty]\,. \label{3eq:MM} \end{equation} \end{definition} In the following we consider a more restrictive setting where $\mu_\varepsilon$ is the area measure of the reduced boundary of a finite perimeter set. In the sharp interface limit of phase fields we will obtain phase indicator functions with a generalized $BV$-regularity. In Proposition \ref{prop:BVstructure} below we will justify the following notion of jump sets for such phase indicator functions. \begin{definition}\label{3def:Ju} Consider a set $E\subset\R^n$ of finite perimeter with inner unit normal $\nu=\nu_E:\partial_* E\to\mathbb{S}^{n-1}$ of $E$, and set $S=\curr{\partial_E,\nu,1}$. When $u\in BV(S)$ with $u(x)\in \{a,b\}$ $\mathcal{H}^{n-1}$-almost everywhere for some $a\neq b$ we write \[ J_u:=\supp \left(\partial\curr{u^{-1}(b),\nu,1}\right) \] and call this set the jump set of $u$. \end{definition} We will now state our main result in this section, which is a generalization of the $\liminf$ statement in the Modica--Mortola Gamma-convergence statement. \begin{theorem} \label{thm:momolb} Consider $p\geq 2$ with \eqref{3eq:dw}. Let a family $(E_\varepsilon)_\varepsilon$ of finite perimeter sets in $\R^n$ with associated perimeter measures $\mu_\e=\mathcal{H}^{n-1}\ecke\partial_ E_\e$ and a sequence $(u_\varepsilon)_\varepsilon$ in $H^{1,p}_{\mu_\e}(\R^n)$ be given. Assume that for some set $E$ of finite perimeter $\chi_{E_\e}\to \chi_{E}$ strictly in $BV(\R^n)$, that is \begin{equation} \chi_{E_\e}\to \chi_{E}\,\text{ in }L^1(\R^n),\quad \nabla \chi_{E_\e}\wsto \nabla \chi_{E},\quad\text{and } \lim_{\e\to 0}\mathcal{H}^{n-1}(\partial_* E_\e)=\mathcal{H}^{n-1}(\partial_* E), \label{3eq:BV-strict} \end{equation} and let $\nu=\nu_E:\partial_* E\to\mathbb{S}^{n-1}$ denote the inner unit normal of $E$, and set $S=[\![\partial_E,\nu_E,1]\!]$, $\mu=\mathcal{H}^{n-1}\ecke\partial_* E$. Let us further assume that for some $\Lambda>0$ \begin{equation} I_\varepsilon(u_\varepsilon,\mu_\varepsilon) < \Lambda. \label{3eq:ass-bound} \end{equation} Then there exists $u\in BV(S)$ and a subsequence $\varepsilon\to 0$ such that the following holds: \begin{align} u(x)\in\{0,1\}\text{ for }\mathcal{H}^{n-1}\text{-almost all } x\in \partial_E, \label{3eq:thm1-1}\\ \big(\mu_\varepsilon,u_\varepsilon\big) \to \big(\mu,u\big) \quad\text{ as measure-function pairs in }L^q \label{3eq:thm1-2} \end{align} for any $1\leq q<p$. Moreover, $\curr{\{u=1\}, \nu,1}$ is an integral $(n-1)$-current and we have the lower estimate \begin{equation} \liminf_{\e\to 0} I_\varepsilon(u_\varepsilon,\mu_\varepsilon) \geq 2k\mathcal{H}^{n-2}\left(J_u\right), \label{3eq:lsc} \end{equation} with the generalized jump set $J_u$ as in Definition \ref{3def:Ju}. \end{theorem} \begin{proof} We first prove that we can assume $u_\varepsilon\in\calD(\R^n)$ for all $\varepsilon>0$. In fact, by the definition of $H^{1,p}_{\mu_\varepsilon}(\R^n)$ we can approximate $(u_\varepsilon)_\varepsilon$ by a family $(\tilde u_\varepsilon)_\varepsilon$ in $\calD(\R^n)$ such that \eqref{3eq:ass-bound} holds for $(\tilde u_\varepsilon)_\varepsilon$, such that \begin{equation} \liminf_{\e\to 0} I_\varepsilon(\tilde u_\varepsilon,\mu_\varepsilon) =\liminf_{\e\to 0} I_\varepsilon(u_\varepsilon,\mu_\varepsilon)\,, \end{equation} and \begin{equation} \\|u_\varepsilon-\tilde u_\varepsilon\\|_{L^p(\mu_\varepsilon)}\to 0 \quad (\varepsilon\to 0)\,. \end{equation} Let us assume we have proved $(\mu_\varepsilon,\tilde u_\varepsilon)\to (\mu, u)$ as measure-function pairs in $L^q$ for some $u$ as in the statement of the present theorem. For any Lipschitz-continuous function $\psi\in C^0_c(\R^n\times\R)$ we then deduce \begin{equation} \Big\|\int \big(\psi(\cdot,u_\varepsilon)-\psi(\cdot,\tilde u_\varepsilon)\big)\,d\mu_\varepsilon\Big\| \leq \\|\psi\\|_{C^{0,1}(\R^{n+1})} \\|u_\varepsilon-\tilde u_\varepsilon\\|_{L^p(\mu_\varepsilon)}\mu_\varepsilon(\R^n)^{1-\frac{1}{p}}\,\to\, 0 \end{equation} with $\varepsilon\to 0$. An approximation argument yields for all $\psi\in C^0_c(\R^n\times\R)$ \begin{equation} \lim_{\varepsilon\to 0}\int \psi(\cdot,u_\varepsilon)\,d\mu_\varepsilon =\lim_{\varepsilon\to 0}\int \psi(\cdot,\tilde u_\varepsilon)\,d\mu_\varepsilon =\int \psi(\cdot,u)\,d\mu \end{equation} and therefore the strong measure-function pair convergence of the original sequence $(\mu_\varepsilon,u_\varepsilon)$ to $(\mu,u)$. Therefore it is sufficient to prove the Theorem for sequences $(u_\varepsilon)_\varepsilon$ in $\calD(\R^n)$, which we assume in the remainder of the proof. We next consider the modified phase fields $v_\varepsilon:=\psi\circ u_\varepsilon$ and the generalized graphs $T_\varepsilon:=T_{v_\varepsilon,E_\e}$. Since $u_\varepsilon\in\calD(\R^n)$ we can apply the chain rule $\nabla_{\mu_\varepsilon}v_\varepsilon=\sqrt{W(u_\varepsilon)}\nabla_{\mu_\varepsilon}u_\varepsilon$ and obtain the representation \begin{equation} T_{v_\varepsilon,S_\e} = (\Phi_\e)_\# (S_\e),\quad \Phi_\e(x):=(x,v_\varepsilon(x))\text{ for }x\in \partial_ E_\varepsilon\,. \end{equation} Therefore we may apply the usual Modica-Mortola trick to obtain \begin{equation}\label{eq:17} \begin{split} \int_{\partial_ E_\e} \|\nabla_{\mu_\e}v_\varepsilon\|\d\mathcal{H}^{n-1}&\leq \int_{\partial_* E_\e} \sqrt{W(u_\e)}\|\nabla_{\mu_\e} u_\e\|\d\mathcal{H}^{n-1}\\ &\leq \frac12 I_\varepsilon(u_\varepsilon,\mu_\varepsilon)\leq \frac{1}{2}\Lambda\,. \end{split} \end{equation} Furthermore we have \begin{equation}\label{3eq:mass-bound} \begin{split} \mass{T_\varepsilon} =\mass{(\Phi_\e)_\# (S_\e)} &=\int_{\partial_E_\e}\sqrt{1+\|\nabla_{\mu_\e}v_\varepsilon\|^2}\d\mathcal{H}^{n-1}\\ &\leq \mathcal{H}^{n-1}(\partial_E_\e)+ \int_{\partial_* E_\e} \|\nabla_{\mu_\e}v_\varepsilon\|\d\mathcal{H}^{n-1}\\ &\leq \mathcal{H}^{n-1}(\partial_E_\e)+ \frac12 I_\varepsilon(u_\varepsilon,\mu_\varepsilon)\leq C(\Lambda) \end{split} \end{equation} and \begin{equation} \mass{y T_{\varepsilon,0}} = \\|v_\varepsilon\\|_{L^1(\mu_\varepsilon)} \leq C(\Lambda,W). \label{3eq:L1-bound} \end{equation} This implies that the currents $T_\varepsilon$ are uniformly bounded in mass and that $(v_\varepsilon)_\varepsilon$ is bounded in an $L^1$ sense. Since $\partial T_\varepsilon=0$, we may use the compactness theorem for integral currents, to obtain a weak limit $T$ of $T_\varepsilon$. By \eqref{3eq:BV-strict}, \eqref{3eq:mass-bound} and \eqref{3eq:L1-bound} we may use \cite[Theorem 4.3]{anzellotti1996bv} to conclude that there exists $v\in BV(S)$ such that \[ T=T_{v,S}\,. \] Furthermore, the assumptions on $W$ induce that $(v_\varepsilon)_\varepsilon$ is bounded in $L^p(\mu_\varepsilon)$. Together with the strict $BV$-convergence of $\chi_{E_\varepsilon}$ and the convergence $T_\varepsilon \rightharpoonup T_{u,S}$ we obtain by Proposition \ref{2pro:mfp-convergence} that we have the strong measure-function pair convergence \begin{equation} \big(\mu_\varepsilon,v_\varepsilon\big) \to \big(\mu,v\big)\quad\text{ in any }L^q,\,1\leq q<p. \label{3eq:mfp-v} \end{equation} Since $\psi$ is continuous we deduce that \eqref{3eq:thm1-2} holds for $u:=\psi^{-1}(v)$. Using the measure-function pair convergence \eqref{3eq:thm1-2} and the energy bound \eqref{3eq:ass-bound} it follows that for any $\eta\in C^0_c(\R^n\times\R)$ \begin{equation} \int_{\R^n} \eta(\cdot,u)W(u)\,\d\mu =\lim_{\varepsilon\to 0} \int_{\R^n} \eta(\cdot,u_\varepsilon)W(u_\varepsilon)\,\d\mu_\varepsilon =0\,. \end{equation} Since $\eta$ was arbitrary \eqref{3eq:thm1-1} holds. Clearly this implies that $v(x)\in\{0,k\}$ for $\mathcal{H}^{n-1}$ almost all $x\in\partial E_$. Using Proposition \ref{prop:BVstructure} below we next obtain that the generalized jump set $J_u$ is $(n-2)$-rectifiable with \begin{equation}\label{3eq:prefin} \begin{split} \mass{T_{v,S}}-\mass{S} &=k\mathcal{H}^{n-2}(J_u)=k(\mass{T_{u,S}}-\mass{S})\,. \end{split} \end{equation} To prove the lower estimate \eqref{3eq:lsc}, we note that mass is weakly lower semicontinuous under weak convergence, and that by strict BV-convergence \eqref{3eq:BV-strict} \[ \lim_{\e\to 0} \mass{S_\e}=\lim_{\e\to 0}\mathcal{H}^{n-1}(\partial_* E_\e)=\mathcal{H}^{n-1}(\partial_* E)=\mass{S}. \] We then deduce from \eqref{eq:17}, \eqref{3eq:mass-bound} and \eqref{3eq:prefin} that \begin{align} k\mathcal{H}^{n-2}(J_u) &=\mass{T_{v,S}}-\mass{S}\\ &\leq \liminf_{\varepsilon\to 0}\Big(\mass{T_\varepsilon}-\mass{S_\varepsilon}\Big)\\ &\leq \liminf_{\varepsilon\to 0}\Big(\big(\mathcal{H}^{n-1}(\partial_ E_\varepsilon)+\frac{1}{2}I_\varepsilon(u_\varepsilon,\mu_\varepsilon)\big) - \mathcal{H}^{n-1}(\partial_* E_\varepsilon)\Big)\\ &= \liminf_{\varepsilon\to 0} \frac{1}{2}I_\varepsilon(u_\varepsilon,\mu_\varepsilon)\,, \end{align} which proves \eqref{3eq:lsc}. \end{proof} \begin{proposition} \label{prop:BVstructure} Let a set $E\subset\R^n$ of finite perimeter and a constant $k>0$ be given, let $\nu=\nu_E:\partial_E\to\mathbb{S}^{n-1}$ denote the inner unit normal, and consider the associated current $S=\curr{\partial_E,\tau,1}$. If $w\in BV(S)$ and $w\in\{0,k\}$ $\mathcal{H}^{n-1}\ecke\partial_ E$ almost everywhere, then $\curr{\{w=k\},\nu,1}$ is an integral $(n-1)$-current, and the generalized jump set \[ J_w=\supp\left(\partial \curr{\{w=k\},\nu,1}\right)\,, \] satisfies \[ k\mathcal{H}^{n-2}(J_w)=\mass{T_{w,S}}-\mathcal{H}^{n-1}(\partial_* E)\,. \] \end{proposition} \begin{proof} Since $T:=T_{w,S}$ is representable by integration, we obtain \begin{equation} \mass{T} = \\|T\\|(\R^{n+1}) = \\|T\\|(\R^n\times\{0\}) + \\|T\\|(\R^n\times (0,k)) + \\|T\\|(\R^n\times\{k\}), \end{equation} where we have also used that the definition of the graph implies $T=T\ecke (\R^n\times [0,k])$. Using \eqref{eq:7}, we observe that \[ \begin{split} &\\|T\ecke\{(x,k):x\in \partial_E\}\\|\\ &\qquad =\sup\Bigg\{\int_{\{x\in \partial_ E:w(x)=k\}}\sum_{\alpha\in\Lambda(n,n-1)}\langle \,\mathrm{d}x^\alpha,\nu\rangle\tilde \varphi_\alpha(x)\d\mathcal{H}^{n-1}(x):\\ &\qquad \qquad\qquad\tilde \varphi_\alpha\in C^1_c(\R^n), \left\\|\sum_{\alpha\in\Lambda(n,n-1)}\tilde\varphi_\alpha\,\mathrm{d}x^\alpha\right\\|\leq 1\Bigg\}\\ &\qquad =\mathcal{H}^{n-1}(\{x\in\partial_ E:w(x)=k\})\,, \end{split} \] with an analogous equality for $k$ replaced by $0$. Therefore \begin{equation} \mass{T}=\mathcal{H}^{n-1}(\partial_E) + \\|T\\|(\R^n\times (0,k))\,. \label{3eq:mass} \end{equation} Let $Y:\R^{n+1}\to \R$ be given by $Y(x,y)=y$, and let $\langle\cdot,\cdot,\cdot\rangle$ denote the slicing operation from \cite[4.3]{MR0257325}. Consider an arbitrary $\varphi\in \calD^{n-2}(\R^{n+1})$ and the stratification $\varphi=\varphi_0+\varphi_1$ with \[ \varphi_0=\sum_{\beta\in \Lambda(n,n-2)}\varphi_\beta \,\mathrm{d}x^\beta,\qquad \varphi_1=\sum_{\gamma\in \Lambda(n,n-3)}\varphi_\gamma \,\mathrm{d}x^\gamma\wedge \d y\,. \] Then, using \eqref{eq:7} and the definition of the slicing operation we calculate that for all $0<s<k$ \begin{equation} \begin{split}\label{eq:9} &\langle T,Y,s\rangle(\varphi)\\ &\qquad =\lim_{\rho\downarrow 0}\frac{(-1)}{2\rho} \left(T\ecke \chi_{\{y:s-\rho<y<s+\rho\}}\d y \right)(\varphi)\\ &\qquad =\lim_{\rho\downarrow 0}\frac{(-1)}{2\rho}\int_{\partial_E}\int_{0}^{w(x)}\langle\d_x\varphi_0(x,y),\nu(x)\rangle\chi_{(s-\rho,s+\rho)}(y)\d y\d\mathcal{H}^{n-1}(x)\\ &\qquad = - \int_{\{w=k\}}\langle\d_x(\varphi_0 (x,s)),\nu(x)\rangle\d\mathcal{H}^{n-1}(x)\\ &\qquad = - \curr{\{w=k\},\nu,1}(\d_x \varphi_0 (\cdot,s))\,. \end{split} \end{equation} This implies that for all $0<s<k$ \begin{align}\label{eq:11} \langle T,Y,s\rangle(\varphi) &= -\partial\curr{\{w=k\},\nu,1}\left(\varphi_0(\cdot,s)\right)\\ &=\partial\curr{\{w=0\},\nu,1}\left(\varphi_0(\cdot,s)\right)\,, \end{align} where in the last equality we have used that $\curr{\partial_E,\nu,1}=\partial \curr{E,e_x,1}$ for $e_x=e_1\wedge\dots\wedge e_n$, hence \begin{equation} 0=\partial\curr{\partial_E,\nu,1} =\partial\curr{\{w=k\},\nu,1}-\partial\curr{\{w=0\},\nu,1}\,. \end{equation} The representation \eqref{eq:11} in particular implies that $\partial\curr{\{w=k\},\nu,1}$ is an integral $(n-2)$-current and that $\curr{\{w=k\},\nu,1}$ is an integal $(n-1)$-current. Furthermore, we have \[ \supp \langle T,Y,s \rangle =\supp \partial\left(\curr{\{w=k\},\nu,1}\right)\times\{s\} =N\times\{s\}\subset\R^{n+1}\,, \] where $N\subset\R^n$ is some $(n-2)$-rectifiable set. It follows that \[ \mass{\langle T,Y,s\rangle}=\mathcal{H}^{n-2}(N)\quad \text{ for } 0<s<k\,. \] By \cite[4.3.2]{MR0257325} this yields \[ (k-2\e)\mathcal{H}^{n-2}(N)=\int_{\e}^{k-\e}\mass{\langle T_{w,S},Y,s\rangle}\d s = \mass{T\ecke \{\e<y<k-\e\}} \] and hence we obtain in the limit $\e\to 0$ that \[ \\|T\\|\big(\R^n\times (0,k)\big) =\mass{T\ecke\{(x,y):0<y<k\}}=k \mathcal{H}^{n-2}(N)\,. \] Together with \eqref{3eq:mass} this proves the claim. \end{proof} \begin{remark} For a corresponding Gamma-convergence result one would need to complement Theorem \ref{thm:momolb} by an upper bound estimate. It is rather straightforward to prove such a statement in a more regular setting: Assume that $M\subset \R^n$ is a smooth oriented $n-1$-dimensional submanifold, $V=\underline{v}(M,\nu,1)$, $S=[\![M,\nu,1]\!]$, $u\in BV(S)$ with $u\in \{0,1\}$ $\mathcal{H}^{n-1}\ecke M$ almost everywhere, $J_u$ a smooth curve on $M$. Then there exists a sequence $(u_\e)_{\e}$ in $H^{1,p}_{\mathcal{H}^{n-1}\ecke M}(\R^n)$ such that $T_{u_\e,S}\wto T_{u,S}$ and \[ \limsup_{\e\to 0} I_\e(S,u_\e)\leq 2k\mathcal{H}^{n-2}(J_u)\,. \] Indeed, one can adapt the proof from \cite[Proposition 2]{modica1987gradient} rather straightforwardly to obtain the above statement. The proof of the upper bound becomes non-trivial as soon as one allows for non-smooth surfaces $M=\supp S$. We do not discuss this question here and leave it open for future research. \end{remark} \section{Application to a two-phase membrane} \label{sec:4} In this section we consider a class of two-phase membrane energies that consist of a bending contribution given by a phase-dependent Willmore functional and a line tension energy. Such kind of energies in particular appear as reductions of the Jülicher--Lipowsky energy discussed in the introduction. The main result of this section connects diffuse and sharp interface description of such energies. \begin{definition}[Two phase membrane energy, sharp interface formulation] \label{4eq:2pW} Let $V\in IV_2^o(\R^3)$ be an integer-rectifiable oriented 2-varifold in $\R^3$ with weak mean curvature $H_V\in L^2_{\\|V\\|}(\R^3;\R^3)$ and consider a $\\|V\\|$-measurable function $u$ on $M=\supp(\\|V\\|)$ with $u(x)\in\{0,1\}$ for $\mathcal{H}^2$-almost every $x\in M$. We then define for given constants $a_1,a_2,k>0$ \begin{align} \mathcal{W}(u,V) &= \frac{1}{4}\int_{\R^3} \big(a_1u + a_2(1-u)\big) \|H_V\|^2\d\\|V\\|\,, \label{4eq:defW}\\ \mathcal{E}(u,V) &:= \mathcal{W}(u,V) + 2k\mathcal{H}^1(J_u)\,, \label{4eq:Eeps} \end{align} where $J_u$ denotes the jump set of $u$ as introduced in Definition \ref{3def:Ju}. \end{definition} In the corresponding diffuse interface description we will use the perimeter approximation from Section \ref{sec:3}. In particular we assume a given nonnegative double-well potential $W$ with $\{W=0\}=\{0,1\}$ that satisfies \eqref{3eq:dw}. We define $\psi$ as in \eqref{eq:12}, the constant $k=\psi(1)$ as in \eqref{3eq:k}, and the Modica--Mortola type functional $I_\varepsilon$ as in \eqref{3eq:MM}. Finally, we fix a smooth interpolation between the phase dependent constants $a_1,a_2$ from Definition \ref{4eq:2pW} of the form \begin{equation} a^{\bar\omega}(r) = \bar\omega(r)a_1+(1-\bar\omega(r))a_2\,, \label{4eq:baromega} \end{equation} where $\bar\omega\in C^\infty_c(\R)$ satisfies $0\leq\bar\omega\leq 1 $ and $\bar\omega(0)=0$, $\bar\omega(1)=1$. \begin{definition}[Two phase membrane energy, diffuse inteface description] Let $V_\varepsilon\in IV_2^o(\R^3)$ be an integer-rectifiable oriented 2-varifold in $\R^3$ with weak mean curvature $H_\varepsilon:=H_{V_\varepsilon}\in L^2_{\\|V_\varepsilon\\|}(\R^3;\R^3)$ and let $u_\varepsilon\in H^{1,p}_{\\|V_\varepsilon\\|}(\R^3)$ be given. Then we define \begin{align} \mathcal{W}_\varepsilon(u_\varepsilon,V_\varepsilon) &:= \frac{1}{4}\int a^{\bar\omega}\circ u_\varepsilon \|H_\varepsilon\|^2\,d\\|V_\varepsilon\\|, \label{4eq:Weps}\\ \mathcal{E}_\varepsilon(u_\varepsilon,V_\varepsilon) &:= \mathcal{W}_\varepsilon(u_\varepsilon,V_\varepsilon) + I_\varepsilon(u_\varepsilon,\\|V_\varepsilon\\|)\,, \label{4eq:Eeps-2} \end{align} where $a^{\bar\omega}$ is as in \eqref{4eq:baromega} and $I_\varepsilon$ as in Definition \ref{3def:MM}. \end{definition} Again we will consider in our main result a more restrictive setting, in particular to enforce the crucial strict convergence property that is needed in Theorem \ref{thm:momolb}. Here we use the Li-Yau inequality \cite[Theorem 6]{li1982new}, that guarantees for any $V\in IV_2^o(\R^3)$ \begin{equation}\label{eq:10} \theta_2(\cdot,\\|V\\|)\leq \frac{\max(a_1^{-1},a_2^{-1})\mathcal{W}(u,V)}{4\pi}\,, \end{equation} where $\theta_2(\cdot,\\|V\\|)$ denotes the two dimensional density of $\\|V\\|$, \[ \theta_2(x,\\|V\\|)=\lim_{r\to 0}\frac{\\|V\\|(B(x,r))}{\pi r^2}\,. \] Since $V$ is rectifiable, the limit in this defintion exists $\\|V\\|$ almost everywhere. From \eqref{eq:10} we deduce in particular that $V$ has unit density if $\mathcal{W}(u,V)<8\pi \min\{a_1,a_2\}$. \begin{theorem}\label{4thm:main} Let $p\geq 2$ be as in \eqref{3eq:dw}. Suppose $(E_\e)_\varepsilon$ is a sequence of finite perimeter sets in $\R^3$ and let $\mu_\varepsilon=\mathcal{H}^2\ecke \partial_E_\varepsilon$, $\nu_\varepsilon:\partial_E_\varepsilon\to\mathbb{S}^2$ the inner normal, and $V_\varepsilon=\ovar{\partial_E_\varepsilon,\nu_\varepsilon,1,0}$. Consider in addition a sequence $(u_\varepsilon)_\varepsilon$ of phase fields $u_\varepsilon\in H^{1,p}_{\mu_\varepsilon}(\R^3)$. Assume that for some $\Lambda>0$ \begin{align} \mathcal{H}^{n-1}(\partial_E_\varepsilon) + \mathcal{E}_\varepsilon(u_\e,V_\e) &\leq \Lambda\quad\text{ for all }\varepsilon>0\,, \label{4ass:1}\\ \mathcal{W}_\varepsilon(u_\varepsilon,V_\varepsilon) &< 8\pi\min\{a_1,a_2\}\,. \label{4ass:2} \end{align} Then there exists a subsequence $\varepsilon\to 0$, a finite perimeter set $E\subset\R^3$ and a function $u:\partial_E\to \{0,1\}$ such that with $V=\ovar{\partial_* E,\nu_E,1,0}$, $S=\underline{c}(V)$ the following holds: $u$ belongs to $BV(S)$ and \begin{align} V_\e&\to V\quad \text{ in }IV^o_2(\R^3)\,, \label{4eq:thm2-1}\\ (\mu_\varepsilon,u_\varepsilon)&\to (\mu,u)\quad\text{ as measure-function pairs in }L^q\,, \label{4eq:thm2-2} \end{align} for any $1\leq q<p$. Moreover, it holds the lower bound estimate \begin{equation} \liminf_{\e\to 0} \mathcal{E}_\varepsilon(u_\e,V_\e) \geq \mathcal{E}(u,V)\,. \end{equation} \end{theorem} The proof will be given below after some preparations. \begin{remark} In the proof we will obtain some additional properties. In particular, we show the lower semicontinuity of both the bending and the line tension energy contribution to $\mathcal{E}_\varepsilon$ separately, and we obtain the strict convergence of the graphs of $\psi\circ u_\varepsilon$ to the generalized graph of $ku$. \end{remark} The next lemma is used to show the strict convergence of generalized graphs associated to $u_\varepsilon$. \begin{lemma} \label{lem:multiplicity_one} Let $V_\e\to V$ in $IV^o_m(\R^n)$ such that \[ \begin{split} \theta^_m(x,\\|V\\|)&=1\quad\text{ for }\\|V\\|\text{-almost every }x\in \R^n\,. \end{split} \] Then we have that \[ \mass{\underline{c}(V_\e)} \to \mass{\underline{c}(V)}\,. \] \end{lemma} \begin{proof} By \eqref{2eq:orvcurr} convergence as oriented varifolds implies weak convergence as currents, $\underline{c}(V_\e) \rightharpoonup \underline{c}(V)$. The lower semicontinuity of the mass under weak convergence of currents yields \begin{equation} \mass{\underline{c}(V)}\leq\liminf_{\e\to 0}\mass{\underline{c}(V_\e)}\,. \label{4eq:massb} \end{equation} Let us write \[ V_\e =\ovar{M_\e,\tau_\e,\theta_{\e,\pm}},\qquad V =\ovar{M,\tau,\theta_{\pm}}\,. \] By assumption \[ \begin{split} \theta_++\theta_- =\theta_+-\theta_-=1\quad \\|V\\|\text{-almost everywhere.} \end{split} \] Since mass is continuous under varifold convergence we deduce \[ \begin{split} \limsup_{\e\to 0} \mass{\underline{c}(V_\e)} &\leq \limsup_{\e\to 0}\\|V_\e\\|(\R^n)\\ &=\\|V\\|(\R^n) =\int_{M} (\theta_{+}+\theta_{-}) \d\mathcal{H}^m =\int_{M} (\theta_{+}-\theta_{-}\big) \d\mathcal{H}^m =\mass{\underline{c}(V)}\,. \end{split} \] Together with \eqref{4eq:massb} this completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{4thm:main}] By \eqref{4ass:1} we may pass to a further subsequence such that there exists a set of finite perimeter $E\subset\R^3$ and an integral varifold $V\in IV_2^o(\R^3)$ with \begin{align} \chi_{E_\e}&\to \chi_{E}\quad\text{ in }L^1(\R^3),\qquad \chi_{E_\e}\wsto \chi_{E}\quad\text{ in }BV\,, \label{4eq:Eeps-conv}\\ V_\varepsilon & \rightharpoonup V\quad\text{ as oriented varifolds,}\qquad \\|V\\| \geq \|\nabla\chi_E\|\,. \label{4eq:Veps-conv} \end{align} By \eqref{4ass:1} and the Li-Yau inequality \eqref{eq:10}, we have that $\\|V\\|$ almost everywhere $\theta_2(\cdot,\\|V\\|)=1$. Hence $S:=\underline c(V)=[\![\partial_ E,\nu_E,1]\!]$ and by Lemma \ref{lem:multiplicity_one} it follows that \[ \lim_{\e\to 0}\mathcal{H}^{n-1}(\partial_ E_\e)=\lim_{\e\to 0}\mass{\underline c(V_\e)}=\mass{\underline c(V)}=\mathcal{H}^{n-1}(\partial_* E)\,, \] which shows the strict BV-convergence \eqref{3eq:BV-strict} of the sets $E_\varepsilon$, $\varepsilon>0$. Therefore, we can apply Theorem \ref{thm:momolb} (see also its proof) and obtain the existence of some $u\in BV(S)$ with $u\in\{0,1\}$ $\\|S\\|$-almost everywhere such that \[ T_{v_\varepsilon,S_\e} \stackrel{c^}{\wto} T_{\psi(u),S}\, \] and such that the measure-function pair convergence \eqref{4eq:thm2-2} and the lower estimate \[ \liminf_{\e\to 0} I_\e(u_\e,V_\e)\geq 2k\mathcal{H}^{n-2}(J_u) \] holds. It remains to show the lower semicontinuity statement for $\mathcal{W}$. By $V_\e\wsto V$ as varifolds and by the uniform bound on $\\|H_\e\\|_{L^2_{\mu_\e}(\R^3;\R^3)}$, we have the weak convergence of measure-function pairs \begin{equation}\label{eq:15} (\\|V_\e\\|,H_\e)\wto (\\|V\\|,H) \quad\text{ in }L^2\,. \end{equation} By \eqref{4eq:thm2-2} we further deduce the strong measure-function pair convergence \begin{equation}\label{eq:16} \big(\\|V_\e\\|,\sqrt{a^{\bar\omega}(u_\e)}\big) \to \big(\\|V\\|,\sqrt{a^{\bar\omega}(u)}\big)\quad \text{ in }L^2\,. \end{equation} By \cite[Proposition 3.2]{moser2001generalization}, \eqref{eq:15} and \eqref{eq:16} may be combined to yield the weak convergence \begin{equation} \Big(\\|V_\e\\|, H_\e \sqrt{a^{\bar\omega}(u_\e)}\Big)\wto \Big(\\|V\\|,\sqrt{a^{\bar\omega}(u)}\Big) \quad\text{ in }L^1\,. \end{equation} But clearly $\\|H_\e \sqrt{a^{\bar\omega}(u_\e)}\\|_{L^2_{\mu_\e}(\R^3;\R^3)}\leq \\|\sqrt{a^{\bar\omega}}\\|_{L^\infty}\\|H_\e \\|_{L^2_{\mu_\e}(\R^3;\R^3)}$ yields a uniform bound on the $L^2$ norms, and the weak convergence in $L^1$ can be upgraded to the weak convergence in $L^2$ \begin{equation}\label{eq:14} \big(\\|V_\e\\|, H_\e \sqrt{a^{\bar\omega}(u_\e)}\big)\wto \big(\\|V\\|,\sqrt{a^{\bar\omega}(u)}\big) \quad\text{ in }L^2\,. \end{equation} By \cite[Theorem 4.4.2 (ii)]{hutchinson1986second} we obtain \begin{align} \mathcal{W}(u,V)&=\int \big(a_1u + a_2(1-u)\big)\|H_V\|^2\,d\\|V\\|\\ &= \int \|\sqrt{a^{\bar\omega}(u)}H_V\|^2\,d\\|V\\|\\ &\leq \liminf_{\varepsilon\to 0}\int \|\sqrt{a^{\bar\omega}(u_\varepsilon)}H_\e\|^2\,d\\|V_\varepsilon\\| =\liminf_{\varepsilon\to 0}\mathcal{W}_\varepsilon(u_\varepsilon,V_\varepsilon). \end{align*} This completes the proof of the theorem. \end{proof} \end{document}	arXiv
DOI:10.4310/CNTP.2009.V3.N1.A2 Borcherds-Kac-Moody Symmetry of N=4 Dyons @article{Cheng2008BorcherdsKacMoodySO, title={Borcherds-Kac-Moody Symmetry of N=4 Dyons}, author={Miranda C N Cheng and Atish Dabholkar}, journal={Communications in Number Theory and Physics}, Miranda C N Cheng, A. Dabholkar Communications in Number Theory and Physics We consider compactifications of heterotic string theory to four dimensions on CHL (Chaudhuri-Hockney-Lykken) orbifolds of the type T /ZN with N = 4 supersymmetry. The exact partition functions of the quarter-BPS (Bogomol'nyi-Prasad-Sommerfeld) dyons in these models are given in terms of genus-two Siegel modular forms. Only the N = 1, 2, 3 models satisfy a certain finiteness condition, and in these cases one can identify a Borcherds–Kac–Moody superalgebra underlying the symmetry structure of… BKM superalgebras from counting dyons in N=4 supersymmetric type II compactifications S. Govindarajan, Dileep P. Jatkar, K. Krishna BKM Lie superalgebras from counting twisted CHL dyons S. Govindarajan Nuclear Physics B Following Sen, we study the counting of ('twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with $ \mathcal{N} = 4 $ supersymmetry. The generating… No more walls! A tale of modularity, symmetry, and wall crossing for 1/4 BPS dyons Natalie M. Paquette, R. Volpato, M. Zimet A bstractWe determine the generating functions of 1/4 BPS dyons in a class of 4d N$$ \mathcal{N} $$ = 4 string vacua arising as CHL orbifolds of K3 × T2, a classification of which has been recently… N ov 2 01 0 HRI / ST / 1012 BKM Lie superalgebra for the Z 5-orbifolded CHL string K. Krishna We study the Z5-orbifolding of the CHL string theory by explicitly constructing the modular form Φ̃2 generating the degeneracies of the 1 4 BPS states in the theory. Since the additive seed for the… K3 Surfaces, N=4 Dyons, and the Mathieu Group M24 Miranda C N Cheng A close relationship between K3 surfaces and the Mathieu groups has been established in the last century. Furthermore, it has been observed recently that the elliptic genus of K3 has a natural… BPS-saturated string amplitudes: K3 elliptic genus and Igusa cusp form χ10 S. Hohenegger, S. Stieberger $\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras -- II S. Govindarajan, M. Shabbir The square-root of Siegel modular forms of CHL Z N orbifolds of type II compactifications are denominator formulae for some Borcherds-Kac-Moody Lie superalgebras for N = 1 , 2 , 3 , 4. We study the… Automorphic products, generalized Kac-Moody algebras and string amplitudes S. Hohenegger We review automorphic products and generalized Kac-Moody algebras from a physics point of view. We discuss the appearance of automorphic products in BPS-saturated one-loop quantities in heterotic… K3 string theory, lattices and moonshine Miranda C N Cheng, Sarah M. Harrison, R. Volpato, M. Zimet Research in the Mathematical Sciences In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on $$K3 \times {\mathbb {R}}^6$$K3×R6,… UvA-DARE (Digital Academic Repository) K3 string theory, lattices and moonshine In this paper, we address the following two closely related questions. First, we complete the classification of finite symmetry groups of type IIA string theory on K 3 × R 6 , where Niemeier lattices… Wall crossing, discrete attractor flow and Borcherds algebra Miranda C N Cheng, E. Verlinde The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root… Comments on the spectrum of CHL dyons A. Dabholkar, D. Gaiotto, S. Nampuri We address a number of puzzles relating to the proposed formulae for the degeneracies of dyons in orbifold compactifications of the heterotic string to four dimensions with = 4 supersymmetry. The… Moduli space of Chaudhuri-Hockney-Lykken strings. Chaudhuri, Polchinski Physical review. D, Particles and fields An orbifold of the toroidally compactified heterotic string is discussed which gives a global reduction of the dimension of the moduli space while preserving the supersymmetry, and the set of allowed gauge groups is invariant under electric-magnetic duality, providing further evidence for the {ital D}=4 heteroticString duality. Partition functions of torsion >1 dyons in heterotic string theory on T 6 Shamik Banerjee, A. Sen, Y. Srivastava The original proposal of Dijkgraaf, Verlinde and Verlinde for the quarter BPS dyon partition function in heterotic string theory on T 6 is known to correctly produce the degeneracy of dyons of… Generalities of quarter BPS dyon partition function and dyons of torsion two We propose a general set of constraints on the partition function of quarter BPS dyons in any N=4 supersymmetric string theory by drawing insight from known examples, and study the consequences of… Spectrum of dyons and black holes in CHL orbifolds using Borcherds lift A. Dabholkar, S. Nampuri The degeneracies of supersymmetric quarter BPS dyons in four dimensions and of spinning black holes in five dimensions in a CHL compactification are computed exactly using Borcherds lift. The Hodge… Counting all dyons in $ \mathcal{N} = 4 $ string theory A. Dabholkar, João Gomes, S. Murthy For dyons in heterotic string theory compactified on a six-torus, with electric charge vector Q and magnetic charge vector P, the positive integer I ≡ gcd(Q ∧ P) is an invariant of the U-duality… View 1 excerpt, references results Spacelike Singularities and Hidden Symmetries of Gravity M. Henneaux, D. Persson, P. Spindel Living reviews in relativity It is shown that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Counting Dyons in N =4 String Theory R. Dijkgraaf, E. Verlinde, H. Verlinde View 7 excerpts, references background and methods Hyperbolic Kac Moody Algebras and Chaos in Kaluza Klein Models T. Damour, M. Henneaux, B. Julia, H. Nicolai	CommonCrawl
TWiki> GRM Web>BaumWelch>BaumWelchDocs (2022-04-06, KyleGorman) EditAttach Baum-Welch documentation Training is performed using the Train function. It takes as arguments a FAR or WFST representing the plaintext, and channel model and a FAR representing the ciphertext. Normalization strategies Training takes a template argument argument representing the desired expectation table type. Two types are available. The state expectation table normalizes so that all arcs leaving a state sum to semiring one. The state/input-label expectation table normalizes so that the weights of all arcs leaving a state with the same input label sum to semiring one; this is arguably the most intuitive normalization strategy and so it used as the default. Semirings and count collection strategies In a semiring without the _ path property _, like the log semiring, the E-step sums over all state sequences and thus performs true Baum-Welch training. In the case the input is a FST language model with backoffs, attach a $$\phi$$-matcher (fstspecial --fst_type=phi) to ensure proper interpretation. In a semiring with the path property, like the tropical semiring, the E-step sums over only the most likely state sequence, an approximation known as Viterbi training (Brown et al. 1993:293). In the case the input is an FST language model with backoffs, backoffs can be encoded exactly using a $$\phi$$-matcher or approximated by $$\epsilon$$ using the default matcher. In practice, Baum-Welch training is slightly slower than Viterbi training, though somewhat more accurate (Jansche 2003). Train takes an struct representing options for training. These define the size of the batch, the maximum number of iterations for training, the learning rate $$\alpha$$, and the comparison/quantization $$\delta$$ used to detect convergence. Since expectation maximization training is only locally optimal, _random (re)starts_ can be used to avoid local minima. For particularly difficult problems, it may be beneficial to perform a very large number of random starts (e.g., Berg-Kilpatrick and Klein 2013). In this library, we randomly initialize weights by sampling from the uniform distribution $$(\texttt{kDelta}, 1]$$ in the real numbers and then mapping these values onto the appropriate values in the desired semiring. Randomize can be used to generate random restarts. Stepwise interpolation The actual training method uses a form of stepwise interpolation due to Liang and Klein (2009). At time $$k$$ a weight $$W_k$$ is given by $$W_k = (1 - \nu_k) W_{k - 1} + \nu_k M_k$$ where $$\nu_k = (k - 2)^{-\alpha}$$, $$\alpha$$ is a fixed hyperparameter controlling exponential decay of $$\nu_k$$, and $$M_k$$ is the maximized expectation at time $$k$$. `Decode` is the primary driver for decoding. Decoding with the trained model is essentially a process of constructing a weighted lattice followed by a shortest path computation. In a semiring with the path property, decoding is simple: Compose the input, channel model, and output, and materialize the cascade. Compute the shortest path using the Viterbi algorithm. Input-project (for decipherment only), then remove epsilons and weights. In a semiring without the path property, exact decoding is much more challenging, and in fact we do not have an algorithm for the pair case since we normally need to preserve the input-output correspondences. In the decipherment case, it is possible to perform with a DFA lattice, but determinizing the entire lattice is rarely feasible for interesting problems. Therefore we use the following strategy: Compose the plaintext model, channel model, and ciphertext, and materialize the cascade, then input-project and remove epsilons to produce an acyclic, epsilon-free non-deterministic weighted acceptor (henceforth, the NFA). Compute $$\beta_n$$, the costs to the future in the NFA. Compute, on the fly, a deterministic acceptor (henceforth, the DFA); expansion of the DFA converts the NFA future costs $$\beta_n$$ to the DFA future costs $$\beta_d$$. Compute, on the fly, a mapping of the DFA to a semiring with the path property by converting the weights to tropical. Compute the shortest path of the on-the-fly tropical-semiring DFA using Asearch (Hart et al. 1968) with $$\beta_d$$ as a heuristic, halting immediately after the shortest path is found. Convert the shortest path back to the input semiring and remove weights. Pair modeling In the pair scenario, we observe a set of paired input/output strings $$(i, o)$$ where $$i \in {I}^{}$$ and $$o \in {O}^{}$$ and wish to construct a conditional model. This is useful for many monotonic string-to-string transduction problems, such as grapheme-to-phoneme conversion or abbreviation expansion. Probabilistic formulation We would like to estimate a conditional probability model over input-output pairs, henceforth $$\textrm{P}(o \mid i)$$. Finite-state formulation We first construct FARs containing all $$i$$ and all $$o$$ pairs, respectively. We then construct a model of the alignment $$ \Gamma \subseteq {I}^{} \times {O}^{}$$. This serves as the initial (usually uniform) model for $$(i, o)$$; we henceforth refer to it as the channel model. Given an estimate for the channel model, $$\hat{\Gamma}$$, we can compute the best alignment by decoding $$\mathrm{ShortestPath}\left[i \circ \hat{\Gamma} \circ o\right]$$ where $$\textrm{ShortestPath}$$ represents the Viterbi algorithm. Pair language modeling Once the alignment model $$\hat{\Gamma}$$ is estimated, we can use it to construct a so-called pair language model (Novak et al. 2012, 2016). This is much like a classic language model but the actual symbols represent pairs in $${I} \times {O}$$; unlike $$\hat{\Gamma}$$, it is a joint model. To construct such a model: Compute the best $$i, o$$ alignments by decoding with $$\hat{\Gamma}$$. Rewrite the FSTs in the decoded alignments FAR as acceptors over a pair symbol vocabulary $${I} \times {O}$$. Compute a higher-order language model (e.g., using the NGram tools), with standard smoothing and shrinking options, producing a weighted finite-state acceptor. Convert the weighted finite-state acceptor to a finite-state transducer by rewriting the "acceptor" $${I} \times {O}$$ arcs with the corresponding "transducer" $${I} : {O}$$ arcs. Then the pair language model can be applied using composition and decoded using the Viterbi algorithm. Decipherment In the decipherment scenario, we observe a corpus of ciphertext $$c \in {C}^{}$$. We imagine that there exists a corpus of plaintext $$p \in {P}^{}$$ which was used to generate the ciphertext $$c$$. Our goal is to decode the ciphertext; i.e., to uncover the corresponding plaintext. We assume that the ciphertext $$c$$ has been generated (i.e., encoded) by $$\pi_o\left(p \circ \gamma\right)$$ where $$\pi_o$$ is the output projection operator and $$\gamma$$ is the encoder (i.e., inverse key). We further assume that the ciphertext can be recovered (i.e., decoded) using $$\pi_i\left(\gamma \circ c\right)$$ where $$\pi_i$$ is the input projection operator. However, we do not observe either $$\gamma$$, and must instead estimate it from data. We would like to estimate a conditional probability model which predicts the plaintext $$p$$ given $$c$$; i.e., $$\textrm{P}(p \mid c)$$. By Bayes' rule $$\textrm{P}(p \mid c) \propto \textrm{P}(p) \textrm{P}(c \mid p).$$ That is; the conditional probability of the plaintext given observed ciphertext is proportional to the product of the probability of the plaintext and the conditional probability of the ciphertext given the plaintext; for any corpus of ciphertext the denominator $$\textrm{P}(c)$$ is constant and thus can be ignored. We then express decoding as $$\hat{p} = \arg\max_p \textrm{P}(p) \textrm{P}(c \mid p).$$ We first construct a probabilistic model $$\Lambda \subseteq {P}^{}$$, a superset of the true plaintext $$p$$ and a model of $$(p)$$. We henceforth refer to this as the language model. We then construct a model of the inverse keyspace, $$ \Gamma \subseteq {P}^{} \times {C}^{}$$. This is a superset of the true encoder relation $$\gamma$$ and serves as an initial (usually uniform) model for $$(c \mid p)$$; we henceforth refer to it as the channel model. Given an estimate for the channel model, $$\hat{\Gamma}$$, we can express decoding as $$\mathrm{ShortestPath}\left[\pi_i\left(\Lambda \circ \hat{\Gamma} \circ c\right)\right]$$ Sparsity penalties Knight et al. (2006) suggest maximizing $$\hat{p} = \arg\max_p \textrm{P}(p) \textrm{P}(c \mid p)^k$$ where $$k$$ is some real number $$> 1$$ during decoding. This strategy increases the sparsity of the hypotheses generated by the trained model. This penalty can be applied using fstmap --map_type=power --power=$K before decoding, if desired. Berg-Kilpatrick, T., and Klein, D. 2013. Decipherment with a million random restarts. In Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing, pages 874-878. Brown, P., Della Pietra, S. A., Della Pietra, V. J., and Mercer, R. L. 1993. The mathematics of statistical machine translation: parameter estimation. Computational Linguistics 19(2): 263-311. Dempster, A. P., Laird, N. M., and Rubin, D. B. 1977. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39(1): 1-38. Hart, P. E., Nilsson, N. J., Raphael, B. 1968. A formal basis for the heuristic determination of minimal cost paths. IEEE Transactions on Systems Science and Cybernetics 4(2): 100-107. Jansche, M. 2003. Inference of string mappings for language technology. Doctoral dissertation, Ohio State University. Knight, K., Nair, A., Rashod, N., and Yamada, K. 2006. Unsupervised analysis for decipherment problems. In Proceedings of the COLING/ACL 2006 Main Conference Poster Sessions, pages 499-506. Liang, P., and Klein, D. 2009. Online EM for unsupervised models. In Proceedings of Human Language Technologies: The 2009 Annual Conference of the North American Chapter of the Association for Computational Linguistics, pages 611-619. Novak, J. R., Minematsu, N., and Hirose, K. 2012. WFST-based grapheme-to-phoneme conversion: Open source tools for alignment, model-building and decoding. In Proceedings of the 10th International Workshop on Finite State Methods and Natural Language Processing, pages 45-49. Novak, J. R., Minematsu, N., and Hirose, K. 2016. Phonetisaurus: exploring grapheme-to-phoneme conversion with joint n-gram models in the WFST framework. Natural Language Engineering 22(6): 907-938. Edit \| Attach \| Watch \| Print version \| History: r4 < r3 < r2 < r1 \| Backlinks \| Raw View \| WYSIWYG \| More topic actions Topic revision: r4 - 2022-04-06 - KyleGorman GRM Web	CommonCrawl
International Journal of Science and Mathematics Education The International Journal of Science and Mathematics Education is a bimonthly peer-reviewed academic journal published by Springer Science+Business Media on behalf of the National Science Council of Taiwan. It covers science and mathematics education topics and research methods, particularly ones with cross-curricular dimensions or which explore the area from different cultural perspectives.[1] The journal was established in 2004 with Fou-Lai Lin (National Taiwan Normal University). The current editor-in-chief is Hsin-Kai Wu (National Taiwan Normal University). International Journal of Science and Mathematics Education DisciplineScience education, mathematics education LanguageEnglish Edited byHsin-Kai Wu Publication details History2004–present Publisher Springer Science+Business Media on behalf of the National Science Council of Taiwan FrequencyBimonthly Impact factor 2.073 (2020) Standard abbreviations ISO 4 (alt) · Bluebook (alt1 · alt2) NLM (alt) · MathSciNet (alt ) ISO 4Int. J. Sci. Math. Educ. Indexing CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt) MIAR · NLM (alt) · Scopus ISSN1571-0068 (print) 1573-1774 (web) OCLC no.474774121 Links • Journal homepage • Online access • Online archive Abstracting and indexing The journal is abstracted and indexed in: • Current Contents/Social & Behavioral Sciences • EBSCO • MathEDUC • Scopus • Social Sciences Citation Index According to the Journal Citation Reports, the journal has a 2020 impact factor of 2.073.[2] References 1. "About this journal". International Journal of Science and Mathematics Education. Springer Science+Business Media. Retrieved July 19, 2013. 2. "Journals Ranked by Impact: Education and Educational Research". 2020 Journal Citation Reports. Web of Science (Social Science ed.). Thomson Reuters. 2021. External links • Official website	Wikipedia
\begin{document} \title[Autoduality of compactified Jacobians]{Autoduality of compactified Jacobians for curves with plane singularities} \author{Dima Arinkin} \address{Department of mathematics, University of North Carolina, Chapel Hill, NC} \email{[email protected]} \begin{abstract} Let $C$ be an integral projective curve with planar singularities. Consider its Jacobian $J$ and the compactified Jacobian $\oJ$. We construct a flat family $\oP$ of Cohen-Macaulay sheaves on $\oJ$ parametrized by $\oJ$; its restriction to $J\times\oJ$ is the Poincar\'e line bundle. We prove that the Fourier-Mukai transform given by $\oP$ is an auto-equivalence of the derived category of $\oJ$. \end{abstract} \maketitle \section{Introduction} Let $C$ be a smooth irreducible projective curve over a field $\kkk$, and let $J$ be the Jacobian of $C$. As an abelian variety, $J$ is self-dual. More precisely, $J\times J$ carries a natural line bundle (the Poincar\'e bundle) $P$ that is universal as a family of topologically trivial line bundles on $J$. The Poincar\'e bundle defines the Fourier-Mukai functor $${\mathfrak F}:D^b(J)\to D^b(J):\cG\mapsto Rp_{2,}(p_1^(\cG)\otimes P).$$ Here $D^b(J)$ is the bounded derived category of quasi-coherent sheaves on $J$ and $p_{1,2}:J\times J\to J$ are the projections. Mukai proved that $\mathfrak F$ is an equivalence of categories (\cite{Mukai}). Now suppose that $C$ is a singular curve, assumed to be projective and integral. The Jacobian $J$ is no longer projective, but it admits a natural compactification $\oJ\supset J$ (\cite{CJ,irreducibility}). By definition, $\oJ$ is the moduli space of torsion-free sheaves $F$ on $C$ such that $F$ has generic rank one and $\chi(F)=\chi(O_C)$; $J$ is identified with the open subset of locally free sheaves. It is natural to ask whether $\oJ$ is self-dual. In this paper, we prove such self-duality assuming that $C$ is an integral projective curve with planar singularities over a field $\kkk$ of characteristic zero. We construct a Poincar\'e sheaf $\oP$ on $\oJ\times\oJ$. The sheaf is flat over each copy of $\oJ$; we can therefore view it as a $\oJ$-family of sheaves on $\oJ$. We prove that this family is universal in the sense that it identifies $\oJ$ with a connected component of the moduli space of torsion-free sheaves of generic rank one on $\oJ$. This generalizes autoduality results of \cite{autoduality,compactified,Ar} and answers the question posed in \cite{compactified}. We also prove that the corresponding Fourier-Mukai functor \[ {\mathfrak F}:D^b(\oJ)\to D^b(\oJ):\cG\mapsto Rp_{2,}(p_1^(\cG)\otimes \oP) \] is an equivalence. \begin{remarks} \begin{myenum} \item If $C$ is a plane cubic (nodal or cuspidal), these results are known: see the remark after Theorem~\ref{th:FM}. \item It is easy to write a formula for the Poincar\'e line bundle $P$ on $J\times\oJ$: see \eqref{eq:Poincare}. Our result is thus a construction of an extension of $P$ to a sheaf on $\oJ\times\oJ$ satisfying certain natural properties. We studied $P$ in \cite{Ar}. There, we prove weaker versions of the results of the present paper: that $P$ is a universal family of topologically trivial line bundles on $\oJ$, and that the corresponding Fourier-Mukai functor \[{\mathfrak F}_J:D^b(J)\to D^b(\oJ)\] is fully faithful. Note that ${\mathfrak F}_J={\mathfrak F}\circ Rj_$, where $j:J\hookrightarrow\oJ$ is the open embedding. \item Compactified Jacobians appear as (singular) fibers of the Hitchin fibration for group $\GL(n)$; therefore, our results imply a kind of autoduality of the Hitchin fibration for $\GL(n)$. Such autoduality can be viewed as a `classical limit' of the (conjectural) Langlands transform. For arbitrary reductive group $G$, one expects a duality between the Hitching fibrations of $G$ and its Langlands dual ${}^LG$. For smooth fibers of the Hitchin fibration, such duality is proved by R.~Donagi and T.~Pantev in \cite{DP} (assuming some non-degeneracy conditions). \item Our construction of $\oP$ can be obtained as a `classical limit' of V.~Drinfeld's construction of automorphic sheaves for group $\GL(2)$ (\cite{Dr}), see Section~\ref{sc:langlands} for more details. \end{myenum} \end{remarks} \subsection{Acknowledgments} I am very grateful to V.~Drinfeld for sharing his ideas on this subject. Thanks to him, I abandoned my original, much clumsier, approach based on presentation schemes. I would also like to thank R.~Donagi, V.~Ginzburg, S.~Kumar, and T.~Pantev for their remarks and suggestions. This work is supported in part by Alfred P.~Sloan Foundation under the Sloan Research Fellowship program. \section{Main results} \subsection{Summary of main results} Fix a ground field $\kkk$ of characteristic zero. To simplify notation, we also assume that $\kkk$ is algebraically closed; this assumption is not necessary for the argument. Let $C$ be an integral projective curve over $\kkk$. Denote by $g$ the arithmetic genus of $C$, and let $J$ be the Jacobian of $C$, that is, $J$ is the moduli space of line bundles on $C$ of degree zero. Denote by $\oJ$ the compactified Jacobian; in other words, $\oJ$ is the moduli space of torsion-free sheaves on $C$ of generic rank one and degree zero. (For a sheaf $F$ of generic rank one, the degree is $\deg(F)=\chi(F)-\chi(O_C)$.) Then $\oJ$ is an irreducible projective variety; it is locally a complete intersection of dimension $g$ (\cite{irreducibility}). Clearly, $J\subset\oJ$ is an open smooth subvariety. Let $P$ be the Poincar\'e bundle; it is a line bundle on $J\times\oJ$. Its fiber over $(L,F)\in J\times\oJ$ equals \begin{equation} \label{eq:Poincare} P_{(L,F)}=\detrg(L\otimes F)\otimes\detrg (O_C)\otimes\detrg(L)^{-1}\otimes\detrg(F)^{-1}. \end{equation} More explicitly, we can write $L\simeq O(\sum a_ix_i)$ for a divisor $\sum a_ix_i$ supported by the smooth locus of $C$, and then $$P_{(L,F)}=\bigotimes(F_{x_i})^{\otimes a_i}.$$ The same formula \eqref{eq:Poincare} defines $P_{(F_1,F_2)}$ for any pair $(F_1,F_2)\in\oJ\times\oJ$ with either $F_1\in J$ or $F_2\in J$. Equivalently, $P\|_{J\times J}$ is symmetric under the permutation of factors in $J\times J$; and therefore $P$ naturally extends to a line bundle on $J\times\oJ\cup\oJ\times J\subset\oJ\times\oJ$. We denote this extension by the same letter $P$. For the rest of the paper, we assume that $C$ has planar singularities; in other words, the tangent space to $C$ at any point is at most two-dimensional. \begin{THEOREM} \label{th:oP} There exists a coherent sheaf $\oP$ on $\oJ\times\oJ$ with the following properties: \begin{enumerate} \item\label{th:oP1} $\oP\|_{J\times\oJ\cup\oJ\times J}\simeq P$; \item\label{th:oP3} $\oP$ is flat for the projection $p_2:\oJ\times\oJ\to\oJ$, and the restriction $\oP\|_{\oJ\times\{F\}}$ is a Cohen-Macaulay sheaf for every $F\in\oJ$. \end{enumerate} \end{THEOREM} \begin{remark} As explained in Section~\ref{sc:properties}, Theorem~\ref{th:oP} uniquely determines $\oP$ as a `Cohen-Macaulay extension' of $P$ under the embedding $j:J\times\oJ\cup\oJ\times J\hookrightarrow\oJ\times\oJ$. In fact, $\oP=j_P$. \end{remark} By Theorem~\ref{th:oP}\eqref{th:oP3}, $\oP$ is a family of coherent (Cohen-Macaulay) sheaves on $\oJ$ parametrized by $\oJ$. For fixed $F\in\oJ$, denote the corresponding coherent sheaf on $\oJ$ by $\oP_F$. In other words, $\oP_F$ is the restriction $\oP\|_{\oJ\times\{F\}}$. Let $\Pic(\oJ)^=$ be the moduli space of torsion-free sheaves of generic rank one on $\oJ$. A.~Altman and S.~Kleiman proved in \cite{CP1,CP2} that connected components of $\Pic(\oJ)^=$ are proper schemes (\cite[Theorem~3.1]{CP2}). The correspondence $F\mapsto\oP_F$ can be viewed as a morphism \[\rho:\oJ\to\Pic(\oJ)^=.\] Denote by $\overline{\Pic}^0(\oJ)\subset\Pic(\oJ)^=$ the irreducible component of the trivial bundle $O_\oJ\in\Pic(\oJ)\subset\Pic(\oJ)^=$ (since $O_\oJ\in\Pic(\oJ)^=$ is a smooth point, it is contained in a single component). We prove that $\oJ$ is self-dual in the following sense. \begin{THEOREM}\label{th:autoduality} $\rho$ is an isomorphism $\oJ\iso\overline{\Pic}^0(\oJ)$. Moreover, $\overline{\Pic}^0(\oJ)\subset\Pic(\oJ)^=$ is a connected component. \end{THEOREM} \begin{remark} The first statement of the theorem follows immediately from Theorem~\ref{th:oP} using \cite[Theorem~2.6]{compactified}. (Although \cite[Theorem~2.6]{compactified} is formulated for curves with double singularities, the same argument works in the case of planar singularities if we use \cite[Theorem~C]{Ar}.) The second statement relies on Theorem~\ref{th:FM}. \end{remark} Finally, we show that $\oP$ also provides a `categorical autoduality' of $\oJ$ in the sense that the corresponding Fourier-Mukai functor is an equivalence of categories. \begin{THEOREM}\label{th:FM} Let $D^b(\oJ)$ be the bounded derived category of quasicoherent sheaves on $\oJ$. The Fourier-Mukai functor $${\mathfrak F}:D^b(\oJ)\to D^b(\oJ):\cG\mapsto Rp_{1,}(p_2^(\cG)\otimes\oP)$$ is an equivalence of categories. Its quasi-inverse is given by $$D^b(\oJ)\to D^b(\oJ):\cG\mapsto Rp_{2,}(p_1^(\cG)\otimes\oP^\vee)\otimes\det(H^1(C,O_C))^{-1}[g].$$ Here $\oP^\vee=\HOM(\oP,O_{\oJ\times\oJ})$. \end{THEOREM} \begin{remarks} \begin{myenum} \item\label{rm:cubic} These results are known in the case of singular plane cubics (nodal or cuspidal). The sheaf $\overline P$ is constructed by E.~Esteves and S.~Kleiman in \cite{compactified} for curves $C$ with any number of nodes and cusps; they also prove that $\overline{P}$ is universal (the first statement of Theorem~\ref {th:autoduality}). If $C$ is a singular plane cubic, Theorem~\ref{th:FM} is proved by I.~Burban and B.~Kreussler (\cite{BK1}). J.~Sawon proves it for nodal or cuspidal curves of genus two in \cite{Sawon}. \item For simplicity, we consider a single curve $C$ in this section, but all our results hold for families of curves. Actually, the universal family of curves is used in the proof of Theorem~\ref{th:FM}. \end{myenum} \end{remarks} \subsection{Organization} The rest of the paper is organized as follows. Sections~\ref{sc:CM} and \ref{sc:PHilbS} contain preliminary results on Cohen-Macaulay sheaves and punctual Hilbert schemes of surfaces. These are used in the construction of the Poincar\'e sheaf $\oP$ in Section~\ref{sc:Q}. The proof of the key step in the construction is contained in Section~\ref{sc:proof}. This completes the proof of Theorem~\ref{th:oP}. Theorem~\ref{th:oP} easily implies certain simple properties of $\oP$ that are given in Section~\ref{sc:properties}. In Section~\ref{sc:FM}, we derive Theorem~\ref{th:FM} from these properties, and show that Theorem~\ref{th:FM} implies Theorem~\ref{th:autoduality}. \section{Cohen-Macaulay sheaves}\label{sc:CM} Our argument is based on certain properties of Cohen-Macaulay sheaves. Let us summarize these properties. Let $X$ be a scheme (all schemes are assumed to be of finite type over $\kkk$). Denote by $D^b_{coh}(X)\subset D^b(X)$ the coherent derived category of $X$. Suppose that $X$ has pure dimension. Let us normalize the dualizing complex $\DC_X\in D^b_{coh}(X)$ by the condition that its stalk at generic points of $X$ is concentrated in cohomological degree $0$. If $X$ is Gorenstein, $\DC_X$ is an invertible sheaf, which we denote $\omega_X$. Consider the duality functor \[\D:D^b_{coh}(X)\to D^b_{coh}(X):\cG\mapsto R\HOM(\cG,\DC_X).\] Let $M$ be a coherent sheaf on $X$. Set $d:=\codim(\supp(M))$. Then $H^i(\D(M))=0$ for $i<d$. Recall that $M$ is Cohen-Macaulay (of codimension $d$) if and only if $H^i(\D(M))=0$ for all $i\ne d$, so that $\D(M)[d]$ is a coherent sheaf. \subsection{Families of Cohen-Macaulay sheaves} \begin{lemma} \label{lm:CMFamily} Let $X$ and $Y$ be schemes of pure dimension, and suppose that $Y$ is Cohen-Macaulay. Suppose that a coherent sheaf $M$ on $X\times Y$ is flat over $Y$, and that for every point $y\in Y$, the restriction $M\|_{X\times\{y\}}$ is Cohen-Macaulay of some fixed codimension $d$. \begin{enumerate} \item\label{lm:CMFamily1} $M$ is Cohen-Macaulay of codimension $d$. \item\label{lm:CMFamily2} If in addition $Y$ is Gorenstein, then $\D(M)[d]$ is also flat over $Y$, and \[\D(M)[d]\|_{X\times\{y\}}\simeq \D(M\|_{X\times\{y\}})[d]\] for all points $y\in Y$. \end{enumerate} \end{lemma} \begin{proof} For \eqref{lm:CMFamily1}, we need to show that $M_z$ is a Cohen-Macaulay $O_z$-module for all $z\in X\times Y$. Set $y=p_2(z)\in Y$. The claim then follows from \cite[Corollary~6.3.3]{EGAIV} applied to morphism $O_y\to O_z$, the $O_y$-module $O_y$ and the $O_z$-module $M_z$. \eqref{lm:CMFamily2} follows from the identity \[L\iota^(R\HOM(M,\DC_{X\times Y}))=R\HOM(L\iota^M,L\iota^\DC_{X\times Y}),\] where $\iota:X\times\{y\}\hookrightarrow X\times Y$ is the embedding. \end{proof} \begin{remark} In principle, the lemma can be stated for all (not necessarily closed) points $y\in Y$; then the restriction $M\|_{X\times\{y\}}$ should be understood as an appropriate inverse image. This form of Lemma~\ref{lm:CMFamily} is more natural, and it suffices for our purposes. On the other hand, it is easy to see that it suffices to check the Cohen-Macaulay property of $M_z$ for closed points $z\in X\times Y$. \end{remark} \subsection{Extension of Cohen-Macaulay sheaves} Recall that a Cohen-Macaulay sheaf is \emph{maximal} if it has codimension zero. Maximal Cohen-Macaulay sheaves are normal in the sense that their sections extend across subsets of codimension two. \begin{lemma}\label{lm:CMExt} As before, let $X$ be a scheme of pure dimension. Let $M$ be a maximal Cohen-Macaulay sheaf on $X$. Then for any closed subset $Z\subset X$ of codimension at least two, we have $M=j_(M\|_{X-Z})$ for the embedding $j:X-Z\hookrightarrow X$. \end{lemma} \begin{proof} This is a special case of \cite[Theorem~5.10.5]{EGAIV}. (Actually, it suffices to require that $M$ has property $(S_2)$.) \end{proof} \subsection{Acyclicity} \begin{lemma} \label{lm:CMAcyclic} Let $f:Y\to X$ be a morphism of schemes. Suppose that $X$ is a Gorenstein scheme of pure dimension, and that $f$ has finite Tor-dimension. Let $M$ be a maximal Cohen-Macaulay sheaf on $X$. \begin{enumerate} \item\label{lm:CMAcyclic1} $Lf^M=f^M$. \item\label{lm:CMAcyclic2} In addition, suppose $Y$ is Cohen-Macaulay. Then $f^M$ is maximal Cohen-Macaulay. \end{enumerate} \end{lemma} \begin{proof} The statement is local on $X$, so we may assume that it is affine without losing generality. Essentially, the statement follows because $M$ is an $\infty$-syzygy sheaf. Indeed, we can include $M$ into a short exact sequence \[0\to M\to E\to M'\to 0,\] for some vector bundle $E$ and a maximal Cohen-Macaulay sheaf $M'$. Then \[L_if^M\simeq L_{i+1}f^M'\quad\text{for all }i>0,\] and \eqref{lm:CMAcyclic1} follows by induction. If $Y$ is Cohen-Macaulay, we can assume that it is of pure dimension (since this is true locally). Then a similar argument shows that \[\EXT^i(f^M,\DC_Y)\simeq\EXT^{i+1}(f^M',\DC_Y)\quad\text{for all }i>0,\] which implies \eqref{lm:CMAcyclic2}. \end{proof} \section{Punctual Hilbert schemes of surfaces}\label{sc:PHilbS} Let $S$ be a smooth surface. Let us review some properties of the Hilbert scheme of $S$. Fix an integer $n>0$. \subsection{Hilbert scheme of points}\label{sc:Hilb} Let $\Hilb_S=\Hilb_S^n$ be the Hilbert scheme of finite subschemes $D\subset S$ of length $n$. It is well known that $\Hilb_S$ is smooth of dimension $2n$, and that it is connected if $S$ is connected. This statement is due to J.~Fogarty (\cite{Fog}). Fix a point $0\in S$, and let $\Hilb_{S,0}\subset\Hilb_S$ be the closed subset of $D\in\Hilb_S$ such that $D$ is (set-theoretically) supported at $0$. If we fix local coordinates $(x,y)$ at $0$, we can identify $\Hilb_{S,0}$ with the scheme of codimension $n$ ideals in $\kkk[[x,y]]$. \begin{lemma}[J.~Brian{\c{c}}on] \label{lm:HilbS0} $\dim(\Hilb_{S,0})=n-1$; $\Hilb_{S,0}$ has a unique component of maximal dimension. \end{lemma} \begin{proof} See \cite{Brian}, \cite{Iarrobino}, \cite{Nakajima}, or \cite{Baranovsky}. \end{proof} \begin{remark} In fact, $\Hilb_{S,0}$ is irreducible. However, we do not need this claim; without it, Lemma~\ref{lm:HilbS0} is much easier, see \cite[Theorem 2]{Baranovsky}. \end{remark} Consider the symmetric power $\Sym^n S$. We write its elements as $0$-cycles \[\zeta=\sum_{x\in S}\zeta_x\cdot x\qquad (\zeta_x\ge 0,\sum_x\zeta_x=n).\] Set \[\supp(\zeta):=\{x\in S\st \zeta_x\ne 0\}\qquad (\zeta=\sum_x\zeta_x\cdot x\in\Sym^n S).\] \begin{corollary} \label{co:HilbS0} Consider the \emph{Hilbert-Chow morphism} \[\HC:\Hilb_S\to\Sym^n S:D\to\sum_{x\in S}(\length_xD)\cdot x.\] For any $\zeta\in\Sym^n S$, the preimage $\HC^{-1}(\zeta)$ has a unique component of maximal dimension; its dimension equals $n-\|\supp(\zeta)\|$. \end{corollary} \begin{proof} The preimage equals $\prod_x\Hilb_{S,x}^{\zeta_x}$. \end{proof} Denote by $\Hilb'_S\subset\Hilb_S$ the open subscheme parametrizing $D\in\Hilb_S$ such that $D$ can be embedded into a smooth curve (which can be assumed to be $\A1$ without loss of generality). Equivalently, $D\in\Hilb'_S$ if and only if the tangent space to $D$ at every point is at most one-dimensional. \begin{lemma} $\codim(\Hilb_S-\Hilb'_S)\ge 2$. \end{lemma} \begin{proof} If $D\in\Hilb_S-\Hilb'_S$, then $\|\supp(\HC(D))\|\le n-2$, and the lemma follows from Corollary~\ref{co:HilbS0}. \end{proof} \subsection{Flags of finite schemes}\label{sc:Flag} Let $\Flag'_S$ be the moduli space of flags \[\emptyset=D_0\subset D_1\subset\dots\subset D_n\subset S,\] where each $D_i$ is a finite scheme of length $i$ and $D_n\in\Hilb'_S$. It is equipped with maps \[\psi:\Flag'_S\to\Hilb'_S:(\emptyset=D_0\subset D_1\subset\dots\subset D_n)\mapsto D_n\] and \[\sigma:\Flag'_S\to S^n:(\emptyset=D_0\subset D_1\subset\dots\subset D_n)\mapsto(\supp(\ker(O_{D_i}\to O_{D_{i-1}})))_{i=1}^n.\] Moreover, $\Flag'_S$ carries an action of $S_n$. \begin{example} \label{ex:T} For $D\in\Hilb'_S$, choose an embedding $t:D\hookrightarrow\A1$. Then $t(D)=Z(f)$ for a monic degree $n$ polynomial $f\in\kkk[t]$. The fiber $\psi^{-1}(D)$ is identified with the scheme \[\Flag_f:=\{(t_1,\dots,t_n)\in\A{n}\st f(t)=(t-t_1)\dots(t-t_n)\};\] the identification sends $(t_1,\dots,t_n)\in\Flag_f$ to the flag \[\emptyset\subset Z(t-t_1)\subset Z((t-t_1)(t-t_2))\subset\dots\subset Z(f).\] The group $S_n$ acts on $\Flag_f$ by permuting $t_i$'s. \end{example} The following claim is well known. \begin{proposition} \begin{enumerate} \item $\psi$ is a degree $n!$ finite flat morphism. \item There exists a unique action of $S_n$ on $\Hilb'_S$ such that $\psi$ and $\sigma$ are equivariant. Here $S_n$ acts on $\Hilb'_S$ trivially and on $S^n$ by permutation of factors. \item The fiber of $\psi_(O_{\Flag'_S})$ over every point of $\Hilb'_S$ is isomorphic to the regular representation of $S_n$. \end{enumerate} \end{proposition} \begin{proof} Locally on $\Hilb'_S$, we can embed $D\in\Hilb'_S$ into $\A1$; the claim then follows from direct calculation (Example~\ref{ex:T}). \end{proof} There is also a more explicit description of $\psi_(O_{\Flag'_S})$. Let $\cD\subset\Hilb_S\times S$ be the universal family of degree $n$ subschemes of $S$. Let $h:\cD\to\Hilb_S$ and $g:\cD\to S$ be the restrictions of projections. Notice that $h$ is a finite flat morphism of degree $n$. Set $\cA:=h_O_\cD$. Then $\cA$ is a coherent sheaf of algebras on $\Hilb_S$ that is locally free of rank $n$. Let $\cA^\times\subset\cA$ be the subsheaf of invertible elements. We can view $\cA^\times$ as the sheaf of sections of a flat abelian group scheme over $\Hilb_S$: the fiber of this group scheme over $D\in\Hilb_S$ is $\kkk[D]^\times$. Clearly, $\cA^\times$ acts on $\cA$, and therefore also on the line bundle $\det(\cA)$. The action of $\cA^\times$ on $\det(\cA)$ is given by the norm character $\cN:\cA^\times\to O^\times$. \begin{lemma}\label{lm:Norm} There is a natural $S_n$-equivariant identification \[\psi_(O_{\Flag'_S})=\left((\cA\|_{\Hilb'_S})^{\otimes n}\right)_\cN,\] where the lower index $\cN$ denotes the maximal quotient on which $\cA^\times$ acts via the character $\cN$. \end{lemma} \begin{remarks} \begin{myenum} \item Let us describe the map $(\cA\|_{\Hilb'_S})^{\otimes n}\to\psi_(O_{\Flag'_S})$ of Lemma~\ref{lm:Norm}. Consider the $n$-fold fiber product \[\cD_n:=\cD\times_{\Hilb_S}\cD\dots\times_{\Hilb_S}\cD=\{(D,s_1,\dots,s_n)\in\Hilb_S\times S^n\st s_i\in D\text{ for all }i\}.\] The projection $h_n:\cD_n\to\Hilb_S$ is finite and flat of degree $n^n$ over $\Hilb_S$, and $h_{n,}(O_{\cD_n})=\cA^{\otimes n}$. Since the image of the map \[(\psi,\sigma):\Flag'_S\to \Hilb_S\times S^n\] is contained in $\cD_n$, we obtain a morphism of sheaves of algebras \[h_{n,}(O_{\cD_n})\|_{\Hilb'_S}\to\psi_(O_{\Flag'_S}).\] \item Lemma~\ref{lm:Norm} is similar to the description of $\psi_(O_{\Flag'_S})$ given in \cite{Haiman}. Namely, $\psi_(O_{\Flag'_S})$ is the quotient of $(\cA\|_{\Hilb'_S})^{\otimes n}$ by the kernel of the symmetric form \[\cA^{\otimes n}\times\cA^{\otimes n}\to\det\cA:(f_1\otimes\dots\otimes f_n,g_1\otimes\dots\otimes g_n) \mapsto\bigwedge_{i=1}^n(f_i g_i).\] This identification extends to $\Hilb_S$ and provides a description of the isospectral Hilbert scheme (see Section~\ref{sc:ISHilb}). I do not know whether the identification of Lemma~\ref{lm:Norm} also extends to $\Hilb_S$. Such an extension would provide another formula for the Poincar\'e sheaf. \end{myenum} \end{remarks} \begin{proof}[Proof of Lemma~\ref{lm:Norm}] Take $D\in\Hilb'_S$, and choose an embedding $t:D\hookrightarrow\A1$. According to Example~\ref{ex:T}, we have to identify $\kkk[X_f]$ and $(\kkk[D]^{\otimes n})_\cN$. Explicitly, for \[f=t^n-a_1t^{n-1}+\dots+(-1)^na_n,\] we have \[\kkk[X_f]=k[t_1,\dots,t_n]/(a_1-(t_1+\dots+t_n),\dots,a_n-(t_1\cdots t_n))\] and \[\kkk[D]^{\otimes n}=k[t_1,\dots,t_n]/(f(t_1),\dots,f(t_n)).\] The multiplicative group $\kkk[D]^\times$ is generated by linear polynomials. The polynomial $a(b-t)$ acts $\kkk[D]^{\otimes n}$ as $a^n(b-t_1)\cdots(b-t_n)$, while $\cN(a(b-t))=a^nf(b)$. Therefore, $(\kkk[D]^{\otimes n})_\cN$ is the quotient of $\kkk[t_1,\dots,t_n]$ modulo the ideal generated by $f(t_1),\dots, f(t_n)$ and $(b-t_1)\dots(b-t_n)-f(b)$ for all $b\in\kkk$ such that $f(b)\ne 0$. The statement follows. \end{proof} \subsection{Isospectral Hilbert scheme}\label{sc:ISHilb} Let us keep the notation of Section~\ref{sc:Flag}. The following result is a part of M.~Haiman's $n!$ Conjecture. \begin{proposition}\label{pp:IS} There exists a scheme $\IS_S$ and a commutative diagram \[ \xymatrix{S^n\ar@{=}[d]& \Flag'_S\ar[l]_-\sigma\ar[r]^-\psi\ar@{^{(}->}[d]&\Hilb'_S\ar@{^{(}->}[d]\\ S^n&\IS_S\ar[l]\ar[r]&\Hilb_S,} \] in which the right square is Cartesian and $\IS_S\to\Hilb_S$ is a degree $n!$ finite flat morphism. The scheme and the diagram are unique up to a unique isomorphism. \end{proposition} \begin{proof} The scheme $\IS_S$ can be constructed by extending $\psi_O_{\Flag'_S}$ to a rank $n!$ sheaf of algebras on $\Hilb_S$. Since $\codim(\Hilb_S-\Hilb'_S)\ge 2$, such extension is unique if it exists. This implies the uniqueness claim. The existence claim is local in the \'etale topology on $\Hilb_S$; it therefore reduces to the case $S=\A2$ proved by M.~Haiman in \cite{Haiman}. \end{proof} Following \cite{Haiman}, we call $\IS_S$ the \emph{isospectral Hilbert scheme} of $S$. We keep the notation $\psi$ and $\sigma$ for the extended morphisms $\IS_S\to\Hilb_S$ and $\IS_S\to S^n$. Finally, note that the action of $S_n$ on $\Flag'_S$ extends to its action on $\IS_S$ (because $\IS_S$ is unique), and that $\psi$ and $\sigma$ are equivariant. \begin{remark} In \cite{Haiman}, it is shown that the map $(\psi,\sigma):\IS_S\to\Hilb_S\times S^n$ is an embedding, so that $\IS_S$ can be defined as the closure of the image of $\Flag'_S$ in $\Hilb_S\times S^n$. We do not use this property. This gives us a choice of two possible references for Proposition~\ref{pp:IS}: the original argument of \cite{Haiman} and V.~Ginzburg's paper \cite{Ginzburg}, which provides a construction of $\IS_S$ based on Hodge $D$-modules. \end{remark} \subsection{Remark: stack of finite schemes}\label{sc:Tsch} Let us define universal versions of $\Flag'_S$ and $\IS_S$. Let $\FS$ be the algebraic stack parametrizing finite schemes of length $n$. Denote by $\FS_1\subset\FS$ the open substack of schemes $D\in\FS$ that are isomorphic to a closed subscheme of $\A1$. Denote by $\Flag'_{univ}$ the stack of flags \[(\emptyset=D_0\subset D_1\subset\dots\subset D_n),\] where $D_i$ is a finite scheme of length $i$ and $D_n\in\FS_1$. The natural morphism $\Flag'_{univ}\to\FS_1$ has an action of $S_n$, and the map $\psi$ is obtained from it by base change via $\Hilb'_S\to\FS_1$. The morphism $\Flag'_S\to\Hilb'_S$ is a cameral cover for the group $\GL(n)$ in the sense of \cite{DG}. Moreover, $\FS_1$ is identified with the stack of cameral covers for $\GL(n)$, and $\Flag'_{univ}\to\FS_1$ is the universal cameral cover. Consider $\Hilb_{\A1}$ (the Hilbert scheme of finite subschemes $D\subset\A1$ of length $n$). The natural map $\Hilb_{\A1}\to\FS_1$ is a presentation, so $\FS_1$ is a quotient of $\Hilb_{\A1}$ by an action of a groupoid. We can identify $\Hilb_{\A1}$ with the affine space of monic degree $n$ polynomials in $\kkk[t]$ (as in Example~\ref{ex:T}). The elements of the groupoid acting on $\Hilb_{\A1}$ are then interpreted as \emph{Tschirnhaus transformations} of polynomials. In this way, the stack $\FS_1$ goes back to the seventeenth century \cite{Tsch}. This relation was pointed out to me by V.~Drinfeld. Now let $\FS_2\subset\FS$ be the open substack of schemes $D$ that admit an embedding into a smooth surface, which may be assumed to be $\A2$ without loss of generality. The natural morphism $\Hilb_{\A2}\to\FS_2$ is a presentation, and the scheme $\IS_{\A2}$ defined by M.~Haiman descend to a flat finite stack $\IS_{univ}$ over $\FS_2$. We can view $\IS_{univ}$ as the universal isospectral Hilbert scheme: for every smooth surface $S$, we have \[\IS_S=\IS_{univ}\times_{\FS_2}\Hilb_S.\] \section{Poincar\'e sheaf}\label{sc:Q} We prove Theorem~\ref{th:oP} by constructing $\oP$. Actually, we construct a sheaf not on $\oJ\times\oJ$ but on its smooth cover, and then show that the sheaf descends to $\oJ\times\oJ$. This is similar to the construction of automorphic sheaves (\cite{Dr}). \subsection{Construction of the Poincar\'e sheaf} Fix an integer $n>0$, and let $\Hilb_C$ be the Hilbert scheme of finite subschemes $D\subset C$ of degree $n$. Recall that $\Hilb_C$ is an irreducible locally complete intersection of dimension $n$ (\cite{irreducibility}). For $n\gg 0$, $\Hilb_C$ is a smooth cover of $\oJ$. More precisely, fix a smooth point $p_0\in C$. It defines an Abel-Jacobi map \[\alpha:\Hilb_C\to\oJ:D\mapsto \cI_D^\vee(-np_0)=\HOM(\cI_D,O_C(-np_0)).\] Here $\cI_D$ is the ideal sheaf of $D\subset C$. For $n\gg0$, the map $\alpha:\Hilb_C\to\oJ$ is smooth and surjective. Our goal is to construct a sheaf $Q$ on $\Hilb_C\times\oJ$, and then show that it descends to a sheaf $\oP$ on $\oJ\times\oJ$ when $n\gg 0$. The construction of $Q$ makes sense even if $n$ is not assumed to be large. Let $\cF$ be the universal sheaf on $C\times\oJ$. Thus, for every $F\in\oJ$, the restriction $\cF\|_{C\times\{F\}}$ is identified with $F$. We normalize $\cF$ by framing it over $p_0$, so we have \[\cF\|_{\{p_0\}\times\oJ}=O_{\{p_0\}\times\oJ}.\] Now consider the sheaf \[\cF_n:=p_{1,n+1}^\cF\otimes\dots\otimes p_{n,n+1}^\cF\] on $C^n\times\oJ$. In other words, $\cF_n$ is the family of sheaves $F^{\boxtimes n}$ on $C^n$ parametrized by $F\in\oJ$. The sheaf $\cF_n$ is $S_n$-equivariant in the obvious way. Choose a closed embedding $\iota:C\hookrightarrow S$ into a smooth surface $S$, and consider the diagram \[ \xymatrix{ \Hilb_S\times\oJ\ar[d]^-{p_1}&\IS_S\times\oJ\ar[l]_-{\psi\times\id_\oJ}\ar[r]^-{\sigma\times\id_\oJ}&S^n\times\oJ&C^n\times\oJ \ar[l]_-{\iota^n\times\id_\oJ}\\ \Hilb_S.} \] Set \begin{equation}\label{eq:Q} Q:=\left((\psi\times\id_\oJ)_(\sigma\times\id_\oJ)^(\iota^n\times \id_\oJ)_\cF_n\right)^{sign}\otimes p_1^\det(\cA)^{-1}. \end{equation} Here the upper index `sign' stands for the space of anti-invariants with respect to the action of $S_n$. Recall that $\det\cA$ is the line bundle on $\Hilb_S$ whose fiber over $D\in\Hilb_S$ is $\det(\kkk[D])$. Note that \eqref{eq:Q} defines a sheaf on $\Hilb_S\times\oJ$. Let us identify $\Hilb_C$ with a closed subscheme of $\Hilb_S$ using $\iota$. The following claim shows that \[\supp(Q)\subset\Hilb_C\times\oJ.\] This is not immediate, because $\psi(\sigma^{-1}(\iota(C)^n))\not\subset\Hilb_C$. \begin{proposition}\label{pp:FCM} As above, $\iota:C\hookrightarrow S$ is a closed embedding of a reduced curve $C$ into a smooth surface $S$ (it is not necessary to assume that $C$ is projective or irreducible). Let $F$ be a torsion-free sheaf of generic rank one on $C$, and consider the sheaf $(\iota_F)^{\boxtimes n}$ on $S^n$. Then \begin{enumerate} \item\label{pp:FCM1} $L\sigma^ (\iota_F)^{\boxtimes n}=\sigma^(\iota_F)^{\boxtimes n}$; \item\label{pp:FCM2} $\sigma^(\iota_F)^{\boxtimes n}$ is Cohen-Macaulay of codimension $n$; \item\label{pp:FCM3} $\psi_(\sigma^(\iota_F)^{\boxtimes n})^{sign}$ is supported by the subscheme $\Hilb_C\subset\Hilb_S$. \end{enumerate} \end{proposition} It is also not hard to check that $Q$ given by formula \eqref{eq:Q} agrees with $P$ in the following sense. Let $\Hilb'_C\subset\Hilb_C$ (resp. $\Hilb''_C\subset\Hilb_C$) be the open subscheme parametrizing $D\in\Hilb_C$ such that $D$ is isomorphic to a subscheme of $\A1$ (resp. $D$ is reduced and contained in the smooth locus $C^{sm}\subset C$). Note that $\Hilb'_C=\Hilb'_S\cap\Hilb_C$ and $\Hilb''_C\subset\Hilb'_C$. Also, note that $\Hilb''_C$ is dense in $\Hilb_C$ (because $\Hilb_C$ is irreducible) and $\Hilb'_C\subset\Hilb_C$ is a complement of subset of codimension at least two (this is easy to see from Corollary~\ref{co:HilbS0}). \begin{lemma}\label{lm:QandP} The restrictions of the sheaves $Q$ and $(\alpha\times\id_\oJ)^P$ to $(\Hilb''_C\times\oJ)\cup(\Hilb'_C\times J)$ are naturally isomorphic. \end{lemma} We postpone the proof of Proposition~\ref{pp:FCM} until Section~\ref{sc:proof}; Lemma~\ref{lm:QandP} follows from Proposition~\ref{pp:QandQ'} below. Let us show that Proposition~\ref{pp:FCM} and Lemma~\ref{lm:QandP} imply Theorem~\ref{th:oP}. \begin{proof}[Proof of Theorem~\ref{th:oP}] Proposition~\ref{pp:FCM}\eqref{pp:FCM1} implies that $Q$ is flat over $\oJ$. By Proposition~\ref{pp:FCM}\eqref{pp:FCM2}, $Q$ is a flat $\oJ$-family of Cohen-Macaulay sheaves of codimension $n$ on $\Hilb_S$. Therefore, $Q$ is a Cohen-Macaulay sheaf of codimension $n$ on $\Hilb_S\times\oJ$ by Lemma~\ref{lm:CMFamily}. By Proposition~\ref{pp:FCM}\eqref{pp:FCM3}, the restriction $Q\|_{\Hilb_S\times\{F\}}$ is supported by $\Hilb_C\times\{F\}$ for every $F\in\oJ$. Therefore, $Q$ is a maximal Cohen-Macaulay sheaf on $\Hilb_C\times\oJ$. Finally, consider the Abel-Jacobi map $\alpha:\Hilb_C\to\oJ$ for $n\gg 0$. By Lemma~\ref{lm:QandP}, $Q$ coincides with the pullback $(\alpha\times\id_\oJ)^P$ on the complement of a subset of codimension at least two. It follows that $Q$ descends to $\oJ\times\oJ$: we can extend the descent data across codimension two using Lemma~\ref{lm:CMExt}. We thus obtain a sheaf $\oP$ on $\oJ\times\oJ$. It is clear that $\oP$ has the properties required by Theorem~\ref{th:oP}. \end{proof} \subsection{Restriction to $\Hilb'_S\subset\Hilb_S$}\label{sc:QonHilb'} The rest of this section contains comments on the formula \eqref{eq:Q}. Recall that $\Hilb'_S\subset\Hilb_S$ is the open subscheme of $D\in\Hilb_S$ such that $D$ is isomorphic to a subscheme of $\A1$. Over $\Hilb'_S$, we can identify $\IS_S$ with the space of flags $\Flag'_S$. In this way, \eqref{eq:Q} is more explicit if we are only interested in the restriction $Q\|_{\Hilb'_S\times\oJ}$. To make the formula more concrete, let us fix $F\in\oJ$. \begin{lemma}\label{lm:QviaFlag'} Consider the diagram \[ \xymatrix{ \Hilb'_S&\Flag'_S\ar[l]_-\psi\ar[r]^-\sigma&S^n&C^n\ar[l]_-{\iota^n}.} \] For every $F\in\oJ$, we have \[ Q\|_{\Hilb'_S\times\{F\}}=\left(\psi_\sigma^(\iota^n)_* F^{\boxtimes n}\right)^{sign}\otimes \det(\cA)^{-1}. \] \end{lemma} \begin{proof} Clear. \end{proof} We can also rewrite the formula for $Q\|_{\Hilb'_S\times\oJ}$ using Lemma~\ref{lm:Norm}. Use the diagram \[ \xymatrix{ \Hilb_S\times\oJ\ar[d]^-{p_1}&\cD\times\oJ\ar[l]_-{h\times\id_\oJ}\ar[r]^-{g\times\id_\oJ}&S\times\oJ&C\times\oJ \ar[l]_-{\iota\times\id_\oJ}\\ \Hilb_S} \] to define the sheaf \[Q':=\left((\bigwedge\nolimits^n (h\times\id_\oJ)_(g\times\id_\oJ)^(\iota\times\id_\oJ)_\cF))\otimes p_1^\det(\cA)^{-1}\right)_{p_1^{-1}(\cA^\times)}.\] Recall that $\cF$ is the universal sheaf on $C\times\oJ$; see Section~\ref{sc:Flag} for definitions of the remaining objects. Explicitly, the fiber of $Q'$ over $(D,F)\in\Hilb_S\times\oJ$ equals \[Q'_{(D,F)}=\left((\bigwedge\nolimits^n H^0(D,\iota_F))\otimes(\det\kkk[D])^{-1}\right)_{\kkk[D]^\times}.\] Up to the twist by $\det\kkk[D]^{-1}$, the fiber is the largest quotient of $\bigwedge^n H^0(D,\iota_F)$ on which $\kkk[D]^\times$ acts by the norm character. (Notice the similarity with \cite[Lemma~3]{Dr}.) \begin{proposition} \label{pp:QandQ'} The restrictions of $Q$ and $Q'$ to $\Hilb'_S\times\oJ$ are naturally isomorphic. \end{proposition} \begin{proof} Follows from Lemma~\ref{lm:Norm}. \end{proof} Lemma~\ref{lm:QandP} follows immediately from Proposition~\ref{pp:QandQ'}. \subsection{Curves with double singular points} As explained in Section~\ref{sc:QonHilb'}, the restriction $Q\|_{\Hilb'_S\times\oJ}$ is more explicit than $Q$ itself. Suppose that all singularities of $C$ are at most double points. (So that at every point $c\in C$, there exists $f\in O_c$ such that $\length(O_c/fO_c)=2$.) \begin{proposition} \label{pp:alpha'} For $n\gg 0$, the morphism $\alpha:\Hilb'_C\to\oJ$ is surjective. \end{proposition} \begin{proof} Let $C^{sing}$ be the singular locus of $C$. For every $p\in C^{sing}$, choose an invertible subsheaf $\cI^{(2)}_p\subset O_C$ of degree $-2$ such that $\cI^{(2)}_p\supset\cI_p^2$. Such $\cI^{(2)}_p$ exists because $C$ has only double singular points. Consider $D\in\Hilb_C$. Then $D\in\Hilb'_C$ if and only if $\cI_D\not\subset\cI_p^2$ for every $p\in C^{sing}$. In particular, $D\in\Hilb'_C$ provided $\cI_D\not\subset\cI^{(2)}_p$ for all $p\in C^{sing}$. Now take $F\in\oJ$ and $n\ge 2g+1$. Recall that $p_0\in C$ is the smooth point used to define the Abel-Jacobi map $\alpha:\Hilb_C\to\oJ$. Choose a non-zero morphism $\phi:O(-np_0)\to F$. By the Riemann-Roch Theorem, the space of such morphisms $\phi$ has dimension $n-g+1$. Then $F=\alpha(D)$, where $\cI_D\subset O_C$ is the image of \[\phi^\vee:F^\vee(-np_0)\to O_C.\] For fixed $p\in C^{sing}$, the space of morphisms $\phi$ such that $\phi(O(-np_0))\subset F\otimes\cI^{(2)}_p$ has dimension $n-g-1$ by the Riemann-Roch Theorem. Thus, if $\phi$ is generic, we have $\phi(O(-np_0))\not\subset F\otimes\cI^{(2)}_p$ for all $p\in C^{sing}$, and $D\in\Hilb'_C$. \end{proof} By Proposition~\ref{pp:alpha'}, we see that the sheaf $\oP$ on $\oJ\times\oJ$ can be constructed as the descent of $Q\|_{\Hilb'_S\times\oJ}$, assuming $C$ has at most double singular points. Thus, in this case it is possible to describe $\oP$ without using isospectral Hilbert schemes. \begin{remark} Suppose the singularities of $C$ are arbitrary planar, and denote by $\oJ'\subset\oJ$ the image $\alpha(\Hilb'_S)$ for $n\gg 0$. It is easy to see that for $n\gg 0$, the image does not depend on $n$ or the choice of $p_0$. The restriction $\oP\|_{\oJ'\times\oJ}$ can be constructed without using isospectral Hilbert schemes. \end{remark} \subsection{Independence of embedding} The definition of $Q$ involves the embedding $\iota:C\to S$, and the argument of Section~\ref{sc:proof} relies on the properties of the Hilbert scheme $\Hilb_S$. On the other hand, it is not hard to see that the restriction $Q\|_{\Hilb_C\times\oJ}$ is independent of this embedding. By Proposition~\ref{pp:FCM}\eqref{pp:FCM3} this restriction coincides with $Q$. Let us provide a formula for $Q\|_{\Hilb_C\times\oJ}$. Set $\IS_C:=\psi^{-1}\Hilb_C\subset\IS_S$. The morphisms $\psi:\IS_C\to\Hilb_C$ and $\sigma:\IS_C\to C^n$ are $S_n$-equivariant for the natural actions, and $\psi$ is a degree $n!$ finite flat morphism. It is easy to see that $\IS_C$ does not depend on $\iota:C\to S$. Indeed, $\IS_C$ is obtained from $\IS_{univ}$ by base change $\Hilb_C\to\FS_2$ (see Section~\ref{sc:Tsch}). For another explanation, note that the preimage $\psi^{-1}(\Hilb'_C)$ is identified with the moduli space $\Flag'_C$ of flags of finite subschemes in $C$, and $\IS_{univ}$ can be viewed as its extension to a finite flat scheme over $\Hilb_C$. Such an extension is unique because $\codim(\Hilb_C-\Hilb'_C)\ge 2$ and $\Hilb_C$ is Gorenstein. Now consider the diagram \[ \xymatrix{ \Hilb_C\times\oJ\ar[d]^-{p_1}&\IS_C\times\oJ\ar[l]_-{\psi\times\id_\oJ}\ar[r]^-{\sigma\times\id_\oJ}&C^n\times\oJ\\ \Hilb_C.} \] Clearly, \[ Q\|_{\Hilb_C\times\oJ}:=\left((\psi\times\id_\oJ)_(\sigma\times\id_\oJ)^\cF_n\right)^{sign}\otimes p_1^\det(\cA)^{-1}. \] Similarly, one can describe the restriction $Q\|_{\Hilb'_C\times\oJ}$ without choosing $\iota:C\hookrightarrow S$ by rewriting the formulas of Section~\ref{sc:QonHilb'}. We leave this description to the reader. \subsection{Poincar\'e sheaf and automorphic sheaves}\label{sc:langlands} As we already mentioned, the formula for $\oP$ can be interpreted as a classical limit of V.~Drinfeld's formula for automorphic sheaves for $\GL(2)$ (\cite{Dr}). Let us sketch the relation. Let $X$ be a smooth projective absolutely irreducible curve over a finite field, and let $\cE$ be a geometrically irreducible $\ell$-adic local system on $X$. In \cite{Dr}, V.~Drinfeld constructs an automorphic perverse sheaf $\Aut_\cE$ on the moduli stack $\Bun_2$ of rank two vector bundles on $X$. We now apply two `transformations' to this construction. Firstly, let us assume that $X$ is a (smooth projective connected) curve over a field $\kkk$ of characteristic zero, which we assume to be algebraically closed for simplicity. Also, let us replace perverse sheaves with $D$-modules. Now the input of the construction is a rank two bundle with connection $\cE$ on $X$, and its output is a $D$-module $\Aut_\cE$ on $\Bun_2$. The second transformation is a `classical limit' (`classical' here refers to the relation between classical and quantum mechanics). This involves replacing $D$-modules on a smooth variety (or stack) $Z$ with $O$-modules on the cotangent bundle $T^Z$. Now the input $\cE$ of the construction is a rank two Higgs bundle on $X$; equivalently, $\cE$ is an $O$-module on $T^X$ whose direct image to $X$ is locally free of rank two. The output is an $O$-module $\Aut_\cE$ on $T^\Bun_2$. In particular, let $\iota:C\hookrightarrow T^X$ be an irreducible reduced \emph{spectral curve} for $\GL(2)$: that is, the map $C\to X$ is finite of degree two. Note that $C$ has at most double singular points. We can then take $\cE=\iota_F$ for a torsion-free sheaf $F$ on $C$ of generic rank one. By interpreting Drinfeld's construction in these settings, we obtain a formula for a sheaf $\Aut_\cE$ on $T^\Bun_2$ (more precisely, on the cotangent space to a smooth cover of $\Bun_2$). From the point of view of geometric Langlands program, it is natural to expect that the sheaf is supported on the compactified Jacobian $\oJ$ of $C$, which is embedded in $T^\Bun_2$ as a fiber of the Hitchin fibration. Thus, given $F\in\oJ$, we have a conjectural construction of a sheaf on $\oJ$. This is the construction of $\oP_F$ provided by Lemma~\ref{lm:QviaFlag'}, with surface $S$ being $T^X$. From this point of view, the general formula \eqref{eq:Q} is obtained by extending Lemma~\ref{lm:QviaFlag'} from $\Hilb'_S$ to $\Hilb_S$. Presumably, it is the classical limit of the formula for automorphic sheaves for $\GL(n)$ suggested by G.~Laumon (\cite{Lau1,Lau2}) and proved by E.~Frenkel, D.~Gaitsgory, and K.~Vilonen (\cite{FGV,Vanishing}). Although \eqref{eq:Q} is inspired by results in the area of geometric Langlands conjecture, it is not clear whether the proofs from \cite{Dr,FGV,Vanishing} can be adapted to our settings. The argument of this paper is based on different ideas. \section{Proof of Theorem~\ref{th:oP}}\label{sc:proof} It remains to prove Proposition~\ref{pp:FCM}. \begin{proof}[Proof of Proposition~\ref{pp:FCM}] The morphism $\sigma:\IS_S\to S^n$ has finite Tor-dimension, because $S^n$ is smooth. The subvariety $C^n\subset S^n$ is locally a complete intersection of codimension $n$. By Corollary~\ref{co:HilbS0}, its preimage $\sigma^{-1}(C^n)\subset\IS_S$ also has codimension $n$. In other words, $C^n\subset S^n$ is locally cut out by a length $n$ regular sequence of functions on $S^n$, and the pull-back of this regular sequence to $\IS_S$ remains regular. This implies that the restriction $\sigma:\sigma^{-1}(C^n)\to C^n$ has finite Tor-dimension. Now \eqref{pp:FCM1} follows from Lemma~\ref{lm:CMAcyclic}\eqref{lm:CMAcyclic1}. Recall that $\IS_S$ is Cohen-Macaulay (it is finite and flat over $\Hilb_S$, which is smooth). As we saw, $\sigma^{-1}(C^n)\subset\Flag'_S$ is locally a complete intersection. Therefore, $\sigma^{-1}(C^n)$ is also Cohen-Macaulay. Lemma~\ref{lm:CMAcyclic}\eqref{lm:CMAcyclic2} implies \eqref{pp:FCM2}. Let us prove \eqref{pp:FCM3}. Since $\psi$ is finite and flat, it follows that $\psi_(\sigma^(\iota_ F)^{\boxtimes n})$ is a Cohen-Macaulay sheaf of codimension $n$. The same is true for its direct summand \[M:=\psi_(\sigma^(\iota_* F)^{\boxtimes n})^{sign}.\] Clearly, $\supp(M)\subset \psi(\sigma^{-1}(C^n))$, which is a reducible scheme of dimension $n$. One of its irreducible components is $\Hilb_C$, and we need to show that $M$ is supported by this component. Let us first verify this on the level of sets. Set \[Z:=\Hilb'_S\cap \psi(\sigma^{-1}((C^{sm})^n)).\] Corollary~\ref{co:HilbS0} implies that $\dim(\psi(\sigma^{-1}(C^n))-Z)<n$. Since $M$ is Cohen-Macaulay, it suffices to check that \[\supp(M\|_Z)\subset\Hilb_C\cap Z.\] But this follows from Proposition~\ref{pp:QandQ'}. Indeed, for $D\in Z$, we have \[H^0(D,\iota_* F)\simeq H^0(D,\iota_* O_C),\] because $F$ and $O_C$ are isomorphic in a neighborhood of $D$. Therefore, if $D\not\subset C$, we have $\dim H^0(D,\iota_* F)<n$, and $M_D=0$ by Proposition~\ref{pp:QandQ'}. Thus, $M$ is supported by $\Hilb_C$ in the set-theoretic sense. Note that $\psi(\sigma^{-1}(C^n))$ is reduced at the generic point of $\Hilb_C$ (in fact, it contains $\Hilb''_C$ as an open set). Since $M$ is Cohen-Macaulay, we see that its support is reduced, as required. \end{proof} \begin{remarks} \begin{myenum} \item Suppose that $S$ admits a symplectic form (in fact, Proposition~\ref{pp:FCM} is local on $S$, so we can make this assumption without losing generality). Fix a symplectic form on $S$; it is well known that it induces a symplectic form on $\Hilb_S$. One can check that the image $\psi(\sigma^{-1}(C^n))\subset\Hilb_S$ is Lagrangian. (Here one can assume that $C$ is smooth, in which case the observation is due to I.~Grojnowski \cite[Proposition~3]{Groj}.) This provides a conceptual explanation why its dimension equals $n$. \item The irreducible components of $\psi(\sigma^{-1}(C^n))$ have the following description (also contained in \cite{Groj}). Consider the `diagonal stratification' of $C^n$. The strata are indexed by the set $\Sigma$ of all equivalence relations $\sim$ on the set $\{1,\dots,n\}$ (in other words, $\Sigma$ is the set of partitions of the set $\{1,\dots,n\}$ into disjoint subsets). Given $\sim\in\Sigma$, the corresponding stratum is \[ C^n_\sim:=\{(x_1,\dots,x_n)\in C^n\st x_i=x_j \text{ if and only if }i\sim j\}.\] In particular, the open stratum $C^n_=$ corresponds to the usual equality relation $=\in\Sigma$. For every $\sim\in\Sigma$, the preimage $\sigma^{-1}(C^n_\sim)$ is irreducible of dimension $n$ (see Corollary~\ref{co:HilbS0}). The irreducible components of $\psi(\sigma^{-1}(C^n))$ are of the form $\overline{\psi(\sigma^{-1}(C^n_\sim))}$; they are indexed by $\Sigma/S_n$ (which is the set of partitions of $n$). \item Let us keep the notation of the previous remark. It is not hard to check that at the generic point of the component corresponding to $\sim\in\Sigma$, the fiber of $\psi_\sigma^(\iota_(F)^{\boxtimes n})$ is isomorphic to the space of functions on $S_n/S_\sim$ as a $S_n$-module. Here \[S_\sim:=\{\tau\in S_n\st \tau(i)\sim i\text{ for all }i\}\subset S_n\] is the subgroup given by the partition $\sim$. In particular, if $\sim$ is not discrete, the generic fiber has no anti-invariants under the action of $S_n$. This provides another explanation of Proposition~\ref{pp:FCM}\eqref{pp:FCM3}. \end{myenum} \end{remarks} \section{Properties of the Poincar\'e sheaf}\label{sc:properties} Consider the Poincar\'e sheaf $\oP$ on $\oJ\times\oJ$ provided by Theorem~\ref{th:oP}. \subsection{$\oP$ as an extension} \begin{lemma}\label{lm:oPCM} Let $j:J\times\oJ\cup\oJ\times J\hookrightarrow\oJ\times\oJ$ be the open embedding. \begin{enumerate} \item\label{lm:oPCM1} $\oP$ is a maximal Cohen-Macaulay sheaf on $\oJ\times\oJ$; \item\label{lm:oPCM2} $\oP=j_P$; \item\label{lm:oPCM3} $\oP$ is equivariant with respect to the permutation of the factors $p_{21}:\oJ\times\oJ\to\oJ\times\oJ$. \end{enumerate} \end{lemma} \begin{proof} \eqref{lm:oPCM1} follows from Lemma~\ref{lm:CMFamily}. Now Lemma~\ref{lm:CMExt} implies \eqref{lm:oPCM2}. Finally, \eqref{lm:oPCM3} is also clear, because $P$ is equivariant under $p_{21}$. \end{proof} In particular, Theorem~\ref{th:oP}\eqref{th:oP3} and Lemma~\ref{lm:oPCM}\eqref{lm:oPCM3} imply that $\oP$ is flat with respect to both projections $\oJ\times\oJ\to\oJ$, and that for every $F\in\oJ$, the restrictions $\oP\|_{\{F\}\times\oJ}$ and $\oP\|_{\oJ\times\{F\}}$ give the same sheaf on $\oJ$, which we denoted by $\oP_F$. \subsection{$\oP$ and duality} Consider now the duality involution \[\nu:\oJ\to\oJ:F\to F^\vee:=\HOM(F,O_C).\] Note that $\nu$ is an algebraic map by Lemma~\ref{lm:CMFamily}. \begin{lemma}\label{lm:oPvee} \begin{enumerate} \item $(\nu\times\id_\oJ)^\oP=(\id_\oJ\times\nu)^\oP=\oP^\vee$; \item $(\nu\times\nu)^\oP=\oP$. \end{enumerate} \end{lemma} \begin{proof} By Lemma~\ref{lm:oPCM}, all of the sheaves in the statement are maximal Cohen-Macaulay. It remains to notice that over $(J\times\oJ\cup\oJ\times J)\subset\oJ\times\oJ$ the required identifications are clear from the definition of $\oP$. \end{proof} \begin{corollary} Let ${\mathfrak F}:D^b(\oJ)\to D^b(\oJ)$ be the Fourier-Mukai functor of Theorem~\ref{th:FM}. Its restriction to $D^b_{coh}(\oJ)$ satisfies \[{\mathfrak F}\circ\D\simeq(\nu^\circ\D\circ{\mathfrak F})[-g].\] \end{corollary} \begin{proof} By Serre's duality, the functor $\D\circ{\mathfrak F}\circ\D$ is given by \[D^b_{coh}(\oJ)\to D^b_{coh}(\oJ):\cG\mapsto Rp_{1,}(p_2^(\cG)\otimes\oP^\vee)\otimes\omega_\oJ[-g].\] Now Lemma~\ref{lm:oPvee} implies the statement. \end{proof} \subsection{Theorem of the Square} Consider the universal Abel-Jacobi map \[A:J\times C\to\oJ:(L,c)\mapsto L(c-p_0):=\HOM(\cI_{c},L(-p_0)).\] (Recall that $p_0\in C$ is a fixed smooth point.) It is easy to see that $\oP$ agrees with it in the following sense: \begin{lemma} \label{lm:AJ} Consider the diagram \[ \xymatrix{J\times\oJ& J\times C\times\oJ\ar[r]^-{p_{23}}\ar[l]_-{p_{13}}\ar[d]^-{A\times\id_\oJ}& C\times\oJ\\ &\oJ\times\oJ.} \] We have $(A\times\id_\oJ)^\oP=p_{23}^(\cF)\otimes p_{13}^P$. Recall that $\cF$ is the universal sheaf on $C\times\oJ$. \end{lemma} \begin{proof} Both sides are maximal Cohen-Macaulay, and their restrictions to $J\times(C^{sm}\times\oJ\cup C\times J)$ are identified. Here $C^{sm}\subset C$ is the smooth locus of $C$. \end{proof} \begin{remark} Set $Y:=(\oJ\times C^{sm}\cap J\times C)\subset\oJ\times C$. The map $A:J\times C\to\oJ$ extends to a regular map $Y\to\oJ$. Lemma~\ref{lm:AJ} remains true for this extension: it provides an isomorphism of sheaves on $Y\times\oJ$. \end{remark} Similarly, we check that $\oP$ satisfies the Theorem of the Square. Namely, consider the action \[\mu:J\times\oJ\to\oJ:(L,F)\to L\otimes F.\] Clearly, $\mu$ is a smooth algebraic morphism. \begin{lemma} \label{lm:Square} Consider the diagram \[ \xymatrix{J\times\oJ& J\times\oJ\times\oJ\ar[r]^-{p_{23}}\ar[l]_-{p_{13}}\ar[d]^-{\mu\times\id_\oJ}&\oJ\times\oJ\\ &\oJ\times\oJ.} \] We have $(\mu\times\id_\oJ)^\oP=p_{13}^(P)\otimes p_{23}^\oP$. \end{lemma} \begin{proof} Both sides are maximal Cohen-Macaulay, and their restrictions to $J\times(J\times\oJ\cup\oJ\times J)$ are identified. \end{proof} \begin{remark} Lemmas~\ref{lm:AJ} and \ref{lm:Square} are contained, in a form, in \cite{compactified}. Namely, Lemma~\ref{lm:AJ} is contained in the proof of \cite[Theorem~2.6]{compactified}, while Lemma~\ref{lm:Square} is equivalent to \cite[Proposition~2.5]{compactified}. Both statements are formulated for curves with double singularities, but this assumption can be removed using \cite[Theorem~C]{Ar}. \end{remark} \subsection{Hecke eigenproperty} Let us also state a less obvious property of $\oP$, which is motivated by the Langlands program. Essentially, we claim that $\oP$ is an `eigenobject' with respect to natural `Hecke endofunctors'. The proof of the property will be given elsewhere; it is not used in this paper. Let $\Heck$ be the moduli space of collections $(F_1,F_2,f)$, where $F_1,F_2\in\oJ$ and $f$ is a non-zero map $F_1\hookrightarrow F_2(p_0)$, defined up to scaling. Informally, $F_2(p_0)$ is an \emph{elementary upper modification} of $F_1$ at the point $\supp(\coker(f))\in C$. The space $\Heck$ is equipped with maps $\phi_1,\phi_2:\Heck\to\oJ$ and $\gamma:\Heck\to C$ that send $(F_1,F_2,f)$ to $F_1\in\oJ$, $F_2\in\oJ$, and $\supp(\coker(f))$ respectively. The Hecke eigenproperty claims that for every $F\in\oJ$, we have \[R(\phi_1,\gamma)_\phi_2^(\oP_F)\simeq\oP_F\boxtimes F.\] We also have a `unversal' Hecke property for the sheaf $\oP$; its precise statement is left to the reader. The Hecke eigenproperty generalizes Lemma~\ref{lm:AJ}. \section{Fourier-Mukai transform}\label{sc:FM} Set \[\Psi:=Rp_{13}(p_{12}^\oP^\vee\otimes p_{23}^\oP)\in D^b(\oJ\times\oJ).\] Our goal is to prove \begin{proposition} \label{pp:FM} $\Psi\simeq O_\Delta[-g]\otimes_\kkk\det H^1(C,O_C)$. \end{proposition} Proposition~\ref{pp:FM} implies Theorem~\ref{th:FM} by the argument completely analogous to the proof of \cite[Theorem~2.2]{Mukai}. The proof of Proposition~\ref{pp:FM} follows the same pattern as the proof of \cite[Theorem~A]{Ar}. \subsection{Upper bound on the support of $\Psi$} Denote by $\tilde g$ the geometric genus of $C$. For any $F\in\oJ$, consider the restriction $P_F=(\oP_F)\|_J$. It is a line bundle on $J$. \begin{proposition}[{cf. \cite[Proposition~1]{Ar}}] \label{pp:1} If $(F_1,F_2)\in\supp(\Psi)$, then $P_{F_1}\simeq P_{F_2}$. \end{proposition} \begin{proof} By base change, $(F_1,F_2)\in\supp(\Psi)$ if and only if $\HH^i(\oJ,\oP_{F_1}\otimes^L\oP_{F_2}^\vee)\ne 0$ for some $i$. (Note that the hypercohomology may be non-zero even if $i$ is negative.) Let $T_i\to J$ be the $\gm$-torsor corresponding to $P_{F_i}$ for $i=1,2$. From definition \eqref{eq:Poincare}, we see that $T_i$ is naturally an abelian group which is an extension of $J$ by $\gm$. Lemma~\ref{lm:Square} implies that the action of $J$ on $\oJ$ lifts to an action of $T_i$ on $P_{F_i}$. Now let $T$ be the $\gm$-torsor corresponding to $P_{F_1}\otimes P_{F_2}^\vee$. It is also an extension of $J$ by $\gm$ (the difference of extensions $T_1$ and $T_2$). It follows that $T$ acts on $\HH^i(\oJ,\oP_{F_1}\otimes^L\oP_{F_2}^\vee)$ for all $i$, with $\gm\subset T$ acting via the tautological character. Finally, if $\HH^i(\oJ,\oP_{F_1}\otimes^L\oP_{F_2}^\vee)\ne 0$, it contains an irreducible $T$ submodule $V$. Since $T$ is abelian, $\dim(V)=1$, and $T$ acts by a character $\chi:T\to\gm$. Then $\chi$ provides a splitting $T\simeq J\times\gm$. This implies the statement. \end{proof} \begin{corollary}\label{co:2} If $(F_1,F_2)\in\supp(\Psi)$, then $F_1\|_{C^{sm}}\simeq F_2\|_{C^{sm}}$. \end{corollary} \begin{proof} Same proof as \cite[Corollary~2]{Ar}: pull back the isomorphism of Proposition~\ref{pp:1} by the Abel-Jacobi map $C\to\oJ$. \end{proof} \begin{proposition}[{cf. \cite[Corollary~3]{Ar}}]\label{pp:3} Consider the map $\mu\times\id_\oJ: J\times\oJ\times\oJ\to\oJ$. For any $F_1,F_2\in\oJ$, the intersection \[Z:=(\mu\times\id_\oJ)^{-1}(\supp(\Psi))\cap J\times\{F_1\}\times\{F_2\}\] is of dimension at most $g-\tilde g$. \end{proposition} \begin{proof} By Corollary~\ref{co:2}, if $(L,F_1,F_2)\in Z$, then \begin{equation}\label{eq:Csm} (L\otimes F_1)\|_{C^{sm}}\simeq F_2^\vee\|_{C^{sm}}. \end{equation} Looking at normalization of $C$, it is easy to see that the set of $L$ satisfying \eqref{eq:Csm} is a countable union of subvarieties of dimension $g-\tilde g$. (In particular, it does not contain generic points of subschemes of higher dimension.) \end{proof} \begin{corollary}\label{co:bound} Suppose $C$ is singular, so $\tilde g<g$. Then $\dim(\supp(\Psi))<2g-\tilde g$. \end{corollary} \begin{proof} Since $\mu:J\to\oJ\to\oJ$ is smooth of relative dimension $g$, it suffices to show that \[\dim(\mu\times\id_\oJ)^{-1}(\supp(\Psi))<3g-\tilde g.\] Proposition~\ref{pp:3} implies that the fibers of the projection $(\mu\times\id_\oJ)^{-1}(\supp(\Psi))\to\oJ\times\oJ$ have dimension at most $g-\tilde g$, while \cite[Theorem~A]{Ar} shows that over $J\times J$, the fibers are zero-dimensional. \end{proof} \subsection{Proof of Theorem~\ref{th:FM}} \begin{lemma}\label{lm:Serre} Let $X$ be a scheme of pure dimension. Suppose $\cG\in D^b_{coh}(X)$ satisfies the following conditions: \begin{enumerate} \item\label{lm:Serre1} $\codim(\supp(\cG))\ge d$; \item\label{lm:Serre2} $H^i(\cG)=0$ for $i>0$; \item\label{lm:Serre3} $H^i(\D\cG)=0$ for $i>d$. \end{enumerate} Then $\cG$ is a Cohen-Macaulay sheaf of codimension $d$. \end{lemma} \begin{proof} The proof is naturally given in the language of perverse coherent sheaves (\cite{AB}). Indeed, \eqref{lm:Serre3} claims that $\D\cG\in D^{p,\le0}(X)$ for perversity $p:\|X\|\to\Z$ given by $p(x)=d$. Here $\|X\|$ is the set of (not necessarily closed) points of $X$. Therefore, $\cG\in D^{\overline{p},\ge0}(X)$ for the dual perversity \[\overline{p}:\|X\|\to\Z:p(x)=\codim(\overline{\{x\}})-d.\] Now \eqref{lm:Serre1} implies that $\cG\in D^{\ge 0}(X)$, and \eqref{lm:Serre2} implies that $\cG$ is a sheaf. Since $\D\cG[d]$ also satisfies conditions \eqref{lm:Serre1}--\eqref{lm:Serre3}, it is also a sheaf, and therefore $\cG$ is Cohen-Macaulay of codimension $d$. \end{proof} Note that the statement of Lemma~\ref{lm:Serre} is local in smooth topology on $X$; therefore, the lemma still holds if $X$ is an algebraic stacks (locally of finite type over $\kkk$). \begin{proof}[Proof of Proposition~\ref{pp:FM}] Both $\oP$ and $\Psi$ are defined for families of curves with plane singularities. Let us consider the universal family. Let $\cM$ be the moduli stack of (reduced irreducible projective) curves of fixed arithmetic genus $g$ with plane singularities. Let $\cC\to\cM$ be the universal curve, and let $\overline\cJ$ (resp. $\cJ\subset\overline\cJ$) be the relative compactified Jacobian (resp. the relative Jacobian) of $\cC$ over $\cM$. The properties of these objects are summarized in \cite{Ar}. As $C\in\cM$ varies, the family of Poincar\'e sheaves gives a Cohen-Macaulay sheaf $\oP_{univ}$ on $\overline\cJ\times_\cM\overline\cJ$; similarly, $\Psi$ is naturally defined as an object of the derived category \[\Psi_{univ}\in D^b(\overline\cJ\times_\cM\overline\cJ).\] The restriction of $\Psi_{univ}$ to the fiber over a particular curve $C\in\cM$ is $\Psi$. Denote by $\frj$ the rank $g$ vector bundle on $\cM$ whose fiber over $C\in\cM$ is $H^1(C,O_C)$. Alternatively, $\frj$ can be viewed as the bundle of (commutative) Lie algebras corresponding to the group scheme $\cJ\to\cM$. Denote the projection $\overline\cJ\times_\cM\overline\cJ\to\cM$ by $\pi$ and diagonal in $\overline\cJ\times_\cM\overline\cJ$ by $\Delta$. Our goal is to prove that \begin{equation}\label{eq:Psi} \Psi_{univ}[g]\simeq O_\Delta\otimes\pi^\det(\frj). \end{equation} Consider on $\cM$ the stratification $\cM^{(\tilde g)}$, where $\cM^{(\tilde g)}$ parametrizes curves of geometric genus $\tilde g$. By \cite[Proposition~6]{Ar}, $\codim(\cM^{(\tilde g)})\ge g-\tilde g$. Now Corollary~\ref{co:bound} implies that $\dim(\supp(\Psi))=\dim(\cM)+g$, and moreover, every maximal-dimensional component of $\supp(\Psi)$ meets $\pi^{-1}(\cM^{(g)})$. Since $\cM^{(g)}$ parametrizes smooth curves, \[\supp(\Psi)\cap\pi^{-1}(\cM^{(g)})=\Delta\cap\pi^{-1}(\cM^{(g)}).\] This is a reformulation of Mumford's result \cite[Section II.8.(vii)]{Mumford}. Finally, $H^i(\Psi_{univ})=0$ for $i>g$ and Serre's duality implies that $H^i(\D\Psi_{univ})=0$ for $i>0$. (For instance, we have \[\D\Psi\simeq p_{21}^*\Psi[g]\] over a fixed curve $C\in\cM$.) Thus, Lemma~\ref{lm:Serre} shows that $\Psi_{univ}[g]$ is a Cohen-Macaulay sheaf of codimension $g$. Outside of a set of codimension $g+1$, we see that $\supp(\Psi_{univ})$ coincides with $\Delta$, therefore, \[\supp(\Psi_{univ})\subset\Delta.\] Also, \cite[Theorem~10]{Ar} provides the required isomorphism \eqref{eq:Psi} over $\cJ\times_\cM\cJ$. Thus both sides of \eqref{eq:Psi} are Cohen-Macaulay sheaves on $\Delta$ that are isomorphic outside of a subset of codimension two. By Lemma~\ref{lm:CMExt}, they are isomorphic. \end{proof} As we have seen, Proposition~\ref{pp:FM} implies Theorem~\ref{th:FM}. \subsection{Autoduality of the compactified Jacobian} It remains to prove Theorem~\ref{th:autoduality}. As we already mentioned, the first statement follows from Theorem~\ref{th:oP} and the results of \cite{compactified}. On the other hand, both statements easily follow from Theorem~\ref{th:FM}. Note that $\Pic(\oJ)^=$ is not claimed to be a fine moduli space; that is, there may be no universal family of torsion-free sheaves on $\oJ$ of generic rank one parametrized by $\Pic(\oJ)^=$. However, locally in the \'etale topology of $\Pic(\oJ)^=$, such a family exists and is unique (\cite[Theorem~3.1]{CP2}). In particular, points of $\Pic(\oJ)^=$ are in bijection with isomorphism classes of torsion-free sheaves on $\oJ$ of generic rank one. We will make no distinction between these two objects, so that $M\in\Pic(\oJ)^=$ means ``$M$ is a torsion-free sheaf on $\oJ$ of generic rank one, defined up to non-canonical isomorphism''. \begin{proof}[Proof of Theorem~\ref{th:autoduality}] Given $M\in \Pic(\oJ)^=$, consider its Fourier-Mukai transform $\FF(M)$. Its cohomology sheaves are concentrated in degrees between $0$ and $g$. Fix an ample line bundle on $\oJ$, and denote by $h^i(\FF(M))\in\Q[t]$ the Hilbert polynomial of the cohomology sheaf $H^i(\FF(M))$ for $0\le i\le g$. Proposition~\ref{pp:FM} implies that if $M=\rho(F)=\oP_F$ for $F\in\oJ$, then $\FF(M)\simeq O_{F^\vee}[-g]$. Here $O_{F^\vee}$ is the structure sheaf of the point $F^\vee\in\oJ$. In particular, \begin{equation}\label{eq:FMvanishing} h^i(\FF(M))=\begin{cases}1,& i=g\\0,& i\ne g\end{cases}. \end{equation} On the other hand, consider $h^i(\FF(M))$ as functions of $M\in\Pic(\oJ)^=$. They are semicontinuous with respect to the order on $\Q[t]$ given by \[f>g\quad\text{if}\quad f(t)>g(t)\quad\text{for $t\gg0$}\quad(f,g\in\Q[t]).\] Therefore, \eqref{eq:FMvanishing} also holds for $M$ in a neighborhood of $\rho(\oJ)\subset\Pic(\oJ)^=$. However, if $M$ satisfies \eqref{eq:FMvanishing}, then $\FF(M)\simeq O_F[-g]$ for some point $F\in\oJ$. Therefore, $\FF(M)\simeq\FF(\oP_{F^\vee})$, and Theorem~\ref{th:FM} implies $M\simeq\oP_{F^\vee}$. Hence $\rho(\oJ)$ is a connected component of $\Pic(\oJ)^=$. We also see that the inverse of the map $\rho:\oJ\to\rho(\oJ)$ is given by \[M\mapsto\nu(\supp(\FF(M))):\rho(\oJ)\to\oJ.\] Clearly, this is an algebraic map. This completes the proof. \end{proof} \end{document} \end{document}	arXiv
Moufang plane In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a translation line, that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line.[1] A translation plane is Moufang if every line of the plane is a translation line.[2] Characterizations A Moufang plane can also be described as a projective plane in which the little Desargues theorem holds.[3] This theorem states that a restricted form of Desargues' theorem holds for every line in the plane.[4] For example, every Desarguesian plane is a Moufang plane.[5] In algebraic terms, a projective plane over any alternative division ring is a Moufang plane,[6] and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and of Moufang planes. As a consequence of the algebraic Artin–Zorn theorem, that every finite alternative division ring is a field, every finite Moufang plane is Desarguesian, but some infinite Moufang planes are non-Desarguesian planes. In particular, the Cayley plane, an infinite Moufang projective plane over the octonions, is one of these because the octonions do not form a division ring.[7] Properties The following conditions on a projective plane P are equivalent:[8] • P is a Moufang plane. • The group of automorphisms fixing all points of any given line acts transitively on the points not on the line. • Some ternary ring of the plane is an alternative division ring. • P is isomorphic to the projective plane over an alternative division ring. Also, in a Moufang plane: • The group of automorphisms acts transitively on quadrangles.[9][10] • Any two ternary rings of the plane are isomorphic. See also • Moufang loop • Moufang polygon Notes 1. That is, the group acts transitively on the affine plane formed by removing this line and all its points from the projective plane. 2. Hughes & Piper 1973, p. 101 3. Pickert 1975, p. 186 4. This restricted version states that if two triangles are perspective from a point on a given line, and two pairs of corresponding sides also meet on this line, then the third pair of corresponding sides meet on the line as well. 5. Hughes & Piper 1973, p. 153 6. Hughes & Piper 1973, p. 139 7. Weibel, Charles (2007), "Survey of Non-Desarguesian Planes", Notices of the AMS, 54 (10): 1294–1303 8. H. Klein Moufang planes 9. Stevenson 1972, p. 392 Stevenson refers to Moufang planes as alternative planes. 10. If transitive is replaced by sharply transitive, the plane is pappian. References • Hughes, Daniel R.; Piper, Fred C. (1973), Projective Planes, Springer-Verlag, ISBN 0-387-90044-6 • Pickert, Günter (1975), Projektive Ebenen (Zweite Auflage ed.), Springer-Verlag, ISBN 0-387-07280-2 • Stevenson, Frederick W. (1972), Projective Planes, W.H. Freeman & Co., ISBN 0-7167-0443-9 Further reading • Tits, Jacques; Weiss, Richard M. (2002), Moufang polygons, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-43714-7, MR 1938841	Wikipedia
1Tax credits for conservation 2Gas Guzzler Tax 2.1Economic impact 2.2Market impact Energy Tax Act The Energy Tax Act (Pub. L. 95–618, 92 Stat. 3174, enacted November 9, 1978) is a law passed by the U.S. Congress as part of the National Energy Act. The objective of this law was to shift from oil and gas supply toward energy conservation; thus, to promote fuel efficiency and renewable energy through taxes and tax credits.[1] Tax credits for conservation[edit] This law gave an income tax credit to private residents who use solar, wind, or geothermal sources of energy. The credit is equal to 30% of the cost of the equipment up to $2000, as well as 20% of costs greater than $2000, up to a maximum of $10,000. There were also tax credits to businesses for renewable energy equipment, amounting to a maximum of 25% of the cost of the equipment.[2] The renewable energy credits of this law were increased by the Crude Oil Windfall Profits Tax Act of 1980. Gas Guzzler Tax[edit] See also: Gas-guzzler The Act also created the Gas Guzzler Tax[3] which applies to the sales of vehicles with official EPA-estimated gas mileage below certain specified levels. In 1980, the tax was $200 for a fuel efficiency of 14 to 15 miles per gallon, and was increased to $1800 in 1985. In 1980, the tax was $550 for fuel efficiencies of 13 mpg and below, and was changed in 1986 to $3,850 for ratings below 12.5 mpg. The Gas Guzzler Tax applies only to passenger cars. Trucks, sport utility vehicles (SUV), and minivans are not covered because these vehicle types were not widely available in 1978 and were rarely used for non-commercial purposes. The tax is collected by the Internal Revenue Service (IRS) and normally paid by the manufacturer or importer. The following chart shows the current tax for various levels of MPG that have been in effect since January 1, 1991. Unadjusted MPG (combined) at least 22.5 No tax at least 21.5, but less than 22.5 $1000 less than 12.5 $7700 The combined fuel economy MPG value (55% city, 45% highway) is used to determine tax liability. The MPG value is also adjusted slightly to account for differences in test procedures made since the base year, but it is not adjusted for in-use short fall. The unadjusted combined MPG of a vehicle can be approximated from the city and highway values provided in the Fuel Economy Guide by the following equation[citation needed]: U n a d j u s t e d M P G ( c o m b i n e d ) = 1 .495 C i t y M P G + .351 H i g h w a y M P G + .15 {\displaystyle \mathrm {UnadjustedMPG(combined)} ={\frac {\mathrm {1} }{\mathrm {{\frac {\mathrm {.495} }{\mathrm {CityMPG} }}+{\frac {\mathrm {.351} }{\mathrm {HighwayMPG} }}} }}+.15} Since this is an approximate calculation, the actual gas guzzler tax may be off by one tax bracket. We can then find out how much penalty, p t {\displaystyle {\mathit {p_{t}}}} , the manufacturer has to pay for that particular vehicle by using the following equation. p i {\displaystyle {\mathit {p_{i}}}} needs to be looked up on the table above[clarification needed] and q i {\displaystyle {\mathit {q_{i}}}} is the numbers of cars that are found to be under the set Gas Guzzler standard, p t = ∑ i p i q i {\displaystyle p_{t}=\sum _{i}{p_{i}q_{i}}\,} Economic impact[edit] Gas guzzler tax creates incentive to meet the minimum MPG requirement by manufacturer. Due to elimination of vehicles that are below minimum MPG which is 22.5 MPG, vehicle sales have decreased approximately 0.5 percent. However, sales revenues increase by a greater amount due to the added value in vehicles making greater use of fuel economy technology.[4] Currently, the additional cost of efficient hybrid systems can range from $2,000-$10,000 on the vehicle sticker price.[5] Manufacturers benefit from the increase in price of products. However, the fuel sector may lose revenue if the increase in sales and production of fuel efficient vehicles doesn't just encourage people to drive more. Market impact[edit] The Gas Guzzler Tax led to the successive downsizing of most major American passenger autos, and the combination of the tax and late-'70s/early-'80s economic woes effectively killed the American full-size car as it had been known up to that point. Coincidentally, it only took one product cycle before the first modern SUVs were introduced, the Cherokee XJ and the S-10 Blazer (in 1984). By the time Ford introduced the Explorer, the SUV had become the common man's luxury vehicle and Ford capitalized on this using extensive cross-marketing, most notably with Northwest clothier Eddie Bauer. Critics of the Gas Guzzler Tax contend that the increased fuel economy of the US passenger car fleet observed since 1978 must be considered in the context of the increased market share of mid-size and full-size SUVs.[6] Many consumers' stated reasons for SUV purchase (comfort, interior room, and a perception of safety based on the vehicle's size) also apply to the now-obsolete American full-size car as produced from the 1920s through the 70s; critics contend that the dominance of the modern SUV is a direct result of the Gas Guzzler Tax, which could have applied to all consumer vehicles but does not. Energy law - United States Vehicle Efficiency Initiative ^ Lazzari, Salvatore; "Energy Tax Policy", Congressional Research Service of The Library of Congress, Updated April 22, 2005; page 6 Archived May 31, 2007, at the Wayback Machine ^ Solar Energy, Photovoltaics, and Domestic Hot Water: A Technical and ..., by Russell H. Plante ^ fueleconomy.gov ^ David Green, Phillip Patterson, Margaret Singh, Jia Li (2005). "Feebates, Rebates, and Gas Guzzler Taxes: A study of incentives for increased fuel economy" (PDF). Energy Policy. {{cite web}}: CS1 maint: uses authors parameter (link) ^ Gordon Hard and Jake Fisher (Aug 10, 2011). "Why a gas-guzzler tax break makes total sense". ConsumerReports.org. Retrieved 7 April 2012. {{cite web}}: CS1 maint: uses authors parameter (link) ^ "Loophole in gas-guzzler tax saves SUV makers billions, study finds \| The Seattle Times". archive.seattletimes.com. Retrieved 2022-11-27. Energy tax credits, colby.edu Legislation affecting the renewable energy market, doe.gov Congressional energy brief, ncseonline.org FuelEconomy.gov Frequently Asked Questions Gas Guzzler Tax: Program Overview Environmental Protection Agency Feebates, rebates and gas-guzzler taxes: a study of incentives for increased fuel economy Gas Guzzler Tax Table Retrieved from "https://en.wikipedia.org/w/index.php?title=Energy_Tax_Act&oldid=1125243315" 1978 in law United States federal taxation legislation United States federal energy legislation 95th United States Congress Environmental tax CS1 maint: uses authors parameter Articles with unsourced statements from October 2010 Wikipedia articles needing clarification from June 2010 United States federal legislation articles without infoboxes	CommonCrawl
Power quality disturbance classification based on time-frequency domain multi-feature and decision tree Wenjing Zhao ORCID: orcid.org/0000-0002-0046-52091, Liqun Shang1 & Jinfan Sun2 Protection and Control of Modern Power Systems volume 4, Article number: 27 (2019) Cite this article Accurate classification of power quality disturbance is the premise and basis for improving and governing power quality. A method for power quality disturbance classification based on time-frequency domain multi-feature and decision tree is presented. Wavelet transform and S-transform are used to extract the feature quantity of each power quality disturbance signal, and a decision tree with classification rules is then constructed for classification and recognition based on the extracted feature quantity. The classification rules and decision tree classifier are established by combining the energy spectrum feature quantity extracted by wavelet transform and other seven time-frequency domain feature quantities extracted by S-transform. Simulation results show that the proposed method can effectively identify six types of common single disturbance signals and two mixed disturbance signals, with fast classification speed and adequate noise resistance. Its classification accuracy is also higher than those of support vector machine (SVM) and k-nearest neighbor (KNN) algorithms. Compared with the method that only uses S-transform, the proposed feature extraction method has more abundant features and higher classification accuracy for power quality disturbance. With the development of grid interconnection, grid-connection of new energy generation, extensive application of power electronic equipment and access of impact load, the problem of power quality disturbance has attracted more and more attention [1]. In-depth study of the power quality influencing factors, accurate extraction of feature quantities, and accurate classification of power quality disturbance are required for improving and controlling power quality [2]. The processes of power quality disturbance classification consist of feature extraction and classification recognition. The methods of feature extraction mainly include Fast Fourier transform (FFT), Short-time Fourier transform (STFT), wavelet transform, S-transform, Hilbert yellow transform (HHT), etc. FFT is a conversion from the time domain to the frequency domain, and has orthogonal and complete features. The frequency analysis of a signal is considered from the perspective of the overall composition of frequency, but the local frequency characteristics of the signal cannot be analyzed. Thus, it is only suitable for the analysis of steady-state disturbance [3]. STFT has fast computation speed and the algorithm is easy to implement. It can detect and analyze the signal's local spectrum features, but its window function is fixed with no ability of self-adaptation [4, 5]. Wavelet transform has the ability of multi-scale time-frequency resolution, which can be used for local analysis of signals, but signal analysis can be easily influenced by wavelet base and decomposition layer [6, 7]. S-transform is developed on the basis of wavelet transform and STFT. It not only overcomes their shortcomings, but also enables the analysis of amplitude change with time of a certain frequency component of the signal. Its window function changes with frequency, resulting in higher frequency resolution but also large amount of calculation [8, 9]. HHT is suitable for time-frequency analysis detecting methods of non-stationary and nonlinear signals, but is easy to generate modal aliasing during analysis [10]. At present, the main methods for power quality disturbance classification are artificial neural network, support vector machine (SVM), decision tree, K-neighbor (KNN), etc. Artificial neural network has long training time and is easy to fall into local optimal solution [11, 12] whereas SVM is susceptible to kernel function and cannot take into account both learning ability and generalization ability [13]. KNN classification requires large amount of computation and large memory [14], while the decision tree has the advantages of simple structure, convenient expansion and fast classification [15, 16]. Based on this, this paper mainly analyzes 6 types of single power quality disturbances, and the compound disturbances of swell + harmonic and sag + harmonic. Wavelet transform and S-transform are combined to extract more abundant feature quantities in time and frequency domains. According to the extracted feature quantities, the classification rules suitable for the 8 disturbance signals are established, and thus, accurate classification effect of disturbance signals can be obtained quickly by constructing decision tree classifier. Due to noise interference in actual power systems, the noise resistance of the proposed method is verified by adding gaussian white noise. The classification speed and accuracy of the proposed method are verified by simulation comparison. This paper classifies 8 different disturbances, including the standard voltage(C0), voltage swell(C1), voltage sag(C2), voltage interruption(C3), transient oscillation(C4), flicker(C5), harmonic(C6), swell + harmonic (C7) and sag + harmonic(C8). 2.1 Feature extraction based on wavelet transform and S-transform 2.1.1 Wavelet transform extracts energy feature quantity Wavelet transform is used for multi-scale decomposition of power quality disturbance signals. The obtained wavelet coefficients reflect the distribution of signals on different decomposition scales, and their differences after decomposition of different disturbance signals can be used to represent the signals' feature quantities. Due to the large amount of data of the wavelet coefficients as the feature quantities, the wavelet energy of different decomposition scales can be calculated through the wavelet coefficients to significantly reduce the feature dimension. Power quality disturbances are nonlinear mutation signals. In order to analyze the disturbances such as transient oscillations and harmonics, which mainly cause high frequency band mutation, and improve the operation speed of the wavelet transform, the wavelet basis function needs to satisfy the requirement of tight support, orthogonality, higher vanishing moments and calculation speed. The Daubechies (dbN) series wavelets in the basic wavelet have the above characteristics and are most commonly used in the Mallat algorithm. With the increase of N (the wavelet order), dbN wavelet in time domain increases the support interval of wavelet, while reduces the overlap of windows between different scales and the spectrum leakage between frequency bands. Higher-order db wavelets have higher vanishing moments, though larger N is not necessary better as the vanishing moments are opposite to the characteristics of tight support. Therefore, combined with the characteristics of power quality disturbance signals, db4 wavelet basis is selected in this paper for wavelet analysis. In order to reduce the spectrum leakage of wavelet transform, the main frequency components of transient components are distributed as far as possible in the center of the wavelet frequency band, and the disturbance signals are decomposed by 10 layers. The wavelet energy of each decomposition layer [17] is: $$ {E}_{Dj}={\sum}_{k=1}^N cdj{(k)}^2 $$ where, cdj is the detail coefficient of layer j, and N is the number of detail coefficient of layer j. The normalized value distribution of wavelet energy for the 8 disturbances is shown in Fig. 1. Comparison of wavelet energy distribution of different power quality disturbance signals It can be observed from Fig. 1 that the wavelet energy of each power quality disturbance signal is mainly concentrated in the 6th and 7th layers, while the 7th layer wavelet energy of harmonic, swell + harmonic and sag + harmonic is noticeably lower compared to the others. To avoid contingency, each disturbance randomly generates 300 samples that are superimposed by 30 dB of noise for testing. Finally, the 7th layer energy is set to the feature F1, and when the threshold value is 0.56, the harmonic and harmonic-containing disturbances are recognized with high precision. 2.2 S-transform extracts feature quantity The S-transform is a reversible time-frequency analysis method proposed by Stockwell. Because the height and width of window function vary with frequency, it has the advantages of both the WT and STFT, and thus is widely used. Continuous S-transformation is defined as: $$ S\left(\tau, f\right)={\int}_{-\infty}^{\infty }h(t)g\left(\tau -t,f\right){e}^{-j2\pi ft} dt $$ $$ g\left(\tau -t,f\right)=\frac{\mid f\mid }{\sqrt{2\pi }}e\frac{-{f}^2{\left(\tau -t\right)}^2}{2} $$ where h(t) is the disturbance signal, and g(τ − t, f) is the Gaussian window. In practical applications, the signal is obtained by sampling. Let the sampled signal be h[kT](k = 0,1,2…N − 1), where N is the number of sampling points and T is the sampling period, then the expression of discrete S-transformation is given as: $$ \Big\{{}_{S\left[ KT,0\right]=\frac{1}{N}{\sum}_{m=0}^{N-1}H\left(\frac{m}{NT}\right),n=0}^{S\left[ KT,\frac{n}{NT}\right]={\sum}_{m=0}^{N-1}H\left(\frac{m+n}{NT}\right)G\left(m,n\right){e}^{j\frac{2\pi mk}{N}},n\ne 0} $$ where j, m, n = 0,1,2... N − 1, H(n/NT) and G(m, n) are the FFT of signal H[kT] and Gaussian window respectively, And are given as: $$ \Big\{{}_{G\left(m,n\right)={e}^{-\frac{2{\pi}^2{m}^2}{n^2}}}^{H\left[ kT\right]=\frac{1}{N}{\sum}_{k=0}^{N-1}h\left[ kT\right]\kern0.28em {e}^{-\frac{j2\pi nk}{N}}} $$ The result of S-transformation is a two-dimensional complex matrix, which is modeled to obtain the modulus matrix. Its row vectors represent the change of amplitude of a certain frequency with time, and the column vectors represent the change of amplitude with frequency at a sampling moment. It reflects the time-frequency characteristics of the signal. If there is disturbance, it must be shown in the modulus matrix. According to the time-frequency matrix, the amplitude and frequency mutation of the disturbance can be detected. According to IEEE's relevant standards for power quality disturbances and the principles of their generation, it is concluded that swell, sag, interruption and flicker occur mainly in amplitude mutation, with the amplitude of flicker changing periodically. Harmonic and transient oscillations occur mainly in frequency mutation. S-transform has been carried out on standard signals and the 8 disturbances. Based on the differences in time, amplitude and frequency of each disturbance signal, the following characteristics are extracted. (1) The maximum value F2, minimum value F3 and standard deviation F4 of the maximum amplitude vector of time can be calculated as: $$ {F}_2=\max \left[{V}_{1t-A}(kT)\right] $$ $$ {F}_3=\min \left[{V}_{1t-A}(kT)\right] $$ $$ {F}_4=\sqrt{\frac{1}{N}{\sum}_{k=0}^{N-1}{\left[{V}_{1t-A}(kT)-{\overline{V}}_1\right]}^2} $$ where, V1t − A is the largest amplitude vector of time, k = 0,1,2... N − 1. (2) Standard deviation F5 of the maximum frequency amplitude vector of 100 − 600 Hz frequency band is calculated as: $$ {F}_5=\sqrt{\frac{1}{N-1}{\sum}_{f=100}^{600}{\left[{V}_{f-\max A}-\overline{V_m}\right]}^2} $$ where Vf − maxA is the maximum frequency amplitude vector, and $ \overline{V_m} $ is the average value of the maximum amplitude vector of the frequency band of 100 − 600 Hz. N is a sampling point in the range of 100 to 600 Hz. (3) Mean value F6 and standard deviation F7 of the maximum frequency amplitude vector in the 700 − 1500 Hz frequency band are calculated as: $$ {F}_6=\frac{fo}{\varDelta f}{\sum}_{f=700}^{1500}\left[{V}_{f-\max A}(f)\right] $$ $$ {F}_7=\sqrt{\frac{1}{N-1}{\sum}_{f=700}^{1500}{\left[{V}_{f-\max A}-\overline{V_h}\right]}^2} $$ where f0 refers to the frequency resolution of 5 Hz. The frequency range ∆f is 700 − 1500 Hz, and $ \overline{V_h} $ is the average value of the maximum amplitude vector of the frequency band of 700 − 1500 Hz, N is the sampling point in the 700-1500 Hz frequency band. (4) The fluctuation time of the maximum amplitude curve is F8. Amplitude varying from small to large or from large to small is regarded as a fluctuation. 2.3 The establish of decision tree classifier model Decision tree is a supervised learning algorithm, a kind of classifier similar to tree structure. The decision tree has the advantages of simple structure, convenient expansion and fast classification speed. It overcomes the disadvantages of SVM which is affected by kernel function, and is unable to give consideration to learning ability and generalization ability, and large computation and memory demand in KNN classification. Through classification rules to build the classification decision tree model. The structure of decision tree plays an important role in the accuracy of classification. In order to reduce the selection requirement of classification threshold and improve the classification accuracy, binary tree structure is adopted for classification. Decision tree is usually used to recursively select the best feature and segment of the training data according to the feature, so as to optimize the classification process of each sub-data set. This process corresponds to the division of feature space and the construction of decision tree classification rules. According to the feature extraction method in Section 2.1, 300 groups of random disturbance samples are generated and 30 dB noise superimposed. The feature quantities F1~F8 are calculated and statistically analyzed. The different disturbance types and feature quantities are compared in Table 1. Table 1 Analytical comparison of the disturbances and feature quantities According to Table 1, the optimal feature is selected recursively and the following 14 classification rules are established: (1) F1 > 0.56 and F5 < 0.01; (2) F1 < 0.56 and 0.01 < F5 < 0.06; (3) F4 > 0.002 and F7 < 0.02; (4) F4 < 0.002; (5) F4 < 0.002 and F7 < 0.02; (6) F4 > 0.002 and F8 < 2; (7) F2 > 1.1; (8) F2 < 1.1; (9) F6 < 0.045 or F7 < 0.02; (10) F6 > 0.045 and F7 > 0.02; (11) F8 < 2 and F3 > 0.9; (12) F8 ≥ 2 and F3 < 0.9; (13) 0.1 < F3 < 0.9; (14) F3 < 0.1. The corresponding power quality disturbance classification decision tree is constructed as shown in Fig. 2. Decision tree of power quality disturbance classification 2.4 Simulation experiment results and discussion The mathematical model for power quality disturbances described in [9] is considered. Random noise-free disturbance samples with parameters are generated through MATLAB simulation for testing (300 samples of C0 ~ C8 each). The sampling frequency is 6.4 kHz, the fundamental frequency is 50 Hz, the weekly wave sampling points are 128, and the data length is 1280 points. Wavelet transform is used to extract the normalized wavelet energy of the 7th layer, and S-transform is used to extract the feature quantities of the disturbances in time and frequency domains. The extracted feature variables are inputted into the constructed decision tree classifier to realize the recognition of the power quality disturbance signals. Considering that actual power system is affected by noise, 20 dB, 30 dB, 40 dB, 50 dB Gaussian white noises are superimposed onto the samples respectively to generate a total of 13,500 samples. To verify the effectiveness of the power quality disturbance classification method based on time-frequency domain multi-feature and decision tree, Table 2 compares the classification accuracy of the disturbance signals by using only S-transform (method 1) and the combination of wavelet transform and S-transform (method 2) to extract feature quantities under different noise conditions. Table 3 shows the classification effect of decision tree, SVM and KNN on disturbance signals under different noise conditions, whereas Table 4 shows the time required for classification of each detection algorithm under the condition of SNR = 30 dB. Table 2 Comparison of recognition results of different feature extraction methods Table 3 Comparison of recognition accuracy of different detection algorithms Table 4 Classification time comparison of different detection algorithms It can be seen from Table 2 that the classification effect of method 2 is better than that of method 1. The classification accuracies of both feature extraction methods decrease with the reduction of SNR, though the reduction of classification accuracy of method 1 is more significant than that of method 2. When SNR = 20 dB, the accuracy of method 2 is 97.1%, which is 4.5% higher than that of method 1. It indicates that the feature extraction method of wavelet transform + S-transform has better noise resistance and richer feature quantity than those of S-transform. As shown in Table 3, the classification accuracies of DT, SVM and KNN algorithms decrease with the increase of noise intensity. When SNR = 20 dB, the accuracy of DT is 5.29% higher than that of SVM and 1.03% higher than that of KNN. In addition, as can be seen from Table 4, DT classification is faster than the other two methods. For the various types of power quality disturbance, this paper proposes a power quality disturbance classification method based on time-frequency domain multi-feature and decision tree, for power quality improvement and governance. By combining the advantages of wavelet and S-transform, 8 time-frequency domain eigenvalues are extracted from 6 single disturbances and 2 compound disturbances. According to the extracted feature quantities, the classification rules of decision tree are established, and the decision tree model for classification is constructed. Simulation results show that the method is effective, and the extracted feature quantities can be effectively used for the classification, and classification of decision tree. Compared with only using S-transform, the proposed feature extraction method has richer feature quantities, higher classification accuracy and robustness to noise. For the feature quantities extracted in this paper, the classification accuracy of decision tree classifier is higher and the calculation speed faster than those of SVM and KNN. The example in this paper is based on MATLAB simulation platform. Further research will try to apply the proposed method to practical power quality disturbance classification. Additional types of power quality disturbance will be included and the classification method will be made more universal. The power quality disturbance samples in this paper are generated by MATLAB using the power quality disturbance mathematical model. Xiao, X.N. (2010). Analysis and control of power quality [M]. Beijing: China Electric Power Press, 124–128. Zhang, Y., & Liu., Z.G. (2012). A new power quality hybrid disturbance classification method based on time-frequency domain multi-characteristic quantities [J]. Proceedings of the CSEE, 32(34), 83–90. Zhang, B. (2010). Power quality analysis method based on Mallat algorithm and fast Fourier transform [C]. Power Quality Seminar, 35–40. Jurado, F., & Saenz, J. R. (2002). Comparison between discrete STFT and wavelets for the analysis of power quality events[J]. Electr Power Syst Res, 62(3), 183–190. Huang, J.M, Qu, H.Z, & Li, X.M. (2016). Classification of mixed disturbance of power quality based on short-time Fourier transform and spectral kurdiness [J]. Power System Technology, 40(10), 3184–3191. Qu, H.Z, Liu, H., Li, X.M, et al. (2017). A feature combination optimization method for multi-disturbance classification of power quality [J]. Electric Powcr Automation Equipment, 37(3), 146–152. Luciano C.M. Andrade, Mário Oleskovicz, & Ricardo A.S. Fernandes. (2016). Adaptive threshold based on wavelet transform applied to the segmentation of single and combined power quality disturbances[J]. Neurocomputing, 38, 967–977. Wu, Y., Tang, Q., Teng Z.S, et al. (2016). Power quality disturbance signal feature extraction method based on improved S transform [J]. Proceedings of the CSEE, 36(10), 2682–2689. Huang, N.T, Peng, H., Cai, G.W, et al. (2017). Composite disturbance feature selection and optimal decision tree construction of power quality [J]. Proceedings of the CSEE, 37(3), 776–786. LI, X.N. (2017). Power quality disturbance detection and recognition based on hilbert-huang transformation [D], China University of Mining and Technology. Anamika Yadav, Yajnaseni Dash & V. Ashok. (2016). ANN based directional relaying scheme for protection of Korba-Bhilai transmission line of Chhattisgarh state[J]. Protection and Control of Modern Power Systems, 1(1), 15. He, J.L, Wang, G.P, Liu, D., et al. (2017). Location and identification of power quality disturbance in distribution network system based on lifting wavelet and improving BP neural network [J]. Power System Protection and Control, 45(10), 69–76. Ren, Z.H, & Wang, Q. (2008). Power quality disturbance identification based on optimal DDAGSVM multi-class classification strategy [J]. Power System Protection and Control, 46(5), 82–88. Panigrahi B.K., Pandi V.R. optimal feature selection for classification of power quality disturbances using wavelet packet-based fuzzy k-nearest neighbour algorithm[J]. Neurocomputing, 2009, 3(3): 296–306. Biswal, M., & Dash, P.K. (2013). Measurement and classification of simultaneous power signal patterns with an S-transform variant and fuzzy decision tree[J]. IEEE Transactions on Industrial Informatics, 9(4), 1819–1827. Zhou, Z.N. (2017). Research on power quality disturbance identification algorithm based on S transform [D]. Harbin Institute of Technology. Han, G., Zhao, J.W, Zhu, X., et al. (2015). Power quality disturbance identification based on multi-feature combination [J]. Proceedings of the CSU-EPSA, 27(8), 71–77. The authors would like to thank Natural Science Basic Research Plan in Shaanxi Province of China for supporting the project. The Project is supported by Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2019JM-544). School of Electrical and Control Engineering, Engineering, Xi'an University of Science and Technology, No.58, Yanta Road, Bei Lin District, Xi'an, Shaanxi Province, China Wenjing Zhao & Liqun Shang Shaanxi Power Generation co., LTD. Weihe Thermal Power Plant, Xianyang, 712000, China Jinfan Sun Wenjing Zhao Liqun Shang Zhao WJ and Sun JF performed the simulation examination, analyzed and interpreted the simulation results. Shang LQ designed and supervised the experiment, prepared and revised the manuscript. All authors read and approved the final manuscript. Correspondence to Wenjing Zhao. Zhao, W., Shang, L. & Sun, J. Power quality disturbance classification based on time-frequency domain multi-feature and decision tree. Prot Control Mod Power Syst 4, 27 (2019). https://doi.org/10.1186/s41601-019-0139-z Disturbance classification;wavelet transform S-transform Classification rules Power System Protection and Control	CommonCrawl
Elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers,[1] whilst algebra introduces variables (quantities without fixed values).[2] ${\overset {}{\underset {}{x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}}}$ The quadratic formula, which is the solution to the quadratic equation $ax^{2}+bx+c=0$ where $a\neq 0$. Here the symbols a, b, and c represent arbitrary numbers, and x is a variable which represents the solution of the equation. This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. It is typically taught to secondary school students and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations. Algebraic notation Main article: Mathematical notation Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression $3x^{2}-2xy+c$ has the following components: A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.[3] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. $a,b,c$) are typically used to represent constants, and those toward the end of the alphabet (e.g. $x,y$ and z) are used to represent variables.[4] They are usually printed in italics.[5] Algebraic operations work in the same way as arithmetic operations,[6] such as addition, subtraction, multiplication, division and exponentiation.[7] and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, $3\times x^{2}$ is written as $3x^{2}$, and $2\times x\times y$ may be written $2xy$.[8] Usually terms with the highest power (exponent), are written on the left, for example, $x^{2}$ is written to the left of x. When a coefficient is one, it is usually omitted (e.g. $1x^{2}$ is written $x^{2}$).[9] Likewise when the exponent (power) is one, (e.g. $3x^{1}$ is written $3x$).[10] When the exponent is zero, the result is always 1 (e.g. $x^{0}$ is always rewritten to 1).[11] However $0^{0}$, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. Alternative notation Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., $x^{2}$, in plain text, and in the TeX mark-up language, the caret symbol ^ represents exponentiation, so $x^{2}$ is written as "x^2".[12][13] This also applies to some programming languages such as Lua. In programming languages such as Ada,[14] Fortran,[15] Perl,[16] Python[17] and Ruby,[18] a double asterisk is used, so $x^{2}$ is written as "x*2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,[19] and it must be explicitly used, for example, $3x$ is written "3x". Concepts Variables Main article: Variable (mathematics) Elementary algebra builds on and extends arithmetic[20] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons. 1. Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as $C=P+20$.[21] 2. Variables allow one to describe general problems,[22] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to $60\times 5=300$ seconds. A more general (algebraic) description may state that the number of seconds, $s=60\times m$, where m is the number of minutes. 3. Variables allow one to describe mathematical relationships between quantities that may vary.[23] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by $\pi =c/d$. 4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as $(a+b)=(b+a)$.[24] Simplifying expressions Main articles: Expression (mathematics) and Computer algebra § Simplification Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example, • Added terms are simplified using coefficients. For example, $x+x+x$ can be simplified as $3x$ (where 3 is a numerical coefficient). • Multiplied terms are simplified using exponents. For example, $x\times x\times x$ is represented as $x^{3}$ • Like terms are added together,[25] for example, $2x^{2}+3ab-x^{2}+ab$ is written as $x^{2}+4ab$, because the terms containing $x^{2}$ are added together, and, the terms containing $ab$ are added together. • Brackets can be "multiplied out", using the distributive property. For example, $x(2x+3)$ can be written as $(x\times 2x)+(x\times 3)$ which can be written as $2x^{2}+3x$ • Expressions can be factored. For example, $6x^{5}+3x^{2}$, by dividing both terms by $3x^{2}$ can be written as $3x^{2}(2x^{3}+1)$ Equations Main article: Equation An equation states that two expressions are equal using the symbol for equality, = (the equals sign).[26] One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:[27] $c^{2}=a^{2}+b^{2}$ This equation states that $c^{2}$, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by a and b. An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as $a+b=b+a$); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. $x^{2}-1=8$ is true only for $x=3$ and $x=-3$. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving. Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: $a>b$ where $>$ represents 'greater than', and $a<b$ where $<$ represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped. Properties of equality By definition, equality is an equivalence relation, meaning it is reflexive (i.e. $b=b$), symmetric (i.e. if $a=b$ then $b=a$), and transitive (i.e. if $a=b$ and $b=c$ then $a=c$).[28] It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties: • if $a=b$ and $c=d$ then $a+c=b+d$ and $ac=bd$; • if $a=b$ then $a+c=b+c$ and $ac=bc$; • more generally, for any function f, if $a=b$ then $f(a)=f(b)$. Properties of inequality The relations less than $<$ and greater than $>$ have the property of transitivity:[29] • If $a<b$ and $b<c$ then $a<c$; • If $a<b$ and $c<d$ then $a+c<b+d$;[30] • If $a<b$ and $c>0$ then $ac<bc$; • If $a<b$ and $c<0$ then $bc<ac$. By reversing the inequation, $<$ and $>$ can be swapped,[31] for example: • $a<b$ is equivalent to $b>a$ Substitution Main article: Substitution (algebra) See also: Substitution (logic) Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expression a5 makes a new expression 35 with meaning 15. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if $a^{2}:=a\times a$ is meant as the definition of $a^{2},$ as the product of a with itself, substituting 3 for a informs the reader of this statement that $3^{2}$ means 3 × 3 = 9. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1. If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0. Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies a = 0 or b = 0 or c = 0. If the original fact were stated as "ab = 0 implies a = 0 or b = 0", then when saying "consider abc = 0," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if abc = 0 then a = 0 or b = 0 or c = 0 if, instead of letting a = a and b = bc, one substitutes a for a and b for bc (and with bc = 0, substituting b for a and c for b). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression a into the a term of the original equation, the a substituted does not refer to the a in the statement "ab = 0 implies a = 0 or b = 0." Solving algebraic equations See also: Equation solving The following sections lay out examples of some of the types of algebraic equations that may be encountered. Linear equations with one variable Main article: Linear equation Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider: Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child? Equivalent equation: $2x+4=12$ where x represent the child's age To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.[32] This problem and its solution are as follows: 1. Equation to solve: $2x+4=12$ 2. Subtract 4 from both sides: $2x+4-4=12-4$ 3. This simplifies to: $2x=8$ 4. Divide both sides by 2: ${\frac {2x}{2}}={\frac {8}{2}}$ 5. This simplifies to the solution: $x=4$ In words: the child is 4 years old. The general form of a linear equation with one variable, can be written as: $ax+b=c$ Following the same procedure (i.e. subtract b from both sides, and then divide by a), the general solution is given by $x={\frac {c-b}{a}}$ Linear equations with two variables A linear equation with two variables has many (i.e. an infinite number of) solutions.[33] For example: Problem in words: A father is 22 years older than his son. How old are they? Equivalent equation: $y=x+22$ where y is the father's age, x is the son's age. That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above. To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that: Problem in words In 10 years, the father will be twice as old as his son. Equivalent equation ${\begin{aligned}y+10&=2\times (x+10)\\y&=2\times (x+10)-10&&{\text{Subtract 10 from both sides}}\\y&=2x+20-10&&{\text{Multiple out brackets}}\\y&=2x+10&&{\text{Simplify}}\end{aligned}}$ Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):[34] ${\begin{cases}y=x+22&{\text{First equation}}\\y=2x+10&{\text{Second equation}}\end{cases}}$ ${\begin{aligned}&&&{\text{Subtract the first equation from}}\\(y-y)&=(2x-x)+10-22&&{\text{the second in order to remove }}y\\0&=x-12&&{\text{Simplify}}\\12&=x&&{\text{Add 12 to both sides}}\\x&=12&&{\text{Rearrange}}\end{aligned}}$ In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations. For other ways to solve this kind of equations, see below, System of linear equations. Quadratic equations Main article: Quadratic equation A quadratic equation is one which includes a term with an exponent of 2, for example, $x^{2}$,[35] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.[36] In general, a quadratic equation can be expressed in the form $ax^{2}+bx+c=0$,[37] where a is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term $ax^{2}$, which is known as the quadratic term. Hence $a\neq 0$, and so we may divide by a and rearrange the equation into the standard form $x^{2}+px+q=0$ where $p={\frac {b}{a}}$ and $q={\frac {c}{a}}$. Solving this, by a process known as completing the square, leads to the quadratic formula $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}},$ where the symbol "±" indicates that both $x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}$ are solutions of the quadratic equation. Quadratic equations can also be solved using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling). As an example of factoring: $x^{2}+3x-10=0,$ which is the same thing as $(x+5)(x-2)=0.$ It follows from the zero-product property that either $x=2$ or $x=-5$ are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example, $x^{2}+1=0$ has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as: $(x+1)^{2}=0.$ For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as $[x-(-1)][x-(-1)]=0.$ Complex numbers All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation $x^{2}+x+1=0$ has solutions $x={\frac {-1+{\sqrt {-3}}}{2}}\quad \quad {\text{and}}\quad \quad x={\frac {-1-{\sqrt {-3}}}{2}}.$ Since ${\sqrt {-3}}$ is not any real number, both of these solutions for x are complex numbers. Exponential and logarithmic equations Main article: Logarithm An exponential equation is one which has the form $a^{x}=b$ for $a>0$,[38] which has solution $X=\log _{a}b={\frac {\ln b}{\ln a}}$ when $b>0$. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if $3\cdot 2^{x-1}+1=10$ then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain $2^{x-1}=3$ whence $x-1=\log _{2}3$ or $x=\log _{2}3+1.$ A logarithmic equation is an equation of the form $log_{a}(x)=b$ for $a>0$, which has solution $X=a^{b}.$ For example, if $4\log _{5}(x-3)-2=6$ then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get $\log _{5}(x-3)=2$ whence $x-3=5^{2}=25$ from which we obtain $x=28.$ Radical equations ${\overset {}{\underset {}{{\sqrt[{2}]{x^{3}}}\equiv x^{\frac {3}{2}}}}}$ Radical equation showing two ways to represent the same expression. The triple bar means the equation is true for all values of x A radical equation is one that includes a radical sign, which includes square roots, ${\sqrt {x}},$ cube roots, ${\sqrt[{3}]{x}}$, and nth roots, ${\sqrt[{n}]{x}}$. Recall that an nth root can be rewritten in exponential format, so that ${\sqrt[{n}]{x}}$ is equivalent to $x^{\frac {1}{n}}$. Combined with regular exponents (powers), then ${\sqrt[{2}]{x^{3}}}$ (the square root of x cubed), can be rewritten as $x^{\frac {3}{2}}$.[39] So a common form of a radical equation is ${\sqrt[{n}]{x^{m}}}=a$ (equivalent to $x^{\frac {m}{n}}=a$) where m and n are integers. It has real solution(s): n is odd n is even and $a\geq 0$ n and m are even and $a<0$ n is even, m is odd, and $a<0$ $x={\sqrt[{n}]{a^{m}}}$ equivalently $x=\left({\sqrt[{n}]{a}}\right)^{m}$ $x=\pm {\sqrt[{n}]{a^{m}}}$ equivalently $x=\pm \left({\sqrt[{n}]{a}}\right)^{m}$ $x=\pm {\sqrt[{n}]{a^{m}}}$ no real solution For example, if: $(x+5)^{2/3}=4$ then ${\begin{aligned}x+5&=\pm ({\sqrt {4}})^{3},\\x+5&=\pm 8,\\x&=-5\pm 8,\end{aligned}}$ and thus $x=3\quad {\text{or}}\quad x=-13$ System of linear equations Main article: System of linear equations There are different methods to solve a system of linear equations with two variables. Elimination method An example of solving a system of linear equations is by using the elimination method: ${\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}$ Multiplying the terms in the second equation by 2: $4x+2y=14$ $4x-2y=2.$ Adding the two equations together to get: $8x=16$ which simplifies to $x=2.$ Since the fact that $x=2$ is known, it is then possible to deduce that $y=3$ by either of the original two equations (by using 2 instead of x ) The full solution to this problem is then ${\begin{cases}x=2\\y=3.\end{cases}}$ This is not the only way to solve this specific system; y could have been resolved before x. Substitution method Another way of solving the same system of linear equations is by substitution. ${\begin{cases}4x+2y&=14\\2x-y&=1.\end{cases}}$ An equivalent for y can be deduced by using one of the two equations. Using the second equation: $2x-y=1$ Subtracting $2x$ from each side of the equation: ${\begin{aligned}2x-2x-y&=1-2x\\-y&=1-2x\end{aligned}}$ and multiplying by −1: $y=2x-1.$ Using this y value in the first equation in the original system: ${\begin{aligned}4x+2(2x-1)&=14\\4x+4x-2&=14\\8x-2&=14\end{aligned}}$ Adding 2 on each side of the equation: ${\begin{aligned}8x-2+2&=14+2\\8x&=16\end{aligned}}$ which simplifies to $x=2$ Using this value in one of the equations, the same solution as in the previous method is obtained. ${\begin{cases}x=2\\y=3.\end{cases}}$ This is not the only way to solve this specific system; in this case as well, y could have been solved before x. Inconsistent systems In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is ${\begin{cases}{\begin{aligned}x+y&=1\\0x+0y&=2\,.\end{aligned}}\end{cases}}$ As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, consider the system ${\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-4\,.\end{aligned}}\end{cases}}$ Multiplying by 2 both sides of the second equation, and adding it to the first one results in $0x+0y=4\,,$ which clearly has no solution. Undetermined systems There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for x and y) For example: ${\begin{cases}{\begin{aligned}4x+2y&=12\\-2x-y&=-6\end{aligned}}\end{cases}}$ Isolating y in the second equation: $y=-2x+6$ And using this value in the first equation in the system: ${\begin{aligned}4x+2(-2x+6)=12\\4x-4x+12=12\\12=12\end{aligned}}$ The equality is true, but it does not provide a value for x. Indeed, one can easily verify (by just filling in some values of x) that for any x there is a solution as long as $y=-2x+6$. There is an infinite number of solutions for this system. Over- and underdetermined systems Systems with more variables than the number of linear equations are called underdetermined. Such a system, if it has any solutions, does not have a unique one but rather an infinitude of them. An example of such a system is ${\begin{cases}{\begin{aligned}x+2y&=10\\y-z&=2.\end{aligned}}\end{cases}}$ When trying to solve it, one is led to express some variables as functions of the other ones if any solutions exist, but cannot express all solutions numerically because there are an infinite number of them if there are any. A system with a higher number of equations than variables is called overdetermined. If an overdetermined system has any solutions, necessarily some equations are linear combinations of the others. See also • History of algebra • Binary operation • Gaussian elimination • Mathematics education • Number line • Polynomial • Cancelling out • Tarski's high school algebra problem References • Leonhard Euler, Elements of Algebra, 1770. English translation Tarquin Press, 2007, ISBN 978-1-899618-79-8, also online digitized editions[40] 2006,[41] 1822. • Charles Smith, A Treatise on Algebra, in Cornell University Library Historical Math Monographs. • Redden, John. 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McKeague, Elementary Algebra, Publisher: Cengage Learning, 2011, ISBN 0840064217, 9780840064219, 571 pages, page 49 25. Andrew Marx, Shortcut Algebra I: A Quick and Easy Way to Increase Your Algebra I Knowledge and Test Scores, Publisher Kaplan Publishing, 2007, ISBN 1419552880, 9781419552885, 288 pages, page 51 26. Mark Clark, Cynthia Anfinson, Beginning Algebra: Connecting Concepts Through Applications, Publisher Cengage Learning, 2011, ISBN 0534419380, 9780534419387, 793 pages, page 134 27. Alan S. Tussy, R. David Gustafson, Elementary and Intermediate Algebra, Publisher Cengage Learning, 2012, ISBN 1111567689, 9781111567682, 1163 pages, page 493 28. Douglas Downing, Algebra the Easy Way, Publisher Barron's Educational Series, 2003, ISBN 0764119729, 9780764119729, 392 pages, page 20 29. Ron Larson, Robert Hostetler, Intermediate Algebra, Publisher Cengage Learning, 2008, ISBN 0618753524, 9780618753529, 857 pages, page 96 30. "What is the following property of inequality called?". Stack Exchange. November 29, 2014. Retrieved 4 May 2018. 31. Chris Carter, Physics: Facts and Practice for A Level, Publisher Oxford University Press, 2001, ISBN 019914768X, 9780199147687, 144 pages, page 50 32. Slavin, Steve (1989). All the Math You'll Ever Need. John Wiley & Sons. p. 72. ISBN 0-471-50636-2. 33. Sinha, The Pearson Guide to Quantitative Aptitude for CAT 2/ePublisher: Pearson Education India, 2010, ISBN 8131723666, 9788131723661, 599 pages, page 195 34. Cynthia Y. Young, Precalculus, Publisher John Wiley & Sons, 2010, ISBN 0471756849, 9780471756842, 1175 pages, page 699 35. Mary Jane Sterling, Algebra II For Dummies, Publisher: John Wiley & Sons, 2006, ISBN 0471775819, 9780471775812, 384 pages, page 37 36. John T. Irwin, The Mystery to a Solution: Poe, Borges, and the Analytic Detective Story, Publisher JHU Press, 1996, ISBN 0801854660, 9780801854668, 512 pages, page 372 37. Sharma/khattar, The Pearson Guide To Objective Mathematics For Engineering Entrance Examinations, 3/E, Publisher Pearson Education India, 2010, ISBN 8131723631, 9788131723630, 1248 pages, page 621 38. Aven Choo, LMAN OL Additional Maths Revision Guide 3, Publisher Pearson Education South Asia, 2007, ISBN 9810600011, 9789810600013, page 105 39. John C. Peterson, Technical Mathematics With Calculus, Publisher Cengage Learning, 2003, ISBN 0766861899, 9780766861893, 1613 pages, page 525 40. Euler's Elements of Algebra Archived 2011-04-13 at the Wayback Machine 41. Euler, Leonhard; Hewlett, John; Horner, Francis; Bernoulli, Jean; Lagrange, Joseph Louis (4 May 2018). "Elements of Algebra". Longman, Orme. Retrieved 4 May 2018 – via Google Books. 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The research literature, while copious, is messy and varied: methodologies and devices vary substantially, sample sizes are tiny, the study designs vary from paper to paper, metrics are sometimes comically limited (one study measured speed of finishing a RAPM IQ test but not scores), blinding is rare and unclear how successful, etc. Relevant papers include Chung et al 2012, Rojas & Gonzalez-Lima 2013, & Gonzalez-Lima & Barrett 2014. Another Longecity user ran a self-experiment, with some design advice from me, where he performed a few cognitive tests over several periods of LLLT usage (the blocks turned out to be ABBA), using his father and towels to try to blind himself as to condition. I analyzed his data, and his scores did seem to improve, but his scores improved so much in the last part of the self-experiment I found myself dubious as to what was going on - possibly a failure of randomness given too few blocks and an temporal exogenous factor in the last quarter which was responsible for the improvement. Schroeder, Mann-Koepke, Gualtieri, Eckerman, and Breese (1987) assessed the performance of subjects on placebo and MPH in a game that allowed subjects to switch between two different sectors seeking targets to shoot. They did not observe an effect of the drug on overall level of performance, but they did find fewer switches between sectors among subjects who took MPH, and perhaps because of this, these subjects did not develop a preference for the more fruitful sector. Since my experiment had a number of flaws (non-blind, varying doses at varying times of day), I wound up doing a second better experiment using blind standardized smaller doses in the morning. The negative effect was much smaller, but there was still no mood/productivity benefit. Having used up my first batch of potassium citrate in these 2 experiments, I will not be ordering again since it clearly doesn't work for me. It's basic economics: the price of a good must be greater than cost of producing said good, but only under perfect competition will price = cost. Otherwise, the price is simply whatever maximizes profit for the seller. (Bottled water doesn't really cost $2 to produce.) This can lead to apparently counter-intuitive consequences involving price discrimination & market segmentation - such as damaged goods which are the premium product which has been deliberately degraded and sold for less (some Intel CPUs, some headphones etc.). The most famous examples were railroads; one notable passage by French engineer-economist Jules Dupuit describes the motivation for the conditions in 1849: So what's the catch? Well, it's potentially addictive for one. Anything that messes with your dopamine levels can be. And Patel says there are few long-term studies on it yet, so we don't know how it will affect your brain chemistry down the road, or after prolonged, regular use. Also, you can't get it very easily, or legally for that matter, if you live in the U.S. It's classified as a schedule IV controlled substance. That's where Adrafinil comes in. When I spoke with Jesse Lawler, who hosts the podcast Smart Drugs Smarts, about breakthroughs in brain health and neuroscience, he was unsurprised to hear of my disappointing experience. Many nootropics are supposed to take time to build up in the body before users begin to feel their impact. But even then, says Barry Gordon, a neurology professor at the Johns Hopkins Medical Center, positive results wouldn't necessarily constitute evidence of a pharmacological benefit. According to clinical psychiatrist and Harvard Medical School Professor, Emily Deans, "there's probably nothing dangerous about the occasional course of nootropics...beyond that, it's possible to build up a tolerance if you use them often enough." Her recommendation is to seek pharmaceutical-grade products which she says are more accurate regarding dosage and less likely to be contaminated. Sulbutiamine, mentioned earlier as a cholinergic smart drug, can also be classed a dopaminergic, although its mechanism is counterintuitive: by reducing the release of dopamine in the brain's prefrontal cortex, the density of dopamine receptors actually increase after continued Sulbutiamine exposure, through a compensatory mechanism. (This provides an interesting example of how dividing smart drugs into sensible "classes" is a matter of taste as well as science, especially since many of them create their discernable neural effects through still undefined mechanisms.) Too much caffeine may be bad for bone health because it can deplete calcium. Overdoing the caffeine also may affect the vitamin D in your body, which plays a critical role in your body's bone metabolism. However, the roles of vitamin D as well as caffeine in the development of osteoporosis continue to be a source of debate. Significance: Caffeine may interfere with your body's metabolism of vitamin D, according to a 2007 Journal of Steroid Biochemistry & Molecular Biology study. You have vitamin D receptors, or VDRs, in your osteoblast cells. These large cells are responsible for the mineralization and synthesis of bone in your body. They create a sheet on the surface of your bones. The D receptors are nuclear hormone receptors that control the action of vitamin D-3 by controlling hormone-sensitive gene expression. These receptors are critical to good bone health. For example, a vitamin D metabolism disorder in which these receptors don't work properly causes rickets. Yes, according to a new policy at Duke University, which says that the "unauthorized use of prescription medicine to enhance academic performance" should be treated as cheating." And no, according to law professor Nita Farahany, herself based at Duke University, who has called the policy "ill-conceived," arguing that "banning smart drugs disempowers students from making educated choices for themselves." The majority of nonmedical users reported obtaining prescription stimulants from a peer with a prescription (Barrett et al., 2005; Carroll et al., 2006; DeSantis et al., 2008, 2009; DuPont et al., 2008; McCabe & Boyd, 2005; Novak et al., 2007; Rabiner et al., 2009; White et al., 2006). Consistent with nonmedical user reports, McCabe, Teter, and Boyd (2006) found 54% of prescribed college students had been approached to divert (sell, exchange, or give) their medication. Studies of secondary school students supported a similar conclusion (McCabe et al., 2004; Poulin, 2001, 2007). In Poulin's (2007) sample, 26% of students with prescribed stimulants reported giving or selling some of their medication to other students in the past month. She also found that the number of students in a class with medically prescribed stimulants was predictive of the prevalence of nonmedical stimulant use in the class (Poulin, 2001). In McCabe et al.'s (2004) middle and high school sample, 23% of students with prescriptions reported being asked to sell or trade or give away their pills over their lifetime. A "smart pill" is a drug that increases the cognitive ability of anyone taking it, whether the user is cognitively impaired or normal. The Romanian neuroscientist Corneliu Giurgea is often credited with first proposing, in the 1960s, that smart pills should be developed to increase the intelligence of the general population (see Giurgea, 1984). He is quoted as saying, "Man is not going to wait passively for millions of years before evolution offers him a better brain" (Gazzaniga, 2005, p. 71). In their best-selling book, Smart Drugs and Nutrients, Dean and Morgenthaler (1990) reviewed a large number of substances that have been used by healthy individuals with the goal of increasing cognitive ability. These include synthetic and natural products that affect neurotransmitter levels, neurogenesis, and blood flow to the brain. Although many of these substances have their adherents, none have become widely used. Caffeine and nicotine may be exceptions to this generalization, as one motivation among many for their use is cognitive enhancement (Julien, 2001). Finally, two tasks measuring subjects' ability to control their responses to monetary rewards were used by de Wit et al. (2002) to assess the effects of d-AMP. When subjects were offered the choice between waiting 10 s between button presses for high-probability rewards, which would ultimately result in more money, and pressing a button immediately for lower probability rewards, d-AMP did not affect performance. However, when subjects were offered choices between smaller rewards delivered immediately and larger rewards to be delivered at later times, the normal preference for immediate rewards was weakened by d-AMP. That is, subjects were more able to resist the impulse to choose the immediate reward in favor of the larger reward. Harrisburg, NC -- (SBWIRE) -- 02/18/2019 -- Global Smart Pills Technology Market - Segmented by Technology, Disease Indication, and Geography - Growth, Trends, and Forecast (2019 - 2023) The smart pill is a wireless capsule that can be swallowed, and with the help of a receiver (worn by patients) and software that analyzes the pictures captured by the smart pill, the physician is effectively able to examine the gastrointestinal tract. Gastrointestinal disorders have become very common, but recently, there has been increasing incidence of colorectal cancer, inflammatory bowel disease, and Crohns disease as well. Your mileage will vary. There are so many parameters and interactions in the brain that any of them could be the bottleneck or responsible pathway, and one could fall prey to the common U-shaped dose-response curve (eg. Yerkes-Dodson law; see also Chemistry of the adaptive mind & de Jongh et al 2007) which may imply that the smartest are those who benefit least23 but ultimately they all cash out in a very few subjective assessments like energetic or motivated, with even apparently precise descriptions like working memory or verbal fluency not telling you much about what the nootropic actually did. It's tempting to list the nootropics that worked for you and tell everyone to go use them, but that is merely generalizing from one example (and the more nootropics - or meditation styles, or self-help books, or getting things done systems - you try, the stronger the temptation is to evangelize). The best you can do is read all the testimonials and studies and use that to prioritize your list of nootropics to try. You don't know in advance which ones will pay off and which will be wasted. You can't know in advance. And wasted some must be; to coin a Umeshism: if all your experiments work, you're just fooling yourself. (And the corollary - if someone else's experiments always work, they're not telling you everything.) Barbaresi WJ, Katusic SK, Colligan RC, Weaver AL, Jacobsen SJ. Modifiers of long-term school outcomes for children with attention-deficit/hyperactivity disorder: Does treatment with stimulant medication make a difference? Results from a population-based study. Journal of Developmental and Behavioral Pediatrics. 2007;28:274–287. doi: 10.1097/DBP.0b013e3180cabc28. [PubMed] [CrossRef] Fortunately, there are some performance-enhancing habits that have held up under rigorous scientific scrutiny. They are free, and easy to pronounce. Unfortunately, they are also the habits you were perhaps hoping to forego by using nootropics instead. "Of all the things that are supposed to be 'good for the brain,'" says Stanford neurology professor Sharon Sha, "there is more evidence for exercise than anything else." Next time you're facing a long day, you could take a pill and see what happens. At dose #9, I've decided to give up on kratom. It is possible that it is helping me in some way that careful testing (eg. dual n-back over weeks) would reveal, but I don't have a strong belief that kratom would help me (I seem to benefit more from stimulants, and I'm not clear on how an opiate-bearer like kratom could stimulate me). So I have no reason to do careful testing. Oh well. I take my piracetam in the form of capped pills consisting (in descending order) of piracetam, choline bitartrate, anhydrous caffeine, and l-tyrosine. On 8 December 2012, I happened to run out of them and couldn't fetch more from my stock until 27 December. This forms a sort of (non-randomized, non-blind) short natural experiment: did my daily 1-5 mood/productivity ratings fall during 8-27 December compared to November 2012 & January 2013? The graphed data28 suggests to me a decline: Noopept is a nootropic that belongs to the ampakine family. It is known for promoting learning, boosting mood, and improving logical thinking. It has been popular as a study drug for a long time but has recently become a popular supplement for improving vision. Users report seeing colors more brightly and feeling as if their vision is more vivid after taking noopept. There is evidence to suggest that modafinil, methylphenidate, and amphetamine enhance cognitive processes such as learning and working memory...at least on certain laboratory tasks. One study found that modafinil improved cognitive task performance in sleep-deprived doctors. Even in non-sleep deprived healthy volunteers, modafinil improved planning and accuracy on certain cognitive tasks. Similarly, methylphenidate and amphetamine also enhanced performance of healthy subjects in certain cognitive tasks. Upon examining the photographs, I noticed no difference in eye color, but it seems that my move had changed the ambient lighting in the morning and so there was a clear difference between the two sets of photographs! The before photographs had brighter lighting than the after photographs. Regardless, I decided to run a small survey on QuickSurveys/Toluna to confirm my diagnosis of no-change; the survey was 11 forced-choice pairs of photographs (before-after), with the instructions as follows: The truth is, taking a smart pill will not allow you to access information that you have not already learned. If you speak English, a smart drug cannot embed the Spanish dictionary into your brain. In other words, they won't make you smarter or more intelligent. We need to throttle back our expectations and explore reality. What advantage can smart drugs provide? Brain enhancing substances have excellent health and cognitive benefits that are worth exploring. The evidence? In small studies, healthy people taking modafinil showed improved planning and working memory, and better reaction time, spatial planning, and visual pattern recognition. A 2015 meta-analysis claimed that "when more complex assessments are used, modafinil appears to consistently engender enhancement of attention, executive functions, and learning" without affecting a user's mood. In a study from earlier this year involving 39 male chess players, subjects taking modafinil were found to perform better in chess games played against a computer. The flanker task is designed to tax cognitive control by requiring subjects to respond based on the identity of a target stimulus (H or S) and not the more numerous and visually salient stimuli that flank the target (as in a display such as HHHSHHH). Servan-Schreiber, Carter, Bruno, and Cohen (1998) administered the flanker task to subjects on placebo and d-AMP. They found an overall speeding of responses but, more importantly, an increase in accuracy that was disproportionate for the incongruent conditions, that is, the conditions in which the target and flankers did not match and cognitive control was needed. This calculation - reaping only \frac{7}{9} of the naive expectation - gives one pause. How serious is the sleep rebound? In another article, I point to a mice study that sleep deficits can take 28 days to repay. What if the gain from modafinil is entirely wiped out by repayment and all it did was defer sleep? Would that render modafinil a waste of money? Perhaps. Thinking on it, I believe deferring sleep is of some value, but I cannot decide whether it is a net profit. The majority of smart pills target a limited number of cognitive functions, which is why a group of experts gathered to discover a formula which will empower the entire brain and satisfy the needs of students, athletes, and professionals. Mind Lab Pro® combines 11 natural nootropics to affect all 4 areas of mental performance, unlocking the full potential of your brain. Its carefully designed formula will provide an instant boost, while also delivering long-term benefits. Instead of buying expensive supplements, Lebowitz recommends eating heart-healthy foods, like those found in the MIND diet. Created by researchers at Rush University, MIND combines the Mediterranean and DASH eating plans, which have been shown to reduce the risk of heart problems. Fish, nuts, berries, green leafy vegetables and whole grains are MIND diet staples. Lebowitz says these foods likely improve your cognitive health by keeping your heart healthy. Disclaimer: No statements on this website have been reviewed by the Food and Drug Administration. No products mentioned on this website are intended to diagnose, treat, cure or prevent any diseases. brs.brainreference.com is sponsored by BRS Publishers. All editorials on this site were written by editors compensated by BRS Publishers and do not claim or state to be medical professionals giving medical advice. This website is only for the purpose of providing information. Please consult with your doctor before starting any mental health program or dietary supplement. All product pictures were photographed by us and used in conjunction with stock photos who are representing lab technicians and not doctors. If you feel any of this information is inaccurate contact us and we will verify and implement your correction within about 48 business hours. Also note that we have multiple affiliates and we are paid commission on various products by different companies. If you wish to advertise with us, please contact us. Any and all trademarks, logos and service marks displayed on this site are registered or unregistered Trademarks of their respective owners. Capsule Connection sells 1000 00 pills (the largest pills) for $9. I already have a pill machine, so that doesn't count (a sunk cost). If we sum the grams per day column from the first table, we get 9.75 grams a day. Each 00 pill can take around 0.75 grams, so we need 13 pills. (Creatine is very bulky, alas.) 13 pills per day for 1000 days is 13,000 pills, and 1,000 pills is $9 so we need 13 units and 13 times 9 is $117. "In 183 pages, Cavin Balaster's new book, How to Feed A Brain provides an outline and plan for how to maximize one's brain performance. The "Citation Notes" provide all the scientific and academic documentation for further understanding. The "Additional Resources and Tips" listing takes you to Cavin's website for more detail than could be covered in 183 pages. Cavin came to this knowledge through the need to recover from a severe traumatic brain injury and he did not keep his lessons learned to himself. This book is enlightening for anyone with a brain. We all want to function optimally, even to take exams, stay dynamic, and make positive contributions to our communities. Bravo Cavin for sharing your lessons learned!" If you happen to purchase anything recommended on this or affiliated websites, we will likely receive some kind of affiliate compensation. We only recommend stuff that we truly believe in and share with our friends and family. If you ever have an issue with anything we recommend please let us know. We want to make sure we are always serving you at the highest level. If you are purchasing using our affiliate link, you will not pay a different price for the products and/or services, but your purchase helps support our ongoing work. Thanks for your support! The stimulant now most popular in news articles as a legitimate "smart drug" is Modafinil, which came to market as an anti-narcolepsy drug, but gained a following within the military, doctors on long shifts, and college students pulling all-nighters who needed a drug to improve alertness without the "wired" feeling associated with caffeine. Modafinil is a relatively new smart drug, having gained widespread use only in the past 15 years. More research is needed before scientists understand this drug's function within the brain – but the increase in alertness it provides is uncontested. These statements have not been evaluated by the Food and Drug Administration. The products and information on this website are not intended to diagnose, treat, cure or prevent any disease. The information on this site is for educational purposes only and should not be considered medical advice. Please speak with an appropriate healthcare professional when evaluating any wellness related therapy. Please read the full medical disclaimer before taking any of the products offered on this site. There is much to be appreciated in a brain supplement like BrainPill (never mind the confusion that may stem from the generic-sounding name) that combines tried-and-tested ingredients in a single one-a-day formulation. The consistency in claims and what users see in real life is an exemplary one, which convinces us to rate this powerhouse as the second on this review list. Feeding one's brain with nootropics and related supplements entails due diligence in research and seeking the highest quality, and we think BrainPill is up to task. Learn More... The information on this website has not been evaluated by the Food & Drug Administration or any other medical body. We do not aim to diagnose, treat, cure or prevent any illness or disease. Information is shared for educational purposes only. You must consult your doctor before acting on any content on this website, especially if you are pregnant, nursing, taking medication, or have a medical condition. Adderall is a mix of 4 amphetamine salts (FDA adverse events), and not much better than the others (but perhaps less addictive); as such, like caffeine or methamphetamine, it is not strictly a nootropic but a cognitive enhancer and can be tricky to use right (for how one should use stimulants, see How To Take Ritalin Correctly). I ordered 10x10mg Adderall IR off Silk Road (Wikipedia). On the 4th day after confirmation from seller, the package arrived. It was a harmless looking little padded mailer. Adderall as promised: 10 blue pills with markings, in a double ziplock baggy (reasonable, it's not cocaine or anything). They matched pretty much exactly the descriptions of the generic I had found online. (Surprisingly, apparently both the brand name and the generic are manufactured by the same pharmacorp.) I took 1.5mg of melatonin, and went to bed at ~1:30AM; I woke up around 6:30, took a modafinil pill/200mg, and felt pretty reasonable. By noon my mind started to feel a bit fuzzy, and lunch didn't make much of it go away. I've been looking at studies, and users seem to degrade after 30 hours; I started on mid-Thursday, so call that 10 hours, then 24 (Friday), 24 (Saturday), and 14 (Sunday), totaling 72hrs with <20hrs sleep; this might be equivalent to 52hrs with no sleep, and Wikipedia writes: Tyrosine (Examine.com) is an amino acid; people on the Imminst.org forums (as well as Wikipedia) suggest that it helps with energy and coping with stress. I ordered 4oz (bought from Smart Powders) to try it out, and I began taking 1g with my usual caffeine+piracetam+choline mix. It does not dissolve easily in hot water, and is very chalky and not especially tasty. I have not noticed any particular effects from it. As mentioned earlier, cognitive control is needed not only for inhibiting actions, but also for shifting from one kind of action or mental set to another. The WCST taxes cognitive control by requiring the subject to shift from sorting cards by one dimension (e.g., shape) to another (e.g., color); failures of cognitive control in this task are manifest as perseverative errors in which subjects continue sorting by the previously successful dimension. Three studies included the WCST in their investigations of the effects of d-AMP on cognition (Fleming et al., 1995; Mattay et al., 1996, 2003), and none revealed overall effects of facilitation. However, Mattay et al. (2003) subdivided their subjects according to COMT genotype and found differences in both placebo performance and effects of the drug. Subjects who were homozygous for the val allele (associated with lower prefrontal dopamine activity) made more perseverative errors on placebo than other subjects and improved significantly with d-AMP. Subjects who were homozygous for the met allele performed best on placebo and made more errors on d-AMP. The blood half-life is 12-36 hours; hence two or three days ought to be enough to build up and wash out. A week-long block is reasonable since that gives 5 days for effects to manifest, although month-long blocks would not be a bad choice either. (I prefer blocks which fit in round periods because it makes self-experiments easier to run if the blocks fit in normal time-cycles like day/week/month. The most useless self-experiment is the one abandoned halfway.) One symptom of Alzheimer's disease is a reduced brain level of the neurotransmitter called acetylcholine. It is thought that an effective treatment for Alzheimer's disease might be to increase brain levels of acetylcholine. Another possible treatment would be to slow the death of neurons that contain acetylcholine. Two drugs, Tacrine and Donepezil, are both inhibitors of the enzyme (acetylcholinesterase) that breaks down acetylcholine. These drugs are approved in the US for treatment of Alzheimer's disease. If you want to focus on boosting your brain power, Lebowitz says you should primarily focus on improving your cardiovascular health, which is "the key to good thinking." For example, high blood pressure and cholesterol, which raise the risk of heart disease, can cause arteries to harden, which can decrease blood flow to the brain. The brain relies on blood to function normally. At this point, I began thinking about what I was doing. Black-market Adderall is fairly expensive; $4-10 a pill vs prescription prices which run more like $60 for 120 20mg pills. It would be a bad idea to become a fan without being quite sure that it is delivering bang for the buck. Now, why the piracetam mix as the placebo as opposed to my other available powder, creatine powder, which has much smaller mental effects? Because the question for me is not whether the Adderall works (I am quite sure that the amphetamines have effects!) but whether it works better for me than my cheap legal standbys (piracetam & caffeine)? (Does Adderall have marginal advantage for me?) Hence, I want to know whether Adderall is better than my piracetam mix. People frequently underestimate the power of placebo effects, so it's worth testing. (Unfortunately, it seems that there is experimental evidence that people on Adderall know they are on Adderall and also believe they have improved performance, when they do not5. So the blind testing does not buy me as much as it could.) Sleep itself is an underrated cognition enhancer. It is involved in enhancing long-term memories as well as creativity. For instance, it is well established that during sleep memories are consolidated-a process that "fixes" newly formed memories and determines how they are shaped. Indeed, not only does lack of sleep make most of us moody and low on energy, cutting back on those precious hours also greatly impairs cognitive performance. Exercise and eating well also enhance aspects of cognition. It turns out that both drugs and "natural" enhancers produce similar physiological changes in the brain, including increased blood flow and neuronal growth in structures such as the hippocampus. Thus, cognition enhancers should be welcomed but not at the expense of our health and well being. Vitamin B12 is also known as Cobalamin and is a water-soluble essential vitamin. A (large) deficiency of Vitamin B12 will ultimately lead to cognitive impairment [52]. Older people and people who don't eat meat are at a higher risk than young people who eat more meat. And people with depression have less Vitamin B12 than the average population [53]. Nondrug cognitive-enhancement methods include the high tech and the low. An example of the former is transcranial magnetic stimulation (TMS), whereby weak currents are induced in specific brain areas by magnetic fields generated outside the head. TMS is currently being explored as a therapeutic modality for neuropsychiatric conditions as diverse as depression and ADHD and is capable of enhancing the cognition of normal healthy people (e.g., Kirschen, Davis-Ratner, Jerde, Schraedley-Desmond, & Desmond, 2006). An older technique, transcranial direct current stimulation (tDCS), has become the subject of renewed research interest and has proven capable of enhancing the cognitive performance of normal healthy individuals in a variety of tasks. For example, Flöel, Rösser, Michka, Knecht, and Breitenstein (2008) reported enhancement of learning and Dockery, Hueckel-Weng, Birbaumer, and Plewnia (2009) reported enhancement of planning with tDCS. Brain focus pills mostly contain chemical components like L-theanine which is naturally found in green and black tea. It's associated with enhancing alertness, cognition, relaxation, arousal, and reducing anxiety to a large extent. Theanine is an amino and glutamic acid that has been proven to be a safe psychoactive substance. Some studies suggest that this compound influences, the expression in the genes present in the brain which is responsible for aggression, fear, and memory. This, in turn, helps in balancing the behavioral responses to stress and also helps in improving specific conditions, like Post Traumatic Stress Disorder (PTSD). There's been a lot of talk about the ketogenic diet recently—proponents say that minimizing the carbohydrates you eat and ingesting lots of fat can train your body to burn fat more effectively. It's meant to help you both lose weight and keep your energy levels constant. The diet was first studied and used in patients with epilepsy, who suffered fewer seizures when their bodies were in a state of ketosis. Because seizures originate in the brain, this discovery showed researchers that a ketogenic diet can definitely affect the way the brain works. Brain hackers naturally started experimenting with diets to enhance their cognitive abilities, and now a company called HVMN even sells ketone esters in a bottle; to achieve these compounds naturally, you'd have to avoid bread and cake. Here are 6 ways exercise makes your brain better. Supplements, medications, and coffee certainly might play a role in keeping our brains running smoothly at work or when we're trying to remember where we left our keys. But the long-term effects of basic lifestyle practices can't be ignored. "For good brain health across the life span, you should keep your brain active," Sahakian says. "There is good evidence for 'use it or lose it.'" She suggests brain-training apps to improve memory, as well as physical exercise. "You should ensure you have a healthy diet and not overeat. It is also important to have good-quality sleep. Finally, having a good work-life balance is important for well-being." Try these 8 ways to get smarter while you sleep. I've been actively benefitting from nootropics since 1997, when I was struggling with cognitive performance and ordered almost $1000 worth of smart drugs from Europe (the only place where you could get them at the time). I remember opening the unmarked brown package and wondering whether the pharmaceuticals and natural substances would really enhance my brain. Contact us at [email protected] \| Sitemap xml \| Sitemap txt \| Sitemap	CommonCrawl
\begin{definition}[Definition:Euler Phi Function] Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Euler $\phi$ (phi) function''' is the arithmetic function $\phi: \Z_{>0} \to \Z_{>0}$ defined as: :$\map \phi n = $ the number of strictly positive integers less than or equal to $n$ which are prime to $n$ That is: :$\map \phi n = \card {S_n}: S_n = \set {k: 1 \le k \le n, k \perp n}$ \end{definition}	ProofWiki
At each basketball practice last week, Jenny made twice as many free throws as she made at the previous practice. At her fifth practice she made 48 free throws. How many free throws did she make at the first practice? At Jenny's fourth practice she made $\frac{1}{2}(48)=24$ free throws. At her third practice she made 12, at her second practice she made 6, and at her first practice she made $\boxed{3}$.	Math Dataset
Hofstadter points In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting.[1] They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X(360) and the Hofstafter one-point is the center denoted as X(359) in Clark Kimberling's Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992.[1] Hofstadter triangles Let ABC be a given triangle. Let r be a positive real constant. Rotate the line segment BC about B through an angle rB towards A and let LBC be the line containing this line segment. Next rotate the line segment BC about C through an angle rC towards A. Let L'BC be the line containing this line segment. Let the lines LBC and L'BC intersect at A(r). In a similar way the points B(r) and C(r) are constructed. The triangle whose vertices are A(r), B(r), C(r) is the Hofstadter r-triangle (or, the r-Hofstadter triangle) of triangle ABC.[2][1] Special case • The Hofstadter 1/3-triangle of triangle ABC is the first Morley's triangle of triangle ABC. Morley's triangle is always an equilateral triangle. • The Hofstadter 1/2-triangle is simply the incentre of the triangle. Trilinear coordinates of the vertices of Hofstadter triangles The trilinear coordinates of the vertices of the Hofstadter r-triangle are given below: A(r) = ( 1 , sin rB / sin (1 − r)B , sin rC / sin (1 − r)C ) B(r) = ( sin rA / sin (1 − r)A , 1 , sin rC / sin (1 − r)C ) C(r) = ( sin rA / sin (1 − r)A , sin (1 − r)B / sin rB , 1 ) Hofstadter points For a positive real constant r > 0, let A(r) B(r) C(r) be the Hofstadter r-triangle of triangle ABC. Then the lines AA(r), BB(r), CC(r) are concurrent.[3] The point of concurrence is the Hofstdter r-point of triangle ABC. Trilinear coordinates of Hofstadter r-point The trilinear coordinates of Hofstadter r-point are given below. ( sin rA / sin ( A − rA) , sin rB / sin ( B − rB ) , sin rC / sin ( C − rC) ) Hofstadter zero- and one-points The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for r in the expressions for the trilinear coordinates for the Hofstdter r-point. Hofstadter zero-point is the limit of the Hofstadter r-point as r approaches zero. Hofstadter one-point is the limit of the Hofstadter r-point as r approaches one. Trilinear coordinates of Hofstadter zero-point = lim r → 0 ( sin rA / sin ( A − rA) , sin rB / sin ( B − rB ) , sin rC / sin ( C − rC) ) = lim r → 0 ( sin rA / r sin ( A − rA) , sin rB / r sin ( B − rB ) , sin rC / r sin ( C − rC) ) = lim r → 0 ( A sin rA / rA sin ( A − rA) , B sin rB / rB sin ( B − rB ) , C sin rC / rC sin ( C − rC) ) = ( A / sin A , B / sin B , C / sin C ) ), as lim r → 0 sin rA / rA = 1, etc. = ( A / a, B / b, C / c ) Trilinear coordinates of Hofstadter one-point = lim r → 1 ( sin rA / sin ( A − rA) , sin rB / sin ( B − rB ) , sin rC / sin ( C − rC) ) = lim r → 1 ( ( 1 − r ) sin rA / sin ( A − rA) , ( 1 - r ) sin rB / sin ( B − rB ) , ( 1 − r )sin rC / sin ( C − rC) ) = lim r → 1 ( ( 1 − r ) A sin rA / A sin ( A − rA) , ( 1 − r ) B sin rB / B sin ( B − rB ) , ( 1 − r ) C sin rC / C sin ( C − rC) ) = ( sin A / A , sin B / B , sin C / C ) ) as lim r → 1 ( 1 − r ) A / sin ( A − rA ) = 1, etc. = ( a / A, b / B, c / C ) References 1. Kimberling, Clark. "Hofstadter points". Retrieved 11 May 2012. 2. Weisstein, Eric W. "Hofstadter Triangle". MathWorld--A Wolfram Web Resource. Retrieved 11 May 2012. 3. C. Kimberling (1994). "Hofstadter points". Nieuw Archief voor Wiskunde. 12: 109–114. Douglas Hofstadter Books • Gödel, Escher, Bach (1979) • The Mind's I (1981)1 • Metamagical Themas (1985) • Fluid Concepts and Creative Analogies (1995)2 • Le Ton beau de Marot (1997) • I Am a Strange Loop (2007) • Surfaces and Essences (2013) Concepts and projects • BlooP and FlooP • Copycat • Hofstadter's butterfly • Hofstadter's law • Hofstadter points • MU puzzle • Platonia dilemma • Six nines in pi • Strange loop • Superrationality Related • Robert Hofstadter (father) • Egbert B. Gebstadter • Indiana University Bloomington • Victim of the Brain • 1 Edited by Hofstadter and Daniel C. Dennett • 2 By Hofstadter and the Fluid Analogies Research Group	Wikipedia
Publications of Domokos, M. Export all publications as EndNote XML Type of Publication BookBook ChapterBook reviewCommentaryConference PaperConference ProceedingsJournal ArticleJournal EditorMagazine ArticleMiscellaneousNewspaper ArticlePolicy BriefPublication reviewReportStudySummaryThesisWorking Paper Cziszter K, Domokos M. The Noether number for the groups with a cyclic subgroup of index two. Journal of Algebra. 2014;399:546-60. EndNote XML BibTex Google Scholar AbstractGoogle Scholar The Noether number for the groups with a cyclic subgroup of index two The exact degree bound for the generators of rings of polynomial invariants is determined for the finite, non-cyclic groups having a cyclic subgroup of index two. It is proved that the Noether number of these groups equals one half the order of the group plus 1 or 2. Cziszter K, Domokos M. On the generalized Davenport constant and the Noether number.. 2012. Publisher linkGoogle Scholar Domokos M. Covariants and the no-name lemma. Journal of Lie Theory. 2008;18(4):839-49. Domokos M. Typical separating invariants. Transformation Groups. 2007;12(1):49-63. Domokos M. A Quantum Homogeneous Space of Nilpotent Matrices. Letters in Mathematical Physics. 2005;72(1):39-50. Publisher linkAbstractGoogle Scholar A Quantum Homogeneous Space of Nilpotent Matrices Copyright of Letters in Mathematical Physics is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) Domokos M, Frenkel PE. Mod 2 indecomposable orthogonal invariants. Advances in Mathematics. 2005;192(1):209-17. Mod 2 indecomposable orthogonal invariants Copyright of Advances in Mathematics is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) Domokos M, Lenagan TH. Quantized trace rings. Quarterly Journal of Mathematics. 2005;56(4):507-23. Domokos M, Frenkel PE. On orthogonal invariants in characteristic 2. Journal of Algebra. 2004;274(2):662. On orthogonal invariants in characteristic 2 Copyright of Journal of Algebra is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) Domokos M, Lenagan TH. Representation rings of quantum groups. Journal of Algebra. 2004;282(1):103-28. Representation rings of quantum groups Domokos M. Matrix invariants and the failure of Weyl's theorem. In: Giambruno A, Regev A, Zaicev M, editors. Polynomial Identities and Combinatorial Methods. Vol 235. New York: Dekker; 2003. p. 215-36. (Lecture Notes in Pure and Applied Mathematics; vol 235). Matrix invariants and the failure of Weyl's theorem Conference publication. Domokos M. On the dimension of faithful modules over finite dimensional basic algebras. Linear Algebra and Its Applications. 2003;365:155-7. Domokos M, Fioresi R, Lenagan TH. Orbits for the adjoint coaction on quantum matrices. Journal of Geometry & Physics. 2003;47(4):447. Orbits for the adjoint coaction on quantum matrices Copyright of Journal of Geometry & Physics is the property of Elsevier Science Publishers B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) Domokos M, Lenagan TH. Conjugation coinvariants of quantum matrices. Bulletin of the London Mathematical Society. 2003;35(1):117-27. Domokos M, Lenagan TH. Weakly multiplicative coactions of quantized function algebras. Journal of Pure & Applied Algebra. 2003;183(1-3):45. Weakly multiplicative coactions of quantized function algebras Copyright of Journal of Pure & Applied Algebra is the property of Elsevier Science Publishers B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) Domokos M. Relative invariants for representations of finite dimensional algebras. Manuscripta Mathematica. 2002;108(1):123-33. Domokos M. Finite generating system of matrix invariants. Mathematica Pannonica. 2002;13(2):175-81. Domokos M, Kuzmin SG, Zubkov AN. Rings of matrix invariants in positive characteristic. Journal of Pure & Applied Algebra. 2002;176(1):61. Rings of matrix invariants in positive characteristic Domokos M, Lenzing H. Moduli Spaces for Representations of Concealed-Canonical Algebras. Journal of Algebra. 2002;251(1):371. Moduli Spaces for Representations of Concealed-Canonical Algebras Domokos M, Zubkov AN. Semisimple Representations of Quivers in Characteristic p. Algebras and Representation Theory. 2002;5(3):305-17. Semisimple Representations of Quivers in Characteristic p</it> We prove that the results of Le Bruyn and Procesi on the varieties parameterizing semisimple representations of quivers hold over an algebraically closed base field of arbitrary characteristic. Domokos M. Invariant theory of algebra representations. In: Roggenkamp KW, Stefanescu M, editors. Algebra : representation theory. Vol 28. Dordrecht: Kluwer; 2001. p. 47-61. (NATO Science Series II. Mathematics, Physics and Chemistry; vol 28). Invariant theory of algebra representations Proceedings of the NATO Advanced Study Institute on Algebra – Representation Theory, Constanta, Romania, 2 – 12 August 2000 Domokos M, Drensky V. Gröbner Bases for the Rings of Special Orthogonal and 2×2 Matrix Invariants. Journal of Algebra. 2001;243(2):706-16. Gröbner Bases for the Rings of Special Orthogonal and 2×2 Matrix Invariants We present a Gröbner basis for the ideal of relations among the standard generators of the algebra of invariants of the special orthogonal group acting on k-tuples of vectors. The cases of SO3 and SO4 are interpreted in terms of the algebras of invariants and semi-invariants of k-tuples of 2×2 matrices. In particular, we present in an explicit form a Gröbner basis for the 2×2 matrix invariants. Finally we use a Sagbi basis to show that the algebra of SO2 invariants is a Koszul algebra. Domokos M, Zubkov AN. Semi-invariants of quivers as determinants. Transformation Groups. 2001;6(1):9-24. Semi-invariants of quivers as determinants A representation of a quiver is given by a collection of matrices. Semi-invariantsof quivers can be constructed by taking admissible partial polarizations of the determinant ofmatrices containing sums of matrix components of the representation and the identity matrix asblocks. We prove that these determinantal semi-invariants span the space of all semi-invariantsfor any quiver and any infinite base field. In the course of the proof we show that one canreduce the study of generating semi-invariants to the case when the quiver has no orientedpaths of length greater than one. Domokos M. Poincaré series of semi-invariants of 2x2 matrices. Linear Algebra and its Applications. 2000;310(1-3):183-94. Domokos M. Relative invariants of 3x3 matrix triples. Linear and Multilinear Algebra. 2000;47(2):175-90. Relative invariants of 3x3 matrix triples We present primary and secondary generators for the algebra of polynomial invariantsof the direct product of two copies of the special linear group S13 acting naturally ontriples of 3 x 3 matrices over a field of characteristic zero. We handle also the analogousproblem for triples and quadruples of 2 x 2 matrices Domokos M, Hegedűs P. Noether's bound for polynomial invariants of finite groups. Archiv der Mathematik. 2000;74(3):161-7. Domokos M, Lenzig H. Invariant theory of canonical algebras. Journal of Algebra. 2000;228(2):738-62. Invariant theory of canonical algebras Summary: "Based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations. This is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces (together with the quotient morphisms from the corresponding representation spaces) for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory. By means of a tilting process we extend these results to the invariant theory of concealed-canonical algebras, thus covering the cases of tame hereditary, tame concealed, and tubular algebras, respectively. Our approach yields, in particular, a uniform treatment of an essential part of the invariant theory of extended Dynkin quivers, a topic popular over the years, but stretches far beyond since concealed-canonical algebras of tubular or wild representation type are also covered." Domokos M. Gröbner bases of certain determinantal ideals. Beiträge zur Algebra und Geometrie. 1999;40(2):479-93. Domokos M. Polynomial ideals and identities of matrices. In: Drensky VS, Giambruno A, Sehgal SK, editors. Methods in ring theory : proceedings of the Trento conference. Vol 198. New York: Dekker; 1998. p. 83-95. (Lecture notes in pure and applied mathematics; vol 198). Domokos M. Invariants of quivers and wreath products. Communications in Algebra. 1998;26(9):2807-19. Domokos M. Cayley-Hamilton theorem for 2x2 matrices over the Grassmann algebra Ring theory (Miskolc, 1996). Journal of Pure and Applied Algebra. 1998;133(1-2):69-81. Cayley-Hamilton theorem for 2x2 matrices over the Grassmann algebra Ring theory (Miskolc, 1996) It is shown that the characteristic polynomial of matrices over a Lie nilpotent ring introduced recently by Szigeti is invariant with respect to the conjugation action of the general linear group. Explicit generators of the corresponding algebra of invariants in the case of 2 × 2 matrices over an algebra over a field of characteristic zero satisfying the identity [[x, y], z] = 0 are described. In this case the coefficients of the characteristic polynomial are expressed by traces of powers of the matrix, yielding a compact form of the Cayley-Hamilton equation of 2 × 2 matrices over the Grassmann algebra. Domokos M, Drensky V. A Hilbert-Nagata theorem in noncommutative invariant theory. Transactions of the American Mathematical Society. 1998;350(7):2797-811. A Hilbert-Nagata theorem in noncommutative invariant theory Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata. Domokos M, Popov A. On the degree of nilpotency of the radical of relatively free algebras. Mathematica Pannonica. 1997;8(1):11-6. Domokos M. Criteria for vanishing of Eulerian polynomials on nxn matrices. Linear Algebra and its Applications. 1996;234:181-95. Criteria for vanishing of Eulerian polynomials on nxn matrices Eulerian polynomial identities on n × n matrices were introduced by Szigeti, Tuza and Révész. Here we prove that some Eulerian polynomials are contained in the T-ideal generated by polynomials corresponding to graphs with less vertices. As a by-product we obtain a generalization of a graph-theoretic result of Swan. Then we give a complete description of Eulerian graphs with two vertices such that the corresponding identity is satisfied by the n × n matrix ring over a unitary commutative ring of characteristic 0. Domokos M. Relatively free invariant algebras of finite reflection groups. Transactions of the American Mathematical Society. 1996;348(6):2217-34. Relatively free invariant algebras of finite reflection groups Let G be a finite subgroup of Gln(K) (K is a field of characteristic 0 and n ≥ 2) acting by linear substitution on a relatively free algebra $K\langle x_1, \ldots, x_n\rangle/I$ of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if G is a pseudo-reflection group and I contains the polynomial [[ x2, x1], x1]. Domokos M. Correction to "On algebras satisfying symmetric identities". Archiv der Mathematik. 1995;64(6):552. Domokos M. A Generalization of a theorem of Chang. Communications in Algebra. 1995;23(12):4333-42. A Generalization of a theorem of Chang Szigeti, Tuza and Révész have developed a method in [6] to obtain polynomial identities for the n×n matrix ring over a commutative ring starting from directed Eulerian graphs. These polynomials are called Euler-ian. In the first part of this paper we show some polynomials that are in the T-ideal generated by a certain set of Eulerian polynomials, hence we get some identities of the n×n matrices. This result is a generalization of a theorem of Chang [l]. After that, using this theorem, we show that any Eulerian identity arising from a graph which lias d-fold multiple edges follows from the standard identity of degree d Domokos M. New identities for 3 × 3 Matrices. Linear and Multilinear Algebra. 1995;38(3):207-13. New identities for 3 × 3 Matrices Eulerian polynomial identities on n×nmatrices were introduced by Szigeti, Tuza and Révész in [12]. In this paper we exhibit two Eulerian identities on 3×3 matrices which are not consequences of the earlier known identities. Domokos M. On algebras satisfying symmetric identities. Archiv der Mathematik. 1994;63(5):407-13. Domokos M. Goldie's theorems for involution rings. Communications in Algebra. 1994;22(2):371-80. Goldie's theorems for involution rings In this paper we shall formulate necessary and sufficient conditions for a semiprime ring to be both left and right Goldie in terms of the symmetric notion of biideal instead of one-sided ideals. We use this result to characterize semiprime Goldie rings with involution by conditions consistent with the notion of involution rings, that is, in terms of ascending chain condition for annihilator -biideals and of maximum condition for -biideal direct sums. Domokos M. Eulerian Polynomial Identities and Algebras Satisfying a Standard Identity. Journal of Algebra. 1994;169(3):913-28. Eulerian Polynomial Identities and Algebras Satisfying a Standard Identity Using the trivial observation that one can get polynomial identities on R from the ones of M[sub:k](R) we derive from the Amitsur-Levitzki theorem a subset of the identities on n × n matrices, obtained recently by Szigeti, Tuza, and Révész starting from directed Eulerian graphs, which generate the same T-ideal of the free algebra. After that we show that by this method we get a generating set of the T-ideal of identities on the 2 × 2 matrix ring over a field of characteristic 0. We reformulate a problem on algebras satisfying a standard identity in terms of Eulerian identities and use this equivalence in both directions. We apply a result of Braun to Eulerian identities on 3 × 3 matrices, and we give a simpler example which answers the question investigated by Braun. Finally we give an upper estimation on the minimal degree of the standard identity which is satisfied by the matrix algebra over an algebra satisfying some standard identity.Copyright 1994, 1999 Academic Press, Inc.	CommonCrawl
Mathematical Biosciences & Engineering 2006 , Volume 3 , Issue 4 Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain and Glenn W.A. Rowe 2006, 3(4): 571-582 doi: 10.3934/mbe.2006.3.571 +[Abstract](2160) +[PDF](1589.3KB) Numerical analysis and computational simulation of partial differential equation models in mathematical biology are now an integral part of the research in this field. Increasingly we are seeing the development of partial differential equation models in more than one space dimension, and it is therefore necessary to generate a clear and effective visualisation platform between the mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualisation of the results is more complicated. The scope of this paper is to apply the established marching cubes volume rendering technique to the study of solid tumour growth and invasion, and present an adaptation of the algorithm to speed up the surface rendering from numerical simulation data. As a specific example, in this paper we examine the computational solutions arising from numerical simulation results of a mathematical model of malignant solid tumour growth and invasion in an irregular heterogeneous three-dimensional domain, i.e., the female breast. Due to the different variables that interact with each other, more than one data set may have to be displayed simultaneously, which can be realized through transparency blending. The usefulness of the proposed method for visualisation in a more general context will also be discussed. Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3(4): 571-582. doi: 10.3934/mbe.2006.3.571. Noise-sensitive measure for stochastic resonance in biological oscillators Ying-Cheng Lai and Kwangho Park There has been ample experimental evidence that a variety of biological systems use the mechanism of stochastic resonance for tasks such as prey capture and sensory information processing. Traditional quantities for the characterization of stochastic resonance, such as the signal-to-noise ratio, possess a low noise sensitivity in the sense that they vary slowly about the optimal noise level. To tune to this level for improved system performance in a noisy environment, a high sensitivity to noise is required. Here we show that, when the resonance is understood as a manifestation of phase synchronization, the average synchronization time between the input and the output signal has an extremely high sensitivity in that it exhibits a cusp-like behavior about the optimal noise level. We use a class of biological oscillators to demonstrate this phenomenon, and provide a theoretical analysis to establish its generality. Whether a biological system actually takes advantage of phase synchronization and the cusp-like behavior to tune to optimal noise level presents an interesting issue of further theoretical and experimental research. Ying-Cheng Lai, Kwangho Park. Noise-sensitive measure for stochastic resonance in biological oscillators. Mathematical Biosciences & Engineering, 2006, 3(4): 583-602. doi: 10.3934/mbe.2006.3.583. Lyapunov functions for tuberculosis models with fast and slow progression C. Connell Mccluskey 2006, 3(4): 603-614 doi: 10.3934/mbe.2006.3.603 +[Abstract](3397) +[PDF](222.0KB) The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable. C. Connell Mccluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Mathematical Biosciences & Engineering, 2006, 3(4): 603-614. doi: 10.3934/mbe.2006.3.603. Mathematical modeling of biowall reactors for in-situ groundwater treatment Donna J. Cedio-Fengya and John G. Stevens In this paper we develop a comprehensive model for the remediation of contaminated groundwater in a passive, in-ground reactor, generally known as a biowall. The model is based on our understanding of the component transport and biokinetic processes that occur as water passes through a bed of inert particles on which a biofilm containing active microbial degraders, typically aerobic bacteria, is developing. We give a detailed derivation of the model based on accepted engineering formulations that account for the mass transport of the contaminant (substrate) to the surface of the biofilm, its diffusion into the biofilm to the proximity of a microbe, and its subsequent destruction within that degrader. The model has been solved numerically and incorporated in a robust computer code. Based on representative input values, the results of varying key parameters in the model are presented. The relation between biofilm growth and biowall performance is explored, revealing that the amount of biomass and its distribution within the biowall are key parameters affecting contaminant removal. Donna J. Cedio-Fengya, John G. Stevens. Mathematical modeling of biowall reactors for in-situ groundwater treatment. Mathematical Biosciences & Engineering, 2006, 3(4): 615-634. doi: 10.3934/mbe.2006.3.615. Modeling shrimp biomass and viral infection for production of biological countermeasures H. Thomas Banks, V. A. Bokil, Shuhua Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy and F.C.T. Allnutt In this paper we develop a mathematical model for the rapid production of large quantities of therapeutic and preventive countermeasures. We couple equations for biomass production with those for vaccine production in shrimp that have been infected with a recombinant viral vector expressing a foreign antigen. The model system entails both size and class-age structure. H. Thomas Banks, V. A. Bokil, Shuhua Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy, F.C.T. Allnutt. Modeling shrimp biomass and viral infection for production of biological countermeasures. Mathematical Biosciences & Engineering, 2006, 3(4): 635-660. doi: 10.3934/mbe.2006.3.635. Modelling the human immune response mechanisms to mycobacterium tuberculosis infection in the lungs Gesham Magombedze, Winston Garira and Eddie Mwenje This work elaborates on the effects of cytotoxic lymphocytes (CTLs) and other immune mechanisms in determining whether a TB-infected individual will develop active or latent TB. It answers one intriguing question: why do individuals infected with Mycobacterium tuberculosis (Mtb) experience different clinical outcomes? In addressing this question, we have developed a model that captures the effects of CTLs and the combined effects of CD4+ helper T cells (Th1 and Th2) immune response mechanisms to TB infection. The occurrence of active or latent infection is shown to depend on a number of factors that include effector function and levels of CTLs. We use the model to predict disease progression scenarios, including primary, latency or clearance. Model analysis shows that occurrence of active disease is much attributed to the Mtb pathogen ability to persist outside the intracellular environment and that high levels of CTLs result in latent TB, while low levels of CTLs result in active TB. This is attributed to the CTLs' ability to directly kill infected macrophages and the bacteria inside the infected macrophages. The study suggests directions for further basic studies and potential new treatment strategies. Gesham Magombedze, Winston Garira, Eddie Mwenje. Modelling the human immune response mechanisms to mycobacterium tuberculosis infection in the lungs. Mathematical Biosciences & Engineering, 2006, 3(4): 661-682. doi: 10.3934/mbe.2006.3.661. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong and Qiang Du In this paper, we develop a population balance model for cell aggregation and adhesion process in a nonuniform shear flow. Some Monte Carlo simulation results based on the model are presented for the heterotypic cell-cell collision and adhesion to a substrate under dynamic shear forces. In particular, we focus on leukocyte (PMN)-melanoma cell emboli formation and subsequent tethering to the vascular endothelium (EC) as a result of cell-cell aggregation. The simulation results are compared with the results of experimental measurement. Discussions are made on how we could further improve the accuracy of the population balance type modelling. Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3(4): 683-696. doi: 10.3934/mbe.2006.3.683. Complex spatio-temporal features in meg data Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna and Maide Bucolo Magnetoencephalography (MEG) brain signals are studied using a method for characterizing complex nonlinear dynamics. This approach uses the value of $d_\infty$ (d-infinite) to characterize the system's asymptotic chaotic behavior. A novel procedure has been developed to extract this parameter from time series when the system's structure and laws are unknown. The implementation of the algorithm was proven to be general and computationally efficient. The information characterized by this parameter is furthermore independent and complementary to the signal power since it considers signals normalized with respect to their amplitude. The algorithm implemented here is applied to whole-head 148 channel MEG data during two highly structured yogic breathing meditation techniques. Results are presented for the spatiotemporal distributions of the calculated $d_\infty$ on the MEG channels, and they are compared for the different phases of the yogic protocol. The algorithm was applied to six MEG data sets recorded over a three-month period. This provides the opportunity of verifying the consistency of unique spatio-temporal features found in specific protocol phases and the chance to investigate the potential long term effects of these yogic techniques. Differences among the spatio-temporal patterns related to each phase were found, and they were independent of the power spatio-temporal distributions that are based on conventional analysis. This approach also provides an opportunity to compare both methods and possibly gain complementary information. Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna, Maide Bucolo. Complex spatio-temporal features in meg data. Mathematical Biosciences & Engineering, 2006, 3(4): 697-716. doi: 10.3934/mbe.2006.3.697. On the stabilizing effect of cannibalism in stage-structured population models Bruno Buonomo and Deborah Lacitignola In this paper we give a contribution to the systematic investigation of cannibalism in predator-prey models commenced since the publication of the paper by Kohlmeier and Ebenhöh in 1995. We present a stage-structured predator-prey model and study its dynamics. We use a Hopf bifurcation analysis to prove that cycles are possible and that cannibalism suppresses these cycles; that is, when cannibalism attack rate is increased so that it passes a critical value, the coexistence steady state changes from being unstable to being stable. Numerical simulations are provided together with the mathematical analysis. Our modelling approach is based on balance arguments and a comparison with some early models which predict that a destabilizing effect of cannibalism is performed. Our results agree with the output of growth simulation for some cannibalistic copepods. Bruno Buonomo, Deborah Lacitignola. On the stabilizing effect of cannibalism in stage-structured population models. Mathematical Biosciences & Engineering, 2006, 3(4): 717-731. doi: 10.3934/mbe.2006.3.717.	CommonCrawl
\begin{document} \title{A single cold atom as efficient stationary source of EPR-entangled light} \author{David Vitali} \affiliation{CNISM and Dipartimento di Fisica, Universit\`a di Camerino, 62032 Camerino, Italy} \author{Giovanna Morigi} \affiliation{Grup d'Optica, Departament de Fisica, Universitat Autonoma de Barcelona, 08193 Bellaterra, Spain} \author{J\"urgen Eschner} \affiliation{ICFO - Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain} \date{\today} \begin{abstract} The Stokes and anti-Stokes components of the spectrum of resonance fluorescence of a single trapped atom, which originate from the mechanical coupling between the scattered photons and the quantized motion of the atomic center of mass, exhibit quantum correlations which are of two-mode-squeezing type. We study and demonstrate the build-up of such correlations in a specific setup, which is experimentally accessible, and where the atom acts as efficient and continuous source of EPR-entangled, two-mode squeezed light. \end{abstract} \pacs {42.50.Dv, 32.80.Qk, 32.80.Lg } \maketitle \section{Introduction} The control of atom-photon interaction is object of intensive research for its potentialities in quantum networking. In fact, several experimental realizations have accessed novel regimes of engineering atom-photon interactions and have opened promising perspectives for implementing controlled nonlinear dynamics with simple quantum optical systems. Fundamental steps in this direction have been, amongst others, the generation of entangled light in atomic ensembles \cite{Kimble03, Giacobino}, atomic memory for quantum states of light \cite{LukinScience03, Lukin03, Polzik04, KuzmichNature05, LukinNature05}, and entanglement of remote ensembles \cite{Julsgaard01, Chou2005, Kuzmich06}. At the single atom level, entanglement between a single atom and its emitted photon \cite{Eberly} has been demonstrated in \cite{Monroe04, Weinfurter2005}, while in cavity quantum electrodynamics generation of quantum light has been achieved, like lasing at the single atom level \cite{An94, Kimble-atomlaser}, controlled single-photon generation \cite{Kuhn02, Kimble-photon, Keller04, Grangier05}, as well as quantum state and entanglement engineering in the microwave regime~\cite{MicroCQED}. Quantum networking with single trapped atoms or ions shows several advantages, due to the high degree of control one can achieve on these systems~\cite{Grangier05, Leibfried03, NeutralAtoms}. Control can be gained on the internal as well as on the external degrees of freedom, which can both be interfaced with light by exchange of angular and linear momentum. In particular, by coupling the atomic external degrees of freedom with photons via the mechanical effect of light, atom-photon interfaces for continuous variables can be implemented even at the level of a single atom~\cite{Ze-Parkins99, Parkins02, PRL, Morigi06}. This concept has been specifically applied in~\cite{PRL, Morigi06}, where the realization of a pulsed optical parametric amplifier based on a single cold trapped atom inside a high-finesse optical cavity was proposed, and it was shown theoretically that this system allows for the controlled, quantum-coherent generation of entangled light pulses by exploiting the mechanical effects of atom-photon interaction. In this manuscript we investigate the quantum correlations between the Stokes and anti-Stokes sidebands of the resonance fluorescence of a trapped atom, i.e.\ between the spectral components which are due to the coupling of the electromagnetic field to the atom's oscillatory motion~\cite{Lindberg86, Cirac93, Raab00, Bienert06}. The spectrum is studied for an atom tightly confined inside a resonator and continuously driven by a laser, in the setup sketched in Fig.~\ref{Fig:1}. This setup has been considered in~\cite{PRL, Morigi06} for the case of pulsed excitation, where scattering could be considered coherent. In the present work, the atom is continuously driven and hence both coherent and incoherent scattering processes determine the dynamics of the system. We find that in a suitable parameter regime the Stokes and anti-Stokes spectral components of the resonance fluorescence are two-mode squeezed, that is, their amplitude and phase quadratures are quantum correlated. In fact, the variance of the difference of the amplitude quadratures of the two sideband modes, as well as the variance of the sum of their phase quadratures, are squeezed below the shot noise limit, hence reproducing the salient properties of the entangled, simultaneous eigenstate of relative distance and total momentum of two particles, as considered in the original EPR paradox \cite{EPR, milwal}. In our model, entanglement between the modes originates from the mechanical coupling of the electromagnetic field with the quantum motion of the atom, and it is endorsed by a specific setup, which achieves resonant emission of the Stokes and anti-Stokes photons. In this regime, the single atom acts as an efficient continuous source of EPR-entangled, two-mode squeezed light. Conventionally, two-mode squeezed states emerge from the nonlinear optical interaction of a laser with a crystal, i.e.\ from parametric amplification or oscillation. As such, the phenomenon is the result of many-atom dynamics, often described by a simple nonlinear polarization model. In the single-atom case novel features appear which are due to the coherent microscopic dynamics. Our study allows us to identify the dependence of these features on the external parameters, thereby giving us insight into how macroscopic properties arise from microscopic dynamics in this particular non-linear process. Moreover, we find peculiar spectral characteristics of the squeezing which are unique to this system, and which we trace back to the interplay of the various time scales of the dynamics. In a more general context, our study is connected to previous work on the quantum features of the spectrum of resonance fluorescence~\cite{Aspect80, Walls81, Mandel82, Schrama92, Nienhuis93, Narducci93, Matos94, Jakob99a, Lindberg86, Cirac93}, and to recent experimental and theoretical studies on quantum correlations in the light scattered by atoms~\cite{LukinScience03, Kimble03, Polzik04, Giacobino, Raizen87, Jakob99b, Serra05}, by semiconductor microcavities~\cite{Polaritons}, and by macroscopic mirrors~\cite{Mancini,Pirandola03}. This article is organized as follows. In Sec.~\ref{Sec:Review} the basic coherent dynamics, giving rise to quantum correlations between the Stokes and anti-Stokes components of the spectrum of resonance fluorescence, are briefly reviewed, and the important time scales are introduced. In Sec.~\ref{Sec:Scattering} the theoretical model is described in detail and the relevant scattering processes in the system are identified and discussed. In Sec.~\ref{Sec:Spectrum} the spectrum of squeezing is evaluated using Quantum Langevin Equations; for a quick overview of the main results without the full theoretical elaboration, the reader may first skip this part and jump to Sec.~\ref{Sec:Results} where the squeezing characteristics are calculated for a specific, experimentally achievable physical system. Finally, Sec.~\ref{Sec:Conclusions} presents the conclusions and an outlook. \section{EPR-entanglement of light at the cavity output} \label{Sec:Review} In this section we briefly review the coherent dynamics, described previously in Refs.~\cite{PRL, Morigi06}, which lead to two-mode squeezing between the Stokes and anti-Stokes modes in the light scattered by a trapped, laser-driven atom. We thus first ignore incoherent processes and focus on the pulsed dynamics which can be obtained in a suitable parameter regime with a setup like the one shown in Fig.~\ref{Fig:1}. \begin{center} \begin{figure} \caption{ Layout of the system. A single atom is confined by an external potential inside an optical cavity and is driven by a laser. The cavity is resonant with the motion-induced Stokes and anti-Stokes components of the resonance fluorescence. Correlations between these spectral component are measured in the cavity output. The orientation of the considered vibrational mode has non-zero projection onto the laser direction. A possible geometry to implement the system would be an $F=0$ to $F'=1$ atomic transition with the quantization axis $\vec{B}$ along the cavity axis, and $\vec{B}$, laser wave vector, and laser polarisation mutually orthogonal, and a motional mode parallel to the laser direction. More details can be found in Refs.~~\cite{PRL, Morigi06} where pulsed coherent excitation was considered. In the present paper we deal with continuous laser excitation. } \label{Fig:1} \end{figure} \end{center} The trapped atom is coupled to an optical cavity of which two modes are resonant with the Stokes and anti-Stokes sidebands, respectively. For short times the laser-induced resonant interaction between the center-of-mass oscillation, denoted by annihilation and creation operators $b$ and $b^{\dagger}$, and the two cavity modes, represented by operators $a_j$ and $a_j^{\dagger}$ ($j=1,2$), is described by the effective Hamiltonian in the interaction picture \begin{eqnarray} \label{W-eff} W_{\rm eff}={\rm i}\hbar\chi_1 a_1^{\dagger}b^{\dagger}+{\rm i}\hbar\chi_2a_2^{\dagger}b+{\rm H.c}~, \end{eqnarray} where the scalars $\chi_j$ indicate the strength of the coupling. This Hamiltonian generates periodic dynamics, provided that $\|\chi_2\|>\|\chi_1\|$, with an angular frequency \begin{equation} \Theta=\sqrt{\|\chi_2\|^2-\|\chi _1\|^2}~. \label{thetabig} \end{equation} The time-evolution of the operators, in the Heisenberg representation, is given by~\cite{Pirandola03} \begin{eqnarray} \label{a_1} a_1(t) &=&\frac{\chi_1}{\Theta}b^{\dagger}(0)\sin\Theta t + \frac{1}{\Theta^2}\left[\|\chi_2\|^2-\|\chi_1\|^2\cos\Theta t\right]a_1(0)~,\nonumber \\ & &-\frac{\chi_1\chi_2}{\Theta^2}\left[1-\cos\Theta t\right]a_2^{\dagger}(0)\\ a_2(t) &=&\frac{\chi_2}{\Theta}b(0)\sin\Theta t+ \frac{\chi_1\chi_2}{\Theta^2}\left[1-\cos\Theta t\right]a_1^{\dagger}(0)\nonumber\\ & &-\frac{1}{\Theta^2}\left[\|\chi_1\|^2-\|\chi_2\|^2\cos\Theta t\right]a_2(0)~,\\ b(t) &=&b(0)\cos\Theta t+\frac{1}{\Theta}\left[-\chi_2^a_2(0)+\chi_1 a_1^{\dagger}(0 )\right]\sin\Theta t~.\nonumber\\ \label{b} \end{eqnarray} In general these solutions describe tripartite entanglement among cavity modes and center-of-mass oscillator~\cite{Pirandola03}. An interesting situation is found after half a period, for $T_{\pi}=\pi/\Theta$. At this time (modulus $2\pi$) the center-of-mass oscillator is uncorrelated with the cavity modes, which exhibit EPR-type entanglement~\cite{PRL,Morigi06}. Clearly, this description is approximate, and valid only when incoherent processes can be neglected. In the present work we consider the situation in which the atom is continuously driven by the laser field, such that quantum noise and dissipative processes affect the dynamics relevantly. We show that steady state entanglement, i.e.\ quantum-correlated spectral fluctuations in the two-mode cavity output field, is found also under these conditions. The details of this entanglement will depend on the comparison between the time scale set by the coherent dynamics, $\Theta^{-1}$, and the time scales of the dissipative processes, $\kappa^{-1}$ for loss of photons from the cavity, and $\gamma^{-1}$ for spontaneous scattering from the atom. In particular, we will show that the squeezing spectrum shows distinct, qualitatively different features in the regimes $\Theta<\kappa$, $\Theta=\kappa$, and $\Theta>\kappa$. The reader is referred to Sec.~\ref{Sec:Results}, where the spectra for different parameter regimes are reported. \section{Scattering processes} \label{Sec:Scattering} The purpose of this section is to discuss the coherent and incoherent scattering processes determining the dynamics of the system. We will present these processes using physical pictures derived from the scattering matrix under moderate simplifications, in order to illustrate the more rigorous derivations presented in the subsequent section. We first introduce the model, and then identify the scattering processes and determine the corresponding rates. \subsection{Model} \label{Sec:II} We consider an atom of mass $M$ inside an optical resonator and driven by a laser. The atomic motion is confined by an external potential, which we assume sufficiently steep in the radial direction so that the motion in this plane can be considered frozen out. We denote by $x$ the axis of the remaining one-dimensional atomic center-of-mass motion. Moreover, we assume that only the atomic dipole transition between ground state $\|g\rangle$ and excited state $\|e\rangle$ couples relevantly to the fields, such that we can restrict the electronic dynamics to these two states. The atomic dipole is laser-driven, and it couples to two modes ($j=1,2$) of the resonator, as well as to the external modes of the electromagnetic field. The cavity modes couple also to the external modes of the electromagnetic field through the imperfect mirrors of the resonator. The total dynamics is governed by the Hamiltonian $$H=H_0+W,$$ where $H_0$ is the self-energy of the system of atom and fields, and $W$ describes their mutual interaction, as well as the coupling between the cavity modes and the external modes through the finite transmission at the cavity mirrors. We now introduce each term in detail, and discuss the dynamics in the reference frame of the laser at the angular frequency $\omega_L$. We decompose $H_0$ according to \begin{equation} H_0=H_a+H_c+H_{\rm emf}~. \end{equation} Here, $H_a$ is the Hamiltonian for the relevant atomic degrees of freedom, \begin{equation} H_a=-\hbar\Delta\|e\rangle\langle e\|+H_{\rm mec}~, \label{H:atom} \end{equation} where $\Delta=\omega_L-\omega_0$ is the detuning of the laser from the dipole transition at the angular frequency $\omega_0$, and \begin{equation} \label{H:mec} H_{\rm mec}=\hbar\nu\left(b^{\dagger}b+\frac{1}{2}\right) \end{equation} describes the harmonic motion of the atomic center of mass at angular frequency $\nu$, as determined by an external potential, where $b,b^{\dagger}$ are the annihilation and creation operators, respectively, of a quantum of vibrational energy $\hbar\nu$. In particular, the atomic position is given by $x=\sqrt{\hbar/2M\nu}(b+b^{\dagger})$. We denote by $\|n\rangle$ the eigenstates of $H_{\rm mec}$ at energy $\hbar\nu (n+1/2)$. The Hamiltonian for the cavity modes, which couple appreciably to the dipole transition, is \begin{equation} H_c=-\sum_{j=1,2}\hbar\delta_ja_j^{\dagger}a_j~, \end{equation} where $\delta_j=\omega_L-\omega_j$ are the detunings of the laser from the frequencies $\omega_j$ of two optical modes, and $a_j,a_j^{\dagger}$ are the respective annihilation and creation operators of a quantum of energy $\hbar\omega_j$, i.e.\ a photon in mode $j$. We denote by $\|n_1,n_2\rangle$ the eigenstates of $H_c$ at energy $-\hbar\delta_1n_1-\hbar\delta_2n_2$, and consider the situation in which the mode frequencies fulfill the relation \begin{equation} \omega_2 - \omega_1 = 2\nu^{\prime}~. \end{equation} where \begin{equation} \nu^{\prime}=\nu+\delta\nu \end{equation} and $\delta\nu$ takes into account radiative shifts, such that cavity modes 1 and 2 can be simultaneously resonant with the Stokes and the anti-Stokes transitions. This contribution will be discussed in Sec.~\ref{Sec:ac-Shift} and determined in Sec.~\ref{Sec:QLE}. Finally, the modes of the electromagnetic field external to the cavity possess the free Hamiltonian $$H_{\rm emf}=-\hbar\sum_{\bf k_j}\delta_{\bf k_j}r^{\dagger}_{\bf k_j}r_{\bf k_j} -\hbar\sum_{\bf k_s}\delta_{\bf k_s}r^{\dagger}_{\bf k_s}r_{\bf k_s}~,$$ where $r_{\lambda}$, $r^{\dagger}_{\lambda}$ are annihilation and creation operators, respectively, of a photon at angular frequency $\omega_{\lambda}=\omega_L-\delta_{\lambda}$, wavevector ${\bf k_{\lambda}}$ and polarization ${\bf e_{\lambda}}$. Here, the subscripts $\lambda={\bf k_s}$ and $\lambda={\bf k_j}$ indicate the modes of the field which couple to the dipole and to the cavity modes (through the mirrors), respectively. The interaction term \begin{equation} W=H_{aL}+H_{ac}+W_{\bf k_s}+W_{\bf k_j} \end{equation} describes the couplings among atom and fields, decomposed into four terms which correspond to the coupling between atom and laser ($H_{aL}$), atom and cavity modes ($H_{ac}$), atom and modes of the external electromagnetic field ($W_{\bf k_s}$), and cavity modes and external electromagnetic field ($W_{\bf k_j}$). We discuss these terms in the Lamb-Dicke regime, when the atomic motion is well localized over the wavelengths of the fields, such that the Lamb-Dicke parameter $\eta=\sqrt{\hbar k^2/2M\nu}$ is small, $\eta\ll 1$. At lowest order in $\eta$, the coupling between laser and dipole has the form~\cite{FootnoteEta} \begin{eqnarray} \label{HL} H_{aL} &=&\hbar\Omega\sigma^{\dagger} \Bigl[\left(1-\frac{\eta^2}{2}\cos^2\theta_L(2b^{\dagger}b+1)\right)\\ &+&{\rm i}\eta\cos\theta_L(b^{\dagger}+b) +{\rm O}(\eta^2)\Bigr] +{\rm H.c.}~,\nonumber \end{eqnarray} with $\sigma=\|g\rangle\langle e\|$ the dipole lowering operator and $\sigma^{\dagger}$ its adjoint, $\Omega$ the Rabi frequency, and $\theta_L$ the angle between the direction of propagation of the laser and the motional axis $\hat{x}$. In what follows we denote the moduli of all relevant wave vectors by $k$, as their differences are negligible. The coupling between the dipole and the cavity modes is represented by \begin{eqnarray} \label{Hint} H_{ac} &=& \hbar\sum_{j=1,2} g_j\cos\phi_ja_j\sigma^{\dagger} \Bigl[\left(1-\frac{\eta^2}{2}\cos^2\theta_c(2b^{\dagger}b+1)\right) \nonumber \\ &-&\eta\cos\theta_c\tan\phi_j(b^{\dagger}+b)\Bigr]+{\rm H.c. +{\rm O}(\eta^2)}~, \end{eqnarray} where $g_j$ is the coupling strength of the dipole to mode $j$, and the cavity axis forms an angle $\theta_c$ with the axis $\hat{x}$ of the motion. The angle $\phi_j$ takes into account the position of the trap center inside the standing wave of the cavity. Finally, the terms \begin{eqnarray} W_{\bf k_s} &=&\sum_{\bf k_s}\hbar g_{\bf k_s}\sigma^{\dagger}r_{\bf k_s}\Bigl[\left(1- \frac{\eta^2}{2}\cos^2\theta_{\bf k_s}(2b^{\dagger}b+1)\right)\\ &+&{\rm i}\eta\cos\theta_{\bf k_s}(b+b^{\dagger})+{\rm O}(\eta^2)\Bigr]+{\rm H.c.})~,\nonumber\\ W_{\bf k_j} &=&\sum_{\bf k_j}\hbar g_{\bf k_j}(a_j^{\dagger}r_{\bf k_j}+{\rm H.c.}) \end{eqnarray} describe the coupling of atom and cavity to the modes of the external e.m.-field. Here, $W_{\bf k_s}$ is the coupling of the dipole, at Rabi frequencies $g_{\bf k_s}$, with the external modes, whose wave vectors form angles $\theta_{\bf k_s}$ with the motional axis. This coupling gives rise to the finite linewidth $\gamma$ of the excited state, $\gamma=2\pi\rho_{\bf k_s}(\omega_0)\|g_{\bf k_s}(\omega_0)\|^2$, with $\rho_{\bf k_s}(\omega_0)$ density of states of the e.m.-field coupling to the atomic dipole at angular frequency $\omega_0$. The term $W_{\bf k_j}$ describes the coupling of the cavity modes with the external modes at strength $g_{\bf k_j}$. This coupling gives rise to the linewidth of the cavity modes $\kappa_j=\pi \|g_{\bf k_j}\|^2\rho_{\bf k_j}(\omega_j)$, with $\rho_{\bf k_j}(\omega_j)$ density of states of the e.m.-field coupling to the cavity modes at angular frequency $\omega_j$. \subsection{Basic scattering processes} \label{Sec:Scattering:I} We consider the limit in which the atom is far-detuned from cavity modes and laser, $\|\Delta\| \gg \gamma, \delta_j, g_j, \Omega$. In this limit all terms of $W$ are weak perturbations to the dynamics. We assume that the system is in the initial state \begin{equation} \|\psi_i\rangle=\|g,n;0_1,0_2;0_{\bf k_j};0_{\bf k_s}\rangle, \end{equation} with energy $E_i=\hbar\nu n$, where the atom is in the ground state $\|g\rangle$, the center-of-mass oscillator is in the number state $\|n\rangle$, and the cavity modes and the external e.m.-field are in the vacuum state, $\|0_1,0_2;0_{\bf k_j};0_{\bf k_s}\rangle$. The scattering matrix elements between the initial state and all possible final states $\|\psi_f\rangle$, with energy $E_f$, have the form \begin{equation} {\cal S}_{if}=\delta_{if}-2\pi{\rm i}\delta(E_f-E_i){\cal T}_{if} \end{equation} where $\delta_{if}$ is the Kronecker-delta, $\delta(E_f-E_i)$ is a delta-function giving energy conservation between initial and final states, and ${\cal T}_{if}$ is the transition matrix to be evaluated in lowest order in perturbation theory, \begin{eqnarray} {\cal T}_{if}=\langle \psi_f\|W\|\psi_i\rangle+\langle \psi_f\|W\frac{1}{E_i-H_{\rm eff}}W\|\psi_i\rangle \end{eqnarray} with \begin{equation} H_{\rm eff}=-\hbar\left(\Delta+{\rm i}\frac{\gamma}{2}\right) \|e\rangle\langle e\|+\hbar\nu b^{\dagger}b -\hbar\sum_{j=1,2}\left(\delta_j+{\rm i}\kappa_j\right)a_j^{\dagger}a_j \end{equation} We now consider all possible scattering transitions to resonant states, i.e.\ to final states $\|\psi_f\rangle$ at energy $E_f=E_i$. \begin{center} \begin{figure} \caption{Basic scattering processes. (a): A laser photon is absorbed and emitted by the atom, without coupling to the cavity mode. (b) A laser photon is scattered into the cavity mode and then rescattered by the atom into the external modes of the electromagnetic field. (c) A laser photon is scattered by the atom into the cavity mode, and then it is transmitted by the cavity mirror into the modes of the external electromagnetic field.} \label{Fig:0b} \end{figure} \end{center} \subsubsection{Scattering of laser photons into the external e.m.-field} \label{Laser:Free} We consider the scattering of a laser photon into the external e.m.-field by spontaneous emission, hence coupling of $\|\psi_i\rangle$ to the final states $\|\psi_{\bf k_s}\rangle=\|g,n^{\prime};0_1,0_2;0_{\bf k_j};1_{\bf k_s}\rangle$. This process is sketched in Fig.~\ref{Fig:0b}(a). Here, the coupling with the cavity mode is neglected, as the cavity is far-detuned from the dipole, and the rate of this process can be approximated by the scattering rate of the atom in free space, \begin{eqnarray} \Gamma_{if}^{\rm sp} &\approx&\frac{1}{\gamma}\Bigl(\|t_{0}^{\rm sp}\|^2\delta_{n^{\prime},n}+\|t_{+1}^{\rm sp}\|^2(n+1)~\delta_{n^{\prime},n+1}\\ & & +\|t_{-1}^{\rm sp}\|^2n~\delta_{n^{\prime},n-1}\Bigr), \nonumber \end{eqnarray} where \begin{eqnarray} &&t_{0}^{\rm sp}=\frac{\gamma\Omega}{\Delta+{\rm i}\gamma/2},\label{T:sp:0}\\ &&t_{+1}^{\rm sp}=\eta\gamma\Omega\left(\frac{\cos\theta_L}{\Delta-\nu+{\rm i}\gamma/2}+\frac{\cos \theta_{\bf k_s}}{\Delta+{\rm i}\gamma/2}\right)\label{T:sp:+},\\ &&t_{-1}^{\rm sp}=\eta\gamma\Omega\left(\frac{\cos\theta_L}{\Delta+\nu+{\rm i}\gamma/2}+\frac{\cos\theta_{\bf k_s}}{\Delta+{\rm i}\gamma/2}\right)~. \label{T:sp:-} \end{eqnarray} The process described by amplitude~(\ref{T:sp:0}) does not affect the dynamics of the cavity modes nor that of the center-of-mass motion. In contrast, the amplitudes~(\ref{T:sp:+}) and~(\ref{T:sp:-}) are coherent superpositions of scattering processes involving, respectively, the mechanical effect of the laser and of the emitted photon on the atomic motion~\cite{Cirac93, Bienert06}, thereby affecting the coherence of the motional state. Their rate is $\gamma_b \approx \eta^2 (\cos^2\theta_L + \alpha) \gamma \Omega^2/ \Delta^2$, where $\alpha$ describes the angular dispersion of the spontaneously emitted photons, determined by the quantum numbers of the atomic transition \cite{Stenholm86}. \subsubsection{Scattering of laser photons into the cavity modes} \label{Sec:Chi} Next we discuss the processes in which a laser photon is scattered into one of the cavity modes, thereby coupling the initial state $\|\psi_i\rangle$ to the states $\|\psi_1\rangle=\|g,n^{\prime};1_1,0_2;0_{\bf k_j};0_{\bf k_s}\rangle$ or $\|\psi_2\rangle=\|g,n^{\prime};0_1,1_2;0_{\bf k_j};0_{\bf k_s}\rangle$. As these states are not stable, but resonantly coupled to the continuum of states $\|g,n^{\prime}; 0_1,0_2; 1_{\bf k_j}; 0_{\bf k_s}\rangle$ by cavity decay, the correct final states of these scattering processes describe the processes sketched in Fig.~\ref{Fig:0b}(c) and have the form \begin{equation} \|\psi_{\bf k_j}\rangle=\sqrt{Z_{\bf k_j}}\left(1+\frac{Q_j}{E_{\bf k_j}-H}W_{\bf k_j}\right)\|\psi_j\rangle, \end{equation} where $Q_j$ projects onto the subspace orthogonal to $\|\psi_j\rangle$, and $Z_{\bf k_j}$ ensures the normalization of the state. Furthermore, $Z_{\bf k_j}$ gives the occupation probability of state $\|\psi_j\rangle$, since $Z_{\bf k_j}=\|\langle\psi_j\|\psi_{\bf k_j}\rangle\|^2$. The coupling rate between state $\|\psi_i\rangle$ and states $\|\psi_{\bf k_j}\rangle$ takes the form \begin{eqnarray} \Gamma_{if_j}^{\rm cav} &\approx&\frac{2\kappa_j}{\delta_j^2+\kappa_j^2}~\|t_{0}^{\rm cav}\|^2~\delta_{n^{\prime},n}\label{sum}\\ & &+\frac{2\kappa_j}{(\delta_j-\nu)^2+\kappa_j^2}~ \|t_{j,+}^{\rm cav}\|^2(n+1)~\delta_{n^{\prime},n+1}\nonumber\\ & &+\frac{2\kappa_j}{(\delta_j+\nu)^2+\kappa_j^2}~\|t_{j,-}^{\rm cav}\|^2n~\delta_{n^{\prime},n-1}\nonumber, \end{eqnarray} where \begin{eqnarray} &&t_{0}^{\rm cav}=\Omega g_j^{}\cos\phi_j\frac{1}{\Delta+{\rm i}\gamma/2},\\ &&t_{j,+}^{\rm cav}=\eta \Omega g_j^{}\cos\phi_j\left[\frac{{\rm i}\cos\theta_L}{\Delta-\nu+{\rm i}\gamma/2} -\frac{\cos\theta_c\tan\phi_j}{\Delta+{\rm i}\gamma/2}\right],\label{T:int:1}\\ &&t_{j,-}^{\rm cav}=\eta \Omega g_j^{}\cos\phi_j\left[\frac{{\rm i}\cos\theta_L}{\Delta+\nu+{\rm i}\gamma/2}-\frac{\cos\theta_c\tan\phi_j}{\Delta+{\rm i}\gamma/2}\right].\label{T:int:2}\end{eqnarray} Like in Eqs.~(\ref{T:sp:+}) and (\ref{T:sp:-}), we recognize on the RHS of Eqs.~(\ref{T:int:1}) and~(\ref{T:int:2}) the coherent addition of two scattering amplitudes, here representing the mechanical effects of the laser and of the cavity, respectively~\cite{Cirac95, Zippilli05}. These processes are at the basis of the coherent coupling between the atomic motion and the cavity modes described by Hamiltonian~(\ref{W-eff}), where $\chi_1=-{\rm i}t_{1,+}^{\rm cav}$ and $\chi_2=-{\rm i}t_{2,-}^{\rm cav}$. We are interested in the regime where energy can be stored in the cavity modes through this coupling, which requires $\|\chi_1\|,\|\chi_2\|\gg \gamma_b$ as a necessary condition. In this situation, it is visible from the equations that in the limit $\kappa_j\ll\nu$, by choosing $\delta_1=\nu$ and $\delta_2=-\nu$ one can achieve the optimum enhancement of the scattering of a laser photon into mode 1 accompanied by the excitation of the motion by one vibrational quantum, and of the scattering of a laser photon into mode 2 accompanied by the de-excitation of the motion by one vibrational quantum. Note that these scattering terms, $t_{j,\pm}^{\rm cav}$, have an incoherent component which scales with $\gamma/\|\Delta\pm\nu\|$. Therefore, in general coherent dynamics can only be achieved when $\gamma\ll\|\Delta\|$, on a time scale such that incoherent terms are negligible. Moreover, the condition $\gamma\ll\nu$ is also required in order to create quantum correlations between the two cavity modes, since the difference between the two coupling strengths $\chi_1$ and $\chi_2$ determines the typical time scale on which entanglement is established, see Sec.~\ref{Sec:Review} and~\cite{Morigi06}. \subsubsection{Scattering of cavity photons into the external e.m.-field} \label{Sec:cav-noise} Assuming that photons have been coherently scattered into the cavity modes, they can be re-absorbed by the atom and emitted spontaneously into the external e.m.-field, as sketched in Fig.~\ref{Fig:0b}(b). In order to focus on the evaluation of the corresponding element of the scattering matrix, we consider the regime of very small cavity loss rate, i.e.\ we assume stable cavity modes and ignore, for the clarity of the picture, cavity decay. Be the initial state \begin{equation} \label{Initial:2} \|\psi_{i,m}\rangle=\|g,n;m_1,m_2;0_{\bf k_j};0_{\bf k_s}\rangle \end{equation} at energy $E_{i,m} = \hbar\nu n - \hbar m_1 \delta_1 - \hbar m_2 \delta_2$, with the atom in $\|g\rangle$, the center-of-mass oscillator in the number state $\|n\rangle$, the cavity modes in the Fock states $\|m_1\rangle$ and $\|m_2\rangle$, and the external e.m.-field in the vacuum state, $\|0_{\bf k_j};0_{\bf k_s}\rangle$. This state is coupled to the states \begin{eqnarray} \|\psi_{f,m_1^{\prime}}\rangle=\|g,n;m_1-1,m_2;0_{\bf k_j};1_{\bf k_s}\rangle\\ \|\psi_{f,m_2^{\prime}}\rangle=\|g,n;m_1,m_2-1;0_{\bf k_j};1_{\bf k_s}\rangle \end{eqnarray} by absorption of a cavity photon and spontaneous emission. We evaluate the corresponding rate under the assumption, that $\tan\phi_j=0$, i.e., there are no mechanical effects of the resonator on the atom at first order in $\eta$, and find an effective loss rate of the cavity modes \begin{eqnarray} \Gamma_{if_j}^{\rm cav-sp} &=&\gamma\|g_j\|^2\left\|\frac{\sqrt{m_j}}{\Delta-\delta_j+{\rm i}\gamma/2}\right\|^2~. \label{cav-sp} \end{eqnarray} It should be noted that these processes arise from atomic scattering of a laser photon into the cavity modes, which is then rescattered by atomic emission into the external modes of the e.m.-field. Hence, these processes can interfere with atomic scattering of a laser photon, in the limit discussed in Sec.~\ref{Laser:Free}, in which the coupling to the cavity plays no role. In these calculations we have not considered the coherent addition of these two noise effects, but we will consider phase relations and possible interference in these noise sources when studying the dynamics with the quantum Langevin equations in Sec.~\ref{Sec:QLE}. \subsubsection{a.c.-Stark shift of the ground state energy} \label{Sec:ac-Shift} Since the efficiency of production of two-mode squeezed light is based on the resonant enhancement of two-photon processes, it is important to consider systematically radiative corrections to the resonance frequencies in the implementation of the dynamics in Sec.~\ref{Sec:Review}. Therefore, we now evaluate corrections to the energy of state $\|\psi_{i,m}\rangle$, Eq.~(\ref{Initial:2}), due to far-off resonance coupling in the limit of very small cavity decay rates. When considering the a.c.-Stark shift of state $\|\psi_{i,m}\rangle$, we find three contributions, each associated to a different kind of coupling: (i) the a.c.-Stark shift due to the off-resonant laser coupling with the excited state at zero order in the mechanical effects, $\delta\omega_0\sim\Omega^2/\Delta$ for $\|\Delta\|\gg\gamma$. It leads to a shift $\delta\omega_0$ of the dipole resonance frequency. The mechanical effects of the laser on the atoms give rise to (ii) a contribution which is linear in the number of vibrational excitation, and can hence be considered a renormalization of the trap frequency. This a.c.-Stark shift reads \begin{eqnarray} \label{deltanu_b} \delta\nu_b &\approx&\eta^2\cos^2\theta_L\Omega^2 b^{\dagger}b \\ &\times &{\rm Re}\left\{ \frac{1}{\Delta+\nu+{\rm i}\gamma/2}+\frac{1}{\Delta-\nu+{\rm i}\gamma/2}-\frac{1}{\Delta+{\rm i}\gamma/2}\right\}\nonumber \\ &= &\eta^2\cos^2\theta_L\Omega^2~b^{\dagger}b \nonumber\\ &\times &\left(\frac{2\Delta(\Delta^2-\nu^2+\gamma^2/4)}{(\Delta^2-\nu^2+\gamma^2/4)^2+\gamma^2\nu^2} -\frac{\Delta}{\Delta^2+\gamma^2/4}\right).\nonumber \end{eqnarray} Finally, off-resonant coupling of the cavity mode with the dipole transition gives rise to an a.c.-Stark shift of the cavity mode levels, which reads at leading order \begin{equation} \label{deltaomega_j} \delta\omega_j\approx\frac{\|g_j\|^2\cos^2\phi_j (\Delta-\delta_j)}{(\Delta-\delta_j)^2+\gamma^2/4}a_j^{\dagger}a_j+{\rm O}(\eta^2). \end{equation} These shifts are in general not small and should be taken into account, when aiming at the resonant enhancement of certain processes over others. It should be remarked that the correction to $\delta\omega_j$ in Eq.~(\ref{deltaomega_j}) which is at second order in $\eta$ arises from the mechanical effects of the interaction between resonator and center-of-mass motion. This term is non-linear, as it is a shift which depends on the number of vibrational excitation, but is a negligible contribution to $\delta\omega_j$. On the other hand, this term gives rise to an additional contribution to the a.c.-Stark shift of the center-of-mass motion, which is of the same order as $\delta\nu_b$ and depends on the number of photons. Its effect is detrimental, as the resulting spectrum of the center-of-mass excitations deviates from the one of a harmonic oscillator. In the system we consider we will neglect this contribution, focussing onto the regime in which the mechanical effects of the cavity mode can be neglected. This corresponds to situations, where the motion, for instance, is almost orthogonal to the cavity wave vector, $\|\cos\theta_c\|\ll 1$. \section{Spectrum of light at the cavity output} \label{Sec:Spectrum} In this section we evaluate the spectrum of the light transmitted by the cavity mirror. The spectrum is best evaluated using the quantum Langevin equations for the operators $a_j$, $a_j^{\dagger}$ and $b$. The equations we obtain are rather involved, however the physical meaning of each term can be identified by comparison with the rates of the scattering processes discussed in the previous section. \subsection{Quantum Langevin Equations} \label{Sec:QLE} We shall study the dynamics using the quantum Langevin equations (QLE) of the system. For convenience, we write the interaction Hamiltonian of the atom with the laser and the cavity fields as \begin{equation}\label{hinteff} H_{int}=H_{aL}+H_{ac}=\hbar\left(\sigma^{\dagger}B+\sigma B^{\dagger}\right), \end{equation} where \begin{eqnarray} && B=\Omega\left(1-\frac{\eta^2}{2}\cos^2 \theta_L (2 b^{\dagger}b+1)\right)+i\eta\Omega\cos\theta_L (b^{\dagger}+b) \nonumber \\ &&+\sum_{j=1,2}g_j \cos\phi_j a_j\left(1-\frac{\eta^2}{2}\cos^2 \theta_c (2 b^{\dagger}b+1)\right)\nonumber\\ &&-\eta \cos\theta_c g_j \sin\phi_j a_j (b^{\dagger}+b), \label{beff} \end{eqnarray} The QLE read \begin{eqnarray} \dot{a}_1(t)&=&i\delta_1 a_1(t)+i\sigma(t)\left[B(t)^{\dagger}, a_1(t)\right] \nonumber \\ && -\kappa_1 a_1(t)+\sqrt{2\kappa_1}a_1^{in}(t), \\ \dot{a}_2(t)&=&i\delta_2 a_1(t)+i\sigma(t)\left[B(t)^{\dagger}, a_2(t)\right] \nonumber \\ && -\kappa_2 a_2(t)+\sqrt{2\kappa_2}a_2^{in}(t), \\ \dot{b}(t)&=&-i\nu b(t)+i\sigma(t)\left[B(t)^{\dagger}, b(t)\right]+ i\sigma(t)^{\dagger}\left[B(t), b(t)\right] \nonumber \\ && -\kappa_b b(t)+\sqrt{2\kappa_b}b^{in}(t), \\ \dot{\sigma}(t)&=&\left[i\Delta -\frac{\gamma}{2}\right] \sigma(t)+\sigma_z(t) \left( i B(t)+\sqrt{\gamma}f^{in}(t)\right), \label{penlastqle}\\ \dot{\sigma}_z(t)&=& 2i\sigma(t) B(t)^{\dagger}-2i\sigma^{\dagger}(t) B(t)-\gamma \left[\sigma_z(t)+1\right] \nonumber \\ &&-2\sigma^{\dagger}(t)\sqrt{\gamma}f^{in}(t)-2\sigma(t)\sqrt{\gamma}f^{in}(t)^{\dagger}, \label{lastqle} \end{eqnarray} where $\sigma_z=\sigma^{\dagger}\sigma-\sigma \sigma^{\dagger}$, and we have introduced the vacuum input noises $a_j^{in}(t)$ ($j=1,2$) of the cavity modes with corresponding decay rate $\kappa_j$, the spontaneous emission noise $f^{in}(t)$ at rate $\gamma$, and we also added a phenomenological input noise $b^{in}(t)$ acting on the atom's motion, describing the heating at rate $\kappa_b$ due to the fluctuations of the trap potential. These four noise sources are mutually uncorrelated and have zero mean, while their second-order correlations have the form \begin{eqnarray} &&\langle a_1^{in}(t)a_1^{in}(t')^{\dagger}\rangle = \langle a_2^{in}(t)a_2^{in}(t')^{\dagger}\rangle = \delta(t-t'), \\ && \langle f^{in}(t)f^{in}(t')^{\dagger}\rangle = \delta(t-t'), \\ && \langle b^{in}(t)b^{in}(t')^{\dagger}\rangle = \left(\bar{N}+1\right)\delta(t-t'), \\ && \langle b^{in}(t)^{\dagger} b^{in}(t')\rangle =\bar{N} \delta(t-t'), \end{eqnarray} where $\bar{N}$ is mean thermal vibrational number of the effective thermal reservoir coupling to the atom center-of-mass motion~\cite{Footnote2}. We assume that the laser is red-detuned and far-off resonance from the atomic transition, i.e., $\Delta$ is negative and $\|\Delta\|$ is much larger than all the other parameters. This allows us to eliminate adiabatically the atomic internal degrees of freedom, and to assume that the atom always remains in the ground state $\|g \rangle $, that is, $\sigma_z(t) \approx -1$. Therefore we neglect the time evolution of $\sigma_z$, Eq.~(\ref{lastqle}), while Eq.~(\ref{penlastqle}) becomes \begin{equation} \dot{\sigma}(t)=-\left(\frac{\gamma}{2}-i\Delta \right)\sigma(t)-i B(t)- \sqrt{\gamma}f^{in}(t), \label{lastqle2} \end{equation} whose formal solution is \begin{eqnarray} && \sigma(t)=e^{-\left(\frac{\gamma}{2}-i\Delta \right)t}\sigma(0) \label{lastqlesol} \\ &&-\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[i B(t-s)+ \sqrt{\gamma}f^{in}(t-s)\right]. \nonumber \end{eqnarray} We now insert solution~(\ref{lastqlesol}) into the other QLE and neglect the transient term because we are interested in the dynamics at times which are much larger than $1/\|\Delta\|$. We obtain \begin{widetext} \begin{eqnarray} \dot{a}_1(t)&=&i\delta_1 a_1(t)+\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[B(t-s)-i\sqrt{\gamma}f^{in}(t-s)\right]\left[B(t)^{\dagger}, a_1(t)\right] -\kappa_1 a_1(t)+\sqrt{2\kappa_1}a_1^{in}(t), \\ \dot{a}_2(t)&=&i\delta_2 a_1(t)+\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[B(t-s)-i\sqrt{\gamma}f^{in}(t-s)\right]\left[B(t)^{\dagger}, a_2(t)\right] -\kappa_2 a_2(t)+\sqrt{2\kappa_2}a_2^{in}(t), \\ \dot{b}(t)&=&-i\nu b(t)+\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[B(t-s)-i \sqrt{\gamma}f^{in}(t-s)\right]\left[B(t)^{\dagger}, b(t)\right] \nonumber \\ &&-\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}+i\Delta \right)s} \left[B(t-s)^{\dagger}+ i\sqrt{\gamma}f^{in}(t-s)^{\dagger}\right]\left[B(t), b(t)\right] -\kappa_b b(t)+\sqrt{2\kappa_b}b^{in}(t), \end{eqnarray} \end{widetext} where we have not taken care of operator ordering, since, as we shall see, within the validity limit of our treatment these integral terms will generate only linear contributions. At this point, we choose the laser angular frequency $\omega_L$ so that \begin{eqnarray} \delta_1 =\nu^{\prime}~~;~~\delta_2 = -\nu^{\prime} \end{eqnarray} namely, the laser frequency is tuned symmetrically between the mode frequencies, which are spaced by a quantity $2\nu^{\prime}$. The angular frequency $\nu^{\prime}\simeq\nu$, and takes into account the a.c.-Stark shifts due to the mechanical coupling with laser and cavity modes, see Sec.~\ref{Sec:ac-Shift}, so that the two cavity modes are resonant with the motional sidebands of the laser light. Together with this choice of the laser frequency, we assume that the motional sidebands are well resolved, that is, $\nu \gg \|g_j\|,\Omega, \kappa_j$. In order to identify the resonant process, we move to a frame rotating at the effective vibrational angular frequency $\nu' \simeq \nu$, (which has to be determined by solving the QLE) and we will neglect in the QLE all the terms oscillating at $\nu'$ or larger. Denoting the slowly varying quantities by $\tilde{a}_1^{\dagger}(t)\equiv e^{i\nu 't} a_1^{\dagger}(t)$, $\tilde{a}_2(t)\equiv e^{i\nu 't} a_2(t)$, $\tilde{b}(t)\equiv e^{i\nu 't} b(t)$, after explicitly evaluating the commutators we obtain \begin{widetext} \begin{eqnarray} && \label{QLE_01} \dot{\tilde{a}}_1^{\dagger}(t)=i\left(\nu'-\delta_1 \right) \tilde{a}_1^{\dagger}(t) +\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}+i\Delta \right)s} \left[B(t-s)^{\dagger}e^{i\nu 't}+i\sqrt{\gamma}f^{in}(t-s)^{\dagger}e^{i\nu 't}\right] \\ && \times \left[-g_1 \cos \phi_1 \left(1-\frac{\eta^2}{2}\cos^2 \theta_c (2 \tilde{b}^{\dagger}\tilde{b}+1)\right)+\eta g_1 \sin\phi_1 \cos\theta_c\left(\tilde{b}(t)e^{-i\nu 't}+\tilde{b}^{\dagger}(t)e^{i\nu 't}\right)\right] -\kappa_1 \tilde{a}_1^{\dagger}(t)+\sqrt{2\kappa_1}\tilde{a}_1^{in}(t)^{\dagger}, \nonumber \\ && \dot{\tilde{a}}_2(t)=i\left(\nu'+\delta_2 \right) \tilde{a}_2(t) +\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[B(t-s)e^{i\nu 't}-i\sqrt{\gamma}f^{in}(t-s)e^{i\nu 't}\right] \\ && \times \left[-g_2^* \cos \phi_2 \left(1-\frac{\eta^2}{2}\cos^2 \theta_c (2 \tilde{b}^{\dagger}\tilde{b}+1)\right) +\eta g_2^* \sin\phi_2 \cos\theta_c\left(\tilde{b}(t)e^{-i\nu 't}+\tilde{b}^{\dagger}(t)e^{i\nu 't}\right)\right] -\kappa_2 \tilde{a}_2(t)+\sqrt{2\kappa_2}\tilde{a}_2^{in}(t), \nonumber \\ && \dot{\tilde{b}}(t)=i\left(\nu'-\nu \right) \tilde{b}(t) +\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}-i\Delta \right)s} \left[B(t-s)e^{i\nu 't}-i\sqrt{\gamma}f^{in}(t-s)e^{i\nu 't}\right] \\ && \times \left[i\eta \Omega^* \cos \theta_L +\eta g_1^* \sin\phi_1 \cos\theta_c\tilde{a}_1^{\dagger}(t)e^{-i\nu 't}+ \eta g_2^* \sin\phi_2 \cos\theta_c\tilde{a}_2^{\dagger}(t)e^{i\nu 't}\right] \nonumber \\ && -\int_0^t {\rm ds} e^{-\left(\frac{\gamma}{2}+i\Delta \right)s} \left[B(t-s)^{\dagger}e^{i\nu 't}+i\sqrt{\gamma}f^{in}(t-s)^{\dagger}e^{i\nu 't}\right] \\ && \times \left[-i\eta \Omega \cos \theta_L +\eta g_1 \sin\phi_1 \cos\theta_c\tilde{a}_1(t)e^{i\nu 't}+ \eta g_2 \sin\phi_2 \cos\theta_c\tilde{a}_2(t)e^{-i\nu 't}\right] -\kappa_b \tilde{b}(t)+\sqrt{2\kappa_b}\tilde{b}^{in}(t), \label{QLE_0N} \end{eqnarray} \end{widetext} where we have introduced the noise operators $\tilde{a}_1^{in}(t)\equiv e^{-i\nu 't} a_1^{in}(t)$, $\tilde{a}_2^{in}(t)\equiv e^{i\nu 't} a_2^{in}(t)$, and $\tilde{b}^{in}(t)\equiv e^{i\nu 't} b^{in}(t)$, which are still delta-correlated. We insert in these equations the explicit expression for $B(t-s)$, thereby neglecting the terms oscillating at $\nu '$ or faster. We finally perform the time integrals by making the Markovian approximation $\exp\{-(\gamma/2\pm i\Delta +i m \nu')s\} \approx \delta(s)/(\gamma/2\pm i\Delta +i m \nu')$, for $m=-1,0,1$. After long, but straightforward calculations we get the final, effective QLE at leading order in the Lamb-Dicke parameter, which read \begin{widetext} \begin{eqnarray} \label{QLE:a1} \dot{\tilde{a}}_1^{\dagger}(t)&=&i\left(\nu'-\delta_1 \right) \tilde{a}_1^{\dagger}(t) +\chi_1^\tilde{b}(t) -\left(\kappa_1 +\kappa_{1L}-i\delta_{1L}\right) \tilde{a}_1^{\dagger}(t)+\sqrt{2\kappa_1}\tilde{a}_1^{in}(t)^{\dagger} +\sqrt{2}\bar{\kappa}_{1L}\tilde{a}_{1L}^{in}(t)^{\dagger}+F_1, \\ \label{QLE:a2} \dot{\tilde{a}}_2(t)&=&i\left(\nu'+\delta_2 \right) \tilde{a}_2(t) +\chi_2\tilde{b}(t) -\left(\kappa_2 +\kappa_{2L}+i\delta_{2L}\right) \tilde{a}_2(t)+\sqrt{2\kappa_2}\tilde{a}_2^{in}(t) +\sqrt{2}\bar{\kappa}_{2L}\tilde{a}_{2L}^{in}(t)+F_2, \\ \label{QLE:b} \dot{\tilde{b}}(t)&=&i\left(\nu'-\nu \right) \tilde{b}(t) +\bar{\chi}_1 \tilde{a}_1^{\dagger}(t)-\bar{\chi}_2^ \tilde{a}_2(t) -\left(\kappa_b+\kappa_{2b}-\kappa_{1b}+i\delta_b\right) \tilde{b}(t)\\ & &+\sqrt{2\kappa_b}\tilde{b}^{in}(t) +\sqrt{2}\bar{\kappa}_{2b}\tilde{a}_{2L}^{in}(t)-\sqrt{2}\bar{\kappa}_{1b}\tilde{a}_{1L}^{in}(t)^{\dagger}+F_b~.\nonumber \end{eqnarray} \end{widetext} Let us now discuss each term appearing in the equations. The coupling coefficients are given by \begin{eqnarray} &&\chi_1=\eta \Omega g_1^\cos\phi_1\left(\frac{\cos\theta_L}{\Delta-\nu'+{\rm i}\gamma/2} +\frac{{\rm i}\tan\phi_1\cos\theta_c}{\Delta+{\rm i}\gamma/2}\right)~,\nonumber\\ &&\label{Chi:1}\\ &&\chi_2=\eta \Omega g_2^\cos\phi_2\left(\frac{\cos\theta_L}{\Delta+\nu'+{\rm i}\gamma/2} +\frac{{\rm i}\tan\phi_2\cos\theta_c}{\Delta+{\rm i}\gamma/2}\right)~,\nonumber\\ &&\label{Chi:2} \\ &&\bar{\chi}_1=\eta \Omega g_1^\cos\phi_1\left(\frac{\cos\theta_L}{\Delta-\nu'-{\rm i}\gamma/2} +\frac{{\rm i}\tan\phi_1\cos\theta_c}{\Delta+{\rm i}\gamma/2}\right)~,\nonumber\\ &&\label{Chi:1_t}\\ &&\bar{\chi}_2=\eta \Omega g_2^\cos\phi_2\left(\frac{\cos\theta_L}{\Delta+\nu'-{\rm i}\gamma/2} +\frac{{\rm i}\tan\phi_2\cos\theta_c}{\Delta+{\rm i}\gamma/2}\right)~,\nonumber\\ &&\label{Chi:2:t} \end{eqnarray} and correspond to the Raman processes, in which laser photons are scattered into the cavity mode with a change in the center-of-mass excitation, see Sec.~\ref{Sec:Chi}. New fluctuation-dissipation sources appear in the equations. We first discuss noise terms appearing in Eqs.~(\ref{QLE:a1}) and~(\ref{QLE:a2}). In addition to cavity decay with rates $\kappa_j$ we find processes described by the decay terms with rate $\kappa_{1L}$ and $\kappa_{2L}$, and the corresponding Langevin noises $\tilde{a}_{1L}^{in}(t)$ and $\tilde{a}_{2L}^{in}(t)$, where \begin{eqnarray} &&\kappa_{1L}=\frac{\gamma}{2}\frac{\|g_1\|^2\cos^2\phi_1}{\gamma^2/4+(\Delta-\nu')^2}, \label{los1} \\ &&\kappa_{2L}=\frac{\gamma}{2}\frac{\|g_2\|^2\cos^2\phi_2}{\gamma^2/4+(\Delta+\nu')^2}, \label{los2} \end{eqnarray} and \begin{eqnarray} &&\tilde{a}_{1L}^{in}(t)=f^{in}(t)e^{-i\nu 't}, \label{nos1} \\ &&\tilde{a}_{2L}^{in}(t)=f^{in}(t)e^{i\nu 't}. \label{nos2} \end{eqnarray} These noises describe input-output processes between the cavity modes and external modes, mediated by the atom. They possess the same correlation functions of the spontaneous emission noise $f^{in}(t)$, and at the timescales of interest, $\nu't \gg 1$, they are uncorrelated from each other, thanks to the oscillating factors. Note that \begin{eqnarray} &&\bar{\kappa}_{1L}=-{\rm i}\sqrt{\frac{\gamma}{2}}\frac{g_1\cos\phi_1}{\gamma/2+{\rm i}(\Delta-\nu')},\\ &&\bar{\kappa}_{2L}={\rm i}\sqrt{\frac{\gamma}{2}}\frac{g_2^\cos\phi_2}{\gamma/2-{\rm i}(\Delta+\nu')}, \end{eqnarray} with $\kappa_{jL}=\|\bar{\kappa}_{jL}\|^2$. They originate from the scattering processes in which cavity photons are lost because they are absorbed and then spontaneously emitted by the atom, as has been discussed in Sec.~\ref{Sec:cav-noise}. The noise and dissipation terms in Eq.~(\ref{QLE:b}), in addition to the noise terms of the trap, are described by the decay terms with rate $\kappa_{1b}$ and $\kappa_{2b}$, and the corresponding Langevin noise operators $\tilde{a}_{1L}^{in}(t)$ and $\tilde{a}_{2L}^{in}(t)$. These processes originate from incoherent emission or absorption of a vibrational quantum accompanied by absorption and subsequent spontaneous emission of a laser photon. The emission of vibrational quanta takes place at rate \begin{equation}\label{absphon2} \kappa_{2b}=\frac{\gamma}{2}\frac{\eta^2\|\Omega\|^2\cos^2\theta_L}{\gamma^2/4+(\Delta+\nu')^2}, \label{losphon1} \end{equation} while the rate of incoherent absorption of vibrational quanta is given by \begin{equation}\label{absphon1} \kappa_{1b}=\frac{\gamma}{2}\frac{\eta^2\|\Omega\|^2\cos^2\theta_L}{\gamma^2/4+(\Delta-\nu')^2}. \label{losphon2} \end{equation} In particular, when $\Delta <0$ then $\kappa_{2b} > \kappa_{1b}$ and the motion is cooled. Moreover, \begin{eqnarray} &&\bar{\kappa}_{1b}=\sqrt{\frac{\gamma}{2}}\frac{\eta\Omega\cos\theta_L}{\gamma/2+{\rm i}(\Delta-\nu')},\\ &&\bar{\kappa}_{2b}=\sqrt{\frac{\gamma}{2}}\frac{\eta\Omega^\cos\theta_L}{\gamma/2-{\rm i}(\Delta+\nu')}, \end{eqnarray} with $\kappa_{jb}=\|\bar{\kappa}_{jb}\|^2$. If we consider the dynamics described by these terms only, these incoherent phonon absorption and emission processes lead to thermalization of the atomic motion at rate $\kappa_{2b}-\kappa_{1b}$, to a final effective mean vibrational number $n_{th}=\kappa_{1b}/(\kappa_{2b}-\kappa_{1b})\simeq \|\Delta \|/4\nu'$, as in standard cooling~\cite{Stenholm86}. However, the noise associated with these incoherent phonon absorptions and emissions is \emph{correlated} with the noise terms $\tilde{a}_{1L}^{in}(t)$ and $\tilde{a}_{2L}^{in}(t)$ describing scattering of cavity photons, because all these processes ultimately originate from spontaneous emission. This is why the noise terms in the Langevin equation for the atomic motion are directly expressed in terms of $\tilde{a}_{1L}^{in}(t)$ and $\tilde{a}_{2L}^{in}(t)$, making therefore this correlation evident. The operators $F_j$ in Eqs.~(\ref{QLE_01})-(\ref{QLE_0N}) represent non-linear terms, which describe the noise associated with the incoherent part of the scattering processes discussed in Sec.~\ref{Sec:Chi}. These terms can be neglected with respect to the coherent processes, provided that $\gamma\ll\|\Delta\|$ and $\gamma\ll\nu$. In particular, the second inequality ensures that rates $\chi_1$ and $\chi_2$ differ appreciably, such that entanglement between the cavity modes can be established in a finite time~\cite{Morigi06}. We will focus on this regime, $\gamma\ll\nu$, in which we can thus neglect $F_j$ in the effective QLE when evaluating the spectrum of squeezing. Finally, the frequency shifts of the two cavity modes and of the vibrational motion read \begin{eqnarray} &&\delta_{1L}=\frac{(\Delta-\nu')\|g_1\|^2\cos^2\phi_1}{\gamma^2/4+(\Delta-\nu')^2} \label{eqfornu1} \\ &&\delta_{2L}=\frac{(\Delta+\nu')\|g_2\|^2\cos^2\phi_2}{\gamma^2/4+(\Delta+\nu')^2} \label{eqfornu2} \\ &&\delta_{b}= \frac{2\Delta \eta^2 \|\Omega\|^2 \cos^2\theta_L\left(\gamma^2/4+\Delta^2-\nu'^2\right)}{\left(\gamma^2/4+\Delta^2-\nu'^2\right)^2 +\nu'^2\gamma^2}\label{nu_b} \label{eqfornu}\\ &&~-\eta^2 \|\Omega\|^2 \cos^2\theta_L\frac{\Delta}{\Delta^2+\gamma^2/4}\nonumber \end{eqnarray} and from their form one can recognize the a.c.-Stark shifts reported in Sec.~\ref{Sec:ac-Shift}, with $\delta\omega_j=\delta_{jL}a^{\dagger}_ja_j$, Eq.~(\ref{deltaomega_j}), and $\delta\nu_b=\delta_bb^{\dagger}b$, Eq.~(\ref{deltanu_b}), where now $\nu\to\nu^{\prime}$. Note that we have omitted a non-linear shift at second order in the Lamb-Dicke parameter, which affects both cavity modes and motion. As discussed in Sec.~\ref{Sec:ac-Shift}, this is a small correction to $\delta_{jL}$, as it scales with $\eta^2$, while it may have a relevant effect on the center-of-mass dynamics. It can be neglected in the limit $\Omega\cos^2\theta_L\gg g_j\cos^2\theta_c$. Under this assumption, which we will consider in the rest of this manuscript, the spectrum of the center-of-mass is the spectrum of a harmonic oscillator, characterized by equidistant energy levels. As the dynamics we seek relies on resonant interaction between the cavity modes and the vibrational motion, the two cavity modes should be exactly at resonance with the sidebands of the driving laser. Equation~(\ref{nu_b}) provides an implicit equation for the actual vibrational angular frequency $\nu'$. In the parameter regime $\eta \|\Omega \| \ll \|\Delta\|, \nu$ we find with good approximation \begin{eqnarray}\label{renfreq} \nu'&\approx& \nu+\frac{2\Delta \eta^2 \|\Omega\|^2 \cos^2\theta_L\left(\gamma^2/4+\Delta^2-\nu^2\right)}{\left(\gamma^2/4+\Delta^2-\nu^2\right)^2+\nu^2\gamma^2}\nonumber\\ & &~-\eta^2 \|\Omega\|^2 \cos^2\theta_L\frac{\Delta}{\Delta^2+\gamma^2/4}. \end{eqnarray} Taking also into account the frequency shifts of Eqs.~(\ref{eqfornu1})-(\ref{eqfornu2}), the resonance conditions are finally \begin{eqnarray} &&\delta_1=\delta_{1L}+\nu' \label{res1} \\ &&\delta_2=\delta_{2L}-\nu' \label{res2} . \end{eqnarray} In the parameter regime $\gamma\ll\nu$, using conditions (\ref{res1})-(\ref{res2}), we arrive therefore to the final QLE, describing the coherent interaction between the two cavity modes and the vibrational motion, competing with losses and noise processes due to spontaneous emission, cavity decay, and vibrational heating, \begin{widetext} \begin{eqnarray} && \dot{\tilde{a}}_1^{\dagger}(t)=\chi_1^\tilde{b}(t) -\left(\kappa_1 +\kappa_{1L}\right) \tilde{a}_1^{\dagger}(t)+\sqrt{2\kappa_1}\tilde{a}_1^{in}(t)^{\dagger} +\sqrt{2}\bar{\kappa}_{1L}\tilde{a}_{1L}^{in}(t)^{\dagger}~, \label{qlefin1}\\ && \dot{\tilde{a}}_2(t)=\chi_2\tilde{b}(t) -\left(\kappa_2 +\kappa_{2L}\right) \tilde{a}_2(t)+\sqrt{2\kappa_2}\tilde{a}_2^{in}(t) +\sqrt{2}\bar{\kappa}_{2L}\tilde{a}_{2L}^{in}(t)~, \label{qlefin2} \\ && \dot{\tilde{b}}(t)=\bar{\chi}_1 \tilde{a}_1^{\dagger}(t)-\bar{\chi}_2^ \tilde{a}_2(t) -\left(\kappa_b+\kappa_{2b}-\kappa_{1b}\right) \tilde{b}(t)+\sqrt{2\kappa_b}\tilde{b}^{in}(t) +\sqrt{2}\bar{\kappa}_{2b}\tilde{a}_{2L}^{in}(t)-\sqrt{2}\bar{\kappa}_{1b}\tilde{a}_{1L}^{in}(t)^{\dagger}~. \label{qlefin3} \end{eqnarray} \end{widetext} \subsection{Evaluation of the spectrum of squeezing} We now use Eqs.~(\ref{qlefin1})-(\ref{qlefin3}) in order to determine the stationary spectrum of squeezing of the light at the cavity output. We consider \begin{eqnarray}\label{imeno} I_-^{out}(t)&=& a_1^{out}(t)+a_1^{out}(t)^{\dagger}-a_2^{out}(t)-a_2^{out}(t)^{\dagger}, \\ I_+^{out}(t)&=&-i\left[ a_1^{out}(t)-a_1^{out}(t)^{\dagger}+a_2^{out}(t)-a_2^{out}(t)^{\dagger}\right],\label{ipiu} \end{eqnarray} corresponding respectively to the difference between the amplitude quadratures, and the sum of the phase quadratures of the two sideband modes. These are the quadratures exhibiting two-mode squeezing in the case of pulsed excitation in this setup, see \cite{PRL,Morigi06}. The output cavity fields $a_j^{out}(t)$ in Eqs.~(\ref{imeno})-(\ref{ipiu}) are given by the usual input-output relation \begin{equation} \label{aout} a_j^{out}(t)=\sqrt{2\kappa_j}a_j(t)-a_j^{in}(t), \;\;\;\;j=1,2. \end{equation} The spectrum of squeezing can be calculated by evaluating the Fourier transforms \begin{equation} \hat{I}_{\pm}^{out}(\omega)=\int{\rm d}t{\rm e}^{{\rm i}\omega t}I_{\pm}^{out}(t), \end{equation} and using the fact that at the stationary state it is \begin{equation}\label{defspec} \langle \hat{I}_{\pm}^{out}(\omega) \hat{I}_{\pm}^{out}(\omega')+ \hat{I}_{\pm}^{out}(\omega') \hat{I}_{\pm}^{out}(\omega)\rangle = 8\pi S_{\pm}(\omega)\delta(\omega+\omega'), \end{equation} where we have normalized the spectrum so that the shot noise level corresponds to $S_{\pm}(\omega)=1$. Two-mode squeezing is found when one spectrum of squeezing takes values below the shot noise limit at some $\omega$. From the Fourier transform of Eqs.~(\ref{qlefin1})-(\ref{qlefin3}) one can see that $S_{+}(\omega)=S_{-}(\omega)\equiv S(\omega)$, which implies that in the present case two-mode squeezing is equivalent to EPR-like entanglement between the two output cavity modes. This is easily verified by applying a sufficient criterion for entanglement, such as the ``sum'' criterion of Duan \textit{et al.} \cite{Duan00}, or the product criterion of Ref.~\cite{Mancini02,GIOV03}. With the chosen normalization for the output cavity modes at $\omega$, the sum criterion reads \begin{equation} S_{+}(\omega)+S_{-}(\omega) < 2, \label{duan} \end{equation} while the product criterion gives \begin{equation} S_{+}(\omega)S_{-}(\omega) < 1, \label{noi} \end{equation} so that in our case both criteria imply that the two output modes are EPR-like entangled as soon as $S(\omega) < 1$. The squeezing spectrum $S(\omega)$ can be obtained from the Fourier transform of the Langevin equations after long but straightforward algebra, yielding a cumbersome expression which will not be reported here. This expression becomes considerably simpler in the limit $\|\Omega\|, \|g_j\|, \gamma \ll \|\Delta\|$ and $\eta \ll 1$. In this limit the additional loss processes due to spontaneous emission, associated with the rates $\kappa_{jL}$ and $\kappa_{jb}$ ($j=1,2$), are typically negligible, that is, $\kappa_{jL},\kappa_{jb} \ll\kappa$. Moreover, we consider the case of ion traps, where heating of the atomic motion is negligible with respect to all radiative noise sources~\cite{Trap:stability}. Finally, as the two cavity modes are very close in frequency, they will have very similar properties, in particular we can take $\kappa_1=\kappa_2=\kappa$. In this parameter regime the main aspects of the squeezing spectrum can be grasped from its analytical expression. One finds \begin{equation}\label{specsimple} S(\omega)=1-\frac{\kappa^2\left(\Theta^4-\Sigma^4\right)}{\left(\kappa^2+\omega^2\right) \left[\left(\omega^2-\Theta^2\right)^2+\omega^2\kappa^2\right]}, \end{equation} where $\Theta=\sqrt{\|\chi_2\|^2-\|\chi _1\|^2}$ as given in Eq.~(\ref{thetabig}), and \begin{equation} \Sigma = \sqrt{\left\|\left\|\chi_2\right\|^2+\left\|\chi_1\right\|^2-2\chi_1\chi_2\right\|}, \end{equation} and we have used that $\chi_j=\bar{\chi}_j$ when $\gamma \ll \|\Delta\|$ (see Eqs.~(\ref{Chi:1})-(\ref{Chi:2:t})). Note that due to the transformations which we have applied, the results which appear around $\omega=0$ in $S(\omega)$ describe quantum correlations of noise components in the optical signal at $\omega_L - \nu' \pm \omega$ with those at $\omega_L + \nu' \pm \omega$, i.e.\ correlated fluctuations of the two modes at the same offset from their center frequencies. From Eq.~(\ref{specsimple}) one notes that the properties of the spectrum are mainly determined by the ratio $\Theta/\kappa$. In fact, the denominator in Eq.~(\ref{specsimple}) has always three poles in the lower complex half-plane, one which is always imaginary at $\omega_0=-i\kappa$, and two poles at $\omega_{\pm}=-i\kappa/2\pm \sqrt{\Theta^2-\kappa^2/4}$. Therefore, when $\Theta/\kappa \gg 1$ the two poles at $\omega_{\pm}$ have a nonzero real part and the spectrum is characterized by three well-separated inverted Lorentzian peaks, one at $\omega=0$ with width (FWHM) $2\kappa$ and the other two symmetrically placed at $\omega\approx \pm \Theta$, with FWHM $\kappa$. At the center of these peaks one has $S=(\Sigma/\Theta)^4 \simeq \left[\left(\|\chi_2\|-\|\chi_1\|\right)/\left(\|\chi_2\|+\|\chi_1\|\right)\right]^2$, approaching $S(\omega)=0$, i.e., infinite two-mode squeezing, for $\|\chi_2\|\simeq \|\chi_1\|$. Therefore, when $\Theta>\kappa$ we find two-mode squeezing within three narrow bandwidths around $\omega=0$ and $\omega =\pm \Theta$. In the opposite case of $\Theta \leq \kappa/2$, the three poles are all on the imaginary axis, and the spectrum shows only one inverted Lorentzian peak at $\omega=0$. When $\Theta/\kappa \ll 1$, this peak becomes very narrow, with FWHM $\sim 2\Theta^2/\kappa$. It is remarkable that even for $\kappa>\Theta$ one finds almost perfect squeezing in the difference of amplitude quadratures at $\omega=0$. This can be understood, as in the regime we consider the scattered photons due to spontaneous emission are negligible with respect to those lost through the output cavity mirror ($\kappa_{jL},\kappa_{jb} \ll \kappa)$. This implies that most of the intracavity photons are detected at the output. These photons are almost perfectly correlated at $\omega=0$ and therefore would give $S(0) \simeq 0$. In this regime, a large cavity decay rate $\kappa$ has only the effect of narrowing the squeezing bandwidth. On the contrary, if the photon scattering by spontaneous emission is no more negligible, two-mode squeezing soon degrades, even at $\omega=0$ (see for example \cite{milwal}). The presence of the three-pole structure in the squeezing spectrum is novel with respect to the spectral features usually encountered in the parametric oscillator (either below and above threshold, see e.g.\ Ref.~\cite{milwal}). This structure is due to the coherent interaction of the two cavity modes with the quantized atomic motion, i.e.\ it arises from the coherent microscopic processes underlying the dynamic establishment of quantum correlations. The peculiar spectral properties can be exploited to achieve \emph{optimal broadband two-mode squeezing} when the three peaks merge, which happens when $\Theta =\kappa$. In this case one easily sees from Eq.~(\ref{specsimple}) that \begin{equation}\label{specsimpleflat} S(\omega)=1-\frac{\kappa^2\left(\Theta^4-\Sigma^4\right)}{\Theta^6 +\omega^6}, \end{equation} i.e., one has large, uniform squeezing for a wide bandwidth of frequencies. The fact that $\Theta = \kappa$ is the condition for the best two-mode squeezing in the output can be easily understood noticing that $\Theta$ is the angular frequency at which two-mode squeezing inside the cavity is periodically built up (see Ref.~\cite{PRL, Morigi06}). Therefore, when $\Theta =\kappa$, squeezing is generated inside the cavity at the same rate at which it is transferred to the output field. In contrast, in the other two cases, squeezing is not efficiently generated in the output because either the output coupling happens too fast, i.e.\ before full intra-cavity squeezing is established, or the output coupling is sufficiently slow to allow that energy is stored inside the cavity and the two-mode squeezing is coherently re-converted into independent states before it is coupled out. \subsection{Results} \label{Sec:Results} We now consider the exact squeezing spectrum $S(\omega)$, defined in Eq.~(\ref{defspec}), where the operator~(\ref{imeno}) is evaluated using the output relation~(\ref{aout}), such that the operators $a_j(t)$, $a_j^{\dagger}(t)$ are the solutions of the QLE~(\ref{qlefin1})-(\ref{qlefin3}) including all noise and loss terms. The parameter regime we consider has been discussed in detail in~\cite{Morigi06}. We take a ${F}\!=0 \leftrightarrow {F^{\prime}}\!=\!1$ atomic transition with the quantization axis $\vec{B}$ along the cavity axis, and $\vec{B}$, $\vec{k}_L$, and laser polarization $\vec{E}_L$ mutually orthogonal. Relation $\gamma\ll\nu$ can be fulfilled by the intercombination line of an Indium ion at $\gamma=2\pi\times 360$~kHz in a trap at $\nu=2\pi\times 3$~MHz, for which $\eta \simeq 0.1$. We consider a geometrical configuration corresponding to $\theta_L=0$, $\theta_c=\pi/2$, and $\phi_1=\phi_2=0$, which means that the ion motion takes place along the direction of the laser beam and orthogonally to the cavity axis, and that the trap center coincides with an antinode of the cavity modes. In such a case, the coupling coefficients of Eqs.~(\ref{Chi:1})-(\ref{Chi:2:t}) are determined by the first term only. Moreover in this case $\Omega\cos^2\theta_L\gg g_j\cos^2\theta_c \simeq 0$, and therefore, as discussed in Secs.~\ref{Sec:ac-Shift} and \ref{Sec:QLE}, the small Stark-shift correction to $\delta_{jL}$, scaling with $\eta^2$ can be neglected. If we consider that the ion couples to two non-degenerate polarization modes of a resonator with vacuum Rabi couplings $g\approx 2\pi\times 0.6$~MHz, and we take laser Rabi frequency $\Omega=2\pi\times 18$~MHz and detuning $\Delta=2\pi\times 60$~MHz, we obtain $\Theta/2\pi \simeq 7.9$~kHz, see~\cite{Morigi06}. The condition $\Theta \gg \kappa$ is found for a finesse ${\cal F}\simeq 10^6$ and free spectral range $\delta\omega= 2\pi\times 1$~GHz, so that $\kappa=2\pi\times 1$~KHz. The corresponding spectrum of squeezing is displayed in Fig.~\ref{Fig:2} (full line). It exhibits three minima at $\omega=0,\pm\Theta$, which correspond to three separated regions of narrow-band squeezing. The two bands around $\omega = \pm\Theta$ have width $\kappa$, while the central one has width $2 \kappa$ and shows almost $100\%$ squeezing. These features are well reproduced by the analytical expression~(\ref{specsimple}) (see dashed line in Fig.~\ref{Fig:2}), except that the latter predicts very large squeezing also for the peaks at $\omega = \pm\Theta$. The success of the simplified expression~(\ref{specsimple}) is due to the fact that, with the chosen parameter values, the loss rates due to the various scattering processes are at least ten times smaller than the cavity output loss rates $\kappa_1=\kappa_2=\kappa$, and therefore do not have a relevant effect on the spectrum. We have also considered a realistic ion vibrational heating rate $\kappa_h=\kappa_b\bar{N}=2\pi \times 0.1$~kHz, which however gives an effect which is negligible even with respect to that due to photon scattering. The appearance of three minima is a novel behavior to our knowledge, and it arises from the coherent microscopic dynamics, as $\Theta$ modulates the exchange of excitations and correlations between the cavity modes and the center-of-mass motion. \begin{center} \begin{figure} \caption{Squeezing spectrum $S(\omega)$ as a function of the sideband frequency $\omega$ (in units of $\Theta$) when $\Theta \simeq 8 \kappa$. Its behavior is well reproduced by the approximate analytical expression of Eq.~(\protect\ref{specsimple}) (dashed line). The parameters are discussed in the text. } \label{Fig:2} \end{figure} \end{center} The most interesting regime of broadband two-mode squeezing, when $\Theta\sim\kappa$, is shown in Fig.~\ref{Fig:3}, which displays the squeezing spectrum in the case of the same parameter values of Fig.~\ref{Fig:2} except for a lower cavity finesse, ${\cal F}\simeq 10^5$, implying $\kappa=2\pi\times 10$~kHz. The three minima merge into a single broad one, centered around $\omega=0$, whose width is determined by $\kappa=\Theta$. Also in this case one gets almost perfect squeezing at the center, and these features are well reproduced by the simple analytical expression of Eq.~(\ref{specsimpleflat}) (dashed line in Fig.~\ref{Fig:3}). \begin{center} \begin{figure} \caption{Squeezing spectrum $S(\omega)$ when $\Theta = \kappa$. Its behavior is well reproduced by the approximate analytical expression of Eq.~(\protect\ref{specsimple}) (dashed line). The parameters are discussed in the text.} \label{Fig:3} \end{figure} \end{center} Finally, in Fig.~\ref{Fig:4} we consider the case $\kappa>\Theta$. We have still kept the parameter values of Fig.~\ref{Fig:2}, but we have now considered a cavity with finesse ${\cal F} \sim 10^4$, implying $\kappa=2\pi\times 100$~kHz. The squeezing features are visibly worsened, as in this regime losses are faster than the typical time scale in which correlations between the field modes are established. One has still two-mode squeezing around $\omega=0$, but with a very narrow bandwidth which is roughly given by $\Theta^2/2\kappa$. \begin{center} \begin{figure} \caption{Squeezing spectrum $S(\omega)$ when $\Theta \simeq \kappa/10$. Its behavior is well reproduced by the approximate analytical expression of Eq.~(\protect\ref{specsimple}) (dashed line). The parameters are discussed in the text.} \label{Fig:4} \end{figure} \end{center} \section{Conclusions} \label{Sec:Conclusions} The resonance fluorescence of a confined, single, laser-driven atom exhibits EPR-entanglement, or two-mode squeezing, in the field modes which interact resonantly with the Stokes and anti-Stokes transitions created by the atomic motion. By coupling these sidebands to a high-finesse optical cavity, we have shown how to create continuous-wave (cw) two-mode squeezed light output from the cavity. At the microscopic level, the process is based on the mechanical effect of light, which allows for quantum-coherent generation and control of entanglement between the motion and the cavity modes. The scattering processes have been characterized and described in simple physical pictures using scattering matrix theory, and the squeezing spectrum has been calculated using Quantum Langevin Equations. Peculiar novel spectral properties are predicted for the squeezing spectrum of the cavity output. They may be divided into three regimes of the cavity output rate $\kappa$ relative to the frequency $\Theta$ of creation of two-mode squeezing inside the cavity. The squeezing spectrum can consist of a single peak ($\kappa > \Theta$), three peaks ($\kappa < \Theta$), or one broad, homogeneous band ($\kappa = \Theta$). Simple analytical approximations have been derived for the three relevant regimes. The squeezing spectrum in the different parameter regimes has been calculated for an experimentally accessible case of a single trapped ion as a specific example. The results for this specific system show all the features predicted by the general derivations, exhibiting novel spectral properties of two-mode squeezing which are novel when compared with conventional Optical-parameter-amplifier-type of sources. In particular, starting from the most fundamental individual quantum systems, a single atom and an optical cavity, we have designed a nonlinear optical source. This is therefore a paradigmatic model system exhibiting the connection between microscopic, quantum-coherent dynamics and macroscopic nonlinear device properties. Its efficiency and the high-degree of control one can achieve on its dynamics offer promising perspectives for the realization of quantum light sources for quantum networking~\cite{CiracKimble, Kraus04}. \end{document}	arXiv
Article Info. Asian-Australasian Journal of Animal Sciences (아세아태평양축산학회지) Pages.977-983 Asian Australasian Association of Animal Production Societies (아세아태평양축산학회) DOI QR Code Bacterial Inoculant Effects on Corn Silage Fermentation and Nutrient Composition Jalc, D. (Institute of Animal Physiology, Slovak Academy of Sciences) ; Laukova, Andrea (Institute of Animal Physiology, Slovak Academy of Sciences) ; Pogany Simonova, M. (Institute of Animal Physiology, Slovak Academy of Sciences) ; Varadyova, Z. (Institute of Animal Physiology, Slovak Academy of Sciences) ; Homolka, P. (Research Institute of Animal Production) Received : 2008.05.15 Accepted : 2008.02.23 Published : 2009.07.01 https://doi.org/10.5713/ajas.2009.80282 Citation PDF KSCI The survival and effect of three new probiotic inoculants (Lactobacillus plantarum CCM 4000, L. fermentum LF2, and Enterococcus faecium CCM 4231) on the nutritive value and fermentation parameters of corn silage was studied under laboratory conditions. Whole corn plants (288.3 g/kg DM) were cut and ensiled at $21^{\circ}C$ for 105 days. The inoculants were applied at a concentration of $1.0{\times}10^{9}$ cfu/ml. Uninoculated silage was used as the control. The chopped corn was ensiled in 40 plastic jars (1 L) divided into four groups (4${\times}$10 per treatment). All corn silages had a low pH (below 3.55) and 83-85% of total silage acids comprised lactic acid after 105 days of ensiling. The probiotic inoculants in the corn silages affected corn silage characteristics in terms of significantly (p<0.05-0.001) higher pH, numerically lower crude protein content and ratio of lactic to acetic acid compared to control silage. However, the inoculants did not affect the concentration of total silage acids (acetic, propionic, lactic acids) as well as dry matter digestibility (IVDMD) of corn silages in vitro. 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