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author:
- 'K. E. Ballantine'
- 'J. Ruostekoski'
title: |
Supplemental Material to\
“Optical Magnetism and Huygens’ Surfaces in Arrays of Atoms Induced by Cooperative Responses”
---
In this supplemental material we briefly recap the basic relations for the electrodynamics of light and atoms, and further illustrate the role of collective excitations for the example of synthesizing optical magnetism. We provide some additional details of the spherical harmonics used to decompose the far-field radiation and the scattered light and excitations of the Huygens’ surface.
Electrodynamics of light and atoms
==================================
Equations of motion
-------------------
In the main section we characterize the optical response of both the unit-cell and the many-atom array by writing the equations of motion as $\dot{{\mathbf{b}}} = i\mathcal{H}{\mathbf{b}}+{\mathbf{F}}$ where ${\mathbf{b}}$ is the vector of polarization amplitudes $\mathcal{P}_\sigma^{(j)}$ and ${\mathbf{F}}$ represents the external driving by the incident light. To see the origin of these equations we note that in the limit of low light intensity, the polarization amplitudes obey [@Jenkins2012a; @Lee16] $$\label{eq:Peoms}
\frac{d}{dt} {\mathcal{P}}_{\mu}^{(j)}
= \left( i \Delta_\mu^{(j)} - \gamma \right)
{\mathcal{P}}^{(j)}_\mu + i\frac{\xi}{\mathcal{D}}\hat{{\mathbf{e}}}_\mu^{\ast}\cdot\epsilon_0{\mathbf{E}}_{\rm ext}({\mathbf{r}}_j),$$ where $\xi=6\pi\gamma/k^3$, the single atom linewidth is $\gamma=\mathcal{D}^2k^3/6\pi\hbar\epsilon_0$, and $\Delta_\mu^{(j)}=\omega-\omega^{(j)}_\mu$ are the the detunings of the $m=\mu$ level of atom $j$ from resonance. The light and atomic field amplitudes here refer to the slowly varying positive frequency components, where the rapid oscillations $\exp(-i \omega t)$ at the laser frequency have been factored out. Each amplitude $\mathcal{P}_\mu^{(j)}$ at position ${\mathbf{r}}_j$ is driven by the field, $$\label{eq:Eext}
{\mathbf{E}}_{\rm ext}({\mathbf{r}}_j) = {\boldsymbol{\mathbf{\cal E}}}({\mathbf{r}}_j) + \sum_{l\neq j} {\mathbf{E}}_s^{(l)}({\mathbf{r}}_j),$$ which consists of the the incident field ${\boldsymbol{\mathbf{\cal E}}}({\mathbf{r}})$ and the scattered field $
\epsilon_0 {\mathbf{E}}_s^{(l)}({\mathbf{r}})=\mathsf{G}({\mathbf{r}}-{\mathbf{r}}_l){\mathbf{d}}_l
$ from the dipole moment ${\mathbf{d}}_l$ of each other atom $l$. The scattered field expression equals the usual positive-frequency component of the electric field from a monochromatic dipole ${{\bf d}}$, given that the dipole resides at the origin and the field is observed at ${{\bf r}}$ [@Jackson]: $$\begin{aligned}
{\sf G}({\bf r})&{{\bf d}}=
{1\over4\pi\epsilon_0}
\big\{ k^2(\hat{\bf n}\!\times\!{{\bf d}})\!\times\!\hat{\bf n}{e^{ikr}\over r} \nonumber\\&+[3\hat{\bf n}(\hat{\bf
n}\cdot{{\bf d}})-{{\bf d}}]
\big( {1\over r^3} - {ik\over r^2}\big) e^{ikr}
\big\}-{{{\bf d}}\over3\epsilon_0}\,\delta({\bf r})\,,
\label{eq:DOL}\end{aligned}$$ with $\hat{\bf n} = {{\bf r}/ r}$ and $k = {\omega / c}$.
Inserting Eq. (\[eq:Eext\]) into Eq. (\[eq:Peoms\]), with the dipole moment expressed in terms of the polarization as ${\mathbf{d}}_j=\mathcal{D}\sum_{\mu}\hat{{\mathbf{e}}}_\mu\mathcal{P}_\mu^{(j)}$, gives $$\begin{aligned}
\frac{d}{dt} {\mathcal{P}}_{\mu}^{(j)}
= \left( i \Delta_\mu^{(j)} - \gamma \right)
{\mathcal{P}}^{(j)}_\mu + i\xi\sum_{l\neq j}\mathcal{G}^{(jl)}_{\mu\nu}{\mathcal{P}}^{(l)}_\nu \nonumber \\ \label{eq:poleoms}
+ i\frac{\xi}{{\cal D}} \hat{{\mathbf{e}}}_{\mu}^{\ast} \cdot
\epsilon_0 {\boldsymbol{\mathbf{\cal E}}}({\mathbf{r}}_j),\end{aligned}$$ with $\mathcal{G}^{(jl)}_{\mu\nu}=\hat{{\mathbf{e}}}^*_\mu\cdot \mathsf{G}({\mathbf{r}}_j-{\mathbf{r}}_l)\hat{{\mathbf{e}}}_\nu$. The linear equations of motion can then be written in matrix form as above.
This equation also describes the decay of a single photon excitation. The full quantum dynamics of the atomic system for a given initial excitation and in the absence of a driving laser follows from the quantum master equation for the many-atom density matrix $\rho$, $$\begin{multlined}
\label{eq:rhoeom}
\dot{\rho} = i\sum_{j,\nu}\Delta_{\nu}\left[\hat{\sigma}_{j\nu}^{+}\hat{\sigma}_{j\nu}^{-},\rho\right] +i\sum_{jl\nu\mu (l\neq j)}\Omega^{(jl)}_{\nu\mu}\left[\hat{\sigma}_{j\nu}^{+}\hat{\sigma}_{l\mu}^{-},\rho\right] \\
+\sum_{jl\nu\mu}\gamma^{(jl)}_{\nu\mu}\left(
2\hat{\sigma}^{-}_{l\mu}\rho\hat{\sigma}^{+}_{j\nu}-\hat{\sigma}_{j\nu}^{+}\hat{\sigma}_{l\mu}^{-}\rho -\rho\hat{\sigma}_{j\nu}^{+}\hat{\sigma}_{l\mu}^{-}\right) \,,
\end{multlined}$$ where $\hat{\sigma}_{j\nu}^{+}=(\hat{\sigma}_{j\nu}^{-})^\dagger={\ensuremath{\left\vert e_{j\nu} \right\rangle}}{\ensuremath{\left\langle g_j \right\vert}}$ is the raising operator to the excited state $\nu$ on atom $j$. The diagonal terms of the dissipative matrix are $\gamma^{(jj)}_{\nu\nu}=\gamma$, while the off-diagonal elements of the dissipation and interaction terms are given by the real and imaginary parts of $$\Omega^{(jl)}_{\nu\mu}+i\gamma^{(jl)}_{\nu\mu}=
\xi {\cal G}^{(jl)}_{\nu\mu} \, .$$ Restricting to the subspace of at most a single excitation, and assuming a pure initial state, the density matrix splits into one excitation and zero excitation parts, $$\rho = {\ensuremath{\left\vert \psi \right\rangle}}{\ensuremath{\left\langle \psi \right\vert}} + p_g{\ensuremath{\left\vert G \right\rangle}}{\ensuremath{\left\langle G \right\vert}},$$ where ${\ensuremath{\left\vert \psi \right\rangle}}$ represents a single excitation and can be expanded over the atomic sites, $${\ensuremath{\left\vert \psi \right\rangle}}= \sum_{j,\nu} {\mathcal{P}}^{(j)}_{\nu}(t)\,\hat{\sigma}^{+}_{j\nu}{\ensuremath{\left\vert G \right\rangle}},$$ and ${\ensuremath{\left\vert G \right\rangle}}$ is the ground state. Regarding single-particle expectation values, the dynamics can equally be described by the evolution of a vector ${\mathbf{b}}$ of these amplitudes ${\mathcal{P}}^{(j)}_{\nu}$ [@SVI10], which obey the same equations as the classical polarization in the absence of drive, $\dot{{\mathbf{b}}}=i\mathcal{H}{\mathbf{b}}$.
Collective excitation eigenmodes
--------------------------------
In both the case of the evolution of a single atomic excitation and the case of driven classical polarization, the optical response of the lattice is then characterized by the eigenvectors ${\mathbf{v}}_n$ and eigenvalues $\delta_n+i\upsilon_n$, of the evolution matrix $\mathcal{H}$ [@Jenkins_long16]. While the eigenmodes are not orthogonal, due to $\mathcal{H}$ being non-Hermitian, we find in our numerics that they always form a basis. Hence, the state at a time $t$ can be expanded in this basis ${\mathbf{b}}(t)=\sum_n c_n{\mathbf{v}}_n$. We define the quantity $$L_j = \frac{|{\mathbf{v}}_j^T {\mathbf{b}}(t)|^2}{\sum_i|{\mathbf{v}}_i^T {\mathbf{b}}(t)|^2},$$ as a measure of the relative occupation of each collective mode [@Facchinetti16].
Eigenmodes of a single unit-cell
================================
We display in Fig. \[figmodes\] the excitation eigenmodes of a single isolated unit-cell of a planar array used to generate optically active magnetism \[Fig. 1(a) in the main section\]. Each unit-cell consists of four atoms and therefore 12 eigenmodes of which three are degenerate with some of the other modes and are obtained by trivial symmetry transformations of the dipole orientations. The properties of the eigenmodes are listed in Table 1 of the main section.
![The orientation of the atomic dipoles for the eigenmodes of an isolated square unit-cell, with the same ordering as Table 1 of the main section, for the spacing $a=0.15\lambda$. Top and middle row: in-plane modes consisting of (a) electric dipole, (b) magnetic dipole ($n=4$), (c-e) electric quadrupole modes, and (f) mixed multipole character. Bottom row: modes with $x$ polarization consisting of (a) Electric dipole and (b-c) mixed multipole. Note the modes shown in (a), (f), and (i) are each doubly degenerate due to lattice symmetry. The ordering of the sites given in the text for the atomic level shifts is illustrated in (b).[]{data-label="figmodes"}](modes.pdf){width="\columnwidth"}
Coupling to collective magnetic mode
====================================
![The collective linewidths $\upsilon_e$ and $\upsilon_m$ of the electric and magnetic dipole modes as a function of total atom number $N$ for a lattice with $a=0.15\lambda$ and $d=0.5\lambda$. While the linewidth of the magnetic mode depends strongly on $N$, the linewidth of the electric mode is almost constant, starting at $\upsilon_m=3.4$ for a single unit-cell and rising to $3.7$ at $N=1600$. []{data-label="fig:collectivelinewidth"}](collectivelinewidthnew.pdf){width="\columnwidth"}
The excitation of a collective mode consisting of magnetic dipoles in a planar array via the coupling between electric and magnetic dipole eigenmodes is described by a two-mode model \[Eq. (2) in the main section\]. The incident field with the polarization $({\hat{\mathbf{{e}}}}_y+{\hat{\mathbf{{e}}}}_z)/\sqrt{2}$ drives the electric dipole mode. The electric dipole modes in the $y$ and $z$ direction are clearly degenerate due to symmetry, and so the excited symmetric combination is also an eigenmode, with amplitude $\mathcal{P}_{e}$. When all the electronically excited levels are degenerate, this mode evolves independently of the magnetic dipole mode of amplitude $\mathcal{P}_{m}$ which is not excited.
When the level shifts are not equal ($\Delta_{\pm,0}^{(j)}\neq 0$), $\mathcal{P}_{e}$ and $\mathcal{P}_{m}$ no longer describe eigenmodes, and these are coupled together (as generally could the other eigenmodes of the $\Delta_{\pm,0}^{(j)} = 0$ system). As shown in Fig. \[figmodes\], the array of atoms in the $yz$ plane \[Fig. 1(a) in the main section\] has two separate families of modes, those with polarization in the $x$ direction out of the plane and those with polarizations in the plane. To avoid coupling the drive to those out of plane modes we choose $\Delta_{+}^{(j)}=\Delta_{-}^{(j)}$ on each atom. Then, to couple the electric dipole mode to the alternating phases of the $y$ and $z$ components of the magnetic dipole mode, we choose the remaining level shifts to have similarly alternating signs:
$$\begin{aligned}
(\delta^{(\pm)}_{(4j+1)},\delta^{(\pm)}_{(4j+2)},\delta^{(\pm)}_{(4j+3)},\delta^{(\pm)}_{(4j+4)}) &=
\delta(2,0,2,0),\\
(\delta^{(0)}_{(4j+1)},\delta^{(0)}_{(4j+2)},\delta^{(0)}_{(4j+3)},\delta^{(0)}_{(4j+4)}) &=
\delta(0,0,2,2),\end{aligned}$$
\[shifts\]
where the ordering of the atoms is shown in Fig. \[figmodes\](b).
The periodic variation of the level shifts can be produced by the AC Stark shift of an external standing-wave laser with the intensity varying along the principal axes of the array, such that the levels $m=\pm1$ are shifted using an intensity variation along the $z$ direction and the shift of the $m=0$ state with the intensity variation along the $y$ direction. In the both cases the intensity maxima are then separated by the distance $d$ between the adjacent unit-cells. Suitable transitions could be found, e.g., with Sr or Yb. For example, for the $^3P_0\rightarrow \mbox{}^3D_1$ transition of $^{88}$Sr the resonance wavelength $\lambda\simeq 2.6\mu$m and the linewidth $2.9\times 10^{5}$/s [@Olmos13]. For the case of optical lattices the periodicity of the sinusoidal potential for the same transition with a magic wavelength may be chosen as 206.4nm and can also be modified to achieve the right periodicity by tilting the propagation direction of the lasers forming the lattice. Alternatively, atoms in different hyperfine states could occupy different lattice sites [@mandel03], with the associated description of the atom-light dynamics [@Jenkins_long16; @Lee16], or the trapping potential strength for tweezers could possibly be spatially varied.
With this choice of level shifts given by Eq. we obtain \[Eq. (2) in the main section\],
$$\begin{aligned}
\label{zeeman1}
\partial_t \mathcal{P}_{e}^{(j)} &= (i\delta_e+i\Delta-\upsilon_e)\mathcal{P}_{e}^{(j)} + \delta\mathcal{P}_{m}^{(j)} +f, \\
\label{zeeman2}
\partial_t \mathcal{P}_{m}^{(j)} &= (i\delta_m+i\Delta-\upsilon_m)\mathcal{P}_{m}^{(j)} + \delta\mathcal{P}_{e}^{(j)}.\end{aligned}$$
\[zeemanboth\]
The effective dynamics of Eqs. can represent both a single unit-cell in isolation and the entire array of multiple unit-cells, but the collective resonance line shifts and linewidths of the electric and magnetic dipole modes, $\delta_{e,m}$ and $\upsilon_{e,m}$, respectively, can considerably differ in the two cases, and generally vary with the number of unit-cells (Fig. \[fig:collectivelinewidth\]). The driving field is denoted by $f=i\xi\epsilon_0{\mathcal{E}}_0/\mathcal{D}$. The alternating signs of $\Delta_{-,0}^{(j)}$ mean there is no coupling to other unit-cell eigenmodes with different symmetries.
The steady state of Eqs. is easily solved and we find the ratio of the magnetic and electric dipole mode amplitudes to be $$\left|\frac{\mathcal{P}_m}{\mathcal{P}_e}\right|=\left|\frac{\delta}{i\delta_m+i\Delta-\upsilon_m}\right|.$$ On resonance, $\Delta=-\delta_m$, the ratio of amplitudes is determined by the collective linewidth $\upsilon_m$. As shown in Fig. \[fig:collectivelinewidth\] the linewidth of the magnetic dipole mode rapidly narrows as a function of the array size and becomes strongly subradiant. It is this collective resonance narrowing which allows the amplitude of the magnetic dipole mode to become large compared with the electric dipole mode amplitude, even for relatively small detuning strength $\delta$.
Equations are similar to those describing the electromagnetically-induced transparency [@FleischhauerEtAlRMP2005] of ‘dark’ and ‘bright’ states of noninteracting atoms in which case the atom population can be trapped in the dark state. In the present case, the dark state is represented by a collective eigenmode, resulting from the resonant dipole-dipole interactions. It is this collective subradiant nature of the mode that drives the excitation into the magnetic dipole mode.
Spherical harmonics
===================
The multipole moments of the lattice unit-cells are characterized from the far-field radiation by decomposing the field in terms of vector spherical harmonics [@Jackson] \[Eq. (1) in the main section\]. The vector spherical harmonics are defined in terms of the ordinary spherical harmonics $Y_{lm}(\theta,\phi)$ as ${\mathbf{\Psi}}_{lm}=r\nabla Y_{lm}$ and ${\mathbf{\Phi}}_{lm}={\mathbf{r}}\times \nabla Y_{lm}$ where ${\mathbf{r}}$ is the vector from the origin to the observation point, $\theta$ is the polar angle with the $x$ axis, and $\phi$ is the azimuthal angle from the $y$ axis in the $yz$ plane. They are orthogonal; $\int{\mathbf{\Psi}}_{lm}^\ast {\mathbf{\Psi}}_{l^\prime m^\prime}\mathrm{d}\Omega=\int{\mathbf{\Phi}}_{lm}^\ast {\mathbf{\Phi}}_{l^\prime m^\prime}\mathrm{d}\Omega=l(l+1)\delta_{l l^\prime}\delta_{m m^\prime}$, $\int{\mathbf{\Psi}}_{lm}^\ast {\mathbf{\Phi}}_{l^\prime m^\prime}\mathrm{d}\Omega=0$, and so the coefficients $\alpha$ can be found by projecting onto the corresponding vector harmonic.
Huygens’ surface
================
For analyzing the properties of the Huygens’ surface, the contributions of the both incident and scattered light are included. The scattered light can be calculated by summing up all the light scattered from all the atoms in the array. We have verified that the transmission of light at distances $\lambda\alt \xi\ll \sqrt{{\cal A}}$ from a planar array of uniform excitations, where ${\cal A}$ denotes the area of the array, can also be estimated by [@Chomaz12; @Javanainen17; @Facchinetti18; @Javanainen19] $$\begin{aligned}
\epsilon_0 {\mathbf{E}} &= {\mathcal{E}}_0\hat{{\mathbf{e}}}_y e^{ik\xi} + \frac{ik}{2 {\cal A}}\sum_k \left[{\mathbf{d}}_k-\hat{{\mathbf{e}}}_x\cdot {\mathbf{d}}_k \hat{{\mathbf{e}}}_x\right]e^{i(\xi-x_k)},\end{aligned}$$ where the second term denotes the scattered field ${\mathbf{E}}_S$ and the first term is the incident field. This expression has been used together with the microscopic calculation to analyze Huygens’ surface, e.g., in Fig. 3 in the main section.
![(a) The magnitude and (b) the phase of the total transmitted light and the scattered light for the same parameters, as Fig. 3(a) in the main section. []{data-label="fig:camparg"}](ampargE.pdf){width="\columnwidth"}
In Fig. \[fig:camparg\] we show the magnitude and the phase of the total transmitted field, and also the contribution of the scattered field alone. Fig. \[fig:camparg\](b) shows that while the phase of the scattered field covers a range of $\pi$, the total field has a full range of $2\pi$. The contribution from electric and magnetic dipoles in a Huygens’ surface add to give a scattered field with magnitude up to twice the incident field, as shown in Fig. \[fig:camparg\](a), allowing for close to total transmission even when the scattered field is $\pi$ out of phase with the incident light.
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---
abstract: 'For analysing and/or understanding languages having no word boundaries based on morphological analysis such as Japanese, Chinese, and Thai, it is desirable to perform appropriate word segmentation before word embeddings. But it is inherently difficult in these languages. In recent years, various language models based on deep learning have made remarkable progress, and some of these methodologies utilizing character-level features have successfully avoided such a difficult problem. However, when a model is fed character-level features of the above languages, it often causes overfitting due to a large number of character types. In this paper, we propose a CE-CLCNN, character-level convolutional neural networks using a character encoder to tackle these problems. The proposed CE-CLCNN is an end-to-end learning model and has an image-based character encoder, i.e. the CE-CLCNN handles each character in the target document as an image. Through various experiments, we found and confirmed that our CE-CLCNN captured closely embedded features for visually and semantically similar characters and achieves state-of-the-art results on several open document classification tasks. In this paper we report the performance of our CE-CLCNN with the Wikipedia title estimation task and analyse the internal behaviour.'
author:
-
bibliography:
- 'reference.bib'
title: 'End-to-End Text Classification via Image-based Embedding using Character-level Networks'
---
text classification, image-based character embedding, convolutional neural networks
Introduction
============
Overfitting is one of the most essential problems in machine learning. Various regularization methods have been proposed in order to improve generalization performance, especially in deep networks. Data augmentation is the most common way to improve generalization performance of the system by increasing the training dataset in a pseudo manner. In natural language processing (NLP) tasks, various data augmentation methods have also been proposed, such as synonym lists [@zhang2015character], grammar induction [@jia2016data], task-specific heuristic rules [@silfverberg2017data], and contextual augmentation [@Kobayashi2018ContextualAD]. However, these methods basically require appropriate word segmentation and semantic analysis of the context in advance, and they are inherently difficult in the Asian languages, especially Japanese, Chinese or Thai, etc. In recent years, various language models based on deep learning have made remarkable progress, and some of these methodologies utilizing character-level features have successfully avoided such problems [@zhang2015character; @kim2016character]. From the model selection point of view, recurrent neural networks (RNN) have been widely applied in NLP tasks, however, they have a significant problem in learning long text sequences. The recent introduction of Long-short term memory (LSTM) [@hochreiter1997long] and gated recurrent units (GRU) [@chung2014empirical] alleviated this issue and are commonly used in the NLP field. However, it still has drawbacks, such as difficulty in parallelization. Character level convolutional neural networks (CLCNN) [@zhang2015character], i.e. one-dimensional convolutional neural networks (CNN), also accept long text sequences, and, in addition, its training speed is generally faster than LSTM and GRU thanks to its native property and the ease of parallelization [@dauphin2016language; @bradbury2016quasi]. However, there are still problems remaining when dealing with the aforementioned languages. When a model is fed character-level features (e.g. in one-hot vector or other common embeddings) of the language above, it often causes overfitting due to a large number of unique characters. For example, Japanese[^1] and Chinese[^2] have over 2,000 type of characters in common use. We need to tackle this problem as well. Fortunately, a not insignificant number of Kanji and Han characters used in Japanese and Chinese are ideograms, which means its character shape represents its meaning. Therefore, capturing the shape feature of characters in the document is meaningful for better understanding the contents. Based on this hypothesis, several studies have been proposed recently. Shimada et al. [@shimada2016document] proposed epoch making schematics called image-based character embedding, in which they treat each character in the target document as an image. Their model learns a low-dimensional character embedding by a convolutional auto-encoder (CAE) [@masci2011stacked] and then the relationship between the sequence of embeddings and the document category is trained with the following CLCNN. They also proposed a simple and very effective data augmentation technique called wildcard training. The wildcard training randomly dropouts [@hinton2012improving] arbitrary elements in the embedded domain at the time of training of CLCNN. This data augmentation method greatly improved system generalization performance without requiring morphological analysis. They confirmed that the effect of this wildcard training improves the document classification accuracy by about 10% on their evaluation, using open and private datasets. Lately, Zhang et al. [@zhang2018word] also proposed similar semantic dropout for word representations and reported its effectiveness. On the other hand, however, since their model learns CAE and CLCNN separately, it cannot fully exploit the merits of image-based character embedding. Further improvement in performance can be expected. Liu et al. [@liu2017learning] proposed an end-to-end document classification model that learns character embedding using a CNN-based character encoder and classifies documents using GRU on Chinese, Japanese, and Korean documents. Unlike Shimada’s model [@shimada2016document], their model does not train to preserve the shape feature of characters explicitly. But they demonstrated that characters with similar shape features are embedded closer to the character representation. Su et al. [@DBLP:conf/emnlp/SuL17] proposed glyph-enhanced word embedding (GWE) to focus on the shape of Kanji. The basic strategy is the same with [@shimada2016document]: GWE extracts the shape information of the character and uses it for training the word representation. They also performed image-based character embedding on a Chinese document with CAE and showed that characters with similar shape features are represented by close character representation. In these studies [@liu2017learning; @DBLP:conf/emnlp/SuL17], image-based character embedding showed promising performance, while there is still room for improvement from the viewpoint of introducing data augmentation in which the model inputs take advantage of the features of character image. Based on these backgrounds, in this paper, we propose a new “character encoder character-level convolutional neural networks” (CLCNN) model. The proposed CE-CLCNN is an end-to-end learning model and has an image-based character encoder. Due to this architecture, our CE-CLCNN has the following desirable features:
1. It is freed from intractable morphological analysis.
2. It learns and obtains character embedding associating with character appearance.
3. It is capable of a suitable data augmentation method both for image and embedded feature spaces.
By introducing two essentially different types of data augmentation, the robustness of the model is enhanced and the performance of document classification task is significantly improved.
![Schematics of our CE-CLCNN model. The CE-CLCNN is made up of character encoder (CE) component and document classification component. These two parts are basically composed of CNN and CLCNN respectively and are directly concatenated.[]{data-label="fig:proposed_classification_system"}](proposed_architecture.eps){width="0.9\linewidth"}
CE-CLCNN
========
The outline of the proposed CE-CLCNN is shown in [Fig.\[fig:proposed\_classification\_system\]]{}. CE-CLCNN is made up of two different CNN consolidations. The first CNN acts as a character encoder (CE) that learns character representations from character images, and the second CNN, CLCNN, performs document classification. The parameters of these two consecutive networks are optimized by the backpropagation with the cross entropy error function as the objective function.
Character encoder by CNN
------------------------
Firstly, each character of the target document is converted to an image having 36 $\times$ 36 pixels. The CE embeds (i.e. encodes) each character image into a $d_{CE}$ dimensional feature vector. [Table\[tab:architecture\_of\_character\_encoder\]]{} shows the architecture of CE used in this instance. Here, let $k$ be the kernel size and $o$ be the number of filters. In the training of CE, continuous $C$ characters in the document are treated as a chunk. The convolution is performed with depth-wise manner, and, in fact, each input character is embedded in 128 dimensional vector. Thus, the input dimension of the CE is 36$\times$36$\times C$, and the output of that is 1$\times$128$\times C$. The CE trains with the batch size of $\mathcal{B}$.
Layer \# CE configuration
---------- ---------------------------------------------
1 Conv($k$=(3, 3), $o$=32) $\rightarrow$ ReLU
2 Maxpool($k$=(2, 2))
3 Conv($k$=(3, 3), $o$=32) $\rightarrow$ ReLU
4 Maxpool($k$=(2, 2))
5 Conv($k$=(3, 3), $o$=32) $\rightarrow$ ReLU
6 Linear(800, 128) $\rightarrow$ ReLU
7 Linear(128, 128) $\rightarrow$ ReLU
: Architecture of Character Encoder (CE)[]{data-label="tab:architecture_of_character_encoder"}
Document classifier by CLCNN
----------------------------
The character representation of the $d_{CE}$ bit/character encoded from the CE is reshaped to be the batch size $\mathcal{B}$ again with the character string length of $\mathcal{C}$. Then the representations are inputted to the CLCNN. Note that we use convolutions with stride $s$ rather than pooling operations which are widely used in natural language processing, with reference to prior work [@zhang2017deconvolutional]. [Table\[tab:architecture\_of\_clcnn\]]{} shows the architecture of CLCNN used in this instance.
Layer \# CLCNN configuration
---------- -----------------------------------------------------
1 Conv($k$=(1, 3), $o$=512, $s$=3) $\rightarrow$ ReLU
2 Conv($k$=(1, 3), $o$=512, $s$=3) $\rightarrow$ ReLU
3 Conv($k$=(1, 3), $o$=512) $\rightarrow$ ReLU
4 Conv($k$=(1, 3), $o$=512)
5 Linear(5120, 1024)
6 Linear(1024, \# classes)
: Architecture of Character-level Convolutional Neural Network (CLCNN)[]{data-label="tab:architecture_of_clcnn"}
Data augmentation on input space and feature space
--------------------------------------------------
![Example of data augmentation on image domain (random erasing data augmentation [@zhong2017random]). Note that this is an example of an implementation on this experiment and augmentation is not limited to this method.[]{data-label="fig:visualize_random_erasing_chars"}](visualize_random_erasing_chars.eps){width="\linewidth"}
Convolutional neural networks are known to require a large amount of diverse training data. Our CE-CLCNN model has a capability to perform data augmentation both in the input space and the feature space thanks to its end-to-end structure. In the input space, we apply random erasing data augmentation (RE) [@zhong2017random] to the character image that will be fed to the CE. Each character image is randomly masked with noise on the rectangular area, and thus a part of the character is occluded as shown in [Fig.\[fig:visualize\_random\_erasing\_chars\]]{}. In the feature embedded space, we apply wildcard training (WT) [@shimada2016document] that randomly drops out some of embedded expression (i.e. some element of encoded vector) with the ratio of $\gamma_w$.
Parameter Scale
------------------------- -------
Erasing probability $p$ 0.3
Max area ratio $s_l$ 0.4
Min area ratio $s_h$ 0.02
Max aspect ratio $r_1$ 2.0
Min aspect ratio $r_2$ 0.3
: Parameters of Random erasing data augmentation[]{data-label="tab:parameters_of_random_erasing_data_augmentation"}
Experiments
===========
Implementation
--------------
The number of embedding dimensions and the chunk size of characters were set to $d_{CE} = 128$ and $C = 10$, respectively. [Table\[tab:parameters\_of\_random\_erasing\_data\_augmentation\]]{} summarizes the parameters used in random erasing data augmentation. The ratio in the wildcard training was set to $\gamma = 0.1$. In the CLCNN, the batch size of the embedded characters in the training $B = 256$, and Adam [@kingma2014adam] was used for parameter optimization.
Category estimation of Wikipedia titles
---------------------------------------
In this paper, we evaluate our proposed CE-CLCNN using an open dataset for category estimation of Wikipedia titles. The Wikipedia title dataset [@liu2017learning] contains the article titles acquired from Wikipedia and the related topic class label. This dataset includes 12 classes: Geography, Sports, Arts, Military, Economics, Transportation, Health Science, Education, Food Culture, Religion and Belief, Agriculture and Electronics. In this experiment, we used the Japanese data subset this time (total 206,313 titles). For training of the model, we split the dataset into the training and testing set with an 8:2 ratio, respectively. Zero padding was performed for titles with less than 10 characters so that the input sentence would be 10 characters or more. [Table\[tab:results\_of\_wikipedia\_titles\_ja\]]{} shows the results. To the best of our knowledge, the proposed CE-CLCNN showed state-of-the art performance on this dataset. Shimada’s method [@shimada2016document] showed about 4% better performance than Liu’s method [@liu2017learning] (proposed later thanks to their WT with highly effective generalization). The proposed CE-CLCNN with RE and WT showed even better performance by about 4%. According to [Table\[tab:results\_of\_wikipedia\_titles\_ja\]]{}, the performance of the native CE-CLCNN (i.e. without RE and WT) was equivalent to Shimada’s CLCNN+WT. Since the performance improvement of CE-CLCNN by introduction of WT was limited, we can speculate CE-CLCNN has sufficient model versatility in the embedded space. While on the other hand, the effect of introducing RE was certain, with a 3-3.5% gain.
[lr]{} Method & Accuracy\[%\]\
(lr)[1-1]{} (lr)[2-2]{} **(Proposed)** RE + CE-CLCNN + WT & **58.4**\
**(Proposed)** RE + CE-CLCNN & 58.0\
**(Proposed)** CE-CLCNN + WT & 55.3\
**(Proposed)** CE-CLCNN & 54.4\
CLCNN + WT$\dagger$ [@shimada2016document] & 54.7\
CLCNN$\dagger$ [@shimada2016document] & 36.2\
VISUAL model$\ddag$ [@liu2017learning] & 47.8\
LOOKUP model$\ddag$ [@liu2017learning] & 49.1\
Ensemble (VISUAL + LOOKUP)$\ddag$ [@liu2017learning] & 50.3\
\
Analysis of Character Encoder
-----------------------------
[c]{}
{width="\linewidth"}
{width="\linewidth"}
[UTF8]{}[ipxm]{}
[ccc]{} Query character & Neighbouring character &
----------------------
Euclidean distance
with query character
----------------------
: Top five characters near the character representation for the query character[]{data-label="tab:result_character_encoder"}
\
& 鰭 & 370.1\
& 駮 & 403.7\
& 鮪 & 405.2\
& 鰐 & 409.4\
& 鰤 & 409.6\
& 癨 & 317.2\
& 癜 & 388.3\
& 瘻 & 398.3\
& 痕 & 398.9\
& 痴 & 399.2\
& 彼 & 452.8\
& 擅 & 491.5\
& 擔 & 520.5\
& 擒 & 533.8\
& 捗 & 536.8\
[Table\[tab:result\_character\_encoder\]]{} shows an example of similar characters in the CE-encoded feature embedding domain with the 5-nearest neighbouring method. Many of the neighbouring characters for the query character were similar in shape features of letters, such as radicals (i.e. character components). Therefore, it was confirmed that the character encoder learned by capturing the shape feature of the character.
[UTF8]{}[ipxm]{} Furthermore, we extracted the character representation of Chinese characters by using learned CE, and then projected the representation on 2 dimensional space using t-SNE [@maaten2008visualizing]. A part of the visualization result is shown in [Fig.\[fig:example\_of\_t-sne\_visualization\]]{}. We can see that characters with the same components are clustered. Note that Su et. al. [@DBLP:conf/emnlp/SuL17] explicitly learned to preserve character shape features by CAE, but our CE-CLCNN does not explicitly learn character representation that preserves the shape features of characters explicitly. In CE-CLCNN, since the loss of document classification backpropagates to the CE which learns character representation, we found that clusters that are semantically similar in character representation are close clusters. For example, it can be seen that the character cluster having “舟” component representing “boat” and the character cluster having “魚” component representing “fish” are closely related.
Conclusion
==========
In this paper, we propose the new and promising text analysis model “CE-CLCNN” to solve several conventional problems for languages such as Japanese and Chinese. We confirm not only its excellent document classification performance, but also its readability in terms of how the model works. In near future, we would like to investigate our model more and apply it to other languages whose character shapes are related to the meaning.
[^1]: F. Sakade, Guide to Reading and Writing Japanese, 4th ed., J. Ikeda, Ed. Tuttle Publishing, 2013.
[^2]: Table of General Standard Chinese Characters (<http://www.gov.cn/gzdt/att/att/site1/20130819/tygfhzb.pdf>)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we define and study a new task called *Context-Aware Semantic Expansion* (CASE). Given a *seed term* in a sentential context, we aim to suggest other terms that well fit the context as the seed. CASE has many interesting applications such as query suggestion, computer-assisted writing, and word sense disambiguation, to name a few. Previous explorations, if any, only involve some similar tasks, and all require human annotations for evaluation. In this study, we demonstrate that annotations for this task can be harvested from existing corpora, in a fully automatic manner. On a dataset of 1.8 million sentences thus derived, we propose a network architecture that encodes the context and seed term separately before suggesting alternative terms. The context encoder in this architecture can be easily extended by incorporating seed-aware attention. Our experiments demonstrate that competitive results are achieved with appropriate choices of context encoder and attention scoring function.'
author:
- |
Jialong Han^1^[^1], Aixin Sun^2^, Haisong Zhang^3^, Chenliang Li^4^, Shuming Shi^3^\
^1^Amazon, USA, ^2^Nanyang Technological University, Singapore, ^3^Tencent AI Lab, China, ^4^Wuhan University, China\
^1^[email protected], ^2^[email protected], ^3^{hansonzhang, shumingshi}@tencent.com, ^4^[email protected]
bibliography:
- 'CASE.bib'
title: 'CASE: Context-Aware Semantic Expansion'
---
Introduction {#sec:intro}
============
Have you ever googled “*Lionel Messi championships*”, browsed the results, and wanted more soccer stars with comparable championships? Have you ever wanted to know types of nutrients rich in barley grass, but were only able to remember amino acid? In this paper, we study *context-aware semantic expansion* (or CASE for short). In CASE, user provides a *seed term* wrapped in a sentential *context* as in Figure \[fig:example\]. The system returns a list of *expansion terms*, each of which is a valid substitute for the seed, [*i.e.,*]{}the substitution is supported by some sentence in a (testing) corpus. This task is not easy due to the large number of potential expansions, as well as the necessity of modeling their interactions with both the context and the seed. Despite the challenge, the task is of practical importance and benefits many applications. We list a few examples here.
[**Query suggestion**]{} [@wen2001clustering]**.** In the aforementioned query “*Lionel Messi championships*”, keywords “*Lionel Messi*” can be a seed term to expand, and a CASE system may suggest related entities, [*e.g.,*]{}“*Christiano Ronaldo*”, as expansion terms. Those terms may be used to suggest queries like “*Christiano Ronaldo championships*”.
[**Computer-assisted writing**]{} [@liu2011computer]**.** For casual or academic writing, exemplifications often help to explain and convince. It is desirable to suggest contextually appropriate alternative words when an author can think of only one.
[**Other NLP tasks.**]{} CASE can potentially enhance natural language processing (NLP) tasks. For example, in word sense disambiguation [@navigli2009word], an ambiguous word like “apple” can be first expanded *w.r.t.* its context. The suggested context-aware terms ([*e.g.,*]{}fruits or companies) provide cues for the disambiguation task.
**Seed in context**: “Young barley grass is high in .”\
**Expansion terms**: vitamin, antioxidant, enzyme, mineral, chlorophyll, …
Comparison with Related Tasks
-----------------------------
Despite its significance, explorations on CASE remain limited. *Lexical substitution* [@mccarthy2007semeval] is the most similar task to CASE. Given a word in a sentential context, [*e.g.,*]{}“the girl is reading a book”, lexical substitution predicts synonyms fitting the context, [*e.g.,*]{}“wise” or “clever” rather than “shining”. The synonym candidates generally come from high-quality but relatively small dictionaries like WordNet [@fellbaum1998wordnet]. Compared with lexical substitution, candidate expansion terms of CASE, [*e.g.,*]{}entity names, are not required to be aliases of the seeds but could be far more in number and less organized.
Besides lexical substitution, another task similar to CASE is *set expansion* [@tong2008system; @wang2007language; @he2011seisa; @chen2016long; @shen2017setexpan; @shi2010corpus]. It is to expand a few seeds ([*e.g.,*]{}amino acid and vitamin) to more terms in the same semantic class ([*i.e.,*]{}nutrition). However, set expansion does not involve possible textual contexts with the seeds. This may cause “fat” to appear in the results of Figure \[fig:example\], which is irrelevant to barley grass. While the above two tasks differ considerably from CASE by task definition, we further note that their model tuning and evaluation require manual annotations, which are hard to collect at scale. Fortunately, CASE benefits from large-scale *natural* annotations, as described below.
Dataset and Formal Task Definition {#sec:task}
----------------------------------
A first step toward CASE with today’s deep learning machinery is to accumulate large-scale annotations. For this task, an ideal piece of annotation would be different terms appearing separately in identical contexts. While this form of annotations is hard to obtain manually and rare in natural corpora, we note that people often list examples, which effectively serve as natural annotations. In a general corpus, lists of examples usually follow Hearst patterns [@hearst1992automatic; @snow2005learning], [*e.g.,*]{}“ such as , , …”, “, , …, and other ”, [*etc.*]{}Here denotes a *hypernym*, and {} are *hyponyms*. We note that, in the context, if all hyponyms other than one is removed, the sentence is still “correct” in the sense of the corpus.
By post-processing a web-scale corpus (detailed in experiments), we derive a collection of 1.8 million naturally annotated sentences. All of them are of the form $\langle C,T\rangle$ as below.
[p[3in]{}]{}\
**Context $C$**: “Young barley grass is high in and other phyto-nutrients.”\
**Terms $T$**: {vitamin, antioxidant, enzyme, mineral, amino acid, chlorophyll}\
Here $C$ is the sentential context with a *placeholder* “”. $T=\{t_i\}$ are hyponyms appearing at the placeholder in Hearst patterns. The CASE task is to use a *seed* term $s\in T$ and the context $C$ to recover the remaining terms $T\setminus\{s\}$.
Taking advantage of the large dataset derived, we propose a neural network architecture with attention mechanism [@bahdanau2014neural] to learn supervised expansion models. Readers may notice that, due to the use of Hearst patterns, the context $C$ above has an additional hypernym “phyto-nutrient” compared with Figure \[fig:example\]. In experiments, in addition to comparisons among solutions, we will also study the impact of this gap.
To summarize, our contributions are:
- We define and study a novel task, [*i.e.,*]{}CASE, which supports many interesting and important applications.
- We identify an easy yet effective method to collect natural annotations.
- We propose a neural network architecture for CASE, and further enhance it with the attention mechanism. On millions of naturally annotated sentences, we experimentally verify the superiority of our model.
Model Overview {#sec:model}
==============
Given the inherent variability of natural language and the sufficient annotations, we tackle CASE by a supervised neural-network-based approach.
Our network to model $P(.|s,C)$ is shown in Figure \[fig:network\]. The network consists of three parts: a *context encoder*, a *seed encoder*, and a *prediction layer*. Given a seed $s$ in a context $C$, the network encodes them into two vectors $\mathbf{v}_s$ and $\mathbf{v}_C$ with the seed and context encoders, respectively. The two vectors are then concatenated as input to the prediction layer to predict potential expansion terms.
![The network architecture of CASE.[]{data-label="fig:network"}](architecture.pdf){width="\columnwidth"}
On training sentences $\mathcal{T}$, we aim to optimize: $$\label{eq:obj}
\max \sum_{\langle C,T\rangle \in \mathcal{T}} \sum_{s\in T} \log P(T\setminus\{s\}|s,C)$$ Note that, a sentence $\langle C,T\rangle$ is regarded as $|T|$ training samples. Each sample treats one term as the seed, and predicts the other terms within the context. In the remainder of this section, we briefly describe each of the three components.
Encoding Sentential Contexts
----------------------------
Given one or two seed terms, the traditional set expansion task simply finds other terms in the same semantic class. However, the sentential context $C$ may contain additional descriptions or restrictions, thus narrowing down the scope of the listed terms. Therefore, it is vital to appropriately model $C$ to capture its underlying information in CASE.
In Figure \[fig:network\], we employ the context encoder component to encode a variable-length context $C$ into a fixed-length vector $\mathbf{v}_C$. There are various off-the-shelf neural models to encode sentences or sentential contexts. On the one hand, by treating $C$ as a bag or a sequence of words, conventional sentence encoders may be applied, [*e.g.,*]{}Neural Bag-of-Words (<span style="font-variant:small-caps;">NBoW</span>) [@kalchbrenner2014convolutional], <span style="font-variant:small-caps;">RNN</span> [@pearlmutter1989learning], and <span style="font-variant:small-caps;">CNN</span> [@lecun1989backpropagation]. On the other hand, there are also techniques that explicitly model placeholders, [*e.g.,*]{}CNN with positional features [@zeng2014relation] and <span style="font-variant:small-caps;">context2vec</span> [@melamud2016context2vec]. In this paper, we mainly investigate <span style="font-variant:small-caps;">NBoW</span>-based and <span style="font-variant:small-caps;">RNN</span>-based encoders. We also involve other encoders for comparison, [*e.g.,*]{}<span style="font-variant:small-caps;">CNN</span>-based and placeholder-aware encoders.
[**Neural Bag-of-Words Encoder.**]{} Given words $\{c_i\}_{i=1}^n$ in a context $C$, an <span style="font-variant:small-caps;">NBoW</span> encoder looks up their vectors $\mathbf{c}_i\in \mathbb{R}^d$ in an embedding matrix, and average the vectors as $\mathbf{v}_C=\frac{1}{n} \sum_{i=1}^{n} \mathbf{c}_i$. The word embedding matrix is initialized with embeddings pre-trained on the original sentences in training set, and is updated during training. Due to its simplicity, <span style="font-variant:small-caps;">NBoW</span> is efficient to train. However, it ignores the order of context words. [**RNN- and CNN-Based Encoders.**]{} To study the impact of word order on context encoding, we consider <span style="font-variant:small-caps;">RNN</span>-based encoders as alternatives to <span style="font-variant:small-caps;">NBoW</span>. <span style="font-variant:small-caps;">RNN</span>s take a sequence of context word vectors $(\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_n)$, and iteratively encodes information before each position $i$ as a sequence of *hidden vectors* $(\mathbf{h}_1,\mathbf{h}_2,\dots,\mathbf{h}_n)$, $\mathbf{h}_i \in \mathbb{R}^d$. Following @wang2016attention, we take the last hidden vector $\mathbf{h}_n$ as the context vector $\mathbf{v}_C$: $$\begin{aligned}
\mathbf{h}_i&=\text{RNN}(\mathbf{c}_i, \mathbf{h}_{i-1})\text{,}\quad i=1\dots n\text{,}\\
\mathbf{v}_C&=\mathbf{h}_n\text{.}\end{aligned}$$
Besides the vanilla version of <span style="font-variant:small-caps;">RNN</span>, other <span style="font-variant:small-caps;">RNN</span> variations like <span style="font-variant:small-caps;">LSTM</span> [@hochreiter1997long], <span style="font-variant:small-caps;">GRU</span> [@chung2014empirical], and bi-directional <span style="font-variant:small-caps;">LSTM</span> (<span style="font-variant:small-caps;">BiLSTM</span>) [@graves2005framewise] have proven effective in various NLP tasks. In our experiments, we compare all these RNN variations.
Other than the <span style="font-variant:small-caps;">NBoW</span>- and <span style="font-variant:small-caps;">RNN</span>-based encoders described above, <span style="font-variant:small-caps;">CNN</span>s [@lecun1989backpropagation] have also been used as sentence encoders [@kalchbrenner2014convolutional; @kim2014convolutional; @hu2014convolutional]. Specifically, we perform the convolution operation on the input vector sequence $(\mathbf{c}_1,\mathbf{c}_2,\dots,\mathbf{c}_n)$, and apply max-pooling to get the context representation $\mathbf{v}_C$.
[**Position-Aware Encoders.**]{} All above encoders ignore the the position of the placeholder, [*i.e.,*]{}where the seed term appears. For CASE, one may hypothesize that words at different distances to the placeholder contributes differently to $\mathbf{v}_C$. [[@zeng2014relation ]{}]{} propose <span style="font-variant:small-caps;">CNN</span> with positional features (<span style="font-variant:small-caps;">CNN+PF</span>) as a counterpart for <span style="font-variant:small-caps;">CNN</span>. Each context word vector $\mathbf{c}_i$ fed into <span style="font-variant:small-caps;">CNN</span> is concatenated with a *position vector* $\mathbf{p}_i$ that models its distance to the placeholder. The positional vectors are treated as parameters and updated during training. In <span style="font-variant:small-caps;">context2vec</span> [@melamud2016context2vec], two <span style="font-variant:small-caps;">LSTM</span>s are used to encode the left and right contexts of placeholders, respectively. The output are concatenated as the final context representation $\mathbf{v}_C$. We implement and compare it with <span style="font-variant:small-caps;">BiLSTM</span> as a counterpart.
Encoding the Seed Term
----------------------
Due to its short length, we simply adopt the same <span style="font-variant:small-caps;">NBoW</span> to encode seed term, for it is less prone to overfitting [@shimaoka2017neural]. Given words $\{s_i\}_{i=1}^m$ of a seed term $s$, we obtain $\mathbf{v}_s$ by $\mathbf{v}_s=\frac{1}{m} \sum_{i=1}^{m} \mathbf{s}_i$. Because of their different role, seed word embeddings $\mathbf{s}_i\in \mathbb{R}^d$ are from another embedding matrix, but are initialized and updated in the same manner with context word embeddings.
Predicting Expansion Terms
--------------------------
After encoding the seed and the context into $\mathbf{v}_s$ and $\mathbf{v}_C$, respectively, we feed their concatenation $\mathbf{x}=\mathbf{v}_s\oplus \mathbf{v}_C$ to the *prediction layer* for expansion terms. We treat the prediction as a classification problem, and each candidate term as a classification label. Given a sufficiently large $\mathcal{T}$, we consider all terms appearing in Hearst pattern lists in $\mathcal{T}$ as candidates, and constitute the label set $L$ by pooling them, [*i.e.,*]{}$L=\cup_{\langle C,T\rangle \in \mathcal{T}}T$. The prediction layer is then instantiated by a fully connected layer (with bias) followed by a softmax layer over $L$. The probability of a term $t$ is then $$P(t|s,C)=\frac{\exp(\mathbf{w}_t^\top\mathbf{x}+b_t)}{\sum_{t'\in L}\exp(\mathbf{w}_{t'}^\top\mathbf{x}+b_{t'})}\quad\text{.}\label{eq:soft_max}\\
$$ Here $\{\textbf{w}_t,b_t\}_{t\in L}$ are weight and bias parameters of the fully connected layer.
Note that we simultaneously predict multiple terms, [*i.e.,*]{}$T\setminus\{s\}$, so the classification is essentially multi-labeled. Moreover, the softmax layer introduces summed exponentials on the denominator of $P(t|s,C)$. This makes training inefficient on a large $L$ (over 180k on our dataset). To relieve both issues, we use a multi-label implementation[^2] of sampled softmax loss [@jean2015using]. That is, a much smaller *candidate set* from $L$ is sampled to approximate gradients related to $L$.
Incorporating Attention on Contexts
===================================
So far, we have detailed various encoders for context $C$. They all essentially aggregate the information in every word with or without position information in $C$. Given potentially long input $C$ and the fixed output dimension, it is vital for encoders to capture the most useful information into $\mathbf{v}_C$.
Recent studies [@bahdanau2014neural; @luong2015effective; @wang2016attention; @shimaoka2017neural] suggest that attention-based encoders can focus on more important parts of sentences, thus achieving better representations. In this section, we explore approaches to incorporate attention into the context encoders. Based on whether they exploit information in the seed term, we categorize them as *seed-oblivious* or *seed-aware*.
Seed-Oblivious Attention
------------------------
By seed-oblivious attention, we aim to model the importance of different words or positions in a sentential context. Following conventional approaches [@bahdanau2014neural], we use a feed-forward network to estimate the importance of each word or position. For the <span style="font-variant:small-caps;">NBoW</span> encoder, the importance score of word $i$ is defined by $$\begin{aligned}
\label{eq:seed-oblivious}
f(i)&=\mathbf{w}_a^\top \tanh(\mathbf{W}_a \mathbf{c}_i)\text{.}\end{aligned}$$ Here $\mathbf{w}_a\in \mathbb{R}^{d'}$ and $\mathbf{W}_a\in \mathbb{R}^{d'\times d}$ are parameters of the feed-forward network. The score $f(i)$ is then fed through a softmax layer and used as weights to combine the word vectors $\mathbf{c}_i$: $$\begin{aligned}
\alpha_i&=\frac{\exp(f(i))}{\sum_{i'}\exp(f(i'))}\text{,}\label{eq:softmax}\\
\mathbf{v}_C&=\sum_{i=1}^{n} \alpha_i \mathbf{c}_i\text{.}\label{eq:attn_combine}\end{aligned}$$ For <span style="font-variant:small-caps;">RNN</span>, attention is applied in a similar manner, except that $\mathbf{c}_i$ is substituted by hidden vector $\mathbf{h}_i$.
Seed-Aware Attention
--------------------
So far, we have discussed various context encoders and an attention-based improvement. For them, the seed does not contribute to context encoding. However, a seed like “amino acid” may conversely indicate informative words or parts in the context, [*e.g.,*]{}“barley” and “grass”, to further narrow down the semantic scope of expansion. Following this observation, we propose involving the seed vector $\mathbf{v}_s$ to compute a seed-aware importance score $f(s,i)$ instead of $f(i)$. Inspired by [[@luong2015effective ]{}]{}, we consider the following instantiations of seed-aware attention.
[**<span style="font-variant:small-caps;">dot</span>**]{}In this variant, we estimate the word importance with the inner product of the seed vector $\mathbf{v}_s$ and each word vector $\mathbf{c}_i$. Formally, the score is $$\begin{aligned}
f(s,i)=\mathbf{v}_s^{\top}\mathbf{c}_i\text{.}\end{aligned}$$
[**<span style="font-variant:small-caps;">concat</span>**]{}Instead of directly taking the inner product of $\mathbf{v}_s$ and $\mathbf{c}_i$, this variant feeds their concatenation through a feed-forward network: $$\begin{aligned}
f(s,i)=\mathbf{w}_a^{\top} \tanh (\mathbf{W}_a[\mathbf{v}_s;\mathbf{c}_i])\text{.}\end{aligned}$$ Here $\mathbf{w}_a\in \mathbb{R}^{d'}$ and $\mathbf{W}_a\in \mathbb{R}^{d'\times 2d}$ are parameters of the feed-forward network. By involving additional parameters $\mathbf{w}_a$ and $\mathbf{W}_a$, we expect the <span style="font-variant:small-caps;">concat</span> variant to be more capable than <span style="font-variant:small-caps;">dot</span>.
[**<span style="font-variant:small-caps;">trans-dot</span>**]{}In <span style="font-variant:small-caps;">dot</span>, we multiply the seed and word vectors $\mathbf{v}_s$ and $\mathbf{c}_i$. Note that the context word vectors $\mathbf{c}_i$ need to both interact with the seed vector $\mathbf{v}_s$ and constitute the context representation $\mathbf{v}_C$. To distinguish between the two potentially different roles, we additionally consider the following <span style="font-variant:small-caps;">trans-dot</span> scoring function: $$\begin{aligned}
\label{eq:trans-dot}
f(s,i)=\mathbf{v}_s^{\top} \tanh (\mathbf{W}_a \mathbf{c}_i)\text{.}\end{aligned}$$ Here, we use a fully connected layer with parameters $\mathbf{W}_a\in \mathbb{R}^{d\times d}$ to transform $\mathbf{c}_i$ before taking a dot product with $\mathbf{v}_s$. Compared with <span style="font-variant:small-caps;">dot</span>, the <span style="font-variant:small-caps;">trans-dot</span> scoring function only introduce a medium-sized parameter space, which is smaller than that of <span style="font-variant:small-caps;">concat</span>.
In order to apply seed-aware attention to our network structure, we use the respective scoring functions $f(s,i)$ to replace $f(i)$ in Eq. \[eq:softmax\]. The resulted attention weights $\alpha_{s,i}$ are fed to Eq. \[eq:attn\_combine\], and make the context vector $\mathbf{v}_C$ seed-aware.
Experimental Settings {#sec:settings}
=====================
Dataset Processing
------------------
**Item** **Count**
---------------------------------------------- -----------
Number of sentences 1,847,717
Number of training sentences $|\mathcal{T}|$ 1,478,173
Number of testing sentences 369,544
Average number of context words $|C|$ 31.39
Average number of hyponym terms $|T|$ 3.46
Number of unique terms 182,167
Number of unique terms on training set $|L|$ 180,684
Vocabulary size of all contexts 941,603
Vocabulary size of all training contexts
(discarding words with freq $<5$)
: Summary of the derived dataset.[]{data-label="tab:dataset"}
We earlier briefed that CASE exploits sentences with Hearst patterns for training and evaluation. For this reason, large-scale natural annotations can be easily obtained without manual effort.
Specifically, we employ an existing web-scale dataset, WebIsA[^3] [@seitner2016large], to derive large-scale annotated sentences. This dataset has 400 million hypernymy relations, extracted from 2.1 billion web pages. For each hyponym-hypernym pair, the dataset provides IDs of source sentences and matched patterns where the pair occurs. For example, a sentence “*Young barley grass is high in vitamin, antioxidant, enzyme, mineral, amino acid, chlorophyll and other phyto-nutrients.*” leaves its ID and pattern “…and other …” in the lists of hypernymy pairs “vitamin $\rightarrow$ phyto-nutrient”, “antioxidant $\rightarrow$ phyto-nutrient”, [*etc.*]{}Precisions of all patterns are also summarized as global information. We use the information to decompose the sentence, obtaining the example in the Dataset and Formal Task Definition section. Specifically, we follow the below steps.
1. We convert all words to lowercase and lemmatize them.
2. We then filter the dataset with the pattern precision information, due to the noisy web pages and the error-prone hypernymy extraction procedure. That is, we identify and keep *high-quality sentences* where a hypernym is extracted with at least three hyponyms by a pattern with precision $\ge 0.5$.
3. We regard hyponym terms appearing in at least ten high-quality sentences as *high-quality terms*. We select high-quality sentences with at least three high-quality terms in the final dataset.
Finally, our dataset contains 1,847,717 naturally labeled sentences, involving over 180k hyponym terms. From them, we sample 20% of sentences to form the test set, and use the remainder for training. Table \[tab:dataset\] summarizes our dataset.
Baseline Approaches {#sec:baseline}
-------------------
Since no previous study addresses the exact CASE task, we evaluate our models against the solutions proposed for the most similar task, [*i.e.,*]{}lexical substitution. Specifically, we compare with [[@melamud2015simple ]{}]{}’s unsupervised method and one of its variants. We also evaluate a supervised method by [[@roller2016pic ]{}]{}.
<span style="font-variant:small-caps;">**L**exical **S**ubstitution</span> (<span style="font-variant:small-caps;">**LS**</span>). Word embedding models such as [[@mikolov2013distributed ]{}]{} compute two types of word vectors, [*i.e.,*]{}IN and OUT. [[@melamud2015simple ]{}]{}’s analysis suggests that the IN-IN similarity favors synonyms or words with similar functions, while the IN-OUT similarity characterizes word compatibility or co-occurrence. By promoting terms $t$ having the same meaning with the seed $s$ and good compatibility with the context $C$, they score a term $t$ by $$\label{eq:LS}
\textsc{LS}=\lambda_1 cos(\mathbf{s}^{I},\mathbf{t}^{I})+\frac{1-\lambda_1}{|C|}\sum_{c\in C} cos(\mathbf{c}^{I},\mathbf{t}^{O})
$$ Here the superscripts $I$ and $O$ stands for IN and OUT, respectively. We train word vectors on all sentences in $\mathcal{T}$, and use averaged vectors to represent multi-word terms. We follow the original paper and set $\lambda_1=0.5$.
<span style="font-variant:small-caps;">**LS** with Term **Co**-occurrence</span> (<span style="font-variant:small-caps;">**LSCo**</span>). Considering that expansion terms are not simply synonyms of seeds, and tend to co-occur with seeds (in Hearst patterns), we also study a modified version of Eq. \[eq:LS\]: $$\label{eq:LSCo}
\textsc{LSCo}=\lambda_2 \textsc{LS}+(1-\lambda_2)cos(\mathbf{s}^{I},\mathbf{t}^{O})
$$ We tune $\lambda_2$ and adopt the best-effort results.
<span style="font-variant:small-caps;">**P**robability **I**n **C**ontext</span> (<span style="font-variant:small-caps;">**PIC**</span>) [@roller2016pic]. Different from the second term of Eq. \[eq:LS\], <span style="font-variant:small-caps;">PIC</span> models the context compatibility by introducing a parameterized linear transformation on $\mathbf{c}^{I}$. Therefore, it needs data to train the additional parameters and is inherently supervised.
Parameters and Evaluation Metrics
---------------------------------
We trim or pad all contexts to length 100, and treat words occurring less than 5 times as OOVs. Word vectors are pre-trained with `cbow` [@mikolov2013distributed]. Their dimensions $d$ as well as encoded contexts’ and seeds’ are set to 100. The intermediate dimension $d'$ of attention-related network is set to 10. Each batch is of size 128 with 1,000 negative samples to compose the sampled candidates. We iterate for 10 epoches with the Adam optimizer. All other hyper-parameters are found to work well by default and not tuned.
![Tuning $\lambda_2$ in the <span style="font-variant:small-caps;">LSCo</span> baseline.[]{data-label="fig:LS_LSCo"}](ls_lsco){width="0.69\columnwidth"}
**Model** **Recall** **MAP** **MRR** **nDCG**
-------------------------------------------------------------------------------------------------------------------- ------------ ----------- ----------- -----------
<span style="font-variant:small-caps;">LS</span> [@melamud2015simple] 2.84 1.80 1.79 1.66
<span style="font-variant:small-caps;">LSCo</span> ($\lambda_2=0.1$) 3.54 2.65 2.69 2.23
<span style="font-variant:small-caps;">PIC</span> [@roller2016pic] 19.62 17.78 18.84 14.71
<span style="font-variant:small-caps;">CASE</span> (ours, with <span style="font-variant:small-caps;">NBoW</span>) **23.42** **21.30** **22.71** **17.80**
: Comparison with LS baselines (top-10 results).[]{data-label="tab:ls_results"}
For all approaches, we uniformly rank all $t\in L$ according to the corresponding probability. We concentrate on top-10 results. Note that, due to the nature of natural language, ground-truth term lists may not be exhaustive. This is an intrinsic limitation of the original dataset, and our processed dataset is probably the best we can access. To this end, we use Recall as the main metric and do not involve Precision. We also report MAP, MRR, and nDCG for reference.
Experimental Results {#sec:results}
====================
In this section, we aim to experimentally answer the following questions: **1)** Are lexical substitution solutions applicable to CASE? **2)** Do contexts have impact on semantic expansion? **3)** Is seed-aware attention superior as expected? **4)** Do additional hypernyms make the experiments biased?
Comparison with LS Baselines
----------------------------
When introducing [[@melamud2015simple ]{}]{}’s lexical substitution baseline, we mention that expansion terms should co-occur with, rather than be synonyms of, the seed term. In Figure \[fig:LS\_LSCo\], we compare the Recall@10 scores of baselines <span style="font-variant:small-caps;">LS</span> and <span style="font-variant:small-caps;">LSCo</span>, *w.r.t.* different $\lambda_2$. Note that <span style="font-variant:small-caps;">LSCo</span> degenerates to <span style="font-variant:small-caps;">LS</span> when $\lambda_2=1$, so their lines overlap at this point. The figure demonstrates that, when $\lambda_2<1$, the <span style="font-variant:small-caps;">LSCo</span> baseline outperforms <span style="font-variant:small-caps;">LS</span>, and achieves optimum when $\lambda_2=0.1$.
In Table \[tab:ls\_results\], we report the top-10 metrics of all three lexical substitution baselines, as well as those of our approach with the preliminary <span style="font-variant:small-caps;">NBoW</span> encoder. By additionally advocating co-occurrence between $s$ and $t$, <span style="font-variant:small-caps;">LSCo</span> outperforms <span style="font-variant:small-caps;">LS</span> on all metrics. However, it is remarkably inferior due to its unsupervised nature.
By parameterizing the context compatibility in <span style="font-variant:small-caps;">LS</span>, <span style="font-variant:small-caps;">PIC</span> achieves reasonably better results. However, <span style="font-variant:small-caps;">PIC</span> only models the similarity of seed and expansion terms through non-parameterized IN-IN similarity like the first term in Eq. \[eq:LS\]. This may be inadequate, with reasons similar to the inferiority of <span style="font-variant:small-caps;">LS</span> to <span style="font-variant:small-caps;">LSCo</span>. In our solution, the embedding-initialized parameters allow our seed encoder and prediction layer to capture type-based similarity beyond IN-IN and IN-OUT through training. With the simplest <span style="font-variant:small-caps;">NBoW</span> encoder, the joint training of the two components helps our approach outperform <span style="font-variant:small-caps;">PIC</span> by a large margin.
**Context Encoder** **Recall** **MAP** **MRR** **nDCG**
----------------------------------------------------------- ------------ ----------- ----------- -----------
No Encoder 15.81 14.02 14.85 11.62
<span style="font-variant:small-caps;">RNN-vanilla</span> 17.51 16.17 17.08 13.26
<span style="font-variant:small-caps;">GRU</span> 18.99 17.28 18.31 14.24
<span style="font-variant:small-caps;">LSTM</span> 19.02 17.40 18.43 14.31
<span style="font-variant:small-caps;">BiLSTM</span> 14.59 13.45 14.22 10.96
<span style="font-variant:small-caps;">CNN</span> 20.97 19.40 20.61 15.94
<span style="font-variant:small-caps;">CNN+PF</span> 20.88 19.04 20.20 15.70
<span style="font-variant:small-caps;">context2vec</span> 20.21 18.53 19.66 15.29
<span style="font-variant:small-caps;">NBoW</span> **23.42** **21.30** **22.71** **17.80**
: Performance of context encoders (top-10 results).[]{data-label="tab:context_results"}
---------------------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
@5 @10 @20 @5 @10 @20 @5 @10 @20 @5 @10 @20
<span style="font-variant:small-caps;">LSTM</span> 13.08 19.02 26.19 16.80 17.40 17.14 17.15 18.43 19.13 11.88 14.31 16.73
<span style="font-variant:small-caps;">+attn</span> 13.73 19.85 27.15 17.72 18.29 17.96 18.11 19.41 20.11 12.55 15.04 17.51
<span style="font-variant:small-caps;">NBoW</span> 16.32 23.42 31.64 20.78 21.30 20.79 21.29 22.71 23.45 14.91 17.80 20.58
<span style="font-variant:small-caps;">+attn</span> 16.69 23.88 32.24 21.36 21.87 21.30 21.89 23.32 24.06 15.29 18.22 21.06
<span style="font-variant:small-caps;">+dot</span> 15.54 22.13 29.89 20.12 20.60 20.12 20.61 21.95 22.66 14.36 17.03 19.66
<span style="font-variant:small-caps;">+concat</span> 16.85 24.12 32.53 21.57 22.04 21.47 22.10 23.54 24.28 15.46 18.41 21.27
<span style="font-variant:small-caps;">+trans-dot</span> **17.20** **24.51** **33.01** **21.97** **22.41** **21.80** **22.53** **23.96** **24.70** **15.80** **18.77** **21.65**
---------------------------------------------------------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
Comparison of Context Encoders
------------------------------
The Introduction section mentioned that set expansion is similar to CASE without context. We find that one seed is usually sufficient to retrieve terms of the same type. The result thus heavily depend on the context to pick the right terms out of many others with the same type. Table \[tab:context\_results\] reflects this by the inferior results of the “No Encoder” setting, where contexts are removed in both training and testing.
Although contexts are important, complex encoders do not necessarily lead to better results. In Table \[tab:context\_results\], encoders at lower semantic levels, [*i.e.,*]{}<span style="font-variant:small-caps;">NBoW</span> at the word level and <span style="font-variant:small-caps;">CNN</span> at the phrase level, are the most effective. Among them, the simpler <span style="font-variant:small-caps;">NBoW</span> achieves better scores. Moreover, RNN-based ones are not very competitive, with the best <span style="font-variant:small-caps;">LSTM</span> variation poorer than <span style="font-variant:small-caps;">CNN</span>. This may be due to that RNNs are only effective where predictions are sensitive to word orders, [*e.g.,*]{}in POS tagging and dependency parsing. Finally, being placeholder-aware, the <span style="font-variant:small-caps;">context2vec</span> encoder performs better than its <span style="font-variant:small-caps;">LSTM</span> counterpart. However, <span style="font-variant:small-caps;">CNN</span> with positional embedding, the stronger placeholder-aware encoder, is inferior to its counterpart. This indicates that CASE is inherently different from tasks like relation classification and aspect/targeted sentiment analysis, which rely on relative position between the placeholder and some key words.
---------------------------------------------------- ---------------------------------------------------------- ---------------------------------------------------- ----------------------------------------------------------
<span style="font-variant:small-caps;">NBoW</span> <span style="font-variant:small-caps;">+trans-dot</span> <span style="font-variant:small-caps;">NBoW</span> <span style="font-variant:small-caps;">+trans-dot</span>
protein **mineral** protein **mineral**
sugar sugar calcium etc
**vitamin** protein salt sugar
**mineral** **vitamin** sugar **vitamin**
carbohydrate b vitamin **vitamin** protein
herb **enzyme** **enzyme** herb
**enzyme** amino acid herb carbohydrate
fat herb potassium salt
fiber **antioxidant** **mineral** fat
salt salt etc vitamin c
---------------------------------------------------- ---------------------------------------------------------- ---------------------------------------------------- ----------------------------------------------------------
: Case study on attention and hypernyms.[]{data-label="tab:case"}
Based on the above observations, we confirm that contexts have major impacts on CASE and deserve appropriate modeling. However, complex encoders are inferior because CASE is insensitive to either word orders or seed term positions. Modeling these signals leads to more unnecessary parameters to learn and brings in noises.
Effectiveness of the Attention Mechanism
----------------------------------------
In previous sections, we proposed two types of scoring functions to incorporate the attention mechanism in the context encoder. In Table \[tab:attn\_results\], we denote the vanilla seed-oblivious attention by <span style="font-variant:small-caps;">attn</span>, and the three seed-aware functions by their names, respectively. Due to the relatively small margin between the scores of different functions, we report the metrics for top-5 and 20 results in addition to top-10. Although seed-aware attention is applicable to <span style="font-variant:small-caps;">LSTM</span>, we do not include the results since they do not outperform the corresponding combinations of <span style="font-variant:small-caps;">NBoW</span>. The limited improvement may be due to the low potential of the base <span style="font-variant:small-caps;">LSTM</span> encoder.
Table \[tab:attn\_results\] shows that seed-oblivious attention can improve both <span style="font-variant:small-caps;">LSTM</span> and <span style="font-variant:small-caps;">NBoW</span>. Although seed-aware, the <span style="font-variant:small-caps;">dot</span> scoring function turns out to adversely affect the quality of expansion terms. We speculate that the two different roles of context word vectors $\mathbf{c}$ render the simple dot function insufficient to characterize its interactions with $\mathbf{v}_s$. The <span style="font-variant:small-caps;">concat</span> function, on the other hand, partially demonstrates superiority of seed-aware attention with limited improvement over <span style="font-variant:small-caps;">attn</span>. By slightly modifying <span style="font-variant:small-caps;">dot</span> with even fewer additional parameters than <span style="font-variant:small-caps;">concat</span>, <span style="font-variant:small-caps;">trans-dot</span> outperforms all competitors. Further paired t-tests show that the superiority of <span style="font-variant:small-caps;">trans-dot</span> (as well as the most competitive runs in Tables \[tab:ls\_results\] and \[tab:context\_results\]) to all competitors is significant at $p<0.01$. We attribute the statistical significance to the huge size of our testing set, [*i.e.,*]{}369,544 sentences.
To illustrate the impact of <span style="font-variant:small-caps;">trans-dot</span>, we show expansion terms of “amino acid” for the example in the Dataset and Formal Task Definition section, in the first two columns of Table \[tab:case\]. Observe that <span style="font-variant:small-caps;">trans-dot</span>-based attention helps promote the ground truth terms (in bold) in the ranking. It also removes nutrition “fat” from the top results, which is irrelevant to barley grass.
Impacts of Hypernyms
--------------------
**Model** (w/o Hypernym) **Recall** **MAP** **MRR** **nDCG**
---------------------------------------------------------- ------------ --------- --------- ----------
<span style="font-variant:small-caps;">NBoW</span> 22.64 20.68 22.03 17.22
<span style="font-variant:small-caps;">+trans-dot</span> 23.41 21.52 22.98 17.94
: Scores after removing hypernyms (top-10 results).[]{data-label="tab:no_hypernym"}
The contexts from WebIsA always contain hypernyms, [*e.g.,*]{}“phyto-nutrients” in the example of the Dataset and Formal Task Definition section. However, practical scenarios may involve sentences without hypernyms as in Figure \[fig:example\]. To study the potential impact, we remove all hypernyms in contexts, retrain and test <span style="font-variant:small-caps;">NBoW</span> with or without <span style="font-variant:small-caps;">trans-dot</span>. The last two columns in Table \[tab:case\] show the results of our running example without the suffix “and other phyto-nutrients”. It is observed that removing hypernyms causes some non-nutrient or noisy terms ([*e.g.,*]{}“salt” and “etc”) to rise. Table \[tab:no\_hypernym\] reports the overall scores for top-10 results. Compared with the corresponding results in Table \[tab:attn\_results\], all scores slightly decrease by around one point. This comparison suggests that, trained with sufficient term co-occurrences, our model is able to find terms of the same types, without the help of hypernyms in most cases. To conclude, the hypernym bias introduced by the data harvesting approach has very small impacts on the practical use of our solution.
Related Work
============
[**Lexical Substitution**]{} This task has been investigated for over a decade [@mccarthy2007semeval]. It differs from CASE in that the substitutes are required to preserve the *same* meaning with the original word. Previous solutions follow two stages, [*i.e.,*]{}*candidate generation* and *candidate ranking*. Synonym candidates are generally generated from external dictionaries or by pooling the testing data. The ranking stage then boils down to estimating the compatibility between candidates and the context.
[[@giuliano2007fbk ]{}]{} rely on n-grams to model candidates’ compatibility. [[@erk2008structured ]{}]{} argues that syntactic relations in contexts are crucial, [*e.g.,*]{}“a horse draws something” and “someone draws a horse”. In [[@melamud2015simple ]{}]{}, word vectors [@mikolov2013distributed] are applied to score candidates’ similarity with the original word and their context compatibility. Their method is nearly state-of-the-art, yet remains relatively simple. Besides unsupervised approaches, supervised methods [@szarvas2013supervised; @szarvas2013learning; @roller2016pic] prove superior at the cost of requiring more annotations. We have experimentally compared with representative ones from both categories.
[**Set Expansion**]{} This task aims to expand a couple of seeds to more terms in the underlying semantic class. Most existing approaches involve bootstrapping on a large corpus of web pages [@tong2008system; @wang2007language; @he2011seisa; @chen2016long] or free text [@shi2010corpus; @shen2017setexpan; @shi2014probabilistic; @thelen2002bootstrapping]. HTML-tag-based or lexical patterns covering a few seeds are extracted, which are then applied to the same corpus for new terms. The process is iterated until certain stopping criterion is met.
Both this task and ours face the challenge of ambiguous terms, [*e.g.,*]{}“apple”. With multiple seeds, set expansion may rely on the other seeds, [*e.g.,*]{}“samsung” or “orange”, for disambiguation. However, since CASE accepts only one seed as input, it is essential to model the additional context to make up for the scarce information. To this end, we resort to neural networks, where many off-the-shelf context modeling architectures are available.
[**Multi-Sense or Contextualized Word Representation**]{} This technique deals with sense-mixing in traditional word representation. Traditional word representations assign a single vector to each word. They mix different senses of polysemous words, and block downstream tasks from exploiting the sense information. [[@reisinger2010multi ]{}]{} cluster the contexts of polysemous words and represent senses by the cluster centroids. By sequentially carrying out context clustering, sense labeling, and representation learning, [[@huang2012improving ]{}]{} obtain low-dimensional sense embeddings. Non-parametric [@neelakantan2014efficient] and probabilistic models with fewer parameters [@tian2014probabilistic] are proposed later to accelerate training.
In multi-sense embedding, polysemous words get static embeddings for coarse-grained senses. Some recent efforts explore dynamic embeddings that vary with the context. [[@melamud2015modeling ]{}]{} use context-aware substitutions of target words to obtain contextualized embeddings. [[@peters2018deep ]{}]{} employ multi-layered bi-directional language models on words in contexts. Embeddings are obtained by aggregating different hidden layers with task-specific weights. CASE separately models contexts and seed terms, because the model needs to generalize to unseen multi-word seeds. For more studies, we refer readers to a survey [@camacho2018word].
Conclusion {#sec:conclusion}
==========
We define and address context-aware semantic expansion. To the best of our knowledge, this is the first study on this task. To facilitate training and evaluation without human annotations, we derive a large dataset with about 1.8 million naturally annotated sentences from WebIsA. We propose a network structure, and study different alternatives of the context encoder. Experiments show that solutions for lexical substitution are not competitive on CASE. Comparisons on various context encoders indicate that, the simplest <span style="font-variant:small-caps;">NBoW</span> encoder achieves surprisingly good performance. Based on <span style="font-variant:small-caps;">NBoW</span>, seed-aware attention, which models the interaction between seed and context words, further improves the performance. The <span style="font-variant:small-caps;">trans-dot</span> scoring function finally shows its capability to focus on indicative words, and outperforms other seed-oblivious or -aware competitors. In further analysis, we also confirm small impacts of a bias introduced when harvesting our data.
[^1]: Work done when Jialong Han was with Tencent AI Lab.
[^2]: <https://www.tensorflow.org/api_docs/python/tf/nn/sampled_softmax_loss>
[^3]: <http://webdatacommons.org/isadb/>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Correlations of $\alpha$-particles are studied on statistics of 400 events of splitting $^{12}$C $\rightarrow$ 3$\alpha$ in nuclear track emulsion exposed to $14.1 MeV$ neutrons. The ranges and emission angles of the $\alpha$-particles are measured. Distributions over energy of $\alpha$-particle pairs and triples are obtained.'
author:
- '**R. R. Kattabekov$^{\textbf{1), 2)}}$, K. Z. Mamatkulov$^{\textbf{1), 3)}}$, D. A. Artemenkov$^{\textbf{1)}}$, V. Bradnova$^{\textbf{1)}}$, P. I. Zarubin$^{\textbf{1)~*}}$, I. G. Zarubina$^{\textbf{1)}}$, L. Majling$^{\textbf{4)}}$, V. V. Rusakova$^{\textbf{1)}}$, A. B. Sadovsky$^{\textbf{1)}}$**'
title: |
Correlations of $\alpha$-particles in splitting of $^{12}$C nuclei by neutrons\
of energy of 14.1 MeV
---
Nuclear track emulsion (NTE) exposed to neutrons of energy of $14.1 MeV$ produced in a low energy reaction $d + t \rightarrow n + \alpha$ allows one to study the ensembles of triples of $\alpha$-particles produced in disintegrations of carbon nuclei of NTE composition. Energy transferred to $\alpha$-particles is sufficient to measure their ranges and directions and, at the same time, it remains below the thresholds of background channels. Such an approach to the experimental study emerged with the advent of neutron generators. Most completely the reaction $^{12}C(n,n')3\alpha$ was studied quite a long time ago \[1\]. An initial objective of this analysis was limited to $\alpha$-calibration of NTE, recently reproduced by TD Slavich. A significant number of $\alpha$-triples of the reaction $^{12}C(n,n')3\alpha$ reached 1200 in a short time made it possible to analyze it on a large statistics as well as to create a commonly available bulk of experimental data. This bulk is useful for a direct comparison with the $\alpha$-cluster models of the $^{12}$C nucleus.
A topical physical interest to the reaction $^{12}C(n,n')3\alpha$ is as follows. Information about a probability of presence of configurations of $\alpha$-particle clusters with different angular momenta is of fundamental importance for description of the structure of light nuclei. Studying the dissociation of relativistic nuclei $^{9}$Be in NTE \[3, 4-6\] the BECQUEREL Collaboration confirmed the two-body model of the nucleus $^{9}$Be in which is dominating the superposition of the neutron and an unstable $^{8}$Be nucleus in states with spin and parity $0^{+}$ and $2^{+}$ presenting with similar weights is dominant. In this way a basis appears for asking questions about the contributions $\alpha$-cluster configurations in the angular momenta of the ground states of heavier nuclei.
Traditionally the nucleus $^{12}$C is regarded as a laboratoryfor the development the $\alpha$-particle clustering concepts. It is a possible that in the ground state of $^{12}$C$_{g.s.}$ there are two pairs of $\alpha$-clusters with orbital angular momenta equal to 2 (D-wave). In this case the basic configurations are $^{8}$Be nuclei in the first excited state 2$^{+}$. In a classical pattern one may imagine a rotation in opposite directions of two $\alpha$-clusters with angular momenta equal to 2 around a common center represented by a third $\alpha$-cluster. Then the remaining combination of two $\alpha$-clusters should correspond to the ground state of the nucleus $^{8}$Be with spin and parity $0^{+}$ (S-wave). As a result the superposition of the pair states in the ensemble of three $\alpha$-clusters leads to a zero spin in $^{12}$C$_{g.s.}$. Naturally, this simplified model requires a quantum-mechanical consideration. Nevertheless, its validity should be confirmed by an intensive formation of states $^{8}$Be$_{2+}$ and $^{8}$Be$_{g.s.}$ with a predominance of the former one in reactions of knocking of $\alpha$-particles from $^{12}$C nuclei.
Such a concept does not contradict the mechanism of the synthesis of the nucleus $^{12}$C accepted in nuclear astrophysics. Fusion of a triple of $\alpha$-particles occurs through its second excited state $0^{+}_{2}$ (the Hoyle state) located on 270 keV above the breakup threshold $^{12}C \rightarrow 3\alpha$. Basically, each pair of $\alpha$-particles in it corresponds to $^{8}$Be$_{g.s.}$. In the transition $0^{+}_{2} \rightarrow 2^{+}_{1}$ with emission of a photon to the first excited state of $^{12}$C, which is bound one, an $\alpha$-pair in the D-wave should arise in a 3$\alpha$ ensemble in order to provide conservation of the angular momentum. The subsequent transition to $^{12}$C$_{g.s.}$, which is also accompanied by emission of a photon leads to the formation of another $\alpha$-particle pair in the D-wave state. This pair should have an opposite angular momentum with respect to the first pair to ensure zero spin value of the ground state $^{12}$C$_{g.s.}$. Thus, the nucleus $^{12}$C$_{g.s.}$ does acquire polarization. Figuratively being expressed it does conserve an invisible rotation.
![Distribution of $\alpha$-particles over ranges $L_{\alpha}$.[]{data-label="fig:1"}](l_alpha){width="45.00000%"}
![Distribution of $\alpha$-particles over energy $E_{\alpha}$.[]{data-label="fig:2"}](E_alpha){width="45.00000%"}
The ratio of the yields of $\alpha$-particle pairs through the states $^{8}$Be$_{2+}$ and $^{8}$Be$_{g.s.}$ in disintegrations of nuclei $^{12}$C not accompanied by a transfer of the angular momentum is a key parameter which should reflect the spin-cluster structure $^{12}$C$_{g.s.}$. Analysis of interactions in NTE exposed to neutrons of energy near the threshold of the $^{12}$C splitting allows one to determine this and other characteristics of the reaction $^{12}C(n,n')3\alpha$.
Exposure of NTE to neutrons of energy $14.1 MeV$ was performed on one of devices DVIN of an applied destination \[7\]. A neutron generator of the device provided a flow of $5 \times 10 ^{7}$ neutrons/s in the full solid angle. The NTE stack was placed on a top cover of the device DVIN approximately 10$cm$ above a tritium target. The stack consisted of several layers of NTE BR-2 of size of $9$ to $12 cm^{2}$ at thickness of 107 microns poured onto glass plates of thickness of 2$mm$. The neutron generator gave rise to an irreducible background of X-ray radiation. This background was detected by the NTE layers with decreasing brightness as the absorption in the glasses grows which allowed one to select layers with a low X-ray backlighting. NTE is comparable to a liquid hydrogen target on density of hydrogen. Therefore, the main background in NTE exposed to neutrons, is presented by recoil protons. Overlaying of tracks that would be imitating 3$\alpha$-disintegrations were reduced to a negligible level by choice of the exposure time of 40 min.
![Distribution triples of $\alpha$-particles over energy $Q_{3\alpha}$.[]{data-label="fig:3"}](Q3alpha){width="45.00000%"}
![Correlation over energy $Q_{2\alpha}$ and opening angles $\Theta_{2\alpha}$ in $\alpha$-particle pairs.[]{data-label="fig:4"}](theta_Q2alpha){width="45.00000%"}
![Distribution pairs of $\alpha$-particles over energy $Q_{2\alpha}$.[]{data-label="fig:5"}](Q2alpha){width="45.00000%"}
Scanning of layers performed on microscopes MBI-9 was aimed at 3$\alpha$-disintegrations. In 400 events of 3$\alpha$-disintegration selected among the found 1200 ones measurements of angles relative to plane of a NTE layer and its surface as well as their lengths were at a KSM microscope performed for all $\alpha$-particle tracks. The only condition for the selection of the events was fullness of measure. Distribution over ranges of $\alpha$-particles L$_{\alpha}$ (Fig. 1) has an average value $<L_{\alpha}> = (5.8 \pm 0.2)$ $\mu m$ at RMS $(3.3 \pm 0.1)$ $\mu m$. This distribution has an asymmetric shape described by the Landau distribution. Directly associated with it the distribution over energy of $\alpha$-particles E$_{\alpha}$ (Fig. 2) defined by ranges $L_{\alpha}$ in the SRIM model \[8\] has an average value $<E_{\alpha}> = (1.86 \pm 0.05) MeV$ with RMS $(0.85 \pm 0.03) MeV$.
Determination of angles and energy values versus ranges allows one to determine the energy $Q_{2\alpha}$ of pairs and triples $Q_{3\alpha}$ of $\alpha$-particles. Distribution over $Q_{3\alpha}$ (Fig. 3) is concentrated in the range of excitations of the $^{12}$C nucleus which is below thresholds of separation of nucleons. The used method does not resolve levels of $^{12}$C while the Hoyle state is not shown, as expected, for the reaction of $\alpha$-particle knocking out.
Correlation over energy $Q_{2\alpha}$ and opening angles $\Theta_{2\alpha}$ in $\alpha$-particle pairs reveals features of the $^{8}$Be nucleus (Fig. 4). The region of large opening angles $\Theta_{2\alpha} > 90^{\circ}$ is corresponding to $Q_{2\alpha}$ of $^{8}$Be$_{2+}$, while $\Theta_{2\alpha} < 30^{\circ} - ^{8}$Be$_{g.s.}$. Distribution over $Q_{2\alpha}$ points to these states (Fig. 5). Its right side meets the shape expected from the decay through $^{8}$Be$_{2+}$. Condition $Q_{2\alpha} < 200 keV$ has allowed to allocate 56 decays $^{8}$Be$_{g.s.}$. For $^{8}$Be$_{g.s.}$ the total momentum distribution is rather narrow and characterized an average value of ($208 \pm 4) MeV/c$ with RMS $(30 \pm 3) MeV/c$. Estimate of the average total momentum for 212 pairs of $\alpha$-particles which are the most appropriate to $^{8}$Be$_{2+}$ is $(130 \pm 3) MeV/c$ with RMS $(43 \pm 2) MeV/c$. Thus, the distribution over the total momentum for $^{8}$Be$_{2+}$ is much softer and relatively wider.
Importance of the discussed structure is determined not only by interest to describe $^{12}$C$_{g.s.}$, but also the fact that it is the starting configuration for the reverse process of generating 3$\alpha$-particle ensembles in the Hoyle state. It is assumed that this state after $^{8}$Be$_{g.s.}$ is a Bose-Einstein condensate consisting of $\alpha$-particles with zero angular momentum \[9\]. Its identification in breakups of $^{12}$C allows one to advance to generation of condensate states of larger number of $\alpha$-particles. Fundamental aspect seems related to the fact that in order to recreate the condensate it is necessary to evacuatetwo hidden rotations $^{12}$C$_{g.s.}$. We note that in this respect the Coulomb dissociation of a nucleus on a heavy nucleus appears to be the most suitable process since few photon exchanges in it are possible.
In general, these data indicate the presence of a superposition of states $0^{+}$ and $2^{+}$ of the nucleus $^{8}$Be in the ground state of $^{12}$C, and $^{8}$Be$_{2+}$ dominates. Deeper consideration of manifestation of $\alpha$-cluster structure of $^{12}$C in disintegrations caused by neutrons requires the use of the Dalitz plot and the theory of nuclear reactions. Presented measurements of the reaction $^{12}C(n,n')3\alpha$ and videos of these events are available on the BECQUEREL project website \[10\]. The authors express gratitude to S. P. Kharlamov (LPI) for a discussion of the results. This work was supported by grant from the Russian Foundation for Basic Research 12-02-00067 and grants Plenary representatives of Bulgaria, Romania and the Czech Republic in JINR.
[99]{} B. Antolkovic et al., Nucl. Phys. A 394, 87 (1983). The TD Slavich, www.slavich.ru. The BECQUEREL Project, http://becquerel.jinr.ru/. D. A. Artemenkov et al., Phys. Atom. Nucl. 70, 1222 (2007)\]; nucl-ex/0605018. D. A. Artemenkov et al., Few Body Syst 44, 273 (2008). D. A. Artemenkov et al., Int. J. Mod. Phys. E 20, 993 (2011); arXiv: 1106.1749. V. M. Bystritsky et al. Phys. Part. Nucl. Lett. 6, 505(2009)\]. J. F. Ziegler, J. P. Biersack and M. D. Ziegler, SRIM - The Stopping and Range of Ions in Matter, 2008, ISBN 0-9654207-1-X., SRIM Co; http://srim.org/. T. Yamada, Y. Funaki, H. Horiuchi, G. Roepke, P. Schuck, A. Tohsaki, Clusters in Nuclei, Lect. Notes Phys. 848, 109 (2012), Springer. The BECQUEREL Project, http://becquerel.jinr.ru/\
miscellanea/DVIN/dvin11.html.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Tahira\*'
- 'Mehboob\*'
- Rahman
- Arif
bibliography:
- 'references.bib'
title: 'CrowdFix: An Eyetracking Dataset of Real Life Crowd Videos'
---
[Memoona]{} [National University of Sciences and]{} [Technology (NUST), Islamabad, Pakistan]{} [http://]{}[[email protected]]{}
[Sobas]{} [National University of Sciences and]{} [Technology (NUST), Islamabad, Pakistan]{} [http://]{}[[email protected]]{}
[Anis U.]{} [National University of Sciences and]{} [Technology (NUST), Islamabad, Pakistan]{} [http://]{}[[email protected]]{}
[Omar]{} [National University of Sciences and]{} [Technology (NUST), Islamabad, Pakistan]{} [http://]{}[[email protected]]{}
Introduction {#sec:intro}
============
Saliency studies form the intersection between natural and computer vision. A quantitative study of saliency provides a structured insight into the human mind on what it perceives to be important in a scene. Visual attention then guides gaze to focus on and further explore that region of interest. To achieve near human accuracy in predicting gaze locations, Saliency models need to be able to approximate gaze over a wide variety of stimuli [@borji2019saliency]. We approach this problem in two ways: first we discuss static and dynamic stimuli used for modelling saliency as well as the need for specialized datasets to boost saliency modelling. Traditionally most of the active research has been on images, but in the recent years, using dynamic content as the subject of saliency studies has picked up pace. The pace of this research is determined by publicly available, diverse datasets of videos covering multitudes of natural scenes. Datasets such as DIEM [@mital2011clustering], HOLLYWOOD-2 [@mathe2014actions], and UCFSports [@mathe2014actions], LEDOV [@jiang2018deepvs] and DHFK [@wang2018revisiting] are dynamic datasets that cover a range of natural scenes. However, there is an obvious gap for specialized datasets targeting a category of natural scenes. Our study focuses on the category of crowded scenes because it presents an interesting use case: The number of stimuli competing for attention in crowd scenes are larger in number and the crowd activity is far more random and attention grabbing than normal scenes containing one or two object of interest [@yoo2016visual]. This insight proves useful for monitoring, managing and securing crowds [@gupta2014design]. To date, there has been only one crowd saliency dataset, namely EyeCrowd, consisting of 500 natural images [@jiang2014people].
Our research contributes by adding a first saliency dataset of crowd videos called ’CrowdFix’ and its corresponding saliency information to the pool of publicly available saliency datasets. Crowdfix is a real-life, moving crowds high definition (720p) videos dataset collected in in RGB. Eyetracking results benefit from higher quality datasets [@vigier2016new]. For this reason we chose not to include videos from pre-existing crowd video datasets due to the lower quality of those videos, i.e. below 720p. The dataset has been further annotated into three different crowd density levels to facilitate understanding of attention modulation within different each level. This also helps in the producing better, more generalized saliency models, particularly deep models by providing a finer categorization of salient images and videos [@he2019understanding]. We assess the attentional impact of different levels of these crowds on individuals and further evaluate three state-of-art deep learning based saliency models on our datasets to judge how well general saliency models perform for crowd saliency prediction. This analysis serves as a baseline for future design of a crowd saliency model.
Related Work
------------
Gaze, a synchronized act of the eyes and head, has frequently been used as an intermediary for attention in natural conduct. For example, a human or a robot has to cooperate with contiguous objects and regulate the gaze to accomplish a task while moving in the surroundings. In this sense, gaze control involves vision, response, and attention concurrently to achieve sensory-motor arrangement necessary for the preferred behavior (e.g., reaching and grasping) [@borji2012state].
Our human visual system is designed to automatically filter the incoming visual information from our gaze. This is done passively based on verdict of visual attention. Visual attention is a mechanism that intervenes between competing aspects of a visual scene and assists in selecting the most important regions while diminishing the importance of others. It is vital to understand how visual attention works to determine how our vision will be directed to the objects presented in front of it. [@jiang2014people]. Attention is a umbrella term which includes all factors that influence selection. The active selection is expected to be suppressed by two major channels called bottom-up and top-down control. Bottom-up attention is spontaneous attention. It is fast, uncontrolled, and stimulus-driven. Our attention is naturally drawn to salient regions in visual field. The term “salient” is interchangeably used for bottom-up attention [@borji2012state]. Human visual attention is supposed to look at the salient stimuli in the environment.
Crowds and Visual Attention
---------------------------
Crowds represent a unique challenge for visual attention selection. A crowd is a big cluster of people assembled together and has attributes like density and movement. Crowds exhibit a distinct category of scenes. Crowd scenes can be categorized as complex scenes, like cross-streets in which several objects interconnect with each other that consists of different movement patterns, for example walking straight and then turning left [@yoo2016visual]. We know that the crowds have an impact on the attention of an individual and can be tested by relating the physical stimuli with the contents of consciousness [@mancas2010dense]. We can then further correlate the different crowd levels with the visual attention to learn the behaviour of an individual while free viewing real life crowds. This information is fundamental for crowd management, safety and surveillance in handling and avoiding emergencies due to rush and congestion [@chiappino2015bio]. Analysis of complex situations like dense crowds can extremely benefit from algorithms which can encode human attention [@mancas2010dense]. This serves many applications in human computer interaction, graphics and user interface design, particularly for small displays, by comprehending where humans look at in a scene. Additionally, knowledge of visual attention is beneficial for automatic image cropping, thumb nailing, image search, image and video compression. [@judd2009learning].
Computational Models for Visual Attention
-----------------------------------------
Older models integrated complicated characteristics of the Human Visual System (HVS) and and reconstruct the visual input through hierarchically combining low level features. The bottom-up mechanism is the maximum occurring feature found in these models [@le2006coherent]. The core indication which implicates bottom-up attention is the uncommonness and distinction of a feature in a given circumstance [@mancas2010attention]. Bottom-up use a feed-forward method to process visual input. They apply sequential transformations to visual features collected over the entire visual field, to highlight regions which are the most attention-grabbing, significant, eye-catching, or so-called salient information [@borji2012state] However, the existing models of visual attention present a reductionist of visual attention. This is because fixations are not only influenced by bottom-up saliency as determined by the models, but also by various top-down influences. Consequently, comparing bottom-up saliency maps to eye fixations is demanding and requires that one attempts to minimize top-down impacts. [@volokitin2016predicting] One way is to focus on early fixations when top-down influences have not yet come into affect, such as by use of jump cuts in videos, in our case, and MTV style video stimulus [@carmi2006role].
Our Contribution: The CrowdFix Dataset {#sec:db}
======================================
In the only crowd eye tracking experiments that have been done before, images were used. There is no HD (720p), FHD (1080p) or 4K crowd video dataset that exist instead all of the existing datasets have low resolutions. A higher quality dataset leads to better eye fixation information. This is because HD and FHD show a better level of detail in the video and allows for more possibilities of visual exploration. [@vigier2016new]. Most datasets cater exclusively to high-density crowds or abnormal crowds which established the need to have a diversity in the dataset according to crowd density levels. To the best of our knowledge, no such categorization has been performed on existing crowd video datasets.
We collected a crowd dataset consisting of videos that depict real life scenarios. The dataset is categorized into three distinct density levels of the crowds named as sparse, dense free-flowing and dense congested. This dataset is built for studying the influence and saliency in crowds. Therefore, this dataset consists of diverse real-life, moving crowds. It has a total of 89 videos cut into 434 clips for MTV style videos. Having high resolution of the videos as the starting key point, our dataset comes with the resolution of 1280$\times$720 with 30 frames per second. None of the videos in this dataset are taken from any previously existing datasets. For maintaining the clarity and simplicity of the videos, none of them is in a fast forward motion nor any of them has a watermark on it while all of the videos being in RGB. For generating the dataset we picked the crowd videos under the Creative Commons depicting multiple real life crowded scenes. The stimulated crowd videos were not considered at all. We collected a wide variety of moving crowd scenes while assessing the varying densities of crowds. The videos were then later finalized. The categorization of the crowd density levels is concluded from the results of the participants. The major step in creating the stimulus was to maximize the bottom up attention. Since bottom up attention is the involuntary attention it follows that the stimulus should change frequently and abruptly. In terms of videos this can be achieved by using jump cuts by combining videos of a very short length back to back together. We call these very short videos as a ’clip’. Based on the research of [@carmi2006role], each snippet duration varies from 1 second to up to 3 seconds. Any clip of length greater than this would invoke top down attention. To create the clips from the crowd videos we take all the videos from each density level and randomly shuffle them. This ensures there is no sequence based on the crowd density to make it predictable. Each video has a duration of 1 - 3 seconds. These snippets are again randomly combined into two videos of approximately 10 minutes each. We then present the stimulus to the participants. Table \[tab:datasetsummary\] shows the attributes of the real life crowd dataset.
[350pt]{}[@llcccccccccclD[.]{}[.]{}[3]{}l@]{} Attributes & Details\
Stimuli type & Outdoor daytime/nighttime human moving crowds\
Sources & 05 (Flickr, Pexels, Pixabay, Vimeo and Youtube)\
Licence & under Creative Commons\
Number of videos& 89 video clips\
Categories & Sparse, dense free-flowing, and dense congested\
Videos per category & Sparse (15), dense free-flowing (41), and dense congested (33)\
Total video frames & 37,493\
Video frame size & 1280 $\times$ 720\
Audio & No\
Video snippets & 485 (1–3s each)\
Video clippet & Randomly selected snippets\
Video clippet duration & $\sim$10 mins\
Dataset Annotation
------------------
The objective behind dataset annotation is to divide the dataset into distinct crowd density levels. All the previously available crowd videos datasets lacked the density feature. The major attributes of crowds include density, orientation, time and location of event, type of event, demographics and organization within the crowd. Hence we choose to focus on different crowd density levels as well to perform better from social, psychological and computational point of view. 1 - 1.5 humans per square meter is treated as sparse, 2 humans per square meter is treated as dense free-flowing and if 3 - 4 humans per square meter then it is treated as dense congested; this is all done for moving crowds [@crowden]. 23 annotators (5 males and 18 females, in between the age group of 17 and 40) free-viewed the crowd videos. After viewing each video they were given some time to mark the video as one of the level explained in the beginning. Figure \[fig:categorydistribution\] shows the distribution of the categories chosen by the annotators that were further assigned to all the videos in the dataset.
\[ ytick=[1,2,3]{}, yticklabels=[Sparse, Dense free-flowing, Dense congested]{}, xlabel=[Number of annotators]{}, \] +\[boxplot,mark=none,draw=black,fill=black!25\] table\[row sep=\
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Since we saved the decision about the density of the video right after showing the video to the participants we can be fairly certain that the participant’s judgment was not influenced by other videos. And the participant could pause as long as they wished before moving on to the next video. Figure \[fig:sampleframes\] shows the sample images of different levels of crowd density. The rows represent sparse, dense free-flowing, and dense congested crowds respectively.
\
\
Eyetracking Data Acquisition
----------------------------
Eye tracking, general motivation and process
--------------------------------------------
We chose eyetracking as our ground truth collection approach to harvest good data. [@tavakoli2017saliency] Ground truth refers to human eye movement data obtained from real life observers who viewed the stimulus. This also works out well because our stimulus are videos where each frame moves rapidly and only stays on screen for a split second.
Videos were displayed on a 23.8” HP 24es LCD monitor (resolution 1920 x 1080) with the person resting his face on the head and chin rest to minimize any kind of ambiguity and shakiness in eye movement tracking. The distance of the viewer from the screen was kept as 60 cm. 32 participants volunteered for the free viewing of the videos while their gaze points were being recorded. Figure \[fig:experiment\] shows the experimental setup and the experiment being performed by a volunteer.
![Conducting the Experiment with a participant[]{data-label="fig:experiment"}](Capture.PNG)
All participants were shown the same set of videos in the same order. Free viewing allows participants to involve in natural visual expedition, while reassuring them to pay solid attention on the screen during the session. Therefore, some instructions kept the participants naive to the objective of experiment. Also, no one had seen the stimulus before. The instructions given were as follows:
1. You have to thoroughly watch the videos that are going to be played in front of you
2. Try to follow the main things in the video as some general questions can be asked at the end of the session
3. Make sure your eye sight is normal or corrected and you’re not wearing any polarized glasses or mascara
The EyeTribe eye tracker is used to perform the experiment with our dataset. The company ’The Eye Tribe’ endorses their eye tracker to be “the world’s first 99 eye tracker with full SDK”. It has two software suites that supplements the device i.e. EyeTribe UI and EyeTribe Server. It has a sampling rate of 60 Hz and standard precision of 0.5$^\circ$ to 1.0$^\circ$. Eye tracking is a measurement of eye movement or activity. Near infra-red light is fixed towards the focal point of the eyes namely pupils, instigating visible reflections in the cornea (the outer-most optical part of the eye), and tracked by a camera. The results provides us with the fixation data that is a time in which our eyes are locked towards a particular object in a visual angle [@dalrymple2018examination].
An eye tracker’s efficiency is commonly assessed by two metrics: accuracy and precision. Systematic error or accuracy replicates the eye tracker’s capability to assess the point of regard. It is also defined as the mean difference between a test stimulus position and the measured gaze position [@holmqvist2012eye]. Whereas the precision invokes the eye tracker’s ability to deliver steady measurements, and is appraised by calculating the root mean square noise [@holmqvist2012eye].
The eye tracker is controlled by the python based PyGaze toolbox (an alternative of the Psychtoolbox from the MATLAB) script running on Lenovo 320-15IKB (Intel Core i7-8550U CPU- @ 1.80 GHz, 8 GB, Windows 10), using a HP 24es LED monitor (23.8 inch, 60 Hz, 1920 $\times$ 1080 pixels, with dimensions 52.7x29.6 cm degrees of visual angle). Calibrations are performed using a nine point grid scripted in Python. Table \[tab:properties\] shows the the properties of the eye tribe eye tracker used for the experiment.
[350pt]{}[@llcccccccccclD[.]{}[.]{}[3]{}l@]{} Property & Value\
Eye tracking principle & Non-invasive, image based eye tracking\
Sampling rate & 30 Hz or 60 Hz\
Accuracy & 0.5–1.0$^\circ$\
Spatial resolution & 1.0$^\circ$ (RMS)\
Latency & $<$20 ms at 60 Hz\
Calibration & 9, 12 or 16 points\
Operating range & 45–75 cm\
Tracking area & 40$\times$30 cm at 65 cm distance\
Gaze tracking range & Up to 24"\
API/SDK & C++/C\#/Java\
Data output & Binocular gaze data\
To establish the aforementioned measures, gaze position is recorded during two-second periods of fixating a target stimulus. The targets appear consecutively, with an inter-trial interval of one second, on locations that were different from the calibration grid. The target grid spans 25.81 degrees of visual angle horizontally, and 19.50 degrees vertically (centred around the display centre). This is done to ensure that the tracker is feasible enough for performing the experiment in terms of: systematic error (spatial accuracy in degrees of visual angle), precision (Temporal accuracy in degrees of visual angle), and sampling accuracy.
Experimental design
-------------------
We use convenience sampling for conducting the eye tracking experiment. This means that we search for volunteers amongst colleagues and people around us in the university only. Data cleaning is also performed by comparing the results from calibration and validation from the accuracy and precision that were calculated at both the times. It regards the systematic error of less than 1.7$^\circ$ as being acceptable [@blignaut2014eye]. Based on the data cleaning process 6 participant’s data was discarded, leaving us with 26 participants - 16 females and 10 males aged between 17 - 40 years. Since the research was being held at the graduate level hence the participants were mostly graduates with normal or corrected vision. Since eye tracking falls under human behavioural research, we choose elements from commonly used behavioural experiments. The design of the experiment is motivated by the need to quantify the response of participants to the stimuli in an objective and reliable way. To maintain reliability, the stimulus duration and sequence for each participant is fixed. The experiment is divided into two identical blocks with a break of 3 - 5 minutes and starting again with a re-calibration process. The video sequence within each block remains the same for each participant. The MTV style sequence in each of the blocks keep the stimulus unpredictable and preserves the objectivity. On the same note, the stimuli did not overlap as it is a mixed design. It includes longitudinal data (by collecting a sample at the rate of 60 Hz) and cross-sectional data across several participants. The hypothesis of underlying the design of the eye tracking experiment is defined by cause, effect and goal. The cause is our stimuli, that is the crowd videos shown on a monitor. The effect is the change in the visual attention of the participant. And out goal is to analyze visual attention in crowd videos. The independent variable in our experiment are the crowd videos. The well defined density levels of a crowd ensures that we provide sufficiently diverse stimuli to our participants. The dependent variables therefore, are the raw gaze data and fixation data. To allow the fixation data to accurately represent actual eye fixations that rest on salient regions only however, is tough and requires the attempt to reduce top-down influences by concentrating on initial fixations on a stimulus [@volokitin2016predicting]. One way to keep the focus on early fixation is the use of jump cuts in videos, in our case, and MTV style video stimulus. This is in line with the understanding that salient parts of a scene consist of an unexpected commencements or local singularity [@le2006coherent]. Early attention is learnt from initial interactions, later viewing involves task/memory and other complex processes. Hence the reassembling is done into two MTV style videos named MTV1 and MTV2 of 10:12 and 10:37 minutes of duration respectively. These videos for bottom up attention helps in reducing the time for a participant to think. Therefore, the recorded data is objective.
Database Location, Structure and Accessibility
----------------------------------------------
The dataset is organized into stimuli containing the video frame, the fixation maps which is a binary map pinpointing the exact location of the fixations, and saliency maps that are the Gaussian blurred fixations. The sigma for the Gaussian was set equal to one degree of visual angle according to standard procedure, which was 38 in our dataset.
A brief description of the dataset along with the download link can be found at: https://github.com/MemoonaTahira/CrowdFix.
Results and Analysis
====================
Fixation Overview
-----------------
All the resulted log files from the eye tracking experiment had the data of participant’s fixations across all the videos. We performed general analysis on those log files to come up with some information about the experiment. Total number of fixations and average fixations were computed from MTV1 and MTV2 both. The count of minimum and maximum number of fixations was also calculated on both the parts. The values clearly show that MTV1 has more number of fixations than MTV2 therefore having more average fixations in the first part as well. Table \[tab:attributes\] shows the results of fixation data gathered by all the participants in MTV part 1 and 2.
[450pt]{}[@llcccccccccclD[.]{}[.]{}[3]{}l@]{} Attributes & MTV 1 & MTV 2\
Total fixations & 11829 & 11518\
Average fixations & 454.9615 & 443\
Median of fixations & 455 & 443\
Maximum fixations & 455 & 443\
Minimum fixations & 454 & 443\
Gaze Data Visualization
-----------------------
The final analysis is done with respect to crowd density levels against their number of videos.Figure \[fig:categoryvsvideos\] shows the distribution of videos over different levels of crowds. Each level has different number of videos as presented in the figure.
coordinates [(116,Sparse) (163,Dense free-flowing) (155,Dense congested)]{};
Different density levels of the crowd were evaluated on two things being, number of fixations and duration of fixation. Table \[tab:insights\] shows the results of the evaluation on sparse, Dense free-flowing and dense congested crowd levels. From the levels that we have, dense congested has the highest number of fixations since there are more people to look at as compared to dense free-flowing and sparse. But even if sparse has less number of fixations, it has the highest duration of fixations on the screen.
[450pt]{}[@llcccccccccclD[.]{}[.]{}[4]{}l@]{} Crowd Level & Number of Fixations & Average Fixation Duration\
Sparse & 3016 & 232.3886 ms\
Dense free-flowing & 4238 & 229.4085 ms\
Dense congested & 4030 & 228.7178 ms\
Fixation duration explains for how long the fixation of an individual lied on the screen. Hence, sparse having the highest fixation duration shows us that it catches most attention of the individuals looking at the crowd as they have lesser people to look at so they spend longer time on viewing such scenes as compared to dense free-flowing and dense congested.
Fixation location is also one of an important aspect to look at while interpreting the results. It reveals the areas where the participants fixate on the screen. The images below are the graphs representing fixation locations across all the participants of different crowd levels. It can be seen that all the fixations lie closer to the center and form a big cluster with all the points tightly loaded. Figure \[fig:fixations\] shows the fixation locations of the participants throughout the experiment on different crowd density levels. It represents the graphs for sparse, dense free-flowing and dense-congested categories from left to right respectively.
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The fixation coordinates of all the participants were extracted and used for calculating the distance from the center. Figure \[fig:alldist\] shows the results of different crowd levels. A peak at the first frame can be seen due to the bottom-up saliency influences. The fixations from sparse density level seem to be more close to the center as there are lesser things to be looked at in those scenes that revolve around the center of the screen. Hence the lesser entities in the scene catch attention for longer periods of time being consistent towards the center of the screen. While the fixations from dense free-flowing and dense congested seem to be more distributed than sparse. Dense free-flowing seems to have more distance from the center as compared to both the categories. Reasons being having more number of videos along with having the attention on both the entities as well as the salient regions of the scene. Dense congested has lesser distribution of distance from dense free-flowing but more than sparse because the scene is so congested that the person is unable to focus on something but rather struggles to explore the screen during which the scene changes already.
![Distance of Fixations from the Center[]{data-label="fig:alldist"}](all_distances.png){width="0.4\linewidth"}
The spread of the recorded data samples was also evaluated to judge the closeness of agreement between the results. Mostly the standard deviation measure is used for estimating the variability. The fixation coordinates were again used to assess the dispersion of the data around the mean which was later averaged across the participants for all the density levels. Figure \[fig:alldisp\] shows the results of different crowd levels.
![Dispersion of Fixations from the Center[]{data-label="fig:alldisp"}](images/all_dispersion.png){width="0.4\linewidth"}
While evaluating the dispersion we can again see the peak at the first frame. The spread remains consistent and around the center showing center-bias. As the attention system lies on perceptual memory, the availability of continuous factors is possible. The small peaks represent the impacts that occurred immediately after jump cuts.
Performance Evaluation over Existing Saliency Models
----------------------------------------------------
Deep Learning models are trained by combining tasks such as feature extraction, integration and saliency value prediction in an end to end manner. Their performance is superior in contrast to classic saliency models. Keeping this in mind, we select two of the latest state-of-the-art dynamic deep learning models [@borji2019saliency]. The models were selected based on best performance on pre-existing dynamic saliency datasets. These models are ACL (resnet variant) [@wang2018revisiting] and DeepVS [@jiang2018deepvs]. The third model, SAM [@cornia2018predicting] is one of the top performing deep static saliency model.
[450pt]{}[@llcccccccccclD[.]{}[.]{}[4]{}l@]{} Model & Crowd level & AUC-J & NSS & KL & CC\
DeepVS & Sparse & 0.779 & 0.929 & 1.713 & 0.339\
& Dense free-flowing & 0.790 & 1.000 & 1.652 & 0.364\
& Dense congested & 0.795 & 1.026 & 1.648 & 0.375\
& **Average** & 0.788 & 0.985 & 1.671 & 0.401\
& *Baseline* & 0.90 & 2.94 & 1.24 & 0.57\
ACL & Sparse & 0.801 & 1.083 & 1.594 & 0.395\
& Dense free-flowing & 0.822 & 1.272 & 1.388 & 0.457\
& Dense congested & 0.830 & 1.395 & 1.497 & 0.499\
& **Average** & 0.817 & 1.250 & 1.493 & 0.450\
& *Baseline* & 0.890 & 2.354 & — & 0.434\
SAM & Sparse & 0.773 & 0.858 & 1.871 & 0.312\
& Dense free-flowing & 0.779 & 0.906 & 1.727 & 0.329\
& Dense congested & 0.780 & 0.919 & 1.740 & 0.335\
&**Average** & 0.777 & 0.894 & 1.779 & 0.325\
& *Baseline* & 0.886 & 3.260 & — & 0.884\
We create a benchmark of these models over our dataset. The three models were tested over videos from each crowd category. We choose 4 of the most common saliency evaluation metrics AUC-J, NSS, KLdiv and CC to provide an easy comparison to other saliency benchmarks such as MIT@saliency [@mit-saliency-benchmark] and the DHF1K video saliency leaderboard. [@cheng_2019] We also provide a baseline from the model’s own performance results over their original datasets. We average our results as well to make a comparison with the baseline results and evaluate the performance difference. Figure \[fig:df\] shows the original image and its ground truth saliency map for dense free-flowing crowd category. Table \[tbl:model\_eval\] shows the results of evaluation with DeepVS, ACL and SAM model over different crowd density levels.
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Based on the results, ACL performs the best out of three models over all three categories of videos individually and on average. However, the difference between these and ACL’s original results is enough to prompt for improvements in model parameter design and architecture to bring saliency prediction in crowds up to par to general saliency prediction. Even in the other two models, the difference between average results and the baseline shows crowd videos need customized saliency prediction models to reach state-of-art-performance.
Discussion and Conclusion
=========================
Crowd Scenes provide a richer set of dynamics and stimuli. These can be used to formulate and test the accuracy of general saliency judgments and models if they hold true fro crowd scenes as well provide insights on how to bring about improvements.
In this work, we studied the crowd characteristics and categorised the crowds into different density levels. The fixation and dispersion analysis shows that attention does vary with the number of people in the crowd. As the crowd gets bigger, most of the time is spent viewing more objects in the scene rather than paying attention to any one particular object. With decrease in the number of entities, salient features are more spontaneously noticeable in individual objects. As s future avenue, to bridge the gap in human performance and predicted saliency, it would be prudent to include more cognitive information about crowded stimuli into the computational models [@feng2016fixation]. The importance of different features particularly facial features in the context of crowd videos is still an unexplored area. General saliency datasets’ evaluation metrics results and ours reflect a quite a big gap in performance. There is an obvious gap for improving deep saliency model to work equally well, if not better for crowds. We reiterate the need to investigate which features should be reinforced for crowd videos in the design of the model to predict better crowd scene saliency.
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{
"pile_set_name": "ArXiv"
}
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bibliography:
- 'main.bib'
- 'LHCb-PAPER.bib'
- 'LHCb-CONF.bib'
- 'LHCb-DP.bib'
---
=1
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{
"pile_set_name": "ArXiv"
}
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abstract: |
We propose consistent locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of non-autonomous parabolic evolution problems under the assumption of maximal parabolic regularity. We present new a priori estimates for low-regularity solutions. In order to avoid reduced convergence rates appearing in the case of uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by algebraic multigrid.\
Keywords: Parabolic initial-boundary-value problems; Space-time finite element methods; Unstructured meshes; Adaptivity
author:
- 'Ulrich Langer[^1]'
- 'Andreas Schafelner[^2]'
bibliography:
- 'references.bib'
title: 'Space-Time Finite Element Methods for Parabolic Evolution Problems with Non-smooth Solutions[^3]'
---
Introduction {#sec:intro}
============
Parabolic initial-boundary value problems of the form $$\label{LS:eq:modelproblem}
\partial_{t}u - \mathrm{div}_x(\nu\, \nabla_{x} u ) = f \;\mbox{in}\; Q, %= \Omega \times (0,T),
\quad
u = 0 \;\mbox{on}\; \Sigma, %= \partial \Omega \times (0,T)
\quad
u = u_0 \;\mbox{on}\; \Sigma_0 %= \Omega\times\{0\}$$ describe not only heat conduction and diffusion processes but also 2D eddy current problems in electromagnetics and many other evolution processes, where $Q = \Omega \times (0,T)$, $\Sigma = \partial \Omega \times (0,T)$, and $\Sigma_0 = \Omega\times\{0\}$ denote the space-time cylinder, its lateral boundary, and the bottom face, respectively. The spatial computational domain $ \Omega \subset \mathbb{R}^d $, $ d = 1,2,3 $, is supposed to be bounded and Lipschitz. The final time is denoted by $T$. The right-hand side $f$ is a given source function from $L_2(Q)$. The given coefficient $\nu$ may depend on the spatial variable $x$ as well as the time variable $t$. In the latter case, the problem is called non-autonomous. We suppose at least that $\nu$ is uniformly positive and bounded almost everywhere. We here consider homogeneous Dirichlet boundary conditions for the sake of simplicity. In practice, we often meet mixed boundary conditions. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions.
In contrast to the conventional time-stepping methods in combination with some spatial discretization method, or the more advanced, but closely related discontinuous Galerkin (dG) methods based on time slices, we here consider space-time finite element discretizations treating time as just another variable and the term $\partial_t u$ in (\[LS:eq:modelproblem\]) as convection term in time. Following [@LS:LangerNeumuellerSchafelner:2019a], we derive consistent, locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes under the assumption of maximal parabolic regularity; see, e.g., [@LS:Fackler:2017a]. Unstructured space-time schemes have clear advantages with respect to adaptivity, parallelization, and the numerical treatment of moving interfaces or special domains. We refer the reader to the survey paper [@LS:SteinbachYang:2018a] that provides an excellent overview of completely unstructured space-time methods and simultaneous space-time adaptivity. In particular, we would like to mention the papers [@LS:Steinbach:2015a] that is based on an inf-sup-condition, [@LS:DevaudSchwab:2018a] that uses mesh-grading in time, and [@LS:BankVassilevskiZikatanov:2016a] that also uses stabilization techniques. All three papers treat the autonomous case.
We here present new a priori discretization error estimates for low-regularity solutions. In order to avoid reduced convergence rates appearing in the case of uniform mesh refinement, we also consider adaptive refinement procedures in the numerical experiments presented in Section \[sec:num\]. The adaptive refinement procedures are based on residual a posteriori error indicators. The huge system of space-time finite element equations is then solved by means of Generalized Minimal Residual Method (GMRES) preconditioned by an algebraic multigrid cycle. In particular, in the 4D space-time case that is 3D in space, simultaneous space-time adaptivity and parallelization can considerably reduce the computational time. The space-time finite element solver was implemented in the framework of MFEM. The numerical results nicely confirm our theoretical findings. The parallel version of the code shows an excellent parallel performance.
Weak formulation and maximal parabolic regularity
=================================================
The weak formulation of the model problem reads as follows: find $ u\in H^{1,0}_0(Q):=\{u\in L_2(Q):\nabla_x u\in [L_2(Q)]^d,\, u=0\ \text{on}\ \Sigma \} $ such that (s.t.) $$\label{LS:eq:weakformulation}
\int_{Q} \bigl( -u\,\partial_{t} v +\nu\, \nabla_{x}u\cdot\nabla_{x}v\bigr)\;\mathrm{d}(x,t)
= \int_{Q}\!f\,v\;\mathrm{d}(x,t) + \int_{\Omega}\!u_0\,v|_{t=0}\;\mathrm{d}x
%\quad\forall v\in \hat{H}^{1}_0(Q),$$ for all $ v \in \hat{H}^{1}_0(Q) = \{ u\in H^1(Q) : u=0\ \text{on}\ \Sigma\cup\Sigma_T \} $, where $ \Sigma_T:=\Omega\times\{T\} $. The existence and uniqueness of weak solutions is well understood; see, e.g., [@LS:Ladyzhenskaya:1985a]. It was already shown in [@LS:Ladyzhenskaya:1985a] that $\partial_t u \in L^2(Q)$ and $\Delta u \in L^2(Q)$ provided that $\nu =1$, $f \in L^2(Q)$, and $u_0 = 0$. This case is called maximal parabolic regularity. Similar results can be obtained under more general assumptions imposed on the data; see, e.g., [@LS:Fackler:2017a] for some very recent results on the non-autonomous case.
Locally stabilized space-time finite element\
methods
=============================================
In order to derive the space-time finite element scheme, we need an admissible, shape regular decomposition $ \mathcal{T}_h = \{K\}$ of the space-time cylinder $ Q = \bigcup_{K\in\mathcal{T}_h}\overline{K}$ into finite elements $K$. On $ \mathcal{T}_h$, we define a $H^1$ conforming finite element space $V_{h}$ by means of polynomial simplicial finite elements of the degree $p$ in the usual way; see, e.g., [@LS:BrennerScott:2008a]. Let us assume that the solution $u$ of (\[LS:eq:weakformulation\]) belongs to the space $V_0 = H^{\mathcal{L},1}_{0,\underline{0}}(\mathcal{T}_h) := \{ u\in L_2(Q) : \partial_t u\in L_2(K),\ \mathcal{L}u:=\mathrm{div}_x(\nu\nabla_{x}u)\in L_2(K)\ \forall K\in\mathcal{T}_h,\ \text{and}\ u|_{\Sigma\cup\Sigma_0}=0 \}$, i.e., we only need some local version of maximal parabolic regularity, and, for simplicity, we assume homogeneous initial conditions, i.e., $u_0 = 0$. Multiplying the PDE (\[LS:eq:modelproblem\]) on $K$ by a local time-upwind test function $ v_h + \theta_K h_K\partial_{t}v_h $, with $ v_h\in V_{0h} = \{v_h \in V_{h}: v_h = 0 \, \mbox{on} \, \Sigma \cup \Sigma_0\}$, $ h_K = \mathrm{diam}(K) $, and a parameter $ \theta_K > 0 $ which we will specify later, integrating over $K$, integrating by parts, and summing up over all elements $K\in\mathcal{T}_h$, we arrive at the following consistent space-time finite element scheme: find $u_h \in V_{0h}$ s.t. $$\label{eq:fe-scheme}
a_h(u_h,v_h) = l_h(v_h),\quad \forall v_h\in V_{0h},$$ with $$\begin{aligned}
a_h(u_h,v_h) =& \sum_{K\in\mathcal{T}_h} \int_{K} \big[ \partial_t u_h v_h + \theta_K h_K \partial_t u_h \partial_t v_h \label{LS:eq:bilinearform} \\
&\qquad\qquad + \nu \nabla_{x}u_h\cdot\nabla_{x}v_h - \theta_K h_K \mathrm{div}_x(\nu \nabla_{x} u_h)\partial_t v_h \big] \mathrm{d}(x,t),\notag\\
l_h(v_h) =& \sum_{K\in\mathcal{T}_h} \int_{K} f v_h + \theta_K h_K f \partial_t v_h\mathrm{d}(x,t). \label{LS:eq:linearform}%\notag\\\end{aligned}$$ The bilinearform $ a_h(\,.\,,\,.\,) $ is coercive on $ V_{0h}\times V_{0h} $ wrt to the norm $$\| v \|_{h}^2 = \frac{1}{2}\| v \|_{L_2(\Sigma_T)}^2 + \sum_{K\in\mathcal{T}_h} \Bigl[ \theta_{K} h_K\| \partial_{t} v \|_{L_2(K)}^2 + \| \nabla_x v \|_{L_2^{\nu}(K)}^2 \Bigr], \label{LNS:eq:norm:h}$$ i.e., $ a_h(v_h,v_h) \ge \mu_c \|v_h\|_h^2$, $\forall v_h\in V_{0h}$, and bounded on $ V_{0h,*}\times V_{0h} $ wrt to the norm $$\begin{aligned}
\| v \|_{h,*}^2 &=
\|v\|_{h}^2 + \sum_{K\in\mathcal{T}_h} \Bigl[ (\theta_{K} h_K)^{-1}\| v \|_{L_2(K)}^2 + \theta_{K} h_K \|\mathrm{div}_x (\nu \nabla_{x} v) \|_{L_2(K)}^2 \Bigr],\label{LNS:eq:norm:h,*}\end{aligned}$$ i.e., $a_h(u_h,v_h) \le \mu_b \|u_h\|_{h,*}\|v_h\|_{h}$, $\forall u_h\in V_{0h,*},\forall v_h\in V_{0h}$, where $ V_{0h,*}:= H^{\mathcal{L},1}_{0,\underline{0}}(Q) + V_{0h} $; see [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.8] and [@LS:LangerNeumuellerSchafelner:2019a Remark 3.13], respectively. The coercivity constant $ \mu_c $ is robust in $ h_K $ provided that we choose $ \theta_{K} = \mathcal{O}(h_K) $; see Section \[sec:num\] or [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.8] for the explicit choice. From the above derivation of the scheme, we get consistency $a_h(u,v_h) = l_h(v_h), \; \forall v_h\in V_{0h}$, provided that the solution $u$ belongs to $H^{\mathcal{L},1}_{0,\underline{0}}(Q) $ that is ensured in the case of maximal parabolic regularity. The space-time finite element scheme (\[eq:fe-scheme\]) and the consistency relation immediately yield Galerkin orthogonality $$\label{eq:GalerkinOrthogonality}
a_h(u - u_h,v_h) = 0, \quad \forall v_h\in V_{0h}.$$ We deduce that is nothing but a huge linear system of algebraic equations. Indeed, let $ V_{0h} = \mathrm{span}\{ p^{(j)}, j=1,\dots,N_h \} $, where $ \{ p^{(j)}, j=1,\dots,N_h \} $ is the nodal finite element basis and $ N_h $ is the total number of space-time degrees of freedom (dofs). Then we can express each function in $ V_{0h} $ in terms of this basis, i.e., we can identify each finite element function $ v_h\in V_{0h} $ with its coefficient vector $ \mathbf{v}_h\in \mathbb{R}^{N_h} $. Moreover, each basis function $ p^{(j)} $ is also a valid test function. Hence, we obtain $ N_h $ equations from , which we rewrite as a system of linear algebraic equations, i.e. $$\mathbf{K}_h\,\mathbf{u}_h = \mathbf{f}_h,$$ with the solution vector $\mathbf{u}_{h}= (u_j)_{j=1,\ldots,N_h} $, the vector $ \mathbf{f}_{h} = \bigl(l_h(p^{(i)})\bigr)_{i=1,\ldots,N_h}$, and system matrix $ \mathbf{K}_{h} = \bigl(a_h(p^{(j)},p^{(i)})\bigr)_{i,j=1,\ldots,N_h} $ that is non-symmetric, but positive definite.
A priori discretization error estimates
=======================================
Galerkin orthogonality (\[eq:GalerkinOrthogonality\]), together with coercivity and boundedness, enables us to prove a Cèa-like estimate, where we bound the discretization error in the $ \|\,.\,\|_h $-norm by the best-approximation error in the $ \|\,.\,\|_{h,*} $-norm.
\[LS:lem:cea-like\] Let the bilinearform be coercive [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.8] with constant $ \mu_c $ and bounded [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.11, Remark 3.13] with constant $ \mu_b $, and let $ u\in H^{\mathcal{L},1}_{0,\underline{0}}(\mathcal{T}_h) $ be the solution of the space-time variational problem . Then there holds $$\label{LS:eq:cea-like}
\|u-u_h\|_h\leq \biggl(1+\frac{\mu_b}{\mu_c}\biggr) \inf_{v_h\in V_{0h}} \|u-v_h\|_{h,*},$$ where $ u_h\in V_{0h} $ is the solution to the space-time finite element scheme .
Estimate (\[LS:eq:cea-like\]) easily follows from triangle inequality and Galerkin-orthogonality; see [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.15, Remark 3.16] for details.
Next, we estimate the best approximation error by the interpolation error, where we have to choose a proper interpolation operator $ \mathfrak{I}_* $. For smooth solutions, i.e., $ u\in H^l(Q) $ with $ l > (d+1)/2 $, we obtained a localized a priori error estimate, see [@LS:LangerNeumuellerSchafelner:2019a Theorem 3.17], where we used the standard Lagrange interpolation operator $ \mathcal{I}_h $; see e.g. [@LS:BrennerScott:2008a]. In this paper, we are interested in non-smooth solutions, which means that we only require $ u\in H^{l}(Q) $, with some real $ l > 1 $. Hence, we cannot use the Lagrange interpolator. We can, however, use so-called quasi-interpolators, e.g. Clément [@LS:Clement:1975a] or Scott-Zhang [@LS:BrennerScott:2008a]. For this kind of operators, we need a neighborhood $ S_K $ of an element $ K\in\mathcal{T}_h $ which is defined as $ S_K := \{ K'\in\mathcal{T}_h : \overline{K}\cap\overline{K}'\neq\emptyset\}. $ Let $ u\in H^{l}(Q) $, with some real $ l>1 $, then, for the Scott-Zhang quasi-interpolation operator $ \mathfrak{I}_{S\!Z} : L_2(Q) \rightarrow V_{0h} $, we have the local estimate (see e.g. [@LS:BrennerScott:2008a (4.8.10)]) $$\label{LS:eq:quasi-interpolation}
\| v-\mathfrak{I}_{S\!Z} v \|_{H^k(K)} \leq C_{\mathfrak{I}_{S\!Z}} h_K^{l-k} |v|_{H^l(S_K)},\ k=0,1.$$ For details on the construction of such a quasi-interpolator, we refer to [@LS:BrennerScott:2008a] and the references therein. For simplicity, we now assume that the diffusion coefficient $ \nu $ is piecewise constant, i.e., $ \nu|_K = \nu_K $, for all $ K\in\mathcal{T}_h $. Then we can show the following lemma.
\[LS:lem:Quasi-InterpolationerrorEstimates\] Let $ l > 1 $ and $ v\in V_0\cap H^l(\mathcal T_h) $. Then the following interpolation error estimates are valid: $$\begin{aligned}
\| v-{\mathfrak{I}_{S\!Z}} v \|_{L_2(\Sigma_T)} &\leq c_1\Bigl(\sum_{\mathclap{\substack{
K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}}h_K^{2s-1}| v |_{H^{s}(K)}^2\Bigr)^{1/2}, \label{LNS:eq:interpolation:boundary}\\
\| v-{\mathfrak{I}_{S\!Z}} v \|_h &\leq c_2 \Bigl(\sum_{K\in\mathcal{T}_h}h_K^{2(s-1)} |v|_{H^s(S_K)}^2\Bigr)^{1/2}, \label{LNS:eq:interpolation:h} \\
\| v-{\mathfrak{I}_{S\!Z}} v \|_{h,*} &\leq c_3 \Bigl(\sum_{K\in\mathcal{T}_h}h_K^{2(s-1)} \bigl(|v|_{H^s(S_K)}^2 + \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2\bigr)\Bigr)^{1/2}, \label{LNS:eq:interpolation:h,*}
\end{aligned}$$ with $ s = \min\{l,p+1\} $ and positive constants $ c_1, c_2$ and $c_3 $, that do not depend on $ v $ or $ h_K $ provided that $ \theta_K=\mathcal{O}(h_K) $ for all $K \in \mathcal{T}_h$. Here, $ p $ denotes the polynomial degree of the finite element shape functions on the reference element.
For the first estimate, we use the scaled trace inequality and the quasi-interpolation estimate with $ k = 0,1 $ $$\begin{aligned}
\| v-\mathfrak{I}_{S\!Z} v \|_{L_2(\Sigma_T)}^2 &= \sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} \| v-\mathfrak{I}_{S\!Z} v \|_{L_2(\partial K\cap\Sigma_T)}^2 \leq \sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} \| v-\mathfrak{I}_{S\!Z} v \|_{L_2(\partial K)}^2 \\
&\leq \sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} \bigl[ c_{Tr}^2h_K^{-1}(\| v-\mathfrak{I}_{S\!Z} v \|_{L_2(K)}^2 + h_K^2 \| \nabla (v-\mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2) \bigr] \\
&\leq c_{Tr}^2 \sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} \bigl[C_{\mathfrak{I}_{S\!Z}}^2\ h_K^{-1}h_K^{2l} | v |_{H^{l}(S_K)}^2 +C_{\mathfrak{I}_{S\!Z}}^2\ h_K\,h_K^{2(l-1)} | v |_{H^{l}(S_K)}^2 \bigr]\\
& \leq \max_{K\in\mathcal{T}_h}\bigl(2\,c_{Tr}^2 C_{\mathfrak I_{S\!Z}}^2\bigr)\sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} \bigl[h_K^{2l-1} | v |_{H^{l}(S_K)}^2 \bigr],
\end{aligned}$$ which corresponds to with $ c_1 = \max_{K\in\mathcal{T}_h}\bigl(2\,c_{Tr}^2 C_{\mathfrak I_{S\!Z}}^2\bigr) $. To show the second estimate , we use definition and that $ \nu $ is piecewise constant, the quasi-interpolation error estimate with $ k = 1 $, and the above estimate , and obtain $$\begin{aligned}
\| v - &\mathfrak{I}_{S\!Z} v \|_h^2 \\
&= \sum_{K\in\mathcal{T}_h} \Bigl[\theta_{K}h_K\,\|\partial_t (v-\mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2 + \|\nu^{1/2}\nabla_{x}(v-\mathfrak{I}_{S\!Z}v) \|_{L_2(K)}^2\Bigr]\\
&\quad+ \frac{1}{2} \|v-\mathfrak{I}_{S\!Z}v\|_{L_2(\Sigma_T)}^2\\
&\leq \sum_{K\in\mathcal{T}_h}\Bigl[\theta_K h_K C_{\mathfrak{I}_{S\!Z}}^2 h_K^{2(l-1)} |v|_{H^l(S_K)}^2 + {\nu}_K C_{\mathfrak{I}_{S\!Z}}^2 h_K^{2(l-1)}|v|_{H^l(S_K)}\Bigr] \\
&\quad+ c_1 \sum_{\mathclap{\substack{K\in\mathcal{T}_h\\ \partial K\cap\Sigma_T\neq\emptyset}}} h_K^{2l-1} | v |_{H^{l}(S_K)}^2\\
&\leq \sum_{K\in\mathcal{T}_h} \bigl(C_{\mathfrak{I}_{S\!Z}}^2(\theta_K h_K + {\nu}_K) + c_1 h_K \bigr) h_K^{2(l-1)} | v |_{H^{l}(S_K)}^2 \leq c_2 \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} | v |_{H^{l}(S_K)}^2,
\end{aligned}$$ with $ c_2 = \max_{K\in\mathcal{T}_h}\bigl(C_{\mathfrak{I}_{S\!Z}}^2(\theta_K h_K + {\nu}_K) + c_1 h_K \bigr) $. For the third estimate, we deduce from that we only have to estimate the additional sum $$\sum_{K\in\mathcal{T}_h}\Bigl[(\theta_K h_K)^{-1} \|v - \mathfrak{I}_{S\!Z} v\|_{L_2(K)} ^2 + \theta_K h_K \|\mathrm{div}_x(\nu\nabla_{x}(v-\mathfrak{I}_{S\!Z} v))\|_{L_2(K)}^2\Bigr].$$ We start with the first $ L_2 $-term. We apply the quasi-interpolation estimate with $ k = 0 $ and obtain $$\begin{aligned}
\sum_{K\in\mathcal{T}_h}(\theta_K h_K)^{-1} \|v - \mathfrak{I}_{S\!Z} v\|_{L_2(K)} ^2 &\leq \sum_{K\in\mathcal{T}_h} (\theta_K h_K)^{-1} C_{\mathfrak{I}_{S\!Z}}^2 h_K^{2l} |v|_{H^l(S_K)}^2\\
&\leq \sum_{K\in\mathcal{T}_h}C_{\mathfrak{I}_{S\!Z}}^2 \Bigl(\frac{h_K}{\theta_K}\Bigr) h_K^{2(l-1)} |v|_{H^l(S_K)}^2.
\end{aligned}$$ Note that the term $ (h_K/\theta_K) $ is bounded for $ \theta_K = \mathcal{O}(h_K) $. For the $ L_2 $-norm of the spatial divergence, we can distinguish between two cases: linear basis functions ($ p=1 $) and higher order basis functions ($ p\geq2 $). For linear basis functions, we split the divergence of the gradient, obtaining $$\|\mathrm{div}_x(\nu\nabla_{x}(v-\mathfrak{I}_{S\!Z} v))\|_{L_2(K)}^2 =
\|\mathrm{div}_x(\nu\nabla_{x}v) - \mathrm{div}_x(\nu\nabla_{x}(\mathfrak{I}_{S\!Z} v))\|_{L_2(K)}^2$$ for each element $ K\in\mathcal{T}_h $. Since $ \mathfrak{I}_{S\!Z}v $ is a linear polynomial and $ \nu $ is piecewise constant, we deduce $$\mathrm{div}_x(\nu\nabla_{x}(\mathfrak{I}_{S\!Z}v)) = \nu_K\,\mathrm{div}_x(\nabla_{x}(\mathfrak{I}_{S\!Z}v)) = 0$$ for each element $ K\in\mathcal{T}_h $. Hence, we get $$\begin{aligned}
\sum_{K\in\mathcal{T}_h} \theta_K h_K \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2}^2 = \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} \theta_K h_K^{1-2(l-1)} \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2 ,
\end{aligned}$$ where the norm is bounded since $ v \in V_0 $. Moreover, for $ \theta_K = \mathcal{O}(h_K) $, the term $ \theta_K h_K^{1-2(l-1)} $ is bounded for $ 1 \leq l \leq 2 $, and the case $ l > 2 $ is already treated in [@LS:LangerNeumuellerSchafelner:2019a]. Combining all above estimates, we obtain $$\begin{aligned}
\|v-\mathfrak{I}_{S\!Z} v\|_{h,*}^2 \leq&c_2 \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} |v|_{H^l(S_K)}^2 + \sum_{K\in\mathcal{T}_h} C_{\mathfrak{I}_{S\!Z}}^2 \Bigl(\frac{h_K}{\theta_K}\Bigr) h_K^{2(l-1)} |v|_{H^l(S_K)}^2 \\
&+ \sum_{K\in\mathcal{T}_h} (\theta_{K}h_K^{3-2l}) h_K^{2(l-1)}\|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2\\
\leq& c_3 \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} \Bigl(|v|_{H^l(S_K)}^2 + \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2\Bigr),
\end{aligned}$$ with $ c_3 = c_2 + \max_{K\in\mathcal{T}_h}\bigl\{ C_{\mathfrak{I}_{S\!Z}}^2(h_K/\theta_{K}), \theta_{K}h_K^{3-2l} \bigr\} $. For the general case of higher order basis functions, i.e., $ p\geq2 $, the divergence of the gradient does not vanish. First we split the divergence of the gradient and also the norms, obtaining $$\begin{aligned}
\sum_{K\in\mathcal{T}_h} \theta_K h_K &\|\mathrm{div}_x(\nu\nabla_{x}v) - \mathrm{div}_x(\nu\nabla_{x}(\mathfrak{I}_{S\!Z} v))\|_{L_2}^2 \\
&\leq \sum_{K\in\mathcal{T}_h} 2\,\theta_K h_K \left( \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2}^2 + \| \mathrm{div}_x(\nu\nabla_{x}(\mathfrak{I}_{S\!Z} v))\|_{L_2}^2 \right)
\end{aligned}$$ The first term in the sum we have already estimated above, where we now replace $ l $ by $ s = \min\{l,p+1\} $, which is in our case ($ l\leq2 $) again just $ l $. For the second term, we insert $ \nu_K \mathrm{div}_x (\nabla_{x}(\mathfrak{I}_{S\!Z}^1 v)) $ into the norm, where $ \mathfrak{I}_{S\!Z}^1 v $ is the linear quasi-interpolation of $ v $. We observe that $ \mathfrak{I}_{S\!Z}^1 v - \mathfrak{I}_{S\!Z} v $ is a finite element function, i.e., we can apply the inverse equality for the $ H(\mathrm{div}_x) $-norm [@LS:LangerNeumuellerSchafelner:2019a Lemma 3.5]. Since the diffusion coefficient is piecewise constant, we obtain $$\begin{aligned}
\sum_{K\in\mathcal{T}_h} 2\,\theta_K h_K &\| \mathrm{div}_x(\nu\nabla_{x}(\mathfrak{I}_{S\!Z}^1 v - \mathfrak{I}_{S\!Z} v))\|_{L_2}^2 \\
&\leq \sum_{K\in\mathcal{T}_h} 2\,\theta_K h_K h_K^{-2} c_{I,3}^2 \|\nu\nabla_{x}(\mathfrak{I}_{S\!Z}^1 v - \mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2 \\
&\leq \sum_{K\in\mathcal{T}_h} 2\,\theta_K h_K^{-1} c_{I,3}^2 {\nu}_K^2 \|\nabla_{x}(\mathfrak{I}_{S\!Z}^1 v - \mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2
\end{aligned}$$ Now we insert and subtract $ \nabla_{x} v $ and use the triangle inequality, which yields $$\begin{aligned}
\sum_{K\in\mathcal{T}_h} 2\,&\theta_K h_K^{-1} c_{I,3}^2 \overline{\nu}_K^2 \|\nabla_{x}(\mathfrak{I}_{S\!Z}^1 v - \mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2 \\
&\leq \sum_{K\in\mathcal{T}_h} 4\,\theta_K h_K^{-1} c_{I,3}^2 \overline{\nu}_K^2\left( \|\nabla_{x}(v - \mathfrak{I}_{S\!Z}^1 v)\|_{L_2(K)}^2 + \| \nabla_{x}( v- \mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2 \right).
\end{aligned}$$ For both terms we can apply (quasi-interpolation) and obtain $$\begin{aligned}
\sum_{K\in\mathcal{T}_h}& 4\,\theta_K h_K^{-1} c_{I,3}^2{\nu}_K^2 \Bigl( \|\nabla_{x}(v - \mathfrak{I}_{S\!Z}^1 v)\|_{L_2(K)}^2 + \| \nabla_{x}( v- \mathfrak{I}_{S\!Z} v) \|_{L_2(K)}^2 \Bigr)\\
&\leq \sum_{K\in\mathcal{T}_h} 4\,\theta_K h_K^{-1} c_{I,3}^2{\nu}_K^2 \Bigl( C_{\mathfrak{I}_{S\!Z}}^2 h_K^{2(l-1)} |v|_{H^l(S_K)}^2 + C_{\mathfrak{I}_{S\!Z}}^2 h_K^{2(l-1)}| v|_{H^l(S_K)}^2 \Bigr)\\
&\leq \sum_{K\in\mathcal{T}_h} {8\,\theta_K h_K^{-1} c_{I,3}^2{\nu}_K^2 C_{\mathfrak{I}_{S\!Z}}^2}%_{=\tilde{C}_2(\theta_K)}
h_K^{2(l-1)} |v|_{H^l(S_K)}^2.
\end{aligned}$$ Combining all of the above, $$\begin{aligned}
\|v-&\mathfrak{I}_{S\!Z} v\|_{h,*}^2 \\
&\leq c_2 \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} |v|_{H^l(S_K)} + \sum_{K\in\mathcal{T}_h} 2(\theta_{K}h_K^{3-2l}) h_K^{2(l-1)}\|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2 \\
&\qquad+ \sum_{K\in\mathcal{T}_h} \bigl({8\,\theta_K h_K^{-1} c_{I,3}^2{\nu}_K^2 C_{\mathfrak{I}_{S\!Z}}^2}\bigr) h_K^{2(l-1)}|v|_{H^l(S_K)}^2\\
&\leq c_{3,p} \sum_{K\in\mathcal{T}_h} h_K^{2(l-1)} \left(|v|_{H^l(S_K)}^2 + \|\mathrm{div}_x(\nu\nabla_{x}v)\|_{L_2(K)}^2\right),
\end{aligned}$$ where $ c_{3,p} = c_2 + \max_{K\in\mathcal{T}_h}\bigl\{{8\,\theta_K h_K^{-1} c_{I,3}^2{\nu}_K^2 C_{\mathfrak{I}_{S\!Z}}^2},2(\theta_{K}h_K^{3-2l}) \bigr\}$ .
Now we are in the position to prove our main theorem.
Let $ p $ be the polynomial degree used, and let $ u\in H^{l}(Q)\cap V_{0} $, with $ l>1 $, be the exact solution, and $ u_h\in V_{0h} $ be the approximate solution of the finite element scheme . Furthermore, let the assumptions of Lemma \[LS:lem:cea-like\] (Céa-like estimate) and \[LS:lem:Quasi-InterpolationerrorEstimates\] (quasi-interpolation estimates) hold. Then the a priori discretization error estimate $$\label{LS:eq:discretization-error}
\|u-u_h\|_h \le C\,\Bigl(\sum_{K\in\mathcal{T}_h} h_K^{2(s-1)}\bigl(|u|_{H^s(S_K)} + \|\mathrm{div}_x(\nu\nabla_{x}u)\|_{L_2(K)}^2\bigr) \Bigr)^{1/2},$$ holds, with $ s=\min\{l,p+1\} $ and a positive generic constant $ C $.
Choosing the quasi-interpolant $ v_h = \mathfrak{I}_{S\!Z}u $ in , we can apply the quasi-interpolation estimate to obtain $$\begin{aligned}
\|u-u_h\|_h &\leq \Bigl(1+\frac{\mu_b}{\mu_c}\Bigr) \|u-\mathfrak I_{S\!Z}u\|_{h,*}\\
&\le c_3\Bigl(1+\frac{\mu_b}{\mu_c}\Bigr) \Bigl(\sum_{K\in\mathcal{T}_h}h_K^{2(s-1)} \bigl(|u|_{H^s(S_K)}^2 + \|\mathrm{div}_x(\nu\nabla_{x}u)\|_{L_2(K)}^2\bigr)\Bigr)^{1/2}.
\end{aligned}$$ We set $ C = c_3(1+\mu_b/\mu_c) $ to obtain , which closes the proof.
Numerical results {#sec:num}
=================
We implemented the space-time finite element scheme in `C++`, where we used the finite element library MFEM[^4] and the solver library *hypre*[^5]. The linear system was solved by means of the GMRES method, preconditioned by the algebraic multigrid *BoomerAMG*. We stopped the iterative procedure if the initial residual was reduced by a factor of $ 10^{-8} $. Both libraries are already fully parallelized with the Message Passing Interface (MPI). Therefore, we performed all numerical tests on the distributed memory cluster RADON1[^6] in Linz. For each element $ K\in\mathcal{T}_h $, we choose $ \theta_{K} = h_K/(\tilde{c}^2 \overline{\nu}_K) $, where $ \tilde{c} $ is computed by solving a local generalized eigenvalue problem which comes from an inverse inequality, see [@LS:LangerNeumuellerSchafelner:2019a] for further details.
Example: Highly oscillatory solution {#sec:num:1}
------------------------------------
As first test example, we consider the unit (hyper-)cube $ Q = (0,1)^{d+1} $, with $ d=2,3 $, as space-time cylinder, and $ \nu\equiv1 $. The manufactured function $$u(x,t) = \sin\!\left(\frac{1}{\frac{1}{10\pi}+\sqrt{\sum_{i=1}^{d}x_i^2 + t^2}}\right)$$ serves as the exact solution, where we compute the right hand side $ f $ accordingly. This solution is highly oscillatory. Hence, we do not expect optimal rates for uniform refinement in the pre-asymptotic range. However, using adaptive refinement, we may be able to recover the optimal rates. We used the residual based error indicator proposed by Steinbach and Yang in [@LS:SteinbachYang:2018a]. For each element $ K\in\mathcal{T}_h $, we compute $$\eta_K^2 := h_K^2 \|R_h(u_h)\|_{L_2(K)}^2 + h_K \|J_h(u_h) \|_{L_2(\partial K)}^2,$$ where $ u_h $ is the solution of , $ R_h(u_h) := f + \mathrm{div}_x(\nu \nabla_x u_h)-\partial_t u_h $ in $K$, and $ J_h(u_h) := [\nu \nabla_x u_h]_e$ on $ e\subset\partial K$, with $ [\,.\,]_e $ denoting the jump across one face $ e\subset \partial K $. We mark each element where the condition $ \eta_K \geq \sigma \max_{K\in\mathcal{T}_h} \eta_K $ is fulfilled, with $ \sigma $ an a priori chosen threshold, e.g., $ \sigma = 0.5 $. Note that $ \sigma = 0 $ results in uniform refinement. In Figure \[fig:example\], we observe indeed reduced convergence rates for all polynomial degrees and dimensions tested. However, using an adaptive procedure, we are able to recover the optimal rates. Moreover, we significantly reduce the number of dofs required to reach a certain error threshold. For instance, in the case $ d=3 $ and $ p=2 $, we need $ 276\,922\,881 $ dofs to get an error in the $ \|\,.\,\|_h $-norm of $ \sim10^{-1} $, whereas we only need $ 26\,359 $ dofs with adaptive refinement. In terms of runtime, the uniform refinement needed $ 478.57 s $ for assembling and solving, while the complete adaptive procedure took $ 156.5 s $ only. The parallel performance is also shown in Figure \[fig:example\], where we obtain a nice strong scaling up to 64 cores. Then the local problems are too small (only $ \sim 10\,000 $ dofs for 128 cores) and the communication overhead becomes too large.
![Example \[sec:num:1\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $ (upper left) and $ d=3 $ (upper right), the dotted lines indicate the optimal rate; Plot of the approximate solution $ u_h $ centered at the origin $ (0,0,0) $ [@LS:LangerNeumuellerSchafelner:2019a] (lower left); Strong scaling on a mesh with $ 1\,185\,921 $ dofs for $ p=1,2 $ and $ 5\,764\,801 $ dofs for $ p=3 $ (lower right).[]{data-label="fig:example"}](figure1a.pdf){width="\textwidth"}
![Example \[sec:num:1\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $ (upper left) and $ d=3 $ (upper right), the dotted lines indicate the optimal rate; Plot of the approximate solution $ u_h $ centered at the origin $ (0,0,0) $ [@LS:LangerNeumuellerSchafelner:2019a] (lower left); Strong scaling on a mesh with $ 1\,185\,921 $ dofs for $ p=1,2 $ and $ 5\,764\,801 $ dofs for $ p=3 $ (lower right).[]{data-label="fig:example"}](figure1b.pdf){width="\textwidth"}
![Example \[sec:num:1\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $ (upper left) and $ d=3 $ (upper right), the dotted lines indicate the optimal rate; Plot of the approximate solution $ u_h $ centered at the origin $ (0,0,0) $ [@LS:LangerNeumuellerSchafelner:2019a] (lower left); Strong scaling on a mesh with $ 1\,185\,921 $ dofs for $ p=1,2 $ and $ 5\,764\,801 $ dofs for $ p=3 $ (lower right).[]{data-label="fig:example"}](figure1c){width=".6\textwidth"}
![Example \[sec:num:1\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $ (upper left) and $ d=3 $ (upper right), the dotted lines indicate the optimal rate; Plot of the approximate solution $ u_h $ centered at the origin $ (0,0,0) $ [@LS:LangerNeumuellerSchafelner:2019a] (lower left); Strong scaling on a mesh with $ 1\,185\,921 $ dofs for $ p=1,2 $ and $ 5\,764\,801 $ dofs for $ p=3 $ (lower right).[]{data-label="fig:example"}](figure1d.pdf){width="\textwidth"}
Example: Moving peak {#sec:num:2}
--------------------
For the second example, we consider the unit-cube $ Q = (0,1)^3 $, i.e. $ d=2 $. As diffusion coefficient, the choice $ \nu\equiv 1 $ is made. We choose the function $$u(x,t) = (x_1^2-x_1)(x_2^2-x_2)e^{-100((x_1-t)^2+(x_2-t)^2)},$$ as exact solution and compute all data accordingly. This function is smooth, and almost zero everywhere, except in a small region around the line from the origin $ (0,0,0) $ to $ (1,1,1) $. This motivates again the use of an adaptive method. We use the residual based indicator $ \eta_K $ introduced in Example \[sec:num:1\]. In Figure \[fig:example2\], we can observe that we indeed obtain optimal rates for both uniform and adaptive refinement. However, using the a posteriori error indicator, we reduce the number of dofs needed to reach a certain threshold by one order of magnitude. For instance, in the case $ p=2 $, we need $ 16\,974\,593 $ dofs to obtain an error of $ \sim 7\times10^{-5} $ with uniform refinement. Using adaptive refinement, we need $ 1\,066\,777 $ dofs only.
![Example \[sec:num:2\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $, the dotted lines indicate the optimal rates (left); Diagonal cut through the space-time mesh $ \mathcal{T}_h $ along the line from $ (0,0,0) $ to $ (1,1,1) $ after 8 adaptive refinements (right). []{data-label="fig:example2"}](figure2a.pdf){width="\textwidth"}
![Example \[sec:num:2\]: Error rates in the $ \|\,.\,\|_h $-norm for $ d=2 $, the dotted lines indicate the optimal rates (left); Diagonal cut through the space-time mesh $ \mathcal{T}_h $ along the line from $ (0,0,0) $ to $ (1,1,1) $ after 8 adaptive refinements (right). []{data-label="fig:example2"}](figure2b){width=".65\textwidth"}
Conclusions {#sec:conc}
===========
Following [@LS:LangerNeumuellerSchafelner:2019a], we introduced a space-time finite element solver for non-autonomous parabolic evolution problems on completely unstructured simplicial meshes. We only assumed that we have so-called maximal parabolic regularity, i.e., the PDE is well posed in $ L_2 $. We note that this property is only required locally in order to derive a consistent space-time finite element scheme. We extended the a priori error estimate in the mesh-dependent energy norm to the case of non-smooth solutions, i.e. $ u\in H^{1+\epsilon}(Q) $, with $ 0 < \epsilon \le 1 $. This is necessary, since we cannot expect a smooth solution, especially when dealing with discontinuous diffusion coefficients. For simplicity, we considered only piecewise constant diffusion coefficients. The extension to piecewise smooth coefficients is straight-forward, but more technical. In comparison to the previous result for sufficiently smooth solutions, we no longer have a completely localized estimate. We also have to include the neighborhood of an element $ K\in\mathcal{T}_h $. This may not be sufficient if we have to deal with solutions that have different regularity in different subdomains of the whole space-time cylinder. In applications, these subdomains correspond, for instance, to different materials. However, Duan et al. [@LS:DuanLiTanZheng:2012a] have shown a quasi-interpolation estimate that fits into this setting.\
We performed two numerical examples with known solutions. The first example had a highly oscillatory solution, and the second one was almost zero everywhere except along a line through the space-time cylinder. Using a high-performance cluster, we solved both problems on a sequence of uniformly refined meshes, where we also obtained good strong scaling rates. In order to reduce the computational cost, we also applied an adaptive procedure, using a residual based error indicator. Indeed, using a simultaneously in space and time adaptive procedure reduced the computational as well as the memory cost by a large factor. Moreover, we could observe that, especially for $ d=3 $, the AMG preconditioned GMRES method solves the problem quite efficiently.\
[^1]: Institute for Computational Mathematics, Johannes Kepler University Linz.
[^2]: Doctoral Program “Computational Mathematics”, Johannes Kepler University Linz.
[^3]: Supported by the Austrian Science Fund (FWF) under the grant W1214, project DK4.
[^4]: <http://mfem.org/>
[^5]: <https://www.llnl.gov/casc/hypre/>
[^6]: <https://www.ricam.oeaw.ac.at/hpc/>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the previous article “Hearts of twin cotorsion pairs on exact categories”, we introduced the notion of the heart for any cotorsion pair on an exact category with enough projectives and injectives, and showed that it is an abelian category. In this paper, we construct a half exact functor from the exact category to the heart. This is analog of the construction of Abe and Nakaoka for triangulated categories. We will also use this half exact functor to find out a sufficient condition when two different hearts are equivalent.'
address: |
Graduate School of Mathematics\
Nagoya University\
464-8602 Nagoya, Japan
author:
- Yu Liu
title: Half exact functors associated with general hearts on exact categories
---
Introduction
============
Cotorsion pairs play an important role in representation theory (see [@AI] and see [@HI] for more recent examples). The heart of cotorsion pair on triangulated categories, which is abelian by Nakaoka’s result [@N], is a generalization of both the heart of t-structure and the quotient by cluster tilting subcategory. In [@L], we define hearts $\underline {\mathcal H}$ of cotorsion pairs $({\mathcal U},{\mathcal V})$ on exact categories ${\mathcal B}$ and proved that they are abelian. Abe and Nakaoka constructed a cohomological functor in the case of triangulated categories [@AN], which is a generalization of cohomological functor for t-structure. It is natural to ask whether we can find the relationship between the hearts and the original exact categories. In this paper we will answer this question by constructing an associated half exact functor $H$ from the exact category ${\mathcal B}$ to the heart $\underline {\mathcal H}$.
Throughout this paper, let ${\mathcal B}$ be a Krull-Schmidt exact category with enough projectives and injectives. Let $\mathcal P$ (resp. $\mathcal I$) be the full subcategory of projectives (resp. injectives) of ${\mathcal B}$.
We recall the definition of a cotorsion pair on ${\mathcal B}$ [@L [Definition 2.3]{}]:
\[2\] Let ${\mathcal U}$ and ${\mathcal V}$ be full additive subcategories of ${\mathcal B}$ which are closed under direct summands. We call $({\mathcal U},{\mathcal V})$ a *cotorsion pair* if it satisfies the following conditions:
- ${\operatorname{Ext}\nolimits}^1_{\mathcal B}({\mathcal U},{\mathcal V})=0$.
- For any object $B\in {\mathcal B}$, there exits two short exact sequences $$\begin{aligned}
V_B\rightarrowtail U_B\twoheadrightarrow B,\quad
B\rightarrowtail V^B\twoheadrightarrow U^B\end{aligned}$$ satisfying $U_B,U^B\in {\mathcal U}$ and $V_B,V^B\in {\mathcal V}$.
For any cotorsion pairs $({\mathcal U},{\mathcal V})$, let ${\mathcal W}:={\mathcal U}\cap {\mathcal V}$. Let $$\begin{aligned}
(1) \text{ } {\mathcal B}^+:=\{B\in {\mathcal B}\text{ } | \text{ } U_B\in {\mathcal W}\}, \quad
(2) \text{ } {\mathcal B}^-:=\{B\in {\mathcal B}\text{ } | \text{ } V^B\in {\mathcal W}\}.\end{aligned}$$ Let $${\mathcal H}:={\mathcal B}^+\cap{\mathcal B}^-$$ we call the additive subcategory ${\mathcal H}/{\mathcal W}$ of ${\mathcal B}/{\mathcal W}$ the *heart* of cotorsion pair $({\mathcal U},{\mathcal V})$.\
For convenience, we denote the quotient of ${\mathcal B}$ by ${\mathcal W}$ as ${\underline{{\mathcal B}}}:={\mathcal B}/{\mathcal W}$. For any morphism $f\in {\operatorname{Hom}\nolimits}_{\mathcal B}(X,Y)$, we denote its image in $ {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(X,Y)$ by $\underline f$. For any subcategory $\mathcal C\supseteq{\mathcal W}$ of ${\mathcal B}$, we denote by $\underline {\mathcal C}$ the full subcategory of ${\underline{{\mathcal B}}}$ consisting of the same objects as $\mathcal C$.
According to [@O p.24], we introduce the definition of the half exact functor on ${\mathcal B}$.
\[8.1\] A covariant functor $F$ from ${\mathcal B}$ to an abelian category $\mathcal A$ is called *half exact* if for any short exact sequence $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ in ${\mathcal B}$, the sequence $$F(A)\xrightarrow{F(f)} F(B)\xrightarrow{F(g)} F(C)$$ is exact in $\mathcal A$.
We will prove the following theorem (see Theorem \[8.11\] and Proposition \[8.7\] for details).
For any cotorsion pair $({\mathcal U},{\mathcal V})$ on ${\mathcal B}$
- There exists an associated half exact functor $H:{\mathcal B}\rightarrow \underline {\mathcal H}.$
- $H({\mathcal U})=0$ and $H({\mathcal V})=0$ hold. In particular, $H(\mathcal P)=0$ and $H(\mathcal I)=0$ hold.
We denote by $\Omega: {\mathcal B}/{\mathcal P}\rightarrow {\mathcal B}/{\mathcal P}$ the syzygy functor and by $\Omega^-: {\mathcal B}/{\mathcal I}\rightarrow {\mathcal B}/{\mathcal I}$ the cosyzygy functor. We will prove in Proposition \[8.15\] that any half exact functor $F$ which satisfies $F(\mathcal P)=0$ and $F(\mathcal I)=0$ has a similar property as cohomological functors on triangulated categories. In particular, we have the following corollary (see Corollary \[8.15\] for details).
For any short exact sequence $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ in ${\mathcal B}$, there exist morphisms $h: C\rightarrow \Omega^- A$ and $h': \Omega C\rightarrow A$ such that the sequence $$\begin{aligned}
\cdots \xrightarrow{H(\Omega h')} H(\Omega A) \xrightarrow{H(\Omega f)} H(\Omega B) \xrightarrow{H(\Omega g)} H(\Omega C) \xrightarrow{H(h')} H(A) \xrightarrow{H(f)} H(B)\\
\xrightarrow{H(g)} H(C) \xrightarrow{H(h)} H(\Omega^- A) \xrightarrow{H(\Omega^- f)} H(\Omega^- B) \xrightarrow{H(\Omega^- g)} H(\Omega^- C) \xrightarrow{H(\Omega^- h)} \cdots\end{aligned}$$ is exact in $\underline {\mathcal H}$.
The half exact functor we construct gives us a way to find out the relationship between different hearts. Let $k\in \{ 1,2 \}$, $({\mathcal U}_k,V_k)$ be a cotorsion pair on ${\mathcal B}$ and ${\mathcal W}_k= {\mathcal U}_k \cap {\mathcal V}_k$. Let ${\mathcal H}_k/{\mathcal W}_k$ be the heart of $({\mathcal U}_k,V_k)$ and $H_k$ be the associated half exact functor. If ${\mathcal W}_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, then $H_2$ induces a functor $\beta_{12}:{\mathcal H}_1/{\mathcal W}_1\rightarrow {\mathcal H}_2/{\mathcal W}_2$, and we have the following proposition (see Proposition \[61\], \[nameless\] and Theorem \[serre\] for details).
Let $({\mathcal U}_1,{\mathcal V}_1)$, $({\mathcal U}_2,{\mathcal V}_2)$ be cotorsion pairs on ${\mathcal B}$. If $W_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$, then
- We have a natural isomorphism $\beta_{21}\beta_{12}\simeq {\operatorname{id}\nolimits}_{{\mathcal H}_1/{\mathcal W}_1}$ of functors.
- $({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2$ is a Serre subcategory.
- Let $\overline {\mathcal H}_2$ be the localization of ${\mathcal H}_2/{\mathcal W}_2$ by $({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2$, then we have an equivalence ${\mathcal H}_1/{\mathcal W}_1 \simeq \overline {\mathcal H}_2 $.
This implies the following corollary which gives a sufficient condition when two different hearts( see Corollary \[suf\]).
If ${\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)={\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, then we have an equivalence ${\mathcal H}_1/{\mathcal W}_1\simeq {\mathcal H}_2/{\mathcal W}_2$ between two hearts.
To construct the associated half exact functor $H$, we first introduce two functors $\sigma^+:{\underline{{\mathcal B}}}\rightarrow {\underline{{\mathcal B}}}^+$ and $\sigma^-:{\underline{{\mathcal B}}}\rightarrow {\underline{{\mathcal B}}}^-$ in section 2, which are analogs of function functors associated with t-structures. In section 3, we show that these two functors commute. We prove the property of the half exact functor in section 4. The relationship between different hearts are studied in section 5. The last section contains several examples of our results.
Preliminaries
=============
For briefly review of the important properties of exact categories, we refer to [@L [§2]{}]. For more details, we refer to [@B]. We introduce the following properties used a lot in this paper, the proofs can be found in [@B [§2]{}]:
\[PO\] Consider a commutative square $$\xymatrix{
A \;\ar@{>->}[r]^{i} \ar[d]_f &B \ar[d]^{f'}\\
A' \;\ar@{>->}[r]_{i'} &B'
}$$ in which $i$ and $i'$ are inflations. The following conditions are equivalent:
- The square is a push-out.
- The sequence $\xymatrix{A \;\ar@{>->}[r]^-{{\left(\begin{smallmatrix}
i \\
-f
\end{smallmatrix}\right)}} &B\oplus A' \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
f' & i'
\end{smallmatrix}\right)}} &B'}$ is short exact.
- The square is both a push-out and a pull-back.
- The square is a part of a commutative diagram $$\xymatrix{
A\; \ar@{>->}[r]^i \ar[d]_f &B \ar[d]^{f'} \ar@{->>}[r] &C \ar@{=}[d]\\
A'\; \ar@{>->}[r]_{i'} &B' \ar@{->>}[r] &C
}$$ with short exact rows.
\[7\]
- If $\xymatrix{X \ar@{ >->}[r]^{i} &Y \ar@{->>}[r]^{d} &Z}$ and $\xymatrix{N \ar@{ >->}[r]^g &M\ar@{->>}[r]^{f} &Y}$ are two short exact sequences, then there is a commutative diagram of short exact sequences $$\xymatrix{
N \ar@{ >->}[d] \ar@{=}[r] &N \ar@{ >->}[d]^g\\
Q \ar@{->>}[d] \;\ar@{>->}[r] &M \ar@{->>}[d]^f \ar@{->>}[r] &Z \ar@{=}[d]\\
X \;\ar@{>->}[r]_i &Y \ar@{->>}[r]_d &Z
}$$ where the lower-left square is both a push-out and a pull-back.
- If $\xymatrix{X \ar@{ >->}[r]^{i} &Y \ar@{->>}[r]^{d} &Z}$ and $\xymatrix{Y \ar@{ >->}[r]^{g} &K \ar@{->>}[r]^f &L}$ are two short exact sequences, then there is a commutative diagram of short exact sequences $$\xymatrix{
X \ar@{=}[d] \ar@{ >->}[r]^{i} &Y \ar@{ >->}[d]^g \ar@{->>}[r]^{d} &Z \ar@{ >->}[d]\\
X \ar@{ >->}[r] &K \ar@{->>}[r] \ar@{->>}[d]^f &R \ar@{->>}[d]\\
&L \ar@{=}[r] &L
}$$ where the upper-right square is both a push-out and a pull-back.
We recall some important definitions and results of [@L], which also work for a single cotorsion pair.
\[re\] For any $B\in{\mathcal B}$, we define $B^+$ and $\alpha_B:B\rightarrow B^+$ as follows:\
Take two short exact sequences: $$\begin{aligned}
\xymatrix{V_B \ar@{ >->}[r] &U_B \ar@{->>}[r]^{u_B} &B},\quad
\xymatrix{U_B \ar@{ >->}[r]^{w'} &W^0 \ar@{->>}[r] &U^0}\end{aligned}$$ where $U_B,U^0 \in {\mathcal U}$, $W^0$,${\mathcal V}_B\in {\mathcal V}$. In fact, $W^0\in {\mathcal W}$ since ${\mathcal U}$ is closed under extension. By Proposition \[7\], we get the following commutative diagram $$\label{F1}
$$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\xymatrix{
V_B \ar@{=}[d] \ar@{ >->}[r] &U_B \ar@{ >->}[d]_{w'} \ar@{->>}[r]^{u_B} &B \ar@{ >->}[d]^{\alpha_B}\\
V_B \ar@{ >->}[r] &W^0 \ar@{->>}[r]_w \ar@{->>}[d] &B^+ \ar@{->>}[d]\\
&U^0 \ar@{=}[r] &U^0
}$$$$ where the upper-right square is both a push-out and a pull-back.
By definition, $B^+\in {\mathcal B}^+$. We recall the following useful proposition.
[@L [Lemma 3.2, ]{}[Proposition 3.3]{}]\[8\] For any $B\in {\mathcal B}$
- If $B\in {\mathcal B}^-$, then $B^+\in {\mathcal H}$.
- $\alpha_B$ is a left ${\mathcal B}^+$-approximation, and for an object $Y\in {\mathcal B}^+$, ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline {\alpha_B},Y):{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B^+,Y)\rightarrow {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B,Y)$ is bijective.
By Proposition \[8\], we can define a functor $\sigma^+$ from ${\underline{{\mathcal B}}}$ to $\underline {\mathcal B}^+$ as follows:\
For any object $B\in {\mathcal B}$, since all the ${B^+}'s$ are isomorphic to each other in ${\underline{{\mathcal B}}}$ by Proposition \[8\], we fix a $B^+$ for $B$. Let $$\begin{aligned}
\sigma^+:{\underline{{\mathcal B}}}\rightarrow \underline {\mathcal B}^+\\
B \mapsto B^+\end{aligned}$$ and for any morphism $\underline f:B\rightarrow C$, we define $\sigma^+(\underline f)$ as the unique morphism given by Proposition \[8\] $$\xymatrix{
B \ar[r]^{\underline f} \ar[d]_{\underline {\alpha_B}} &C \ar[d]^{\underline {\alpha_C}}\\
B^+ \ar@{.>}[r]_{\sigma^+(\underline f)} &C^+.
}$$
Let $i^+:{\underline{{\mathcal B}}}^+\hookrightarrow {\underline{{\mathcal B}}}$ be the inclusion functor, then $(\sigma^+,i^+)$ is an adjoint pair by Proposition \[8\].
\[8.2\] The functor $\sigma^+$ has the following properties:
- $\sigma^+$ is an additive functor.
- $\sigma^+|_{{\underline{{\mathcal B}}}^+}={\operatorname{id}\nolimits}_{{\underline{{\mathcal B}}}^+}$.
- For any morphism $f:A\rightarrow B$, $\sigma^+(\underline f)=0$ in ${\underline{{\mathcal B}}}$ if and only if $f$ factors through ${\mathcal U}$. In particular, $\sigma^+(B)=0$ if and only if $B\in \underline {\mathcal U}$.
(a), (b) can be concluded easily by definition, we only prove (c).\
The “if” part is followed by [@L [Lemma 3.4]{}].\
Now suppose $\sigma^+(\underline f)=0$ in ${\underline{{\mathcal B}}}$. By Proposition \[8\], we have the following commutative diagram $$\xymatrix{
A \ar@{ >->}[d]_{\alpha_A} \ar[r]^f &B \ar@{ >->}[d]^{\alpha_B} &U_B \ar@{ >->}[d]^{w'} \ar@{->>}[l]_{u_B} &V_B \ar@{=}[d] \ar@{ >->}[l]\\
A^+ \ar@{->>}[d] \ar[r]_{f^+} &B^+ \ar@{->>}[d] &W^0 \ar@{->>}[d] \ar@{->>}[l]^w &V_B \ar@{ >->}[l]\\
U^0_A \ar[r] &U^0 \ar@{=}[r] &U^0
}$$ where $\underline f^+=\sigma^+(\underline f)$. Then $f^+$ factors through an object $W\in {\mathcal W}$. $$\xymatrix{
A^+ \ar[dr]_a \ar[rr]^{f^+} &&B^+\\
&W \ar[ur]_b
}$$ Since $w$ is a right ${\mathcal U}$-approximation of $B^+$, there exists a morphism $c:W\rightarrow W^0$ such that $b=wc$. Thus $\alpha_Bf=f^+\alpha_A=ba\alpha_A=w(ca\alpha_A)$. By the definition of pull-back, there exists a morphism $d:A\rightarrow U_B$ such that $f=u_Bd$. Thus $f$ factors through ${\mathcal U}$.
\[10\] For any object $B\in {\mathcal B}$, we define $B^-$ and $\gamma_B:B^- \rightarrow B$ as follows:
Take the following two short exact sequences $$\begin{aligned}
\xymatrix{B \ar@{ >->}[r]^{v^B} &V^B \ar@{->>}[r] &U^B},\quad
\xymatrix{V_0 \ar@{ >->}[r] &W_0 \ar@{->>}[r] &V^B}\end{aligned}$$ where $V^B,V_0\in {\mathcal V}$, and $W_0$,$U^B\in {\mathcal U}$. Then $W_0\in {\mathcal W}$ holds since ${\mathcal V}$ is closed under extension. By Proposition \[7\], we get the following commutative diagram: $$\label{F2}
$$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\xymatrix{
V_0 \ar@{ >->}[d]_v \ar@{=}[r] &{V_0} \ar@{ >->}[d]\\
B^- \ar@{->>}[d]_{\gamma_B} \ar@{ >->}[r] &{W_0} \ar@{->>}[d] \ar@{->>}[r] &{U^B} \ar@{=}[d]\\
B \ar@{ >->}[r]_{v^B} &{V^B} \ar@{->>}[r] &{U^B}.}
$$$$
By definition $B^-\in {\mathcal B}^-$ and we have:
[@L [Proposition 3.6]{}]\[81\] For any object $B\in {\mathcal B}$
- $B\in {\mathcal B}^+$ implies $B^- \in {\mathcal H}$.
- $\gamma_B$ is a right ${\mathcal B}^-$-approximation. For any $X\in {\mathcal B}^-$, ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}( X,\underline{\gamma_B}):{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(X,B^-)\rightarrow {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(X,B)$ is bijective.
we define a functor $\sigma^-$ from ${\underline{{\mathcal B}}}$ to $\underline {\mathcal B}^-$ as the dual of $\sigma^+$: $$\begin{aligned}
\sigma^-:{\underline{{\mathcal B}}}\rightarrow \underline {\mathcal B}^-\\
B \mapsto B^-.\end{aligned}$$ For any morphism $\underline f:B\rightarrow C$, we define $\sigma^-(\underline f)$ as the unique morphism given by Proposition \[81\] $$\xymatrix{
B^- \ar@{.>}[r]^{\sigma^-(\underline f)} \ar[d]_{\underline {\gamma_B}} &C^- \ar[d]^{\underline {\gamma_C}}\\
B \ar[r]_{\underline f} &C.
}$$
Let $i^-:{\underline{{\mathcal B}}}^-\hookrightarrow {\underline{{\mathcal B}}}$ be the inclusion functor, then $(i^-,\sigma^-)$ is an adjoint pair by Proposition \[81\].
\[8.3\] The functor $\sigma^-$ has the following properties:
- $\sigma^-$ is an additive functor.
- $\sigma^-|_{{\underline{{\mathcal B}}}^-}={\operatorname{id}\nolimits}_{{\underline{{\mathcal B}}}^-}$.
- For any morphism $f:A\rightarrow B$, $\sigma^-(\underline f)=0$ in ${\underline{{\mathcal B}}}$ if and only if $f$ factors through ${\mathcal V}$. In particular, $\sigma^-(B)=0$ if and only if $B\in \underline {\mathcal V}$.
Reflection sequences and coreflection sequences
===============================================
In the following two sections we fix a cotorsion pair $({\mathcal U},{\mathcal V})$. The reflection (resp. coreflection) sequences [@AN] are defined on triangulated categories, but the definitions of the similar concepts on exact categories are not simple.
Let $\mathcal C$ be a subcategory of ${\mathcal B}$, denote by $\Omega \mathcal C$ (resp. $\Omega^- \mathcal C$) the subcategory of ${\mathcal B}$ consisting of objects $\Omega C$ (resp. $\Omega^- C$) such that there exists a short exact sequence $$\begin{aligned}
\Omega C\rightarrowtail P_C\twoheadrightarrow C \text{ } (P_C\in \mathcal P, C\in \mathcal C)\\
(\text{resp. } C\rightarrowtail I^C\twoheadrightarrow \Omega^- C \text{ } (I^C\in \mathcal I, C\in \mathcal C)).\end{aligned}$$
\[8.00\] $\Omega {\mathcal U}\subseteq {\mathcal B}^-$ and $\Omega^- {\mathcal V}\subseteq {\mathcal B}^+$.
We only prove the first one, the second is dual.\
An object $\Omega U\in \Omega {\mathcal U}$ admits two short exact sequences $$\xymatrix{
\Omega U \ar@{ >->}[r]^q &P_{U} \ar@{->>}[r] &U,\\
} \quad \xymatrix{
\Omega U \ar@{ >->}[r]^{v'} &V^{\Omega U} \ar@{->>}[r] &U^{\Omega U}
}$$ where $U,U^{\Omega U}\in {\mathcal U}$, $V^{\Omega U}\in {\mathcal V}$ and $P_U\in \mathcal P$. It is enough to show that $V^{\Omega U}\in {\mathcal U}$. Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U,V^{\Omega U})=0$, there exists a morphism $p:P_U\rightarrow V^{\Omega U}$ such that $pq=v'$. $$\xymatrix{
\Omega U \ar@{ >->}[r]^q \ar@{=}[d] &P_U \ar[d]^p \ar@{->>}[r] &U \ar[d]\\
\Omega U \ar@{ >->}[r]_{v'} &V^{\Omega U} \ar@{->>}[r] &U^{\Omega U}
}$$ Now we get a short exact sequence $\xymatrix{P_U \ar@{ >->}[r] &V^{\Omega U}\oplus U \ar@{->>}[r] &U^{\Omega U}}$. Since ${\mathcal U}$ is closed under extension and direct summands, $V^{\Omega U}\in {\mathcal U}$. Thus $\Omega U\in {\mathcal B}^-$.
\[8.4\] Let $B$ be any object in ${\mathcal B}$.
- A *reflection sequence* for $B$ is a short exact sequence $$\xymatrix{B \ar@{ >->}[r]^{z} &Z \ar@{->>}[r] &U}$$ where $U\in {\mathcal U}$, $Z\in {\mathcal B}^+$ and there exists a commutative diagram $$\xymatrix{
\Omega U \ar@{ >->}[r]^q \ar[d]_x &P_U \ar@{ ->>}[r] \ar[d]^p &U \ar@{=}[d]\\
B \ar@{ >->}[r]_z &Z \ar@{->>}[r] &U}$$ with $P_U\in \mathcal P$ and $x$ factoring through ${\mathcal U}$.
- A *coreflection sequence* for $B$ is a short exact sequence $$\xymatrix{V \ar@{ >->}[r] &K \ar@{->>}[r]^{k} &B}$$ where $V\in {\mathcal V}$, $K\in {\mathcal B}^-$ and there exists a commutative diagram $$\xymatrix{
V \ar@{ >->}[r] \ar@{=}[d] &K \ar@{->>}[r]^k \ar[d] &B \ar[d]^y\\
V \ar@{ >->}[r] &I^V \ar@{->>}[r] &\Omega^-V}$$ with $I^V\in \mathcal I$ and $y$ factoring through ${\mathcal V}$.
\[8.5\] Let $B$ be an object in ${\mathcal B}$. Then
- The short exact seqeunce $\xymatrix{B \ar@{ >->}[r]^{\alpha_B} &B^+ \ar@{->>}[r] &U^0}$ in (2.1) is a reflection sequence for $B$.
- The short exact seqeunce $\xymatrix{V_0 \ar@{ >->}[r] &B^- \ar@{->>}[r]^{\gamma_B} &B}$ in (2.2) is a coreflection sequence for $B$.
- For any reflection sequence $\xymatrix{B \ar@{ >->}[r]^{z} &Z \ar@{->>}[r] &U}$ for $B$, we have $Z\simeq B^+$ in ${\underline{{\mathcal B}}}$.
- For any coreflection sequence $\xymatrix{V \ar@{ >->}[r] &K \ar@{->>}[r]^{k} &B}$ for $B$, we have $K\simeq B^-$ in ${\underline{{\mathcal B}}}$.
We only prove (a) and (c), the other two are dual.\
(a) Since $U^0$ admits the following short exact sequence $$\xymatrix{\Omega U^0 \ar@{ >->}[r]^{q_0} &P_{U^0} \ar@{->>}[r] &U^0}$$ we get the following commutative diagram $$\xymatrix{
\Omega U^0 \ar@{ >->}[r]^{q_0} \ar[d]_{x_0} &P_{U^0} \ar@{->>}[r] \ar[d]^{p_0} &U^0 \ar@{=}[d]\\
B \ar@{ >->}[r]_{\alpha_B} &B^+ \ar@{->>}[r] &U^0.
}$$ Since $P_{U^0}$ is projective, there exists a morphism $p'_0:P_{U^0}\rightarrow W^0$ such that $wp'_0=p_0$, we get $\alpha_Bx_0=p_0q_0=wp'_0q_0$. Then $x_0$ factors through $U_B\in {\mathcal U}$ since (2.1) is a pull-back diagram. $$\xymatrix{
&\Omega U^0 \ar@{.>}[ddl] \ar@{ >->}[r]^{q_0} \ar[d]^{x_0} &P_{U^0} \ar[ddl]_(.25){p_0'} \ar@{ ->>}[r] \ar[d]^{p_0} &U^0 \ar@{=}[d]\\
&B \ar@{ >->}[r]^(.35){\alpha_B} &B^+ \ar@{->>}[r] &U^0\\
U_B \ar@{ >->}[r]_{w'} \ar[ur]_{u_B} &W^0 \ar[ur]_w \ar@{->>}[r] &U^0 \ar@{=}[ur]
}$$ Hence by definition $\xymatrix{B \ar@{ >->}[r]^{\alpha_B} &B^+ \ar@{->>}[r] &U^0}$ is a reflection sequence for $B$.\
(c) We first show that there exists a morphism $\underline f: Z\rightarrow B^+$ such that $\alpha_B=fz$.\
The reflection sequence admits a commutative diagram $$\xymatrix{
\Omega U \ar@{ >->}[r]^q \ar[d]_x &P_U \ar@{ ->>}[r] \ar[d]^p &U \ar@{=}[d]\\
B \ar@{ >->}[r]_z &Z \ar@{->>}[r] &U}$$ where the left square is a push-out by Proposition 2.1. Since $x$ factors through ${\mathcal U}$, and $u_B$ is a right ${\mathcal U}$-approximation of $B$, there exists a morphism $x':\Omega U\rightarrow U_B$ such that $x=u_Bx'$. Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U,W^0)=0$, there exists a morphism $p':P_U\rightarrow W^0$ such that $w'x'=p'q$, thus $\alpha_Bx=\alpha_Bu_Bx'=ww'x'=wp'q$. Then by the definition of push-out, there exists a morphism $f:Z\rightarrow B^+$ such that $\alpha_B=fz$. $$\xymatrix{
&\Omega U \ar[dddl]_{x'} \ar@{ >->}[r]^q \ar[d]^x &P_U \ar[dddl]^{p'} \ar@{ ->>}[r] \ar[d]^p &U \ar@{=}[d]\\
&B \ar@{ >->}[r]^z \ar@{=}[d] &Z \ar@{->>}[r]^a \ar@{.>}[d]^f &U\\
&B \ar@{ >->}[r]_{\alpha_B} &B^+ \ar@{->>}[r] &U^0\\
U_B \ar@{ >->}[r]_{w'} \ar[ur]_{u_B} &W^0 \ar[ur]_w \ar@{->>}[r] &U^0 \ar@{=}[ur]
}$$ Since By Proposition \[8\], there is a morphism $g:B^+\rightarrow Z$ such that $g\alpha_B=z$, we have a morphism $\underline {fg}:B^+\rightarrow B^+$ such that $\underline {fg\alpha}=\underline \alpha$, which implies that $\underline {fg}=\underline {\operatorname{id}\nolimits}_{B^+}$.\
Now we prove that $\underline {gf}=\underline {\operatorname{id}\nolimits}_Z$.\
Since $(gf-{\operatorname{id}\nolimits}_Z)z=0$, we get a morphism $b:U\rightarrow B^+$ such that $gf-{\operatorname{id}\nolimits}_Z=ba$. Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U,V_B)=0$, $b$ factors through $W^0$, hence $\underline {gf}=\underline {\operatorname{id}\nolimits}_Z$.\
Thus $B^+\simeq Z$ in ${\underline{{\mathcal B}}}$.
\[8.6\] There exists an isomorphism of functors from ${\underline{{\mathcal B}}}$ to $\underline {\mathcal H}$ $$\eta:\sigma^+ \circ \sigma^-\xrightarrow{\simeq}\sigma^- \circ \sigma^+.$$
By Proposition \[8\] and \[81\] both $\sigma^+ \circ \sigma^-$ and $\sigma^- \circ \sigma^+$ are functors from ${\underline{{\mathcal B}}}$ to $\underline {\mathcal H}$.\
By Lemma \[8.5\], We can take the following commutative diagram of short exact sequences $$\xymatrix{
V_0 \ar@{ >->}[r]^v \ar@{=}[d] &B^- \ar@{->>}[r]^{\gamma_B} \ar[d]_d &B \ar[d]^{y_0}\\
V_0 \ar@{ >->}[r]_j &I^0 \ar@{->>}[r]_i &\Omega^-V_0
}$$ where $y_0$ factors through $V^B$ since $v^B$ is a left ${\mathcal V}$-approximation of ${\mathcal B}$. $$\xymatrix{
B \ar[rr]^{y_0} \ar[dr]_{v^B} &&\Omega^-V_0\\
&V^B \ar[ur]_{v'}
}$$ By Lemma \[8.00\] and Proposition \[8\], there exists a morphism $t:B^+\rightarrow \Omega^-V_0$ such that $y_0=t\alpha_B$. Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U^0,V^B)=0$, there exists a morphism $v_0:B^+\rightarrow V^B$ such that $v^B=v_0\alpha_B$. Thus $t\alpha_B=v'v^B=v'v_0\alpha_B$, then we obtain that $t-v'v_0$ factors through $U^0$. $$\xymatrix{
B \ar@{ >->}[r]^{\alpha_B} \ar[d]_{v^B} &B^+ \ar[dl]^{v_0}_{\circlearrowright} \ar[d]^-{t-v'v_0} \ar@{->>}[rr] &&U^0 \ar@{.>}@/^/[dll]^u_{\circlearrowright}\\
V^B \ar[r]_{v'} &\Omega^-V_0
}$$ Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U^0,V_0)=0$, $u$ factors through $I^0\in {\mathcal V}$. Hence $t$ factors through ${\mathcal V}$.\
Take a pull-back of $t$ and $i$, we get the following commutative diagram $$\xymatrix{
V_0 \ar@{ >->}[r] \ar@{=}[d] &Q \ar@{}[dr]|{PB} \ar@{->>}[r]^s \ar[d]_{d'} &B^+ \ar[d]^t\\
V_0 \ar@{ >->}[r]_j &I^0 \ar@{->>}[r]_i &\Omega^-V_0.
}$$ By [@L [Lemma 2.11]{}], we obtain $Q\in {\mathcal B}^+$. Now by Proposition \[7\], we get the following commutative diagram $$\xymatrix{
V_0 \ar@{ >->}[d] \ar@{=}[r] &V_0 \ar@{ >->}[d]\\
Q' \ar@{ >->}[r] \ar@{->>}[d] &Q \ar@{->>}[r] \ar@{->>}[d]^s &U^0 \ar@{=}[d]\\
B \ar@{ >->}[r]_{\alpha_B} &B^+ \ar@{->>}[r] &U^0.
}$$ By the definition of pull-back, there exists a morphism $k:B\rightarrow Q$ such that $sk=\alpha_B\gamma_B$ and $d'k=d$. Hence we have the following diagram $$\xymatrix{
V_0 \ar@/_15pt/[dd]_{\text{id}_{V_0}} \ar@{.>}[d]^{v_0} \ar@{ >->}[r]^v &B^- \ar@{.>}[d]^k \ar@/_15pt/[dd]_(.75)d \ar@{->>}[r]^{\gamma_B} &B \ar[d]^{\alpha_B}\\
V_0 \ar@{ >->}[r] \ar@{=}[d] &Q \ar@{->>}[r]^s \ar@{}[dr]|{PB} \ar[d]^{d'} &B^+ \ar[d]^t\\
V_0 \ar@{ >->}[r]_j &I^0 \ar@{->>}[r]_i &\Omega^-V_0
}$$ where the upper-left square commutes. Hence $jv_0=d'kv=dv=j$, we can conclude that $v_0=\text{id}_{V_0}$ since $j$ is monomorphic. By the same method we can get the following commutative diagram $$\xymatrix{
V_0 \ar@/_15pt/[dd]_{\text{id}_{V_0}} \ar@{.>}[d]^{v_0'} \ar@{ >->}[r]^v &B^- \ar@{.>}[d]^{k'} \ar@/_15pt/[dd]_(.75)k \ar@{->>}[r]^{\gamma_B} &B \ar@{=}[d]\\
V_0 \ar@{ >->}[r] \ar@{=}[d] &Q' \ar@{->>}[r] \ar@{ >->}[d] &B \ar@{ >->}[d]^{\alpha_B}\\
V_0 \ar@{ >->}[r] &Q \ar@{->>}[r] &B^+
}$$ where $v_0'=\text{id}_{V_0}$. Therefore $k'$ is isomorphic by [@B [Corollary 3.2]{}]. We obtain the following commutative diagram $$\xymatrix{
V_0 \ar@{ >->}[d]_v \ar@{=}[r] &V_0 \ar@{ >->}[d]\\
B^- \ar@{ >->}[r]^k \ar@{->>}[d]_{\gamma_B} &Q \ar@{->>}[r] \ar@{->>}[d]^s &U^0 \ar@{=}[d]\\
B \ar@{ >->}[r]_{\alpha_B} &B^+ \ar@{->>}[r] &U^0.
}$$ We get $Q\in {\mathcal B}^-$ by [@L [Lemma 2.10]{}], hence $Q\in {\mathcal H}$. Since $t$ factors through ${\mathcal V}$, $\xymatrix{V_0 \ar@{ >->}[r] &Q \ar@{->>}[r]^{s} &B^+}$ is a coreflection sequence for $B^+$. By Lemma \[8.5\], we have the following commutative diagram $$\xymatrix{
&Q \ar[dr]^{\underline s} \ar@{.>}[dl]_-{\alpha'}\\
\sigma^-(B^+) \ar[rr]_\alpha &&B^+
}$$ in ${\underline{{\mathcal B}}}$ where $\alpha'$ is isomorphic. By duality we conclude that $\xymatrix{B^- \ar@{ >->}[r]^{k} &Q \ar@{->>}[r] &U^0}$ is a reflection sequence for $B^-$. By Lemma \[8.5\], we have the following commutative diagram $$\xymatrix{
B^- \ar[rr]^{\beta} \ar[dr]_{\underline k} &&\sigma^+(B^-) \ar@{.>}[dl]^{\beta'}\\
&Q
}$$ in ${\underline{{\mathcal B}}}$ where $\beta'$ is isomorphic. By Proposition \[81\], there exists a morphism $\theta:B^-\rightarrow \sigma^-\sigma^+(B)$ in ${\underline{{\mathcal B}}}$ such that $\alpha \theta=\underline {\alpha_B\gamma_B}$. Then by Proposition \[8\], there exists a unique morphism $\eta_B:\sigma^+\sigma^-(B) \rightarrow \sigma^-\sigma^+(B)$ such that $\eta_B\beta=\theta$. Hence we get the following commutative diagram $$\xymatrix{
\sigma^+\sigma^-(B) \ar@{.>}[dd]_{\eta_B} &B^- \ar[l]_-{\beta} \ar@{.>}[ddl]^\theta \ar[d]^{\underline {\gamma_B}}\\
&B \ar[d]^{\underline {\alpha_B}}\\
\sigma^-\sigma^+(B) \ar[r]_-{\alpha} &B^+.
}$$ Then $\alpha\eta_B\beta=\underline {\alpha_B\gamma_B}=\underline {sk}=\alpha\alpha'\beta'\beta$, and we have $\eta_B=\alpha'\beta'$ by Proposition \[8\] and \[81\]. Thus $\eta_B$ is isomorphic. Let $\underline f:B\rightarrow C$ be a morphism in ${\underline{{\mathcal B}}}$, then we can get the following diagram by Proposition \[8\] and \[81\]. $$\xymatrix{
\sigma^+\sigma^-(B) \ar@{}[ddr]|{\circlearrowright} \ar@/^15pt/[rrr]^{\sigma^+\sigma^-(\underline f)} \ar[dd]_{\eta_B} &B^- \ar@{}[dr]|{\circlearrowright} \ar[l]^-{\beta} \ar[d]_{\underline {\gamma_B}} \ar[r]^{\sigma^-(\underline f)} &C^- \ar@{}[ddr]|{\circlearrowright} \ar[r]_-{\gamma} \ar[d]^{\underline {\gamma_C}} &\sigma^+\sigma^-(C) \ar[dd]^{\eta_C}\\
&B \ar@{}[dr]|{\circlearrowright} \ar[d]_{\underline {\alpha_B}} \ar[r]^{\underline f} &C \ar[d]^{\underline {\alpha_C}}\\
\sigma^-\sigma^+(B) \ar@/_15pt/[rrr]_{\sigma^-\sigma^+(\underline f)} \ar[r]^-{\alpha} &B^+ \ar[r]_{\sigma^+(\underline f)} &C^+ &\sigma^-\sigma^+(C) \ar[l]_-{\delta}
}$$ Since $$\delta(\sigma^-\sigma^+(\underline f))\eta_B\beta=(\sigma^+(\underline f))\underline {\alpha_B\gamma_B}=\underline {\alpha_C\gamma_C}(\sigma^-(\underline f))=\delta\eta_C(\sigma^+\sigma^-(\underline f))\beta$$ we get $(\sigma^-\sigma^+(\underline f))\eta_B=\eta_C(\sigma^+\sigma^-(\underline f))$ by Proposition \[8\] and \[81\]. Thus $\eta$ is a natural isomorphism.
Half exact functor
==================
By Proposition \[8.6\], we have a natural isomorphism of functors from ${\mathcal B}$ to $\underline {\mathcal H}$ $$\sigma^+ \circ \sigma^- \circ \pi \simeq \sigma^- \circ \sigma^+ \circ \pi$$ where $\pi:{\mathcal B}\rightarrow {\underline{{\mathcal B}}}$ denotes the canonical functor. We denote $\sigma^- \circ \sigma^+ \circ \pi$ by $$H:{\mathcal B}\rightarrow \underline {\mathcal H}.$$ The aim of this section is to show the following theorem.
\[8.11\] For any cotorsion pair $({\mathcal U},{\mathcal V})$ in ${\mathcal B}$, the functor $$H:{\mathcal B}\rightarrow \underline {\mathcal H}$$ is half exact.
We call $H$ the associated *half exact* functor to $({\mathcal U},{\mathcal V})$.
\[8.7\] The functor $H$ has the following properties:
- $H$ is an additive functor.
- $H|_{{\mathcal H}}=\pi|_{{\mathcal H}}$.
- $H({\mathcal U})=0$ and $H({\mathcal V})=0$ hold. In particular, $H(\mathcal P)=0$ and $H(\mathcal I)=0$.
- For any reflection sequence $\xymatrix{B \ar@{ >->}[r]^{z} &Z \ar@{->>}[r] &U}$ for $B$, $H(z)$ is an isomorphism in $\underline {\mathcal H}$.
- For any coreflection sequence $\xymatrix{V \ar@{ >->}[r] &K \ar@{->>}[r]^{k} &B}$ for $B$, $H(k)$ is an isomorphism in $\underline {\mathcal H}$.
\(a) is followed by the definition of $H$ and Proposition \[8.2\], \[8.3\] directly. Since ${\mathcal H}={\mathcal B}^+\cap {\mathcal B}^-$, by Proposition \[8.2\], \[8.3\], we get (b). By Proposition \[8.2\], $\sigma^+({\underline{{\mathcal B}}}^+)=0$, hence $H({\mathcal U})$, $H(\mathcal P)=0$ since $\mathcal P\subseteq {\mathcal U}$, dually we have $H({\mathcal V})=0=H(\mathcal I)$. Hence (c) holds. For any reflection sequence, we have $H(z)=\sigma^-\circ\sigma^+(\underline z)=\sigma^-(\underline g)$ where $g:B^+\rightarrow Z$ is the morphism in the proof of Lemma \[8.5\]. Since $\underline g$ is an isomorphism, we get $H(z)$ is an isomorphism in $\underline {\mathcal H}$. Thus (d) holds and by dual, (e) also holds.
\[1\] Let $B$ be any object in ${\mathcal B}$, ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}({\mathcal U},{\mathcal B}^+)=0$ and ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B^-,{\mathcal V})=0$ hold.
We only show ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}({\mathcal U},{\mathcal B}^+)=0$, the other one is dual.\
Since $B\in {\mathcal B}^+$, it admits a short exact sequence $V_B \rightarrowtail W_B\twoheadrightarrow B$ where $W_B\in {\mathcal W}$. Then any morphism from an object in ${\mathcal U}$ to $B$ factors through $W_B$, and the assertion follows.
\[8.9\] Let $$\label{F3}
$$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\xymatrix{
\Omega U\ar@{ >->}[r]^q \ar[d]_f &P_U \ar@{->>}[r] \ar[d]^p &U \ar@{=}[d]\\
A \ar@{ >->}[r]_g &B \ar@{->>}[r]_h &U
}
$$$$ be a commutative diagram satisfying $U\in {\mathcal U}$ and $P_U\in \mathcal P$. Then the sequence $$H(\Omega U)\xrightarrow{H(f)} H(A)\xrightarrow{H(g)} H(B)\rightarrow 0$$ is exact in $\underline {\mathcal H}$.
By Proposition \[7\], we get a commutative diagram by taking a pull-back of $g$ and $\gamma_B$ $$\xymatrix{
V_0 \ar@{=}[r] \ar@{ >->}[d] &V_0 \ar@{ >->}[d]\\
L \ar@{ >->}[r]^{g'} \ar@{->>}[d]_l &B^- \ar@{->>}[d]^{\gamma_B} \ar@{->>}[r] &U \ar@{=}[d]\\
A \ar@{ >->}[r]_g &B \ar@{->>}[r]_h &U.
}$$ By [@L [Lemma 2.10]{}], $L\in {\mathcal B}^-$. We can obtain a commutative diagram of short exact sequences $$\xymatrix{
V_0 \ar@{=}[d] \ar@{ >->}[r] &L \ar[d] \ar@{->>}[r]^l &A \ar[d]^g\\
V_0 \ar@{=}[d] \ar@{ >->}[r] &B^- \ar[d] \ar@{->>}[r]^{\gamma_B} &B \ar[d]^j\\
V_0 \ar@{ >->}[r] &I^0 \ar@{->>}[r] &\Omega^-V_0
}$$ where $j$ factors through ${\mathcal V}$ by Lemma \[8.5\], hence $$\xymatrix{V_0 \ar@{ >->}[r] &L \ar@{->>}[r]^{l} &A}$$ is a coreflection sequence for $A$. By Proposition \[8.7\], $H(l)$ and $H(\gamma_B)$ are isomorphic in $\underline {\mathcal H}$. Thus, replacing $A$ by $L$ and $B$ by $B^-$, we may assume that $A,B\in {\mathcal B}^-$. Under this assumption, we show $H(g)$ is the cokernel of $H(f)$. We have $\Omega U\in {\mathcal B}^-$ by Lemma \[8.00\]. For any $Q\in {\mathcal H}$, we have a commutative diagram $$\xymatrix{
{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(H(B),Q) \ar[d]^{\simeq} \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(H(g),Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(H(A),Q) \ar[d]^{\simeq} \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(H(f),Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(H(\Omega U),Q) \ar[d]^{\simeq}\\
{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\sigma^+(B),Q) \ar[d]^{\simeq} \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\sigma^+(\underline g),Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\sigma^+(A),Q) \ar[d]^{\simeq} \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\sigma^+(\underline f),Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\sigma^+(\Omega U),Q) \ar[d]^{\simeq}\\
{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B,Q) \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline g,Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(A,Q) \ar[rr]^-{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline f,Q)} &&{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\Omega U,Q).
}$$ So it suffices to show the following sequence $$0\rightarrow {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B,Q)\xrightarrow{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline g,Q)} {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(A,Q) \xrightarrow{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline f,Q)} {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\Omega U,Q)$$ is exact.\
We first show that ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline g,Q)$ is injective. Let $r:B\rightarrow Q$ be any morphism such that $\underline {rg}=0$. Take a commutative diagram of short exact sequences $$\xymatrix{
\Omega U^A \ar@{ >->}[r]^{q_A} \ar[d]_a &P_{U^A} \ar[d]^{p_A} \ar@{->>}[r] &U^A \ar@{=}[d]\\
A \ar@{ >->}[r]_{w^A} &W^A \ar@{->>}[r] &U^A.
}$$ Since $rga$ factors through ${\mathcal W}$ and ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U^A,{\mathcal W})=0$, it factors through $q_A$. Thus there exists $c:W^A\rightarrow Q$ such that $cw^A=rg$. $$\xymatrix{
\Omega U^A \ar[r]^{q_A} \ar[d]_a &P_{U^A} \ar[d]_{p_A} \ar@/^/[ddr]\\
A \ar[r]^{w^A} \ar@/_/[drr]_{rg} &W^A \ar@{.>}[dr]^c\\
&&Q
}$$ As ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U,W^A)=0$, there exists $d:B\rightarrow W^A$ such that $w^A=dg$. Hence $rg=cw^A=cdg$, then $r-cd$ factors through $U$. $$\xymatrix{
A \ar[d]_{w^A} \ar@{ >->}[r]^g &B \ar@{->>}[rr] \ar[dl]^d_{\circlearrowright} \ar[d]^{r-cd} &&U \ar@{.>}@/^/[dll]_{\circlearrowright}\\
W^A \ar[r]_c &Q
}$$ Since ${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(U,Q)=0$ by Lemma \[1\], we get that $\underline r=0$.\
Assume $r':A\rightarrow Q$ satisfies $\underline {r'f}=0$, since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U,{\mathcal W})=0$, $r'f$ factors through $q$. As the left square of (3) is a push-out, we get the following commutative diagram. $$\xymatrix{
\Omega U \ar[r]^q \ar[d]_f &P_U \ar[d] \ar@/^/[ddr]\\
A \ar[r]^g \ar@/_/[drr]_{r'} &B \ar@{.>}[dr]\\
&&Q
}$$ Hence $r'$ factors through $g$. This shows the exactness of $${\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(B,Q)\xrightarrow{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline g,Q)} {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(A,Q) \xrightarrow{{\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\underline f,Q)} {\operatorname{Hom}\nolimits}_{{\underline{{\mathcal B}}}}(\Omega U,Q).$$
Dually, we have the following:
\[8.10\] Let $$\xymatrix{
V \ar@{ >->}[r]^f \ar@{=}[d] &A \ar@{->>}[r]^g \ar[d] &B \ar[d]^h\\
V \ar@{ >->}[r] &I^V \ar@{->>}[r] &\Omega^-V
}$$ be a commutative diagram satisfying $V\in {\mathcal V}$ and $I^V\in \mathcal I$. Then the sequence $$0\rightarrow H(A)\xrightarrow{H(g)} H(B)\xrightarrow{H(h)} H(\Omega^-V)$$ is exact in $\underline {\mathcal H}$.
Now we are ready to prove Theorem \[8.11\].
Let $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ be any short exact sequence in ${\mathcal B}$. By Proposition \[PO\], we can get the following commutative diagram: $$\xymatrix{
\Omega U^A \ar@{ >->}[r]^b \ar[d]_a &P_{U^A} \ar[d] \ar@{->>}[r] &U^A \ar@{=}[d]\\
A \ar@{ >->}[r]^{v^A} \ar@{}[dr]|{PO} \ar@{ >->}[d]_f &V^A \ar@{->>}[r] \ar@{ >->}[d]^e &U^A \ar@{=}[d]\\
B \ar@{ >->}[r]_c \ar@{->>}[d]_g &D \ar@{->>}[r] \ar@{->>}[d]^d &U^A\\
C \ar@{=}[r] &C.
}$$ From the first and second row from the top, we get an exact sequence $H(\Omega U^A) \xrightarrow{H(a)} H(A) \rightarrow 0$ by Lemma \[8.9\]. From the first and the third row from the top, we get an exact sequence $H(\Omega U^A) \xrightarrow{H(fa)} H(B) \xrightarrow{H(c)} H(D) \rightarrow 0$ by Lemma \[8.9\]. From the middle column, we get an exact sequence $0\rightarrow H(D)\xrightarrow{H(d)} H(C)$ by Lemma \[8.10\]. Now we can obtain an exact sequence $H(A)\xrightarrow{H(f)} H(B)\xrightarrow{H(g)} H(C)$.
Now we prove the following general observation on half exact functors.
\[8.15\] Let $\mathcal A$ be an abelian category and $F:{\mathcal B}\rightarrow \mathcal A$ be a half exact functor satisfying $F(\mathcal P)=0$ and $F(\mathcal I)=0$. Then for any short exact sequence $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ in ${\mathcal B}$, there exist morphisms $h: C\rightarrow \Omega^- A$ and $h': \Omega C\rightarrow A$ such that the following sequence $$\begin{aligned}
\cdots \xrightarrow{F(\Omega h')} F(\Omega A) \xrightarrow{F(\Omega f)} F(\Omega B) \xrightarrow{F(\Omega g)} F(\Omega C) \xrightarrow{F(h')} F(A) \xrightarrow{F(f)} F(B)\\
\xrightarrow{F(g)} F(C) \xrightarrow{F(h)} F(\Omega^- A) \xrightarrow{F(\Omega^- f)} F(\Omega^- B) \xrightarrow{F(\Omega^- g)} F(\Omega^- C) \xrightarrow{F(\Omega^- h)} \cdots\end{aligned}$$ is exact in $\mathcal A$.
Since $F(\mathcal P)=0$ (resp. $F(\mathcal I)=0$), the functor $F$ can be regarded as a functor from ${\mathcal B}/\mathcal P$ (resp. ${\mathcal B}/\mathcal I$) to $\mathcal A$.\
For convenience, we fix the following commutative diagram: $$\xymatrix{
\Omega A \ar@{ >->}[r]^{q_A} \ar[d]_{\Omega f} &P_A \ar@{->>}[r]^{p_A} \ar[d]^r &A \ar[d]^f\\
\Omega B \ar@{ >->}[r]^{q_B} \ar[d]_{\Omega g} &P_B \ar@{->>}[r]^{p_B} \ar[d]^k &B \ar[d]^g\\
\Omega C \ar@{ >->}[r]_{q_C} &P_C \ar@{->>}[r]_{p_C} &C.
}$$ Since $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ admits two commutative diagrams $$\xymatrix{\Omega C \ar@{ >->}[r]^{q_C} \ar[d]_{h'} &P_C \ar@{->>}[r]^{p_C} \ar[d]^{l} &C \ar@{=}[d]\\
A \ar@{ >->}[r]_{f} &B \ar@{->>}[r]_{g} &C,
} \quad \xymatrix{
A \ar@{ >->}[r]^{f} \ar@{=}[d] &B \ar@{->>}[r]^{g} \ar[d]^i &C \ar[d]^h\\
A \ar@{ >->}[r] &I^A \ar@{->>}[r]_{j} &\Omega^-A
}$$ we get two short exact sequences by Proposition \[PO\]: $$\xymatrix{
\Omega C \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
-q_C \\
h'
\end{smallmatrix}\right)}} &P_C\oplus A \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
l & f
\end{smallmatrix}\right)}} &B,\\
} \quad \xymatrix{
B \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
i \\
g
\end{smallmatrix}\right)}} &I^A\oplus C \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
-j & h
\end{smallmatrix}\right)}} &\Omega^-A.
}$$ They induce two exact sequences $$\xymatrix{
F(\Omega C) \ar[r]^{F(h')} &F(A) \ar[r]^{F(f)} &F(B),\\
} \quad \xymatrix{
F(B) \ar[r]^{F(g)} &F(C) \ar[r]^-{F(h)} &F(\Omega^-A).
}$$ by Theorem \[8.11\]. Now it is enough to show that
- $\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$ induces an exact sequence $$F(\Omega A) \xrightarrow{F(\Omega f)} F(\Omega B) \xrightarrow{F(\Omega g)} F(\Omega C)\xrightarrow{F(h')} F(A).$$
- $\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$ induces an exact sequence $$F(C) \xrightarrow{F(h)} F(\Omega^- A) \xrightarrow{F(\Omega^- f)} F(\Omega^- B) \xrightarrow{F(\Omega^- g)} F(\Omega^- C).$$
We only show the first one, the second is by dual.\
The short exact sequence $\xymatrix{\Omega C \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
-q_C \\
h'
\end{smallmatrix}\right)}} &P_C\oplus A \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
l & f
\end{smallmatrix}\right)}} &B}$ admits the following commutative diagram $$\xymatrix{\Omega B \ar@{ >->}[r]^{q_B} \ar[d]_x &P_B \ar@{->>}[r]^{p_B} \ar[d]^-{{\left(\begin{smallmatrix}
k' \\
m
\end{smallmatrix}\right)}} &B \ar@{=}[d]\\
\Omega C \ar@{ >->}[r]_-{{\left(\begin{smallmatrix}
-q_C \\
h'
\end{smallmatrix}\right)}} &P_C\oplus A \ar@{->>}[r]_-{{\left(\begin{smallmatrix}
l & f
\end{smallmatrix}\right)}} &B
}$$ which induces the following exact sequence $$\xymatrix{\Omega B \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
-q_B \\
x
\end{smallmatrix}\right)}} &P_A\oplus \Omega C \ar@{->>}[rr]^-{\left(\begin{smallmatrix}
k' &-q_C\\
m &h'
\end{smallmatrix}\right)} &&P_C\oplus A.}$$ We prove that $x+\Omega g$ factors through $\mathcal P$.\
Since $fm+lk'=p_B\Rightarrow gfm+glk'=gp_B\Rightarrow p_Ck'=p_Ck$, there exists a morphism $n:P_B\rightarrow \Omega C$ such that $k-k'=q_Cn$. Thus we have $q_Cnq_B=kq_B-k'q_B=q_C\Omega g+q_Cx$, which implies that $x+\Omega g=nq_B$.\
Hence we obtain an exact sequence $F(\Omega B) \xrightarrow{F(\Omega g)} F(\Omega C) \xrightarrow{F(h')} F(A).$\
Since we have the following commutative diagram $$\xymatrix{
\Omega A \ar@{ >->}[r]^{q_A} \ar[d]_{x'} &P_A \ar@{->>}[rr]^{p_A} \ar[d]^-{{\left(\begin{smallmatrix}
s \\
t
\end{smallmatrix}\right)}} &&A \ar[d]^-{{\left(\begin{smallmatrix}
0 \\
1
\end{smallmatrix}\right)}}\\
\Omega B \ar@{ >->}[r]_-{{\left(\begin{smallmatrix}
-q_B \\
x
\end{smallmatrix}\right)}} &P_B\oplus \Omega C \ar@{->>}[rr]_-{\left(\begin{smallmatrix}
k' &-q_C\\
m &h'
\end{smallmatrix}\right)} &&P_C\oplus A
}$$ we can show that $x'+\Omega f$ factors through $\mathcal P$ using the same method.\
Hence we get the following exact sequence $$F(\Omega A) \xrightarrow{F(\Omega f)} F(\Omega B) \xrightarrow{F(\Omega g)} F(\Omega C) \xrightarrow{F(h')} F(A).$$ Now we obtain a long exact sequence $$\begin{aligned}
\cdots \xrightarrow{F(\Omega h')} F(\Omega A) \xrightarrow{F(\Omega f)} F(\Omega B) \xrightarrow{F(\Omega g)} F(\Omega C) \xrightarrow{F(h')} F(A) \xrightarrow{F(f)} F(B)\\
\xrightarrow{F(g)} F(C) \xrightarrow{F(h)} F(\Omega^- A) \xrightarrow{F(\Omega^- f)} F(\Omega^- B) \xrightarrow{F(\Omega^- g)} F(\Omega^- C) \xrightarrow{F(\Omega^- h)} \cdots\end{aligned}$$ in $\underline {\mathcal H}$.
Since $H(\mathcal P)=H(\mathcal I)=0$, we can see from this proposition that $H$ has the property we claimed in the introduction.
For two subcategories ${\mathcal B}_1,{\mathcal B}_2\subseteq {\mathcal B}$, we denote ${\operatorname{add}\nolimits}({\mathcal B}_1*{\mathcal B}_2)$ by the subcategory which consists by the objects $X$ which admits a short exact sequence $$\xymatrix{B_1\ar@{ >->}[r] &X\oplus Y \ar@{->>}[r] &B_2}$$ where $B_1\in {\mathcal B}_1$ and $B_2\in {\mathcal B}_2$.
\[8.14\] For any cotorsion pair $({\mathcal U},{\mathcal V})$ on ${\mathcal B}$ and any object $B\in {\mathcal B}$, the following are equivalent.
- $H(B)=0$.
- $B\in {\operatorname{add}\nolimits}({\mathcal U}*{\mathcal V})$.
We first prove that (a) implies (b).\
By Proposition \[8.3\], since $H(B)=\sigma^-\circ\sigma^+(B)=0$, we get that $B^+\in{\mathcal V}$, hence from the following commutative diagram $$\xymatrix{
V_B \ar@{=}[d] \ar@{ >->}[r] &U_B \ar@{ >->}[d]_{w'} \ar@{->>}[r]^{u_B} &B \ar@{ >->}[d]^{\alpha_B}\\
V_B \ar@{ >->}[r] &W^0 \ar@{->>}[r]_w \ar@{->>}[d] &B^+ \ar@{->>}[d]\\
&U^0 \ar@{=}[r] &U^0
}$$ we get a short exact sequence $\xymatrix{U_B \ar@{ >->}[r] &B\oplus W^0 \ar@{->>}[r] &B^+}$, which implies that $B\in {\operatorname{add}\nolimits}({\mathcal U}*{\mathcal V})$.\
We show that (b) implies (a).\
This is followed by Theorem \[8.11\] and Proposition \[8.7\].
The kernel of $H$ becomes simple in the following cases.
\[8.16\] Let $({\mathcal U},{\mathcal V})$ be a cotorsion pair on ${\mathcal B}$, then
- If ${\mathcal U}\subseteq {\mathcal V}$, then $H(B)=0$ if and only if $B\in {\mathcal V}$.
- If ${\mathcal V}\subseteq {\mathcal U}$, then $H(B)=0$ if and only if $B\in {\mathcal U}$.
This is followed by Proposition \[8.14\] directly.
Relationship between different hearts
=====================================
The half exact functor constructed in the previous section gives a useful to study the relationship between the hearts of different cotorsion pairs on ${\mathcal B}$. First, we start with fixing some notations
Let $i\in \{ 1,2\}$. Let $({\mathcal U}_i,{\mathcal V}_i)$ be a cotorsion pair on ${\mathcal B}$ and ${\mathcal W}_i={\mathcal U}_i \cap {\mathcal V}_i$. Let ${\mathcal B}^+_i$ and ${\mathcal B}^-_i$ be the subcategories of $B$ defined in (1.1) and (1.2). Let ${\mathcal H}_i:={\mathcal B}^+_i\cap{\mathcal B}^-_i$, then ${\mathcal H}_i/{\mathcal W}_i$ is the heart of $({\mathcal U}_i,{\mathcal V}_i)$. Let $\pi_i:{\mathcal B}\rightarrow {\mathcal B}/{\mathcal W}_i$ be the canonical functor and $\iota_i: {\mathcal H}_i/{\mathcal W}_i \hookrightarrow {\mathcal B}/{\mathcal W}_i$ be the inclusion functor.
If $H_2({\mathcal W}_1)=0$, which means ${\mathcal W}_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$ by Proposition \[8.14\], then there exists a functor $h_{12}:{\mathcal B}/{\mathcal W}_1\rightarrow {\mathcal H}_2/{\mathcal W}_2$ such that $H_2=h_{12}\pi_1$. $$\xymatrix{
{\mathcal B}\ar[dr]_{H_2} \ar[rr]^{\pi_1} &&{\mathcal B}/{\mathcal W}_1 \ar@{.>}[dl]^{h_{12}}\\
&{\mathcal H}_2/{\mathcal W}_2
}$$ Hence we get a functor $\beta_{12}:=h_{12}\iota_1:{\mathcal H}_1/{\mathcal W}_1\rightarrow {\mathcal H}_2/{\mathcal W}_2$.
\[equi\] The following conditions are equivalent to each other.
- $H_1({\mathcal U}_2)=H_1({\mathcal V}_2)=0$.
- ${\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$.
By Proposition \[8.7\] and Theorem \[8.11\], (b) implies (a). Now we prove that (a) implies (b).\
By Proposition \[8.14\], we get ${\mathcal U}_2\subseteq {\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$ and ${\mathcal V}_2\subseteq {\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$. Let $X\in {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, then by definition, it admits a short exact sequence $$U_2\rightarrowtail X\oplus Y \twoheadrightarrow V_2$$ where $U_2\in {\mathcal U}_2$ and $V_2 \in {\mathcal V}_2$. Since $U_2,V_2\in {\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$, by definition, there exist two objects $A$ and $B$ such that $U_2\oplus A, V_2\oplus B\in {\mathcal U}_1*{\mathcal V}_1$. Thus we get a short exact sequence $$U_2\oplus A\rightarrowtail X\oplus Y\oplus A \oplus B \twoheadrightarrow V_2\oplus B.$$ Hence $X\in {\operatorname{add}\nolimits}(({\mathcal U}_1*{\mathcal V}_1)*({\mathcal U}_1*{\mathcal V}_1))={\operatorname{add}\nolimits}({\mathcal U}_1*({\mathcal V}_1*{\mathcal U}_1)*{\mathcal V}_1)={\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal U}_1*{\mathcal V}_1*{\mathcal V}_1)={\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$, which implies that ${\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$.
\[nameless\] The functor $\beta_{12}$ is half exact. Moreover, if ${\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, then $\beta_{12}$ is exact and $({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$ is a Serre subcategory of ${\mathcal H}_1/{\mathcal W}_1$.
Let $0\rightarrow A \xrightarrow{\rho} B \xrightarrow{\mu} C\rightarrow 0$ be a short exact sequence in ${\mathcal H}_1/{\mathcal W}_1$, then $\mu$ admits a morphism $g:B\twoheadrightarrow C$ such that $\pi_1(g)=\beta$. We get the following commutative diagram $$\xymatrix{
V_C \ar@{ >->}[r] \ar@{=}[d] &{K_g} \ar@{->>}[r]^{k_g} \ar[d]^a &B \ar[d]^g\\
V_C \ar@{ >->}[r] &{W_C} \ar@{->>}[r]_{w_C} &C}$$ where $V_C\in {\mathcal V}_1$ and $W_C\in {\mathcal W}_1$. Then we obtain a short exact sequence $$\xymatrix{K_g \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
-a \\
k_g
\end{smallmatrix}\right)}} &B\oplus W_C \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
g & w_C
\end{smallmatrix}\right)}} &C.}$$ By [@L [Lemma 4.1]{}], $K_g\in {\mathcal B}^-_j$. By [@L [Definition 3.8]{}], $K_g\in {\mathcal B}^+_1$. Hence $K_g\in {\mathcal H}_1$. By [@L [Theorem 4.3]{}], $\mu$ is the cokernel of $\pi_1(k_g)$. By dual of [@L [Theorem 3.10]{}], $\pi_1(k_g)$ is the kernel of $\mu$. Hence $K_g\simeq A$ in ${\mathcal H}_1/W_1$. By Theorem \[8.11\], We get the an exact sequence $$H_2(K_g)\xrightarrow{H_2(k_g)} H_2(B) \xrightarrow{H_2(g)} H_2(C)$$ which implies the following following exact sequence $$\beta_{12}(A) \xrightarrow{\beta_{12}(\rho)} \beta_{12}(B) \xrightarrow{\beta_{12}(\mu)} \beta_{12}(C).$$ Hence $\beta_{12}$ is half exact. Now we prove that if ${\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, which means $H_2({\mathcal U}_1)=0=H_2({\mathcal V}_1)$, then $\beta_{12}$ is exact.\
In this case, we only need to show that $\beta_{12}(\rho)$ is a monomorphism and $\beta_{12}(\mu)$ is an epimorphism. We show that $\beta_{12}(\mu)$ is an epimorphism, the other part is by dual.\
Since we have the following commutative diagram $$\xymatrix{
B \ar@{ >->}[r]^{w^B} \ar[d]_{g} &{W^B} \ar@{->>}[r] \ar[d]^b &{U^B} \ar@{=}[d]\\
C \ar@{ >->}[r]_{c_g} &{C_g} \ar@{->>}[r]_s &{U^B}}$$ where $W_B\in {\mathcal W}_1$ and $U^B\in {\mathcal U}_1$. Since $\mu$ is epimorphism, by [@L [Corollary 3.11]{}], $C_g\in {\mathcal U}_1$. Since we have the following short exact sequence $$\xymatrix{B \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
g \\
-h
\end{smallmatrix}\right)}} &C\oplus W^B \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
c_g & b
\end{smallmatrix}\right)}} &C_g.}$$ By Theorem \[8.11\], We have an exact sequence $H_2(B) \xrightarrow{H_2(g)} H_2(C) \rightarrow 0$, which induces the following exact sequence $$\beta_{12}(B) \xrightarrow{\beta_{12}(\mu)} \beta_{12}(C)\rightarrow 0.$$ Now we prove that $({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$ is a Serre subcategory of ${\mathcal H}_1/{\mathcal W}_1$.\
Let $0\rightarrow A \xrightarrow{\rho} B \xrightarrow{\mu} C\rightarrow 0$ be a short exact sequence in ${\mathcal H}_1/{\mathcal W}_1$.\
If $B\in ({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$, since $\beta_{12}$ is exact and $\beta_{12}(B)=0$ by Proposition \[8.14\], we have $\beta_{12}(A)=0=\beta_{12}(C)$, which implies that $A,C\in ({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$.\
If $A,C\in ({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$, since we have the following short exact sequence $$\xymatrix{K_g \ar@{ >->}[r]^-{{\left(\begin{smallmatrix}
-a \\
k_g
\end{smallmatrix}\right)}} &B\oplus W_C \ar@{->>}[r]^-{{\left(\begin{smallmatrix}
g & w_C
\end{smallmatrix}\right)}} &C.}$$ in ${\mathcal B}$ such that $K_g\simeq A$ in ${\mathcal H}_1/W_1$, we get that ${\mathcal B}\in{\operatorname{add}\nolimits}(({\mathcal U}_1*{\mathcal V}_1)*({\mathcal U}_1*{\mathcal V}_1))={\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$. Hence $B\in ({\mathcal H}_1\cap{\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2))/{\mathcal W}_1$.
We prove the following proposition, and we recall that a similar property has been proved for triangulated case in [@ZZ Lemma 6.3].
\[61\] Let $({\mathcal U}_1,{\mathcal V}_1)$, $({\mathcal U}_2,{\mathcal V}_2)$ be cotorsion pairs on ${\mathcal B}$. If ${\mathcal W}_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$, then we have a natural isomorphism $\beta_{21}\beta_{12}\simeq {\operatorname{id}\nolimits}_{{\mathcal H}_1/{\mathcal W}_1}$ of functors.
Let $B\in {\mathcal H}_1$. By Definition \[re\] and \[10\], we get the following commutative diagrams $$\xymatrix{
V_B \ar@{=}[d] \ar@{ >->}[r] &U_B \ar@{ >->}[d] \ar@{->>}[r] &B \ar@{ >->}[d]^{s_B}\\
V_B \ar@{ >->}[r] &W^0 \ar@{->>}[r] \ar@{->>}[d] &B^+_2 \ar@{->>}[d]\\
&U^0 \ar@{=}[r] &U^0,\\
} \quad
\xymatrix{
V_0 \ar@{ >->}[d] \ar@{=}[r] &{V_0} \ar@{ >->}[d]\\
(B^+_2)^-_2 \ar@{->>}[d]_{t_B} \ar@{ >->}[r] &{W_0} \ar@{->>}[d] \ar@{->>}[r] &{U^{B_2}} \ar@{=}[d]\\
B^+_2 \ar@{ >->}[r] &{V^{B_2}} \ar@{->>}[r] &{U^{B_2}}}$$ where $U_B,U^0,U^{B_2}\in {\mathcal U}_2$, $V_B,V^{B_2},V_0\in {\mathcal V}_2$, $W_0,W^0\in {\mathcal W}_2$ and $(B^+_2)^-_2=H_2(B)$ in ${\mathcal B}/{\mathcal W}_2$. By Lemma \[equi\], we get $H_1({\mathcal U}_2)=H_1({\mathcal V}_2)=0$, by Lemma \[8.5\] and Theorem \[8.11\], we get two isomorphisms $B\xrightarrow{H_1(s_B)} H_1(B^+_2)$ and $H_1((B^+_2)^-_2)\xrightarrow{H_1(t_B)} H_1(B^+_2)$ in ${\mathcal H}_1/{\mathcal W}_1$. Since $H_1((B^+_2)^-_2)=\beta_{21}\beta_{12}(B)$, we get a isomorphism $\rho_B:=H_1(t_B)^{-1}H_1(s_B):B\rightarrow \beta_{21}\beta_{12}(B)$ on ${\mathcal H}_1/{\mathcal W}_1$. Let $f:B\rightarrow C$ be a morphism in ${\mathcal H}_1$, we also denote it image in ${\mathcal H}_1/{\mathcal W}_1$ by $f$. By the definition of $H_2$, we get the following commutative diagrams in ${\mathcal B}$ $$\xymatrix{
B \ar[r]^{s_B} \ar[d]_f &B^+_2 \ar[d]^{f^+}\\
C \ar[r]_{s_C} &C^+_2,\\
} \quad
\xymatrix{
(B^+_2)^-_2 \ar[r]^{t_B} \ar[d]_{(f^+)^-} &B^+_2 \ar[d]^{f^+}\\
(C^+_2)^-_2 \ar[r]_{t_C} &C^+_2}$$ where $\pi_2((f^+)^-)=H_2(f)$. Hence we obtain the following commutative diagram in ${\mathcal H}_1/{\mathcal W}_1$ $$\xymatrix{
B \ar[r]^-{\rho_B} \ar[d]_f &\beta_{21}\beta_{12}(B) \ar[d]^{\beta_{21}\beta_{12}(f)}\\
C \ar[r]_-{\rho_C} &\beta_{21}\beta_{12}(C)
}$$ which implies that $\beta_{21}\beta_{12}\simeq {\operatorname{id}\nolimits}_{{\mathcal H}_1/{\mathcal W}_1}$.
According to Proposition \[61\], we obtain the following corollary immediately.
\[suf\] If ${\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)={\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)$, then we have an equivalence ${\mathcal H}_1/{\mathcal W}_1\simeq {\mathcal H}_2/{\mathcal W}_2$ between two hearts.
Let $S=\{ \alpha \in {\operatorname{Mor}\nolimits}({\mathcal H}_2/{\mathcal W}_2) \text{ }|\text{ } {\operatorname{Ker}\nolimits}(\alpha), {\operatorname{Coker}\nolimits}(\alpha) \in ({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2\}$ and let $\overline {\mathcal H}_2$ be localization of ${\mathcal H}_2/{\mathcal W}_2$ respect to $({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2$, then $\overline {\mathcal H}_2$ is abelian. Since $\beta_{21}$ is exact and ${\operatorname{Ker}\nolimits}(\beta_{21})=({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2$, we get the following commutative diagram $$\xymatrix{
{\mathcal H}_2/{\mathcal W}_2 \ar[dr]_{L} \ar[rr]^{\beta_{21}} &&{\mathcal H}_1/{\mathcal W}_1\\
&\overline {\mathcal H}_2 \ar@{.>}[ur]_-{\overline {\beta_{21}}}
}$$ where $L$ is the localization functor which is exact and $\overline {\beta_{21}}$ is a faithful exact functor. Since $\overline {\beta_{21}}L\beta_{12}\simeq {\operatorname{id}\nolimits}_{{\mathcal H}_1/{\mathcal W}_1}$, we get that $L\beta_{12}$ is fully-faithful. Now we prove that $L\beta_{12}$ is dense under the assumption of Proposition \[61\].\
Let $B\in {\mathcal H}_2$, by Definition \[re\] and \[10\], we get the following commutative diagrams $$\xymatrix{
V_B \ar@{=}[d] \ar@{ >->}[r] &U_B \ar@{ >->}[d] \ar@{->>}[r] &B \ar@{ >->}[d]^{s_B}\\
V_B \ar@{ >->}[r] &W^0 \ar@{->>}[r] \ar@{->>}[d] &B^+_1 \ar@{->>}[d]\\
&U^0 \ar@{=}[r] &U^0,\\
} \quad
\xymatrix{
V_0 \ar@{ >->}[d] \ar@{=}[r] &{V_0} \ar@{ >->}[d]\\
(B^+_1)^-_1 \ar@{->>}[d]_{t_B} \ar@{ >->}[r] &{W_0} \ar@{->>}[d] \ar@{->>}[r] &{U^{B_1}} \ar@{=}[d]\\
B^+_1 \ar@{ >->}[r] &{V^{B_1}} \ar@{->>}[r] &{U^{B_1}}}$$ where $U_B,U^0,U^{B_1}\in {\mathcal U}_1$, $V_B,V^{B_1},V_0\in {\mathcal V}_1$, $W_0,W^0\in {\mathcal W}_1$ and $(B^+_1)^-_1=H_1(B)$ in ${\mathcal B}/{\mathcal W}_1$. Since $H_2(W_1)=0$, we get the following exact sequences by Theorem \[8.11\] $$H_2(U_B)\rightarrow B \rightarrow H_2(B^+_1) \rightarrow H_2(U^0),$$ $$H_2(V_0)\rightarrow H_2((B^+_1)^-_1) \rightarrow H_2(B^+_1) \rightarrow H_2(V^{B_1}).$$ One can check that $H_2({\mathcal U}_1),H_2({\mathcal V}_1)\subseteq ({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2$ by definition. Since $\overline {\mathcal H}_2$ is abelian and $L$ is exact, we get $B \simeq H_2(B^+_1) \simeq H_2((B^+_1)^-_1)=L\beta_{12}\beta_{21}(B)$ in $\overline {\mathcal H}_2$, which implies that $L\beta_{12}$ is dense.
Now we get the following theorem.
\[serre\] Let $({\mathcal U}_1,{\mathcal V}_1)$, $({\mathcal U}_2,{\mathcal V}_2)$ be cotorsion pairs on ${\mathcal B}$. If ${\mathcal W}_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$, then we have an equivalence $L\beta_{12}: {\mathcal H}_1/{\mathcal W}_1 \rightarrow \overline {\mathcal H}_2 $.
In the rest of this section, we discuss about the relationship between the heart of a twin cotorsion pair and the hearts of its two components.
First we recall the definition of the twin cotorsion pair. A pair of cotorsion pairs $({\mathcal U}_1,{\mathcal V}_1)$, $({\mathcal U}_2,{\mathcal V}_2)$ is called a twin cotorsion pair if ${\mathcal U}_1\subseteq {\mathcal U}_2$. This condition is equivalent to ${\mathcal V}_2\subseteq {\mathcal V}_1$ and also equivalent to ${\operatorname{Ext}\nolimits}^1_{\mathcal B}({\mathcal U}_1,{\mathcal V}_2)=0$. We introduce some notations.
Let ${\mathcal W}_t:={\mathcal V}_1\cap {\mathcal U}_2$.
- ${\mathcal B}^+_t$ is defined to be the full subcategory of ${\mathcal B}$, consisting of objects $B$ which admits a short exact sequence $$V_B\rightarrowtail U_B\twoheadrightarrow B$$ where $U_B\in {\mathcal W}_t$ and $V_B\in {\mathcal V}_2$.
- ${\mathcal B}^-_t$ is defined to be the full subcategory of ${\mathcal B}$, consisting of objects $B$ which admits a short exact sequence $$B\rightarrowtail V^B\twoheadrightarrow U^B$$ where $V^B\in {\mathcal W}_t$ and $U^B\in {\mathcal U}_1$.
Denote $${\mathcal H}_t:={\mathcal B}^+_t\cap{\mathcal B}^-_t.$$ Then ${\mathcal H}_t/{\mathcal W}_t$ is called the *heart* of $({\mathcal U}_1,{\mathcal V}_1),({\mathcal U}_2,{\mathcal V}_2)$.
\[63\] Let $({\mathcal U}_1,{\mathcal V}_1),({\mathcal U}_2,{\mathcal V}_2)$ be a twin cotorsion pair on ${\mathcal B}$ and $f:A\rightarrow B$ be a morphism in ${\mathcal H}_t$, then $H_k(f)=0$ $(k=1 \text{ or } 2)$ if and only if $f$ factors through ${\mathcal W}_t$.
We only prove the case $k=2$, the other case is by dual.\
The “if” is followed directly by Proposition \[8.2\]. Now we prove the “only if” part.\
Since $H_k(f)=0$, by Proposition \[8.3\] and \[8.6\], we get in the following commutative diagram $$\xymatrix{
A \ar@{ >->}[d]_{\alpha_A} \ar[r]^f &B \ar@{ >->}[d]^{\alpha_B} &U_B \ar@{ >->}[d]^{w'} \ar@{->>}[l]_{u_B} &V_B \ar@{=}[d] \ar@{ >->}[l]\\
A^+ \ar@{->>}[d] \ar[r]_{f^+} &B^+ \ar@{->>}[d] &W^0 \ar@{->>}[d] \ar@{->>}[l]^w &V_B \ar@{ >->}[l]\\
U^0_A \ar[r] &U^0 \ar@{=}[r] &U^0
}$$ which is similar as in Proposition \[8.2\], where $U^0_A,U^0\in {\mathcal U}_2$, $V_B\in {\mathcal V}_2$, $U_B\in {\mathcal W}_t$ and $W^0\in {\mathcal W}_2$, $f^+$ factors through an object $V\in {\mathcal V}_2$. Since $A,B\in {\mathcal H}_t$, by [@L Lemma 2.10], $A^+,B^+\in {\mathcal B}^-_t$. Hence there exits a diagram $$\xymatrix{
A^+ \ar[dr]^a \ar[dd]_{f^+} \ar@{ >->}[rr]^-{w^A} &&W^A \ar@{->>}[r] &U^A\\
&V \ar[dl]^b\\
B^+ \ar@{ >->}[rr] &&W^B \ar@{->>}[r] &U^B
}$$ where $W^A,W^B\in {\mathcal W}_t$ and $U^A,U^B\in {\mathcal U}_1$. Since ${\operatorname{Ext}\nolimits}^1_{\mathcal B}(U^A,V)=0$, there exists a morphism $c:W^A\rightarrow V$ such that $f^+=bcw^A$. Now using the same argument as in Proposition \[8.2\], we get that $f$ factors through $U_B\in {\mathcal W}_t$.
Let $\pi_t:{\mathcal B}\rightarrow {\mathcal B}/{\mathcal W}_t$ be the canonical functor and $\iota_t: {\mathcal H}_t/{\mathcal W}_t \hookrightarrow {\mathcal B}/{\mathcal W}_t$ be the inclusion functor.
Let $k \in \{ 1,2 \}$, since $H_k(W_t)=0$ by Proposition \[8.7\], there exists a functor $h_k:{\mathcal B}/{\mathcal W}_t\rightarrow {\mathcal H}_k/{\mathcal W}_k$ such that $H_k=h_k\pi_t$. $$\xymatrix{
{\mathcal B}\ar[dr]_{H_k} \ar[rr]^{\pi_t} &&{\mathcal B}/{\mathcal W}_t \ar@{.>}[dl]^{h_k}\\
&{\mathcal H}_k/{\mathcal W}_k
}$$ Hence we get a functor $\beta_k:=h_k\iota_t:{\mathcal H}_t/{\mathcal W}_t\rightarrow {\mathcal H}_k/{\mathcal W}_k$ and the following corollary.
Let $({\mathcal U}_1,{\mathcal V}_1),({\mathcal U}_2,{\mathcal V}_2)$ be a twin cotorsion pair on ${\mathcal B}$, then $\beta_k:{\mathcal H}_t/{\mathcal W}_t\rightarrow {\mathcal H}_k/{\mathcal W}_k$ $(k \in \{ 1,2 \})$ is faithful.
This corollary also implies that if ${\mathcal H}_1/{\mathcal W}_1=0$ or ${\mathcal H}_2/{\mathcal W}_2=0$, ${\mathcal H}_t/{\mathcal W}_t$ is also zero.
Moreover, we have the following proposition.
Let $({\mathcal U}_1,{\mathcal V}_1),({\mathcal U}_2,{\mathcal V}_2)$ be a twin cotorsion pair on ${\mathcal B}$. If ${\mathcal H}_t/{\mathcal W}_t=0$, then ${\mathcal H}_1\subseteq {\mathcal U}_2$ and ${\mathcal H}_2\subseteq {\mathcal V}_1$.
We only prove that ${\mathcal H}_t/{\mathcal W}_t=0$ implies ${\mathcal H}_1\subseteq {\mathcal U}_2$, the other one is by dual.\
Let $B\in {\mathcal H}_1$, since $B^-_1\subseteq B^-_t$ by definition, in the following diagram $$\xymatrix{
V_B \ar@{=}[d] \ar@{ >->}[r] &U_B \ar@{ >->}[d] \ar@{->>}[r] &B \ar@{ >->}[d]\\
V_B \ar@{ >->}[r] &W^0 \ar@{->>}[r] \ar@{->>}[d] &B^+ \ar@{->>}[d]\\
&U^0 \ar@{=}[r] &U^0
}$$ where $U_B\in {\mathcal U}_2$, $V_B\in {\mathcal V}_2$, $U^0\in {\mathcal U}_1$ and $W^0\in {\mathcal W}_t$, we get $B^+\in {\mathcal H}_t$ by [@L Lemma 2.10]. If ${\mathcal H}_t/{\mathcal W}_t=0$, then $B^+\in {\mathcal W}_t$. By [@L [Lemma 3.4]{}], $B\in {\mathcal U}_2$.
Examples
========
Let $\Lambda$ be the $k$-algebra given by the quiver $$\xymatrix{
1 \ar@/^10pt/[r]^a &2 \ar@/^10pt/[l]^{a^*} \ar@/^10pt/[r]^b &3 \ar@/^10pt/[l]^{b^*}}$$ and bounded by the relations $a^*a=0=bb^*$, $aa^*=b^*b$. The AR-quiver of ${\mathcal B}={\operatorname{mod}\nolimits}\Lambda$ is given by $$\xymatrix@[email protected]{
&&&{\begin{smallmatrix}
1\\
2\\
3
\end{smallmatrix}} \ar[dr]\\
{\begin{smallmatrix}
1
\end{smallmatrix}} \ar[dr]
&&{\begin{smallmatrix}
2\\
3
\end{smallmatrix}} \ar[ur] \ar[dr]
&&{\begin{smallmatrix}
1\\
2
\end{smallmatrix}} \ar[dr] &&{\begin{smallmatrix}
3
\end{smallmatrix}}\\
{\begin{smallmatrix}
&2&\\
1&&3\\
&2&
\end{smallmatrix}} \ar[r]
&{\begin{smallmatrix}
&2&\\
1&&3
\end{smallmatrix}} \ar[dr] \ar[ur]
&&{\begin{smallmatrix}
2
\end{smallmatrix}} \ar[dr] \ar[ur]
&&{\begin{smallmatrix}
1&&3\\
&2&
\end{smallmatrix}} \ar[dr] \ar[ur] \ar[r] &{\begin{smallmatrix}
&2&\\
1&&3\\
&2&
\end{smallmatrix}}\\
{\begin{smallmatrix}
3
\end{smallmatrix}} \ar[ur]
&&{\begin{smallmatrix}
2\\
1
\end{smallmatrix}} \ar[ur] \ar[dr]
&&{\begin{smallmatrix}
3\\
2
\end{smallmatrix}} \ar[ur] &&{\begin{smallmatrix}
1
\end{smallmatrix}.}\\
&&&{\begin{smallmatrix}
3\\
2\\
1
\end{smallmatrix}} \ar[ur]
}$$ We denote by “$\circ$” in the AR-quiver the indecomposable objects belong to a subcategory and by “$\cdot$” the indecomposable objects do not.\
Let ${\mathcal U}_1$ and ${\mathcal V}_1$ be the full subcategories of ${\operatorname{mod}\nolimits}\Lambda$ given by the following diagram. $$\xymatrix@[email protected]{
&&&&\circ \ar[dr]\\
&\cdot \ar[dr]
&&\cdot \ar[ur] \ar[dr]
&&\cdot \ar[dr] &&\cdot\\
{{\mathcal U}_1=}&\circ \ar[r]
&\circ \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur] \ar[r]
&\circ\\
&\cdot \ar[ur]
&&\cdot \ar[ur] \ar[dr]
&&\cdot \ar[ur]
&&\cdot\\
&&&&\circ \ar[ur]\\} \quad \quad
\xymatrix@[email protected]{
&&&&\circ \ar[dr]\\
&\circ \ar[dr]
&&\circ \ar[ur] \ar[dr]
&&\cdot \ar[dr] &&\circ\\
{{\mathcal V}_1=}&\circ \ar[r]
&\circ \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur] \ar[r]
&\circ\\
&\circ \ar[ur]
&&\circ \ar[ur] \ar[dr]
&&\cdot \ar[ur]
&&\circ\\
&&&&\circ \ar[ur]
}$$ The heart ${\mathcal H}_1/{\mathcal W}_1={\operatorname{add}\nolimits}({\begin{smallmatrix}
2
\end{smallmatrix}})$ and ${\mathcal H}_1\simeq {\operatorname{mod}\nolimits}({\mathcal U}_1/\mathcal P)$ by [@DL [Theorem 3.2]{}]. Now let ${\mathcal U}_2$ and ${\mathcal V}_2$ be the full subcategories of ${\operatorname{mod}\nolimits}\Lambda$ given by the following diagram. $$\xymatrix@[email protected]{
&&&&\circ \ar[dr]\\
&\cdot \ar[dr]
&&\circ \ar[ur] \ar[dr]
&&\cdot \ar[dr] &&\cdot\\
{{\mathcal U}_2=}&\circ \ar[r]
&\circ \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur] \ar[r]
&\circ\\
&\cdot \ar[ur]
&&\cdot \ar[ur] \ar[dr]
&&\cdot \ar[ur]
&&\cdot\\
&&&&\circ \ar[ur]\\} \quad \quad
\xymatrix@[email protected]{
&&&&\circ \ar[dr]\\
&\cdot \ar[dr]
&&\circ \ar[ur] \ar[dr]
&&\cdot \ar[dr] &&\circ\\
{{\mathcal V}_2=}&\circ \ar[r]
&\circ \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur]
&&\cdot \ar[dr] \ar[ur] \ar[r]
&\circ\\
&\circ \ar[ur]
&&\circ \ar[ur] \ar[dr]
&&\cdot \ar[ur]
&&\cdot\\
&&&&\circ \ar[ur]
}$$ The heart ${\mathcal H}_2/{\mathcal W}_2={\operatorname{add}\nolimits}({\begin{smallmatrix}
1
\end{smallmatrix}},{\begin{smallmatrix}
2
\end{smallmatrix}})$. Since ${\mathcal W}_1={\mathcal U}_1\subseteq {\mathcal U}_2\subseteq {\mathcal V}_2 \subseteq {\mathcal V}_1$, by Theorem \[serre\], $\overline {\mathcal H}_2\simeq {\mathcal H}_1/{\mathcal W}_1$.\
Moreover, ${\mathcal V}_1/{\mathcal U}_1$ has a triangulated category structure, and $({\mathcal U}_2/{\mathcal U}_1,{\mathcal V}_2/{\mathcal U}_1)$ is a cotorsion pair on it. The Serre subcategory $({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1))/{\mathcal W}_2={\operatorname{add}\nolimits}({\begin{smallmatrix}
1
\end{smallmatrix}})$ is the heart of $({\mathcal U}_2/{\mathcal U}_1,{\mathcal V}_2/{\mathcal U}_1)$.
Recall that a subcategory ${\mathcal M}$ of ${\mathcal B}$ is called rigid if ${\operatorname{Ext}\nolimits}^1_{\mathcal B}({\mathcal M},{\mathcal M})=0$, ${\mathcal M}$ is cluster tilting if it satisfies
- ${\mathcal M}$ is contravariantly finite and covariantly finite in ${\mathcal B}$.
- $X\in {\mathcal M}$ if and only if ${\operatorname{Ext}\nolimits}^1_{{\mathcal B}}(X,{\mathcal M})=0$.
- $X\in {\mathcal M}$ if and only if ${\operatorname{Ext}\nolimits}^1_{{\mathcal B}}({\mathcal M},X)=0$.
If ${\mathcal M}$ is a cluster tilting subcategory of ${\mathcal B}$, then $({\mathcal M},{\mathcal M})$ is a cotorsion pair on ${\mathcal B}$ (see [@L [Proposition 10.5]{}]).. In this case we have ${\mathcal H}={\mathcal B}^-={\mathcal B}^+={\mathcal B}$, $\sigma^-=\sigma^+=\text{id}$ and $H=\pi$.
Let $\Lambda$ be the $k$-algebra given by the quiver $$\xymatrix@[email protected]{
&&3 \ar[dl]\\
&5 \ar[dl] \ar@{.}[rr] &&2 \ar[dl] \ar[ul]\\
6 \ar@{.}[rr] &&4 \ar[ul] \ar@{.}[rr] &&1 \ar[ul]}$$ with mesh relations. The AR-quiver of ${\mathcal B}:={\operatorname{mod}\nolimits}\Lambda$ is given by $$\xymatrix@[email protected]{
&&{\begin{smallmatrix}
3&&\\
&5&\\
&&6
\end{smallmatrix}} \ar[dr] &&&&&&{\begin{smallmatrix}
1&&\\
&2&\\
&&3
\end{smallmatrix}} \ar[dr]\\
&{\begin{smallmatrix}
5&&\\
&6&
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
3&&\\
&5&
\end{smallmatrix}} \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
4
\end{smallmatrix}} \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
2&&\\
&3&
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
1&&\\
&2&
\end{smallmatrix}} \ar[dr]\\
{\begin{smallmatrix}
6
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] &&{\begin{smallmatrix}
5
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
3&&4\\
&5&
\end{smallmatrix}} \ar[ur] \ar[r] \ar[dr] \ar@{.}@/^15pt/[rr] &{\begin{smallmatrix}
&2&\\
3&&4\\
&5&
\end{smallmatrix}} \ar[r] &{\begin{smallmatrix}
&2&\\
3&&4
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] \ar[dr] &&{\begin{smallmatrix}
2
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] &&{\begin{smallmatrix}
1
\end{smallmatrix}}.\\
&&&{\begin{smallmatrix}
4&&\\
&5&
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] &&{\begin{smallmatrix}
3
\end{smallmatrix}} \ar[ur] \ar@{.}[rr] &&{\begin{smallmatrix}
2&&\\
&4&
\end{smallmatrix}} \ar[ur]
}$$ Let ${\mathcal U}_1$ and ${\mathcal V}_1$ be the full subcategories of ${\operatorname{mod}\nolimits}\Lambda$ given by the following diagram. $$\xymatrix@[email protected]{
&&&\circ \ar[dr] &&&&&&\circ \ar[dr]\\
{{\mathcal U}_1=} &&\circ \ar[ur] \ar[dr] &&\cdot \ar[dr] &&\cdot \ar[dr] &&\cdot \ar[ur] \ar[dr] &&\circ \ar[dr]\\
&\circ \ar[ur] &&\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] \ar[r] \ar[dr] &\circ \ar[r] &\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] &&\circ\\
&&&&\circ \ar[ur] &&\cdot \ar[ur] &&\circ \ar[ur]
\\} \quad
\xymatrix@[email protected]{
&&&\circ \ar[dr] &&&&&&\circ \ar[dr]\\
{{\mathcal V}_1=} &&\circ \ar[ur] \ar[dr] &&\cdot \ar[dr] &&\circ \ar[dr] &&\cdot \ar[ur] \ar[dr] &&\circ \ar[dr]\\
&\circ \ar[ur] &&\circ \ar[ur] \ar[dr] &&\cdot \ar[ur] \ar[r] \ar[dr] &\circ \ar[r] &\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] &&\circ\\
&&&&\circ \ar[ur] &&\cdot \ar[ur] &&\circ \ar[ur]
}$$ Then $({\mathcal U}_1,{\mathcal V}_1)$ is a cotorsion pair on ${\operatorname{mod}\nolimits}\Lambda$. The heart ${\mathcal H}_1/{\mathcal W}_1$ is the following. $$\xymatrix@[email protected]{
&&&{\begin{smallmatrix}
2&\ \\
&3
\end{smallmatrix}}\ar[dr]\\
{\begin{smallmatrix}
3&&\ \\
&5&
\end{smallmatrix}}\ar[dr] \ar@{.}[rr]
&&{\begin{smallmatrix}
&2&\ \\
3&&4
\end{smallmatrix}}\ar[ur] \ar@{.}[rr]
&&{\begin{smallmatrix}
\ &2&\
\end{smallmatrix}}\\
&{\begin{smallmatrix}
\ &3&\
\end{smallmatrix}} \ar[ur]}$$ The only indecomposable object which does not lie in ${\mathcal H}_1$ or ${\mathcal U}_1$,${\mathcal V}_1$ is ${\begin{smallmatrix}
3&&4\\
&5&
\end{smallmatrix}}$, since we have the following commutative diagram $$\xymatrix{
{\begin{smallmatrix}
5
\end{smallmatrix}} \ar@{=}[r] \ar@{ >->}[d] &{\begin{smallmatrix}
5
\end{smallmatrix}} \ar@{ >->}[d]\\
{\begin{smallmatrix}
&4&\\
&&5
\end{smallmatrix}}\oplus{\begin{smallmatrix}
&3&\\
&&5
\end{smallmatrix}} \ar@{ >->}[r] \ar@{->>}[d] &{\begin{smallmatrix}
&4&\\
&&5
\end{smallmatrix}}\oplus{\begin{smallmatrix}
&2&\\
3&&4\\
&5&
\end{smallmatrix}} \ar@{->>}[r] \ar@{->>}[d] &{\begin{smallmatrix}
&2&\\
&&4
\end{smallmatrix}} \ar@{=}[d]\\
{\begin{smallmatrix}
3&&4\\
&5&
\end{smallmatrix}} \ar@{ >->}[r] &{\begin{smallmatrix}
4
\end{smallmatrix}}\oplus{\begin{smallmatrix}
&2&\\
3&&4\\
&5&
\end{smallmatrix}} \ar@{->>}[r] &{\begin{smallmatrix}
&2&\\
&&4\\
\end{smallmatrix}}.
}$$ We get $H_1({\begin{smallmatrix}
3&&4\\
&5&
\end{smallmatrix}})={\begin{smallmatrix}
3&&\\
&5&
\end{smallmatrix}}$ since ${\begin{smallmatrix}
&4&\\
&&5
\end{smallmatrix}}\in \mathcal P$. Let $$\xymatrix@[email protected]{
&&&\circ \ar[dr] &&&&&&\circ \ar[dr]\\
{{\mathcal M}=} &&\circ \ar[ur] \ar[dr] &&\cdot \ar[dr] &&\circ \ar[dr] &&\cdot \ar[ur] \ar[dr] &&\circ \ar[dr]\\
&\circ \ar[ur] &&\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] \ar[r] \ar[dr] &\circ \ar[r] &\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] &&\circ\\
&&&&\circ \ar[ur] &&\cdot \ar[ur] &&\circ \ar[ur]
}$$ Since ${\mathcal M}$ is a cluster tilting subcategory of ${\mathcal B}$, $({\mathcal U}_2,{\mathcal V}_2)=({\mathcal M},{\mathcal M})$ is a cotorsion pair. The heart ${\mathcal H}_2/{\mathcal W}_2={\operatorname{mod}\nolimits}\Lambda/{\mathcal M}$ is the following. $$\xymatrix@[email protected]{
&{\begin{smallmatrix}
&3&\ \\
&&5
\end{smallmatrix}}\ar[dr]
&&&&{\begin{smallmatrix}
2&\ \\
&3
\end{smallmatrix}}\ar[dr]\\
{\begin{smallmatrix}
\ &5&\
\end{smallmatrix}} \ar[ur] \ar@{.}[rr]
&&{\begin{smallmatrix}
3&&4\ \\
&5&
\end{smallmatrix}}\ar[dr] \ar@{.}[rr]
&&{\begin{smallmatrix}
&2&\ \\
3&&4
\end{smallmatrix}}\ar[ur] \ar@{.}[rr]
&&{\begin{smallmatrix}
\ &2&\
\end{smallmatrix}}\\
&&&{\begin{smallmatrix}
\ &3&\
\end{smallmatrix}} \ar[ur]}$$ Since ${\mathcal U}_1\subseteq {\mathcal M}\subseteq {\mathcal V}_1$, we have ${\mathcal W}_1\subseteq {\operatorname{add}\nolimits}({\mathcal U}_2*{\mathcal V}_2)\subseteq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$. Since We get $H_1({\begin{smallmatrix}
4&&3\\
&5&
\end{smallmatrix}})={\begin{smallmatrix}
3&&\\
&5&
\end{smallmatrix}}$, we get that $\beta_{21}$ is exact. But $\beta_{12}$ is not exact, since $\xymatrix@[email protected]{
{\begin{smallmatrix}
&3&\ \\
&&5
\end{smallmatrix}} \ar[r] &{\begin{smallmatrix}
\ &3&\
\end{smallmatrix}} \ar[r] &{\begin{smallmatrix}
&2&\ \\
3&&4
\end{smallmatrix}}}$ is a short exact sequence in ${\mathcal H}_1/{\mathcal W}_1$ but not a short exact sequence in ${\mathcal H}_2/{\mathcal W}_2$. In this case, $({\mathcal H}_2\cap{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)/{\mathcal W}_2)$ is ${\operatorname{add}\nolimits}({\begin{smallmatrix}
\ &5&\
\end{smallmatrix}})$, we can see that $\overline {\mathcal H}_2\simeq {\mathcal H}_1/{\mathcal W}_1$.\
Let ${\mathcal U}_3$ and ${\mathcal V}_3$ be the full subcategories of ${\operatorname{mod}\nolimits}\Lambda$ given by the following diagram. $$\xymatrix@[email protected]{
&&&\circ \ar[dr] &&&&&&\circ \ar[dr]\\
{{\mathcal U}_3=} &&\circ \ar[ur] \ar[dr] &&\cdot \ar[dr] &&\circ \ar[dr] &&\cdot \ar[ur] \ar[dr] &&\circ \ar[dr]\\
&\circ \ar[ur] &&\cdot\ar[ur] \ar[dr] &&\cdot \ar[ur] \ar[r] \ar[dr] &\circ \ar[r] &\cdot \ar[ur] \ar[dr] &&\circ \ar[ur] &&\circ\\
&&&&\circ \ar[ur] &&\cdot \ar[ur] &&\circ \ar[ur]\\}\quad
\xymatrix@[email protected]{
&&&\circ \ar[dr] &&&&&&\circ \ar[dr]\\
{{\mathcal V}_3=} &&\circ \ar[ur] \ar[dr] &&\cdot \ar[dr] &&\cdot \ar[dr] &&\cdot \ar[ur] \ar[dr] &&\circ \ar[dr]\\
&\circ \ar[ur] &&\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] \ar[r] \ar[dr] &\circ \ar[r] &\cdot \ar[ur] \ar[dr] &&\cdot \ar[ur] &&\circ\\
&&&&\circ \ar[ur] &&\cdot \ar[ur] &&\circ \ar[ur]}$$ and the heart ${\mathcal H}_3/{\mathcal W}_3$ is the following. $$\xymatrix@[email protected]{
&{\begin{smallmatrix}
&3&\ \\
&&5
\end{smallmatrix}}\ar[dr]\\
{\begin{smallmatrix}
\ &5&\
\end{smallmatrix}} \ar[ur] \ar@{.}[rr]
&&{\begin{smallmatrix}
3&&4\ \\
&5&
\end{smallmatrix}}\ar[dr] \ar@{.}[rr]
&&{\begin{smallmatrix}
&2&\ \\
&&3
\end{smallmatrix}}\\
&&&{\begin{smallmatrix}
\ &3&\
\end{smallmatrix}} \ar[ur]}$$ Hence we get ${\mathcal H}_1/{\mathcal W}_1\simeq {\mathcal H}_3/{\mathcal W}_3$. But we find that ${\mathcal U}_3\nsubseteqq{\operatorname{add}\nolimits}({\mathcal U}_1*{\mathcal V}_1)$ and ${\mathcal V}_1\nsubseteqq{\operatorname{add}\nolimits}({\mathcal U}_3*{\mathcal V}_3)$, which implies that the condition Corollary \[suf\] is not necessary for the equivalence of two hearts.
By Theorem \[8.11\] and Proposition \[8.15\], we get:
\[5.00\] Let ${\mathcal M}$ be a cluster tilting subcategory of ${\mathcal B}$. Then the canonical functor $$\pi:{\mathcal B}\rightarrow {\mathcal B}/{\mathcal M}$$ is half exact. Moreover, every short exact sequence $$\xymatrix{A \ar@{ >->}[r]^{f} &B \ar@{->>}[r]^{g} &C}$$ in $B$ induces a long exact sequence $$\cdots \xrightarrow{\underline {\Omega h'}} \Omega A \xrightarrow{\underline {\Omega f}} \Omega B \xrightarrow{\underline {\Omega g}} \Omega C \xrightarrow{\underline h'} A \xrightarrow{\underline f} B \xrightarrow{\underline g} C \xrightarrow{\underline h} \Omega^- A \xrightarrow{\underline {\Omega^- f}} \Omega^- B \xrightarrow{\underline {\Omega^- g}} \Omega^- C \xrightarrow{\underline {\Omega^- h}} \cdots$$ in the abelian category ${\mathcal B}/{\mathcal M}$.
Let ${\mathcal M}$ be a cluster tilting subcategory of ${\mathcal B}$ (for instance, see [@DL [Example 4.2]{}]). Then we have a half exact functor $$\begin{aligned}
G: \text{ }&{\mathcal B}\rightarrow {\operatorname{mod}\nolimits}{{\mathcal M}/\mathcal P}\\
&X\mapsto {\operatorname{Ext}\nolimits}^1_{{\mathcal B}}(-,X)|_{{\mathcal M}}.\end{aligned}$$ This is a composition of the half exact functor $\pi:{\mathcal B}\rightarrow {\mathcal B}/{\mathcal M}$ given by Proposition \[5.00\] and an equivalence $$\begin{aligned}
\text{ }&{\mathcal B}/{\mathcal M}\xrightarrow{\simeq} {\operatorname{mod}\nolimits}{{\mathcal M}/\mathcal P}\\
&X\mapsto {\operatorname{Ext}\nolimits}^1_{{\mathcal B}}(-,X)|_{{\mathcal M}}.\end{aligned}$$ given by [@DL [Theorem 3.2]{}]. By Proposition \[8.16\], $G(X)=0$ if and only if $X\in {\mathcal M}$.
A more general case is given as follows. If ${\mathcal M}$ is a rigid subcategory of ${\mathcal B}$ which is contravariantly finite and contains $\mathcal P$, then by [@L [Proposition 2.12]{}], $({\mathcal M},{{\mathcal M}}^{\bot_1})$ is a cotorsion pair where ${{\mathcal M}}^{\bot_1}=\{X\in {\mathcal B}\text{ }| \text{ }{\operatorname{Ext}\nolimits}^1_{\mathcal B}({\mathcal M},X)=0 \}$. Since ${\mathcal M}$ is rigid, we have ${\mathcal M}\subseteq {{\mathcal M}}^{\bot_1}$. In this case we have ${\mathcal B}^+={\mathcal B}$, ${\mathcal B}^-={\mathcal H}$, $\sigma^+=\text{id}$ and $H=\sigma^-\circ \pi$. By [@DL [Theorem 3.2]{}], there exists an equivalence between $\underline {\mathcal H}$ and ${\operatorname{mod}\nolimits}({\mathcal M}/\mathcal P)$. Hence by Theorem \[8.11\], we get the following example:
Let ${\mathcal M}$ be a rigid subcategory of ${\mathcal B}$ which is contravariantly finite and contains $\mathcal P$ (for instance, see [@DL [Example 4.3]{}]). Then there exists a half exact functor $$\begin{aligned}
G: \text{ }&{\mathcal B}\rightarrow {\operatorname{mod}\nolimits}{{\mathcal M}/\mathcal P}\\
&X\mapsto {\operatorname{Ext}\nolimits}^1_{{\mathcal B}}(-,\sigma^-(X))|_{{\mathcal M}}\end{aligned}$$ which is a composition of $H$ and the equivalence $$\begin{aligned}
\text{ }&\underline {\mathcal H}\xrightarrow{\simeq} {\operatorname{mod}\nolimits}{{\mathcal M}/\mathcal P}\\
&Y\mapsto {\operatorname{Ext}\nolimits}^1_{{\mathcal B}}(-,Y)|_{{\mathcal M}}\end{aligned}$$ given by [@DL [Theorem 3.2]{}]. By Proposition \[8.16\], $G(X)=0$ if and only if $X\in {{\mathcal M}}^{\bot_1}$.
[99]{}
M. Auslander, I. Reiten. Application of contravariantly finite subcategories. Adv. Math. 86, 111–152(1991).
N. Abe, H. Nakaoka. General heart construction on a triangulated category (II): Associated cohomological functor. Appl. Categ. Structures 20 (2012), no. 2, 162–174.
T. Bühler. Exact categories. Expo. Math. 28 (2010), no. 1, 1–69.
L. Demonet, Y. Liu. Quotients of exact categories by cluster tilting subcategories as module categories. Journal of Pure and Applied Algebra. 217 (2013), 2282–2297
Z. Huang, O. Iyama. Auslander-type conditions and cotorsion pairs. J. Algebra. 318 (2007), 93–110.
Y. Liu. Hearts of twin cotorsion pairs on exact categories. J. Algebra. 394 (2013), 245–284.
H. Nakaoka. General heart construction on a triangulated category (I): unifying $t$-structures and cluster tilting subcategories. Appl. Categ. Structures 19 (2011), no. 6, 879–899.
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Y. Zhou, B. Zhu. Cotorsion pairs and t-structure in a 2-Calabi-Yau triangulated category. arXiv: 1210.6424v2.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Geodesic incompleteness is a problem in both general relativity and string theory. The Weyl invariant Standard Model coupled to General Relativity (SM+GR), and a similar treatment of string theory, are improved theories that are geodesically complete. A notable prediction of this approach is that there must be antigravity regions of spacetime connected to gravity regions through gravitational singularities such as those that occur in black holes and cosmological bang/crunch. Antigravity regions introduce apparent problems of ghosts that raise several questions of physical interpretation. It was shown that unitarity is not violated but there may be an instability associated with negative kinetic energies in the antigravity regions. In this paper we show that the apparent problems can be resolved with the interpretation of the theory from the perspective of observers strictly in the gravity region. Such observers cannot experience the negative kinetic energy in antigravity directly, but can only detect in and out signals that interact with the antigravity region. This is no different than a spacetime black box for which the information about its interior is encoded in scattering amplitudes for in/out states at its exterior. Through examples we show that negative kinetic energy in antigravity presents no problems of principles but is an interesting topic for physical investigations of fundamental significance.'
author:
- Itzhak Bars and Albin James
title: 'Physical Interpretation of Antigravity[^1]'
---
[^2]
Why Antigravity?
================
The Lagrangian for the geodesically complete version of the Standard Model coupled to General Relativity (SM+GR) is [@BST-conf] $$\mathcal{L}\left( x\right) =\sqrt{-g}\left(
\begin{array}
[c]{c}L_{\text{SM}}\left( A_{\mu}^{\gamma,W,Z,g},\;\psi_{q,l},\;\nu_{R},\;\chi\right) \\
+g^{\mu\nu}\left( \frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-D_{\mu
}H^{\dagger}D_{\nu}H\right) \\
-\left( \frac{\lambda}{4}\left( H^{\dagger}H-\omega^{2}\phi^{2}\right)
^{2}+\frac{\lambda^{\prime}}{4}\phi^{4}\right) \\
+\frac{1}{12}\left( \phi^{2}-2H^{\dagger}H\right) R\left( g\right)
\end{array}
\right) \label{action}$$ In the first line, $L_{SM}$ contains all the familiar degrees of freedom in the properly extended conventional Standard Model, including gauge bosons $(A_{\mu}^{\gamma,W,Z,g}),$ quarks & leptons $\left( \psi_{q,l}\right) ,$ right-handed neutrinos $\nu_{R},$ dark matter $\chi,$ and their SU$\left(
3\right) \times$SU$\left( 2\right) \times$U$\left( 1\right) $ gauge invariant interactions among themselves and with the spin-0 fields $\left(
H,\phi\right) $, where $H$=electroweak Higgs doublet, $\phi=$ a singlet. In $L_{SM}$ all fields are minimally coupled to gravity. The second and third lines describe the kinetic energy terms and interactions of the scalars among themselves. The last term is the unique non-minimal coupling of conformal scalars to the scalar curvature $R\left( g\right) $ that is required by invariance of the full $\mathcal{L}\left( x\right) $ under local rescaling (Weyl) with an arbitrary local parameter $\Omega\left( x\right) $ $$\begin{array}
[c]{c}g_{\mu\nu}\rightarrow\Omega^{-2}g_{\mu\nu},~\phi\rightarrow\Omega
\phi,\;H\rightarrow\Omega H\\
\psi_{q,l}\rightarrow\Omega^{3/2}\psi_{q,l},\;A_{\mu}^{\gamma,W,Z,g}~\text{{\scriptsize unchanged.}}\end{array}$$ If dark matter $\chi$ is a spin-0 field, then lines 2-4 in (\[action\]) should be modified to treat $\chi$ as another conformally coupled scalar.
This theory has several pleasing features. There are no dimensionful parameters, so all of those arise from a unique source, namely the gauge fixing of the Weyl symmetry such as, $\phi\left( x\right) \rightarrow
\phi_{0},$ where $\phi_{0}$ is a dimensionful constant of the order of the Planck scale. Then the gravitational constant is $\left( 16\pi G_{N}\right)
^{-1}=\phi_{0}^{2}/12$, the electroweak scale is $\langle\left\vert
H\right\vert \rangle=\omega\phi_{0},$ while dark energy, and masses for quarks, leptons, gauge bosons, neutrinos and dark matter arise from interactions with the scalars $\left( \phi,H\right) .$ The hierarchy of mass scales is put in by hand through a hierarchy of dimensionless parameters. A deeper theory is needed to explain this hierarchy, but in the present effective theory it is at least possible to maintain it under renormalization since dimensionless constants receive only logarithmic quantum corrections (no need for low energy supersymmetry for the purpose of naturalness). To preserve the local scale symmetry in the quantum theory one must adopt a Weyl invariant renormalization scheme in which $\phi$ is the only renormalization scale, and consequently dimensionless constants receive only Weyl invariant logarithmic renormalizations of the form $\ln\left( H/\phi\right) ,$ etc. With such a renormalization scheme the scale anomaly of all matter cancels against the scale anomaly of $\phi$ [@Percacci], thus not spoiling the local symmetry. Then the unbroken Weyl symmetry in the renormalized theory plays a central role in explaining the smallness of dark energy as shown in [@IB-darkEnergy]. This also suggests a definite relation between the electroweak vacuum and dark energy both of which fill the entire universe.
The scalar $\phi$ is compensated by the Weyl symmetry, so $\phi$ is not a true additional physical degree of freedom but, as a conformally coupled scalar, participates in an important structure of the Weyl symmetry that has further physical consequences involving antigravity spacetime regions in cosmology and black holes as discussed in the following sections. The structure of interest, that leads to the central discussion in the rest of this paper, is the relative minus sign in $\left( \phi^{2}-2H^{\dagger}H\right) R$ and in the scalar kinetic terms in (\[action\]). These signs are compulsory and play an important role in the geodesic completeness of the theory. With the given sign patterns, $H$ has the correct sign for its kinetic term but $\phi$ has the wrong sign. If $\phi$ had the same sign of kinetic energy as $H,$ then the conformal coupling to $R$ would become purely negative which would lead to a negative gravitational constant. So, to generate a positive gravitational constant, $\phi$ must come with the opposite sign to $H.$ This makes $\phi$ a ghost, but this is harmless since the Weyl symmetry can remove this ghost by a gauge fixing.
This scheme has a straightforward generalization to supersymmetry/supergravity and grand unification, but all scalars $\vec{s}$ must be conformally coupled, $\left( \phi^{2}-\vec{s}^{2}\right) R,$ although some generalization is permitted as long as the geodesically complete feature (related to signs) is maintained [@BST-conf]. Furthermore, we point out that in all supergravity theories, the curvature term has the form $\left( 1-K\left( \varphi_{i},\bar{\varphi}_{i}\right) /3\right) R,$ where $K$ is the Kähler potential and $1$ represents the Einstein-Hilbert term [@Weinberg]. This is again of the form $(\left\vert \phi\right\vert ^{2}-\left\vert \vec
{s}\right\vert ^{2})R$ with complex $\left( \phi,\vec{s}\right) ,$ where a complex version of $\phi$ has been gauge fixed to 1 in a Weyl invariant formulation of supergravity [@IB-sugra] (see also [@FerraraKallosh]). Finally we emphasize that the same relative minus sign occurs also in a Weyl invariant reformulation of low energy string theory (ST) but with a different interpretation of $s$ related to the dilaton [@BST-string]. Hence the structure $\left( \phi^{2}-\vec{s}^{2}\right) R$ is ubiquitous, but was overlooked because it was commonly assumed that the gravitational constant, or an effective structure that replaces it, could not or should not become negative .
At the outset of this approach in 2008 [@IB-2Tgravity] the immediate question was whether the dynamics would allow $\left( \phi^{2}-s^{2}\right)
$ to remain always positive. It was eventually determined by Bars, Chen, Steinhardt and Turok, in a series of papers during 2010-2012 (summary in [@IB-cosmoSummary]), that the solutions of the field equations that do not switch sign for this quantity are non-generic and of measure zero in the phase space of initial conditions for the fields $\left( \phi,s\right) $. So, according to the dynamics, it is untenable to insist on a limited patch $\left\vert \phi\right\vert >\left\vert \vec{s}\right\vert $ of field space. By contrast, it was found that the theory becomes geodesically complete when all field configurations are included, thus solving generally the basic problem of geodesic incompleteness.
The other side of the coin is that solving geodesic incompleteness comes with the prediction that there would be antigravity sectors in the theory since the effective gravitational constant that is proportional to $\left( \phi
^{2}\left( x\right) -s^{2}\left( x\right) \right) ^{-1}$ would dynamically become negative in some spacetime regions. In view of the pleasing features of the theory outlined in the second paragraph above, these antigravity sectors must then be taken seriously and the corresponding new physics must be understood. In our investigations so far, we discovered that the antigravity sectors are geodesically connected to our own gravity sector at gravitational singularities, like the big bang/crunch or black holes, which occur precisely at the same spacetime points where $\left( \phi^{2}\left(
x\right) -s^{2}\left( x\right) \right) $ vanishes or goes to infinity. The related dynamical string tension [@BST-string] $$T\left( \phi,s\right) \sim\left( \phi+s\right) ^{2\frac{1+\sqrt{d-1}}{d-2}}\left( \phi-s\right) ^{2\frac{1-\sqrt{d-1}}{d-2}},$$ goes to zero or infinity simultaneously. So we need to figure out the physical effects that can be observed in our universe due to the presence of antigravity sectors behind cosmological [@BST-antigravity] and black hole singularities [@ABJ-blackhole]. After overcoming several conceptual as well as technical challenges we have been able to discuss some new physics problems and developed new cosmological scenarios that involve an antigravity period in the history of the universe [@BST-Higgs][@turok]. A remaining conceptual puzzle is an apparent possible instability in the antigravity sector that is addressed and resolved in the remainder of this paper. Our conclusion is that there are no fundamental problems but only interesting physics of crucial significance.
Geodesic completeness in the Einstein or string frames \[frames\]
=================================================================
The classical or quantum analysis of this theory is best conducted in a Weyl gauge we called the $\gamma$-gauge [@BST-antigravity][@BST-conf][@BST-string] which amounts to $\det\left( -g\right) =1.$ This allows the Sign$\left( \phi^{2}-s^{2}\right) $ to be determined by the dynamics. Note that the sign is gauge invariant, so if the sign switches dynamically in one gauge it has to also switch in all gauges. If one wishes to use the traditional Einstein gauge (E) or the string gauge (s) one can err by choosing an illegitimate gauge that corresponds to a geodesically incomplete patch, such as$$\begin{array}
[c]{l}\text{E}^{+}\text{-gauge:\ }\frac{1}{12}\left( \phi_{E+}^{2}-s_{E+}^{2}\right) =\frac{+1}{16\pi G_{N}},\\
\text{s}^{+}\text{-gauge:\ }\frac{d-2}{8\left( d-1\right) }\left( \phi
_{s+}^{2}-s_{s+}^{2}\right) =\frac{+1}{2\kappa_{d}^{2}}e^{-2\Phi},\;\Phi=\text{dilaton.}\end{array}
\label{EsGuages}$$ The $E$ or $s$ subscripts on the fields indicate the gauge fixed form of the corresponding field. If this were all, then there would be nothing new, and the Weyl symmetry could be regarded as fake [@Jackiw]. However, the fact is that conventional general relativity and string theory are geodesically incomplete because the gauge choices just shown are valid only in the field patch in which $\left\vert \phi\right\vert >\left\vert s\right\vert $. The dynamics contradict the assumption of gauge fixing to only the positive patch. In the negative regions one may choose again the Einstein or string gauge, but now with a negative gravitational constant, $\frac{1}{12}\left( \phi_{E-}^{2}-s_{E-}^{2}\right) =\frac{-1}{16\pi G_{N}},$ or $\frac{d-2}{8\left(
d-1\right) }\left( \phi_{s-}^{2}-s_{s-}^{2}\right) =\frac{-1}{2\kappa
_{d}^{2}}e^{-2\Phi}$. In those spacetime regions gravity is repulsive (antigravity).
The same situation arises in string theory. In the worldsheet formulation of string theory the string tension is promoted to a background field $T\left(
\phi,s\right) $ by connecting it directly to the features of the Weyl invariant low energy string theory [@BST-string]. Then the string tension $T\left( \phi,s\right) $ switches sign together with the corresponding gravitational constant [@BST-string]. Thus the Weyl symmetric (SM+GR) and ST predict that, in the Einstein or string gauges, one should expect a *sudden sign switch* of the effective Planck mass $\frac{1}{12}\left(
\phi^{2}-s^{2}\right) $ at certain spacetime points that typically correspond to singularities (e.g. big bang, black holes) encountered in the Einstein or string frames.
One may choose better Weyl gauges (e.g. $\gamma
$-gauge, choose $\det\left( -g\right) \rightarrow1$, or $c$-gauge, choose $\phi\rightarrow$ constant$)$ that cover globally all the positive and negative patches. Then the sign switch of the effective Planck mass $\frac{1}{12}\left( \phi
^{2}-s^{2}\right) $ is smooth rather than abrupt.
However, if one wishes to work in the more familiar Einstein or string frames, to recover the geodesically complete theory one must allow for the gravitational constant to switch sign at singularities, and connect solutions for fields across gravity/antigravity patches. In the $\pm$ Einstein gauges shown above, the last term in Eq.(\[action\]) becomes $$\frac{\left( \phi_{E\pm}^{2}-s_{E\pm}^{2}\right) R\left( g_{E\pm}\right)
}{12}=\frac{R\left( g_{E\pm}\right) }{\pm16\pi G_{N}}=\frac{R\left( \pm
g_{E\pm}\right) }{16\pi G_{N}}. \label{pmg}$$ where the $\pm$ for the gravity/antigravity regions can be absorbed into a redefinition of the signature of the metric, $$\hat{g}_{\mu\nu}^{E}=\pm g_{\mu\nu}^{E\pm}, \label{g-hat}$$ where the *continuous* $\hat{g}_{\mu\nu}^{E}$ is the geometry in the *union* of the gravity/antigravity patches.
The same $\pm$ gauge choice is applied to every term in the SM+GR action in Eq.(\[action\]). Under the replacement $g_{\mu\nu}^{E-}\rightarrow-g_{\mu
\nu}^{E-}$ in the *antigravity sector* some terms in the action flip sign and some don’t [@Duff], e.g. $F_{\mu\nu}F^{\mu\nu}$ does not, but $R\left( g\right) $ does as in Eq.(\[pmg\]). One may be concerned that the sign switches of the gravitational constant or the string tension may lead to problems like unitarity or negative kinetic energy ghosts. We mention that [@BST-string] has already argued that there are no unitarity problems due to sign flips in field/string theories. There remains the question of possible instability due to negative kinetic energy in the antigravity region. We show in this paper that its presence is not a problem of principle for observers in the gravity region and that those observers can detect interesting physical effects related to the geodesically connected regions of antigravity.
Unitarity and antigravity in cosmology
======================================
There is a general impression that negative kinetic energy in field or string theory implies ghosts associated with negative norm states. It is not generally appreciated that negative norms (hence negative probabilities) are automatically avoided by insisting on a strictly unitary quantization of the theory. This has been illustrated in the quantization of the relativistic harmonic oscillator [@IB-relativisticHO] with a timelike direction that appears with the opposite sign to the spacelike directions, just like the $\phi$ field as compared to the $H$ field in the SM+GR action in Eq.(\[action\]). Similar situations occur in the antigravity region where some fields may appear with the wrong sign as described after Eq.(\[pmg\]). The first duty in quantization should be maintaining sanity in the meaning of probability, as in [@IB-relativisticHO], by avoiding a quantization procedure that introduces negative norm states. Of course, there exist successful cases, such as string theory in the covariant quantization procedure, that at first admits negative norms to later kill them by applying constraints that select the positive norm states. In principle, the relativistic oscillators in string theory could also be treated as in [@IB-relativisticHO] and very likely still recover the same gauge invariant physical states without ever introducing negative norm states in string theory. It would be preferable to quantize without negative norm states at all from the very beginning.
When there is not enough gauge symmetry to remove a degree of freedom that has the wrong sign of kinetic energy, a unitary quantization procedure like [@IB-relativisticHO] maintains unitarity. However, the effect of the negative kinetic energy is to cause an instability (not unlike a tachyonic mass term, or a bottomless potential, would), so that there may not be a ground state for that degree of freedom while it propagates in the antigravity region. This is the negative kinetic energy issue in the antigravity sector. Perhaps some complete theory as a whole conspires to have a ground state even in antigravity. Although this would be reassuring, it appears that this is not necessary in order to make sense of the physics as detected by observers in the gravity sector. Such observers can verify that the same degree of freedom does have a ground state in the gravity region while they can never experience directly the negative kinetic energy in the antigravity sector. The only physics questions that make sense for those observers is what can be learned about the existence of antigravity through scattering experiments that involve in/out states as defined in the gravity region. For those questions the issue of whether there is a ground state in the antigravity region does not matter, but unitarity continues to matter. Therefore we point out how this works in the case of cosmology that admits an antigravity region.
WdW equation and unitarity in mini superspace
---------------------------------------------
The Wheeler de Witt (WdW) equation is the quantum version of the $\mu=0$ and $\nu=0$ component of the Einstein equation, $\left( G_{00}-T_{00}\right)
\psi=0.$ This is a constraint applied on physical states in covariant quantization of general relativity [@deWitt]. The mini superspace consists of only time dependent (homogeneous) scalar fields $\left( \phi\left( x^{0}\right) ,s\left( x^{0}\right)
\right) $ and the FRW metric, $ds^{2}=a^{2}\left( x^{0}\right) \left(
-\left( dx^{0}\right) ^{2}+\gamma_{ij}\left( x^{0},\vec{x}\right)
dx^{i}dx^{j}\right) ,$ with $\gamma_{ij}$ describing spacial curvature and anisotropies, while $T_{00}$ includes the radiation density, $\rho_{r}\left(
x^{0}\right) /a^{4}\left( x^{0}\right) .$ From the action in Eq.(\[action\]) we can derive a Wheeler de Witt equation that is invariant under Weyl rescalings $\left( \phi,s,a\right) \rightarrow\left( \Omega
\phi,\Omega s,\Omega^{-1}a\right) $ with a time dependent $\Omega\left(
x^{0}\right) $; this allows us to choose a gauge. To allow $\left( \phi
^{2}-s^{2}\right) $ to have any sign dynamically, we prefer the $\gamma
$-gauge given by $$\left( \phi,s,a\right) \rightarrow\left( \phi_{\gamma},s_{\gamma},1\right)
,\text{ or }a_{\gamma}\left( x^{0}\right) =1.$$ We concentrate here on the simplest FRW geometry in the $\gamma$-gauge, $$ds_{\gamma}^{2}=-\left( dx^{0}\right) ^{2}+\frac{dr^{2}}{1-Kr^{2}}+r^{2}d\Omega^{2}, \label{K}$$ with no anisotropy or inhomogeneities, but with a positive constant spatial curvature $K>0$. This is not realistic, but it is the easiest case to illustrate the unitarity properties of the quantum theory that includes antigravity regions (more degrees of freedom, and negative or zero $K$ would be treated in a similar manner). The mini superspace is just $\left(
\phi_{\gamma},s_{\gamma}\right) ,$ while the constraint $\left(
T_{00}-G_{00}\right) =0$ derived from (\[action\]) is, $-\frac{1}{2}\dot{\phi}_{\gamma}^{2}+\frac{1}{2}\dot{s}_{\gamma}^{2}+\frac{1}{2}K\left(
-\phi_{\gamma}^{2}+s_{\gamma}^{2}\right) +\rho_{r}=0.$ This is recognized as the Hamiltonian for the relativistic harmonic oscillator, $H=\frac{1}{2}\left( \dot{x}^{2}+Kx^{2}\right) ,$ with $x^{\mu}\left( \tau\right)
=\left( \phi_{\gamma}\left( \tau\right) ,s_{\gamma}\left( \tau\right)
\right) ,$ subject to the constraint, $H+\rho_{r}=0,$ where $\rho_{r}$ is a constant. Note that this Hamiltonian contains negative energy for the (time-like) $\phi_{\gamma}$ degree of freedom. Recall that we have already used up the Weyl symmetry so this degree of freedom cannot be removed and its negative energy must be dealt with. The naive quantization of the relativistic harmonic oscillator would introduce negative norm states for the $\phi
_{\gamma}$ degree of freedom (as in string theory), so it appears there may be trouble with unitarity. However, this is not the case, because this system (and similar cases) can be quantized by respecting unitarity without ever introducing negative norms as shown in [@IB-relativisticHO]. This goes as follows: the quantum system obeys the constraint equation $\left( H+\rho
_{r}\right) \Psi=0.$ This is the WdW equation that takes the form$$\left( \frac{1}{2}\partial_{\phi_{\gamma}}^{2}-\frac{1}{2}\partial
_{s_{\gamma}}^{2}+\frac{K}{2}\left( -\phi_{\gamma}^{2}+s_{\gamma}^{2}\right)
+\rho_{r}\right) \Psi\left( \phi_{\gamma},s_{\gamma}\right) =0. \label{WdW}$$ This is recognized as the Klein-Gordon equation for the quantized relativistic harmonic oscillator. The eigenstates and eigenvalues of the independent $\phi_{\gamma}$ and $s_{\gamma}$ oscillators are $$\begin{array}
[c]{l}\frac{1}{2}\left( -\partial_{\phi_{\gamma}}^{2}+K\phi_{\gamma}^{2}\right)
\psi_{n_{\phi}}\left( \phi_{\gamma}\right) =\sqrt{K}\left( n_{\phi}+\frac{1}{2}\right) \psi_{n_{\phi}}\left( \phi_{\gamma}\right) ,\\
\frac{1}{2}\left( -\partial_{s_{\gamma}}^{2}+Ks_{\gamma}^{2}\right)
\psi_{n_{s}}\left( s_{\gamma}\right) =\sqrt{K}\left( n_{s}+\frac{1}{2}\right) \psi_{n_{s}}\left( s_{\gamma}\right) ,
\end{array}$$ where $\left( n_{\phi},n_{s}\right) $ are positive integers, $0,1,2,3,\cdots
,$ and the explicit *positive norm* complete set of off-shell solutions are$$\begin{array}
[c]{c}\Psi_{n_{\phi}n_{s}}\left( \phi_{\gamma},s_{\gamma}\right) =\psi_{n_{\phi}}\left( \phi_{\gamma}\right) \psi_{n_{s}}\left( s_{\gamma}\right) ,\\
\psi_{n_{\phi}}\left( \phi_{\gamma}\right) =A_{n_{\phi}}e^{-\frac{1}{2}\sqrt{K}\phi_{\gamma}^{2}}H_{n_{\phi}}\left( \phi_{\gamma}\right) ,\\
\psi_{n_{s}}\left( s_{\gamma}\right) =A_{n_{s}}e^{-\frac{1}{2}\sqrt
{K}s_{\gamma}^{2}}H_{n_{s}}\left( s_{\gamma}\right) ,
\end{array}
\label{complete}$$ where $H_{n}\left( z\right) $ are the Hermite polynomials and $A_{n_{\phi}},A_{n_{s}}$ are normalization constants. Then the WdW equation (\[WdW\]) is solved by constraining the eigenvalues, $\sqrt{K}\left( -n_{\phi}+n_{s}\right) +\rho_{r}=0.$ Hence the complete on-shell basis that satisfies the constraint is $$\Psi_{n}\left( \phi_{\gamma},s_{\gamma}\right) =A_{n+r}A_{n}e^{-\frac
{\sqrt{K}}{2}\left( \phi_{\gamma}^{2}+s_{\gamma}^{2}\right) }H_{n+r}\left(
\phi_{\gamma}\right) H_{n}\left( s_{\gamma}\right) , \label{RelHO}$$ with $n=0,1,2,\cdots,$ where we defined $$n_{s}\equiv n,\text{ \ }n_{\phi}\equiv n+r,\;\text{ and }\frac{\rho_{r}}{\sqrt{K}}\equiv r\;\text{a fixed integer.}$$ If $\frac{\rho_{r}}{\sqrt{K}}$ is not an integer there is no solution to the constraint, hence radiation must be quantized for this system to be non-trivial at the quantum level. The general on-shell solution of the WdW equation is an arbitrary superposition of this basis$$\Psi\left( \phi_{\gamma},s_{\gamma}\right) =\sum_{n=0}^{\infty}c_{n}\Psi
_{n}\left( \phi_{\gamma},s_{\gamma}\right)$$ The complex coefficients $c_{n}$ are chosen to insure that $\Psi\left(
\phi_{\gamma},s_{\gamma}\right) $ is normalized.
All quantum states have positive norm and unitarity is satisfied. $\Psi\left(
\phi_{\gamma},s_{\gamma}\right) $ is the probability amplitude for where the system is in the $\left( \phi_{\gamma},s_{\gamma}\right) $ plane. The gravity/antigravity regions are $\phi_{\gamma}^{2}\lessgtr s_{\gamma}^{2}.$ Evidently there is no way of preventing the generic wavefunctions from being non-zero in the antigravity region, so the system generically evolves through both the gravity and antigravity regions.
We emphasize that the quantization method in [@IB-relativisticHO] that we used to maintain unitarity is very different than the quantization of the relativistic oscillator used in string theory. In string theory one defines relativistic creation/annihilation operators $a_{\mu},a_{\mu}^{\dagger}$ and a vacuum state that satisfies $a_{\mu}|0\rangle=0.$ Then the quantum states at level $l$ are given by applying $l$ creation operators, $a_{\mu_{1}}^{\dagger
}a_{\mu_{2}}^{\dagger}\cdots a_{\mu_{l}}^{\dagger}|0\rangle.$ The vacuum state is Lorentz invariant, while the states at level $l$ form a collection of *finite dimensional* irreducible representations of the Lorentz group. All the states at level $l$ have *positive energy*, $E_{l}=\sqrt
{K}\left( l+1\right) .$ The constraint $H+\rho_{r}=0$ (WdW equation) can be satisfied only for negative quantized $\rho_{r}$ at only one level $l=-1+\left\vert \rho_{r}\right\vert /\sqrt{K}.$ In position space the vacuum state takes the Lorentz invariant form $\psi_{0}\left( x^{\mu}\right) \sim
e^{-\sqrt{K}x^{2}}=e^{-\sqrt{K}\left( -\phi_{\gamma}^{2}+s^{2}\right) }$, while the the states at level $l$ are of the form of a polynomial of $x^{\mu}$ of degree $l$ multiplied by the same exponential $e^{-\sqrt{K}x^{2}}.$ A subset of the level-$l$ states have negative norm because finite dimensional representations are not unitary representations of the Lorentz group, so this method of quantization gets into trouble with unitarity. We contrast this result to ours in Eq.(\[RelHO\]) where we have displayed an infinite, rather than finite, number of states and a Gaussian factor $e^{-\sqrt{K}\left(
\phi_{\gamma}^{2}+s^{2}\right) }$ that converges in all directions, rather than the non-convergent Lorentz invariant form $e^{-\sqrt{K}\left(
-\phi_{\gamma}^{2}+s^{2}\right) }.$ There is no Lorentz invariant vacuum state. As shown in [@IB-relativisticHO], our states in Eq.(\[RelHO\]) form an *infinite dimensional unitary representation* of the Lorentz group for which all the states have positive norm. Furthermore, those that satisfy the constraint have positive total energy, $H=\rho_{r},$ as long as $\rho_{r}$ is positive. However, as seen in Eq.(\[complete\]), there are *off-shell states* of positive as well as negative energy. These remarks make it clear that the price for maintaining unitarity (which is the first duty in quantization) is the presence of regions of spacetime with negative kinetic energy, which, in our case, amounts to regions of antigravity. Our task in this paper is to explain that negative kinetic energy in the antigravity sector does not necessarily imply a problem by interpreting the physical significance of antigravity.
Feynman propagator in mini superspace
-------------------------------------
The Feynman propagator associated with this WdW equation is$$G\left( \phi^{\prime},s^{\prime};\phi,s\right) =\langle\phi^{\prime
},s^{\prime}|\frac{i}{H+\rho_{r}+i\varepsilon}|\phi,s\rangle.$$ We can use the complete basis $|n_{\phi},n_{s}\rangle$ to insert identity in terms of the eigenstates of the off-shell $H=-H_{\phi}+H_{s}$ operator without any constraints on the integers $\left( n_{\phi},n_{s}\right) .$ Then we compute$$\begin{array}
[c]{l}G\left( \phi^{\prime},s^{\prime};\phi,s\right) =i\sum_{n_{\phi},n_{s}\geq
0}\frac{\langle\phi^{\prime},s^{\prime}|n_{\phi},n_{s}\rangle\langle n_{\phi
},n_{s}|\phi,s\rangle}{-n_{\phi}+n_{s}+\rho_{r}+i\varepsilon}\\
\;\;=i\sum_{n_{\phi},n_{s}\geq0}\psi_{n_{\phi}}\left( \phi^{\prime}\right)
\psi_{n_{s}}\left( s^{\prime}\right) \psi_{n_{\phi}}^{\ast}\left(
\phi\right) \psi_{n_{s}}^{\ast}\left( s\right) \left( -n_{\phi}+n_{s}+\rho_{r}+i\varepsilon\right) ^{-1}\\
\;\;=i\sum_{n_{\phi},n_{s}\geq0}\int_{0}^{\infty}d\tau~\psi_{n_{\phi}}\left(
\phi^{\prime}\right) \psi_{n_{s}}\left( s^{\prime}\right) \psi_{n_{\phi}}^{\ast}\left( \phi\right) \psi_{n_{s}}^{\ast}\left( s\right) \left(
-i~e^{i\tau\left( -n_{\phi}+n_{s}+\rho_{r}+i\varepsilon\right) }\right) \\
\;\;=\int_{0}^{\infty}d\tau e^{i\tau\left( \rho_{r}+i\varepsilon\right)
}\langle\phi^{\prime}|e^{-i\tau H_{\phi}}|\phi\rangle~\langle s^{\prime
}|e^{i\tau H_{s}}|s\rangle\\
\;\;=\int_{0}^{\infty}d\tau\frac{\sqrt{K}~e^{i\tau\left( \rho_{r}+i\varepsilon\right) }}{2\pi\sin\left( \sqrt{K}\tau\right) }\exp\left(
\frac{-i\sqrt{K}}{2\sin\left( \sqrt{K}\tau\right) }\left[ \left(
x^{2}+x^{\prime2}\right) \cos\sqrt{K}\tau-2x\cdot x^{\prime}\right] \right)
\end{array}
\label{Feynman}$$ In the last step we used the propagator $\langle\phi^{\prime}|e^{-i\tau
H_{\phi}}|\phi\rangle$ for the $1$-dimensional harmonic oscillator, and then substituted $x^{2}=-\phi^{2}+s^{2}$ and $x\cdot x^{\prime}=-\phi\phi^{\prime
}+ss^{\prime}.$ This quantum computation in the Hamiltonian formalism agrees with the path integral computation in [@turok].
The Feynman propagator is a measure of the probability that the system that starts in some initial state will be found in some final state. For observers outside of the antigravity region the initial and final states $|\phi
,s\rangle,|\phi^{\prime},s^{\prime}\rangle$ are both in the gravity region, $\left\vert \phi\right\vert >\left\vert s\right\vert $ and $\left\vert
\phi^{\prime}\right\vert >\left\vert s^{\prime}\right\vert ,$ although during the propagation from initial to final state the antigravity region is probed as seen from the sums over $\left( n_{\phi},n_{s}\right) $ where both positive and negative energy states of the off-shell Hamiltonian $H=-H_{\phi
}+H_{s}$ enter in the calculation. We see from the last expression in Eq.(\[Feynman\]) that $G\left( \phi^{\prime},s^{\prime};\phi,s\right) $ is a perfectly reasonable function indicating that there are no issues with fundamental principles in this calculation which involves an intermediate period of antigravity in the evolution of the universe.
This was the case of a radiation dominated spatially curved spacetime which is far from being a generic configuration in the early universe close to the singularity. The generic dominant terms in the Einstein frame are the kinetic energy of the scalar and anisotropy (in the spatial metric) and the next non-leading term is radiation. The sub-dominant terms, including curvature, inhomogeneities, potential energy, dark energy, etc. are negligible near the singularity. The dominant generic behavior near the singularity was computed classically in [@BST-antigravity] where it was discovered that there *must* be an inescapable excursion into the antigravity regime before coming back to the gravity sector, as outlined in the previous paragraph. Hence, a similar computation to Eq.(\[Feynman\]), by using the dominant terms in the WdW equation (instead of (\[WdW\])) should replace our computation here. Unpublished work along these lines dating back to 2011 [@IB-cosmoSummary] indicate that the physical picture already obtained through classical solutions in [@BST-antigravity] continues to hold in mini-superspace at the quantum level.
More general WdW equation
-------------------------
As mentioned in the previous paragraph, we can generalize the WdW equation above (\[WdW\]) by including the physical features that would make it more realistic for a description of the early universe in terms of a mini superspace. This includes the kinetic energy for anisotropy degrees of freedom that cannot be neglected when $s^{2}/\phi^{2}\simeq1$ close to the singularity, and the potential energy terms for both the scalars and anisotropies that tend to become important when $\left\vert 1-s^{2}/\phi
^{2}\right\vert \gtrsim1.$ The action for the mini-superspace that includes these features is given in Eq.(8) in [@IB-cosmoSummary]. The corresponding WdW equation in the $\gamma$-gauge modifies Eq.(\[WdW\]) as follows $$\left(
\begin{array}
[c]{c}\frac{1}{2}\partial_{\phi_{\gamma}}^{2}-\frac{1}{2}\partial_{s_{\gamma}}^{2}-\frac{1}{2}\frac{1}{\phi_{\gamma}^{2}-s_{\gamma}^{2}}\left(
\partial_{\alpha_{1}}^{2}+\partial_{\alpha_{2}}^{2}\right) +\rho_{r}\\
+V\left( \phi_{\gamma},s_{\gamma}\right) -\frac{1}{2}\left( \phi_{\gamma
}^{2}-s_{\gamma}^{2}\right) v\left( \alpha_{1},\alpha_{2}\right)
\end{array}
\right) \Psi\left( \phi_{\gamma},s_{\gamma},\alpha_{1},\alpha_{2}\right)
=0.$$ The additional anisotropy degrees of freedom $\left( \alpha_{1},\alpha
_{2}\right) $ are part of the 3-dimensional metrics of types Kasner, Bondi-VIII, and Bondi-IX, as shown in Eq.(7) of Ref.[@IB-cosmoSummary]. The term containing the anisotropy potential $v\left( \alpha_{1},\alpha
_{2}\right) $ above vanishes for the Kasner metric, while for Bondi-VIII and IX it simplifies to a constant, $v\left( \alpha_{1},\alpha_{2}\right)
\rightarrow K,$ in the zero anisotropy limit (as in Eq.(\[WdW\])). The details of the anisotropy potential $v\left( \alpha_{1},\alpha_{2}\right) $ are given in Eq.(9) of Ref.[@IB-cosmoSummary].
With these additional features the WdW equation is no longer separable in the $\left( \phi_{\gamma},s_{\gamma}\right) $ degrees of freedom. To make progress we make a change of variables by defining $$z=\phi_{\gamma}^{2}-s_{\gamma}^{2},\;\sigma=\frac{1}{2}\ln\left\vert
\frac{\phi_{\gamma}+s_{\gamma}}{\phi_{\gamma}-s_{\gamma}}\right\vert .$$ Note that $z\gtrless0$ corresponds to gravity/antigravity. These variables were used in the classical analysis of the same system in [@BST-antigravity] where the classical equations that follow from the same action were studied. Weyl invariance requires $V\left( \phi,s\right) $ to be a homogeneous function of degree four, $V\left( t\phi,ts\right)
=t^{4}V\left( \phi,s\right) ,$ so without loss of generality we may write $$V\left( \phi_{\gamma},s_{\gamma}\right) =z^{2}v\left( \sigma\right) ,$$ where $v\left( \sigma\right) $ is any function that is specified by some physical model. The WdW equation above takes the following form in the $z,\sigma,\alpha_{1},\alpha_{2}$ basis $$\left(
\begin{array}
[c]{c}\partial_{z}^{2}+\frac{1}{4z^{2}}\left( -\partial_{1}^{2}-\partial_{2}^{2}-\partial_{\sigma}^{2}+1\right) \\
+\frac{z}{2}v\left( \sigma\right) -\frac{1}{4}v\left( \alpha_{1},\alpha
_{2}\right) +\frac{\rho_{r}}{2z}\end{array}
\right) \left( z^{1/2}\Psi\left( z,\sigma,\alpha_{1},\alpha_{2}\right)
\right) =0.\label{TotPsi}$$ Near the singularity, $z=0,$ assuming the potentials are neglected compared to the dominant and subdominant $z^{-2},z^{-1}$ terms, the wavefunction becomes separable in the form of a 3-dimensional plane wave, $\Psi=\exp\left(
ip_{1}\alpha_{1}+ip_{2}\alpha_{2}+ip_{3}\sigma\right) \psi\left( z\right) $ with constant momenta $\left(
p_{1},p_{2},p_{3}\right) ,$ thus reducing (\[TotPsi\]) to an ordinary second order differential equation $$\left( \partial_{z}^{2}+\frac{1}{4z^{2}}\left( p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+1\right) +\frac{\rho_{r}}{2z}\right) \left( z^{1/2}\psi\left(
z\right) \right) \simeq0.\label{closeToZero}$$ This is recognized as a Hydrogen-atom type differential equation; its solutions are given analytically in terms of special functions related to the representations of SL$\left( 2,R\right) $ as discussed in [@IB-QM]. From these wavefunctions we learn that the behavior of the probability distributions near the singularity, $z\sim0$ where the gravity/antigravity transition occurs, matches closely the behavior of the unique analytic classical attractor solution that corresponds to the antigravity loop described in [@BST-antigravity]. Namely, with a non-zero parameter, $p\equiv\sqrt{p_{1}^{2}+p_{2}^{2}+p_{3}^{2}},$ the cosmological evolution of the universe cannot avoid to pass temporarily though an antigravity sector, $z<0$. Meanwhile, as seen here, the wavefunctions are normalizable and fully consistent with unitarity in both the gravity and antigravity sectors $z\gtrless0$. This conclusion, in the presence of the dominant anisotropy terms, is in agreement with the lessons learned above with the simpler form of the WdW equation in (\[WdW\]).
As $\left\vert z\right\vert $ increases beyond the singularity and reaches $\left\vert z\right\vert \sim1$, in either the gravity or antigravity sectors, the potentials $v\left( \sigma\right) $ and $v\left( \alpha
_{1},\alpha_{2}\right) $ can no longer be neglected. We may still reduce the 4-variable partial differential equation to a single-variable ordinary differential equation as follows. Define the wavefunctions $\Phi_{n}\left(
\sigma\right) $ and $\xi_{m_{1}m_{2}}\left( \alpha_{1},\alpha_{2}\right) $ as follows $$\begin{array}
[c]{c}\left( -\partial_{\sigma}^{2}+2z^{3}v\left( \sigma\right) \right) \Phi
_{n}\left( \sigma\right) =E_{n}\left( z\right) \Phi_{n}\left(
\sigma\right) \\
\left( -\partial_{1}^{2}-\partial_{2}^{2}-z^{2}v\left( \alpha_{1},\alpha
_{2}\right) \right) \xi_{m_{1}m_{2}}\left( \alpha_{1},\alpha_{2}\right)
=E_{m_{1}m_{2}}\left( z\right) \xi_{m_{1}m_{2}}\left( \alpha_{1},\alpha
_{2}\right)
\end{array}
\label{PhiXi}$$ In solving these equations the parameter $z$ is considered a constant parameter, but the eigenvalues $E_{n}\left( z\right) $ and $E_{m_{1}m_{2}}\left( z\right) $ clearly depend on $z.$ Then, writing the wavefunction in separable form, $$\Psi\left( z,\sigma,\alpha_{1},\alpha_{2}\right) \sim\Psi_{n,m_{1},m_{2}}\left( z\right) \times\Phi_{n}\left( \sigma\right) \times\xi_{m_{1}m_{2}}\left( \alpha_{1},\alpha_{2}\right) ,$$ Eq.(\[TotPsi\]) reduces to $$\left( \partial_{z}^{2}+\frac{1}{4z^{2}}\left( E_{n}\left( z\right)
+E_{m_{1}m_{2}}\left( z\right) +1\right) +\frac{\rho_{r}}{2z}\right)
\left( z^{1/2}\Psi_{n,m_{1},m_{2}}\left( z\right) \right) =0.\label{zEq}$$ while the general solution takes the form $$\Psi\left( z,\sigma,\alpha_{1},\alpha_{2}\right) =\sum_{n,m_{1},m_{2}}c_{n,m_{1},m_{2}}\Psi_{n,m_{1},m_{2}}\left( z\right) \times\Phi_{n}\left(
\sigma\right) \times\xi_{m_{1}m_{2}}\left( \alpha_{1},\alpha_{2}\right)$$ with arbitrary constant coefficients $c_{n,m_{1},m_{2}}.$
As we saw above, when $v\left( \sigma\right) ,v\left( \alpha_{1},\alpha
_{2}\right) $ are zero, the corresponding energies tend to constants $E_{n}\left( z\right) \rightarrow p_{3}^{2}$ and $E_{m_{1}m_{2}}\left(
z\right) \rightarrow p_{1}^{2}+p_{2}^{2},$ so this can be used as a guide to the role played by $E_{n}\left( z\right) $ and $E_{m_{1}m_{2}}\left(
z\right) $ in Eq.(\[zEq\]). Using simple models for $v\left(
\sigma\right) ,v\left( \alpha_{1},\alpha_{2}\right) $ to extract properties of $E_{n}\left( z\right) $ and $E_{m_{1}m_{2}}\left( z\right) $ and also using semi-classical WKB approximation methods for more complicated cases, we may estimate the behavior of $E_{n}\left( z\right) +E_{m_{1}m_{2}}\left(
z\right) $ for small as well as large $z.$ This can then be used to discuss the behavior of the quantum universe beyond the approximations described above. In such attempts, with non-trivial $v\left( \sigma\right) ,v\left(
\alpha_{1},\alpha_{2}\right) ,$ we find that $E_{n}\left( z\right)
,E_{m_{1}m_{2}}\left( z\right) $ are generically not analytic near $z=0$ in the complex $z$-plane (in the sense of cuts that extend to $z=0)$ so this complicates the use of analyticity methods [@turok] to extract information from these equations. We continue to investigate this approach and hope to report more results in the future.
To conclude this section, an important remark is that unitarity is maintained in the WdW treatment throughout gravity and antigravity, while the presence of negative energy during antigravity is not of concern regarding fundamental principles as already illustrated in the previous sections, especially with the simpler computation based on Eq.(\[WdW\]).
Negative energy in antigravity and observers in gravity
=======================================================
To develop a physical understanding of negative kinetic energy we will discuss several toy models that will include the analog of a background gravitational field that switches sign between positive and negative kinetic energy. The physical question is, what do observers in the gravity region detect about the presence of a negative kinetic energy sector? Conceptually this is the analog of a black box being probed by in/out signals detected at the exterior of the box.
In the field theory or particle examples discussed below a simple sign function that is modeled after the antigravity loop in [@BST-antigravity] captures the main effect of antigravity. This sign function is a simple device to answer questions that arose repeatedly on unitarity and possible instability and is not necessarily a solution to the gravitational field equations of some specific model. Rather, it is used here only to capture the main effect of an antigravity sector in a simple and solvable model. In the case of realistic applications one would need to use a self consistent solution of matter and gravitational equations (as in [@BST-antigravity]) as long as it captures the main features of antigravity as in the simplified model background discussed here.
Particle with time dependent kinetic energy flips \[particleFlip\]
------------------------------------------------------------------
A free particle with a relativistic (or non-relativistic) Hamiltonian that switches sign as a function of time provides an example of a system propagating in a background gravitational field that switches sign as in Eq.(\[g-hat\]) $$H=\varepsilon\left( \left\vert t\right\vert -\frac{\Delta}{2}\right)
\times\sqrt{p^{2}+m^{2}}\;\text{ or \ }H=\varepsilon\left( \left\vert
t\right\vert -\frac{\Delta}{2}\right) \times\frac{p^{2}}{2m},
\label{Hparticle}$$ where $\varepsilon\left( u\right) \equiv$Sign$\left( u\right) .$ Such a background captures some of the properties of the antigravity loop of Bars-Steinhardt-Turok [@BST-antigravity]. The particle’s phase space $\left( x,p\right) $ can also represent more generally a typical generalized degree of freedom in field theory or string field theory.
The momentum $p$ is conserved since $H$ is independent of $x,$ but the velocity $\dot{x}=\partial H/\partial p=\varepsilon\left( \left\vert
t\right\vert -\frac{\Delta}{2}\right) \frac{p}{\sqrt{p^{2}+m^{2}}}$ alternates signs as shown below. The Hamiltonian is time dependent, so it is not conserved.$$\begin{tabular}
[c]{|l|c|c|c|}\hline
$t:$ & $t<-\frac{\Delta}{2}$ & $-\frac{\Delta}{2}<t<\frac{\Delta}{2}\;\;$ &
$\;\;t>\frac{\Delta}{2}\;\;$\\\hline
$H_{\pm}:$ & $\sqrt{p^{2}+m^{2}}$ & $-\sqrt{p^{2}+m^{2}}$ & $\sqrt{p^{2}+m^{2}}$\\\hline
$x:$ & $\dot{x}=\frac{p}{\sqrt{p^{2}+m^{2}}}$ & $\dot{x}=-\frac{p}{\sqrt
{p^{2}+m^{2}}}$ & $\dot{x}=\frac{p}{\sqrt{p^{2}+m^{2}}}$\\\hline
\end{tabular}
\ \ \ \ \ \ $$ At the $t=\pm\Delta/2$ kinks the velocity vanishes if we define $\varepsilon
\left( 0\right) =0.$ It is possible to make other models of what happens to the velocity by replacing the sign function $\varepsilon\left( z\right) $ by other time dependent kinky or smooth models; for example, if we replace $\varepsilon\left( z\right) $ by $\left( \varepsilon\left( z\right)
\right) ^{-1},$ then the velocity at the kinks changes sign at an infinite value rather than at zero, while the momentum remains a constant in all cases$.$
If the initial position before entering antigravity is $x_{i}\left(
t_{i}\right) ,$ we compute the evolution at any time as follows (see Fig.1) $$\begin{array}
[c]{cc}t<-\frac{\Delta}{2}\;\;\;\;\;: & \;x\left( t\right) =x_{i}\left(
t_{i}\right) +\frac{p}{\sqrt{p^{2}+m^{2}}}\left( t-t_{i}\right) \\
-\frac{\Delta}{2}<t<\frac{\Delta}{2}: & \;x\left( t\right) =x\left(
-\frac{\Delta}{2}\right) -\frac{p}{\sqrt{p^{2}+m^{2}}}\left( t+\frac{\Delta
}{2}\right) \\
t>\frac{\Delta}{2}\;\;\;\;\;\;: & \;x\left( t\right) =x\left( \frac{\Delta
}{2}\right) +\frac{p}{\sqrt{p^{2}+m^{2}}}\left( t-\frac{\Delta}{2}\right)
\end{array}
\label{method}$$ where $x\left( -\frac{\Delta}{2}\right) =x_{i}\left( t_{i}\right)
+\frac{p}{\sqrt{p^{2}+m^{2}}}\left( -\frac{\Delta}{2}-t_{i}\right) ,$ and $x\left( \frac{\Delta}{2}\right) =x_{i}\left( t_{i}\right) -\frac{p}{\sqrt{p^{2}+m^{2}}}\left( 3\Delta/2+t_{i}\right) .$ The final position $x_{f}\left( t_{f}\right) ,$ at a time $t_{f}$ after waiting long enough to exit from antigravity, $t_{f}>\Delta/2,$ is$$x_{f}\left( t_{f}\right) =x_{i}\left( t_{i}\right) +\frac{p}{\sqrt
{p^{2}+m^{2}}}\left( t_{f}-t_{i}-2\Delta\right) . \label{delayedParticle}$$ The effect of antigravity during the interval, $-\frac{\Delta}{2}<t<\frac{\Delta}{2},$ is the backward excursion between the two kinks shown in Fig.1 $.$ For observers waiting for the arrival of the particle at some position $x_{f}\left( t_{f}\right) ,$ we see from Eq.(\[delayedParticle\]) that antigravity causes a time delay by the amount of $2\Delta$ as compared to the absence of antigravity. Hence there is a measurable signal, namely a time delay, as an observable effect in comparing the presence and absence of antigravity.
[Fig-particleAntigravity.eps]{}\
Fig.1 - Propagation through antigravity.
A similar problem is analyzed at the quantum level by computing the transition amplitude from an initial state $|x_{i},t_{i}\rangle$ to a final state $|x_{f},t_{f}\rangle,$ requiring that the final observation is in the gravity period, *after* passing through the antigravity period. This is given by $$\begin{array}
[c]{l}A_{fi}=\langle x_{f},t_{f}|e^{-\frac{i}{\hbar}H_{+}\left( t_{f}-\frac{\Delta
}{2}\right) }e^{-\frac{i}{\hbar}H_{-}\left( \frac{\Delta}{2}-\frac{-\Delta
}{2}\right) }e^{-\frac{i}{\hbar}H_{+}\left( \frac{-\Delta}{2}-t_{i}\right)
}|x_{i},t_{i}\rangle\\
=\langle x_{f},t_{f}|e^{-\frac{i}{\hbar}H\left( t_{f}-t_{i}-2\Delta\right)
}|x_{i},t_{i}\rangle\\
=\sqrt{\frac{m}{2\pi i\hbar\left( t_{f}-t_{i}-2\Delta\right) }}\exp\left(
\frac{i~m~\left( x_{f}-x_{i}-2\Delta\right) ^{2}}{2\hbar\left( t_{f}-t_{i}-2\Delta\right) }\right)
\end{array}$$ The last expression is for the case of a non-relativistic particle with $H_{\pm}=\pm H=\pm p^{2}/2m.$ The exponentials involving $H_{\pm}$ are simplified because $H_{\pm}$ commute with each other, allowing the combination of the exponentials into a single exponential. Thus, the effect of the intermediate antigravity period is to cause only a time delay just as in the classical solution above. Note also that there are no unitarity problems; the evolution operator is unitary, and norms of states are positive, at all stages.
Particle with space dependent kinetic energy flips
--------------------------------------------------
Consider a non-relativistic particle with a total energy Hamiltonian that switches sign in different regions of space, for example $$H=\varepsilon\left( \left\vert x\right\vert -\frac{\Delta}{2}\right)
\times\left( \frac{p^{2}}{2m}+V\left( x\right) \right) .$$ This is another example of a system propagating in a background gravitational field that switches sign as in Eq.(\[g-hat\]). In this case energy is conserved since there is no explicit time dependence in $H$. Therefore at generic energies, $E=\left( \frac{p^{2}}{2m}+V\left( x\right) \right) $, the particle cannot cross the boundaries at $\left\vert x\right\vert
=\frac{\Delta}{2}$ since the Hamiltonian would flip sign and this would contradict the energy conservation. Hence if the particle is in the the gravity region, $\left\vert x\right\vert >\frac{\Delta}{2},$ it stays there, and if it is in the antigravity region, $\left\vert x\right\vert <\frac
{\Delta}{2},$ it stays there. However, the particle can cross from gravity to antigravity and back again to gravity at zero energy $\frac{p^{2}}{2m}+V\left( x\right) =0.$ This is similar to the geodesics in a black hole that cross from gravity to antigravity [@ABJ-blackhole].
Free massless scalar field with sign flipping kinetic energy \[NoMass\]
-----------------------------------------------------------------------
Consider a free massless scalar field in flat space with a time dependent background field that causes sign flips of the kinetic energy as a function of time$$S=-\frac{1}{2}\int d^{4}x\;\varepsilon\left( \left\vert x^{0}\right\vert
-\frac{\Delta}{2}\right) \;\partial^{\mu}\phi\left( x\right) \partial_{\mu
}\phi\left( x\right) .$$ The factor $\varepsilon\left( \left\vert x^{0}\right\vert -\frac{\Delta}{2}\right) $ can be viewed as a gravitational background field of the form, $\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,$ with $g^{\mu\nu
}\left( x\right) =\varepsilon\left( \left\vert x^{0}\right\vert
-\frac{\Delta}{2}\right) \eta^{\mu\nu}$ and $\sqrt{-g}=1.$ This sign flipping metric should be regarded as an example of a geometry that spans the union of the gravity and antigravity regions, as in Eq.(\[g-hat\]). We proceed to analyze the time evolution of this system. Let the on-shell initial field configuration at time $x_{i}^{0}<\left( -\Delta/2\right) $ be defined by $$\phi_{i}\left( \vec{x}_{i},x_{i}^{0}\right) =\int\frac{d^{3}p}{\left(
2\pi\right) ^{3/2}2\left\vert p\right\vert }\left( a\left( \vec{p}\right)
e^{-i\left\vert p\right\vert x_{i}^{0}+i\vec{p}\cdot\vec{x}_{i}}+\bar
{a}\left( \vec{p}\right) e^{i\left\vert p\right\vert x_{i}^{0}-i\vec{p}\cdot\vec{x}_{i}}\right)$$ The general solution for $\phi\left( \vec{x},x^{0}\right) $ evolved up to a final time $x_{f}^{0}>\Delta/2$ is then given by (using the method in Eq.(\[method\]))$$\phi_{f}\left( \vec{x}_{f},x_{f}^{0}\right) =\int\frac{d^{3}p}{\left(
2\pi\right) ^{3/2}2\left\vert p\right\vert }\left(
\begin{array}
[c]{c}a\left( \vec{p}\right) e^{-i\left\vert p\right\vert \left( x_{f}^{0}-x_{i}^{0}-2\Delta\right) +i\vec{p}\cdot\vec{x}_{f}}\\
+\bar{a}\left( \vec{p}\right) e^{i\left\vert p\right\vert \left( x_{f}^{0}-x_{i}^{0}-2\Delta\right) -i\vec{p}\cdot\vec{x}_{f}}\end{array}
\right) .$$ This shows that for initial/final observations, that are strictly outside of the antigravity period, the effect of the antigravity period is only a time delay as compared to the complete absence of antigravity. The time evolution of the field in the interim period is just like the time evolution of the particle as shown in Fig.1. For more details on the classical evolution of the field in the interim period see the case of the massive field in section (\[Mass\]), and take the zero mass limit.
An important remark is that the multiparticle Hilbert space $\left\{ |\vec
{p}_{1},\vec{p}_{2}\cdots\vec{p}_{n}\rangle\right\} $ is the Fock space constructed from the creation operators applied on the vacuum defined by $a\left( \vec{p}\right) |0\rangle=0,$ namely $|\vec{p}_{1},\vec{p}_{2}\cdots\vec{p}_{n}\rangle\equiv\bar{a}\left( \vec{p}_{1}\right) \bar
{a}\left( \vec{p}_{2}\right) \cdots\bar{a}\left( \vec{p}_{n}\right)
|0\rangle.$ This time independent Fock space is the complete Hilbert space that can be used during gravity or antigravity. It is clearly unitary since it is the same Hilbert space that is independent of the existence of an antigravity period (i.e. same as the $\Delta=0$ case). This shows that there is no unitarity problem due to the presence of the antigravity period.
However, there is negative kinetic energy during antigravity, seen as follows. The time dependent Hamiltonian for this system is $$H\left( x^{0}\right) =\left\{
\begin{array}
[c]{l}+H,\;\text{for }t<-\frac{\Delta}{2}\\
-H,\;\text{for }-\frac{\Delta}{2}<t<\frac{\Delta}{2}\\
+H,\;\text{for }t>\frac{\Delta}{2}\end{array}
\right.$$ where $H,$ which is constructed from the quantum creation-annihilation operators as usual, is time independent. So there seems to be a possible source of instability due to negative energy during antigravity. For freely propagating particles there are no transitions that alter the energy, so no questions arise, it is only when there are interactions that an effect may be observed due to transitions created by the negative energy sector. The effect of interactions, as observed by detectors in the gravity sector, is analogous to the case of a time dependent Hamiltonian as discussed in simple examples below in section (\[interact\]). Hence, the presence of a sector with negative kinetic energy is not a fundamental problem in the quantum theory.
Nevertheless, the antigravity sector, with or without interactions, is the source of interesting physical signals for the observers in the gravity sectors. For example, in the absence of additional interactions, consider the quantum propagator that corresponds to initial/final states in the two gravity sectors $\left\vert x^{0}\right\vert >\Delta/2$. The transition amplitude from an initial state in gravity $\left( x_{i}^{0}<-\Delta/2\right) $ to a final state in gravity $\left( x_{f}^{0}>\Delta/2\right) $, after the field evolves through antigravity, is given by$$\begin{aligned}
A_{fi} & =\langle\phi_{f}\left( x_{f}\right) |e^{-\frac{i}{\hbar}H_{+}\left( t_{f}-\frac{\Delta}{2}\right) }e^{-\frac{i}{\hbar}H_{-}\left(
\frac{\Delta}{2}-\frac{-\Delta}{2}\right) }e^{-\frac{i}{\hbar}H_{+}\left(
\frac{-\Delta}{2}-t_{i}\right) }|\phi_{i}\left( x_{i}\right) \rangle\\
& =\langle\phi_{f}\left( \vec{x}_{f},x_{f}^{0}\right) |e^{-\frac{i}{\hbar
}H\left( t_{f}-t_{i}-2\Delta\right) }|\phi_{i}\left( \vec{x}_{i},x_{i}^{0}\right) \rangle\end{aligned}$$ Here, $|\phi\left( x\right) \rangle$ is defined as the 1-particle state in the quantum theory which is created by applying the quantum field $\hat{\phi
}\left( x\right) $ on the oscillator vacuum $a\left( \vec{p}\right)
|0\rangle=0,$ $$|\phi\left( \vec{x},x^{0}\right) \rangle=\hat{\phi}\left( x\right)
|0\rangle=\int d^{3}p\frac{e^{i\left\vert p\right\vert x^{0}-i\vec{p}\cdot
\vec{x}}}{\left( 2\pi\right) ^{3/2}2\left\vert p\right\vert }\bar{a}\left(
\vec{p}\right) |0\rangle.$$ Then we obtain $$A_{fi}=\int d^{3}p\frac{e^{i\left\vert p\right\vert \left( x_{f}^{0}-x_{i}^{0}-2\Delta\right) -i\vec{p}\cdot\left( \vec{x}_{f}-\vec{x}_{i}\right) }}{\left( 2\pi\right) ^{3/2}2\left\vert p\right\vert }.$$ This is the propagator for a free massless particle. From this expression it is clear that the effect of antigravity on the result for the transition amplitude $A_{fi}$ is only a time delay by an amount of $2\Delta$ as compared to the same quantity in the complete absence of antigravity$.$ The same general statement holds true for the transition amplitudes for multi-particle states. Clearly there is no particle production due to antigravity in the case of free massless particles. This will be contrasted with the case of massive particles in section (\[Mass\]).
Of course, if there are field interactions, there will be additional effects, but none of those are à priori problematic from the point of view of fundamental principles.
Particle with flipping kinetic energy while interacting in a potential \[interact\]
-----------------------------------------------------------------------------------
To learn more about the effects of antigravity we now add an interaction term that does not flip sign during antigravity. We first investigate the case of a single degree of freedom whose kinetic energy flips sign during antigravity. This phase space $\left( x,p\right) $ should be thought of as a generalized coordinate associated with any single degree of freedom within local field theory or string field theory (after integrating out all other degrees of freedom), but in its simplest form it can be regarded as representing a particle moving in one dimension.
We discuss a simple model described by the Hamiltonian $$H=\varepsilon\left( \left\vert t\right\vert -\frac{\Delta}{2}\right)
\times\frac{p^{2}}{2m}+\frac{m\omega^{2}}{2}x^{2}.$$ This is a time dependent Hamiltonian that has two different forms, $H_{\pm},$ during different periods of time as follows$$\begin{tabular}
[c]{|l|l|l|l|}\hline
$t:$ & $t<-\frac{\Delta}{2}$ & $-\frac{\Delta}{2}<t<\frac{\Delta}{2}\;\;$ &
$\;\;t>\frac{\Delta}{2}\;\;$\\\hline
$H_{\pm}:$ & $\left( \frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}\right) $ &
$\left( -\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}\right) $ & $\left(
\frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}\right) $\\\hline
\end{tabular}$$ During gravity, $\left\vert t\right\vert >\frac{\Delta}{2},$ the Hamiltonian $H_{+}$ is the familiar harmonic oscillator Hamiltonian with a well defined quantum state, so all energies are positive. But during antigravity, $-\frac{\Delta}{2}<t<\frac{\Delta}{2},$ the Hamiltonian $H_{-}$ has no bottom, so all positive and negative energies are permitted. Does this pose an instability problem for the entire system? The answer is that, as in the simpler cases already illustrated above, there is no such problem from the perspective of observers in gravity.
A complete basis for a unitary Hilbert space may be defined to be the positive norm complete Fock space associated with the usual harmonic oscillator Hamiltonian $H_{+}$ whose energy eigenvalues are strictly positive. The eigenstates of $H_{-}$ are also positive norm and define another complete unitary basis. Clearly one complete basis may be expanded in terms of another complete basis, so the usual Fock space basis is sufficient to analyze the complete system, including its evolution through antigravity. This shows that the interacting problem that includes antigravity is an ordinary time dependent problem in quantum mechanics. There are no unitarity problems, and the presence of antigravity is analyzed below as a regular problem of a time dependent Hamiltonian, without encountering any fundamental problems of principles.
A technical remark may be useful: this model can be treated group theoretically by using the properties of SL$(2,R)$ representations. Note that the three Hermitian quantum operators $\left( x^{2},p^{2},\frac{1}{2}\left(
xp+px\right) \right) $ form the algebra of SL$\left( 2,R\right) $ under quantum commutation rules $\left[ x,p\right] =i\hbar$. The Hamiltonian $H_{+}$ is proportional to the compact generator $J_{0}$ of SL$\left(
2,R\right) ,$ while $H_{-}$ is proportional to one of the non-compact generators $J_{1}$. The second non-compact generator $J_{2}$ appears in the commutator $\left[ H_{+},H_{-}\right] .$ Explicitly, $$J_{0}\equiv\frac{1}{2\hbar\omega}H_{+},\;J_{1}\equiv\frac{1}{2\hbar\omega
}H_{-},\;J_{2}=\frac{1}{4\hbar}\left( xp+px\right) .$$ The $\left( J_{0},J_{1},J_{2}\right) $ form the standard Lie algebra of SL$\left( 2,R\right) .$ Since these $J_{0,1,,2}$ are Hermitian operators, the corresponding quantum states which are labelled as $|j,\mu\rangle$ form a unitary representation of SL$\left( 2,R\right) .$ The quantum number $\mu$ is associated with the eigenvalues of $J_{0}$ (which is basically the eigenvalues of $H_{+})$ while $j\left( j+1\right) $ is associated with the eigenvalues of Casimir operator $C_{2}$ that commutes with all the generators, $C_{2}\equiv J_{0}^{2}-J_{1}^{2}-J_{2}^{2}.$ For the present construction, keeping track of the orders of operators $\left( x,p\right) $ one finds that $C_{2}$ is a constant, $C_{2}=-3/16=j\left( j+1\right) ,$ which yields two solutions $j=-\frac{3}{4}$ or $j=-\frac{1}{4}.$ Hence the spectrum of this theory, including the properties of $H_{\pm}$ can be thought of consisting of two infinite dimensional irreducible unitary representations of SL$\left(
2,R\right) .$ For $j=-\frac{3}{4}$ or $j=-\frac{1}{4}$ these are positive discrete series representations. The allowed values of $\mu$ are given by $\mu=j+1+k$ where $k=0,1,2,\cdots$ is an integer. We see that the two representations taken together correspond to the spectrum of $H_{+}$, which is the spectrum of the harmonic oscillator given by $E_{n}=\omega\left(
n+\frac{1}{2}\right) \Leftrightarrow2\omega\mu,$ with even $n=2k$ corresponding to $j=-3/4$ and odd $n=2k+1$ corresponding to $j=-1/4.$ Hence the basis $|j,\mu\rangle$ form a complete set of eigenstates for the observers in the gravity sector of the theory.
How about the antigravity sector? Since the corresponding Hamiltonian is $H_{-},$ a complete set of eigenstates corresponds to diagonalizing the non-compact generator $J_{1}$ instead of the compact generator $J_{0}.$ Either way the Casimir operator is the same; hence diagonalizing $J_{1}\rightarrow q$ provides another unitary basis $|j,q\rangle$ for the same unitary representations of SL$\left( 2,R\right) .$ The spectrum of $J_{1},$ $J_{1}|j,q\rangle=q$ $|j,q\rangle,$ is continuous $q$ on the real line since this is a non-compact generator of SL$\left( 2,R\right) .$ This antigravity basis is also a complete unitary basis for this Hamiltonian that includes both sectors $H_{\pm}$. One basis can be expanded in terms of the other, $|j,q\rangle=\sum_{\mu=j+1}^{\infty}|j,\mu\rangle\langle j,\mu|j,q\rangle,$ where the expansion coefficients $\langle j,\mu|j,q\rangle=U_{\mu,q}^{\left(
j\right) }$ is a unitary transformation for each $j=-\frac{3}{4}$ or $-\frac{1}{4}$.
Therefore it doesn’t matter which basis we use to analyze the quantum properties of this Hamiltonian. Using the discrete basis $|j,\mu\rangle$ which is more convenient to analyze the physics in the gravity sector, in no way excludes the antigravity sector from making its effects felt for observers in the gravity sector.
With this understanding of this simple quantum system, we now analyze the transition amplitudes $A_{fi}$ for an initial state $|i\rangle$ to propagate to a final state both in the gravity sector. We define $|i\rangle,|f\rangle$ at the two edges of the antigravity sector, at times $t_{i}=-\Delta/2$ and $t_{f}=\Delta/2$. Moving $t_{i},t_{f}$ to other arbitrary times in the gravity sector is trivial since we can write $|i\rangle=e^{-iH_{+}\left(
-\Delta/2-t_{i}\right) }|i,t_{i}\rangle$ and $|f\rangle=e^{-iH_{+}\left(
t_{f}-\Delta/2\right) }|f,t_{f}\rangle$ and we know how $H_{+}$ acts on any linear combination of harmonic oscillator states $|i\rangle,|f\rangle$. Hence we have $$A_{fi}=\langle f|e^{-\frac{i}{\hbar}\Delta H_{-}}|i\rangle$$ where $|i\rangle,|f\rangle$ are arbitrary states in the gravity sector. If we take any two states in the SL$\left( 2,R\right) $ basis $|j,\mu\rangle$, this becomes$$A_{fi}=\langle j,\mu_{f}|e^{-i\frac{\Delta}{2\omega}J_{1}}|j,\mu_{i}\rangle.$$ This is just the matrix representation of a group element of SL$\left(
2,R\right) $ in a unitary representation labelled by $j=-\frac{3}{4}$ or $-\frac{1}{4}.$ It must be the same $j$ for both the initial and final states, i.e. there is a selection rule because there can be no transitions at all from $j=-\frac{3}{4}$ to $j=-\frac{1}{4}$ and vice-versa.
This quantity can be computed by using purely group theoretical means, but it is perhaps more instructive to use the standard harmonic oscillator creation/annihilation operators to evaluate it. Then we can write $$H_{+}=\hbar\omega\left( a^{\dagger}a+\frac{1}{2}\right) ,\;\;H_{-}=\frac{\hbar\omega}{2}\left( a^{\dagger2}+a^{2}\right) .$$ We have used this form to compute the transition amplitude $$A_{fi}=\langle f|e^{-\frac{i}{\hbar}\Delta H_{-}}|i\rangle=\langle
f|e^{-i\frac{\omega\Delta}{2}\left( a^{\dagger2}+a^{2}\right) }|i\rangle,$$ by taking initial/final states to be the number states or the coherent states of the harmonic oscillator. To perform the computation we use the following identity$$e^{-i\frac{\omega\Delta}{2}\left( a^{\dagger2}+a^{2}\right) }=e^{-\frac
{i}{2}\tanh\left( \omega\Delta\right) a^{\dagger2}}\left( \cosh\left(
\omega\Delta\right) \right) ^{-\left( a^{\dagger}a+\frac{1}{2}\right)
}e^{-\frac{i}{2}\tanh\left( \omega\Delta\right) a^{2}}.$$ For initial/final coherent states, $|z_{i}\rangle$ & $|z_{f}\rangle$ for observers in gravity, we define the transition amplitude for normalized states as, $A\left( z_{f},z_{i}\right) =\langle z_{f}|e^{-\frac{i}{\hbar}\Delta
H_{-}}|z_{i}\rangle/\sqrt{\langle z_{f}|z_{f}\rangle\langle z_{i}|z_{i}\rangle},$ which yields $$\left\vert A\left( z_{f},z_{i}\right) \right\vert ^{2}=\frac{e^{-\left\vert
z_{f}\right\vert ^{2}-\left\vert z_{i}\right\vert ^{2}+\frac
{2\operatorname{Re}\left( z_{i}\bar{z}_{f}\right) }{\cosh\left(
\omega\Delta\right) }}e^{\tanh\left( \omega\Delta\right) \operatorname{Im}\left( \bar{z}_{f}^{2}e^{-i\omega\Delta}+z_{i}^{2}e^{i\omega\Delta}\right)
}}{\cosh\left( \omega\Delta\right) }.$$ This should be compared to the absence of antigravity when $\Delta=0$, namely $\left\vert A\left( z_{f},z_{i}\right) \right\vert ^{2}\overset
{\Delta\rightarrow0}{=}e^{-\left\vert z_{f}-z_{i}\right\vert ^{2}}.$
Similarly, for initial/final number eigenstates $|n\rangle$ & $|m\rangle$ of the Hamiltonian $H_{+}=\left( \frac{p^{2}}{2m}+\frac{m\omega^{2}x^{2}}{2}\right) =\hbar\omega\left( a^{\dagger}a+\frac{1}{2}\right) \;$for observers in gravity, we obtain $$A_{mn}=\sqrt{\frac{m!n!e^{i\omega\Delta(n+m+1)}}{\left( \cosh\left(
\omega\Delta\right) \right) ^{m+n+1}}}\sum_{k=0}^{\min\left( m,n\right)
}\frac{\left( \frac{1}{2i}\sinh\left( \omega\Delta\right) \right)
^{\frac{m+n}{2}-k}}{k!\left( \frac{m-k}{2}\right) !\left( \frac{n-k}{2}\right) !}\text{,}$$ where $\left( m,n,k\right) $ are all even or all odd. This gives $$\left\vert A_{mn}\right\vert ^{2}=\left(
\begin{array}
[c]{c}\left( _{2}F_{1}(-\left[ \frac{m}{2}\right] ,-\left[ \frac{n}{2}\right]
;\left( 1-\frac{\left( -1\right) ^{m}}{2}\right) ;\frac{-1}{\sinh
^{2}\left( \omega\Delta\right) })\right) ^{2}\\
\times\frac{\left( m!\right) \left( n!\right) \left( \frac{1}{2}\tanh\left( \omega\Delta\right) \right) ^{2\left( \left[ \frac{m}{2}\right] +\left[ \frac{n}{2}\right] \right) }}{\left( \left[ \frac
{m}{2}\right] !\left[ \frac{n}{2}\right] !\right) ^{2}~\left(
\cosh\left( \omega\Delta\right) \right) ^{2-\left( -1\right) ^{m}}}\end{array}
\right)$$ where $_{2}F_{1}\left( a,b;c;z\right) $ is the hypergeometric function, $\left[ \frac{m}{2}\right] $ means the integer part of $m/2$, and $\left(
m,n\right) $ are both even or both odd. Special cases are$$\begin{array}
[c]{c}\left\vert A_{00}\right\vert ^{2}=\frac{1}{\cosh\left( \omega\Delta\right)
},\ \;\left\vert A_{2M,0}\right\vert ^{2}=\frac{\left( 2M\right) !}{2^{2M}\left( M!\right) ^{2}}\frac{\left( \tanh\left( \omega\Delta\right)
\right) ^{2M}}{\cosh\left( \omega\Delta\right) },\\
\left\vert A_{11}\right\vert ^{2}=\frac{1}{\cosh^{3}\left( \omega
\Delta\right) },\;A_{2M+1,1}=\frac{\left( 2M+1\right) !}{2^{2M}\left(
M!\right) ^{2}}\frac{\left( \tanh\left( \omega\Delta\right) \right)
^{2M}}{\cosh^{3}\left( \omega\Delta\right) }.
\end{array}$$ As compared to the absence of antigravity, $\Delta=0,$ when there are no transitions, we see that antigravity causes an observable effect. Clearly, these transition amplitudes are well behaved, and do not blow up for large $\Delta.$ Unitarity is obeyed: one may verify explicitly that the sum over all states is 100% probability, $\sum_{m}\left\vert A_{mn}\right\vert ^{2}=1,$ for all fixed $n,$ and similarly $\int\frac{d^{2}z_{f}}{\pi}~\left\vert
A\left( z_{f},z_{i}\right) \right\vert ^{2}=1$ for all fixed $z_{i}.$
Massive scalar field with sign flipping kinetic energy \[Mass\]
---------------------------------------------------------------
This system has some similarities to the interacting particle above but it is not quite the same. The action is$$S=\frac{1}{2}\int d^{d}x\;\left[ -\varepsilon\left( \left\vert
x^{0}\right\vert -\frac{\Delta}{2}\right) \;\partial^{\mu}\phi\left(
x\right) \partial_{\mu}\phi\left( x\right) -m^{2}\phi^{2}\left( x\right)
\right]$$ As in the case of the massless field in section (\[NoMass\]), the factor $\varepsilon\left( \left\vert x^{0}\right\vert -\frac{\Delta}{2}\right) $ can be viewed as a gravitational background field of the form, $\sqrt
{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi,$ with $g^{\mu\nu}\left(
x\right) =\varepsilon\left( \left\vert x^{0}\right\vert -\frac{\Delta}{2}\right) \eta^{\mu\nu}$ and $\sqrt{-g}=1,$ that spans the union of the gravity and antigravity regions, as explained in Eq.(\[g-hat\]). The mass term does not flip sign. Note that, due to the non-zero mass, this is *not* a Weyl invariant action, but we will investigate it anyway to learn about the properties of such a system.
In momentum space, using the notation $x^{0}=t$, we have$$\phi\left( \vec{x},t\right) =\int\frac{d^{d-1}p}{\left( 2\pi\right)
^{\left( d-1\right) /2}}\phi_{p}\left( t\right) e^{i\vec{p}\cdot\vec{x}}$$ We rewrite the action in momentum space as$$S=\frac{1}{2}\int dt\int d^{d-1}p\;\left[
\begin{array}
[c]{c}\varepsilon\left( \left\vert t\right\vert -\frac{\Delta}{2}\right) \left[
\begin{array}
[c]{c}\dot{\phi}_{p}\left( t\right) \dot{\phi}_{-p}\left( t\right) \\
-\vec{p}^{2}\phi_{p}\left( t\right) \phi_{-p}\left( t\right)
\end{array}
\right] \\
-m^{2}\phi_{p}\left( t\right) \phi_{-p}\left( t\right)
\end{array}
\right]$$ The equation of motion is $$\partial_{t}\left( \varepsilon\left( \left\vert t\right\vert -\frac{\Delta
}{2}\right) \partial_{t}\phi_{p}\left( t\right) \right) +\left[
\varepsilon\left( \left\vert t\right\vert -\frac{\Delta}{2}\right) \vec
{p}^{2}+m^{2}\right] \phi_{p}\left( t\right) =0$$ The solutions in separate regions of time are (similar to (\[method\]))$$\begin{array}
[c]{c}t<-\frac{\Delta}{2}:\;\phi_{p}^{A}\left( t\right) =\left(
\begin{array}
[c]{c}A_{p}^{+}e^{-i\sqrt{\vec{p}^{2}+m^{2}}\left( t+\frac{\Delta}{2}\right) }\\
+A_{p}^{-}~e^{i\sqrt{\vec{p}^{2}+m^{2}}\left( t+\frac{\Delta}{2}\right) }\end{array}
\right) \\
-\frac{\Delta}{2}<t<\frac{\Delta}{2}:\phi_{p}^{B}\left( t\right) =\left(
\begin{array}
[c]{c}B_{p}^{+}~e^{-i\sqrt{\vec{p}^{2}-m^{2}}\left( t+\frac{\Delta}{2}\right) }\\
+B_{p}^{-}e^{i\sqrt{\vec{p}^{2}-m^{2}}\left( t+\frac{\Delta}{2}\right) }\end{array}
\right) \\
t>\frac{\Delta}{2}:\;\phi_{p}^{C}\left( t\right) =\left(
\begin{array}
[c]{c}C_{p}^{+}~e^{-i\sqrt{\vec{p}^{2}+m^{2}}\left( t-\frac{\Delta}{2}\right) }\\
+C_{p}^{-}e^{i\sqrt{\vec{p}^{2}+m^{2}}\left( t-\frac{\Delta}{2}\right) }\end{array}
\right)
\end{array}$$ We need to match the field $\phi_{p}\left( t\right) $ and its canonical momentum, $\varepsilon\left( \left\vert t\right\vert -\frac{\Delta}{2}\right) \partial_{t}\phi_{p}\left( t\right) ,$ at each boundary $t=\pm\Delta/2$$$\begin{array}
[c]{c}\phi_{p}^{A}\left( -\frac{\Delta}{2}\right) =\phi_{p}^{B}\left(
-\frac{\Delta}{2}\right) ,\text{ and }\dot{\phi}_{p}^{A}\left( -\frac
{\Delta}{2}\right) =-\dot{\phi}_{p}^{B}\left( -\frac{\Delta}{2}\right) ,\\
\phi_{p}^{C}\left( +\frac{\Delta}{2}\right) =\phi_{p}^{B}\left(
+\frac{\Delta}{2}\right) ,\text{ and }\dot{\phi}_{p}^{C}\left( +\frac
{\Delta}{2}\right) =-\dot{\phi}_{p}^{B}\left( +\frac{\Delta}{2}\right) ,
\end{array}$$ Note the sign flip of $\dot{\phi}$ at $t=\pm\Delta/2$ although the canonical momentum does not flip. This gives four equations to relate $C_{p}^{\pm}$ and $B_{p}^{\pm}$ to $A_{p}^{\pm}$ as follows$$\begin{array}
[c]{l}A_{p}^{+}+A_{p}^{-}=B_{p}^{+}+B_{p}^{-}\\
A_{p}^{+}-A_{p}^{-}=-\left( B_{p}^{+}~-B_{p}^{-}\right) \frac{\sqrt{\vec
{p}^{2}-m^{2}}}{\sqrt{\vec{p}^{2}+m^{2}}}\\
C_{p}^{+}+C_{p}^{-}=B_{p}^{+}e^{-i\sqrt{\vec{p}^{2}-m^{2}}\Delta}+B_{p}^{-}e^{i\sqrt{\vec{p}^{2}-m^{2}}\Delta}\\
C_{p}^{+}-C_{p}^{-}=-\left(
\begin{array}
[c]{c}B_{p}^{+}e^{-i\sqrt{\vec{p}^{2}-m^{2}}\Delta}\\
-B_{p}^{-}e^{i\sqrt{\vec{p}^{2}-m^{2}}\Delta}\end{array}
\right) \frac{\sqrt{\vec{p}^{2}-m^{2}}}{\sqrt{\vec{p}^{2}+m^{2}}}\end{array}$$ The solution determines $B_{p}^{\pm}$ and $C_{p}^{\pm}$ in terms of $A_{p}^{\pm},$ $$\begin{aligned}
\left(
\begin{array}
[c]{c}C_{p}^{+}\\
C_{p}^{-}\end{array}
\right) & =\left(
\begin{array}
[c]{cc}\alpha & \beta^{\ast}\\
\beta & \alpha^{\ast}\end{array}
\right) \left(
\begin{array}
[c]{c}A_{p}^{+}\\
A_{p}^{-}\end{array}
\right) \label{transBeta}\\
\left(
\begin{array}
[c]{c}B_{p}^{+}\\
B_{p}^{-}\end{array}
\right) & =\left(
\begin{array}
[c]{cc}\frac{1}{2}-\frac{\sqrt{\vec{p}^{2}+m^{2}}}{2\sqrt{\vec{p}^{2}-m^{2}}} &
\;\;\frac{1}{2}+\frac{\sqrt{\vec{p}^{2}+m^{2}}}{2\sqrt{\vec{p}^{2}-m^{2}}}\\
\frac{1}{2}+\frac{\sqrt{\vec{p}^{2}+m^{2}}}{2\sqrt{\vec{p}^{2}-m^{2}}} &
\;\;\frac{1}{2}-\frac{\sqrt{\vec{p}^{2}+m^{2}}}{2\sqrt{\vec{p}^{2}-m^{2}}}\end{array}
\right) \left(
\begin{array}
[c]{c}A_{p}^{+}\\
A_{p}^{-}\end{array}
\right)\end{aligned}$$ where $\left( \alpha,\beta\right) $ are the parameters of a Bogoliubov transformation (an SU$\left( 1,1\right) $ group transformation)$$\begin{array}
[c]{l}\alpha=\cos\left( \Delta\sqrt{\vec{p}^{2}-m^{2}}\right) +i\frac{\vec{p}^{2}\sin\left( \Delta\sqrt{\vec{p}^{2}-m^{2}}\right) }{\sqrt{\left( \vec
{p}^{2}\right) ^{2}-m^{4}}},\\
\beta=i\frac{m^{2}\sin\left( \Delta\sqrt{\vec{p}^{2}-m^{2}}\right) }{\sqrt{\left( p^{2}\right) ^{2}-m^{4}}},\\
\left\vert \alpha\right\vert ^{2}-\left\vert \beta\right\vert ^{2}=1.
\end{array}$$ Assume the incoming state $\phi_{p}^{A}\left( t\right) $ has only positive frequency, meaning $A_{p}^{-}=0.$ Then we see that (unlike the massless case in section (\[NoMass\])) negative frequency fluctuations are produced in the final state $\phi_{p}^{C}\left( t\right) $ since according to Eq.(\[transBeta\]), $C_{p}^{-}=\beta A_{p}^{+}.$ The corresponding probability amplitude for particle production is $$\left( C_{p}^{-}/A_{p}^{+}\right) =\beta=i\frac{\sin\left( m\Delta
\sqrt{\vec{p}^{2}/m^{2}-1}\right) }{\sqrt{\left( \vec{p}^{2}/m^{2}\right)
^{2}-1}}.$$ The produced particle number density (particles per unit volume) is the integral of $\left\vert \beta\right\vert ^{2}$ over all momenta$$\begin{array}
[c]{c}n\left( m,\Delta\right) =\int d^{d-1}p\left\vert \beta\right\vert ^{2}=\int
d^{d-1}p\frac{\sin^{2}\left( m\Delta\sqrt{\left( \vec{p}^{2}/m^{2}\right)
-1}\right) }{\left\vert \left( \vec{p}^{2}/m^{2}\right) ^{2}-1\right\vert
},\\
=m^{d-1}\Omega_{d-1}\int_{0}^{\infty}\frac{x^{d-2}\sin^{2}\left( \left(
m\Delta\right) \sqrt{x^{2}-1}\right) }{\left\vert x^{4}-1\right\vert }dx,
\end{array}$$ where $x^{2}=\vec{p}^{2}/m^{2},$ while $\Omega_{d-1}$ is the volume of the solid angle in $d-1$ dimensions, $\Omega_{2}=2\pi,\;\Omega_{3}=4\pi,$ etc.. This is a convergent integral for $d<\left( 5-\varepsilon\right) $ dimensions, hence $n\left( m,\Delta\right) $ is finite for $d=1,2,3,4$ dimensions. We note that the number density $n\left( m,\Delta\right) $ increases monotonically at fixed $m$ as $\Delta$ increases. The energy density per unit volume for the produced particles for all momenta is$$\begin{array}
[c]{c}\rho\left( m,\Delta\right) =\int\frac{d^{d-1}p}{\left( 2\pi\right) ^{d-1}}\sqrt{\vec{p}^{2}+m^{2}}\left\vert \beta\right\vert ^{2}\\
\;\;\;\;=\frac{m^{d}\Omega_{d-1}}{\left( 2\pi\right) ^{d-1}}\int_{0}^{\infty}\frac{x^{d-2}\sqrt{x^{2}+1}\sin^{2}\left( \left( m\Delta\right)
\sqrt{x^{2}-1}\right) }{\left\vert x^{4}-1\right\vert }dx
\end{array}$$ $\rho\left( m,\Delta\right) $ is convergent for $d<\left( 4-\varepsilon
\right) $ dimensions, and is logarithmically divergent at $d=4$ despite the rapid oscillations at the ultraviolet limit.
Recall that the massive field is not a scale invariant model. In the Weyl symmetric limit, $m\rightarrow0,$ there is no particle production at all in any dimension. In the scale invariant theory masses for fields must come from interactions, such as interactions with the Higgs field. In a cosmological context the Higgs field is not just a constant, and therefore in the type of investigation above, the parameter $m$ should be replaced by the cosmological behavior of the Higgs field (see [@BCT-geodesics] for an example). This very different behavior in a Weyl invariant theory should be the more serious approach for investigating effectively massive fields to answer the type of questions discussed in this section.
Conformally exact sign-flipping backgrounds in string theory
============================================================
We consider the worldsheet formulation of the relativistic string, but we make string theory consistent with target space Weyl symmetry as suggested in [@BST-string]. This requires promoting the string tension to a dynamical field, $\left( 2\pi\alpha^{\prime}\right) ^{-1}\rightarrow T\left( X^{\mu
}\left( \tau,\sigma\right) \right) $. The background field $T\left(
X\right) ,$ along with any other additional background fields, must be restricted to satisfy exact worldsheet conformal symmetry at the quantum level. In the worldsheet formalism, typically the tension appears together with the metric $g_{\mu\nu}\left( X\left( \tau,\sigma\right) \right) $ or antisymmetric tensor $b_{\mu\nu}\left( X\left( \tau,\sigma\right) \right)
$ in the Weyl invariant combination, $Tg_{\mu\nu}$ or $Tb_{\mu\nu}.$ The requirement of exact *worldsheet* conformal symmetry constrains these target-space Weyl invariant combinations. Perturbative worldsheet conformal symmetry (vanishing beta functions) is captured by the properties of the low energy effective string action. From the study of the Weyl invariant and geodesically complete formalism of the low energy string action [@BST-string] we have learned that the tension (closely connected to the gravitational constant) switches sign generically near the singularities in the classical solutions of this theory. If we fix the target space Weyl symmetry by choosing the string gauge as in Eq.(\[EsGuages\]), then in those generic solutions, the tension becomes $T\left( X^{\mu}\left( \tau
,\sigma\right) \right) =\pm\left( 2\pi\alpha^{\prime}\right) ^{-1}$ on the two sides of the singularity as it appears *in the string gauge*. Those two sides are identified as the gravity/antigravity sectors of the low energy theory as discussed in section (\[frames\]). From the perspective of the worldsheet string theory these observations lead to a simple prescription to capture all these effects in the string gauge, namely replace the Weyl invariant structures $\left( Tg_{\mu\nu},Tb_{\mu\nu}\right) $ by $\left(
\pm\left( 2\pi\alpha^{\prime}\right) ^{-1}G_{\mu\nu}^{\pm},\pm\left(
2\pi\alpha^{\prime}\right) ^{-1}B_{\mu\nu}^{\pm}\right) ,$ where the capital $\left( G_{\mu\nu}^{\pm}\left( X\right) ,B_{\mu\nu}^{\pm}\left( X\right)
\right) $ are the background fields on the gravity/antigravity patches that are joined at the singularities *as they appear in the string gauge*. We may absorb the overall $\pm$ due to the signs of the tension into a redefinition of the background fields, and as we did for the Einstein gauge in Eq.(\[g-hat\]), define $$\left( \hat{G}_{\mu\nu}\left( X\right) ,\hat{B}_{\mu\nu}\left( X\right)
\right) =\left( \pm G_{\mu\nu}^{\pm}\left( X\right) ,\pm B_{\mu\nu}^{\pm
}\left( X\right) \right) ,$$ as the full set of background fields in the union of the gravity/antigravity sectors of the worldsheet string theory. Of course, $\left( \hat{G}_{\mu\nu
}\left( X\right) ,\hat{B}_{\mu\nu}\left( X\right) \right) $ are required to satisfy worldsheet conformal invariance at the quantum level as usual. What is new is the geodesic completeness of the background fields $\left( \hat
{G}_{\mu\nu}\left( X\right) ,\hat{B}_{\mu\nu}\left( X\right) \right) $ which is achieved by the sign flipping tension and the union of the corresponding gravity/antigravity sectors.
String in flat background with tension that flips sign \[flat\]
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A simple example of a conformally exact worldsheet CFT, that includes a dynamical string tension that flips signs, is the flat string background $\eta_{\mu\nu}$ modified only by a time dependent string tension $T\left(
X\right) =\frac{1}{2\pi\alpha^{\prime}}$Sign$\left( \left\vert X^{0}\left(
\tau,\sigma\right) \right\vert -\frac{\Delta}{2}\right) .$ This can also be presented in the string gauge by absorbing the sign of the tension into a redefined metric$$\hat{G}_{\mu\nu}\left( X\right) =\eta_{\mu\nu}\text{Sign}\left( \left\vert
X^{0}\left( \tau,\sigma\right) \right\vert -\frac{\Delta}{2}\right) ,\text{
\ }\hat{B}_{\mu\nu}\left( X\right) =0,$$ where $\Delta$ is a constant. Note the similarity to Eq.(\[Hparticle\]) or sections (\[NoMass\],\[Mass\]). Thus the tension is positive when $\left\vert X^{0}\left( \tau,\sigma\right) \right\vert >\frac{\Delta}{2}$ and negative when $-\frac{\Delta}{2}<X^{0}\left( \tau,\sigma\right)
<\frac{\Delta}{2}.$ This is also similar to the cosmological example with an antigravity loop given in [@BST-string], but we have greatly simplified it here by keeping only the signs but not the magnitude of the tension, thus defining a conformally exact rather than a conformally approximate CFT on the worldsheet. The corresponding worldsheet string model is $$S=-\frac{1}{4\pi\alpha^{\prime}}\int d^{2}\sigma\sqrt{-h}h^{ab}\partial
_{a}X^{\mu}\partial_{b}X^{\nu}\eta_{\mu\nu}\text{Sign}\left( \left\vert
X^{0}\left( \tau,\sigma\right) \right\vert -\frac{\Delta}{2}\right) .
\label{newFlatString}$$
We should mention that it is also possible to consider a model, at least at the classical level, by inserting in the action (\[newFlatString\]) the inverse of the Sign function $\left( \text{Sign}\left( \left\vert
X^{0}\left( \tau,\sigma\right) \right\vert -\frac{\Delta}{2}\right)
\right) ^{-1}.$ In this case the tension flips sign when it is infinite rather than zero. Both of these possibilities occur smoothly rather than suddenly in cosmological backgrounds in string theory (see Eq.(30) in [@BST-string] or its generalizations). Both behaviors are significant from the perspective of string theory because perturbative versus non-perturbative methods would be needed to understand fully the physics in the vicinity of the gravity/antigravity transitions. Namely, when the tension at the transition is large the string would be close to being pointlike, so the stringy corrections would be small and perturbative in the vicinity of the gravity/antigravity transitions; by contrast when the tension at the transition is small the string would be floppy so stringy corrections could be significant. In the latter case, high spin fields [@Vasiliev] may be an interesting tool to investigate the gravity/antigravity transition in our setting.
From the form of the action in Eq.(\[newFlatString\]) it is evident that the string action is invariant under reparametrizations of the worldsheet at the classical level. We will use this symmetry to choose a gauge to perform the classical analysis below. But eventually we also need to know if this symmetry is valid also at the quantum level. The generator of this gauge symmetry is the stress tensor, so the stress tensor vanishes as a constraint to impose the gauge invariance. At the classical level the stress tensor does vanish as part of the solution of the classical equations and constraints (see below). At the quantum level, in ‘covariant quantization, the stress tensor does not vanish on all states but only on the gauge invariant physical states. For consistency of covariant quantization one must verify that the constraints form a set of first class constraints that close under quantum operator products. In our case the stress tensor derived from (\[newFlatString\]) has the form $T_{\pm\pm}=($Sign)$\times T_{\pm\pm}^{0}$ , where $T_{\pm\pm}^{0}$ is the usual worldsheet stress tensor in the flat background $\eta_{\mu\nu}$, while the sign factor switches signs at the kinks $\left\vert X^{0}\left( \tau,\sigma\right) \right\vert =\frac{\Delta}{2}$. In the positive (gravity, Sign=+) region, we have symbolically the operator products, $T_{\pm\pm}^{0}\times T_{\pm\pm}^{0}\sim T_{\pm\pm}^{0},$ where the standard CFT result on the right hand side is computed exactly for the flat string. Similarly, in the negative (antigravity, Sign=$-$) region we have, $\left( -T_{\pm\pm}^{0}\right) \times\left( -T_{\pm\pm}^{0}\right)
\sim-\left( -T_{\pm\pm}^{0}\right) .$ So the algebra is closed like the standard CFT locally in the positive and negative regions away from the kinks. There remains analyzing the operator products at the kinks $\left\vert
X^{0}\left( \tau,\sigma\right) \right\vert =\frac{\Delta}{2}$ (worldsheet analogs of the kinks in Fig.1). The operator products involving the Sign factor non-trivially introduce delta functions and derivatives of delta functions multiplied by the sign factor or its derivatives that have support only at the kinks. At one contraction (order $\hbar$ effects) the coefficient of the delta function includes the flat $T_{\pm\pm}^{0}$ or its derivatives evaluated at the kinks. Since $T_{\pm\pm}^{0}$ or its derivatives are in the list of first class constraints (Virasoro operators), this is still a closed algebra of first class constraints all of which vanish on physical states. At two contractions (order $\hbar^{2}$ in quantum effects) there are again some terms that contain $T_{\pm\pm}^{0}$ or its derivatives, which again are of no concern since these still vanish on physical states. However, there are also additional operators of the form of $\left( \partial X^{0}\right) $ multiplying products of the sign function, delta function, or its derivatives, all evaluated at the kinks. We have analyzed these complicated distributions and found that they vanish when integrated with $\left( \partial
X^{0}\right) $, so they do not seem to contribute. Similarly we can drop several similar terms due to the properties of the distributions. The analysis at the kinks becomes harder at higher contractions ($\hbar^{3}$ and beyond in quantum effects), and we leave this for future analysis to be reported at a later stage. The main point is that if there are additional constraints that must be imposed at the kinks they will show up in this type of operator product analysis. So far, we have not found new constraints up to two contractions in the operator products. Thus, the algebra of the operator products is basically the standard algebra of a conformal field theory (CFT) locally in the positive and negative regions away from the kinks. The modification of the CFT algebra at the kinks with terms that are proportional to Virasoro operators does not change the validity of the gauge symmetry at the quantum level, since those terms vanish on physical states anyway. Although we have not yet found other operator modifications of the algebra at the kinks, conceptually it is possible that such terms may arise at higher contractions or in other models that include gravity/antigravity transitions. When and if such terms appear, they must be included in an enlarged list of constraints that should form a closed algebra under operator products, then this will define the proper quantum theory.
In this paper our aim is to first understand the classical theory of a string described by the action in (\[newFlatString\]), so we do not need to be concerned here about the subtleties described in the previous paragraph. In fact the classical analysis that we give below is helpful in further developing the right approach for the quantum theory. Thus, setting aside temporarily the possible stringy corrections, we are at first interested in the classical behavior of strings as they propagate in the union of the gravity/antigravity regions, and later try to figure out the possible additional effects due to interactions at those transitions by using more sophisticated methods, such as string field theory, or others, as outlined in section \[sft\].
### General string propagating classically through antigravity
In this section we will discuss the properties of the model in Eq.(\[newFlatString\]). The main objective is to show that there are no problems due to the negative tension during antigravity from the point of view of fundamental principles, such as unitarity or possible instability due to negative kinetic energy. The unitarity of this string model was already established in [@BST-string] more generally for any time dependent tension $T\left( X^{0}\right) ,$ and more general metric, so we will not repeat it here. We will concentrate on the effect of the antigravity period on the propagation of the string and the corresponding signals that observers in gravity may detect. As we will demonstrate, as compared to the complete absence of antigravity, the presence of an antigravity period for a certain amount of time causes only a time delay in the propagation of an open or closed free string of any configuration. This may seem surprising since, at first thought, one may think that string bits would fly apart under an instability caused by a negative string tension. In fact, this does not happen because a negative tension is simply an overall sign in the action of a free string, and this does not change the equations of motion and constraints of a free string during antigravity.
We work in the conformal gauge at the classical level. There is a remaining reparametrization symmetry that permits the further choice of the following time-like gauge$$X^{0}\left( \tau,\sigma\right) =\left\vert H\right\vert \tau,$$ where $H$ is the total time dependent Hamiltonian of the string while $\left\vert H\right\vert $ is time independent. This is similar to the massless free field in section (\[NoMass\]). In this gauge the remaining degrees of freedom satisfy the following equations of motion and constraints $$\begin{array}
[c]{c}\left( \partial_{\tau}^{2}-\partial_{\sigma}^{2}\right) \vec{X}\left(
\tau,\sigma\right) =0,\\
H^{2}=\left( \partial_{\tau}\vec{X}\pm\partial_{\sigma}\vec{X}\right) ^{2},
\end{array}$$ to be solved in each time region $A,B,C$ defined by $$\text{ }A:\text{\ }\tau\left\vert H\right\vert <-\Delta/2,\;B:-\Delta
/2<\tau\left\vert H\right\vert <\Delta/2,\ \;C:\tau\left\vert H\right\vert
>\Delta/2.$$ Furthermore, the solutions for $\vec{X}_{A,B,C}\left( \tau,\sigma\right) $ and the canonical momenta $\vec{P}_{A,B,C}\left( \tau,\sigma\right)
=\partial_{\tau}\vec{X}_{A,B,C}\left( \tau,\sigma\right) \times$Sign$\left(
\left\vert H\tau\right\vert -\frac{\Delta}{2}\right) $ should be continuous at the boundaries $\tau\left\vert H\right\vert =\pm\Delta/2.$ The method of solution follows the simple model in Eq.(\[method\]) or the massive field in Eq.(\[Mass\]).
We will discuss the case of an open string; the closed string is treated similarly. The general solution in each region is given in terms of the center of mass $\left( \vec{q},\vec{p}\right) $ and oscillator $\left( \vec
{\alpha}_{n},n=\pm1,\pm2,\cdots\right) $ degrees of freedom. The general configuration of the string in the positive tension region $A$, at a time $\tau<-\Delta/2,$ is a general solution $\vec{X}_{A}\left( \tau
,\sigma\right) $ given by $$\vec{X}_{A}\left( \tau,\sigma\right) =\vec{q}_{0}+\vec{p}\tau+\sum
_{n=-\infty,\neq0}^{\infty}\frac{i}{n}\vec{\alpha}_{n}\cos n\sigma~e^{-in\tau
}.$$ The time independent parameters $\left( \vec{q}_{0},\vec{p}\right) $ and $\left( \vec{\alpha}_{n},n=\pm1,\pm2,\cdots\right) $ determine the initial configuration of the string at the time $\tau=\tau_{0}.$ From the constraint equations we compute the time independent $\left\vert H\right\vert $ and the remaining constraint $$\begin{array}
[c]{c}\left\vert H\right\vert =\sqrt{\vec{p}^{2}+\sum_{n=1}^{\infty}\vec{\alpha
}_{-n}\cdot\vec{\alpha}_{n}}\\
0=\vec{p}\cdot\vec{\alpha}_{n}+\frac{1}{2}\sum_{m=-\infty,\neq0}^{\infty}\vec{\alpha}_{-m}\cdot\vec{\alpha}_{n+m}\end{array}
\label{constr-string}$$ Thus the time dependent Hamiltonian that switches sign is $$H\left( \tau\right) =\text{Sign}\left( \left\vert H\right\vert \left\vert
\tau\right\vert -\frac{\Delta}{2}\right) \sqrt{\vec{p}^{2}+\sum_{n=1}^{\infty}\vec{\alpha}_{-n}\cdot\vec{\alpha}_{n}}.$$ Assuming the constraints (\[constr-string\]) are satisfied at the classical level by some set of parameters $\left( \vec{\alpha}_{n},\vec{p}\right) ,$ the momentum, $\overrightarrow{P}_{A}=\overrightarrow{\dot{X}}_{A},$ in region $A$ is $$\overrightarrow{P}_{A}\left( \tau,\sigma\right) =\vec{p}+\sum_{n=-\infty
,\neq0}^{\infty}\vec{\alpha}_{n}\cos n\sigma~e^{-in\tau}.$$
In region $B,$ $-\frac{\Delta}{2}<\tau\left\vert H\right\vert <\frac{\Delta
}{2},$ the solution $\left( \vec{X}_{B},\vec{P}_{B}\right) $ takes the same form as above, but with a new set of parameters $\left( \vec{q}_{B},\vec
{p}_{B},\vec{\alpha}_{nB}\right) .$ Note that in this region there is a non-trivial minus sign in the relation between momentum and velocity, $\vec
{P}_{B}\left( \tau,\sigma\right) =-\partial_{\tau}\vec{X}_{B}\left(
\tau,\sigma\right) .$ At the transition time, $\tau_{\ast}\equiv-\frac
{\Delta}{2\left\vert H\right\vert },$ we must match the position and momentum, therefore $\vec{X}_{A}\left( \tau_{\ast},\sigma\right) =\vec{X}_{B}\left(
\tau_{\ast},\sigma\right) $ and $\partial_{\tau}\vec{X}_{A}\left( \tau
_{\ast},\sigma\right) =-\partial_{\tau}\vec{X}_{B}\left( \tau_{\ast},\sigma\right) ,$ noting the negative sign in the case of velocities. Because the matching is for every value of $\sigma$ we find that all the parameters $\left( \vec{q}_{B},\vec{p}_{B},\vec{\alpha}_{nB}\right) $ are uniquely determined in terms of the initial parameters $\left( \vec{q}_{0},\vec
{p},\vec{\alpha}_{n}\right) $ in region $A$. So the solution in region $B$ is $$\begin{aligned}
\vec{X}_{B}\left( \tau,\sigma\right) & =\left(
\begin{array}
[c]{c}\vec{q}_{0}+\vec{p}\left( -\tau-\frac{\Delta}{\left\vert H\right\vert
}\right) \\
+\sum_{n=-\infty,\neq0}^{\infty}\frac{i}{n}\vec{\alpha}_{n}\cos n\sigma
~e^{-in\left( -\tau-\frac{\Delta}{\left\vert H\right\vert }\right) }\end{array}
\right) \\
\vec{P}_{B}\left( \tau,\sigma\right) & =-\overrightarrow{\dot{X}}_{B}\left( \tau,\sigma\right) =\vec{p}+\sum_{n=-\infty,\neq0}^{\infty}\vec{\alpha}_{n}\cos n\sigma~e^{-in\left( -\tau-\frac{\Delta}{\left\vert
H\right\vert }\right) }$$ There are no new constraints beyond those that are already assumed to have been satisfied in region $A$ by the parameters $\left( \vec{\alpha}_{n},\vec{p}\right) $. Note the structure $(-\tau-\frac{\Delta}{\left\vert
H\right\vert })$ that indicates a backward propagation similar to Fig.1 as $\tau$ increases beyond $\tau_{\ast}$.
At the next transition time, $\tau_{\ast\ast}\equiv+\frac{\Delta}{2\left\vert
H\right\vert },$ we must connect the solution $\left( \vec{X}_{B},\vec{P}_{B}\right) $ above to the solution $\left( \vec{X}_{C},\vec{P}_{C}\right)
$ in region $C,$ $\tau>\tau_{\ast\ast},$ which is given in terms of a new set of parameters $\left( \vec{q}_{C},\vec{p}_{C},\vec{\alpha}_{nC}\right) .$ Using the matching conditions $\vec{X}_{C}\left( \tau_{\ast},\sigma\right)
=\vec{X}_{B}\left( \tau_{\ast},\sigma\right) $ and $\overrightarrow{\dot{X}}_{C}\left( \tau_{\ast},\sigma\right) =-\overrightarrow{\dot{X}}_{B}\left(
\tau_{\ast},\sigma\right) $ that include the extra minus sign for velocities (as discussed above), we find that $\left( \vec{q}_{C},\vec{p}_{C},\vec{\alpha}_{nC}\right) $ are all determined again uniquely in terms of the initial parameters $\left( \vec{q}_{0},\vec{p},\vec{\alpha}_{n}\right) $ introduced in region $A.$ $$\begin{array}
[c]{c}\vec{X}_{C}\left( \tau,\sigma\right) =\left(
\begin{array}
[c]{c}\vec{q}_{0}+\vec{p}\left( \tau-2\frac{\Delta}{\left\vert H\right\vert
}\right) \\
+\sum_{n=-\infty,\neq0}^{\infty}\frac{i}{n}\vec{\alpha}_{n}\cos n\sigma
~e^{-in\left( \tau-2\frac{\Delta}{\left\vert H\right\vert }\right) }\end{array}
\right) \\
\vec{P}_{C}\left( \tau,\sigma\right) =\vec{p}+\sum_{n=-\infty,\neq0}^{\infty}\vec{\alpha}_{n}\cos n\sigma~e^{-in\left( \tau-2\frac{\Delta
}{\left\vert H\right\vert }\right) }\end{array}$$
For closed strings we find a similar result but with some additional information for region $B.$ Namely, given some solution in region $A$ that satisfies the string equations of motion and constraints, then the solution in regions $B,C$ are obtained by the following substitutions of $\tau$ and $\sigma$ $$\begin{array}
[c]{l}\vec{X}_{B}\left( \tau,\sigma\right) =\vec{X}_{A}\left( -\tau-\frac{\Delta
}{\left\vert H\right\vert },~-\sigma\right) \\
\vec{X}_{C}\left( \tau,\sigma\right) =\vec{X}_{A}\left( \tau-2\frac{\Delta
}{\left\vert H\right\vert },~\sigma\right)
\end{array}$$ Note the extra minus sign in $\sigma\rightarrow-\sigma$ in region $B$. Namely, for the closed string the left and right movers get scrambled during antigravity. For the open string with Neumann boundary conditions discussed above, the sign flip $\sigma\rightarrow-\sigma$ in region $B$ has no effect since $\cos\left( -n\sigma\right) =+\cos\left( n\sigma\right) $, but if the open string had Dirichlet boundary conditions then $\sin\left(
-n\sigma\right) =-\sin\left( n\sigma\right) $ would induce an overall sign flip of the oscillations during the antigravity period.
Putting it all together, we see that after the antigravity period$,$ the emergent string experiences only a time delay $2\Delta/\left\vert H\right\vert
$ as compared to the string that propagates in the complete absence of antigravity. This is the same conclusion that was reached for the free particle or the free massless field.
### Rotating rod propagating through antigravity
As a concrete example of a string configuration that satisfies all the constraints, we present the rotating rod solution that is modified by a tension that flips sign during antigravity as in Eq.(\[newFlatString\]). We begin with a straight string lying along the $\hat{x}$ axis with its center of mass located at $\vec{q}_{0},$ as given by, $\vec{X}_{0}\left( \sigma\right)
=\vec{q}_{0}+\hat{x}\;R_{0}\cos\sigma.$ Let this string rotate in the $\left(
\hat{x},\hat{y}\right) $ plane and translate in the $\hat{z}$ direction as follows $$\vec{X}_{A}\left( \tau,\sigma\right) =\vec{q}_{0}+\hat{z}p\tau+R_{0}\cos\sigma~\left( \hat{x}\cos\tau+\hat{y}\sin\tau\right) .$$ This satisfies the constraints in Eq.(\[constr-string\]), since $\partial_{\tau}\vec{X}\cdot\partial_{\sigma}\vec{X}=0,$ and gives $\left\vert
H\right\vert =\left( p^{2}+R_{0}^{2}\right) ^{1/2}.$ Following the steps above we compute the matching string configuration during the antigravity period $-\frac{\Delta}{2\left\vert H\right\vert }<\tau<\frac{\Delta
}{2\left\vert H\right\vert }$$$\vec{X}_{B}\left( \tau,\sigma\right) =\left(
\begin{array}
[c]{c}\vec{q}_{0}+\hat{z}p\left( -\tau-\theta\right) \\
+R_{0}\cos\sigma\left(
\begin{array}
[c]{c}\hat{x}~\cos\left( -\tau-\theta\right) \\
+\hat{y}~\sin\left( -\tau-\theta\right)
\end{array}
\right)
\end{array}
\right)$$ where $\theta=\Delta\left( p^{2}+R_{0}^{2}\right) ^{-1/2},$ noting that this describes a backward propagation similar to Fig.1. Finally the matching string configuration in the time period $\tau>\frac{\Delta}{2\left\vert H\right\vert
}$ is $$\vec{X}_{C}\left( \tau,\sigma\right) =\left(
\begin{array}
[c]{c}\vec{q}_{0}+\hat{z}p\left( \tau-2\theta\right) \\
+R_{0}\cos\sigma~\left( \hat{x}\cos\left( \tau-2\theta\right) +\hat{y}\sin\left( \tau-2\theta\right) \right)
\end{array}
\right) .$$
As promised, as compared to the complete absence of antigravity, the presence of an antigravity period for a certain amount of time causes only a time delay in the propagation of a string of any configuration. The string bits of a freely propagating string do not fly apart during antigravity when the string tension is negative!
2D black hole including antigravity \[2dBH\]
--------------------------------------------
Another simple example is the 2-dimensional black hole [@WittenBH] based on the SL$\left( 2,R\right) /$R gauged WZW model [@IB-SL2R-R]. The well known string background metric in this case is, $ds^{2}=-2\left( 1-uv\right)
^{-1}dudv,$ with $uv<1,$ where $\left( u,v\right) $ are the string coordinates $X^{\mu}\left( \tau,\sigma\right) $ in the Kruskal-Szekeres basis. This space is geodesically incomplete similar to the case of the four dimensional Schwarzschild blackhole [@ABJ-blackhole].
The geodesically complete modification consists of allowing the string tension to flip sign precisely at the singularity, namely $T\left( X\right) =\left(
2\pi\alpha^{\prime}\right) ^{-1}$Sign$\left( 1-uv\right) .$ Then the new geodesically complete 2D-blackhole action is$$S=\frac{1}{2\pi\alpha^{\prime}}\int d^{2}\sigma\sqrt{-h}h^{ab}\frac
{\partial_{a}u\partial_{b}v}{\left\vert 1-uv\right\vert }.$$ This differs from the old 2D black hole action by the absolute value sign, and includes the antigravity region $uv>1$ just as the 4-dimensional case [@ABJ-blackhole]. Despite the extra sign, this model is an exact CFT on the worldsheet as can be argued in the same way following Eq.(\[newFlatString\]). Properties of the new 2D black hole, including the related dilaton and all orders quantum corrections in powers of $\alpha
^{\prime},$ will be investigated in detail in a separate paper [@ABJ2-2Dblackhole].
String field theory with antigravity \[sft\]
============================================
In the neighborhood of the gravity/antigravity transition, which occurs typically at a gravitational singularity, a proper understanding of the physics would be incomplete without the input of quantum gravity that may possibly contribute large quantum effects. How should we estimate the effects of quantum gravity?
We first point out that attempting to use an effective low energy field theory that includes higher powers of curvature, such as those computed from string theory, is the wrong approach. Higher powers of curvature capture approximations to quantum gravity that are valid at momenta much smaller than the Planck scale; those cannot be used to investigate the phenomena of interest that are at the Planck scale close to the singularity. For investigating the gravity/antigravity transition more closely, we do not see an alternative to using directly an appropriate theory of quantum gravity that can incorporate the geodesically complete spacetime that includes both gravity and antigravity regions. Hence we first need to define the proper theory of quantum gravity that is consistent with geodesic completeness. As far as we know this notion of quantum gravity was first considered in [@BST-string].
Assuming that quantum string theory is a suitable approach to quantum gravity, we outline here how string field theory may be modified to take into account geodesic completeness and the presence of an antigravity sector, so that it can be used as a proper tool to answer the relevant questions.
Open and closed string field theory (SFT) is a formalism for computing string-string interactions, including those that involve stringy gravitons. As in standard field theory, in principle the SFT formalism is suitable for both perturbative and non-perturbative computations. Technically SFT is hard to compute with, but it has the advantage of being a self consistent and conceptually complete definition of quantum gravity and the interactions with matter. It is therefore crucial to see how antigravity fits in SFT and therefore how the pertinent questions involving antigravity can be addressed in a self consistent manner.
In the context of SFT, gravitational and other backgrounds in which strings propagate are incorporated through the BRST operator $Q$ that appears in the quadratic part of the action [@WittenBH] $$S_{open}=Tr\left[ \frac{1}{2}AQA+\frac{g}{3}A\star A\star A\right] .
\label{SFTaction}$$ The complete SFT action must also include closed strings, $S_{closed}.$ The supersymmetric versions of these may also be considered. Here $A\left(
X\right) $ is the string field, the product $\star$ describes string joining or splitting, and the BRST operator $Q$ is given by$$Q=\int d\sigma\sum_{\pm}\left\{ c_{\pm}~T_{\pm\pm}\left( X\right) +b_{\pm
}c_{\pm}\partial c_{\pm}\right\} .$$ where $\left( b_{\pm},c_{\pm}\right) $ are the Fadeev-Popov ghosts, which is a device of covariant quantization, while $T_{\pm\pm}\left( X\left( \sigma\right) \right) $ is the stress tensor for left/right moving strings, associated to any conformal field theory (CFT) on the worldsheet that is conformally exact at the quantum level.
The gravitational and other backgrounds, including a dynamical tension that flips signs (i.e. incorporating antigravity) of the type we discussed in the previous sections, are included in the stress tensor $T_{\pm\pm}\left(
X\right) .$ If these backgrounds are not geodesically complete we expect that the SFT theory is incomplete since even at the classical level on the worldsheet there would be string solutions that would be incomplete just like particle geodesics that would be incomplete. Thus for a geodesically complete SFT we need to make sure that $T_{\pm\pm}\left( X\right) $ belongs to a geodesically complete worldsheet string model as described in the previous section. Examples of such string models were provided in sections (\[flat\],\[2dBH\]). Similarly one can construct many more geodesically complete backgrounds by allowing the string tension to change sign at singularities (and perhaps more generally) as long as the CFT conditions, that amount to $Q^{2}=0,$ are satisfied.
If the interactions in the SFT action (\[SFTaction\]) are neglected we do not expect dramatic effects due to the presence of antigravity since we have seen in the previous section the effect is only a time delay as compared to the complete absence of antigravity (as in sections \[delayedParticle\],\[NoMass\],\[flat\]). By including the interactions either perturbatively or non-perturbatively we can explore the effects of antigravity in the context of the quantum theory. In the previous sections, we have obtained a glimpse of the phenomena that could happen, including particle (or string) production (as in section \[Mass\]), excitations of various string states (as in section \[interact\]), and more dramatic phenomena that remain to be explored.
From the discussion in the first part of section (\[flat\]) one may gather that we are still in the process of addressing some technicalities in the construction of the BRST operator $Q$ for the simple model in that section. So we are not yet in a position to perform explicit computations, but we hope we have provided an outline of how one may formulate an appropriate theory to address and answer the relevant questions.
There may be alternative formalisms that could provide answers more easily than SFT, and of course those should be explored, but the advantage of SFT for being a conceptually complete and self consistent definition of the system, including the presence of antigravity as outlined above, is likely to remain as an important feature of this approach because of the overall perspective that it provides.
Comments
========
We have argued that a fundamental theory that could address the physical phenomena close to gravitational singularities, either in the form of field theory or string theory, is unlikely to be complete without incorporating geodesic completeness. The Weyl symmetric approach to the standard model coupled to gravity in Eq.(\[action\]), and the similar treatment of string theory [@BST-string], generally solves this problem and naturally requires that antigravity regions of spacetime should appear on the other side of gravitational singularities as integral parts of the spacetime described by a fundamental theory. There are other views that the notion of spacetime may not even exist at the extremes close to singularities. While acknowledging that there may be other scenarios that are little understood at this time, we believe that our concrete proposal merits further investigation.
While emphasizing that there are nicer Weyl gauges, we have shown how gravitational theories and string theories can be formulated in their traditional Einstein or string frames to include effects of a Weyl symmetry that renders them geodesically complete. A prediction of the Weyl symmetry is to naturally include an antigravity region of field space and spacetime that is geodesically connected to the traditional gravity spacetime at gravitational singularities. Precisely at the singularities that appear in the Einstein or string frames the gravitational constant or string tension flip sign suddenly (but smoothly in nicer Weyl gauges). As shown in section (\[frames\]), this sign can be absorbed into a redefinition of the metric in the Einstein or string frame, $\hat{g}_{\mu\nu}=\pm g_{\mu\nu}^{\pm},$ where $\hat{g}_{\mu\nu}$ describes the spacetime in the union of the gravity and antigravity regions. This definition of the complete spacetime may then be used to perform computations in the geodesically complete theory.
The appearance of negative kinetic energy terms for some degrees of freedom during antigravity was a source of concern. The arguments presented here show that this was a false alarm. We argued that unitarity is not an issue either in gravity or antigravity and that negative energy does not imply an instability of the theory as seen by observers in the gravity region (namely, observers like us, analyzing the universe). We made this point by studying many simple examples and we showed that observers in the gravity sector can deduce the existence and at least some properties of antigravity.
We have thus eliminated the initial concerns regarding unitarity or instability of the complete theory when there is an antigravity sector with negative kinetic energy. We have also demonstrated that there are very interesting physical phenomena associated with antigravity that remain to be explored concerning fundamental physics at the extremities of spacetime. These will have applications in cosmology as in [@BST-antigravity][@BST-conf][@BST-Higgs][@BST-string] and black hole physics as in [@ABJ-blackhole][@ABJ2-2Dblackhole].
We thank Edward Witten for encouraging us to investigate antigravity for a single degree of freedom, Ignacio J. Araya for discussions on all topics presented in this paper, and Neil Turok for useful remarks and comments.
[99]{}
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A. Codello, G. D’Odorico, C. Pagani, R. Percacci, Class. Quant. Grav. **30** (2013) 115015 \[arXiv:1210.3284\].
I. Bars, in preparation.
See for example S. Weinberg, The Quantum Theory of Fields, Volume III, Cambridge 2000, page 351.
I. Bars, Phys.Rev. **D82** (2010) 125025 \[arXiv:1008.1540\]
S. Ferrara, R. Kallosh, A. Linde, A. Marrani and A. Van Proeyen, Phys. Rev. **D83** (2011) 025008 \[arXiv:1008.2942\]; R. Kallosh and A.Linde JCAP **1306** (2013) 027 \[arXiv:1306.3211\] and JCAP **1306** (2013) 028 \[arXiv:1306.3214\].
I. Bars, P. Steinhardt, N. Turok, Fortsch. Phys. **62** (2014) 901 \[arXiv:1407.0992\].
I. Bars, Phys.Rev. **D77** (2008) 125027 \[arXiv:0804.1585\], see last part of Sec.(8).
I. Bars, Traversing Cosmological Singularities, Complete Journeys Through Spacetime Including Antigravity, arXiv:1209.1068.
I. Bars, P. Steinhardt, N. Turok, Phys.Lett. **B715** (2014) 278-281 \[arXiv:1112.2470\].
I. Bars, online book on Quantum Mechanics, page 358.
S. Gielen, N. Turok, A perfect bounce, arXiv: 1510.00699.
I. J. Araya, I. Bars, A. James, Journey Beyond the Schwarzschild Black Hole Singularity, arXiv:1510.03396.
I. Bars, P. Steinhardt, N. Turok, Phys.Lett. **B726** (2013) 50 \[arXiv:1307.8106\].
R. Jackiw and So-Young Pi, Phys.Rev. **D91** (2015) 6, 067501 \[arXiv:1407.8545\].
M. J. Duff and J. Kalkkinen Nucl. Phys.**B758** (2006) 161; *ibid* Nucl. Phys. **B760** (2007) 64.
I. Bars, Phys.Rev. **D79** (2009) 045009 \[arXiv:0810.2075\].
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[^1]: Partly based on conference lectures at String Field Theory-2015 (Sichuan Univ.), and Convergence (Perimeter Inst.).
[^2]: Partly based on conference lectures at String Field Theory-2015, Sichuan University, China, and at Convergence, Perimeter Institute, Waterloo, ON, Canada.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'WiFi technology has been used pervasively in fine-grained indoor localization, gesture recognition, and adaptive communication. Achieving better performance in these tasks generally boils down to differentiating Line-Of-Sight (LOS) from Non-Line-Of-Sight (NLOS) signal propagation reliably which generally requires expensive/specialized hardware due to the complex nature of indoor environments. Hence, the development of low-cost accurate positioning systems that exploit available infrastructure is not entirely solved. In this paper, we develop a framework for indoor localization and tracking of ubiquitous mobile devices such as smartphones using on-board sensors. We present a novel deep LOS/NLOS classifier which uses the Received Signal Strength Indicator (RSSI), and can classify the input signal with an accuracy of 85%. The proposed algorithm can globally localize and track a smartphone (or robot) with *a priori* unknown location, and with a semi-accurate prior map (error within $0.8 {\mathop{\mathrm{m}}}$) of the WiFi Access Points (AP). Through simultaneously solving for the trajectory and the map of access points, we recover a trajectory of the device and corrected locations for the access points. Experimental evaluations of the framework show that localization accuracy is increased by using the trained deep network; furthermore, the system becomes robust to any error in the map of APs.'
author:
- 'Sahib Singh Dhanjal, Maani Ghaffari, and Ryan M. Eustice[^1]'
bibliography:
- 'strings-abrv.bib'
- 'ieee-abrv.bib'
- 'refs.bib'
title: '**DeepLocNet: Deep Observation Classification and Ranging Bias Regression for Radio Positioning Systems** '
---
Introduction
============
Low-cost indoor localization solutions using radio signals such as WiFi and Bluetooth have long been studied. Radio signals are easily distorted by the presence of dynamic objects, the room temperature, dust, and even humidity. Furthermore, shadow fading and multipath propagation severely hinder the reliability of signal strength for ranging [@amiot2013pylayers]. The current state-of-art of radio-based positioning techniques [@pap0; @pap1] are broadly based on the following four distinct categories: (i) Received Signal Strength Indicator (RSSI), (ii) Angle Of Arrival (AOA), (iii) Time Of Arrival (TOA), and (iv) Physical Layer Information (PHY). For further details of the above approaches please see [@pap0].
![The Fetch Mobile Manipulator and the environment in which we performed the experiments. We manually operated the robot while recording its pose computed using a 2D laser range-finder and the WiFi signatures of all available APs throughout the trial. The recorded poses were used as ground truth and WiFi signatures used as measurements. Since we knew positions of all APs, we manually labeled all data as LOS/NLOS and calculated the Euclidean distances using the floor plan.[]{data-label="fig:fetch"}](fetch.jpg){width="0.99\columnwidth"}
Except (i), all of the above methods require specialized hardware for obtaining range measurements from WiFi Access Points (APs). This requirement limits the applicability of these methods in non-commercial applications. Once range measurements are obtained, positioning techniques such as spherical or hyperbolic positioning can be used to localize the device [@pap0]. In order to improve the positioning accuracy, one can combine measurements from multiple sensors such as GPS, magnetometer, and camera using filtering methods such as Kalman filtering or particle filtering [@thrun2005probabilistic]. In this work, we are concerned with scenarios where only the RSSI information using a commodity WiFi receiver is available. Such information is ubiquitous and available on even the cheapest smartphones nowadays. We develop a “light-weight” deep neural network for accurately classifying between Line-Of-Sight (LOS) and Non-LOS (NLOS) with the capability of easily being deployed on a smartphone.
The ranging methods mentioned earlier also heavily rely on one of the following two prerequisites about the environment: (i) construction of a radio fingerprint of the entire environment, or (ii) accurate map of the position of each AP. Both of these methods have their associated pros and cons. Radio fingerprinting requires an initial dry run in the environment to record information. The entire area is divided into a grid, and two data points are recorded for each location: (i) the RSSI from each AP, (ii) the standard deviation of each signal. This process is more robust than using a map; however, can be very cumbersome and is not conveniently scalable. Similarly, getting accurate maps of the environment is not always possible.
In this work, a semi-accurate initial map (location of APs within $0.8 {\mathop{\mathrm{m}}}$ from the ground-truth location) suffices for localization, relaxing the assumption of requiring an accurate initial map. The algorithm estimates the location of the APs, as well as the trajectory of the robot/device. In addition, we introduce a novel deep learning-based LOS/NLOS observation classification and ranging bias regression that is integrated within the widely used particle filtering-based localization and Simultaneous Localization And Mapping (SLAM) frameworks.
Contributions
-------------
The main contributions of this paper are as follows. First, we design a deep network to classify LOS/NLOS signal propagation with $85\%$ accuracy using the passive RSSI information only. The use of passive RSSI information assures that we will not require any custom hardware and the framework will work on any commodity smartphone/robot. The network is designed in a way that it can easily run on embedded devices such as most modern smartphones, without a lot of memory overhead. Second, we use the WiFi sensor model in a position estimation and mapping framework to provide a map of the APs as well as a trajectory of smartphone or robot. By utilizing our approach, the tedious process of WiFi fingerprinting, or the cost of additional specialized hardware is eliminated. Finally, we make the code for the framework, the deep network architecture, and wave propagation simulation environments open source:\
Outline
-------
A review of related works is given in the following section. Section III describes the problem statement and formulation. An overview of the proposed framework is given in Section IV, followed by experimental methodology and results in Section V. Finally, Section VI concludes the paper by discussing limitations and achievements of the proposed framework and providing ideas for future work.
Related Work
============
One of the earliest Radio Frequency (RF)-based localization methods, *RADAR* [@pap9], used fingerprinting and environment profiling with commodity hardware to provide for indoor localization. Building on top of that, *Horus*[@pap8] introduced a client-based probabilistic technique aiming to identify and address channel loss in a light-weight package. Even though accurate up to $0.6 {\mathop{\mathrm{m}}}$, the major drawback of this method is that localization occurs in two phases: (i) offline map building and clustering phase, (ii) online localization phase.
Similarly, *WiFi iLocate*[@pap19] describes a method based on Gaussian process regression and develops a press-to-go package, where an initial run is made for training the model and then online localization takes place. Going a step further, [@pap16] tracks footstep data and integrates it with RSSI based range measurements, IMU, and magnetometer to provide an accuracy of up to $0.9 {\mathop{\mathrm{m}}}$. In [@jadidi2017gaussian], the authors use a Gaussian processing-based method to classify RSSI data from Bluetooth Low Energy (BLE) beacons into LOS/NLOS which is then used in conjunction with an IMU and a particle filter for localization.
Most recent methods have started using the Physical Layer (PHY) information instead of the MAC Layer RSSI information. In [@pap7], the authors devise a new method called *PinLoc*, which is able to localize a device within a $1 {\mathop{\mathrm{m}}}\times 1 {\mathop{\mathrm{m}}}$ box. The main observation they made was that dynamic obstructions in the environment can be statistically reproduced. This fact, in turn, was used to detect LOS signals using Bayesian Inference.
A similar method is described in [@pap5], where the authors extract phase, transmission time, and strength information from the PHY layer and incorporate a classifier to localize. Most of the above methods are based on clustering approaches, which limits the capabilities of the system as there is no guarantee that the dataset will inherently contain clusters. To tackle this, the authors in [@pap14] came up with a unique statistic called the *Hopkins Statistic*, which measures the clustering tendency to recognize environments. Based on this clustering tendency, they were able to model the environments better, and in turn the location of the APs, leading to improved ranging. However, the method heavily relies on Multiple-Input-Multiple-Output (MIMO) techniques, with a number of antennas, and hence suffers from considerable hardware modification, limiting the practical applications.
Other methods such as AmpN [@pap13], leverage the use of amplitude information of the Channel State Information (CSI) from the PHY Layer. These methods measure standard properties such as the *kurtosis*, *Rician K-factor*, *skewness*, and variation [@pap13] among others for each of the CSI amplitudes, and then train a neural network for dynamic classification and recognition of LOS/NLOS signals. The authors of [@pap15] add another layer of filtering to better understand the property of LOS/NLOS propagation. In their work, *Bi-Loc*, they use phase errors in conjunction with the the CSI amplitude data, to propose a deep learning approach for fingerprinting. Using this, they have two modalities: Angle of Arrival and CSI, which is then used for fingerprinting. The authors of [@pap12] go as far as visualizing the CSI heatmap as an image, and then running a deep convolutional network on it to differentiate between LOS and NLOS.
Although the use of PHY level information has now provided means for more accurate LOS/NLOS classification, and in turn localization, the use of customized hardware *such as Network Interface Cards (NICs)* limits its application to truly mobile and ubiquitous devices (such as smartphones). Some of the issues, identified by these works, to enable real-time LOS identification are as listed below:
- Commodity WiFi devices fail to support precise Channel Impulse Response measurements due to limited operating bandwidth.
- Existing channel statistics-based features require large amount of samples, impeding real-time performance.
- Most LOS identification schemes are designed for stationary scenarios. Even those incorporating slight mobility fail in truly mobile cases.
- Requirement of custom hardware limits the application of these algorithms for truly mobile cases.
In this work, we bring the advances in SLAM to efficiently solve the indoor localization and tracking problem using sensors available in commonly used mobile devices. The main features that distinguishes this work from the available radio signal-based indoor positioning literature are as follows. We develop a deep neural network that can classify between LOS/NLOS using only RSSI information. Our framework also does not require the tedious process of fingerprinting (site survey), hence is more scalable. Moreover, the system is robust to errors ($\leq 0.8m$) in the map of access points due to the usage of FastSLAM to localize the device, as well as obtain a map of the environment.
Problem Formulation and Preliminaries
=====================================
We now define the problems we study in this paper and then briefly explain required preliminaries to solve these problems. Let $x_t \in \mathbb{R}^3$ be the device position at time $t$. The device is initially located at $x_0$ which is unknown and can only receive the RSSI of a broadcasted signal. Let $Z_t$ be the set of possible range measurements obtained from converting RSSI at time $t$. Given the set of known APs, we wish to solve the following problems:
*Problem 1 (Measurement Model):* The measurement model $p(z_t|x_t)$ is a conditional probability distribution that represents the likelihood of range measurements. We want to find the mapping from signal, $s_t$, to range measurements, $z_t$, and the likelihood function describing the measurement noise.
*Problem 2 (Positioning):* Let $z_{1:t} = \{ z_1,\dots, z_t\} $ be a sequence of range measurements up to time $t$. Let $x_t$ be a Markov process of initial distribution $p(x_0)$ and transition model $p(x_t|x_{t-1})$. Given $p(z_t|x_t)$, estimate recursively in time the posterior distribution $p(x_{0:t}|z_{1:t})$.
*Problem 3 (Access Point Locations):* Let $M = \{ m^{[j]}|j = 1,\dots, n_m \}$ be a set of unknown and partially observable features whose elements, $m^{[j]} \in \mathbb{R}^3$ represent WiFi access point locations with respect to a global frame of reference. Given $p(z_t|x_t)$ and $p(x_{0:t}|z_{1:t})$ recursively estimate $p(m^{[j]} | x_t, z_t)$.
In the first problem, we try to characterize the received signal, and through an appropriate model, transform it to a range measurement. Furthermore, we need to find a likelihood function that describes the measurement noise. The second problem can be seen as a range-only self-localization problem. Finally, the last problem is to estimate the locations of the access points given the location of the device. We now state the main assumptions we use to solve the defined problems:
*Assumption 1 (Known Data Association):* Each access point has a unique hardware identifier that is available to the receiver device. This assumption is usually satisfied in practice as each device has a unique MAC-address that is broadcasted together with the RSSI.
*Assumption 2 (only RSSI available):* We assume that the only available information to the receiver is the RSSI. This is the common case for existing wireless routers and commercial NICs.
WiFi Technology
---------------
WiFi is a technology for radio wireless local area networking of devices based on the IEEE 802.11 standards. It most commonly operates on the $2.4 {\mathop{\mathrm{GHz}}}$ Ultra-High Frequency (UHF) and $5.8 {\mathop{\mathrm{GHz}}}$ Super-High Frequency (SHF) Industrial, Scientific and Medical (ISM) radio bands. These wavelengths work best for line-of-sight. Many common materials absorb or reflect them, which further restricts the range, but can minimize interference between different networks in crowded environments.
![Neural Network Architecture. $x_1$ and $x_2$ represent the Euclidean distance and the distance calculated using Free Space Path Loss (FSPL) [@pap1] formula, respectively. ReLU [@agarap2018deep] activation is used between each of the hidden layers. As the network depth increases, it learns the underlying representation of the data, in turn compressing it to learn the underlying structure. The final layer consists of two neurons and the SoftMax activation which gives the probabilities of LOS/NLOS signal propagation as output.[]{data-label="fig:nn"}](DNN2_s){width="0.99\columnwidth"}
Particle Filtering
------------------
In the problem of localization using RSSI, the observation space is nonlinear, and the posterior density is often multimodal. Particle filters are a non-parametric implementation of the Bayes filter that are suitable for tracking and localization problems where dealing with global uncertainty is crucial. Hence, in this work, we use a particle filter and its extension FastSLAM to solve the positioning and mapping problems [@thrun2005probabilistic; @jadidi2017gaussian; @GhaffariJadidi2018].
The DeepLocNet Framework
========================
The proposed framework can be divided into two distinct parts as follows:
LOS/NLOS Classifier
-------------------
The first part of the framework comprises of a deep neural network which is trained to classify between LOS and NLOS signal propagation. The network includes a total of 12 fully connected (FCN) layers which first expand and then contract. The architecture of the network we are using can be seen in Fig. \[fig:nn\]. The structure of the neural network is inspired by an autoencoder, where we learn the low-level features representing the underlying data by scaling up, and then scaling down to compress these low-level features into output dimensions. The only difference here is instead of using the input as the output, we are using the class as the output to compute classification loss. `ReLU` [@agarap2018deep] activation is used between every layer and is found out to work better than `tanh` activation during the training and experimentation phases. One of the reasons is that the `tanh` activation function causes saturation of multiple neurons during the training process. The network is trained using the cross entropy loss function given by
$$H(y,\hat{y}) = -y \log(\hat{y}) - (1-y) \log(1-\hat{y}),$$
where $y$ is the probability of the true label being $1$ *(that is, $p_{\text{label}=1} = y$)* and $\hat{y}$ is the probability of the label predicted by the network being $1$. Cross entropy was chosen over other losses such as MSE or Hinge loss as they are mathematically ill-defined for the classification problem. Before finalizing the network architecture, we tried several different architectures to learn a valid representation of the signal propagation. We also tried going deeper than the current depth, but that led to overfitting on the training data, causing a substantially less testing accuracy than what we obtained using this network. Another reason going deeper did not help is that the number of parameters in the network increased without a substantial increase in classification accuracy.
The number of neurons used per hidden layer was decided based on trial-and-error and the complexity of the signal propagation we wanted to represent. The input data is highly non-linear and signal propagation with shadowing and multi-path effects cause further non-linearity. Because of this complexity, we increased the number of hidden layers until we obtained an increase in both training and testing accuracy. A summary of a few of the well-performing architectures, number of network parameters, and classification accuracies are shown in Table \[table:archs\].
Localization Framework
----------------------
The localization framework is responsible for the positioning of the device given an estimate of the motion and the received signal strengths of the WiFi signal. We divide this framework into two categories as follows.
### Map accurately known
In the case where the map is accurately known, we use a particle filter [@thrun2005probabilistic] to estimate the position of the robot/device. The Sample Importance Resampling (SIR) particle filter we use consists of three main modules: (i) Motion model (sample), (ii) Measurement model (importance), and (iii) resampling.
**Motion model** is responsible for generating a set of hypothesis for the current position, based on the previous position and the action taken. More specifically, it specifies a probability $p(x_t | x_{t-1},u_t)$, that action $u_t$ carries the robot from state $x_{t-1}$ to $x_t$ . Let the number of particles generated be $n_{p}$, then the motion model generates a position hypothesis for each of the particles.
**Measurement model** is responsible for assigning the weights (or *importance weights*) to each of the particles sampled from the motion model. Since we only obtain range measurements from the sensor, the measurement function can be given as the distance $z_t$ between the current position $x_t$ and the location $m^{[j]}$ of the $j^{th}$ access point as calculated using the Free Space Path Loss (FSPL) [@pap1] equation. Hence, our measurement model using only WiFi can be given as follows: $$d_{Euc}(x_t, m^{[j]}) = \left( (x_{t} - m^{[j]})^{\mathsf{T}} (x_{t} - m^{[j]}) \right)^{\frac{1}{2}}
\label{deuc}$$ $$d_{rssi}(RSSI^{[j]}) = \frac{1}{20} 10^{\lvert RSSI^{[j]} \rvert - K - 20 \log_{10}(f)},
\label{drssi}$$ where $d_{Euc}$ is the Euclidean distance between the current position $x_t$ and the position $m^{[j]}$ of the $j^{th}$ access point. Similarly, $d_{rssi}$ is the distance of the $j^{th}$ access point based on the RSSI value that the device receives at position $x_t$. $f$ is the frequency of the signal in ${\mathop{\mathrm{MHz}}}$. $K$ is a constant that depends on the units for $d_{rssi}$ and $f$. For $f$ in ${\mathop{\mathrm{MHz}}}$ and $d$ in $\mathrm{km}$, $K=32.44$ [@amiot2013pylayers].
Set of particles $\{X_t, w_t\}$ sampled from motion model, $RSSI_j$ from each AP $m^{[j]}$, measurement noise variance $sz_j$ for each AP $m^{[j]}$; $\sigma_n \gets 3$ $w_{total} \gets 0$ $de \gets d_{Euc}(x_t^{[i]}, m^{[j]})$ $dr \gets d_{rssi}(RSSI_j)$ $label \gets \texttt{getLabel}(de, dr)$ $dz \gets \lvert dr-de \rvert$ $p_{LOS}, p_{NLOS} \gets \texttt{getProbs}(de, dr)$ $d_{LOS} \gets \lvert dr-de \rvert$ $d_{NLOS} \gets \lvert dr - dn \rvert$ $dz \gets p_{LOS} \cdot d_{LOS} + p_{NLOS} \cdot d_{NLOS}$ $dz \gets \lvert dr-de \rvert$ $w_t^{[i]} \gets w_t^{[i]} \cdot f(dz; 0, sz)$ $w_{total} \gets w_{total} + w_t^{[i]}$ $w_t^{[i]} \gets w_t^{[i]}/w_{total}$ $\{X_t, w_t\}$
\[measure\_model\]
We implement three methods in the measurement model: (i) No Classification (NC), (ii) Hard (acceptance/rejection) Classification (HC), and (iii) Soft (probabilistic) Classification (SC). The first case represents the naive implementation of the measurement model. In the second case, we only consider LOS signal propagation for ranging (hence hard classification), whereas in the third case, we use probabilities of the signal being LOS or NLOS to calculate an importance weight for the particle. For soft classification, $\sigma_n$ can be given as the standard deviation in the maximum range the device is able to sense. We assign it an arbitrary value of $3$ assuming that the changes in power transmission would not change the distance ($d_{Euc}$) more than $3 \mathrm{m}$. We use two functions which access the classifier, `getLabel()` and `getProbs()`. Both of them take the Euclidean distance and RSSI based distance using FSPL as inputs. While `getLabel()` returns the label predicted by the network, `getProbs()` returns the probabilities that the given inputs are either LOS or NLOS. The overall measurement model we use for the particle filter is as given in Algorithm \[measure\_model\].
**Importance Sampling** draws, with replacement, $n_{p}$ particles from the set $X_t$ of generated and weighted particles using the above two steps. The probability of drawing each particle is given by its importance weight. The resampling essentially transforms the particle set of size $n_{p}$ into another particle set of the same size by replicating particles with higher weights and, in the end, setting all weights uniformly. The resulting sample set usually possesses many duplicates, since particles are drawn with replacement.
### Map partially known
Particle filtering is not effective when the locations of the access points are partially known. Since we are not sure of the access point locations, ranging using the measurement model, as discussed in the previous section, becomes inaccurate. To tackle such cases, the FastSLAM [@thrun2005probabilistic] algorithm is used to provide us with an effective means of localization, as well as a method of rectifying the locations of the access points using the sensor measurements.
The FastSLAM algorithm, in essence, is a particle filter where each particle comprises a map of the locations of each detected access point in addition to the weight and the position of the device. The access point locations are tracked using an Extended Kalman Filter, whereas the robot position is tracked using Particle Filtering. Analogous to the previous case, we divide the measurement model into 3 cases here as well. We refer the reader to [@thrun2005probabilistic] for implementation and other details.
Experimentation and Evaluation
==============================
In this section, we define our experimentation apparatus and methods. We divide this section into the following subsections:
Deep Neural Network Training:
------------------------------
The focus of this part was to collect data to train our network. The training was done in 2 phases: *(i)* initial training on simulated data, and *(ii)* network weight refinements on real-world data. Since obtaining floor plans and blueprints with access point locations may not always be a feasible task, we used *Pylayers* [@amiot2013pylayers], an open-source tool, to simulate wave propagation in complex and dense environments. Using this tool, we defined 30+ environments with varying temperatures, humidity, AP locations, AP characteristics (transmission power, antenna directions, etc) and floor plans. Some of the environments are shown in Fig. \[fig:envs\]. Results of Motley Keenan Path Loss and 3D Ray tracing for one of the environments can be seen in Fig. \[fig:defstr\].
A random walk algorithm was implemented which was able to navigate through a given environment in three dimensions given random start and goal points. We used this algorithm to generate waypoints and calculate the RSSI, Euclidean distance, and label (LOS/NLOS) of each AP at each waypoint in the path. We used Bresenham’s line algorithm to calculate if there was any obstruction between the said AP and the current waypoint. If there is an obstruction, we label it as NLOS, otherwise LOS. We simulated over 10 million data points and initially trained our neural network on this data.
We also had access to the blueprint of one of the buildings on our campus. We implemented software which aided us in data-collection given these blueprints as the robot was moved through the environment. We collected data from this building and used it to refine the weights of our deep neural network.
From the several architectures we tested, network D in Table \[table:archs\] performed the best both in simulation as well as hardware experiments. The classifier was able to classify LOS from NLOS signal propagation with an accuracy of $\approx 85\%$. From multiple experiments, we found out that the maximum misclassification occurs when the ground truth is NLOS but the prediction is LOS. The reason for this happening is that in cases when the device is near the access point (distance $<10$ map units or $2.5 m$), the signal propagation might be NLOS because of the presence of an obstruction in between the AP and the device (for instance a wall), the Euclidean distance and the distance calculated using the path-loss formula is approximately the same. Hence, though the signal is NLOS, it is predicted as LOS.
Experimentation in Simulation:
-------------------------------
We developed a simulator for testing the performance of the deep network. The random walk algorithm used in the training phase is used to generate ground truth data between a start and goal point. We start simulating the motion of the robot along this path. We then calculate the RSSI value at each of the waypoints in the path, as stated in the previous section. We also calculate the Euclidean distance of the estimated position of the robot from each of the access points (as we know the map within reasonable accuracy). We input these 2 values to the classifier, which then gives us the label (or probabilities) of the signal. We then use this data to update the measurement model accordingly. An example of the localization obtained with the FastSLAM algorithm and soft classification can be seen in Fig. \[fig:3dlocfs\]. We also attach results for 2D localization in a different environment using FastSLAM and the 3 cases of classifier usage in Fig. \[fig:2dlocfs\]. We refer the reader to our Github repository for similar results in both 2D and 3D for more complex environments *(such as the Office and W2PTIN)*.
![3D Localization in the Defstr environment[@amiot2013pylayers] using FastSLAM and soft classification (top to bottom - orthographic view, top view). The yellow spheres represent actual location of access points, black spheres the localized estimates of the access points, the white pipe the ground truth path and the black pipe the localized path. The green and red spheres represent the start and goal positions respectively. One AP is not detected in this case because it is out of sensing range (2m).The red wall is a door, hence the device is allowed to pass through it.[]{data-label="fig:3dlocfs"}](3D-FS.png){width="0.75\linewidth"}
The results obtained in simulation for both the 2D and 3D case for the office environment are given in Table \[tab:results\]. A set of 50 experiments each was run for each of the scenarios presented in the given table and the mean root mean square error (RMSE) was calculated. As can be seen in Fig. \[fig:rmse\], the RMSE shows substantial improvement for localization using both FastSLAM and Particle Filter.
The parameters used for generating these results are given in Table \[tab:params\]. The number of particles defines $n_{p}$ used in the particle filter/FastSLAM algorithm. The step size is the length of step taken during the random walk. The motion standard deviation is used to produce white gaussian noise *(zero-centered gaussian distribution with given standard deviation)* [@thrun2005probabilistic] in each dimension (x,y,z) which is taken to be a percentage of the step size. The higher the percentage, the noisier the motion model is considered and a more scattered distribution of hypothesis is produced. Similarly, the measurement standard deviation is used to produce white gaussian noise in the range measurements obtained using the RSSI values of access points. Since RSSI based ranging is very inaccurate, we consider the noise to be anywhere between 10m to 100m. In practice, the noise can be higher than 100m, however, in that case, the localization algorithm would totally rely on the motion model for tracking the location rendering the use of a measurement model ineffective. The sensing range defines the maximum range the device is able to sense for WiFi APs. We take the sensing range of the device to be a safe estimate of 15m. Lastly, the AP location noise determines the noise in the location of every AP in each dimension *(only valid for FastSLAM)*.
![2D Localization in the Office environment using FastSLAM and Soft Classification *(RMSE=0.5258 m)*. No Classification *(RMSE=2.5737 m)* and Hard Classification *(RMSE=0.5377 m)* results are less accurate. All access points have been detected in all 3 cases. The colors show the ground truth as blue and the estimates as red.[]{data-label="fig:2dlocfs"}](fs_soft_classifier){width="0.95\columnwidth"}
\
Hardware Experiments
--------------------
We used the Fetch Mobile Manipulator for our experiments in one of the buildings of our campus, whose floor plan we had access to. The environment and the robot is as given in Fig. \[fig:fetch\]. We manually moved the robot around using a remote controller and constructed a map using an open-source implementation of gmapping [@thrun2005probabilistic]. On obtaining the map, we used an open-source implementation of Adaptive Monte Carlo Localization [@amcl] for obtaining the ground truth of the robot. We broke down the entire continuous path into waypoints with each segment at least 2.5m apart. At each waypoint, we recorded the WiFi signatures (includes signal strength and MAC address) of all the access points present and the visual odometry calculated using ORB-SLAM [@mur2015orb].
Once we gathered all the WiFi, odometry and waypoint information, we ran the FastSLAM algorithm for all three cases. Particle Filter wasn’t run as we had a certain measure of ambiguity in the location of all the access points. Localization is only performed in two dimensions because of the sensing capabilities of the robot.
We assumed the motion standard deviation of the robot to be $1 m$ in both $x$ and $y$ axes based on experiments in our domain using ORB-SLAM [@mur2015orb]. All other parameters were same as in the case of simulation. Step size and sensing range are not applicable in this case. The access point locations as well weren’t accurate as the floor plan wasn’t to scale. Hence we assume a certain error ($\leq 0.8 {\mathop{\mathrm{m}}}$) in their locations as well. The results obtained using FastSLAM for one of our trials is given in Table \[tab:expres\]. The result for 2D localization using soft classification in FastSLAM can be seen in Fig. \[fig:errexpFSSC\].
{width="1.88\columnwidth"}
Conclusion and Future Work
==========================
We presented a deep learning-based approach for classifying LOS/NLOS signal propagation, aiding in better localization estimates solely based on signal strength measurements. In particular, the proposed method can provide effective localization using the available infrastructure without the use of any custom hardware. We further incorporated the proposed deep learning-based classifier into the measurement model of particle filtering and FastSLAM. Our experiments show that the resulting system can track a person moving through a large and highly-structured indoor environment with accuracy within $2 {\mathop{\mathrm{m}}}$. We also present results in indoor environments using signal strength from multiple access points. The presented results in this work show that the DeepLocNet framework performs promisingly better than available wireless-based positioning systems which have an accuracy of $1-10{\mathop{\mathrm{m}}}$ [@liu2007].
Future work includes modifying the network to incorporate information from the PHY layer as well, further improving the accuracy of localization. However, for this to happen, mobile devices would have to be fitted with the required hardware.
Acknowledgment {#acknowledgment .unnumbered}
==============
[ This work was partially supported by the Toyota Research Institute (TRI), partly under award number N021515, however, this article solely reflects the opinions and conclusions of its authors and not TRI or any other Toyota entity. We also thank Dr. Corina Barbalata and Deep Robot Optical Perception Lab for providing us with the help and support for all hardware experiments.]{}
[^1]: The authors are with the Robotics Institute, University of Michigan, Ann Arbor, MI 48109 USA [{[sdhanjal, maanigj, eustice}]{}@umich.edu]{}.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: '[In this letter, we characterize experimentally the diffusiophoretic motion of colloids and $\lambda$-DNA toward higher concentration of solutes, using microfluidic technology to build spatially- and temporally-controlled concentration gradients. We then demonstrate that segregation and spatial patterning of the particles can be achieved from temporal variations of the solute concentration profile. This segregation takes the form of a strong trapping potential, stemming from an osmotically induced rectification mechanism of the solute time-dependent variations. Depending on the spatial and temporal symmetry of the solute signal, localization patterns with various shapes can be achieved. These results highlight the role of solute contrasts in out-of-equilibrium processes occuring in soft matter. ]{}'
address: |
$^\dag$ LPMCN; Université de Lyon; Université Lyon 1 and CNRS, UMR 5586; F-69622 Villeurbanne, France\
$^\ddag$ CEA, LETI, MINATEC, F38054 Grenoble, France
author:
- 'Jérémie Palacci $^\dag$, Benjamin Abécassis $^\ddag$, Cécile Cottin-Bizonne $^\dag$, Christophe Ybert $^\dag$ Lydéric Bocquet $^\dag$'
title:
- Colloidal motility and localization under rectified diffusiophoresis
- Colloidal motility and pattern formation under rectified diffusiophoresis
---
Diffusiophoresis is the mechanism by which particles and molecules drift along solute concentration gradients [@anderson; @abecassis2008; @Prieve87]. It belongs to the general class of phoretic transport phenomena, such as electro- and thermo- phoresis [@Prost2009; @Duhr06; @Piazza08; @Jiang09; @Wurger09], [*i.e.*]{} the migration of particles under gradients of an external thermodynamic variable (electric potential, temperature, ...). Physically, the diffusio-phoretic migration under solute gradients takes its origin in an osmotic pressure gradient occurring [*within*]{} the diffuse interface at the surface of the particle [@anderson; @AjdariBocquet2006; @abecassis2008]: this leads to a drift velocity proportional to the solute gradient $ \nabla c$, in the form $V_{DP}=\mu \nabla c$, with $\mu$ a mobility. The diffusiophoretic transport was recognized since the pioneering work of Derjaguin and later by Anderson and Prieve [@anderson]. However its implication in out-of-equilibrium phenomena occuring in soft matter systems – in which solute contrasts are ubiquituous – has been barely explored up to now. Several works recently revealed the role played by transport induced by solute contrasts in out-of-equilibrium processes. It was [*e.g.*]{} shown to be at the origin of the strongly enhanced effective diffusion of colloids under solute gradients [@abecassis2008] with potential implications in microfluidics [@abecassis2009]. Its possible interplay with other transport phenomena was also demonstrated, concuring for example to induce a sign reversal of thermophoretic transport of colloids in the presence of polymers [@Jiang09]. Its role in skin formation during evaporation of mixtures was suggested [@Kabalnov09]. Finally, in the context of the efforts to develop autonomous microsystems, diffusiophoresis was used to harness chemical power to produce self-propelled artificial swimmers [@Golestanian], using chemical gradients as driving forces.
![[*(a)*]{} Experimental setup [*(b)-(Left)*]{} A 3-channel and gel matrix setup, superimposed on fluorescence intensity image measured for the stationary gradient of fluorescein with $C_1-C_2=10^{-4}$M ($w=300{\mu\text{m}}$, $\ell=800\,\mu$m, scale bar 300${\mu\text{m}}$). [*(b)-(Right)*]{} Fluorescence intensity profiles measured in stationary state, showing the expected linear profile. Actuation of the microfluidic switch allows to invert the concentration gradient. \[figure-1\] ](figure-1){width="8cm"}
In this paper we pursue the exploration of this phenomenon and demonstrate segregation and pattern formation of colloids and macromolecules on the basis of diffusiophoretic transport. We evidence a localization process of particles –here, colloids and $\lambda$-DNA–, taking its origin in the rectification of the underlying time-dependent concentration variations of a solute specie. The shape of the pattern is shown to be strongly dependent on the temporal symmetry of the underlying solute concentration signal. A theoretical model based on a Smoluchowski description allows to reproduce the experimental observations.
The thorough exploration of diffusiophoresis requires the building-up of controlled salt concentration gradients. To this end, we have developed a microfluidic experimental setup sketched in Fig.\[figure-1\], inspired from Ref. [@Wu06]. This set-up allows to impose stationary, as well as temporally switchable, solute gradients. A three channels device is molded in agarose gel, allowing free diffusion of solute without convective flow. A double syringe pump with two solutions of salt of concentration $C_1$ and $C_2$ fills the two side channels. A microfluidic switch (Upchurch) allows to exchange the solute solutions in the side channels, leading to a time-dependent tuning of the gradient. Particles under investigation (colloids or $\lambda$-DNA) are located in the center channel. We use polystyren-carboxylate fluorescent colloids (F8888 200nm, Molecular Probes) in a Tris buffer (1mM, pH9), and DNA (48 kbp, Fermentas, Germany) labelled with YOYO-1 (Invitrogen), in Tris-EDTA buffer (1mM, pH7.6). As a solute, we have considered here either fluorescein or bare salts –LiCl, NaCl and KCl–. In a stationary regime, a linear profile of the solute concentration is achieved, as illustrated in Fig. \[figure-1\]-(b), with fluorescein as a benchmark.
![Diffusiophoretic transport of fluorescent colloids and $\lambda$-DNA under a LiCl gradient ($\Delta C_s[LiCl]= |C_2-C_1| = 100$mM. $\ell=800{\mu\text{m}}$, scale bar 100$\mu$m). [*(a)-(c)*]{}). Motion of particles under a salt gradient, sketched by the lateral bar, towards higher salt concentrations. Images separated by 90s for the colloids and images at $t=$100, 150, 200, 300s for $\lambda$-DNA. [*(b)-(d)*]{} Time evolution of the particles population location. Experimental data (symbols) are fitted according to the theoretical description (dashed lines), with $D_{DP}$ as the only fitting parameter (see text). In (b), open symbols correspond to a subsequent migration, and fully superimposes on the previous results. The solid straight line in (b) is a guide line corresponding to constant drift velocity. Shaded regions correspond to time periods where wall effects prevent from proper fluorescence measurements. \[figure-2\] ](figure-2){width="8cm"}
Let us first explore the response of colloidal particles under stationary salt gradients. Starting from a configuration with all particles gathered on one side of the channel, the salt gradient is switched toward the opposite channel wall: colloids are accordingly observed to drift toward the higher solute concentration, see Fig. \[figure-2\]-(a). In order to quantify this motion, we plot in Fig. \[figure-2\]-(b) the time dependent location of the colloidal population, defined as the maximum of the distribution. The observed drift is close to linear, as would be expected for a constant mobility $\mu=V_{DP}/\nabla c$. However a slight deviation from this linear expectation can be observed at long times. This result can be understood by going more into details of the diffusiophoretic transport. Indeed, in the case of electrolytes as a driving solute, the mobility $\mu$ is expected to depend on the salt concentration $c$, scaling as $\mu(c) \sim c^{-1}$, and the diffusio-phoretic velocity under a salt concentration gradient $\nabla c(x,t)$ can be written as [@Prieve87]: $$V_{DP}={D_{\rm DP}}\nabla \log c
\label{VDP}$$ with ${D_{\rm DP}}$ a diffusio-phoretic mobility [@anderson]. Physically, this dependence originates in the balance between osmotic forces and visous stresses occuring within the Debye layer at the particle surface [@AjdariBocquet2006; @abecassis2008]. This leads to ${D_{\rm DP}}$ scaling typically as ${D_{\rm DP}}\sim {k_BT/ \eta\,
\ell_B}$, with $\eta$ the water viscosity and $\ell_B$ the Bjerrum length ($\ell_B =0.7$nm in water). This non-linear dependence on solute concentration is at the origin of the deviation from constant drift observed experimentally. Indeed under a linear salt profile $c(x)={c_0 / 2} (1 \pm 2 {x/ \ell})$, with $\ell^{-1}$ the slope of the concentration gradient ($\ell^{-1}=\nabla c/c_0$), and $x$ the distance to mid-channel, Eq. (\[VDP\]) predicts that the position $X_0$ of a colloid obeys ${dX_0/ dt}= \pm {{D_{\rm DP}}}/{( {\ell}/{2} \pm X_0)}$. This equation can be solved analytically and provides a very good fitting expression for experimental data, see Fig. \[figure-2\]-(b), allowing for the determination of the colloids diffusio-phoretic mobility $D_{DP}$. In the case of colloids, the diffusiophoretic mobility was measured for three different salts, exhibiting salt specificity effects with values for the mobility in the order LiCl$>$NaCl$>$KCl, see Table I, in line with previous results [@abecassis2008]. Furthermore, we performed additional experiments with $\lambda$-DNA in place of spherical colloids: Figs. \[figure-2\]-(c,d) demonstrate a comparable motion of DNA molecules resulting from the diffusiophoretic migration towards higher concentrations of salt, Table I.
DNA
-- ------------- -------------- -------------- -------------- -------------- --
LiCl NaCl KCl LiCl
290 $\pm$ 5 150 $\pm$ 10 70 $\pm$ 10 150 $\pm$ 20
Exp. 22 $\pm$ 2 16$ \pm$ 1.5 N.A. 12.5 $\pm$ 3
Theo. 23$\pm$ 0.5 16.5$\pm$1 13$\pm$2 16.5 $\pm$ 1
: Values for the diffusiophoretic mobility ${D_{\rm DP}}$ extracted from Fig. \[figure-2\] (see text). Experimental trap strength $K$ extracted from the trapping experiments in Fig. \[figure-3\], and compared to the predicted value from Eq. (\[gaussian\]).\
We now turn to the effect of solute time-dependent oscillations: while in the previous section migration was studied under a stationary gradient, we now generate concentration gradients oscillations – Fig. \[figure-1\]-(b) – (with period $T_0$ in the range $T_0/2\sim 180 -300$s in the following). Typically, experiments show that a linear solute gradient establishes in the system within a time $T_t \sim 50$s for $\ell=800\mu$m, in agreement with numerical estimates [@Supp]. The result is shown in Fig. \[figure-3\]-(a) for colloids: starting from a homogeneous distribution, the particles population evolves after a few cycles towards a band of [*gaussian shape*]{} with stationary width, Fig. \[figure-3\]-(a) [@Supp]. Note that the position of the band keeps oscillating around the center with the salt cycles, but its width remains stationary. Moreover, the process is robust: starting from inhomogeneous particles distribution, or changing the oscillation frequency yields the same width for the trapped band. Finally as for the colloids motility, the width of the trapped band is dependent on the salt nature: LiCl $>$ NaCl $>$ KCl in terms of trapping efficiency. Additionally, we also performed similar experiments with $\lambda$-DNA as motile particles: Fig. \[figure-3\]-(b) shows that trapping is equally achieved with macromolecules, the latter gathering also in a narrow band with gaussian profile.
![Trapping under oscillatory gradients: [*(Top)*]{}: trapping of colloids, (a), and $\lambda$-DNA, (b), under salt gradient oscillations. Buffers and salt concentration are identical to Fig. \[figure-2\] with $T_0=600$s (scale bar 100$\mu$m). ([*Bottom*]{}): experimental particles density profile in stationary state, as obtained from the fluorescence images (symbols), fitted to a gaussian curve (dashed line). \[figure-3\] ](figure-3){width="8cm"}
Going further, a physical interpretation for the segregation can be proposed: as we now show, trapping of the particles indeed results from the rectification of their motion under the oscillating driving field of the salt concentration. The particles’ population obeys the Smoluchowski equation $$\partial_t \rho = -{\nabla}\cdot \left( - D_c \nabla\rho_0 + {D_{\rm DP}}\nabla[\log c] \times \rho_0 \right),
\label{Smolu}$$ coupled to the solute diffusive dynamics on $c(x,t)$. Neglecting transients in the salt dynamics, its concentration profile can be written as $c(x,t)=c_0(1/2 + f(t)\, x/\ell)$, with $f(t)$ a time-dependent function, oscillating with the forcing periodicity, $T_0$. Typically, $f(t)$ can be approximated as a $\pm 1$ step-function, so that $\langle f(t) \rangle=0$ while $\langle f(t)^2 \rangle=1 \ne 0$, with $\langle\cdot\rangle$ the time average over $T_0$. Accordingly the position $X_0(t)$ of the particle population is expected to oscillate with the gradients following the description above and Eq. (\[VDP\]). In the limit of small excursions around the center ($X_0\ll\ell$), this reduces to $dX_0/dt \simeq 2{D_{\rm DP}}f(t)/\ell \left[1 - 2 f(t) (X_0/\ell) + \ldots\right]$.
Now, in the limit of fast salt oscillations, one expects the distribution in steady state to behave to leading order as $\rho(x,t)\simeq\bar\rho_0(x-X_0(t))$. Inserting this guess into the Schmoluchowski equation, Eq. (\[Smolu\]) and using the above equation for $X_0(t)$ leads to the following equation for the particle density profile $\bar\rho_0$: $$\langle J\rangle= - D_c \nabla_{x} \bar\rho_0 + {4{D_{\rm DP}}\over\ell^2}\langle f^2\rangle\times \delta x\, \times \bar\rho_0 \simeq 0$$ with $\delta x=x-X_0(t)$. In deriving this equation, we have furthermore averaged out over the fast salt variables. Solving this equation in the stationary state predicts a gaussian distribution for the particles: $$\bar\rho_0(\delta x) \propto e^{-{\delta x^2\over 2\sigma^2}}\,\,\,{\rm with}\,\,\, \sigma=\frac{1}{2 }\sqrt{{D_c\over {D_{\rm DP}}}} \times \ell,
\label{gaussian}$$ that oscillates as a whole around the channel central position. The model thus reproduces the experimental observations: trapping towards a [*gaussian distribution*]{}, with a [*frequency independent width*]{}. We furthermore assessed the validity of this description by performing a full numerical resolution of the coupled particle and solute dynamics [@Supp].
Physically the origin of the focusing lies in the non-linear dependency of the diffusio-phoretic phenomenon versus the solute concentration, Eq. (\[VDP\]). Under a constant solute gradient, the velocity of particles is larger in regions with smaller solute concentration. Accordingly, the front particles move slower than the back ones: iterated over the oscillations, such a process leads to the observed focusing. The balance with Brownian motion leads to an harmonic “osmotic” trapping potential, ${\cal V}_{\rm trap} (x)= {1\over 2} K x^2$, with $K={k_{\text{B}}T}/\sigma^2$. The experimental values for the measured trapping strength $K$ are gathered in Table I for various salts. These values are in good agreement with theoretical predictions, in which ${D_{\rm DP}}$ was set to the mobility measured independently in channel-crossing experiments (Fig. \[figure-2\]). A final important remark is that the trapping potential is indeed strong: over the scale of the system $\ell$, the trapping free-energy well has a depth of $\Delta {\cal F} \approx K \ell^2 = {k_{\text{B}}T}\times {{D_{\rm DP}}\over D_c}$ up to hundreds of ${k_{\text{B}}T}$ ! (and independent of $\ell$).
![Trapping of colloids in a circular chamber: *(a)-(b)* Initial distribution of the fluorescent colloids and sketch of the experimental set-up. The central circular well is 650${\mu\text{m}}$ in diameter with gel walls 125 ${\mu\text{m}}$ wide ($\ell=900{\mu\text{m}}$; scale bar 200${\mu\text{m}}$). The salt (LiCl) concentration oscillates in the two side channels (period $T_0=480$s) either anti-symmetrically, $C_1(t)-\langle{C}_1\rangle= -[C_2(t)-\langle{C}_2\rangle]$, or symmetrically $C_1(t)=C_2(t)$, between $c_0=100mM$ and $0$ (in addition to a TRIS buffer). *(c)-(d)*: Stationary colloidal distribution under an *antisymmetric* driving (c) and under *symmetric* driving (d). *(c’)-(d’)*: Theoretical predictions under corresponding experimental conditions. The predicted profiles are those obtained by avering out the oscillating salt distribution (see text). The values for ${D_{\rm DP}}$ were taken from experiments in Fig. \[figure-2\], see Table I, while $\ell$ and $T_0$ take their experimental values. \[figure-5\] ](figure-4){width="8cm"}
We now generalize the previous results to more complex geometries in order to demonstrate the robustness and generic character of the above scenario. To this end, we tested the localization phenomenon in a circular, “cell shaped”, chamber, Fig. \[figure-5\]-(a)-(b). Two different drivings are explored: an [*antisymmetric*]{} driving, where the salt boundary concentrations, $C_1$ and $C_2$ are switched periodically between 0 and $c_0$ with antisymmetric phase; and a [*symmetric*]{} driving, where $C_1=C_2$ switches periodically between 0 and $c_0$. As demonstrated in Figs. \[figure-5\]-(c),(d), the solute oscillations again produce a localization of the particles in the cell chamber [@Supp]. Furthermore the symmetry of the pattern depends directly on the symmetry of the driving: linear –“cat eye” shape– for the antisymmetric driving, and circular –cell-shape– for the symmetric driving, thus demonstrating the versatility of the trapping process. A further remark is that this segregation process is also robust w.r.t. the initial distribution of the particles.
A rationalization of the different patterns observed can be obtained along similar lines as above. First the solute diffusion equation is solved with the appropriate boundary conditions under periodic inversion. In the case of the antisymmetric geometry, the salt concentration profile $c(r,\theta,t)$ (with $r,\theta$ the orthoradial coordinates) is found to take the general form $c(r,\theta,t)={c_0/ 2}\left[ 1+\delta_{\rm anti}(r,\theta) \,f(t) \right]$ with $\delta_{\rm anti}(r,\theta)={4\over \pi} \sum_{k, odd} {k^{-1}} \left({r/\ell}\right)^{k} \sin[k \theta]$, and $f(t)$ the periodic step-like function with period $T_0$. In the case of a symmetric driving, one finds $c(r,\theta,t)={c_0/ 2}\left[ 1+\delta_{\rm sym}(r,\theta,t) \right]$ where $\delta_{\rm sym}(r,t)={4\over \pi} \sum_{k, odd} (1/ k) {\rm Im}\left[f_{k}(r) \exp( j\, k \omega\, t)\right]$ with $f_k(r)=I_0(r/\delta_k)/I_0(\ell/2\delta_k)$, $\delta_k=\sqrt{ {j D_s/ k\, \omega}}$ the salt diffusive length, $\omega=2\pi/T_0$ and $I_0(x)$ is the Bessel function of order 0. In both cases, averaging out the fast salt variables, the trapping potential is then obtained in the form ${\cal V}_{\rm trap}(r,\theta)={1\over 2} ({D_{\rm DP}}/ D_c)\times \phi(r,\theta)$: $\phi(r,\theta)=\delta_{\rm anti}(r,\theta)^2$ for the antisymmetric case, while $\phi(r, \theta)\simeq \kappa_0\, r^4$ for the symmetric driving, with $\kappa_0^{-1}= 16\pi \delta_1^5\cosh(\ell/\sqrt{2}\delta_1)/\ell$. These predicted localization patterns are exhibited in Figs. \[figure-5\]-(c$^\prime$),(d$^\prime$), showing a good qualitative and semi-quantitative agreement with the experimental results.
To conclude, we have demonstrated a new mechanism leading to segregation and pattern formation of colloids and macromolecules, originating in rectified time-dependent solute contrasts. A key ingredient of the process is the so-called [*diffusiophoresis*]{}, a passive transport phenomenon leading to motion of particles under chemical potential gradients. Our results highlight the importance of solute contrast induced transport in out-of-equibrium processes, with potential implications in soft matter, as well as in living and chemical systems, where concentration gradients are ubiquituous.
We thank H. Ayari, F. Jülicher, J.-F. Joanny and P. Jop for highlighting discussions and H. Feret for technical support. We acknowledge support from Région Rhône-Alpes under CIBLE program.
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See supplementary material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.104.138302.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
By retracing research on *coexistent magnitudes* (*grandeurs coexistantes*) by <span style="font-variant:small-caps;">Cauchy</span> [@cauchy1841 (1841)], <span style="font-variant:small-caps;">Peano</span> in *Applicazioni geometriche del calcolo infinitesimale* [@peano87 (1887)] defines the “density” (strict derivative) of a “mass" (a distributive set function) with respect to a “volume” (a positive distributive set function), proves its continuity (whenever the strict derivative exists) and shows the validity of the *mass-density paradigm*: “mass” is recovered from “density” by integration with respect to “volume”. It is remarkable that <span style="font-variant:small-caps;">Peano</span>’s strict derivative provides a consistent mathematical ground to the concept of “infinitesimal ratio” between two magnitudes, successfully used since <span style="font-variant:small-caps;">Kepler</span>. In this way the classical (i.e., pre-Lebesgue) measure theory reaches a complete and definitive form in <span style="font-variant:small-caps;">Peano</span>’s *Applicazioni geometriche*.
A primary aim of the present paper is a detailed exposition of <span style="font-variant:small-caps;">Peano</span>’s work of 1887 leading to the concept of strict derivative of distributive set functions and their use. Moreover, we compare <span style="font-variant:small-caps;">Peano</span>’s work and <span style="font-variant:small-caps;">Lebesgue</span>’s *La mesure des grandeurs* [@lebesgue1935 (1935)]: in this memoir <span style="font-variant:small-caps;">Lebesgue</span>, motivated by coexistent magnitudes of <span style="font-variant:small-caps;">Cauchy</span>, introduces a uniform-derivative of certain additive set functions, a concept that coincides with <span style="font-variant:small-caps;">Peano</span>’s strict derivative. Intriguing questions are whether <span style="font-variant:small-caps;">Lebesgue</span> was aware of the contributions of <span style="font-variant:small-caps;">Peano</span> and which role is played by the notions of strict derivative or of uniform-derivative in today mathematical practice.
address:
- |
Dipartimento di Matematica\
Università di Trento, 38050 Povo (TN), Italy
- |
Dipartimento di Matematica\
Università di Trento, 38050 Povo (TN), Italy
- |
Dipartimento di Matematica\
Università di Trento, 38050 Povo (TN), Italy
author:
- 'Gabriele H. Greco'
- Sonia Mazzucchi
- 'Enrico M. Pagani'
date: 'February 22, 2010'
title: |
Peano on derivative of measures:\
strict derivative of distributive set functions
---
Introduction
============
By referring to <span style="font-variant:small-caps;">Cauchy</span> [@cauchy1841 (1841)] <span style="font-variant:small-caps;">Peano</span> introduces in *Applicazioni geometriche del calcolo infinitesimale* [@peano87 (1887)] the concept of *strict derivative* of set functions. The set functions considered by him are not precisely finite additive measures. The modern concept of finite additivity is based on partitions by disjoint sets, while <span style="font-variant:small-caps;">Peano</span>’s additivity property coincides with a traditional supple concept of “decompositions of magnitudes”, which <span style="font-variant:small-caps;">Peano</span> implements in his proofs as *distributive set functions*.
Contrary to <span style="font-variant:small-caps;">Peano</span>’s strict derivative (*rapporto*), <span style="font-variant:small-caps;">Cauchy</span>’s derivative (*rapport différentiel*) of a set function corresponds to the usual derivative of functions of one variable. In <span style="font-variant:small-caps;">Peano</span>’s Theorem \[peanodev\] on strict derivative of distributive set functions the (physical) *mass-density paradigm* is realized: the “mass” (a distributive set function) is recovered from the “density” (strict derivative) by integration with respect to the “volume” (a positive distributive set function of reference).
<span style="font-variant:small-caps;">Peano</span> expresses <span style="font-variant:small-caps;">Cauchy</span>’s ideas in a more precise and modern language and completes the program proposed by <span style="font-variant:small-caps;">Cauchy</span>, who, at the end of his article [@cauchy1841 (1841) p.229], writes:
> Dans un autre Mémoire nous donnerons de nouveaux développements aux principes ci-dessus exposés \[on coexistent magnitudes\], en les appliquant d’une manière spéciale l’évalutation des longueurs, des aires et des volumes.[^1]
Among numerous applications of <span style="font-variant:small-caps;">Peano</span>’s strict derivatives of set functions which can be found in *Applicazioni geometriche*, there are formulae on oriented integrals, in which the geometric vector calculus by <span style="font-variant:small-caps;">Grassmann</span> plays an important role. For instance, <span style="font-variant:small-caps;">Peano</span> proves the formula of area starting by his definition of area of a surface, that he proposed in order to solve the drawbacks of <span style="font-variant:small-caps;">Serret</span>’s definition of area [@serret (1879)].
The didactic value of <span style="font-variant:small-caps;">Peano</span>’s strict derivative of set function is transparent: in *La mesure des grandeurs* [@lebesgue1935 (1935)] <span style="font-variant:small-caps;">Lebesgue</span> himself uses a similar approach to differentiation of measures in order to simplify the exposition of his measure theory.
In Section \[sez-paradigma\], <span style="font-variant:small-caps;">Peano</span>’s and <span style="font-variant:small-caps;">Lebesgue</span>’s derivative are compared in view of the paradigm of mass-density and of the paradigm of primitives, that motivated mathematical research between $19^{th}$ century and the beginning of $20^{th}$ century. In the celebrated paper *L’intégration des fonctions discontinues* [@lebesgue1910 (1910)] <span style="font-variant:small-caps;">Lebesgue</span> defines a derivative of $\sigma$-additive measures with respect to the volume. He proves its existence and its measurability. In the case of absolute continuity of the $\sigma$-additive measures, <span style="font-variant:small-caps;">Lebesgue</span> proves that the measure is given by the integral of his derivative with respect to the volume. As it will be seen later in details, <span style="font-variant:small-caps;">Peano</span>’s *strict derivative* of distributive set functions does not necessarily exist and, moreover, whenever it exists, <span style="font-variant:small-caps;">Peano</span>’s strict derivative is continuous, while <span style="font-variant:small-caps;">Lebesgue</span>’s derivative in general is not.
Section \[sez-misura\] presents an overview of <span style="font-variant:small-caps;">Peano</span>’s work on pre-Lebesgue classical measure theory which is completed in Sections \[sez-distributive\]-\[sez-derivata\].
Section \[sez-cauchy\] is devoted to an analysis of <span style="font-variant:small-caps;">Cauchy</span>’s *Coexistent magnitudes* [@cauchy1841 (1841)][^2], by emphasizing the results that will be found, in a different language, in <span style="font-variant:small-caps;">Peano</span>’s *Applicazioni geometriche* or in <span style="font-variant:small-caps;">Lebesgue</span>’s *La mesure des grandeurs*.
Section \[sez-distributive\] concerns the concept of “distributive families" and of “distributive set functions” as presented by <span style="font-variant:small-caps;">Peano</span> in *Applicazioni geometriche* and in his paper *Le grandezze coesistenti di Cauchy* [@peano1915 (1915)].
Section \[sez-derivata\] presents a definition of strict derivative of set functions, main results and some applications, while in Section \[sez-massdensity\] we discuss <span style="font-variant:small-caps;">Peano</span>’s definition of integral of set functions and a related theorem that realizes the mentioned physical paradigm of mass-density.
Section \[sez-comments\] presents the approach of <span style="font-variant:small-caps;">Lebesgue</span> in *La mesure des grandeurs* to <span style="font-variant:small-caps;">Cauchy</span>’s coexistent magnitudes, leading to introduction of a new notion of derivative: the uniform-derivative.
We observe that this paper is meanly historical. From a methodological point of view, we are focussed on primary sources, that is, on mathematical facts and not on the elaborations or interpretations of these facts by other scholars of history of mathematics. For convenience of the reader, original statements and, in some case, terminology are presented in a modern form, preserving, of course, their content.
Historical investigations on forgotten mathematical achievements are not useless (from the point of view of mathematics), because some of them carry ideas that remain innovative today. This thought was very well expressed by <span style="font-variant:small-caps;">Mascheroni</span> before the beginning of the study of the geometrical problems leading to the *Geometria del compasso* (1797):
> \[…\] mentre si trovano tante cose nuove progredendo nelle matematiche, non si potrebbe forse trovare qualche luogo ancora incognito retrocedendo?[^3]
By respect for historical sources and for the reader’s convenience, the quotations in the sequel will appear in the original tongue with a translation in square brackets, placed in footnote.
The physical paradigm of mass-density\
versus the paradigm of primitives {#sez-paradigma}
======================================
In [*Philosophiae Naturalis Principia Mathematica*]{} (1687) the first definition concerns mass and density:
> Quantitas materiae est mensura ejusdem orta ex ilius densitate et magnitudine conjunctim \[…\]. Hanc autem quantitem sub nomine corporis vel massa in sequentibus passim intelligo.[^4]
In this sentence <span style="font-variant:small-caps;">Newton</span> presents the *mass-density paradigm* (i.e., the mass can be computed in terms of the density and, conversely, the density can be obtained from the mass) as a fundament of Physics.
In *Coexistent magnitudes* [@cauchy1841 (1841)] <span style="font-variant:small-caps;">Cauchy</span>, with a clear didactic aim, uses the mass-density paradigm in order to give a unitary exposition of several problems related to differential calculus.
From a mathematical point of view the implementation of this physical paradigm presents some difficulties and it does not assure a univocal answer. The first difficulty is in defining what is a “mass”, the second is in choosing a procedure for evaluating “density” and, finally, in determining under what condition and how it is possible “to recover” the mass from the density.
All these critical aspects that we find in <span style="font-variant:small-caps;">Cauchy</span> [@cauchy1841 (1841)], are overcome in a precise and clear way by <span style="font-variant:small-caps;">Peano</span> in *Applicazioni geometriche* [@peano87 (1887)].
Natural properties that connect density and mass are the following:
1. *The density of a homogenous body is constant.*[\[hom\]]{}
2. *The greater is the density, the greater is the mass.* [\[gre\]]{}
3. *The mass of a body, as well as its volume, is the sum of its parts.*[\[sum\]]{}
The realization of the physical paradigm can be mathematically expressed by the following formula $$\label{mass-volume}
\mu(A)=\int_A g \,{{\mathrm{d}}}({\rm vol}_n)$$ where $\mu$ is the “mass”, $g$ is the “density” and ${\rm vol}_n$ is the $n-$dimensional volume.[^5]
The properties , and do not allow for a direct derivation of without further conditions depending on the meaning of integral; for instance, having in mind the Riemann integral, an obvious necessary condition is the Riemann integrability of the density $g$.
In <span style="font-variant:small-caps;">Peano</span>’s *Applicazioni Geometriche* [@peano87 (1887)]:
- the “masses” and the “volumes” are represented by *distributive set functions*, as it will be shown in detail in §\[sez-distributive\],
- the “densities” (strict derivatives) are computed using a limit procedure, as we shall see in the sequel (see formula ),
- the “mass” is recovered by integration using . This final step is strengthened by the fact that <span style="font-variant:small-caps;">Peano</span>’s strict derivative is continuous.
The mathematical realization of mass-density paradigm is directly connected with mathematical paradigm of primitives, that is with the study of conditions assuring that integration is the inverse operation of differentiation.
At the beginning of the $20^{th}$ century the problem of looking for primitives is the cornerstone of the new theory of measure, founded by <span style="font-variant:small-caps;">Lebesgue</span> [@lebesgue1904 (1904)]. The problem of primitives becomes arduous when one has to pass from functions of one variable to functions of more variables. <span style="font-variant:small-caps;">Lebesgue</span> in *L’intégration des fonctions discontinues* [@lebesgue1910 (1910)] overcomes these difficulties by substituting the integral of a generic function $g$ with a set function $\mu$ described by formula .
The paradigm of primitives gives more importance to the operations (of differentiation and integration) than to the set functions. On the contrary, in the mass-density paradigm the primary aim is the evaluation of the infinitesimal ratio between two set functions (for instance, mass and volume) in order to recover the “mass” by integrating the “density” with respect to “volume”. On the other hand in the paradigm of primitives the main problem is an extension of the notion of integral in order to describe a primitive of a given function and, consequently, to preserve fundamental theorem of calculus.
In <span style="font-variant:small-caps;">Lebesgue</span>’s works the two paradigms appear simultaneously for the first time in the second edition of his famous book [*Le[ç]{}ons sur l’int[é]{}gration et la recherche des fonctions primitives*]{} [@lebesgue1928 (1928) pp.196-198]. In 1921 (see [@lebesgue_opere vol.I, p.177]) <span style="font-variant:small-caps;">Lebesgue</span> has already used some physical concept in order to make the notion of set function intuitive; analogously in [@lebesgue1926 (1926)] and [@lebesgue1928 (1928) pp.290-296] he uses the mass-density paradigm in order to make more natural the operations of differentiation and integration. In his lectures *Sur la mesure des grandeurs* [@lebesgue1935 (1935)], the physical paradigm leads <span style="font-variant:small-caps;">Lebesgue</span> to an alternative definition of derivative: he replaces his derivative of 1910 with the new uniform-derivative (equivalent to the strict derivative introduced by <span style="font-variant:small-caps;">Peano</span>), thus allowing him to get continuity of the derivative.
Before comparing <span style="font-variant:small-caps;">Peano</span>’s and <span style="font-variant:small-caps;">Lebesgue</span>’s derivative of set functions, we recall the definitions of derivatives given by <span style="font-variant:small-caps;">Peano</span> and <span style="font-variant:small-caps;">Cauchy</span>.
<span style="font-variant:small-caps;">Peano</span>’s strict derivative of a set function (for instance, the “density” of a “mass” $\mu$ with respect to the “volume”) at a point $\bar x$ is computed, when it exists, as the limit of the quotient of the “mass” with respect to the “volume” of a cube $Q$, when the supremum of the distances of the points of the cube from $\bar x$ tends to 0 (in symbols $Q\to\bar x$). In formula, <span style="font-variant:small-caps;">Peano</span>’s strict derivative $g_{P}(\bar x)$ of a mass $\mu$ at $\bar x$ is given by: $$\label{der-f-peano}
g_P(\bar x):=\lim_{Q\to \bar x}\frac{\mu (Q)}{{{\mathrm{vol}}}_n(Q)} \,\,\,.$$ Every limit procedure of a quotient of the form $\frac{\mu (Q)}{{{\mathrm{vol}}}_n(Q)}$ with $Q\to\bar x$ and the point $\bar x$ not necessarily belonging to $Q$, will be referred to as *derivative à la Peano*.
On the other hand, <span style="font-variant:small-caps;">Cauchy</span>’s derivative [@cauchy1841 (1841)] is obtained as the limit between “mass" and “volume” of a cube $Q$ [*including*]{} the point $\bar x$, when $Q\to\bar x$. In formula, <span style="font-variant:small-caps;">Cauchy</span>’s derivative $g_{C}(\bar x)$ of a mass $\mu$ at $\bar x$ is given by: $$\label{der-f-lebesgue}
g_C(\bar x):=\lim_{\begin{subarray}{c}Q\to \bar x\\ \bar x\in Q\end{subarray}}\frac{\mu (Q)}{{{\mathrm{vol}}}_n(Q)} \,\,\,.$$ Every limit procedure of a quotient of the form $\frac{\mu (Q)}{{{\mathrm{vol}}}_n(Q)}$ with $Q\to\bar x$ and the point $\bar x$ belonging to $Q$, will be referred to as *derivative à la Cauchy*.
<span style="font-variant:small-caps;">Lebesgue</span>’s derivative of set functions is computed *à la Cauchy*. Notice that <span style="font-variant:small-caps;">Lebesgue</span> considers finite $\sigma$-additive and absolutely continuous measures as “masses", while <span style="font-variant:small-caps;">Peano</span> considers distributive set functions. <span style="font-variant:small-caps;">Lebesgue</span>’s derivative exists (i.e., the limit $(2.6)$ there exists for *almost every* $\bar x$), it is measurable and the reconstruction of a “mass" as the integral of the derivative is assured by absolute continuity of the “mass” with respect to volume. On the contrary, <span style="font-variant:small-caps;">Peano</span>’s strict derivative does not necessarily exist, but when it exists, it is continuous and the mass-density paradigm holds.[^6]
The constructive approaches to differentiation of set functions corresponding to the two limits and are opposed to the approach given by <span style="font-variant:small-caps;">Radon</span> [@radon1913 (1913)] and <span style="font-variant:small-caps;">Nikodym</span> [@nikodym1930 (1930)], who define the derivative in a more abstract and wider context than those of <span style="font-variant:small-caps;">Lebesgue</span> and <span style="font-variant:small-caps;">Peano</span>. As in the case of <span style="font-variant:small-caps;">Lebesgue</span>, a Radon-Nikodym derivative exists; its existence is assured by assuming absolute continuity and $\sigma$-additivity of the measures.
In concluding this Section, let us remark that the physical properties , and , that stand at the basis of the mass-density paradigm, lead to the following direct characterization of the Radon-Nikodym derivative. Let $\mu$ and $\nu$ be finite $\sigma$-additive measures on a $\sigma$-algebra ${{\mathcal{A}}}$ of subsets of $X$ and let $\nu$ be positive and $\mu$ be absolutely continuous with respect to $\nu$. A function $g:X\to{{\mathbb{R}}}$ is a *Radon-Nikodym derivative* of $\mu$ with respect to $\nu$ (i.e., $\mu (A)=\int_A g\,{{\mathrm{d}}}\nu$ for every $A\in{{\mathcal{A}}}$) if and only if the following two properties hold for every real number $a$:
1. $\mu(A)\geq a\,\nu(A)$ for every $A\subset \{g\geq a\}$ and $A\in{{\mathcal{A}}}$, [\[Hahn\_1\]]{}
2. $\mu(A)\leq a\,\nu(A)$ for every $A\subset \{g\leq a\}$ and $A\in{{\mathcal{A}}}$, [\[Hahn\_2\]]{}
where $\{g\le a\}:=\{x\in X: g(x)\le a\}$ and, dually, $\{g\ge a\}:=\{x\in X: g(x)\ge a\}$. These properties and , expressed by <span style="font-variant:small-caps;">Nikodym</span> [@nikodym1930 (1930)] in terms of Hahn decomposition of measures, are a natural translation of properties , and .
Peano on (pre-Lebesgue) classical measure theory {#sez-misura}
================================================
The interest of <span style="font-variant:small-caps;">Peano</span> in measure theory is rooted in his criticism of the *definition of area* (1882), of the *definition of integral* (1883) and of the *definition of derivative* (1884). This criticism leads him to an innovative measure theory, which is extensively exposed in Chapter V of *Applicazioni geometriche* [@peano87 (1887)].
The definition of area given by <span style="font-variant:small-caps;">Serret</span> in [@serret (1879)] contrasted with the traditional definition of area: in 1882 <span style="font-variant:small-caps;">Peano</span>, independently of <span style="font-variant:small-caps;">Schwarz</span>, observed (see [@peano_area1890 (1890)]) that the area of a cylindrical surface cannot be evaluated as the limit of inscribed polyhedral surfaces, as prescribed by <span style="font-variant:small-caps;">Serret</span>’s definition. In *Applicazioni geometriche*, <span style="font-variant:small-caps;">Peano</span> provides a consistent definition of area and proves the integral formula of area.[^7]
<span style="font-variant:small-caps;">Peano</span>’s criticism of the definition of Riemann integral of a function and of its relation with the area of the *ordinate-set* (i.e., hypograph of the function) [@peano1883 (1883)], forces him to introduce outer/inner measure as the set-theoretic counterparts of upper/lower integral: he defines the latter in terms of infimum/supremum (instead of limits, as done traditionally) of the Darboux sums.[^8] <span style="font-variant:small-caps;">Peano</span>, in introducing the inner and outer measure as well as in defining area [@peano_area1890 (1890)], is also influenced by <span style="font-variant:small-caps;">Archimedes</span>’s approach on calculus of area, length and volume of convex figures.
In 1884, by analyzing the proof of mean value theorem, given by <span style="font-variant:small-caps;">Jordan</span> [^9] in the first edition of *Cours d’analyse*, <span style="font-variant:small-caps;">Peano</span> stresses the difference between differentiable functions and functions with continuous derivative. The continuity of derivative is expressed by <span style="font-variant:small-caps;">Peano</span> in terms of the existence of the limit $$\label{str-diff}\lim_{\begin{subarray}{c}x,y\to \bar x \\ x\neq y\end{subarray}}\frac{f(x)-f(y)}{x-y}$$ for any $\bar x$ in the domain of $f$.[^10] Moreover, <span style="font-variant:small-caps;">Peano</span>, in his correspondence with <span style="font-variant:small-caps;">Jordan</span> [@peano_jordan; @peano_gilbert (1884)], observes that uniform convergence of the difference quotient is equivalent to the continuity of the derivative.[^11] This notion of continuous derivative will be the basis of <span style="font-variant:small-caps;">Peano</span>’s strict derivative of distributive set functions.
*Applicazioni geometriche* is a detailed exposition (more than 300 pages) of several topics of geometric applications of infinitesimal calculus.[^12] In *Applicazioni geometriche* <span style="font-variant:small-caps;">Peano</span> refounds the notion of Riemann integral by means of inner and outer measures[^13], and extends it to abstract measures. The development of the theory is based on solid topological and logical ground and on a deep knowledge of set theory. He introduces the notions of closure, interior and boundary of sets.
<span style="font-variant:small-caps;">Peano</span> in *Applicazioni geometriche* [@peano87 (1887)], and later <span style="font-variant:small-caps;">Jordan</span> in the paper [@jordan1892 (1892)] and in the second edition of *Cours d’Analyse* [@jordan1893 (1893)], develop the well known concepts of classical measure theory, namely, measurability, change of variables, fundamental theorems of calculus, with some methodological differences between them.[^14]
The mathematical tools employed by <span style="font-variant:small-caps;">Peano</span> were really innovative at that time (and maybe are even nowadays), both on a geometrical and a topological level. <span style="font-variant:small-caps;">Peano</span> used extensively the geometric vector calculus introduced by <span style="font-variant:small-caps;">Grassmann</span>. The geometric notions include oriented areas and volumes (called *geometric forms*).
Our main interest concerns Chapter V of <span style="font-variant:small-caps;">Peano</span>’s *Applicazioni geometriche*, where we find differentiation of distributive set functions.
*Applicazioni geometriche* is widely cited, but we have the feeling that the work is not sufficiently known. The revolutionary character of <span style="font-variant:small-caps;">Peano</span>’s book is remarked by J. <span style="font-variant:small-caps;">Tannery</span> [@tannery1887 (1887)]:
> Le Chapitre V porte ce titre: [*Grandeurs géométriques*]{}. C’est peut-être le plus important et le plus intéressant, celui, du moins, par lequel le Livre de M. Peano se distingue davantage des Traités classiques: les définitions qui se rapportent aux [*champs de points*]{}, aux points extérieurs, intérieurs ou limites par rapport un champ, aux fonctions distributives (coexistantes d’après Cauchy), la longueur (l’aire ou au volume) externe, interne ou propre d’un champ, la notion d’intégrale étendue a un champ sont présentées sous une forme abstraite, très précise et très claire.[^15]
Only a few authors fully realized the innovative value of Chapter V of *Applicazioni geometriche*. As an instance, <span style="font-variant:small-caps;">Ascoli</span> says:
> In \[*Applicazioni geometriche*\] vi sono profusi, in forma così semplice da parere definitiva, idee e risultati divenuti poi classici, come quelli sulla misura degli insiemi, sulla rettificazione delle curve, sulla definizione dell’area di una superficie, sull’integrazione di campo, sulle funzioni additive di insieme; ed altri che sono tutt’ora poco noti o poco studiati \[…\].[^16]
Most of the modern historians are aware of the contributions to measure theory given by <span style="font-variant:small-caps;">Peano</span> and <span style="font-variant:small-caps;">Jordan</span> concerning inner and outer measure and measurability.[^17]
Only a few historians mention <span style="font-variant:small-caps;">Peano</span>’s contributions to derivative of set functions: <span style="font-variant:small-caps;">Pesin</span> [@pesin], <span style="font-variant:small-caps;">Medvedev</span> [@medvedev] and <span style="font-variant:small-caps;">Hawkins</span> [@hawkins] and others.
<span style="font-variant:small-caps;">Pesin</span> [@pesin (1970) pp.32-33], who does “not intend to overestimate the importance of <span style="font-variant:small-caps;">Peano</span>’s results”, recalls some results of <span style="font-variant:small-caps;">Peano</span>’s work without giving details or appropriate definitions.
<span style="font-variant:small-caps;">Medvedev</span> in [@medvedev (1983)] recalls <span style="font-variant:small-caps;">Peano</span>’s contributions giving detailed information both on the integral as a set function and on the <span style="font-variant:small-caps;">Peano</span>’s derivative. In our opinion he gives an excessive importance to mathematical priorities without pointing out the differences between <span style="font-variant:small-caps;">Peano</span>’s contribution of 1887 and <span style="font-variant:small-caps;">Lebesgue</span>’s contribution of 1910.[^18]
<span style="font-variant:small-caps;">Hawkins</span> does not describe <span style="font-variant:small-caps;">Peano</span>’s results on differentiation and integration in detail, as they are too far from the main aim of his book, but he is aware of <span style="font-variant:small-caps;">Peano</span>’s contributions to differentiation of set functions [@hawkins p.88,185], and appraises <span style="font-variant:small-caps;">Peano</span>’s book *Applicazioni geometriche*:
> the theory is surprisingly elegant and abstract for a work of 1887 and strikingly modern in his approach [@hawkins p.88].
None of the historian quoted above, establishes a link between <span style="font-variant:small-caps;">Peano</span>’s work on differentiation of measure in *Applicazioni geometriche* with his paper *Grandezze coesistenti* [@peano1915] and with Lebesgue’s comments on differentiation presented in *La mesures des grandeurs* [@lebesgue1935 (1935)].
Main primary sources on which our paper is based are [@cauchy1841; @peano1915; @lebesgue1910; @lebesgue1935; @vitali1915; @vitali1916; @fubini1915b; @fubini1915a].
Cauchy’s coexistent magnitudes {#sez-cauchy}
==============================
<span style="font-variant:small-caps;">Cauchy</span>’s seminal paper *Coexistent magnitudes* [@cauchy1841 (1841)] presents some difficulties for the modern reader: the terms he introduces are rather obscure (for instance, *grandeurs*, *coexistantes*, *éléments*, …), and the reasonings are based on vague geometric language, accordingly to the <span style="font-variant:small-caps;">Cauchy</span>’s taste. Actually, <span style="font-variant:small-caps;">Cauchy</span>’s aim was to make mathematical analysis as well rigorous as geometry [@cauchy1821 (1821) p.ii]:
> Quant aux méthodes, j’ai cherché leur donner toute la rigueur qu’on exige en géométrie, de manière à ne jamais recourir aux raisons tirées de la généralité de l’algèbre.[^19]
In his *Leçons de mécanique analytique* [@moigno1868 (1868) pp.172-205] <span style="font-variant:small-caps;">Moigno</span>, a follower of <span style="font-variant:small-caps;">Cauchy</span>, reprints the paper *Coexistent magnitudes*. He puts into evidence the vagueness of some terms of <span style="font-variant:small-caps;">Cauchy</span>, unfortunately without adding any comment that may help the reader to a better understanding of <span style="font-variant:small-caps;">Cauchy</span>’s paper itself.
The meaning of the terms “*grandeurs*” and “*coexistantes*” can be made precise by analyzing the list of examples given by <span style="font-variant:small-caps;">Cauchy</span>. He implicitly postulates the following properties of “*grandeurs*”:
1. [\[cond2\]]{} a magnitude can be divided into finitely many infinitesimal equal elements (using the terminology of <span style="font-variant:small-caps;">Cauchy</span>), where infinitesimal is related to magnitude and diameter;
2. the ratio between coexistent magnitudes (not necessarily homogeneous) is a numerical quantity.
Concerning the term “*coexistantes*”, coexistent magnitudes are defined by <span style="font-variant:small-caps;">Cauchy</span> as “magnitudes which exist together, change simultaneously and the parts of one magnitude exist and change in the same way as the parts of the other magnitude”.[^20] Despite of the vagueness of this definition, the meaning of “*coexistantes*" is partially clarified by many examples of coexistent magnitudes given by <span style="font-variant:small-caps;">Cauchy</span> [@cauchy1841 (1841) pp.188–189], such as the volume and the mass of a body, the time and the displacement of a moving point, the radius and the surface of a circle, the radius and the volume of a sphere, the height and the area of a triangle, the height and the volume of a prism, the base and the volume of a cylinder, and so on.
Vagueness of the <span style="font-variant:small-caps;">Cauchy</span>’s definition of “*grandeurs coexistantes*” was pointed out by <span style="font-variant:small-caps;">Peano</span>. In *Applicazioni geometriche* [@peano87 (1887)] and in *Grandezze coesistenti* [@peano1915 (1915)], <span style="font-variant:small-caps;">Peano</span> defines them as set functions over the same given domain, satisfying additivity properties in a suitable sense.
The primary aim of <span style="font-variant:small-caps;">Cauchy</span> is pedagogic: he wants to write a paper making easier the study of infinitesimal calculus and its applications. As it is easy to understand, <span style="font-variant:small-caps;">Cauchy</span> bases himself on the mass-density paradigm and introduces the limit of the average of two coexistent magnitudes, calling it [*differential ratio*]{}. In a modern language we could say that the coexistent magnitudes are set functions, while the differential ratio is a point function. <span style="font-variant:small-caps;">Cauchy</span> points out that the differential ratio is termed in different ways depending on the context, namely, on the nature of the magnitudes themselves (for instance, mass density of a body at a given point, velocity of a moving point at a given time, hydrostatic pressure at a point of a given surface, …).
Now we list the most significant theorems that are present in the paper of <span style="font-variant:small-caps;">Cauchy</span>, preserving, as much as possible, his terminology.
\[mediaintegrale\][@cauchy1841 Theorem 1, p.190] The average between two coexistent magnitudes is bounded between the supremum and the infimum of the values of the differential ratio.
\[densitanulla\][@cauchy1841 Theorem 4, p.192] A magnitude vanishes whenever its differential ratio, with respect to another coexistent magnitude, is a null function.
\[teomedia\] [@cauchy1841 Theorem 5, p.198] If the differential ratio between two coexistent magnitudes is a continuous function, then the “mean value property” holds.[^21]
\[densitauguale\] [@cauchy1841 Theorem 13, p.202] If two magnitudes have the same differential ratio with respect to another magnitude, then they are equal.
Even if <span style="font-variant:small-caps;">Cauchy</span> presents proofs that are rather “vanishing”, his statements (see theorems listed above) and his use of the differential ratio allow <span style="font-variant:small-caps;">Peano</span> to rebuild his arguments on solid grounds. <span style="font-variant:small-caps;">Peano</span> translates the coexistent magnitudes into the concept of distributive set functions, restating the theorems presented by <span style="font-variant:small-caps;">Cauchy</span> and proving them rigorously.
In <span style="font-variant:small-caps;">Peano</span>, the property of continuity of the differential ratio (whenever it exists) is a consequence of its definition. On the contrary, <span style="font-variant:small-caps;">Cauchy</span>’s definition of differential ratio does not guarantee its continuity. <span style="font-variant:small-caps;">Cauchy</span> is aware of the fact that the differential ratio can be discontinuous, nevertheless he thinks that, in the most common “real” cases, it may be assumed to be continuous; see [@cauchy1841 (1841), p.196]:
> Le plus souvent, ce rapport différentiel sera une fonction continue de la variable dont il dépend, c’est-à-dire qu’il changera de valeur avec elle par degrés insensibles.[^22]
and [@cauchy1841 (1841) p.197]:
> Dans un grand nombre de cas, le rapport différentiel $\rho$ est une fonction continue \[…\].[^23]
In evaluating the differential ratio as a “limit of average values $\frac{\mu(A)}{\nu(A)}$ at a point $P$”, for <span style="font-variant:small-caps;">Peano</span> the set $A$ does not necessarily include the point $P$, while for <span style="font-variant:small-caps;">Cauchy</span> $A$ includes $P$ (as <span style="font-variant:small-caps;">Cauchy</span> says: $A$ *renferme le point* $P$).
This difference is fundamental also in case of linearly distributed masses. Indeed a linear mass distribution, described in terms of a function of a real variable, admits a differential ratio in the sense of <span style="font-variant:small-caps;">Peano</span> if the derivative exists and is continuous, whilst it admits a differential ratio in the sense of <span style="font-variant:small-caps;">Cauchy</span> [^24] only if the function is differentiable [@cauchy1841 (1841) p.208]:
> Lorsque deux grandeurs ou quantités coexistantes se réduisent à une variable $x$ et à une fonction $y$ de cette variable, le rapport différentiel de fonction à la variable est précisément ce qu’on nomme la *dérivée* de la fonction ou le *coefficient différentiel*.[^25]
Concerning the existence of the differential ratio, <span style="font-variant:small-caps;">Cauchy</span> is rather obscure; indeed whenever he defines the differential ratio, he specifies that “it will converge in general to a certain limit different from $0$”. As <span style="font-variant:small-caps;">Cauchy</span> does not clarify the meaning of the expression “in general”, the conditions assuring the existence of the differential ratio are not given explicitly. On the other hand, <span style="font-variant:small-caps;">Cauchy</span> himself is aware of this lack, as in several theorems he explicitly assumes that the differential ratio is “completely determined at every point”.
Concerning the mass-density paradigm, in <span style="font-variant:small-caps;">Cauchy</span>’s *Coexistent magnitudes* an explicit formula allowing for constructing the mass of a body in terms of its density is also lacking. In spite of this, <span style="font-variant:small-caps;">Cauchy</span> provides a large amount of theorems and corollaries giving an approximate calculation of the mass under the assumption of continuity of the density. We can envisage this approach as a first step toward the modern notion of integral with respect to a general abstract measure.
We can summarize further <span style="font-variant:small-caps;">Cauchy</span>’s results into the following theorem:
[@cauchy1841 (1841) pp.208–215] Let us assume that the differential ratio $g$ between two coexistent magnitudes $\mu$ and $\nu$ exists and is continuous. Then $\mu$ can be computed in terms of the integral of $g$ with respect to $\nu$.
<span style="font-variant:small-caps;">Cauchy</span> concludes his memoir [@cauchy1841 (1841) pp.215–229] with a second section in which he states the following theorem in order to evaluate lengths, areas and volumes of homothetic elementary figures.
\[teo-prop\] [@cauchy1841 Theorem 1, p.216] Two coexistent magnitudes are proportional, whenever to equal parts of one magnitude there correspond equal parts of the other.[^26]
Even if the <span style="font-variant:small-caps;">Cauchy</span>’s paper contains several innovative procedures, to our knowledge only a few authors (<span style="font-variant:small-caps;">Moigno</span>, <span style="font-variant:small-caps;">Peano</span>, <span style="font-variant:small-caps;">Vitali</span>, <span style="font-variant:small-caps;">Picone</span> and <span style="font-variant:small-caps;">Lebesgue</span>) quote it, and only <span style="font-variant:small-caps;">Peano</span> and <span style="font-variant:small-caps;">Lebesgue</span> analyze it in details.
Distributive families, decompositions and Peano additivity {#sez-distributive}
==========================================================
In his paper *Le grandezze coesistenti* [@peano1915 (1915)], <span style="font-variant:small-caps;">Peano</span> introduces a general concept of *distributive function*, namely a function $f:A\to B$, where $(A,+), (B,+)$ are two sets endowed with binary operations, denoted by the same symbol $+$, satisfying the equality $$\label{distr}f(x+y)=f(x)+f(y)$$ for all $x,y$ belonging to $A$ and, if necessary, verifying suitable assumptions.[^27] <span style="font-variant:small-caps;">Peano</span> presents several examples of distributive functions. As a special instance, $A$ stands for the family ${{\mathcal{P}}}(X)$ of all subsets of a finite dimensional Euclidean space $X$, “$+$” in the left hand side of (\[distr\]) is the union operation, and “$+$” in the right hand side of (\[distr\]) is the logical OR (denoted in <span style="font-variant:small-caps;">Peano</span>’s ideography by the same symbol of set-union); therefore, equation becomes: $$\label{distr2}f(x\cup y)=f(x) \cup f(y).$$
To make (\[distr2\]) significant, <span style="font-variant:small-caps;">Peano</span> chooses a family ${\mathcal U}\subset {{\mathcal{P}}}(X)$ and defines “$f(x)$” as “$x\in \mathcal{U}$”. Consequently (\[distr2\]) becomes: $$\label{griglia}x\cup y\in {\mathcal U}\Longleftrightarrow x\in {\mathcal U}\text{ or } y\in {\mathcal U}$$ for all $x,y \in {{\mathcal{P}}}(X)$. A family $\mathcal U$ satisfying (\[griglia\]) is called by <span style="font-variant:small-caps;">Peano</span> a *distributive family*.[^28]
Moreover, <span style="font-variant:small-caps;">Peano</span> considers *semi-distributive* families ${{\mathcal{F}}}\subset {{\mathcal{P}}}(X)$, i.e., families of sets such that $$x\cup y\in {{\mathcal{F}}}\Longrightarrow x\in {{\mathcal{F}}}\text{ or } y\in {{\mathcal{F}}}$$ for all $x, y \in {{\mathcal{P}}}(X)$.
A distributive family of subsets of $X$ is obtained by a semi-distributive family ${{\mathcal{F}}}$ by adding to ${{\mathcal{F}}}$ any supersets of its elements. <span style="font-variant:small-caps;">Peano</span> states the following theorem, and attributes to <span style="font-variant:small-caps;">Cantor</span> [@cantor1884 (1884) p.454] both its statement and its proof.
\[teo-F1\] Let ${{\mathcal{F}}}$ be a semi-distributive family of a finite-dimensional Euclidean space, and let $S$ be a bounded non-empty set belonging to ${{\mathcal{F}}}$. Then there exists a point $\bar x$, belonging to the closure of $S$, such that any neighborhood of $\bar x$ contains a set belonging to ${{\mathcal{F}}}$.
The notion of distributive family is essential in the study of the derivation of distributive set functions by <span style="font-variant:small-caps;">Peano</span>. Distributive families have been introduced by <span style="font-variant:small-caps;">Peano</span> in *Applicazioni geometriche* in 1887. Moreover, he uses them in his famous paper on the existence of solutions of differential equations [@peano1890 (1890) pp.201–202] and, later, in his textbook *Lezioni di analisi infinitesimale* [@peano1893 (1893) vol.2, pp.46–53]. The role played by this notion is nowadays recovered by “compactness by coverings” or by “existence of accumulation points”.[^29]
In proving Theorem \[teo-F1\], <span style="font-variant:small-caps;">Peano</span> decomposes a subset of the Euclidean space ${{\mathbb{R}}}^n$ following a grid of n-intervals implemented by cutting sets along hyperplanes parallel to coordinate axis. We may formalize this procedure in the following way.
Let us denote by $H$ a hyperplane of the form $H:=\{x\in{{\mathbb{R}}}^n\,:\, \langle x,e_i\rangle =a\}$ where $e_i$ is a vector of the canonical basis of ${{\mathbb{R}}}^n$ and $a\in{{\mathbb{R}}}$. Let us denote by $H^+$ and $H^-$ the two closed half-spaces delimited by $H$.
A family ${{\mathcal{F}}}$ of subsets of ${{\mathbb{R}}}^n$ is called *semi-distributive by cutting along hyperplanes* if $$A\cap H^+ \in {{\mathcal{F}}}\text{ or } A\cap H^-\in {{\mathcal{F}}}$$ for every $A \in {{\mathcal{F}}}$ and for every hyperplane $H$ of ${{\mathbb{R}}}^n$ of the form indicated above. Under this restrictions a new version of Theorem \[teo-F1\] still holds:
\[teo-F2\] Let ${{\mathcal{F}}}$ be semi-distributive by cutting along hyperplanes and let $S$ be a bounded non-empty set belonging to ${{\mathcal{F}}}$. Then there exists a point $\bar x$ belonging to the closure of $S$ such that any neighborhood of $\bar x$ contains a set belonging to ${{\mathcal{F}}}$.
To express additivity properties of set functions, <span style="font-variant:small-caps;">Peano</span>, as it was common at his time [^30], uses the term *decomposition*. <span style="font-variant:small-caps;">Peano</span> writes in *Applicazioni geometriche* [@peano87 (1887) p.164, 167]:
> Se un campo $A$ è decomposto in parti $A_1,A_2, \dots, A_n$ esso si dirà *somma* delle sue parti, e si scriverà $$A=A_1+A_2+\dots+A_n$$ \[…\] Una grandezza dicesi *funzione distributiva* di un campo, se il valore di quella grandezza corrispondente ad un campo è la somma dei valori di essa corrispondenti alle parti in cui si può decomporre un campo dato.[^31]
In order to formalize in modern language both the operation of “decomposing” and his use in <span style="font-variant:small-caps;">Peano</span>’s works, we can pursuit a “minimal” way, leading to “families of interval-decompositions”, and a “proof-driven” way, leading to “families of finite decompositions”.
First, the minimal way consists in implementing the procedure of decomposing by cutting along hyperplanes used by <span style="font-variant:small-caps;">Peano</span> in proving Theorem \[teo-F1\]. More precisely, let ${{\mathcal{A}}}$ be a family of subsets of the Euclidean space ${{\mathbb{R}}}^n$; a finite family $\{A_i\}_{i=1}^m$ of elements of ${{\mathcal{A}}}$ is called an *interval-decomposition* of $A \in {{\mathcal{A}}}$ if it is obtained by iterating the procedure of cutting by hyperplanes. In other words, an interval-decomposition $\{A_i\}_{i=1}^m$ of a set $A$ is a finite sub-family of ${{\mathcal{A}}}$ defined recursively as follows:
- for $m=1$, $A_1=A$;
- for $m=2$, there exists a hyperplane $H$ such that $A_1= A\cap H^-$ and $A_2= A\cap H^+$;
- for $m>2$, there exist two distinct indices $i_0, i_1 \le n$ such that $\tilde A := A_{i_0} \cup A_{i_1} \in {{\mathcal{A}}}$ and the families $\{A_i : 1 \le i \le m, i \ne i_0, i \ne i_1 \}\cup \{\tilde A \}$ and $\{A_{i_0}, A_{i_1} \}$ are interval-decompositions of $A$ and $\tilde A$, respectively.
The totality of these interval-decompositions will be denoted by ${{\mathbb{D}}}_{\textrm{int}}({{\mathcal{A}}})$. In the case where ${{\mathcal{A}}}$ is the family of all the closed bounded subintervals of a given closed interval $[a,b]$ of the real line, an arbitrary interval-decomposition of an interval $[a',b']\subset[a,b]$ is a family $\{ [a_{i-1},a_i] \}_{i=1}^m$ where $a' = a_0 \le a_1 \le \dots \le a_{m-1} \le a_m= b'$. The totality of these interval-decompositions are denoted by ${{\mathbb{D}}}_{\textrm{int}}(a,b)$.
The second way consists in summarizing explicitly the properties of the decompositions themselves, as used by <span style="font-variant:small-caps;">Peano</span> in defining the integral and in proving related theorems [^32], as it will be seen in Section \[sez-massdensity\]. This leads to the following definitions of *family of finite decompositions* and of the related *semi-distributive family*, *Cantor compactness property* and *distributive set functions*.
Let ${{\mathcal{A}}}$ be again a family of subsets of an Euclidean space ${{\mathbb{R}}}^n$ and let us denote by ${{\mathcal{P}}}_{f}({{\mathcal{A}}})$ the set of all non-empty finite subfamily of ${{\mathcal{A}}}$. Define ${{\mathbb{U}}}({{\mathcal{A}}})$ by $${{\mathbb{U}}}({{\mathcal{A}}}):=\{{{\mathcal{H}}}\in {{\mathcal{P}}}_{f}({{\mathcal{A}}}): \cup {{\mathcal{H}}}\in {{\mathcal{A}}}\}.$$ Let ${{\mathbb{D}}}$ be a subset of ${{\mathbb{U}}}({{\mathcal{A}}})$; we will say that ${{\mathcal{H}}}$ is a ${{\mathbb{D}}}$-*decomposition* of $A$ if ${{\mathcal{H}}}\in {{\mathbb{D}}}$ and $A=\cup{{\mathcal{H}}}$.
${{\mathbb{D}}}\subset {{\mathbb{U}}}({{\mathcal{A}}})$ is called a *family of finite decompositions relative to ${{\mathcal{A}}}$* if the following properties are satisfied:
1. $\{A\}\in {{\mathbb{D}}}$ for every $A\in{{\mathcal{A}}}$;
2. [\[dec\_1\]]{} if ${{\mathcal{H}}}$ and ${{\mathcal{G}}}$ are ${{\mathbb{D}}}$-decompositions of a set $A$, then $$\{H\cap G: H\in{{\mathcal{H}}},\,G\in{{\mathcal{G}}}\}$$ is a ${{\mathbb{D}}}$-decomposition of $A$;
3. [\[dec\_2\]]{} if ${{\mathcal{H}}}$ and ${{\mathcal{G}}}$ are ${{\mathbb{D}}}$-decompositions of $A$, then for every $G\in {{\mathcal{G}}}$ the family $${{\mathcal{H}}}_G := \{H\cap G : H\in{{\mathcal{H}}}\}$$ is a ${{\mathbb{D}}}$-decomposition of $G$;
4. [\[dec\_3\]]{} if ${{\mathcal{H}}}$ is a ${{\mathbb{D}}}$-decomposition of $A$ and, moreover, for every $H \in {{\mathcal{H}}}$ the family ${{\mathcal{G}}}_H$ is a ${{\mathbb{D}}}$-decomposition of $H$, then $$\cup \{{{\mathcal{G}}}_H : H \in {{\mathcal{H}}}\}$$ is a ${{\mathbb{D}}}$-decomposition of $A$.
\[def\_inf\] A family ${{\mathbb{D}}}$ of finite decompositions relative to ${{\mathcal{A}}}$ is called *infinitesimal* if, for every bounded set $A \in {{\mathcal{A}}}$ and for every real number $\varepsilon >0$, there is a ${{\mathbb{D}}}$-decomposition ${{\mathcal{H}}}$ of $A$ such that the diameter of every $H \in {{\mathcal{H}}}$ is less than $\varepsilon$.
Let ${{\mathbb{D}}}$ be a family of finite decompositions relative to ${{\mathcal{A}}}$. Then a *set function* $\mu \colon {{\mathcal{A}}}\to {{\mathbb{R}}}$ is said to be *distributive with respect to* ${{\mathbb{D}}}$, if $$\mu (\cup {{\mathcal{H}}}) = \sum_{H \in {{\mathcal{H}}}} \mu(H) \text{ for every }{{\mathcal{H}}}\in {{\mathbb{D}}}.$$
Consequently,
\[def\_semi\] Let ${{\mathbb{D}}}$ be a family of finite decompositions relative to ${{\mathcal{A}}}$. A family ${{\mathcal{F}}}$ of subsets of the Euclidean space ${{\mathbb{R}}}^{n}$ is said to be *semi-distributive with respect to* ${{\mathbb{D}}}$, if $${{\mathcal{H}}}\in{{\mathbb{D}}}\text{ and } \cup{{\mathcal{H}}}\in {{\mathcal{F}}}\Longrightarrow \exists H\in{{\mathcal{H}}}\text{ such that } H\in{{\mathcal{F}}}.$$
\[teo-F3\] Let ${{\mathbb{D}}}$ be an infinitesimal family of finite decompositions relative to ${{\mathcal{A}}}$ and let ${{\mathcal{F}}}$ be a semi-distributive family with respect to ${{\mathbb{D}}}$. If $S$ is a bounded non-empty set belonging to ${{\mathcal{F}}}$. Then there exists a point $\bar x$ belonging to the closure of $S$ such that any neighborhood of $\bar x$ contains a set belonging to ${{\mathcal{F}}}$.
In the following, an expression of type “$\mu \colon({{\mathcal{A}}},{{\mathbb{D}}}) \to {{\mathbb{R}}}$ is a distributive set function” stands for “${{\mathbb{D}}}$ is a family of finite decompositions relative to ${{\mathcal{A}}}$ and $\mu\colon{{\mathcal{A}}}\to {{\mathbb{R}}}$ is a distributive set function with respect to ${{\mathbb{D}}}$.
Examples of families of decompositions are ${{\mathbb{U}}}({{\mathcal{A}}})$, and
1. [\[dec-inter\]]{} the family ${{\mathbb{D}}}_{\textrm{int}}({{\mathcal{A}}})$ of all interval-decompositions introduced above;[^33]
2. [\[dec-picone\]]{} the family of all ${{\mathcal{H}}}\in {{\mathbb{U}}}({{\mathcal{A}}})$ such that the interiors of two arbitrary distinct elements of ${{\mathcal{H}}}$ have empty intersection and every $H \in {{\mathcal{H}}}$ is Peano-Jordan measurable;
3. [\[dec-picone2\]]{} the family of all ${{\mathcal{H}}}\in {{\mathbb{U}}}({{\mathcal{A}}})$ such that the intersection of the closure of two arbitrary distinct elements of ${{\mathcal{H}}}$ have null Peano-Jordan measure and every $H\in {{\mathcal{H}}}$ is bounded;
4. [\[last\_ex\]]{} the family of all ${{\mathcal{H}}}\in {{\mathbb{U}}}({{\mathcal{A}}})$ such that two arbitrary distinct elements of ${{\mathcal{H}}}$ have empty intersection.
The interval-decompositions (in particular ${{\mathbb{D}}}(a,b)$) occurs frequently in <span style="font-variant:small-caps;">Peano</span>’s works. Distributive set functions related to the last example (\[last\_ex\]) are well known as *finitely additive set functions*; this type of additivity, expressed in terms of partitions of sets, was introduced for the first time in <span style="font-variant:small-caps;">Borel</span> [@borel1898 (1898), pp.46-50], and, more clearly, in <span style="font-variant:small-caps;">Lebesgue</span> [@lebesgue1902 (1902), p.6].
As far as we know, all historians interpreted <span style="font-variant:small-caps;">Peano</span>’s distributive set functions as “finitely additive” set functions.[^34] For instance, in the proof of the integrability of functions [@peano87 (1887) p.188], <span style="font-variant:small-caps;">Peano</span> uses distributivity properties of the upper and lower integral with respect to the domain of integration; clearly neither the upper nor the lower integral are finitely additive.
Peano’s strict derivative of distributive functions\
and its applications {#sez-derivata}
====================================================
In *Applicazioni geometriche* [@peano87 (1887)] <span style="font-variant:small-caps;">Peano</span> translates in terms of “distributive functions” the “magnitudes” of <span style="font-variant:small-caps;">Cauchy</span>, so that two <span style="font-variant:small-caps;">Cauchy</span>’s magnitudes are “coexistent” if they are distributive functions with the same domain.
<span style="font-variant:small-caps;">Peano</span>’s distributive set functions are called *positive* if their values are positive. <span style="font-variant:small-caps;">Peano</span>’s *strict derivative* is defined by [^35]
Let $\mu, \nu : ({{\mathcal{A}}},{{\mathbb{D}}}) \to {{\mathbb{R}}}$ be distributive set functions, and let $\nu$ be positive. A real function $g$ over a set $S$ is called a “strict derivative of $\mu$ with respect to $\nu$” on $S$ (denoted by $\frac{d\,\mu}{d\,\nu}$ and termed *rapporto* in *Applicazioni geometriche*) if, for every point $x \in S$ and for every $\epsilon >0$, there exists $\delta >0$ such that [^36] $$\left|\frac{\mu (A)}{\nu (A)}-g(x)\right|<\epsilon \quad \text{for every } A\in {{\mathcal{A}}}, \, \text{with } \nu (A)\neq 0, \,
A\subset B_\delta(x).$$
It is worth noticing that the concept of strict derivative given by <span style="font-variant:small-caps;">Peano</span> provides a consistent mathematical ground to the concept of “infinitesimal ratio” between two magnitudes, successfully used since <span style="font-variant:small-caps;">Kepler</span>. A remarkable example given by <span style="font-variant:small-caps;">Peano</span> is the evaluation of a rectifiable arc length by integrating the “infinitesimal arc length” $ds$. Notice that, whenever $ds$ exists in the sense of <span style="font-variant:small-caps;">Peano</span>, the corresponding integral provides the length of the arc. On the contrary, the integration of the infinitesimal arc length $ds$, evaluated in the sense of <span style="font-variant:small-caps;">Lebesgue</span> (1910), provides the length of the arc only in case of absolute continuity of the arc parametrization (see <span style="font-variant:small-caps;">Tonelli</span> [@tonelli (1908)])
The existence of <span style="font-variant:small-caps;">Peano</span>’s strict derivative is not assured in general; its characterizing properties are clearly presented in *Applicazioni geometriche* and can be summarized in the following theorems.
First, <span style="font-variant:small-caps;">Peano</span> gives a precise form to <span style="font-variant:small-caps;">Cauchy</span>’s Theorem \[mediaintegrale\], stating the following:
\[teorema-fondamentale\] Let $\mu, \nu : ({{\mathcal{A}}},{{\mathbb{D}}}) \to {{\mathbb{R}}}$ be distributive set functions with ${{\mathbb{D}}}$ infinitesimal and $\nu$ positive. If $S \in {{\mathcal{A}}}$ is a closed and bounded non-empty set and $g$ is the strict derivative of $\mu$ with respect to $\nu$ on $S$, then $$\label{mediafor}
\inf_{S} g\leq \frac{\mu (A)}{\nu(A)}\leq \sup_{S} g$$ for all $A\in {{\mathcal{A}}}$ with $A\subset S$ and $\nu (A)>0$.
In the case ${{\mathbb{D}}}={{\mathbb{D}}}_{\textrm{int}}$, <span style="font-variant:small-caps;">Peano</span> proves this fundamental theorem by applying Theorem \[teo-F2\] to the semi-distributive families ${{\mathcal{F}}}_a:=\{A\in{{\mathcal{A}}}\, :\,\mu(A)>a\, \nu(A)\}$ and ${{\mathcal{G}}}_a:=\{A\in{{\mathcal{A}}}\, :\,\mu(A)<a\, \nu(A)\}$, for real numbers $a$. Observe that (\[mediafor\]) amounts to (\[Hahn\_1\])-(\[Hahn\_2\]) and also, indirectly, to (\[hom\])-(\[sum\]).
In *Applicazioni geometriche*, Theorem \[teorema-fondamentale\] is followed by three corollaries, which we summarize into the following:
\[cor-fond\] [@peano87 (1987) p.171] Under the same hypothesis as in the previous theorem:
1. [\[cor-1\]]{} if the strict derivative $\frac{d\,\mu}{d\,\nu}$ is a constant $b$ on $S$, then $\mu (A)=b\, \nu(A)$, for all $A\in {{\mathcal{A}}}$ with $A\subset S$;
2. [\[cor-2\]]{} if the strict derivative $\frac{d\,\mu}{d\,\nu}$ vanishes at every point of $S$, then $\mu(A)=0$, for all $A\in {{\mathcal{A}}}$ with $A\subset S$;
3. [\[due-mis\]]{} if two distributive set functions have equal strict derivatives with respect to $\nu$ on $S$, then they are equal on subsets of $S$ belonging to ${{\mathcal{A}}}$.[^37]
The following fundamental <span style="font-variant:small-caps;">Peano</span>’s result point out the difference of <span style="font-variant:small-caps;">Peano</span>’s approach with respect to both approaches of <span style="font-variant:small-caps;">Cauchy</span> and of <span style="font-variant:small-caps;">Lebesgue</span> (1910).
Under the same hypothesis as in the previous theorem, if the strict derivative of $\mu$ with respect to $\nu$ exists on $S$, then it is continuous on $S$.
The importance of these results is emphasized in *Applicazioni geometriche* by a large amount of evaluations of derivatives of distributive set functions. As a consequence of the existence of the strict derivative, <span style="font-variant:small-caps;">Peano</span> gives, for the first time, several examples of measurable sets. The most significant examples, observations and results are listed below.
1. [\[hypo-der\]]{} [*Measurability of the hypograph of a continuous function*]{} [@peano87 (1887) pp.172-174]. Let $f$ be a continuous positive real function defined on an interval $[a,b]$, let ${{\mathcal{A}}}$ be the family of all sub-intervals of $[a,b]$ and let $\nu$ be the Euclidean measure on $1$-dimensional intervals. Define $\mu_f : {{\mathcal{A}}}\to {{\mathbb{R}}}$ on every $A$ belonging to ${{\mathcal{A}}}$, by the inner (respectively, the outer) $2$-dimensional measure (in the sense of Peano-Jordan) of the *positive-hypograph* of $f$, restricted to $A$.[^38] In any case, independently of the choice of inner or outer measure, we have that $\mu_f$ and $\nu$ are distributive set functions with respect to ${{\mathbb{D}}}(a,b)$, and that $\frac{{{\mathrm{d}}}\mu_f}{{{\mathrm{d}}}\nu}(x)=f(x)$ for every $x \in [a,b]$. From (\[due-mis\]) of Corollary \[cor-fond\] it follows that the inner measure of the positive-hypograph of the continuous function $f$ coincides with its outer measure; therefore it is measurable in the sense of Peano-Jordan.
2. Analogously, <span style="font-variant:small-caps;">Peano</span> considers continuous functions of two variables and the volume of the *positive-hypograph* [@peano87 (1887) p.175].
3. [\[area-star\]]{} *Area of a plane star-shaped subset delimited by a continuous closed curve* [@peano87 (1887) pp.175-176]. Consider a continuous closed curve that can be described in polar coordinates in terms of a continuous function $\rho:[0,2\pi] \to {{\mathbb{R}}}_+$, with $\rho(0)=\rho(2\pi)$. Let ${{\mathcal{A}}}$ be the family of all subintervals of $[0,2\pi]$; and for every $A\in{{\mathcal{A}}}$, let $\nu(A)$ denote the Euclidean measure of the area of the circular sector $\{(\rho \cos(\theta),\rho\sin(\theta)) : \theta \in A, \rho\in [0,1]\}$. Moreover, let $\mu(A)$ denote inner (or outer, indifferently) Peano-Jordan $2$-dimensional measure of the set $\{(\rho\cos(\theta),\rho\sin(\theta)) : \theta \in A, \rho\in [0,\tilde\rho (\theta)]\}$. Then the strict derivative $\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(\theta )$ is equal to $\rho^2 (\theta)$. From the fact that this derivative does not depend on the choice of inner or outer measure, it follows Peano-Jordan measurability of plane star-shaped sets delimited by continuous closed curves.
4. Analogously, <span style="font-variant:small-caps;">Peano</span> considers the volume of a star-shaped set bounded by simple continuous closed surface [@peano87 (1887) p.177].
5. [\[cav-pri\]]{}[*Cavalieri’s principle between a prism and a spatial figure*]{} [@peano87 (1887) pp.177-179]. Consider a straight line $r$ in the tri-dimensional space, an unbounded cylinder $P$ parallel to $r$ with polygonal section, and a spatial figure $F$. Let $\pi_x$ denote the plane perpendicular to $r$ at the point $x\in r$. Assume Peano-Jordan measurability of all sections of $F$ perpendicular to $r$, namely $$\mu_e(\partial F\cap \pi_x)=0 \quad \text{ for all } x \in r
\leqno \quad \qquad (*)$$ where $\mu_e$ denotes $2$-dimensional Peano-Jordan outer measure and $\partial F$ denotes the boundary of $F$. Let ${{\mathcal{A}}}$ be the family of all segments of $r$. Given a set $A\in{{\mathcal{A}}}$, let $\mu : {{\mathcal{A}}}\to {{\mathbb{R}}}$ denote the outer (or inner, indifferently) $3$-dimensional measure of the set $\cup_{x\in A}(F\cap \pi _x)$, and $\nu (A)$ denote Peano-Jordan $3$-dimensional measure of the set $\cup_{x\in A}(P\cap \pi_x)$. The set functions $\mu$ and $\nu$ are distributive with respect to the family ${{\mathbb{D}}}(r)$ of interval-decompositions of $r$ and $$\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(x )=\frac{\mu_e(F\cap \pi _x)}{\mu_e(P\cap \pi _x)} \quad \text{for every } x \in r.$$ From the fact that this derivative does not depend on the choice of the inner or outer measure involved in defining $\mu$, it follows Peano-Jordan measurability of the spatial figure $F$.
6. [*Cavalieri’s principle between two spatial figures*]{} [@peano87 (1887) p.180]. Consider two spatial figures $F$ and $G$ such that all their sections with planes perpendicular to a given straight line $r$ are Peano-Jordan measurable. Let ${{\mathcal{A}}}$ be the family of all segments of $r$. Given a set $A\in{{\mathcal{A}}}$, let $\mu(A)$ and $\nu(A)$ denote outer (or inner, indifferently) Peano-Jordan $3$-dimensional measures of the sets $\cup_{x\in A}(F\cap \pi _x)$ and $\cup_{x\in A}(G\cap \pi _x)$, respectively. The set functions $\mu$ and $\nu$ are distributive with respect to the family ${{\mathbb{D}}}(r)$ of interval-decompositions of $r$ and $$\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(x)=\frac{\mu_e(F\cap \pi _x)}{\mu_e(G\cap \pi _x)} \quad \text{for every } x \in r.$$ Hence, from (\[cor-1\]) it follows the classical Cavalieri’s principle: two figures whose corresponding sections have equal areas, have the same volume.
7. [\[cav-3\]]{} [*Cavalieri’s principle for 3 dimensional figures with respect to one dimensional sections*]{} [@peano87 (1887) p.180]. Consider a plane $\pi$. Let ${{\mathcal{A}}}$ be the family of all rectangles of $\pi$ and let $r_x$ be the straight line perpendicular to $\pi$ at $x\in\pi$. Moreover, consider a spatial figures $F$ such that for any $x\in \pi$ $$\mu_e(\partial F\cap r_x)=0 \quad \text{ for every } x \in \pi \leqno \quad \qquad (**)$$ where $\mu_e$ denotes the Peano-Jordan $1$-dimensional outer measure and $\partial F$ denotes the boundary of $F$. Given a set $Q\in{{\mathcal{A}}}$, let $\mu (Q)$ denote the outer (or inner, indifferently) measure of the set $\cup_{x\in Q}(F\cap r_x)$, and $\nu(Q)$ denote the elementary usual measure of $Q$. Then $\mu$ and $\nu$ are distributive with respect interval-decompositions of rectangles of $\pi$ and $$\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(x )= \mu_e(F\cap r_x) \quad \text{ for every } x \in \pi.$$
8. [\[cav-pri-pia\]]{} [*Cavalieri’s principle for 2 dimensional figures*]{} [@peano87 (1887) p.180]. Analogously to (\[cav-pri\]), <span style="font-variant:small-caps;">Peano</span> considers Cavalieri’s principle for planar figures.
9. [*Derivative of the length of an arc*]{} [@peano87 (1887) p.181]. In order to compare the length of an arc with the length of its orthogonal projection on a straight line $r$, <span style="font-variant:small-caps;">Peano</span> assumes that the orthogonal projection is bijective on a segment $\rho$ of $ r$, and that the arc can be parametrized through a function with continuous non null derivative.[^39] Let ${{\mathcal{A}}}$ be the family of all closed bounded segments of $\rho$. For every segment $A \in {{\mathcal{A}}}$, let $\mu(A)$ denote the length of the arc whose orthogonal projection over $r$ is $A$ and let $\nu(A)$ denote the length of $A$. Then $$\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(x )=\frac{1}{\cos\theta_x} \leqno \qquad \quad (***)$$ where $\theta_x$ is the angle between $r$ and the straight line that is tangent to the arc at the point (of the arc) corresponding to $x \in \rho$.[^40]
10. [*Derivative of the area of a surface*]{} [@peano87 (1887) pp.182-184]. By adapting the previous argument, <span style="font-variant:small-caps;">Peano</span> shows that the strict derivative between the area of a surface and its projection on a plane is given by (\*\*\*), where $\cos \theta$ is the cosinus of the angle between the tangent plane and the projection plane.
Distributive set functions: integral and strict derivative {#sez-massdensity}
==========================================================
<span style="font-variant:small-caps;">Peano</span> introduces also the notion of integral with respect to a positive distributive set function. The *proper integral* of a bounded function $\rho $ on a set $A\in {{\mathcal{A}}}$ with respect to a positive distributive set function $\nu \colon ({{\mathcal{A}}}, {{\mathbb{D}}}) \to {{\mathbb{R}}}$, is denoted by $\int _A\rho \,{{\mathrm{d}}}\nu$ and is defined as the real number such that for any ${{\mathbb{D}}}$-decomposition $\{A_i\}_{i=1}^m$ of the set $A$, one has $$\int _A\rho \, {{\mathrm{d}}}\nu \ge \rho_1'\nu (A_1)+\rho_2'\nu (A_2)+\dots+\rho_n'\nu (A_m)$$ $$\int _A\rho \,{{\mathrm{d}}}\nu \le \rho_1''\nu (A_1)+\rho_2''\nu (A_2)+\dots+\rho_n''\nu (A_m)$$ where $\rho_1',\rho_2',\dots,\rho_m'$ (respectively $\rho_1'',\rho_2'',\dots,\rho_m''$), are numbers defined by $$\label{darboux}
\rho'_i :=\inf_{x\in A_i}\rho(x) \quad\text{and} \quad \rho_i'':=\sup_{x\in A_i}\rho (x),$$ for all $i=1,...,m$.[^41]
<span style="font-variant:small-caps;">Peano</span> defines also the [*lower*]{} $ \underline\int _{A}\rho \, {{\mathrm{d}}}\nu$ and the [*upper*]{} integral $ \overline\int _{A}\rho \, {{\mathrm{d}}}\nu$ of a bounded function $\rho$ on a set $A\in{{\mathcal{A}}}$ by $$\underline\int _{A}\rho \, {{\mathrm{d}}}\nu :=\sup s' \quad \text{ and } \quad
\overline\int _{A}\rho \, {{\mathrm{d}}}\nu := \inf s''$$ where $s'= \rho_1'\nu (A_1)+\rho_2'\nu (A_2)+\dots+\rho_m'\nu (A_m)$ and $s''=\rho_1''\nu (A_1)+\rho_2''\nu (A_2)+\dots+\rho_m''\nu (A_m)$, where $\rho_i'$ and $\rho_i''$ are defined as in (\[darboux\]) and $\{A_i\}_{i=1}^m$ runs over ${{\mathbb{D}}}$-decompositions of $A$.
In <span style="font-variant:small-caps;">Peano</span>’s terminology, the integrals defined above are called *geometric integrals*. <span style="font-variant:small-caps;">Peano</span> stresses the analogy among these integrals and the usual *elementary integral* $\int_a^b f(x) \, {{\mathrm{d}}}x$ of functions $f$ defined over intervals of ${{\mathbb{R}}}$.
Using property of ${{\mathbb{D}}}$-decompositions, <span style="font-variant:small-caps;">Peano</span> shows that the lower integral is always less or equal than the upper integral. When these values coincide, their common value is called a proper integral and is denoted by $\int_A\rho\, {{\mathrm{d}}}\nu$.
Moreover, using properties and of ${{\mathbb{D}}}$-decompositions, <span style="font-variant:small-caps;">Peano</span> shows that the lower integral $A \mapsto \underline\int _{A}\rho \, {{\mathrm{d}}}\nu $ and the upper integral $A \mapsto \bar\int _{A}\rho \, {{\mathrm{d}}}\nu$ are distributive set functions on ${{\mathcal{A}}}$ with respect to the same family ${{\mathbb{D}}}$ of decompositions [@peano87 (1887) TheoremI, p.187].
In case of $\rho$ continuous, using the property of “infinitesimality” of ${{\mathbb{D}}}$ (see Definition \[def\_inf\]), <span style="font-variant:small-caps;">Peano</span> shows that the derivative of both lower and upper integrals with respect to $\nu$ is $\rho$ [@peano87 (1887) TheoremII, p.189]; consequently the proper integral $\int_A\rho\, {{\mathrm{d}}}\nu$ of a continuous $\rho$ exists whenever $A$ is closed and bounded [@peano87 (1887) Cor. of TheoremII, p.189].
The definitions introduced above allow <span style="font-variant:small-caps;">Peano</span> to realize the mass-density paradigm, i.e., to prove that it is possible to recover a distributive function $\mu$ as the integral of the strict derivative $\frac{d\mu}{d \nu}$ with respect to a positive distributive function $\nu$. <span style="font-variant:small-caps;">Peano</span>’s results can be formulated into the following
\[peanodev\] Let $\mu,\nu : ({{\mathcal{A}}}, {{\mathbb{D}}}) \to {{\mathbb{R}}}$ be distributive set functions, with $\nu$ positive and ${{\mathbb{D}}}$ infinitesimal. Let $S\in{{\mathcal{A}}}$ be a closed bounded set and $\rho:S\to{{\mathbb{R}}}$ a function. The following properties are equivalent:
1. $\rho$ is the strict derivative $\frac{d\,\mu}{d\,\nu}$ of $\mu$ with respect to $\nu$ on $S$;
2. $\rho$ is continuous and $\mu (A)=\int_A\rho \, {{\mathrm{d}}}\nu$ for any $A\subset S$, $A\in{{\mathcal{A}}}$.
<span style="font-variant:small-caps;">Peano</span> applies Theorem \[peanodev\] to the list of examples of strict derivatives of distributive set functions of §\[sez-derivata\] and obtains the following results.
1. *Fundamental theorem of integral calculus for continuous functions* [@peano87 (1887) pp.191-193]. Consider a continuous function $f$ on ${{\mathbb{R}}}$ and let $F$ be a primitive of $f$. Define $\mu$ and $\nu$ over the family ${{\mathcal{A}}}$ of closed bounded intervals $[a,b]$ of ${{\mathbb{R}}}$ by $\mu([a,b]):=F(b) - F(a)$ and $\nu([a,b]):=b-a$. Observe that both $\mu$ and $\nu$ are distributive set functions with respect to ${{\mathbb{D}}}({{\mathbb{R}}})$ and $$\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}(x) =
\lim_{\begin{subarray}{c}a,b\to x \\ a\neq b\end{subarray}}
\frac{F(b) - F(a)}{b-a} = f(x)$$ since $F$ has continuous derivative.[^42] Therefore, by Theorem \[peanodev\], <span style="font-variant:small-caps;">Peano</span> obtains $$F(b)-F(a)=\mu([a,b]) =\int_{[a,b]} f\, d\, \nu = \int_a^b f(x)\, {{\mathrm{d}}}x \,.$$
2. [*Calculus of an integral as a planar area*]{} [@peano87 (1887) pp.193-195]. The elementary integral of a continuous positive function is Peano-Jordan measure of the positive hypograph of the function. This is an immediate application of Theorem \[peanodev\] to the setting (\[hypo-der\]).
3. [*Cavalieri’s formula for planar figures*]{} [@peano87 (1887) p.195]. Let us suppose that $C\subset{{\mathbb{R}}}^2$, $C_x :=\{y\in{{\mathbb{R}}}\, :\; (x,y)\in C\}$ and $(\partial C)_x := \{y\in{{\mathbb{R}}}\, :\; (x,y)\in \partial C\}$ for every $x \in {{\mathbb{R}}}$. Assume that for any $x$ the set $(\partial C)_x$ has vanishing outer measure. As a consequence of Theorem \[peanodev\] and the two-dimensional version of *Cavalieri’s principle* (\[cav-pri-pia\]) (see [@peano87 (1887) p.180]), it follows that the measure of the part of the figure $C$, bounded by the abscissas $a$ and $b$, is equal to $$\int_a^b \mu_e (C_x) \, {{\mathrm{d}}}x$$ where $\mu_e$ denotes outer Peano-Jordan one-dimensional measure.
4. [*Area of a plane star-shaped subset delimited by a continuous closed curve*]{} [@peano87 (1887) p.199]. In the setting of example (\[area-star\]), <span style="font-variant:small-caps;">Peano</span> shows that the area of the sector between the angles $\theta_0$ and $\theta_1$, delimited by a curve described in polar coordinates by $\rho$, is equal to $$\frac{1}{2}\int_{\theta_0}^{\theta_1} \rho(\theta)^2{{\mathrm{d}}}\theta \,.$$
5. [*Cavalieri’s formula for volumes*]{} [@peano87 (1887) p.221]. In the setting (\[cav-3\]), let’s define $F_x := \{(y,z) \in{{\mathbb{R}}}^2 : (x,y,z)\in F\}$ and $(\partial F)_x := \{(y,z)\in{{\mathbb{R}}}^2 : (x,y,z)\in \partial F\}$. Assume that for any $x$, the set $(\partial F)_x$ has vanishing outer measure. From Theorem \[peanodev\], <span style="font-variant:small-caps;">Peano</span> shows that the volume of the part of the figure $F$, delimited by the planes $x=a$ and $x=b$, is equal to $$\int_a^b \mu_e(F_x)\,{{\mathrm{d}}}x \,$$ where $\mu_e$ denotes outer Peano-Jordan two-dimensional measure.
Coexistent magnitudes in Lebesgue and Peano’s derivative {#sez-comments}
========================================================
<span style="font-variant:small-caps;">Lebesgue</span> gives a final pedagogical [^43] exposition of his measure theory in *La mesure des grandeurs* [@lebesgue1935 (1935) p.176], by referring directly to <span style="font-variant:small-caps;">Cauchy</span>’s *Coexistent magnitudes*:[^44]
> La théorie des grandeurs qui constitue le précédent chapitre avait été préparée par des recherches de Cauchy, sur ce qu’il appelait des grandeurs concomitantes \[*sic*\], par les travaux destinés à éclaircir les notions d’aire, de volume, de mesure \[…\].[^45]
<span style="font-variant:small-caps;">Lebesgue</span> is aware of the obscurity of the concepts that are present in <span style="font-variant:small-caps;">Cauchy</span>’s *Coexistent magnitudes*, starting by the meaning of the term *magnitude* itself. In this respect, in order to put on a solid ground the ideas of <span style="font-variant:small-caps;">Cauchy</span>, <span style="font-variant:small-caps;">Lebesgue</span> was compelled to pursuit an approach similar to that <span style="font-variant:small-caps;">Peano</span>: in fact he defines a “magnitude” as a set function on a family of sets ${{\mathcal{A}}}$, requires infinitesimality of ${{\mathcal{A}}}$ (in the sense that every element of ${{\mathcal{A}}}$ can be *réduit à un point par diminutions successives*), and additivity properties that he express in *La mesure des grandeurs* [@lebesgue1934 (1934) p.275] in these words:
> Si l’on divise un corps $C$ en un certain nombre de corps partiels $C_1, C_2,$ $ \dots, C_p$, et si la grandeur $G$ est, pour ces corps, $g$ d’une part, $g_1, g_2, \dots, g_p$ d’autre part, on doit avoir: $g=g_1 + g_2 + \dots + g_p$.[^46]
In *La mesure des grandeurs* <span style="font-variant:small-caps;">Lebesgue</span> considers the operations of integration and differentiation by presenting these topics in a new form with respect to his fundamental and celebrated paper *L’int[é]{}gration des fonctions discontinues* [@lebesgue1910 (1910)].
<span style="font-variant:small-caps;">Lebesgue</span> theory of differentiation of 1910 concerns absolutely continuous $\sigma$-additive measures on Lebesgue measurable sets. On the contrary, twenty-five years later in *La mesure des grandeurs* of 1935
- $\sigma$-additive set functions are replaced by continuous [^47] additive [^48] measures;
- absolutely continuous measures become set functions with bounded-derivative [^49] (*à nombres dérivés bornés*);
- Lebesgue measurable sets are replaced by Jordan-Peano measurable subsets of a given bounded set.
Let $K$ be a bounded closed subset of Euclidean space ${{\mathbb{R}}}^n$, let ${{\mathcal{A}}}_K$ be the family of Jordan-Peano measurable (*quarrables*) subsets of $K$ and let $V$ be a positive, continuous, additive set function on ${{\mathcal{A}}}_K$ with bounded-derivative. Then <span style="font-variant:small-caps;">Lebesgue</span> introduces a definition of derivative. The *uniform-derivative* (*dérivée convergence uniforme*) $\varphi$ of a set function $f$ with respect to $V$, is defined as the function $\varphi:K\to {{\mathbb{R}}}$ such that, for every $\epsilon>0$, there exists $\eta >0$ such that $$\Big|\frac{f(\Delta)}{V(\Delta)}-\varphi (x)\Big|<\epsilon$$ for all $x\in K$ and $\Delta \in {{\mathcal{A}}}_K$ with $x\in \Delta \subset B_\eta (x)$. It is clear that <span style="font-variant:small-caps;">Lebesgue</span>’s new notion of uniform-derivative is strictly related to <span style="font-variant:small-caps;">Peano</span>’s one. In fact, <span style="font-variant:small-caps;">Lebesgue</span> observes that the uniform-derivative is continuous whenever it exists; moreover, he defines the integral $$\int_K \varphi \,{{\mathrm{d}}}V$$ of a *continuous function* $\varphi$ with respect to $V$. His definition of integral [@lebesgue1935 (1935) pp.188-191] is rather intricate with respect to that of <span style="font-variant:small-caps;">Peano</span>.
It is worthwhile noticing that <span style="font-variant:small-caps;">Lebesgue</span> recognizes the relevance of the notion of an integral with respect to set functions. <span style="font-variant:small-caps;">Lebesgue</span>, not acquainted with previous <span style="font-variant:small-caps;">Peano</span>’s contributions, assigns the priority of this notion to <span style="font-variant:small-caps;">Radon</span> [@radon1913 (1913)]. On the other hand, <span style="font-variant:small-caps;">Lebesgue</span> notices that the integral with respect to set functions was already present in Physics [^50] and express his great surprise in recovering in <span style="font-variant:small-caps;">Stieltjes</span>’s integral [@stieltjes (1894)] an instance of integral with respect to set functions; Lebesgue writes [@lebesgue1926 (1926) p.69-70]:
> Mais son premier inventeur, Stieltjès, y avait été conduit par des recherches d’analyse et d’arithmétique et il l’avait présentée sous une forme purement analytique qui masquait sa signification physique; sì bien qu’il a fallu beaucoup d’efforts pour comprendre et connaître ce qui est maintenant évident. L’historique de ces efforts citerait les nom de F. Riesz, H. Lebesgue, W.H. Young, M. Fréchet, C. de la Vallé-Poussin; il montrerait que nous avons rivalisé en ingéniosité, en perspicacité, mais aussi en aveuglement.[^51]
The first important theorem presented by <span style="font-variant:small-caps;">Lebesgue</span> is the following
Let $K$ be a bounded closed subset of ${{\mathbb{R}}}^n$, $\varphi:K\to {{\mathbb{R}}}$ a continuous function and $V$ a positive additive continuous set function with bounded-derivative. Then the integral $\Delta \mapsto \int_\Delta \varphi \,{{\mathrm{d}}}V$ with $\Delta \in {{\mathcal{A}}}$ is the unique additive set function with bounded-derivative which has $\varphi$ as uniform-derivative with respect to $V$.[^52]
The main applications of this theorem, given by <span style="font-variant:small-caps;">Lebesgue</span> in *La mesure des grandeurs* [@lebesgue1935 (1935) p.176], concern:
1. the proof that multiple integrals can be given in terms of simple integrals;
2. the formula of change of variables;[^53]
3. several formulae for oriented integrals (Green’s formula, length of curves and area of surfaces).
The uniform-derivative defined by <span style="font-variant:small-caps;">Lebesgue</span> is, as observed above, a continuous function, and coincides exactly with <span style="font-variant:small-caps;">Peano</span>’s strict derivative. Through a different and more difficult path [^54] than <span style="font-variant:small-caps;">Peano</span>’s one, <span style="font-variant:small-caps;">Lebesgue</span> rediscovers the importance of the continuity of the derivative. In <span style="font-variant:small-caps;">Lebesgue</span>’s works there are no references to the contributions of <span style="font-variant:small-caps;">Peano</span> concerning differentiation of set functions.
Several years before *La mesure des grandeurs* of 1935, <span style="font-variant:small-caps;">Lebesgue</span> in [@lebesgue1926 (1926)] outlines his contribution to the notion of integral. In the same paper he mentions <span style="font-variant:small-caps;">Cauchy</span>’s *Coexistent magnitudes* in the setting of derivative of measures. Moreover he cites <span style="font-variant:small-caps;">Fubini</span>’s and <span style="font-variant:small-caps;">Vitali</span>’s works of 1915 and 1916 (published by Academies of Turin and of Lincei) in the context of the general problem of primitive functions.
More precisely, in 1915, the year of publication of <span style="font-variant:small-caps;">Peano</span>’s paper *Le grandezze coesistenti* [@peano1915], <span style="font-variant:small-caps;">Fubini</span> [@fubini1915b; @fubini1915a (1915)] and <span style="font-variant:small-caps;">Vitali</span> [@vitali1915; @vitali1916 (1915, 1916)] introduce a definition of derivative of “finitely additive measures” [^55], oscillating themselves between definitions *à la Cauchy* and *à la Peano*.
<span style="font-variant:small-caps;">Vitali</span>, in his second paper [@vitali1916], refers to the *Coexistent magnitudes* of <span style="font-variant:small-caps;">Cauchy</span>, and presents a comparison among the notions of derivative given by <span style="font-variant:small-caps;">Fubini</span>, himself, <span style="font-variant:small-caps;">Peano</span> and the one of <span style="font-variant:small-caps;">Lebesgue</span> of 1910, emphasizing the continuity of the <span style="font-variant:small-caps;">Peano</span>’s strict derivative. <span style="font-variant:small-caps;">Vitali</span> writes in [@vitali1916 (1916)]:
> Il Prof.G.Peano nella Nota citata \[*Le grandezze coesistenti*\] e in un’altra sua pubblicazione anteriore \[*Applicazioni geometriche*\], si occupa dei teoremi di Rolle e della media e ne indica la semplice dimostrazione nel caso in cui la derivata \[della funzione di insieme $f$\] in $P$ sia intesa come il limite del rapporto di $\frac{f(\tau)}{\tau}$, dove $\tau$ è un campo qualunque che può anche non contenere il punto $P$.
>
> L’esistenza di tale simile derivata finita in ogni punto porta difatti la continuità \[della derivata medesima\].[^56]
This proves that since 1926 <span style="font-variant:small-caps;">Lebesgue</span> should have been aware of <span style="font-variant:small-caps;">Peano</span>’s derivative and of its continuity.[^57]
Undoubtably, the contributions of <span style="font-variant:small-caps;">Peano</span> and <span style="font-variant:small-caps;">Lebesgue</span> have a pedagogical and mathematical relevance in formulating a definition of derivative having the property of continuity whenever it exists. Surprisingly these contributions are not known.
Rarely the notion of derivative of set functions is presented and used in educational texts.
An example is provided by *Lezioni di analisi matematica* of <span style="font-variant:small-caps;">Fubini</span>. There are several editions of these *Lezioni*: starting by the second edition [@fubiniB1915 (1915)], <span style="font-variant:small-caps;">Fubini</span> introduces a derivative *à la Peano* of additive set functions in order to build a basis for integral calculus in one or several variables. Nevertheless, in his *Lezioni*, <span style="font-variant:small-caps;">Fubini</span> assumes continuity of its derivative as an additional property. Ironically, <span style="font-variant:small-caps;">Fubini</span> is aware of continuity of <span style="font-variant:small-caps;">Peano</span>’s derivative, whenever it exists; this is clear from two letters of 1916 that he sent to <span style="font-variant:small-caps;">Vitali</span> [@vitali-opere p.518-520]; in particular, in the second letter, about the <span style="font-variant:small-caps;">Peano</span>’s paper *Grandezze coesistenti* [@peano1915 (1915)], he writes:
> Sarebbe bene citare \[l’articolo di\] Peano e dire che, se la derivata esiste e per calcolarla in \[un punto\] $A$ si adottano anche dominii che tendono ad $A$, pur non contenendo $A$ all’interno, allora la derivata è continua.[^58]
The notion of derivative of set function is also exposed in the textbooks *Lezioni di analisi infinitesimale* of <span style="font-variant:small-caps;">Picone</span> [@picone (1923) vol.II, p.465–506], in *Lezioni di analisi matematica* of <span style="font-variant:small-caps;">Zwirner</span> [@zwirner (1969), pp.327-335] and in *Advanced Calculus* of R.C. and E.F. <span style="font-variant:small-caps;">Buck</span> [@buck (1965)]. In the book of <span style="font-variant:small-caps;">Picone</span>, a definition of derivative *à la Cauchy* of “additive” set functions is given;[^59] it represents an improvement of <span style="font-variant:small-caps;">Cauchy</span>, <span style="font-variant:small-caps;">Fubini</span> and <span style="font-variant:small-caps;">Vitali</span> definitions. Of course, his derivative is not necessarily a continuous function. Whenever the derivative is continuous, <span style="font-variant:small-caps;">Picone</span> states a fundamental theorem of calculus, and applies it to the change of variables in multiple integrals. In the book of <span style="font-variant:small-caps;">Zwirner</span> the notion of derivative *à la Peano* of set functions is introduced, without mentioning <span style="font-variant:small-caps;">Peano</span> and, unfortunately, without providing any application. In the third book, R.C. and E.F. <span style="font-variant:small-caps;">Buck</span> introduce in a clear way a simplified notion of the uniform-derivative of <span style="font-variant:small-caps;">Lebesgue</span> (without mentioning him), and they apply it to obtain the basic formula for the change of variables in multiple integrals.
Appendix
========
All articles of <span style="font-variant:small-caps;">Peano</span> are collected in *Opera omnia* [@peano_omnia], a CD-ROM edited by C. S. <span style="font-variant:small-caps;">Roero</span>. Selected works of <span style="font-variant:small-caps;">Peano</span> were assembled and commented in *Opere scelte* [@peano_opere] by <span style="font-variant:small-caps;">Cassina</span>, a pupil of <span style="font-variant:small-caps;">Peano</span>. For a few works there are English translations in *Selected Works* [@peano_english]. Regrettably, fewer <span style="font-variant:small-caps;">Peano</span>’s papers have a public URL and are freely downloadable.
For reader’s convenience, we provide a chronological list of some mathematicians mentioned in the paper, together with biographical sources.
`Html` files with biographies of mathematicians listed below with an asterisk can be attained at University of St Andrews’s web-page
`http://www-history.mcs.st-and.ac.uk/history/{Name}.html`
<span style="font-variant:small-caps;">Kepler</span>, Johannes (1571-1630)\*
<span style="font-variant:small-caps;">Cavalieri</span>, Bonaventura (1598-1647)\*
<span style="font-variant:small-caps;">Newton</span>, Isaac (1643-1727)\*
<span style="font-variant:small-caps;">Mascheroni</span>, Lorenzo (1750-1800)\*
<span style="font-variant:small-caps;">Cauchy</span>, Augustin L. (1789-1857)\*
<span style="font-variant:small-caps;">Lobachevsky</span>, Nikolai I. (1792-1856)\*
<span style="font-variant:small-caps;">Moigno</span> François N. M. (1804-1884), see *Enc. Italiana*, Treccani, Roma, 1934
<span style="font-variant:small-caps;">Grassmann</span>, Hermann (1809-1877)\*
<span style="font-variant:small-caps;">Serret</span>, Joseph A. (1819-1885)\*
<span style="font-variant:small-caps;">Riemann</span>, Bernhard (1826-1866)\*
<span style="font-variant:small-caps;">Jordan</span>, Camille (1838-1922)\*
<span style="font-variant:small-caps;">Darboux</span>, Gaston (1842-1917)\*
<span style="font-variant:small-caps;">Stolz</span>, Otto (1842-1905)\*
<span style="font-variant:small-caps;">Schwarz</span>, Hermann A. (1843-1921)\*
<span style="font-variant:small-caps;">Cantor</span>, Georg (1845-1918)\*
<span style="font-variant:small-caps;">Tannery</span>, Jules (1848-1910)\*
<span style="font-variant:small-caps;">Harnack</span>, Carl (1851-1888), see May [@may1973 (1973) p.186]
<span style="font-variant:small-caps;">Stieltjes</span>, Thomas J. (1856-1894)\*
<span style="font-variant:small-caps;">Peano</span>, Giuseppe (1858-1932)\*, see [@peano_vita]
<span style="font-variant:small-caps;">Young</span>, William H. (1863-1942)\*
<span style="font-variant:small-caps;">Segre</span>, Corrado (1863-1924)\*
<span style="font-variant:small-caps;">Vallée Poussin</span> (de la), Charles (1866-1962)\*
<span style="font-variant:small-caps;">Hausdorff</span>, Felix (1868-1942)\*
<span style="font-variant:small-caps;">Borel</span>, Emile (1871-1956)\*
<span style="font-variant:small-caps;">Vacca</span>, Giovanni (1872-1953)\*
<span style="font-variant:small-caps;">Carathéodory</span>, Constantin (1873-1950)\*
<span style="font-variant:small-caps;">Lebesgue</span>, Henri (1875-1941)\*
<span style="font-variant:small-caps;">Vitali</span>, Giuseppe (1875-1932)\*
<span style="font-variant:small-caps;">Fréchet</span>, Maurice (1878-1973)\*
<span style="font-variant:small-caps;">Fubini</span>, Guido (1879-1943)\*
<span style="font-variant:small-caps;">Riesz</span>, Frigyes (1880-1956)\*
<span style="font-variant:small-caps;">Tonelli</span>, Leonida (1885-1946)\*
<span style="font-variant:small-caps;">Picone</span>, Mauro (1885-1977), see `http://web.math.unifi.it`
<span style="font-variant:small-caps;">Ascoli</span>, Guido (1887-1957), see May [@may1973 (1973) p.63]
<span style="font-variant:small-caps;">Radon</span>, Johann (1887-1956)\*
<span style="font-variant:small-caps;">Nikodym</span>, Otton (1887-1974)\*
<span style="font-variant:small-caps;">Bouligand</span>, George (1889-1979), *see* `http://catalogue.bnf.fr`
<span style="font-variant:small-caps;">Banach</span>, Stefan (1892-1945)\*
<span style="font-variant:small-caps;">Kuratowski</span>, Kazimierz (1896-1980)\*
<span style="font-variant:small-caps;">Cassina</span>, Ugo (1897-1964), see Kennedy [@peano_vita (1980)]
<span style="font-variant:small-caps;">Cartan</span>, Henri (1904-2008)\*
<span style="font-variant:small-caps;">Dieudonné</span>, Jean A. E. (1906-1992)\*
<span style="font-variant:small-caps;">Choquet</span>, Gustave (1915-2006), see *Gazette des Math.* v111:74-76, 2007
<span style="font-variant:small-caps;">May</span> Kenneth O. (1915-1977), see [@dauben_scriba p.479]
<span style="font-variant:small-caps;">Medvedev</span> Fëdor A. (1923-1993), see [@dauben_scriba p.482]
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[^1]: [$[\![$In another memoir we will give new developments to the above mentioned statements \[on coexistent magnitudes\], and we will apply them to evaluate lengths, areas and volumes.$]\!]$]{}
[^2]: From now on we refer to <span style="font-variant:small-caps;">Cauchy</span>’s paper *M[é]{}moire sur le rapport diff[é]{}rentiel de deux grandeurs qui varient simultan[é]{}ment* [@cauchy1841 (1841)] as to *Coexistent magnitudes.*
[^3]: [$[\![$While we can find so many new things by moving forward in mathematics, why can’t we find some still unknown place by retroceding?$]\!]$]{}
[^4]: [$[\![$The quantity of matter is a measure of the matter itself, arising from its density and magnitude conjunctly \[…\]. It is this quantity that I mean hereafter everywhere under the name of body or mass.$]\!]$]{}
[^5]: In today terminology, the realization of (\[mass-volume\]) is expressed by saying that $g$ is the *Radon-Nikodym derivative* of $\mu$ with respect to ${\rm vol}_n$.
[^6]: Clearly, if <span style="font-variant:small-caps;">Peano</span>’s strict derivative of a finite $\sigma$-additive measure exists, then it coincides with <span style="font-variant:small-caps;">Lebesgue</span> derivative and the “mass” is absolutely continuous.
Nowadays it is not surprising that <span style="font-variant:small-caps;">Lebesgue</span>’s derivative can be seen as <span style="font-variant:small-caps;">Peano</span>’s strict derivative by lifting measures on a $\sigma$-algebra ${{\mathcal{A}}}$ and ${{\mathcal{A}}}$-measurable functions to measures on the Stone space associated to ${{\mathcal{A}}}$ and the related continuous functions, respectively.
[^7]: This topic will be extensively analyzed in a forthcoming paper by <span style="font-variant:small-caps;">Greco, Mazzucchi, Pagani</span> [@gremazpagAREA].
[^8]: According with <span style="font-variant:small-caps;">Letta</span> [@letta], the notion of negligible set is introduced after an arduous process of investigation on “similar” notions related to cardinality and topology, between 1870 and 1882. Afterward the definition of [*Inhalt*]{} (content) appears in the works by <span style="font-variant:small-caps;">Stolz</span> [@stolz1884 (1884)], <span style="font-variant:small-caps;">Cantor</span> [@cantor1884 (1884)], <span style="font-variant:small-caps;">Harnack</span> [@harnack1885 (1885)]. The notions of inner and outer measure are introduced by <span style="font-variant:small-caps;">Peano</span> in [@peano1883 (1883) p.446] and in [@peano87 (1887)], and later by <span style="font-variant:small-caps;">Jordan</span> [@jordan1892 (1892)]. In the following we will refer to the inner and to the outer measures as to *Peano-Jordan measures*.
[^9]: <span style="font-variant:small-caps;">Jordan</span>, famous geometer and algebraist, publishes only a few papers on mathematical analysis. His most famous work is the [*Cours d’analyse*]{}, published in several editions. To our knowledge the relationship between <span style="font-variant:small-caps;">Peano</span> and <span style="font-variant:small-caps;">Jordan</span> was good and based on reciprocal appreciation, as one can deduce from two letters conserved in Archives de la Bibliothèque Centrale de l’Ecole Polytechnique (Paris).
[^10]: Later, in a paper with didactic value [@peano_der1892 (1892)], <span style="font-variant:small-caps;">Peano</span> re-proposes the distinction between Definition and the usual derivative of a function, and underlines the correspondence of with the definition of density in Physics.
Nowadays the function $f$ is said *strictly differentiable* at the point $\bar x$ if the limit exists; consequently, the value of the limit $(3.1)$ is called *strict derivative* of $f$ at $\bar x$.
[^11]: Section 80 of <span style="font-variant:small-caps;">Jordan</span>’s *Cours d’analyse* [@jordan1893 (1893) p.68], titled “*Cas où $\frac{f(x+h)-f(x)}{h}$ tend uniformément vers $f'(x)$*”, contains a trace of it.
[^12]: As detailed in <span style="font-variant:small-caps;">Dolecki, Greco</span> [@greco-dolecki], between several interesting concepts studied in *Applicazioni geometriche* that are not directly connected with measure theory, we recall the limit of sequences of sets (now called *Kuratowski limits*), the introduction of the concept of differentiability of functions (nowadays called *Fréchet differentiability*), the definition of tangent cone (nowadays called *Bouligand cone*), the necessary condition of optimality (nowadays called *Fermat conditions*) and a detailed study of problems of maximun and minimun.
[^13]: The simultaneous construction of inner and outer measure is the basis of the evolution of the theory leading to Lebesgue measure. Fortunately, <span style="font-variant:small-caps;">Carathéodory</span> [@caratheodory1914 (1914)] and <span style="font-variant:small-caps;">Hausdorff</span> [@hausdorff1919 (1919)] put an end to the intoxication due to the presence of inner measure, as <span style="font-variant:small-caps;">Carathéodory</span> writes:
> Borel and Lebesgue (as well as Peano and Jordan) assigned an outer measure $m^*(A)$ and an inner measure $m_* (A)$ to every point set $A$ \[...\]. The main advantage, however, is that the new definition \[i.e., the exterior measure of Charathéodory\] is independent of the concept of an *inner measure* [@edgar (2004) p.72].
[^14]: In a first paper of <span style="font-variant:small-caps;">Jordan</span> and in a more extensive way in his *Cours d’analyse* [@jordan1893 (1893)], we find several <span style="font-variant:small-caps;">Peano</span>’s results. There are, however, methodological differences between their approaches: <span style="font-variant:small-caps;">Peano</span> constructs his measure by starting from polygons, while <span style="font-variant:small-caps;">Jordan</span> considers (in the 2-dimensional case) squares. The definition proposed by <span style="font-variant:small-caps;">Peano</span> does not have the simplicity of that of <span style="font-variant:small-caps;">Jordan</span>, but it is independent of the reference frame and it is, by definition, invariant under isometries, without any need of further proof. Moreover, <span style="font-variant:small-caps;">Peano</span>’s definition allows for a direct computation of the proportionality factor appearing under the action of affine transformation (in previous works <span style="font-variant:small-caps;">Peano</span> had developed a formalism allowing for computation of areas of polygons in a simple way, see [@gremazpagAREA]) for details.
[^15]: [$[\![$Chapter V is titled: Geometric magnitudes. This chapter is probably the most relevant and interesting, the one that marks the difference of the Book of Peano with respect to other classical Treatises: definitions concerning *sets of points*, exterior, interior and limit points of a given set, distributive functions (coexistent magnitudes in the sense of Cauchy), exterior, interior and proper length (or area or volume) of a set, the extension of the notion of integral to a set, are stated in an abstract, very precise and very clear way.$]\!]$]{}
[^16]: [$[\![$In *Applicazioni geometriche* it is possible to find a clear and definitive exposition of many mathematical concepts and results, nowadays of common knowledge: results on measure of sets, on length of arcs, on the definition of area of a surface, on the integration on a set, on additive set functions; and other results that are not well known \[…\] $]\!]$]{}
[^17]: To our knowledge the latest example of historian who forgot to quote any <span style="font-variant:small-caps;">Peano</span>’s contributions, is <span style="font-variant:small-caps;">Hochkirchen</span> [@hochkirchen2003 (2003)]. Ironically, the symbols $\underline\int$ and $\overline\int$ which <span style="font-variant:small-caps;">Volterra</span> introduced for denoting lower and upper integral, were ascribed to <span style="font-variant:small-caps;">Peano</span> by <span style="font-variant:small-caps;">Hochkirchen</span>.
[^18]: <span style="font-variant:small-caps;">Dieudonné</span>, reviewing in [@dieudonne (1983)] the <span style="font-variant:small-caps;">Medvedev</span>’s paper [@medvedev (1983)], with his usual sarcasm denies any logical value of <span style="font-variant:small-caps;">Peano</span>’s definitions concerning limits and sets. Against any historical evidence, <span style="font-variant:small-caps;">Dieudonné</span> forgets several <span style="font-variant:small-caps;">Peano</span>’s papers on several notions of limit, and ignores the *Formulario mathematico* where <span style="font-variant:small-caps;">Peano</span> presents a large amount of mathematical results, including set axiomatization, through his logical *ideography*. Besides, <span style="font-variant:small-caps;">Dieudonné</span> forgets <span style="font-variant:small-caps;">Bourbaki</span>’s *Elements of the history of mathematics* and ignores that the building blocks of <span style="font-variant:small-caps;">Peano</span>’s ideography are the atomic propositions: $x\in X$ and $x=y$.
[^19]: [$[\![$About methods, I have tried to be rigorous as required in geometry, in order to avoid the general reasonings occurring in algebra.$]\!]$]{}
Not all mathematicians at that time considered geometry as a model of rigor. Indeed <span style="font-variant:small-caps;">Lobachevsky</span> starts his famous book “Theory of parallels” [@Loba1829 (1829) p.11] with the following sentence:
> In geometry I find certain imperfections which I hold to be the reason why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid.
[^20]: <span style="font-variant:small-caps;">Cauchy</span> says in [@cauchy1841 (1841) p.188]:
> Nous appellons *grandeurs* ou quantités *coexistantes* deux grandeurs ou quantités qui existent ensemble et varient simultanément, de telle sorte que les éléments de l’une existent et varient, ou s’évanouissent, en même temps que les éléments de l’autre.
[^21]: Let $\mu,\nu : {{\mathcal{A}}}\to {{\mathbb{R}}}$ be two magnitudes and let $g$ be the differential ratio of $\mu$ with respect to $\nu$. We say that the *mean value property* holds if, for any set $A \in {{\mathcal{A}}}$, with $\nu(A) \ne 0$, there exists a point $P\in A$ such that $g(P)=\frac{\mu(A)}{\nu (A)}$.
[^22]: [$[\![$Almost always, this differential ratio is a continuous function of the independent variable, i.e., its values change in a smooth way.$]\!]$]{}
[^23]: [$[\![$Almost always, the differential ratio $\rho$ is a continuous function \[…\].$]\!]$]{}
[^24]: Using the identity $$\frac{f(x+h) - f(x-k)}{h+k} =
\frac{h\frac{f(x+h)-f(x)}{h} +k\frac{f(x)-f(x-k)}{k}}{h+k}
\quad \text{ for every } k, h > 0$$ the reader can easily verify that the differential ratio in the sense of <span style="font-variant:small-caps;">Cauchy</span> exists (i.e., the limit of $\frac{f(x+h) - f(x-k)}{h+k}$ exists for $k\to 0^+$ and $h \to 0^+$, with $h+k > 0$) whenever $f'(x)$ exists.
[^25]: [$[\![$When two coexistent magnitudes are a variable $x$ and a function $y$ of $x$, the differential ratio of the function with respect to the variable $x$ coincides with the *derivative* of the function.$]\!]$]{}
[^26]: One can observe that this theorem holds true by imposing condition .
[^27]: Among distributive functions considered by <span style="font-variant:small-caps;">Peano</span>, there are the usual linear functions and particular set functions. The reader has to pay attention in order to avoid the interpretation of distributive set functions as finitely additive set functions.
[^28]: This notion of distributive family will be rediscovered later by <span style="font-variant:small-caps;">Choquet</span> [@choquet (1947)], who called it [*grill*]{} and recognized it as the dual notion of <span style="font-variant:small-caps;">Cartan</span>’s [*filter*]{} [@cartan (1937)].
[^29]: Two examples of distributive families considered by <span style="font-variant:small-caps;">Peano</span> are ${\mathcal U}:=\{ A\subset {{\mathbb{R}}}^n\, :\; \text{ card}( A)=\infty \}$, and ${\mathcal U}_h:=\{A\subset {{\mathbb{R}}}^n\, :\, \sup_A h=\sup_{{{\mathbb{R}}}^n}h\}$, where $h:{{\mathbb{R}}}^n\to{{\mathbb{R}}}$ is a given real function.
[^30]: A similar expression is used also by <span style="font-variant:small-caps;">Jordan</span> [@jordan1892]:
> \[C\]haque champ $E$ a une étendue déterminée; \[…\] si on le décompose en plusieurs parties $E_1$, $E_2$, …, la somme des étendues de ces parties est égale l’étendue totale de $E$. [$[\![$Every set $E$ has a defined extension; \[…\] if $E$ is decomposed into parts $E_1$, $E_2$, …, the sum of the extensions of these parts is equal to the extension of $E$.$]\!]$]{}
[^31]: [$[\![$If a set $A$ is decomposed into the parts $A_1,A_2,\dots,A_n$, it will be called *sum* of its parts, and it will be denoted by $A=A_1+A_2+\dots+A_n$. \[…\] A magnitude is said to be a *distributive set function* if its value on a given set is the sum of the corresponding values of the function on the parts decomposing the set itself.$]\!]$]{}
[^32]: See pages 165 and 186-188 of *Applicazioni geometriche* [@peano87 (1887)].
[^33]: Notice that a real valued set function $\mu:{{\mathcal{A}}}\to {{\mathbb{R}}}$ is distributive with respect to ${{\mathbb{D}}}_{\textrm{int}}({{\mathcal{A}}})$, if $\mu (A)=\mu (A\cap H^+)+\mu (A\cap H^-)$ for every $A\in {{\mathcal{A}}}$ and for every hyperplanes $H$ such that $A\cap H^+ \in {{\mathcal{A}}}$ and $A\cap H^-\in {{\mathcal{A}}}$. Inner and upper Peano-Jordan measures are both distributive in this sense, but they are not finitely additive.
[^34]: Observe that inner and outer Peano-Jordan measures on Euclidean spaces are not finitely additive, but they are distributive set functions with respect to the families of decomposition of type (\[dec-inter\]) or (\[dec-picone\]). Moreover, notice that outer Peano-Jordan measure is a distributive set function with respect to a family of decompositions of type (\[dec-picone2\]).
[^35]: In <span style="font-variant:small-caps;">Peano</span>’s words [@peano87 (1987) p.169]:
> Diremo che, in un punto $P$, il *rapporto* delle due funzioni distributive $y$ ed $x$ d’un campo vale $\rho$, se $\rho$ è il limite verso cui tende il rapporto dei valori di queste funzioni, corrispondenti ad un campo di cui tutti i punti si avvicinano indefinitamente a $P$.
[$[\![$Given two distributive functions $y$ an $x$ defined over a given set, we say that their *ratio*, at a given point $P$, is $\rho$, if $\rho$ is the limit of the ratio between the values of the two functions, taken along sets for which all its points approach the point $P$.$]\!]$]{}
[^36]: One can note that for the definition of strict derivative at a point $x$, the point $x$ itself must be an accumulation point with respect to the family ${{\mathcal{A}}}$ and the measure $\nu$, that is, for all $\delta >0$, there exists a $A\in{{\mathcal{A}}}$ such that $\nu(A)\neq 0$ and $A\subset B_\delta (x)$, where $B_\delta (x)$ denotes the Euclidean ball of center $x$ and radius $\delta$.
[^37]: It is evident that properties (\[cor-1\])-(\[due-mis\]) are equivalent. To prove (\[due-mis\]), <span style="font-variant:small-caps;">Peano</span> shows that the strict derivative of a sum of two distributive set functions is the sum of their derivatives.
[^38]: By *positive-hypograph* of $f$ restricted to $A$ we mean the set $\{(x,y)\in [a,b]\times R_+ : x\in A$ and $y\leq f(x), \}$, where ${{\mathbb{R}}}_+ := \{ x\in {{\mathbb{R}}}: x\ge 0\}$.
[^39]: The requirement that the derivative of the arc with respect to a parameter be continuous and non null is expressed by <span style="font-variant:small-caps;">Peano</span> in geometrical terms, namely by requiring that “the tangent straight line exists at every point $P$ of the arc, and it is the limit of the straight lines passing through two points of the arc, when they tend to $P$”. <span style="font-variant:small-caps;">Peano</span> was aware that these geometrical conditions are implied by the existence of a parametrization with a continuous non-null derivative [@peano87 (1987) p.59, 184].
[^40]: Of course, to avoid $\cos\theta_x = 0$ along the arc, <span style="font-variant:small-caps;">Peano</span> assumes that the tangent straight line at every point of the arc is not orthogonal to $r$.
[^41]: This clear, simple and general definition of integral with respect to an abstract positive distributive set function is ignored until the year 1915, when <span style="font-variant:small-caps;">Fréchet</span> re-discovers it in the setting of “finitely additive” measures [@frechet1915 (1915)].
[^42]: <span style="font-variant:small-caps;">Peano</span> observes that continuity of derivative of $F$ is a necessary and sufficient condition to have the existence of $\frac{{{\mathrm{d}}}\mu}{{{\mathrm{d}}}\nu}$.
[^43]: <span style="font-variant:small-caps;">Lebesgue</span> says in [@lebesgue1931 (1931) p.174]:
> \[…\] depuis trente ans \[d’ enseignement\] \[…\] on ne s’étonnera pas que l’idée me soit venue d’écrire des articles de nature pédagogique; si j’ose employer ce qualificatif que suffit ordinairement pour faire fuir les mathématiciens. [$[\![$\[…\] in the thirty years \[of teaching\] \[…\] it is not at all surprising that the idea should occur to me of writing articles on a pedagogical vein; if I may use an expression which usually puts mathematicians to flight. (transl. <span style="font-variant:small-caps;">May</span> [@lebesgue_may (1966) p.10])$]\!]$]{}
[^44]: The five parts of the essay *La mesure des grandeurs* have been published in *L’Enseignement mathèmatique* during the years 1931-1935. An english translation *Measure and the Integral* of *La mesure des grandeurs* is due to Kenneth O. May [@lebesgue_may (1966)].
[^45]: [$[\![$The theory of magnitudes forming the subject of the preceding chapter was prepared by researches of Cauchy on what he called concomitant magnitudes, by studies destined to clarify the concepts of area, volume, and measure \[…\] (transl. <span style="font-variant:small-caps;">May</span> [@lebesgue_may (1966) p.138])$]\!]$]{}
[^46]: [$[\![$If a body $C$ is partitioned into a certain number of sub-bodies $C_1, C_2, \dots, C_p$ and if for these bodies the magnitude $G$ is $g$ on the one hand and $g_1, g_2, \dots, g_p$ on the other, we must have $g= g_1 + g_2 + \cdots + g_p$. (transl. <span style="font-variant:small-caps;">May</span> [@lebesgue_may (1966) p.129])$]\!]$]{}
<span style="font-variant:small-caps;">Lebesgue</span> observes that in order to make this condition rigorous, it would be necessary to give a precise meaning to the words *corp* and *partage de la figure totale en parties* [@lebesgue1934 (1934) p.275-276]. Moreover he observes that *diviser un corps* may be interpreted in different ways [@lebesgue1934 (1934) p.279].
[^47]: It is not easy to give in a few words a definition of the concept of continuity according to <span style="font-variant:small-caps;">Lebesgue</span>: such a continuity is based on a convergence of sequences of sets that in the relevant cases coincides with the convergence in the sense of Hausdorff. We recall that a sequence of sets $\Delta_n$ *converges to $\Delta$ in the sense of Hausdorff* if for all $\epsilon >0$ there exists $n_0$ such that $\Delta_n\subset B_\epsilon (\Delta)$ and $\Delta\subset B_\epsilon (\Delta_n)$ for all $n>n_0$, where $B_\epsilon (A):= \{x \in {{\mathbb{R}}}^n : \text{ there exists } a \in A
\text{ such that } \|x-a\|<\epsilon \}$. Therefore, a set function $f$ is said to be *continuous* if for any $\Delta_n$ and $\Delta$ Peano-Jordan measurable sets, we have that $\lim_{n\to \infty} f (\Delta_n) = f(\Delta)$, whenever $\Delta_n$ converges to $\Delta$ in Hausdorff sense.
[^48]: <span style="font-variant:small-caps;">Lebesgue</span> writes in [@lebesgue1935 (1935) p.185]:
> \[…\] nous supposerons cette fonction \[$f$\] *additive*, c’est-a-dire telle que, si l’on divise $\Delta$ en deux domaines quarrables $\Delta_1$ et $\Delta_2$ on ait $f(\Delta) = f(\Delta_1) + f(\Delta_2)$. [$[\![$\[…\] let us assume that this function is *additive*; that is, it is such that, if we partition $\Delta$ into two quadrable domains $\Delta_1$ and $\Delta_2$, we have $f(\Delta) = f(\Delta_1) + f(\Delta_2)$. (transl. <span style="font-variant:small-caps;">May</span> [@lebesgue_may (1966) p.146])$]\!]$]{}
[^49]: A set function $f$ has a *bounded-derivative* with respect to Peano-Jordan $n$-dimensional measure ${{{\mathrm{vol}}}}_n$ if there exists a constant $M$ such that $|f(\Delta)|\leq M \, {{{\mathrm{vol}}}}_n(\Delta)$ for any Peano-Jordan measurable set $\Delta$. A set function with bounded-derivatives is called *uniformly Lipschitzian* by <span style="font-variant:small-caps;">Picone</span> [@picone (1923) vol.2, p.467].
[^50]: <span style="font-variant:small-caps;">Lebesgue</span> gives several examples of this. For instance, the evaluation of the heath quantity necessary to increase the temperature of a body as integral of the specific heath with respect to the mass.
[^51]: [$[\![$But its original inventor, Stieltjes, was led to it by researches in analysis and theory of number and he presented it in a purely analytical form which masked its physical significance, so much so that it required a much effort to understand and recognizes what is nowadays obvious. The history of these efforts includes the works of F. Riesz, H. Lebesgue, W.H. Young, M. Fréchet, C. de la Vallé-Poussin. It shows that we were rivals in ingenuity, in insight, but also in blindness. (transl. <span style="font-variant:small-caps;">May</span> [@lebesgue_may (1966) p.190]) $]\!]$]{}
[^52]: The proof is rather lengthy, as <span style="font-variant:small-caps;">Lebesgue</span> included in it the definition of integral as well as the theorem of average value.
[^53]: Lebesgue uses the implicit function theorem.
[^54]: The exposition of 1935 is elementary, but more lengthy and difficult than those presented by <span style="font-variant:small-caps;">Lebesgue</span> in 1910. Surprisingly, the terms *domain, decomposition, limit, additive, continuous* are used by <span style="font-variant:small-caps;">Lebesgue</span> in a supple way.
[^55]: <span style="font-variant:small-caps;">Fubini</span>’s first paper [@fubini1915a] is presented by C.<span style="font-variant:small-caps;">Segre</span> at the Academy of Sciences of Turin on January 10, 1915. In the same session, <span style="font-variant:small-caps;">Peano</span>, Member of the Academy, presents a multilingual dictionary and a paper written by one of his students, <span style="font-variant:small-caps;">Vacca</span>. <span style="font-variant:small-caps;">Segre</span>, on April 11, 1915, presents, as a Member, a second paper of <span style="font-variant:small-caps;">Fubini</span> [@fubini1915b] to [*Accademia dei Lincei*]{}. In the session of the Academy of Turin of June 13, 1915, <span style="font-variant:small-caps;">Peano</span> presents his paper *Le grandezze coesistenti*. Moreover <span style="font-variant:small-caps;">Segre</span> presents two papers by <span style="font-variant:small-caps;">Vitali</span> [@vitali1915 (1915)] and [@vitali1916 (1916)] to Academy of Turin on November 28, 1915 and to Academy of Lincei on May 21, 1916, respectively.
There is a rich correspondence between <span style="font-variant:small-caps;">Vitali</span> and <span style="font-variant:small-caps;">Fubini</span>. In the period March-May 1916 <span style="font-variant:small-caps;">Fubini</span> sends three letters to <span style="font-variant:small-caps;">Vitali</span> (transcribed in *Selected papers* of <span style="font-variant:small-caps;">Vitali</span> [@vitali-opere pp.519-520]), concerning differentiation of finitely additive measures and related theorems. In particular <span style="font-variant:small-caps;">Fubini</span> suggests <span style="font-variant:small-caps;">Vitali</span> to quote <span style="font-variant:small-caps;">Peano</span>’s paper [@peano1915 (1915)] and to compare alternative definitions of derivative. In *Selected papers* of <span style="font-variant:small-caps;">Vitali</span> it is also possible to find six letters by <span style="font-variant:small-caps;">Peano</span> to <span style="font-variant:small-caps;">Vitali</span>. Among them, there is letter of March 21, 1916 concerning <span style="font-variant:small-caps;">Cauchy</span>’s coexistent magnitudes; <span style="font-variant:small-caps;">Peano</span> writes:
> Grazie della sua nota [@vitali1915 (1915)]. Mi pare che la dimostrazione che Ella dà, sia proprio quella di Cauchy, come fu rimodernata da G. Cantor, e poi da me, e di cui trattasi nel mio articolo, Le grandezze coesistenti di Cauchy, giugno 1915, e di cui debbo avere inviato copia.
\[\]
To our knowledge, <span style="font-variant:small-caps;">Fubini</span> [@fubini1915b; @fubini1915a (1915)] and <span style="font-variant:small-caps;">Vitali</span> [@vitali1915; @vitali1916 (1915, 1916)] are not cited by other authors, with the exception of <span style="font-variant:small-caps;">Banach</span> [@banach1924 (1924) p.186], who refers to <span style="font-variant:small-caps;">Fubini</span> [@fubini1915b (1915)].
[^56]: [$[\![$Prof.Peano, in the cited Paper \[*Le grandezze coesistenti*\] and in a previous publication \[*Applicazioni geometriche*\] deals with Rolle’s and mean value theorems, pointing out a simple proof, valid in the case in which the derivative \[of the set function $f$\], in a given point $P$, is the limit of the ratio $\frac{f(\tau)}{\tau}$, where $\tau$ is a set that might not contain the point $P$.$]\!]$]{}
[^57]: We can ask how much <span style="font-variant:small-caps;">Lebesgue</span> was aware of the contributions of <span style="font-variant:small-caps;">Peano</span>. In many historical papers the comment of <span style="font-variant:small-caps;">Kennedy</span> [@peano_vita (1980) p.174], a well known biographer of <span style="font-variant:small-caps;">Peano</span>, occurs:
> Lebesgue acknowledged Peano’s influence on his own development.
In our opinion <span style="font-variant:small-caps;">Peano</span>’s influence on <span style="font-variant:small-caps;">Lebesgue</span> is relevant but sporadic. After a reading of <span style="font-variant:small-caps;">Lebesgue</span>’s works, we have got the feeling that his knowledge of <span style="font-variant:small-caps;">Peano</span>’s contributions was restricted to two papers on the definition of area and on Peano’s curve.
[^58]: \[\]
[^59]: Significant instances of additive set functions in the sense of <span style="font-variant:small-caps;">Picone</span> are outer measure of Peano-Jordan on all subsets of ${{\mathbb{R}}}^n$ and lower/upper integrals of functions with respect to arbitrary domain of integration [@picone (1923) vol. II, p.356-357, 370-371]. The family of decompositions that leads to the notion of additive set function in the sense of <span style="font-variant:small-caps;">Picone</span> is clearly defined on page 356-357 of his book [@picone] and includes the family of decompositions (\[dec-picone\]) and (\[dec-picone2\]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We review the main results and ideas showing that quantum correlations at finite temperatures ($T$), in particular quantum discord, are useful tools in characterizing quantum phase transitions that only occur, in principle, at the unattainable absolute zero temperature. We first review some interesting results about the behavior of thermal quantum discord for small spin-1/2 chains and show that they already give us important hints of the infinite chain behavior. We then study in detail and in the thermodynamic limit (infinite chains) the thermal quantum correlations for the XXZ and XY models, where one can clearly appreciate that the behavior of thermal quantum discord at finite $T$ is a useful tool to spotlight the critical point of a quantum phase transition.'
address: 'Departamento de Física, Universidade Federal de São Carlos, São Carlos, SP 13565-905, Brazil'
author:
- 'T. Werlang, G. A. P. Ribeiro, and Gustavo Rigolin[^1]'
title: Interplay between quantum phase transitions and the behavior of quantum correlations at finite temperatures
---
Introduction
============
A deeper understanding of the low temperature macroscopic phases of a many-body system can only be achieved with the aid of a quantum theory. Of particular interest is the modeling and description of how the system goes from one phase to another, a process known as “phase transition”. In ordinary phase transitions, which occur at finite temperatures, the phase change is driven by thermal fluctuations. For example, if we heat a magnet we arrive at a temperature (Curie temperature) above which it loses its magnetism. In other words, the magnet initially in the ferromagnetic phase, where all spins are aligned, changes to the paramagnetic phase, where the magnetic moments are in a disordered state.
However, a phase transition can also occur at or near absolute zero temperature ($T=0$), where thermal fluctuations are negligible. This process is called a quantum phase transition (QPT)[@sac99] and is attained varying a tuning parameter in the Hamiltonian (e.g. an external magnetic field) while keeping the temperature fixed. When the tuning parameter reaches a particular value, the so called critical point (CP), the system’s Hamiltonian ground state changes drastically, which reflects in an abrupt modification of the macroscopic properties of the system. In this scenario, quantum fluctuations, which loosely speaking are governed by the Heisenberg uncertainty principle, are responsible for the phase transition. Although it is impossible to achieve $T=0$ due to the third law of thermodynamics, the effects of QPTs can be observed at finite temperatures whenever the de Broglie wavelength is greater than the correlation length of thermal fluctuations[@sac99]. Important examples of QPTs are the paramagnetic-ferromagnetic transition in some metals[@row10], the superconductor-insulator transition[@dol10], and superfluid-Mott insulator transition[@gre02].
The ground state of a many-body system at $T=0$ near a CP is often described by a non-trivial wave function due to the long-range correlations among the system’s constituents. In Ref. Preskill argued that quantum entanglement could be responsible for these correlations and therefore the methods developed by quantum information theory (QIT) could be useful in studying the critical behavior of many-body systems. Besides, new protocols for quantum computation and communication could be formulated based on such systems. Along these lines many theoretical tools (in particular entanglement quantifiers) originally developed to tackle QIT problems were employed to determine the CPs of QPTs at $T=0$ [@Nie99]. Later L.-A. Wu et al.[@lidar] proved a general connection between non-analyticities in bipartite entanglement measures and QPTs while in Ref. this connection was extended to multipartite entanglement.
Recently another quantity, namely, quantum discord (QD), has attracted the attention of the quantum information community. QD goes beyond the concept of entanglement and captures in a certain sense the “quantumness” of the correlations between two parts of a system. It was built by noting the fact that two classically equivalent versions of the mutual information are inequivalent in the quantum domain [@zurek; @vedral]. Using QD one can show that quantum entanglement does not describe all quantum correlations existing in a correlated quantum state. In other words, it is possible to create quantum correlations other than entanglement between two quantum systems via local operations and classical communication (LOCC).
The first study showing a possible connection between QD and QPT was done by R. Dillenschneider[@Dil08] in the context of spin chains at $T=0$, where a CP associated to a QPT was well characterized by QD. Further studies subsequently have confirmed the usefulness of QD in describing other types of QPTs at zero temperature[@disT0]. Moreover, it is important to note that such quantum informational approaches to spotlight CPs of QPTs do not require the knowledge of an order parameter (a macroscopic quantity that changes abruptly during the QPT); only the extremal values or the behavior of the derivatives of either the entanglement or QD is sufficient.
The previous theoretical studies, however, were restricted to the zero temperature regime, which is experimentally unattainable ; the third law of thermodynamics dictates that it is impossible to drive a system to $T=0$ by a finite amount of thermodynamic operations. Due to this limitation it is therefore not straightforward to directly compare those theoretical results with experimental data obtained at finite $T$. In order to overcome this problem one has to study the behavior of quantum correlations in a system at thermal equilibrium, which is described by the canonical ensemble $\rho_T=e^{-H/kT}/Z$, with $H$ being the system’s Hamiltonian, $k$ the Boltzmann’s constant and $Z=\mbox{Tr}\left(e^{-H/kT}\right)$ the partition function. [*Thermal entanglement*]{}, i.e. the entanglement computed for states described by $\rho_T$, was first studied by M. C. Arnesen [*et al.*]{}[@Ved01] for a finite unidimensional Heisenberg chain. Other interesting works followed this one, where thermal entanglement has been considered for other Hamiltonians, both for finite[@Kam02; @Rig03] and infinite chains[@ltte]. See Ref. for extensive reviews on entanglement and QPT. However, the focus of the aforementioned works was not the study of the ability of thermal entanglement to point out CPs when $T>0$. In Ref. , two of us introduced the analogous to thermal entanglement, namely, [*thermal quantum discord*]{} (TQD), and we studied the behavior of this quantity in a system consisting of two spins described by the $XYZ$ model in the presence of an external magnetic field. In this work it was observed for the first time that TQD could be able to signal a QPT at *finite* $T$.
In order to fully explore the previous possibility, highlighted by solving a simple two-body problem[@Wer10], we tackled the XXZ Hamiltonian in the thermodynamic limit for several values of $T>0$[@werPRL]. Now, working with infinite chains, we were able to show for the first time that TQD keeps its ability to detect the CPs associated to the QPTs for the XXZ model even if $T\neq 0$, while entanglement and other thermodynamic quantities are not as good as TQD. Also, we showed that these quantities when contrasted to TQD lose for increasing $T$ their CP-detection property faster than TQD. Later[@werPRA] we generalized those results considering $(i)$ the XXZ Hamiltonian and $(ii)$ the XY Hamiltonian both in the presence of an external magnetic field. For these two models it was shown that among the usual quantities employed to detect CPs, TQD was the best suited to properly estimate them when $T>0$.
Our goal in this paper is to present a short but self-contained review of our aforementioned results about TQD and its application as a CP detector of QPTs. To this end we structure this paper as follows. In Sec. 2 we present a brief review about quantum correlations where QD and the entanglement of formation take on a prominent role. In Sec. 3 we review the behavior of TQD in the context of simple two-qubit models. We then move on to the analysis of the ability of quantum correlations, in particular TQD, to spotlight the CPs of QPTs when the system is at $T>0$ and in the thermodynamic limit. In this context we study the XXZ and XY models with and without an external magnetic field. Finally, we conclude and discuss future directions in Sec. 4.
Quantum Correlations
====================
The superposition principle of quantum mechanics is directly related to the existence of (quantum) correlations that are not seen in classical objects. This principle together with the tensorial nature of combining different quantum systems (Hilbert spaces) lead to entanglement, which implies intriguing correlations among the many constituents of a composite system that puzzle our classical minds. It is worth mentioning that this tensorial nature from which a composite quantum system is described in terms of its parts is not a truism. Indeed, the principle of superposition is also present in classical physics, for example in the classical theory of electromagnetism. However, in classical physics this tensorial nature for combining systems is missing, which helps in understanding why “weird” quantum effects such as non-locality[@EPR] are not seen in a classical world.
During many decades after the birth of quantum mechanics quantum correlations were thought to be necessarily linked to non-locality, or more quantitatively, to the violation of a Bell-like inequality[@Bel64]. The non-violation of a Bell-like inequality implies that the correlations among the parts of a composite system can be described by a local realistic theory. This fact led many to call a state not violating any Bell-like inequality a classical state.
This situation changed by the seminal work of R. F. Werner[@Wer89], who showed that there are mixed entangled states that do not violate any Bell-like inequality. Therefore, according to the Bell/non-locality paradigm these states should be considered examples of classical states although possessing entanglement. This state of affairs was unsatisfactory and the notion of classical states was expanded. A classical (non-entangled) state was then defined as any state that can be created only by local operations on the subsystems and classical communication among its many parts (LOCC)[@nielsen; @Hor09]. For a bipartite system described by the density operator $\rho_{AB}$, the states created via LOCC (separable states) can be generally written as $$\begin{aligned}
\label{sstate}
\rho_{AB}=\sum_jp_j\rho_j^A\otimes\rho_j^B,\end{aligned}$$ where $p_j\geq 0$, $\sum_jp_j=1$, and $\rho_j^{A,B}$ are legitimate density matrices. If a quantum state cannot be written as (\[sstate\]) then it is an entangled state.
At this point one may wonder if this is a definitive characterization of a classical state. Or one may ask: Isn’t there any “quantumness” in the correlations for some sort of separable (non-entangled) quantum states? Can we go beyond the entanglement paradigm? As observed in refs. there exist some states written as (\[sstate\]) that possess non-classical features. This fact led the authors of refs. to push further our definition of classical states. Now, instead of eq. (\[sstate\]), we call a bipartite state classical if it can be written as $$\begin{aligned}
\rho_{AB}=\sum_{jk}p_{jk}|j \rangle_A \langle j|\otimes |k\rangle_B\langle k|,
\label{two}\end{aligned}$$ where $|j\rangle_A$ and $|k\rangle_B$ span two sets of orthonormal states. States described by (\[two\]) are a subset of those described by (\[sstate\]) and they are built via mixtures of locally distinguishable states[@ved10b]. Intuitively, classical states are those where the superposition principle does not manifest itself either on the level of different Hilbert spaces (zero entanglement) or on the level of single Hilbert spaces (no Schrödinger cat states leading to a mixture of non-orthogonal states). Such states have null QD.
Let us be more quantitative and define QD for a bipartite quantum state [@zurek; @vedral] divided into parts $A$ and $B$. In the paradigm of classical information theory[@nielsen] the total correlation between $A$ and $B$ is quantified by the mutual information (MI), $$\begin{aligned}
\label{mi1}
\mathcal{I}_1(A:B)= \mathcal{H}(A)+\mathcal{H}(B)-\mathcal{H}(A,B),\end{aligned}$$ where $\mathcal{H}(X)=-\sum_xp_x\log_2p_x$ is the Shannon entropy with $p_x$ the probability distribution of the random variable $X$. The conditional probability for classical variables $p_{a|b}$ is defined by the Baye’s rule, that is, $p_{a|b}=p_{a,b}/p_{b}$, with $p_{a,b}$ denoting the joint probability distribution of variables $a$ and $b$. Using the conditional probability one can show that $\mathcal{H}(A,B) = \mathcal{H}(A|B) + H(B)$, where $\mathcal{H}(A|B)=-\sum_{a,b}p_{a,b}\log_2(p_{a|b})$. This last result allows one to write MI, eq. (\[mi1\]), as $$\begin{aligned}
\label{mi2}
\mathcal{I}_2(A:B)= \mathcal{H}(A)-\mathcal{H}(A|B).\end{aligned}$$ Note that $\mathcal{H}(A|B)\geq0$ is the conditional entropy, which quantifies how much uncertainty is left on average about $A$ when one knows $B$. The quantum version of eq. (\[mi1\]), denoted by $\mathcal{I}^q_1(A:B)$, can be obtained replacing the Shannon entropy by the von-Neumann entropy $\mathcal{S}(X)=\mathcal{S}(\rho_X)=-\mbox{Tr}\left(\rho_X\log_2\rho_X\right)$, where $\rho_X$ is the density operator of the system $X=A,B$. On the other hand, a quantum version of eq. (\[mi2\]) is not so straightforward because the Bayes’ rule is not always valid for quantum systems[@brule]. For instance, this rule is violated for a pure entangled state. Indeed, one can show that if we naively extend the usual quantum conditional entropy to quantum systems as $\mathcal{S}(A|B)\equiv\mathcal{S}(A,B)-\mathcal{S}(B)$, it becomes negative for pure entangled states. QD is built in a way to circumvent this limitation.
In order to build a meaningful quantum version of the conditional entropy $\mathcal{H}(A|B)$ it is necessary to take into account the fact that knowledge about system B is related to measurements on $B$. And now, differently from the classical case, a measurement in the quantum domain can be performed in many non-equivalent ways (different set of projectors, for instance). If a general quantum measurement, i.e. a POVM (positive operator valued measure)[@nielsen], $\{M_b\}$ is performed in the quantum state $\rho_{AB}$, then after the measurement the state is described by $\sum_bM_b\rho_{AB}M_b^\dagger$. The probability of the outcome $b$ of $B$ is $p_b=\mbox{Tr}\left[M_b\rho_{AB}M_b^\dagger\right]$ and the conditional state of $A$ in this case is $\rho_{A|b}=\left(M_b\rho_{AB}M_b^\dagger\right)/p_b$. Thus, the conditional entropy with respect to the POVM $\{M_b\}$ is $\mathcal{S}(A|\{M_b\})\equiv\sum_bp_b\mathcal{S}(\rho_{A|b})$. Therefore, in order to quantify the uncertainty left on $A$ after a measurement on $B$ one has to minimize over all POVMs. This leads to the following definition of the quantum conditional entropy[@zurek; @vedral]: $$\begin{aligned}
\label{qce}
\mathcal{S}_q(A|B)\equiv\min_{\left\{M_b\right\}}\mathcal{S}(A|\{M_b\}).
\label{cond}\end{aligned}$$
At this point, the quantum version of the mutual information (\[mi2\]), denoted by $\mathcal{I}^q_2(A:B)$, is obtained replacing the classical conditional entropy $\mathcal{H}(A|B)$ by its quantum analog $\mathcal{S}_q(A|B)$, which does not assume negative values. The [*quantum discord*]{} is defined as the difference between these two versions of the quantum mutual information[@zurek; @vedral]: $$\begin{aligned}
\label{qd}
D(A|B)&\equiv&\mathcal{I}^q_1(A:B)-\mathcal{I}^q_2(A:B)\nonumber\\
&=&\mathcal{S}_q(A|B)-\mathcal{S}(A|B).\end{aligned}$$ Note that QD is not necessarily a symmetric quantity, because the conditional entropy (\[qce\]) depends on the system in which the measurement is performed. However, for the density operators studied in this paper QD is always symmetric. Furthermore, while for mixed states there are states with null entanglement and QD $>0$, for pure states QD is essentially equivalent to entanglement. In other words, only for mixed states there might be quantum correlations other than entanglement. As demonstrated recently[@ved10], a quantum state $\rho_{AB}$ has $D(A|B)=0$ if, and only if, it can be written as $\rho_{AB}=\sum_jp_j\rho_j^A\otimes\left|\psi_j^B\right\rangle\left\langle \psi_j^B\right|$, with $\sum_jp_j=1$ and $\{\left|\psi_j^B\right\rangle\}$ a set of orthogonal states. This result shows the importance of the superposition principle to explain the origin of the quantum correlations. It is due to this principle that one can generate a set $\{\left|\psi_j^B\right\rangle\}$ of non-orthogonal states leading to states with nonzero QD.
For arbitrary $N\times M$-dimensional bipartite states the computation of QD involves a complicated minimization procedure whose origin can be traced back to the evaluation of the conditional entropy $\mathcal{S}_q(A|B)$, eq. (\[cond\]). In general one must then rely on numerical procedures to get QD and it is not even known whether a general efficient algorithm exists. For two-qubit systems, however, the minimization over generalized measurements can be replaced by the minimization over projective measurements (von Neumann measurements)[@minDIS]. In this case the minimization procedure can be efficiently implemented numerically[@Dnum] for arbitrary two-qubit states and some analytical results can be achieved for a restricted class of states[@Danalitico]. In this work we will be dealing with density matrices $\rho$ in the X-form, that is, $\rho_{12}=\rho_{13}= \rho_{24}=\rho_{34}=0$. Moreover, in our models $\rho_{22}=\rho_{33}$ and all matrix elements are real, making the numerical evaluation of QD simple and fast.
To close this section, we introduce the measure of entanglement used in this paper, the [*Entanglement of Formation*]{} (EoF)[@Woo98]. EoF quantifies, at least for pure states, how many singlets are needed per copy of $\rho_{AB}$ to prepare many copies of $\rho_{AB}$ using only LOCC. For an $X$-form density matrix we have $$\begin{aligned}
\label{eof}
EoF(\rho_{AB})&=&-g\log_2g-(1-g)\log_2(1-g),\end{aligned}$$ with $g=(1+\sqrt{1-C^2})/2$ and the concurrence[@Woo98] given by $C=2\max\left\{0,\Lambda_1,\Lambda_2 \right\}$, where $\Lambda_1=|\rho_{14}|-\sqrt{\rho_{22}\rho_{33}}$ and $\Lambda_2=|\rho_{23}|-\sqrt{\rho_{11}\rho_{44}}$.
Results and Discussions
=======================
Two interacting spins
---------------------
In this section we consider a two spin system described by the XYZ model with an external magnetic field acting on both spins.[@Wer10] The Hamiltonian of this model is $$\begin{aligned}
\label{hxyz}
H_{XYZ}=\frac{J_x}{4}\sigma^x_1\sigma^x_{2}+\frac{J_y}{4}\sigma^y_1\sigma^y_{2}+\frac{J_z}{4}\sigma^z_1\sigma^z_{2}
+ \frac{B}{2}\left(\sigma^z_1+\sigma^z_2\right),\end{aligned}$$ where $\sigma_j^\alpha$ ($\alpha=x,y,z$) are the usual Pauli matrices acting on the $j$-th site and we have assumed $\hbar=1$. As mentioned above, the density matrix describing a system in equilibrium with a thermal reservoir at temperature $T$ is $\rho_T=e^{-H/kT}/Z$, where $Z=\mbox{Tr}\left(e^{-H/kT}\right)$ is the partition function. Therefore, the thermal state for the Hamiltonian (\[hxyz\]) assumes the following form $$\rho = \frac{1}{Z} \left(
\begin{array}{cccc}
A_{11} & 0 & 0 & A_{12}\\
0 & B_{11} & B_{12} & 0 \\
0 & B_{12} & B_{11} & 0 \\
A_{12} & 0 & 0 & A_{22} \\
\end{array}
\right), \label{rho2}$$ with $A_{11}$ $=$ $\mathrm{e}^{-\alpha}$ $(\cosh(\beta)$ $-$ $4B$ $\sinh(\beta)/\eta)$, $A_{12}$ $=$ $-$ $\Delta$ $\mathrm{e}^{-\alpha}$ $\sinh(\beta)/\eta$, $A_{22}$ $=$ $\mathrm{e}^{-\alpha}$ $(\cosh(\beta)$ $+$ $4$ $B$ $\sinh(\beta)/\eta)$, $B_{11}$ $=$ $\mathrm{e}^{\alpha}$ $\cosh(\gamma)$, $B_{12}$ $=$ $-$ $\mathrm{e}^{\alpha}$ $\sinh(\gamma)$, and $Z = 2\left( \exp{(-\alpha)}\cosh(\beta) + \exp{(\alpha)}\cosh(\gamma) \right)$, where $\Delta = J_{x} - J_{y}$, $\Sigma = J_{x} + J_{y}$, $\eta = \sqrt{\Delta^{2} + 16B^2}$, $\alpha = J_{z}/(4kT)$, $\beta = \eta/(4kT)$, and $\gamma = \Sigma/(4kT)$.
The first important result appears in the absence of an external field. As shown in Ref. , when $B=0$, the entanglement does not increase with increasing temperature. On the other hand, as can be seen in Fig. \[fig1\] (panels $a$ and $b$) for the XXZ model $\left(J_x=J_y=J \quad\mbox{and}\quad J_z\neq0 \right)$, TQD begins with a non-null value at $T=0$ and increases as $T$ increases before decreasing with $T$, while EoF is always zero[@Rig03]. Note that such effect is observed for different configurations of coupling constants.
The possibility of TQD to point out a QPT even when the system is at finite temperatures emerged from our study about the XXX model for two spins[@Wer10]. The XXX model is obtained from Hamiltonian (\[hxyz\]) making $J_x=J_y=J_z=J$. When $J\rightarrow\infty$ the density operator (\[rho2\]) will be the Bell state $\rho=\left|\psi\right\rangle\left\langle\psi\right|$, with $\left|\psi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle-\left|10\right\rangle\right)$, for any $T$. For the opposite limit $J\rightarrow-\infty$ the density operator is the mixed state $\rho=\frac{1}{3}\left(\left|00\right\rangle\left\langle
00\right|+\left|11\right\rangle\left\langle11\right|+\left|\phi\right\rangle\left\langle \phi\right|\right)$ with $\left|\phi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|01\right\rangle +\left|10\right\rangle\right)$. In this case EoF is zero, while TQD assumes the value $1/3$. Furthermore, as shown in Fig. \[fig2\], EoF is zero in the ferromagnetic region $(J<0)$ and non-zero in the antiferromagnetic region $(J>0)$ only when $T=0$.
For $T>0$ EoF becomes non-zero only for $J>J_c(T)=kT\ln(3)$. On the other hand, TQD is equal to zero only at the trivial point $J=0$, even at finite $T$. Although we are considering here only two spins, such result suggests that TQD may possibly signal a QPT for $T>0$.
Let us now analyze the case where $B\neq 0$ and focus on the XY model in a transverse magnetic field ($J_x$, $J_y\neq 0$, and $J_z=0$). As noted in Ref. EoF shows a sudden death and then a revival (see Fig. \[fig3\], panel b).
However, TQD does not suddenly disappear as Fig. \[fig3\], panel a, depicts. Actually, TQD decreases with $T$ to a non-null value and after a critical temperature $T_c$ it starts increasing again. This effect is called [*regrowth*]{}[@Wer10]. Although the regrowth of EoF with temperature is not observed for two spin chains, we showed in Ref. that such interesting behavior is possible in the thermodynamic limit. Also, if we carefully look at Fig. \[fig3\] the distinctive aspects of these two types of quantum correlations become more evident. For example, comparing panels a and b we see regions where TQD increases while EoF decreases. Finally, in a very interesting and recent work, X. Rong *et al.*[@Ron12] experimentally verified some of the predictions here revised and reported in refs. , namely, the sudden change of TQD at finite temperatures while changing the anisotropy parameter of a two spin XXZ Hamiltonian.
XXZ Model
---------
Turning our attention to infinite chains ($L\rightarrow \infty$), let us start working with the one-dimensional anisotropic spin-$1/2$ XXZ model subjected to a magnetic field in the $z$-direction. Its Hamiltonian is $$\begin{aligned}
\label{hxxz}
H_{xxz}&=&J\sum_{j=1}^L\left(\sigma_j^x\sigma_{j+1}^x+\sigma_j^y\sigma_{j+1}^y
+\Delta\sigma_j^z\sigma_{j+1}^z\right) - \frac{h}{2}\sum_{j=1}^L\sigma_j^z,\end{aligned}$$ where $\Delta$ is the anisotropy parameter, $h$ is the external magnetic field, and $J$ is the exchange constant ($J=1$). We have assumed periodic boundary conditions $\left(\sigma_{L+1}^\alpha=\sigma_1^\alpha\right)$. The nearest neighbor two spin state is obtained by tracing all but the first two spins, $\rho_{1,2}=\mbox{Tr}_{L-2}(\rho)$, where $\rho=\exp{\left(-\beta H_{xxz}\right)}/Z$. The Hamiltonian (\[hxxz\]) exhibits both translational invariance and $U(1)$ invariance $\left(\left[H_{xxz},
\sum_{j=1}^L\sigma_j^z\right]=0\right)$ leading to the following nearest neighbor two spin state $$\begin{aligned}
\label{rho}
\rho_{1,2} = \frac{1}{4}\left(
\begin{array}{cccc}
\rho_{11} & 0 & 0 & 0\\
0 & \rho_{22} & \rho_{23} & 0 \\
0 & \rho_{23} & \rho_{22} & 0 \\
0 & 0 & 0 & \rho_{44} \\
\end{array}
\right),\end{aligned}$$ where $$\begin{aligned}
\label{rhoe}
\rho_{11} &=& 1+2\left\langle \sigma^z\right\rangle+{\left\langle \sigma_1^z\sigma_2^z \right\rangle},\nonumber\\
\rho_{22} &=& 1-{\left\langle \sigma_1^z\sigma_2^z \right\rangle},\\
\rho_{44} &=& 1-2\left\langle \sigma^z\right\rangle+{\left\langle \sigma_1^z\sigma_2^z \right\rangle},\nonumber\\
\rho_{23} &=& 2{\left\langle \sigma_1^x\sigma_2^x \right\rangle}\nonumber.\end{aligned}$$
The magnetization and the two-point correlations above are obtained in terms of the derivatives of the free-energy[@NLIE], $f=-\frac{1}{\beta}
\lim_{L\rightarrow \infty} \frac{\ln{Z}}{L}$, $$\begin{aligned}
{\left\langle \sigma^z \right\rangle}&=& -2\partial_h f/J, \\
{\left\langle \sigma_j^{z}\sigma_{j+1}^{z} \right\rangle}&=&\partial_{\Delta}f/J, \\
{\left\langle \sigma_j^{x}\sigma_{j+1}^{x} \right\rangle}&=&\frac{u-\Delta \partial_{\Delta}f+
h {\left\langle \sigma^z \right\rangle}}{2J}, \\
{\left\langle \sigma_j^{z}\sigma_{j+1}^{z} \right\rangle}&=&{\left\langle \sigma_j^{x}\sigma_{j+1}^{x} \right\rangle}=
\frac{u + h {\left\langle \sigma^z \right\rangle}}{3J}, ~~ \Delta=1,\end{aligned}$$ where $u=\partial_{\beta} (\beta f)$ is the internal energy. We explained in details the procedures to determine the free-energy $f$ in Ref. . It is not a simple task and it involves the application of complicated analytical and numerical computations. The CPs of the XXZ model depend on the value of the magnetic field $h$[@GAUDIN]. One of them, called $\Delta_{inf}$, is an infinite-order QPT determined by solving the following equation, $$\begin{aligned}
\label{cpinf}
h=4J\sinh(\eta)\sum_{n=-\infty}^\infty\frac{(-1)^n}{\cosh(n\eta)},\end{aligned}$$ with $\eta = \cosh^{-1}(\Delta_{inf})$. The other CP, called $\Delta_1$, is a first-order QPT given by $$\begin{aligned}
\label{cp1}
\Delta_1 = \frac{h}{4J}-1.\end{aligned}$$
The behavior of TQD and EoF for the XXZ model at finite $T$ and $h=0$ was initially studied in ref. . For $h=0$ the XXZ model has two CPs[@TAKAHASHI]. At $\Delta_{inf}=1$ the ground state changes from an XY-like phase ($-1<\Delta<1$) to an Ising-like antiferromagnetic phase ($\Delta>1$). At $\Delta_1=-1$ it changes from a ferromagnetic phase ($\Delta<-1$) to the critical antiferromagnetic phase ($-1<\Delta<1$). In ref. we analyzed the behavior of TQD and EoF only for $\Delta>0$. Here we extend those results by computing the correlation functions for the remaining values of the anisotropy parameter, namely, $\Delta<0$. In Fig. \[fig4\] we plot TQD (panel a) and EoF (panel c) as a function of $\Delta$ for different values of $kT$ and $h=0$.
For $T=0$ we can see that both TQD and EoF are able to detect the CPs. TQD is discontinuous at $\Delta_1$ while at $\Delta_{inf}$ the first-order derivative of TQD presents a discontinuity. Furthermore, EoF is zero for $\Delta<\Delta_1$ and non-zero for $\Delta>\Delta_1$ reaching a maximum value at $\Delta=\Delta_{inf}$. However, as the temperature increases the maximum value of EoF is shifted to the right. Besides, EoF becomes zero also for $\Delta>\Delta_1$ as we increase $T$. On the other hand, the first-order derivative of TQD is still discontinuous at $\Delta_{inf}=1$ for finite $T$. We can also observe that TQD increases for $\Delta<-1$ as $T$ increases while its first-order derivative diverges at the CP $\Delta=-1$. As mentioned in Ref. the cusp-like behavior at CPs $\Delta_1=1$ and $\Delta_{inf}=-1$ is due to an exchange in the set of projectors that minimizes the quantum conditional entropy (\[qce\]).
The study about the XXZ model was further explored in Ref. , with the addition of an external field. The effects of the magnetic field $h$ on the quantum correlations are exemplified in Fig. \[fig4\] (panels b and d), where we set $h=12$. The values of the CPs for $h=12$ are calculated employing Eqs. (\[cpinf\]) and (\[cp1\]), resulting in $\Delta_{inf}\approx4.88$ and $\Delta_1=2$. Again, for $T=0$ the CPs are detected by both TQD and EoF and, differently from the case $h=0$, the behavior of these two quantities is quite similar. Both quantities are zero for $\Delta<2$ and non-zero for $\Delta>2$, with their first-order derivatives diverging at the CP $\Delta_1=2$. However, the infinite-order QPT is no longer characterized by a global maximum of TQD or EoF, but by a discontinuity in their first-order derivatives. Note also that TQD presents a cusp-like behavior between the CPs. This behavior is once again related to the minimization procedure of the quantum conditional entropy and so far it is not associated to any known QPT for this model. It is important to mention that entanglement measures may also have a discontinuity and/or a divergence in their derivatives that are not related to a QPT[@yang].
Now we move on to the cases where $T>0$. When $T$ increases both curves of TQD and EoF become smoother and broader, with well defined derivatives in the CPs. Besides, the cusp-like behavior of TQD previously mentioned tends to disappear while both maximums of TQD and EoF decrease[@werPRA]. We noted in Ref. that some features of the derivatives of these quantities remain for a finite, but not too high temperature. To illustrate this fact, we plotted in Fig. \[fig5\] the first-order (panel a) and second-order (panel b) derivatives of TQD with respect to the anisotropy parameter $\Delta$ for $h=12$ and $kT=0.02, 0.1, 0.5$. To plot the curves for different temperatures in the same graph we normalized the derivatives of TQD, that is, for each $T$ we plotted the derivative of TQD divided by the maximum value of the respective derivative. For $T=0$ the divergence in the first-order derivative of both TQD and EoF spotlights the CP $\Delta_1$ while the CP $\Delta_{inf}$ is characterized by a divergence in the second-order derivative. As can be seen in Fig. \[fig5\], although the divergence at the CPs disappears as $T$ increases, the derivatives reach their maximum values around the CPs. We used these maximum values to estimate the CPs at finite temperatures. The same analysis was applied to estimate the CPs using EoF instead of TQD. In Ref. we compared the ability of TQD, EoF, and pairwise correlations (${\left\langle \sigma_1^z\sigma_2^z \right\rangle}$ and ${\left\langle \sigma_1^x\sigma_2^x \right\rangle}$) to point out the CPs for $T>0$ and we showed that TQD is the best candidate to estimate the CPs.
To illustrate such result we compared in Fig. \[fig6\] the difference between the correct CP $\Delta_c$ and the CP estimated by our method $\Delta_e$ for $h=6$ and $h=12$. One can see in this figure that from zero to $kT\approx 1$ the CPs estimated by TQD are closer to the correct ones than the estimated CPs coming from other quantities.
XY Model
--------
The Hamiltonian of the one-dimensional XY model in a transverse field is given by $$\begin{aligned}
\label{HXY}
H_{xy}&=& - \frac{\lambda}{2} \sum_{j=1}^L
\left[(1+\gamma)\sigma^x_j\sigma^x_{j+1}
+(1-\gamma)\sigma^y_j\sigma^y_{j+1}\right]
-\sum_{j=1}^L\sigma^z_j,\end{aligned}$$ where $\lambda$ is the strength of the inverse of the external transverse magnetic field and $\gamma$ is the anisotropy parameter. The transverse Ising model is obtained for $\gamma=\pm1$ while $\gamma=0$ corresponds to the XX model in a transverse field[@LSM]. At $\lambda_c=1$ the XY model undergoes a second-order QPT (Ising transition[@isingQPT]) that separates a ferromagnetic ordered phase from a quantum paramagnetic phase. Another second order QPT is observed for $\lambda>1$ at the CP $\gamma_c=0$ (anisotropy transition[@LSM; @anisQPT]). This transition is driven by the anisotropy parameter $\gamma$ and separates a ferromagnet ordered along the $x$ direction and a ferromagnet ordered along the $y$ direction. These two transitions are of the same order but belong to different universality classes [@LSM; @anisQPT].
The XY Hamiltonian is $Z_2$-symmetric and can be exactly diagonalized [@LSM] in the thermodynamic limit $L\rightarrow\infty$. Due to translational invariance the two spin density operator $\rho_{i,j}$ for spins $i$ and $j$ at thermal equilibrium is [@osborne] $$\begin{aligned}
\label{doXY}
\rho_{0,k}&=&\frac{1}{4}\left[I_{0,k}
+\left\langle \sigma^z\right\rangle\left(\sigma^z_0+\sigma^z_k\right)\right]
+ \frac{1}{4} \sum_{\alpha=x,y,z}
\left\langle \sigma^\alpha_0\sigma^\alpha_k\right\rangle \sigma^\alpha_0\sigma^\alpha_k,\end{aligned}$$ where $k=\left|j-i\right|$ and $I_{0,k}$ is the identity operator of dimension four. The transverse magnetization $\left\langle \sigma^z_k\right\rangle$ $=$ $\left\langle \sigma^z\right\rangle$ is $$\begin{aligned}
\label{tmag}
\left\langle \sigma^z\right\rangle=-\int_0^\pi
(1+\lambda\cos{\phi})\tanh{(\beta\omega_\phi)}\frac{d\phi}{2\pi\omega_\phi},\end{aligned}$$ with $\omega_\phi=\sqrt{(\gamma\lambda\sin{\phi})^2+(1+\lambda\cos{\phi})^2}/2$. The two-point correlation functions are given by $$\begin{aligned}
\label{tpcf}
\left\langle \sigma^x_0\sigma^x_k\right\rangle &=& \left|
\begin{array}{cccc}
G_{-1} & G_{-2} & \cdots & G_{-k}\\
G_{0} & G_{-1} & \cdots & G_{-k+1} \\
\vdots & \vdots & \ddots & \vdots \\
G_{k-2} & G_{k-3} & \cdots & G_{-1} \\
\end{array}
\right|,\\
\left\langle \sigma^y_0\sigma^y_k\right\rangle &=& \left|
\begin{array}{cccc}
G_{1} & G_{0} & \cdots & G_{-k+2}\\
G_{2} & G_{1} & \cdots & G_{-k+3} \\
\vdots & \vdots & \ddots & \vdots \\
G_{k} & G_{k-1} & \cdots & G_{1} \\
\end{array}
\right|,\\
\left\langle \sigma^z_0\sigma^z_k\right\rangle &=&
\left\langle \sigma^z\right\rangle^2 - G_k G_{-k},\end{aligned}$$ where $$\begin{aligned}
G_k&=&\int_0^\pi \tanh{(\beta\omega_\phi)}\cos{(k\phi)}(1+\lambda\cos{\phi})
\frac{d\phi}{2\pi\omega_\phi}\\
&-&\gamma\lambda\int_0^\pi \tanh{(\beta\omega_\phi)} \sin{(k\phi)\sin{\phi}}
\frac{d\phi}{2\pi\omega_\phi}.\end{aligned}$$
The relation between TQD and QPT for the Ising model (XY model with $\gamma=1$) at $T=0$ was investigated initially by Dillenschneider[@Dil08] for first and second nearest-neighbors. More general results were obtained in Ref. where TQD and EoF from first to fourth nearest-neighbors was computed for different values of $\gamma$. This study at $T=0$ showed that while EoF between far neighbors becomes zero, QD is not null and detects the QPT. The effects of the symmetry breaking process in entanglement and QD for the XY and the XXZ models were discussed in Refs. , where the low temperature regime was taken into account. In Ref. we compared the ability of TQD and EoF for first and second nearest-neighbors to detect the CPs for the XY model at finite temperature.
The behavior of TQD and EoF for first nearest-neighbors as a function of $\lambda$ for $kT=0.01,0.1,0.5$ and $\gamma=0,0.5,1.0$ can be seen in Fig. \[fig7\]. First, note that TQD is more robust to temperature increase than EoF. For $kT=0.5$ TQD is always non-zero while EoF is zero or close to zero for almost all $\lambda$ (see the blue/solid curves in Fig. \[fig7\]).
As showed in Ref. , for second nearest-neighbors the situation is more drastic since EoF is always zero for $kT=0.5$. Now, to estimate the CPs at finite $T$ we used the same procedure adopted for the XXZ model. If the first-order derivative of TQD or EoF is divergent at $T=0$ then the CP is pointed out by a local maximum or minimum at $T>0$; if the first-order derivative is discontinuous at $T=0$ then we look after local maximum or minimum in the second derivative for $T>0$. These extreme values act as indicators of QPTs. The CPs estimated with such method are denoted by $\lambda_e$ while the correct CPs are denoted by $\lambda_c$. The differences between $\lambda_c$ and $\lambda_e$ as a function of kT for $\gamma=0,0.5,1.0$ are plotted in Fig. \[fig8\]. We can see in this figure that TQD provides a better estimate of the CP $\lambda_c=1$ than EoF.
For $\gamma=0.5$ EoF and TQD give almost the same estimation of the CP and for $\gamma=1.0$ TQD is better than EoF with predictions differing at the second decimal place. For $\gamma = 0$, TQD outperforms EoF already in the first decimal place. Moreover, for $\gamma=0$ TQD is able to correctly estimate the CP for higher temperatures than for $\gamma=0.5$ and $\gamma=1.0$.
So far we have studied a QPT driven by the magnetic field. However, for $\lambda>1$ the XY model undergoes a QPT driven by the anisotropy parameter $\gamma$, whose critical point is $\gamma_c=0$. To study such transition we fixed $\lambda=1.5$. In Fig. \[fig9\] we plotted TQD and EoF for the first-neighbors as functions of $\gamma$ and for $kT = 0.001, 0.1, 0.5, 1.0,$ and $2.0$. Note that the maximum of TQD and EoF is reached at the CP $\gamma_c=0$. However, only TQD has a cusp-like behavior at the CP.
This pattern of TQD (maximum with a cusp-like behavior) remains up to $kT=2.0$. On the other hand, the maximum of EoF at the CP can only be seen as far as $kT<1$ for above this temperature EoF becomes zero. In Ref. we computed TQD and EoF for second-neighbors. In this case TQD is able to detect the CP even for values near $kT=1.0$ while EoF is nonzero only for $kT\lesssim 0.1$.
Conclusions
===========
In this article we presented a review of our studies about the behavior of quantum correlations in the context of spin chains at finite temperatures. The main goal of this paper was to analyze the quantum correlation’s ability to pinpoint the critical points associated to quantum phase transitions assuming the system’s temperature is greater than the absolute zero. The two measures of quantum correlations studied here were quantum discord and entanglement, with the former producing the best results.
We first reviewed a work of two of us about a simple but illustrative two spin-1/2 system described by the XYZ model with an external magnetic field[@Wer10]. We showed many surprising results about the thermal quantum discord’s behavior. For example, and differently from entanglement, thermal quantum discord can increase with temperature in the absence of an external field even for such a small system; and that there are situations where thermal quantum discord increases while entanglement decreases with temperature. Furthermore, for the XXX model we observed for the first time that quantum discord could be a good candidate to signal a critical point at finite $T$.
To check whether quantum discord is indeed a good critical point detector for temperatures higher than absolute zero, we analyzed its behavior for an infinite chain described by the XXZ model without[@werPRL] and with[@werPRA] an external magnetic field and in equilibrium with a thermal reservoir at temperature $T$. Here we also extended our previous results by computing the quantum correlations between the first nearest-neighbor spins for the whole range of the anisotropy parameter (positive and negative values). In this way we were able to describe the two critical points of the XXZ model in the thermodynamic limit and the behavior of quantum discord near them. The results presented here and in refs. showed that quantum discord is far better the best critical point detector for $T>0$ with respect to all quantities tested (entanglement, entropy, specific heat, magnetic susceptibility, and the two-point correlation functions).
Another model considered in this work was the XY model in a transverse magnetic field[@werPRA]. Again, we computed the quantum correlations between the first nearest-neighbors assuming the system in equilibrium with a thermal reservoir at temperature $T$. This model has two second-order quantum phase transitions, namely, an Ising transition and an anisotropy transition. For the Ising transition we observed that the critical point is better estimated by quantum discord. For the anisotropy transition both quantities, entanglement and discord, provide an excellent estimate for the critical point at low temperatures. However, since for increasing temperatures quantum discord is more robust than entanglement, the former was able to spotlight the quantum critical point for a wider range of temperatures, even for values of temperature where entanglement was absent.
In conclusion, we showed that for the spin models studied here and in Refs. quantum discord was the best quantum critical point estimator among all quantities tested when the system assumes a finite temperature. It is also important to mention that the knowledge of the order parameter was not needed to estimate the critical points. Therefore, we strongly believe that our results suggest that quantum correlations - mainly quantum discord - are important tools to study quantum phase transitions in realistic scenarios, where the temperature is always above absolute zero.
Acknowledgements {#acknowledgements .unnumbered}
================
TW and GR thank the Brazilian agency CNPq (National Council for Scientific and Technological Development) for funding and GAPR thanks CNPq and FAPESP (State of São Paulo Research Foundation) for funding. GR also thanks CNPq/FAPESP for financial support through the National Institute of Science and Technology for Quantum Information.
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O. F. Syljuåsen, Phys. Rev. A **68**, 060301(R) (2003); O. F. Syljuåsen, Phys. Lett. A **322**, 25 (2004); A. Osterloh, G. Palacios, and S. Montangero, Phys. Rev. Lett. **97**, 257201 (2006); T. R. de Oliveira et al., Phys. Rev. A **77**, 032325 (2008); B. Tomasello *et al*, Europhys. Lett. **96**, 27002 (2011).
[^1]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Based on analytical estimation and lattice simulation, a proposal is made that magnetic skyrmions can be generated through the pinning effect in 2D chiral magnetic materials, in absence of an external magnetic field or magnetic anisotropy. In our simulation, stable magnetic skyrmions can be generated in the pinning areas. The properties of the skyrmions are studied for various values of ferromagnetic exchange strength and the Dzyaloshinskii-Moriya interaction strength.'
author:
- 'Ji-Chong Yang'
- 'Qing-Qing Mao'
- Yu Shi
title: Generation of magnetic skyrmions through pinning effect
---
0.25cm
\[sec:1\]Introduction
=====================
The topologically protected structure called skyrmion can be formed in a chiral magnet [@nagaosa; @Fert]. Magnetic skyrmions have been discovered in the bulk MnSi by using neutron scattering [@Muhlbauer], and have also been observed by using Lorentz transmission electron microscopy [@Yu1] and by using spin-resolved scanning tunnelling microscopy (STM) [@2DSkyrmion2]. They can be created in magnetic materials with Dzyaloshinskii-Moriya (DM) interactions [@DMI]. They can be driven by spin current with critical current density lower than that for magnetic domain walls [@ultralow], and are thus promising as future information carriers in magnetic information storage and processing devices.
Therefore it is interesting to find efficient methods of creation and manipulation of magnetic skyrmions. In the presence of a magnetic field, single skyrmions can be created and deleted by using local spin-polarized STM [@Romming]. A large number of magnetic skyrmons were created with the aid of a special geometrical constriction in presence of interfacial DM interaction [@bubble; @thetarho]. In the absence of a magnetic field, skyrmions can be generated with the aid of a circulating current [@Tchoe], or magnetic anisotropy [@anisotropy; @bilayer1], or perpendicular anisotropy energy [@anisotropyenergy], or DC current together with inhomogeneous magnetization but without DM interactions [@dccurrent].
Pinning effect refers to the inhomogeneities of the ferromagnetic exchange coupling, the DM interaction and the magnetic anisotropy, which may be caused by defects and impurities [@pin; @pin2]. In this paper, we propose a novel method to generate magnetic skyrmions by exploiting the inhomogeneities of the ferromagnetic coupling and the DMI strength.
The rest of the paper is organized as the following. In Sec. \[sec:2\], we briefly review the 2D magnetic skyrmions and introduce our basic idea. In Sec. \[sec:4\], we report a lattice simulation and study the properties of the skyrmions generated in the simulation. A summary is made in Sec. \[sec:5\] .
\[sec:2\]Basic Idea
===================
In consideration of that the skyrmions can be generated by using magnetic anisotropy, here we only consider the case without magnetic anisotropy. In this case, in the presence of an external magnetic field, the Hamiltonian can be written as [@nagaosa; @FreeEnergy] $$\begin{split}
&{\mathcal H}_{\rm tot}({\bf r})=\frac{J({\bf r})}{2}\left(\nabla {\bf n}\right)^2+D({\bf r}) {\bf n}\cdot (\nabla \times {\bf n})-{\bf B}(\bf r)\cdot {\bf n},\\
\end{split}
\label{eq.2.1}$$ where ${\bf n}$ is the orientation of the magnetic moment, $J({\bf r})$ is the ferromagnetic exchange strength, $D({\bf r})$ is the strength of DM interaction, ${\bf B}=B\mathbf{e}_z$ is the external magnetic field along ${\bf e}_z$ direction. We use dimensionless parameters.
The local magnetic moment of a skyrmion or anti-skyrmion can be parameterized as [@nagaosa]. $$\begin{split}
&{\bf n}(x,y)=\left(\cos(\gamma +m \phi)\sin(\theta(\rho)),\sin(\gamma +m \phi)\sin(\theta(\rho)),g \cos (\theta (\rho))\right),\\
\end{split}
\label{eq.2.2}$$ where $\rho$ and $\phi$ are polar coordinates of the 2D position vector $(x, y)$, with the origin at the center of the skyrmion, $m=\pm 1$ is the vorticity of the skyrmion or anti-skyrmion, $g=\pm 1$ is the polarization, $\gamma$ is helicity and distinguishes between N$\rm \acute{e}$el-type and Bloch-type skyrmions, $\theta (\rho)$ is a function describing the shape of a skyrmion, with $\theta(0)=\pi$ and $\theta(\infty)=0$. The skyrmion charge is defined as [@nagaosa; @Tchoe] $$\begin{split}
&Q=\frac{1}{4\pi}\int dxdy {\bf n}\cdot \left(\frac{\partial {\bf n}}{\partial x}\times \frac{\partial {\bf n}}{\partial y}\right)=-mg.\\
\end{split}
\label{eq.2.3}$$ The anti-skyrmions usually result from anisotropic DM interaction [@antiskyrmion], but here we only consider isotropic DM interaction, hence we only consider $m=1$.
It has been found that for a skyrmion, $\theta(\rho)$ can be approximated as [@arctan] $$\begin{split}
&\theta(\rho)\approx 4\tan^{-1}(\exp(-a\rho)).\\
\end{split}
\label{eq.2.5}$$ Obtained from this expression with $a=0.05$, some examples of ${\bf n}({\bf r})$ are shown in Fig. \[fig:examples\].
Two kinds of pinning were considered previously. In Ref. [@pin], both $J$ and $D$ are inhomogeneous, while $D/J$ is kept constant. In Ref. [@pin2], $D$ is constant while $J$ is inhomogeneous. For simplicity, we only consider the latter case in this section. The former case will be studied in a lattice simulation in next section.
From Eq. (\[eq.2.1\]), we obtain the Euler-Lagrange equation $$\begin{split}
&\frac{ \sin(\theta)\cos(\theta)}{\rho}-\theta ' - \rho \theta ''-2\frac{\tilde{D}}{J(\rho)}\sin^2(\theta)+\frac{\tilde{B}}{J(\rho)} \rho \sin (\theta)-\rho \frac{J'(\rho)}{J(\rho)}\theta '=0,
\end{split}
\label{eq.2.6}$$ where $\tilde{D} \equiv g\sin(\gamma)D$, $\tilde{B}\equiv gB$, $J(\rho)$ is a pinning function, simply assumed to depend only on $\rho$ so that the skyrmion center is at the center of the pinning, which is rotationally symmetric. When there is no pinning while an external magnetic field is applied, Eq. (\[eq.2.6\]) becomes $$\begin{split}
&\frac{ \sin(\theta)\cos(\theta)}{\rho}-\theta ' - \rho \theta ''-2\frac{\tilde{D}}{J}\sin^2(\theta)+\frac{\tilde{B}}{J} \rho \sin (\theta)=0.\\
\end{split}
\label{eq.2.7}$$ It is known that the skyrmions can be generated in this case.
We note that when pinning is present, $J'$ term in Eq. (\[eq.2.6\]) plays a role similar to that of $\tilde{B}$ term. Using the ansatz $\theta(\rho)$ in Eq. (\[eq.2.5\]) for $b \sin(\theta(\rho))=J'(\rho)\theta'(\rho)$, where $b$ is some value of $\tilde{B}$ , we find that if $$\begin{split}
&J(\rho)=J_0+b\frac{a\rho+\log \left(e^{-2a\rho}+1\right)}{a^2},
\end{split}
\label{eq.2.8}$$ where $a$ as given in Eq. (\[eq.2.5\]), $J_0$ is an undetermined coefficient, then Eq. (\[eq.2.6\]) without $\tilde{B} $ term becomes $$\begin{split}
&\frac{ \sin(\theta)\cos(\theta)}{\rho}-\theta ' - \rho \theta ''-2\frac{\tilde{D}}{J(\rho)}\sin^2(\theta)+\frac{b}{J(\rho)} \rho \sin (\theta)=0,\\
\end{split}
\label{eq.2.9}$$ which is very close to Eq. (\[eq.2.7\]) except that the $J$ is now position dependent.
With $J(\rho)$ in Eq. (\[eq.2.8\]) as an ansatz, we solve Eq. (\[eq.2.6\]) in the absence of $\tilde{B} $ term, using the numerical method in Ref. [@pin], which proposed a mechanism of pinning skyrmions. We choose the parameter values to be $J_0=1$, $\tilde{B}=0$, $\tilde{D}=0.05$, $b=0.005$, $a=0.05$. As shown in Fig. \[fig:ansatz\], the solution is very close to the skyrmion ansatz (\[eq.2.5\]), suggesting that it is possible to create skyrmions by using the pinning effect.
![The dotted line represents the numerical result of Euler-Lagrange Equation (\[eq.2.6\]) with $\tilde{B}=0$. The dotted-dashed line represents the skyrmion ansatz (\[eq.2.5\]).[]{data-label="fig:ansatz"}](ansatz){width="70.00000%"}
\[sec:4\]Lattice simulation
===========================
The above estimation with an artificial pinning effect suggests the possibility of really creating a skyrmion using pinning effect only. Since the effective parameter $\tilde{D}$ depends not only on $D$ but also on the parameters of the skyrmions, whether or not a solution of $\theta(\rho)$ can be identified as a skyrmion is technically subtle. Therefore, rather than solving the Euler-Lagrange equation, we perform a lattice simulation, which directly provides evidence of skyrmions.
We now consider more realistic pinning effect. As a local structure, the effect of pinning should be suppressed very quickly in deviating away from the pinning center. An exponentially decaying function $J(\rho)$ is assumed in Ref. [@pin2], while a Gaussian function $J(\rho)$ is assumed in Ref. [@pin]. We follow Ref. [@pin] to assume $J(\rho)$ to be Gaussian, $$\begin{split}
&J(\rho)=J_0+J_1e^{-J_2\rho^2},
\end{split}
\label{eq.3.1}$$ where $J_0$, $J_1$ and $J_2$ are undetermined coefficients, with $J_0>0$, $J_1>-J_0$ and $J_2>0$. The radius of the pinning is denoted as $R_p$, and $R_p \sim 1/\sqrt{J_2}$.
In the simulation, as dimensionless parameters, $J=1$ is used as the definition of the energy unit, $J(\infty)=J_0=1$ is assumed for simplicity [@unitJ; @alpha3; @bilayer]. The dimensionless quantities can be rescaled to physical ones as the following [@Tchoe; @unitJ; @pin]. The time unit $\Delta t = 1$ in the simulation represents $t=\hbar / J $ in physical time. The rescaling factor $r$ can be determined from the helical wavelength $\lambda$ and the lattice spacing $a$, as $r=(D/J)\lambda/(2\pi \sqrt{2}a)$. Then the time is rescaled as $t'=r^2 t$. For example, if we adopt the real material such that $\lambda \approx 60 \;{\rm nm}$ and $a\approx 4\;{\AA}$ and with $D/J=0.5$, we find $r\approx 8.44$. Then, if we adopt the energy unit as $J=3\;{\rm meV}$, then $t'=r^2\hbar/J \approx 220r^2\;{\rm fs} \approx 0.016\;{\rm ns}$.
The lattice simulation is based on the Landau-Lifshitz-Gilbert (LLG) equation [@nagaosa; @pin; @LLG; @LLG2] $$\begin{split}
&\frac{d}{dt} {\bf n}_{\bf r}=-{\bf B}_{\rm eff}({\bf r})\times {\bf n}_{\bf r}-\alpha {\bf n}_{\bf r}\times \frac{d}{dt} {\bf n}_{\bf r},\\
\end{split}
\label{eq.4.1}$$ where ${\bf n}_{\bf r}$ is the local magnetic moment at site ${\bf r}$, $\alpha$ is the Gilbert damping constant, ${\bf B}_{\rm eff}$ is the effective magnetic field, $$\begin{split}
&{\bf B}_{\rm eff}({\bf r})=-\frac{\delta H}{\delta {\bf n}_{\bf r}},
\end{split}
\label{eq.4.2}$$ with the discrete Hamiltonian [@unitJ; @discreteH] $$\begin{split}
&H=\sum _{{\bf r},i=x,y}\left[-J({\bf r}){\bf n}_{{\bf r}+\delta_i}-D({\bf r}) {\bf n}_{{\bf r}+\delta_i}\times {{\bf e}}_i-{\bf B}\right]\cdot {\bf n}_{\bf r},\\
\end{split}
\label{eq.4.3}$$ where $ \delta _{i}$ refers to each neighbour, and $\delta _{i}= {\bf e}_{i}$ on a square lattice. So [@pin] $$\begin{split}
&{\bf B}_{\rm eff}({\bf r})=\sum _{i=x,y}\left[J({\bf r}){\bf n}_{{\bf r}+\delta_i}+J({\bf r}-\delta_i){\bf n}_{{\bf r}-\delta_i}\right]\\
&+\sum _{i=x,y}\left[D({\bf r}){\bf n}_{{\bf r}+\delta_i}\times {{\bf e}}_i-D({\bf r}-\delta _i){\bf n}_{{\bf r}-\delta_i}\times {{\bf e}}_i\right]+{\bf B}({\bf r}).
\end{split}
\label{eq.4.4}$$
Unless specified otherwise, the simulation is run on a $512\times 512$ square lattice with open boundary condition and ${\bf B}=0$, and with the pinning center set to be at the point $(256,256)$. In the following, we denote the time step as $\Delta t$, and the time in unit of $\Delta t$ the simulation takes is denoted as $\tau$. The number of step is $\tau/\Delta t$. The simulation is run on the GPU, which has a great advantage over CPU on this problem. Because ${\bf n}_{{\bf r}}$’s for different sites at a same instant are independent of each other, we use GPU to do parallel computing. Within the simulation, the LLG is numerically integrated by using fourth-order Runge-Kutta method.
\[sec:4.1\] Skyrmion generation from random initial configurations
------------------------------------------------------------------
We first run the simulation for $J_1=3$, $J_2=0.001$ and various values of $D$. The simulation starts from randomized ${\bf n}_{\bf r}$’s and stops when ${\bf n}_{\bf r}$’s become stable. Previously, the Gilbert constant is taken to be $\alpha=0.01$ to $1$ [@pin; @LLG2; @antiskyrmion; @arctan; @alpha1; @alpha2; @alpha3; @bilayer]. We find that the larger the value of $\alpha$, the more rapid the simulation is completed, while the smaller the value of $D$, the longer the simulation time. Hence we use $\alpha=0.1$ [@LLG2; @alpha2] and $\Delta t=0.002$ for $D\geq 0.1$, while $\alpha=0.2$ [@pin; @alpha3] and $\Delta t=0.01$ for $D< 0.1$.
The results are shown in Fig. \[fig:results1\]. We run the simulation for $D=1, 0.8, 0.5, 0.2, 0.1, 0.08, 0.05, 0.03, 0.02, 0.01$. For each value of $D$ except the smallest ones $D=0.02$ or $0.01$, a skyrmion can be generated at the pinning center. When $D$ is smaller, it takes longer time to generate the skyrmion. Generally speaking, the time needed to generate a skyrmion by using pinning is longer than generation of a skyrmion by using an external magnetic field.
We have also studied the properties of the skyrmions generated in the simulation. Only those skyrmions near the pinning centers are considered. For each skyrmion considered, $g$ is determined from $n_z$ at the center. The radius $R_s$ of each skyrmion can be determined from the iso-height contour with $n_z=\mp 1$ for $g=\pm 1$ respectively, because $n_z$ varies from $\pm 1$ at the center to $\mp 1$ on the edge. In the actual simulation, however, $n_z$ can only approach $\mp 1$. Hence, $R_s$ is estimated from the radius of the iso-height contour with $n_z=\mp 0.9$ for $g=\pm 1$, respectively. $\gamma$ is determined by using $n_x$ and $n_y$ at the iso-height contour with $n_z=0$.
We first investigate the relation between the skyrmion radius $R_s$ and the DM interaction strength $D$. We find that the larger the value of $D$, the smaller $R_s$. As shown in Fig. \[fig:rsrelation\], the relation between $R_s$ and $D$ is about $$a+b\frac{1}{\sqrt{D}},$$ which explains why skyrmions are not generated for $D=0.02$ or $D=0.01$, as for small values of $D$, the radius of the skyrmion is too large for the lattice size.
![The relation between skyrmion radius $R_s$ and the DM interaction strength $D$.[]{data-label="fig:rsrelation"}](rsrelation){width="70.00000%"}
All the skyrmions appearing in the simulation are of $m=1$. Then the skyrmion number $Q=-mg$ is determined from the sign of $g$. As shown in Fig. \[fig:results1\], skyrmions with $g= 1$ and $g=-1$ can both be generated. For a skyrmion generated in an external magnetic field, $g$ is determined by the direction of ${\bf B}$. But for a skyrmion generated by using the pinning effect in absence of a magnetic field, the sign of $g$ becomes a free choice and depends on the initial state. This is verified in simulations starting with different randomized initial states but with the same values of various parameters. For $D>0$, we find that $\gamma \approx \pi/2$ for $g=1$, while $\gamma \approx -\pi/2$ for $g=-1$, hence $\tilde{D}>0$ in both cases. This can be understood by considering $$\begin{split}
&H_{\rm DM}=D {\bf n}\cdot \left(\nabla \times {\bf n}\right)=gD\sin \left((m-1)\phi+\gamma\right)\left(\frac{m}{2\rho}\sin (2\theta (\rho))+\theta '(\rho)\right), \\
\end{split}
\label{eq.4.5}$$ which differs from the special case of $g=1$ [@nagaosa] in replacing $D$ as $gD$. As a result, the sign of $g$ is a free choice, while the sign of $\gamma$ is determined by $gD$. Hence the energy is lowest when $\tilde{D} >0 $ and $\gamma = \pm \pi / 2$ with the sign of $\gamma$ determined by $gD$, that is, $\gamma = \pi / 2$ when $gD>0$, while $\gamma = - \pi / 2$ when $gD<0$.
We have also studied the case of $J_1<0$. Using $J_1=-0.5$, $J_2=0.0001$, $\alpha=0.2$, and $\Delta t=0.01$, skyrmions are generated in the simulation (Fig. \[fig:results3\]). In this case, the range of the value of $D$ in which skyrmions can be generated is narrower than the case of $J_1=3$ and $J_2=0.001$. Similar to the case of $J_1>0$, the radius of the skyrmion also increases with the decrease of $D$, and $\tilde{D}>0$ for all the skyrmions generated.
\[sec:4.2\] Skyrmion generation from helical phase
---------------------------------------------------
We have also studied a more realistic situation. Starting with the helical phase, we suddenly switch on the pinning, by substituting the constant $J$ with an inhomogeneous $J(\rho)$. In previous works, $D/J\approx 0.09\sim 0.5$ [@Tchoe; @pin; @unitJ; @alpha3; @bilayer; @LLG2; @discreteH; @dvalue], therefore we use $D/J_0=0.08,0.1,0.2,0.3,0.5$. In absence of an external magnetic field and with constant $J$, the initial state is prepared by relaxing from a saturated state with ${\bf n}_{\bf r}={\bf e}_z$ until the stability is reached. Thus the helical phase is obtained. Then we substitute the constant $J$ with inhomogeneous $J(\rho)$ at $\tau = 0$ and continue the simulation. For $D=0.08$, we use $\alpha = 0.2$ and $\Delta t = 0.01$. For $D=0.1,0.2,0.3,0.5$, we use $\alpha =0.1$ and $\Delta t = 0.002$.
The results for $D=0.5$, $J_1=3$ and $J_2=0.0001$ are shown in Fig. \[fig:impD05\]. The strip width increases with the decrease of $D/J$. So the stripes are wider near the pinning center, where $D/J$ is smaller. The stripes start to grow at $\tau = 0$. Then a kind of turbulence is created, and small skyrmions are generated, some of which finally become stable.
The cases for $D=0.08,0.1,0.2,0.3$ are shown in Fig. \[fig:imps\]. For $D=0.2, 0.3$, we still use $J_2=0.0001$, as in the case of $D=0.5$. The size of the pinning should be much larger than the width of the stripes. Hence for smaller values of $D$, $J_2$ is chosen to be smaller, consequently the lattice is set to be larger. Therefore, for $D=0.08, 0.1$, we run the simulation on a $1024\times 1024$ lattice. We choose $J_2=0.000025$ and set the pinning center to be at $(512,512)$. For all these cases, skyrmions are generated in the pinning region (Fig. \[fig:imps\]), although they are not at the pinning center, as in the cases starting with randomized initial state.
For $D=0.3$, we run the simulation on a $1024\times 1024$ lattice, with $5$ pinning centers locating at sites ${\bf r}_1= (512,512)$, ${\bf r}_2=(256,512)$, ${\bf r}_3=(768,512)$, ${\bf r}_4=(512,256)$ and ${\bf r}_5=(512,768)$, respectively, $$\begin{split}
&J({\bf r})=1+3\sum _{i=1}^5 \exp \left(-0.00025 ({\bf r} -{\bf r} _i)^2\right),
\end{split}
\label{eq.4.6}$$ with $J_2=0.00025$. The result (Fig. \[fig:impD03\]) shows that skyrmions are generated near some but not all of the pinning centers.
We also find that, for $J_1<0$, skyrmions cannot be generated by using this method.
\[sec:4.3\] Skyrmion generation with both J and D inhomogeneous
----------------------------------------------------------------
Although the origins of the ferromagnetic exchange and DM interaction are different, it is difficult to vary $J$ while keeping $D$ constant in real experiments. We also simulate a scenario in which both $J$ and $D$ inhomogeneous while $D/J$ constant, that is, $D$ has the same $\rho$-dependence as $J$ [@pin]. When $D/J$ is homogeneous, the widths of the stripes are not changed. In this case, skyrmions cannot be generated from helical phases. However, skyrmions can still be generated at the pinning center through the pinning effect with the aid of the boundary condition, in the following way.
In our simulation, when an external magnetic field is applied, ${\bf n}_{\bf r}$ can be saturated to ${\bf n}_{\rm r}={\bf e}_z$ no matter whether pinning is presented. If we suddenly switch off the external magnetic field and start the simulation from a saturated initial configuration, the boundary condition plays an important role. Because of DM interaction, ${\bf n}_{\bf r}$ tends to tilt to its neighbours. If $D$ is homogenous, the tilt starts from the sites on the boundary (Fig. \[fig:tilt\]). When $D$ is inhomogeneous, the tilt starts from both the boundary and the sites with inhomogeneous $D$. This can be understood by using LLG in Eq. (\[eq.4.2\]), which can be rewritten as $$\begin{split}
&\frac{d{\bf n}}{dt}=\frac{1}{1+\alpha^2}\left({\bf N}+\alpha {\bf N}\times {\bf n}\right),
\end{split}
\label{eq.4.7}$$ with ${\bf N}={\bf B}_{\rm eff}\times {\bf n}$. When ${\bf n}_{\bf r}={\bf e}_z$ and $D$ is homogeneous, ${\bf B}_{\rm eff}\propto {\bf n}_{\bf r}$ and ${\bf N}=0$. However, when $D$ is inhomogeneous, ${\bf N}\neq 0$, therefore ${\bf n}_{\bf r}$ of the sites at the boundary or at the sites with inhomogeneous $D$ start to tilt first.
![When the simulation starts from ${\bf n}_{\bf r}={\bf e}_z$, the boundary plays an important role. With DM interaction, ${\bf n}_{\bf r}$ at a site tends to tilt to its neighbours. If $D$ is homogenous, the tilt starts from the boundary.[]{data-label="fig:tilt"}](figtilt2){width="80.00000%"}
For $D/J=0.08$, we use $J_1=3$, $J_2=0.001$, $\alpha=0.2$, $\Delta t=0.002$. The result is shown in Fig. \[fig:satD008\]. Because of the inhomogeneous $D$, a ring domain wall is generated around the pinning and keeps shrinking to the center of the ring. The center of the ring is a skyrmion. The ring keeps shrinking until about $\tau = 200$, then the skyrmion at the center starts to grow until it is constrained by the domain walls generated from the boundary.
For $D/J=0.1$, $0.2$, $0.3$, $0.5$, with the same values of other parameters as for $D/J=0.08$, stable skyrmions cannot be generated, because the skyrmion generated at the center keeps expanding and becomes very large before it meets the domain walls generated from the boundary, so its border splits into stripes. However, if we use $J_1<0$, domain walls are generated as several rings around the pinning and expand, as shown in Fig. \[fig:satD03\] for $D/J=0.3$. Then if the pinning radius is close to the stripe widths, the skyrmions can be generated at the pinning center inside the smallest ring.
We run the simulation for $D/J=0.1,0.2,0.3,0.5$ with $\alpha=0.1,\Delta t=0.002$ and $J_2=0.00156,0.00625,0.012,0.045\sim 1/D^2$. For $J_1<0$, $J_1$ is related to the gap between the rings. For $D/J=0.1,0.2,0.3$, we use $J_1=-0.5$. For $D/J=0.5$, the gap is too large compared with the stripe widths, so we use $J_1=-0.85$ to make gap narrower. The result for $D/J=0.3$ is shown in Fig. \[fig:satD03\] and the results for $D/J=0.1,0.2,0.5$ are shown in Fig. \[fig:sats\].
Comparing the final states in Sec.\[sec:4.3\] with the initial states in Sec.\[sec:4.2\], we can conclude that, with boundary condition alone, the skyrmions are not generated at the center. Therefore, the effect of the pinning is essential. We also run a simulation to testify how important the boundary condition is. In real experiments, the material could be much larger, and the pinning could be located away from the center of the material. We run a simulation on a $2048\times 2048$ lattice with the pinning located at $(1200,700)$ to investigate the generation of the skyrmion in this situation. We choose $D/J=0.2$, and follow the rescaling method in Refs. [@Tchoe; @pin]. If for the real material, $\lambda \approx 60 \;{\rm nm}$, $a\approx 4\;{\AA}$, the rescaling factor is $r\approx 3.38$, therefore, a $2048 \times 2048$ lattice corresponds to about $2.77\;{\rm \mu m}\times 2.77\; {\rm \mu m}$. For $J=3 \;{\rm meV}$, the dimensionless time $\tau=15000$ corresponds to $37.6\; {\rm n s}$. We run the simulation with $\alpha = 0.2$, $\Delta t =0.01$, $J_1=-0.8$ and $J_2=0.005$. The result is shown in Fig.\[fig:satLarge\]. We find that, a skyrmion is generated on the left of the pinning center, and the distance between the skyrmion center and the pinning center is about $50$, corresponding to $67.5\;{\rm nm}$ if we use the above realistic parameter values.
\[sec:4.4\]Bound states
-----------------------
In the simulation starting from randomized initial state, we also observe an interesting phenomenon when $D$ is constant and very small while $J_1<0$ (Figs. \[fig:special1\] and \[fig:special2\]). It does not show up when $J_1>0$ or $J_1=0$ (no pinning), so it is a special case in presence of pinning with $J_1<0$. When $D$ is very small, a skyrmion and an anti-skyrmion with $g=\pm 1$ and $m=\pm 1$ are generated and move to the pinning center. They keep rotating around each other with the distance shrinking till annihilation. Previously, it was found that the interaction between two skyrmions on two layers with opposite skyrmion charge can form a bound state [@bilayer].
On a single layer, in contrast, if the skyrmions are generated in an external magnetic field, the sign of $g$ is determined by the external magnetic field, and whether a skyrmions or an anti-skyrmions is generated depends on $D$, then the bound states of skyrmions and anti-skyrmions are usually difficult to realize. The situation we consider may provide a novel avenue to study such bound states in a single layer.
\[sec:5\] Summary
==================
In this paper, we propose a novel mechanism to generate magnetic skyrmions without the need of an external magnetic field or magnetic anisotropy. We find that skyrmions can be generated through the pinning effect only, i.e., with magnetic exchange strength $J$ inhomogeneous, or with $J$ and DM interaction $D$ both inhomogeneous. Our lattice simulation has verified this idea. In the simulation, we study the properties of the skyrmions generated under various parameter values. We find that the radius of the skyrmion increases when $D$ decrease. We also find that all the skyrmions generated have $m=1$ and $\tilde{D}>0$, while the sign of $g$ depends on the initial state. For $J_1<0$, we also find the generation of a pair of skyrmions with opposite $g$ and $m$ at the pinning centers.
That the skyrmions can be generated by using the pinning effect only is useful for practical applications of the magnetic skyrmions. Through the engineering of pinning in the designated site, we can generate a skyrmion on this site. It is hoped that experiments and applications be made by using this method.
This work is supported by National Natural Science Foundation of China (Grant No. 11374060 and No. 11574054).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate a model that the Yukawa coupling form is constructed by two kinds of matrix ($M_0$ and $M_1$). For example, in the $SO(10)$ GUT model, $M_0$ and $M_1$ are Yukawa couplings generated by the $10$ and $126$ Higgs scalars. We study how this model can give the observed mass and mixings of quarks and leptons. Parameter fitting is fully scanned by assuming all the input data to be normally distributed around the center value.'
address: ' Graduate school of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan'
author:
- Koichi MATSUDA
title: 'Phenomenological analysis of lepton and quark Yukawa couplings in SO(10) two Higgs model'
---
Introduction
============
The grand unification theory (GUT) is very attractive as an unified description of the fundamental forces in the nature. However, in order to reproduce the observed quark and charged-lepton masses and mixings, a lot of Yukawa couplings are usually brought into the model. We think that the nature is simple. So it is the very crucial problem to know the minimum number of Yukawa couplings which can give the observed fermion mass spectra and mixings. However, if the quark and lepton Yukawa couplings are composed by only one matrix $$M_u = c_u M_0 , \quad M_d = c_d M_0 , \quad M_e = c_e M_0,$$ the CKM Matrix must be diagonalized, these model is obviously ruled out for the description of realistic quark and lepton mass spectra. Therefore, at the unification scale $\mu=\Lambda_X$, we assume the Yukawa coupling of up quark, down quark and charged-lepton ($M_u^0$, $M_d^0$ and $M_e^0$) are composed by two matrices, $$M_f^0= c_{f0} M_0^0+ c_{f1} M_1^0.
\qquad (f=u,\ d,\ e%,\ \nu_D \mbox{ and } \nu_R
)\label{eq90101}$$ Here, $c_{f0}$ and $c_{f1}$ are real numbers which can be associated with the vacuum expectation values (VEV). For example, in the SO(10) GUT model with one 10 and one 126 Higgs scalars, the Yukawa couplings of quarks and charged leptons are expressed in the following forms [@twohiggs1][@twohiggs2]: $$\begin{aligned}
M_u^0&=&c_0M_0^0+c_1M_1^0,~ M_d^0=M_0^0+M_1^0,~ M_e^0=M_0^0- 3 M_1^0%, \\
. \label{eq110501}
%M_{\nu_D}&=&c_0M_0-3 c_1M_1,~ M_{\nu_R}=c_R M_1,~
%M_\nu=M_{\nu_D} M_{\nu_D}^{-1} M_{\nu_D}^T.\label{eq120901}\end{aligned}$$ where $M_0^0$ and $M_1^0$ are symmetric Yukawa couplings. In the previous paper [@twohiggs1], eliminating $M_0^0$ and $M_1^0$ from Eq.(\[eq90101\]), we obtain the relation $$M_e^0 = c_u M_u^0 + c_d M_d^0 \equiv c_u (M_u^0 + \kappa M_d^0), \label{eq82515}$$ where $$c_d=\frac{c_{u0}c_{e1} - c_{e0} c_{u1}}{c_{u0}c_{d1} - c_{d0} c_{u1}}
\qquad \mbox{ and } \qquad
c_u=\frac{c_{e0} c_{d1} - c_{d0} c_{e1}}{c_{u0}c_{d1} - c_{d0} c_{u1}}.$$ These relations are realized at the GUT scale, but each value of the Yukawa couplings is given by the experiment at the weak scale $\mu=m_Z$. Therefore, we must investigate how the mass ratios and CKM matrix parameters change from $\mu=\Lambda_X$ down to $\mu=m_Z$. [@evol] In this paper, we distinguish between the values at $\mu=\Lambda_X$ and $\mu=m_Z$ by using the superscript “0” or not.
Numerical study
===============
Because $M_u^0$, $M_d^0$, and $M_e^0$ are symmetric at the unification scale $\mu=\Lambda_X$ in the model with one 10 and one 126 Higgs scalars, they are diagonalized by unitary matrices $U_u^0$, $U_d^0$, and $U_e^0$, respectively, as $$U_u^{0\dagger}M_u^0 U_u^{0\star}=D_u^0 \ , \ \
U_d^{0\dagger}M_d^0 U_d^{0\star}=D_d^0 \ \mbox{\ and \ }
U_e^{0\dagger}M_e^0 U_e^{0\star}=D_e^0 \ , \label{diag}$$ where $D_u^0$, $D_d^0$, and $D_e^0$ are diagonal matrices which are given by $$\begin{aligned}
D_u^0 &\equiv& \frac{1+\tan\beta^{-2}}{v^2} \mbox{diag}(m_u^0,m_c^0,m_t^0) \ , \ \
D_d^0 \equiv \frac{1+\tan\beta^{ 2}}{v^2} \mbox{diag}(m_d^0,m_s^0,m_b^0), \nonumber\\
D_e^0 &\equiv& \frac{1+\tan\beta^{ 2}}{v^2} \mbox{diag}(m_e^0,m_\mu^0,m_\tau^0) \ ,
\label{diag}\end{aligned}$$ Here, $v$($=174$ GeV) is VEV of Higgs , and it is divided into up and down quark (neutrino and charged lepton) in the ratio $\tan\beta$. Using the Cabibbo-Kobayashi-Maskawa (CKM) matrix $V_q^0$ which is expressed as $V_q^0=U_u^{0\dagger} U_d^0$, the relation (\[eq82515\]) is rewritten as follows: $$(U_e^{0\dagger}U_u^0)^\dagger D_e^0(U_e^{0\dagger}U_u^0)^\star
= c_uD_u^0
+ c_{d}V_q^0 D_{d}V_q^{0 T}
= c_u(D_u^0
+ \kappa V_q^0 D_{d}V_q^{0 T}).\label{eq071703}$$ We take a basis on which the up-quark Yukawa coupling is diagonal in order to compare with the experiment values and obtain the independent two equations: $$\begin{aligned}
A \left( \kappa \right)
&\equiv& \frac{{\left( {\left( {m_e^0 m_\mu ^0 } \right)^2 + \left( {m_\mu ^0 m_\tau ^0 } \right)^2
+ \left( {m_\tau ^0 m_e^0 } \right)^2 } \right)}}{{\left( {\left( {m_e^0 } \right)^2
+ \left( {m_\mu ^0 } \right)^2 + \left( {m_\tau ^0 } \right)^2 } \right)^2 }}
\frac{{2\left[ {{\mathop{\rm Tr}\nolimits}
\left\{ {H_q\left( \kappa \right)} \right\}} \right]^2 }}
{{\left\{ {{\mathop{\rm Tr}\nolimits} \left( {H_q\left( \kappa \right)} \right)} \right\}^2
- {\mathop{\rm Tr}\nolimits} \left\{ {\left( {H_q\left( \kappa \right)} \right)^2 } \right\}}}
\to 1 \nonumber \\
B\left( \kappa \right) &\equiv& \frac{{\left( {m_e^0 m_\mu ^0 m_\tau ^0 } \right)^2 }}
{{\left( {\left( {m_e^0 } \right)^2 + \left( {m_\mu ^0 } \right)^2
+ \left( {m_\tau ^0 } \right)^2 } \right)^3 }}
\frac{{\left[ {{\mathop{\rm Tr}\nolimits}
\left\{ {H_q\left( \kappa \right)} \right\}} \right]^3 }}
{{\det \left\{ {H_q\left( \kappa \right)} \right\}}} \to 1
\label{eq1012-02}\end{aligned}$$ Here $H_q(\kappa)$ is the following hermite matrix which is defined by the Yukawa couplings of quark: $$H_q(\kappa) \equiv (D_u^0+{\kappa}V^0D_{d}^0V^{0\dagger})
(D_u^0+{\kappa}V^0D_{d}^0V^{0\dagger})^\dagger .$$ If we find the $\kappa$ which sets $A(\kappa)$ and $B(\kappa)$ to $1$ simultaneously, the three Yukawa couplings $M_u^0$, $M_d^0$ and $M_e^0$ can be unified into two matrices. However we don’t know precisely how to determine these data, especially quark masses. And above procedures depend on these ambiguities. So in this paper, we substitute the random numbers which becomes following normal distributions [@expmass]: $$\begin{aligned}
\left| {m_u \left( {2{\rm{GeV}}} \right)} \right| &=& 2.9 \pm 0.6{\rm{MeV}}, \quad
\left| {m_d \left( {2{\rm{GeV}}} \right)} \right| = 5.2 \pm 0.9{\rm{MeV}}, \label{eq011901} \\
\left| {m_s \left( {2{\rm{GeV}}} \right)} \right| &=& 80 - 155{\rm{MeV}}, \quad
\left| {m_c \left( {m_c } \right)} \right| = 1.0 - 1.4{\rm{GeV}},\\
m_b \left( {m_b } \right) &=& 4.0 - 4.5{\rm{GeV}}, \quad
m_t^{{\rm{direct}}} = 174.3 \pm 5.1{\rm{GeV}},\end{aligned}$$ $$\begin{aligned}
\left| {m_e^{{\rm{pole}}} } \right| &=& {\rm{0.510998902}} \pm {\rm{0.000000021MeV}},\\
\left| {m_\mu ^{{\rm{pole}}} } \right| &=& {\rm{105.658357}} \pm {\rm{0.00005,}} \quad
m_\tau ^{{\rm{pole}}} = {\rm{1776.99}} \pm {\rm{0.29MeV}} \end{aligned}$$ $$\begin{aligned}
\sin \theta _{12} &=& 0.2229 \pm 0.0022, \quad
\sin \theta _{23} = 0.0412 \pm 0.0020,\\
\sin \theta _{13} &=& 0.0036 \pm 0.0007, \quad
\delta = \left( {59 \pm 13} \right)^\circ \label{eq011902}\end{aligned}$$ for each mass and CKM mixing parameter 10,000 times. And we estimate the evolution effect about the values in Eqs. (\[eq011901\]) - (\[eq011902\]) from $\mu=m_Z$ to $\mu=\Lambda_X$ by using of RGE.[@evol] In this work, we suppose MSSM for $\mbox{tan}\beta=10$. Without loss of generality, we can make the masses of third generation positive real number. Although the remaining masses are complex under the ordinary circumstances, we assume that all masses are real in order to simplify the problem. Therefore, there are 16 combinations of the signs of the masses as shown in table \[tab1\].
As shown in Fig.1, we scan the range $A(\kappa)=1$ by changing $\mbox{Im}(\kappa)$ from -100 to 100 at 2000 equal intervals. Moreover, we get the maximum and minimum of $B(\kappa)$ on the line of $A(\kappa)=1$ by changing $\mbox{Im}(\kappa)$ at 5000 equal intervals. Because $B(\kappa)$ is continuous, there is the $\kappa$ which sets $A(\kappa)$ and $B(\kappa)$ to $1$ simultaneously when $\mbox{Min}(B(\kappa))<1<\mbox{Max}(B(\kappa))$ as explained in Fig.2. In this way, we draw the histograms in Figs 3,4,5 and 6 which show the distribution of input values conforming to the requirements $\mbox{Min}(B(\kappa))<1<\mbox{Max}(B(\kappa))$. The each summation of the conforming case is tabulated in Table 2 after the 10,000 substitutions. Expressed in another way, Table 2 shows the number of dots in the white area in Fig.2. In Fig. 7, each circle in the complex plane shows the value of $\kappa$ to meet the requirement $A(\kappa)=B(\kappa)=1$, and the total number of circle in each figure corresponds to the number in Table 2, obviously. From these figures and tables, it is understandable that the sign of $m_u$ is not important. Perhaps the reason is that $m_u$ is very small, and it is almost negligible in comparison with other masses.
conclusion and discussion
=========================
In conclusion, we have discussed the probability that the following model will be realized without fine tuning. The random numbers which become normal distributions have been substituted for each physical value at $\mu=m_Z$. And we have taken the RGE effect between $\mu=m_Z$ and $\Lambda_X$ into consideration. In this way, the search for $\kappa$ which sets $A(\kappa)$ and $B(\kappa)$ to 1 simultaneously has been repeated 10,000 times. By this way, we have arrived at three conclusions: (1) The probability that the model will be realized without fine tuning is about $5\%$ if we select the appropriate signs (14) or (15) of the masses. (2) This probability will increase if the signs of $m_d$, and $m_s$ are same. This gives the suggestion to the texture model. For example, a model with a texture $(M_d)_{11}=0$ on the nearly diagonal basis of the up-quark Yukawa coupling $M_u$ is denied because these model leads to $m_d/m_s < 0$. (3) From Fig.3-Fig.6, this probability will increase if we make $m_s$ somewhat larger or smaller than the present experiment value properly.
In the present paper, we have demonstrate that the quark and charged lepton Yukawa coupling can be unified into only two matrices. However, we have not referred to the neutrino masses and lepton flavor mixings. The neutrino Yukawa coupling is given by $$\begin{aligned}
M_D^0 &=& \frac{{c_{e0} }}{{c_{d0} }}c_{u0} M_0^0 + \frac{{c_{e1} }}{{c_{d1} }}c_{u1} M_1^0, \\
M_\nu ^0 &=& c_{R0}^{ - 1} M_D^{0} M_1^{0 - 1} M_D^{0\, T}.\end{aligned}$$ Concerning this problem, we have not been able to find the positive solutions within $3\sigma$ which is written by the paper [@neutrino] for the present. However, since there are many possibilities for the neutrino mass generation mechanism, we are optimistic about this problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author is grateful to Y.Koide, H.Fusaoka, T.Fukuyama, T.Kikuchi and H.Nishiura for the useful comments. This work is supported by the JSPS Research Fellowships for Young Scientists, No.3700.
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($m_u$, $m_c$, $m_t$) ($m_d$, $m_s$, $m_b$) ($m_u$, $m_c$, $m_t$) ($m_d$, $m_s$, $m_b$)
----- ----------------------- ----------------------- ------ ----------------------- -----------------------
(0) $(+++)$ $(+++)$ (8) $(+++)$ $(+-+)$
(1) $(-++)$ $(+++)$ (9) $(-++)$ $(+-+)$
(2) $(+-+)$ $(+++)$ (10) $(+-+)$ $(+-+)$
(3) $(--+)$ $(+++)$ (11) $(--+)$ $(+-+)$
(4) $(+++)$ $(-++)$ (12) $(+++)$ $(--+)$
(5) $(-++)$ $(-++)$ (13) $(-++)$ $(--+)$
(6) $(+-+)$ $(-++)$ (14) $(+-+)$ $(--+)$
(7) $(--+)$ $(-++)$ (15) $(--+)$ $(--+)$
: The combinations of the signs of ($m_u$, $m_c$, $m_t$) and ($m_d$, $m_s$, $m_b$). The signs of the charged lepton are negligible in Eq.(\[eq1012-02\]). []{data-label="tab1"}
sum sum sum sum
----- ----- ----- ----- ------ ----- ------ -----
(0) 344 (4) 34 (8) 56 (12) 283
(1) 328 (5) 30 (9) 60 (13) 294
(2) 225 (6) 35 (10) 54 (14) 470
(3) 209 (7) 35 (11) 56 (15) 482
: The total number of the cases conforming to the requirements Min$(B(\kappa))$ $<1<$ Max$(B(\kappa))$ after the 10,000 substitutions.[]{data-label="tab2"}
![The relations in Eq.(\[eq1012-02\]) on the complex plane of $\kappa$. The solid line show $A(\kappa)=1$ and the dotted line $B(\kappa)=1$. This is an example which is given as follows: $D_u^0$ = diag($-4.5116*10^{-6}$, $-0.0011789$, $0.5028 $), $D_d^0$ = diag($-5.6008*10^{-5}$, $-0.0007776$, $0.038776$), $D_e^0$ = diag($ 1.8697*10^{-5}$, $ 0.0039461$, $0.067375$), $\theta_{12}^0$ = $0.22695$, $\theta_{23}^0$ = $0.035057$, $\theta_{31}^0$ = $0.0023936$, and $\delta^0$ = $1.3173$. []{data-label="fig1"}](ron2bit.eps){width="5cm"}
![The maximum and minimum of $B(\kappa)$ on the line of $A(\kappa)=1$ in the case of (15) in Table 1. There are 10000 dots in all area, and 482 dots in the white area as tabulated in Table 2. []{data-label="fig2"}](ron1bit.eps){width="7cm"}
![The histograms show the distribution of data values conforming to the requirements $\mbox{Min}(B(\kappa))<1<\mbox{Max}(B(\kappa))$. Each number in parentheses show the signs of the mass in Table 1. We bins the data values into 20 equally spaced containers, and show the number of elements in each container as a bar graph. The vertical solid and dotted lines show the center value and range of error in Eqs.(\[eq011901\]) - (\[eq011902\]), respectively. []{data-label="fig3"}](fig100bit.eps){width="16cm"}
![The histograms show the distribution of data values as Fig.3. []{data-label="fig4"}](fig104bit.eps){width="16cm"}
![The histograms show the distribution of data values as Fig.3.[]{data-label="fig5"}](fig108bit.eps){width="16cm"}
![The histograms show the distribution of data values as Fig.3.[]{data-label="fig6"}](fig112bit.eps){width="16cm"}
![The distribution of $\kappa$ in the complex plane. Each circle shows the value of $\kappa$ to meet the requirement $A(\kappa)=B(\kappa)=1$[]{data-label="fig7"}](ron4bit.eps){width="16cm"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A three-step laser excitation scheme is used to make absolute frequency measurements of highly excited $n$F$_{7/2}$ Rydberg states in $^{85}$Rb for principal quantum numbers $n$=33-100. This work demonstrates the first absolute frequency measurements of rubidium Rydberg levels using a purely optical detection scheme. The Rydberg states are excited in a heated Rb vapour cell and Doppler free signals are detected via purely optical means. All of the frequency measurements are made using a wavemeter which is calibrated against a GPS disciplined self-referenced optical frequency comb. We find that the measured levels have a very high frequency stability, and are especially robust to electric fields. The apparatus has allowed measurements of the states to an accuracy of 8.0MHz. The new measurements are analysed by extracting the modified Rydberg-Ritz series parameters.'
address:
- '$^1$ School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK'
- '$^2$ Max Planck Institute of Quantum Optics, Hans Kopfermann Str. 1 85748 Garching, Germany'
author:
- 'L A M Johnson$^1$, H O Majeed$^1$, B Sanguinetti$^1$, Th Becker$^2$ and B T H Varcoe$^1$'
bibliography:
- 'LukeJohnson.bib'
title: 'Absolute frequency measurements of $\mathbf{^{85}}$Rb ***n***F$\mathbf{_{7/2}}$ Rydberg states using purely optical detection'
---
Introduction {#intro}
============
The accurate measurement of highly excited Rydberg level energies in the alkali atoms plays an important role in improving the accuracy of atomic models [@drake1991]. In most Rydberg spectroscopy experiments the atoms are detected via field ionization. However, in this study we use a method of purely optical detection in an ordinary vapour cell, which has been demonstrated in [@brandenberger2002; @Mohapatra2008; @thoumany2009; @kubler10]. A vapour cell is a convenient and straightforward solution for finding Rydberg levels, that could potentially permit rapid advances in Rydberg spectroscopy. This technique presents a method of finding Rydberg states quickly, with a large signal to noise ratio and an apparent insensitivity to electric fields [@Mohapatra2008; @thoumany2009], which makes it particularly suited to studying high $\ell$ Rydberg states with large polarisabilities. It is therefore important to verify the ability to perform precision spectroscopy in such a setup.\
Although there is a large body of work on precision interval and fine structure measurements of the different rubidium Rydberg series’ [@harvey77; @meschede87; @li2003; @han2006; @afrousheh06], measurements of the absolute energies of these levels are more difficult to carry out, and are therefore mainly limited to the lower $\ell$ states [@lee78; @stoicheff79; @Lorenzen83; @sanguinetti2009]. It appears that absolute measurements of the $^{85}$Rb $n$F series have only been made once by Johansson in 1961 [@johansson61] for $n$= 4-8. However, as new tools are now available in laser spectroscopy, such as the optical frequency-comb technique, it is interesting to return to such measurements. In this work we wanted to demonstrate that precision laser spectroscopy measurements of Rydberg states could be effectively made using purely optical detection with a vapour cell sample.\
During the experiment $n$F$_{7/2}$ Rydberg states between $n$=33-100 were excited in $^{85}$Rb using a three step laser excitation scheme identical to that outlined in [@thoumany2009; @sanguinetti2009]. The three step level system, shown in figure \[level\], consists of a 780.24nm transition 5S$_{1/2}$ $F=3$ to 5P$_{3/2}$ $F=4$, a 775.98nm transition 5P$_{3/2}$ $F=4$ to 5D$_{5/2}$ $F=5$ and finally a 1260nm transition 5D$_{5/2}$ to $n$F$_{7/2}$.\
![The three step level scheme used to excite $^{85}$Rb $n$F$_{7/2}$ Rydberg states in this experiment.[]{data-label="level"}](figure1){width="4.5cm"}
To observe excitations to Rydberg states, the first two step lasers are fixed at their respective transition frequencies and the absorption of the 780nm laser is monitored whilst the 1260nm laser is swept across the transition of interest. This technique involves the quantum amplification effect; due to the large differences in decay lifetimes of the three excited states of the system, the excitation of a single atom by the third step laser will hinder many absorption-emission cycles on the second step transition. This in turn will hinder a large amount of cycles on the strong first step cycling transition which can cause a measurable decrease in the first step absorption. Even for Rydberg atoms confined in a room temperature vapour cell, with the associated limitations of interaction time and interatomic collisions, this amplification factor can be large enough to observe significant changes in absorption [@thoumany2009].\
Optical pumping is applied on all three steps with $\sigma^{+}$ polarised light. Optical pumping on the first step transition ensures the second step laser only excites to the $m_{F}=5$ sublevel of the 5D$_{5/2}$ $F$=5 hyperfine state. Therefore the third step laser can only excite a single transition, the 5D$_{5/2}$ $F$=5 to $n$F$_{7/2}$ $F$=6. Having a well defined pathway to the Rydberg states is important because of the relatively small $\sim$10MHz hyperfine splitting of the 5D$_{5/2}$ level [@Nez1993].
Apparatus {#app}
=========
![The experimental setup used for measuring Rydberg state frequencies. The first step is phase locked to a self-referenced optical frequency comb and the second step is frequency locked using a separate rubidium reference cell. The first and third step laser light is transported to the comb and wavemeter using single mode optical fibres.[]{data-label="Setup"}](figure2){width="8cm"}
In the experimental setup, shown in figure \[Setup\], all three steps are excited using commercial tunable external cavity diode lasers, and associated electronics. The third step laser is broadly tunable across a range of 110nm using a precision stepper motor, this allows a large range of $n$F states to be accessed. The first and third step lasers are superimposed and co-propagate through a rubidium vapour cell of length 80mm. The second step laser travels through the same cell; it counter-propagates and overlaps with the first and third step lasers. Absorption of the first step laser is monitored using a conventional photodiode as the third step laser is swept across the 5D$_{5/2}$ to $n$F$_{7/2}$ transition of interest. Removal of the first step laser from the other two laser paths is carried out using a polarising beam splitter. The first two steps are circularly polarised using quarter wave plates, and the third step laser is circularly polarised using a broadband Fresnel rhomb. All three lasers are focused to a beam waist of $\sim$100$\mu$m inside the cell, which increases the available third step laser power density. The vapour cell is heated to a temperature of 60$^{\circ}$C to increase the atomic density in the cell and to therefore enhance the first step absorption.\
In this experiment the first step laser Doppler selects those atoms which take part in the subsequent excitations, therefore it is important that the first step frequency is well known and well stabilised. Hence we stabilise this laser to a self-referenced frequency comb, by phase locking the beat note between the laser and a comb line to a stable direct digital synthesiser. The frequency comb repetition rate is adjusted such that the laser frequency is stabilised to 384229242.8 MHz, corresponding to the first step transition frequency from reference [@barwood1991]. All locking circuits are referenced to a GPS disciplined rubidium frequency standard. The comb system allows laser frequencies to be measured with an absolute accuracy of 10$^{-11}$. Fast feedback for the offset lock is supplied using a field-effect transistor connected to the laser diode. The stability of the first step lock was measured as less than 100Hz over all time scales relevant to this experiment. However, the absolute accuracy is limited to the measurement uncertainty of 750kHz from Barwood *et al* [@barwood1991].\
Before adding the third step laser to the system, we verified that efficient optical pumping was occurring on the first step transition by scanning the second step laser across the 5D$_{5/2}$ manifold, with the first step laser locked. The first step laser selects only zero velocity atoms, and therefore the second step laser scan showed a single and symmetric Doppler free peak in the first step absorption. This single peak, with a FWHM of 11.5MHz, corresponds to the reduced absorption of the first step laser as the second step laser excites the 5P$_{3/2}$ $F=4$ to 5D$_{5/2}$ $F=5$ transition. To confirm this we measured the absolute frequency of this transition using our frequency comb and added it to the first step locked frequency to get 770570284(1)MHz. This agrees with 770570284734(8)kHz from [@Nez1993], obtained from two photon spectroscopy. This therefore demonstrates that the pathway to the Rydberg states is well understood. This scheme is also used to stabilise the second step laser with a separate room temperature vapour cell. By adding a small frequency modulation to the second step laser, and monitoring the first step absorption via a lock-in amplifier, an error signal is extracted. Using our frequency comb we verified that this second step frequency lock was repeatable to an absolute accuracy of 1MHz on a daily basis.\
We found that it is possible to detect lower $n$ states with a very good signal to noise ratio. Therefore to verify the line shape of the detected third step transitions the photodiode was monitored directly on an oscilloscope during a fast scan across the 5D$_{5/2}$ to 33F$_{7/2}$ transition. The trace is displayed in figure \[Lorentzian\]. The scan was carried out in 10ms and the frequency axis was calibrated using a Fabry-Pérot resonator. The data fits a Lorentzian function with a linewidth of 20MHz.\
![A scan of the third step laser across the 33F$_{7/2}$ Rydberg state from an oscilloscope. The vertical axis is the first step transmitted intensity from the photodiode. The frequency axis was calibrated with a Fabry-Pérot resonator at 1268nm. The fitted curve is a Lorentzian with FWHM of 20MHz.[]{data-label="Lorentzian"}](figure3){width="7cm"}
To improve the detection sensitivity of third step transitions, a frequency modulation is added to the third step laser via the injection current, with a modulation amplitude of 15MHz and frequency of 6kHz. Detection of the first step absorption is carried out at the first harmonic using a lock-in amplifier with a time constant of 1 second. The free running third step laser is scanned by applying a linear voltage ramp to the laser Piezo using computer software and a Digital to Analogue converter interface. The free running laser stability was measured as less than 1MHz over one second, which is sufficient to carry out slow scans across the Rydberg transitions. As the third step laser is scanned, its absolute frequency is monitored using a WS7 High Finesse wavemeter. The wavemeter readings are recorded simultaneously using the same computer software.\
We used our frequency comb to check the wavemeter’s accuracy and stability across the range of third step laser wavelengths used in this experiment. We found that the wavemeter’s stability stayed below 2MHz for times of $\sim$1000s. We also found that the wavemeter was able to maintain a day-to-day absolute accuracy of 6.2MHz across the 1254nm-1268nm range, when regularly calibrated at 780nm. Therefore, throughout this experiment the wavemeter is calibrated every 30 minutes to the comb-locked first step laser, to supply a direct frequency link with the comb.
Results {#res}
=======
The third step transition absolute frequencies were collected for $n$=33-50 in intervals of one, and from $n$=50-100 in larger intervals of five. Fitting to the transition data was done using a Wahlquist first derivative function [@wahlquist1961]. The function is given by
$$\label{wahlquist}
f(H_{\mathrm{\delta}}) = \frac{H_{\mathrm{\delta}}}{\vert H_{\mathrm{\delta}}\vert} \left(\frac{2}{H_{\mathrm{\omega}}}\right)^{2} \frac{\sqrt{2 \gamma-u}} {2\sqrt{u-2}(u-\gamma)},$$
where $\gamma= 1 + \beta^{2} + \alpha^{2}$, $u= \gamma + \sqrt{\gamma^{2} - 4 \alpha^{2}}$, $\alpha= H_{\mathrm{\delta}}/H_{\mathrm{\omega}}$ and $\beta = (\frac{1}{2} H_{1/2}/H_{\mathrm{\omega}})$. $H_{1/2}$, $H_{\mathrm{\omega}}$ and $H_{\mathrm{\delta}}$ are the FWHM, modulation amplitude and frequency detuning respectively. Figure \[Example\] shows a typical scan across a Rydberg transition with the fitted profile from . We found that the linewidths of the detected third step transitions prevented resolving the $n$F$_{7/2}$ and $n$F$_{5/2}$ fine structure splitting in this experiment, which for $n$=33 to 100 is 4.35MHz to 0.16MHz respectively [@han2006]. However, the use of $\sigma^{+}$ light for the third step laser ensures only the $n$F$_{7/2}$ level is excited in this case.\
![A typical scan of the third step laser across the 33F$_{7/2}$ Rydberg state. The plot displays the demodulated first step absorption from the lock-in amplifier against the absolute frequency of the third step laser from the WS7 wavemeter.[]{data-label="Example"}](figure4){width="7cm"}
Ten traces were taken for each state in order to understand the repeatability of the measurements. It was found that on average the standard deviation of each set of ten scans was 2MHz with an accuracy limited by the short term drift of the wavemeter during the time taken to collect each set. The mean transition frequencies of the third step are summarised in the second column of table \[data\]. The third column of this table displays the total 5S$_{1/2}$ to $n$F$_{7/2}$ frequency, measured from the center of mass of the 5S$_{1/2}$ ground states. These values were calculated by adding a constant value of 770571549.6MHz to the third step transition frequencies in column two, this frequency was computed from references [@Nez1993] and [@arimondo1977].\
[@llll]{} $n$ & $\nu_{3}$ & $ E_{n}$\
& (MHz) & (MHz)\
33 & 236429214 & 1007000764\
34 & 236604549 & 1007176099\
35 & 236765078 & 1007336627\
36 & 236912402 & 1007483952\
37 & 237047954 & 1007619503\
38 & 237172932 & 1007744481\
39 & 237288417 & 1007859967\
40 & 237395343 & 1007966892\
41 & 237494542 & 1008066092\
42 & 237586734 & 1008158283\
43 & 237672570 & 1008244119\
44 & 237752610 & 1008324159\
45 & 237827379 & 1008398929\
46 & 237897325 & 1008468875\
47 & 237962850 & 1008534399\
48 & 238024325 & 1008595874\
49 & 238082056 & 1008653605\
50 & 238136367 & 1008707917\
55 & 238364972 & 1008936522\
60 & 238538826 & 1009110376\
65 & 238674124 & 1009245673\
70 & 238781461 & 1009353011\
75 & 238868053 & 1009439602\
80 & 238938927 & 1009510477\
85 & 238997658 & 1009569208\
90 & 239046866 & 1009618416\
95 & 239088516 & 1009660066\
100 & 239124074 & 1009695624\
To study potential frequency offsets of the transitions caused by power shifts, pressure shifts or Zeeman shifts we took measurements of both high and low $n$ states with a range of different first, second and third step laser powers, cell temperatures and opposite circular polarisations respectively. We also checked for errors from time delays in the data acquisition process by scanning the third step laser across the same transition in opposing directions. No repeatable shifts of the transition frequencies were found with increased laser powers or cell temperature and therefore potential offsets from these effects were not added as corrections but instead the spread of measurements were used to estimate a maximum error in each individual case. Neither Zeeman shifts nor time delay errors were detectable within the short term accuracy of the wavemeter and therefore these effects were assumed to give a negligible contribution to the uncertainty. The summarised error estimates are displayed in table \[errors\]. The errors add in quadrature to give a total error of 8.0MHz.\
Rydberg $n$F states are highly polarisable in external electric fields, with polarisabilities scaling as $n^7$ [@gallagher88]. To measure potential DC Stark shifts of the Rydberg states we applied electric fields of up to 30Vcm$^{-1}$ across the vapour cell and checked for frequency shifts of both the 33F$_{7/2}$ and 100F$_{7/2}$ transitions. In each case there was no measurable deviation. This unexpected observation was also made in references [@Mohapatra2008] and [@thoumany2009] when detecting Rydberg states in a cell. A screening of the Rydberg atoms inside the cell seems to be present, which makes them resilient to electric fields. This is a very positive effect as it allows precision spectroscopy of high $\ell$ states with no DC Stark shifts.
[@llll]{} Source & Error\
wavemeter calibration & 6.2MHz\
first step frequency & 750kHz\
second step frequency & 1.0MHz\
pressure shifts & 2.7MHz\
power shifts & 4.0MHz\
TOTAL & 8.0MHz\
Analysis
========
Rydberg level energies are very well described by the Rydberg formula
$$\label{rydberg}
E_{n}= E_{\rmi} - \frac{R_{X}}{[n-\delta(n)]^{2}} = E_{\rmi} - \frac{R_{X}}{n^{*2}} \ ,$$
where $E_{\rmi}$ is the ionisation energy, $E_{n}$ is the excitation energy from the ground state to a state with principal quantum number $n$, $R_{X}$ is the Rydberg constant for the atom of interest, $\delta(n)$ is the quantum defect and $n^{*}$ is the effective quantum number. The quantum defect can also be written as a Ritz expansion
$$\label{ritz}
\delta(n) = \delta_{0} + \delta_{2}t_{n} + \delta_{4}t_{n}^{2} + ... \ ,$$
where
$$\label{ritzparam}
t_{n} = \frac{1}{[n-\delta(n)]^{2}} = \frac{E_{\rmi}-E_{n}}{R_{X}} .$$
The data from this experiment was analysed using three different fitting methods. The first two methods follow the same theme as [@martin79], whilst the third method is a consistency check of the data with previous work. These methods are outlined in sections \[sec:meth1\], \[sec:meth2\] and \[sec:meth3\]. To aid in the analysis, five values of $E_{n}$ for $n$=4-8 were added to the data set from [@johansson61]. Weighted fitting was important to take account of the larger uncertainties on these older measurements. Throughout the analysis the Rydberg constant for rubidium 85 was taken as $R_{\mathrm{Rb}}$=10973660.672249$\times c$ from [@sanguinetti2009].
Method 1 {#sec:meth1}
--------
In method 1 the energy levels $E_{n}$ were fitted using a least squares fitting procedure to the formula:
$$\label{modified}
E_{n}= E_{\rmi} - \frac{R_{\mathrm{Rb}}}{ \left[ n- \delta_{0} - \delta_{2}t_{n} - \delta_{4}t_{n}^{2} - ...\right] ^{2}} \ .$$
The fit algorithm balanced both sides of to find the optimum parameters for $E_{\rmi}$, $\delta_{0}$, $\delta_{2}$, $\delta_{4}$,... The results from this fit are displayed in table \[fits\] and the residuals are shown in figure \[Residuals\]. Reference [@drake1991] describes in great detail how the series parameters extracted from this type of fit can explain physical properties of the Rydberg atom, such as the core polarisation.
Method 2 {#sec:meth2}
--------
To remove the recursive nature of it is common to make the approximation
$$\label{approx}
t_{n} \approx \frac{1}{(n-\delta_{0})^{2}} \ ,$$
which when substituted into gives a Rydberg-Ritz expression that can be evaluated with greater simplicity [@martin79]:
$$\label{extended}
E_{n}= E_{\rmi} - \frac{R_{\mathrm{Rb}}}{[n-\delta_{0} - \frac{a}{(n-\delta_{0})^{2}} - \frac{b}{(n-\delta_{0})^{4}} - ... ]^{2}} \ .$$
The method 2 fit involved a direct least squares fit of to the energy levels $E_{n}$. The results from this fit are displayed in table \[fits\] where the $a$ and $b$ parameters are placed underneath the equivalent $\delta_{2}$ and $\delta_{4}$ parameters from the method 1 fit. It can be seen that the values of $E_{\rmi}$ and the series parameters extracted from the first two fitting methods agree to well within the uncertainties. The value of $E_{\rmi}$ from this work also lies within 2$\sigma$ of the previous value from [@sanguinetti2009]. An analysis of the residuals shown in figure \[Residuals\], from the method 1 fit, shows that the points are scattered around a mean of zero with a standard deviation of 4.4MHz. The states were measured across several days and therefore this spread comes mainly from the long term accuracy of the wavemeter.\
The Rydberg-Ritz formula in has the significant advantage that it allows any energy level $E_{n}$ to be calculated with knowledge only of the principle quantum number $n$. In this manner can be used with the relevant parameters in table \[fits\] to predict the absolute energies of other rubidium $n$F$_{7/2}$ states outside the range of this experiment.
Method 3 {#sec:meth3}
--------
As a consistency check of this data we compared the Ritz series parameters extracted from our absolute measurements with those from the most recent relative interval measurements [@han2006]. For this fit we used an abridged version of :
$$\label{abridged}
E_{n}= E_{\rmi} - \frac{R_{\mathrm{Rb}}}{[n-\delta_{0} - \frac{a}{(n-\delta_{0})^{2}}]^{2}} \ .$$
This is the equivalent function which was used for fitting in reference [@han2006] and is an accurate approximation for $n \geq$20. For this reason we restricted this fit to the $n\geq$33 levels. The parameters from this fitting method are shown in table \[fits\] with the values from [@han2006]. The $a$ parameter is placed underneath the equivalent $\delta_{2}$ parameter from the method 1 fit.\
[@lllll]{} & $E_{\rmi}$ (MHz) & $\delta_{0}$ & $\delta_{2}$ & $\delta_{4}$\
Method 1 & 1010024719(8) & 0.016473(14) & -0.0783(7) & 0.028(7)\
Method 2 & 1010024719(8) & 0.016473(14) & -0.0784(7) & 0.032(7)\
Method 3 & 1010024717(8) & 0.01640(8) & 0.00(9)& -\
Reference [@han2006] & - & 0.0165437(7) & -0.086(7) & -\
![The residuals for the $n$=33-100 states from the method 1 fitting routine. The error bars show the total accumulated error on each data point of 8.0MHz.[]{data-label="Residuals"}](figure5){width="7cm"}
It can be seen that the $\delta_{0}$ and $a$ parameters from this fit agree at the 2$\sigma$ level with those from the previous work [@han2006]. Because our parameters are extracted from absolute measurements one does not expect as high an accuracy as from interval measurements, however absolute measurements do have the advantage that the ionisation energy $E_{\rmi}$ can also be extracted. The larger errors on the series parameters from this fit, as compared to the method 1 and 2 fits, arise because of the absence of lower $n$ states. This makes extracting higher order series parameters more difficult. For example, in , for lower $n$ states the parameters $\delta_{0}$ and $a$ make a bigger contribution than for higher $n$ states, where $E_{n}$ becomes dominated by $E_{\rmi}$. As displayed in table \[fits\], the addition of the lower $n$ states from reference [@johansson61] greatly aided in the reliable extraction of the higher order parameters in the method 1 and 2 fitting routines.
Conclusion {#conc}
==========
We have presented absolute frequency measurements of $n$F$_{7/2}$ Rydberg states in rubidium 85 to an accuracy of 8.0MHz. This is a factor 40 improvement over previous measurements of the $n$=4-7 $n$F$_{7/2}$ states [@johansson61] and gives measurements for a range of $n$F$_{7/2}$ states between $n$=33-100 for the first time. The Rydberg-Ritz series parameters which have been extracted from this work allow absolute energies of $n$F$_{7/2}$ states with higher or lower principle quantum number $n$ to be predicted with a comparable accuracy. Our new measurements also show consistency with results from recent microwave spectroscopy experiments [@han2006]. This work demonstrates that methods of Rydberg spectroscopy involving purely optical detection can be used very effectively to carry out precision measurements of Rydberg states in a simple way, and with extraordinary robustness to DC stark shifts. Not only is the set up simple to construct and maintain but it is easier to use than beam experiments, and Rydberg signals can be monitored in real-time on an oscilloscope. We believe that this experiment could be readily adapted to study other alkali metal atoms and could even be used to study such unusual features as Rydberg-Rydberg interactions and molecular states. In future work we hope to stabilise the third step laser to the transitions and directly count the laser frequency against a frequency comb. We estimate a potential 80$\times$ improvement in absolute accuracy can be made with this new approach. We also plan to study states with lower $n$ and $\ell$ by modifying the laser system. Carrying out these types of precision measurement on lower $n$ states would also allow quantum defects to be extracted with much greater accuracy.
References {#references .unnumbered}
==========
|
{
"pile_set_name": "ArXiv"
}
|
[ ]{}
Introduction
============
The recent discovery of carrier-mediated ferromagnetism in diluted magnetic semiconductors (DMS) has generated intense interest, in part because it suggests the prospect of developing devices which combine information processing and storage functionalities in one material.[@Prinz95; @Prinz98; @Furdyna; @Dietl94; @Pashitski; @Story86; @Haury97; @Ohno92; @Ohno96; @Ohno98; @Ohno99; @Hayashi98; @Pekarek98; @VanEsch97/1; @VanEsch97/2; @Oiwa99; @Matsukura98; @Ohno98.2; @Okabayashi98; @Omiya00; @Beschoten99] Ferromagnetism has been observed in Mn doped GaAs up to critical temperatures $T_c$ of $110 {\rm K}$ (see Ref. ). Doping a III-V compound semiconductor with Mn introduces both local magnetic moments, with concentration $N_{\rm Mn}$ and spin $S=5/2$, and itinerant valence-band carriers with density $p$ and spin $s=1/2$. An antiferromagnetic interaction between both kinds of spin mediates an effective ferromagnetic interaction between ${\rm Mn}^{2+}$ spins.
A phenomenological long-wavelength description of ferromagnets usually requires only a small set of characteristic parameters. For example, a ferromagnet which possesses an uniaxial anisotropy can be modeled by a classical micromagnetic energy functional $E[{\bf \hat n}({\bf r})]$, $$E[{\bf \hat n}({\bf r})] = E_0 + \int d^3 r \,
\left[ K \sin^2 \theta + A \left( {\bf \nabla} {\bf \hat n} \right)^2
\right] \, ,
\label{uniaxial}$$ where ${\bf \hat n}({\bf r})$ is the unit vector that specifies the (space-dependent) local Mn spin orientation, and $\theta$ is its angle with respect to the easy axis. The dependence on the orientation of the magnetic moment is parametrized by the anisotropy constant $K$. The exchange constant (or spin stiffness) $A$ governs the energy cost to twist the orientations of adjacent spins relative to each other. Together with the saturation moment $\mu_0 M_s$ and the magnetostatic energy, dropped for convenience in Eq. (\[uniaxial\]), the parameters $K$ and $A$ determine a whole variety of magnetic properties such as[@Skomski] domain-wall width $\delta_{\rm B}$, domain-wall energy per area $\gamma$, exchange length $l_{\rm ex}$, hardness parameter $\kappa$, single-domain radius $R_{\rm sd}$, and anisotropy field $\mu_0 H_0$. In addition they determine the energy cost $\Omega_{k}$ of collective long-wavelength spin excitations, spin waves. For uniaxial anisotropy the quantized energy is $$\Omega_{k} = {2K \over M_s/(g \mu_B)} + {2A \over M_s/(g \mu_B)} k^2 \, ,$$ at wavevector $k$. Here, $g$ is the $g$-factor and $\mu_B$ the Bohr magneton. There is an energy gap which is determined by the anisotropy $K$. The spin exchange constant $A$ defines the curvature of the spin-wave dispersion.
The purpose of this paper is to present a microscopic theory for the phenomenological parameters which characterize (Ga,Mn)As as a ferromagnet. The form of the micromagnetic energy functional appropriate to the symmetry of the crystal will be addressed in Sec. \[Section\_classical\]. We expect that these predictions will be useful in interpreting experimental studies of these semiconductor’s magnetic properties. The search for the carrier density, Mn concentration and (III,Mn)V compound semiconductor material for which the critical temperature has its maximum value is, perhaps, the most important endeavor in this research area. It has been guided to date by mean-field-theory[@Dietl97; @Jungwirth99; @Dietl00; @Lee00; @Abolfath01; @Dietl00.2] considerations that neglect the low-energy correlated magnetization fluctuations characterized by $M_s$, $K$, and $A$. However, as we have emphasized earlier,[@Koenig1; @Koenig2; @Koenig3; @Schliemann00.1] small exchange can also limit the temperature at which long-range magnetic order exists.
In recent work[@Koenig1; @Koenig2; @Koenig3] we developed a theory of diluted magnetic semiconductor ferromagnetism which accounts for dynamic correlations in the ordered state. For simplicity we used parabolic bands for the itinerant carriers, leading to isotropic ferromagnetism. We found, in addition to the usual spin-wave mode, a continuum of Stoner excitations, and another collective branch of excitations, optical spin waves. As we have shown,[@Koenig1; @Schliemann00.1] the low-energy Goldstone modes can suppress the critical temperature in comparison to mean-field estimates and can even change trends in $T_c$ as a function of the system’s parameters.
To address anisotropy one has to go beyond a parabolic-band model and take details of the band structure into account. The itinerant-carrier bands are $p$-type and reflect the crystal symmetry of the underlying lattice. Due to spin-orbit coupling the spin degrees of freedom also feel the crystal anisotropy. We will use a six-band description based on the Kohn-Luttinger Hamiltonian to model the (Ga,Mn)As valence bands. States near the Fermi energy in our model include substantial mixing with the split-off valence band making the six-band model we employ the minimal band model. Mean-field calculations[@Dietl00; @Abolfath01; @Dietl00.2] as well as Monte Carlo studies[@Schliemann00.2] based on this more realistic band structure have been performed recently to address magnetic anisotropy effects and explore trends in the critical temperature.
In Sec. \[Section\_Hamiltonian\] we set the starting point of our theory by deriving a formal expression for the effective action for the Mn impurity spins after integration out the itinerant carriers. This formal development generalizes earlier work[@Koenig1; @Koenig2; @Koenig3] to general spin-orbit coupled band models. Using the effective action we numerically determine the zero-temperature spin-wave dispersion in Sec. \[Section\_SWT\]. Then, in Sec. \[Section\_classical\] we establish the connection between our spin-wave results and the classical micromagnetic energy functional, adjusted to the symmetry defined by the crystal structure. Finally, in Sec. \[Section\_results\], we present numerical results for the spin-wave dispersion and the exchange constant. We find that for the parameter range of interest the exchange constant is enhanced by up to an order of magnitude compared to naive results obtained earlier, in which light-hole bands were neglected and the heavy-hole bands were approximated as parabolic.
Hamiltonian and effective action {#Section_Hamiltonian}
================================
Our theory is based on the following model. Magnetic ions with spin $S=5/2$ at positions ${\bf R}_I$ are antiferromagnetically coupled to valence-band carriers described by an envelope-function approach, $$H = H_0 + J_{\rm pd} \int \hspace{-1mm}d^3 r \;
{\bf S}({\bf r}) \cdot {\bf s}({\bf r}) ,$$ where ${\bf S}({\bf r}) = \sum_I {\bf S}_I \delta( {\bf r} - {\bf R}_I)$ is the impurity-spin density and $J_{\rm pd}>0$. We approximate the impurity-spin density by a continuous functions (instead of a sum of delta functions). Provided that $J_{\rm pd}$ is not strong enough to localize valence-band carriers near the Mn sites and the average distance between two Mn ions is small in comparison to the Fermi wavelength of the itinerant carriers, this approximation can be justified. The itinerant-carrier spin density is expressed in terms of carrier field operators by ${\bf s}({\bf r})= \sum_{ij} \Psi_i^\dagger
({\bf r}) {\bf s}_{ij} \Psi_j({\bf r})$ where $i$ and $j$ label the basis in Hilbert space of spin and orbital angular momentum, and ${\bf s}_{ij}$ is the corresponding representation of the spin operator. The envelope-function Hamiltonian $H_0$ for the valence bands can be parametrized by a small number of symmetry-adapted parameters, the Luttinger parameters $\gamma_1$, $\gamma_2$, $\gamma_3$, and the spin-orbit coupling $\Delta_{\rm so}$ which splits the six states at the band edge into a quartet and a doublet. For explicit expressions of $H_0$ as well as representations of the spin matrices in a coordinate system in which the $x$, $y$, and $z$-axis are along the crystal axes, we refer to Eqs. (A8)-(A10) of Ref. , based on the Kohn-Luttinger Hamiltonian.[@Luttinger55]
Phenomenological Hamiltonians of this form have proven successful in understanding optical properties of the closely related magnetically doped II-VI semiconductors.[@Furdyna; @Dietl94] In that case however, the inclusion of direct antiferromagnetic interactions between Mn spins that lie on neighboring lattice sites proved to be important. This interactions appear to be much weaker in the III-V case, although this difference is not fully understood.[@Dietl00.2] We do not include these spin-spin interactions in our calculations, because we have no knowledge of the strength of the coupling coefficient. If included, they would mainly influence short-distance spin excitations which are not in any event the focus of this work.
We note that local-spin-density-approximation electronic structure calculations,[@Sanvito00] taken at face value, find a strong hybridization between $p$ and $d$ carriers, which is not completely consistent with the model we use. However, given the overwhelming success[@Furdyna; @Dietl94] of the present model in the case of paramagnetic (II,Mn)VI materials, we find it unlikely that the Mn $d$ electrons are itinerant. If they were, the phenomenological approach we study here would be incomplete.
Our treatment of the Mn spin system as a continuum greatly simplifies our calculation by eliminating disorder associated with randomness in the Mn positions. We do anticipate that the disorder can influence both anisotropy and exchange constant at the low- and high-carrier-concentration extremes. These effects are, however, outside the scope of the present study. We will identify several important features of the microscopic physics in the disorder-free model that we expect to be robust.
Our first goal is to integrate out the itinerant carriers and arrive at an effective description for the impurity-spin degrees of freedom. This procedure follows the analysis presented in Ref. for the (simpler) two-band model. For small fluctuations around its mean-field polarization, we can approximate the spin operators by $S^+ ({\bf r}) \approx b({\bf r}) \sqrt{2N_{\rm Mn}S}$, $ S^- ({\bf r}) = \left( S^+ ({\bf r}) \right)^\dagger$, and $S^z ({\bf r}) = N_{\rm Mn}S - b^{\dag}({\bf r}) b({\bf r})$, where $b^{\dag}({\bf r}), b({\bf r})$ are bosonic Holstein-Primakoff[@Auerbach94] fields. The quantization axis $z$ is chosen here along the zero-temperature spin orientation. After integrating out the itinerant carriers the partition function, $Z = \int {\cal D} [\bar z z] \exp (- S_{\rm eff} [\bar z z])$, is governed by the effective action for the impurity spins, $$S_{\rm eff} [\bar z z] = S_{\rm BP} [\bar z z]
- \ln \det \left[ (G^{\rm MF})^{-1} + \delta G^{-1}(\bar zz) \right] \, ,
\label{effective action}$$ where $S_{\rm BP} [\bar z z] = \int_0^\beta d\tau \int d^3 r \, \bar z
\partial_\tau z$ is the usual Berry’s phase term, and the complex numbers $\bar z$ and $z$ label the bosonic degrees of freedom. In Eq. (\[effective action\]), we have already split the total kernel $G^{-1}$ into a mean-field part $(G^{\rm MF})^{-1}$ and a fluctuating part $\delta G^{-1}$, $$\begin{aligned}
(G^{\rm MF})^{-1}_{ij} &=&
\left( \partial_\tau -\mu \right)\delta_{ij} + \langle i| H_0 | j \rangle
+ N_{\rm Mn}J_{\rm pd}S s^z_{ij}
\\
\delta G^{-1}_{ij}(\bar zz) &=& {J_{\rm pd}\over 2} \left[ \left(
z s^-_{ij} + \bar z s^+_{ij} \right) \sqrt{2N_{\rm Mn}S}
- 2 \bar z z s^z_{ij} \right]\end{aligned}$$ where $\mu$ denotes the chemical potential, and $i$ and $j$ range over a complete set of hole-band states. In the following we define the mean-field energy $\Delta = N_{\rm Mn}J_{\rm pd}S$ to flip the spin of an itinerant carrier. The physics of the itinerant carriers is embedded in the effective action of the magnetic ions. It is responsible for the retarded and non-local character of the interactions between magnetic ions.
Independent spin-wave theory {#Section_SWT}
============================
Independent spin-wave theory is obtained by expanding Eq. (\[effective action\]) up to quadratic order in $z$ and performing Matsubara imaginary time and space Fourier transforms. Since $\delta G^{-1}$ is at least linear in $z$ the series $$S_{\rm eff} [\bar z z] = \sum_{n=0}^\infty S^{(n)}_{\rm eff}[\bar zz] \, ,
\label{series}$$ can be truncated after $n=2$, where $n$ denotes the order in $\delta G^{-1}$. The zeroth-order contribution, $ S^{(0)}_{\rm eff} [\bar z z] =
S_{\rm BP}[\bar zz] - \ln \det (G^{\rm MF})^{-1}$ contains the Berry’s phase term $$S_{\rm BP}[\bar zz] = \sum_{m,{\bf k}} (-i \nu_m)
\bar z({\bf k},\nu_m) z({\bf k},\nu_m) \, ,
\label{SB}$$ and the mean-field contribution from the itinerant carriers, which is independent of the bosonic fields $z$ and $\bar z$. Here, $\nu_m$ are the bosonic Matsubara frequencies.
The next term of the expansion, $S^{(1)}[\bar zz] = - {\rm tr} \left( G^{\rm MF} \delta G^{-1} \right)$, reads in Fourier representation $$\begin{aligned}
S^{(1)}_{\rm eff}[\bar zz] &=&
{J_{\rm pd} \over (\beta V)^2} \sum_{n,{\bf q}} \sum_{ij} \,
G^{\rm MF}_{ij} ({\bf q},\omega_n) s^z_{ji}
\nonumber \\ && \times
\sum_{m,{\bf k}} \bar z({\bf k},\nu_m) z({\bf k},\nu_m) \, ,\end{aligned}$$ plus terms linear in $z$ and $\bar z$. Here, $\omega_n$ and $\nu_m$ are fermionic and bosonic Matsubara frequencies, respectively. To determine the mean-field Green’s functions we diagonalize the matrix $\langle i| H_0({\bf q}) + \Delta s^z | j \rangle$ for each wavevector ${\bf q}$ and denote eigenvalues and eigenstates by $\epsilon_{\alpha}({\bf q})$ and $|\alpha \rangle$, respectively. Since the itinerant-carrier spin density $\langle {\bf s} \rangle = (1/V) \sum_{\bf q} \sum_{\alpha}
\, f [\epsilon_\alpha({\bf q})] \langle \alpha | {\bf s} | \alpha \rangle$ is aligned antiparallel to the impurity spin-polarization axis (otherwise this would not be an easy axis), the terms linear in $z$ and $\bar z$ drop out, and we get $$S^{(1)}_{\rm eff}[\bar zz] =
{J_{\rm pd} p\xi \over 2\beta V}
\sum_{m,{\bf k}} \bar z({\bf k},\nu_m) z({\bf k},\nu_m) \, ,
\label{S1}$$ where $\xi = -2 \langle s^z \rangle /p$ is the fractional itinerant-carrier polarization, $0\le \xi \le 1$.
For the second-order term of the expansion, $S^{(2)}_{\rm eff}[\bar zz] = {1\over 2}\, {\rm tr} \left( G^{\rm MF}
\delta G^{-1} G^{\rm MF} \delta G^{-1} \right)$, we find $$\begin{aligned}
S^{(2)}_{\rm eff}[\bar zz] &=& {N_{\rm Mn} J_{\rm pd}^2 S \over 4\beta V^2}
\sum_{m,{\bf q},{\bf k}} \sum_{\alpha\beta}
{ f [\epsilon_\alpha({\bf q})] -
f [\epsilon_\beta({\bf q}+{\bf k})] \over
i\nu_m + \epsilon_\alpha({\bf q})
- \epsilon_\beta({\bf q}+{\bf k}) }
\nonumber \\ &&
\left[
2 s^+_{\alpha\beta} s^-_{\beta\alpha}
\bar z({\bf k},\nu_m) z({\bf k},\nu_m)
\nonumber \right. \\ && \left.
+ s^+_{\alpha\beta} s^+_{\beta\alpha}
\bar z({\bf k},\nu_m) \bar z(- {\bf k},-\nu_m)
\nonumber \right. \\ && \left.
+ s^-_{\alpha\beta} s^-_{\beta\alpha}
z({\bf k},\nu_m) z(- {\bf k},-\nu_m)
\right] + {\cal O}\left( z^3 \right) \, ,
\label{S2}\end{aligned}$$ with $s^\pm_{\alpha\beta} = \langle \alpha | s^\pm | \beta \rangle$ and $s^\pm_{\beta\alpha} = \langle \beta | s^\pm | \alpha \rangle$. Note, that the indices $\alpha$ and $\beta$ label the single-particle eigenstates for valence-band carriers with [*different*]{} wavevectors, namely [**q**]{} and [**q**]{}+[**k**]{}, respectively.
The Matsubara frequency $\nu_m$ in the denominator on the r.h.s of Eq. (\[S2\]) accounts for the dynamics of the itinerant carriers. This frequency dependence is crucial to account for the existence of the Stoner spin-flip continuum and the optical spin-wave mode. On the other hand, the existence of the usual spin wave follows already from the static limit (i.e., when the frequency dependence in the denominator in Eq. (\[S2\]) is dropped), and the spin-wave dispersion is described rather accurately.
The sum of Eqs. (\[SB\]), (\[S1\]), and (\[S2\]) is a quadratic form in the bosonic fields $z$ and $\bar z$. The zeros of the kernel define the spin-wave energies $\Omega_{\bf k}$ as a function of momentum ${\bf k}$ (after analytic continuation $i\nu_m \rightarrow \Omega +i0^+$). In the following we go the static limit as discussed above. We define the quantities $$E_{\bf k}^{\sigma \sigma'} = - {1\over V} \sum_{\bf q}
\sum_{\alpha\beta}
{ f [\epsilon_\alpha({\bf q})] -
f [\epsilon_\beta({\bf q}+{\bf k})] \over
\epsilon_\alpha({\bf q}) - \epsilon_\beta ({\bf q}+{\bf k}) }
s^\sigma_{\alpha\beta} s^{\sigma'}_{\beta\alpha}
\label{E}$$ with $\sigma,\sigma'=\pm$, and perform a Bogoliubov transformation, which eventually yields $${\Omega_{\bf k}\over \Delta} = { J_{\rm pd} \over 2}
\sqrt{ \left( { p\xi \over \Delta }
- E_{\bf k}^{+-} \right)^2
- \left|E_{\bf k}^{++}\right|^2 } \, .
\label{dispersion}$$ From the definition Eq. (\[E\]) we see that $E_{\bf k}^{+-}$ is real and $E_{\bf k}^{--}= \left( E_{\bf k}^{++}\right)^*$. Equation (\[dispersion\]) is the central result of this section. The remaining task is to evaluate the fractional itinerant-carrier polarization $\xi$ and the quantities $E_{\bf k}^{+-}$ and $E_{\bf k}^{++}$ numerically.
Before we carry on with establishing the relation between Eq. (\[dispersion\]) and micromagnetic parameters of a classical energy functional, we make three remarks:
\(i) Correlation effects among the Mn spins, which are not described by the mean-field picture, enter our theory via the contribution $S^{(2)}_{\rm eff}[\bar zz]$. To reduce our theory to the mean-field level we would have to neglect this term, i.e., truncate the series Eq. (\[series\]) already after $n=1$. In this case, the energy $\Omega^{\rm MF} = J_{\rm pd} p\xi/2$ of a Mn spin excitation would be dispersionless and by a factor of $p\xi/(2N_{\rm Mn}S)$ smaller than the mean-field energy $\Delta$ to flip an itinerant-carrier spin. Due to correlations between Mn and band-spin orientations, however, the spin-wave energy $\Omega_{\bf k}$ is always smaller than $\Omega^{\rm MF}$.
\(ii) In the absence of spin-orbit coupling all products of the form $s^+_{\alpha\beta} s^+_{\beta\alpha}$ or $s^-_{\alpha\beta} s^-_{\beta\alpha}$ vanish. As a consequence, $E_{\bf k}^{++}=E_{\bf k}^{--}=0$ for all ${\bf k}$. This statement is even true for finite spin-orbit coupling in case when the valence bands are isotropic.
\(iii) To go beyond the static limit, we may expand the fraction on the r.h.s of Eq. (\[S2\]) up to linear order in $i\nu_m$. As shown in Appendix \[append\_berry\], this linear correction simply amounts to a renormalization of the Berry’s phase term by replacing $\Omega \rightarrow \Omega (1-x)$. In the absence of spin-orbit coupling we find that $x$ is just the ratio of the spin densities, $x = \langle s^z \rangle /(N_{\rm Mn}S)$, where $\langle \ldots \rangle = (1/V) \sum_{\bf q} \sum_\alpha
f[\epsilon_\alpha({\bf q})] \, \langle \alpha| \ldots | \alpha \rangle$. For finite spin-orbit coupling but with a band Hamiltonian that is invariant under rotation in space, the spin $s^z$ has to be replaced by the total angular momentum $s^z + l^z$, i.e., we find $x = \langle s^z+l^z \rangle /(N_{\rm Mn}S)$.
The renormalization factor $(1-x)$ indicates that in a semiclassical picture, as employed in Sec. \[Section\_classical\], the effective spin density is not quite given by the Mn impurities, but has to be reduced due to coupling to the valence-band carriers. On the other hand, $x$ is always small since the impurity spin density $N_{\rm Mn}$ is larger than the itinerant-carrier concentration $p$ and the Mn spin $S=5/2$ is comparatively large. We, therefore, stick to the static limit in the following discussion.
Easy axis, energy gap and spin stiffness {#Section_classical}
========================================
The purpose of this section is to establish the connection between the spin-wave dispersions evaluated later and the micromagnetic energy functional. The non-local magnetostatic contribution which is omitted from our theory can be added as needed in applications. The short-range part of the functional is a symmetry-adapted gradient expansion of the energy density $e[{\bf \hat n}]$. In magnetism literature the non-constant portion of the zeroth-order term $e^{\rm ani}[{\bf \hat n}]$ is known as the magnetic anisotropy energy, and the leading gradient term $e^{\rm ex}[{\bf \hat n}]$ is known as the exchange energy. As shown in Ref. magnetic anisotropy effect in ferromagnetic semiconductors are, in the absence of strain, very well described by a cubic harmonic expansion which is truncated after sixth order, $$\label{aniso}
e^{\rm ani}[{\bf \hat n}] =
K_1 \left( n_x^2 n_y^2 + n_x^2 n_z^2 + n_y^2 n_z^2 \right)
+ K_2 \left( n_x n_y n_z \right) ^2$$ with anisotropy parameters $K_1$ and $K_2$. Correlation of spin polarizations at different positions are described by the gradient term $e^{\rm ex}[{\bf \hat n}]$. In order to address long-wavelength spatial fluctuations, we expand the gradient term up to lowest nonvanishing order, $$\label{gradient}
e^{\rm ex}[{\bf \hat n}] =
\sum_{a,b \in \{x,y,z\}} A_{ab} |\partial_a n_b|^2 \, ,$$ with exchange parameters $A_{ab}$. We find in our numerical calculations that anisotropy in the exchange constant is negligibly small, i.e., we can choose $A_{ab} = A$ for all $a,b$, as generally assumed in the magnetism literature.
To establish the connection of the energy functional to our microscopic spin-wave calculation, we first have to determine the direction of the mean-field spin polarization and then to study small fluctuations. The first step is achieved by minimizing the energy Eq. (\[aniso\]). It is easy to show that the mean-field orientation ${\bf \hat n}^{\rm MF}$ can only point along a high-symmetry axis $\langle 100 \rangle$, $\langle 110 \rangle$, $\langle 111 \rangle$, or an equivalent direction (except for the special case $K_2=0$ and $K_1>0$, where we find an easy-plane anisotropy in the planes $n_x=0$, $n_y=0$ or $n_z=0$, and, of course, the isotropic case $K_1=K_2=0$). In Fig. \[fig1\] we show how the mean-field polarization direction depends on $K_1$ and $K_2$.
Now we consider small fluctuations around the mean-field orientation ${\bf \hat n}^{\rm MF}$. The kernel of the quadratic form $e^{\rm ani}[{\bf \hat n}] - e^{\rm ani}[{\bf \hat n}^{\rm MF}]$ has two eigenvalues $\lambda_1$ and $\lambda_2$. We find for ${\bf \hat n}^{\rm MF}$ along $\langle 100 \rangle$ that $\lambda_1 = \lambda_2 = K_1$, for $\langle 110 \rangle$ we get $\lambda_1 = -K_1$ and $\lambda_2 = (2K_1+K_2)/4$, and for $\langle 111 \rangle$ we obtain $\lambda_1 = \lambda_2 = - (6K_1+2K_2)/9$. In all cases, $\lambda_1$ and $\lambda_2$ are positive. Quantizing the collective spin coordinate at long wavelengths, it follows that $$\Omega_{k} = {2\sqrt{\lambda_1 \lambda_2}\over N_{\rm Mn}S}
+ {2A\over N_{\rm Mn}S} \, k^2 + {\cal O}(k^4) \, .
\label{gap+stiffness}$$ There are two alternative ways to determine the energy gap $\Omega_{k=0}$. One can either perform the $k=0$ limit of the spin-wave dispersion Eq. (\[dispersion\]) or evaluate the coefficients $\lambda_1$ and $\lambda_2$ from calculating the energy for mean-field orientation ${\bf \hat n}^{\rm MF}$ along the three high-symmetry axes $\langle 100 \rangle$, $\langle 110 \rangle$, and $\langle 111 \rangle$. The numerical effort of the latter procedure, which has been used previously in Ref. , is lesser for given accuracy. The virtue of the spin-wave calculation is to determine the exchange constant $A$.
We conclude this section with two remarks:
\(i) In case that the easy axis is along $\langle 100 \rangle$ or $\langle 111 \rangle$ (as it is for all parameter sets considered in Sec. \[Section\_results\]), the energy cost of tilting the polarization axis by small angles is independent of the direction of the deflection. This is required by symmetry and indicated by the fact that $\lambda_1$ equals $\lambda_2$. As shown in Appendix \[append\_gap\], the term $E^{++}_{k=0}$ then vanishes, and the $k=0$ limit of Eq. (\[dispersion\]) is identical to the energy gap calculated from standard perturbation theory where the perturbation describes the deviation of the spin polarization from the mean-field direction.
\(ii) The denominators $N_{\rm Mn}S$ in Eq. (\[gap+stiffness\]) corresponds to the static limit employed in our calculation. The renormalization of the Berry’s phase term due to corrections to the static limit (see discussion in the previous section) could be accounted for by multiplying $N_{\rm Mn}S$ with $(1-x)$.
Numerical results for the spin-wave dispersion {#Section_results}
==============================================
In this section we present numerical results for the spin-wave dispersion of (Ga,Mn)As. From these calculations we extract the spin stiffness as a function of the itinerant-carrier concentration $p$ and on the exchange coupling $J_{\rm pd}$. To model the sample which showed the highest transition temperature of $110 {\rm K}$ so far, we choose as parameters[@Omiya00] $N_{\rm Mn} = 1 \, {\rm nm}^{-3}$, $p = 0.35 \, {\rm nm}^{-3}$, and $J_{\rm pd} = 0.068 \, {\rm eV \, nm}^{-3}$. As a consequence, the mean-field spin-splitting gap for the itinerant carriers is $\Delta = N_{\rm Mn} J_{\rm pd} S = 0.17 \, {\rm eV}$.
Isotropic vs. six-band model
----------------------------
The origin of ferromagnetism, the nature of the spin excitations, and trends in the critical temperatures can be explained within a simple model which describes the itinerant carriers by parabolic bands.[@Koenig1; @Koenig2; @Koenig3; @Lyu01] For more quantitative statements, a more realistic description of the band structure should be employed.
In Fig. \[fig2\] we show results for the isotropic model with two parabolic band with effective mass $m^* = 0.5 m_e$ (a Debye cutoff $k_D$ with $k_D^3 = 6\pi^2 N_{\rm Mn}$ ensures that we include the correct number of magnetic ion degrees of freedom). We find for the majority-spin Fermi energy $\epsilon_F = 0.44 \, {\rm eV}$ measured from the bottom of the band. This yields the itinerant-carrier polarization $\xi = 0.35$. The dashed line in Fig. \[fig2\] marks the mean-field spin-flip energy, obtained by neglecting correlation. Since the spin-wave energies are far below the mean-field result, the isotropic model suggest that for these parameters correlation is very important and the critical temperature is limited by collective fluctuations. According to the classification scheme introduced in Ref. , the system would be in the ”RKKY collective regime”.
By fitting a parabola at small momenta we find $\Omega_k/\Delta = 0.0068 (k/k_D)^2$ which yields $A = 0.095 \, {\rm meV \, nm}^{-1} = 0.015 \, {\rm pJ \, m}^{-1}$.
Now we use the six-band Kohn-Luttinger Hamiltonian with Luttinger parameters $\gamma_1 = 6.85$, $\gamma_2 = 2.1$, and $\gamma_3 = 2.9$ and spin-orbit coupling $\Delta_{\rm so} = 0.34 \, {\rm eV}$. In the absence of the Mn ion, this model has anisotropic heavy- and light-hole bands with masses $m_h\approx 0.498m_e$ and $m_l \approx 0.086m_e$. The spirit of the naive parabolic band model is the hope that only the band with the larger density of states matters and the anisotropy is unimportant. We will see that these hopes are not fulfilled.
We find a much lower Fermi energy $\epsilon_F = 0.25 \, {\rm eV}$ than obtained for the isotropic model, and, therefore, a much higher itinerant-carrier spin polarization $\xi = 0.73$. By calculating the mean-field energy for a Mn spin polarization along the three high-symmetry axes we determine the anisotropy parameters as $K_1 = 19.6 \times 10^{-6} \, {\rm eV \, nm}^{-3}$ and $K_2 = 1.6 \times 10^{-6} \, {\rm eV \, nm}^{-3}$. As a consequence, the easy axis is $\langle 100 \rangle$ and the energy gap, $\Omega_{k=0}/\Delta = 9.2 \times 10^{-5}$, is very small in comparison to the bandwidth of the spin-wave dispersion.
In Fig. \[fig3\] we show the spin-wave dispersion for wavevectors ${\bf k}$ along the easy axis. We observe that the effect of $E_{\bf k}^{++}$ in Eq. (\[dispersion\]) is negligibly small and can, therefore, be dropped. Furthermore, we find that the dispersion for a spin wave perpendicular to the easy axis cannot be distinguished from the dispersion of spin waves along the easy axis within numerical accuracy. By fitting a parabola at small momenta we find $\Omega_{k}/\Delta = \Omega_{k=0}/\Delta + 0.16 (k/k_D)^2$ which yields $A = 2.2 \, {\rm meV \, nm}^{-1} = 0.36\,{\rm pJ \, m}^{-1}$ (see inset of Fig. \[fig3\]). Furthermore, we see that the energy gap $\Omega_{k=0}/\Delta$ determined in the previous paragraph is consistent with our spin-wave results. Employing the six-band model we see that for the given parameters the spin-wave energies are much closer to the mean-field value than for the isotropic model. According to our classification given in Ref. the system is rather in the ”mean-field regime”, which explains the success of mean-field theory to reproduce the critical temperature.
Six-band model plus strain
--------------------------
The lattice constants of (Ga,Mn)As and GaAs do not match. Since low-temperature molecular beam epitaxy (MBE) has to be used to overcome the low solubility of Mn in GaAs, even thick films of (Ga,Mn)As grown on GaAs cannot relax to their equilibrium. The lattice of (Ga,Mn)As is instead locked to that of the underlying substrate. This induces strain which breaks the cubic symmetry. The influence of MBE growth lattice-matching strain on hole bands of cubic semiconductors is well understood.[@Chow99; @Bir74] This effect can easily be accounted for adding a strain term to the Hamiltonian (we use Eq. (32) of Ref. with strain parameters $e_0 = -0.0028$ and $\Gamma = -3.24 \, {\rm eV}$).
We choose the growth direction to be along $\langle 001 \rangle$ and compute the energy for five different directions $\langle 100 \rangle$, $\langle 001 \rangle$, $\langle 110 \rangle$, $\langle 011 \rangle$, and $\langle 111 \rangle$. The lowest energy is found for $\langle 100 \rangle$, i.e., we find an easy-axis anisotropy where the easy axis is in the plane perpendicular to the growth direction, in accordance with experiments.[@Ohno.proc] From the expansion of the mean-field anisotropy energy for small fluctuations around $\langle 100 \rangle$ and determining the eigenvalues for small fluctuations we estimate that the energy gap in the dispersion is larger than in the absence of strain by a factor of 1.4, i.e., still small. The spin stiffness derived from the curvature of the spin-wave dispersion is identical to that in the absence of strain within the accuracy of our numerical calculations. We will, therefore, ignore the effect of strain in the following.
Spin stiffness
--------------
In Fig. \[fig4\] we show the exchange constant (or spin stiffness) $A$ as a function of the itinerant-carrier density for both the isotropic two-band and the full six-band model. We find that the spin stiffness is much larger for the six-band calculation than for the two-band model. Furthermore, for the chosen range of itinerant-carrier densities the trend is different: in the two-band model the exchange constant decreases with increasing density, while for the six-band description we observe an increase with a subsequent saturation.
Experimental estimates for $J_{\rm pd}$ vary from $0.054 \, {\rm eV \, nm}^3$ to $0.15 \, {\rm eV \, nm}^3$, with more recent work suggesting a value toward the lower end of this range.[@Ohno98.2; @Okabayashi98; @Omiya00] To address the dependence of the spin stiffness on $J_{\rm pd}$, we show in Fig. \[fig4\] results for a two values of $J_{\rm pd}$ which differ by a factor two.
To understand the different behavior of the two-band and the six-band model we recall[@Koenig2; @Koenig3] that the two-band model predicts $A = p\hbar^2 /(8m^*)$ for low densities $p$, while at high densities $A$ decreases as a function of $p$ (note that in Refs. and the spin stiffness was characterized by $\rho = 2A$). The crossover occurs near $\Delta \sim \epsilon_F$. The difference in the trends seen for the two- and six-band model in Fig. \[fig4\] is consistent with the observation that, at given itinerant-carrier concentration $p$, the Fermi energy $\epsilon_F$ is much smaller when the six-band model is employed, where more bands are available for the carriers, than in the two-band case. Furthermore, we emphasize that, even in the limit of low carrier concentration, it is not only the (heavy-hole) mass of the lowest band which is important for the spin stiffness. Instead, a collective state in which the spins of the itinerant carriers follow the spatial variation of a Mn spin-wave configuration will involve the light-hole band, too.
Conclusion {#conclusion .unnumbered}
==========
In conclusion we present a microscopic calculation of micromagnetic parameters for ferromagnetic (Ga,Mn)As. We draw the connection of the anisotropy and exchange constant of a classical energy functional to the gap and curvature of the spin-wave dispersion. Numerical results for the spin stiffness as a function of itinerant-carrier concentration and $p-d$ exchange coupling are shown. We find that the energy gap is much smaller than the bandwidth, the spin stiffness is nearly isotropic, and strain does not effect the dispersion much. Furthermore, we see that a model with isotropic valence bands underestimates the spin stiffness considerably.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge useful discussions with M. Abolfath, B. Beschoten, T. Dietl, J. Furdyna, H.H. Lin, and J. Schliemann. This work was supported by the Deutsche Forschungsgemeinschaft, the Ministry of Education of the Czech Republic, the EU COST Program, the Welch Foundation, the Indiana $21^{\rm st}$ Century Fund, and the Office of Naval Research and the Research Foundation of the State University of New York under grant number N000140010951.
Renormalization of Berry’s phase term {#append_berry}
=====================================
In this section, we show that the linear correction beyond the static limit yields a correction to the Berry’s phase. This accounts for the fact that the effective spin in a semiclassical approach does not have the length $S$ of the Mn impurities by is reduced by a factor $(1-x)$ due to coupling to the itinerant-carrier spin degree of freedom. We expand the r.h.s of Eq. (\[S2\]) up to linear order in $i\nu_m$. The terms with $\bar z \bar z$ and $z z$ vanish since it is an odd function under the following operation: we shift ${\bf q} \rightarrow -{\bf q} - {\bf k}$, use the fact that $\epsilon_\alpha( -{\bf q} - {\bf k}) = \epsilon_\alpha( {\bf q} + {\bf k})$ and $\epsilon_\beta( -{\bf q}) = \epsilon_\beta( {\bf q})$ and, eventually, exchange $\alpha \leftrightarrow \beta$. Hence, we only have to deal with the term which involves $\bar z z$.
Assume that there is an operator $Q$ such that $N_{\rm Mn}J_{\rm pd}S s^+ =
[H,Q]$ and $N_{\rm Mn}J_{\rm pd}S s^- = - [H,Q^\dagger]$. Then the renormalization for the Berry’s phase is given by $(1-x)$ with $$x = {1\over 2N_{\rm Mn}SV} \sum_{\bf q} \sum_{\alpha}
f [\epsilon_\alpha({\bf q})]
\langle \alpha | [Q^\dagger,Q] | \alpha \rangle$$ independent of ${\bf k}$. In the absence of spin-orbit coupling we choose $Q=s^+$, which yields $$x = {\langle s^z \rangle \over N_{\rm Mn}S } = { p\xi \over 2N_{\rm Mn}S }
\, ,
\label{Berry_1}$$ i.e., $x$ is just the ratio of itinerant-carrier spin concentration to Mn spin density. For finite spin-orbit coupling the spin of the itinerant carrier is coupled to the orbital angular momentum. If the band Hamiltonian is invariant under rotation in space, we can choose $Q=s^++l^+$ and find $$x = {\langle s^z+l^z \rangle \over N_{\rm Mn}S } \, .
\label{Berry_2}$$ In this case the correction is given by the ratio of the total angular momentum density of the itinerant carriers and the Mn concentration.
If the valence bands are described by the Kohn-Luttinger Hamiltonian, however, the total angular momentum is no longer a conserved quantity. Or, equivalently, the orbital angular momentum of the valence-band carriers couples to the crystal lattice, and the simple form Eq. (\[Berry\_2\]) no longer holds.
Spin-wave dispersion gap {#append_gap}
========================
The goal of this section is to rederive the $k=0$ limit of the spin-wave energy Eq. (\[dispersion\]) by standard perturbation theory where the perturbation describes the deviation of the spin polarization from the mean-field direction $\langle 100 \rangle$ or $\langle 111 \rangle$.
Proof that $E_{k = 0}^{++} = 0$ for $\langle 100 \rangle$ or $\langle 111 \rangle$ easy axis
--------------------------------------------------------------------------------------------
We start by showing that $$E_{k = 0}^{++} = - {1\over V} \sum_{\bf q}
\sum_{\alpha\beta}
{ f [\epsilon_\alpha({\bf q})] - f [\epsilon_\beta({\bf q})] \over
\epsilon_\alpha({\bf q}) - \epsilon_\beta ({\bf q}) }
s^+_{\alpha\beta} s^+_{\beta\alpha}
\label{E++0}$$ vanishes, if the mean-field polarization is along $\langle 100 \rangle$ or $\langle 111 \rangle$. Note that $\alpha$ and $\beta$ now label the [*same*]{} basis states.
Let ${\bf \tilde q}$ be a wavevector which is obtained from ${\bf q}$ by rotation about the $z$-axis (which is defined by the mean-field Mn spin polarization direction) by an angle $\varphi$ which respects the symmetry of the crystal. If the $z$-axis is $\langle 100 \rangle$ or $\langle 111 \rangle$, then the allowed angles are $\varphi \in \{ 0,\pi/2,\pi,3\pi/2 \} $ or $\{ 0,2\pi/3,\pi,4\pi/3 \} $, respectively. For $\langle 110 \rangle$ it would be $\varphi \in \{ 0,\pi \} $. Due to the symmetry of the crystal, the spectrum at ${\bf \tilde q}$, labeled by $\tilde \alpha$ (or $\tilde \beta$), is identical to that at ${\bf q}$, i.e., $\epsilon_{\tilde \alpha} = \epsilon_{\alpha}$. The corresponding eigenstates are connected by $|\tilde \alpha \rangle = U |\alpha \rangle$ with $U=\exp[i (s^z+l^z)\varphi]$.
Since the spin operator $s^+$ and the operator for the orbital angular momentum $l^z$ commute, the following relation is satisfied, $$\begin{aligned}
s^+_{\tilde \alpha \tilde \beta} s^+_{\tilde \beta \tilde \alpha} &=&
\langle \alpha | U^{-1} s^+ U | \beta \rangle
\langle \beta | U^{-1} s^+ U | \alpha \rangle
\nonumber \\
&=&
e^{-2 i\varphi} s^+_{\alpha \beta} s^+_{\beta \alpha} \, .\end{aligned}$$ As a consequence, the partial summation in Eq. (\[E++0\]) over all wavevectors ${\bf q}$ which are equivalent due to symmetry yields a factor $(1+e^{-i\pi}+e^{-2i\pi}+e^{-3i\pi})=0$ for the easy axis $\langle 100 \rangle$ and $(1+e^{-4i\pi/3}+e^{-8i\pi/3})=0$ for $\langle 111 \rangle$, i.e., $E^{++}_{k=0}=0$ in both cases. In contrast, the corresponding factor $(1+e^{-2i\pi})$ for $\langle 110 \rangle$ is nonzero, and $E^{++}_{k=0}$ is, in general, finite.
Relation between energy gap and $E_{k = 0}^{+-}$
------------------------------------------------
To determine the energy cost of tilting the spin polarization by small angle $\theta$ out of the mean-field direction, we add to the Hamiltonian the perturbation $$H' = \Delta \left[ -{\theta^2\over 2}s^z +
\theta \left( s^x \cos \varphi + s^y \sin \varphi \right) \right] \, ,
\label{pert}$$ and use standard perturbation theory. Here, $\varphi\in [0,2\pi)$ is the azimuth, and $\Delta=N_{\rm Mn}J_{\rm pd}S$. The linear order in $\theta$ does not contribute since $\langle s^x \rangle = \langle s^y \rangle = 0$. To obtain the quadratic order in $\theta$ we use first-order perturbation theory for the $s^z$ term in Eq. (\[pert\]) and second-order perturbation theory for the $s^x \cos \varphi + s^y \sin \varphi =
(s^+ e^{-i\varphi} + s^- e^{i\varphi})/2$ contribution. For the former we get $$\delta E' = - {\theta^2 \Delta \over 2V} \sum_{\bf q} \sum_{\alpha}
f [\epsilon_\alpha({\bf q})] \langle \alpha | s^z | \alpha \rangle
= {\theta^2 \Delta p\xi \over 4} \, ,$$ and the result for latter reads $$\begin{aligned}
\delta E'' &=& {\theta^2 \Delta^2 \over 4V} \sum_{\bf q}
\sum_{\alpha\beta} f [\epsilon_\alpha({\bf q})]
{ | \langle \alpha | s^+ e^{-i\varphi} + s^- e^{i\varphi} | \beta
\rangle |^2 \over \epsilon_\alpha({\bf q}) - \epsilon_\beta ({\bf q})}
\nonumber \\
&=& -{\theta^2\Delta^2\over 8}\left( 2 E_{k = 0}^{+-}
+ e^{-2i\varphi} E_{k = 0}^{++} + e^{2i\varphi} E_{k = 0}^{--} \right)
\, .\end{aligned}$$ If the easy axis is along $\langle 100 \rangle$ or $\langle 111 \rangle$, then $E_{k = 0}^{++} = E_{k = 0}^{--} =0$, and the energy is independent of $\varphi$. The spin wave energy at $k=0$ can now be obtained from the ratio of the energy change $\delta E'+\delta E''$ and the change of the spin $\delta S = \theta^2N_{\rm Mn}S/2$, which yields $${\Omega_{k=0}\over \Delta} = { J_{\rm pd} \over 2}
\left( { p\xi \over \Delta } - E_{k=0}^{+-} \right) \, ,$$ and, as desired, we recover the $k=0$ limit of Eq. (\[dispersion\]) for an easy-axis direction $\langle 100 \rangle$ or $\langle 111 \rangle$.
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![Easy-axis direction as a function of $K_1$ and $K_2$.[]{data-label="fig1"}](fig1.eps){width="8.cm"}
![Spin-wave dispersion for the isotropic model for itinerant-carrier density $p=0.35 \, {\rm nm}^{-3}$, impurity-spin concentration $N_{\rm Mn} = 1.0 \, {\rm nm}^{-3}$ and exchange coupling $J_{\rm pd} = 0.068 \, {\rm eV \, nm}^{-3}$ (which yields $\Delta = 0.17 \, {\rm eV}$).[]{data-label="fig2"}](fig2.eps){width="8.cm"}
![Main panel: Spin-wave dispersion for the 6-band model for itinerant-carrier density $p=0.35 \, {\rm nm}^{-3}$, impurity-spin concentration $N_{\rm Mn} = 1.0 \, {\rm nm}^{-3}$ and exchange coupling $J_{\rm pd} = 0.068 \, {\rm eV \, nm}^{-3}$ (which yields $\Delta = 0.17 \, {\rm eV}$). The momentum $k$ is chosen to be parallel to the easy axis $\langle 100 \rangle$. Inset: Spin-wave dispersion on a log-log plot (circles) and the fit $9.2\times 10^{-5} + 0.16 (k/k_D)^2$ (solid line).[]{data-label="fig3"}](fig3.eps){width="8.cm"}
![Exchange constant $A$ as a function of itinerant-carrier density $p$ for the six-band and the two-band model for two different values of $J_{\rm pd} = 0.068 \, {\rm eV \, nm}^{-3}$ (solid lines) and $0.136 \, {\rm eV \, nm}^{-3}$ (dashed lines). The impurity-spin concentration is chosen as $N_{\rm Mn} = 1.0 \, {\rm nm}^{-3}$, which yields $\Delta = 0.17 \, {\rm eV}$ (solid lines) and $\Delta = 0.34 \, {\rm eV}$ (dashed lines), respectively.[]{data-label="fig4"}](fig4.eps){width="8.cm"}
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{
"pile_set_name": "ArXiv"
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---
title: 'Cosmic-ray spectral anomaly at GeV-TeV energies as due to re-acceleration by weak shocks in the Galaxy'
---
Introduction
============
Recent measurements of cosmic rays by the ATIC [@bib:Panov2007], CREAM [@bib:Yoon2011], and PAMELA [@bib:Adriani2011] experiments have found a spectral anomaly at GeV-TeV energies. The spectrum in the TeV region is found to be harder than at GeV energies. The spectral anomaly is difficult to explain under the standard models of cosmic-ray acceleration and their propagation in the Galaxy, which predict a single power-law spectrum over a wide range in energy.
Various explanations for the spectral anomaly have been proposed. These include hardening in the cosmic-ray source spectrum at high energies [@bib:Biermann2010; @bib:Ohira2011; @bib:Yuan2011; @bib:Ptuskin2013], change in the cosmic-ray propagation properties in the Galaxy [@bib:Tomassetti2012; @bib:Blasi2012], and the effect of nearby sources [@bib:Thoudam2012; @bib:Thoudam2013; @bib:Erlykin2012; @bib:Zatsepin2013].
In this contribution, we discuss the possibility that the anomaly can be an effect of re-acceleration of cosmic rays by weak shocks in the Galaxy. This scenario was also shortly discussed recently by Ptuskin et al. 2011 [@bib:Ptuskin2011]. After acceleration by strong supernova remnant shock waves, cosmic rays escape from the remnants and undergo diffusive propagation in the Galaxy. The propagation can be accompanied by some level of re-acceleration due to repeated encounters with expanding supernova remnant shock waves [@bib:Wandel1988; @bib:Berezhko2003]. As older remnants occupy a larger volume in the Galaxy, cosmic rays are expected to encounter older remnants more often than the younger ones. Thus, this process of re-acceleration is expected to be produced mainly by weaker shocks. As weaker shocks generate a softer particle spectrum, the resulting re-accelerated component will have a spectrum steeper than the initial cosmic-ray source spectrum produced by strong shocks. As will be shown later, the re-accelerated component can dominate at GeV energies, and the non-reaccelerated component (hereafter referred to as the “normal component") dominates at higher energies.
Cosmic rays can also be re-accelerated by the same magnetic turbulence responsible for their scattering and spatial diffusion in the Galaxy. This process, which is commonly known as the distributed re-acceleration, has been studied quite extensively, and it is known that it can produce strong features on some of the observed properties of cosmic rays at low energies. For instance, the peak in the secondary-to-primary ratios at $\sim 1$ GeV/n can be attributed to this effect [@bib:Seo1994]. Earlier studies suggest that a strong amount of re-acceleration of this kind can produce unwanted bumps in the cosmic-ray proton and helium spectra at few GeV/n [@bib:Cesarsky1987; @bib:Stephens1990]. However, it was later shown that for some mild re-acceleration which is sufficient to reproduce the observed boron-to-carbon ratio, the resulting proton spectrum does not show any noticeable bumpy structures [@bib:Seo1994]. In fact, the efficiency of distributed re-acceleration is expected to decrease with energy, and its effect becomes negligible at energies above $\sim 20$ GeV/n.
On the other hand, for the case of encounters with old supernova remnants mentioned earlier, the re-acceleration efficiency does not depend on energy. It depends only on the rate of supernova explosions and the fractional volume occupied by supernova remnants in the Galaxy. Hence, its effect can be extended to higher energies compared to that of the distributed re-acceleration, as also noted in Ref. [@bib:Ptuskin2011]. In the present study, we will first determine the maximum amount of re-acceleration permitted by the available measurements on the boron-to-carbon ratio. Then, we will apply the same re-acceleration strength to the proton and helium nuclei, and check if it can explain the observed spectral hardening for a reasonable set of model parameters.
Transport equation with re-acceleration
=======================================
Following Ref. [@bib:Wandel1987], the re-acceleration of cosmic rays in the Galaxy is incorporated in the cosmic-ray transport equation as an additional source term with a power-law momentum spectrum. Then, the steady-state transport equation for cosmic-ray nuclei undergoing diffusion, re-acceleration and interaction losses can be written as, $$\begin{aligned}
\nabla\cdot(D\nabla N)&-\left[\left(\bar{n} v\sigma+\xi\right)N+\xi sp^{-s}\int^p_{p_0}du\;N(u)u^{s-1}\right]\delta(z)\nonumber\\
&=-Q\delta(z)\end{aligned}$$ where we use cylindrical spatial coordinates with the radial and vertical distance represented by $r$ and $z$ respectively, $p$ is the momentum/nucleon of the nuclei, $N(\textbf{r},p)$ represents the differential number density, $D(p)$ is the diffusion coefficient, and $Q(r,p)$ represents the source term. The first term on the left-hand side of Eq. (1) represents diffusion. The second and third terms represent losses due to inelastic interactions with the interstellar matter and due to re-acceleration to higher energies respectively, where $\bar{n}$ represents the averaged surface density of interstellar atoms, $v(p)$ the particle velocity, $\sigma(p)$ the inelastic collision cross-section, and $\xi$ corresponds to the rate of re-acceleration. The fourth term with the integral represents the generation of particles via re-acceleration of lower energy particles. It assumes that a given cosmic-ray population is instantaneously re-accelerated to form a power-law distribution with an index $s$. We neglect ionization losses and the effect of convection due to the Galactic wind as these processes are important mostly at energies below $\sim 1$ GeV/nucleon. The present study concentrates only at energies above $1$ GeV/nucleon.
The cosmic-ray propagation region is assumed as a cylindrical region bounded in the vertical direction at $z=\pm H$, and unbounded in the radial direction. Both the matter and the sources are assumed to be uniformly distributed in an infinitely thin disk of radius $R$ located at $z=0$. This assumption is based on the known high concentration of supernova remnants, and atomic and molecular hydrogens near the Galactic plane. For cosmic-ray primaries, the source term is taken as $Q(r,p)=\bar{\nu} Q(p)$, where $\bar{\nu}$ denotes the rate of supernova explosions (SNe) per unit surface area on the disk. The source spectrum is assumed to follow a power-law in total momentum with a high-momentum exponential cut-off. In terms of momentum/nucleon, it can be expressed as $$Q(p)=AQ_0 (Ap)^{-q}\exp\left(-\frac{Ap}{Zp_c}\right)$$ where $A$ and $Z$ represents the mass number and charge of the nuclei respectively, $Q_0$ is a constant related to the amount of energy $f$ injected into a cosmic ray species by a single supernova event, $q$ is the source spectral index, and $p_c$ is the high-momentum cut-off for protons. In writing Eq. (2), we assume that the maximum total momentum (or energy) for a cosmic-ray nuclei produced by a supernova remnant is $Z$ times that of the protons. We further assume that the source spectrum has a low-momentum/nucleon cut-off at $p_0$ which also serves as the lower limit in the integral in Eq. (1). Moreover, the diffusion coefficient as a function of particle rigidity is assumed to follow $D(\rho)=D_0\beta(\rho/\rho_0)^a$, where $\rho=Apc/Ze$ is the particle rigidity, $\beta=v/c$, and $c$ is the velocity of light.
In the present model, since the re-acceleration of cosmic rays is considered to be produced by their encounters with supernova remnants, it follows that re-acceleration occurs only in the Galactic disk. If $V=4\pi \Re^3/3$ is the volume occupied by a supernova remnant of radius $\Re$, then in Eq. (1), $\xi=\eta V\bar{\nu}$, where $\eta$ is a correction factor for $V$ we have introduced to take care of the unknown actual volume of the supernova remnants that re-accelerate cosmic rays. We keep $\eta$ as a parameter, and we take $\Re=100$ pc which is roughly the typical radius of a supernova remnant of age $10^5$ yr expanding in the interstellar medium with an initial shock velocity of $10^9$ cm s$^{-1}$.
The solution of Eq. (1) is obtained using the standard Green’s function technique. For sources uniformly distributed in the Galactic disk, the solution at $r=0$ is obtained as, $$\begin{aligned}
N(z,p)&=\bar{\nu} R\int^{\infty}_0 dk\; \frac{\sinh\left[k(H-z)\right]}{\sinh(kH)}\times \frac{\mathrm{J_1}(kR)}{L(p)}
\times F(p) \end{aligned}$$ where $\mathrm{J_1}$ is a Bessel function of order 1, $$L(p)=2D(p)k\coth(kH)+\bar{n}v(p)\sigma(p)+\xi,$$ $$\begin{aligned}
F(p)=Q(p)+\xi sp^{-s}\int^p_{p_0}u^s du\;Q(u)A(u)\nonumber\\
\times\exp\left(\xi s\int^p_u A(w)dw\right) \end{aligned}$$ and, the function $A$ is given by $$A(x)=\frac{1}{xL(x)}$$
Considering that the position of our Sun is very close to the Galactic plane, the cosmic-ray density at the Earth can be calculated from Eq. (3) taking $z=0$.
The first term on the right-hand side of Eq. (5) is the normal cosmic-ray component which have not suffered re-acceleration, and the second term is purely the re-accelerated component. For a given diffusion index, the high-energy spectra of the two components are shaped by their respective injection indices $q$ and $s$. As re-acceleration takes out particles from the low-energy region and puts them into the higher energy part of the spectrum, for re-acceleration by weak shocks for which $s>q$, the re-accelerated component might become visible as a bump or enhancement in the energy spectrum at a certain energy range. In the case of re-acceleration by strong shocks which produces a harder particle spectrum, say $s=q$, the effect of re-acceleration will be hard to notice as both the components will have the same spectra in the Galaxy. These have been extensively discussed in Ref. [@bib:Wandel1987].
For cosmic-ray secondaries, their equilibrium density $N_2(\textbf{r},p)$ in the Galaxy is obtained following the same procedure as for their primaries described above, but with the source term replaced by $$Q_2(\textbf{r},p)=\bar{n} v_1(p)\sigma_{12}(p)N_1(\textbf{r},p) \delta(z)$$ where $v_1$ represents the velocity of the secondary nuclei, $\sigma_{12}$ represents the total fragmentation cross-section of the primary to the secondary, and $N_1$ is the primary nuclei density. The subscripts $1$ and $2$ have been introduced to denote primary and secondary nuclei respectively.
The secondary-to-primary ratio can be calculated by taking the ratio $N_2/N_1$. For the case of no re-acceleration $\xi=0$, it can be checked that Eq. (3) reduces to the standard solution of pure-diffusion equation (see e.g., [@bib:Thoudam2008]), and also that the secondary-to-primary ratio becomes inversely proportional to the diffusion coefficient at high energies. It can be mentioned that a steeper re-acceleration index $s>q$ will produce an enhancement in the ratio at lower energies, and a harder index $s=q$ will result into significant flattening of the ratio at high energies [@bib:Berezhko2003; @bib:Wandel1987]. Thus, the effect of re-acceleration on cosmic-ray properties in the Galaxy depends strongly on the index of re-acceleration. In the present study, since we assume that re-acceleration is produced mainly by the interactions with old supernova remnants, we will only consider the case of $s>q$ with $s\gtrsim 4$. This value of $s$ corresponds to a Mach number of $\sim 1.7$ of the shocks that re-accelerate the cosmic rays.
Results and discussions
=======================
For the interstellar matter density, we consider the averaged surface density on the Galactic disk within a radius equivalent to the halo height $H$. We take $H=5$ kpc for our study, and the averaged surface density of atomic hydrogen as $\bar{n}=7.24\times 10^{20}$ atoms cm$^{-2}$ [@bib:Thoudam2013]. We assume that the interstellar medium consists of $10\%$ helium. The inelastic interaction cross-sections are taken as the same used in the calculation in Ref. [@bib:Thoudam2013].
We take the size of the source distribution $R=20$ kpc, the proton low and high-momentum cut-offs as $p_0=100$ MeV/c and $p_c=1$ PeV/c respectively, and the supernova explosion rate as $\bar{\nu}=25$ SNe Myr$^{-1}$ kpc$^{-1}$. The latter corresponds to a rate of $\sim 3$ SNe per century in the Galaxy. The cosmic-ray propagation parameters $(D_0, \rho_0, a)$, the re-acceleration parameters $(\eta, s)$ and the source parameters $(q, f)$ are taken as model parameters. They are determined based on the measured data.
![Boron-to-Carbon (B/C) ratio. *Solid line*: Present work including re-acceleration. *Dashed line*: Pure diffusion model without re-acceleration [@bib:Thoudam2013].](icrc2013-1022-01){width="47.00000%"}
We first determine $(D_0, \rho_0, a, \eta, s)$ based on the measurement data for the boron-to-carbon ratio, and the spectra for the carbon, oxygen, and boron nuclei. Their values are found to be $D_0=9\times 10^{28}$ cm$^2$ s$^{-1}$, $\rho=3$ GV, $a=0.33$, $\eta=1.02$, $s=4.5$. These values correspond to the maximum amount of re-acceleration permitted by the available boron-to-carbon data, while at the same time produces a reasonable good fit to the measured primary and secondary spectra. Figure 1 shows the result on the boron-to-carbon ratio (solid line) along with the measurement data. For comparison, we have also shown the result for the case of pure diffusion (dashed line) with no re-acceleration $(\eta=0)$ taken from Ref. [@bib:Thoudam2013]. The good fit carbon and oxygen source parameters are found to be $q_C=2.24, f_C=0.024\%$, and $q_O=2.26$, $f_O=0.025\%$ respectively, where the $f$’s are given in units of $10^{51}$ ergs. The present calculation assumes a force-field solar modulation parameter of $\phi=450$ MV.
![*Top*: Proton spectrum. *Bottom*: Helium spectrum. The lines represent our results. For the data, see the experiments listed in Ref. [@bib:Thoudam2013].](icrc2013-1022-02 "fig:"){width="47.00000%"}\
![*Top*: Proton spectrum. *Bottom*: Helium spectrum. The lines represent our results. For the data, see the experiments listed in Ref. [@bib:Thoudam2013].](icrc2013-1022-03 "fig:"){width="47.00000%"}
Using the same values of $(D_0, \rho_0, a, \eta, s)$ obtained above, we calculate the spectra for the proton and helium nuclei. The results are shown in Figure 2, where the top panel represents proton and the bottom panel represents helium. The lines represent our results, and the data are the same as used in Ref. [@bib:Thoudam2013]. The source parameters used are $q_p=2.21, f_p=6.95\%$ for protons, and $q_{He}=2.18, f_{He}=0.79\%$ for helium, and we use the same solar modulation parameter as given above. It can be seen that our results are in good agreement with the measured data, and explain the observed spectral anomaly between the GeV and TeV energy regions. Below $\sim 200$ GeV/n, our model spectrum is dominated by the re-accelerated component while above, it is dominated by the normal component.
![Iron spectrum. The line represents our result. For the data, see the experiments listed in Ref. [@bib:Thoudam2013].](icrc2013-1022-04){width="47.00000%"}
The effect of re-acceleration is stronger in the case of protons than helium which is due to the larger inelastic collision losses for helium. This result into more prominent spectral differences in the GeV-TeV region for protons than for helium. For heavier nuclei for which the inelastic cross-sections are much larger, the re-acceleration effect is expected to be negligible. Figure 3 shows our result for iron nuclei. The calculation assumes $q_{Fe}=2.28$, and $f_{Fe}=4.9\times 10^{-3}\%$. As expected, the re-acceleration effect is hard to notice in Figure 3, and the model spectrum above $\sim 20$ GeV/n follow approximately a single power-law unlike the proton and helium spectra.
Conclusions
===========
We have shown that the spectral anomaly at GeV-TeV energies, observed for the proton and helium nuclei, can be an effect of re-acceleration by weak shocks associated with old supernova remnants in the Galaxy. The re-acceleration effect is shown to be important for light nuclei, and negligible for heavier nuclei such as iron. Our prediction of decreasing effect of re-acceleration with the elemental mass can be checked by future sensitive measurements of heavier nuclei at TeV/n energies.
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{
"pile_set_name": "ArXiv"
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abstract: '[**The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.**]{} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$'
address:
- |
Institut Camille Jordan\
Université de Saint-Étienne\
42023 Saint-Étienne Cedex 2, France
- |
Institut Camille Jordan\
Université de Saint-Étienne\
42023 Saint-Étienne Cedex 2, France
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Department of Mathematics\
Indian Institute of Technology\
Roorkee–247667, India
- |
School of Mathematics\
Tata Institute of Fundamental Research\
Mumbai–400005, India
author:
- Alain Faisant
- Georges Grekos
- Ram Krishna Pandey
- Sai Teja Somu
date: 'April, 9, 2019'
title: Additive complements for a given asymptotic density
---
Introduction
============
The purpose of this paper is to introduce a new, up to our knowledge, subject of research, resolving a few particular cases.
Two subsets, not necessarily distinct, $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ are called [*additive complements*]{} if $A+B$ contains all, except finitely many, positive integers; that is, if the set $\mathbb{N}\setminus(A+B)$ is finite. From the huge literature on this topic, we just mention the papers [@L], [@Nar] and [@D] among the initial ones, and the papers [@FC] and [@R] among the last ones. The reader may find there information and more references. A set $A$ being additive complement of itself, that is a set $A$ such that $\mathbb{N}\setminus(2A)$ be finite (where we put $2A:=A+A),$ is called an [*asymptotic basis*]{} of [*order*]{} 2.
In this paper we are interested in what happens when one asks that the density of the sumset $A+B$ is equal to a given value $\alpha,$ $0\leq\alpha\leq 1.$ As density concept we use the asymptotic density, defined below.
\[definition1\] Let $X$ be a subset of $\mathbb{N}$ and $x$ a real number. For $x\geq1,$ we put $X(x):=|X\cap[1,x]|.$ For $x<1,$ we put $X(x)=0.$ We define the [*asymptotic*]{} (also called [*natural*]{}) [*density*]{} of $X$ as $$dX:=\lim_{x\rightarrow+\infty}{{X(x)}\over x}$$ provided that the above limit exists. The [*lower*]{} and the [*upper asymptotic densities*]{}, denoted by ${\underline d}X$ and ${\overline d}X,$ respectively, are defined by taking in the above formula the lower and the upper limits, respectively, which always exist.
In the classical situation the existence of additive complements is obvious and the main problem is to find “thin" subsets of $\mathbb{N}$ being additive complements or asymptotic bases. Our first question in this study is to guarantee the existence of such sets $A$ and $B$. Precisely, in this paper, we establish the existence of $A$ for a given finite set $B$. Furthemore, we also consider the case $B = A$, in this existence. Once the existence has been established, two questions naturally arising are to find [*thin*]{} (similarly [*thick*]{}) sets verifying the required conditions.
#### Notations
$|S|$ denotes, according to the context, either the cardinality of the finite set $S$ or the length of the interval $S.$ All small letters, except $f,g, c, d, x, \alpha$ and $\theta,$ represent nonnegative integers. Let $x$ be a real number. We denote by $\left \lfloor{x}\right \rfloor$ the “integer part" of $x$ and by $\{x\}$ the “fractional part" of $x.$ Thus $$x=\left \lfloor{x}\right \rfloor + \{x\}$$ where $\left \lfloor{x}\right \rfloor$ is an integer and $0\leq\{x\}<1,$ this writing being [*unique*]{}.\
In Section 2, we indicate or prove some properties of asymptotic density and in Section 3, we prove the next theorem.
\[thm1\] Let $\alpha$ be a real number, $0\leq\alpha\leq1,$ and $B$ a finite subset of $\mathbb{N}.$ Then there exists a set $A\subset \mathbb{N}$ such that $d(A+B)=\alpha.$
Let us now suppose that $\alpha$ is given and $A=B.$ Our goal was to prove the existence of a set $A$ such that $d(2A)=\alpha.$ The proof(s) can be generalized to sumsets of more summands $A$ ; precisely, we prove the following more general result, showing that, for the constructed set $A,$ the density of the sumsets $jA,$ $1\leq j\leq k,$ increases regularly with $j.$ Proofs are of different form, according to the nature of $\alpha$ : rational or irrational. Here is the formulation of the theorem proved in Section 4.
\[thm2\] Let $\alpha$ be a real number, $0\leq\alpha\leq1,$ and $k$ an integer, $k\geq2.$ Then there exists a subset $A$ of $\mathbb{N}$ such that for every $j,$ $1\leq j\leq k,$ one has $$d(jA)={{j\alpha}\over k}.$$ In particular, $ d(kA)=\alpha.$
We give a list of open questions and prospects for further research in Section 5.\
Auxiliary results
=================
We mention next lemma without proof as it is a basic result concerning the density. Although the lemma is elementary in nature, it is quite useful for our study.
Let the set $X$ in Definition \[definition1\] be infinite, $X=:\{a_1<a_2<\dots\}.$ Then $${\underline d}X:=\liminf_{x\rightarrow+\infty}{{X(x)}\over x}=\liminf_{k\rightarrow+\infty} {k\over{a_k}},$$ and $${\overline d}X:=\limsup_{x\rightarrow+\infty}{{X(x)}\over x}=\limsup_{k\rightarrow+\infty} {k\over{a_k}}.$$
As a consequence, we get the next property.
[**Property (i)**]{} Let $X$ be a subset of $\mathbb {N}$ and $\theta$ a real number, $\theta\geq1.$ Put $\theta . X:=\{ \left \lfloor{\theta a}\right \rfloor ;a\in X\}.$ Then ${\underline d}(\theta . X)=\theta^{-1}~{\underline d} X$ and ${\overline d}(\theta . X)=\theta^{-1}~{\overline d} X.$
Put $\mathbb{N}_+:=\mathbb{N}\setminus\{0\}=\{1,2,\dots\}.$ The uniform distribution of the sequence $(\{n\theta\})_{n\in \mathbb{N}_+}$ when $\theta$ is irrational ([@KN], p. 8; [@SP], p. 2 - 72) amounts to:
[**Property (ii)**]{} Let $I$ be a subinterval of $[0,1[$ and $\theta$ an irrational real number. Then $$d\{n\in \mathbb{N}_+;\{n\theta\}\in I\}=|I|.$$
As a corollary of the previous properties, we get:
\[cor1\] Let $I$ be a subinterval of $[0,1[$ and $\theta$ an irrational real number, $\theta>1.$ Then $$d\{\lfloor{n\theta}\rfloor;n\in\mathbb{N}_+,\{n\theta\}\in I\}=|I|/\theta.$$
Proof of Theorem \[thm1\]
=========================
If $\alpha$ is 0 or 1, the answer is easy: take, respectively, $A$ to be the set of powers of 2 or $A=\mathbb{N}.$
[We shall suppose in the sequel that]{} $0<\alpha<1.$
Without loss of generality, we can suppose that $\min B=0.$
Suppose that the theorem is proved when $b_1:=\min B=0.$ Now let $b_1>0.$ Put $B':=B-b_1.$ By the case $\min B=0,$ there is $A'\subset\mathbb{N}$ such that $d(A'+B')=\alpha.$ Put $A_1:=A'-b_1\subset\{-b_1,-b_1+1,\dots,-1\}\cup \mathbb{N}$ and $A:=\{a\in A_1;a\geq0\}.$ Notice that $$A+B\subset A_1+B=A'+B'\subset\mathbb{N}$$ so $A_1+B$ has asymptotic density $\alpha.$ Then notice that $$(A_1+B)\setminus(A+B)\subset\cup_{k=-1}^{-b_1} \{k+B\}.$$ The set in the right hand member is finite. It yields that $d(A+B)=d(A_1+B)=\alpha$ which proves the Lemma.
[**Continuation of the proof of Theorem \[thm1\].**]{} We suppose from now on that $\min B=0.$ Let also $b:=\max B,$ $k:=|B|.$ Of course the theorem is trivial if $k=1.$ We suppose in the sequel that $k\geq2;$ consequently $b\ge1.$ The set $A=\{a_1,a_2,\dots\},$ $a_1<a_2<\cdots,$ will be defined recursively. We define $a_1:=\min A$ in the following manner: $$a_1:=\min\{a\ge0;~\max_{n\geq1}{{(a+B)(n)}\over n}\leq\alpha\}.\leqno(1)$$ [*Remark on*]{} (1). Notice that $a_1=0$ if and only if, for all $n\geq1,$ $B(n)\leq \alpha n.$ For given $a$ and all $n\geq1,$ let $f(n):={{(a+B)(n)}\over n}.$ The function $f$ takes nonnegative values and is decreasing when $n\ge{a+b};$ so, for fixed $a,$ the maximum exists. This maximum is attained for (at least) an $n=n_1,$ $a\le{n_1}\le{a+b},$ and verifies, when $a\neq 0,$ $$\max_{n\geq1}f(n)=f(n_1)\le{k\over{n_1}}\le{k\over a}.$$ So this maximum is less than or equal to $\alpha$ provided that $a\ge{k\over\alpha}.$ It follows that the minimum in formula (1) exists and hence $a_1$ is well defined and $a_1\le\lceil{k\over\alpha}\rceil.$
[*Recursion.*]{} Suppose that $a_1,a_2,\dots,a_m$ have been defined, and let $A_m:=\{a_1,a_2,\dots,a_m\}.$ We suppose that, for all $n\geq1,$ $${(A_m+B)(n)\over n}\leq\alpha.$$ Then we define $a_{m+1}$ in the following manner: $$a_{m+1}:=\min\{a>{a_m};~\max_{n\geq a}{{((A_m\cup\{a\})+B)(n)}\over n}\leq\alpha\}.\leqno(2)$$ [*Remark on*]{} (2). Notice that $a_{m+1}=a_m+1$ if and only if $((A_m\cup\{a_m+1\})+B)(n)\leq \alpha n$ for all $n\geq{a_m+1}.$ Otherwise, $a_{m+1}>a_m+1.$ For fixed $a>a_m,$ let $g(n):={{((A_m\cup\{a\})+B)(n)}\over n},$ for all $n\geq a. $ The function $g$ takes nonnegative values and is decreasing when $n\ge{a+b};$ so, for fixed $a,$ the maximum exists. This maximum is attained for (at least) an $n=n_{m+1},$ $a\le{n_{m+1}}\le{a+b},$ and verifies $$\max_{n\geq a}g(n)=g(n_{m+1})\le{{k(m+1)}\over{n_{m+1}}}\le{{k(m+1)}\over a}.$$ So this maximum is less than or equal to $\alpha$ provided that $a\ge{{k(m+1)}\over\alpha}.$ It follows that the minimum in formula (2) exists and hence $a_{m+1}$ is well defined and $a_{m+1}\le\lceil{{k(m+1)}\over\alpha}\rceil.$\
Let us now observe that, for all $n\in\mathbb{N},$ we have $$(A+B)(n)\le\alpha n.$$ This is clear when $n=0$ or $n<a_1.$ Otherwise, there is $m\ge1$ such that $a_m\le n<a_{m+1}.$ Then we have $(A+B)(n)=(A_m+B)(n)\le\alpha n.$
This implies that ${\overline d}(A+B)\le\alpha$.\
It remains to prove that ${\underline d}(A+B)\ge\alpha.$ To do that, it is sufficient to prove that, for any $\varepsilon>0,$ we have ${\underline d}(A+B)\ge\alpha-\varepsilon.$
In what follows, we shall need the next property of the counting function $n\mapsto(A+B)(n)$ of the set $A+B.$ In its (short) proof, we shall need the fact that $0\in B.$
[**Property (iii)**]{} If $a\in A,$ $a>1,$ then $${{(A+B)(a)}\over a}\ge{{(A+B)(a-1)}\over {a-1}}.$$
Let $y:=(A+B)(a-1).$ Since $a\in A$ and $0\in B,$ we get that $a=a+0\in A+B$ and so $(A+B)(a)=y+1.$ We have to show that $(y+1)/a\geq y/(a-1).$ The verification is straightforward and this proves Property (iii).
[**Continuation of the proof of Theorem \[thm1\].**]{} Let $0<\varepsilon<\alpha.$ We shall prove that ${\underline d}(A+B)\ge\alpha-\varepsilon$ by contradiction. Suppose that ${\underline d}(A+B)<\alpha-\varepsilon.$ Then the set $$S:=\{n\in\mathbb{N}; (A+B)(n)<(\alpha-\varepsilon)n\}$$ is infinite.
We observe that, for $0<\alpha<1,$ the constructed set $A$ is neither finite nor cofinite. $A$ is a collection of finite “blocks", each block consisting of one or of a finite number of consecutive integers, and two consecutive blocks are separated by a “hole" of length at least 2.
The preceding Property (iii) implies that if an element $a$ of $A$ , $a>1,$ belongs to $S$, then $a-1$ also belongs to $S.$ And since $a$ belongs to a block of $A,$ the last (the biggest) element of the hole just before the block to which $a$ belongs, is also an element of $S.$ We conclude that the set $S':=S\setminus A$ is infinite.
From $S'$ we can extract an infinite, strictly increasing, sequence of positive integers $(N_t)_{t\ge1}$ such that to each $t\ge1$ there corresponds an index $m_t,$ verifying :
\(i) $a_{m_t}<N_t<a_{m_t+1},$ and
\(ii) $1\le{m_1} < m_2 < \cdots$ .
Let us fix now an index $t\ge1.$ Recall that $$(A+B)(N_t)=(A_{m_t}+B)(N_t)<(\alpha-\varepsilon)N_t~. \leqno(3)$$ Let $A':=A_{m_t}\cup\{N_t\}.$ By the formula (2), we get that $$\max_{n\ge N_t}{1\over n}(A'+B)(n)>\alpha.$$ So there is an integer $n',$ $N_t\le n'\le N_t+b,$ such that $$(A'+B)(n')>\alpha n' ~.\leqno(4)$$ By the construction, $$(A_{m_t}+B)(n')\le \alpha n' ~.\leqno(5)$$ We observe that $$(A_{m_t}+B)(n')-(A_{m_t}+B)(N_t)\le n'- N_t\le b ~,$$ which implies, using (3), that $$(A_{m_t}+B)(n')\le (A_{m_t}+B)(N_t)+b<(\alpha-\varepsilon)N_t+b ~.\leqno(6)$$ We also observe that $$(A'+B)(n')-(A_{m_t}+B)(n')\le k ~,$$ which implies, using (4), that $$(A_{m_t}+B)(n')\ge(A'+B)(n')-k>\alpha n'-k\ge\alpha N_t-k~.\leqno(7)$$ The left members of (6) and (7) are the same. Comparing their right members, we get that $\varepsilon N_t < k+b.$
This is not true for any $t$, since $N_t$ tends to infinity. This implies that the hypothesis ${\underline d}(A+B)<\alpha-\varepsilon$ is false and completes the proof of Theorem \[thm1\].
Proof of Theorem \[thm2\]
=========================
If $\alpha$ is 0 or 1, the answer is easy: take, respectively, $A$ to be the set of powers of 2 or $A=\mathbb{N}.$
[We shall suppose in the sequel that]{} $0<\alpha<1.$\
We distinguish two cases
$\clubsuit$ [**Case A**]{}: $\alpha$ rational; say, $\alpha={m\over n},$ where $m,n$ are integers, $1\leq m\leq n-1.$ It is not necessary that $\gcd(m,n)=1.$ Our construction of such a set $A$ is [simpler]{} (see also the remark at the end of the proof of Case A) when $m\geq3.$ So multiplying, if necessary, the terms of the fraction for $\alpha$ by 2 or by 3, [we will suppose in the sequel that]{} $3\leq m\leq n-1.$
Let $H:=\{0,1,\dots,m-2,m\}.$ We shall prove that the set $$A:=\cup_{h\in H}(nk\cdot \mathbb{N}+h)$$ verifies $d(jA)=j\alpha/k,$ for all $j,$ $1\leq j\leq k.$ To prove that, let us first observe that $$jA=nk\cdot \mathbb{N}+jH;$$ hereon it is easy to verify that each element of the left member belongs to the right member and vice versa.
Notice that $k~{\max H}=km<nk.$ The set $jA$ being a finite union of mutually disjoint arithmetic progressions of difference $nk,$ we have $$d(jA)=\sum_{t\in jH}d(nk\cdot \mathbb{N}+t)=|jH|{1\over{nk}}.$$ But $jH=\{0,1,\dots,jm-2,jm\}$ because $jm=j~{\max H}\in jH,$ $jm-1\not\in jH,$ and every nonnegative integer less than $jm-1$ belongs to $jH.$ For example, $jm-2=(j-1)m+(m-2)\in jH$ ; or $jm-3=(j-1)m+(m-3)\in jH.$ So $$d(jA)=|jH|{1\over{nk}}={{jm}\over{nk}}={{j\alpha}\over k}~,~ 1\leq j\leq k~.$$
It is possible to invent specific constructions for $m=1$ and for $m=2.$
$\clubsuit$ [**Case B**]{}: $\alpha$ irrational, $0<\alpha<1.$
We put $\theta:=1/\alpha.$ We recall our notation $\mathbb{N}_+:=\mathbb{N}\setminus\{0\}=\{1,2,\dots\}.$ We shall prove that the set $$A:=\{\lfloor{n\theta}\rfloor;n\in \mathbb{N}_+, \{n\theta\}<{1\over k}\}$$ verifies $d(jA)=j\alpha/k,$ for all $j,$ $1\leq j\leq k.$
For $j=1,$ this follows from Corollary \[cor1\].\
[We suppose in the sequel that]{} $2\leq j\leq k.$\
${\spadesuit}$ We firstly prove that ${\overline d}(jA)\leq j\alpha/k.$\
To do that, we begin by proving that $jA\subset T_j$ where $$T_j:=\{\lfloor{m\theta}\rfloor; m\in \mathbb{N}_+, \{m\theta\}<j/k \}.\leqno (8)$$ An element of $jA$ is of the form $a_1+\cdots+a_j$ where $a_i=\lfloor{n_i\theta}\rfloor\in A,$ $1\leq i\leq j.$ This yields
$a_1+\cdots+a_j=\lfloor{n_1\theta}\rfloor
+\cdots+\lfloor{n_j\theta}\rfloor={n_1\theta}
-\{n_1\theta\}+\cdots+{n_j\theta}-\{n_j\theta\}$
and consequently
$(n_1+\cdots+n_j)\theta=a_1+\cdots+a_j + \{n_1\theta\}+\cdots+\{n_j\theta\}.$
We have
$0<\{n_1\theta\}+\cdots+\{n_j\theta\}<j{1\over k}\leq1$,
and so $\{n_1\theta\}+\cdots+\{n_j\theta\}$ is the fractional part and $a_1+\cdots+a_j$ is the integer part of $(n_1+\cdots+n_j)\theta.$ In other terms and according to Definition (8), $a_1+\cdots+a_j$ belongs to $T_j.$ Since, by Corollary \[cor1\], $T_j$ has asymptotic density $j\alpha/k,$ the desired inequality follows.
${\spadesuit}$ We shall now prove that ${\underline d}(jA)\geq j\alpha/k.$\
This will be done in the following way. We fix a real number $\varepsilon,$ $0<\varepsilon<{1\over{4k}},$ and we shall prove that $${\underline d}(jA)\geq\alpha({j\over k}-\varepsilon).\leqno (9)$$ Taking the limit for $\varepsilon$ tending to zero in (9) gives the desired inequality for ${\underline d}(jA).$
To prove (9), we introduce the set $$B:=\{\lfloor{N\theta}\rfloor;N\in \mathbb{N}_+,{\varepsilon\over2}\leq\{N\theta\}<{j\over k}-{\varepsilon\over2}\}\leqno (10)$$ which, by Corollary \[cor1\], has asymptotic density $({j\over k}-{\varepsilon\over2}-{\varepsilon\over2})/\theta=\alpha({j\over k}-\varepsilon)$ and we will verify that [almost all]{} (that is, all except a finite number) [elements of]{} $B$ [belong to]{} $jA;$ this implies (9).
Let $\ell:={j\over k}-\varepsilon,$ which is a positive real number $(\ell>{2\over k}-{1\over{4k}}>0)$ less than 1 $(\ell\leq{k\over k}-\varepsilon<1).$ We split the interval $[{\varepsilon\over2},{j\over k}-{\varepsilon\over2})$ into $j$ intervals of equal length $\ell/j$ : $$[{\varepsilon\over2},{j\over k}-{\varepsilon\over2})=\cup_{i=0}^{j-1}I_i$$ where $$I_i:=[{\varepsilon\over2}+{i\ell\over j},{\varepsilon\over2}+{(i+1)\ell\over j})~~,~~0\leq i\leq j-1~.$$ The set $B$ splits into $j$ sets $B=\cup_{i=0}^{j-1}B_i$, where $B_i:=\{\lfloor{N\theta}\rfloor;N\in \mathbb{N}_+, \{N\theta\}\in I_i\},$ and it will be sufficient (and necessary!) to prove that, for each $i,$ all except finitely many elements of $B_i$ lie in $jA.$ Here is the procedure.
First, one can easily verify that $$0 \le{\varepsilon\over2}+{(i+1)\ell\over j}-{1\over k}<{\varepsilon\over2}+{i\ell\over j}.$$ By the uniform distribution modulo 1 of the sequence $(\{n\theta\})_n$ (Property (ii)), there is a positive integer $m_i$ such that $${\varepsilon\over2}+{(i+1)\ell\over j}-{1\over k}<(j-1)\{m_i\theta\}<
{\varepsilon\over2}+{i\ell\over j}.\leqno (11)$$ We have that $\lfloor{m_i\theta}\rfloor$ belongs to $A$ since $$\{m_i\theta\}<{1\over{j-1}}({\varepsilon\over2}+{i\ell\over j})\leq{1\over{j-1}}({\varepsilon\over2}+{(j-1)\ell\over j})={\varepsilon\over{2(j-1)}}+{1\over j}\ell=$$ $$={\varepsilon\over{2(j-1)}}+{1\over j}({j\over k}-\varepsilon)={1\over k}-\varepsilon({1\over j}-{1\over{2(j-1)}})= {1\over k}-\varepsilon{{j-2}\over{2j(j-1)}}\leq {1\over k}.$$ We shall prove that, for every $N>(j-1)m_i$ such that $\lfloor{N\theta}\rfloor$ belongs to $B_i,$ $\lfloor{N\theta}\rfloor$ belongs also to $jA.$
Since $\lfloor{N\theta}\rfloor$ belongs to $B_i,$ we have that $${\varepsilon\over2}+{i\ell\over j}\leq\{N\theta\}<{\varepsilon\over2}+{(i+1)\ell\over j}.\leqno(12)$$ Putting (11) and (12) together, gives $$0<\{N\theta\}-(j-1)\{m_i\theta\}<{1\over k}. \leqno (13)$$ From the equality $N\theta=(N-(j-1)m_i)\theta+(j-1)m_i\theta,$ taking the integer part and the fractional part of each multiple of $\theta$, we get $$\lfloor{N\theta}\rfloor+ \{N\theta\}-(j-1)\{m_i\theta\}=
\lfloor{(N-(j-1)m_i)\theta}\rfloor+\{(N-(j-1)m_i)\theta\}+(j-1)\lfloor{m_i\theta}\rfloor. \leqno(14)$$
By the uniqueness of decomposition of a real number into its integer part and its fractional part, the inequality (13) implies that the fractional parts appearing in (14) verify
$$\{N\theta\}-(j-1)\{m_i\theta\}=\{(N-(j-1)m_i)\theta\} \leqno(15)$$ and this, combined with (14), gives that $$\lfloor{N\theta}\rfloor= \lfloor{(N-(j-1)m_i)\theta}\rfloor +(j-1)\lfloor{m_i\theta}\rfloor. \leqno(16)$$ As observed before, $\lfloor{m_i\theta}\rfloor\in A.$ Because of (15) and (13), $\lfloor{(N-(j-1)m_i)\theta}\rfloor$ belongs to $A$ and (16) gives us that $\lfloor{N\theta}\rfloor\in jA.$ This completes the proof of (9) and of the whole Theorem \[thm2\].\
[**Added in proof.-**]{} In [@V] the author resolves in a more general context ($\mathbb{Z}^t$ instead of $\mathbb{N}$) a problem which, in some sense, contains as special case the problem solved in the above theorem. When $k=2,$ the meaning of our sentence “in some sense" is as follows: Given two positive real numbers $\alpha_1$ and $\alpha_2$ such that $\alpha_1+\alpha_2\leq1$ and a third real number $\gamma,$ $\alpha_1+\alpha_2\leq\gamma\leq1,$ Bodo Volkmann [@V] constructs sets $A_1, A_2$ of natural numbers satisfying $d(A_1)=\alpha_1$, $d(A_2)=\alpha_2$ and $d(A_1+A_2)=\gamma.$ The construction uses ideas similar to the ours. The principle is the same: to use uniform distribution of fractional parts in order to obtain sets of integers with prescribed density. In the case when $\alpha_1=\alpha_2,$ the sets $A_1,A_2$ are not equal. The set $A_1$ is constructed in a way similar to the one used in our proof, but the set $A_2$ is constructed in a specific way in order to obtain $d(A_1+A_2)=\gamma.$ Even in the case when $\gamma=2\alpha_1=2\alpha_2$, the set $A_2$ is different from $A_1.$ We observed that in this particular case Volkmann’s construction can be slightly modified to give $A_2=A_1$ thus providing another proof, based on Lemma 2 of [@V], of our theorem. A similar remark is valid when $k\geq3.$
Future prospects
================
We separate this section into questions: [*Q1*]{} to [*Q7*]{}.
#### Q1 - Thin sets
Concerning the case $B=A,$ it would be interesting to study the existence of thin sets $A$ verifying $d(kA)=\alpha.$ For [*additive bases*]{} $A$ (that is, for sets $A$ verifying $kA= \mathbb{N}$ for some $k,$ called “order" of the basis $A)$ this was done by Cassels [@C] (see also [@HR], p. 35-43) where, for every $k\geq2,$ a “thin" basis of order $k$ was found. As in the case of additive bases, in our situation, with $\alpha>0,$ the condition $A(x)\geq c x^{1/k}$ is necessary. In Cassels’ construction, the basis corresponding to the order $k$ verifies $A(x)\leq c' x^{1/k}.$ The question here is to find sets $A$ verifying $d(kA)=\alpha>0$ and $A(x)\leq c'' x^{1/k}.$ Here is a related question: is it possible to extract from Cassels’ thin basis $A$ (such that $kA= \mathbb{N})$ a set $A'\subset A$ such that $d(kA')=\alpha?$ If this is possible, the condition $A(x)\leq c'' x^{1/k}$ is automatically verified.
[**Remark.-**]{} The above mentioned Cassels’ thin bases allow to answer the above question when $\alpha=1/n.$ Take the Cassels asymptotic basis $C=\{c_1<c_2<\dots\}$ [@C] (see also [@HR] , p. 37) of order $n$. It verifies $c_m=\beta m^n+O(m^{n-1}).$ Now a solution to the above asked question is to take $A:=\{nx;x\in C\}.$ For other values of $\alpha$ the question remains open.
Concerning the case of a given finite set $B$ considered in this paper, we see two questions:
1. To evaluate the density (or the upper and the lower densities) of the greedily constructed set $A.$ The constructed set seems to be “the thickest".
2. To determine the more thin set $A$ that verifies $d(A+B)=\alpha$: A necessary condition is that ${\underline d}A\geq\alpha/|B|.$
#### Q2 - Thick sets
In the case of bases, the thickest set $A$ verifying $kA=\mathbb{N}$ is $A=\mathbb{N}.$ What are thick sets $A$ satisfying $d(kA)=\alpha$ when $\alpha<1?$ In the case $\alpha={1\over r},$ where $r$ is an integer, $r\geq 2,$ the answer is trivial: take $A=\{0,r,2r, 3r,\dots\}.$ But in general the answer is not obvious. It may depend on the nature of $\alpha$: rational or irrational.
#### Q3 - $B$ infinite
What happens for given infinite $B?$ Find necessary or sufficient (or both!) conditions on $B$ (on the upper and lower densities of $B$) such that $A$ exists. Our greedy method of Section 2, with some supplementary considerations on formulas (1) and (2), allows to construct a set $A$ such that ${\overline d}(A+B)\leq\alpha$ when $dB=0.$ But we are not able to prove that $d(A+B)=\alpha;$ nor to disprove it for a specific set $B.$
#### Q4 - Other densities
Replace asymptotic density by other densities. See [@G] for a list of definitions. For instance, the [*exponential density*]{}, defined as \[compare with Definition 1.1 of asymptotic density in Section 1\] $${\varepsilon}X:=\lim_{x\rightarrow+\infty}{{\log X(x)}\over {\log x}},$$ could be of interest. That is, given $B$ and $\alpha,$ is there $A$ such that the exponential density of $A+B$ is equal to $\alpha?$ Instead of using specific (concrete) definitions of density, one could use axiomatically defined densities which generalize some of the usual concepts of density; see [@FS], [@LT1] and [@LT2]. Concerning the analog of Theorem 1.3, the cases of Schniremann density and of [*lower*]{} asymptotic density were already considered (but only as for existence of a set $A$) in [@Le], [@Ch] and [@Nat], where best possible results to Mann’s and Kneser’s theorems are proved.
#### Q5 - Couple of densities
It is possible to consider the initial problem and to ask all the above questions replacing $\alpha$ by two real numbers $\alpha'$ and $\alpha'',$ $0\leq\alpha'\leq\alpha''\leq1,$ that will be, respectively, the lower and the upper densities. For instance, given $\alpha'$ and $\alpha''$ as above and $k\geq2,$ find a set $A$ such that ${\underline d}(kA)=\alpha'$ and ${\overline d}(kA)=\alpha''.$
#### Q6 - Generalization
Given a subset $B$ of $\mathbb{N}$, finite or of zero asymptotic density, a real number $\alpha$, $0\leq\alpha\leq1,$ and an integer $k\geq2$, is there a set $A\subset \mathbb{N}$ such that $d(B+kA)=\alpha$? Search for “thin" and for “thick" such sets $A.$
#### Q7 - Last but not least: $A$ and $B$
The more studied question on classical additive complements is to compare the functions $(A+B)(x)$ and $A(x)B(x).$ Obviously $(A+B)(x)\leq A(x)B(x).$ So to have “thin" sets $A$ and $B$ means that $A(x)B(x)$ is not much bigger than $(A+B)(x).$ In the classical case, $(A+B)(x)$ is equal, up to a constant, to $x.$ In our case, with $\alpha>0,$ $(A+B)(x)$ is “equivalent" to $\alpha x:$ $\lim_{x\rightarrow+\infty}(A+B)(x)/{\alpha x}=1.$ So questions studied in [@Nar], [@D] or [@FC] and finally in [@R] can be formulated with $\alpha x$ in place of $x.$
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors are thankful
- to Salvatore Tringali for having suggested the idea used in the proof of Case A of Theorem \[thm2\];
- to H[é]{}di Daboussi, Władysław Narkiewicz, Jo[ë]{}l Rivat, Andrzej Schinzel and Bodo Volkmann for fruitful discussions;
- to the anonymous referee for useful observations on the initial version.
[HD]{}
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M. B. Nathanson, *Best possible results on the density of subsets*, in “Analytic number theory", Proc. Conf. in honor of Paul T. Bateman, Urbana/IL (USA) 1989, Progr. Math. [85]{}, Birkh[ä]{}user (1990), 395-403.
I. Z. Ruzsa, *Exact additive complements*, The Quarterly Journal of Mathematics [68]{} (2017), 227–235.
O. Strauch and [Š]{}. Porubsk[ý]{}, *Distribution of sequences: a sampler*, Peter Lang, 2005; corrected version on-line available at [https://math.boku.ac.at/udt/]{}
B. Volkmann, *On uniform distribution and the density of sum sets*, Proc. Amer. Math. Soc. [8]{} (1957), 130–136.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present RCSED[^1], the value-added Reference Catalog of Spectral Energy Distributions of galaxies, which contains homogenized spectrophotometric data for 800,299 low and intermediate redshift galaxies ($0.007<z<0.6$) selected from the Sloan Digital Sky Survey spectroscopic sample. Accessible from the Virtual Observatory (VO) and complemented with detailed information on galaxy properties obtained with the state-of-the-art data analysis, RCSED enables direct studies of galaxy formation and evolution during the last 5 Gyr. We provide tabulated color transformations for galaxies of different morphologies and luminosities and analytic expressions for the red sequence shape in different colors. RCSED comprises integrated $k$-corrected photometry in up-to 11 ultraviolet, optical, and near-infrared bands published by the GALEX, SDSS, and UKIDSS wide-field imaging surveys; results of the stellar population fitting of SDSS spectra including best-fitting templates, velocity dispersions, parameterized star formation histories, and stellar metallicities computed for instantaneous starburst and exponentially declining star formation models; parametric and non-parametric emission line fluxes and profiles; and gas phase metallicities. We link RCSED to the Galaxy Zoo morphological classification and galaxy bulge+disk decomposition results by Simard et al. We construct the color–magnitude, Faber–Jackson, mass–metallicity relations, compare them with the literature and discuss systematic errors of galaxy properties presented in our catalog. RCSED is accessible from the project web-site and via VO simple spectrum access and table access services using VO compliant applications. We describe several SQL query examples against the database. Finally, we briefly discuss existing and future scientific applications of RCSED and prospectives for the catalog extension to higher redshifts and different wavelengths.'
author:
- 'Igor V. Chilingarian, Ivan Yu. Zolotukhin, Ivan Yu. Katkov, Anne-Laure Melchior, Evgeniy V. Rubtsov, Kirill A. Grishin'
bibliography:
- 'biblib.bib'
title: 'RCSED – A Value-Added Reference Catalog of Spectral Energy Distributions of 800,299 Galaxies in 11 Ultraviolet, Optical, and Near-Infrared Bands: Morphologies, Colors, Ionized Gas and Stellar Populations Properties'
---
Introduction and Motivation
===========================
During the last decade we witnessed a breakthrough in wide field imaging surveys across the electromagnetic spectrum. The new era started with the Sloan Digital Sky Survey (SDSS) that used a 2.5-m telescope and covered over 11,600 sq. deg. of the sky in 5 optical photometric bands ($ugriz$) down to the 22nd AB magnitude in its latest 7th legacy data release [@SDSS_DR7]. It had a spectroscopic follow-up survey that targeted over 1 million galaxies and quasars and half a million stars down to the magnitude limit of $r=17.77$ AB mag. Even though by the end of 2015, the data from SDSS and its successors, SDSS-II, and SDSS-III were used in about 20,000 research papers[^2], the SDSS potential for scientific exploration remains far from exhaustion.
In the late 2000s, deep wide field surveys went beyond the optical spectral domain. The Galaxy Evolution Explorer (GALEX) satellite [@Martin+05] provided nearly all-sky photometric coverage in two ultraviolet bands centered at 154 and 228 nm down to the limiting magnitudes $AB = 20.5$ mag. The SDSS footprint area was observed by GALEX with 15 times longer exposure that yielded a much deeper limit of $AB = 23.5$ mag. The relatively small telescope provided the spatial resolution of a couple of arcsec comparable to the typical image quality level at ground-based facilities.
At the same time, a major effort was undertaken by the international team at the 4-m United Kingdom Infrared Telescope UKIRT to survey a substantial area of the sky largely overlapping with the SDSS footprint in 4 near-infrared (NIR) bands ($YJHK$). The Large Area Survey of the UKIRT Deep Sky Survey (UKIDSS LAS, [@Lawrence+07]) provides a sub-arcsecond resolution and the flux limit comparable to that of SDSS in the optical domain. It reaches $AB \sim 21.2$ mag, 3–4 mag deeper than the first all-sky NIR survey 2MASS [@Skrutskie+06].
Numerous projects studied the entire SDSS spectroscopic sample of galaxies by analyzing both absorption [see e.g. @Kauffmann+03; @GCBW06] and emission lines [@Brinchmann+04; @Tremonti+04; @OSSY11; @Oh+15] in SDSS spectra (MPA-JHU and OSSY catalogs). However, they did not make use of any additional information beyond that available in the SDSS database.
The first successful attempt of providing an added value to SDSS data was done a decade ago in the “New York University Value-Added Galaxy Catalog” (NYU-VAGC) project [@Blanton+05]. It was aimed at statistical studies of galaxy properties and the large scale structure of the Universe and included a compilation of information derived from photometry and spectroscopy in one of the earlier SDSS data releases (that represents about 20% of its final imaging footprint). It also comprised positional cross matches with 2MASS, far-infrared *IRAS* point source catalog [@Saunders+00], the Faint Images of the Radio sky 20 cm survey FIRST [@BWH95], and additional data on galaxies from the 3rd Reference Catalogue of Bright Galaxies [@deVaucouleurs+91] and the Two-Degree Field Galaxy Redshift Survey [@Colless+01]. Now, a decade after the NYU-VAGC has been published, there is a sharp need to assemble a next generation of a value-added galaxy catalog based on modern survey data that were not available back then.
Here we present a new generation and a different flavor of a value-added catalog of galaxies based on a combination of data from SDSS, GALEX, and UKIDSS surveys that also includes comprehensive analysis of absorption and emission lines in galaxy spectra. Our main motivation is to use the synergy provided by the joint panchromatic dataset for extragalactic astrophysics: the optical domain is traditionally the best studied and there exist well calibrated stellar population models; the UV fluxes are sensitive to even small fractions of recently formed stars and therefore contain valuable information on star formation histories; the near-IR band is substantially less sensitive to the internal dust reddening and stellar population ages, and therefore can provide good stellar mass estimates. Our mission is to build a reference multi-wavelength spectrophotometric dataset and complement it with additional detailed information on galaxy properties so that it will allow astronomers to study galaxy formation and evolution at redshifts $z=0.0\textup{--}0.6$ in a transparent way with as little extra manipulations as possible.
We aim to provide: (i) the first homogeneous set of low redshift galaxy FUV-to-NIR spectral energy distributions (SEDs) corrected to rest frame for hundreds of thousands of objects; (ii) the first photometric dataset containing rest frame aperture SEDs with corresponding spectra and their stellar population analysis: velocity dispersions, parameterized star formation histories; (iii) consistent analysis of absorption and emission lines in SDSS galaxy spectra including parametric and non-parametric emission line fitting performed using state-of-the-art stellar population models, which cover a wider range of ages and metallicities and, therefore, help to minimize the template mismatch; (iv) easy and fully Virtual Observatory compliant data access mechanisms for our dataset and several third-party catalogs that include morphological and structural information for galaxies in our sample.
We started this project in 2009 by developing a new approach to convert galaxy SEDs to the rest frame by calculating analytic approximations of $k$-corrections in optical and NIR bands [@CMZ10]. Then we extended our algorithm to GALEX *FUV* and *NUV* bands and discovered a universal 3-dimensional relation of *NUV* and optical galaxy colors and luminosities [@CZ12]. Then, we fitted SDSS spectra using state-of-the-art stellar population models, derived velocity dispersions and stellar ages and metallicities, and provided our measurements to the project that calibrated the fundamental plane of galaxies [@DD87] in SDSS by vigorous statistical analysis [@SMZC13]. Our dataset also helped to find and characterize massive compact early-type galaxies at intermediate redshifts [@DCHG13; @DHGC14]. Finally, we used a complex set of selection criteria and discovered a large sample of previously considered extremely rare compact elliptical galaxies [@CZ15].
The paper is organized as follows: in *Section 2* we describe the construction of the catalog that includes cross-matching of the three surveys, adding third-party catalogs, absorption and emission line analysis of SDSS spectra; in *Section 3* we discuss the photometric properties of the sample and derive mean colors of galaxies of different morphological types across the spectrum; in *Section 4* we explore the information derived from our spectral analysis; *Section 5* contains the description of the catalog access interfaces; *Section 6* provides the summary of our project; and *Appendices* include some technical details on the catalog construction, detailed description of tables included in the database, and discussion of systematic uncertainties of emission line measurements.
Construction of the catalog
===========================
The input sample and data sources used. {#sec_sample}
---------------------------------------
We compiled the photometric catalog by re-processing several publicly available datasets. Our core object list is the SDSS Data Release 7 [@SDSS_DR7] spectral sample of non-active galaxies (marked as “GAL\_EM” or “GALAXY” specclass in the SDSS database) in the redshift range $0.007 \le z < 0.6$. We provide the exact query that we used to select this sample in the SDSS CasJobs Data System[^3] in Appendix \[sec\_sql\]. The query executed in the DR7 CasJobs context returned 800,299 records. We deliberately excluded quasars and Seyfert-1 galaxies (specclass=“QSO”) because neither the $k$-correction technique, nor stellar population analysis algorithm supported that object type. We used the output table as an input list for positional cross-matches against GALEX Data Release 6 [@Martin+05] and UKIDSS Data Release 10 [@Lawrence+07].
For the UKIDSS cross-match we queried the UKIDSS Large Area Survey catalog using the best match criterion within a 3 arcsec radius. In order to perform this query, we employed the WFCAM Science Archive[^4] for the programmatic access to the International Virtual Observatory Alliance (IVOA) ConeSearch service with a multiple cone search (“multi-cone”) capability. The query returned 280,870 UKIDSS objects matching the galaxies from our input sample. We used the [stilts]{} software package [@Taylor06] in order to access the UKIDSS data and merge the tables.
Then we uploaded the input SDSS galaxy list to the GALEX CasJobs web interface[^5] and searched best matches within 3 arcsec similarly to the UKIDSS cross-match. The query returned 485,996 GALEX objects.
As a result of this selection procedure we compiled an input catalog of 800,299 spectroscopically confirmed SDSS galaxies, out of which 90,717 have 11 band photometry (two *GALEX* *FUV* and *NUV*, 5 *SDSS* $ugriz$ bands, 4 UKIDSS $YJHK$ bands), 163,709 have all UKIDSS bands and at least one UV band, 582,534 have at least one additional photometric band to SDSS bands. In Fig. \[fig\_samplemap\] we present the footprint of our catalog on the all-sky aitoff projection marking the regions covered by all three wide field imaging surveys using different colors. The statistics of galaxies measured in different photometric bands is given in Table \[tab\_photnumb\].
![A full sky aitoff projection in equatorial coordinates demonstrating the footprint of our catalog. Green areas denote the availability of all three input photometric datasets, SDSS, UKIDSS, and GALEX; red areas are for SDSS and GALEX; and blue areas are for SDSS only. Note that we include all objects from the input datasets that have at least one flux measurement in them. \[fig\_samplemap\]](fig_aitoff.pdf){width="\hsize"}
[lr]{} SDSS $ugriz$ & 799783\
GALEX $FUV$ + $ugriz$ & 286570\
GALEX $NUV$ + $ugriz$ & 469419\
$FUV$ + $NUV$ + $ugriz$ & 270152\
$ugriz$ + UKIDSS $Y$ & 270603\
$ugriz$ + UKIDSS $J$ & 265316\
$ugriz$ + UKIDSS $H$ & 272028\
$ugriz$ + UKIDSS $K$ & 273050\
$ugriz + YJHK$ & 250608\
$NUV + ugriz + YJHK$ & 157531\
all 11 bands & 90717
Then we linked the following published datasets to our catalog in order to contribute the spectrophotometric information with some of the most widely used galaxy properties: (i) the results of the two-dimensional light profile decomposition of SDSS galaxies by @Simard+11 that include structural properties of all objects in our catalog; (ii) the morphological classification table from the citizen science “Galaxy Zoo” project [@Lintott+08; @Lintott+11] that provides a human eye classification of well spatially resolved SDSS galaxies made by citizen scientists. 661,319 objects in our sample have 10 or more morphological classifications in the Galaxy Zoo catalog ($nvote \ge 10$).
The photometric catalog
-----------------------
### Petrosian and aperture magnitudes
All three photometric surveys used in our study provide extended source photometry along with aperture measurements made in several different aperture sizes (GALEX and UKIDSS).
For the SED photometric analysis and construction of scaling relations involving galaxy luminosity, we need total magnitudes. For this purpose we adopt @Petrosian76 magnitudes available in SDSS and UKIDSS as measurements which do not significantly depend on galaxy light profile shapes conversely to SDSS *modelmags* [see discussion in @CZ12]. The GALEX catalog provides “total” magnitudes that are close to Petrosian magnitudes for exponential surface brightness profiles (i.e. disc galaxies) and up-to 0.2 mag brighter for elliptical galaxies [@Yasuda+01]. However, given the average photometric uncertainty in the GALEX $NUV$ fluxes of red galaxies of $0.3$ mag, we can neglect this difference.
On the other hand, our parent sample of galaxies was derived from the SDSS spectroscopic sample, and all SDSS DR7 spectra were obtained in circular 3-arcsec wide apertures. Therefore, we need 3-arcsec aperture magnitudes in order to make quantitative comparison of spectroscopic and photometric data. Hence, we computed aperture magnitudes for all GALEX and UKIDSS sources with available aperture measurements by interpolating the flux to a 3-arcsec aperture, and used SDSS *fibermags* for the optical SED part. To be noticed, that the spatial resolution of the GALEX survey in the $NUV$ band is about 5 arcsec, therefore 3-arcsec aperture magnitudes will be slightly underestimated for small objects. For compact (point-like) sources a 3-arcsec aperture $NUV$ magnitude can be underestimated by as much as 0.3 mag, however, such objects are very rare in the SDSS DR7 galaxy sample. @DCHG13 [@ZDGC15; @Zahid+16] found a couple of thousands compact sources in SDSS and SDSS-[iii]{} BOSS, only a few hundreds of which were in SDSS DR7. We estimated a number of compact galaxies in our sample by selecting the sources where the average difference of aperture and Petrosian magnitudes in $ugriz$ bands was $<0.3$ mag: this query returned 831 objects or $<0.1$% of the sample.
We corrected the obtained sets of Petrosian and 3-arcsec aperture magnitudes for the Galactic foreground extinction by using the $E(B-V)$ values computed from the @SFD98 extinction maps. Then, we computed $k$-corrections for both sets of photometric points using the analytic approximations presented in @CMZ10 and updated for $GALEX$ bands in @CZ12.
In Fig. \[fig\_sed\] we provide an example of a fully corrected SED for a late type spiral galaxy ($z=0.035$) that has flux measurements in all 11 bands. We show both total and fiber magnitudes and overplot an SDSS spectrum with the wavelength axis converted into the rest frame and fluxes converted into $AB$ magnitudes. One can see a remarkable agreement between the corrected photometric points and the observed spectral flux density, typical for our catalog.
![Example of fully corrected SED in 11 bands for a late type spiral galaxy at redshift 0.035. Blue and red symbols represent total (Petrosian) and 3-arcsec fiber magnitudes correspondingly. The rest framed SDSS spectrum is overplotted and demonstrates a typical excellent agreement with the corrected fiber magnitudes for that galaxy. The inset shows an 36$\times$36 arcsec optical SDSS false color image. \[fig\_sed\]](fig_sed.pdf){width="\hsize"}
### Correcting the SDSS–UKIDSS photometric offset
An important problem of the UKIDSS photometric catalog of extended sources is the observed spread of colors including optical SDSS and NIR UKIDSS photometric measurements (e.g. $g-J$ for red sequence galaxies). We detected this inconsistency in @CMZ10 and applied an empirical correction to UKIDSS magnitudes based on the assumption of continuous SEDs of galaxies. We computed $z-Y$ colors by interpolating over all other available colors approximating the SED with a low order polynomial function. This approach, however, required the availability of the $Y$ band photometry in the UKIDSS catalog. We have analyzed the SDSS–UKIDSS Petrosian magnitude offset amplitude for different galaxies and concluded, that it originates from the surface brightness limitation imposed by relatively short integration time in UKIDSS and by high and variable sky background level in the NIR. Therefore, Petrosian radii and magnitudes become underestimated, and comparison of original UKIDSS extended source magnitudes with SDSS and GALEX integrated photometry becomes impossible, because any color including data from UKIDSS and another data source depends on the galaxy surface brightness and size.
Here we propose a general and simple empirical solution. We exploit the UKIDSS Galactic Cluster Survey photometric catalog that includes the $Z$ band photometry, convert it into SDSS $z$ with the available color transformation [@HWLH06] for both Petrosian and 3-arcsec aperture magnitudes, and compare it to actual SDSS $z$ band measurements from the SDSS DR7 catalog for exactly the same objects. It turns out, that (i) the Petrosian magnitude difference $z_{\mathrm{SDSS,Petro}} -
z_{\mathrm{UKIDSS,Petro}}$ correlates with the galaxy mean surface brightness; (ii) the fiber magnitude difference $z_{\mathrm{SDSS,fib}} -
z_{\mathrm{UKIDSS,3''}}$ is close to zero within 0.02 mag; (iii) differences between Petrosian and fiber magnitudes in all UKIDSS photometric bands ($ZYJHK$) are almost identical that indicates virtually flat NIR color profiles in most galaxies. This suggests that the correction for UKIDSS Petrosian magnitudes should be calculated as: $\Delta
(mag_{\mathrm{UKIDSS,Petro}}) = (z_{\mathrm{SDSS,fib}} -
z_{\mathrm{SDSS,Petro}}) - (Y_{\mathrm{UKIDSS,3''}} -
Y_{\mathrm{UKIDSS,Petro}})$. This transformation adjusts the UKIDSS integrated photometry in a way that the differences between the 3 arcsec and Petrosian magnitudes of a galaxy in $z$ and $Y$ bands become equal. For objects, where $Y$ magnitudes are not available in the UKIDSS survey, we use the next available photometric band ($J$, $H$, or $K$).
In this fashion, we obtained fully corrected *FUV-to-NIR* spectral energy distributions converted into rest-frame magnitudes for a large sample of galaxies in 3-arcsec apertures and integrated over entire galaxies.
The spectral catalog: absorption lines
--------------------------------------
We fitted all SDSS spectra using the [nbursts]{} full spectrum fitting technique [@CPSK07; @CPSA07] and determined their radial velocities $v$, stellar velocity dispersions $\sigma$, and parameterized star formation histories represented by an instantaneous star burst (simple stellar populations, SSP) or an exponentially declining star formation history (exp-SFH) assuming that it started shortly after the Big Bang. We chose these two families of stellar population models because: (i) SSP models are widely used in extragalactic studies by different authors and we wanted our data to be directly comparable to other sources; (ii) exponentially declining SFHs were demonstrated to be a better representation of broadband SEDs of non-active galaxies [@CZ12] than SSPs. We should, however, notice, that exp-SFH models cannot adequately describe young stellar populations with mean ages $t<1.5$ Gyr (see discussion below).
The fitting procedure first convolves a grid of stellar population models with the wavelength dependent spectral line spread function available for every SDSS spectrum in the original data files, then runs a non-linear Levenberg-Markquardt minimization by first choosing a model spectrum from the grid by two-dimensional interpolation in the age–metallicity ($t$–\[Fe/H\]) space, then convolving it with a Gaussian-Hermite representation of the line-of-sight velocity distribution (LOSVD) of stars in a galaxy described by $v, \sigma, h3, h4$, and finally multiplying it by a low order Legendre polynomial continuum (its parameters are determined linearly in a separate loop) in order to absorb flux calibration imperfections and possible internal extinction in a galaxy. Hence, the procedure returns values of $v, \sigma, h3, h4, t, $\[Fe/H\], and coefficients of the multiplicative polynomial continuum. Here we use a pure Gaussian LOSVD shape with $h3=h4=0$.
The [nbursts]{} algorithm is similar to the penalized pixel fitting approach by @CE04. It, however, has some important differences. (i) We use a linear fit of the low order multiplicative polynomial continuum because its parameters are decoupled from galaxy kinematics and stellar populations. (ii) Instead of using a fixed grid of template spectra and interpreting stellar populations using their relative weights in a linear combination, we interpolate in a grid of models inside the minimization loop in order to obtain the best-fitting stellar population parameters of each starburst (or an exponentially declining model). As a result, for the simplest case of a single component SSP model, we obtain the best-fitting SSP-equivalent age and metallicity. These values are usually close to the luminosity weighted ones, however, in cases of complex SFHs approximated by an SSP there might be biases similar to those affecting Lick indices [@ST07]. @CPSA07 [@Chilingarian+08] demonstrated that SSP equivalent ages and metallicity remain unbiased for galaxies with super-solar $\alpha$-element abundances (\[Mg/Fe\]$>$0 dex) and when Balmer line regions are masked in order to fit emission lines.
We excluded the spectral regions affected by bright atmosphere lines (O[i]{}, NaD, OH, etc.) and by the $A$ and $B$ telluric absorption bands from the fitting procedure. We also re-ran the fitting code excluding $8\textup{--}14$ Å-wide regions around locations of bright emission lines for objects, where the reduced $\chi^2$ value of the fit exceeded the threshold $\chi^2/DOF=$0.8, that was selected empirically from a sample of galaxies without and with emission lines of different intensity levels.[^6]
We used three grids of stellar population models all computed with the [pegase.hr]{} evolutionary synthesis code [@LeBorgne+04]:
1. SSP models based on the high resolution (R=10000) ELODIE.3.1 empirical stellar library [@PS04; @PSKLB07] covering the wavelength range $3900 < \lambda < 6800$ Å, the metallicity range $-2.5 <
$\[Fe/H\]$<0.5$ dex, and ages $20 < t < 20000$ Myr.
2. Models with exponentially declining SFH at a constant metallicity computed for the same metallicity and wavelength ranges as those for SSP models, covering the range of exponential decay timescales $10 < \tau
< 20600$ Myr (the latter one effectively being a constant star formation rate model) and starting epochs of star formation between $4.3$ Gyr and $13.8$ Gyr of the age of the Universe corresponding to the redshift range $0<z<1.5$. We used the exponential decay timescale $\tau$ in the same fashion as the SSP age in the minimization procedure. For every galaxy, we first computed a grid of $\tau - $\[Fe/H\] models with the star formation epoch corresponding to its redshift assuming that a galaxy was formed at very high redshift, e.g. for $z=0.2$ with the light travel time of 2.45 Gyr, we computed a grid of models for star formation that started 11.27 Gyr ago assuming the standard WMAP9 cosmology [@Hinshaw+13].
3. Intermediate resolution SSP models ($R=2300$) based on the MILES empirical stellar library [@SanchezBlazquez+06] covering the wavelength range $3600 < \lambda < 7400$ Å, the metallicity range $-2.5 < $\[Fe/H\]$<0.7$ dex, and ages $20 < t < 20000$ Myr.
We stress that the best-fitting stellar population ages $t>14000$ Myr (SSP) and exponential timescales $\tau<1000$ Myr (exp-SFH) should be considered as upper and lower limits for the corresponding parameters.
In the public version of our catalog we provide two sets of stellar population parameters for every galaxy: (1) SSP ages and metallicities obtained from the spectrum fitting in the wavelength range in a galaxy rest-frame $4500 < \lambda < 6795$ Å using the MILES-[pegase.hr]{} models with the 5-th degree of the multiplicative polynomial continuum; (2) [pegase.hr]{} based exponentially declining SFR models in the wavelength range $3915 < \lambda < 6795$ Å with the 19-th degree continuum. The 19-th degree corresponds to the emipirically determined optimal degree of the multiplicative polynomial continuum for SDSS spectra when the $\chi^2$ value reaches a “plateau” as explained in @Chilingarian+08. We performed the SSP fitting with the MILES-[pegase.hr]{} models in the truncated wavelength range with a very low order polynomial continuum in order to minimize the artifacts originating from imperfections in the SSP model grid (see Section 4.2). In the publicly available Simple Spectrum Access Service we provide the results of the MILES-[pegase.hr]{} based SSP fitting in the wavelength range $3600 < \lambda < 6790$ Å in order to enable the emission line analysis from the fitting residuals for all lines including the \[O[ii]{}\] 3727 Å doublet.
The spectral catalog: emission lines
------------------------------------
Our full spectral fitting procedure precisely matches the stellar continuum of each galaxy by the best-fitting stellar population model (see example in Fig. \[fig\_absspec\]). Although the regions of all Balmer absorption lines are age sensitive, they contain at most 20% of the age sensitive information from the entire optical spectral range [@Chilingarian09]. @CPSA07 have demonstrated that masking the H$\beta$ and H$\gamma$ regions biases neither age nor metallicity determinations by the [nbursts]{} procedure. Hence, we do not expect to introduce significant template mismatch by masking the regions of emission lines when fitting SDSS spectra. Having subtracted the best-fitting model we obtain clean emission line spectra unaffected by stellar absorptions that is especially important for the Balmer lines. The precision of our stellar continuum fitting allows us to recover faint emission lines at a few per cent level of the continuum intensity, whereas very often such lines are not detected in the SDSS spectral pipeline results. In Table \[tbl\_linelist\] we provide the statistics of the emission line detection and strength in our sample.
In order to measure fluxes and equivalent widths (EW) of emission lines we applied two different approaches, namely Gaussian and non-parametric fitting of emission lines profiles.
In some galaxies, emission lines profiles cannot be described by a Gaussian. This often becomes a case in galaxies with peculiar gas kinematics, e.g. multi-component bulk gas motions and outflows can produce complex asymmetric lines. Also, this is crucially important for active galactic nuclei (AGN) with broad components in Balmer lines. Approximation of such emission lines by a Gaussian profile results in biased estimates of flux and kinematic parameters. We address this problem by employing a non-parametric fitting approach which allows us to recover arbitrary line profiles and measure their fluxes with higher precision. At the same time, this method requires several lines with sufficiently high signal-to-noise ratio to be present in a spectrum, and may produce biased results when dealing with noisy data. We, therefore, perform a “classical” Gaussian profile fitting too in order to allow for cross-comparison and validation of our line fitting results.
Both non-parametric and Gaussian fitting techniques take into account the SDSS line-spread function computed individually for each spectrum by the standard SDSS pipeline and provided in FITS (Flexible Image Transport System) tables in the RCSED distribution.
### Gaussian fitting {#sec_gaussfit}
This approach consists of simultaneously fitting the entire set of emission lines (see the line list in Table \[tbl\_linelist\]) with Gaussians pre-convolved with the SDSS line-spread function. We allow two different sets of redshifts and intrinsic widths for recombination and forbidden lines. We estimate the kinematic parameters with the non-linear least-square minimization that is implemented by the MPFIT package[^7] [@Markwardt09]. The emission line fluxes are computed linearly for each minimization iteration. When solving the linear problem, we constrain the line fluxes to be non-negative. For this purpose we use the BVLS (bounded-variables least-squares) algorithm [@LH95] and its implementation by M. Cappellari[^8].
[lclrrrrrrrrr]{}
[\[O[ii]{}\] ]{} & 3726.03 & f3727\_oii & 780665 & 543374 & 69.6% & 354307 & 45.4% & 225669 & 28.9% & 92161 & 11.81%\
[\[O[ii]{}\] ]{} & 3728.82 & f3730\_oii & 782417 & 562707 & 71.9% & 387264 & 49.5% & 257713 & 32.9% & 110287 & 14.10%\
[H$\kappa$ ]{} & 3750.15 & f3751\_h\_kappa & 791720 & 233850 & 29.5% & 26219 & 3.3% & 5584 & 0.7% & 535 & 0.07%\
[H$\iota$ ]{} & 3770.63 & f3772\_h\_iota & 796081 & 180306 & 22.6% & 18912 & 2.4% & 4524 & 0.6% & 595 & 0.07%\
[H$\theta$ ]{} & 3797.90 & f3799\_h\_theta & 798386 & 229961 & 28.8% & 31590 & 4.0% & 8466 & 1.1% & 1345 & 0.17%\
[H$\eta$ ]{} & 3835.38 & f3836\_h\_eta & 798565 & 255213 & 32.0% & 52680 & 6.6% & 16876 & 2.1% & 2976 & 0.37%\
[\[Ne[iii]{}\] ]{} & 3868.76 & f3870\_neiii & 798635 & 246202 & 30.8% & 45733 & 5.7% & 19500 & 2.4% & 7579 & 0.95%\
[He[i]{} ]{} & 3887.90 & f3889\_hei & 798674 & 182764 & 22.9% & 37166 & 4.7% & 10644 & 1.3% & 1328 & 0.17%\
[H$\zeta$ ]{} & 3889.07 & f3890\_h\_zeta & 798675 & 223107 & 27.9% & 63940 & 8.0% & 26166 & 3.3% & 6072 & 0.76%\
[H$\epsilon$ ]{} & 3970.08 & f3971\_h\_epsilon & 798835 & 360759 & 45.2% & 156226 & 19.6% & 78430 & 9.8% & 21312 & 2.67%\
[\[S[ii]{}\] ]{} & 4068.60 & f4070\_sii & 799003 & 202235 & 25.3% & 14766 & 1.8% & 2352 & 0.3% & 149 & 0.02%\
[\[S[ii]{}\] ]{} & 4076.35 & f4078\_sii & 799010 & 149342 & 18.7% & 4755 & 0.6% & 517 & 0.1% & 55 & 0.01%\
[H$\delta$ ]{} & 4101.73 & f4103\_h\_delta & 799043 & 342858 & 42.9% & 172913 & 21.6% & 96748 & 12.1% & 32463 & 4.06%\
[H$\gamma$ ]{} & 4340.46 & f4342\_h\_gamma & 799276 & 419668 & 52.5% & 275775 & 34.5% & 192540 & 24.1% & 87637 & 10.96%\
[\[O[iii]{}\] ]{} & 4363.21 & f4364\_oiii & 799293 & 118667 & 14.8% & 8001 & 1.0% & 2569 & 0.3% & 787 & 0.10%\
[He[ii]{} ]{} & 4685.76 & f4687\_heii & 799381 & 109369 & 13.7% & 6779 & 0.8% & 2204 & 0.3% & 614 & 0.08%\
[\[Ar[iv]{}\] ]{} & 4711.37 & f4713\_ariv & 799381 & 79310 & 9.9% & 3477 & 0.4% & 530 & 0.1% & 110 & 0.01%\
[\[Ar[iv]{}\] ]{} & 4740.17 & f4742\_ariv & 799380 & 118077 & 14.8% & 7031 & 0.9% & 1108 & 0.1% & 85 & 0.01%\
[H$\beta$ ]{} & 4861.36 & f4863\_h\_beta & 799375 & 514321 & 64.3% & 381556 & 47.7% & 317350 & 39.7% & 214953 & 26.89%\
[\[O[iii]{}\] ]{} & 4958.91 & f4960\_oiii & 799372 & 449021 & 56.2% & 164285 & 20.6% & 92442 & 11.6% & 47287 & 5.92%\
[\[O[iii]{}\] ]{} & 5006.84 & f5008\_oiii & 799371 & 638852 & 79.9% & 404135 & 50.6% & 244845 & 30.6% & 119215 & 14.91%\
[\[N[i]{}\] ]{} & 5197.90 & f5199\_ni & 799360 & 144430 & 18.1% & 9742 & 1.2% & 1345 & 0.2% & 178 & 0.02%\
[\[N[i]{}\] ]{} & 5200.25 & f5202\_ni & 799360 & 184255 & 23.1% & 16318 & 2.0% & 2676 & 0.3% & 226 & 0.03%\
[\[N[ii]{}\] ]{} & 5754.59 & f5756\_nii & 799131 & 196670 & 24.6% & 12800 & 1.6% & 2763 & 0.3% & 966 & 0.12%\
[He[i]{} ]{} & 5875.62 & f5877\_hei & 798546 & 260312 & 32.6% & 81904 & 10.3% & 38829 & 4.9% & 12209 & 1.53%\
[\[O[i]{}\] ]{} & 6300.30 & f6302\_oi & 784763 & 439640 & 56.0% & 177144 & 22.6% & 77850 & 9.9% & 16856 & 2.15%\
[\[O[i]{}\] ]{} & 6363.78 & f6366\_oi & 780219 & 285886 & 36.6% & 40626 & 5.2% & 9143 & 1.2% & 1395 & 0.18%\
[\[N[ii]{}\] ]{} & 6548.05 & f6550\_nii & 764832 & 596254 & 78.0% & 422810 & 55.3% & 289961 & 37.9% & 133553 & 17.46%\
[H$\alpha$ ]{} & 6562.79 & f6565\_h\_alpha & 763451 & 614029 & 80.4% & 531966 & 69.7% & 479842 & 62.9% & 395722 & 51.83%\
[\[N[ii]{}\] ]{} & 6583.45 & f6585\_nii & 761376 & 641883 & 84.3% & 553212 & 72.7% & 479386 & 63.0% & 334901 & 43.99%\
[He[i]{} ]{} & 6678.15 & f6679\_hei & 750612 & 178330 & 23.8% & 21069 & 2.8% & 6321 & 0.8% & 1408 & 0.19%\
[\[S[ii]{}\] ]{} & 6716.43 & f6718\_sii & 745687 & 571758 & 76.7% & 423064 & 56.7% & 320126 & 42.9% & 186973 & 25.07%\
[\[S[ii]{}\] ]{} & 6730.81 & f6733\_sii & 743742 & 554071 & 74.5% & 374143 & 50.3% & 263155 & 35.4% & 135572 & 18.23%\
### Non-parametric emission line fitting
![An example of the complex emission line profile of a Seyfert galaxy, and results of its fitting with two different techniques. Black stepped line in the upper panel shows the observed spectrum of H$\alpha$ and N[ii]{} lines in relative flux units, green dotted line is a Gaussian fit result, red solid line is a non-parametric fitting result. Individual H$\alpha$ and N[ii]{} profiles recovered by the non-parametric fitting are shown in orange and blue respectively. Lower panel shows fitting residuals. In the case of complex asymmetric emission lines profiles non-parametric fitting method is clearly preferred over Gaussian one. \[fig\_emis\_lines\]](fig_emis_lines_NPvsG.pdf){width="\hsize"}
Our non-parametric emission line fitting method includes two main steps which we repeat several times until the convergence is achieved. First, we derive discretely sampled emission line profiles, i.e. line-of-sight velocity distributions (LOSVDs) of ionized gas. During the second step, we estimate emission lines fluxes. Because allowed and forbidden transitions often originate from different regions of a galaxy having very different physical properties (i.e. density, temperature, mechanism of excitation), however, all emission lines of each type (allowed and forbidden) have similar shapes, our procedure recovers two different non-parametric profiles, one for each type.
The LOSVD derivation is organized as follows. We note that convolution of any logarithmically rebinned observed spectrum $S_{obs}$ of $m$ elements with LOSVD $\mathcal{L}$ having $n$ elements can be expressed as a linear matrix equation $A \ast \mathcal{L} = S_{obs}$, where $A$ is a $m\times n$ matrix of template spectra having lengths of $m$ pixels each. Every template spectrum from the $i$-th row in the matrix $A$ is shifted by the velocity which represents the $i$-th position within the LOSVD vector. Here a template spectrum is a synthetic spectrum made of a set of flux normalized Gaussians with LSF widths representing emission lines detected in the observed spectrum. Such approach allows us to take into account the SDSS instrumental resolution instead of a set of Dirac $\delta$-functions. The continuum level of a template spectrum is set to zero.
Thus, we end up with a linear inverse problem whose solution $\mathcal{L}$ can be derived by a least square technique. The emission line profiles obviously cannot be negative and, therefore, we use the BVLS algorithm mentioned above.
Once the LOSVD has been derived, we compute emission line fluxes by solving a similar linear problem to that described in Section \[sec\_gaussfit\]. This finishes the first iteration.
At the same time, the LOSVD derivation step requires the knowledge of emission line fluxes in order to construct better template spectra. During the first iteration when they are unknown, we set all fluxes to unity and the all template spectra hence are made of equally normalized Gaussians. Typically, 3 iterations of this procedure is enough to reach the convergence.
A linear inversion is an ill-conditioned problem and is sensitive to noise in the data. In order to improve the profile reconstruction quality, we exploit a regularization technique, which minimizes the squared third derivative of the recovered line profile. This approach, however, causes artefacts in sharp narrow line profiles. Therefore, we apply the regularization only in the wings of emission lines where flux levels are generally low and, consequently, noise is higher. The regularization technique yields the dramatic improvement of recovered Balmer line profiles for faint AGNs. In the catalog we provide measurements of non-parametric emission lines with and without regularization.
The comparison between the parametric (Gaussian) and non-parametric fitting results for a complex emission line profile in a Seyfert galaxy is presented at Fig. \[fig\_emis\_lines\]. A Gaussian approximation for such lines is often inadequate and causes serious biases in the kinematics that can reach few hundred km s$^{-1}$.
We ran Monte-Carlo simulations for a random sample of 2,000 objects with emission lines of different intensity levels in order to estimate realistic flux uncertainties obtained with the non-parametric fitting technique. They turned out to be consistent with statistical uncertainties of Gaussian emission line fluxes for most objects and up-to a factor of 2 lower for AGNs with broad line components. The RCSED database will be updated with Monte-Carlo based uncertainties as we compute them: this procedure is very computationally intensive and will take a couple of months to complete.
### Gas phase metallicities
We used our emission line flux measurements in order to estimate the gas phase metallicities for galaxies where emissions originate from the star formation induced excitation. We exploited two different techniques to measure the metallicity, (i) a new calibration by @DKSN16 and (ii) the IZI Bayesian technique [@BKVD15] using a grid of models ($\kappa=\infty$) with $\kappa$-distributed electron energies [@Dopita+13]. We selected star formation dominated and “transition type” galaxies using the standard BPT [@BPT81] diagram that exploits hydrogen, nitrogen, and oxygen emission lines with the criteria defined in @Kauffmann+03.
The @DKSN16 calibration uses only the H$\alpha$, \[N[ii]{}\], and \[S[ii]{}\] emission lines, all located in a very narrow spectral interval and is, therefore, virtually insensitive to the internal extinction within an observed galaxy. This calibration is presented in a form of a simple formula which makes it very easy to use. However, a disadvantage of this approach in application to our dataset is that at redshifts $z>0.1$ the emission lines used for the metallicity determination shift to the spectral region dominated by telluric absorption and airglow emission lines (mostly, OH) which can seriously affect the quality of the emission line flux estimates. Another natural limitation of this approach originates from the SDSS spectral wavelength range ($\lambda<9200$ Å) that corresponds to the upper redshift limit $z=0.36$ when the forbidden sulfur line \[S[ii]{}\] 6730 Å shifts out of the wavelength range. Besides, the calibration critically depends on the \[N/O\] relation and is therefore sensitive to possible galaxy to galaxy \[N/H\] abundance variations. In the catalog we included the metallicity estimates obtained using the @DKSN16 calibration for Gaussian emission line analysis ([rcsed\_gasmet]{} table). We computed the uncertainties of the gas phase metallicities by propagating the statistical flux errors through the calculations according to the formula in @DKSN16.
The IZI technique [@BKVD15] takes advantage of all available emission line measurements and, hence, is more robust and can in principle be used for the entire sample of SDSS galaxies. The algorithm is implemented in an [idl]{} software package distributed by the authors along with 17 grids of photoionization models. However, this technique relies on the external dust attenuation correction which must be applied to emission lines fluxes prior to fitting. It also requires (similar to the @DKSN16 approach) a pre-selection of star forming galaxies. We estimated the internal dust attenuation using the typical value of the Balmer decrement H$\alpha$/H$\beta = 2.83$ [@GBW12] and corrected all emission line fluxes accordingly. In galaxies where the observed H$\alpha$/H$\beta$ ratio fell below that value, we assumed the extinction to be zero. Finally, the fluxes were supplied to the IZI software package with the @Dopita+13 model grid and the resulting \[O/H\] and ionizing parameter values for Gaussian emission line fluxes were included in the gas phase metallicity table [rcsed\_gasmet]{} of the catalog.
Photometric properties of the sample
====================================
![(Top) Optical color–magnitude diagram for extinction- and $k$-corrected Petrosian magnitudes of all galaxies in our sample. (Bottom) Redshift distributions of galaxies in corresponding bins on absolute magnitude.\[fig\_red\_sequence\]](colmag_fig_lf.pdf){width="\hsize"}
Completeness at different redshifts
-----------------------------------
Because our catalog uses the SDSS DR7 spectroscopic galaxy sample as its master list, and the legacy SDSS spectroscopic survey was magnitude limited with the $r=17.77$ mag limit in a 3 arcsec aperture, we sample different parts of the galaxy luminosity function with the redshift dependent completeness. Also, there is an important fiber collision effect, that is when two fibers in the SDSS multi-object spectrograph cannot be put too close to each other: because of this, there is a systematic undersampling of dense clusters and groups of galaxies.
In Fig. \[fig\_red\_sequence\] (top panel) we present a two-dimensional distribution of our galaxies in the $(M_z, g-r)$ color–magnitude space. We identify the regions traditionally referred to as “the red sequence” and “the blue cloud” as well as the locus of typical post-starburst (E+A) galaxies. The density in the plot corresponds to the object number density in our catalog at a given position of the parameter space. We also show by small crosses the tidally stripped systems, compact elliptical galaxies, from the sample of @CZ15 which reside systematically above the red sequence. One can see the bimodality of the galaxy distribution by color for intermediate luminosity and dwarf galaxies ($M_z>-20.5$ mag) while the transition is rather smooth for more luminous systems.
Even though dwarf galaxies are more numerous in the Universe than giants because of the raising low end of the galaxy luminosity function [@Schechter76; @Blanton+03a], we see the apparent decrease of the histogram density at fainter magnitudes. In the bottom panel of Fig. \[fig\_red\_sequence\] we demonstrate the breakdown by redshift for the luminosity distribution of galaxies contributing to the histogram in the top panel. The high luminosity end decline is due to the intrinsic shape of the luminosity function, while the low luminosity tail drops because of the SDSS completeness and target selection biased against very extended (and therefore nearby) galaxies. We clearly see how the magnitude limit constraint of SDSS causes the drop in the number of galaxies further and further up the luminosity function as we move to higher redshifts. Fig. \[fig\_red\_sequence\] confirms that we start probing the dwarf galaxy regime ($M_z>-19.8$ mag) at $z<0.06$, however the selection effects have to be seriously considered for any type of a statistical study.
Red sequence in different bands
-------------------------------
For practical reasons such as selection of candidate early-type members in galaxy clusters using photometric data, it is important to know the shape of the red sequence in different photometric bands. Here we provide the best fitting second degree polynomial approximations of the red sequence shape for a set of galaxy colors spanning optical and NIR bands.
First we created a sample of red sequence galaxies by the following criteria: (1) We selected all objects at redshifts $z<0.27$; (2) we applied a color cut on $NUV-r$ colors by selecting all objects on $(M_r,NUV-r)$ plane that resided above the straight line passing through $p_0=(-16.0,3.5)$ mag and $p_1=(-24.0,5.0)$ mag; (3) we applied a color cut on $g-r$ colors by selecting all objects on $(M_r,g-r)$ plane that resided above the straight line passing through $q_0=(-16.0,0.5)$ mag and $q_1=(-24.0,0.75)$ mag and also satisfying the criterion $(g-r)<0.95$ mag.
Then, in order to account for two orders of magnitude variations of galaxy density along the red sequence, for every combination of colors and magnitudes (e.g. $g-r$, $M_r$): (1) We selected measurements having statistical uncertainties $<0.1$ mag in both bands; (2) binned the distribution on luminosity using 0.5 mag wide bins and computed median color values and outlier resistant standard deviations in every bin; (3) fitted a second degree polynomial into median values only in those bins that contained more than 15 objects. For convenience and because mean absolute $AB$ magnitudes of galaxies in our sample stay mostly within the range $-25<M<-15$ mag in all optical red ($riz$) and NIR filters, we added 20.0 mag to all absolute magnitudes prior to fitting.
(u-r) &= +2.51 -0.065 M\_[20r]{} -0.005 M\_[20r]{}\^2; = 0.16\
(u-i) &= +2.90 -0.069 M\_[20i]{} -0.007 M\_[20i]{}\^2; = 0.17\
(u-z) &= +3.15 -0.050 M\_[20z]{} -0.014 M\_[20z]{}\^2; = 0.19\
(g-r) &= +0.75 -0.026 M\_[20r]{} -0.001 M\_[20r]{}\^2; = 0.045\
(g-i) &= +1.12 -0.038 M\_[20i]{} -0.003 M\_[20i]{}\^2; = 0.074\
(g-z) &= +1.39 -0.044 M\_[20z]{} -0.009 M\_[20z]{}\^2; = 0.10\
(g-Y) &= +1.91 -0.067 M\_[20Y]{} -0.018 M\_[20Y]{}\^2; = 0.14\
(g-J) &= +2.01 -0.073 M\_[20J]{} -0.016 M\_[20J]{}\^2; = 0.18\
(g-H) &= +2.30 -0.094 M\_[20H]{} -0.015 M\_[20H]{}\^2; = 0.19\
(g-K) &= +2.00 -0.108 M\_[20K]{} -0.018 M\_[20K]{}\^2; = 0.22\
&M\_[20]{} = M\_ + 20.0 \[red\_seq\_eq\]
Eqs. \[red\_seq\_eq\] provide the best-fitting polynomials for the red sequence shape in 10 photometric bands. We consider the mean standard deviation value from all bins used in the fitting procedure as the “red sequence width” ($\sigma$ in Eqs. \[red\_seq\_eq\]) and stress that the actual fitting residuals for median values are usually an order of magnitude smaller.
We notice that in the most widely used parameter spaces, $(M_r, g-r)$, $(M_r, u-r)$, $(M_i, g-i)$, and $(M_z, g-z)$, the red sequence does not show any substantial curvature which is indicated by negligible 2nd order polynomial terms. This suggests that there is no “red sequence saturation” at the bright end.
Color transformations for galaxies of different morphologies and luminosities
-----------------------------------------------------------------------------
----------- ------ ------ ------ ------ ------ --------- ------ ------ ------ ------ ------ --------- ------ ------ ------ ------ ------ ------
Sdm Sc Sb Sa S0 E Sdm Sc Sb Sa S0 E Sdm Sc Sb Sa S0 E
$FUV$-$r$ 1.72 2.76 3.46 4.33 5.58 6.86 1.60 2.48 3.22 4.15 5.22 6.72 1.44 2.18 2.87 3.81 4.91 6.78
stdev
$NUV$-$r$ 1.20 2.20 2.89 3.73 4.84 5.64 1.17 2.02 2.69 3.57 4.57 5.44 1.15 1.80 2.41 3.29 4.18 4.98
stdev
$u$-$r$ 0.99 1.48 1.80 2.14 2.44 2.56 0.95 1.34 1.68 2.05 2.32 2.44 0.88 1.19 1.44 1.76 2.02 2.14
stdev
$g$-$r$ 0.26 0.49 0.60 0.70 0.78 0.80 0.24 0.40 0.53 0.65 0.73 0.76 0.22 0.34 0.45 0.57 0.66 0.69
stdev
$g$-$i$ 0.42 0.75 0.92 1.06 1.15 1.17 0.33 0.58 0.81 1.00 1.10 1.12 0.26 0.45 0.64 0.85 0.98 1.02
stdev
$g$-$z$ 0.66 0.97 1.16 1.32 1.42 1.44 0.41 0.72 1.02 1.25 1.37 1.39 0.33 0.55 0.79 1.05 1.18 1.24
stdev
$g$-$Y$ 1.27 1.50 1.70 1.87 1.97 1.98 0.78 1.12 1.50 1.79 1.91 1.92 0.59 0.82 1.12 1.46 1.58 1.68
stdev
$g$-$J$ 1.37 1.56 1.77 1.96 2.05 2.07 0.83 1.15 1.58 1.90 2.01 2.03 0.53 0.77 1.11 1.51 1.66 1.75
stdev
$g$-$H$ 1.63 1.87 2.10 2.30 2.39 2.40 1.05 1.42 1.89 2.22 2.33 2.33 0.77 1.02 1.37 1.80 1.89 1.99
stdev
$g$-$K$ 1.54 1.61 1.83 2.03 2.11 2.11 0.80 1.12 1.60 1.95 2.04 2.03 0.34 0.62 1.00 1.44 1.55 1.63
stdev
----------- ------ ------ ------ ------ ------ --------- ------ ------ ------ ------ ------ --------- ------ ------ ------ ------ ------ ------
@CZ12 demonstrated that the Hubble morphological classification derived by a human eye [@Fukugita+07] correlates very well with the total $NUV-r$ color of a galaxy. With a computed dispersion of $0.8 t$, where $t$ is the Hubble type, it corresponds to the subjective precision of such a classification. Here we use this relation in order to derive median values of galaxy colors across the Hubble sequence for three galaxy luminosity classes defined on a basis of their $r$-band luminosities.
We dissect the $(M_r, NUV-r)$ color–magnitude plane into 18 quadrangular regions by assuming that the morphological type for giant galaxies ($M_r=-24$ mag) can be estimated by linearly varying the $(NUV-r)$ color from $+0.5$ to $+6.5$ mag with a step of 1 mag corresponding to one Hubble type from *Sd* to *E*. At the same time, we assume that in the dwarf regime $(M_r=-16$ mag) the step reduces to 0.75 mag per Hubble type that corresponds to the observed reduction of the $(NUV-r)$ color range. We choose 3 luminosity bins, $-24.0 \le M_r< -22.0$ mag, $-22.0 \le M_r<
-19.0$ mag, and $-19.0 \le M_r <-16.0$ mag, which represent giant, intermediate luminosity, and dwarf galaxies. Then, in every region we compute the median value of a desired color, and the standard deviation of the distribution.
We present our results in Table \[tab\_gal\_col\]. They expand and update the widely used color transformations from @FSI95 by using a very rich dataset properly corrected for systematic effects and using modern prescriptions for $k$-corrections. We extend their results at $z=0$ (see table 3 in [@FSI95]) to near-UV and NIR colors and also towards intermediate and low luminosity galaxies. The direct comparison of our values with those of @FSI95 reveals a good agreement of optical colors except (a) S0 galaxies which are systematically redder in our case and stay really close to the ellipticals; (b) the $u-g$ color of ellipticals that is some 0.25 mag bluer in our case. We assign the latter systematics to our improved $k$-correction prescriptions for the $u$ band photometry and generally higher quality of the $u$ band SDSS photometric data compared to the dataset used in @FSI95. On the other hand, we attribute redder colors of lenticular galaxies in our data to the specificities of the synthetic color estimation technique used in @FSI95 that underestimated colors of 2 of 4 their lenticular galaxies by 0.1–0.15 mag (see their table 1).
Spectroscopic properties of the sample
======================================
Stellar kinematics of galaxies
------------------------------
In comparison to original SDSS measurements of stellar kinematics based on cross-correlation with a limited set of template spectra, our approach yields significantly smaller template mismatch between models and observed spectra for non-active galaxies. We, therefore, achieve on average 30% lower statistical uncertainties of radial velocity and velocity dispersion measurements. Moreover, there is a known degeneracy between stellar metallicity and velocity dispersion estimates when using pixel space fitting techniques [@CPSA07], because an underestimated metallicity (i.e. using a metal poor template for a metal rich galaxy) can be compensated by a lower velocity dispersion that would smear that template spectrum to a lesser degree. Therefore, by using a grid of stellar population models ranging from low (\[Fe/H\]$=-2.0$ dex) to high (\[Fe/H\]$=+0.7$ dex) metallicities and covering the whole range of ages, we reduce the systematic errors of velocity dispersion measurements, especially in the most metal-rich regime including massive elliptical and lenticular galaxies. On the other hand, we accurately take into account the spectral line spread function of the SDSS spectrograph that allows us to measure velocity dispersions down to 50 km s$^{-1}$, thus going far into the dwarf galaxy regime [@Chilingarian09].
As it was already pointed out by @Fabricant+13, stellar velocity dispersion measurements in the SDSS DR7 catalog are systematically underestimated for luminous elliptical galaxies compared to the values obtained by the full spectrum fitting, that is likely caused by the template mismatch and degeneracy with metallicity mentioned above. Here we observe a very similar trend: our SSP velocity dispersion measurements for massive ellipticals ($\sigma \gtrsim 250$ km s$^{-1}$) are up-to 30 km s$^{-1}$ higher than those reported in the SDSS DR7 catalog, and this difference goes down to 7–10 km s$^{-1}$ for low luminosity galaxies ($\sigma \sim 100$ km s$^{-1}$). In Fig. \[figsigsig\] (top panel) we present the comparison made for our entire sample for 361,421 galaxies with velocity dispersion uncertainties better than 7% of the value (i.e. $\Delta \sigma = $7 km s$^{-1}$ for $\sigma =
100$ km s$^{-1}$). The velocity dispersions estimated from the fitting of exponentially declining SFH models computed with [pegase.hr]{}, are a little bit closer to the values in SDSS DR7, however, the general trend looks similar (Fig. \[figsigsig\], bottom panel).
![Comparison of RCSED stellar velocity dispersion measurements with those published by the SDSS DR7. Top and bottom panel correspond to the two sets of stellar population models, SSP and exponentially declining SFHs correspondingly. \[figsigsig\]](fig_veldisp.pdf){width="\hsize"}
{width="\hsize"}
In Fig. \[figFJR\] we present the relation between galaxy luminosities and velocity dispersions or the Faber–Jackson ([-@FJ76]) relation constructed for 52,506 elliptical galaxies which were morphologically selected by the Galaxy Zoo [@Lintott+11] citizen science project and had statistical uncertainties of their velocity dispersion measurements better than 10% of the value. We have corrected velocity dispersion measurements to their global values according to @Cappellari+06 using half-light radii from @Simard+11 included in our catalog. We used the criterion formulated in @SMZC13: in order to be included in our early type galaxy sample, an object has to be classified by at least 10 Galaxy Zoo users of whom at least 70% classify it as an elliptical galaxy. Six panels present measurements in six different redshift intervals shown as dots while the contours display the cumulative distribution at all redshifts. The lowest redshift panel contains the measurements for a sample of galaxies in the Abell 496 cluster ($z=0.033$) obtained from the analysis of intermediate resolution (R=6300) spectra collected with the FLAMES–Giraffe spectrograph at the 8-m Very Large Telescope of the European Southern Observatory [@Chilingarian+08]. This dataset comprises mostly dwarf early-type galaxies and it clearly forms an extension of the low–luminosity part of the relation formed by the SDSS galaxies which demonstrates that our velocity dispersion measurements at the low end do not suffer from systematic errors connected to the spectral line spread function uncertainties. The red dashed line represents the Faber–Jackson relation for giant elliptical galaxies $L_g\propto \sigma^{4.00}$ at $z=0$ presented in @2003AJ....125.1849B. We see that the slope changes to $L_g\propto \sigma^{2.00}$ at fainter luminosities $M_g > -19.5$ mag similar to what was demonstrated for a sample of dwarf galaxies in the Coma galaxy cluster by @MG05.
Because of the correlation of a galaxy luminosity with a stellar velocity dispersion and the magnitude limited input galaxy sample, higher redshift galaxies contribute only to the bins at high stellar velocity dispersions. Our catalog probes dwarf galaxies ($\sigma<100$ km s$^{-1}$) at low redshifts ($0.007<z<0.06$) that includes hundreds of massive galaxy clusters and groups.
RCSED velocity dispersion measurements were used prior to publication by @SMZC13 for the calibration of the Fundamental Plane [@DD87]. We refer to that work for an intensive discussion regarding the FP of elliptical galaxies observed by the SDSS.
Stellar populations from absorption line analysis
-------------------------------------------------
In our catalog we include stellar population parameters obtained by the fitting of galaxy spectra using two stellar population model grids computed with the [pegase.hr]{} evolutionary synthesis code: (i) SSP models based on the intermediate resolution MILES stellar library characterized by ages ($t$) and metallicities (\[Fe/H\]); and (ii) models with exponentially declining SFHs based on the high resolution ELODIE-3.1 stellar library characterized by exponential timescales ($\tau$) and metallicities (\[Fe/H\]). In Fig \[fig\_age\_met\_SSP\] we present distributions of galaxies in the two parameter spaces.
One can clearly see a spotty structure in the SSP best fitting results and the lack of such a structure for the exponentially decaying models. We also performed similar tests for original MILES stellar population models by @Vazdekis+10 and models by @BC03a for a sub-sample of SDSS DR7 spectra. We will provide a complete description and detailed discussion in the forthcoming paper (Katkov et al. in prep.), here we present a brief summary and conclusions of our study.
The observed spotty structure represents artifacts caused by the improper implementation of the interpolation algorithm in the stellar population code most likely on the stellar library interpolation step propagating into stellar population models and not by the [nbursts]{} population fitting procedure. The [nbursts]{} code uses a non-linear minimization technique that requires the second partial derivatives on all parameters to be continuous. Discontinuities will cause the solution to be either attracted to some region of the parameter space, or pushed away from it.
Our conclusion is supported by the following observations: (i) the morphology of the spotty structure remains very similar when using two different sets of SSP models computed with the same [pegase.hr]{} code but with different stellar libraries, MILES and ELODIE; (ii) switching to original MILES models [@Vazdekis+10] where the interpolation procedure is much simpler than in [pegase.hr]{} (linear interpolation between 5 nearest neighbors) changes the structure completely and strengthens the artifacts; (iii) using exponentially decaying star formation models which are constructed of numerous weighted SSPs removes most of the pattern at $\tau \gtrsim 1.0$ Gyr but the structure still holds at $\tau < 1.0$ Gyr where the number of co-added SSPs is small; (iv) smoothing a grid of [pegase.hr]{} MILES based SSP models using basic splines ($b$-splines) on age removes most of the pattern.
We also notice, that extending the working wavelength range to shorter wavelengths ($<4500$ Å) and increasing the multiplicative polynomial degree strengthens the pattern while leaving spot positions in the pattern virtually the same. Therefore, we chose a very low order 5-th degree polynomial continuum and restricted the wavelength range to $\lambda > 4500$ Å for the SSP fitting that produced stellar population parameters presented in our catalog.
In the top panel of Fig. \[fig\_ssp\_vs\_expSFH\], we present the comparison of SSP ages and exponential timescales $\tau$. Despite the artifact structure in ages that extends into horizontal stripes on this plot, there is a 1-to-1 correspondence between $t$ and $\tau$ in a wide range of ages. Short timescales $\tau$ correspond to old stellar populations while $\tau =
20$ Gyr is equivalent to $t \approx 1.8$ Gyr. The relation “saturates” for younger stellar populations because they cannot be represented by exponentially declining SFHs starting at high redshifts: either a later start or multiple star formation episodes are needed to describe them. @CZ12 demonstrated that exponentially declining SFHs much better represent observed broadband optical and UV colors than SSP models. In our current sample about 16% of galaxies ($\sim$131,500) have stellar populations too young too be described by exponentially declining SFHs.
The bottom panel of Fig. \[fig\_ssp\_vs\_expSFH\] displays the comparison of stellar metallicities for the two sets of models. The agreement is very good with a slight systematic difference between $-0.6<$\[Fe/H\]$<-0.2$ dex which we attribute to the degeneracy between the metallicity and velocity dispersion measurements for intermediate signal-to-noise ratios.
![Distributions of galaxies in the age–metallicity space from the fitting of SSP (top panel) and exponentially decaying SFH (bottom panel) models. \[fig\_age\_met\_SSP\]](fig_age_met.pdf){width="\hsize"}
![Comparison of SSP ages to timescales $\tau$ for exponentially decaying models (top) and metallicity measurements (bottom). \[fig\_ssp\_vs\_expSFH\]](fig_ssp_vs_exp.pdf){width="\hsize"}
We clearly see a substantial degeneracy between the metallicity and velocity dispersion estimates which was pointed out in @CPSA07. In order to perform a clear test, free of any effects connected to the usage of different star formation histories, we fitted a subset of $\sim$420,000 spectra using [pegase.hr]{} SSP models in the wavelength range 3910–6790 Å and compared the metallicity and velocity dispersion measurements to those obtained from the fitting of MILES–[pegase]{} models against the same spectra. In Fig. \[fig\_degen\] we present the relation between the differences of velocity dispersions and SSP metallicities obtained using the two sets of models at different signal-to-noise ratios. We can clearly see the degeneracy manifested by the elongated shape of the cloud that decreases with the increasing signal-to-noise ratio up-to the signal-to-noise of 30. Above 30 the improvement becomes insignificant. This result suggest that published velocity dispersion values obtained with the full spectral fitting of intermediate resolution spectra $R=1500-2500$ are subject to serious systematic errors reaching 15% of the measured value.
![Degeneracy between metallicity and velocity dispersion estimates shown as the ratio between velocity dispersions vs SSP metallicities obtained from the fitting of some 420,000 SDSS spectra using [pegase.hr]{} and MILES–PEGASE SSP models. The contours display the 1$\sigma$ values which correspond to the areas containing 68% galaxies with the spectra having signal-to-noise ratios within 20% of a displayed value. The number of galaxies for each contour ranges from $\sim$1800 (S/N=50) to $\sim$111,000 (S/N=10). \[fig\_degen\]](fig_sig_met_degen.pdf){width="\hsize"}
Emission line properties
------------------------
### Comparison of line fluxes with the MPA–JHU and OSSY catalogs and between the two techniques
{width="\textwidth"}
We compare a subset of our catalog containing measurements of emission line fluxes obtained from the parametric Gaussian fitting to the results from the MPA–JHU catalog distributed by the SDSS project [@Brinchmann+04; @Tremonti+04] and with the OSSY catalog [@OSSY11]. We used OSSY emission line measurements prior to the internal extinction correction. Given that the fluxes were computed using very similar methodologies with the main different corresponding to the subtraction of the underlying stellar population and the Galactic extinction correction techniques used, we expect a very good agreement for well detected emission lines. We directly compare fluxes of the \[O[ii]{}\] (3727 Å), \[O[iii]{}\] (5007 Å), H$\alpha$, and \[N[ii]{}\] (6584 Å) emission lines for a sample of galaxies where they were detected at a level exceeding 10$\sigma$ (i.e. Flux/$\sigma($Flux$) >
10$). The results are presented in Fig. \[fig\_emis\_mpajhu\_comp\]. We obtain an excellent agreement with a systematic difference of less than 1% and the standard deviation of residuals of about 2% for the bright end of the H$\alpha$ flux distribution. At the faint end (10$\sigma$ detection), the systematic difference stays within 2% while the standard deviation grows to 3%. Hence, we conclude that our emission line fitting code works as expected and does not introduce any substantial systematic errors to flux measurements.
Compared to H$\alpha$, the H$\beta$ line is much more sensitive to the age of the stellar population being subtracted. In Appendix \[sec\_appsys\] we discuss the systematic errors of the H$\beta$ measurements as a function of the age mismatch. In case of faint emission lines, the systematics dominates the measurements if the age was determined incorrectly, and makes them useless for the emission line diagnostics.
![Comparison of the H$\alpha$ fluxes obtained for the parametric (Gaussian) and non-parametric emission line profile fitting as a function of the $\chi^2$ ratio. An example profile decomposition is shown in the inset for an object with highly discrepant flux estimates.[]{data-label="fig_emis_gaus_vs_np"}](fig_emis_gausVSnonpar.pdf){width="50.00000%"}
The principal difference of our results with those published earlier is the non-parametric approach to the emission line fitting. For galaxies which exhibit some signs of nuclear activity, the Balmer lines fluxes derived non-parametrically significantly exceed the values obtained with the Gaussian fitting. In Fig. \[fig\_emis\_gaus\_vs\_np\] we present the H$\alpha$ flux ratio between the two approaches. The inset contains the same Seyfert galaxy which we presented earlier in Fig. \[fig\_emis\_lines\] and the arrow indicates its position in the diagram that suggests that its non-parametric H$\alpha$ flux estimate is about 20% higher than that obtained with the Gaussian profile fitting.
### BPT diagrams and gas phase metallicities
![Three flavors of a BPT diagram with the color coded $H_\alpha$ equivalent width. In each panel we display only those galaxies where all emission lines used in the corresponding plot have $S/N>3$. The contours correspond to the galaxy density smoothed with a moving average based on a 4$\times$4-pixels window. The number of galaxies kept in the sample is indicated inside each panel. The full and dashed lines correspond to starforming and transitional galaxies in @KGKH06.[]{data-label="fig_BPTdiagram"}](fig12_BPTdiagram.pdf "fig:"){width="\hsize"} ![Three flavors of a BPT diagram with the color coded $H_\alpha$ equivalent width. In each panel we display only those galaxies where all emission lines used in the corresponding plot have $S/N>3$. The contours correspond to the galaxy density smoothed with a moving average based on a 4$\times$4-pixels window. The number of galaxies kept in the sample is indicated inside each panel. The full and dashed lines correspond to starforming and transitional galaxies in @KGKH06.[]{data-label="fig_BPTdiagram"}](fig12_BPTdiagram2.pdf "fig:"){width="\hsize"} ![Three flavors of a BPT diagram with the color coded $H_\alpha$ equivalent width. In each panel we display only those galaxies where all emission lines used in the corresponding plot have $S/N>3$. The contours correspond to the galaxy density smoothed with a moving average based on a 4$\times$4-pixels window. The number of galaxies kept in the sample is indicated inside each panel. The full and dashed lines correspond to starforming and transitional galaxies in @KGKH06.[]{data-label="fig_BPTdiagram"}](fig12_BPTdiagram3.pdf "fig:"){width="\hsize"}
In Fig. \[fig\_BPTdiagram\] we present three flavors of the BPT diagram which use different combinations of emission lines computed using the non-parametric fitting. The points are color-coded corresponding to the H$\alpha$ emission line EW. @2010MNRAS.403.1036C proposed to use the H$\alpha$ EW to discriminate between Seyfert and LINER activity (instead of the traditionally used \[O[iii]{}\]/H$\beta$ ratio), because H$\beta$ is often too weak to be detected and measured. We clearly see the bimodal distribution of non-starforming galaxies in the bottom two panels which corresponds to Seyfert galaxies (cloud to the top) and LINER/shockwave/post-AGB ionization (cloud to the bottom). The top panel displays the original BPT relation. The region between the red solid and the blue dashed lines defines “transitional” galaxies [@KGKH06] which we included in the calculation of metallicities in addition to the star forming galaxies located to the bottom left of the blue dashed line.
As we described above, RCSED includes gas phase metallicity measurements calculated with the Bayesian method implemented in the IZI software package with the @Dopita+13 model grid, which uses all available emission lines in a spectrum; and a recent technique by @DKSN16 that relies on the \[N/O\] calibration and uses only 5 emission lines around H$\alpha$.
@KE08 demonstrated that different emission line calibrations yield largely inconsistent gas phase metallicity estimates when applied to the same input dataset with the differences reaching 0.7 dex (5 times). There is currently no consensus in the astronomical community about which calibrations produce more reliable metallicity estimates with arguments for both direct [@AM13] and indirect [@LopezSanchez+12] methods. Gas and stellar metallicities also seem to strongly disagree [@YKG12]. We notice, however, that all emission line calibrations result in the gas phase \[O/H\] mass–metallicity relations spanning much lower range of metallicities than that of stellar metallicities for a given galaxy stellar mass range. All gas metallicity relations saturate at high metallicities and the saturation occurs at different values [see e.g. fig. 10 in @AM13]. The highest range of metallicities is covered by the calibration used by @Tremonti+04 and provided in the MPA–JHU catalog.
In Fig. \[fig\_gas\_met\_comparison\] we present the comparison of the @DKSN16 calibration with the MPA–JHU metallicities (blue shaded area) for 231,107 galaxies with the signal-to-noise ratios of H$\alpha$, \[N[ii]{}\], \[O[ii]{}\], and \[O[iii]{}\] lines exceeding 10. The agreement is very good at $12+$\[O/H\]$<9$ dex with the standard deviation of the difference $0.08$ dex. At higher abundances @DKSN16 metallicities become slightly higher than those from the MPA–JHU dataset.
We ran the IZI metallicity determination code for a small sub-sample of 20,000 randomly selected starforming galaxies with high signal-to-noise emission lines (S/N$>$10) using all available grids of models and compared the derived metallicities with those obtained with the @DKSN16 calibration for the same galaxy sample. The only model grid that demonstrated a satisfactory agreement was that from @Dopita+13. As expected, it also provides a satisfactory agreement with the MPA–JHU catalog (see Fig. \[fig\_gas\_met\_comparison\], orange shaded areas) with the standard deviation of the difference $0.10$ dex.
In Fig. \[fig\_emis\_metallicities\] we show the luminosity–metallicity relation (left panel) and the comparison of gas phase and SSP stellar metallicities (right panel) for the IZI–based determination using the @Dopita+13 models (orange contours) and the @DKSN16 calibration (blue points). The mass–metallicity relation is well defined and we clearly see that the @Dopita+13 model grid used in IZI yields a flatter shape than the more recent calibration [@DKSN16].
The comparison of gas phase and stellar metallicities reveal a substantial offset ranging from about 0.3 dex at solar stellar metallicities to 0.8 dex at the low end (\[Fe/H\]$_{\mathrm{star}}=-1.1$ dex). Keeping in mind that stellar and gas phase metallicities might have different zero points and should not be directly compared to each other, the observed pattern is exactly what is expected due to the self enrichment of stellar populations happening in galaxies with extended star formation histories. The stars during their evolution form heavy elements which then get ejected into the ISM and recycled in the subsequent generations of stars, hence, increasing their metal abundances [see e.g. @Matteucci94]. Therefore, younger generations of stars become more metal rich. SSP models probe mean stellar metallicities over the entire lifetime of a galaxy weighted with the stellar $M/L$ ratios and the star formation rate while the gas phase metallicity reflects the current chemical abundance pattern in the ISM enriched with metals, therefore, we expect to see the offset in metallicities. On the other hand, for the constant metal production rate per solar mass, the difference at low metallicities will be higher because the metallicity scale is logarithmic, therefore stellar metallicities should span a larger range of value compared to gas phase metallicities and the mass–metallicity relation slopes for gas will be shallower than that for stars.
![Comparison of gas phase metallicities published in the MPA–JHU catalog (horizontal axis) to our measurements (vertical axis). The results of the IZI Bayesian technique are shown in brown and the measurements obtained with the @DKSN16 calibration are shown in blue. \[fig\_gas\_met\_comparison\]](fig_emis_metallicities.pdf){width="\hsize"}
{width="80.00000%"}
Catalog access: web-site and Virtual Observatory access interfaces
==================================================================
Efficient, convenient, and intuitive data access mechanisms and interfaces are essential for a complex project like RCSED. Therefore, we decided to build access interfaces for both interactive and batch access to the data.
RCSED includes several different data types (e.g. spectra and tabular data) and our access infrastructure (see Fig. \[fig\_block\_diagram\]) is organized to simplify their usage through different interfaces. The most natural way to access the catalog is by using the web application at <http://rcsed.sai.msu.ru/>. It provides a single-field [google]{}-style search interface where one can query the catalog by an object identifier, coordinates or object properties, e.g. *select all galaxies with redshifts $z<0.1$ having red colors $g-r>1.5$*. Every object in the sample has its own web page with the summary of all its properties, SED, spectral data available in the catalog, and image cutouts displaying the object at different wavelength provided by GALEX, SDSS, and UKIDSS surveys. An example of a spectrum summary plot presented on such web pages for every object is given in Fig. \[fig\_emspec\].
We developed an Application Programming Interface (API) to UKIDSS data, which allow us to extract image cutouts around an arbitrary position with a given box size in every filter. From cutout images in the *JHK* bands we generate a color composite image and display it in the object web-page. The API implemented in [python]{} is available for download from the project web-site. Another service we present is an interactive spectrum plotter implemented in JavaScript, our alternative to the SDSS spectrum plotter. It contains a number of value-added features, such as the display of best-fitting templates and identification of emission lines.
In addition to the custom web application, our data distribution infrastructure has the open source GAVO DaCHS[^9] data center suite in its core (see Fig. \[fig\_block\_diagram\]) which provides a set of VO data access mechanisms.
The data for SDSS spectra and their best-fitting SSP models are provided as FITS files that can be fetched by direct unique URLs. One can find a URL for every particular object spectrum file either in the object’s web page or by querying the provided IVOA Simple Spectral Access Protocol (SSAP) web service using object coordinates. The SSAP web service answers essentially with a list of spectra URLs and it is convenient to access programmatically or by using VO compatible client applications such as TOPCAT[^10] [@Taylor05], SPLAT-VO[^11] or VO-Spec[^12] which can directly load spectral data for further analysis by analyzing the SSAP web service query result.
For the ultimate flexibility of querying tabular catalog data, we provide a Table Access Protocol (TAP) web service. IVOA TAP is an access interface, which allows a user to query the entire relational database schema (see Fig. \[fig\_er\_diagram\]) using a powerful SQL-like language. It can be considered as an open source equivalent of the SDSS CasJobs service. Again, TAP web service can be used for script-based access as well as by using desktop VO applications. In particular, TOPCAT has a very useful TAP query dialogue with built-in help, query examples, syntax highlighting and given database schema assistance tools. We encourage our users to access the RCSED TAP web service through TOPCAT. We also note that our TAP service has a table upload capability, so that the user can upload his/her own tables and use it in subsequent SQL queries i.e. in [join]{} clauses, that is convenient for cross-identification of user provided object samples with the RCSED objects without the need of downloading our full catalog.
Below we give several query examples that are helpful to start using the RCSED database. More query examples and science case tutorials are provided on the project website <http://rcsed.sai.msu.ru>. Our TAP web service can be used for joining tables from the database schema [specphot]{} presented in Fig. \[fig\_er\_diagram\], so that it is easy to retrieve a single table with the GalaxyZoo morphology, the photometric bulge+disk decomposition and the RCSED basic parameters combined for any galaxy of interest. An example of such a query to select all those data for a particular object would be:
SELECT
r.*, g.*, s2.*
FROM
specphot.rcsed AS r
JOIN specphot.galaxyzoo AS g
ON r.objid = g.objid
JOIN specphot.simard_table2 AS s2
ON r.objid = s2.objid
WHERE
r.objid = 587731891649052703
Note that [specphot]{} prefix for table names corresponds to the name of the database schema where RCSED tables are stored. A query to retreive all data from the RCSED on galaxies, classified as ellipticals in GalaxyZoo is:
SELECT
r.*
FROM
specphot.rcsed AS r
JOIN specphot.galaxyzoo AS g
ON r.objid = g.objid
WHERE
g.elliptical = 1
A query to select the data for a BPT [@BPT81] diagram for 10,000 galaxies with ${\rm S/N} > 10$ in the corresponding line fluxes obtained with the Gaussian fitting looks like this:
SELECT
TOP 10000
f6550_nii_flx / f6565_h_alpha_flx AS BPT_x,
f5008_oiii_flx / f4863_h_beta_flx AS BPT_y
FROM
specphot.rcsed_lines_gauss
WHERE
f6565_h_alpha_flx / f6565_h_alpha_flx_err > 10
AND f5008_oiii_flx / f5008_oiii_flx_err > 10
AND f4863_h_beta_flx / f4863_h_beta_flx_err > 10
AND f6550_nii_flx / f6550_nii_flx_err > 10
Finally, all the catalog tables (see Fig. \[fig\_er\_diagram\]) are available for download as FITS tables from the project’s website for the offline use.
![Block diagram of the catalog data access infrastructure. Data are stored in the relational database (catalog tables and spectra metadata) and on the disk (FITS files with spectra and continuum models). They are accessed by applications (a custom web application and the GAVO DaCHS suite) which in turn expose several public access interfaces suitable for convenient queries and data retrieval by the multitude of user client programs, both VO-compatible and generic. \[fig\_block\_diagram\]](fig13_block_diagram.pdf){width="\hsize"}
{width="\hsize"}
Summary
=======
We presented a reference catalog of homogeneous multi-wavelength spectrophotometric information for some 800,000 low to intermediate redshift galaxies ($0.007<z<0.6$) from the SDSS DR7 spectroscopic galaxy sample with value-added data. For every galaxy we provide:
- a $k$-corrected and Galactic extinction corrected far-UV to NIR broad band SED for integrated fluxes compiled from the SDSS (optical), GALEX (UV), and UKIDSS (NIR) surveys
- a $k$-corrected and Galactic extinction corrected far-UV to NIR broad band SED for fluxes in circular 3-arcsec apertures that correspond to SDSS spectral apertures
- results of the full spectrum fitting of an SDSS spectrum using the [nbursts]{} technique that includes: (a) an original SDSS spectrum; (b) the best-fitting simple stellar population template in the wavelength range $3700<\lambda<6800$ Å and the best-fitting stellar population model with an exponentially declining star formation history in the wavelength range $3900<\lambda<6800$ Å; (c) estimates of stellar radial velocities, velocity dispersions, age, an exponential characteristic timescale for the star formation history, metallicities for two sets of stellar population models
- results of the emission line analysis using parametric (Gaussian) and non-parametric line profiles that include: (a) emission line fluxes corrected for the Galactic extinction; (b) estimates of the reddening inside a galaxy for star formation dominated systems derived from the observed Balmer decrement; (c) radial velocity offsets with respect to stars; (d) intrinsic emission line widths for the parametric fitting
- cross-match of a galaxy with third-party catalogs providing its structural parameters from the two-dimensional light profile fitting [@Simard+11] and galaxy morphology by the Galaxy Zoo project [@Lintott+08]
The catalog is fully integrated into the international Virtual Observatory infrastructure and available via a web application and as a Virtual Observatory resource providing IVOA TAP and IVOA SSAP interfaces in order to programmatically access tabular data and spectra correspondingly.
In addition to that, we presented best-fitting polynomial approximations for the red sequence shape in color–magnitude diagrams that include different colors, and mean colors for galaxies of 6 morphological types, from elliptical to late type spirals and irregulars and 3 luminosity classes (giants, intermediate luminosity, dwarfs).
Our catalog has already been used in several research projects that can be categorized into two groups: (i) statistical studies of galaxy properties; (ii) search and discovery of rare galaxies.
The first interesting result obtained with RCSED was the discovery of a universal 3-dimensional relation of *NUV* and optical galaxy colors and luminosities [@CZ12]. It also demonstrated that the integrated *NUV*$-$*r* color is a good proxy for a morphological type. The spectum fitting results for elliptical galaxies were later used in the re-calibration of the fundamental plane in SDSS [@SMZC13] which allowed us to compute redshift independent distances to early-type galaxies. Finally, we performed a calibration of near-infrared stellar $M/L$ ratios using optical colors and computed stellar masses for a new catalog of groups and clusters combining SDSS and 2MASS redshift survey data [@Saulder+16]. Our non-parametric emission line fitting results will be used to perform massive determinations of virial black hole masses in AGNs (Katkov et al. in prep). Other potential applications of statistical studies based on RCSED include but not limited to: environmental dependence of galaxy scaling relations and stellar population properties; connecting AGNs to stellar populations in galaxy centers; comparing different star formation rate indicators (e.g. emission line fluxes, UV and MIR photometry).
Thanks to the unique combination of photometric and spectral data as well as physical properties of galaxies derived from them, RCSED becomes an efficient search tool for rare or unique galaxies. Our dataset was used to discover and characterize massive compact galaxies at intermediate redshifts $0.2<z<0.8$ [@DCHG13] which were thought to exist only in the early Universe ($z>1.5$) and measure their volume density [@DHGC14]. Then, using their fundamental plane positions, intermediate redshift compact galaxies were shown to be an extension of normal ellipticals to the compact regime [@ZDGC15]. Finally, it was demonstrated that some massive compact early-type galaxies actually stopped forming stars very recently [@Zahid+16]. We also used the universal UV–optical color–color–magnitude relation to define complex selection criteria and discover 195 previously considered extremely rare compact elliptical galaxies [@CZ15]. One can identify other obvious extragalactic rarities easily searchable with RCSED: post-starburst galaxies, candidate double-peaked AGNs, dwarf AGN hosts, “normal” galaxies with peculiarities detectable in multi-wavelength data such as ellipticals with NUV excess.
In the future, we anticipate to release intermediate and high redshift extensions of our catalog that will include the analysis of publicly available spectra from the Smithsonian Astrophysical Observatory Hectospec archive[^13] collected with the Hectospec multi-fiber spectrograph [@Fabricant+05] the 6.5-m MMT and the DEEP2 galaxy redshift survey [@Newman+13] made with the DEIMOS spectrograph at the 10-m Keck telescope. We also plan to expand the wavelength coverage by adding the all-sky infrared data from the Wide-field Infrared Survey Explorer (WISE) satellite [@Wright+10]. A major update to our catalog will be made with the full spectrophotometric fitting the entire sample using the [nbursts+phot]{} algorithm [@CK12] and resolving star formation histories for about $10^5$ galaxies with high quality UV and NIR data.
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge our anonymous referee whose comments helped us to improve this manuscript. IC’s reseach is supported by the Smithsonian Astrophysical Observatory Telescope Data Center. IZ acknowledges the support by the Russian Scientific Foundation grant 14-50-00043 for the catalog assembly tasks and grant 14-12-00146 for the data publication and deployment system. The authors acknowledge partial support from the M.V.Lomonosov Moscow State University Program of Development, and a Russian–French PICS International Laboratory program (no. 6590) co-funded by the RFBR (project 15-52-15050), entitled “Galaxy evolution mechanisms in the Local Universe and at intermediate redshifts”. The statistical studies of galaxy populations by IC, IZ, IK, and ER are supported by the RFBR grant 15-32-21062 and the presidential grant MD-7355.2015.2. The authors are grateful to citizen scientists M. Chernyshov, A. Kilchik, A. Sergeev, R. Tihanovich, and A. Timirgazin for their valuable help with the development of the project website. In 2009–2011 the project was supported by the VO-Paris Data Centre and by the Action Specifique de l’Observatoire Virtuel (VO-France). A substantial progress in our project was achieved during our 2013, 2014, and 2015 annual Chamonix workshops and we are grateful to our host O. Bevan at Châlet des Sapins. This research has made use of TOPCAT, developed by Mark Taylor at the University of Bristol; Aladin developed by the Centre de Données Astronomiques de Strasbourg (CDS); the “exploresdss” script by G. Mamon (IAP); the VizieR catalogue access tool (CDS). Funding for the *SDSS* and *SDSS*-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The *SDSS* Web Site is <http://www.sdss.org/>. *GALEX* (Galaxy Evolution Explorer) is a NASA Small Explorer, launched in April 2003. We gratefully acknowledge NASA’s support for construction, operation, and science analysis for the *GALEX* mission, developed in cooperation with the Centre National d’Etudes Spatiales of France and the Korean Ministry of Science and Technology.
Systematics in emission line measurements due to stellar population template mismatch {#sec_appsys}
=====================================================================================
Absorption lines of the hydrogen Balmer series contain important information about stellar population ages [@Worthey94], they become weaker when stars get older. At the same time, emission Balmer lines are used for the ISM diagnostic and star formation studies [@BPT81]. For the vast majority of galaxies in our sample, we see relatively weak emission lines on top of a stellar continuum. Therefore, in order to accurately measure emission line fluxes, we need to precisely model stellar populations. Hence, when gas emission lines reside on top of a stellar continuum, any systematic uncertainty in the modelling of absorption lines will affect emission line measurements. Specifically, the age mismatch in the stellar population fitting will substantially bias Balmer line fluxes.
In order to quantify this effect, we performed the following procedure: (i) We selected 2,000 spectra from our sample with Balmer emission line intensities ranging from weak to strong based on their equivalent widths; (ii) we fitted those spectra using stellar population model grids fixing the SSP age to 2, 4, 8, and 16 Gyr; (iii) we measured emission line fluxes in the fitting residuals in these four sets of spectra; (iv) we compared them to emission line fluxes obtained for best-fitting stellar populations presented in our catalog.
In Fig. \[fig\_EMLAge\] we present our results. It is clear, that the age mismatch affects emission line fluxes for weak lines: The systematic errors grow when lines become weaker, and the difference between the best fitting and the fixed ages templates gets higher. When ages are underestimated by the fitting procedure (i.e. a galaxy is older than the age of a template), Balmer emission line fluxes are underestimated too. Because forbidden lines often used in the gas state diagnostics (e.g. \[N[ii]{}\] or \[O[iii]{}\]) do not lie on top of strong age sensitive absorption features, their fluxes remain virtually unaffected, hence, moving a galaxy over the diagnostic plots (e.g. BPT) and potentially leading to the ionization mechanism misclassification.
![The stellar population age mismatch effect on H$\beta$ flux measurements. The difference of the H$\beta$ EW computed using the best fitting SSP template and a template with the age fixed to 2 Gyr is plotted against the measured H$\beta$ EW for the best fitting SSP template. The age difference between the best fitting SSP age and 2 Gyr is color coded. \[fig\_EMLAge\]](fig_Appendix_emis_syst.pdf){width="0.5\hsize"}
Catalog compilation: SQL query {#sec_sql}
==============================
When selecting the core sample of galaxies we performed the following SQL query in the SDSS CasJobs service in the DR7 context (see details in Section \[sec\_sample\]):
SELECT
p.objID, p.ra, p.dec,
p.modelMag_u, p.modelMagErr_u, p.modelMag_g, p.modelMagErr_g,
p.modelMag_r, p.modelMagErr_r, p.modelMag_i, p.modelMagErr_i,
p.modelMag_z, p.modelMagErr_z,
petroMag_u, petroMagErr_u, petroMag_g, petroMagErr_g,
petroMag_r, petroMagErr_r, petroMag_i, petroMagErr_i,
petroMag_z, petroMagErr_z,
p.fiberMag_u, p.fiberMagErr_u, p.fiberMag_g, p.fiberMagErr_g,
p.fiberMag_r, p.fiberMagErr_r, p.fiberMag_i, p.fiberMagErr_i,
p.fiberMag_z, p.fiberMagErr_z,
p.petroR50_u, p.petroR50Err_u, p.petroR50_g, p.petroR50Err_g,
p.petroR50_r, p.petroR50Err_r, p.petroR50_i, p.petroR50Err_i,
p.petroR50_z, p.petroR50Err_z,
p.extinction_u, p.extinction_g, p.extinction_r, p.extinction_i, p.extinction_z,
s.specObjID, s.mjd, s.plate, s.fiberID,
s.z, s.zerr, s.zconf, s.objType, s.sn_0, s.sn_1, s.sn_2,
(SELECT stripe FROM dbo.fCoordsFromEq(p.ra,p.dec)) AS stripe,
s.specClass
INTO mydb.RCSED_SDSS
FROM PhotoObj AS p, SpecObj as s
WHERE
s.bestObjid = p.objID
AND s.z >= 0.007
AND s.z < 0.6
AND s.specClass IN (dbo.fSpecClass('GAL_EM'), dbo.fSpecClass('GALAXY'))
This query returned 800,311 rows with 12 duplicate objects for which SDSS [SpecObj]{} table contains 2 records despite it is documented to be clean from duplicates. We discard these duplicate spectra by keeping the record with higher S/N out of each pair of duplicates (and hence having e.g. better redshift estimate). From now we continue with the sample of 800,299 galaxies.
The coordinates of obtained galaxies were then uploaded to the GALEX CasJobs service and the following query was performed there in GALEXGR6Plus7 context:
SELECT
sdss.objid,
galex_objid,
nuv_mag, nuv_magerr, fuv_mag, fuv_magerr,
nuv_mag_aper_1, nuv_magerr_aper_1, nuv_mag_auto,
fuv_mag_aper_1, fuv_magerr_aper_1, fuv_mag_auto,
e_bv
INTO mydb.RCSED_SDSS_GALEX
FROM
(
SELECT
s.objid,
(SELECT objid FROM dbo.fGetNearestObjEq(s.ra, s.dec, 0.05)) AS galex_objid
FROM
mydb.RCSED_SDSS_coords AS s
) AS sdss
JOIN
photoObjAll AS p
ON sdss.galex_objid = p.objid
This query returned 485,996 rows.
\[lastpage\]
Catalog column descriptions
===========================
In Tables \[tab\_metadata\_main\]–\[tab\_metadata\_nonpar\] we provide descriptions and metadata for columns of the original tables of RCSED, which are shown in blue in Fig. \[fig\_er\_diagram\]. The external datasets available in the RCSED database are described in the corresponding original papers (see the text for references).
This column information is identical for FITS tables distribution of the catalog, as well as when accessing the RCSED database through the Table Access Protocol, or using the catalog website <http://rcsed.sai.msu.ru>. For each column name in every table we give: (i) units (dash sign indicates that a column is dimensionless or units are not applicable to it); (ii) data type in the database convention in order to guide a user on the precisionm and puropse of a column; (iii) IVOA Unified Content Descriptor (UCD) that helps one to identify equivalent physical quantities available for comparison in the VO or to associate a column and its uncertainty; and (iv) human readable description of the column contents. When a table includes many similar columns as in the case of spectral lines properties in the [rcsed\_lines\_gauss]{} and [rcsed\_lines\_nonpar]{} database tables, we only give metadata for first group of columns in it and abridge the rest (Table \[tab\_metadata\_gauss\] and Table \[tab\_metadata\_nonpar\]). The complete list of emission lines included in our catalog and the column name prefixes in [rcsed\_lines\_gauss]{} and [rcsed\_lines\_nonpar]{} are given in Table \[tbl\_linelist\].
Column Units Datatype UCD Description
------------------- -------- ---------- ----------------------------------- -----------------------------------------------------------------------------------------------------
objid - bigint meta.id;meta.main SDSS ObjID (unique identifier)
specobjid - bigint meta.id SDSS SpecObjID (unique identifier within spectral galaxies sample)
mjd - integer time.epoch MJD of observation
plate - smallint meta.id SDSS plate ID
fiberid - smallint meta.id SDSS fiber ID
ra deg double pos.eq.ra;meta.main RA (J2000) of galaxy
dec deg double pos.eq.dec;meta.main Dec (J2000) of galaxy
z - real src.redshift Galaxy redshift
zerr - real stat.error;src.redshift Uncertainty of galaxy redshift
zconf - real stat.fit.param;src.redshift SDSS $r$edshift confidence
petror50\_r arcsec real phys.angSize SDSS $r$adius containing 50% of Petrosian flux
e\_bv mag real phot.color.excess E(B-V) at this (l,b) from SFD98
specclass - smallint src.spType SDSS spectral classification
corrmag\_fuv mag real phot.mag;em.UV.FUV Galactic extinction corrected total (Kron-like elliptical aperture) magnitude in GALEX $FUV$ filter
corrmag\_nuv mag real phot.mag;em.UV.NUV Same as above for GALEX $NUV$ filter
corrmag\_u mag real phot.mag;em.opt.U Galactic extinction corrected total (Petrosian) magnitude in SDSS $u$ filter
corrmag\_g mag real phot.mag;em.opt.B Same as above for SDSS $g$ filter
corrmag\_r mag real phot.mag;em.opt.R Same as above for SDSS $r$ filter
corrmag\_i mag real phot.mag;em.opt.I Same as above for SDSS $i$ filter
corrmag\_z mag real phot.mag;em.opt.I Same as above for SDSS $z$ filter
corrmag\_y mag real phot.mag;em.IR.J Same as above for UKIDSS $Y$ filter
corrmag\_j mag real phot.mag;em.IR.J Same as above for UKIDSS $J$ filter
corrmag\_h mag real phot.mag;em.IR.H Same as above for UKIDSS $H$ filter
corrmag\_k mag real phot.mag;em.IR.K Same as above for UKIDSS $K$ filter
corrmag\_fuv\_err mag real stat.error;phot.mag;em.UV.FUV Uncertainty of corrmag\_fuv column
corrmag\_nuv\_err mag real stat.error;phot.mag;em.UV.NUV Uncertainty of corrmag\_nuv column
corrmag\_u\_err mag real stat.error;phot.mag;em.opt.U Uncertainty of corrmag\_u column
corrmag\_g\_err mag real stat.error;phot.mag;em.opt.B Uncertainty of corrmag\_g column
corrmag\_r\_err mag real stat.error;phot.mag;em.opt.R Uncertainty of corrmag\_r column
corrmag\_i\_err mag real stat.error;phot.mag;em.opt.I Uncertainty of corrmag\_i column
corrmag\_z\_err mag real stat.error;phot.mag;em.opt.I Uncertainty of corrmag\_z column
corrmag\_y\_err mag real stat.error;phot.mag;em.IR.J Uncertainty of corrmag\_y column
corrmag\_j\_err mag real stat.error;phot.mag;em.IR.J Uncertainty of corrmag\_j column
corrmag\_h\_err mag real stat.error;phot.mag;em.IR.H Uncertainty of corrmag\_h column
corrmag\_k\_err mag real stat.error;phot.mag;em.IR.K Uncertainty of corrmag\_k column
kcorr\_fuv mag real arith.factor;em.UV.FUV K-correction for GALEX $FUV$ magnitude
kcorr\_nuv mag real arith.factor;em.UV.NUV Same as above for GALEX $NUV$ magnitude
kcorr\_u mag real arith.factor;em.opt.U K-correction for (Petrosian) SDSS $u$ magnitude
kcorr\_g mag real arith.factor;em.opt.B Same as above for SDSS $g$ magnitude
kcorr\_r mag real arith.factor;em.opt.R Same as above for SDSS $r$ magnitude
kcorr\_i mag real arith.factor;em.opt.I Same as above for SDSS $i$ magnitude
kcorr\_z mag real arith.factor;em.opt.I Same as above for SDSS $z$ magnitude
kcorr\_y mag real arith.factor;em.IR.J Same as above for UKIDSS $Y$ magnitude
kcorr\_j mag real arith.factor;em.IR.J Same as above for UKIDSS $J$ magnitude
kcorr\_h mag real arith.factor;em.IR.H Same as above for UKIDSS $H$ magnitude
kcorr\_k mag real arith.factor;em.IR.K Same as above for UKIDSS $K$ magnitude
exp\_radvel km/s real spect.dopplerVeloc.opt Radial velocity (exp SFH)
exp\_radvel\_err km/s real stat.error;spect.dopplerVeloc.opt Radial velocity error (exp SFH)
exp\_veldisp km/s real phys.veloc.dispersion Velocity dispersion (exp SFH)
exp\_veldisp\_err km/s real stat.error;phys.veloc.dispersion Velocity dispersion error (exp SFH)
exp\_tau Myr real time.age Age (exp SFH)
exp\_tau\_err Myr real stat.error;time.age Age error (exp SFH)
exp\_met - real phys.abund.Z Metallicity (exp SFH)
exp\_met\_err - real stat.error;phys.abund.Z Metallicity error (exp SFH)
exp\_chi2 - real stat.fit.chi2 Goodness of fit (exp SFH)
ssp\_radvel km/s real spect.dopplerVeloc.opt Radial velocity (SSP)
ssp\_radvel\_err km/s real stat.error;spect.dopplerVeloc.opt Radial velocity error (SSP)
ssp\_veldisp km/s real phys.veloc.dispersion Velocity dispersion (SSP)
ssp\_veldisp\_err km/s real stat.error;phys.veloc.dispersion Velocity dispersion error (SSP)
ssp\_age Myr real time.age Age (SSP)
ssp\_age\_err Myr real stat.error;time.age Age error (SSP)
ssp\_met - real phys.abund.Z Metallicity (SSP)
ssp\_met\_err - real stat.error;phys.abund.Z Metallicity error (SSP)
ssp\_chi2 - real stat.fit.chi2 Goodness of fit (SSP)
zy\_offset mag real phot.mag;arith.diff Offset applied to UKIDSS magnitudes to correct for mismatch with SDSS ones
spectrum\_snr - real stat.snr Signal-to-noise ratio of SDSS spectrum at 5500A (restframe) in the 20A box
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Column Units Datatype UCD Description
----------------- ------- ---------- ------------------------ ---------------------------------------------------------------------------------------------------------------------------------
objid - bigint meta.id;meta.main SDSS ObjID (unique identifier)
corrfibmag\_fuv mag real phot.mag;em.UV.FUV Galactic extinction corrected 3“ aperture magnitude in GALEX $FUV$ filter\
corrfibmag\_nuv & mag & real & phot.mag;em.UV.NUV & Same as above for GALEX $NUV$ filter\
corrfibmag\_u & mag & real & phot.mag;em.opt.U & Galactic extinction corrected fiber (3” aperture) magnitude in SDSS $u$ filter
corrfibmag\_g mag real phot.mag;em.opt.B Same as above for SDSS $g$ filter
corrfibmag\_r mag real phot.mag;em.opt.R Same as above for SDSS $r$ filter
corrfibmag\_i mag real phot.mag;em.opt.I Same as above for SDSS $i$ filter
corrfibmag\_z mag real phot.mag;em.opt.I Same as above for SDSS $z$ filter
corrfibmag\_y mag real phot.mag;em.IR.J Galactic extinction corrected 3“ aperture magnitude in UKIDSS $Y$ filter\
corrfibmag\_j & mag & real & phot.mag;em.IR.J & Same as above for UKIDSS $J$ filter\
corrfibmag\_h & mag & real & phot.mag;em.IR.H & Same as above for UKIDSS $H$ filter\
corrfibmag\_k & mag & real & phot.mag;em.IR.K & Same as above for UKIDSS $K$ filter\
corrfibmag\_fuv\_err & mag & real & stat.error;phot.mag;em.UV.FUV & Uncertainty of corrfibmag\_fuv column\
corrfibmag\_nuv\_err & mag & real & stat.error;phot.mag;em.UV.NUV & Uncertainty of corrfibmag\_nuv column\
corrfibmag\_u\_err & mag & real & stat.error;phot.mag;em.opt.U & Uncertainty of corrfibmag\_u\
corrfibmag\_g\_err & mag & real & stat.error;phot.mag;em.opt.B & Uncertainty of corrfibmag\_g\
corrfibmag\_r\_err & mag & real & stat.error;phot.mag;em.opt.R & Uncertainty of corrfibmag\_r\
corrfibmag\_i\_err & mag & real & stat.error;phot.mag;em.opt.I & Uncertainty of corrfibmag\_i\
corrfibmag\_z\_err & mag & real & stat.error;phot.mag;em.opt.I & Uncertainty of corrfibmag\_z\
corrfibmag\_y\_err & mag & real & stat.error;phot.mag;em.IR.J & Uncertainty of corrfibmag\_y\
corrfibmag\_j\_err & mag & real & stat.error;phot.mag;em.IR.J & Uncertainty of corrfibmag\_j\
corrfibmag\_h\_err & mag & real & stat.error;phot.mag;em.IR.H & Uncertainty of corrfibmag\_h\
corrfibmag\_k\_err & mag & real & stat.error;phot.mag;em.IR.K & Uncertainty of corrfibmag\_k\
kcorrfib\_fuv & mag & real & arith.factor;em.UV.FUV & K-correction for 3” aperture GALEX $FUV$ magnitude
kcorrfib\_nuv mag real arith.factor;em.UV.NUV Same as above for GALEX $NUV$ magnitude
kcorrfib\_u mag real arith.factor;em.opt.U K-correction for fiber (3“ aperture) SDSS $u$ magnitude\
kcorrfib\_g & mag & real & arith.factor;em.opt.B & Same as above for SDSS $g$ magnitude\
kcorrfib\_r & mag & real & arith.factor;em.opt.R & Same as above for SDSS $r$ magnitude\
kcorrfib\_i & mag & real & arith.factor;em.opt.I & Same as above for SDSS $i$ magnitude\
kcorrfib\_z & mag & real & arith.factor;em.opt.I & Same as above for SDSS $z$ magnitude\
kcorrfib\_y & mag & real & arith.factor;em.IR.J & K-correction for 3” aperture UKIDSS $Y$ magnitude
kcorrfib\_j mag real arith.factor;em.IR.J Same as above for UKIDSS $J$ magnitude
kcorrfib\_h mag real arith.factor;em.IR.H Same as above for UKIDSS $H$ magnitude
kcorrfib\_k mag real arith.factor;em.IR.K Same as above for UKIDSS $K$ magnitude
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Column Units Datatype UCD Description
--------------------- ------- ---------- -------------------------------- ------------------------------------------------------------------------------------------------------------------------------------
id bigint meta.id;meta.main Primary key
objid bigint SDSS ObjID
mjd d integer time.epoch MJD of observation
plate smallint meta.id SDSS plate ID
fiberid smallint meta.id SDSS fiber ID
e\_bv mag real phot.color.excess Intrinsic E(B-V)
gas\_oh\_d16 real phys.abund.Z Oxygen abundance of ionized gas (12 + log O/H) calculated using Dopita+16 calibration from Gaussian fit to emission lines
gas\_oh\_d16\_err real phys.abund.Z Error of oxygen abundance of ionized gas (12 + log O/H) calculated using Dopita+16 calibration from Gaussian fit to emission lines
gas\_oh\_izi real phys.abund.Z Oxygen abundance of ionized gas (12 + log O/H) calculated using IZI calibration from Gaussian fit to emission lines
gas\_oh\_izi\_errlo real stat.error;phys.abund.Z Lower error of oxygen abundance of ionized gas (12 + log O/H) calculated using IZI calibration from Gaussian fit to emission lines
gas\_oh\_izi\_errhi real stat.error;phys.abund.Z Upper error of oxygen abundance of ionized gas (12 + log O/H) calculated using IZI calibration from Gaussian fit to emission lines
q\_izi real phys.ionizParam.rad Ionization parameter calculated using IZI calibration from Gaussian fit to emission lines
q\_izi\_errlo real stat.error;phys.ionizParam.rad Lower error of ionization parameter calculated using IZI calibration from Gaussian fit to emission lines
q\_izi\_errhi real stat.error;phys.ionizParam.rad Upper error of ionization parameter calculated using IZI calibration from Gaussian fit to emission lines
Column Units Datatype UCD Description
---------------------- ----------------------------- ---------- ------------------------------------------------ -----------------------------------------------------------------------------------------------------------------
id bigint meta.id; meta.main Primary key
objid bigint meta.id SDSS ObjID
mjd d integer time.epoch MJD of observation
plate smallint meta.id SDSS plate ID
fiberid smallint meta.id SDSS fiber ID
forbid\_v km/s real phys.veloc Velocity measured simultaneously in all forbidden lines
forbid\_v\_err km/s real stat.error; phys.veloc Uncertainty in the velocity measured simultaneously in all forbidden lines
forbid\_sig km/s real phys.veloc.dispersion Velocity dispersion measured simultaneously in all forbidden lines
forbid\_sig\_err km/s real stat.error; phys.veloc.dispersion Uncertainty in the velocity dispersion measured simultaneously in all forbidden lines
allowed\_v km/s real phys.veloc Velocity measured simultaneously in all allowed lines
allowed\_v\_err km/s real stat.error; phys.veloc Uncertainty in the velocity measured simultaneously in all allowed lines
allowed\_sig km/s real phys.veloc.dispersion Velocity dispersion measured simultaneously in all allowed lines
allowed\_sig\_err km/s real stat.error; phys.veloc.dispersion Uncertainty in the velocity dispersion measured simultaneously in all allowed lines
chi2 real stat.fit.chi2 Reduced goodness of fit
f3727\_oii\_flx 10$^{-17}$ erg/s/cm$^{2}$ real phot.flux; spect.line Flux from Gaussian fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_flx\_err 10$^{-17}$ erg/s/cm$^{2}$ real stat.error; phot.flux; spect.line Uncertainty in the flux from Gaussian fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_cnt 10$^{-17}$ erg/s/cm$^{2}$/Å real phot.flux.density; spect.continuum Continuum level at \[O[ii]{}\] (3727 Å) line center
f3727\_oii\_cnt\_err 10$^{-17}$ erg/s/cm$^{2}$/Å real stat.error; phot.flux.density; spect.continuum Uncertainty in the continuum level at \[O[ii]{}\] (3727 Å) line center
f3727\_oii\_ew Å real spect.line.eqWidth Equivalent width from Gaussian fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_ew\_err Å real stat.error; spect.line.eqWidth Uncertainty in the equivalent width from Gaussian fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
$\dots$ $\dots$ $\dots$ $\dots$ $\dots$
Column Units Datatype UCD Description
---------------------- ----------------------------- ---------- ------------------------------------------------ -----------------------------------------------------------------------------------------------------------------------
id bigint meta.id; meta.main Primary key
objid bigint meta.id SDSS ObjID
mjd d integer time.epoch MJD of observation
plate smallint meta.id SDSS plate ID
fiberid smallint meta.id SDSS fiber ID
forbid\_v km/s real phys.veloc Velocity measured simultaneously in all forbidden lines
forbid\_sig km/s real phys.veloc.dispersion Velocity dispersion measured simultaneously in all forbidden lines
allowed\_v km/s real phys.veloc Velocity measured simultaneously in all allowed lines
allowed\_sig km/s real phys.veloc.dispersion Velocity dispersion measured simultaneously in all allowed lines
chi2 real stat.fit.chi2 Reduced goodness of fit
f3727\_oii\_flx 10$^{-17}$ erg/s/cm$^{2}$ real phot.flux; spect.line Flux from non-parametric fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_flx\_err 10$^{-17}$ erg/s/cm$^{2}$ real stat.error; phot.flux; spect.line Uncertainty in the flux from non-parametric fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_cnt 10$^{-17}$ erg/s/cm$^{2}$/Å real phot.flux.density; spect.continuum Continuum level at \[O[ii]{}\] (3727 Å) line center
f3727\_oii\_cnt\_err 10$^{-17}$ erg/s/cm$^{2}$/Å real stat.error; phot.flux.density; spect.continuum Uncertainty in the continuum level at \[O[ii]{}\] (3727 Å) line center
f3727\_oii\_ew Å real spect.line.eqWidth Equivalent width from non-parametric fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
f3727\_oii\_ew\_err Å real stat.error; spect.line.eqWidth Uncertainty in the equivalent width from non-parametric fit to continuum subtracted data of \[O[ii]{}\] (3727 Å) line
$\dots$ $\dots$ $\dots$ $\dots$ $\dots$
[^1]: The data tables and other supporting technical information are available at the project web-site: <http://rcsed.sai.msu.ru/>
[^2]: According to NASA ADS, <http://ads.harvard.edu/>
[^3]: <http://skyserver.sdss3.org/CasJobs/>
[^4]: <http://surveys.roe.ac.uk/wsa/>
[^5]: <http://galex.stsci.edu/casjobs/>
[^6]: Published SDSS spectra are slightly oversampled in wavelength, therefore, flux uncertainties in neighboring pixels are correlated and, hence, the reduced $\chi^2$ for a spectrum well represented by its model is less than 1 (around 0.6).
[^7]: <http://www.physics.wisc.edu/~craigm/idl/fitting.html>
[^8]: <http://www-astro.physics.ox.ac.uk/~mxc/software/bvls.pro>
[^9]: <http://soft.g-vo.org/dachs>
[^10]: <http://www.star.bris.ac.uk/~mbt/topcat/>
[^11]: <http://www.g-vo.org/pmwiki/About/SPLAT>
[^12]: <http://www.sciops.esa.int/index.php?project=SAT&page=vospec>
[^13]: <http://oirsa.cfa.harvard.edu/>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Fundamental errors in the Chubykalo et al paper \[2\] are highlighted. Contrary to their claim that “... the irrotational component of the electric field has a physical meaning and can propagate exclusively [*instantaneously*]{}," it is shown that this instantaneous component is physically irrelevant because it is always canceled by a term contained into the solenoidal component. This result follows directly from the solution of the wave equation that satisfies the solenoidal component. Therefore the subsequent inference of these authors that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one retarded and the other one instantaneous, has no basis. The example given by these authors in which the full electric field of an oscillating charge equals its instantaneous irrotational component on the axis of oscillations is proved to be false.'
author:
- 'José A. Heras'
title: 'Comment on “Helmholtz theorem and the v-gauge in the problem of superluminal and instantaneous signals in classical electrodynamics," by A. Chubykalo et al'
---
In the context of the so-called velocity gauge (v-gauge) of classical electrodynamics \[1\], in which the scalar potential $\varphi$ and the vector potential $\v A$ satisfy the condition: $\nabla\cdot \v A+[c/v^2]\partial\varphi/\partial t=0,$ and applying the Helmholtz theorem, which states that $\v E=\v E_i+\v E_s$ where $\v E_i$ is irrotational $(\nabla\times\v E_i=0)$ and $\v E_s$ is solenoidal $(\nabla\cdot\v E_s=0)$, Chubykalo et al \[2\] make the claim that “...the irrotational component of the electric field has a physical meaning and can propagate exclusively [*instantaneously*]{}." These authors also claim that “... there are [*two*]{} mechanisms of the energy and momentum transmission in classical electrodynamics: (1) the [*retarded*]{} one by means of a radiation $(\v E_s$ and $\v B)$...; (2) the [*instantaneous*]{} one by means of the irrotational field $\v E_i$." To illustrate their conclusion (2), they discuss the problem of an oscillating charge and claim that in this case the full electric field is equal to its irrotational component on the axis of oscillations. In other words, they claim to have found a full electric field $\v E$ of Maxwell’s theory satisfying the equation $\v E =\v E_i$ where $\v E_i$ is an instantaneous field!
The purpose of this comment is to point out that the above claims are incorrect. This means that the instantaneous component $\v E_i$ has no physical meaning and therefore there is no two transmission mechanisms in the electrodynamics of localized sources in vacuum. We point out that the misleading conclusions of Chubykalo et al \[2\] arise from having done a separated lecture of the equations for the components $\v E_i$ and $\v E_s$, [*i.e.,*]{} these authors treat these equations as if they were independent equations. But the fact is that they are coupled and therefore no physical inference on $\v E_i$ and $\v E_s$ should be extracted from these equations without considering their coupling.
Here we show that the component $\v E_i$ is physically irrelevant because it is always canceled by its additive negative $-\v E_i$ which is contained into the solenoidal component $\v E_s$. The fact that $-\v E_i$ is a part of $\v E_s$ follows directly from the solution of the wave equation that satisfies $\v E_s$. Chubykalo et al \[2\] have also ignored the comment of Jackson \[3\] on a paper of Chubykalo and Vlaev \[4\], in which Jackson has emphasized that the inference of both electromagnetic interactions, instantaneous and retarded ones, cannot be made.
The example of Chubykalo et al \[2\] in which for an oscillating charge they conclude that $\v E=\v E_i$ on the axis of oscillations is proved to be false in the Maxwell theory, but as a consolation for these authors this result is shown to be correct in the context of a Galilei-invariant electromagnetic theory \[5,6\]. Furthermore, not only does Chubykalo et al \[2\] paper display fundamental errors but it misinforms the readers by attributing to me an incorrect inference on the decomposition of a retarded field that I have never made.
Starting with the v-gauge potentials and applying the Helmholtz theorem to: (i) the v-gauge vector potential $\v A=\v A_i+\v A_s$; (ii) the retarded electric field $\v E=\v E_i+\v E_s$ and (iii) the current density $\v j=\v j_i+\v j_s$, Chubykalo et al \[2\] derive the following equations $$\begin{aligned}
\nabla^2\v E_i=4\pi\nabla\rho,\;\\
\nabla^2\v E_s-\frac{1}{c^2}\frac{\partial^2\v E_s}{\partial t^2}=\frac{4\pi}{c^2}\frac{\partial\v j_s}{\partial t},\end{aligned}$$ \[see Eq. (24) and the line below Eq. (23) both in Ref. 2\]. They claim: “Thus we see that the vector fields $\v E_i$ and $\v E_s$ are solutions of [*different*]{} equations with $\v E_i$-“wave“ propagating [*instantaneously*]{} and $\v E_s$-wave propagating with the velocity $c$ respectively.” Clearly, these conclusions are obtained from considering separately Eqs. (1) and (2).
Furthermore, without making use of the v-gauge potentials, the authors of Ref. 2 also apply the Helmholtz theorem directly to both the wave equation of the electric field and the Ampere-Maxwell equation and obtain again Eqs. (1) and (2) \[Eqs. (27) and (28) of Ref. 2\]. In fact, it is not difficult to show that the equations $$\begin{aligned}
\nabla^2\v (\v E_i+\v E_s) -\frac{1}{c^2}\frac{\partial^2(\v E_i+\v E_s)}{\partial t^2}=4\pi\Bigg(\nabla\rho+\frac{1}{c^2}\frac{\partial\v (\v j_i+\v j_s)}{\partial t}\Bigg),\\
\nabla\times(\v B_i+\v B_s) -\frac{1}{c}\frac{\partial(\v E_i+\v E_s)}{\partial t}=\frac{4\pi}{c}(\v j_i+\v j_s),\qquad\qquad\qquad\end{aligned}$$ imply Eqs. (1) and (2) as well as the equation $$\begin{aligned}
-\frac{1}{4\pi}\frac{\partial\v E_i}{\partial t}= \v j_i,\end{aligned}$$ \[Eq. (30 of Ref. 2\]. First of all, we note that Eqs. (1) and (2) are [*different*]{} but not [*independent.*]{} Actually, they are coupled equations. From a formal point of view Eq. (1) states that the field $\v E_i$ propagates instantaneously, but before concluding that this acausal feature of $\v E_i$ is a physical prediction of Maxwell’s theory, we should consider also Eqs. (2) and (5) because the latter involves explicitly $\v E_i$ and the former involves implicitly $\v E_i$ via the current $\v j_s=\v j-\v j_i$. In fact, using this decomposition of the current together with Eq. (5) we can write Eq. (2) as $$\begin{aligned}
\nabla^2\v E_s-\frac{1}{c^2}\frac{\partial^2\v E_s}{\partial t^2}=\frac{4\pi}{c^2}\frac{\partial\v j}{\partial t}+\frac{1}{c^2}\frac{\partial^2\v E_i}{\partial t^2}.\end{aligned}$$ It is now clear that (1) and (6) are coupled equations. From Eq. (6) we see that the component $\v E_i$ may be considered as a source of the component $\v E_s$ and therefore it would not be surprising that the solution of Eq. (6) would involve information on $\v E_i$. This means that the statement that the solenoidal component $\v E_s$ propagates with speed $c$ is simplistic because one of the sources in Eq. (6), namely, $\v E_i$ extends over all space and propagates instantaneously. In order to find what is the exact connection between $\v E_s$ and $\v E_i$ predicted by Eq. (6) \[or equivalently by Eq. (2)\] we must solve this equation. It can be shown (proof below) that the solution of Eq. (6) can be written as $$\begin{aligned}
\v E_s=-\frac{1}{4\pi}\int\int d^3x'dt'G_R\bigg(\nabla'\rho+\frac{1}{c^2}\frac{\partial \v j}{\partial t'}\bigg)-\v E_i,\end{aligned}$$ where $G_R= \delta(t'-t+R/c)/R$ is the retarded Green function satisfying $\Box^2 G_R(\v x,t;\v x',t') = -4\pi\delta(\v x-\v x')\delta(t-t')$ with $\Box^2\equiv\nabla^2-(1/c^2)\partial^2/\partial t^2$ being the D’Alambertian operator. As may be seen, Eq. (7) contains the term $-\v E_i$ and therefore the irrotational component $\v E_i$ appearing in the electric field $\v E=\v E_i+\v E_s$ is exactly canceled by the term $-\v E_i$ appearing in the solenoidal component $\v E_s$ given by Eq. (7). The fact that $\v E_i$ is an instantaneous component is physically irrelevant because it is always eliminated. Any possible interaction of $\v E_i$ with a charge $e$, for example, that given by the force $e\v E_i$ is automatically eliminated in the Lorentz force $\v F=e\v E_i+e\v E_s=e\v E_i + e\v E-e\v E_i=e\v E$. In other words: the field $\v E_i$ is a spurious field and therefore physically undetectable.
After substituting Eq. (7) into $\v E=\v E_i+\v E_s$, we obtain the usual retarded solution of Maxwell’s equations for the electric field: $$\begin{aligned}
\v E=-\frac{1}{4\pi}\int\int d^3x'dt'G_R\bigg(\nabla'\rho+\frac{1}{c^2}\frac{\partial \v j}{\partial t'}\bigg),\end{aligned}$$ which propagates with the speed $c$. We can now answer the question: How does the field $\v E_s$ propagate? Answer: The field $\v E_s$ in Eq. (7) contains two parts, one of which propagates with the speed $c$ \[the first term\] and the other one with infinite speed \[the second term\] which is always canceled by the irrotational component $\v E_i$. This means that causality is never effectively lost in applying the Helmholtz theorem to the electric field of Maxwell’s equations.
As above stated the instantaneous component $\v E_i$ is a spurious quantity which has mathematical but not physical existence. This result emphasizes the fact that the standard Helmholtz decomposition of the electric field involves terms with no physical significance. Yang \[7\] has recently emphasized the difficulties arising from applying the Helmholtz theorem to time-dependent vector fields. He wrote \[7\]: “There are two physics-related problems with this \[Helmholtz\] decomposition that are relevant: It introduces a spurious nonlocal property and spurious propagation behavior into the gradient and curl components. The result of the Helmholtz theorem are not physically consistent with the original vector function because of the spurious properties of its components." He observes that in the Helmholtz decomposition of the current electric $\v j=\v j_i+\v j_s$, the components $\v j_i$ and $\v j_s$ do not in general vanish outside the source region and so they cannot be physically measured. He also notes that in the Helmholtz decomposition of the Lorenz-gauge vector potential $\v A^L=\v A^L_i+\v A^L_s$, the components $\v A^L_i$ and $\v A^L_s$ propagate ahead of its progenitor $\v A^L.$
It is easy to show that Eq. (7) satisfies Eq. (6). We simply take the D’Alambertian to Eq. (7) to obtain $$\begin{aligned}
\Box^2\v E_s= 4\pi\nabla\rho+\frac{4\pi}{c^2}\frac{\partial \v j}{\partial t}-\nabla^2\v E_i+ \frac{1}{c^2}\frac{\partial^2 \v E_i}{\partial t^2}.\end{aligned}$$ If we use Eq. (1) then Eq. (9) reduces to $$\begin{aligned}
\Box^2\v E_s=\frac{4\pi}{c^2}\frac{\partial \v j}{\partial t}+ \frac{1}{c^2}\frac{\partial^2 \v E_i}{\partial t^2},\end{aligned}$$ which is the same as Eq. (6). Furthermore, if we use Eq. (5) and $\v j=\v j_i-\v j_s$ then Eq. (10) becomes $$\begin{aligned}
\Box^2\v E_s=\frac{4\pi}{c^2}\frac{\partial \v j_s}{\partial t},\end{aligned}$$ which is the same as Eq. (2). Alternatively, we can integrate Eq. (10) to obtain the solution (7). This procedure is somewhat laborious. In fact, the solution of (10) can be written as $$\begin{aligned}
\v E_s=-\frac{1}{4\pi c^2}\int\int d^3x'dt'G_R\frac{\partial \v j}{\partial t'}
-\frac{1}{4\pi c^2}\int\int d^3x'dt'G_R\frac{\partial^2 \v E_i}{\partial t'^2}.\end{aligned}$$ The second term on the right-hand side of Eq. (12) can be transformed using the tensor identity: $$\begin{aligned}
-\frac{1}{c^2}G_R\frac{\partial^2E^j}{\partial t'^2}=-
G_R\partial'_k\partial'^kE^j+E^j\bigg(\partial'_k\partial'^k-\frac{1}{c^2}\frac{\partial^2}{\partial t'^2}\bigg)G_R \nonumber\\+\partial'_k(G_R\partial'^kE^j-E^j\partial'^k G_R)\qquad \qquad\quad\;\;\,
\nonumber\\-\frac{1}{c^2}\frac{\partial}{\partial t'}\bigg(G_R \frac{\partial E^j}{\partial t'} -E^j\frac{\partial G_R}{\partial t'}\bigg),\quad\quad\qquad\,\,\end{aligned}$$ where $E^j=(\v E_i)^j$ and $\partial'^j=(\nabla')^j$. Latin indices $k$ and $j$ run from 1 to 3 and the summation convention on repeated indices is adopted. Integrating Eq. (13) over all space and all time (from $t=-\infty$ to $t=\infty$) and using $\partial'_k\partial'^k E^j=4\pi\partial^j\rho$ and $(\partial'_k\partial'^k-(1/c^2)\partial^2/\partial t'^2)G_R=-4\pi\delta(x^j-x'^j)\delta(t-t')$, we obtain $$\begin{aligned}
-\frac{1}{ c^2}\int\int d^3x'dt'G_R\frac{\partial^2 E^j}{\partial t'^2}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\nonumber\\
=-\int d^3x'dt'G_R \partial'^j\rho -4\pi E^j\qquad\qquad\qquad\qquad\;\nonumber\\
+\int dt'\bigg\{\int d^3x'\partial'_k(G_R\partial'^k E^j-E^j\partial'^k G_R)\bigg\}\quad\;\,\,
\nonumber\\-\frac{1}{ c^2}\int d^3x'\bigg\{\int dt'\frac{\partial}{\partial t'}\bigg(G_R \frac{\partial E^j}{\partial t'} -E^j\frac{\partial G_R}{\partial t'}\bigg)\bigg\}\;.\end{aligned}$$ The volume integral within the third term on the right-hand side of Eq. (14) can be transformed into a surface integral which vanishes on account of the behavior of both $G_R$ and $E^i$ at spatial infinity. Similarly, the exact time integration within the fourth term on the right-hand side vanishes on account of the behavior of $G_R$ and $E^j$ at temporal infinity. Thus Eq. (14) reduces to an expression that multiplied by $[1/(4\pi)]$ takes the form $$\begin{aligned}
-\frac{1}{4\pi c^2}\int\int d^3x'dt'G_R\frac{\partial^2 E^j }{\partial t'^2}=
-\frac{1}{4\pi}\int d^3x'dt'G_R \partial'^j\rho -E^j,
\end{aligned}$$ or equivalently, $$\begin{aligned}
-\frac{1}{4\pi c^2}\int\int d^3x'dt'G_R\frac{\partial^2\v E_i }{\partial t'^2}= -\frac{1}{4\pi}\int d^3x'dt'G_R\nabla'\rho -\v E_i.
\end{aligned}$$ From Eqs. (12) and (16) we obtain the solution of (7).
The conclusion of Chubykalo et al \[2\] that there are two mechanisms of transmission of energy and momentum in classical electrodynamics, one instantaneous (via $\v E_i$) and the other one retarded (via $\v E_s$ and $\v B$), has no basis because the mechanism of instantaneous transmission of $\v E_i$ is canceled by a part of $\v E_s$. The remaining part of $\v E_s$ is the well-known retarded contribution. The final result is that there is only one mechanism of transmission in electrodynamics of localized sources in vacuum: the mechanism associated with the retarded fields propagating with the speed $c$. Whatever propagation or nonpropagation characteristics are exhibited by the irrotational and solenoidal components of the electric field, this field always display the experimentally verified properties of causality and propagation at speed of light $c$.
The supposed necessity of considering both instantaneous and retarded interactions in classical electrodynamics was previously suggested by Chubykalo and Vlaev \[4\] on the basis of a supposed failure of the Lienard-Wiechert fields to satisfy the Maxwell equations. The demonstration of the supposed failure was proved to be false by Jackson \[3\] in a comment on the paper of Chubykalo and Vlaev \[4\] (an unconvincing reply \[8\] of these authors has not been published in the pertinent Journal). Jackson wrote \[3\]: “It is of course known that in certain gauges the potentials can contain both retarded and instantaneous contributions. But there is [*no necessity*]{} for such a mixture. And the fields are always retarded..." He also pointed out \[3\]: “Classical electromagnetic theory is complete as usually expressed. One may choose to work in the Lorenz gauge in which all quantities are retarded." The comment of Jackson \[3\] was completely ignored in the paper of Chubykalo et al \[2\].
Paraphrasing Jackson, we can say that it is known that the standard Helmholtz decomposition of the electric field involves instantaneous and retarded components but there is no necessity of considering such a decomposition. After all, the full electric field is always retarded and one may choose to use a retarded Helmholtz’s decomposition in which all quantities are retarded. We will discuss this in section 5.
Chubykalo et al \[2\] claim: “Let us consider the case when exclusively $\v E_i$ can be responsible for a signal transfer from a point charge q to the other point charge Q..." Their argument is as follows. They consider an arbitrarily moving charge for which $\rho(\v r,t)=q\delta(\v r-\v r_q(t))$ and $\v j(\v r,t)=\v v_q\delta(\v r-\v r_q(t))$, where $\v r_q(t)$ and $\v v_q(t)$ are the position and velocity of the particle. From Eqs. (30) and (42) of their paper \[2\] they obtain $$\begin{aligned}
\v E_i=q \frac{\v r-\v r_q(t)}{|\v r-\v r_q(t)|^3},\end{aligned}$$ \[Eq. (49) of Ref. 2\], and state that this field \[2\]: “...is a Coulomb type field: it is conservative ..." They then consider the particular case of a point charge oscillating along the X-axis: $\v r_q=(A_0\sin \omega t)\v i$, and find $$\begin{aligned}
\nabla\times\v E_i=0,\end{aligned}$$ \[Eq. (55) of Ref. 2\]. After some considerations, they conclude \[2\]: “On account of the symmetry of the problem and because of $\v E=\v E_i+\v E_s,\, \v E_s$ must be equal to $zero$ along of the X-axis. It can mean solely the following: The irrotational component of the electric field has a physical meaning and in some case is charged with the instantaneous energy and momentum transmission." In a few words, they pretend to have found an example in which the full field $\v E$ equals its irrotational component $\v E_i$ along the X-axis: $$\begin{aligned}
\v E=\v E_i. \end{aligned}$$ There are several ways to prove the inconsistence of Eq. (19) in Maxwell’s theory. The simplest way is the obvious one: Any time-dependent electric field $\v E$ of Maxwell’s theory must satisfy Faraday’s law: $$\begin{aligned}
\nabla\times\v E=-\frac{1}{c}\frac{\partial\v B}{\partial t}. \end{aligned}$$ But the field $\v E=\v E_i$ satisfies Eq. (18) and then Faraday’s law is not obeyed! Moreover, if for a moment we would consider the possibility that $\v E_s=0$, as Chubykalo et al \[2\] claim to have proved in their particular example, then the solution (7) leads to $$\begin{aligned}
\v E_i=-\frac{1}{4\pi}\int\int d^3x'dt'G_R\bigg(\nabla'\rho+\frac{1}{c^2}\frac{\partial \v j}{\partial t'}\bigg).\end{aligned}$$ The inconsistence of Eq. (21) becomes evident for the case of an arbitrarily moving charge. In this case the left-hand side of Eq. (21) gives the instantaneous field in Eq. (17) while the right-hand side gives the Lienard-Wiechert electric field, [*i.e.,*]{} $$\begin{aligned}
q \frac{\v n}{R^2}= q
\Bigg[\frac{\v n-\v v_q/c}{\gamma^2(1-\v n\cdot\v v_q/c)^3R^2}\Bigg]_{\rm ret}\qquad\;\;\nonumber\\ \;+\frac{q}{c}\Bigg[\frac{\v n\times \{(\v n-\v v_q/c)\times \v a_q/c}{(1-\v n\cdot\v v_q/c)^3R}\Bigg]_{\rm ret},\end{aligned}$$ where the subscript ret means that the quantity enclosed in the square brackets is to be evaluated at the retarded time $t'=t-R(t')/c;\, \v n=\v R/R=\v r-\v r_q(t)/|\v r-\v r_q(t)|$ and $\gamma= (1-v_q^2/c^2).^{-1/2}$ A simple reflection shows that Eq. (22) is manifestly inconsistent. The right-hand side cannot be equal to the left-hand side. The choice $c\to \infty$ is physically forbidden in the right-hand side of Eq. (22).
As a consolation for the authors of Ref. 2, their claim that in some cases the full electric field can equal to its irrotational part is shown to be correct in the context of a Galilean-invariant electromagnetic theory whose field equations are \[5,6\]: $$\begin{aligned}
\nabla\cdot\widetilde{\v E}= 4\pi\rho,\\
\nabla\cdot\widetilde{\v B}=0,\;\;\, \\
\nabla\times\widetilde{\v E}=0,\;\;\,\\
\nabla\times\widetilde{\v B}-\frac{1}{c}\frac{\partial\widetilde{\v E}}{\partial t}=\frac{4\pi}{c}\v j,\end{aligned}$$ where $\widetilde{\v E}$ and $\widetilde{\v B}$ are instantaneous electric and magnetic fields. From Eqs. (23) and (25) we obtain $\nabla^2\widetilde{\v E}=4\pi\nabla\rho$. Using the Helmholtz’s theorem we have $\widetilde{\v E}=\widetilde{\v E}_i$. It follows that $\nabla^2\widetilde{\v E}_i=4\pi\nabla\rho$. The solution of this equation for $\rho(\v r,t)=q\delta(\v r-\v r_q(t))$ gives naturally Eq. (17): $$\begin{aligned}
\widetilde {\v E}_i=q \frac{\v r-\v r_q(t)}{|\v r-\v r_q(t)|^3}.\end{aligned}$$
Chubykalo et al \[2\]: “... although the electric field... can be [*retarded*]{}, it is decomposed into just two parts, one of which is [*pure irrotational*]{} and the other is [*pure solenoidal*]{}: $$\begin{aligned}
\v E=\v E_i+\v E_s, \quad \v E_i=-\nabla\varphi, \quad \v E_s=-\frac{1}{c}\frac{\partial\v A}{\partial t},\end{aligned}$$ (in the Coulomb gauge $\nabla\cdot \v A=0)$ \[Eq. (6) of Ref. 2\]. This alone shows that [**the inference of J. A. Heras**]{} ... that a retarded field cannot be decomposed into [*only*]{} two parts (irrotational and solenoidal) can be insufficiently rigorous." The boldface emphasis is mine. First of all, I have never made this incorrect inference as Chubykalo et al \[2\] claim. Some years ago McQuistan \[9\] and more recently the present author \[10-12\], have formulated the retarded Helmholtz theorem which states that a retarded field vanishing at infinity can be decomposed into irrotational, solenoidal and temporal components: $$\begin{aligned}
\v E=\v E^{\cal R}_i+\v E^{\cal R}_s +\v E^{\cal R}_{\cal T},\end{aligned}$$ where $$\begin{aligned}
\v E^{\cal R}_i=-\nabla\int d^3x'\frac{[\nabla'\cdot\v E]}{4\pi R},\;\;\,\\
\v E^{\cal R}_s=\nabla\times\int d^3x'\frac{[\nabla'\times\v E]}{4\pi R},\\
\v E^{\cal R}_{\cal T}= \frac{1}{c^2}\frac{\partial}{\partial t}\int d^3x'\frac{[\partial \v E/\partial t]}{4\pi R}.\end{aligned}$$ The square brackets $[\;]$ mean that the enclosed quantity is to be evaluated at the retarded time $t'=t-R/c$, and the superscript ${\cal R}$ emphasizes the retarded character of the quantities \[13\]. An alternative version of this retarded Helmholtz theorem has also recently formulated \[14\]. The result that the field $\v E$ can be decomposed in terms of the components $\v E^{\cal R}_i,\,\v E^{\cal R}_s$ and $ \v E^{\cal R}_{\cal T}$ does not exclude the possibility that $\v E$ can also be decomposed in terms of other different irrotational and solenoidal components $\v E_i$ and $\v E_s$. In Ref. 12, I show that if $\v E$ is the retarded electric field of Maxwell’s equations then $$\begin{aligned}
\v E^{\cal R}_i=-\nabla\int d^3x'\frac{[\rho]}{4\pi R},\,\\
\v E^{\cal R}_s+\v E^R_{\cal T}=- \frac 1c\frac{\partial}{\partial t} \int d^3x' \frac{[\v j]}{Rc},\end{aligned}$$ and I show also that these expressions can be written in terms of the Coulomb-gauge potentials $\Phi_C$ and $\v A_C$ as follows: $$\begin{aligned}
\v E^{\cal R}_i=-\nabla\Phi_C+\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\int d^3x'\frac{[\nabla'\Phi_C]}{4\pi R},\;\;\,\\
\v E^{\cal R}_s+\v E^R_{\cal T}=-\frac{1}{c}\frac{\partial\v A_C}{\partial t}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\int d^3x'\frac{[\nabla'\Phi_C]}{4\pi R}.\end{aligned}$$ Therefore, $$\begin{aligned}
\v E=\v E^{\cal R}_i+\v E^{\cal R}_s +\v E^{\cal R}_{\cal T}=-\nabla\Phi_C-\frac{1}{c}\frac{\partial\v A_C}{\partial t}.\end{aligned}$$ This result means that the retarded electric field can $rigorously$ be decomposed either in terms of the irrotational, solenoidal and temporal components: $\v E^{\cal R}_i,\, \v E^{\cal R}_s$ and $\v E^{\cal R}_{\cal T}$, or equivalently in terms of the instantaneous irrotational component: $-\nabla\Phi_C$ and of the solenoidal component: $-(1/c)\partial\v A_C/\partial t.$ I have never inferred that: “... a retarded field cannot be decomposed into [*only*]{} two parts (irrotational and solenoidal)..." as Chubykalo et al \[2\] attribute to me. Furthermore, I have proved in Ref. 12 exactly the opposite: By applying the retarded Helmholtz theorem, I could decompose the retarded electric field into two parts (irrotational and solenoidal) as may be seen in Eq. (37).
The direct application of the retarded Helmholtz theorem to the electric field of Maxwell’s equations leads to the well-known retarded expression of this field \[see Eqs. (33) and (34)\] which of course do not include instantaneous contributions. On the other hand, the direct application of the standard Helmholtz theorem to the electric field of Maxwell’s equations leads to the expression of this field in terms of the Coulomb-gauge potentials \[see Eq. (6) of Ref. 2 or Eq. (28) in the present paper\] with the disadvantage that in such an application a spurious instantaneous electric field is introduced. Accordingly, if we do not want to generally introduce instantaneous fields using the standard Helmholtz theorem then we may use the retarded form of this theorem \[10-12\].
Chubykalo et al \[2\] attempt: “ ... to substantiate the applying of the Helmholtz theorem to vector fields in classical electrodynamics." Unfortunately the physical interpretations given by these authors for the irrotational and solenoidal components of the electric field, obtained from applying the standard Helmholtz theorem, are misleading and add nothing but confusion to the topic of instantaneous and retarded fields. The present author has formally demonstrated \[6\] that the instantaneous fields can be introduced as [*unphysical*]{} objects into classical electrodynamics which can be used to express the retarded fields.
[**Acknowledgements.**]{} The present author is grateful to Professor R. F. O’Connell for the kind hospitality extended to him in the Department of Physics and Astronomy of the Louisiana State University.
[99]{}
J. D. Jackson, “From Lorenz to Coulomb and other explicit gauge transformations," [*Am. J. Phys.*]{} [**70**]{}, 917 (2002).
A. Chubykalo, A. Espinoza, R. Alvarado Flores, and A. Gutierrez Rodriguez, “Helmholtz theorem and the v-gauge in the problems of superluminal and instantaneous signals in classical electrodynamics," [*Found. Phys. Lett.*]{} [**19**]{}, 37 (2006).
J. D. Jackson, “Criticism of ‘Necessity of simultaneous co-existence of instantaneous and retarded interactions in classical electrodynamics, by Chubykalo and Vlaev,’" [*Int. J. Mod. Phys. A*]{} [**17**]{}, 3975 (2002).
A. Chubykalo and S. J. Vlaev, “Necessity of simultaneous co-existence of instantaneous and retarded interactions in classical electrodynamics, by Chubykalo and Vlaev," [*Int. J. Mod. Phys. A*]{} [**14**]{}, 3789 (1999).
M. Jammer and J. Stachel,“If Maxwell had worked between Ampere and Faraday: An historical fable with a pedagogical moral," [*Am. J. Phys.*]{} [**48**]{}, 5 (1980).
J. A. Heras, “Instantaneous fields in classical electrodynamics," [*Europhys. Letts.*]{} [**69**]{}, 1 (2005).
Kuo-Ho Yang, “The physics of gauge transformations," [*Am. J. Phys.*]{} [**73**]{}, 742 (2005).
A. Chubykalo and S. J. Vlaev, “Reply to ’Criticism of ‘Necessity of simultaneous co-Existence of instantaneous and retarded interactions in classical electrodynamics’ by J.D.Jackson," e-print: physics/0205041
R. B. McQuistan, [*Scalar and Vector fields: A Physical Interpretation*]{} (Wiley, New York, 1965), Sec. 12-3.
J. A. Heras, “Jefimenko’s formulas with magnetic monopoles and the Lienard-Wiechert fields of a dual-charged particle," [*Am. J. Phys.*]{} [**62**]{}, 525 (1994).
J. A. Heras, “Time-dependent generalizations of the Biot-Savart and Coulomb laws: A formal derivation," [*Am. J. Phys.*]{} [**63**]{}, 928 (1995).
J. A. Heras, “Comment on ‘Causality, the Coulomb field, and Newton’s law of gravitation’ by F. Rohrlich \[Am. J. Phys. [**70**]{}, 411-414 (2002)\],’" [*Am. J. Phys.*]{} [**71**]{}, 729 (2003).
I pointed out in Refs. 10 and 11 that the retarded Helmholtz theorem \[Eq. (29) in the present comment\] is not very useful in practice because we cannot generally specify $\partial \v E/\partial t$. Despite of this objection, I show in Ref. 10 how the retarded Helmholtz theorem can successfully be applied to find the retarded solutions of Maxwell’s equations for the general case that include magnetic monopoles. Also, I show in Ref. 11 how an equivalent form of theorem \[Eq. (17) of Ref. 11\] leads to the retarded solutions of Maxwell’s equations expressed in the form of time-dependent generalizations of the Coulomb and Biot-Savart laws.
A. M. Davis, “A generalized Helmholtz theorem for time-varying vector fields," [*Am. J. Phys.*]{} [**74**]{}, 72 (2006); J. A. Heras,“Coment on ‘A generalized Helmholtz theorem for time-varying vector fields by A. M. Davis, \[Am. J. Phys. [**74**]{}, 72 (2006)\],’" [*Am. J. Phys.*]{} [**74**]{}, 743 (2006).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We tackle the problem of acting in an unknown finite and discrete Markov Decision Process (MDP) for which the expected shortest path from any state to any other state is bounded by a finite number $D$. An MDP consists of $S$ states and $A$ possible actions per state. Upon choosing an action $a_t$ at state $s_t$, one receives a real value reward $r_t$, then one transits to a next state $s_{t+1}$. The reward $r_t$ is generated from a fixed reward distribution depending only on $(s_t, a_t)$ and similarly, the next state $s_{t+1}$ is generated from a fixed transition distribution depending only on $(s_t, a_t)$. The objective is to maximize the accumulated rewards after $T$ interactions. In this paper, we consider the case where the reward distributions, the transitions, $T$ and $D$ are all unknown. We derive the first polynomial time Bayesian algorithm, that achieves up to logarithm factors, a regret (i.e the difference between the accumulated rewards of the optimal policy and our algorithm) of the optimal order $\TilO(\sqrt{DSAT})$. Importantly, our result holds with high probability for the worst-case (frequentist) regret and not the weaker notion of Bayesian regret. We perform experiments in a variety of environments that demonstrate the superiority of our algorithm over previous techniques. Our work also illustrates several results that will be of independent interest. In particular, we derive a sharper upper bound for the KL-divergence of Bernoulli random variables. We also derive sharper upper and lower bounds for Beta and Binomial quantiles. All the bound are very simple and only use elementary functions.'
author:
- |
Aristide Tossou, Christos Dimitrakakis, Debabrota Basu\
Department of Computer Science and Engineering\
Chalmers University of Technology\
G[ö]{}teborg, Sweden\
`(aristide,chrdimi,basud)@chalmers.se`
bibliography:
- 'rlfastposteriorsampling.bib'
title: 'Near-optimal Bayesian Solution For Unknown Discrete Markov Decision Process'
---
Introduction {#sec:introduction}
============
Markov Decision Process (MDP) is a framework that is of central importance in computer science. Indeed, MDPs are a generalization of (stochastic) shortest path problems and can thus be used for routing problems [@psaraftis2016dynamic], scheduling and resource allocation problems [@gocgun2011markov]. One of its most successful application comes in reinforcement learning where it has been used to achieve human-level performance for a variety of games such as Go [@silver2017mastering], Chess [@silver2017masteringchess]. It is also a generalization for online learning problems (such as multi-armed bandit problems) and as such has been used for online advertisement [@lu2009showing] and movie recommendations [@qin2014contextual].
#### Problem Formulation
In this paper, we focus on the problem of online learning of a near optimal policy for an unknown Markov Decision Process. An MDP consists of $S$ states and $A$ possible actions per state. Upon choosing an action $a_t$ at state $s_t$, one receives a real value reward $r_t$, then one transits to a next state $s_{t+1}$. The reward $r_t$ is generated from a fixed reward distribution depending only on $(s_t, a_t)$ and similarly, the next state $s_{t+1}$ is generated from a fixed transition distribution $p(. | s_t, a_t)$ depending only on $(s_t, a_t)$. The objective is to maximize the accumulated (and undiscounted) rewards after $T$ interactions. An MDP is characterized by a quantity (called $D$) known as the diameter. It indicates an upper bound on the expected shortest path from any state to any other state. When this diameter (formally defined by Definition \[def:diameter\]) is finite, the MDP is called *communicating*.
\[def:diameter\] The diameter $D$ of an MDP $M$ is defined as the minimum number of rounds needed to go from one state $s$ and reach any other state $s'$ while acting using some deterministic policy. Formally, $$D(M) = \max_{s\ne s', s,s' \in \mathcal{S}} \min_{\pi: \mathcal{S} \rightarrow \mathcal{A}} T(s' | s, \pi)$$ where $T(s' | s, \pi)$ is the expected number of rounds it takes to reach state $s'$ from $s$ using policy $\pi$.
In this paper, we consider the case where the reward distributions $r$, the transitions $p$, $T$ and $D$ are all unknown. Given that the rewards are undiscounted, a good measure of performance is the gain, i.e. the infinite horizon average rewards. The gain of a policy $\pi$ starting from state s is defined by: $$\gain(s | \pi) \triangleq \limsup_{T \to \infty}\frac{1}{T} \EX\left[\sum_{t=1}^{T} r(s_t, \pi(s_t)) \mid s_1 = s\right].$$ @puterman2014markov shows that there is a policy $\pi^*$ whose gain, $\gain^*$ is greater than that of any other policy. In addition, this gain is the same for all states in a communicating MDP. We can then characterize the performance of the agent by its regret defined as: $$\regret(T) \triangleq \sum_{t=1}^{T} \left(\gain^* - r(s_t, a_t)\right).$$
Thus our goal is equivalent to obtaining a regret as low as possible.
#### Related Work
It has been shown that any algorithm must incur a regret of $\Omega(DSTA)$ in the worst case. [@jaksch2010near]. Since the establishment of this lower bound on the regret, there has been numerous algorithms for the problem. They can be classified in two ways: Frequentist and Bayesian. The frequentist algorithms usually construct explicit confidence interval while the Bayesian algorithms start with a prior distribution and uses the posterior derived from Bayes Theorem. Following a long line of algorithms KL-UCRL [@filippi2010optimism], REGAL.C [@bartlett2009regal], UCBVI [@azar2017minimax], SCAL [@fruit2018efficient] the authors of [@full_ucrlv_paper] derived a frequentist algorithm that achieved the lower bound up to logarithmic factors.
In contrast, the situation is different for Bayesian algorithms. One of the first to prove theoretical guarantees for posterior sampling is @osband2013more, for their algorithm. However, they only consider reinforcement learning problems with a finite and known episode length[^1] and prove an upper bound of $\BigO(HS\sqrt{TA})$ on the expected Bayesian regret where $H$ is the length of the episode. @ouyang2017learning generalises @osband2013more results to weakly communicating MDP and proves a $\BigO(H_SS\sqrt{TA})$ on the expected Bayesian regret where $H_S$ is a bound on the span of the MDP. Other Bayesian algorithms have also been derived in the litterature however, none of them is able to attain the lower bound for the general communicating MDP considered in this paper. Also many of the previous Bayesian algorithms only provide guarantees about the Bayesian regret (i.e, the regret under the assumption that the true MDP is being sampled from the prior). It was thus an open-ended question whether or not one can design Bayesian algorithms with optimal worst-case regret guarantees[@pmlr-v70-osband17a; @osband2016posterior]. In this work, we provide guarantees for the worst-case (frequentist) regret. We solve the challenge by designing the first Bayesian algorithm with provable upper bound on the regret that matches the lower bound up to logarithmic factors. Our algorithm start with a prior on MDP and computes the posterior similarly to previous works. However, instead of sampling from the posterior, we compute a quantile from the posterior. We then uses all the MDPs possible under the quantile as a set of statistically plausible MDPs and then follow the same steps as the state-of-the art [@full_ucrlv_paper]. The idea of using quantiles have already been explored in the algorithm named *Bayes-UCB* [@Kaufmann12onbayesian] for multi-armed bandit (a special case of MDP where there is only one single state). Our work can also be considered as a generalization to *Bayes-UCB*.
#### Our Contributions.
Hereby, we summarise the contributions of this paper that we elaborate in the upcoming sections.
- We provide a conceptually simple Bayesian algorithm for reinforcement learning that achieves near-optimal worst case regret. Rather than actually sampling from the posterior distribution, we simply construct upper confidence bounds through Bayesian quantiles.
- Based on our analysis, we explain why Bayesian approaches are often superior in performance than ones based on concentration inequalities.
- We perform experiments in a variety of environments that validates the theoretical bounds as well as proves to be better than the state-of-the-art algorithms. (Section \[sec:experiments\])
We conclude by summarising the techniques involved in this paper and discussing the possible future works they can lead to (Section \[sec:conclusion\]).
Algorithms Description and Analysis {#sec:algorithms}
===================================
In this section, we describe our Bayesian algorithm . We combine Bayesian priors and posterior together with optimism in the face of uncertainty to achieve a high probability upper bound of $\TilO(\sqrt{DSAT})$[^2] on the worst-case regret in any finite communicating MDP. Our algorithm can be summarized as follow:
1. Consider a prior distribution over MDPs and update the prior after each observation
2. Construct a set of statistically plausible MDPs using the set of all MDPs inside a Quantile of the posterior distribution.
3. Compute a policy (called *optimistic*) whose gain is the maximum among all MDPs in the plausible set. We used a *modified extended value iteration* algorithm derived in [@full_ucrlv_paper].
4. Play the computed *optimistic* policy for an artificial that lasts until the average number of times state-action pairs has been doubled reaches 1. This is known as the *extended doubling trick* [@full_ucrlv_paper].
They are multiple variants of quantiles definition for MDP (since an unknown MDP can be viewed as a multi-variate random variable). In this paper, we adopt a specific definition of quantiles for multi-variate random variable called *marginal quantiles*. More precisely,
Let $\bm{X} = (\bm{X_1} \ldots \bm{X_m})$ be a multivariate random vector with joint d.f.( distribution function) $F$, the i-th marginal d.f. $F_i$. We denote the ith marginal quantile function by: $$\quant_i(F,q) = \inf\{x:F_i(x) \geq q\}, 0 \leq q \leq 1.$$
Unless otherwise specified, we will refer to marginal quantile as simply *quantile*. For univariate distributions, the subscript $i$ can be omitted, as the quantile and the marginal quantile coincide.
**Input:** Let $\mu_1$ the prior distribution over MDPs. $1-\delta$ are confidence level. **Initialization:** Set $t \gets 1$ and observe initial state $s_1$ Set $N_k, N_k(s,a), N_{t_k}(s,a)$ to zero for all $k \geq 0$ and $(s,a)$.
**Compute optimistic policy $\tilde{\pi}_k$:** */\*Update the bounds on statistically plausible MDPs\*/* $\lowerb{r}(s,a) \gets Q_{s,a, \samples{r}}(\mu_t, \delta^k_r)$ (lower quantile)
$\upperb{r}(s,a) \gets Q_{s,a, \samples{r}}(\mu_t, 1-\delta^k_r)$ (upper quantile)
For any $\mathcal{S}_c \subseteq \mathcal{S}$ use: $\lowerb{p}(\mathcal{S}_c | s,a) \gets Q_{s,a,\mathcal{S}_c, \samples{p}}(\mu_t, \delta^k_p)$ (lower quantile)
$\upperb{p}(\mathcal{S}_c |s,a) \gets Q_{s,a,\mathcal{S}_c, \samples{p}}(\mu_t, 1-\delta^k_p)$ (upper quantile)
*/\*Find $\tilde{\pi}_k$ with value $\frac{1}{\sqrt{t_k}}$-close to the optimal\*/* $\tilde{\pi}_k \gets \Call{ExtendedValueIteration}{\lowerb{r}, \upperb{r}, \lowerb{p}, \upperb{p}, \frac{1}{\sqrt{t_k}}}$ (Algorithm 2 in @full_ucrlv_paper.)
**Execute Policy $\tilde{\pi}_k$:**
Play action $a_t$ and observe $r_t, s_{t+1}$. Let $r_t \gets \Bernoulli(r_t)$
Increase $N_k$ and $N_k(s_t, a_t)$, $N_{t_{k+1}}(s_t, a_t)$ by 1.
Update the posterior $\mu_{t+1}$ using Bayes rule.
$t \gets t + 1$
Our analysis is based on the choice of a specific prior distribution for MDP with bounded rewards.
#### Prior Distribution
We consider two different prior distributions. One for computing lower bound on rewards/transitions, that is when computing $\delta$-marginal quantile. One for computing upper bound on rewards/transitions, that is when computing $1-\delta$-marginal quantile.
For the lower bound, we used independent distribution for the rewards and transitions. We also used independent distribution for the rewards of each state-action $(s,a)$. And independent distribution for the transition from any state-action $(s,a)$ to any next subset of states $\mathcal{S}_c$. The prior distribution for any of those components is a beta distribution of parameter $(0,1)$: $\BetaDis(0, 1)$[^3].
The situation is similar with the upper bound. However, here the prior distribution for any component is a beta distribution of parameter $(1,0)$: $\BetaDis(1, 0)$.
#### Posterior Distribution
Let’s start by assuming that the rewards come from the Bernoulli distribution. For the upper bounds, using Bayes rule, the posterior at $t_k$ are:
For the rewards of any $(s,a)$: $$\BetaDis(\alpha + \sum_{t\leq t_k: s_t = (s,a)} r_t, \beta + N_{t_k}(s,a) - \sum_{t\leq t_k: s_t = (s,a)} r_t)$$
For the transitions from any $(s,a)$ to any subset of next state $\mathcal{S}_c$ are:
$$\BetaDis(\alpha + \sum_{t\leq t_k: s_t = (s,a)} p_t, \beta + N_{t_k}(s,a) - \sum_{t\leq t_k: s_t = (s,a)} p_t)$$ where $p_t = 1$ if $s_{t+1} \in \mathcal{S}_c$; $p_t = 0$ otherwise.
$\alpha = 1, \beta = 0$ for the upper posteriors and $\alpha = 0, \beta = 1$ for the lower posteriors.
#### Dealing with non-Bernoulli rewards
We deal with non-Bernoulli rewards by performing a Bernoulli trials on the observed rewards. In other words, upon observing $r_t$ we used $\Bernoulli(r_t)$ where $\Bernoulli(r_t)$ is a sample from the Bernoulli distribution of parameter $r_t$. This technique is already used in [@agrawal2012analysis] and ensures that our prior remain valid.
#### Quantiles
When $N_{t_k}(s,a) = 0$ the lower and upper quantiles are respectively $0$ and $1$. When the first parameter of the posterior is $0$, the lower quantile is $0$. When the second parameter of the posterior is $0$, the upper quantile is $1$. In all other cases, the $\delta$ quantile corresponds to the inverse cumulative distribution function of the posterior at the point $\delta$. To achieve a high probability bound of $1-\delta$ on our regret, we used the following parameters respectively for the rewards and transitions $\delta^k_r = \frac{\delta}{4 SA \ln\paren*{2t}}$, $\delta^k_p = \frac{\delta}{8S^2A \ln\paren*{2t}}$, where $1-\delta$ is the desired confidence level of the set of plausible MDPs.
\[thm:bayes\_ucrl\] With probability at least $1-\delta$ for any $\delta \in ]0,1[$, any $T \geq 1$, the regret of is bounded by: $$\begin{aligned}
\mathcal{R}(T) &\leq
20\cdot \sqrt{\min\{S,\log_2^2 2D\} D T SA\log T \ln \paren*{\frac{B}{\delta}}} + 9DSA\ln \paren*{\frac{B}{\delta}}\end{aligned}$$ for $B = 9S\sqrt{TDSA}\ln\paren*{TSA}$.
Our proof is based on the generic proof provided in @full_ucrlv_paper. To apply that generic proof, we need to show that with high probability the true rewards/transitions of any state-action is contained in the lower and upper quantiles of the Bayesian Posterior. In other words we need to show that the Bayesian quantiles provide exact coverage probabilities. For that we notice that our prior lead to the same confidence interval as the Clopper-Pearson interval (See Lemma \[lemma:coverage\_beta\]). Furthermore, we need to provide upper and lower bound for the maximum deviation of the Bayesian posterior quantiles from the empirical values. This is a direct consequence of Proposition \[lemma:beta\_quantile\_bound\] and \[lemma:beta\_quantile\_bound\_lower\].
The following results were all useful in establishing our main result in Theorem \[thm:bayes\_ucrl\]. Our main contribution in Proposition \[theo:kl\_bounds\] is the upper bound (the first term of the upper bound) for the KL-divergence of two bernoulli random variables. The last term of the upper bound is a direct derivation from the upper bounds in [@dragomir2000some]. Our result in Proposition \[theo:kl\_bounds\] shows a factor of $2$ improvement in the leading term of the upper bound. The KL divergence of Bernoulli random is useful for many online learning problems and we used it here to bound the quantile of the Binomial distributions in term of simple functions.
[theo:kl\_bounds]{}\[Bernoulli KL-Divergence\]
The main idea to prove the upper bound is by studying the sign of the function in $x$ obtained by taking the difference of the KL-divergence and the upper bound. We used Sturm’ theorem to basically show that this function starts as a decreasing function then after a point becomes increasing for the remaining of its domain. This together with the observation that at the end of its domain the function is non-positive concludes our proof. Full detailed are available in the appendix.
Proposition \[lemma\_binomial\_quantile\_bounds\] provides tight lower and upper bound for the quantile of the binomial distribution in the same simple form as Bernstein inequalities. Binomial distributions and their quantiles are useful for a lot of applications and we use it here to derive the bounds for the quantile from a Beta distribution in Proposition \[lemma:beta\_quantile\_bound\] and \[lemma:beta\_quantile\_bound\_lower\].
[lemma\_binomial\_quantile\_bounds]{}\[Lower and Upper bound on the Binomial Quantile\]
We used the tights bounds for the cdf of Binomial in [@zubkov2013complete]. We inverted those bounds and then use the upper and lower bound for KL divergence in Proposition \[theo:kl\_bounds\] to conclude. Full detailed is available in the appendix.
Proposition \[lemma:beta\_quantile\_bound\] and \[lemma:beta\_quantile\_bound\_lower\] provides lower and upper bound for the Beta quantiles in term of simple functions similar to the one for Bernstein inequalities. We used it to prove our main result in Theorem \[thm:bayes\_ucrl\].
[lemma:beta\_quantile\_bound]{}\[Upper bound on the Beta Quantile\]
These bounds comes directly from the relation between Beta and Binomial cdfs. We apply Proposition \[lemma\_binomial\_quantile\_bounds\] which gives a bounds for the quantile $p$ in term of $p(1-p)$. We then applies again Proposition \[lemma\_binomial\_quantile\_bounds\] to bound $p(1-p)$ in term of $\frac{x}{n} \paren*{1-\frac{x}{n}}$. Full proof is available in the appendix.
[lemma:beta\_quantile\_bound\_lower]{}\[Lower bound on the Beta Quantile\]
The proof comes almost exclusively by performing the same steps as in the proof of Proposition \[lemma:beta\_quantile\_bound\].
Experimental Analysis {#sec:experiments}
=====================
We empirically evaluate the performance of in comparison with that of [@full_ucrlv_paper], [@filippi2010optimism] and [@jaksch2010near]. We also compared against [@ouyang2017learning] which is a variant of posterior sampling for reinforcement learning suited for infinite horizon problems. We used the environments *Bandits*, *Riverswim*, **, ** as described in @full_ucrlv_paper. We also eliminate unintentional bias and variance in the exact way described in [@full_ucrlv_paper]. Figure \[fig:regrets\] illustrates the evolution of the average regret along with confidence region (standard deviation). Figure \[fig:regrets\] is a log-log plot where the ticks represent the actual values.
#### Experimental Setup.
The confidence hyper-parameter $\delta$ of , , and is set to $\ConfidenceDelta{}$. is initialized with independent $\BetaDis(\frac{1}{2}, \frac{1}{2})$ priors for each reward $r(s,a)$ and a Dirichlet prior with parameters $(\alpha_1, \ldots \alpha_S)$ for the transition functions $p(.|s,a)$, where $\alpha_i = \frac{1}{S}$. We plot the average regret of each algorithm over $T = 2^{24}$ computed using independent trials.
#### Implementation Notes on
We note here that the quantiles to any subset of next states can be computed efficiently with of a complexity linear in $SA$ and not the naive exponential complexity. This is because The posterior to any subset of next states only depend on the sum of the rewards of its constituent.
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\
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#### Results and Discussion.
We can see that outperforms over all environments except in the Bandits one. This is in line with the theoretical regret whereby we can see that using the Bernstein bound is a factor times worse than the Bayesian quantile. Note that this is not an artifact of the proof. Indeed, pure optimism can be seen as using the *proof* inside the algorithm whereas the Bayesian version provides a general algorithm that has to be proven separately. Consequently, the actual performance of the Bayesian algorithm can often be much better than the bounds provided.
Conclusion {#sec:conclusion}
==========
In conclusion, using Bayesian quantiles lead to an algorithm with strong performance while enjoying the best of both frequentist and Bayesian view. It also provides a conceptually simple and very general algorithm for different scenarios. Although we were only able to prove its performance for bounded rewards in $[0,1]$ and a specific prior, we believe it should be possible to provide proof for other rewards distribution and prior such as Gaussian. As future work, it would be interesting to explore how one can re-use the idea of for non-tabular settings such as with linear function approximation or deep learning.
Proofs
======
Proof of Theorem \[thm:bayes\_ucrl\]
------------------------------------
Our proof is a direct application of the generic proof provided in Section B.2 of @full_ucrlv_paper. To use that generic proof we need to show that with high probability the true rewards/transitions of any state-action is contained in the lower and upper interval of the Bayesian Posterior. This is a direct consequence of Lemma \[lemma:coverage\_beta\] and the fact that our posterior matches the Beta Distribution used in Lemma \[lemma:coverage\_beta\].
Furthermore, we need to provide lower and upper bounds for the maximum deviation of the Bayesian posteriors from their empirical values. This comes directly from using Proposition \[lemma:beta\_quantile\_bound\] and Proposition \[lemma:beta\_quantile\_bound\_lower\], and bounding $\Phi^{-1}$ using equation (15) in @chiani2003new.
\[lemma:coverage\_beta\] Let $\bm{X}_1, \ldots \bm{X}_n$ be $n$ independent Bernoulli random variable with common parameter $\mu$ such that $0 < \mu < 1$ and $n \geq 1$. Let $\bm{X} = \sum_{i=1}^{n} \bm{X}_i$ denote the corresponding Binomial random variable. Let $U_{1-\delta}(\bm{X})$ the (random) $1-\delta$th quantile of the distribution $\BetaDis(\bm{X} + 1, n-\bm{X})$ and $U_{\delta}(\bm{X})$ the $\delta$th quantile of the distribution $\BetaDis(\bm{X}, n-\bm{X}+1)$. If $0 < \bm{X} < n$, we have: $$\Prob\left[U_{\delta}(\bm{X}) \leq \mu \leq U_{1-\delta}(\bm{X}) | \mu\right] \geq 1-2\delta.$$
Since each $\bm{X_i}$ is a Bernoulli random variable with parameter $\mu$, then $\bm{X} = \sum_{i=1}^{n} \bm{X}_i$ is a Binomial random variable with parameter $(n, \mu)$. According to @thulin2014cost equation (4) the quantile of the Beta distribution used in this lemma corresponds exactly to the upper one sided Clopper–Pearson interval (for Binomial distribution) whose coverage probability is at least $1-\delta$ by construction [@thulin2014cost]. The same argument holds for the lower one sided Clopper–Pearson interval. Combining them concludes the proof.
\[lemma\_binomial\_quantile\_bounds\]
Using basic computation, we can verify that the bounds hold trivially for $p=0$, for $p=1$ and $n=0$. Furthermore, it is known that any median $m$ of the binomial satisfies $\floor{np} \leq m \leq \ceil{np}$ [@kaas1980mean]. So, our bounds also holds for $1-\delta = 0.5$. As a result, we can focus the proof on the case where $0 < p < 1$, $n > 0$ and $0.5 \leq 1-\delta < 1$.
From equation (1) in [@zubkov2013complete] we have: $$\begin{aligned}
&\Phi\left(\sign(\frac{k}{n}-p)\sqrt{2n\kldiv{\frac{k}{n}}{p}}\right)\notag\\
& \leq \Prob\{\bm{X}_{n,p} \leq k\}\label{eq:binom_cdf_bound}\\
& \leq \Phi\left(\sign(\frac{k+1}{n}-p)\sqrt{2n\kldiv{\frac{k+1}{n}}{p}}\right)\notag\end{aligned}$$ for $0 \leq k < n$. Let’s also observe that when $k = n$, the lower bound in trivially holds since
$$\Prob\{\bm{X}_{n,p} \leq k\} = 1 \geq \Phi\left(\sign(\frac{k}{n}-p)\sqrt{2n\kldiv{\frac{k}{n}}{p}}\right).$$
#### Proof of the upper bound
Our upper bound provides a correction to the Theorem 5 in @short2013improved.
Consider any $k$ ($0 \leq k \leq n$) such that: $$\begin{aligned}
\Phi\left(\sign(\frac{k}{n}-p)\sqrt{2n\kldiv{\frac{k}{n}}{p}}\right) \geq 1-\delta. \label{eq:upper_binom_upper_goal}\end{aligned}$$ Combining with the left side of we have that $ \Prob\{\bm{X}_{n,p} \leq k\} \geq 1-\delta$ and as a result: $$\begin{aligned}
\quant(\Binomial(n,p),1-\delta) &= \inf\{x:\Prob\{\bm{X}_{n,p} \leq x\} \geq 1-\delta\} \leq k\end{aligned}$$
So we just need to find a value $k$ satisfying . Remarking that $\Phi^{-1}$ is the CDF of the normal distribution (since it is the inverse of the normal quantile) we can conclude that $\Phi^{-1}$ is continuous and increasing. Applying $\Phi^{-1}$ to , we have: $$\begin{aligned}
\sign(\frac{k}{n}-p)\sqrt{2n\kldiv{\frac{k}{n}}{p}} \geq \Phi^{-1}(1-\delta). \label{eq:goal_inverse_applied}\end{aligned}$$
##### The sign of $\frac{k}{n}-p$:
Assume that $\quant(\Binomial(n,p),1-\delta) \leq \floor{np}$. In that case, we can see that our upper bound trivially holds since $C_u(x,y) \geq \sqrt{\frac{(1-2x)^2y^4}{36}} + \frac{(1-2x)y^2}{6} \geq \abs*{\frac{(1-2x)y^2}{6}} + \frac{(1-2x)y^2}{6} \geq 0$. Then we can focus on the case where $\quant(\Binomial(n,p),1-\delta) > \floor{np}$. Since the binomial distribution is discrete with domain the set of integers, $\quant(\Binomial(n,p),1-\delta) > \floor{np}$ implies that $\quant(\Binomial(n,p),1-\delta) \geq \floor{np} + 1$. As a result we have $k \geq \quant(\Binomial(n,p),1-\delta) \geq \floor{np} + 1 > np$ and $\sign(\frac{k}{n}-p) = 1$.
Let $x$ a number such that $\frac{k}{n} = p + x$. Using this in , we thus need to find an $x$ such that:
$$\kldiv{p+x}{p} \geq \frac{\Phi^{-1}(1-\delta)^2}{2n}$$
Consider a function $g$ such that $\kldiv{p+x}{p} \geq g(x)$. If we find an $x$ such that $g(x) \geq \frac{\Phi^{-1}(1-\delta)^2}{2n}$, then it would mean that $\kldiv{p+x}{p} \geq \frac{\Phi^{-1}(1-\delta)^2}{2n}$. We will pick $g$ to be the lower bound on $\kldiv{p+x}{p}$ in Theorem \[theo:kl\_bounds\]. Now let’s observe that since $\sign(\frac{k}{n}-p) = 1$, it means that $x \geq 0$. Also $q-x = 1-p-x = 1 -\frac{k}{n} \geq 0$ so that $x \leq q$.
So the condition of Theorem \[theo:kl\_bounds\] are satisfied and our goal becomes finding an $x \geq 0$ such that:
$$\frac{x^2}{2(pq+x(q-p)/3)} \geq \frac{\Phi^{-1}(1-\delta)^2}{2n}$$ Solving for this inequality leads to the upper bound part of the Theorem.
#### Proof of the Lower bound
If $\quant(\Binomial(n,p),1-\delta) = n$, it is easy to verify that our lower bound trivially holds. So we can focus on the case where $\quant(\Binomial(n,p),1-\delta) < n$.
Consider any $k$ ($0 \leq k < n$) such that: $$\begin{aligned}
\Phi\left(\sign(\frac{k+1}{n}-p)\sqrt{2n\kldiv{\frac{k+1}{n}}{p}}\right) \leq 1-\delta \label{eq:lower_binom_upper_goal}\end{aligned}$$
Combining with the right side of we have that $ \Prob\{\bm{X}_{n,p} \leq k\} \leq 1-\delta$ and since the CDF of a Binomial is an increasing function, we have: $$\begin{aligned}
\quant(\Binomial(n,p),1-\delta) &\geq k \label{eq:binom_lower}\end{aligned}$$
##### The sign of $\frac{k+1}{n}-p$:
Let’s note that the quantile function of the binomial distribution is increasing (since it is the inverse of the cdf and the cdf is increasing). So, we have: $\quant(\Binomial(n,p),1-\delta) \geq \quant(\Binomial(n,p),\frac{1}{2})$. As a result, there exists a number $k$ satisfying both and:
$$\quant(\Binomial(n,p),\frac{1}{2}) \leq k.$$ We will try to find this number. Let’s observe that $\quant(\Binomial(n,p),\frac{1}{2})$ is the (smallest) median of the binomial distribution and thus we have: $\quant(\Binomial(n,p),\frac{1}{2} \geq \floor{np}$ [@kaas1980mean].
So, $$\begin{aligned}
k &\geq \quant(\Binomial(n,p),1-\delta)\\
&\geq \quant(\Binomial(n,p),\frac{1}{2})\\
&\geq \floor{np}\label{eq:binom_median_bound}\end{aligned}$$
As a result, we have $k+1 \geq \floor{np} + 1 > np$ and $\sign(\frac{k+1}{n}-p) = 1$.
Then our objective is to find a $k\geq \floor{np}$ satisfying . Let $x$ a number such that $\frac{k+1}{n} = p + x$.
Applying the inverse $\Phi^{-1}$ to and replacing $\frac{k+1}{n}$ by $x$, our objective becomes finding an $x \geq 0$ such that: $$\kldiv{p+x}{p} \leq \frac{\Phi^{-1}(1-\delta)^2}{2n}$$
Our objective is equivalent to finding an $x$ such that $g(x) \leq \frac{\Phi^{-1}(1-\delta)^2}{2n}$ for a function $g$ such that $D(p+x,p) \leq g(x)$.
We can easily verify that $x\geq 0$ and $x \leq q$ ($q-x = 1-p-x = 1 - \frac{k+1}{1} \geq 0$). And as a result, we pick $g$ as the first upper bound on $\kldiv{p+x}{p}$ in Theorem \[theo:kl\_bounds\].
Our objective is thus to find $x$ ($ 0 \leq x \leq 1-p$) such that: $$\frac{x^2}{2(pq-xp/2)} \leq \frac{\Phi^{-1}(1-\delta)^2}{2n}$$
Solving for this equation and picking a value for $x$ such that $0 \leq x \leq 1-p$, $k \geq \floor{np}$ leads to the first lower bound part of the Theorem.
\[fact\_beta\_binomial\] Let $\bm{Y}_{a,b} \sim \BetaDis(a,b)$ where $a$ and $b$ some integers such that $a > 0, b>0$ a random variable from the Beta distribution. Then, for any $p \in [0,1]$: $$\begin{aligned}
\Prob(\bm{Y}_{a,b} \leq p) &= \Prob(\bm{X}_{a+b-1, 1-p} \leq b-1)\label{fact_beta_binomial_upper}\\
\Prob(\bm{Y}_{a,b} \geq p) &= \Prob(\bm{X}_{a+b-1, p} \leq a-1)\label{fact_beta_binomial_lower}\end{aligned}$$
where $\bm{X}_{n,x}$ is used to denote a random variable distributed according to the binomial distribution of parameters $(n,x)$ ( $\bm{X}_{n,x} \sim \Binomial(n,x)$).
\[lemma:beta\_quantile\_bound\]
For simplicity, in this proof we used $p = Q(\BetaDis(x+1, n-x), 1-\delta)$ and $y = \Phi^{-1}(1-\delta)$. Using Equation , we have:
$\Prob[\bm{Y}_{x+1, n-x} \leq p] = \Prob[\bm{X}_{n,1-p} \leq n-x-1]$. Since the CDF of the beta distribution is continuous, we know that $\Prob[\bm{Y}_{x+1, n-x} \leq p] = 1-\delta$. So we have $\Prob[\bm{X}_{n,1-p} \leq n-x-1] = 1-\delta$
Using the upper bound for Binomial quantile in Lemma \[lemma\_binomial\_quantile\_bounds\], we have:
$n-x-1 \leq n(1-p) + C_u(1-p,\Phi^{-1}(1-\delta)) + 1$ where $C_u$ is the function defined in .
This leads to: $$\begin{aligned}
\label{eq:upper_bound_p}
p &\leq \frac{x}{n} + \frac{C_u(1-p,y)+2}{n}= \frac{x}{n} + \sqrt{\frac{p(1-p)y^2}{n} + \frac{(2p-1)^2y^4}{36n^2} } +\frac{(2p-1)y^2}{6n} + \frac{2}{n}\end{aligned}$$
We would like to find an upper bound for $p(1-p)$ in that depends on $\frac{x}{n}(1-\frac{x}{n})$.
Using Equation with the lower bound for binomial quantile in Lemma \[lemma\_binomial\_quantile\_bounds\], we have
$$\begin{aligned}
\label{lower_bound_p}
p &\geq \frac{x}{n} + \frac{\max\left\{0,\min\left\{np-1, C_l(1-p,\Phi^{-1}(1-\delta))\right\}\right\}}{n} \geq \frac{x}{n}\end{aligned}$$
Multiplying equations and together (both are all positive) leads to:
$$\begin{aligned}
p(1-p) &\leq \paren*{1-\frac{x}{n}}\paren*{\frac{x}{n} + \sqrt{\frac{p(1-p)y^2}{n} + \frac{(2p-1)^2y^4}{36n^2} } +\frac{(2p-1)y^2}{6n} + \frac{2}{n}}\end{aligned}$$
Using the fact that $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$, $2p-1 \leq 1$ and using $1-\frac{x}{n} \leq 1$ for the terms not involving $\frac{x}{n}$, we have:
$$\begin{aligned}
p(1-p) &\leq \paren*{\frac{x}{n}}\paren*{1-\frac{x}{n}} + \sqrt{\frac{p(1-p)y^2}{n}} + \frac{1}{3} \frac{y^2 + 6}{n}\label{eq:p_1_p_poly}\end{aligned}$$
Letting $z= \sqrt{p(1-p)}$ in leads to an inequality involving a polynomial of degree $2$ in $z$. Solving for this inequality and then using $\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$:
$$\begin{aligned}
\sqrt{p(1-p)} &\leq \sqrt{ \paren*{\frac{x}{n}}\paren*{1-\frac{x}{n}}} + \frac{1}{\sqrt{n}} \paren*{\sqrt{\frac{7y^2+24}{12}} + \sqrt{\frac{y^2}{4}}}\label{lower_bound_quantile_variance}\end{aligned}$$
Replacing into and using the fact that $2p-1 \leq 1$, we have the desired upper bound of the lemma
\[lemma:beta\_quantile\_bound\_lower\]
Let’s denote $p = Q(\BetaDis(x, n-x+1), \delta)$. Using , we have that: $\Prob(\bm{Y}_{x,n-x+1} \geq p) = \Prob(\bm{X}_{n, p} \leq x-1)$.
Since the Beta distribution is continuous and also have a continuous cdf, then there exists a unique $p$ such that:
$\Prob(\bm{Y}_{x,n-x+1} \geq p) = 1 - \Prob(\bm{Y}_{x,n-x+1} \leq p) = 1 -\delta$.
As a result, we have $\Prob(\bm{X}_{n, p} \leq x-1) = 1-\delta$.
Using the upper and lower bound for Binomial quantile in Lemma \[lemma\_binomial\_quantile\_bounds\], we have respectively:
$$\begin{aligned}
\label{eq:upper_bound_p_lower}
p &\geq \frac{x}{n} - \frac{C_u(p,y)+2}{n}= \frac{x}{n} - \sqrt{\frac{p(1-p)y^2}{n} + \frac{(1-2p)^2y^4}{36n^2} } -\frac{(1-2p)y^2}{6n} - \frac{2}{n}\end{aligned}$$
$$\begin{aligned}
\label{lower_bound_p_lower}
p &\leq \frac{x}{n} - \frac{ C_l(p,\Phi^{-1}(1-\delta))}{n} \leq \frac{x}{n}\end{aligned}$$
We would like to find a lower bound for $p(1-p)$ in that depends on $\frac{x}{n}(1-\frac{x}{n})$.
implies that: $$\begin{aligned}
1-p \geq 1- \frac{x}{n} \label{eq:lower_1_p}\end{aligned}$$
Note that we can multiply by to get a lower bound for $p(1-p)$ even if the left hand side of is negative since both $p(1-p)$ and $1-\frac{x}{n}$ are always positive.
After this multiplication, we follow the exact same steps as in the equivalent part of the proof for Lemma \[lemma:beta\_quantile\_bound\]. We can do that since even if we are looking for a lower bound, all the term previously upper bounded in Lemma \[lemma:beta\_quantile\_bound\] are multiplied by $-$.
This completes the proof for this lemma.
Useful Results
--------------
\[theo:kl\_bounds\]
The proof of the lower bound already appear in @janson2016large (after equation (2.1)).
First, let’s observe that:
$$\kldiv{p+x}{p} = p(1 + \frac{x}{p}) \ln(1 + \frac{x}{p}) + h(x)$$
with $$h(x)=
\begin{cases}
q(1-\frac{x}{q}) \ln(1-\frac{x}{q}) & \text{ if }x < q\\
0 & \text{ if } x = q\\
\end{cases}$$
Note that this is a valid definition for the KL-divergence since for any $q \in ]0,1[$, $$\lim_{x \to q^-} (1-\frac{x}{q}) \ln(1-\frac{x}{q}) = 0$$
Let $g(x)$ a parametric function defined by:
$$g(x) = p(1 + \frac{x}{p}) \ln(1 + \frac{x}{p}) + h(x) - \frac{x^2}{2(pq+x(q-p-a)/b)}$$
Where $a$ and $b$ are constants (independent of $x$ but possibly depending on $p$) such that $pq+x(q-p-a)/b > 0$ for all $q \in ]0,1[, x \in [0,q]$.
We can immediately see that $g$ is continuous and differentiable in its domain $[0, q]$ since it is the sum of continuous and differentiable functions. For any $x \in [0,q[$, the derivative $g'(x)$ of $g(x)$ is:
$$g'(x) = \ln(1+ \frac{x}{p}) -\ln(1-\frac{x}{q}) - \frac{4pqx + 2x^2(q-p-a)/b}{\paren*{2(pq+x(q-p-a)/b)}^2}$$
And $g'(0) = 0$.
We can see that $g'$ is a continuous and differentiable in $[0, q[$.
The second derivative for any $x \in [0,q[$ is
$$\begin{aligned}
g''(x) &= \frac{1}{x+p} + \frac{1}{x+q} - \frac{p^2q^2}{(pq+x(q-p-a)/b)^3}\\
&=\frac{\frac{x^3(q-p-a)^3}{b^3} + \frac{3pqx^2(q-p-a)^2}{b^2} +p^2q^2x^2 - p^2q^2xa + (\frac{3}{b}-1)p^2q^2x(q-p-a) }{(x+p) (x+q) (pq+x(q-p-a)/b)^3}\label{eq:second_derivative}\end{aligned}$$
#### Proof of the Lower bound
Let’s set $a = 0$ and $b = 3$. In that case we have for any $x \in [0,q[$:
$$\begin{aligned}
g''(x) &= \frac{x^3(q-p)^3/27 + pqx^2(q-p)^2/3 +p^2q^2x^2}{(x+p) (x+q) (pq+x(q-p)/b)^3}\\
&\geq \frac{-px^3(q-p)^2/27 + pqx^2(q-p)^2/3 +p^2q^2x^2}{(x+p) (x+q) (pq+x(q-p)/b)^3}\\
&\geq \frac{-pqx^2(q-p)^2/27 + pqx^2(q-p)^2/3 +p^2q^2x^2}{(x+p) (x+q) (pq+x(q-p)/b)^3}\geq 0\end{aligned}$$
Furthermore, elementary calculations leads to $g(0) = g'(0) = 0$.
Since $g'$ is continuous in $[0,q[$ and $g''$ is positive in $[0,q[$, we can conclude that $g'$ is increasing in $[0,q[$. Since $g'(0) = 0$, it means $g'(x) \geq 0$ for any $ x \in [0,q[$. Since $g$ is continuous, $g'(x) \geq 0$ for any $ x \in [0,q[$ means that $g$ is increasing in $[0,q[$. Using the fact that $g(0) = 0$ we have that $g(x) \geq 0$ for any $ x \in [0,q[$. We will now show that $g(x)$ is non-negative at $q$ too. Using the fact that $g$ is continuous at $q$, we have that $\lim_{x \to q^-} g(x) = g(q)$. Also $\lim_{x \to q^-} g(x) \geq 0$ since $g(x) \geq 0$ for any $x \in [0,q[$. And as a result, $g(q) = \lim_{x \to q^-} g(x) \geq 0$ which concludes the proof of the lower bound.
#### Proof of the first upper bound
Let $a = q$ and $b = 2$. We want to analyze the sign of the resulting $g''$ over its domain $[0,q[$. For that observe that the denominator of $g''$ is always strictly positive. This means that the sign of $g''$ is the same as the sign of its numerator. Let’s denote $g''_0$ the numerator. We have:
$$\begin{aligned}
g''_0(x) &= \frac{x^3(-p)^3}{8} + \frac{3qx^2p^3}{4} +p^2q^2x^2 - p^2q^3x - \frac{p^3q^2x}{2}\\
&= x \cdot \paren*{\frac{x^2(-p)^3}{8} + \frac{3qxp^3}{4} +p^2q^2x - p^2q^3 - \frac{p^3q^2}{2}}\end{aligned}$$
Let’s denote $f(x) = \frac{x^2(-p)^3}{8} + \frac{3qxp^3}{4} +p^2q^2x - p^2q^3 - \frac{p^3q^2}{2}$.
We will use Sturm’s theorem (Theorem \[theo:sturm\]) to find the number of roots of $f$ in $]0,q]$. The Sturm sequence of $f$ is $\set{f_0, f_1, f_2}$ with: $$\begin{aligned}
f_0(x) &= f(x)\\
f_1(x) &= \frac{x(-p)^3}{4} + \frac{3qp^3}{4} +p^2q^2\\
f_2(x) &= \frac{p(-5p^4 + 26p^3 - 53p^2 + 48p - 16)}{8}\end{aligned}$$
We have: $$f_0(0) = -p^2q^3-\frac{p^3q^2}{2} < 0$$ $$f_1(0) = \frac{3qp^3}{4} +p^2q^2 > 0$$
$$f_0(q) = \frac{p^3q^2}{8} > 0$$
$$f_1(q) = \frac{qp^3+2p^2q^2}{2} > 0$$
And we have $f_2(q) = f_2(0)$
The number of sign alternations in $\set{f_0(0), f_1(0), f_2(0)}$ is: $1+\1_{f_2(0) < 0}$ where $\1_{f_2(0) < 0} = 1$ if $f_2(0) < 0$ and $\1_{f_2(0) < 0} = 0$ otherwise. The number of sign alternations in $\set{f_0(q), f_1(q), f_2(q)}$ is:$\1_{f_2(0) < 0}$. Observing that neither $0$, nor $q$ are roots of $f$, we can conclude by the Sturm’s theorem (Theorem \[theo:sturm\]) that the number of roots of $f$ in $[0, q]$ is exactly 1.
Since $f$ is a polynomial it means that the sign of $f$ changes at most once in the interval $[0,q]$. Let’s $\alpha$ ($0 < \alpha < q$) the unique root of $f$ in $[0,q]$. Then (ignoring zero-values) the function $f$ have the same sign for all values in $[0,\alpha]$ and $f$ have the same sign for all values in $[\alpha, q]$.
Observing that $f(0) < 0$, it means that $f(x) \leq 0$ for any $x \in [0,\alpha]$. Observing that $f(q) > 0$, it means that $f(x) \geq 0$ for any $x \in [\alpha, q]$. Since the second derivative $g''$ is a multiple of a non-negative terms by $f$; it means that $g''(x) \leq 0$ for any $x \in [0,\alpha]$ and $g''(x) \geq 0$ for any $x \in [\alpha, q[$.
We will now derive the sign of $g'$ and $g$ over their domain. Since $g'(0) = 0$, $g''(x) \leq 0$ in $x \in [0, \alpha]$ and $g'$ is continuous in $[0, \alpha]$, we can conclude that $g'$ is decreasing in $[0, \alpha]$ and $g'(x) \leq 0$ for any $x \in [0, \alpha]$. A similar argument for $g$ allows us to conclude that $g(x) \leq 0$ for any $x \in [0, \alpha]$.
Since $g''(x) \geq 0$ for any $x \in [\alpha, q[$, it means that $g'$ is increasing in the interval $[\alpha, q[$. Let $\alpha_0$, the lowest value in $[\alpha, q]$ such that $g'(\alpha_0) = 0$. This means that for any $x \in [\alpha, \alpha_0]$ $g'(x) \leq 0$ and for any $x \in [\alpha_0, q]$, $g'(x) \geq 0$. Since $g'$ is non-positive in $[\alpha, \alpha_0]$ and $g$ is continuous, we have that $g$ is decreasing in $[\alpha, \alpha_0]$ so that $g(x) \leq g(\alpha) \leq 0$.
If $\alpha_0 = q$, our proof is essentially done since it implies $g(x) \leq 0$ for all $x \in [0,q]$.
Assume that $\alpha_0 < q$. We want to identify the sign of $g$ in $[\alpha_0, q]$. In this case, we know that $g'(x) \geq 0$ so that $g$ is increasing in $[\alpha_0, q]$.
Now let’s observe that:
$$\begin{aligned}
g(q) &= p(1 + \frac{q}{p}) \ln(1 + \frac{q}{p}) + h(q) - \frac{q^2}{pq}\\
&= \ln(1 + \frac{q}{p}) - \frac{q}{p}\\
&\leq \frac{q}{p} - \frac{q}{p} = 0 \label{eq:g_q_zero}\end{aligned}$$
We will now show by contradiction that $g(x) \leq 0$ for all $x \in [\alpha_0, q]$. Assume that there exists a number $c \in [\alpha_0, q]$ for which $g(c) > 0$. Since $g$ is increasing (and continuous) in $[\alpha_0, q]$, it means that $g(q) > 0$ which contradicts . As a result, there is no value $c \in [\alpha_0, q]$ such that $g(x) > 0$. And this concludes the proof for the first upper bound.
#### Proof of the second upper bound
The second upper bound comes directly from the first upper bound and the fact that: $$2(pq-xp/2) = pq + p (q-x) \geq pq$$
Previously known results
========================
Sturm’s Theorem
---------------
For any univariate polynomial $f(x)$ of degree $d$ with real coefficients, the sturm sequence for $f$ is a sequence of polynomials $\bm{\bar{f}} = \set{f_0, f_1 \ldots f_d}$ such that: $$\begin{aligned}
f_0 &= f\\
f_1 &= f'\\
f_{i+1} &= - f_{i-1} \rem f_i \;\forall i \in \set{0,\ldots, d-2}\end{aligned}$$ where $f_{i-1} \rem f_i$ denotes the remainder of the euclidian division of $f_{i-1}$ by $f_i$
Let $\bm{\bar{\alpha}} = \set{\alpha_0, \alpha_1, \ldots \alpha_n}$ be a sequence of real numbers. We say that there is a sign alternation at position $i \in \set{1, \ldots n}$, if there exists some $j \in \set{0, \ldots i-1}$ such that the following two conditions are satisfied
1. $\alpha_i \alpha_j < 0$
2. $j = i-1$ or $\alpha_{j+1} = \alpha_{j+2} = \ldots = \alpha_{i-1} = 0$.
The number of sign alternations of $\bm{\bar{\alpha}}$ is the number of positions for which there is a sign alternation.
Let $r \geq 0$ a non-negative integer. Let $f^{(0)} = f$ and $f^{(n)}$ the $n$th derivative of a function $f$ differentiable up to $n$ times. A real number $\alpha$ is called a root of multiplicity $r$ for $f$ if: $$f^{(0)}(\alpha) = \ldots = f^{(r-1)}(\alpha) = 0; \;\; f^{(r)}(\alpha) \ne 0$$
We say a real number *is not a multiple root* if its multiplicity is less or equal to $1$ (i.e. it is either a non-root or it is a root of multiplicity 1).
\[theo:sturm\] For any non-zero univariate polynomial $f(x)$ with real coefficients and any numbers $a\leq b$; let $N_f]a,b]$ the number of distincts real roots of $f$ in $]a,b]$. If neither $a$ nor $b$ are multiple roots, We have:
$$N_f]a,b] = S_f(a) - S_f(b)$$
Where $S_f(x)$ denotes the number of sign alternatives obtained for the sequences $\set{f_0(x), f_1(x), \ldots f_d(x)}$ where $\set{f_0, f_1, \ldots, f_d}$ is the sturm sequence of the polynomial $f$.
[^1]: Informally, it is known that the MDP resets to a starting state after a fixed number of steps.
[^2]: $\TilO$ is used to hide log factors.
[^3]: Technically, beta distributions are only defined for parameter strictly greater than 0. In this paper, when the parameter $\epsilon$ is 0, we compute the posterior and the quantiles by considering the limit when $\epsilon$ tends to 0. \[fn:beta\_valid\]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Nearly 70 Ground Level Enhancements (GLEs) of cosmic rays have been recorded by the worldwide neutron monitor network since the 1950s depicting a big variety of energy spectra of solar energetic particles (SEP). Here we studied a statistical relation between the event-integrated intensity of GLEs (calculated as count-rate relative excess, averaged over all available polar neutron monitors, and expressed in percent-hours) and the hardness of the solar particle energy spectra. For each event the integral omnidirectional event-integrated fluences of particles with energy above 30 MeV ($F_{30}$) and above 200 MeV ($F_{200}$) were computed using the reconstructed spectra, and the ratio between the two fluences was considered as a simple index of the event’s hardness. We also provided a justification of the spectrum estimate in the form of the Band-function, using direct PAMELA data for GLE 71 (17-May-2012). We found that, while there is no clear relation between the intensity and the hardness for weak events, all strong events with the intensity greater 100 %hr are characterized by a very hard spectrum. This implies that a hard spectrum can be securely assumed for all extreme GLE events, e.g., those studied using cosmogenic isotope data in the past.'
address:
- 'ReSoLVE Centre of Excellence, University of Oulu, FIN-90014 Oulu, Finland'
- 'University of Helsinki, FIN-00014 Helsinki, Finland'
- 'Institute of Mathematics and Physics, Siedlce University, Stanislawa Konarskiego 2, 08-110 Siedlce, Poland'
- 'Sodankylä Geophysical Observatory, University of Oulu, FIN-90014 Oulu, Finland'
- 'Ioffe Physical-Technical Institute, Politechnicheskaya st. 26, 194021 St. Petersburg, Russia'
- 'National research nuclear university “MEPhI”, Kashirskoye sh. 31, 115409 Moscow, Russia'
author:
- 'E. Asvestari'
- 'T. Willamo'
- 'A. Gil'
- 'I.G. Usoskin'
- 'G.A. Kovaltsov'
- 'V.V. Mikhailov'
- 'A. Mayorov'
title: 'Analysis of Ground Level Enhancements (GLE): Extreme solar energetic particle events have hard spectra'
---
Solar energetic particles; cosmic rays
=0.5 cm
Introduction
============
Strong energy releases may sporadically occur on the Sun, leading to transient phenomena in the interplanetary space. In particular, solar energetic particle (SEP) events may take place with short but very intensive (by orders of magnitude) increases of the flux of energetic particles. Such events are caused by high fluxes of solar energetic particles (SEPs) accelerated in the solar corona and interplanetary space by shocks driven by coronal mass ejections, and by solar flares to energies sufficiently high to be able to penetrate into the Earth’s atmosphere, where they initiate atmospheric cascades whose nucleonic component can be registered by ground-level detectors [e.g., @shea_SSR_00; @andriopoulou11], that is called a Ground Level Enhancement (GLE). Such events were firstly identified using ground-based ionization chambers [@forbush_three_1946] and since the 1950s they are monitored by the neutron monitor (NM) network. The first and the strongest GLE detected by the NM network took place on 23-Feb-1956 and recorded as GLE number 5. GLE events are numbered consequently since then. The most recent officially accepted GLE was on 17-May-2012, numbered as GLE 71. Several weak events [@thakur14; @belov15] have been detected by a few polar NMs even after that date, called sub-GLE events. However, since they were very weak they are not of interest for this study. All the GLEs, starting from number 5 are archived at the International GLE database http://gle.oulu.fi [@usoskin_GLE_15].
A NM is an energy-integrated device which cannot measure the differential energy spectrum of primary cosmic-ray particles. However, for many applications it is important to know the spectrum. The use of the world-wide NM network makes it possible to assess the integral spectrum, but still an assumption on the spectra shape is needed. First descriptions were based on the assumption that a GLE spectrum can be described by an exponential over rigidity [@freier_radiation_1963] or a power law with an exponential roll-off [@elisson85]. However, these simple approximations often do not work well, especially for high energies above several GeV [@shea_space_2012]. As an alternative, the Band-function [@band_batse_1993] was proposed as a suitable model to parameterize the event-integrated fluence [@tylka09]. The Band-function describes the integral rigidity spectrum by a double power law in rigidity with a smooth exponential junction inbetween [@usoskin_ACP_11]. This approximation describes the integral spectrum by a double power law in rigidity with a smooth roll over in-between. A tremendous work has been performed by @tylka_survey_2008 [@tylka09] to make a Band-function fit to almost all (59) GLE events, using both NM data for high-energy tail and in-situ space-borne measurements for the lower part of the spectrum. We based our present analysis on the result of this extensive work, updated recently (Allan Tylka, personal communication 2015).
The most distinctive feature of GLE events is the hardness of their energy/rigidity spectra [@shea_SSR_00]. As a measure of the intensity of a SEP event, the event-integrated fluence of SEPs with energy above 30 MeV is often used. Although it is intuitively expected that all GLE events should have a hard spectrum, that is not true. For example, GLE 24 (Aug 1972) was moderately strong but it provided the largest fluence (omnidirectional flux integrated over the total duration of the event) of SEPs with energy above 30 MeV (called henceforth $F_{30}$), greater than that of the strongest GLE 5 in February 1956 [@belov05]. The former event had a very soft spectrum, while the latter – a very hard one. All other GLEs have a wide variety of spectra between these two cases. We note that, while ground based NMs are sensitive to relatively high energy part of the SEP spectrum, for many application in atmospheric sciences, climate, dosimetry, etc., it is important to know the fluence of lower energy particles, with energy above 30 MeV, $F_{30}$. In particular, strong SEP events can lead to essential changes in the polar atmosphere and even affect regional climate [see, e.g., @mironova15].
Over the last decades, the lower-energy part of the SEP spectrum, which cannot be assessed by NMs, was evaluated from satellite-borne data [@vainio09; @bazilevskaya14]. On the other hand, when one goes back in time, strong SEP/GLE events can be estimated from indirect proxies – cosmogenic radionuclides produced by cosmic rays in the atmosphere and stored in natural archives, such as tree trunks or ice cores [@beer12]. Looking for spikes in cosmogenic data, one may find extreme SEP events in the past [e.g., @miyake12; @usoskin_ApJ_12]. It has been recently estimated that production of cosmogenic nuclides can be used as a measure of SEP fluence with energy above 200 MeV, called $F_{200}$ [@usoskin_F200_14]. However, while cosmogenic proxy are somewhat more sensitive to SEP comparing to the ground-based NMs, they are still incapable to evaluate the low-energy range of the SEP spectrum, which is most important for atmospheric processes, viz. $F_{30}$. Accordingly, it would be useful to know wether GLE data can provide at least a first order estimate for the low-energy fluence of SEP when the higher-energy fluence $F_{200}$ is known.
Different aspects of the GLE event statistic have been studied earlier [see, e.g. @cliver06; @belov10]. Here we perform a statistical analysis searching for a relation between the event-integrated intensity of GLE events and the hardness of their spectra, using the full database of GLEs for the last 60 years.
Analysis {#Sec1}
========
GLE strength
------------
It is common to characterize the strength of a GLE as the peak intensity in percentage of the increase above the GCR background [e.g., @andriopoulou11]. For example, the event of 23-Feb-1956 (GLE 5) was characterized by the highest 5116 % increase in pseudo[^1] 5-min data of the Leeds NM. However, for some purposes it is more useful to study the integral intensity of the events, so that the same GLE 5 had the largest increase of $\approx$5300 %hr in Ottawa NM, while it was 4450 $\%$\*hr in Leeds station (see Fig. \[Fig:GLE5\]). The event-integrated intensity $I$ is defined as the integral of the excess above the GCR background over the entire duration of the event (see the shaded area in Fig. \[Fig:GLE5\]) and is given here in units of %hr. It corresponds to the total fluence of SEPs with energy sufficient to cause an atmospheric cascade (several hundred MeV). We note that, while the peak intensity is important for the problem related to particle acceleration and transport in the interplanetary medium, the total fluence is more relevant for the terrestrial effect [@usoskin_ACP_11]. Moreover, the integral intensity is much more robustly defined than the peak intensity for the following reasons [e.g., @dorman04]: the peak intensity may be distorted for greatest events by the dead-time of the NM (for example, a NM with long dead-time of 1.2 ms can lost $\approx 40$% of counts for a 50-fold increase of the counting rate), but this is small for the event-integrated intensity; the peak intensity depends on the time resolution, while the intensity is non-dependent on it; the peak intensity, especially at the impulsive phase of the event may depend on exact location of NM.
The values of $I$ were computed for GLEs 5 throughout 71 using data collected at the International GLE database http://gle.oulu.fi. The relative increase of GLE vs. the GCR background depends on the local geomagnetic rigidity cutoff and the altitude of the NM location. Here we consider as the strength of the GLE, the value of $I$ averaged over 30 polar sea-level NMs, in order to consider a homogeneous response of the NM network to each event. For each event we considered all the NMs with the geomagnetic cutoff rigidity $<1.5$ GV and the altitude less than 200 m above the sea level. The $<1.5$ GV limit is smooth because of the long-term variations of the cutoff rigifity [@kudela04b]. The list of GLEs with their strengths and the number of the used NMs is given in Table \[Tab1\]. Typical values of $I$ vary a few %hr to 5300 %hr. Events with the values of $I$ less than 3 %hr are not listed. We arbitrarily divide events into weak ($I<100$ %hr) and strong ($I> 100$ %hr).
Energy spectrum {#S:spec}
---------------
Here we used energy/rigidity spectra of each GLE event as reconstructed by @tylka09 from both satellite-borne and ground-based NM data and updated using the newest data (Allan Tylka, personal communication, 2015). The omnidirectional event-integrated fluence for each event is parameterized using the Band function, see formalism in [@tylka09; @usoskin_F200_14]. For each event we computed, using the spectrum reconstructions, the ratio of two fluences, $F_{30}=J(>30$ MeV) and $F_{200}=J(>200$ MeV), which is listed in Table \[Tab1\]. The error bars of the ratio were estimated from the propagated uncertainties of the Band-function fits [@tylka09]. We note that the highest ratio of $\approx 480$ appears for GLE 24, where the the fit appeared unphysical below 10 MeV, giving a spectrum rising in energy and therefore the parameters of the Band-function fitting were estimated omitting the 10 MeV point (Allan Tylka, personal communication, 2015). We also note that the source region for GLE 24 was near the central meridian. Accordingly, the event might have included a lower-energy energetic storm particle (ESP) event, caused by the CME-driven shock’s arrival at Earth. In this case, the streaming-limit does not apply. For other near-central-meridian events with clear ESP phases (19-Oct-1989, 24-Aug-1998, 14-Jul-2000, 4-Nov-2001, 28-Oct-2003) @tylka09 provided separate fits for the “GLE” and “ESP” phases. We used the sum of the two components to estimate the total fluence of SEP particles for such events.
GLE 71 with PAMELA data {#Sec2}
-----------------------
Here we perform a test of correctness of the Band-function fit to the SEP spectrum using direct space-borne measurements. For some GLEs during the last decade, direct measurements of SEP energy spectra could have been made with space-borne magnetic spectrometers such as PAMELA (Payload for Antimatter Matter Exploration and Light-nuclei Astrophysics) mission [@adriani11], installed onboard the low orbiting satellite Resurs-DK1 with a quasi-polar (inclination $70^\circ$) elliptical orbit (height 350–600 km). PAMELA is in operation since Summer 2006 and continuously measures all energetic ($>80$ MeV) particles in space. Thus, it could potentially measure SEP spectra for two GLE events analyzed here, viz. GLE 70 (13-Dec-2006) and 71 (17-May-2012). No other events can be studied using direct spectral measurements. If more events are measured they will be analyzed in a due course. However, as any low orbiting satellite, PAMELA can measure SEP only for a fraction of time, when it is in [the]{} polar part of its orbit, since SEPs are shielded by the geomagnetic field in the lower latitude part of the orbit.
For the GLE 70 PAMELA detected only the initial phase of the event, until about 10 UT of 13-Dec-2006, followed by a nearly 24-hr long data gap, making it impossible to study the event-integrated fluence [@adriani_apj_11].
[Measurements of GLE 71 were significantly better]{}: spectra were measured about $^1\hspace{-0.1cm}/\hspace{-0.05cm}_6$ of the time, in the wide energy range with roughly half an hour cadence (polar passes of the orbit) [@bazi_icrc_13]. Thus, the spectrum can be interpolated between the measured points to evaluate the total spectral fluence of the SEPs during the event. The differential energy fluence of the GLE 71, as measured by PAMELA and reconstructed from NMs, is shown in Fig. \[Fig:2012\]. One can see that the two spectra agree quite well in the energy range below 1 GeV, but the errors increase for the PAMELA data beyond 1 GeV mostly because of the uncertain contribution of GCR. We note that PAMELA data were not used in the construction of the Band-function fit for this event. Unfortunately, the PAMELA spectrum does not go down to 30 MeV, and we cannot obtain the $F_{30}$ fluence directly from this data, but considering the lower bound of the energy range for PAMELA data (about 80 MeV), we can assess the ratio of $F_{80}/F_{200}$. It appears $6.15\pm 0.9$ for direct PAMELA data and $6.02\pm 1.38$ for the parameterized spectrum.
Accordingly, we conclude that the Band-function parametrization used here is in good agreement with direct measurements of SEPs for the only event, GLE 71, where such a comparison is possible.
Discussion
==========
We show in Figure \[Fig:f\_per\] the $F_{30}/F_{200}$ ratio as a function of the GLE strength for the analyzed events. One can see that for weak GLE events ($I<100$ %hr) the ratio takes a wide range of values, from 10 to 200 (even more for the GLE 24). This implies that these GLEs can be with different hardness of the spectrum – from very hard to very soft. Interestingly, very weak GLEs ($I<10$ %hr) are harder than moderate events ($10<I<100$ %hr). On the other hand, all strong events ($I>100$ %hr) are characterized by a hard spectrum – the ratio is limited [to the range]{} 20–50. The greatest GLE 5 [has a ratio of about ten, implying a very hard spectrum]{}.
Figure \[fig:both\] depicts the two fluences as function of the GLE strength. The $F_{30}$ fluence (panel A), while increasing with $I$, shows only insignificant correlation (correlation coefficient $0.16\pm0.15$). Interestingly, except for GLE 24 (see section \[S:spec\]), the $F_{30}$ fluence does not exceed the value of 3$\cdot 10^9$ particles/cm$^2$. This may be related to a saturation, e.g., the streaming limit [@reames10; @reames13] caused by a possible resonance interaction between the particle flux and the plasma waves at the interplanetary shock so that there is a limit for the flux of SEPs accelerated at one instance. We note that our empirical limit is close to a realistic maximum $F_{30}$ fluence related to the streaming limit, estimated by @mccracken01 as (6–8)$\cdot 10^9$ cm$^{-2}$.
On the other hand, the $F_{200}$ fluence (panel B) varies almost linearly with the value of $I$, [having a Pearson’s linear correlation coefficient to be $0.87^{+0.03}_{-0.04}$.]{} We note that the best correlation ($0.988^{+0.003}_{-0.004}$) is obtained between the GLE strength $I$ and fluence $F_{800}$ [(panel C)]{}, viz. above 800 MeV. The $F_{200}$ fluence shows no sign of saturation, probably, because the 200 MeV protons do not reach the streaming limit because of the falling spectrum. Accordingly, this may lead to ‘hardening’ of the SEP spectrum for strong events so that $F_{200}$ [continues to increase]{} with the event strength, while $F_{30}$ is saturated at the level of (6–8)$\cdot 10^9$ particles per cm$^{2}$. The expected in this case ratio $6\cdot 10^9 /F_{200}$ is shown in Figure \[fig:R\_F200\] as the red dotted line. One can see that it agrees with the observed ‘hardening’ of the spectrum. Although such a saturation was not directly observed for the $F_{200}$ fluence, it may likely reach the streaming limit for extreme events, such as the one of 775 AD, which is recognized as the strongest SEP event over the last millennia [@usoskin_ApJ_12; @mekhaldi15].
Thus, for [the]{} strongest events, the higher-energy tail of the spectrum may increase while the lower-energy part of the spectrum is saturated, leading to the observed hardening of the spectrum.
Summary
=======
We studied here the energy spectra of 59 out of 71 GLEs for which there are reconstructions of the event-integrated spectra in the form of Band-functions in a wide energy range. [Using the direct space-borne PAMELA data for GLE 71 (17-May-2012) we confirmed in Section \[S:spec\] the use of the Band-function reconstructions.]{} As the strength of each GLE event we considered the event-integrated response of polar sea-level NMs to SEP, above the GCR background. As a measure of the hardness of the energy spectrum we considered the ratio of $F_{30}/F_{200}$.
We have shown that all the strong GLE events (the strength $I>100$ %hr) are characterized by a hard spectrum with the $F_{30}/F_{200}$ ratio being 10–50. Moreover, there seems to be a saturation, most likely due to the streaming limit, of the $F_{30}$ fluence at the level of (6–8)$\cdot 10^9$ particle/cm$^2$ so that it does not exceed this level with increasing the strength of the event. This implies that strongest GLE-like events have very hard spectrum [in the energy range below several hundred MeV]{}, and that we do not expect to see an extreme GLE event with a soft spectrum in the energy range of tens-hundreds MeV.
This has a practical application for historical SEP events studied using cosmogenic isotope records [e.g., @usoskin_ApJ_12]. As discussed by @usoskin_F200_14, cosmogenic isotope data allows reconstructions of the $F_{200}$ while a lower-energy range (several tens of MeV) is important for terrestrial effects (ionization of the polar atmosphere or radiation hazards). Accordingly, it is a reasonable assumption that all strong SEP events identified in cosmogenic isotope data, such as the famous event of 775 AD, which was the strongest know over ten millennia [@miyake12; @usoskin_775_13] have very hard spectra [cf., @mekhaldi15].
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Center of Excellence ReSoLVE (project No. 272157).
GLE Date I (%hr) $F_{30}/F_{200}$ N
----- ------------- -------------- ------------------ ----
5 23-Feb-1956 5202$\pm$104 11.0$\pm$2.1 2
8 04-May-1960 58$\pm$14 8.6$\pm$1.9 9
10 12-Nov-1960 677$\pm$25 44.7$\pm$8.4 14
11 15-Nov-1960 552$\pm$106 51.0$\pm$10.0 15
13 18-Jul-1961 51$\pm$5 42.9$\pm$12.5 13
16 28-Jan-1967 110$\pm$3 18.1$\pm$5.5 19
19 18-Nov-1968 6$\pm$1 79.3$\pm$22.3 15
21 30-Mar-1969 81$\pm$7 18.6$\pm$6.2 18
22 24-Jan-1971 25$\pm$2 91.9$\pm$23.1 20
23 01-Sep-1971 88$\pm$6 17.8$\pm$4.7 18
24 04-Aug-1972 35$\pm$2 488.1$\pm$118.2 16
25 07-Aug-1972 20$\pm$2 87.8$\pm$24.6 17
26 29-Apr-1973 6$\pm$1 30.9$\pm$10.2 12
27 30-Apr-1976 6$\pm$1 36.0$\pm$7.3 13
28 19-Sep-1977 4$\pm$1 63.0$\pm$13.2 9
29 24-Sep-1977 52$\pm$8 17.1$\pm$4.2 11
30 22-Nov-1977 37$\pm$3 29.3$\pm$6.7 12
31 07-May-1978 28$\pm$7 31.4$\pm$5.9 12
32 23-Sep-1978 24$\pm$1 166.3$\pm$51.6 13
36 12-Oct-1981 63$\pm$4 130.9$\pm$24.2 11
37 26-Nov-1982 16$\pm$2 40.5$\pm$8.4 11
38 08-Dec-1982 44$\pm$3 48.6$\pm$12.1 12
39 16-Feb-1984 16$\pm$4 23.4$\pm$4.4 13
41 16-Aug-1989 57$\pm$3 54.6$\pm$12.8 12
42 29-Sep-1989 1189$\pm$60 41.5$\pm$3.2 15
43 19-Oct-1989 411$\pm$15 42.1$\pm$4.2 13
44 22-Oct-1989 72$\pm$5 61.8$\pm$16.3 13
45 24-Oct-1989 576$\pm$27 22.9$\pm$6.0 13
46 15-Nov-1989 3$\pm$0 15.5$\pm$3.2 5
47 21-May-1990 33$\pm$2 28.3$\pm$8.0 14
48 24-May-1990 56$\pm$4 18.9$\pm$5.1 14
49 26-May-1990 27$\pm$2 12.1$\pm$2.7 13
52 15-Jun-1991 49$\pm$4 44.1$\pm$11.4 11
53 25-Jun-1992 5$\pm$1 62.8$\pm$12.9 10
55 06-Nov-1997 48$\pm$2 40.0$\pm$11.3 9
56 02-May-1998 4$\pm$1 26.7$\pm$5.7 4
59 14-Jul-2000 92$\pm$5 79.5$\pm$9.2 11
60 15-Apr-2001 170$\pm$15 17.6$\pm$4.2 13
61 18-Apr-2001 26$\pm$4 32.0$\pm$7.4 13
62 04-Nov-2001 14$\pm$1 186.6$\pm$7.9 11
63 26-Dec-2001 7$\pm$2 51.7$\pm$11.0 11
64 24-Aug-2002 8$\pm$1 56.6$\pm$13.0 11
65 28-Oct-2003 110$\pm$7 125.6$\pm$11.7 14
66 29-Oct-2003 36$\pm$4 49.3$\pm$3.3 14
67 02-Nov-2003 13$\pm$3 100.6$\pm$20.9 11
69 20-Jan-2005 385$\pm$55 14.1$\pm$2.7 14
70 13-Dec-2006 62$\pm$4 26.1$\pm$6.8 15
71 17-May-2012 10$\pm$1 22.8$\pm$5.5 11
: Parameters of the studied GLE events. The columns provide: GLE number and date, the GLE strength (event-integrated intensity), the ratio of $F_{30}/F_{200}$, and the number of polar sea-level NMs used to calculate the strength $I$ (see text for details).[]{data-label="Tab1"}
[33]{} natexlab\#1[\#1]{}url \#1[`#1`]{}urlprefix
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[^1]: The 5-min resolution data for the Leeds NM were interpolated from a graph of the original data with 15-min resolution (E. Eroshenko, personal communication 2016).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.'
address: 'Institut für Theoretische Physik, Universität Göttingen, Tammannstr. 1, D-37077 Göttingen, Germany'
author:
- 'R. Hedden, V. Meden, Th. Pruschke, and K. Schönhammer'
title: 'Functional renormalization group approach to zero-dimensional interacting systems'
---
Introduction {#sec:intro}
============
The reliable calculation of physical properties of interacting quantum mechanical systems presents a formidable task. Typically, one has to cope with the interplay of different energy-scales possibly covering several orders of magnitude even for simple situations. Approximate tools like perturbation theory, but even numerically exact techniques can usually handle only a restricted window of energy scales and are furthermore limited in their applicability by the approximations involved or the computational resources available. In addition due to the divergence of certain classes of Feynman diagrams some of the interesting many-particle problems cannot be tackled by straightforward perturbation theory.
A general scheme that is designed to handle such multitude of energy scales and competition of divergences is the renormalization group [@RG_general]. The idea of this approach is to start from high energy scales, leaving out possible infrared divergences and work one’s way down to the desired low-energy region in a systematic way. The precise definition of “systematic way” does in general depend on the problem studied.
In particular for interacting quantum mechanical many-particle systems, two different schemes attempting a unique, problem independent prescription have emerged during the past decade. One is Wegner’s flow equation technique [@wegner], where a given Hamiltonian is diagonalized by continuous unitary transformation. From the final result one can extract detailed information about the structure of the ground state and low-lying excitations. This technique has been applied successfully to both fermionic and bosonic problems [@flow_apps]. However, in general it becomes a rather cumbersome task to really calculate physical quantities, especially dynamics from correlation functions. Here, one typically has to introduce further approximations [@uhrig; @kehrein], again tightly tailored for the problem under investigation.
The second field theoretical approach, which we want to focus on in the following, is based on a functional representation of the partition function of the system under consideration. It has become known as functional renormalization group (fRG) [@polchinski; @wetterich; @morris; @salmhoferbuch]. A detailed description of the fRG will be given in the next section; here we make some principle remarks and discuss previous applications.
Different versions of the fRG have been developed over the last few years. One either generates an exact infinite hierarchy of coupled differential equations for the amputated connected $m$-particle Green functions of the many-body system [@polchinski; @salmhoferbuch], or the one-particle irreducible $m$-particle vertex functions [@wetterich; @morris] respectively. For explicit calculations one has to truncate the set of equations which is the major approximation involved. At what level this truncation is performed to obtain a tractable set of equations depends on the complexity of the problem. We here exclusively study the one-particle irreducible version of the fRG. It has the advantage of including self-energy corrections already in low truncation orders and being formally easily extendable to higher orders [@footnote1].
Up to now, most applications of the fRG in many-body physics concentrate on low-dimensional, interacting fermion systems where it provides a possibility to sum diverging classes of diagrams. The homogeneous two-dimensional Hubbard model [@schulz; @metzner; @honerkamp], the homogeneous one-dimensional Tomonaga-Luttinger model [@peter], and the one-dimensional lattice model of spinless fermions with nearest neighbor interaction and local impurities [@meden] were investigated. The focus was put on properties of the system close to the Fermi surface, for example on the hierarchy of interactions to identify possible instabilities [@schulz; @metzner; @honerkamp] or on Tomonaga-Luttinger liquid exponents [@peter; @meden]. The frequency dependence of the vertex functions was mostly neglected [@honerkamp2].
In this paper we investigate the frequency dependence of the self-energy and the effective interaction. For this purpose, we study two different zero-dimensional (local) models: the quantum anharmonic oscillator and a well-known problem of solid state physics, the single impurity Anderson model (SIAM). The former has quite often been used as a “toy model” to investigate the performance of different approximation schemes of many-particle physics [@oscpaper; @goetz]. For this problem conventional perturbation theory is regular — although it generates a generic example of an asymptotic series [@oscpaper] — and one expects that compared to perturbation theory the fRG leads to a better agreement with the exact solution at larger coupling constants (“renormalization group enhanced perturbation theory”). We calculate the ground state energy and the energy of the first excited state as well as the spectral function of the propagator. Exact results for these observables can numerically be obtained quite easily.
The SIAM has a known hierarchy of energy scales, and presents a true challenge to any many-body tool due to the generation of an exponentially small energy scale, the Kondo scale, leading for example to a sharp resonance in the single-particle spectrum. No exact solutions for dynamical quantities of the model are known. We present fRG results for the one-particle spectral function of the impurity site and compare them to conventional second order perturbation theory (in the interaction $U$ of the impurity site) and Wilson’s numerical renormalization group (NRG). In both models the truncated fRG scheme, which is correct at least to second order perturbation theory in the interaction, leads to a considerable improvement compared to plain second order perturbation theory.
The paper is organized as follows. In the next section we present a detailed discussion of the fRG. The third part contains the application to the quantum anharmonic oscillator, while in the forth section we discuss the fRG for the single impurity Anderson model. A summary and outlook concludes the paper.
Functional renormalization group {#sec:fRG}
================================
Expressed as a functional integral the grand canonical partition function of a system of quantum mechanical particles (either fermions or bosons) interacting via a two-particle potential can be written as [@negele] $$\begin{aligned}
\label{partitionfunction}
\frac{{\mathcal Z}}{{\mathcal Z}_0} = \frac{1}{{\mathcal Z}_0} \int
{\mathcal D} \bar \psi \psi \exp{\left\{ (\bar \psi, \left[{\mathcal
G}^0 \right]^{-1}\psi ) - S_{\rm int}\left(\{ \bar \psi \}, \{
\psi\} \right) \right\} } \; ,\end{aligned}$$ with $$\begin{aligned}
\label{sintallg}
S_{\rm int} \left(\{ \bar \psi \}, \{
\psi\} \right) = \frac{1}{4}
\sum_{k_1',k_2',k_1,k_2}\bar v_{k_1',k_2',k_1, k_2}
\bar \psi_{k_1'} \bar \psi_{k_2'} \psi_{k_2} \psi_{k_1} \; ,\end{aligned}$$ and ${\mathcal Z}_0$ being the non-interacting partition function. Here $\psi$ and $\bar \psi$ denote either Grassmann (fermions) or complex (bosons) fields. The multi-indices $k_j^{(')}$ stand for the quantum numbers of the one-particle basis in which the problem is considered (e.g. momenta and spin directions) and Matsubara frequencies $\omega$. We have introduced the short hand notation $$\begin{aligned}
\left(\bar \psi, \left[{\mathcal
G}^0 \right]^{-1} \psi \right) = \sum_{k,k'} \bar
\psi_{k} \left[{\mathcal
G}^0 \right]^{-1}_{k,k'} \psi_{k'} \; ,\end{aligned}$$ with the propagator ${\mathcal G}^0$ of the related non-interacting problem given as a matrix. The anti-symmetrized (fermions) or symmetrized (bosons) matrix elements of the two-particle interaction are denoted by $\bar v_{k_1',k_2',k_1, k_2}$. They contain the energy conserving factor $\delta_{\omega+\omega',\nu+\nu'}$ and the factor $1/\beta$, with $\beta=1/T$ being the inverse temperature. We consider units such that $\hbar=1$ and $k_B=1$. The generating functional of the $m$-particle Green function is given by $$\begin{aligned}
\label{genfunct}
{\mathcal W } \left(\{\bar \eta \}, \{ \eta\} \right) & = &
\frac{1}{{\mathcal Z}_0}
\int {\mathcal D} \bar \psi \psi \exp \left\{
\left( \bar \psi, \left[ {\mathcal G}^{0}\right]^{-1}
\psi \right) - S_{\rm int}(\{\bar \psi \}, \{ \psi \}) \nonumber
\right. \\
&& \left.
- \left( \bar \psi, \eta\right) - \left(\bar \eta, \psi
\right) \right\} \; ,\end{aligned}$$ with $\left( \bar \psi, \eta \right) = \sum_{k} \bar \psi_{k}
\eta_{k}$ and the external source fields $\eta$ and $\bar \eta$. From this the generating functional of the connected $m$-particle Green function follows as $$\begin{aligned}
\label{wcdef}
{\mathcal W}^c\left(\{\bar \eta \}, \{ \eta\} \right) =
\ln{ \left[ {\mathcal W}
\left(\{\bar \eta \}, \{ \eta\} \right) \right]} \; .\end{aligned}$$ The (connected) $m$-particle Green function $G_m^{(c)}$ can be obtained by taking functional derivatives $$\begin{aligned}
\label{gmdef}
\hspace{-1.5cm} G_m^{(c)}\left(k_1', \ldots, k_m'; k_1, \ldots, k_m \right) =
\left. \frac{\delta^m}{\delta \bar \eta_{k_1'}
\ldots \delta \bar \eta_{k_m'}}
\frac{\delta^m}{\delta \eta_{k_m} \ldots \delta \eta_{k_1}} {\mathcal
W}^{(c)}\left(\{\bar \eta \}, \{ \eta\} \right) \right|_{\eta = 0= \bar
\eta} \; .\end{aligned}$$ By a Legendre transformation $$\begin{aligned}
\label{gammadef}
\Gamma \left(\{\bar \phi \}, \{ \phi \} \right) = - {\mathcal
W}^c\left(\{\bar \eta \}, \{ \eta\} \right) - \left(\bar \phi, \eta
\right) -\left(\bar \eta, \phi \right) + \left(\bar \phi, \left[{\mathcal
G}^{0}\right]^{-1} \phi\right) \; ,\end{aligned}$$ the generating functional of the one-particle irreducible vertex functions $\gamma_m$, with external source fields $\phi$ and $\bar
\phi$ and $$\begin{aligned}
\label{vertexfundef}
\hspace{-1.5cm}\gamma_m\left(k_1', \ldots, k_m'; k_1, \ldots, k_m \right) & = &
\left. \frac{\delta^m}{\delta \bar \phi_{k_1'} \ldots \delta \bar \phi_{k_m'}}
\frac{\delta^m}{\delta \phi_{k_m} \ldots \delta \phi_{k_1}} \Gamma
\left(\{\bar \phi\}, \{ \phi\} \right) \right|_{\phi = 0= \bar
\phi} \; ,\end{aligned}$$ is obtained. Note that in contrast to the usual definition [@negele] of $\Gamma $ we have added a term $\left(\bar \phi, \left[{\mathcal G}^{0}\right]^{-1}
\phi\right)$ in Eq. (\[gammadef\]) for convenience (see below). The relation between the $G_m^{(c)}$ and $\gamma_m$ can be found in text books [@negele]. The 0-particle vertex $\gamma_0$ provides the interacting part of the grand canonical potential $\Omega$ $$\begin{aligned}
%\label{grandpot}
\Omega=- T \ln {\mathcal Z} = T \gamma_0 - T \ln {\mathcal Z}_0 \; .\end{aligned}$$ For the 1-particle Green function we obtain $$\begin{aligned}
G_1(k';k) = G_1^c(k';k) = - \zeta {\mathcal G}_{k',k}
= \left[ \gamma_1 - \zeta \left[ {\mathcal G}^0\right]^{-1}
\right]^{-1}_{k',k} \; ,\end{aligned}$$ where $$\begin{aligned}
{\mathcal G}_{k',k} =
\left[ \left[ {\mathcal G}^0\right]^{-1}
- \Sigma \right]^{-1}_{k',k} \; ,\end{aligned}$$ with the self-energy $\Sigma$, and $\zeta=-1$ for fermions or $\zeta=1$ for bosons, respectively. This implies the relation $\Sigma = \zeta \gamma_1$. The 2-particle vertex $\gamma_2$ is usually referred to as the effective interaction. For $m \geq 3$, the $m$-particle interaction $\gamma_m$ has diagrammatic contributions starting at $m$-th order in the two-particle interaction.
In Eqs. (\[partitionfunction\]) and (\[genfunct\]) we replace the non-interacting propagator by a propagator ${\mathcal
G}^{0,\Lambda}$ depending on a cutoff $\Lambda$ and ${\mathcal Z}_0$ by ${\mathcal Z}_0^{\Lambda}$ determined using ${\mathcal G}^{0,\Lambda}$. The boundary conditions for the cutoff $\Lambda \in [\Lambda_0,0]$ are taken as $$\begin{aligned}
\label{demands}
{\mathcal G}^{0,\Lambda_0} = 0 \;\;\; , \;\;\;
{\mathcal G}^{0,\Lambda=0} = {\mathcal G}^{0} \; , \end{aligned}$$ i.e. at the starting point $\Lambda = \Lambda_0$ no degrees of freedom are “turned on” while at $\Lambda=0$ the cutoff independent problem is recovered. In our applications we use a sharp Matsubara frequency cutoff with $$\begin{aligned}
\label{matsubaracutoff}
{\mathcal G}^{0,\Lambda} = \Theta\left(|\omega|-\Lambda\right)
{\mathcal G}^{0} \end{aligned}$$ and consider $\Lambda_0 \to \infty$ [@meden]. Through ${\mathcal G}^{0,\Lambda}$ the quantities defined in Eqs.(\[partitionfunction\]) to (\[vertexfundef\]) acquire a $\Lambda$-dependence. One now derives a functional differential equation for $\Gamma^{\Lambda}$. From this, by expanding in powers of the external sources, a set of coupled differential equations for the $\gamma_m^\Lambda$ is obtained.
As a first step we differentiate ${\mathcal W}^{c,\Lambda}$ with respect to $\Lambda$, which after straightforward algebra leads to $$\begin{aligned}
\label{flusswc}
\hspace{-1cm}\frac{d}{d \Lambda} {\mathcal W}^{c,\Lambda} & = &
\zeta \mbox{ Tr}\, \left(
{\mathcal Q}^{\Lambda} {\mathcal G}^{0,\Lambda}
\right) +
\mbox{Tr} \,
\left({\mathcal Q}^{\Lambda}
\frac{\delta^2
{\mathcal W}^{c,\Lambda} }{ \delta \bar \eta \delta
\eta} \right) + \zeta \left(
\frac{\delta {\mathcal W}^{c,\Lambda} }{\delta \eta},
{\mathcal Q}^{\Lambda} \frac{\delta {\mathcal
W}^{c,\Lambda} }{\delta \bar \eta} \right) \; ,\end{aligned}$$ with $$\begin{aligned}
\label{Qlambdadef}
{\mathcal Q}^{\Lambda} = \frac{d}{d \Lambda} \left[ {\mathcal
G}^{0,\Lambda}\right]^{-1} \; .\end{aligned}$$ Considering $\phi$ and $\bar \phi$ as the fundamental variables we obtain from Eq. (\[gammadef\]) $$\begin{aligned}
\hspace{-2.3cm}\frac{d}{d \Lambda} \Gamma^{\Lambda}\left(\{\bar \phi \}, \{ \phi \} \right) =
- \frac{d}{d \Lambda} {\mathcal W}^{c,\Lambda} \left(\{\bar
\eta^{\Lambda} \}, \{ \eta^{\Lambda} \} \right)
- \left(\bar \phi, \frac{d}{d \Lambda} \eta^{\Lambda}
\right) -\left( \frac{d}{d \Lambda} {\bar \eta}^{\Lambda} , \phi \right)
+ \left(\bar \phi, {\mathcal Q}^{\Lambda} \phi\right) \; .\end{aligned}$$ Applying the chain rule and using Eq. (\[flusswc\]) this leads to $$\begin{aligned}
\frac{d}{d \Lambda} \Gamma^{\Lambda} = - \zeta \mbox{ Tr}\, \left(
{\mathcal Q}^{\Lambda}
{\mathcal
G}^{0,\Lambda} \right)
- \mbox{Tr} \, \left( {\mathcal Q}^{\Lambda}
\frac{\delta^2
{\mathcal W}^{c,\Lambda} }{ \delta \bar \eta^{\Lambda} \delta
\eta^{\Lambda}} \right) \; .\end{aligned}$$ The last term in Eq. (\[gammadef\]) is exactly cancelled, which a posteriori justifies its inclusion. Using the well known relation [@negele] between the second functional derivatives of $\Gamma$ and ${\mathcal W}^{c}$ we obtain the functional differential equation $$\begin{aligned}
\label{gammafluss}
\frac{d}{d \Lambda} \Gamma^{\Lambda} = - \zeta
\mbox{ Tr}\, \left( {\mathcal Q}^{\Lambda} {\mathcal
G}^{0,\Lambda}
\right) - \mbox{Tr} \, \left({\mathcal Q}^{\Lambda}
{\mathcal V}_{\bar \phi, \phi}^{1,1}(\Gamma^{\Lambda},
{\mathcal G}^{0,\Lambda}) \right) \; ,\end{aligned}$$ where $ {\mathcal V}_{\bar \phi, \phi}^{1,1}$ stand for the upper left block of the matrix $$\begin{aligned}
\label{invers}
{\mathcal V}_{\bar \phi,\phi}(\Gamma^{\Lambda}, {\mathcal G}^{\Lambda}) =
\left( \begin{array}{cc}
\frac{\delta^2 \Gamma^{\Lambda}}{ \delta \bar \phi \delta \phi}
- \zeta \left[ {\mathcal G}^{0,\Lambda} \right]^{-1}
& \frac{\delta^2 \Gamma^{\Lambda}}{\delta \bar \phi \delta \bar \phi} \\
\frac{\delta^2 \Gamma^{\Lambda}}{\delta \phi \delta \phi}
& \frac{\delta^2 \Gamma^{\Lambda}}{\delta \phi \delta \bar \phi} -
\left[ \left[{\mathcal G}^{0,\Lambda} \right]^{-1}\right]^t
\end{array} \right)^{-1} \end{aligned}$$ and the upper index $t$ denotes the transposed matrix. To obtain differential equations for the $\gamma_m^{\Lambda}$ which include self-energy corrections we express $ {\mathcal V}_{\bar \phi,
\phi}$ in terms of ${\mathcal G}^{\Lambda}$ instead of ${\mathcal G}^{\Lambda,0}$. This is achieved by defining $$\begin{aligned}
{\mathcal U}_{\bar \phi, \phi} = \frac{\delta^2 \Gamma^{\Lambda}}{\delta \bar
\phi \delta \phi} - \gamma_1^{\Lambda} \end{aligned}$$ and using $$\begin{aligned}
\label{GG}
{\mathcal G}^{\Lambda} = \left[ \left[ {\mathcal G}^{0,\Lambda}
\right]^{-1} - \zeta \gamma_1^{\Lambda} \right]^{-1} \end{aligned}$$ which leads to $$\begin{aligned}
\label{gammaflussalt}
\frac{d}{d \Lambda} \Gamma^{\Lambda} = - \zeta \mbox{ Tr}\, \left(
{\mathcal Q}^{\Lambda} {\mathcal
G}^{0,\Lambda}
\right) + \zeta \mbox{Tr} \, \left({\mathcal G}^{\Lambda}
{\mathcal Q}^{\Lambda}
\tilde{\mathcal V}_{\bar \phi, \phi}^{1,1}(\Gamma^{\Lambda},{\mathcal
G}^{\Lambda} ) \right)
\; , \end{aligned}$$ with $$\begin{aligned}
\label{Vtildedef}
\tilde {\mathcal V}_{\bar \phi,\phi}\left(\Gamma^{\Lambda}
,{\mathcal G}^{\Lambda} \right) = \left[ {\bf 1} -
\left( \begin{array}{cc}
\zeta {\mathcal G}^{\Lambda}
&
0\\
0
&
\left[ {\mathcal G}^{\Lambda} \right]^t
\end{array} \right)
\left( \begin{array}{cc}
{\mathcal U}_{\bar \phi, \phi}
& \frac{\delta^2 \Gamma^{\Lambda}}{\delta \bar \phi \delta \bar \phi} \\
\frac{\delta^2 \Gamma^{\Lambda}}{\delta \phi \delta \phi}
& \zeta {\mathcal U}^{t}_{\bar \phi, \phi}
\end{array} \right) \right]^{-1} \; .\end{aligned}$$ For later applications it is important to note that ${\mathcal U}_{\bar \phi,
\phi}$ as well as $ \frac{\delta^2
\Gamma^{\Lambda}}{\delta \bar \phi \delta \bar \phi}$ and $\frac{\delta^2
\Gamma^{\Lambda}}{\delta \phi \delta \phi} $ are at least quadratic in the external sources. The initial condition for the exact functional differential equation (\[gammaflussalt\]) can either be obtained by lengthy but straightforward algebra not presented here, or by the following simple argument: at $\Lambda= \Lambda_0$, ${\mathcal G}^{0,\Lambda_0}=0$ (no degrees of freedom are “turned on”) and in a perturbative expansion of the $\gamma_m^{\Lambda_0}$ the only term which does not vanish is the bare 2-particle vertex. We thus find $$\begin{aligned}
\label{anfanggamma}
\Gamma^{\Lambda_0} \left( \{ \bar \phi \} , \left\{ \phi
\right\} \right) = S_{\rm int} \left(\{\bar
\phi \}, \{ \phi \} \right) \; .\end{aligned}$$
An exact infinite hierarchy of flow equations for the $\gamma_m^{\Lambda}$ can be obtained by expanding Eq. (\[Vtildedef\]) in a geometric series and $\Gamma^{\Lambda}$ in the external sources $$\begin{aligned}
\Gamma^{\Lambda}\left(\{\bar \phi \}, \{ \phi \} \right) =
\sum_{m=0}^{\infty} \frac{\zeta^m}{(m !)^2} \sum_{k_1', \ldots, k_m'}
\sum_{k_1, \ldots, k_m } && \gamma_m^{\Lambda}\left(k_1',
\ldots, k_m'; k_1, \ldots, k_m \right) \nonumber \\
&& \times \bar \phi_{k_1'} \ldots \bar
\phi_{k_m'} \phi_{k_m} \ldots \phi_{k_1} \; .\end{aligned}$$ The equation for $\gamma_0^{\Lambda}$ reads $$\begin{aligned}
\label{gamma0fl}
\frac{d}{d \Lambda}\gamma_0^{\Lambda} = - \zeta \mbox{ Tr}\, \left(
{\mathcal Q}^{\Lambda} {\mathcal
G}^{0,\Lambda} \right) +
\zeta \mbox{ Tr}\, \left( {\mathcal Q}^{\Lambda} {\mathcal
G}^{\Lambda} \right) \; .\end{aligned}$$ Via ${\mathcal G}^{\Lambda}$ the derivative of $\gamma_0^{\Lambda}$ couples to the one-particle self-energy. For the flow of the self-energy we obtain $$\begin{aligned}
\label{gamma1fl}
\frac{d}{d \Lambda} \gamma_1^{\Lambda}(k';k) =
\zeta \frac{d}{d \Lambda} \Sigma^{\Lambda}_{k',k}
= \mbox{ Tr}\, \left( {\mathcal S}^{\Lambda}
\gamma_2^{\Lambda}(k', \ldots;
k, \ldots) \right) \; , \end{aligned}$$ with the so-called single scale propagator $$\begin{aligned}
\label{Sdef}
{\mathcal S}^{\Lambda} =
{\mathcal G}^{\Lambda} {\mathcal Q}^{\Lambda}
{\mathcal G}^{\Lambda} \; .\end{aligned}$$ Here $\gamma_2^{\Lambda}(k', \ldots;
k, \ldots)$ is a matrix in the variables not explicitly written, i.e. $\left[\gamma_2^{\Lambda}(k', \ldots ;k,\ldots
)\right]_{q',q} = \gamma_2^{\Lambda}(k',q'; k,q) $. Diagrammatically Eq. (\[gamma1fl\]) is shown in Fig. \[fig1\]. The derivative of $ \gamma_1^{\Lambda}$ is determined by $ \gamma_1^{\Lambda}$ and the 2-particle vertex $ \gamma_2^{\Lambda}$. Thus an equation for $
\gamma_2^{\Lambda}$ is required $$\begin{aligned}
\label{gamma2fl}
\hspace{-0.8cm} \frac{d}{d \Lambda} \gamma_2^{\Lambda}(k_1', k_2'; k_1, k_2)
& = & \mbox{ Tr}\, \left( {\mathcal S}^{\Lambda}
\gamma_3^{\Lambda}(k_1',k_2', \ldots; k_1,k_2,\ldots)
\right)
\nonumber \\
&& + \zeta \mbox{ Tr}\, \Big( {\mathcal S}^{\Lambda}
\gamma_2^{\Lambda}(\ldots , \ldots ; k_1, k_2 )
\left[{\mathcal G}^{\Lambda}\right]^{t}
\gamma_2^{\Lambda}(k_1', k_2' ; \ldots
,\ldots ) \Big)
\nonumber \\
&& + \zeta \mbox{ Tr}\, \Big( {\mathcal S}^{\Lambda}
\gamma_2^{\Lambda}(k_1', \ldots ;
k_1,\ldots )
{\mathcal G}^{\Lambda} \gamma_2^{\Lambda}(k_2', \ldots ;
k_2,\ldots ) \nonumber \\
&& + \zeta \left[ k_1' \leftrightarrow k_2' \right]
+ \zeta \left[ k_1 \leftrightarrow k_2 \right] +
\left[ k_1' \leftrightarrow k_2' , k_1 \leftrightarrow k_2
\right] \Big)
\; .\end{aligned}$$ The corresponding diagrammatic representation is shown in Fig.\[fig2\]. The right hand side (rhs) of Eq. (\[gamma2fl\]) contains $\gamma_1^{\Lambda}$, $\gamma_2^{\Lambda}$, and the 3-particle vertex $\gamma_3^{\Lambda}$. For $m \geq 1$ the equation for $\frac{d}{d \Lambda}
\gamma_m^{\Lambda}$ contains $\gamma_{m'}^{\Lambda}$ with $m'=1,2,\ldots,m, m+1$. The initial condition for the $\gamma_m^{\Lambda_0}$ can be obtained from Eq.(\[anfanggamma\]) and is given by $$\begin{aligned}
\gamma_2^{\Lambda_0}(k_1',k_2';k_1,k_2) = \bar v_{k_1',k_2',k_1, k_2}
\;\;\; , \;\;\; \gamma_m^{\Lambda_0} = 0 \;\; \mbox{for} \;\; m \neq 2 \; .
\label{gammamanfang} \end{aligned}$$ We here refrain from explicitly presenting equations for $\frac{d}{d \Lambda} \gamma_m^{\Lambda}$ with $m \geq 3$ since later on the set of differential equations is truncated by setting $\gamma_3^{\Lambda} = \gamma_3^{\Lambda_0} =0$, which implies that vertices with $m \geq 3$ do not contribute.
![Diagrammatic form of the flow equation for $\gamma_1^{\Lambda} = \zeta \Sigma^{\Lambda}$. The slashed line stands for the single scale propagator ${\mathcal S}^{\Lambda}$. \[fig1\]](fig1){width="28.00000%"}
![Diagrammatic form of the flow equation for $\gamma_2^{\Lambda}$. The slashed line stands for the single scale propagator ${\mathcal S}^{\Lambda}$ the unslashed line for ${\mathcal G}^{\Lambda}$. \[fig2\]](fig2){width="60.00000%"}
Following the above systematics, a truncation scheme emerges quite naturally. If, for $m_c \geq 2$, the vertex $\gamma_{m_c+1}^{\Lambda}$ on the rhs of the coupled flow equations is replaced by its initial condition $\gamma_{m_c+1}^{\Lambda_0}=0$, a closed set of equations for $\gamma_{m}^{\Lambda}$ with $m \leq m_c$ follows. This set of differential equations can then be integrated over $\Lambda$ starting at $\Lambda=\Lambda_0$ down to $\Lambda=0$ providing approximate expressions for the $\gamma_{m}$ of the original (cutoff free) problem with $m \leq m_c$. Expanding $\gamma_{m}^{\Lambda}$ in terms of the bare interaction, conventional perturbation theory for the grand canonical potential, the self-energy, the effective interaction and higher order vertex functions can be recovered from an iterative treatment of the flow equations. It is easy to show that the vertex functions obtained from the truncated equations are at least correct up to order $m_c$ in the bare interaction.
To obtain a manageable set of equations in applications of the fRG to one- and two-dimensional quantum many-body problems [@schulz; @metzner; @honerkamp; @peter; @meden], further approximations in addition to the above truncation scheme were necessary. In the following two sections we will avoid such additional approximations and solve the truncated fRG equations to order $m_c=2$ for two zero-dimensional (local) interacting many-particle problems: the quantum harmonic oscillator with a quartic perturbation and the SIAM.
Very recently a modified version of the flow equation for $\gamma_2^{\Lambda}$ was suggested. As we will consider the truncation order $m_c=2$ we here only describe this modified scheme for this order. Guided by the idea of an improved fulfillment (compared to the scheme described above) of a certain Ward identity Katanin replaced the combined propagator ${\mathcal
S}^{\Lambda}$ in the last five terms of Eq. (\[gamma2fl\]) (the first term does not contribute to order $m_c=2$) by $-d {\mathcal G}^{\Lambda} / d\Lambda$ [@katanin]. Using $$\begin{aligned}
\frac{d {\mathcal G}^{\Lambda}}{d \Lambda} = - {\mathcal S}^{\Lambda}
+ {\mathcal G}^{\Lambda} \frac{d \Sigma^{\Lambda}}{d \Lambda}
{\mathcal G}^{\Lambda} \end{aligned}$$ obtained from Eq. (\[GG\]) it is obvious that the terms added to Eq. (\[gamma2fl\]) are at least of third order in the bare interaction. Using this modification in the order $m_c=2$ equations for $\gamma_1^{\Lambda}$ and $\gamma_2^{\Lambda}$ leads to the exact solution for certain exactly solvable models (reduced BCS model, interacting fermions with forward scattering only). This suggests that the replacement possibly improves the results of the truncated fRG also for other models. In the next section we show that this is indeed the case for the harmonic oscillator with a quartic perturbation. The same holds for the SIAM as discussed in Sect. 4.
Application to the anharmonic oscillator {#sec:ao}
========================================
In appropriate units the Hamiltonian of the harmonic oscillator with a quartic perturbation is given by $$\label{anhaos}
H=\frac{1}{2} x^2 + \frac{1}{2} p^2 + \frac{g}{4!} x^4 \; ,$$ with the position operator $x$, the momentum operator $p$, and the coupling constant $g$. We here focus on $T=0$ and are interested in low-lying eigenenergies $E_n$ as well as the (imaginary) time-ordered propagator $${\mathcal G}(\tau) = \left< E_0 \right| {\mathcal T}
\left[x(\tau) x(0) \right] \left| E_0 \right> \; ,$$ with $\left| E_n \right>$ being the eigenstates of the Hamiltonian in Eq. (\[anhaos\]). The Fourier transform of the propagator can be written as $$\begin{aligned}
{\mathcal G} (i \omega) =
\int_{-\infty}^{\infty} d \tau \, e^{i \omega \tau} {\mathcal G}(\tau)
= \frac{1}{\left[ {\mathcal G}^0 (i \omega)
\right]^{-1}- \Sigma(i\omega)} \; ,\end{aligned}$$ where we have introduced the self-energy $\Sigma$ and the non-interacting propagator $$\begin{aligned}
{\mathcal G}^0 (i \omega) =
\frac{1}{\omega^2 + 1} \; . \end{aligned}$$ In contrast to the more general notation used in the last section, propagators and the self-energy only depend on a single frequency and do no longer contain the energy conserving $\delta$-function ($T=0$) here. The propagator has the Lehmann representation $$\begin{aligned}
\label{lehmann}
\hspace{-1.0cm} {\mathcal G} (i \omega) =
\frac{1}{2} \sum_{n=1}^{\infty} \left[
\frac{1}{i \omega + (E_n-E_0)} - \frac{1}{i \omega - (E_n-E_0)}
\right] \left| \left< E_0 \right| (a +a^{\dag}) \left| E_n \right>
\right|^2 \; ,\end{aligned}$$ where $a$, $a^\dag$ denote the usual raising and lowering operators. The spectral weights $$\begin{aligned}
s_n = \left| \left< E_0 \right| (a +a^{\dag}) \left|
E_n \right> \right|^2\end{aligned}$$ and energies $E_n$ fulfill the f-sum rule $$\begin{aligned}
\label{sumrule}
1 = \sum_{n=1}^{\infty} (E_n-E_0) \; s_n \; .\end{aligned}$$ It turns out that for coupling constants $g \leq 50$ considered here the sums in Eqs. (\[lehmann\]) and (\[sumrule\]) are dominated by the first term. For this reason only the first few eigenstates and eigenenergies are required to obtain accurate (“numerically exact”) results for ${\mathcal G} (i \omega)$. These can quite easily be obtained by expressing $H$ in the basis of eigenstates $\left| n \right>$ of the unperturbed harmonic oscillator and numerically diagonalizing the upper left corner of the (infinite) matrix $\left< n \right| H \left| n' \right>$ with $n,n' \leq n_c$ and a sufficiently large $n_c$. For $g \leq 50$, $n_c=100$ turns out to be large enough to fulfill the sum rule Eq. (\[sumrule\]) to very high precision.
Second order perturbation theory for the $g$-dependent part of the ground state energy yields $$\begin{aligned}
\label{energy2}
e_0^{(2)} = E_0^{(2)} - E_0^0=
\frac{1}{32} \; g - \frac{7}{1536} \; g^2\end{aligned}$$ and for the self-energy one obtains $$\begin{aligned}
\label{selfenergy2}
\Sigma^{(2)}(i\omega) =
- \frac{1}{4} \; g + \frac{1}{32} \; g^2 + \frac{1}{8} \; g^2 \frac{1}{\omega^2
+9} \; .\end{aligned}$$ Within the fRG approximate expressions for $$\begin{aligned}
E_{n,0}= E_n - E_0\end{aligned}$$ and $s_n$ can be obtained from the poles and residues of the propagator ${\mathcal G} (i
\omega)$. Furthermore, since Eqs. (\[lehmann\]) and (\[sumrule\]) are dominated by the first terms we only consider $E_{1,0}$ and $s_1$ [@footnote4]. Second order approximations for these quantities are given by the smallest pole of ${\mathcal G}^{(2)}(i \omega) = \left[ \omega^2+1 -
\Sigma^{(2)}(i\omega) \right]^{-1}$ and the related residue. It is important to note that this approximation $E_{1,0}^{(2)}$ agrees with $E_1^{(2)}-E_0^{(2)}$, where $E_1^{(2)}$ is determined directly from Rayleigh-Schrödinger perturbation theory, only up to second order in $g$, but is closer to the exact $E_{1,0}$.
Within mean field theory, $e_0^{\rm MF}$ and a frequency independent $\Sigma^{\rm MF}$ are given by $$\begin{aligned}
e_0^{\rm MF} = \frac{1}{2} \sqrt{1+\frac{g}{2} \left< X^2
\right>_{\rm MF}} \; - \frac{g}{8} \left< X^2 \right>_{\rm MF}^2 -
\frac{1}{2} \; \;
\; , \; \; \; \Sigma^{\rm MF} = -\frac{g}{2} \left< X^2
\right>_{\rm MF} \end{aligned}$$ and $\left< X^2 \right>_{\rm MF}$ is the solution of the self-consistency equation $$\begin{aligned}
\left< X^2 \right>_{\rm MF} = \frac{1}{2} \; \frac{1}{\sqrt{1+g \left<
X^2 \right>_{\rm MF} /2}} \; . \end{aligned}$$ From the mean field propagator one obtains $$\begin{aligned}
E_{1,0}^{\rm MF} = \sqrt{1 - \Sigma^{\rm MF}} \; \;
\; , \; \; \; s_1^{\rm MF} = \frac{1}{2} \; \frac{1}{ \sqrt{1 -
\Sigma^{\rm MF}}} \; .\end{aligned}$$
The functional integral representation of the grand canonical partition function of the Hamiltonian Eq. (\[anhaos\]) reads $$\begin{aligned}
\label{partitionfunctionao}
\frac{{\mathcal Z}}{{\mathcal Z}_0} = \frac{1}{{\mathcal Z}_0} \int
{\mathcal D} \, \bar x \, x \exp{\left\{ (\bar x, \left[{\mathcal
G}^0 \right]^{-1} x )/2 - S_{\rm int}\left(\{ \bar x \}, \{
x \} \right) \right\} } \; ,\end{aligned}$$ with $$\begin{aligned}
\label{sintao}
S_{\rm int} \left(\{ \bar x \}, \{
x \} \right) = \frac{g}{\beta \, 4!}
\sum_{n_1,\ldots , n_4} \delta_{n_1+n_2+n_3+n_4,0} \;
x(i \omega_1) x(i \omega_2) x(i \omega_3) x(i \omega_4) \; ,\end{aligned}$$ bosonic Matsubara frequencies $\omega_j=2 \pi \, n_j/\beta$, and complex fields $\bar x( i \omega) = x(-i \omega) $. As outlined in the last section and using a frequency cutoff Eq.(\[matsubaracutoff\]) flow equations for the $\gamma_m^{\Lambda}$ can be obtained. Here we focus on the equations in truncation order $m_c=2$. For $T \to 0$ and after introducing $$\begin{aligned}
e_0^{\Lambda} = \lim_{T \to 0} T \gamma_0^{\Lambda}\end{aligned}$$ we find $$\begin{aligned}
\label{gamma0ao}
\frac{d}{d \Lambda} e_0^{\Lambda}
= - \frac{1}{2 \pi} \ln \left[ 1- {\mathcal
G}^0(i \Lambda) \; \Sigma^{\Lambda}(\Lambda) \right] \; ,\end{aligned}$$ with the initial condition $e_0^{\Lambda=\infty}=0$. At the end of the flow, $e_0^{\Lambda=0}$ directly provides the fRG approximation $e_0^{\rm fRG}$ for the $g$-dependent part of the ground state energy. The flow equation for the self-energy follows as $$\begin{aligned}
\label{gamma1ao}
\frac{d }{d \Lambda} \Sigma^{\Lambda} (i\omega)=
\frac{1}{2 \pi} \; \frac{1}{\Lambda^2 + 1 - \Sigma^{\Lambda}(
i \omega)}
\; g^{\Lambda}(i\omega,-i\omega,i\Lambda, -i\Lambda)
\; ,\end{aligned}$$ with the initial condition $\Sigma^{\Lambda=\infty}=0 $ and the fRG approximation for the self-energy $\Sigma^{\rm fRG}(i\omega)=\Sigma^{\Lambda=0}(i\omega)$. Here $g^{\Lambda}$ denotes the totally symmetric 2-particle vertex which, in contrast to the vertex $\gamma_2^{\Lambda}$ introduced in the last section, does not contain an energy conserving $\delta$-function and factors of $\beta$. It depends on only three frequencies, but the fourth will nevertheless always be included in the following. To derive Eqs. (\[gamma0ao\]) and (\[gamma1ao\]) one has to deal with products of delta functions $\delta(|\omega| - \Lambda)$ and terms involving step functions $\Theta(|\omega| - \Lambda)$. These seemingly ambiguous expressions are well defined and unique if the sharp cutoff is implemented as a limit of increasingly sharp broadened cutoff functions $\Theta_{\epsilon}$, with the broadening parameter $\epsilon$ tending to zero. The expressions can then be conveniently evaluated by using the following relation [@morris], valid for arbitrary continuous functions $f$: $$\label{morristrick}
\delta_{\epsilon}(x-\Lambda) \, f[\Theta_{\epsilon}(x-\Lambda)] \to
\delta(x-\Lambda) \int_0^1 f(t) \, dt \; ,$$ where $\delta_{\epsilon} = - d \Theta_{\epsilon}/ d \epsilon$.
For $g^{\Lambda}$ the flow equation reads $$\begin{aligned}
\label{gamma2ao}
&& \hspace{-2.0cm}\frac{d }{d \Lambda} g^{\Lambda} (i\omega_1, i
\omega_2,i\omega_3, -i \omega_1-i \omega_2-i \omega_3 ) =
\frac{1}{2 \pi} \int_{-\infty}^{\infty} d \, \nu \left[
{\mathcal P}(i \nu , i \nu - i\omega_1 - i \omega_2)
\right. \nonumber \\&& \hspace{-1.5cm} \times
g^{\Lambda}
(i\omega_1, i\omega_2,- i \nu, i \nu - i\omega_1 - i \omega_2)
g^{\Lambda} (i\omega_3, -i
\omega_1-i \omega_2-i \omega_3,
-i\nu +i\omega_1 + i \omega_2, i \nu ) \nonumber \\
&& \hspace{-1.5cm} \left. +\left( \omega_2 \leftrightarrow \omega_3 \right)
+\left( \omega_2 \leftrightarrow -\omega_1- \omega_2- \omega_3
\right)
\right]\end{aligned}$$ where ${\mathcal P}(i \nu, i \nu')$ stands for two different products of propagators. In the conventional fRG scheme it is given by $$\label{Pconventional}
{\mathcal P}_{\rm con}(i \nu, i \nu') = {\mathcal S}^{\Lambda}(i \nu)
\; {\mathcal G}^{\Lambda}(i \nu')$$ while in the modified scheme [@katanin] one obtains $$\label{Pmodified}
{\mathcal P}_{\rm mod}(i \nu, i \nu') = - \frac{d {\mathcal G}^{\Lambda}(i
\nu)}{d \Lambda} \;
{\mathcal G}^{\Lambda}(i \nu') \; .$$ To explicitly evaluate ${\mathcal P}(i \nu, i \nu') $ Eq.(\[morristrick\]) has to be used, where special care has to be taken for the case $\nu=\nu'$. In the conventional scheme ${\mathcal P}_{\rm con}(i \nu, i \nu')$ contains a factor $\delta(|\nu|-\Lambda)$ and the integral over $\nu$ in Eq. (\[gamma2ao\]) can be performed analytically.
![Coupling constant dependent part of the groundstate energy $e_0=E_0 - E_0^0$ as a function of $g$. Different approximations (second order perturbation theory \[thin dashed line\], mean field \[dotted line\], conventional fRG \[thick dashed line\], modified fRG \[dashed-dotted line\]) are compared to the exact result (solid line). The inset shows the difference between the exact result and the two fRG approximations. \[fig3\]](oscill1paper){width="80.00000%"}
![As in Fig. \[fig3\], but for the energy difference $E_{1,0}$ of the first excited state and the ground state. \[fig4\]](oscill2paper){width="80.00000%"}
![As in Fig. \[fig3\], but for the spectral weight $s_1$ of the first peak. \[fig5\]](oscill3paper){width="80.00000%"}
To numerically solve the set of differential equations (\[gamma0ao\]), (\[gamma1ao\]), and (\[gamma2ao\]) we have discretized the frequencies (which at $T=0$ are continuous) on a linear mesh $\omega_j = j \delta$ with $j = -j_0,-j_0+1, \ldots, j_0$ [@footnote3]. By increasing $j_0$ and decreasing $\delta$ convergence can be achieved up to the required accuracy. For our purposes $j_0=40$ and $\delta=0.5$ turned out to be appropriate. This leads to a set of roughly $5.3\times 10^5$ coupled equations. The $\Lambda$-integration is started at $\Lambda_0=10^5$ making sure that further increasing $\Lambda_0$ does not lead to significant changes in the results. Figs. \[fig3\] to \[fig5\] show comparisons of $e_0$, $E_{1,0}$, and $s_1$ with the different approximations considered here (second order perturbation theory, mean field, conventional fRG, modified fRG) and the exact results. Although the approximate fRGs correctly reproduce only the first two derivatives with respect to $g$ at $g=0$, they give extremely accurate results even up to $g=50$, while conventional second order perturbation theory can only be trusted for $g<1$. This provides an impressive example of the power of “renormalization group enhanced perturbation theory”. Comparing the two fRG approximations the modified scheme is roughly a factor of two closer to the exact result and thus indeed a substantial improvement [@katanin; @footnote5].
To avoid the problem of analytic continuation (see the next section) the results for $e_{1,0}^{\rm fRG}$ and $s_1^{\rm fRG}$ were obtained by fitting a function $a/(\omega^2 + b^2)$ with $a$ and $b$ as fitting parameters to ${\mathcal G}^{\rm fRG}(i \omega) = \left[ \omega^2+1 -
\Sigma^{\rm fRG}(i\omega) \right]^{-1}$. Assuming this fitting form we have used that the spectral function is dominated by the first peak [@footnote4]. For the problem studied also mean field theory leads to fairly accurate results (but not as good as the fRG). This is related to the fact that low-lying eigenstates of the Hamiltonian in Eq. (\[anhaos\]) can be described quite well by the eigenstates of a harmonic oscillator with a shifted frequency determined self-consistently.
Single-particle dynamics of the single impurity Anderson model {#sec:SIAM}
==============================================================
In contrast to the anharmonic oscillator studied in the previous section, the single impurity Anderson model [@SIAM] $$\label{equ:SIAM}
H=\sum\limits_{\vec k\sigma}\epsilon_{\vec k}
c^\dagger_{\vec k\sigma}c^{\phantom{\dagger}}_{\vec k\sigma}
+
\epsilon_d\sum\limits_\sigma d^\dagger_{\sigma}d^{\phantom{\dagger}}_{\sigma}
+U d^\dagger_{\uparrow}d^{\phantom{\dagger}}_{\uparrow}
d^\dagger_{\downarrow}d^{\phantom{\dagger}}_{\downarrow}
+
\frac{V}{\sqrt{N}}\sum\limits_{\vec k\sigma}\left(
c^\dagger_{\vec k\sigma}d^{\phantom{\dagger}}_{\sigma}+\mbox{h.c.}\right)$$ consists of two subsystems, namely a “conduction band” with continuous energy spectrum described by the first term in Eq. (\[equ:SIAM\]) and a localized level (typically referred to as “$d$”-state) with energy $\epsilon_d$ relative to the chemical potential $\mu$ of the conduction electrons. Two electrons occupying the localized level are in addition subject to a Coulomb repulsion $U>0$. Both subsystems are coupled via the hybridization in the last term. As the two-body interaction is restricted to a single level the SIAM falls into the of class of zero-dimensional interacting systems.
While this model looks rather simple, it contains all ingredients that make it a complicated many-body problem. The bare energy scales of the model are the bandwidth $W$ of the band electrons, the local energy $\epsilon_d$, the Coulomb repulsion $U$, and the bare level width generated by the hybridization, $\Delta_0=\pi V^2\rho_c(\mu)$, where $\rho_c(\epsilon)$ denotes the density of states of the conduction states which is assumed to be slowly varying. In the following we restrict ourselves to the particle-hole symmetric case $2\epsilon_d+U=2\mu=0$. As in the parameter regime $U/\Delta_0\gg1$ the charge fluctuations on the $d$-level from the average value $1$ are small, it can effectively be described by a spin antiferromagnetically coupled to the conduction electron spin density [@swt], the so-called Kondo model [@Kondo]. This antiferromagnetic coupling leads to a screening of the local spin by the conduction electrons with a characteristic energy scale $\ln(T_{\rm K})\propto-1/\Delta_0$, the Kondo temperature. As is apparent from this expression, $T_{\rm K}$ depends non-analytically on $\Delta_0$, signalling the occurrence of infrared divergences in perturbation theories in $ \Delta_0/U$ [@hewson] and a severe problem for computational techniques, namely the task to resolve an exponentially small energy scale.
On the other hand, the physics in the regimes $T,\omega\gg T_{\rm K}$ and $T,\omega\ll T_{\rm K}$ is comparatively simple. For $T,\omega\gg T_{\rm K}$ it is governed by charge excitations with energies $\epsilon_d$ and $\epsilon_d+U$ with a life-time given by $1/(2\Delta_0)$. In the other limit $T,\omega\ll T_{\rm K}$ it has been worked out using Wilson’s NRG [@Wilson] that the system can again be described by a Hamiltonian of the form Eq. (\[equ:SIAM\]), but with $U\to U^\ast=0$, $\epsilon_d\to\epsilon_d^\ast=0$ and $\Delta_0$ replaced by $\Delta_0^\ast\sim T_{\rm K}$. This regime has been coined “local Fermi liquid” by Nozières [@hewson]. This effective description yields a narrow resonance at the chemical potential in the spectral function of the $d$-Green function which in the symmetric case takes the form $$G_{dd}(i\nu)=\frac{1}{i\nu-\Delta (i\nu)-\tilde \Sigma(i\nu)}.$$ Here $\Delta(i\nu)=V^2\int d\epsilon \rho_c(\epsilon)/(i\nu-\epsilon)$ and $\tilde \Sigma$ denotes all self-energy contributions of second and higher order in $U$. As a special feature of the symmetric case, already the approximation to only keep the second order self-energy $\Sigma^{(2)}$ describes the qualitative behavior of the $d$-spectral function for different values of $U/\Delta_0$ correctly [@hewson]. For $U/\Delta_0 \ll 1$ there is a Lorentzian peak at the chemical potential with a width differing little from $\Delta_0$. For $U/\Delta_0 \gg 1$ most of the spectral weight is in the high energy peaks near $\pm U/2$ with a narrow resonance at $\mu=0$. Quantitatively the width and shape of this Kondo (or Abrikosov-Suhl) resonance is described poorly using $\Sigma^{(2)}$, vanishing only $\sim \Delta_0/U$. From the results of Sect. 3 one expects that the application of the functional renormalization group to the SIAM leads to improvements over the direct perturbational description of the Kondo resonance. For the SIAM the fRG results can be compared to the outcome of NRG calculations [@RG_general; @Wilson]. An important aspect of the treatment of the SIAM with the fRG is that, while the numerical effort of the NRG method increases exponentially with the number of impurity degrees of freedom, e.g. in case of additional orbital degrees of freedom or for a system of many magnetic impurities, the increase in computational resources necessary in the fRG approach described below is governed at most by a power-law.
The starting point of our fRG approach are again Eqs. (\[gamma1fl\]) and (\[gamma2fl\]) . We also use $m_c=2$, i.e. we replace the 3-particle-vertex by its initial condition $\gamma_3^\Lambda=0$, and consider $T=0$ only. We again use the frequency cutoff described in Eq. (\[matsubaracutoff\]) and because we presently concentrate on spectral properties only flow equations for the self-energy and for the 2-particle-vertex (and not the ground state energy) will be derived. Note that the calculation of two-particle properties is possible within the same scheme without further difficulties [@metzner] and will be discussed in a forthcoming publication.
For the SIAM all Green functions $G_{i,j}$ different from $G_{dd}$ can be expressed in terms of $G_{dd}$ and Green functions of the non-interacting system. For the calculation of $G_{dd}$ the multi-index $k$ is replaced by a spin-index $\sigma$ and a frequency $\nu$. The 2-particle-vertex can then be written as $$\begin{aligned}
\label{gamma2:SIAMa}
\hspace{-2cm}
\g(k_1',k_2';k_1,k_2)=\D{\nu_1+\nu_2-\nu_1'-\nu_2'}\\\nonumber
\times\Bigl\{\DS \delta_{\sigma_1,\sigma_1'}\delta_{\sigma_2,\sigma_2'}\ \gw(i\nu_1',i\nu_2';i\nu_1,i\nu_2)
-\DS \delta_{\sigma_1,\sigma_2'}\delta_{\sigma_2,\sigma_1'}\ \gw(i\nu_2',i\nu_1';i\nu_1,i\nu_2)
\Bigr\}.\end{aligned}$$ This can be used to perform the sum over the spins in Eq. (\[gamma1fl\]) and we find for the self-energy
$$\label{Self:SIAM}
\hspace{-2cm}
\DDL\Se{i\nu}=-\frac{1}{2\pi}\int_{-\infty}^{\infty}d\nu'
\Gp{i\nu'}\Bigl[
2 \, \gw(i\nu,i\nu';i\nu,i\nu')-
\gw(i\nu',i\nu;i\nu,i\nu')
\Bigr].$$
In order to derive a flow equation for $\gw(i\nu_1',i\nu_2';i\nu_1,i\nu_2)$, with $\nu_2=\nu_1'+\nu_2'-\nu_1$, the spin-sum in Eq. (\[gamma2fl\]) has to be performed as well. This leads to the lengthy expression Eq. (\[gamma2:SIAM\]), presented in Appendix A.
The initial values are given by $\Sigma^{\Lambda=\infty}(i\nu)=0$ and $\mathcal U^{\Lambda=\infty}(i\nu_1',i\nu_2';i\nu_1,i\nu_2)=U$. In the numerical solution of these flow equations we start integrating at finite $\Lambda_0 \gg \max (U,\Delta_0)$. The integration from $\Lambda=\infty$ to $\Lambda_0$ can be performed analytically. Up to corrections of order $\Lambda_0^{-1}$ the new initial conditions are $\Sigma^{\Lambda_0}(i\nu)=U/2$ and $\mathcal U^{\Lambda_0}(i\nu_1',i\nu_2';i\nu_1,i\nu_2)=U$.
$\mathcal S^\Lambda(i\nu)$ and $\mathcal P_{\rm con}^{\Lambda}(i\nu,i\nu')=\mathcal S^\Lambda(i\nu) \mathcal G^\Lambda(i\nu')$ are calculated the same way as for the anharmonic oscillator using Eq. (\[morristrick\]), and are given by $$\label{S:SIAM}
\Gp{i\nu} \to
\D{|\nu|-\Lambda}\frac{1}{\left[\Gn{i\nu}\right]^{-1}-\Se{i\nu}} \; ,$$ with $\left[\Gn{i\nu}\right]^{-1} = i \nu - \Delta(i \nu) - \epsilon_d$ and $$\hspace{-1cm}
\mathcal P_{\rm con}^{\Lambda}(i\nu,i\nu') \to\D{|\nu|-\Lambda}\frac{1}{\left[\Gn{i\nu}\right]^{-1}-\Se{i\nu}}
\frac{\H{\nu'}}{\left[\Gn{i\nu'}\right]^{-1}-\Se{i\nu'}}$$ with $\Theta(0)=1/2$. In our calculations we use the limit of an infinite bandwidth, i.e. $\Delta(i\nu)\to -i \, {\rm sign} (\nu) \, \Delta_0$. The results of second order perturbation theory for the self-energy can be recovered by replacing the self-energy on the rhs of Eqs. (\[Self:SIAM\]) and (\[gamma2:SIAM\]) and $\mathcal
U^\Lambda$ on the rhs of Eq. (\[gamma2:SIAM\]) by their initial values. It turns out that the fRG version using $ \mathcal P_{\rm con}^{\Lambda} $ gives good results for $U/\Delta_0<3$, but fails to provide the expected improvement compared to the use of $\Sigma^{(2)}$ for $U/\Delta_0 > 3$. This will be discussed in more detail in a forthcoming publication. For this reason we follow [@katanin] and replace $\mathcal P^\Lambda_{\rm con}(i\nu,i\nu') $ by $\mathcal
P^\Lambda_{\rm mod}(i\nu,i\nu')=-\mathcal G^\Lambda(i\nu')
\DS\DDL\mathcal G^\Lambda(i\nu)$ in Eq. (\[gamma2:SIAM\]). Applying Eq. (\[morristrick\]) yields $$\label{Kat:SIAM}
\hspace{-2cm}
\begin{array}{lll}
-\Gl{i\nu'}\DS\DDL\Gl{i\nu}
&\to&\DS\D{|\nu|-\Lambda}\frac{1}{\left[\Gn{i\nu}\right]^{-1}-\Se{i\nu}}
\frac{\H{\nu'}}{\left[\Gn{i\nu'}\right]^{-1}-\Se{i\nu'}}\\[8mm]
&-&\DS\DDL\Se{i\nu}~\frac{\H{\nu}}{[\left[\Gn{i\nu}\right]^{-1}-\Se{i\nu}]^2}
\frac{\H{\nu'}}{\left[\Gn{i\nu'}\right]^{-1}-\Se{i\nu'}}.
\end{array}$$
For the integration of the $T=0$-flow equations the continuous frequencies again have to be discretized. However, in contrast to the anharmonic oscillator it is not sufficient to work with a linear mesh. Because for $U/\Delta_0 \gg 1$ the Kondo resonance is visible only on an exponentially small energy scale around the Fermi level, we use a combination of a linear and a logarithmic mesh. This enables us to recover both, the high energy as well as the low-energy physics keeping the number of frequencies to a manageable size. The numerical effort grows with the third power of the number of frequencies in the conventional version and with almost the forth power using the modified version, due to the absence of a $\delta$-function in the last term of Eq. (\[Kat:SIAM\]).
In the fRG one naturally obtains the self-energy for imaginary frequencies. In order to calculate spectral functions, an analytic continuation to the real axis is necessary, which is known to be an ill-posed problem. Since the results from fRG are not subject to statistical errors or noise as for example quantum Monte Carlo data, we have applied the method of Padé approximation [@Pade] to obtain $\Sigma(z)$ from $\Sigma(i\nu)$. We find that especially the low-energy part of the spectral function which contains the Kondo resonance can be reliably extracted if sufficiently many frequencies close to the Fermi level are used (see below).
![Left panel: $\Sigma(i\omega)$ from fRG for $U=\Delta_0$ (circles) compared to NRG (dashed line) and second order perturbation theory (dotted line).\
Right panel: $\Im m\Sigma(\omega+i0^+)$ for fRG obtained from Padé approximants for the data in left panel. The inset shows an enlarged view around $\omega=0$. \[fig:SIAM1\]](SIAM1){width="90.00000%"}
As an example, we present in Fig. \[fig:SIAM1\] the calculated $\Sigma(i\omega)$ (left panel) and the corresponding $\Im m\Sigma(\omega+i0^+)$ from the Padé approximation (right panel) for $U/\Delta_0=1$. The dashed and dotted lines represent results from NRG and second order perturbation theory, respectively. The inset in the right panel shows an enlarged view of the region around $\omega=0$. Apparently, the Padé approximation to the fRG provides reliable results, in particular around $\omega=0$, and recovers the perturbation theory as expected for such small value of $U$. Discrepancies to NRG for large $\omega$ can be traced to broadening effects in the NRG.
The true test for the method however is a comparison of the self-energy to NRG results for larger values of $U$ and in particular the behavior on low-energy scales. Such a comparison is presented in Fig. \[fig:SIAM2\] for $U/\Delta_0=1$, $5$, and $10$. For $U/\Delta_0=5$ the fRG results are still in excellent agreement with the NRG data while second order perturbation theory already deviates. For $U=10\Delta_0$ we observe a significant dependency of the general structures and also of the slope at $\omega\to0$ on details of the discretization mesh. It seems necessary to have an extremely fine resolution around $\omega\to0$ and a sufficient resolution around $\omega\approx U/2$. With an exponentially vanishing low-energy scale these constraints are hard to fulfill with both linear and logarithmic meshes keeping the number of flow equations to a manageable size. The bottom left pannel contains the best fRG data we were able to obtain so far. A more extended discussion of this issue will be presented in a forthcoming publication.
![$\Sigma(i\omega)$ from fRG (circles) for $U/\Delta_0=1$, $5$ and $10$ compared to NRG (dashed line) and second order perturbation theory (dotted line).\
Bottom right: Effective mass $m^\ast$ from fRG (circles), NRG (squares) and second order perturbation theory (triangles).\[fig:SIAM2\]](SIAM2){width="90.00000%"}
In addition to the self-energy the effective mass $m^\ast$ obtained from $$m^\ast=1+\lim_{\omega\searrow0}\frac{\Im m\Sigma(i\omega)}{\omega}$$ as a function of $U$ is depicted, in the bottom right panel of Fig. \[fig:SIAM2\]. This quantity is directly related to the Kondo temperature [@hewson]. Compared to perturbation theory the effective mass is much closer to the very accurate values determined from the NRG.
The evolution of the spectral function of the $d$-level, $\rho_d(\omega)=-\frac{1}{\pi}\Im mG_d(\omega+i 0^+)$ for the above values of $U$ is collected in Fig. \[fig:SIAM3\]. The inset shows a comparison to NRG and perturbation theory for the region around $\omega=0$ at $U/\Delta_0=10$. One can see very nicely the development of the sharp resonance in the spectrum and the formation of the Hubbard bands with increasing $U$.
![The evolution of the fRG spectral function for the $d$-level for the values of $U$ in Fig. \[fig:SIAM2\]. The inset shows a comparison to NRG for $U/\Delta_0=10$ for the region around $\omega=0$.\[fig:SIAM3\]](SIAM3){width="90.00000%"}
Due to the insufficient resolution at larger $\omega$ in the calculation for $U/\Delta_0=10$, the Hubbard bands come out too broad here. We believe that an improved discretization of the energy mesh in the solution of the flow equations will remedy that particular problem. The important region around $\omega=0$ on the other hand is captured rather well by the fRG. It is in particular noteworthy that the functional form at low frequencies (see the inset) apparently does not follow a Lorentzian but rather, like the NRG, the more complex scaling form with logarithmic tails predicted by Logan [*et al.*]{} [@logan].
Summary and outlook {#sec:summary}
===================
We have presented an application of the functional renormalization group technique to solve for the dynamics of zero-dimensional interacting quantum problems. As particular examples, we discussed the anharmonic oscillator and the single impurity Anderson model. In both cases the fRG proved to be a substantial improvement over conventional low-order perturbation theory and rather close to the very accurate results obtained numerically.
We also investigated the differences between conventional fRG and a modification suggested by Katanin [@katanin], which should improve the accuracy of the method further. That this is indeed true was shown directly for the anharmonic oscillator. For the single impurity Anderson model it was actually necessary to use this modified version to obtain sensible results for values $U/\Delta_0>3$.
An important aspect of the fRG is that an extension of the SIAM to more complex systems, like e.g. orbital degrees of freedom or systems of coupled impurities is straightforward. This is in principle also true for Wilson’s NRG. In the latter, however, the exponentially increasing Hilbert space renders a practical application quickly impossible. For the fRG, on the other hand, the major modification will be an increase of the number of equations, which means that the numerical effort will increase at most with a power-law. This feature makes the fRG a possible method to study features of complex impurity systems and in particular a potential “impurity solver” for mean-field theories of interacting lattice models like the Hubbard model in the framework of the dynamical mean-field theory [@rmp_geo] or the dynamical cluster approximation [@rmp_tm]. Furthermore, as it becomes clear from the anharmonic oscillator, the fRG is of equal complexity for bosonic and fermionic systems. This makes opens the possibility to study combinations of such degrees of freedom in impurity models. A question which can be addressed using the fRG is the influence of phonons or magnetic fluctuations on low energy scales. Within the dynamical mean-field theory the metal-insulator transition in the presence of phonons (Kondo volume collapse [@allen]) as well as the problem of non-Fermi liquid formation in for example CeCu$_{6-x}$Au$_x$ [@qimiao] can be studied.
We thank S. Dusuel, W. Metzner, M. Salmhofer, H. Schoeller, and G. Uhrig for useful discussions. This work was supported by the SFB 602 of the Deutsche Forschungsgemeinschaft (R.H. and K.S.). V.M. is grateful to the Bundesministerium für Bildung und Forschung for support.
Depending on the specific parameterization of the 2-particle-vertex flow equations for the relevant frequency dependent parts of the vertex and for the self-energy can be derived by using the spin conservation on the vertex when the sum over spins in Eq. (\[gamma2fl\]) is performed. For the form of the the vertex given in Eq. (\[gamma2:SIAMa\]) the corresponding flow equation for the self-energy is given by Eq. (\[Self:SIAM\]), and the equation for the frequency dependent part of the vertex $\mathcal U^\Lambda(i\nu_1',i\nu_2';\nu_1,i\nu_2)$ with $\nu_2=\nu_1'+\nu_2'-\nu_1$ reads $$\begin{aligned}
\hspace{-2cm}
\label{gamma2:SIAM}
\DDL \gw\begin{array}[t]{l}\!\!\!\DS(i\nu_1',i\nu_2';i\nu_1,i\nu_2)
=-\frac{1}{2\pi}\int_{-\infty}^{\infty}d\nu
\DS\Biggl[\mathcal P^{\Lambda}(i\nu,i\nu_1+i\nu_2-i\nu)
\\[5mm]
\Bigl(-\gw(i\nu,i\nu_1+i\nu_2-i\nu;i\nu_1,i\nu_2)
\gw(i\nu_2',i\nu_1';i\nu_1+i\nu_2-i\nu,i\nu)\\[5mm]
-\gw(i\nu_1+i\nu_2-i\nu,i\nu;i\nu_1,i\nu_2)
\gw(i\nu_1',i\nu_2';i\nu_1+i\nu_2-i\nu,i\nu)\Bigr)\\[5mm]
\DS+\biggl\{\mathcal P^\Lambda(i\nu,-i\nu_1+i\nu_1'+i\nu)
\\[5mm]
\Bigl(2\, \gw(i\nu_1',i\nu;i\nu_1,-i\nu_1+i\nu_1'+i\nu)
\gw(i\nu_2',-i\nu_1+i\nu_1'+i\nu;i\nu_2,i\nu)\\[5mm]
-
\gw(i\nu_1',i\nu;i\nu_1,-i\nu_1+i\nu_1'+i\nu)
\gw(-i\nu_1+i\nu_1'+i\nu,i\nu_2';i\nu_2,i\nu)\\[5mm]
-
\gw(i\nu,i\nu_1';i\nu_1,-i\nu_1+i\nu_1'+i\nu)
\gw(i\nu_2',-i\nu_1+i\nu_1'+i\nu;i\nu_2,i\nu)\Bigr)\\[5mm]
+(1'\leftrightarrow2'; 1\leftrightarrow2)\biggr\}\\[5mm]
\DS-\biggl\{\mathcal P^\Lambda(i\nu,-i\nu_1+i\nu_2'+i\nu)
\\[5mm]
\gw(i\nu,i\nu_2';i\nu_1,-i\nu_1+i\nu_2'+i\nu)
\gw(-i\nu_1+i\nu_2'+i\nu,i\nu_1';i\nu_2,i\nu)\\[5mm]
+(1'\leftrightarrow2'; 1\leftrightarrow2)\biggr\}
\Biggr] \; . \end{array} \end{aligned}$$ Again $\mathcal P$ stands either for ${\mathcal P}_{\rm con}$ or ${\mathcal P}_{\rm mod}$.
Two other possible parameterizations of the 2-particle vertex are $$\begin{aligned}
\hspace{-2cm}
\g(\xi_1',\xi_2';\xi_1,\xi_2)&=&
\begin{array}[t]{l}\D{\nu_1+\nu_2-\nu_1'-\nu_2'}
\DS \Bigl\{\delta_{\sigma_1,\sigma_2}\delta_{\sigma_1',\sigma_2'}\delta_{\sigma_1,\sigma_1'}\ \gp(i\nu_1',i\nu_2';i\nu_1,i\nu_2)\\[5mm]
\hspace{-2cm} +\DS \delta_{\sigma_1,-\sigma_2}\delta_{\sigma_1',-\sigma_2'}\ \Bigl(\delta_{\sigma_1,\sigma_1'}\ga(i\nu_1',i\nu_2';i\nu_1,i\nu_2)-
\delta_{\sigma_1,-\sigma_1'}\ga(i\nu_2',i\nu_1';i\nu_1,i\nu_2)\Bigr)
\Bigr\}.\end{array}\end{aligned}$$\
and $$\hspace{-2cm}
\g(\xi_1',\xi_2';\xi_1,\xi_2)=\D{\nu_1+\nu_2-\nu_1'-\nu_2'}\Bigl\{
\begin{array}[t]{l}\DS S_{\sigma_1', \sigma_2'; \sigma_1, \sigma_2}\ \gs(i\nu_1',i\nu_2';i\nu_1,i\nu_2)\\[5mm]
+\DS T_{\sigma_1', \sigma_2'; \sigma_1, \sigma_2}\ \gt(i\nu_1',i\nu_2';i\nu_1,i\nu_2)\Bigr\}\end{array}$$ with $$S_{\sigma_1', \sigma_2'; \sigma_1, \sigma_2}=\frac{1}{2}\left(\delta_{\sigma_1, \sigma_1'}\delta_{\sigma_2, \sigma_2'}-\delta_{\sigma_1, \sigma_2'}\delta_{\sigma_2, \sigma_2'}\right);\
T_{\sigma_1', \sigma_2'; \sigma_1,
\sigma_2}=\frac{1}{2}\left(\delta_{\sigma_1,
\sigma_1'}\delta_{\sigma_2, \sigma_2'}+\delta_{\sigma_1,
\sigma_2'}\delta_{\sigma_2, \sigma_2'}\right) \; .$$ Both these parameterizations lead to different set of flow equations for the self-energy and to two sets of flow equations for the frequency dependent functions $\gp$ and $\ga$ for the first and $\gs$ and $\gt$ for the second (compared to one set for the parameterization Eq. (\[gamma2:SIAMa\])) implying an increased numerical effort. They are usefull to further investigate the processes occurring during the integration of the flow equations. This way we are able to distinguish the behavior of different channels of the interaction.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We have calculated the effects of structural distortions of armchair carbon nanotubes on their electrical transport properties. We found that the bending of the nanotubes decreases their transmission function in certain energy ranges and leads to an increased electrical resistance. Electronic structure calculations show that these energy ranges contain localized states with significant $\sigma$-$\pi$ hybridization resulting from the increased curvature produced by bending. Our calculations of the contact resistance show that the large contact resistances observed for SWNTs are likely due to the weak coupling of the NT to the metal in side bonded NT-metal configurations.\
address:
- '$^*$ Centre de recherche en calcul appliqué (CERCA), 5160 Boul. Décarie, bureau 400, Montréal, (Québec) Canada H3X-2H9.'
- '$^{**}$ Centre de recherche mathématiques (CRM), Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, (Québec) Canada H3C-3J7.'
- '$^\dagger$ Département de chimie, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, (Québec) Canada H3C-3J7.'
- '$^\ddagger$ IBM Research Division, T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.'
author:
- 'Alain Rochefort$^*$, Frédéric Lesage$^{**}$, Dennis R. Salahub$^{*,\dagger}$, Phaedon Avouris$^\ddagger$'
title: Conductance of Distorted Carbon Nanotubes
---
Carbon nanotubes (NTs) can be metallic or semiconducting. They have high mechanical strength and good thermal conductivity [@dresselhaus], properties that make them potential building blocks of a new, carbon-based, nanoelectronic technology [@tans; @bockrath; @collins; @martel]. Conduction in defect-free NTs, especially at low temperatures, can be ballistic, thus involving little energy dissipation within the NT [@white]. Furthermore, NTs are expected to behave like quasi one-dimensional systems (Q1D) with quantized electrical resistance, which, for metallic armchair nanotubes at low bias should be about 6 k$\Omega$ ($h/4e^2$). The experimentally observed behavior is, however, quite different. The contact resistance of single-wall nanotubes (SWNTs) with metal electrodes is generally quite high. Furthermore, at low temperatures a localization of the wavefunction in the nanotube segment contained between the metal electrodes is observed that leads to Coulomb blockade phenomena [@bezryadin]. The latter observation suggests that a barrier or bad-gap develops along the NT near its contact with the metal. In an effort to understand the origin of these discrepancies we have used Green’s function techniques to calculate the effect of the modification of the NTs by bending on their electronic structure and electric transport properties. We also investigated the effects of the strength of the NT-metal pad interaction on the value of the contact resistance.\
Most discussions on the electronic structure of NTs assume perfect cylindrical symmetry. The introduction of point defects such as vacancies [@chico] or disorder [@white; @anan] has been shown to lead to significant modification of their electrical properties. Here we focus on the effects of structural (axial) distortions on the transport properties of armchair NTs. AFM experiments [@hertel1] and molecular mechanics simulations [@hertel2] have shown that the van der Waals forces between NTs and the substrate on which they are placed can lead to a significant deformation of their structure. To maximize their adhesion energy the NTs tend to follow the topography of the substrate [@hertel1; @hertel2]. Thus, for example, NTs bend to follow the curvature of the metal electrodes on which they are deposited. When the strain due to bending exceeds a certain limit, kinks develop in the nanotube structure [@hertel1; @iijima; @falvo]. It is important to understand how these NT deformations affect the electrical transport properties of the NTs. Could they be responsible for the low temperature localization observed ? [@bezryadin] Early theoretical work on this issue was based on tight-binding model involving only the $\pi$-electrons of the NTs and accounted for the electronic structure changes induced by bending through the changes in $\pi$-orbital overlap at neighboring sites. This study concluded that bending distortions would have a negligible effect on the electrical properties of the NTs [@kane]. The applicability of this approach is limited to weak distortions. Experiments, however, show that strong deformations and kink formation are common. Under such conditions, bending-induced $\sigma$-$\pi$ mixing, which was not considered before, becomes very important in strongly bent NTs [@rochefort]. In this work, the NT electronic structure is computed using the extended Hückel method (EHM) [@yaehmop] that includes both $s$ and $p$ valence electrons. We have previously [@rochefort2] shown that EHM calculations on an armchair $(6,6)$ NT model (96 [Å]{} long) reproduce the electronic properties obtained with more sophisticated [*ab-initio*]{} and band structure computations on NTs. The approach we used in the computation of the electrical properties is similar to that of Datta [*et al.*]{} [@datta; @dattamol].\
The conductance through a molecule or an NT cannot be easily computed; even if the electronic structure of the free molecule or NT is known, the effect of the contacts on it can be substantial [@lang] and needs to be taken into account. Typically, there will be two (or more) leads connected to the NT. We model the measurement system as shown in Figure 1a. The leads are macroscopic gold pads that are coupled to the ends of the NT through matrix elements between the Au surface atoms and the end carbon atoms of the NT. In most experiments to date the NTs are laid on top of the metal pads. As we discussed above, the NTs then tend to bend around the pads. Such bending deformations are modelled in our calculations by introducing a single bend placed at the center of the tube.\
The electrical transport properties of a system can be described in terms of the retarded Green’s function [@datta; @economou]. To evaluate the conductance of the NT we need to compute the transmission function, $T(E)$, from one contact to the other. This can be done following the Landauer-Büttiker formalism as described in [@datta]. The key element of this approach lies in the treatment of the infinite leads which are here described by self-energies. We can write the Green’s function in the form of block matrices separating explicitly the molecular Hamiltonian. After some simplification we obtain: $$G_{NT}=\big[ ES_{NT}-H_{NT}-\Sigma_1-\Sigma_2 \big]^{-1}$$ where $S_{NT}$ and $H_{NT}$ are the overlap and the Hamiltonian matrices, respectively, and $\Sigma_{1,2}$ are self-energy terms that describe the effect of the leads. They have the form $\tau_{i}^\dagger g_{i} \tau_{i}$ with $g_{i}$ the Green’s function of the individual leads [@dattamol; @papa] and $\tau_{i}$ is a matrix describing the interaction between the NT and the leads. The Hamiltonian and overlap matrices are determined using EHM for the system Gold-NT-Gold. The transmission function, $T(E)$, that is obtained from this Green’s function is given by [@datta]:
$$T(E)=T_{21}=Tr [\Gamma_2 G_{NT} \Gamma_1 G_{NT}^\dagger ].$$
In this formula, the matrices have the form:
$$\Gamma_{1,2}=i(\Sigma_{1,2}-\Sigma^\dagger_{1,2}).$$
The summation over all conduction channels in the molecule allows the evaluation of the resistance at the Fermi energy, $R=h/(2e^2T(E_F))$. Transport in the presence of an applied potential is also computed. The differential conductance is computed in this case using the approximation [@datta]: $$\kappa (V)=\frac{\partial I}{\partial V} \approx \frac{2e^2}{h}
[\eta T(\mu_1)+(1-\eta )T(\mu_2)]$$ with $\eta\in [0,1]$ describing the equilibration of the nanotube energy levels with respect to the reservoirs [@dattamol]. As a reference, we use the $E_F$ obtained from EHM for individual nanotubes as the zero of energy. The NT model used in our calculations is a $(6,6)$ carbon nanotube segment containing 948 carbon atoms. The bond distance between carbon atoms in non-deformed regions of the NT is fixed to that in graphite 1.42 [Å]{}, leading to a tube length of 96 [Å]{}. The building of deformed NTs using molecular mechanics minimization schemes [@mm3; @tinker] has been described in detail elsewhere [@rochefort]. The structures of the bent NTs are shown in Figure 1b. The metallic contacts consist each of 22 gold atoms in a (111) crystalline arrangement. The height of the NT over the gold layer is 1.0 [Å]{}, where the Au-C bond distances vary from 1.1 to 1.6 [Å]{}.
[**a**]{}
{width="6cm"}
[**b**]{}
{width="8cm"}
In Figure 2 we present the computed transmission function $T(E)$ for the bent tubes (note that $T(E)$ represents the sum of the transmission probabilities over all contributing NT conduction channels). The upper-right of Figure 2 shows the raw transmission results obtained for the straight NT. The fast oscillations of $T(E)$ are due to the discrete energy levels of the finite segment of the carbon nanotubes used. For clarity, we will use smoothed curves in the description of the results. At $E_F$, $T(E)$ is about 1.2, leading to a resistance ($\approx$ 11 k$\Omega$ ) higher than expected for ballistic transport ($\approx$ 6 k$\Omega$ for $T(E)$ = 2.0). This reduction in transmission is due to the contribution from the contact resistance. The increasing $T(E)$ at higher binding energies is due to the opening of new conduction channels. The asymmmetry in the transmission T(E) is a function of the NT-pad coupling (C-Au distance). A longer NT-Au distance increases the T(E) above $E_F$, while it decreases it below $E_F$, and vise versa. Since the NT-pad geometry is kept fixed in all computations, this behavior does not influence the effects induced by NT bending.\
According to our calculation the contact resistance at $E_F$ is only about 5 k$\Omega$, much smaller than the $\approx$ 1 M$\Omega$ resistance typically observed in experiments on single-wall NTs [@tans; @martel]. The dependence of $T(E)$ and contact resistance at $E_F$ on the Au-NT distance is shown in the upper-left of Figure 2. We see that $T(E_F)$ remains nearly constant between 1-2 [Å]{}, then decreases exponentially. For distances appropriate for van der Waals bonding ($\geq 3$ [Å]{}) the contact resistance is already in the M$\Omega$ range. The above findings suggest that the high NT contact resistance observed experimentally may, in addition to experimental factors such as the presence of asperities at the metal-NT interface, be due to the topology of the contact. In most experiments, the NT is laid on top of the metal pad. The NT is at nearly the van der Waals distance away from the metal surface, and given that transport in the NT involves high $k$-states which decay rapidly perpendicular to the tube axis, the coupling between NT and metal is expected to be weak [@note]. Direct chemical bonding between metal and the NT, or interaction of the metal with the NT cap states [@kim] should lead to stronger coupling. In this respect, it has been found [@bachtold] that high energy electron irradiation of the contacts leads to a drastic reduction of the resistance. Since the irradiation is capable of breaking NT C-C bonds it may be that the resulting dangling bonds lead to a stronger metal-NT coupling.
![Transmission function $T(E)$ for the different bent nanotubes. The upper-right inset gives the raw data obtained from the computation for the straight NT along with the smoothed curve. The upper-left shows the variation of the transmission function $T(E_F)$ with the nanotube-pad distance (R(NT-pad)). In this last figure, one gold-NT distance is fixed at 1.0 [Å]{} while the other is varied.](FIG2.eps){width="9cm"}
The strongest modification of $T(E)$ as a result of bending is observed at around $E$=-0.6 eV where a transmission dip appears. This dip is strongest in the $60^{\circ}$ bent NT. Furthermore, its transmission function at higher binding energies (BE) is lower than those of the 0$^{\circ}$-45$^{\circ}$ bent NTs, indicating that the transmission of higher conduction channels is also decreased. The nature of the dip at about -0.6 eV can be understood by examining the local density-of-states (LDOS) of bent tubes shown in Figure 3 [@rochefort]. A change (increase) in the LDOS is seen in the same energy region (0.5-0.8 eV below $E_F$) as the transmission dip. This change is essentially localized in the vicinity of the deformed region. The new states result from the mixing of $\sigma$ and $\pi$ levels, have a more localized character than pure $\pi$ states leading to a reduction of $T(E)$. As Figure 3 shows, the change in transmission with bending angle is not gradual; the transmission of the 30$^{\circ}$ and 45$^{\circ}$ models is only slightly different from that of the straight tube. Apparently, large changes in DOS and $T(E)$ require the formation of a kink in the NT structure, as is the case in the 60$^{\circ}$ and 90$^{\circ}$ bent NTs.
{width="8cm"}
Once the transmission function is computed, the determination of the differential conductance and resistance is straigtforward. Figure 4 shows the results for two extreme cases of equilibration of the Fermi levels. The first is when $\eta=0$ (Figure 4a), and the symmetric case $\eta=0.5$ (Figure 4b). When $\eta$=0.0, the Fermi level of the NT follows exactly the applied voltage on one gold pad and the conductance spectrum is directly proportional to $T(E)$. As expected, there is no large difference between the 0$^{\circ}$, 30$^{\circ}$ and 45$^{\circ}$ models, while the 60 and 90$^{\circ}$ models show the dip structure at around 0.6 V. The non-linear resistance (NLR) spectra show clearly a sharp increase by almost an order of magnitude at 0.6 V. These features are also observed when $\eta$=0.5, where the Fermi level of the NT is floating at half the voltage applied between the two gold pads. The dip at around 1.2 V in the conductance spectra is now broader, and the NLR of the 60$^{\circ}$ bent tube increases by about a factor of 4 from the computed resistance of the straight tube. These results suggest that there exists a critical bending angle (between 45$^{\circ}$ and 60$^{\circ}$) above which the conduction in armchair carbon nanotubes is drastically altered [@comment].
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{width="9cm"}
In conclusion, we have calculated the effects of structural distortions of armchair carbon nanotubes on their electrical transport properties. We found that bending of the nanotubes decreases their transmission function and leads to an increased electrical resistance. The effect is particularly strong at bending angles higher than 45$^{\circ}$ degrees when the strain is strong enough to lead to kinks in the nanotube structure. The electronic structure calculations show that the reduction in $T(E)$ is correlated with the presence at the same energy of localized states with significant $\sigma$-$\pi$ hybridization due to the increased curvature produced by bending. Resistance peaks near $E_F$ are the likely cause for the experimentally observed low temperature localization in carbon NTs bent over metal electrodes [@bezryadin]. Our calculations of the resistance (including the contact resistance) of a perfect NT give a value close to $h/2.4e^2$ instead of $h/4e^2$. This increase in resistance is solely due to the finite transmission of the contacts. The much larger contact resistances observed in many experiments on SWNTs are likely due to the weaker coupling of the NT to the metal when the NT is simply placed on top of the metal electrodes. We predict that NTs end-bonded to metal pads will have contact resistances of only a few k$\Omega$. Such low contact resistances will greatly improve the performance of NT-based devices and unmask the Q1D transport properties of NTs.
M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, [*Science of Fullerenes and Carbon Nanotubes*]{} (Academic, San Diego, 1996).
S.J. Tans, A.R.M. Verschueren, and C. Dekker, Nature [**393**]{}, 49 (1998). M. Bockrath, D.H. Cobden, P.L. McEuen, N.G. Chopra, A. Zettl, A. Thess, and R. E. Smalley Science [**275**]{}, 1922 (1997). P.G. Collins, A. Zettl, H. Bando, A. Thess, and R.E. Smalley, Science [**278**]{}, 100 (1997). R. Martel, T. Schmidt, H.R. Shea, T. Hertel, and Ph. Avouris, Appl. Phys. Lett. [**73**]{}, 2447 (1998). C.T. White, and T.N. Todorov, Nature [**393**]{}, 240 (1998). A. Bezryadin, A.R.M. Verschueren, S.J. Tans, and C. Dekker, Phys. Rev. Lett. [**80**]{}, 4036 (1998).
L. Chico, L.X. Benedict, S.G. Louie, and M.L. Cohen, Phys. Rev. B [**54**]{}, 2600 (1996). M.P. Anantram, and T.R. Govidan, Phys. Rev. B [**58**]{}, 4882 (1998). T. Hertel, R. Martel, and Ph. Avouris, J. Phys. Chem. B [**102**]{}, 910 (1998). T. Hertel, R.E. Walkup, and Ph. Avouris, Phys. Rev. B [**58**]{}, 13870 (1998). S. Iijima, C. Brabec, A. Maiti, and J. Bernholc, J. Chem. Phys. [**104**]{} 2089 (1996). M.R. Falvo, G.J. Clary, R.M. Taylor II, V. Chi, F.P. Brooks Jr, S. Washburn, and R. Superfine Nature [**389**]{}, 582 (1997). C.L. Kane, and E.J. Mele, Phys. Rev. Lett. [**78**]{}, 1932 (1997). A. Rochefort, D.R. Salahub, and Ph. Avouris, Chem. Phys. Lett. [**297**]{}, 45 (1998). A. Rochefort, D.R. Salahub, and Ph. Avouris, J. Phys. Chem. [**103**]{}, 641 (1999). Landrum, G. [*YAeHMOP*]{} (Yet Another Extended Hückel Molecular Orbital Package), Cornell University, Ithaca, NY, 1995. S. Datta, [*Electronic Transport in Mesoscopic Systems*]{}, (Cambridge University Press, Cambridge, U.K., 1995). W. Tian, S. Datta, S. Hong, R. Reifenberger, J.I. Henderson, and C.P. Kubiak, J. Chem. Phys. [**109**]{}, 2874 (1998). N.D. Lang, and Ph. Avouris, Phys. Rev. Lett. [**81**]{}, 3515 (1998). E.N. Economou, [*Green’s Functions in Quantum Physics*]{}, (Springer-Verlag, New York, 1983). D.A. Papaconstantopoulos, [*Handbook of the Band Structure of Elemental Solids*]{}, (Plenum Press, NY 1986). N.L. Allinger, Y.H. Yuh, and J.-H. Lii, J. Am. Chem. Soc. [**111**]{}, 8551 (1989). Y. Kong, and J.W. Ponder, J. Chem. Phys. [**107**]{}, 481 (1997).
The coupling strength depends on the amount of charge transfer, i.e. the difference in the work functions of the NT and the metal, and the area of the contact. P. Kim, T. W. Odom, J. L. Huang, and C. B. Lieber, Phys. Rev. Lett. [**82**]{}, 1225 (1999).
A. Bachtold, M. Henny, C. Terrier, C. Strunk, C. Schönenberger, J.-P. Salvetat, J.-M. Bonard, and L. Forró , Appl. Phys. Lett. [**73**]{}, 274 (1998). The assumption that the potential drops linearly across the NT is particularly drastic for highly bent models. In reality, in the highly deformed region, the voltage would drop in the vicinity of defects. This voltage drop near defects would decrease the transmission probability of the propagating electrons, and consequently emphasize the magnitude of the changes we have computed with our first assumption.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the tandem duplication distance between binary sequences and their roots. In other words, the quantity of interest is the number of tandem duplication operations of the form ${\boldsymbol{x}_{}} = {\boldsymbol{a}_{}} {\boldsymbol{b}_{}} {\boldsymbol{c}_{}} \to {\boldsymbol{y}_{}} = {\boldsymbol{a}_{}} {\boldsymbol{b}_{}} {\boldsymbol{b}_{}} {\boldsymbol{c}_{}}$, where ${\boldsymbol{x}_{}}$ and ${\boldsymbol{y}_{}}$ are sequences and ${\boldsymbol{a}_{}}$, ${\boldsymbol{b}_{}}$, and ${\boldsymbol{c}_{}}$ are their substrings, needed to generate a binary sequence of length $n$ starting from a square-free sequence from the set $\{0,1,01,10,010,101\}$. This problem is a restricted case of finding the duplication/deduplication distance between two sequences, defined as the minimum number of duplication and deduplication operations required to transform one sequence to the other. We consider both exact and approximate tandem duplications. For exact duplication, denoting the maximum distance to the root of a sequence of length $n$ by $f(n)$, we prove that $f(n)=\Theta(n)$. For the case of approximate duplication, where a $\beta$-fraction of symbols may be duplicated incorrectly, we show that the maximum distance has a sharp transition from linear in $n$ to logarithmic at $\beta=1/2$. We also study the duplication distance to the root for sequences with a given root and for special classes of sequences, namely, the de Bruijn sequences, the Thue-Morse sequence, and the Fibbonaci words. The problem is motivated by genomic tandem duplication mutations and the smallest number of tandem duplication events required to generate a given biological sequence.'
author:
- 'Noga Alon, Jehoshua Bruck, Farzad Farnoud, and Siddharth Jain, [^1][^2][^3][^4] [^5]'
bibliography:
- 'bib.bib'
title: |
Duplication Distance to the Root\
for Binary Sequences
---
Introduction
============
The genome of every organism is subject to mutations resulting from imperfect genome replication as well as environmental factors. These mutations include *tandem duplications*, which create *tandem repeats* by duplicating a substring and inserting it adjacent to the original (e.g., $A\underline{CG}T\to A\underline{CGCG}T$); and *point mutations* or *substitutions*, which substitute one base in the sequence by another (e.g., $A\underline{C}GT\to A\underline{T}GT$). Gaining a better understanding of mutations that modify genomes –thereby creating the variety needed for natural selection– is helpful in many fields including phylogenomics, systems biology, medicine, and bioinformatics.
One aspect of this task is the study of how genomic sequences are generated through mutations. In this paper, we focus on tandem duplication mutations and tandem repeats, which form about $3\%$ of the human genome [@lander2001initial], and study the minimum number of mutation events that can create a given sequence. More specifically, we define distance measures between pairs of sequences based on the number of exact or approximate tandem duplications that are needed to transform one sequence to the other. We then study the distances between sequences of length $n\in\mathbb N$ and their roots, i.e., the shortest sequences from which they can be obtained via these operations. Formally, a (*tandem*) *repeat of length $h$* in a sequence is two identical adjacent blocks, each consisting of $h$ consecutive elements. For example, the sequence $12\underline{134134}51$ contains the repeat $134134$ of length $3$. A repeat of length $h$ may be created through a (*tandem*) *duplication of length $h$*, e.g., $1213451{\,{\xrightarrow{d}}\allowbreak\,}1213413451$, where ${\,{\xrightarrow{d}}\allowbreak\,}$ denotes a duplication operation. On the other hand, a repeat may be removed through a (*tandem*) *deduplication of length $h$*, i.e., by removing one of the two adjacent identical blocks, e.g., $1213413451{\,{\xrightarrow{dd}}\allowbreak\,}1213451$.
The *duplication/deduplication distance* between two sequences ${\boldsymbol{x}_{}}$ and ${\boldsymbol{y}_{}}$ is the smallest number of duplications and deduplications that can turn ${\boldsymbol{x}_{}}$ into ${\boldsymbol{y}_{}}$ (to denote sequences we use bold symbols). We set the distance to infinity if the task is not possible, for example, if ${\boldsymbol{x}_{}}=1$ and ${\boldsymbol{y}_{}} = 0$.
For two sequences ${\boldsymbol{x}_{}}$ and ${\boldsymbol{y}_{}}$, if ${\boldsymbol{y}_{}}$ can be obtained from ${\boldsymbol{x}_{}}$ through duplications, we say that ${\boldsymbol{x}_{}}$ is an *ancestor* of ${\boldsymbol{y}_{}}$ and that ${\boldsymbol{y}_{}}$ is a *descendant* of ${\boldsymbol{x}_{}}$. An ancestor ${\boldsymbol{x}_{}}$ of ${\boldsymbol{y}_{}}$ is a *root* of ${\boldsymbol{y}_{}}$, denoted ${\boldsymbol{x}_{}} = {\operatorname{root}}({\boldsymbol{y}_{}})$, if it is *square-free*, i.e., it does not contain a repeat. We define the *duplication distance* between two sequences as the minimum number of duplications required to convert the shorter sequence to the longer one.[^6] This distance is finite if and only if one sequence is the ancestor of the other. This paper is focused on finding bounds on the duplication distance of sequences to their roots. From an evolutionary point of view, the duplication distance between a sequence and its root is of interest since it corresponds to a likely path through which a root may have evolved into a sequence present in the genome of an organism.
Our attention here is limited to binary sequences for the sake of simplicity, since for the binary alphabet, the root of every sequence is unique and belongs to the set $\{0,1,01,10,010,101\}$. Specifically, the roots of $0^n$ and $1^n$, $n\in \mathbb N$, are $0$ and $1$, respectively. For every other binary sequence ${\boldsymbol{s}_{}}$ of length $n$, the root of ${\boldsymbol{s}_{}}$ is the sequence in the set $\{01,10,010,101\}$ that starts and ends with the same symbols as ${\boldsymbol{s}_{}}$. For example, the root of ${\boldsymbol{s}_{}}=1001011$ is $101$ since $$101{\,{\xrightarrow{d}}\allowbreak\,}\underline{1010}1{\,{\xrightarrow{d}}\allowbreak\,}1010\underline{11}{\,{\xrightarrow{d}}\allowbreak\,}1\underline{00}1011={\boldsymbol{s}_{}}.$$ A *run* in a sequence is a maximal substring consisting of one or more copies of a single symbol. Through duplication, we can generate every binary sequence from its root by first creating the correct number of runs of appropriate symbols. For example, since ${\boldsymbol{s}_{}}=1001011$ has $5$ runs, the first being a run of the symbol $1$, we first generate $10101$ through duplication. It is not difficult to see that this is always possible. The next step is then to extend each run so that it has the appropriate length.
In the proofs in the paper, it is often helpful to think of the distance to the root in terms of converting a sequence to its root via a sequence of deduplications, e.g. the sequence ${\boldsymbol{s}_{}}$ above can be *deduplicated to* its root as $${\boldsymbol{s}_{}} =1\underline{00}1011{\,{\xrightarrow{dd}}\allowbreak\,}1010\underline{11}{\,{\xrightarrow{dd}}\allowbreak\,}\underline{1010}1{\,{\xrightarrow{dd}}\allowbreak\,}101={\operatorname{root}}({\boldsymbol{s}_{}}).$$
We note that a celebrated result by Thue from 1906 [@thue1906] states that for alphabets of size $\ge3$, there is an infinite square-free sequence. Thus, in contrast to the binary alphabet, the set of roots for such alphabets is infinite since each substring of Thue’s sequence is square-free.
For a binary sequence ${\boldsymbol{s}_{}}$, let $f({\boldsymbol{s}_{}})$ denote the duplication distance between ${\boldsymbol{s}_{}}$ and its root and let $f(n)$ be the maximum of $f({\boldsymbol{s}_{}})$ for all sequences ${\boldsymbol{s}_{}}$ of length $n$. Table \[tab:f(n)\], which was obtained through computer search, shows the values of $f(n)$ for $1\le n\le32$.
In this paper, we provide bounds on $f({\boldsymbol{s}_{}})$ and on $f(n)$. We also consider a variation of the duplication distance, referred to as the *approximate-duplication distance*, where the duplication process is imprecise and so the inserted block is not necessarily an exact copy. Specifically, the *$\beta$-approximate-duplication distance* between two sequences ${\boldsymbol{x}_{}}$ and ${\boldsymbol{y}_{}}$ is the smallest number of duplications that can turn the shorter sequence into the longer one, where each duplication may produce a block that differs from the original in at most a $\beta$-fraction of positions. This distance between ${\boldsymbol{s}_{}}$ and any of its roots is denoted by $f_\beta({\boldsymbol{s}_{}})$ and the maximum of $f_\beta({\boldsymbol{s}_{}})$ over all sequences ${\boldsymbol{s}_{}}$ of length $n$ is denoted by $f_\beta(n)$. We provide bounds on $f_\beta(n)$ and in particular show that there is a sharp transition in the behavior of $f_\beta$ at $\beta=1/2$.
$n$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-------- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
$f(n)$ 0 1 2 2 3 4 4 5 6 6 7 7 8 8 9 9
$n$ 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
$f(n)$ 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15
: $f(n)$ for $1\le n\le32$\[tab:f(n)\].
Since each binary sequence has a unique root in the set $\left\{ 0,1,01,10,010,101\right\}$, the set of sequences can be partitioned based on their roots. In the paper, we also study the duplication distance to the root for sequences based on the part they belong to, that is, we consider $f_{{\boldsymbol{\sigma}_{}}}(n)$ for ${\boldsymbol{\sigma}_{}}\in\left\{ 0,1,01,10,010,101\right\}$, where $f_{{\boldsymbol{\sigma}_{}}}(n)=\max\left\{ f({\boldsymbol{s}_{}}):{\operatorname{root}}({\boldsymbol{s}_{}})=\sigma,|{\boldsymbol{s}_{}}|=n\right\}$.
The rest of the paper is structured as follows. In the next two subsections, we summarize the results of the paper and describe the related work. Then, in Section \[sec:genBounds\], we prove the bounds on $f(n)$ and discuss some variants, as well as special classes of sequences. In Section \[sec:Lsystems\], we discuss duplication distance for special class of sequence generating systems called Lindenmayer Systems. In Sections \[sec:mismatch\] and \[sec:stringSystems\], we study the approximate-duplication distance to the root and the duplication distance for different roots, respectively. Finally, several open problems and possible future directions are presented in Section \[sec:conc\].
Results
-------
In this subsection, we present the main results of the paper. The proofs, unless they are very short, are postponed to later sections.
Suppose the root of ${\boldsymbol{s}_{}}$ is ${\boldsymbol{\sigma}_{}}\in\left\{ 0,1,01,10,010,101\right\} $. It is easy to see that $$\log\frac{|{\boldsymbol{s}_{}}|}{|{\boldsymbol{\sigma}_{}}|}\le f({\boldsymbol{s}_{}})\le|{\boldsymbol{s}_{}}|.$$ While the above lower bound is tight in the sense that there exist ${\boldsymbol{\sigma}_{}}$ and ${\boldsymbol{s}_{}}$ that satisfy it with equality, e.g., ${\boldsymbol{s}_{}}=0^{2^k}$ and ${\boldsymbol{\sigma}_{}}=0$, we show there is a positive constant $c$ such that for most sequences of length $n$, the duplication distance to the root is bounded below by $cn$. We also improve the upper bound.
[thm]{}[thmbounds]{} \[thm:bounds\]The limit $\lim_{n\to\infty}f(n)/n$ exists and $$0.045\le \lim_{n\to\infty}\frac{f(n)}n\le\frac{2}{5}\ \cdot$$ Furthermore, for sufficiently large n, $f({\boldsymbol{s}_{}})\ge 0.045 n$ for all but an exponentially small fraction of sequences ${\boldsymbol{s}_{}}$ of length $n$; and $f(n)\le2n/5+15$.
Although the linear lower bound on the duplication distance to the root holds for almost all sequences, finding a specific family of sequences that satisfy it appears to be difficult. The next lemma and its corollary give the best known construction for a family with large distance to the root, namely, this family achieves distance $\Omega(n/\log n)$.
\[lem:SeqDepBound\]Consider a sequence ${\boldsymbol{s}_{}}$ and a positive integer $k\ge4$, and let $K({\boldsymbol{s}_{}})$ denote the number of distinct $k$-mers (sequences of length $k$) occurring in ${\boldsymbol{s}_{}}$. We have $$f({\boldsymbol{s}_{}})\ge\frac{K({\boldsymbol{s}_{}})}{k-1}\ \cdot$$
For two sequences ${\boldsymbol{x}_{}}={\boldsymbol{t}_{}}{\boldsymbol{u}_{}}{\boldsymbol{u}_{}}{\boldsymbol{v}_{}}$ and ${\boldsymbol{y}_{}}={\boldsymbol{t}_{}}{\boldsymbol{u}_{}}{\boldsymbol{v}_{}}$, we have $K({\boldsymbol{y}_{}})\ge K({\boldsymbol{x}_{}})-(k-1)$, since the only case in which a $k$-mer occurs in ${\boldsymbol{x}_{}}$ but not in ${\boldsymbol{y}_{}}$ is when the only instance of that $k$-mer intersects both copies of ${\boldsymbol{u}_{}}$ in ${\boldsymbol{x}_{}}$. There are at most $k-1$ $k$-substrings of ${\boldsymbol{x}_{}}$ that intersect both copies of ${\boldsymbol{u}_{}}$. Finally, no root contains a $k$-mer for $k\ge4$.
A *binary De Bruijn sequence* [@debruijn1946combinatorial] of order $k$ is a binary sequence of length $n=2^k$ that when viewed cyclically contains every possible binary sequence of length $k$ as a substring exactly once. For example, $0011$ and $00010111$ are De Bruijn sequences of order 2 and order 3, respectively. A binary De Bruijn sequence of order $k$ and length $n=2^k$ has precisely $n-k+1$ distinct $k$-mers. Hence, we have the following corollary.
For any binary De Bruijn sequence ${\boldsymbol{s}_{}}$ of order $k$ (which has length $n=2^{k}$), we have $$f({\boldsymbol{s}_{}})\ge\frac{n-\log_2 n}{\log_2 n}\ \cdot$$
It is worth noting that using the same technique as the proof of $f(n)=\Omega(n)$ in Theorem \[thm:bounds\], and the fact that there are at least $\frac{2^{n/2}}{n}$ De Bruijn sequences of length $n$ when $n$ is a power of two,[^7] one can show that the largest duplication distance for De Bruijn sequences grows linearly in their length.
A question arising from observing that $f(n)=\Theta(n)$ is that how does allowing mismatches in the duplication process affect the distance to the root. In particular, for what values of $\beta$, is $f_\beta(n)$ linear in $n$ and for what values is it logarithmic? The next theorem establishes that there is a sharp transition at $\beta=1/2$.
[thm]{}[thmbetalessthanhalf]{}\[thm:betalessthanhalf\] If $\beta<1/2$, then there exists a constant $c=c(\beta)>0$ such that $$f_{\beta}(n)\ge cn.$$ Furthermore, if $\beta>1/2$, for any constant $C>\left\lceil \frac{2\beta+1}{2\beta-1}\right\rceil ^{2}$ and sufficiently large $n$, $$f_{\beta}(n)\le C\ln n.$$
Finally, we establish that the limit of $\frac{f(n)}{n}$ is the same if we consider only sequences with root $10$ or only sequences with root $101$.
[thm]{}[thmdiffrentroots]{} The limits $\lim_{n}\frac{f_{10}(n)}{n}$ and $\lim_{n}\frac{f_{101}(n)}{n}$ exist and are equal to $\lim_n\frac{f(n)}{n}$.
Related Work {#sub:relwork}
------------
Tandem duplications and repeats in sequences have been studied from a variety of points of view. One of the most relevant to this work is the study of estimating the tandem duplication history of a given sequence, i.e., a sequence of duplication events that may have generated the sequence, see e.g., [@benson1999reconstructing; @tang2002zinc; @gascuel2005reconstructing]. While the afforementioned works study the problem from an algorithmic point of view, in this paper, we are focused on extremal distance values for binary sequences. Furthermore, [@tang2002zinc; @gascuel2005reconstructing] have a more restrictive duplication model than that of the present paper.
Another aspect, the study of the ability of duplication mutations to generate diversity, has been recently investigated from an information-theoretic point of view [@farnoud2016capacity; @jain2015capacity]. In particular, [@farnoud2016capacity] models sequences generated from a starting “seed” through different types of duplications as sequence systems and studies their *capacity* and *expressiveness*. The notion of capacity quantifies the ability of the systems to generate diverse families of sequences, and expressiveness is concerned with determining whether every sequence can be generated as a substring of another sequence, if not independently. The results in [@farnoud2016capacity; @jain2015capacity] include lower bounds on the capacity of tandem duplications and establishing that certain systems have nonzero capacity. The aforementioned works focus on the possibility of generating sequences and do not consider the number of duplication steps it takes to do so for any given sequence, which is the subject of the current paper.
Finally, we mention that the stochastic behavior of certain duplication systems has been studied in [@elishco2016capacity; @farnoud2015stochastic], and error-correcting codes for combating duplication errors have been introduced in [@jain2016duplication].
Bounds on $f(n)$\[sec:genBounds\]
=================================
The lower bound of Theorem \[thm:bounds\] is proved with the help of Theorem \[t21\], and its upper bound uses Theorem \[thm:ub\]. These theorems are stated next.
\[t21\]For $0<\alpha<1$, consider the set of the $\left\lfloor 2^{n\alpha}\right\rfloor $ sequences of length $n$ with the smallest duplication distance to the root and let $F_{\alpha}$ be the maximum duplication distance to the root for a sequence in this set. Then $$6n\sum_{f=1}^{F_{\alpha}}\binom{n+f}{f}\binom{2n+f}{f}\binom{2n+f+2}{f}2^{f}\ge2^{n\alpha}-1.\label{eq:lb2}$$
Before stating the proof, we present some background, definitions, and a useful claim, as well as a simpler but weaker result.
Recall that if the sequence ${\boldsymbol{s}_{}}=s_{1}s_{2}\dotsm s_{m}$ contains a repeat, then omitting one of the two blocks of this repeat to obtain a new sequence is called a deduplication. We also refer to the resulting sequence ${\boldsymbol{s}_{}}'$ as a deduplication of ${\boldsymbol{s}_{}}$, and write ${\boldsymbol{s}_{}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{}}'$. A *deduplication process* for a binary sequence ${\boldsymbol{s}_{}}$ is a sequence of sequences ${\boldsymbol{s}_{}}={\boldsymbol{s}_{0}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{1}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{2}}{\,{\xrightarrow{dd}}\allowbreak\,}\dotsm{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{f}}={\operatorname{root}}({\boldsymbol{s}_{}})$, where each ${\boldsymbol{s}_{i+1}}$ is a deduplication of ${\boldsymbol{s}_{i}}$ and the final sequence ${\boldsymbol{s}_{f}}$ is the (square-free) root of ${\boldsymbol{s}_{}}$. The *length* of the deduplication process above is $f$, that is, the number of deduplications in it. A deduplication of ${\boldsymbol{s}_{}}$ is an $(i,h)$-*step* if $i$ is the starting position of (the first block) of a repeat of length $h$ and one of the blocks of this repeat is omitted. For example, if ${\boldsymbol{s}_{}}=123\underline{134}13451$, a $(4,3)$-step produces ${\boldsymbol{s}_{}}'=12313451$. A deduplication process of length $f$ of a sequence ${\boldsymbol{s}_{}}$ can be described by a sequence of pairs $(i_{t},h_{t})_{t=1}^{f}$, where step number $t$ is an $(i_{t},h_{t})$-step. It is not difficult to check that knowing the final sequence in the process, and knowing all the pairs $(i_{t},h_{t})$ of deduplications in the process, in order, we can reconstruct the original sequence ${\boldsymbol{s}_{}}$. From the preceding discussion, each binary sequence ${\boldsymbol{s}_{}}$ can be encoded as the pair $\left({\boldsymbol{\sigma}_{}},(i_{t},h_{t})_{t=1}^{f({\boldsymbol{s}_{}})}\right),$ where ${\boldsymbol{\sigma}_{}}$ is the root of ${\boldsymbol{s}_{}}$ and $(i_{t},h_{t})_{t=1}^{f({\boldsymbol{s}_{}})}$ a deduplication process of ${\boldsymbol{s}_{}}$. Since there are only $6$ possibilities for ${\boldsymbol{\sigma}_{}}$, and less than $n^{2}$ possibilities for each pair $(i_{t},h_{t})$, if $F=f(n)$, then $$6\sum_{f=1}^{F}\left(n^{2}\right)^{f}\geq2^{n},\label{eq:n/logn}$$ which implies that $F=f(n)=\Omega(n/\log n)$.
In the aforementioned encoding, several deduplication processes may map to the same sequence. We improve upon (\[eq:n/logn\]) by defining deduplication processes of a special form that remove some of the redundancy, and by doing so, we obtain (\[eq:lb2\]), which will lead to the linear lower bound of Theorem \[thm:bounds\].
A deduplication process ${\boldsymbol{s}_{}}={\boldsymbol{s}_{0}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{1}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{2}}{\,{\xrightarrow{dd}}\allowbreak\,}\dotsm{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{f}}={\operatorname{root}}({\boldsymbol{s}_{}})$ of a sequence ${\boldsymbol{s}_{}}$, in which the steps are $(i_{1},h_{1}),(i_{2},h_{2}),\ldots,(i_{f},h_{f})$, is *normal* if the following condition holds: For any $1\leq t<f$, if $i_{t+1}<i_{t}$ then $i_{t+1}+2h_{t+1}\ge i_{t}$.
The following claim shows that if we limit ourselves to normal deduplication processes, we can still encode every binary sequence with processes of the same length.
\[clm:normal\]For any deduplication process ${\boldsymbol{s}_{}}={\boldsymbol{s}_{0}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{1}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{2}}{\,{\xrightarrow{dd}}\allowbreak\,}\dotsm{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{f}}={\operatorname{root}}({\boldsymbol{s}_{}})$ of length $f$ of a sequence ${\boldsymbol{s}_{}}$, there is a normal deduplication process ${\boldsymbol{s}_{}}={\boldsymbol{s}_{0}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{1}}'{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{2}}'{\,{\xrightarrow{dd}}\allowbreak\,}\dotsm{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{f}}'={\boldsymbol{s}_{f}}$ of the same length, with the same final sequence.
Among all deduplication processes of length $f$ starting with ${\boldsymbol{s}_{}}$ and ending with ${\boldsymbol{s}_{f}}$, consider the one minimizing the number of pairs $(i_{t},h_{t})$, $(i_{q},h_{q})$ with $1\leq t<q\leq f$, and $i_{q}<i_{t}$. We claim that this process is normal. Indeed, otherwise there is some $t$, $1\leq t<f$ so that $i_{t+1}<i_{t}$ and $i_{t+1}+2h_{t+1}< i_{t}$. But in this case we can switch the steps $(i_{t},h_{t})$ and $(i_{t+1},h_{t+1})$, performing the step $(i_{t+1},h_{t+1})$ just before $(i_{t},h_{t})$. This will clearly leave all sequences ${\boldsymbol{s}_{0}},{\boldsymbol{s}_{1}},\ldots,{\boldsymbol{s}_{f}}$, besides ${{\boldsymbol{s}_{}}}_{t}$, the same, and in particular ${\boldsymbol{s}_{0}}={\boldsymbol{s}_{}}$ and ${\boldsymbol{s}_{f}}={\operatorname{root}}({\boldsymbol{s}_{}})$ stay the same. This contradicts the minimality in the choice of the process, establishing the claim.
We now turn to the proof of Theorem \[t21\].
Let $U_{\alpha}$ denote the set of $\left\lfloor 2^{n\alpha}\right\rfloor $ sequences that have the smallest duplication distances to their roots among binary sequences of length $n$ and recall that $F_{\alpha}=$ $\max\left\{ f({\boldsymbol{s}_{}}):{\boldsymbol{s}_{}}\in U_{\alpha}\right\} $. By Claim \[clm:normal\], for each of the sequences ${\boldsymbol{s}_{}}$ of $U_{\alpha}$, there is a normal deduplication process ${\boldsymbol{s}_{}}={\boldsymbol{s}_{0}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{1}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{2}}{\,{\xrightarrow{dd}}\allowbreak\,}\dotsm{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{s}_{f}}$ of length $f\leq F_{\alpha}$. Let the steps of this process be $(i_{1},h_{1}),$ $(i_{2},h_{2}),\ldots,(i_{f},h_{f})$. As before, it is clear that knowing the final sequence ${\boldsymbol{s}_{f}}$ and all the pairs $(i_{t},h_{t})$, we can reconstruct ${\boldsymbol{s}_{}}$. There are $6$ possibilities for ${\boldsymbol{s}_{f}}$. As each step $(i_{t},h_{t})$ reduces the length of the sequence by $h_{t}$, it follows that $\sum_{i=1}^{f}h_{t}<n$ and therefore there are at most ${{n+f} \choose f}$ possibilities for the sequence $(h_{1},h_{2},h_{3},\dots,h_{f})$. In order to record the sequence $(i_{1},i_{2},\ldots,i_{f})$ it suffices to record $i_{1}$ and all the differences $i_{t}-i_{t+1}$ for all $1\leq t<n$. There are less than $n$ possibilities for $i_{1}$, and there are at most $2^{f}$ possibilities for deciding about the set of all indices $t$ for which the difference $i_{t}-i_{t+1}$ is positive. As the process is normal, for each such positive difference, we know that $i_{t+1}+2h_{t+1}\ge i_{t}$, that is $i_{t}-i_{t+1}\le2h_{t+1}$. It follows that the sum of all positive differences, $\sum_{t:i_{t}-i_{t+1}>0}\left(i_{t}-i_{t+1}\right)$, is at most $2\sum_{t}h_{t}<2n$, and hence the number of choices for these differences is at most ${2n+f \choose f}$.
Since $i_{f}\le3$, we have $i_{1}-i_{f}\ge1-3=-2$. So $$\begin{aligned}
\phantom{xx}\sum_{\mathclap{t:i_{t}-i_{t+1}\le0}}\left(i_{t}-i_{t+1}\right)
=
(i_1-i_f)-\sum_{\mathclap{t:i_{t}-i_{t+1}>0}}\left(i_{t}-i_{t+1}\right)
>-2-2n.\end{aligned}$$ Therefore, the number of choices for all non-positive differences $i_{t}-i_{t+1}$ is at most ${2n+f+2 \choose f}$. Putting all of these together, and noting that $\left|U_{\alpha}\right|\ge2^{n\alpha}-1$, implies the assertion of Theorem \[t21\].
Since $\binom{p}{q}\le2^{pH(q/p)}$ for positive integers $0<q<p$ [@macwilliams1977theory p. 309], Theorem \[t21\] implies that $$3\left(2+\frac{F_{\alpha}}{n}\right)H\left(\frac{F_{\alpha}/n}{2+F_{\alpha}/n}\right)+\frac{F_{\alpha}}{n}\ge\alpha+o(1),$$ where $H$ is the binary entropy function, $H(x) = -x \log_2 x -(1-x) \log_2(1-x)$. The expression on the left side of the inequality is strictly increasing in $\frac{F_{\alpha}}{n}$, and it is less than $0.99$ if we substitute $\frac{F_{\alpha}}{n}$ by $0.045$. If we let $\alpha=0.99$, it follows that for sufficiently large $n$, we have $\frac{F_{\alpha}}{n}\ge0.045$, thereby establishing the lower bound in Theorem \[thm:bounds\].
To prove the upper bound in Theorem \[thm:bounds\], we prove the following theorem.
\[thm:ub\]The limit $\lim_{n\to\infty}f(n)/n$ exists and for all $n$, $f(n)\le\frac{2}{5}n+15$.
Note that for any positive integers $n$ and $m$, $f(n+m)\leq f(n)+f(m)+2$. Indeed, given a sequences of length $n+m$ we can deduplicate separately its first $n$ bits and its last $m$ bits, getting a concatenation of two square-free sequences (of total length at most $6$). It then suffices to check that each such concatenation can be deduplicated to its root through at most $2$ additional deduplication steps. Therefore, the function $g(n)=f(n)+2$ is subadditive: $$g(n+m)=f(n+m)+2\leq f(n)+f(m)+4=g(n)+g(m).$$ Now, by Fekete’s Lemma [@steel1997probability], $g(n)/n$ tends to a limit (which is the infimum over $n$ of $g(n)/n$), and it is clear that the limit of $f(n)/n$ is the same as that of $g(n)/n$. We term this limit the *binary duplicatoin constant*.
This proof of the existence of $\lim_{n\to\infty}f(n)/n$ provides a simple way to derive an upper bound for the limit by computing $f(n)$ precisely for some small $n$. In particular, from Table \[tab:f(n)\], we find $\lim_{n\to\infty}f(n)/n\le(f(32)+2)/32=17/32$. We can improve upon this result as follows.
For positive integers $n,m$, let $f(n,m)$ be the smallest number $k$ such that every sequence of length $n$ can be converted to a sequences of length at most $m$ via $k$ deduplication steps. A sequence of length $n$ can be converted to its root by first repeatedly converting its $a$-substrings to substrings of length at most $b$ via $f(a,b)$ deduplication steps. Thus for integers $a>b>0$, we have $$f(n)\le\frac{f(a,b)}{a-b}n+\max_{i<a}f(i)\label{eq:fnm}$$ With the help of a computer we find the values of $f(n,m)$ for $3\le m<n\le32$. An illustration is given in Figure \[fig:bestRatio\]. In particular we have $\frac{f\left(32,12\right)}{20}=\frac{8}{20}=\frac{2}{5}$ from Figure \[fig:bestRatio\] and $\max_{i<32}f(i)=15$ from Table \[tab:f(n)\], implying $f(n)\le\frac{2}{5}n+15$.
![$\frac{f(n,m)}{n-m}$ for $3\le m<n\le32$.[]{data-label="fig:bestRatio"}](BestRatio){width="70.00000%"}
Weaker upper bounds on $f(n)$ can be obtained without resorting to computation in the following ways. First, to deduplicate a sequence to its root, we first can deduplicate each block of $t$ consecutive identical bits to a single bit by $\lceil\log_{2}t\rceil$ deduplications and then finish in less than $\log_{2}n$ additional steps. This shows that for large $n$ , $f(n)\leq\frac{2}{3}n+o(n)$ (the extremal case for this argument is the one in which each block is of size $3$). Second, it is known that every binary sequence of length at least 19 contains a repeat of length at least 2 [@entringer1974nonrepetitive], implying that $f(n)\leq\frac{1}{2}n+o(n)$.
#### Parallel duplication {#parallel-duplication .unnumbered}
One can also define the parallel duplication distance to the root by allowing non-overlapping duplications to occur simultanously, with $f'(n)$ being the maxmimum parallel duplication distance to the root of a sequence of length $n$. Similar to the normal duplication distance it is helpful to think in terms of deduplications. Since each parallel deduplication step decreases the length of a sequence by at most a factor of $2$, $f'(n)>\log_{2}n-2$ (and in fact $f'({\boldsymbol{s}_{}})\geq\log_{2}n-2$ for every sequence of length $n$.) It is not difficult to see that $f'(n)<2\log_{2}n$ by first deduplicating, in parallel, all blocks of identical elements in the sequence to blocks of size $1$, and then by deduplicating the resulting alternating sequence to its root.
#### Partial deduplication {#partial-deduplication .unnumbered}
The definition of $f(n,m)$ gives rise to the following question: For a fixed $0<\alpha\le1$, what is $\lim_{n}\frac{f\left(n,\left\lfloor \alpha n\right\rfloor \right)}{1-\alpha}$, if it exists? At first glance, one may expect $\lim_{n}\frac{f\left(n,\left\lfloor \alpha n\right\rfloor \right)}{1-\alpha}$ to be decreasing in $\alpha$ since if $\alpha$ is large, one may think it is easier to find enough long repeats to reduce the length of the sequence quickly by a factor of $1-\alpha$. However, we show that $\lim_{n}\frac{f\left(n,\left\lfloor \alpha n\right\rfloor \right)}{n\left(1-\alpha\right)}=\lim_{n}\frac{f(n)}{n}$.
Let $\gamma=\lim_n\frac{f(n)}{n}$. For $\epsilon>0$, there exists $k$ such that for all $n>k$, $f(n)\le(\gamma+\epsilon)n$. Thus $$\label{eq:f1}
f(n,\lfloor\alpha n\rfloor)\le f\left(n-\lfloor\alpha n\rfloor+3\right)\le(\gamma+\epsilon)\left({\left(1-\alpha\right)n}+4\right).$$ On the other hand, let $\delta = \liminf_n \frac{f(n,\lfloor\alpha n\rfloor)}{\left(1-\alpha\right)n}$. For $\epsilon>0$, there exists $k$ such $f(k,\lfloor\alpha k\rfloor)\le (\delta+\epsilon)(1-\alpha)k$. Hence, $$\label{eq:f2}
f(n)\le\frac{f(k,\lfloor\alpha k\rfloor)}{k-\lfloor\alpha k\rfloor}n+k\le (\delta+\epsilon)n+k.$$ The result follows by dividing by $(1-\alpha)n$ and taking a $\limsup_n$ and by dividing by $n$ and taking a $\lim_n$.
Duplication Distance for L-systems\[sec:Lsystems\]
==================================================
*L-systems*, or Lindenmayer systems are sequence rewriting systems developed by Lindemayer in 1968 [@lsystems]. He used them in the context of biology to model the growth process of plant development. He introduced context-free as well as context-sensitive L-systems. Here we will discuss distance to the root for sequences arising in context-free L-systems, also known as 0L-systems. A 0L-system comprises three components:
- Alphabet ($\Sigma$): An alphabet of symbols used to construct sequences.
- Axiom sequence or initiator (${\boldsymbol{\omega}_{}}$): The starting sequence from which a 0L-system is constructed.
- Production rule ($h$): A rule that constructs new sequences by expanding each symbol in a given sequence into a sequence of symbols. The production rule is represented by the function $h: \Sigma^* \rightarrow \Sigma^*$, which for any two sequences ${\boldsymbol{a}_{}}$ and ${\boldsymbol{b}_{}} \in \Sigma^*$ satisfies $$h({\boldsymbol{a}_{}}{\boldsymbol{b}_{}}) = h({\boldsymbol{a}_{}})h({\boldsymbol{b}_{}})$$ where $h({\boldsymbol{a}_{}})h({\boldsymbol{b}_{}})$ represents the concatenation of $h({\boldsymbol{a}_{}})$ and $h({\boldsymbol{b}_{}})$. The production rule $h$ can be deterministic or stochastic. Here we consider only deterministic rules. Such 0L-systems with deterministic $h$ are denoted as D0L-systems [@DOL].
Consider $\Sigma = \{X,Y\}$, ${\boldsymbol{\omega}_{}} = X$, and $$h(X) = XY, \quad h(Y) = X.$$ For this D0L-system, the first $5$ sequences are as follows: $$\begin{aligned}
h^0({\boldsymbol{\omega}_{}}) &= X\\
h^1({\boldsymbol{\omega}_{}}) &= XY\\
h^2({\boldsymbol{\omega}_{}}) &= XYX\\
h^3({\boldsymbol{\omega}_{}}) &= XYXXY\\
h^4({\boldsymbol{\omega}_{}}) &= XYXXYXYX\\
h^5({\boldsymbol{\omega}_{}}) &= XYXXYXYXXYXXY\end{aligned}$$ This can also be represented by the following tree:
These sequences are called Fibonacci words as they satisfy $$h^n({\boldsymbol{\omega}_{}}) = h^{n-1}({\boldsymbol{\omega}_{}})h^{n-2}({\boldsymbol{\omega}_{}})~ \forall ~ n\geq 2.$$
Let $\Sigma = \{0,1\}$, ${\boldsymbol{\omega}_{}} = 0$, and $$h(0) = 01,\quad h(1) = 10.$$ For this D0L-system the tree of sequence generation is given below:
The sequence generated by this D0L-system are called Thue-Morse sequences. Alternatively, the Thue-Morse sequences can be defined recursively by starting with ${\boldsymbol{t}_{0}}=0$ and forming ${\boldsymbol{t}_{i+1}}$ by concatenating ${\boldsymbol{t}_{i}}$ and its complement $\overline{{\boldsymbol{t}_{i}}}$.
We show that binary D0L-systems, which have production rules of the form $h(0)={\boldsymbol{u}_{}}$ and $h(1)={\boldsymbol{v}_{}}$, with ${\boldsymbol{u}_{}},{\boldsymbol{v}_{}}\in\left\{ 0,1\right\} ^{*}$ have a logarithmic distance to their roots.
[lem]{}[lemdzerol]{}\[lem:d0L\] For any binary D0L-system with initiator ${\boldsymbol{\omega}_{}} $ and production rule $h$, we have $$f\left(h^{r}({\boldsymbol{\omega}_{}} )\right)=\Theta\left(\log_2\left|h^{r}({\boldsymbol{\omega}_{}} )\right|\right),\qquad\text{as }r\to\infty.$$
For any sequence ${\boldsymbol{t}_{}}$, since $f({\boldsymbol{t}_{}})\ge\log_2|{\boldsymbol{t}_{}}|$, we have $f\left(h^{r}({\boldsymbol{\omega}_{}})\right)\ge\log_2\left|h^{r}({\boldsymbol{\omega}_{}})\right|$. It remains to show that $f\left(h^{r}({\boldsymbol{\omega}_{}})\right)=O\left(\log_2\left|h^{r}({\boldsymbol{\omega}_{}})\right|\right)$. We start by proving the following claim.
For any binary D0L-system with initiator $\omega$ and production rule $h$, we have $$f\left(h^{r}({\boldsymbol{\omega}_{}})\right)\le f\left(h^{r-1}({\boldsymbol{\omega}_{}})\right)+c\le f({\boldsymbol{\omega}_{}})+rc,\label{eq:L-sys}$$ where $c=\max_{{\boldsymbol{z}_{}}\in\left\{ 0,1,01,10,010,101\right\} }f\left(h({\boldsymbol{z}_{}})\right)$.
To prove the claim, let ${\boldsymbol{x}_{}}=h^{r-1}({\boldsymbol{\omega}_{}})$ and ${\boldsymbol{y}_{}}=h^{r}({\boldsymbol{\omega}_{}})$ and consider the sequence of deduplications that turns $x$ into its root ${\boldsymbol{z}_{}}\in\left\{ 0,1,01,10,010,101\right\} $. We can deduplicate ${\boldsymbol{y}_{}}$ in a similar manner to $h({\boldsymbol{z}_{}})$: For each step in the deduplication process of ${\boldsymbol{x}_{}}$ that deduplicates a substring $a_{1}\dotsm a_{k}a_{1}\dotsm a_{k}$ to $a_{1}\dotsm a_{k}$, we deduplicate $h\left(a_{1}\right)\dotsm h\left(a_{k}\right)h\left(a_{1}\right)\dotsm h\left(a_{k}\right)$ to $h\left(a_{1}\right)\dotsm h\left(a_{k}\right)$ in the deduplication process of ${\boldsymbol{y}_{}}$, resulting eventually in $h({\boldsymbol{z}_{}})$. This completes the proof of the claim.
We now turn to proving $f\left(h^{r}({\boldsymbol{\omega}_{}})\right)=O\left(\log_2\left|h^{r}({\boldsymbol{\omega}_{}})\right|\right)$. If $\left|h^{r}({\boldsymbol{\omega}_{}})\right|=O(1)$, then $f\left(h^{r}({\boldsymbol{\omega}_{}})\right)=O(1)$ as well, and there is nothing to prove. If $\left|h^{r}({\boldsymbol{\omega}_{}})\right|=2^{\Omega(r)}$, then $r=O\left(\log_2\left|h^{r}({\boldsymbol{\omega}_{}})\right|\right)$ and the desired result follows from (\[eq:L-sys\]). The last case that we need to consider is when $\left|h^{r}({\boldsymbol{\omega}_{}})\right|\to\infty$ but $\left|h^{r}({\boldsymbol{\omega}_{}})\right|=2^{o(r)}$. Without loss of generality, assume $\left|h(1)\right|\ge\left|h(0)\right|$. Then the condition $\left|h^{r}({\boldsymbol{\omega}_{}})\right|=2^{o(r)}$ can be shown to occur only if the initiator ${\boldsymbol{\omega}_{}}$ contains at least one occurrence of $1$, $h(0)=0$, and $h(1)$ has exactly one occurrence of $1$ and one or more 0s. In this case, the number of 1s in $h^{r}({\boldsymbol{\omega}_{}})$ is constant and again $f\left(h^{r}({\boldsymbol{\omega}_{}})\right)=O\left(\log_2\left|h^{r}({\boldsymbol{\omega}_{}})\right|\right)$.
The previous lemma shows that the duplication distances to the root for both of Fibonacci words and Thue-Morse sequences are logarithmic in sequence length. This is particularly interesting in the case of the Thue-Morse sequence. Despite the fact that the Thue-Morse sequence grows by taking the complement, it contains enough repeats to allow a logarithmic distance. Note also that the Thue-Morse sequence is used to generate ternary square-free sequences.
In the next lemma, we give better bounds than those that can be obtained from Lemma \[lem:d0L\] or (\[eq:L-sys\]) for Thue-Morse and Fibonacci sequences.
\[lem:TMF\] Let ${\boldsymbol{t}_{r}}$ and ${\boldsymbol{u}_{r}}$ denote the $r$th Thue-Morse and Fibonacci words, respectively. For $r\ge2$, we have $$\begin{aligned}
f\left({\boldsymbol{t}_{r}}\right) & \le2r,\\
f\left({\boldsymbol{u}_{r}}\right) & \le r.\end{aligned}$$
We first prove the upper bound for ${{\boldsymbol{t}_{}}}_{r}$. For $r\ge3$, we have $$\begin{aligned}
f\left({{\boldsymbol{t}_{}}}_{r}\right) & =f\left({{\boldsymbol{t}_{}}}_{r-1}\overline{{{\boldsymbol{t}_{}}}}_{r-1}\right)\\
& =f\left({{\boldsymbol{t}_{}}}_{r-2}\overline{{{\boldsymbol{t}_{}}}}_{r-2}\overline{{{\boldsymbol{t}_{}}}}_{r-2}{{\boldsymbol{t}_{}}}_{r-2}\right)\\
& \le1+f\left({{\boldsymbol{t}_{}}}_{r-2}\overline{{{\boldsymbol{t}_{}}}}_{r-2}{{\boldsymbol{t}_{}}}_{r-2}\right)\\
& =1+f\left({{\boldsymbol{t}_{}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}{{\boldsymbol{t}_{}}}_{r-3}{{\boldsymbol{t}_{}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}\right)\\
& \le3+f\left({{\boldsymbol{t}_{}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}{{\boldsymbol{t}_{}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}\right)\\
& \le4+f\left({{\boldsymbol{t}_{}}}_{r-3}\overline{{{\boldsymbol{t}_{}}}}_{r-3}\right)\\
& =4+f\left({{\boldsymbol{t}_{}}}_{r-2}\right).\end{aligned}$$ If $r\ge3$ is even, then $f\left({{\boldsymbol{t}_{}}}_{r}\right)\le4\frac{r-2}{2}+f\left({{\boldsymbol{t}_{}}}_{2}\right)=2\left(r-2\right)+1=2r-3$; and if $r\ge3$ is odd, then $f\left({{\boldsymbol{t}_{}}}_{r}\right)\le4\frac{r-1}{2}+f\left({{\boldsymbol{t}_{}}}_{1}\right)=2\left(r-1\right)$. This completes the proof of the first claim.
We now turn to $f\left({{\boldsymbol{u}_{}}}_{r}\right)$. The $r$th Fibonacci word can be obtained via the following recursion: ${{\boldsymbol{u}_{}}}_{r}={{\boldsymbol{u}_{}}}_{r-1}{{\boldsymbol{u}_{}}}_{r-2}$ for $r\ge2$ and ${{\boldsymbol{u}_{}}}_{0}=0$, ${{\boldsymbol{u}_{}}}_{1}=01$. If $r\ge5$, then $$\begin{split}{{\boldsymbol{u}_{}}}_{r} & ={{\boldsymbol{u}_{}}}_{r-1}{{\boldsymbol{u}_{}}}_{r-2}\\
& ={{\boldsymbol{u}_{}}}_{r-2}{{\boldsymbol{u}_{}}}_{r-3}{{\boldsymbol{u}_{}}}_{r-3}{{\boldsymbol{u}_{}}}_{r-4}\\
& ={{\boldsymbol{u}_{}}}_{r-2}{{\boldsymbol{u}_{}}}_{r-3}{{\boldsymbol{u}_{}}}_{r-4}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}\\
& ={{\boldsymbol{u}_{}}}_{r-2}^{2}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}.
\end{split}$$ Hence, $f\left({{\boldsymbol{u}_{}}}_{r}\right)\le1+f\left({{\boldsymbol{u}_{}}}_{r-2}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}\right)$. Noting that ${{\boldsymbol{u}_{}}}_{r-2}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}={{\boldsymbol{u}_{}}}_{r-3}{{\boldsymbol{u}_{}}}_{r-4}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}={{\boldsymbol{u}_{}}}_{r-3}^{2}{{\boldsymbol{u}_{}}}_{r-4}$, we write $$\begin{aligned}
f\left({{\boldsymbol{u}_{}}}_{r}\right) & \le1+f\left({{\boldsymbol{u}_{}}}_{r-2}{{\boldsymbol{u}_{}}}_{r-5}{{\boldsymbol{u}_{}}}_{r-4}\right)\\
& =1+f\left({{\boldsymbol{u}_{}}}_{r-3}^{2}{{\boldsymbol{u}_{}}}_{r-4}\right)\\
& \le2+f\left({{\boldsymbol{u}_{}}}_{r-3}{{\boldsymbol{u}_{}}}_{r-4}\right)\\
& =2+f\left({{\boldsymbol{u}_{}}}_{r-2}\right).\end{aligned}$$ Now, if $r\ge5$ is even, then $f\left({{\boldsymbol{u}_{}}}_{r}\right)\le\left(r-4\right)+f\left({{\boldsymbol{u}_{}}}_{4}\right)\le r-2$ since $f\left({{\boldsymbol{u}_{}}}_{4}\right)=f\left(01001010\right)\le2$; and if $r\ge5$ is odd, then $f\left({{\boldsymbol{u}_{}}}_{r}\right)\le\left(r-3\right)+f\left({{\boldsymbol{u}_{}}}_{3}\right)\le r-1$ as $f\left({{\boldsymbol{u}_{}}}_{3}\right)=f\left(01001\right)\le2$.
Approximate-duplication distance {#sec:mismatch}
================================
Recall that $f_{\beta}(n)$ is the least $k$ such that every sequence of length $n$ can be converted to a square-free sequence in $k$ approxmiate deduplication steps, with at most a $\beta$ fraction of mismatches in each step. In this section, we provide bounds on $f_{\beta}(n)$ for $\beta<1/2$ and $\beta>1/2$. We first however present some useful definitions.
For $0\le\beta<1$, a *$\beta$-repeat of length* $h$ in a binary sequence consists of two consecutive blocks in the sequence, each of length $h$, such that the Hamming distance between them is at most $\beta h$. If ${\boldsymbol{u}_{}}{\boldsymbol{v}_{}}{\boldsymbol{v}_{}}'{\boldsymbol{w}_{}}$ is a binary sequence, and ${\boldsymbol{v}_{}}{\boldsymbol{v}_{}}'$ is a $\beta$-repeat, then a $\beta$*-deduplication* produces ${\boldsymbol{u}_{}}{\boldsymbol{v}_{}}{\boldsymbol{w}_{}}$ or ${\boldsymbol{u}_{}}{\boldsymbol{v}_{}}'{\boldsymbol{w}_{}}$. Note that in this case the set of roots of ${\boldsymbol{s}_{}}$ is not necessarily unique, but the length of any root is at most 3, even if $\beta=0.$
The next theorem establishes a sharp phase transition in the behavior of $f_{\beta}(n)$ at $\beta=1/2$. Its proof relies on Theorem \[thm:rep\_beta\], which guarantees the existence of $\beta$-repeats under certain conditions. In what follows, for an integer $m$, we use $[m]$ to denote $\{1,\dotsc,m\}$.
The proof for $\beta<1/2$ is similar to the proof of the lower bound in Theorem \[thm:bounds\]. In this case however, to make the deduplication process reversible, for every deduplication we need to record whether it is of the form ${\boldsymbol{u}_{}} {\boldsymbol{v}_{}}{\boldsymbol{v}_{}}'{\boldsymbol{w}_{}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{u}_{}} {\boldsymbol{v}_{}}{\boldsymbol{w}_{}}$ or of the form ${\boldsymbol{u}_{}} {\boldsymbol{v}_{}}'{\boldsymbol{v}_{}}{\boldsymbol{w}_{}}{\,{\xrightarrow{dd}}\allowbreak\,}{\boldsymbol{u}_{}} {\boldsymbol{v}_{}}{\boldsymbol{w}_{}}$, and we must also encode the sequence ${\boldsymbol{v}_{}}'$. In the $t$th deduplication step, we have $|{\boldsymbol{v}_{}}|=|{\boldsymbol{v}_{}}'|=h_{t}$. Since ${\boldsymbol{v}_{}}'$ is in the Hamming sphere of radius $\beta h_{t}$ around ${\boldsymbol{v}_{}}$, there are at most $2^{h_{t}H(\beta)}$ options for ${\boldsymbol{v}_{}}'$ [@roth2006introduction Lemma 4.7]. Thus $$6n\sum_{f=1}^{F_{\beta}}\binom{n+f}{f}\binom{2n+f}{f}\binom{2n+f+2}{f}2^{nH(\beta)}2^{2f}\ge2^{n},$$ where $F_{\beta}=f_{\beta}(n)$ and we have used $\sum_{t}h_t\le n$. The desired result then follows since $H(\beta)<1$.
Suppose $\beta>1/2$. Let $K=\left\lceil \frac{2\beta+1}{2\beta-1}\right\rceil ^{2}$ and $\epsilon=C-K$. Note that $\epsilon>0$. By appropriately choosing $C_{1}$, we can have $f_{\beta}(i)\le\left(K+\frac{\epsilon}{2}\right)\ln i+C_{1}$ for all $i<M$, where $M$ is sufficiently large and in particular $M>K$. Assuming that this holds also for all $i<n$, where $n\ge M$, we show that it holds for $i=n$. From Theorem \[thm:rep\_beta\], every binary sequence ${\boldsymbol{s}_{}}$ of length $n$ has a $\beta$-repeat of length $\ell\lfloor n/K\rfloor$ for some $\ell\in\left[\sqrt{K}\right]$, implying $$\begin{aligned}
f_{\beta} ({\boldsymbol{s}_{}})
&\le
f_{\beta}\left(n-\ell\left\lfloor\frac{n}{K}\right\rfloor\right)+1\\
& \le\left(K+\frac{\epsilon}{2}\right)\ln\left(n-\left\lfloor\frac{n}{K}\right\rfloor\right)+1+C_{1}\\
& \le\left(K+\frac{\epsilon}{2}\right)\ln n-\frac{\left(K+\frac{\epsilon}{2}\right)\left(n-K\right)}{Kn}+1+C_{1}\\
& \le\left(K+\frac{\epsilon}{2}\right)\ln n+C_{1}\\
&\le C\ln n,\end{aligned}$$ where the last two steps hold for sufficiently large $n$. Hence, $f_\beta(n)\le C\ln n$.
\[thm:rep\_beta\]If $\beta>\frac{1}{2}$, then for any integer $k\ge\frac{2\beta+1}{2\beta-1}$, any binary sequence of length $n$ contains a $\beta$-repeat of length $\ell\lfloor n/k^{2}\rfloor$ for some $\ell\in[k]$.
Let $k$ be a positive integer to be determined later and put $K=k^{2}$. Furthermore, let ${{\boldsymbol{s}_{}}}'={{\boldsymbol{s}_{}}}_{1}\dotsm {{\boldsymbol{s}_{}}}_{K}$ be a partition of the first $KB$ symbols of ${\boldsymbol{s}_{}}$ into blocks of length $B=\left\lfloor\frac{n}{K}\right\rfloor$. We now consider as a code [@macwilliams1977theory] the $k+1$ binary vectors $${{\boldsymbol{t}_{}}}_{i}={{\boldsymbol{s}_{}}}_{i}\dotsm {{\boldsymbol{s}_{}}}_{i+K-k-1},\qquad\left(1\le i\le k+1\right),$$ each of length $m=\left(K-k\right)B$. By Plotkin’s bound [@macwilliams1977theory p. 41], the minimum Hamming distance of this code is at most $\left(\frac{1}{2}+\frac{1}{2k}\right)m$. Thus there exist ${{\boldsymbol{t}_{}}}_{i}$ and ${{\boldsymbol{t}_{}}}_{j}$ with $i<j$ with Hamming distance at most $\left(\frac{1}{2}+\frac{1}{2k}\right)m$.
Put $h=\left(j-i\right)B$ and let $m'=h\lfloor m/h\rfloor$ be the largest integer which is at most $m$ and is divisible by $h$. Let ${{\boldsymbol{t}_{}}}'_{i}$ and ${{\boldsymbol{t}_{}}}'_{j}$ consist of the first $m'$ bits of ${{\boldsymbol{t}_{}}}_{i}$ and ${{\boldsymbol{t}_{}}}_{j}$, respectively. The Hamming distance between ${{\boldsymbol{t}_{}}}'_{i}$ and ${{\boldsymbol{t}_{}}}'_{j}$ is clearly still at most $\left(\frac{1}{2}+\frac{1}{2k}\right)m$. But $\left(\frac{1}{2}+\frac{1}{2k}\right)m\le\left(\frac{1}{2}+\frac{1}{k-1}\right)m'$ since $$\begin{aligned}
\left(\frac{1}{2}+\frac{1}{2k}\right)m & =\left(\frac{1}{2}+\frac{1}{2k}\right)\frac{m}{m'}m'
\stackrel{(*)}{\le}
\left(\frac{1}{2}+\frac{1}{2k}\right)\frac{k}{k-1}m'
=
\left(\frac{1}{2}+\frac{1}{k-1}\right)m',\end{aligned}$$ where $(*)$ can be proved as follows. By the definition of $m'$, we have $m-m'<h$. Additionally, $h\le k B$ since $1\le i< j\le k+1$. So, $$\frac{m-m'}B< k,$$ which since $B$ divides $m,m'$, implies $\frac{m-m'}B\le k-1$ and, in turn, $m'\ge m-\left(k-1\right)B=\left(k-1\right)^{2}B$. Hence $\frac m{m'} \le \frac{k(k-1)B}{(k-1)^2B}=\frac k{k-1}$.
Split ${{\boldsymbol{t}_{}}}'_{i}$ and ${{\boldsymbol{t}_{}}}'_{j}$ into blocks of length $h$ each: ${{\boldsymbol{t}_{}}}'_{i}={{\boldsymbol{z}_{}}}_{1}{{\boldsymbol{z}_{}}}_{2}\cdots {{\boldsymbol{z}_{}}}_{p}$, ${{\boldsymbol{t}_{}}}'_{j}={{\boldsymbol{z}_{}}}_{2}{{\boldsymbol{z}_{}}}_{3}\cdots {{\boldsymbol{z}_{}}}_{p}{{\boldsymbol{z}_{}}}_{p+1}$, where $p=m'/h$. The Hamming distance between ${{\boldsymbol{t}_{}}}'_{i}$ and ${{\boldsymbol{t}_{}}}'_{j}$ is the sum of the Hamming distances between ${{\boldsymbol{z}_{}}}_{q}$ and ${{\boldsymbol{z}_{}}}_{q+1}$ as $q$ ranges from $1$ to $p$. Thus, by averaging, there exists an index $r$ so that the Hamming distance between ${{\boldsymbol{z}_{}}}_{r}$ and ${{\boldsymbol{z}_{}}}_{r+1}$ is at most $\left(\frac{1}{2}+\frac{1}{k-1}\right)h$. Putting $k\ge\frac{2\beta+1}{2\beta-1}$ so that $\frac{1}{2}+\frac{1}{k-1}\le\beta$ ensures that ${{\boldsymbol{z}_{}}}_{r}{{\boldsymbol{z}_{}}}_{r+1}$ is $\beta$-repeat of length $h=\left(j-i\right)B=\left(j-i\right)$$\lfloor n/K\rfloor$.
Let a $\beta_{h}$-repeat be a repeat of length $h$ with at most $h\beta_{h}$ mismatches, i.e., the two blocks are at Hamming distance at most $h\beta_h$. In the preceding theorems and their proofs, in principal, we do not need the maximum number of permitted mismatches to be a linear function of the length of the repeat, so we can apply the same techniques to $\beta_h$-repeats with nonlinear relationships:
[thm]{}[thmbetaabouthalf]{} Let $\beta_{h}^{a}=\frac{1}{2}+\frac{1}{h^{a}}$, where $0<a<1$ is a constant, and let $f_{a}(n)$ be the smallest number $f$ such that any binary sequence of length $n$ can be deduplicated to a root in $f$ steps by deduplicating $\beta_{h}^{a}$-repeats. There exist positive constants $c_{2},c_{3}$ such that $$f_{a}(n)\le c_{2}n^{2a/(1+a)}+c_{3}.\label{eq:f_a}$$
By making appropriate changes to the proof of Theorem \[thm:rep\_beta\], one can show that for $k=\left\lceil 2n^{a/(1+a)}\right\rceil$, every binary sequence of sufficiently long length $n$ contains a $\beta_{h}^{a}$-repeat of length $h=\ell\lfloor n/k^{2}\rfloor$, for some $\ell\in[k]$. To do so, we need to prove $\left(\frac{1}{2}+\frac{1}{k-1}\right)h\le\beta_h^ah$ for all $h$ of the form $h=\ell \lfloor n/k^2\rfloor$, $\ell\in[k]$. This holds since with the aforementioned value of $k$, $$\beta_{\ell\lfloor n/k^{2}\rfloor}^{a}=\frac{1}{2}+\frac{1}{\left(\ell\lfloor n/k^{2}\rfloor\right)^{a}}
\ge\frac{1}{2}+\frac{1}{\left(k\lfloor n/k^{2}\rfloor\right)^{a}}
\ge\frac{1}{2}+\frac{1}{k-1},$$ for all $\ell\in[k]$ and sufficiently large $n$.
We can now prove (\[eq:f\_a\]) by induction. Clearly, for any $M$, there exist constants $c_{2},c_{3}$ such that $f_{a}(i)\le c_{2}i^{2a/(1+a)}+c_{3}$ for all $i\le M$. Choose $M$ to be sufficiently large as to satisfy the requirements of the rest of the proof. Fix $n>M$ and assume that $f_{a}(i)\le c_{2}i^{2a/(1+a)}+c_{3}$ for all $i<n$. Since in every sequence of length $n$, there exists a $\beta_{h}^{a}$-repeat with $h=\ell\lfloor n/k^{2}\rfloor$, for some $\ell\in[k]$ and $k=\left\lceil 2n^{a/(1+a)}\right\rceil $, it holds that $$\begin{aligned}
f_{a} (n)&\le1+c_{2}\left(n-\ell\lfloor n/k^{2}\rfloor\right)^{2a/(1+a)}+c_{3}\\
& \le1+c_{2}\left(n-\frac{1}{5}n^{\frac{1-a}{1+a}}\right)^{2a/(1+a)}+c_{3}\\
& =1+c_{2}n^{2a/(1+a)}\left(1-\frac{1}{5}n^{-\frac{2a}{1+a}}\right)^{2a/(1+a)}+c_{3}\\
& \le1+c_{2}n^{2a/(1+a)}\left(1-\frac{2a}{5\left(1+a\right)}n^{-\frac{2a}{1+a}}\right)+c_{3}\\
& =c_{2}n^{2a/(1+a)}+\left(1-\frac{2ac_{2}}{5\left(1+a\right)}\right)+c_{3}\\
& \le c_{2}n^{2a/(1+a)}+c_{3},\end{aligned}$$ where the inequalities hold for sufficiently large $n$. The third inequality follows from Bernoulli’s inequality and the the last one follows from the fact that we can choose $c_{2}$ to be arbitrarily large.
Duplication distances for different roots\[sec:stringSystems\]
==============================================================
In this section, we study $f_{{\boldsymbol{\sigma}_{}}}$ for ${\boldsymbol{\sigma}_{}}\in\{0,1,01,10,010,101\}$. It is easy to see that $f_{0}(n)=f_{1}(n)=\left\lceil \log_2 n\right\rceil .$ Clearly $f_{10}=f_{01}$ and $f_{101}=f_{010}$. So we limit our attention to roots ${\boldsymbol{\sigma}_{}}=10$ and ${\boldsymbol{\sigma}_{}}=101$. Plots for $f_{10}(n)$ and $f_{101}(n)$, obtained through computer search, are given in Figure \[fig:f10-101(n)\].
![$f_{10}(n)$ and $f_{101}(n)$ for $1\le n\le32$.\[fig:f10-101(n)\]](reductionLength){width="3in"}
The general approach in this proof is similar to that of the proof of Fekete’s lemma in [@steel1997probability]. We prove the theorem for $\lim_{n}\frac{f_{10}(n)}{n}$. The proof for $\frac{f_{101}(n)}{n}$ is similar.
Let $\gamma=\liminf_{n}\frac{f_{10}(n)}{n}$ and let $k\ge3$ be such that $f_{10}(k)+5+2\log_2 k\le k\left(\gamma+\epsilon\right)$ for $\epsilon>0$. Let ${\boldsymbol{s}_{}}$ be a sequence of length $n$. Starting from the beginning of ${\boldsymbol{s}_{}}$, partition it into substrings that are the shortest possible while having length at least $k$ and different symbols at the beginning and the end (so that their root is either 10 or 01). Name these substrings ${{\boldsymbol{s}_{}}}_{1},\dotsc,{{\boldsymbol{s}_{}}}_{m+1}$, where $\left|{{\boldsymbol{s}_{}}}_{i}\right|\ge k$ for $i\le m$ and $1\le\left|{{\boldsymbol{s}_{}}}_{m+1}\right|\le k$. Let $s_{i,j}$ denote the $j$th element of ${{\boldsymbol{s}_{}}}_{i}$. We deduplicate ${{\boldsymbol{s}_{}}}$ to its root by first deduplicating its substrings ${{\boldsymbol{s}_{}}}_{i}$ to their roots.
For each substring ${{\boldsymbol{s}_{}}}_{i}$ of the partition, except the last one, we consider the following cases and deduplicate ${{\boldsymbol{s}_{}}}_{i}$ as indicated, where without loss of generality we assume ${{\boldsymbol{s}_{}}}_{i}$ starts with 1 and ends with 0:
- $\left|{{\boldsymbol{s}_{}}}_{i}\right|=k$: Deduplicate this substring to 10 in $f_{10}(k)$ steps.
- $\left|{{\boldsymbol{s}_{}}}_{i}\right|>k$ and $s_{i,k-1}=1$: In this case, ${{\boldsymbol{s}_{}}}_{i}=1{\boldsymbol{x}_{}}11,\negthinspace1^{*}0$, where ${\boldsymbol{x}_{}}\in\left\{ 0,1\right\} ^{k-3}$, for clarity a comma is placed after the $k$th element of ${{\boldsymbol{s}_{}}}_{i}$, and $a^*$ denotes that the symbol $a$ appears 0 or more times. We reduce the length of the last run of 1s in ${{\boldsymbol{s}_{}}}_{i}$ by $\left|{{\boldsymbol{s}_{}}}_{i}\right|-k$ in $\left\lceil \log_2\left(\left|{{\boldsymbol{s}_{}}}_{i}\right|-k+1\right)\right\rceil $ deduplication steps to obtain $1{\boldsymbol{x}_{}}10$. Then deduplicate the result to 10 in $f_{10}(k)$ steps.
- $\left|{{\boldsymbol{s}_{}}}_{i}\right|>k$ and $s_{i,k-1}=0$: In this case, ${{\boldsymbol{s}_{}}}_{i}=1{\boldsymbol{x}_{}}01,\negthinspace1^{*}0$, where ${\boldsymbol{x}_{}}\in\left\{ 0,1\right\} ^{k-3}$ and where a comma is placed after the $k$th element of ${{\boldsymbol{s}_{}}}_{i}$. We reduce the length of the last run of 1s in ${{\boldsymbol{s}_{}}}_{i}$ by $\left|{{\boldsymbol{s}_{}}}_{i}\right|-k-1$ in $\left\lceil \log_2\left(\left|{{\boldsymbol{s}_{}}}_{i}\right|-k\right)\right\rceil $ deduplication steps to obtain $\hat{{\boldsymbol{s}_{}}}_{i}=1{\boldsymbol{x}_{}}01,0$ and note that $\hat{{\boldsymbol{s}_{}}}_{i}$ has length $k+1$ and ends with $010$. Now either $\hat{{\boldsymbol{s}_{}}}_{i}$ has a run of length at least 2 or not. If it does, we reduce the length of this run by 1 to obtain a sequence of length $k$, which we then convert to $10$ in $f_{10}(k)$ deduplication steps. If not, then $\hat{{\boldsymbol{s}_{}}}_{i}$ is an alternating sequence of the form $101010\dotsm10$ which can be deduplicated to $10$ in no more than $\left\lceil \log_2\frac{k+1}{2}\right\rceil $ steps.
The resulting sequences has length at most $2m+k$ and can be deduplicated to its root in at most as many steps. We thus have $$\begin{aligned}
f(n) & \le mf_{10}(k)+\sum_{i=1}^{m}\left\lceil \log_2\left(\left|{{\boldsymbol{s}_{}}}_{i}\right|-k+1\right)\right\rceil +m\left\lceil \log_2\frac{k+1}{2}\right\rceil +3m+k\\
& \le mf_{10}(k)+\sum_{i=1}^{m}\log_2\left|{{\boldsymbol{s}_{}}}_{i}\right|+m\log_2 k+5m+k\\
& \le\frac{n}{k}f_{10}(k)+\frac{2n}{k}\log_2 k+5\frac{n}{k}+k,\end{aligned}$$ where for the last step we have used the fact that $$\sum_{i=1}^{m}\log_2\left|{{\boldsymbol{s}_{}}}_{i}\right|\le m\log_2\left(n/m\right)\le\frac{n}{k}\log_2 k$$ which holds since $\sum_{i=1}^{m}\left|{{\boldsymbol{s}_{}}}_{i}\right|\le n$, $\frac{d}{dm}m\log_2\frac{n}{m}>0$ and $m\le\frac{n}{k}$. It follows that $$\frac{f(n)}{n}\le\frac{f_{10}(k)}{k}+\frac{2\log_2 k}{k}+\frac{5}{k}+\frac{k}{n}\le\gamma+\epsilon+\frac{k}{n}\ .$$ Taking $\lim$ of both sides and noting that $\epsilon>0$ is arbitrary proves that $\lim_n \frac {f(n)}{n}\le\liminf_n\frac{f_{10}(n)}n$. On the other hand, it is clear that $\limsup_{n}\frac{f_{10}(n)}{n}\le\lim_{n}\frac{f(n)}{n}$. Hence, $\lim_{n}\frac{f(n)}{n}=\lim_{n}\frac{f_{10}(n)}{n}$. Similar arguments hold for $f_{101}(n)$.
Open Problems {#sec:conc}
=============
We now describe some of the open problems related to extremal values of duplication distance to the root. First, the binary duplication constant, $\lim_{n}\frac{f(n)}{n}$ is unknown. It is also interesting to find bounds tighter than the one given in Theorem \[thm:bounds\], namely $0.045\le\lim\frac{f(n)}{n}\le0.4$. Furthermore, although the lower bound $f({\boldsymbol{s}_{}})\ge0.045n$ is valid for all but an exponentially small fraction of sequences of length $n$, we have not been able to find an explicit family of sequences whose distance is linear in $n$. A related problem to identifying sequences with large duplication distance is improving bounds on $f({\boldsymbol{s}_{}})$ that depend on the structure of ${\boldsymbol{s}_{}}$, such as the bound given in Lemma \[lem:SeqDepBound\], relating $f({\boldsymbol{s}_{}})$ to the number of distinct $k$-mers of ${\boldsymbol{s}_{}}$.
While we showed in our study of approximate duplication that at $\beta=1/2$, $f_\beta(n)$ transitions from a linear dependence on $n$ to a logarithmic one, the behavior at $\beta=1/2$ is not known. Furthermore, we can alter the setting by decoupling duplications and substitutions, i.e., we generate the sequence through exact duplications and substitutions, possibly with limitations on the number of substitutions. We can then study the same problems as the ones we have in this paper as well as new problems, e.g., the minimum number substitutions required to generate the sequence via a logarithmic number of duplication steps.
A different strand of problems are algorithmic in nature, including designing an algorithm that can efficiently find or approximate the duplication distance to the root and provide a duplication process of the appropriate length. The computational complexity of these tasks is also not known. Similar questions may be asked for approximate duplication, or duplication along with substitution. These problems are important because of their potential application in determining the sequence of duplications and point mutations that may have resulted in a particular segment of an organism’s DNA.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by the NSF Expeditions in Computing Program (The Molecular Programming Project), by a USA-Israeli BSF grant 2012/107, by an ISF grant 620/13, and by the Israeli I-Core program.
[^1]: Noga Alon is with the Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 6997801, Israel, Email: <[email protected]>.
[^2]: Jehoshua Bruck is with the Electrical Engineering Department, California Institute of Technology, Pasadena, CA, 91125, Email: <[email protected]>.
[^3]: Farzad Farnoud is with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA, 22903, Email: <[email protected]>. He was with the Electrical Engineering Department, California Institute of Technology.
[^4]: Siddharth Jain is with the Electrical Engineering Department, California Institute of Technology, Pasadena, CA, 91125, Email: <[email protected]>.
[^5]: This paper was presented in part at 2016 IEEE International Symposium on Information Theory in Barcelona, Spain.
[^6]: Note that using the term distance here is a slight abuse of notation as the duplication distance does not satisfy the triangle inequality.
[^7]: If De Bruijn seqences are defined cyclically as opposed to linearly, there are exactly $\frac{2^{n/2}}{n}$ De Bruijn sequences of length $n$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that maximizes the probability of correct detection. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.'
author:
- 'Yonina C. Eldar'
- Mihailo Stojnic
- Babak Hassibi
title: Optimal quantum detectors for unambiguous detection of mixed states
---
Introduction
============
The problem of detecting information stored in the state of a quantum system is a fundamental problem in quantum information theory. Several approaches have emerged to distinguishing between a collection of non-orthogonal quantum states. In one approach, a measurement is designed to maximize the probability of correct detection [@H73; @H76; @EMV02; @CBH89; @OBH96; @BKMO97; @EF01; @EMV02s; @EF01; @E03l]. A more recent approach, referred to as unambiguous detection [@I87; @D88; @P88; @JS95; @PT98; @C98; @CB98; @E02; @E03p], is to design a measurement that with a certain probability returns an inconclusive result, but such that if the measurement returns an answer, then the answer is correct with probability $1$. An interesting alternative approach for distinguishing between a collection of quantum states, which is a combination of the previous two approaches, is to allow for a certain probability of an inconclusive result, and then maximize the probability of correct detection [@E03p; @ZLG99; @FJ02].
We consider a quantum state ensemble consisting of $m$ density operators $\{\rho_i, 1 \le i \le m\}$ on an $n$-dimensional complex Hilbert space ${{\mathcal{H}}}$, with prior probabilities $\{p_i>0, 1
\le i \le m\}$. A pure-state ensemble is one in which each density operator $\rho_i$ is a rank-one projector ${{|\phi_i\rangle}}{{\langle\phi_i|}}$, where the vectors ${{|\phi_i\rangle}}$, though evidently normalized to unit length, are not necessarily orthogonal. Our problem is to design a quantum detector to distinguish unambiguously between the states $\{\rho_i\}$.
Chefles [@C98] showed that a necessary and sufficient condition for the existence of unambiguous measurements for distinguishing between a collection of [*pure*]{} quantum states is that the states are linearly independent. Necessary and sufficient conditions on the optimal measurement minimizing the probability of an inconclusive result for pure states were derived in [@E02]. The optimal measurement when distinguishing between a broad class of symmetric pure-state sets was also considered in [@E02].
The problem of unambiguous detection between [*mixed*]{} state ensembles has received considerably less attention. Rudolph [*et al. *]{}[@RST03] showed that unambiguous detection between mixed quantum states is possible as long as one of the density operators in the ensemble has a non-zero overlap with the intersection of the kernels of the other density operators. They then consider the problem of unambiguous detection between two mixed quantum states, and derive upper and lower bounds on the probability of an inconclusive result. They also develop a closed form solution for the optimal measurement in the case in which both states have kernels of dimension $1$.
In this paper we develop a general framework for unambiguous state discrimination between a collection of mixed quantum states, which can be applied to any number of states with arbitrary prior probabilities. For our measurement we consider general positive operator-valued measures [@H76; @P90], consisting of $m+1$ measurement operators. We derive a set of necessary and sufficient conditions for an optimal measurement that minimizes the probability of an inconclusive result, by exploiting principles of duality theory in vector space optimization. We then show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions.
Next, we consider geometrically uniform (GU) and compound GU state sets [@EF01; @EMV02s; @EB01], which are state sets with strong symmetry properties. We show that the optimal measurement operators for unambiguous discrimination between such state sets are also GU and CGU respectively, with generators that can be computed very efficiently by solving a reduced size semidefinite program.
The paper is organized as follows. In Section \[sec:problem\], we provide a statement of our problem. In Section \[sec:conditions\] we develop the necessary and sufficient conditions for optimality using Lagrange duality theory. Some special cases are considered in Section \[sec:sc\]. In Section \[sec:sym\] we consider the problem of distinguishing between a collection of states with a broad class of symmetry properties.
Problem Formulation {#sec:problem}
===================
Assume that a quantum channel is prepared in a quantum state drawn from a collection of mixed states, represented by density operators $\{ \rho_i,1 \leq i \leq m \}$ on an $n$-dimensional complex Hilbert space ${{\mathcal{H}}}$. We assume without loss of generality that the eigenvectors of $\rho_i,1 \leq i \leq m$, collectively span[^1] ${{\mathcal{H}}}$.
To detect the state of the system a measurement is constructed comprising $m+1$ measurement operators $\{\Pi_i,0 \leq i \leq m\}$ that satisfy $$\begin{aligned}
\label{eq:psdt}& \Pi_i \geq 0,\quad 0 \leq i \leq m; &\nonumber
\\
&\sum_{i=0}^m \Pi_i = I.&\end{aligned}$$ The measurement operators are constructed so that either the state is correctly detected, or the measurement returns an inconclusive result. Thus, each of the operators $\Pi_i,1 \leq i \leq m$ correspond to detection of the corresponding states $\rho_i,1 \leq
i \leq m$, and $\Pi_0$ corresponds to an inconclusive result.
Given that the state of the system is $\rho_j$, the probability of obtaining outcome $i$ is ${\mbox{Tr}}(\rho_j\Pi_i)$. Therefore, to ensure that each state is either correctly detected or an inconclusive result is obtained, we must have $$\label{eq:zecond} {\mbox{Tr}}(\rho_j\Pi_i)=\eta_i\delta_{ij},\quad 1 \leq
i,j \leq m,$$ for some $0 \leq \eta_i \leq 1$. Since from (\[eq:psdt\]), $\Pi_0=I-\sum_{i=1}^m \Pi_i$, (\[eq:zecond\]) implies that ${\mbox{Tr}}(\rho_i\Pi_0)=1-\eta_i$, so that given that the state of the system is $\rho_i$, the state is correctly detected with probability $\eta_i$, and an inconclusive result is returned with probability $1-\eta_i$.
It was shown in [@C98] that for pure-state ensembles consisting of rank-one density operators $\rho_i={{|\phi_i\rangle}}{{\langle\phi_i|}}$, (\[eq:zecond\]) can be satisfied if and only if the vectors ${{|\phi_i\rangle}}$ are linearly independent. For mixed states, it was shown in [@RST03] that (\[eq:zecond\]) can be satisfied if and only if one of the density operators $\rho_i$ has a non-zero overlap with the intersection of the kernels of the other density operators. Specifically, denote by ${{\mathcal{K}}}_i$ the null space of $\rho_i$ and let $$\label{eq:Si} {{\mathcal{S}}}_i=\cap_{j=1,j \neq i}^m{{\mathcal{K}}}_j$$ denote the intersection of ${{\mathcal{K}}}_j,1 \leq j \leq m,j \neq i$. Then to satisfy (\[eq:zecond\]) the eigenvectors of $\Pi_i$ must be contained in ${{\mathcal{S}}}_i$ and must not be entirely contained in ${{\mathcal{K}}}_i$. This implies that ${{\mathcal{K}}}_i$ must not be entirely contained in ${{\mathcal{S}}}_i$. Some examples of mixed states for which unambiguous detection is possible are given in [@RST03].
If the state $\rho_i$ is prepared with prior probability $p_i$, then the total probability of correctly detecting the state is $$P_D=\sum_{i=1}^mp_i {\mbox{Tr}}(\rho_i\Pi_i).$$ Our problem therefore is to choose the measurement operators $\Pi_i,0 \leq i \leq m$ to maximize $P_D$, subject to the constraints (\[eq:psdt\]) and $$\label{eq:zec} {\mbox{Tr}}(\rho_j\Pi_i)=0,\quad 1 \leq i,j \leq m,i \neq
j.$$ To satisfy (\[eq:zec\]), $\Pi_i$ must lie in ${{\mathcal{S}}}_i$ defined by (\[eq:Si\]), so that $$\label{eq:pip} \Pi_i=P_i \Pi_iP_i,\quad 1 \leq i \leq m,$$ where $P_i$ is the orthogonal projection onto ${{\mathcal{S}}}_i$. Denoting by $\Theta_i$ an $n \times r$ matrix whose columns form an arbitrary orthonormal basis for ${{\mathcal{S}}}_i$, where $r=\dim({{\mathcal{S}}}_i)$, we can express $P_i$ as $P_i=\Theta_i\Theta_i^*$. From (\[eq:pip\]) and (\[eq:psdt\]) we then have that $$\Pi_i=\Theta_i\Delta_i\Theta_i^*,\quad 1 \leq i \leq m,$$ where $\Delta_i=\Theta_i^* \Pi_i\Theta_i$ is an $r \times r$ matrix satisfying $$\begin{aligned}
\label{eq:psd}& \Delta_i \geq 0,\quad 1 \leq i \leq m;
&\nonumber
\\
&\sum_{i=1}^m \Theta_i\Delta_i\Theta_i^* \leq I.&\end{aligned}$$ Therefore, our problem reduces to maximizing $$\label{eq:Pd} P_D=\sum_{i=1}^mp_i
{\mbox{Tr}}(\rho_i\Theta_i\Delta_i\Theta_i^*),$$ subject to (\[eq:psd\]).
To show that the problem of (\[eq:Pd\]) and (\[eq:psd\]) does not depend on the choice of orthonormal basis $\Theta_i$, we note that any orthonormal basis for ${{\mathcal{S}}}_i$ can be expressed as the columns of $\Psi_i$, where $\Psi_i=\Theta_i U_i$ for some $r
\times r$ unitary matrix $U_i$. Substituting $\Psi_i$ instead of $\Theta_i$ in (\[eq:Pd\]) and (\[eq:psd\]), our problem becomes that of maximizing $$\label{eq:Pdp} P_D=\sum_{i=1}^mp_i
{\mbox{Tr}}(\rho_i\Psi_i\Delta_i\Psi_i^*)= \sum_{i=1}^mp_i
{\mbox{Tr}}(\rho_i\Theta_i\Delta'_i\Theta_i^*),$$ where $\Delta_i'=U_i\Delta_iU_i^*$, subject to $$\begin{aligned}
\label{eq:psdp} & \Delta_i \geq 0,\quad 1 \leq i \leq m;
&\nonumber
\\
&\sum_{i=1}^m \Psi_i\Delta_i\Psi_i^* =\sum_{i=1}^m
\Theta_i\Delta'_i\Theta_i^* \leq I.&\end{aligned}$$ Since $\Delta_i \geq 0$ if and only if $\Delta_i' \geq 0$, the problem of (\[eq:Pdp\]) and (\[eq:psdp\]) is equivalent to that of (\[eq:Pd\]) and (\[eq:psd\]).
Equipped with the standard operations of addition and multiplication by real numbers, the space ${{\mathcal{B}}}$ of all Hermitian $n
\times n$ matrices is an $n^2$-dimensional [*real*]{} vector space. As noted in [@RST03], by choosing an appropriate basis for ${{\mathcal{B}}}$, the problem of maximizing $P_D$ subject to (\[eq:psd\]) can be put in the form of a standard semidefinite programming problem, which is a convex optimization problem; for a detailed treatment of semidefinite programming problems see, [[*e.g.,* ]{}]{}[@A91t; @A92; @NN94; @VB96]. By exploiting the many well known algorithms for solving semidefinite programs [@VB96], [[*e.g.,* ]{}]{}interior point methods[^2] [@NN94; @A91t], the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy.
Using elements of duality theory in vector space optimization, in the next section we derive necessary and sufficient conditions on the measurement operators $\Pi_i=\Theta_i\Delta_i \Theta_i^*$ to maximize $P_D$ of (\[eq:Pd\]) subject to (\[eq:psd\]).
Conditions for optimality {#sec:conditions}
=========================
Dual Problem Formulation {#sec:sdp}
------------------------
To derive necessary and sufficient conditions for optimality on the matrices $\Delta_i$ we first derive a dual problem, using Lagrange duality theory [@B99].
Denote by $\Lambda$ the set of all ordered sets $\Pi=\{\Pi_i=\Theta_i\Delta_i\Theta_i^*\}_{i=1}^m$ satisfying (\[eq:psd\]) and define $J(\Pi)= \sum_{i=1}^m
p_i{\mbox{Tr}}(\rho_i\Theta_i\Delta_i\Theta_i^*)$. Then our problem is $$\label{eq:primal} \max_{\Pi \in \Lambda} J(\Pi).$$ We refer to this problem as the primal problem, and to any $\Pi
\in \Lambda$ as a primal feasible point. The optimal value of $J(\Pi)$ is denoted by $\widehat{J}$.
To develop the dual problem associated with (\[eq:primal\]) we first compute the Lagrange dual function, which is given by $$\begin{aligned}
\label{eq:gf} \lefteqn{g(Z)=} \nonumber \\ & \hspace*{-0.3in}= &
\hspace*{-0.1in} \min_{\Delta_i \geq 0}{\left\{}
\newcommand{\brc}{\right\}}-\sum_{i=1}^m p_i{\mbox{Tr}}(\rho_i\Theta_i\Delta_i\Theta_i^*)\right.+ \nonumber \\
&& + \left. {\mbox{Tr}}{\left(}Z{\left(}\sum_{i=0}^m
\Theta_i\Delta_i\Theta_i^*-I{\right)}{\right)}\brc \nonumber \\
& \hspace*{-0.3in}= & \hspace*{-0.1in}\min_{\Delta_i \geq 0}{\left\{}
\newcommand{\brc}{\right\}}\sum_{i=1}^m {\mbox{Tr}}{\left(}\Delta_i\Theta_i^* {\left(}Z-p_i\rho_i {\right)}\Theta_i{\right)}-{\mbox{Tr}}(Z)\brc,\end{aligned}$$ where $Z \geq 0$ is the Lagrange dual variable. Since $\Delta_i
\geq 0,1 \leq i \leq m$, we have that ${\mbox{Tr}}(\Delta_iX) \geq 0$ for any $X \geq 0$. If $X$ is not positive semidefinite, then we can always choose $\Delta_i$ such that ${\mbox{Tr}}(\Delta_i X)$ is unbounded below. Therefore, $$g(Z)=\left\{
\begin{array}{ll}
-{\mbox{Tr}}(Z), & A_i \geq 0, 1 \leq i \leq m, Z \geq 0; \\
-\infty, & \mbox{otherwise},
\end{array}
\right.$$ where $$A_i=\Theta_{i}^*(Z-p_i\rho_i)\Theta_{i},\quad 1\leq i \leq m.$$ It follows that the dual problem associated with (\[eq:primal\]) is $$\label{eq:dual} \min_{Z} {\mbox{Tr}}(Z)$$ subject to $$\begin{aligned}
\label{eq:dualc} & \Theta_i^*(Z-p_i\rho_i)\Theta_i\geq 0,
\quad 1 \leq i \leq m; & \nonumber \\
& Z \geq 0. &\end{aligned}$$ Denoting by $\Gamma$ the set of all Hermitian operators $Z$ such that $\Theta_{i}^*(Z-p_i\rho_i)\Theta_{i} \geq 0,1 \leq i \leq m$ and $Z \geq 0$, and defining $T(Z)={\mbox{Tr}}(Z)$, the dual problem can be written as $$\min_{Z \in \Gamma} T(Z).
\label{eq:dualc1}$$ We refer to any $Z \in \Gamma$ as a dual feasible point. The optimal value of $T(Z)$ is denoted by $\widehat{T}$.
Optimality Conditions {#sec:opt}
---------------------
We can immediately verify that both the primal and the dual problem are strictly feasible. Therefore, their optimal values are attainable and the duality gap is zero [@VB96], [[*i.e.,* ]{}]{}$$\label{eq:dualitys} \widehat{J}=\widehat{T}.$$ In addition, for any $\Pi=\{\Pi_i=\Theta_i\Delta_i\Theta_i^*\}_{i=1}^m \in \Lambda$ and $Z \in \Gamma$, $$\begin{aligned}
\label{eq:wduality} \lefteqn{\hspace*{-0.1in} T(Z)-J(\Pi) =} \nonumber \\
& = &
{\mbox{Tr}}{\left(}\sum_{i=1}^m\Theta_{i}\Delta_i\Theta_{i}^*(Z-p_i\rho_i)
+\Pi_0 Z {\right)}\nonumber \\
& \ge & 0,\end{aligned}$$ where $\Pi_0=I-\sum_{i=1}^m \Theta_i\Delta_i\Theta_i^* \geq 0$. Note, that (\[eq:wduality\]) can be used to develop an upper bound on the optimal probability of correct detection $\widehat{J}$. Indeed, since for any $Z \in \Gamma$, $T(Z) \geq
J(\Pi)$, we have that $\widehat{J} \leq T(Z)$ for any dual feasible $Z$.
Now, let ${\widehat{\Pi}}_i=\Theta_{i}\widehat{\Delta}_{i}\Theta_{i}^*,1 \leq
i \leq m$ and ${\widehat{\Pi}}_{0}=I-\sum_{i=1}^m {\widehat{\Pi}}_i$ denote the optimal measurement operators that maximize (\[eq:Pd\]) subject to (\[eq:psd\]), and let ${\widehat{Z}}$ denote the optimal $Z$ that minimizes (\[eq:dual\]) subject to (\[eq:dualc\]). From (\[eq:dualitys\]) and (\[eq:wduality\]) we conclude that $$\label{eq:duality}
{\mbox{Tr}}{\left(}\sum_{i=1}^m{\widehat{\Pi}}_i\Theta_{i}^*({\widehat{Z}}-p_i\rho_i)\Theta_{i}+
{\widehat{\Pi}}_0 {\widehat{Z}}{\right)}=0.$$ Since $\widehat{\Delta}_i \geq 0$, ${\widehat{Z}}\geq 0$ and $\Theta_{i}^*
({\widehat{Z}}-p_i\rho_i)\Theta_{i} \geq 0,1\leq i \leq m$, (\[eq:duality\]) is satisfied if and only if
$$\begin{aligned}
\label{eq:condz1a} & {\widehat{Z}}{\widehat{\Pi}}_0=0 & \\
\label{eq:condz1b} & \Theta_{i}^*({\widehat{Z}}-p_i\rho_i)\Theta_{i}
\widehat{\Delta}_i=0, \quad 1 \leq i \leq m. &\end{aligned}$$
Once we find the optimal ${\widehat{Z}}$ that minimizes the dual problem (\[eq:dual\]), the constraints (\[eq:condz1a\]) and (\[eq:condz1b\]) are necessary and sufficient conditions on the optimal measurement operators ${\widehat{\Pi}}_i$. We have already seen that these conditions are necessary. To show that they are sufficient, we note that if a set of feasible measurement operators ${\widehat{\Pi}}_i$ satisfies (\[eq:condz1a\]) and (\[eq:condz1b\]), then ${\mbox{Tr}}{\left(}\sum_{i=1}^m\widehat{\Delta}_i\Theta_{i}^*({\widehat{Z}}-p_i\rho_i)\Theta_{i}+
{\widehat{\Pi}}_0 {\widehat{Z}}{\right)}=0$ so that from (\[eq:wduality\]), $J({\widehat{\Pi}})=T({\widehat{Z}})=\widehat{J}$.
We summarize our results in the following theorem:
\[thm:dual\] Let $\{\rho_i,1 \leq i \leq m\}$ denote a set of density operators with prior probabilities $\{p_i>0,1 \leq i \leq
m\}$, and let $\{\Theta_i,1 \leq i \leq m\}$ denote a set of matrices such that the columns of $\Theta_{i}$ form an orthonormal basis for ${{\mathcal{S}}}_i=\cap_{j=1,j \neq i}^m{{\mathcal{K}}}_j$, where ${{\mathcal{K}}}_i$ the null space of $\rho_i$. Let $\Lambda$ denote the set of all ordered sets of Hermitian measurement operators $\Pi=\{\Pi_i\}_{i=0}^m$ that satisfy $\Pi_i \geq 0$, $\sum_{i=0}^m
\Pi_i=I$, and ${\mbox{Tr}}(\rho_{j}\Pi_{i})=0,1 \leq i \leq m, i\neq j$ and let $\Gamma$ denote the set of Hermitian matrices $Z$ such that $Z \geq 0$, $ \Theta_{i}^*(Z-p_i\rho_i)\Theta_{i},1 \leq i
\leq m$. Consider the problem $\max_{\Pi \in \Lambda} J(\Pi)$ and the dual problem $\min_{Z \in \Gamma} T(Z)$, where $J(\Pi)=\sum_{i=1}^m p_i {\mbox{Tr}}(\rho_i\Pi_i)$ and $T(Z)={\mbox{Tr}}(Z)$. Then
1. For any $Z \in \Gamma$ and $\Pi \in \Lambda$, $T(Z) \geq J(\Pi)$.
2. There is an optimal $\Pi$, denoted ${\widehat{\Pi}}$, such that $\widehat{J}=J({\widehat{\Pi}}) \geq J(\Pi)$ for any $\Pi \in \Lambda$;
3. There is an optimal $Z$, denoted ${\widehat{Z}}$ and such that $\widehat{T}=T({\widehat{Z}}) \leq T(Z)$ for any $Z \in \Gamma$;
4. $\widehat{T}=\widehat{J}$;
5. Necessary and sufficient conditions on the optimal measurement operators ${\widehat{\Pi}}_i$ are that there exists a $Z
\in \Gamma$ such that $$\begin{aligned}
& Z {\widehat{\Pi}}_0=0 & \\
& \Theta_{i}^*(Z-p_i\rho)\Theta_{i} \widehat{\Delta}_i=0, \quad 1
\leq i \leq m, &\end{aligned}$$ where ${\widehat{\Pi}}_i=\Theta_{i}\widehat{\Delta}_i\Theta_{i}^*,1 \leq i
\leq m$, and $\widehat{\Delta}_i \geq 0$.
6. Given ${\widehat{Z}}$, necessary and sufficient conditions on the optimal measurement operators ${\widehat{\Pi}}_i$ are $$\begin{aligned}
& {\widehat{Z}}{\widehat{\Pi}}_0=0 & \\
& \Theta_{i}^*({\widehat{Z}}-p_i\rho_i)\Theta_{i} \widehat{\Delta}_i=0,
\quad 1 \leq i \leq m. &\end{aligned}$$
Although the necessary and sufficient conditions of Theorem \[thm:dual\] are hard to solve, they can be used to verify a solution and to gain some insight into the optimal measurement operators. In the next section we show that the previous optimal measurements that were derived in the literature for certain special cases satisfy these optimality conditions.
Special cases {#sec:sc}
=============
We now consider two special cases that where addressed in [@RST03], for which a closed form solution exists. In Section \[sec:orthog\] we consider the case in which the spaces ${{\mathcal{S}}}_i$ defined by (\[eq:Si\]) are orthogonal, and in Section \[sec:one\] we consider the problem of distinguishing unambiguously between two density operators with $\dim({{\mathcal{S}}}_i)=1,1
\leq i \leq 2$.
Orthogonal Null Spaces ${{\mathcal{S}}}_i$ {#sec:orthog}
------------------------------------------
The first case we consider is the case in which the null spaces ${{\mathcal{S}}}_i$ are orthogonal, so that $$P_iP_j=\delta_{ij},\quad 1 \leq i,j, \leq m,$$ where $P_i$ is an orthogonal projection onto ${{\mathcal{S}}}_i$. It was shown in [@RST03] that in this case the optimal measurement operators are $$\label{eq:pio} {\widehat{\Pi}}_{i} = P_i=\Theta_{i}\Theta_{i}^*,\quad 1 \leq
i \leq m.$$ In Appendix \[sec:a\] we show that the optimal solution of the dual problem can be expressed as $$\label{eq:zo}
{\widehat{Z}}=\sum_{i=1}^{m}p_iP_i\rho_{i}P_i.$$ It can easily be shown that ${\widehat{Z}}$ and ${\widehat{\Pi}}_i$ of (\[eq:zo\]) and (\[eq:pio\]) satisfy the optimality conditions of Theorem \[thm:dual\].
Null Spaces of Dimension $1$ {#sec:one}
----------------------------
We now consider the case of distinguishing between two density operators $\rho_1$ and $\rho_2$, where ${{\mathcal{S}}}_1$ and ${{\mathcal{S}}}_2$ both have dimension equal to $1$. In this case, as we show in Appendix \[sec:b\], the optimal dual solution is $${\widehat{Z}}=
\begin{cases}
d_{1}P_1, & \text{$d_{2}-d_{1}|f|^2\leq 0$;} \\
d_{2}P_2, & \text{$d_{1}-d_{2}|f|^2\leq 0$;} \\
d_{2}(\Theta_{2}+s\Theta_{2}^{\perp})(\Theta_{2}+s\Theta_{2}^{\perp})^*,
& \text{otherwise,}
\end{cases}
\label{eq:dualf10}$$ where $P_i$ is an orthogonal projection onto ${{\mathcal{S}}}_i$, $\Theta_2^{\perp}$ is a unit norm vector in the span of $\Theta_1$ and $\Theta_2$ such that $\Theta_{2}^*\Theta_{2}^{\perp}=0$, and $$\begin{aligned}
\label{eq:def}
& d_i=p_i\Theta_i^*\rho_i\Theta_i, \quad 1 \leq i \leq 2; & \nonumber \\
&s=\frac{f^*}{e^*}{\left(}\sqrt{\frac{d_1}{d_2 |f|^2}}-1{\right)}; & \nonumber \\
& f=\Theta_2^*\Theta_1; & \nonumber \\
& e=(\Theta_2^{\perp})^*\Theta_1. &\end{aligned}$$ The optimal measurement operators for this case were developed in [@RST03], and can be written as $$\{{\widehat{\Pi}}_i\}_{i=1}^2=
\begin{cases}
{\widehat{\Pi}}_{1}=P_1,{\widehat{\Pi}}_{2}=0, &
\text{$d_{2}-d_{1}|f|^2\leq 0$}; \\
{\widehat{\Pi}}_{1}=0,{\widehat{\Pi}}_{2}=P_2, &
\text{$d_{1}-d_{2}|f|^2\leq 0$;} \\
{\widehat{\Pi}}_{1}=\alpha_1P_1,{\widehat{\Pi}}_{2}=\alpha_2P_2, & \text{otherwise},
\end{cases}
\label{eq:dualf11}$$ where $$\begin{aligned}
& \alpha_1=\frac{1-\sqrt{\frac{d_2|f|^2}{d1}}}{1-|f|^2}; & \nonumber \\
& \alpha_2=\frac{1-\sqrt{\frac{d_1|f|^2}{d2}}}{1-|f|^2}. &\end{aligned}$$
We now show that ${\widehat{Z}}$ and ${\widehat{\Pi}}$ of (\[eq:dualf10\]) and (\[eq:dualf11\]) satisfy the optimality conditions of Theorem \[thm:dual\]. To this end we note that from (\[eq:dualf11\]), $$\{\widehat{\Delta}_i\}_{i=1}^2=
\begin{cases}
\widehat{\Delta}_{1}=1,\widehat{\Delta}_{2}=0, &
\text{$d_{2}-d_{1}|f|^2\leq 0$}; \\
\widehat{\Delta}_{1}=0,\widehat{\Delta}_{2}=1, &
\text{$d_{1}-d_{2}|f|^2\leq 0$}; \\
\widehat{\Delta}_{1}=\alpha_1,\widehat{\Delta}_{2}=\alpha_2, &
\text{otherwise}.
\end{cases}
\label{eq:delta1}$$ From (\[eq:dualf10\])–(\[eq:delta1\]) we have that if $d_{2}-d_{1}|f|^2\leq 0$, then $$\begin{aligned}
&\Theta_{1}^*({\widehat{Z}}-p_1\rho_1)\Theta_{1}
\widehat{\Delta}_1 = d_1-\Theta_{1}^*p_1\rho_1\Theta_{1}=0; &\nonumber \\
&\Theta_{2}^*({\widehat{Z}}-p_2\rho_2)\Theta_{2}\widehat{\Delta}_2 = 0; &\nonumber \\
&{\widehat{Z}}{\widehat{\Pi}}_0 =
{\widehat{Z}}(I-{\widehat{\Pi}}_1)=d_1\Theta_1\Theta_{1}^*-d_1\Theta_1\Theta_{1}^*=0. &
\label{eq:satopt1}\end{aligned}$$ Similarly, if $d_{1}-d_{2}|f|^2\leq 0$, then $$\begin{aligned}
&\Theta_{1}^*({\widehat{Z}}-p_1\rho_1)\Theta_{1}\widehat{\Delta}_1 = 0; &\nonumber \\
&\Theta_{2}^*({\widehat{Z}}-p_2\rho_2)\Theta_{2}
\widehat{\Delta}_2 = d_2-\Theta_{2}^*p_2\rho_2\Theta_{2}=0; &\nonumber \\
&{\widehat{Z}}{\widehat{\Pi}}_0 =
{\widehat{Z}}(I-{\widehat{\Pi}}_2)=d_2\Theta_2\Theta_{2}^*-d_2\Theta_2\Theta_{2}^*=0. &
\label{eq:satopt2}\end{aligned}$$ Finally, if neither of the conditions $d_{1}-d_{2}|f|^2\leq 0$, $d_{2}-d_{1}|f|^2\leq 0$ hold, then $$\begin{aligned}
\lefteqn{\Theta_{1}^*({\widehat{Z}}-p_1\rho_1)\Theta_{1}\widehat{\Delta}_1 =
} \nonumber \\
& = & (d_2(f^*+e^*s)(f^*+e^*s)^*-d_1)\frac{1-\sqrt{\frac{d_2|f|^2}{d_1}}}{1-|f|^2}\nonumber \\
& = & {\left(}d_2|f|^2{\left(}\sqrt{\frac{d_1}{d_2|f|^2}}{\right)}^2-d_1 {\right)}\frac{1-\sqrt{\frac{d_2|f|^2}{d_1}}}
{1-|f|^2} \nonumber \\
&= &0, \label{eq:satopt3}\end{aligned}$$ $$\begin{aligned}
\Theta_{2}^*({\widehat{Z}}-p_2\rho_2)\Theta_{2}\widehat{\Delta}_2 & = &
(\Theta_2^*{\widehat{Z}}\Theta_2-d_2)\frac{1-\sqrt{\frac{d_1|f|^2}{d_2}}}{1-|f|^2}
\nonumber \\ & = & 0, \label{eq:eqnarray}\end{aligned}$$ and $$\begin{aligned}
{\widehat{Z}}{\widehat{\Pi}}_0 &=& {\widehat{Z}}-{\widehat{Z}}{\widehat{\Pi}}_{1}-{\widehat{Z}}{\widehat{\Pi}}_{2} \nonumber \\
&=&{\widehat{Z}}-\widehat{\Delta}_1{\widehat{Z}}\Theta_1\Theta_1^*-\widehat{\Delta}_2{\widehat{Z}}\Theta_2\Theta_2^*.
\label{eq:h}\end{aligned}$$ To show that ${\widehat{Z}}{\widehat{\Pi}}_0=0$, we note that $$\begin{aligned}
{\widehat{Z}}\Theta_1\Theta_1^*
&=& d_2(|f|^2+s^*ef^*)\Theta_2\Theta_2^* \nonumber \\
&+& d_2(s|f|^2+ss^*ef^*)\Theta_2^{\perp}\Theta_2^* \nonumber \\
&+& d_2(e^*f+s^*|e|^2)\Theta_2\Theta_2^{\perp *} \nonumber \\
&+& d_2(se^*f+ss^*|e|^2)\Theta_2^{\perp}\Theta_2^{\perp *},
\label{eq:h1}\end{aligned}$$ and $${\widehat{Z}}\Theta_2\Theta_2^*=d_2\Theta_2\Theta_2^*+d_2s\Theta_2^{\perp}\Theta_2^*.
\label{eq:h2}$$ Substituting (\[eq:h1\]) and (\[eq:h2\]) into (\[eq:h\]), and after some algebraic manipulations, we have that $${\widehat{Z}}{\widehat{\Pi}}_0={\widehat{Z}}-\widehat{\Delta}_1{\widehat{Z}}\Theta_1\Theta_1^*-\widehat{\Delta}_2{\widehat{Z}}\Theta_2\Theta_2^*=0.
\label{eq:satopt5}$$ Combining (\[eq:satopt1\])–(\[eq:satopt5\]) we conclude that the optimal measurement operators of [@RST03] satisfy the optimality conditions of Theorem \[thm:dual\].
Optimal Detection of Symmetric States {#sec:sym}
=====================================
We now consider the case in which the quantum state ensemble has symmetry properties referred to as geometric uniformity (GU) and compound geometric uniformity (CGU). These symmetry properties are quite general, and include many cases of practical interest.
Under a variety of different optimality criteria the optimal measurement for distinguishing between GU and CGU state sets was shown to be GU and CGU respectively [@EF01; @EMV02s; @E02; @E03p]. In particular it was shown in [@E02] that the optimal measurement for unambiguous detection between linearly independent GU and CGU pure-states is GU and CGU respectively. We now generalize this result to unambiguous detection of mixed GU and CGU state sets.
GU State Sets {#sec:gu}
=============
A GU state set is defined as a set of density operators $\{\rho_i,
1 \leq i \leq m\}$ such that $\rho_i=U_i\rho U_i^*$ where $\rho$ is an arbitrary [*generating operator*]{} and the matrices $\{U_i,1 \leq i \leq m\}$ are unitary and form an abelian group ${{\mathcal{G}}}$ [@F91; @EMV02s]. For concreteness, we assume that $U_1=I$. The group ${{\mathcal{G}}}$ is the *generating group* of ${{\mathcal{S}}}$. For consistency with the symmetry of ${{\mathcal{S}}}$, we will assume equiprobable prior probabilities on ${{\mathcal{S}}}$.
As we now show, the optimal measurement operators that maximize the probability of correct detection when distinguishing unambiguously between the density operators of a GU state set are also GU with the same generating group. The corresponding generator can be computed very efficiently in polynomial time.
Suppose that the optimal measurement operators that maximize $$J(\{\Pi_i\})=\sum_{i=1}^m {\mbox{Tr}}(\rho_i \Pi_i)$$ subject to (\[eq:psd\]) and (\[eq:zec\]) are ${\widehat{\Pi}}_i$, and let $\widehat{J}=J(\{{\widehat{\Pi}}_i\})=\sum_{i=1}^m{\mbox{Tr}}(\rho_i {\widehat{\Pi}}_i)$. Let $r(j,i)$ be the mapping from ${{\mathcal{I}}}\times {{\mathcal{I}}}$ to ${{\mathcal{I}}}$ with ${{\mathcal{I}}}=\{1,\ldots,m\}$, defined by $r(j,i)=k$ if $U_j^*U_i=U_k$. Then the measurement operators ${\widehat{\Pi}}_i^{(j)}=U_j{\widehat{\Pi}}_{r(j,i)}U_j^*$ and ${\widehat{\Pi}}_0^{(j)}=I-\sum_{i=1}^m {\widehat{\Pi}}_i^{(j)}$ for any $1 \leq j \leq
m$ are also optimal. Indeed, since ${\widehat{\Pi}}_i \geq 0,1 \leq i \leq m$ and $\sum_{i=1}^m {\widehat{\Pi}}_i \leq I$, ${\widehat{\Pi}}^{(j)}_i \geq 0,1 \leq i
\leq m$ and $$\sum_{i=1}^m {\widehat{\Pi}}^{(j)}_i=U_j{\left(}\sum_{i=1}^m {\widehat{\Pi}}_i {\right)}U_j^*
\leq U_jU_j^*=I.$$ Using the fact that $\rho_i=U_i\rho U_i^*$ for some generator $\rho$, $$\begin{aligned}
J(\{{\widehat{\Pi}}^{(j)}_i\})& = & \sum_{i=1}^m{\mbox{Tr}}(\rho U_i^*
U_j{\widehat{\Pi}}_{r(j,i)}U_j^*U_i) \nonumber \\
& = & \sum_{k=1}^m{\mbox{Tr}}(\rho U_k^*{\widehat{\Pi}}_kU_k) \nonumber \\
& = & \sum_{i=1}^m{\mbox{Tr}}(\rho_i{\widehat{\Pi}}_i) \nonumber \\
& = & \widehat{J}.\end{aligned}$$ Finally, for $l \neq i$, $$\begin{aligned}
{\mbox{Tr}}{\left(}\rho_l{\widehat{\Pi}}^{(j)}_i {\right)}& = & {\mbox{Tr}}{\left(}U_l\rho
U_l^*U_j{\widehat{\Pi}}_{r(j,i)}U_j^*
{\right)}\nonumber \\
& = & {\mbox{Tr}}{\left(}U_s\rho U_s^*{\widehat{\Pi}}_{r(j,i)} {\right)}\nonumber \\
& = & {\mbox{Tr}}{\left(}\rho_s {\widehat{\Pi}}_k {\right)}\nonumber \\
& = & 0,\end{aligned}$$ where $U_s=U_j^*U_l$ and $U_k=U_j^*U_i$ and the last equality follows from the fact that since $l \neq i$, $s \neq k$.
It was shown in [@E03p; @EMV02s] that if the measurement operators ${\widehat{\Pi}}_i^{(j)}$ are optimal for any $j$, then $\{\overline{\Pi}_i=(1/m)\sum_{j=1}^m {\widehat{\Pi}}_i^{(j)},1 \leq i \leq
m\}$ and $\overline{\Pi}_0=I-\sum_{i=1}^m \overline{\Pi}_i$ are also optimal. Furthermore, $\overline{\Pi}_i = U_i
\widehat{\Pi}U_i^*$ where $\widehat{\Pi}=(1/m)\sum_{k=1}^mU_k^*{\widehat{\Pi}}_kU_k$.
We therefore conclude that the optimal measurement operators can always be chosen to be GU with the same generating group ${{\mathcal{G}}}$ as the original state set. Thus, to find the optimal measurement operators all we need is to find the optimal generator ${\widehat{\Pi}}$. The remaining operators are obtained by applying the group ${{\mathcal{G}}}$ to ${\widehat{\Pi}}$.
Since the optimal measurement operators satisfy $\Pi_i=U_i \Pi
U_i^*,1 \leq i \leq m$ and $\rho_i=U_i \rho U_i^*$, ${\mbox{Tr}}(\rho_i
\Pi_i)={\mbox{Tr}}(\rho \Pi)$, so that the problem (\[eq:Pd\]) reduces to the maximization problem $$\label{eq:max} \max_{\Pi \in {{\mathcal{B}}}} {\mbox{Tr}}(\rho\Pi),$$ where ${{\mathcal{B}}}$ is the set of $n \times n$ Hermitian operators, subject to the constraints $$\begin{aligned}
\label{eq:condp}
&\Pi \geq 0; &\nonumber \\
&\sum_{i=1}^m U_i \Pi U_i^* \leq I; & \nonumber \\
& {\mbox{Tr}}(\Pi \rho_i)=0, \quad 2 \leq i \leq m. &\end{aligned}$$ The problem of (\[eq:max\]) and (\[eq:condp\]) is a (convex) semidefinite programming problem, and therefore the optimal $\Pi$ can be computed very efficiently in polynomial time within any desired accuracy [@VB96; @A91t; @NN94], for example using the LMI toolbox on Matlab. Note that the problem of (\[eq:max\]) and (\[eq:condp\]) has $n^2$ real unknowns and $m+1$ constraints, in contrast with the original maximization problem (\[eq:Pd\]) subject to (\[eq:psd\]) and (\[eq:zec\]) which has $mn^2$ real unknowns and $m^2+1$ constraints.
CGU State Sets {#sec:cgu}
==============
A CGU state set is defined as a set density operators $\{\rho_{ik},1 \leq i \leq l,1 \leq k\leq r\}$ such that $\rho_{ik}=U_i\phi_kU_i^*$ for some generating density operators $\{\rho_k,1 \leq k \leq r \}$, where the matrices $\{U_i,1 \leq i
\leq l\}$ are unitary and form an abelian group ${{\mathcal{G}}}$ [@EB01; @EMV02s]. A CGU state set is in general not GU. However, for every $k$, the operators $\{\rho_{ik},1 \leq i \leq l\}$ are GU with generating group ${{\mathcal{G}}}$.
Using arguments similar to hose of Section \[sec:gu\] and [@E03p] we can show that the optimal measurement operators corresponding to a CGU state set can always be chosen to be GU with the same generating group ${{\mathcal{G}}}$ as the original state set. Thus, to find the optimal measurement operators all we need is to find the optimal generators ${\widehat{\Pi}}_k$. The remaining operators are obtained by applying the group ${{\mathcal{G}}}$ to each of the generators ${\widehat{\Pi}}_k$.
Since the optimal measurement operators satisfy $\Pi_{ik}=U_i
\Pi_k U_i^*,1 \leq i \leq l,1 \leq k \leq r$ and $\rho_{ik}=U_i
\rho_k U_i^*$, ${\mbox{Tr}}(\rho_{ik} \Pi_{ik})={\mbox{Tr}}(\rho_k \Pi_k)$, so that the problem (\[eq:Pd\]) reduces to the maximization problem $$\label{eq:max2} \max_{\Pi_k \in {{\mathcal{B}}}} \sum_{k=1}^r {\mbox{Tr}}(\rho_k\Pi_k),$$ subject to the constraints $$\begin{aligned}
\label{eq:condp2}
&\Pi_k \geq 0,\quad 1 \leq k \leq r; &\nonumber \\
&\sum_{i=1}^l \sum_{k=1}^r U_{ik} \Pi_k U_{ik}^* \leq I; & \nonumber \\
& {\mbox{Tr}}(\Pi_k \rho_{ik})=0, \quad 1 \leq k \leq r,2 \leq i \leq l. &\end{aligned}$$ Since this problem is a (convex) semidefinite programming problem, the optimal generators $\Pi_k$ can be computed very efficiently in polynomial time within any desired accuracy [@VB96; @A91t; @NN94]. Note that the problem of (\[eq:max2\]) and (\[eq:condp2\]) has $rn^2$ real unknowns and $lr+1$ constraints, in contrast with the original maximization which has $lrn^2$ real unknowns and $(lr)^2+1$ constraints.
Conclusion
==========
We considered the problem of distinguishing unambiguously between a collection of [*mixed*]{} quantum states. Using elements of duality theory in vector space optimization, we derived a set of necessary and sufficient conditions on the optimal measurement operators. We then considered some special cases for which closed form solutions are known, and showed that they satisfy our optimality conditions. We also showed that in the case in which the states to be distinguished have strong symmetry properties, the optimal measurement operators have the same symmetries, and can be determined efficiently by solving a semidefinite programming problem.
An interesting future direction to pursue is to use the optimality conditions we developed in this paper to derive closed form solutions for other special cases.
Proof of (\[eq:zo\]) {#sec:a}
====================
To develop the optimal dual solution in the case of orthogonal null spaces, let $\Theta=\begin{bmatrix}\Theta_1 & \Theta_2 & ... & \Theta_m
\end{bmatrix}$, and define a matrix $\Theta^{\perp}$ such that $\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}$ is a square, unitary matrix, [[*i.e.,* ]{}]{}$\begin{bmatrix} \Theta &
\Theta^{\perp}
\end{bmatrix}^*
\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}=I$. Denoting $Z={\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}Y
\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}^*}$, the dual problem can be expressed as $$\min_Y {\mbox{Tr}}{\left(}\begin{bmatrix} \Theta & \Theta^{\perp}
\end{bmatrix}Y
\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}^*{\right)}\label{eq:p1}$$ subject to $$\begin{aligned}
&\Theta_i^*{\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}Y
\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}^*}\Theta_i
\geq \Theta_i^*p_i\rho_i\Theta_i, \quad 1 \leq i \leq m;& \nonumber \\
&Y\geq 0.& \label{eq:p2}\end{aligned}$$ Using the orthogonality properties of $\Theta_i$ and $\Theta^\perp$, the problem of (\[eq:p1\]) and (\[eq:p2\]) is equivalent to $$\min_Y {\mbox{Tr}}(Y) \label{eq:p3}$$ subject to $$\begin{aligned}
&Y_i
\geq \Theta_i^*p_i\rho_i\Theta_i, \quad 1 \leq i \leq m;& \nonumber \\
&Y\geq 0,& \label{eq:p4}\end{aligned}$$ where $$Y=\begin{bmatrix}
Y_1 & & & &\\
& Y_2 & & & \\
& & \ddots & & \\
& & & Y_m & \\
& & & & \text{0}
\end{bmatrix}.$$ Since ${\mbox{Tr}}(Y)=\sum_{i=1}^m{\mbox{Tr}}(Y_i)$, a solution to (\[eq:p3\]) subject to (\[eq:p4\]) is $$\widehat{Y}=\begin{bmatrix}
\widehat{Y}_1 & & & &\\
& \widehat{Y}_2 & & & \\
& & \ddots & & \\
& & & \widehat{Y}_m & \\
& & & & \text{0}
\end{bmatrix},$$ where $$\widehat{Y}_i=\Theta_i^*p_i\rho_i\Theta_i, \quad 1 \leq i \leq m.$$ Then, $$\begin{aligned}
{\widehat{Z}}&=&{\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}\widehat{Y}
\begin{bmatrix} \Theta & \Theta^{\perp} \end{bmatrix}^*}=
\sum_{i=1}^{m} p_iP_i\rho_i P_i,\end{aligned}$$ as in (\[eq:zo\]).
Proof of (\[eq:dualf10\]) {#sec:b}
=========================
To develop the optimal dual solution ${\widehat{Z}}$ for one-dimensional null spaces, we note that ${\widehat{Z}}$ lies in the space spanned by $\Theta_1$ and $\Theta_2$. Denoting by $\Theta$ a matrix whose columns represent an orthonormal basis for this space, ${\widehat{Z}}$ can be written as ${\widehat{Z}}=\Theta\widehat{Y}\Theta^*$, where the $2 \times
2$ matrix $\widehat{Y}$ is the solution to $$\min_Y {\mbox{Tr}}(Y) \label{eq:p11}$$ subject to $$\begin{aligned}
\label{eq:p1a}
&\Phi_1^* Y \Phi_1\geq d_1;& \\
\label{eq:p2a}
&\Phi_2^* Y \Phi_2\geq d_2;& \\
\label{eq:p3a}
&Y\geq 0.&\end{aligned}$$ Here $\Phi_i=\Theta^*\Theta_i$ and $d_i=p_i\Theta_i^*\rho_i\Theta_i$ for $1 \leq i \leq 2$.
To develop a solution to (\[eq:p11\]) subject to (\[eq:p1a\])–(\[eq:p3a\]), we form the Lagrangian $${{\mathcal{L}}}={\mbox{Tr}}(Y)-\sum_{i=1}^2\gamma_i (\Phi_i^*Y\Phi_i-d_i)-{\mbox{Tr}}(XY),$$ where from the Karush-Kuhn-Tucker (KKT) conditions [@BN01] we must have that $\gamma_i \geq 0, X \geq 0$, and $$\begin{aligned}
\label{eq:cs1}
&\gamma_i (\Phi_i^*Y\Phi_i-d_i)=0,\quad i=1,2; \\
\label{eq:cs2} &{\mbox{Tr}}(XY)=0. &\end{aligned}$$ Differentiating ${{\mathcal{L}}}$ with respect to $Y$ and equating to zero, $$I-\sum_{i=1}^2 \gamma_i \Phi_i\Phi_i^*-X=0.$$ If $X=0$, then we must have that $I=\sum_{i=1}^2 \gamma_i
\Phi_i\Phi_i^*$, which is possible only if $\Phi_1$ and $\Phi_2$ are orthogonal. Therefore, $X \neq 0$, which implies from (\[eq:cs2\]) that (\[eq:p3a\]) is active. Now, suppose that only (\[eq:p3a\]) is active. In this case our problem reduces to minimizing ${\mbox{Tr}}(y^*y)$ whose optimal solution is $y=0$, which does not satisfy (\[eq:p1a\]) and (\[eq:p2a\]).
We conclude that at the optimal solution (\[eq:p3a\]) and at least one of the constraints (\[eq:p1a\]) and (\[eq:p2a\]) are active. Thus, to determine the optimal solution we need to determine the solutions under each of the $3$ possibilities: only (\[eq:p1a\]) is active, only (\[eq:p2a\]) is active, both (\[eq:p1a\]) and (\[eq:p2a\]) are active, and then choose the solution with the smallest objective.
Consider first the case in which (\[eq:p1a\]) and (\[eq:p3a\]) are active. In this case, $\widehat{Y}=\hat{y}\hat{y}^*$ for some vector $\hat{y}$, and without loss of generality we can assume that $$\label{eq:yc1} \Phi_1^*\hat{y}=d_1.$$ To satisfy (\[eq:yc1\]), $\hat{y}$ must have the form $$\label{eq:yf1} \hat{y}=\sqrt{d_1}\Phi_1+\hat{s}\Phi_1^{\perp},$$ where $\Phi_1^{\perp}$ is a unit norm vector orthogonal to $\Phi_1$, so that $\Phi_1^*\Phi_1^{\perp}=0$, and $\hat{s}$ is chosen to minimize ${\mbox{Tr}}(\widehat{Y})$. Since, $${\mbox{Tr}}(\widehat{Y})=\hat{y}^*\hat{y}=d_1+|\hat{s}|^2,$$ $\hat{s}=0$. Thus, $\widehat{Y}=d_1\Phi_1\Phi_1^*$, and ${\mbox{Tr}}(\widehat{Y})=d_1$. This solution is valid only if (\[eq:p2a\]) is satisfied, [[*i.e.,* ]{}]{}only if $$\Phi_2^*\widehat{Y}\Phi_2=d_1 |f|^2 \geq d_2.$$ Here we used the fact that $$\Phi_2^*\Phi_1=\Theta_2^*\Theta\Theta^*\Theta_1=\Theta_2^*\Theta_1=f,$$ since $\Theta\Theta^*$ is an orthogonal projection onto the space spanned by $\Theta_1$ and $\Theta_2$.
Next, suppose that (\[eq:p2a\]) and (\[eq:p3a\]) are active. In this case, $\widehat{Y}=\hat{y}\hat{y}^*$ where without loss of generality we can choose $\hat{y}$ such that $$\Phi_2^*\hat{y}=d_2,$$ and $$\hat{y}=\sqrt{d_2}\Phi_2+\hat{s}\Phi_2^{\perp},$$ where $\Phi_2^{\perp}$ is a unit norm vector orthogonal to $\Phi_2$, and $\hat{s}$ is chosen to minimize ${\mbox{Tr}}(\widehat{Y})$. Since ${\mbox{Tr}}(\widehat{Y})=d_2+|\hat{s}|^2$, $\hat{s}=0$, and ${\mbox{Tr}}(\widehat{Y})=d_2$. This solution is valid only if (\[eq:p1a\]) is satisfied, [[*i.e.,* ]{}]{}$$\Phi_1^*Y\Phi_1=d_2 |f|^2 \geq d_1.$$
Finally, consider the case in which (\[eq:p1a\])–(\[eq:p3a\]) are active. In this case, we can assume without loss of generality that $\Phi_2^*\hat{y}=\sqrt{d_2}$. Then, $$\label{eq:haty} \hat{y}=\sqrt{d_2}\Phi_2+\hat{s}\Phi_2^{\perp},$$ where $\hat{s}$ is chosen such that $$\label{eq:ipc} \Phi_{1}^*\widehat{Y}\Phi_{1}=d_1,$$ and ${\mbox{Tr}}(\widehat{Y})=\hat{y}^*\hat{y}$ is minimized. Now, for $\hat{y}$ given by (\[eq:haty\]), $$\begin{aligned}
\lefteqn{\widehat{Y}=d_2\Phi_2\Phi_2^*+
|\hat{s}|^2\Phi_2^{\perp}\Phi_2^{\perp
*}+}\nonumber \\
& &+ \hat{s}\sqrt{d_2}\Phi_2^{\perp}\Phi_2^{*}+
\hat{s}^*\sqrt{d_2}\Phi_2\Phi_2^{\perp *},\end{aligned}$$ so that $$\begin{aligned}
\Phi_{1}^*\widehat{Y}\Phi_{1} & = &
d_2|f|^2+|\hat{s}|^2|e|^2+\sqrt{d_2}\hat{s}e^*f+
\sqrt{d_2}\hat{s}^*f^*e \nonumber \\
& = & |\sqrt{d_2}f+\hat{s}^*e|^2, \label{eq:ap1}\end{aligned}$$ where we defined $\Theta_2^\perp=\Theta \Psi_2^\perp$, and $e$ and $f$ are given by (\[eq:def\]). Therefore, to satisfy (\[eq:ipc\]), $\hat{s}$ must be of the form $$\hat{s}=\frac{1}{e^*}{\left(}e^{j\varphi}\sqrt{d_1}-f^*\sqrt{d_2}{\right)},$$ for some $\varphi$. The problem of (\[eq:p11\]) then becomes $$\min_\varphi \frac{1}{|e|^2}\left|
e^{j\varphi}\sqrt{d_1}-f^*\sqrt{d_2}\right|^2,$$ which is equivalent to $$\max_\varphi \Re\left\{{e^{j\varphi}f}\right\}.$$ Since $$\Re\left\{{e^{j\varphi}f}\right\} \leq
\left|e^{j\varphi}f\right|=|f|,$$ the optimal choice of $\varphi$ is $e^{j\varphi}=f^*/|f|$, and $$\hat{s}= \frac{f^*\sqrt{d_2}}{e^*}{\left(}\frac{\sqrt{d_1}}{\sqrt{d_2}|f|}-1{\right)}.$$ For this choice of $\hat{s}$, $$\begin{aligned}
{\mbox{Tr}}(\widehat{Y}) & = & d_2+|\hat{s}|^2 \nonumber \\
& = & d_2{\left(}1+\frac{|f|^2}{|e|^2} {\left(}\frac{\sqrt{d_1}}{\sqrt{d_2}|f|}-1{\right)}^2 {\right)}\nonumber \\
& {\ {\stackrel{\triangle}{=}} \ }& \alpha.\end{aligned}$$
Clearly, $\alpha \geq d_2$. Therefore, to complete the proof of (\[eq:dualf10\]) we need to show that $\alpha \geq d_1$. Now, $$\begin{aligned}
\lefteqn{|e|^2(\alpha-d_1) =}\nonumber \\
& = &
|e|^2(d_2-d_1)+|f|^2 {\left(}\frac{\sqrt{d_1}}{|f|}-\sqrt{d_2}{\right)}^2 \nonumber \\
& = & (1-|e|^2)d_1+(|e|^2+|f|^2)d_2-2
\sqrt{d_1}\sqrt{d_2}|f| \nonumber \\
& = & (|f|\sqrt{d_1}-\sqrt{d_2})^2 \nonumber \\
& \geq & 0,\end{aligned}$$ where we used the fact that $$\begin{aligned}
|e|^2+|f|^2 & = & \Theta_1^*\Theta_2\Theta_2^*\Theta_1+
\Theta_1^*\Theta_2^\perp(\Theta_2^\perp)^*\Theta_1 \nonumber \\
& = & \Theta_1^*\Theta_1=1,\end{aligned}$$ since $\Theta_2\Theta_2^*+\Theta_2^\perp(\Theta_2^\perp)^*$ is an orthogonal projection onto the space spanned by $\Theta_1$ and $\Theta_2$.
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[^1]: Otherwise we can transform the problem to a problem equivalent to the one considered in this paper by reformulating the problem on the subspace spanned by the eigenvectors of $\{\rho_i,1 \leq i \leq m\}$.
[^2]: Interior point methods are iterative algorithms that terminate once a pre-specified accuracy has been reached. A worst-case analysis of interior point methods shows that the effort required to solve a semidefinite program to a given accuracy grows no faster than a polynomial of the problem size. In practice, the algorithms behave much better than predicted by the worst case analysis, and in fact in many cases the number of iterations is almost constant in the size of the problem.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
F. J. Alexander, S. Chen and J. D. Sterling[^1]\
Center for Nonlinear Studies and Theoretical Division\
Los Alamos National Laboratory\
Los Alamos, NM 87545
title: Lattice Boltzmann Thermohydrodynamics
---
=6.in =8.5in =.4in =.1in
[**ABSTRACT**]{}
We introduce a lattice Boltzmann computational scheme capable of modeling thermohydrodynamic flows of monatomic gases. The parallel nature of this approach provides a numerically efficient alternative to traditional methods of computational fluid dynamics. The scheme uses a small number of discrete velocity states and a linear, single-time-relaxation collision operator. Numerical simulations in two dimensions agree well with exact solutions for adiabatic sound propagation and Couette flow with heat transfer.
The lattice Boltzmann (LB) method is a discrete, in space and time, microscopic, kinetic equation description for the evolution of the velocity distribution function of a fluid [@zanetti; @hig; @succi1]. Like lattice gas (LG) automata [@fhp], LB methods are well suited for simulating a variety of physical systems in a parallel computing environment. As a result, the LB approach has found recent successes in a host of fluid dynamical problems, including flows in porous media [@guns], magnetohydrodynamics [@syc1], immiscible fluids [@guns] and turbulence [@succi; @syc2]. Its efficiency competes with, and in some cases exceeds, that of traditional numerical methods, while its physical interpretation is transparent.
Noticeably absent, though, from the list of successful applications of LG and LB methods is a model which can simulate the full set of thermohydrodynamic equations. Previous attempts at developing such a model have [*exclusively*]{} involved LG automata [@bzk; @syc3] whose Fermi-Dirac equilibrium distributions do not have sufficient flexibility to guarantee the correct form of the energy equation (3). LB methods are considerably more flexible, but have not, until now, been applied to this problem.
The thermohydrodynamic equations of classical kinetic theory result from a Chapman-Enskog expansion of the [*continuum*]{} Boltzmann equation with the assumption of a Maxwellian equilibrium distribution. Since an exact Maxwellian distribution with a continuous distribution of velocities, both in angle and magnitude, cannot be implemented on a system that is discrete in both space and time, we seek an alternative distribution which will nevertheless give rise to the same macroscopic physics. In this Letter we address this issue and introduce a LB scheme which can simulate the following continuity, momentum, and energy equations for viscous, compressible, and heat-conducting flows: $$\begin{aligned}
\frac{\partial n }{\partial t} + \frac{\partial}
{\partial x_{\alpha}}(nu_{\alpha}) = 0 &,\end{aligned}$$ $$\begin{aligned}
n\frac{\partial u_{\alpha}}{\partial t}+nu_{\beta}\frac{\partial
u_{\alpha}}{\partial x_{\beta}}=
-\frac{\partial p}{\partial x_{\alpha}}+\frac{\partial}
{\partial x_{\alpha}}(\lambda\frac{\partial u_{\gamma}}
{\partial x_{\gamma}})+\frac{\partial}
{\partial x_{\beta}}(\mu(\frac{\partial u_{\beta}}
{\partial x_{\alpha}}+\frac{\partial u_{\alpha}}
{\partial x_{\beta}}))
& ,\end{aligned}$$ and $$\begin{aligned}
n\frac{\partial \epsilon}{\partial t}+
nu_{\alpha}\frac{\partial
\epsilon}{\partial x_{\alpha}}=-p\frac{\partial u_{\gamma}}
{\partial x_{\gamma}}+
\frac{\partial}
{\partial x_{\beta}}(\kappa\frac{\partial T}
{\partial x_{\beta}})+
\mu(\frac{\partial u_{\alpha}}
{\partial x_{\beta}}+\frac{\partial u_{\beta}}
{\partial x_{\alpha}})\frac{\partial u_{\beta}}
{\partial x_{\alpha}}+\lambda
(\frac{\partial u_{\gamma}}{\partial x_{\gamma}})^{2}
&,\end{aligned}$$ where $n$ is the fluid mass density, $\epsilon$ is the internal energy per unit mass and is proportional to the temperature $T$, ${\bf u}$ is the local velocity, $p$ is the pressure, and $\lambda$, $\mu$, and $\kappa$ are the second viscosity, shear viscosity, and thermal conductivity, respectively.
The starting point of the LB method is the kinetic equation for the velocity distribution function, $f_{\sigma i} ({\bf x}, t)$: $$f_{\sigma i} ({\bf x} + {\bf e}_{\sigma i }, t+1) -
f_{\sigma i} ({\bf x},t) =
\Omega_{\sigma i},$$ where the nonnegative, real number $f_{\sigma i} ({\bf x},t)$ is the mass of fluid at lattice node ${\bf x}$ and time $t$, moving in direction $i$ with speed, $ |{\bf e}_{\sigma i }| = \sigma$, $\sigma =1,2,...N$, where $N$ is the number of speeds. The $\sigma = 0$ speed corresponds to the component of the fluid which is at rest. The term $\Omega_{\sigma i}$ represents the rate of change of $f_{\sigma i}$ due to collisions. For computational efficiency, it is desirable to find the minimal set of $\sigma$ and $i$, for which a coarse-graining of the kinetic equation (4) leads to the macroscopic dynamics of interest.
The microscopic dynamics associated with Equation (4) can be viewed as a two step process: free streaming and collision. During the free streaming step, $f_{\sigma i}({\bf x}+{\bf e}_{\sigma i})$ is replaced by $f_{\sigma i}(\bf{x})$. Thus, each site exchanges mass with its neighbors, i.e. sites connected by lattice vectors ${\bf e}_{\sigma i}$. In the collision step the distribution functions at each site then relax toward a state of local equilibrium. For simplicity, we consider the linearized, single-time-relaxation model of Bhatnagar, Gross, and Krook [@BGK], which has recently been applied to LB models [@syc1; @qian; @syc4; @koelman]: $$\Omega_{\sigma i}= -\frac{1}{\tau} (f_{\sigma i}-f_{\sigma i}^{eq}).$$ The collision operator $\Omega_{\sigma i}$ conserves the local mass, momentum and kinetic energy: $ \sum_{\sigma i} \Omega_{\sigma i} = 0,
\sum_{\sigma i} \Omega_{\sigma i}{\bf e}_{\sigma i } = 0,$ and $ \sum_{\sigma i} \Omega_{\sigma i}{\bf e}_{\sigma i}^{2}/2 = 0$, and the parameter, $\tau$, controls the rate at which the system relaxes to the local equilibrium, $f_{\sigma i}^{eq}$.
The LB method, unlike LGs, has considerable flexibility in the choice of the local equilibrium distribution. A general equilibrium distribution is given by a truncated power series in the local velocity $\bf{u}$, valid for $|{\bf u}| \ll 1$, $$f_{\sigma i}^{\rm eq} = A_{\sigma} + B_{\sigma}{\bf e}_{\sigma i}\cdot
{\bf u}
+C_{\sigma}({\bf e}_{\sigma i}\cdot{\bf u})^{2}
+D_{\sigma}u^{2}
+E_{\sigma}({\bf e}_{\sigma i}\cdot{\bf u})^{3}
+F_{\sigma}({\bf e}_{\sigma i}\cdot{\bf u})u^{2},$$ where the velocity is defined by: $n {\bf u} =
\sum_{\sigma i}f_{\sigma i}{\bf e}_{\sigma i}$. The coefficients, $A,B,...,F$, are functions of the local density $n = \sum_{\sigma i}f_{\sigma i}$ and internal energy $n \epsilon = \sum_{\sigma i}f_{\sigma i}({\bf e}_{\sigma i}-{\bf u})^2/2$, and their functional forms depend on the geometry of the underlying lattice.
The long-wavelength, low-frequency behavior of the this system is obtained by a Taylor series expansion of Equation (4) to second order in the lattice spacing and time step: $$\begin{aligned}
\frac{\partial f_{\sigma i}}{\partial t}
+{\bf e}_{\sigma i} \cdot {\bf \nabla}f_{\sigma i}
+\frac{1}{2} {\bf e}_{\sigma i} {\bf e}_{\sigma i} :{\bf \nabla}
{\bf \nabla} f_{\sigma i}
+{\bf e}_{\sigma i}
\cdot {\bf \nabla}\frac{\partial}{\partial t}f_{\sigma i}
+\frac{1}{2}\frac{\partial}{\partial t}\frac{\partial}{\partial t}
f_{\sigma i} = \Omega_{\sigma i}.\end{aligned}$$ In order to derive the macroscopic hydrodynamic equations, we adopt the following Chapman-Enskog multi-scale expansions. We expand the time derivative as $$\frac{\partial}{\partial t} =
\varepsilon \frac{\partial}{\partial t_{1}}+
\varepsilon^{2} \frac{\partial}{\partial t_{2}} + ...,$$ where the lower order terms in $\varepsilon$ vary more rapidly. Because we are interested in small departures from local equilibrium, we expand the distribution function as $$f_{\sigma i} = f_{\sigma i}^{eq}
+\varepsilon f_{\sigma i}^{(1)} +\varepsilon^{2}f_{\sigma i}^{(2)} + ...,$$ and the collision operator as $$\frac{\Omega_{\sigma i}}{\varepsilon} =
-\frac{1}{\tau \varepsilon}
(\varepsilon f_{\sigma i}^{(1)} +\varepsilon^{2}f_{\sigma i}^{(2)} + ...).$$
Substituting the above expansions into the kinetic equation, we find $$\begin{aligned}
\frac{\partial}{\partial t_{1}}f^{eq}_{\sigma i}
+{\bf e}_{\sigma i} \cdot {\bf \nabla} f^{eq}_{\sigma i} = -\frac{1}{\tau}
f^{(1)}_{\sigma i}\end{aligned}$$ to order $\varepsilon$, and $$\begin{aligned}
\frac{\partial}{\partial t_{1}}f^{(1)}_{\sigma i}+
\frac{\partial}{\partial t_{2}}f^{eq}_{\sigma i}
+{\bf e}_{\sigma i} \cdot {\bf \nabla} f^{(1)}_{\sigma i}
+\frac{1}{2} {\bf e}_{\sigma i} {\bf e}_{\sigma i} :{\bf \nabla}
{\bf \nabla} f^{eq}_{\sigma i}+
{\bf e}_{\sigma i} \cdot {\bf \nabla} \frac{\partial}{\partial t_{1}}
f^{eq}_{\sigma i}+ \frac{1}{2} \frac{\partial^{2}}{\partial t_{1}^{2}}
f^{eq}_{\sigma i}
=-\frac{1}{\tau}
f^{(2)}_{\sigma i}\end{aligned}$$ to order $\varepsilon^{2}$. With Equation (11) and some algebra, we can rewrite Equation (12) as: $$\begin{aligned}
\frac{\partial}{\partial t_{2}}f^{eq}_{\sigma i}
+(1-\frac{1}{2\tau})(\frac{\partial}{\partial t_{1}}f^{(1)}_{\sigma i}
+{\bf e}_{\sigma i} \cdot {\bf \nabla} f^{(1)}_{\sigma i}) =-\frac{1}{\tau}
f^{(2)}_{\sigma i}.\end{aligned}$$
Summing moments of Equations (11) and (13), we obtain to order $\varepsilon^2$, the continuity equation, $$\frac{\partial n}{\partial t} +\nabla \cdot n{\bf u} = 0,$$ the momemtum equation, $$\frac{\partial n{\bf u}}{\partial t} + \nabla \cdot {\bf \Pi} = 0,$$ and the energy equation, $$\frac{\partial n\epsilon}{\partial t} + \nabla\cdot {(n\epsilon{\bf u})} +
\nabla\cdot{\bf q} + {\bf P}:\nabla {\bf u} = 0.$$ The momentum flux tensor ${\bf \Pi}
= \sum_{\sigma i}
[f^{eq}_{\sigma i} + (1 - \frac{1}{2\tau})f^{(1)}_{\sigma i}]
{\bf e}_{\sigma i}
{\bf e}_{\sigma i}$; the heat flux, ${\bf q}_{\alpha} = (1/2)\sum_{\sigma i}
[f^{eq}_{\sigma i} + (1 - \frac{1}{2\tau})f^{(1)}_{\sigma i}]
({\bf e}_{\sigma i} - {\bf u})^2
({\bf e}_{\sigma i}- {\bf u})_{\alpha}$, and $\bf P$ is the pressure tensor, ${\bf P}_{\alpha\beta} = (1/2)\sum_{\sigma i}[f^{eq}_{\sigma i} + (1 -
\frac{1}{2\tau})f^{(1)}_{\sigma i}]
({\bf e}_{\sigma i} - {\bf u})_{\alpha}
({\bf e}_{\sigma i}- {\bf u})_{\beta}$.
To recover the Euler equations, we neglect the order $\varepsilon^2$ terms and impose four further constraints on the equilibrium distribution function. The first of these constraints requires that the momentum flux tensor, ${\bf \Pi}_{\alpha \beta}^{eq}$, be isotropic. The velocity independent portion of the tensor is then identified as the pressure, and this immediately results in the equation of state for an ideal gas, $p=n\epsilon$. The remaining two constraints require that the convective terms be Galilean invariant, and that the heat flux vanish to first order in $\varepsilon$, ${\bf q}^{(eq)}=0$. Thus we obtain the equations for compressible, inviscid and nonconducting flow of a monatomic gas.
Retaining terms to order $\varepsilon^2$ and imposing two additional constraints, we recover the Navier-Stokes level equations. These constraints are that the momentum flux tensor, ${\bf \Pi}^{(1)}$, be isotropic and that the heat flux, ${\bf q}^{(1)}$, be proportional to the gradient of the temperature: ${\bf q}^{(1)}
\sim {\bf \nabla}T$. Note that the order $\varepsilon^2$ terms describe diffusive processes, and, as assumed in Equation (8), evolve on a slower time scale than the convective terms associated with the order $\varepsilon$ Euler equations.
To demonstrate the utility of the above LB method, we apply it to a two-dimensional triangular lattice. The model has one rest particle state, $\sigma=0$, for which ${\bf e}_{\sigma i}=0$, and two nonzero speeds for which ${\bf e}_ {\sigma i}= \sigma (\cos{(2\pi i/6)},
\sin{(2\pi i/6)})$ for $i = 0, 1\ldots,6$ and $\sigma = 1,2$. The extension to three dimensions is straightforward and will be discussed elsewhere [@frank2].
For this lattice geometry and the constraints discussed above, we can solve for the coefficients of the distribution function. One possible solution is the following: $$A_{0}= -\frac{5}{2}n\epsilon + n + 2n\epsilon^{2},
A_{1}= \frac{4}{9}n\epsilon -\frac{4}{9}n\epsilon^{2},
A_{2}= \frac{1}{9}n\epsilon^{2} -\frac{1}{36}n\epsilon,$$ $$B_{1}= \frac{4}{9}n - \frac{4}{9}n\epsilon,
B_{2}= \frac{1}{9}n\epsilon - \frac{1}{36}n,$$ $$C_{1}= \frac{8}{9}n - \frac{4}{3}n\epsilon,
C_{2}= -\frac{1}{72}n + \frac{1}{12}n\epsilon,$$ $$D_{0}= -\frac{5}{4}n + 2n\epsilon,
D_{1}= -\frac{2}{9}n + \frac{2}{9}n\epsilon,
D_{2}= \frac{1}{72}n - \frac{1}{18}n\epsilon,$$ $$E_{1}= -\frac{4}{27}n,
E_{2}= \frac{1}{108}n,
F_{1} = 0,
F_{2}=0$$
Identifying the coefficients in Equations (1) - (3) with the corresponding terms from the Chapman-Enskog expansion, we determine the values for the transport coefficients. The shear viscosity and the thermal conductivity are given by, $\mu = n\epsilon(\tau-\frac{1}{2})$ and $\kappa = 2n\epsilon (\tau-\frac{1}{2})$, respectively, and yield a Prandtl number, $Pr=1/2$. As in the case of a monatomic gas, the bulk viscosity vanishes because $\lambda = -\mu$.
We carried out four numerical tests to determine the accuracy of this method for simulating Equations (1)-(3). Each test focused on one aspect of thermohydrodynamic transport.
We determined the viscosity $\mu$ by simulating an isothermal Poiseuille flow. Numerical results demonstate that the model accurately reproduces a parabolic momentum profile (not shown here). The viscosity is related to the momentum at the channel center by $\mu = W^{2}f/8u_{cen}$ where $f$ is the magnitude of the forcing, $u_{cen}$ is the velocity at the center, and W is the channel width [@kad]. In Figure 1, we show the dependence of viscosity on the relaxation time $\tau$ and two internal energies $\epsilon$. The resulting viscosities from measurements agree with the Chapman-Enskog theory to within around 1$\%$ over the entire range of parameters simulated.
We determined the thermal conductivity $\kappa$ by measuring the heat transfer across a temperature gradient, using Fourier’s law $\bf{q} = - \kappa \bf{\nabla}T$. By fixing the temperatures at the channel walls, we obtain a linear temperature profile and thus a constant gradient. Again, numerical results agree with the theoretical predictions quite well. Since the thermal conductivity has the same functional form as the viscosity, we also display the results in Figure 1.
For the two dimensional LB scheme, linearized perturbation theory gives a simple relation between the adiabatic sound speed, $c_s$ and internal energy: $c_s = \sqrt{2\epsilon}$. In Figure 2, we present the sound speed as a function of internal energy for both numerical measurements and theory – the agreement is evident.
We simulated a Couette shear flow with a temperature gradient between the boundaries[@whi]. For small temperature gradients the pressure is essentially constant across the channel, and the temperature profile has an analytic solution given by $\epsilon^* = (\epsilon - \epsilon_{0})
/(\epsilon_{1} - \epsilon_{0}) = (1/2)(1+y^*)+(Br/8)(1-y^{*2})$, where $y^*$ is the normalized distance from the center of the channel, $\epsilon_{1}$ and $\epsilon_{0}$ are the internal energies of the upper and lower walls, respectively. The Brinkman number, $Br$ is the product of the Prandtl and Eckert numbers. The agreement between theory and simulation, as shown in Figure 3, demonstrates the validity of the method in simulating flows in which energy dissipation is an important factor.
In conclusion, we have developed a lattice Boltzmann scheme for the simulation of viscous, compressible, heat-conducting flows of an ideal monatomic gas. The kinetics of this model can be easily implemented on a parallel architecture machine. We have demonstrated theoretically and numerically that the macroscopic behavior of this model corresponds to that of Equations (1) - (3). Several issues remain. First, the current model uses the single-time-relaxation approximation, and this restricts simulations to flows with Prandtl number $Pr= 1/2$. In order to simulate flows with other Prandtl numbers, we should use a full matrix collision operator, which leads to a multi-time scale relaxation [@frank2]. Second, the equation of state in the present model is that of ideal monatomic gas. To simulate non-ideal gases we may incorporate some internal degrees of freedom. An analysis of the numerical stability of the current model and its benchmarking against other computational fluid dynamical schemes are under investigation.
We thank G. D. Doolen, D. W. Grunau, B. Hasslacher, S. A. Janowsky, J. L. Lebowitz, L. Luo, R. Mainieri and W. Matthaeus for encouragement and helpful discussions. This work is supported by the US Department of Energy at Los Alamos National Laboratory. Numerical simulations were performed on the CM-200 at the Advanced Computing Laboratory at Los Alamos National Laboratory.
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Figure Captions {#figure-captions .unnumbered}
===============
Fig.1 : Shear viscosity ($+$) and thermal conductivity ($\Diamond$), as functions of relaxation parameter $\tau$. Upper curve corresponds to internal energy, $\epsilon=0.625$, lower curve $\epsilon=0.5$. The solid lines are the theoretical predictions.
Fig.2 : Numerical simulations of adiabatic sound speed ($+$) as a function of internal energy $\epsilon$. The solid line is the function $\sqrt{2\epsilon}$.
Fig.3 : Normalized internal energy, $\epsilon^*$ for Couette flow with heat transfer for Brinkman numbers $Br=5$ ($+$) and $Br=10$ ($\Box$). The upper wall is moving with speed $U_{1} = 0.1$, and the lower wall is stationary. The solid lines represent analytical results.
[^1]: Permanent Address: Advanced Projects Research Incorporated, 5301 N. Commerce Ave., Suite A, Moorpark, CA 93021
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The NP-hard problem of optimizing a quadratic form over the unimodular vector set arises in radar code design scenarios as well as other active sensing and communication applications. To tackle this problem (which we call unimodular quadratic programming (UQP)), several computational approaches are devised and studied. A specialized local optimization scheme for UQP is introduced and shown to yield superior results compared to general local optimization methods. Furthermore, a **m**onotonically **er**ror-bound **i**mproving **t**echnique (MERIT) is proposed to obtain the global optimum or a local optimum of UQP with good sub-optimality guarantees. The provided sub-optimality guarantees are case-dependent and generally outperform the $\pi/4$ approximation guarantee of semi-definite relaxation. Several numerical examples are presented to illustrate the performance of the proposed method. The examples show that for cases including several matrix structures used in radar code design, MERIT can solve UQP efficiently in the sense of sub-optimality guarantee and computational time.'
author:
- 'Mojtaba Soltanalian\* and Petre Stoica, [^1]'
bibliography:
- 'Des\_uni\_radarcodes\_not\_hard\_1.bib'
nocite:
- '[@sergio-multiusercomplexity]'
- '[@MaVo-blindML]'
- '[@Cui_OFDM_spheredecoing]'
- '[@levanon]'
- '[@seq_book]'
- '[@unimodular_good]'
- '[@soltanalian_comp_sec_design]'
title: Designing Unimodular Codes via Quadratic Optimization is not Always Hard
---
radar codes, unimodular codes, quadratic programming.
Introduction {#sec:intro}
============
Unimodular codes are used in many active sensing and communication systems mainly as a result of the their optimal (i.e. unity) peak-to-average-power ratio (PAR). The design of such codes can be often formulated as the optimization of a quadratic form (see sub-section \[subsec:background\] for examples). Therefore, we will study the problem $$\begin{aligned}
\label{eq:UQP}
\mbox{UQP: }~ \max_{\bs \in \Omega^n} \bs^H \bR \bs\end{aligned}$$ where $\bR \in \complexC^{n \times n}$ is a given Hermitian matrix, $(.)^H$ denotes the vector/matrix Hermitian transpose, $\Omega$ represents the unit circle, i.e. $\Omega=\{s \in \complexC : ~| s | =1 \}$ and UQP stands for Unimodular Quadratic Program(ming).
Motivating Applications {#subsec:background}
-----------------------
To motivate the UQP formulation considered above, we present four scenarios in which a design problem in active sensing or communication boils down to an UQP. $\bullet$ *Designing codes that optimize the SNR or the CRLB:* We consider a monostatic radar which transmits a linearly encoded burst of pulses. The observed backscattered signal $\bv$ can be written as (see, e.g. [@DeMaio-similarity]): $$\begin{aligned}
\bv= a (\bc \odot \bp) + \bw,\end{aligned}$$ where $a$ represents both channel propagation and backscattering effects, $\bw$ is the disturbance/noise component, $\bc$ is the unimodular vector containing the code elements, $\bp=( 1, e^{j2\pi f_d T_r}, \cdots, e^{j2\pi (n-1) f_d T_r})^T$ is the temporal steering vector with $f_d$ and $T_r$ being the target Doppler frequency and pulse repetition time, respectively, and the symbol $\odot$ stands for the Hadamard (element-wise) product of matrices.
Under the assumption that $\bw$ is a zero-mean complex-valued circular Gaussian vector with known positive definite covariance matrix $\mathbb{E}[\bw \bw^H]=\bM$, the signal-to-noise ratio (SNR) is given by [@DeMaio-selection] $$\begin{aligned}
\label{eq:SNR}
\mbox{SNR}=| a |^2 \bc^H \bR \bc\end{aligned}$$ where $\bR = \bM^{-1} \odot (\bp \bp^H)^*$ with $(.)^*$ denoting the vector/matrix complex conjugate. Therefore, the problem of designing codes optimizing the SNR of the radar system can be formulated directly as an UQP. Additionally, the Cramer-Rao lower bound (CRLB) for the target Doppler frequency estimation (which yields a lower bound on the variance of any unbiased target Doppler frequency estimator) is given by [@DeMaio-selection] $$\begin{aligned}
\mbox{CRLB}&=&\left( 2 | a |^2 (\bc \odot \bp \odot \bu)^H \bM^{-1} (\bc \odot \bp \odot \bu) \right)^{-1} \\ \nonumber
&=& \left( 2 | a |^2 \bc^H \bR' \bc \right)^{-1}\end{aligned}$$ where $\bu=(0, j 2 \pi T_r, \cdots, j 2 \pi (n-1) T_r )^T$ and $\bR' = \bM^{-1} \odot (\bp \bp^H)^* \odot (\bu \bu^H)^*$. Therefore the minimization of CRLB can also be formulated as an UQP. For the simultaneous optimization of SNR and CRLB see [@DeMaio-selection].
$\bullet$ *Synthesizing cross ambiguity functions (CAFs):* The ambiguity function (which is widely used in active sensing applications [@seq_book][@levanon]) represents the two-dimensional response of the matched filter to a signal with time delay $\tau$ and Doppler frequency shift $f$. The more general concept of cross ambiguity function occurs when the match filter is replaced by a mismatched filter. The cross ambiguity function (CAF) is defined as $$\begin{aligned}
\label{eq:CAF}
\chi(\tau,f)=\int_{-\infty}^{\infty} u(t) v^*(t+\tau) e^{j 2 \pi f t} dt\end{aligned}$$ where $u(t)$ and $v(t)$ are the transmit signal and the receiver filter, respectively (the ambiguity function is obtained from (\[eq:CAF\]) with $v(t) = u(t)$). In several applications $u(t)$ and $v(t)$ are given by: $$\begin{aligned}
\label{eqWaveDef}
u(t) = \sum_{k=1}^n x_k p_k(t),~~~v(t) = \sum_{k=1}^n y_k p_k(t)\end{aligned}$$ where $\{p_k(t)\}$ are pulse-shaping functions (with the rectangular pulse as a common example), and $$\begin{aligned}
\bx=( x_1 \cdots x_n )^T,~~\by=( y_1 \cdots y_n)^T\end{aligned}$$ are the code and, respectively, the filter vectors. The design problem of synthesizing a desired CAF has a small number of free variables (i.e. the entries of the vectors $\bx$ and $\by$) compared to the large number of constraints arising from two-dimensional matching criteria (to a given $|\chi(\tau,f)|$). Therefore, the problem is generally considered to be difficult and there are not many methods to synthesize a desired (cross) ambiguity function. Below, we describe briefly the cyclic approach of [@Hao_CAF] for CAF design.
The problem of matching a desired $|\chi(\tau,f)| = d(\tau,f)$ can be formulated as the minimization of the criterion [@Hao_CAF] $$\begin{aligned}
\label{eq:CAF_crit}
g(\bx,\by,\phi)= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} w(\tau,f) \left| d(\tau,f) e^{j \phi(\tau,f)} - \by^H \bJ(\tau,f) \bx \right|^2 \ d\tau df \end{aligned}$$ where $\bJ(\tau,f) \in \complexC^{n \times n}$ is given, $w(\tau,f)$ is a weighting function that specifies the CAF area which needs to be emphasized and $\phi(\tau,f)$ represent auxiliary phase variables. It is not difficult to see that for fixed $\bx$ and $\by$, the minimizer $\phi(\tau,f)$ is given by $\phi(\tau,f)= \arg \{ \by^H \bJ(\tau,f) \bx \}$. For fixed $\phi(\tau,f)$ and $\bx$, the criterion $g$ can be written as $$\begin{aligned}
g(\by)&=&\by^H \bD_1 \by - \by^H \bB^H \bx - \bx^H \bB \by + \mbox{const}_1 \\ \nonumber
&=& (\by - \bD_1^{-1} \bB^H \bx)^H \bD_1 (\by - \bD_1^{-1} \bB^H \bx) + \mbox{const}_2\end{aligned}$$ where $\bB$ and $\bD_1$ are given matrices in $\complexC^{n \times n}$ [@Hao_CAF]. Due to practical considerations, the transmit coefficients $\{x_k\}$ must have low PAR values. However, the receiver coefficients $\{y_k\}$ need not be constrained in such a way. Therefore, the minimizer $\by$ of $g(\by)$ is given by $\by=\bD_1^{-1} \bB^H \bx$. Similarly, for fixed $\phi(\tau,f)$ and $\by$, the criterion $g$ can be written as $$\begin{aligned}
\label{eq:CAF_UQP1}
g(\bx)= \bx^H \bD_2 \bx - \bx^H \bB \by - \by^H \bB^H \bx + \mbox{const}_3\end{aligned}$$ where $\bD_2 \in \complexC^{n \times n}$ is given [@Hao_CAF]. If a unimodular code vector $\bx$ is desired then the optimization of $g(\bx)$ is an UQP as $g(\bx)$ can be written as $$\begin{aligned}
\label{eq:CAF_UQP2}
g(\bx)= \left( \begin{array}{c}
e^{j \varphi} \bx \\
e^{j \varphi} \\
\end{array} \right)^H
\left( \begin{array}{cc}
\bD_2 & -\bB \by \\
-(\bB \by)^H & 0 \\
\end{array} \right)\left( \begin{array}{c}
e^{j \varphi} \bx \\
e^{j \varphi} \\
\end{array} \right)+ \mbox{const}_3\end{aligned}$$ where $\varphi \in [0, 2 \pi)$ is a free phase variable.
$\bullet$ *Steering vector estimation in adaptive beamforming:* Consider a linear array with $n$ antennas. The output of the array at time instant $k$ can be expressed as [@Petre-beamformig] $$\begin{aligned}
\bx_k= s_k \ba +\bn_k\end{aligned}$$ with $\{s_k\}$ being the signal waveform, $\ba$ the associated steering vector (with $|[\ba]_l|=1$, $1 \leq l \leq n$), and $\bn_k$ the vector accounting for all independent interferences.
The true steering vector is usually unknown in practice, and it can therefore be considered as an unimodular vector to be determined [@steer-unique]. Define the sample covariance matrix of $\{\bx_k\}$ as $%\begin{eqnarray}
\widehat{\bR}= \frac{1}{T} \sum_{k=1}^{T} \bx_k \bx^H_k
$ where $T$ is the number of training data samples. Assuming some prior knowledge on $\ba$ (which can be represented by $\arg(\ba)$ being in a given sector $\Theta$), the problem of estimating the steering vector can be formulated as [@Khabbazibasmenj] $$\begin{aligned}
\label{eq:UQP-related}
\min_{\ba}~~ \ba^H \widehat{\bR}^{-1} \ba~ \\ \nonumber
\mbox{s.t. } \arg(\ba) \in \Theta,\end{aligned}$$ hence an UQP-type problem. Such problems can be tackled using general local optimization techniques or the optimization scheme introduced in Section \[sec:local\]. $\bullet$ *Maximum likelihood (ML) detection of unimodular codes:* Assume the linear model $$\begin{aligned}
\by = \bQ \bs + \bn\end{aligned}$$ where $\bQ$ represents a multiple-input multiple-output (MIMO) channel, $\by$ is the received signal, $\bn$ is the additive white Gaussian noise and $\bs$ contains the unimodular symbols which are to be estimated. The ML detection of $\bs$ may be stated as $$\begin{aligned}
\widehat{\bs}_{ML}= \arg \min_{\bs \in \Omega^n} \| \by - \bQ \bs \|_2\end{aligned}$$ It is straightforward to verify that the above optimization problem is equivalent to the UQP [@Jalden]: $$\begin{aligned}
\min_{\overline{\bs} \in \Omega^{n+1}}~ \overline{\bs}^H \bR \overline{\bs}\end{aligned}$$ where $$\begin{aligned}
\bR=\left( \begin{array}{cc}
\bQ^H \bQ & -\bQ^H \by \\
-\by^H \bQ & 0 \\
\end{array} \right) \mbox{, } \overline{\bs} = \left( \begin{array}{c}
e^{j \varphi} \bs \\
e^{j \varphi} \\
\end{array} \right)\end{aligned}$$ and $\varphi \in [0, 2 \pi)$ is a free phase variable.
Related Work {#subsec:previous}
------------
In [@zhang-complexquad], the NP-hardness of UQP is proven by employing a reduction from an NP-complete matrix partitioning problem. The UQP in (\[eq:UQP\]) is often studied along with the following (still NP-hard) related problem in which the decision variables are discrete: $$\begin{aligned}
\label{eq:MUQP}
\mbox{$m$-UQP:}~ \max_{\bs \in \Omega^n_m} \bs^H \bR \bs \end{aligned}$$ where $\Omega_m=\{ 1, e^{j\frac{2 \pi}{m}},\cdots,e^{j\frac{2 \pi}{m}(m-1)} \}$. Note that the latter problem coincides with the UQP in (\[eq:UQP\]) as $m \rightarrow \infty$. The authors of [@low-rank] show that when the matrix $\bR$ is rank-deficient (more precisely, when $d=$rank$(\bR)$ behaves like $\mathcal{O}(1)$ with respect to the problem dimension) the $m$-UQP problem can be solved in polynomial-time and they propose a $\mathcal{O}((mn/2)^{2d})$-complexity algorithm to solve (\[eq:MUQP\]). However, such algorithms are not applicable to the UQP which corresponds to an infinite $m$.
Studies on polynomial-time algorithms for UQP (and $m$-UQP) have been extensive (e.g. see \[9\]-\[19\] and the references therein). In particular, the semi-definite relaxation (SDR) technique has been one of the most appealing approaches to the researchers. To derive an SDR, we note that $\bs^H \bR \bs = \mbox{tr} (\bs^H \bR \bs) = \mbox{tr} (\bR \bs \bs^H )$. Hence, the UQP can be rewritten as $$\begin{aligned}
\label{eq:rUQP0}
\max_{\bS} \, \mbox{tr} (\bR \bS ) ~~~~~~~ \\ \nonumber
\mbox{s.t. } \bS= \bs \bs^H,~ \bs \in \Omega^n.\end{aligned}$$ If we relax (\[eq:rUQP0\]) by removing the rank constraint on $\bS$ and the unimodularity constraint on $\bs$ then the result is a semi-definite program: $$\begin{aligned}
\label{eq:rUQP}
\mbox{SDP: }~ \max_{\bS} \mbox{tr} (\bR \bS )~~~~~~~ \\ \nonumber
\mbox{s.t. } [\bS]_{k,k}=1,~~ 1 \leq k \leq n, ~~~\\ \nonumber
~~~~~~ \bS \mbox{ is positive semi-definite.}\end{aligned}$$ The above SDP can be solved in polynomial time using interior-point methods [@convex_boyd]. The approximation of the UQP solution based on the SDP solution can be accomplished in several ways. For example, we can approximate the phase values of the solution $\bs$ using a rank-one approximation of $\bS$. A more effective approach for guessing $\bs$ is based on randomized approximations (see [@zhang-complexquad], [@Goemans] and [@DeMaio-PAR]). A detailed guideline for randomized approximation of the UQP solution can be found in [@DeMaio-PAR]. In addition, we refer the interested reader to the survey of the rich literature on SDR in [@SDR_mag].
Analytical assessments of the quality of the UQP solutions obtained by SDR and randomized approximation are available. Let $v_{SDR}$ be the expected value of the UQP objective at the obtained randomized solution. Let $v_{opt}$ represent the optimal value of the UQP objective. We have $$\begin{aligned}
\gamma v_{opt} \leq v_{SDR} \leq v_{opt}\end{aligned}$$ with the sub-optimality guarantee coefficient $\gamma=\pi /4$ [@zhang-complexquad][@so-approxSDR]. Note that the sub-optimality coefficient of the solution obtained by SDR can be arbitrarily close to $\pi /4$ (e.g., see [@so-approxSDR]).
Contributions of this Work {#subsec:contributions}
--------------------------
Besides SDR, the literature does not offer many other numerical approaches to tackle UQP. In this paper, a specialized local optimization scheme for UQP is proposed. The proposed computationally efficient local optimization approach can be used to tackle UQP as well as improve upon the solutions obtained by other methods such as SDR. Furthermore, a **m**onotonically **er**ror-bound **i**mproving **t**echnique (called MERIT) is introduced to obtain the global optimum or a local optimum of UQP with good sub-optimality guarantees. Note that:
- MERIT provides case-dependent sub-optimality guarantees. To the best of our knowledge, such guarantees for UQP were not known prior to this work. Using the proposed method one can generally obtain better performance guarantees compared to the analytical worst-case guarantees (such as $\gamma=\pi /4$ for SDR).
- The provided case-dependent sub-optimality guarantees are of practical importance in decision making scenarios. For instance in some cases the UQP solution obtained by SDR (or other optimization methods) might achieve good objective values. However, unless the goodness of the obtained solution is known (this goodness can be determined using the proposed bounds), the solution cannot be trusted.
- Using MERIT, numerical evidence is provided to show that several UQPs (particularly those which occur in active sensing code design) can be solved efficiently without sacrificing the solution accuracy.
Finally, we believe that the general ideas of this work can be adopted to tackle $m$-UQP as the finite alphabet case of UQP. However, a detailed study of $m$-UQP is beyond the scope of this paper.
Organization of the Paper {#subsec:organization}
-------------------------
The rest of this work is organized as follows. Section \[sec:properties\] discusses several properties of UQP. Section \[sec:local\] introduces a specialized local optimization method. Section \[sec:cone\] presents a cone approximation that is used in Section \[sec:global\] to derive the algorithmic form of MERIT for UQP. Several numerical examples are provided in section \[sec:numerical\]. Finally, Section \[sec:conclusion\] concludes the paper.
*Notation:* We use bold lowercase letters for vectors/sequences and bold uppercase letters for matrices. $(.)^T$ denotes the vector/matrix transpose. $\bone$ and $\bzero$ are the all-one and all-zero vectors/matrices. $\be_k$ is the $k^{th}$ standard basis vector in $\complexC^n$. $\| \bx \|_n$ or the $l_n$-norm of the vector $\bx$ is defined as $\left( \sum_k |\bx(k)|^n \right)^\frac{1}{n}$ where $\{ \bx(k) \}$ are the entries of $\bx$. The Frobenius norm of a matrix $\bX$ (denoted by $\| \bX \|_F$) with entries $\{ \bX(k,l) \}$ is equal to $\left( \sum_{k,l} |\bX(k,l)|^2 \right)^\frac{1}{2}$. We use $\Re(\bX)$ to denote the matrix obtained by collecting the real parts of the entries of $\bX$. The matrix $e^{j \bX}$ is defined element-wisely as $\left[e^{j \bX}\right]_{k,l}=e^{j [\bX]_{k,l}}$. $\arg(.)$ denotes the phase angle (in radians) of the vector/matrix argument. $\mathbb{E}[.]$ stands for the expectation operator. $\mathbf{Diag}(.)$ denotes the diagonal matrix formed by the entries of the vector argument, whereas $\mathbf{diag}(.)$ denotes the vector formed by collecting the diagonal entries of the matrix argument. $\sigma_k(\bX)$ represents the $k^{th}$ maximal eigenvalue of $\bX$. Finally, $\realR$ and $ \complexC $ represent the set of real and complex numbers, respectively.
Some Properties of UQP {#sec:properties}
======================
In this section, we study several properties of UQP. The discussed properties lay the grounds for a better understanding of UQP as well as the tools proposed to tackle it in the following sections.
Basic Properties {#subsec:basic}
----------------
The UQP formulation in (\[eq:UQP\]) covers both maximization and minimization of quadratic forms (one can obtain the minimization of the quadratic form in (\[eq:UQP\]) by considering $-\bR$ in lieu of $\bR$). In addition, without loss of generality, the Hermitian matrix $\bR$ can be assumed to be positive (semi)definite. If $\bR$ is not positive (semi)definite, we can make it so using the diagonal loading technique (i.e. $\bR \leftarrow \bR + \lambda \bI$ where $\lambda \geq -\sigma_n(\bR )$). Note that such a diagonal loading does not change the solution of UQP as $\bs^H (\bR + \lambda \bI) \bs = \bs^H \bR \bs + \lambda n$. Next, we note that if $\widetilde{\bs}$ is a solution to UQP then $e^{j \phi} \widetilde{\bs}$ (for any $\phi \in [0, 2 \pi)$) is also a valid solution. To establish connections among different UQPs, Theorem 1 presents a bijection among the set of matrices leading to the same solution.
\[th:K(s)\] Let $\mathcal{K}(\bs) $ represent the set of matrices $\bR$ for which a given $\bs \in \Omega^n$ is the global optimizer of UQP. Then
1. $\mathcal{K}(\bs)$ is a convex cone.
2. For any two vectors $\bs_1,\bs_2 \in \Omega^n$, the one-to-one mapping (where $\bs_0=\bs_1^* \odot \bs_2$) $$\begin{aligned}
\bR \in \mathcal{K}(\bs_1) \Longleftrightarrow \bR \odot (\bs_0 \bs_0^H) \in \mathcal{K}(\bs_2)\end{aligned}$$ holds among the matrices in $\mathcal{K}(\bs_1)$ and $\mathcal{K}(\bs_2)$.
See the Appendix.
It is interesting to note that in light of the above result, the characterization of the cone $\mathcal{K}(\bs)$ for any given $\bs=\widetilde{\bs}$ leads to a complete characterization of all $\mathcal{K}(\bs)$, $\bs \in \Omega^n$, and thus solving any UQP. However, the NP-hardness of UQP suggests that such a characterization cannot be expected. Further discussions regarding the characterization of $\mathcal{K}(\bs)$ are deferred to Section \[sec:cone\].
Analytical Solutions to UQP {#subsec:accurate}
---------------------------
There exist cases for which the analytical global optima of UQP are easy to obtain. In this sub-section, we consider two such cases which will be used later to provide an approximate characterization of $\mathcal{K}(\bs)$. A special example is the case in which $e^{j \arg(\bR)}$ (see the notation definition in \[subsec:organization\]) is a rank-one matrix. More precisely, let $%\begin{eqnarray}
\bR=\bR_1 \odot (\widetilde{\bs} \widetilde{\bs}^H)
$ where $\bR_1$ is a real-valued Hermitian matrix with non-negative entries and $\widetilde{\bs} \in \Omega^n$ (a simple special case of this example is when $\bR$ is a rank-one matrix itself). In this case, it can be easily verified that $\bR_1 \in \mathcal{K}(\bone_{n \times 1})$. Therefore, using Theorem 1 one concludes that $\bR \in \mathcal{K}(\widetilde{\bs})$ i.e. $\bs=\widetilde{\bs}$ yields the global optimum of UQP. As another example, Theorem 2 considers the case for which several largest eigenvalues of the matrix $\bR$ are identical.
\[th:uniqueness\] Let $\bR$ be a Hermitian matrix with eigenvalue decomposition $\bR=\bU \bSig \bU^H$. Suppose $\bSig$ is of the form $$\begin{aligned}
\bSig=\mathbf{Diag}([\underbrace{\sigma_1~ \cdots ~\sigma_1}_{m~\mbox{times}}~ \sigma_2~\cdots~ \sigma_{n-m+1}]^T)\end{aligned}$$ $$\begin{aligned}
\sigma_1 > \sigma_2 \geq \cdots \geq \sigma_{n-m+1} \nonumber\end{aligned}$$ and let $\bU_m$ be the matrix made from the first $m$ columns of $\bU$. Now suppose $\widetilde{\bs} \in \Omega^n$ lies in the linear space spanned by the columns of $\bU_m$, i.e. there exists a vector $\balpha \in \complexC^m$ such that $$\begin{aligned}
\label{eq:sualpha}
\widetilde{\bs}=\bU_m \balpha.\end{aligned}$$ Then $\widetilde{\bs}$ is a global optimizer of UQP.
Refer to the Appendix.
We end this section by noting that the solution to an UQP is not necessarily unique. For any set of unimodular vectors $ \{ \bs_1,\bs_2, \cdots , \bs_k \}$, $k \leq n$, we can use the Gram-Schmidt process to obtain a unitary matrix $\bU $ the first $k$ columns of which span the same linear space as $\bs_1,\bs_2, \cdots , \bs_k$. In this case, Theorem 2 suggests a method to construct a matrix $\bR$ (by choosing a $\bSig$ with $k$ identical largest eigenvalues) for which all $\bs_1,\bs_2, \cdots , \bs_k$ are global optimizers of the corresponding UQP.
Specialized Local Optimization of UQP {#sec:local}
======================================
Due to its NP-hard nature, UQP has in general a highly multi-modal optimization objective. Finding and studying the local optima of UQP is not only useful to tackle the problem itself (particularly for UQP-related problems such as (\[eq:UQP-related\])), but also to improve the UQP approximate solutions obtained by SDR or other optimization techniques. In this section, we introduce a computationally efficient procedure to obtain a local optimum of UQP. Note that, while the risk for this to happen in practice is nearly zero, local optimization methods can in theory converge to a saddle point. Consequently, in the sequel we let $L$ represent the set of all local optima and saddle points of UQP. Moreover, we assume that $\bR$ is positive definite. Consider the following relaxed version of UQP: $$\begin{aligned}
\label{eq:UQP-relaxed}
\mbox{(RUQP) }~ \max_{\bs_1,\bs_2 \in \Omega^n} \Re (\bs_1^H \bR \bs_2)\end{aligned}$$ We note that for fixed $\bs_2$ the maximizer of RUQP is given by $$\begin{aligned}
\label{eq:s1}
\bs_1=e^{j \arg(\bR \bs_2)}.\end{aligned}$$ Similarly, for any fixed $\bs_1$ the maximizer of RUQP is given by $$\begin{aligned}
\label{eq:s2}
\bs_2=e^{j \arg(\bR \bs_1)}.\end{aligned}$$ In the following, we show that such a cyclic maximization of (\[eq:UQP-relaxed\]) can be used to find local optima of UQP. It is not difficult to see that the criterion in (\[eq:UQP-relaxed\]) increases and is upper bounded (by $\sum_{k,l} | \bR(k,l)|$) through the iterations in (\[eq:s1\])-(\[eq:s2\]), thus the said iterations are convergent in the sense of associated objective value. Next consider the identity $$\begin{aligned}
\label{eq:ident}
2 \Re (\bs_1^H \bR \bs_2) &=& \bs_1^H \bR \bs_1 + \bs_2^H \bR \bs_2 \\ \nonumber &-& (\bs_1-\bs_2)^H \bR (\bs_1-\bs_2).\end{aligned}$$ Define $\varepsilon_\bs = \| \bs_1 - \bs_2\|_2^2$ and suppose that $\bs_2$ is fixed and its associated optimal $\bs_1$ is obtained by (\[eq:s1\]). It follows from (\[eq:ident\]) that $$\begin{aligned}
\label{eq:conv0}
\Re (\bs_1^H \bR \bs_2) \leq \frac{1}{2} \left(\bs_1^H \bR \bs_1 + \bs_2^H \bR \bs_2 \right) - \frac{\varepsilon_\bs}{2} \sigma_n (\bR).\end{aligned}$$ Now suppose $\bs'_2$ is the optimal vector in $\Omega^n$ obtained by (\[eq:s2\]) for the above $\bs_1$. Observe that $\Re (\bs_1^H \bR \bs'_2) \geq \Re (\bs_1^H \bR \bs_1)$ and that $\Re (\bs_1^H \bR \bs'_2) \geq \Re (\bs_1^H \bR \bs_2) \geq \Re (\bs_2^H \bR \bs_2) $ which imply $$\begin{aligned}
\label{eq:conv1}
\Re (\bs_1^H \bR \bs'_2) %&\geq& \max \{ \bs_1^H \bR \bs_1, \bs_2^H \bR \bs_2 \} \\ \nonumber
&\geq& \frac{1}{2} \left(\bs_1^H \bR \bs_1 + \bs_2^H \bR \bs_2 \right) \\ \nonumber
&\geq& \Re (\bs_1^H \bR \bs_2) + \frac{\varepsilon_\bs}{2} \sigma_n (\bR).\end{aligned}$$ It follows from (\[eq:conv1\]) that $$\begin{aligned}
\label{eq:sandwich}
\varepsilon_\bs \leq \frac{2}{\sigma_n (\bR)} \left| \Re (\bs_1^H \bR \bs'_2) - \Re (\bs_1^H \bR \bs_2) \right|.\end{aligned}$$ The right-hand side of (\[eq:sandwich\]) vanishes through the cyclic minimization in (\[eq:s1\])-(\[eq:s2\]) which implies that $\varepsilon_\bs$ converges to zero at the same time. Note that the above arguments can be repeated for fixed $\bs_1$. We conclude that the iterations in (\[eq:s1\])-(\[eq:s2\]) are convergent and also that they cannot converge to $(\bs_1,\bs_2)$ with $\bs_1 \neq \bs_2 $. Moreover, as $\bs^H \bR \bs = \Re (\bs_1^H \bR \bs_2)$ for any $\bs_1=\bs_2=\bs$ then any local optimum $(\bs_1,\bs_2)$ of RUQP satisfying $\bs_1=\bs_2=\bs$ yields a local optimum $\bs$ of UQP. Based on the above discussions, the cyclic optimization of RUQP can be used to find local optima of UQP. Particularly, starting from any vector $\bs^{(0)} \in \Omega^n$, the *power method*-like iterations $$\begin{aligned}
\label{eq:powermethod-like}
\bs^{(t+1)}= e^{j \arg(\bR \bs^{(t)})}\end{aligned}$$ converge to an element in $L$. As an aside remark, we show that the objective of UQP is also increasing through the iterations of (\[eq:powermethod-like\]). Using (\[eq:ident\]) with $\bs_1=\bs^{(t+1)}$, and $\bs_2=\bs^{(t)}$ ($\bs^{(t+1)} \neq \bs^{(t)}$) implies that $$\begin{aligned}
\bs^{(t+1) \, H } \bR \bs^{(t+1)} &>& - \bs^{(t) \, H} \bR \bs^{(t)} + 2 \Re (\bs^{(t+1) \, H} \bR \bs^{(t)}) \nonumber \\
&\geq& \bs^{(t) \, H} \bR \bs^{(t)}.\end{aligned}$$
Note that while (\[eq:powermethod-like\]) can obtain the local optima of UQP, it might not converge to every of them. To observe this, let $\widetilde{\bs_1}$ be a local optimum of UQP and initialize (\[eq:powermethod-like\]) with $\bs^{(0)}=\widetilde{\bs_1}$. Let $\widetilde{\bs_2}$ be another local optimum of UQP but with a larger value of UQP than that at $\widetilde{\bs_1}$. Now one can observe from (\[eq:ident\]) that if $\widetilde{\bs_2}$ is sufficiently close to $\widetilde{\bs_1}$ then the above power method-like iterations can move away from $\widetilde{\bs_1}$, meaning that they can converge to another local optimum of UQP with a larger value of the UQP objective than that at $\widetilde{\bs_1}$. Therefore, (\[eq:powermethod-like\]) bypasses some local optima of UQP with relatively small UQP objective values (which can be considered as an advantage compared to a general local optimization method). Moreover, one can note that there exist initializations for which (\[eq:powermethod-like\]) leads to the global optimum of UQP (i.e. the global optimum is not excluded from the local optima to which (\[eq:powermethod-like\]) can converge).
Next, we observe that any $\widetilde{\bs} \in L$ obtained by the above local optimization can be characterized by the equation $$\begin{aligned}
\label{eq:arg}
\arg(\widetilde{\bs})=\arg(\bR \widetilde{\bs}).\end{aligned}$$ We refer to the subset of $L$ satisfying (\[eq:arg\]) as the hyper points of UQP. Note that if $\widetilde{\bs} \in \Omega^n$ is a hyper point of UQP, then (\[eq:arg\]) follows from the convergence of (\[eq:powermethod-like\]). On the other hand, if (\[eq:arg\]) is satisfied, it implies the convergence of the iterations in (\[eq:powermethod-like\]) and as a result $\widetilde{\bs}$ being a hyper point of UQP. The characterization given in (\[eq:arg\]) is used below to motivate the characterization approach of Theorem 3.
Results on the cone $\mathcal{K}(\bs)$ {#sec:cone}
======================================
While a complete characterization of $\mathcal{K}(\bs)$ cannot be expected (due to the NP-hardness of UQP), approximate characterizations of $\mathcal{K}(\bs)$ are possible. The goal of this section is to provide an approximate characterization of the cone $\mathcal{K}(\bs)$ which can be used to tackle the UQP problem. Our main result is as follows:
\[th:cone-characterization\] For any given $\bs= ( e^{j \phi_1} , \cdots , e^{j \phi_n} )^T \in \Omega^n$, let $\{\bB_{k,l}\}$ be a set of matrices defined as $$\begin{aligned}
\bB_{k,l} &=& (\be_k \be_l^H + \be_l \be_k^H) \odot (\bs \bs^H)\end{aligned}$$ and $V_\bs=\{ \bB_{k,l}: 1 \leq k \leq l \leq n\} \cup \{- \bI_n \}$. Let $\mathcal{C}(V_\bs)$ represent the convex cone associated with the basis matrices in $V_\bs$. Also let $\mathcal{C}_\bs$ represent the convex cone of matrices with $\bs$ being their dominant eigenvector (i.e the eigenvector corresponding to the maximal eigenvalue). Then for any $\bR \in \mathcal{K}(\bs)$, there exists $\alpha_0 \geq 0$ such that for all $\alpha \geq \alpha_0$, $$\begin{aligned}
\label{eq:cone_eq}
\bR+ \alpha \bs \bs^H \in \mathcal{C}(V_\bs) \cup \mathcal{C}_\bs.\end{aligned}$$
The proof of Theorem 3 will be presented in several steps (Theorems 4-7 and thereafter). Note that we show that (\[eq:cone\_eq\]) can be satisfied even if $\bs$ is a hyper point of UQP (satisfying (\[eq:arg\])). However, since $\bs$ is the global optimum of UQP for all matrices in $\mathcal{C}_\bs$ and $\mathcal{C}(V_\bs)$, the case of $\alpha_0=0$ can occur only when $\bs$ is a global optimum of UQP associated with $\bR$. Suppose $\bs$ is a hyper point of UQP associated with a given positive definite matrix $\bR$, and let $\theta_{k,l} = [\arg(\bR)]_{k,l}$. We define the matrix $\bR_+$ as $$\begin{aligned}
\bR_+ (k,l) \! = \! \left\{ \begin{array}{ll}
\! | \bR (k,l) | \! \cos(\theta_{k,l}-(\phi_k - \phi_l)) \! & \!(k,l)\! \in \Theta, \\
0 & otherwise
\end{array} \right. \end{aligned}$$ where $\Theta$ represents the set of all $(k,l)$ such that $|\theta_{k,l}-(\phi_k - \phi_l)|< \pi / 2$. Now, let $\rho$ be a positive real number such that $$\begin{aligned}
\label{eq:lambda_ineq}
\rho > \max_{(k,l) \notin \Theta } \left\{ | \bR (k,l)\cos(\theta_{k,l}-(\phi_k - \phi_l)) | \right\}\end{aligned}$$ and consider the sequence of matrices $\{\bR^{(t)} \}$ defined (in an iterative manner) by $\bR^{(0)}=\bR$, and $$\begin{aligned}
\bR^{(t+1)}=\bR^{(t)} - (\bR_+^{(t)} - \rho \bone_{n \times n}) \odot (\bs \bs^H)\end{aligned}$$ for $t\geq 0$. The next two theorems (whose proofs are given in the Appendix) study some useful properties of the sequence $\{\bR^{(t)} \}$.
$\{\bR^{(t)} \}$ is convergent in at most two iterations: $$\begin{aligned}
\bR^{(t)} = \bR^{(2)},~~\forall~ t \geq 2.\end{aligned}$$
$\bR^{(t)}$ is a function of $\rho$. Let $\rho$ and $\rho'$ both satisfy the criterion (\[eq:lambda\_ineq\]). At the convergence of $\{\bR^{(t)}\}$ (which is attained for $t=2$) we have: $$\begin{aligned}
\label{eq:lambda_diff}
\bR^{(2)} (\rho') = \bR^{(2)} (\rho) + (\rho'-\rho) (\bs \bs^H).\end{aligned}$$
Using the above results, Theorems 6 (whose proof is given in the Appendix) and 7 pave the way for a constructive proof of Theorem 3.
If $\bs$ is a hyper point of the UQP associated with $\bR^{(0)}=\bR$ then it is also a hyper point of the UQPs associated with $\bR^{(1)} $ and $\bR^{(2)} $. Furthermore, $\bs$ is an eigenvector of $\bR^{(2)} $ corresponding to the eigenvalue $n \rho$.
If $\bs$ is a hyper point of UQP for $\bR^{(0)}=\bR$ then it will be the dominant eigenvector of $\bR^{(2)} $ if $\rho$ is sufficiently large. In particular, let $\mu$ be the largest eigenvalue of $\bR^{(2)}$ which belongs to an eigenvector other than $\bs$. Then for any $\rho \geq \mu / n$, $\bs$ is a dominant eigenvector of $\bR^{(2)} $.
We know from Theorem 6 that $\bs$ is an eigenvector of $\bR^{(2)} $ corresponding to the eigenvalue $n \rho$. However, if $\bs$ is not the dominant eigenvector of $\bR^{(2)} $, Theorem 5 implies that increasing $\rho$ would not change any of the eigenvalues/vectors of $\bR^{(2)} $ except that it increases the eigenvalue corresponding to $\bs$. As a result, for $\bs$ to be the dominant eigenvector of $\bR^{(2)}$ we only need $\rho$ to satisfy $n \rho \geq \mu$ or equivalently $\rho \geq \mu / n$, which concludes the proof.
Returning to Theorem 3, note that $\bR$ can be written as $$\begin{aligned}
\bR&=& \bR^{(0)} \\ \nonumber &=& \bR^{(2)} + ( \bR_+^{(0)} +\bR_+^{(1)} ) \odot (\bs \bs^H) - 2 \rho \bs \bs^H.\end{aligned}$$ For sufficiently large $\rho$ (satisfying both (\[eq:lambda\_ineq\]) and the condition of Theorem 7) we have that $$\begin{aligned}
\label{eq:lasteq}
\bR + 2 \rho \bs \bs^H = \bR^{(2)} + ( \bR_+^{(0)} +\bR_+^{(1)} ) \odot (\bs \bs^H)\end{aligned}$$ where $\bR^{(2)} \in \mathcal{C}_\bs$ and $(\bR_+^{(0)} +\bR_+^{(1)} ) \odot (\bs \bs^H) \in \mathcal{C}(V_\bs)$. Theorem 3 can thus be directly satisfied using Eq. (\[eq:lasteq\]) with $\alpha_0= 2 \rho $.
We conclude this section with two remarks. First of all, the above proof of Theorem 3 does not attempt to derive the minimal $\alpha_0$. In the following section we study a computational method to obtain an $\alpha_0$ which is as small as possible. Secondly, we can use $ \mathcal{C}(V_\bs) \cup \mathcal{C}_\bs$ as an approximate characterization of $\mathcal{K}(\bs)$ noting that the accuracy of such a characterization can be measured by the minimal value of $\alpha_0$. An explicit formulation of a sub-optimality guarantee for a solution of UQP based on the above $\mathcal{K}(\bs)$ approximation is derived in the following section.
MERIT for UQP {#sec:global}
=============
Using the previous results, namely the one-to-one mapping introduced in Theorem 1 and the approximation of $\mathcal{K}(\bs)$ derived in Section \[sec:cone\], we build a sequence of matrices (for which the UQP global optima are known) whose distance from a given matrix is decreasing. The proposed iterative approach can be used to solve for the global optimum of UQP or at least to obtain a local optimum (with an upper bound on the sub-optimality of the solution). The sub-optimality guarantees are derived noting that the proposed method decreases an upper bound on the sub-optimality of the obtained UQP solution in each iteration.
We know from Theorem 3 that if $\bs$ is a hyper point of the UQP associated with $\bR$ then there exist matrices $\bQ_\bs \in \mathcal{C}_\bs$, $\bP_\bs \in \mathcal{C}(V_\bs)$ and a scalar $\alpha_0 \geq 0$ such that $$\begin{aligned}
\label{eq:decomposition}
\bR+ \alpha_0 \bs \bs^H = \bQ_\bs+ \bP_\bs.\end{aligned}$$ Eq. (\[eq:decomposition\]) can be rewritten as $$\begin{aligned}
\bR+ \alpha_0 \bs \bs^H = (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H)\end{aligned}$$ where $\bQ_\bone \in \mathcal{C}_\bone$, $\bP_\bone \in \mathcal{C}(V_\bone)$. We first consider the case of $\alpha_0=0$ which corresponds to the global optimality of $\bs$.
Global Optimization of UQP (the Case of $\alpha_0=0$) {#subsec:alpha00}
-----------------------------------------------------
Consider the optimization problem: $$\begin{aligned}
\label{eq:opt_imp}
\min_{\bs \in \Omega^n, \bQ_\bone \in \mathcal{C}_\bone, \bP_\bone \in \mathcal{C}(V_\bone)} \| \bR - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F\end{aligned}$$ Note that, as $\mathcal{C}_\bone \cup \mathcal{C}(V_\bone) $ is a convex cone, the global optimizers $\bQ_\bone $ and $\bP_\bone$ of (\[eq:opt\_imp\]) for any given $\bs$ can be easily found. On the other hand, the problem of finding an optimal $\bs$ for fixed $\bR_\bone =\bQ_\bone+ \bP_\bone$ is non-convex and hence more difficult to solve globally (see below for details).
We will assume that $\bR_\bone$ is a positive definite matrix. To justify this assumption let $\overline{\bR}=\bR \odot (\bs \bs^H)^* $ and note that the eigenvalues of $\overline{\bR}$ are exactly the same as those of $\bR$, hence $\overline{\bR}$ is positive definite. Suppose that we have $$\begin{aligned}
\label{eq:cond}
\left\{ \begin{array}{l}
\bx^H \overline{\bR}\bx> \varepsilon, ~~ \forall \mbox{ unit-norm } \bx \in \complexC^{n \times 1}\\
\| \overline{\bR} - \bR_\bone \|_F \leq \varepsilon
\end{array} \right. \end{aligned}$$ for some $\varepsilon \geq 0$. It follows from (\[eq:cond\]) that $$\begin{aligned}
\bx^H \bR_\bone \bx &\geq& \bx^H \overline{\bR} \bx - | \bx^H \overline{\bR} \bx - \bx^H \bR_\bone \bx | \\ \nonumber
&>& \varepsilon - | \bx^H (\overline{\bR} - \bR_\bone ) \bx | \\ \nonumber
&\geq& \varepsilon - |\sigma_1(\overline{\bR} - \bR_\bone) | \\ \nonumber
&\geq& \varepsilon - \| \overline{\bR} - \bR_\bone \|_F \geq 0\end{aligned}$$ which implies that $\bR_\bone$ is also a positive definite matrix. The conditions in (\[eq:cond\]) can be met as follows. By considering only the component of $\bR_\bone$ in $\mathcal{C}(V_\bone)$ (namely $\bP_\bone$) we observe that any positive (i.e. with $\lambda>0$) diagonal loading of $\bR$, which leads to the same diagonal loading of $\overline{\bR}$ (as $\overline{\bR} + \lambda \bI = \bR \odot (\bs \bs^H)^* + \lambda \bI = (\bR+ \lambda \bI ) \odot (\bs \bs^H)^* $), will be absorbed in $\bP_\bone$. Therefore, a positive diagonal loading of $\bR$ does not change $\| \overline{\bR} - \bR_\bone \|_F$ but increases $\bx^H \overline{\bR} \bx$ by $\lambda$. We also note that due to $\| \overline{\bR} - \bR_\bone \|_F$ being monotonically decreasing through the iterations of the method, if the conditions in (\[eq:cond\]) hold for the solution obtained in any iteration, it will hold for all the iterations afterward. In the following, we study a suitable diagonal loading of $\bR$ that ensures meeting the conditions in (\[eq:cond\]). Next the optimization of the function in (\[eq:opt\_imp\]) is discussed through a separate optimization over the three variables of the problem.\
$\bullet$ *Diagonal loading of $\bR$:* As will be explained later, we can compute $\bQ_\bone$ and $\bP_\bone$, (hence $\bR_\bone=\bQ_\bone+\bP_\bone$) for any initialization of $\bs$. In order to guarantee the positive definiteness of $\bR_\bone$, define $$\begin{aligned}
\label{eq:dl1}
\varepsilon_0 \triangleq \| \overline{\bR} - \bR_\bone \|_F.\end{aligned}$$ Then we suggest to diagonally load $\bR$ with $\lambda > \lambda_0 = -\sigma_n(\bR) + \varepsilon_0$: $$\begin{aligned}
\label{eq:dl2}
\bR \leftarrow \bR + \lambda \bI.\end{aligned}$$ $\bullet$ *Optimization with respect to $\bQ_\bone$:* We restate the objective function of (\[eq:opt\_imp\]) as $$\begin{aligned}
&~& \| \bR - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F \\ \nonumber
&=& \| \underbrace{\left( \bR \odot (\bs \bs^{H})^* - \bP_\bone \right)}_{\bR_Q} - \bQ_\bone \|_F.\end{aligned}$$ Given $\bR_Q$, (\[eq:opt\_imp\]) can be written as $$\begin{aligned}
\label{eq:opt_q}
\min_{\bQ_\bone \in \mathcal{C}_\bone} \| \bR_Q - \bQ_\bone \|_F.\end{aligned}$$ In [@nearest-doubly-stochastic], the authors have derived an explicit solution for the optimization problem $$\begin{aligned}
\label{eq:exp_sol}
\min_{\bQ_\bone} \| \bR_Q - \bQ_\bone \|_F ~~~~\\ \nonumber
\mbox{s.t. } \bQ_\bone \bone = \rho \bone. \mbox{ ($\rho=$given)}\end{aligned}$$ The explicit solution of (\[eq:exp\_sol\]) is given by $$\begin{aligned}
\label{eq:frob_imply}
\bQ_\bone (\rho) &=& \rho \bI_n %\\ \nonumber &+&
+(\bI_n - \frac{\bone_{n \times n}}{n}) (\bR_Q - \rho \bI_n) (\bI_n - \frac{\bone_{n \times n}}{n}) \\ \nonumber
&=& \bR_Q + \frac{\rho}{n} \bone_{n \times n} %\\ \nonumber &-&
-\frac{2}{n} (\bR_Q \bone_{n \times n}) + \frac{1}{n^2} (\bone_{n \times n} \bR_Q \bone_{n \times n} )\end{aligned}$$ Note that $$\begin{aligned}
\bQ_\bone (\rho') - \bQ_\bone (\rho)= (\rho'-\rho) (\bone_{n \times 1}/\sqrt{n}) (\bone_{n \times 1}/\sqrt{n})^T %\nonumber \end{aligned}$$ which implies that except for the eigenpair $(\bone_{n \times 1}/\sqrt{n},\rho)$, all other eigenvalue/vectors are independent of $\rho$. Let $\rho_0$ represent the maximal eigenvalue of $\bQ_\bone (0)$ corresponding to an eigenvector other than $\bone_{n \times 1}/\sqrt{n}$. Therefore, (\[eq:opt\_q\]) is equivalent to $$\begin{aligned}
\label{eq:boils}
\min_{\rho} \| \bR_Q - \bQ_\bone (\rho) \|_F \\ \nonumber
\mbox{s.t. } \rho \geq \rho_0.~~~~~~~\end{aligned}$$ It follows from (\[eq:frob\_imply\]) that $$\begin{aligned}
\label{eq:75}
\| \bR_Q - \bQ_\bone (\rho) \|^2_F = \sum_{k=1}^{n} n \left| \frac{\rho}{n}- \frac{2 G_k}{n} + \frac{H}{n^2} \right|^2\end{aligned}$$ where $G_k$ and $H$ are the sum of the $k^{th}$ row and, respectively, the sum of all entries of $\bR_Q $. The $\rho$ that minimizes (\[eq:75\]) is given by $$\begin{aligned}
\rho = \frac{1}{n} \sum_{k=1}^{n} \Re \left(2 G_k - \frac{H}{n} \right) = \frac{H}{n}\end{aligned}$$ which implies that the minimizer $\rho=\rho_\star$ of (\[eq:boils\]) is equal to $$\begin{aligned}
\rho_\star = \left\{ \begin{array}{ll}
\frac{H}{n} & \frac{H}{n} \geq \rho_0, \\
\rho_0 & \mbox{otherwise.}
\end{array} \right.\end{aligned}$$ Finally, the optimal solution $\bQ_\bone$ to (\[eq:opt\_q\]) is given by $$\begin{aligned}
\label{eq:bq1_opt}
\bQ_\bone = \bQ_\bone (\rho_\star).\end{aligned}$$ $\bullet$ *Optimization with respect to $\bP_\bone$:* Similar to the previous case, (\[eq:opt\_imp\]) can be rephrased as $$\begin{aligned}
\label{eq:opt_p}
\min_{\bQ_\bone \in \mathcal{C}(V_\bone)} \| \bR_P - \bP_\bone \|_F\end{aligned}$$ where $\bR_P = \bR \odot (\bs \bs^H)^* - \bQ_\bone$. The solution of (\[eq:opt\_p\]) is simply given by $$\begin{aligned}
\label{eq:bp1_opt}
\bP_\bone (k,l)= \left\{ \begin{array}{ll}
\bR'_P (k,l) & \bR'_P(k,l) \geq 0 \mbox{ or } k=l, \\
0 & \mbox{otherwise}
\end{array} \right.\end{aligned}$$ where $\bR'_P = \Re\{\bR_P\}$.\
$\bullet$ *Optimization with respect to $\bs$:* Suppose that $\bQ_\bone $ and $\bP_\bone$ are given and that $\bR_\bone=\bQ_\bone+\bP_\bone$ is a positive definite matrix (see the discussion on this aspect following Eq. (\[eq:opt\_imp\])). We consider a relaxed version of (\[eq:opt\_imp\]), $$\begin{aligned}
\label{eq:opt_imp_relaxed}
\min_{\bs_1,\bs_2 \in \Omega^n } \| \bR - \bR_\bone \odot (\bs_1 \bs_2^H) \|_F \end{aligned}$$ The objective function in (\[eq:opt\_imp\_relaxed\]) can be re-written as $$\begin{aligned}
\label{eq:s1s2}
&~& \| \bR - \bR_\bone \odot (\bs_1 \bs_2^H) \|^2_F \\ \nonumber
&=& \| \bR - \mathbf{Diag}(\bs_1) \, \bR_\bone \, \mathbf{Diag}( \bs_2^*) \|^2_F \\ \nonumber
&=& \mbox{tr}(\bR^2) + \mbox{tr}( \bR_\bone^2) %\\ \nonumber &-&
- 2 \Re \{\mbox{tr}(\bR ~\mathbf{Diag}(\bs_1) \, \bR_\bone \, \mathbf{Diag}( \bs_2^*))\}.\end{aligned}$$ Note that only the third term of (\[eq:s1s2\]) is a function of $\bs_1$ and $\bs_2$. Moreover, it can be verified that [@horn1990matrix] $$\begin{aligned}
\mbox{tr}(\bR ~\mathbf{Diag}(\bs_1) \, \bR_\bone \, \mathbf{Diag}( \bs_2^*)) = \bs_2^H (\bR \odot \bR_\bone^T) \bs_1.\end{aligned}$$ As $\bR \odot \bR_\bone^T$ is positive definite, we can employ the power method-like iterations introduced in (\[eq:powermethod-like\]) to obtain a solution to (\[eq:opt\_imp\]) i.e. starting from the current $\bs=\bs^{(0)}$, a local optimum of the problem can be obtained by the iterations $$\begin{aligned}
\label{eq:respect_s}
\bs^{(t+1)}= e^{j \arg((\bR \odot \bR_\bone^T) \bs^{(t)})}.\end{aligned}$$
Finally, the proposed algorithmic optimization of (\[eq:opt\_imp\]) based on the above results is summarized in Table \[table:alpha\_0\_0\]-A.
[p[4.0in]{}]{} (A) The case of $\alpha_0=0$\
**Step 0**: Initialize the variables $\bQ_\bone$ and $\bP_\bone$ with $\bI$. Let $\bs$ be a random vector in $ \Omega^n$.\
**Step 1**: Perform the diagonal loading of $\bR$ as in (\[eq:dl1\])-(\[eq:dl2\]) (note that this diagonal loading is sufficient to keep $\bR_\bone=\bQ_\bone+\bP_\bone$ always positive definite).\
**Step 2**: Obtain the minimum of (\[eq:opt\_imp\]) with respect to $\bQ_\bone$ as in (\[eq:bq1\_opt\]).\
**Step 3**: Obtain the minimum of (\[eq:opt\_imp\]) with respect to $\bP_\bone$ using (\[eq:bp1\_opt\]).\
**Step 4**: Minimize (\[eq:opt\_imp\]) with respect to $\bs$ using (\[eq:respect\_s\]).\
**Step 5**: Goto step 2 until a stop criterion is satisfied, e.g. $\| \bR - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F \leq \epsilon_0$ (or if the number of iterations exceeded a predefined maximum number).\
(B) The case of $\alpha_0>0$\
**Step 0**: Initialize the variables $(\bs, \bQ_\bone ,\bP_\bone) $ using the results obtained by the optimization of (\[eq:opt\_imp\]) as in Table \[table:alpha\_0\_0\]-A.\
**Step 1**: Set $\delta$ (the step size for increasing $\alpha_0$ in each iteration). Let $\delta_0$ be the minimal $\delta$ to be considered and $\alpha_0=0$.\
**Step 2**: Let $\alpha_0^{pre}=\alpha_0$, $\alpha_0^{new}=\alpha_0+\delta$ and $\bR' = \bR + \alpha_0^{new} \bs \bs^H$.\
**Step 3**: Solve (\[eq:opt\_imp2\]) using the steps 2-5 in Table \[table:alpha\_0\_0\]-A (particularly step 4 must be applied to (\[eq:local\_s\])).\
**Step 4**: If $\| \bR' - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F \leq \epsilon_0$ do:\
- **Step 4-1**: If $\delta \geq \delta_0$, let $\delta \leftarrow \delta/2$ and initialize (\[eq:opt\_imp2\]) with the previously obtained variables $(\bs, \bQ_\bone ,\bP_\bone) $ for $\alpha_0=\alpha_0^{pre}$. Goto step 2.
- **Step 4-2**: If $\delta < \delta_0$, stop.
Else, let $\alpha_0=\alpha_0^{new}$ and goto step 2.\
Achieving a Local Optimum of UQP (the Case of $\alpha_0>0$) {#subsec:alpha0p}
-----------------------------------------------------------
There exist examples for which the objective function in (\[eq:opt\_imp\]) does not converge to zero. As a result, the proposed method cannot obtain a global optimum of UQP in such cases. However, it is still possible to obtain a local optimum of UQP for some $\alpha_0>0$. To do so, we solve the optimization problem, $$\begin{aligned}
\label{eq:opt_imp2}
\min_{\bs \in \Omega, \bQ_\bone \in \mathcal{C}_\bone, \bP_\bone \in \mathcal{C}(V_\bone)} \| \bR' - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F\end{aligned}$$ with $\bR' = \bR + \alpha_0 \bs \bs^H$, for increasing $\alpha_0$. The above optimization problem can be tackled using the same tools as proposed for (\[eq:opt\_imp\]). In particular, note that increasing $\alpha_0$ decreases (\[eq:opt\_imp2\]). To observe this, suppose that the solution $(\bs,\bQ_\bone,\bP_\bone)$ of (\[eq:opt\_imp2\]) is given for an $\alpha_0 \geq 0$. The minimization of (\[eq:opt\_imp2\]) with respect to $\bQ_\bone$ for $\alpha_0^{new}=\alpha_0+ \delta$ ($\delta>0$) yields $\widetilde{\bQ}_\bone \in \mathcal{C}_\bone$ such that $$\begin{aligned}
&& \| \bR+ \alpha_0^{new} \bs \bs^H - (\widetilde{\bQ}_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F \\ \nonumber
&\leq& \| \bR+ \alpha_0^{new} \bs \bs^H - ((\bQ_\bone+ \delta \bone \bone^T)+
\bP_\bone) \odot (\bs \bs^H) \|_F \\ \nonumber
&=& \| \bR+ \alpha_0 \bs \bs^H - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F\end{aligned}$$ where $\bQ_\bone+ \delta \bone \bone^T \in \mathcal{C}_\bone$. The optimization of (\[eq:opt\_imp2\]) with respect to $\bP_\bone$ can be dealt with as before (see (\[eq:opt\_imp\]) and it leads to a further decrease of the objective function. Furthermore, $$\begin{aligned}
\label{eq:local_s}
&& \| \bR+ \alpha_0 \bs \bs^H - (\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H) \|_F \\ \nonumber
&=& \| \bR+ \lambda' \bI - (\bQ_\bone +\bP_\bone - \alpha_0 \bone \bone^T + \lambda' \bI ) \odot (\bs \bs^H) \|_F\end{aligned}$$ which implies that a solution $\bs$ of (\[eq:opt\_imp2\]) can be obtained via optimizing (\[eq:local\_s\]) with respect to $\bs$ in a similar way as we described for (\[eq:opt\_imp\]) provided that $\lambda' \geq 0$ is such that $\bQ_\bone +\bP_\bone - \alpha_0 \bone \bone^T + \lambda' \bI $ is positive definite. Finally, note that the obtained solution $(\bs, \bQ_\bone ,\bP_\bone) $ of (\[eq:opt\_imp\]) can be used to initialize the corresponding variables in (\[eq:opt\_imp2\]). In effect, the solution of (\[eq:opt\_imp2\]) for any $\alpha_0$ can be used for the initialization of (\[eq:opt\_imp2\]) with an increased $\alpha_0$.
Based on the above discussion and the fact that small values of $\alpha_0$ are of interest, a bisection approach can be used to obtain $\alpha_0$. The proposed method for obtaining a local optimum of UQP along with the corresponding $\alpha_0$ is described in Table \[table:alpha\_0\_0\]-B.
Sub-optimality Analysis {#subsec:subopt}
-----------------------
In this sub-section, we show that the proposed method can provide a sub-optimality guarantee ($\gamma$) that is close to $1$. Let $\alpha_0=0$ (as a result $\bR'=\bR$) and define $$\begin{aligned}
\bE \triangleq \bR' - \underbrace{(\bQ_\bone+ \bP_\bone) \odot (\bs \bs^H)}_{\bR_\bs}\end{aligned}$$ where $\bQ_\bone \in \mathcal{C}_\bone$ and $\bP_\bone \in \mathcal{C}(V_\bone)$. The global optimum of the UQP associated with $\bR_\bs$ is $\bs$. We have that $$\begin{aligned}
\label{eq:up-bound}
\max_{\bs' \in \Omega^n} \bs'^H \bR \bs' &\leq& \max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs' + \max_{\bs' \in \Omega^n} \bs'^H \bE \bs'
\\ \nonumber
&\leq & \max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs' + n \sigma_1(\bE)
\\ \nonumber
&=& \bs^H \bR_\bs \bs + n \sigma_1(\bE)\end{aligned}$$ Furthermore, $$\begin{aligned}
\label{eq:low-bound}
\max_{\bs' \in \Omega^n} \bs'^H \bR \bs' &\geq& \max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs' + \min_{\bs' \in \Omega^n} \bs'^H \bE \bs'
\\ \nonumber
&\geq & \max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs' + n \sigma_n(\bE)
\\ \nonumber
&= & \bs^H \bR_\bs \bs + n \sigma_n(\bE)\end{aligned}$$ As a result, an upper bound and a lower bound on the objective function for the global optimum of (\[eq:opt\_imp\]) can be obtained *at each iteration*. Furthermore, as $$\begin{aligned}
| \sigma_1(\bE)| \leq \| \bE \|_F, ~| \sigma_n(\bE)| \leq \| \bE \|_F\end{aligned}$$ if $\| \bE \|_F $ converges to zero we conclude for (\[eq:up-bound\]) and (\[eq:low-bound\]) that $$\begin{aligned}
\max_{\bs' \in \Omega^n} \bs'^H \bR \bs' = \bs^H \bR_\bs \bs = \bs^H \bR \bs %\max_{\bs' \in \Omega} \bs'^H \bR_\bs \bs'\end{aligned}$$ and hence $\bs$ is the global optimum of the UQP associated with $\bR$ (i.e. a sub-optimality guarantee of $\gamma=1$ is achieved).
Next, suppose that we have to increase $\alpha_0$ in order to obtain the convergence of $\| \bE \|_F $ to zero. In such a case, we have that $%\begin{eqnarray}
\bR = \bR_\bs - \alpha_0 \bs \bs^H
$ and as a result, $\max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs' - \alpha_0 n^2 \leq \ \max_{\bs' \in \Omega^n} \bs'^H \bR \bs' \leq \max_{\bs' \in \Omega^n} \bs'^H \bR_\bs \bs'$ or equivalently, $$\begin{aligned}
\bs^H \bR_\bs \bs - \alpha_0 n^2 \leq \max_{\bs' \in \Omega^n} \bs'^H \bR \bs' \leq \bs^H \bR_\bs \bs. \end{aligned}$$ The provided sub-optimality guarantee is thus given by $$\begin{aligned}
\label{eq:subopt}
\gamma= \frac{\bs^H \bR \bs}{\bs^H \bR_\bs \bs} = 1- \frac{\alpha_0 n^2}{\bs^H \bR_\bs \bs}. \end{aligned}$$
Note that while solving the optimization problem (\[eq:opt\_imp2\]) does not necessarily yield the exact optimal solution to UQP, the so-obtained solution can be still optimal. We also note that (\[eq:subopt\]) generally yields tighter sub-optimality guarantees than the currently known approximation guarantee (i.e. $\pi /4$ for SDR). The following section provides empirical evidence for such a fact.
Numerical Examples {#sec:numerical}
==================
In order to examine the performance of the proposed method, several numerical examples will be presented. Random Hermitian matrices $\bR$ are generated using the formula $$\begin{aligned}
\label{eq:xdef}
\bR = \sum_{k=1}^{n} \bx_k \bx_k^H\end{aligned}$$ where $\{\bx_k\}$ are random vectors in $\complexC^n$ whose real-part and imaginary-part elements are i.i.d. with a standard Gaussian distribution $\mathcal{N}(0,1)$. In all cases, we stopped the iterations when $\| \bE \|_F \leq 10^{-9}$.
We use the MERIT algorithm to solve the UQP for a random positive definite matrix of size $n=16$. The obtained values of the UQP objective for the true matrix ($\bR$) and the approximated matrix ($\bR_\bs$) as well as the sub-optimality bounds (derived in (\[eq:up-bound\]) and (\[eq:low-bound\])) are depicted in Fig. 1 versus the iteration number. In this example, a sub-optimality guarantee of $\gamma=1$ is achieved which implies that the method has successfully obtained the global optimum of the considered UQP. A computational time of 3.653 sec was required on a standard PC to accomplish the task.
Next, we solve the UQP for $20$ full-rank random positive definite matrices of sizes $n \in \{ 8,16,32,64 \}$. Inspired by [@low-rank] and [@rankdef_binary], we also consider rank-deficient matrices $\bR = \sum_{k=1}^{d} \bx_k \bx_k^H$ where $\{\bx_k\}$ are as in (\[eq:xdef\]), but $d \ll n$. The performance of MERIT for different values of $d$ is shown in Table \[table\_rank\_def\]. Interestingly, the solution of UQP for rank-deficient matrices appears to be obtained more efficiently than for the full-rank matrices. For each problem solved by MERIT, we also let the SDR algorithm of [@DeMaio-PAR] use the same computational time for solving the problem. The SDR algorithm is able to solve the problem only if its core semi-definite program can be solved within the available time. Any remaining time is dedicated to the randomization procedure. The results can be found in Table \[table\_rank\_def\]. Note that the maximum UQP objective values obtained by MERIT and SDR were nearly identical in those cases in which SDR was able to solve the UQPs in the same amount of time as MERIT. Note also that given the solutions obtained by MERIT and SDR as well as the sub-optimality guarantee of MERIT, a case-dependent sub-optimality guarantee for SDR can be computed as $$\begin{aligned}
\gamma_{\mbox{SDR}} \triangleq \gamma_{\mbox{MERIT}} \left(\frac{v_{SDR}}{v_{MERIT}} \right).\end{aligned}$$ This can be used to examine the goodness of the solutions obtained by SDR.
$n$ Rank ($d$) \#[problems for which $\gamma=1$]{} [Average $\gamma$]{} [Minimum $\gamma$]{} [Average CPU time (sec)]{} \#[problems solved by SDR]{}
-------- ------------ ------------------------------------- ---------------------- ---------------------- ---------------------------- ------------------------------
**8** *2* $17$ $0.9841$ $0.8184$ $0.13$ $4$
*8* $16$ $0.9912$ $0.9117$ $0.69$ $7$
*2* $15$ $0.9789$ $0.8301$ $1.06$ $2$
**16** *4* $13$ $0.9773$ $0.8692$ $1.58$ $10$
*16* $4$ $0.9610$ $0.8693$ $3.54$ $13$
*2* $9$ $0.9536$ $0.8190$ $47.04$ $3$
**32** *6* $4$ $0.9077$ $0.8106$ $55.59$ $7$
*32* $2$ $0.9031$ $0.8021$ $94.90$ $16$
*2* $3$ $0.8893$ $0.8177$ $406.56$ $4$
**64** *8* $1$ $0.8567$ $0.7727$ $560.35$ $10$
*64* $0$ $0.8369$ $0.7811$ $1017.69$ $15$
Besides random matrices, we also consider several other matrix structures for which solving the UQP using the proposed method is not “hard", as explained below.
- *Case 1:* An exponentially shaped disturbance matrix [@DeMaio-similarity] with correlation coefficient $\eta = 0.8$, $$\begin{aligned}
\bM(k,l)=\eta^{|k-l|},~~~1\leq k,l \leq n .\end{aligned}$$
- *Case 2:* A disturbance matrix with the structure $$\begin{aligned}
\bM(k,l)=\eta_1^{|k-l|} e^{j2\pi \rho (k-l)} + 10 \eta_2 ^{|k-l|} + 10^{-2} \bI(k,l),~~~1\leq k,l \leq n \end{aligned}$$ whose terms represent the effects of sea clutter, land clutter and thermal noise, respectively. The values of $(\eta_1,\eta_2,\rho)$ are set to $(0.8,0.9,0.2)$ in accordance to an example provided in [@Demaio-maxmin].
- *Case 3:* A disturbance matrix accounting for both discrete clutter scatterers and thermal noise [@DeMaio-PAR], $$\begin{aligned}
\bM= \sum_{k=1}^{n_c} \eta_k \bp_{v_{d,k}} \bp_{v_{d,k}}^H + \eta \bI\end{aligned}$$ where $n_c=10$, $\eta_k=10^3$, $v_{d,k}=(k-1)/2$, $$\begin{aligned}
\label{eq:p_struct}
\bp_{v_{d,k}}=( 1, e^{j2\pi v_{d,k}}, \cdots, e^{j2\pi (n-1) v_{d,k}})^T, ~~~ 1\leq k \leq n_c,\end{aligned}$$ and $\eta=10^{-2}$. The chosen values are the same as those considered in [@DeMaio-PAR].
We let $\bR=\bM^{-1} \odot (\bp \bp^H)^*$ (see (\[eq:SNR\]) and the following discussion) where $\bp$ is an unimodular vector with a structure similar to that of $\{ \bp_{v_{d,k}}\}$ in (\[eq:p\_struct\]). The UQP for the above cases is solved via MERIT using $20$ different random initializations for sizes $n \in \{ 8,16,32,64 \}$. Similar to the previous example, we also used SDR to solve the same UQPs. The results are shown in Table \[table\_cases\]. The obtained solutions can be considered to be quite accurate in the sense of a sub-optimality guarantee $\gamma$ close to one.
$n$ Rank ($d$) \#[problems for which $\gamma=1$]{} [Average $\gamma$]{} [Minimum $\gamma$]{} [Average CPU time (sec)]{} \#[problems solved by SDR]{}
-------- ------------ ------------------------------------- ---------------------- ---------------------- ---------------------------- ------------------------------
*Case 1* $20$ $1.0000$ $1.0000$ $2.82$ $17$
**8** *Case 2* $20$ $1.0000$ $1.0000$ $0.60$ $20$
*Case 3* $20$ $1.0000$ $1.0000$ $0.27$ $10$
*Case 1* $20$ $1.0000$ $1.0000$ $42.83$ $20$
**16** *Case 2* $18$ $0.9812$ $0.8075$ $21.58$ $20$
*Case 3* $20$ $1.0000$ $1.0000$ $2.01$ $12$
*Case 1* $20$ $1.0000$ $1.0000$ $990.90$ $20$
**32** *Case 2* $19$ $0.9995$ $0.9913$ $525.34$ $20$
*Case 3* $20$ $1.0000$ $1.0000$ $7.52$ $7$
*Case 1* $17$ $0.9901$ $0.9862$ $5574.98$ $20$
**64** *Case 2* $16$ $0.9540$ $0.8359$ $2053.26$ $20$
*Case 3* $20$ $1.0000$ $1.0000$ $22.78$ $9$
A different code design problem arises when synthesizing waveforms that have good resolution properties in range and Doppler \[3\]-\[5\],\[24\]-[@Phasecoded]. In the following, we consider the design of a thumbtack CAF (see the definitions in sub-section \[subsec:background\]): $$\begin{aligned}
d(\tau,f) = \left\{ \begin{array}{ll}
n & (\tau,f)=(0,0), \\
0 & \mbox{otherwise.}
\end{array} \right.\end{aligned}$$ Suppose $n=53$, let $T$ be the time duration of the total waveform, and let $t_p= T / n$ represent the time duration of each sub-pulse. Define the weighting function as $$\begin{aligned}
w(\tau,f) = \left\{ \begin{array}{ll}
1 & (\tau,f)\in \Psi \backslash \Psi_{ml}, \\
0 & \mbox{otherwise,}
\end{array} \right.\end{aligned}$$ where $\Psi=[-10 t_p, 10 t_p] \times [-2/ T, 2/ T]$ is the region of interest and $\Psi_{ml}=([- t_p, t_p] \backslash \{ 0\}) \times ([-1/ T, 1/ T] \\ \backslash \{ 0\}) $ is the mainlobe area which is excluded due to the sharp changes near the origin of $d(\tau,f)$. Note that the time delay $\tau$ and the Doppler frequency $f$ are typically normalized by $T$ and $1/T$, respectively, and as a result the value of $t_p$ can be chosen freely without changing the performance of CAF design. The synthesis of the desired CAF is accomplished via the cyclic minimization of (\[eq:CAF\_crit\]) with respect to $\bx$ and $\by$ (see sub-section \[subsec:background\]). In particular, we use MERIT to obtain a unimodular $\bx$ in each iteration. A Björck code is used to initialize both vectors $\bx$ and $\by$. The Björck code of length $n=p$ (where $p$ is a prime number for which $p \equiv 1~(\bmod~4)$) is given by $\bb(k)=e^{j (\frac{k}{p}) \arccos\left(1/(1+\sqrt{p})\right)}$, $0 \leq k <p$, with $(\frac{k}{p})$ denoting the Legendre symbol. Fig. 2 depicts the normalized CAF modulus of the Björck code (i.e. the initial CAF) and the obtained CAF using the UQP formulation in (\[eq:CAF\_UQP2\]) and the proposed method. Despite the fact that designing CAF with a unimodular transmit vector $\bx$ is a rather constrained problem, MERIT is able to efficiently suppress the CAF sidelobes in the region of interest.
\
Concluding Remarks {#sec:conclusion}
==================
A computational approach to the NP-hard problem of optimizing a quadratic form over the unimodular vector set (called UQP) has been introduced. The main results can be summarized as follows:
- Some applications of the UQP were reviewed. It was shown that the solution to UQP is not necessarily unique. Several examples were provided for which an accurate global optimum of UQP can be obtained efficiently.
- Using a relaxed version of UQP, a specialized local optimization scheme for UQP was devised and was shown to yield superior results compared to any general local optimization of UQP.
- It was shown that the set of matrices ($\mathcal{K}(\bs)$) leading to the same solution ($\bs$) as the global optimum of UQP is a convex cone. An one-to-one mapping between any two such convex cones was introduced and an approximate characterization of $\mathcal{K}(\bs)$ was proposed.
- Using the approximate characterization of $\mathcal{K}(\bs)$, an iterative approach (called MERIT) to the UQP was proposed. It was shown that MERIT provides case-dependent sub-optimality guarantees. The available numerical evidence shows that the sub-optimality guarantees obtained by MERIT are generally better than the currently known approximation guarantee (of $\pi /4$ for SDR).
- Numerical examples were provided to examine the potential of MERIT for different UQPs. In particular, it was shown that the UQP solutions for certain matrices used in active sensing code design can be obtained efficiently via MERIT.
We should note that no theoretical efficiency assessment of the method was provided. It is clear that $\mathcal{C}(V_\bs) \cup \mathcal{C}_\bs \subset \mathcal{K}(\bs)$. A possible approach would be to determine how large is the part of $\mathcal{K}(\bs)$ that is “covered" by $\mathcal{C}(V_\bs) \cup \mathcal{C}_\bs$. However, this problem is left for future work. Furthermore, a study of $m$-UQP using the ideas in this paper will be the subject of another paper.
Proof of Theorem 1 {#subsec:th1}
------------------
In order to verify the first part of the theorem, consider any two matrices $\bR_1 ,\bR_2 \in \mathcal{K}(\widetilde{\bs})$. For any two non-negative scalars $\gamma_1, \gamma_2$ we have that $$\begin{aligned}
\bs^H (\gamma_1 \bR_1 + \gamma_2 \bR_2) \bs = \gamma_1 \bs^H \bR_1 \bs + \gamma_2 \bs^H \bR_2 \bs.\end{aligned}$$ Clearly, if some $\bs=\widetilde{\bs}$ is the global maximizer of both $\bs^H \bR_1 \bs$ and $\bs^H \bR_2 \bs$ then it is the global maximizer of $\bs^H (\gamma_1 \bR_1 + \gamma_2 \bR_2) \bs$ which implies $\gamma_1 \bR_1 + \gamma_2 \bR_2 \in \mathcal{K}(\widetilde{\bs}) $.
The second part of the theorem can be shown noting that $$\begin{aligned}
\bs_2^H (\bR \odot (\bs_0 \bs_0^H)) \bs_2 &=& (\bs_0^* \odot \bs_2)^H \bR (\bs_0^* \odot \bs_2) \\ \nonumber
&=& \bs_1^H \bR \bs_1\end{aligned}$$ for all $\bs_1,\bs_2 \in \Omega^n$ and $\bs_0=\bs_1^* \odot \bs_2$. Therefore, if $\bR \in \mathcal{K}(\widetilde{\bs}_1)$ then $\bR \odot (\widetilde{\bs}_0 \widetilde{\bs}_0^H) \in \mathcal{K}(\widetilde{\bs}_2)$ (for $\widetilde{\bs}_0=\widetilde{\bs}_1^* \odot \widetilde{\bs}_2$) and vice versa.
Proof of Theorem 2 {#subsec:th2}
------------------
It is well-known that $\bx^H \bR \bx \leq \sigma_1 \| \bx \|_2^2$ for all vectors $\bx \in \complexC^n$. Let $$\begin{aligned}
\balpha'=\left( \begin{array}{c}
\balpha \\
\bzero_{(n-m) \times 1}
\end{array} \right).\end{aligned}$$ It follows from (\[eq:sualpha\]) that $\widetilde{\bs} = \bU \balpha' $ and therefore $$\begin{aligned}
\widetilde{\bs}^H \bR \widetilde{\bs} &=& \balpha'^H \bSig \balpha' = \sigma_1 \| \balpha' \|_2^2 \\ \nonumber
&=& \sigma_1 \| \widetilde{\bs} \|_2^2 = n \sigma_1\end{aligned}$$ which implies the global optimality of $\widetilde{\bs}$ for the considered UQP.
Proof of Theorem 4 {#subsec:th4}
------------------
It is worthwhile to observe that the convergence rate of $\{\bR^{(t)} \}$ is not dependent on the problem dimension ($n$), as each entry of $\{\bR^{(t)} \}$ is treated independently from the other entries (i.e. all the operations are element-wise). Therefore, without loss of generality we study the convergence of one entry (say $\{\bR^{(t)} (k,l) \}=\{ r_t e^{j \theta_t} \}$) in the following.
Note that in cases for which $|\theta_t - (\phi_k - \phi_l)| > \pi / 2$, the next element of the sequence $\{ r_t e^{j \theta_t} \}$ can be written as $$\begin{aligned}
r_{t+1} e^{j \theta_{t+1}} = r_t e^{j \theta_t} + \rho e^{j(\phi_k - \phi_l)}\end{aligned}$$ which implies that the proposed operation tends to make $\theta_t$ closer to $(\phi_k - \phi_l)$ in each iteration, and finally puts $\theta_t$ within the $ \pi / 2$ distance from $(\phi_k - \phi_l)$.
Let us suppose that $|\theta_0 - (\phi_k - \phi_l)| > \pi / 2$, and that the latter phase criterion remains satisfied for all $\theta_t$, $t<T$. We have that $$\begin{aligned}
r_{T} e^{j \theta_{T}} = r_0 e^{j \theta_0} + T \rho e^{j(\phi_k - \phi_l)}\end{aligned}$$ which yields $$\begin{aligned}
r_{T} \cos(\theta_T - (\phi_k - \phi_l))= r_0 \cos(\theta_0 - (\phi_k - \phi_l)) + T \rho.\end{aligned}$$ Therefore it takes only $T=\left\lceil - r_0 \cos(\theta_0 - (\phi_k - \phi_l)) / \rho \right\rceil =1$ iteration for $\theta_t$ to stand within the $ \pi / 2$ distance from $(\phi_k - \phi_l)$.
Now, suppose that $|\theta_0 - (\phi_k - \phi_l)| \leq \pi / 2$. For every $t\geq 1$ we can write that $$\begin{aligned}
\label{eq:rec}
r_{t+1} e^{j \theta_{t+1}} &=& r_t e^{j \theta_t} + \rho e^{j(\phi_k - \phi_l)} \\ \nonumber &-& r_t \cos(\theta_t - (\phi_k - \phi_l)) e^{j(\phi_k - \phi_l)} \\ \nonumber &=&e^{j(\phi_k - \phi_l)} \left( \rho + j r_t \sin(\theta_t - (\phi_k - \phi_l)) \right).
\end{aligned}$$ Let $\delta_{t+1}= r_{t+1} e^{j \theta_{t+1}} - r_{t} e^{j \theta_{t}}$. The first equality in (\[eq:rec\]) implies that $$\begin{aligned}
\label{eq:deltaarg1}
\delta_{t+1}=e^{j(\phi_k - \phi_l)} (\rho - r_t \cos(\theta_t - (\phi_k - \phi_l)) ) .\end{aligned}$$ On the other hand, the second equality in (\[eq:rec\]) implies that $$\begin{aligned}
\label{eq:deltaarg2}
\delta_{t+1}= j e^{j(\phi_k - \phi_l)} ( r_t \sin(\theta_t - (\phi_k - \phi_l)) ~~~~~~~~~~~~\\ \nonumber - r_{t-1} \sin(\theta_{t-1} - (\phi_k - \phi_l)) )\end{aligned}$$ for all $t\geq 1$. Note that in (\[eq:deltaarg1\]) and (\[eq:deltaarg2\]), $\delta_{t+1}$ is a complex number having different phases. We conclude $$\begin{aligned}
\delta_{t+1}=0, ~~ \forall ~t \geq 1\end{aligned}$$ which shows that the sequence $\{ r_{t} e^{j \theta_{t}} \}$ is convergent in one iteration. In sum, every entry of the matrix $R$ will converge in at most two iterations (i.e. at most one to achieve a phase value within the $ \pi / 2$ distance from $(\phi_k - \phi_l)$, and one iteration thereafter).
Proof of Theorem 5 {#subsec:th5}
------------------
We use the same notations as in the proof of Theorem 4. If $|\theta_0 - (\phi_k - \phi_l)| \leq \pi / 2$ then $$\begin{aligned}
\label{eq:lambda_diff1}
r_{2} e^{j \theta_{2}}&=& r_{1} e^{j \theta_{1}} \\ \nonumber &=& r_0 e^{j \theta_0} + \rho e^{j(\phi_k - \phi_l)} \\ \nonumber &-& r_0 \cos(\theta_0 - (\phi_k - \phi_l)) e^{j(\phi_k - \phi_l)}.\end{aligned}$$ On the other hand, if $|\theta_0 - (\phi_k - \phi_l)| > \pi / 2$ we have that $r_{1} e^{j \theta_{1}} = r_0 e^{j \theta_0} + \rho e^{j(\phi_k - \phi_l)}$. As a result, $r_1 \cos(\theta_1 - (\phi_k - \phi_l)) = \rho + r_0 \cos(\theta_0 - (\phi_k - \phi_l)) $ which implies $$\begin{aligned}
\label{eq:lambda_diff2}
r_{2} e^{j \theta_{2}} &=& r_1 e^{j \theta_1} + \rho e^{j(\phi_k - \phi_l)} \\ \nonumber &-& r_1 \cos(\theta_1 - (\phi_k - \phi_l)) e^{j(\phi_k - \phi_l)}
\\ \nonumber &=& r_0 e^{j \theta_0} + \rho e^{j(\phi_k - \phi_l)} \\ \nonumber &-& r_0 \cos(\theta_0 - (\phi_k - \phi_l)) e^{j(\phi_k - \phi_l)}.\end{aligned}$$ Now, it is easy to verify that (\[eq:lambda\_diff\]) follows directly from (\[eq:lambda\_diff1\]) and (\[eq:lambda\_diff2\]).
Proof of Theorem 6 {#subsec:th6}
------------------
If $\bs$ is a hyper point of UQP associated with $\bR^{(0)}=\bR$ then we have that $\arg(\bs)=\arg(\bR \bs)$. Let $\bR \bs= \bv \odot \bs$ where $\bv$ is a non-negative real-valued vector in $\realR^n$. It follows that $$\begin{aligned}
\bv(k) e^{j \phi_k} = \sum_{l=1}^{n} |\bR (k,l)| e^{j \theta_{k,l}} e^{j \phi_l}\end{aligned}$$ or equivalently $$\begin{aligned}
\bv(k) = \sum_{l=1}^{n} |\bR (k,l)| e^{j (\theta_{k,l}-( \phi_k - \phi_l))}\end{aligned}$$ which implies that $$\begin{aligned}
\left\{ \begin{array}{l}
\sum_{l=1}^{n} |\bR (k,l)| \cos( \theta_{k,l}-( \phi_k - \phi_l)) \geq 0 \\
\sum_{l=1}^{n} |\bR (k,l)| \sin( \theta_{k,l}-( \phi_k - \phi_l)) = 0
\end{array} \right.\end{aligned}$$ for all $1 \leq k \leq n$. Now, note that the recursive formula of the sequence $\{\bR^{(t)} \}$ can be rewritten as $$\begin{aligned}
\bR^{(t+1)}=\bR^{(t)} - \mathbf{Diag} (\bs) ~(\bR_+^{(t)} - \rho \bone_{n \times n}) ~\mathbf{Diag}(\bs^*)\end{aligned}$$ and as a result, $$\begin{aligned}
\label{eq:recursive}
\bR^{(t+1)} \bs = \bR^{(t)} \bs - \mathbf{Diag} (\bs) ~(\bR_+^{(t)} - \rho \bone_{n \times n}) ~\bone_{n \times 1}.\end{aligned}$$ It follows from (\[eq:recursive\]) that if $\bs$ is a hyper point of the UQP associated with $\bR^{(t)}$ (which implies the existence of non-negative real-valued vector $\bv^{(t)}$ such that $\bR^{(t)} \bs = \bv^{(t)} \odot \bs$), then there exists $\bv^{(t+1)} \in \realR^n$ for which $\bR^{(t+1)} \bs = \bv^{(t+1)} \odot \bs$ and therefore, $$\begin{aligned}
\label{eq:55}
\bv^{(t+1)}(k) \, e^{j \phi_k} &=& \sum_{l=1}^{n} |\bR^{(t)} (k,l)| e^{j \theta_{k,l}} e^{j \phi_l} \\ \nonumber
&-& \left( \left( \sum_{l=1}^{n} \bR^{(t)}_{+}(k,l) \right) - n \rho \right) e^{j \phi_k}.\end{aligned}$$ Eq. (\[eq:55\]) can be rewritten as $$\begin{aligned}
\label{eq:56}
\bv^{(t+1)} (k) &=& \sum_{l=1}^{n} |\bR^{(t)}(k,l)| e^{j (\theta_{k,l}-( \phi_k - \phi_l))} \\ \nonumber
&-& \left( \sum_{l=1}^{n} \bR^{(t)}_{+}(k,l) \right) + n \rho \end{aligned}$$ As indicated earlier, $\bs$ being a hyper point for $\bR^{(0)}$ assures that the imaginary part of (\[eq:56\]) is zero. To show that $\bs$ is a hyper point of the UQP associated with $\bR^{(t+1)}$, we only need to verify that $\bv^{(t+1)}(k) \geq 0$: $$\begin{aligned}
\bv^{(t+1)}(k)&=& \sum_{l=1}^{n} |\bR^{(t)}(k,l)| \cos (\theta_{k,l}-( \phi_k - \phi_l)) \\ \nonumber
&-& \left( \sum_{l=1}^{n} \bR^{(t)}_{+}(k,l) \right) + n \rho \\ \nonumber
&=& n \rho \\ \nonumber &+& \sum_{l:~(k,l) \notin \Theta} |\bR^{(t)}(k,l)| \cos (\theta_{k,l}-( \phi_k - \phi_l)) \end{aligned}$$ Now note that the positivity of $\bv^{(t+1)}(k)$ is concluded from (\[eq:lambda\_ineq\]). In particular, based on the discussions in the proof of Theorem 4, for $t=1$, there is no $\theta_{k,l}$ such that $|\theta_{k,l} - (\phi_k - \phi_l)| \geq \pi / 2$ and therefore $\bv^{(2)}(k)= n \rho$ for all $1 \leq k \leq n$. As a result, $$\begin{aligned}
\bR^{(2)} \bs = n \rho \bs\end{aligned}$$ which implies that $\bs$ is an eigenvector of $\bR^{(2)}$ corresponding to the eigenvalue $n \rho$.
Acknowledgement {#acknowledgement .unnumbered}
===============
We would like to thank Prof. Antonio De Maio for providing us with the MATLAB code for SDR.
[^1]: This work was supported in part by the European Research Council (ERC) under Grant \#228044 and the Swedish Research Council. The authors are with the Dept. of Information Technology, Uppsala University, Uppsala, SE 75105, Sweden.
\* Please address all the correspondence to Mojtaba Soltanalian, Phone: (+46) 18-471-3168; Fax: (+46) 18-511925; Email: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'With declining sequencing costs a promising and affordable tool is emerging in cancer diagnostics: genomics [@Schwarze:2020dr]. By using association studies, genomic variants that predispose patients to specific cancers can be identified, while by using tumor genomics cancer types can be characterized for targeted treatment. However, a severe disparity is rapidly emerging in this new area of precision cancer diagnosis and treatment planning, one which separates a few genetically well-characterized populations (predominantly European) from all other global populations. Here we discuss the problem of population-specific genetic associations, which is driving this disparity, and present a novel solution–coordinate-based local ancestry–for helping to address it. We demonstrate our boosting-based method on whole genome data from divergent groups across Africa and in the process observe signals that may stem from the transcontinental Bantu-expansion.'
author:
- |
Daniel Mas Montserrat[^1]\
Purdue University\
Arvind Kumar\
Stanford University\
Carlos Bustamante\
Stanford University\
Alexander Ioannidis\
Stanford University\
bibliography:
- 'mybiblio.bib'
title: 'Addressing Ancestry Disparities in Genomic Medicine: A Geographic-aware Algorithm'
---
Introduction
============
Cancer genomics depends upon the identification of variants that are associated with particular types of cancers. Because such variants are deleterious, they are not typically part of the ancient standing variation spread across all humans; instead they are more recent mutations specific to particular populations. Indeed, such variants are often present prominently only in particular ethnic groups due to genetic drift [@Foulkes:2002cc]. In addition, most associations are mapped not to causal variants, but to more common neighboring variants that are present on genotyping arrays. Since these neighboring variants are linked to the causal variant via correlation structures (linkage) that are specific to each population, the ancestry of the genomic segment in which the correlated variant is found becomes crucial. Indeed, as a result of linkage and epistatic effects, genomic variants that are associated with cancer in one ancestry maybe have no association [@Wang:2018js], or may even have an opposite association [@Rajabli:2018cb], in another ancestry. This phenomenon persists even in admixed individuals possessing multiple ancestries, such as African Americans; in such individuals the ancestry (European or African) of the specific genomic fragment containing the associated variant has been found to reverse the association [@shortRajabli:2018cb]. This phenomenon dubbed “flip-flop,” is not an unusual case, rather ancestry-specific effects in genetic association studies are the rule. For this reason, polygenic-risk scores (PRS), increasingly important to genomic cancer prediction [@Mavaddat:2019ix], have been found to be several times less accurate when used on populations of different ancestry from the one on which they were trained [@shortMartin:2019bm].
As a result of these ancestry specific effects, accurately identifying the ancestry of each segment of the genome is becoming increasingly crucial for genomic medicine. Such algorithms, known as local ancestry inference, have been developed both for historical population genetics [@Tang2006; @Sundquist2008; @ShortPrice:2009bga; @sankararaman2008estimating; @Durand:2014hj; @maples2013rfmix; @vaegan2019; @lainet2020] and for recreational consumer ancestry products [@Durand:2015jx], but none have been developed to date for the particular demands of clinical genomic medicine. Such an algorithm would need to provide ancestry not as a culturally defined label, but as continuous genetic coordinates that could be used as a covariate in predication and association algorithms. This method is also important for deconvolving ancestry effects in genetic association studies. To date, most genome-wide association studies (GWAS) are conducted in populations of single ancestry (typically European) to avoid confounding effects of ancestry on reversing associations. Researchers often avoid admixed populations, for instance African Americans or Hispanics, who encompass more than one ancestry, and avoid populations with too much genetic variation or too many diverse sub-populations, as is common within Africa. This has resulted in over 80% of the individuals in GWAS studies to date stemming from European ancestry (and only 2% from African ancestry) [@Sirugo:2019ie; @Popejoy:2016di]. A reliable coordinate-based local ancestry algorithm would allow such studies to embrace diversity, rather than intentionally eschewing it, by allowing an additional covariate along the genome to be used (ancestry) to remove the confounding effects of ancestry-dependent genomic associations. With such a tool, medical researchers would no longer need to avoid admixed and globally diverse genetic study cohorts.
Ancestry Inference
==================
Here were present an accurate coordinate-based local ancestry inference algorithm, XGMix, that can be used for addressing ancestry-specific associations and predictions. XGMix uses modern single ancestry reference populations to accurately predict the latitude and longitude of the closest modern source population for each segment of an individual’s genome. These coordinate annotations along the genome can then be used as covariates for genome-wide association studies (GWAS) and for polygenic risk score (PRS) predictions.
Estimation of an individual’s ancestry, both globally and locally (i.e. assigning an ancestry estimate to each region of the chromosomal sequence), has been tackled with a wide range of methods and technologies [@Tang2006; @Sundquist2008; @ShortPrice:2009bga; @sankararaman2008estimating; @Durand:2014hj; @maples2013rfmix; @vaegan2019; @lainet2020]. Local ancestry inference has traditionally been framed as a classification problem using pre-defined ancestries. Classification approaches provide discrete ancestry labels but can be highly inaccurate for neighboring populations (or population gradients) and intractable for genetically diverse populations with multiple sources. Geographical regression along the genome, although a much more challenging problem, could provide a continuous representation of ancestry capable of capturing the complexities of worldwide populations.
XGMix consists of two layers of stacked gradient boosted trees (a genomic window-specific layer and a window aggregating smoother) and can infer local-ancestry with both classification probabilities and geographical coordinates along each phased chromosome. Here we demonstrate XGMix by training on whole genomes from real individuals from the five African populations included in the 1000 genomes project [@10002015global]. We simulate admixed individuals of various generations using Wright-Fisher simulation [@maples2013rfmix] to create ground truth labels of ancestry along the genome and split this data for training and testing. As these reference African populations lie close to a single arc along the globe we estimate along this arc, getting geographic assignments for each genomic segment.
![(a) The inferred coordinates for each genomic segment of an admixed Kenyan-Nigerian individual. The model was trained on all indicated African reference populations. (b-c) The inferred location of each genomic segment of a Kenyan-Nigerian (b) and Kenyan-Gambian (c) individual using the principal coordinate arc of the reference populations’ locations. The bimodal distribution of Kenyan segments (green) may reflect the historical Bantu expansion from Cameroon into Kenya.[]{data-label="fig:map"}](plot_2.pdf){width="85.00000%"}
[^1]: Work conducted during an internship at Stanford University.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Reading comprehension has recently seen rapid progress, with systems matching humans on the most popular datasets for the task. However, a large body of work has highlighted the brittleness of these systems, showing that there is much work left to be done. We introduce a new English reading comprehension benchmark, DROP, which requires [**D**]{}iscrete [**R**]{}easoning [**O**]{}ver the content of [**P**]{}aragraphs. In this crowdsourced, adversarially-created, 96k-question benchmark, a system must resolve references in a question, perhaps to multiple input positions, and perform discrete operations over them (such as addition, counting, or sorting). These operations require a much more comprehensive understanding of the content of paragraphs than what was necessary for prior datasets. We apply state-of-the-art methods from both the reading comprehension and semantic parsing literatures on this dataset and show that the best systems only achieve 32.7% $F_1$ on our generalized accuracy metric, while expert human performance is 96.4%. We additionally present a new model that combines reading comprehension methods with simple numerical reasoning to achieve 47.0% $F_1$.'
author:
- |
Dheeru Dua^$\clubsuit$^, Yizhong Wang^$\diamondsuit$^[^1], Pradeep Dasigi^$\heartsuit$^,\
**Gabriel Stanovsky**^$\heartsuit$$+$^, **Sameer Singh**^$\clubsuit$^, and **Matt Gardner**^$\spadesuit$^\
^$\clubsuit$^University of California, Irvine, USA\
^$\diamondsuit$^Peking University, Beijing, China\
^$\heartsuit$^Allen Institute for Artificial Intelligence, Seattle, Washington, USA\
^$\spadesuit$^Allen Institute for Artificial Intelligence, Irvine, California, USA\
^$+$^University of Washington, Seattle, Washington, USA\
[[email protected]]{}\
bibliography:
- 'paper.bib'
title: |
DROP: A Reading Comprehension Benchmark\
Requiring Discrete Reasoning Over Paragraphs
---
Introduction {#sec:intro}
============
Related Work {#sec:related_work}
============
DROP Data Collection {#sec:data_collection}
====================
DROP Data Analysis {#sec:data_analysis}
==================
Baseline Systems {#sec:baselines}
================
NAQANet {#sec:model}
=======
[lrrrr]{} & &\
(lr)[2-3]{} (lr)[4-5]{} & EM & F$_1$ & EM& F$_1$\
\
Majority & 0.09 & 1.38 & 0.07 & 1.44\
Q-only & 4.28 & 8.07 & 4.18 & 8.59\
P-only & 0.13 & 2.27 & 0.14 & 2.26\
\
Syn Dep & 9.38 & 11.64 & 8.51 & 10.84\
OpenIE & 8.80 & 11.31 & 8.53 & 10.77\
SRL & 9.28 & 11.72 & 8.98 & 11.45\
\
BiDAF & 26.06 & 28.85 & 24.75 & 27.49\
QANet & 27.50 & 30.44 & 25.50 & 28.36\
QANet+ELMo & 27.71 & 30.33 & 27.08 & 29.67\
BERT & 30.10 & 33.36 & 29.45 & 32.70\
\
+ Q Span & 25.94 & 29.17 & 24.98 & 28.18\
+ Count & 30.09 & 33.92 & 30.04 & 32.75\
+ Add/Sub & 43.07 & 45.71 & 40.40 & 42.96\
Complete Model & [**46.20**]{} & [**49.24**]{} & [**44.07**]{} & [**47.01**]{}\
**Human & - & - & 94.09 & 96.42\
**
---------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------- ------ ---------------
[**Our**]{}
[**model**]{}
Subtraction + Coreference … Twenty-five of his 150 men were sick, and his advance stalled … How many of Bartolomé de Amésqueta’s 150 men were not sick? 125 145
Count + Filter … Macedonians were the largest ethnic group in Skopje, with 338,358 inhabitants … Then came … Serbs (14,298 inhabitants), Turks (8,595), Bosniaks (7,585) and Vlachs (2,557) … How many ethnicities had less than 10000 people? 3 2
Domain knowledge … Smith was sidelined by a torn pectoral muscle suffered during practice … How many quarters did Smith play? 0 2
Addition … culminating in the Battle of Vienna of 1683, which marked the start of the 15-year-long Great Turkish War … What year did the Great Turkish War end? 1698 1668
---------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------- ------ ---------------
Results and Discussion {#sec:results}
======================
Conclusion
==========
We have presented DROP, a dataset of complex reading comprehension questions that require **D**iscrete **R**easoning **O**ver **P**aragraphs. This dataset is substantially more challenging than existing datasets, with the best baseline achieving only 32.7% F1, while humans achieve 96%. We hope this dataset will spur research into more comprehensive analysis of paragraphs, and into methods that combine distributed representations with symbolic reasoning. We have additionally presented initial work in this direction, with a model that augments QANet with limited numerical reasoning capability, achieving 47% F1 on DROP.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Noah Smith, Yoav Goldberg, and Jonathan Berant for insightful discussions that informed the direction of this work. The computations on [beaker.org](beaker.org) were supported in part by credits from Google Cloud.
[^1]: Work done as an intern at the Allen Institute for Artificial Intelligence in Irvine, California.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Zurek has provided a simple picture for the onset of the $\lambda$-transition in $^{4}He$, not currently supported by vortex density experiments. However, we argue that the seemingly similar argument by Zurek that superflow in an annulus of $^{4}He$ at a quench will be measurable is still valid.'
address: 'Blackett Laboratory, Imperial College, London SW7 2BZ'
author:
- 'E. Kavoussanaki and R. J. Rivers'
title: 'Can we Measure Superflow on Quenching $^{4}He$?'
---
As the early universe cooled it underwent a series of phase transitions, whose inhomogeneities have observable consequences. To understand how such transitions occur it is necessary to go beyond the methods of equilibrium thermal field theory that identified the transitions in the first instance.
In practice, we often know remarkably little about the dynamics of quantum field theories. A simple question to ask is the following: In principle, the field correlation length diverges at a continuous transition. In reality, it does not. What happens? Using simple causal arguments Kibble[@kibble1; @kibble2] made estimates of this early field ordering, because of the implications for astrophysics.
There are great difficulties in converting predictions for the early universe into experimental observations. Zurek suggested[@zurek1] that similar arguments were applicable to condensed matter systems for which direct experiments could be performed. In particular, for $^{4}He$ he argued that the measurement of superflow at a quench provided a simple test of these ideas. We present a brief summary of his argument.
Assume that the dynamics of the $^{4}He$ lambda-transition can be derived from an explicitly time-dependent Landau-Ginzburg free energy of the form $$F(T) = \int d^{3}x\,\,\bigg(\frac{\hbar^{2}}{2m}|\nabla\phi |^{2}
+\alpha (T)|\phi |^{2} + \frac{1}{4}\beta |\phi |^{4}\bigg),
\label{FNR}$$ in which $\alpha (T)$ vanishes at the critical temperature $T_{c}$. Explicitly, let us assume the mean-field result $\alpha (T) = \alpha_{0}\epsilon (T_{c})$, where $\epsilon = (T/T_{c}
-1)$, remains valid as $T/T_{c}$ varies with time $t$. In particular, we first take $\alpha (t)=\alpha (T(t))=-\alpha_{0}
t/\tau_{Q}$ in the vicinity of $T_{c}$. Then the fundamental length and time scales $\xi_{0}$ and $t_0$ are given from Eq.\[FNR\] as $\xi_{0}^{2} = \hbar^{2}/2m\alpha_{0}$ and $\tau_{0} = \hbar
/\alpha_{0}$. It follows that the equilibrium correlation length $\xi_{eq} (t)=\xi_{eq} (T(t)) $ and the relaxation time $\tau (t)$ diverge at $T_c$, which we take to be when $t$ vanishes, as $$\xi_{eq} (t) = \xi_{0}\bigg|\frac{t}{\tau_{Q}}\bigg|^{-1/2},
\,\,\tau (t) = \tau_{0}\bigg|\frac{t}{\tau_{Q}}\bigg|^{-1}.$$
Although $\xi_{eq} (t)$ diverges at $t=0$ this is not the case for the true correlation length $\xi (t)$, which can only grow so far in a finite time. Initially, for $t<0$, when we are far from the transition, we can assume that the field correlation length $\xi (t)$ tracks $\xi_{eq}(t)$ approximately. However, as we get closer to the transition $\xi_{eq}(t)$ begins to increase arbitrarily fast. As a crude upper bound, the true correlation length fails to keep up with $\xi_{eq}(t)$ by the time $-{\bar t}$ at which $\xi_{eq}$ is growing at the speed of sound $c(t) =\xi_{eq} (t)/\tau (t)$, which determines the rate at which the order-parameter can change. The condition $d\xi_{eq}
(t)/dt = c(t)$ is satisfied at $t=-{\bar t}$, where ${\bar t}
=\sqrt{\tau_{Q}\tau_{0}}$, with corresponding correlation length $${\bar\xi } =\xi_{eq}(-{\bar t}) = \xi_{0}\bigg(\frac{\tau_{Q}}{\tau_{0}}\bigg)^{1/4}.
\label{xiZ}$$ After this time it is assumed that the relaxation time is so long that $\xi (t)$ is essentially frozen in at ${\bar\xi}$ until time $t\approx +{\bar t}$, when it sets the scale for the onset of the broken phase.
A concrete realisation of how the freezing sets in is provided by the time-dependent Landau-Ginzburg (TDLG) equation for $F$ of (\[FNR\])[@zurek2], $$\frac{1}{\Gamma}\frac{\partial\phi_{a}}{\partial t} = -\frac{\delta
F}{\delta\phi_{a}} + \eta_{a},
\label{tdlg}$$ for $\phi = (\phi_{1} +i\phi_{2})/\sqrt{2}$, where $\eta_{a}$ is Gaussian noise. We can show self-consistently[@ray] that, for the relevant time-interval $-{\bar t}\leq t\leq {\bar t}$ the self-interaction term can be neglected ($\beta =0$), whereby a simple calculation finds $\xi\approx{\bar\xi}$ in this interval, as predicted. It thus happens that, at the onset of the phase transition, the field fluctuations are approximately Gaussian. The field phases $e^{i\theta({\bf r})}$, where $\phi ({\bf r}) =|\phi ({\bf r})| e^{i\theta({\bf r})}
$, are then correlated on the same scale as the fields.
Consider a closed path in the bulk superfluid with circumference $C\gg \xi (t)$. Naively, the number of ’regions’ through which this path passes in which the phase is correlated is ${\cal N} = O(C/\xi
(t))$. Assuming an independent choice of phase in each ’region’, the r.m.s phase difference along the path is $$\Delta\theta_{C} \approx\sqrt{{\cal N}} = O(\sqrt{C/\xi (t)}).
\label{rphase}$$
If we now consider a quench in an annular container of similar circumference $C$ of superfluid $^{4}He$ and radius $l\ll C$, Zurek suggested that the phase locked in is [*also*]{} given by Eq.\[rphase\], with ${\bar \xi}$ of Eq.\[xiZ\]. Since the phase gradient is directly proportional to the superflow velocity we expect a flow after the quench with r.m.s velocity $$\Delta v = O\bigg(\frac{\hbar}{m}\sqrt{\frac{1}{C{\bar\xi}}}\bigg).
\label{v1}$$ provided $l = O({\bar\xi})$. Although in bulk fluid this superflow will disperse, if it is constrained to a narrow annulus it should persist, and although not large is measurable.
In addition to this experiment, Zurek also suggested that the same correlation length ${\bar\xi}$ should characterise the separation of vortices in a quench. In an earlier paper[@ray] one of us showed that this is too simple. Causality arguments are not enough, and whether vortices form on this scale is also determined by the thermal activation of the Ginzberg regime, in which all $^{4}He$ experiments take place. Experimentally, this seems to be the case[@lancaster2].
Our aim in this paper is to see whether thermal fluctuations interfere with the prediction Eq.\[v1\], for which experiments have yet to be performed. Again consider a circular path in the bulk fluid (in the 1-2 plane), circumference $C$, the boundary of a surface $S$. For given field configurations $\phi_{a}({\bf x})$ the phase change $\theta_{C}$ along the path can be expressed as the surface integral $$\theta_{C} = 2\pi\int_{{\bf x}\in S} d^{2}x\,\,\rho({\bf x}),$$ where the topological density $\rho ({\bf x})$ is given by $$\rho ({\bf x}) = \delta^{2}[\phi ({\bf x})]\epsilon_{jk}\partial_{j}
\phi_{1}({\bf x}) \partial_{k}\phi_{2}({\bf x}),\,\,\,i,j=1,2
\label{rhof2}$$ where $\epsilon_{12}=-\epsilon_{21}=1$, otherwise zero.
The ensemble average $\langle\rho ({\bf x})\rangle_{t}$ is taken to be zero at all times $t$, guaranteed by taking $\langle\phi_{a}({\bf x})\rangle_{t} = 0 =
\langle\phi_{a}({\bf x})\partial_{j}\phi_{b}({\bf x})\rangle_{t}$. That is, we quench from an initial state with no rotation. For the Gaussian fluctuations that are relevant for the times of interest[@ray; @cal], all correlations are given in terms of the diagonal [*equal-time*]{} correlation function $G(r, t)$, defined by $$\langle \phi_{a}({\bf x})\phi_{b}({\bf 0})\rangle_{ t} =
\delta_{ab}G(r, t)\,\,\,\,\, r = |{\bf x}|.
\label{diag}$$ The correlation length $\xi (t)$ is defined by $G(r,t) = o(e^{-r/\xi (t)})$, for large $r >\xi (t)$. The TDLG does not lead to simple exponential behaviour, but there is no difficulty in defining $\xi (t)$ in practice[@ray; @cal].
The variance in the phase change around $C$, $\Delta\theta_{C}$ is determined from $$(\Delta\theta_{C})^{2}
=4\pi^{2}\int_{{\bf x}\in S}d^{2}x\int_{{\bf y}\in S}d^{2}y\,\langle\rho
({\bf x})\rho ({\bf y})\rangle_{t}.$$ The properties of densities for Gaussian fields have been studied in detail[@halperin; @maz]. Define $f(r,t)$ by $f(r,t) = G(r,t)/G(0,t)$. On using the conservation of charge $$\int\,d^{2}x\,\langle\rho ({\bf x})\rho ({\bf 0})\rangle_{t} = 0$$ it is not difficult to show, from the results of[@halperin; @maz], that $\Delta\theta_{C}$ satisfies $$(\Delta\theta_{C})^{2}
=-\int_{{\bf x}\not\in S}d^{2}x\int_{{\bf y}\in
S}d^{2}y\,\,{\cal C}(|{\bf x}-{\bf y}|,t),
\label{per}$$ where ${\bf x}$ and ${\bf y}$ are in the plane of $S$, and $${\cal C}(r,t) = \frac{1}{r}\frac{\partial}{\partial r}\bigg(\frac{f'^{2}(r,t)}{1 - f^{2}(r,t)}\bigg).
\label{C2}$$
Since $G(r,t)$ is short-ranged ${\cal C}(r,t)$ is short-ranged also. With ${\bf x}$ outside $S$, and ${\bf y}$ inside $S$, all the contribution to $(\Delta\theta_{C})^{2}$ comes from the vicinity of the boundary of $S$, rather than the whole area. That is, if we removed all fluid except for a strip from the neighbourhood of the contour $C$ we would still have the same result. This supports the assertion by Zurek that the correlation length for phase variation in bulk fluid is also appropriate for annular flow. The purpose of the annulus (more exactly, a circular capillary of circumference $C$ with radius $l\ll
C$) is to stop this flow dissipating into the bulk fluid.
More precisely, suppose that $C\gg\xi (t)$. Then, if we take the width $2l$ of the strip around the contour to be larger than the correlation length of ${\cal C}(r,t)$, Eq.\[per\] can be written as $$(\Delta\theta_{C})^{2}\approx -2
\,C\int_{0}^{\infty}dr\,r^{2}\,\,{\cal C}(r,t).
\label{per2}$$ The linear dependence on $C$ is purely a result of Gaussian fluctuations.
Insofar as we can identify the bulk correlation with the annular correlation, instead of Eq.\[v1\], we have $$\Delta v = \frac{\hbar}{m}\sqrt{\frac{1}{C\xi_{s}(t)}}.
\label{v3}$$ The step length $\xi_{s}(t)$ is given by $$\frac{1}{\xi_{s}(t)} = 2\int_{0}^{\infty}dr\frac{f'^{2}(r,t)}{1 - f^{2}(r,t)}.
\label{step}$$
There are two important differences between Eq.\[v3\] and Eq.\[v1\]. The first is in the choice of time for which $\Delta v$ of Eq.\[v3\] is to be evaluated. In Eq.\[v1\] the time is the time $-{\bar
t}$ of freezing in of the field correlation. Since $\xi (t)$ does not change much in the interval $-{\bar t}<t<{\bar t}$ we can as well take $t = 0$. We shall argue below that for Eq.\[v3\] a more appropriate time is the spinodal time $t_{sp}$ at which the transition has completed itself in the sense that the fields have begun to populate the ground states.
Secondly, a priori there is no reason to identify $\xi_{s}(t_{sp})$ with either ${\bar\xi}$ (or even $\xi (t_{sp})$). In particular, because ${\bar
\xi}$ in Eq.\[v1\] is defined from the large-distance behaviour of $G(r,t)$, and thereby on the position of the nearest singularity of $G(k,t)$ in the $k$-plane, it does [*not*]{} depend on the scale at which we observe the fluid. This is not the case for $\xi_{s}(t)$ which, from Eq.\[step\], explores all distance scales. Because of the fractal nature of the short wavelength fluctuations, $\xi_{s}(t)$ will depend on how many are included, i.e. the scale at which we look. If we quench in an annular capillary of radius $l$ much smaller than its circumference, we are, essentially, coarsegraining to that scale. That is, the observed variance in the flux along the annulus is $\pi l^{2}\Delta v$ for $\Delta v$ averaged on a scale $l$. We make the approximation that that is the [*major*]{} effect of quenching in an annulus. This cannot be wholly true, but it is plausible if the annulus is not too narrow for boundary effects to be important.
Provisionally we introduce a coarsegraining by hand, modifying $G(r,t)$ by damping short wavelengths $O(l)$ as $$G(r,t;l) = \int d \! \! \! / ^3 k\,
e^{i{\bf k}.{\bf x}}G(k,t)\,e^{-k^{2}l^{2}}.
\label{Gl}$$ We shall denote the value of $\xi_{s}$ obtained from Eq.\[Gl\] as $\xi_{s}(t;l)$. It permits an expansion in terms of the moments of $G(k,t)\,e^{-k^{2}l^{2}}$, $$G_{n}(t;l) = \int_{0}^{\infty}dk\,k^{2n}\,G(k,t)\,e^{-k^{2}l^{2}}.$$ For small $r$ it follows that $f'^{2}(r,t;l)/(1 - f^{2}(r,t;l))$ $$= \frac{G_{2}}{3G_{1}}\bigg[1 -
\bigg(\frac{3G_{3}}{20G_{2}}-\frac{G_{2}}{12G_{1}}\bigg)r^{2} + O(r^{4})\bigg].
\label{stepint}$$
Although, for large $r$, $f'(r,t;l)^{2} = o(e^{-2r/\xi (t)})$, we find that the bulk of the integral Eq.\[step\] lies in the forward peak, and that a good [*upper*]{} bound for $\xi_{s}$ is given by just integrating the quadratic term, whence $$\frac{1}{\xi_{s}(t;l)}\geq\frac{1}{\xi^{min}_{s}(t;l)} =\frac{4G_{2}}
{9G_{1}}\bigg( \frac{3G_{3}}{20G_{2}}-\frac{G_{2}}{12G_{1}}\bigg)^{-1/2},
\label{xis}$$ with the equality slightly overestimated. In units of $\xi_{0}$ and $\tau_{0}$ we have, in the linear regime[@ray], $$G_{n}(t;l)\approx\frac{I_{n}}{2^{n + 1/2}}\,e^{(t/{\bar t})^{2}}
\int_{0}^{\infty}dt'\,\frac{e^{-(t'-t)^{2}/{\bar t}^{2}}}{[t'
+l^{2}/2]^{n +1/2}}\frac{T(t')}{T_{c}},
\label{Gt}$$ where $I_{n} = \int_{0}dk k^{2n}\, e^{-k^{2}}$. The presence of the $T(t')/T_{c}$ term is a reminder that the strength of the noise $\eta$ is proportional to temperature. However, for the time scales $O({\bar t})\ll \tau_{Q}$ of interest to us this ratio remains near to unity and we ignore it. For small relative times the integrand gets a large contribution from the ultraviolet [*cutoff dependent*]{} lower endpoint, increasing as $n$ increases.
If we return to the Landau-Ginzberg equation Eq.\[tdlg\] we find that $\langle |\phi |^{2}\rangle_{t}\ll\alpha_{0}/\beta$ in the interval $ {-\bar t}\leq t\leq {\bar t}$. Although the field has frozen in, the fluctuations have amplitudes that are more or less uniform across all wavelengths. As a result, what we see depends totally on the scale at which we look. Specifically, from Eq.\[Gt\] $\xi^{min}_{s}(0;l) = O(l)$, as shown in the lowest curve of Fig.1.
If, as suggested by Zurek, we take $l =O({\bar \xi})$ we recover Eq.\[v1\] qualitatively, although a wider bore would give a correspondingly smaller flow. However, this is not the time at which to look for superflow since, although the field correlation length $\xi (t)$ may have frozen in by $t = 0$, the symmetry breaking has not begun.
Assuming the [*linearised*]{}[@ray] Eq.\[tdlg\] for small times $t >0$ we see that, as the unfreezing occurs, long wavelength modes with $k^{2} < t/\tau_Q$ grow exponentially and soon begin to dominate the correlation functions. How long a time we have depends on the self-coupling $\beta$ which, through $G_{1}$, sets the shortest time scale. This is because, at the absolute latest, $G_{1}$ must stop its exponential growth at $t = t_{sp}$, when $\langle |\phi |^{2}\rangle_{t_{sp}}$, satisfies $\langle |\phi |^{2}\rangle_{t_{sp}} = \alpha_{0}/\beta$. We further suppose that the effect of the backreaction that stops the growth initially freezes in any structure. In Fig.1 we also show $\xi^{min}_{s}(t;l)$ for $t= 3{\bar t}$ and $t= 4{\bar t}$, increasing as $t$ increases.
For $^{4}He$ with quenches of milliseconds the field magnitude has grown to its equilibrium value before the scale-dependence has stopped[@ray]. For vortex formation, for which the scale is $O(\xi_{0})$, the thickness of a vortex, the dependence of the density on scale makes the interpretation of observations problematic. This is not the same here. That the incoherent $\xi_{s}$ depends on radius $l$ is immaterial. The end result is that $$\Delta v = \frac{\hbar}{m}\sqrt{\frac{1}{C\xi_{s}(t_{sp};l)}}.
\label{v4}$$
We saw that the expression Eq.\[xis\] for $\xi_{s}$ assumed that $2l$ is larger than $$\xi_{eff}(t;l)=\bigg( \frac{3G_{3}}{20G_{2}}-\frac{G_{2}}{12G_{1}}\bigg)^{-1/2}.$$ Otherwise the correlations in the bulk fluid from which we want to extract annular behaviour are of longer range than the annulus thickness. Numerically, we find that $\xi_{eff}(0,l)=2l$ very accurately at $t=0$, but that $\xi_{eff}(t,l)\geq 2l$ for all $t>0$. A crude way to accomodate this is to cut off the integral Eq.\[per2\]. With a little effort, we see that the effect of this is that $\xi^{min}_{s}(t_{sp},l)$ of Eq.\[xis\] is replaced by $$\xi^{max}_{s}(t_{sp},l) =\xi^{min}_{s}(t_{sp},l)[1-(1-4l^{2}/\xi_{eff}(t_{sp},l)^{2})^{3/2}]^{-1},
\label{ximax}$$ greater than $\xi^{min}_{s}(t_{sp},l)$ and thereby [*reducing*]{} the flow velocity for narrower annuli. These are the dashed curves in Fig.1. The effect is largest for small radii $l\leq {\bar\xi}$, for which the approximation of trying to read the behaviour of annular flow from bulk behaviour is most suspect. A more realistic approach for such narrow capillaries is to treat the system as one-dimensional[@zurek1]. For this reason we have only considered $l\geq
{\bar\xi}$ in Fig.1. We would expect, from Eq.\[xis\], that $\xi_{s}(t_{sp};l)$ has an [*upper*]{} bound that lies somewhere between the curves.
Once $l$ is very large, so that the power in the fluctuations is distributed strongly across all wavelengths we recover our earlier result, that $\xi_{s}(t_{sp};l) = O(l)$. In Fig.1 this corresponds to the curves becoming parallel as $l$ increases for fixed $t$. However, the change is sufficiently slow that annuli, significantly wider than ${\bar\xi}$, for which experiments are more accessible, will give almost the same flow as narrower annuli. This would seem to extend the original Zurek prediction of Eq.\[v1\] to thicker annuli, despite our expectations for incoherent flow. However, we stress again that caution is necessary, since in the approximation to characterise an annulus by a coarse-grained ring without boundaries we have ignored effects in the direction perpendicular to the annulus. In particular, the circular cross-section of the tube has not been taken into account. One consequence of this is that infinite (non-selfintersecting) vortices in the bulk fluid have no counterpart in an annulus. Removing such strings will have an effect on $\Delta\theta_{C}$, since the typical fraction of vortices in infinite vortices is at the level of $70\%$. However, at the spinodal time the fluctuations in $^{4}He$ are relatively enhanced in the long wavelengths, and such an enhancement is known to reduce the amount of infinite vortices, perhaps to something nearer to $20\%$. The details of this effect (being pursued elsewhere) are unclear but, for the sake of argument we take the predictions of the curves in Fig.1 as a rough guide in the vicinity of their minima.
So far we have avoided the question as to which time curves we should follow. This is because $t_{sp}$ itself depends on the scale $l$ of the spatial volume for which the field average achieves its ground state value. In practice variation is small, with $t_{sp}$ for $^{4}He$ varying from about $3{\bar t}$ to $4{\bar t}$ as $l$ varies from $\xi_{0}\ll{\bar\xi}$ to $l = 10{\bar\xi}$. Since the curves for $\xi_{s}(t_{sp};l)$ lie so close to one another in Fig.1 once $l\geq 4{\bar\xi}$ the scale at which the coarse-grained field begins to occupy the ground states becomes largely irrelevant.
Since $\Delta v$ only depends on $\xi_{s}^{-1/2}$ it is not sensitive to choice of $l > 2{\bar\xi}$ at the relevant $t$. Given all these approximations our final estimate is (in the cm/sec units of Zurek[@zurek1]) $$\Delta v\approx 0.2(\tau_{Q}[\mu s])^{-\nu /4}/\sqrt{C[cm]}
\label{vf}$$ for radii of $2{\bar\xi} - 4{\bar\xi}$, $\tau_{Q}$ of the order of milliseconds and $C$ of the order of centimetres. $\nu = 1/2$ is the mean-field critical exponent above. In principle $\nu$ should be renormalised to $\nu =
2/3$, but the difference to $\Delta v$ is sufficiently small that we shall not bother. Given the uncertainties in its derivation the result Eq.\[vf\] is indistinguishable from Zurek’s[@zurek1] (with prefactor $0.4$), but for the possibility of using somewhat larger annuli. The agreement is, ultimately, one of dimensional analysis, but the coefficient could not have been anticipated. How experiments can be performed, even with the wider annuli that Eq.\[vf\] and Fig.1 suggest, is another matter.
We thank Glykeria Karra, with whom some of this work was done. This work is the result of a network supported by the European Science Foundation .
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We propose a theoretical method to enhance the coherent dipole coupling between two atoms in an optical cavity via parametrically squeezing the cavity mode. In the present scheme, conditions for coherent coupling are derived in detail and diverse dynamics of the system can be obtained by regulating system parameters. In the presence of environmental noise, an auxiliary squeezed field is employed to suppress, and even completely eliminate the additional noise induced by squeezing. In addition, we demonstrate that our scheme enables the effective suppression of atomic spontaneous emission. The results in our investigation could be used for diverse applications in quantum technologies.'
author:
- Yan Wang
- Chuang Li
- 'Elijah M. Sampuli'
- Jie Song
- Yongyuan Jiang
- Yan Xia
bibliography:
- 'reference.bib'
title: Enhancement of coherent dipole coupling between two atoms via squeezing a cavity mode
---
Introduction
============
Transport of excitation via resonant interaction between dipoles is fundamental to numerous disciplines, ranging from life sciences to quantum computing [@PhysRevLett.100.243201]. In quantum physics, the interaction between two atoms, one initially in the excited state and the other in the ground state, through virtual photon exchange is generally known as dipole-dipole interaction [@PhysRevA.89.043838]. It enables the atoms to resonantly exchange their energies by virtue of homenergic transition between energy levels. The availability of a strong long-range dipole-dipole interaction is an enabling resource for a wide range of studies and applications [@RevModPhys.82.2313; @PhysRevA.96.042306; @Quinten:98]. Typically, the interaction between atoms in highly excited Rydberg states attracts a great deal of research interests and has been elaborately studied both in theoretical schemes and in experimental realization [@0022-3700-14-21-003; @PhysRevA.72.022347; @PhysRevLett.93.153001; @PhysRevLett.93.233001; @PhysRevA.61.062309; @labuhn2016tunable]. This strong interaction can give rise to the so-called dipole blockade mechanism [@PhysRevLett.100.113003; @Comparat:10], which has been demonstrated experimentally [@PhysRevLett.99.073002; @gaetan2009observation; @PhysRevLett.99.163601; @PhysRevLett.105.193603; @PhysRevLett.97.083003] and offers possibilities for generating entanglement of several atoms [@urban2009observation; @PhysRevA.84.013831; @PhysRevA.72.042302], or implementing scalable quantum logic gates [@PhysRevLett.85.2208; @PhysRevLett.87.037901] as well as other applications in quantum information processing [@PhysRevA.66.065403; @PhysRevLett.99.260501].
Cavity quantum electrodynamics (QED) studies the light-matter interactions at the quantum level in terms of single atoms coupled to quantized cavity field [@PhysRevA.97.043820; @PhysRevLett.85.2392; @PhysRevLett.83.5166]. Realization of the strong coupling regime (SCR) via high manipulation and control of quantum systems [@PhysRevLett.117.043601] is requisite for implementing quantum information tasks. However, strong strength of light-matter coupling requires resonators with high quality factors and small mode volumes simultaneously [@PhysRevLett.112.213602], which remains extremely challenging to implement in experiments [@petersson2012circuit; @PhysRevLett.105.140501]. Alternatively, flexible schemes for effectively enhancing atom-cavity coupling have been proposed. Recent studies have made it easy to reach the ultrastrong coupling (USC) or deep strong coupling (DSC) regime [@kockum2018ultrastrong; @PhysRevLett.105.263603; @sanchez2018resolution]. For instance, Leroux and co-workers proposed a method to enhance the qubit-cavity coupling via parametric driving of the cavity [@PhysRevLett.120.093602]. Qin *et al.* exploited optical parametric amplification to enhance the atom-field coupling as well as the cooperativity of the system [@PhysRevLett.120.093601]. Upon parametrically squeezing the cavity mode, exponentially enhanced coupling strength has been achieved. This makes the squeezing-based scheme a powerful tool for enhancing coupling, while the dynamics of the system becomes complicated as a consequence. In-depth studies on the diverse dynamics are of importance and need to be promoted.
For controlled quantum dynamics, it is prerequisite to achieve strong interactions between single pairs of atoms. Motivated by the recent advances mentioned above, here, we propose a scheme to enhance the coherent dipole coupling between two atoms confined in an optical cavity. The general idea is based on parametrically squeezing the cavity mode. We show that the resonance interaction between atoms can be achieved in the presence of squeezing, for the atoms coupled to cavity with both identical and different strengths. In both cases, we derive in detail the effective Hamiltonian to describe the dynamics and verify the parametric conditions for resonance interaction, as well as enhancement of coupling. For consideration of the environmental noise, the dynamics of the system will be destroyed due to the additional noise induced by squeezing. We show that the undesired noise can be suppressed and even completely eliminated by employing a squeezed field with proper parameters. In particular, we demonstrate that our scheme is capable of effectively suppressing the atomic spontaneous emission.
The remainder of the paper is structured as follows. In Sec.\[sec:model\], we describe the physical model of the system and give the Hamiltonian in the presence of squeezing. In Sec.\[sec:dynamics\], we show how squeezing the cavity mode enables the dramatical enhancement of the coherent coupling between atoms. In Sec.\[sec:noise\], the environmental noise is considered into the system. We present an approach to suppress the squeezing-induced noises by simply employing a squeezed field. Finally, we give a brief discussion on experimental implementations and summarize our conclusions in Sec.\[sec:conclusion\].
The system and Hamiltonian {#sec:model}
==========================
We consider a quantum system consisting of two identical two-level atoms and a nonlinear medium, which are confined into a single mode cavity (see Fig.\[Fig1\]). The cavity mode can be squeezed while the nonlinear medium is pumped at frequency $\omega_p$, amplitude $\Omega_p$, and phase $\theta_p$. Working in a frame rotating at half the squeeze frequency $\omega_p/2$, the Hamiltonian of this system is given by (hereafter, $\hbar$=1) $$\label{Eq:original H}
H=\Delta_ca^\dag a+
\sum_{i=1,2}[\frac{\Delta_i}{2}\sigma_z^i+g_i(\sigma_+^ia+a^\dag\sigma_-^i)]+\frac{\Omega_p}{2}(e^{\mathrm{i}\theta_p}a^2+e^{-\mathrm{i}\theta_p}a^{\dag2}).$$
![Schematics of the system. (a) Two atoms and a nonlinear medium are trapped in an optical cavity. The nonlinear medium is pumped at frequency $\omega_p$, amplitude $\Omega_p$ and phase $\theta_p$. The coupling strengths between two atoms and cavity mode are $g_1$ and $g_2$, respectively. The decay rates of atoms and cavity are $\gamma$ and $\kappa$, respectively. (b) Level scheme of two atoms interacting with cavity field with detunings $\Delta_1'$ and $\Delta_2'$ ($\Delta_i'=\omega_i-\omega_c$), respectively. []{data-label="Fig1"}](fig1-eps-converted-to.pdf){width="0.8\linewidth"}
Here, $a$($a^\dag$) is the annihilation(creation) operator of the cavity mode with frequency $\omega_c$. The two-level atom is described by the Pauli operator $\sigma_z^i=\vert e\rangle_i\langle e\vert-\vert g\rangle_i\langle g\vert$ ($i$=1,2 labels the atoms) and the transition operators $\sigma_+^i={\sigma_-^i}^\dag=\vert e\rangle_i\langle g\vert$, where $\vert e\rangle_i$ and $\vert g\rangle_i$ are the excited state and the ground state, respectively. $g_i$ is the atom-cavity coupling strength. $\Delta_c=\omega_c-\omega_p/2$ and $\Delta_i=\omega_i-\omega_p/2$ are the cavity and atom detunings (with $\omega_i$ being the frequency of atom $i$). The Hamiltonian can be diagonalized by introducing the Bogoliubov squeezing transformation $a=a_s\cosh(r_p)-e^{-\mathrm{i}\theta_p}\sinh(r_p)a_s^\dag$ [@scully1999quantum], where the squeezing parameter $r_p$ is controllable and defined via $r_p=(1/2)\arctan(\alpha)$ ($\alpha=\Omega_p/\Delta_c$). The resulting Hamiltonian can be expressed as follows $$\label{Eq2}
H'=\Delta_sa_s^\dag a_s+\sum_{i=1,2}[\frac{\Delta_i}{2}\sigma_z^i+g_i\cosh(r_p)(\sigma_+^ia_s+a_s^\dag\sigma_-^i)
-g_i\sinh(r_p)(e^{-\mathrm{i}\theta_p}\sigma_+^ia_s^\dag+e^{\mathrm{i}\theta_p}a_s\sigma_-^i)].$$ The Hamiltonian (\[Eq2\]) has the form of the usual Rabi Hamiltonian, where $\Delta_s=\Delta_c\sqrt{1-\alpha^2}$ denotes the squeezed cavity frequency. The effective coupling strengths in $H'$ show an enhancement of atom-cavity coupling, which has been verified for the single atom case in [@PhysRevLett.120.093601]. In contrast, our system focuses on the dipole transitions, i.e., state transfer between two atoms using the cavity mode as a quantum bus [@PhysRevA.97.032341], and we intend to validate the possibility of enhancement of coherent dipole coupling between two atoms. To illuminate this, we then transform the Hamiltonian $H'$ into the interaction picture and obtain ($\theta_p$ is set to zero for simplicity) $$\label{Eq3}
\begin{split}
H_I=&g_1\cosh(r_p)a_s\sigma_+^1e^{\mathrm{i}\Delta_xt}+g_2\cosh(r_p)a_s\sigma_+^2e^{\mathrm{i}\Delta_yt}\\
&-g_1\sinh(r_p)a_s^\dag\sigma_+^1e^{\mathrm{i}\Delta_zt}-g_2\sinh(r_p)a_s^\dag\sigma_+^2e^{\mathrm{i}\Delta_wt}
+\mathrm{H.c.},
\end{split}$$ where $$\label{Eq4}
\begin{split}
&\Delta_x=\Delta_1-\Delta_s,\ \Delta_y=\Delta_2-\Delta_s,\\
&\Delta_z=\Delta_1+\Delta_s,\ \Delta_w=\Delta_2+\Delta_s.
\end{split}$$
We consider the states $\vert e_1g_2,n\rangle$ and $\vert g_1e_2,n\rangle$ with one excited atom and $n$ photons in the cavity. For large detunings with $\vert\Delta_x\vert,\vert\Delta_y\vert,\vert\Delta_z\vert,\vert\Delta_w\vert\gg \vert g_i\cosh(r_p)\vert,\vert g_i\sinh(r_p)\vert$, there is no energy exchange between atomic system and cavity. We can adiabatically eliminate the non-resonant states and obtain the effective Hamiltonian by using the perturbation theory. The coherent coupling between two atoms under different parametric conditions will be discussed separately in the following section.
Dynamics and enhancement of coherent coupling {#sec:dynamics}
=============================================
Different atoms-cavity coupling strengths
-----------------------------------------
We first examine the dipole-dipole interaction or state transfer between the two atoms coupled to the cavity mode with unequal strengths, $g_1\neq g_2$. In addition, we assume that the detunings $\Delta_1$ and $\Delta_2$ are different; the cavity mode is initially in the vacuum state, which is only virtually excited due to the large detuning conditions. Thus the cavity mode will be always in the vacuum state. By adiabatically eliminating the cavity mode, we obtain the following effective atomic Hamiltonian [@doi:10.1139/p07-060]: $$\label{Eq5}
\begin{split}
H_{\mathrm{eff}}=&\frac{g_1^2\cosh^2(r_p)}{\Delta_x}\vert e\rangle_1\langle e\vert
+\frac{g_2^2\cosh^2(r_p)}{\Delta_y}\vert e\rangle_2\langle e\vert
-\frac{g_1^2\sinh^2(r_p)}{\Delta_z}\vert g\rangle_1\langle g\vert
-\frac{g_2^2\sinh^2(r_p)}{\Delta_w}\vert g\rangle_2\langle g\vert\\
&+\big[\frac{g_1g_2\cosh^2(r_p)}{2}(\frac{1}{\Delta_x}+\frac{1}{\Delta_y})\sigma_+^1\sigma_-^2e^{\mathrm{i}(\Delta_x-\Delta_y)t}
-\frac{g_1g_2\sinh^2(r_p)}{2}(\frac{1}{\Delta_z}+\frac{1}{\Delta_w})\sigma_+^1\sigma_-^2e^{\mathrm{i}(\Delta_z-\Delta_w)t}\\
&+\mathrm{H.c.}\big].
\end{split}$$ The first four terms describe the photo-number-dependent Stark shifts, and the rest describe the dipole coupling between the two atoms. To realize efficient energy transition between states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$, the following condition should be satisfied $$\label{Eq6}
\begin{split}
\frac{g_1^2\cosh^2(r_p)}{\Delta_x}+\frac{g_1^2\sinh^2(r_p)}{\Delta_z}+\Delta_1=\frac{g_2^2\cosh^2(r_p)}{\Delta_y}+\frac{g_2^2\sinh^2(r_p)}{\Delta_w}+\Delta_2.
\end{split}$$
The accuracy of this condition can be examined by calculating the eigenenergies of the states, which are obtained as the real part of the eigenvalues. In Fig.\[Fig2\](a), we plot the real part of the eigenvalues $E_{1,2}$ obtained from exact Hamiltonian (\[Eq2\]) as a function of detuning $\Delta_2$. Here, only the lowest two energy levels are shown since others are well separated from the subspace. The eigenenergies of the two states avoid crossing at $\Delta_2=199.82g_1$ (where resonance occurs) and are split by $2g_{\mathrm{eff}}=0.096g_1$, as magnified in the inset in Fig.\[Fig2\](a). $g_{\mathrm{eff}}$ is expressed as $\frac{g_1g_2}{2}[\cosh^2(r_p)(\frac{1}{\Delta_x}+\frac{1}{\Delta_y})-\sinh^2(r_p)(\frac{1}{\Delta_z}+\frac{1}{\Delta_w})]$, which is obtained from effective Hamiltonian (\[Eq5\]). Simultaneously, by inserting the corresponding parameters into Eq.(\[Eq6\]), the detuning $\Delta_2$ is calculated to be $199.82g_1$, which is in good accordance with the result shown in Fig.\[Fig2\](a). Thus, we confirm that Eq.(\[Eq6\]) gives the parametric condition for effective resonance between the two atoms coupled to the cavity mode with different strengths.
![(a) Eigenenergies (real part of the eigenvalues $E_{1,2}$) as a function of detuning $\Delta_2$. Only the lowest two energy levels are shown. The inset shows a clear view of the avoided crossing phenomenon. The avoided crossing point corresponds to $\Delta_2=199.82g_1$, indicating the position where resonance occurs. The difference between the two levels is 0.096$g_1$ which is approximately equal to 2$g_{\mathrm{eff}}$. Here, the squeezing parameter $r_p$ is set as 2. (b)–(d) Oscillations between states $\vert e_1g_2\rangle$ (red curves) and $\vert g_1e_2\rangle$ (blue curves) obtained from the exact Hamiltonian (solid curves) and the effective Hamiltonian (square points) with different values of $r_p$: (b) $r_p=1$, (c) $r_p=2$, and (d) $r_p=3$. The shrink of oscillation period with increasing $r_p$ indicates the enhancement of coupling strength. (e) Enhancement of coupling as a function of $r_p$ ranging from 0 to 3. The three dots correspond to the cases in (b), (c), and (d), respectively. The common parameters in all figures are $g_2=1.5g_1$, $\Delta_1=200g_1$, and $\Delta_c=10g_1\cosh(r_p)/\sqrt{1-\alpha^2}$, and various values of $\Delta_2$ are obtained from Eq.(\[Eq6\]). The initial state of the system is $\vert e_1g_2,0\rangle$.[]{data-label="Fig2"}](fig2-eps-converted-to.pdf){width="0.65\linewidth"}
In what follows, we plot the oscillations between the states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ of the two atoms [@JOHANSSON20121760], with squeezing parameter $r_p$ taken as several values, as depicted in Figs.\[Fig2\](b)–\[Fig2\](d). The solid curves correspond to the exact results obtained from the total Hamiltonian (\[Eq2\]), whereas the square points denote the approximate results obtained from the effective Hamiltonian (\[Eq5\]). Clearly, the approximate results agree well with the exact results when our proposed condition (\[Eq6\]) is satisfied, which is a further evidence of the validity of the effective Hamiltonian $H_{\mathrm{eff}}$. Note that the effective coupling strength in Hamiltonian (\[Eq5\]) depends on the squeezing parameter $r_p$. Therefore, we predict that the effective coupling will be largely enhanced with the increasing of $r_p$. By comparing the dynamics shown in Figs.\[Fig2\](b)–\[Fig2\](d), we can see that the period of oscillation decreases apparently with the increasing of $r_p$, corresponding to the enhancement of effective coupling strength. It is worth mentioning that the ideal Rabi-like oscillations exist for $r_p$ as large as 3 under the proposed parameters. With a higher value of $r_p$, the ideal oscillations will be destroyed because the large detuning conditions are no longer satisfied. This can be surmounted by properly increasing the atomic detuning $\Delta_1$, e.g., the ideal oscillations occur with $\Delta_1=600g_1$ for $r_p=4$. We further plot the enhancement of coupling as a function of $r_p$ ranging from 0 to 3, as shown in Fig.\[Fig2\](e). A strong enhancement of coupling exceeding $10^2$ can be achieved in the present parameter range.
Identical atoms-cavity coupling strength
----------------------------------------
We now discuss the dynamics of our system when the coupling strengths between the two atoms and the cavity are identical, namely, $g_1=g_2=g$. Under the parametric condition $\Delta_1=\Delta_2=\Delta$ (or $\Delta_x=\Delta_y$, $\Delta_z=\Delta_w$), the effective Hamiltonian (\[Eq5\]) can be simplified to $$\label{Eq7}
\begin{split}
H_{\mathrm{eff}}=&\frac{g^2\cosh^2(r_p)}{\Delta_x}\sum_{i=1,2}\vert e\rangle_i\langle e\vert-\frac{g^2\sinh^2(r_p)}{\Delta_z}\sum_{i=1,2}\vert g\rangle_i\langle g\vert\\&+\big[\big(\frac{g^2\cosh^2(r_p)}{\Delta_x}-\frac{g^2\sinh^2(r_p)}{\Delta_z}\big)\sigma_+^1\sigma_-^2+\mathrm{H.c.}\big].
\end{split}$$
For different values of detuning $\Delta_s$, the efficient energy transfer between the two atoms can be realized. Here, we mainly consider two conditions associated with $\Delta_s$: (1) $\Delta_s$ is far less than $\Delta$; (2) the difference between $\Delta_s$ and $\Delta$ is equal to a certain value $\delta$ that is much larger than $\{g\cosh(r_p),g\sinh(r_p)\}$, and the sum of $\Delta_s$ and $\Delta$ is much larger than $\delta$. The oscillations between the states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ of the two atoms under conditions (1) and (2) are plotted in Figs.\[Fig3\](a) and \[Fig3\](b), respectively. The solid curves and square points are obtained by using the total Hamiltonian (\[Eq2\]) and effective Hamiltonian (\[Eq7\]), respectively. As clearly shown in the figures, the dynamics of the system exhibit ideal Rabi-like oscillations for $\Delta_s$ in different parametric regimes. To verify whether the effective coupling strengths in Hamiltonian (\[Eq7\]) (under the above-mentioned two conditions) can be enhanced by increasing the squeezing parameter $r_p$, we plot the corresponding dynamics with $r_p=2$, as depicted in Figs.\[Fig3\](c) and \[Fig3\](d). By separately comparing the results shown in the left or right panel of Fig.\[Fig3\], it can be directly found that the period of oscillation decreases almost a half as $r_p$ increases from 1 to 2. These results reveal that we indeed can realize the enhancement of dipole coupling by simply increasing the squeezing parameter. Overall, the effect of enhancement is valid in broad parametric space where the atoms-cavity coupling strengths and detunings take various values. Furthermore, we note that Eq.(\[Eq6\]) is a versatile condition because it is also valid for $g_1=g_2$ and $\Delta_1=\Delta_2$.
![Oscillations between states $\vert e_1g_2\rangle$ (red curves) and $\vert g_1e_2\rangle$ (blue curves) obtained from the exact Hamiltonian (solid curves) and the effective Hamiltonian (square points). The squeezing parameters $r_p$ in (a),(b) and (c),(d) are set as 1 and 2, respectively. Other parameters in (a),(c) are $\Delta=50g$, $\Delta_s=0.05\Delta$; in (b),(d), $\delta=20g\cosh(r_p)$, and the values of $\Delta_s$ and $\Delta$ fulfill $\Delta_s-\Delta=\delta$ and $\Delta_s+\Delta=20\delta$. The initial state of the system in all figures is $\vert e_1g_2,0\rangle$.[]{data-label="Fig3"}](fig3-eps-converted-to.pdf){width="0.8\linewidth"}
Next, we consider a special condition $\Delta=0$. In this case, the Hamiltonian (\[Eq3\]) transforms to $$H_I=g\cosh(r_p)a_s\sum_{i=1,2}\sigma_+^ie^{-\mathrm{i}\Delta_st}
-g\sinh(r_p)a_s^\dag\sum_{i=1,2}\sigma_+^ie^{\mathrm{i}\Delta_st}+\mathrm{H.c.}.$$
By using the perturbation theory and adiabatically eliminating the cavity mode, the effective atomic Hamiltonian can be written as $$\label{Eq9}
\begin{split}
H_{\mathrm{eff}}=&-\frac{g^2}{\Delta_s}\big[\cosh^2(r_p)\sum_{i=1,2}\vert e\rangle_i\langle e\vert+\sinh^2(r_p)\sum_{i=1,2}\vert g\rangle_i\langle g\vert\big]\\
&+\frac{g^2}{\Delta_s}\big[\cosh(2r_p)\sigma_+^1\sigma_-^2-\sinh(2r_p)\sigma_+^1\sigma_+^2+\mathrm{H.c.}\big].
\end{split}$$ As expected, we can see the terms describing the transitions between $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ in the Hamiltonian (\[Eq9\]). More interestingly, the specific condition $\Delta=0$ results in new terms $\sigma_+^1\sigma_+^2$ and $\sigma_-^1\sigma_-^2$ appearing in the Hamiltonian, corresponding to the transition path $\vert e_1e_2\rangle$ $\leftrightarrow$ $\vert g_1g_2\rangle$. The dynamics of these two transition paths with different squeezing parameters are shown in Fig.\[Fig4\]. For the path $\vert e_1g_2\rangle$ $\leftrightarrow$ $\vert g_1e_2\rangle$, coherent population oscillation occurs for $r_p=1$ \[see Fig.\[Fig4\](a)\] and the period of it shrinks markedly when $r_p$ increases to 2 \[see Fig.\[Fig4\](c)\]. However, for the path $\vert e_1e_2\rangle$ $\leftrightarrow$ $\vert g_1g_2\rangle$, an inadequate oscillation between $\vert e_1e_2\rangle$ and $\vert g_1g_2\rangle$ is observed for $r_p=1$ \[see Fig.\[Fig4\](b)\], arising from the asymmetrical energy shifts associated with $r_p$ in the Hamiltonian (\[Eq9\]). With the increasing of $r_p$, the effective coupling can be enhanced to much larger than the difference between the energy shifts. This compensates the inadequate oscillation and therefore the ideal Rabi-like oscillation reappears in the dynamics, as shown in Fig.\[Fig4\](d).
![Oscillations between states (a),(c) $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ (b),(d) $\vert e_1e_2\rangle$ and $\vert g_1g_2\rangle$ obtained from the exact Hamiltonian (solid curves) and the effective Hamiltonian (square points). The squeezing parameters $r_p$ in (a),(b) and (c),(d) are set as 1 and 2, respectively. Other common parameters in (a–d) are $\Delta=0$ and $\Delta_c=1500g$. The initial state of the system in (a),(c) and (b),(d) are $\vert e_1g_2,0\rangle$ and $\vert g_1g_2,0\rangle$, respectively.[]{data-label="Fig4"}](fig4-eps-converted-to.pdf){width="0.8\linewidth"}
The above discussions are valid under the condition $\Delta_c=1500g$. When, however, the squeezing parameter $r_p$ further increases to over 2, the ideal Rabi-like oscillations will be destroyed. This is reasonable because the large detuning condition $\Delta_s\gg\{g\cosh(r_p),g\sinh(r_p)\}$ cannot be satisfied perfectly with the increasing of $r_p$. To give more insights into this, we take the transition $\vert e_1g_2\rangle$ $\leftrightarrow$ $\vert g_1e_2\rangle$ as an example, and plot the detuning $\Delta_s/g$ and the enhancement of coupling as functions of $r_p$ in Fig.\[Fig5\](a). Clearly, $\Delta_s/g$ drops off rapidly with the increasing of $r_p$ and down to near zero when $r_p$ increases to 3. By checking the numerical results of the dynamical evolutions, it can be directly seen that the ideal oscillations will be destroyed when $r_p$ reaches and exceeds 2.2 (not shown here). Accordingly for $r_p=2.2$, $\Delta_s/g$ is calculated to be 36.8, which is close to tenfold of $\{\cosh(r_p),\sinh(r_p)\}$. This indicates that the enhancement of coupling is valid in the region $r_p<2.2$ while invalid in the region $r_p\ge2.2$, as labeled in Fig.\[Fig5\](a). To enlarge the valid region of the enhancement of coupling, a possible strategy is to change the detuning $\Delta_s$ to $10g\sqrt{\cosh(2r_p)}$ ($\Delta_{s}^{'}$). In Fig.\[Fig5\](b), we plot the detuning $\Delta_{s}^{'}/g$ and the corresponding enhancement of coupling as functions of $r_p$ ranging from 0 to 5. Upon the adjustment of detuning, the large detuning condition can be always satisfied due to the incremental increase of $\Delta_{s}^{'}/g$ with increases in $r_p$. Therefore, the valid region of the enhancement of coupling, where the ideal Rabi-like oscillations occur (verified by the numerical simulations of the dynamics), is greatly enlarged as labeled in Fig.\[Fig5\](b).
![Validity of the enhancement of effective coupling. (a) The detuning $\Delta_s/g$ (blue curve) and the enhancement of coupling (red curve, expressed as $\cosh(2r_p)/\sqrt{1-\alpha^2}$) as functions of $r_p$, when $\Delta_c$ is set as $1500g$. The yellow and gray shaded areas represent the valid ($r_p<2.2$, where ideal oscillations can be obtained) and the invalid region ($r_p\ge2.2$, where ideal oscillations will be destroyed) of the enhancement of coupling, respectively. The black dot corresponds to $r_p=2.2$ and $\Delta_s/g=36.8$. (b) The detuning $\Delta_{s}^{'}/g$ (blue curve, set as $10\sqrt{\cosh(2r_p)}$) and the enhancement of coupling (red curve, expressed as $\sqrt{\cosh(2r_p)}$) as functions of $r_p$. Based on the adjustment of detuning, the valid region of the enhancement of coupling is enlarged to $r_p=5$ (and even more).[]{data-label="Fig5"}](fig5-eps-converted-to.pdf){width="0.6\linewidth"}
Consideration of squeezing-induced noise {#sec:noise}
========================================
{width="0.9\linewidth"}
We have demonstrated the enhancement of coherent dipole coupling between the two atoms by parametrically squeezing the cavity mode. However, the introduction of squeezing will lead to additional noise into the system. To demonstrate explicitly the squeezing-induced noise, it is straightforward to derive the master equation in the presence of squeezing. Here, we assume that the cavity mode is coupled to a thermal reservoir. When the cavity mode is squeezed with squeezing parameter $r_p$ and reference phase $\theta_p$, the system master equation in terms of $a_s$ is derived as (see Appendix for details) $$\label{Eq:master equation without squeezed field}
\begin{split}
\dot{\rho}(t)=&i[\rho(t),H(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)
+[(2N+1)\bar{n}_{\mathrm{th}}+N+1]\mathcal{L}(L_{as})\rho(t)\\
&+[(2N+1)\bar{n}_{\mathrm{th}}+N]\mathcal{L}(L_{as}^\dag)\rho(t)
-[M(2\bar{n}_{\mathrm{th}}+1)]\mathcal{L'}(L_{as}^\dag)\rho(t)
-[M^{\ast}(2\bar{n}_{\mathrm{th}}+1)]\mathcal{L'}(L_{as})\rho(t)
\Big\},
\end{split}$$ where $H(t)$ is the Hamiltonian given by Eq.(\[Eq:original H\]); $N$ and $M$ are derived as
$$N=\sinh^2(r_p),$$
$$M=\cosh(r_p)\sinh(r_p)e^{\mathrm{-i}\theta_p}.$$
These two parameters describe the effective thermal noise and two-photon correlation of the squeezed cavity mode [@PhysRevLett.114.093602; @breuer2002theory]. The introduction of these noises will destroy the regular system dynamics, i.e., suppressing the amplitude of oscillations. To illustrate this, we take the oscillations between states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ \[described by Hamiltonian (\[Eq5\])\] as an example. In Fig.\[Fig6\](a), we numerically solve the master equation (\[Eq:master equation without squeezed field\]) and plot the oscillations between states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$, which are immensely suppressed due to the additional noise induced by cavity mode squeezing.
It is demonstrated in Ref.[@PhysRevLett.120.093601] that the squeezing-induced noise can be suppressed by introducing an auxiliary squeezed-vacuum field to drive the cavity. Motivated by this, we explore the noise-resisted scheme in our system by virtue of the squeezed field. The squeezed field with the squeezing parameter $r_e$ and reference phase $\theta_e$, which has a much larger linewidth than the cavity mode, can be regarded as a squeezed reservoir. In this situation, we can assume that the cavity mode is coupled to a squeezed thermal reservoir. By choosing appropriate matching conditions, e.g., $r_e=r_p$ and $\theta_e+\theta_p=\pi$, the squeezing-induced noise can be completely eliminated (i.e., $N_s=\bar{n}_{\mathrm{th}}$ and $M_s=0$, see Appendix for details) and the dynamic of system is therefore governed by the master equation in the standard Lindblad form $$\label{Eq:master equation with squeezed field}
\dot{\rho}(t)=i[\rho(t),H'(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+(\bar{n}_{\mathrm{th}}+1)\mathcal{L}(L_{as})\rho(t)+\bar{n}_{\mathrm{th}}\mathcal{L}(L_{as}^\dag)\rho(t)
\Big\}.$$ From the point of view of system-reservoir coupling, squeezing the cavity mode induces an increase in system-reservoir coupling strength. Employing an auxiliary squeezed field can, in principle, be equivalent to offset the increase in system-reservoir coupling via reservoir manipulation (squeezing), as proper matching conditions are satisfied. In Fig.\[Fig6\](b), we plot the oscillations between states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ by numerically solving the master equation (\[Eq:master equation with squeezed field\]). As expected, the populations of states $\vert e_1g_2\rangle$ and $\vert g_1e_2\rangle$ exhibit rather strong oscillations.
For further insight into the parametric ranges for suppressing the squeezing-induced noise, we plot the dynamics of the same model as dependent on $r_e/r_p$ ranging from 0 to 4 in Fig.\[Fig6\](c). It reveals that the oscillations occur in a broad region except for the above-mentioned condition $r_e=r_p$. Specifically, the amplitude of oscillation during the whole evolution process reaches a peak for $r_e=r_p$ \[marked with a solid line in Fig.\[Fig6\](c)\], which gradually drops off with increasing or decreasing the value of $r_e/r_p$. A clearly view of the oscillations for several values of $r_e/r_p$ (1, 1.5, and 2) can be seen in Fig.\[Fig6\](e). Likewise, we also plot the dynamics as a function of $\theta_e+\theta_p$ in Fig.\[Fig6\](d). The oscillations periodically occur when $\theta_e+\theta_p=\pm n\pi\ (n=1, 3, 5 \cdots)$, and exhibit almost identical evolution behavior. To verify this, in Fig.\[Fig6\](f) we plot the oscillations of state $\vert g_1e_2\rangle$ for $\theta_e+\theta_p=-3\pi, -\pi, \pi$ and $3\pi$. It is clearly to see the exactly overlap of the oscillations.
Another prominent feature of our method is the effective suppression of atomic spontaneous emission. As mentioned above, the coherent coupling between atoms can be greatly enhanced by squeezing the cavity mode, with the additional noise in the cavity being suppressed by applying the squeezed field. With the enhancement of coupling, the atomic dissipation cannot be affected and thus unchanged, which is equivalent to an effective suppression of atomic dissipation. This allows the observation of oscillations even for a large atomic decay rate $\gamma$, for instance, $\gamma/g_1=0.1$ \[Fig.\[Fig7\](a)\]. When we further increase $\gamma/g_1$ to 0.5, it can be seen that the oscillations also occur for several periods \[Fig.\[Fig7\](b)\]. Note that the cavity decay rate ($\kappa/g_1$) and thermal photon number ($\bar{n}_{\mathrm{th}}$) in Figs.\[Fig7\](a) and \[Fig7\](b) have been set as 1 and 0, respectively. The capacity of our scheme for resisting strong cavity dissipation is attributed to the smaller population of photons arising from the adiabatical elimination of the non-resonant intermediate state $\vert g_1g_2,1\rangle$. For comparison, we also plot the oscillations in the case of $\bar{n}_{\mathrm{th}}=0.1$, as shown in Figs.\[Fig7\](c) and \[Fig7\](d). Clearly, injection of the thermal photons results in a slight depopulation of states.
![Oscillations between states $\vert e_1g_2\rangle$ (red curves) and $\vert g_1e_2\rangle$ (blue curves) obtained by solving the master equation (\[Eq:master equation with squeezed field\]) at different values of decay rates and thermal photons (as labeled in figures). The parameters are $r_p=r_e=3$, $\theta_e=\pi$, and other common parameters are the same as Fig.\[Fig2\](d).[]{data-label="Fig7"}](fig7-eps-converted-to.pdf){width="0.8\linewidth"}
Discussion and conclusion {#sec:conclusion}
=========================
We give a brief discussion regarding the experimental implementation. The promising pathway for realizing our proposal is based on the platform of cavity QED. The configuration of the two-level atom can be realized in alkali-metal atoms, e.g., cesium [@PhysRevLett.97.083003; @PhysRevLett.93.233603] and rubidium [@PhysRevLett.112.043601; @PhysRevLett.94.033002]. The single-mode cavity can be implemented typically using a high-finesse Fabry-Perot resonator [@PhysRevLett.118.133604; @PhysRevLett.110.223003]. An atom-cavity coupling strength ($g$) of tens of MHz is generally available. The cavity mode squeezing can be generated via the process of optical parametric amplification, e.g., pumping a second-order nonlinear medium \[periodically-poled potassium titanyl phosphate (PPKTP) crystal [@PhysRevLett.117.110801; @schnabel2017squeezed]\]. For producing the squeezed field with high bandwidth up to GHz, similar methods with generating cavity-field squeezing have been demonstrated in [@Ast:13; @Serikawa:16] by using the PPKTP crystal. The squeezing parameter and reference phase can be controlled via adjusting the amplitude and phase of the pumped laser. On the basis of the above setups, we assume the following parameters: $g_1/2\pi=5$ MHz, $g_2/2\pi=7.5$ MHz, $r_p=r_e=2$, $\theta_p=0$, $\theta_e=\pi$, $\kappa/2\pi=500$ kHz, $\gamma/2\pi=5$ kHz, and $\bar{n}_{\mathrm{th}}=0.1$. We then choose the detunings $\Delta_1=200g_1$, $\Delta_2=199.822g_1$, and $\Delta_c=10g_1\cosh(r_p)/\sqrt{1-\alpha^2}$ to satisfy the resonant condition between the atoms. The resulting period of oscillation is about 2.1 $\mu$s and the population of state $\vert g_1e_2\rangle$ for the first period reaches 0.9, which we have verified numerically. In addition, our scheme is also promising for application in solid-state systems, in particular, superconducting quantum circuits that combine superconducting qubits with microwave-frequency cavity [@PhysRevLett.101.253602; @PhysRevLett.119.023602].
In conclusion, we have demonstrated that parametric squeezing of the cavity mode enables a strong enhancement of coherent dipole coupling between two atoms. The resonance between atoms occurs for atoms coupled to the cavity with both identical and different strengths, manifesting as the observation of coherent population oscillations in the dynamics. In both cases, we have derived in detail the effective Hamiltonian to describe the dynamics of various systems and verified the parametric conditions for effective state transfer, as well as the enhancement of coupling. It is demonstrated that the shrink of the period of oscillations with increasing the squeezing parameters can act as an indication of the enhancement of coupling. We hereby anticipate that by modulating the squeezing parameter, the enhancement can be achieved in broad parametric regimes where the atoms-cavity coupling strength and detuning take various values. For consideration of the environmental noise, the dynamics of the system will be destroyed due to the additional noise induced by squeezing, which can be completely eliminated by employing a squeezed field. The parametric conditions for resisting noise have been investigated and demonstrated here in detail. In addition, we also shown that our scheme can effectively suppress atomic spontaneous emission and cavity decay. The oscillations occur for several periods even for strong atomic and cavity dissipations. Our method for enhancing dipole-dipole interaction between atoms can be applicable to a wide range of physical systems, and will find various applications in quantum information processing.
ACKNOWLEDGMENTS
===============
This work was supported by National Natural Science Foundation of China (NSFC) (11675046), Program for Innovation Research of Science in Harbin Institute of Technology (A201412), and Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).
APPENDIX: Effective master equations {#appendix-effective-master-equations .unnumbered}
====================================
Here, we first assume that the cavity mode is coupled to a thermal reservoir. Considering several relaxation processes (including the decay of atoms and cavity field) affecting the system, the dynamics of the system can be described by the quantum master equation in the standard Lindblad form [@scully1999quantum] $$\label{Eq:standard Lindblad form}
\dot{\rho}(t)=i[\rho(t),H(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+(\bar{n}_{\mathrm{th}}+1)\mathcal{L}(L_{a})\rho(t)+\bar{n}_{\mathrm{th}}\mathcal{L}(L_{a}^\dag)\rho(t)
\Big\},$$ where $\rho(t)$ is the density operator of the system, $H(t)$ is the Hamiltonian given by Eq.(\[Eq:original H\]), $L_a=\sqrt{\kappa}a$ is the Lindblad operator describing the cavity decay with rate $\kappa$, $L_{1,2}=\sqrt{\gamma}\vert g\rangle_{1,2}\langle e\vert$ are the Lindblad operators describing the atomic spontaneous emissions with identical rate $\gamma$, $\bar{n}_{\mathrm{th}}=(e^{\hbar\omega/k_BT}-1)^{-1}$ is the mean number of thermal photons in the cavity at temperature $T$, $\mathcal{L}(O)$ and $\mathcal{L}'(O)$ are defined by
$$\mathcal{L}(O)\rho(t)=O^\dag O\rho(t)-2O\rho(t) O^\dag +\rho(t) O^\dag O,$$
$$\mathcal{L}'(O)\rho(t)=O O\rho(t)-2O\rho(t) O+\rho(t)OO.$$
When the cavity mode is squeezed with a squeezing parameter $r_p$ and a reference phase $\theta_p$, the master equation can be rewritten by simply performing Bogoliubov transformation $a=a_s\cosh(r_p)-e^{-\mathrm{i}\theta_p}\sinh(r_p)a_s^\dag$, given by $$\label{Eq in Appen:master equation without squeezed field}
\begin{split}
\dot{\rho}(t)=&i[\rho(t),H'(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+[(2N+1)\bar{n}_{\mathrm{th}}+N+1]\mathcal{L}(L_{as})\rho(t)\\
&+[(2N+1)\bar{n}_{\mathrm{th}}+N]\mathcal{L}(L_{as}^\dag)\rho(t)-[M(2\bar{n}_{\mathrm{th}}+1)]\mathcal{L'}(L_{as}^\dag)\rho(t)-[M^{\ast}(2\bar{n}_{\mathrm{th}}+1)]\mathcal{L'}(L_{as})\rho(t)
\Big\},
\end{split}$$ where $H'(t)$ is given by Eq.(\[Eq2\]), $\mathcal{L}_{as}=\sqrt{\kappa}a_s$ is the Lindblad operator describing the decay of the squeezed-cavity mode, $N$ and $M$ describe the effective thermal noise and two-photon correlation [@PhysRevLett.114.093602; @breuer2002theory], respectively, given by
$$N=\sinh^2(r_p),$$
$$M=\cosh(r_p)\sinh(r_p)e^{\mathrm{-i}\theta_p}.$$
In the following, we demonstrate the squeezing-induced noise, i.e., thermal noise and two-photon correlation, can be suppressed by employing a squeezed field to drive the cavity. In view of a high-bandwidth squeezed field (up to GHz), which has been realized in experiments [@Ast:13; @Serikawa:16], it can be regarded as a squeezed reservoir due to a relatively small linewidth of typical optical cavity mode. Therefore, we assume that the cavity mode is coupled to a squeezed thermal reservoir with a squeezing parameter $r_e$ and phase $\theta_e$. The dynamics of system can be described by the following master equation [@doi:10.1080/09500349808230898]
$$\label{EqA1}
\begin{split}
\dot{\rho}(t)=&i[\rho(t),H(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+(N'+1)\mathcal{L}(L_a)\rho(t)+N'\mathcal{L}(L_a^\dag)\rho(t)\\
&-M'\mathcal{L}'(L_a^\dag)\rho(t)-M'^\ast\mathcal{L}'(L_a)\rho(t)\Big\},
\end{split}$$
where $N'$ and $M'$ are parameters that describe the squeezed thermal reservoir and are given by
$$N'=\bar{n}_{\mathrm{th}}\cosh(2r_e)+\sinh^2(r_e),$$
$$M'=(2\bar{n}_{\mathrm{th}}+1)\cosh(r_e)\sinh(r_e)e^{\mathrm{i}\theta_e}.$$
Following the same method as before, the master equation after squeezing the cavity mode is re-expressed as $$\label{EqA5}
\begin{split}
\dot{\rho}(t)=&i[\rho(t),H'(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+(N_s+1)\mathcal{L}(L_{as})\rho(t)+N_s\mathcal{L}(L_{as}^\dag)\rho(t)\\
&-M_s\mathcal{L}'(L_{as}^\dag)\rho(t)-M_s^\ast\mathcal{L}'(L_{as})\rho(t)\Big\},
\end{split}$$ where $N_s$ and $M_s$ are given by
$$\begin{split}
N_s=&[\bar{n}_{\mathrm{th}}\cosh(2r_e)+\sinh^2(r_e)]\cosh(2r_p)+\sinh^2(r_p)\\&+(\bar{n}_{\mathrm{th}}+\frac{1}{2})\sinh(2r_e)\sinh(2r_p)\cos(\theta_e+\theta_p),
\end{split}$$
$$\begin{split}
M_s=&-\exp(-\mathrm{i}\theta_p)(2\bar{n}_{\mathrm{th}}+1)\Big\{\frac{1}{2}\sinh(2r_p)\cosh(2r_e)\\&+\frac{1}{2}\sinh(2r_e)\{\exp[\mathrm{i}(\theta_e+\theta_p)]\cosh^2(r_p)\\&+\exp[-\mathrm{i}(\theta_e+\theta_p)]\sinh^2(r_p)\}\Big\}.
\end{split}$$
These two terms indicate the undesired noise in the cavity induced by squeezing, which can be removed by choosing appropriate condition. For instance, when choosing $r_e=r_p$ and $\theta_e+\theta_p=\pi$, the $N_s$ and $M_s$ can be reduced to $\bar{n}_{\mathrm{th}}$ and 0, respectively. In this case, the master equation (\[EqA5\]) is simplified to the standard Lindblad form $$\dot{\rho}(t)=i[\rho(t),H'(t)]-\frac{1}{2}\Big\{\sum_{x=1,2}\mathcal{L}(L_{x})\rho(t)+(\bar{n}_{\mathrm{th}}+1)\mathcal{L}(L_{as})\rho(t)+\bar{n}_{\mathrm{th}}\mathcal{L}(L_{as}^\dag)\rho(t)
\Big\}.$$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We find new constructions of infinite families of skew Hadamard difference sets in elementary abelian groups under the assumption of the existence of cyclotomic strongly regular graphs. Our construction is based on choosing cyclotomic classes in finite fields.'
author:
- Koji Momihara
title: Skew Hadamard difference sets from cyclotomic strongly regular graphs
---
[Keywords: skew Hadamard difference set, cyclotomic strongly regular graph, Gauss sum ]{}
Introduction
============
We assume that the reader is familiar with the basic theories of difference sets and strongly regular graphs (srg) as can found in [@bjl; @bh].
A difference set $D$ in an (additively written) finite group $G$ is called [*skew Hadamard*]{} if $G$ is the disjoint union of $D$, $-D$, and $\{0\}$. The primary example (and for many years, the only known example in abelian groups) of skew Hadamard difference sets is the classical Paley (quadratic residue) difference set in $(\F_q,+)$ consisting of the nonzero squares of $\F_q$, where $\F_q$ is the finite field of order $q$, a prime power congruent to 3 modulo 4. Skew Hadamard difference sets are currently under intensive study, see [@CP; @DY06; @DWX07; @F11; @FX113; @FMX11; @M; @WH09; @WQWX]. There were two major conjectures in this area: (i) If an abelian group $G$ contains a skew Hadamard difference set, then $G$ is necessarily elementary abelian. (ii) Up to equivalence the Paley difference sets mentioned above are the only skew Hadamard difference sets in abelian groups. The former conjecture is still open in general. The latter conjecture turned out to be false: Ding and Yuan [@DY06] constructed a family of skew Hadamard difference sets in $(\F_{3^m},+)$, where $m\ge 3$ is odd, and showed that two examples in the family are inequivalent to the Paley difference sets. Very recently, Muzychuk [@M] constructed infinitely many inequivalent skew Hadamard difference sets in an elementary abelian group of order $q^3$. The reader may check the introduction of [@FX113] for a good short survey of known constructions of skew Hadamard difference sets and related problems.
A classical method for constructing both connection sets of strongly regular Cayley graphs (called [*partial difference sets*]{}) and ordinary difference sets in the additive groups of finite fields is to use cyclotomic classes of finite fields. Let $p$ be a prime, $f$ a positive integer, and let $q=p^f$. Let $k>1$ be an integer such that $k|(q-1)$, and $\gamma$ be a primitive root of $\F_q$. Then the cosets $C_i^{(k,q)}=\gamma^i \langle \gamma^k\rangle$, $0\leq i\leq k-1$, are called the [*cyclotomic classes of order $k$*]{} of $\F_q$. Many authors have studied the problem of determining when a union $D$ of some cyclotomic classes forms a (partial) difference set. Especially, when $D$ consists of only a subgroup of $\F_q$, many authors have studied extensively [@BM73; @BMW82; @FX111; @FX113; @FMX11; @GXY11; @M75; @SW02; @S67; @VLSch]. We call such a strongly regular Cayley graph $\Cay(\F_q,D)$ [*cyclotomic*]{}. The well known Paley graphs are primary examples of cyclotomic srgs. Also, if $D$ is the multiplicative group of a subfield of $\F_q$, then it is clear that $\Cay(\F_q , D)$ is strongly regular. These cyclotomic srgs are usually called [*subfield examples*]{}. Next, if there exists a positive integer $t$ such that $p^t\equiv -1\,(\mod{k})$, then $\Cay(\F_q , D)$ is strongly regular. This case is usually called [*semi-primitive*]{}. In [@SW02], Schmidt and White conjectured that if $k\,|\,\frac{q-1}{p-1}$ and $\Cay(\F_{p^f},C_0^{(k,q)})$ is strongly regular, then one of the following holds:
1. (subfield case) $C_0=\F_{p^d}^\ast$ where $d\,|\,f$,
2. (semi-primitive case) $-1\in \langle p\rangle\le (\Z/k\Z)^\ast$,
3. (exceptional case) $\Cay(\F_{p^f},C_0)$ has one of the parameters given in Table \[Tab1\].
$$\begin{array}{|c||c|c|c|c|}
\hline
\mbox{No.}&k&p&f&e:=[(\Z/k\Z)^\ast:\langle p\rangle]\\
\hline
1&11&3&5&2\\
2&19&5&9&2\\
3&35&3&12&2\\
4&37&7&9&4\\
5&43&11&7&6\\
6&67&17&33&2\\
7&107&3&53&2\\
8&133&5&18&6\\
9&163&41&81&2\\
10&323&3&144&2\\
11&499&5&249&2\\
\hline
\end{array}$$
Recently, in [@FX111; @FMX11; @GXY11; @M12], it was succeeded to generalize the sporadic examples of Table \[Tab1\] except for the srg of No. 1 and several subfield examples into infinite families using “index $2$ or $4$ Gauss sums” and “relative Gauss sums.” Also, Wu [@Wu12] gave a necessary and sufficient condition for $\Cay(\F_{p^{(k-1)/e}},C_0^{(p_1,p^{(k-1)/e})})$ to be strongly regular by generalizing the method of [@GXY11] when $k$ is a prime. On the other hand, in [@FX113; @FMX11], Feng, Xiang, and this author found new constructions of skew Hadamard difference sets via a computation of a character sum involving index $2$ Gauss sums. In particular, in [@FX113; @FMX11], it was shown that $D=\bigcup_{i\in \{0\}\cup \langle p\rangle\cup 2\langle p\rangle}C_i^{(k,p^f)} $ is a skew Hadamard difference sets or a Paley type partial difference sets for the triples $(k,p,f)$ of Table \[Subsec2Tab1\] and these examples can be generalized into infinite families. (A partial difference set $D$ in a group $G$ is said to be of [*Paley type*]{} if the parameters of the corresponding strongly regular Cayley graph are $(v, (v-1)/2,(v-5)/4,(v-1)/4)$.)
$$\begin{array}{|c||c|c|c|}
\hline
\mbox{No.} &\mbox{$k$}&\mbox{$p$} & \mbox{$f$}\\
\hline \hline
\mbox{1} & \mbox{$2\cdot 11$}&\mbox{$3$} & \mbox{$5$}\\
\mbox{2} & \mbox{$2\cdot 19$}&\mbox{$5$} &\mbox{$9$}\\
\mbox{3} & \mbox{$2\cdot 67$}&\mbox{$17$} &\mbox{$33$}\\
\mbox{4} &\mbox{$2\cdot 107$}&\mbox{$3$}& \mbox{$53$}\\
\mbox{5} &\mbox{$2\cdot 163$}&\mbox{$41$}& \mbox{$81$}\\
\mbox{6} &\mbox{$2\cdot 499$}&\mbox{$5$}& \mbox{$249$} \\
\hline
\end{array}$$
Now, one may recognize an interesting interaction between cyclotomic srgs and skew Hadamard difference sets of Tables \[Tab1\] and \[Subsec2Tab1\]: for odd primes $p$ and $p_1$ such that $p$ is of index $2$ modulo $p_1$, the graph $\Cay(\F_{q},C_0^{(p_1,p^{(p_1-1)/2})})$ is strongly regular if and only if $D=\bigcup_{i\in \{0\}\cup \langle p\rangle\cup 2\langle p\rangle}C_i^{(2p_1,p^{(p_1-1)/2})} $ is a skew Hadamard difference set or a Paley type partial difference set in $\F_q$.
In this paper, we investigate such a relation between cyclotomic srgs and skew Hadamard difference sets, and find new constructions of infinite families of skew Hadamard difference sets from known cyclotomic srgs.
Background
==========
Let $p$ be a prime, $f$ a positive integer, and $q=p^f$. The canonical additive character $\psi$ of $\F_q$ is defined by $$\psi\colon\F_q\to \C^{\ast},\qquad\psi(x)=\zeta_p^{\Tr _{q/p}(x)},$$ where $\zeta_p={\rm exp}(\frac {2\pi i}{p})$ and $\Tr _{q/p}$ is the trace from $\F_q$ to $\F_p$. For a multiplicative character $\chi_k$ of order $k$ of $\F_q$, we define the [*Gauss sum*]{} $$G_f(\chi_k)=\sum_{x\in \F_q^\ast}\chi_k(x)\psi(x),$$ which belongs to the ring $\Z[\zeta_{kp}]$ of integers in the cyclotomic field $\Q(\zeta_{kp})$. Let $\sigma_{a,b}$ be the automorphism of $\Q(\zeta_{kp})$ determined by $$\sigma_{a,b}(\zeta_k)=\zeta_{k}^a, \qquad
\sigma_{a,b}(\zeta_p)=\zeta_{p}^b$$ for $\gcd{(a,k)}=\gcd{(b,p)}=1$. Below are several basic properties of Gauss sums [@BEW97]:
- $G_f(\chi_k)\overline{G_f(\chi_k)}=q$ if $\chi$ is nontrivial;
- $G_f(\chi_k^p)=G_f(\chi_k)$, where $p$ is the characteristic of $\F_q$;
- $G_f(\chi_k^{-1})=\chi_k(-1)\overline{G_f(\chi_k)}$;
- $G_f(\chi_k)=-1$ if $\chi_k$ is trivial;
- $\sigma_{a,b}(G_f(\chi_k))=\chi_k^{-a}(b)G_f(\chi_k^a)$.
In general, to explicitly evaluate Gauss sums is very difficult. There are only a few cases where the Gauss sums have been evaluated. The most well known case is [*quadratic*]{} case, in other words, the order of $\chi$ is two. In this case, as can found in [@BEW97 Theorem 11.5.4], it holds that $$\label{eq:quad}
G_f(\chi_k)=(-1)^{f-1}\left(\sqrt{(-1)^{\frac{p-1}{2}}p}\right)^f.$$ The next simple case is the so-called [*semi-primitive case*]{} (also referred to as [*uniform cyclotomy*]{} or [*pure Gauss sum*]{}), where there exists an integer $j$ such that $p^j\equiv -1\,(\mod{k})$, where $k$ is the order of the multiplicative character $\chi$ involved. The explicit evaluation of Gauss sums in this case is given in [@BEW97]. The next interesting case is the index $2$ case where the subgroup $\langle p\rangle$ generated by $p\in (\Z/{k}\Z)^\ast$ is of index $2$ in $(\Z/{k}\Z)^\ast$ and $-1\not\in \langle p\rangle $. In this case, it is known that $k$ can have at most two odd prime divisors. Many authors have investigated this case, see [@YX10] for the complete solution to the problem of evaluating index $2$ Gauss sums. Recently, these index $2$ Gauss sums were applied to show the existence of infinite families of new strongly regular graphs and skew Hadamard difference sets in $\F_q$ [@FX111; @FX113; @FMX11].
Now we recall the following well-known lemmas in the theories of difference sets and strongly regular graphs (see e.g., [@bh; @M94]).
\[Sec3Le1\] Let $(G, +)$ be an abelian group of odd order $v$, $D$ a subset of $G$ of size $\frac{v-1}{2}$. Assume that $D\cap -D=\emptyset$. Then, $D$ is a skew Hadamard difference set in $G$ if and only if $$\psi(D)=\frac{-1\pm \sqrt{-v}}{2}$$ for all nontrivial characters $\psi$ of $G$. On the other hand, assume that $0\not\in D$ and $-D=D$. Then $D$ is a Paley type partial difference set in $G$ if and only if $$\psi(D)=\frac{-1\pm \sqrt{v}}{2}$$ for all nontrivial characters $\psi$ of $G$.
\[Sec3Le2\] Let $(G, +)$ be an abelian group and $D$ a subset of $G$. Then, $\Cay(G,D)$ is a strongly regular graph if and only if the size of the set $$\{\psi(D)\,|\,\psi\in \widehat{G}\setminus \{\psi_0\}\}$$ is exactly two, where $\widehat{G}$ is the character group of $G$ and $\psi_0$ is the trivial character.
Let $q$ be a prime power and let $C_i^{(k,q)}=\gamma^i \langle \gamma^k\rangle$, $0\le i\le k-1$, be the cyclotomic classes of order $k$ of $\F_q$, where $\gamma$ is a fixed primitive root of $\F_q$. In this paper, we assume that $D$ is a union of cyclotomic classes of order $k$ of $\F_q$. In order to check whether a candidate subset $D=\bigcup_{i\in I}C_i^{(k,q)}$ is a skew Hadamard difference set or a Paley type partial difference set, we will compute the sums $\psi(aD)=\sum_{x\in D}\psi(ax)$ for all $a\in \F_q^\ast$, where $\psi$ is the canonical additive character of $\F_q$, because of Lemma \[Sec3Le1\]. Similarly, to check whether $D$ is a connection set of a strongly regular Cayley graph, we should compute the sums $\psi(aD)$ for all $a\in \F_q^\ast$ by Lemma \[Sec3Le2\]. Note that the sum $\psi(aD)$ can be expressed as a linear combination of Gauss sums (cf. [@Wu12 Lemma 3.1]) by using the orthogonality of characters: $$\begin{aligned}
\psi(aD)=\frac{1}{k}
\sum_{\chi\in C_0^{\perp}}G_f(\chi^{-1})
\sum_{i\in I}\chi(a\gamma^i ), \end{aligned}$$ where $C_0^{\perp}$ is the subgroup of $\widehat{\F_q^\ast}$ consisting of all $\chi$ which are trivial on $C_0^{(k,q)}$. Thus, the computation to know whether a candidate subset $D=\bigcup_{i\in I}C_i^{(k,q)}$ is a skew Hadamard difference set or a Paley type partial difference set is essentially reduced to evaluating Gauss sums. In fact, in [@FX113; @FMX11], known evaluation of index $2$ Gauss sums are used. However, as previously said, to explicitly evaluate Gauss sums is very difficult. In this paper, we will show the existence of skew Hadamard difference sets and Paley type partial difference sets without computing explicit values of Gauss sums. Instead, we use the following theorem, called the [*Davenport-Hasse product formula*]{}
\[thm:Stickel2\]([@BEW97]) Let $\eta$ be a multiplicative character of order $\ell>1$ of $\F_q=\F_{p^f}$. For every nontrivial character $\chi$ on $\F_q$, $$G_f(\chi)=\frac{G_f(\chi^\ell)}{\chi^\ell(\ell)}
\prod_{i=1}^{\ell-1}
\frac{G_f(\eta^i)}{G_f(\chi\eta^i)}.$$
Construction of skew Hadamard difference sets
=============================================
To show our main theorem, we use the following result of [@SW02]. (They gave this result in terms of irreducible cyclic codes.)
([@SW02 Lemma 2.8, Corollary 3.2])\[prop\] Let $m$ be the order of $p$ modulo $k$ and set $q=p^f=p^{sm}$. Assume that $k\,|\,\frac{p^f-1}{p-1}$ is odd and $\Cay(\F_{p^f},C_0^{(k,q)})$ is strongly regular. Then, for a system $L$ of coset representatives of $\F_{p^f}^\ast/C_0^{(k,p^f)}$, there exists a partition $L_1\cup L_2=L$ such that $$G_f(\chi_k)=\epsilon p^{s\theta}\sum_{x\in L_1}\chi_{k}(x)=
-\epsilon p^{s\theta}\sum_{x\in L_2}\chi_{k}(x),$$ where $\epsilon=\pm 1$ and $\theta$ is the integer such that $p^\theta|| G_{m}(\chi_k)$. (In this case, $p^{s\theta}||G_f(\chi_k)$ also holds.) Furthermore, if $|L_1|=k-d$ and $|L_2|=d$, then it holds that $$\begin{aligned}
\label{eq:Wu1}
k\cdot \psi(\gamma^a C_0^{(k,q)})+1&=&
\sum_{\chi\in C_0^{\perp\ast}}\chi(\gamma^a)G_f(\chi^{-1})\nonumber\\
&=&p^{s\theta}\epsilon d \mbox{\, \, or\, \, }
p^{s\theta}\epsilon (d-k). \end{aligned}$$
*Note that $L_1$ and $L_2$ are cyclic difference sets in $\F_{p^f}^\ast/C_0^{(k,p^f)}$ since $\chi_k(L_i)\overline{\chi_k(L_i)}=G_f(\chi_k)
\overline{G_f(\chi_k)}/p^{2s\theta}=p^{s(f-2\theta)}$. As determined in [@EHKX; @SW02], the corresponding cyclic $(v,k,\lambda)$ difference sets with $k\le (v-1)/2$ to cyclotomic strongly regular graphs of the Schmidt-White conjecture are as follows:*
1. (subfield case) the Singer $(\frac{p^f-1}{p^d-1},\frac{p^{f-d}-1}{p^d-1},\frac{p^{f-2d}-1}{p^d-1})$ difference set;
2. (semi-primitive case) the trivial $(v,1,0)$ difference set;
3. (exceptional case) see Table \[T6\].
$$\begin{array}{|c||c|c|c|c|}
\hline
\mbox{No.}&v&k&\lambda&\mbox{Name}\\
\hline
1&11&5&2&\mbox{Quadratic residue difference set~\cite[Theorem 1.12]{Beth}}\\
2&19&9&4&\mbox{Quadratic residue difference set}\\
3&35&17&8&\mbox{Twin-prime difference set~\cite[Theorem 8.2]{Beth}}\\
4&37&9&2&\mbox{Biquadratic residue difference set~\cite[Theorem 8.11]{Beth}}\\
5&43&21&10&\mbox{Hall's sextic difference set~\cite[Theorem 8.3]{Beth}}\\
6&67&33&16&\mbox{Quadratic residue difference set}\\
7&107&53&26&\mbox{Quadratic residue difference set}\\
8&133&33&8&\mbox{Quadratic residue difference set}\\
9&163&81&40&\mbox{Hall's sporadic difference set~\cite[Remarks 8.21(b)]{Beth}}\\
10&323&161&80&\mbox{Twin-prime difference set}\\
11&499&249&124&\mbox{Quadratic residue difference set}\\
\hline
\end{array}$$
The following is our main theorem of this paper.
\[thm:difmain1\] We assume that $ L_i\cap C_0^{(k,q)}=\emptyset$. Let $L'=\{y\,(\mod{k})\,|\,\gamma^{-y}\in L_i;0\le y\le q-2\}$ and let $I$ be the $|L_i|$-element set of odd integers modulo $2k$ such that $I\,(\mod{k})=L'$. Set $$J=\{0\}\cup I \cup
2\left((\Z/k\Z)\setminus 2^{-1}\cdot (L'\cup \{0\})\right) \, (\mod{2k}).$$ Then, $D=\bigcup_{j\in J}C_j^{(2k,q)}$ in $\F_q$ is a skew Hadamard difference set or a Paley type partial difference set according to $q\equiv 3\,(\mod{4})$ or $\equiv 1\,(\mod{4})$, i.e., it holds that $$\psi(D)=\frac{-1\pm \sqrt{\pm q}}{2}.$$
First of all, we observe the following facts:
(1)
: It is clear that $J\,(\mod{k})=\{0,1,\ldots,k-1\}$. In particular, if $q\equiv 3\,(\mod{4})$, i.e., $-1\in C_{k}^{(2k,q)}$, it follows that $\F_q=\{0\}\cup D\cup -D$.
(2)
: By the Davenport-Hasse product formula, it holds that $$G_f(\chi_{2k})=\frac{G_f(\chi_{k})G_f(\chi_2)}{\chi_{k}(2)G_f(\chi_{k}^{2^{-1}})}.$$ Then, by noting that $G_f(\chi_{k}^{2^{-1}})G_f(\chi_{k}^{-2^{-1}})=\chi_{k}^{2^{-1}}(-1)q$ and the restriction of $\chi_{k}$ to $\F_p$ is trivial, it follows that $$\label{eq:Gauss2p1}
G_f(\chi_{2k})=\frac{1}{q}G_f(\chi_2)G_f(\chi_{k})G_f(\chi_{k}^{-2^{-1}}).$$
(3)
: The sum $\sum_{y\in J}\chi_{2k}^{x}(\gamma^y)$ for any $x$ such that $2,k\not |x$ is computable by using Proposition \[prop\] as follows: $$\begin{aligned}
\sum_{y\in J}\chi_{2k}^{x}(\gamma^y)&=&
\sum_{y\in J}(-1)^y\chi_{k}^{x 2^{-1}}(\gamma^y)\nonumber\\
&=&1-\sum_{y\in L'}\chi_{k}^{x 2^{-1}}(\gamma^y)
+
\sum_{y\in (\Z/k\Z)\setminus (L'\cup \{0\})}\chi_{k}^{x 2^{-1}}(\gamma^y)\nonumber \\
&=&-2\sum_{y\in L'}\chi_{k}^{x2^{-1}}(\gamma^y)\nonumber\\
&=&-2\sum_{\omega\in L_i}\chi_{k}^{-x2^{-1}}(\omega)\nonumber\\
&=&(-1)^{i}2\epsilon G(\chi_k^{-x2^{-1}})/p^{s\theta}\label{eq:partialsum2}. \end{aligned}$$
Now, we compute the sum $$T_a=\sum_{2,k\not |x}G_f(\chi_{2k}^{-x})
\sum_{y\in J}\chi_{2k}^{x}(\gamma^{a+y}).$$ By (\[eq:Gauss2p1\]) and (\[eq:partialsum2\]), we have $$\begin{aligned}
T_a
&=&(-1)^{a+i}\epsilon\frac{2}{p^{s\theta}}\sum_{2,k\not |x}G_f(\chi_{2k}^{-x})G(\chi_k^{-x2^{-1}})\chi_{k}^{x2^{-1}}(\gamma^{a})\\
&=&(-1)^{a+i}\epsilon 2\frac{G_f(\chi_2)}{qp^{s\theta}}\sum_{x=1}^{k-1}
G_f(\chi_{k}^{-x})G_f(\chi_{k}^{x2^{-1}})
G(\chi_k^{-x2^{-1}})
\chi_{k}^{x2^{-1}}(\gamma^{a})\nonumber\\
&=&(-1)^{a+i} \epsilon 2\frac{G_f(\chi_2)}{p^{s\theta}}\sum_{x=1}^{k-1}
G_f(\chi_{k}^{-x})
\chi_{k}^{x}(\gamma^{2^{-1}a})\\
&=&(-1)^{a+i} \epsilon 2\frac{G_f(\chi_2)}{p^{s\theta}}
(k\cdot \psi(\gamma^{2^{-1}a} C_0^{(k,q)})+1),\end{aligned}$$ where we used $G_f(\chi_{k}^{x2^{-1}})G_f(\chi_{k}^{-x2^{-1}})
=\chi_{k}^{x2^{-1}}(-1)q$. Then, by (\[eq:Wu1\]), we obtain $$\begin{aligned}
k(2\cdot \psi(aD)+1)&=&
\sum_{\ell=1}^{2k-1}G_f(\chi_{2k}^{-\ell})\sum_{y\in J}
\chi_{2k}^\ell(\gamma^{a+y})\\
&=&G_f(\chi_2)\sum_{y\in J}
\chi_{2}(\gamma^{a+y})+T_a\\
&=&(-1)^aG_f(\chi_2)\left(k-2|L_i|\right)\\
& &\hspace{1cm}+\epsilon 2(-1)^{a+i}\frac{G_f(\chi_2)}{p^{s\theta}}(k\cdot \psi(\gamma^{2^{-1}a} C_0^{(k,q)})+1)\\
&=&\pm (-1)^a k G_f(\chi_2). \end{aligned}$$ By (\[eq:quad\]), we obtain $$\psi(aD)=\frac{-1\pm \sqrt{\delta q}}{2},$$ where $\delta=1$ or $-1$ according to $q\equiv 3\,(\mod{4})$ or $\equiv 1\,(\mod{4})$. This completes the proof.
Applying our theorem to subfield examples of cyclotomic strongly regular graphs, we obtain skew Hadamard difference sets in $\F_q$ for any $q=p^{st}$ with $st\ge 3$ by a nontrivial and different cyclotomic construction from that of the Paley difference sets although we do not know the constructed difference sets are inequivalent or not.
Also, we may obtain an infinite family of skew Hadamard difference sets starting from each skew Hadamard difference set of Theorem \[thm:difmain1\] by applying the following theorem.
([@M12])\[cor:sHd1\] Let $h=2p_1$ with an odd prime $p_1$ and let $p$ be a prime such that $\langle p\rangle$ is of index $e$ modulo $h$. Furthermore, let $k=2p_1^{m}$ and assume that $\langle p\rangle$ is again of index $e$ modulo $k$. Put $q=p^{(p_1-1)/e}$ and $q'=p^{p_1^{m-1}(p_1-1)/e}$. Define $J$ as any subset of $\{0,1,\ldots,h-1\}$ such that $J\,(\mod{p_1})=\{0,1,\ldots,p_1-1\}$. Let $$D=\bigcup_{i\in J}C_{i}^{(h,q)} \mbox{\, and \, }
D'=\bigcup_{i_1=0}^{p_1^{m-1}}\bigcup_{i\in J}C_{2i_1+ik/h}^{(k,q')}.$$ If $D$ is a skew Hadamard difference set or a Paley type partial difference set in $\F_{q}$, then so does $D'$ in $\F_{q'}$.
By combining Theorems \[thm:difmain1\] and \[cor:sHd1\], we immediately have the following corollary, which yields an infinite family of skew Hadamard difference sets from a cyclotomic strongly regular graph.
\[maincor\] Let $k=p_1^m$ and let $p$ be of index $e$ both of modulo $p_1$ and $k$. Put $q=p^{(p_1-1)/e}$, $q'=p^{p_1^{m-1}(p_1-1)/e}$, and $$D=\bigcup_{i=0}^{p_1^{m-1}-1}\bigcup_{j\in J}C_{2i+p_1^{m-1}j}^{(2k,q)},$$ where $J$ is defined as in Theorem \[thm:difmain1\]. If $\Cay(\F_q,C_0^{(p_1,q)})$ is strongly regular, then $D$ in $\F_{q'}$ is a skew Hadamard difference set or a Paley type partial difference set according to $q\equiv 3\,(\mod{4})$ or $\equiv 1\,(\mod{4})$.
\[exam\]
*By Corollary \[maincor\], we obtain new constructions of infinite families of skew Hadamard difference sets and Paley type partial difference sets for the quadruples $(p_1,p,f,e)$ of No. 2, 4, 5, 6, 7, 9, and 11 in Table \[Tab1\]. Note that we can not obtain an infinite family of skew Hadamard difference sets from the cyclotomic srg of No. 1 because $p$ is not of index $2$ in $\Z/2p_1^m\Z$ for $m\ge 2$ while $\bigcup_{j\in \{0\}\cup \langle p\rangle \cup 2\langle p\rangle}C_j^{(2p_1,p^f)}$ forms a skew Hadamard difference set.*
Also, there are a lot of subfield examples satisfying $[(\Z/p_1\Z)^\ast:\langle p\rangle]=e$ and $p_1=\frac{p^{(p_1-1)/e}-1}{p^t-1}$ for some $t\,|\,(p_1-1)/e$. We list ten examples satisfying these conditions in Table \[Tab4\].
$$\begin{array}{|c|c|c|c||c|c|c|c|}
\hline
p_1&p&f&e&p_1&p&f&e\\
\hline
13&3&3&4&1723&41&3&574\\
31&5&3&10&2801&7&5&560\\
307&17&3&102&3541&59&3&1180\\
757&3&9&84&5113&71&3&1704\\
1093&3&7&156&8011&89&3&2670\\
\hline
\end{array}$$
From these examples, we obtain infinite families of skew Hadamard difference sets and Paley type partial difference sets by Corollary \[maincor\].
Concluding remarks and open problems
====================================
In this section, we give important remarks and open problems related to our results.
*In [@M12], the author found two examples of skew Hadamard difference sets of index $4$, those are, $\bigcup_{j\in \{p_1\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^f)}$ for $(p_1,p,f)=(13,3,3)$ and $\bigcup_{j\in \{0\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^f)}$ for $(p_1,p,f)=(29,7,7)$, where $Q$ is the subgroup of index $2$ of $(\Z/2p_1\Z)^\ast$. These two examples are not covered by Theorem \[thm:difmain1\], i.e., there do not exist corresponding cyclotomic strongly regular graphs and cyclic difference sets. More generally, via a computation similar to [@GXY11] involving known evaluations of index $4$ Gauss sums, one can prove that either of $\bigcup_{j\in \{0\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^{(p_1-1)/4})}$ or $\bigcup_{j\in \{p_1\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^{(p_1-1)/4})}$ is a skew Hadamard difference set or a Paley type partial difference set in $\F_{p^{(p_1-1)/4}}$ if the following conditions are fulfilled:*
- $p$ is of index $4$ modulo $p_1$,
- $p_1=4p^{(p_1-1)/4-2b}+1$, where $b$ is defined as $$b=\min\left\{\frac{1}{p_1}\sum_{x\in S}x\,|\,S\in (\Z/p_1\Z)^\ast/\langle p\rangle \right\},$$
- $p_1=A^2+4$ for some integer $A\equiv 3\,(\mod{4})$.
The author found only three examples satisfying these conditions, which are $$(p_1,p,f)=(13,3,3),(29,7,7),(53,13,13).$$ For each of these three examples, we obtain an infinite family of skew Hadamard difference sets or Paley type partial difference sets by applying Theorem \[cor:sHd1\]. Here, we have the following natural question.
Determine for which $(p,p_1,e)$ either $\bigcup_{j\in \{0\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^{(p_1-1)/e})}$ or $\bigcup_{j\in \{p_1\}\cup Q \cup 2Q}C_{j}^{(2p_1,p^{(p_1-1)/e})}$ forms a skew Hadamard difference set or a Paley type partial difference set.
Also, by computer, the author found an interesting example of a skew Hadamard difference set in the case where $(p,f,p_1)=(7,3,19)$ and $e=6$: $$D=\bigcup_{x\in I}C_{i}^{(2p_1,p^f)},$$ where $$I=\{p_1\}\cup \langle p\rangle\cup 3\langle p\rangle\cup
3^3\langle p\rangle\cup 2\cdot 3\langle p\rangle\cup 2\cdot 3^3\langle p\rangle\cup 2\cdot 3^4\langle p\rangle\, (\mod{2p_1}).$$ One can use a computer to find that the automorphism group of the symmetric design $Dev(D)$ derived from $D$ has size $3^4\cdot 7^3$. (We will write the size as $\#Aut(Dev(D))$.) On the other hand, $\#Aut(Dev(P))=3^3\cdot 7^3\cdot 19$ for the Paley difference set $P$ with the same parameter. Thus, the skew Hadamard difference set $D$ is inequivalent to the Paley difference set. Furthermore, since the size of the Sylow $p$-subgroup of the automorphism group of the design derived from a difference set constructed by Muzychuk [@M] is strictly greater than $q$, we conclude that $D$ is also inequivalent to the corresponding skew Hadamard difference sets of [@M]. Also, since the set $I$ satisfies $I\,(\mod{p_1})=\{0,1,\ldots,p_1-1\}$, we obtain an infinite family of skew Hadamard difference sets including this example by Theorem \[cor:sHd1\].
*As described in Introduction, to check whether obtained skew Hadamard difference sets and Paley type partial difference sets are equivalent or not to the classical Paley (partial) difference sets is very important. Although the problem is in general difficult and the author could not prove that our construction always yields inequivalent skew Hadamard difference sets and Paley type partial difference sets to the Paley (partial) difference sets, the author still believes that our infinite families include inequivalent ones abundantly. As an evidence for my believe, we can see by computer that the skew Hadamard difference set $D=\bigcup_{x\in J}C_{i}^{(2p_1,p^f)}$ with $$J=\{0\}\cup \left(\bigcup_{i\in I}g^{i}\langle p\rangle\right)\cup
\left(2\bigcup_{i\in (\Z/e\Z)\setminus I}g^{i-s}\langle p\rangle\right)\hspace{0.5cm}(\mod{2p_1}),$$ where assume that $(\Z/k\Z)^\ast/\langle p\rangle $ is a cyclic group of order $e$ and let $g$ be a representative of a generator of $(\Z/k\Z)^\ast/\langle p\rangle $, is inequivalent to the Paley difference set in the following cases:*
- $(p,f,p_1)=(3,5,11)$, $(g,s)=(-1,1)$ and $I=\{0\}$: In this case, $\# Aut(Dev(D))=3^5\cdot 5\cdot 11$ and $\# Aut(Dev(P))=3^5\cdot 5\cdot 11^2$ for the corresponding Paley difference set $P$.
- $(p,f,p_1)=(3,7,1093)$, $(g,s)=(5,63)$ and take $I$ as $\bigcup_{i\in I}g^i \langle p\rangle =5\cdot (S+948))$ for the Singer difference set $S$ of PG$(6,3)$: In this case, $\# Aut(Dev(D))=3^7\cdot 7$ and $\# Aut(Dev(P))=3^7\cdot 7\cdot 1093$ for the corresponding Paley difference set $P$.
- $(p,f,p_1)=(7,5,2801)$, $(g,s)=(3,58)$ and take $I$ as $\bigcup_{i\in I}g^i \langle p\rangle =3^{58}\cdot (S+292))$ for the Singer difference set $S$ of PG$(4,7)$: In this case, $\# Aut(Dev(D))=3\cdot 5\cdot 7^5$ and $\# Aut(Dev(P))=3\cdot 5\cdot 7^5\cdot 2801$ for the corresponding Paley difference set $P$.
Furthermore, the author checked by computer that the Paley type srgs with parameters $(p_1,p,f)=(31,5,3)$ and $(307,17,3)$ of Example \[exam\] are not isomorphic (as graph isomorphism) to the classical Paley graphs. (Note that in these cases there is no factor $m>2$ of $p^f-1$ such that $p$ is semi-primitive modulo $m$.)
Determine whether or not skew Hadamard difference sets and Paley type partial difference sets obtained in this paper are equivalent to the classical Paley (partial) difference sets.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank Tao Feng, Zhejian University, for his helpful comments on computations of the automorphism groups of symmetric designs derived from our skew Hadamard difference sets.
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|
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---
abstract: 'We provably compute rational points on 1403 Picard curves defined over ${\mathbb{Q}}$ with Jacobians of Mordell–Weil rank $1$ using the Chabauty–Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes Coleman integrals on curves to points defined over number fields and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel-Jacobi map. We discuss several interesting examples of points defined over number fields that occur in the Chabauty–Coleman set.'
address:
- 'Sachi Hashimoto, Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA'
- 'Travis Morrison, Institute for Quantum Computing, The University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada'
author:
- Sachi Hashimoto and Travis Morrison
title: 'Chabauty–Coleman computations on rank 1 Picard curves'
---
Introduction
============
Let $X/{\mathbb{Q}}$ be a smooth projective curve of genus at least $2$. Faltings’s theorem [@Faltings], proved in 1983, establishes that the set of rational points $X({\mathbb{Q}})$ of $X$ is finite, but falls short of giving an explicit method for computing this set. Our goal in this paper is to compute this set explicitly for certain curves of genus $3$.
We focus on curves amenable to Chabauty–Coleman calculations, which use $p$-adic (Coleman) integrals to carve out a finite set of $p$-adic points containing the rational points. The Chabauty–Coleman method is one attempt at an effective Faltings’s theorem. The method dates back to work of Chabauty in 1941 [@Chabauty], who showed finiteness of $X({\mathbb{Q}})$, for certain curves $X$, using its embedding into the $p$-adic closure of $J({\mathbb{Q}})$, where $J$ is the Jacobian variety of $X$. In the 1980s, Coleman [@EffectiveChab; @Torsion] made Chabauty’s approach to Diophantine problems effective with the development of the Coleman integral. With this computational tool, he was able to bound the number of points on curves.
The Chabauty–Coleman method requires that, for a given curve $X$, the Mordell-Weil rank of its Jacobian be computable and smaller than the genus of $X$. While there exist techniques to compute the rank of smooth plane quartics [@descent], practical implementations of rank-bounding techniques only exist for superelliptic curves, based on work of Poonen and Schaefer [@PoonenSchaefer]. Therefore we focus on Picard curves whose Jacobians have Mordell-Weil rank $1$. A is a smooth plane quartic $X/{\mathbb{Q}}$ that admits a Galois cover of degree $3$ of ${\mathbb{P}}^1$. Every Picard curve has an affine equation of the form $y^3 = f(x)$ where $f(x) \in {\mathbb{Q}}[x]$ is degree $4$ and without multiple roots in $\overline{{\mathbb{Q}}}$. While for some particular genus $3$ curves $X$ of rank $2$ one can determine the rational points using the Chabauty–Coleman method, when the rank is one less than genus we expect the method to fail to determine $X({\mathbb{Q}})$ for most $X$, and so we focus on the rank $1$ case, where we can work with at least two linearly independent functionals on the $p$-adic closure of $J({\mathbb{Q}})$.
Recently, Balakrishnan and Tuitman [@balatuitman] have made Chabauty–Coleman computations explicit and practical for machine computation of rational point sets on general curves satisfying the rank hypothesis, opening up the ability to compute rational points on entire databases of curves.
Sutherland has computed a database of genus 3 curves of bounded discriminant [@sutherlanddatabase; @g3database], providing the necessary data for large-scale computational experiments in genus 3. Balakrishnan–Bianchi–Cantoral-Farfán–[Ç]{}iperiani–Etropolski [@CCExp] computed rational points on a database of genus 3 rank 1 hyperelliptic curves using Chabauty–Coleman methods and de Frutos-Fernández–Hashimoto [@MariaSachi] computed rational points on genus 3 rank 0 hyperelliptic curves. This work is the first instance of such computations being carried out on non-hyperelliptic curves.
The Chabauty–Coleman method cuts out a set of $p$-adic points $X({\mathbb{Q}}_p)_1$ on a curve $X$ that contains the rational points $X({\mathbb{Q}})$; sometimes this set is strictly larger than the set of rational points. We classify the set of extra points $X({\mathbb{Q}}_p)_1 \setminus X({\mathbb{Q}})$ in the Chabauty–Coleman set, showing that they fall into one of three categories, and are defined over a number field $K$:
- $P \in X(K)$ such that $[P-\infty]$ is in $J(K)_{tors}$;
- $P \in X(K)$ which possess a relation with rational points in the Jacobian $J({\mathbb{Q}})$;
- $P \in X(K)$ that appear due to the existence of extra automorphisms of $X_{\overline{{\mathbb{Q}}}}$.
We exhibit examples of all three possibilities in Section 4.
The Chabauty–Coleman method
===========================
Chabauty–Coleman
----------------
In this section we introduce the Coleman integral as a $p$-adic tool for computing the rational points on a curve. We follow the exposition of the Coleman integral in [@Wetherell] and [@McCallumPoonen]. Let $X$ be a smooth projective curve of genus $g>1$ over ${\mathbb{Q}}$. Fix $b \in X({\mathbb{Q}})$ and let $$\begin{aligned}
\iota\colon X &\hookrightarrow J \\
Q &\mapsto [Q - b]\end{aligned}$$ be an embedding of $X$ into its Jacobian $J$. Let $p$ be a prime of good reduction for $X$. There are two ways to consider the $p$-adic Coleman integral, one as an integral on $J_{{\mathbb{Q}}_p}$ thought of as a $p$-adic Lie group, and the other as defined by Coleman directly on $X$ as a rigid analytic space. We first describe the $p$-adic Lie group approach and do not address the equivalence here; Coleman discusses the details in [@Torsion].
Let $\omega \in H^0(J_{{\mathbb{Q}}_p}, \Omega^1)$. Since $J({\mathbb{Q}}_p)$ is a $p$-adic Lie group, $\omega$ is translation-invariant, and we can define an anti-derivative homomorphism $\int: J({\mathbb{Q}}_p)\to {\mathbb{Q}}_p$ which sends $\omega$ to its local anti-derivative. This homomorphism sends a point $Q \in J({\mathbb{Q}}_p)$ to $\int_0^Q \omega$. For any open $U \subset J({\mathbb{Q}}_p)$, the anti-derivative of $\omega$ can be computed by expanding $\omega$ in local coordinates at $Q \in U$ and evaluating formally as a power series; this converges on small enough open sets.
The anti-derivative gives a pairing $$H^0(J_{{\mathbb{Q}}_p}, \Omega^1) \times J({\mathbb{Q}}_p) \to {\mathbb{Q}}_p$$ sending $(\omega, Q)$ to $\int_0^Q\omega$.
The restriction map $$\iota^* \colon H^0(J_{{\mathbb{Q}}_p}, \Omega^1) \to H^0(X_{{\mathbb{Q}}_p}, \Omega^1)$$ induced by $\iota$ is an isomorphism, independent of the choice of $b$ [@MilneJacobian]\*[Proposition 2.2]{}. This gives us a way to think of the integral on the curve $X_{{\mathbb{Q}}_p}$: for any $\omega \in H^0(X_{{\mathbb{Q}}_p}, \Omega^1)$ and any $P, Q\in X({\mathbb{Q}}_p)$, we can define $\int_P^{Q}\omega$ by $\int_0^{[Q-P]} (\iota^*)^{-1}\omega$. Conversely, we can write an integral on $J_{{\mathbb{Q}}_p}$ in terms of integrals on $X_{{\mathbb{Q}}_p}$: for any $D \in J({\mathbb{Q}})$ of degree zero and $\omega \in H^0(J_{{\mathbb{Q}}_p}, \Omega^1)$, we can write $D = \sum_i [Q_i- P_i]$ where $P_i \in J(L)$ for $L / {\mathbb{Q}}$ a finite extension. Then $$\int_0^D \omega = \sum_i \int_{P_i}^{Q_i} \iota^*\omega.$$ Note $\int_0^D \omega =0$ when $D\in J({\mathbb{Q}}_p)_{tors}$ because the anti-derivative is a homomorphism.
Coleman gives a definition of the integral $\int_P^Q \eta$ on the rigid analytic space $X_{{\mathbb{Q}}_p}$; he defines an integral for $P, Q \in X(\overline{{\mathbb{Q}}_p})$ and $\eta$ any differential of the second kind [@Torsion]. Coleman shows the two definitions agree when both are applicable. For all points $P, Q, R \in X(\overline{{\mathbb{Q}}_p})$ and one-forms $\omega, \eta$ of the second kind on $X_{{\mathbb{Q}}_p}$, the Coleman integral enjoys the following properties:
(Coleman) \[thm:properties\]
1. Linearity: $\int_P^{Q}(\alpha\omega +\beta\eta)=\alpha\int_P^{Q}\omega + \beta\int_P^{Q}\eta$, for all $\alpha, \beta \in \overline{{\mathbb{Q}}_p}$.
2. Additivity: $\int_{P}^Q \omega = \int_P^{R} \omega + \int_{R}^Q$.
3. Fundamental theorem of calculus: $\int_P^Q {\ensuremath{\operatorname{d}\!{f}}} = f(Q) - f(P)$.
4. Change of variables: for any rigid analytic map $\phi\colon X\to X^\prime$, we have $\int_{\phi(P)}^{\phi(Q)}\omega = \int_P^{Q}\phi^*(\omega)$.
5. For any divisor $D \coloneqq \sum_i [Q_i- P_i] \in J_{{\mathbb{Q}}_p}$ of degree zero, then $\int_D \omega \colonequals \sum_i \int_{P_i}^{Q_i} \omega $ is well-defined, and $\int_D \omega = 0$ when $D$ is principal. \[item:sumD\]
When the rank $r$ of $J({\mathbb{Q}})$ is less than the genus $g$ of $X$, the space of $${\text{Van}}(X({\mathbb{Q}}))\coloneqq \left \{\omega \in H^0(X_{{\mathbb{Q}}_p}, \Omega^1)\colon \int_D \omega = 0 \text{ for all } D \in J({\mathbb{Q}})\right\}$$ is at least $(g -r)$-dimensional, and thus nonzero. In particular, when $r = 1$, the space of vanishing differentials is $(g-1)$-dimensional.
We will think of the $p$-adic integral of $\omega$ in ${\text{Van}}(X({\mathbb{Q}}))$ as the Coleman integral $\int_D \omega \colonequals \sum_i \int_{P_i}^{Q_i} \omega $ taking place directly on the curve $X_{{\mathbb{Q}}_p}$ as a rigid analytic space (as in Theorem \[thm:properties\] part \[item:sumD\]). While we could consider both the $p$-adic Lie group integral $\int_0^D \omega$ on the Jacobian and the Coleman integral when $\omega \in H^0(X_{{\mathbb{Q}}_p}, \Omega^1)$, in the process of explicitly computing the Coleman integral, we compute integrals of differentials of the second kind that are not everywhere regular. Therefore throughout the paper we consider all $p$-adic integrals to take place on the curve as a rigid analytic space.
Coleman [@EffectiveChab] proves a bound on the number of rational points $\# X({\mathbb{Q}}) \leq \#X({\mathbb{F}}_p) + 2g-2$ by bounding the number of zeros of a nonzero differential in ${\text{Van}}(X({\mathbb{Q}}))$, assuming $p > 2g$.
The proof suggests an algorithm for computing $X({\mathbb{Q}})$ by computing the set $$X({\mathbb{Q}}_p)_1 \coloneqq \left \{Q \in X({\mathbb{Q}}_p): \int_b^Q \omega = 0\text{ for all } \omega \in {\text{Van}}(X({\mathbb{Q}}))\right \}.$$ Since the Coleman integral is locally analytic, it can be expressed as the formal integral of a $p$-adic power series on each residue disk. Computing $X({\mathbb{Q}}_p)_1$ yields finitely many solutions in each disk, and so finitely many in total. Therefore $X({\mathbb{Q}})\subseteq X({\mathbb{Q}}_p)_1$ is finite.
Explicit Coleman integration after Balakrishnan and Tuitman
-----------------------------------------------------------
To carry out the Chabauty–Coleman method in practice, we need an explicit method for computing the set $X({\mathbb{Q}}_p)_1$, which is defined as the vanishing locus of some $p$-adic integrals. In this section, we give an overview of a practical algorithm to compute $p$-adic integrals on curves, following the method of Balakrishnan and Tuitman [@balatuitman]. For a more detailed description of the algorithm, which applies in more generality than described here, we refer the reader to their paper, as well as the zeta function algorithm of Tuitman described in [@Tuitman1; @Tuitman2] which develops the reduction in cohomology.
Let $X/{\mathbb{Q}}$ be a smooth projective curve with affine plane model given by the equation $Q(x,y)=0$ and let $p$ a prime of good reduction. Tuitman’s algorithm exploits the existence of a low-degree map $x:X \to {\mathbb{P}}^1$ of degree $d_x$. After removing the ramification locus of this map, the algorithm chooses a lift of Frobenius which sends $x \mapsto x^p$ and Hensel lifts $y$. Then, we compute the action of Frobenius on a certain basis of differentials for $H^1_{rig}(X)$, and reduce using Lauder’s fibration method which is given by explicit and fast linear algebra.
To specify a point $P$ on $X$, Balakrishnan and Tuitman’s algorithm uses the following data:
- $\tt{ P `x}$ the $x$-coordinate of $P$ (or $1/x$ if $x$ is infinite).
- $\tt{ P` b}$ the values of $[b^0_0, \dots, b^0_{d_x-1}]$ that form an integral basis for the function field ${\mathbb{Q}}(X)/ {\mathbb{Q}}[x]$ at $x$, ($[b^\infty_0, \dots, b^\infty_{d_x-1}]$ an integral basis for ${\mathbb{Q}}(X)/ {\mathbb{Q}}[1/x]$ if $P$ is infinite).
- $\tt{ P`inf }$ a boolean value specifying whether $P$ is infinite.
In the case where $X$ is a Picard curve $y^3=f(x)$, the map $x:X\to {\mathbb{P}}^1$ is projection to the $x$-coordinate, $b^0 =[1,y,y^2] $, $b^\infty=[y, y/x^2, y/x^3]$, and we have a 6-dimensional basis of differentials for $H^1_{rig}(X)$, where the regular differentials are $\omega_1 ={\ensuremath{\operatorname{d}\!{x}}}/y^2$, $\omega_2 = x {\ensuremath{\operatorname{d}\!{x}}}/y^2$, and $\omega_3 = {\ensuremath{\operatorname{d}\!{x}}}/ y$.
The $p$-adic points $X({\mathbb{Q}}_p)$ are a disjoint union of , which are the preimages of points in $X({\mathbb{F}}_p)$. After Balakrishnan and Tuitman, we classify the residue disks based on their ramification with respect to the map $x: X\to {\mathbb{P}}^1.$ Let $\Delta(x)$ be the discriminant of $Q$ thought of as a polynomial in $y$, and $r(x)$ be the squarefree polynomial with the same zeros as $\Delta$. Then our residue disks partition as follows:
- are residue disks that contain a point whose $x$-coordinate is a zero of $r(x)$ or is infinite. Such points are called . If the very bad point of a disk has infinite $x$-coordinate, the disk is also an , and the point is also a . Otherwise the disk is a .
- comprise the remaining residue disks. All good residue disks are finite.
Note that each residue disk contains at most one very bad point $P=(x,y)$ which will be defined over an unramified extension of ${\mathbb{Q}}_p$. We denote by $e_P$ the ramification index of the map by $x$ at a very bad point $P$. When $Q(x,y) = y^3 - f(x)$ is a Picard curve, then $r(x) = f(x)$, the very bad points are the ramification points (where $y=0$), the ramification point residue disks are the bad disks, and these very bad points have ramification index $3$.
There are two kinds of Coleman integrals on $X$: are integrals between two points $P$ and $Q$ in $X ({\mathbb{Q}}_p)$ in the same residue disk. These are computed using uniformizing parameters. A uniformizing parameter gives an isomorphism from a residue disk to $p {\mathbb{Z}}_p$ and transforms a Coleman integral into a formal integral. For the second kind of Coleman integral, between points in different residue disks, we turn to the full machinery of Tuitman’s zeta function algorithm which computes the action of Frobenius on cohomology.
\[tinyintegrals\]
Let $P, Q \in X({\mathbb{Q}}_p)$ and let $\omega$ be a differential of the second kind on $X$.
1. If $P, Q$ are in a bad disk, find $P{^\prime}$, the very bad point in the disk, and break up the integral as $\int_P^{P{^\prime}} \omega + \int_{P{^\prime}}^Q \omega$. Otherwise proceed with $P = P{^\prime}$.
2. Compute a uniformizing parameter $t$ at $P{^\prime}$: in a good disk, $t$ is given by $x- x(P{^\prime})$ ; at a very bad point $t=b_i^0-b_i^0(P)$ for some $i$ (replace with $b_i^\infty$ for very infinite $P{^\prime}$).
3. Compute $x$ and the $b^0$ vector (or $b^\infty$) as a function of $t$ to desired precision, and using these expansions compute $\omega$ as a power series in $t$.
For more details on the computations of the tiny integral and the uniformizing parameters see [@balatuitman]\*[Algorithm 3.4, Proposition 3.2]{}.
For $P$ and $Q$ in $X({\mathbb{Q}}_p)$ good points in different residue disks, we use a lift of the Frobenius morphism $\phi$. Fix a basis differential $\omega_i$ of $H^1_{rig}(X)$ for some $i = 1, \dots, 2g$. By applying Frobenius to $P$ and $Q$, we obtain$$\int_{P}^Q \omega_i = \int_P ^{\phi(P)} \omega_i + \int_{\phi(P)}^{\phi(Q)} \omega_i + \int _{\phi(Q)}^{Q} \omega_i =\int_P ^{\phi(P)} \omega_i + \int_{P}^{Q} \phi^*\omega_i + \int _{\phi(Q)}^{Q} \omega_i$$ where $P$ and $ \phi(P)$ are in the same residue disk, so define a tiny integral, and similarly for $Q$ and $\phi(Q)$.
So, we expressed the unknown integral as a sum of two computable quantities and the integral of $\phi^* \omega_i$. This motivates computing $\phi^*\omega_i$; Balakrishnan and Tuitman compute the reduction in cohomology of $\phi^* \omega_i$ as a sum of basis elements $\sum_j M_{ij} \omega_j$ plus an exact differential ${\ensuremath{\operatorname{d}\!{f}}}_i$. We compute the reductions of all $\phi^* \omega_i$ and consider the linear system of equations to compute the Coleman integral. For more discussion see [@balatuitman]\*[Algorithm 3.6, Algorithm 3.9]{} as well as [@Tuitman2]\*[Section 4.1]{} which discusses how to construct the $2g$ basis differentials of $H^1_{rig}(X)$. If $P$ is a bad point, some $f_i$ might not converge at $P$, but near the boundary of the disk of $P$ they do; we compute the integral $\int_P^Q \omega$ by taking $S \in X({\mathbb{Q}}_p(p^{1/e}))$ on the boundary for $e$ large enough, and breaking up the integral as $\int_P^S \omega + \int_S^Q\omega$.
In practice, to compute $X({\mathbb{Q}}_p)_1$, we integrate to arbitrary endpoints $\int_b^z \omega$; to do this we iterate over residue disks, breaking up the integral by fixing $P$ in each residue disk and integrating $\int_b^P \omega + \int_P^z \omega$ where $\int_b^P \omega $ sets the constant of integration and $\int_P^z \omega$ is a tiny integral which can be expanded as a $p$-adic power series to arbitrary precision and solved using analytic techniques. Balakrishnan and Tuitman implement the Coleman integral as well some of the process of computing the Chabauty–Coleman set $X({\mathbb{Q}}_p)_1$ in Magma [@BalaTuitgit] under the assumption that we can find rational points $P_1, P_2, \dots, P_r \in X({\mathbb{Q}})$ such that the classes $[P_i - b]$ are infinite order and linearly independent in $J({\mathbb{Q}})$, where $r$ is the rank of $J({\mathbb{Q}})$.
The algorithm
=============
In this section, we describe the algorithm we implemented to compute $X({\mathbb{Q}})_1$ for each $X$ a rank 1 Picard curve in our database. These curves come from Sutherland’s database of genus 3 curves of small discriminant [@sutherlanddatabase] and have discriminant bounded by $10^{12}$. Each curve in the database is given by the data of its affine model $y^3 = f(x)$ with $f(x)$ degree four, along with its discriminant $\Delta$.
We began by filtering the database for those curves whose Jacobians have Mordell-Weil rank 1. We ran the Magma function $\tt RankBounds(f,3:ReturnGenerators:=true)$ which computes the rank based on work of Poonen and Schaefer [@PoonenSchaefer] and Creutz [@Creutz] and is implemented in Magma by Creutz. It gives a lower and upper bound for the rank of the Jacobian of the cyclic cover of ${\mathbb{P}}^1$ defined by $y^3 = f(x)$ along with a list of polynomials $g(T)$ which define divisors ${\operatorname{div}}(g)$ on ${\mathbb{P}}^1$ which lift to rational divisors on $X$. Each polynomial is either $3$-torsion or defines an infinite order divisor on $X$.
If $\tt RankBounds$ returns upper and lower bounds that prove $X$ is rank 1, we proceed. Otherwise, we discard the curve and move on to the next one.[^1] For $X$ rank 1, we consider the non-torsion generators $g(T)$; if there is some non-torsion $g(T)$ of degree one, then we have a rational point $P \in X({\mathbb{Q}})$ such that $[P - \infty]$ is infinite order in $J({\mathbb{Q}})$. Otherwise, we are left with $g(T)$ of higher degree, which correspond to rational divisors that are supported on Galois conjugates of points defined over a number field $K$. To handle this second case, which comprises 544 of the 1403 Picard curves in our rank 1 database, we extended functionality in Balakrishnan and Tuitman’s code to compute Coleman integrals to points defined over number fields on general curves as well as specify divisors on curves when running effective Chabauty computations.
With a rational point whose class in the Jacobian is of infinite order
----------------------------------------------------------------------
\[RatPoint\]
Input: a Picard curve $X$ with affine equation $y^3= f(x)$, discriminant $\Delta$ and Jacobian $J$ such that $J({\mathbb{Q}})$ is rank 1, along with a point $P\in X({\mathbb{Q}})$ such that $[P-\infty]\in J({\mathbb{Q}})$ has infinite order.\
Output: $X({\mathbb{Q}})$
1. Find the first [^2] prime $p > 3$ such that $p \nmid \Delta$.
2. Set parameters $(N,e)$ for precision and ramification degrees. We must also choose $N$ large enough to ensure the $p$-adic and $t$-adic truncations of the integrals in Step \[Integrate\] contain all zeros in each residue disk. See Section \[prec\] for a discussion of the precision. For some heuristics for choosing $e$ given a fixed $N$ see the Appendix.
3. Compute a list of known points in $X({\mathbb{Q}})$ by searching in a box up to naive height $1000$.
4. Compute a basis $v_1, v_2$ of vanishing differentials ${\text{Van}}(X({\mathbb{Q}}))$ by computing a basis for the kernel of the matrix of integrals $[\int_{\infty}^P \omega_i]$ for $i = 1, 2, 3$ regular basis differentials.
5. \[Integrate\] Solve for $X({\mathbb{Q}}_p)_1 = \{ Q \in X({\mathbb{Q}}_p) | \int_\infty^Q v_i = 0, i= 1,2 \}$. (If we fail because some integral does not converge or has multiple zeros, fix the parameters in Step 2.) See Algorithm \[SolveTiny\] for a details.
6. For each $Q \in X({\mathbb{Q}}_p)_1$, compare to the list of known points in $X({\mathbb{Q}})$ and match up points.
7. For any remaining points $Q$, run the Magma function $\tt PowerRelation$ to see if we can recognize them as number field points.
i. First check if $f(Q) =0$; if so, $Q$ is a ramification point, and therefore 3-torsion.
ii. Compute $\int_\infty^Q \omega_i$, $i = 1, 2, 3$ on regular basis differentials. If these are $0$ to our $p$-adic precision, then $[Q- \infty]$ is torsion.
iii. Search for relations of small height with known rational points in a box on $J({\mathbb{Q}})$. If we succeed, we have explained $Q \in X({\mathbb{Q}}_p)_1$ by linearity.
iv. If there are points $Q$ still remaining, then more work must be done to explain their presence in $X({\mathbb{Q}}_p)_1$ and provably determine $X({\mathbb{Q}})$.
Precision Bounds {#prec}
----------------
In Algorithm \[RatPoint\] Step \[Integrate\] we solve for the zeros of the integral of a regular differential on $X$. To do this, we solve for the zeros in each residue disk of $\bar{Q} \in X({\mathbb{F}}_p)$, expanding the integral $\int_b^z v = \int_b^Q v + \int_{Q(t)}^{z(t)} v(t) {\ensuremath{\operatorname{d}\!{t}}}$ where $Q \in X({\mathbb{Q}}_p)$ reduces to $\bar{Q}$, using Algorithm \[tinyintegrals\]. In practice, we must truncate the $p$-adic power series to be a polynomial with finite $p$-adic precision. Therefore we need to determine how much $p$-adic and $t$-adic precision we need to determine all of the zeros of $\int_b^Q v+\int_{Q(t)}^{z(t)} v(t) {\ensuremath{\operatorname{d}\!{t}}}$ on the residue disk of $\bar{Q}$.
\[finalprec\] Let $f(t) \in {\mathbb{Q}}_p[[t]]$ be a $p$-adic power series such that $f{^\prime}(t) \in {\mathbb{Z}}_p[[t]]$. Write $f{^\prime}(t) = a_0 + a_1t + a_2t^2 + \dots $ and $f(t) = a_0 t+ a_1t^2/2 + a_2t^3/3 + \dots $, and define the normalized power series $F(x)$ by substituting $t = px$ in $f(t)$: $$F(x) \colonequals f(px) = \sum_{i=1}^\infty \frac{a_{i-1} x^i p^{i} }{i}.$$ Let $N{^\prime}\in {\mathbb{N}}$. Define $F(x)_{N{^\prime}} = F(x) \mod p^{N{^\prime}}$. Let $$M(N{^\prime},m) \colonequals \min_M \{M -\log_p (M) > N{^\prime}\}.$$ Write $F(x)_{N{^\prime}} = F(x)_{M} + F(x)_\infty$ where $F(x)_{M} \in ({\mathbb{Z}}/p^{N{^\prime}}{\mathbb{Z}})[x]$ contains only terms of degree less than $M(N{^\prime}, m)$ and $F(x)_\infty \in {\mathbb{Z}}_p[[x]]$ is a power series with the higher order terms.
Then each simple zero of $F_{N{^\prime}}(x)$ of nonnegative valuation in ${\mathbb{Z}}/p^{N{^\prime}}{\mathbb{Z}}$ is the reduction modulo $p^{N{^\prime}}$ of a root of $F(x)$ of nonnegative valuation. If all such roots are simple they are the full set of roots of $f(t)$ in the disk $p{\mathbb{Z}}_p$.
We first remark that $F(x)$ is indeed in ${\mathbb{Z}}_p[[x]]$. The coefficient of $x^{i}$ is $b_i \colonequals \frac{ma_{i-1} p^{i} }{i}$, which has positive valuation for all $i$.
Then it suffices to show that for $i \geq M(m, N{^\prime})$, $v_p(b_i) > N{^\prime}$. Then $v_p(b_i) \geq v_p (p^{i} /i) \geq i -\log_p(i)$, and for $i \geq M(m, N{^\prime})$, this is at least $ N{^\prime}$.
\[SolveTiny\]
Input: $\omega \in {\text{Van}}(X({\mathbb{Q}}))$, $Q \in X({\mathbb{Q}}_p)$ with uniformizer $t$ computed as in Algorithm \[tinyintegrals\], $N_{in}\in {\mathbb{N}}$ an initial precision.\
Output: The zeros in the residue disk of $Q$ of the antiderivative of $\omega$ with constant term $\int_b^Q \omega$.
1. Compute $c=\int_b^Q \omega$ the constant term (if $Q$ is a bad point, $c = \int_b^S \omega + \int_S^Q \omega$ where $S \in X({\mathbb{Q}}_p(p^{1/e}))$ is a point on the boundary of the residue disk of $Q$ with $e$ large enough so that the integral from $b$ to $S$ converges).
2. Compute $f{^\prime}(t) \colonequals \omega(t)$ to precision $N_{in}$ using Algorithm \[tinyintegrals\].
3. Compute $f(t)$ the antiderivative of $f{^\prime}(t)$ with constant term $c$, accurate to precision $N{^\prime}= N_{in} - \delta$, where $\delta$ is the precision loss when integrating discussed in [@balatuitman]\*[Section 4]{}.
4. Compute $M(m, N{^\prime})$. Let $F_{N{^\prime}}(x)$ as in Lemma \[finalprec\].
5. Check if $F_{N{^\prime}}(x)$ has only simple zeros. If so, these correspond to the zeros of $f(t)$. If not, increase $N_{in}$ until there are only simple zeros.
If $f(t)$ has double roots, then $F_{N{^\prime}}(x)$ always has double roots. In Algorithm \[RatPoint\], we solve for common roots of two power series; we check the roots are single roots of at least one of the truncations. In the 1403 Picard curves in our database, at $N = 15$ we never find common double roots.
According to [@balatuitman]\*[Remark 4.2]{} the precision $M$ in Balakrishnan and Tuitman’s implementation used to truncate $F(x)$ is chosen so that for all $i \geq M$, $v_p(b_i) \geq N_{in}$ for $b_i$ the coefficient of $x^i$. The chosen $M \geq M(m, N{^\prime})$ is sufficient.
Beyond rational points
----------------------
If we cannot find $P \in X({\mathbb{Q}})$ such that $[P - \infty]$ is infinite order in $J({\mathbb{Q}})$, then we are left with a rational divisor on $X$. Furthermore, as $X$ is a genus $3$ curve we have a surjection ${\operatorname{Sym}}^{3}(X) \twoheadrightarrow J$ [@MilneJacobian], the support of the divisor is defined over an extension of degree at most $3$.
Using our extension of Balakrishnan and Tuitman’s code, it is possible to integrate over any rational divisor of the form $D - \deg(D)b$ for rational primes $p$ split completely in $L$, the number field over which the support of $D$ is defined, and also to points defined over a number field $K$ on a general curve, for $p$ completely split in $K$. However, we focus our discussion on the situation for superelliptic curves where it is practical to compute rank bounds. Our code is tailored to this case. For non-hyperelliptic superelliptic curves, when computing infinite order points in the Mordell-Weil group of $X$, $\tt RankBounds$ outputs polynomials $g(T)$ such that the divisor ${\operatorname{div}}(g)$ on ${\mathbb{P}}^1$ lifts to a rational divisor on $X$ under the map $X \to {\mathbb{P}}^1$.[^3] Then $\deg(g(T))$ is at most the genus of $X$, which is $3$ in our case of Picard curves. (For odd degree hyperelliptic curves $\tt RankBounds$ returns the Mumford representation of the divisor, and one can recover $g$ by taking the polynomial for the $x$-coordinate. Our code does not handle the even degree hyperelliptic case.)
Let $D$ be the rational divisor obtained by lifting ${\operatorname{div}}(g)$ to $X$. Then $D = P_1 + \dots + P_d - d \infty$. We write $L= {\mathbb{Q}}[T]/g(T)$ to denote the number field $L$ over which the points $P_j$ are defined. To compute the Chabauty–Coleman set for $X$ we need to compute the set of vanishing differentials, which is done by taking a basis for the kernel of the $1 \times 3$ matrix of integrals of the regular differentials over $D$. The integral $\int_D \omega_i$ breaks up as $\sum_{j=1}^d\int_\infty^{P_j}\omega_i$. By choosing a prime $p$ that splits completely in $L$, for any ${\mathfrak{p}}|p$ in $L$ we have $L_{{\mathfrak{p}}} \cong {\mathbb{Q}}_p$. Therefore, we can proceed as before and compute these integrals in $L_{{\mathfrak{p}}}$.
There are some implementation details we must discuss. First, we can no longer compute our integral bases $b^0$ and $b^\infty$ of ${\mathbb{Q}}(X) / {\mathbb{Q}}[x]$ and ${\mathbb{Q}}(X)/{\mathbb{Q}}[1/x]$ respectively, as we need to evaluate $b^0_i$ and $b^\infty_i$ at the points $P_j$ in the support of $D$ which are defined over $L$. Tuitman [@Tuitman2] alternatively defines $b^0$ and $b^\infty$ as integral bases for ${\mathbb{Q}}_q(X)/{\mathbb{Q}}_q[x]$ and ${\mathbb{Q}}_q(X)/{\mathbb{Q}}_q[1/x]$, where $q=p^k$ and ${\mathbb{Q}}_q$ is the unramified extension of ${\mathbb{Q}}_p$ of degree $k$. There are efficient algorithms to construct integral bases in function fields over a number field [@Bauch; @Hess]. Therefore we compute integral bases for the function field $L(X)/L[x]$ and $L(X)/L[1/x]$, since $L(X) \otimes L_{\mathfrak{p}}\cong {\mathbb{Q}}_p(X)$, $L[x] \otimes L_{\mathfrak{p}}\cong {\mathbb{Q}}_p[x]$, and $L[1/x] \otimes L_{\mathfrak{p}}\cong {\mathbb{Q}}_p[1/x]$. Thus $b^0$ and $b^\infty$ will give local coordinates for the points $P_j$ on $X _{{\mathbb{Q}}_p}$. Following Tuitman’s algorithm, we must compute matrices $W^0 \in {\operatorname{GL}}_{d_x} ({\mathbb{Z}}_p [x, 1/r])$ and $W^\infty \in {\operatorname{GL}}_{d_x} ({\mathbb{Z}}_p [x,1/x, 1/r])$ such that $$b^0_j = \sum_{i=0}^{d_x-1} W^0 _{i+1,j+1}y^i \text{ and } b^\infty_j = \sum_{i=0}^{d_x-1} W^\infty _{i+1,j+1}y^i, \text{ for all } 0 \leq j \leq d_x - 1.$$ However, we compute $W^0 \in {\operatorname{GL}}_{d_x}(L[x,1/r])$ and $W^\infty \in {\operatorname{GL}}_{d_x} (L[x,1/x, 1/r])$. There are no theoretical obstructions to also extending the code to use inert primes $p$ in $L$, but it would require more extensive changes to the base code. Note that when the plane model $Q$ for $X$ is non-singular then $W^0$ will be the identity c.f. [@Tuitman2]\*[Remark 2.2]{} as is the case for all curves $Q=y^3-f$ in our database. The code directly extending Balakrishnan and Tuitman’s code for Coleman integrals and the code for batch computations of rational points and testing algebraic points and Jacobian relations are available on GitHub [@TravisSachiextensions; @TravisSachiexperiments].
The data
========
We computed the Chabauty–Coleman sets $X({\mathbb{Q}}_p)_1$ of $1403$ rank $1$ Picard curves of discriminant bounded by $10^{12}$ and provably determined rational points on all curves. Of these curves, on $859$ we found a rational point $P$ whose class $[P - \infty]$ under the Abel-Jacobi map is infinite order. For the other $544$ curves, we used infinite order divisors not given by the image of a rational point under the Abel-Jacobi map.[^4]
On all curves, the rational points were of small height, and the naive search for points of height at most $1000$ yielded all of the rational points. In Figure \[figure1\] we display the number of Picard curves with $n$ rational points; every Picard curve has a point at infinity, and the curves in our database had at most six rational points. In Figure \[figure2\] we show the primes used for the curves with divisors not given by the image of a rational point under the Abel-Jacobi map. These are the first split prime in the number field ${\mathbb{Q}}[T]/ g(T)$ where the divisor $D$ is the rational lift of ${\operatorname{div}}(g)$, and $g$ is chosen from a list of possible $g$. For ease of reading, we leave out the primes used once: those primes are 61, 83, and 107.
For all but $14$ curves, the only extra points in the Chabauty–Coleman set were the 3-torsion ramification points. For six curves, we found global torsion points of higher order. These points were $9$-torsion and $4$-torsion and of the form $(a,b^{1/3})$ with $a, b \in {\mathbb{Z}}$. For five curves, we found extra points that have relations with points in $J({\mathbb{Q}})$. For three curves, we found an extra point in the Chabauty–Coleman set whose presence is explained by an automorphism of the curve.
Examples
--------
For some Picard curves $X$ and choices of prime $p$, we found interesting global points in $X({\mathbb{Q}}_p)_1$ that were not ramification points or rational points. We discuss four notable examples here.
Consider the curve $X: y^3 = x^4 + 6 x^3 - 48 x-64$ with discriminant $1289945088$. Then $X$ has the infinite order divisor $[(-3, -1) - \infty]$ so we compute the Chabauty–Coleman set at $p = 5$ and find an extra global point $T=(t, 1/2 t^2 - 4)$ over ${\mathbb{Q}}[t] / (t^3 - 24 t - 48)$. However, if we instead compute the Chabauty–Coleman set at the prime $p = 17$ we find $X({\mathbb{Q}}_{17})_1 = X({\mathbb{Q}}) \cup W \cup \{ T, S\}$ where $W$ is the pair of $17$-adic ramification points. The point $S$ is a different global point $S = (s, s^2 + 6s + 8)$ over ${\mathbb{Q}}[s]/(s^3 + 9s^2 + 24 s + 24)$.
In the Jacobian of $X$, we have that $18 [T - \infty] = 9[S- \infty]$. Furthermore, relating this back to the known class of infinite order in $J({\mathbb{Q}})$, $[(-3, -1)- \infty]$, we can show that $18[T - \infty ] = 3 [(-3, -1)- \infty]$. So $S$ and $T$ appear in the Chabauty–Coleman set by linearity of the Coleman integral.
The curve $X:y^3 = x^4 + 25x^3 - 78x^2 + 76x - 24$ with discriminant $411210307584$ is rank $1$ and possesses the infinite order divisor $D = {\operatorname{div}}(x^2 - 6x + 4)$ in $J({\mathbb{Q}})$.
The first prime that splits completely in ${\mathbb{Q}}[x]/(x^2 - 6x + 4)$ is $11$, so we compute $X({\mathbb{Q}}_{11})_1$. We obtain the rational points, one ramification point, as well as the $9$-torsion point $(2, 32^{1/3})$.
\[endoexample1\] In their Chabauty–Coleman experiments on genus 3 rank 1 hyperelliptic curves, Balakrishnan–Bianchi–Cantoral-Farfán–[Ç]{}iperiani–Etropolski [@CCExp] find a novel reason for extra points in the Chabauty–Coleman set. They exhibit a curve with two extra points coming from the presence of extra endomorphisms of the Jacobian, which are not explained by linearity or torsion. Motivated by their example, we searched Sutherland’s Picard curve database for similar examples and found several. This first example of a curve with an extra point in the Chabauty–Coleman set explained by the automorphism group of the curve comes from searching outside of our rank 1 database.
Let $X: y^3 =2x^4-5$ be the Picard curve with discriminant $10319560704000000$. This is a rank 2 curve with linearly independent non-torsion classes $[(2,3)- \infty]$ and $[(-2,3) - \infty]$ in $J({\mathbb{Q}})$.
The $\overline{{\mathbb{Q}}}$-automorphism group of $X$ is a group of order $48$ defined over the degree $12$ number field ${\mathbb{Q}}[x]/(x^{12}-6x^{11}+21x^{10}-50x^9+90x^8-126x^7+191x^6-276x^5-285x^4+950x^3-354x^2-156x+676)$ (LMFDB label [](https://www.lmfdb.org/NumberField/12.0.7652750400000000.2)). The group is isomorphic to ${\operatorname{SL}}_2({\mathbb{F}}_3) \rtimes_{\rho} C_2$ where the image of $\rho: C_2 \to {\operatorname{Aut}}({\operatorname{SL}}_2({\mathbb{F}}_3))$ is given by conjugation by an element of order $2$. Let $p = 13$ be the first prime that splits completely in this extension. Computing the Chabauty–Coleman set at $p$, we find $X({\mathbb{Q}}_{13})_1 = X({\mathbb{Q}}) \cup W \cup T \cup A$ where $B$ is the set of ramification points, $T$ is the set of torsion points, and $A$ is a set of points that will be explained only by automorphisms of the curve, as shown in Table $\ref{pointsinXqp1}$.
Set Points
-------------------- ------------------------------------------------------------------------------------------------------
$X({\mathbb{Q}}) $ $(2,3), (-2,3), \infty$
$W$ Four Galois conjugates of $((5/2)^{1/4}, 0)$
$T$ Twelve Galois conjugates of $((45/2)^{1/4}, 40^{1/3})$, Three Galois conjugates of $(0, (-5)^{1/3})$
$A$ Two Galois conjugates of $((-4)^{1/2}, 3)$
: Points in $X({\mathbb{Q}}_{13})_1$[]{data-label="pointsinXqp1"}
To verify that the points in $T$ are torsion, and those in $A$ are not, we compute the Coleman integrals $\int_\infty^P \omega_i$, $i = 1, 2, 3$ on regular differentials for each $P$ in $T$ and $A$ using the new functionality for integrating to points defined over number fields. These integrals are $0$ on torsion points to our $p$-adic precision. The torsion points of the form $((45/2)^{1/4}, 40^{1/3})$ are $12$-torsion, and those of the form $(0, (-5)^{1/3})$ are $4$-torsion.
On the other hand, we show that the classes $A_1 \coloneqq[((-4)^{1/2}, 3)- \infty] $ and $A_2 \coloneqq [(-(-4)^{1/2}, 3)- \infty]$ do not have linear relations with $P_1 \coloneqq [(2,3) - \infty]$ and $P_2 \coloneqq [(-2,3) - \infty]$ in $J({\mathbb{Q}})$. Suppose $n A_1 = a P_1 + b P_2$ for some $n,a,b \in {\mathbb{Z}}$. Then $nA_1^c = a P_1 + bP_2$ where $c$ is complex conjugation. But $A_1^c = A_2$, and subtracting we would have $n(A_1 - A_2) = 0$ in $J({\mathbb{Q}})$. However, computing the Coleman integrals to $A_1$ minus the Coleman integrals to $A_2$ we get a nonzero value, so $A_1 - A_2$ cannot be torsion.
Therefore we need a new reason to explain the presence of the points in $A$ in the Chabauty–Coleman set. This reason will come from examining the automorphisms of $X$. Recall for $X$ a Picard curve we have a basis for $H^0(X_{{\mathbb{Q}}_p}, \Omega^1)$, given by $\omega_1 = y{\ensuremath{\operatorname{d}\!{x}}}/r, \omega_2 = xy{\ensuremath{\operatorname{d}\!{x}}}/r, \omega_3 = y^2 {\ensuremath{\operatorname{d}\!{x}}}/r$ where $r=x^4 -2$. Note $X$ has an order $4$ automorphism given by $\varphi: x \mapsto i x$ that acts by pullback on the differentials: $\varphi^* \omega_1 = i \omega_1, \varphi^* \omega_2 = - \omega_2, \varphi^* \omega_3 = i\omega_3$. Let $v$ be the vanishing differential in the computation of the Chabauty–Coleman set. We will show $v$ is an eigenvector for $\varphi^*$. When the vanishing differential is an eigenvector for an automorphism of the curve, there is potential for extra points to appear in the Chabauty–Coleman set because of the automorphism.
First we will explain this phenomenon by showing why $A_j$ is in the Chabauty–Coleman set, assuming $v$ is an eigenvector for $\varphi^*$, for $j = 1,2$. It is enough to show that integrating $A_j$ against the vanishing differential $v$ is zero. Note that choosing square roots, $\varphi(P_1) =A_1$ and $\varphi(P_2) = A_2$. Using change of variables and applying the fact that our vanishing differential is an eigenvector, we have $$\int_\infty^{A_j} v = \int_{\varphi(\infty)}^{\varphi(P_j)} v = \int_\infty^{P_j} \varphi^* v = i \int_\infty^{P_j} v = 0 .$$ Thus $A_j$ is in the Chabauty–Coleman set because of $\varphi$.
Now we prove that $v$ is an eigenvector for $\varphi^*$. Write $v = A \omega_1 + B \omega_2 + C \omega_3$, with $A, B, C \in {\mathbb{Q}}_p$. The vanishing differential is defined by the equation $\int_\infty^{P_j} v= 0$ for $j = 1, 2$. Acting by $\varphi^2$ we have $$\int_{\varphi^2(\infty)}^{\varphi^2(P_1)} v = \int_{\infty}^{P_2}\varphi^* v = \int_\infty^{P_2} -A \omega_1 +B \omega_2 -C\omega_3 = 0.$$ Adding this to the fact that $\int_\infty^{P_2} v = \int_\infty^{P_2} A\omega_1 +B \omega_2 +C \omega_3 =0$ we see that $$2 \int_\infty^{P_2} B \omega_2 = 0.$$ Since the integral from infinity to $P_2$ of $\omega_2$ is not zero, and neither is 2 times it, then $B$ must be, and $v = A \omega_1 + C \omega_3$ is an eigenvector for $\varphi^*$ with eigenvalue $i$.
In addition to the previous example, we found three examples of extra points coming from automorphisms of the curve that arose during the computation of the Chabauty–Coleman sets in our rank 1 database. We describe one here. Let $X: y^3 = x^4 + 2x^3 + 6x^2 + 5x + 2$ be the rank $1$ Picard curve with discriminant $277826509467$. We have an infinite order divisor $D = {\operatorname{div}}(x^2 + x - 1)$, and we compute the Chabauty–Coleman set for $X$ at $p = 11$, the first split prime in ${\mathbb{Q}}[x]/(x^2+ x -1)$. We obtain: $$X({\mathbb{Q}}_{11})_1 = \{ \infty, (-1/2, \sqrt[3]{13/16})\}.$$ Let $R \coloneqq(-1/2, \sqrt[3]{13/16})$. To check if $R$ is torsion, we compute the Coleman integrals on the regular basis differentials $\int_\infty^R \omega_i$, $i=1, 2, 3$, which are not all zero. As infinity is the only rational point on $X$, the presence of $R$ in $X({\mathbb{Q}}_{11})_1$ also cannot be explained by relations in $J({\mathbb{Q}})$. Computing the automorphism group of $X$ over ${\mathbb{Q}}[x]/(16x^3 - 13)$, $X$ has an order $2$ automorphism $\varphi$ sending $x \mapsto -x -1 $ and fixing $y$. The pullback $\varphi^*$ acts on the regular differentials by $\varphi^* \omega_1 = - \omega_1$, $\varphi^* \omega_2 = \omega_2$, and $\varphi^* \omega_3 = -\omega_3$. By a similar method used in Example \[endoexample1\], comparing the Coleman integral over $D$ against $v$, and the Coleman integral over $D$ against $\varphi^*v$, we can show that the vanishing differentials are of the form $v = A\omega_1 + C\omega_3$, and are eigenvectors for $\varphi^*$ with eigenvalue $-1$. Furthermore, $\varphi(R) = R$ and $\varphi(\infty) = \infty$ and therefore $$\int_\infty^R v = \int_{\varphi(\infty)}^{\varphi(R)} v = \int_\infty^R \varphi^* v = -\int_\infty^R v.$$ So $2 \int_\infty^R v = 0$, showing $R \in X({\mathbb{Q}}_{11})_1$.
Acknowledgments
===============
We are very grateful to Jennifer Balakrishnan for suggesting this project and for numerous helpful discussions and suggestions. We are also grateful to Drew Sutherland for providing us with data, and to Alex Best and Borys Kadets for their generous help and advice.
Appendix: Precision heuristics {#appendix-precision-heuristics .unnumbered}
==============================
This section gives tables of the smallest $e$ needed for $\tt effective\_chabauty$ rounded up to the nearest multiple of five, fixing $p$ and $N$, for $10$ arbitrarily chosen curves of rank $1$ with a rational point whose class in the Jacobian is of infinite order taken from Sutherland’s database. Entries “bad” denote that a curve has bad reduction at that prime. For the tables, after small primes $p \leq 13$ we chose a sample of primes less than $100$ spaced roughly by $10$.
[315pt]{}[ l l l l l ]{} Discriminant : Curve & $5$ & $7$ & $11$ & $13$\
$31492800:y^3=x^4 + 3x^3 - 3x + 1 $& bad &$50$ & $85$ & $100$\
$47258883:y^3=x^4 + 2x^3 - x^2 - x$& $40$ & bad & $85$ & $100$\
$70858800:y^3=x^4 + 3x^3 - x^2 - 4x + 2$& bad &$ 50$& $85$ & $100$\
$106288200:y^3=x^4 + x^3 - 66x^2 - 324x - 432$ & bad & $30$& $50$& $55$\
$151322904:y^3=x^4 + 11x^3 - 32x^2 + 28x - 8$& $30$ & $35$ & $50$ & $100$\
$212891328:y^3=x^4 + 5x^3 + 4x^2 - 5x + 1$& $10$ & $50$ & $85$ & bad\
$105303439827:y^3=x^4 + 4x^3 + x^2 - 3x - 1$&$30$ & $50$ & $85$ & $95$\
$108095630841:y^3=x^4 + 4x^3 + 6x^2 - 9x$& $35$ & $40$ & bad & $55$\
$113232992256:y^3=x^4 + x^3 - 4x^2 - 8x$& $30$ & $50$ & $80$ & $100$\
$988868881152:y^3=x^4 + x^2 - 2x + 3$& $10$& $55$& $85$ & $100$\
[360pt]{}[ l l l l l l l l l ]{} Curve & $23 $& $31$&$41$&$53$&$61$ &$71$&$83$&$97$\
$y^3=x^4 + 3x^3 - 3x + 1$ &$180$ & $245$& $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + 2x^3 - x^2 - x$ & $180$ & $245$& $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + 3x^3 - x^2 - 4x + 2$&$180$ & $245$& $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + x^3 - 66x^2 - 324x - 432$ & $95$ & $105$ & $155$ &$200$ & $205$ & $240$ &$280$&$325$\
$y^3=x^4 + 11x^3 - 32x^2 + 28x - 8$ &$180$ & bad & $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + 5x^3 + 4x^2 - 5x + 1$ &$180$ & $240$& $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + 4x^3 + x^2 - 3x - 1$ &$180$ & $245$& $325$ &$420$&$485$&$565$&$660$& $770$\
$y^3=x^4 + 4x^3 + 6x^2 - 9x$ & $95$ & $120$ & bad & $200$ & $230$ & $270$ & $315$ & $365$\
$y^3=x^4 + x^3 - 4x^2 - 8x$ & $180$ & $245$ & $320$ &bad &$485$ &$565$ &$660$&$770$\
$y^3=x^4 + x^2 - 2x + 3$& $180$ &$245$ & $325$ &$420$& $485$ &$565$ &$660$ &$770$\
We also recorded the $e$ value used to compute $X({\mathbb{Q}}_p)_1$ and select a non-torsion divisor when $N = 15$ curve $X$ in the full database of $1403$ Picard curves. We initialized $e$ at $40$ and then incremented by $20$ until the integrals to compute $X({\mathbb{Q}}_p)_1$ converged. These values can be found in the master list of Chabauty data on the 1403 Picard curves `masteralldata.txt` available in the repository [@TravisSachiexperiments].
[^1]: For practical purposes, we terminated RankBounds if it ran for 120 seconds without returning an answer.
[^2]: Due to a minor error in the code of Balakrishnan and Tuitman for computing local coordinates at very infinite points, we sometimes choose a larger prime.
[^3]: For a given curve there are often multiple choices of $g$, and we pick the one with the smallest first split prime.
[^4]: For the remaining 544 curves, it can be quite computationally expensive to compute $X({\mathbb{Q}})$ depending on the parameters $N$, $e$, and $p$. Individual curves can require up to several hours. This computation was run a single core of a $28$-core $2.2$ GHz Intel 2 Xeon Gold server with $256$GB RAM.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'It is well known [@Erd] that a random subgraph of the complete graph $K_n$ has chromatic number $\Theta(n/\log n)$ w.h.p. Boris Bukh asked whether the same holds for a random subgraph of any $n$-chromatic graph, at least in expectation. In this paper it is shown that for every graph, whose fractional chromatic number is at least $n$, the fractional chromatic number of its random subgraph is at least $n/(8\log_2(4n))$ with probability more than $1-\frac{1}{2n}$. This gives the affirmative answer for a strengthening of Bukh’s question for the fractional chromatic number.'
author:
- |
Bojan Mohar[^1]\
Department of Mathematics\
Simon Fraser University\
Burnaby, BC, Canada\
[[email protected]]{}
- |
Hehui Wu[^2]\
Shanghai Center for Mathematical Sciences\
Fudan University\
Shanghai, China\
[[email protected]]{}
title: Fractional chromatic number of a random subgraph
---
Introduction
============
If $G$ is a graph and $p \in (0,1)$, we let $G_p$ denote a *random subgraph* of $G$ where each edge of $G$ appears in $G_p$ independently with probability $p$. The standard Erdős-Renyi random graph $G_{n,p}$ can be viewed as a random subgraph of the complete graph $K_n$. A lot is known about properties of random graphs, but not so much about the generalized notion of random subgraphs. In particular, it is known that the chromatic number of $G_{n,p}$ is $\Theta(n/\log_{1/p}n)$, almost surely, for a wide range of values $p=p(n)$. Boris Bukh [@Bukh] asked whether the same phenomenon occurs for random subgraphs of any $n$-chromatic graph, at least in expectation, when $p$ is constant.
\[prb:Bukh\] Does there exist a constant $c$ so that for every graph $G$, the expected chromatic number of its random subgraphs satisfies: $${\mathbb E}(\chi(G_{1/2})) > c \, \frac{\chi(G)}{\log \chi(G)}\,?$$
Only very special cases have been considered in the past. Bogolyubskiy et al. [@BGPR1; @BGPR2] considered random subgraphs of certain distance graphs and their chromatic number. Kupavskii [@Ku16] studied the chromatic number of a random subgraph of Kneser and Schrijver graphs $KG(n,k)$ and $SG(n,k)$, as $n$ grows. For a wide range of parameters $k=k(n)$ and $p=p(n)$, he proved that $\chi(KG_p(n,k))$ is very close to $\chi(KG_p(n,k))$ w.h.p., differing by at most 4 in many cases. His work was preceded by pioneering work of Bollobás, Narayanan and Raigorodskii [@BNR] and Balogh, Bollobás, and Narayanan [@BBN], who studied independent sets in random subgraphs of Kneser graphs $KG(n,k)$.
While Problem \[prb:Bukh\] remains open, we found evidence to answer Bukh’s question in the affirmative when the chromatic number is replaced by the fractional chromatic number. In fact, we show that this holds not only in expectation but holds for subgraphs of $G$ with high probability, see Theorem \[thm:main\] below.
Let us first recall the definition of the fractional chromatic number. Let $\mathcal{I}(G)$ be the family of all independent sets of $G$. For each vertex $v\in V(G)$, let $\mathcal{I}(G,v)$ be the family of all those independent sets which contain $v$. For each independent set $I$, consider a nonnegative real variable $y_I$. The *fractional chromatic number* of $G$, denoted by $\chi_f(G)$, is the minimum value of $$\sum_{I\in\mathcal{I}(G)} y_I, \quad \hbox{ subject to } \quad \sum_{I\in\mathcal{I}(G,v)} y_I \ge 1 \quad \hbox{ for each } v\in V(G).
\label{eq:chif_LP}$$
In this paper, we prove in the affirmative a strengthening of Bukh’s question for the fractional chromatic number.
\[thm:main\] Let $t\ge2$ be a rational number and let $G$ be a graph with $\chi_f(G)=t$. Then for every $p\in (0,1)$ and every $c>0$ we have: $$Pr\Bigl(\chi_f(G_p)\ge \frac{t}{4\log_{1/p}(et)+4+4c}\Bigr) > \frac{1-2p^c}{1-p^c}.$$
By taking $c=\log_{1/p}(et)$, we obtain the following:
\[cor:chif(Gp)\] If $\chi_f(G)=t$ and $p\in (0,1)$, then $$\chi_f(G_p) \ge \frac{t}{8\log_{1/p}(et)+4}$$ with probability at least $1-\tfrac{1}{2t}$.
Fractional weight and principal vertex-sets
===========================================
The proof of Theorem \[thm:main\] uses the tools presented in this section. They are based on two concepts, that of a principal subset of vertices and that of a sparse subset. These two notions were used previously in our fractional versions of the Erdős-Neumann Lara conjecture (see [@MW]) and the Erdős-Hajnal conjecture from [@EH] for triangle-free subgraphs (see [@MW_trianglefree]) that every graph with large chromatic number contains a triangle-free subgraph whose chromatic number is still large.
We let $V=V(G)$, $n=|V|$, and $t = \chi_f(G)$. By the linear program duality for the definition (\[eq:chif\_LP\]) of $\chi_f(G)$, there exists a non-negative *weight function* $w:V\to{\mathbb{R}}^+$, such that $w(V)=t$, and for any $I\in \mathcal{I}(G)$, $w(I)\le 1$. See [@GR] for more details. Here and in the sequel we write $w(A) = \sum_{v\in A} w(v)$ for any vertex set $A\subseteq V$, and call this value the *weight* of $A$.
From now on we fix $w$ and assume that the vertices of $G$ are listed as $\VEC v1n$ in the non-increasing order of their weights, i.e. $w(v_{i+1})\le w(v_i)$ for $i=1,\dots,n-1$. For any subset $X$ of $V$, we also rank the elements in $X$ according to the ordering of $V$, and we denote by $X_k$ the subset of the first $k$ elements in $X$. In particular, $V_k=\{\VEC v1k\}$. We extend this notion to any real number $s\ge1$ by setting $X_s := X_{\lfloor s\rfloor}$.
For a real number $s\ge 1$, a nonempty subset $X$ of a vertex set $Y$ is said to be *$s$-principal* in $Y$ if $X\subseteq Y_{s|X|}$. That is, if $X$ has size $m$, then all elements of $X$ are among the first $\lfloor sm\rfloor$ vertices in $Y$. As a kind of opposite property, we say that a subset $X$ of $Y$ is *$s$-sparse* in $Y$ if $X$ contains no $s$-principal subset in $Y$. Note that every subset of an $s$-sparse set in $Y$ is also $s$-sparse in $Y$. When the hosting set $Y$ for $s$-principal or $s$-sparse is not specified, by default it is the whole vertex-set $V$.
The following condition gives another description of sparse sets that is easier to deal with computationally.
\[lem:sparse characterization\] $X$ is an $s$-sparse set in $Y$ if and only if $|Y_k\cap X|< k/s$ for every integer $k = 1,2,\dots |Y|$. In particular, if $X$ is $s$-sparse in $Y$, then $|X|<|Y|/s$.
It is clear that $X$ is $s$-sparse if and only if for each $r=1,\dots,|X|$, $|Y_{sr}\cap X| < r$. If $|Y_k\cap X|< k/s$ for every integer $k$, then this holds also for $k=\lfloor sr\rfloor$, implying that $|Y_{sr}\cap X| < \lfloor sr\rfloor / s \le r$. Conversely, if $|Y_{sr}\cap X| < r$, then $|Y_{sr}\cap X| \le r-1$. Let $s(r-1) < k \le sr$. Then $|Y_k\cap X| \le |Y_{sr}\cap X| \le r-1 < k/s$.
The next claim about the total weight of an $s$-sparse set will be essential for us.
\[lem:sparseislight\] Let $s\ge1$ be a real number. If $X$ is an $s$-sparse subset of $Y$, then $w(X)\le \frac{1}{s}\,w(Y)$.
Let $\VEC y1r$ be the non-decreasing order of the elements of $Y$ with $r=|Y|$, and let $\VEC x1m$ be the ordering of $X$ with $m=|X|$. Since $X$ is an $s$-sparse subset of $Y$, we have $x_i\not\in Y_{si}$. Hence for $1\le i\le m$, $w(x_i)\le w(y_j)$ if $1\le j\le \lfloor si\rfloor$. Moreover, since $x_i\in Y\setminus Y_{si}$, we also have $w(x_i)\le w(y_j)$ for $j=\lceil si\rceil$.
For a real parameter $z\in(0,|Y|]$, define $f(z)=y_{\lceil z\rceil}$. Then $f(z)\ge w(x_1)$ for $0<z\le s$, $f(z)\ge w(x_2)$ for $s<z\le 2s$, …, $f(z)\ge w(x_m)$ for $(m-1)s < z \le ms$. Therefore, $$s\,w(X) = s\sum_{i=1}^m w(x_i) \le \int_0^{sm} f(z)dz \le \sum_{j=1}^{\lceil sm\rceil} w(y_j) \le w(Y),$$ which gives what we were aiming to prove.
Fractional Chromatic number of a random subgraph
================================================
\[lem:5\] Let $p\in (0,1)$, $c>0$, and $s\ge1$ be real numbers. With probability at least $\frac{1-2p^{c}}{1-p^{c}}$ no $s$-principal set in $V(G)$ with average degree at least $2\log_{1/p}(es)+2c$ in $G$ is independent in $G_p$.
Let $A$ be an $s$-principal vertex-set of cardinality $k$ and with average degree at least $2\log_{1/p}(es)+2c$. Note that $A$ is independent in $G_p$ if and only if none of the edges of $G(A)$ is present in $G_p$. In other words, $$Pr(A\mbox{ is independent}) = p^{e(G(A))} \le p^{(\log_{1/p}(es)+c)k}=
\Bigl(\frac{p^c}{es}\Bigr)^k.$$ Also, as an $s$-principal set, $A$ is a $k$-set contained in $V_{sk}$. Thus, there are at most ${sk\choose k}$ $s$-principal sets with $k$ elements. Let $B_k^s$ be the event that some $s$-principal $k$-set with average degree at least $2\log_{1/p}(es)+2c$ is independent. By the above, the probability of $B_k^s$ is at most $$\label{eq:B_s}
\biggl(\frac{p^c}{es}\biggr)^k{sk\choose k}\le \biggl(\frac{p^c}{es}\biggr)^k(es)^k = p^{ck}.$$ Using (\[eq:B\_s\]) we see that the probability that some $s$-principal set with average degree at least $2\log_{1/p}(es)+2c$ is independent in $G_p$ is at most $$\sum_{k=1}^n Pr(B_k^s) \le \sum_{k=1}^n p^{ck} < \frac{p^c}{1-p^c}.$$ This implies that with probability more than $\frac{1-2p^c}{1-p^c}$ no $s$-principal set of $G$ with average degree at least $2\log_{1/p}(es)+2c$ is independent in $G_p$.
\[lem:6\] Suppose that $A\subseteq V(G)$ is a vertex-set that contains no $s$-principal sets whose average degree in $G$ is at least $x$. If every independent subset of $A$ has weight at most $1$, then $A$ has weight at most $2\lfloor x+1\rfloor + \frac{2w(V)}{s}$.
For any vertex-set $X$, we will denote by $e(X)$ the number of edges in the induced subgraph $G(X)$. For a vertex $v \in A$, let $d^>(v)$ be the number of neighbors of $v$ in $G(A)$ that appear before $v$ in the ordering $\VEC v1n$. For each $i$, we have $e(A_i)=\sum_{v\in A_i} d^>(v)$. Let $L=\{v\in A: d^>(v)\ge x\}$, and assume $L=\{v_{i_1},\dots,v_{i_l}\}$, where $l=|L|$. Then $e(V_{i_j}\cap A)\ge x\cdot j $ for $1\le j\le l$. In particular, if $|V_{i_j}\cap A|\le 2j$, then the average degree $\bar d(V_{i_j}\cap A)$ will be at least $x$. Since $A$ contains no $s$-principal sets with average degree at least $x$, the set $V_{i_j}\cap A$ cannot be $s$-principal in this case. Thus, we have one of the following:
\(1) $|V_{i_j}\cap A|> 2j$, or
\(2) $|V_{i_j}|> s\,|V_{i_j}\cap A|$.
Let $L_1=\{v_{i_j}\in L:|V_{i_j}\cap A|> 2j\}$, and let $L_2=\{v_{i_j}\in L: |V_{i_j}|\ge t|V_{i_j}\cap A|\}$. Then $L=L_1\cup L_2$. If $v_{i_j}\in L_1$, then $v_{i_j}$ is not among the first $2j$ elements of $A$. As the $j$-th element in $L_1$ does not appear before $v_{i_j}$, the $j$th element of $L_1$ is not among the first $2j$ elements of $A$. Therefore $L_1$ is a 2-sparse subset of $A$. By Lemma \[lem:sparseislight\], $w(L_1)\le \frac{1}{2}w(A)$.
For each $v_{i_j}\in L_2$, we have $|V_{i_j}\cap A|<\frac{1}{t}|V_{i_j}|$. As $|V_{i_j}\cap L_2|\le |V_{i_j}\cap A|<\frac{1}{t}|V_{i_j}|$, we see by Lemma \[lem:sparse characterization\] that $L_2$ is an $s$-sparse subset of $V$. By Lemma \[lem:sparseislight\], $w(L_2)\le \frac{w(V)}{s}$.
Let $S=A-L=\{v\in A: d^>(v)<x\}$. Then we have $w(S)\ge w(A)-w(L_1)-w(L_2)\ge \frac{w(A)}{2}-\frac{w(V)}{s}$. Also, $G(S)$ is an $\lfloor x\rfloor$-degenerate graph, hence $S$ is $\lfloor x+1\rfloor$-colorable. There is at least one independent set $I\subseteq S$ with weight $$w(I) \ge \frac{w(S)}{\lfloor x+1\rfloor}\ge \frac{w(A)/2-w(V)/s}{\lfloor x+1\rfloor}.$$ As every independent set contained in $A$ has weight at most $1$, we have $w(A)\le 2\lfloor x+1\rfloor + 2w(V)/s$.
We are going to use Lemmas \[lem:5\] and \[lem:6\] with $s=t=\chi_f(G)$, and $x=2\log_{1/p}(et)+2c$. With probability at least $(1-2p^c)/(1-p^c)$, no principal set with average degree in $G$ at least $x$ is independent in $G_p$ by the first lemma. Thus it is sufficient to see that every such subgraph $G_p$ has fractional chromatic number at least $\tfrac{t}{2x+4}$. To see this, we will use the second lemma.
Note that the weight function $w$ defines $\chi_f(G)$, i.e., $w(V)=t$. Define the weight function $w' = w/(2x+4)$ and consider any independent vertex-set $A$ in $G_p$. By Lemma \[lem:6\], $$w'(A) = \frac{w(A)}{2x+4} \le \frac{2\lfloor x+1\rfloor + 2w(V)/t}{2x+4} \le 1.$$ This weight function thus justifies that $\chi_f(G_p) \le w'(V) = t/(2x+4)$.
[9]{}
József Balogh, Béla Bollobás, Bhargav P. Narayanan, Transference for the Erdős-Ko-Rado theorem, Forum Math. Sigma 3 (2015), e23, 18 pp.
L. I. Bogolyubskiy, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, The independence numbers and the chromatic numbers of random subgraphs of some distance graphs, Dokl. Math. 457 (2014) 383–387.
L. I. Bogolyubskiy, A. S. Gusev, M. M. Pyaderkin, and A. M. Raigorodskii, The independence numbers and the chromatic numbers of random subgraphs of some distance graphs, Mat. Sb. (2015), in press.
Béla Bollobás, Bhargav P. Narayanan, and Andrei M. Raigorodskii, On the stability of the Erdős-Ko-Rado theorem, J. Combin. Theory Ser. A 137 (2016), 64–78.
Boris Bukh, Interesting problems, <http://www.borisbukh.org/problems.html>. See also <http://www.openproblemgarden.org/category/bukh_boris>
P. Erdős, Graph theory and probability, Canad. J. Math. 11(1) (1959), 34–38.
P. Erdős, Problems and results in combinatorial analysis and graph theory, in “Proof Techniques in Graph Theory” (ed. F. Harary), Academic Press, New York, 1969, pp. 27–35.
C. Godsil, G. Royle, Algebraic Graph Theory, Springer, 2001.
Andrey Kupavskii, On random subgraphs of Kneser and Schrijver graphs, J. Combin. Theory, Ser. A 141 (2016) 8–15. http://dx.doi.org/10.1016/j.jcta.2016.02.003
Bojan Mohar, Hehui Wu, Dichromatic number and fractional chromatic number, Forum of Mathematics, Sigma 4 (2016) e32, 14 pages. <https://doi.org/10.1017/fms.2016.28>
Bojan Mohar, Hehui Wu, Triangle-free subgraphs with large fractional chromatic number, submitted.
[^1]: Supported in part by the NSERC Discovery Grant R611450 (Canada), by the Canada Research Chairs program, and by the Research Project J1-8130 of ARRS (Slovenia).
[^2]: Part of this work was done while the author was a PIMS Postdoctoral Fellow at the Department of Mathematics, Simon Fraser University, Burnaby, B.C.
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---
abstract: 'We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result also holds for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such $M$ the moduli space of complete nonnegative sectional curvature metrics on the open simply connected manifold $M\times\R$ also has infinitely many path components.'
author:
- 'Anand Dessai[^1], Stephan Klaus, and Wilderich Tuschmann'
title: |
Nonconnected Moduli Spaces\
of Nonnegative Sectional Curvature Metrics\
on Simply Connected Manifolds
---
\[section\] \[theorem\][Proposition]{} \[theorem\][Lemma]{} \[theorem\][Remark]{} \[theorem\][Remarks]{} \[theorem\][Definition]{} \[theorem\][Corollary]{} \[theorem\][Example]{} \[theorem\][Assumption]{} \[theorem\][Problem]{} \[theorem\][Question]{} \[theorem\][Conjecture]{} \[theorem\][Rigidity Theorem]{} \[theorem\][Main Lemma]{} \[theorem\][Claim]{}
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\#1[ (\#1)]{} \#1\#2[[R]{}\_[\#2]{}(\#1)]{}
Introduction {#intro}
============
A central question in Riemannian geometry concerns the existence of complete metrics on smooth manifolds which satisfy certain prescribed curvature properties such as, e.g., positivity of scalar or Ricci curvature, nonnegativity or negativity of sectional curvature, etc. On the other hand, once the respective existence question is settled, there is an equally important second one, namely: How ‘many’ metrics of the given type are there, and how ‘many’ different geometries of this kind does the manifold actually allow?
To answer these questions, one is naturally led to study the corresponding spaces of metrics that satisfy the curvature characteristics under investigation, as well as their respective moduli spaces, i.e., the quotients of these spaces by the action of the diffeomorphism group given by pulling back metrics. When equipped with the topology of smooth convergence on compact subsets and the respective quotient topology, the topological properties and complexity of these objects constitute the appropiate means to measure the ‘number’ of different metrics and geometries, and we will adapt to this viewpoint.
There has been much activity and profound progress on these issues in the last decades, compare, e.g., \[BER\], \[BFK\], \[BG\], \[BH\], \[BHSW\], \[BKS\], \[Ca\], \[Ch\], \[CM\], \[CS\], \[FO1\], \[FO2\], \[FO3\], \[Ga\], \[GL\], \[Hit\], \[HSS\], \[LM\], \[KPT\], \[KS\], \[Wa1\], \[Wa2\], \[Wa3\], \[Wr\]. Since the very beginning, special importance has been given to the study of connectedness properties of spaces and moduli spaces of metrics with lower curvature bounds on closed as well as open manifolds. As these are also the main issue of the present note, let us now shortly review and comment on the most relevant developments in this respect, starting with closed manifolds:
Hitchin (\[Hit\]) showed first that there are closed manifolds for which the space of positive scalar curvature metrics is disconnected. Unfortunately, there is no explicit information on the actual number of path components available in Hitchin’s work. But for the standard spheres of dimension $4n+3$, where $n\ge 1$, Carr (\[Ca\]) then proved that their respective spaces of positive scalar curvature metrics indeed have an [*infinite*]{} number of path components. Since $\pi _0({\rm Diff}(S^{4n+3}))$ is finite, the corresponding statement also holds for their moduli spaces (\[LM, IV §7\]).
Results about the disconnectedness of moduli spaces of Riemannian metrics on a wide variety of examples are due to Kreck and Stolz (\[KS\]). They showed that for every closed simply connected spin manifold $M$ of dimension $4n+3$, where $n\ge 1$, with vanishing real Pontrjagin classes and $H^1 (M; \Z /2)=0$, the moduli space of metrics of positive scalar curvature, if not empty, has infinitely many path components. To achieve this, they introduced in this setting an invariant, the so-called $s$- (or also, nowadays Kreck-Stolz) invariant, whose absolute value is constant on path components of the moduli space.
Using the $s$-invariant together with work of Wang and Ziller about Einstein metrics on principal torus bundles over products of Kähler-Einstein manifolds (\[WZ\]), Kreck and Stolz went on to show that there are closed seven-manifolds for which the moduli space of metrics with positive Ricci curvature has infinitely many path components, and that in this dimension there are also closed manifolds which exhibit a disconnected moduli space of positive sectional curvature metrics.
Considering different metrics on the Kreck-Stolz examples, it was subsequently observed in \[KPT\] that the Kreck-Stolz result for positive Ricci curvature is actually also true for moduli spaces of nonnegative sectional curvature metrics. Notice, however, that both these results only apply and are indeed confined to dimension seven.
David Wraith (\[Wr\]) was then the first to show that there is an infinite number of dimensions with closed manifolds for which the moduli space of Ricci positive metrics has infinitely many path components. Namely, he proved that for any homotopy sphere $\Sigma^{4n+3}$, $n\ge 1$, bounding a parallelizable manifold, the moduli space of metrics with positive Ricci curvature has an infinite number of path components distinguished by the above Kreck-Stolz invariant.
The main result of this note establishes a corresponding statement for moduli spaces of nonnegative sectional curvature metrics on infinite numbers of manifolds in an infinite range of dimensions:
\[main theorem\] In each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components.
Moreover, the manifolds figuring here also complement Wraith’s homotopy sphere results. Indeed, they do actually also show that there is an infinite number of dimensions in which there exist infinite sequences of closed smooth simply connected manifolds of pairwise distinct homotopy type for which the moduli space of metrics with positive Ricci curvature has an infinite number of path components. The same holds true if one restricts to metrics with positive Ricci and nonnegative sectional curvature.
The manifolds in Theorem \[main theorem\] are total spaces of principal circle bundles over $\C P^{2n}\times \C P^1$ and agree in dimension $7$ with the examples considered by Kreck and Stolz. Their topology has been studied by Wang and Ziller in their work on Einstein metrics on principal torus bundles mentioned above. These manifolds can also be viewed as quotients of the product of round spheres $S^{4n+1}\times S^3$ by free isometric circle actions. As in \[KPT\] we equip these manifolds with the submersion metric. We show that in any fixed dimension $4n+3$ there exist among these Riemannian manifolds infinite sequences with the following properties: these manifolds belong to a fixed diffeomorphism type and their metrics have nonnegative sectional and positive Ricci curvature, but pairwise distinct absolute $s$-invariants. Since there are indeed infinite sequences of manifolds of pairwise distinct homotopy types with these properties, this will imply Theorem \[main theorem\].
The study of moduli spaces of nonnegative sectional curvature metrics on [*open*]{} manifolds was inititated in the paper \[KPT\], with follow-ups in particular in the works \[BFK\], \[BH\] and \[BKS\].
Using \[BKS, Proposition 2.8\] one obtains by means of stabilizing the manifolds in Theorem \[main theorem\] with $\R$, in each dimension $4n$, where $n\ge 2$, many new examples of simply connected open manifolds whose moduli spaces of nonnegative sectional curvature metrics have infinitely many path components. The following result generalizes Corollary 1.11 in \[BKS\] to an infinite number of dimensions.
\[Corollary\] For each of the manifolds $M$ in Theorem \[main theorem\], the moduli space of complete Riemannian metrics with nonnegative sectional curvature on $M\times\R$ has infinitely many path components.
The remaining parts of the present note are structured as follows: Section $2$ contains the relevant preliminaries needed for the proofs of Theorem \[main theorem\] and the corollary and the latter are presented in Section \[proof\].
We gratefully acknowledge the support and hospitality of CIRM at Luminy where a major part of this work was carried out during a research stay within the “Research in Pairs" program in March 2015. Moreover, it is our pleasure to thank Igor Belegradek and David Wraith for several helpful comments and the referee for useful suggestions concerning the exposition of the paper.
Preliminaries
=============
Spaces and moduli spaces of metrics {#moduli spaces}
-----------------------------------
Let $M$ be an $n$-dimensional smooth manifold (without boundary), let $S^2 \, T^*M$ denote the second symmetric power of the cotangent bundle of $M$, and let $C^\infty (M,S^2 \, T^*M)$ be the real vector space of smooth symmetric $(0,2)$ tensor fields on $M$. We always topologize $C^\infty (M,S^2 \, T^*M)$ and its subsets with the smooth topology of uniform convergence on compact subsets (compare, e.g., \[KrMi\], \[Ru\]). If $M$ is compact, then $C^\infty (M,S^2 \, T^*M)$ is a Fréchet space.
The [*space ${\cal R}(M)$ of all (complete) Riemannian metrics on $M$*]{} is the subspace of $C^\infty (M,S^2 \, T^*M)$ consisting of all sections which are complete Riemannian metrics on $M$. Notice that ${\cal R}(M)$ is a convex cone in $C^\infty (M,S^2 \, T^*M)$, i.e., if $a, b > 0$ and $g_1, g_2 \in {\cal R}(M)$, then $ag_1 + bg_2 \in {\cal R}(M)$. In particular, ${\cal R}(M)$ is contractible and any open subset of ${\cal R}(M)$ is locally path-connected.
Let us now define moduli spaces of Riemannian metrics. If $M$ is a finite-dimensional smooth manifold, let ${\rm Diff}(M)$ be the group of self-diffeomorphisms of $M$ (which is a Fréchet Lie group if $M$ is compact). Then ${\rm Diff}(M)$ acts on ${\cal R}(M)$ by pulling back metrics, i.e., one has the action $${\rm Diff}(M) \times {\cal R}(M) \to {\cal R}(M), \quad (g,\phi)\mapsto \phi^*(g) \, .$$
The [*moduli space*]{} $${\cal M}(M):={\cal R}(M)/\hbox{Diff}(M)$$ of (complete) Riemannian metrics on $M$ is the quotient space of ${\cal R}(M)$ by the above action of the diffeomorphism group ${\rm Diff}(M)$, equipped with the quotient topology.
Notice that usually ${\rm Diff}(M)$ will not act freely on ${\cal R}(M)$. Moreover, due to the fact that different Riemannian metrics may have isometry groups of different dimension, the moduli space ${\cal M}(M)$ will in general not have any kind of manifold structure. Note that ${\cal M}(M)$ is locally path-connected since it is the quotient of the locally path-connected space ${\cal R}(M)$. Thus there is no difference between connected and path connected components of open subsets of ${\cal M}(M)$. Notice also that a lower bound on the number of components in a moduli space is also a lower bound on the number of components for the respective space of metrics.
One can similarly form spaces and moduli spaces of metrics satisfying various curvature conditions, as these conditions are invariant under the action of $\hbox{Diff}(M)$. We will here employ the following notation (and always tacitly assume that our metrics are complete): The space of all metrics with positive scalar curvature on $M$ shall be denoted ${\cal R}_{scal >0}(M)$. The corresponding spaces of positive Ricci and nonnegative sectional curvature will be respectively denoted ${\cal R}_{Ric >0}(M)$ and ${\cal R}_{sec \ge 0}(M)$. The respective moduli spaces will be denoted as ${\cal M}_{scal >0}(M):={\cal R}_{scal >0}(M)/\hbox{Diff}(M)$, ${\cal M}_{Ric >0}(M):={\cal R}_{Ric >0}(M)/\hbox{Diff}(M)$, etc.
Following \[KS\] we will consider in Section \[proof\] closed smooth simply connected manifolds $M$ for which the absolute $s$-invariant is well-defined and constant on path-connected components of the moduli space ${\cal M}_{scal >0}(M)$. Note that in this situation the invariant is also constant on path components of the moduli space ${\cal M}_{Ric >0}(M)$. In other words, two metrics $g_0, g_1\in {\cal R}_{Ric >0}(M)$ with $\vert s\vert (g_0)\neq \vert s\vert (g_1)$ belong to different path components of ${\cal M}_{Ric >0}(M)$.
If, in addition, $g_0, g_1$ are metrics of nonnegative sectional curvature then they also belong to different path components of the moduli space ${\cal M}_{sec\geq 0}(M)$. An elegant way to see this (compare also \[BKS, Prop. 2.7\]) uses the Ricci flow: Suppose that $\gamma $ is a path in ${\cal M}_{sec\geq 0}(M)$ with end points represented by $g_0$ and $g_1$. Consider the Ricci flow on ${\cal R}(M)$. Since the Ricci flow is invariant under diffeomorphisms it descends to a local flow on the moduli space. As shown by Böhm and Wilking (\[BW\]), $\gamma$ evolves under this flow instantly to a path in ${\cal M}_{Ric >0}(M)$. Concatenation of the evolved path and the trajectories of the end points of $\gamma$ then yields a path in ${\cal M}_{Ric >0}(M)$ connecting the end points of $\gamma $, thereby contradicting $\vert s\vert (g_0)\neq \vert s\vert (g_1)$.
The Atiyah-Patodi-Singer Index Theorem {#APS-index theorem}
--------------------------------------
An essential ingredient for the definition of the Kreck-Stolz invariant is the index theorem of Atiyah, Patodi and Singer for manifolds with boundary. In this section we briefly recall the index theorem for the Dirac and the signature operator. For more details and the general discussion we refer to \[APS1\],\[APS2\].
Let $W$ be a $4m$-dimensional compact spin manifold with boundary $\partial W = M$ and let $g_W$ be a Riemannian metric on $W$ which is a product metric near $M$. Let $g_M$ denote the induced metric on $M$. Then one can define the Dirac operator $D^+(W,g_W):C^\infty (W,S^+)\to C^\infty (W,S^-)$, where $S^{\pm }$ are the half-spinor bundles. This operator is Fredholm after imposing the APS boundary conditions, i.e. after restricting to spinors $\varphi$ on $W$ for which the restriction to $M$ is in the kernel of the orthogonal projection $P$ onto the space spanned by eigenfunctions for nonnegative eigenvalues \[APS1, p. 55\]. Its index will be denoted by $\mathrm{ind} \; D^+(W,g_W)$.
In general, the projection $P$ and the APS boundary condition do not depend continuously on the metric due to potential zero eigenvalues of the Dirac operator on the boundary. Hence, the index $\mathrm{ind}\; D^+(W,g_W)\in \Z$ may jump under variations of $g_W$. However, if $g_W(t)$ is a path of metrics as above such that the induced metrics $g_M(t)$ on $M$ have [*positive*]{} scalar curvature for all $t$ then, by Lichnerowicz’s argument, the APS-boundary condition does depend continuously on $t$ and $\mathrm{ind}\; D^+(W,g_W(t))$ is constant in $t$ (see \[APSII, p. 417\]).
Taking a slightly different point of view, suppose that $(M,g_M)$ is a $(4m-1)$-dimensional spin manifold of positive scalar curvature which bounds a spin manifold $W$. Let $g_W$ be any extension of $g_M$ to $W$ which is a product metric near the boundary. From the above it follows that $\mathrm{ind}\; D^+(W,g_W)$ is independent of the chosen extension and only depends on the bordism $W$ and the path-component of $g_M$ in the space of metrics on $M$ of positive scalar curvature $\riemmetric {M}{scal>0}$. Moreover the index vanishes if $g_W$ can be chosen to be of positive scalar curvature (see also \[KS, Rem. 2.2\]).
The index $\mathrm{ind} \; D^+(W,g_W)$ can be computed with the index theorem of Atiyah, Patodi and Singer \[APS1, Th. 4.2\]. If $(M,g_M)$ has positive scalar curvature one obtains $$\label{APS Dirac}\mathrm{ind}\; D^+(W,g_W)=\int _W {\cal \hat A}(p_1(W,g_W),\ldots ,p_m(W,g_W))-\frac {\eta (D(M,g_M))}2,$$ where $p_i(W,g_W)$ are the Pontrjagin forms of $(W,g_W)$, $\cal \hat A$ is the multiplicative sequence for the $\hat A$-genus, $D(M,g_M)$ is the Dirac operator on $M$ and $\eta(D(M,g_M))$ is its $\eta$-invariant.
Next we recall the signature theorem for manifolds with boundary. Let $W$ be a $4m$-dimensional compact oriented manifold with boundary $\partial W = M$ and let $g_W$ be a Riemannian metric on $W$ which is a product metric near $M$. Let $g_M$ denote the induced metric on $M$. Then one can consider the signature operator on $W$ which becomes a Fredholm operator after imposing the APS-boundary conditions. Applying their index theorem to this operator Atiyah, Patodi and Singer proved the following signature theorem \[APS1, Th. 4.14\]:
$$\label{APS signature}\mathrm{sign}\; W = \int _W {\cal L}(p_i(W,g_W))-\eta (B(M,g_M)).$$
Here ${\cal L}$ is the multiplicative sequence for the $L$-genus and $\eta(B(M,g_M))$ is the $\eta$-invariant of the signature operator on the boundary. Recall that the signature of $W$, $\mathrm{sign}\; W $, is by definition the signature of the (non-degenerate) quadratic form defined by the cup product on the image of $H^*(W,M)$ in $H^*(W)$. In particular, the right hand side of equation (\[APS signature\]) is a purely topological invariant.
The Kreck-Stolz $s$-invariant {#s invariant}
-----------------------------
The index of the Dirac operator, the signature and the integrals involving multiplicative sequences considered in the previous subsection depend on the choice of the bordism. However, under favorable circumstances one can combine the data to define a non-trivial invariant of the boundary itself (cf. \[EK\], \[KS\]). In this section we give a brief introduction to the so-called $s$-invariant which was used by Kreck and Stolz in their study of moduli space of metrics of positive scalar curvature.
A starting point in \[EK\] and \[KS\] is to consider a certain linear combination ${\cal \hat A} + a_m\cdot \cal L$ of the multiplicative sequences for the Dirac and signature operator. Recall that the degree $\leq 4m$-part of a multiplicative sequence is a polynomial in the variables $p_1,\ldots ,p_m$, where the $p_i$’s are variables which may be thought of as universal rational Pontrjagin classes. By chosing $a_m=1/ (2^{2m+1}\cdot (2^{2m-1}-1))$ in the linear combination above one obtains in degrees $\leq 4m$, a polynomial $N_m(p_1,\ldots ,p_{m-1})$ not involving $p_m$ (cf. also \[Hir\]).
Now suppose that $W$ is a $4m$-dimensional spin manifold with boundary $M$. In \[EK\] Eells and Kuiper considered the situation where the real Pontrjagin classes $p_i(W)$, $i<m$, can be lifted uniquely to $H^*(W,M;\R)$ and the natural homomorphism $H^1(W;\Z /2)\to H^1(M;\Z /2)$ is surjective. This is, for example, the case if $M$ is simply connected with $b_{2m-1}(M)=0$ and $b_{4i-1}(M)=0$, where $0<i<m$. In this situation Eells and Kuiper showed that the modulo $1$ reduction of $$\label{EK invariant}\langle j^{-1}(N_m(p_1(W),\ldots ,p_{m-1}(W))),[W,M]\rangle- a_m\cdot \mathrm{sign}\; (W)$$ (or half of it if $m$ is odd) is independent of the choice of $W$ and can be used to detect different smooth structures of $M$. Here $j:H^*(W,M;\R)\to H^*(W;\R )$ is the natural homomorphism induced by inclusion and $\langle \quad ,[W,M]\rangle$ is the Kronecker product with the fundamental homology class.
In \[KS\] Kreck and Stolz used the APS-index theorem to define an invariant, the so-called $s$-invariant, which refines the Eells-Kuiper invariant. Its relevance for the study of moduli spaces is summarized in Proposition \[s moduli proposition\] below.
Suppose, as in the last subsection, that $g_W$ is a metric of $W$ which is a product metric near the boundary and let $g_M$ denote the induced metric on $M$. The integral $\int_W N_m(p_1(W,g_W),\ldots ,p_{m-1}(W,g_W))$ can be computed via the APS-index theorem. Suppose $(M,g_M)$ has positive scalar curvature. Then, using equations (\[APS Dirac\]) and (\[APS signature\]), one obtains $${\mathrm{ind}\; D^+(W,g_W)+a_m\; \mathrm{sign}\; W =\int _W N_m(p_i(W,g_W))-\frac {\eta (D(M,g_M))}2 -a_m \eta (B(M,g_M)).}$$Under favorable circumstances, e.g., if all real Pontrjagin classes of $M$ vanish, Kreck and Stolz show that the integral $\int _W N_m(p_1(W,g_W),\ldots ,p_{m-1}(W,g_W))$ is equal to $$\int _M d^{-1}(N_m(p_i(M,g_M))) + \langle j^{-1}(N_m(p_i(W)),[W,M]\rangle,$$ where $d^{-1}(N_m(p_i(M,g_M)))$ is a term which only depends on $(M,g_M)$, i.e., is independent of the choice of the bordism $W$ and the metric $g_W$ (see \[KS, p. 829). Collecting the summands involving $(M,g_M)$ one obtains the $s$-invariant of Kreck and Stolz $$\label{s invariant formula} s(M,g_M):= -\frac {\eta (D(M,g_M))}2 -a_m \eta (B(M,g_M)) + \int _M d^{-1}(N_m(p_i(M,g_M)))$$
Let $t(W):=-(\langle j^{-1}(N_m(p_i(W))),[W,M]\rangle -a_m\cdot \mathrm{sign}\; W)$. Note that the rational number $t(W)$ only depends on the topology of $W$ and reduces modulo $1$ to the negative of the invariant considered by Eells and Kuiper (see equation (\[EK invariant\])). Now, still asuming that $(M,g_M)$ has positive scalar curvature equations (\[APS Dirac\]) and (\[APS signature\]) give $$\label{s equation}s(M,g_M)= \mathrm{ind}\; D^+(W,g_W) +t(W).$$
Note that the right hand side of (\[s equation\]) only depends on the bordism $W$ and on the path component of $g_M$ in the space $\riemmetric{M}{scal>0}$ of metrics on $M$ of positive scalar curvature, whereas the left hand side only depends on $(M,g_M)$, i.e. is independent of the chosen bordism. Hence, both sides depend only on the path component of $g_M$ in $\riemmetric{M}{scal>0}$. If $(W,g_W)$ has positive scalar curvature as well then, by Lichnerowicz’s argument, $\mathrm{ind}\; D^+(W,g_W)$ vanishes and $s(M,g_M)$ provides a refinement of the Eells-Kuiper invariant (for all of this see \[KS, Section 2\]).
If $H^1(M;\Z / 2)=0$, then the spin structure of $M$ is uniquely determined up to isomorphism by the orientation. In this case the absolute $s$-invariant does not change under the action of the diffeomorphism group $\diff{M}$ on $\riemmetric{M}{scal>0}$. In summary, one has
\[s moduli proposition\] If $M$ is a closed connected spin manifold of dimension $4m-1$ with vanishing real Pontrjagin classes and $H^1(M;\Z /2)=0$, then $s$ induces a map $$\vert s\vert :\pi _0(\riemmetric{M}{scal>0}/\diff {M})\to \Q.$$
Kreck and Stolz also derive an explicit formula for $s(M,g_M)$ in the case where $M$ is the total space of an $S^1$-principal bundle. We will give this formula in the next section in a particular case.
Circle bundles over products of projective spaces {#manifolds subsection}
-------------------------------------------------
This subsection describes the Riemannian manifolds which will be used in the proof of Theorem \[main theorem\]. As in \[KS\] we will consider $S^1$-principal bundles over the product of two complex projective spaces. The total spaces of these bundles have been studied by Wang and Ziller \[WZ\] in their work about Einstein metrics on principal torus bundles, and we will employ some of their cohomological computations.
Let $x$ (resp. $y$) be the positive generators of the integral cohomology ring of $\C P^{2n}$ (resp. $\C P^1$). Let $M_{k,l}$ be the total space of the $S^1$-principal bundle $P$ over $B:=\C P^{2n}\times \C P^1$ with Euler class $c:=lx +ky$, where $k,l$ are coprime positive integers.[^2] We summarize the relevant topological properties of the manifolds $M_{k,l}$ (see also \[WZ, Prop. 2.1 and Prop. 2.3\]).
\[topology prop\]
1. The spaces $M_{k,l}$ are closed smooth simply connected $(4n+3)$-dimensional manifolds.
2. The integral cohomology ring of $M_{k,l}$ is given by $$H^*(M_{k,l};\Z )\cong \Z[u,v]/((lv)^{2}, v^{2n+1},uv^2,u^2)$$ where $\deg u=4n+1$ and $\deg v=2$. In particular, $H^*(M_{k,l};\Z )$ does not depend on $k$.
3. The rational cohomology ring of $M_{k,l}$ is isomorphic to the one of $S^{4n+1}\times \C P^1$ and all rational Pontrjagin classes of $M_{k,l}$ vanish.
4. The manifolds $M_{k,l}$ are all formal.
5. For fixed $l$, the manifolds $M_{k,l}$ fall into finitely many diffeomorphism types.
[**Proof:**]{} The first assertion follows from $k$ and $l$ being coprime. The second one can be verified using a spectral sequence argument (cf. \[WZ, Prop. 2.1\]). The third statement is a direct consequence of the second. The fourth statement holds since $H^*(S^{4n+1}\times \C P^1;\Q )$ is intrinsically formal, which follows by direct computation. The last statement is a consequence of results from Sullivan’s surgery theory, compare (\[Su\], Theorem 13.1 and the proof of Theorem 12.5\]). These imply that if a collection of simply connected closed smooth manifolds of dimension $\ge 5$ all have isomorphic integral cohomology rings, the same rational Pontrjagin classes, and if their minimal model is a formal consequence of their rational cohomology ring, then there are only finitely many diffeomorphism types among them.
Let $L$ denote the complex line bundle associated to the $S^1$-principal bundle $P\to B$ and let $W=D(L)$ be the total space of the disk bundle with boundary $\partial W=S(L)=M_{k,l}$. Note that the tangent bundle of $W$ is isomorphic to $\pi ^*(TB\oplus L)$, where $\pi :W\to B$ is the projection. Hence, the total Stiefel-Whitney class of $W$ is equal to $$\label{SW for W}w(W)\equiv \pi ^*((1+x)^{2n+1}\cdot (1+y)^2\cdot (1+l x +k y))\bmod 2.$$ Since the restriction of $\pi ^*(L)$ to $M_{k,l}$ is trivial, the tangent bundle of $M_{k,l}$ is stably isomorphic to $\pi ^*(TB)$.
The classes $x$ and $y$ of $H^2(B;\Z )$ pull back under $P\to B$ to $-kv$ and $lv$, respectively (cf. \[WZ\]). Hence, $$\label{SW for P}w(M_{k,l})=(1+lv)^2\cdot (1-kv)^{2n+1}.$$ Restricting formulas (\[SW for W\]) and (\[SW for P\]) to $w_2$ and recalling that $M_{k,l}$ and $W$ are simply connected, one obtains the following
$M_{k,l}$ is spin if and only if $k$ is even. In this case $l$ is odd because of $(k,l)=1$, and, for a fixed orientation, $M_{k,l}$ as well as $W$ admit a unique spin structure.
From now on we will assume that $k$ is even (and thus $l$ is odd). We equip $W$ with the orientation induced from the standard orientation of $B$ and the complex structure of $L$. From the lemma above we see that $M_{k,l}$ and $W$ admit unique spin structures and that $W$ is a spin bordism for $M_{k,l}$.
Recall from equation (\[s equation\]) that the computation of the $s$-invariant of $M_{k,l}$ involves the topological term $t(W):=-(\langle j^{-1}(N_{n+1}(p_i(W)),[W,M]\rangle -a_{n+1}\cdot \mathrm{sign}\; W)$. This term has already been computed by Kreck and Stolz. For the spin bordism $W=D(L)$ as above we thus obtain
\[t(W) lemma\]$$t(W)=-\left\langle \frac 1 c \cdot \left( {\cal \hat A}(TB)\cdot \frac {c/2}{\sinh c/2}+a_{n+1}\cdot {\cal L}(TB)\cdot \frac {c}{\tanh c}\right ),[B]\right \rangle$$
[**Proof:**]{} By \[KS, Lemma 4.2, part 2\] $$t(W)=-\left\langle {\cal \hat A}(TB)\cdot \frac {1}{2\sinh c/2}+a_{n+1}\cdot {\cal L}(TB)\cdot \frac {1}{\tanh c},[B]\right \rangle +a_{n+1}\cdot \mathrm{sign} (B_c).$$
Here ${\cal \hat A}$ and ${\cal L}$ are the polynomials in the Pontrjagin classes associated to the $\hat A$- and $L$-genus, $a_{n+1}:=1/(2^{2n+3}\cdot (2^{2n+1}-1))$ and $\mathrm{sign} (B_c)$ is the signature of the symmetric bilinear form $$\label{bilinear form}H^{2n}(B)\otimes H^{2n}(B)\to \Q ,\quad (u,v)\mapsto \langle u\cdot v\cdot c, [B]\rangle .$$ It remains to show that the signature term $\mathrm{sign} (B_c)$ vanishes. We note that the bilinear form (\[bilinear form\]) is represented with respect to the basis $(x^n,x^{n-1}y)$ by the matrix $\left (\begin{smallmatrix}
k & l\\ l & 0\end{smallmatrix}\right )$. Hence, $\mathrm{sign} (B_c)$ vanishes for $l\neq 0$.
Proofs of Theorem \[main theorem\] and Corollary \[Corollary\] {#proof}
==============================================================
In this section we prove our main theorem and the corollary. Using the manifolds $M_{k,l}$ from above, we will employ the absolute Kreck-Stolz invariant $\vert s\vert $ to differentiate between the path components of the moduli spaces of nonnegative sectional curvature as well as positive Ricci curvature metrics.
We will first show that each $M_{k,l}$ admits a metric $g_M=g_{k,l}$ of nonnegative sectional curvature and positive Ricci curvature which is connected in ${\cal M}_{scal >0}(M)$ to a metric that extends to a metric $g_W$ of positive scalar curvature on the associated disk bundle $W$ such that $g_W$ is a product metric near the boundary. This will imply that the $s$-invariant of $(M_{k,l},g_M)$ is equal to the topological term $t(W)$ given in Lemma \[t(W) lemma\] (see equation (\[s equation\])).
Next we consider, for fixed $n$, a certain infinite sequence of $S^1$-principal bundles $P_m\to \C P^{2n}\times \C P^1$ with diffeomorphic total spaces, where each total space $P_m$ can also be described as the quotient of $S^{4n+1}\times S^3$, equipped with the standard nonnegatively curved product metric, by the action of an isometric free $S^1$-action. Notice also that, in fact, there is a free and isometric $T^2=S^1 \times S^1$ - action on $S^{4n+1}\times S^3$ which is given by the product of the circle Hopf actions on the respective sphere factors, and all $S^1$-actions that yield the $P_m$ are subactions of this fixed one.
The induced metric $g_m$ on $P_m$ has nonnegative sectional and positive Ricci curvature by the O’Neill formulas, and an upper sectional curvature bound which is independent of $m$. We show that the absolute $s$-invariants of the $g_m$ are pairwise different (for $n=1$ this was already shown by Kreck and Stolz) and, hence, belong to different connected components of the moduli space of positive scalar (and positive Ricci) curvature metrics. As explained above, this implies that the metrics $g_m$ also belong to different connected components of the moduli space of nonnegative sectional curvature metrics (see Subsection \[moduli spaces\]).
Let us now discuss the metrics on $M_{k,l}$ and its associated disk bundle $W$.
Each manifold $M:=M_{k,l}$ admits a metric $g_M$ of simultaneously nonnegative sectional curvature and positive Ricci curvature which is connected in ${\cal M}_{scal >0}(M)$ to a metric that extends to a metric $g_W$ of positive scalar curvature on the associated disk bundle $W$ such that $g_W$ is a product metric near the boundary.
[**Proof:**]{} Consider the circle Hopf action on odd dimensional spheres $S^{2r+1}\subset \C^ {r+1}$ given by multiplication of unit complex numbers from the right. This gives rise to an isometric and free action from the right of a two-dimensional torus $T$ on any product of round unit spheres. Here we consider the $T$-action on $S^{4n+1}\times S^3$ with quotient $\C P^{2n}\times \C P^1$. We note that the $T$-orbits are products of geodesics and, hence, totally geodesic flat tori in $S^{4n+1}\times S^3$.
The manifold $M_{k,l}$ can be described as the quotient of $S^{4n+1}\times S^3$ by a certain circle subaction of $T$ (the subaction depends on $(k,l)$). Let $g_M$ be the submersion metric with respect to this action. Then $(M_{k,l}, g_M)$ has nonnegative sectional and positive Ricci curvature by the O’Neill formulas. The (non-effective) action of $T$ on $M_{k,l}$ gives rise also to the $S^1$-principal bundle $M_{k,l}\to M_{k,l}/T=\C P^{2n}\times \C P^1$. When we equip the base of this fibration with the submersion metric, it is not difficult to see that this Riemannian submersion has totally geodesic fibers. Now, after shrinking our metric along the fibers if necessary, following the arguments in \[KS, Section 4\], we see that the metric can be extended to a metric $g_W$ of positive scalar curvature on the associated disk bundle $W$ such that $g_W$ is a product metric near the boundary.
Thus, combining the proposition above with Lemma \[t(W) lemma\] and equation (\[s equation\]), it follows that $s(M_{k,l},g_M)$ is equal to $$-\left\langle \frac 1 c \cdot \left( {\cal \hat A}(TB)\cdot \frac {c/2}{\sinh c/2}+a_{n+1}\cdot {\cal L}(TB)\cdot \frac {c}{\tanh c}\right ),[B]\right \rangle .$$ Let us now compute the invariant $s(k,l):=s(M_{k,l},g_M)$. Recall that $M_{k,l}$ is the total space of the $S^1$-principal bundle over $B=\C P^{2n}\times \C P^1$ with Euler class $c:=lx +k y$ where $x$ and $y$ are the positive generators of the integral cohomology rings of $\C P^{2n}$ and $\C P^1$, respectively, $k,l$ are coprime positive integers, $k$ is even and $l$ is odd.
The expression $s(k,l)/k := s(M^{4n+3}_{k,l},g_M)/k$ is a Laurent polynomial in $l$ of degree $2n$.
[**Proof:**]{} Since the total Pontrjagin class of $B=\C P^{2n}\times \C P^1$ is given by $p(B)=(1+x^2)^{2n+1}$, we see that $s(k,l)$ is equal to $$\label{s polynomial}-\left \langle \frac 1 c \left(\left(\frac {x/2}{\sinh x/2}\right)^{2n+1}\cdot \frac {c/2}{\sinh c/2}+a_{n+1}\left( \frac {x}{\tanh x}\right )^{2n+1}\cdot \frac {c}{\tanh c}\right ),[B]\right \rangle .$$ Here $$\frac {t/2}{\sinh t/2}=1+\sum_{j\geq 1}(-1)^{j}\frac {(2^{2j-1}-1)}{(2j)!\cdot 2^{2j-1}}B_j\cdot t^{2j} =: 1 +\sum _{j\geq 1}\hat a_{2j}\cdot t^{2j}$$ and $$\frac {t}{\tanh t}=1+\sum _{j\geq 1}(-1)^{j-1}\frac {2^{2j}}{(2j)!}B_j\cdot t^{2j}=:1 +\sum _{j\geq 1}b_{2j}\cdot t^{2j}$$ are the characteristic power series associated to the $\hat A$- and $L$-genus (cf. \[Hir\], §1.5, \[MS\], Appendix B). The $B_j$ are the Bernoulli numbers, $B_1=1/6$, $B_2=1/30$, $B_3=1/42$, $B_4=1/30$, $B_5=5/66$,…, and are all non-zero. In particular, $\hat a_{2j}=\frac {1-2^{2j-1}}{2^{4j-1}}\cdot b_{2j}$.
Note that $x^2$ is equal to $c\cdot \frac {lx-ky}{l^2}$ and, hence, divisible by $c$. To evaluate the expression in (\[s polynomial\]) on the fundamental cycle $[B]$, one first divides out in the inner expression the terms of cohomological degree $4n+4$ by $c$ and then computes the coefficients of $x^{2n}\cdot y$. We claim that the result is $k$ times a Laurent polynomial in $l$ of degree $2n$. To see this we rewrite the formula for $s(k,l)$ in the following form:
$$-s(k,l)=\left \langle \frac 1 c \left(\left(\frac {x/2}{\sinh x/2}\right)^{2n+1}+a_{n+1}\left( \frac {x}{\tanh x}\right )^{2n+1}\right ),[B]\right \rangle$$ $$+\left \langle \left(\frac {x/2}{\sinh x/2}\right)^{2n+1}\cdot \left (\sum _{j\geq 1}\hat a_{2j} c^{2j-1}\right ) +a_{n+1}\left( \frac {x}{\tanh x}\right )^{2n+1}\cdot \left (\sum _{j\geq 1}b_{2j} c^{2j-1}\right ),[B]\right \rangle$$
Using $x^2/c=\frac {lx-ky}{l^2}$ we see that the first summand is a rational multiple of $k\cdot l^{-2}$ and the second summand is of the form $k\cdot p(l)$, where $p(l)$ is a polynomial in $l$ of degree $\leq 2n$. The term of degree $2n$ in $k\cdot p(l)$ is equal to $$\left \langle \hat a_{2n+2}\cdot c^{2n+1}+a_{n+1}\cdot b_{2n+2}\cdot c^{2n+1},[B]\right \rangle=(2n+1)\cdot k\cdot ( \hat a_{2n+2}+a_{n+1}\cdot b_{2n+2})\cdot l^{2n}$$
Using $a_{n+1}:=1/(2^{2n+3}\cdot (2^{2n+1}-1))$ and $\hat a_{2j}=(1-2^{2j-1})/(2^{4j-1})\cdot b_{2j}$, it then follows that the coefficient of $l^{2n}$ in $p(l)$ is, indeed, non-zero.
[**Proof of Theorem \[main theorem\]:**]{} We fix a positive odd integer $l_0$ for which the Laurent polynomial $p(l):=s(k,l)/k$ does not vanish. Note that there are infinitely many of such integers. Notice also that $s(k,l_0)=k\cdot p(l_0)$ takes pairwise different absolute values for positive even integers $k$. It follows from Sullivan’s surgery theory (see Proposition \[topology prop\]) that there exists an infinite sequence of such integers $(k_m)_m$ with diffeomorphic $M_{k_m,l}$. As before let $P_m:=M_{k_m,l}$ be equipped with the submersion metric.
Recall that $P_m$ has nonnegative sectional and positive Ricci curvature. Since the $P_m$’s have pairwise different absolute $s$-invariant, they belong to pairwise different connected components of the moduli spaces ${\cal M}_{sec \geq 0}(M)$ and ${\cal M}_{Ric >0}(M)$ (see Subsections \[moduli spaces\] and \[s invariant\]).
From the classification results of Section $2$ we conclude that there exists, in each relevant dimension, an infinite sequence of manifolds of pairwise distinct homotopy type which satisfy these properties.
[**Proof of Corollary \[Corollary\]:**]{} Consider any manifold $M$ as in Theorem \[main theorem\]. From the results in Section 2.4 we know that there is an infinite sequence of nonnegatively curved Riemannian manifolds $(M_k,g_k)$ that are all diffeomorphic to $M$ but have pairwise distinct absolute $s$-invariants. For each $k$ we fix a diffeomorphism $\varphi_k:M\to M_k$ and denote by $\overline g_k$ the pull-back metric $\varphi_k ^*(g_k)$ on $M$. We obtain an infinite sequence of nonnegatively curved metrics $(\overline g_k)_k$ on $M$ which represent pairwise different path components of ${\cal M}_{sec \ge0}(M)$.
Now crossing with $(\R, dt^2)$ gives us nonnegatively curved complete metrics $\overline g_k \times dt^2$ on the simply connected manifold $M\times \R$ with souls $(M, \overline g_k)$ belonging to different path components of ${\cal M}_{sec \ge0}(M)$. By \[BKS, Prop. 2.8\] the moduli space of complete Riemannian metrics with nonnegative sectional curvature on $M\times\R$ is homeomorphic to the disjoint union of the moduli spaces of nonnegative sectional curvature metrics of all possible pairwise nondiffeomorphic souls of metrics in ${\cal M}_{sec \ge0}(M\times\R)$.
In particular, the Riemannian manifolds $(M, \overline g_k)\times (\R,dt^2)$ belong to pairwise different path components of the moduli space of complete Riemannian metrics with nonnegative sectional curvature on $M\times\R$.
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[Email:]{} [[email protected], [email protected], [email protected]]{}
Mathematisches Forschungsinstitut Oberwolfach (MFO), Schwarzwaldstr. 9-11,\
D-77709 Oberwolfach-Walke, Germany
Département de Mathématiques, Chemin du Musée 23, Faculté des sciences, Université de Fribourg, Pérolles, CH-1700 Fribourg, Switzerland
Karlsruher Institut für Technologie (KIT), Fakultät für Mathematik, Institut für Algebra und Geometrie, Arbeitsgruppe Differentialgeometrie, Englerstr. 2,\
D-76131 Karlsruhe, Germany
[^1]: The first author acknowledges support by SNF grant $200020\_149761$
[^2]: We follow here the notation of Kreck and Stolz. In the notation of Wang and Ziller \[WZ\] $M_{k,l}$ is denoted by $M^{2n,1}_{l,k}\cong M^{1,2n}_{k,l}$.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We reveal how the vector field links are untied under the influence of anti-parity-time-symmetric couplings in a dissipative sublattice-symmetric topological photonic crystal lattice. The topology of the quasi-one-dimensional two-band system is encoded in the geometric topology of the vector fields associated with the Bloch Hamiltonian. The linked vector fields reflect the topology of the nontrivial phase. The topological phase transition occurs concomitantly with the untying of the vector field link at the exceptional points. Counterintuitively, more dissipation constructively creates a nontrivial topology. The linking number predicts the number of topological photonic zero modes.'
author:
- 'H. C. Wu'
- 'X. M. Yang'
- 'L. Jin'
- 'Z. Song'
title: 'Untying links through anti-parity-time-symmetric coupling'
---
*Introduction.*—The monopole, skyrmion, and vortex are robust topological structures that have attracted much research interests in optics, quantum physics, and condensed matter physics. The zero-intensity lines of light field, in the form of isolated knotted nodal lines, are observed as optical vortices in experiments [@Dennis10; @IrvinePRL13]. The creation, observation, and investigation on the static and dynamical properties of quantum knots in Bose-Einstein condensates are reported [Ueda08,Hall16,Hall19]{}. Topological phases are extremely stable due to topological protection, even in non-Hermitian topological systems [Hasan,XLQi,YFChen,DWZhang,Cooper,Ozawa,YXu,LonghiPRBSR,Hirsbrunner,SLZhu20,DWZhang20A,CHLee20]{}. In topological systems, zero-energy Fermi surfaces can form knotted or linked nodal lines [SCZhang,JMHou,GChang,Ezawa,ZWang,ZWang2,Ackerman,DLD,Carlstrom,Bergholtz,JPHu19L]{}. Alternatively, the fictitious magnetic field of a topological system with topological defects, associated with vortex or anti-vortex textures, reflects nontrivial topology [@SLin; @PXue19]. The fictitious magnetic field is a geometrical analogy of the magnetic field. A nontrivial topology can be extracted from the spin polarization, which is associated with a two-band topological system and defines a vector field. The vector field exhibits a link or knot topology in the topologically nontrivial phase. We visualize the vortex links, present alterations to the vector field topology under the influence of anti-parity-time-symmetric (anti-$\mathcal{PT}$-symmetric) couplings, and reveal the untying of vector field links associated with a topological phase transition. Counterintuitively, we demonstrate that more dissipation constructively creates nontrivial topology.
*Dissipative photonic crystal lattice.—*Photonic crystal is an excellent platform for the study of topological physics [@Hafezi]. The non-Hermitian quasicrystal [@Longhi19L; @YXu20B], high-order topological phase [@XLuo19; @TLiu19; @CHLeePRL19; @Ezawa19], robust edge state [Esaki,Schomerus,JLPRA,Weimann,MPan,Poli,ZKL1905,XDZhang]{}, topological lasing [Feng,Conti,LonghiAnnPhys,Segev,Khajavikhan,Skryabin,Carusotto,Jean,Parto,Iwamoto]{} in photonic crystals were discovered. Here, we consider a photonic crystal lattice of coupled resonators, as schematically illustrated in Fig. [fig1]{}(a). The colored resonators are the primary resonators, and the white and gray resonators are the auxiliary resonators, evanescently coupled to the primary resonators. The primary resonators have an identical resonant frequency, $\omega _{c}$. The auxiliary resonators induce the effective couplings between primary resonators. The blue, red, and white resonators are passive; the gray resonators have dissipation. The dissipation rate is much larger than the coupling strength between the gray auxiliary resonators and the primary resonators. The large dissipation enables the adiabatical elimination of the auxiliary resonator light field and results in a reciprocal anti-$\mathcal{PT}$-symmetric coupling $-i\gamma $ between two primary resonators; in the on-resonance condition, the strength $\gamma $ equals the product of two evanescent couplings between the gray auxiliary resonator and its two adjacent primary resonators divided by the dissipation rate of the gray auxiliary resonator [@LYou]. The other two effective coupling strengths induced by the white auxiliary resonators are denoted as $%
J$ and $\kappa $.
In coupled mode theory [@CMT], the equations of motion for the coupled resonator array are$$\begin{aligned}
i\frac{\mathrm{d}}{\mathrm{d}t}\psi _{A,j} &=&\omega _{c}\psi _{A,j}-i\gamma
\psi _{B,j}+J\psi _{B,j-1}+\kappa \psi _{B,j-2}, \\
i\frac{\mathrm{d}}{\mathrm{d}t}\psi _{B,j} &=&\omega _{c}\psi _{B,j}-i\gamma
\psi _{A,j}+J\psi _{A,j+1}+\kappa \psi _{A,j+2},\end{aligned}$$where $\psi _{A,j}$ and $\psi _{B,j}$ are the wavefunction amplitudes for the corresponding sites $j$ of the sublattices $A$ and $B$. The schematic of the quasi-one-dimensional (quasi-1D) bipartite lattice is shown in Fig. [fig1]{}(b).
Applying the Fourier transformation, the equations of motion for the eigenmode with momentum $k$ reduce to$$\begin{aligned}
i\frac{\mathrm{d}}{\mathrm{d}t}\psi _{A,k} &=&\omega _{c}\psi _{A,k}+\left(
-i\gamma +Je^{-ik}+\kappa e^{-2ik}\right) \psi _{B,k}, \\
i\frac{\mathrm{d}}{\mathrm{d}t}\psi _{B,k} &=&\omega _{c}\psi _{B,k}+\left(
-i\gamma +Je^{ik}+\kappa e^{2ik}\right) \psi _{A,k}.\end{aligned}$$After the common resonant frequency $\omega _{c}$ is removed for both sublattices, the two-band system is described by the Bloch Hamiltonian $%
h_{k}=(\vec{d}-i\vec{\Gamma})\cdot \vec{\sigma}$, where $\vec{\sigma}=\left(
\sigma _{x},\sigma _{y},\sigma _{z}\right) $ is the Pauli matrix for spin-$%
1/2$ and $\vec{d}-i\vec{\Gamma}$ is the fictitious magnetic field including the real part $\vec{d}=\left( J\cos k+\kappa \cos \left( 2k\right) ,J\sin
k+\kappa \sin \left( 2k\right) ,0\right) $ and the imaginary part $\vec{%
\Gamma}=(\gamma ,0,0)$ [@XZhang20L; @Hofmann20]. The two energy bands are $$\varepsilon _{k}^{\pm }=\pm \lbrack (d_{x}-i\gamma )^{2}+d_{y}^{2}]^{-1/2}.$$The corresponding eigenmodes are$$|\psi _{k}^{\pm }\rangle =[1,(d_{x}-i\gamma +id_{y})/\varepsilon _{k}^{\pm
}]^{T}/\mathcal{N},$$which are normalized to unity $\langle \psi _{k}^{\pm }|\psi _{k}^{\pm
}\rangle =1$. The average values of Pauli matrices under the eigenmode yield the spin polarization [@SLin] and define a momentum dependent *unity* 3D vector field $\mathbf{F}_{\pm }\left( k\right) =\left( F_{\pm
,x}\left( k\right) ,F_{\pm ,y}\left( k\right) ,F_{\pm ,z}\left( k\right)
\right) $. All of the three components $F_{\pm ,x,y,z}\left( k\right)
=\langle \psi _{k}^{\pm }|\sigma _{x,y,z}|\psi _{k}^{\pm }\rangle $ are gauge independent periodic functions of the momentum $k$ and can be expressed in the form $$\mathbf{F}_{\pm }\left( k\right) =\left( \sin \theta _{\pm }\cos \phi _{\pm
},\sin \theta _{\pm }\sin \phi _{\pm },\cos \theta _{\pm }\right) .$$
*Linking topology*.*—*In geometry, 1D topology describes 1D closed curves in 3D space described by knot theory [@KnotTheory]. Here, the vector field information is encoded in $\left( \theta _{\pm },\phi _{\pm
}\right) $, and Fig. \[fig2\] shows the vector fields $\mathbf{F}%
_{+}\left( k\right) $ and $\mathbf{F}_{-}\left( k\right) $ in different topological phases; they are mapped onto a torus as two closed curves, and their interrelation provides a visualization of the band topology of $h_{k}$: The linked vector field curves represent the nontrivial topology. In Fig. \[fig2\], the torus has a major radius of $R_{0}$, which is the distance from the center of the tube to the center of the torus. $r_{0}$ is the minor radius of the tube. $k$ is taken as the toroidal direction, and $\mathbf{R}%
_{\pm }(k)=(R_{0}+r_{\pm }\cos \phi _{\pm })\vec{e}_{r}+r_{\pm }\sin \phi
_{\pm }\vec{e}_{z}$; $r_{\pm }=r_{0}\sin \theta _{\pm }$ is selected to ensure that $\mathbf{R}_{\pm }(k)$ forms a closed curve. The curve detaches from the surface of the torus when $\left\vert r_{\pm }\right\vert \neq
r_{0} $. The braiding of the two vector field curves is characterized by the linking number$$L=\frac{1}{4\pi }\oint\nolimits_{k}\oint\nolimits_{k^{\prime }}\frac{\mathbf{%
R}_{+}(k)-\mathbf{R}_{-}(k^{\prime })}{\left\vert \mathbf{R}_{+}(k)-\mathbf{R%
}_{-}(k^{\prime })\right\vert ^{3}}\cdot \lbrack \mathrm{d}\mathbf{R}%
_{+}(k)\times \mathrm{d}\mathbf{R}_{-}(k^{\prime })], \label{L1}$$which is a topological invariant [@Alexander]. Different links are topologically inequivalent and cannot be continuously deformed into one another without a topological phase transition.
In the Hermitian case ($\gamma =0$), the band energy is real and $F_{\pm
,z}\left( k\right) =0$; subsequently, $\theta _{\pm }=\pm \pi /2$ and $%
\left\vert r_{\pm }(k)\right\vert =r_{0}$. Both vector field curves are located on surface of the torus. The orthogonality of eigenmodes in the Hermitian system yields $r_{\pm }(k)=\pm r_{0}$. In the presence of non-Hermiticity ($\gamma \neq 0$), the energy band is complex, and $\theta
_{\pm }\neq \pm \pi /2$. The minor radius for the vector field curve is $%
\left\vert r_{\pm }(k)\right\vert \neq r_{0}$, because of the nonorthogonality of eigenmodes in the non-Hermitian system, and the two vector field curves detach from the torus surface.
In Fig. \[fig2\](a), the two vector field curves form a Solomon’s knot with the linking number $L=-2$. As non-Hermiticity increases, two independent vector field curves meet at the exceptional points (EPs) [Midya,Miri,Kawabata19L,Schomerus18,CTChan18X]{} and form a network. The EPs are the nodes of the network with $\mathbf{F}_{+}\left( k\right) =\mathbf{F}%
_{-}\left( k\right) $ as a consequence of eigenmode coalescence. As non-Hermiticity increases, the curves reform as two independent closed loops; the vector field link changes through the reconnection associated with the EPs, and the linking number of the vector field curves is reduced by one, forming a Hopf link with linking number $L=-1$ \[Fig. \[fig2\](c)\]. This exhibits the untying of the link topology through the anti-$\mathcal{PT}
$-symmetric coupling. After another EP occurs at even greater anti-$\mathcal{%
PT}$-symmetric coupling strength \[Fig. \[fig2\](d)\], the two vector field curves are unlinked \[Fig. \[fig2\](e)\]. The linking number associated with this phase is $L=0$ and the band is topologically trivial. Notably, the anti-$\mathcal{PT}$-symmetric coupling induces the topological phase transition at the EP, which dramatically differs from the topological phase transitions in other non-Hermitian systems with $\mathcal{PT}$-symmetric gain and loss [@JLPRA; @Poli; @Schomerus; @Weimann; @MPan; @ZKL1905] as well as in non-Hermitian systems with asymmetric coupling strengths [ZGong,ZWang18,Kunst,LJinPRB,WHCPRB,YFChen20R]{}, where topological phase transition is irrelevant to the EP. Counterintuitively, increasing the dissipation can create nontrivial topology. More dissipation reduces the strength of the anti-$\mathcal{PT}$-symmetric coupling, tying vector field link or increasing the linking number instead of behaving destructively.
The linking topology is closely related to the vortex associated with the topological defects in the vector field. For the photonic crystal lattice under consideration, the real part of the fictitious magnetic field satisfies $d_{x}\left( k\right) =d_{x}\left( -k\right) $, $d_{y}\left(
k\right) =-d_{y}\left( -k\right) $, and $d_{z}\left( k\right) =0$. The types of non-Hermiticity and their appearance manners are important for non-Hermitian topological systems [@LJinPRB; @ZKL1905]. If the non-Hermiticity is $\vec{\Gamma}=\left( 0,0,\gamma \right) $, the topological phase transition and the existence of the edge state are unaltered because of the pseudo-anti-Hermiticity protection [Esaki,ZKL1905]{}; the topological properties of the non-Hermitian system are inherited by the EPs (exceptional rings or exceptional surfaces in 2D or 3D) [ZhouH,Okugawa,Budich,Cerjan,LMDuan17L,QZhong19L,XFZhang19L,SHFan20B,Yamamoto19L,LLi]{}. If the non-Hermiticity is $\vec{\Gamma}=\left( 0,\gamma ,0\right) $, the non-Hermitian skin effect occurs under open boundary condition [TLee16L,ZGong,ZWang18,Kunst,ZWang19L,Longhi19R,SChen19B,HJZhang19B,CHLee19B,LJinPRB,WHCPRB,Longhi20L,YFChen20R,Sato20L,XZZhang20B,WXTeo,SPKou20B,SPKou20B2,SPKouIJPB20,ELee]{}, the non-Hermitian Aharonov-Bohm effect under periodical boundary condition invalidates the conventional bulk-boundary correspondence [LJinPRB,WHCPRB]{}, and the non-Bloch band theory is developed for topological characterization [Murakami19L,ZWang19L2,Borgnia,Kawabata20B,KawabataPRX,PXue20,Thomale20]{}. Here, anti-$\mathcal{PT}$-symmetric coupling induces the imaginary part $%
\vec{\Gamma}=\left( \gamma ,0,0\right) $. $h_{k}$ lacks pseudo-anti-Hermiticity protection. The inversion symmetry of $h_{k}$ ensures the absence of a non-Hermitian Aharonov-Bohm effect and the validity of conventional bulk-boundary correspondence. The anti-$\mathcal{PT}$-symmetric coupling $\gamma $ in $h_{k}$ splits the diabolic point ($%
d_{x}=d_{y}=0$) in the Hermitian case ($\gamma =0$) into a pair of EPs \[cyan crosses in Fig. \[fig3\](a)\]. As $\gamma $ changes, the system switches among topologically trivial phases and different topologically nontrivial phases; therefore, the anti-$\mathcal{PT}$-symmetric coupling induces topological phase transition and unties the vortex link.
Figure \[fig3\](a) depicts the complex energy bands for $%
h_{k}=[d_{x}\left( k\right) -i\gamma ]\sigma _{x}+d_{y}\left( k\right)
\sigma _{y}$ as functions of $d_{x}$ and $d_{y}$ when $\gamma =2$ is fixed. The Riemann surface spectrum has two EPs connected by the Fermi arc, where the real part of the band gap closes. Two EPs located at $\left( 0,\pm
\gamma \right) $ in the $d_{x}$-$d_{y}$ plane are split from the diabolic point \[at the origin $\left( 0,0\right) $\] for $\gamma =0$. The integer topological charge associated with the diabolic point spits into two equal half-integer topological charges in the presence of anti-$\mathcal{PT}$-symmetric coupling. The two-component planer vector field $(F_{+,x}\left(
k\right) ,F_{+,y}\left( k\right) )$ is depicted in Fig. \[fig3\](b) [VectorField]{}. The linking number of the vector field curves $\mathbf{F}%
_{\pm }\left( k\right) $ equals to the winding number of the planar vector field, $w_{\pm }=\left( 2\pi \right) ^{-1}\int_{0}^{2\pi }\nabla _{k}\phi
_{\pm }\left( k\right) \mathrm{d}k=-L$, where $\phi _{\pm }(k)=\arctan
[F_{\pm ,y}\left( k\right) /F_{\pm ,x}\left( k\right) ]$. The half-integer topological charge associated with EP is revealed from the argument of the planar vector field $\mathbf{F}_{\pm }\left( k\right) $. The topological charge associated with the diabolic point or EP is equal to the global Zak phase [@Huang; @SChen; @Lieu; @MPan], which is the summation of the winding numbers of the real fictitious magnetic field loop encircling each individual EP. Notably, the global Zak phase can always be zero for an inappropriate gauge of the eigenmodes [@Nesterov], which is invalid for topological characterization and not relevant to the winding number associated with the gauge independent vector field for each band.
The topological phase transition occurs at the EPs, where the band gap closes at exact zero energy. The topological phase transition points are $$\gamma ^{2}=\kappa ^{2}+J^{2}/2\pm \left( J/2\right) \sqrt{J^{2}+8\kappa ^{2}%
}.$$The phase diagram in the $\kappa $-$\gamma $ plane is shown in Fig. [fig3]{}(c). The black curve in Fig. \[fig3\](a) indicates one energy branch of the photonic crystal lattice. The projection of the black curve in the $%
d_{x}$-$d_{y}$ plane is the real fictitious magnetic field loop $\vec{d}$ \[Fig. \[fig3\](b)\], which exhibits time-reversal symmetry and mirror symmetry with respect to the horizontal axis, $d_{x}\left( k\right)
=d_{x}\left( -k\right) $ and $d_{y}\left( k\right) =-d_{y}\left( -k\right) $. The real fictitious magnetic field loop $\vec{d}$ passes the EPs $\left(
0,\pm \gamma \right) $ at the gapless phase. In Fig. \[fig3\](b), the colored crosses indicate the EPs, and the colored lines indicate the Fermi arcs at different $\gamma $. Each time the loop $\vec{d}$ comes across the Fermi arc, the real part of the energy gap closes and two eigenmodes switch [@QZhong19NC; @SHFan18B]. For weak non-Hermiticity, the real part of the energy bands is gapped; the fictitious magnetic field loop passes [$%
d_{x}\left( k\right) =0$]{} four times as a result of the winding number $%
w_{\pm }=2$. Consequently, the magnetic field loop encircles the two EPs (green crosses) twice, $F_{\pm }\left( k\right) $ form a Solomon’s knot, and the imaginary part of the energy band intersects four times \[Fig. \[fig2\](a)\] at fixed momenta [$d_{x}\left( k\right) =0$]{} independent of non-Hermiticity. As non-Hermiticity increases, the band gap diminishes; the vector field loop passing [$\left( 0,\pm d_{y}\left( k_{0}\right) \right) $]{} may cross the Fermi arc when [$\left\vert \gamma \right\vert >\left\vert
d_{y}\left( k_{0}\right) \right\vert $]{}. The crossing indicates that the band gap of the real energy closes, and the real part instead of the imaginary part of the energy band crosses at [$k=\pm k_{0}$]{}; this occurs for large non-Hermiticity. We notice [two]{} and [four]{} crossings in the real part of the energy bands in Figs. \[fig2\](c) at $\gamma =2$ and \[fig2\](e) at $\gamma =3$, respectively. In Fig. \[fig2\](c), the magnetic field loop encircles the two EPs (cyan crosses) once; in Fig. \[fig2\](e), the magnetic field loop does not encircle either of the two EPs (blue crosses).
*Topological photonic zero modes.—*Topological phase transition is inextricably related to the change of topological photonic edge modes. Untying the link by $1$ reduces one pair of edge modes. Figure \[fig1\] is a quasi-1D coupled Su-Schrieffer-Heeger chain; the termination under the open boundary condition is indicated by the red lines, the open boundary spectrum as a function of the anti-$\mathcal{PT}$-symmetric coupling strength is depicted in Fig. \[fig3\](d). The edge modes are zero modes, which are analytically solvable at the limitation of infinite size, and satisfy the steady-state equations of motion$$\begin{aligned}
0 &=&-i\gamma \psi _{A,j}+J\psi _{A,j+1}+\kappa \psi _{A,j+2}, \\
0 &=&-i\gamma \psi _{B,j}+J\psi _{B,j-1}+\kappa \psi _{B,j-2}.\end{aligned}$$The edge modes decay to zero from one boundary to the other. The left (right) edge modes localize in the sublattice $A$ ($B$) in blue (red). The steady-state equations of motion are recurrence formulas. The localized edge modes can be regarded as plane waves $e^{\pm ikj}$ with complex momenta $k$, which are zeros of the complex function $$f\left( \rho \right) \equiv -i\gamma +J\rho +\kappa \rho ^{2}=0.$$where $\rho =e^{\pm ik_{0}}=-(J\pm \sqrt{4i\kappa \gamma +J^{2}})/\left(
2\kappa \right) $ yields the localization length of the edge mode $\xi
=\left\vert \mathrm{Im}(k_{0})\right\vert ^{-1}$ [@Pollmann]. The number of edge modes at either boundary is equal to the number of zeros $f\left(
\rho \right) =0$ within the region $\left\vert \rho \right\vert <1$, which is identical to the linking number of the two vector field links. The gapless phase is the boundary of two gapped phases, where topological phase transition occurs. The number of topologically protected edge modes at the gapless phase is equal to that at the gapped phase with a small winding number; two (zero) topological edge modes present in the gapless phase between topological phases $L=-2$ ($L=0$) and $L=-1$. The left edge mode amplitude is $\psi _{A,j}=\rho ^{j}$ and the right edge mode amplitude is $%
\psi _{B,N+1-j}=\rho ^{j}$. The two pairs of edge modes at $\gamma =1$ are depicted in Figs. \[figES\] (a) and \[figES\](b), respectively. The pair of edge modes at $\gamma =1.274$ and $\gamma =2$ are depicted in Figs. [figES]{}(c) and \[figES\](d).
*Conclusion.*—We investigate a dissipative quasi-1D topological photonic crystal lattice of coupled resonators. Counterintuitively, more dissipation constructively generate nontrivial topology from trivial topology. The coupled resonator array exhibits sublattice symmetry with alternatively presented anti-$\mathcal{PT}$-symmetric coupling. The nontrivial topology of the two-band lattice is established from the linking geometry of the vector fields, which are the average values of the Pauli matrices under the eigenmodes of the Bloch Hamiltonian. The anti-$\mathcal{PT%
}$-symmetric coupling alters the band topology and induces topological phase transition at the EPs, associated with the untying of vector field links. The linking number determines the number of topologically protected photonic zero modes. Our findings provide novel insight for the influence of non-Hermitian anti-$\mathcal{PT}$-symmetric coupling in the topological photonics and deepen the understanding of 1D topology in non-Hermitian systems.
This work was supported by National Natural Science Foundation of China (Grants No. 11975128 and No. 11874225).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'PHL 6625 is a luminous quasi-stellar object (QSO) at $z = 0.3954$ located behind the nearby galaxy NGC 247 ($z = 0.0005$). Hubble Space Telescope (HST) observations revealed an arc structure associated with it. We report on spectroscopic observations with the Very Large Telescope (VLT) and multiwavelength observations from the radio to the X-ray band for the system, suggesting that PHL 6625 and the arc are a close pair of merging galaxies, instead of a strong gravitational lens system. The QSO host galaxy is estimated to be $(4-28) \times 10^{10}$ $M_\sun$, and the mass of the companion galaxy of is estimated to be $M_\ast = (6.8 \pm 2.4) \times 10^{9}$ $M_\sun$, suggesting that this is a minor merger system. The QSO displays typical broad emission lines, from which a black hole mass of about $(2-5) \times 10^8$ $M_\sun$ and an Eddington ratio of about 0.01–0.05 can be inferred. The system represents an interesting and rare case where a QSO is associated with an ongoing minor merger, analogous to Arp 142.'
author:
- 'Lian Tao, Hua Feng, Yue Shen, Luis C. Ho, Junqiang Ge, Philip Kaaret, Shude Mao, Xin Liu'
title: 'PHL 6625: A Minor Merger-Associated QSO Behind NGC 247'
---
Introduction {#sec:intro}
============
Quasi-stellar objects (QSOs) or quasars are believed to be powered by accretion onto supermassive black holes in the centers of galaxies. How quasars are triggered is still not clear and under investigation. It is generally accepted that major mergers can trigger substantial star formation and possibly accretion onto the central black hole. @Sanders1988 proposed that major mergers, particularly between gas-rich disk galaxies, might drive gas to flow toward the nuclear region and initiate starburst, and then the triggering of the quasar phase. This picture is in good agreement with numerical simulations [e.g., @Hernquist1989; @Hopkins2006]. Observations of the ultraluminous infrared galaxies (ULIRGs) indicate that merging features [@Sanders1996] and the quasar fraction [@Kartaltepe2010] are strongly correlated with their IR luminosities, implying that major mergers and quasar activity may have a connection.
Minor mergers have been proposed to induce the fueling of low-luminosity AGNs and explain some observational features, such as the random orientation of narrow-line regions with respect to the host disks, the excess of ring-like structures and their off-center locations, and their amorphous morphology [e.g., @Taniguchi1999; @Combes2009]. Several authors suggested separating the fueling mechanisms for quasars from that for low-luminosity AGNs: major mergers trigger quasars, while minor mergers trigger low-luminosity AGNs [@Hopkins2009; @Taniguchi2013]. However, it is still uncertain if minor mergers could trigger high-luminosity AGN (quasar) activity. Significant fine structures such as shells and tidal tails were observed in deep Hubble Space Telescope (HST) images of four out of five elliptical low-redshift quasar host galaxies, which can be explained as due to minor mergers between a dwarf galaxy and a giant elliptical galaxy [@Bennert2008]. Thus, @Bennert2008 suggested that minor mergers might trigger the observed quasar activity. Moreover, @Tadhunter2014 examined 32 quasar-like AGN host galaxies and found that their dust masses were intermediate between those of quiescent elliptical galaxies and ULIRGs, suggesting that most of these AGNs were triggered in mergers between giant elliptical galaxies and relatively low gas mass companion galaxies.
It is also debatable whether the central black holes can be ignited in an interacting close pair at the early stage of merging when they start to have tidal interactions but are still spatially separate; some observations lead to a positive answer [e.g., @Ellison2011; @Silverman2011; @Liu2012] while others do not [e.g., @Ellison2008].
This study is unable to address these questions from a statistical point of view, but it presents an interesting case where a luminous QSO (PHL 6625) is found in a close pair of a merging system in the local universe. PHL 6625 ($z = 0.3954$, see § \[sec:vlt\_spec\] for details) is a radio-quiet QSO projected on the outskirts of a nearby spiral galaxy, NGC 247 [$z = 0.0005$, @Karachentsev2013]. It was detected as a redshifted object behind NGC 247 by @Margon1985. In 2011 October, @Tao2012 serendipitously discovered that PHL 6625 was associated with an arc structure on an HST image. We thus conducted new observations to further investigate its nature.
[cllccl]{} 1a & PSF & S[é]{}rsic ($n=0.6$) & $-22.01$ & $-21.15 $ & 1.803 (31954)\
1b & PSF + S[é]{}rsic ($n=0.3$) & S[é]{}rsic ($n=8.3$) & $-22.28$ & $-20.58 $ & 1.318 (31947)\
2a & PSF & Exponential & $-22.42$ & $-19.32 $ & 2.402 (31955)\
2b & PSF + S[é]{}rsic ($n=0.2$) & Exponential & $-22.30$ & $-19.52 $ & 1.393 (31948)\
3a & PSF & S[é]{}rsic ($n=0.3$) + S[é]{}rsic ($n=8.3$) & $-21.87$ & $-21.62 $ & 1.318 (31947)\
3b & PSF + S[é]{}rsic ($n=0.3$) & S[é]{}rsic ($n=1.0$) + S[é]{}rsic ($n=8.4$) & $-22.28$ & $-20.59 $ & 1.316 (31940)\
4a & PSF & S[é]{}rsic ($n=0.2$) + Exponential & $-21.90$ & $-21.33 $ & 1.393 (31948)\
4b & PSF + S[é]{}rsic ($n=0.3$) & S[é]{}rsic ($n=8.3$) + Exponential & $-22.28$ & $-20.59 $ & 1.316 (31941)
The projected distance of the quasar from NGC 247 at the distance of NGC 247 [3.4 Mpc, @Gieren2009] is about 4.4 kpc. Thanks to its spatial location and strong X-ray emission, PHL 6625 is of interest in probing the halo of NGC 247 and acts as a useful probe to detect the “missing baryons,” which have an observational deficit [e.g., @Shull2012] compared with cosmological predictions. Some of the missing baryons in the local universe are thought to be locked in the warm-hot intergalactic medium, which can be detected via X-ray absorption lines in the presence of a background QSO using next generation telescopes [@Yao2012].
In this paper, we report spectroscopic observations for PHL 6625 and its nearby arc structure with the Very Large Telescope (VLT) of the European Southern Observatory (ESO), along with multiwavelength observations from the radio to the X-ray band. We adopt a cosmology with $h = 0.7$, $\Omega_m = 0.3$, and $\Omega_\Lambda = 0.7$ and a luminosity distance of 2.14 Gpc to PHL 6625 ($z = 0.3954$).
Observations and data analysis {#sec:obs}
==============================
*HST imaging*
-------------
HST observed the northwestern region of NGC 247 on 2011 October 11 (proposal ID 12375), using the broadband filter F606W of the Wide Field Channel (WFC) on the Advanced Camera for Surveys (ACS) with two sub-exposures for a total exposure of 846 s. The observation was designed to use the QSO PHL 6625, which was known to be bright in both the X-ray and optical bands, to align the Chandra and HST images to improve their relative astrometry [@Tao2012]. However, it serendipitously found that the QSO was associated with an arc structure to its southeast; see Figure \[fig:hst\]. The QSO is projected at about 0.4 times the $R_{25}$ radius of NGC 247 [@de; @Vaucouleurs1991] in a relatively uncrowded environment.
![HST image around PHL 6625. The red lines indicate the VLT slit positions for the arc structure. The three objects from east to west are an anonymous galaxy (not NGC 247), the arc structure, and the QSO PHL 6625, respectively. The arrows have a length of 2 (1 = 5.4 kpc at a redshift of 0.3970 assuming $h = 0.7$, $\Omega_m = 0.3$, and $\Omega_\Lambda = 0.7$). \[fig:hst\]](qso_hst.eps){width="1.0\columnwidth"}
![Azimuthally averaged radial profiles of PHL 6625 and the TinyTim PSF. MAG is the mean isophotal magnitude. Crosses show the profile of PHL 6625 shifted to match the peak of the TinyTim PSF and diamonds show the PHL 6625 profile shifted to match the TinyTim PSF at a radius of 1.5 pixels. In either case, the profile of PHL 6625 is significantly broader than that of the PSF, indicating that there is a host galaxy component. \[fig:profile\]](qso_psf_radial.eps){width="\columnwidth"}
A reasonable point-spread function (PSF) is needed to analyze the QSO image. Looking through the archival HST images[^1], isolated bright stars with a flat background cannot be found at a similar chip position. We thus derived a synthetic PSF by synthesizing PSF models generated by the TinyTim tool [@Krist1995] onto the flat-fielded calibrated (\_flc) data and drizzling them into science images using the [astrodrizzle]{} task. The azimuthally averaged radial profiles of the QSO and the TinyTim PSF are computed using the [ellipse]{} task in the IRAF/STSDAS package and plotted in Figure \[fig:profile\] for comparison. They are shifted to have the same brightness at the center (0.5 pixel). As the core ($3\times3$ pixels) of the QSO image is saturated due to high brightness and may be not useful, we also compared the two profiles by matching their magnitudes at a radius of 1.5 pixels, beyond which there is no saturation. In either way, the PSF profile is significantly narrower than that for the QSO, suggestive of an additional component (likely the host galaxy) underneath the QSO component.
The QSO image is decomposed into a QSO component and a host galaxy component using GALFIT [@Peng2002; @Peng2010]. The arc structure, central saturated $3 \times 3$ pixels, and nearby stars are masked away during the fit. Without information from the central pixels, it is hard to reconstruct the bulge component unless it is sufficiently extended. We thereby experimented with several models to explore the systematics in the modeling. For the QSO, we also tried a PSF with the addition of a small-index S[é]{}rsic profile to account for PSF artifacts. For the host galaxy, we tested with either a single-component model (a S[é]{}rsic or an exponential disk) or a two-component model (two S[é]{}rsics or a S[é]{}rsic + an exponential disk). These lead to eight combinations of models, tabulated in Table \[tab:galfit\]. The simplest models with only two components (model 1a and 2a) do not provide adequate fits, while any model with three or four components can fit the image similarly well. The residuals are shown in Figure \[fig:galfit\] for comparison. Therefore, we discard the simplest (two-component) models and adopt the magnitude range derived from other models as a conservative estimate of its uncertainty, for both the QSO and the host galaxy.
The flux of the companion arc galaxy is measured using aperture photometry, with a visually defined polygon aperture and nearby source-free regions for background estimate. Assuming a flat spectrum ($F_\lambda \propto \lambda^{0}$ ) for the arc and the QSO host galaxy, and a power-law spectrum for the QSO (see § \[sec:qso\_gfit\] for details), the measured count rates can be translated to dereddened, K-corrected, absolute $B$ magnitudes of \[$-$21.87, $-$22.30\], \[$-$19.52, $-$21.62\], and $-$19.86, respectively, for the QSO, the QSO host galaxy, and the arc galaxy. Assuming a solar $B$-band magnitude of 5.48 [@Binney1998], the total luminosity in the $B$-band is, respectively, $(0.87-1.30) \times 10^{11}$, $(1.00-6.93) \times 10^{10}$, and $1.37 \times 10^{10}$ $L_\sun$ for the three objects in the same order. The projected size of the arc galaxy is roughly 16 by 4 kpc.
{width="\textwidth"}
*VLT spectroscopy* {#sec:vlt_spec}
------------------
To unveil the nature of the QSO and the arc galaxy, we conducted spectroscopic observations with the ESO 8.2-m diameter VLT at Paranal in Chile (program ID 091.A-0149(A)), using the FORS2 long-slit spectrograph mounted on the unit telescope 1 (UT1). A red (GRIS\_300I) and blue (GRIS\_600B) grism is used, respectively, to cover a wavelength range from $\sim$3700–10000Å. The blue setup of the arc is spilt into two identical observations. Each observation consists of two (for the QSO) or three (for the arc) observation blocks (OBs) with an offset of 3 along the spatial direction between successive ones for bad pixel and cosmic-ray removal. We used a 1 slit and a $2 \times 2$ binning of pixels, resulting in a sampling of 2.8Å/pixel in the red and 1.32Å/pixel in the blue. The spectral resolution in FWHM found from the lamp lines varies from 11.3–12.9Å in the red and 5.3–5.9Å in the blue. For the QSO observations, the slit is oriented across both the QSO and the central region of the arc, while for the arc observations, the slit is placed along its major axis. The observational log is listed in Table \[tab:vlt\] and the location of the slit for the arc observation is displayed in Figure \[fig:hst\].
[llllll]{} QSO & Jul 17 09:04:08 & 300I & $150 \times 2$ & 1.009 & 0.6\
QSO & Jul 18 09:30:59 & 600B & $150 \times 2$ & 1.003 & 0.7\
arc & Aug 04 05:54:06 & 300I & $940 \times 3$ & 1.169 & 0.7\
arc & Aug 04 06:51:06 & 600B & $810 \times 3$ & 1.054 & 0.9\
arc & Aug 04 07:44:30 & 600B & $810 \times 3$ & 1.012 & 0.7
{width="\textwidth"}
The [esorex]{} package was used to create bias-subtracted, flat-fielded, and wavelength-calibrated 2D spectra for each OB, using calibration files obtained in the same night. The [imcombine]{} task in IRAF was then used to combine different OBs and remove cosmic rays with the option [crreject]{}. The 1D spectra were extracted using the [apall]{} task. The trace information was obtained by fitting the QSO spectra and was applied for the extraction of the arc spectra. The source aperture size is around 10–11 pixels for the QSO and 18–20 pixels for the arc, and the background was estimated by fitting fluxes from two source-free regions on each side. The standard star LTT 1020 observed in the same night as the arc observations was used for flux calibration. An extinction table created on 2011 January 18 was used for atmospheric extinction correction.
The red or blue spectra from individual OBs for either the QSO or arc are averaged. The red spectra have a flux higher than the blue spectra in their overlapping region, which is due to a smaller PSF image size in the red. We thus scale the blue flux by a constant factor of $\sim$1.2 to yield a consistent flux in their overlapping wavelength range. Spectral smoothing is done with a 5 pixel median filter.
The observed QSO and arc spectra are shown in Figures \[fig:qso\_spec\] and \[fig:arc\_spec\], respectively. The QSO spectrum exhibits characteristic emission lines, such as broad [Mg[ii]{}]{}, [H[$\beta$]{}]{} and [H[$\alpha$]{}]{}, narrow [\[O[iii]{}\]$\lambda\lambda$4959,5007]{} and Balmer lines. The absorption features around 5450Å are telluric. The arc spectrum shows some narrow emission lines, such as [\[O[ii]{}\]$\lambda$3728]{}, [H[$\beta$]{}]{}, [\[O[iii]{}\]$\lambda$5007]{}, [H[$\alpha$]{}]{} and [\[N[ii]{}\]$\lambda$6583]{} and some weak absorption lines, such as [Ca[ii]{}K$\lambda$3933]{} and [Ca[ii]{}H$\lambda$3968]{}. The arc spectrum is contaminated by the QSO, showing broad [H[$\alpha$]{}]{} and [Mg[ii]{}]{} lines (see § \[sec:arc\_fit\] for details). The redshift of the QSO is measured to be $0.3954\pm0.0001$ from the [\[O[iii]{}\]$\lambda\lambda$4959,5007]{} emission lines. The redshift of the arc, using the QSO-contamination-subtracted spectrum, is measured to be $0.3970\pm0.0001$ via [\[O[ii]{}\]]{} and [H[$\beta$]{}]{} emission lines, and consistent with the value of $0.3968\pm0.0003$ measured via the [Ca[ii]{}K$\lambda$3933]{} and [Ca[ii]{}H$\lambda$3968]{} absorption lines. The arc is likely located nearer than the QSO and moving toward it with a line-of-sight velocity of roughly 340 km s$^{-1}$.
### Decomposition of the QSO spectrum {#sec:qso_gfit}
The QSO spectrum is decomposed into multiple emission components following @Shen2008. The fitting is performed in the rest-frame wavelength range of 2750–7100Å, where the S/N is sufficiently high. First, a pseudo-continuum was fitted to the spectrum in some continuum windows[^2], consisting of a power-law component and [Fe[ii]{}]{} templates in both the [Mg[ii]{}]{} region [@Salviander2007] and the [H[$\beta$]{}]{} region [@Boroson1992]. The continuum-subtracted line spectrum was then fitted with multiple Gaussian components: three Gaussians for each of the [H[$\alpha$]{}]{}, [H[$\beta$]{}]{}, and [Mg[ii]{}]{} broad components; five Gaussians for narrow lines in the [H[$\alpha$]{}]{} region[^3], three near [H[$\beta$]{}]{}[^4], and one for narrow [Mg[ii]{}]{}. All of the narrow lines are imposed to have the same shift and width.
[cll]{} $\alpha$ & $-2.171 \pm 0.010$ & Power-law spectral index, $f_\lambda \propto \lambda^\alpha$\
$L_{3000}$ & $4.54 \times 10^{44}$ [erg s$^{-1}$]{}& $\lambda L_\lambda(3000{\text{\normalfont\AA}})$ of the power-law component\
$L_{5100}$ & $2.44 \times 10^{44}$ [erg s$^{-1}$]{}& $\lambda L_\lambda(5100{\text{\normalfont\AA}})$ of the power-law component\
FWHM([Mg[ii]{}]{}) & 6269 km s$^{-1}$ & Broad component\
FWHM([H[$\beta$]{}]{}) & 6447 km s$^{-1}$ & Broad component\
FWHM([H[$\alpha$]{}]{}) & 7314 km s$^{-1}$ & Broad component\
$\log (M/M_\sun)$ & 8.43 & Based on $L_{5100}$ and FWHM([H[$\beta$]{}]{}), $\alpha = 0.5$ and $\beta = 2$ (Ref. 1)\
$\log (M/M_\sun)$ & 8.74 & Based on $L_{5100}$ and FWHM([H[$\beta$]{}]{}), $\alpha = 0.533$ and $\beta = 2$ (Ref. 2)\
$\log (M/M_\sun)$ & 8.47 & Based on $L_{5100}$ and FWHM([H[$\beta$]{}]{}), $\alpha = 0.5$ and $\beta = 1.09$ (Ref. 3)\
$\log (M/M_\sun)$ & 8.29 & Based on $L_{5100}$ and FWHM([H[$\beta$]{}]{}), $\alpha = 0.572$ and $\beta = 1.200$ (Ref. 1)\
$\log (M/M_\sun)$ & 8.25 & Based on filtered luminosities (Ref. 1)
The monochromatic continuum luminosity $\lambda L_\lambda$ of the power-law component at the rest frame 3000Å and 5100Å, the power-law spectral index, and the broad-line width derived from the global fitting are listed in Table \[tab:gfit\]. Based on the radius-luminosity ($R-L$) relation, the black hole mass in the QSO can be estimated from the line width and the continuum luminosity with a single-epoch spectrum, i.e., $M_{\rm BH} \propto L^\alpha {\rm FWHM}^\beta$ [@Shen2013]. Here we use five recipes that were calibrated against [H[$\beta$]{}]{} reverberation-mapped masses: the updated @Vestergaard2006 formula described in @Feng2014 that assumes theoretical slopes ($\alpha = 0.5$ and $\beta = 2$) on the luminosity and line width, the calibration of @Ho2015 based on a best-fit slope for the $L-R$ relation [$\alpha = 0.533$; @Bentz2013] and $\beta = 2$, the calibration of @Wang2009 with $\alpha = 0.5$ and a best-fit slope for the single-epoch FWHM versus rms line dispersion relation ($\beta = 1.09$), one based on best-fit slopes for both the luminosity and line width [$\alpha = 0.572$ and $\beta = 1.200$; @Feng2014], and a novel technique [@Feng2014] that establishes a correlation between the black hole mass and filtered luminosities (luminosities extracted in two wavelength bands). All the recipes give consistent results for a black hole mass of about $(2-5) \times 10^8$ $M_\sun$, also listed in Table \[tab:gfit\]. If we use the second moment [line dispersion; @Peterson2004] instead of the FWHM, the inferred black hole mass is consistent with the result above within the intrinsic scatter (a factor of $\sim$2).
[cccccc]{} $(6.8 \pm 2.4) \times 10^9$ & $0.24 \pm 0.06$ & $8.44 \pm 0.16$ & $9.8 \pm 0.2$ & $0.08 \pm0.02$ & $0.7 \pm 0.5$
{width="\textwidth"}
[ccccc]{} $13.6 \pm 2.7$ & $4.1 \pm 0.7$ & $3.6 \pm 0.6$ & $13.2 \pm 1.4$ & $5.0 \pm 0.6$
![Decomposed star formation history of the arc galaxy. \[fig:arc\_sfh\]](arc_spec_sfh.eps){width="\columnwidth"}
### Population synthesis for the Arc galaxy {#sec:arc_fit}
The stellar population synthesis code STARLIGHT [@CidFernandes2005] was used to fit the arc spectrum. As the arc spectra may be contaminated by flux from the QSO, we added the QSO spectrum into the simple stellar population (SSP) fitting library as a model template. We then fit the arc spectrum with the stellar components using the BC03 theoretical library [150 SSPs with 25 ages and 6 metallicities, @Bruzual2003] with Chabrier’s initial mass function [@Chabrier2003]. Assuming that the flux error follows a Gaussian distribution, we generated 50 mock spectra to estimate the parameter uncertainties.
The scaling factor for a QSO contribution to the arc spectrum is found to be 3.5% from the fit. We fit a Moffat function to the QSO on the acquisition image, and estimate that the QSO roughly contributes 3.4% of its flux to the arc aperture, in reasonable agreement with the result from the spectral fitting. After the QSO contribution is removed, the arc spectrum barely shows broad-line components ([H[$\alpha$]{}]{} and [Mg[ii]{}]{}).
The STARLIGHT fitting results are listed in Table \[tab:pop\], including the stellar mass, star-derived extinction, and luminosity-weighted and mass-weighted ages and metallicities. The best-fit model spectrum is shown in Figure \[fig:arc\_spec\] and the decomposed star formation history (SFH) is shown in Figure \[fig:arc\_sfh\]. The emission line luminosities are measured from the galaxy model and QSO template subtracted spectrum. The gas-derived extinction was derived from the star-derived extinction, assuming the ratio between the gas-derived and star-derived extinction to be 0.44 [@Calzetti2001]. The dereddened luminosities are listed in Table \[tab:emi\]. The ratio of ${\rm H}\alpha / {\rm H}\beta$, $3.2\pm0.6$, is consistent with the Balmer decrement [2.86, @Osterbrock1989]. The luminosity of the [\[O[iii]{}\]$\lambda$5007]{} line only accounts for 0.1% of the observed flux in the HST F606W filter, suggesting that the arc is truly made of stars rather than some extended ionized gas.
A single Gaussian component is able to fit each of the narrow lines in the contamination-subtracted arc spectrum, and no obvious residuals are seen. The observed FWHM of the emission/absorption lines corrected for instrumental broadening is consistent with zero within errors, which agrees with the result that the stellar velocity dispersion ($\sim$200 km s$^{-1}$) derived from the fundamental plane [summarized in @Kormendy2013] is smaller than the instrument dispersion (> 300 km s$^{-1}$) and hence unresolved.
The star formation rate (SFR) is estimated from the strongest emission line [H[$\alpha$]{}]{} [@Kennicutt1994; @Madau1998; @Kennicutt1998] assuming solar abundance and @Salpeter1955’s IMF,
$$\label{eq:SFR}
{\rm SFR} = 7.9 \times 10^{-42} \; \frac{\; L({\rm H}\alpha)\; }{{\rm erg \; s}^{-1}} \quad {M_\sun \; {\rm yr}^{-1}}.$$
We derived an SFR for the arc galaxy, ${\rm SFR} = (1.04 \pm 0.11)$ $M_\sun$ yr$^{-1}$, and a specific SFR (SFR per stellar-mass unit), ${\rm sSFR} = {\rm SFR} / M_\ast$ = 0.15 Gyr$^{-1}$. Using @Kennicutt1998’s calibration, the SFR derived from the [\[O[ii]{}\]$\lambda$3728]{} line gives a marginally consistent result, $(1.9 \pm 0.7)$ $M_\sun$ yr$^{-1}$. Given a redshift of 0.3954 and the ${\rm SFR}$ range estimated above, the stellar mass is estimated to be $(1.7-7.9) \times 10^{9}$ $M_\sun$ if the source lies on the main sequence of star-forming galaxies [@Whitaker2012], consistent with $M_\ast = (6.8\pm2.4) \times 10^{9}$ $M_\sun$ derived from the population synthesis.
Given the metallicities and ages in Table \[tab:pop\], we can predict a mass-to-light ratio ($M_\ast/L_{\rm B}$) of $\sim0.1-0.2$ and $\sim1.3-5.0$ in the $B$-band following @Maraston2005, for luminosity-weighted and mass-weighted measurements, respectively. Using the mass obtained from the stellar population synthesis and the blue luminosity measured from the HST image, this ratio is $0.5\pm0.2$, larger than the luminosity-weighted estimate but smaller than the mass-weighted estimate.
The $\log \,($[\[N[ii]{}\]]{}$/$[\[O[ii]{}\]]{}$)$ of the arc galaxy is $-0.4$, which meets the criterion for the upper $R_{23}$ branch [@Kewley2008]. Using the metallicity calibration of @Zaritsky1994, we obtained $12 + \log(\rm O/ \rm H)$ to be 8.88. Assuming a solar metallicity of $12 + \log(\rm O/ \rm H)=8.86$ [@Delahaye2006], the metallicity is estimated to be $Z/Z_{\odot} \sim 1.0$, similar to the value derived from the STARLIGHT mass-weighted measurement and consistent with the value of $Z/Z_{\odot}=(0.3-1.3)$ estimated from 10 different mass-metallicity relations listed in @Kewley2008. These results are not sensitive to QSO contamination; consistent results are obtained without removing the QSO contamination.
X-Ray spectra with *XMM-Newton*
-------------------------------
[lll]{} $N_{\rm H,Gal}$ ($10^{20}$ cm$^{-2}$) & 2.07 fixed & 2.07 fixed\
$N_{\rm H,ext}$ ($10^{20}$ cm$^{-2}$) & $3.6_{-1.2}^{+1.4}$ & $1.608_{-0.009}^{+0.010}$\
PL photon index & $2.24_{-0.10}^{+0.12}$ & $1.96 \pm 0.08$\
PL norm & $2.13_{-0.19}^{+0.23}$ & $1.88_{-0.13}^{+0.15}$\
$E_{\rm line}$ (keV) & $2.12 \pm 0.03$ &\
$A_{\rm line}$ ($10^{-6}~$ph cm$^{-2}$ s$^{-1}$) & $-3.9 \pm 0.13$ &\
EW (eV) & 71 &\
$f_{\rm 0.3-10\;keV}$ ($10^{-13}~$[erg cm$^{-2}$ s$^{-1}$]{}) & $4.04 \pm 0.24$ & $5.08 \pm 0.26$\
$f_{\rm 2-10\;keV}$ ($10^{-13}~$[erg cm$^{-2}$ s$^{-1}$]{}) & $1.81 \pm 0.23$ & $2.70 \pm 0.25$\
$L_{\rm 0.3-10\;keV}$ ($10^{44}~$[erg s$^{-1}$]{}) & $3.01_{-0.19}^{+0.24}$ & $3.07 \pm 0.14$\
$L_{\rm 2-10\;keV}$ ($10^{44}~$[erg s$^{-1}$]{}) & $1.05 \pm 0.10$ & $1.47 \pm 0.10$\
$\chi^2$ / degree of freedom & 96.5/93 & 135.9/121
{width="49.00000%"}{width="49.00000%"}
{width="98.00000%"}
The [*Chandra*]{} observation suggests that the X-ray emission arises from a point-like source spatially coincident with the QSO; there is no X-ray emission detected in the arc region [@Tao2012]. [*XMM-Newton*]{} observed the galaxy on 2009 December 27 and 2014 July 1 (ObsID 0601010101 and 0728190101, respectively). An earlier observation made on 2001 July 8 was not used due to heavy background contamination. Only data from the PN CCD were used for analysis. New events files were created with up-to-date calibration files. Events were selected from low background intervals, where the background flux is within $\pm 3 \sigma$ of the mean quiescent level, adding up to an effective exposure of 19.0 ks and 24.3 ks, respectively, for the two observations. The source energy spectra were extracted from a circular region of 32radius, and the background spectra were extracted from nearby circular regions on the same chip at a similar readout distance. The spectral bins were grouped such that each new bin is 1/4 of the local FWHM and has at least 15 counts, from 0.2 keV to 10 keV.
We tried to fit the energy spectra with a redshifted power-law model subject to interstellar absorption. The [TBabs]{} model [@Wilms2000] is used to account for Galactic absorption, and the column density is fixed at the Galactic value $2.07 \times 10^{20}$ cm$^{-2}$ [@Kalberla2005], while [phabs]{} is adopted for additional extragalactic absorption. For the 2009 observation, we obtained consistent results with those reported by @Jin2011, if the same model and energy range are used. However, we found that a simple power-law model is insufficient to fit the data. An absorption feature near 2.12 keV in the rest frame (or 1.54 keV in the observed frame) and excessive emission above 10 keV in the rest frame (or 7 keV in the observed frame) are possibly shown in the residual. We further added a zero-width Gaussian component to fit the absorption feature. The addition of the absorption line reduced the $\chi^2$ by 24.6, corresponding to a chance probability of $2.6\times10^{-5}$. The hard excess could be due to a reflection component, which is often seen in the spectra of AGNs and Galactic accreting black holes, but the current data quality and energy coverage do not allow us to quantify it. For the 2014 observation, a simple power-law model can adequately fit the data. The spectra are shown in Figure \[fig:xmm\] and the best-fit parameters are listed in Table \[tab:xmm\].
Multiwavelength SED
-------------------
NGC 247 was observed by the Wide-field Infrared Survey Explorer ([*WISE*]{}) in 2010 with the passbands W1 (3.4 $\mu$m), W2 (4.6 $\mu$m), W3 (12 $\mu$m) and W4 (22 $\mu$m). The QSO was detected in the W1, W2 and W3 bands with a signal-to-noise ratio (SNR) larger than 9, while in the W4 band, the source was not detected. From the AllWISE Source Catalog in NASA/IPAC Infrared Science Archive (IRSA)[^5], we obtained the profile-fitting photometry for the W1, W2, and W3 bands in VEGA magnitudes, and a 95% upper limit for the W4 band, which are $14.23\pm0.03$, $13.17\pm0.03$, $10.88\pm0.12$ and $<8.261$ for the W1, W2, W3 and W4 bands, respectively. Using the zero magnitude flux density and the color corrections from @Wright2010, the observed VEGA magnitudes were translated to flux density, which are $(6.26\pm0.18)\times10^{-27}$, $(9.3\pm0.3)\times10^{-27}$, $(1.37\pm0.15)\times10^{-26}$ and $ < 4.1\times10^{-26}$ [erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$]{} for the four bands in the same order. Moreover, the QSO was not detected in the 1.4 GHz NRAO VLA Sky Survey (NVSS), suggesting a flux less than 2.5 mJy [@Elvis1997]. The multiwavelength spectral energy distribution (SED) from the radio to the X-ray band with a QSO SED template renormalized in the optical band [@Hopkins2007] is shown in Figure \[fig:sed\].
Discussion
==========
The consistent redshifts of the two galaxies suggest that they are a merger event, between a luminous QSO (PHL 6625) and a tidally distorted companion galaxy, instead of a strong gravitational lens system. The stellar content of the two galaxies are not in contact yet, and the mass fraction of young stars is less than 1% (Figure \[fig:arc\_sfh\]), suggesting they are at the early stage of a merging process. Such a system seems to be an analogue of the nearby event Arp 142 (NGC 2936/37) in morphology [e.g., @Romano2008], except that the central black hole in PHL 6625 is an active quasar.
A major or minor merger?
------------------------
The broadening of the QSO image (Figure \[fig:profile\]) suggests that an underlying component possibly due to its host galaxy is detected. However, due to the saturation of the central pixels, the galaxy bulge cannot be spatially resolved. A conservative estimate of the $B$-band luminosity of the QSO host galaxy is $(1.00 - 6.93) \times 10^{10}$ $L_\sun$, corresponding to a mass of $(4-28) \times 10^{10}$ $M_\sun$ assuming a typical mass-to-light ratio of 4 [@Faber1979]. The black hole mass for the QSO is estimated to be $(2-5) \times 10^8$ $M_\sun$ via different techniques. Assuming an $M_{\rm BH}$-$M_{\rm bulge}$ relation [@Kormendy2013], $$\frac{M_{\rm BH}}{10^{9} \; M_\odot} =
0.49
\left( \frac{M_{\rm bulge}}{10^{11} \; M_\odot} \right)^{1.17},$$ we can derive the bulge mass to be $(4-11) \times 10^{10}$ $M_\sun$ with an intrinsic scatter of 0.28 dex (a factor of $\sim$2). This is consistent with the mass range estimated from image decomposition, and suggests that the QSO host galaxy may be an elliptical or a bulge-dominated system. If we adopt the total mass range of the host galaxy, PHL 6625 also follows the distribution of AGNs at $z=0.1-1.0$ in the $M_{\rm BH}$-$M_{\rm bulge}$ plane [Figure 38 of @Kormendy2013], and is consistent with the $M_{\rm BH}$-$M_{\rm bulge}$ relation at $z=0.4$ [@Kormendy2013]. In summary, the QSO and its host seem to be a canonical example on the co-evolution path.
The stellar mass of the arc galaxy, using the population synthesis measurement, is about $6.8 \times 10^{9}$ $M_\sun$, indicating that the mass ratio of the QSO host galaxy and the arc galaxy is around 10, suggesting that the system is a minor merger. Given the $B$-band luminosity of the arc galaxy of $1.4 \times 10^{10}$ $L_\sun$ and $M_\ast/L_{\rm B} \sim 0.1-0.2$ derived from the luminosity-weighted metallicity of the population synthesis model, the stellar mass of the arc galaxy is on the order of $10^{9}$ $M_\sun$, and the system is also likely to be a minor merger. But if $M_\ast/L_{\rm B}$ derived from the mass-weighted metallicity is used, the stellar mass of the arc galaxy will be more than 10 times higher, and the system could be a major merger. However, during the population synthesis fit, the mass-to-light ratio of young stars (age $\sim 10^{8.5}$ year) is about one-sixth that of old stars (age $\sim 10^{10}$ year), then any small light-fraction variations of young stellar populations will make large mass-fraction variations on old stellar populations and result in huge uncertainties in the mass-weighted metallicity and mass-to-light ratio. Thus, the luminosity-weighted measurement is more reliable than the mass-weighted measurement, and the system is more likely to be a minor merger, although a major merger cannot be excluded.
Multiwavelength properties of the QSO
-------------------------------------
While there is some weak evidence for mild spectral variability, the QSO outputs a consistent luminosity in the X-ray band ($\sim 3\times10^{44}$ [erg s$^{-1}$]{} in 0.3-10 keV, rest frame). The bolometric luminosities, $L_{\rm bol}$, calculated from the unabsorbed flux in the 0.01–100 keV range, are, respectively, $1.1\times10^{45}$ [erg s$^{-1}$]{} and $8.0\times10^{44}$ [erg s$^{-1}$]{} for the 2009 and 2014 observations, corresponding to an Eddington ratios ($L_{\rm bol}/L_{\rm Edd}$) of about $0.01-0.05$. This implies a bolometric correction factor of 3–5 for the $L_{5100}$ luminosity or 5–11 for the 2-10 keV X-ray luminosity.
The observed power-law photon index and the Eddington ratio are consistent with the $\Gamma - L_{\rm bol}/L_{\rm Edd}$ relation for AGNs [@Brightman2013]. The hardening of the spectrum along with the decrease of the bolometric luminosity between the 2009 to 2014 observations, if true, is also consistent with the above relation.
The X-ray to optical/UV ratio, $\alpha_{\rm ox}$, is defined as $${\alpha_{\rm ox}} = 0.3838\log(L_{\rm 2~keV}/L_{\rm 2500~{\text{\normalfont\AA}}}),$$ where $L_{\rm 2~keV}$ and $L_{\rm 2500~{\text{\normalfont\AA}}}$ are the monochromatic luminosities at 2 keV and 2500 Å in the rest frame, respectively. With $L_{\rm 2500~{\text{\normalfont\AA}}}=4.68 \times 10^{29}$ [erg s$^{-1}$ Hz$^{-1}$]{} and $L_{\rm 2~keV}=1.82 \times 10^{26}$ [erg s$^{-1}$ Hz$^{-1}$]{} for the 2014 [*XMM-Newton*]{} observation, we derive $\alpha_{\rm ox} = -1.3$, which is consistent with the $\alpha_{\rm ox} = -1.4\pm0.3$ derived from the $\alpha_{\rm ox} - L_{\rm 2500~{\text{\normalfont\AA}}}$ relation of @Just2007.
The [Fe[ii]{}]{} strength of the QSO, defined as the ratio of the equivalent width for the optical [Fe[ii]{}]{}$\lambda$4570 blend and the broad [H[$\beta$]{}]{} ($R_{\rm Fe\,{\sc II}} \equiv {\rm EW_{Fe\,{\sc II}}}/{\rm EW_{H\beta}}$), is $\sim 0.14$. The FWHM of the broad [H[$\beta$]{}]{} is 6447 km s$^{-1}$. The measured log\[EW$_{\rm [O\,{\sc III}]\,\lambda 5007}$ (Å)\] is about 0.8. These properties are not typical for SDSS quasars [@Shen2014] on the eigenvector 1 plane [@Boroson1992], although the QSO presents a typical luminosity among the SDSS DR7 quasars [@Shen2011]. Quasars similar to PHL 6625 that have a small $R_{\rm Fe\,{\sc II}}$ and a low equivalent width of [\[O[iii]{}\]$\lambda$5007]{} compose only 3.4% of the SDSS DR7 quasars with a small $R_{\rm Fe\,{\sc II}}$, regardless of [\[O[iii]{}\]$\lambda$5007]{} strength, but the physical explanation is not clear.
To conclude, this system gives us a case where a luminous quasar is associated with a minor merger in the close pair phase, although there is no conclusive evidence to show a link between the quasar activity and the merger event. High-resolution and high-sensitivity observations with a large sample of nearby quasars may address the question whether the case like PHL 6625 is rare or ubiquitous, and whether quasar activity can be triggered by minor mergers.
We thank the anonymous referee for useful comments that have improved the paper. We also thank Chien Y. Peng, Minjin Kim, Ning Jiang, and Yulin Zhao for their help in using of the GALFIT software. H.F. acknowledges funding support from the National Natural Science Foundation of China under grant No. 11633003, and the National Program on Key Research and Development Project (grant No. 2016YFA040080X). L.C.H. was supported by the National Key Program for Science and Technology Research and Development grant 2016YFA0400702. J.Q.G. and S.M. were partially supported by the Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences grant No. XDB09000000 and by the National Natural Science Foundation of China (NSFC) under grant numbers 11333003 and 11390372.
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[^1]: The observing date of proposal ID 12375 was later than the Servicing Mission 4 (SM4) of HST. In SM4, the ACS was repaired, and the PSF might be affected. Therefore, the PSF stars are searched in the observations after SM4.
[^2]: 2925–3500Å, 4200–4230Å, 4435–4700Å, 5100–5535Å, 6000–6250Å, and 6800–6900Å at the rest frame.
[^3]: Two for [\[S[ii]{}\]]{}, two for [\[N[ii]{}\]]{}, and one for narrow [H[$\alpha$]{}]{}.
[^4]: Two for [\[O[iii]{}\]]{} and one for narrow [H[$\beta$]{}]{}.
[^5]: http://irsa.ipac.caltech.edu/applications/Radar/
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{
"pile_set_name": "ArXiv"
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---
address: 'Department of Mathematics, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France'
author:
- Teodor Banica
title: Spectral measures of free quantum groups
---
This is a report on joint work with Julien Bichon, Dietmar Bisch, Benoit Collins and Sergiu Moroianu, part of which is in preparation [@banica-bichon; @banica-bisch; @banica-collins; @banica-moroianu].
A compact quantum group is an abstract object, dual to a Hopf ${\mathbb C}^*$-algebra. Such a Hopf algebra is by definition a pair $(A,\Delta)$, where $A$ is a ${\mathbb C}^*$-algebra with unit, and $\Delta$ is a morphism of ${\mathbb C}^*$-algebras $$\Delta:A\to A\otimes A$$ subject to certain axioms, discovered by Woronowicz in the late 80’s.
The very first example is $A={\mathbb C}(G)$, where $G$ is a compact group. Here the comultiplication map $\Delta$ comes from the multiplication map $m:G\times G\to G$.
The other basic example is $A={\mathbb C}^*(\Gamma)$, where $\Gamma$ is a discrete group. Here the comultiplication is defined on generators by $\Delta(g)=g\otimes g$.
In general, a Hopf ${\mathbb C}^*$-algebra $A$ can be thought of as being of the form $$A={\mathbb C}(G)={\mathbb C}^*(\Gamma)$$ where $G$ is a compact quantum group, and $\Gamma$ is a discrete quantum group.
The free analogues of ${\mathbb C}(U(n))$, ${\mathbb C}(O(n))$, ${\mathbb C}(S_n)$ are the universal Hopf ${\mathbb C}^*$-algebras $A_u(n)$, $A_o(n)$, $A_s(n)$, constructed by Wang in the 90’s: $$\begin{aligned}
A_u(n)&=&{\mathbb C}^*\left( u_{ij}\mid u=\mbox{ unitary},\,\, \bar{u}=\mbox{ unitary}\right)\cr
A_o(n)&=&{\mathbb C}^*\left( u_{ij}\mid u=\bar{u}=\mbox{ unitary}\right)\cr
A_s(n)&=&{\mathbb C}^*\left( u_{ij}\mid u=\mbox{ magic biunitary}\right)\end{aligned}$$
Here $u=u_{ij}$ is a $n\times n$ matrix, and $\bar{u}=u_{ij}^*$. The magic biunitarity condition says that all entries $u_{ij}$ are projections, and on each row and each column of $u$ these projections are orthogonal, and sum up to $1$. See [@banica-bichon] for details.
The fundamental question is: who are these algebras?
In other words, we would like to have models for $A_u(n)$, $A_o(n)$, $A_s(n)$, where generators $u_{ij}$ correspond to explicit operators, say in some known ${\mathbb C}^*$-algebra.
We have the following models for Wang’s algebras:
– an embedding $A_u(n)\subset {\mathbb C}^*({\mathbb Z})*A_o(n)$.
– an isomorphism $A_o(2)={\mathbb C}(SU(2))_{-1}$.
– an isomorphism $A_s(n)={\mathbb C}(S_n)$ for $n=1,2,3$.
– an inner faithful representation $A_s(4)\to {\mathbb C}(SU(2),M_4({\mathbb C}))$.
Here the middle assertions are easy, and provide models for $A_o(2)$, $A_s(2)$, $A_s(3)$. The first assertion is proved in my thesis, and reduces study of $A_u(n)$ to that of $A_o(n)$ (in particular, we get a model for $A_u(2)$). As for the last assertion, this is based on a realisation of the universal $4\times 4$ magic biunitary matrix, obtained with S. Moroianu by using the magics of Pauli matrices [@banica-moroianu].
All proofs are based on the following key lemma.
A morphism $(A,u)\to (B,v)$ is faithful if and only if the spectral measures of $\chi(u)$ and of $\chi(v)$ are the same.
Here $\chi(w)=w_{11}+w_{22}+\ldots +w_{nn}$ is the character of $w=w_{ij}$. In the self-adjoint case the spectral measure of $\chi(w)$ is the real probability measure coming from Haar integration; in the general case, it is the $*$-distribution.
The key lemma tells us that in order to find models for $A_u(n)$, $A_o(n)$, $A_s(n)$, the very first thing to be done is to compute the spectral measure of $\chi(u)$. This was done by myself in the 90’s, with the following conclusion.
We have the following spectral measures:
– for $A_u(n)$ the variable $\chi(u)$ is circular.
– for $A_o(n)$ the variable $\chi(u)$ is semicircular.
– for $A_s(n)$ the variable $\chi(u)$ is free Poisson.
This gives some indication about where to look for models (and that is how theorem 1 was found!), but in general, the fundamental problem is still there.
The recent work [@banica-bichon; @banica-bisch; @banica-collins] focuses on three related problems.
A first natural question is to find analogues of theorem 2, for other classes of universal Hopf algebras. One would like of course to investigate “simplest” such Hopf algebras, and according to general theory of Jones and Bisch-Jones (the “2-box” philosophy), these are algebras $A(X)$, with $X$ finite graph.
If $X$ is a finite graph having $n$ vertices, the algebra $A(X)$ is by definition the quotient of $A_s(n)$ by the commutation relation $[u,d]=0$, where $d$ is the Laplacian of $X$. This algebra $A(X)$ corresponds to a so-called quantum permutation group.
In joint work with J. Bichon [@banica-bichon] we investigate several formulae of type $\mu(X\times Y)=\mu(X)\times\mu (Y)$, where $\mu(Z)$ is the spectral measure of the character of $A(Z)$. In particular we obtain evidence for the following conjecture.
We have an equality of spectral measures $$\mu(X*Y)=\mu(X)\boxtimes\mu(Y)$$ where $X,Y$ are colored graphs, $X*Y$ is obtained by “putting a copy of $X$ at each vertex of $Y$”, and $\boxtimes$ is Voiculescu’s free multiplicative convolution.
We prove this statement in two simple situations: one using work of Bisch-Jones and Landau, the other one using work of Nica-Speicher and Voiculescu. In the general case we have no proof, but we suspect that our free product operation \* is a a graph-theoretic version of the free product operation for planar algebras discovered by Bisch and Jones. See the report of Bisch in these Proceedings.
A second natural question is to find finer versions of theorem 2, with $\chi(u)$ replaced by arbitrary coefficients of $u$. This would no doubt give more indication about what models for $A_u(n)$, $A_o(n)$, $A_s(n)$ should look like.
We are currently investigating this problem, in joint work with B. Collins [@banica-collins]. The idea is to use an old idea of Weingarten, recently studied in much detail by Collins and Sniady, for classical groups. So far, we have several results for $A_o(n)$, including a general integration formula, and a formula for moments of diagonal coefficients of the form $o_{sn}=u_{11}+\ldots +u_{ss}$.
The odd moments of $o_{sn}$ are all $0$, and the even ones are given by $$\int o_{sn}^{2k}=Tr(A_{kn}^{-1}A_{ks})$$ where $A_{kn}$ is the Gram matrix of Temperley-Lieb diagrams in $TL(k,n)$.
As a corollary, the normalised variable $(n/s)^{1/2}o_{sn}$ with $n\to\infty$ is asymptotically semicircular. We are trying now to get more information about $o_{sn}$, along with some similar results for variables $u_{sn},s_{sn}$, corresponding to $A_u(n)$, $A_s(n)$.
A third natural question is whether similar problems can be asked about subfactors. A much studied invariant of subfactors (Jones, Bisch-Jones) is a series with integer coefficients, called Poincaré series. In case the subfactor comes from a Hopf ${\mathbb C}^*$-algebra $(A,v)$, the Poincaré series is nothing but the Stieltjes transform of the spectral measure of $\chi(v)$. In other words, the question is to compute the measure-theoretic version of the Poincaré series, for various subfactors.
We investigate this problem for subfactors of index $\leq 4$, in joint work with D. Bisch [@banica-bisch]. These are known to be classified by ADE graphs. We use Jones’ change of variables $z=q/(1+q)^2$: at level of measures, this leads to consideration of a certain probability measure $\varepsilon$ supported by the unit circle, that we call spectral measure of the graph. Our key remark is that $\varepsilon$ is given by a nice formula.
The spectral measures of $AD$ graphs are given by $$\begin{aligned}
A_{n-1}&\rightarrow&\alpha\,d_n\cr
D_{n+1}&\rightarrow&\alpha\,d^\prime_n\cr
A_\infty&\rightarrow&\alpha\,d\cr
A_{2n}^{(1)}&\rightarrow&d_n\cr
A_{-\infty,\infty}&\rightarrow&d\cr
D_{n+2}^{(1)}&\rightarrow&d^\prime_1/2+d_n/2\cr
D_\infty&\rightarrow&d^\prime_1/2+d/2\end{aligned}$$ where $d,d_n,d^\prime_n$ are the uniform measures on the unit circle, on $2n$-th roots of unity, and on $4n$-th roots of unity of odd order, and $\alpha(u)=2Im(u)^2$.
This is closely related to work of Reznikoff; she computes moments of the spectral measures of ADE graphs by counting planar modules, via a theorem of Jones.
We are trying now to find a nice formula for $E$ graphs, plus of course to formulate some kind of relevant question regarding graphs of small index $>4$.
[99]{} T. Banica and J. Bichon, *Free product formulae for quantum permutation groups*, submitted for publication, arxiv:math.QA/0503461.
T. Banica and D. Bisch, *Spectral measures of small index graphs*, in preparation.
T. Banica and B. Collins, *Integration over compact quantum groups*, in preparation.
T. Banica and S. Moroianu, *On the structure of quantum permutation groups*, Proc. Amer. Math. Soc., to appear, arxiv:math.QA/0411576.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'Despite recent progress, the complete understanding of the perturbations of charged, rotating black holes as described by the Kerr-Newman metric remains an open and fundamental problem in relativity. In this study, we explore the existence of families of quasinormal modes of Kerr-Newman black holes whose decay rates limit to zero at extremality, called zero-damped modes in past studies. We review the nearly extremal and WKB approximation methods for spin-weighted scalar fields (governed by the Dudley-Finley equation) and give an accounting of the regimes where scalar zero-damped and damped modes exist. Using Leaver’s continued fraction method, we verify that these approximations give accurate predictions for the frequencies in their regimes of validity. In the nonrotating limit, we argue that gravito-electromagnetic perturbations of nearly extremal Reissner-Nordström black holes have zero-damped modes in addition to the well-known spectrum of damped modes. We provide an analytic formula for the frequencies of these modes, verify their existence using a numerical search, and demonstrate the accuracy of our formula. These results, along with recent numerical studies, point to the existence of a simple universal equation for the frequencies of zero-damped gravito-electromagnetic modes of Kerr-Newman black holes, whose precise form remains an open question.'
author:
- Aaron Zimmerman
- Zachary Mark
bibliography:
- 'KNArxiv2.bib'
title: 'Damped and zero-damped quasinormal modes of charged, nearly extremal black holes'
---
Introduction
============
The Kerr-Newman (KN) black hole [@Newman:1965my; @Adamo:2014baa; @Chrusciel:2012jk] is the most general four-dimensional black hole solution to the electro-vacuum Einstein field equations, provided that the unphysical magnetic and NUT charges are set to zero. While astrophysical black holes cannot maintain significant charge [@Gibbons1974; @Blandford:1977ds], charged black holes remain fundamental objects of study in gravitational and quantum theories. KN black holes are the simplest charged, rotating objects allowed by relativity, and so provide a natural arena to study the interplay of electromagnetism and gravity. However, perturbations of these black holes have until recently been poorly understood, even many years after their discovery.
As in other black hole solutions, perturbed KN black holes possess a spectrum of decaying, resonant oscillations. These quasinormal modes (QNMs) [@Kokkotas1999; @Berti2009] are excited by transient sources, and they decay as energy flows into the black hole horizon and outward to asymptotic infinity. In simpler black hole solutions, such as the rotating Kerr black hole [@Kerr:1963ud; @Teukolsky:2014vca], the quasinormal modes can be understood as the eigensolutions to systems of ordinary differential equations, with the QNM frequencies given by the eigenvalues. The study of QNMs is an essential topic in understanding the structure of black hole spacetimes. QNMs play a role in gravitational wave astrophysics (e.g. [@Dreyer2004; @Berti:2005ys; @Barausse:2014tra]) where they make up the “ringdown” following the birth of a black hole or the merger of two black holes; have connections to quantum field theories through the AdS/CFT correspondence [@Son:2007vk]; and are potentially linked to quantum mechanical excitations of black holes (see the extensive references in [@Berti2009]).
While much is known about the spectrum of Kerr black holes and the nonrotating, charged Reissner-Nordstr[ö]{}m (RN) black holes, the investigation of the QNMs of KN black holes has proven difficult outside of the scalar case discussed below. Gravitational and electromagnetic perturbations of the Kerr black hole can be tackled by studying spin-weighted scalar fields propagating on the black hole background. These scalars obey a master equation [@Teukolsky1973], which separates into coupled ordinary differential equations. When these same methods are applied to applied to the KN black hole, the result is a system of equations describing coupled gravitational and electromagnetic perturbations [@ChandraBook; @Dias:2015wqa]. While these equations can still be expanded in frequency and azimuthal harmonics due to the symmetries of the spacetime, they have not been separated in the remaining coordinates. In contrast, for the spherically symmetric RN black hole, separation is possible and equations for coupled “gravito-electromagnetic” (GEM) perturbations can be derived [@Moncrief:1974ng; @Moncrief:1974gw; @Zerilli:1974ai]. Using these equations, the QNMs of RN have been extensively studied [@Berti2009].
The difficulties in analyzing the perturbations of KN black holes led to the exploration of simpler wave equations on the KN background, in the hope that they might provide a reasonable approximation to the full problem. In particular, Kokkotas and Berti studied the QNMs of the Dudley-Finley (DF) equation on the KN backgrounds [@Kokkotas:1993ef; @Berti:2005eb]. The DF equation [@Dudley:1977zz; @Dudley:1978vd] describes the propagation of spin-weighted test fields in various spacetimes. Of particular interest is type D spacetimes such as KN, where the DF equation separates. In this case, the DF equation can be reduced to a coupled eigenvalue problem for the QNMs, just as for the perturbations of the Kerr black hole. In the spin-zero case, $s=0$, the DF equation reduces to the wave equation for a massless, uncharged scalar field, $\nabla^\mu \nabla_\mu \psi = 0$. Thus the analysis of the $s = 0$ DF equation yields the true scalar QNMs of the KN black hole. For $s\neq 0$, the QNMs can only be an approximation (and possibly a poor one) to the QNMs of the gravitational and electromagnetic perturbations of KN. The DF equations assume that each test field is treated independently[^1], which does not correctly capture the coupling between the electromagnetic and gravitational perturbations of KN. Although there has been some confusion in the literature on this point, there is no reason [*a priori*]{} to expect that the DF equation provides anything more than a qualitative description of the QNMs of KN. Recently, a great deal of progress has been made in understanding the true GEM modes of KN black holes. New approximation techniques have allowed for the investigation of KN black holes which deviate from Kerr and RN black holes by small amounts. Slowly rotating, charged black holes were treated using using a matrix-valued continued fraction method [@Pani:2013ija; @Pani:2013wsa]. Following this, Mark [*et al.*]{} [@Mark:2014aja] tackled the case of a weakly charged KN black hole, using an eigenvalue perturbation method adapted for quasinormal modes [@Yang:2014tla; @Zimmerman:2014aha]. Most excitingly, challenging numerical studies have allowed the exploration of the QNMs for the full range of angular momentum and charge parameters of KN black holes for the first time [@Zilhao:2014wqa; @Dias:2015wqa]. The work of Dias, Godazgar, and Santos [@Dias:2015wqa] is especially noteworthy, presenting a complete scan of the $(a,\,Q)$ parameter space of KN black holes, for the lowest overtone and $l\leq 3$. While this comprehensive information is now in principle available, an analytic understanding of these QNMs remains an important goal.
One region of parameter space which is of special interest and which may be amenable to analytic techniques is the nearly extremal KN (NEKN) black holes. Kerr-Newman black holes have an extremal combination of charge and angular momentum, which causes their inner and outer horizons to coalesce, and where the surface gravity of the black hole vanishes. Beyond these extremal combinations of charge and spin the singularity within the black hole is naked to asymptotic infinity, destroying any notion of causality. As black holes enter the nearly extremal regime a new approximation scheme becomes available, in terms of the small distance from extremality. For the Kerr spacetime, the Teukolsky equation simplifies in this limit and approximate formulas for the QNM frequencies are known. These formulas describe weakly damped QNMs with the real part of the frequency $\omega$ given by $m \Omega_H$ and the imaginary part proportional to $\sqrt{\epsilon}$ [@TeukolskyPress1974; @Detweiler1980; @Andersson2000; @Glampedakis2001; @Hod2008a; @Yang:2012he; @Yang:2012pj; @Yang:2013uba], where $\epsilon = 1- a/M \ll 1$ is the small expansion parameter. These modes have been well studied, and were called the zero-damped modes (ZDMs) in [@Yang:2012pj; @Yang:2013uba] to distinguish them from a second family of modes, the damped modes (DMs) whose decay remains nonzero in the limit $\epsilon \to 0$. The ZDMs can be analyzed in the nearly extremal limit using a matched asymptotic expansion of the radial equation, and appear to be related to the mathematical horizon instability of extremal black holes [@Aretakis:2012ei; @Aretakis:2013dpa; @Reiris:2013efa]. Meanwhile, the DMs must be treated using a different method, either through numerical exploration or the use of a WKB analysis in Kerr [@Yang:2012he].
Mark [*et al.*]{} [@Mark:2014aja] were able to deal with the case of rapidly rotatng, weakly charged KN black holes. The results of that study hint at the existence of ZDMs in these spacetimes, and surprisingly also show that the DF equation alone provides the correct small-charge perturbation to the ZDM frequencies of Kerr. In addition, the recent numerical studies of KN QNMs [@Zilhao:2014wqa; @Dias:2015wqa] provide strong evidence that for NEKN black holes, the QNM frequencies are described by an equation like that obeyed by the ZDMs of Kerr. With these results, one might hope that all the modes of NEKN might be described by a simple equation.
These facts motivate the exploration of both the DF equation and the full GEM equations of KN in the nearly extremal limit. The goal of this study is more modest; we give an accounting of those cases where the full problem simplifies. In Sec. \[sec:DF\] we study the DF equation for NEKN black holes. We identify ZDMs which exist for any value of $a$, including the nonrotating case of the nearly extremal RN (NERN) black hole, by combining analytic approximations and numerical mode searches. A WKB analysis valid for scalar fields on KN [@HodEikonal2012; @Zhao:2015pqa] identifies where modes with nonzero decay exist in the parameter space of extremal KN black holes, implying that in these cases the QNM spectrum bifurcates into two branches as the black hole approaches extremality at fixed angular momentum. This bifurcation occurs for certain QNM families of Kerr, as discussed in [@Yang:2012pj; @Yang:2013uba]. We discuss this WKB analysis in Sec. \[sec:WKBanalysis\]. In Sec \[sec:DFNumerics\] we use a numerical search to determine the accuracy of the nearly extremal and WKB approximations, and we also find separated families of DMs and ZDMs. Note that the frequency formula we derive here for scalar modes of NEKN black holes was presented without derivation in recent note [@Hod:2014uqa]. Motivated by these results for scalar fields, in Sec. \[sec:RN\] we show that ZDMs also exist in the case of the GEM modes of NERN, using an analytic approximation and matching ansatz. We confirm the existence of these modes with an explicit numerical search. These modes are purely decaying with a small decay rate, and they appear to have been overlooked despite a long history of study of the QNMs of RN. Such modes give additional support to the conjecture that ZDMs exist for all nearly extremal black holes, but they also demonstrate the failure of the DF equation to accurately capture the GEM frequencies in this limit.
Finally, we discuss prospects for the open problem of the coupled GEM perturbations of NEKN in Sec. \[sec:Conclusions\]. In this paper, we do not attempt to discuss the topic of charged and massive fields on KN backgrounds where superradiant instabilities are found to arise (see e.g. [@Brito:2015oca] for a recent review).
Throughout this paper we use geometric units so that charge and mass have units of length, setting $G = c =1$. We provide a reference for the definitions of some of the important variables used in Table \[tab:VarList\].
------------------------------------------------------------ ------------------------------------------------------- -------------------------
$r_\pm$ $\displaystyle M \pm \sqrt{M^2 - a^2 -Q^2}$ Horizon positions
$\sigma$ $\displaystyle \frac{r_+ - r_-}{r_+}$ Near-extremal parameter
$k$ $\omega - m \Omega_H$ Corotating frequency
$\hat{\omega}$ $\omega r_+$ Dimensionless frequency
$ \Omega_H$ $\displaystyle \frac{a}{r_+^2 +a^2}$ Horizon frequency
$\kappa$ $\displaystyle \frac{\sigma r_+}{2(r_+^2 +a^2)}$ Surface gravity
$\mathcal J^2$ $(m \Omega_H)^2(6M^2 +a^2) - A$ WKB indicator of DMs
$\delta^2$ $\displaystyle 4 \hat{\omega}^2- (s+1/2)^2 - \lambda$ DF matching parameter
$\varpi$ $\displaystyle \frac{k}{\kappa} - i s $ DF matching parameter
$\zeta$ $2m r_+ \Omega_H - i s$ DF matching parameter
$x$ $\displaystyle \frac{r-r_+}{r_+} $ Radial parameter
$\displaystyle{\genfrac{}{}{0pt}{}{\omega_R,} {\omega_I}}$ $\omega = \omega_R + i \omega_I$ Frequency components
------------------------------------------------------------ ------------------------------------------------------- -------------------------
: \[tab:VarList\] List of relevant definitions
The Dudley-Finely equation for nearly extremal Kerr-Newman black holes {#sec:DF}
======================================================================
In this section we discuss the QNMs of the Dudley-Finley equation for nearly extremal Kerr-Newman black holes.
Due to the similarity between the Teukolsky equation and the DF equation, Leaver’s continued fraction method [@Leaver1986], used to accurately compute the QNMs of Kerr, extends easily to the DF equation in the KN background [@Berti:2005eb]. This method can in principle provide accurate QNM frequencies for any spin angular momentum, charge, and harmonic. Since our interest is to develop further analytic understanding of the QNM frequencies in the limit of nearly extremal black holes, we use the nearly extremal and WKB approximations to explore the ZDM and DM frequencies. We then use Leaver’s method to confirm these approximations, and to measure their error.
Kerr-Newman black holes
-----------------------
Kerr-Newman black holes are parametrized by their mass $M$, specific angular momentum $a$, and charge $Q$ when magnetic and NUT charges are neglected. One convenient way to represent the metric for the KN spacetime in Boyer-Lindquist coordinates [@BoyerLindquist1967] is obtained by writing the line element of Kerr in terms of the second degree polynomial $\Delta = r^2 - 2 M r+a^2$. The roots of this polynomial are the inner ($r_-$) and outer $(r_+)$ horizons of Kerr, $\Delta = (r-r_+)(r-r_-)$. The Kerr-Newman metric follows from using the appropriate definition of $\Delta$ when charge is included. This form of the line element is [@MTW] $$\begin{aligned}
ds^2 &= - \frac{\Delta}{\rho^2 } \left(dt - a \sin^2\theta d \phi \right)^2 + \frac{\rho^2}{\Delta} dr^2 + \rho^2 d\theta^2 \notag \\
& \, + \frac{\sin^2 \theta}{\rho^2} \left[a dt - (r^2+a^2)d\phi\right]^2
\,, \\
\Delta &= r^2 - 2 M r +a^2 +Q^2 \,, \\
\rho^2 &= r^2 +a^2 \cos^2 \theta\,.\end{aligned}$$ The corresponding vector potential takes the form $$\begin{aligned}
A_\mu dx^\mu =\frac{Qr}{\rho^2}\left(dt -a\sin^2\theta d\phi \right)\,.\end{aligned}$$ The outer and inner horizons are located at $$\begin{aligned}
r_\pm = M \pm \sqrt{M^2 - a^2 - Q^2} \,.\end{aligned}$$ In the nearly extremal limit, where the inner and outer horizons approach each other, we define the small parameter[^2] $\sigma \ll1$ as
$$\begin{aligned}
\sigma = \frac{r_+ - r_-}{r_+} \approx 2 \sqrt{1 - a^2/M^2 - Q^2/M^2} \,.\end{aligned}$$
It is also useful to recall the expression for the surface gravity $\kappa$ of the KN black hole [@Wald1984] $$\begin{aligned}
\kappa = \frac{r_+ - r_-}{2(r_+^2 +a^2)} = \frac{\sigma r_+}{2(r_+^2 +a^2)} \,.\end{aligned}$$ We see that $r_+ \kappa \ll 1$ in the nearly extremal limit.
The Dudley-Finley equation
--------------------------
We turn to the analysis of the DF equations in the KN spacetime. These equations and their analysis closely parallels the treatment of scalar, electromagnetic, and gravitational perturbations of the Kerr spacetime by spin-weight $s$ scalars ${}_s \psi$, which obey a separable master equation [@Teukolsky1973]. Just as in Kerr, we expand the spin-weighted scalars ${}_s \psi$ in frequency and azimuthal harmonics as $$\begin{aligned}
{}_s \psi & = \sum_{lm} \int d\omega \, e^{-i(\omega t - m \phi)} {}_s R_{lm\omega}(r) {}_sS_{lm\omega}(\theta) \,.\end{aligned}$$ With this expansion, the DF wave equations in the KN spacetime separate [@Kokkotas:1993ef] and are nearly identical to the corresponding equations in Kerr.
The angular functions $ {}_sS_{lm\omega}(\theta)$ are spin-weighted spheroidal harmonics and obey the angular Teukolsky equation [@Teukolsky1973; @Fackerell1977; @Berti2009], $$\begin{aligned}
\label{eq:TeukS}
\csc \theta & \frac{d}{d\theta} \left( \sin \theta \frac{dS}{d\theta}\right) + V_\theta S = 0 \,, \\
V_\theta = & \, a^2 \omega^2 \cos^2 \theta - m^2 \csc^2 \theta - 2 a \omega s \cos \theta \notag \\ &
- 2 m s \cot \theta \csc \theta -s^2 \cot^2 \theta + s + {}_s A_{lm \omega} \,.\end{aligned}$$ Here the angular separation constants for each harmonic are denoted ${}_s A_{lm\omega}$. In the limit of a Schwarzschild black hole ($a \to 0$) they simplify, $A \to l(l+1) - s(s+1)$.
The radial functions ${}_s R_{lm\omega}(r)$ obey a second order differential equation. In the source-free case it is $$\begin{aligned}
\label{eq:TeukR}
\Delta^{-s} & \frac{d}{dr} \Delta^{s+1} \frac{dR}{dr} + V_r R = 0 \,, \\
\label{eq:PotR}
V_r & = \frac{K^2 + i s K \partial_r \Delta }{\Delta} - 2 i s \partial_r K - {}_s\lambda_{lm\omega} \,, \\
K & = - \omega(r^2+a^2) + am \,, \\
{}_s \lambda_{lm \omega} & = {}_sA_{lm\omega} - 2 a m \omega +a^2 \omega^2\,.\end{aligned}$$ Here and elsewhere we suppress spin-weight and harmonic indices where there is no danger of confusion.
It is useful to define a tortoise coordinate $r_*$ and rewrite the radial equation in terms of a different function ${}_s u_{lm\omega}$. These are defined as follows: $$\begin{aligned}
\frac{dr_*} {dr}& = \frac{r^2 +a ^2} {\Delta} \,, & u & = \Delta^{s/2} \sqrt{r^2 +a^2} R \,.\end{aligned}$$ With these substitutions, the radial equation becomes $$\begin{aligned}
\label{eq:Teuku}
\frac{d^2 u}{dr_*^2}& + V_u u = 0 \,,\\
V_u & = \frac{K^2 + 2 i s K (r- M) + \Delta(4 i s \omega r - \lambda)}{(r^2+a^2)^2} - G^2 -\frac{dG}{dr_*} \,,\\
G & = \frac{\Delta r}{(r^2+a^2)^2} + \frac{s \partial_r \Delta}{2(r^2+a^2)} \,.\end{aligned}$$ Equation makes it apparent that the asymptotic solution as $r_* \to - \infty$ (as $r\to r_+$) is the same as in Kerr but with a change in the definition of $k$ and $\Delta$, $$\begin{aligned}
\label{eq:Ingoingu}
u &\sim \exp\left[\pm \frac{s}{2} \frac{\sigma r_+}{r_+^2 +a^2} r_* \right]e^{\pm i k r_*} = \Delta^{\pm s/2} e^{\pm i k r_*}
\,, \\
\label{eq:Defk}
k& = \omega - m \frac{a}{r_+^2 +a^2} = \omega - m \Omega_H \,,\end{aligned}$$ where we have identified the horizon frequency $\Omega_H$ in the final expression. The asymptotic solution $u \sim \Delta^{-s/2} e^{-ikr_*}$ corresponds to waves traveling into the horizon [@Teukolsky1973]. The second asymptotic solution corresponds to waves which are directed out of the horizon, and we discard this unphysical solution. Meanwhile, the outer asymptotic solutions remain unchanged from Kerr to KN, $$\begin{aligned}
\label{eq:Outgoingu}
u \sim r^{\pm s} e^{\mp i \omega r_*} && r \to \infty \,.\end{aligned}$$ These solutions correspond to ingoing waves for $u\sim e^{+i\omega r_*}$ and outgoing waves for $u\sim e^{-i\omega r_*}$. The QNMs of the DF equation can be found by solving the radial equation with an outgoing boundary condition; only certain frequencies allow for solutions which also obey the boundary condition at the horizon.
Both the matched asymptotic expansion and the WKB analysis of Kerr carry directly over to the case of the DF equation in KN. In the remainder of this section, we discuss these results, which demonstrate the bifurcation of the scalar spectrum of the NEKN black hole into ZDMs and DMs. These results also set the stage for an understanding of the existence of both ZDMs and DMs for the coupled GEM perturbations of NERN black holes as discussed in Sec. \[sec:RN\]. For NEKN black holes, either the spin parameter $a$ or the charge $Q$ can be eliminated in favor of $\sigma$. We choose to retain an explicit dependence on $a$ in our equations.
Matching analysis of the Dudley-Finley equation in nearly extremal Kerr Newman {#sec:Matching}
------------------------------------------------------------------------------
To investigate the nearly extremal modes, we must use the technique of matched asymptotic expansions[^3]. Our analysis closely follows the analysis of the Kerr case in Yang [*et al.*]{} [@Yang:2013uba] and the references therein. For this we define a new coordinate variable $x$ and dimensionless frequency $\hat \omega$ via $$\begin{aligned}
x & = \frac{r - r_+}{r_+} \,, & \hat \omega &= \omega r_+ \,.\end{aligned}$$ The method splits the region exterior to the black hole into an outer, far-field region $x \gg \sigma$, and an inner, near-horizon region $x \ll 1$. The method only works for ZDMs; that is, we assume from the beginning that the frequency has the form $\omega = m \Omega_H + O(\sigma)$. First we look at radial equation in the outer region.
### The outer solution
We rewrite the Teukolsky equation for $R$, Eq. , in terms of $x,\, \hat \omega$ and substitute $k$ from Eq. . Using the approximation $\sigma\ll x$, we arrive at the same outer solution as in Kerr, $$\begin{aligned}
\label{eq:TeukFar}
x^2 R'' + 2 (s+1) x R' + \left[\hat \omega^2 (x+2)^2 + 2 i s \hat \omega x - \lambda \right] R = 0 \,.\end{aligned}$$ We define $\delta$ through $$\begin{aligned}
{}_s\delta_{lm\omega}^2 = 4 \hat \omega^2 - (s+1/2)^2 - {}_s\lambda_{lm\omega} \,,\end{aligned}$$ and we find the solution to Eq. in terms of confluent hypergeometric functions ${}_1F_1$ [@nist], $$\begin{aligned}
\label{eq:OuterSln}
R = A &e^{-i \hat \omega x} x^{ -1/2 - s + i \delta} \notag \\
& \times {}_1F_1(1/2-s + i \delta + 2i \hat \omega, 1 + 2 i \delta, 2 i \hat \omega x) \notag \\
& + B (\delta \to - \delta) \,,\end{aligned}$$ where $(\delta \to - \delta)$ indicates that the same functions are repeated with the sign of $\delta$ reversed. In NEKN $\delta$ no longer takes the explicit form it has in Kerr, $\delta_{\rm K}^2 = 7m^2 / 4 -(s+1/2)^2 - A$, because while the frequency still becomes nearly proportional to the horizon frequency, $\omega \to m \Omega_H$, here the horizon frequency is a varying function of $a$.
The outgoing wave condition for QNM frequencies imposes one constraint on $A$ and $B$. This condition is derived by expanding the ${}_1F_1$ functions at large $x$ into ingoing and outgoing parts, and requiring a cancellation of the ingoing waves. We provide the condition in Appendix \[sec:MatchingApp\], since there are a few minor sign errors[^4] in [@Yang:2013uba]. In order to get a second condition and derive the QNMs frequencies, we turn to the inner solution in the near-horizon region.
### The inner solution
The next step is to assume that both $x \ll1$ and $\sigma \ll 1$ but make no assumptions about their relative size. We use Eq. as our starting point, make the substitutions for $x$ and $\sigma$, and further note that the quantity $M k / \sigma$ can be order unity. Expanding the potential $V_u$ to second order in the small quantities while keeping factors of $k/\sigma$ intact gives our desired potential. A second coordinate transform brings the radial equation into a simpler, more tractable form.
The second transformation is performed by noting that in the near-horizon region, we can approximate $r_*$ by $$\begin{aligned}
r_* \approx \frac{1}{2\kappa} \ln \left(\frac{x}{x+\sigma} \right) = \frac{r_+^2 +a^2}{\sigma r_+} \ln \left(\frac{x}{x+\sigma} \right) \,.\end{aligned}$$ We then define $y$ through $$\begin{aligned}
y & = e^{2 \kappa r_* } \approx \frac{x}{x+\sigma} \,, &
\frac{d}{dr_*} & = 2 \kappa y \frac{d}{dy} \,,\end{aligned}$$ which in the Kerr spacetime limits to $y = \exp(\sqrt{2 \epsilon} r_*)$. After transforming to the $y$ coordinate the second derivatives in the radial equation become $$\begin{aligned}
\frac{d^2 u}{dr_*^2} & = \left( 2 \kappa \right)^2 \left(y^2 \frac{d^2 u}{dy^2} + y \frac{d u}{dy} \right) \,.\end{aligned}$$ Substituting $y$ for $x$ in the expansion of $V_u$ and dividing out the prefactor gives us our differential equation for $u$. It is useful to make the following definitions[^5], $$\begin{aligned}
\varpi & = \frac{k}{\kappa} - i s \,, & \zeta = 2 m r_+ \Omega_H - i s \,.\end{aligned}$$ Then we have $$\begin{aligned}
\label{eq:NHdiffeq}
0 & = y^2 u'' + y u' + V_y u \,, \\
\label{eq:Ypot}
V_y & = \frac{\varpi^2}{4} + \frac{y \zeta (\varpi -\zeta)}{1-y} + \frac{y(\delta^2 + 1/4)}{(1-y)^2} \,.\end{aligned}$$ The form of these equations is identical to the ones derived in the Kerr spacetime [@Yang:2013uba] and the parameters here have the appropriate form in the $Q \to 0$ limit[^6].
The inner solution is written in terms of hypergeometric functions ${}_2 F_1$, $$\begin{aligned}
\label{eq:NHsln}
u & = y^{-p} (1-y)^{-q} {}_2 F_1 ( \alpha, \beta, \gamma, y)\,,\end{aligned}$$ with $$\begin{aligned}
p & = i \varpi/2 \,, & q & = -1/2-i\delta \,, \\
\alpha & = 1/2 + i \zeta - i \varpi + i \delta \,, & \beta & = 1/2 -i \zeta + i \delta \,, \\
\gamma & = 1 - i \varpi \,.\end{aligned}$$ Because of the form of this solution and the fact that the outer solution is the same as for Kerr (save for the fact that $\delta$ depends on $a$) the matching proceeds identically to that case. This allows us to calculate the ZDM frequencies for KN.
### Matching and zero-damped modes {#sec:Matching}
The two approximate radial solutions are matched in the region $\sigma \ll x \ll 1$ where both approximations are valid. In this regime the confluent hypergeometric functions simplify, ${}_1 F_1 \to 1$. We must apply an inversion to the hypergeometric function ${}_2 F_1$ using the variable $z = 1-y$, and then take $z \to 0$ as $y\to 1$; this limit essentially takes us to the outer edge of the near-horizon region. We equate the two expressions for the radial wave functions, taking care to match the coordinates and the particular wave function used ($u$ versus $R$). We give further details in Appendix \[sec:MatchingApp\]. With the outgoing wave condition, the matching can be achieved if the argument of a particular Gamma function is near its poles at the negative integers; specifically we require $$\begin{aligned}
\Gamma[\gamma - \beta] = \Gamma[-n - i \eta] \,,\end{aligned}$$ where $\eta$ is a small correction which guarantees that the matching holds. Plugging in all the preceding expressions gives $$\begin{aligned}
\label{eq:DFfreq}
\omega = & \, m \Omega_H + \kappa \left[2 m r_+ \Omega_H - \delta - i \left( n + \frac 12\right) + \eta \right] + O(\sigma^2) \notag \\
=& \frac{ma}{M^2+a^2} - \frac{M \sigma}{2(M^2+a^2)} \left[\delta + i \left(n + \frac12\right) \right]
\notag \\ &
+ O(\sigma^2, \sigma \eta) \,.\end{aligned}$$ To get to the final line, we used the definition of $\kappa$ and expanded $\Omega_H$ in small $\sigma$. Our final expression for $\omega$ matches the Kerr result in the limit $Q =0$, where $\epsilon = 1- a/M \ll 1$, $$\begin{aligned}
M \omega_K & = \frac{m}{2} - \sqrt{\frac{\epsilon}{2}} \left[ \delta + i \left(n + \frac 12\right) \right]\,.\end{aligned}$$
For KN black holes, the smaller horizon frequency makes a smaller positive contribution to $\delta^2$ than in Kerr, and it is easier for $\delta^2$ to become negative. In the Kerr spacetime, the implication of an imaginary value of $\delta$ is the existence of DMs (due to the close relation between $\delta^2 > 0$ and the criterion for a WKB peak outside the horizon, discussed below in \[sec:DampedModes\]). An imaginary value of $\delta$ also turns various oscillatory terms in the radial wave function into decaying terms, and suppresses the collective excitation of many ZDM overtones, as discussed in [@Andersson2000; @Glampedakis2001; @Yang:2013uba]. Further consequences include a suppression of multipolar fluxes from a particle orbiting at the innermost stable circular orbit of a nearly extremal Kerr black hole for those multipoles with imaginary $\delta$ [@Gralla:2015rpa], but the full implications of an imaginary $\delta$ have not yet been explored. Finally, we note that while usually $\eta$ is extremely small, it can be significant when $\delta$ is very near zero [@Yang:2013uba]. When $\eta > \sigma$, the $O(\eta \sigma)$ corrections dominate over the $O(\sigma^2)$ terms and the explicit expression for $\eta$ given in Appendix \[sec:MatchingApp\] should be incorporated into $\omega$. For even larger values of $\eta$ the matching analysis may break down. We explore these considerations further in Sec. \[sec:DFNumerics\].
The results presented so far show that ZDMs are present in NEKN in the case of spin-$s$ test fields. We turn next to a discussion of the DMs of these test fields.
WKB analysis of the Dudley-Finley equation: Damped modes and spectrum bifurcation {#sec:WKBanalysis}
---------------------------------------------------------------------------------
The DMs cannot be accessed by the nearly extremal matching analysis discussed here, because they violate the assumption that $M k \ll 1$. To see how these DMs behave in the KN spacetime, we require a different tool. The WKB analysis of the modes of the DF equation, valid for any $(a,\ Q)$ at high frequencies, provides approximate DF frequencies, gives a criteria for their existence, and provides a unifying picture of the spectrum bifurcation of scalar waves in KN. This analysis is a straightforward generalization of the results discussed in [@Yang:2012he], and was also recently presented in [@Zhao:2015pqa], which focused on the connection between the WKB frequency formulas and the behavior of unstable null geodesics at the light sphere (see also [@Berti:2005eb]). The study [@HodEikonal2012] also extended the WKB results of [@Yang:2012pj; @Yang:2013uba] to the case of scalar modes of NEKN, deriving a condition for when the WKB formulas describe ZDMs (a condition also derived in [@Zhao:2015pqa]).
Our focus is on the insights that the WKB analysis provides us in the extremal limit, and we include it here to present a complete picture of the spectrum of scalar modes of NEKN. We give simplified WKB frequency formulas in Appendix \[sec:WKBApp\] not given in [@Zhao:2015pqa], for reference when we take the nearly extremal limit. We discuss the WKB formulas for the DMs in this limit, in addition to explicitly deriving the ZDM frequency formula given in [@HodEikonal2012]. The WKB analysis is brief enough that we include all the essential details here, with some additional formulas in Appendix \[sec:WKBApp\].
The WKB approximation is an expansion in large frequency. We define the large parameter $L = (l+1/2) \gg 1$, and we also distinguish the real and imaginary parts of $\omega$ as $\omega_R$ and $\omega_I$. With these definitions, the leading order WKB analysis dictates that the eigenvalues of the DF equation have the scalings $A_{lm}\sim O(L^2)$, $\omega_R \sim O(L)$, and $\omega_I \sim O(1)$. These quantities are parametrized in terms of $a,\, Q$, and an “inclination parameter” $\mu = m/L$ with $-1 < \mu <1$. These facts, and the dependence of the decay rate on the overtone number $n$ derived in [@Schutz:1985zz], lead to the definition of convenient rescaled quantities $$\begin{aligned}
A_{lm} = L^2 \alpha(\mu,a,Q)\,, & & \omega &= \omega_R + i \omega_I \,, \notag \\
\omega_R = L \Omega_R(\mu,a,Q)\,, & & \omega_I &= - \left(n + \frac 12\right)\Omega_I(\mu,a,Q) \,.\end{aligned}$$ A key insight drawn from the high-frequency approximation is that the QNMs correspond to rays on the unstable photon orbits of the spacetime [@Goebel1972; @FerrariMashhoon1984; @Cardoso2009; @Dolan2010; @Yang:2012he; @HodEikonal2012; @Zhao:2015pqa]. Especially relevant to this viewpoint in the KN spacetime is the work of Mashhoon [@Mashhoon1985], who studied equatorial null orbits under the assumption that they correspond to the $m=l$ QNMs, although only a full analysis of the wave equation justifies this correspondence [@Yang:2012he; @Zhao:2015pqa]. As such, the extremum of the radial potential $V_r$ takes on a central role, where it gives the radius of unstable orbits. We denote the position of the extremum as $r_0$. Although $r_0$ is in fact a minimum of $V_r$, when the problem is recast into form similar to the Schrödinger equation, it is $-V_u$ that appears as the potential for the wave function, and so we call $r_0$ the “peak” of the potential.
The WKB analysis provides an integral condition for the angular eigenvalues $A_{lm}$, along with algebraic conditions for the real and imaginary part of the frequencies $\omega$. These conditions are the Bohr-Sommerfeld quantization condition $$\begin{aligned}
\int_{\theta_-}^{\theta_+}&d\theta\sqrt{a^2\omega_R^2\cos^2\theta-\frac{m^2}{\sin^2\theta}+A^R_{l m}}=(L-|m|)\pi \,, \\
\sin^2 \theta_\pm &=\frac{2m^2}{A_{l m}+a^2 \omega^{2}_{R}\mp\sqrt{(A_{l m}+a^2 \omega^2_{R})^2+4m^2}} \,,\end{aligned}$$ and $$\begin{aligned}
\label{eq:WKBSolve}
V_u(r_0, \omega_R) = 0\,, & & \left. \partial_r V_u \right|_{r_0,\omega_R} = 0 \,,\end{aligned}$$ where $V_u$ is taken to be the leading order WKB potential $$\begin{aligned}
V_u \approx \frac{K^2 - \Delta \lambda}{(r^2 +a^2)^2} \,.\end{aligned}$$ Generally these conditions must be solved jointly, but the analysis in [@Yang:2012he] reveals that a simple secondary approximation for $A_{lm}$ provides algebraic relations for the mode frequencies which are quite accurate even in the extremal limit. We use this approximation for the WKB analysis of DF, which reads $$\begin{aligned}
\label{eq:AppxA}
\alpha \approx 1 - \frac{a^2 \Omega_R^2}{2}\left( 1- \mu^2 \right) \,.\end{aligned}$$ Then Eqs. can be reduced to a polynomial equation for $r_0$ and algebraic expressions for $\Omega_R$ and $\Omega_I$ in terms of $r_0$ and the remaining parameters. With $\omega_R$ and $r_0$, the imaginary part of the frequency can be found by evaluating the curvature of the potential at the peak, $$\begin{aligned}
\label{eq:WKBOmegaI}
\Omega_I & = \left. \frac{\sqrt{2 d^2 V_u/dr_*^2 }}{\partial_\omega V_u }\right|_{r_0,\omega_R} \,.\end{aligned}$$ This expression shows that $r_0$ must be a peak of $-V_u$, so that $V_u$ has nonnegative curvature at $r_0$.
We present the general formulas for $r_0$, $\Omega_R$, and $\Omega_I$ in Appendix \[sec:WKBApp\]. We verify in Sec. \[sec:DFNumerics\] that these formulas in general have residual errors of order $O(L^{-1})$ and $O(L^{-2})$ respectively, as occurs in Kerr [@Yang:2012he]. Focusing on the extremal and near-extremal cases, we find that the expressions simplify. As before, we eliminate $Q$ in favor of $\sigma$.
### WKB analysis in the extremal limit
We focus on the extremal case first, where $\sigma = 0$. Figure \[fig:ExWKB\] illustrates $r_0$, $\Omega_R$, and $\Omega_I$ as a function of $\mu$ for a few chosen values of $a$. We see that the features found in the WKB analysis of extremal Kerr carry through: at sufficiently high $\mu$, the peak $r_0$ is at the horizon, the frequency asymptotes to $\mu$ times the horizon frequency $\Omega_H$, and the decay rate falls to zero. We can understand this behavior by noting that the polynomial for $r_0$ derived from Eqs. reduces in the limit $\sigma = 0$ to $$\begin{aligned}
\label{eq:FullExPoly}
(r-M)^2 &[2r^2(r-2M)^2 - 4a^2r(r-2M + \mu^2(r+2M)) \notag \\
&+ a^4 (2 - 3 \mu^2 + \mu^4)] =0\,.\end{aligned}$$ This has two roots at the horizon, and for sufficiently large $\mu$ there is no other root outside of the horizon. This means that the peak of $V_u$ lies on the horizon, and we can take $r_0 = M$. With this the frequency is $$\begin{aligned}
\Omega_R = \frac{\mu a}{M^2+a^2} \,,\end{aligned}$$ which is the horizon frequency for the extremal black hole. In addition, $\Omega_I \propto \Delta(r_0)^{1/2}$, so that $\Omega_I = 0 $ when we evaluate it at the horizon radius.
Meanwhile, when $\mu$ is small enough that an additional root of Eq. lies outside of the horizon we get a nonzero decay rate even in the extremal limit. These are the DM frequencies. For smaller values of $a$ (larger values of $Q$), larger values of the inclination parameter $\mu$ are required for $r_0$ to occur at the horizon. This in turn corresponds to corotating photon orbits which lie closer to the equatorial plane.
\
\
Together with our matching results on the existence of ZDMs for all values of $a$ in the nearly extremal case, we see the spectrum bifurcation found in Kerr occurs also for the scalar QNMs of KN. An important difference between the KN black hole and Kerr is that even when $\mu = 1$, for sufficiently small values of $a$, DMs exist. This is expected: extremal RN black holes, where $a=0$, are known to possess damped scalar modes even for $m=l$.
We can derive an approximate formula for the critical $\mu$ above which the WKB results predict ZDMs by investigating the potential $V_u$ in the extremal case. For this, we set $\Omega_R = \mu \Omega_H$. We know that when $a \to M$ (Kerr), a second peak appears outside the horizon for $\mu < \mu_c \approx 0.74$. For KN, $\mu_c$ is a function of $a$. The top panel of Fig. \[fig:CritMu\] shows this extremal potential for $\mu =1$ and various values of $a$. We see the second peak emerge when $a < 0.5$ ($Q > \sqrt{3}/2$). With the above simplifications applied to $V_u$ we find that a peak exists outside the horizon when $$\begin{aligned}
\label{eq:ExPoly}
a^2 \mu^2 (r^2 + 2Mr +a^2 +3M^2) - \alpha(M^2+a^2)^2 = 0\end{aligned}$$ has a solution for $r>M$. Inserting $r=M$ into the above polynomial, we solve for the critical inclination parameter $$\begin{aligned}
\label{eq:MuCrit}
\mu_c^2 = \frac12 \left(3 + \frac{12 - \sqrt{136 +56(a/M)^2 +(a/M)^4}}{(a/M)^2} \right) \,.\end{aligned}$$ This formula was previously given in [@HodEikonal2012; @Zhao:2015pqa], and it reduces to the known result in Kerr, $\mu_c = [(15 -\sqrt{193})/2]^{1/2}\approx0.74$ [@HodEikonal2012; @Yang:2013uba]. We plot $\mu_c$ as a function of $a$ in Fig. \[fig:CritMu\]. Further setting $\mu_c^2 = 1$ we arrive at the value $a/M = 0.5$ beyond which there are no values of $\mu$ where the WKB peak remains on the horizon.
We see from this that the nearly extremal WKB analysis splits into two cases where we expect simple expressions for the frequencies: when the WKB peak is near the horizon and we have ZDMs, and when the WKB peak is supported away from the horizon we have DMs. We briefly treat each case.
### Zero-damped modes
Now we consider the case where $\sigma \ll1$, and $\mu >\mu_c(a)$. This is the case where we expect that the WKB approximation describes ZDMs. Inspired by the form of Eq. and previous work in Kerr, we define $$\begin{aligned}
\mathcal J^2 & = (m \Omega_H)^2(6M^2 +a^2) - A \,,\end{aligned}$$ so that $\mathcal J^2 > 0$ is the condition for $\mu > \mu_c$ in the extremal limit. Next, we make the guess that $r_0$ approaches the horizon at a rate controlled by $\sigma$, $r_0 = M( 1+ c \sigma)$. The solution for the peak at leading order in $\sigma$ is then $$\begin{aligned}
r_0 \approx M\left(1 + \sigma \frac{M m \Omega_H}{\mathcal J} \right) \,.\end{aligned}$$ For this peak, $\Omega_R$ becomes $$\begin{aligned}
\label{eq:WKBOmegaZDM}
\Omega_R \approx \frac{\mu a}{M^2+a^2} - \sigma \frac{M (\mathcal J/L)}{2(M^2+a^2)} \,.\end{aligned}$$ Finally, inserting these results into the expression for $\Omega_I$ gives $$\begin{aligned}
\Omega_I & \approx \frac{M \sigma}{2(M^2+a^2)}\,.\end{aligned}$$ Collecting these, the WKB approximation for $\omega$ is $$\begin{aligned}
\label{eq:WKBFreq}
\omega & = \frac{m a}{M^2 +a^2} - \frac{M \sigma}{2(M^2+a^2)} \left[ \mathcal J + i \left(n + \frac 12 \right) \right]\,.\end{aligned}$$ Equation for $\omega$ matches the Kerr limit derived in [@Yang:2012he; @Yang:2013uba]. In addition, it is the correct WKB limit of the ZDM expression since the only difference between $\delta$ and $\mathcal J$ is at subleading order in $L$.
### Damped modes {#sec:DampedModes}
When the WKB peak is outside the horizon in the extremal limit, the roots of the quartic in the square bracket of Eq. have involved analytic forms, and the expressions for $\Omega_R$ and $\Omega_I$ do not appear to admit useful simplifications.
The exception is for $\mu = \pm 1$, which gives corotating and counterrotating orbits, respectively. Due to the symmetries of the QNMs, we focus on the case $\mu = 1$ and allow $a$ to vary between positive and negative values, which interpolates between the two cases as $a$ passes through zero. The position of the peak and corresponding frequency are then $$\begin{aligned}
\label{eq:WKBFreqOuterPeak}
r_0 = 2(M-a)\,, & & \Omega_{\rm peak} = \frac{1}{4M-3a}\,. \end{aligned}$$ Here we denote the frequency at the WKB peak $\Omega_{\rm peak}$ to distinguish it from the limiting value $\Omega_H$ when both $\mu = 1$ and the peak is at the horizon. These two limits match smoothly when $a = M/2,$ $Q = \sqrt{3} M/2$. These give a decay rate $$\begin{aligned}
\label{eq:DMExDecay}
\Omega_I = \frac{(M-2a)\sqrt{2a^2-2Ma +M^2}}{\sqrt{2} (M-a)^2(4M-3a)}\,.\end{aligned}$$ This decay rate joins onto the $\Omega_I = 0$ solution as $a \to M/2$.
Numerical results for the Dudley-Finley equation {#sec:DFNumerics}
------------------------------------------------
We turn to the problem of determining the accuracy of our analytic results, Eqs. and , and Eq. , valid in the regime of $L\gg1$ and $\sigma\ll1$ respectively. We examine the residual errors in these approximations $\Delta\omega \equiv |\omega_\text{A}-\omega_\text{N}|$, where $\omega_\text{A}$ is computed with the appropriate analytic approximation and $\omega_\text{N}$ is computed numerically with sufficiently small error that we can take $\omega_\text{N}$ to be the “true” QNM frequency. To numerically compute the QNMs we use Leaver’s method [@Cook:2014cta; @Berti:2005eb; @Leaver1985; @LeaverRN]. Leaver’s method turns the coupled eigenvalue problem posed by the radial and angular equations and into a root finding problem (see [@Cook:2014cta] for a nice discussion). The eigenvalues, $\omega$ and $A_{lm}$, are reported as simultaneous roots of two infinite, convergent continued fractions, the values of which we denote $\mathcal C^r$ and $\mathcal C^\theta$: $$\begin{aligned}
\label{eq:cfform}
{\mathcal C^r} = \beta_0^r-\frac{\alpha_0^r\gamma_1^r}{\beta_1^r-}\frac{\alpha_1^r\gamma_2^r}{\beta_2^r- \dots} \,.\end{aligned}$$ The indexed greek letters $\alpha^r_i$, $\beta^r_i$, $\gamma^r_i$ are functions of $\omega$, $A$, $a$, $Q$, $l$, and $m$, and the same equation describes $\mathcal C^\theta$ in terms of $\alpha^\theta_i$, $\beta^\theta_i$, $\gamma^\theta_i$. These functions are given in [@Berti:2005eb]. To implement Leaver’s method, $\mathcal C^r$ and $\mathcal C^\theta$ must be truncated, and the resulting expressions are subjected to a numerical root finding algorithm. We use $500+$ terms in the continued fractions and *Mathematica’s* FindRoot routine. For the purposes of analyzing the accuracy of our analytic formula, we ensure that our numerical errors are orders of magnitude smaller than the errors in the analytic approximations.
### Confirmation of the DF WKB results {#sec:WKBnum}
\[fig:eikonal\]\
To confirm the WKB predictions for the QNMs of the DF equations, we calculate the analytic $\omega_{\rm A}$ by solving Eqs. \[yielding Eq. \] and . Our closed form expressions assume the approximation for $A_{lm}$ given by Eq. . When using Leaver’s method to find numerical values $\omega_{\rm N}$, we find that seeding the root search at large $l$ is challenging. To overcome this, we use an accurate approximation for the angular eigenvalue $A_{lm}$ presented in [@Berti:2005gp] (which is especially good at large $l$), leaving only a one-dimensional, numerical root search of $\mathcal C^r$. In all the cases we have checked, the frequencies calculated in this way are negligibly different than those computed from the coupled root search.
To analyze the error in $\omega_{\rm A}$, we examine modes with $l$ up to $l=14$ for all allowed, discrete values of $\mu=m/(l+1/2)$, with the parameters $Q$, $a$, $s$, and $n=0$ fixed. We calculate the scaled residuals $L\Delta \omega_R$ and $L^2\Delta \omega_I$. These are finite as $L\to\infty$ if $\omega_{\rm A}$ has errors of orders $O(L^{-1})$ and $O(L^{-2})$ in its real and imaginary parts, respectively. We join residuals with the same $l$ with lines, so that as $l$ grows the scaled residuals illustrate the limit curve which depends continuously on $\mu$. Four examples are shown in Fig. \[fig:DFeik\], where we increase $l$ from $l = 2$ to $l = 14$. The parameters for these plots are $s = 1$, $a = 0.2M$, $Q = 0.8M$; $s = 2$, $a = 0.2M$, $Q = 0.8M$; $s = 1$, $a = 0.9M$, $Q = 0.1M$; and $s = 2, a = 0.9M, Q = 0.1M$. We find that the residual errors do generally scale as $\Delta\omega_R = O(L^{-1})$ and $\Delta\omega_I = O(L^{-2})$, which is actually one power of $L$ better than expected by the WKB theory presented in [@Schutz:1985zz]. This unexpected accuracy is also seen in Kerr [@Yang:2012he].
It can be difficult to determine visually whether or not the lines are converging to the limit curve. For the scaling we claim, the spacing between each successive $l$-curve must decrease as they approach the limit curve. We have checked that this is true for all the cases in Fig. \[fig:DFeik\] except for $s = 1$, $a = 0.9M$, $Q = 0.1M$ (top right panels of Fig. \[fig:DFeik\]). There, the spacing between the curves appears to be small and constant. The magnitude of the residual errors are 10 times smaller than the $s = 2$, $a = 0.9M$, $Q = 0.1M$ case, and we believe it is likely that we are probing errors introduced by using Eq. for $A_{lm}$ in the DF WKB implementation, or from using the $A_{lm}$ expansion in the continued fraction $\mathcal C^r$. Because the residual is quite small, and still an order of $L$ below the leading WKB prediction, we conclude that the WKB formulas here should be accurate enough for most purposes.
### Confirmation of the matched asymptotic expansion results
In this section we investigate the scalar ZDMs $(s=0)$ of the KN spacetime and compute $\omega_{\rm A}$ using Eq. , obtained from the matched asymptotic expansion[^7]. We verify that the residual error in the analytic formula scales as $\sigma^{2}$, which can be taken as an independent check of the validity of the matched asymptotic calculation.
In certain regions of parameter space, it can be difficult to apply Leaver’s technique because $\mathcal C^r$ becomes a rapidly varying function of $\omega$ and the success of the root-finding scheme becomes heavily dependent on the accuracy of the initial seed. We find in practice that this can occur when many QNM frequencies bunch together, which is the qualitative behavior we expect for the ZDMs. To get a sense of where the roots are, we borrow a technique from [@Yang:2012pj] where contours of the logarithm of $|\mathcal C^r|$ are plotted as a function of complex $\omega$. QNM frequencies appear as clusters of contours forming circles around places where $\mathcal C^r$ is zero. Nonphysical poles [@LeaverPoles] of $\mathcal C^r$ also appear as clusters of contours forming circles; however these can be distinguished by examining the value of the continued fraction. In our plots, blue (dark) regions correspond to smaller values of $|\mathcal C^r|$, while red (light) regions correspond to larger values.
Such plots can immediately demonstrate the existence of separated families of DMs and ZDMs. In these plots we fix $a$, $s=0$, $l$, and $m$, and examine two different values of $\sigma$. The ZDMs appear in a roughly vertical line with $\omega_R \approx m \Omega_H$. In the more extreme case, the line shifts toward the real axis and stacks more neatly, as seen in the left column of Fig. \[fig:cfplot\] where $l=2$, $m=1$, and $a = 0.9 M$. DMs can be distinguished in these figures, and move only slightly as $\sigma$ is decreased. A single DM can be seen in the top left panel of Fig. \[fig:cfplot\].
To illustrate the accuracy of our analytic approximations, we fix $s = 0$, $l = 2$ and test Eq. for chosen values of $m$ and $a$. Together with a choice of $\sigma \ll1 $, this fixes $Q$. Since the QNMs with $\delta^2>0$ are expected to be qualitatively different from those with $\delta^2<0$, we choose a value of $a$ covering each case. We start with $\delta^2<0$, and examine QNMs with $a = 0.1 M$, $m =2$ while varying $\sigma$. The center column of Fig. \[fig:cfplot\] presents the contour plots for two values of $\sigma$, and we observe the qualitative signatures of ZDMs. Figure \[fig:a1res\] contains a more detailed look at the $\sigma$-dependence of $\omega$ for the eight lowest overtones, and demonstrates the excellent agreement with the analytic formula. For the $\delta^2>0$ regime, we fix $a = 0.9 M$ and $m =2$. In Figs. \[fig:cfplot\] (right column) and \[fig:a9res\], we present the similar plots to Figs. \[fig:cfplot\] (center column) and \[fig:a1res\], except with $a=0.9M$. Again we observe ZDMs in good agreement with the analytic prediction.
We expect the analytic formula to have residual errors of $O(\sigma^2)$. Hence we expect the quantity $\sigma^{-2}\Delta \omega$ is $O(1)$ as $\sigma \to 0$, representing the next order in the nearly extremal expansion. In the left panel of Fig. \[fig:DFsigcon\], we return to the $\delta^2 <0$ case $a = 0.1M$, $l = m =2$. We plot the scaled residuals errors $M (n+1/2)^{-1}\sigma^{-2}\Delta \omega$ for the real and imaginary parts of $\omega$ of the lowest eight overtones as we vary $\sigma$. For each overtone, we can follow the curve from right to left and observe that the scaled residuals become constant, demonstrating $M\Delta\omega = O(\sigma^{-2})$. We can also follow the curves from top to bottom and observe that they cluster around a limit curve, indicating $M\Delta \omega=O(n+1/2)$ at large $n$.
\
In the right panel of Fig. \[fig:DFsigcon\], we return to the $\delta^2 >0$ case $a=0.9 M$, and $l = m =2$. We observe that the residual errors scale are $O(\sigma)$, since the quantity $\sigma^{-1}\Delta \omega$ approaches a nonzero finite number as $\sigma \to 0$. In our case studies, all of the modes with $\delta^2 >0$ had residual errors one power larger than the modes with $\delta^2 < 0$ at these small values of $\sigma$. This indicates that in these cases, where only ZDMs are present, the additional term $\eta$ in Eq. (discussed further in Appendix \[sec:MatchingApp\]) is not completely negligible, with $\eta \sim 10^{-3}$. When $\sigma \sim 10^{-3}$, the $O(\eta\sigma)$ correction is not negligible relative to the $O(\sigma^2)$ term in Eq. and the $O(\sigma^2)$ convergence is not seen.
Meanwhile, in Fig. \[fig:errscale\] we show that $\eta$ is so small that $\Delta\omega = O(\sigma^{-2})$ in practice when there are DMs ($\delta^2<0$). Here we first fix a value of $a$ and $n$ and calculate $M (n+1/2)^{-1}\sigma^{-2}\Delta \omega$ for several values of $\sigma$, spaced by roughly $\Delta \sigma \approx 2 \times 10^{-3}$. We plot these points in Fig. \[fig:errscale\] above the corresponding value of $a$. For each of the three overtones, the lines corresponding to the same value of $\sigma$ are plotted with the same color and the limit $\mathcal \sigma \to 0$ is taken by following the curves from top to bottom. We plot the data for six values of $a$, evenly spaced from $a = 0.1 M$ to $a = 0.6 M$, and connect data points with the same values of $\sigma$ to allow for a rough interpolation to other values of $a$. The exception is $a = 0.7M$ (not shown), where $\delta^2$ is negative and close to zero, and we expect a larger value of $\eta$. In this case, we find that the residuals do not scale as $\sigma^{2}$.
Overall, the numerical results indicate that the simple expression can be used over a large range of the parameter space. However, care must be taken to include the correction $\eta$ when $\delta^2$ is close to zero, and when the hole is very close to extremality, e.g. $\sigma \sim 10^{-3}$ (when $Q = 0$, $\sigma \sim 10^{-3}$ gives $a/M \sim 1 - 10^{-6}$).
Gravito-electromagnetic modes of Reissner-Nordström {#sec:RN}
===================================================
The existence of ZDMs for spin-weighted scalar perturbations of NEKN black holes naturally raises the question of whether ZDMs also exist for gravitational and electromagnetic perturbations of KN. As discussed previously, the equations for GEM perturbations of KN are coupled, and so electromagnetic perturbations cannot be considered separately from gravitational perturbations. We can begin to approach this challenging problem by considering first the simpler case of RN. The results of Sec. \[sec:DF\] hold for any spin parameter $a$, including the limit of $a \to 0$, and this shows that even in the well-studied case of the RN black hole, there are scalar ZDMs which reduce to zero decay in the extremal limit $Q \to M$. In this section, we examine the separated, decoupled GEM equations in the NERN background, and show that ZDMs exist for these perturbations as well.
The ZDMs of RN are purely decaying, like the $m=0$ ZDM modes of Kerr and the DF equation, and so they do not fit with the usual intuition into the nature of QNMs. Purely decaying perturbations of Schwarzschild have been discussed by Price [@Price:1971fb; @Price:1972pw; @wheeler1972magic], although the connection between this exponential decay and quasinormal modes remains unclear. Purely decaying modes in RN have been described in [@Andersson:1996xw; @Andersson:2003fh], and include the algebraically special modes [@Chandra:1984a], but to our knowledge none of these exhibit the slow decay rate we find, despite a large literature exploring the QNMs of extremal and nearly extremal RN black holes [@LeaverRN; @Onozawa:1995vu; @Berti2009]. Before exploring the existence of ZDMs for the NERN black hole, we review the fundamental equations for the perturbations of this spacetime.
Perturbations of Reissner-Nordström
-----------------------------------
The problem of GEM perturbations for the RN spacetime closely parallels the investigation of perturbations of Schwarzschild using the Regge-Wheeler-Zerilli equations. The equations come in two sets, according to the parity of the perturbations, and it is known that the QNM spectrum of both sets is the same [@ChandraBook; @Dias:2015wqa]. This means that we can focus on the magnetic-parity perturbations \[those which are multiplied by $(-1)^{l+1}$ under a parity transform\][^8], which gives two equations indexed by $j,k = 1,2$: $$\begin{aligned}
\label{eq:RNwave}
\frac{d^2 Z_j}{dr_*^2}& + (\omega^2 - V_j) Z_j = 0 \,, \\
\label{eq:RNpot}
V_j & = \frac{\Delta}{r^5}\left[ l(l+1) r - q_k +\frac{4 Q^2}{r} \right] \,,\end{aligned}$$ where $q_k$, $k \neq j$ indicates that $q_2\, (q_1)$ be used for $Z_1\, (Z_2)$, and $$\begin{aligned}
q_1 &= 3 M + \sqrt{9 M^2 +4 Q^2 [l(l+1)-2]} \,, \\
q_2 & = 6M - q_1 \,.\end{aligned}$$ When $Q \to 0$, $Z_1$ obeys the equations for magnetic-parity electromagnetic perturbations and $Z_2$ obeys the Regge-Wheeler equation for gravitational perturbations. In the above equations, $r_*$ and $\Delta$ are defined in the same way as in the KN spacetime, in the limit $a \to 0$. In particular, $\Delta$ has two roots which give the coordinate positions of the outer and inner event horizons, $r_\pm = M \pm M \sqrt{1 - Q^2/M^2}$. We are interested in the nearly extremal limit, where $\sigma = (r_+ - r_-)/r_+ \ll 1$. We maintain the same notation as in Sec. \[sec:DF\], which highlights many parallels between two analyses.
To search for ZDMs in the NERN spacetime analytically, we repeat the steps of the matched asymptotic expansion used for the DF equation in Sec. \[sec:DF\]. First we discuss the inner solution.
The inner solution
------------------
In the near horizon, nearly extremal limit \[$x = (r-r_+)/r_+\ll 1$ and $\sigma \ll 1$, but without assuming that $\sigma/x$ is small\], Eqs. and reduce at leading order to $$\begin{aligned}
y^2 & Z_j''(y) + y Z_j'(y) + V^y_j Z_j (y)= 0\,, \\
V^y_j & = \left(\frac{\hat \omega}{\sigma}\right)^2 +\frac{y[q_k /M -4 - l(l+1)]}{(1-y)^2} \,.\end{aligned}$$ We recall the variable $y$ used in Sec. \[sec:DF\], $$\begin{aligned}
y = \exp \left( \frac{\sigma r_*} {r_+} \right) \approx \frac{x}{x+\sigma} \,,\end{aligned}$$ By making the replacements $ \hat \omega/\sigma= \varpi / 2 $ and $$\begin{aligned}
q_k /M -4 - l(l+1) & = \delta_j^2 + 1/4 \,,
\end{aligned}$$ we see that the near-horizon approximation for $Z_j$ reduces to the same equations as in the DF analysis, Eqs. and , with $\zeta = 0$, $s = 0$, and $$\begin{aligned}
\delta_j & = i \left [L + (-1)^{j-1} \right]\,, && j = 1,2\,,\end{aligned}$$ recalling that $L = l+ 1/2$. Note however that the form of $\delta_j$ differs slightly from the factor $\delta$ for the DF equation in the $a \to 0$ limit. In that case, $\delta = i L$ (and is independent of the spin $s$ of the test scalar field). For RN, $\delta_j$ is purely imaginary, and we take as our convention that $\delta_j$ is a positive imaginary number. Again selecting the solution to Eq. which has no waves emerging from the horizon and normalizing the amplitude of the solution to unity at the horizon, we have $$\begin{aligned}
\label{eq:RNnearsln}
Z_j & = y^{- i \varpi} (1 - y)^{1/2 + i \delta_j} \, {}_2 F_1\left(\alpha,\beta,\gamma, y \right) \,, \\
\alpha & = 1/2 - i \varpi + i \delta_j \,, \qquad \beta = 1/2 + i \delta_j, \qquad \gamma = 1 - i\varpi \,.\end{aligned}$$ As with the DF equation, we next wish to match this solution onto a solution in the outer region, where $x \gg \sigma$.
Ansatz for matching
-------------------
We turn to the approximation of Eq. when we can take $(x + \sigma) \approx x$. Substituting in our definitions, we have after some manipulation, $$\begin{aligned}
&x^2 \frac{d^2 Z_j}{dx^2} + \frac{2x}{(x+1)} \frac{d Z_j}{dx} \notag \\ & + \left[\frac{(x+1)^4}{x^2} \omega^2 - l(l+1) + \frac{q_k}{x+1} + \frac{4}{(x+1)^2}\right] Z_j = 0 \,,\end{aligned}$$ We have not yet found a simple analytic solution that gives a convenient matching condition for small $x$. Using transformations such as $Z_j = \Delta^{s/2} r R$ yields promising forms of the equation for various choices of $s$, but none which allow for a straightforward matching analysis.
Instead, motivated by past experience, we make an ansatz to complete the matching. By expanding the inner solution as in Sec. \[sec:Matching\] and Appendix \[sec:MatchingApp\], we have $$\begin{aligned}
\label{eq:RNmatch}
Z_j \to &\frac{\Gamma(2 i \delta_j) \Gamma(1-i \varpi)}{\Gamma(1/2 + i \delta_j)\Gamma(1/2 - i \varpi + i \delta_j)} \left( \frac{\sigma}{x} \right)^{1/2-i\delta_j}\notag \\
&+ (\delta_j \to - \delta_j) \,,\end{aligned}$$ In the case of the DF equation in NEKN, the ZDM solutions correspond to the near vanishing of one of the two coefficients of $(\sigma/x)^{1/2 \pm i \delta}$ in the above expansion. This occurs at the zeros of one of the $1/\Gamma(w)$ factors, since $\Gamma(w)$ has poles at the negative integers. We make the ansatz that the corresponding Gamma function is also near its pole in RN. Investigating Eq. , it is apparent that for the convention where $\delta_j$ is a positive imaginary number, the only possibility for fulfilling this criteria here is by taking $$\begin{aligned}
\label{eq:RNprematch}
1/2 - i \varpi - i \delta_j = - n \,,\end{aligned}$$ which gives $$\begin{aligned}
\label{eq:RNfreq}
\omega = -i \frac{\sigma}{2r_+} \left[ |\delta_j| + \left(n + \frac 12 \right)\right] \,.\end{aligned}$$ The fact that $\delta_j$ is pure imaginary indicates the presence of DMs in addition to the ZDMs with frequencies given by Eq. , which is of course what is observed. These DMs are the usual QNMs of extremal and nearly extremal RN which have been the subject of past study.
We can provide a heuristic argument in support of our ansatz. When $x \gg \sigma$ and $x \lesssim 1$, the two terms in the wave function have distinctly different behavior while approaching the edge of the near-horizon region (corresponding to a decreasing $\sigma/x$). In the first term, displayed explicitly in Eq. , $\sigma /x$ has an exponent $1/2 + |\delta_j| >0$ and so is decaying as $x$ increases. In the second term, where the first term has the replacement $\delta_j \to - \delta_j$, the exponent of $\sigma/x$ is negative and so the term is growing. Our matching condition sets this growing term to zero, which is reasonable: when $x \sim 1$, the growing term will have a size $\sim (1/\sigma)^{|\delta_j| -1/2} \gg 1$. In the DF case, if the amplitude of this term is not suppressed, this term is too large to match onto the outer solution. It seems reasonable that in general an outer solution, which is regular as $\sigma \to 0$, cannot match onto this growing term if the amplitude is not suppressed. Unless the perturbations have nearly zero amplitude in the near-horizon region, we would encounter unnaturally strong perturbations in the matching region, invalidating the approximate formalism even for QNMs sourced by small initial data. At the very least, a large amplitude in the matching region goes against the intuition that the ZDMs are concentrated or trapped near the horizon [@Andersson2000]. This possibility can be avoided for frequencies which eliminate this troublesome growing term.
Note that if an outer solution were available to us, we would find that Eq. is corrected by a small $\eta$ as in the case of the ZDM frequencies predicted by the DF matching analysis of Sec \[sec:DF\]. This correction would mean that the second term of Eq. is not precisely zero, but its large size is compensated by the small amplitude factor. The balance of these two effects would allow for a matching onto the outer solution to take place, as is done for the DF equation.
In Sec. \[sec:RNNumerics\] we show that Eq. gives the correct decay rates for the RN ZDMs. The fact that this equation differs from the $a \to 0$ limit of the DF prediction, Eq. (recalling that $\delta_j \neq \delta$) shows that the DF equation fails to describe the QNMs of the nearly extremal RN spacetime outside of the scalar case. This is in contrast to the situation where the charge of the NEKN spacetime is small, where the DF equation provides the leading frequency corrections to the Kerr ZDMs [@Mark:2014aja], and shows that we cannot hope that the DF equation is accurate for all NEKN black holes.
WKB analysis
------------
For completeness, we include the WKB analysis of Eqs. and . The form of Eq. allows us to immediately use the methods of [@Schutz:1985zz; @IyerWill1987]. We recall our definitions $L = l + 1/2 \gg 1$, $\Omega_R = \omega_R/L$, and in the notation of [@Schutz:1985zz] we define $$\begin{aligned}
\mathcal Q = \omega^2 - V_j \approx L^2 \left(\Omega_R^2 - \frac{\Delta}{r^4} \right) \,.\end{aligned}$$ The conditions that $\mathcal Q = 0$ and $d \mathcal Q/dr_* = 0$ at the WKB frequency identifies the peak $r_0$ and gives $\Omega_R$, $$\begin{aligned}
r_0 & = \frac 12 \left( 3 M + \sqrt{9 M^2 - 8 Q^2}\right) \,, &
\Omega_R & = \left. \frac{\sqrt{\Delta}}{r^2} \right|_{r_0} \,,\end{aligned}$$ The curvature at the extrema of the potential determines the decay rate of the mode, Eq. . The result for RN is $$\begin{aligned}
\Omega_I = \frac{\sqrt{3 M^2 r_0 - Q^2 (M+2r_0)}}{r_0^{5/2}} \,.\end{aligned}$$ These results agree with the literature [@Kokkotas:1988fm; @Andersson:1996xw; @Mashhoon1985; @Cardoso2009]. In the extremal limit, $$\begin{aligned}
r_0 &= 2 M \,, & \Omega_R & = \frac{1}{4M} \,, & \Omega_I & = \frac{1}{4M \sqrt{2}}\,. \end{aligned}$$ In the context of this study, the important point is that the decay rate remains nonzero for all $Q$, and so this analysis can approximate only the usual DMs of RN in the extremal limit. It does not access the ZDMs.
Numerical results {#sec:RNNumerics}
-----------------
While we consider the ansatz for the outer solution to be well motivated, we must numerically check the expression for the ZDM frequencies of NERN, Eq. . In this section, we show that Eq. is accurate using the methods of Sec. \[sec:DFNumerics\], namely via contour plots of the logarithm of the continued fraction $\mathcal C^r$ and numerical calculations of the residual error $\Delta \omega$. We find that residual error in Eq. scales identically to the residual error in the DF ZDM formula; $\Delta \omega \sim O[(n+1/2)\sigma^{2}]$. Together with the WKB results for the damped modes, this analysis demonstrates that the RN QNM spectrum also undergoes a bifurcation as $\sigma \to 0$ .
To numerically compute the QNM frequencies, we once again use Leaver’s method, which can be extended with some effort to RN black holes [@LeaverRN]. For RN, the angular problem decouples from the radial problem and is solved by scalar, vector, and tensor spherical harmonics with known eigenvalues. The radial wave functions can be expanded as a power series whose coefficients obey a four-term recurrence relation. Through Gaussian elimination, these can be converted into three-term recurrence relations and then into a radial continued fraction $\mathcal C^r$ whose roots give the QNM frequencies.
In Fig. \[fig:RNClose\], we present two contour plots of the logarithm of $|\mathcal C^r|$ in the complex-$\omega$ plane, for the case $j=1$ (electromagnetic-type) and $l=1$, for nearly extremal values of charge $Q = 0.999M$ and $Q= 0.9999M$. These plots demonstrate the existence of $j=1$, $l=1$ ZDMs lying on the imaginary axis. Visually, we again see agreement with the prediction of Eq. (black crosses).
\
Figure \[fig:slRN\] presents a broader and more quantitative analysis. Each panel corresponds to a different value of $l$ and $j$ (mode type). The top of each panel plots the lowest eight overtones of the ZDMs, found using Leaver’s method, along with the predictions of Eq. . In the bottom of each panel, we plot the scaled residual error $M(n+1/2)^{-1}\sigma^{-2}\Delta\omega$ versus $\sigma$. For each overtone, one can check the $\sigma^2$ scaling of the residual by following the corresponding line. For each value of $\sigma$, one can check the $(n+1/2)^{-1}$ scaling by following the calculations vertically downward. While these results do not represent a comprehensive search for the ZDMs of RN, they give us confidence that the matching ansatz gives the correct expression for these frequencies. Importantly, our analysis establishes the existence of ZDMs of the GEM perturbations of RN black holes for the first time to our knowledge[^9].
Conclusions {#sec:Conclusions}
===========
In this paper we have given an overview of the QNMs of nearly extremal Kerr-Newman black holes. While many of the results in Sec. \[sec:DF\] have appeared elsewhere, there are many contradictory results in the literature. We have reviewed the derivation of the ZDM frequencies for NEKN black holes, using matched asymptotic expansions. Using the WKB approximation for scalar fields in KN, we have discussed the existence of damped modes and given approximate formulas for these frequencies. Finally, using Leaver’s method, we have validated these approximations, and effectively measured the higher order corrections to the nearly extremal and WKB approximations.
By carrying out this analysis using the DF equation for spin-weighted scalars, the results of Sec. \[sec:DF\] can be compared to results for the true GEM modes of NEKN, in order to see how this simplistic model performs, for example by a careful comparison to numerical results. This is left for future studies, although we reiterate that the DF equation correctly predicts the small-charge corrections to the ZDMs of nearly extremal Kerr black holes [@Mark:2014aja].
Since the case of scalar QNMs in NERN follows immediately from the results of Sec. \[sec:DF\], in Sec. \[sec:RN\] we have investigated the coupled GEM equations of NERN. In this case, we have shown that ZDMs exist alongside the well known DMs, and given a frequency formula for these modes using a matching ansatz. A numerical study using Leaver’s method confirms this ansatz and again the residual errors provide higher order corrections. The ZDM frequency formula differs from that of the spin-weighted scalars found using the DF equation, indicating that the DF equation cannot accurately describe the ZDM frequencies for all spinning, charged black holes. For completeness, we have provided the WKB formulas for RN, and examined its extremal limit, concluding that the technique only describes damped modes.
While this work demonstrates the existence of a family of purely damped QNMs for RN, it is unclear what the implications of these modes are. They may assist in the shedding of black hole hair following collapse to an RN black hole, as is the case for exponentially decaying modes in Schwarzschild [@Price:1971fb; @Price:1972pw; @wheeler1972magic]. A careful analysis of the excitation of QNMs would be required to assess the importance of these modes and their physical meaning.
Looking ahead, the daunting problem of gaining an analytic understanding of coupled GEM equations in the nearly extremal case remains. Two lines of evidence indicate that the GEM perturbations of NEKN admit ZDMs. The first is the weakly charged, rapidly rotating case discussed in Mark [*et al.*]{} [@Mark:2014aja]. That study computed the QNM frequencies of weakly charged Kerr black holes in the form $\omega \approx \omega^{(0)} + Q^2 \omega^{(1)}$, where $\omega^{(0)}$ is the Kerr value. Mark [*et al.*]{} showed that the DF equation provides a complete accounting of the frequency corrections $\omega^{(1)}$ to the gravitational and electromagnetic ZDMs in Kerr as the black hole angular momentum increases towards extremality. The coefficients $\omega^{(1)}$ also begin to diverge in this limit, although they are controlled by the smallness of $Q/Q_{\rm max}$, where $Q_{\rm max}$ is the charge of the extremal KN black hole at a given $a$. Both of these behaviors can be explained if the full KN QNMs are ZDMs, whose frequencies are $m\Omega_H$ at leading order, with corrections proportional to the surface gravity $\kappa$. Reexpanding such frequencies in small charge compared to $Q_{\rm max}$ recovers the apparent divergences of $\omega^{(1)}$ seen in that study, naturally suppressing them by $Q/Q_{\rm max}$. Meanwhile the increasing accuracy of the DF equation can be understood by examining the near-horizon, near-extremal scalings of the ZDM wave functions of Kerr [@Mark:2014aja], although the reason for these scalings remains a puzzle.
The other, even more compelling line of evidence is provided by the recent numerical investigations of the QNMs of KN. The numerical results of [@Dias:2015wqa] show definitively the existence of ZDMs in KN. In that study, a GEM mode with $l=m=2$ showed the behavior $\omega_R \sim m \Omega_H$ and $\omega_I \to 0$ in the extremal limit, for all values of $a$. The fact that this occurs even when $Q \geq 0.5 M$ indicates that the search of [@Dias:2015wqa] identifies the ZDMs, even in the regime where we expect DMs and where spectrum bifurcation might confuse a numerical search. Since [@Dias:2015wqa] focused on only the lowest overtones (defined as having the smallest decay rate), future numerical studies will be required to understand the existence and behavior of the GEM damped modes of the KN black hole.
The dependence of the ZDM frequency on spin and charge also explains the frequency behavior seen in the numerical simulations presented in [@Zilhao:2014wqa], as pointed out in a remark by Hod [@Hod:2014uqa]. Those numerical simulations evolved perturbations of NEKN black holes using the full Einstein-Maxwell equations, and argued that for a range of values of $Q$ the perturbations had frequencies and decay rates dependent primarily on the combination $a/a_{\rm max} = a/ \sqrt{1 - Q^2/M^2}$. At first glance, this is in contradiction with our formula . In fact the proposed relation gives almost the same frequencies as for the cases $a\gtrsim0.9M, \, Q \lesssim 0.4 M$. Only a precise analysis of the frequencies and decay rates, such as that given in [@Dias:2015wqa], can differentiate the proposed relation from the one derived here. Hod also notes that in the case presented in [@Zilhao:2014wqa] where $Q$ is large and the KN black hole is nearly extremal, $\Omega_H$ gives a good accounting for the observed frequency of the oscillations.
The success of the nearly extremal approximation for describing ZDMs of scalar modes in RN and both scalar and GEM perturbations of RN begs the question of whether such methods can be applied to the full, coupled GEM perturbations of Kerr-Newman. Unfortunately, a naive near-horizon, nearly extremal scaling analysis indicates that these equations remain coupled in this limit, and this coupling in turn obstructs separation of the differential equations. Nevertheless, the results of this paper, many previous studies, and recent comprehensive numerical results [@Zilhao:2014wqa; @Dias:2015wqa] all indicate that the ZDMs of the full coupled perturbations of NEKN obey a simple frequency formula like Eq. . The challenge is to show that this is so, and provide an analytic expression for the factor of $\delta(a)$. The wealth of progress in studying perturbations of KN black holes in the past several years places this goal within reach. It may be that the connection to conformal field theories available in the near-horizon region of NEKN [@Bardeen:1999px; @Guica2009; @Hartman:2008pb; @Hartman:2009nz; @Porfyriadis:2014fja] will allow for the solution of this problem, or at least to further connections to quantum theories. Another promising avenue is the application of WKB techniques to the coupled GEM equations. In the WKB limit, the differences between the DF and full GEM predictions for RN vanish, although the equations describe very different kinds of perturbations. It may be that this fact carries through to the rotating KN black hole, in which case the DF WKB predictions would give an accurate accounting of the high-frequency GEM modes of KN. We leave the investigation of this possibility to future studies.
We thank Emanuele Berti, Yanbei Chen, David Nichols, Huan Yang, An[i]{}l Zenginoğlu and Fan Zhang for previous collaboration and valuable discussion on the topic of the QNMs of nearly extremal black holes. We are especially grateful to Huan Yang and Yanbei Chen for collaboration on additional studies of Kerr-Newman black holes and the WKB approximation for QNMs, which this study grew out of, and to Yanbei Chen and Fan Zhang for past collaboration on the implementation of Leaver’s method in the Kerr spacetime. We also thank Sam Gralla and Alexandru Lupsasca for insights into the near-horizon region of nearly extremal Kerr. A. Z. was supported by the Beatrice and Vincent Tremaine Postdoctoral Fellowship at the Canadian Institute for Theoretical Astrophysics during much of this work. Z. M. is supported by NSF Grant No. PHY-1404569, CAREER Grant No. 0956189, the David and Barbara Groce Startup Fund at Caltech, and the Brinson Foundation. A portion of this work was carried out at the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the government of Canada and by the Province of Ontario though Ministry of Research and Innovation.
Details of the matching calculation {#sec:MatchingApp}
===================================
In this Appendix we provide some supplementary equations related to the matching analysis of Sec. \[sec:Matching\]. First we consider the outer solution, Eq. . In the limit $x \to \infty$, we use the expansion [@nist] $$\begin{aligned}
{}_1 F_1(a, b, 2 i \hat \omega x) \to \frac{\Gamma[b]}{\Gamma[a]} e^{2 i \hat \omega x}(2 i \hat \omega x)^{a-b} + \frac{\Gamma[b]}{\Gamma[b -a ]} (-2 i \hat \omega x)^{-a} \,. \end{aligned}$$ The first term above contributes to the part of the radial function which behaves as $R \propto e^{i \omega r_*}$ and so is an outgoing solution. Similarly, the second term contributes to the ingoing solution, which can be eliminated by a particular choice of $A$ and $B$. The requirement that we have only outgoing waves provides the condition $$\begin{aligned}
\label{eq:Ratio1}
\mathcal R = \frac{A}{B} & = \frac{\Gamma[-2 i \delta] \Gamma[1/2 + s + i \delta - 2 i \hat \omega]}{\Gamma[2 i \delta] \Gamma[1/2 + s - i \delta - 2 i \hat \omega]} e^{\pi \delta+ 2 i \delta \log 2 \hat \omega} \,.\end{aligned}$$ We can also identify the outgoing and ingoing wave amplitudes in the general scattering problem. We have $$\begin{aligned}
\label{eq:OuterAmp1}
\frac{Z^{\rm out}}{Z^{\rm hole}} & = A (2 i \hat \omega)^{-1/2 -s - i \delta + 2 i \hat \omega} \frac{\Gamma[1 + 2 i \delta]}{\Gamma[1/2 - s + i \delta + 2 i \hat \omega]} +B (\delta \to \delta) \,, \\
\label{eq:OuterAmp2}
\frac{Z^{\rm in}}{Z^{\rm hole}} & = A (-2 i \hat \omega)^{-1/2 + s - i \delta - 2 i \hat \omega} \frac{\Gamma[1 + 2 i \delta]}{\Gamma[1/2 + s + i \delta - 2 i \hat \omega]} + B (\delta \to - \delta)\,.\end{aligned}$$ Here we have normalized each amplitude by the amplitude of the wave function at the horizon, $Z^{\rm hole}$. Elsewhere in this paper, we have assumed that $Z^{\rm hole} = 1$.
Meanwhile, in the limit of small $x$, the outer solution takes the simple form $$\begin{aligned}
R = A x^{-1/2 - s + i \delta} + B x^{-1/2 -s - i \delta} \,.\end{aligned}$$ The inner region provides a second condition by matching this to the inner solution. For this we transform the domain of the hypergeometric function ${}_2 F_1$ by taking $z = 1-y$ and using the identity [@nist] $$\begin{aligned}
\label{eq:Inversion}
{}_2 F_1( \alpha, \beta, \gamma, y) & = \frac{\Gamma[\gamma] \Gamma[-2 i \delta]}{\Gamma[\gamma- \alpha]\Gamma[\gamma-\beta]} \,{}_2 F_1(\alpha, \beta, 1+ 2 i \delta, z) + \frac{\Gamma[\gamma]\Gamma[2 i \delta]}{\Gamma[\alpha]\Gamma[\beta]} z^{-2 i \delta} \,{}_2 F_1(\gamma - \alpha, \gamma -\beta, 1 -2 i \delta, z) \,.\end{aligned}$$ It is important that in KN the parameters of the hypergeometric function obey $\gamma - \alpha - \beta = - 2 i \delta$ just as in Kerr, which allows the matching to occur. This has been used to simplify some of the terms in the above identity. Note also that $\gamma - \alpha = \beta|_{\delta \to -\delta}$ and $ \gamma - \beta = \alpha|_{\delta \to -\delta}$, demonstrating the symmetry of these equations under the change of the sign of $\delta$. This means that we can use the convention that $\delta$ is a positive real or imaginary number without loss of generality.
Next we take the limit $z \to 0$, which sets the hypergeometric functions in Eq. to unity. Some useful identities for the matching are $$\begin{aligned}
R \approx \frac{r_+^{-s} \, x^{-s}}{\sqrt{r_+^2 +a^2}} u \,, && z \approx \frac{\sigma}{x} \,,\end{aligned}$$ and we recall that $u = y^{-p} (1-y)^{-q} {}_2 F_1(\alpha, \beta, \gamma, y)$. Combining all of these equations gives us an expression for the inner solution, $$\begin{aligned}
\label{eq:innmatch}
R & \approx \frac{r_+^{-s}}{\sqrt{r_+^2 +a^2}} x^{-s} \left(\frac{\sigma}{x} \right)^{1/2+i\delta} \left [\frac{\Gamma[\gamma] \Gamma[-2 i \delta]}{\Gamma[\gamma- \alpha]\Gamma[\gamma-\beta]} + \frac{\Gamma[\gamma]\Gamma[2 i \delta]}{\Gamma[\alpha]\Gamma[\beta]}\left( \frac{\sigma}{x}\right)^{-2 i \delta} \right] \,,\end{aligned}$$
The matching gives $$\begin{aligned}
\label{eq:InnerAmp}
A & = \frac{r_+^{-s}\sigma^{1/2 - i \delta}}{\sqrt{r_+^2 + a^2}} \frac{\Gamma[\gamma] \Gamma[2 i \delta]}{\Gamma[\alpha] \Gamma[\beta]}\,, &
B & = \frac{r_+^{-s}\sigma^{1/2 + i \delta}}{\sqrt{r_+^2 + a^2}} \frac{\Gamma[\gamma] \Gamma[-2i\delta]}{\Gamma[\gamma - \alpha] \Gamma[\gamma - \beta]} \,,\end{aligned}$$ so that $$\begin{aligned}
\label{eq:Ratio2}
\frac{A}{B} & = \sigma^{- 2 i \delta} \frac{\Gamma[2i\delta]}{\Gamma[-2 i \delta]}\frac{\Gamma[\gamma - \alpha]\Gamma[\gamma - \beta]}{\Gamma[\alpha]\Gamma[\beta]} \,.\end{aligned}$$ In the general scattering problem, Eqs. , , and give the full expressions for the amplitudes, or equivalently the reflection and transmission coefficients of scalar waves in NEKN. The scattering amplitudes given here also allow for the calculation of QNM excitation factors, following the steps in [@Yang:2013uba]. Since we do not deal with excitation of QNMs here, we omit these lengthy expressions.
The conditions from Eqs. and can be satisfied if one of the Gamma functions in is near to a pole at the negative integers. In the case where $\delta$ is pure imaginary with our convention, Eq. is suppressed by the smallness of $\sigma$, whereas Eq. has no sensitive dependence on $\sigma$. This difference in behavior can be compensated by having one of the Gamma functions take a large value, i.e. be near its poles. Meanwhile, when $\delta$ is real, then Eq. exhibits rapid oscillation in phase when $\sigma$ is changed by a small amount; there is no such sensitive phase dependence on $\sigma$ in $\mathcal R$. The same assumption, that one of the Gamma functions is near its pole, can be used to compensate for this phase dependence, if the shift of the argument of the Gamma from its pole absorbs this rapid phase variation. These considerations motivate the condition $\Gamma[\gamma - \beta] = \Gamma[-n - i \eta]$. Since $\gamma - \beta$ depends on the frequency through $\varpi$, this condition selects a particular QNM frequency. Expanding this condition in small $\eta$, we have $1/\Gamma[-n - i \eta] = (-1)^n (n!) (-i \eta) + O(\eta^2)$. We can solve the matching condition for $\eta$, giving an expression for the correction to $\omega$, $$\begin{aligned}
\label{eq:EtaEqn}
\eta & = \mathcal R^{-1} (-1)^n \frac{i \sigma^{- 2 i \delta}}{n!} \frac{\Gamma[2i\delta]}{\Gamma[-2 i \delta]}\frac{\Gamma[\gamma - \alpha]}{\Gamma[\alpha]\Gamma[\beta]}\,.\end{aligned}$$ As discussed in [@Yang:2013uba], $\eta$ is generally quite small, with the exception of cases where $\delta^2$ is small and negative, in which case it can be order unity or greater. Although the possibility has not been explored in detail, there may even be situations where $\eta$ could be large enough to invalidate the approximation used above to find a closed form solution for the ZDM frequencies. We do not incorporate the correction $\eta$ into our frequency formula in this paper, but in cases where $\eta$ is a significant contribution to the $O(\sigma)$ frequency corrections, Eq. can be used to augment the ZDM frequency formula. In addition, whenever $\eta \gtrsim \sigma$, it dominates the residual error in our frequency formula. In Sec. \[sec:DFNumerics\] for $l=m=2$, $s=1$ and $\delta^2>0$, we find that the term $\sigma \eta$ prevents $O(\sigma^2)$ scaling of the residual errors at small values of $\sigma$.
The WKB analysis of the Dudley-Finley equations {#sec:WKBApp}
===============================================
The WKB analysis of the Dudley-Finley equation in the Kerr-Newman spacetime is a straightforward extension of the methods discussed in [@Schutz:1985zz; @IyerWill1987] and later extended for generic orbits in Kerr [@Yang:2012he]. The result of this analysis gives Eqs. and for the WKB frequency and decay rates, using the leading order parts of the radial potential for the DF equation. We present the relevant equations here in full, and specialize to the near-extremal case in Sec. \[sec:WKBanalysis\]. For convenience in this section we set $M=1$. Solving both of the equations of while eliminating $\lambda$ gives our useble formula for $\Omega_R$, $$\begin{aligned}
\label{eq:WKBOmegaR}
\Omega_R & = \frac{\mu a (r - 1)}{(r^2 +a^2)(r -1) - 2 r \Delta}\,,\end{aligned}$$ which must be evaluated at the peak $r_0$ to give a consistent solution to the WKB equations. The above is only an implicit equation for $\Omega_R$ unless we can determine $r_0$ independently of $\Omega_R$. Equation shows that if $r_0$ approaches the horizon, $\Omega_R$ approaches the horizon frequency $\Omega_H$. This is in agreement with the situation in Kerr, and is only modified by the fact that $\Omega_H$ depends on both $a$ and $Q$.
Using both conditions of Eqs. together with the approximate analytic expression for $\alpha(\mu,a,Q)$ lets us eliminate $\Omega_R$, yielding a sixth order polynomial equation, $$\begin{aligned}
\label{eq:WKBPoly}
2 r^2[r(r-3)+2Q^2]^2 + 4 r\left(r[r^2(1-\mu^2) - 2r -3 (1-\mu^2)]+2Q^2(1-\mu^2 +r) \right)a^2
\notag \\
+(1-\mu^2)[r^2(2-\mu^2) + 2r(2+\mu^2) + 2-\mu^2]a^4 =0 \,.\end{aligned}$$ The outermost root of this polynomial gives the position of the peak $r_0$, and when Eq. is evaluated at $r_0$ we attain a self-consistent solution to the equations. Note that in the $a \to 0$ case, both the numerator and denominator of Eq. vanish. A better behaved expression in this limit can be found by using the polynomial to eliminate the vanishing denominator; after some simplification we find $$\begin{aligned}
\label{eq:WKBOmegaRSecond}
\Omega_R = \frac{\sqrt{2}(r - M)}{\sqrt{4r(r^3 - 3 r +2 Q^2) + a^2[(r^2+M)(3-\mu^2)+2r(1+\mu^2)]}}\,.\end{aligned}$$ This equation also behaves correctly in the case $\mu = 0$, for which additional, closed form analytic expressions can be derived for $r_0$ and the frequencies (see e.g [@Teo2003; @Dolan2010; @Yang:2012he] for the Kerr case). Meanwhile, it is poorly behaved when $r_0$ approaches the horizon in the extremal case, and so is not used outside of this Appendix.
The WKB analysis gives an equation for $\Omega_I = \omega_I/(n+1/2)$, Eq. . Using some algebraic tricks that rely on the conditions in Eq. , we get $$\begin{aligned}
\Omega_I = \left. \Delta\frac{\sqrt{ 4(6 \Omega_R^2 r^2 -1) + 2 a^2 \Omega_R^2(3-\mu^2) }}{2 \Omega_R(r^2+a^2)^2 - 2\mu a (r^2 +a^2) -a \Delta [a \Omega_R (1+\mu^2) - 2 \mu] } \right|_{r_0} \,.\end{aligned}$$ We see immediately that if $r_0$ goes to the horizon in the extremal limit, the WKB analysis predicts a vanishing $\omega_I$, which indicates the existence of ZDMs.
In Fig. \[fig:WKBFreqFixQ\] we plot some representative values of $\Omega_R$ and $\Omega_I$ at fixed charge $Q$ and maximum inclination parameter $\mu = 1$, for values of $a$ varying between each extremal case. For positive values of $a$, the WKB modes correspond to corotating equatorial photon orbits, while for negative values of $a$ they correspond to counterrotating equatorial orbits.
[^1]: For example, one can show using the methods in [@Teukolsky1973] that the DF equation is obeyed by a second fictitious set of Maxwell fields propagating on the KN background, which have no leading order background contributions.
[^2]: In [@Yang:2013uba] the small parameter $\epsilon$ is used, while older studies use $\sigma$ as the small parameter. Using $\sigma$, the analysis of nearly extremal Kerr carries over naturally to KN black holes.
[^3]: A regular perturbation analysis where the wave function is a power series in $\sigma$ does not work because, in the language of matched asymptotic expansions, there is a boundary layer at the horizon.
[^4]: Specifically, the factors of $s$ in Eq. (3.9) have the wrong signs, which is countered by an identical error in Eq. (3.16).
[^5]: When comparing these to the results in [@Yang:2013uba] it is useful to note that $\varpi = \sqrt{2} \bar \omega$ and $\zeta = \bar m$, in the notation of that paper.
[^6]: Although note that Eqs. (3.11) and (3.12) of [@Yang:2013uba] suffer from two typos: overall, $V_y$ needs to be divided by $2$, and the factor of $y$ is absent from the third term in $V_y$ which involves $\mathcal F_0$, which is the same as the third term here involving $\delta$.
[^7]: We note that charged, massive scalar QNMs were investigated using Leaver’s method in [@KonoplyaNEKN]. That study provided some results in the massless, uncharged limit, which is the problem we investigate here. However, that study found that in the extremal limit, QNMs which were ZDMs in Kerr limited to a finite decay rate for $Q \neq 0$. This is in conflict with the results we present here.
[^8]: We follow the convention of [@Zerilli:1970wd]. Other studies refer to these modes as odd-parity and even-parity, or axial and polar, see e.g. [@Pani:2013ija; @Pani:2013wsa].
[^9]: During the completion of this work we became aware of the study [@Couch:2012zz], which uses methods analogous to Leaver’s method on the nearly extremal DF equation in the RN limit. That study identifies the scalar ZDMs, but incorrectly claims that the results apply to electromagnetic and gravitational perturbations.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The leading order “temperature” of a dense two dimensional granular material fluidised by external vibrations is determined. The grain interactions are characterised by inelastic collisions, but the coefficient of restitution is considered to be close to $1$, so that the dissipation of energy during a collision is small compared to the average energy of a particle. An asymptotic solution is obtained where the particles are considered to be elastic in the leading approximation. The velocity distribution is a Maxwell-Boltzmann distribution in the leading approximation. The density profile is determined by solving the momentum balance equation in the vertical direction, where the relation between the pressure and density is provided by the virial equation of state. The temperature is determined by relating the source of energy due to the vibrating surface and the energy dissipation due to inelastic collisions. The predictions of the present analysis show good agreement with simulation results at higher densities where theories for a dilute vibrated granular material, with the pressure-density relation provided by the ideal gas law, are in error.'
address: |
Department of Chemical Engineering,\
Indian Institute of Science, Bangalore, 560 012, India.
author:
- P Sunthar and V Kumaran
title: 'Temperature scaling in a dense vibro-fluidised granular material'
---
Introduction
============
Recent developments in the physics of granular matter [@jae-nag96] have illustrated that the dissipative nature of the interactions between grains can result in a variety of different phenomena. Of particular interest in recent years has been the dynamics of vibrated granular materials [@warretal95; @meloetal95], which exhibit stationary states as well as waves and complex patterns. In order to describe these diverse states of the material, it is necessary to derive macroscopic descriptions by averaging over the microscopic details of the motion and interactions between individual grains. This goal has proved elusive, however, because a vibrated granular material is a driven dissipative system, and the interactions between the particles are characterised by a loss of energy due to inelastic collisions. The statistical mechanics framework developed for equilibrium or near equilibrium systems cannot be used in this case. Consequently, phenomenological models [@shrin97; @tsim-aran97; @venkat-ott98] have been used to describe the dynamics of granular materials. The kinetic theories developed for granular flows [@jen-sav83; @kum98:vib] usually assume that the system is close to “equilibrium” and the velocity distribution function is close to the Maxwell-Boltzmann distribution.
Experimental studies and computer simulations have reported the presence of a uniformly fluidised state in a vibrated bed of granular material. Luding, Herrmann and Blumen [@ludetal94] carried out ‘Event Driven’ (ED) simulations of a two dimensional system of inelastic disks in a gravitational field vibrated from below, and obtained scaling laws for the density variations in the bed. An experimental study of a vibrated fluidised bed was carried out by Warr, Huntley and Jacques [@warretal95]. Their experimental set up consisted of steel spheres confined between two glass plates that are separated by a distance slightly larger than the diameter of the spheres. The particles were fluidised by a vibrating surface at the bottom of the bed, and the statistics of the velocity distribution of the particles were obtained using visualisation techniques. Profiles for the density and the mean square velocity were obtained, and the particle velocity distributions were also determined at certain positions in the bed. Both of these studies reported that there is an exponential dependence of the density on the height near the top of the bed, similar to the Boltzmann distribution for the density of a gas in a gravitational field. However, the dependence of the density deviates from the exponential behaviour near the bottom. The dependence of the mean square velocity on the vibration frequency and amplitude were found to be different in the two studies.
A theoretical calculation of the distribution function in a vibro-fluidised bed was carried out by Kumaran [@kum98:vib; @kum98:vibscal]. The limit of low dissipation, where the coefficient of restitution $e$ is close to $1$ was considered. In this limit, the mean square velocity of the particles is large compared to the mean square of the velocity of the vibrating surface, and the dissipation of energy during a binary collision is small compared to the energy of a particle. A perturbation approximation is used, where the energy dissipation is neglected in the leading order approximation, and the system resembles a gas at equilibrium in a gravitational field. The velocity distribution function is a Maxwell-Boltzmann distribution, and the density decreases exponentially from the vibrating surface. The first order correction to the distribution due to dissipative effects was calculated using the moment expansion method, and the results were found to be in qualitative agreement with the experiments of Warr et. al. [@warretal95].
The theoretical predictions [@kum98:vib; @kum98:vibscal] were compared with previous experimental and simulation studies by McNamara and Luding [@mclud98]. They found that the theory was in good agreement with experiments for dilute beds, where the area fraction of the particles is low, but there were systematic deviations from the theoretical predictions as the area fraction increases. This is to be expected, since the analysis assumed that the density is small and the pair distribution function was set equal to $1$ and therefore the pressure is related to the density by the ideal gas law. These assumptions become inaccurate as the area fraction of the bed increases. An approximate method for including the correction to the pair distribution function was suggested by Huntley [@hunt98].
In the present analysis, the correction to the low density theory of Kumaran [@kum98:vib; @kum98:vibscal] is determined for a vibro-fluidised bed where the coefficient of restitution is close to $1$. An asymptotic analysis is used, where the dissipation is neglected in the leading approximation. The leading order density and velocity profiles are determined using the momentum balance equation in the vertical direction. In contrast to the earlier theory [@kum98:vib; @kum98:vibscal], the virial equation of state for a non-ideal two dimensional gas is used to determine the leading order density profile. The density profile differs from the Boltzmann distribution, but the velocity distribution function is still a Maxwell-Boltzmann distribution. The leading order temperature is determined by a balance between the source and dissipation of energy as before. The complete equilibrium pair distribution function is used to determine the rate of dissipation of energy due to inelastic collisions. The results are compared with hard sphere MD simulations, and also with earlier theoretical and simulation studies.
Analysis
========
The system consists of a bed of circular disks (of diameter $\sigma$) in a gravitational field driven by a vibrating surface. The vibrating surface has a periodic amplitude function but no assumption is made regarding the form of the function. There is a source of energy at the vibrating surface due to particle collisions with the surface, and the dissipation is due to inelastic collisions. A balance between the two determines the “temperature”, which is the mean square velocity of the particles.
The limit of low dissipation, where the coefficient of restitution $e$ is close to $1$, is considered. In this limit, it can be shown that the mean square velocity of the particles is large compared to the mean square velocity of the vibrating surface. An asymptotic expansion in the parameter $\epsilon \equiv U_0^2/T_0$ is used [@kum98:vib]. If the source and dissipation of energy are neglected in the leading approximation, the system resembles a gas of hard disks at equilibrium in a gravitational field. The velocity distribution function is a Maxwell-Boltzmann distribution at equilibrium $$F({{\bf u}}) = \frac{1}{2 \pi T_{0}} \exp{ \left( - \frac{u^{2}}{2
T_{0}} \right)},$$ where $T_{0}$ is the leading order temperature. The density profile is determined by solving the momentum balance equation in the vertical direction $$\label{eq:mombal}
{{\relax}{\frac{\partialp}{\partialz}}} - \rho g = 0,$$ where $p$ is the pressure, $\rho$ is the density (number of particles per area) and $g$ is the acceleration due to gravity. For a gas at equilibrium, the pressure is related to the density by the virial equation of state, which in the case of inelastic circular disks is $$p = \rho T_{0} \left[\frac{1 + e}{2} + (1 + e) g_0(\nu) \, \nu \right],$$ where $g_{0}(\nu)$ is the pair distribution function at contact, which for circular disks is given by [@verlet82] $$g_{0}(\nu) = \frac{1}{16 (1 - \nu)^{2}} \left[ 16 - 7 \nu -
\frac{\nu^{3}}{4 (1 - \nu)^{2}} \right],$$ and $\nu$ is the area fraction corresponding to $\rho$. If the coefficient of restitution is set equal to $1$ in the leading approximation, the equation for the pressure reduces to the standard virial equation of state $$p = \rho T_{0} \left[ 1 + 2 g_{0}(\nu) \, \nu\right].$$ The resulting equation from [Eq. [(\[eq:mombal\])]{}]{} for the density profile is a first order ordinary differential equation, which can be solved using the mass conservation condition $$\label{eq:masscons}
{\int\limits_{0}^{\infty}\!\!\!{d}z\,} \rho = N,$$ where $N$ is the number of particles per unit width of the bed. Note that the leading order temperature $T_{0}$ is still unknown at this stage. This is determined using a balance between the source and dissipation of energy. The source of energy due to particle collisions with the vibrating surface is determined using an equilibrium average over the increase in energy due to particle collisions with the vibrating surface [@kum98:vib; @kum98:vibscal] $$\label{eq:s0}
S_{0} = 2 \sqrt{\frac{2}{\pi}} \, T_{0}^{1/2}
{\left\langleU^2\right\rangle} \, g_0(\nu) \, \rho \, \Big|_{z=0}.$$ Here ${\left\langleU^2\right\rangle}$ represents the mean square velocity of the vibrating surface. The rate of dissipation of energy per unit width is calculated by averaging over the energy loss over all the collisions between particles and integrating over the height of the bed [@kum98:vib] $$\label{eq:d0e}
D_0 = \sqrt{\pi} \, \sigma (1-e^2)\, T_0^{3/2}
{\int\limits_{0}^{\infty}\!\!\!{d}z\,} g_0(\nu) \, \rho^2.$$ Note that the $g_0$ appearing in $S_0$ and $D_0$ is the Enskog factor which accounts for the increase in the frequency of collision for hard disks at high densities. The temperature $T_{0}$ can now be determined from the relation $$\label{eq:s0d0}
S_0 = D_0$$ An analytical solution to the density variation [Eq. [(\[eq:mombal\])]{}]{} can be determined in the low density limit using the equation of state for an ideal gas for the pressure [@kum98:vib]. $$\label{eq:nulow}
\rho = \frac{N g}{T_{0}} \exp{ \left( - \frac{g z}{T_{0}}
\right)}$$ where the leading order temperature is given by, $$T_{0} = \frac{4 \sqrt{2}}{\pi} \frac{{\left\langleU^2\right\rangle}}{N \sigma (1 -
e^{2})}.$$ In the low-density limit the density decays exponentially from the bottom of the bed. At higher densities the solution to the density variation is no longer exponential throughout, and has to be obtained numerically by an iterative scheme. However, at large distances from the bottom, the bed is dilute and the ideal gas law holds good, hence the decay is exponential, even though near the bottom it is not. This gives a convenient starting point for the numerical integration from a *finite* height, above which we assume the asymptotic solution ($z \rightarrow \infty$) to be given by an exponential decay known to within two undetermined constants. A value for the density and the temperature is assumed at this height and the integration is carried out up to the vibrating plate ($z=0$). The complete density profile is obtained by combining the numerical and the asymptotic solutions. If the conditions [Eq. [(\[eq:masscons\])]{}]{} and [Eq. [(\[eq:s0d0\])]{}]{} are not satisfied after one such integration, a new value is determined for the density and temperature using the Newton-Raphson method, and the iteration is repeated till convergence. In cases where the convergence is poor, the solution is obtained by *continuing* a low density solution in a parameter such as $N \sigma$ or $U_0$.
**Viscous dissipation:** The above analysis can be easily extended to the case of dissipation purely due to viscous drag. The expression for the source of energy remains the same as given by [Eq. [(\[eq:s0\])]{}]{}. A drag law given by $a_i = -\mu u_i$ is assumed. The total leading order rate of dissipation per unit width will then be $$\begin{aligned}
\label{eq:d0v}
D_{D0} & = & \mu {\int\limits_{0}^{\infty}\!\!\!{d}z\,} \, \rho {\int\limits_{}^{}\!\!\!{d}{{\bf u}}\,} \, F({{\bf u}})
\,\, {{\bf u}}\cdot{{\bf u}}\nonumber\\
& = & 2 \mu N T_0\end{aligned}$$ Unlike [Eq. [(\[eq:d0e\])]{}]{}, the leading order dissipation is the same for the low density and the high density cases. Nevertheless, the density profile has to be obtained numerically in the manner outlined above, with [Eq. [(\[eq:d0v\])]{}]{} substituted for [Eq. [(\[eq:d0e\])]{}]{} in [Eq. [(\[eq:s0d0\])]{}]{}.
Simulation and Results
======================
The hard sphere molecular dynamics (MD), also known as event driven (ED) method [@ludetal94] is used for the simulations of the vibro-fluidised bed. Periodic boundary conditions are used in the horizontal direction and the vibrating surface at the bottom has a sawtooth form for the amplitude function. The simulations are carried out only for the case of inelastic collisions, since the viscous drag requires a different treatment than the ED method.
The density profiles obtained using the present analysis, as well as the earlier low density approximations of Kumaran [@kum98:vib], are compared with the simulation results in [Figs. \[fig:ldlenu\]]{} and \[fig:hdlenu\]. It is seen that the density profiles of the present analysis are in good agreement with the simulation results even when the density near the bottom of the bed becomes large, while the profiles from the low density approximation have significant errors. [Fig. \[fig:hvis\]]{} shows the nature of the density profile in the high density limit in the case of dissipation due to viscous drag. Here too the present analysis gives reasonable values for packing fraction near the bottom, while the low density theory predicts physically incorrect values.
[\[t\]\[\][$z/\sigma$]{}]{} [\[B\]\[\][$\nu$]{}]{} [\
]{}
[\[t\]\[\][$z/\sigma$]{}]{} [\[B\]\[\][$\nu$]{}]{} [\
]{}
[\[t\]\[\][$z/\sigma$]{}]{} [\[B\]\[\][$\nu$]{}]{} [\
]{}
In a recent work, McNamara and Luding [@mclud98] reported the scaling of dissipation with the center of mass obtained from simulations. The results agreed with the low density theory of [@kum98:vibscal] but a systematic deviation was observed at high densities in all the cases. This deviation is captured in the present analysis. The leading order dissipation at low densities in the bed is given by [@kum98:vib] $$\label{eq:d0}
D_0 = \frac{\sqrt{\pi}}{2}\, (1-e^2) \, N^2 \sigma g \sqrt{T_0}.$$ In [@mclud98] the total dissipation obtained from the simulation was normalised by a factor taken out from this leading order dissipation and a non dimensional number was defined as $$\label{e:cpp}
C_{pp} \equiv \frac{D_0}{(1-e) N^2 \sigma g \sqrt{T_0/2}}.$$ The scaling of this factor with the height of the center of mass ($h$) above the position at rest ($h_0$) was studied. This factor was found out for different parameter sets by varying the bottom wall velocity $U_0$ over several decades such that the bed is taken from a densely packed regime to a very low density regime. They chose a central data set and varied the parameters one at a time. It was found that in all the cases considered, the scaling relation collapsed to a single curve. The central parameter set has the following values $N =
3.2$, $\sigma=1$, $g=1$, $e=0.95$.
The present analysis is valid when $\epsilon \equiv U_0^2/T_0 \ll 1$ and when the frequency of particle-particle collision is much greater than the frequency of particle-wall collisions. It can be shown that in the leading order the ratio of the frequency of particle-particle collisions to the frequency particle-wall collisions is $ \sqrt{2}\pi
\,{N \sigma}$. Hence the present analysis will hold good when ${N \sigma}\gg
1/\sqrt{2}\pi$. The central set corresponds to $\epsilon = 0.35$, ${N \sigma}= 3.2$ and therefore we expect the present analysis to hold good for this case. Most of the parameter sets used in [@mclud98] also fall within the limits of the theory derived here.
[\[t\]\[\][$2(h-h_0)/\sigma$]{}]{} [\[B\]\[\][$C_{pp}$]{}]{} [\
]{}
[\[t\]\[\][$2(h-h_0)/\sigma$]{}]{} [\[B\]\[\][$C_{pp}$]{}]{} [\
]{}
[Fig. \[fig:theoscal\]]{} shows the theoretical predictions of the total dissipation for the different cases reported in Fig. 2 in [@mclud98]. It is compared with the results of two simulations in [Fig. \[fig:simtheo\]]{}. It is seen that the present analysis correctly predicts the lowering of the coefficient $C_{pp}$ at high densities. This reduction in the dissipation from the constant value at low densities is the net result of two opposing factors: (i) decrease in the density from the exponential behaviour near the vibrating bottom (see [Fig. \[fig:hdlenu\]]{}), hence reducing the total value of the dissipation, and (ii) increase in frequency of collisions at high densities, increasing the dissipation.
It is also seen that not all the theoretical predictions collapse on to a curve as is the case with the data from the simulation. In two of the cases the theory does not agree with the simulations because (i) in one the value of the perturbation parameter is high ($\epsilon =
1.73$) and the leading order theory is valid only for low $\epsilon$, and (ii) in the other case the value of ${N \sigma}= 0.65 $ is low.
In [Fig. \[fig:theoscal\]]{}, the apparent mismatch with ‘e-’ is not a discrepancy with the model, but has got to do with the formula chosen used in [@mclud98] for the normalisation of the dissipation factor $C_{pp}$. They had chosen to normalise the dissipation by a factor $(1-e)$. While this might have given a better fit for high densities (low center of mass), the correct factor for very low densities is $(1-e^2)$ as given by [Eq. [(\[eq:d0\])]{}]{}. The difference is more pronounced in the case of $e\ll 1$, which, here, has a value $e=0.75$. A close inspection of the curves ‘e-’ in [Fig. \[fig:theoscal\]]{} and [Fig. \[fig:simtheo\]]{} show that the theory and simulation do indeed agree with each other.
We also note here that the data taken from the reported simulation [@mclud98] is for asymmetric sawtooth vibration, whereas our simulation is for the symmetric sawtooth. Both these give similar results for the scaling of $C_{pp}$. Also the theoretical predictions for the symmetric and the asymmetric sawtooth are identical, indicating that the form of the bottom wall vibration does not affect the scaling of the dissipation with the center of mass.
Conclusion
==========
In summary, a theory to describe the state of a vibro-fluidised bed in the dense limit was derived. This is different from the earlier theory of Kumaran [@kum98:vib; @kum98:vibscal], which is valid for low densities where the ideal gas equation was used and the pair distribution function was set equal to $1$. We have made use of the virial equation of state to obtain a correction to the exponential density profile obtained in low densities and the pair distribution function is used to calculate the increased frequency of collisions in the source and the dissipation of energy. The theoretical predictions of density and temperature were compared with the results obtained from MD simulation of two dimensional disks. The theory correctly predicts the lowering of the density from the exponential value at high densities near the bottom. The theory also predicts the scaling relations of the total dissipation in the bed reported in [@mclud98].
[10]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Number counts of galaxies are re-analyzed using a semi-analytic model (SAM) of galaxy formation based on the hierarchical clustering scenario. We have determined the astrophysical parameters in the SAM that reproduce observations of nearby galaxies, and used them to predict the number counts and redshifts of faint galaxies for three cosmological models for (1) the standard cold dark matter (CDM) universe, (2) a low-density flat universe with nonzero cosmological constant, and (3) a low-density open universe with zero cosmological constant. The novelty of our SAM analysis is the inclusion of selection effects arising from the cosmological dimming of surface brightness of high-redshift galaxies, and also from the absorption of visible light by internal dust and intergalactic clouds. Contrary to previous SAM analyses which do not take into account such selection effects, we find, from comparison with observed counts and redshifts of faint galaxies in the Hubble Deep Field (HDF), that the standard CDM universe is [*not*]{} preferred, and a low-density universe either with or without cosmological constant is favorable, as suggested by other recent studies. Moreover, we find that a simple prescription for the time scale of star formation (SF), being proportional to the dynamical time scale of the formation of the galactic disk, is unable to reproduce the observed number- redshift relation for HDF galaxies, and that the SF time scale should be nearly independent of redshift, as suggested by other SAM analyses for the formation of quasars and the evolution of damped Ly-$\alpha$ systems.'
author:
- 'Masahiro Nagashima, Tomonori Totani and Naoteru Gouda'
- Yuzuru Yoshii
title: Galaxy number counts in the Hubble Deep Field as a strong constraint on a hierarchical galaxy formation model
---
INTRODUCTION
============
In the field of observational cosmology, counts of the number of faint galaxies in a given area of sky provide one of the most fundamental observables from which the cosmological parameters, and hence the geometry of the Universe, can be determined (e.g., Peebles 1993). The predicted number counts of galaxies as a function of apparent magnitude and redshift, which should be compared with the data, is obtained from summing up the derived luminosity functions over all redshifts and morphological types, then multiplied by a redshift-dependent cosmological volume element. While the local luminosity function of galaxies of individual types is known from redshift surveys, the luminosity function at any high redshift still has to be deduced with the help of morphological-type-dependent evolution models of galaxies. Accordingly, the predicted number counts rely directly on how well the evolution of galaxies is modeled from their formation until the present (e.g., Yoshii & Takahara 1988).
With the usual assumption of monolithic collapse, the wind model for elliptical galaxies and the infall model for spiral galaxies are able to reproduce many of the observed properties of nearby galaxies, and provide a strong theoretical tool for understanding their evolution (Arimoto & Yoshii 1986, 1987; Arimoto, Yoshii,& Takahara 1991). However, when using these traditional evolution models, it has been claimed that the standard cold dark matter (CDM) universe, or the Einstein-de Sitter (EdS) universe, is not reconciled with the observed high counts of faint galaxies, and that a low-density universe is preferred (Yoshii & Takahara 1988; Yoshii & Peterson 1991, 1995; Yoshii 1993). Most recently, Totani & Yoshii (2000, hereafter TY00) compared their predictions against the number counts observed to the faint limits in the Hubble Deep Field (HDF, Williams et al. 1996) by taking into account various selection effects for the first time. Allowing for the possibility of number evolution of galaxies in a phenomenological way, and after a comprehensive check of systematic model uncertainties, they strengthened previous claims and further demonstrated that mild or negligible merging of high-redshift galaxies in a low-density flat universe with nonzero cosmological constant is a likely solution to simultaneously reproduce the observed high counts and redshift distribution of faint galaxies. On the other hand, the EdS universe is in serious contradiction with the data even if a strong number evolution is invoked.
However, there is a growing evidence from recent observations of the large-scale structure of the universe that gravitationally bound objects, such as galaxy clusters, are formed through continuous mergers of dark halos. Some authors have tried to construct a galaxy formation model based on this scenario of hierarchical clustering in a CDM universe, referred to as a semi-analytic model (SAM) of galaxy formation (e.g., Kauffmann, White, & Guiderdoni 1993; Cole et al. 1994). In fact, SAMs successfully reproduced a variety of observed features of local galaxies such as their luminosity function, color distribution, and so on.
The number counts of faint galaxies have also been analyzed using SAMs (Cole et al. 1994; Kauffmann, Guiderdoni & White 1994; Heyl et al. 1995; Baugh, Cole & Frenk 1996). The Durham group claimed that their model agrees with the observed number counts in the EdS universe rather than in a low-density universe. The Munich group reached a similar conclusion by halting the process of star formation in smaller halos with circular velocities less than 100 km s$^{-1}$. These conclusions by both groups are apparently at variance with those obtained using the traditional evolution models of galaxies. This apparent discrepancy needs to be immediately resolved.
In this paper we examine the importance of selection effects in observations of faint galaxies which have been ignored in previous SAM analyses of galaxy number counts, such as the cosmological dimming of surface brightness and the absorption of emitted light by internal dust and intergalactic clouds. Taking into account these effects in the same way as in TY00, our SAM analysis obtains, for the first time, the predicted number counts that can be consistently compared with the HDF counts to faint magnitude limits.
This paper is outlined as follows. In §2 we briefly describe our SAM. In §3 we constrain the astrophysical parameters in our SAM analysis against local observations. In §4 we compare theoretical number counts and redshift distributions of faint galaxies with the data, and discuss the range of uncertainties in our calculations of galaxy counts. In §5 we provide a summary and discussion.
MODEL {#sec:model}
=====
The SAM we employ involves known physical processes connected with the process of galaxy formation. It is therefore straightforward to understand how galaxies form and evolve within the context of this model. In the CDM universe, dark matter halos cluster gravitationally and are merged in a manner that depends on the adopted power spectrum of initial density fluctuations. In each of the merged dark halos, radiative gas cooling, star formation, and gas reheating by supernovae occur. The cooled dense gas and stars constitute [*galaxies*]{}. These galaxies sometimes merge together in a common dark halo and more massive galaxies form.
For the purpose of comparison with observation, we use a stellar population synthesis approach, from which the luminosities and colors of model galaxies are calculated. The SAM well explains the local luminosity function of galaxies, the color distribution, and so on. Our present SAM analysis obtains essentially the same results of previous SAM analyses, with minor differences in a number of details. In this section we only briefly describe our model; its full description will be given in Nagashima & Gouda (2001, in preparation).
Scheme of Galaxy Formation {#sec:scheme}
--------------------------
First, based on the method of Somerville & Kolatt (1999), we determine the merging history of dark matter halos by extending the Press-Schechter formalism (Press & Schechter 1974; Bower 1991; Bond et al. 1991; Lacey & Cole 1993). We adopt the power spectrum for a specific cosmology from Bardeen et al. (1986), and assume a halo with circular velocity $V_{\rm circ}<$40km s$^{-1}$ as a diffuse accretion entity. The evolution of the baryonic component is followed until the output redshift with the redshift interval of $\Delta z=0.06(1+z)$, corresponding to the dynamical time scale of halos which collapse at that time. In order to minimize the artificial effect of dividing the history of galaxy formation into discrete redshift intervals, we fix $\Delta z=0.06$ until just prior to the output redshift. This manipulation is important especially at high redshift. In our analysis, the highest output redshift is $z\simeq 9$ with the interval of the output redshift $\sim$0.05-0.4.
If a dark matter halo has no progenitor halos, the mass fraction of the gas is given by $\Omega_{\rm b}/\Omega_{\rm 0}$ , where $\Omega_{0}$ is the density parameter for the total baryonic and non-baryonic components, and $\Omega_{\rm b}=0.015h^{-2}$ is the baryonic density parameter constrained by primordial nucleosynthesis calculations (e.g., Suzuki, Yoshii, & Beers 2000). In this work $h$ represents the Hubble parameter given by $h=H_{0}/100$ km s$^{-1}$Mpc$^{-1}$.
When a dark matter halo collapses, the gas in the halo is shock-heated to the virial temperature of the halo. We refer to this heated gas as the [*hot gas*]{}. At the same time, the gas in dense regions of the halo is cooled due to efficient radiative cooling. We call this cooled gas the [*cold gas*]{}. Assuming an isothermal density distribution of the entire halo and using the metallicity-dependent cooling function by Sutherland & Dopita (1991), we calculate the amount of cold gas which eventually falls onto a central galaxy in the halo. In order to avoid the formation of unphysically large galaxies, the above cooling process is applied only to halos with $V_{\rm circ}<$500 km s$^{-1}$ in the standard CDM and 400 km s$^{-1}$ in low-density universes. The physical reason for this restriction is not clear, but Cole et al. (2000a) recently reported that the formation of enormous galaxies is hindered in the isothermal halo having a large core. Nevertheless, in this paper we adopt a simple isothermal distribution and prevent the formation of so-called “monster galaxies” by hand.
Stars are formed from the cold gas at a rate of $\dot{M}_{*}={M_{\rm
cold}}/{\tau_{*}}$, where $M_{\rm cold}$ is the mass of cold gas and $\tau_{*}$ is the time scale of star formation. We assume that $\tau_{*}$ is independent of $z$, but dependent on $V_{\rm circ}$ as follows:
$$\tau_{*}=\tau_{*}^{0}\left(\frac{V_{\rm circ}}{300\mbox{km~s}^{-1}}\right)
^{\alpha_{*}}\label{eqn:taustar}.$$
This form is referred to as the “Durham model” by Somerville & Primack (1999). It should be noted that Cole et al. (2000a) modified their original form (equation \[eqn:taustar\]) as $$\tau_*=\tau_*^0\left(\frac{V_{\rm circ}}{200\mbox{km~s}^{-1}}\right)^{\alpha_*}
\left(\frac{\tau_{\rm dyn}(z)}{\tau_{\rm dyn}(0)}\right),
\label{eqn:taustdyn}$$ by multiplying a factor proportional to the dynamical time scale in the galactic disk. This form is similar to “Munich model” (Kauffman, White & Guiderdoni 1993).
For convenience, the cases in equations (\[eqn:taustar\]) and (\[eqn:taustdyn\]) are hereafter referred to as “constant star formation (CSF)” and “dynamical star formation (DSF),” respectively. The free parameters of $\tau_{*}^{0}$ and $\alpha_{*}$ in CSF and DSF are fixed by matching the observed mass fraction of cold gas in neutral form in the disks of spiral galaxies. Cole et al. (2000a) stressed that the $V_{\rm circ}$-dependence is needed to reproduce the observed ratio of gas mass relative to the $B$-band luminosity of a galaxy. The effect of introducing the $\tau_{\rm dyn}$-dependence in DSF will be discussed in §\[sec:sft\].
In our SAM, stars with masses larger than $10M_\odot$ explode as Type II supernovae (SNe) and heat up the surrounding cold gas. This SN feedback reheats the cold gas to hot gas at rate of $\dot{M}_{\rm reheat}={M_{\rm
cold}}/{\tau_{\rm reheat}}$, where the time scale of reheating is given by
$$\tau_{\rm reheat}=\left(\frac{V_{\rm circ}}{V_{\rm hot}}
\right)^{\alpha_{\rm hot}} \tau_{*}.$$
The free parameters of $V_{\rm hot}$ and $\alpha_{\rm hot}$ are determined by matching the local luminosity function of galaxies.
With these $\dot{M}_{*}$ and $\dot{M}_{\rm reheat}$ thus determined, we obtain the masses of hot gas, cold gas, and disk stars as a function of time during the evolution of galaxies. Chemical enrichment is also taken into account adopting [*heavy-element yield*]{} of $y=0.038=2Z_{\odot}$, but changing this value of $y$ has a minimal effect on the results described below.
When two or more progenitor halos have merged, the newly formed larger halo should contain at least two or more galaxies which had originally resided in the individual progenitor halos. By definition, we identify the central galaxy in the new common halo with the central galaxy contained in the most massive of the progenitor halos. Other galaxies are regarded as satellite galaxies.
These satellites merge by either dynamical friction or random collision. The time scale of merging by dynamical friction is given by $$\tau_{\rm fric}=\frac{260}{\ln\Lambda_{\rm c}}\left(\frac{R}{\rm Mpc}\right)^{2}
\left(\frac{V_{\rm circ}}{10^{3}{\rm km~s}^{-1}}\right)
\left(\frac{M_{\rm sat}}{10^{12}M_{\odot}}\right)^{-1}{\rm Gyr},$$ where $R$ and $V_{\rm circ}$ are the radius and the circular velocity of the new common halo, respectively, $\ln\Lambda_{\rm c}$ is the Coulomb logarithm, and $M_{\rm sat}$ is the mass of the satellite galaxies including the mass of dark matter (Binney & Tremain 1987). When the time elapsed after a galaxy becomes a satellite exceeds $\tau_{\rm
fric}$, a satellite galaxy infalls onto the central galaxy. On the other hand, the mean free time scale of random collision is given by $$\begin{aligned}
\tau_{\rm coll}&=&\frac{500}{N^{2}}\left(\frac{R}{\mbox{Mpc}}
\right)^{3}\left(\frac{r_{\rm gal}}{0.12\mbox{Mpc}}\right)^{-2}
\nonumber\\
&&\qquad\times\left(\frac{\sigma_{\rm gal}}{100\mbox{km~s}^{-1}}
\right)^{-4}\left(\frac{\sigma_{\rm halo}}{300\mbox{km~s}^{-1}}
\right)^{3}\mbox{Gyr},\end{aligned}$$ where $N$ is the number of satellite galaxies, $r_{\rm gal}$ is their radius, and $\sigma_{\rm halo}$ and $\sigma_{\rm gal}$ are the 1D velocity dispersions of the common halo and satellite galaxies, respectively (Makino & Hut 1997). For simplicity, the satellite radius $r_{\rm gal}$ is set to be one tenth of the virial radius of a progenitor halo to which the satellite once belonged as a central galaxy. With a probability $\Delta t/\tau_{\rm coll}$, where $\Delta t$ is the time step corresponding to the redshift interval $\Delta z$, a satellite galaxy merges another satellite picked out randomly. This process was first introduced in a SAM by Somerville & Primack (1999).
Consider the case that two galaxies of masses $m_1$ and $m_2 (>m_1)$ merge together. If the mass ratio $f=m_1/m_2$ is larger than a certain critical value of $f_{\rm bulge}$, we assume that a starburst occurs and all the cold gas turns into hot gas, which fills in the halo, and the stars populate the bulge of a new galaxy. On the other hand, if $f<f_{\rm bulge}$, no starburst occurs and a smaller galaxy is simply absorbed into the disk of a larger galaxy. These processes are repeated until the output redshift.
Given the SF rate as a function of time or redshift, the absolute luminosity and colors of individual galaxies are calculated using a population synthesis code by Kodama & Arimoto (1997). The stellar metallicity grids in the code cover a range of $Z_{*}=$0.0001-0.05. The initial stellar mass function (IMF) that we adopt is the power-law IMF of Salpeter form with lower and upper mass limits of $0.1$M$_{\odot}$ and $60$M$_{\odot}$, respectively. Since our knowledge of the lower mass limit is incomplete, there is the possibility that many brown dwarf-like objects are formed. Therefore, following Cole et al. (1994), we introduce a parameter defined as $\Upsilon=(M_{\rm lum}+M_{\rm
BD})/M_{\rm lum}$, where $M_{\rm lum}$ is the total mass of luminous stars with $m\geq 0.1M_\odot$ and $M_{\rm BD}$ is that of invisible brown dwarfs.
To account for extinction by internal dust we adopt a simple model by Wang & Heckman (1996) in which the optical depth in $B$-band is related to the luminosity as $\tau_{B}=0.8(L_{B}/1.3\times
10^{10}L_{\sun})^{0.5}$. Optical depths in other bands are calculated by using the Galactic extinction curve, and the dust distribution in disks is assumed to be the slab model considered by Somerville & Primack (1999). It is not trivial that such an empirical dust model can be extraporated to very bright galaxies. However, since the extinction is typically $A_{B}\sim 1$ mag at the bright-end of the local luminosity function, our results especially in the $I$-band are not significantly affected by the details of the dust model.
Emitted light from distant galaxies is absorbed by Lyman lines and Lyman continuum in intervening intergalactic clouds. The redshift at which this effect becomes important is different for different photometric passbands, because the absorption occurs around 1000 Å in the rest frame of the clouds. Fig. \[fig:h1clouds\] shows the expected extinction as a function of redshift for four passband filters of the HDF (Yoshii & Peterson 1994). This effect has been included into a SAM by Baugh et al. (1998) in order to pick out the Lyman-break galaxies at high redshift by color selection criteria.
We classify galaxies into different morphological types according to the $B$-band bulge-to-disk luminosity ratio $B/D$. In this paper, following Simien & de Vaucouleurs (1986), galaxies with $B/D\geq 1.52$, $0.68\leq
B/D<1.52$, and $B/D<0.68$ are classified as ellipticals, S0s, and spirals, respectively. Kauffmann et al. (1993) and Baugh, Cole & Frenk (1996) showed that this method of type classification well reproduces the observed type mix.
The above procedure is a standard one in the SAM of galaxy formation. In the next subsection, in order to investigate the properties of high-$z$ galaxies properly, we introduce two important effects into our SAM analysis.
New ingredients of the model: selection effects {#sec:luminosity}
-----------------------------------------------
We judge whether the surface brightness of the galaxies in our SAM is above the detection threshold of the HDF observations. The intrinsic size of spiral or disk-dominated galaxies is estimated by adopting the dimensionless spin parameter $\lambda_{\rm H}=0.05$, and by conserving the specific angular momentum during the gas cooling. On the other hand, the intrinsic size of early-type galaxies is estimated from their virial radii, adjusted by a scaling parameter $f_{\rm b}$ to match with the observed size of early-type galaxies. Note that the selection effects on the local luminosity function have been considered by Cole et al. (2000a) in a simple way in which the isophotal magnitude within 25 mag arcsec$^{-2}$ was used.
The selection effects in predicting the HST number counts in our SAM analysis are evaluated as follows. Using the intrinsic size of model galaxies as obtained above, and adopting a Gaussian point-spread function $f(x)$ for the HST observations, the surface brightness profile $\tilde{g}(x)$ in the observer frame is given by $$\tilde{g}(\vert{{\bf x}}\vert)=\int{\rm d}{{\bf x}}'
f(\vert{{\bf x}}'-{{\bf x}}\vert)g(\vert{{\bf x}}'\vert),$$ where $\vert{{\bf x}}\vert=x=r/r_{\rm e}$ is the normalized radius away from the galaxy center relative to the effective radius $r_{\rm e}$, and $g(x)=\exp(-a_nx^{1/n})$ is the intrinsic surface brightness profile. We adopt $n=1$ for spirals and $n=4$ for ellipticals. The coefficient $a_n$ is given by $a_1=1.68$ and $a_4=7.67$. Then, for each of our model galaxies, we determine their surface brightness profile $S(\theta)$, where $\theta=xr_{\rm e}/d_{\rm A}$ and $d_{\rm A}$ is the angular-diameter distance. We note that model galaxies with surface brightness brighter than the threshold $S_{\rm th}$ and with an isophotal diameter larger than the minimum diameter $D_{min}$ are actually detected as galaxies (Yoshii 1993). In order to be consistent with the HDF observations, we use the isophotal magnitude scheme, $S_{\rm th}=27.5$ mag arcsec$^{-2}$ in $V_{606}$, $S_{\rm th}=27.0$ mag arcsec$^{-2}$ in $I_{814}, U_{300}$ and $B_{450}$, and $D_{min}\sim$0.2 arcsec. More details are described in TY00.
THE SETTING OF PARAMETERS IN OUR SAM ANALYSIS {#sec:norm}
=============================================
We consider the predicted number counts in four models – SC, OC, LC, and LD (Table \[tab:astro\]). The first three models are for CSF (equation \[eqn:taustar\]), and the fourth model is for DSF (equation \[eqn:taustdyn\]). The capitals S, O, and L refer to the standard CDM universe, a low-density open universe and a low-density flat universe with non-zero cosmological constant ($\Lambda$), respectively.
The cosmological parameters ($\Omega_0$, $\Omega_\Lambda$, $h$, $\sigma_8$) are tabulated in Table \[tab:astro\]. For all the models the baryon density parameter $\Omega_{\rm b}=0.015h^{-2}$ is used in common. For the low-density open and flat universes the value of $\sigma_8$ is determined from observed cluster abundances (Eke, Cole & Frenk 1996).
The astrophysical parameters ($V_{\rm hot}$, $\alpha_{\rm hot}$, $\tau_*^0$, $\alpha_*$, $f_{\rm b}$, $f_{\rm bulge}$, $\Upsilon$) are constrained from local observations. However, since the parameter of $f_{\rm bulge}$, among others, do not affect our result, we set $f_{\rm bulge}=0.2$ for the standard CDM universe, and $f_{\rm bulge}=0.5$ for the low-density open and flat universes. Other parameters are discussed below.
The Local Luminosity Function of Galaxies
-----------------------------------------
[\[sec:lf\]]{}
The SN feedback-related parameters of $V_{\rm hot}$ and $\alpha_{\rm
hot}$ determine the location of the knee of the luminosity function and the faint-end slope, respectively. It should be noted that the mass fraction $\Upsilon$ of invisible stars determines the magnitude scale of galaxies, so that changing $\Upsilon$ moves the luminosity function horizontally without changing its overall shape. Therefore, coupled with $V_{\rm hot}$, $\Upsilon$ determines the bright portion of the luminosity function.
Fig. \[fig:lf\] shows theoretical results represented by thick lines for the four models tabulated in Table \[tab:astro\]. Symbols with errorbars indicate observational results from the $B$-band redshift surveys such as APM (Loveday et al. 1992), ESP (Zucca et al. 1997), Durham/UKST (Ratcliffe et al. 1997), 2dF (Folkes et al. 1999) and SDSS (Blanton et al. 2000), and from the $K$-band redshift surveys (Mobasher et al. 1993; Gardner et al. 1997; 2MASS, Cole et al. 2000b). It is evident that while most of $B$-band redshift surveys give a rather flat slope in the faint end, the ESP and SDSS surveys give an extreme case showing a much steeper slope. Note that the SDSS luminosity function shown in Fig. \[fig:lf\] is that with the same detection limit as employed in the 2dF survey.
In this paper we have chosen the values of SN feedback-related parameters so as to reproduce a flat faint-end slope, but in §\[sec:fb\] we will investigate the effect of using the steepest ESP slope in predicting the number counts of faint galaxies.
The Mass Fraction of Cold Gas in Spiral Galaxies
------------------------------------------------
[\[sec:gas\]]{}
The SF rate-related parameters of $\tau_*^0$ and $\alpha_*$ determine the overall mass fraction of cold gas in galaxies with given luminosity and its luminosity dependence, respectively. In some previous SAM analyses, the mass fraction of gas in the Milky Way is exclusively used to fix $\tau_*^0$ for all other spiral galaxies. Then, when $\alpha_*=0$, Cole et al. (2000a) found that dwarf-size galaxies have too little gas to be consistent with observations. According to Cole et al. (2000a), we here constrain both $\tau_*^0$ and $\alpha_*$ in a combined manner to reproduce the mass fraction of cold gas in galaxies spanning a wide range of luminosity.
Fig. \[fig:gas\] shows the ratio of cold gas mass relative to $B$-band luminosity of spiral galaxies as a function of their luminosity. Solid curves show theoretical results for the four models tabulated in Table \[tab:astro\]. We here assume that 75% of the cold gas in these models is comprised of hydrogen, i.e., $M_{\rm
HI}=0.75M_{\rm cold}$. Filled diamonds with errorbars indicate the data taken from Huchtmeier & Richter (1988).
It should be noted that use of $\tau_*^0$ for either CSF (equation \[eqn:taustar\]) or DSF (equation \[eqn:taustdyn\]) hardly affects the resulting luminosity function, as seen from the difference between LC and LD in Fig. \[fig:lf\]. However, $\alpha_*$ depends on the strength of the SN feedback, and the strength should be different for different universe models (see Table \[tab:astro\]). Therefore, in the case of $\alpha_*$, we can only fix its value after the SN feedback-related parameters are constrained in a specified universe model.
The Intrinsic Sizes of Galaxies
-------------------------------
[\[sec:size\]]{}
The scaling parameter $f_{\rm b}$ is determined by matching the effective radius of early-type $L_*$ galaxies, while the actual $L$-dependence in $f_{\rm b}$ is ignored for simplicity. We discuss the effect of this assumption in §\[sec:sizedis\].
Knowledge of the intrinsic size of galaxies is a necessary quantity in our SAM analysis, because it is the surface brightness of galaxies, not their luminosity, that is relevant to evaluating the selection effects for model galaxies. For spiral galaxies, the key process is the conservation of specific angular momentum during the cooling of hot gas. The effective disk radius $r_{\rm e}$ is given by $r_{\rm
e}=(1.68/\sqrt{2})\lambda_{\rm H}r_{\rm i}$, where $r_{\rm i}$ is an initial radius of the progenitor gas sphere (Fall 1983). However, the definition of disk size during the cooling phase is not readily apparent from this formula. Furthermore, the angular momentum transfer during mergers of galaxies in a merged halo is very complicated. Without entering into all these complexities, in this paper we re-estimate the disk size by the above equation when the disk mass increases twice. This approach is simple but reproduces the observations rather well.
Fig. \[fig:rad\] shows the effective disk radii of spiral galaxies as a function of their luminosity. Thick solid lines show the theoretical results for the four models tabulated in Table \[tab:astro\]. Dotted lines indicate the best-fit relation to the observational data given by Impey et al. (1996). All models provide reasonable disk sizes for dwarf galaxies. Slight deviations from the observations of bright galaxies may have occurred partly because dust obscuration becomes effective for such galaxies (cf. §\[sec:luminosity\]), and partly because our method of defining the disk size is too simple. We discuss the effect of changing the disk size in §\[sec:sizedis\].
In this paper, the effective radius of elliptical galaxies is estimated from scaling the virial radius by a parameter $f_{\rm b}$, i.e., $r_{\rm
e}=f_{\rm b}GM_{\rm b}/V_{\rm circ}^2$, where $M_{\rm b}$ is the mass of stars and cold gas, and $f_{\rm b}$ is fixed to reproduce the effective radius of an $L_*$ galaxy. In Fig. \[fig:rad\], thick dashed lines indicate the effective radius of elliptical galaxies for the four models tabulated in Table \[tab:astro\]. Dashed and dot-dashed lines show the best-fit relations for dwarf and compact ellipticals, respectively, based on the observational data given by Bender et al. (1992). Unfortunately, our SAM analysis cannot reproduce two distinct sequences simultaneously; the theoretical $r_{\rm e}-L_B$ relation becomes coincident with the bright portion of the giant-dwarf sequence, whereas the same relation changes its slope becoming coincident with the faint portion of giant-compact sequence. We find that this changeover magnitude is determined by the strength of SN feedback (see Appendix \[sec:app\]), and that the extreme manipulation that reproduce the dwarf sequence affects the number counts of model galaxies only weakly.
Results
=======
Galaxy Number Counts {#sec:counts}
--------------------
Fig. \[fig:counts\] shows the number counts of galaxies as a function of apparent isophotal magnitude for the SC, OC, and LC models in the HST ${UBVI}$ bands in the AB magnitude system. The thick lines are the theoretical predictions, based on the HST observational conditions, including the selection effects from the cosmological dimming of surface brightness and also from the absorption of visible light by internal dust and intergalactic clouds. The thin lines are the predictions ignoring the selection effects except for the effect of dust absorption, because the dust absorption is already taken into account in reproducing the local luminosity function. Open circles with errorbars show the HDF data (Williams et al. 1996), and other symbols show ground-based data after transformation to AB magnitudes to be consistent with the HST.
In order to discriminate favorable universe models, the predictions shown by the thick lines should be compared directly with the observational data. It is evident from this figure that the SC model falls short of the HST data. We note that the discrepancy between the SC model and the data is seen at $B_{450}\gtrsim 25$, even if we do not consider the selection effects. On the other hand, both of the LC and OC models agree well with the data, owing to the inclusion of selection effects. We will discuss the uncertainty of estimating these effects in the next section.
Fig. \[fig:zdist\] shows the redshift distribution of galaxies. The thick lines denote the SC, OC and LC models, including all the selection effects as in Fig. \[fig:counts\]. The histogram indicates the observed redshift distribution based on photometric redshifts of HDF galaxies estimated by Furusawa et al. (2000), in which they improved the method of redshift estimation compared to the method by Fern[á]{}ndez-Soto, Lanzetta & Yahil (1999).
In the LC model, both the peak height and the distribution towards higher redshift agree well with the observation. However, the peak height of the SC model falls significantly short of that observed at $z\simeq 1-1.5$. This is a direct reflection of the lack of galaxies in the theoretical number-magnitude relation for the SC model (see Fig.\[fig:counts\]). In the OC model, the relative number between the peak and high-$z$ tail at $z\gtrsim 3$ is smaller than that observed and is inconsistent with the data. This is a direct reflection of the $z$-dependence of the comoving volume element $dV/dz$, which is shown in the middle panel of Fig.\[fig:zdist\]. While in the LC model the matter density dominates over the cosmological constant at high redshift, which leads to a similar $z$-dependence of $dV/dz$ to that in the EdS universe, the negative curvature effect makes $dV/dz$ decline more slowly at such high redshift in the OC model. Thus the redshift distribution in the OC model declines toward higher redshift more slowly than that in the LC model. In order for the OC model to agree with the data, we would need to halt the star formation at $z\gtrsim 3$ by hand, for unknown reasons.
We conclude that as far as that the astrophysical parameters in our SAM analysis are constrained by local luminosity function of galaxies, the standard CDM universe does not agree with both the observed number-magnitude and number-redshift relations, and that the $\Lambda$-dominated flat universe is best able to reproduce these relations simultaneously.
Uncertainties in the SAM counts
-------------------------------
[\[sec:discussion\]]{}
In this section we discuss uncertainties in predicting the number counts of galaxies. Sources of such uncertainties considered here include the time scale of star formation (§\[sec:sft\]), the galaxy size (§\[sec:sizedis\]), and the adopted SN feedback (§\[sec:fb\]).
### Star Formation Time-Scale {#sec:sft}
Here we evaluate the effects of changing the time-scale of star formation from CSF (equation \[eqn:taustar\]) to DSF (equation \[eqn:taustdyn\]). Fig. \[fig:sft\] shows predicted number counts for the corresponding $\Lambda$-dominated LC and LD models. The difference between these models is apparently small but its magnitude-dependence is different among the predictions in the $UBVI$ bands. In the case of longer wavelength such as the $I_{814}$ band, the number of faint galaxies in the LD model becomes larger than that in the LC model, because in the case of DSF more stars are formed at high redshift according to much a shorter $\tau_*({\rm DSF})$ as compared with CSF. \[In the particular case of the EdS universe, not shown here, the DSF gives $\tau_*({\rm DSF})\propto (1+z)^{-3/2}$, so that the present $\tau_*({\rm DSF})$ is more than 10 times longer than that at $z\sim 5$.\] However, in the case of shorter wavelengths, particularly the $U_{300}$ band, because the apparent luminosity from galaxies is dominated by instantaneous SF, and because many stars have already been formed at higher redshift, the number of faint galaxies in the LD model becomes slightly smaller than that in the LC model.
The difference between the LC and LD models is most prominent when we consider the redshift distribution of faint galaxies. Fig. \[fig:z\_sft\] shows such predictions for the LC and LD models. Clearly the LD model predicts too many high-$z$ galaxies to be consistent with observation. We may be able to remedy this defect by imposing a more efficient internal dust absorption in order to decrease the number of high-$z$ galaxies. However, this manipulation simply decreases the number of faint galaxies below that observed in the $U_{300}$. Thus, we suggest that the $z$-dependence of SF should be negligible and the CSF is a reasonable option in the framework of our SAM.
### Galaxy Size {#sec:sizedis}
In this paper, it is assumed according to the usual SAM analysis that the physical mechanism for determining the galaxy size is the conservation of specific angular momentum for spiral galaxies and the virial theorem for the baryon component of elliptical galaxies. While this assumption is considered to be reasonable, it is important to note the uncertainties in the normalization of the above relations. In this section we consider two LC variants and compare them with the original LC model. Figs. \[fig:rad\_l\] and \[fig:z\_rad\_lcdm\] present theoretical predictions from these three LC models. We rather arbitrarily decrease and increase the original galaxy size by a factor of 2, and refer to these variants as “high surface brightness” and “low surface brightness,” respectively. Note that the 1$\sigma$ scatter in the observational data is about a factor of 1.7, so the range of changing the radius is sufficient large to check the uncertainty in the galaxy size. Given the threshold of surface brightness for detection used in the HST observation, model galaxies with “high surface brightness” are more easily detected, while those with “low surface brightness” remain undetected. The number counts of these two variants differ only by a factor of 1.5 at $B_{450}\sim 28$, as well as at the corresponding magnitudes in the other bands. We found that the uncertainties from changing the galaxy size in the SC and OC models are almost the same as those in the LC model.
### Supernova Feedback {#sec:fb}
The SN feedback-related parameters essentially determine the resulting shape of the local luminosity function of galaxies (§\[sec:lf\]). Therefore, changing these parameters corresponds to changing the number density of galaxies in the local universe. In this section, we constrain these parameters against the ESP luminosity function (Zucca et al. 1997) which gives the steepest faint-end slope among other observations. This is done by weakening the strength of SN feedback in three CSF variants such as SCw, OCw, and LCw. Other parameters such as the SF rate-related are determined by the same way in §\[sec:norm\]. All the parameters for these models are tabulated in Table \[tab:astro2\], and their $B$-band luminosity functions are shown in Fig. \[fig:lf\_fb\].
Straightforward calculations predict the number counts of galaxies which exceed the bright counts at $B\sim 20$, and this excess alone may invalidate the extreme assumption for faint-end slope of the local luminosity function. However, considering the possible existence of systematic uncertainties that could affect current observations of the local luminosity function, we adjust the normalization in such a way as to reproduce the observed bright counts. Then we examine the effect of adopting the steepest ESP slope instead of our standard choice. Fig. \[fig:fb\] shows the number counts of galaxies for the SCw, OCw, and LCw models. It is evident from this figure that even the steepest ESP slope does not save the standard CDM universe, which confirms the claim by TY00.
SUMMARY AND DISCUSSION
======================
[\[sec:conclusion\]]{}
We have calculated the number counts of faint galaxies in the framework of a SAM for three cosmological models of the standard CDM (EdS) universe, a low-density open universe, and a low-density flat universe with nonzero $\Lambda$. The novelty of our SAM analysis is that theoretical predictions are made by fully taking into account the selection effects from the cosmological dimming of surface brightness of galaxies and also from the absorption of visible light by internal dust and intergalactic clouds.
Comparison of theoretical predictions with the observed number counts and photometric redshift distribution of HST galaxies, as well as other ground-based observations, indicates that the standard CDM is ruled out and a $\Lambda$-dominated flat universe is most favorable, while a low-density open universe is marginally favored. This result is in sharp contrast with previous SAM analyses on galaxy number counts where many of the conceivable selection effects in faint observations have been ignored. It is only recently that the SAM analyses have included the effects of internal dust absorption (Somerville & Primack 1999; Cole et al. 2000a), and intergalactic absorption (Baugh et al. 1998), and the isophotal selection effect in a very simple way (Cole et al. 2000a). However, as stressed by TY00, any predictions based on number count analyses will be seriously compromised unless all of the selection effects considered in this paper are taken into account simultaneously.
Based on a hierarchical clustering scenario in the CDM universe, the SAM naturally involves the merger-driven number evolution of galaxies, which has been introduced only phenomenologically in traditional models of galaxy evolution (Yoshii 1993; Yoshii & Peterson 1995; TY00). The fact that we have essentially reached the same conclusion from phenomenological approach confirms that previous simple prescriptions of the number evolution of galaxies are still useful in studying the global evolution of faint galaxies. It should be noted that in TY00 the adopted number evolution law which reproduces the observational data is $\phi_{*}\propto (1+z)$ and $L_{*}\propto (1+z)^{-1}$, and is consistent with the observational estimate of the merger rate of the Canada-France redshift survey (CFRS) galaxies (Le F[é]{}vre et al. 2000).
The basic ingredients in the SAM analysis include the SF process and SN feedback. Although the constant and dynamical SFs can equally reproduce the local luminosity function by adjustment of their free parameters, we find that the dynamical SF predicts the formation of too many high-$z$ galaxies to be consistent with the photometric redshift distribution of faint HST galaxies. Thus, our SAM analysis prefers the constant SF, as is also supported from other recent SAM analyses on the formation of quasars (Kauffmann & Haehnelt 2000) and the evolution of damped Ly-$\alpha$ systems (Somerville, Primack & Faber 2001).
The SN feedback, associated with the virial equilibrium for the baryonic component, controls the resulting size versus luminosity relation of elliptical galaxies. However, our SAM analysis is unable to reproduce the observed relation bifurcating into the giant-dwarf and giant-compact sequences; our brighter ellipticals reside on the bright portion of giant-dwarf sequence and our fainter ellipticals on the faint portion of giant-compact sequence. Although this limitation of our SAM analysis is found to hardly affect the conclusion in this paper, it is of urgent importance for the SAM to be equipped with some mechanism enabling the bifurcation of early-type galaxies into two distinct sequences as observed.
BULGE SIZE AND SUPERNOVA FEEDBACK {#sec:app}
=================================
In §\[sec:sizedis\] we mentioned that the intrinsic size of early-type galaxies is related to the strength of SN feedback. In §2.1 describing SN re-heating the baryonic mass in a halo is given by $M_{\rm
b}\lesssim \Omega_{\rm b}M_{\rm H}/(1+\beta)$, where $\beta\equiv
\tau_*/\tau_{\rm reheat}$ and $M_{\rm H}$ is the halo mass. The left- and right-hand sides are not equal to each other, because this relation depends on the merging history of halos. The spherical collapse model gives $M_{\rm H}\propto V_{\rm circ}^3$, ignoring the dependence on the formation redshift. Thus, we obtain $r_{\rm e}\propto M_{\rm b}/V_{\rm
circ}^2\propto V_{\rm circ}/(1+\beta)$. Assuming a constant ratio of baryonic mass $M_{\rm b}$ relative to luminosity $L$, two limiting cases of $\beta\ll 1$ and $\beta\gg 1$ give $$\log r_{\rm e}=\left\{
\begin{array}{ll}
\displaystyle{-\frac{1}{7.5}{\cal M}}+\mbox{const.}& \mbox{for~~~}
\beta\ll 1\\
\displaystyle{-\frac{(1+\alpha_{\rm hot})}{2.5(3+\alpha_{\rm hot})}{\cal
M}}+\mbox{const.}& \mbox{for~~~} \beta\gg 1,\\
\end{array}\right.\label{eqn:rad}$$ where $\cal M$ is the absolute magnitude. Fig. \[fig:rad\_ana\] shows the size of early-type galaxies for the SC model ([*upper thick curve*]{}) and the LC model ([*lower thick curve*]{}). The vertical scale is chosen arbitrarily to avoid overlapping. Solid lines are for $\beta\ll 1$ in equation (\[eqn:rad\]). Dot-dashed and dashed lines are for $\beta\gg 1$ with $\alpha_{\rm hot}=5.5$ and 2.5, respectively (see Table \[tab:astro\]). Although we fixed the baryon fraction, the baryonic mass-to-light ratio, and the formation redshift of halos, theoretical results ([*thick curves*]{}) can readily fit the scaling relations. This indicates that the SN feedback process essentially determines the size of early-type galaxies. In other words, we can constrain the SN feedback from reproducing their observed size versus luminosity relation. It should be noted that the importance of SN feedback in galaxy evolution has also been highlighted in other recent SAM analyses of the color versus magnitude relation of early-type galaxies (Kauffmann & Charlot 1998; Nagashima & Gouda 1999).
We would like to thank T. C. Beers for his critical reading of the manuscript. This work has been supported in part by the Grant-in-Aid for the Center-of-Excellence research (07CE2002) and for the Scientific Research Funds (10640229 and 12047233) of the Ministry of Education, Science, Sports and Culture of Japan.
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----------- -------------- -------------------- ----- -------------- -- ----------------------------- -------------------- ---------------------- ------------ ----------------- ------------- ------------
CDM Model $\Omega_{0}$ $\Omega_{\Lambda}$ $h$ $\sigma_{8}$ $V_{\rm hot}$ (km s$^{-1}$) $\alpha_{\rm hot}$ $\tau_{*}^{0}$ (Gyr) $\alpha_*$ $f_{\rm bulge}$ $f_{\rm b}$ $\Upsilon$
SC 1 0 0.5 0.67 320 5.5 4 -3.5 0.2 1 1.
OC 0.3 0 0.6 1 220 4 1 -3 0.5 0.5 1.5
LC 0.3 0.7 0.7 1 280 2.5 1.5 -2 0.5 0.5 1.5
LD 0.3 0.7 0.7 1 280 2.5 4 -2 0.5 0.5 1.5
----------- -------------- -------------------- ----- -------------- -- ----------------------------- -------------------- ---------------------- ------------ ----------------- ------------- ------------
----------- -------------- -------------------- ----- -------------- -- ----------------------------- -------------------- ---------------------- ------------ ----------------- ------------- ------------
CDM Model $\Omega_{0}$ $\Omega_{\Lambda}$ $h$ $\sigma_{8}$ $V_{\rm hot}$ (km s$^{-1}$) $\alpha_{\rm hot}$ $\tau_{*}^{0}$ (Gyr) $\alpha_*$ $f_{\rm bulge}$ $f_{\rm b}$ $\Upsilon$
SCw 1 0 0.5 0.67 300 4 4 -2.5 0.2 1 1
OCw 0.3 0 0.6 1 160 2.5 1.5 -1.5 0.5 0.5 1.8
LCw 0.3 0.7 0.7 1 160 2 1.5 -1.5 0.5 0.5 1.5
----------- -------------- -------------------- ----- -------------- -- ----------------------------- -------------------- ---------------------- ------------ ----------------- ------------- ------------
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.'
author:
- |
Laurent DESVILLETTES\
[CMLA, ENS Cachan,]{}\
[61, avenue du Président Wilson, 94235 Cachan Cedex, France]{}\
[E-mail:]{} `[email protected]`\
\
Giulia FURIOLI\
[Dipartimento di Ingegneria Gestionale e dell’Informazione, Università di Bergamo,]{}\
[Viale Marconi 5, I–24044 Dalmine (BG), Italy]{}\
[E-mail:]{} `[email protected]`\
\
Elide TERRANEO\
[Dipartimento di Matematica F. Enriques, Università degli studi di Milano,]{}\
[Via Saldini 50 , I–20133 Milano, Italy]{}\
[E-mail:]{} `[email protected]`
title: Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules
---
Introduction
============
This paper deals with a propagation property for the solution of the following Cauchy problem for the spatially homogenous Boltzmann equation for Maxwellian molecules $$\left\{
\begin{aligned}
&\partial_t f(v,t)= Q(f,f)(v,t)\\
& f(v,0)= f_0(v).
\end{aligned}
\right .$$ Here, $f(v,t) : \R^3 \times \R^+ \freccia \R$ is the probability density of a gas which depends only on the velocity $v\in\R^3$ at the time $t\geq 0$ and $Q$ is the quadratic Boltzmann collision operator in the case of Maxwellian molecules: $$\label{Q}
Q(f,f)(v,t)=\int_{w\in\R^3}\int_{n\in S^2}
\left (f(v_*, t) f(w_*, t) -f(v,t)f(w,t)\right ) b\left(\frac{v-w}{|v-w|}\cdot n
\right ) \, \d n \, \d w.$$ Due to the physical assumptions that the gas evolves through binary, elastic collisions which are localized both in space and time, the relations between the velocities $(v_*,w_*)$ of two particles before the collision and $(v, w)$ after it are the following : $$\left\{
\begin{aligned}
& v_*= \frac {v+w} 2 + \frac {|v-w|}2 \, n,\\
& w_* = \frac {v+w} 2 - \frac {|v-w|}2 \, n,
\end{aligned}
\right .$$ where $n$ is a vector in $S^2$, the unit sphere in $\R^3$, and parametrizes all the possible pre-collisional velocities.
The collision kernel $b$, which is supposed to be nonnegative, is the function which selects in which way the pre-collisional velocities contribute to produce particles with velocity $v$ after the collision and is supposed (this is precisely the assumption of Maxwellian molecules) to depend only on the cosine of the deviation angle $\theta$, namely $$\cos\theta= \frac{v-w}{|v-w|}\cdot n.$$ Finally, we will make the so-called non cut-off assumption, which means that $b\notin L^1_{\rm loc} (]-1,1[)$ and, more precisely, we shall consider $$b(\cos \theta ) \sim \frac 1{ (1-\cos \theta)^{\frac 54}},\quad \theta \to 0.$$
From a physical point of view, that means that the gas molecules repel each other with a force proportional to the fifth power of their distance and a great contribution to the integral collision term is given by the grazing collisions ($\theta \sim 0$). The assumption that the collision kernel $b$ is instead integrable on $]-1,1[$ is called a cut-off assumption. For more information about Boltzmann equation and its physical meaning, the reader can consult for instance the review article by Villani [@vil].
Due to the singularity of the collision kernel at the origin, the integral term is not meaningful if $f$ is not smooth and so it is convenient to consider the weak form of the Boltzmann equation: for $\phi \in C^\infty_c (\R^3)$, $$\begin{aligned}
& \int_{v\in \R^3} \partial_t f(v,t) \phi(v) \, \d v \\
&= \int_{v\in\R^3}
Q(f,f)(v,t) \phi(v) \, \d v\\
& = \int_{v\in\R^3}\int_{w\in\R^3}
\int_{n\in S^2}
f(v,t) f(w,t)\left (\phi (v_*) - \phi (v)\right ) b\left(\frac{v-w}{|v-w|}\cdot n
\right) \, \d n \, \d w\, \d v,
\end{aligned}$$ or even, with another point of simplification, in the Fourier variable, \[bolt-fourier\] \_t f(, t)= \_[nS\^2]{} (f(\^+,t) f(\^-,t)-f(,t)f(0,t)) b(n ) n= (,t), as was firstly done by Bobylev (see for instance [@Bo88]). Here we have used the standard notations $$\xi^+ = \frac \xi 2 + \frac {|\xi|}2 n, \quad \xi^- =
\frac \xi 2 - \frac {|\xi|}2 n.$$ The first results about the non cut-off case for the weak equation go back to Arkeryd [@A81], in a more general setting. In [@PT], A. Pulvirenti and Toscani reformulated the existence theory starting from the equation in the Fourier variable both for the cut-off and non cut-off cases. We briefly recall their method and their result, because they will be useful in the following. The classical approach is to find a solution of equation through a limiting process on the solutions of a sequence of cut-off approximating problems in the following way. Let us consider the following sequence of bounded functions obtained by cutting out the singularity of $b$ at the origin \[alpha\] |b\_l= (b, l), l, and let \[beta\] b\_l\^\* = \_[nS\^2]{} |b\_l(n) n. Then, define \[gamma\] \_l = , so that $$\int_{n\in S^2} \beta_l\left (\frac{\xi}{|\xi|}\cdot n\right )\, \d n =1,$$ and then consider the sequence of Cauchy problems \[bolt-cut-off\] {
&\_\_l (,)= \_[nS\^2]{} (\_l (\^+ ,) \_l ( \^- ,)-\_l (,)\_l (0,)) \_l(n) n,\
& \_l (,0)= f\_0().
. A. Pulvirenti and Toscani proved first the existence and uniqueness of a solution $\varphi_l$ of the Cauchy problems . Then, letting $$\hat f_l(\xi, t):= \varphi_l (\xi, b_l^* t),$$ they proved the convergence in a suitable setting of a subsequence of $\hat f_l$ to a solution of the Cauchy problem for the non cut-off equation. More precisely, the result is as follows :
\[A. Pulvirenti, Toscani [@PT]\]\[esist\] We consider an initial datum $f_0 \geq 0$ satisfying the following assumptions: $$\begin{gathered}
\int_{\R^3} f_0(v) \, \d v =1,\quad
\int_{\R^3} f_0(v)\, v_i \,\d v=0,\ i=1,2,3,\quad
\int_{\R^3} f_0(v)|v|^2\, \d v =3,\\
\int_{\R^3} f_0(v) |\log f_0(v)|\, \d v <\infty,
\end{gathered}$$ and the following Cauchy problem: \[cauchy-bolt\] {
&\_t f(,t)= \_[nS\^2]{} (f( \^+,t) f( \^- ,t)-f(,t)f(0,t)) b( n ) n, t>0,\
& f(, 0)= f\_0()
. where $ b$ is a nonnegative function of $L^1_{{\rm loc}}([-1,1[)$ satisfying $ b(\cos \theta) = O\left( \frac 1{
(1-\cos \theta)^{\frac 54}}\right),\ \theta \to 0$. Then, there exists a nonnegative solution $f\in C^1 \left( [0, +\infty),
L^1(\R^3)\right )$ to eq. (\[cauchy-bolt\]) satisfying for all $t>0$ : $$\begin{gathered}
\int_{\R^3} f(v, t) \, \d v =1,\quad
\int_{\R^3} f(v,t)\, v_i\,\d v=0,\ i=1,2,3, \quad
\int_{\R^3} f(v,t)|v|^2\, \d v =3,\\\
\int_{\R^3} f(v,t) |\log f(v,t)|\, \d v <\infty .
\end{gathered}$$ Moreover, for all $t>0$, the Fourier transform $\hat f(\cdot, t)$ of the solution is obtained as the (uniform on compact sets) limit of a subsequence of the functions $\varphi_l (\cdot, b_l^* t) \in C^1 \left( [0, +\infty), C_b(\R^3)\right )$, solutions of the cut-off Cauchy problems , which have the following explicit representation (called Wild’s expansion): $$\varphi_l (\xi, \tau)= \e^{-\tau} \sum_{k=0}^\infty \varphi_l^{(k)}(\xi)
(1-e^{-\tau})^k,$$ where $$\begin{aligned}
&\varphi_l^{(0)}(\xi) = \hat f_0(\xi),\\
&\varphi_l^{(k+1)}(\xi) = \frac 1{k+1} \sum_{j=0}^k
\int_{n\in S^2}
\varphi_l^{(j)}( \xi^+ ) \varphi_l^{(k-j)} ( \xi^-) \, \beta_l \left
(\frac{\xi}{|\xi|}\cdot n \right )\, \d n.
\end{aligned}$$
In [@TV], Toscani and Villani proved that under the hypotheses of Theorem \[esist\], the solution for the non cut-off equation is indeed unique, whereas the uniqueness of the solution for the cut-off problem was already known.
The question whether an extra property satisfied by the initial datum $f_0$ propagates along the solution has been already addressed concerning Sobolev or Lebesgue regularity. In [@d-m] Desvillettes and Mouhot proved the uniform propagation of $L^p$ moments for both the cut-off and non cut-off equations (Cf. also [@gus1], [@gus2], [@mvv] for earlier works on the propagation of $L^p$ regularity). In [@CGT], Carlen, Gabetta and Toscani proved for the cut-off equation that also all the $H^s$ Sobolev norms remain uniformely bounded if they exist initially (Cf. also [@mvv] for related results in the case of hard potentials and hard spheres). When the non cut-off equation is considered, the same is true, and moreover the $H^s$ norms are immediately created (Cf. [@des1; @des2]). In [@U84], Ukai proved for both the cut-off and non cut-off equations that a regularity property of Gevrey type satisfied by the initial datum keeps on being satisfied at least for a finite time by the solution. We shall come back to this result later.
Note finally that many papers address the important question of the propagation of the behavior of the solution with respect to large $v$ (that is, propagation of moments, evolution of Maxwellian tales, etc.). We do not investigate in this direction in this work.
This paper is devoted to the discussion of the following question: if the initial datum $f_0$ satisfies the upper bound $$|\hat f_0(\xi)| \leq K_1 e^{-K_2|\xi|^s},\quad \xi\in\R^3, \ K_1 \geq 1,\ K_2>0, \ s>0,$$ does the solution (of the cut-off or non cut-off Boltzmann equation with Maxwellian molecules) keep on satisfying the same property? The answer is positive, provided that $s \in ]0,2]$ and we allow the constants $K_1, K_2$ to be different from those of the initial datum.
More precisely, the result we are going to prove is the following:
\[7\] Let $f_0$ be a nonnegative function satisfying $$\begin{gathered}
\int_{\R^3} f_0(v) \, \d v =1,\quad \int_{\R^3} f_0(v)\, v_i \,\d
v=0,\ i=1,2,3,\quad
\int_{\R^3} f_0(v)|v|^2\, \d v =3,\\
\int_{\R^3} f_0(v) |\log f_0(v)|\, \d v <\infty.
\end{gathered}$$ If $f_0$ is such that $$\sup_{{\xi}\in \R^3}|\hat f_0({\xi})|{\rm
e}^{K_2\psi(|{\xi}|^2)}\leq K_1,$$ for some $K_1\geq 1$, $K_2>0$, and for some concave function $\psi:[0,+\infty)\to [0,+\infty)$, such that $\psi(0)=0$, $\psi(r)\leq r$ for $r$ large enough and $\psi(r)\to +\infty $ for $r\to +\infty$, then there exist $R_0>0$, $K>0$ such that the unique solution of the Cauchy problem with $f_0$ as initial datum satisfies $$\begin{aligned}
&\sup_{|{\xi}|< R_0}|\hat f({ \xi},t)|{\rm e}^{K
|{ \xi}|^2}\leq 1,\quad t\geq 0,\\
&\sup_{|{\xi}|\geq R_0}|\hat f({\xi},t)|{\rm e}^{K \psi(|{
\xi}|^2)}\leq 1, \quad t\geq 0.
\end{aligned}$$
Denoting by $G^\nu(\R^3)$ the space of Gevrey functions and by $G^\nu_0(\R^3)$ the space of Gevrey functions with compact support (we shall recall in Section 4 their definition), we are able to deduce from the previous result the propagation along the solution of a Gevrey-type regularity satisfied by the initial datum.
\[condiz\] Let $f_0$ be a nonnegative function satisfying \[integ\]
\_[\^3]{} f\_0(v) v =1,\_[\^3]{} f\_0(v) v\_i v=0, i=1, 2, 3,\_[\^3]{} f\_0(v)|v|\^2 v =3,\
\_[\^3]{} f\_0(v) |f\_0(v)| v <.
i) If $\nu>1$ and $f_0\in G^\nu_0(\R^3)$, then the solution $f(\cdot, t)$ of the Cauchy problem (\[cauchy-bolt\]) is in $G^\nu(\R^3)$, uniformly for all $t\geq
0$.
ii) If $\nu\geq 1$, $f_0\in G^\nu (\R^3) \cap {\cal
S'}(\R^3)$, and moreover satisfies $\sup_{\xi\in \R^3} |\hat
f_0(\xi) | \leq K_1 e^{-K_2 |\xi|^{\frac 1\nu}}$ for $K_1 \geq 1$, $K_2>0$, then the solution $f(\cdot, t)$ of the Cauchy problem (\[cauchy-bolt\]) is in $G^\nu(\R^3)$, uniformly for all $t\geq 0$.
The plan of the paper is the following:
in Section 2, we shall present the result in a simpler form and for the so-called Kac model (which is 1-dimensional and describes radially symmetric solutions of the Boltzmann equation);
in Section 3, we shall generalize the result both to Boltzmann equation and to more general bounds on the initial datum;
finally, in Section 4, we shall recall the main definitions of Gevrey functions, and we shall state the propagation result of a Gevrey-type regularity.
[**Acknowledgements:**]{} The authors would like to thank G. Toscani for useful discussions about this problem.
The Kac equation
================
In this section, we present our result in the simpler case of the Kac equation. This equation, in its cut-off or non cut-off version, is obtained when one considers radially symmetric solutions of the homogenous Boltzmann equation for Maxwellian molecules. It reads: $$\partial_t f(v,t)= \int_{w\in\R}\int_{\theta \in [-\frac {\pi} 2,
\frac {\pi} 2]} \left (f(\tilde{v}_*, t) f(\tilde{w}_*, t) -f(v,t)f(w,t)\right )
b(\theta) \, \d\theta \, \d w = \tilde{Q}(f,f)(v,t).$$ Here, $f(v,t) : \R
\times \R^+ \freccia \R$ is the probability density of a gas of one dimensional particles which depends only on the velocity $v\in\R$ at the time $t\geq 0$, and which evolves through collisions which conserve energy but not momentum. The relations between the velocities $(\tilde{v}_*,\tilde{w}_*)$ of two particles before the collision and $(v, w)$ after it are the following $$\left\{
\begin{aligned}
& \tilde{v}_*= v\cos \theta + w\sin \theta,\\
& \tilde{w}_* = v\sin \theta -w\cos \theta.
\end{aligned}
\right .$$ We shall make the following non cut-off assumption on the collision kernel $b$: $$b(\theta) = O_{\theta \to 0} \bigg(\frac {\cos \theta}{|\sin \theta|^\gamma} \bigg),\quad \gamma
\in ]1,3[.$$ Actually, this kind of assumption for the Kac equation was introduced by Desvillettes in [@des1] whereas, in the original equation, $b(\theta)$ is a strictly positive constant. In the same way as for the Boltzmann equation, it is useful to consider the Cauchy problem in the Fourier variable \[kac-fourier\] {
&\_t f(,t)= \_ (f(, t) f(, t) -f(,t)f(0,t)) b() = (,t),\
&f(, 0)= f\_0(),
. where the even initial datum $ f_0 \geq 0$ satisfies the assumptions: \[in-cond\] \_ f\_0(v) v =1, \_ v\^2 f\_0(v) v =1, \_f\_0(v) |f\_0(v)| v <. The Kac equation shares with the homogenous Boltzmann equation for Maxwellian molecules the existence and uniqueness theory for the solutions. By considering the sequence of cut-off approximating problems \[kac-cut-off\] {
&\_\_l (,)= \_ (\_l (, ) \_l (, ) - \_l (,)\_l (0,)) \_l () ,\
& \_l (,0)= f\_0(),
. where each $\beta_l (\theta)$ is a bounded function defined as in , and , it is possible to prove that each Cauchy problem has a unique solution $\varphi_l$, which has the following explicit representation, called Wild’s expansion: $$\varphi_l (\xi, \tau)= \e^{-\tau} \sum_{n=0}^\infty
\varphi_l^{(n)}(\xi) (1-e^{-\tau})^n,$$ where $$\begin{aligned}
&\varphi_l^{(0)}(\xi) = \hat f_0(\xi),\\
&\varphi_l^{(n+1)}(\xi) = \frac 1{n+1} \sum_{j=0}^n \int_{\theta \in
[-\frac {\pi} 2, \frac {\pi} 2]}
\varphi_l^{(j)}(\xi\cos \theta) \varphi_l^{(n-j)} (\xi \sin \theta) \,\beta_l
(\theta)\,
\d \theta.
\end{aligned}$$ Finally, letting $$\hat f_l(\xi, t):= \varphi_l (\xi, b_l^* t),$$ it is possible to establish the (uniform on compact sets) convergence of a subsequence of $\hat f_l$ to a solution $\hat f$ of the Cauchy problem for the (non necessarily cut-off) equation .
Let us suppose now that the initial datum $f_0$ satisfies the extra property: $$|\hat f_0(\xi)|\leq {\e}^{-K|\xi|^s},\quad \xi\in \R,\ K>0, \ s\in (0,2].$$ Thanks to the representation of the solution of the cut-off equation in Wild’s expansion, it is straightforward to prove that the solution itself satisfies the same upper bound. Indeed, by a direct computation, we have that $| \varphi_l^{(1)}(\xi)|\leq
{\e}^{-K|\xi|^s}$, since $$\varphi_l^{(1)}(\xi){\e}^{K|\xi|^s} = \int_{\theta } {\e}^{K|\xi|^s-K|\xi\cos
\theta|^s-K|\xi\sin \theta|^s}
\hat f_{0}(\xi\cos \theta){\e}^{K|\xi\cos\theta|^s}\hat f_0(\xi
\sin\theta){\e}^{K|\xi\sin\theta|^s}\,\beta_l(\theta)\, \d \theta,$$ and $1-|\cos \theta|^s-|\sin \theta|^s\leq 0$, for $s\in (0,2]$. Hence by an immediate iteration argument, the same inequality holds for any $\varphi_l^{(n)}(\xi)$, and finally for the solution of the cut-off equation for any $t\geq 0$. Passing to the limit when $l\to +\infty$ in the estimate $\varphi_l (\xi, b_l^* t) \le
{\e}^{-K|\xi|^s} $, we see that the inequality also holds for the solution of the (non necessarily cut-off) equation .
Due to the non-linearity of the collision operator, if we now consider the weaker assumption $$|\hat f_0(\xi)|\leq
K_1{\e}^{-K_2|\xi|^s}, \quad \xi\in \R,\, K_1>1,\, K_2>0,\, s\in (0,2],$$ the same argument allows to prove that the solution of each cut-off equation satisfies the same upper bound, but only for a finite interval of time. In this case, by letting $l$ go to infinity, the interval of time where the estimate is true can reduce to nothing.
In spite of this, we prove in this section that the condition $|\hat f_0(\xi)|\leq K_1{\e}^{-K_2|\xi|^s}$ propagates (though possibly with different constants $K_1$ and $K_2$ ) along the solution of the (non necessarily cut-off) Kac equation.
Some preliminary properties of initial data
-------------------------------------------
In this section we emphasize some useful properties satisfied by any even, nonnegative function $g$ such that $\int_{\R}g(v)\, \d
v=1$ and $\int_{\R}v^2\, g(v)\, \d v=1$.
\[lemma3-c\] Let $g$ be a nonnegative, even function, satisfying $$\int_{\R} g(v) \, \d v =1,\quad \int_{\R} g(v)\, v^2 \, \d v
=1.$$ Then, there exist $\rho >0$ and $\tilde K>0$ such that for all $|\xi|\leq \rho$: $$|\hat g(\xi)| \leq \e^{-\tilde K |\xi|^2}.$$
We observe that under the hypotheses of the lemma, $\hat{g}$ is of class $C^2$ and satisfies the following property : $\hat{g}(0) =
1$, $\hat{g}'(0) = 0$, and $\hat{g}''(0) = -1$. Using a Taylor expansion of $\hat{g}$ at order $2$, we obtain $\hat{g}(\xi) = 1 -
\frac 12 \xi^2 + o(\xi^2)$ when $\xi \to 0$. Then, the estimate of the lemma holds for any $\tilde K \in ]0, \frac 12[$.
Then, we prove the:
\[lemma1-c\] Let $g \geq 0$ such that $\int_\R g(v) \, \d v =1$. Then, for all $r>0$, there exist $C_r\in(0,\frac 12)$ and $\tilde C_r\in(0,\frac 12)$ such that $$\begin{aligned}
&\int_\R g(v) \sin^2\left(\frac {v\xi}2\right ) \, \d v \geq C_r,\quad |\xi| >r,\\
&\int_\R g(v) \cos^2\left(\frac {v\xi}2\right ) \, \d v \geq \tilde C_r, \quad |\xi|
>r.
\end{aligned}$$
We only prove the first inequality, since the second one can be proven in exactly the same way.
Thanks to Lebesgue’s dominated convergence theorem and thanks to the absolute continuity of the measure $\nu(E):= \int_E g(v)\, \d v$ with respect to the Lebesgue measure, there exist $R>0$ and $\delta>0$ such that for all measurable set $A \subset \R$ such that $|A| \le \delta$, we have \[prima\] \_[A\^c B(0,R)]{} g(v) v 12. Let $\xi \in \R$ be fixed. For $\mu \in ]0, \pi/2[$, we define $$K_{\mu, R} := \left \{v\in \R, |v| \leq R\ {\rm and\ }\exists\ k\in\Z, \left| \frac
{v\xi}2 -k\pi\right | \leq \mu \right \}.$$ It is clear that $$\left| K_{\mu, R}\right | \leq \left( \frac {|\xi|R}\pi +1\right ) \frac {4\mu}
{|\xi|} = 4\mu\left ( \frac R \pi +\frac 1{|\xi|}\right )$$ so that, when $|\xi| \geq r$ we have $ \left| K_{\mu, R}\right | \leq 4\mu\left (
\frac R \pi +\frac 1r\right )$. When $\mu= \frac {\delta}{4\left( \frac R\pi +\frac 1r\right)}$, we see thanks to that $$\int_{K_{\mu,R}^c\cap B(0,R)} g(v)\, \d v \geq \frac 12.$$ We can therefore conclude that $$\int_\R g(v) \sin^2\left(\frac {v\xi}2\right ) \, \d v \geq \int_{K_{\mu,R}^c\cap
B(0,R)} g(v)\sin^2\left(\frac {v\xi}2\right ) \, \d v \geq \frac 12 \sin^2 \mu :=C_r.$$
The propagation theorem
-----------------------
We are now in position to state the theorem.
\[5\] Let $f_0$ be a nonnegative, even function, satisfying $$\int_{\R} f_0(v) \, \d v =1,
\quad
\int_{\R} f_0(v)\, v^2\, \d v =1,\quad
\int_{\R} f_0(v) |\log f_0(v)|\, \d v <\infty.$$ We suppose that $f_0$ is such that $$\sup_{\xi\in \R}|\hat f_0(\xi)|{\rm e}^{K_2|\xi|^s}\leq K_1$$ for some $K_1\geq 1$, $K_2>0$, and $0<s\leq 2$. Then there exist $R_0>0$, $K>0$ such that the unique solution of the Cauchy problem satisfies: \[finale\]
&\_[||< R\_0]{}|f(,t)|[e]{}\^[K ||\^2]{}1,t0,\
&\_[||R\_0]{}|f(,t)|[e]{}\^[K ||\^s]{}1, t0.
We begin by proving the following proposition.
\[espo\] Let $g$ be a nonnegative, even function, satisfying $$\int_\R g(v) \, \d v =1,\quad \int_\R g(v)\, v^2 \, \d v =1.$$ Let us suppose moreover that, for given $s\in (0, 2]$, $K_1 >
1$ and $K_2>0$, $g$ satisfies the following bound: $$|\hat g(\xi)| \leq K_1\e^{-K_2 |\xi|^s}, \quad \xi\in \R.$$ Then, there exists $\eta >0$ such that for all $R > \eta $, there exists $K>0$ (depending on $R$) such that $$|\hat g(\xi)| \leq \begin{cases}
\e^{-K |\xi|^2}, & |\xi| < R,\\
\e^{-K |\xi|^s}, & |\xi| \geq R.
\end{cases}$$
We have already proven in Lemma \[lemma3-c\] that $|\hat g(\xi)|
\leq \e^{- \tilde K |\xi|^2}$ for $|\xi| \leq \rho$, where $\tilde K$ and $\rho$ are suitably chosen. Now, let $\eta=\left(\frac{\log {K_1}}{K_2}\right)^{\frac 1s}$. For every $R>\eta$, we can find $0<K_3 <K_2$ such that $$K_1\e^{-K_2 |\xi|^s} \leq \e^{-K_3 |\xi|^s}, \quad |\xi| \geq R,$$ so that $$|\hat g(\xi)| \leq \e^{-K_3 |\xi|^s}, \quad |\xi| \geq R.$$ It is now enough to find $K_4 >0$ such that $$|\hat g(\xi)| \leq \e^{-K_4 |\xi|^2}, \quad \rho <|\xi| < R.$$ Since $g$ is an even function, we have $$\hat g(\xi) = \int_{\R} g(v) \e^{-i \xi v} \, \d v= \int_\R g(v)
\left( \frac{\e^{-i\xi v} +\e^{i\xi v}} 2\right )\, \d v =
\int_{\R} g(v) \cos ( \xi v) \, \d v.$$ Then, $\hat g$ is real and $|\hat g (\xi)| \leq 1$ for all $\xi
\in \R$. Moreover $$\begin{aligned}
&1-\hat g(\xi) = \int_\R g(v) (1-\cos (\xi v) )\, \d v = 2\int_\R g(v) \sin^2
\left(\frac {\xi v} 2\right )\, \d v,\\
&\hat g(\xi) +1 = \int_\R g(v) (1+\cos (\xi v) )\, \d v = 2\int_\R
g(v) \cos^2 \left(\frac {\xi v} 2\right )\, \d v.
\end{aligned}$$ According to Lemma \[lemma1-c\], we know that $$\begin{aligned}
&1-\hat g(\xi) \geq 2\,C_\rho,\quad |\xi|> \rho,\\
&\hat g(\xi) +1\geq 2\,\tilde{C_\rho},\quad |\xi| >\rho.
\end{aligned}$$ Therefore, $$|\hat g(\xi)| \leq 1-\min(2\, C_\rho,2\, \tilde C_\rho),\quad |\xi| > \rho.$$ Now, there exists $K_4>0$ such that $$1-\min(2\,C_\rho,2\,\tilde{C_\rho}) \leq \e^{-K_4 R^2},$$ which implies $$|\hat g(\xi)| \leq 1-\min(2\,C_\rho,2\,\tilde{C_\rho}) \leq \e^{-K_4 R^2} \leq
\e^{-K_4 |\xi|^2}, \quad \rho < |\xi| \leq R.$$ We can conclude letting $K= \min(\tilde K, K_3, K_4)$.
[**Proof of Theorem \[5\]: the cut off case.**]{} Thanks to Proposition \[espo\], there exists $\eta >0$ such that for any $R_0>\eta$, there exists a strictly positive $K$ such that the initial datum $\hat f_0$ satisfies $$\label{(tre)}
\begin{aligned}
&\sup_{|\xi|< R_0}|\hat f_0(\xi)|{\rm e}^{K
|\xi|^2}\leq 1,\\
&\sup_{|\xi|\geq R_0}|\hat f_0(\xi)|{\rm e}^{K
|\xi|^s}\leq 1.\\
\end{aligned}$$ In order to prove the theorem for the cut-off case, it is enough to establish that any $\varphi_l^{(n)} $ in Wild’s sums satisfies . Let us check that this is true for $\varphi_l^{(1)}$. Let us define $$H(|\xi|)=\begin{cases}
{K |\xi|^2}, &|\xi|< R_0,\\
{K |\xi|^ s}, & |\xi|\geq R_0.
\end{cases}$$ Condition on the initial datum $f_0$ reads therefore $$\sup_{\xi \in \R}|\hat f_0(\xi)|{\rm e}^{H(|\xi|)}\leq 1.$$ Then $$\begin{aligned}
& \left| {\rm e}^{H(|\xi|)}\varphi_l^{(1)}(\xi)\right|\\
\leq & \int_{\theta \in [-\frac \pi 2, \frac \pi 2]}{\rm
e}^{H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos \theta|)} {\rm
e}^{H(|\xi\sin \theta|)} |\hat f_0(\xi\sin \theta)|{\rm
e}^{H(|\xi\cos \theta|) }|\hat f_0(\xi\cos \theta)|
\beta_l(\theta)\d \theta\\
\leq &\int_{\theta \in [-\frac \pi 2, \frac \pi 2]} {\rm
e}^{H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos \theta|)}
\beta_l(\theta)\d \theta.
\end{aligned}$$ Since $\int_\theta \beta_l(\theta) \d \theta=1$, we end the estimate by proving that $H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos
\theta|)\leq 0$ for $\xi\in\R$ and $\theta \in [-\frac {\pi}2,
\frac {\pi}2]$ if $R_0 \ge 1$. Thanks to the symmetries of the function $H$ with respect to $\theta$, we can restrict ourselves to the interval $[0,\frac \pi 4]$. Now, when $|\xi|<
R_0$, we have $$H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos \theta|)=K |\xi|^2(1-(\sin
\theta)^2-(\cos \theta)^2)=0.$$ If $|\xi|\geq R_0$ and $|\xi\sin \theta|\geq R_0$, $|\xi\cos
\theta|\geq R_0$, then $$H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos \theta|)=K |\xi|^s(1-(\sin
\theta)^s-(\cos \theta)^s)\leq 0$$ for $0<s\leq 2$. Whenever $|\xi|\geq R_0$ and $|\xi\sin \theta|<
R_0$, $|\xi\cos \theta|< R_0$ we have $$H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos
\theta|)=K \left(|\xi|^s-|\xi|^2\left((\sin \theta)^2+(\cos
\theta)^2\right)\right )=K \left(|\xi|^s-|\xi|^2\right ).$$ If we choose $R_0\geq 1$, we can conclude since $|\xi|\geq R_0$ that $$|\xi|^s-|\xi|^2\leq 0.$$ If now $|\xi|\geq R_0$ and $|\xi\sin \theta|< R_0$, $|\xi\cos \theta|\geq
R_0$, we have $$H(|\xi|)-H(|\xi\sin \theta|)-H(|\xi\cos
\theta|)=K \left(|\xi|^s-|\xi|^2(\sin \theta)^2-|\xi|^s(\cos \theta)^s\right)$$ $$\le K\, \left( |\xi|^s - |\xi|^s\,(\sin\theta)^2 - |\xi|^s\,(\cos\theta)^2 \right)=0.$$ Note that since $0\le \theta \le \pi/4$, there is no other case to treat.
0.5truecm [**The non cut-off case.**]{} As we have recalled in the introduction of section 2, the solution of in the non cut-off case is obtained as the limit of a subsequence of the solutions of the Cauchy problems . Since the estimate on $\varphi_l (\xi, \tau)$ holds true (for any $\tau\geq 0$), the same is valid for $\hat f_l(\xi,t)$ and hence for $\hat f(\xi,t)$. $\scriptstyle\square$
In Theorem \[5\], the hypothesis that $f_0$ is even could be replaced by the weaker hypothesis that $\int_\R f_0(v)\,v\, \d v=0$. Since the Kac equation comes from the Boltzmann equation when one considers radially symmetric solutions, it is however natural to study only even initial data.
Propagation for the Boltzmann equation
======================================
We would like to extend to the solution of the Boltzmann equation the results proven in the previous section for the solution of the Kac equation. Two kinds of extensions are in order: first, we have to pass from the one-dimensional to the three-dimensional setting; second we would like to state the result considering not only functions like $\e ^ {-|\xi|^s}$, but also like $\e^{-\psi(|\xi|^2)}$, where $\psi$ is a suitable concave function. We now begin by restating the lemmas of the previous section in three dimensions. We shall only indicate the major modifications in the proofs.
\[lemma3-b\] Let $g : \R^3 \to \R$ be a nonnegative function satisfying $$\int_{\R^3} g(v) \, \d v =1,\quad \int_{\R^3} g(v) \,v_i \d v =0, \ i=1,2,3, \quad
\int_{\R^3} g(v)\, |v|^2 \, \d v =3.$$ Then there exist $\rho >0$ and $\tilde K>0$ such that for all $|\xi|\leq \rho$: $$|\hat g(\xi)| \leq \e^{-\tilde K |\xi|^2}.$$
We observe that the result of the lemma is not changed when $g$ is replaced by $g \circ R$, where $R$ is any rotation of $\R^3$. As a consequence, we can suppose that the symmetric matrix $(\int_{\R^3} g(v)\,v_i\,v_j\, \d v)_{i,j \in \{1,2,3\}} $ is diagonal. Moreover, since $g \in L^1(\R^3)$ and $ \int_{v\in \R^3}
g(v)\,|v|^2\, \d v = 3$, we see that $\int_{v\in \R^3} g(v)\,
v_i^2\, \d v >0$ for $i=1,2,3$.
Then, $\hat g(\xi) = 1 -
\sum_{j=1}^3 \lambda_j \,\xi_j^2 + o_{\xi \to 0}(|\xi|^2)$, with $\lambda_j > 0$ for $j=1,2,3$, and we conclude like in Lemma \[lemma3-c\].
\[lemma1-b\] Let $g : \R^3 \to \R$ be a nonnegative function satisfying $\int_{\R^3} g(v) \, \d v =1$. Then, for all $r>0$, there exists $C_r\in(0,\frac 12)$ such that for all $\theta \in\R$, $$\int_{\R^3} g(v) \sin^2\left(\frac {v\cdot \xi + \theta}2\right ) \, \d v
\geq C_r,\quad |\xi| >r .$$
The proof follows the same lines as that of Lemma \[lemma1-c\]. Let $\xi \in \R^3$ be fixed. We start by choosing in $\R^3$ an orthogonal system in which the unitary vector along the z-axis is $\frac \xi {|\xi|}$. As in the one-dimensional case, there exist $R>0$ and $\delta>0$ such that for all measurable set $A \subset \R^3$ such that $|A| < \delta$, we have $$\int_{A^c \cap Q(0,R)} g(v)\, \d v \geq
\frac 12,$$ where $Q(0,R)$ is the cube centered at the origin: $$Q(0,R)= \{v\in \R^3:\, |v_i| \leq R,\, i=1,2,3\}.$$ Now, for $\mu \in\ ]0, \frac \pi 2[$, we define $$K_{\mu, R, \theta} := \left \{v\in \R^3, v\in Q(0,R)\ {\rm and\ }\exists\
k\in\Z, \left| \frac {v\cdot \xi + \theta}2 -k\pi\right | \leq \mu \right
\}.$$ Thanks to the choice of the coordinate system, we have $$K_{\mu, R, \theta} = \left\{v\in \R^3, |v_i| \leq R, i=1,2,3\ {\rm and\ }
\exists\ k\in\Z, \left| \frac {v_3|\xi| + \theta}2 -k\pi\right | \leq \mu
\right \}.$$ So, it is easy to see that $$\left| K_{\mu, R, \theta}\right | \leq \left( \frac {|\xi|R}\pi +1\right )
\frac {4\mu} {|\xi|} R^{2} = 4\mu R^{2}\left ( \frac R \pi +\frac
1{|\xi|}\right ),$$ and we can conclude as in the one-dimensional case.
We are now in position to prove the main theorem of our paper, namely Theorem \[7\]. The proof of this theorem relies on the following proposition:
\[espo-b\] Let $g$ be a nonnegative function satisfying $$\int_{\R^3} g(v) \, \d v =1,\quad \int_{\R^3} g(v) v_i \,\d
v=0,\ i=1,2,3, \quad \int_{\R^3} g(v)\, |v|^2 \, \d v =3.$$ Let us suppose moreover that $$|\hat g(\xi)| \leq K_1\e^{- K_2 \phi( |\xi|)}, \quad \xi\in \R^3,$$ where $K_1\geq 1$, $K_2>0$ and $\phi:[0,+\infty)\to [0,+\infty)$ satisfies $\lim_{t\to +\infty}
\phi(t) = +\infty$. Then, there exists $\eta >0$ such that for all $R > \eta $, there exists $ K>0$ (depending on $R$) such that $$|\hat g(\xi)| \leq \begin{cases}
\e^{- K |\xi|^2}, & |\xi| < R,\\
\e^{- K \phi(|\xi|)}, & |\xi| \geq R.
\end{cases}$$
The proof of this proposition is only a slight modification of that of Proposition \[espo\]. We only point out the few differences.
First, we explain how to find $\eta >0$: we can fix $K_3 \in (0,
K_2)$ and let $\eta = \eta (K_3)$ be a positive constant such that $\varphi(|\xi|)\geq \frac{1}{K_2-K_3}\log K_1$ for every $|\xi|\geq \eta$; then of course for every $R\geq\eta$, $$K_1\e^{-K_2 \varphi(|\xi|)} \leq \e^{-K_3 \varphi(|\xi|)}, \quad
|\xi| \geq R.$$
Second, we observe that for all $\xi \in \R^3$ it is possible to find $\theta\in\R$ (depending on $\xi$) such that $$|\hat{g}(\xi)| = \hat{g}(\xi)\,e^{i\theta} = \int_{\R^3} g(v)\, \cos \left(
\xi\cdot v + \theta\right ) \, \d v.$$ So, we have $$1-|\hat{g}(\xi)| = 2\int_{\R^3} g(v) \sin^2\left(\frac{\xi\cdot v + \theta}2\right)
\, \d v .$$ Thanks to Lemma \[lemma1-b\], there exists $C_\rho$ such that $$2\int_{\R^3} g(v)\,
\sin^2\left(\frac{\xi\cdot v + \theta}2\right) \, \d v \ge 2\,C_\rho,\quad |\xi| >
\rho.$$ Then, we can conclude as in the proof of Proposition \[espo\].
[**Proof of Theorem \[7\]**]{}. As we did for Kac equation, for each of the Cauchy problems , we write the solution $\phi_l$ under the form of a Wild’s expansion.
In order to prove the bound for the solution, it is enough to prove it for every term $\varphi_l^{(n)}$ in the sum. We define $$H(|\xi|)=\begin{cases}
{K |\xi|^2}, &|\xi|< R_0,\\
{K \psi(|\xi|^ 2)}, & |\xi|\geq R_0,
\end{cases}$$ where $R_0$ will be chosen (large enough) later, and $K$ is given by Proposition \[espo-b\]. Thanks to this proposition, the initial datum satisfies $$\sup_{\xi \in \R^3}|\hat f_0(\xi)|{\rm e}^{H(|\xi|)}\leq 1.$$ We recall the identities ([@des2], page 56) $$\begin{aligned}
&\left|\xi^+\right |= |\xi| \cos \frac \theta 2,\\
&\left|\xi^-\right | = |\xi| \sin \frac \theta 2.
\end{aligned}$$ For the first term $\varphi_l^{(1)}$ we have: $$\begin{aligned}
\left|\varphi_l^{(1)}({\xi}){\rm e}^{\;H(|{\xi}|)}\right| &=
\left|\int_{S^2} {\rm e}^{\;H(|{\xi}|)
-\;H ( |\xi^+|)
-\; H(|\xi^-|) }
\hat f_0(\xi^+)
{\rm e}^{\;H(|\xi^+ | )}
\hat f_0(\xi^-)
{\rm e}^{\;H(|\xi^-| )}
\beta_l \left( \frac \xi {|\xi|} \cdot n\right )
\, \d n\right |\\
&\leq \int_{\theta\in (0, \pi )}\int_{\varphi \in (0,2\pi)}{\rm
e}^{\;H(|{ \xi}|)- \; H(|{\xi}| \cos\frac{\theta}{2})- \;
H(|{\xi}| \sin\frac{\theta}{2})} \beta_l(\cos \theta) \sin
\theta\, \d\theta\, \d\varphi,
\end{aligned}$$ where in the last integral, we have used the spherical coordinates with $\frac{\xi}{|\xi|}$ as $z$-axis. Then, in order to establish $\left|\varphi^{(1)}_l({\xi}){\rm e}^{\;H(|{ \xi}|)}\right|\leq 1$, we show that for $R_0$ large enough, $$\;H(|{\xi}|)- \; H\left(\left|{\xi}
\cos\frac{\theta}{2}\right|\right)- \;
H\left(\left|{\xi}\sin\frac{\theta}{2}\right|\right) \leq 0,\quad \theta \in (0,
{\pi}),\ \xi \in \R^3.$$ We denote $\tilde \theta=\frac{\theta}{2}$. Thanks to the symmetries of the functions $H(|\xi|)-H(|\xi\sin \tilde\theta|)-
H(|\xi\cos \tilde\theta|)$ with respect to $\tilde\theta$, we can restrict ourselves to the interval $(0,\frac \pi 4)$. Now, the case $|\xi|< R_0$ is the same as in Theorem \[5\]. If $|\xi|\geq R_0$ and $|\xi\sin
\tilde\theta|\geq R_0$, $|\xi\cos \tilde\theta|\geq R_0$, then $$H(|\xi|)-H(|\xi\sin \tilde\theta|)-H(|\xi\cos \tilde\theta|)=K
(\psi(|\xi|^2)-\psi(|\xi\sin \tilde\theta|^2)-\psi(|\xi\cos
\tilde\theta|^2)).$$ Thanks to the concavity property of $\psi$ and the fact that $\psi(0)=0$, we have $\psi(|\xi|^2(\sin\tilde\theta)^2)\geq
(\sin\tilde\theta)^2\psi(|\xi|^2)$ and $\psi(|\xi|^2(\cos\tilde\theta)^2)\geq
(\cos\tilde\theta)^2\psi(|\xi|^2)$. Hence $$\psi(|\xi|^2)-\psi(|\xi\sin\tilde {\theta}|^2)-\psi(|\xi\cos
\tilde\theta|^2)\leq \psi(|\xi|^2)-(\sin
\tilde\theta)^2\psi(|\xi|^2)-(\cos\tilde{ \theta})^2\psi(|\xi|^2)=
0.$$ Whenever $|\xi|\geq R_0$ and $|\xi\sin\tilde\theta|< R_0$, $|\xi\cos\tilde\theta|< R_0$ we have $$H(|\xi|)-H(|\xi\sin\tilde \theta|)-H(|\xi\cos\tilde\theta |)=K
(\psi(|\xi|^2)-|\xi|^2((\sin \tilde\theta)^2+(\cos
\tilde\theta)^2))=K (\psi(|\xi|^2)-|\xi|^2).$$ If we choose $R_0$ large enough, thanks to the assumption that $\psi(r) \le r$ for $r$ large enough, and since $|\xi|\geq R_0$, we can conclude $$\psi(|\xi|^2)-|\xi|^2\leq 0.$$ By using now the concavity property of $\psi$ and the fact that $\psi(r)\leq r$, if now $|\xi|\geq R_0$ and $|\xi\sin\tilde\theta |< R_0$, $|\xi\cos \tilde\theta|\geq R_0$ we have $$\begin{aligned}
H(|\xi|)-H(|\xi\sin \tilde\theta|)-H(|\xi\cos \tilde\theta|)&=K
(\psi(|\xi|^2)-|\xi|^2(\sin\tilde\theta
)^2-\psi(|\xi\cos\tilde\theta
|^2))\\
&\leq K (\psi(|\xi|^2) - \psi(|\xi|^2)\,(\sin \tilde\theta)^2 -
\psi(|\xi\cos\tilde\theta
|^2))\\
&\leq K\psi(|\xi|^2)(1-(\sin
\tilde\theta)^2-(\cos\tilde\theta)^2)=0.
\end{aligned}$$ We end the proof by first noticing that a simple induction shows the estimate $\left|\varphi^{(n)}_l({\xi}){\rm e}^{\;H(|{ \xi}|)}\right|\leq 1$ when $n\ge 1$, and then we may pass to the limit when $l\to +\infty$ if necessary (that is, in the non cut-off case). $\scriptstyle\square$
We would like to point out that the assumption of concavity for the function $\psi(|\xi|^2)$ is not mandatory and that the argument exploited in the proof of Theorem \[7\] could work also in a more general framework.
By analysing the proof, one can see that, instead of assuming that $\psi$ is concave, it is in fact enough to assume that for some $R_0$ large enough, $$\psi(\lambda^2\, |\xi|^2) \ge \lambda^2\,\psi(|\xi|^2)$$ when $0 \le \lambda \le 1$, $\lambda\,|\xi| \ge R_0$. This is true for example when $\psi(t) = \frac12\, \sqrt{t}\, |\log t|$.
Gevrey spaces
=============
In this section, we translate the propagation result obtained in the previous sections in terms of Gevrey regularity for the solutions of Boltzmann equation. Let us begin by recalling the classical definitions of Gevrey functions and a useful characterisation of these functions through their Fourier transform. For more information, the interested reader can consult for instance the book by Rodino [@rod], from where we have taken the following recalls.
Let $\Omega \subseteq \R^n$ be an open set and let $\nu\geq 1$ be a fixed real number.
\[defgevrey-b\] The class $G^\nu(\Omega)$ of Gevrey functions of order $\nu$ in $\Omega$ is the set of functions $f \in C^\infty (\Omega)$ satisfying the following property: for every compact subset $K$ of $\Omega$, there exists a positive constant $C=C(K)$ such that for all $l \in \N^n$ and all $x\in K$, \[defgevrey\] |\^l f(x)| C\^[|l|+1]{} (l !)\^.
Assumption can be replaced by other equivalent assumptions, for example $$|\partial^l f(x)| \leq R C^{|l|} (l !)^\nu,$$ where $R$ and $C$ are two positive constants independent of $l$ and $x\in K$. It is easy to recognize that $G^1(\Omega)= A(\Omega)$, the space of all analytic functions in $\Omega$, and that for $\nu\leq \tau$, one has $G^\nu(\Omega) \subseteq G^\tau(\Omega)$. Moreover, it is interesting to underline the following inclusions: $$A(\Omega) \subset \bigcap_{\nu>1} G^\nu(\Omega),\quad \bigcup_{\nu\geq 1}
G^\nu(\Omega) \subset C^\infty(\Omega),$$ which are strict in both cases. We also recall that the Gevrey class $G^\nu(\Omega)$ is closed under differentiation.
In what follows, we shall also need the following
\[gev-comp\] Assume $\nu>1$. We shall denote by $G^\nu_0 (\Omega)$ the vector space of all $f\in G^\nu(\Omega)$ with compact support in $\Omega$.
The exclusion of $\nu=1$ in the previous definition is mandatory, because there are no analytic test functions other than the zero function. As for the other values $\nu>1$, one could wonder whether such compact supported functions do exist. An example in $\R$ is the following: let $r>0$, $\nu >1$, $d= \frac 1 {1-\nu}$ and $$\varphi(t)=
\begin{cases}
\e^{-t^d} & t>0,\\
0 & t\leq 0.
\end{cases}$$ The function $$f(x)= \varphi (x+r) \varphi (x-r)$$ is then in $G^\nu_0(\R)$ ([@rod]).
The result that we are going to use in order to relate our propagation result to Gevrey regularity is the following.
\[caratt\]
i) Let $\nu>1$. If $\phi \in G^\nu_0(\R^n)$, then there exist positive constants $C$ and $\eps$ such that \[diret\] |()| C \^ [-||\^[1]{}]{}, \^n.
ii) Let $\nu\geq 1$. If the Fourier transform of $\phi\in\S'(\R^n)$ satisfies , then $\phi\in
G^\nu(\R^n)$.
We can therefore deduce Corollary \[condiz\] concerning the regularity of the solutions of Boltzmann equation.
[**Proof of Corollary \[condiz\]**]{}. The result is straightforward from Theorem \[caratt\] and Theorem \[7\]. It is enough to notice that one can replace the uniform estimate $$\begin{aligned}
&\sup_{|\xi|< R_0}|\hat f(\xi,t)|{\rm e}^{K
|\xi|^2}\leq 1,\quad t\geq 0,\\
&\sup_{|\xi|\geq R_0}|\hat f(\xi,t)|{\rm e}^{K
|\xi|^s}\leq 1, \quad t\geq 0,
\end{aligned}$$ obtained in Theorem \[7\], by the following uniform estimate: $$\sup_{\xi\in \R^3}|\hat f(\xi,t)|{\rm e}^{\tilde K_2 |\xi|^s}\leq
\tilde K_1,\quad t\geq 0,$$ for $\tilde K_1\geq 1$ and $\tilde K_2 >0$ properly chosen. Letting now $\nu=\frac 1 s$, one immediately gets the result. This ends the proof.$\scriptstyle\square$
We end up this section by comparing the regularity result we have just obtained with the one obtained by Ukai in [@U84]. In his work, he considered among others a Cauchy problem for the homogeneous Boltzmann equation for Maxwellian molecules both in the cut-off and non cut-off settings. He considered only initial data $f_0$ satisfying a strong regularity assumption: for $\alpha \geq 0$, $\rho\geq 0$ and $\nu \geq 1$ he supposed $f_0\in \gamma_{\alpha, \rho}^\nu$, where $$\gamma_{\alpha, \rho}^\nu = \{ g: \|g\|_{\alpha, \rho, \nu}=
\sum_{l\in \N^3} \frac{\rho^{|l|}}{(l!)^\nu} \sup_{v\in\R^3} \e
^{\alpha(1+|v|^2)^{\frac 12}} |\partial^l g(v)| <\infty\}.$$ Comparing this space with the spaces in Definition \[defgevrey-b\], one can deduce that initial data in Ukai setting are indeed in the Gevrey space $G^\nu (\R^3)$, but also decay very strongly at infinity together with all their derivatives. Ukai was able to prove by a fixed point argument that there exists a unique, local in time solution $f$ belonging at every time to a functional space of the same kind as the initial datum but in which the indices change with $t$. More precisely, he proved that there exist $T>0$, $\beta >0$, $\sigma>0$ such that \[ukai-2\] f(, t)\_[-t , -t , ]{} 2 f\_0\_[, , ]{}, t . Since initial data which belong to $G^\nu_0(\R^3)$ also belong to Ukai space, the question of comparing the two results is meaningful. Let us consider a nonnegative function $f_0 \in G^\nu_0(\R^3)$. If we suppose moreover that $f_0$ satisfies assumptions , then by Theorem \[esist\] we know that there is a solution $f\in
C^1 \left( [0, +\infty), L^1(\R^3)\right )$ which is unique by Toscani–Villani’s result. Since $G^\nu_0(\R^3) \subset
\gamma_{\alpha, \rho}^\nu$ for all $\alpha\geq 0$, we can deduce from Ukai result that for all $\alpha \geq 0$ there is a time $T=T(\alpha)>0$ (possibily finite) such that this solution stays in the class for $t\in [0,T(\alpha)]$ but this space is not uniform in time (in addition to the fact that for all $\alpha$ it is difficult to compare the life times $T(\alpha)$). Our result says instead that the solution stays for $t\in
[0,\infty)$ in the same Gevrey class as its initial datum, without any information about the decay at infinity and moreover that all the estimates on the Gevrey seminorms are uniform in time.
[CGT99]{}
L. Arkeryd. Intermolecular forces of infinite range and the [B]{}oltzmann equation. , 77(1):11–21, 1981.
A. V. Bobyl[ë]{}v. The theory of the nonlinear spatially uniform [B]{}oltzmann equation for [M]{}axwell molecules. In [*Mathematical physics reviews, Vol. 7*]{}, volume 7 of [ *Soviet Sci. Rev. Sect. C Math. Phys. Rev.*]{}, pages 111–233. Harwood Academic Publ., Chur, 1988.
E. A. Carlen, E. Gabetta, and G. Toscani. Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous [M]{}axwellian gas. , 199(3):521–546, 1999.
L. Desvillettes. About the regularizing properties of the non-cut-off [K]{}ac equation. , 168(2):417–440, 1995.
L. Desvillettes. About the use of the [F]{}ourier transform for the [B]{}oltzmann equation. , 2\*:1–99, 2003. Summer School on “Methods and Models of Kinetic Theory” (M&MKT 2002).
L. Desvillettes and C. Mouhot. About [$L\sp p$]{} estimates for the spatially homogeneous [B]{}oltzmann equation. , 22(2):127–142, 2005.
T. Gustafsson. -estimates for the nonlinear spatially homogeneous [B]{}oltzmann equation. , 92(1):23–57, 1986.
T. Gustafsson. Global [$L\sp p$]{}-properties for the spatially homogeneous [B]{}oltzmann equation. , 103(1):1–38, 1988.
C. Mouhot and C. Villani. Regularity theory for the spatially homogeneous boltzmann equation with cut-off. , 173(2):169–212, 2004.
A. Pulvirenti and G. Toscani. The theory of the nonlinear [B]{}oltzmann equation for [M]{}axwell molecules in [F]{}ourier representation. , 171:181–204, 1996.
L. Rodino. . World Scientific Publishing Co. Inc., River Edge, NJ, 1993.
G. Toscani and C. Villani. Probability metrics and uniqueness of the solution to the [B]{}oltzmann equation for a [M]{}axwell gas. , 94(3-4):619–637, 1999.
S. Ukai. Local solutions in [G]{}evrey classes to the nonlinear [B]{}oltzmann equation without cutoff. , 1(1):141–156, 1984.
C. Villani. A review of mathematical topics in collisional kinetic theory. In [*Handbook of mathematical fluid dynamics, Vol. I*]{}, pages 71–305. North-Holland, Amsterdam, 2002.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite dimensional Hopf algebras. In this paper, we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg’s 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev’s invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. We speculate that relaxing semisimplicity will lead to even richer invariants.'
author:
- 'Julian Chaidez$^{\,1}$, Jordan Cotler$^{\,2}$, Shawn X. Cui$^{\,3,4}$'
bibliography:
- 'trisection\_inv\_bib.bib'
date: |
\
.2cm\
.2cm\
.2cm\
title: '4-Manifold Invariants From Hopf Algebras'
---
Introduction
============
Since the discovery of the Jones polynomial of knots [@jones1997polynomial] and its interpretation in terms of a topological quantum field theory (TQFT) [@witten1989quantum; @witten1988topological] in the 1980s, the field of quantum topology and TQFTs has seen substantial progress. Deep connections have been discovered between quantum topology and such disparate areas as knot theory, low dimensional topology, quantum groups, tensor categories, conformal field theories, and topological quantum computing.
Roughly, for a non-negative integer $d$, a TQFT in dimension $d+1$, or a $(d+1)$-TQFT, is an assignment of vector spaces to $d$-manifolds and vectors/scalars to $(d+1)$-manifolds subject to certain compatibility conditions. All manifolds involved are assumed to be smooth. In particular, each $(d+1)$-TQFT provides an invariant, called a quantum invariant, for smooth $(d+1)$-manifolds. Quantum invariants have important applications in smooth topology as they can be used to distinguish different manifolds. A fundamental and well-established family of TQFTs in dimension $(2+1)$ is the Reshetikhin-Turaev/Witten-Chern-Simons theory [@turaev1994quantum; @turaev1994quantum; @witten1989quantum]. $(3+1)$-TQFTs are especially interesting from the perspective of 4-manifolds. Below the notions of TQFTs and quantum invariants are referred to interchangeably.
The construction of TQFTs is closely related to algebraic structures such as quantum groups, tensor categories, and more generally higher categories. In dimension $(3+1)$, perhaps the simplest example of a TQFT is the Dijkgraaf-Witten theory based on finite groups [@dijkgraaf1990topological]. This was generalized in one direction to the Yetter TQFT based on finite categorical groups or 2-groups [@yetter1993tqft], and was generalized in another direction to the Crane-Yetter/Walker-Wang TQFT based on ribbon fusion categories [@crane1993categorical; @crane1997state; @walker2012top]. More recently, the third author of the current paper proposed a construction [@cui2016higher] based on crossed braided fusion categories which simultaneously generalized the Yetter and Crane-Yetter/Walker-Wang TQFTs. Finally, Douglas and Reutter [@douglas2018fusion] pinned down the notion of spherical fusion 2-categories and used it to define invariants of 4-manifolds, further generalizing the invariants from crossed braided fusion categories. There are also a few other invariants of 4-manifolds such as the dichromatic invariant [@petit2008dichromatic; @barenz2016dichromatic] based on pivotal functors and the Kashaev invariant indexed by a finite cyclic group [@kashaev2014asimple]. These are speculated to be special cases of the invariants mentioned above, but a proof of this is not known.
The Douglas-Reutter invariant from spherical fusion 2-categories is believed to be the most general state-sum type invariant. However, it has several known limitations. To the authors’ best knowledge, there are not many examples of spherical fusion 2-categories apart from the ones constructed from crossed braided fusion categories (plus some cohomology twistings) and the ones arising as the module categories of braided fusion categories which should correspond to the Crane-Yetter theory. Moreover, from a practical point of view, both the data encoding a spherical fusion 2-category and the state-sum formulation of the invariant have very large complexity, which makes calculations intractable beyond a few simple examples. An alternative formulation of the invariant in terms of handlebody decomposition may help with the calculations. But most importantly, it is speculated (at a non-rigorous level) that all invariants of 4-manifolds of state-sum type or those from fully extended TQFTs that are based on semisimple algebraic data are not sensitive to smooth structures. None of the invariants mentioned above are known to distinguish smooth structures. Thus, it may be necessary to construct invariants from non-semisimple data.
In one dimension lower, the Kuperberg invariant [@kuperberg1996noninvolutory], which is constructed from a finite dimensional Hopf algebra, is a fundamental invariant of 3-manifolds. When the Hopf algebra is semisimple, the invariant recovers the Turaev-Viro-Barrett-Westbury theory [@turaev1992state; @barrett1996invariants]. However, the invariant is more powerful when the Hopf algebra is non-semisimple. In this case, the invariant contains information about some additional structures of the 3-manifold, such as combings and framings. A generalization of the invariant from Hopf algebras in the category of vector spaces to those in a symmetric fusion category is also possible [@kashaev2018generalized]. L[ó]{}pez-Neumann [@lopez2019kuperberg] studied the invariant associated with involutory (possibly non-semisimple) Hopf algebras in the category of super vector spaces and showed that the invariant specializes to the Reidemeister torsion invariant (cf. [@turaev2001introduction]) which is closely related to the Seiberg-Witten theory.
Main Results
------------
In this paper, we initiate a program of constructing 4-manifold invariants in the spirit of Kuperberg’s 3-manifold invariant.
The algebraic data used in our construction is a structure called a [*Hopf triplet*]{}, which consists of three finite dimensional Hopf algebras and a bilinear form on each pair of them satisfying certain compatibility conditions (see Definition \[def:hopf\_triplet\]). As a starting point in this paper, we assume that all the Hopf algebras involved are semisimple. There are several ways of producing examples of Hopf triplets. For instance, any quasi-triangular Hopf algebra gives rise to a Hopf triplet (see Example \[ex:basic\_examples\_of\_triplets\]).
The topological data used in the construction is a presentation of a 4-manifold, in the form of a trisection diagram [@gk2016]. A trisection diagram is a 4-dimensional analog of a Heegaard diagram, consisting of three families of circles on a closed surface.
The first main result of this paper addresses the construction and well-definedness of our invariant. Informally, it can be stated as follows.
Given a Hopf triplet $\mathcal{H}$ over a field $k$, we may associate a scalar $\tau_{\mathcal{H}}(X) \in k$ to any $4$-manifold $X$, and this scalar is a diffeomorphism invariant.
The invariant is constructed by associating, to each circle in a trisection diagram, a generalized comultiplication tensor chosen from an appropriate Hopf algebra and contracting the tensors via the bilinear forms in a similar manner as the Kuperberg invariant.
The second main result of this paper addresses the relationship between our invariant and existing invariants. In particular, we prove that (again, informally stated here)
\[thm:trisection\_vs\_crane\_yetter\_informal\] Let $\mathcal{H}$ be the Hopf triplet associated to a quasi-triangular Hopf algebra $H$, and let $\mathcal{C}$ be the ribbon fusion category of representations of the $H$
Then the trisection invariant $\tau_{\mathcal{H}}(X)$ equals the Crane-Yetter invariant $CY_{\mathcal{C}}(X)$ up to a factor depending on ${\operatorname}{dim}(H)$ and the Euler characteristics of $X$.
We show more generally that for certain triplets, our invariants recover some cases of dichromatic invariants.
For each positive integer $N$, the group algebra ${{\mathbb C}}[{{\mathbb Z}}/N]$ of the cyclic group ${{\mathbb Z}}/N$ has a quasi-triangular structure. The corresponding trisection invariant agrees with the Kashaev invariant associated with ${{\mathbb Z}}/N$ for some examples of 4-manifolds (up to Euler characteristics).
Combined with Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\], this provides supporting evidence for the conjecture that the Kashaev invariant is a special case of the Crane-Yetter invariant, which was first proposed in [@williamson2016hamiltonian] when studying the Hamiltonian models of the two theories.
In sequel projects, we plan to generalize the trisection invariant in several directions, e.g.: 1) to 4-manifolds endowed with a Spin$^c$ structure, 2) to 4-manifolds with an embedded closed surface (a 2-knot for instance), and 3) to Hopf triplets in a general symmetric fusion category. We also aim to produce invariants from non-semisimple Hopf triplets.
**Organization.** The rest of the paper is organized as follows.
- In **§\[sec:preliminaries\]**, we review tensor algebras and Hopf algebras, emphasizing a diagrammatic point of view. We define Hopf triplets, and derive their essential structural properties. Then we review trisections of 4-manifolds, and their corresponding diagrammatics.
- In **§\[sec:trisection\_kuperberg\_invt\]**, we define the input data to our invariant, culminating in the definition of the trisection invariant itself. We prove that it is a diffeomorphism invariant of smooth closed 4-manifolds, along with other structural properties. The section concludes with a generalization of the involutory Kuperberg invariant for 3-manifolds using an insight from the 4-manifold setting.
- In **§\[sec:examples\_and\_calculations\]**, we give examples of trisection diagrams and their corresponding tensor diagrams which evaluate to the trisection invariant. We detail computational methods we devised to evaluate trisection invariants, and provide examples. Two examples of particular interest are cyclic triplets which we conjecture give rise to Kashaev’s invariants, and triplets of 8-dimensional Hopf algebras. Some of the latter appear inequivalent to any known 4-manifold invariant.
- In **§\[sec:CY\_dichro\]**, we prove a relationship between special cases of the trisection invariant, and both the Crane-Yetter and generalized dichromatic invariants.
**Acknowledgements.** JCh was supported by the NSF Graduate Research Fellowship under Grant No. 1752814. JCo is supported by the Fannie and John Hertz Foundation and the Stanford Graduate Fellowship program. SCui is partially supported by the startup fund from Purdue University. SCui also acknowledges the support from Simons Foundation and Virginia Tech.
Preliminaries {#sec:preliminaries}
=============
In this section, we review the background required to construct and compute our invariant. In §\[subsec:tensor\_algebra\], we discuss the algebraic objects involved in the construction, namely Hopf algebras and certain assemblages of Hopf algebras called Hopf triplets. In §\[subsec:trisections\], we discuss trisections of 4-manifolds and trisection diagrams.
Tensor Algebra {#subsec:tensor_algebra}
--------------
Here we review the algebraic preliminaries for our invariant. We begin by discussing tensor diagrams, which provide a diagrammatic notation for tensor calculations (§\[subsubsec:tensor\_diagrams\]). We then review the basic theory of Hopf algebras (§\[subsubsec:hopf\_algebra\]) using the diagrammatic notation. Finally, we introduce the fundamental notion of a Hopf triplet (§\[subsubsec:doublets\_and\_triplets\]), which will serve as the algebraic input for our invariant.
### Tensor Diagrams {#subsubsec:tensor_diagrams}
Let us begin by reviewing tensor diagram notation. This will be our main tool for defining all tensorial quantities in this section, §\[subsec:tensor\_algebra\], for defining our invariant and proving its properties in §\[sec:trisection\_kuperberg\_invt\], and for performing calculations in §\[sec:examples\_and\_calculations\].
\[not:general\_tensor\_diagrams\] (Tensor Diagrams) Tensor diagram notation is a basic pictoral tool for performing formal manipulations with tensors. We now explain a very general version of this notation.
Fix a collection ${{\mathcal V}}$ of vector spaces over a field $k$ and let ${{\mathcal F}}$ be a collection of tensors $f \in \bigotimes_{i=1}^a U^*_i \otimes \bigotimes_{j=1}^b V_j$. Here the vector spaces $U_i,V_j \in {{\mathcal V}}$ and the indices $a,b \ge 0$ can vary with $f \in {{\mathcal F}}$. Any tensor $g \in \bigotimes_{i=1}^c U^*_i \otimes \bigotimes_{j=1}^d V_j$ that can be obtained from the maps in ${{\mathcal F}}$ by a sequence of tensor products and dual pairings can be represented as a *tensor diagram*, which is a directed graph equipped with decorations and immersed in the plane, in the following manner.
- (Base Maps) Any tensor $f \in \bigotimes_{i=1}^a U^*_i \otimes \bigotimes_{j=1}^b V_j$ in the collection ${{\mathcal F}}$ may be represented by a node $f$ with $a$ incoming edges and $b$ outgoing edges. The in and out edges are ordered, with the $i$th incoming edge denoting the $U^*_i$ tensor index and the $j$th outgoing edges denoting the $V_j$ tensor index. $$\begin{tikzpicture}
\node at (-1.8,0) (lbrack) {$a$ inputs $\Big\{$};
\node at (3.9,0) (rbrack) {$\Big\}$ $b$ outputs};
\node at (1,0) [draw,rectangle,minimum size=30pt] (f) {$f$};
\draw[->] (-.7,.5)--(.4,.5);
\draw[->] (-.7,.3)--(.4,.3);
\draw (-.4,.1) node (vdts1) {\vdots};
\draw (-.2,.1) node (vdts2) {\vdots};
\draw (0,.1) node (vdts3) {\vdots};
\draw[->] (-.7,-.3)--(.4,-.3);
\draw[->] (-.7,-.5)--(.4,-.5);
\draw[->] (1.6,.5)--(2.7,.5);
\draw[->] (1.6,.3)--(2.7,.3);
\draw (1.9,.1) node (vdts1) {\vdots};
\draw (2.1,.1) node (vdts2) {\vdots};
\draw (2.3,.1) node (vdts3) {\vdots};
\draw[->] (1.6,-.3)--(2.7,-.3);
\draw[->] (1.6,-.5)--(2.7,-.5);
\end{tikzpicture}$$
- (Tensor Products) Given two tensors $g$ and $g'$, the tensor product $g \otimes g'$ is denoted by taking the disjoint union of the corresponding graphs for $g$ and $g'$. $$\begin{tikzpicture}
\node at (1,0) [draw,rectangle,minimum size=30pt] (gg') {$g \otimes g'$};
\node at (3.8,0) (=) {$=$};
\node at (6,.4) [draw,rectangle,minimum size=20pt] (g) {$g$};
\node at (6,-.4) [draw,rectangle,minimum size=20pt] (g') {$g'$};
\draw[->] (-.9,.5)--(.2,.5);
\draw[->] (-.9,.3)--(.2,.3);
\draw (-.6,.1) node (vdts1) {\vdots};
\draw (-.4,.1) node (vdts2) {\vdots};
\draw (-.2,.1) node (vdts3) {\vdots};
\draw[->] (-.9,-.3)--(.2,-.3);
\draw[->] (-.9,-.5)--(.2,-.5);
\draw[->] (1.8,.5)--(2.9,.5);
\draw[->] (1.8,.3)--(2.9,.3);
\draw (2.1,.1) node (vdts1) {\vdots};
\draw (2.3,.1) node (vdts2) {\vdots};
\draw (2.5,.1) node (vdts3) {\vdots};
\draw[->] (1.8,-.3)--(2.9,-.3);
\draw[->] (1.8,-.5)--(2.9,-.5);
\draw[->] (4.8,.5)--(5.6,.5);
\draw[->] (4.8,.3)--(5.6,.3);
\draw[->] (4.8,-.3)--(5.6,-.3);
\draw[->] (4.8,-.5)--(5.6,-.5);
\draw[->] (6.4,.5)--(7.2,.5);
\draw[->] (6.4,.3)--(7.2,.3);
\draw[->] (6.4,-.3)--(7.2,-.3);
\draw[->] (6.4,-.5)--(7.2,-.5);
\end{tikzpicture}$$
- (Dual Pairing) Given a tensor $g \in \bigotimes_{i=1}^a U^*_i \otimes \bigotimes_{j=1}^b V_j$ and a choice of $c$ and $d$ so that $U_c = V_d \in {{\mathcal V}}$, the dual pairing ${\operatorname}{Tr}_c^d(g)$ along $c$ and $d$ is denoted by taking the diagram for $g$ and connecting the $d$-th output to the $c$-th input. $$\hspace{30pt}\begin{tikzpicture}
\node at (.8,-.2) [draw,rectangle,minimum size=20pt] (Trg) {${\operatorname}{Tr}_c^d(g)$};
\draw[->] (-1.1,.1)--(0,.1);
\draw[->] (-1.1,-.1)--(0,-.1);
\draw[->] (-1.1,-.3)--(0,-.3);
\draw[->] (-1.1,-.5)--(0,-.5);
\draw[->] (1.6,.1)--(2.7,.1);
\draw[->] (1.6,-.1)--(2.7,-.1);
\draw[->] (1.6,-.3)--(2.7,-.3);
\draw[->] (1.6,-.5)--(2.7,-.5);
\node at (3.4,-.2) (=) {$=$};
\node at (6,0) [draw,rectangle,minimum size=30pt] (g) {$g$};
\draw[->] (6.7,.3) to [out=30,in=150,looseness=2] (5.3,.3);
\draw[->] (4.2,.1)--(5.3,.1);
\draw[->] (4.2,-.1)--(5.3,-.1);
\draw[->] (4.2,-.3)--(5.3,-.3);
\draw[->] (4.2,-.5)--(5.3,-.5);
\draw[->] (6.7,.1)--(7.8,.1);
\draw[->] (6.7,-.1)--(7.8,-.1);
\draw[->] (6.7,-.3)--(7.8,-.3);
\draw[->] (6.7,-.5)--(7.8,-.5);
\end{tikzpicture}$$ Here the map ${\operatorname}{Tr}^d_c:{\operatorname}{Hom}(\bigotimes_i U_i,\bigotimes_j V_j) \to {\operatorname}{Hom}(\bigotimes_{i\neq c} U_i,\bigotimes_{j\neq d} V_j)$ is the linear map extending the standard dual pairing ${\operatorname}{Tr}:U_c^* \otimes V_d \to k$.
[**Important Convention.**]{} In order to avoid ambiguity as to which edges of a tensor $f \in \bigotimes_{i=1}^a U^*_i \otimes \bigotimes_{j=1}^b V_j$ (as in (a) above) correspond to which tensor product terms, we must adopt an order convention for the edges of $f$. More precisely, let $e^*_i$ and $e_j$ denote the edges representing $U_i^*$ and $V_j$ respectively, for all $1 \le i \le a$ and $1 \le j \le b$. Then we adopt the convention that the *counter-clockwise ordering* of these edges about the node $f$ beginning at $e_0$ must *always* be $e_0,e_1,\dots,e_a,e^*_b,e^*_{b-1},\dots,e_1^*$.
When a tensor $f$ has only input edges or only output edges (corresponding to a single vector space) and the tensor is *not* invariant under cyclic permutation of the input/output tensor factors, the ordering convention specificed here is still ambiguous. In this case, we will *always* draw a box about the node for $f$. If $f$ has only inputs, we draw the edges entering the left side of the box ordered top to bottom. Likewise, if $f$ has only output edges, we draw the edges exiting the right side of the box ordered top to bottom.
### Hopf Algebra {#subsubsec:hopf_algebra}
A Hopf algebra is a bialgebra (i.e., a simultaneous algebra and coalgebra where the two structures interact nicely) equipped with a canonical antipode. More precisely, we have the following.
\[def:hopf\_algebra\] An *Hopf algebra* $H = (H,M,\eta,\Delta,\epsilon,S)$ over a ring $k$ is a module $H$ over $k$ equipped with structure tensors of the form $$\begin{tikzpicture}
\draw (-1.5,0) node (label_product) {(Product)};
\draw (1,0) node (M) {$M$};
\draw[->] (0,-.25)--(M);
\draw[->] (0,.25)--(M);
\draw[->] (M)--(2,0);
\draw (5.5,0) node (label_unit) {(Unit)};
\draw (7.5,0) node (N) {$\eta$};
\draw[->] (N)--(6.5,0);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (-1.5,0) node (label_coproduct) {(Coproduct)};
\draw (1,0) node (D) {$\Delta$};
\draw[->] (D)--(2,-.25);
\draw[->] (D)--(2,.25);
\draw[->] (0,0)--(D);
\draw (5.5,0) node (label_counit) {(Counit)};
\draw (7.5,0) node (E) {$\epsilon$};
\draw[->] (6.5,0)--(E);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (-1.5,0) node (label_antipode) {(Antipode)};
\draw (1,0) node (S) {$S$};
\draw[->] (0,0)--(S);
\draw[->] (S)--(2,0);
\end{tikzpicture}$$ These structure tensors must satisfy a series of compatibility properties, which we now specify using tensor diagram notation.
- (Algebra) $(H,M,\eta)$ must define a unital $k$-algebra. In tensor diagrams, we have the following identities.
$$\hspace{-20pt}\begin{tikzpicture}
\draw (0,.25) node (M1) {$M$};
\draw (1,0) node (M2) {$M$};
\draw (2.4,-1) node (label_assoc) {(Associativity)};
\draw[->] (-1,.5)--(M1);
\draw[->] (-1,0)--(M1);
\draw[->] (M1)--(M2);
\draw[->] (0,-.25)--(M2);
\draw[->] (M2)--(1.9,0);
\draw (2.4,0) node (=) {$=$};
\draw (3.9,-.25) node (M1) {$M$};
\draw (4.9,0) node (M2) {$M$};
\draw[->] (3.1,-.5)--(M1);
\draw[->] (3.1,-0)--(M1);
\draw[->] (M1)--(M2);
\draw[->] (3.9,.25)--(M2);
\draw[->] (M2)--(5.8,0);
\draw (7.4,.25) node (n) {$\eta$};
\draw (8.4,0) node (M) {$M$};
\draw[->] (7.4,-.25)--(M);
\draw[->] (n)--(M);
\draw[->] (M)--(9.3,0);
\draw (9.7,0) node (=) {$=$};
\draw (10.1,-.25) node (n) {$\eta$};
\draw (11.1,0) node (M) {$M$};
\draw[->] (10.1,.25)--(M);
\draw[->] (n)--(M);
\draw[->] (M)--(12,0);
\draw (12.5,0) node (=) {$=$};
\draw[->] (12.9,0)--(13.5,0);
\draw (10.4,-1) node (label_unit) {(Unitality)};
\end{tikzpicture}$$
- (Coalgebra) $(H,\Delta,\epsilon)$ must define a counital $k$-coalgebra. In tensor diagrams, we have the following identities. $$\hspace{-20pt}\begin{tikzpicture}
\draw (0,.25) node (D1) {$\Delta$};
\draw (1,0) node (D2) {$\Delta$};
\draw (2.4,-1) node (label_assoc) {(Co-associativity)};
\draw[->] (D1)--(-1,.5);
\draw[->] (D1)--(-1,0);
\draw[->] (D2)--(D1);
\draw[->] (D2)--(0,-.25);
\draw[->] (1.9,0)--(D2);
\draw (2.4,0) node (=) {$=$};
\draw (3.9,-.25) node (D1) {$\Delta$};
\draw (4.9,0) node (D2) {$\Delta$};
\draw[->] (D1)--(3.1,-.5);
\draw[->] (D1)--(3.1,-0);
\draw[->] (D2)--(D1);
\draw[->] (D2)--(3.9,.25);
\draw[->] (5.8,0)--(D2);
\draw (7.4,.25) node (e) {$\epsilon$};
\draw (8.4,0) node (D) {$\Delta$};
\draw[->] (D)--(7.4,-.25);
\draw[->] (D)--(e);
\draw[->] (9.3,0)--(D);
\draw (9.7,0) node (=) {$=$};
\draw (10.1,-.25) node (e) {$\epsilon$};
\draw (11.1,0) node (D) {$\Delta$};
\draw[->] (D)--(10.1,.25);
\draw[->] (D)--(e);
\draw[->] (12,0)--(D);
\draw (12.5,0) node (=) {$=$};
\draw[->] (13.5,0)--(12.9,0);
\draw (10.4,-1) node (label_unit) {(Co-unitality)};
\end{tikzpicture}$$
- (Bialgebra/Antipode) The coalgebra and algebra structures must be compatible, in the sense that they define a bialgebra, and the antipode must satisfy a standard antipode identity involving the product and coproduct. $$\hspace{-20pt}\begin{tikzpicture}
\draw (0,0) node (D1) {$\Delta$};
\draw (0,1) node (D2) {$\Delta$};
\draw (1,0) node (M1) {$M$};
\draw (1,1) node (M2) {$M$};
\draw[->] (-.6,0)--(D1);
\draw[->] (-.6,1)--(D2);
\draw[->] (M1)--(1.6,0);
\draw[->] (M2)--(1.6,1);
\draw[->] (D1)--(M1);
\draw[->] (D1)--(M2);
\draw[->] (D2)--(M1);
\draw[->] (D2)--(M2);
\draw (2.3,.35) node (=) {$=$};
\draw (3.4,.5) node (M) {$M$};
\draw (4.4,.5) node (D) {$\Delta$};
\draw[->] (2.8,0)--(M);
\draw[->] (2.8,1)--(M);
\draw[->] (D)--(5,0);
\draw[->] (D)--(5,1);
\draw[->] (M)--(D);
\draw (2.8,-1) node (label_assoc) {(Bi-Algebra)};
\draw (7,0) node (D1) {$\Delta$};
\draw (7.5,.8) node (S1) {$S$};
\draw (8,0) node (M1) {$M$};
\draw[->] (6.4,0)--(D1);
\draw[->] (D1)--(M1);
\draw[->] (D1)--(S1);
\draw[->] (S1)--(M1);
\draw[->] (M1)--(8.7,0);
\draw (9,.35) node (=) {$=$};
\draw (10,.8) node (D2) {$\Delta$};
\draw (10.5,0) node (S2) {$S$};
\draw (11,.8) node (M2) {$M$};
\draw[->] (9.4,.8)--(D2);
\draw[->] (D2)--(M2);
\draw[->] (D2)--(S2);
\draw[->] (S2)--(M2);
\draw[->] (M2)--(11.7,.8);
\draw (12,.35) node (=) {$=$};
\draw (13,.4) node (e) {$\epsilon$};
\draw (13.3,.4) node (n) {$\eta$};
\draw[->] (12.4,.4)--(e);
\draw[->] (n)--(13.8,.4);
\draw (10.8,-1) node (label_unit) {(Antipode)};
\end{tikzpicture}$$
A map $f:H \to I$ of Hopf algebras is a linear map intertwining the product, unit, coproduct, counit and antipode. The tensor diagram identities for $f$ are clear.
In this paper, we will restrict to the following special class of Hopf algebras.
A Hopf algebra $H$ is *involutory* if the antipode squares to the identity. $$\label{eqn:involutory_condition}
\to S \to S \to \quad = \quad \to$$
In the case where $H$ is a Hopf algebra over a field of characteristic $0$, $H$ is involutory if and only if $H$ is semisimple by a theorem of Larson and Radford (see [@lr1987]).
In addition to the above structure maps, the following maps arise frequently in the study of Hopf algebras, and in this paper.
\[def:trace\_cotrace\] Let $(H,M,\eta,\Delta,\epsilon)$ be an involutory Hopf algebra.
- The *trace* $T:H \to k$ and *cotrace* $C:k \to H$ of $H$ are defined by $$\begin{tikzpicture}
\draw (9,0) node (C) {$C$};
\draw[->] (C)--(10,0);
\draw (10.5,0) node (=1) {$:=$};
\draw (12,0) node (D) {$\Delta$};
\draw (D) edge [loop left] node {} (D);
\draw[->] (D)--(13,0);
\draw (2,0) node (T) {$T$};
\draw[->] (1,0)--(T);
\draw (3,0) node (=1) {$:=$};
\draw (4.5,0) node (M) {$M$};
\draw (M) edge [loop right] node {} (M);
\draw[->] (3.5,0)--(M);
\end{tikzpicture}$$
- An *integral* $\mu:H \to k$ and a *cointegral* $e: k \to H$ of $H$ are maps such that $$\begin{tikzpicture}
\draw (1,0) node (D1) {$\Delta$};
\draw (2,-.25) node (i1) {$\mu$};
\draw[->] (D1)--(i1);
\draw[->] (D1)--(2,.25);
\draw[->] (0,0)--(D1);
\draw (2.5,0) node (=1) {$=$};
\draw (4,0) node (D2) {$\Delta$};
\draw (5,.25) node (i2) {$\mu$};
\draw[->] (D2)--(5,-.25);
\draw[->] (D2)--(i2);
\draw[->] (3,0)--(D2);
\draw (5.5,0) node (=2) {$=$};
\draw (6.5,0) node (i3) {$\mu$};
\draw (7,0) node (N) {$\eta$};
\draw[->] (6,0)--(i3);
\draw[->] (N)--(7.5,0);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (1,0) node (M1) {$M$};
\draw (2,-.25) node (m1) {$e$};
\draw[<-] (M1)--(m1);
\draw[<-] (M1)--(2,.25);
\draw[<-] (0,0)--(M1);
\draw (2.5,0) node (=1) {$=$};
\draw (4,0) node (M2) {$M$};
\draw (5,.25) node (m2) {$e$};
\draw[<-] (M2)--(5,-.25);
\draw[<-] (M2)--(m2);
\draw[<-] (3,0)--(M2);
\draw (5.5,0) node (=2) {$=$};
\draw (6.5,0) node (m3) {$e$};
\draw (7,0) node (E) {$\epsilon$};
\draw[<-] (6,0)--(m3);
\draw[<-] (E)--(7.5,0);
\end{tikzpicture}$$ We remark that there are notions of left and right (co)integrals for non-involutory Hopf algebras, which will not appear in this paper.
It is a basic fact that integrals/co-integrals are unique up to scalar multiplication. It can also be checked directly that the antipode fixes both $T$ and $C$, namely, $T \circ S = T, \ S\circ C = C$.
\[lem:trace\_is\_integral\] Let $H$ be an involutory Hopf algebra. Then the trace $T$ and cotrace $C$ are, respectively, an integral and a cointegral.
\[not:product\_coproduct\_abbreviated\_notation\] The associativity and coassociativity axioms permit us to adopt the following abbreviated notation for iterated products and coproducts. $$\begin{tikzpicture}
\draw (0,0) node (M1) {$M$};
\draw (-.6,-.08) node (dots1) {$\dots$};
\draw[->] (-.8,.5)--(M1);
\draw[->] (-.8,.2)--(M1);
\draw[->] (-.8,-.5)--(M1);
\draw[->] (M1)--(.8,0);
\draw (1.2,0) node (=1) {$:=$};
\draw (2.6,0) node [draw,rectangle] (TM1) {$TM$};
\draw (1.8,-.08) node (dots2) {$\dots$};
\draw[-] (1.6,.5)--(TM1);
\draw[-] (1.6,.2)--(TM1);
\draw[-] (1.6,-.5)--(TM1);
\draw[->] (TM1)--(3.4,0);
\draw (5,0) node (M2) {$M$};
\draw[->] (4.2,0)--(M2);
\draw[->] (M2)--(5.8,0);
\draw (6.2,0) node (=2) {$:=$};
\draw[->] (6.6,0)--(7.4,0);
\draw (8.4,0) node (M3) {$M$};
\draw (10.1,0) node (n3) {$\eta$};
\draw[->] (M3)--(9.2,0);
\draw (9.6,0) node (=3) {$:=$};
\draw[->] (n3)--(10.9,0);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (0,0) node (D1) {$\Delta$};
\draw (-.6,-.08) node (dots1) {$\dots$};
\draw[<-] (-.8,.5)--(D1);
\draw[<-] (-.8,.2)--(D1);
\draw[<-] (-.8,-.5)--(D1);
\draw[<-] (D1)--(.8,0);
\draw (1.2,0) node (=1) {$:=$};
\draw (2.6,0) node [draw,rectangle] (TD1) {$T\Delta$};
\draw (1.8,-.08) node (dots2) {$\dots$};
\draw[<-] (1.6,.5)--(TD1);
\draw[<-] (1.6,.2)--(TD1);
\draw[<-] (1.6,-.5)--(TD1);
\draw[-] (TD1)--(3.4,0);
\draw (5,0) node (D2) {$\Delta$};
\draw[<-] (4.2,0)--(D2);
\draw[<-] (D2)--(5.8,0);
\draw (6.2,0) node (=2) {$:=$};
\draw[<-] (6.6,0)--(7.4,0);
\draw (8.4,0) node (D3) {$\Delta$};
\draw (10.1,0) node (e3) {$\epsilon$};
\draw[<-] (D3)--(9.2,0);
\draw (9.6,0) node (=3) {$:=$};
\draw[<-] (e3)--(10.9,0);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (0,0) node (M1) {$M$};
\draw (-.6,0) node (dots1) {$\dots$};
\draw (.65,0) node (dots2) {$\dots$};
\draw[->] (-.6,.6)--(M1);
\draw[->] (0,.8)--(M1);
\draw[->] (.6,.6)--(M1);
\draw[->] (.6,-.6)--(M1);
\draw[->] (0,-.8)--(M1);
\draw[->] (-.6,-.6)--(M1);
\draw (1.6,0) node (=1) {$:=$};
\draw (3,0) node (M2) {$M$};
\draw (2.4,-.08) node (dots3) {$\dots$};
\draw (4,0) node (T2) {$T$};
\draw[->] (2.2,.5)--(M2);
\draw[->] (2.2,.2)--(M2);
\draw[->] (2.2,-.5)--(M2);
\draw[->] (M2)--(T2);
\draw (6,0) node (D1) {$\Delta$};
\draw (5.4,0) node (dots4) {$\dots$};
\draw (6.65,0) node (dots5) {$\dots$};
\draw[<-] (5.4,.6)--(D1);
\draw[<-] (6,.8)--(D1);
\draw[<-] (6.6,.6)--(D1);
\draw[<-] (6.6,-.6)--(D1);
\draw[<-] (6,-.8)--(D1);
\draw[<-] (5.4,-.6)--(D1);
\draw (7.6,0) node (=2) {$:=$};
\draw (9,0) node (D2) {$\Delta$};
\draw (8.4,-.08) node (dots6) {$\dots$};
\draw (10,0) node (C2) {$C$};
\draw[<-] (8.2,.5)--(D2);
\draw[<-] (8.2,.2)--(D2);
\draw[<-] (8.2,-.5)--(D2);
\draw[<-] (D2)--(C2);
\end{tikzpicture}$$ Here $TM$ denotes an arbitrary tree with $i$ in edges, $1$ out edge and only $M$ nodes, and similarly $T\Delta$ denotes an arbitrary tree with $i$ out edges, $1$ in edge and only $\Delta$ nodes. The multiplication-trace and comultiplication-cotrace compositions are symmetric under cyclic permutation, as suggested by the notation.
Any fixed Hopf algebra gives rise to a number of associated Hopf algebras acquirable by simple alterations of the structure tensors.
\[def:dual\_op\_cop\_algebras\] Let $H$ be a finite dimensional Hopf algebra. Then
- (Dual) The *dual Hopf algebra* $H^*$ is the linear dual $H^*$ equipped multiplication $\Delta$, comultiplication $M$ and antipode $S$, interpreted as tensors for the dual space $H^*$. Written left to right, these are $$\begin{tikzpicture}
\draw (1,0) node (D) {$\Delta$};
\draw[<-] (0,-.25) to [out=0,in=150,looseness=1] (D);
\draw[<-] (0,.25) to [out=0,in=210,looseness=1] (D);
\draw[<-] (D)--(2,0);
\draw (4,0) node (M) {$M$};
\draw[<-] (M) to [out=30,in=180,looseness=1] (5,-.25);
\draw[<-] (M) to [out=-30,in=180,looseness=1] (5,.25);
\draw[->] (M)--(3,0);
\draw (7,0) node (S) {$S$};
\draw[<-] (6,0)--(S);
\draw[<-] (S)--(8,0);
\end{tikzpicture}$$ Note the change in the order of the inputs and outputs in the multiplication and comultiplication tensors. This is necessary due to our input and output ordering convention (see Notation \[not:general\_tensor\_diagrams\]).
- (Op) The *op Hopf algebra* $H^{{\operatorname}{op}}$ is $H$ equipped with coproduct $\Delta^{{\operatorname}{op}} := \Delta$, antipode $S^{{\operatorname}{op}} := S$ and product $M^{{\operatorname}{op}}$ given by the tensor $$\begin{tikzpicture}
\draw (1,0) node (Mop) {$M^{{\operatorname}{op}}$};
\draw[->] (0,-.25)--(Mop);
\draw[->] (0,.25)--(Mop);
\draw[->] (Mop)--(2,0);
\draw (2.5,0) node (=) {$:=$};
\draw (4,0) node (M) {$M$};
\draw[->] (3,-.25) to [out=0,in=150,looseness=1] (M);
\draw[->] (3,.25) to [out=0,in=210,looseness=1] (M);
\draw[->] (M)--(5,0); \end{tikzpicture}$$
- (Cop) The *cop Hopf algebra* $H^{{\operatorname}{cop}}$ is $H$ equipped with product $M^{{\operatorname}{cop}} := M$, antipode $S^{{\operatorname}{op}} := S$ and coproduct $\Delta^{{\operatorname}{cop}}$ given by the tensor $$\begin{tikzpicture}
\draw (1,0) node (Dcop) {$\Delta^{{\operatorname}{cop}}$};
\draw[<-] (0,-.25)--(Dcop);
\draw[<-] (0,.25)--(Dcop);
\draw[<-] (Dcop)--(2,0);
\draw (2.5,0) node (=) {$:=$};
\draw (4,0) node (D) {$\Delta$};
\draw[<-] (3,-.25) to [out=0,in=150,looseness=1] (D);
\draw[<-] (3,.25) to [out=0,in=210,looseness=1] (D);
\draw[<-] (D)--(5,0); \end{tikzpicture}$$
All of these constructions are functorial, i.e. a Hopf algebra map $f:H \to I$ induces maps $f^*:I^* \to H^*$, $f^{{\operatorname}{op}}:H^{{\operatorname}{op}} \to I^{{\operatorname}{op}}$ and $f^{{\operatorname}{cop}}:H^{{\operatorname}{cop}} \to I^{{\operatorname}{cop}}$.
### Hopf Doublets And Triplets {#subsubsec:doublets_and_triplets}
We are now ready to introduce the Hopf algebra data that are used to formulate $3$-manifold and $4$-manifold invariants.
The data for $3$-manifold invariants (used for the Kuperberg invariant and its generalization at the end of §\[sec:trisection\_kuperberg\_invt\]) can be formulated in terms of Hopf doublets, which we now define.
\[def:hopf\_doublet\] An (involutory) *Hopf doublet* $(H,\langle-\rangle)$ consists of two (involutory) Hopf algebras $H_\alpha$ and $H_\beta$, and a bilinear form $$\langle-\rangle:H_\alpha \otimes H_\beta \to k$$ The form $\langle-\rangle$ must satisfy the following properties.
- The linear map $H_\alpha \to H_\beta^{*,{\operatorname}{cop}}$ induced by $\langle-\rangle$ must be a Hopf algebra map.
A map of Hopf doublets $f:(H,\langle-\rangle) \to (I,(-))$ is a set of maps of Hopf algebras $f_*:H_* \to I_*$ for $* \in \{\alpha,\beta\}$ intertwining the bilinear forms $\langle-\rangle$ and $(-)$.
There are several ways of making new Hopf doublets from a single Hopf doublet by applying the various operations of Definition \[def:dual\_op\_cop\_algebras\]. In particular, we have the following lemma.
\[lem:reverse\_doublet\] Let $(H_\alpha,H_\beta,\langle-\rangle)$ be a Hopf doublet. Then
- $(H^{{\operatorname}{op}}_\alpha,H^{{\operatorname}{cop}}_\beta,\langle-\rangle)$ is a Hopf doublet.
- $(H^{{\operatorname}{cop}}_\alpha,H^{{\operatorname}{op}}_\beta,\langle-\rangle)$ is a Hopf doublet.
- $(H^{{\operatorname}{op},{\operatorname}{cop}}_\beta,H_\alpha,\langle-\rangle)$ is a Hopf doublet.
It will be convenient, for later constructions, to introduce the following shorthand notation involving the tensors in a doublet.
\[not:TUV\_notation\] Let $(H_\alpha,H_\beta,\langle-\rangle)$ be a Hopf doublet. We fix the following notation $$\begin{tikzpicture}
\draw (1,0) node (Tab) {$T_{\alpha\beta}$};
\draw[->] (0,.25)--(Tab);
\draw[->] (0,-.25)--(Tab);
\draw[->] (Tab)--(2,.25);
\draw[->] (Tab)--(2,-.25);
\draw (3,0) node (=1) {$:=$};
\draw (5,1) node (Db) {$\Delta_\beta$};
\draw (5,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (5,-1) node (Da) {$\Delta_\alpha$};
\draw[->] (4,1)--(Db);
\draw[->] (Db)--(6,1);
\draw[->] (4,-1)--(Da);
\draw[->] (Da)--(6,-1);
\draw[->] (Db)--(Pab);
\draw[->] (Da)--(Pab);
\draw (9,0) node (Tab_inv) {$T^{-1}_{\alpha\beta}$};
\draw[->] (8,.25)--(Tab_inv);
\draw[->] (8,-.25)--(Tab_inv);
\draw[->] (Tab_inv)--(10,.25);
\draw[->] (Tab_inv)--(10,-.25);
\draw (11,0) node (=2) {$=$};
\draw (13,1) node (Db2) {$\Delta_\beta$};
\draw (13.6,0) node (Sa2) {$S_\alpha$};
\draw (12.4,0) node (Pab2) {$\langle-\rangle_{\alpha\beta}$};
\draw (13,-1) node (Da2) {$\Delta_\alpha$};
\draw[->] (12,1)--(Db2);
\draw[->] (Db2)--(14,1);
\draw[->] (12,-1)--(Da2);
\draw[->] (Da2)--(14,-1);
\draw[->] (Db2)--(Pab2);
\draw[->] (Sa2)--(Pab2);
\draw[->] (Da2)--(Sa2);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (-2,0) node (Uab) {$U_{\alpha\beta}$};
\draw[->] (-3,.25)--(Uab);
\draw[->] (-3,-.25)--(Uab);
\draw[->] (Uab)--(-1,.25);
\draw[->] (Uab)--(-1,-.25);
\draw (-.4,0) node (=1) {$:=$};
\draw (0,1) node (in_b) {};
\draw (0,-1) node (in_a) {};
\draw (1,1) node (Sb1) {$S_\beta$};
\draw (2,1) node (Db2) {$\Delta_\beta$};
\draw (2.6,0) node (Sa2) {$S_\alpha$};
\draw (1.4,0) node (Pab2) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,-1) node (Da2) {$\Delta_\alpha$};
\draw (3,1) node (Sb) {$S_\beta$};
\draw (3,-1) node (Sa) {$S_\alpha$};
\draw (4,1) node (Db) {$\Delta_\beta$};
\draw (4,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (4,-1) node (Da) {$\Delta_\alpha$};
\draw (5,-1) node (Sa3) {$S_\alpha$};
\draw (6,1) node (out_b) {};
\draw (6,-1) node (out_a) {};
\draw[->] (in_b)--(Sb1);
\draw[->] (Sb1)--(Db2);
\draw[->] (in_a)--(Da2);
\draw[->] (Db2)--(Pab2);
\draw[->] (Sa2)--(Pab2);
\draw[->] (Da2)--(Sa2);
\draw[->] (Db2)--(Sb);
\draw[->] (Sb)--(Db);
\draw[->] (Da2)--(Sa);
\draw[->] (Sa)--(Da);
\draw[->] (Db)--(Pab);
\draw[->] (Da)--(Pab);
\draw[->] (Db)--(out_b);
\draw[->] (Da)--(Sa3);
\draw[->] (Sa3)--(out_a);
\draw (7,0) node (=1) {$=$};
\draw (7.5,1) node (in_b4) {};
\draw (7.5,-1) node (in_a4) {};
\draw (8.5,1) node (Sb4) {$S_\beta$};
\draw (9,0) node (Tab_inv2) {$T^{-1}_{\alpha\beta}$};
\draw (9.5,1) node (Sb5) {$S_\beta$};
\draw (9.5,-1) node (Sa5) {$S_\alpha$};
\draw (10,0) node (Tab_2) {$T_{\alpha\beta}$};
\draw (10.5,-1) node (Sa6) {$S_\alpha$};
\draw (11.5,1) node (out_b4) {};
\draw (11.5,-1) node (out_a4) {};
\draw[->] (in_b4)--(Sb4);
\draw[->] (Sb4)--(Tab_inv2);
\draw[->] (in_a4)--(Tab_inv2);
\draw[->] (Tab_inv2)--(Sb5);
\draw[->] (Tab_inv2)--(Sa5);
\draw[->] (Sb5)--(Tab_2);
\draw[->] (Sa5)--(Tab_2);
\draw[->] (Tab_2)--(Sa6);
\draw[->] (Sa6)--(out_a4);
\draw[->] (Tab_2)--(out_b4);
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (-4,.5) node (in_b0) {};
\draw (-4,-.5) node (in_a0) {};
\draw (-3,0) node (Vab) {$V_{\alpha\beta}$};
\draw (-2,.5) node (out_b0) {};
\draw (-2,-.5) node (out_a0) {};
\draw[->] (in_b0)--(Vab);
\draw[->] (in_a0)--(Vab);
\draw[->] (Vab)--(out_b0);
\draw[->] (Vab)--(out_a0);
\draw (-1,0) node (=1) {:=};
\draw (0,-.5) node (in_a1) {};
\draw (0,.5) node (in_b1) {};
\draw (1,-.5) node (Sa1) {$S_\alpha$};
\draw (1,.5) node (Sb1) {$S_\beta$};
\draw (2,0) node (Tab1) {$T_{\alpha\beta}$};
\draw (3,.5) node (Sb2) {$S_\beta$};
\draw (4,-.5) node (out_a1) {};
\draw (4,.5) node (out_b1) {};
\draw[->] (in_a1)--(Sa1);
\draw[->] (in_b1)--(Sb1);
\draw[->] (Sa1)--(Tab1);
\draw[->] (Sb1)--(Tab1);
\draw[->] (Tab1)--(Sb2);
\draw[->] (Tab1)--(out_a1);
\draw[->] (Sb2)--(out_b1);
\end{tikzpicture}$$
We will make use of the following two identities relating the above tensors.
\[lem:UTV\_identity\] Let $(H_\alpha,H_\beta,\langle-\rangle_{\alpha\beta})$ be an involutory Hopf doublet. Then the tensors $T_{\alpha\beta}, U_{\alpha\beta}$ and $V_{\alpha\beta}$ satisfy the following relations. $$\label{eqn:TUV_identity_1}
\begin{tikzpicture}
\draw (0,-.5) node (in_a1) {};
\draw (0,.5) node (in_b1) {};
\draw (1,0) node (Vab) {$V_{\alpha\beta}$};
\draw (3,0) node (Uab) {$U_{\alpha\beta}$};
\draw (4,-.5) node (out_a1) {};
\draw (4,.5) node (out_b1) {};
\draw[->] (in_a1)--(Vab);
\draw[->] (in_b1)--(Vab);
\draw[->] (Vab) to [out=30,in=150,looseness=1] (Uab);
\draw[->] (Vab) to [out=-30,in=210,looseness=1] (Uab);
\draw[->] (Uab)--(out_a1);
\draw[->] (Uab)--(out_b1);
\draw (4.5,0) node (=1) {=};
\draw (5,-.5) node (in_a2) {};
\draw (5,.5) node (in_b2) {};
\draw (6,0) node (Tab) {$T_{\alpha\beta}$};
\draw (7,-.5) node (Sa) {$S_\alpha$};
\draw (8,-.5) node (out_a2) {};
\draw (8,.5) node (out_b2) {};
\draw[->] (in_a2)--(Tab);
\draw[->] (in_b2)--(Tab);
\draw[->] (Tab)--(out_b2);
\draw[->] (Tab)--(Sa);
\draw[->] (Sa)--(out_a2);
\end{tikzpicture}$$ $$\label{eqn:TUV_identity_2}
\begin{tikzpicture}
\draw (0,.5) node (in_b) {};
\draw (0,-.5) node (in_a) {};
\draw (1,0) node (Tab_1) {$T_{\alpha\beta}$};
\draw (2,.5) node (Sb) {$S_\beta$};
\draw (2,-.5) node (Sa) {$S_\alpha$};
\draw (3,0) node (Tab_2) {$T^{-1}_{\alpha\beta}$};
\draw (4,.5) node (Sb2) {$S_\beta$};
\draw (4,-.5) node (Sa2) {$S_\alpha$};
\draw (5,.5) node (out_b) {};
\draw (5,-.5) node (out_a) {};
\draw[->] (in_b)--(Tab_1);
\draw[->] (in_a)--(Tab_1);
\draw[->] (Tab_1)--(Sb);
\draw[->] (Tab_1)--(Sa);
\draw[->] (Sb)--(Tab_2);
\draw[->] (Sa)--(Tab_2);
\draw[->] (Tab_2)--(Sb2);
\draw[->] (Tab_2)--(Sa2);
\draw[->] (Sb2)--(out_b);
\draw[->] (Sa2)--(out_a);
\end{tikzpicture}
\begin{tikzpicture}
\draw (.5,0) node (=1) {$=$};
\draw (1,.5) node (in_b) {};
\draw (1,-.5) node (in_a) {};
\draw (2,.5) node (Sb) {$S_\beta$};
\draw (2,-.5) node (Sa) {$S_\alpha$};
\draw (3,0) node (Tab_2) {$T^{-1}_{\alpha\beta}$};
\draw (4,.5) node (Sb2) {$S_\beta$};
\draw (4,-.5) node (Sa2) {$S_\alpha$};
\draw (5,0) node (Tab_3) {$T_{\alpha\beta}$};
\draw (6,.5) node (out_b) {};
\draw (6,-.5) node (out_a) {};
\draw[->] (in_b)--(Sb);
\draw[->] (in_a)--(Sa);
\draw[->] (Sb)--(Tab_2);
\draw[->] (Sa)--(Tab_2);
\draw[->] (Tab_2)--(Sb2);
\draw[->] (Tab_2)--(Sa2);
\draw[->] (Sb2)--(Tab_3);
\draw[->] (Sa2)--(Tab_3);
\draw[->] (Tab_3)--(out_b);
\draw[->] (Tab_3)--(out_a);
\end{tikzpicture}$$
A given Hopf doublet can be used to construct a type of twisted tensor product Hopf algebra, called the Drinfeld double of the pair.
The *Drinfeld double* $D(H_\alpha,H_\beta)$ of a Hopf doublet $(H_\alpha,H_\beta,\langle-\rangle)$ is the involutory Hopf algebra defined as follows.
The underlying $k$-module is $D(H_\alpha,H_\beta) = H_\alpha \otimes H_\beta$. The coproduct $\Delta_{\alpha\beta}$, counit $\epsilon_{\alpha\beta}$ and unit $\eta_{\alpha\beta}$ are given by tensor products of the corresponding tensors of $H_\alpha$ and $H_\beta$. On the other hand, the product $M_{\alpha\beta}$ and antipode $S_{\alpha\beta}$ are given by the following tensor diagrams $$\begin{tikzpicture}
\draw (0,1) node (in_Ma1) {};
\draw (0,.5) node (in_Mb1) {};
\draw (0,-.5) node (in_Ma2) {};
\draw (0,-1) node (in_Mb2) {};
\draw (1,0) node (Mab) {$M_{\alpha\beta}$};
\draw (2,.3) node (out_Ma1) {};
\draw (2,-.3) node (out_Mb1) {};
\draw[->] (in_Ma1)--(Mab);
\draw[->] (in_Mb1)--(Mab);
\draw[->] (in_Ma2)--(Mab);
\draw[->] (in_Mb2)--(Mab);
\draw[->] (Mab)--(out_Ma1);
\draw[->] (Mab)--(out_Mb1);
\draw (3,0) node (=1) {$:=$};
\draw (4,1) node (in_Ma3) {};
\draw (4,.5) node (in_Mb3) {};
\draw (4,-.5) node (in_Ma4) {};
\draw (4,-1) node (in_Mb4) {};
\draw (5,0) node (Uab1) {$U_{\alpha\beta}$};
\draw (6,1) node (Ma1) {$M_\alpha$};
\draw (6,-1) node (Mb1) {$M_\beta$};
\draw (7,.5) node (out_Ma2) {};
\draw (7,-.5) node (out_Mb2) {};
\draw[->] (in_Ma3)--(Ma1);
\draw[->] (in_Mb3)--(Uab1);
\draw[->] (in_Ma4)--(Uab1);
\draw[->] (in_Mb4)--(Mb1);
\draw[->] (Uab1) to [out=-30,in=-90,looseness=1] (Ma1);
\draw[->] (Uab1) to [out=30,in=90,looseness=1] (Mb1);
\draw[->] (Ma1) to (out_Ma2) {};
\draw[->] (Mb1) to (out_Mb2) {};
\draw (8,.3) node (Sab_a_in) {};
\draw (8,-.3) node (Sab_b_in) {};
\draw (9,0) node (Sab) {$S_{\alpha\beta}$};
\draw (10,.3) node (Sab_a_out) {};
\draw (10,-.3) node (Sab_b_out) {};
\draw[->] (Sab_a_in)--(Sab);
\draw[->] (Sab_b_in)--(Sab);
\draw[->] (Sab)--(Sab_a_out);
\draw[->] (Sab)--(Sab_b_out);
\draw (10.5,0) node (=2) {$:=$};
\draw (11,.3) node (Sab_a_in2) {};
\draw (11,-.3) node (Sab_b_in2) {};
\draw (12,.3) node (Sa1) {$S_\alpha$};
\draw (12,-.3) node (Sb1) {$S_\beta$};
\draw (13,0) node (Uab2) {$U_{\alpha\beta}$};
\draw (14,.3) node (Sab_a_out2) {};
\draw (14,-.3) node (Sab_b_out2) {};
\draw[->] (Sab_a_in2)--(Sa1);
\draw[->] (Sab_b_in2)--(Sb1);
\draw[->] (Sa1)--(Uab2);
\draw[->] (Sb1)--(Uab2);
\draw[->] (Uab2)--(Sab_a_out2);
\draw[->] (Uab2)--(Sab_b_out2);
\end{tikzpicture}$$
The Drinfeld double defines a functor, in the sense that a map $f:(H_*,\langle-\rangle) \to (I_*,(-))$ induces a map of doubles. $$D(f):D(H_\alpha,H_\beta) \to D(I_\alpha,I_\beta)$$
For the tautological doublet $(H_\alpha,H_\beta,\langle-\rangle)$ of a single Hopf algebra $H$, where $H_\alpha = H^{*,{\operatorname}{cop}}$, $H_\beta = H$ and $\langle-\rangle$ is the usual dual pairing, we will use the notation $D(H)$ to denote the Drinfeld double.
The data needed to formulate the $4$-manifold invariants discussed in this paper can be formulated in terms of Hopf triplets, in analogy with the $3$-manifold case.
\[def:hopf\_triplet\] An (involutory) *Hopf triplet* $\mathcal{H} = (H_\alpha,H_\beta,H_\kappa,\langle-\rangle)$ consists of three (involutory) Hopf algebras $H_\alpha,H_\beta$ and $H_\kappa$, and three pairings $$\langle-\rangle_{\alpha\beta}:H_\alpha \otimes H_\beta \to k \qquad \langle-\rangle_{\beta\kappa}:H_\beta \otimes H_\kappa \to k \qquad \langle-\rangle_{\kappa\alpha}:H_\kappa \otimes H_\alpha \to k$$ The bilinear forms $\langle-\rangle_*$ must satisfy the following properties.
- Each of the pairs $(H_\alpha,H_\beta,\langle-\rangle_{\alpha\beta})$, $(H_\beta,H_\kappa,\langle-\rangle_{\beta\kappa})$ and $(H_\kappa,H_\alpha,\langle-\rangle_{\kappa\alpha})$ is a Hopf doublet in the sense of Definition \[def:hopf\_doublet\].
- The three $k$-linear maps form the following doubles $$D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H_\kappa^* \qquad D(H_\beta^{{\operatorname}{op}},H_\kappa^{{\operatorname}{cop}}) \to H^*_\alpha \qquad D(H_\kappa^{{\operatorname}{op}},H_\alpha^{{\operatorname}{cop}}) \to H^*_\beta$$ which we define, respectively, via the following tensor diagrams $$\hspace{-24pt}\begin{tikzpicture}
\draw (-.3,1) node (i_a1) {};
\draw (-.3,0) node (i_b1) {};
\draw (1,1) node (Pa1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,0) node (Pb1) {$\langle-\rangle_{\beta\kappa}$};
\draw (2.5,.5) node (Dc1) {$\Delta_\kappa$};
\draw (3.5,.5) node (o_c1) {};
\draw[->] (i_a1)--(Pa1);
\draw[->] (i_b1)--(Pb1);
\draw[->] (Dc1) to [out=200,in=-30,looseness=1] (Pa1);
\draw[->] (Dc1) to [out=160,in=30,looseness=1] (Pb1);
\draw[<-] (Dc1)--(o_c1);
\draw (4.2,1) node (i_b2) {};
\draw (4.2,0) node (i_c2) {};
\draw (5.5,1) node (Pb2) {$\langle-\rangle_{\alpha\beta}$};
\draw (5.5,0) node (Pc2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (7,.5) node (Da2) {$\Delta_\alpha$};
\draw (8,.5) node (o_a2) {};
\draw[->] (i_b2)--(Pb2);
\draw[->] (i_c2)--(Pc2);
\draw[->] (Da2) to [out=200,in=-30,looseness=1] (Pb2);
\draw[->] (Da2) to [out=160,in=30,looseness=1] (Pc2);
\draw[<-] (Da2)--(o_a2);
\draw (8.7,1) node (i_c3) {};
\draw (8.7,0) node (i_a3) {};
\draw (10,1) node (Pc3) {$\langle-\rangle_{\beta\kappa}$};
\draw (10,0) node (Pa3) {$\langle-\rangle_{\alpha\beta}$};
\draw (11.5,.5) node (Db3) {$\Delta_\beta$};
\draw (12.5,.5) node (o_b3) {};
\draw[->] (i_c3)--(Pc3);
\draw[->] (i_a3)--(Pa3);
\draw[->] (Db3) to [out=200,in=-30,looseness=1] (Pc3);
\draw[->] (Db3) to [out=160,in=30,looseness=1] (Pa3);
\draw[<-] (Db3)--(o_b3);
\end{tikzpicture}$$ are maps of Hopf algebras.
A map of Hopf triplets $f:\mathcal{H} \to \mathcal{I}$ is a triple of maps of Hopf algebras $f_*:H_* \to I_*$ for $* \in \{\alpha,\beta,\kappa\}$ intertwining the pairwise bilinear forms on both sides.
We will use two notations for the tensor diagrams of the pairings in a Hopf triplet. These notations are $$\rightarrow \langle-\rangle_{\alpha\beta} \leftarrow \qquad \text{or} \qquad \rightarrow \bullet \leftarrow$$ The first notation is the obvious one, while the second (which we deem *bullet notation*) will be a helpful abbreviation that we will use exclusively in the more elaborate tensor diagrams in §\[sec:trisection\_kuperberg\_invt\].
We now prove a fundamental Lemma giving alternate formulations of Definition \[def:hopf\_triplet\](b). This lemma will be used for many purposes, including the construction of examples of Hopf triplets and at a key point in the proof of invariance of our invariant in §\[subsec:main\_definition\_and\_properties\].
\[lem:fundamental\_triplet\_lemma\] Let $(H,\langle-\rangle)$ be a set of three Hopf algebras $H_*$, $* \in \{\alpha,\beta,\kappa\}$, as in Definition \[def:hopf\_triplet\] along with three pairings $\langle-\rangle$ satisfying Definition \[def:hopf\_triplet\](a).
Then the following are equivalent.
- $(H,\langle-\rangle)$ is a Hopf triplet, i.e. the three of the maps in Definition \[def:hopf\_triplet\](b) $$D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H_\kappa^* \qquad D(H_\beta^{{\operatorname}{op}},H_\kappa^{{\operatorname}{cop}}) \to H^*_\alpha \qquad D(H_\kappa^{{\operatorname}{op}},H_\alpha^{{\operatorname}{cop}}) \to H^*_\beta$$ are Hopf algebra maps.
- The map $D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H_\kappa^*$ of Definition \[def:hopf\_triplet\](b) is a Hopf algebra map.
- The following identity relates the structure tensors of the triples. $$\label{eqn:symmetric_triplet_identity_c}
\begin{tikzpicture}
\draw (0,-3) node (ia1) {};
\draw (0,0) node (ib1) {};
\draw (4,-1.5) node (ic1) {};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1.5) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw[->] (ia1)--(Da);
\draw[->] (ib1)--(Db);
\draw[->] (ic1)--(Dc);
\draw[->] (Da)--(Pab);
\draw[->] (Da)--(Pca);
\draw[->] (Db)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-1,-1.5) node (=) {$=$};
\draw (-1,-3) node (ia1) {};
\draw (-1,-0) node (ib1) {};
\draw (5,-1.5) node (ic1) {};
\draw (0,-3) node (Sa1) {$S_\alpha$};
\draw (0,-0) node (Sb1) {$S_\beta$};
\draw (4.2,-1.5) node (Sc1) {$S_\kappa$};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,-0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1.5) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,-0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw[->] (ia1)--(Sa1);
\draw[->] (ib1)--(Sb1);
\draw[->] (ic1)--(Sc1);
\draw[->] (Sa1)--(Da);
\draw[->] (Sb1)--(Db);
\draw[->] (Sc1)--(Dc);
\draw[->] (Da)--(Pab);
\draw[->] (Da)--(Pca);
\draw[->] (Db)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}$$
- The following identity relates the structure tensors of the triples. $$\label{eqn:symmetric_triplet_identity_d}
\begin{tikzpicture}
\draw (0,-3) node (ia1) {};
\draw (0,-0) node (ib1) {};
\draw (4,-1.5) node (ic1) {};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,-0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,-0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2,-3) node (Sa2) {$S_\alpha$};
\draw (2,-0) node (Sb2) {$S_\beta$};
\draw (1,-2) node (Sa3) {$S_\alpha$};
\draw[->] (ia1)--(Da);
\draw[->] (ib1)--(Db);
\draw[->] (ic1)--(Dc);
\draw[->] (Da)--(Sa3);
\draw[->] (Sa3)--(Pab);
\draw[->] (Da)--(Sa2);
\draw[->] (Sa2)--(Pca);
\draw[->] (Db)--(Sb2);
\draw[->] (Sb2)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-1,-1.5) node (=) {$=$};
\draw (-1,-3) node (ia1) {};
\draw (-1,-0) node (ib1) {};
\draw (5,-1.5) node (ic1) {};
\draw (0,-3) node (Sa1) {$S_\alpha$};
\draw (0,-0) node (Sb1) {$S_\beta$};
\draw (4.2,-1.5) node (Sc1) {$S_\kappa$};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,-0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,-0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2,-3) node (Sa2) {$S_\alpha$};
\draw (2,-0) node (Sb2) {$S_\beta$};
\draw (1,-2) node (Sa3) {$S_\alpha$};
\draw[->] (ia1)--(Sa1);
\draw[->] (ib1)--(Sb1);
\draw[->] (ic1)--(Sc1);
\draw[->] (Sa1)--(Da);
\draw[->] (Sb1)--(Db);
\draw[->] (Sc1)--(Dc);
\draw[->] (Da)--(Sa3);
\draw[->] (Sa3)--(Pab);
\draw[->] (Da)--(Sa2);
\draw[->] (Sa2)--(Pca);
\draw[->] (Db)--(Sb2);
\draw[->] (Sb2)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}$$
We will demonstrate the following four equivalences between the statements $$(a) \implies (b)\,, \qquad (b) \iff (d)\,, \qquad (c) \iff (d)\,, \qquad (c) \implies (a)$$ Taken together, these equivalences prove the desired equivalence of (a)-(d). The fact that $(a) \implies (b)$ is trivial, and the fact that $(b) \iff (d) \iff (c)$ along with the symmetry of the tensor diagrams (\[eqn:symmetric\_triplet\_identity\_d\]) and (\[eqn:symmetric\_triplet\_identity\_c\]) will imply $(c) \implies (a)$. Thus, we prove the middle two equivalences: $(b) \iff (d)$ and $(c) \iff (d)$.
*$(b) \iff (d)$.* Let $\Phi^\kappa_{\alpha\beta}:D(H^{{\operatorname}{op}}_\alpha,H^{{\operatorname}{cop}}_\beta) \to H^{*}_\kappa$ denote the linear map of (a). We must check that $\Phi^\kappa_{\alpha\beta}$ intertwines the product, unit, coproduct and counit. Below, we will denote the $T,U$ and $V$ tensors of Notation \[not:TUV\_notation\] for the doublet $(H^{{\operatorname}{op}}_\alpha,H^{{\operatorname}{cop}}_\beta,\langle-\rangle)$ by $T'_{\alpha\beta}$, $U'_{\alpha\beta}$ and $V'_{\alpha\beta}$, and we denote the structure tensors (multiplication, comultiplication, etc.) similarly.
We will in fact show that $\Phi^\kappa_{\alpha\beta}$ automatically intertwines the coproduct, counit and unit (without assuming either (b) or (d)). We then show that $\Phi^\kappa_{\alpha\beta}$ intertwines comultiplication (i.e. satisfies (b)) if and only if (d) is satisfied.
(Counit/Comultiplication/Unit) Recall that the counit, coproduct and unit of $D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}})$ agree with those of $H_\alpha^{{\operatorname}{op}} \otimes H_\beta^{{\operatorname}{cop}}$, which are simply the tensor products of the corresponding tensors of $H^{{\operatorname}{op}}_\alpha$ and $H^{{\operatorname}{cop}}_\beta$. Thus it suffices to show that $\Phi^\kappa_{\alpha\beta}:H_\alpha^{{\operatorname}{op}} \otimes H_\beta^{{\operatorname}{cop}} \to H_\kappa^*$ intertwines the counit, coproduct and unit. The map $\Phi^\kappa_{\alpha\beta}$ is defined as the composition of two maps $$H^{{\operatorname}{op}}_\alpha \otimes H^{{\operatorname}{cop}}_\beta \xrightarrow{\langle-\rangle_{\kappa\beta} \otimes \langle-\rangle_{\beta\kappa}} H^*_\kappa \otimes H^*_\kappa \xrightarrow{\Delta_\kappa} H^*_\kappa$$ Above, the first map is the tensor product of the Hopf algebra maps $H^{{\operatorname}{op}}_\alpha \to H^*_\kappa$ and $H^{{\operatorname}{cop}}_\beta \to H^*_\kappa$ induced by the pairings $\langle-\rangle_{\kappa\alpha}$ and $\langle-\rangle_{\beta\kappa}$, and the second map is the product in $H^*_\kappa$ (i.e. the coproduct in $H_\kappa$).
The first map is a Hopf algebra map because it is the tensor product of Hopf algebra maps. Thus, it suffices to check that the product $H^*_\kappa \otimes H^*_\kappa \to H^*_\kappa$ intertwines the counit, coproduct and unit. The counit and coproduct are intertwined because multiplication is a coalgebra homomorphism. The unit is intertwined because this is equivalent to the fact that the unit squares to itself.
(Multiplication) Consider the following homomorphism identity. $$\label{eqn:homomorphism_identity_1}\begin{tikzpicture}
\draw (0,1) node (ib1) {};
\draw (0,.5) node (ia1) {};
\draw (0,-.5) node (ib2) {};
\draw (0,-1) node (ia2) {};
\draw (1,.5) node (F1) {$\Phi^\kappa_{\alpha\beta}$};
\draw (1,-.5) node (F2) {$\Phi^\kappa_{\alpha\beta}$};
\draw (2.5,0) node (Dc) {$\Delta_\kappa$};
\draw (3.5,0) node (oc1) {};
\draw[->] (ib1)--(F1);
\draw[->] (ia1)--(F1);
\draw[->] (ib2)--(F2);
\draw[->] (ia2)--(F2);
\draw[->] (Dc) to [out=200,in=0,looseness=1] (F1);
\draw[->] (Dc) to [out=160,in=0,looseness=1] (F2);
\draw[<-] (Dc)--(oc1);
\draw (4.25,0) node (=1) {$=$};
\draw (5,1) node (ib3) {};
\draw (5,.5) node (ia3) {};
\draw (5,-.5) node (ib4) {};
\draw (5,-1) node (ia4) {};
\draw (6,0) node (Mab) {$M'_{\alpha\beta}$};
\draw (8,0) node (F3) {$\Phi^\kappa_{\alpha\beta}$};
\draw (9,0) node (oc2) {};
\draw[->] (ib3)--(Mab);
\draw[->] (ia3)--(Mab);
\draw[->] (ib4)--(Mab);
\draw[->] (ia4)--(Mab);
\draw[->] (Mab) to [out=20,in=160,looseness=1] (F3);
\draw[->] (Mab) to [out=-20,in=200,looseness=1] (F3);
\draw[<-] (F3)--(oc2);
\end{tikzpicture}$$ We will now show that this identity is equivalent to (\[eqn:symmetric\_triplet\_identity\_d\]).
First, consider the left-hand side of (\[eqn:homomorphism\_identity\_1\]). By precomposing the middle two inputs with $V_{\alpha\beta}'$, we can convert that tensor into the following one. $$\label{eqn:homomorphism_identity_2}
\begin{tikzpicture}
\draw (0,2) node (in_a2) {};
\draw (0,1) node (in_b1) {};
\draw (0,-1) node (in_a1) {};
\draw (0,-2) node (in_b2) {};
\draw (.8,-1) node (Sa1) {$S_\alpha$};
\draw (.8,1) node (Sb1) {$S_\beta$};
\draw (4,2) node (Pca2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (4,-2) node (Pbc2) {$\langle-\rangle_{\beta\kappa}$};
\draw (3,1) node (Sb2) {$S_\beta$};
\draw (4.2,1) node (Pbc1) {$\langle-\rangle_{\beta\kappa}$};
\draw (4.2,-1) node (Pca1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (5.3,1.5) node (Dc1) {$\Delta_\kappa$};
\draw (5.3,-1.5) node (Dc2) {$\Delta_\kappa$};
\draw (6,0) node (Dc3) {$\Delta_\kappa$};
\draw (6.8,0) node (out_1) {};
\draw[->] (in_a1)--(Sa1);
\draw[->] (in_b1)--(Sb1);
\draw[->] (in_a2)--(Pca2);
\draw[->] (Sa1)--(Da1);
\draw[->] (Sb1)--(Db1);
\draw[->] (in_b2)--(Pbc2);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-70,in=90,looseness=1] (Pab);
\draw[->] (Da1)--(Pca1);
\draw[->] (Db1) to [out=-90,in=210,looseness=1] (Sb2);
\draw[->] (Sb2)--(Pbc1);
\draw[->] (Dc1) to [out=200,in=-20,looseness=1] (Pca2);
\draw[->] (Dc1) to [out=180,in=90,looseness=1] (Pbc1);
\draw[->] (Dc2) to [out=180,in=-90,looseness=1] (Pca1);
\draw[->] (Dc2) to [out=160,in=20,looseness=1] (Pbc2);
\draw[->] (Dc3) to [out=200,in=-110,looseness=1] (Dc1);
\draw[->] (Dc3) to [out=160,in=110,looseness=1] (Dc2);
\draw[->] (out_1)--(Dc3);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-.5,0) node (=1) {$=$};
\draw (0,2) node (in_a2) {};
\draw (0,1) node (in_b1) {};
\draw (0,-1) node (in_a1) {};
\draw (0,-2) node (in_b2) {};
\draw (.8,-1) node (Sa1) {$S_\alpha$};
\draw (.8,1) node (Sb1) {$S_\beta$};
\draw (4,2) node (Pca2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (4,-2) node (Pbc2) {$\langle-\rangle_{\beta\kappa}$};
\draw (3,1) node (Sb2) {$S_\beta$};
\draw (4.2,1) node (Pbc1) {$\langle-\rangle_{\beta\kappa}$};
\draw (4.2,-1) node (Pca1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4.6,0) node (Dc1) {$\Delta_\kappa$};
\draw (6,0) node (Dc2) {$\Delta_\kappa$};
\draw (6.8,0) node (out_1) {};
\draw[->] (in_a1)--(Sa1);
\draw[->] (in_b1)--(Sb1);
\draw[->] (in_a2)--(Pca2);
\draw[->] (Sa1)--(Da1);
\draw[->] (Sb1)--(Db1);
\draw[->] (in_b2)--(Pbc2);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-70,in=90,looseness=1] (Pab);
\draw[->] (Da1)--(Pca1);
\draw[->] (Db1) to [out=-90,in=210,looseness=1] (Sb2);
\draw[->] (Sb2)--(Pbc1);
\draw[->] (Dc2) to [out=200,in=0,looseness=1] (Pca2);
\draw[->] (Dc1) to [out=190,in=-110,looseness=1] (Pbc1);
\draw[->] (Dc1) to [out=170,in=110,looseness=1] (Pca1);
\draw[->] (Dc2) to [out=160,in=0,looseness=1] (Pbc2);
\draw[->] (Dc2)--(Dc1);
\draw[->] (out_1)--(Dc2);
\end{tikzpicture}$$ Likewise, consider the right-hand side of (\[eqn:homomorphism\_identity\_1\]). By similarly precomposing the middle two inputs with $V_{\alpha\beta}'$ and applying (\[eqn:TUV\_identity\_1\]) from Lemma \[lem:UTV\_identity\], we acquire the following tensor. $$\label{eqn:homomorphism_identity_3}
\begin{tikzpicture}
\draw (0,2) node (in_a2) {};
\draw (0,1) node (in_b1) {};
\draw (0,-1) node (in_a1) {};
\draw (0,-2) node (in_b2) {};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (3.5,.5) node (Sa3) {$S_\alpha$};
\draw (4,2) node (Ma2) {$M_\alpha$};
\draw (4,-2) node (Mb2) {$M_\beta$};
\draw (5,1) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (5,-1) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (6,0) node (Dc) {$\Delta_\kappa$};
\draw (6.8,0) node (out1) {};
\draw[->] (in_b2)--(Mb2);
\draw[->] (in_a1)--(Da1);
\draw[->] (in_b1)--(Db1);
\draw[->] (in_a2) to [out=0,in=200,looseness=1] (Ma2);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-50,in=70,looseness=1] (Pab);
\draw[->] (Db1) to [out=-70,in=110,looseness=1] (Mb2);
\draw[->] (Da1)--(Sa3);
\draw[->] (Sa3) to [out=90,in=180,looseness=1] (Ma2);
\draw[->] (Mb2)--(Pbc);
\draw[->] (Ma2)--(Pca);
\draw[->] (Dc) to [out=160,in=90,looseness=1] (Pbc);
\draw[->] (Dc) to [out=200,in=-90,looseness=1] (Pca);
\draw[->] (out_1)--(Dc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (.5,0) node (=1) {$=$};
\draw (1,2) node (in_a2) {};
\draw (1,1) node (in_b1) {};
\draw (1,-1) node (in_a1) {};
\draw (1,-2) node (in_b2) {};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (3,-1) node (Sa3) {$S_\alpha$};
\draw (4,2) node (Pca2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4.5,1) node (Pbc1) {$\langle-\rangle_{\beta\kappa}$};
\draw (4.5,-1) node (Pca1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4,-2) node (Pbc2) {$\langle-\rangle_{\beta\kappa}$};
\draw (5.5,0) node (D1) {$\Delta_\kappa$};
\draw (7,0) node (D2) {$\Delta_\kappa$};
\draw (7.8,0) node (out_1) {};
\draw[->] (in_a2)--(Pca2);
\draw[->] (in_b1)--(Db1);
\draw[->] (in_a1)--(Da1);
\draw[->] (in_b2)--(Pbc2);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-70,in=90,looseness=1] (Pab);
\draw[->] (Db1) to [out=-90,in=180,looseness=1] (Pbc1);
\draw[->] (Da1)--(Sa3);
\draw[->] (Sa3)--(Pca1);
\draw[->] (D1)--(Pbc1);
\draw[->] (D1)--(Pca1);
\draw[->] (D2) to [out=270,in=0,looseness=1] (Pca2);
\draw[->] (D2) to [out=90,in=0,looseness=1] (Pbc2);
\draw[->] (D2)--(D1);
\draw[->] (out_1)--(D2);
\end{tikzpicture}$$ Examining the middle portions of (\[eqn:homomorphism\_identity\_2\]) and (\[eqn:homomorphism\_identity\_3\]), we find that the equality (\[eqn:homomorphism\_identity\_1\]) is equivalent to the identity $$\label{eqn:homomorphism_identity_4}
\begin{tikzpicture}
\draw (0,1) node (in_b1) {};
\draw (0,-1) node (in_a1) {};
\draw (.8,-1) node (Sa1) {$S_\alpha$};
\draw (.8,1) node (Sb1) {$S_\beta$};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (3,1) node (Sb2) {$S_\beta$};
\draw (4.2,1) node (Pbc1) {$\langle-\rangle_{\beta\kappa}$};
\draw (4.2,-1) node (Pca1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4.6,0) node (Dc1) {$\Delta_\kappa$};
\draw (5.8,0) node (out_1) {};
\draw[->] (in_a1)--(Sa1);
\draw[->] (in_b1)--(Sb1);
\draw[->] (Sa1)--(Da1);
\draw[->] (Sb1)--(Db1);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-70,in=90,looseness=1] (Pab);
\draw[->] (Da1)--(Pca1);
\draw[->] (Db1) to [out=-90,in=210,looseness=1] (Sb2);
\draw[->] (Sb2)--(Pbc1);
\draw[->] (Dc1) to [out=190,in=-110,looseness=1] (Pbc1);
\draw[->] (Dc1) to [out=170,in=110,looseness=1] (Pca1);
\draw[->] (out_1)--(Dc1);
\end{tikzpicture}
\begin{tikzpicture}
\draw (.5,0) node (=1) {$=$};
\draw (1,1) node (in_b1) {};
\draw (1,-1) node (in_a1) {};
\draw (2,-1) node (Da1) {$\Delta_\alpha$};
\draw (2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,1) node (Db1) {$\Delta_\beta$};
\draw (3,-1) node (Sa3) {$S_\alpha$};
\draw (4.5,1) node (Pbc1) {$\langle-\rangle_{\beta\kappa}$};
\draw (4.5,-1) node (Pca1) {$\langle-\rangle_{\kappa\alpha}$};
\draw (5.5,0) node (D1) {$\Delta_\kappa$};
\draw (6.8,0) node (out_1) {};
\draw[->] (in_b1)--(Db1);
\draw[->] (in_a1)--(Da1);
\draw[->] (Da1)--(Pab);
\draw[->] (Db1) to [out=-70,in=90,looseness=1] (Pab);
\draw[->] (Db1) to [out=-90,in=180,looseness=1] (Pbc1);
\draw[->] (Da1)--(Sa3);
\draw[->] (Sa3)--(Pca1);
\draw[->] (D1)--(Pbc1);
\draw[->] (D1)--(Pca1);
\draw[->] (out_1)--(D1);
\end{tikzpicture}$$ By applying the anti-homomorphism property of the antipodes $S_\alpha,S_\beta$ and $S_\kappa$ at all of the coproduct nodes in (\[eqn:homomorphism\_identity\_4\]) that have twisted output edges, we can convert that identity into the following one. $$\label{eqn:homomorphism_identity_5}
\begin{tikzpicture}
\draw (-1,-3) node (ia1) {};
\draw (-1,-0) node (ib1) {};
\draw (5,-1.5) node (ic1) {};
\draw (0,-3) node (Sa1) {$S_\alpha$};
\draw (4.2,-1.5) node (Sc1) {$S_\kappa$};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,-0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,-0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2,-3) node (Sa2) {$S_\alpha$};
\draw (2,-0) node (Sb2) {$S_\beta$};
\draw (1,-2) node (Sa3) {$S_\alpha$};
\draw[->] (ia1)--(Sa1);
\draw[->] (ic1)--(Sc1);
\draw[->] (Sa1)--(Da);
\draw[->] (ib1)--(Db);
\draw[->] (Sc1)--(Dc);
\draw[->] (Da)--(Sa3);
\draw[->] (Sa3)--(Pab);
\draw[->] (Da)--(Sa2);
\draw[->] (Sa2)--(Pca);
\draw[->] (Db)--(Sb2);
\draw[->] (Sb2)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-1.5,-1.5) node (=1) {$=$};
\draw (-1,-3) node (ia1) {};
\draw (-1,0) node (ib1) {};
\draw (4,-1.5) node (ic1) {};
\draw (0,0) node (Sb1) {$S_\beta$};
\draw (1,-3) node (Da) {$\Delta_\alpha$};
\draw (1,-0) node (Db) {$\Delta_\beta$};
\draw (3.2,-1.5) node (Dc) {$\Delta_\kappa$};
\draw (3.2,-3) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (1,-1) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (3.2,-0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2,-3) node (Sa2) {$S_\alpha$};
\draw (2,-0) node (Sb2) {$S_\beta$};
\draw (1,-2) node (Sa3) {$S_\alpha$};
\draw[->] (ia1)--(Da);
\draw[->] (ib1)--(Sb1);
\draw[->] (Sb1)--(Db);
\draw[->] (ic1)--(Dc);
\draw[->] (Da)--(Sa3);
\draw[->] (Sa3)--(Pab);
\draw[->] (Da)--(Sa2);
\draw[->] (Sa2)--(Pca);
\draw[->] (Db)--(Sb2);
\draw[->] (Sb2)--(Pbc);
\draw[->] (Db)--(Pab);
\draw[->] (Dc)--(Pca);
\draw[->] (Dc)--(Pbc);
\end{tikzpicture}$$ This identity, (\[eqn:homomorphism\_identity\_5\]), is evidently equivalent to (\[eqn:symmetric\_triplet\_identity\_d\]), modulo composition at the inputs with some antipodes and commutation of some antipodes across pairings. Thus we have shown that $\Phi^\kappa_{\alpha\beta}$ intertwines multiplication if and only if the identity (\[eqn:symmetric\_triplet\_identity\_d\]), and thus (d), is satisfied. This concludes the proof that $(b) \iff (d)$.
*$(c) \iff (d)$.* We want to show that (\[eqn:symmetric\_triplet\_identity\_c\]) and (\[eqn:symmetric\_triplet\_identity\_d\]) are equivalent. It is convenient for us to rewrite these two identities in terms of the tensors $T_{\alpha\beta}$ and $T^{-1}_{\alpha\beta}$, as well as the antipode tensors. These identities are, respectively $$\label{eqn:symmetric_triplet_identity_c_using_T}
\begin{tikzpicture}
\draw (0,-.5) node (in_a) {};
\draw (0,.5) node (in_b) {};
\draw (1,0) node (Tab) {$T_{\alpha\beta}$};
\draw (2.5,.5) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2.5,-.5) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4,0) node (Dc) {$\Delta_\kappa$};
\draw (4.8,0) node (in_c) {};
\draw[->] (in_a)--(Tab);
\draw[->] (in_b)--(Tab);
\draw[->] (Tab)--(Pbc);
\draw[->] (Tab)--(Pca);
\draw[->] (Dc)--(Pbc);
\draw[->] (Dc)--(Pca);
\draw[->] (in_c)--(Dc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-1.5,0) node (=1) {$=$};
\draw (-1,-.5) node (in_a) {};
\draw (-1,.5) node (in_b) {};
\draw (0,-.5) node (Sa) {$S_\alpha$};
\draw (0,.5) node (Sb) {$S_\beta$};
\draw (1,0) node (Tab) {$T_{\alpha\beta}$};
\draw (2.5,.5) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (2.5,-.5) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (4,0) node (Dc) {$\Delta_\kappa$};
\draw (5,0) node (Sc) {$S_\kappa$};
\draw (5.8,0) node (in_c) {};
\draw[->] (in_a)--(Sa);
\draw[->] (in_b)--(Sb);
\draw[->] (Sa)--(Tab);
\draw[->] (Sb)--(Tab);
\draw[->] (Tab)--(Pbc);
\draw[->] (Tab)--(Pca);
\draw[->] (Dc)--(Pbc);
\draw[->] (Dc)--(Pca);
\draw[->] (Sc)--(Dc);
\draw[->] (in_c)--(Sc);
\end{tikzpicture}$$ .05cm $$\label{eqn:symmetric_triplet_identity_d_using_T}
\begin{tikzpicture}
\draw (0,-.5) node (in_a) {};
\draw (0,.5) node (in_b) {};
\draw (1,0) node (Tab_inv) {$T^{-1}_{\alpha\beta}$};
\draw (2,-.5) node (Sa1) {$S_\alpha$};
\draw (2,.5) node (Sb1) {$S_\beta$};
\draw (3.5,.5) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (3.5,-.5) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (5,0) node (Dc) {$\Delta_\kappa$};
\draw (5.8,0) node (in_c) {};
\draw[->] (in_a)--(Tab_inv);
\draw[->] (in_b)--(Tab_inv);
\draw[->] (Tab_inv)--(Sb1);
\draw[->] (Sb1)--(Pbc);
\draw[->] (Tab_inv)--(Sa1);
\draw[->] (Sa1)--(Pca);
\draw[->] (Dc)--(Pbc);
\draw[->] (Dc)--(Pca);
\draw[->] (in_c)--(Dc);
\end{tikzpicture}
\begin{tikzpicture}
\draw (-1.5,0) node (=1) {$=$};
\draw (-1,-.5) node (in_a) {};
\draw (-1,.5) node (in_b) {};
\draw (0,-.5) node (Sa) {$S_\alpha$};
\draw (0,.5) node (Sb) {$S_\beta$};
\draw (1,0) node (Tab_inv) {$T^{-1}_{\alpha\beta}$};
\draw (2,-.5) node (Sa1) {$S_\alpha$};
\draw (2,.5) node (Sb1) {$S_\beta$};
\draw (3.5,.5) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (3.5,-.5) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (5,0) node (Dc) {$\Delta_\kappa$};
\draw (6,0) node (Sc) {$S_\kappa$};
\draw (6.8,0) node (in_c) {};
\draw[->] (in_a)--(Sa);
\draw[->] (in_b)--(Sb);
\draw[->] (Sa)--(Tab_inv);
\draw[->] (Sb)--(Tab_inv);
\draw[->] (Tab_inv)--(Sb1);
\draw[->] (Sb1)--(Pbc);
\draw[->] (Tab_inv)--(Sa1);
\draw[->] (Sa1)--(Pca);
\draw[->] (Dc)--(Pbc);
\draw[->] (Dc)--(Pca);
\draw[->] (Sc)--(Dc);
\draw[->] (in_c)--(Sc);
\end{tikzpicture}$$ With the above rewriting in mind, we consider the following tensor diagram. $$\label{eqn:main_triplet_lemma_cancelling_tensor}
\begin{tikzpicture}
\draw (0,.5) node (in_b) {};
\draw (0,-.5) node (in_a) {};
\draw (1,0) node (Tab_1) {$T^{-1}_{\alpha\beta}$};
\draw (2,.5) node (Sb) {$S_\beta$};
\draw (2,-.5) node (Sa) {$S_\alpha$};
\draw (3,0) node (Tab_2) {$T^{-1}_{\alpha\beta}$};
\draw (4,.5) node (out_b) {};
\draw (4,-.5) node (out_a) {};
\draw[->] (in_b)--(Tab_1);
\draw[->] (in_a)--(Tab_1);
\draw[->] (Tab_1)--(Sb);
\draw[->] (Tab_1)--(Sa);
\draw[->] (Sb)--(Tab_2);
\draw[->] (Sa)--(Tab_2);
\draw[->] (Tab_2)--(out_b);
\draw[->] (Tab_2)--(out_a);
\end{tikzpicture}$$ Note that this tensor is invertible. This follows by expressing it in terms of $T_{\alpha\beta}^{-1}$ (see Notation \[not:TUV\_notation\]) and the antipodes $S_\alpha$ and $S_\beta$.
Now observe that composing the tensor (\[eqn:main\_triplet\_lemma\_cancelling\_tensor\]) on the left with the left-most tensor of (\[eqn:symmetric\_triplet\_identity\_c\_using\_T\]) produces (using involutarity and the identities in Notation \[not:TUV\_notation\]) the left-most tensor of (\[eqn:symmetric\_triplet\_identity\_d\_using\_T\]). Thus we must show that the right-most tensor of (\[eqn:symmetric\_triplet\_identity\_c\_using\_T\]) becomes the right-most tensor of (\[eqn:symmetric\_triplet\_identity\_d\_using\_T\]). In particular, it suffices to check that $$\label{eqn:main_triplet_lemma_cancelling_tensor_final_id}
\begin{tikzpicture}
\draw (0,.5) node (in_b) {};
\draw (0,-.5) node (in_a) {};
\draw (1,0) node (Tab_1) {$T^{-1}_{\alpha\beta}$};
\draw (2,.5) node (Sb) {$S_\beta$};
\draw (2,-.5) node (Sa) {$S_\alpha$};
\draw (3,0) node (Tab_2) {$T^{-1}_{\alpha\beta}$};
\draw (4,.5) node (Sb2) {$S_\beta$};
\draw (4,-.5) node (Sa2) {$S_\alpha$};
\draw (5,0) node (Tab_3) {$T_{\alpha\beta}$};
\draw (6,.5) node (out_b) {};
\draw (6,-.5) node (out_a) {};
\draw[->] (in_b)--(Tab_1);
\draw[->] (in_a)--(Tab_1);
\draw[->] (Tab_1)--(Sb);
\draw[->] (Tab_1)--(Sa);
\draw[->] (Sb)--(Tab_2);
\draw[->] (Sa)--(Tab_2);
\draw[->] (Tab_2)--(Sb2);
\draw[->] (Tab_2)--(Sa2);
\draw[->] (Sb2)--(Tab_3);
\draw[->] (Sa2)--(Tab_3);
\draw[->] (Tab_3)--(out_b);
\draw[->] (Tab_3)--(out_a);
\end{tikzpicture}
\begin{tikzpicture}
\draw (.5,0) node (=1) {$=$};
\draw (1,.5) node (in_b) {};
\draw (1,-.5) node (in_a) {};
\draw (2,.5) node (Sb) {$S_\beta$};
\draw (2,-.5) node (Sa) {$S_\alpha$};
\draw (3,0) node (Tab_2) {$T^{-1}_{\alpha\beta}$};
\draw (4,.5) node (Sb2) {$S_\beta$};
\draw (4,-.5) node (Sa2) {$S_\alpha$};
\draw (5,.5) node (out_b) {};
\draw (5,-.5) node (out_a) {};
\draw[->] (in_b)--(Sb);
\draw[->] (in_a)--(Sa);
\draw[->] (Sb)--(Tab_2);
\draw[->] (Sa)--(Tab_2);
\draw[->] (Tab_2)--(Sb2);
\draw[->] (Tab_2)--(Sa2);
\draw[->] (Sb2)--(out_b);
\draw[->] (Sa2)--(out_a);
\end{tikzpicture}$$ This identity is the second identity (\[eqn:TUV\_identity\_2\]) in Lemma \[lem:UTV\_identity\].
An immediate corollary of the fundamental lemma for triplets is the following, which provides a rich source of examples.
\[cor:map\_version\_of\_triplet\] Let $(H_\alpha,H_\beta,(-))$ be a Hopf doublet and let $\pi:D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H^*_\kappa$ be a Hopf algebra map to a third Hopf algebra. Denote by $\iota_{\alpha}: H_\alpha \to D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}})$ the embedding sending $v$ to $v \otimes 1$ and similarly by $\iota_\beta: H_\beta \to D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) $ sending $w$ to $1 \otimes w$.
Then $\mathcal{H} = (H_\alpha, H_\beta, H_\kappa;\langle-\rangle)$ has the structure of a Hopf triplet with the pairings $$\begin{tikzpicture}
\draw (.2,0) node (Pab) {$\langle-\rangle_{\alpha\beta}$};
\draw (1,0) node (=ab) {$:=$};
\draw (1.5,0) node (i_a1) {};
\draw (4,0) node (i_b1) {};
\draw (2.75,0) node (Qab) {$(-)$};
\draw[->] (i_a1)--(Qab);
\draw[->] (i_b1)--(Qab);
\draw (5.2,0) node (Pbc) {$\langle-\rangle_{\beta\kappa}$};
\draw (6,0) node (=bc) {$:=$};
\draw (6.3,0) node (i_b2) {};
\draw (9.2,0) node (i_c2) {};
\draw (7.2,0) node (Ibc) {$\iota_\beta$};
\draw (8.3,0) node (Qbc) {$\pi$};
\draw[->] (i_b2)--(Ibc);
\draw[->] (Ibc)--(Qbc);
\draw[->] (i_c2)--(Qbc);
\draw (10.2,0) node (Pca) {$\langle-\rangle_{\kappa\alpha}$};
\draw (11,0) node (=ca) {$:=$};
\draw (14.2,0) node (i_a3) {};
\draw (11.3,0) node (i_c3) {};
\draw (13.2,0) node (Ica) {$\iota_\alpha$};
\draw (12.3,0) node (Qca) {$\pi$};
\draw[->] (i_a3)--(Ica);
\draw[->] (Ica)--(Qca);
\draw[->] (i_c3)--(Qca);
\end{tikzpicture}$$
We must check Definition \[def:hopf\_triplet\](a) and (b). Clearly $(H_\alpha,H_\beta,\langle-\rangle_{\alpha\beta})$ is a Hopf doublet. The brackets $\langle-\rangle_{\beta\kappa}$ and $\langle-\rangle_{\kappa\alpha}$ as defined above provide Hopf algebra maps $$H_\alpha^{{\operatorname}{op}} \to H_\kappa^* \qquad H_\beta^{{\operatorname}{cop}} \to H_\kappa^*$$ Dualizing the map on the left and cop-ing the map on the right, we find that the pairings are equivalently Hopf algebra maps $$H_\kappa \to (H^{{\operatorname}{op}}_\alpha)^* = H^{*,{\operatorname}{cop}}_\alpha \qquad H_\beta \to H_\kappa^{*,{\operatorname}{cop}}$$ Thus every pair of indices determines a Hopf doublet, and Definition \[def:hopf\_triplet\](a) is satisfied. Definition \[def:hopf\_triplet\](b) follows from the fact that the map $D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H^*_\kappa$ defined as Definition \[def:hopf\_triplet\](b) agrees with $\pi$, which is a Hopf algebra map by hypothesis.
To conclude this part, we provide a few more specific examples of Hopf triplets.
\[ex:basic\_examples\_of\_triplets\] Here are some basic examples of Hopf triplets.
1. (Tautological) Let $(H_\alpha,H_\beta,(-))$ be a finite-dimensional, involutory Hopf doublet. Then we may form a tautological triplet by letting $H_\kappa := D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}})^*$. We can define a map $$\pi:D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H_\kappa^* = D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}})$$ to be the identity. Corollary \[cor:map\_version\_of\_triplet\] then gives a Hopf triplet structure on these three Hopf algebras using $\pi$.
2. (Quasi-triangular) Let $(H,R)$ be a quasi-triangular Hopf algebra $H$ equipped with an R-matrix $R \in H \otimes H$. Then the following linear map is a Hopf algebra morphism (c.f. Radford [@radford2011hopfalgebras]). $$\begin{tikzpicture}
\draw (-5,.5) node (label) {$\pi:D(H) = D(H^{*,{\operatorname}{cop}},H) \to H \quad\text{given by}$};
\draw (0,0) node (i1) {};
\draw (0,1) node (i2) {};
\draw (1,.5) node [draw,rectangle] (R) {$R$};
\draw (2,0) node (M) {$M$};
\draw (3,.2) node (o) {};
\draw[->] (R) to [out=20,in=20,looseness=1.5] (i2);
\draw[->] (i1)--(M);
\draw[->] (R) to [out=-20,in=150,looseness=1.5] (M);
\draw[->] (M)--(o);
\end{tikzpicture}$$ This map may be alternativly written (in the less diagrammatic notation of [@radford2011hopfalgebras]) as $\pi = M \circ (f_R \otimes \text{Id}) $ with $f_R(q):= (q \otimes Id)R : H^{*, {\text{cop}}} \to H$. Now consider the following Hopf algebras and pairing derived from $H$. $$H_\alpha := (H^{{\operatorname}{cop}})^{*,{\operatorname}{cop}} \quad H_\beta := H^{{\operatorname}{cop}} \quad\text{and}\quad H_\kappa := H^*$$ $$(-):H_\alpha \otimes H_\beta \to k \qquad (a,b) = b(a)$$ We see that $D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}})$ is simply the usual double $D(H)$ of $H$ and $\pi$ is a Hopf algebra map $D(H_\alpha^{{\operatorname}{op}},H_\beta^{{\operatorname}{cop}}) \to H_\kappa^{*}$.
We thus acquire a Hopf triplet $\mathcal{H}_H = (H_\alpha,H_\beta,H_\kappa,\langle-\rangle)$ by Corollary \[cor:map\_version\_of\_triplet\], which we call the *quasi-triangular triplet* associated to $(H,R)$. Note that the pairings $\langle-\rangle_{\alpha\beta}$ and $\langle-\rangle_{\beta\kappa}$ are simply the standard dual pairings, while the pairing $\langle-\rangle_{\kappa\alpha}$ agrees with $R$ interpreted as a map $H^* \otimes H^* \to k$. In terms of the notation of [@radford2011hopfalgebras], the $\kappa\alpha$ pairing sends a pair $p,q \in H^*$ to $\langle p,q \rangle_{\kappa\alpha} = p(f_R(q)) = (q \otimes p)R$.
Hopf doublets $(H_\alpha, H_\beta, \langle - \rangle)$ more broadly can sometimes be equipped with a quasi-triangular structure, given by a generalized $R$-matrix $R \in H_\alpha^{*,\text{cop}} \otimes H_\beta$ satisfying suitably modified axioms (now involving the pairing $\langle - \rangle$). Analogously, $\pi = M \circ (f_R \otimes \text{Id}) : D(H_\alpha, H_\beta) \to H_\beta$ is a Hopf algebra morphism, and this morphism can be used to construct a triplet as in Example \[ex:basic\_examples\_of\_triplets\](b). However, we will not use any Hopf triplets arising in this way in the current paper.
Trisection Diagrams {#subsec:trisections}
-------------------
Now we review the theory of trisections and trisection diagrams. Trisections diagrams for 4-manifolds were introduced in [@gk2016] (also see [@agk2018grouptrisections] and [@cgpc2018relativetrisections]).
Like Kirby diagrams, which are essentially handlebody diagrams, a trisection specifies some set of instructions for constructing a 4-manifold. However, trisections are more directly similar to Heegaard diagrams in the sense that the data for the 4-manifold is specified by a surface along with some curves on that surface.
\[def:trisection\_diagram\] An *oriented trisection diagram* $T$ is a triple $(\Sigma,\alpha,\beta,\kappa)$ consisting of the following data.
- (Surface) A closed, oriented 2-manifold $\Sigma$ of genus $g$.
- (Curves) Three sets of $g$ non-separating, embedded curves $\{\alpha_i\},\{\beta_i\}$ and $\{\kappa_i\}$ on $\Sigma$ such that:
- All curves from a single set are disjoint, i.e. $\alpha_i \cap \alpha_j$ is empty when $i \neq j$.
- Any pair of the three curve sets form a Heegaard diagram for $\#_{i=1}^k S^1 \times S^2$, for some $k$ independent of which two curve sets are used. By convention, we say that $\#_{i=1}^k S^1 \times S^2 = S^3$ in the case where $k = 0$.
![A simple trisection diagram. On the left, we have a visualization of the trisection on a surface in ${{\mathbb R}}^3$ with the relevant curves. On the right, we have a Heegaard diagram type visualization of the same trisection.[]{data-label="fig:example_trisection_diagram"}](trisection_pictures){width="\textwidth"}
\[def:basic\_trisection\_constructions\] We adopt the following terminology for the most basic topological operations on trisections.
- (Diffeomorphism) A *diffeomorphism* $\varphi:T \simeq T'$ of trisections is a diffeomorphism $\varphi:\Sigma \simeq \Sigma'$ of the underlying surfaces that intertwines the curve sets, i.e. $\varphi(\alpha) = \alpha'$, $\varphi(\beta) = \beta'$ and $\varphi(\kappa) = \kappa'$.
- (Isotopy) An *isotopy* between trisections $T$ and $T'$ with the same underlying surface $\Sigma = \Sigma'$ is simply an isotopy of the corresponding curve sets $\alpha^t, \beta^t$ and $\kappa^t$ so that $\alpha^0 = \alpha$, $\alpha^1 = \alpha'$, $\beta^0 = \beta$ etc.
- (Connect Sum) A *connect sum* $T \# T'$ of two trisections $T$ and $T'$ (along disks $D \subset \Sigma$ and $D' \subset \Sigma'$ disjoint from the $\alpha,\beta$ and $\kappa$ curves) is defined as follows. The surface of $T \# T'$ is given by a connect sum $\Sigma \# \Sigma'$ along the boundary created by removing $D$ from $\Sigma$ and $D'$ from $\Sigma'$. Note that the diffeomorphism type of the result of this operation depends on the choice of $D$ and $D'$.
- (Orientation Reversal) The *orientation reversed* trisection $\overline{T}$ of a trisection $T$ has underlying surface given by $\overline{\Sigma}$, i.e. $\Sigma$ with the opposite orientation, and the same curve sets $\alpha$,$\beta$ and $\kappa$.
General isotopies, even of immersed curves in surfaces, can be quite complicated. In order to prove invariance of our invariants in §\[sec:trisection\_kuperberg\_invt\], we need to work with a more restricted, combinatorial class of isotopies.
\[def:two\_three\_point\_moves\] Let $T$ be a trisection.
A *two-point move* on $T$ is a new trisection $T'$ acquired as so. First, identify a disk $D \subset \Sigma$ possessing a diffeomorphism of pairs $$(D,D \cap (\alpha \cup \beta \cup \kappa)) \simeq (D_+,\gamma_+ \cup \eta_+) \text{ or }(D_-,\gamma_- \cup \eta_-)$$ Then replace $(D_+,\gamma_+ \cup \eta_+) \subset \Sigma$ with $(D_-,\gamma_- \cup \eta_-)$ in the $+$ case, or alternatively replace $(D_-,\gamma_- \cup \eta_-) \subset \Sigma$ with $(D_+,\gamma_+ \cup \eta_+)$ in the $-$ case. Here $(D_+,\gamma_+ \cup \eta_+)$ and $(D_-,\gamma_- \cup \eta_-)$ are given by the following pictures. $$\includegraphics[width=.6\textwidth]{two_point_move.png}$$
A *three-point move* on $T$ is a new trisection $T'$ acquired as so. First identify a disk $D \subset \Sigma$ possessing a diffeomorphism of pairs $$(D,D \cap (\alpha \cup \beta \cup \kappa)) \simeq (D_+,\gamma_+ \cup \eta_+ \cup \xi_+) \text{ or }(D_-,\gamma_- \cup \eta_- \cup \xi_-)$$ Then replace $(D_+,\gamma_+ \cup \eta_+ \cup \xi_+) \subset \Sigma$ with $(D_-,\gamma_- \cup \eta_- \cup \xi_-)$ in the $+$ case, or alternatively replace $(D_-,\gamma_- \cup \eta_- \cup \xi_-) \subset \Sigma$ with $(D_+,\gamma_+ \cup \eta_+ \cup \xi_+)$ in the $-$ case. Here $(D_+,\gamma_+ \cup \eta_+ \cup \xi_+)$ and $(D_-,\gamma_- \cup \eta_- \cup \xi_-)$ are given by the following pictures. $$\includegraphics[width=.6\textwidth]{three_point_move.png}$$
Note that the two and three point moves above are analogues of Redemeister 2 and 3 moves from knot theory. As in knot theory, we are essentially allowed to reduce to the case of such isotopies by the following lemma.
\[lem:curve\_isotopy\] Let $\Gamma$ be a closed $1$-manifold and let $\Sigma$ be a closed $2$-manifold. Let $\iota_0,\iota_1:\Gamma \to \Sigma$ be a pair of homotopic immersions of $\Gamma$ such that
- Each component $C$ of $\Gamma$ is embedded by $\iota_i$.
- The components of $\iota_i(\Gamma)$ only intersect transversely at double points.
Then $\iota_0$ and $\iota_1$ are diffeomorphic after sequence of two-point and three-point moves.
First assume that $\iota_0(\Gamma)$ and $\iota_0(\Gamma)$ both have the minimal possible number of transverse double points in their homotopy class. Any two such minimal immersions are ambiently isotopic after a sequence of three-point moves (see p. 231-232 and Lemma 3.4 of [@patterson2002loops]).
So it suffices to show that $\iota_0$ and $\iota_1$ can be isotoped to minimal immersions $\iota_0'$ and $\iota_1'$ satisfying (a) and (b) using two-point moves. This is implied immediately by Lemma 3.1 of [@hs1985], which states that if $\iota_0$ (for instance) does not minimize intersections, then there is an inner-most 2-gon (i.e. a copy of $(D_+,\gamma_+ \cup \eta_+)$ as above) on which one can perform a two-point move to decrease the self-intersections by $1$.
By applying Lemma \[lem:curve\_isotopy\] to the $\alpha$, $\beta$ and $\gamma$ curves of trisection diagrams, we acquire the following corollary.
\[lem:two\_three\_point\_moves\] Let $T$ and $T'$ be isotopic trisection diagrams. Then $T$ and $T'$ are diffeomorphic after a sequence of two-point and three-point moves.
There are three types of operations beyond diffeomorphism and isotopy that emerge in the study of trisection diagrams as presentations of $4$-manifolds. These are the following.
\[def:trisection\_moves\] (Trisection Moves) Let $T = (\Sigma,\alpha,\beta,\kappa)$ be a trisection diagram. We now describe three special operations for producing a new trisection diagram $T \rightsquigarrow T'$, collectively called *trisection moves*.
1. (Handle Slides) Given two distinct $\alpha$ curves $\alpha_0$ and $\alpha_1$ along with an arc $\gamma$ connecting $\alpha_0$ to $\alpha_1$, one may alter $T$ to a new trisection $T'$ by replacing $\alpha_0$ by the *handle-slide* $\alpha_0 \#_\gamma \alpha_1$ of $\alpha_0$ over $\alpha_1$ via $\gamma$. Here $\alpha_0 \#_\gamma \alpha_1$ is defined as so. Let $U \subset \Sigma$ be a ribbon neighborhood of $\alpha_0 \sqcup \gamma \sqcup \alpha_1$. The boundary $\partial U$ then decomposes into three closed curves: a normal push-off of $\alpha_0$, a normal push-off of $\alpha_1$ and a third piece, which is precisely the handle-slide $\alpha_0 \#_\gamma \alpha_1$.
2. (Stabilization) A *stabilization* $T'$ of $T$ is the trisection given by the connect sum $T \# T_{{\operatorname}{st}}$ where $T_{{\operatorname}{st}}$ is the genus $3$ *stabilized sphere trisection* in Figure \[fig:stabilized\_sphere\_trisection\].
3. (De-Stabilization) A *destabilization* $T'$ of a trisection $T$ which is diffeomorphic $T \simeq T' \# T_{{\operatorname}{st}}$ with a stabilization is simply the summand $T'$.
![The standard stabilized sphere trisection. As in Figure \[fig:example\_trisection\_diagram\], we include an embedded surface depiction and a Heegaard diagram type depiction.[]{data-label="fig:stabilized_sphere_trisection"}](standard_stabilized_sphere){width="\textwidth"}
The significance of trisection diagrams comes from their usefulness for specifying a particular diffeomorphism class of 4-manifolds, via the following construction.
\[def:4\_manifold\_of\_trisection\] (4-Manifold of a Trisection) Let $T = (\Sigma,\alpha,\beta,\kappa)$ be a trisection diagram. The *4-manifold $X(T)$ of the trisection $T$* is the oriented 4-manifold constructed by the procedure below.
To construct $X(T)$, we proceed as follows. Let $\Sigma \times D^2$ be the surface $\Sigma$ thickened by a 2-disk. For each $* \in \{\alpha,\beta,\kappa\}$, let $H_*$ denote a genus $g$ handlebody (i.e., the boundary sum $H_* \simeq \natural_{i=1}^g S^1 \times D^2$) and let $H_* \times D^1$ denote that handlebody thickened by a 1-disk. Divide the boundary $\partial D^2$ into a union $D^1_\alpha \cup D^1_\beta \cup D^1_\kappa$ of three cyclically ordered intervals $D^1_*$ meeting at their boundaries (with an explicit oriented diffeomorphism given by $\iota_*:D^1 \simeq D_*$).
For each $* \in \alpha,\beta,\kappa$, the $*$-curves in $\Sigma$ determine a unique oriented diffeomorphism $\varphi_*:\partial H_* \simeq \Sigma$ up to isotopy sending the belt spheres $\bigsqcup_{i=1}^g \{1/2\} \times S^1\subset \natural_{i=1}^g S^1 \times D^2 \simeq H_*$ of $H_*$ to the $*$-curves. We thus get a unique oriented embedding $\varphi_\alpha \times \iota_\alpha:\partial H_\alpha \times D^1 \to \Sigma \times D^1_*$. If we glue $\Sigma \times D^2$ to $H_\alpha \times D^1$ via each of the maps $\varphi_\alpha$, the boundary $\partial W(T)$ of the resulting glued space $W(T)$ decomposes into 3 connected pieces $\partial_{**} W(T)$ with $** \in \{\alpha\beta,\beta\kappa,\kappa\alpha\}$, where each piece admits an oriented diffeomorphism $\phi_{**}:\partial_{**}W(T) \simeq \partial(\natural_{i=1}^k S^1 \times D^3)$.
To get $X(T)$, we simply glue $W(T)$ and three copies of $\natural_{i=1}^k S^1 \times D^3$ along their boundaries via the maps $\phi_{**}$. The orientation of $X(T)$ is induced by the product orientation on $\Sigma \times D^2$, where we take the standard orientation on $D^2$.
A 4-manifold $X$ along with a diffeomorphism $X \simeq X(T)$ is said to be *trisected*.
[@gk2016] The 4-manifold $X(T)$ of a trisection diagram $T$ is independent of the choices made in Definition \[def:4\_manifold\_of\_trisection\] up to oriented diffeomorphism.
The construction $T \to X(T)$ has the following naturality properties with respect to the operations in Definition \[def:basic\_trisection\_constructions\] and Definition \[def:trisection\_moves\]. Let $T$ and $T'$ be a pair of trisection diagrams.
1. (Diffeomorphism) If $T$ and $T'$ are oriented diffeomorphic, then $X(T) \simeq X(T')$.
2. (Trisection Moves) If $T$ and $T'$ are diffeomorphic after a sequence of trisection moves and isotopies, then $X(T) \simeq X(T')$.
3. (Connect Sum) There is an oriented diffeomorphism $X(T \# T') \simeq X(T) \# X(T')$.
4. (Orientation Reversal) There is an oriented diffeomorphism $\overline{X(T)} \simeq X(\overline{T})$.
The fundamental theorem of trisections is that any closed, oriented 4-manifold can be trisected and that this trisection is unique modulo the trisection moves.
[@gk2016] Let $X$ be a closed oriented $4$-manifold. Then:
- (Existence) $X$ admits a trisection, i.e. there exists a trisection diagram $T$ and an oriented diffeomorphism $\varphi:X \simeq X(T)$.
- (Uniqueness) Any two trisection diagrams $T$ and $T'$ of $X$ are oriented diffeomorphic after a series of trisection moves and isotopies are applied to $T$.
4-Manifold Invariants {#sec:trisection_kuperberg_invt}
=====================
In this section, we describe the construction of our family of 4-manifold invariants and demonstrate its basic properties. In §\[subsec:trisection\_bracket\], we construct an auxiliary (non-invariant) number called the trisection bracket, and prove its essential properties. In §\[subsec:main\_definition\_and\_properties\], we apply the results of the previous section to quickly define the $4$-manifold invariants of interest.
Trisection Bracket {#subsec:trisection_bracket}
------------------
We begin this section by introducing the following bracket.
\[def:trisection\_bracket\] (Trisection Bracket) Let $\mathcal{H} = (H_\alpha,H_\beta,H_\kappa,\langle - \rangle)$ be a Hopf triplet over a field $k$ and $T = (\Sigma,\alpha,\beta,\kappa)$ be a trisection diagram.
The *trisection bracket* $\langle T\rangle_{\mathcal{H}} \in k$ is defined to be the scalar specified by a particular tensor diagram, which is constructed according to the following procedure.
- Begin by setting $\langle T\rangle_{\mathcal{H}}$ to be the empty tensor diagram. Fix arbitrary orientations $o_\alpha,o_\beta$ and $o_\kappa$ of the $\alpha,\beta$ and $\kappa$ curves of the trisection $T$.
- For each $\gamma \in \{\alpha,\beta,\kappa\}$ and each $\gamma$-curve $\gamma_i$, add a comultiplication node to the diagram $\langle T\rangle_{\mathcal{H}}$ as so. Let $m = m^\gamma_i$ denote the number of intersections of $\gamma_i$ with the other curves on $T$, i.e. $m = |\gamma_i \cap (\alpha \sqcup \beta \sqcup \kappa - \gamma_i)|$. Let $\{\iota^\gamma_{i,j}\}_{j=1}^m$ denote the sequence of intersection points between $\gamma$ and the other curves. We order the sequence $\{\iota^\gamma_{i,j}\}_{j=1}^m$ according to the cyclic ordering induced by the orientation $o_\gamma$ on $\gamma_i$.
In terms of the above notation, we include a comultiplication $C_\gamma \rightarrow \Delta_\gamma \rightrightarrows$ in $\langle T\rangle_{\mathcal{H}}$ from the Hopf algebra $H_\gamma$ with $1$ input (from a cotrace $C_\gamma$) and $m^\gamma_i$ outputs. We also label the outputs by the intersection points $\iota^\gamma_{i,1}, \iota^\gamma_{i,2},\dots$ in counter-clockwise cyclic order. In tensor diagram notation, we are performing the following move.$$\includegraphics[width=.2\linewidth]{alpha_curve_w_intersections.png}
\begin{tikzpicture}
\draw (0,-1.2) node (space) { };
\draw (-2.8,0) node (space) { };
\draw (-2.2,0) node (=>) {$\mapsto$};
\draw (0,0) node (D1) {$\Delta_\alpha$};
\draw (-1,0) node (C1) {$C_\alpha$};
\draw (0,-1) node (i1) {$\iota^\alpha_{i,1}$};
\draw (.8,-.8) node (i2) {$\iota^\alpha_{i,2}$};
\draw (1,0) node (i3) {$\iota^\alpha_{i,3}$};
\draw (.8,.8) node (i4) {$\iota^\alpha_{i,4}$};
\draw (0,1) node (i5) {$\iota^\alpha_{i,5}$};
\draw (-.6,.6) node (dots1) {$\dots$};
\draw[->] (C1)--(D1);
\draw[<-] (i1)--(D1);
\draw[<-] (i2)--(D1);
\draw[<-] (i3)--(D1);
\draw[<-] (i4)--(D1);
\draw[<-] (i5)--(D1);
\end{tikzpicture}$$
- On each pair of outgoing edges $\Delta_\gamma \rightarrow \iota^\gamma_{i,a}$ and $\Delta_\eta \rightarrow \iota^\eta_{j,b}$ labelled by the same geometric intersection $\iota := \iota^\gamma_{i,a} = \iota^\eta_{j,b}$, we perform the following contraction within $\langle T\rangle_{\mathcal{H}}$.
First assign a sign, denoted by ${\operatorname}{sgn}(\iota) \in \{+,-\}$, to the intersection $\iota$ according to the following rule. Let $\gamma,\eta \in \{\alpha,\beta,\kappa\}$ be the type of the intersecting curves $\gamma_i$ and $\eta_j$ as above. Relabel the curves $\gamma_i$ and $\eta_j$ in the pair so that $\gamma \prec \eta$ with respect to the cyclic ordering $\alpha \prec \beta \prec \kappa \prec \alpha$. The orientations $o_\gamma$ and $o_\eta$ induce orientations of the tangent spaces $T_\iota\gamma_i$ and $T_\iota\eta_j$ to the curves at $\iota$. This in turn induces an orientation $o_\gamma \otimes o_\eta$ on $T_\iota\Sigma = T_\iota \gamma_i \oplus T_\iota \eta_j$. On the other hand, $T_\iota\Sigma$ is oriented by a background orientation $o_\Sigma$, since $\Sigma$ is an oriented surface. The sign of $\iota$ is thus defined by the relation $o_\gamma \otimes o_\eta = {\operatorname}{sgn}(\iota) \cdot o_\Sigma$.
Pictorally, this amounts to the following sign assignments when the plane is given the standard orientation. $$\includegraphics[width=.4\linewidth]{sign_rules_intersections_pos.png}\hspace{40pt}
\includegraphics[width=.4\linewidth]{sign_rules_intersections_neg.png}$$Finally, we perform the following subtitution. If ${\operatorname}{sgn}(\iota)$ is positive, we pair the out edges $\rightarrow \iota^\gamma_{i,a}$ and $\rightarrow^\eta_{j,b}$ via a pairing node $\rightarrow \langle -\rangle_{\gamma\eta} \leftarrow$. If ${\operatorname}{sgn}(\iota)$ is negative, we pair the out edges $\rightarrow \iota^\gamma_{i,a}$ and $\rightarrow^\eta_{j,b}$ via a pairing node $\rightarrow S_\gamma \rightarrow \langle -\rangle_{\gamma\eta} \leftarrow$. Pictorally (with the bullet notation) this can be written as: $$\begin{tikzpicture}
\draw (0,0) node (D1) {$\Delta_\gamma$};
\draw (3,0) node (D2) {$\Delta_\eta$};
\draw (1,0) node (i1) {$\iota^\gamma_{i,a}$};
\draw (2,0) node (i2) {$\iota^\eta_{j,b}$};
\draw (-.6,0) node (dots1) {$\dots$};
\draw (3.6,0) node (dots2) {$\dots$};
\draw[->] (D1)--(i1);
\draw[->] (D2)--(i2);
\draw (4.5,0) node (=>) {$\mapsto$};
\draw (6,0) node (D1) {$\Delta_\gamma$};
\draw (9,0) node (D2) {$\Delta_\eta$};
\draw (7.5,0) node (P1) {$\bullet$};
\draw (5.4,0) node (dots1) {$\dots$};
\draw (9.6,0) node (dots2) {$\dots$};
\draw (11.5,0) node (label1) {if ${\operatorname}{sgn}(\iota) = +$};
\draw[->] (D1)--(P1);
\draw[->] (D2)--(P1);
\end{tikzpicture}$$$$\begin{tikzpicture}
\draw (0,0) node (D1) {$\Delta_\gamma$};
\draw (3,0) node (D2) {$\Delta_\eta$};
\draw (1,0) node (i1) {$\iota^{\gamma}_{i,a}$};
\draw (2,0) node (i2) {$\iota^{\eta}_{j,b}$};
\draw (-.6,0) node (dots1) {$\dots$};
\draw (3.6,0) node (dots2) {$\dots$};
\draw[->] (D1)--(i1);
\draw[->] (D2)--(i2);
\draw (4.5,0) node (=>) {$\mapsto$};
\draw (6,0) node (D3) {$\Delta_\gamma$};
\draw (9,0) node (D4) {$\Delta_\eta$};
\draw (8,0) node (P3) {$\bullet$};
\draw (7,0) node (S3) {$S_\gamma$};
\draw (5.4,0) node (dots3) {$\dots$};
\draw (9.6,0) node (dots4) {$\dots$};
\draw (11.5,0) node (label1) {if ${\operatorname}{sgn}(\iota) = -$};
\draw[->] (D3)--(S3);
\draw[->] (S3)--(P3);
\draw[->] (D4)--(P3);
\end{tikzpicture}$$
The tensor diagram $\langle T\rangle_{\mathcal{H}}$ acquired after performing steps (a)-(c) above will have no input or output edges by construction, and will therefore define a scalar as claimed.
The following lemma demonstrates that the bracket $\langle T\rangle_{\mathcal{H}}$ depends only on $T$ and $H$, and not on the extraneous choices made in the definition.
\[lem:bracket\_well\_defined\] The bracket $\langle T\rangle_{\mathcal{H}} \in k$ of a trisection $T$ is independent on the orientations of the $\alpha,\beta$ and $\kappa$ curves chosen in step (a) of Definition \[def:trisection\_bracket\].
Let $\gamma \in \{\alpha,\beta,\kappa\}$ and let $\gamma_i$ be a $\gamma$-curve. It suffices to show that $\langle T\rangle_{\mathcal{H}}$ is invariant under changes of choices of orientation for $\gamma_i$.
Thus let $T_+$ and $T_-$ be the trisections with curve orientations chosen to match on all $\alpha,\beta$ and $\kappa$ curves except at $\gamma_i$, where the orientations are opposite. Label the $m$ intersections of $\gamma_i$ with other curves as $\iota_1,\dots,\iota_m$ in the cyclic order determined by the $T_+$ orientation. Let $\sigma:\{1,\dots,m\} \to \{0,1\}$ be defined by $\sigma_i = 0$ if ${\operatorname}{sgn}(\iota_i) = +$ and $\sigma_i = 1$ if ${\operatorname}{sgn}(\iota_i) = -$. Then the tensor diagrams in the two cases can be written as $$\raisebox{3pt}{\includegraphics[width=.25\linewidth]{curve_orientation_a.png}}
\begin{tikzpicture}
\draw (-1.4,0) node (Sp1) { };
\draw (-.9,0) node (=>) {$\mapsto$};
\draw (0,2) node (Sp2) { };
\draw (0,0) node (C) {$C_\gamma$};
\draw (1,0) node (D) {$\Delta_\gamma$};
\draw (2.5,0) node (dots) {$\dots$};
\node at (4.3,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Id}^{\otimes m}$};
\node at (6.2,0) [draw,circle,dashed] (A) {$A$};
\draw (2.2,-1.2) node (S1) {$S_\gamma^{\sigma_1}$};
\draw (2.2,-.6) node (S2) {$S_\gamma^{\sigma_2}$};
\draw (2.2,.6) node (S3) {$S_\gamma^{\sigma_{m-1}}$};
\draw (2.2,1.2) node (S4) {$S_\gamma^{\sigma_m}$};
\draw (3.4,-1.2) node (P1) {$\bullet$};
\draw (3.4,-.6) node (P2) {$\bullet$};
\draw (3.4,.6) node (P3) {$\bullet$};
\draw (3.4,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (D) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (D)--(1.6,.2);
\draw [->] (D)--(1.6,0);
\draw [->] (D)--(1.6,-.2);
\draw[->] (D) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (D) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (3.4,.2)--(Id);
\draw [->] (3.4,0)--(Id);
\draw [->] (3.4,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id) to [out=-30,in=210,looseness=1] (A);
\draw[->] (Id) to [out=-15,in=195,looseness=1] (A);
\draw (5.3,0) node (dots) {$\dots$};
\draw[->] (Id) to [out=15,in=165,looseness=1] (A);
\draw[->] (Id) to [out=30,in=150,looseness=1] (A);
\end{tikzpicture}$$ $$\raisebox{3pt}{\includegraphics[width=.25\linewidth]{curve_orientation_b.png}}
\begin{tikzpicture}
\draw (-1.4,0) node (Sp1) { };
\draw (-.9,0) node (=>) {$\mapsto$};
\draw (0,2) node (Sp2) { };
\draw (0,0) node (C) {$C_\gamma$};
\draw (1,0) node (D) {$\Delta_\gamma$};
\draw (2.5,0) node (dots) {$\dots$};
\node at (4.3,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Fl}^{(m)}$};
\node at (6.2,0) [draw,circle,dashed] (A) {$A$};
\draw (2.2,-1.2) node (S1) {$S_\gamma^{\sigma_m+1}$};
\draw (2.2,-.6) node (S2) {$S_\gamma^{\sigma_{m-1}+1}$};
\draw (2.2,.6) node (S3) {$S_\gamma^{\sigma_2+1}$};
\draw (2.2,1.2) node (S4) {$S_\gamma^{\sigma_1+1}$};
\draw (3.4,-1.2) node (P1) {$\bullet$};
\draw (3.4,-.6) node (P2) {$\bullet$};
\draw (3.4,.6) node (P3) {$\bullet$};
\draw (3.4,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (D) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (D)--(1.6,.2);
\draw [->] (D)--(1.6,0);
\draw [->] (D)--(1.6,-.2);
\draw[->] (D) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (D) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (3.4,.2)--(Id);
\draw [->] (3.4,0)--(Id);
\draw [->] (3.4,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id) to [out=-30,in=210,looseness=1] (A);
\draw[->] (Id) to [out=-15,in=195,looseness=1] (A);
\draw (5.3,0) node (dots) {$\dots$};
\draw[->] (Id) to [out=15,in=165,looseness=1] (A);
\draw[->] (Id) to [out=30,in=150,looseness=1] (A);
\end{tikzpicture}$$Here $A$ is the same $m$-input tensor sub-diagram in both of the right-most diagrams and ${\operatorname}{Fl}^{(m)}$ denotes the $m$ input and $m$ output tensor permuting the $i$-th input to the $(m-i)$-th output. We use the fact that $S_\gamma^2 = {\operatorname}{Id}$, so that $S_\gamma^k$ depends only on $k \mod 2$. Now we compute that:$$\begin{tikzpicture}
\draw (0,0) node (C) {$C_\gamma$};
\draw (1,0) node (D) {$\Delta_\gamma$};
\draw (2.5,0) node (dots) {$\dots$};
\node at (4.3,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Fl}^{(m)}$};
\draw (2.2,-1.2) node (S1) {$S_\gamma^{\sigma_m+1}$};
\draw (2.2,-.6) node (S2) {$S_\gamma^{\sigma_{m-1}+1}$};
\draw (2.2,.6) node (S3) {$S_\gamma^{\sigma_2+1}$};
\draw (2.2,1.2) node (S4) {$S_\gamma^{\sigma_1+1}$};
\draw (3.4,-1.2) node (P1) {$\bullet$};
\draw (3.4,-.6) node (P2) {$\bullet$};
\draw (3.4,.6) node (P3) {$\bullet$};
\draw (3.4,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (D) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (D)--(1.6,.2);
\draw [->] (D)--(1.6,0);
\draw [->] (D)--(1.6,-.2);
\draw[->] (D) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (D) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (3.4,.2)--(Id);
\draw [->] (3.4,0)--(Id);
\draw [->] (3.4,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id)--(5,.5);
\draw[->] (Id)--(5.2,.25);
\draw (5.3,0) node (dots) {$\dots$};
\draw[->] (Id)--(5.2,-.25);
\draw[->] (Id)--(5,-.5);
\draw (5.9,0) node (=1) {$=$};
\draw (6.6,0) node (C) {$C_\gamma$};
\draw (7.6,0) node (D) {$\Delta_\gamma$};
\node at (9.1,0) [draw,rectangle,dashed] (F1) {${\operatorname}{Fl}^{(m)}$};
\draw (10.9,0) node (dots) {$\dots$};
\node at (12.7,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Fl}^{(m)}$};
\draw (10.6,-1.2) node (S1) {$S_\gamma^{\sigma_m}$};
\draw (10.6,-.6) node (S2) {$S_\gamma^{\sigma_{m-1}}$};
\draw (10.6,.6) node (S3) {$S_\gamma^{\sigma_2}$};
\draw (10.6,1.2) node (S4) {$S_\gamma^{\sigma_1}$};
\draw (11.8,-1.2) node (P1) {$\bullet$};
\draw (11.8,-.6) node (P2) {$\bullet$};
\draw (11.8,.6) node (P3) {$\bullet$};
\draw (11.8,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=40,in=140,looseness=1.5] (F1);
\draw[->] (D) to [out=20,in=170,looseness=1.5] (F1);
\draw (8.2,0) node (dots) {$\dots$};
\draw[->] (D) to [out=-20,in=190,looseness=1.5] (F1);
\draw[->] (D) to [out=-40,in=220,looseness=1.5] (F1);
\draw[->] (F1) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (F1) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (F1)--(10,.2);
\draw [->] (F1)--(10,0);
\draw [->] (F1)--(10,-.2);
\draw[->] (F1) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (F1) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (11.8,.2)--(Id);
\draw [->] (11.8,0)--(Id);
\draw [->] (11.8,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id)--(13.4,.5);
\draw[->] (Id)--(13.6,.25);
\draw (13.7,0) node (dots) {$\dots$};
\draw[->] (Id)--(13.6,-.25);
\draw[->] (Id)--(13.4,-.5);
\draw (14.4,0) node (=2) {$=$};
\end{tikzpicture}$$$$\begin{tikzpicture}
\draw (0,0) node (C) {$C_\gamma$};
\draw (1,0) node (D) {$\Delta_\gamma$};
\draw (2.5,0) node (dots) {$\dots$};
\node at (4.3,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Fl}^{(m)}$};
\node at (6.2,0) [draw,rectangle,dashed] (Id2) {${\operatorname}{Fl}^{(m)}$};
\draw (2.2,-1.2) node (S1) {$S_\gamma^{\sigma_1}$};
\draw (2.2,-.6) node (S2) {$S_\gamma^{\sigma_2}$};
\draw (2.2,.6) node (S3) {$S_\gamma^{\sigma_{m-1}}$};
\draw (2.2,1.2) node (S4) {$S_\gamma^{\sigma_m}$};
\draw (3.4,-1.2) node (P1) {$\bullet$};
\draw (3.4,-.6) node (P2) {$\bullet$};
\draw (3.4,.6) node (P3) {$\bullet$};
\draw (3.4,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (D) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (D)--(1.6,.2);
\draw [->] (D)--(1.6,0);
\draw [->] (D)--(1.6,-.2);
\draw[->] (D) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (D) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (3.4,.2)--(Id);
\draw [->] (3.4,0)--(Id);
\draw [->] (3.4,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id) to [out=-30,in=210,looseness=1] (Id2);
\draw[->] (Id) to [out=-15,in=195,looseness=1] (Id2);
\draw (5.3,0) node (dots) {$\dots$};
\draw[->] (Id) to [out=15,in=165,looseness=1] (Id2);
\draw[->] (Id) to [out=30,in=150,looseness=1] (Id2);
\draw[->] (Id2)--(6.8,.5);
\draw[->] (Id2)--(7,.25);
\draw (7.2,0) node (dots) {$\dots$};
\draw[->] (Id2)--(7,-.25);
\draw[->] (Id2)--(6.8,-.5);
\draw (7.8,0) node (=1) {$=$};
\draw (8.4,0) node (C) {$C_\gamma$};
\draw (9.4,0) node (D) {$\Delta_\gamma$};
\draw (10.9,0) node (dots) {$\dots$};
\node at (12.7,0) [draw,rectangle,dashed] (Id) {${\operatorname}{Id}^{(m)}$};
\draw (10.6,-1.2) node (S1) {$S_\gamma^{\sigma_1}$};
\draw (10.6,-.6) node (S2) {$S_\gamma^{\sigma_2}$};
\draw (10.6,.6) node (S3) {$S_\gamma^{\sigma_{m-1}}$};
\draw (10.6,1.2) node (S4) {$S_\gamma^{\sigma_m}$};
\draw (11.7,-1.2) node (P1) {$\bullet$};
\draw (11.7,-.6) node (P2) {$\bullet$};
\draw (11.7,.6) node (P3) {$\bullet$};
\draw (11.7,1.2) node (P4) {$\bullet$};
\draw[->] (C)--(D);
\draw[->] (D) to [out=-90,in=180,looseness=1.5] (S1);
\draw[->] (D) to [out=-80,in=180,looseness=1.5] (S2);
\draw [->] (D)--(10,.2);
\draw [->] (D)--(10,0);
\draw [->] (D)--(10,-.2);
\draw[->] (D) to [out=80,in=180,looseness=1.5] (S3);
\draw[->] (D) to [out=90,in=180,looseness=1.5] (S4);
\draw[->] (S1)--(P1);
\draw[->] (S2)--(P2);
\draw[->] (S3)--(P3);
\draw[->] (S4)--(P4);
\draw[->] (P1) to [in=-90,out=0,looseness=1.5] (Id);
\draw[->] (P2) to [in=-120,out=0,looseness=1.5] (Id);
\draw [->] (11.8,.2)--(Id);
\draw [->] (11.8,0)--(Id);
\draw [->] (11.8,-.2)--(Id);
\draw[->] (P3) to [in=120,out=0,looseness=1.5] (Id);
\draw[->] (P4) to [in=90,out=0,looseness=1.5] (Id);
\draw[->] (Id)--(13.4,.5);
\draw[->] (Id)--(13.6,.25);
\draw (13.7,0) node (dots) {$\dots$};
\draw[->] (Id)--(13.6,-.25);
\draw[->] (Id)--(13.4,-.5);
\end{tikzpicture}$$ Here we are using the cotrace/antipode identity $C_\gamma \rightarrow S_\gamma \rightarrow = C_\gamma \rightarrow $ and the coproduct/antipode identity $\rightarrow \Delta_\gamma \rightrightarrows S_\gamma^{\otimes m} \rightrightarrows \,\,=\,\, \rightarrow S_\gamma \rightarrow \Delta_\gamma \rightrightarrows {\operatorname}{Fl}^{(m)}\rightrightarrows$. This yields the desired tensorial equality.
Next, we illustrate the various ways that $\langle -\rangle_{\mathcal{H}}$ transforms under the elementary operations on trisections, discussed in Definition \[def:basic\_trisection\_constructions\].
\[prop:properties\_of\_bracket\] (Properties of Bracket) Let $\mathcal{H}$ be a Hopf triplet and let $T,T'$ be trisections. The trisection bracket $\langle -\rangle_{\mathcal{H}}$ has the following properties.
- (Diffeomorphism) $\langle -\rangle_{\mathcal{H}}$ is invariant under oriented diffeomorphism.
- (Isotopy) $\langle -\rangle_{\mathcal{H}}$ is invariant under isotopy of trisections.
- (Connect Sum) $\langle -\rangle_{\mathcal{H}}$ satisfies $\langle T \# T'\rangle_{\mathcal{H}} = \langle T\rangle_{\mathcal{H}} \cdot \langle T'\rangle_{\mathcal{H}}$.
- (Handle Slides) $\langle -\rangle_{\mathcal{H}}$ is invariant under handle-slides.
*(a) - Diffeomorphism.* The number of $\alpha,\beta$ and $\kappa$ curves as well as the number, order and sign of the pairwise intersections are all preserved under the diffeomorphism. Thus the tensor diagrams defining $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$ are the same when $T$ and $T'$ are diffeomorphic.
*(b) - Isotopy.* For isotopies, let $T_+$ and $T_-$ be isotopic. By Lemma \[lem:two\_three\_point\_moves\] and diffeomorphism invariance, we simply need to show that $\langle T_+\rangle_{\mathcal{H}} = \langle T_-\rangle_{\mathcal{H}}$ if $T_+$ and $T_-$ are related by a two-point move or a three-point move (see Definition \[def:two\_three\_point\_moves\]). We proceed with these two cases.
*(b)(i) - Two-Point Move.* Let $T_+$ and $T_-$ be two diagrams related by a two-point move. After orienting and relabelling $T_+$ and $T_-$, we have the following pair of sub-diagrams of the trisection diagrams, along with their corresponding tensor diagrams. $$\includegraphics[width=.25\linewidth]{two_point_move_2p.png}
\begin{tikzpicture}
\draw (-1.8,0) node (=>) {$\mapsto$};
\draw (0,-1.2) node (Sp2) { };
\draw (0,0) node (D1) {$\Delta_\gamma$};
\draw (1,1) node (S1) {$S_\gamma$};
\draw (2,1) node (P1) {$\bullet$};
\draw (1.5,-1) node (P2) {$\bullet$};
\draw (3,0) node (D2) {$\Delta_\eta$};
\draw (-.6,0) node (dots1) {$\dots$};
\draw (3.6,0) node (dots2) {$\dots$};
\draw[->] (D1)--(0,1);
\draw[->] (D1)--(0,-1);
\draw[->] (D1)--(-.5,1);
\draw[->] (D1)--(-.5,-1);
\draw[->] (D2)--(3,1);
\draw[->] (D2)--(3,-1);
\draw[->] (D2)--(3.5,1);
\draw[->] (D2)--(3.5,-1);
\draw[->] (D1)--(S1);
\draw[->] (S1)--(P1);
\draw[->] (D2)--(P1);
\draw[->] (D1)--(P2);
\draw[->] (D2)--(P2);
\draw (4.5,0) node (=) {$=$};
\draw (5.5,0) node (D3) {$\Delta_\gamma$};
\draw (6.5,1) node (S3) {$S_\gamma$};
\draw (7.5,1) node (P3) {$\bullet$};
\draw (7,-1) node (P4) {$\bullet$};
\draw (8.5,0) node (D4) {$\Delta_\eta$};
\node at (7,0) [draw,circle,dashed] (T) {$T$};
\draw[->] (D3)--(S3);
\draw[->] (S3)--(P3);
\draw[->] (D4)--(P3);
\draw[->] (D3)--(P4);
\draw[->] (D4)--(P4);
\draw[->] (T) to [out=220,in=270,looseness=1.5] (D3);
\draw[->] (T) to [out=320,in=270,looseness=1.5] (D4);
\end{tikzpicture}$$$$\includegraphics[width=.25\linewidth]{two_point_move_2n.png}
\begin{tikzpicture}
\draw (-1.8,0) node (=>) {$\mapsto$};
\draw (0,-1.2) node (Sp2) { };
\draw (0,0) node (D1) {$\Delta_\gamma$};
\draw (3,0) node (D2) {$\Delta_\eta$};
\draw (-.6,0) node (dots1) {$\dots$};
\draw (3.6,0) node (dots2) {$\dots$};
\draw[->] (D1)--(0,1);
\draw[->] (D1)--(0,-1);
\draw[->] (D1)--(-.5,1);
\draw[->] (D1)--(-.5,-1);
\draw[->] (D2)--(3,1);
\draw[->] (D2)--(3,-1);
\draw[->] (D2)--(3.5,1);
\draw[->] (D2)--(3.5,-1);
\draw (4.5,0) node (=) {$=$};
\draw (5.5,0) node (e3) {$\epsilon_\gamma$};
\draw (8.5,0) node (e4) {$\epsilon_\eta$};
\node at (7,0) [draw,circle,dashed] (T) {$T$};
\draw[->] (T)--(e3);
\draw[->] (T)--(e4);
\end{tikzpicture}$$ Here the diagrams are equal outside of the region depicted, and $T$ denotes the same tensor sub-diagrams in both of the right-most diagrams. We now compute that: $$\begin{tikzpicture}
\draw (0,0) node (D3) {$\Delta_\gamma$};
\draw (1,1) node (S3) {$S_\gamma$};
\draw (2,1) node (P3) {$\bullet$};
\draw (1.5,0) node (P4) {$\bullet$};
\draw (3,0) node (D4) {$\Delta_\eta$};
\draw[->] (0,1)--(D3);
\draw[->] (3,1)--(D4);
\draw[->] (D3)--(S3);
\draw[->] (S3)--(P3);
\draw[->] (D4)--(P3);
\draw[->] (D3)--(P4);
\draw[->] (D4)--(P4);
\draw (4,.5) node (=) {$=$};
\draw (5,0) node (D5) {$\Delta_\gamma$};
\draw (6,1) node (S5) {$S_\gamma$};
\draw (6,0) node (M5) {$M_\gamma$};
\draw (7,0) node (P5) {$\bullet$};
\draw[->] (5,1)--(D5);
\draw[->] (D5)--(S5);
\draw[->] (D5)--(M5);
\draw[->] (S5)--(M5);
\draw[->] (M5)--(P5);
\draw[->] (7,1)--(P5);
\draw (8,.5) node (=) {$=$};
\draw (9,0) node (e5) {$\epsilon_\gamma$};
\draw (10,1) node (P5) {$\bullet$};
\draw (10,0) node (n5) {$\eta_\gamma$};
\draw[->] (9,1)--(e5);
\draw[->] (n5)--(P5);
\draw[->] (11,1)--(P5);
\draw (12,.5) node (=) {$=$};
\draw (13,0) node (e7) {$\epsilon_\gamma$};
\draw (14,0) node (e8) {$\epsilon_\eta$};
\draw[->] (13,1)--(e7);
\draw[->] (14,1)--(e8);
\end{tikzpicture}$$ This proves that the diagrams computing $\langle T_+\rangle_{\mathcal{H}}$ and $\langle T_-\rangle_{\mathcal{H}}$ specify the same tensor.
*(b)(ii) - Three-Point Move.* Let $T_+$ and $T_-$ be two diagrams related by a 3-point move. By choosing our curve orientations properly, we can ensure that the diagrams $T_+$ and $T_-$ will be modelled (locally, near the move region) by one of the following pairs of diagrams.
The first local model gives a counter-clockwise order to the curves in the $+$ diagram. $$\includegraphics[width=.25\linewidth]{three_point_move_p_a.png}
\qquad
\includegraphics[width=.25\linewidth]{three_point_move_n_a.png}$$ The second local model gives a clockwise cyclic order to the curves in the $+$ diagram. $$\includegraphics[width=.25\linewidth]{three_point_move_p_b.png}
\qquad
\includegraphics[width=.25\linewidth]{three_point_move_n_b.png}$$ We focus on the first case, the second being exactly analogous in a manner that we will remark on near the end of the proof.
Proceeding, we write the local contribution of each of these regions to their respective brackets. $$\includegraphics[width=.25\linewidth]{three_point_move_p_a.png}
\begin{tikzpicture}
\draw (-.8,0) node (=>) {$\mapsto$};
\draw (0,-1.2) node (Sp2) { };
\draw (1,0) node (Db) {$\Delta_\beta$};
\draw (2,1) node (Da) {$\Delta_\alpha$};
\draw (2,-1) node (Dc) {$\Delta_\kappa$};
\draw (1,1) node (Pab) {$\bullet$};
\draw (1,-1) node (Pbc) {$\bullet$};
\draw (2,0) node (Pca) {$\bullet$};
\draw (3,1) node (in_a) {};
\draw (0,0) node (in_b) {};
\draw (3,-1) node (in_c) {};
\draw[->] (in_a)--(Da);
\draw[->] (in_b)--(Db);
\draw[->] (in_c)--(Dc);
\draw[->] (Da)--(Pab);
\draw[->] (Da)--(Pca);
\draw[->] (Db)--(Pab);
\draw[->] (Db)--(Pbc);
\draw[->] (Dc)--(Pbc);
\draw[->] (Dc)--(Pca);
\end{tikzpicture}$$ $$\includegraphics[width=.25\linewidth]{three_point_move_n_a.png}
\begin{tikzpicture}
\draw (-1.8,0) node (=>) {$\mapsto$};
\draw (0,-1.2) node (Sp2) { };
\draw (0,0) node (Db) {$\Delta_\beta$};
\draw (2.5,1) node (Da) {$\Delta_\alpha$};
\draw (2.5,-1) node (Dc) {$\Delta_\kappa$};
\draw (1,1) node (Pab) {$\bullet$};
\draw (1,-1) node (Pbc) {$\bullet$};
\draw (2,0) node (Pca) {$\bullet$};
\draw (3.5,1) node (in_a) {};
\draw (-1,0) node (in_b) {};
\draw (3.5,-1) node (in_c) {};
\draw[->] (in_a)--(Da);
\draw[->] (in_b)--(Db);
\draw[->] (in_c)--(Dc);
\draw[->] (Da) to [out=180,in=90,looseness=1] (Pca);
\draw[->] (Da) to [out=200,in=0,looseness=1] (Pab);
\draw[->] (Db) to [out=-20,in=-90,looseness=1] (Pab);
\draw[->] (Db) to [out=20,in=90,looseness=1] (Pbc);
\draw[->] (Dc) to [out=160,in=0,looseness=1] (Pbc);
\draw[->] (Dc) to [out=180,in=-90,looseness=1] (Pca);
\draw (3.7,0) node (=1) {$=$};
\draw (8,1) node (Sa2) {$S_\alpha$};
\draw (5,0) node (Sb2) {$S_\beta$};
\draw (8,-1) node (Sc2) {$S_\kappa$};
\draw (6,0) node (Db2) {$\Delta_\beta$};
\draw (7,1) node (Da2) {$\Delta_\alpha$};
\draw (7,-1) node (Dc2) {$\Delta_\kappa$};
\draw (6,1) node (Pab2) {$\bullet$};
\draw (6,-1) node (Pbc2) {$\bullet$};
\draw (7,0) node (Pca2) {$\bullet$};
\draw (9,1) node (in_a2) {};
\draw (4,0) node (in_b2) {};
\draw (9,-1) node (in_c2) {};
\draw[->] (in_a2)--(Sa2);
\draw[->] (Sa2)--(Da2);
\draw[->] (in_b2)--(Sb2);
\draw[->] (Sb2)--(Db2);
\draw[->] (in_c2)--(Sc2);
\draw[->] (Sc2)--(Dc2);
\draw[->] (Da2)--(Pab2);
\draw[->] (Da2)--(Pca2);
\draw[->] (Db2)--(Pab2);
\draw[->] (Db2)--(Pbc2);
\draw[->] (Dc2)--(Pbc2);
\draw[->] (Dc2)--(Pca2);
\end{tikzpicture}$$ Now we simply observe that the equality of these two tensor sub-diagrams is implied by the Hopf triplet axioms, specifically Definition \[def:hopf\_triplet\](b). This is due to Lemma \[lem:fundamental\_triplet\_lemma\], more precisely the equivalence between Lemma \[lem:fundamental\_triplet\_lemma\](a) and Lemma \[lem:fundamental\_triplet\_lemma\](c).
For the case of the second local model, we use the same argument and appeal to the equivalence of the conditions Lemma \[lem:fundamental\_triplet\_lemma\](a) and Lemma \[lem:fundamental\_triplet\_lemma\](d).
*(c) - Connect Sum.* If $T$ and $T'$ are two trisection diagrams, then the oriented connect sum $T \# T'$ has curve sets $\alpha \sqcup \alpha'$, $\beta \sqcup \beta'$ and $\kappa \sqcup \kappa'$. The set of intersections $\mathcal{I}(T \# T')$ is the disjoint union $\mathcal{I}(T) \sqcup \mathcal{I}(T')$ of the intersections of $T$ and $T'$, and the signs of the intersections remain unchanged. Thus the tensor diagram $\langle T \# T'\rangle_{\mathcal{H}}$ is the disjoint union of the diagrams $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$, and from Notation \[not:general\_tensor\_diagrams\](b) we deduce that $\langle T \# T'\rangle_{\mathcal{H}} = \langle T\rangle_{\mathcal{H}} \otimes \langle T'\rangle_{\mathcal{H}} = \langle T\rangle_{\mathcal{H}} \cdot \langle T'\rangle_{\mathcal{H}} \in k$
*(d) - Handle Slides.* Let $T$ denote a trisection, and let $\gamma \in \{\alpha,\beta,\kappa\}$. Furthermore, let $\gamma_i$ and $\gamma_j$ be distinct $\gamma$-curves and $\xi$ be an arc connecting $\gamma_i$ to $\gamma_j$ in $\Sigma$. Finally, let $T'$ denote the trisection acquired by a handle-slide of $\gamma_i$ over $\gamma_j$ via $\xi$. Before we proceed, we fix some additional notation.
First, fix orientations $o_\alpha,o_\beta$ and $o_\kappa$ of the curves, such that the orientation $o_j$ on $\gamma_j$ is induced by the orientation $o_i$ on $\gamma_i$ and the arc $\xi$, in the following sense. Consider the surface $\Sigma - (\gamma_i \cup \xi \cup \gamma_j)$, which is the interior of a (topological) compact surface with two circle boundary components $C$ and $C'$. $C$ contains a copy of $\gamma_i - \gamma_i \cap \xi$ and a copy of $\gamma_j - \gamma_j \cap \xi$. Any orientation $o$ of $\gamma_i$ thus induces orientations $C$, $\gamma_j - \gamma_j \cap \xi$ and therefore $\gamma_j$. The orientation induced by $o$ and $\xi$ is simply $o'$.
Second, label the curves intersecting $\gamma_i$ by $\nu_1,\dots,\nu_a$ and label the curves intersecting $\gamma_j$ by $\eta_1,\dots,\eta_b$. Here we use the orderings such that the points $\xi \cap \gamma_i,{\nu}_1 \cap \gamma_i,\dots,\nu_a \cap \gamma_i$ and $\xi \cap \gamma_j, \eta_1 \cap \gamma_j, \dots,\eta_b \cap \gamma_j$ occur in cyclic order about $\gamma_i$ and $\gamma_j$, respectively. Also define $s:\{1,\dots,a\} \to \{0,1\}$ to be $0$ if ${\operatorname}{sgn}(\gamma_i \cap {\nu}_l) = +$ and $1$ if ${\operatorname}{sgn}(\gamma_i \cap {\nu}_l) = -$, and define $t:\{1,\dots,b\} \to \{0,1\}$ similarly for intersections of $\gamma_j$.
Under this setup, the (curve oriented) trisection diagrams $T$ and $T'$, along with the corresponding tensor diagrams $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$, are the following.
$$\raisebox{20pt}{\includegraphics[width=.27\linewidth]{handle_slide_a.png}}
\begin{tikzpicture}
\draw (-1.4,0) node (Sp1) { };
\draw (-1.4,0) node (=>) {$\mapsto$};
\draw (0,-1.7) node (Sp2) { };
\draw (0,1.6) node (C1) {$C_\gamma$};
\draw (1,1.4) node (D1) {$\Delta_\gamma$};
\draw (-.6,-.3) node (P1a) {$\bullet$};
\draw (.1,-.3) node (P1b) {$\bullet$};
\draw (.8,-.3) node (dots1c) {$\dots$};
\draw (1.5,-.3) node (P1d) {$\bullet$};
\draw (-.6,.5) node (S1a) {$S_\gamma^{s_1}$};
\draw (.1,.5) node (S1b) {$S_\gamma^{s_2}$};
\draw (.8,.5) node (dots1c2) {$\dots$};
\draw (1.5,.5) node (S1d) {$S_\gamma^{s_a}$};
\draw (2,-1.7) node (s1a) {};
\draw (2,-1.6) node (s1b) {};
\draw (2,-1.5) node (s1d) {};
\draw (7,1.6) node (C2) {$C_\gamma$};
\draw (8,1.4) node (D2) {$\Delta_\gamma$};
\draw (4.6,-.3) node (P2a) {$\bullet$};
\draw (5.8,-.3) node (P2b) {$\bullet$};
\draw (7,-.3) node (dots2c1) {$\dots$};
\draw (8.2,-.3) node (P2d) {$\bullet$};
\draw (4.6,.5) node (S2a) {$S_\gamma^{t_1}$};
\draw (5.8,.5) node (S2b) {$S_\gamma^{t_2}$};
\draw (7,.5) node (dots2c2) {$\dots$};
\draw (8.2,.5) node (S2d) {$S_\gamma^{t_b}$};
\draw (4.6,-1.1) node (D2a) {$\Delta_{\eta_1}$};
\draw (5.8,-1.1) node (D2b) {$\Delta_{\eta_2}$};
\draw (7,-1.1) node (dots2c) {$\dots$};
\draw (8.2,-1.1) node (D2d) {$\Delta_{\eta_b}$};
\node at (3,-1) [draw,rectangle,dashed,minimum height=45pt] (A) {$A$};
\draw (-1,1.4) node (R1) {$R_H$};
\draw[-,dashed] (-.5,2)--(8.7,2);
\draw[-,dashed] (8.7,2)--(8.7,-1.7);
\draw[-,dashed] (8.7,-1.7)--(4.1,-1.7);
\draw[-,dashed] (4.1,-1.7)--(4.1,1);
\draw[-,dashed] (4.1,1)--(-.5,1);
\draw[-,dashed] (-.5,1)--(-.5,2);
\draw[->] (C1)--(D1);
\draw[->] (D1)--(S1a);
\draw[->] (D1)--(S1b);
\draw[->] (D1)--(S1d);
\draw[->] (S1a)--(P1a);
\draw[->] (S1b)--(P1b);
\draw[->] (S1d)--(P1d);
\draw[->] (1.6,-2) to [in=270,out=180,looseness=1] (P1a);
\draw[->] (1.6,-1.9) to [in=270,out=180,looseness=1] (P1b);
\draw[->] (1.6,-1.8) to [in=270,out=180,looseness=1,dotted] (dots1c);
\draw[->] (1.6,-1.7) to [in=270,out=180,looseness=1] (P1d);
\draw[-] (1.6,-2) to [out=0,in=180,looseness=1] (2.7,-1.6);
\draw[-] (1.6,-1.9) to [out=0,in=180,looseness=1] (2.7,-1.2);
\draw[-] (1.6,-1.8) to [out=0,in=180,looseness=1,dotted] (2.7,-.8);
\draw[-] (1.6,-1.7) to [out=0,in=180,looseness=1] (2.7,-.4);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(S2a);
\draw[->] (D2)--(S2b);
\draw[->] (D2)--(S2d);
\draw[->] (S2a)--(P2a);
\draw[->] (S2b)--(P2b);
\draw[->] (S2d)--(P2d);
\draw[<-] (P2a)--(D2a);
\draw[<-] (P2b)--(D2b);
\draw[<-] (P2d)--(D2d);
\draw[->] (4,-2) to [in=270,out=0,looseness=.6] (D2d);
\draw[->] (4,-1.9) to [in=270,out=0,looseness=.6] (dots2c);
\draw[->] (4,-1.8) to [in=260,out=0,looseness=.6] (D2b);
\draw[->] (4,-1.7) to [in=250,out=0,looseness=.6] (D2a);
\draw[-] (4,-2) to [out=180,in=0,looseness=1] (3.3,-1.6);
\draw[-] (4,-1.9) to [out=180,in=0,looseness=1] (3.3,-1.2);
\draw[-] (4,-1.8) to [out=180,in=0,looseness=1,dotted] (3.3,-.8);
\draw[-] (4,-1.7) to [out=180,in=0,looseness=1] (3.3,-.4);
\end{tikzpicture}$$ $$\raisebox{20pt}{\includegraphics[width=.27\linewidth]{handle_slide_b.png}}
\begin{tikzpicture}
\draw (-1.4,0) node (Sp1) { };
\draw (-1.4,0) node (=>) {$\mapsto$};
\draw (0,-1.7) node (Sp2) { };
\draw (0,1.6) node (C1) {$C_\gamma$};
\draw (1,1.4) node (D1) {$\Delta_\gamma$};
\draw (-.6,-.3) node (P1a) {$\bullet$};
\draw (.1,-.3) node (P1b) {$\bullet$};
\draw (.8,-.3) node (dots1c) {$\dots$};
\draw (1.5,-.3) node (P1d) {$\bullet$};
\draw (-.6,.5) node (S1a) {$S_\gamma^{s_1}$};
\draw (.1,.5) node (S1b) {$S_\gamma^{s_2}$};
\draw (.8,.5) node (dots1c2) {$\dots$};
\draw (1.5,.5) node (S1d) {$S_\gamma^{s_a}$};
\draw (2,-1.7) node (s1a) {};
\draw (2,-1.6) node (s1b) {};
\draw (2,-1.5) node (s1d) {};
\draw (7,1.6) node (C2) {$C_\gamma$};
\draw (8,1.4) node (D2) {$\Delta_\gamma$};
\draw (4.6,1.4) node (D3) {$\Delta_\gamma$};
\draw (4.6,-1.1) node (D2a) {$\Delta_{\eta_1}$};
\draw (5.8,-1.1) node (D2b) {$\Delta_{\eta_2}$};
\draw (7,-1.1) node (dots2c) {$\dots$};
\draw (8.2,-1.1) node (D2d) {$\Delta_{\eta_b}$};
\draw (4.8,-.3) node (P2a) {$\bullet$};
\draw (6,-.3) node (P2b) {$\bullet$};
\draw (7,-.3) node (dots2c1) {$\dots$};
\draw (8.4,-.3) node (P2d) {$\bullet$};
\draw (4.4,-.3) node (P3a) {$\bullet$};
\draw (5.6,-.3) node (P3b) {$\bullet$};
\draw (8,-.3) node (P3d) {$\bullet$};
\draw (4.6,.5) node (F2a) {$E_{1}$};
\draw (5.8,.5) node (F2b) {$E_{2}$};
\draw (7,.5) node (dots2c2) {$\dots$};
\draw (8.2,.5) node (F2d) {$E_{b}$};
\node at (3,-1) [draw,rectangle,dashed,minimum height=45pt] (A) {$A$};
\draw (-1,1.4) node (R1) {$R'_H$};
\draw[-,dashed] (-.5,2)--(8.7,2);
\draw[-,dashed] (8.7,2)--(8.7,-1.7);
\draw[-,dashed] (8.7,-1.7)--(4.1,-1.7);
\draw[-,dashed] (4.1,-1.7)--(4.1,1);
\draw[-,dashed] (4.1,1)--(-.5,1);
\draw[-,dashed] (-.5,1)--(-.5,2);
\draw[->] (C1)--(D1);
\draw[->] (D1)--(S1a);
\draw[->] (D1)--(S1b);
\draw[->] (D1)--(S1d);
\draw[->] (S1a)--(P1a);
\draw[->] (S1b)--(P1b);
\draw[->] (S1d)--(P1d);
\draw[->] (1.6,-2) to [in=270,out=180,looseness=1] (P1a);
\draw[->] (1.6,-1.9) to [in=270,out=180,looseness=1] (P1b);
\draw[->] (1.6,-1.8) to [in=270,out=180,looseness=1,dotted] (dots1c);
\draw[->] (1.6,-1.7) to [in=270,out=180,looseness=1] (P1d);
\draw[-] (1.6,-2) to [out=0,in=180,looseness=1] (2.7,-1.6);
\draw[-] (1.6,-1.9) to [out=0,in=180,looseness=1] (2.7,-1.2);
\draw[-] (1.6,-1.8) to [out=0,in=180,looseness=1,dotted] (2.7,-.8);
\draw[-] (1.6,-1.7) to [out=0,in=180,looseness=1] (2.7,-.4);
\draw[->] (C2)--(D2);
\draw[->] (D1)--(D3);
\draw[->] (D2)--(F2a);
\draw[->] (D2)--(F2b);
\draw[->] (D2)--(F2d);
\draw[->] (D3)--(F2a);
\draw[->] (D3)--(F2b);
\draw[->] (D3)--(F2d);
\draw[->] (F2a)--(P2a);
\draw[->] (F2b)--(P2b);
\draw[->] (F2d)--(P2d);
\draw[->] (F2a)--(P3a);
\draw[->] (F2b)--(P3b);
\draw[->] (F2d)--(P3d);
\draw[<-] (P2a)--(D2a);
\draw[<-] (P2b)--(D2b);
\draw[<-] (P2d)--(D2d);
\draw[<-] (P3a)--(D2a);
\draw[<-] (P3b)--(D2b);
\draw[<-] (P3d)--(D2d);
\draw[->] (4,-2) to [in=270,out=0,looseness=.6] (D2d);
\draw[->] (4,-1.9) to [in=270,out=0,looseness=.6] (dots2c);
\draw[->] (4,-1.8) to [in=260,out=0,looseness=.6] (D2b);
\draw[->] (4,-1.7) to [in=250,out=0,looseness=.6] (D2a);
\draw[-] (4,-2) to [out=180,in=0,looseness=1] (3.3,-1.6);
\draw[-] (4,-1.9) to [out=180,in=0,looseness=1] (3.3,-1.2);
\draw[-] (4,-1.8) to [out=180,in=0,looseness=1,dotted] (3.3,-.8);
\draw[-] (4,-1.7) to [out=180,in=0,looseness=1] (3.3,-.4);
\end{tikzpicture}$$
Let us elaborate on the various notational components of the two tensor diagrams above. Above, $A$ specifies the contribution of intersections other than $\nu_l$ and $\eta_l$ (for all $l$). This contribution is the same in both $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$. The symbols $R_H$ and $R'_H$ respectively denote the tensors formed by the circled (or rather, boxed) regions in $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$. By convention, the diagram $\to \Delta_{\eta_i} \to$ simply denotes the identity (see Notation \[not:product\_coproduct\_abbreviated\_notation\]). The tensor $S^l_\gamma$ denotes $l$-th powers with respect to composition of the tensor $S_\gamma$.
The tensors $E_k$ for $k \in \{1,\dots,b\}$ are abbreviations, and require some careful elaboration. First, fix the notation ${\operatorname}{col}(\eta) \in \{\alpha,\beta,\kappa\}$ for the color of a curve $\eta$. Given another curve $\xi$, we say that ${\operatorname}{col}(\xi) - {\operatorname}{col}(\eta) \equiv 1 \mod 3$ if the color of $\xi$ occurs to the right of the color of $\eta$ with respect to the standard cyclic ordering $(\alpha,\beta,\kappa)$ of the colors. So for instance, ${\operatorname}{col}(\beta_j) - {\operatorname}{col}(\alpha_i) \equiv 1$. Likewise, ${\operatorname}{col}(\xi) - {\operatorname}{col}(\eta) \equiv -1 \mod 3$ if the color of $\xi$ occurs to the left of the color of $\eta$, and ${\operatorname}{col}(\xi) - {\operatorname}{col}(\eta) \equiv 0 \mod 3$ if the colors are equal. Using this notation, we define $E_k$ case by case depending on the colors of $\gamma$ and $\eta_j$, and the sign of $\gamma \cap \eta_j$. $$\begin{tikzpicture}
\draw (7,0) node (E1) {$E_k$};
\draw (8.5,0) node (=0) {$:=$};
\draw[->] (6.2,-.3)--(E1);
\draw[->] (6.2,.3)--(E1);
\draw[->] (E1)--(7.8,-.3);
\draw[->] (E1)--(7.8,.3);
\draw[->] (9,-.3)--(10.8,.3);
\draw[->] (9,.3)--(10.8,-.3);
\draw (15,0) node (lb1) {if ${\operatorname}{sgn}(\gamma_j \cap \eta_k) = +$ and ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv 1$};
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (7,0) node (E1) {$E_k$};
\draw (8.5,0) node (=0) {$:=$};
\draw[->] (6.2,-.3)--(E1);
\draw[->] (6.2,.3)--(E1);
\draw[->] (E1)--(7.8,-.3);
\draw[->] (E1)--(7.8,.3);
\draw (9.8,-.3) node (Sa) {$S_\gamma$};
\draw (9.8,.3) node (Sb) {$S_\gamma$};
\draw[->] (9,-.3)--(Sa);
\draw[->] (9,.3)--(Sb);
\draw[->] (Sa)--(10.8,-.3);
\draw[->] (Sb)--(10.8,.3);
\draw (15,0) node (lb1) {if ${\operatorname}{sgn}(\gamma_j \cap \eta_k) = -$ and ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv 1$};
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (7,0) node (E1) {$E_k$};
\draw (8.5,0) node (=0) {$:=$};
\draw[->] (6.2,-.3)--(E1);
\draw[->] (6.2,.3)--(E1);
\draw[->] (E1)--(7.8,-.3);
\draw[->] (E1)--(7.8,.3);
\draw[->] (9,-.3)--(10.8,-.3);
\draw[->] (9,.3)--(10.8,.3);
\draw (15,0) node (lb1) {if ${\operatorname}{sgn}(\gamma_j \cap \eta_k) = +$ and ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv -1$};
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (7,0) node (E1) {$E_k$};
\draw (8.5,0) node (=0) {$:=$};
\draw[->] (6.2,-.3)--(E1);
\draw[->] (6.2,.3)--(E1);
\draw[->] (E1)--(7.8,-.3);
\draw[->] (E1)--(7.8,.3);
\draw (9.8,-.3) node (Sa) {$S_\gamma$};
\draw (9.8,.3) node (Sb) {$S_\gamma$};
\draw[->] (9,-.3)--(Sa);
\draw[->] (9,.3)--(Sb);
\draw[->] (Sa) to [out=0,in=180,looseness=1] (10.8,.3);
\draw[->] (Sb) to [out=0,in=180,looseness=1] (10.8,-.3);
\draw (15,0) node (lb1) {if ${\operatorname}{sgn}(\gamma_j \cap \eta_k) = -$ and ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv -1$};
\end{tikzpicture}$$ The presence of these cases comes from the relative order of the output legs of $\Delta_{\eta_k}$ corresponding to the original intersection $\gamma_i \cap \eta_k$ and the cloned intersection $(\gamma_i \# \gamma_j) \cap \eta_k$, and the relation of this order to the sign of the original intersection.
Returning to the proof of (d), to show that the tensor diagrams $\langle T\rangle_{\mathcal{H}}$ and $\langle T'\rangle_{\mathcal{H}}$ specify the same scalar, it suffices to demonstrate that $R_H$ and $R'_H$ denote the same tensor. To accomplish this, we first observe the following identities. $$\label{eqn:handle_slide_id_1}
\begin{tikzpicture}
\draw (0,0) node (S1) {$S_\gamma^{t_l}$};
\draw (1,0) node (P1) {$\bullet$};
\draw (2,0) node (D1) {$\Delta_{\eta_l}$};
\draw[->] (-.8,0)--(S1);
\draw[->] (S1)--(P1);
\draw[<-] (P1)--(D1);
\draw[<-] (D1)--(2.8,0);
\draw (3.5,0) node (=1) {$=$};
\draw (5,0) node (M3) {$M_\gamma$};
\draw (6,0) node (P3) {$\bullet$};
\draw (7,0) node (S3) {$S_{\eta_l}^{t_l}$};
\draw[->] (4.2,0)--(M3);
\draw[->] (M3)--(P3);
\draw[<-] (P3)--(S3);
\draw[<-] (S3)--(7.8,0);
\end{tikzpicture}$$ $$\label{eqn:handle_slide_id_2}
\begin{tikzpicture}
\draw (0,0) node (E1) {$E_l$};
\draw (1,.3) node (P1a) {$\bullet$};
\draw (1,-.3) node (P1b) {$\bullet$};
\draw (2,0) node (D1) {$\Delta_{\eta_l}$};
\draw[->] (-.8,.3)--(E1);
\draw[->] (-.8,-.3)--(E1);
\draw[->] (E1)--(P1a);
\draw[->] (E1)--(P1b);
\draw[<-] (P1a)--(D1);
\draw[<-] (P1b)--(D1);
\draw[<-] (D1)--(2.8,0);
\draw (3.5,0) node (=1) {$=$};
\draw (5,0) node (M2) {$M_\gamma$};
\draw (6,0) node (P2) {$\bullet$};
\draw (7,0) node (S2) {$S_{\eta_l}^{t_l}$};
\draw[->] (4.2,.3)--(M2);
\draw[->] (4.2,-.3)--(M2);
\draw[->] (M2)--(P2);
\draw[<-] (P2)--(S2);
\draw[<-] (S2)--(7.8,0);
\end{tikzpicture}$$ $$\label{eqn:handle_slide_id_3}
\begin{tikzpicture}
\draw (1,-.6) node (C1) {$C_\gamma$};
\draw (0,-1) node (D1) {$\Delta_\gamma$};
\draw (-.6,-1) node (dots1) {$\dots$};
\draw (3,-.6) node (C2) {$C_\gamma$};
\draw (4,-1) node (D2) {$\Delta_\gamma$};
\draw (0,-2) node (Ma) {$M_\gamma$};
\draw (1,-2) node (Mb) {$M_\gamma$};
\draw (2,-2) node (Mc) {$M_\gamma$};
\draw (3,-2) node (dotsd) {$\dots$};
\draw (4,-2) node (Me) {$M_\gamma$};
\draw[->] (D1)--(-.5,-1.6);
\draw[->] (D1)--(-.7,-1.3);
\draw[->] (C1)--(D1);
\draw[->] (D1)--(Ma);
\draw[->] (D1)--(Mb);
\draw[->] (D1)--(Mc);
\draw[->] (D1)--(Me);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(0,-2.7);
\draw[->] (Mb)--(1,-2.7);
\draw[->] (Mc)--(2,-2.7);
\draw[->] (dotsd)--(3,-2.7);
\draw[->] (Me)--(4,-2.7);
\draw (6,-1.5) node (=) {$=$};
\draw (9,-.6) node (C1) {$C_\gamma$};
\draw (8,-1) node (D1) {$\Delta_\gamma$};
\draw (7.4,-1) node (dots1) {$\dots$};
\draw (11,-.6) node (C2) {$C_\gamma$};
\draw (12,-1) node (D2) {$\Delta_\gamma$};
\draw (8,-2) node (Ma) {$M_\gamma$};
\draw (9,-2) node (Mb) {$M_\gamma$};
\draw (10,-2) node (Mc) {$M_\gamma$};
\draw (11,-2) node (dotsd) {$\dots$};
\draw (12,-2) node (Me) {$M_\gamma$};
\draw[->] (D1)--(7.5,-1.6);
\draw[->] (D1)--(7.3,-1.3);
\draw[->] (C1)--(D1);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(8,-2.7);
\draw[->] (Mb)--(9,-2.7);
\draw[->] (Mc)--(10,-2.7);
\draw[->] (dotsd)--(11,-2.7);
\draw[->] (Me)--(12,-2.7);
\end{tikzpicture}$$ Here we emphasize again that $\to M_\gamma \to$ and $\to \Delta_{\eta_l} \to$ denote, by convention, the identity tensor (see Notation \[not:product\_coproduct\_abbreviated\_notation\]). The equations (\[eqn:handle\_slide\_id\_1\])-(\[eqn:handle\_slide\_id\_3\]) are consequences of the diagrammatic Hopf algebra and Hopf triplet axioms from Definition \[def:hopf\_algebra\] and \[def:hopf\_triplet\]. We will explain them in detail in Lemma \[lem:handle\_slide\_lemma\] below.
We may now apply (\[eqn:handle\_slide\_id\_1\]), (\[eqn:handle\_slide\_id\_3\]) and (\[eqn:handle\_slide\_id\_2\]) in that order to perform the following manipulation transforming $R_H$ into $R'_H$. This finishes the proof of the handle slide property. $$\begin{tikzpicture}
\draw (0,1.6) node (C1) {$C_\gamma$};
\draw (0,-.3) node (D1) {$\Delta_\gamma$};
\draw (-.3,-1.2) node (dots1) {$\dots$};
\draw (3.8,1.6) node (C2) {$C_\gamma$};
\draw (4.8,1.4) node (D2) {$\Delta_\gamma$};
\draw (1.4,-.3) node (P2a) {$\bullet$};
\draw (2.6,-.3) node (P2b) {$\bullet$};
\draw (3.8,-.3) node (dots2c1) {$\dots$};
\draw (5,-.3) node (P2d) {$\bullet$};
\draw (1.4,-1.1) node (D2a) {$\Delta_{\eta_1}$};
\draw (2.6,-1.1) node (D2b) {$\Delta_{\eta_2}$};
\draw (3.8,-1.1) node (dots2c) {$\dots$};
\draw (5,-1.1) node (D2d) {$\Delta_{\eta_b}$};
\draw (1.4,.5) node (S2a) {$S_\gamma^{t_1}$};
\draw (2.6,.5) node (S2b) {$S_\gamma^{t_2}$};
\draw (3.8,.5) node (dots2c2) {$\dots$};
\draw (5,.5) node (S2d) {$S_\gamma^{t_b}$};
\draw[->] (C1)--(D1);
\draw[->] (D1)--(-.8,-.7);
\draw[->] (D1)--(-.7,-1.1);
\draw[->] (D1)--(0,-1.2);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(S2a);
\draw[->] (D2)--(S2b);
\draw[->] (D2)--(S2d);
\draw[->] (S2a)--(P2a);
\draw[->] (S2b)--(P2b);
\draw[->] (S2d)--(P2d);
\draw[<-] (P2a)--(D2a);
\draw[<-] (P2b)--(D2b);
\draw[<-] (P2d)--(D2d);
\draw[<-] (D2a)--(1.4,-1.7);
\draw[<-] (D2b)--(2.6,-1.7);
\draw[<-] (D2d)--(5,-1.7);
\draw (6,.2) node (=1) {$=$};
\draw (7.5,1.6) node (C1) {$C_\gamma$};
\draw (7.5,-.3) node (D1) {$\Delta_\gamma$};
\draw (7.2,-1.2) node (dots1) {$\dots$};
\draw (11.3,1.6) node (C2) {$C_\gamma$};
\draw (12.3,1.4) node (D2) {$\Delta_\gamma$};
\draw (8.9,-.3) node (P2a) {$\bullet$};
\draw (10.1,-.3) node (P2b) {$\bullet$};
\draw (11.3,-.3) node (dots2c1) {$\dots$};
\draw (12.5,-.3) node (P2d) {$\bullet$};
\draw (8.9,.5) node (M2a) {$M_\gamma$};
\draw (10.1,.5) node (M2b) {$M_\gamma$};
\draw (11.3,.5) node (dots2c) {$\dots$};
\draw (12.5,.5) node (M2d) {$M_\gamma$};
\draw (8.9,-1.1) node (S2a) {$S_{\eta_1}^{t_1}$};
\draw (10.1,-1.1) node (S2b) {$S_{\eta_2}^{t_2}$};
\draw (11.3,-1.1) node (dots2c2) {$\dots$};
\draw (12.5,-1.1) node (S2d) {$S_{\eta_b}^{t_b}$};
\draw[->] (C1)--(D1);
\draw[->] (D1)--(6.7,-.7);
\draw[->] (D1)--(6.8,-1.1);
\draw[->] (D1)--(7.5,-1.2);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(M2a);
\draw[->] (D2)--(M2b);
\draw[->] (D2)--(M2d);
\draw[->] (M2a)--(P2a);
\draw[->] (M2b)--(P2b);
\draw[->] (M2d)--(P2d);
\draw[<-] (P2a)--(S2a);
\draw[<-] (P2b)--(S2b);
\draw[<-] (P2d)--(S2d);
\draw[<-] (S2a)--(8.9,-1.7);
\draw[<-] (S2b)--(10.1,-1.7);
\draw[<-] (S2d)--(12.5,-1.7);
\draw (13.4,.2) node (=1) {$=$};
\end{tikzpicture}$$ $$\begin{tikzpicture}
\draw (6,.2) node (=1) {$=$};
\draw (0,1.6) node (C1) {$C_\gamma$};
\draw (0,-.3) node (D1) {$\Delta_\gamma$};
\draw (-.3,-1.2) node (dots1) {$\dots$};
\draw (3.8,1.6) node (C2) {$C_\gamma$};
\draw (4.8,1.4) node (D2) {$\Delta_\gamma$};
\draw (1.3,1.4) node (D3) {$\Delta_\gamma$};
\draw (1.4,-.3) node (P2a) {$\bullet$};
\draw (2.6,-.3) node (P2b) {$\bullet$};
\draw (3.8,-.3) node (dots2c1) {$\dots$};
\draw (5,-.3) node (P2d) {$\bullet$};
\draw (1.4,.5) node (M2a) {$M_\gamma$};
\draw (2.6,.5) node (M2b) {$M_\gamma$};
\draw (3.8,.5) node (dots2c) {$\dots$};
\draw (5,.5) node (M2d) {$M_\gamma$};
\draw (1.4,-1.1) node (S2a) {$S_{\eta_1}^{t_1}$};
\draw (2.6,-1.1) node (S2b) {$S_{\eta_2}^{t_2}$};
\draw (3.8,-1.1) node (dots2c2) {$\dots$};
\draw (5,-1.1) node (S2d) {$S_{\eta_b}^{t_b}$};
\draw[->] (C1)--(D1);
\draw[->] (D1)--(-.8,-.7);
\draw[->] (D1)--(-.7,-1.1);
\draw[->] (D1)--(0,-1.2);
\draw[->] (C2)--(D2);
\draw[->] (D1)--(D3);
\draw[->] (D2)--(M2a);
\draw[->] (D2)--(M2b);
\draw[->] (D2)--(M2d);
\draw[->] (M2a)--(P2a);
\draw[->] (M2b)--(P2b);
\draw[->] (M2d)--(P2d);
\draw[<-] (P2a)--(S2a);
\draw[<-] (P2b)--(S2b);
\draw[<-] (P2d)--(S2d);
\draw[->] (D3)--(M2a);
\draw[->] (D3)--(M2b);
\draw[->] (D3)--(M2d);
\draw[<-] (S2a)--(1.4,-1.7);
\draw[<-] (S2b)--(2.6,-1.7);
\draw[<-] (S2d)--(5,-1.7);
\draw (7.5,1.6) node (C1) {$C_\gamma$};
\draw (7.5,-.3) node (D1) {$\Delta_\gamma$};
\draw (7.2,-1.2) node (dots1) {$\dots$};
\draw (11.3,1.6) node (C2) {$C_\gamma$};
\draw (12.3,1.4) node (D2) {$\Delta_\gamma$};
\draw (8.9,1.4) node (D3) {$\Delta_\gamma$};
\draw (9.1,-.3) node (P2a) {$\bullet$};
\draw (10.3,-.3) node (P2b) {$\bullet$};
\draw (11.3,-.3) node (dots2c1) {$\dots$};
\draw (12.7,-.3) node (P2d) {$\bullet$};
\draw (8.7,-.3) node (P3a) {$\bullet$};
\draw (9.9,-.3) node (P3b) {$\bullet$};
\draw (12.3,-.3) node (P3d) {$\bullet$};
\draw (8.9,-1.1) node (D2a) {$\Delta_{\eta_1}$};
\draw (10.1,-1.1) node (D2b) {$\Delta_{\eta_{2}}$};
\draw (11.3,-1.1) node (dots2c) {$\dots$};
\draw (12.5,-1.1) node (D2d) {$\Delta_{\eta_b}$};
\draw (8.9,.5) node (F2a) {$E_1$};
\draw (10.1,.5) node (F2b) {$E_2$};
\draw (11.3,.5) node (dots2c2) {$\dots$};
\draw (12.5,.5) node (F2d) {$E_b$};
\draw[->] (C1)--(D1);
\draw[->] (D1)--(6.7,-.7);
\draw[->] (D1)--(6.8,-1.1);
\draw[->] (D1)--(7.5,-1.2);
\draw[->] (C2)--(D2);
\draw[->] (D1)--(D3);
\draw[->] (D3)--(F2a);
\draw[->] (D3)--(F2b);
\draw[->] (D3)--(F2d);
\draw[->] (D2)--(F2a);
\draw[->] (D2)--(F2b);
\draw[->] (D2)--(F2d);
\draw[->] (F2a)--(P2a);
\draw[->] (F2b)--(P2b);
\draw[->] (F2d)--(P2d);
\draw[->] (F2a)--(P3a);
\draw[->] (F2b)--(P3b);
\draw[->] (F2d)--(P3d);
\draw[<-] (P2a)--(D2a);
\draw[<-] (P2b)--(D2b);
\draw[<-] (P2d)--(D2d);
\draw[<-] (P3a)--(D2a);
\draw[<-] (P3b)--(D2b);
\draw[<-] (P3d)--(D2d);
\draw[<-] (D2a)--(8.9,-1.7);
\draw[<-] (D2b)--(10.1,-1.7);
\draw[<-] (D2d)--(12.5,-1.7);
\end{tikzpicture}$$ Having proven the handle slide property, we have demonstrated properties (a)-(d) listed in the proposition statement and so concluded the proof of Proposition \[prop:properties\_of\_bracket\].
\[lem:handle\_slide\_lemma\] The formulas (\[eqn:handle\_slide\_id\_1\]), (\[eqn:handle\_slide\_id\_2\]) and (\[eqn:handle\_slide\_id\_3\]) in the proof of Proposition \[prop:properties\_of\_bracket\](d) are valid and follow from Definition \[def:hopf\_algebra\] and \[def:hopf\_triplet\].
For (\[eqn:handle\_slide\_id\_1\]), again recall that $\to M_\gamma \to$ and $\to \Delta_\gamma \to $ both denote the identity and $S^{t_l}_\gamma$ is either ${\operatorname}{Id}$ and $S_\gamma$. Thus (\[eqn:handle\_slide\_id\_1\]) is (in the non-trivial case) simply the fact that skew pairing intertwine antipodes.
Next we handle (\[eqn:handle\_slide\_id\_2\]). Assume (without loss of generality) that $\gamma_j = \alpha_j$ is an $\alpha$ curve, so that ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv 1 \mod 3$ implies $\eta_k = \beta_k$ is a $\beta$ curve, and similarly ${\operatorname}{col}(\eta_k) - {\operatorname}{col}(\gamma_j) \equiv - 1 \mod 3$ implies $\eta_k = \kappa_k$ is a $\kappa$ curve. Then the identity (\[eqn:handle\_slide\_id\_2\]) for the four cases for $E_i$ translates to the following identities.
$$\begin{tikzpicture}
\draw (-2,.3) node (i1a) {};
\draw (-2,-.3) node (i1b) {};
\draw (.6,.3) node (P1a) {$\langle-\rangle_{\alpha\beta}$};
\draw (.6,-.3) node (P1b) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,0) node (D1) {$\Delta_\beta$};
\draw (2.8,0) node (o1) {};
\draw[->] (i1b)--(P1a);
\draw[->] (i1a)--(P1b);
\draw[<-] (P1a)--(D1);
\draw[<-] (P1b)--(D1);
\draw[<-] (D1)--(o1);
\draw (3.5,0) node (=1) {$=$};
\draw (4,.3) node (i2a) {};
\draw (4,-.3) node (i2b) {};
\draw (5,0) node (M2) {$M_\alpha$};
\draw (6.4,0) node (P2) {$\langle-\rangle_{\alpha\beta}$};
\draw (8.4,0) node (o2) {};
\draw[->] (i2a)--(M2);
\draw[->] (i2b)--(M2);
\draw[->] (M2)--(P2);
\draw[<-] (P2)--(o2);
\end{tikzpicture}$$
$$\begin{tikzpicture}
\draw (-2,.3) node (i1a) {};
\draw (-2,-.3) node (i1b) {};
\draw (-1,.3) node (S1a) {$S_\alpha$};
\draw (-1,-.3) node (S1b) {$S_\alpha$};
\draw (.6,.3) node (P1a) {$\langle-\rangle_{\alpha\beta}$};
\draw (.6,-.3) node (P1b) {$\langle-\rangle_{\alpha\beta}$};
\draw (2,0) node (D1) {$\Delta_\beta$};
\draw (2.8,0) node (o1) {};
\draw[->] (i1a)--(S1a);
\draw[->] (i1b)--(S1b);
\draw[->] (S1a) to [out=0,in=180,looseness=1] (P1b);
\draw[->] (S1b) to [out=0,in=180,looseness=1] (P1a);
\draw[<-] (P1a)--(D1);
\draw[<-] (P1b)--(D1);
\draw[<-] (D1)--(o1);
\draw (3.5,0) node (=1) {$=$};
\draw (4,.3) node (i2a) {};
\draw (4,-.3) node (i2b) {};
\draw (5,0) node (M2) {$M_\alpha$};
\draw (6.4,0) node (P2) {$\langle-\rangle_{\alpha\beta}$};
\draw (7.7,0) node (S2) {$S_\beta$};
\draw (8.4,0) node (o2) {};
\draw[->] (i2a)--(M2);
\draw[->] (i2b)--(M2);
\draw[->] (M2)--(P2);
\draw[->] (P2)--(S2);
\draw[<-] (S2)--(o2);
\end{tikzpicture}$$
$$\begin{tikzpicture}
\draw (-2,.3) node (i1a) {};
\draw (-2,-.3) node (i1b) {};
\draw (.6,.3) node (P1a) {$\langle-\rangle_{\kappa\alpha}$};
\draw (.6,-.3) node (P1b) {$\langle-\rangle_{\kappa\alpha}$};
\draw (2,0) node (D1) {$\Delta_\kappa$};
\draw (2.8,0) node (o1) {};
\draw[->] (i1a)--(P1a);
\draw[->] (i1b)--(P1b);
\draw[<-] (P1a)--(D1);
\draw[<-] (P1b)--(D1);
\draw[<-] (D1)--(o1);
\draw (3.5,0) node (=1) {$=$};
\draw (4,.3) node (i2a) {};
\draw (4,-.3) node (i2b) {};
\draw (5,0) node (M2) {$M_\alpha$};
\draw (6.4,0) node (P2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (8.4,0) node (o2) {};
\draw[->] (i2a)--(M2);
\draw[->] (i2b)--(M2);
\draw[->] (M2)--(P2);
\draw[<-] (P2)--(o2);
\end{tikzpicture}$$
$$\begin{tikzpicture}
\draw (-2,.3) node (i1a) {};
\draw (-2,-.3) node (i1b) {};
\draw (-1,.3) node (S1a) {$S_\alpha$};
\draw (-1,-.3) node (S1b) {$S_\alpha$};
\draw (.6,.3) node (P1a) {$\langle-\rangle_{\kappa\alpha}$};
\draw (.6,-.3) node (P1b) {$\langle-\rangle_{\kappa\alpha}$};
\draw (2,0) node (D1) {$\Delta_\kappa$};
\draw (2.8,0) node (o1) {};
\draw[->] (i1a)--(S1a);
\draw[->] (i1b)--(S1b);
\draw[->] (S1a) to [out=0,in=180,looseness=1] (P1a);
\draw[->] (S1b) to [out=0,in=180,looseness=1] (P1b);
\draw[<-] (P1a)--(D1);
\draw[<-] (P1b)--(D1);
\draw[<-] (D1)--(o1);
\draw (3.5,0) node (=1) {$=$};
\draw (4,.3) node (i2a) {};
\draw (4,-.3) node (i2b) {};
\draw (5,0) node (M2) {$M_\alpha$};
\draw (6.4,0) node (P2) {$\langle-\rangle_{\kappa\alpha}$};
\draw (7.7,0) node (S2) {$S_\kappa$};
\draw (8.4,0) node (o2) {};
\draw[->] (i2a)--(M2);
\draw[->] (i2b)--(M2);
\draw[->] (M2)--(P2);
\draw[->] (P2)--(S2);
\draw[<-] (S2)--(o2);
\end{tikzpicture}$$ The first and third identities follow from the fact that the pairings induce Hopf algebra morphisms $H_\alpha \to H_\beta^{*,{\operatorname}{cop}}$ and $H_\kappa \to H_\alpha^{*,{\operatorname}{cop}}$ (see Definition \[def:hopf\_triplet\](a)). The second and the fourth also follow from this fact, after commuting the antipodes past the pairings and using the anti-homomorphism property of the antipodes.
Finally, we address (\[eqn:handle\_slide\_id\_3\]). This is just a fact about Hopf algebras, so let $H = (H,M,\eta,\Delta,\epsilon,S)$ be a Hopf algebra. Then we compute as follows.
$$\begin{tikzpicture}
\draw (-.2,-.6) node (C1) {$C$};
\draw (0,-1) node (D1) {$\Delta$};
\draw (-1.6,-1) node (dots1) {$\dots$};
\draw (-1,-1) node (D3) {$\Delta$};
\draw (3,-.6) node (C2) {$C$};
\draw (4,-1) node (D2) {$\Delta$};
\draw (0,-2) node (Ma) {$M$};
\draw (1,-2) node (Mb) {$M$};
\draw (2,-2) node (Mc) {$M$};
\draw (3,-2) node (dotsd) {$\dots$};
\draw (4,-2) node (Me) {$M$};
\draw[->] (D3)--(-1.5,-1.6);
\draw[->] (D3)--(-1.7,-1.3);
\draw[->] (C1)--(D3);
\draw[->] (D3)--(D1);
\draw[->] (D1)--(Ma);
\draw[->] (D1)--(Mb);
\draw[->] (D1)--(Mc);
\draw[->] (D1)--(Me);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(0,-2.7);
\draw[->] (Mb)--(1,-2.7);
\draw[->] (Mc)--(2,-2.7);
\draw[->] (dotsd)--(3,-2.7);
\draw[->] (Me)--(4,-2.7);
\draw (5,-1.5) node (=) {$=$};
\end{tikzpicture}\begin{tikzpicture}
\draw (-.2,-.6) node (C1) {$C$};
\draw (1,-1) node (M1) {$M$};
\draw (-1.6,-1) node (dots1) {$\dots$};
\draw (-1,-1) node (D3) {$\Delta$};
\draw (3,-.6) node (C2) {$C$};
\draw (4,-1) node (D2) {$\Delta$};
\draw (0,-2) node (Ma) {$M$};
\draw (1,-2) node (Mb) {$M$};
\draw (2,-2) node (Mc) {$M$};
\draw (3,-2) node (dotsd) {$\dots$};
\draw (4,-2) node (Me) {$M$};
\draw[->] (D3)--(-1.5,-1.6);
\draw[->] (D3)--(-1.7,-1.3);
\draw[->] (C1)--(D3);
\draw[->] (D3)--(M1);
\draw[->] (M1)--(D2);
\draw[->] (C2)--(M1);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(0,-2.7);
\draw[->] (Mb)--(1,-2.7);
\draw[->] (Mc)--(2,-2.7);
\draw[->] (dotsd)--(3,-2.7);
\draw[->] (Me)--(4,-2.7);
\draw (5,-1.5) node (=) {$=$};
\end{tikzpicture}$$
$$\begin{tikzpicture}
\draw (-.2,-.6) node (C1) {$C$};
\draw (1,-1) node (e1) {$\epsilon$};
\draw (-1.6,-1) node (dots1) {$\dots$};
\draw (-1,-1) node (D3) {$\Delta$};
\draw (3,-.6) node (C2) {$C$};
\draw (4,-1) node (D2) {$\Delta$};
\draw (0,-2) node (Ma) {$M$};
\draw (1,-2) node (Mb) {$M$};
\draw (2,-2) node (Mc) {$M$};
\draw (3,-2) node (dotsd) {$\dots$};
\draw (4,-2) node (Me) {$M$};
\draw[->] (D3)--(-1.5,-1.6);
\draw[->] (D3)--(-1.7,-1.3);
\draw[->] (C1)--(D3);
\draw[->] (D3)--(e1);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(0,-2.7);
\draw[->] (Mb)--(1,-2.7);
\draw[->] (Mc)--(2,-2.7);
\draw[->] (dotsd)--(3,-2.7);
\draw[->] (Me)--(4,-2.7);
\draw (5,-1.5) node (=) {$=$};
\end{tikzpicture}
\begin{tikzpicture}
\draw (-.2,-.6) node (C1) {$C$};
\draw (-1.6,-1) node (dots1) {$\dots$};
\draw (-1,-1) node (D3) {$\Delta$};
\draw (3,-.6) node (C2) {$C$};
\draw (4,-1) node (D2) {$\Delta$};
\draw (0,-2) node (Ma) {$M$};
\draw (1,-2) node (Mb) {$M$};
\draw (2,-2) node (Mc) {$M$};
\draw (3,-2) node (dotsd) {$\dots$};
\draw (4,-2) node (Me) {$M$};
\draw[->] (D3)--(-1.5,-1.6);
\draw[->] (D3)--(-1.7,-1.3);
\draw[->] (C1)--(D3);
\draw[->] (C2)--(D2);
\draw[->] (D2)--(Ma);
\draw[->] (D2)--(Mb);
\draw[->] (D2)--(Mc);
\draw[->] (D2)--(Me);
\draw[->] (Ma)--(0,-2.7);
\draw[->] (Mb)--(1,-2.7);
\draw[->] (Mc)--(2,-2.7);
\draw[->] (dotsd)--(3,-2.7);
\draw[->] (Me)--(4,-2.7);
\end{tikzpicture}$$ Here we use the bialgebra property of Hopf algebras (see Definition \[def:hopf\_algebra\](c)) on the first line and the fact that the cotrace $C$ is a cointegral (see Definition \[def:trace\_cotrace\] and Lemma \[lem:trace\_is\_integral\]) to move from the first to second line. This concludes the proof of the Lemma.
Main Definition and Properties {#subsec:main_definition_and_properties}
------------------------------
The definition of the invariant itself is straightforward, as it is simply a normalization of the bracket discussed in §\[subsec:trisection\_bracket\]. However, the invariant requires a small constraint on the Hopf triplets we use, which is easily verified in practice.
Let $\mathcal{H}$ be a Hopf triplet over a unital ring $k$. We say that $H$ is *trisection admissible* if the equation $\xi^3 - \langle T_{{\operatorname}{st}}\rangle_{\mathcal{H}} = 0$ admits a solution $\xi \in k^\times$ in the group of units $k^\times$ of $k$, where $T_{{\operatorname}{st}}$ is the standard genus-$3$ (stabilizing) trisection of $S^4$ (see Figure \[fig:stabilized\_sphere\_trisection\]).
(Trisection Invariant) \[def:trisection\_kuperberg\_invariant\] Let $\mathcal{H}$ be a trisection admissible Hopf triplet and fix a root $\xi \in k^\times$ of $\langle T_{{\operatorname}{st}}\rangle_{\mathcal{H}}$. Let $X$ be a smooth, closed, oriented 4-manifold and $T$ be a trisection diagram for $X$. The *trisection invariant* $\tau_{\mathcal{H},\xi}(X;T) \in k$ is defined to be $$\begin{aligned}
\tau_{\mathcal{H},\xi}(X;T) = \xi^{-g(T)} \langle T \rangle_{\mathcal{H}}\,.\end{aligned}$$
As a consequence of Proposition \[prop:properties\_of\_bracket\], we have the following invariance result, which is one of the main theorems of the paper.
\[thm:main\_invariance\_thm\] (Invariance) The trisection invariant $\tau_{\mathcal{H},\xi}(X;T)$ is invariant under all trisection moves and isotopy of $T$.
The genus $g(T)$ is obviously invariant under isotopy and handle-slides, and is additive under connect sum. Thus Proposition \[prop:properties\_of\_bracket\](b) and (e) imply that $\tau_{\mathcal{H}}(X;T)$ is invariant under isotopies and handle-slides. For stabilizations, we observe that: $$\xi^{-g(T \# T_{{\operatorname}{st}})} \langle T \# T_{{\operatorname}{st}} \rangle_{\mathcal{H}} = \xi^{-g(T)} \langle T_{{\operatorname}{st}}\rangle^{-1} \langle T \rangle_{\mathcal{H}} \langle T_{{\operatorname}{st}}\rangle_{\mathcal{H}} = \xi^{-g(T)}\langle T \rangle_{\mathcal{H}}$$ Here we use the multiplicativity of $\langle -\rangle_{\mathcal{H}}$ under connect sum, see Proposition \[prop:properties\_of\_bracket\](d). Thus $\tau_{\mathcal{H},\xi}(X;T \# T_{{\operatorname}{st}}) = \tau_{\mathcal{H},\xi}(X;T)$, and $\tau_{\mathcal{H},\xi}$ is invariant under stabilization.
\[cor:invariance\] $\tau_{\mathcal{H},\xi}(X) := \tau_{\mathcal{H},\xi}(X;T)$ is an oriented diffeomorphism invariant.
Moreover, the connect sum property of the trisection bracket implies the same property for $\tau_{\mathcal{H},\xi}(X)$.
Let $X$ and $X'$ be smooth, closed $4$-manifolds and let $\mathcal{H}$ be a Hopf triplet. Then $\tau_{\mathcal{H},\xi}(X \# X') = \tau_{\mathcal{H},\xi}(X)\tau_{\mathcal{H},\xi}(X')$.
In this paper, we will only compute examples of this invariant for Hopf triplets over $k = {{\mathbb R}}$ or ${{\mathbb C}}$. Thus we will use the following abbreviation for the rest of the paper.
\[con:choice\_of\_root\_over\_R\_or\_C\] Let $\mathcal{H}$ be a trisection admissible Hopf triplet over ${{\mathbb R}}$ or ${{\mathbb C}}$ such that $\langle T_{{\operatorname}{st}}\rangle_{\mathcal{H}} \in {{\mathbb R}}_+$ is positive and real. We fix the convention that $$\tau_{\mathcal{H}}(X) := \tau_{\mathcal{H},\xi}(X)$$ where $\xi$ is the unique real cube root of $\langle T_{{\operatorname}{st}}\rangle_{\mathcal{H}}$.
In all of the cases calculated in §\[sec:examples\_and\_calculations\] and the setting of §\[sec:CY\_dichro\], the conditions for Convention \[con:choice\_of\_root\_over\_R\_or\_C\] hold.
Generalized 3-Manifold Invariant {#subsec:generalized_kuperberg_invt}
--------------------------------
A generalized version of Kuperberg’s invariant for involutory Hopf algebras can be defined using Hopf doublets. The original Kuperberg invariant can be recovered by using the standard doublet $\mathcal{H} = (H,H^{*,{\operatorname}{op}},\langle-\rangle)$.
Let $\mathcal{H}$ be an involutory Hopf doublet, $Y$ be a $3$-manifold and $S$ be a Heegaard diagram for $Y$.
The *(generalized) Kuperberg bracket* $\langle S\rangle_{\mathcal{H}}$ is defined as in Definition \[def:trisection\_bracket\], using only the $\alpha$ and $\beta$ curves in that discussion. The *(generalized) Kuperberg invariant* $\tau_{\mathcal{H}}(Y;S)$ for a Hopf doublet $\mathcal{H}$ with $\langle S_{{\operatorname}{st}}\rangle_{\mathcal{H}} \neq 0$ is the normalization $$\tau_{\mathcal{H}}(Y;S) := \langle S_{{\operatorname}{st}}\rangle_{\mathcal{H}}^{-g(S)} \cdot \langle S\rangle_{\mathcal{H}}$$ Here $S_{{\operatorname}{st}}$ denotes the standard (stabilizing) genus $1$ Heegaard splitting of the $3$-sphere.
Using the arguments in §\[subsec:trisection\_bracket\] (more specifically, Proposition \[prop:properties\_of\_bracket\]) and §\[subsec:main\_definition\_and\_properties\], we can prove an invariance theorem.
The generalized Kuperberg invariant $\tau_{\mathcal{H}}(Y) := \tau_{\mathcal{H}}(Y;S)$ is independent of the choice of Heegaard splitting, and satisfies $\tau_{\mathcal{H}}(Y \# Y') = \tau_{\mathcal{H}}(Y) \cdot \tau_{\mathcal{H}}(Y')$.
We expect that a similar treatment is possible for the invariant incorporating ${\operatorname}{Spin}^c$-structures (see [@lopez2019kuperberg]) and the invariant for non-involutory Hopf algebras (see [@kuperberg1996noninvolutory]). It would be interesting to explore the degree to which the invariant $\tau_{\mathcal{H}}$ goes beyond Kuperberg’s original invariant in its sensitivity to topological phenomena.
Examples and Calculations {#sec:examples_and_calculations}
=========================
The trisection invariant formulated in §\[sec:trisection\_kuperberg\_invt\] is very explicit and computer friendly. To demonstrate this, we will now perform some example calculations of the trisection invariant.
We start (§\[subsec:trisection\_diagrams\_of\_examples\]) by providing a menagerie of trisection diagrams and bracket tensor diagrams for various examples of $4$-manifolds. For more difficult trisection diagrams, we wrote a Python script (available at [@pythonscript2019]) to calculate the invariant using a simple combinatorial description of the trisection diagram in use (§\[subsec:computational\_methods\_and\_scripting\]). We then compute the trisection invariant for triplets arising from cyclic group algebras (\[subsec:cyclic\_triplets\_and\_kashaev\]), in particular demonstrating that they coincide with a slight modification of Kashaev’s invariant. Finally, we discuss computations for a class of Hopf triplet whose corresponding invariants do not coincide with Crane-Yetter or dichromatic invariants via Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\] (§\[subsec:triplets\_from\_8d\_algebra\]).
Trisection and Tensor Diagrams of Examples {#subsec:trisection_diagrams_of_examples}
------------------------------------------
Here is a list of trisection diagrams for a number of standard (and some exotic) $4$-manifolds, along with the corresponding tensor diagram for the trisection bracket. These diagrams are drawn primarily from [@gk2016], [@co2017lefschetztrisections] and [@lcm2018rationalsurfaces].
\[rmk:efficient\_diagrams\] Consider a simply connected, closed $4$-manifold $X$. The genus $g(T)$ of any trisection $T$ admits a lower bound of the form $g(T) \ge b_2(X)$ where $b_2(X)$ is the rank of $H_2(X)$. An *efficient trisection* $T$ is a trisection for which this lower bound is an equality, i.e. $g(T) = b_2(X)$ (see [@lcm2018rationalsurfaces]).
Many of the trisection diagrams given in this section are efficient in this sense, making them particularly suitable for computations of the trisection invariant.
The first four examples that we introduce here, namely ${{\mathbb C}}P^2$, $S^1 \times S^3$ and sphere bundles over $S^2$, are all very simple and provide easy sources of example calculations.
\[ex:CP2\_trisection\] Complex projective space admits a standard $(1,0)$-trisection, which can be written as in Figure \[fig:CP2\_trisection\_1\] below.
![A trisection diagram for ${{\mathbb C}}P^2$.[]{data-label="fig:CP2_trisection_1"}](CP2_trisection_a){width=".25\textwidth"}
at (0,0) (Da) [$\Delta_\alpha$]{}; at (2,1) (Db) [$\Delta_\beta$]{}; at (2,-1) (Dc) [$\Delta_\kappa$]{}; at (0,1) (Pab) [$\bullet$]{}; at (0,-1) (Pca) [$\bullet$]{}; at (2,0) (Pbc) [$\bullet$]{};
(Da)–(Pab); (Da)–(Pca); (Db)–(Pab); (Db)–(Pbc); (Dc)–(Pca); (Dc)–(Pbc);
\[ex:S1xS3\_trisection\] Another very simple trisection is that of the product manifold $S^1 \times S^3$, which admits a $(1,0)$-trisection as in Figure \[fig:S1xS3\_trisection\_1\] below.
![A trisection diagram for $S^1 \times S^3$.[]{data-label="fig:S1xS3_trisection_1"}](S1xS3_trisection){width=".25\textwidth"}
at (0,1) (Ca) [$C_\alpha$]{}; at (1,1) (Cb) [$C_\beta$]{}; at (2,1) (Cc) [$C_\kappa$]{}; at (0,0) (Ea) [$\epsilon_\alpha$]{}; at (1,0) (Eb) [$\epsilon_\beta$]{}; at (2,0) (Ec) [$\epsilon_\kappa$]{};
(Ca)–(Ea); (Cb)–(Eb); (Cc)–(Ec);
\[ex:S2xS2\_trisection\] The product $S^2 \times S^2$ of two $2$-spheres admits a genus $2$ trisection diagram as in Figure \[fig:S2xS2\_trisection\] below.
![A trisection diagram for $S^2 \times S^2$.[]{data-label="fig:S2xS2_trisection"}](S2xS2_trisection_a){width=".25\textwidth"}
at (0,1) (Da1) [$\Delta_\alpha$]{}; at (0,-1) (Da2) [$\Delta_\alpha$]{}; at (2,1) (Db1) [$\Delta_\beta$]{}; at (2,-1) (Db2) [$\Delta_\beta$]{}; at (4,1) (Dc1) [$\Delta_\kappa$]{}; at (4,-1) (Dc2) [$\Delta_\kappa$]{}; at (1,1) (Pab1) [$\bullet$]{}; at (1,-1) (Pab2) [$\bullet$]{}; at (3,1) (Pbc1) [$\bullet$]{}; at (3,-1) (Pbc2) [$\bullet$]{}; at (1,0) (Pca1) [$\bullet$]{}; at (3,0) (Pca2) [$\bullet$]{};
(Da1)–(Pab1); (Da2)–(Pab2); (Db1)–(Pab1); (Db2)–(Pab2);
(Db1)–(Pbc1); (Db2)–(Pbc2); (Dc1)–(Pbc1); (Dc2)–(Pbc2);
(Da1)–(Pca1); (Da2)–(Pca2); (Dc2)–(Pca1); (Dc1)–(Pca2);
\[ex:S2tildexS2\_trisection\] The twisted product $S^2 \tilde{\times} S^2$ (that is, the total space of the non-trivial oriented sphere bundle over $S^2$) admits a genus $2$ trisection diagram as in Figure \[fig:twS2xS2\_trisection\] below.
![A trisection diagram for $S^2 \tilde{\times} S^2$.[]{data-label="fig:twS2xS2_trisection"}](twS2xS2_trisection_a){width=".25\textwidth"}
at (0,1) (Da1) [$\Delta_\alpha$]{}; at (0,-1) (Da2) [$\Delta_\alpha$]{}; at (2,1) (Db1) [$\Delta_\beta$]{}; at (2,-1) (Db2) [$\Delta_\beta$]{}; at (4,1) (Dc1) [$\Delta_\kappa$]{}; at (4,-1) (Dc2) [$\Delta_\kappa$]{}; at (1,1) (Pab1) [$\bullet$]{}; at (1,-1) (Pab2) [$\bullet$]{}; at (3,1) (Pbc1) [$\bullet$]{}; at (3,-1) (Pbc2) [$\bullet$]{}; at (1,0) (Pca1) [$\bullet$]{}; at (3,0) (Pca2) [$\bullet$]{};
at (2,.3) (Pbc3) [$\bullet$]{};
(Da1)–(Pab1); (Da2)–(Pab2); (Db1)–(Pab1); (Db2)–(Pab2);
(Db1)–(Pbc1); (Db2)–(Pbc2); (Dc1)–(Pbc1); (Dc2)–(Pbc2);
(Da1)–(Pca1); (Da2)–(Pca2); (Dc2)–(Pca1); (Dc1)–(Pca2);
(Dc1)–(Pbc3); (Db2)–(Pbc3);
\[ex:T2xS2\_trisection\] The product $T^2 \times S^2$ admits a genus $4$ trisection diagram as in Figure \[fig:T2xS2\_trisection\] below. We omit the corresponding trisection bracket, as it is quite complicated in this case.
![A trisection diagram for $T^2 \times S^2$ (see [@co2017lefschetztrisections Fig 19]).[]{data-label="fig:T2xS2_trisection"}](T2xS2_trisection_a){width=".8\textwidth"}
The final (and most non-trivial) example of a trisection that we will include in this section is the following, of the Kummer surface.
\[ex:K3\_trisection\] The Kummer surface (or K3 surface) $K$ is the unique Calabi-Yau surface aside from $T^4$, up to deformation. By representing the Kummer surface as a certain branched cover of ${{\mathbb C}}P^2$, Lambert-Cole and Meier give an efficient trisection diagram for $K$ in [@lcm2018rationalsurfaces]. See Figure \[fig:K3\_trisection\].
![A trisection diagram for the Kummer surface $K$ (see [@lcm2018rationalsurfaces Fig 16]).[]{data-label="fig:K3_trisection"}](K3_trisection_square){width=".7\textwidth"}
Although this trisection is currently beyond the computation abilities of our script [@pythonscript2019], the diagram can be tabulated and easily stored as a trisection datum (see Definition \[def:trisection\_datum\] below) and is included in [@pythonscript2019]. Improvements in the efficiency of [@pythonscript2019] or enhancements to the properties of the invariant (e.g. gluing formulae) may make calculations with this trisection tractable in the near future.
Computational Methods {#subsec:computational_methods_and_scripting}
---------------------
Most trisection diagrams produce tensor diagram expressions for the corresponding trisection invariant that are too large and complicated to evaluate by hand. However, it is relatively straightforward to write computer code to calculate the invariant, essentially directly from the definition. Here we briefly outline how this is done.
\[def:trisection\_datum\] A *trisection datum* $D = (N, g, \sigma, I)$ consists of the following data.
- A genus $g \in {{\mathbb Z}}_{\ge 0}$ and an intersection number $N \in {{\mathbb Z}}_{\ge 0}$. The list $\{1,\dots,N\}$ is called the list of intersections.
- A map $\sigma:\{1,\dots,N\} \to \{\pm 1\}$ or equivalently an ordered list of $N$ signs.
- For each $\gamma \in \{\alpha,\beta,\kappa\}$ and $i \in \{1,\dots,g\}$, a list $I^\gamma_i = (i_1,\dots,i_m)$ of integers $1 \le i_j \le N$ where each intersection occurs exactly twice across all lists $I^\gamma_i$.
A trisection $T$ determines a trisection datum $D(T) = (N,g,\sigma,I)$ as so. First, order the intersections $\mathcal{I}(T)$ and the $\alpha/\beta/\kappa$-curves. Then define $N := \# \mathcal{I}(T)$ and $g := g(T)$, take the sign map $\sigma:\{1,\dots,N\} \simeq \mathcal{I}(T) \to \{\pm 1\}$ to be the intersection sign and form $I^\gamma_i$ by listing the intersections along each curve $\gamma_i$.
A trisection datum is essentially a combinatorial data type containing all of the data necessary to calculate $\tau_{\mathcal{H}}(T)$, given the additional data of a Hopf triplet. The Hopf triplet itself can be stored as a set of tensors, namely the structure tensors of the three Hopf algebras and the pairing tensors.
The procedure for computing the trisection invariant with this data is essentially a direct application of the definition.
\[proc:computing\_trisection\] Let $\mathcal{H} = (H_\alpha,H_\beta,H_\kappa,\langle-\rangle)$ be a Hopf triplet and $D = (N,g,\sigma,I)$ be a datum for a trisection $T$. The procedure for computing the trisection invariant $\tau_{\mathcal{H}}(T)$ goes like this.
- For each $\gamma \in \{\alpha,\beta,\kappa\}$ and $i \in \{1,\dots,g\}$, form a copy $\Delta_{\gamma,i}$ of the $0$-input, $k$-output coproduct tensor from the Hopf algebra $H_\gamma$.
- Label the outputs of $\Delta_{\gamma,i}$ by the intersections in the list $I^\gamma_i$.
- For each intersection $i$, find the two pairs $(\gamma,j)$ and $(\eta,k)$ so that $I^\gamma_j$ and $I^\eta_k$ contain the intersection $i$. Then contract $\Delta^\gamma_j$ and $\Delta^\eta_k$ using the pairing $\langle-\rangle_{\gamma\eta}$ if $\sigma(i) = +1$ or $\langle S-\rangle_{\gamma\eta} = -1$.
An implementation of this procedure as a Python script, written by the authors of this paper, can be found at [@pythonscript2019]. To conclude our discussion of computational methods, let us include a brief discussion of optimization.
Here is a list of optimizations that are useful in implementing Procedure \[proc:computing\_trisection\].
- It is often more efficient to implement the structure tensors of $\mathcal{H}$ as sparse matrices, since (for instance) the structure tensors for many naturally occuring Hopf algebras (like group algebras) are sparse.
- Relatedly, it is generally advantageous from an efficiency perspective to minimize the dimension of the vector-spaces being used in tensor calculations. This can be accomplished, for instance, by performing the contractions in step (c) in stages so that the maximum number of outputs are paired at each stage.
Cyclic Triplets and Kashaev’s Invariant {#subsec:cyclic_triplets_and_kashaev}
---------------------------------------
We now explain a first set of example calculations using the trisection diagrams of §\[subsec:trisection\_diagrams\_of\_examples\] and the computational methods described in §\[subsec:computational\_methods\_and\_scripting\]. Namely, we compute the trisection invariant for Hopf triplets living in the following simple family.
The *cyclic Hopf triplet* $\mathcal{Z}[N]$ for a positive integer $N \ge 2$ is the (trisection admissible, involutory) Hopf triplet defined as follows.
Consider the Hopf algebra $\mathbb{C}[\mathbb{Z}/N]$ which is the group Hopf algebra of the cyclic group $\mathbb{Z}/N$. This Hopf algebra has a well-known quasi-triangular structure (c.f. [@majid2000]), and the $R$-matrix can be written explicitly as follows. $$R = \frac{1}{N} \sum_{k,\ell=0}^{N-1} \exp\left(2\pi i\, \frac{k \ell}{N}\right) [k] \otimes [\ell] \,,$$ We may thus construct a triple $\mathcal{Z}[N] = (Z_\alpha,Z_\beta,Z_\kappa,\langle-\rangle)$ as in Example \[ex:basic\_examples\_of\_triplets\]. Namely, we define the constituent Hopf algebras by $$Z_\alpha = {{\mathbb C}}[\mathbb{Z}/N]^*\,, \qquad Z_\beta = {{\mathbb C}}[\mathbb{Z}/N]\,, \qquad Z_\kappa = {{\mathbb C}}[\mathbb{Z}/N]^*$$ The pairings between $Z_\alpha = Z_\kappa$ and $Z_\beta$ are given by the dual pairing, while the final pairing is constructed using the $R$-matrix as in Example \[ex:basic\_examples\_of\_triplets\](b). Note that we are omiting all applications of $(-)^{{\operatorname}{op}}$ and $(-)^{{\operatorname}{cop}}$, since all the Hopf algebras here are commutative and co-commutative and thus these operations have no effect.
Empirically, the trisection invariant associated to $\mathcal{Z}[N]$ seems to be essentially equivalent to the numerical invariants arising from a family of simple 4D TQFTs introduced by Kashaev in [@kashaev2014asimple]. Let us briefly recount Kashaev’s construction.
Let $X$ be a closed $4$-manifold and fix an integer $N \ge 2$. The *Kashaev invariant* $\mathcal{K}_N(X)$ is defined as follows.
We start by fixing some auxiliary data. Let $V = \mathbb{C}^N$ be the standard $N$-dimensional Hilbert space wit the standard Hermitian inner product, and let $\{e_k\}_{k=0}^{N-1}$ and $\{\overline{e}_k\}_{k=0}^{N-1}$ denote the standard bases of $V$ and the dual basis of $V^*$ respectively. Let $Q$ denote the $5$ index tensor $$\label{eqn:Kashaev_Q_matrix}
Q := \frac{1}{\sqrt{N}} \sum_{k,\ell,m = 0}^{N-1} {\operatorname}{exp}\left(2\pi i \,\frac{km}{N}\right) \cdot e_k \otimes \overline{e}_{k + \ell} \otimes e_\ell \otimes \overline{e}_{\ell + m} \otimes e_m$$ Note that the Hermitian conjugate tensor $Q^\dag$ of $Q$ may be written as follows. $$\label{eqn:Kashaev_Qbar_matrix}
Q^\dag := \frac{1}{\sqrt{N}} \sum_{k,\ell,m = 0}^{N-1} {\operatorname}{exp}\left(-2\pi i \,\frac{km}{N}\right) \cdot \overline{e}_k \otimes e_{k + \ell} \otimes \overline{e}_\ell \otimes e_{\ell + m} \otimes \overline{e}_m$$ Finally, choose an arbitrary triangulation $\mathcal{T}$ of $X$ and order the vertices of $\mathcal{T}$. To compute $\mathcal{K}_N(X)$, proceed as so.
First, assign a copy $Q[\Delta]$ of either $Q$ or $Q^\dag$ to each $4$-dimensional simplex $\Delta \in \mathcal{T}$ in the triangulation $\mathcal{T}$. We use $Q$ if the orientation on $\Delta$ induced by $X$ agrees with that induced by the vertex order, and $Q^\dag$ if the orientations disagree. Then, label the five indices of $Q[\Delta]$ by the $3$-dimensional facets of $\partial\Delta$, using the dictionary ordering induced by the vertex ordering. Finally, contract all pairs of indices sharing a label on any pair of tensors $Q[\Delta]$ and $Q[\Delta']$.
The Kashaev invariant $\mathcal{K}_N(X) \in {{\mathbb C}}$ is defined to be the resulting scalar acquired by this final contraction times the normalization factor $N^{-|\mathcal{T}_0|}$, where $|\mathcal{T}_0|$ is the number of vertices of $\mathcal{T}$.
In Table 1 of [@kashaev2014asimple], Kashaev presents a calculation of the Kashaev invariant for the spaces $S^4, {{\mathbb C}}P^2, S^2 \times S^2$, $S^1 \times S^3$ and $T^2 \times S^2$. By utilizing the diagrams presented in Examples \[ex:CP2\_trisection\]-\[ex:T2xS2\_trisection\] and (in some cases) the computation methods discussed in §\[subsec:computational\_methods\_and\_scripting\], we computed the following table comparing the Kashaev invariant $\mathcal{K}_N(-)$ to $\tau_{\mathcal{Z}[N]}(-)$ in these cases.
\[table:trisection\_vs\_Kashaev\]
$X$ $\chi(X)$ $N^{\chi(X) + 1} \cdot \mathcal{K}_N(X)$ $\tau_{\mathcal{Z}[N]}(X)$
------------------ ----------- ------------------------------------------ ------------------------------------- --
$S^4$ 2 1 1
$S^2 \times S^2$ 4 $N^{-1}(3 + (-1)^N)/2$ $N^{-1}(3 + (-1)^N)/2 \quad (\dag)$
$\mathbb{C}P^2$ 3 $N^{-1}\sum_{k=1}^N \omega_N^{k^2}$ $N^{-1}\sum_{k=1}^N \omega_N^{k^2}$
$S^3 \times S^1$ 0 $N$ $N$
$S^2 \times T^2$ 0 $N(3 + (-1)^N)/2$ $N(3 + (-1)^N)/2 \quad (\dag\dag)$
: Comparing a family of trisection invariants, and the Kashaev invariants.
Above, $\chi(X)$ denotes the Euler characteristic. Note that the formula $(\dag)$ for $\tau_{\mathcal{Z}[N]}(S^2 \times S^2)$ has been verified for $2 \le N \le 100$ and the formula $(\dag\dag)$ for $\tau_{\mathcal{Z}[N]}(T^2 \times S^2)$ has been verified for $2 \le N \le 4$. The remaining cases can be checked exactly. In light of these empirical results, we formulate the following conjecture.
\[conj:Kashaev\_is\_trisection\] For all oriented closed $4$-manifolds $X$ and any $N \ge 2$, we have $$\tau_{\mathcal{Z}[N]}(X) = N^{\chi(X) + 1} \cdot \mathcal{K}_N(X)$$
Due to Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\] (or the formal version Theorem \[thm:tri=dichro\]) and the fact that $\tau_{\mathcal{Z}[N]}(X)$ is built using a Hopf triplet arising from a quasi-triangular Hopf algebra, Conjecture \[conj:Kashaev\_is\_trisection\] would imply that the Kashaev invariants are equivalent to Crane-Yetter invariants.
Triplets from the $8$-Dimensional Algebra {#subsec:triplets_from_8d_algebra}
-----------------------------------------
As a final computation for this section, we tabulate the value of the trisection invariant on some of the simple spaces in §\[subsec:trisection\_diagrams\_of\_examples\] for a Hopf triplet that is beyond the purview of the dichromatic invariants via Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\] (and the more formally stated Theorem \[thm:tri=dichro\]). That is, these invariants do *not* arise from the Hopf triplet associated to a quasi-triangular Hopf algebra.
Each of the constituent Hopf algebras in the triplets of interest in this section will be isomorphic to a fixed Hopf algebra $H_8$, which admits the following description.
\[not:8d\_Hopf\_algebra\] Denote by $H_8$ the unique semisimple Hopf algebra over ${{\mathbb C}}$ with ${\operatorname}{dim}(H) = 8$ that is neither commutative nor cocommutative.
More explicitly, $H_8$ may be presented as a quotient algebra $H_8 = {{\mathbb C}}\langle x,y,z\rangle/I$ of the free, unital associative algebra ${{\mathbb C}}\langle x,y,z\rangle$ generated by $3$ variables. The ideal $I$ in the quotient is generated by the relations $$I = \langle xy - yx, xz - zy, yz - zx, x^2 -1, y^2 - 1, \ z^2 - \frac{1}{2}(1+x+y-xy)\rangle$$ This defines the algebra structure on $H_8$ and implies that the set of elements $B = \{1,x,y,xy,z,xz,yz,xyz\}$ form a basis of $H_8$ as a ${{\mathbb C}}$ vector space. The coalgebra structure $(\Delta,\epsilon)$ can be specified as follows. $$\label{eqn:8_dim_coproduct}
\Delta(x) = x \otimes x, \ \Delta(y) = y \otimes y, \ \Delta(z) = \frac{1}{2}(z \otimes z + yz \otimes z + z \otimes xz - yz \otimes xz)$$ The coproduct of the remaining basis elements of $B$ can be deduced from (\[eqn:8\_dim\_coproduct\]) and the bialgebra property. The counit may likewise be specified as follows. $$\epsilon(w) = 1 \qquad \text{for} \qquad w \in B$$ Finally, the antipode tensor $S$ can be specified by $$S(w) = w \qquad \text{for} \qquad w \in \{x,y,z\}$$ The antipode of the remaining basis elements of $B$ can be deduced from (\[eqn:8\_dim\_coproduct\]) and the anti-homomorphism property of $S$.
Next, we fix notation for a curated collection of skew pairings on $H_8$, each of which give the pair $(H_8,H_8)$ the structure of a Hopf doublet.
\[not:skew\_pairing\_a\] We denote by $\langle-\rangle_i$ for $i \in \{0,1,2,3\}$ the pairings $H_8 \times H_8 \to {{\mathbb C}}$ specified by the following matrices $M_i$ in the basis $B$ of Notation \[not:8d\_Hopf\_algebra\]. $$M_0 := \left(
\begin{array}{cccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & i & -i & -i & i \\
1 & -1 & -1 & 1 & i & -i & -i & i \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & i & i & 1 & -1-i & 0 & 0 & -1-i \\
1 & -i & -i & 1 & 0 & -1+i & -1+i & 0 \\
1 & -i & -i & 1 & 0 & -1+i & -1+i & 0 \\
1 & i & i & 1 & -1-i & 0 & 0 & -1-i \\
\end{array}
\right)$$
$$M_1 := \left(
\begin{array}{cccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & i & -i & -i & i \\
1 & -1 & -1 & 1 & -i & i & i & -i \\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\
1 & -i & i & -1 & -\sqrt{2} & 0 & 0 & \sqrt{2} \\
1 & i & -i & -1 & 0 & i \sqrt{2} & -i \sqrt{2} & 0 \\
1 & i & -i & -1 & 0 & -i \sqrt{2} & i \sqrt{2} & 0 \\
1 & -i & i & -1 & \sqrt{2} & 0 & 0 & -\sqrt{2} \\
\end{array}
\right)$$
$$M_2 := \left(
\begin{array}{cccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & i & -i & -i & i \\
1 & -1 & -1 & 1 & -i & i & i & -i \\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\
1 & -i & i & -1 & \sqrt{2} & 0 & 0 & -\sqrt{2} \\
1 & i & -i & -1 & 0 & -i \sqrt{2} & i \sqrt{2} & 0 \\
1 & i & -i & -1 & 0 & i \sqrt{2} & -i \sqrt{2} & 0 \\
1 & -i & i & -1 & -\sqrt{2} & 0 & 0 & \sqrt{2} \\
\end{array}
\right)$$
$$M_3 := \left(
\begin{array}{cccccccc}
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & -1 & -1 & 1 & -i & i & i & -i \\
1 & -1 & -1 & 1 & i & -i & -i & i \\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \\
1 & i & -i & -1 & -\sqrt{2} & 0 & 0 & \sqrt{2} \\
1 & -i & i & -1 & 0 & -i \sqrt{2} & i \sqrt{2} & 0 \\
1 & -i & i & -1 & 0 & i \sqrt{2} & -i \sqrt{2} & 0 \\
1 & i & -i & -1 & \sqrt{2} & 0 & 0 & -\sqrt{2} \\
\end{array}
\right)$$
Finally, we construct three triplets by combining the pairings given above. We emphasize that these triplets are just a selection of examples, and there are many more pairings and pairing combinations that are possible.
We denote by $\mathcal{H}_*$ for $* \in \{{\operatorname}{A},{\operatorname}{B},{\operatorname}{C}\}$ the Hopf triplet defined as follows. The consistituent $\alpha,\beta$ and $\kappa$ Hopf algebras are, for each triplet, simply equal to the $8$-dimensional algebra $H_8$. The pairings are given as so.
- For $\mathcal{H}_{{\operatorname}{A}}$, the pairings are defined to be the pairings $\langle-\rangle_1$. That is $$\langle-\rangle_{\alpha\beta} = \langle-\rangle_{\beta\kappa} = \langle-\rangle_{\kappa\alpha} = \langle-\rangle_1$$
- For $\mathcal{H}_{{\operatorname}{B}}$, the pairings are defined to vary as follows. $$\langle-\rangle_{\alpha\beta} = \langle-\rangle_1 \qquad \langle-\rangle_{\beta\kappa} = \langle-\rangle_2 \qquad \langle-\rangle_{\kappa\alpha} = \langle-\rangle_3$$
- For $\mathcal{H}_{{\operatorname}{C}}$, the pairings are defined to vary as follows. $$\langle-\rangle_{\alpha\beta} = \langle-\rangle_0 \qquad \langle-\rangle_{\beta\kappa} = \langle-\rangle_1 \qquad \langle-\rangle_{\kappa\alpha} = \langle-\rangle_1$$
Of course, one must verify that the pairings and triplets satisfy the necessary properties. We verified this Lemma computationally using a script available at [@pythonscript2019].
The tuple $(H_8,H_8,\langle-\rangle_i)$ is a Hopf doublet for each $i \in \{0,1,2,3\}$. Furthermore, the tuple $\mathcal{H}_*$ for each $* \in \{{\operatorname}{A},{\operatorname}{B},{\operatorname}{C}\}$ is an (involutory and trisection admissible) Hopf triplet.
We now conclude this section with Table \[table:trisection\_invariant\_with\_8d\_examples\], where we record the trisection invariants $\tau_{\mathcal{H}_{*}}(-)$ for $* \in \{{\operatorname}{A},{\operatorname}{B},{\operatorname}{C}\}$ and the trisections in Examples \[ex:CP2\_trisection\]-\[ex:S1xS3\_trisection\]. These invariants were calculated using the methods described in §\[subsec:computational\_methods\_and\_scripting\].
\[table:trisection\_invariant\_with\_8d\_examples\]
$X$ $\chi(X)$ $\mathcal{H}_{{\operatorname}{A}}$ $\mathcal{H}_{{\operatorname}{B}}$ $\mathcal{H}_{{\operatorname}{C}}$
-------------------------- ----------- ------------------------------------ ------------------------------------ ------------------------------------
$S^4$ 2 1 1 1
$\mathbb{C}P^2$ 3 $\frac{-1 + i}{2\sqrt{2}}$ $\frac{1 + i}{2\sqrt{2}}$ $0$
$S^3 \times S^1$ $0$ $8$ $8$ $2^{8/3}$
$S^2 \times S^2$ $4$ $\frac{1}{4}$ $\frac{1}{4}$ $2^{-2/3}$
$S^2 \tilde{\times} S^2$ $4$ $\frac{1}{4}$ $\frac{1}{4}$ $0$
: Computations of trisection invariant for $\mathcal{H}_{{\operatorname}{A}}, \mathcal{H}_{{\operatorname}{B}}$ and $\mathcal{H}_{{\operatorname}{C}}$.
Relation to the Dichromatic Invariant {#sec:CY_dichro}
=====================================
In this section, we review the dichromatic invariant (§\[subsec:dichro\]). We then (§\[subsec:trisection\_dichro\]) formally restate and prove Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\] (as Theorem \[thm:tri=dichro\]).
Before beginning, we provide the reader with some historical discussion of the dichromatic invariant and its relation to the Crane-Yetter invariant.
The Crane-Yetter invariant is an invariant of closed oriented 4-manifolds first defined in [@crane1993categorical] based on a semisimple quotient of ${\text{Rep}}(U_q(sl_2))$ at some root of unity, which is a special example of modular tensor categories. The invariant was later generalized to take as input any ribbon fusion category [@crane1997state], not necessary modular. In both cases, the invariant takes the form of a weighted state-sum on a triangulation.
Using skein-theoretical methods, Roberts [@roberts1995skein] introduced a Broda-type invariant of 4-manifolds again based on the semisimple quotient of ${\text{Rep}}(U_q(sl_2))$. In [@roberts1995skein], Roberts showed that his invariant is equal to the Crane-Yetter invariant associated to ${\text{Rep}}(U_q(sl_2))$, up to a factor involving Euler characteristics. He also showed that his invariant can be expressed in terms of the signature of the 4-manifold. In fact, Roberts’ definition extends in a straightforward way to take as input any modular tensor category and the resulting invariant has the same relation with the Crane-Yetter invariant as in the ${\text{Rep}}(U_q(sl_2))$ case. This implies that the modular Crane-Yetter invariant only involves only the signature and the Euler characteristic.
The existence of a Broda-type reformulation of the Crane-Yetter invariant for premodular categories (i.e., ribbon fusion categories that are not modular) remained open until the recent progress in [@barenz2016dichromatic]. Generalizing the work of [@petit2008dichromatic; @roberts1995skein], the authors of [@barenz2016dichromatic] defined a Broda-type invariant (called the dichromatic invariant) of 4-manifolds based on a pivotal functor $F: {\mathcal{C}}\rightarrow {\mathcal{D}}$ where ${\mathcal{C}}$ is a spherical fusion category and ${\mathcal{D}}$ is a ribbon fusion category.
Among other properties, the authors of [@barenz2016dichromatic] showed that if ${\mathcal{C}}$ is a ribbon fusion category, ${\mathcal{D}}$ is modular, and $F$ is a full inclusion, then the corresponding invariant depends only on ${\mathcal{C}}$ and it recovers the Crane-Yetter invariant associated with ${\mathcal{C}}$. They also showed that the premodular Crane-Yetter invariant contains strictly more information than the signature and the Euler characteristic combined.
Review of the Dichromatic Invariant {#subsec:dichro}
-----------------------------------
We now present a brief description of the dichromatic invariant. We refer the reader to [@barenz2016dichromatic] for a more detailed treatment.
For basics on fusion categories (spherical, ribbon, modular), see for instance [@bojko2001lectures; @Wang2010topological]. For a detailed introduction of picture calculus in ribbon categories see [@turaev1994quantum]. In particular, we follow the conventions in [@turaev1994quantum] for the evaluation of ribbon graphs. The graphs are evaluated from bottom to top. A strand labeled by an object $V$ is interpreted as $\text{Id}_V$ if it is directed downwards and as $\text{Id}_{V^*}$ if directed upwards. A positive crossing denotes the braiding and a negative crossing denotes the inverse of the braiding. See Figure \[fig:ribbongraph\] for examples of the evaluation of some ribbon graphs.
(0,0)node\[left\][$V$]{} – (0,2); (0,-0.5) node[$\text{Id}_V$]{};
(2,0)node\[left\][$V$]{} – (2,2); (2,-0.5) node[$\text{Id}_{V^*}$]{};
(2,0)node\[right\][$W$]{} – (0,2); (0,0) – (2,2); (0,0)node\[left\][$V$]{} – (2,2); (1,-0.5) node[$c_{V, W^*}$]{};
(0,0)node\[left\][$V$]{} – (2,2); (2,0)node\[right\][$W$]{} – (0,2); (2,0)node\[right\][$W$]{} – (0,2); (1,-0.5) node[$c_{V, W}^{-1}$]{};
Let ${\mathcal{C}}$ be spherical fusion category and $S({\mathcal{C}})$ be a complete set of representatives of simple objects of ${\mathcal{C}}$, namely, $S({\mathcal{C}})$ contains a representative for each isomorphism class of simple objects. For an object $a \in {\mathcal{C}}$, let $d(a):= d_a$ be the quantum dimension of $a$. Next we introduce the formal object, sometimes called the Kirby color, $\omega_{{\mathcal{C}}}:= \sum_{a \in S({\mathcal{C}})} d_a \, a$, and call $d(\omega_{{\mathcal{C}}}) := \sum_{a \in S({\mathcal{C}})}d_a^2$ the dimension of ${\mathcal{C}}$. Let ${\mathcal{D}}$ be a braided fusion category and denote by ${\mathcal{D}}'$ the symmetric center (or Muger center) of ${\mathcal{D}}$. If $A \in {\mathcal{D}}$ is an object or a formal object, denote by $A'$ the subobject of $A$ which lies in ${\mathcal{D}}'$.
Let $F: {\mathcal{C}}\rightarrow {\mathcal{D}}$ be a pivotal functor such that all objects in ${\mathcal{D}}'$ have trivial twists. For a closed 4-manifold $X$, choose for $X$ a surgery link $L = L_1 \sqcup L_2$, where $L_1$ is a 0-framed unlink (of dotted circles) representing the 1-handles and $L_2$ is a framed link for attaching 2-handles. Label each component of $L_1$ by $\omega_{{\mathcal{D}}}$ and each component of $L_2$ by $F\omega_{{\mathcal{C}}}$. Then evaluate $L$ with the above labels to a complex number $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}})$. Lastly, denote by $|L_i|$ the number of components of $L_i$. The dichromatic invariant $I_F(X)$ is defined by [@barenz2016dichromatic]: $$I_F(X) := \frac{1}{d(\omega_{{\mathcal{C}}})^{|L_2|-|L_1|} \left(d(\omega_{{\mathcal{D}}})d((F\omega_{{\mathcal{C}}})')\right)^{|L_1|}} L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}}).$$
If ${\mathcal{C}}$ is ribbon fusion, ${\mathcal{D}}$ is modular, and $F$ is a full braided inclusion, then $I_{{\mathcal{C}}}:=I_F$ turns out to depend only on ${\mathcal{C}}$ and furthermore, it recovers the Crane-Yetter invariant: $$\label{equ:dichro=CY}
I_{{\mathcal{C}}}(X) = \text{CY}_{{\mathcal{C}}}(X) d(\omega_{{\mathcal{C}}})^{1-\chi(X)}.$$
Given two semisimple Hopf algebras $H$ and $ K$, and a Hopf algebra morphism $\phi: D(H) \to K$, there is an induced pivotal functor ${\text{Rep}}(\phi): {\text{Rep}}(K) \to {\text{Rep}}(D(H))$, where, for a representation $V$ of $K$, ${\text{Rep}}(\phi)(V)$ is the same as $V$ but viewed as a representation of $D(H)$ via $\phi$. It is well known that ${\text{Rep}}(D(H))$ is equivalent to the Drinfeld center of ${\text{Rep}}(H)$ and thus is modular. In the following we give a description of the invariant $I_{{\text{Rep}}(\phi)}$ in terms of Hopf algebras.
For any semisimple $H$, choose a two-sided integral $e \in H, \mu \in H^*$ such that $\mu(e) = 1,\ \epsilon(e) = 1$. Then $e$ is central, $e^2 = e$ is a projector, and $\mu(1)=\dim(H)$. In fact, for $a \in H, f \in H^*$, $\mu(a) = {{\operatorname}{Tr}}(L_a)$ and $f(e) = \frac{1}{\dim(H)}{{\operatorname}{Tr}}(L_{f})$, where $L_a$ (respectfully $L_f$) denotes the left multiplication by $a$ (respectfully by $f$).
\[lem:proj\_trivial\] Let $e \in H$ be the two-sided integral as above and $V$ be a representation of $H$ with the action given by $\rho: H \rightarrow {{\operatorname}{End}}(V)$. Then $V$ has a decomposition $V = {{\operatorname}{Im}}\rho_e \oplus {{\operatorname}{Ker}}\rho_e$ as representations, and $\rho_e$ is a projection onto the trivial subrepresentation of $V$.
This is straightforward by noting that the the trivial irreducible representation of $H$ is ${{\mathbb C}}$ with the action given by $\epsilon$ and that $e$ is a central projector.
\[lem:encircling\] Let ${\mathcal{C}}= {\text{Rep}}(H)$ and $F: {\mathcal{C}}\rightarrow {\mathcal{D}}$ be a full braided inclusion of ${\mathcal{C}}$ into a modular tensor category ${\mathcal{D}}$. Let $e \in H$ be the two-sided integral as above. For any object $V \in {\mathcal{C}}$ with the action $\rho: H \rightarrow {{\operatorname}{End}}(V)$, the following equality holds:
(1,0) arc(0:180: 1cm and 0.4cm); (0,0) – (0,1); (0,0) – (0,1);
(0,0) – (0,-1)node\[left\][$F(V)$]{}; (1,0) arc(0:-180: 1cm and 0.4cm); (1,0) arc(0:-180: 1cm and 0.4cm) node\[left, pos = 1\][$\omega_{\mathcal{D}}$]{};
(2,0) node[$=$]{};
(0,-1) – (0,1); (-0.5, -0.3) rectangle (0.5, 0.3) node\[pos = .5\][$F(\rho_e)$]{}; (-1.2,0) node[$d(\omega_{\mathcal{D}})$]{};
Decompose $V$ as $V = V_1 \oplus V_2$, where $V_1$ contains all copies of the unit object in $V$. Choose $\pi_i: V \to V_i, \ \iota_i: V_i \to V$ such that $\pi_j \circ \iota_i = \delta_{ij} \text{Id}_{V_i}$ and $\iota_1\circ \pi_1+ \iota_2\circ \pi_2 = \text{Id}_{V}$. Since $F$ is a full inclusion, $F(V)$ decomposes as $F(V) = F(V_1) \oplus F(V_2)$ where $F(V_1)$ corresponds to the subobject of $F(V)$ containing all copies of the unit object, and $F(\pi_i)$ and $F(\iota_i)$ satisfy similar relations as above.
It is well known (see for instance [@lickorish1993skein; @bojko2001lectures]) that the left-hand side of the equality in this lemma is equal to $d(\omega_{{\mathcal{D}}})$ times $ F(\iota_1) \circ F(\pi_1)$ which is the identity on $F(V_1)$ and the zero map on $F(V_2)$. By Lemma \[lem:proj\_trivial\], $\iota_1 \circ \pi_1 = \rho_e$, and hence the equality follows.
For simplicity, we start with the special case of $H$ being quasi-triangular, $K = H,$ and $ \phi : D(H) \to H$ is given by $\phi:= \phi_H:= M \circ (f_R \otimes \text{Id})$, where $f_R(q):= (q \otimes \text{Id})R$. See Example \[ex:basic\_examples\_of\_triplets\] from Section \[subsubsec:doublets\_and\_triplets\]. Given a surgery link $L = L_1 \sqcup L_2$ for some 4-manifold, present $L$ as a planar link diagram with respect to a height function such that the crossings are not critical points and that the framing of $L$ is given by the blackboard framing. We associate to $L$ the following tensors. Orient the components of $L_2$ arbitrarily. For a non-critical point $p \in L_2$, let $c(p) = 0$ if the orientation of $L_2$ near $p$ is downwards and $c(p) = 1$ otherwise.
For each dotted circle in $L_1$, there can be some strands of $L_2$ intersecting the bounding disk of it. Denote the intersection points from left to right by $p_1,..., p_n$. Then assign to the dotted circle the tensor $(S^{c(p_1)} \otimes \cdots \otimes S^{c(p_n)})\Delta^n(e)$, where $\Delta^n(x) = \sum x^{(1)} \otimes \cdots \otimes x^{(n)}$. See Figure \[fig:dotted\_circle\_tensor\] (Left). The $i$-th outgoing leg of the tensor is associated with $p_i$. For each crossing of $L_2$, pick a point $q_1$ (respectfully $q_2$) on the overcrossing (respectfully undercrossing) strand near the crossing. If the crossing is positive (ignoring the orientation), then assign to it the tensor $(S^{c(q_1)} \otimes S^{c(q_2)})R$ where the first outgoing leg is associated with $q_1$ and the second with $q_2$. If the crossing is negative, replace $R$ by $R^{-1}$ in the above tensor. See Figure \[fig:crossing\_tensor\]. Call all the previously chosen points “labeled points”. Finally, for each component of $L_2$, assume there are $m$ labeled points on it. Then assign to it the tensor $\mu \circ M^m$, where the incoming legs are arranged according to the orientation of the link component and each of them corresponds to a labeled point. No base point is needed since $\mu \circ M^m$ is cyclically invariant. See Figure \[fig:dotted\_circle\_tensor\] (Right). We define $L(H)$ to be the contraction of all the tensors assigned above.
(2,0) arc(0:180: 2cm and 0.4cm);
(-1,0) – (-1,1); (0,0) – (0,1); (1,0) – (1,1); (-1,0) – (-1,1); (0,0) – (0,1); (1,0) – (1,1);
(-1,0) – (-1,-1); (0,0) – (0,-1); (1,0) – (1,-1); (2,0) arc(0:-180: 2cm and 0.4cm); (2,0) arc(0:-180: 2cm and 0.4cm); (0.5, 0.8) node[$\cdots$]{};
(-1,0) circle(2pt); (0,-0.1) circle(2pt); (0,0) circle(2pt); (1,0) circle(2pt);
(-1,0)node\[left\][$p_1$]{}; (0,0)node\[left\][$p_2$]{}; (1,0)node\[left\][$p_n$]{};
(-3,-2) node(Delta)[$\Delta$]{}; (-4,-2) node(e) [$e$]{}; (-2,-1) node(S1)[$S^{c(p_1)}$]{}; (-1.7,-2) node(S2)[$S^{c(p_2)}$]{}; (-2,-3) node(S3)[$S^{c(p_n)}$]{}; (-1.5,-2.3) node[$\vdots$]{}; (-1,0) node(p1); (0,0) node(p2); (1,0) node(p3);
\(e) to (Delta); (Delta) to (S1); (Delta) to (S2); (Delta) to (S3);
(S1) to (p1); (S1) to (p1);
(S2) to\[out = 0, in = -120\] (p2); (S2) to\[out = 0, in = -120\] (p2);
(S3) to\[out = 0, in = -120\] (p3); (S3) to\[out = 0, in = -120\] (p3);
(0,0) circle(1.5cm); ([1.5\*cos(135)]{}, [1.5\*sin(135)]{}) circle(2pt); ([1.5\*cos(165)]{}, [1.5\*sin(165)]{}) circle(2pt); ([1.5\*cos(210)]{}, [1.5\*sin(210)]{}) circle(2pt);
([1.5\*cos(135)]{}, [1.5\*sin(135)]{}) node(p1); ([1.5\*cos(165)]{}, [1.5\*sin(165)]{}) node(p2); ([1.5\*cos(210)]{}, [1.5\*sin(210)]{}) node(p3);
(0,0) node(M)[$M$]{}; (1,0) node(mu)[$\mu$]{};
(p1) to (M); (p2) to (M); (p3) to (M); (M) to (mu);
(-1, 0) node[$\vdots$]{};
(3,0) – (1,3); (0,0) – (2,3); (0,0) – (2,3);
(1.5, -1) node(R)[$R$]{}; (1, 0) node(S1)[$S^{c(q_1)}$]{}; (2, 0) node(S2)[$S^{c(q_2)}$]{};
(1.3, [1.3\*1.5]{}) circle(2pt); (1.7, [1.3\*1.5]{}) circle(2pt);
(1.3, [1.3\*1.5]{}) node\[left\] [$q_1$]{}; (1.7, [1.3\*1.5]{}) node\[right\] [$q_2$]{};
(1.3, [1.3\*1.5]{}) node(q1); (1.7, [1.3\*1.5]{}) node(q2) ;
\(R) to (S1); (R) to (S2); (S1) to\[out = 90, in = -90\] (q1); (S2) to\[out = 90, in = -90\] (q2);
(0,0) – (2,3); (3,0) – (1,3); (3,0) – (1,3);
(1.5, -1) node(R)[$R^{-1}$]{}; (2, 0) node(S1)[$S^{c(q_1)}$]{}; (1, 0) node(S2)[$S^{c(q_2)}$]{};
(1.3, [1.3\*1.5]{}) circle(2pt); (1.7, [1.3\*1.5]{}) circle(2pt);
(1.3, [1.3\*1.5]{}) node\[left\] [$q_2$]{}; (1.7, [1.3\*1.5]{}) node\[right\] [$q_1$]{};
(1.3, [1.3\*1.5]{}) node(q2); (1.7, [1.3\*1.5]{}) node(q1) ;
\(R) to (S1); (R) to (S2); (S1) to\[out = 90, in = -90\] (q1); (S2) to\[out = 90, in = -90\] (q2);
By the properties $\mu \circ S = \mu$ and using that $S$ is an anti-algebra morphism, it is direct to check that $L(H)$ is independent of the choice of orientations of $L$.
When $H$ is factorizable, $L(H)$ is essentially the link invariant introduced in [@kauffman1995invariants] in reformulating the Hennings invariant [@hennings1996invariants]. However, when $H$ is not factorizable, these two are different. See also [@chang2019two] for an exposition of the link invariant.
Let ${\mathcal{C}}= {\text{Rep}}(H)$. Then $\omega_{{\mathcal{C}}} = H$ where $H$ is viewed as a representation of $H$ by left multiplication, and $d(\omega_{{\mathcal{C}}}) = \dim(H)$. Choose any modular tensor category ${\mathcal{D}}$ such that there is a full braided inclusion $F: {\mathcal{C}}\to {\mathcal{D}}$. For instance, one can take ${\mathcal{D}}= {\text{Rep}}(D(H))$ and $F = {\text{Rep}}(\phi_H)$ as above. Then $(F\omega_{{\mathcal{C}}})'$ contains only the unit object and hence $d((F\omega_{{\mathcal{C}}})') = 1$.
\[thm:dichro\_reform\] Let $L, H, {\mathcal{C}}, {\mathcal{D}}, F$ be as above, then $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}}) = d(\omega_{{\mathcal{D}}})^{|L_1|} L(H)$. Therefore for a 4-manifold $X$ with surgery link $L$, $$I_{{\text{Rep}}(H)}(X) = \frac{1}{\dim (H)^{|L_2|-|L_1|}}\,L(H)\,.$$
Label each component of $L_1$ by $\omega_{{\mathcal{D}}}$ and each component of $L_2$ by $F\omega_{{\mathcal{C}}} = F(H)$. Since both $\omega_{{\mathcal{D}}}$ and $F\omega_{{\mathcal{C}}}$ are self-dual, the orientation of $L$ does not change its evaluation. Hence, we orient each component of $L$ arbitrarily. Also, present $L$ as a planar link diagram as before.
Recall that a strand of $L_2$ directed downwards represents $F(H)$ and a strand directed upwards represents $F(H)^* = F(H^*)$. To compute $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}})$, by Lemma \[lem:encircling\], we can replace each dotted circle by $d(\omega_{{\mathcal{D}}}) F(\rho_e)$, where $\rho_e$ denotes the action of $e \in H$ on the strands intersecting the disk bounded by the dotted circle. Specifically, arrange all the intersecting strands from left to right and denote them by $K_1, K_2, \cdots, K_n$. See Figure \[fig:remove\_dotted\_circle\]. Assume $\Delta^{n}(e) = \sum e^{(1)} \otimes \cdots \otimes e^{(n)} $ and denote the left multiplication of $e$ on $H$ by $L_e$. Then, $$\rho_e = \sum_{e^{(i)}} \tilde{L}_{e^{(1)}} \otimes \cdots \otimes \tilde{L}_{e^{(n)}},$$ where $\tilde{L}_{e^{(i)}} = L_{e^{(i)}}$ if $K_i$ is directed downwards near the disk, and otherwise $\tilde{L}_{e^{(i)}} = L_{S(e^{(i)})}^*$, the linear dual of $L_{S(e^{(i)})}$.
(2,0) arc(0:180: 2cm and 0.4cm); (-1.5,0) – (-1.5,1); (0,0) – (0,1); (1.5,0) – (1.5,1); (-1.5,0) – (-1.5,1); (0,0) – (0,1); (1.5,0) – (1.5,1);
(-1.5,0) – (-1.5,-1)node\[below\][$K_1$]{}; (0,0) – (0,-1)node\[below\][$K_2$]{}; (1.5,0) – (1.5,-1)node\[below\][$K_n$]{}; (2,0) arc(0:-180: 2cm and 0.4cm); (2,0) arc(0:-180: 2cm and 0.4cm) node\[left, pos = 1\][$\omega_{\mathcal{D}}$]{}; (0.75, -0.8) node[$\cdots$]{};
(3,0) – (4,0);
(-1.5,0) – (-1.5,1); (0,0) – (0,1); (1.5,0) – (1.5,1); (-1.5,0) – (-1.5,-1)node\[below\][$K_1$]{}; (0,0) – (0,-1)node\[below\][$K_2$]{}; (1.5,0) – (1.5,-1)node\[below\][$K_n$]{}; (0.75, -0.8) node[$\cdots$]{};
(-0.5-1.5, -0.3) rectangle (0.5-1.5, 0.3) node\[pos = .5\][$L_{e^{(1)}}$]{}; (-0.5, -0.3) rectangle (0.5, 0.3) node\[pos = .5\][$L^*_{e^{(2)}}$]{}; (-0.5+1.5, -0.3) rectangle (0.5+1.5, 0.3) node\[pos = .5\][$L_{e^{(n)}}$]{};
(-2.5,0) node[$\sum\limits_{e^{(i)}}$]{};
(-2.7, -1) to\[round left paren\] (-2.7, 1); (2.2, -1) to\[round right paren\] (2.2, 1);
(-3.3,0) node[$F$]{};
Away from the dotted circles, the constituents of the link consist of crossings, caps, and cups, all from $L_2$. Since $F$ is a braided inclusion, we can hence forget about $F$, label all components of $L_2$ by $H$, and replace each dotted circle by $d(\omega_{{\mathcal{D}}}) \rho_e$ as discussed in the previous paragraph. We can then evaluate $L_2$ entirely within ${\mathcal{C}}$. Note that after removing all the dotting circles, we obtain an extra scalar factor $d(\omega_{{\mathcal{D}}})^{|L_1|}$. (This also shows that $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}}) $ depends on ${\mathcal{D}}$ only for the factor $d(\omega_{{\mathcal{D}}})^{|L_1|}$ which is canceled later in the renormalization of $I_F$ and hence $I_F$ depends only on ${\mathcal{C}}$.)
At each crossing of $L_2$, denote the over-crossing strand by $K_1$ and the under-crossing strand by $K_2$. If the crossing is positive, then the morphism it represents is $\sum \tilde{L}_{R_1} \otimes \tilde{L}_{R_2}$ where $R = \sum R_1 \otimes R_2$ and $\tilde{L}_{R_i}$ acts on the strand $K_i$ with the same convention as before. If the crossing is negative, then replace $R$ by $R^{-1}$.
Now for each component of $L_2$, the maps on it (fixing a summation term) have the form of either $L_{a}$ or $L_{S(b)}^*$ depending on the orientation. Also note that $\mu(a) = {{\operatorname}{Tr}}L_{a}$. Then it follows that the evaluation of that component is obtained by multiplying the $a\,'$s and $S(b)\,'$s along the orientation followed by the action of $\mu$. In other words, we apply the tensor as in Figure \[fig:dotted\_circle\_tensor\] (Right) to get the evaluation. Hence we have $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}}) = d(\omega_{{\mathcal{D}}})^{|L_1|} L(H)$. The second part in the proposition follows immediately.
In the general case $\phi: D(H) \to K$ for two Hopf algebras $H, K$ not necessarily quasi-triangular, the description of $I_{{\text{Rep}}(\phi)}$ for the induced functor ${\text{Rep}}(\phi)$ can be given analogously. Choose two-sided integrals $e_H \in H, \mu_H \in H^*$ as above. To avoid confusion, we have introduced subscripts to indicate which Hopf algebra we are working with. Similarly choose integrals $e_K \in K, \mu_K \in K^*, e_D \in D(H), \mu_D \in D(H)^*$. It can be checked that $e_D = \frac{1}{\dim (H)}\, \mu_H \otimes e_H, \ \mu_D = \dim (H) \,e_H \otimes \mu_H$. Let $\phi_1$ (respectfully $\phi_2$) be the restriction of $\phi$ on $H^{*,{\text{cop}}}$ (respectfully $H$), then $\phi = M_K \circ (\phi_1 \otimes \phi_2)$. Also the universal $R$-matrix for $D(H)$ is given by $R_D = \sum_{i} (\epsilon \otimes v_i) \otimes (v_i^* \otimes 1)$, where $\{v_1, v_2,...\}$ is a basis of $H$.
Given a surgery link $L = L_1 \sqcup L_2$ representing a manifold $X$, we introduce $L(H, K; \phi)$ which is similar to $L(H)$, and we only point out the relevant modifications based on $L(H)$. Firstly, for the tensor assigned to each dotted circle, replace $e$ by $\tilde{e}:= \phi(e_D) = \phi(\frac{1}{\dim(H)} \mu_H \otimes e_H)$. Secondly, for each crossing of $L_2$, replace the $R$-matrix by $\tilde{R}:= (\phi \otimes \phi)R_D = \sum_{i} \phi_2(v_i) \otimes \phi_1(v_i^*)$. It can be checked that $\tilde{R}^{-1} =\sum_{i} \phi_2(S(v_i)) \otimes \phi_1(v_i^*) $. Lastly, due to the use of notations, replace all operations taking in $H$ with the relevant operations in $K$. Letting ${\mathcal{C}}= {\text{Rep}}(K), {\mathcal{D}}= {\text{Rep}}(D(H))$, then we have $L(\omega_{{\mathcal{D}}}, F\omega_{{\mathcal{C}}}) = d(\omega_{{\mathcal{D}}})^{|L_1|} L(H,K;\phi)$. Note that $\omega_{{\mathcal{C}}} = K, \ d(\omega_{{\mathcal{C}}}) = \dim(K), $ and $ d(({\text{Rep}}(\phi) \omega_{{\mathcal{C}}})')$ is the rank of map $L_{\tilde{e}}$. Since $L_{\tilde{e}}$ is a projector, its rank is equal to its trace. Hence $d(({\text{Rep}}(\phi) \omega_{{\mathcal{C}}})') = {{\operatorname}{Tr}}(L_{\tilde{e}}) = \mu_K(\tilde{e})$. We have $$I_{{\text{Rep}}(\phi)}(X) = \frac{1}{\dim (K)^{|L_2|-|L_1|}} \frac{1}{(\mu_K(\tilde{e}))^{|L_1|}}\,L(H,K;\phi).$$
Proof of Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\]/\[thm:tri=dichro\] {#subsec:trisection_dichro}
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Let $H, K$ be any semisimple Hopf algebras and $\phi: D(H) \to K$ be a Hopf algebra morphism. Denote the restriction of $\phi$ on $H^{*,{\text{cop}}}$ by $\phi_1: H^{*,{\text{cop}}} \to K$ and that on $H$ by $\phi_2: H \to K$. Then both $\phi_1 $ and $\phi_2$ are Hopf algebra morphisms and $\phi = M \circ (\phi_1 \otimes \phi_2)$. According to Corollary \[cor:map\_version\_of\_triplet\], we can construct a Hopf triplet ${\mathcal{H}_{H, K; \phi}} = (H_{\alpha}, H_{\beta}, H_{\kappa}; \langle\;, \;\rangle)$ by letting $H_{\alpha} = H^{*, {\text{cop}}, {\text{op}}},\ H_{\beta} = H^{{\text{cop}}}$ and $H_{\kappa} = K^*$ with the pairings defined as follows. The pairing ${\langle - \rangle}_{\alpha,\beta}$ on $H_{\alpha} \otimes H_{\beta}$ is the canonical one, ${\langle - \rangle}_{\beta,\kappa}$ on $H_{\beta} \otimes H_{\kappa}$ is given by $\langle h, p\rangle_{\beta,\kappa} = p \circ \phi_2(h)$, and ${\langle - \rangle}_{\kappa,\alpha}$ on $H_{\kappa} \otimes H_{\alpha}$ is given by $\langle p, q\rangle_{\kappa,\alpha} = p \circ \phi_1(q)$.
In particular, if $(H, R)$ is a quasi-triangular Hopf algebra with the universal $R$-matrix $R \in H \otimes H$. Then $\phi_H: D(H) \to H$ defined by $\phi_H = M \circ (f_R \otimes Id)$ is a Hopf algebra morphism, where $f_R(q) := (q \otimes Id)R$. Hence one can obtain the Hopf triplet $\mathcal{H}_H := {\mathcal{H}_{H, H; \phi_H}}$ (see also Example \[ex:basic\_examples\_of\_triplets\]).
Let ${\text{Rep}}(\phi): {\text{Rep}}(K) \to {\text{Rep}}(D(H))$ be the induced functor. Recall that $\tilde{e} = \phi(e_D) = \phi(\frac{1}{\dim(H)} \,\mu_H \otimes e_H)$, where $e_H, \mu_H$ are integrals defined in §\[subsec:dichro\]. In this subsection, we prove the following theorem, which is the promised refinement of Theorem \[thm:trisection\_vs\_crane\_yetter\_informal\] which we stated in the Introduction.
\[thm:tri=dichro\] For the above setup and for any closed $4$-manifold $X$, the following equality holds, $$\tau_{{\mathcal{H}_{H,K;\phi}}}(X) = \left[\frac{\dim(H)\mu_K(\tilde{e})}{\dim(K)}\right]^{\frac{2-\chi(X)}{3}}\,I_{{\text{Rep}}(\phi)}(X),$$ where $\chi(X)$ is the Euler characteristic of $X$.
If $H$ is quasi-triangular and we take $K = H, \phi = M \circ (f_R \otimes \text{Id})$, it is direct to check that $\mu_K(\tilde{e}) = 1$, and hence $\tau_{{\mathcal{H}_{H}}}(X) = I_{{\text{Rep}}(H)}(X)$.
The following corollary follows immediately from Theorem \[thm:tri=dichro\] and Equation \[equ:dichro=CY\].
Let $H$ be any quasi-triangular semisimple Hopf algebra. Then for any closed $4$-manifold $X$, $\tau_{{\mathcal{H}_{H}}}(X) = \text{CY}_{{\text{Rep}}(H)}(X) \dim(H)^{1-\chi(X)}$.
Given a generalized Drinfeld double $D_{\varphi}(H_{\alpha}, H_{\beta})$ where $\varphi: H_{\alpha} \to H_{\beta}^{*,{\text{cop}}}$ is a Hopf algebra morphism, in general ${\text{Rep}}(D_{\varphi}(H_{\alpha}, H_{\beta}))$ is not a braided fusion category. By [@burciu2012irreducible], it is equivalent to $\mathcal{Z}_{{\text{Rep}}(\varphi^{*,{\text{op}}})}({\text{Rep}}(H_{\beta}))$, the relative center of ${\text{Rep}}(\varphi^{*,{\text{op}}})$, where ${\text{Rep}}(\varphi^{*,{\text{op}}}): {\text{Rep}}(H_{\alpha}^{*,{\text{op}}}) \to {\text{Rep}}(H_{\beta})$ is the induced functor of $\varphi^{*,{\text{op}}}: H_{\beta} \to H_{\alpha}^{*,{\text{op}}} $. Therefore, given a Hopf algebra morphism $\phi: D_{\varphi}(H_{\alpha}, H_{\beta}) \to H_{\kappa}$, the induced functor ${\text{Rep}}(\phi): {\text{Rep}}(H_{\kappa}) \to {\text{Rep}}(D_{\varphi}(H_{\alpha}, H_{\beta})) $ can not be used to define the dichromatic invariant. This suggests that the trisection invariant arising from the most general Hopf triplet $(H_{\alpha}, H_{\beta}, H_{\kappa}; \langle-\rangle)$ is different from the dichromatic invariant.
The rest of the subsection is devoted to the proof of Theorem \[thm:tri=dichro\]. For readers who are familiar with quasi-triangular Hopf algebras, they may find it easier to follow the proof first for the special case of $H$ being quasi-triangular and $\phi = M \circ (f_R \otimes \text{Id})$.
The trisection invariant is defined in terms of trisection diagrams while the dichromatic invariant is defined in terms of surgery links. So we need a translation between the trisection diagrams and surgery links. We provide a concrete translation.
Think of ${{S}}^3 = {{\mathbb R}}^3 \cup \{\infty\}$, ${{S}}^2 = ({{\mathbb R}}^2 \times \{0\}) \cup \{\infty\}$. We identify ${{\mathbb R}}^2$ with ${{\mathbb R}}^2 \times \{0\}$. Denote by $D_{(x,y)}(r)$ the $2$-ball centered at $(x,y)$ with radius $r$. Remove the balls $D_{(\pm 1, i)}(\epsilon) \subset {{S}}^2$, $i = 1,...,g$, for some small $\epsilon \ll 1$. Let ${{S}}_{(\pm 1,i)}(\epsilon) = \partial D_{(\pm 1, i)}(\epsilon) $. We identify ${{S}}_{( 1,i)}(\epsilon)$ with ${{S}}_{( -1,i)}(\epsilon)$ by the reflection about the $y$-axis. The resulting surface is clearly $\Sigma_g$, a closed surface of genus $g$. Equivalently, $\Sigma_g$ can be thought as being obtained by removing the $D_{(\pm 1, i)}(\epsilon)\,'$s from ${{S}}^2$ and then for each $i$, gluing a bridge’ $h_i = {{S}}^1 \times [0,1]$ such that ${{S}}^1 \times \{0\}$ is identified with $S_{(-1,i)}(\epsilon)$ and ${{S}}^1 \times \{1\}$ with $S_{(1,i)}(\epsilon)$. To be explicit, we embed the interior of the bridges $h_i$ in ${{\mathbb R}}^2 \times (-\infty, 0)$ so that each of them is unlinked with the rest.
Let $0 \leq k \leq g$. For $1 \leq i \leq k$, let $\alpha_i = {{S}}_{(1,i)}(2\epsilon),\ \beta_i = {{S}}_{(1,i)}(3\epsilon)$. For $k+1 \leq i \leq g$, let $\alpha_i = {{S}}_{(1,i)}(2\epsilon)$ and $\beta_i$ be the longitude of $\Sigma_g$ passing through the $i$-th bridge $h_i$. Let $\alpha = \{\alpha_1, \cdots, \alpha_g\}$ and $\beta = \{\beta_1, \cdots, \beta_g\}$. Then $(\Sigma_g, \alpha,\beta)$ is the standard Heegaard diagram of $\#^k\; {{S}}^1 \times {{S}}^2$. See Figure \[fig:special\_trisection\] (Left).
For any closed 4-manifold $X$, we can always, for some $g,k$, choose a $(g,k)$ trisection diagram $T = (\Sigma_g, \alpha,\beta, \kappa)$ such that $(\Sigma_g, \alpha,\beta)$ is given as above. We can arrange the $\kappa$ curves so that they travel through the bridges $h_i\,'$s in parallel with the longitude. Moreover, every time a $\kappa$ curve travels through $h_i$, it crosses $\alpha_i$ once and if $i \leq k$ it also crosses $\beta_i$ once. The $\kappa$ curves are otherwise away from the $D_{(\pm 1, i)}(3\epsilon)$ disks.
(4,0) circle(0.5cm); (4,2) circle(0.5cm); (4,4) circle(0.5cm);
(4,0) circle(0.7cm); (4,2) circle(0.7cm); (0,4) to (4,4);
(0,0) to\[out = 180, in = 180, looseness = 1.5\] (0,2+0.2); (0,2-0.2) to\[out = 180, in = 180, looseness = 1.5\] (1,1) to\[out = 0,in = 0\] (1,3) to\[out = 180, in = 180,looseness = 1.5\] (0,4); (4,0) to\[out = 30, in = 0, looseness = 2\] (4, -1) to\[out = 180, in = 180\] (4,5) to\[out = 0, in = -30, looseness = 2\] (4,4); (4,2-0.2) to\[out = 0, in = 0, looseness = 2\] (4, -1.3) to\[out = 180, in = 180,looseness = 1.1\] (4,5.3) to\[out = 0, in = 0, looseness = 2\] (4,2+0.2);
(0,0) circle(0.3cm); (4,0) circle(0.3cm); (0,2) circle(0.3cm); (4,2) circle(0.3cm); (0,4) circle(0.3cm); (4,4) circle(0.3cm);
(0.5,0.5) to \[out = 0, in = 0,looseness = 0.7\] (0.5,-0.5); (0.5,2.5) to \[out = 0, in = 0,looseness = 0.7\] (0.5,1.5);
(0,0) to (4,0); (0,2-0.2) to (4,2-0.2); (0,2+0.2) to (4,2+0.2); (0,4) to (4,4);
(0,0) to (4,0); (0,2-0.2) to (4,2-0.2); (0,2+0.2) to (4,2+0.2); (0,4) to (4,4);
(0,0) to\[out = 180, in = 180, looseness = 1.5\] (0,2+0.2); (0,2-0.2) to\[out = 180, in = 180, looseness = 1.5\] (1,1) to\[out = 0,in = 0\] (1,3) to\[out = 180, in = 180,looseness = 1.5\] (0,4); (4,0) to\[out = 30, in = 0, looseness = 2\] (4, -1) to\[out = 180, in = 180\] (4,5) to\[out = 0, in = -30, looseness = 2\] (4,4); (4,2-0.2) to\[out = 0, in = 0, looseness = 2\] (4, -1.3) to\[out = 180, in = 180,looseness = 1.1\] (4,5.3) to\[out = 0, in = 0, looseness = 2\] (4,2+0.2);
(0,0) to\[out = 180, in = 180, looseness = 1.5\] (0,2+0.2); (0,2-0.2) to\[out = 180, in = 180, looseness = 1.5\] (1,1) to\[out = 0,in = 0\] (1,3) to\[out = 180, in = 180,looseness = 1.5\] (0,4); (4,0) to\[out = 30, in = 0, looseness = 2\] (4, -1) to\[out = 180, in = 180\] (4,5) to\[out = 0, in = -30, looseness = 2\] (4,4); (4,2-0.2) to\[out = 0, in = 0, looseness = 2\] (4, -1.3) to\[out = 180, in = 180,looseness = 1.1\] (4,5.3) to\[out = 0, in = 0, looseness = 2\] (4,2+0.2);
(0.5,0.5) to \[out = 180, in = 180, looseness = 0.5\] (0.5,-0.5); (0.5,2.5) to \[out = 180, in = 180, looseness = 0.5\] (0.5,1.5);
(0.5,0.5) to \[out = 180, in = 180, looseness = 0.5\] (0.5,-0.5); (0.5,2.5) to \[out = 180, in = 180, looseness = 0.5\] (0.5,1.5);
According to [@gk2016], a surgery link $L^T = L_1 \sqcup L_2$ for $X$ can be obtained as follows. For each $1 \leq i\leq k$, push the $\alpha_i\,'$s slightly off $\Sigma_g$ and designate those deformed curves as the dotted circles $L_1$. The attaching maps of 2-handles are simply given by $L_2 := \kappa$ with the framing determined by the surface $\Sigma_g$. We project $L^T$ to the plane ${{\mathbb R}}^2 \times \{0\}$ to get a link diagram. See Figure \[fig:special\_trisection\] (Right).
Let ${\mathcal{H}_{}} = {\mathcal{H}_{H,K;\phi}}$. We now compare $\langle T \rangle_{{\mathcal{H}_{}}} $ with $L^T(H,K;\phi)$ defined in §\[subsec:dichro\]. Note that each $\kappa$ curve, viewed as either a trisection component or a link component, is assigned the same tensor, $\mu\circ M^{m}$, in their respective evaluations. Also note that the $\alpha$ and $\beta$ curves are fixed and their intersections with the $\kappa$ curves can be easily characterized. In particular, the intersections of the $\alpha$ curves and the $\kappa$ curves are all located near the $D_{(1, i)}(2\epsilon)$ disks. Therefore, in evaluating $\langle T \rangle_{{\mathcal{H}_{}}} $, we first contract the tensors along $\alpha$ and $\beta$ curves resulting in some tensors only associated with $\kappa$ curves, and then we compare those tensors with the tensors in the definition of $L^T(H,K;\phi)$. This process is divided into two cases depending on whether $i \leq k$ or $i > k$ for $\alpha_i$.
For $i \leq k$, see Figure \[fig:trisection\_link\_1\] (Left) for a configuration of the trisection $T$ involving $\alpha_i$ and $\beta_i$, and the same figure (Right) for the corresponding configuration in the surgery link $L^T$. Contracting the tensors along $\alpha_i$ and $\beta_i$ is carried out in Figure \[fig:trisection\_contr\_1\]. Respectively, the tensors associated with the corresponding configuration in the surgery link are also contracted, as shown in Figure \[fig:link\_contr\_1\]. As we can see, the two calculations result in the same tensor except the first has an extra factor $\dim (H)$. It should be noted that although the tensor contractions in Figures \[fig:trisection\_contr\_1\] and \[fig:link\_contr\_1\] are for the specific trisection/link configuration in Figure \[fig:trisection\_link\_1\], the contractions for general configurations are exactly analogous. In particular, one can try to reverse some of the orientations in Figure \[fig:trisection\_link\_1\] and compare the calculations.
(4,0) circle(0.5cm); (4,0) circle(0.7cm);
(4, -0.5) circle(2pt); (4, 0.7) circle(2pt);
(-1,0.7) to \[out = 0, in =150\] (0,0); (-1,0) to \[out = 0, in =180\] (0,0); (-1,-0.7) to \[out = 0, in =-150\] (0,0);
(4,0) to \[out = 30, in =180\] (5,0.7) to (5.3,0.7); (4,0) to \[out = 0, in =180\] (5,0) to (5.3,0); (4,0) to \[out = -30, in =180\] (5,-0.7) to (5.3,-0.7);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(0,0) circle(0.3cm); (4,0) circle(0.3cm);
(0.5,0.5) to \[out = 0, in = 0,looseness = 0.7\] (0.5,-0.5);
(-1,0.7) to \[out = 0, in =180\] (0,0.3) to (3,0.3) to\[out = 0, in = 180\] (4, 0.7); (-1,0) to (4,0); (-1,-0.7) to \[out = 0, in =180\] (0,-0.3) to (3,-0.3) to\[out = 0, in = 180\] (4, -0.7);
(-1,0.7) to \[out = 0, in =180\] (0,0.3) to (3,0.3) to\[out = 0, in = 180\] (4, 0.7); (-1,0) to (4,0); (-1,-0.7) to \[out = 0, in =180\] (0,-0.3) to (3,-0.3) to\[out = 0, in = 180\] (4, -0.7);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(0.5,0.5) to \[out = 180, in = 180,looseness = 0.5\] (0.5,-0.5); (0.5,0.5) to \[out = 180, in = 180, looseness = 0.5\] (0.5,-0.5);
(0,0) node(eH)[$e_H$]{}; (0,2) node(muH)[$\mu_H$]{};
(1,0) node(DeltaH)[$\Delta_H$]{}; (1,2) node(MH)[$M_H$]{};
(2,-0.5) node(phi2bot)[$\phi_2$]{}; (2,0) node(phi2mid)[$\phi_2$]{}; (2,0.5) node(phi2top)[$\phi_2$]{};
(2,2-0.5) node(phi1bot)[$\phi_1$]{}; (2,2-0) node(phi1mid)[$\phi_1$]{}; (2,2+0.5) node(phi1top)[$\phi_1$]{};
(4,-0.5) node(MKbot)[$M_K$]{}; (4,1) node(MKmid)[$M_K$]{}; (4,2.5) node(MKtop)[$M_K$]{};
(5,-0.5) node(Nullbot); (5,1) node(Nullmid); (5,2.5) node(Nulltop);
(eH) to (DeltaH); (MH) to (muH); (DeltaH) to (phi2bot); (DeltaH) to (phi2mid); (DeltaH) to (phi2top); (MH) to (phi1bot); (MH) to (phi1mid); (MH) to (phi1top); (phi2bot) to (MKbot); (phi2mid) to (MKmid); (phi2top) to (MKtop); (phi1bot) to (MKbot); (phi1mid) to (MKmid); (phi1top) to (MKtop); (MKbot) to (Nullbot); (MKmid) to (Nullmid); (MKtop) to (Nulltop);
(6, 1) node[$=$]{};
(0,0) node(eH)[$e_H$]{}; (0,2) node(muH)[$\mu_H$]{};
(1,0) node(phi2)[$\phi_2$]{}; (1,2) node(phi1)[$\phi_1$]{};
(2,0) node(DeltaKbot)[$\Delta_K$]{}; (2,2) node(DeltaKtop)[$\Delta_K$]{};
(4,-0.5) node(MKbot)[$M_K$]{}; (4,1) node(MKmid)[$M_K$]{}; (4,2.5) node(MKtop)[$M_K$]{};
(5,-0.5) node(Nullbot); (5,1) node(Nullmid); (5,2.5) node(Nulltop);
(eH) to (phi2); (muH) to (phi1); (phi2) to (DeltaKbot); (phi1) to (DeltaKtop); (DeltaKbot) to (MKbot); (DeltaKbot) to (MKmid); (DeltaKbot) to (MKtop); (DeltaKtop) to (MKbot); (DeltaKtop) to (MKmid); (DeltaKtop) to (MKtop); (MKbot) to (Nullbot); (MKmid) to (Nullmid); (MKtop) to (Nulltop);
(-1, 1) node[$=$]{};
(0,0) node(eH)[$e_H$]{}; (0,2) node(muH)[$\mu_H$]{};
(1,0) node(phi2)[$\phi_2$]{}; (1,2) node(phi1)[$\phi_1$]{};
(2,1) node(MK)[$M_K$]{};
(4,1) node(DeltaK)[$\Delta_K$]{};
(5,0) node(Nullbot); (5,1) node(Nullmid); (5,2) node(Nulltop);
(eH) to (phi2); (muH) to (phi1); (phi2) to (MK); (phi1) to (MK); (MK) to (DeltaK); (DeltaK) to\[out = -30, in = 180\] (Nullbot); (DeltaK) to (Nullmid); (DeltaK) to\[out = 30 ,in = 180\] (Nulltop);
(6, 1) node[$=$]{};
(0,1) node[$\text{dim}(H)$]{};
(1,1) node(e)[$\tilde{e}$]{};
(2,1) node(DeltaK)[$\Delta_K$]{};
(3,0) node(Nullbot); (3,1) node(Nullmid); (3,2) node(Nulltop);
\(e) to (DeltaK); (DeltaK) to\[out = -30, in = 180\] (Nullbot); (DeltaK) to (Nullmid); (DeltaK) to\[out = 30 ,in = 180\] (Nulltop);
(0,0) node(phi1)[$\phi_1$]{}; (0,1) node(phi2)[$\phi_2$]{};
(1,2) node(DeltaK)[$\Delta_K$]{};
(1,1) node(e)[$\tilde{e}$]{}; (2,2.3) node(Nulltop1); (2,1.7) node(Nullbot1);
(2,0) node(DeltaKbot)[$\Delta_K$]{}; (2,1) node(DeltaKtop)[$\Delta_K$]{};
(4,0) node(MKbot)[$M_K$]{}; (4,1) node(MKmid)[$M_K$]{}; (4,2) node(MKtop)[$M_K$]{};
(5,0) node(Nullbot); (5,1) node(Nullmid); (5,2) node(Nulltop);
(phi1) to\[out = 180, in = 180, looseness = 2\] (phi2); (phi1) to (DeltaKbot); (phi2) to (DeltaK); (DeltaK) to (Nullbot1); (DeltaK) to (Nulltop1); (e) to (DeltaKtop); (DeltaKbot) to (MKbot); (DeltaKbot) to (MKmid); (DeltaKbot) to (MKtop); (DeltaKtop) to (MKbot); (DeltaKtop) to (MKmid); (DeltaKtop) to (MKtop); (MKbot) to (Nullbot); (MKmid) to (Nullmid); (MKtop) to (Nulltop);
(-2,1) node[$=$]{};
(0,0) node(phi1)[$\phi_1$]{}; (0,1) node(phi2)[$\phi_2$]{};
(1,2) node(DeltaK)[$\Delta_K$]{};
(1,1) node(e)[$\tilde{e}$]{}; (2,2.3) node(Nulltop1); (2,1.7) node(Nullbot1);
(2,1) node(MK)[$M_K$]{};
(3,1) node(DeltaK2)[$\Delta_K$]{};
(4,0) node(Nullbot); (4,1) node(Nullmid); (4,2) node(Nulltop);
(phi1) to\[out = 180, in = 180, looseness = 2\] (phi2); (phi1) to\[out = 0, in = -135\] (MK); (phi2) to (DeltaK); (DeltaK) to (Nullbot1); (DeltaK) to (Nulltop1); (e) to (MK); (MK) to (DeltaK2); (DeltaK2) to\[out = -45, in = 180\] (Nullbot); (DeltaK2) to (Nullmid); (DeltaK2) to\[out = 45, in = 180\] (Nulltop);
(5,1) node[$=$]{};
(1,1.7) node(unitbot)[$1_K$]{}; (1,2.3) node(unittop)[$1_K$]{};
(1,1) node(e)[$\tilde{e}$]{}; (2,2.3) node(Nulltop1); (2,1.7) node(Nullbot1);
(2,1) node(DeltaK)[$\Delta_K$]{};
(3,0) node(Nullbot); (3,1) node(Nullmid); (3,2) node(Nulltop);
(unitbot) to (Nullbot1); (unittop) to (Nulltop1); (e) to (DeltaK); (DeltaK) to\[out = -45, in = 180\] (Nullbot); (DeltaK) to (Nullmid); (DeltaK) to\[out = 45, in = 180\] (Nulltop);
Similarly, see Figures \[fig:trisection\_link\_2\] and \[fig:trisection\_contr\_2\] for the case of $i>k$. Note that the last term in Figure \[fig:trisection\_contr\_2\] is precisely the tensor associated with the link configuration in Figure \[fig:trisection\_link\_2\] (Right). In this case, the two configurations in the trisection and in the surgery link result in the same tensor.
(4,0) circle(0.7cm); (4,0) to (0,0);
(4, 0.7) circle(2pt); (3.5, 0) circle(2pt);
(-1,0.7) to \[out = 0, in =150\] (0,0); (-1,0) to \[out = 0, in =180\] (0,0); (-1,-0.7) to \[out = 0, in =-150\] (0,0);
(4,0) to \[out = 30, in =180\] (5,0.7) to (5.3,0.7); (4,0) to \[out = 0, in =180\] (5,0) to (5.3,0); (4,0) to \[out = -30, in =180\] (5,-0.7) to (5.3,-0.7);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(0,0) circle(0.3cm); (4,0) circle(0.3cm);
(-1,0.7) to \[out = 0, in =180\] (0,0.3) to (3,0.3) to\[out = 0, in = 180\] (4, 0.7); (-1,0) to (4,0); (-1,-0.7) to \[out = 0, in =180\] (0,-0.3) to (3,-0.3) to\[out = 0, in = 180\] (4, -0.7);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(1.5,1) to (1.5, -1); (2.5,1) to (2.5, -1);
(0,0) node(muH) [$\mu_H$]{}; (0,2) node(eH) [$e_H$]{};
(1,0) node(MH) [$M_H$]{}; (1,2) node(DeltaH) [$\Delta_H$]{};
(2,1.5) node(S) [$S$]{};
(2.5,-0.5) node(phi1bot) [$\phi_1$]{}; (2.5,0) node(phi1mid) [$\phi_1$]{}; (2.5,0.5) node(phi1top) [$\phi_1$]{}; (2.5,2) node(phi2bot) [$\phi_2$]{}; (2.5,2.5) node(phi2top) [$\phi_2$]{};
(3.5, -0.5) node(Null1); (3.5, 0) node(Null2); (3.5, 0.5) node(Null3); (3.5, 2) node(Null4); (3.5, 2.5) node(Null5);
(muH) to (MH); (eH) to (DeltaH); (MH) to (phi1bot); (MH) to (phi1mid); (MH) to (phi1top); (DeltaH) to (phi2bot); (DeltaH) to (phi2top); (DeltaH) to (S); (S) to\[out = -45, in = 45\] (MH); (phi1bot) to (Null1); (phi1mid) to (Null2); (phi1top) to (Null3); (phi2bot) to (Null4); (phi2top) to (Null5);
(4.5,1) node[$=$]{};
(1,0) node(MH) [$M_H$]{}; (1,2) node(DeltaH) [$\Delta_H$]{};
(2.5,-0.5) node(phi1bot) [$\phi_1$]{}; (2.5,0) node(phi1mid) [$\phi_1$]{}; (2.5,0.5) node(phi1top) [$\phi_1$]{}; (2.5,2) node(phi2bot) [$\phi_2$]{}; (2.5,2.5) node(phi2top) [$\phi_2$]{};
(3.5, -0.5) node(Null1); (3.5, 0) node(Null2); (3.5, 0.5) node(Null3); (3.5, 2) node(Null4); (3.5, 2.5) node(Null5);
(MH) to\[out = 180, in = 180, looseness = 1\] (DeltaH); (MH) to (phi1bot); (MH) to (phi1mid); (MH) to (phi1top); (DeltaH) to (phi2bot); (DeltaH) to (phi2top); (phi1bot) to (Null1); (phi1mid) to (Null2); (phi1top) to (Null3); (phi2bot) to (Null4); (phi2top) to (Null5);
(10,1) node[$=$]{};
(0,0) node(phi1) [$\phi_1$]{}; (0,2) node(phi2) [$\phi_2$]{};
(1,0) node(DeltaKbot) [$\Delta_K$]{}; (1,2) node(DeltaKtop) [$\Delta_K$]{};
(2, -0.5) node(Null1); (2, 0) node(Null2); (2, 0.5) node(Null3); (2, 2) node(Null4); (2, 2.5) node(Null5);
(phi1) to\[out = 180, in = 180, looseness = 1\] (phi2); (phi1) to (DeltaKbot); (phi2) to (DeltaKtop); (DeltaKbot) to (Null1); (DeltaKbot) to (Null2); (DeltaKbot) to (Null3); (DeltaKtop) to (Null4); (DeltaKtop) to (Null5);
Therefore, we have $\langle T \rangle_{{\mathcal{H}_{}}} = \dim (H)^k L^T(H,K;\phi)$. By direct calculations, the standard trisection diagram of $S^4$ has the invariant $\langle T_{{\operatorname}{st}} \rangle_{{\mathcal{H}_{}}} = \dim(K)^2 \dim(H) \mu_K(\tilde{e})$. So, $$\begin{aligned}
\langle T \rangle_{{\mathcal{H}_{}}} &= \dim (K)^g \left[\frac{\dim(H)\mu_K(\tilde{e})}{\dim(K)}\right]^{\frac{g}{3}}\tau_{{\mathcal{H}_{}}}(X).\end{aligned}$$ On the the other hand, $$\begin{aligned}
\dim (H)^k L^T(H,K;\phi) &= \dim (H)^k \dim(K)^{g-k} \mu_K(\tilde{e})^{k} I_{{\text{Rep}}(\phi)}(X) \\
&= \dim (K)^g \left[\frac{\dim(H)\mu_K(\tilde{e})}{\dim(K)}\right]^{k}I_{{\text{Rep}}(\phi)}(X).\end{aligned}$$ The theorem follows by combining the above two equations and noting that $\chi(X) = 2 + g - 3k$.
We make a final remark. In this paper, we have focused on Hopf algebras in constructing invariants. There is also a weaker notion of Hopf algebras, namely, [*weak*]{} Hopf algebras. One can similarly formulate the notion of weak Hopf triplets using weak Hopf algebras. We believe that it is possible to define invariants of 4-manifolds more generally from weak Hopf triplets. It is known that every braided fusion category is equivalent to the category of representations of a weak Hopf algebra. Hence, it is expected that the resulting invariants from weak Hopf triplets would recover all cases of the Crane-Yetter invariants and a larger class of the dichromatic invariant than what is stated in Theorem \[thm:tri=dichro\].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The dynamical characterization of proteins is crucial to understand protein function. From a microscopic point of view, protein dynamics is governed by the local atomic interactions that, in turn, trigger the functional conformational changes. Unfortunately, the relationship between local atomic fluctuations and global protein rearrangements is still elusive. Here, atomistic molecular dynamics simulations in conjunction with complex network analysis show that fast peptide relaxations effectively build the backbone of the global free-energy landscape, providing a connection between local and global atomic rearrangements. A minimum-spanning-tree representation, built on the base of transition gradients networks, results in a high resolution mapping of the system dynamics and thermodynamics without requiring any a priori knowledge of the relevant degrees of freedom. These results suggest the presence of a local mechanism for the high communication efficiency generally observed in complex systems.'
address: 'Laboratoire de Chimie Biophysique/ISIS, Universite de Strasbourg, 8, alle G. Monge, Strasbourg, France'
author:
- Francesco Rao
bibliography:
- '/Users/ruvido/work/documents/biblio/ruvido.bib'
title: Local transition gradients determine the global attributes of protein energy landscapes
---
In complex systems the behavior of the whole is hardly predictable from the fundamental laws of interactions of the single components [@anderson1972]. Indeed, it is hard to reveal the intimate connection between the local properties of a complex system and its global behavior. This problem has been formalized in the context of complex networks [@boguna2008] as a question of “navigability"; i.e., the mechanism for the efficient flow of information when the single network nodes do not have a global view of the overall topology. As a matter of fact, many collective dynamical processes are driven by the presence of (usually hidden) local gradients [@toroczkai2004].
The study of protein dynamics involve a similar problem – the coupling between the fast atomic fluctuations, which are local, and the slow conformational changes, which are global [@frauenfelder1991]. Those dynamical aspects have been recently recognized to be crucial for protein function, playing an important role in signalling, allosteric pathways and enzymatic reactions [@kern2003; @eisenmesser2005]. Molecular dynamics (MD) simulations are playing an increasing role in complementing the experimental results [@boehr2006; @schuler2008; @colletier2008] which supply useful , but limited, information to this question. The recent combination of computational and experimental studies of a protein enzyme has pointed out that the fast atomic fluctuations are partly correlated to the displacements occuring in the catalytic reaction [@henzler2007]. As yet, the coupling between the fast nanosecond timescales and the functional relevant transitions occuring in the microsecond to millisecond range largely remains obscure.
Most descriptions of the free-energy surface governing protein dynamics have been rather qualitative because of the lack of proper order parameters and the intrinsic multidimensionality of the problem [@du1998; @pande1998]. These limitations have triggered the development of a completely new arsenal of tools inspired by network theory [@caflisch2006]. The essential idea is to map the protein trajectory, obtained by computer simulations or experiments, on a conformation space network (CSN), whose nodes represent the different microstates and whose links correspond to direct transitions between them [@rao2004; @caflisch2006; @gfeller2007]. This approach has been successfully applied to the study of peptide folding and structural transitions [@rao2004; @krivov2004; @gfeller2007; @settanni2008; @yang2008; @prada2009], as well as to interpret electron transfer experiments [@li2008] and time-resolved IR measurements [@ihalainen2007; @ihalainen2008].
In this letter, the relation between the local properties of the free-energy landscape and its global architecture is investigated by MD simulations of a 4 residues peptide, (GlySer)$_2$, and complex network analysis . In particular, when the CSN of a fully-atomistic peptide is reduced to the subgraph containing only one link per microstate pointing towards the most probable transition (i.e. following the transition gradient), the presence of energy valleys and subvalleys and their equilibrium populations is naturally extracted as well as the hierarchy of transitions between them. Hence, the fast local motions build up the backbone of the global communication. The observed coupling between local and global dynamical properties is expected to occur in a large class of complex systems.
GlySer peptides have been used for quite some time as flexible linkers (and are known to show poor secondary structure) for polypeptide dynamics [@bieri1999; @moglich2006]. MD simulations using the Langevin algorithm [^1] and the implicit solvation FACTS [@haberthur2008; @charmm2009] have been performed. A trajectory of 280 ns at 340 K was obtained and snapshots were saved every 140 steps for a total of $10^6$ conformations. During the simulation the peptide visits several different conformations characterized by an end-to-end distance between 3.1 and 12.7 Å (Fig. \[fig:timeseries\]) indicating large structural fluctuations. The peptide microstates are defined as the inherent structures (IS), i.e. the potential energy minima, of the system [@stillinger1982; @stillinger1983]. They are calculated minimizing all the $10^6$ snapshots along the trajectory, resulting in 3044 different IS. The IS are a natural, physically meaningfull, partition into microstates [@baumketner2003; @kim2007; @rao2010]. The conformation space network (CSN) is built on top of this microstate definition: the nodes and the links are the microstates (i.e. the IS) and their direct transitions observed during the MD trajectory, respectively. The obtained network is weighted and it is equivalent to a classical transition matrix when the columns of the adjacency matrix (i.e. the network links) are appropriately normalized to one.
![Timeseries window of the end-to-end distance of the (GlySer)$_2$ peptide. The peptide is extremely flexible, alternating between compact states ($d_{ee}\sim 3.4$ Å) and extended conformations ($d_{ee}\sim 10.8$ Å).[]{data-label="fig:timeseries"}](trj-340_ete){width="80mm"}
The relation between fast local modes and global chain rearrangements is investigated by constructing from the full CSN, a new network with a reduced number of links. For each network node, the transition with the highest probability (excluding self interactions) is kept, and all others are deleted. These transitions define a gradient in the network dynamics and result in a partitioning of the network into several disconnected minimum spanning trees [@carmi2008], called here gradient-clusters for convenience. (Gradient networks were originally introduced by Toroczkai and Bassler to study jamming [@toroczkai2004], though in their case the gradient is defined on the nodes as a quenched scalar field). Following the pathway defined by the most probable transitions leads to microstates lower and lower in free energy, resulting in a kind of “steepest descent pathway" on the free-energy surface. High energy microstates, in the neighbor of a free-energy barrier, would connect either to one valley or another. As a matter of fact, the network characterizing the free-energy surface is split into a set of disconnected minimum-spanning-trees representing the local attractors of the system dynamics.
To better understand the nature of the gradient-clusters (161 in total), a cut-based free-energy profile (CFEP) [@krivov2006] is calculated on the CSN and compared to the output of the gradient-partition. The CFEP is based on a network flux analysis following the idea that the network regions of minimum flow correspond to transition states [@krivov2006; @krivov2008; @muff2009]. In Fig. \[fig:cfep-gradient\] the calculated CFEP is shown. The profile reveals that the peptide free-energy landscape is rugged. Remarkably, the obtained gradient-clusters represent either a valley or a subvalley of the free-energy landscape (Fig. \[fig:cfep-gradient\]). In Table \[populations\] the population of the four most relevant free-energy basins detected by the gradient-approach is compared against the populations calculated by the minimum-cut method [@krivov2004], which is one of the most accurate approaches for this type of calculation [@caflisch2006]. The results indicate that the populations of the gradient-clusters are accurate, effectively reproducing the correct thermodynamics of the system. This is particularly relevant since the gradient-approach does not use any global property of the system, neither in terms of barrier heights or microstates energies. On the other hand, the CFEP and the minimum-cut method do perform a global analysis of the network. These observations suggest an interesting application of the proposed approach to automatically detect the presence of metastable states sampled by multiple short MD runs with the aim of building simplified Markov models [@chodera2007].
![(Color online) Cut-based free-energy profile for the (GlySer)$_2$ peptide. Microstates assigned by the gradient-approach to the four most populated valleys are shown on the profile in different colors. The CFEP represents the free-energy surface projected on the cumulative partition function reaction coordinate $Z$ [@krivov2006], relative to a given reference microstate (in this case the most populated one). In the inset a profile is calculated on a subgraph made by the microstates belonging to $V_4$ (triangles) and a smaller gradient-cluster (circles).[]{data-label="fig:cfep-gradient"}](cfep-gradient){width="80mm"}
Valley Gradient population Mincutpopulation $<d_{ee}>$ \[Å\]
-------- --------------------- ------------------ ------------------
$V_1$ 0.429 0.428 4.167
$V_2$ 0.120 0.120 4.253
$V_3$ 0.068 0.064 10.960
$V_4$ 0.037 0.051 4.160
: Comparison of the populations of the found valleys calculated by the gradient-approach or by the minimum-cut method [@krivov2004]. Populations are defined as the sum of the number of occurencies calculated along the trajectory of the microstates assigned to a given valley. In the last column the average value of the end-to-end distance $d_{ee}$ inside a valley is given.[]{data-label="populations"}
The gradient-approach can be applied in an iterative, heirarchical fashion, when the gradient-clusters are considered themselves as nodes of a higher level CSN. In this case, the nodes and the links of the network are the gradient-clusters and the connections between them, respectively. The iteration can be applied recursively until all the microstates are merged into one cluster, which is represented as a minimum-spanning-forest. The tree structure obtained for the (GlySer)$_2$ peptide is shown in Fig. \[fig:gradient-forest\]. Link widths represent at which iteration step the edge was introduced. For example, the $V_1$ and $V_2$ valleys are merged at the first iteration, indicating that they interconvert rapidly. On the other hand, $V_4$ is merged to $V_1$ at the fourth iteration indicating that this is the slowest transition to $V_1$. The gradient-minimum-spanning-forest represents, in an intrinsically multidimensional fashion, the valleys of the free-energy landscape and their dynamical organization in fast and slow relaxations. The iteration step at which two gradient-clusters are merged together is a kinetic mesaure of the dynamical distance between the two valleys.
![(Color online) Gradient-minimum-spanning-forest of the (GlySer)$_2$ peptide energy landscape. This free-energy representation consistently represents the fast relaxations between the microstates inside a valley (star-like structures) as well as the dynamical separation between different valleys (expressed by link thickness). Larger link widths indicate slower relaxations (see main text for details). The size of the nodes is proportional to the microstate population (to avoid overcrowiding only nodes with populations larger than $10^{-3}$ are shown). []{data-label="fig:gradient-forest"}](gmtx-340){width="80mm"}
Concluding, we found that, for an all-atom peptide MD simulation, the fast local relaxations produce a partition of the conformational space into disconnected minimum-spanning-trees. This partition reproduces the organization of the free-energy landscape into valleys and subvalleys with the correct populations. The iterative connection of those valleys into a minimum-spanning-forest recovers the global backbone architecture of the dynamics occuring on the free-energy landscape. A similar coupling between local and global dynamics is expected to take place in other complex systems. These results are relevant to investigate the still unclear relationship between network structure and dynamics in transport processes ranging from metabolic pathways to air-travelling and the internet.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I am grateful to Prof. A. Caflisch, Prof. M. Karplus, Dr S. Muff, Dr. S. Krivov, Dr. M. Cecchini for useful comments and discussions and Prof. P. Hamm, Dr. S. Garrett-Roe for a critical reading of the manuscript.
[^1]: MD simulations, using the Langevin algorithm with a friction coefficient equal to 0.6 $ps^{-1}$, were calculated with the CHARMM program [@charmm2009], the polar hydrogen energy function (PARAM19) was used. The effects of water have been included using the generalized born FACTS implicit solvation model [@haberthur2008]. SHAKE was employed so that an integration step of 2 fs could be used.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '$1/f^\alpha$ noises are ubiquitous and affect many measurements. These noises are both a nuisance and a peculiarity of several physical systems; in dielectrics, glasses and networked liquids it is very common to study this noise to gather useful information. Sometimes it happens that the noise has a power-law shape only in a certain frequency range, and contains other important features, that are however difficult to study because simple fits often fail. Here I propose a model-based fit procedure that performs well on spectra obtained in a molecular dynamics simulation.'
address: |
Dipartimento di Fisica, Università di Udine and I.N.F.N. – Sezione di Trieste\
Via delle Scienze, 208 – I-33100 Udine, Italy
author:
- Edoardo Milotti
title: 'Model-based fit procedure for power-law-like spectra'
---
Introduction {#intro}
============
$1/f^\alpha$ noises are very common and affect many measurements; the literature on this subject keeps growing and the apparent ubiquity of these noises has always drawn a great deal of attention. In the experimental practice, they are both a nuisance and a peculiarity of several physical systems; in dielectrics, glasses and networked liquids it is very common to study these noises to gather useful information [@sor; @isra; @CC1; @cm]. Sometimes it happens that the noise has a power-law shape only in a certain frequency range which spans several decades, and at the same time contains other important features, that are however difficult to study because simple fits often fail. The main reason of this failure is that the prominent low-frequency peak biases the fit so much that the minute and mostly high-frequency features are neglected. Here I propose a model-based fit procedure that bypasses this problem and that performs well on spectra obtained in a molecular dynamics simulation of water.
In the rest of this introduction I review the classic superposition argument that relates power-law spectra to the single exponential relaxation processes; in section \[seccorr\] I analyze the properties of the autocorrelation function of $1/f^\alpha$ spectra; in section \[secgallery\] I consider the spectral behavior associated to some well-defined distributions of relaxation rates; finally in the last section I show the results of a model-based fit in the case of a molecular dynamics simulation of liquid water, and I summarize my conclusions.
A mathematical mechanism for producing $1/f^\alpha$ noise was proposed long ago by Bernamont [@bernamont], who observed that the superposition of many Lorentzian spectra with a certain distribution of different rates could produce a spectral density with a $1/f$ region. The Bernamont superposition argument can be made rigorous with a slight modification of the standard proof of Campbell’s theorem [@campbell], and it goes as follows. Take a signal $x(t)$ originated by the linear superposition of many random pulses, i.e., pulses that are random in time and can be described by a memoryless process with a Poisson distribution, have random amplitude $A$ drawn from a distribution with finite variance and probability density $g_A(A)$, and such that their pulse response function $h(t,\lambda) = \exp(-\lambda t)$ (if $t>0$, otherwise $h(t,\lambda)=0$) is drawn from a distribution with probability density $g_\lambda(\lambda)$. The pulses are received and detected with a rate $n(A,\lambda)$ which in general depends both on the amplitude $A$ and on the decay rate $\lambda$. The pulse arrival process is Poissonian and thus one detects on average $\left[n(A,\lambda) dA d\lambda\right] dt$ pulses in the time interval $(t',t'+dt)$ (and in the amplitude-$\lambda$ range $dA d\lambda$); for the same reason the variance of the number of detected pulses is also equal to $\left[n(A,\lambda) dA d\lambda\right] dt$. This means that the mean square fluctuation of the output signal at time $t$ is given by the integral $$\label{msq}
\langle (\Delta x)^2 \rangle = \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) d\lambda \int_{A_{min}}^{A_{max}} g_A(A) dA \int_{-\infty}^t dt' n(A,\lambda) \left[A h(t-t',\lambda) \right]^2$$ If we assume that the rate of occurrence $n$ does not depend on $A$ and $\lambda$, and rearrange the time integration, then the integral (\[msq\]) simplifies to $$\label{msq2}
\langle (\Delta x)^2 \rangle = n \langle A^2 \rangle \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) d\lambda \int_0^\infty dt \left[ h(t,\lambda) \right]^2$$ Now let $H(\omega,\lambda)$ be the Fourier transform of $h(t,\lambda)$, then from the causality constraint on $h(t,\lambda)$ and Parseval’s theorem we find that the mean square fluctuation (\[msq2\]) can be trasformed into $$\begin{aligned}
\nonumber
\langle (\Delta x)^2 \rangle &=& \frac{n \langle A^2 \rangle}{2\pi} \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) d\lambda \int_{-\infty}^\infty d\omega \left| H(\omega,\lambda) \right|^2\\
\label{msq3}
&=& \frac{n \langle A^2 \rangle}{2\pi} \int_{-\infty}^\infty d\omega \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) d\lambda \left| H(\omega,\lambda) \right|^2\end{aligned}$$ The right-hand expression in equation (\[msq3\]) shows that the spectral density is $$\label{psd}
S(\omega) = \frac{n \langle A^2 \rangle}{2\pi} \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) d\lambda \left| H(\omega,\lambda) \right|^2$$ and since $\left| H(\omega,\lambda) \right|^2=(\omega^2 + \lambda^2)^{-1}$ we obtain eventually $$\label{psd2}
S(\omega) = \frac{n \langle A^2 \rangle}{2\pi} \int_{\lambda_{min}}^{\lambda_{max}} \frac{g_\lambda(\lambda)}{\omega^2 + \lambda^2} d\lambda$$ If we assume that the decay rates $\lambda$ are uniformly distributed between $\lambda_{min}$ and $\lambda_{max}$ (i.e., $g_\lambda(\lambda) = (\lambda_{max}-\lambda_{min})^{-1}$ ) the spectral density becomes $$\label{psdflat}
S(\omega) = \frac{n \langle A^2 \rangle}{2\pi(\lambda_{max}-\lambda_{min})} \frac{1}{\omega} \left(\arctan\frac{\lambda_{max}}{\omega}-\arctan\frac{\lambda_{min}}{\omega}\right)$$ so that $S(\omega)$ is approximately constant if $0<\omega\ll\lambda_{min}\ll\lambda_{max}$, and it is approximately equal to $$\frac{n \langle A^2 \rangle}{2\pi (\lambda_{max}-\lambda_{min})} \frac{1}{\omega^2}$$ if $\lambda_{min}\ll\lambda_{max}\ll\omega$, while it is approximately equal to $$\frac{n \langle A^2 \rangle}{4 (\lambda_{max}-\lambda_{min})} \frac{1}{\omega}$$ in the region in between the extreme rates ($\lambda_{min}\ll\omega\ll\lambda_{max}$).
The spectral density (\[psdflat\]) has an intermediate region with a $1/f$ behavior, however most observed spectra are not quite $1/f$ but rather $1/f^\alpha$ with $\alpha$ ranging from about 0.5 to nearly 2: how can we obtain such spectra using a superposition as above, i.e., sampling a distribution of relaxation processes? We could take, e.g., a nonuniform distribution of relaxation processes like $g_\lambda \propto \lambda^{-\beta}$, then in the region $\lambda_{min}\ll\omega\ll\lambda_{max}$ we would find $$\begin{aligned}
\label{psdbeta}
S(\omega) & \propto & \int_{\lambda_{min}}^{\lambda_{max}} \frac{1}{\omega^2 + \lambda^2} \frac{d\lambda}{\lambda^\beta} = \frac{1}{\omega^{1+\beta}} \int_{\lambda_{min}/\omega}^{\lambda_{max}/\omega} \frac{1}{1 + (\lambda/\omega)^2} \frac{d(\lambda/\omega)}{(\lambda/\omega)^\beta}\\
& & \approx \frac{1}{\omega^{1+\beta}} \int_0^\infty \frac{1}{1+x^2}\frac{dx}{x^\beta}\end{aligned}$$
We shall return to these distributions in section \[secgallery\].
The rate distribution from the correlation function {#seccorr}
===================================================
We see that from a given rate distribution we obtain a certain spectral density: can we do the reverse and obtain the rate distribution from a given spectral density? This is not obvious because the spectral density is only a second-order statistics, and does not contain phase information (nor is it possible to preserve it for a noise process). However the answer is yes, the rate distribution can be recovered from the spectral density. This can easily be seen from the formal Taylor expansion of the denominator in the integral (\[psd2\]): $$\label{taylor}
S(\omega) = \frac{n \langle A^2 \rangle}{2\pi\omega^2} \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda \sum_{k=0,\infty} \left(-\frac{\lambda}{\omega}\right)^{2k} d\lambda=\frac{n \langle A^2 \rangle}{2\pi\omega^2} \sum_{k=0,\infty} \left(\frac{-1}{\omega}\right)^{2k} \langle \lambda^{2k}\rangle$$ This expansion is only formal inasmuch as it does not converge everywhere, however it shows unequivocally that the shape of $S(\omega)$ depends only on the even moments about the origin of the probability density $g_\lambda$. A probability density function is uniquely determined by the knowledge of [*all*]{} the moments $\langle \lambda^n\rangle$ (see, e.g., [@feller]), and the even moments alone are not enough, but we could still do without the odd moments if the probability density function were an even function. This is not so, because the decay rates $\lambda$ must be non-negative, and thus the associated probability density function does not have any definite parity. However a probability density function which is non-zero only for positive values of the decay rates can be written in a unique way as the sum of an even and an odd function $g_\lambda(\lambda) = g_\lambda^{(odd)}(\lambda) + g_\lambda^{(even)}(\lambda)$, where $g_\lambda^{(odd)}(\lambda) = g_\lambda^{(even)}(\lambda) = g_\lambda(\lambda)/2$ if $\lambda \ge 0$ and $g_\lambda^{(odd)}(\lambda) = -g_\lambda^{(even)}(\lambda) = -g_\lambda(-\lambda)/2$ if $\lambda < 0$, therefore the odd moments can be computed from the even moments of the distribution, and the even moments alone uniquely identify the rate distribution.
The previous result is only formal and does not yield a practical inversion formula; the actual inversion can be performed in the time domain when we recall that the spectral density $S(\omega)$ is related to the correlation function $R(\tau)$ by the Wiener-Kintchine theorem $$\begin{aligned}
\label{wk}
R(\tau)&=&\frac{1}{2\pi}\int_{-\infty}^{+\infty} S(\omega) e^{i\omega\tau} d\omega\\
\nonumber
&=&\frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{i \omega \tau} \frac{n\langle A^2\rangle}{2\pi}\int_{\lambda_{min}}^{\lambda_{max}} \frac{g_\lambda(\lambda)}{\omega^2+\lambda^2} d\lambda d\omega\\
\nonumber
&=&\frac{n\langle A^2\rangle}{2\pi}\int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) \frac{1}{2\pi}\int_{-\infty}^{+\infty} \frac{e^{i \omega \tau}}{\omega^2+\lambda^2} d\omega d\lambda\\
\label{laplace}
&=&\frac{n\langle A^2\rangle}{2\pi} \int_{\lambda_{min}}^{\lambda_{max}} g_\lambda(\lambda) \frac{e^{-\lambda |\tau|}}{2\lambda} d\lambda\end{aligned}$$ then we see from equation (\[laplace\]) that the correlation function is also the Laplace transform of $g_\lambda(\lambda)/(2\lambda)$, and the rate distribution function is uniquely determined by the spectral density and can be retrieved by means of a numerical inverse Laplace transform. In practice, rather than a numerical evaluation of the inverse Laplace transform, one is forced to fit a discrete set of decaying exponentials, and moreover from the correspondence between the Bromwich inversion integral and the inverse Fourier transform, and from the sampling theorem, we see that we must sample the time correlation function, and therefore the noise signal, at a frequency at least twice as high as $\lambda_{max}$ to retrieve $g_\lambda$. Notice that because of the $\lambda$ in the denominator of the integrand in (\[laplace\]), the slow relaxations are more heavily weighted in the integral, and the high-frequency parts of the decay rate distribution are much harder to recover than the low-frequency parts; this makes even harder an inversion task which is already known to be very difficult [@davismartin].
The mixtures of decaying exponentials that characterize many experimental measurements differ significantly only at very short times, while for longer times all the exponentials are equally buried in noise. Disentangling the mixture and finding the relative weights of the different components is possible only if sampling times are very closely spaced at the beginning (and one common strategy is to use logarithmically spaced sampling times (see, e.g. [@mx])) and only if one includes some form of prior or assumed knowledge of the distribution of decay rates. There are very few well-established procedures to do this, and the best known are the programs CONTIN and UPEN. CONTIN [@sp] uses the following strategies: a) it takes into account [*absolute prior knowledge*]{}, i.e. whichever exact information that may be available at the beginning, like the non-negativity of decay rates; b) it assumes some [*statistical prior knowledge*]{} as well, which is essentially the knowledge of the statistics of the measurement noise; c) it uses a [*principle of parsimony*]{}, which is similar to the principle of maximum entropy, though not as well defined. UPEN (Uniform PENalty) [@upen] assumes instead [*a priori*]{} that the distribution of decay rates is a continuous function and penalizes distributions which are either discontinuous or have wildly varying curvature.
In addition to constraints on the shape of the distribution function it is common to use some well-defined standard functions that appear to fit very well many sets of experimental data; the Kohlrausch-Williams-Watts function describes stretched exponentials and works well for relaxations in the time domain and similarly the Havrilijak-Negami (HN) function provides good fits to spectral data. These empirical functions are well-known, and in particular from the HN spectral shape it is possible to compute analytically the distribution of relaxation rates [@aac]. However, even though these functions often give satisfactory fits, it would be much better to connect data from experiments or numerical simulations to some well-defined, simple distribution of relaxation rates, just like the spectral density in equation (\[psdflat\]) can be directly related to a flat distribution of relaxation rates: in the following section I give a list of such spectral shapes.
A gallery of spectral densities {#secgallery}
===============================
The spectral density in equation (\[psdflat\]) produces an intermediate region with a $1/f$ behavior, and includes both a minimum and a maximum relaxation rate: at a frequency lower than the minimum relaxation rate the spectral density whitens and becomes nearly flat, while at a frequency higher than the maximum relaxation rate the spectral density bends downward and assumes a $1/f^2$ behavior, and for fitting purposes we define the standard spectral density $$\label{psdflatflat}
S_{\rm flat}(\omega; \lambda_{min}, \lambda_{max}) = \frac{1}{\omega} \left(\arctan\frac{\lambda_{max}}{\omega}-\arctan\frac{\lambda_{min}}{\omega}\right)$$ However either the minimum or the maximum relaxation rate (or both) may be out of the experimental or numerical simulation range: in these cases the bends at low- and high-frequency become invisible, and a fit with the spectral density (\[psdflatflat\]) is unstable (at least one of the range parameters is invisible and the chi-square hypersurface flattens out in that direction, adversely influencing the fit). This can be corrected using the modified spectral density $$\label{psdflatlow}
S_{\rm flat,A}(\omega; \lambda_{min}) = \frac{1}{\omega} \left[\frac{\pi}{2} - \arctan\left(\frac{\lambda_{min}}{\omega}\right)\right]$$ when the maximum observable frequency is smaller than the maximum relaxation rate (and the minimum relaxation rate is in the observable range). We should use instead the spectral density $$\label{psdflathigh}
S_{\rm flat,B}(\omega; \lambda_{max}) = \frac{1}{\omega} \arctan\left(\frac{\lambda_{max}}{\omega}\right)$$ when the minimum observable frequency is higher than the minimum relaxation rate (and the maximum observable rate is in the observable range), and finally the spectral density $$\label{psd1overf}
S_{\rm 1overf}(\omega) = \frac{1}{\omega}$$ when both the minimum and the maximum relaxation rates are out of range; the spectral densities (\[psdflat\]), (\[psdflatlow\]), and (\[psdflathigh\]) are shown in figures \[fig1\] to \[fig4\]. Using (\[psdflatlow\]), (\[psdflathigh\]) or (\[psd1overf\]) improves the fit stability but means that the final description of the relaxation rate distribution is incomplete.
We have already given a simple argument that shows that a nonuniform distribution of relaxation processes like $g_\lambda \propto \lambda^{-\beta}$ between the maximum and minimum relaxation rates $\lambda_{min}$, $\lambda_{max}$, produces a spectral density with an intermediate $1/f^{1+\beta}$ region: an exact integration yields the spectral density $$\begin{aligned}
\label{psdbetaexact}
\nonumber
S_{\rm pl}(\omega; \lambda_{min}, \lambda_{max}, \beta) &=& \frac{1}{(1-\beta)\omega^2}\left[ \lambda_{max}^{1-\beta} F\left(\frac{1-\beta}{2},1;\frac{1-\beta}{2};\frac{-\lambda_{max}^2}{\omega^2}\right) \right. \\ && \left.- \lambda_{min}^{1-\beta} F\left(\frac{1-\beta}{2},1;\frac{1-\beta}{2};\frac{-\lambda_{min}^2}{\omega^2}\right) \right]\end{aligned}$$ where $F(a,b;c;z) = \sum_{k=0}^{\infty} \frac{(a)_k (b)_k}{(c)_k} \frac{x^k}{k!}$ is the hypergeometric function and $\beta \in (-1,1)$. Just as in the $1/f$ case either the minimum or the maximum relaxation rate (or both) may be out of the experimental or numerical simulation range and a fit with the spectral density (\[psdbetaexact\]) becomes unstable, and this can be corrected with the modified spectral densities $$\label{psdbetalow}
S_{\rm pl,A}(\omega; \lambda_{min}, \beta) = L(\omega,\beta) - \frac{1}{(1-\beta)\omega^2}\left[ \lambda_{min}^{1-\beta} F\left(\frac{1-\beta}{2},1;\frac{1-\beta}{2};\frac{-\lambda_{min}^2}{\omega^2}\right) \right]$$ when the maximum observable frequency is smaller than the maximum relaxation rate (and the minimum relaxation rate is in the observable range) and where the function $$\label{limitL}
L(\omega,\beta) = \lim\limits_{\lambda_{max} \to \infty} \frac{\lambda_{max}^{1-\beta}}{(1-\beta)\omega^2} F\left(\frac{1-\beta}{2},1;\frac{1-\beta}{2};\frac{-\lambda_{max}^2}{\omega^2}\right)$$ is shown in figure \[fig4\] and is well approximated by the rational function $$\frac{\pi}{2\omega^{1+beta}} \frac{1}{\left( 1+ c_2 \beta^2 + c_4 \beta^4 + c_6 \beta^6 + c_8 \beta^8 + c_{10} \beta^{10} \right)}$$ with
1. $c_2 \approx -1.2337$
2. $c_4 \approx 0.253669$
3. $c_6 \approx -0.0208621$
4. $c_8 \approx 0.000917057$
5. $c_{10} \approx -0.0000235759$
The spectral density $$\label{psdbetahigh}
S_{\rm pl,B}(\omega; \lambda_{max}, \beta) = \frac{\lambda_{max}^{1-\beta} }{(1-\beta)\omega^2} F\left(\frac{1-\beta}{2},1;\frac{1-\beta}{2};\frac{-\lambda_{max}^2}{\omega^2}\right)$$ works when the minimum observable frequency is higher than the minimum relaxation rate (and the maximum observable rate is in the observable range), and finally the spectral density $$\label{psd1overfbeta}
S_{\rm 1overf}(\omega; \beta) \propto \frac{1}{\omega^{1+\beta}}$$
when both the minimum and the maximum relaxation rates are out of range (here I extend the notation of definition (\[psd1overf\]) ); the spectral densities (\[psdbetaexact\]), (\[psdbetalow\]), and (\[psdbetahigh\]) are shown in figures \[fig5\] to \[fig7\].
In addition to these distributions, it is possible to consider other shapes like $g_\lambda(\lambda) \propto a + b \lambda$ so that the resulting spectral density from equation (\[psd2\]) is the sum of a spectral density like the one in equation (\[psdflat\]) plus a term proportional to $$\int_{\lambda_{min}}^{\lambda_{max}} \frac{\lambda}{\lambda^2+\omega^2} \propto \ln\frac{\lambda^2_{max}+\omega^2}{\lambda^2_{min}+\omega^2};$$ but I shall not consider them here, since these shapes seem to be far less common than the cases discussed above.
The integral (\[psd2\]) is a sum of functions that decrease for positive, increasing $\omega$ and therefore cannot be an increasing function and therefore no distribution of relaxation rates can possibly describe bumps and other small local features such as those that are observed in the spectral densities of glassy systems. These features can be described by resonances or by groups of close resonances; the simplest choices are a) fixed resonance frequency and flat distribution of relaxation rates; b) fixed relaxation rate and flat distribution of resonance frequencies. In the case of a flat distribution of relaxation rates between the maximum and minimum rates $\lambda_{min}$, $\lambda_{max}$ we find $$\begin{aligned}
\nonumber
S_{\rm fr}(\omega; \lambda_{min}, \lambda_{max}, \omega_0) &=& \int_{\lambda_{min}}^{\lambda_{max}} \frac{d\lambda}{\lambda^2 + (\omega-\omega_0)^2} \\
\label{resof}
&=& \frac{1}{\omega-\omega_0}\left[ \arctan\frac{\lambda_{max}}{\omega-\omega_0}-\arctan\frac{\lambda_{min}}{\omega-\omega_0} \right]\end{aligned}$$ and similarly in the case of a flat distribution of resonance frequencies between the maximum and minimum frequencies $\omega_{min}$, $\omega_{max}$ we find the spectral densities (\[resof\]) and (\[reslf\]) are shown in figures \[fig8\] and \[fig9\]. $$\begin{aligned}
\nonumber
S_{\rm fw}(\omega; \omega_{min}, \omega_{max}, \lambda) &=& \int_{\omega_{min}}^{\omega_{max}} \frac{d\omega_0}{\lambda^2 + (\omega-\omega_0)^2} \\
\label{reslf}
&=& \frac{1}{\lambda}\left[ \arctan\frac{\omega - \omega_{min}}{\lambda}-\arctan\frac{\omega - \omega_{max}}{\lambda} \right];\end{aligned}$$
Model-based fit of a simulated spectral density {#secresults}
===============================================
When fitting spectra it is important to include the variance of spectral data: if $S_k$ is the spectral estimate at the [*k*]{}-th frequency, and if the time-domain data are affected by Gaussian white noise, then the spectral estimate of the white noise background has standard deviation $S_k$ [@specvar]; this estimate of the standard deviation is usually assumed for simplicity even when there are deterministic components or the noise is not white. Moreover if the final spectral density is the average of $M$ uncorrelated spectra, then the estimate of the standard deviation at the [*k*]{}-th frequency is $S_k/\sqrt{M}$. I wish to stress that this treatment of the spectral variance is only approximate in the case of colored noises, but it is assumed nonetheless, because of the complexity of a calculation that includes the correlation between different samples in the time domain (see, e.g. [@tm]).
I have tested the simple model-derived spectral densities described in section \[secgallery\] on data kindly provided by C. Chakravarty and A. Mudi [@chmupri]: the original spectral data are shown in figure \[fig10\] and correspond to the 230 K curve in figure 1a of reference [@chmu] (see also [@chmu1; @chmu2; @chmu3] for full simulation details).
At very low frequency the spectrum is rather steep: a simple fit of the low-frequency data shows a $1/f^2$ behavior, and thus we can surmise that this is just the high-frequency tail of a simple relaxation with a very low relaxation constant (this accounts for 2 fit parameters: amplitude and relaxation rate). At higher frequency the slope is smaller and Mudi and Chakravarty estimate a spectral index slightly higher than 1 [@chmu]: since there is no hint of a downward bend, I exclude the full spectral shape (\[psdbetaexact\]) and also the reduced form (\[psdbetahigh\]), and I choose (\[psdbetalow\]) instead, i.e. I include the possibility of a low-frequency flattening, made invisible by the high-frequency tail of the simple relaxation (this adds three more parameters to the fit: an amplitude, a minimum relaxation rate, and a spectral index $\beta$). The high-frequency bump resembles rather closely the shape in figure \[fig9\], and thus it is reasonable to assume that both the low-frequency and the high-frequency bumps correspond to flat superpositions of resonances like in equation (\[reslf\]) (each bump accounts for 4 more parameters: an amplitude, a relaxation rate, a minimum and a maximum resonance frequency, but the relaxation rate is taken to be the same in both bumps). The resulting 12 parameter model is: $$\begin{aligned}
\nonumber
S(\omega) &=& \frac{a_1^2}{\omega^2 + \lambda_1^2} + a_2^2 S_{\rm pl,A}(\omega; \lambda_{min,2}, \beta) \\
\label{model}
&& + a_3^2 S_{\rm fw}(\omega; \omega_{min,3}, \omega_{max,3}, \lambda_{34}) + a_4^2 S_{\rm fw}(\omega; \omega_{min,4}, \omega_{max,4}, \lambda_{34})\end{aligned}$$ Notice that the assumptions on the relaxation rate distributions help keep the number of fit parameters rather low. If we tried to fit with a superposition of $N$ simple relaxations we would have $2N$ parameters (one amplitude plus one relaxation rate for each relaxation component): with 12 parameters we could fit only 6 simple relaxation components, therefore the assumed shapes (that correspond to given distributions of relaxation rates and resonance frequencies) allow for a much more economical fit procedure. In this case the spectral data are averages of $M=448$ spectra; table \[tab1\] lists the fit parameters to the data [@chmupri] obtained with a standard Levenberg-Marquardt chi-square minimization procedure, and figure \[fig11\] compares the fit with the data (the $a$ amplitude values in the table are in the spectral amplitude units of fig. \[fig10\], the $\lambda$’s and the $\omega$’s are in cm$^{-1}$, and $\beta$ is dimensionless).
------------------- -----------------------
$a_1$ $11.435 \pm 0.880$
$\lambda_1$ $0.144 \pm 0.012$
$a_2$ $1.351 \pm 0.0059$
$\lambda_{min,2}$ $5.226 \pm 0.099$
$\beta$ $0.327 \pm 0.013$
$a_3$ $0.102 \pm 0.002$
$\omega_{min,3}$ $32.1 \pm 1.0$
$\omega_{max,3}$ $64.4 \pm 0.5$
$a_4$ $0.02534 \pm 0.00002$
$\omega_{min,4}$ $421.0 \pm 0.2$
$\omega_{max,4}$ $947.2 \pm 0.1$
$\lambda_{34}$ $22.77 \pm 0.11$
------------------- -----------------------
The model (\[model\]) is a function of both relaxation rate and resonance frequency and should thus be described by a two-parameter distribution $g(\lambda, \omega_0)$ rather than $g_\lambda(\lambda)$, however if we concentrate on the projection on the $\lambda$ axis, then we can consider only the first two terms: the (reduced) $\lambda$ distribution is shown in figure \[fig12\], and is the sum of a delta-function plus an (unbounded) continuous distribution. Notice that such a distribution is quite challenging for other fit methods, like those implemented by CONTIN and UPEN.
Conclusion
==========
In this paper I have described a model-based fit of power-law-like spectral densities. Like other similar methods, it embodies [*a priori*]{} information on the shape of the distribution, but unlike the other methods, the shape is physically motivated, and the fits can be efficiently performed with a reduced number of parameters.
I wish to thank Giorgio Careri, Giuseppe Consolini, and Charusita Chakravarty for useful discussions. I also wish to thank Charusita Chakravarty and Anirban Mudi for allowing me to use the spectral data from their extensive molecular dynamics simulations of water.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The field of Machine Learning research is divided into subject areas, where each area tries to solve a specific problem, using specific methods. In recent years, borders have almost been erased, and many areas inherit methods from other areas. This trend leads to better results and the number of papers in the field is growing every year. The problem is that the amount of information is also growing, and many methods remain unknown in a large number of papers. In this work, we propose the concept of inheritance between machine learning models, which allows conducting research, processing much less information, and pay attention to previously unnoticed models. We hope that this project will allow researchers to find ways to improve their ideas. In addition, it can be used by researchers to publish their methods too. Project is available by link: https://www.infornopolitan.xyz/backronym'
author:
- |
Arip Asadulaev\
INFORNOPOLITAN\
`[email protected]`\
title: Backronym
---
Motivation
==========
Machine Learning researchers analyzing hundreds of papers every year, and it is rather difficult to structure such type of information. On average, the most valuable information extracted from a Machine Learning paper is method components and their interaction. To remember all processed methods is problematic, and it is necessary to somehow save it.
It is important to note that today, almost any new method is the development of an older method or compilation of a set of methods. To represent such type of information as a table or a list it not optimal. In our opinion, a strictly fixed graph, with edges between methods and their components is the best way to do it.
We thought about cite graph and unfortunately, the citation does not reflect very well which methods are directly used in the architecture. We decided to hand annotate it and write down from which components every method consists, to give an opportunity to researchers see preprocessed info about methods.
During papers analyzing, we realized how many methods with excellent results are still not used, due to the fact that they were lost among many other papers. There are a lot of models being developed, and often research’s on a new conference do not inherit most of the improvements proposed a year ago.
There are many amazing services that allow you to monitor the emergence of new articles in the field. But all these methods were aimed primarily at the acquisition of knowledge, but not for help in research. Our platform aims to diversify research. Creativity is more important than the experience, so the tools that can help you to be more creative should be an integral part of the research process.
We have tools that make it easy to conduct experiments, run and evaluate models, but we do not have methods that allow us to extend our model in a ideas level. In Machine learning, and Artificial Intelligence research as a whole, fresh and elegant ideas play a key role.
Other fields of science cannot boast of such tools too, but we are absolutely sure that Machine Learning and other areas that solve difficult problems, needs an extraordinary approach for ideas improvements.
The Graph
=========
In the graph each paper can be presented as one single method, for example, Autoencoder (AE), or separately, Autoencoder (AE) -> Encoder (ENCDR), Decoder (DCDR). Models inherit from each other but, some method may not use another method fully, but only some part of it. For example, GAN\[1\] consist of Generator(GEN), Discriminator (DIS), Adversarial Autoencoder (AAE)\[2\] based on Autoencoder (AE)\[3\] and DIS. Models have the edge to other methods, if it based on this method or include it directly in architecture, for example for AAE we will have an edge to AE and DIS. We call this graph BACKRONYM.
In just a few months, we took out about 250 articles from the NeurIPS 2019 conference and 250 other papers on which they are based. We preprocess papers and presented info in a matrix with 10 columns:
- Paper title
- Link to paper
- Names of authors
- Release date
- Place of publication
- Method name
- Subject area (Using For)
- Acronym of method name
- A brief description of the method
- “Based on”:the list of methods on which this method is based (list of acronyms which are available in table).
The graph is built using “Based on” column, where each row consists of a list of methods on which the method is based, Fig. 1.
The way we analyzed papers is very far from ideal. Most of the areas were completely unfamiliar to us, to understand them took several days. Sometimes we could not find the right description for some methods, and the abstract of the article was used for this. The graph is interactive, clicking on the node opens a list with meta-information from columns, Fig. 2.
The “Subject area (Using For)” column allows creating subgraphs with inheritances inside the field, where the method is marked in red if it using in other areas, see Fig. 3.
The visualization is built using the 3d-force-graph javascript library: https://github.com/vasturiano/3d-force-graph
Now the simplest way to use the graph it skips connections. For example, you use CNN\[4\] in your model, and in a graph, you can see another method that inherited from CNN. You can make little research and try just to replace your CNN with the advanced version.
It is very difficult to create an inheritance graph that would be fully consistent with the truth. Based on this, we created an opportunity for authors to add their own methods and make changes to existing ones in the graph. After all, no one except the authors of the article knows how to disassemble and describe their method in the best way. https://www.infornopolitan.xyz/add-research
Discussion and Future Work
==========================
Nevertheless, We believe that the main thing that the graph can give is associations. We were excited to know how GAN idea occurred to Ian Goodfellow. Talking on Artificial Intelligence podcast by Lex Fridman\[5\], Ian said that it was motivated by Boltzmann machine “positive” and “negative” training phases. This is a great example when a model inherits the properties of another not directly but very abstractly.
We would like very much to see this type of connections in this graph, and it depends solely on authors of papers, will or not they share what was motivated them to create such type of model to solve some problem. We think that the story not only about what methods were used specifically in their architecture but also ideologically on which concepts and knowledge the solution was formed, can allow us to be much more resourceful.
This project is community-driven. We want to make it better and motivate more peoples to add their models to the graph. More accurate information about methods and better visualization technologies can really make it a very useful tool.
In the future, with community support, we can level up BACKRONYM and built the system which will recommend us how to extend research. For this scenario, we plan to give to users the ability to visualize their own knowledge, publicly or privately, irrespective of the main graph. For example, user can build a graph of all methods which he knows or even graph of one model components, and the system will recommend him the most useful paper or method. Also, we plan to add: 1) 2D visualization. 2) Ability to build and save individual subgraphs. 3) Search by method name.
It seems that even today this graph take a place to be because it probably allows someone to get associations that may help to create some new method or extend current. Today, it’s just a graph, but tomorrow we will have the tools to objectively get machine recommendations and automatically evaluate the impact the proposed idea on the way to General Machine Intelligence systems.
[1]{}
Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio. Generative Adversarial Nets. https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf.
Alireza Makhzani, Jonathon Shlens, Navdeep Jaitly, Ian Goodfellow, Brendan Frey. Adversarial Autoencoders. https://arxiv.org/abs/1511.05644
Dana H. Ballard. Autoencoder. https://www.aaai.org/Papers/AAAI/1987/AAAI87-050.pdf.
Yann Lecun, Patrick Haffer, Leon Bottou and Yoshua Bengio. Convolutional Neural Network. http://yann.lecun.com/exdb/publis/pdf/lecun-99.pdf
Ian Goodfellow: Artificial Intelligence podcast at MIT. https://www.youtube.com/watch?v=Z6rxFNMGdn0
Images
======
![[]{data-label="fig:fig1"}](fig1.png)
![[]{data-label="fig:fig2"}](fig3.png)
![[]{data-label="fig:fig3"}](fig2.png)
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'V. L. Kashevarov[^1] for Crystal Ball at MAMI, TAPS, and A2 Collaborations'
title: 'Target and beam-target asymmetry measurements at MAMI'
---
Introduction
============
Main features of $\pi^0\eta$ photoproduction on protons were studied in works [@Doring; @HornEPJ; @Horn1940; @FKLO; @Anis12; @W_LM], were the reaction mechanism was explained through the dominance of the $D_{33}$ wave with a small admixture of positive parity resonances and an insignificant Born terms contribution. Further steps toward investigation of this reaction are related to polarisation observables that are especially sensitive to small reaction amplitudes. There are 64 polarization observables for double pseudoscalar meson photoproduction. Different relationships among these observables and their symmetries decrease this value to 15 independent quantities [@RobertsOed]. Some of these were already measured and analyzed [@Ajaka; @Gutz1; @Gutz2; @Aphi; @DorMeiss].
In the present work we measured at the first time the target and the beam-target asymmetries. For the totally exclusive five-fold cross section there are two independent transversal target asymmetries ($P_x$ and $P_y$) and two independent circular beam-target asymmetries ($P_x^\odot$ and $P_y^\odot$) [@FiAr11]. The observables $P_y$ and $P_x^\odot$ integrated over the phase space of two from three final state particles are equivalent to $T$ and $F$ asymmetries of single pseudoscalar meson photoproduction.
Experimental setup and data analysis
====================================
The experiment was performed at the MAMI C accelerator in Mainz [@MAMIC] using the Glasgow-Mainz tagged photon facility [@TAGGER]. In the present measurement the longitudinally polarized electron beam with energy of 1557 MeV and polarization degree of 80% was used. The longitudinal polarization of electrons is transferred to circular polarization of the photons during the bremsstrahlung process in a radiator. The reaction $\gamma p\to \pi^0\eta p$ was measured using the Crystal Ball [@CB] as the central spectrometer and TAPS [@TAPS] as a forward spectrometer. The solid angle of the combined Crystal Ball and TAPS detection system is nearly $97\%$ of $4\pi$ sr.
The experiment requires transversely polarized protons, which were provided by a frozen-spin butanol ($C_4H_9OH$) target. A specially designed $^3He/^4He$ dilution refrigerator was built in cooperation with JINR (Dubna). For transverse polarization a 4-layer saddle coil was installed as the holding magnet, which operated at a field of 0.45 Tesla. The target container with 2-cm length and 2-cm diameter was filled by 2-mm diameter butanol spheres with the filling factor of around $60\%$. The average proton polarization during beam time period May-June 2010 and April 2011 was $70\%$ with relaxation times of around 1500 hours. More details about the target are given in Ref.[@Thomas].
The event-selection procedure was similar to the one that was described in Ref. [@PiEtaEPJA], where meson pairs were identified via their decay into 2 photons. For the case of 4 detected photons the best solution for the meson pair was found using the $\chi^2$ minimization. This was then followed by application of the invariant and missing mass cuts providing good identification of the reaction.
Using the butanol target has an essential disadvantage because of an additional background coming from the reactions on $^{12}C$ and $^{16}O$. Detection of the outgoing protons and application of the coplanarity cut suppress the background significantly, but the effect of this procedure is still insufficient. To subtract the residual background events we used the results of the analysis of $\pi^0\eta$ photoproduction on the carbon and the liquid hydrogen targets, which were fitted to the butanol data. Because of the magnitude and the shape of the background depend on the initial beam energy and momenta of the final particles, the background subtraction procedure was performed for each bin, where the asymmetries were measured. The procedure of the background subtraction is illustrated on Fig.\[fig1\] for two different examples, which are typical for the presented data analysis. Missing mass spectra for the reaction $\gamma p\to \pi^0\eta p$ with the butanol target are shown on Fig.\[fig1\] (a) and (b) by the full black circles. Spectra measured with the hydrogen and carbon targets are presented on the same plots by the green triangles and the blue squares correspondingly. Their absolute values were fitted to the butanol data. The red opened cycles, representing the sum of the hydrogen and carbon contribution, is our result of the fit. To minimize the uncertainties of the background subtraction, the procedure was used only within the missing mass energy interval, which is indicated on the plots by the vertical solid lines.
The systematic uncertainties come mainly from the determination of the proton polarization degree (4%), circular photon beam polarization degree (2%), and the background subtraction procedure (3-4%) and were estimated to be less than 6%. In order to reduce the systematic errors coming from target and detector conditions, the proton polarization direction was regularly reversed during experiment.
Preliminary results
===================
Figs.\[fig2\] and \[fig3\] show our preliminary results together with different theoretical predictions. The solid and dashed lines are predictions of the isobar model from Ref.[@FKLO]. In this model the reaction amplitude contains the sum of $s$-channel Breit-Wigner resonances with the total spin $J\leq 5/2$ and the Born terms. Parameters of the resonances were fitted to the experimental data [@PiEtaEPJA]. For the solid line we used the fit (I) which gives the best description of the measured linear beam asymmetry $\Sigma$ [@Ajaka; @Gutz1]. Dashed line include only $\Delta 3/2^-$ resonances.
The dash-dotted line shows predictions of the Bonn-Gatchina multichannel partial wave analysis [@Anis12] (solution BG2011-02). Within this approach the positions of resonances, their partial decay widths, and relative strengths are fitted simultaneously to the data sets in different channels, including single and double meson production as well as strangeness production.
As we can see from Figs.\[fig2\]-\[fig3\], both $P_y$ and $P_x^\odot$ demonstrate more complicated behavior, than the one predicted by the single $\Delta 3/2^-$ model. At the same time, the deviation is not large, thus indicating that the role of the states besides $\Delta 3/2^-$ remains restricted. Interference between $\Delta 3/2^-$ and the positive parity states $\Delta 1/2^+$ and $\Delta 3/2^+$ is responsible for the nontrivial angular and energy dependence of all asymmetries presented.
Summary
=======
Preliminary data for the target and the beam-target asymmetry of the cross section for $\gamma p\to\pi^0\eta p$ obtained with circular polarized photons and transversally polarized protons were presented. The measurements were performed at the MAMI C accelerator using the Crystal Ball/TAPS spectrometer. The polarization observables are sensitive to the contribution of the small components in the reaction amplitude. Obtained data could be usefully for further study of the partial wave content of $\pi\eta$ photoproduction.
This work was supported by the Deutsche Forschungsgemeinschaft (SFB 1044).
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[^1]:
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A theorem of Swan states that the locally free class group of a maximal order in a central simple algebra is isomorphic to a restricted ideal class group of the center. In this article we discuss this theorem and its generalization to separable algebras for which it is more applicable to integral representations of finite groups. This is an expository article with aim to introduce the topic for non-specialists.'
address:
- |
Institute of Mathematics, Academia Sinica and NCTS\
6th Floor, Astronomy Mathematics Building\
No. 1, Roosevelt Rd. Sec. 4\
Taipei, Taiwan, 10617
- |
The Max-Planck-Institut für Mathematik\
Vivatsgasse 7, Bonn\
Germany 53111
author:
- 'Chia-Fu Yu'
title: Notes on locally free class groups
---
Introduction {#sec:intro}
============
A more precise title of this article should be “Notes on locally free class groups of orders in separable algebras over global fields”. Our goal is to introduce the locally free class group of an $R$-order $\Lambda$ in a separable $K$-algebra. Here $R$ is a Dedekind domain and $K$ is its fraction field, which we shall assume to be a global field later. We refer to Section \[sec:03\] for definitions of separable $K$-algebras and $R$-orders.
The notion of locally free class groups can be defined in a more general setting. However, since results discussed here are only restricted to the case where the ground field $K$ is a global field, we do not attempt to discuss its definition as general as it could be. Instead we shall illustrate the essential idea of this notion (see Section \[sec:cancel\]).
After illustrating the notion, we present the main theorem on locally free class groups. We then explain how to deduce a theorem of Swan [@swan:1962 Theorem 1, p. 56] from the main theorem. The strong approximation theorem (SAT) plays an important role in the proof of the main theorem, and we give a short exposition of the SAT. Our another goal to take this opportunity to introduce the reader (mainly for graduate students and non-specialists) some useful tools and results in Algebraic Number Theory and show how to apply them together.
The cancellation law {#sec:cancel}
====================
Let us first motivate the notion of locally free class groups by the classical theorem of Steinitz. Let $R$ be a Dedekind domain with fraction field $K$ and we will always suppose that $R\neq K$. An [*$R$-lattice*]{} is a finite $R$-module $M$ which does not have non-zero torsion elements, that is, $M$ is isomorphic to a finite $R$-submodule in a (finite-dimensional) $K$-vector space. We have the following results concerning the classification of $R$-lattices (cf. [@curtis-reiner:1 Theorem (4.13), p. 85]):
1. Every $R$-lattice $M$ is $R$-projective, and $M\simeq
\oplus_{i=i}^n J_i$ for some non-zero ideals $J_i$ of $R$, where $n$ is the $R$-rank of $M$.
2. Two $R$-lattices $M=\oplus_{i=i}^n J_i$ and $M'=\oplus_{i=i}^m J'_i$ of the form in (1) are isomorphic if and only if $n=m$ and the products $J_1\cdots J_n$ and $J_1'\cdots J_n'$ are isomorphic.
From the statement (2) one can easily deduce the following result: If $M$ and $M'$ are two $R$-lattices, then we have $$\label{eq:cancel}
M\oplus R\simeq M'\oplus R \iff M\simeq M'$$ The property (\[eq:cancel\]) is called the [*cancellation law*]{}. As we learned from Algebra, a useful and easier way of studying rings to study their modules, instead of their underlying ring structures. Using this approach, the cancellation law then can be used to distinguish certain rings which share the same good properties. For example, consider the quaternion ${\mathbb Q}$-algebras $B_{p,\infty}$, which are those ramified exactly the two places $\{p,\infty\}$ of ${\mathbb Q}$, for primes $p$. Choose a maximal order $\Lambda(p)$ in each $B_{p,\infty}$, that is, $\Lambda(p)$ is not strictly contained in another ${\mathbb Z}$-order in $B_{p,\infty}$. Then one can show that the cancellation law for ideals of $\Lambda(p)$ holds true if and only if $p\in
\{2,3,5,7,13\}$. We will also give a proof of this fact later.
Now let $\Lambda$ be an (not necessarily commutative) $R$-algebra which is finitely generated as an $R$-module. The above example shows that the cancellation law for (right) projective $\Lambda$-modules need not hold. In Mathematics, we often encounter a situation that a nice property we are looking for turns out to be impossible. In that situation one usually remedies it by creating a more flexible notion so that the desired nice property remains valid in a slightly weaker setting. In the present case, one can for example consider the following weaker equivalence relation: $$\label{eq:weak}
\text{Define\ } M\sim M' \quad \text{if $M\oplus
\Lambda^r\simeq M'\oplus \Lambda^r$
for some integer $r\ge 0$.}$$ Then it follows from the definition that the cancellation law holds true for this new equivalence relation, that is, we have $$\label{eq:can1}
M\oplus \Lambda \sim M'\oplus \Lambda \iff M\sim M'.$$ The modules $M$ and $M'$ satisfying the above property are said to be [*stably isomorphic*]{}. The reader familiar with algebraic topology would immediately find that the way of defining a “stable” notion here is similar to that in the definition of stable homotopy groups. It is also similar to that in the definitions of stable freeness and stable rationality. These are reminiscent of the definition of groups $K_0$ and $K_1$ in Algebraic $K$-theory using an inductive limit procedure.
Locally free class groups {#sec:03}
=========================
For the rest of this article we assume that the ground field $K$ is a global field; that is, $K$ is a finite separable extension of ${\mathbb Q}$ or $\Fp(t)$. Thus, $R$ is the ring of $S$-integers of $K$ for a finite set $S$ of places which contains all archimedean ones when $K$ is a number field. Let $A$ denote a finite-dimensional separable $K$-algebra. That is, $A$ is a finite-dimensional semi-simple $K$-algebra such that the center $C$ of $A$ is a product of finite separable field extensions $K_i$ of $K$. Recall that an [*$R$-order*]{} in $A$ is an $R$-subring of $A$ which is finitely generated as an $R$-module and generates $A$ over $K$. We let $\Lambda$ denote an $R$-order in $A$. A [*$\Lambda$-lattice*]{} $M$ is a $R$-torsion free finitely generated $\Lambda$-module.\
[**Example.**]{} Let $G$ be a finite group with ${\rm char} K \nmid
|G|$. Then the group algebra $A=KG$ is a separable $K$-algebra. We can see this by Maschke’s Theorem (cf. ): Every finite-dimensional representation of $G$ over $K$ is a direct sum of irreducible representations. Then by definition $KG$ is a semi-simple $K$-algebra. Applying Maschke’s Theorem again to an algebraic closure $\ol
K$ of $K$, we see that the algebra $\ol K\otimes_K KG=\ol K G$ is also semi-simple. Therefore, $A$ is a separable $K$-algebra. Clearly, the group ring $\Lambda=RG$ is an $R$-order in $A$, and any representation $M$ of $G$ over $R$ is a $\Lambda$-lattice.\
For any integer $n\ge 1$, denote by ${\rm LF}_n(\Lambda)$ the set of isomorphism classes of locally free right $\Lambda$-modules of rank $n$. Two locally free right $\Lambda$-modules $M$ and $M'$ are said to be [*stably isomorphic*]{}, denoted by $M\sim_{s}M'$, if $$M\oplus \Lambda^{r} \simeq M'\oplus \Lambda^r$$ as $\Lambda$-modules for some integer $r\ge 0$. The stable class of $M$ will be denoted by $[M]_s$, while the isomorphism class is denoted by $[M]$. By a $\Lambda$-ideal we mean a $\Lambda$-lattice in $A$, that is, it is an $R$-lattice which is also a $\Lambda$-module. Let $\Cl(\Lambda)$ denote the set of stable classes of locally free right $\Lambda$-ideals in $A$. The Jordan-Zassenhaus Theorem (cf. [@curtis-reiner:1 Theorem 24.1, p. 534]) states that ${\rm LF_1}(\Lambda)$ is a finite set, and hence so the set $\Cl(\Lambda)$ is. We define the group structure on $\Cl(\Lambda)$ as follows. Let $J$ and $J'$ be two locally free $\Lambda$-ideals. Define $$\label{eq:group}
[J]_s+[J']_s=[J'']_s,$$ where $J''$ is any locally free $\Lambda$-ideal satisfying $$\label{eq:com}
J\oplus J'=J''\oplus \Lambda$$ as $\Lambda$-modules. Such a $\Lambda$-ideal $J''$ always exists and we will see this in Section \[sec:04\]. The following basic lemma shows that $\Cl(\Lambda)$ is an abelian group, called the [*locally free class group*]{} of $\Lambda$.
The finite set $\Cl(\Lambda)$ with the binary operation defined in (\[eq:group\]) forms an abelian group.
By (\[eq:com\]), the commutativity holds true. We prove the associativity. Let $J_1,J_2,J_3$ be three locally free ideals of $\Lambda$. Suppose we have $[J_1]_s + [J_2]_s=[J']_s$ and $[J']_s+
[J_3]_s=[J'']_s$. Then $$(J_1 \oplus J_2)\oplus J_3 \simeq \Lambda\oplus J'\oplus J_3\simeq
J'' \oplus \Lambda^2.$$ Similarly if $[J_2]_s + [J_3]_s=[G']_s$ and $[J_1]_s+
[G']_s=[G'']_s$, then $J_1 \oplus (J_2\oplus J_3) \simeq G'' \oplus \Lambda^2$. This shows $[J'']_s=[G'']_s$ and the associativity holds true.
We introduce some more notations. Denote by $C$ the center of $A$. One has $C=\prod^s_i K_i$ and $A=\prod^s_i A_i$, where each $A_i$ is a central simple algebra over $K_i$. For any place $v$ of $K$, let $K_v$ denote the completion of $K$ at $v$, and $O_v$ the valuation ring if $v$ is non-archimedean. We also write $R_v$ for $O_v$ when $v\not\in S$. Let $A_v:=K_v\otimes_K A$, $C_v:=K_v\otimes_K A$ and $\Lambda_v:=R_v\otimes_R \Lambda$ be the completions of $A$, $C$ and $\Lambda$ at $v$, respectively. By a place $w$ of $C$ we mean a place $w$ of $K_i$ for some $i$; that the algebra $A$ splits (resp. is ramified) at the place $w$ of $C$ means that $A_i$ splits (resp. is ramified) at the place $w$. Let $\wh R=\prod_{v\not\in S} R_v$ be the profinite completion of $R$, and let $\wh K=K\otimes_{R} \wh R$ be the finite $S$-adele ring of $K$; one also writes ${\mathbb A}_K^S$ for $\wh K$. Put $\wh A:=\wh K \otimes_K A$, $\wh C:=\wh K \otimes_K C$ and $\wh \Lambda:=
\wh R\otimes_R \Lambda=\prod_{v\not\in S} \Lambda_v$.
It is a basic fact that the set ${\rm LF}_1(\Lambda)$ is isomorphic to the double coset space $A^\times \backslash \wh A^\times/\wh \Lambda^\times$. There is a natural surjective map $$\label{eq:LF1}
{\rm LF}_1(\Lambda)\to \Cl(\Lambda)$$ by sending $[J]\mapsto [J]_s$. Let $N_{A_i/K_i}: A_i\to K_i$ denote the reduced norm map. It induces a surjective map $N_i: \wh A_i^\times \to \wh K^\times_i$ because we have the surjectivity $A_{i}\otimes K_v\to K_{i}\otimes K_v$ for any finite place $v$ of $K$. The reduced norm map $N:A=\prod_i A_i\to C=\prod_i K_i$ is simply defined as the product $N=(N_{A_i/K_i})_i$. Then we have a surjective map $N: \wh A^\times\to \wh C^\times$, and it gives rise to surjective map (again denoted by) $$\label{eq:norm}
N :{\rm LF}_1(\Lambda)\simeq
A^\times \backslash \wh A^\times /\wh \Lambda^\times \to
N(A^\times)\backslash \wh C^\times /N(\wh \Lambda^\times).$$ We will see that $N(A^\times)=C^\times_{+,A}$, where $C^\times_{+,A}\subset C^\times$ consists of all elements $a\in C^\times$ with $r(a)>0$ for all real embeddings(places) $r$ which are ramified in $A$. The main theorem for the locally free class groups is as follows.
\[2\] The map (\[eq:norm\]) factors through ${\rm LF}_1(\Lambda)\to
\Cl(\Lambda)$ and it induces an isomorphism of finite abelian groups $$\label{eq:nu}
\nu: \Cl(\Lambda)\simeq \wh K^\times /C^\times_{+,A}
N(\wh \Lambda^\times).$$
We now describe the theorem of Swan on locally free class groups. Assume that $A$ is a central simple algebra and $\Lambda$ is a maximal $R$-order in $A$. Define the ray class group $\Cl_A(R)$ of $K$ by $$\Cl_A(R):=I(R)/P_A(R),$$ where $I(R)$ be the ideal group of $R$ and $P_A(R)$ be the subgroup generated by the principal ideals $(a)$ for $a\in K^\times_{+,A}$. Here $K^\times_{+,A}\subset K^\times$ is the subgroup of $K^\times$ defined as above. In terms of the adelic language, the group $\Cl_A(R)$ is nothing but the group $\wh K^\times/K^\times_{+,A} \wh
R^\times$.
\[3\] Let $K$ be a global field and $R$ the ring of $S$-integers of $K$ for a finite set $S$ of places containing all archimedean ones. Let $A$ be a central simple algebra and $\Lambda$ a maximal $R$-order in $A$. Then theres is an isomorphism of finite abelian groups $\Cl(\Lambda)\simeq \Cl_A(R)$.
To see Theorem \[3\] is an immediate consequence of Theorem \[2\], we just need to check that $N(\Lambda_v^\times)=R_v^\times$ for $v\not\in S$ ($\Lambda_v$ here is a maximal $R_v$-order). There exists a maximal subfield $E\subset A_v$ which is unramified over $K_v$. Since any two maximal orders in $A_v$ are conjugate, $\Lambda_v$ contains a copy of the ring of integers $O_E$ of $E$. As $E$ is unramified over $K_v$, the successive approximation shows that $N_{E/K_v}(O_E^\times)=R_v^\times$. It follows that $N(\Lambda_v^\times)=R_v^\times$.
\[4\] Let $B_{p,\infty}$ and $\Lambda(p)$ for primes $p$ be as in Section \[sec:cancel\]. Then the cancellation law for ideals of $\Lambda(p)$ holds true if and only if $p\in \{2,3,5,7,13\}$.
The cancellation law holds if and only if the map ${\rm LF}_1(\Lambda(p))\to \Cl(\Lambda(p))$ is a bijection. By Swan’s theorem, the locally free class group $\Cl(\Lambda(p))\simeq
\Cl_{B_{p,\infty}}({\mathbb Z})$ is trivial. Thus, the cancellation law holds if and only if the class number $h(\Lambda(p))=|{\rm LF}_1(\Lambda(p))|$ is one. On the other hand we have the class number formula [@eichler:1938] $$\label{eq:class_no}
h(\Lambda(p))=\frac{p-1}{12}+\frac{1}{3}\left
(1-\left(\frac{-3}{p}\right )\right )+\frac{1}{4}\left
(1-\left(\frac{-4}{p}\right )\right ),$$ where $\left( \frac{\cdot}{p}\right )$ denotes the Legendre symbol. From this one easily sees that $h(\Lambda(p))=1$ if and only if $p\in \{2,3,5,7,13\}$.
For the rest of this section we give a proof of the following basic fact.
\[5\] Let $A$ is a separable $K$-algebra and $C$ its center. Then $N(A^\times)=C^\times_{+,A}$.
Since $A=\prod_i A_i$ and $C^\times_{+,A}=\prod_i
K^\times_{i,+,A_i}$, it suffices to show $N(A^\times)=K^\times_{+,A}$ for any central simple $K$-algebra $A$. We can use the Hasse-Schilling norm theorem (the local-global principle for the reduced norm map) to describe $N(A^\times)$: $$N(A^\times)=\{x\in K^\times ; x\in N(A_v^\times) \ \forall\, v\};$$ see [@reiner:mo (32.9) Theorem, p. 275] and [@reiner:mo (32.20) Theorem, p. 280]). Clearly $N(A_v^\times)=K_v^\times$ when $v$ is complex, non-archimedean, or a real split place for $A$. It remains to show that if $v$ is a real ramified place for $A$, then one has $v(a)>0$ if and only if $a\in N(A_v^\times)$.
\[7\] Let ${\mathbb H}$ be the real Hamilton quaternion and $n\in \bbN$. Then $N(\GL_n({\mathbb H}))={\mathbb R}_{+}$.
We give two proofs of this result. One is topological and the other one is algebraic. As ${\mathbb R}_+=N({\mathbb H}^\times)\subset N(\GL_n({\mathbb H}))$, it suffices to show $N(\GL_n({\mathbb H}))\subset {\mathbb R}_{+}$.
\(1) The set $\GL_n({\mathbb H})^{ss}\subset\GL_n({\mathbb H})$ of semi-simple elements is open and dense in the classical topology. By continuity it suffices to show $N(x)>0$ for any $x\in \GL_n({\mathbb H})^{ss}$. Since such $x$ is contained in a maximal commutative semi-simple subalgebra, which is isomorphic to ${\mathbb C}^n$, we have $N(x)>0$. (2) The algebraic proof relies on the existence of the [Dieudonné ]{} (non-commutative) determinant (cf. [@curtis-reiner:1 p. 165]). Suppose that $D$ is a central division algebra over any field $K$. There is a group homomorphism (called the [Dieudonné ]{}determinant) $$\det: \GL_n(D) \to D^\#,$$ where $D^\#=D^\times/[D^\times, D^\times]$. The reduced norm map $N:
D^\times \to K^\times$ gives rise to a map $\nr: D^\# \to K^\times$. The reduced norm map $N: \GL_n(D) \to K^\times$ can be also described as $$N (a)= \nr (\det a). \quad \forall \, a\in \GL_n(D).$$ It follows that $N(\GL_n(D))\subset
N(D^\times)$. Particularly $N(\GL_n({\mathbb H}))\subset
N({\mathbb H}^\times)={\mathbb R}_+$.
One can show a slightly stronger result that the Lie group $\GL_n({\mathbb H})$ is connected. Let $G_1$ be the kernel of the reduced norm map $N:\GL_n({\mathbb H})\to {\mathbb R}^\times$. Then $G_1=\ul G_1({\mathbb R})$ for a connected, semi-simple and simply connected algebraic ${\mathbb R}$-group $\ul G_1$, and hence that $G_1$ is connected. Then the fibers of the reduced norm map $N$ are all connected as they are principal homogeneous spaces under $G_1$. We just showed that the image of the map $N$ is also connected (Lemma \[7\]). Thus, $\GL_n({\mathbb H})$ is connected.
Proof of Theorem \[2\] {#sec:04}
======================
For any integer $n\ge 1$ and any ring $L$ not necessarily commutative, let $\Mat_n(L)$ denote the matrix ring over $L$ and let $\GL_n(L)$ denote the group of units in $\Mat_n(L)$. Let $N_n:\Mat_n(A)\to C$ be the reduced norm map, which induces a surjective homomorphism $N_n: \GL_n(\wh A)\to \wh C^\times$. For any integer $r\ge 1$, let $I_r\in \Mat_r({\mathbb Z})$ be the identity matrix. Let $\varphi_r:\GL_n \to \GL_{n+r}$ be the morphism of algebraic groups which sends $$a \mapsto \varphi_r(a)=
\begin{pmatrix}
a & \\
& I_r
\end{pmatrix}.$$
Clearly any locally free right $\Lambda$-module $M$ of rank $n$ is isomorphic to a $\Lambda$-submodule in $A^n$. Therefore, the set ${\rm LF}_n(\Lambda)$ is in bijection with the set of global equivalence classes of the genus of the standard lattice $\Lambda^n$ in $A^n$. The latter is naturally isomorphic to $\GL_n(A)\backslash \GL_n(\wh
A)/\GL_n(\wh \Lambda)$. If $n\ge 2$, then it follows from the strong approximation theorem (see Kneser [@kneser:sa] and Prasad [@prasad:sa1977], also see Theorem \[SAT\]) that the induced map $$\label{eq:strong}
N_n: \GL_n(A)\backslash \GL_n(\wh A)/\GL_n(\wh \Lambda){\stackrel{\sim}{\longrightarrow}}\wh C^\times /N_n(\GL_n(A))N_n(\GL_n(\wh \Lambda))$$ is a bijection.
\[4.1\] We have $$\label{eq:reduction}
\wh C^\times /N_n(\GL_n(A))N_n(\GL_n(\wh \Lambda))=
\wh C^\times /N(A^\times)N(\wh \Lambda)=\wh C^\times /C^\times_{+,A}
N(\wh \Lambda).$$
We have seen in Lemma \[5\] that $N_n(\GL_n(A))=N(A^\times)=C^\times_{+,A}$. We now prove $N_n(\GL_n(\Lambda_v))=N(\Lambda_v^\times)$ for $v\not \in S$ since the statement is local. The group $\GL_n(\Lambda_v)$ contains as a subgroup the group $E_n(\Lambda_v)$ of elementary matrices with values in $\Lambda_v$. Since $\Lambda_v$ is semi-local, we have a result of Bass [@swan:K_order Proposition 8.5] that $\GL_n(\Lambda_v)$ is generated by the subgroup $E_n(\Lambda_v)$ and the image $\varphi_{n-1}(\GL_1(\Lambda_v))$. Since $E_n(\Lambda_v)$ is contained in the kernel of $N$, we have $N_n(\GL_n(\Lambda_v))=N_n(\varphi_{n-1}(\Lambda_v^\times))=
N(\Lambda_v^\times)$.
For any integer $r\ge 1$, we say two locally free right $\Lambda$-ideals $J$ and $J'$ are [*$r$-stably isomorphic*]{} if $J\oplus \Lambda^r\simeq J'\oplus \Lambda^r$ as $\Lambda$-modules. Let $\hat c\in \wh A^\times$ be an element such that $\hat c \Lambda=J$; we have $\varphi_r(\hat c)
\Lambda^{r+1}=J\oplus \Lambda^r$.
The morphism $\varphi_r$ induces the following commutative diagram: $$\label{eq:21}
\begin{CD}
A^\times \backslash \wh A^\times/\wh \Lambda^\times @>\varphi_r>>
\GL_{r+1}(A)\backslash \GL_{r+1}(\wh A)/\GL_{r+1}(\wh \Lambda) \\
@VNVV @V N_{r+1}VV \\
\wh C^\times/C^\times_{+,A} N(\wh \Lambda^\times) @>{\rm id}>> \wh
C^\times/C^\times_{+,A} N(\wh \Lambda^\times), \\
\end{CD}$$ where the reduced norm map $N_{r+1}$ is known be a bijection. Two isomorphism classes $[J]$ and $[J']$ in $A^\times \backslash \wh
A^\times/\wh \Lambda^\times$ are $r$-stably isomorphic if and only if $\varphi_r([J])=\varphi_r([J'])$. As $N_{r+1}$ is an isomorphism, this is equivalent to $N([J])=N([J'])$. The latter condition is independent of $r$. Therefore, we conclude the following statement.
\[4.2\] Let $J$ and $J'$ be two locally free right $\Lambda$-ideals. The following statements are equivalent.
1. $J$ and $J'$ are stably isomorphic.
2. $J$ and $J'$ are $r$-stably isomorphic for some $r\ge 1$.
3. $J$ and $J'$ are $r$-stably isomorphic for all $r\ge 1$.
4. One has $N([J])=N([J'])$ in $\wh K^\times/C^\times_{+,A} N(\wh
\Lambda^\times)$.
Thus, the reduced norm map $N$ induces an isomorphism $$\label{eq:22}
\nu: \Cl(\Lambda)\simeq \wh C^\times/C^\times_{+,A} N(\wh
\Lambda^\times).$$
We now check that $\nu$ is a group homomorphism. Let $J$ and $J'$ be two locally free $\Lambda$-ideals. Let $\hat c$ and $\hat c'$ be elements in $\hat
A^\times$ such that $\hat c \Lambda=J$ and $\hat c' \Lambda=J'$. Put $J'':=\hat c \hat c' \Lambda$. We claim that
- $J\oplus J'\simeq J''\oplus \Lambda$ as $\Lambda$-modules;
- $\nu([J]_s)\nu([J']_s)=\nu([J'']_s)$.
Statement (a) follows from $$\begin{bmatrix}
\hat c \hat c' & 0 \\
0 & 1 \\
\end{bmatrix}\cdot \Lambda^2=J''\oplus \Lambda,\quad \text{and} \quad
N_2\left ( \begin{bmatrix}
\hat c \hat c' & 0 \\
0 & 1 \\
\end{bmatrix} \right )=N_2\left ( \begin{bmatrix}
\hat c & 0 \\
0 & \hat c' \\
\end{bmatrix} \right )$$ in $\wh C^\times/C^\times_{+,A} N(\wh \Lambda^\times)$. Statement (b) follows from $$\nu([J]_s)\nu([J']_s)=N([\hat c])N([\hat c'])=N([\hat c \hat
c'])=\nu([J'']_s).$$
This completes the proof of Theorem \[2\].\
Strong approximation and remarks {#sec:strong_app}
================================
In this supplementary section we give a short exposition of the strong approximation theorem and explain how (\[eq:strong\]) follows immediately from this. We keep the notations of Section \[sec:03\]. In particular $K$ denotes a global field and $S$ is a nonempty finite set of places of $K$.
\[SAT\] Let $G$ be a connected, semi-simple and simply connected algebraic group over $K$. Suppose that
- $G$ does not contain any $K$-simple factor $H$ such that the topological group $H_S:=\prod_{v\in S} H(K_v)$ is compact.
Then the group $G(K)$ is dense in $G({\mathbb A}^S_K)$.
See Kneser [@kneser:sa] when $K$ is a number field and Prasad [@prasad:sa1977] when $K$ is a global function field. The results were proved upon the Hasse principle, i.e. the map $$H^1(K,G) \to \prod_{v} H^1(K_v, G)$$ is injective. The Hasse principle was known to hold for any simply-connected group at that time except possibly for those of type $E_8$. The last case of type $E_8$ was finally completed by Chernousov in 1989.
The strong approximation theorem is a strong version of “class number one” result.
\[11\] Let $G$ be as in Theorem \[SAT\] satisfying the condition ($*$) and assume that $S$ contains all archimedean places of $K$. Then for any open compact subgroup $U\subset G({\mathbb A}^S_K)$, the double coset space $G(K)\backslash G({\mathbb A}^S_K)/U$ consists of a single element.
Let $A$, $C$ and $R$ be as in Section \[sec:03\]. Now we let $G$ and $\ul C^\times$ denote the algebraic groups $K$ associated to the multiplicative groups of $A$ and $C$, respectively. For any commutative $K$-algebra $L$, one has $$G(L)=(A\otimes_K L)^\times, \quad
\ul C^\times(L)=(C\otimes_K L)^\times.$$ We denote again by $N:G\to \ul C^\times$ the homomorphism of algebraic $K$-groups induced by the reduced norm map $N:A \to C$, and let $G_1=\ker N$ denote reduced norm-one subgroup. It is easy to see that the base change $G_1\otimes \ol K$ is a finite product of simple groups of the form $\SL_{m}$, and hence $G_1$ is semi-simple and simply connected.
Recall that $A$ is said to satisfy the [*Eichler condition with respect to $S$*]{}, if for any simple factor $A_i$ of $A$ there is one place $w$ of the center $K_i$ over some place $v$ in $S$ such that the completion $A_{i,w}$ at $w$ is not a [*division*]{} $K_{i,w}$-algebra. Another way to rephrase the last condition for $A_i$ is that the kernel of the reduced norm map $$N_{A_i/A_i}: \prod_{v\in S} (A_i\otimes K_v)^\times \to \prod_{v\in
S} (K_i\otimes K_v)^\times$$ is not compact. In other words, the algebra $A$ satisfies the Eichler condition with respect to $S$ (also denote $A$=Eichler$/R$, where $R$ is the ring of $S$-integers of $K$) if and only if the reduced norm-one subgroup $G_1$ satisfies the condition ($*$) in Theorem \[SAT\]. In particular, we have the following special case of Theorem \[SAT\] for $G_1$.
\[eichler\] Let $A$ be a separable $K$-algebra and $G_1$ the associated reduced norm-one subgroup defined as above. If $A$ satisfies the Eichler condition with respect to $S$, then $G_1(K)$ is dense in $G_1({\mathbb A}_K^S)$.
Theorem \[eichler\] is what we use in the proof of Theorem \[2\]. When $K$ is a number field, this is the first case of the strong approximation theorem, proved by Eichler [@eichler:sa]. Swan [@swan:1980] gives a more elementary proof of this theorem.
Suppose that $A$=Eichler$/R$, and let $U$ be an open compact subgroup of $G({\mathbb A}_K^S)=\wh A^\times$. We want to show that the induced surjective map $$\label{eq:5.1}
N:G(K)\backslash G({\mathbb A}^S_K)/U \to N(G(K))\backslash \wh C^\times
/N(U)$$ is injective. Let $\hat c\in \wh C^\times$ be an element and $\hat g\in \wh
A^\times$ with $N(\hat g)=\hat c$. Then the fiber of the class $[\hat
c]$ is $$\label{eq:5.2}
N^{-1}([\hat c])=G(K)\backslash G(K)G_1({\mathbb A}_K^S) \hat g U/U.$$ If $x_1,x_2\in G_1(A^S_K)$ be two elements, then $$\label{eq:5.3}
G(K)x_1\hat g U=G(K) x_2 \hat g U \iff
G_1(K)x_1 (\hat g U \hat g^{-1})=G_1(K)x_2
(\hat g U \hat g^{-1}).$$ Thus, we get a surjective map $$\label{eq:5.4}
G_1(K)\backslash G_1({\mathbb A}^S_K)/G_1({\mathbb A}^S_K)\cap \hat g U \hat g^{-1}
\to G(K)\backslash G(K)G_1({\mathbb A}_K^S) \hat g U/U=N^{-1}([\hat c]).$$ As we know the source of (\[eq:5.4\]) consists of one element (Corollary \[11\]), the fiber $N^{-1}([c])$ also consists of one element. This shows that (\[eq:strong\]) (or (\[eq:5.1\])) is a bijection.
We end with this article by a few remarks. Theorem \[3\] was first proved by Swan [@swan:1962 Theorem 1, p. 56] when $K$ is a number field. Fröhlich [@frohlich:crelle1975 Theorem 2, p. 118] gave another proof of Swan’s theorem using the ideles. The proof given here is the same as that of Fröhlich and of Swan; all uses Theorem \[eichler\]. The statement for Swan’s theorem over global fields (Theorem \[3\]) can be found in Curtis-Reiner [@curtis-reiner:2 Theorem (49.32), p. 233] and Reiner [@reiner:mo (35.14) Theorem p. 313]. Note that in Reiner [@reiner:mo (35.14) Theorem p. 313] there is an assumption of $A$=Eichler$/R$ when $K$ is a global function field, but that is superfluous. Notice that Prasad’s theorem, though for most general cases, came a few years after Reiner wrote his book [*Maximal Orders*]{} (published in 1975). This could be the reason why the result [@reiner:mo (35.14) Theorem p. 313] is limited to those satisfying the Eichler condition in the function field case.
The updated version of Swan’s Theorem (Theorem \[3\]) is then presented in the later books by Curtis and Reiner. They also give a more general variant (Theorem \[2\]); see [@curtis-reiner:2 (49.17) Theorem, p. 225]. The proof of Theorem \[2\] given in Curtis-Reiner [@curtis-reiner:2] is different from the original proof of Swan and Frölich; it is proved based on Algebraic $K$-theory. This of course brings in more insights to the topic. Nevertheless, the original proof may be more accessible for non-specialists as it is much shorter and also conceptual. A very nice exposition for the proof of Theorem \[eichler\] can be found in Section 51 of Curtis-Reiner [@curtis-reiner:2], which follows Swan [@swan:1980]. The paper [@swan:1980] contains some minor errors; see [@swan:crelle1983 Appendix A] for corrections and improvements.
Acknowledgments {#acknowledgments .unnumbered}
===============
The manuscript is prepared while the author’s stay at the Max-Planck-Institut für Mathematik in Bonn. He is grateful to the Institute for kind hospitality and excellent working environment. The author thanks JK Yu and Professor Ming-Chang Kang for pointing out the references [@prasad:sa1977] and [@swan:crelle1983] to him. The author is partially supported by the grants MoST 100-2628-M-001-006-MY4 and 103-2918-I-001-009.
[10]{}
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}
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---
abstract: 'We develop a theory of *modulus sheaves with transfers*, which generalizes Voevodsky’s theory of sheaves with transfers. This repairs part of the flaw in a previous preprint by three of the authors (<https://arxiv.org/abs/1511.07124>).'
address:
- |
IMJ-PRG\
Case 247\
4 place Jussieu\
75252 Paris Cedex 05\
France
- 'RIKEN iTHEMS, Wako, Saitama 351-0198, Japan'
- |
Graduate School of Mathematical Sciences\
University of Tokyo\
3-8-1 Komaba,\
Tokyo 153-8941\
Japan
- |
Institute of Mathematics\
Tohoku University\
Aoba\
Sendai 980-8578\
Japan
author:
- Bruno Kahn
- Hiroyasu Miyazaki
- Shuji Saito
- Takao Yamazaki
date: 'July 21, 2020'
title: |
Motives with modulus, I:\
Modulus sheaves with transfers for non-proper modulus pairs
---
[^1]
Introduction {#introduction .unnumbered}
============
The aim of this paper is to lay a foundation for a theory of *motives with modulus* generalizing Voevodsky’s theory of motives. Voevodsky’s construction is based on ${\mathbf{A}}^1$-invariance. It captures many important invariants such as Bloch’s higher Chow groups, but not their natural generalisations like additive Chow groups [@BE; @Pa] or higher Chow groups with modulus [@BS]. Our basic motivation is to build a theory that captures such non ${\mathbf{A}}^1$-invariant phenomena, as an extension of [@rec].
Let ${\operatorname{\mathbf{Sm}}}$ be the category of smooth separated schemes of finite type over a field $k$. Voevodsky’s construction starts from an additive category ${\operatorname{\mathbf{Cor}}}$, whose objects are those of ${\operatorname{\mathbf{Sm}}}$ and morphisms are finite correspondences. We define ${{\operatorname{\mathbf{PST}}}}$ as the category of additive presheaves of abelian groups on ${\operatorname{\mathbf{Cor}}}$ (i.e. functors ${\operatorname{\mathbf{Cor}}}\to {\operatorname{\mathbf{Ab}}}$ that commute with finite sums). Let ${\operatorname{\mathbf{NST}}}\subset {{\operatorname{\mathbf{PST}}}}$ be the full subcategory of those objects $F\in {{\operatorname{\mathbf{PST}}}}$ whose restrictions $F_X$ to $X_{{\operatorname{Nis}}}$ is a sheaf for any $X \in {\operatorname{\mathbf{Sm}}}$, where $X_{{\operatorname{Nis}}}$ denotes the small Nisnevich site of $X$, that is, the category of all étale schemes over $X$ equipped with the Nisnevich topology. Objects of ${\operatorname{\mathbf{NST}}}$ are called *(Nisnevich) sheaves with transfers*. For $F\in {\operatorname{\mathbf{NST}}}$, we write $$H^i_{{\operatorname{Nis}}}(X,F) = H^i(X_{{\operatorname{Nis}}}, F_X).$$
The following result of Voevodksy [@voetri Th. 3.1.4] plays a fundamental rôle in his theory of motives.
\[thm;Voevodksy\] The following assertions hold.
- The inclusion ${\operatorname{\mathbf{NST}}}\to {{\operatorname{\mathbf{PST}}}}$ has an exact left adjoint $a^V_{{\operatorname{Nis}}}$ such that for any $F\in {{\operatorname{\mathbf{PST}}}}$ and $X\in {\operatorname{\mathbf{Sm}}}$, $(a^V_{{\operatorname{Nis}}}F)_X$ is the Nisnevich sheafication of $F_X$ as a presheaf on $X_{{\operatorname{Nis}}}$. In particular ${\operatorname{\mathbf{NST}}}$ is a Grothendieck abelian category.
- For $X\in {\operatorname{\mathbf{Sm}}}$, let $\Ztr(X)={\operatorname{\mathbf{Cor}}}(-,X) \in {{\operatorname{\mathbf{PST}}}}$ be the associated representable additive presheaf. Then we have $\Ztr(X)\in {\operatorname{\mathbf{NST}}}$ and there is a canonical isomorphism for any $i\ge 0$ and $F\in {\operatorname{\mathbf{NST}}}$: $$H^i_{{\operatorname{Nis}}}(X,F)\simeq {\operatorname{Ext}}^i_{{\operatorname{\mathbf{NST}}}}(\Ztr(X),F) .$$
Our basic principle for generalizing Voevodsky’s theory of sheaves with transfers is that the category ${\operatorname{\mathbf{Cor}}}$ should be replaced by the larger category of *modulus pairs*, ${\operatorname{\mathbf{\underline{M}Cor}}}$: Objects are pairs $M=({\overline}{M}, \Minf)$ consisting of a separated $k$-scheme of finite type ${\overline}{M}$ and an effective (possibly empty) Cartier divisor $\Minf$ on it such that the complement $M^\circ:={\overline}{M}\setminus \Minf$ is smooth over $k$. The group ${\operatorname{\mathbf{\underline{M}Cor}}}(M,N)$ of morphisms is defined as the subgroup of ${\operatorname{\mathbf{Cor}}}(M^\circ,N^\circ)$ consisting of finite correspondences between $M^{{\operatorname{o}}}$ and $N^{{\operatorname{o}}}$ whose closures in ${\overline}{M} \times_k {\overline}{N}$ are proper[^2] over ${\overline}{M}$ and satisfy certain admissibility conditions with respect to $\Minf$ and $\Ninf$ (see Definition \[d2.2\]). Let ${\operatorname{\mathbf{MCor}}}\subset {\operatorname{\mathbf{\underline{M}Cor}}}$ be the full subcategory consisting of objects $({\overline}{M},\Minf)$ with ${\overline}{M}$ proper over $k$.
We then define ${\operatorname{\mathbf{\underline{M}PST}}}$ (resp. ${\operatorname{\mathbf{MPST}}}$) as the category of additive presheaves of abelian groups on ${\operatorname{\mathbf{\underline{M}Cor}}}$ (resp. ${\operatorname{\mathbf{MCor}}}$). We have a functor $$\omega:{\operatorname{\mathbf{\underline{M}Cor}}}\to {\operatorname{\mathbf{Cor}}}, \quad ({{\overline{M}}},\Minf) \mapsto {{\overline{M}}}- |\Minf|,$$ and two pairs of adjunctions $${\operatorname{\mathbf{MPST}}}\begin{smallmatrix} \tau^*\\ \longleftarrow\\ \tau_!\\ \longrightarrow\\
\end{smallmatrix}{\operatorname{\mathbf{\underline{M}PST}}},
\quad
{\operatorname{\mathbf{MPST}}}\begin{smallmatrix} \omega^*\\ \longleftarrow\\ \omega_!\\ \longrightarrow\\
\end{smallmatrix}{{\operatorname{\mathbf{PST}}}},$$ where $\tau^*$ is induced by the inclusion $\tau:{\operatorname{\mathbf{MCor}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}$ and $\tau_!$ is its left Kan extension, and $\omega^*$ is induced by $\omega$ and $\omega_!$ is its left Kan extension (see Propositions \[eq.tau\] and \[lem:counit\]).
The main aim of this paper is to develop a *sheaf theory* on ${\operatorname{\mathbf{\underline{M}Cor}}}$ generalizing Voevodsky’s theory. For $M=({{\overline{M}}},\Minf)\in {\operatorname{\mathbf{\underline{M}Cor}}}$, let $F_M$ be the presheaf on ${{\overline{M}}}_{{\operatorname{Nis}}}$ which associates $F(U,\Minf\times_{{{\overline{M}}}} U)$ to an étale map $U\to {{\overline{M}}}$.
\[defintro;sheavesulMCor\] We define ${\operatorname{\mathbf{\underline{M}NST}}}$ to be the full subcategory of ${\operatorname{\mathbf{\underline{M}PST}}}$ of such objects $F$ that $F_M$ is a Nisnevich sheaf on ${{\overline{M}}}$ for any $M \in {\operatorname{\mathbf{\underline{M}Cor}}}$.
For $F\in {\operatorname{\mathbf{\underline{M}PST}}}$ and $M=({{\overline{M}}},\Minf)$, let $(F_M)_{{\operatorname{Nis}}}$ be the Nisnevich sheafication of the preshseaf $F_M$ on ${{\overline{M}}}_{{\operatorname{Nis}}}$. Let ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ be the subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$ which has the same objects as ${\operatorname{\mathbf{\underline{M}Cor}}}$ and such that a morphism $f \in {\operatorname{\mathbf{\underline{M}Cor}}}(M, N)$ belongs to ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ if and only if $f^{{\operatorname{o}}}\in {\operatorname{\mathbf{Cor}}}(M^{{\operatorname{o}}}, N^{{\operatorname{o}}})$ is the graph of an isomorphism $M^{{\operatorname{o}}}{{\xrightarrow}{\sim}}N^{{\operatorname{o}}}$ in ${\operatorname{\mathbf{Sm}}}$ that extends to a proper morphism ${\overline}{f} : {\overline}{M} \to {\overline}{N}$ of $k$-schemes such that $\Minf={\overline}{f}^*\Ninf$. (See Theorems \[thm:sheafification-ulMNST\], \[c3.1v\] and Lemma \[lcom3-2\].)
\[thmintro;ulMNST\] The following assertions hold.
- The inclusion ${\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}PST}}}$ has an exact left adjoint $\ulaNis$ such that $$(\ulaNis F)(M) ={\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M} (F_N)_{{\operatorname{Nis}}}(N)$$ for every $F\in {\operatorname{\mathbf{\underline{M}PST}}}$ and $M \in {\operatorname{\mathbf{\underline{M}Cor}}}$. In particular ${\operatorname{\mathbf{\underline{M}NST}}}$ is a Grothendieck abelian category. (See §\[sec:pro-obj\] for the comma category ${{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M$.)
- For $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$, let $\Ztr(M)={\operatorname{\mathbf{\underline{M}Cor}}}(-,M) \in {\operatorname{\mathbf{\underline{M}PST}}}$ be the associated representable presheaf. Then we have $\Ztr(M)\in {\operatorname{\mathbf{\underline{M}NST}}}$ and there is a canonical isomorphism for any $i\ge 0$ and $F\in {\operatorname{\mathbf{\underline{M}NST}}}$: $${\operatorname{Ext}}^i_{{\operatorname{\mathbf{\underline{M}NST}}}}({\mathbb{Z}}_{{\operatorname{tr}}}(M),F)\simeq {\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M}
H_{{\operatorname{Nis}}}^i({{\overline{N}}},F_N).$$
\[rem;thmintro;ulMNST\] Theorem \[thmintro;ulMNST\] implies that the formation $$M \mapsto {\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M}
H^i_{{\operatorname{Nis}}}({{\overline{N}}},F_N)$$ is contravariantly functorial for morphisms in ${\operatorname{\mathbf{\underline{M}Cor}}}$, which does not follow immediately from the definition.
The preprint [@motmod] contained a mistake, pointed out by Joseph Ayoub: namely, Proposition 3.5.3 of loc. cit. is false. Theorem \[thmintro;ulMNST\] (1) shows that the only false thing in that proposition is that the functor ${{\underline{b}}}^{{\operatorname{Nis}}}$ of loc. cit. is not exact, but only left exact (see Proposition \[lem;b!ulMNST\] of the present paper.) This weakens [@motmod Prop. 3.6.2] into Theorem \[thmintro;ulMNST\] (2); see however Question \[qn;cdhdescnet\] below. What we gain in the present correction is that the notion of sheaf, which was artificially developed in [@motmod] for ${\operatorname{\mathbf{\underline{M}Cor}}}$, corresponds now to a genuine Grothendieck topology.
Another proposition incorrectly proven in [@motmod] was Proposition 3.7.3. In Part II of this work [@modsheafII], we correct this proof and recover the proposition in full, hence get a good sheaf theory also for *proper* modulus pairs. We expect that this will allow us to develop the categories of motives again as in [@motmod], which is the subject of our ongoing project.
In the last part of this introduction, we raise the following question.
\[qn;cdhdescnet\] Assume that $F\in {\operatorname{\mathbf{\underline{M}NST}}}$ satisfies the following conditions:
- $F$ is ${{{\overline}{\square}}}$-invariant, namely the map $F(M) \to F(M\otimes {{{\overline}{\square}}})$ is an isomorphism for any $M=({{\overline{M}}},\Minf)\in {\operatorname{\mathbf{\underline{M}Cor}}}$, where $$\begin{aligned}
{{{\overline}{\square}}}= ({\mathbf{P}}^1 ,\infty ), \ \ M\otimes {{{\overline}{\square}}}=({\overline}{M} \times {\mathbf{P}}^1, M^\infty \times {\mathbf{P}}^1 + {\overline}{M} \times (\infty)).\end{aligned}$$
- $F$ lies in the essential image of $\tau_!:{\operatorname{\mathbf{MPST}}}\to {\operatorname{\mathbf{\underline{M}PST}}}$.
Then, is the map $$H^q({\overline}{M}_{{\operatorname{Nis}}},F_M) \to {\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M}
H^q({\overline}{N}_{{\operatorname{Nis}}},F_N)$$ an isomorphism for $M\in {\operatorname{\mathbf{\underline{M}Cor}}}_{ls}$? Here ${\operatorname{\mathbf{\underline{M}Cor}}}_{ls}$ denotes the full subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$ consisting of the objects $M=({{\overline{M}}},\Minf)$ such that ${{\overline{M}}}\in {\operatorname{\mathbf{Sm}}}$ and $|\Minf|$ is a simple normal crossing divisor on ${{\overline{M}}}$.
If ${{\operatorname{ch}}}(k)=0$, by resolution of marked ideals ([@BM the case $d=1$ of Th.1.3]), the above question is reduced to the following.
\[qn2;cdhdescnet\] Let the assumptions be as in Question \[qn;cdhdescnet\] and $M=({{\overline{M}}},\Minf)\allowbreak \in {\operatorname{\mathbf{\underline{M}Cor}}}_{ls}$. Let $Z\subset \Minf$ be a regular closed subscheme such that, for any point $x$ of $Z$, there exists a system $z_1, \cdots, z_d$ of regular parameters of ${{\overline{M}}}$ at $x$ (with $d=\dim_x {{\overline{M}}}$) satisfying the following conditions:
- Locally at $x$, $Z =\{z_1= \dots = z_r=0\}$ for $r={{\operatorname{codim}}}_{{\overline}{M}} Z$.
- Locally at $x$, $|\Minf| = \{\prod_{j \in J} z_j=0 \}$ for some $J\subset \{1,\dots,r\}$.
Consider $\pi: {{\overline{N}}}={{\mathbf{Bl}}}_Z({{\overline{M}}})\to {{\overline{M}}}$ and $\Ninf={{\overline{N}}}\times_{{{\overline{M}}}} \Minf$. Then, is the map $$H^q({\overline}{M}_{{\operatorname{Nis}}},F_M) \to H^q({\overline}{N}_{{\operatorname{Nis}}},F_N).$$ an isomorphism?
Acknowledgements {#acknowledgements .unnumbered}
----------------
Part of this work was done while the authors stayed at the university of Regensburg supported by the SFB grant “Higher Invariants". Another part was done in a Research in trio in CIRM, Luminy. Yet another part was done while the fourth author was visiting IMJ-PRG supported by the Foundation Sciences Mathématiques de Paris. We are grateful to the support and hospitality received in all places.
We are very grateful to J. Ayoub for pointing out a flaw on the computation of the sheafification functor ${{\underline{a}}}_{{{\operatorname{Nis}}}}$ and on the non-exactness of the functor ${{\underline{b}}}^{{\operatorname{Nis}}}$ in the earlier version. The authors believe that the whole theory has been deepened by the effort to fix it. We also thank the referee for a careful reading and many useful comments.
Finally, the influence of Voevodsky’s ideas is all-pervasive, as will be evident when reading this paper.
Notation and conventions {#notation-and-conventions .unnumbered}
------------------------
In the whole paper we fix a base field $k$. Let ${\operatorname{\mathbf{Sm}}}$ be the category of separated smooth schemes of finite type over $k$, and let ${\operatorname{\mathbf{Sch}}}$ be the category of separated schemes of finite type over $k$. We write ${\operatorname{\mathbf{Cor}}}$ for Voevodsky’s category of finite correspondences [@voetri].
Modulus pairs and admissible correspondences
============================================
Admissible correspondences
--------------------------
\[d2.2\]
1. A *modulus pair* $M$ consists of ${\overline}{M} \in {\operatorname{\mathbf{Sch}}}$ and an effective Cartier divisor $M^\infty \subset {\overline}{M}$ such that the open subset $M^{{\operatorname{o}}}:={\overline}{M} - |M^\infty|$ is smooth over $k$. (The case $|M^\infty|=\emptyset$ is allowed.) We say that $M$ is *proper* if ${{\overline{M}}}$ is proper over $k$.
We write $M=({\overline}{M}, M^{\infty})$, since $M$ is completely determined by the pair, although we regard $M^{{\operatorname{o}}}$ as the main part of $M$. We call ${\overline}{M}$ the *ambient space* of $M$ and $M^{{\operatorname{o}}}$ the *interior* of $M$.
2. Let $M_1, M_2$ be modulus pairs. Let $Z \in {\operatorname{\mathbf{Cor}}}(M_1^{{\operatorname{o}}}, M_2^{{\operatorname{o}}})$ be an elementary correspondence (i.e. an integral closed subscheme of $M_1^{{\operatorname{o}}}\times M_2^{{\operatorname{o}}}$ which is finite and surjective over an irreducible component of $M_1^{{\operatorname{o}}}$). We write ${\overline}{Z}^N$ for the normalization of the closure ${\overline}{Z}$ of $Z$ in ${\overline}{M}_1 \times {\overline}{M}_2$ and $p_i : {\overline}{Z}^N \to {\overline}{M}_i$ for the canonical morphisms for $i=1, 2$. We say $Z$ is *admissible* for $(M_1, M_2)$ if $p_1^* M_1^\infty \geq p_2^* M_2^\infty$. An element of ${\operatorname{\mathbf{Cor}}}(M_1^{{\operatorname{o}}}, M_2^{{\operatorname{o}}})$ is called admissible if all of its irreducible components are admissible. We write ${\operatorname{\mathbf{Cor}}}_{{\operatorname{adm}}}(M_1, M_2)$ for the subgroup of ${\operatorname{\mathbf{Cor}}}(M_1^{{\operatorname{o}}}, M_2^{{\operatorname{o}}})$ consisting of all admissible correspondences.
\[r2.1\]
1. In [@rec Def. 2.1.1], we used a different notion of modulus pair, where ${{\overline{M}}}$ is supposed proper, $M^{{\operatorname{o}}}$ smooth quasi-affine and $M^\infty$ is any closed subscheme of ${\overline}{M}$. Definition \[d2.2\] (1) is the right one for the present work. Definition \[d2.2\] (2) is the same as [@rec Def. 2.6.1], mutatis mutandis.
2. In the first version of this paper, we imposed the condition that ${\overline}{M}$ is locally integral; it is now removed. The main reason for this change is that this condition is not stable under products or extension of the base field. The next remark shows that this removal is reasonable (see also Remark \[l3.3\]).
3. Let $M$ be a modulus pair. Then $M^{{\operatorname{o}}}$ is dense in ${{\overline{M}}}$, since the Cartier divisor $M^\infty$ is everywhere of codimension $1$. Moreover, ${\overline}{M}$ is reduced. (In particular, ${\overline}{M}$ has no embedded component.) Indeed, take $x \in {\overline}{M}$ and let $f \in {\mathcal{O}}_{{\overline}{M}, x}$ be a local equation for $M^\infty$. Then $f$ is not a zero-divisor (since $M^\infty$ is Cartier), and hence ${\mathcal{O}}_{{\overline}{M}, x} \to {\mathcal{O}}_{{\overline}{M}, x}[1/f]$ is injective, but ${\mathcal{O}}_{{\overline}{M}, x}[1/f]$ is reduced as $M^{{\operatorname{o}}}$ is smooth. In particular, ${{\overline{M}}}$ is integral if $M^{{\operatorname{o}}}$ is.
4. Let $M$ be a modulus pair, and let $f:{\overline}{M}_1\to {\overline}{M}$ be a morphism such that $f(T)\not\subset |M^\infty|$ for any irreducible component $T$ of ${\overline}{M}_1$ and $M_1^{{\operatorname{o}}}:={\overline}{M}_1-|f^*M^\infty|$ is smooth. Then $M_1=({\overline}{M}_1,f^*M^\infty)$ defines a modulus pair. We call it the *minimal modulus structure induced by $f$*. We shall use this construction several times. Also, $f$ defines a minimal morphism $f:M_1\to M$ in the sense of Def. \[def:minimality\] below.
5. If $Z$ is an admissible elementary correspondence as in Definition \[d2.2\] (2), then $|M_1^\infty\times {\overline}{M}_2|\cap {\overline}{Z}
\supseteq |{\overline}{M}_1\times M_2^\infty|\cap {\overline}{Z}$ since ${\overline}{Z}^N\to {\overline}{Z}$ is surjective. On the other hand, the inequality $(M_1^\infty\times {\overline}{M}_2)|_{{\overline}{Z}}
\ge
({\overline}{M}_1\times M_2^\infty)|_{{\overline}{Z}}$ may fail. As an example, let $C$ be the affine cusp curve ${\operatorname{Spec}}k[x,y]/(x^2-y^3)$. Its normalization is ${\mathbf{A}}^1$, via the morphism $t\mapsto (t^3,t^2)$. Let $M_1=(C,(x))$ and $M_2=(C,(y))$. Then $1_C$ defines an admissible correspondence $M_1\to M_2$, even though $(x)\not\ge (y)$ on $C$.
The following lemma will play a key rôle:
\[lem:mod-exists\] Let ${{\overline{X}}}\in {\operatorname{\mathbf{Sch}}}$ and let $X$ be an open dense subscheme of ${\overline}{X}$. Assume that $X \in {\operatorname{\mathbf{Sm}}}$ and that ${{\overline{X}}}-X$ is the support of a Cartier divisor. Then for any modulus pair $N$ we have $$\bigcup_M {\operatorname{\mathbf{Cor}}}_{{\operatorname{adm}}}(M, N) = {\operatorname{\mathbf{Cor}}}(X, N^{{\operatorname{o}}}),$$ where $M$ ranges over all modulus pairs such that ${{\overline{M}}}={{\overline{X}}}$ and $M^{{\operatorname{o}}}= X$. (Note that by definition we have ${\operatorname{\mathbf{Cor}}}_{{\operatorname{adm}}}(M, N) \subset {\operatorname{\mathbf{Cor}}}(X, N^{{\operatorname{o}}})$.)
This is proven in [[@rec Lemma 2.6.2]]{}. In loc. cit. $X$ and $N^{{\operatorname{o}}}$ are assumed to be quasi-affine, and ${\overline}{X}$ and ${\overline}{N}$ proper and normal (see Remark \[r2.1\]). But these assumptions are not used in the proof. (Nor is the assumption on Cartier divisors, but the latter is essential for the proof of Proposition \[lem:comp-admcorr\] below.)
Composition
-----------
To discuss composability of admissible correspondences, we need the following lemma of Krishna and Park [@KP Lemma 2.2].
\[lKL\] Let $f:X\to Y$ be a surjective morphism of normal integral schemes, and let $D,D'$ be two Cartier divisors on $Y$. If $f^*D'\le f^*D$, then $D'\le D$.
\[dcomp\] Let $M_1, M_2, M_3$ be three modulus pairs, and let $\alpha \in {\operatorname{\mathbf{Cor}}}_{{\operatorname{adm}}}(M_1, M_2), \beta \in {\operatorname{\mathbf{Cor}}}_{{\operatorname{adm}}}(M_2, M_3)$. We say that $\alpha$ and $\beta$ are *composable* if their composition $\beta \alpha$ in ${\operatorname{\mathbf{Cor}}}(M_1^{{\operatorname{o}}}, M_3^{{\operatorname{o}}})$ is admissible.
\[lem:comp-admcorr\] With the above notations, assume $\alpha$ and $\beta$ are integral and let $\bar\alpha$ and $\bar\beta$ be their closures in ${{\overline{M}}}_1\times {{\overline{M}}}_2$ and ${{\overline{M}}}_2\times {{\overline{M}}}_3$ respectively. Then $\alpha$ and $\beta$ are composable provided the projection $\bar\alpha\times_{{{\overline{M}}}_2}\bar\beta\to {{\overline{M}}}_1\times {{\overline{M}}}_3$ is proper. This happens in the following cases:
$\bar\alpha\to {{\overline{M}}}_1$ is proper.
$\bar\beta\to {{\overline{M}}}_3$ is proper.
${{\overline{M}}}_2$ is proper over $k$.
Note that $\alpha \times_{M_2^{{\operatorname{o}}}} \beta$ is a closed subscheme of $(M_1^{{\operatorname{o}}}\times M_2^{{\operatorname{o}}}) \times_{M_2^{{\operatorname{o}}}} (M_2^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}})
=M_1^{{\operatorname{o}}}\times M_2^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}}$; we have $|\beta \alpha| = |p_{13*}( \alpha \times_{M_2^{{\operatorname{o}}}} \beta)|$ where $p_{13} : M_1^{{\operatorname{o}}}\times M_2^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}}\to M_1^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}}$ is the projection. Let $\gamma$ be a component of $\alpha \times_{M_2^{{\operatorname{o}}}} \beta$. We have a commutative diagram $$\xymatrix{
\gamma \ar@{^{(}->}[r] \ar@{^{(}->}[d]& \alpha \times_{M_2^{{\operatorname{o}}}} \beta \ar@{^{(}->}[r] \ar@{^{(}->}[d]& M_1^{{\operatorname{o}}}\times M_2^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}}\ar[r]^(0.6){p_{13}} \ar@{^{(}->}[d] &M_1^{{\operatorname{o}}}\times M_3^{{\operatorname{o}}}\ar@{^{(}->}[d]& \delta \ar@{_{(}->}[l] \ar@{^{(}->}[d]\\
\bar\gamma
\ar@{^{(}->}[r] & \bar\alpha \times_{{{\overline{M}}}_2} \bar\beta \ar@{^{(}->}[r]& {{\overline{M}}}_1\times {{\overline{M}}}_2 \times {{\overline{M}}}_3 \ar@{->}[r]^(0.6){{\overline}{p}_{13}} &{{\overline{M}}}_1\times {{\overline{M}}}_3& \bar\delta \ar@{_{(}->}[l]
}$$ where ${\overline}{p}_{ij} : {{\overline{M}}}_1\times {{\overline{M}}}_2 \times {{\overline{M}}}_3 \to {{\overline{M}}}_i\times {{\overline{M}}}_j$ denotes the projection, $\delta=p_{13}(\gamma)$, and $\bar{}$ denotes closure. The hypothesis implies that $\bar \gamma\to \bar\delta$ is proper surjective. The same holds for $\pi_{\gamma \delta}^N$ appearing in the second of the two other commutative diagrams: $$\xymatrix{
\bar\alpha& \bar\alpha^N \ar[r]^{{\varphi}_{\alpha}}\ar[l]
& {{\overline{M}}}_1 \times {{\overline{M}}}_2
\\
\bar\gamma\ar[u]_{\pi_{\gamma \alpha}}\ar[d]^{\pi_{\gamma\beta}} & \bar\gamma^N
\ar[r]^(0.3){{\varphi}_\gamma}\ar[l] &{{\overline{M}}}_1\times {{\overline{M}}}_2\times {{\overline{M}}}_3\ar[d]^{\bar p_{23}}
\ar[u]_{\bar p_{12}}
\\
\bar \beta&\bar\beta^N \ar[r]^{{\varphi}_{\beta}}\ar[l] & {{\overline{M}}}_2 \times {{\overline{M}}}_3
}
\quad
\xymatrix{
\bar\gamma^N
\ar[r]^(0.3){{\varphi}_\gamma}\ar[d]^{\pi_{\gamma\delta}^N} &{{\overline{M}}}_1\times {{\overline{M}}}_2\times {{\overline{M}}}_3\ar[d]^{\bar p_{13}}
\\
\bar\delta^N \ar[r]^{{\varphi}_\delta} &{{\overline{M}}}_1\times {{\overline{M}}}_3
}$$ where $^N$ means normalisation. (Note that $\pi_{\gamma\alpha}$ and $\pi_{\gamma\beta}$ need not extend to the normalisations, as they need not be dominant.) We have the admissibility conditions for $\alpha$ and $\beta$: $$\begin{aligned}
{\varphi}_\alpha^*({{\overline{M}}}_1\times M_2^\infty)&\le {\varphi}_\alpha^*(M_1^\infty\times {{\overline{M}}}_2)\label{eq1a}\\
{\varphi}_\beta^*({{\overline{M}}}_2\times M_3^\infty)&\le {\varphi}_\beta^*(M_2^\infty\times {{\overline{M}}}_3).\label{eq1b}\end{aligned}$$ Applying [@Mi Lemma 2.4][^3], we get an inequality $${\varphi}_\gamma^*({{\overline{M}}}_1\times {{\overline{M}}}_2\times M_3^\infty)\le {\varphi}_\gamma^*({{\overline{M}}}_1\times M_2^\infty\times {{\overline{M}}}_3)\le {\varphi}_\gamma^*(M_1^\infty\times {{\overline{M}}}_2\times {{\overline{M}}}_3),$$ which implies by the right half of the above diagram $$\label{eq1c}
(\pi_{\gamma\delta}^N)^* {\varphi}_\delta^*({{\overline{M}}}_1\times M_3^\infty)\le (\pi_{\gamma\delta}^N)^*{\varphi}_\delta^*(M_1^\infty\times {{\overline{M}}}_3)$$ hence ${\varphi}_\delta^*({{\overline{M}}}_1\times M_3^\infty)\le {\varphi}_\delta^*(M_1^\infty\times {{\overline{M}}}_3) $ by Lemma \[lKL\].
Finally, one trivially checks that (i) or (ii) implies that the projection ${\overline}{\alpha} \times_{{\overline}{M}_2} {\overline}{\beta}
\to {\overline}{M}_1 \times {\overline}{M}_3$ is proper, and that (iii) implies both of (i) and (ii).
\[ex1.1\] Let ${{\overline{M}}}_1={{\overline{M}}}_3={\mathbf{P}}^1$, ${{\overline{M}}}_2={\mathbf{A}}^1$, ${{\overline{M}}}_i^{{\operatorname{o}}}={\mathbf{A}}^1$, $M_1^\infty =\infty$, $M_2^\infty=\emptyset$, $M_3^\infty=2 \cdot \infty$, $\alpha=\beta=$ graph of the identity on ${\mathbf{A}}^1$. Then $\alpha$ and $\beta$ are admissible but $\beta\circ \alpha$ is not admissible because $\infty \not\ge 2 \cdot \infty$. (Note that neither of ${\overline}{\alpha}=\alpha$ or ${\overline}{\beta}=\beta$ is proper over ${\mathbf{P}}^1$.)
\[d2.4\] Let $M,N$ be two modulus pairs. A correspondence $\alpha\in {\operatorname{\mathbf{Cor}}}(M^{{\operatorname{o}}}, N^{{\operatorname{o}}})$ is *left proper* (relatively to $M,N$) if the closures of all components of $\alpha$ are proper over ${{\overline{M}}}$; this is automatic if ${{\overline{N}}}$ is proper.
\[p1a\] Let $M_1,M_2,M_3$ be three modulus pairs and let $\alpha\in {\operatorname{\mathbf{Cor}}}(M_1^{{\operatorname{o}}},M_2^{{\operatorname{o}}})$, $ \beta\in {\operatorname{\mathbf{Cor}}}(M_2^{{\operatorname{o}}},M_3^{{\operatorname{o}}})$ be left proper. Then $\beta\alpha$ is left proper.
We may assume $\alpha$ and $\beta$ are irreducible. The assumption on $\beta$ means ${\overline}{\beta} \to {\overline}{M}_2$ is proper, hence so is its base change ${\overline}{\alpha} \times_{{{\overline{M}}}_2} {\overline}{\beta} \to {\overline}{\alpha}$. The assumption on $\alpha$ means ${\overline}{\alpha} \to {\overline}{M}_1$ is proper, hence so is ${\overline}{\alpha} \times_{{{\overline{M}}}_2} {\overline}{\beta} \to {{\overline{M}}}_1$ as a composition of proper morphisms. This implies the left properness of $\beta\alpha$, since ${\overline}{\beta\alpha}$ is the image of ${\overline}{\alpha} \times_{{{\overline{M}}}_2} {\overline}{\beta}$ in ${{\overline{M}}}_1\times {{\overline{M}}}_3$.
Categories of modulus pairs
---------------------------
\[d2.4a\] By Propositions \[lem:comp-admcorr\] and \[p1a\], modulus pairs and left proper admissible correspondences define an additive category that we denote by ${\operatorname{\mathbf{\underline{M}Cor}}}$. We write ${\operatorname{\mathbf{MCor}}}$ for the full subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$ whose objects are proper modulus pairs (see Definition \[d2.2\] (1)).
In the context of modulus pairs, the category ${\operatorname{\mathbf{Sm}}}$ and the graph functor ${\operatorname{\mathbf{Sm}}}\to {\operatorname{\mathbf{Cor}}}$ are replaced by the following:
\[d2.5\] We write ${\operatorname{\mathbf{\underline{M}Sm}}}$ for the category with the same objects as ${\operatorname{\mathbf{\underline{M}Cor}}}$, and a morphism of ${\operatorname{\mathbf{\underline{M}Sm}}}(M_1,M_2)$ is given by a (scheme-theoretic) $k$-morphism $f:M_1^{{\operatorname{o}}}\to M_2^{{\operatorname{o}}}$ whose graph belongs to ${\operatorname{\mathbf{\underline{M}Cor}}}(M_1,M_2)$. We write ${\operatorname{\mathbf{MSm}}}$ for the full subcategory of ${\operatorname{\mathbf{\underline{M}Sm}}}$ whose objects are proper modulus pairs.
We will need some variants of these categories.
\[d1.1\]
1. We write ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ for the subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$ with the same objects and the following condition on morphisms: $\alpha\in {\operatorname{\mathbf{\underline{M}Cor}}}(M,N)$ belongs to ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(M,N)$ if and only if, for any component $Z$ of $\alpha$, the projection ${{\overline{Z}}}\to {{\overline{M}}}$ is *finite*, where ${\overline}{Z}$ is the closure of $Z$ in ${\overline}{M} \times {\overline}{N}$. The same argument as in the proof of Proposition \[p1a\] shows that ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ is indeed a subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$. We write ${\operatorname{\mathbf{MCor}}}^{{\operatorname{fin}}}$ for the full subcategory of ${\operatorname{\mathbf{\underline{M}Cor}}}$ whose objects are proper modulus pairs.
2. We write ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ for the subcategory of ${\operatorname{\mathbf{\underline{M}Sm}}}$ with the same objects and such that a morphism $f:M\to N$ belongs to ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ if and only if $f^{{\operatorname{o}}}: M^{{\operatorname{o}}}\to N^{{\operatorname{o}}}$ extends to a $k$-morphism ${\overline}{f} : {\overline}{M}\to {\overline}{N}$. Such extension ${\overline}{f}$ is unique because $M^{{\operatorname{o}}}$ is dense in a reduced scheme ${\overline}{M}$ and ${\overline}{N}$ is separated. This yields a forgetful functor ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{Sch}}}$, which sends $M$ to ${\overline}{M}$.
We write ${\operatorname{\mathbf{MSm}}}^{{\operatorname{fin}}}$ for the full subcategory of ${\operatorname{\mathbf{\underline{M}Sm}}}$ whose objects are proper modulus pairs.
3. We write $$\begin{aligned}
\notag
&c : {\operatorname{\mathbf{MSm}}}\to {\operatorname{\mathbf{MCor}}},\quad
\\
\label{eq:def-c}
&{{\underline{c}}} : {\operatorname{\mathbf{\underline{M}Sm}}}\to {\operatorname{\mathbf{\underline{M}Cor}}},\quad
\\
\notag
&{{\underline{c}}}^{{\operatorname{fin}}}: {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\end{aligned}$$ for the functors which are the identity on objects and which carry a morphism $f$ to the graph of $f^{{\operatorname{o}}}$.
Let $f : M \to N$ be a morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Since ${\overline}{f}(M^{{\operatorname{o}}})\subseteq N^{{\operatorname{o}}}$, none of the images of the generic points of the irreducible components of ${\overline}{M}$ is contained in $|N^\infty|$, hence the pullback of Cartier divisor ${\overline}{f}^\ast N^\infty$ is well-defined. For ease of notation, we simply write it $f^*N^\infty$.
\[def:minimality\] A morphism $f : M \to N$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ is *minimal* if we have $f^\ast N^\infty = M^\infty$.
\[rk-graph-trick\] We remark the following.
1. \[gt1\] Assume that ${{\overline{M}}}$ is normal. Then Zariski’s connectedness theorem implies that for any $N$ $${\operatorname{\mathbf{\underline{M}Sm}}}(M,N)\cap {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(M,N)={\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}(M,N).$$ (Indeed, given an elementary correspondence belonging to the left hand side, its closure in ${\overline}{M} \times {\overline}{N}$ is birational and finite over an irreducible component of ${\overline}{M}$, but such a morphism is an isomorphism if ${\overline}{M}$ is normal by [@EGA3 Corollaire 4.4.9]). If $f^{{\operatorname{o}}}: M^{{\operatorname{o}}}\to N^{{\operatorname{o}}}$ extends to a morphism between ambient spaces ${\overline}{f}:{\overline}{M} \to {\overline}{N}$, then the graph of $f^{{\operatorname{o}}}$ is admissible if and only if we have $M^\infty \geq {\overline}{f}^\ast N^\infty$.
2. \[gt2\] For $M \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, set $M^N:=({\overline}{M}^N, M^\infty |_{{\overline}{M}^N})$ where $p:{\overline}{M}^N \to {\overline}{M}$ is the normalization and $M^\infty |_{{\overline}{M}^N}$ is the pull-back of $M^\infty$ to ${\overline}{M}^N$. Then $p : M^N \to M$ is an isomorphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ and ${\operatorname{\mathbf{\underline{M}Sm}}}$ (but not in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ in general).
3. \[gt4\] Let $M=({\overline}{M},M^\infty)$ and $N=({\overline}{N},N^\infty)$ be two modulus pairs and let ${\overline}{Z}\subset {\overline}{M}\times {\overline}{N}$ be an integral closed subscheme which is finite and surjective over an irreducible component of ${\overline}{M}$, such that ${\overline}{Z}\not \subset {\overline}{M}\times N^\infty$ and that $M^\infty|_{{\overline}{Z}^N}\ge N^\infty|_{{\overline}{Z}^N}$, where ${\overline}{Z}^N$ is the normalization of ${\overline}{Z}$. Then $Z={\overline}{Z}\cap (M^{{\operatorname{o}}}\times {\overline}{N})$ belongs to ${\operatorname{\mathbf{Cor}}}(M^{{\operatorname{o}}},N^{{\operatorname{o}}})$ and its closure in ${\overline}{M}\times {\overline}{N}$ is ${\overline}{Z}$: this follows from Remark \[r2.1\] (4).
4. \[gt3\] For any morphism $f : M \to N$ in ${\operatorname{\mathbf{\underline{M}Sm}}}$, there exists a morphism $M' \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ which is invertible in ${\operatorname{\mathbf{\underline{M}Sm}}}$ such that the induced morphism $M' \to N$ is in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. More generally, we have the following lemma.
\[l-gt\] Let $f : M \to N$ be a morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}$. Then there exists a minimal morphism $p : M_1 \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ such that it is invertible in ${\operatorname{\mathbf{\underline{M}Sm}}}$ and the composite $f \circ p : M_1 \to M \to N$ is a morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Moreover, if $f^{{\operatorname{o}}}: M^{{\operatorname{o}}}\to N^{{\operatorname{o}}}$ extends to a morphism ${\overline}{U} \to {\overline}{N}$ for an open subset ${\overline}{U} \subset {\overline}{M}$, then we can choose such $M_1$ that we have ${\overline}{M}_1 \to {\overline}{M}$ is an isomorphism over ${\overline}{U}$ (note that we can always take ${\overline}{U}=M^{{\operatorname{o}}}$).
Let $\Gamma$ be the graph of the morphism ${\overline}{U} \to {\overline}{N}$, and let ${\overline}{\Gamma}$ be its closure in ${\overline}{M} \times {\overline}{N}$. Then we have natural projections $p_1 : {\overline}{\Gamma} \to {\overline}{M}$ and $p_2 : {\overline}{\Gamma} \to {\overline}{N}$. Since we have $\Gamma \cong {\overline}{U}$, Lemma \[nexfib\] below implies that $p_1$ is an isomorphism over ${\overline}{U}$ and we have $p_1^{-1}({\overline}{U}) = \Gamma$. Defining $M_1 := ({\overline}{\Gamma}, p_1^\ast M^\infty)$, the morphism $p_1$ induces a morphism $p_1 : M_1 \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ such that $f \circ p_1 : M_1 \to M \to N$ comes from ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ defined by $p_2$. Also note that ${\overline}{\Gamma} \to {\overline}{M}$ is proper since $f$ is, which implies that $p_1 : M_1 \to M$ is an isomorphism in ${\operatorname{\mathbf{\underline{M}Sm}}}$. This finishes the proof.
\[nexfib\] Let $f : X \to Y$ be a separated morphism of schemes, and let $U \subset X$ be an open dense subset. Assume that the image $f(U)$ of $U$ is open in $Y$, and the induced morphism $U \to f(U)$ is proper (e.g., an isomorphism). Then, we have $f^{-1}(f(U)) = U$.
Consider the commutative diagram $$\begin{xymatrix}{
U \ar[rd]_{\mathrm{proper}} \ar[r]^(0.38)j & f^{-1}(f(U)) \ar[r] \ar@{}[rd]|\square \ar[d]^{\mathrm{sep.}} & X \ar[d]_f^{\mathrm{sep.}} \\
& f(U) \ar[r] & Y}\end{xymatrix}$$ where all the horizontal arrows are open immersions, the square is cartesian and the two vertical morphisms are separated. The triangle diagram on the left implies that $j$ is proper, hence it is a closed (and open) immersion. Since $U$ is dense in $X$, it is dense in $f^{-1}(f(U))$ as well, hence the conclusion.
\[l3.3\] Let $M\in {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$. Assume that $M^{{\operatorname{o}}}=M_1^{{\operatorname{o}}}\coprod M_2^{{\operatorname{o}}}$; let ${\overline}{M}_i$ be the closure of $M_i^{{\operatorname{o}}}$ in ${\overline}{M}$ and $M_i^\infty$ be the pull-back of $M^\infty$ to ${\overline}{M}_i$. Then $M_i=({\overline}{M}_i,M_i^\infty)$ are modulus pairs, the inclusions $M_i^{{\operatorname{o}}}{\hookrightarrow}M^{{\operatorname{o}}}$ yield morphisms $M_i\to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, and the induced morphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ $$M_1\oplus M_2\to M$$ is an isomorphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$. The proof is easy and left to the reader.
This remark may help in reducing some reasonings to the case where $M^{{\operatorname{o}}}$ is irreducible (but will not be used in this paper).
The functors $(-)^{(n)}$
------------------------
\[dn1\] Let $n\ge 1$ and $M=({\overline}M,M^\infty)\in {\operatorname{\mathbf{\underline{M}Cor}}}$. We write $$M^{(n)} = ({\overline}M,nM^\infty).$$ This defines an endofunctor of ${\operatorname{\mathbf{\underline{M}Cor}}}$. Those come with natural transformations $$\label{eqn1}
M^{(n)}\to M^{(m)}\quad \text{if } m\le n.$$
\[ln1\] The functor $(-)^{(n)}$ is fully faithful.
This follows from the definition and the fact that if $A$ is an integral domain with quotient field $K$, then $a \in K$ is integral over $A$ if and only if so is $a^n$.
Changes of categories {#s1.2}
---------------------
We now have a basic diagram of additive categories and functors $$\label{eq.taulambda}
\xymatrix{
{\operatorname{\mathbf{MCor}}}\ar[rr]^\tau\ar[rd]^\omega && {\operatorname{\mathbf{\underline{M}Cor}}}\ar@<4pt>[ld]^{\underline{\omega}}\\
&{\operatorname{\mathbf{Cor}}}\ar[ur]^\lambda
}$$ with $$\tau(M)=M;\quad \omega(M)=M^{{\operatorname{o}}};\quad {\underline{\omega}}(M)=M^{{\operatorname{o}}}; \quad \lambda(X)=(X,\emptyset).$$
All these functors are faithful, and $\tau$ is fully faithful; they “restrict” to analogous functors $\tau_s,\omega_s,{\underline{\omega}}_s,\lambda_s$ between ${\operatorname{\mathbf{MSm}}}$, ${\operatorname{\mathbf{\underline{M}Sm}}}$ and ${\operatorname{\mathbf{Sm}}}$. Note that ${\underline{\omega}}\circ (-)^{(n)}={\underline{\omega}}$ for any $n$. Moreover:
\[l1.2\] We have ${\underline{\omega}}\tau=\omega$. Moreover, $\lambda$ is left adjoint to ${\underline{\omega}}$. Finally, the restriction of $\lambda$ to ${\operatorname{\mathbf{Cor}}}^{{\operatorname{prop}}}$ (finite correspondences on smooth proper schemes over $k$) is “right adjoint” to ${\underline{\omega}}$. (That is, ${\operatorname{\mathbf{Cor}}}({\underline{\omega}}(M), X)={\operatorname{\mathbf{\underline{M}Cor}}}(M, \lambda(X))$ for $M \in {\operatorname{\mathbf{\underline{M}Cor}}}$ and $X \in {\operatorname{\mathbf{Cor}}}^{{\operatorname{prop}}}$). The same statements are valid for $\tau_s,\omega_s,{\underline{\omega}}_s,\lambda_s$ when restricted to ${\operatorname{\mathbf{MSm}}}$, ${\operatorname{\mathbf{\underline{M}Sm}}}$ and ${\operatorname{\mathbf{Sm}}}$.
The first identity is obvious. For the adjointness, let $X\in {\operatorname{\mathbf{Cor}}}$, $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$ and $\alpha\in {\operatorname{\mathbf{Cor}}}(X,M^{{\operatorname{o}}})$ be an integral finite correspondence. Then $\alpha$ is closed in $X\times {{\overline{M}}}$, since it is finite over $X$ and ${{\overline{M}}}$ is separated; it is evidently finite (hence proper) over $X$ and $q^*M^\infty=0$ where $q$ is the composition $\alpha^N\to \alpha\to M^{{\operatorname{o}}}\to {{\overline{M}}}$. Therefore $\alpha\in {\operatorname{\mathbf{\underline{M}Cor}}}(\lambda(X),M)$.
For the second statement, assume $X$ proper and let $\beta\in {\operatorname{\mathbf{Cor}}}(M^{{\operatorname{o}}},X)$ be an integral finite correspondence. Then $\beta$ is trivially admissible, and its closure in ${{\overline{M}}}\times X$ is proper over ${{\overline{M}}}$, so $\beta\in {\operatorname{\mathbf{\underline{M}Cor}}}(M,\lambda(X))$. The last claim is immediate.
The following theorem is an important refinement of Lemma \[l1.2\]. The proof starts from §\[starts-pf-1.5.2\] and is completed in §\[end-pf-1.5.2\].
\[t2.1\] The functors $\omega$, $\tau$, $\omega_s$ and $\tau_s$ have pro-left adjoints $\omega^!$, $\tau^!$, $\omega_s^!$ and $\tau_s^!$ (see §\[s1.1\]).
General definitions and results on pro-objects and pro-adjoints are gathered in the Appendix. We shall freely use results from there.
The closure of a finite correspondence
--------------------------------------
We shall need the following result for the proof of Theorem \[t2.1\].
\[lrg\] Let $X$ be a Noetherian scheme, $(\pi_i:Z_i\to X)_{1\le i\le n}$ a finite set of proper surjective morphisms with $Z_i$ integral, and let $U\subseteq X$ be a normal open subset. Suppose that $\pi_i:\pi_i^{-1}(U)\to U$ is finite for every $i$. Then there exists a proper birational morphism $X'\to X$ which is an isomorphism over $U$, such that the closure of $\pi_i^{-1}(U)$ in $Z_i\times_X X'$ is finite over $X'$ for every $i$.
By induction, we reduce to $n=1$; then this follows from [@rg Cor. 5.7.10] applied with $(S,X,U)\equiv (X,Z_1,U)$ and $n=0$ (note that quasi-finite + proper $\iff$ finite, and that an admissible blow-up of an algebraic space is a scheme if the algebraic space happens to be a scheme).
\[trg\] Let $X,Y\in {\operatorname{\mathbf{Sch}}}$. Let $U$ be a normal dense open subscheme of $X$, and let $\alpha$ be a finite correspondence from $U$ to $Y$. Suppose that the closure ${{\overline{Z}}}$ of $Z$ in $X\times Y$ is proper over $X$ for any component $Z$ of $\alpha$. Then there is a proper birational morphism $X'\to X$ which is an isomorphism over $U$, such that $\alpha$ extends to a finite correspondence from $X'$ to $Y$.
Apply Lemma \[lrg\], noting that $Z= {{\overline{Z}}}\times_X U$ by [@rec Lemma 2.6.3].
The following lemma also relies on [@rg]: it will be used several times in the sequel.
\[mainlem;blowup\] Let $f: U\to X$ be an étale morphism of quasi-compact and quasi-separated integral schemes. Let $g: V\to U$ be a proper birational morphism, $T \subset U$ a closed subset such that $g$ is an isomorphism over $U-T$ and $S$ the closure of $f(T)$ in $X$. Then there exists a closed subscheme $Z\subset X$ supported in $S$ such that $U\times_X {{\mathbf{Bl}}}_Z(X) \to U$ factors through $V$.
The following argument is taken from the proof of [@sv Pr. 5.9]. Noting $V$ is étale over $X-S$, we apply the platification theorem [@rg Cor. 5.7.11] to $V \to X$ and conclude that there exists a closed subscheme $Z$ supported in $S$ such that the proper transform $V'$ of $V$ under $X'={{\mathbf{Bl}}}_Z(X)\to X$ is flat over $X'$. By the construction the induced morphism ${\varphi}: V'\to U\times_X X'$ is proper birational. On the other hand ${\varphi}$ is flat since it becomes flat being composed with the étale morphism $ U\times_X X'\to X'$. Hence it is an isomorphism. This proves the lemma since $V' \to U$ factors $V\to U$.
Proof of Theorem \[t2.1\]: case of $\omega$ and $\omega_s$ {#starts-pf-1.5.2}
----------------------------------------------------------
We need a definition:
\[d.sigma\] Let $\Sigma$ be the class of all morphisms $M_1\to M_2$ in ${\operatorname{\mathbf{MCor}}}$ given by the graph of an isomorphism $M_1^{{\operatorname{o}}}{{\xrightarrow}{\sim}}M_2^{{\operatorname{o}}}$ in ${\operatorname{\mathbf{Sm}}}$.
In view of Proposition \[p2.6\], the existence of the pro-left adjoint of $\omega$ is a consequence of the following more precise result:
\[prop:localization\] a) The class $\Sigma$ enjoys a calculus of right fractions.\
b) The functor $\omega$ induces equivalences of categories $$\Sigma^{-1}{\operatorname{\mathbf{MCor}}}{{\xrightarrow}{\sim}}{\operatorname{\mathbf{Cor}}}.$$ The same statement holds for $\omega_s:{\operatorname{\mathbf{MSm}}}\to {\operatorname{\mathbf{Sm}}}$.
a\) We check the axioms of Definition \[d.cf\]:
1. Identities, stability under composition: obvious.
2. Given a diagram in ${\operatorname{\mathbf{MCor}}}$ $$\begin{CD}
&& M'_2\\
&&@VVV\\
M_1@>\alpha>> M_2
\end{CD}$$ with $M_2^{{\operatorname{o}}}\cong{M_2'}^{{\operatorname{o}}}$, Lemma \[lem:mod-exists\] provides a $M_1''\in {\operatorname{\mathbf{MCor}}}$ such that ${M_1''}^{{\operatorname{o}}}=M_1^{{\operatorname{o}}}$ and $\alpha\in {\operatorname{\mathbf{MCor}}}(M''_1,M'_2)$. We may choose $M''_1$ such that ${\overline}{M''_1} = {\overline}{M_1}$. Then $M'_1=({\overline}{M_1}, {M_1'}^\infty)$ with any ${M_1'}^\infty$ such that ${M_1'}^\infty \geq M_1^\infty$, ${M_1'}^\infty \geq {M''_1}^\infty$ allows us to the square in ${\operatorname{\mathbf{MCor}}}$.
3. Given a diagram $$M_1\begin{smallmatrix}f\\ \rightrightarrows\\ g
\end{smallmatrix} M_2{\xrightarrow}{s} M'_2$$ with $M_1,M_2,M'_2$ as in (2) and such that $sf=sg$, the underlying correspondences to $f$ and $g$ are equal since the one underlying $s$ is $1_{M_2^{{\operatorname{o}}}}$. Hence $f=g$.
The above proof of (2) also shows that we have $${\operatornamewithlimits{\varinjlim}}_{M' \in \Sigma \downarrow M} {\operatorname{\mathbf{MCor}}}(M', N)
={\operatorname{\mathbf{Cor}}}(M, N).$$ for any $M, N \in {\operatorname{\mathbf{MCor}}}$. b) now follows from a) and Corollary \[c.cf\], noting that $\omega$ is essentially surjective. The case of $\omega_s$ is exactly parallel.
Let $\omega^!:{\operatorname{\mathbf{Cor}}}\to {\text{\rm pro}_{}\text{\rm--}}{\operatorname{\mathbf{MCor}}}$ be the pro-left adjoint of $\omega$. By Proposition \[p2.6\], we have for $X\in {\operatorname{\mathbf{Cor}}}$: $$\omega^! X = ``{\operatornamewithlimits{\varprojlim}}_{M\in \Sigma\downarrow X}" M.$$ and the same formula for the pro-left adjoint $\omega_s^!$ of $\omega_s$. Let us spell out the indexing set ${\operatorname{\mathbf{MSm}}}(X)$ of these pro-objects, and refine them:
\[def:mpx\]
1. For $X \in {\operatorname{\mathbf{Sm}}}$, we define a subcategory ${\operatorname{\mathbf{MSm}}}(X)$ of ${\operatorname{\mathbf{MSm}}}$ as follows. The objects are those $M \in {\operatorname{\mathbf{MSm}}}$ such that $M^{{\operatorname{o}}}=X$. Given $M_1, M_2 \in {\operatorname{\mathbf{MSm}}}(X)$, we define ${\operatorname{\mathbf{MSm}}}(X)(M_1, M_2)$ to be $\{ 1_X \}$ if $1_X$ belongs to ${\operatorname{\mathbf{MSm}}}$ and $\emptyset$ otherwise.
2. Let $X \in {\operatorname{\mathbf{Sm}}}$ and fix a compactification ${{\overline{X}}}$ such that ${{\overline{X}}}-X$ is the support of a Cartier divisor (for short, a *Cartier compactification*). Define ${\operatorname{\mathbf{MSm}}}({\overline}{X}!X)$ to be the full subcategory of ${\operatorname{\mathbf{MSm}}}(X)$ consisting of objects $M \in {\operatorname{\mathbf{MSm}}}(X)$ such that ${\overline}{M}={\overline}{X}$.
\[ln2\] a) For any $X\in {\operatorname{\mathbf{Sm}}}$ and any Cartier compactification ${{\overline{X}}}$, ${\operatorname{\mathbf{MSm}}}(X)$ is a cofiltered ordered set, and ${\operatorname{\mathbf{MSm}}}({\overline}{X}!X)$ is cofinal in ${\operatorname{\mathbf{MSm}}}(X)$.\
b) Let $X\in {\operatorname{\mathbf{Cor}}}$, and let $M\in {\operatorname{\mathbf{MSm}}}(X)$. Then $(M^{(n)})_{n\ge 1}$ defines a cofinal subcategory of ${\operatorname{\mathbf{MSm}}}(X)$.
a\) “Ordered” is obvious and “cofiltered” follows from Propositions \[prop:localization\] and \[p.cf\] a); the cofinality follows again from Lemma \[lem:mod-exists\].
b\) Let $M=({{\overline{X}}},X^\infty)$. By a) it suffices to show that $(M^{(n)})_{n\ge 1}$ defines a cofinal subcategory of ${\operatorname{\mathbf{MSm}}}({\overline}{X}!X)$. If $({{\overline{X}}},Y)\in {\operatorname{\mathbf{MSm}}}({\overline}{X}!X)$, $Y$ and $X^\infty$ both have support ${{\overline{X}}}-X$, so there exists $n>0$ such that $nX^\infty \ge Y$.
Proof of Theorem \[t2.1\]: case of $\tau$ {#end-pf-1.5.2}
-----------------------------------------
We need a definition:
\[d2.6\] Take $M=({{\overline{M}}},M^\infty)\in {\operatorname{\mathbf{\underline{M}Sm}}}$. Let ${{\operatorname{\mathbf{Comp}}}(M)}$ be the category whose objects are pairs $(N,j)$ consisting of a modulus pair $N=({{\overline{N}}},N^\infty)\allowbreak\in{\operatorname{\mathbf{MSm}}}$ equipped with a dense open immersion $j:{{\overline{M}}}\hookrightarrow {{\overline{N}}}$ such that $N^\infty=M_N^\infty+C$ for some effective Cartier divisors $M_N^\infty, C$ on ${\overline}{N}$ satisfying ${\overline}{N} \setminus |C| = j({\overline}{M})$ and $j$ induces a minimal morphism $M \to N$ in the sense of Def. \[def:minimality\]. Note that for $N\in {{\operatorname{\mathbf{Comp}}}(M)}$ we have $j(M^{{\operatorname{o}}})=N^{{\operatorname{o}}}$ and $N$ is equipped with $j_N\in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}(M, N) \subset {\operatorname{\mathbf{\underline{M}Sm}}}(M,N)$ which is the graph of $j|_{M^{{\operatorname{o}}}} : M^{{\operatorname{o}}}\cong N^{{\operatorname{o}}}$. For $N_1,N_2\in {{\operatorname{\mathbf{Comp}}}(M)}$ we define $${{\operatorname{\mathbf{Comp}}}(M)}(N_1,N_2)=\{\gamma\in {\operatorname{\mathbf{MSm}}}(N_1,N_2)\;|\; \gamma\circ j_{N_1} =j_{N_2}\}.$$ Note that any $\gamma$ as above induces an isomorphism $N_1^{{\operatorname{o}}}{{\xrightarrow}{\sim}}N_2^{{\operatorname{o}}}$ in ${\operatorname{\mathbf{Sm}}}$.
\[c1.1\] The category ${{\operatorname{\mathbf{Comp}}}(M)}$ is a cofiltered ordered set.
That it is ordered is obvious as ${{\operatorname{\mathbf{Comp}}}(M)}(N_1,N_2)$ has at most 1 element for any $(N_1,N_2)$. For cofiltering, we first show that ${{\operatorname{\mathbf{Comp}}}(M)}$ is nonempty. For this, choose a compactification $j_0:{\overline}{M}{\hookrightarrow}{\overline}{N}_0$, with ${\overline}{N}_0\in {\operatorname{\mathbf{Sch}}}$ proper. Let ${\overline}{N}_1={{\mathbf{Bl}}}_{({\overline}{N}_0-{\overline}{M})_{{\operatorname{red}}}}({\overline}{N_0})$; then $j_0$ lifts to $j_1:{\overline}{M}{\hookrightarrow}{\overline}{N}_1$ by the universality of the blowup [@hartshorne Ch. II, Prop. 7.14], and ${\overline}{N}_1-{\overline}{M}$ is the support of an effective Cartier divisor $C_1$. Consider now the scheme-theoretic closure $N_1^\infty$ of $M^\infty$ in ${\overline}{N}_1$, and define ${\overline}{N}={{\mathbf{Bl}}}_{N_1^\infty}({\overline}{N}_1)$, $M_N^\infty =$ pull-back of $N_1^\infty$, $C=$ pull-back of $C_1$, $N^\infty=M_N^\infty+C$ and $N=({\overline}{N},N^\infty)$: then $j_1$ lifts to $j:{\overline}{M}{\hookrightarrow}{\overline}{N}$ (by the same reason as $j_0$), which defines an object of ${{\operatorname{\mathbf{Comp}}}(M)}$.
Let $N_1$ and $N_2$ be two objects in ${\operatorname{\mathbf{Comp}}}(M)$. Let $\Gamma$ be the graph of the rational map ${\overline}{N}_1 \dashrightarrow
{\overline}{N}_2$ given by $1_{M^{{\operatorname{o}}}}$. Then we have morphisms of schemes $p : \Gamma \to {\overline}{N}_1$ and $q : \Gamma
\to {\overline}{N}_2$, and there exists a natural open immersion ${\overline}{M} \to
\Gamma$. Note that $(\Gamma,p^\ast N_1^\infty)$ and $(\Gamma,q^\ast N_2^\infty)$ are objects of ${\operatorname{\mathbf{Comp}}}(M)$. Since $(\Gamma,p^\ast N_1^\infty)$ dominates $N_1$ and $(\Gamma,q^\ast
N_2^\infty)$ dominates $N_2$, we are reduced to the case that $N_1$ and $N_2$ have the same ambient space ${\overline}{N}$. Let $C$ be the effective Cartier divisor on ${\overline}{N}$ such that $|C|={\overline}{N} -
{\overline}{M}$, which exists since $N_1 \in {\operatorname{\mathbf{Comp}}}(M)$. Then for a sufficiently large $n$ we have $N_1^\infty + nC \geq N_2^\infty$ since $N_1^\infty \cap {\overline}{M} = N_2^\infty \cap {\overline}{M} = M^\infty$. Therefore $N_3 = ({\overline}{N},N_1^\infty + nC)$ dominates both $N_1$ and $N_2$. This finishes the proof.
For $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$ and $L\in {\operatorname{\mathbf{MCor}}}$ we have a natural map $$\begin{aligned}
&\Phi: \underset{N\in {{\operatorname{\mathbf{Comp}}}(M)}}{{\operatornamewithlimits{\varinjlim}}} {\operatorname{\mathbf{MCor}}}(N,L) \to {\operatorname{\mathbf{\underline{M}Cor}}}(M,\tau L),\end{aligned}$$ which maps a representative $\alpha_N\in {\operatorname{\mathbf{MCor}}}(N,L)$ to $\alpha_N\circ j_N$. We also have a natural map for $M, L'\in {\operatorname{\mathbf{\underline{M}Cor}}}$ $$\begin{aligned}
&\Psi: {\operatorname{\mathbf{\underline{M}Cor}}}(L',M) \to \underset{N\in {{\operatorname{\mathbf{Comp}}}(M)}}{{\operatornamewithlimits{\varprojlim}}} {\operatorname{\mathbf{\underline{M}Cor}}}(L',\tau N),\end{aligned}$$ which maps $\beta$ to $(j_N \circ \beta)_N$.
The following is an analogue to Lemma \[lem:mod-exists\]:
\[l1.1\] The maps $\Phi$ and $\Psi$ are isomorphisms. In other words, the formula $$\tau^! M = ``{\operatornamewithlimits{\varprojlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)}" N,$$ defines a pro-left adjoint to $\tau$, which is fully faithful.
We start with $\Phi$. Injectivity is obvious since both sides are subgroups of ${\operatorname{\mathbf{Cor}}}(M^{{\operatorname{o}}},L^{{\operatorname{o}}})$. We prove surjectivity. Choose a dense open immersion $j_1: {{\overline{M}}}\hookrightarrow {{\overline{N}}}_1$ with ${{\overline{N}}}_1$ proper such that ${{\overline{N}}}_1-{{\overline{M}}}$ is the support of an effective Cartier divisor $C_1$. Let $M_1^\infty$ be the scheme-theoretic closure of $M^\infty$ in ${{\overline{N}}}_1$. (This may not be Cartier.) Let $\pi:{{\overline{N}}}_2\to {{\overline{N}}}_1$ be the blowup with center in $M_1^\infty$ and put $M_2^\infty=M_1^\infty \times_{{\overline}{N}_1} {\overline}{N}_2$ and $C_2=C_1 \times_{{\overline}{N}_1} {\overline}{N}_2$. Note that $M_2^\infty$ and $C_2$ are effective Cartier divisors on ${\overline}{N}_2$. By the universal property of the blowup [@hartshorne Ch. II, Prop. 7.14], $j_1$ extends to an open immersion $j_2: {{\overline{M}}}\to {{\overline{N}}}_2$ so that $j_1=\pi j_2$. Then ${{\overline{N}}}_2-M^{{\operatorname{o}}}$ is the support of the Cartier divisor $N_2^\infty:=M_2^\infty+ C_2$ so that $$(({{\overline{N}}}_2, N_2^\infty), j_2) \in {\operatorname{\mathbf{Comp}}}(M).$$ Now the claim for $\Phi$ follows from the following:
\[cl2.1\] For any $\alpha\in {\operatorname{\mathbf{\underline{M}Cor}}}(M,L)$, there exists an integer $n>0$ such that $\alpha\in {\operatorname{\mathbf{MCor}}}(({{\overline{N}}}_2,M_2^\infty +nC_2),L)$.
Indeed we may assume $\alpha$ is an integral closed subscheme of $M^{{\operatorname{o}}}\times L^{{\operatorname{o}}}$. We have a commutative diagram $$\xymatrix{
{{\overline}{\alpha}}^N \ar[r]^{j_1} \ar[d]_{{\varphi}_\alpha} & {{\overline}{\alpha}}_1^N \ar[d]_{{\varphi}_{\alpha_1}}
& \ar[l]_{\pi}{{\overline}{\alpha}}_2^N \ar[d]_{{\varphi}_{\alpha_2}}\\
{{\overline{M}}}\times{{\overline{L}}}\ar[r]^{j_1} & {{\overline{N}}}_1\times{{\overline{L}}}& \ar[l]_{\pi}{{\overline{N}}}_2\times{{\overline{L}}}\\
}$$ where ${{\overline}{\alpha}}^N$ (resp. ${{\overline}{\alpha}}^N_1$, resp. ${{\overline}{\alpha}}^N_2$) is the normalization of the closure of $\alpha\subset M^{{\operatorname{o}}}\times L^0$ in ${{\overline{M}}}\times{{\overline{L}}}$ (resp. ${{\overline{N}}}_1\times{{\overline{L}}}$, resp. ${{\overline{N}}}_2\times{{\overline{L}}}$), and $j_1$ and $\pi$ are induced by $j_1:{{\overline{M}}}\to {{\overline{N}}}_1$ and $\pi:{{\overline{N}}}_2\to {{\overline{N}}}_1$ respectively. Now the admissibility of $\alpha\in {\operatorname{\mathbf{\underline{M}Cor}}}(M,L)$ implies $${\varphi}^*_\alpha({{\overline{M}}}\times L^\infty) \leq {\varphi}^*_\alpha(M^\infty \times {{\overline{L}}}).$$ Since ${{\overline}{\alpha}}_1^N-j_1({{\overline}{\alpha}}^N)$ is supported on ${\varphi}^{-1}_{\alpha_1}(C_1\times {{\overline{L}}})$, this yields an inclusion of closed subschemes $${\varphi}^*_{\alpha_1}({{\overline{N}}}_1\times L^\infty) \subseteq {\varphi}^*_{\alpha_1}((M_1^\infty + n C_1) \times {{\overline{L}}})$$ for a sufficiently large $n>0$. Applying $\pi^*$ to this inclusion, we get an inequality of Cartier divisors $${\varphi}^*_{\alpha_2}({{\overline{N}}}_2\times L^\infty) \leq {\varphi}^*_{\alpha_2}((M_2^\infty + n C_2) \times {{\overline{L}}})$$ which proves the claim.
Next we prove that $\Psi$ is an isomorphism. Injectivity is obvious since both sides are subgroups of ${\operatorname{\mathbf{Cor}}}(L^{{\operatorname{o}}},M^{{\operatorname{o}}})$. We prove surjectivity. Take $\gamma\in \underset{N\in {{\operatorname{\mathbf{Comp}}}(M)}}{{\operatornamewithlimits{\varprojlim}}} {\operatorname{\mathbf{MCor}}}(L,N)$. Then $\gamma\in {\operatorname{\mathbf{Cor}}}(L^{{\operatorname{o}}},M^{{\operatorname{o}}})$ is such that any component $\delta\subset L^{{\operatorname{o}}}\times M^{{\operatorname{o}}}$ of $\gamma$ satisfies the following condition: take any $(N, j) \in {\operatorname{\mathbf{Comp}}}(M)$ and write $N^\infty = M_N^\infty+C$ as in Definition \[d2.6\]. Let ${{\overline}{\delta}}^N$ be the normalization of the closure of $\delta$ in ${{\overline{L}}}\times {{\overline{N}}}$ with the natural map ${\varphi}_{\delta}:{{\overline}{\delta}}^N \to {{\overline{L}}}\times {{\overline{N}}}$. Then we have $${\varphi}_{\delta}^*({{\overline{L}}}\times (M^\infty_N + n C))\leq {\varphi}_{\delta}^*(L^\infty\times {{\overline{N}}})$$ for any integer $n>0$. Clearly this implies that $|{{\overline}{\delta}}|$ does not intersect with ${\overline}{L} \times |C|$ so that ${{\overline}{\delta}}\subset {{\overline{L}}}\times{{\overline{M}}}$. Since ${{\overline}{\delta}}$ is proper over ${\overline}{L}$ by assumption, this implies $\delta\in {\operatorname{\mathbf{\underline{M}Cor}}}(L,M)$ which proves the surjectivity of $\Psi$ as desired.
We come back to the proof of Theorem \[t2.1\]. It remains to consider $\tau_s$. The natural maps $$\begin{aligned}
&{\varphi}: \underset{N\in {\operatorname{\mathbf{Comp}}}(M)}{{\operatornamewithlimits{\varinjlim}}} {\operatorname{\mathbf{MSm}}}(N,L) \to {\operatorname{\mathbf{\underline{M}Sm}}}(M,\tau L),
\\
&\psi: {\operatorname{\mathbf{\underline{M}Sm}}}(L',M) \to \underset{N\in {\operatorname{\mathbf{Comp}}}(M)}{{\operatornamewithlimits{\varprojlim}}} {\operatorname{\mathbf{\underline{M}Sm}}}(L',\tau N)\end{aligned}$$ are also bijective for any $M, L'\in {\operatorname{\mathbf{\underline{M}Cor}}}$ and $L\in {\operatorname{\mathbf{MCor}}}$. The proof is identical to Lemma \[l1.1\]. In particular, the inclusion functor $\tau_s : {\operatorname{\mathbf{MSm}}}\to {\operatorname{\mathbf{\underline{M}Sm}}}$ admits a pro-left adjoint given by $\tau_s^{!} M = ``{\operatornamewithlimits{\varprojlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)}" N$, which commutes with the inclusions ${\operatorname{\mathbf{MSm}}}{\hookrightarrow}{\operatorname{\mathbf{MCor}}}$ and ${\operatorname{\mathbf{\underline{M}Sm}}}{\hookrightarrow}{\operatorname{\mathbf{\underline{M}Cor}}}$. This completes the proof of Theorem \[t2.1\].
More on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}^\protect{{\operatorname{fin}}}$ and $\protect{\operatorname{\mathbf{\underline{M}Cor}}}^\protect{{\operatorname{fin}}}$
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\[deff\] A morphism $f:M\to N$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ is in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ if it is minimal (Def. \[def:minimality\]), ${\overline}{f}:{{\overline{M}}}\to {{\overline{N}}}$ is a proper morphism and $f^{{\operatorname{o}}}$ is an isomorphism in ${\operatorname{\mathbf{Sm}}}$. We write $\Sigma^{{\operatorname{fin}}}$ for the class of morphisms in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ that belong to ${\operatorname{\mathbf{MSm}}}$.
In particular, we have ${\Sigma}^{{\operatorname{fin}}}\subset {\Sigma}$ (see Definition \[d.sigma\]) and ${{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M = {\Sigma}^{{\operatorname{fin}}}\downarrow M$ for $M \in {\operatorname{\mathbf{MSm}}}$. Let us consider the inclusion functors $$\begin{aligned}
\label{eq:def-b}
&{{\underline{b}}}_s : {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Sm}}}, \quad
{{\underline{b}}} : {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}.\end{aligned}$$
The following commutative diagram of categories will become fundamental (cf. ): $$\label{eq:six-cat-diag0}
\xymatrix{
{\operatorname{\mathbf{MCor}}}\ar[r]^{\tau} &
{\operatorname{\mathbf{\underline{M}Cor}}}&
{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\ar[l]_{{{\underline{b}}}}
\\
{\operatorname{\mathbf{MSm}}}\ar[r]^{\tau_s} \ar[u]^{c}&
{\operatorname{\mathbf{\underline{M}Sm}}}\ar[u]^{{{\underline{c}}}} &
{\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}. \ar[l]_{{{\underline{b}}}_s} \ar[u]^{{{\underline{c}}}^{{\operatorname{fin}}}}
}$$
\[peff1\] a) The class ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ enjoys a calculus of right fractions within ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ and ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$.\
b) The functors ${{\underline{b}}}_s$ and ${{\underline{b}}}$ are localisations having left pro-adjoints $b_s^!$ and $b^!$. They induce equivalences of categories $$({{\underline{\Sigma}}}^{{\operatorname{fin}}})^{-1} {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\cong {\operatorname{\mathbf{\underline{M}Sm}}}\quad \text{and} \quad
({{\underline{\Sigma}}}^{{\operatorname{fin}}})^{-1} {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\cong {\operatorname{\mathbf{\underline{M}Cor}}}.$$ c) A morphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ (resp. ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$) is invertible in ${\operatorname{\mathbf{\underline{M}Cor}}}$ (resp. ${\operatorname{\mathbf{\underline{M}Sm}}}$) if and only if it belongs to ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$. A morphism $f$ in ${\operatorname{\mathbf{\underline{M}Cor}}}$ (resp. ${\operatorname{\mathbf{\underline{M}Sm}}}$) is an isomorphism if and only if it can be written as $s=s_1 s_2^{-1}$ for some $s_1, s_2 \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$.\
All statements hold for $\Sigma^{{\operatorname{fin}}}$ (without an underline) as well.
a\) Same as the proof of Proposition \[prop:localization\] a), except for (2): consider a diagram in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ $$\begin{CD}
&& M'_2\\
&&@VfVV\\
M_1@>\alpha >> M_2
\end{CD}$$ with $f\in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$ (in particular $f^{{\operatorname{o}}}$ is an isomorphism). By the properness of $f$, the finite correspondence $\alpha^{{\operatorname{o}}}:M_1^{{\operatorname{o}}}\to {M'_2}^{{\operatorname{o}}}$ satisfies the hypothesis of Theorem \[trg\]. Applying this theorem, we find a proper birational morphism $f':{{\overline{M}}}'_1\to {{\overline{M}}}_1$ which is an isomorphism over $M_1^{{\operatorname{o}}}$ and such that $\alpha^{{\operatorname{o}}}$ defines a finite correspondence $\alpha':{{\overline{M}}}'_1\to {{\overline{M}}}'_2$. If we define ${M'_1}^\infty={f'}^*M_1^\infty$, then $f'\in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$ and $\alpha'\in {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(M'_1,M'_2)$.
If $\alpha\in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}(M_1,M_2)$, then $\alpha'$ is not in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}(M'_1,M'_2)$ in general (unless ${\overline}{M'_1}$ is normal, see Remark \[rk-graph-trick\] ). However, write ${\overline}{M''_1}$ for the closure of the graph of the rational map $\alpha':{\overline}{M'_1}{\dashrightarrow}{\overline}{M'_2}$, and $\pi$ for the projection ${\overline}{M''_1}\to {\overline}{M'_1}$: by hypothesis, $\pi$ is finite birational. Define a modulus pair $M''_1=({\overline}{M''_1}, {M''_1}^\infty)$ by putting ${M''_1}^\infty:=\pi^*{M'_1}^\infty$. Then $\pi$ defines a minimal morphism $M''_1\to M'_1$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, hence the morphism $\alpha'':M''_1\to M'_2$ determined by $\alpha'$ is in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$.
For b), all assertions are obvious except for the equivalences, for which it suffices as in Corollary \[c.cf\] to show that for any $M,N\in {\operatorname{\mathbf{\underline{M}Cor}}}$, the obvious maps $${\operatornamewithlimits{\varinjlim}}_{M'\in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M} {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(M',N)\to {\operatorname{\mathbf{\underline{M}Cor}}}(M,N)$$ and the corresponding map for ${{\underline{b}}}_s$ are isomorphisms. These maps are clearly injective, and its surjectivity follows again from Theorem \[trg\]. It then follows from Proposition \[p2.6\] they have pro-left adjoints.
The first statement of c) is clear, and the second follows from b).
The same proof works for $\Sigma^{{\operatorname{fin}}}$.
\[cor:sigma-fin-cofil\] For any $M \in {\operatorname{\mathbf{\underline{M}Cor}}}$, the category ${{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M$ is cofiltered.
This follows from Propositions \[peff1\] and \[p.cf\].
\[cor:fiber-cat\] Let ${\mathcal{C}}$ be a category and let $F:{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\to {\mathcal{C}}$, $G:{\operatorname{\mathbf{\underline{M}Sm}}}\to {\mathcal{C}}$ be two functors whose restrictions to the common subcategory ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ are equal. Then $(F,G)$ extends (uniquely) to a functor $H:{\operatorname{\mathbf{\underline{M}Cor}}}\to {\mathcal{C}}$.
The hypothesis implies that $F$ inverts the morphisms in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$; the conclusion now follows from Proposition \[peff1\] b).
\[c2.1\] Any modulus pair in ${\operatorname{\mathbf{\underline{M}Sm}}}$ is isomorphic to a modulus pair $M$ in which ${\overline}{M}$ is normal. Under resolution of singularities, we may even choose ${\overline}{M}$ smooth and the support of $M^\infty$ to be a divisor with normal crossings.
Let $M_0\in {\operatorname{\mathbf{\underline{M}Sm}}}$. Consider a proper morphism $\pi:{\overline}{M}\to {\overline}{M_0}$ which is an isomorphism over $M_0^{{\operatorname{o}}}$. Define $M^\infty:=\pi^*M_0^\infty$. Then the induced morphism $\pi:M\to M_0$ of ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ is in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$, hence invertible in ${\operatorname{\mathbf{\underline{M}Sm}}}$. The corollary readily follows.
We also have the following important lemma:
\[l1.3\] Let $M, L, N \in {\operatorname{\mathbf{\underline{M}Sm}}}$. Let $f: L \to N$ be a minimal morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ such that ${\overline}{f} : {\overline}{L} \to {\overline}{N}$ is faithfully flat. Then the diagram $$\xymatrix{
{\operatorname{\mathbf{\underline{M}Cor}}}(N, M) \ar[r]^{f^*} \ar@{^{(}->}[d]
&{\operatorname{\mathbf{\underline{M}Cor}}}(L, M) \ar@{^{(}->}[d]
\\
{\operatorname{\mathbf{Cor}}}(N^{{\operatorname{o}}}, M^{{\operatorname{o}}}) \ar[r]_{(f^{{\operatorname{o}}})^*}
&{\operatorname{\mathbf{Cor}}}(L^{{\operatorname{o}}}, M^{{\operatorname{o}}})
}$$ is cartesian. The same holds when ${\operatorname{\mathbf{\underline{M}Cor}}}$ is replaced by ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$.
As the second statement is proven in a completely parallel way, we only prove the first one. Take $\alpha \in {\operatorname{\mathbf{Cor}}}(N^{{\operatorname{o}}}, M^{{\operatorname{o}}})$ such that $(f^{{\operatorname{o}}})^*(\alpha) \in {\operatorname{\mathbf{\underline{M}Cor}}}(L, M)$. We need to show $\alpha \in {\operatorname{\mathbf{\underline{M}Cor}}}(N, M)$.
We first reduce to the case where $\alpha$ is integral. To do this, it suffices to show that for two distinct integral finite correspondences $V, V' \in {\operatorname{\mathbf{Cor}}}(N^{{\operatorname{o}}}, M^{{\operatorname{o}}})$, $(f^{{\operatorname{o}}})^*(V)$ and $(f^{{\operatorname{o}}})^*(V')$ have no common component. For this, we may assume $M^{{\operatorname{o}}}$ and $N^{{\operatorname{o}}}$ integral. By the injectivity of ${\operatorname{\mathbf{Cor}}}(N^{{\operatorname{o}}}, M^{{\operatorname{o}}}) \to {\operatorname{\mathbf{Cor}}}(k(N^{{\operatorname{o}}}), M^{{\operatorname{o}}})$, this can be reduced to the case where $N^{{\operatorname{o}}}$ and $L^{{\operatorname{o}}}$ are fields, and then the claim is obvious.
Now assume $\alpha$ is integral and put $\beta := (f^{{\operatorname{o}}})^*(\alpha)$. We have a commutative diagram $$\xymatrix{
{\overline}{\beta}^N
\ar@/^3ex/[rr]^{{\varphi}_{\beta}}
\ar[r]^{} \ar[d]_{f^N}
&
{\overline}{\beta}
\ar@/^3ex/[rr]^{a'}
\ar[r] \ar[d]
&
{\overline}{L} \times {\overline}{M}
\ar[r] \ar[d]
&
{\overline}{L} \ar[d]_{{\overline}{f}}
\\
{\overline}{\alpha}^N
\ar@/_3ex/[rr]_{{\varphi}_{\alpha}}
\ar[r]^{}
&
{\overline}{{\alpha}}
\ar@/_3ex/[rr]_{a}
\ar[r]
&
{\overline}{N} \times {\overline}{M}
\ar[r]
&
{\overline}{N}.
}$$ Here ${\overline}{\alpha}$ (resp. ${\overline}{\beta}$) is the closure of $\alpha$ (resp. $\beta$) in ${\overline}{N} \times {\overline}{M}$ (resp. ${\overline}{L} \times {\overline}{M}$) and ${\overline}{\alpha}^N$ (resp. ${\overline}{\beta}^N$) is the normalization of ${\overline}{\alpha}$ (resp. ${\overline}{\beta}$). By hypothesis $a'$ is proper and ${\overline}{f}$ is faithfully flat. This implies that $a$ is proper [@SGA1 Exp. VIII, Cor. 4.8]. We also have $$\begin{aligned}
(f^N)^*&({\varphi}_{\alpha}^*(N^\infty \times {\overline}{M}))
=
{\varphi}_{\beta}^*({\overline}{f}^*(N^\infty) \times {\overline}{M}))
\\
&={\varphi}_{\beta}^*(L^\infty \times {\overline}{M}) \geq
{\varphi}_{\beta}^*({\overline}{L} \times M^\infty)
=(f^N)^* ({\varphi}_{\alpha}^*({\overline}{N} \times M^\infty))\end{aligned}$$ (the second equality by the minimality of $f$). Note that $f^N$ is surjective since ${\overline}{f}$ is. Hence Lemma \[lKL\] shows that ${\varphi}_{\alpha}^*(N^\infty \times {\overline}{M}) \geq
{\varphi}_{\alpha}^*({\overline}{N} \times M^\infty)$, and we are done.
Fiber products and quarrable morphisms {#sect:quarrable}
--------------------------------------
We need the following elementary lemma.
\[lem:have-sup\] Let $X$ be a scheme. For two effective Cartier divisors $D$ and $E$ on $X$, the following conditions are equivalent:
1. $D \times_X E$ is an effective Cartier divisor on $X$.
2. There exist effective Cartier divisors $D', E'$ and $F$ on $X$ such that $D=D'+F, ~E=E'+F$ and $|D'| \cap |E'|=\emptyset$.
Moreover, the divisors $D', E'$ and $F$ in (2) are unique.
We may suppose $X={\operatorname{Spec}}A$ is affine and $D, E$ are defined by non-zero-divisors $d, e \in A$, respectively.
Suppose (1). This means that $(d, e)=(f)$ for some non-zero-divisor $f \in A$, because $D \times_X E = {\operatorname{Spec}}A/(d, e)$. Thus there are $d', e' \in A$ such that $d=d'f$ and $e=e'f$. Since $(f)=(d', e')(f)$ and $f$ is a non-zero-divisor, we have $(d', e')=A$. Now (2) holds by taking $D', E', F$ to be the Cartier divisors defined by $(d'), (e')$ and $(f)$.
\(2) immediately implies $F=D \times_X E$, whence (1). The last statement also follows from this.
Let $D$ and $E$ be effective Cartier divisors on a scheme $X$. If the conditions of Lemma \[lem:have-sup\] hold, we say that $D$ and $E$ *have a universal supremum*, and write $$\sup(D, E):=D'+E'+F (= D+E-F).$$
\[rem:sup-div\] Let $D$ and $E$ be effective Cartier divisors on $X$ having a universal supremum. The following are obvious from the definition.
1. We have $|\sup(D, E)|=|D| \cup |E|$.
2. If $f : Y \to X$ is a morphism such that $f(T) \not\subset |D|\cup |E|$ for any irreducible component $T$ of $Y$, then $f^*D$ and $f^*E$ have a universal supremum which is equal to $f^* \sup(D, E)$ (hence the name “universal”).
3. If moreover $Y$ is normal, then $f^* \sup(D, E)$ agrees with the supremum of $f^*D$ and $f^*E$ computed as a Weil divisor on $Y$.
Let $u_i : U_i \to M$ be morphisms in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ for $i=1, 2$ with projections $p_i : {\overline}{W}_0 := {\overline}{U}_1 \times_{{\overline}{M}} {\overline}{U}_2 \to {\overline}{U}_i$. Denote by ${\overline}{W}_1$ the union of irreducible components $T$ of ${\overline}{W}_0$ such that $p_i(T) \not\subset |U_i^\infty|$ for each $i=1, 2$. Observe that ${\overline}{W}_1$ is the closure of $U:=U_1^{{\operatorname{o}}}\times_{M^{{\operatorname{o}}}} U_2^{{\operatorname{o}}}$ in ${\overline}{W}_0$. Indeed, let $Z$ be the closure of $U$ in ${\overline}{W}_0$. Then any irreducible component $T$ of $Z$ meets $U$, which implies that $T\subset {\overline}{W}_1$. Conversely, any irreducible component $T$ of ${\overline}{W}_1$ meets $U$, hence $T \cap U$ is dense in $T$ and thus $T \subset Z$.
We write $q_i : {\overline}{W}_1 \to {\overline}{U}_i$ for the composition of the inclusion ${\overline}{W}_1 \to {\overline}{W}_0$ and $p_i$. By definition, we have effective Cartier divisors $q_i^*(U^\infty_i)$ on ${\overline}{W}_1$ and $q_1 \times q_2$ restricts to an isomorphism $$\label{eq:inside}
{\overline}{W}_1 \setminus |q_1^*(U_1^\infty)+q_2^*(U_2^\infty)|
\simeq U_1^{{\operatorname{o}}}\times_{M^{{\operatorname{o}}}} U_2^{{\operatorname{o}}}.$$
\[prop:fiber-prod\] Suppose that $U_1^{{\operatorname{o}}}\times_{M^{{\operatorname{o}}}} U_2^{{\operatorname{o}}}$ is smooth over $k$.
1. If $q_1^*U_1^\infty$ and $q_2^*U_2^\infty$ have a universal supremum, then $$W_1:=({\overline}{W}_1, \sup(q_1^*U_1^\infty, q_2^*U_2^\infty))
\in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$$ represents the fiber product of $U_1$ and $U_2$ over $M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ as well as in ${\operatorname{\mathbf{\underline{M}Sm}}}$. If further $U_1, U_2, M \in {\operatorname{\mathbf{MSm}}}^{{\operatorname{fin}}}$, then it holds in ${\operatorname{\mathbf{MSm}}}^{{\operatorname{fin}}}$ as well as in ${\operatorname{\mathbf{MSm}}}$.
2. If $u_1$ is minimal and ${\overline}{U}_2$ is normal, then $q_1^\ast U_1^\infty$ and $q_2^\ast U_2^\infty$ have a universal supremum, namely $q_2^\ast U_2^\infty$, and the morphism $W_1 \to U_2$ is a minimal morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. If moreover ${\overline}{u}_1$ is flat[^4], we have ${\overline}{W}_1={\overline}{W}_0$.
3. In general, there is a proper birational morphism $\pi : {\overline}{W}_2 \to {\overline}{W}_1$ which restricts to an isomorphism over ${\overline}{W}_1 \setminus |q_1^*(U_1^\infty)+q_2^*(U_2^\infty)|$, and such that $r_1^*U_1^\infty$ and $r_2^*U_2^\infty$ have a universal supremum, where $r_i := q_i \pi$ for $i=1, 2$. For such ${\overline}{W}_2$, $$W_2:=({\overline}{W}_2, \sup(r_1^*U_1^\infty, r_2^*U_2^\infty))
\in {\operatorname{\mathbf{\underline{M}Sm}}}$$ represents the fiber product of $U_1$ and $U_2$ over $M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}$. If further $U_1, U_2, M \in {\operatorname{\mathbf{MSm}}}$, then it holds in ${\operatorname{\mathbf{MSm}}}$.
\(1) Let $f_i : N \to U_i$ be morphisms in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ for $i=1, 2$ such that $u_1f_1=u_2f_2$. Then the morphisms ${\overline}{f}_i : {\overline}{N} \to {\overline}{U}_i$ for $i=1, 2$ induce a unique morphism ${\overline}{h} :{\overline}{N} \to {\overline}{W}_0$ with ${\overline}{f}_i = p_i{\overline}{h}$ for $i=1, 2$. Since $f_i$ are morphisms in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, for any irreducible component $T$ of ${\overline}{N}$ we have ${\overline}{f}_i(T) \not\subset |U_i^\infty|$, and hence ${\overline}{h}$ factors though ${\overline}{g} : {\overline}{N} \to {\overline}{W}_1$ so that we have ${\overline}{f}_i=q_i {\overline}{g}$. It remains to prove $\nu^*N^\infty \ge \nu^* {\overline}{g}^* W_1^\infty$, where $\nu : {\overline}{N}^N \to {\overline}{N}$ is the normalization. As we have $\nu^* {\overline}{g}^*W_1^\infty
= \nu^*{\overline}{g}^*\sup(q_1^* U_1^\infty, q_2^* U_2^\infty)
= \sup(\nu^*{\overline}{f}_1^* U_1^\infty, \nu^*{\overline}{f}_1^* U_2^\infty)$ by definition and Remark \[rem:sup-div\], this follows from the admissibility of $f_i$, that is, $\nu^* {\overline}{f}_i^\ast U_i^\infty \leq \nu^* N^\infty$. We have shown that $W_1$ represents the fiber product in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Propositions \[peff1\] and \[p1.10\] show that the same holds in ${\operatorname{\mathbf{\underline{M}Sm}}}$ as well. (This also follows from (3) below.) The last statement is an immediate consequence of the first.
\(2) Let $p_W : {\overline}{W}_1^N \to {\overline}{W}_1$ and $p_{U_1} : {\overline}{U}_1^N \to {\overline}{U}_1$ be the normalizations. By the minimality of $u_1$, we have $q_1^\ast U_1^\infty = q_1^\ast {\overline}{u}_1^\ast M^\infty = q_2^\ast {\overline}{u}_2^\ast M^\infty \leq q_2^\ast U_2^\infty$, where the last inequality holds by the admissibility of $u_2$ and the normality of ${\overline}{U}_2$. Then $q_1^\ast U_1^\infty$ and $q_2^\ast U_2^\infty$ have a universal supremum since $q_1^\ast U_1^\infty \subset q_2^\ast U_2^\infty$ implies Condition (1) of Lemma \[lem:have-sup\], which also implies that $W_1^\infty = \sup (q_1^\ast U_1^\infty,q_2^\ast U_2^\infty) = q_2^\ast U_2^\infty$. This shows the minimality of $W_1 \to U_2$.
Suppose now ${\overline}{u}_1$ flat, and let $T$ be an irreducible component of ${\overline}{W}_0$. Then $p_2:{\overline}{W}_0\to {\overline}{U}_2$ is also flat, hence $T$ dominates an irreducible component $E$ of ${\overline}{U}_2$ [@hartshorne Prop. III.9.5] and we cannot have $p_2(T)\subset |U_2^\infty|$ since $U_2^\infty$ is everywhere of codimension $1$ in ${\overline}{U}_2$. Suppose that $p_1(T)\subset |U_1^\infty|$. By the minimality of $u_1$, this implies $u_2p_2(T)=u_1p_1(T)\subset |M^\infty|$, hence $u_2(E)\subset |M^\infty|$, contradicting the admissibility of $u_2$.
\(3) If $\pi$ is the blow-up of ${\overline}{W}_1$ with center $q_1^*(U_1^\infty) \times_{{\overline}{W}_1} q_2^*(U_1^\infty)$, then $r_1^*U^\infty_1 \times_{{\overline}{W}_2} r_2^*U^\infty_2$ is precisely the exceptional divisor by definition, which is therefore an effective Cartier divisor, showing the first assertion. Note that $W_2^{{\operatorname{o}}}\cong U_1^{{\operatorname{o}}}\times_{M^{{\operatorname{o}}}} U_2^{{\operatorname{o}}}$ by .
Now let $f_i : N \to U_i$ be morphisms in ${\operatorname{\mathbf{\underline{M}Sm}}}$ for $i=1, 2$ such that $u_1f_1=u_2f_2$. Then the morphisms $f_i^{{\operatorname{o}}}: N^{{\operatorname{o}}}\to U_i^{{\operatorname{o}}}$ for $i=1, 2$ induce a unique morphism $h^{{\operatorname{o}}}:N^{{\operatorname{o}}}\to W_2^{{\operatorname{o}}}$ with $f_i^{{\operatorname{o}}}= p_ih^{{\operatorname{o}}}$ for $i=1, 2$. It suffices to prove that $h^{{\operatorname{o}}}$ defines a \[unique\] morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}$. By the graph trick (Lemma \[l-gt\]), we may assume that $f_i^{{\operatorname{o}}}$ and $h^{{\operatorname{o}}}$ extend to morphisms ${\overline}{f}_i: {\overline}{N} \to {\overline}{U}_i$ and ${\overline}{h} : {\overline}{N} \to {\overline}{W}_2$. Moreover we may assume that ${\overline}{N}$ is normal by Remark \[rk-graph-trick\] . It remains to prove $N^\infty \ge {\overline}{h}^* W_2^\infty$. As we have ${\overline}{h}^*W_2^\infty
= \sup({\overline}{f}_1^* U_1^\infty , {\overline}{f}_1^*U_2^\infty)$ by the assumption and Remark \[rem:sup-div\], this follows from the admissibility of $f_i$, that is, ${\overline}{f}_i^\ast U_i^\infty \leq N^\infty$.
\[rem:fiberprod-int\] If $W$ represents a fiber product $U_1 \times_M U_2$ (either in ${\operatorname{\mathbf{\underline{M}Sm}}}$ or in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$), then we have $W^{{\operatorname{o}}}= U_1^{{\operatorname{o}}}\times_{M^{{\operatorname{o}}}} U_2^{{\operatorname{o}}}$. Indeed, the functors ${\operatorname{\mathbf{\underline{M}Sm}}}\to {\operatorname{\mathbf{Sm}}}$ and ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{Sm}}}$ given by $M \mapsto M^{{\operatorname{o}}}$ have the left adjoint $X \mapsto (X, \emptyset)$ (Lemma \[l1.2\]), hence commute with limits.
Let $B=k[x_1,x_2]$, ${\mathbf{A}}^2={\operatorname{Spec}}B$, $D_i={\operatorname{Spec}}(B/x_iB)$ and $P=D_1\cap D_2$. Let now $M=(D_1\cup D_2, P)$ and $U_i=(D_i,P)$ for $i=1, 2$. Then ${\overline}{W}_0$ is a point but ${\overline}{W}_1=\emptyset$, and $W_1=(\emptyset, \emptyset)$ indeed represents the fiber product $U_1 \times_M U_2$. In particular, fiber products do not commute with the forgetful functor $M\mapsto {\overline}{M}$ from ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ to ${\operatorname{\mathbf{Sch}}}$ of Definition \[d1.1\] (2). Another counterexample: let $M=({\mathbf{A}}^2,D_1)$, ${\overline}{U}_1={{\mathbf{Bl}}}_{P}({\mathbf{A}}^2)$, $u_1:U_1\to M$ be the minimal induced modulus structure, $U_2=(D_2,P)$ and $U_2\to M$ be given by the inclusion. Then ${\overline}{W}_1\subsetneq {\overline}{W}_0$ is the proper transform of $u_1$. See however Corollary \[exist-pullback\] (1).
Recall [@SGA3 IV.1.4.0] that a morphism $f:M\to N$ in a category ${\mathcal{C}}$ is *quarrable* if, for any $g:N'\to N$, the fibred product $N'\times_N M$ is representable in ${\mathcal{C}}$. We have:
\[exist-pullback\] The following assertions hold.
1. If $f : U \to M$ is a minimal morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ (see Definition \[def:minimality\]) such that $f^{{\operatorname{o}}}$ is smooth, then $f$ is quarrable in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. If $f\in {\operatorname{\mathbf{MSm}}}^{{\operatorname{fin}}}$, it is quarrable in this category. If moreover ${\overline}{f}$ is flat, then the pull-back by $f$ of morphisms from normal modulus pairs commutes with the forgetful functor $M\mapsto {\overline}{M}$ from ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ to ${\operatorname{\mathbf{Sch}}}$ of Definition \[d1.1\] (2).
2. If $f : U \to M$ is a morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}$ such that $f^{{\operatorname{o}}}$ is smooth, then $f$ is quarrable in ${\operatorname{\mathbf{\underline{M}Sm}}}$. If $f\in {\operatorname{\mathbf{MSm}}}$, it is quarrable in this category.
\(1) follows from Proposition \[prop:fiber-prod\] (1) and (2); (2) follows from Proposition \[prop:fiber-prod\] (3).
Finite products exist in ${\operatorname{\mathbf{\underline{M}Sm}}}$ and ${\operatorname{\mathbf{MSm}}}$.
This is the special case $M=({\operatorname{Spec}}k,\emptyset)$ in Corollary \[exist-pullback\] (2).
Presheaf theory
===============
Modulus presheaves with transfers
---------------------------------
\[d2.7\] By a presheaf we mean a contravariant functor to the category of abelian groups.
1. The category of presheaves on ${\operatorname{\mathbf{MSm}}}$ (resp. ${\operatorname{\mathbf{\underline{M}Sm}}}$, ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$) is denoted by ${\operatorname{\mathbf{MPS}}}$ (resp. ${\operatorname{\mathbf{\underline{M}PS}}}$, ${\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}$).
2. The category of additive presheaves on ${\operatorname{\mathbf{MCor}}}$ (resp. ${\operatorname{\mathbf{\underline{M}Cor}}}$, ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$) is denoted by ${\operatorname{\mathbf{MPST}}}$ (resp. ${\operatorname{\mathbf{\underline{M}PST}}}$, ${\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$.)
All these categories are abelian Grothendieck, with projective sets of generators: this is classical for those of (1) and follows from Theorem \[t.mon\] for those of (2). (See also proof of Proposition \[eq:c-functor\] below.)
\[n2.1\] We write $$\begin{aligned}
{\mathbb{Z}}_{{\operatorname{tr}}}:& {\operatorname{\mathbf{\underline{M}Cor}}}\to{\operatorname{\mathbf{\underline{M}PST}}},
\quad {\operatorname{\mathbf{MCor}}}\to {\operatorname{\mathbf{MPST}}},
\\
{\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}}:& {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}},
\\
{\mathbb{Z}}_{{\operatorname{tr}}}:&{\operatorname{\mathbf{Cor}}}\to {{\operatorname{\mathbf{PST}}}}\end{aligned}$$ for the associated representable presheaves (i.e. ${\mathbb{Z}}_{{\operatorname{tr}}}(M) \in {\operatorname{\mathbf{\underline{M}PST}}}$ is given by ${\mathbb{Z}}_{{\operatorname{tr}}}(M)(N)={\operatorname{\mathbf{\underline{M}Cor}}}(N, M)$, etc.). We shall use the common notation ${\mathbb{Z}}_{{\operatorname{tr}}}$ but they will be distinguished by the context.
We now briefly describe the main properties of the functors induced by those of the previous section.
$\protect{\operatorname{\mathbf{MPST}}}$ and $\protect{{\operatorname{\mathbf{PST}}}}$
--------------------------------------------------------------------------------------
\[lem:counit\] The functor $\omega:{\operatorname{\mathbf{MCor}}}\to {\operatorname{\mathbf{Cor}}}$ of §\[s1.2\] yields a string of $3$ adjoint functors $(\omega_!,\omega^*,\omega_*)$: $${\operatorname{\mathbf{MPST}}}\begin{smallmatrix}\omega_!\\\longrightarrow\\\omega^*\\\longleftarrow\\\omega_*\\ \longrightarrow\end{smallmatrix}{{\operatorname{\mathbf{PST}}}},$$ where $\omega^*$ is fully faithful and $\omega_!,\omega_*$ are localisations; $\omega_!$ has a pro-left adjoint $\omega^!$, hence is exact.
Similarly, $\omega_s:{\operatorname{\mathbf{MSm}}}\to {\operatorname{\mathbf{Sm}}}$ yields a string of $3$ adjoint functors $(\omega_{s!},\omega_s^*,\omega_{s*})$; $\omega_s^*$ is fully faithful and $\omega_{s!},\omega_{s*}$ are localisations; $\omega_{s!}$ has a pro-left adjoint $\omega_s^!$, hence is exact.
This follows from Theorems \[prop:localization\] and \[lem:omega-sh\].
Let $X \in {\operatorname{\mathbf{Sm}}}$ and let $M \in {\operatorname{\mathbf{MSm}}}(X)$. Lemma \[ln2\] and Proposition \[p.funct\] show that the inclusions $\{M^{(n)}\mid n>0\}\subset {\operatorname{\mathbf{MSm}}}({\overline}{M}!X) \subset {\operatorname{\mathbf{MSm}}}(X)$ induce isomorphisms (see Def. \[def:mpx\]) $$\label{rem:single-cpt}
\omega_!(F)(X)\simeq{\operatornamewithlimits{\varinjlim}}_{N \in {\operatorname{\mathbf{MSm}}}(X)} F(N)\simeq {\operatornamewithlimits{\varinjlim}}_{N \in {\operatorname{\mathbf{MSm}}}({\overline}{M}!X)} F(N)\simeq {\operatornamewithlimits{\varinjlim}}_{n>0} F(M^{(n)}).$$
$\protect{\operatorname{\mathbf{\underline{M}PST}}}$ and $\protect{{\operatorname{\mathbf{PST}}}}$
--------------------------------------------------------------------------------------------------
\[p4.2\] The adjoint functors $(\lambda,{\underline{\omega}})$ of Lemma \[l1.2\] induce a string of $4$ adjoint functors $(\lambda_!={\underline{\omega}}^!,\lambda^*={\underline{\omega}}_!,\lambda_*={\underline{\omega}}^*,{\underline{\omega}}_*)$: $${\operatorname{\mathbf{\underline{M}PST}}}\begin{smallmatrix}{\underline{\omega}}^!\\\longleftarrow\\{\underline{\omega}}_!\\\longrightarrow\\{\underline{\omega}}^*\\\longleftarrow\\{\underline{\omega}}_*\\ \longrightarrow\end{smallmatrix}{{\operatorname{\mathbf{PST}}}},$$ where ${\underline{\omega}}_!,{\underline{\omega}}_*$ are localisations while ${\underline{\omega}}^!$ and ${\underline{\omega}}^*$ are fully faithful. Moreover, if $X\in {\operatorname{\mathbf{Cor}}}$ is proper, we have a canonical isomorphism ${\underline{\omega}}^*{\mathbb{Z}}_{{\operatorname{tr}}}(X)\allowbreak\simeq {\mathbb{Z}}_{{\operatorname{tr}}}(X,\emptyset)$.
The only non obvious statement is the last claim, which follows from Lemma \[l1.2\].
With and without underlines
---------------------------
\[eq.tau\] The functor $\tau:{\operatorname{\mathbf{MCor}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}$ of yields a string of $3$ adjoint functors $(\tau_!,\tau^*,\tau_*)$: $${\operatorname{\mathbf{MPST}}}\begin{smallmatrix}\tau_!\\\longrightarrow\\\tau^*\\\longleftarrow\\\tau_*\\ \longrightarrow\end{smallmatrix}{\operatorname{\mathbf{\underline{M}PST}}},$$ where $\tau_!,\tau_*$ are fully faithful and $\tau^*$ is a localisation; $\tau_!$ has a pro-left adjoint $\tau^!$, hence is exact. There are natural isomorphisms $$\omega_!\simeq {\underline{\omega}}_!\tau_!, \quad\omega_* \simeq {\underline{\omega}}_* \tau_*, \quad \omega^! \simeq \tau^! {\underline{\omega}}^!.$$ The same holds for the functor $\tau_s$ from Theorem \[t2.1\]. Namely, we have a string of $3$ adjoint functors $(\tau_{s!},\tau_s^*,\tau_{s*})$ and they satisfy $$\omega_{s!}\simeq {\underline{\omega}}_{s!}\tau_{s!},
\quad \omega_{s*} \simeq {\underline{\omega}}_{s*} \tau_{s*},
\quad \omega_s^! \simeq \tau_s^! {\underline{\omega}}_s^!.$$
This follows from Theorem \[t2.1\] and Proposition \[p.funct\].
\[lem:tau-colim\]
1. For $G \in {\operatorname{\mathbf{MPST}}}, G' \in {\operatorname{\mathbf{MPS}}}$ and $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$, we have $${\operatornamewithlimits{\varinjlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)} G(N) \simeq \tau_!G(M),
\quad
{\operatornamewithlimits{\varinjlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)} G'(N) \simeq \tau_{s !}G'(M).$$
2. The adjunction maps ${{\operatorname{Id}}}\to \tau^* \tau_!$ and ${{\operatorname{Id}}}\to \tau_s^* \tau_{s !}$ are isomorphisms.
3. There is an natural isomorphism $\tau_! \omega^*\simeq {\underline{\omega}}^*$.
\(1) This follows from Lemma \[l1.1\], Theorem \[t2.1\] and Proposition \[p.funct\].
\(2) This follows from (1) since ${\operatorname{\mathbf{Comp}}}(M)=\{ M \}$ for $M \in {\operatorname{\mathbf{MSm}}}$.
\(3) For $F\in {{\operatorname{\mathbf{PST}}}}$ and $M\in {\operatorname{\mathbf{MCor}}}$, we compute $$\begin{gathered}
\tau_! \omega^* F(M) = {\operatornamewithlimits{\varinjlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)} \omega^*F(N)\\
={\operatornamewithlimits{\varinjlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)} F(N^{{\operatorname{o}}}) = F(M^{{\operatorname{o}}})={\underline{\omega}}^*F(M).\end{gathered}$$ We are done.
\[rem:formula-tau\] By Lemma \[l1.1\] we have the formulas $$\tau^! {\mathbb{Z}}_{{\operatorname{tr}}}(M) = \underset{N\in {\operatorname{\mathbf{Comp}}}(M)}{``{\operatornamewithlimits{\varprojlim}}"} {\mathbb{Z}}_{{\operatorname{tr}}}(N), \quad \tau^* {\mathbb{Z}}_{{\operatorname{tr}}}(M) = {\operatornamewithlimits{\varprojlim}}_{N\in {\operatorname{\mathbf{Comp}}}(M)} {\mathbb{Z}}_{{\operatorname{tr}}}(N),$$ where the latter inverse limit is computed in ${\operatorname{\mathbf{MPST}}}$.
\[q.exact\] Is $\tau^!$ exact?
With and without $\protect{{\operatorname{fin}}}$
-------------------------------------------------
\[eq:bruno-functor\] Let ${{\underline{b}}}_s : {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Sm}}}$ and ${{\underline{b}}}:{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}$ be the inclusion functors from . Then ${{\underline{b}}}_s$ and ${{\underline{b}}}$ yield strings of $3$ adjoint functors $({{\underline{b}}}_{s !},{{\underline{b}}}_s^*,{{\underline{b}}}_{s *})$ and $({{\underline{b}}}_!,{{\underline{b}}}^*,{{\underline{b}}}_*)$: $${\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}\begin{smallmatrix}{{\underline{b}}}_{s !}\\\longrightarrow\\ {{\underline{b}}}_s^*\\\longleftarrow\\ {{\underline{b}}}_{s *}\\ \longrightarrow\end{smallmatrix}{\operatorname{\mathbf{\underline{M}PS}}},
\quad
{\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}\begin{smallmatrix}{{\underline{b}}}_!\\\longrightarrow\\ {{\underline{b}}}^*\\\longleftarrow\\ {{\underline{b}}}_*\\ \longrightarrow\end{smallmatrix}{\operatorname{\mathbf{\underline{M}PST}}},$$ where ${{\underline{b}}}_{s !},{{\underline{b}}}_{s *}$, ${{\underline{b}}}_!,{{\underline{b}}}_*$ are localisations; ${{\underline{b}}}_s^*$, ${{\underline{b}}}^*$ are exact and fully faithful; ${{\underline{b}}}_{s !}$, ${{\underline{b}}}_!$ have pro-left adjoints, hence are exact. The counit maps ${{\underline{b}}}_{s !} {{\underline{b}}}_s^\ast \to {{\operatorname{Id}}}$ and ${{\underline{b}}}_! {{\underline{b}}}^\ast \to {{\operatorname{Id}}}$ are isomorphisms. For $F_s \in {\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}$, $F\in {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ and $M\in \mathrm{Ob}({\operatorname{\mathbf{\underline{M}Sm}}}) = \mathrm{Ob}({\operatorname{\mathbf{\underline{M}Cor}}})$, we have (see Def. \[deff\]) $$\label{eq:b-sh-explicit}
{{\underline{b}}}_{s !} F_s (M) = {\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} F_s (N),
\quad
{{\underline{b}}}_! F(M) = {\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} F(N).$$
This follows from the usual yoga applied with Proposition \[peff1\] and Lemma \[lA.6\].
With and without transfers
--------------------------
\[eq:c-functor\] Let ${{\underline{c}}} :{\operatorname{\mathbf{\underline{M}Sm}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}$ be the functor from . Then ${{\underline{c}}}$ yields a string of $3$ adjoint functors $({{\underline{c}}}_!,{{\underline{c}}}^*,{{\underline{c}}}_*)$: $${\operatorname{\mathbf{\underline{M}PS}}}\begin{smallmatrix}{{\underline{c}}}_!\\\longrightarrow\\ {{\underline{c}}}^*\\\longleftarrow\\ {{\underline{c}}}_*\\ \longrightarrow\end{smallmatrix}{\operatorname{\mathbf{\underline{M}PST}}},$$ where ${{\underline{c}}}^*$ is exact and faithful (but not full). We have $$\label{eq2.6}
{{\underline{c}}}_! {\mathbb{Z}}^p(M) = {\mathbb{Z}}_{{\operatorname{tr}}}(M)$$ for any $M\in {\operatorname{\mathbf{\underline{M}Sm}}}$, where ${\mathbb{Z}}^p(M)$ is [^5] the presheaf $N\mapsto {\mathbb{Z}}[{\operatorname{\mathbf{\underline{M}Sm}}}(N,M)]$. The same statements hold for $c : {\operatorname{\mathbf{MSm}}}\to {\operatorname{\mathbf{MCor}}}$ and ${{\underline{c}}}^{{\operatorname{fin}}}; {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ from . Precisely, they yield strings of $3$ adjoint functors $(c_!, c^*, c_*)$ and $({{\underline{c}}}^{{\operatorname{fin}}}_!,{{\underline{c}}}^{{{\operatorname{fin}}}*},{{\underline{c}}}^{{\operatorname{fin}}}_*)$; $c^*$ and ${{\underline{c}}}^{{{\operatorname{fin}}}*}$ are exact and faithful. (The analogue of also holds for $c$ and ${{\underline{c}}}^{{\operatorname{fin}}}$, but we will not need it.)
To define ${{\underline{c}}}_!,{{\underline{c}}}^*$ and ${{\underline{c}}}_*$, we use the free additive category ${\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}$ on ${\operatorname{\mathbf{\underline{M}Sm}}}$ [@mcl VIII.3, Ex. 5 & 6]: it comes with a canonical functor $\gamma: {\operatorname{\mathbf{\underline{M}Sm}}}\to {\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}$ and is $2$-universal for contravariant functors to additive categories. In particular:
- The functor ${{\underline{c}}}$ induces an additive functor $\tilde{{{\underline{c}}}}:{\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}\to {\operatorname{\mathbf{\underline{M}Cor}}}$.
- By the $2$-universality, the functor $\gamma$ induces an equivalence $\gamma^* : {\operatorname{Mod}\hbox{--}}{\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}\cong {\operatorname{\mathbf{\underline{M}PS}}}$, where ${\operatorname{Mod}\hbox{--}}{\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}$ denotes the category of additive contravariant functors ${\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}\to {\operatorname{\mathbf{Ab}}}$
- For $M,N\in {\operatorname{\mathbf{\underline{M}Sm}}}$, we have a canonical isomorphism $${\mathbb{Z}}{\operatorname{\mathbf{\underline{M}Sm}}}(\gamma(N),\gamma(M))\simeq {\mathbb{Z}}[{\operatorname{\mathbf{\underline{M}Sm}}}(N,M)].$$
As usual, $\tilde{{{\underline{c}}}}$ induces a string of $3$ adjoint functors $(\tilde{{{\underline{c}}}}_!, \tilde{{{\underline{c}}}}^*, \tilde{{{\underline{c}}}}_*)$ (see §\[s.presh\]). We then define ${{\underline{c}}}_!$ as $\tilde{{{\underline{c}}}}_! \circ (\gamma^*)^{-1}$, etc. Everything follows from this except the faithfulness of ${{\underline{c}}}^*$, which is a consequence of the essential surjectivity of ${{\underline{c}}}$. The cases of ${{\underline{c}}}^{{\operatorname{fin}}}$ and $c$ are dealt with similarly.
\[lem:b-c-tau\]
1. We have $$\label{eq:b-and-c}
{{\underline{c}}}^{{{\operatorname{fin}}}*} {{\underline{b}}}^* = {{\underline{b}}}_s^* {{\underline{c}}}^*,
\quad
{{\underline{b}}}_! {{\underline{c}}}^{{\operatorname{fin}}}_! = {{\underline{c}}}_! {{\underline{b}}}_{s !},
\quad
{{\underline{c}}}^* {{\underline{b}}}_!= {{\underline{b}}}_{s !} {{\underline{c}}}^{{{\operatorname{fin}}}*}.$$
2. We have $$\label{eq:c-and-tau}
c^* \tau^* = \tau_s^* {{\underline{c}}}^*,
\quad
{{\underline{c}}}^* \tau_! = \tau_{s !} c^*,
\quad
{{\underline{c}}}^{{{\operatorname{fin}}}*} \tau_!^{{\operatorname{fin}}}= \tau_{s !}^{{\operatorname{fin}}}c^{{{\operatorname{fin}}}*}.$$
The first two equalities of (1) follows from the equality ${{\underline{b}}} ~{{\underline{c}}}^{{\operatorname{fin}}}= {{\underline{c}}} ~{{\underline{b}}}_s$ (see ). Similarly, the first equality of (2) follows from $\tau c = {{\underline{c}}} \tau_s$. By , we have $${{\underline{c}}}^* {{\underline{b}}}_! F(M)
\cong {\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M} F(N)
\cong {{\underline{b}}}_{s !} {{\underline{c}}}^{{{\operatorname{fin}}}*}F(M)$$ for any $F \in {\operatorname{\mathbf{MPST}}}^{{\operatorname{fin}}}$ and $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. (Note that all morphisms of ${{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M$ are in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, and that both of ${{\underline{b}}}_!$ and ${{\underline{b}}}_{s !}$ can be computed by using the same ${{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M$.) This proves the last formula of (1). Lemma \[lem:tau-colim\] (1) shows that $${{\underline{c}}}^*\tau_!F(M)
\cong {\operatornamewithlimits{\varinjlim}}_{N \in {\operatorname{\mathbf{Comp}}}(M)} F(N)
\cong \tau_{s !}c^*F(M)$$ for any $F \in {\operatorname{\mathbf{MPST}}}$ and $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. The last one of (2) is similar.
A patching lemma
----------------
By the previous lemma, we obtain a commutative diagram of categories (cf. ): $$\label{eq:six-cat-diag}\vcenter{
\xymatrix{
{\operatorname{\mathbf{MPST}}}\ar[r]^{\tau_!} \ar[d]^{c^*} &
{\operatorname{\mathbf{\underline{M}PST}}}\ar[r]^{{{\underline{b}}}^*} \ar[d]^{{{\underline{c}}}^*} &
{\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}} \ar[d]^{{{\underline{c}}}^{{{\operatorname{fin}}}*}}
\\
{\operatorname{\mathbf{MPS}}}\ar[r]^{\tau_{s!}} &
{\operatorname{\mathbf{\underline{M}PS}}}\ar[r]^{{{\underline{b}}}_s^*} &
{\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}.
}}$$ All vertical arrows are faithful and horizontal ones fully faithful.
\[lem:patching\] Both squares of are “$2$-Cartesian”. More precisely, the following assertions hold.
1. Let ${\operatorname{\mathbf{\underline{M}PS}}}\times_{{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}} {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ be the category of pairs $(F_s, F_t)$ consisting of $F_s \in {\operatorname{\mathbf{\underline{M}PS}}}$ and $F_t \in {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ such that their restriction to the common subcategory ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ are equal. The functor $${\operatorname{\mathbf{\underline{M}PST}}}\to {\operatorname{\mathbf{\underline{M}PS}}}\times_{{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}} {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}},$$ defined by $F \mapsto ({{\underline{c}}}^*F, {{\underline{b}}}^* F)$ is an equivalence of categories.
2. Let ${\operatorname{\mathbf{MPS}}}\times_{{\operatorname{\mathbf{\underline{M}PS}}}} {\operatorname{\mathbf{\underline{M}PST}}}$ be the category of triples $(F_s, F_t, {\varphi})$ consisting of $F_s \in {\operatorname{\mathbf{MPS}}}, ~F_t \in {\operatorname{\mathbf{\underline{M}PST}}}$ and an isomorphism ${\varphi}: \tau_{s!} F_s \cong {{\underline{c}}}^* F_t$ in ${\operatorname{\mathbf{\underline{M}PS}}}$. The functor $${\operatorname{\mathbf{MPST}}}\to {\operatorname{\mathbf{MPS}}}\times_{{\operatorname{\mathbf{\underline{M}PS}}}} {\operatorname{\mathbf{\underline{M}PST}}},$$ defined by $F \mapsto (c^*F, \tau_! F, \theta_F)$, where $\theta_F : \tau_{s!} c^*F \cong {{\underline{c}}}^* \tau_! F$ is from , is an equivalence of categories.
\(1) is the content of Corollary \[cor:fiber-cat\]. We show (2). Given $(F_s, F_t, {\varphi})$, we shall construct $F \in {\operatorname{\mathbf{MPST}}}$ as follows. Set $F(M):=F_s(cM)$ for any $M \in {\operatorname{\mathbf{MCor}}}$. Since $M$ is proper, we have an isomorphism $$F(M)=F_s(cM) = \tau_{s!}F_s(\tau_s cM)
\overset{{\varphi}_M}{\longrightarrow}
{{\underline{c}}}^*F_t({{\underline{c}}}\tau M) = F_t(\tau M),$$ which we denote by ${\widetilde}{{\varphi}}_M$. For $\gamma \in {\operatorname{\mathbf{MCor}}}(M, N)$, we define $F(\gamma):={\widetilde}{{\varphi}}_M^{-1} F_t(\gamma) {\widetilde}{{\varphi}}_N$. It is straightforward to see that $(F_s, F_t, {\varphi}) \mapsto F$ gives a quasi-inverse.
The functors $n_!$ and $n^*$
----------------------------
As in §\[s.presh\], the functor $(-)^{(n)}$ of Definition \[dn1\] induces a string of adjoint endofunctors $(n_!,n^*,n_*)$ of ${\operatorname{\mathbf{MPST}}}$, where $n^*$ is given by $n^*(F)(M)=F(M^{(n)})$. We shall not use $n_*$ in the sequel.
\[ln3\] The functor $n_!$ is fully faithful.
This follows formally from the same properties of $(-)^{(n)}$.
\[p4.1\] For any $F\in {\operatorname{\mathbf{MPST}}}$, there is a natural isomorphism $$\omega^*\omega_! F \simeq \infty^*F,$$ where $\infty^* F(M):= {\operatornamewithlimits{\varinjlim}}_n F(M^{(n)})$ (for the natural transformations ).
Let $M\in {\operatorname{\mathbf{MCor}}}$ and $X=\omega M$. Then $$\omega^*\omega_! F(M) = {\operatornamewithlimits{\varinjlim}}_{M'\in {\operatorname{\mathbf{MSm}}}(X)} F(M'),$$ and the claim follows from Lemma \[ln2\].
\[pn1\] For all $n\ge 1$, the natural transformation $\omega_!\to\omega_!n^*$ stemming from is an isomorphism.
Let $F\in {\operatorname{\mathbf{MPST}}}$. For $X\in {\operatorname{\mathbf{Cor}}}$, we have $$\omega_!n^*F(X)={\operatornamewithlimits{\varinjlim}}_{M\in {\operatorname{\mathbf{MSm}}}(X)} n^*F(M)
={\operatornamewithlimits{\varinjlim}}_{M\in {\operatorname{\mathbf{MSm}}}(X)} F(M^{(n)})={\operatornamewithlimits{\varinjlim}}_{M\in {\operatorname{\mathbf{MSm}}}(X)} F(M),$$ where the last isomorphism follows from Lemma \[ln2\].
Sheaves on [$\protect{\operatorname{\mathbf{\underline{M}Sm}}}^\protect{{\operatorname{fin}}}$ and $\protect{\operatorname{\mathbf{\underline{M}Cor}}}^\protect{{\operatorname{fin}}}$]{}
==========================================================================================================================================================================================
Nisnevich topology on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}^{\protect{{\operatorname{fin}}}}$
----------------------------------------------------------------------------------------------------------
\[def:groth-top-mp\] We call a morphism $p : U \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$ a Nisnevich cover if
${\overline}{p} : {\overline}{U} \to {\overline}{M}$ is a Nisnevich cover of ${\overline}{M}$ in the usual sense;
$p$ is minimal (that is, $U^\infty = {\overline}{p}^*(M^\infty)$).
Since the morphisms appearing in the Nisnevich covers are quarrable by Corollary \[exist-pullback\] (1), we obtain a Grothendieck topology on ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$. The category ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$ endowed with this topology will be called the big Nisnevich site of ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$ and denoted by ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}_{{\operatorname{Nis}}}$.
\[d3.3\] Let us fix $M \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$. Let $M_{{{\operatorname{Nis}}}}$ be the category of minimal morphisms $f : N \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$ such that ${\overline}{f}$ is étale, endowed with the topology induced by ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}_{{{\operatorname{Nis}}}}$.
The following lemma is obvious from the definitions:
\[lem:equiv-smallsites\] Let $M \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$. Let $({\overline}{M})_{{\operatorname{Nis}}}$ be the (usual) small Nisnevich site on ${\overline}{M}$. Then we have an isomorphism of sites $$M_{{\operatorname{Nis}}}\to ({\overline}{M})_{{\operatorname{Nis}}}, \qquad N \mapsto {\overline}{N},$$ whose inverse is given by $(p : X \to {\overline}{M}) \mapsto (X, p^*(M^\infty))$. (This isomorphism of sites depends on the choice of $M^\infty$.)
\[lem:refine-cover\] Let $\alpha : M \to N$ be a morphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ and let $p : U \to N$ be a Nisnevich over in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Then there is a commutative diagram $$\begin{CD}
V @>\alpha'>> U \\
@V{p'}VV @VV{p}V \\
M @>>\alpha> N,
\end{CD}$$ where $\alpha' : V \to U$ is a morphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ and $p' : V \to M$ is a Nisnevich cover in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$.
We may assume $\alpha$ is integral. Let ${\overline}{\alpha}$ be the closure of $\alpha$ in ${\overline}{M} \times {\overline}{N}$. Since ${\overline}{\alpha}$ is finite over ${\overline}{M}$, we may find a Nisnevich cover $p' : {\overline}{V} \to {\overline}{M}$ such that $\tilde{p}$ in the diagram (all squares being cartesian) $$\begin{CD}
{\overline}{V} \times_{{\overline}{M}} \times ({\overline}{\alpha} \times {\overline}{U})
@>>> {\overline}{\alpha} \times_{{\overline}{N}} {\overline}{U} @>>> {\overline}{U}
\\
@V{\tilde{p}}VV @VVV @VV{{\overline}{p}}V
\\
{\overline}{V} \times_{{\overline}{M}} \times {\overline}{\alpha}
@>>> {\overline}{\alpha} @>>> {\overline}{N}
\\
@VVV @VVV
\\
{\overline}{V}
@>>{p'}> {\overline}{M}
\end{CD}$$ has a splitting $s$. Put $V:=({\overline}{V}, {p'}^*(M^\infty)) \in {\operatorname{\mathbf{\underline{M}Sm}}}$. The image of $s$ gives us a desired correspondence $\alpha'$.
One can also define the Zariski and étale topologies on ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Most results of this section (notably Theorems \[thm:cech\], \[thm:sheaf-transfer\], and Corollary \[rem:faith-exact-c\]) remain true for the étale topology, but not for the Zariski topology (e.g. Lemma \[lem:refine-cover\] already fails for it).
However, from the next section onward we will make essential use of cd-structures. As the étale topology cannot be defined by a cd-structure, we decide to stick to the Nisnevich topology from the beginning.
A cd-structure on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}^{\protect{{\operatorname{fin}}}}$ {#s3.2}
------------------------------------------------------------------------------------------------------
Let ${{\operatorname{\mathbf{Sq}}}}$ be the product category of $[0]=\{ 0 \to 1 \}$ with itself, depicted as $$\xymatrix{
00 \ar[r] \ar[d] &01 \ar[d]\\
10 \ar[r] & 11.
}$$ For any category ${\mathcal{C}}$, denote by ${\mathcal{C}}^{{\operatorname{\mathbf{Sq}}}}$ for the category of functors from ${{\operatorname{\mathbf{Sq}}}}$ to ${\mathcal{C}}$. A functor $f : {\mathcal{C}}\to {\mathcal{C}}'$ induces a functor $f^{{\operatorname{\mathbf{Sq}}}}: {\mathcal{C}}^{{\operatorname{\mathbf{Sq}}}}\to {{\mathcal{C}}'}^{{\operatorname{\mathbf{Sq}}}}$.
We refer to §\[sa-cd\] for the notion of cd-structure, and its properties.
\[d3.2\]
1. A Cartesian square $$\label{eq.cd}
\begin{CD}
W@>v>> V\\
@VqVV @Vp VV\\
U@>u>> X
\end{CD}$$ in ${\operatorname{\mathbf{Sch}}}$ is called an *elementary Nisnevich square* if $u$ is an open embedding, $p$ is étale and $p^{-1}(X \setminus U)_{{\operatorname{red}}}\to (X \setminus U)_{{\operatorname{red}}}$ is an isomorphism. In this situation, we say $U \sqcup V \to X$ is an *elementary Nisnevich cover*. Recall that an additive presheaf is a Nisnevich sheaf if and only if it transforms any elementary Nisnevich square into a cartesian square [@cdstructures Cor. 2.17], [@unstableJPAA Thm. 2.2].
2. A diagram in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ is called an *${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square* if all morphisms are minimal and it becomes an elementary Nisnevich square (in ${\operatorname{\mathbf{Sch}}}$) after applying the forgetful functor of Definition \[d1.1\] (2).
\[lem;d3.2\] A ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square is cartesian in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$.
The last part of Proposition \[prop:fiber-prod\] (2) shows that no irreducible component of ${\overline}{X}$ has its image inside $|U^\infty|$ or $|V^\infty|$ (i.e. ${\overline}{W}_1$ in loc. cit. agrees with ${\overline}{W}$), and then Proposition \[prop:fiber-prod\] (1) shows that $X$ is the fiber product since $q^*U^\infty=v^*V^\infty=W^\infty$ by minimality.
\[p3.1\] The following assertions hold.
1. The topology on ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}}$ (cf. Def. \[def:groth-top-mp\]) coincides with the topology associated with the cd-structure $P_{{\operatorname{\underline{MV}^{\mathrm{fin}}}}}$ consisting of ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-squares.
2. The cd-structure $P_{{\operatorname{\underline{MV}}}^{{\operatorname{fin}}}}$ is strongly complete and strongly regular in the sense of Definition \[dA.5\], hence complete and regular in the sense of [@cdstructures] (cf. Definition \[d3.1\]).
\(1) follows from Lemma \[lem:equiv-smallsites\] and [@unstableJPAA remark after Prop. 2.17]. The first assertion of (2) follows from (the proof of) [@unstableJPAA Th. 2.2]. The second assertion of (2) follows from [@cdstructures Lemma 2.5, Lemma 2.11]
Sheaves on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}^{\protect{{\operatorname{fin}}}}$
-----------------------------------------------------------------------------------------------
We define ${\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$ to be the full subcategory of ${\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}$ consisting of Nisnevich sheaves.
\[t.cd\] Let $F\in{\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$. Then $H^i_{{\operatorname{Nis}}}(X,F)=0$ for any $X \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ and $i>\dim X(:=\dim X^{{\operatorname{o}}}=\dim {{\overline{X}}})$.
This is clear from Lemma \[lem:equiv-smallsites\] and the known properties of the Nisnevich site.
\[dA.2\] An additive functor $F$ between additive categories is *strongly additive* if it commutes with infinite direct sums.
(This property is not used in the present paper, but will be essential in its sequel to deal with unbounded derived categories.)
\[lcom1\] The category ${\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$ is closed under infinite direct sums and the inclusion functor ${{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}} : {\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}} \to {\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}$ is strongly additive.
Indeed, the sheaf condition is tested on finite diagrams, hence the presheaf given by a direct sum of sheaves is a sheaf (small filtered colimits commute with finite limits, [@mcl IX.2, Th. 1]).
\[prop:rep-sheaf\] For any $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$ we have $${{\underline{c}}}^{{{\operatorname{fin}}}*} {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M), \quad
{{\underline{c}}}^{{{\operatorname{fin}}}*} {{\underline{b}}}^* {\mathbb{Z}}_{{{\operatorname{tr}}}}(M)
\in {\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}},$$ where ${\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}},{\mathbb{Z}}_{{\operatorname{tr}}}$ are the representable presheaves (notation \[n2.1\]) and the functors ${{\underline{b}}}^*$ and ${{\underline{c}}}^{{{\operatorname{fin}}}*}$ are from Propositions \[eq:bruno-functor\], \[eq:c-functor\].
We show a stronger statement that ${\mathbb{Z}}_{{\operatorname{tr}}}(M)$ restricts to an étale sheaf on ${{\underline{N}}}_{{\operatorname{\acute{e}t}}}$ for any $N \in {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$. Let $p : {\overline}{U} \to {\overline}{N}$ be an étale cover and let $U:=({\overline}{U}, p^*N^\infty)$. We have a commutative diagram $$\xymatrix{
0 \ar[r] &
{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(N, M) \ar[r] \ar@{^{(}->}[d] &
{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(U, M) \ar[r] \ar@{^{(}->}[d] &
{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(U \times_N U, M) \ar@{^{(}->}[d]
\\
0 \ar[r] &
{\operatorname{\mathbf{\underline{M}Cor}}}(N, M) \ar[r] \ar@{^{(}->}[d] &
{\operatorname{\mathbf{\underline{M}Cor}}}(U, M) \ar[r] \ar@{^{(}->}[d] &
{\operatorname{\mathbf{\underline{M}Cor}}}(U \times_N U, M) \ar@{^{(}->}[d]
\\
0 \ar[r] &
{\operatorname{\mathbf{Cor}}}(N^{{\operatorname{o}}}, M^{{\operatorname{o}}}) \ar[r] &
{\operatorname{\mathbf{Cor}}}(U^{{\operatorname{o}}}, M^{{\operatorname{o}}}) \ar[r] &
{\operatorname{\mathbf{Cor}}}(U^{{\operatorname{o}}}\times_{N^{{\operatorname{o}}}} U^{{\operatorname{o}}}, M^{{\operatorname{o}}}).
}$$ The bottom row is exact by [@mvw Lemma 6.2]. The exactness of the top and middle row now follows from Lemma \[l1.3\].
Čech complex
------------
Let $p : U \to M$ be a Nisnevich cover in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$. We write $U \times_M U$ for the modulus pair corresponding to ${\overline}{U}\times_{{\overline}{M}}{\overline}{U}$ under the isomorphism of sites from Lemma \[lem:equiv-smallsites\]. Note that it is a fibre product in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ and in ${\operatorname{\mathbf{\underline{M}Sm}}}$, thanks to Proposition \[prop:fiber-prod\]. Iterating this construction, we obtain the Čech complex $$\label{eq:ceck}
\dots \to {{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U \times_M U)
\to {{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U)
\to {{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M)
\to 0.$$ in ${\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$.
\[thm:cech\] The complex is exact in ${\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$.
This result will be refined several times, see Corollary \[rem:faith-exact-c\] and Theorem \[thm:cech2\]. Its proof is adapted from [@voetri Prop. 3.1.3].
Before starting it, it is convenient to generalize the notion of relative cycles to the modulus setting.
\[d3.4\] Let $S=({\overline}{S},D)$, $Z=({\overline}{Z},Z^\infty)$ be two pairs formed of a scheme and an effective Cartier divisor, and let $f:{\overline}{Z}\to{\overline}{S}$ be a morphism. (We don’t put any regularity requirement on ${\overline}{S}-|D|$ or ${\overline}{Z}-|Z^\infty|$.) We write $L(Z/S)$ for the free abelian group with basis the closed integral subschemes $T\subset {\overline}{Z}$ such that $T$ is finite and surjective over an irreducible component of ${\overline}{S}$ and $D|_{T^N} \ge Z^\infty|_{T^N}$, where $T^N \to T$ is normalization and $(-)|_{T^N}$ denotes pull-back of Cartier divisors.
If $S$ is a modulus pair and $M=({\overline}{M},M^\infty)$ is another modulus pair, then we have a canonical isomorphism ${\operatorname{\mathbf{MCor}}}^{{\operatorname{fin}}}(S,M) \simeq L({\overline}{S}\times M/S)$, where ${\overline}{S}\times M$ is the modulus pair $({\overline}{S}\times {\overline}{M},{\overline}{S}\times M^\infty)$: this follows from Remark \[rk-graph-trick\] .
Define a category ${\mathcal{D}}(S)$ as follows: objects are pairs $(Z,f)$ as in Definition \[d3.4\]. A morphism $(Z,f)\to (Z',f')$ is a minimal morphism ${\varphi}:Z\to Z'$ such that $f=f'\circ {\overline}{{\varphi}}$. Composition is obvious.
\[l3.4\] The push-forward of cycles makes $(Z,f)\mapsto L(Z/S)$ a covariant functor on ${\mathcal{D}}(S)$.
Let ${\varphi}:(Z,f)\to (Z',f')$ be a morphism in ${\mathcal{D}}(S)$, and let $T\in L(Z/S)$. Then ${\varphi}(T)$ is still finite and surjective over a component of ${\overline}{S}$ [@mvw Lemma 1.4]. Moreover, it still verifies the modulus condition: this follows from the minimality of ${\varphi}$ and from Lemma \[lKL\]. We set as usual ${\varphi}_* T = [k(T):k({\varphi}(T))]{\varphi}(T)$: this defines ${\varphi}_*:L(Z/S)\to L(Z'/S)$, and the functoriality $(\psi\circ {\varphi})_*=\psi_*\circ {\varphi}_*$ is obvious.
In view of Lemma \[lem:equiv-smallsites\], it suffices to show the exactness of evaluated at $S$ $$\begin{gathered}
\label{eq:ceck2}
\dots
\to {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S, U \times_M U)
\to {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S, U)\\
\to {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S, M)
\to 0\end{gathered}$$ for the henselisation $S=({\overline}{S}, D)$ of any modulus pair $N=({\overline}{N},N^\infty)$ at any point of ${\overline}{N}$. As in [@voetri], the strategy is to write as a filtered colimit of contractible chain complexes.
Write ${\mathcal{E}}(S,M)$ for the collection of integral closed subsets of $S^{{\operatorname{o}}}\times M^{{\operatorname{o}}}$ which belong to ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S,M)$ (this is the canonical basis of ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S,M)$). Let ${\mathcal{C}}(M)$ be the set of closed subschemes of ${\overline}{S} \times {\overline}{M}$ that are quasi-finite over ${\overline}{S}$ and not contained in ${\overline}{S}\times M^\infty$, viewed as an (ordered, cofiltering) category. To $Z\in {\mathcal{C}}(M)$ we associate the subset ${\mathcal{E}}(Z)\subset {\mathcal{E}}(S,M)$ of those $F$ such that $F\subset Z$.
Provide $Z\in {\mathcal{C}}(M)$ with the minimal modulus structure induced by the projection $Z\to {\overline}{M}$ (in a sense slightly generalized from Remark \[r2.1\] (4), as in Definition \[d3.4\]: the open subset $Z-Z^\infty$ is not necessarily smooth). This yields a functor $${\mathcal{C}}(M)\to {\mathcal{D}}(S)$$ where ${\mathcal{D}}(S)$ is the category defined above. In particular, we have a subgroup $L(Z/S)\subset {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S,M)$: it is the free abelian group on ${\mathcal{E}}(Z)$.
Let $u:M'\to M$ be an étale morphism in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, as in Definition \[d3.3\]. For $Z\in {\mathcal{C}}(M)$, define $u^*Z = Z\times_{{\overline}{M}} {\overline}{M}' $. Then $u^*Z\in {\mathcal{C}}(M')$, and there is a commutative diagram $$\begin{CD}
L(u^*Z/S)@>>> L(Z/S)\\
@VVV @VVV\\
{\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S,M')@>u_*>> {\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}(S,M)
\end{CD}$$ where the bottom horizontal map is composition by \[the graph of\] $u$. This yields subcomplexes $$\label{eq:ceck3}
\dots \to
L(Z \times_{{\overline}{M}} ({\overline}{U} \times_{{\overline}{M}} {\overline}{U})) \to
L(Z \times_{{\overline}{M}} {\overline}{U}) \to
L(Z) \to 0$$ of , for $Z\in {\mathcal{C}}(M)$.
Let ${\mathcal{C}}_f(M)\subset {\mathcal{C}}(M)$ be the subset of those $Z$ which are finite over ${\overline}{S}$. It is a filtering subcategory, and we have $${\mathcal{E}}(S,M')= \bigcup_{Z\in {\mathcal{C}}_f(M)} {\mathcal{E}}(u^*Z).$$
Indeed, for $Z'\in {\mathcal{C}}(M')$, let $Z={\overline}{({{\operatorname{Id}}}_{{\overline}{S}} \times {\overline}{u})(Z')}$ and let $Z_f= \bigcup_{F\in {\mathcal{E}}(Z)} F$. Then ${\mathcal{E}}(Z')\subset {\mathcal{E}}(u^*Z_f)$ since $({{\operatorname{Id}}}_{{\overline}{S}} \times {\overline}{u})(F)$ is finite over ${\overline}{S}$ for $F\in {\mathcal{E}}(Z')$.
This proves that is obtained as the filtering inductive limit of the complexes when $Z$ ranges over ${\mathcal{C}}_f(M)$. It suffices to show the exactness of for such a $Z$.
Since $Z$ is finite over the henselian local scheme ${\overline}{S}$, $Z$ is a disjoint union of henselian local schemes. Thus the Nisnevich cover $Z \times_{{\overline}{M}} {\overline}{U} \to Z$ admits a section $s_0 : Z \to Z \times_{{\overline}{M}} {\overline}{U}$. Define for $k \geq 1$ $$s_k := s_0 \times_{{\overline}{M}} {{\operatorname{Id}}}_{{\overline}{U}^k} :
Z \times_{{\overline}{M}} {\overline}{U}^k \to
Z \times_{{\overline}{M}} {\overline}{U} \times_{{\overline}{M}} {\overline}{U}^k =
Z \times_{{\overline}{M}} {\overline}{U}^{k+1}$$ where ${\overline}{U}^{k} = {\overline}{U} \times_{{\overline}{M}} \dots \times_{{\overline}{M}} {\overline}{U}$. Then the maps $$L(Z \times_{{\overline}{M}} {\overline}{U}^k)\to
L(Z \times_{{\overline}{M}} {\overline}{U}^{k+1})$$ induced by $s_k$ via Lemma \[l3.4\] give us a homotopy from the identity to zero.
Sheafification preserves finite transfers
-----------------------------------------
Let ${{\underline{a}}}^{{{\operatorname{fin}}}}_{s {{\operatorname{Nis}}}} : {\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}\allowbreak \to {\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$ be the sheafification functor, that is, the left adjoint of the inclusion functor ${{\underline{i}}}^{{{\operatorname{fin}}}}_{s {{\operatorname{Nis}}}} : {\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}} \hookrightarrow {\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}$. It exists for general reasons and is exact [@SGA4 II.3.4].
\[def:mpst-fin\] Let ${\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}$ be the full subcategory of ${\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ consisting of all objects $F \in {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ such that ${{\underline{c}}}^{{{\operatorname{fin}}}*} F \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ (see Propositions \[eq:c-functor\] for ${{\underline{c}}}^{{{\operatorname{fin}}}*}$).
\[lcom2\] The category ${\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$ is closed under infinite direct sums in ${\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}}$, and the inclusion functor ${{\underline{i}}}_{{{\operatorname{Nis}}}}^{{{\operatorname{fin}}}} : {\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}} \to {\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}}$ is strongly additive. The objects ${\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}}(M)$ and $b^*{\mathbb{Z}}_{{\operatorname{tr}}}(M)$ belong to ${\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$ for any $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$.
This follows from Lemma \[lcom1\], because ${{\underline{c}}}^{{{\operatorname{fin}}}*}$ is strongly additive as a left adjoint. The last claim follows from Proposition \[prop:rep-sheaf\].
We write ${{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}} : {\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}NS}}}$ for the functor induced by ${{\underline{c}}}^{{{\operatorname{fin}}}*}$. By definition, we have $$\label{eq:c-i-fin}
{{\underline{c}}}^{{{\operatorname{fin}}}*} {{\underline{i}}}_{{{\operatorname{Nis}}}}^{{{\operatorname{fin}}}}
={{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}} {{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}.$$
\[thm:sheaf-transfer\] The following assertions hold.
1. Let $F \in {\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}}$. There exists a unique object $F_{{\operatorname{Nis}}}\in {\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}}$ such that ${{\underline{c}}}^{{{\operatorname{fin}}}*}(F_{{\operatorname{Nis}}})
={{\underline{a}}}^{{{\operatorname{fin}}}}_{s {{\operatorname{Nis}}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}(F))$ and such that the canonical morphism $u: {{\underline{c}}}^{{{\operatorname{fin}}}*}(F) \to {{\underline{a}}}^{{{\operatorname{fin}}}}_{s {{\operatorname{Nis}}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}(F))
={{\underline{c}}}^{{{\operatorname{fin}}}*}(F_{{\operatorname{Nis}}})$ extends to a morphism in ${\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}}$.
2. The functor ${{\underline{i}}}_{ {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}}$ has an exact left adjoint ${{\underline{a}}}_{ {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}} : {\operatorname{\mathbf{\underline{M}PST}}}^{{{\operatorname{fin}}}} \to {\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$ satisfying $$\label{eq:c-a-fin}
{{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}} {{\underline{a}}}_{ {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}}
= {{\underline{a}}}_{s {{\operatorname{Nis}}}}^{{{\operatorname{fin}}}} {{\underline{c}}}^{{{\operatorname{fin}}}*}.$$ In particular the category ${\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$ is Grothendieck (§\[s.groth\]).
3. The functor ${{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}$ has a left adjoint ${{\underline{c}}}^{{{\operatorname{fin}}}}_{{\operatorname{Nis}}}={{\underline{a}}}^{{\operatorname{fin}}}_{{{\operatorname{Nis}}}}{{\underline{c}}}^{{{\operatorname{fin}}}}_! {{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}$ Moreover, ${{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}$ is exact, strongly additive, and faithful.
This can be shown by a rather trivial modification of [@voetri Th. 3.1.4], but for the sake of completeness we include a proof. To ease the notation, put $F':={{{\underline{a}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}}{{\underline{c}}}^{{{\operatorname{fin}}}*}F \in {\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}$. First we construct a homomorphism $$\Phi_M:
F'(M) \to {\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M), F')$$ for any $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. Take $f \in F'(M)$. There exists a Nisnevich cover $p : U \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$ and $g \in {{\underline{c}}}^{{{\operatorname{fin}}}*}F(U)=F(U)$ such that $f|_U = u(g)$ in $F'(U)$. There also exists a Nisnevich cover $W \to U \times_M U$ such that $g|_W=0$ in $F(W)$. We have ${{\underline{a}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}{{\underline{c}}}^{{{\operatorname{fin}}}*} {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M)
={{\underline{c}}}^{{{\operatorname{fin}}}*} {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M)$ because ${{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M) \in {\operatorname{\mathbf{\underline{M}PS}}}^{{{\operatorname{fin}}}}_{{\operatorname{Nis}}}$ by Prop. \[prop:rep-sheaf\]. Thus we get a commutative diagram in which the horizontal maps are induced by ${{\underline{a}}}^{{{\operatorname{fin}}}}_{s {{\operatorname{Nis}}}}{{\underline{c}}}^{{{\operatorname{fin}}}*}$ $$\xymatrix{
0 \ar[d]
\\
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M), F')
\ar[d]_{s}
\\
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U), F')
\ar[d]
&
{\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}({\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U), F)
\ar[l]_{s'} \ar[d] \ar@/^17ex/[dd]^{s''}
\\
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U \times_M U), F')
\ar@{^{(}->}[d]_{l}
&
{\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}({\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U \times_M U), F)
\ar[l] \ar[d]
\\
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(W), F')
&
{\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}({\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(W), F).
\ar[l]
}$$ Since $F'$ is a sheaf, Theorem \[thm:cech\] implies that the left vertical column is exact except at the last spot, and that the map $l$ is injective. Since $g \in F(U)=
{\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}({\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U), F)$ satisfies $s''(g)=g|_W=0$, there exists a unique $h \in
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}(c^*{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M), F')$ such that $s(h)=s'(g)$. One checks that $h$ does not depend on the choices we made. We define $\Phi_M(f):=h$.
Now we define $G$. On objects we put $G(M) = F'(M)$ for $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. For $\alpha \in {\operatorname{\mathbf{\underline{M}Cor}}}^{{{\operatorname{fin}}}}(M, N)$, we define $\alpha^* : F'(N) \to F'(M)$ as the composition of $$\begin{aligned}
F'(N)
&\overset{\Phi_N}{\longrightarrow}
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(N), F')
\\
&\overset{}{\longrightarrow}
{\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}({{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M), F')
\to F'(M),\end{aligned}$$ where the middle map is induced by ${{\underline{c}}}^{{{\operatorname{fin}}}*}(\alpha) :
{{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}}(M) \to {{\underline{c}}}^{{{\operatorname{fin}}}*}{\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}}(N)$, and the last map is given by $f \mapsto f_M({{\operatorname{Id}}}_M)$. One checks that, with this definition, $G$ becomes an object of ${\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$.
To prove uniqueness, take $G, G' \in {\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ which enjoy the stated properties. We have $G(M)=G'(M)=F'(M)$ for any $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. We also have $G({{\underline{c}}}^{{{\operatorname{fin}}}*}(q))=G'({{\underline{c}}}^{{{\operatorname{fin}}}*}(q))=F'(q)$ for any morphism $q$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. Let $\alpha : M \to N$ be a morphism in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ and let $f \in F'(N)$. Take a Nisnevich cover $p : U \to N$ of ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ and $g \in {{\underline{c}}}^{{{\operatorname{fin}}}*}F(U)=F(U)$ such that $f|_U = u(g)$ in $F'(U)$. Apply Lemma \[lem:refine-cover\] to get a morphism $\alpha' : V \to U$ in ${\operatorname{\mathbf{\underline{M}Cor}}}^{{\operatorname{fin}}}$ and a ${{\operatorname{Nis}}}$-cover $p' : V \to M$ of ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ such that $\alpha p'=p \alpha'$. Then we have $$\begin{aligned}
G&(p')G(\alpha)(f) =
G(\alpha')G(p)(f) =
G(\alpha')(u(g)) =
u(F(\alpha')(g))
\\
&=
G'(\alpha')(u(g)) =
G'(\alpha')G'(p)(f) =
G'(p')G'(\alpha)(f) =
G(p')G'(\alpha)(f).\end{aligned}$$ Since $p': V \to M$ is a Nisnevich cover and $G$ is separated, this implies $G(\alpha)(f) =G'(\alpha)(f)$. This completes the proof or (1).
\(2) is a consequence of (1) and the fact that ${\operatorname{\mathbf{\underline{M}PST}}}^{{\operatorname{fin}}}$ is Grothendieck as a category of modules (see Theorem \[t.groth\] d)). Then (3) follows from Lemma \[lem:lr-adjoint\].
A different argument may be given by mimicking the proof of [@ayoubrig Cor. 2.2.26].
An additive functor ${\varphi}$ between abelian categories is *faithfully exact* if a complex $F'\to F\to F''$ is exact if and only if ${\varphi}F'\to {\varphi}F\to {\varphi}F''$ is.
This happens if ${\varphi}$ is exact and either faithful or conservative. By Theorems \[thm:sheaf-transfer\] and \[thm:cech\], we get:
\[rem:faith-exact-c\] The functor ${{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}$ is faithfully exact. In particular, if $p : U \to M$ is a Nisnevich cover in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{{\operatorname{fin}}}}$, then the Čech complex $$\label{eq:ceck-without-c}
\dots \to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U \times_M U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M)
\to 0$$ is exact in ${\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$.
Cohomology in $\protect{\operatorname{\mathbf{\underline{M}NST}}}^\protect{{\operatorname{fin}}}$
-------------------------------------------------------------------------------------------------
\[n3.7a\] Let $M \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ and let $F \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ (resp. $F \in {\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}$). We write $F_M$ for the sheaf on $({\overline}{M})_{{\operatorname{Nis}}}$ induced from $F$ (resp. ${{\underline{c}}}^{{{\operatorname{fin}}}\sigma} F$) via the isomorphism of sites from Lemma \[lem:equiv-smallsites\]. (Note that $F_M$ depends not only on ${\overline}{M}$, but also on $M^\infty$.) We thus have canonical isomorphisms $$\begin{aligned}
\label{eq3.62}
&H^i_{{\operatorname{Nis}}}(M, F)\simeq H^i_{{\operatorname{Nis}}}({\overline}{M}, F_M),
\\
&
\label{eq3.6}
H^i_{{\operatorname{Nis}}}(M, {{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}} F)\simeq H^i_{{\operatorname{Nis}}}({\overline}{M}, F_M),\end{aligned}$$ where the right hand sides denote the cohomology of the (usual) small site $({\overline}{M})_{{\operatorname{Nis}}}$.
1. Let $S$ be a scheme. We say a sheaf $F$ on $S_{{\operatorname{Nis}}}$ is *flasque* if $F(V) \to F(U)$ is surjective for any open dense immersion $U \to V$. Flasque sheaves are flabby in the sense of Definition \[def:flabby\] (see [@riou lemme 1.40]).
2. We say $F \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ flasque if $F_M$ is flasque for any $M \in {\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ (see Notation \[n3.7a\]). Again, flasque sheaves are flabby by .
\[lif1\] Let $I\in {\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}$ be an injective object. Then ${{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}(I)\allowbreak\in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ is flasque, and hence flabby.
Let $j:U{\hookrightarrow}M$ be a minimal open immersion of modulus pairs in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$. The morphism of sheaves ${\mathbb{Z}}_{{\operatorname{tr}}}^{{\operatorname{fin}}}(j)$ is a monomorphism, hence $j^*:I(M)\to I(U)$ is surjective. Alternatively, one can apply Lemma \[lem:inj-flabby\] with to show that ${{\underline{c}}}^{{{\operatorname{fin}}}\sigma}(I)$ is flabby. (This proof also works for étale topology.)
Sheaves on [$\protect{\operatorname{\mathbf{\underline{M}Sm}}}$ and $\protect{\operatorname{\mathbf{\underline{M}Cor}}}$]{}
============================================================================================================================
A cd-structure on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}$ {#sec:sd-str-ulMSm}
---------------------------------------------------------------------
Let $P_{{{\underline{{\operatorname{MV}}}}}}$ be the collection of commutative squares in ${\operatorname{\mathbf{\underline{M}Sm}}}$ which are isomorphic in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{\mathbf{Sq}}}}$ to ${{\underline{b}}}_s^{{\operatorname{\mathbf{Sq}}}}(Q)$ for some ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square $Q$ in Definition \[d3.2\]. Then $P_{{{\underline{{\operatorname{MV}}}}}}$ defines a cd-structure on ${\operatorname{\mathbf{\underline{M}Sm}}}$ (see §\[s3.2\]).
\[d4.1\] The squares which belong to $P_{{{\underline{{\operatorname{MV}}}}}}$ are called *${\operatorname{\underline{MV}}}$-squares*.
\[thm;cd-str-ulMSm\] The cd-structure $P_{{{\underline{{\operatorname{MV}}}}}}$ is strongly complete and strongly regular, in particular complete and regular (see Definitions \[d3.1\] and \[dA.5\]).
This follows from Propositions \[p3.1\] and \[pA.4\].
Sheaves on $\protect{\operatorname{\mathbf{\underline{M}Sm}}}$
--------------------------------------------------------------
\[def;sheavesulMSmul\] Consider the Grothendieck topology on ${\operatorname{\mathbf{\underline{M}Sm}}}$ generated by the squares in $P_{{{\underline{{\operatorname{MV}}}}}}$. The resulting site will be denoted by ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$. We write ${\operatorname{\mathbf{\underline{M}NS}}}$ for the full subcategory of sheaves in ${\operatorname{\mathbf{\underline{M}PS}}}$. We denote by ${{\underline{i}}}_{s {{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NS}}}\to {\operatorname{\mathbf{\underline{M}PS}}}$ the inclusion functor.
By the general properties of Grothendieck topologies [@SGA4 Exp. 2], we have:
\[thm:sheafification-ulMNS\] The inclusion functor ${{\underline{i}}}_{s {{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NS}}}\to {\operatorname{\mathbf{\underline{M}PS}}}$ has an exact left adjoint ${{\underline{a}}}_{s {{\operatorname{Nis}}}}$. The category ${\operatorname{\mathbf{\underline{M}NS}}}$ is Grothendieck (§\[s.groth\]).
\[lem:sheaves-MNS\] The following conditions are equivalent for $F \in {\operatorname{\mathbf{\underline{M}PS}}}$.
$F\in {\operatorname{\mathbf{\underline{M}NS}}}$.
${{\underline{b}}}_s^*F \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$; in other words, $({{\underline{b}}}_s^*F)_M$ is a Nisnevich sheaf for any $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$ (see for ${{\underline{b}}}_s$ and Notation \[n3.7a\] for $(-)_M$).
$F$ transforms any ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square $$\label{Q0} Q_0 : \vcenter{\xymatrix{
W_0 \ar[r]^{} \ar[d]_{} & V_0 \ar[d]^{} \\
U_0 \ar[r]^{} & M
}}$$ into an exact sequence $$0\ \to F(M) \to F(U_0) \times F(V_0) \to F(W_0).$$
In view of Theorem \[thm;cd-str-ulMSm\] and [@cdstructures Cor. 2.17], we have (i) $\iff$ (iii). On the other hand, (ii) $\iff$ (iii) by adjunction and Proposition \[p3.1\].
\[lcom3\] The category ${\operatorname{\mathbf{\underline{M}NS}}}$ is closed under infinite direct sums in ${\operatorname{\mathbf{\underline{M}PS}}}$ and ${{\underline{i}}}_{s {{\operatorname{Nis}}}}$ is strongly additive.
This follows from Lemmas \[lcom1\], \[lem:sheaves-MNS\] ((i) $\iff$ (ii)) and \[lem:lr-adjoint\] (2) because ${{\underline{b}}}_s^*$ is strongly additive as a left adjoint.
The adjunction $(\protect{{\underline{b}}}_{ s, \protect{{\operatorname{Nis}}}},\protect{{\underline{b}}}_s^\protect{{\operatorname{Nis}}})$
--------------------------------------------------------------------------------------------------------------------------------------------
A cover in ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$ is called a *strict Nisnevich cover* if it is the image of a cover of ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}_{{\operatorname{Nis}}}$ by ${{\underline{b}}}_s:{\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}Sm}}}$.
By definition, a strict Nisnevich cover is evidently a cover in ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$. Up to isomorphism, any cover of ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$ can be refined to such a cover. More precisely, we have the following lemma.
\[lem:cov-ulMSm\] Any cover $U \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$ admits a refinement of the form $V \to N \to M$, where $V \to N$ is a strict Nisnevich cover and $N \to M$ is a morphism in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ (see Definition \[deff\]).
By Definition \[def;sheavesulMSmul\] and Proposition \[peff1\], there is a refinement of $U \to M$ of the form $$U_n \overset{f_n}{\to} U_{n-1} \overset{f_{n-1}}{\to}
\cdots \overset{f_1}{\to} U_0=M,$$ where for each $i$ we have either (i) $f_i \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$, (ii) $f_i=g^{-1}$ for some $g \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$, or (iii) $f_i$ is a strict Nisnevich cover. We proceed by induction on $n$, the case $n=0$ being trivial. Suppose $n>0$. By induction, we have a refinement of $U_n \to U_1$ of the form $V' \to N' \to U_1$ where $V' \to N'$ is a strict Nisnevich cover and $N' \to U_1$ is in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$.
If $f_1 \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$, then we can take $V=V'$ and $N=N'$, as the composition $N' \to U_1 \to U_0$ belongs to ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$. Next, suppose $f_1=g^{-1}$ with $g \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}$. Then we can take $V=V' \times_{U_1} U_0$ and $N=N' \times_{U_1} U_0$, where $U_0$ is regarded as a $U_1$-scheme by $g$. Finally, suppose $f_1$ is a strict Nisnevich cover. By Lemma \[mainlem;blowup\], we may find a morphism $N \to U_0$ in ${{\underline{\Sigma}}}^{{\operatorname{fin}}}$ such that $N'':=N \times_{U_0} U_1 \to U_1$ factors through $N'$. Then we can take $V=V' \times_{N'} N''$. This completes the proof.
We define ${{\underline{b}}}_s^{{{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NS}}}\to {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ to be the restriction of ${{\underline{b}}}_s^*$, cf. Lemma \[lem:sheaves-MNS\] (ii). By definition, we have $$\label{eq:b-and-i}
{{\underline{b}}}_s^* {{\underline{i}}}_{s {{\operatorname{Nis}}}} = {{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}{{\underline{b}}}_s^{{\operatorname{Nis}}}.$$
\[lem;b!ulMNS\] The following assertions hold.
1. We have ${{\underline{b}}}_{s !}({\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}) \subset {\operatorname{\mathbf{\underline{M}NS}}}$. In particular, ${{\underline{b}}}_{s !}$ restricts to ${{\underline{b}}}_{s {{\operatorname{Nis}}}} : {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}NS}}}$ so that we have $$\label{eq:b-and-i2}
{{\underline{b}}}_{s !} {{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}= {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}}.$$
2. The functor ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$ is an exact left adjoint of ${{\underline{b}}}_s^{{\operatorname{Nis}}}$. The functor ${{\underline{b}}}_s^{{\operatorname{Nis}}}$ is fully faithful and preserves injectives. The counit map ${{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_s^{{\operatorname{Nis}}}\to {{\operatorname{Id}}}$ is an isomorphism and ${{\underline{b}}}_{s {{\operatorname{Nis}}}} R^q{{\underline{b}}}_s^{{\operatorname{Nis}}}=0$ for $q>0$.
Let $F \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ and take $M \in {\operatorname{\mathbf{\underline{M}Sm}}}$. We shall show that $({{\underline{b}}}_s^*{{\underline{b}}}_{s !}F)_M$ is a Nisnevich sheaf on ${\overline}{M}$. For a given ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$ $$\xymatrix{
W \ar[r]\ar[d] & V \ar[d]\\
U\ar[r] & M\\}$$ its pullback via $(N\to M) \in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M$ (which exists by Corollary \[exist-pullback\] (1)) $$\xymatrix{
W\times_M N \ar[r]\ar[d] & V\times_M N \ar[d]\\
U\times_M N\ar[r] & N\\}$$ is also an ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square. By Proposition \[p3.1\] (2) and by [@cdstructures Cor. 2.17], the sequence $$0\to F(N) \to F(U\times_M N) \oplus F(V\times_M N) \to F(W\times_M N)$$ is exact. By Lemma \[mainlem;blowup\], the pullback of ${{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M$ via $U \to M$ is cofinal in ${{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow U$, and similarly for $V\to M$ and $W\to M$. Hence, by taking its colimit over $N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M$, the above exact sequences and imply the desired exact sequence $$0\to b_{s !} F(M) \to b_{s !} F(U) \oplus b_{s !} F(V) \to b_{s !} F(W).$$ In view of Lemma \[lem:sheaves-MNS\], this finishes the proof of (1).
\(2) The adjunction $({{\underline{b}}}_{s {{\operatorname{Nis}}}} , {{\underline{b}}}_{s}^{{{\operatorname{Nis}}}})$ follows from the adjunction $({{\underline{b}}}_{s !} , {{\underline{b}}}_s^\ast)$ (see Proposition \[eq:bruno-functor\]), by the full faithfullness of ${{\underline{i}}}_{s {{\operatorname{Nis}}}}$ and ${{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}$, and by the formulas and . The full faithfulness of ${{\underline{b}}}_s^{{\operatorname{Nis}}}$ follows from that of ${{\underline{b}}}_s^\ast$ (see Proposition \[eq:bruno-functor\]), ${{\underline{i}}}_{s {{\operatorname{Nis}}}}$ and ${{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}$. Then the counit map ${{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_s^{{\operatorname{Nis}}}\to {{\operatorname{Id}}}$ is an isomorphism by Lemma \[lA.6\].
We prove the exactness of ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$ as follows. Since it is right exact as a left adjoint, it suffices to show its left exactness.
Assume given an exact sequence in ${\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$: $$0\to F\to G\to H \to 0.$$ Applying the left exact functor ${{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}:{\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}$ and the exact functor ${{\underline{b}}}_{s !}: {\operatorname{\mathbf{\underline{M}PS}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}PS}}}$ and using , we get an exact sequence $$0\to {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} F\to {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} G\to {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} H.$$ For every $Q\in {\operatorname{\mathbf{\underline{M}NS}}}$, this gives rise to an exact sequence $$\begin{aligned}
0&\to {\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}PS}}}}({{\underline{i}}}_{s {{\operatorname{Nis}}}} Q, {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} F)
\to {\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}PS}}}}({{\underline{i}}}_{s {{\operatorname{Nis}}}} Q, {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} G)
\\
&\to
{\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}PS}}}}({{\underline{i}}}_{s {{\operatorname{Nis}}}} Q, {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_{s {{\operatorname{Nis}}}} H ).\end{aligned}$$ Since ${{\underline{i}}}_{s {{\operatorname{Nis}}}}$ is fully faithful, this gives an exact sequence $$0\to {\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}NS}}}}(Q, {{\underline{b}}}_{s {{\operatorname{Nis}}}} F)
\to {\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}NS}}}}(Q, {{\underline{b}}}_{s {{\operatorname{Nis}}}} G)\to
{\operatorname{Hom}}_{{\operatorname{\mathbf{\underline{M}NS}}}}(Q, {{\underline{b}}}_{s {{\operatorname{Nis}}}} H),$$ which shows the exactness of $$0\to {{\underline{b}}}_{s {{\operatorname{Nis}}}} F \to {{\underline{b}}}_{s {{\operatorname{Nis}}}} G \to {{\underline{b}}}_{s {{\operatorname{Nis}}}} H,$$ as desired. Therefore, ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$ is exact.
Then ${{\underline{b}}}_s^{{{\operatorname{Nis}}}}$ preserves injectives since it has an exact left adjoint ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$. Moreover, applying $R^q$ ($q>0$) to the counit isomorphism ${{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_s^{{\operatorname{Nis}}}\to {{\operatorname{Id}}}$, we have $${{\underline{b}}}_{s {{\operatorname{Nis}}}} R^q {{\underline{b}}}_s^{{\operatorname{Nis}}}\simeq R^q ({{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{b}}}_s^{{\operatorname{Nis}}}) \simeq R^q {{\operatorname{Id}}}\simeq 0,$$ by Ex. \[exA.3\] and the exactness of ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$. This concludes the proof.
We have a natural isomorphism ${{\underline{a}}}_{s {{\operatorname{Nis}}}}\simeq {{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{a}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}{{\underline{b}}}_s^*$.
By the uniqueness of left adjoints, it suffices to check that the right hand side is also left adjoint to ${{\underline{i}}}_{s {{\operatorname{Nis}}}}$. We first apply double adjunction by $({{\underline{b}}}_{s {{\operatorname{Nis}}}}, {{\underline{b}}}_s^{{\operatorname{Nis}}})$ (Prop. \[lem;b!ulMNS\] (2)) and $({{\underline{a}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}}, {{\underline{i}}}_{s {{\operatorname{Nis}}}}^{{\operatorname{fin}}})$, then use and the full faithfulness of ${{\underline{b}}}_s^*$ (Prop. \[eq:bruno-functor\]).
Cohomology in $\protect{\operatorname{\mathbf{\underline{M}NS}}}$
-----------------------------------------------------------------
\[n:pre\]
1. Let $M\in {\operatorname{\mathbf{\underline{M}Sm}}}$ and $F\in {\operatorname{\mathbf{\underline{M}NS}}}$. Using Notation \[n3.7a\], we define $F_M :=({{\underline{b}}}_s^{{\operatorname{Nis}}}F)_M$ which is a sheaf on $({\overline}{M})_{{\operatorname{Nis}}}$.
2. For $M\in {\operatorname{\mathbf{\underline{M}Sm}}}$, let $\Zp(M)\in {\operatorname{\mathbf{\underline{M}PS}}}$ be the associated representable additive presheaf (see ) and let $$\label{Z}
{\mathbb{Z}}(M) ={{\underline{a}}}_{s {{\operatorname{Nis}}}} \Zp(M) \in {\operatorname{\mathbf{\underline{M}NS}}}$$ be the associated sheaf.
\[lem;cohMsigmaS\] For $M\in {\operatorname{\mathbf{\underline{M}Sm}}}$, $F\in {\operatorname{\mathbf{\underline{M}NS}}}$ and $i\ge 0$, we have a natural isomorphism $$\begin{gathered}
\label{eq:coh-ulMNS}
{\operatorname{Ext}}_{{\operatorname{\mathbf{\underline{M}NS}}}}^i({\mathbb{Z}}(M), F)
\simeq
{\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} H_{{\operatorname{Nis}}}^i({\overline}{N},F_N)\\
:={\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} H_{{\operatorname{Nis}}}^i({\overline}{N},({{\underline{b}}}_s^{{\operatorname{Nis}}}F)_N).\end{gathered}$$ Moreover, we have $$\label{eq:coh-ulMNS1}
{\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} H_{{\operatorname{Nis}}}^i({\overline}{N},(R^q{{\underline{b}}}_s^{{\operatorname{Nis}}}F)_N)=0\text{ for all $q>0$.}$$
Define functors $\Gamma_M^\downarrow : {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{Ab}}}$ and ${{\underline{\Gamma}}}_M : {\operatorname{\mathbf{\underline{M}NS}}}\to {\operatorname{\mathbf{Ab}}}$ by $$\Gamma_M^\downarrow(G)
={\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M} G(N),
\quad
{{\underline{\Gamma}}}_M(F)=F(M).$$ We have $\Gamma_M^\downarrow={{\underline{\Gamma}}}_M {{\underline{b}}}_{s {{\operatorname{Nis}}}}$. By Theorem \[tA.2\] and Lemma \[lem:claim\] below, we get $(R^p {{\underline{\Gamma}}}_M){{\underline{b}}}_{s {{\operatorname{Nis}}}}=R^p \Gamma_M^\downarrow$ for any $p \ge 0$ since ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$ is exact. Thus, by Lemma \[lem:claim2\] below we obtain $${\operatorname{Ext}}_{{\operatorname{\mathbf{\underline{M}NS}}}}^p({\mathbb{Z}}(M), {{\underline{b}}}_{s {{\operatorname{Nis}}}} G)
\cong {\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M}
H^p_{{\operatorname{Nis}}}({\overline}{N}, G_N)$$ for any $G \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ and $p \ge 0$. Setting $G=R^q{{\underline{b}}}_s^{{\operatorname{Nis}}}F$, we get for $q=0$ (resp. for $q>0$) thanks to Proposition \[lem;b!ulMNS\] (2).
\[lem:claim\] For an injective $I\in \ulMNSfin$, ${{\underline{b}}}_{s {{\operatorname{Nis}}}} I\in {\operatorname{\mathbf{\underline{M}NS}}}$ is flabby (see Definition \[def:flabby\]).
\#1[\^[\#1]{}]{}
Write $F=\ulb_{{\operatorname{Nis}}}I$. By Lemma \[lem:milne\], it suffices to show the vanishing of the canonical map $\check{H}^q(U/M, F) \to \check{H}^q(M, F)$ for any cover $U \to M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$ and any $q>0$. By Lemma \[lem:cov-ulMSm\], we may assume $U \to M$ is a strict Nisnevich cover (as any morphism in ${{{\underline{\Sigma}}}}^{{\operatorname{fin}}}$ is an isomorphism in ${\operatorname{\mathbf{\underline{M}Sm}}}$). Denote by $U^n_M \in {\operatorname{\mathbf{\underline{M}Sm}}}$ the $n$-fold fiber product of $U$ over $M$ in ${\operatorname{\mathbf{\underline{M}Sm}}}$ (which exists by Corollary \[exist-pullback\] (1)). Then $\check{H}^q(U/M, F)$ is computed as the cohomology of the complex whose term in degree $q$ is given by $${\operatornamewithlimits{\varinjlim}}_{L_q\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow U_M^{q+1}} I(L_q).$$ By Lemma \[mainlem;blowup\], for any integer $n>0$ and given $L_q\in {{{\underline{\Sigma}}}}^{{{\operatorname{fin}}}}\downarrow U_M^{q+1}$ for $0\leq q\leq n$, there exists $L\in {{{\underline{\Sigma}}}}^{{{\operatorname{fin}}}}\downarrow M$ in such that $L\times_M U_M^{q+1} \to U_M^{q+1}$ factor through $L_q$ for all $q=0, \dots, n$. This implies that for $0\leq q\leq n-1$ the canonical map $\chH q(U/M, {{\underline{b}}}_{s {{\operatorname{Nis}}}} I) \to \chH q(M, {{\underline{b}}}_{s {{\operatorname{Nis}}}} I)$ factors through $${\operatornamewithlimits{\varinjlim}}_{L\in {{{\underline{\Sigma}}}}^{{{\operatorname{fin}}}}\downarrow M} \chH q (U\times_M L/L, I),$$ where $\chH q (U\times_M L/L, I)$ is the Čech cohomology of $I$ with respect to the cover $U\times_M L \to L$ in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$, but it vanishes since $I$ is injective in ${\operatorname{\mathbf{\underline{M}NS}}}^{{{\operatorname{fin}}}}$. This proves the desired vanishing and completes the proof of Lemma \[lem:claim\].
\[lem:claim2\] For any $G \in {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ and $p \ge 0$, we have $$R^p\Gamma_M^\downarrow G \cong
{\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M}
H^p_{{\operatorname{Nis}}}({\overline}{N}, G_N).$$
Take an injective resolution $G \to I^\bullet$ in $\ulMNSfin$. Then we have $$\begin{aligned}
R^p\Gamma_M^\downarrow G
&= H^p(\Gamma_M^\downarrow I^\bullet)
= H^p({\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M} I^\bullet)
\\
&\cong {\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M}H^p(I^\bullet(N))
\cong {\operatornamewithlimits{\varinjlim}}_{N \in {{\underline{\Sigma}}}^{{\operatorname{fin}}}\downarrow M}H^p_{{\operatorname{Nis}}}({\overline}{N}, G_N),\end{aligned}$$ where we used Corollary \[cor:sigma-fin-cofil\] for the last-but-one isomorphism, and for the last one.
Sheaves on $\protect{\operatorname{\mathbf{\underline{M}Cor}}}$
---------------------------------------------------------------
\[lem:mnst-condition\] For $F \in {\operatorname{\mathbf{\underline{M}PST}}}$, one has ${{\underline{c}}}^*F \in {\operatorname{\mathbf{\underline{M}NS}}}$ if and only if $b^*F \in {\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}$.
This follows from and Definitions \[def:mpst-fin\] and \[def;sheavesulMSmul\].
\[def;sheavesulMCor\] We define ${\operatorname{\mathbf{\underline{M}NST}}}$ to be the full subcategory of ${\operatorname{\mathbf{\underline{M}PST}}}$ consisting of those $F$ enjoying the conditions of Lemma \[lem:mnst-condition\]. We denote by ${{\underline{i}}}_{{{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}PST}}}$ the inclusion functor.
\[lcom3-2\] The category ${\operatorname{\mathbf{\underline{M}NST}}}$ is closed under infinite direct sums in ${\operatorname{\mathbf{\underline{M}PST}}}$, and ${{\underline{i}}}_{{\operatorname{Nis}}}$ is strongly additive. It contains ${\mathbb{Z}}_{{\operatorname{tr}}}(M)$ for any $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$.
This follows from Lemma \[lcom2\], because ${{\underline{b}}}^*$ is strongly additive as a left adjoint. The last statement follows from Lemma \[lcom2\].
By Definition \[def;sheavesulMCor\] and Lemma \[lem:mnst-condition\], the functors ${{\underline{b}}}^*$ and ${{\underline{c}}}^*$ restricts to ${{\underline{b}}}^{{{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}$ and ${{\underline{c}}}^{{\operatorname{Nis}}}: {\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}NS}}}$. It holds that $$\begin{aligned}
\label{eq:b-and-i4}
&{{\underline{b}}}^* {{\underline{i}}}_{{\operatorname{Nis}}}= {{\underline{i}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}}{{\underline{b}}}^{{\operatorname{Nis}}},
\quad
{{\underline{c}}}^* {{\underline{i}}}_{{\operatorname{Nis}}}= {{\underline{i}}}_{s {{\operatorname{Nis}}}} {{\underline{c}}}^{{\operatorname{Nis}}},
\\
\notag
&{{\underline{b}}}_s^{{\operatorname{Nis}}}{{\underline{c}}}^{{{\operatorname{Nis}}}}
= {{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}} {{\underline{b}}}^{{\operatorname{Nis}}},
\quad
{{\underline{b}}}_{s {{\operatorname{Nis}}}} {{\underline{c}}}^{{{\operatorname{fin}}}{{\operatorname{Nis}}}}
={{\underline{c}}}^{{\operatorname{Nis}}}{{\underline{b}}}_{{\operatorname{Nis}}},\end{aligned}$$ where for the last two formulas we used .
\[lem;b!ulMNST\] The following assertions hold.
1. We have ${{\underline{b}}}_{!}({\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}) \subset {\operatorname{\mathbf{\underline{M}NST}}}$.
2. Let ${{\underline{b}}}_{{{\operatorname{Nis}}}} : {\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}\to {\operatorname{\mathbf{\underline{M}NST}}}$ be the restriction of ${{\underline{b}}}_{!}$ so that we have $$\label{eq:b-and-i3}
{{\underline{b}}}_{!} {{\underline{i}}}_{{{\operatorname{Nis}}}}^{{\operatorname{fin}}}= {{\underline{i}}}_{{{\operatorname{Nis}}}} {{\underline{b}}}_{{{\operatorname{Nis}}}}.$$ Then, the functor ${{\underline{b}}}_{{{\operatorname{Nis}}}}$ is an exact left adjoint of ${{\underline{b}}}^{{\operatorname{Nis}}}$, which is fully faithful.
3. The functor ${{\underline{b}}}^{{\operatorname{Nis}}}$ preserves injectives.
\(1) It suffices to show that ${{\underline{c}}}^*{{\underline{b}}}_{!}({\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}) \subset {\operatorname{\mathbf{\underline{M}NS}}}$. By , we have ${{\underline{c}}}^*{{\underline{b}}}_{!}={{\underline{b}}}_{s!}{{\underline{c}}}^{{{\operatorname{fin}}}*}$. Moreover, ${{\underline{c}}}^{{{\operatorname{fin}}}*}{\operatorname{\mathbf{\underline{M}NST}}}^{{\operatorname{fin}}}\subset {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}$ by Definition \[def:mpst-fin\] and ${{\underline{b}}}_{s!} {\operatorname{\mathbf{\underline{M}NS}}}^{{\operatorname{fin}}}\subset {\operatorname{\mathbf{\underline{M}NS}}}$ by Proposition \[lem;b!ulMNS\] (1). In (2), the adjointness and the full faithfulness are seen by using Proposition \[eq:bruno-functor\], and . This proves that ${{\underline{b}}}_{{\operatorname{Nis}}}$ is right exact, and it is also exact by and Proposition \[eq:bruno-functor\] (see also the proof of the exactness of ${{\underline{b}}}_{s {{\operatorname{Nis}}}}$ in Proposition \[lem;b!ulMNS\] (2)). (3) is a consequence of (2).
\[thm:sheafification-ulMNST\] The inclusion functor ${{\underline{i}}}_{{{\operatorname{Nis}}}}: {\operatorname{\mathbf{\underline{M}NST}}}\to {\operatorname{\mathbf{\underline{M}PST}}}$ has an exact left adjoint ${{\underline{a}}}_{{\operatorname{Nis}}}$ given by ${{\underline{a}}}_{{\operatorname{Nis}}}={{\underline{b}}}_{{\operatorname{Nis}}}{{\underline{a}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}}{{\underline{b}}}^*$. In particular, ${\operatorname{\mathbf{\underline{M}NST}}}$ is Grothendieck.
The formula defining ${{\underline{a}}}_{{\operatorname{Nis}}}$ yields a left adjoint to ${{\underline{i}}}_{{\operatorname{Nis}}}$ by the full faithfulness of ${{\underline{b}}}^*$ (Proposition \[eq:bruno-functor\]) and the adjunctions $({{\underline{a}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}},{{\underline{i}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}})$ and $({{\underline{b}}}_{{\operatorname{Nis}}},{{\underline{b}}}^{{\operatorname{Nis}}})$ (use ). Its exactness follows from the exactness of the three functors.
\[prop:c-Nis\] We have $$\label{eq:a-c2}
{{\underline{b}}}_{{\operatorname{Nis}}}{{\underline{a}}}_{{\operatorname{Nis}}}^{{\operatorname{fin}}}= {{\underline{a}}}_{{\operatorname{Nis}}}{{\underline{b}}}_!,
\qquad
{{\underline{a}}}_{s {{\operatorname{Nis}}}} {{\underline{c}}}^* = {{\underline{c}}}^{{\operatorname{Nis}}}{{\underline{a}}}_{{\operatorname{Nis}}},$$ Moreover, ${{\underline{c}}}^{{{\operatorname{Nis}}}}$ is faithful, exact, strongly additive and has a left adjoint ${{\underline{c}}}_{{\operatorname{Nis}}}={{\underline{a}}}_{{{\operatorname{Nis}}}}{{\underline{c}}}_! {{\underline{i}}}_{s {{\operatorname{Nis}}}}$ such that ${{\underline{c}}}_{{\operatorname{Nis}}}{{\underline{a}}}_{s {{\operatorname{Nis}}}} = {{\underline{a}}}_{{\operatorname{Nis}}}{{\underline{c}}}_!$.
The first equality follows from the first formula of by adjunction. For the second, we use Theorems \[thm:sheafification-ulMNS\] and \[thm:sheafification-ulMNST\], together with , and . The last statement follows from Lemma \[lem:lr-adjoint\] (3).
\[thm:cech2\] If $p : U \to M$ is a cover in ${\operatorname{\mathbf{\underline{M}Sm}}}_{{\operatorname{Nis}}}$, then the Čech complex $$\label{eq:ceck3-2}
\dots \to {\mathbb{Z}}_{{{\operatorname{tr}}}}(U \times_M U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}(U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}(M)
\to 0$$ is exact in ${\operatorname{\mathbf{\underline{M}NST}}}$. (Note that the fiber products exist in ${\operatorname{\mathbf{\underline{M}Sm}}}$ by Corollaty \[exist-pullback\] (1).) Moreover, the sequence $$0\to {\mathbb{Z}}_{{\operatorname{tr}}}(W)\to {\mathbb{Z}}_{{\operatorname{tr}}}(U)\oplus {\mathbb{Z}}_{{\operatorname{tr}}}(V)\to {\mathbb{Z}}_{{\operatorname{tr}}}(X)\to 0$$ is exact in ${\operatorname{\mathbf{\underline{M}NST}}}$ for any ${\operatorname{\underline{MV}^{\mathrm{fin}}}}$-square in ${\operatorname{\mathbf{\underline{M}Sm}}}^{{\operatorname{fin}}}$.
By Lemma \[lem:cov-ulMSm\], we may assume $M \to U$ is a strict Nisnevich cover. Then, by the complex $$\dots \to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U \times_M U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(U)
\to {\mathbb{Z}}_{{{\operatorname{tr}}}}^{{\operatorname{fin}}}(M)
\to 0$$ is exact in ${\operatorname{\mathbf{\underline{M}NST}}}^{{{\operatorname{fin}}}}$. Applying the exact functor ${{\underline{b}}}_{{\operatorname{Nis}}}$, we get . The second statement follows from the first and a small computation (cf. [@mvw Prop. 6.14]).
Cohomology in $\protect{\operatorname{\mathbf{\underline{M}NST}}}$
------------------------------------------------------------------
\[lif1v\] Let $I\in {\operatorname{\mathbf{\underline{M}NST}}}$ be an injective object. Then ${{\underline{c}}}^{{{\operatorname{Nis}}}}(I)\in {\operatorname{\mathbf{\underline{M}NS}}}$ is flabby.
This follows from Lemma \[lem:inj-flabby\] and Theorem \[thm:cech2\].
\[not4\] Let $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$ and $F \in {\operatorname{\mathbf{\underline{M}NST}}}$. Using Notation \[n3.7a\], we define $F_M :=({{\underline{b}}}^{{\operatorname{Nis}}}F)_M$, which is a sheaf on $({\overline}{M})_{{\operatorname{Nis}}}$.
\[c3.1v\] Let $F\in {\operatorname{\mathbf{\underline{M}NST}}}$, and let $M\in {\operatorname{\mathbf{\underline{M}Cor}}}$. Then there are canonical isomorphisms for any $i\ge 0$: $${\operatorname{Ext}}^i_{{\operatorname{\mathbf{\underline{M}NST}}}}({\mathbb{Z}}_{{\operatorname{tr}}}(M),F)\simeq
{\operatorname{Ext}}^i_{{\operatorname{\mathbf{\underline{M}NS}}}}({\mathbb{Z}}(M), {{\underline{c}}}^{{\operatorname{Nis}}}F) \simeq
{\operatornamewithlimits{\varinjlim}}_{N\in {{{\underline{\Sigma}}}}^{{{\operatorname{fin}}}}\downarrow M}H^i_{{\operatorname{Nis}}}({{\overline{N}}},F_N).$$ (See for ${\mathbb{Z}}(M)$.) Moreover, we have $${\operatornamewithlimits{\varinjlim}}_{N\in {{\underline{\Sigma}}}^{{{\operatorname{fin}}}}\downarrow M} H_{{\operatorname{Nis}}}^i({\overline}{N},(R^q({{\underline{b}}}_s^{{\operatorname{Nis}}}){{\underline{c}}}^{{\operatorname{Nis}}}F)_N)=0\text{ for all $q>0$.}$$
Applying the last identity of Proposition \[prop:c-Nis\] to ${\mathbb{Z}}^p(M)$, we get $${{\underline{c}}}_{{\operatorname{Nis}}}{\mathbb{Z}}(M) = {{\underline{a}}}_{{\operatorname{Nis}}}{{\underline{c}}}_! {\mathbb{Z}}^p(M)
= {{\underline{a}}}_{{\operatorname{Nis}}}{\mathbb{Z}}_{{\operatorname{tr}}}(M)
= {\mathbb{Z}}_{{\operatorname{tr}}}(M)$$ where the second equality follows from , and the third one holds by Lemma \[lcom3-2\]. This yields an isomorphism $${\operatorname{\mathbf{\underline{M}NS}}}({\mathbb{Z}}(M), {{\underline{c}}}^{{\operatorname{Nis}}}F)\\
\simeq {\operatorname{\mathbf{\underline{M}NST}}}({\mathbb{Z}}_{{\operatorname{tr}}}(M), F)$$ which is the case $i=0$ of the first isomorphism in the proposition. The general case $i\ge 0$ then follows from Theorem \[tA.2\], Lemma \[lif1v\] and the exactness of ${{\underline{c}}}^{{\operatorname{Nis}}}$ (Proposition \[prop:c-Nis\]), and the second isomorphism follows from Proposition \[lem;cohMsigmaS\] and . The last assertion follows from .
Categorical toolbox, I {#sect:app}
======================
This appendix gathers known and less-known results that we use constantly.
Pro-objects ([@SGA4 I.8], [@am App. 2]) {#sec:pro-obj}
---------------------------------------
Recall that a *pro-object* of a category ${\mathcal{C}}$ is a functor $F:A\to {\mathcal{C}}$, where $A$ is a small cofiltered category (dual of [@mcl IX.1]). They are denoted by $(X_\alpha)_{\alpha\in A}$ or by $``{\operatornamewithlimits{\varprojlim}}_{\alpha\in A}" X_\alpha$ (Deligne’s notation), with $X_\alpha=F(\alpha)$. Pro-objects of ${\mathcal{C}}$ form a category ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$, with morphisms given by the formula $${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}((X_\alpha)_{\alpha\in A},(Y_\beta)_{\beta\in B})={\operatornamewithlimits{\varprojlim}}_{\beta\in B} {\operatornamewithlimits{\varinjlim}}_{\alpha\in A} {\mathcal{C}}(X_\alpha,Y_\beta).$$
There is a canonical full embedding $c:{\mathcal{C}}{\hookrightarrow}{\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$, sending an object to the corresponding constant pro-object ($A=\{*\}$).
For the next lemma, we recall a special case of comma categories from Mac Lane [@mcl II.6]. If $F:{\mathcal{A}}\to {\mathcal{B}}$ is a functor and $b\in{\mathcal{B}}$, we write $b\downarrow F$ for the category whose objects are pairs $(a,f)\in {\mathcal{A}}\times {\mathcal{B}}(b,F(a))$; a morphism $(a_1,f_1)\to (a_2,f_2)$ is a morphism $g\in {\mathcal{A}}(a_1,a_2)$ such that $f_2=F(g)f_1$. The category $F\downarrow b$ is defined dually (objects: systems $F(a){\xrightarrow}{f} b$, etc.) According to [@mcl IX.3], $F$ is *final* if, for any $b\in {\mathcal{B}}$, the category $F\downarrow b$ is nonempty and connected; here we shall use the dual property *cofinal* (same conditions for $b\downarrow F$). As usual, we abbreviate $Id_{\mathcal{A}}\downarrow a$ and $a\downarrow Id_{\mathcal{A}}$ by ${\mathcal{A}}\downarrow a$ and $a\downarrow {\mathcal{A}}$.
Let $F:A\to {\mathcal{C}}$ be a pro-object. For each $\alpha\in A$, we have a “projection” morphism $\pi_\alpha:F\to c(X_\alpha)$ in ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$. This yields an isomorphism in ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$ $$F{{\xrightarrow}{\sim}}{\operatornamewithlimits{\varprojlim}}_{\alpha\in A} c(X_\alpha)$$ (explaining Deligne’s notation) and a functor $$\theta:A\to F\downarrow c.$$
\[l.cof\] The functor $\theta$ is cofinal.
This is a tautology: let $F{\xrightarrow}{f} c(X)$ be an object of $F\downarrow c$. An object of $\theta\downarrow (F{\xrightarrow}{f} c(X))$ is a pair $(\alpha,{\varphi})$, with $\alpha\in A$ and ${\varphi}:F(\alpha)\to X$ such that $f=c({\varphi})\pi_\alpha$. The definition of morphisms in ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$ shows that this category is nonempty. Since $A$ is cofiltering, it suffices by the dual of [@ks Prop. 3.2.2 (iii)] to show that for any pair $(t_1,t_2)$ of morphisms $(\alpha_1,{\varphi}_1)\to (\alpha_2,{\varphi}_2)$, there exists $t:\alpha\to \alpha_1$ such that $t_1t =t_2t$ : this condition is verified thanks to the definition of ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}(F,X)$.
(Warning: the use of co in (co)final and (co)filtered is opposite in [@mcl] and in [@ks]. We use the convention of [@mcl].)
Pro-adjoints [@SGA4 I.8.11.5] {#s1.1}
-----------------------------
Let $u:{\mathcal{C}}\to {\mathcal{D}}$ be a functor: it induces a functor ${\text{\rm pro}_{}\text{\rm--}}u:{\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}\to {\text{\rm pro}_{}\text{\rm--}}{\mathcal{D}}$.
Recall standard terminology for the functoriality of limits (=inverse limits) and colimits (= direct limits):
A functor $u:{\mathcal{C}}\to {\mathcal{D}}$ is *left exact* (resp. *right exact*, resp. *exact*) if it commutes with finite limits (resp. finite colimits, resp. finite limits and colimits).
\[p.proadj\] Consider the following conditions:
The functor ${\text{\rm pro}_{}\text{\rm--}}u$ has a left adjoint.
There exists a functor $v:{\mathcal{D}}\to {\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$ and an isomorphism $${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}(v(d),c)\simeq {\mathcal{D}}(d,u(c))$$ contravariant in $d\in {\mathcal{D}}$ and covariant in $c\in {\mathcal{C}}$.
$u$ is left exact.
Then [(i)]{} $\iff$ [(ii)]{} $\Rightarrow$ [(iii)]{}, and [(iii)]{} $\Rightarrow$ [(i)]{} if ${\mathcal{C}}$ is essentially small and closed under finite inverse limits.
(The condition on finite inverse limits appears in [@am p. 158], but is skipped in [@SGA4 I.8.11.4].)
\[d.proadj\] In Condition (ii) of Proposition \[p.proadj\], we say that $v$ is *pro-left adjoint* to $u$.
Localisation ([@gz Ch. I], see also [@ks Ch. 7])
------------------------------------------------
Let ${\mathcal{C}}$ be a category, and let $\Sigma\subset Ar({\mathcal{C}})$ be a class of morphisms: following Grothendieck and Maltsiniotis, we call $({\mathcal{C}},\Sigma)$ a *localiser*. Consider the functors $F:{\mathcal{C}}\to {\mathcal{D}}$ such that $F(s)$ is invertible for all $s\in \Sigma$. This “$2$-universal problem” has a solution $Q:{\mathcal{C}}\to {\mathcal{C}}[\Sigma^{-1}]$. One may choose ${\mathcal{C}}[\Sigma^{-1}]$ to have the same objects as ${\mathcal{C}}$ and $Q$ to be the identity on objects; then ${\mathcal{C}}[\Sigma^{-1}]$ is unique (not just up to unique equivalence of categories). If ${\mathcal{C}}$ is essentially small, then ${\mathcal{C}}[\Sigma^{-1}]$ is small, but in general the sets ${\mathcal{C}}[\Sigma^{-1}](X,Y)$ may be “large”; one can sometimes show that it is not the case (Corollary \[c.small\]). A functor of the form $Q:{\mathcal{C}}\to {\mathcal{C}}[\Sigma^{-1}]$ will be called a *localisation*. We have a basic result on adjoint functors [@gz Prop. I.1.3]:
\[lA.6\] Let $G:{\mathcal{C}}\leftrightarrows {\mathcal{D}}:D$ be a pair of adjoint functors ($G$ is left adjoint to $D$). Then the following conditions are equivalent:
$D$ is fully faithful.
The counit $GD\Rightarrow Id_{\mathcal{D}}$ is a natural isomorphism.
$G$ is a localisation.
The same holds if $G$ is right adjoint to $D$ (replacing the counit by the unit).
\[d.sat\] Let $({\mathcal{C}},\Sigma)$ be a localiser, and let $Q:{\mathcal{C}}\to {\mathcal{C}}[\Sigma^{-1}]$ be the corresponding localisation functor. We write $${{\operatorname{sat}}}(\Sigma)=\{s\in Ar({\mathcal{C}})\mid Q(s) \text{ is invertible}\}.$$ This is the *saturation* of $\Sigma$; we say that $\Sigma$ is *saturated* if ${{\operatorname{sat}}}(\Sigma)=\Sigma$.
\[lA.5\] Let $({\mathcal{C}},\Sigma)$ be a localiser, ${\mathcal{D}}$ a category, $F,G:{\mathcal{C}}[\Sigma^{-1}]\to {\mathcal{D}}$ two functors and $u:F\circ Q\Rightarrow G\circ Q$ a natural transformation, where $Q:{\mathcal{C}}\to {\mathcal{C}}[\Sigma^{-1}]$ is the localisation functor. Then $u$ induces a unique natural transformation $\bar u:F\Rightarrow G$.
Define $\bar u_X=u_X:F(X)\to G(X)$ for $X\in Ob {\mathcal{C}}[\Sigma^{-1}]=Ob {\mathcal{C}}$. We must show that $\bar u$ commutes with the morphisms of ${\mathcal{C}}[\Sigma^{-1}]$. This is obvious, since $u$ commutes with the morphisms of ${\mathcal{C}}$ and the morphisms of ${\mathcal{C}}[\Sigma^{-1}]$ are expressed as fractions in the morphisms of ${\mathcal{C}}$.
Presheaves and pro-adjoints {#s.presh}
---------------------------
Let ${\mathcal{C}}$ be a category. We write $\hat{{\mathcal{C}}}$ for the category of presheaves of sets on ${\mathcal{C}}$ (functors ${\mathcal{C}}^{{\operatorname{op}}}\to {{\operatorname{\mathbf{Set}}}}$); it comes with the Yoneda embedding $$y:{\mathcal{C}}\to \hat{{\mathcal{C}}}$$ which sends an object to the corresponding representable presheaf. If $u:{\mathcal{C}}\to {\mathcal{D}}$ is a functor, we have the standard sequence of $3$ adjoint functors $$\begin{CD}
{\mathcal{C}}@>y_{\mathcal{C}}>> \hat{{\mathcal{C}}}\\
@V{u}VV \begin{smallmatrix}u_!\Big\downarrow u^*\Big\uparrow u_*\Big\downarrow \end{smallmatrix}\\
{\mathcal{D}}@>y_{\mathcal{D}}>> \hat{{\mathcal{D}}}
\end{CD}$$ where $u_!$ extends $u$ through the Yoneda embeddings [@SGA4 Exp. I, Prop. 5.4]; $u_!$ and $u_*$ are computed by the usual formulas for left and right Kan extensions (loc. cit., (5.1.1)). If $u$ has a left adjoint $v$, the sequence $(u_!,u^*,u_*)$ extends to $$(v_!,v^*=u_!,v_*=u^*,u_*)$$ (ibid., Rk. 5.5.2).
Let ${\mathcal{A}}$ be an essentially small additive category. Instead of presheaves of sets on ${\mathcal{A}}$, one usually uses the category ${\operatorname{Mod}\hbox{--}}{\mathcal{A}}$ of *additive presheaves of abelian groups*; the above results transfer to this context, mutatis mutandis.
\[p.funct\] a) The functor $u_!$ (resp. $u_*$, $u^*$) commutes with all representable colimits (resp. limits, limits and colimits). If $u$ has a left adjoint, then $u_!$ also commutes with all limits. If $u$ has a pro-left adjoint $v$, so does $u_!$ which is therefore exact. Moreover, $u_!$ is then given by the formula $$(u_!F)(Y) = {\operatornamewithlimits{\varinjlim}}(F\circ v(Y)), \quad F\in \hat{{\mathcal{C}}}, Y\in {\mathcal{D}}.$$ b) If $u$ is fully faithful, so is $u_!$.\
c) If $u$ is a localisation or is full and essentially surjective, then $u_!$ is a localisation.\
d) In the case of c), for $C\in {\mathcal{C}}$ the following conditions are equivalent:
The representable functor $y_{\mathcal{C}}(C)\in \hat{{\mathcal{C}}}$ induces a functor on ${\mathcal{D}}$ via $u$.
The unit map $y_{\mathcal{C}}(C)\to u^*u_! y_{\mathcal{C}}(C)\simeq u^*y_{\mathcal{D}}(u(C))$ is an isomorphism.
For any $C'\in {\mathcal{C}}$, the map ${\mathcal{C}}(C',C)\to {\mathcal{D}}(u(C'),u(C))$ induced by $u$ is bijective.
a\) follows from general properties of adjoint functors, except for the case of a pro-left adjoint. Let $u$ admit a pro-left adjoint $v$, and let $Y\in {\mathcal{D}}$: so there is an isomorphism of categories $Y\downarrow u\simeq v(Y)\downarrow c$. Hence, we get by Lemma \[l.cof\] a cofinal functor $$A\to Y\downarrow u,$$ where $A$ is the indexing set of $v(Y)$. Thus, for $F\in \hat{{\mathcal{C}}}$, $u_! F(Y)$ may be computed as $$u_!F(Y)={\operatornamewithlimits{\varinjlim}}_{\alpha\in A} F(v(Y)(\alpha))={\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}(y_{\mathcal{C}}(v(Y)),c(F)).$$
The first equality is the formula in the proposition. The second one shows that the pro-left adjoint $v_!$ of $u_!$ is defined at $Y$ by $y_{\mathcal{C}}(v(Y))$; since any object of $\hat{{\mathcal{D}}}$ is a colimit of representable objects, this shows that $v_!$ is defined everywhere.
For b), see [@SGA4 Exp. I, Prop. 5.6]. In c), it is equivalent to show that $u^*$ is fully faithful. Let $F,G\in \hat{{\mathcal{D}}}$, and let ${\varphi}:u^*F\to u^*G$ be a morphism of functors. In both cases, $u$ is essentially surjective: given $X\in {\mathcal{D}}$ and an isomorphism $\alpha:X{{\xrightarrow}{\sim}}u(Y)$, we get a morphism $$\psi_X:F(X){\xrightarrow}{{\alpha^*}^{-1}} F(u(Y)){\xrightarrow}{{\varphi}_Y} G(u(Y)){\xrightarrow}{\alpha^*} G(X).$$
The fact that $\psi_X$ is independent of $(Y,\alpha)$ and is natural in $X$ is an easy consequence of each hypothesis (see Lemma \[lA.5\] in the first case).
In d), the equivalence (ii) $\iff$ (iii) is tautological and (iii) $\Rightarrow$ (i) is obvious. The implication (i) $\Rightarrow$ (iii) was proven in [@gz I.4.1.2] assuming that $u$ is a localisation enjoying a calculus of left fractions; let us prove (i) $\Rightarrow$ (ii) in general. Under (i), we have $y_{\mathcal{C}}(C)\simeq u^* F$ for some $F\in \hat{D}$; the unit map becomes $$\eta_{u^*F}:u^*F\to u^*u_!u^*F.$$
On the other hand, the counit map ${\varepsilon}_F:u_!u^*F\to F$ is invertible by the full faithfulness of $u^*$. By the adjunction identities, we have $u^*({\varepsilon}_F)\circ \eta_{u^*F}=1_{u^*F}$. Hence the conclusion.
We shall usually write $u^!$ for the pro-left adjoint of $u_!$, when it exists.
Calculus of fractions
---------------------
\[d.cf\] A localiser $({\mathcal{C}},\Sigma)$ (or simply $\Sigma$) enjoys a *calculus of right fractions* if:
The identities of ${\mathcal{C}}$ are in $\Sigma$.
$\Sigma$ is stable under composition.
(Ore condition.) For each diagram $X'{\xrightarrow}{s} X{\xleftarrow}{u} Y$ where $s\in \Sigma$, there exists a commutative square $$\begin{CD}
Y'@>u'>> X'\\
@Vt VV @Vs VV\\
Y@>u>> X
\end{CD}
\qquad \text{where } t\in\Sigma.$$
(Cancellation.) If $f,g:X\rightrightarrows Y$ are morphisms in ${\mathcal{C}}$ and $s:Y\to Y'$ is a morphism of $\Sigma$ such that $sf=sg$, there exists a morphism $t:X'\to X$ in $\Sigma$ such that $ft=gt$.
\[p.cf\] Suppose that $\Sigma$ enjoys a calculus of right fractions. For $c\in {\mathcal{C}}$, let $\Sigma\downarrow c$ denote the full subcategory of the comma category ${\mathcal{C}}\downarrow c$ given by the objects $c'{\xrightarrow}{s}c$ with $s\in \Sigma$. Then\
a) $\Sigma\downarrow c$ is cofiltered.\
b) [@gz I.1.2.3] For any $d\in {\mathcal{C}}$, the obvious map $$\label{eq.cf}
{\operatornamewithlimits{\varinjlim}}_{c'\in \Sigma\downarrow c}{\mathcal{C}}(c',d)\to {\mathcal{C}}[\Sigma^{-1}](c,d)$$ is an isomorphism.\
c) Any morphism in ${\mathcal{C}}[\Sigma^{-1}]$ is of the form $Q(f)Q(s)^{-1}$ for $f\in Ar({\mathcal{C}})$ and $s\in \Sigma$; if $f_1,f_2$ are two parallel arrows in ${\mathcal{C}}$, then $Q(f_1)=Q(f_2)$ if and only if there exists $s\in \Sigma$ such that $f_1s=f_2s$.
a\) The dual of Condition (a) in [@mcl p. 211] (supremum of two objects) follows from Axioms (iii) and (ii) of Definition \[d.cf\]; the dual of Condition (b) (equalizing parallel arrows) follows from Axioms (iv) and (ii).
b\) First let us specify the “obvious map” : it sends a pair $(c'{\xrightarrow}{s} c,c'{\xrightarrow}{f}d)$ with $s\in \Sigma$ and $f\in {\mathcal{C}}(c',d)$ to $Q(f)Q(s)^{-1}$. We now follow the strategy of [@gz pp. 13/14]: using Axioms (ii) and (iii), we define for 3 objects $c,d,e\in {\mathcal{C}}$ a composition $${\operatornamewithlimits{\varinjlim}}_{c'\in \Sigma\downarrow c}{\mathcal{C}}(c',d)\times {\operatornamewithlimits{\varinjlim}}_{d'\in \Sigma\downarrow d}{\mathcal{C}}(d',e)\to {\operatornamewithlimits{\varinjlim}}_{c'\in \Sigma\downarrow c}{\mathcal{C}}(c',e)$$ which is shown to be well-defined and associative thanks to Axiom (iv). Hence we get a category $\Sigma^{-1}{\mathcal{C}}$ with the same objects as ${\mathcal{C}}$ and Hom sets as above, and yields a functor $\Sigma^{-1}{\mathcal{C}}\to {\mathcal{C}}[\Sigma^{-1}]$. But the obvious map ${\mathcal{C}}(c,d)\to {\operatornamewithlimits{\varinjlim}}_{c'\in \Sigma\downarrow c}{\mathcal{C}}(c',d)$ also yields a functor ${\mathcal{C}}\to \Sigma^{-1}{\mathcal{C}}$, which is the identity on objects and is easily seen to have the universal property of ${\mathcal{C}}[\Sigma^{-1}]$. Hence is an isomorphism for all $(c,d)$.
c\) The first statement has already been observed; the second one follows readily from .
We shall write $\Sigma^{-1}{\mathcal{C}}$ instead of ${\mathcal{C}}[\Sigma^{-1}]$ if $\Sigma$ enjoys a calculus of fractions.
\[c.small\] If $\Sigma$ admits a calculus of right fractions and if for any $c\in {\mathcal{C}}$, the category $\Sigma\downarrow c$ contains a small cofinal subcategory, then the ${\operatorname{Hom}}$ sets of $\Sigma^{-1}{\mathcal{C}}$ are small.
\[c.cf\] Let $({\mathcal{C}},\Sigma)$ be a localiser such that $\Sigma$ enjoys a calculus of right fractions. Let $F:{\mathcal{C}}\to {\mathcal{D}}$ be a functor. Suppose that $F$ inverts the morphisms of $\Sigma$ and that, for any $c,d\in {\mathcal{C}}$, the obvious map $${\operatornamewithlimits{\varinjlim}}_{c'\in \Sigma\downarrow c}{\mathcal{C}}(c',d)\to {\mathcal{D}}(F(c),F(d))$$ is an isomorphism. Then the functor $\Sigma^{-1}F:\Sigma^{-1}{\mathcal{C}}\to {\mathcal{D}}$ induced by $F$ is fully faithful.
\[p1.10\] a) Let $({\mathcal{C}},\Sigma)$ be a localiser. Assume that $\Sigma$ enjoys a calculus of right fractions. Then the localisation functor $Q:{\mathcal{C}}\to \Sigma^{-1}{\mathcal{C}}$ is left exact; if limits indexed by a finite category $I$ exist in ${\mathcal{C}}$, they also exist in $\Sigma^{-1}{\mathcal{C}}$.\
b) Let ${\mathcal{C}}$ be an essentially small category closed under finite limits, and let $G:{\mathcal{C}}\to {\mathcal{D}}$ be a left exact functor. Let $\Sigma=\{s\in Ar({\mathcal{C}})\mid G(s) \text{ is invertible}\}$. Then $\Sigma$ enjoys a calculus of right fractions; the induced functor $\Sigma^{-1}{\mathcal{C}}\to {\mathcal{D}}$ is conservative and left exact.
After passing to the opposite categories, a) is [@gz Prop. I.3.1 and Cor. I.3.2] and b) is [@gz Prop. I.3.4].
We also have a useful lemma:
\[l4.1a\] Let $G:{\mathcal{C}}\to {\mathcal{D}}$ be an exact functor between abelian categories. Then ${\mathcal{B}}={\operatorname{Ker}}G$ is a Serre subcategory of ${\mathcal{C}}$; if $G$ is a localisation, the induced functor ${\mathcal{C}}/{\mathcal{B}}\to {\mathcal{D}}$ is an equivalence of categories.
The first statement is obvious. For the second one, let $f$ be a morphism in ${\mathcal{C}}$. The exactness of $G$ shows that $G(f)$ is an isomorphism if and only if ${\operatorname{Ker}}f,{\operatorname{Coker}}f\in {\mathcal{B}}$.
Pro-$\Sigma$-objects
--------------------
\[d2.1\] Let $({\mathcal{C}},\Sigma)$ be a localiser. We write ${\text{\rm pro}_{\Sigma}\text{\rm--}}{\mathcal{C}}$ for the full subcategory of the category ${\text{\rm pro}_{}\text{\rm--}}{\mathcal{C}}$ of pro-objects of ${\mathcal{C}}$ consisting of filtering inverse systems whose transition morphisms belong to $\Sigma$. An object of ${\text{\rm pro}_{\Sigma}\text{\rm--}}{\mathcal{C}}$ is called a *pro-$\Sigma$-object*.
\[p2.6\] Suppose that $\Sigma$ has a calculus of right fractions and, for any $c\in {\mathcal{C}}$, the category $\Sigma\downarrow c$ contains a small cofinal subcategory. Then $Q:{\mathcal{C}}\to \Sigma^{-1}{\mathcal{C}}$ has a pro-left adjoint $Q^!$, which takes an object $X\in \Sigma^{-1} {\mathcal{C}}$ to $``{\operatornamewithlimits{\varprojlim}}_{M\in \Sigma\downarrow X}" M$. In particular, $Q^!(\Sigma^{-1}{\mathcal{C}})\subset {\text{\rm pro}_{{{\operatorname{sat}}}(\Sigma)}\text{\rm--}}{\mathcal{C}}$, where ${{\operatorname{sat}}}(\Sigma)$ is the saturation of $\Sigma$ (Definition \[d.sat\]).
In view of Corollary \[c.small\] and Proposition \[p1.10\], this follows from Proposition \[p.cf\] b).
\[r.cf\] Consider the localisation functor $Q:{\mathcal{C}}\to \Sigma^{-1} {\mathcal{C}}$: it has a left Kan extension $\hat Q:{\text{\rm pro}_{\Sigma}\text{\rm--}}{\mathcal{C}}\to \Sigma^{-1}{\mathcal{C}}$ [@mcl Ch. X] along the constant functor ${\mathcal{C}}\to {\text{\rm pro}_{\Sigma}\text{\rm--}}{\mathcal{C}}$, given by the formula $$\hat Q(``{\operatornamewithlimits{\varprojlim}}\nolimits" C_\alpha)= {\operatornamewithlimits{\varprojlim}}Q(C_\alpha).$$
(The right hand side makes sense as an inverse limit of isomorphisms.) Then one checks easily that $Q^!$ is left adjoint to $\hat Q$.
\[lem:omega-sh0\] Let $({\mathcal{C}},\Sigma)$ be a localiser verifying the conditions of Proposition \[p2.6\]. Let $Q:{\mathcal{C}}\to \Sigma^{-1}{\mathcal{C}}$ denote the localisation functor, and consider the string of adjoint functors $(Q_!,Q^*,Q_*)$ between $\hat{{\mathcal{C}}}$ and $\widehat{\Sigma^{-1}{\mathcal{C}}}$ from §\[s.presh\]. Then:
1. $Q_!$ has a pro-left adjoint, and is therefore exact.
2. For ${\mathcal{F}}\in \hat{{\mathcal{C}}}$ and $Y \in \Sigma^{-1}{\mathcal{C}}$, we have $$Q_!{\mathcal{F}}(Y) = {\operatornamewithlimits{\varinjlim}}_{X \in \Sigma\downarrow Y} F(Y).$$
This follows from Propositions \[p.funct\] a) and \[p2.6\].
If $({\mathcal{A}},\Sigma)$ is a localiser with ${\mathcal{A}}$ additive and $\Sigma$ enjoys a calculus of right fractions, then $\Sigma^{-1}{\mathcal{A}}$ is additive and so is the functor $Q:{\mathcal{A}}\to \Sigma^{-1}{\mathcal{A}}$ [@gz I.3.3]. For future reference, we give the additive analogue of Theorem \[lem:omega-sh0\]:
\[lem:omega-sh\] Let $({\mathcal{A}},\Sigma)$ be a localiser; assume that ${\mathcal{A}}$ is an additive category and that $\Sigma$ has a calculus of right fractions. Let $Q:{\mathcal{A}}\to \Sigma^{-1}{\mathcal{A}}$ denote the localisation functor, as well as the string of adjoint functors $(Q_!,Q^*,Q_*)$ between ${\operatorname{Mod}\hbox{--}}{\mathcal{A}}$ and ${\operatorname{Mod}\hbox{--}}\Sigma^{-1}{\mathcal{A}}$. Then:
1. $Q_!$ has a pro-left adjoint, and is therefore exact.
2. For ${\mathcal{F}}\in {\operatorname{Mod}\hbox{--}}{\mathcal{A}}$ and $Y \in \Sigma^{-1}{\mathcal{A}}$, we have $$Q_!{\mathcal{F}}(Y) = {\operatornamewithlimits{\varinjlim}}_{X \in \Sigma\downarrow Y} F(Y).$$
cd-structures {#sa-cd}
-------------
Let ${\mathcal{C}}$ be a category with an initial object. According to [@cdstructures], a *cd-structure* on ${\mathcal{C}}$ is given by a collection of commutative squares stable under isomorphisms, called *distinguished squares*. Any cd-structure defines a topology on ${\mathcal{C}}$: the smallest Grothendieck topology such that for a distinguished square of the form the sieve $(p, u)$ generated by the morphisms $${p:V \to X,u:U\to X}$$ is a cover sieve and such that the empty sieve is a cover sieve of the initial object $\emptyset$.
Recall from [@cdstructures] some important properties of cd-structures.
\[d3.1\] Let ${\mathcal{C}}$ be a category with an initial object $\emptyset$.
1. Let $P$ be a cd-structure on ${\mathcal{C}}$. The class ${\mathcal{S}}_P$ of *simple covers* is the smallest class of families of morphisms of the form $\{U_i \to X\}_{i \in I}$ satisfying the following two conditions:
- for any isomorphism $f$, $\{f\}$ is in ${\mathcal{S}}_P$
- for a distinguished square $Q$ of the form and families $\{p_i : V_i \to V\}_{i \in I}$ and $\{U_j \to U\}_{j \in J}$ in ${\mathcal{S}}_P$ the family $\{p \circ p_i , u \circ q_j\}_{i \in I, j \in J}$ is in ${\mathcal{S}}_P$.
2. A cd-structure on ${\mathcal{C}}$ is called *complete* if any cover sieve of an object $X \in {\mathcal{C}}$ which is not isomorphic to $\emptyset$ contains a sieve generated by a simple cover.
3. A cd-structure $P$ is called *regular* if for $Q \in P$ of the form one has
- $Q$ is a pullback square (i.e., is cartesian)
- $u$ is a monomorphism
- the morphisms of sheaves $$\Delta \bigsqcup \rho (v) : \rho (V) \bigsqcup \rho (W) \times_{\rho (U)} \rho (W) \to \rho (V) \times_{\rho (X)} \rho (V)$$ is surjective, where for $C \in {\mathcal{C}}$ we denote by $\rho (C)$ the sheaf associated with the presheaf represented by $C$.
\[lA.7\] A cd-structure is complete provided:
1. any morphism with values in $\emptyset$ is an isomorphism, and
2. for any distinguished square $S$ of the form and for any morphism $X ' \to X$, the square $S' = X' \times_X S$ is defined and distinguished.
\[lA.8\] A cd-structure is regular provided, for any distinguished square $S$ of the form we have
1. $S$ is cartesian,
2. $u$ is a monomorphism, and
3. the objects $V \times_X V$ and $W \times_{U} W$ exist in ${\mathcal{C}}$ and the derived square $$\label{uld(S)} d(S) : \vcenter{\xymatrix{
W \ar[r]^v \ar[d]_{\Delta_{W/U}} & V \ar[d]^{\Delta_{V/X}} \\
W \times_{U} W \ar[r]^{} & V \times_{X} V
}}$$ is distinguished.
\[dA.5\] A cd-structure verifying the conditions of Lemma \[lA.7\] (resp. \[lA.8\]) is *strongly complete* (resp. *strongly regular*).
\[lA.1\] Under Hypotheses (1) and (3) of Lemma \[lA.8\], the square is cartesian.
This is a logical consequence of [@cdstructures Lemma 2.11], but let us provide a quick proof: let $Z{\xrightarrow}{a} V$, $Z{\xrightarrow}{b} W \times_{U} W$ be two morphisms making the corresponding square commutative. Then $b$ amounts to two morphisms $b_1, b_2:Z\to W$ such that (with the notation of ) $qb_1=qb_2$ and $a=vb_1=vb_2$. Since $S$ is cartesian, we have $b_1=b_2$, and this morphism $Z\to W$ is a solution to the universal problem.
\[pA.4\] Let $({\mathcal{C}},\Sigma)$ be a localiser such that $\Sigma$ admits a calculus of right fractions.
1. If ${\mathcal{C}}$ has an initial object verifying Conditon (1) of Lemma \[lA.7\], so does $\Sigma^{-1} {\mathcal{C}}$.
2. Assume (1) and let $Q:{\mathcal{C}}\to \Sigma^{-1} {\mathcal{C}}$ be the localisation functor. Suppose given a cd-structure $P$ on ${\mathcal{C}}$, and let $P'$ be the cd-structure on $\Sigma^{-1} {\mathcal{C}}$ given by all squares isomorphic to a square of the form $Q(S)$, where $S\in P$. If $P$ is strongly complete (resp. strongly regular), so is $P'$.
\(1) Let $\emptyset$ be an initial object of ${\mathcal{C}}$. Since $Q$ is (essentially) surjective, $Q(\emptyset)$ admits a morphism to any object; Condition (1) of Lemma \[lA.7\] for $\emptyset$ implies that this morphism is unique, and this in turn implies the same condition for $Q(\emptyset)$.
\(2) By Proposition \[p1.10\] a), $Q$ commutes with finite limits. This implies Condition (2) of Lemma \[lA.7\]. Conditions (1), (3) of Lemma \[lA.8\] for $P'$ follow from the same conditions for $P$ (note that the diagonals are preserved by $Q$, since they are finite limits). It remains to show that $Q$ carries a monomorphism $u:U\to X$ to a monomorphism. Let $f,g:V\to U$ be two morphisms in $\Sigma^{-1} {\mathcal{C}}$ such that $Q(u)f=Q(u)g$. By calculus of fractions, we may write $f=Q(\tilde f)Q(s)^{-1}$ and $g=Q(\tilde g)Q(s)^{-1}$ for some $\tilde f, \tilde g\in Ar({\mathcal{C}})$ and $s\in \Sigma$. Then $Q(u\tilde f)=Q(u \tilde g)$. By Proposition \[p.cf\] c), we may find $t\in \Sigma$ such that $u\tilde ft = u\tilde g t$, which implies $\tilde f t=\tilde gt$ since $u$ is a monomorphism. This shows $f=g$, as desired.
A pull-back lemma
-----------------
We shall use the following elementary lemma several times.
\[lem:lr-adjoint\] Let ${\mathcal{C}}, {\mathcal{D}}$ be abelian categories and let ${\mathcal{C}}' \subset {\mathcal{C}}, {\mathcal{D}}' \subset {\mathcal{D}}$ be full abelian subcategories. Let $c :{\mathcal{C}}\to {\mathcal{D}}$ and $c' :{\mathcal{C}}' \to {\mathcal{D}}'$ be additive functors satisfying $c i_C = i_D c'$, where $i_C : {\mathcal{C}}' \to {\mathcal{C}}$ and $i_D : {\mathcal{D}}' \to {\mathcal{D}}$ are inclusion functors.
1. If $c$ is faithful, so is $c'$.
2. Suppose that $i_D$ is strongly additive or has a strongly additive left inverse (for example, a left adjoint). If $c$ and $i_C$ are strongly additive, so is $c'$.
3. Suppose that $i_C$ has a left adjoint $a_C$. If $c$ has a left adjoint $d$, then $d'=a_C d i_D$ is a left adjoint of $c'$. If $d$ and $a_{\mathcal{C}}$ are exact, so is $d'$. Moreover, $a_{\mathcal{C}}d=d'a_{\mathcal{D}}$ if $i_{\mathcal{D}}$ has a left adjoint $a_{\mathcal{D}}$.
4. Suppose that $i_C$ and $i_D$ have left adjoints $a_C$ and $a_D$, that $a_D$ is exact, and that $a_D c= c' a_C$. If $c$ is exact, then so is $c'$.
\(1) is obvious. (2) Let $\{F_i \}_{i \in I}$ be a family of objects of ${\mathcal{C}}'$. We must show that the natural map $$f:\bigoplus_{i \in I} c'(F_i )\to c'\big(\bigoplus_{i \in I} F_i\big)$$ is an isomorphism. It suffices to see that $i_D f$ is an isomorphism. The composition $$\bigoplus_{i \in I} i_D c' F_i{\xrightarrow}{g} i_D\big(\bigoplus_{i \in I} c'(F_i) \big){\xrightarrow}{i_Df} i_D c'\big(\bigoplus_{i \in I} F_i\big)$$ is an isomorphism by the strong additivity of $c$ and $i_C$. If $i_D$ is strongly additive, $g$ is also an isomorphism and we are done. If now $i_D$ has a strongly additive left inverse $a_D$, we apply it to the diagram and get a composition $$\bigoplus_{i \in I} a_D\ i_D c' F_i{{\xrightarrow}{\sim}}a_D\bigoplus_{i \in I} i_D c' F_i{\xrightarrow}{a_D g} a_Di_D\big(\bigoplus_{i \in I} c'(F_i) \big){\xrightarrow}{a_Di_Df} a_Di_D c'\big(\bigoplus_{i \in I} F_i\big)$$ which is an isomorphism and naturally isomorphic to $f$. This concludes the proof of (2).
\(3) For $F \in {\mathcal{C}}'$ and $G \in {\mathcal{D}}'$, we have ${\mathcal{C}}'(a_C d i_D F, G)
={\mathcal{D}}(i_D F, c i_C G)
={\mathcal{D}}(i_D F, i_D c' G)
={\mathcal{D}}'(F, c'G)$. This proves the first claim; therefore if $d$ and $a_{\mathcal{C}}$ are exact, $d'$ is left exact, hence exact since it is right exact as a left adjoint. The last isomorphism follows from taking left adjoints of the isomorphism $ci_{\mathcal{C}}=i_{\mathcal{D}}c'$.
\(4) Let us take an exact sequence $0 \to F \to G \to H \to 0$ in ${\mathcal{C}}'$. Put $K:={\operatorname{Coker}}(i_C G \to i_C H) \in {\mathcal{C}}$. Since $a_D c K = c' a_C K = 0$, we get an exact sequence $0 \to a_D c i_C F \to
a_D c i_C G \to a_D c i_C H \to 0$ by the exactness of $c$ and $a_D$. Using $a_D c= c' a_C$ and $a_C i_C = {{\operatorname{Id}}}$ (Lemma \[lA.6\]), we conclude $0 \to c' F \to c' G \to c' H \to 0$ is exact.
For the sake of clarity, we add:
\[lA.9\] Let ${\mathcal{D}}\subseteq {\mathcal{C}}$ be a full embedding of categories. Suppose that a direct (resp. inverse) system $(d_\alpha)$ of objects of ${\mathcal{D}}$ has a colimit (resp. a limit) in ${\mathcal{C}}$, which is isomorphic to an object $d$ of ${\mathcal{D}}$. Then $d$ represents the (co)limit of $(d_\alpha)$ in ${\mathcal{D}}$.
Trivial.
Homological algebra
-------------------
Recall Grothendieck’s theorem [@tohoku Th. 2.4.1]:
\[tA.2\] Let ${\mathcal{A}}{\xrightarrow}{F}{\mathcal{B}}{\xrightarrow}{G}{\mathcal{C}}$ be a string of left exact functors between abelian categories. Suppose that ${\mathcal{A}}$ and ${\mathcal{B}}$ have enough injectives and that $F$ carries injectives of ${\mathcal{A}}$ to $G$-acyclics. Then, for any $A\in {\mathcal{A}}$, there is a convergent spectral sequence $$E_2^{p,q}=R^pG R^qF(A)\Rightarrow R^{p+q}(GF)(A).$$
\[exA.3\] If $F$ has an exact left adjoint, it carries injectives to injectives. If $G$ is exact, the hypothesis on $F$ is automatically verified.
We shall also use the following standard result:
\[pA.2\] Let $a:{\mathcal{B}}\leftrightarrows{\mathcal{A}}:i$ be a pair of adjoint functors between abelian categories ($a$ is left adjoint to $i$). Suppose that ${\mathcal{A}}$ has enough injectives and that $a$ is exact. Then, for any $(A,B)\in {\mathcal{A}}\times {\mathcal{B}}$, there is a convergent spectral sequence $${\operatorname{Ext}}_{\mathcal{B}}^p(B,R^qi A)\Rightarrow {\operatorname{Ext}}_{\mathcal{A}}^{p+q}(aB,A).$$ If $B$ is projective, this spectral sequence collapses to isomorphisms $${\mathcal{B}}(B,R^qi A)\simeq {\operatorname{Ext}}_{\mathcal{A}}^{q}(aB,A).$$
Fix $B$. By adjunction, the composition of functors $${\mathcal{A}}{\xrightarrow}{i}{\mathcal{B}}{\xrightarrow}{{\mathcal{B}}(B,-)}{\operatorname{\mathbf{Ab}}}$$ is isomorphic to ${\mathcal{A}}(aB,-)$. We then get the spectral sequence from Theorem \[tA.2\] and Example \[exA.3\]. The last fact is obvious.
The following is a slight generalization of [@milne III 2.12], (where the underlying category of ${\mathcal{S}}$ is supposed to be a category of schemes).
\[lem:milne\] Let $F$ be a sheaf on a site ${\mathcal{S}}$. The following conditions are equivalent.
1. We have $H^q(X, F)=0$ for any $X \in {\mathcal{S}}$ and $q>0$.
2. We have $\check{H}^q(X, F)=0$ for any $X \in {\mathcal{S}}$ and $q>0$.
3. We have $\check{H}^q(U/X, F)=0$ for any cover $U \to X$ in ${\mathcal{S}}$ and $q>0$.
4. The sheaf $F$ is $i_{{\mathcal{S}}}$-acyclic, where $i_{{\mathcal{S}}}$ is the inclusion functor of the category of sheaves to that of presheaves.
For $X \in {\mathcal{S}}$, we write $\Gamma_X$ (resp. $\Gamma^{{\operatorname{pr}}}_X$) for the functor $F \mapsto F(X)$ from the category of sheaves (resp. presheaves) to ${\operatorname{\mathbf{Ab}}}$. We have $\Gamma_X = \Gamma_X^{{\operatorname{pr}}}i_{{\mathcal{S}}}$. Since $\Gamma_X^{{\operatorname{pr}}}$ is exact, Theorem \[tA.2\] implies $R^q \Gamma_X = \Gamma_X^{{\operatorname{pr}}}R^q i_{{\mathcal{S}}}$, and hence $H^q(X, F)=R^q i_{{\mathcal{S}}}F(X)$. This proves the equivalence of (1) and (4). The rest is shown in the same way as [@milne III 2.12].
\[def:flabby\] We say $F$ is *flabby* if the conditions of Lemma \[lem:milne\] are satisfied.
\[lem:inj-flabby\] Let ${\mathcal{S}}$ be the category of abelian sheaves on a site ${\mathcal{C}}$, ${\mathcal{T}}$ an abelian category, and $c^* : {\mathcal{T}}\to {\mathcal{S}}$ an additive functor which has a left adjoint $c_! : {\mathcal{S}}\to {\mathcal{T}}$. Suppose that any cover in ${\mathcal{C}}$ admits a refinement $U \to X$ such that $c_!(\check{C}(U/X))$ is exact in ${\mathcal{T}}$, where $$\check{C}(U/X) = ( \dots \to y(U \times_X U)
\to y(U) \to y(X) \to 0)$$ is the Čech complex associated to $U \to X$ ($y$ denotes the Yoneda functor). Then $c^*I$ is flabby for any injective object $I \in {\mathcal{T}}$.
(Compare [@voetri 3.1.7].) It suffices to show $\check{H}^q(U/X, c^*I)=0$ for any $q>0$ and for any $U \to X$ as in the assumption. If we denote by $U_X^n$ the $n$-fold fiber product of $U$ over $X$, then $\check{H}^q(U/X, c^*I)$ is computed as the cohomology of the complex $$\begin{aligned}
c^*I(U_X^{\bullet+1})
= {\mathcal{S}}(y(U_X^{\bullet+1}), c^*I)
= {\mathcal{T}}(c_! y(U_X^{\bullet+1}), I),\end{aligned}$$ which is acyclic by the assumption and the injectivity of $I$.
Grothendieck categories {#s.groth}
-----------------------
Recall that a *Grothendieck abelian category* (for short, a Grothendieck category) is an abelian category verifying Axiom AB5 of [@tohoku]: small colimits are representable and exact, and having a set of generators (equivalently, a generator). These generators are generators by strict epimorphisms. We have the following basic facts:
\[t.groth\] a) Any Grothendieck category is complete and has enough injectives.\
b) Let ${\mathcal{F}}:{\mathcal{C}}\to {\mathcal{D}}$ be a functor, where ${\mathcal{C}}$ is a Grothendieck category. Then $F$ has a right adjoint if and only if it commutes with all colimits.\
c) Let ${\mathcal{C}}$ be a Grothendieck category, ${\mathcal{B}}\subset {\mathcal{C}}$ be a Serre subcategory, ${\mathcal{D}}={\mathcal{C}}/{\mathcal{B}}$ and $G:{\mathcal{C}}\to {\mathcal{D}}$ the (exact) localisation functor. Then $G$ has a right adjoint $D$ if and only if ${\mathcal{B}}$ is stable under infinite direct sums. In this case, ${\mathcal{B}}$ and ${\mathcal{D}}$ are Grothendieck.\
d) Let $G:{\mathcal{C}}\leftrightarrows {\mathcal{D}}:D$ be a pair of adjoint additive functors between additive categories, with $D$ fully faithful. If ${\mathcal{C}}$ is Grothendieck and $G$ is exact, ${\mathcal{D}}$ is Grothendieck.
a\) See [@tohoku Th. 1.10.1], [@SGA4 V.0.2.1] or [@ks Th. 8.3.27 (i) and 9.6.2]. b) See [@ks Prop. 8.3.27 (iii)]. c) See [@gabriel Ch. III, Prop. 8 and 9]. d) Let ${\mathcal{B}}$ be the kernel of $G$. Then ${\mathcal{B}}$ is easily seen to be a Serre subcategory (e.g. [@gabriel Ch. III, Prop. 5]), so the claim follows from c).
\[t.mon\] For any additive category ${\mathcal{A}}$, ${\operatorname{Mod}\hbox{--}}{\mathcal{A}}$ is a Grothendieck category with a set of projective generators.
See e.g. [@ak Prop. 1.3.6] for the first statement; the projective generators are given by ${\mathcal{E}}=\{y(A)\mid A\in {\mathcal{A}}\}$.
[SGA4]{} Y. André, B. Kahn, [*Nilpotence, radicaux et structures monoïdales*]{} (with an appendix by P. O’Sullivan), Rend. Sem. Math. Univ. Padova 108 (2002), 107–291. M. Artin, B. Mazur, Étale homotopy, Lect. Notes in Math. 100, Springer, 1969. J. Ayoub, Motifs de variétés analytiques rigides, Mém. SMF 140-141 (2015), vi+386. S. Bloch, H. Esnault, [*An additive version of higher Chow groups*]{}, Ann. Sci. Éc. Norm. Sup. 36, 463–477. E. Bierstone, P. Milman, [*Functoriality in resolution of singularities*]{}, Publ. RIMS, Kyoto Univ. 44 (2008), 609-639. F. Binda, S. Saito, [*Relative cycles with moduli and regulator maps*]{}, J. Inst. Math. Jussieu 18 (2019), no. 6, 1233–1293. P. Gabriel, Des catégories abéliennes, Bull. SMF 90 (1962), 323–448. P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer, 1967. A. Grothendieck, [*Sur quelques points d’algèbre homologique*]{}, Tohôku Math. J. 9 (1957), 119–221. R. Hartshorne, Algebraic Geometry, Springer, 1977. B. Kahn, S. Saito, T. Yamazaki, [*Reciprocity sheaves*]{} (with two appendices by Kay Rülling), Compositio Math. 152 (2016), 1851–1898. B. Kahn, S. Saito, T. Yamazaki, [*Motives with modulus*]{}, preprint, 2015, <https://arxiv.org/abs/1511.07124>, withdrawn. B. Kahn, H. Miyazaki, S. Saito, T. Yamazaki, [*Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs*]{}, preprint, 2019, <https://arxiv.org/abs/1910.14534>. M. Kashiwara, P. Schapira, Categories and sheaves, Springer, 2006. A. Krishna, J. Park, [*Moving lemma for additive higher Chow groups*]{}, Algebra Number Theory 6 (2012), no. 2, 293–326. C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, Clay Math. Monographs 2, AMS, 2006. S. Mac Lane, Categories for the working mathematician, Grad. Texts in Math. 5, Springer (2nd ed.), 1998. J. Milne, Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 1980. H. Miyazaki, [*Cube invariance of higher Chow groups with modulus*]{}, J. Algebraic Geom. 28 (2019), no. 2, 339–390. J. Park, [*Regulators on additive higher Chow groups*]{}, Amer. J. Math. 131 (2009), 257–276. M. Raynaud, L. Gruson, [*Critères de platitude et de projectivité*]{}, Invent. Math. 13 (1971), 1–89. J. Riou, [*Théorie homotopique des $S$-schémas*]{}, mémoire de DEA, Paris 7, 2002, <http://www.math.u-psud.fr/~riou/dea/dea.pdf>. A. Sulin, V. Voevodsky, [*Bloch-Kato Conjecture and motivic cohomology with finite coefficients*]{}, in : The Arithmetic and Geometry of Algebraic Cycles. eds. B. Gordon, J. Lewis, S. M'’uller-Stach, S. Saito, N. Yui, NATO Science Series 548. V. Voevodsky, [*Triangulated categories of motives over a field*]{}, [*in*]{} E. Friedlander, A. Suslin, V. Voevodsky Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 2000, 188–238. V. Voevodsky, [*Homotopy theory of simplicial sheaves in completely decomposable topologies*]{}, J. pure appl. Algebra 214 (2010) 1384–1398. V. Voevodsky, [*Unstable motivic homotopy categories in Nisnevich and cdh-topologies*]{}, J. pure appl. Algebra 214 (2010) 1399–1406.
Acronyms
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math. No. 11 (1961), 167 pp. A. Grothendieck et al, Revêtements étales et groupe fondamental (SGA 1), Séminaire de Géométrie Algébrique du Bois-Marie 1960 – 1961, new edition: Documents mathématiques 3, SMF, 2003. M. Demazure, A. Grothendieck, Schémas en groupes, new edition, SMF, 2012. E. Artin, A. Grothendieck, J.-L. Verdier, Théorie des topos et cohomologie étale des schémas (SGA4), Lect. Notes in Math. 269, 270, 305, Springer, 1972–73.
[^1]: The first author acknowledges the support of Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005. The work of the second author is supported in part by Fondation Sciences Mathématiques de Paris (FSMP), and in part by RIKEN Special Postdoctoral Researchers (SPDR) Program and RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) Program. The third author is supported by JSPS KAKENHI Grant (15H03606). The fourth author is supported by JSPS KAKENHI Grant (15K04773).
[^2]: Here we stress that we do not assume it is finite over ${\overline}{M}$.
[^3]: To apply this lemma, factor $\pi_{\gamma\alpha}$ and $\pi_{\gamma\beta}$ into dominant morphisms followed by closed immersions.
[^4]: By the local criterion of flatness [@hartshorne Lemma III.10.3.A], this is equivalent to the flatness of $U_1^\infty\to M^\infty$.
[^5]: We put a superscript $p$ to distinguish it from its associated sheaf ${\mathbb{Z}}(M)$, to be introduced in .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Gaussian process (GP) predictors are an important component of many Bayesian approaches to machine learning. However, even a straightforward implementation of Gaussian process regression (GPR) requires $O(n^2)$ space and $O(n^3)$ time for a dataset of $n$ examples. Several approximation methods have been proposed, but there is a lack of understanding of the relative merits of the different approximations, and in what situations they are most useful. We recommend assessing the quality of the predictions obtained as a function of the compute time taken, and comparing to standard baselines (e.g., Subset of Data and FITC). We empirically investigate four different approximation algorithms on four different prediction problems, and make our code available to encourage future comparisons.'
author:
- |
Krzysztof [email protected]\
Computation and Neural Systems\
California Institute of Technology\
1200 E. California Boulevard\
Pasadena, CA 91125, USA Christopher K. I. Williams [email protected]\
Iain Murray [email protected]\
School of Informatics\
University of Edinburgh\
10 Crichton St, Edinburgh EH8 9AB, UK
bibliography:
- 'approxgp.bib'
title: A Framework for Evaluating Approximation Methods for Gaussian Process Regression
---
Gaussian process regression, subset of data, FITC, local GP.
Gaussian process (GP) predictors are widely used in non-parametric Bayesian approaches to supervised learning problems [@rasmussen-williams-06]. They can also be used as components for other tasks including unsupervised learning [@lawrence2004], and dependent processes for a variety of applications (e.g., @sudderth2009 [@adams2010]). The basic model on which these are based is Gaussian process regression (GPR), for which a standard implementation requires $O(n^2)$ space and $O(n^3)$ time for a dataset of $n$ examples, see e.g. @rasmussen-williams-06 [ch. 2]. Several approximation methods have now been proposed, as detailed below. Typically the approximation methods are compared to the basic GPR algorithm. However, as there are now a range of different approximations, the user is faced with the problem of understanding their relative merits, and in what situations they are most useful.
Most approximation algorithms have a tunable complexity parameter, which we denote as $m$. Our key recommendation is to study the quality of the predictions obtained as a function of the *compute time* taken as $m$ is varied, as times can be compared across different methods. New approximation methods should be compared against current baselines like Subset of Data and FITC (described in Secs. \[sec:sod\]–\[sec:fitc\]). The time decomposes into that needed for training the predictor (including setting hyperparameters), and test time; the user needs to understand which will dominate in their application. We illustrate this process by studying four different approximation algorithms on four different prediction problems. We have published our code in order to encourage comparisons of other methods against these baselines.
The structure of the paper is as follows: In Section \[sec:theory\] we outline the complexity of the full GP algorithm and various approximations, and give some specific details needed to apply them in practice. Section \[sec:compare\] outlines issues that should be considered when selecting or developing a GP approximation algorithm. Section \[sec:expts\] describes the experimental setup for comparisons, and the results of these experiments. We conclude with future directions and a discussion.
Approximation algorithms for Gaussian Process Regression (GPR) \[sec:theory\]
=============================================================================
A regression task has a training set ${\cal D}{\!=\!}\{{\mathbf{x}}_i,y_i\}_{i=1}^n$ with $D$-dimensional inputs ${\mathbf{x}}_i$ and scalar outputs $y_i$. Assuming that the outputs are noisy observations of a latent function $f$ at values $f_i{\!=\!}f({\mathbf{x}}_i)$, the goal is to compute a predictive distribution over the latent function value $f_*$ at a test location ${\mathbf{x}}_*$.
Assuming a Gaussian process prior over functions $f$ with zero mean, and covariance or kernel function $k(\cdot, \cdot)$, and Gaussian observations, $y_i = f_i + \epsilon_i$ where $\epsilon_i\sim{{\mathcal N}}(0,\sigma^2)$, gives Gaussian predictions $p(f_*{\!\mid\!}{\mathbf{x}}_*,{\cal
D}){\!=\!}{{\mathcal N}}(\overline{f}_*,\mathbbm{V}[f_*])$, with predictive mean and variance [see e.g., @rasmussen-williams-06 Sec. 2.2]: $$\begin{aligned}
\overline{f}_* &=& {\mathbf{k}}^\top({\mathbf{x}}_*) (K + \sigma^2 I)^{-1} {\mathbf{y}}\;\;{{\stackrel{\mathit{def}}{=}}}\;\;
{\mathbf{k}}^\top({\mathbf{x}}_*) {\mbox{\boldmath{$\alpha$}}}, \label{eq:meanpred}\\
\mathbbm{V}[f_*] &=& k({\mathbf{x}}_*,{\mathbf{x}}_*) - {\mathbf{k}}^\top({\mathbf{x}}_*) (K + \sigma^2 I)^{-1}
{\mathbf{k}}({\mathbf{x}}_*), \label{eq:varpred}\end{aligned}$$ where $K$ is the $n \times n$ matrix with $K_{ij} =
k({\mathbf{x}}_i,{\mathbf{x}}_j)$, ${\mathbf{k}}({\mathbf{x}}_*)$ is the $n \times 1$ column vector with the $i$th entry being $k({\mathbf{x}}_*,{\mathbf{x}}_i)$, ${\mathbf{y}}$ is the column vector of the $n$ target values, and ${\mbox{\boldmath{$\alpha$}}}=
(K + \sigma^2 I)^{-1} {\mathbf{y}}$.
The log marginal likelihood of the GPR model is also available in closed form: $$L = \log p({\mathbf{y}}|X) = -\tfrac{1}{2}\, {\mathbf{y}}^\top (K+\sigma^2_n I)^{-1}{\mathbf{y}}-\tfrac{1}{2}\log|K+\sigma^2 I|-\tfrac{n}{2}\log 2\pi.
\label{eq:logmarglike}$$ Typically $L$ is viewed as a function of a set of parameters $\theta$ that specify the kernel. Below we assume that $\theta$ is set by numerically maximizing $L$ with a routine like conjugate gradients. Computation of $L$ and the gradient $\nabla_{\theta} L$ can be carried out in $O(n^3)$. Optimizing $L$ is a maximum-likelihood type II or ML-II procedure for $\theta$; alternatively one might sample over $p(\theta|{\cal D})$ using e.g. MCMC. Equations \[eq:meanpred\]–\[eq:logmarglike\] form the basis of GPR prediction.
We identify three computational phases in carrying out GPR:
hyperparameter learning:
: The hyperparameters are learned, by for example maximizing the log marginal likelihood. This is often the most computationally expensive phase.
training:
: Given the hyperparameters, all computations that don’t involve test inputs are performed, such as computing ${\mbox{\boldmath{$\alpha$}}}$ above, and/or computing the Cholesky decomposition of $K + \sigma^2_n I$. This phase was called “precomputation” by @candela2007 [Sec. 9.6].
testing:
: Only the computations involving the test inputs are carried out, those which could not have been done previously. This phase may be significant if there is a very large test set, or if deploying a trained model on a machine with limited resources.
Table \[t:approxgpr\] lists the computational complexity of training and testing full GPR as a function of $n$. Evaluating the marginal likelihood $L$ and its gradient takes more operations than ‘training’ (i.e. computing the parts of and that don’t depend on ${\mathbf{x}}_*$), but has the same scaling with $n$. Hyperparameter learning involves evaluating $L$ for all values of the hyperparameters $\theta$ that are searched over, and so is more expensive than training for fixed hyperparameters.
Method Storage Training Mean Variance
-------- ---------- ----------- -------- ----------
Full $O(n^2)$ $O(n^3)$ $O(n)$ $O(n^2)$
SoD $O(m^2)$ $O(m^3)$ $O(m)$ $O(m^2)$
FITC $O(mn)$ $O(m^2n)$ $O(m)$ $O(m^2)$
Local $O(mn)$ $O(m^2n)$ $O(m)$ $O(m^2)$
: A comparison of the space and time complexity of the Full, SoD, FITC and Local methods, ignoring the time taken to select the $m$ subset/inducing points/clusters from the $n$ datapoints. Training: the time required for preliminary computations before the test point ${\mathbf{x}}_*$ is known, for each hyperparameter setting considered. Mean (resp. variance): the time needed to compute the predictive mean (variance) at test point ${\mathbf{x}}_*$. \[t:approxgpr\]
These complexities can be reduced in special cases, e.g.for stationary covariance functions and grid designs, as may be found e.g.in geoscience problems. In this case the eigenvectors of $K$ are the Fourier basis, and matrix inversions etc can be computed analytically. See e.g. @wikle2001 [@paciorek2007; @fritz-neuweiler-nowak-09] for more details.
Common methods for approximate GPR include Subset of Data (SoD), where data points are simply thrown away; inducing point methods [@candela2005], where $K$ is approximated by a low-rank plus diagonal form; Local methods where nearby data is used to make predictions in a given region of space; and fast matrix-vector multiplication (MVM) methods, which can be used with iterative methods to speed up the solution of linear systems. We discuss these in turn, so as to give coverage to the wide variety methods that have been proposed. We use the Fully Independent Training Conditional (FITC) method as it is recommended over other inducing point methods in @candela2007, and the Improved Fast Gauss Transform (IFGT) of @yang2004 as a representative of fast MVM methods.
Subset of Data {#sec:sod}
--------------
The simplest way of dealing with large amounts of data is simply to ignore some or most of it. The ‘Subset of Data (SoD) approximation’ simply applies the full GP prediction method to a subset of size $m <
n$. Therefore the computational complexities of SoD result from replacing $n$ with $m$ in the expressions for the full method (Table \[t:approxgpr\]). Despite the ‘obvious’ nature of SoD, most papers on approximate GP methods only compare to a GP applied to the full dataset of size $n$.
To complete the description of the SoD method we must also specify how the subset is selected. We consider two of the possible alternatives: 1) Selecting $m$ points randomly costs $O(m)$ if we need not look at the other points. 2) We select $m$ cluster centres from a Farthest Point Clustering (FPC, @gonzales-85) of the dataset; using the algorithm proposed by Gonzales this has computational complexity of $O(mn)$. In theory, FPC can be sped up to $O(n \log m)$ using suitable data structures [@feder-greene-88], although in practice the original algorithm can be faster for machine learning problems of moderate dimensionality. FPC has a random aspect as the first point can be chosen randomly. Our SoD implementation is based on `gp.m` in the [[Matlab]{}]{} `gpml` toolbox:\
<http://www.gaussianprocess.org/gpml/code/matlab/doc/>.
Rather than selecting the subset randomly, it is also possible to make a more informed choice. For example @lawrence-seeger-herbrich-03 came up with a fast selection scheme (the “informative vector machine”) that takes only $O(m^2
n)$. @keerthi-chu-06 also proposed a matching pursuit approach which has similar asymptotic complexity, although the associated constant is larger.
Inducing point methods: FITC {#sec:fitc}
----------------------------
A number of GP approximation algorithms use alternative kernel matrices based on *inducing points*, ${\mathbf{u}}$, in the $D$-dimensional input space [@candela2005]. Here we restrict the $m$ inducing points to be a subset of the training inputs. The Subset of Regressors (SoR) kernel function is given by $k_{SoR}({\mathbf{x}}_i,{\mathbf{x}}_j) = {\mathbf{k}}({\mathbf{x}}_i,{\mathbf{u}}) K^{-1}_{{\mathbf{u}}{\mathbf{u}}} {\mathbf{k}}({\mathbf{u}}, {\mathbf{x}}_j)$, and the Fully Independent Training Conditional (FITC) method uses $$k_{FITC}({\mathbf{x}}_i,{\mathbf{x}}_j) = k_{SoR}({\mathbf{x}}_i,{\mathbf{x}}_j) + \delta_{ij} [k({\mathbf{x}}_i,{\mathbf{x}}_j) -
k_{SoR}({\mathbf{x}}_i,{\mathbf{x}}_j)].$$ FITC approximates the matrix $K$ as a rank-$m$ plus diagonal matrix. An attractive property of FITC, not shared by all approximations, is that it corresponds to exact inference for a GP with the given $k_{FITC}$ kernel [@candela2007]. Other inducing point approximations (e.g. SoR, deterministic training conditionals) have similar complexity but @candela2007 recommend FITC over them. Since then there have been further developments [@titsias-09; @lazarogredilla2010], which would also be interesting to compare.
To make predictions with FITC, and to evaluate its marginal likelihood, simply substitute $k_{FITC}$ for the original kernel in Equations \[eq:meanpred\]–\[eq:logmarglike\]. This substitution gives a mean predictor of the form $\overline{f}_* = \sum_{i=1}^m \beta_i k({\mathbf{x}}_*, {\mathbf{x}}_i)$, where $i = 1, \ldots, m$ indexes the selected subset of training points, and the $\beta$s are obtained by solving a linear system. @snelson2007 [pp 60-62] showed that in the limit of zero noise FITC reduces to SoD.
We again choose a set of inducing points of size $m$ from the training inputs either randomly or using FPC, and use the FITC implementation from the `gpml` toolbox.
It is possible to “mix and match” the SoD and FITC methods, adapting the hyperparmeters to optimize the SoD approximation to the marginal likelihood, then using the FITC algorithm to make predictions using the same data subset and the SoD-trained hyperparameters. We refer to this procedure as the Hybrid method[^1]. We expect that saving time on the hyperparameter learning phase, $O(m^3)$ instead of $O(m^2n)$, will come at the cost of reducing the predictive performance of FITC for a given $m$.
Local GPR {#sec:localgpr}
---------
The basic idea here is of divide-and-conquer, although without any guarantees of correctness. We divide the $n$ training points into $k=\lceil \frac{n}{m} \rceil$ clusters each of size $m$, and run GPR in each cluster, ignoring the training data outside of the given cluster. At test time we assign a test input ${\mathbf{x}}_*$ to the closest cluster. This method has been discussed by @snelson-ghahramani-07. The hard cluster boundaries can lead to ugly discontinuities in the predictions, which are unacceptable if a smooth surface is required, for example in some physical simulations.
One important issue is how the clustering is done. We found that FPC tended to produce clusters of very unequal size, which limited the speedups obtained by Local GPR. Thus we devised a method we call Recursive Projection Clustering (RPC), which works as follows. We start off with all the data in one cluster $C$. Choose two data points at random from $C$, draw a line through these points and calculate the orthogonal projection of all points from $C$ onto the line. Split $C$ into two equal-sized subsets $C_L$ and $C_R$ depending on whether points are to the left or right of the median. Now repeat recursively in each cluster until the cluster size is no larger than $m$. In our implementation we make use of [[Matlab]{}]{}’s `sort` function to find the median value, taking time $O(n \log n)$ for $n$ datapoints, although it is possible to reduce median finding to $O(n)$ [@blum-etal-73]. Thus overall the complexity of RPC is $O(ns \log n)$, where $s=\lceil \log_2(n/m)
\rceil$. A test point ${\mathbf{x}}_*$ is assigned to the appropriate cluster by descending the tree of splits constructed by RPC.
Another issue concerns hyperparameter learning. $L$ is approximated by the sum of terms like Eq. \[eq:logmarglike\] over all clusters. Hyperparameters can either be tied across all clusters (“joint” training), or unique to each cluster (“separate” training). Joint training is likely to be useful for small $m$. We implemented Local GPR using the `gpml` toolbox with small modifications to sum gradients for joint training.
Iterative methods and IFGT matrix-vector multiplies {#sec:itmvm}
---------------------------------------------------
The Conjugate Gradients (CG) method (e.g., @golub1996) can be used at training time to solve the linear system $(K + \sigma^2 I)
{\mbox{\boldmath{$\alpha$}}}= {\mathbf{y}}$. Indeed, all GPR computations can be based on iterative methods [@gibbs1997]. CG and several other iterative methods (e.g., @li2007 [@liberty2007]) for solving linear systems require the ability to multiply a matrix of kernel values with an arbitrary vector.
Standard dense matrix-vector multiplication (MVM) costs $O(n^2)$. It has been argued (e.g., @gibbs1997 [@li2007]) that iterative methods alone provide a cost saving if terminated after $k\ll n$ matrix-vector multiplies. Papers often don’t state how CG was terminated [[e.g. @shen2005; @freitas2005]]{}, although some are explicit about using a small fixed number of iterations based on preliminary runs [[e.g. @gray2004a]]{}. Ad-hoc termination rules, or those using the ‘relative residual’ [@golub1996] (see Section \[sec:itresults\]) do not necessarily give the best trade-off between time and test-error. In Section \[sec:itresults\] we examine the progression of test error throughout training, to see what error/time trade-offs might be achieved by different termination rules.
Iterative methods are not used routinely for dense linear system solving, they are usually only recommended when the cost of MVMs is reduced by exploiting sparsity or other matrix structure. Whether iterative methods can provide a speedup for GPR or not, fast MVM methods will certainly be required to scale to huge datasets. Firstly, while other methods can be made linear in the size of the dataset size ($O(m^2n)$, see Table \[t:approxgpr\]), a standard MVM costs $O(n^2)$. Most importantly, explicitly constructing the $K$ matrix uses $O(n^2)$ memory, which sets a hard ceiling on dataset size. Storing the kernel elements on disk, or reproducing the kernel computations on the fly, is prohibitively expensive. Fast MVM methods potentially reduce the storage required, as well as the computation time of the standard dense implementation.
We have previously demonstrated some negative results concerning speeding up MVMs [@murray-09]: 1) if the kernel matrix were approximately sparse (i.e. many entries near zero) it would be possible to speed up MVMs using sparse matrix techniques, but in the hyperparameter regimes identified in practice this does not usually occur; 2) the piecewise constant approximations used by simple kd-tree approximations to GPR [@shen2005; @gray2004a; @freitas2005] cannot safely provide meaningful speedups.
The Improved Fast Gauss Transform (IFGT) is a MVM method that can be applied when using a squared-exponential kernel. The IFGT is based on a truncated multivariate Taylor series around a number of cluster centres. It has been applied to kernel machines in a number of publications, e.g. [@yang2004; @morariu-etal-09]. Our experiments use the IFGT implementation from the Figtree C++ package with [[Matlab]{}]{} wrappers available from <http://www.umiacs.umd.edu/~morariu/figtree/>. This software provides automatic choices for a number of parameters within IFGT. The time complexity of IFGT depends on a number of factors as described in [@morariu-etal-09], and we focus below on empirical results.
There are open problems with making iterative methods and fast MVMs for GPR work routinely. Firstly, unlike standard dense linear algebra routines, the number of operations depends on the hyperparameter settings. Sometimes the programs can take a very long time, or even crash due to numerical problems. Methods to diagnose and handle these situations automatically are required. Secondly, iterative methods for GPR are usually only applied to mean prediction, Eq. \[eq:meanpred\]; finding variances $\mathbbm{V}[f_*]$ would require solving a new linear system for each ${\mathbf{k}}({\mathbf{x}}_*)$. In principle, an iterative method could approximately factorize $(K +
\sigma^2 I)$ for variance prediction. To our knowledge, no one has demonstrated the use of such a method for GPR with good scaling in practice.
Comparing the Approximation Methods
-----------------------------------
Above we have reviewed the SoD, FITC, Hybrid, Local and Iterative MVM methods for speeding up GP regression for large $n$. The space and time complexities for the SoD, FITC, and Local methods are given in Table \[t:approxgpr\]; as explained above there are open problems with making iterative methods and fast MVMs work routinely for GPR, see also Secs. \[sec:itresults\] and \[sec:ifgt\_res\].
Comparing FITC to SoD, we note that the mean predictor contains the same basis functions as the SoD predictor, but that the coefficients are (in general) different as FITC has “absorbed” the effect of the remaining $n-m$ datapoints. Hence for fixed $m$ we might expect FITC to obtain better results. Comparing Local to SoD, we might expect that using training points lying nearer to the test point would help, so that for fixed $m$ Local would beat SoD. However, both FITC and Local have $O(m^2 n)$ training times (although the associated constants may differ), compared to $O(m^3)$ for SoD. So if equal training time was allowed, a larger $m$ could be afforded for SoD than the others. This is the key to the comparisons in Sec. \[sec:compsodfitclocal\] below. The Hybrid method has the same hyperparameter learning time as SoD by definition, but the training phase will take longer than SoD with the same $m$, because of the need for a final $O(m^2n)$ phase of FITC training, as compared to the $O(m^3)$ for SoD. However, as per the argument above, we would expect the FITC predictions to be superior to the SoD ones, even if the hyperparameters have not been optimized explicitly for FITC prediction; this is explored experimentally in Sec.\[sec:compsodfitclocal\].
At test time Table \[t:approxgpr\] shows that the SoD, FITC, Hybrid and Local approximations are $O(m)$ for mean prediction, and $O(m^2)$ for predictive variances. This means that the method which has obtained the best “$m$-size” predictor will win on test-time performance.
A Basis for Comparing Approximations \[sec:compare\]
====================================================
For fixed hyperparameters, comparing an approximate method to the full GPR is relatively straightforward: we can evaluate the predictive error made by the approximate method, and compare that against the “gold standard” of full GPR. The ‘best’ method could be the approximation with best predictions for a given computational cost, or alternatively the smallest computational cost for a given predictive performance. However, there are still some options e.g. different performance criteria to choose from (mean squared error, mean predictive log likelihood). Also there are different possible relevant computational costs (hyperparameter learning, training, testing) and definitions of cost itself (CPU time, ‘flops’ or other operation counts). It should also be borne in mind that any error measure compresses the predictive mean and variance functions into a single number; for low-dimensional problems it is possible to visualize these functions, see e.g. Fig. 9.4 in @candela2007, in order to help understand the differences between approximations.
It is rare that the appropriate hyperparameters are known for a given problem, unless it is a synthetic problem drawn from a GP. For real-world data we are faced with two alternatives: (i) compare approximate methods using the same set of hyperparameters as obtained by full GPR, or (ii) allow the approximate methods freedom to determine their own hyperparameters, e.g. by using approximate marginal likelihoods consistent with the approximations. Below we follow the second approach as it is more realistic, although it does complicate comparisons by changing both the approximation method and the hyperparameters.
In terms of computational cost we use the CPU time in seconds, based on [[Matlab]{}]{} implementations of the algorithms (except for the IFGT where the Figtree C++ code is used with [[Matlab]{}]{} wrappers). The core GPR calculations are well suited to efficient implementation in [[Matlab]{}]{}. Our SoD, FITC, Hybrid and Local GP implementations are all derived from the standard `gpml` toolbox of Rasmussen and Nickisch.
Before making empirical comparisons on particular datasets, we identify aspects of regression problems, models and approximations that affect the appropriateness of using a particular method:
**The nature of the underlying problem:** We usually standardize the inputs to have zero mean and unit variance on each dimension. Then clearly we would expect to require more datapoints to pin down accurately a higher frequency (more “wiggly”) function than a lower frequency one.
For multivariate input spaces there will also be issues of dimensionality, either wrt the intrinsic dimensionality of ${\mathbf{x}}$ (for example if the data lies on a manifold of lower dimensionality) or the apparent dimensionality. Note that if there are irrelevant inputs these can potentially be detected by a kernel equipped with “Automatic Relevance Determination” (ARD) ([@neal-96]; [@rasmussen-williams-06 p. 106]).
Another factor is the noise level on the data. An eigenanalysis of the problem (see e.g.@rasmussen-williams-06 [ Sec. 2.6]) shows that it is more difficult to discover low-amplitude components in the underlying function if there is high noise. It is relatively easy to get an upper bound on the noise level by computing the variance of the $y$’s around a given ${\mathbf{x}}$ location (or an average of such calculations), particularly if the lengthscale of variation of function is much larger than inter-datapoint distances (i.e. high sampling density); this provides a useful sanity check on the noise level returned during hyperparameter optimization.
**The choice of kernel function:** Selecting an appropriate family of kernel functions is an important part of modelling a particular problem. For example, poor results can be obtained when using an isotropic kernel on a problem where there are irrelevant input dimensions, while an ARD parameterization would be a better choice. Some approximation methods (e.g., the IFGT) have only been derived for particular kernel functions. For simplicity of comparison we consider only the SE-ARD kernel [@rasmussen-williams-06 p. 106], as that is the kernel most widely used in practice.
**The practical usability of a method:** Finally, some more mundane issues contribute significantly to the usability of a method, such as: (a) Is the method numerically robust? If there are problems it should be clear how to diagnose and deal with them. (b) Is it clear how to set tweak parameters e.g. termination criteria? Difficulties with these issues don’t just make it difficult to make fair comparisons, but reflect real difficulties with using the methods. (c) Does the method work efficiently for a wide range of hyperparameter settings? If not, hyperparameter searching must be performed much more carefully and one has to ask if the method will work well on good hyperparameter settings.
Experiments \[sec:expts\]
=========================
[**Datasets**]{}: We use four datasets for comparison. The first two are synthetic datasets, <span style="font-variant:small-caps;">synth2</span> and <span style="font-variant:small-caps;">synth8</span>, with $D{\!=\!}2$ and $D{\!=\!}8$ input dimensions. The inputs were drawn from a $N(0,I)$ Gaussian, and the function was drawn from a GP with zero mean and isotropic SE kernel with unit lengthscale. There are 30,543 training points and 30,544 test points in each dataset.[^2] The noise variance is $10^{-6}$ for <span style="font-variant:small-caps;">synth2</span>, and $10^{-3}$ for <span style="font-variant:small-caps;">synth8</span>. The <span style="font-variant:small-caps;">chem</span> dataset is derived from physical simulations relating to electron energies in molecules[^3] [@malshe-etal-05]. The input dimensionality is 15, and the data is split into 31,535 training cases and 31,536 test cases. Additional results on this dataset have been reported by @manzhos2008. The <span style="font-variant:small-caps;">sarcos</span> dataset concerns the inverse kinematics of a robot arm, and is used e.g. in @rasmussen-williams-06 [Sec. 2.5]. It has 21 input dimensions, 44,484 training cases and 4,449 test cases (the same split as in @rasmussen-williams-06). The <span style="font-variant:small-caps;">sarcos</span> dataset is already publicly available from <http://www.gaussianprocess.org>. All four datasets are included in the code and data tarfile associated with this paper.
[**Error measures**]{}: We measured the accuracy of the methods’ predictions on the test sets using the Standardized Mean Squared Error (SMSE), and Mean Standardized Log Loss (MSLL), as defined in [@rasmussen-williams-06 Sec. 2.5]. The SMSE is the mean squared error normalized by the MSE of the dumb predictor that always predicts the mean of the training set. The MSLL is obtained by averaging $-\log p(y_*|{\cal D}, {\mathbf{x}}_*)$ over the test set and subtracting the same score for a trivial model which always predicts the mean and variance of the training set. Notice that MSLL involves the predictive variances while SMSE does not.
Each experiment was carried out on a 3.47GHz core with at least 10GB available memory, except for Section \[sec:itresults\] which used 3GHz cores with 12GB memory. Approximate log marginal likelihoods were optimized wrt $\theta$ using Carl Rasmussen’s `minimize.m` routine from the `gpml` toolbox, using a maximum of 100 iterations. The code and data used to run the experiments is available from <http://homepages.inf.ed.ac.uk/ckiw/code/gpr_approx.html> .
In Section \[sec:itresults\] we provide results investigating the efficacy of iterative methods for GPR. In Section \[sec:ifgt\_res\] we investigate the utility of IFGT to speed up MVMs. Section \[sec:compsodfitclocal\] compares the SoD, FITC and Local approximations on the four datasets, and Section \[sec:compare\_gen\] compares predictions made with the learned hyperparameters and the generative hyperparameters on the synthetic datasets.
Results for iterative methods \[sec:itresults\]
-----------------------------------------------
Most attempts to use iterative methods for Gaussian processes have used conjugate gradient (CG) methods [@gibbs1997; @gray2004a; @shen2005; @freitas2005]. However, @li2007 introduced a method, which they called Domain Decomposition (DD), that over 50 iterations appeared to converge faster than CG. We have compared CG and DD for training a GP mean predictor based on 16,384 points from the <span style="font-variant:small-caps;">sarcos</span> data, with the same fixed hyperparameters used by @rasmussen-williams-06.
![Experiments with 16,384 training points. CG: conjugate gradients; DD: ‘domain decomposition’ with 16 randomly chosen clusters; CG-init: CG initialized with one iteration of DD (CG’s starting point of zero isn’t responsible for bad early behaviour); DD-RPC: clusters were chosen with recursive projection clustering (Section \[sec:localgpr\]). The horizontal lines give test performance for SoD with 4,096, 8,192 and 16,384 training points. Crosses on these lines also show the time taken. []{data-label="fig:iter"}](results/iter_rel_res_sarcos "fig:"){width="1.01\linewidth"}\
a\) <span style="font-variant:small-caps;">sarcos</span>
![Experiments with 16,384 training points. CG: conjugate gradients; DD: ‘domain decomposition’ with 16 randomly chosen clusters; CG-init: CG initialized with one iteration of DD (CG’s starting point of zero isn’t responsible for bad early behaviour); DD-RPC: clusters were chosen with recursive projection clustering (Section \[sec:localgpr\]). The horizontal lines give test performance for SoD with 4,096, 8,192 and 16,384 training points. Crosses on these lines also show the time taken. []{data-label="fig:iter"}](results/iter_smse_detail_sarcos "fig:"){width="1.01\linewidth"}\
b\) <span style="font-variant:small-caps;">sarcos</span>
![Experiments with 16,384 training points. CG: conjugate gradients; DD: ‘domain decomposition’ with 16 randomly chosen clusters; CG-init: CG initialized with one iteration of DD (CG’s starting point of zero isn’t responsible for bad early behaviour); DD-RPC: clusters were chosen with recursive projection clustering (Section \[sec:localgpr\]). The horizontal lines give test performance for SoD with 4,096, 8,192 and 16,384 training points. Crosses on these lines also show the time taken. []{data-label="fig:iter"}](results/smse_time_sarcos "fig:"){width="1.01\linewidth"}\
c\) <span style="font-variant:small-caps;">sarcos</span>
![Experiments with 16,384 training points. CG: conjugate gradients; DD: ‘domain decomposition’ with 16 randomly chosen clusters; CG-init: CG initialized with one iteration of DD (CG’s starting point of zero isn’t responsible for bad early behaviour); DD-RPC: clusters were chosen with recursive projection clustering (Section \[sec:localgpr\]). The horizontal lines give test performance for SoD with 4,096, 8,192 and 16,384 training points. Crosses on these lines also show the time taken. []{data-label="fig:iter"}](results/smse_time_synth8 "fig:"){width="1.01\linewidth"}\
d\) <span style="font-variant:small-caps;">synth8</span>
Figure \[fig:iter\]a) plots the ‘relative residual’, $\|(K {\!+\!}\sigma^2 I) {\mbox{\boldmath{$\alpha$}}}_t - {\mathbf{y}}\| / \|{\mathbf{y}}\|$, the convergence diagnostic used by @li2007 [Fig. 2], against iteration number for both their method and CG, where ${\mbox{\boldmath{$\alpha$}}}_t$ is the approximation to ${\mbox{\boldmath{$\alpha$}}}$ obtained at iteration $t$. We reproduce the result that CG gives higher and fluctuating residuals for early iterations. However, by running the simulation for longer, and plotting on a log scale, we see that CG converges, according to this measure, much faster at later iterations. Figure \[fig:iter\]a) is not directly useful for choosing between the methods however, because we do not know how many iterations are required for a competitive test-error.
Figure \[fig:iter\]b) instead plots test-set SMSE, and adds reference lines for the SMSEs obtained by subsets with 4,096, 8,192 and 16,384 training points. We now see that 50 iterations are insufficient for meaningful convergence on this problem. Figure \[fig:iter\]c) plots the SMSE against computer time taken on our machine[^4]. SoD performs better than the iterative methods.
These results depend on the dataset and hyperparameters. Figure \[fig:iter\]d) shows the test-set SMSE progression against time for 16,384 points from <span style="font-variant:small-caps;">synth8</span> using the true hyperparameters. Here CG takes a similar time to direct Cholesky solving. However, there is now a part of the error-time plot where the DD approach has better SMSEs at smaller times than either CG or SoD.
The timing results are heavily implementation and architecture dependent. For example, the results reported so far were run on a single 3GHz core. On our machines, the iterative methods scale less well when deployed on multiple CPU cores. Increasing the number of cores to four (using [[Matlab]{}]{}, which uses Intel’s MKL), the time to perform a $16384\!\times\!16384$ Cholesky decomposition decreased by a factor of 3.1, whereas a matrix vector multiply improved by only a factor of 1.7.
Results for IFGT \[sec:ifgt\_res\]
----------------------------------
We focus here on whether the IFGT provides fast MVMs for the datasets in our comparison. We used the isotropic squared-exponential kernel (which has one lengthscale parameter shared over all dimensions). For each of the four datasets we randomly chose 5000 datapoints to construct a kernel matrix, and a 5000-element random vector (with elements sampled from $U[0,1]$). Figure \[fig:mvm\] shows the MVM time as a function of lengthscale. For <span style="font-variant:small-caps;">synth2</span> and <span style="font-variant:small-caps;">synth8</span> the known lengthscale is 1. For the two other problems, and indeed many standardized regression problems, lengthscales of $\approx\!1$ (the width of the input distribution) are also appropriate. Figure \[fig:mvm\] shows that useful MVM speedups over a direct implementation are only obtained for <span style="font-variant:small-caps;">synth2</span>. The result on <span style="font-variant:small-caps;">sarcos</span> is consistent with @raykar2007’s result that IFGT does not accelerate GPR on this dataset.
![Plot of time vs lengthscale using IFGT for matrix-vector multiplication (MVM) on the four datasets. The Auto method was introduced in [@raykar2007] as a way to speed up IFGT in some regimes. \[fig:mvm\]](IFGT/Times2_SYNTH2.pdf "fig:"){width="1.01\linewidth"}\
<span style="font-variant:small-caps;">synth2</span>
![Plot of time vs lengthscale using IFGT for matrix-vector multiplication (MVM) on the four datasets. The Auto method was introduced in [@raykar2007] as a way to speed up IFGT in some regimes. \[fig:mvm\]](IFGT/Times2_SYNTH8.pdf "fig:"){width="1.01\linewidth"}\
<span style="font-variant:small-caps;">synth8</span>
![Plot of time vs lengthscale using IFGT for matrix-vector multiplication (MVM) on the four datasets. The Auto method was introduced in [@raykar2007] as a way to speed up IFGT in some regimes. \[fig:mvm\]](IFGT/Times2_CHEM.pdf "fig:"){width="1.01\linewidth"}\
<span style="font-variant:small-caps;">chem</span>
![Plot of time vs lengthscale using IFGT for matrix-vector multiplication (MVM) on the four datasets. The Auto method was introduced in [@raykar2007] as a way to speed up IFGT in some regimes. \[fig:mvm\]](IFGT/Times2_SARCOS.pdf "fig:"){width="1.01\linewidth"}\
<span style="font-variant:small-caps;">sarcos</span>
Comparison of SoD, FITC, Hybrid and Local GPR \[sec:compsodfitclocal\]
----------------------------------------------------------------------
All of the experiments below used the squared exponential kernel with ARD parameterization . The test times given below include computation of the predictive variances.
SoD was run with $m$ ascending in powers of 2 from $32, 64 \ldots$ up to $4096$. FITC was run with $m$ ranging from $8$ to $512$ in powers of two; this is smaller than for SoD as FITC is much more memory intensive. Local was run with $m$ ranging from $16$ to $2048$ in powers of two. For all experiments the selection of the subset/inducing points/clusters has a random aspect, and we performed five runs.
In Figure \[fig:timeres\] we plot the test set SMSE against hyperparameter training time (left column), and test time (right column) for the four methods on the four datasets. Figure \[fig:mslltimeres\] shows similar plots for the test set MSLL. When there are further choices to be made (e.g. subset selection methods, joint/separate estimation of hyperparameters), we generally present the best results obtained by the method; these choices are detailed at the end of this section for each dataset individually. Further details including tables of learned hyperparameters can be found in @chalupka-11, although the experiments were re-run for this paper, so there are some differences between the two.
The empirical times deviate from theory (Table \[t:approxgpr\]) most for the Local method for small $m$. There is overhead due to the creation of many small matrices in [[Matlab]{}]{}, so that (for example) $m=32$ is always slower (on our four datasets) than $m=64$ and $m=128$. This effect is demonstrated explicitly in [@chalupka-11 Fig. 4.1], and accounts for the bending back observed in the plots for Local. (This is present on all four datasets, but can be difficult to see in some of the plots.)
Looking at the hyperparameter training plots (left column), it is noticeable that SoD and FITC reduce monotonically with increasing time, and that SoD outperforms FITC on all datasets (i.e. for the same amount of time, the SoD performance is better). On the test time plots (right column) the pattern between SoD and FITC is reversed, with FITC being superior. These results are consistent with theoretical scalings (Table \[t:approxgpr\]): at training time FITC has worse scaling, at test time its scaling is the same[^5], and it turns out that its more sophisticated approximation does give better results.
Comparing Hybrid to SoD for hyperparameter learning, we note a general improvement in performance for very similar time; this is because the additional cost of one FITC training step at the end is small relative to the time taken to optimize the hyperparameters using the SoD approximation of the marginal likelihood. At test time the Hybrid results are inferior to FITC for the same $m$ as expected, but the faster hyperparameter learning time means that larger subset sizes can be used with Hybrid.
For Local, the most noticeable pattern is that the run time does not change monotonically with $m$. We also note that for small $m$ the other methods can make faster approximations than Local can for any value of $m$. For Local there is a general trend for larger $m$ to produce better results, although on <span style="font-variant:small-caps;">sarcos</span> the error actually increases with $m$, and for <span style="font-variant:small-caps;">synth2</span> the SMSE error rises for $m=1024, \; 2048$. However, Local often gives better performance than the other methods in the time regimes where it operates.
We now comment on the specific datasets:
<span style="font-variant:small-caps;">synth2</span>: This function was fairly easy to learn and all methods were able to obtain good performance (with SMSE close to the noise level of $10^{-6}$) for sufficiently large $m$. For SoD and FITC, it turned out that FPC gave significantly better results than random subset selection. FPC distributes the inducing points in a more regular fashion in the space, instead of having multiple close by in regions of high density. For Local, the joint estimation of hyperparameters was found to be significantly better than separate; this result makes sense as the target function is actually drawn from a single GP. For FITC and Hybrid the plots are cut off at $m=128$ and $m=256$ respectively, as numerical instabilities in the `gpml` FITC code for larger $m$ values gave larger errors.
<span style="font-variant:small-caps;">synth8</span>: This function was difficult for all methods to learn, notice the slow decrease in error as a function of time. The SMSE obtained is far above the noise level of $10^{-3}$. Both SoD and FITC did slightly better when selecting the inducing points randomly. For the Local method, again joint estimation of hyperparameters was found to be superior, as for <span style="font-variant:small-caps;">synth2</span>. For both <span style="font-variant:small-caps;">synth2</span> and <span style="font-variant:small-caps;">synth8</span> we note that the lengthscales learned by the FITC approximation did not converge to the true values even for the largest $m$, while convergence was observed for SoD and Local; see Appendix 1 in @chalupka-11 for full details.
<span style="font-variant:small-caps;">chem</span>: Both SoD and FITC did slightly better when selecting the inducing points randomly. Local with joint and separate hyperparameter training gave similar results. We report results on the joint method, for consistency with the other datasets.
<span style="font-variant:small-caps;">sarcos</span>: For SoD and FITC, FPC gave very slightly better results than random. Local with joint hyperparameter training did better than separate training.
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![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SYNTH2_hyp_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SYNTH2HypSMSE\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SYNTH2_test_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SYNTH2TestSMSE\]
\[-1mm\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SYNTH8_hyp_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SYNTH8HypSMSE\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SYNTH8_test_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SYNTH8TestSMSE\]
\[-1mm\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/CHEM_hyp_SMSE.pdf "fig:"){width="5.5cm"} \[fig:CHEMHypSMSE\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/CHEM_test_SMSE.pdf "fig:"){width="5.5cm"} \[fig:CHEMTestSMSE\]
\[-1mm\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SARCOS_hyp_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SARCOSHypSMSE\] ![SMSE (log scale) as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point (including variance computations, despite not being needed to report SMSE). Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:timeres\]](allPlots3/SARCOS_test_SMSE.pdf "fig:"){width="5.5cm"} \[fig:SARCOSTestSMSE\]
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![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SYNTH2_hyp_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SYNTH2HypMSLL\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SYNTH2_test_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SYNTH2TestMSLL\]
\[-1mm\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SYNTH8_hyp_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SYNTH8HypMSLL\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SYNTH8_test_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SYNTH8TestMSLL\]
\[-1mm\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/CHEM_hyp_MSLL.pdf "fig:"){width="5.5cm"} \[fig:CHEMHypMSLL\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/CHEM_test_MSLL.pdf "fig:"){width="5.5cm"} \[fig:CHEMTestMSLL\]
\[-1mm\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SARCOS_hyp_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SARCOSHypMSLL\] ![MSLL as a function of time (log scale) for the four datasets. Left: hyperparameter training time. Right: test time per test point. Points give the result for each run; lines connect the means of the 5 runs at each $m$. \[fig:mslltimeres\]](allPlots3/SARCOS_test_MSLL.pdf "fig:"){width="5.5cm"} \[fig:SARCOSTestMSLL\]
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Comparison with Prediction using the Generative Hyperparameters \[sec:compare\_gen\]
------------------------------------------------------------------------------------
For the <span style="font-variant:small-caps;">synth2</span> and <span style="font-variant:small-caps;">synth8</span> datasets it is possible to compare the results with learned hyperparameters against those obtained with hyperparameters fixed to the true generative values. We refer to these as the learned and fixed hyperparameter settings.
For the SoD and Local methods there is good agreement between the learned and fixed settings, although for SoD the learned setting generally performs worse on both SMSE and MSLL for small $m$, as would be expected given the small data sizes. The learned and fixed settings are noticeably different for SoD for $m \le 128$ on <span style="font-variant:small-caps;">synth2</span>, and $m \le 512$ on <span style="font-variant:small-caps;">synth8</span>.
For FITC there is also good agreement between the learned and fixed settings, although on <span style="font-variant:small-caps;">synth8</span> we observed that the learned model slightly outperformed the fixed model by around 0.05 nats for MSLL, and by up to 0.05 for SMSE. This may suggest that for FITC the hyperparameters that produce optimal performance may not be the generative ones.
Future directions
=================
We have seen that Local GPR can sometimes make better predictions than the other methods for some ranges of available computer time. However, our implementation suffers from unusual scaling behaviour at small $m$ due to the book-keeping overhead required to keep track of thousands of small matrices. More careful, lower-level programming than our [[Matlab]{}]{} code might reduce these problems.
It is possible to combine the SoD with other methods. As a dataset’s size tends to infinity, SoD (with random selection) will always beat the other approximations that we have considered, as SoD is the only method with no $n$-dependence (Table \[t:approxgpr\]). Of course the other approximate methods, such as FITC, could also be run on a subset. Investigating how to simultaneously choose the dataset size to consider, $n$, and the control parameter of an approximation, $m$, has received no attention in the literature to our knowledge.
Some methods will have more choices than a single control parameter $m$. For example, @snelson2006 optimized the locations of the $m$ inducing points, potentially improving test-time performance at the expense of a longer training time. A potential future area of research is working out how to intelligently balance the computer time spent on selecting and moving inducing points, while performing hyperparameter training, and choosing a subset size. Developing methods that work well in a wide variety of contexts without tweaking might be challenging, but success could be measured using the framework of this paper.
Conclusions
===========
We have advocated the comparison of GPR approximation methods on the basis of prediction quality obtained vs compute time. We have explored the times required for the hyperparameter learning, training and testing phases, and also addressed other factors that are relevant for comparing approximations. We believe that future evaluations of GP approximations should consider these factors (Sec. \[sec:compare\]), and compare error-time curves with standard approximations such as SoD and FITC. To this end we have made our data and code available to facilitate comparisons. Most papers that have proposed GP approximations have not compared to SoD, and on trying the methods it is often difficult to get appreciably below SoD’s error-time curve for the learning phase. Yet these methods are often more difficult to run and more limited in applicability than SoD.
On the datasets we considered, SoD and Hybrid dominate FITC in terms of hyperparameter learning. However, FITC (for as long as we ran it) gave better accuracy for a given test time. SoD, Hybrid and FITC behaved monotonically with subset/inducing-set size $m$, making $m$ a useful control parameter. The Local method produces more varied results, but can provide a win for some problems and cluster sizes. Comparison of the iterative methods, CG and DD, to SoD revealed that they shouldn’t be run for a small fixed number of iterations, and that performance can be comparable with simpler, more stable approaches. Faster MVM methods might make iterative methods more compelling, although the IFGT method only provided a speedup on the <span style="font-variant:small-caps;">synth2</span> problem out of our datasets. Assuming that hyperparameter learning is the dominant factor in computation time, the results presented above point to the very simple Subset of Data method (or the Hybrid variant) as being the leading contender. We hope this will act as a rallying cry to those working on GP approximations to beat this “dumb” method. This can be addressed both by empirical evaluations (as presented here), and by theoretical work.
Many approximate methods require choosing subsets of partitions of the data. Although farthest point clustering (FPC) improved SoD and FITC on the low-dimensional (easiest) problem, simple random subset selection worked similarly or better on all other datasets. Random selection also has better scaling (no $n$-dependence) for the largest-scale problems. The choice of partitioning scheme was important for Local regression: Our preliminary experiments showed that performance was severely hampered by many small clusters produced by FPC; we recommend our recursive partitioning scheme (RPC).
### Acknowledgments {#acknowledgments .unnumbered}
We thank the anonymous referees whose comments helped improve the paper. We also thank Carl Rasmussen, Ed Snelson and Joaquin Quiñinero-Candela for many discussions on the comparison of GP approximation methods.
This work is supported in part by the IST Programme of the European Community, under the PASCAL2 Network of Excellence, IST-2007-216886. This publication only reflects the authors’ views.
0.2in
[^1]: We thank one of the anonymous reviewers for suggesting this method.
[^2]: We thank Carl Rasmussen for providing these datasets.
[^3]: We thank Prof. Lionel Raff of Oklahoma State University and colleagues for permission to distribute this data.
[^4]: The time per iteration was measured on a separate run that wasn’t slowed down by storing the intermediate results required for these plots.
[^5]: In fact, careful comparison of the test time plots show that FITC takes longer than SoD; this constant-factor performance difference is due to an implementation detail in `gpml`, which represents the FITC and SoD predictors differently, although they could be manipulated into the same form.
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{
"pile_set_name": "ArXiv"
}
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---
author:
- 'N. J. Simm'
bibliography:
- 'fbm\_project\_refs3.bib'
title: Central limit theorems for the real eigenvalues of large Gaussian random matrices
---
Introduction
============
How many eigenvalues of a random matrix are real? This very natural and fundamental question was asked in 1994 by Edelman, Kostlan and Shub [@EKS94] who proved that if $G$ is an $N \times N$ matrix of independent identically distributed standard normal variables, and $N_{\mathbb{R}}$ is the number of real eigenvalues of $G$, then $$\mathbb{E}(N_{\mathbb{R}}) = \sqrt{2N/\pi}+O(1), \qquad N \to \infty \label{edelman}.$$ Note that we are not assuming $G$ is symmetric, in the usual parlance we say that $G$ belongs to the so-called Ginibre ensemble of real non-Hermitian random matrices, first considered by Ginibre in 1965 [@Gin65].
In addition to being of instrinsic mathematical interest, the statistics of non-Hermitian matrices also have important applications. The earliest such application is probably due to May [@M72] who showed that real random matrices describe the stability properties of large biological systems. Very recently it was shown [@FK15] that the counting of the average number of equilibria in a non-linear analogue of May’s model can be mapped to the problem of $N_{\mathbb{R}}$ and to the density of real eigenvalues in the Ginibre type ensembles. See also [@FD14; @MHNSS15] for further applications of $N_{\mathbb{R}}$ to the enumeration of equilibria in complex systems. The question of *fluctuations* in such contexts is usually extremely difficult and has only recently begun to receive attention [@S15].
The purpose of this article is to describe the asymptotic central limit theorem fluctuations around Edelman and company’s estimate . In other words, thinking of as a law of large numbers, what happens when one recenters $N_{\mathbb{R}}$ with respect to its expectation and studies the convergence in law of the fluctuating remainder?
Our approach to this problem is based on a formalism recently developed in [@KPTTZ15], which allowed the authors to characterize the large deviation behaviour for the probability of an anomalously small number of real eigenvalues of $G$. We will show how it is possible to adapt their methods to prove a central limit theorem for the number of real eigenvalues, in addition to the following generalization. From now on let $N=2n$ be even and denote the real eigenvalues of $G$ by $\lambda_{1},\lambda_{2},\ldots,\lambda_{N_{\mathbb{R}}}$. The quantity, $$X_{n}^{\mathbb{R}}(P) = \sum_{j=1}^{N_{\mathbb{R}}}P(\lambda_{j}/\sqrt{N}), \label{linstat}$$ is known as a *linear statistic* (but crucially, note that we only sum the real eigenvalues). The count of real eigenvalues is the special case $X^{\mathbb{R}}_{n}(1) = N_{\mathbb{R}}$.
\[th:var\] The variance of the total number of real eigenvalues of the standard $2n \times 2n$ Ginibre real random matrix is given by $$\mathrm{Var}(N_{\mathbb{R}}) = \frac{2\sqrt{2}}{\sqrt{\pi}}\sum_{k=1}^{n}\frac{\Gamma(2k-3/2)}{\Gamma(2k-1)}-\frac{2}{\pi}\sum_{k_{1}=1}^{n}\sum_{k_{2}=1}^{n}\frac{\Gamma(k_{1}+k_{2}-3/2)^{2}}{\Gamma(2k_{1}-1)\Gamma(2k_{2}-1)} \label{exact}$$ and has $n \to \infty$ asymptotics given by $$\mathrm{Var}(N_{\mathbb{R}}) = (2-\sqrt{2})\mathbb{E}(N_{\mathbb{R}})+O(1), \qquad n \to \infty \label{asympt}$$
Let us note that the asymptotics also appear in [@FN07] and weaker variance estimates (without the constant $2-\sqrt{2}$) were obtained in [@TV15] for non-Gaussian matrices. The same asymptotics (including the constant) apply to the generalized eigenvalue problem of real Ginibre matrices [@FM12]. Formulae and are proved in Section \[sec:cov\], including a generalization to the variance of for $P$ an even polynomial, see Proposition \[prop:cov\]. We also have a central limit theorem for linear statistics:
\[th:main\] Let $P(x)$ be any even polynomial with real coefficients and let $N=2n$ be even. Then in the limit $n \to \infty$, we have the convergence in distribution $$\frac{1}{\sqrt{\mathbb{E}(N_{\mathbb{R}})}}(X_{n}^{\mathbb{R}}(P) - \mathbb{E}(X_{n}^{\mathbb{R}}(P)) \to \mathcal{N}(0,\sigma^{2}(P)) \label{normconv}$$ where $\mathcal{N}(0,\sigma^{2}(P))$ denotes the normal distribution with mean $0$ and variance $$\sigma^{2}(P) := \frac{2-\sqrt{2}}{2}\int_{-1}^{1}P(x)^{2}\,dx \label{limvar}$$
For *Hermitian* random matrices, results of this type continue to occupy a major industry in the field, since at least the 1980s [@Jon82] with work continuing unabated to the present day. A quite comprehensive treatment was given by Johansson [@Joh98], who proved that for a general class of Hermitian ensembles, the linear statistic converges as $N \to \infty$, *without normalization*, to a normal random variable with finite variance. The lack of any normalization is usually interpreted as a consequence of strong correlations between the eigenvalues; indeed, for Hermitian matrices, the variance of remains bounded in $N$. In the non-Hermitian case, including all complex eigenvalues in the sum leads again to a bounded variance central limit theorem which is closely related to the Gaussian free field (GFF) [@F99; @RS06; @VR07; @AHM11; @OR14], a log-correlated field of great importance in mathematical physics and probability, see [@GFF] for a survey. See also [@FKS13; @LS15; @CW15] for further relations between linear statistics of random matrices and log-correlated fields. An important question for future work could be to determine if there a process interpolating between the Poisson fluctuations of Theorem \[th:main\] and the GFF obtained in [@VR07].
The $N \to \infty$ fate of the sum is therefore quite different to that typically encountered in random matrix theory, requiring a normalization of order $N^{-1/4}$ to ensure distributional convergence. Furthermore, linear statistics of random matrix eigenvalues involving a *random number of terms* have not been studied so widely. However, the Poissonian structure of the limiting Gaussian process can be guessed at in the following way. Viewed as a point process, it is known [@TZ11; @TKZ12; @TZ14] that the unscaled law of the real Ginibre eigenvalues converges as $N \to \infty$ to a system of annihilating Brownian motions taken at time $t=1$. Since the particles move independently, except for annihilation, the terms in the sum are approximately independent. Combined with Edelman’s law we may expect that is close to a sum of $O(\sqrt{N})$ independent random variables, for which the classical central limit theorem is applicable. These heuristics are enough to guess , but do not seem to explain the constant[^1] $2-\sqrt{2}$ in .
For finite $N$, the real spectrum of a Ginibre matrix is not completely independent and therefore requires its own proof. The results of [@For13] and [@Bee13] indicate that the real eigenvalues have quite interesting statistics, with linear repulsion at close range and Poisson behaviour at large spacings. Specifically, it is shown in [@For13] that if $p_{\mathrm{GinOE}}(s)$ is the probability density of real eigenvalue spacings, then $$\begin{split}
&p_{\mathrm{GinOE}}(s) \sim c_{0}s, \qquad \qquad s\to 0\\
&p_{\mathrm{GinOE}}(s) \sim c_{1}^{2}e^{-c_{1}s}, \qquad \hspace{2pt} s \to \infty \label{mermaid}
\end{split}$$ where $c_{0} = 1/(2\sqrt{2\pi})$ and $c_{1} = \zeta(3/2)/c_{0}$. This should be contrasted with the case of random symmetric matrices which have the Wigner-Dyson form (see [@Meh04]) $$\begin{split}
&p_{\mathrm{GOE}}(s) \sim (\pi^{2}/6)s, \qquad \hspace{10pt} s \to 0\\ &p_{\mathrm{GOE}}(s) \sim e^{-(\pi s)^{2}/16}, \qquad s \to \infty
\end{split}$$ In [@Bee13], the real eigenvalues of non-Hermitian matrices are shown to characterize level crossings in a superconducting quantum dot. Although not of the Ginibre type, the ensembles considered in [@Bee13] seem to share the same ‘mermaid statistics’ as .
Finally, as noted in [@FN07], the real eigenvalues of Ginibre matrices bare a close analogy to the study of real roots of random polynomials of high degree. For a quite general class of random polynomials, variance estimates and central limit theorems for the number of real roots were obtained by Maslova [@Masvar; @Masclt]. See [@TV13] for further references and recent progress in the field of random polynomials. An ensemble of random polynomials closely related to the present study are the $SO(2)$ polynomials defined by $p(x) = \sum_{j=0}^{N}c_{j}x^{j}$ where $c_{j}$ are i.i.d. Gaussian variables with mean zero and variance $\binom{N}{j}$. As for the Ginibre ensemble, the mean and variance of the number of real roots scale as $\sqrt{N}$ [@BD97] $$\mathrm{Var}(N_{\mathbb{R}}^{\mathrm{SO(2)}}) \sim c\sqrt{N}$$ where the constant $c=0.57173\ldots$ is close to the Ginibre constant $2-\sqrt{2}=0.5857\ldots$ in . We do not yet have a good explanation for this closeness.
To prove Theorem \[th:main\], we rely on the fact that the Ginibre ensemble is a Pfaffian point process. This means that all real and complex correlation functions of the eigenvalues can be written as a Pfaffian [@BS09; @FN07; @FN08; @SW08], in addition to the class of ensemble averages described in [@Sin07]. These results rely on the explicit knowledge of the joint probability density function of real and complex eigenvalues [@LS91; @E93]. In fact, for $f$ even, the moment generating function of the random variable is actually a *determinant* of size $n \times n$. In general, if $f$ is not even it is a Pfaffian of size $2n \times 2n$ that seems more difficult to analyze. From the determinantal formulae, the cumulants of can easily be extracted, and further analysis of their asymptotic behaviour is made possible by appropriately modifying the method used in [@KPTTZ15].\
\
*Note*: During the preparation of this article, the arXiv submission [@Kop15] appeared, which proves Theorem \[th:main\] under the different condition that $P$ is compactly supported inside $(-1,1)$. It is likely that combining the methods of [@Kop15] and the present article would yield an improved regularity condition on $P$.
Proof of the main result
========================
In the first section we compute the joint cumulant generating function of linear statistics of real and complex eigenvalues. In the second section we calculate the variance and prove Theorem \[th:var\]. In the final section we bound the higher order cumulants and establish our main result, Theorem \[th:main\].
Pfaffian and determinantal structures
-------------------------------------
The first step towards proving is to calculate the moment generating function of the statistic . A key role (see [@KG05] and [@Sin07]) is played by the real and complex integrals $$\begin{aligned}
A[h(x)h(y)]_{jk} &= \frac{1}{2}\int_{\mathbb{R}}dx\,\int_{\mathbb{R}}dy\,h(x)h(y)e^{-x^{2}/2-y^{2}/2}P_{j-1}(x)P_{k-1}(y)\mathrm{sign}(y-x) \label{realint}\\
B[g(z)g(\overline{z})]_{jk} &= -2i\int_{\mathbb{C}}g(z)g(\overline{z})P_{j-1}(z)P_{k-1}(\overline{z})\mathrm{sign}(\Im(z))e^{-z^{2}/2-\overline{z}^{2}/2}\mathrm{erfc}(\sqrt{2}|\Im(z)|)\,d^{2}z \label{complexint}\end{aligned}$$ where $\{P_{j}(x)\}_{j \geq 0}$ are a family of degree $j$ monic polynomials. We will choose them to be skew-orthogonal with respect to and , as in [@FN07] where they were calculated to be $$P_{2j}(x) = x^{2j}, \qquad P_{2j+1}(x) = x^{2j+1}-2jx^{2j-1} \label{skew-orthog}$$ With these polynomials specified, the following skew-orthogonality relation is satisfied: $$A[1]+B[1] = \mathrm{diag}\bigg\{\begin{pmatrix} 0 & r_{j-1}\\-r_{j-1} & 0 \end{pmatrix}\bigg\}_{j=1}^{n}$$ where $r_{j-1} = \sqrt{2\pi}\Gamma(2j-1)$.
\[prop:jointgen\] Let $f \in L^{2}(\mathbb{R})$ and $g \in L^{2}(\mathbb{C})$ be integrable functions and consider the linear statistics $$X^{\mathbb{R}}_{N}(f) = \sum_{j=1}^{N_{\mathbb{R}}}f(\lambda_{j}), \qquad X^{\mathbb{C}}_{N}(g) = \sum_{j=1}^{N_{\mathbb{C}}}g(z_{j}) \label{realstat}$$ Then the joint cumulant generating function of is given by: $$\log\mathbb{E}\left(\mathrm{exp}\left(sX^{\mathbb{R}}_{N}(f)+tX^{\mathbb{C}}_{N}(g)\right)\right) = \frac{1}{2}\log\mathrm{det}\left(I_{2n}+M^{\mathbb{R}}[e^{sf(x)+sf(y)}-1]+M^{\mathbb{C}}[e^{tg(z)+tg(\overline{z})}-1]\right) \label{det}$$ where $I_{2n}$ is the $2n \times 2n$ identity matrix and $M^{\mathbb{R}/\mathbb{C}}[h(x,y)]$ are $2n \times 2n$ block matrices, where block $(j,k)$ is given by $$\begin{split}
M^{\mathbb{R}}[h(x,y)]_{jk} &= \frac{1}{\sqrt{2\pi}\Gamma(2j-1)}\begin{pmatrix} -A[h(x,y)]_{2j,2k-1} & -A[h(x,y)]_{2j,2k}\\ A[h(x,y)]_{2j-1,2k-1} & A[h(x,y)]_{2j-1,2k}. \label{mmatrix}
\end{pmatrix}\\
M^{\mathbb{C}}[g(z,\overline{z})]_{jk} &= \frac{1}{\sqrt{2\pi}\Gamma(2j-1)}\begin{pmatrix} -B[g(z,\overline{z})]_{2j,2k-1} & -B[g(z,\overline{z})]_{2j,2k}\\ B[g(z,\overline{z})]_{2j-1,2k-1} & B[g(z,\overline{z})]_{2j-1,2k}
\end{pmatrix}
\end{split}$$
The resulting structure of Proposition \[prop:jointgen\] is reminiscent of formula (3.1) in Tracy and Widom [@TW98], which proved to be extremely useful for the $\beta=2$ Hermitian ensembles.
This follows from a result of Sinclair [@Sin07] combined with an important observation of Forrester and Nagao [@FN07]. Namely, we apply Theorem 2.1 of [@Sin07] but as was observed in [@FN07] the proof continues to hold separately for the real and complex eigenvalues. Namely, if we define $$X^{\mathbb{C}}_{N}(g) = \sum_{j=1}^{N_{\mathbb{C}}}g(z_{j})$$ where $z_{j}$ are the purely complex eigenvalues, then one has a slightly more general statement $$\mathbb{E}(\mathrm{exp}(sX_{N}^{\mathbb{R}}(f)+tX_{N}^{\mathbb{C}}(g))) = \frac{\mathrm{Pf}(A[e^{sf(x)+sf(y)}]+B[e^{tg(z)+tg(\overline{z})}])}{2^{N(N+1)/4}\prod_{j=1}^{N}(\Gamma(j/2))}.$$ By normalization of the generating function and linearity of the scalar products $A$ and $B$, we have $$\mathbb{E}(\mathrm{exp}(sX_{N}^{\mathbb{R}}(f)+tX_{N}^{\mathbb{C}}(g))) = \frac{\mathrm{Pf}(A[1]+B[1]+A[e^{sf(x)+sf(y)}-1]+B[e^{tg(z)+tg(\overline{z})}-1])}{\mathrm{Pf}(A[1]+B[1])} \label{normgen}$$ Due to the skew-orthogonality of the $P_{j}'s$, the matrix $A[1]+B[1]$ is block diagonal and skew-symmetric: $$A[1]+B[1] = \mathbf{r} \otimes J,\qquad \mathbf{r} = \mathrm{diag}(r_{0},\ldots,r_{n-1}),\qquad J = \begin{pmatrix} 0 & 1\\ -1 & 0 \end{pmatrix}$$ with $r_{j} = \sqrt{2\pi}\Gamma(2j+1)$. Taking logarithms and writing the Pfaffians as square roots of determinants gives after elementary algebra.
A further simplification occurs whenever the functions $f$ and $g$ are both even. In this case the Pfaffian has a checkerboard structure of zeros and the Pfaffians reduce to determinants of half the size. We then have a *bona fide* determinant
$$\mathbb{E}(\mathrm{exp}(sX_{N}^{\mathbb{R}}(f)+tX_{N}^{\mathbb{C}}(g))) = \mathrm{det}\bigg\{\delta_{jk}+\frac{(A[e^{sf(x)+sf(y)-1}]+B[e^{tg(z)+tg(\overline{z})}-1])_{2j-1,2k}}{\sqrt{2\pi\Gamma(2j-1)\Gamma(2k-1)}}\bigg\}_{j,k=1}^{N} \label{detrem}$$
which is a generalization of formula (6) in [@KPTTZ15] (setting $g=0$ and $f=1$). See also [@FN08] for similar calculations.
To proceed, we will focus our attention on the real eigenvalues and set $g \equiv 0$ from now on. To prove the central limit theorem we will calculate the cumulants of $X^{\mathbb{R}}_{N}(f)$, for which the determinantal formula is quite well-suited.
\[lem:cumu\] The $l^{\mathrm{th}}$ order cumulant $\kappa_{l}$ of any even linear statistic $X_{N}^{\mathbb{R}}(f)$ is given by $$\kappa_{l}(f) = l!\sum_{m=1}^{l}\frac{(-1)^{m+1}}{m}\sum_{\substack{\nu_{1}+\ldots+\nu_{m}=l\\\nu_{i} \geq 1}}\frac{\mathrm{Tr}M^{(\nu_{1})}[f]\ldots M^{(\nu_{m})}[f]}{\nu_{1}!\ldots \nu_{m}!} \label{kapp}$$ where $$M^{(\nu)}[f]_{jk} := \frac{A[(f(x)+f(y))^{\nu}]_{2j-1,2k}}{\sqrt{2\pi \Gamma(2j-1)\Gamma(2k-1)}}$$ and $A[f(x,y)]$ is given by .
From formula with $g=0$, we get $$\begin{aligned}
&[s^{l}]\log\mathbb{E}(\mathrm{exp}(sX_{N}^{\mathbb{R}}(f))) = [s^{l}]\log\mathrm{det}\bigg\{\delta_{jk}+\frac{(A[e^{sf(x)+sf(y)}-1]_{2j-1,2k}}{\sqrt{2\pi\Gamma(2j-1)\Gamma(2k-1)}}\bigg\}_{j,k=1}^{n}\\
&=[s^{l}]\mathrm{Tr}\log\bigg\{\delta_{jk}+\frac{(A[e^{sf(x)+sf(y)}-1]_{2j-1,2k}}{\sqrt{2\pi\Gamma(2j-1)\Gamma(2k-1)}}\bigg\}_{j,k=1}^{n}\\
&=[s^{l}]\sum_{m=1}^{\infty}\frac{(-1)^{m+1}}{m}\mathrm{Tr}\left(\bigg\{\frac{A[e^{sf(x)+sf(y)}-1]_{2j-1,2k}}{\sqrt{2\pi\Gamma(2j-1)\Gamma(2k-1)}}\bigg\}_{j,k=1}^{n}\right)^{m}\end{aligned}$$ Expanding the term $e^{s(f(x)+f(y))}-1$ in a Taylor series and re-ordering the sum gives .
The covariance {#sec:cov}
--------------
The main purpose of this section is to prove the following
\[prop:cov\] Let $P(x)$ and $Q(x)$ be any even polynomials with real coefficients. Then the covariance of the linear statistics $X^{\mathbb{R}}_{n}[P]$ and $X^{\mathbb{R}}_{n}[Q]$ satisfies the asymptotic formula $$\lim_{n \to \infty}\mathrm{Cov}\bigg\{n^{-1/4}X^{\mathbb{R}}_{n}[P],n^{-1/4}X^{\mathbb{R}}_{n}[Q]\bigg\} = \frac{(2-\sqrt{2})}{\sqrt{\pi}}\int_{-1}^{1}P(x)Q(x)\,dx$$
To compute the covariance of a general polynomial linear statistic, it suffices to just consider the case of monomials $$C_{p,q} := \mathrm{Cov}(X^{\mathbb{R}}_{n}(\lambda^{p}),X^{\mathbb{R}}_{n}(\lambda^{q})) \label{covmon}$$ Our goal in what follows will be to first find an exact formula for $C_{p,q}$ in Lemmas \[lem:cpq\] and \[lem:exact\], and then compute the large-$n$ asymptotics, which is done in Proposition \[prop:dblsum\]. Throughout the paper we will make use of the notation $$f^{(r,s)}_{j,k} := A[x^{r}y^{s}]_{jk}. \label{fpqdef}$$
\[lem:cpq\] The covariance of two even monomial linear statistics is given for any even matrix dimension $N=2n$ by the formula: $$\begin{split}
C_{p,q} &= n^{-(p+q)/2)}\sum_{k_{1}=1}^{n}\frac{f^{(p,q)}_{2k_{1}-2,2k_{1}-1}+f^{(q,p)}_{2k_{1}-2,2k_{1}-1}+f^{(0,p+q)}_{2k_{1}-2,2k_{1}-1}+f^{(p+q,0)}_{2k_{1}-2,2k_{1}-1}}{\sqrt{2\pi}\Gamma(2k_{1}-1)}\\
&-n^{-(p+q)/2}\sum_{k_{1},k_{2}=1}^{n}\frac{f^{(0,p)}_{2k_{1}-2,2k_{2}-1}f^{(0,q)}_{2k_{2}-2,2k_{1}-1}+f^{(p,0)}_{2k_{1}-2,2k_{2}-1}f^{(q,0)}_{2k_{2}-2,2k_{1}-1}}{2\pi\Gamma(2k_{1}-1)\Gamma(2k_{2}-1)} \label{cpq}
\end{split}$$
This follows from expressing $C_{p,q}$ in terms of variances using the identity $$2C_{p,q} = \kappa_{2}(\lambda^{p}+\lambda^{q})-\kappa_{2}(\lambda^{p})-\kappa_{2}(\lambda^{q}) \label{varident}$$ The variance terms are then calculated from Lemma \[lem:cumu\] with $l=2$.
The coefficients in appearing in can be evaluated in the following convenient form.
\[lem:exact\] For any even $p$ and $q$, the following exact formula holds: $$f^{(p,q)}_{2k_{1}-2,2k_{2}-1} = \Gamma(k_{1}+k_{2}+(p+q)/2-3/2)+qE(k_{1}+p/2,k_{2}+q/2-1) \label{fpq}$$ where $$E(j,k) := (k-1)!2^{k-1}\sum_{i=0}^{k-1}\frac{\Gamma(i+j-1/2)}{2^{i}i!}.$$ The second term is an error term that satisfies the inequality $$\begin{split}
E(k_{1}+p/2,k_{2}+q/2-1) &\leq c(k_{2}+q/2-2)!2^{k_{2}}\sum_{i=0}^{\infty}\frac{\Gamma(i+k_{1}+p/2-1/2)}{2^{i}i!}\\
&\leq c\sqrt{n}2^{k_{1}+k_{2}}\Gamma(k_{2}+q/2-3/2)\Gamma(k_{1}+p/2-1/2) \label{facbound}
\end{split}$$ where $c$ is a constant independent of $k_{1},k_{2}$ and $n$.
The first term in $\Gamma(k_{1}+k_{2}+(p+q)/2-3/2)$ in is a natural generalization of the case $p=q=0$ found in [@KPTTZ15] and will play just as important a role here in determining the $n \to \infty$ asymptotics.
From the identities $P_{2k_{1}-2}(x)x^{p} = P_{r+2k_{1}-2}(x)$ and $P_{2k_{2}-1}(y)y^{q} = P_{q+2k_{2}-1}+qy^{q+2k_{2}-3}$ we have $$f^{(p,q)}_{2k_{1}-2,2k_{2}-1} = f^{(0,0)}_{2k_{1}+p-2,2k_{2}+q-1} + qf^{(2k_{1}+p-2,2k_{2}+q-3)}_{0,0}$$ The proof is completed by verifying the following identities which are a simple integration exercise: $$\begin{split}
&f^{(0,0)}_{2k_{1}+p-2,2k_{2}+q-1} = \Gamma(k_{1}+k_{2}+(p+q)/2-3/2)\\
&f^{(2k_{1}+p-2,2k_{2}+q-3)}_{0,0} = E(k_{1}+p/2,k_{2}+q/2-1)
\end{split}$$
The key point is that to prove Theorem \[th:main\], it will suffice to only consider the contribution from the first term in . This is proved more generally for all cumulants in Proposition \[prop:errors\].
To extract the asymptotics based on just the first term in , we have the following
\[prop:dblsum\] Consider the sum $$S_{p,q} := N^{-(p+q+1)/2}\sum_{k_{1},k_{2}=1}^{N}\frac{\Gamma(k_{1}+k_{2}+\frac{q}{2}-3/2)\Gamma(k_{1}+k_{2}+\frac{p}{2}-3/2)}{\Gamma(2k_{1}-1)\Gamma(2k_{2}-1)} \label{spq}$$ Then the following limit holds: $$\lim_{n \to \infty}S_{p,q} = \sqrt{\pi}\frac{2^{(p+q+1)/2}}{p+q+1}$$
Our strategy will be to bound the sum from above and below. An upper bound can be obtained by extending the $k_{2}$ range of summation to $\infty$: $$\begin{split}
S_{p,q} &\leq n^{-(p+q+1)/2}\sum_{k_{1}=1}^{n}\frac{\Gamma(k_{1}+(p-1)/2)\Gamma(k_{1}+(q-1)/2)}{\Gamma(2k_{1}-1)}\\
&\times {}_2 F_1([k_{1}+(p-1)/2,k_{1}+(q-1)/2],[1/2],1/4) \label{hypergeom}
\end{split}$$ where ${}_2 F_1$ is the classical Gauss hypergeometric function. Since the summand is independent of $n$, it suffices to substitute the $k_{1} \to \infty$ asymptotics in . Hence we need the asymptotics of the hypergeometric function with fixed argument and large parameters. These were calculated by several authors using the method of steepest descent, see e.g. [@P13]. Indeed, the main result in Section $4$ of [@P13] and Stirling’s formula imply that $$\begin{split}
&\frac{\Gamma(k_{1}+(\chi_{p}-1)/2)\Gamma(k_{1}+(q-1)/2)}{\Gamma(2k_{1}-1)}{}_2 F_1([k_{1}+(p-1)/2,k_{1}+(q-1)/2],[1/2],1/4)\\
&\sim \sqrt{\pi}(2k_{1})^{(p+q-1)/2} = \sqrt{\pi}(2k_{1})^{(p+q-1)/2}, \qquad k_{1} \to \infty
\end{split}$$ Inserting this into the summand of shows that $$\begin{split}
\lim_{n \to \infty}S_{p,q} &\leq \lim_{n \to \infty}n^{-(p+q+1)/2}\sqrt{\pi}\sum_{k_{1}=1}^{n}(2k_{1})^{(p+q-1)/2}\\
&=\sqrt{\pi}\frac{2^{(p+q+1)/2}}{p+q+1}
\end{split}$$
To obtain a lower bound, we will use the techniques of [@KPTTZ15]. The main idea is to write the Gamma functions in the numerator of as Gaussian integrals. For $a\geq0$ even, we have $$\Gamma(k_{1}+k_{2}+a/2-3/2) = 2\int_{\mathbb{R}_{+}}x^{a}\,x^{2k_{1}-2}\,x^{2k_{2}-2}\,e^{-x^{2}}\,dx$$ Substituting this expression for the numerator in and summing over $k_{1}$ and $k_{2}$ leads to an integral representation $$S_{p,q} = n^{-(p+q+1)/2}4\int_{\mathbb{R}^{2}_{+}}dx_{1}\,dx_{2}\, x_{1}^{p}x_{2}^{q}\cosh_{n-1}(x_{1}x_{2})^{2}e^{-x_{1}^{2}-x_{2}^{2}}$$ where we have employed the hyperbolic cosine series $\cosh_{n-1}(x) = \sum_{k=0}^{n-1}\frac{x^{2k}}{(2k)!}$. By Lemma $4$ of [@KPTTZ15], we have the lower bound $$\cosh_{n-1}(x_{1}x_{2}n) \geq h_{n}e^{x_{1}x_{2}n}1(x_{1}x_{2}<T_{n}) \label{lowerbnd}$$ where $\lim_{n \to \infty}T_{n}=2$ and $\lim_{n \to \infty}h_{n}=1/2$. Changing variables $x_{i} \to \sqrt{n}x_{i}$ for $i=1,2$ in and inserting , we get $$\begin{split}
S_{p,q} &\geq 4\sqrt{n}h_{n}^{2}\int_{\mathbb{R}_{+}^{2}}dx_{1}\,dx_{2}\,x_{1}^{p}x_{2}^{q}1(x_{1}x_{2} < S_{n})e^{-n(x_{1}-x_{2})^{2}}\\
&\geq 4\sqrt{n}h_{n}^{2}\int_{0}^{\sqrt{T_{n}}}\int_{0}^{\sqrt{T_{n}}}dx_{1}\,dx_{2}\,x_{1}^{p}x_{2}^{q}e^{-n(x_{1}-x_{2})^{2}}\\
&=4\sqrt{n}h_{n}^{2}\frac{1}{2}\int_{0}^{\sqrt{T_{n}}}dR\,\int_{-R}^{R}dz\,\left(\left(\frac{R+z}{2}\right)^{p}\left(\frac{R-z}{2}\right)^{q}\right.\\
&\qquad \qquad \qquad \qquad+\left.\left(\frac{2\sqrt{2}-R-z}{2}\right)^{p}\left(\frac{2\sqrt{2}-R+z}{2}\right)^{q}\right)e^{-nz^{2}}\\
&\sim 4\sqrt{n}h_{n}^{2}\frac{1}{2}\int_{0}^{\sqrt{T_{n}}}dR\,\left((R/2)^{p+q}+((2\sqrt{2}-R)/2)^{p+q}\right)\int_{-R}^{R}dz\,e^{-nz^{2}}\\
&\sim \sqrt{\pi}\frac{2^{(p+q+1)/2}}{(p+q+1)}
\end{split}$$ where we used that the domain $\{x_{1}x_{2}<T_{n}\}\cap \mathbb{R}^{2}_{+}$ contains the square $[0,\sqrt{T_{n}}]^{2}$. The subsequent estimates follow from integration by parts.
To complete the proof of Proposition \[prop:cov\], it is enough to observe that the first line of is asymptotic to $$4\sum_{k_{1}=1}^{n}\frac{\Gamma(2k_{1}+\frac{p+q}{2}-3/2)}{\Gamma(2k_{1}-1)\sqrt{2\pi}} \sim \frac{2\sqrt{2}(2n)^{(p+q+1)/2}}{p+q+1}$$ By Proposition \[prop:dblsum\], the second line is asymptotic to $\frac{(2n)^{(p+q+1)/2}2}{\sqrt{\pi}(p+q+1)}$. The difference of these two terms divided by the normalizing factor $(2n)^{(p+q+1)/2}$ is equal to $\frac{(\sqrt{2}-1)}{\sqrt{\pi}}\int_{-1}^{1}x^{p+q}\,dx$. The fact that nothing contributes from the second term in is proved for all cumulants in Proposition \[prop:errors\].
Higher cumulants and Gaussian fluctuations
------------------------------------------
Specialising now to the case $f \equiv P$ of an even polynomial, we will prove in this section that the cumulants of with any order $l \geq 3$ are $O(\sqrt{n})$ as $n \to \infty$. Due to the normalization of order $n^{-1/4}$ in , this bound will be sufficient to conclude the central limit theorem and completes the proof of our main result, Theorem \[th:main\].
By Lemma \[lem:cumu\] it will suffice to prove that the trace in satisfies the bound $$\mathrm{Tr}M^{(\nu_{1})}[P]\ldots M^{(\nu_{m})}[P] = O(\sqrt{n}), \qquad n \to \infty \label{cumubound}$$ If $P$ is an even polynomial, the above trace is a finite linear combination of terms of the form $$\mathcal{Z}_{n,m} := n^{-\mathcal{M}_{m}}\sum_{k_{1},\ldots,k_{m}}f^{(2r_{1},2s_{1})}_{2k_{1}-2,2k_{2}-1}\ldots f^{(2r_{m-1},2s_{m-1})}_{2k_{m-1}-2,2k_{m}-1}f^{(2r_{m},2s_{m})}_{2k_{m}-2,2k_{1}-1} \label{partitionsum}$$ where $\mathcal{M}_{m} = \sum_{i=1}^{m}(r_{i}+s_{i})$ and as before we have $f^{(2r_{i},2s_{i})}_{2k-2,2j-1} = A[x^{2r_{i}}y^{2s_{i}}]_{2k-1,2j}$ which are explicitly evaluated in . This follows by definition of the trace and expanding $P$ in a basis of monomials. We will prove with two Propositions. First we estimate the leading term in by substituting just the first factor from , denoted $\Gamma^{(2r_{i},2s_{i})}_{2k_{i}-2,2k_{i+1}-1}$. Then in the second Proposition we deal with the error term in using the bound .
Define $$\Gamma^{(2r_{i},2s_{i})}_{2k_{i}-2,2k_{i+1}-1} = \Gamma(k_{i}+k_{i+1}+r_{i}+s_{i}-3/2)$$ for $i=1,\ldots,m$, where $k_{m+1} \equiv k_{1}$ and define exponents $$\mathcal{M}_{m} = \sum_{i=1}^{m}(r_{i}+s_{i})$$ Then the sum $$\mathcal{Z}^{(0)}_{n,m} := \sum_{k_{1},\ldots,k_{m}}\Gamma^{(2r_{1},2s_{1})}_{2k_{1}-2,2k_{2}-1}\ldots \Gamma^{(2r_{m-1},2s_{m-1})}_{2k_{m-1}-2,2k_{m}-1}\Gamma^{(2r_{m},2s_{m})}_{2k_{m}-2,2k_{1}-1}$$ is $O(n^{1/2+\mathcal{M}_{m}})$ as $n \to \infty$.
As for the covariance calculation, we write the Gamma factors as Gaussian integrals: $$\Gamma^{(r_{i},s_{i})}_{2k_{i}-2,2k_{i+1}-1} = \int_{\mathbb{R}}dx\,x^{2r_{i}+2s_{i}}\,x^{2k_{i}-2}\,x^{2k_{i+1}-2}\,e^{-x^{2}}$$ shows that $$Z_{n,m}^{(0)} = n^{-\mathcal{M}_{m}}\int_{\mathbb{R}^{m}}\,\prod_{j=1}^{m}dx_{j}\,x_{j}^{2r_{j}+2s_{j}}\cosh_{n-1}(x_{j}x_{j+1})e^{-x_{j}^{2}}. \label{zngauss}$$ We now use the obvious bound $\cosh_{n-1}(x) \leq \cosh(x)$ on every factor except one, which we write as a contour integral: $$\cosh_{n-1}(x_{j}x_{1}) = \oint_{\mathcal{C}}\frac{dz}{2\pi i }\frac{z^{-2n+1}}{1-z^{2}}e^{zx_{j}x_{1}}$$ where $\mathcal{C}$ is a small loop around $z=0$. Writing the other $\cosh$ factors as exponentials leads to a finite linear combination of terms of the form $$Z_{n,m}^{(0)} \leq c_{m}\oint_{}\frac{dz}{2\pi i }\frac{z^{-2n+1}}{1-z^{2}}\int_{\mathbb{R}^{m}}\left(\prod_{j=1}^{m}dx_{j}\,x_{j}^{2r_{j}+2s_{j}}\right)\mathrm{exp}\left(-\mathbf{x}^{\mathrm{T}}A(z)\mathbf{x}\right) \label{gaussint}$$ where $\mathbf{x} = (x_{1},\ldots,x_{m})$ and $$\label{cyctridiag}
A(z) = \begin{pmatrix} 1 & -\alpha_1/2 & 0 & 0 & \ldots & 0 & -z/2\\
-\alpha_1/2 & 1 & -\alpha_2/2 & 0 & 0 & \ldots & 0\\
0 & -\alpha_{2}/2 & 1 & -\alpha_{3}/2 & 0 & \ldots & 0\\
\vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \vdots\\
0 & \ldots & 0 & -\alpha_{m-3}/2 & 1 & -\alpha_{m-2}/2 & 0\\
0 & \ldots & 0 & 0 & -\alpha_{m-2}/2 & 1 & -\alpha_{m-1}/2\\
-z/2 & 0 & \ldots & 0 & 0 & -\alpha_{m-1}/2 & 1
\end{pmatrix}$$ where $\alpha_{i} \in \{1,-1\}$. According to Wick’s formula, the integral , which is essentially the moments of a multivariate Gaussian, can be evaluated explicitly in terms of the determinant and inverse of $A(z)$. We have $$\det(A(z)) = (m-1)2^{-m}(z-A_{m})((m-1)z+(m+1)A_{m}) \label{detz}$$ and $$\label{invelements}
\sigma_{jk}(z) := (A^{-1}(z))_{jk} = \frac{a_{jk}z^{2}+b_{jk}z+c_{jk}}{\det(A(z))}$$ where $A_{m} = \pm1$ and $a_{jk}$, $b_{jk}$ and $c_{jk}$ are constants. The calculation of $\sigma_{jk}$ is given in Lemma \[lem:inv\]. Now let $\mathcal{P}_{2}$ be the set of all pairings of elements of the set $\{1,2,\ldots,2\mathcal{M}_{m}\}$. Then Wick’s formula tells us that $$\int_{\mathbb{R}^{m}}\left(\prod_{j=1}^{m}dx_{j}\,x_{j}^{2r_{j}+2s_{j}}\right)\mathrm{exp}\left(-\mathbf{x}^{\mathrm{T}}A(z)\mathbf{x}\right) = \mathrm{det}^{-1/2}\{A(z)\}\sum_{\pi \in \mathcal{P}_{2}}\prod_{(r,s) \in \pi}\sigma_{\chi(r),\chi(s)}(z) \label{wick}$$ Inserting into , it is apparent that one can set $A_{m}=1$ in , as can be seen by changing variables $z \to zA_{m}$. The number of terms in the product in is clearly just $\mathcal{M}_{m}$, so that the integral can be bounded by $$Z^{(0)}_{n,m} \leq c_{m}\oint_{}\,\frac{z^{-2n+1}}{1-z^{2}}\frac{K(z)}{(z-1)^{\mathcal{M}_{m}+3/2}((m-1)z+(m+1))^{\mathcal{M}_{m}+1/2}}$$ for some other constant $c_{m}>0$. Here $K(z)$ is a polynomial of degree $2\mathcal{M}_{m}$ with no dependence on $n$. As in [@KPTTZ15], deforming contours away from $z=0$ and out to $\infty$, encircling the branch cuts at $(1,\infty)$, $(-\infty,-(m+1)/(m-1))$ and the simple pole at $z=-1$ using that $K(z)$ is analytic shows that the leading contribution for large $n$ comes from the branch point singularity at $z=1$. Integrating by parts $\mathcal{M}_{m}+1$ times, the contribution from the integral along the branch cut $(1,\infty)$ is bounded by a constant times $$\begin{split}
&\frac{(2n)!}{(2n-\mathcal{M}_{m}-1)!}\int_{1}^{\infty}dy\,\frac{y^{-2n}}{(y-1)^{1/2}}\\
&=\frac{(2n)!}{(2n-\mathcal{M}_{m}-1)!}n^{-1/2}\int_{0}^{\infty}du\,\frac{(1+u/n)^{-2n}}{u^{1/2}} = O(n^{1/2+\mathcal{M}_{m}})
\end{split}$$ where we changed variables $u=n(y-1)$ and used the fact that the limit $n \to \infty$ of the last integral is finite.
It remains to show that the error terms in only give rise to sub-leading contributions in the summation .
\[prop:errors\] Consider the sum $Z_{n,m}$ in , the summands of which consist of a product of $m$ factors. Suppose that $1 \leq c \leq m$ factors are replaced with the error bound in , while the remaining factors are replaced with the leading $\Gamma$-factor in . Denoting the resulting sum by $Z^{(c)}_{n,m}$, we have $$Z^{(c)}_{n,m} = O(n^{\mathcal{M}_{m}}), \qquad n \to \infty.$$
Due to the factorized form of , the sum is a product of $c$ terms of the form $$E_{v,\sigma} := \sum_{\substack{k_{1},\ldots,k_{v}}}\frac{b^{(2s_{\sigma(1)})}_{2k_{1}-1}\Gamma^{(2s_{\sigma(2)},2r_{\sigma(2)})}_{2k_{1}-2,2k_{2}-1}\ldots \Gamma^{(2s_{\sigma(v)},2r_{\sigma(v)})}_{2k_{v-1}-2,2k_{v}-1}a^{(2r_{\sigma(v+1)})}_{2k_{v}-2}}{\Gamma(2k_{1}-1)\Gamma(2k_{2}-1)\ldots \Gamma(2k_{v+1}-1)}$$ for some permutation $\sigma$ (corresponding to a re-labelling of the $k_{i}'s$) and $1 \leq v \leq j$. The boundary terms $a$ and $b$ come directly from the error term and are given by $$\begin{aligned}
a^{(2r_{\sigma(v+1)})}_{2k_{v+1}-2} &= \sqrt{n}2^{k_{v+1}}\Gamma\left(k_{v+1}+r_{\sigma(v+1)}-3/2\right)\\
b^{(2s_{\sigma(1)})}_{2k_{1}-1} &= 2^{k_{1}}\Gamma\left(k_{1}+s_{\sigma(1)}-1/2\right)\end{aligned}$$ The asymptotic behaviour of $E_{v,\sigma}$ as $n \to \infty$ can be estimated according to the programme already outlined for the leading term. We get
$$|E_{v,\sigma}| \leq c\sqrt{n}\oint_{}\frac{dz}{2\pi i}\frac{z^{-2n+1}}{1-z^{2}}\,\int_{\mathbb{R}^{v+1}}x_{1}^{2s_{\sigma(1)}}x_{v+1}^{2r_{\sigma(v+1}-2}\prod_{i=2}^{v}x_{i}^{2s_{\sigma(i)}+2r_{\sigma(i)}}\,\mathrm{exp}\left(-\mathbf{x}^{\mathrm{T}}\tilde{A}(z)\mathbf{x}\right)\,dx_{1}\ldots dx_{v+1} \label{gausserror}$$
This time $\tilde{A}(z)$ is symmetric and tridiagonal: $$\tilde{A}(z) = \begin{pmatrix}1 & -\sqrt{2}z/2 & 0 & 0 & \ldots & 0 & 0\\
-\sqrt{2}z/2 & 1 & -\alpha_{1}/2 & 0 & 0 & \ldots & 0\\
0 & -\alpha_{1}/2 & 1 & -\alpha_{2}/2 & 0 & \ldots & 0\\
\vdots & \vdots & \vdots & \ldots & \vdots & \vdots & \vdots\\
0 & \ldots & 0 & -\alpha_{v-2}/2 & 1 & -\alpha_{v-1}/2 & 0\\
0 & \ldots & 0 & 0 & -\alpha_{v-1}/2 & 1 & -\sqrt{2}\alpha_{v}/2\\
0 & 0 & \ldots & 0 & 0 & -\sqrt{2}\alpha_{v}/2 & 1
\end{pmatrix}$$ The Gaussian integral is again evaluated by Wick’s theorem. It’s easy to show that $$\begin{split}
\mathrm{det}(\tilde{A}(z)) &= (1-z^{2})2^{1-v}\\
\tilde{A}^{-1}(z)_{jk} &= \frac{a_{jk}+b_{jk}z+c_{jk}z^{2}}{1-z^{2}}
\end{split}$$ where $a_{jk}$, $b_{jk}$ and $c_{jk}$ are constants independent of $n$ and $z$. Therefore gives a finite linear combination of terms of the form $$\sqrt{n}\oint_{}\,\frac{dz}{2\pi i}\frac{z^{-2n+1}P(\alpha z)}{(z^{2}-1)^{3/2+\mathcal{E}_{v}}}$$ where $\mathcal{E}_{v} = s_{\sigma(1)}+\sum_{i=2}^{v}(s_{\sigma(i)}+r_{\sigma(i)})+r_{\sigma(v+1)}-1$ and $P(\alpha z)$ is an $n$-independent polynomial. The asymptotics are now dominated by the two branch cuts along $(\pm 1,\infty)$ and one easily sees that the integrals along these cuts are both $O(n^{\mathcal{E}_{v}+1})$ as $n \to \infty$. Taking the product over all such factors gives a bound of order $O(n^{\mathcal{M}_{m}})$, which is what we wanted to show.
Miscellaneous Lemmas
====================
In [@KPTTZ15] the determinant of the matrix $A(z)$ in was evaluated explicitly. Due to the application of Wick’s theorem, we need the inverse too.
\[lem:inv\] The inverse of the cyclic tridiagonal matrix $A(z)$ in is given by $$(A(z)^{-1})_{rs} := \sigma_{rs}(z) = \sigma(0)_{rs} - \frac{p_{rs}(z)}{D_{m}(z)}$$ where $D_{m}(z) = \mathrm{det}(A(z))$, $A_{s} = \prod_{j=1}^{s}\alpha_{j}$ and $$\sigma(0)_{rs} = (-1)^{r+s}\frac{A_{s}}{A_{r}}2r\frac{j-s+1}{j+1}, \qquad r \leq s$$ and $$p_{rs}(z) = 2z^{2}\sigma(0)_{mm}\sigma(0)_{1s}\sigma(0)_{r1}-2z(z\sigma(0)_{m1}-2)\sigma(0)_{rm}\sigma(0)_{1s}$$
The matrix $A(z)$ in is called a cyclic tri-diagonal matrix. Its inverse can be calculated by noting that it is a rank $2$ perturbation of the tridiagonal matrix $A(0)$: $$A(z) = A(0)+R(z)S^{\mathrm{T}}$$ where $R(z)$ is a $j \times 2$ matrix of zeros except the corners $R(z)_{12} = R(z)_{j2} = -z/2$. Similarly $S$ is a $j \times 2$ matrix of zeros except the corners $S_{11}=1$ and $S_{j2}=1$. The inverse now follows from the algebraic identity $$A(z)^{-1} = A(0)^{-1}-A(0)^{-1}R(z)(I_{2}+S^{\mathrm{T}}A(0)^{-1}R(z))^{-1}S^{\mathrm{T}}A(0)^{-1}$$ The important part for us is the $2 \times 2$ matrix $$\begin{split}
F &:= (I_{2}+S^{\mathrm{T}}A(0)^{-1}R(z))^{-1}\\
&= \frac{1}{\mathcal{D}_{j}(z)}\begin{pmatrix} z(A(0)^{-1})_{j1}-2 & -z(A(0)^{-1})_{11}\\ -z(A(0)^{-1})_{jj} & z(A(0)^{-1})_{j1}-2
\end{pmatrix}
\end{split}$$ where $D_{j}(z) = z^{2}((A(0)^{-1})_{11}(A(0)^{-1})_{jj}-(A(0)^{-1})_{j,1}^{2})+4z(A(0)^{-1})_{j,1}-4$. The inverse of the tridiagonal matrix $A(0)$ can be calculated via classical recurrence relations which can be solved explicitly in this case: $$(A(0))^{-1}_{rs} =: \sigma(0)_{rs} = (-1)^{r+s}\frac{A_{s}}{A_{r}}2r\frac{j-s+1}{j+1}, \qquad r \leq s$$ This completes the proof of the Lemma.
**Acknowledgements:** I gratefully acknowledge the support of the Leverhulme Trust Early Career Fellowship (ECF-2014-309).
[^1]: Interestingly, in the parlance of log-gases, the $2-\sqrt{2}$ prefactor has the physical interpretation as the *compressibility* of the particle system [@For13].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Successful deployment of cognitive radios requires efficient sensing of the spectrum and dynamic adaptation of the available resources according to the sensed (imperfect) information. While most works design these two tasks separately, in this paper we address them jointly. In particular, we investigate an overlay cognitive radio with multiple secondary users that access orthogonally a set of frequency bands originally devoted to primary users. The schemes are designed to minimize the cost of sensing, maximize the performance of the secondary users (weighted sum rate), and limit the probability of interfering the primary users. The joint design is addressed using dynamic programming and nonlinear optimization techniques. A two-step strategy that first finds the optimal resource allocation for any sensing scheme and then uses that solution as input to solve for the optimal sensing policy is implemented. The two-step strategy is optimal, gives rise to intuitive optimal policies, and entails a computational complexity much lower than that required to solve the original formulation.'
author:
- '[^1] [^2]'
title: '[Jointly Optimal Sensing and Resource Allocation for Multiuser Overlay Cognitive Radios]{}'
---
Cognitive radios, sequential decision making, dual decomposition, partially observable Markov decision processes
Introduction {#s:introduction}
============
are viewed as the next-generation solution to alleviate the perceived spectrum scarcity. When s are deployed, the have to sense their radio environment to optimize their communication performance while avoiding (limiting) the interference to the . As a result, effective operation of s requires the implementation of two critical tasks: i) sensing the spectrum and ii) dynamic adaptation of the available resources according to the sensed information [@Haykin05]. To carry out the *sensing task* two important challenges are: C1) the presence of errors in the measurements that lead to errors on the channel occupancy detection and thus render harmless SU transmissions impossible; and C2) the inability to sense the totality of the time-frequency lattice due to scarcity of resources (time, energy, or sensing devices). Two additional challenges that arise to carry out the *task* are: C3) the need of the algorithms to deal with channel imperfections; and C4) the selection of metrics that properly quantify the reward for the s and the damage for the s.
Many alternatives have been proposed in the literature to deal with these challenges. Different forms of imperfect , such as quantized or noisy , have been used to deal with C1 [@refInterf2]. However, in the context of , fewer works have considered the fact that the may be not only noisy but also outdated, or have incorporated those imperfections into the design of algorithms [@zhao_separation]. The inherent trade-off between sensing cost and throughput gains in C2 has been investigated [@liang_ST_tradeoff], and designs that account for it based on convex optimization [@xin_convex] and [@zhao_separation] for specific system setups have been proposed. Regarding C3, many works consider that the is imperfect, but only a few exploit the statistical model of these imperfections (especially for the time correlation) to mitigate them; see, e.g., [@zhao_separation; @JSAC]. Finally, different alternatives have been considered to deal with C4 and limit the harm that the s cause to the s [@Sergiy_icassp11]. The most widely used is to set limits on the peak (instantaneous) and average interfering power. Some works also have imposed limits on the rate loss that s experience [@quantizedCR; @marques_cap_crown12], while others look at limiting the instantaneous or average probability of interfering the (bounds on the short-term or long-term outage probability) [@NeelyCR09; @JSAC].
Regardless of the challenges addressed and the formulation chosen, the sensing and policies have been traditionally designed separately. Each of the tasks has been investigated thoroughly and relevant results are available in the literature. However, a globally optimum design requires designing those tasks jointly, so that the interactions among them can be properly exploited. Clearly, more accurate sensing enables more efficient , but at the expense of higher time and/or energy consumption. Early works dealing with joint design of sensing and are [@zhao_framework] and [@zhao_separation]. In such works, imperfections in the sensors, and also time correlation of the state of the primary channel, are considered. As a result, the sensing design is modeled as a [@pomdpsurvey], which can be viewed as a specific class of . The design of the in these works amounts to select the user transmitting on each channel (also known as user scheduling). Under mild conditions, the authors establish that a separation principle holds in the design of the optimal access and sensing policies. Additional works addressing the joint design of sensing and , and considering more complex operating conditions, have been published recently [@xin_convex; @kim_sequential]. For a single operating multiple fading channels, [@xin_convex] relies on convex optimization to optimally design both the and the indexes of the channels to be sensed at every time instant. Assuming that the number of channels that can be sensed at every instant is fixed and that the primary activity is independent across time, the author establishes that the channels to sense are the ones that can potentially yield a higher reward for the secondary user. Joint optimal design is also pursued in [@kim_sequential], although for a very different setup. Specifically, [@kim_sequential] postulates that at each slot, the must calculate the fraction of time devoted to sense the channel and the fraction devoted to transmit in the bands which are found to be unoccupied. Clearly, a trade-off between sensing accuracy and transmission rate emerges. The design is formulated as an optimal stopping problem, and solved by means of Lagrange relaxation of [@castanon]. However, none of these two works takes into account the temporal correlation of the .
The objective of this work is to design the sensing and the policies *jointly* while accounting for the challenges C1-C4. The specific operating conditions considered in the paper are described next. We analyze an *overlay*[^3] with multiple s and s. s are able to adapt their power and rate loadings and access orthogonally a set of frequency bands. Those bands are originally devoted to s transmissions. *Orthogonally* here means that if a is transmitting, no other can be active in the same band. The schemes are designed to maximize the sum-average rate of the s while adhering to constraints that *limit* the maximum “average power” that s transmit and the average “probability of interfering” the s. It is assumed that the of the links is instantaneous and free of errors, while the of the s activity is outdated and noisy. A simple first-order hidden Markov model is used to characterize such imperfections. Sensing a channel band entails a given cost, and at each instant the system has to decide which channels (if any) are sensed.
The jointly optimal sensing and schemes will be designed using and nonlinear optimization techniques. techniques are required because the activity of s is assumed to be correlated across time, so that sensing a channel has an impact not only for the current instant, but also for future time instants [@zhao_framework]. To solve the joint design, a two-step strategy is implemented. In the first step, the sensing is considered given and the optimal is found for *any* fixed sensing scheme. This problem was recently solved in [@marques_icassp; @JSAC]. In the second step, the results of the first step are used as input to obtain the optimal sensing policy. The motivation for using this two-step strategy is twofold. First, while the joint design is non convex and has to be solved using techniques, the problem in the first step (optimal for a fixed sensing scheme) can be recast as a convex one. Second, when the optimal is substituted back into the original joint design, the resulting problem (which does need to be solved using techniques) has a more favorable structure. More specifically, while the original design problem was a constrained , the updated one is an unconstrained problem which can be solved separately for each of the channels.
The rest of the paper is organized as follows. Sec. II describes the system setup and introduces notation. The optimization problem that gives rise to the optimal sensing and schemes is formulated in Sec. \[s:problem\_statement\]. The solution for the optimal given the sensing scheme is presented in Sec. \[s:RA\]. The optimization of the sensing scheme is addressed in Sec. \[s:sensing\]. The section begins with a brief review of and s. Then, the problem is formulated in the context of and its solution is developed. Numerical simulations validating the theoretical claims and providing insights on our optimal schemes are presented in Sec. \[s:simulations\]. Sec. \[s:analyzing\_future\] analyzes the main properties of our jointly optimal and sensing policies, provides insights on the operation of such policies, and points out future lines of work.[^4]
System setup and state information {#s:SystemSetup}
==================================
This section is devoted to describe the basic setup of the system. We begin by briefly describing the system setup and the operation of the system (tasks that the system runs at every time slot). Then, we explain in detail the model for the , which will play a critical role in the problem formulation. The resources that s will adapt as a function of the are described in the last part of the section.
We consider a scenario with several s and s. The frequency band of interest (the portion of spectrum that is licensed to s, or the subset of this shared with the s, if not all) is divided into $K$ frequency-flat orthogonal subchannels (indexed by $k$). Each of the $M$ secondary users (indexed by $m$) opportunistically accesses any number of these channels during a time slot (indexed by $n$). Opportunistic here means that the user accessing each channel will vary with time, with the objective of optimally utilizing the available channel resources. For simplicity, we assume that there is a which acts as a central scheduler and will also perform the task of sensing the medium for primary presence. The scheduling information will be forwarded to the mobile stations through a parallel feedback channel. The results hold for one-hop (either cellular or any-to-any) setups.
Next, we briefly describe the operation of the system. A more detailed description will be given in Sec. \[s:problem\_statement\], which will rely on the notation and problem formulation introduced in the following sections. Before starting, it is important to clarify that we focus on systems where the is more difficult to acquire than the . As a result, we will assume that is error-free and acquired at every slot $n$, while is not. With these considerations in mind, the operates as follows. At every slot $n$ the following tasks are run sequentially: T1) the acquires the ; T2) the relies on the output of T1 (and on previous measurements) to decide which channels to sense (if any), then the output of the sensing is used to update the ; and T3) the uses the outputs of T1 and T2 to find the optimal for instant $n$. Overheads associated with acquisition of the and notification of the optimal to the s are considered negligible. Such an assumption facilitates the analysis, and it is reasonable for scenarios where the s are deployed in a relatively small area which allows for low-cost signaling transmissions.
State information and sensing scheme {#ss:SystemSetup:CSI}
------------------------------------
We begin by introducing the model for the . The noise-normalized square magnitude of the fading coefficient (power gain) of the channel between the $m$th secondary user and its intended receiver on frequency $k$ during slot $n$ is denoted as $h_k^m[n]$. Channels are random, so that $h_k^m[n]$ is a stochastic process, which is assumed to be independent across time. The values of $h_m^k[n]$ for all $m$ and $k$ form the at slot $n$. We assume that the is perfect, so that the values of $h_m^k[n]$ at every time slot $n$ are know perfectly (error-free). While comprises the power gains of the secondary links, the accounts for the channel occupancy. We will assume that the primary system contains one user per channel. This assumption is made to simplify the analysis and it is reasonable for certain primary systems, e.g. mobile telephony where a single narrow-band channel is assigned to a single user during the course of a call. Since we consider an overlay scenario, it suffices to know whether a given channel is occupied or not [@GoldsmithCR]. This way, when a is not active, opportunities for s to transmit in the corresponding channel arise. The primary system is not assumed to collaborate with the secondary system. Hence, from the point of view of the s, the behavior of s is a stochastic process independent of $h_k^m[n]$. With these considerations in mind, the presence of the primary user in channel $k$ at time $n$ is represented by the binary state variable $a_k[n]$ (0/idle, 1/busy). Each primary user’s behavior will be modeled as a simple Gilbert-Elliot channel model, so that $a_k[n]$ is assumed to remain constant during the whole time slot, and then change according to a two-state, time invariant Markov chain. The Markovian property will be useful to keep the modeling simple and will also be exploited to recursively keep track of the . Nonetheless, more advanced models can be considered without paying a big computational price [@JSAC; @nonMarkovPU_Act]. With $P_k^{xy} := \Pr (a_k[n]=x\vert a_k[n-1]=y)$, the dynamics for the Gilbert-Elliot model are fully described by the $2\times 2$ Markov transition matrix $\mathbf{P}_k := [P_k^{00}, P_k^{01}; P_k^{10}, P_k^{11}]$. Sec. \[s:analyzing\_future\] discusses the implications of relaxing some of these assumptions.
While knowledge of $h_k^m[n]$ at instant $n$ was assumed to be perfect (deterministic), knowledge of $a_k[n]$ at instant $n$ is assumed to be imperfect (probabilistic). Two important sources of imperfections are: i) errors in the sensing process and ii) outdated information (because the channels are not always sensed). For that purpose, let $s_k[n]$ denote a binary design variable which is 1 if the $k$th channel is sensed at time $n$, and 0 otherwise. Moreover, let $z_k[n]$ denote the output of the sensor if indeed $s_k[n]=1$; i.e., if the $k$th channel has been sensed. We will assume that the output of the sensor is binary and may contain errors. To account for asymmetric errors, the probabilities of false alarm $P^{FA}_k = \Pr(z_k[n]=1\vert a_k[n]=0)$ and miss detection $P^{MD}_k = \Pr(z_k[n]=0\vert a_k[n]=1)$ are considered. Clearly, the specific values of $P^{FA}_k$ and $P^{MD}_k$ will depend on the detection technique the sensors implement (matched filter, energy detector, cyclostationary detector, etc.) and the working point of the curve, which is usually controlled by selecting a threshold [@arslan_sensing]. In our model, this operation point is chosen beforehand and it is fixed during the system operation, so that the values of $P^{FA}_k$ and $P^{MD}_k$ are assumed known. As already mentioned, the sensing imperfections render the knowledge of $a_k[n]$ at instant $n$ probabilistic. In other words, $a_k[n]$ is a partially observable state variable. The knowledge about the value of $a_k[n]$ at instant $n$ will be referred to as (instantaneous) belief, also known as the information process. For a given instant $n$, two different beliefs are considered: the *pre-decision* belief $B_k[n]$ and the *post-decision* belief $B_k^S[n]$. Intuitively, $B_k[n]$ contains the information about $a_k[n]$ before the sensing decision has been made (i.e., at the beginning of task T2), while $B_k^S[n]$ contains the information about $a_k[n]$ once $s_k[n]$ and $z_k[n]$ (if $s_k[n]=1$) are known (i.e., at the end of task T2). Mathematically, if $\mathcal{H}_n$ represents the history of all sensing decisions and measurements, i.e., $\mathcal{H}_n:=\{s_k[0],z_k[0],\ldots,s_k[n],z_k[n]\}$; then $B_k[n] := \Pr(a_k[n]=1 \vert \mathcal{H}_{n-1} )$ and $B_k^S[n] := \Pr(a_k[n]=1 \vert \mathcal{H}_{n})$. For notational convenience, the beliefs will also be expressed as vectors, with $\mathbf{b}_k[n] := \Big[ 1-B_k[n], B_k[n]\Big]^T$ and $\mathbf{b}^S_k[n] = \Big[1- B^S_k[n], B^S_k[n]\Big]^T$. Using basic results from Markov chain theory and provided that $\mathbf{P}_k$ (time-correlation model) is known, the expression to get the pre-decision belief at time slot $n$ is $$\label{E:gilbert_predict}
\mathbf{b}_k[n]= \mathbf{P}_k\mathbf{b}_k^S[n-1].$$ Differently, the expression to get $\mathbf{b}_k^S[n]$ depends on the sensing decision $s_k[n]$. If $s_k[n]=0$, no additional information is available, so that $$\label{E:gilbert_correct_s0}
\mathbf{b}_k^S[n] = \mathbf{b}_k[n].$$ If $s_k[n]=1$, the belief is corrected as $\mathbf{b}_k^S[n]=\mathbf{b}_k^S \Big(\mathbf{b}_k[n], z_k[n]\Big)$, with $$\label{E:gilbert_correct_s1}
\mathbf{b}_k^S \Big(\mathbf{b}_k[n], z\Big) :=
\frac{\mathbf{D}_{z}\mathbf{b}_k[n]}{\Pr(z_k[n] = z \big\vert \mathbf{b}_k[n])}, \hspace*{-0.5mm}$$ where $\mathbf{D}_{z}$ with $z\in\{0,1\}$ is a $2\times2$ diagonal matrix with entries $[\mathbf{D}_{z}]_{1,1}:=\Pr(z_k[n]=z\vert a_k=0)$ and $[\mathbf{D}_{z}]_{2,2}:=\Pr(z_k[n]=z \vert a_k=1)$. Note that the denominator is the probability of an outcome conditioned to a specified belief: $ \Pr(z_k[n]=z \big\vert \mathbf{b}_k[n]) = \mathbf{1}^T\mathbf{D}_{z}\mathbf{b}_k[n] $, so that corresponds to the correction step of a Bayesian recursive estimator. If no information about the initial state of the is available, the best choice is to initialize $\mathbf{b}_k[0]$ to the stationary distribution of the Markov chain associated with channel $k$ (i.e., the principal eigenvector of $\mathbf{P}_k$).
In a nutshell, the actual state of the primary and secondary networks is given by the random processes $a_k[n]$ and $h_k^m[n]$, which are assumed to be independent. The operating conditions of our are such that at instant $n$, the value of $h_k^m[n]$ is perfectly known, while the value of $a_k[n]$ is not. As a result, the is not formed by $a_k[n]$, but by $\mathbf{b}_k[n]$ and $\mathbf{b}_k^S[n]$ which are a probabilistic description of $a_k[n]$. The system will perform the sensing and tasks based on the available and . In particular, the sensing decision will be made based on $h_k^m[n]$ and $\mathbf{b}_k[n]$, while the will be implemented based on $h_k^m[n]$ and $\mathbf{b}_k^S[n]$.
Resources at the secondary network {#ss:SystemSetup:ResourcesSN}
----------------------------------
We consider a secondary network where users are able to implement adaptive modulation and power control, and share orthogonally the available channels. To describe the channel access scheme (scheduling) rigorously, let $w_k^m[n]$ be a boolean variable so that $w_k^m[n]=1$ if $m$ accesses channel $k$ and zero otherwise. Moreover, let $p_k^m[n]$ be a nonnegative variable denoting the nominal power $m$ transmits in channel $k$, and let $C_k^m[n]$ be its corresponding rate. We say that the $p_m^k[n]$ is a nominal power in the sense that power is consumed only if the user is actually accessing the channel. Otherwise the power is zero, so that the actual (effective) power user $m$ loads in channel $k$ can be written as $w_k^m[n]p_k^m[n]$.
The transmission bit rate is obtained through Shannon’s capacity formula [@Li-Goldsmith01a]: $C_k^m[n]:=\log_2(1+h_k^m[n] p_k^m[n]/\Gamma)$ where $\Gamma$ is a gap that accounts for the difference between the theoretical capacity and the actual rate achieved by the modulation and coding scheme the implements. This is a bijective, nondecreasing, concave function with $p_k^m[n]$ and it establishes a relationship between power and rate in the sense that controlling $p_k^m[n]$ implies also controlling $C_k^m[n]$.
The fact of the access being orthogonal implies that, at any time instant, at most one can access the channel. Mathematically, $$\label{E:c_inst_sched}
\sum_m w_k^m[n]\leq 1 \; \; \forall k,n.$$ Note that allows for the event of all $w_k^m[n]$ being zero for a given channel $k$. That would happen, if, for example, the system thinks that it is very likely that channel $k$ is occupied by a .
Problem statement {#s:problem_statement}
=================
The approach in this paper is to design the sensing and schemes as the solution of a judiciously formulated optimization problem. Consequently, it is critical to identify: i) the design (optimization) variables, ii) the state variables, iii) the constraints that design and state variables must obey, and iv) the objective of the optimization problem.
The first two steps were accomplished in the previous section, stating that the design variables are $s_k[n]$, $w_k^m[n]$ and $p_k^m[n]$ (recall that there is no need to optimize over $C_k^m[n]$); and that the state variables are $h_k^m[n]$ (), and $\mathbf{b}_k[n]$ and $\mathbf{b}_k^S[n]$ ().
Moving to step iii), the constraints that the variables need to satisfy can be grouped into two classes. The first class is formed by constraints that account for the system setup. This class includes constraint as well as the following constraints that were implicitly introduced in the previous section: $s_k[n]\in\{0,1\}$, $w_k^m[n]\in\{0,1\}$ and $p_k^m[n]\geq 0$. The second class is formed by constraints that account for . In particular, we consider the following two constraints. The first one is a limit on the maximum average (long-term) power a can transmit. By enforcing an average consumption constraint, opportunistic strategies are favored because energy can be saved during deep fadings (or when the channel is known to be occupied) and used during transmission opportunities. Transmission opportunities are time slots where the channel is certainly known to be idle and the fading conditions are favorable. Mathematically, with $\check{p}^m$ denoting such maximum value, the average power constraint is written as: $$\label{E:c_power}
\mathbb{E}\left[ \lim_{N \to \infty} (1-\gamma)\sum_{n=0}^{N-1} \gamma^n \sum\nolimits_{ k}w_k^m[n]p_k^m[n]\right]\leq\check{p}^m,~~\forall m,$$ where $0 < \gamma < 1$ is a discount factor such that more emphasis is placed in near future instants. The factor $(1-\gamma)$ ensures that the averaging operator is normalized; i.e., that $\lim_{N \to \infty}\sum_{n=0}^{N-1} (1-\gamma)\gamma^n=1$. As explained in more detail in Sec. \[s:sensing\], using an exponentially decaying average is also useful from a mathematical perspective (convergence and existence of stationary policies are guaranteed).
While the previous constraint guarantees for the s, we also need to guarantee a level of for the s. As explained in the introduction, there are different strategies to limit the interference that s cause to s; e.g., by imposing limits on the interfering power at the s, or on the rate loss that such interference generates [@JSAC]. In this paper, we will guarantee that the *long-term* probability of a being interfered by s is below a certain prespecified threshold $\check{o}_k$. Mathematically, we require $\Pr\{\sum\nolimits_{m}w_k^m =1|a_k =1\}\leq \check{o}_k$ for each band $k=1,\ldots,K$. Using Bayes’ theorem, and capitalizing on the fact that both $a_k$ and $\sum\nolimits_{m}w_k^m$ are boolean variables, the constraint can be re-written as: $$\label{E:c_prob_int}
\mathbbm{E}\left[ \lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n a_k[n] \sum\nolimits_{m}w_k^m[n]\right]/A_k\leq\check{o}_k,~~\forall k,$$ where $A_k$, which is assumed known, denotes the stationary probability of the $k$th band being occupied by the corresponding primary user. Writing the constraint in this form reveals its underlying convexity. Before moving to the next step, two clarifications are in order. The first one is on the practicality of . Constraints that allow for a certain level of interference are reasonable because error-free sensing is unrealistic. Indeed, our model assumes that even if channel $k$ is sensed as idle, there is a probability $P_k^{MD}$ of being occupied. Moreover, when the interference limit is formulated as long-term constraint (as it is in our case), there is an additional motivation for the constraint. The system is able to exploit the so-called interference diversity [@interference_diversity]. Such diversity allows s to take advantage of very good channel realizations even if they are likely to interfere s. To balance the outcome, s will be conservative when channel realizations are not that good and may remain silent even if it is likely that the is not present. The second clarification is that we implicitly assumed that transmissions are possible even if the is present. The reason is twofold. First, the fact that a transmitter is interfering a receiver, does not necessarily imply that the reciprocal is true. Second, since the does not have any control over the power that primary transmitters use, the interfering power at the secondary receiver is a state variable. As such, it could be incorporated into $h_k^m[n]$ as an additional source of noise.
The fourth (and last) step to formulate the optimization problem is to design the metric (objective) to be maximized. Different utility (reward) and cost functions can be used to such purpose. As mentioned in the introduction, in this work we are interested in schemes that maximize the weighted sum rate of the s and minimize the cost associated with sensing. Specifically, we consider that every time that channel $k$ is sensed, the system has to pay a price $\xi_k>0$. We assume that such a price is fixed and known beforehand, but time-varying prices can be accommodated into our formulation too (see Sec. \[ss:analyzing\_future:sensing\_cost\] for additional details). This way, the sensing cost at time $n$ is $U_S[n]:=\sum_{k}\xi_k s_k[n]$. Similarly, we define the utility for the s at time $n$ as $U_{SU}[n] :=\sum_{k} \left(\sum_{m} \beta^m w_k^m[n] C_k^m(h_k^m[n],p_k^{m}[n]) \right)$, where $\beta^m>0$ is a user-priority coefficient. Based on these definitions, the utility for our at time $n$ is $U_T[n]:= U_{SU}[n]-U_S[n]$. Finally, we aim to maximize the long-term utility of the system denoted by $\bar{U}_T$ and defined as $$\label{E:ubar}
\bar{U}_T := \mathbbm{E} \left[ \lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n U[n] \right].$$
With these notational conventions, the optimal $s_k^*[n]$, $w_k^{m*}[n]$ and $p_k^{m*}[n]$ will be obtained as the solution of the following constrained optimization problem.
\[E:constrained\_original\] $$\begin{aligned}
{2} \label{E:constrained_original_obj}
\max_{\{s_k[n], w_k^m[n], p_k^m[n]\}} & ~~\bar{U}_T\\
~\mathrm{s.~to:}~~~~& ~\eqref{E:c_inst_sched},~ w_k^m[n]\in \{0,1\},~p_k^m[n]\geq 0 \label{E:constrained_original_shortterm_RA}\\
& ~\eqref{E:c_power},~\eqref{E:c_prob_int} \label{E:constrained_original_longterm_RA}\\
& ~ s_k[n]\in \{0,1\} \label{E:constrained_original_shortterm_Sensing}.\end{aligned}$$
Note that constraints in and affect the design variables involved in the task ($w_k^m[n]$ and $p_k^m[n]$), while affects the design variables involved in the sensing task ($s_k[n]$). Moreover, the reason for writing and separately is that refers to constraints that need to hold for each and every time instant $n$, while refers to constraints that need to hold in the long-term.
The main difficulty in solving is that the solution for all time instants has to be found jointly. The reason is that sensing decisions at instant $n$ have an impact not only at that instant, but at future instants too. As a result, a separate per-slot optimization approach is not optimal, and techniques have to be used instead. Since problems generally have exponential complexity, we will use a two step-strategy to solve which will considerably reduce the computational burden without sacrificing optimality. To explain such a strategy, it is convenient to further clarify the operation of the system. In Sec. \[s:SystemSetup\] we explained that at each slot $n$, our had to implement three main tasks: T1) acquisition of the , T2) sensing and update of the , and T3) allocation of resources. In what follows, task T2 is split into 3 subtasks, so that the runs five sequential steps:
- T1) At the beginning of the slot, the system acquires the exact value of the channel gains $h_k^m[n]$;
- T2.1) the Markov transition matrix and the post-decision beliefs $\mathbf{b}_k^S[n-1]$ of the previous instant are used to obtain pre-decision beliefs $\mathbf{b}_k[n]$ via ;
- T2.2) $h_k^m[n]$ and $\mathbf{b}_k[n]$ are used to find $s_k^*[n]$;
- T2.3) $s_k^*[n]$ and $z_k[n]$ (for the channels for which $s_k^*[n] = 1$) are used to get the post-decision beliefs $\mathbf{b}_k^S[n]$ via and ;
- T3) $h_k^m[n]$ and $\mathbf{b}_k^S[n]$ are used to find the optimal value of $w_k^{m*}[n]$ and $p_k^{m*}[n]$, and the s transmit accordingly.
The two-step strategy to solve will proceed as follows. In the first step, we will find the optimal $w_k^m[n]$ and $p_k^m[n]$ for any sensing scheme. Such a problem is simpler than the original one in not only because the dimensionality of the optimization space is smaller, but also because we can ignore (drop) all the terms in that depend only on $s_k[n]$. This will be critical, because if the sensing is not optimized, a per-slot optimization the remaining design variables is feasible. In the second step, we will substitute the output of the first step into and solve for the optimal $s_k[n]$. Clearly, the output of the first step will be used in T3 while the output of the second step will be used in T2.2. The optimization in the first step () is addressed next, while the optimization in the second step (sensing) is addressed in Sec. \[s:sensing\].
Optimal RA for the secondary network {#s:RA}
====================================
According to what we just explained, the objective of this section is to design the optimal (scheduling and powers) for a fixed sensing policy. It is worth stressing that solving this problem is convenient because: i) it corresponds to one of the tasks our has to implement; ii) it is a much simpler problem than the original problem in , indeed the problem in this section has a smaller dimensionality and, more importantly, can be recast as a convex optimization problem; and iii) it will serve as an input for the design of the optimal sensing, simplifying the task of finding the global solution of .
Because in this section the sensing policy is considered given (fixed), $s_k[n]$ is not a design variable, and all the terms that depend only on $s_k[n]$ can be ignored. Specifically, the sensing cost $U_S[n]$ in and the constraint in can be dropped. The former implies that the new objective to optimize is $\bar{U}_{SU}:=\sum\nolimits_{ k,m}\mathbbm{E}[\lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)
\gamma^n \beta^mw_k^m[n] C_k^m(h_k^m[n],p_k^{m}[n])]$. With these considerations in mind, we aim to solve the following problem \[cf. \]
\[E:optimization\_problem\] $$\begin{aligned}
{2} \label{E:optRA_obj}
\hspace{-.3cm}\underset{\{w_k^m[n],p_k^m[n]\}}{\max}
~~&\bar{U}_{SU}\hspace{5.04cm}\\
\label{E:optRA_const}~\mathrm{s.~to:}
~~&\eqref{E:constrained_original_shortterm_RA},~ \eqref{E:constrained_original_longterm_RA}.\end{aligned}$$
A slightly modified version of this problem was recently posed and solved in [@JSAC]. For this reason we organize the remaining of this section into two parts. The first one summarizes (and adapts) the results in [@JSAC], presenting the optimal . The second part is devoted to introduce new variables that will serve as input for the design of the optimal sensing in Sec. \[s:sensing\].
Solving for the RA {#ss:RA:presenting_opt_RA}
------------------
It can be shown that after introducing some auxiliary (dummy) variables and relaxing the constraint $w_k^m[n] \in \{0,1\}$ to $w_k^m[n] \in [0,1]$, the resultant problem in is convex. Moreover, with probability one the solution to the relaxed problem is the same than that of the original problem; see [@JSAC] as well as [@amggjr_tsp11] for details on how to obtain the solution for this problem. The approach to solve is to dualize the long-term constraints in . For such a purpose, let $\pi^m$ and $\theta_k$ be the Lagrange multipliers associated with constraints and , respectively. It can be shown then that the optimal solution to is $$\begin{aligned}
\label{E:opt_pow}p_k^{m*}[n] &:=& \left[(\dot{C}_k^m)^{-1}\left(h_k^m[n],\pi^m/\beta^m\right)\right]_+;\hspace{.3cm}\\
\label{E:opt_sched}w_k^{m*}[n] &:=& \mathbbm{1}_{\{ (L_k^m[n]=\max_{q} L_k^q[n])~
\wedge~(L_k^m[n]>0)\}},~ \mathrm{with}\hspace{.3cm}\\
\label{E:opt_phi}L_k^m[n] &:=& L_{SU,k}^m[n] - \theta_k B_k^S[n],~ \mathrm{and}\hspace{.3cm} \\
\label{E:opt_ind}L_{SU,k}^m[n] &:=& \beta^m C_k^m(h_k^m[n],p_k^{m*}[n])-\pi^mp_k^{m*}[n].\hspace{.3cm}\end{aligned}$$ Two auxiliary variables $L_{SU,k}^m[n]$ and $L_k^m[n]$ have been defined. Such variables are useful to express the optimal but also to gain insights on how the optimal operates. Both variables can be viewed as which represent the reward that can be obtained if $w_k^m[n]$ is set to one. The indicator $L_{SU,k}^m[n]$ considers information only of the secondary network and represents the best achievable trade-off between the rate and power transmitted by the . The risk of interfering the is considered in $L_k^m[n]$, which is obtained by adding an interference-related term to $L_{SU,k}^m[n]$. Clearly, the (positive) multipliers $\pi^m$ and $\theta_k$ can be viewed as power and interference *prices*, respectively. Note that dictates that only the user with highest can access the channel. Moreover, it also establishes that if all users obtain a negative , then none of them should access the channel (in other words, an idle with zero would be the winner during that time slot). This is likely to happen if, for example, the probability of the $k$th being present is close to one, so that the value of $\theta_kB_k^S[n]$ in is high, rendering $L_k^m[n]$ negative *for all* $m$.
The expressions in - also reveal the favorable structure of the optimal . The only parameters linking users, channels and instants are the multipliers $\pi^m$ and $\theta_k$. Once they are known, the optimal can be found separately. Specifically: i) the power for a given user-channel pair, which is the one that maximizes the corresponding (setting the derivative of to zero yields ), is found separately from the power for other users and channels; and ii) the optimal scheduling for a given channel, which is the one that maximizes the within the corresponding channel, is found separately from that in other channels. Since once the multipliers are known, the s depend only on information at time $n$, the two previous properties imply that the optimal can be found separately for each time instant $n$. Additional insights on the optimal schemes will be given in Sec. \[ss:analyzing\_future:analyzing\].
Several methods to set the value of the dual variables $\pi^m$ and $\theta_k$ are available. Since, after relaxation, the problem has zero duality gap, there exists a *constant* (stationary) optimal value for each multiplier, denoted as $\pi^{m*}$ and $\theta_k^*$, such that substituting $\pi^m = \pi^{m*}$ and $\theta_k = \theta_{k}^*$ into and yields the optimal solution to the problem. Optimal Lagrange multipliers are rarely available in closed form and they have to be found through numerical search, for example by using a dual subgradient method aimed to maximize the dual function associated with [@BertsekasNP]. A different approach is to rely on stochastic approximation tools. Under this approach, the dual variables are rendered time variant, i.e., $\pi_m=\pi_m[n]$ and $\theta_k=\theta_k[n]$. The objective now is not necessarily trying to find the exact value of $\pi^{m*}$ and $\theta_k^*$, but online estimates of them that remain inside a neighborhood of the optimal value. See Sec. \[ss:analyzing\_future:stationary\_and\_stochastic\] and [@xin_convex; @JSAC] for further discussion on this issue.
RA as input for the design of the optimal sensing {#ss:RA:input_for_sensing}
-------------------------------------------------
The optimal solution in - will serve as input for the algorithms that design the optimal sensing scheme. For this reason, we introduce some auxiliary notation that will simplify the mathematical derivations in the next section. On top of being useful for the design of the optimal sensing, the results in this section will help us to gain insights and intuition on the properties of the optimal . Specifically, let $L[n]$ be an auxiliary variable referred to as global , which is defined as $$\label{E:J}
L[n] := \sum\nolimits_{k} L_k[n], \;\mathrm{with} \;L_k[n]:= \sum\nolimits_{m} w_k^{m\ast}[n] L_k^m[n]$$ Due to the structure of the optimal , the for channel $k$ can be rewritten as \[cf. , \]: $$\label{E:J_max}
L_k[n] := \big[ \max_q {L_k^{q}[n]}\big]_{+}$$ Mathematically, $L[n]$ represents the contribution to the Lagrangian of at instant $n$ when $p_k^m[n]= p_k^{m*}[n]$ and $w_k^m[n] = w_k^{m*}[n]$ for all $k$ and $m$. Intuitively, one can view $L[n]$ as the *instantaneous* functional that the optimal maximizes at instant $n$.
Key for the design of the optimal sensing is to understand the effect of the belief on the performance of the secondary network, thus, on $L[n]$. For such a purpose, we first define the for the s in channel $k$ as $L_{SU,k}[n]:=\max_q L_{SU,k}^q[n]$. Then, we use $L_{SU,k}[n]$ to define the nominal vector $\boldsymbol{l}_k[n]$ as $$\label{E:F}
\boldsymbol{l}_k[n] := \binom{L_{SU,k}[n]}{L_{SU,k}[n] - \theta_k[n]}.$$ Such a vector can be used to write $L_k[n]$ as a function of the belief $\mathbf{b}^S_k[n]$. Specifically, $$\label{E:J_compacto}
L_k[n] = \left[ \boldsymbol{l}^T_k [n] \mathbf{b}^S_k[n] \right]_+.$$ This suggests that the optimization of the sensing (which affects the value of $\mathbf{b}^S_k[n]$) can be performed separately for each of the channels. Moreover, also reveals that $L_k[n]$ can be viewed as the expected : the second entry of $\boldsymbol{l}_k[n]$ is the if the is present, the first entry of $\boldsymbol{l}_k[n]$ is the if it is not, and the entries of $\mathbf{b}^S_k[n]$ account for the corresponding probabilities, so that the expectation is carried over the uncertainties. Equally important, while the value of $\mathbf{b}^S_k[n]$ is only available after making the sensing decision, the value of $\boldsymbol{l}^T_k [n]$ is available before making such a decision. In other words, sensing decisions do not have an impact on $\boldsymbol{l}^T_k [n]$, but only on $\mathbf{b}^S_k[n]$. These properties will be exploited in the next section.
Optimal sensing {#s:sensing}
===============
The aim of this section is leveraging the results of Secs. \[s:problem\_statement\] and \[s:RA\] to design the optimal sensing scheme. Recall that current sensing decisions have an impact not only on the current reward (cost) of the system, but also on future rewards. This in turn implies that future sensing decisions are affected by the current decision, so that the sensing decisions across time form a string of events that has to be optimized jointly. Consequently, the optimization problem has to be posed as a . The section is organized as follows. First, we present a brief summary of the relevant concepts related to and which will be important to address the design of the optimal sensing for the system setup considered in this paper (Sec. \[ss:sensing:ReviewDP\]). Readers familiar with and can skip that section. Then, we substitute the optimal policy obtained in Sec. \[s:RA\] into the original optimization problem presented in Sec. \[s:problem\_statement\] and show that the design of the optimal sensing amounts to solving a set of separate unconstrained problems (Sec. \[ss:sensing:sensing as a DP\]). Lastly, we obtain the solution to each of the problems formulated (Sec. \[ss:sensing:solution for sensing\]). It turns out that the optimal sensing leverages: $\xi_k$, the sensing cost at time $n$; the expected channel at time $n$, which basically depends on $\boldsymbol{l}_k[n]$ () and the pre-decision belief (); and the future reward for time slots $n'>n$. The future reward is quantified by the value function associated with each channel’s , which plays a fundamental role in the design of our sensing policies. Intuitively, a channel is sensed if there is uncertainty on the actual channel occupancy () and the potential reward for the secondary network is high enough (). The expression for the optimal sensing provided at the end of this section will corroborate this intuition.
Basic concepts about DP {#ss:sensing:ReviewDP}
-----------------------
DP is a set of techniques and strategies used to optimize the operation of discrete-time complex systems, where decisions have to be made sequentially and there is a dependency among decisions in different time instants. These systems are modeled as state-space models composed of: a set of state variables $u[n]\in\mathcal{U}$; a set of actions which are available to the controller and which can depend on the state $\alpha[n]\in \mathcal{A}(u[n])$; a transition function that describes the dynamics of the system as a function of the current state and the action taken $u[n+1] = U^{\prime} (u[n], \alpha[n], \omega[n+1])$, where $\omega[n+1]$ is a random (innovation) variable; and a function that defines the reward associated with a state transition or a state-action pair $R(u[n],\alpha[n])$. In general, finding the optimal solution of a is computationally demanding. Unless the structure of the specific problem can be exploited, complexity grows exponentially with the size of the state space, the size of the action space, and the length of the temporal horizon. This is commonly referred to as the triple curse of dimensionality [@powell_book]. Two classical strategies to mitigate such a problem are: i) framing the problem into a specific, previously studied model and ii) find approximate solutions that allow to reduce the computational cost in exchange for a small loss of optimality.
DP problems can be classified into finite-horizon and infinite-horizon problems. For the latter class, which is the one corresponding to the problem in this paper, it is assumed that the system is going to be operated during a very large time lapse, so that actions at any time instant are chosen to maximize the expected long-term reward, i.e., $$\label{E:optimalactionDP}
\max_{\alpha[n]}{{\mathbbm{E}\left[{\sum_{t=n}^\infty{\gamma^t R(u[t], \alpha[t])}}\right]}}.$$ The role of the discount factor $\gamma\in(0,1)$ is twofold: i) it encourages solutions which are focused on early rewards; and ii) it contributes to stabilize the numerical calculation of the optimal policies. In particular, the presence of $\gamma$ guarantees the existence of a stationary policy, i.e. a policy where the action at a given instant is a function of the system state and not the time instant. Note that multiplying by factor $(1-\gamma)$, so that the objective resembles the one used through paper, does not change the optimal policy.
Key to solve a problem is defining the so-called *value function* that associates a real number with a state and a time instant. This number represents the expected sum reward that can be obtained, provided that we operate the system optimally from the current time instant until the operating horizon. If a minimization formulation is chosen, the value function is also known as *cost-to-go function* [@BertsekasDP]. The relationship between the optimal action at time $n$ and the value function at time $n$, denoted as $V_n(\cdot)$, is given by Bellman’s equations [@BertsekasDP; @powell_book]:
\[E:standard\_Belman\] $$\begin{aligned}
V_n(u[n]) = &\max_\alpha \left\{ R(u[n],\alpha) + \mathbbm{E}_\omega \left[ V_{n+1}\left(U^{\prime}(u[n], \alpha, \omega)\right)\right]\right\}
\\
\alpha^*[n]=\alpha^*(u[n]) = \arg &\max_\alpha \left\{ R(u[n],\alpha) + \mathbbm{E}_\omega \left[ V_{n+1}\left(U^{\prime}(u[n], \alpha, \omega)\right)\right]\right\}\end{aligned}$$
where $\omega$ is the information that arrives at time $n+1$ and thus we have to take the expectation over $\omega$. The value function for different time instants can be recursively computed by using backwards induction. Moreover, for infinite horizon formulations with $\gamma<1$, it holds that the value function is stationary. As a result, the dependence of $V_n(\cdot)$ on $n$ can be dropped and can be rewritten using the stationary value function $V(\cdot)$. In this scenario, alternative techniques that exploit the fact of the value function being stationary (such as “value iteration” and “policy iteration” [@powell_book Ch. 2]) can be used to compute $V(\cdot)$.
### Partially Observable Markov Decision Processes
are an important class within problems. For such problems, the state transition probabilities depend only on the current state-action pair, the average reward in each step only depends on the state-action pair, and the system state is fully observable. s can have finite or infinite state-action spaces. s with finite state-action spaces can be solved exactly for finite-horizon problems. For infinite horizon problems, the solution can be approximated with arbitrary precision. A partially Observable (i.e. a ) can be viewed as a generalization of for which the state is not always known perfectly. Only an observation of the state (which may be affected by errors, missing data or ambiguity) is available. To deal with these problems, it is assumed that an *observation function*, which assigns a probability to each observation depending on the current state and action, is known. When dealing with s, there is no distinction between actions taken to change the state of the system under operation and *actions taken to gain information*. This is important because, in general, every action has both types of effect.
The framework provides a systematic method of using the history of the system (actions and observations) to aid in the disambiguation of the current observation. The key point is the definition of an internal *belief state* accounting for previous actions and observations. The belief state is useful to infer the most probable state of the system. Formally, the belief state is a probability distribution over the states of the system. Furthermore, for s this probability distribution comprises a sufficient statistic for the past history of the system. In other words, the process over belief states is Markov, and no additional data about the past would help to increase the agent’s expected reward [@Astrom]. The optimal policy of a agent must map the current belief state into an action. This implies that a discrete state-space can be re-formulated (and viewed) as a continuous-space . This equivalent is defined such that the state space is the set of possible belief spaces of the –the probability simplex of the original state space. The set of actions remains the same; and the state-transition function and the reward functions are redefined over the belief states. More details about how these functions are redefined in general cases can be found at [@kaelbling_POMDP]. Clearly, our problem falls into this class. The actual is Markovian, while the errors in the render the partially observable. These specific functions corresponding to our problem are presented in the following sections.
Formulating the optimal sensing problem {#ss:sensing:sensing as a DP}
---------------------------------------
The aim of this section is to formulate the optimal decision problem as a standard (unconstrained) . The main task is to substitute the optimal into the original optimization problem in . Recall that optimization in involved variables $s_k[n]$, $w_k^m[n]$ and $p_k^m[n]$, and the sets of constraints in , and , the latter requiring $s_k[n]\in \{0,1\}$. When the optimal solution for $w_k^{m*}[n]$, $p_k^{m*}[n]$ presented in Sec. \[s:RA\] is substituted into , the resulting optimization problem is
\[E:constrained\_dp\] $$\begin{aligned}
{2} \label{E:constrained_bellman}
\max_{\{s_k[n]\}} & ~~\bar{U}_{T|RA^*}\\
~\mathrm{s.~to:}~~~~& ~s_k[n]\in \{0,1\}, \label{E:constrained_bellman_binary_const}\end{aligned}$$
where $\bar{U}_{T|RA^*}$ stands for the total utility given the optimal and is defined as $$\label{E:unconstrained_dp_agm}
\bar{U}_{T|RA^*} := \mathbbm{E} \left[
\lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n \sum_{k}
\Big(-\xi_k s_k[n] + \sum_{m} w_k^{m*}[n]L_k^m[n]\Big) \right],$$ which, using the definitions introduced in Sec. \[ss:RA:input\_for\_sensing\], can be rewritten as \[cf. and \] $$\begin{aligned}
\label{E:unconstrained_dp_agm_bis}
\bar{U}_{T|RA^*} &:=& \mathbbm{E} \left[
\lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n \sum_{k} -\xi_k s_k[n] + L_k[n] \right] \\
&=& \mathbbm{E} \left[
\lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n \sum_{k} -\xi_k s_k[n] + \Big[
\boldsymbol{l}^T_k [n] \mathbf{b}^S_k[n]
\Big]_+ \right].\end{aligned}$$ The three main differences between and the original formulation in are that now: i) the only optimization variables are $s_k[n]$; ii) because the optimal fulfills the constraints in and , the only constraints that need to be enforced are , which simply require $s_k[n]\in \{0,1\}$ \[cf. \]; and iii) as a result of the Lagrangian relaxation of the DP, the objective has been augmented with the terms accounting for the dualized constraints.
Key to find the solution of will be the facts that: i) $a_k[n]$ is independent of $h_k^m[n]$, and ii) that $a_k[n]$ is independent of $a_{k'}[n]$ for $k\neq k'$. The former implies that the state transition functions for $a_k[n]$ do not depend on $h_k^m[n]$, while the latter allows to solve for each of the channels separately. Therefore, we will be able to obtain the optimal sensing policy by solving separate s (s), which will rely only on state information of the corresponding channel. Specifically, the optimal sensing can be found as the solution of the following : $$\label{E:unconstrained_DP_score}
\max_{\{
s_k[n]\in\{0,1\}\}
}\sum_{k} \mathbbm{E} \left[
\lim_{N \to \infty} \sum_{n=0}^{N-1}(1-\gamma)\gamma^n \left(
-\xi_k s_k[n] + \Big[
\boldsymbol{l}^T_k [n] \mathbf{b}^S_k[n]
\Big]_+
\right)
\right],$$ which can be separated channel-wise. Clearly, the reward function for the $k$th is $$\label{E:reward_pomdp}
R_k [n] = -\xi_k s_k[n] + \Big[
\boldsymbol{l}^T_k [n] \mathbf{b}^S_k[n]
\Big]_+.$$ The structure of manifests clearly that this is a joint design because $s_k[n]$ affects the two terms in . The first term (which accounts for the cost of the sensing scheme) is just the product of constant $\xi_k$ and the sensing variable $s_k[n]$. The second term (which accounts for the reward of the ) is the dot product of vectors $\boldsymbol{l}_k [n]$ (which does not depend on $s_k [n]$) and $\mathbf{b}^S_k[n]$ (which does depend on $s_k[n]$). The expression in also reveals that $\boldsymbol{l}_k [n]$ encapsulates all the information pertaining the s which is relevant to find $s_k^*[n]$. In other words, in lieu of knowing $h_k^m[n]$, $w_k^{m*}[n]$ and $p_k^{m*}[n]$, it suffices to know $\boldsymbol{l}_k[n]$.
Relying on and , and taking into account that the problem can be separated across channels, *at each time slot* $n$ the optimal sensing for channel $k$ can be obtained as \[cf. \] $$\label{E:compare}
s_k^*[n]= \arg \max_{s \in \{0,1\}} \left\{ \lim_{N \to \infty} \sum_{t=n}^{N-1}(1-\gamma )\gamma^t \mathbbm{E} \Big[ R_k[t]|s_k[n]=s \Big] \right\}.$$
Bellman’s equations and optimal solution {#ss:sensing:solution for sensing}
----------------------------------------
To find $s_k^*[n]$, we will derive the Belmman’s equations associated with . For such a purpose, we split the objective in into the present and future rewards and drop the constant factor $(1-\gamma )\gamma^n$. Then, can be rewritten as $$\label{E:compare_2}
s_k^*[n]= \arg \max_{s \in \{0,1\}} \left\{ \mathbbm{E} \Big[ R_k[n]|s_k[n]=s\Big] + \gamma\lim_{N \to \infty} \sum_{t=n+1}^{N-1}\gamma^{t-n-1} \mathbbm{E} \Big[ R_k[t]|s_k[n]=s \Big] \right\}.$$ It is clear that the expected reward at time slot $t=n$ depends on $s_k[n]$ –recall that both terms in depend on $s_k[n]$. Moreover, the expected reward at time slots $t>n$ also depend on the current $s_k[n]$. The reason is that $\mathbf{b}^S_k[t]$ for $t>n$ depend on the $s_k[n]$ \[cf. \]. This is testament to the fact that our problem is indeed a : current actions that improve the information about the current state have also an impact on the information about the state in future instants.
To account for that effect in the formulation, we need to introduce the value function $V_k(\cdot)$ that quantifies the expected sum reward on channel $k$ for all future instants. Recall that due to the fact of being and infinite horizon problem with $\gamma <1$, the value function is stationary and its existence is guaranteed \[cf. Sec. \[ss:sensing:ReviewDP\]\]. Stationarity implies that the expression for $V_k(\cdot)$ does not depend on the specific time instant, but only on the state of the system. Since in our problem the state information is formed by the and the , $V_k(\cdot)$ should be written as $V_k({B}_k[n], \mathbf{h}_k[n])$. However, since $\mathbf{h}_k[n]$ is i.i.d. across time and independent of $s_k[n]$, the alternative value function $\bar{V}_k({B}_k[n]):=$${\mathbbm{E}_\mathbf{h}}[V_k({B}_k[n], \mathbf{h}_k[n])]$, where ${\mathbbm{E}_\mathbf{h}}$ denotes that the expectation is taken over all possible values of $\mathbf{h}_k[n]$, can also be considered. The motivation for using $\bar{V}_k({B}_k[n])$ instead of $V_k({B}_k[n], \mathbf{h}_k[n])$ is twofold: it emphasizes the fact that the impact of the sensing decisions on the future reward is encapsulated into ${B}_k[n]$, and $\bar{V}_k(\cdot)$ is a one-dimensional function, so that the numerical methods to compute it require lower computational burden.
Based on the previous notation, the standard Bellman’s equations that drive the optimal sensing decision and the value function are \[cf. \] $$\begin{aligned}
\label{E:bellman_decision}
s_k^*[n]= \arg \max_{s \in \{0,1\}} \left\{ {\mathbbm{E}_\mathbf{z}}\left[R_k[n] \big\vert s_k[n] = s \right] + \gamma {\mathbbm{E}_\mathbf{z}}\left[ \bar{V}_k ({B}_k[n+1]) \big\vert s_k[n] = s \right] \right \}&
\\
\label{E:bellman_value}
\bar{V}_k({B}_k[n])= {\mathbbm{E}_\mathbf{h}}\Big[\max_{s\in \{0,1\}} \left\{ {\mathbbm{E}_\mathbf{z}}\left[R_k[n] \big\vert s_k[n]=s \right] + \gamma {\mathbbm{E}_\mathbf{z}}\left[ \bar{V}_k ({B}_k[n+1]) \big\vert s_k[n]=s \right]\right \}&\Big],\end{aligned}$$ where ${\mathbbm{E}_\mathbf{z}}$ denotes taking the expectation over the sensor outcomes. Equation exploits the fact of the value function being stationary, manifests the dynamic nature of our problem, and provides further intuition about how sensing decisions have to be designed. The first term in , ${\mathbbm{E}_\mathbf{z}}\left[R_k[n] \big\vert s_k[n] =s\right]$, is the expected short-term reward conditioned to $s_k[n]=s$, while the second term, ${\mathbbm{E}_\mathbf{z}}\big[ \bar{V}_k ({B}_k[n+1]) \big\vert s_k[n]=s \big]$, is the expected long-term sum reward to be obtained in all future time instants, conditioned to $s_k[n]=s$ and that every future decision is optimal. Equation expresses the condition that the value function $\bar{V}_k(\cdot)$ must satisfy in order to be optimal (and stationary) and provides a way to compute it iteratively.
Since obtaining the optimal sensing decision $s_k^{\ast}[n]$ at time slot $n$ (and also evaluating the stationarity condition for the value function) boils down to evaluate the objective in for $s_k[n]=0$ and $s_k[n] = 1$, in the following we obtain the expressions for each of the two terms in for both $s_k[n]=0$ and $s_k[n]=1$. Key for this purpose will be the expressions to update the belief presented in Sec. \[ss:SystemSetup:CSI\]. Specifically, expressions in - describe how the future beliefs depend on the current belief, on the set of possible actions (sensing decision), and on the random variables associated with those actions (outcome of the sensing process if the channel is indeed sensed).
The expressions for the expected *short-term reward* \[cf. first summand in \] are the following. If $s_k[n] = 0$, the channel is not sensed, there is no correction step, and the post-decision belief coincides with the pre-decision belief \[cf. \]. The expected short-term reward in this case is: $$\label{E:erkns0}
{{{\mathbbm{E}_\mathbf{z}}\left[{R_k[n] \big\vert s_k[n]Ê= 0}\right]}} = \Big[ \boldsymbol{l}_k[n]^T \mathbf{b}_k[n]\Big]_+.$$ On the other hand, if $s_k[n] = 1$, the expected short-term reward is found by averaging over the probability mass of the sensor outcome $z_k[n]$ and subtracting the cost of sensing $$\label{E:erkns1_pre}
{\mathbbm{E}_\mathbf{z}}[R_k[n] \big\vert s_k[n]Ê= 1] = -\xi_k + \sum_{z\in\{0,1\}}\Pr(z_k[n] = z \big\vert\mathbf{b}_k[n])\Big[ \boldsymbol{l}_k[n]^T \mathbf{b}_k^S(\mathbf{b}_k[n], z)\Big]_+,$$ which, by substituting into , yields $$\label{E:erkns1}
{\mathbbm{E}_\mathbf{z}}[R_k[n] \big\vert s_k[n]Ê= 1] = -\xi_k + \sum_{z\in\{0,1\}}\Big[ \boldsymbol{l}_k[n]^T \mathbf{D}_z \mathbf{b}_k[n]\Big]_+.$$
Once the expressions for the expected short-term reward are known, we find the expressions for the expected *long-term* sum *reward* \[cf. second summand in \] for both $s_k[n] = 0$ and $s_k[n] = 1$. If $s_k[n] = 0$, then there is no correction step \[cf. \], and using On the other hand, if $s_k[n] = 1$, the belief for instant $n$ is corrected according to , and updated for instant $n+1$ using the prediction step in as: Clearly, the expressions for the expected *long-term reward* in and account for the expected value of $\bar{V}_k$ at time $n+1$. Substituting , , and into yields $$\begin{aligned}
\label{E:final_value}
\nonumber \bar{V}_k({B}_k[n]) &=& {\mathbbm{E}_\mathbf{h}}\Bigg[ \max \Bigg\{ \Big[ \boldsymbol{l}_k[n]^T \mathbf{b}_k[n]\Big]_+ + \gamma \bar{V}_k\left({\big[\mathbf{P}_k \mathbf{b}_k[n]\big]_2}\right); \\
&-&\xi_k + \sum_{z\in\{0,1\}} \left(\Big[ \boldsymbol{l}_k[n]^T \mathbf{D}_z \mathbf{b}_k[n]\Big]_+ + \gamma \Pr(z_k[n] \big\vert \mathbf{b}_k[n]) \bar{V}_k \bigg(
\frac {{\big[\mathbf{P}_k \mathbf{D}_z \mathbf{b}_k[n]\big]_2}}{\mathbf{1}^T \mathbf{D}_z \mathbf{b}_k[n]}
\bigg)\right) \Bigg\} \Bigg].\hspace{1cm}\end{aligned}$$ where for the last term we have used the expression for $\mathbf{b}^S_k(\mathbf{b}_k[n], z)$ in . Equation is useful not only because it reveals the structure of $\bar{V}_k({B}_k[n])$ but also because it provides a mean to compute the value function numerically (e.g., by using the value iteration algorithm [@powell_book Ch. 3]).
Similarly, we can substitute the expressions - into and get the optimal solution for our sensing problem. Specifically, the sensing decision at time $n$ is $$\label{E:final_decision}
\begin{split}
\Big[\boldsymbol{l}_k^T[n] \mathbf{b}_k[n]\Big]_+ &+ \gamma \bar{V}_k\left({\big[\mathbf{P}_k \mathbf{b}_k[n]\big]_2}\right)
\overset{{s_k^*[n]}=0}{\underset{{s_k^*}[n]=1}\gtrless} \\
- \xi_k + \sum_{z\in\{0,1\}} \Bigg(\Big[ \boldsymbol{l}^T_k[n] \mathbf{D}_z \mathbf{b}_k[n]\Big]_+ &+ \gamma \Pr(z_k[n] \big\vert \mathbf{b}_k[n]) \bar{V}_k \bigg( \frac {{\big[\mathbf{P}_k \mathbf{D}_z \mathbf{b}_k[n]\big]_2}}{\mathbf{1}^T \mathbf{D}_z \mathbf{b}_k[n]} \bigg)\Bigg).
\end{split}$$ The most relevant properties of optimal sensing policy (several of them have been already pointed out) are summarized next: i) it can be found separately for each of the channels; ii) since it amounts to a decision problem, we only have to evaluate the long-term aggregate reward if $s_k[n] = 1$ (the channel is sensed at time $n$) and that if $s_k[n] = 0$ (the channel is sensed at time $n$), and make the decision which gives rise to a higher reward; iii) the reward takes into account not only the sensing cost but also the utility and for the s (joint design); iv) the sensing at instant $n$ is found as a function of both the instantaneous and the future reward (the problem is a DP); vi) the instantaneous reward depends on both the current and the current , while the future reward depends on the current and not on the current ; and vii) to quantify the future reward, we need to rely on the value function $\bar{V}_k(\cdot)$. The input of this function is the . Additional insights on the optimal sensing policy will be given in Sec. \[ss:analyzing\_future:analyzing\].
Numerical results {#s:simulations}
=================
Numerical experiments to corroborate the theoretical findings and gain insights on the optimal policies are implemented in this section. Since an *RA scheme* similar to the one presented in this paper was analyzed in [@JSAC], the focus is on analyzing the properties of the optimal *sensing scheme*. The readers interested can find additional simulations as well as the Matlab codes used to run them in [$\mathrm{http://www.tsc.urjc.es/\sim amarques/simulations/NumSimulations\_lramjr12.html}$]{}.
The experiments are grouped into two test cases. In the first one, we compare the performance of our algorithms with that of other existing (suboptimal) alternatives. Moreover, we analyze the behavior of the sensing schemes and assess the impact of variation of different parameters (correlation of the s activity, sensing cost, sensor quality, and average ). In the second test case, we provide a graphical representation of the sensing functions in the form of two-dimensional decision maps. Such representation will help us to understand the behavior of the optimal schemes.
The parameters for the default test case are listed in Table \[T:params\]. Four channels are considered, each of them with different values for the sensor quality, the sensing cost and the requirements. In most cases, the value of $\check{o}_k$ has been chosen to be larger than the value of $P^{MD}_k$ (so that the cognitive diversity can be effectively exploited), while the values of the remaining parameters have been chosen so that the test-case yields illustrative results. The secondary links follow a Rayleigh model and the frequency selectivity is such that the gains are uncorrelated across channels. The parameters not listed in the table are set to one.
$k$ $P^{FA}_k$ $P^{MD}_k$ $\mathbf{P}_k$ $\xi_k$ $\check{o}_k$ $m$ $\check{p}_m$
----- ------ ------------ ------------ ---------------------------- --------- --------------- ----- ---------------
1 5 dB 0.09 0.08 \[0.95, 0.05; 0.02, 0.98\] 1.00 0.30 1 20.0
2 5 dB 0.09 0.08 \[0.95, 0.05; 0.02, 0.98\] 1.80 0.05 2 16.0
3 5 dB 0.05 0.03 \[0.95, 0.05; 0.02, 0.98\] 1.00 0.10 3 18.0
4 5 dB 0.05 0.03 \[0.95, 0.05; 0.02, 0.98\] 1.80 0.10 4 10.0
: Parameters of the system under test.[]{data-label="T:params"}
**Test case 1: Optimality and performance analysis.** The objective here is twofold. First, we want to numerically demonstrate that our schemes are indeed optimal. Second, we are also interested in assessing the loss of optimality incurred by suboptimal schemes with low computational burden. Specifically, the optimal sensing scheme is compared with the three suboptimal alternatives described next. A) A myopic policy, which is implemented by setting $V(B)=0 \; \forall B $. This is equivalent to the greedy sensing and technique proposed in [@xin_convex], since it only accounts for the reward of sensing in the current time slot and not in the subsequent time slots. B) A policy which replaces the infinite horizon value function with a horizon-1 value function. In other words, a sensing policy that makes the sensing decision at time $n$ considering the (expected) reward for instants $n$ and $n+1$. C) A rule-of-thumb sensing scheme implementing the simple (separable) decision function: $s_k[n] = \mathbbm{1}_{\{L_k[n] \in [\xi_k, \theta_k-\xi_k]\}}\mathbbm{1}_{\{B_k[n] \in [\mathbf{b}^S_k(A_k, 0), \mathbf{b}^S_k(A_k, 1)]\}}$. In words, the channel is sensed if and only if the following two conditions are satisfied: a) the channel’s is greater than the sensing cost and less than the interfering cost minus the sensing cost; and b) the uncertainty on the primary occupancy is higher than that obtained from a unique, isolated measurement.
\
\[f:comparison\]
Results are plotted in Fig. \[f:comparison\]. The slight lack of monotonicity observed in the curves is due to the fact that simulations have been run using a Monte-Carlo approach. As expected, the optimal sensing scheme achieves the best performance for all test cases. Moreover, Figs. \[f:1\_corr\] and \[f:2\_cost\] reveal that the *horizon-1* value function approximation constitutes a good approximation to the optimal value function in two cases: i) when the expected transition time is short (low time correlation) and ii) when the sensing cost is relatively small. The performance of the myopic policy is shown to be far from the optimal. This finding is in disagreement with the results obtained for simpler models in the opportunistic spectrum access literature [@zhao_framework] where it was suggested that the myopic policy could be a good approximation to solve the associated efficiently. The reason can be that the models considered were substantially different (the schemes in this paper are more complex and the interference constraints are formulated differently). In fact, the only cases where the myopic policy seems to approximate the optimal performance are: i) if $\xi_k \to 0$, this is expected because then the optimal policy is to sense at every time instant; and ii) if the s activity is not correlated across time (which was the assumption in [@xin_convex]).
Fig. \[f:3\_err\] suggests that the benefits of implementing the optimal sensing policies are stronger when sensors are inaccurate. In other words, the proposed schemes can help to soften the negative impact of deploying low quality (cheap) sensing devices. Finally, results in Fig. \[f:4\_SNR\] also suggest that changes in the average between and , have similar effects on the performance of all analyzed schemes.
**Test case 2: Sensing decision maps.** To gain insights on the behavior of the optimal sensing schemes, Fig. \[f:maps\] plots the sensing decisions as a function of $B_k[n]$ and $L_k[n]$. Simulations are run using the parameters for the default test case (see Table \[T:params\]) and each subplot corresponds to a different channel $k$. Since the domain of the sensing decision function is two dimensional, the function itself can be efficiently represented as an image (map). To primary regions are identified, one corresponding to the pairs $(B_k[n],L_k[n])$ which give rise to $s_k[n]=1$, and one corresponding to the pairs giving rise to $s_k[n]=0$. Moreover, the region where $s_k[n]=0$ is split into two subregions, the first one corresponding to $\sum_m w_k^{m}[n]=1$ (i.e., when there is a user accessing the channel) and the second one when $\sum_m w_k^{m}[n]=0$ (i.e., when the system decides that no user will access the channel). Note that for the region where $s_k[n]=1$, the access decision basically depends on the outcome of the sensing process $z_k[n]$ (if fact, it can be rigorously shown that $\sum_m w_k^{m}[n]=1$ if and only if $z_k[n] = 1$).
Upon comparing the different subplots, one can easily conclude that the size and shape of the $s_k[n]=1$ region depend on $\mathbf{P}_k$, $P_k^{FA}$, $P_k^{MD}$, $\xi_k$, and $\check{o}_k$. For example, the simulations reveal that channels with stricter interference constraint need to be more frequently sensed and thus the sensing region is larger: Fig. \[sf:hoja11\] vs. Fig. \[sf:hoja12\]. They also reveal that if the sensing cost $\xi_k$ increases, then the sensing region becomes smaller: Fig. \[sf:hoja21\] vs. Fig. \[sf:hoja22\]. This was certainly expected because if sensing is more expensive, then resources have to be saved and used only when the available information is scarce.
\
\[f:maps\]
Analyzing the joint schemes and future lines of work {#s:analyzing_future}
====================================================
This section is intended to summarize the main results of the paper, analyze the properties of the optimal and sensing schemes, and briefly discuss extensions and future lines of work.
Jointly optimal RA and sensing schemes {#ss:analyzing_future:analyzing}
--------------------------------------
The aim of this paper was to design jointly optimal and sensing schemes for an overlay cognitive radio with multiple primary and secondary users. The main challenge was the fact that sensing decisions at a given instant do not only affect the state of the system during the instant they are made, but also the state of the future instants. As a result, our problem falls into the class of , which typically requires a very high computational complexity to be solved. To address this challenge efficiently, we formulated the problem as an optimization over an infinite horizon, so that the objective to be optimized and the constraints to be guaranteed were formulated as long-term time averages. The reason was twofold: i) short-term constraints are more restrictive than their long-term counterparts, so that the latter give rise to a better objective, and ii) long-term formulations are in general easier to handle, because they give rise to stationary solutions. Leveraging the long-term formulation and using dual methods to solve our *constrained* sum-utility maximization, we designed optimal schemes whose input turned out to be: a) the current ; b) the current ; c) the stationary Lagrange multipliers associated with the long-term constraints; and d) the stationary value (reward-to-go) function associated with the future long-term objective. While a) and b) accounted for the state of the system at the current time instant $n$, c) and d) accounted for the effect of sensing and in the long-term (i.e., for instants other than $n$). In particular, the Lagrange multipliers $\pi^m$ and $\theta_k$ accounted for the long-term cost of satisfying the corresponding constraints. This cost clearly involves all time instants and cannot be computed based only on the instantaneous . Similarly, the value function $\bar{V}_k(\cdot)$ quantified the *future* long-term reward.
Due to a judiciously chosen formulation, our problem could be separated across channels (and partially across users), giving rise to simple and intuitive expressions for the optimal and the optimal sensing policies. Specifically, for each time instant $n$, their most relevant properties were:
- The optimal depends on: the at instant $n$, the at instant $n$ (post-decision belief), and the Lagrange multipliers \[cf. -\]. The effect of the sensing policy on the is encapsulated into the post-decision belief vector (testament to the fact that this is a joint design). The effect of other time instants is encapsulated into $\pi^m$ (long-term price of the transmission power) and $\theta_k$ (long-term price for interfering the $k$th ). decisions are made so that the instantaneous is maximized. The is a trade-off between a reward (rate transmitted by the ) and a cost (compound of the power consumed by the and the probability of interfering the ). The is accomplished in a rather intuitive way: each user selects its power to optimize its own , and then in each of the channel the system picks the who achieves the highest (so that the for that channel is maximized).
- The optimum sensing depends on: the at instant $n$, the at instant $n$ (pre-decision belief), the Lagrange multipliers, and the value function \[cf. \]. The optimum sensing is a trade-off between the expected instantaneous (which depends on the current and ), the instantaneous sensing cost, and the future reward (which is given by the value function $\bar{V}_k(\cdot)$ and the current ). Both the instantaneous and the value function depend on the Lagrange multipliers and the policies, testament to the fact that this is a joint design.
For each time instant, the had to run five consecutive steps that were described in detail in Sec. \[s:problem\_statement\]. The expressions for the optimum sensing in had to be used in step T2.2, while the expressions for the optimal in - had to be used in step T3. Once the values of $\pi^m$, $\theta_k$ and $\bar{V}_k(\cdot)$ were found (during the initialization phase of the system), all five steps entailed very low computational complexity.
Sensing cost {#ss:analyzing_future:sensing_cost}
------------
To account for the cost of sensing a given channel, the additive and constant cost $\xi_k$ was introduced. So far, we considered that the value of $\xi_k$ was pre-specified by the system. However, the value of $\xi_k$ can be tuned to represent physical properties of the . Some examples follow. Example 1: Suppose that to sense channel $k$, the spends a power $P_k^{NC}$. In this case, $\xi_k$ can be set to $\xi_k=\pi^{NC} P_k^{NC}$, where $\pi^{NC}$ stands for the Lagrange multiplier associated with a long-term power constraint on the NC. Example 2: Consider a setup for which the long-term rate of sensing is limited, mathematically, this can be accomplished by imposing that $\mathbbm{E} [
\lim_{N \to \infty} \sum_{n=0}^{N-1} (1-\gamma)\gamma^n s_k[n]]\leq \eta$, where $\eta$ represents the maximum sensing rate (say $10\%$). Let $\rho_k$ be the Lagrange multiplier associated with such a constraint, in this scenario $\xi_k$ should be set to $\xi_k=\rho_k$. Example 3: Suppose that if the senses a channel, one fraction of the slot (say 25$\%$) is lost. In this scenario $\xi_k[n]=0.25 L_k[n]$ (time-varying opportunity cost). Linear combinations and stochastic versions of any of those costs are possible too. Similarly, if a collaborative sensing scheme is assumed, aggregation of costs across users can also be considered.
Computing the multipliers and the value functions {#ss:analyzing_future:stationary_and_stochastic}
-------------------------------------------------
In this paper, both the objective to be optimized as well as the requirements were formulated as long-term (infinite-horizon) metrics, cf. , and . As a result, the value function associated with the objective in and the Lagrange multipliers associated with constraints and are stationary (time invariant). Obtaining $\bar{V}_k(\cdot)$, $\pi^m$ and $\theta_k$ is much easier than obtaining their counterparts for short-term (finite-horizon) formulations. In fact, the optimum value of $\bar{V}_k(\cdot)$, $\pi^m$ and $\theta_k$ for the short-term formulations would vary with time, so that at every time instant a numerical search would have to be implemented. Differently, for the long-term formulation, the numerical search has to be implemented only once. Such a search can be performed with iterative methods which are known to converge. At each iteration those methods perform an average over the random (channel) state variables, which is typically implemented using a Montecarlo approach. Such a procedure may be challenging not only from a computational perspective, but also because there may be cases where the statistics of the random processes are not known or they are not stationary. For all those reasons, low-complexity stochastic estimations of $\bar{V}_k(\cdot)$, $\pi^m$ and $\theta_k$ are also of interest. Regarding the Lagrange multipliers, dual stochastic subgradient methods can be used as low-complexity alternative with guaranteed performance (see, e.g., [@xin_convex; @JSAC] and references therein for examples in the field of resource allocation in wireless networks). Development of stochastic schemes to estimate $\bar{V}_k(\cdot)$ is more challenging because the problem follows into the category of functional estimation. Methods such as Q-learning [@powell_book Ch.8] or existing alternatives in the reinforcement learning literature can be considered for the problem at hand. Although design and analytical characterization of stochastic implementations of the schemes derived in this paper are of interest, they are out of the scope of the manuscript and left as future work.
Extending the results to other CR setups {#ss:analyzing_future:future}
----------------------------------------
There are multiple meaningful ways to extend our results. One of them is to consider more complex models for the . Imperfect can be easily accommodated into our formulation. Non-Markovian models for the activity can be used too. The main problem here is to rely on models that give rise to efficient ways to update the belief, e.g., by using recursive Bayesian estimation; see [@JSAC] and references therein for further discussion on this issue. Finally, additional sources of correlation (correlation across time for the and correlation across channels for the ) can be considered too, rendering the more challenging to solve. Another line of work is to address the optimal design for layouts different from the one in this paper. An overlay was considered here, but underlay networks are of interest too. In such a case, information about the channel gains between the s and s would be required. Similarly, in this paper we limit the interference to the by bounding the average probability of interference. Formulations limiting the average interfering power or the average rate loss due to the interfering power are other reasonable options. Last but not least, developing distributed implementations for our novel schemes is also a relevant line of work. Distributed solutions should address the problem of cooperative sensing as well as the problem of distributed . Distributed schemes should be able to cope with noise and delay in the (state) information the nodes exchange, so that a previous step which is key for developing distributed schemes is the design of stochastic versions for the sensing and allocation policies. For some of this extensions, designs based on suboptimal but low complexity solutions may be a worth exploring alternative.
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[^1]: This work is supported by the Spanish Ministry of Science, under FPU Grant AP2010-1050. This paper has been submitted for publication to the IEEE Journal on Selected Areas in Communications. Parts of this paper were presented at CROWNCOM 2012.
[^2]: All authors are with the Dept. of Signal Theory and Comms., King Juan Carlos Univ., Camino del Molino s/n, Fuenlabrada, Madrid 28943, Spain. Contact info available at: http://www.tsc.urjc.es/
[^3]: Some authors refer to overlay networks as interweave networks, see, e.g., [@GoldsmithCR].
[^4]: *Notation:* $x^*$ denotes the optimal value of variable $x$; $\mathbbm{E}[\cdot]$ expectation; $\wedge$ the boolean “and” operator; $\mathbbm{1}_{\{\cdot\}}$ the indicator function ($\mathbbm{1}_{\{x\}}=1$ if $x$ is true and zero otherwise); and $[x]_+$ the projection of $x$ onto the non-negative orthant, i.e., $[x]_+:=\max\{x,0\}$.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We examine the role of complexity on arterial tree structures, determining globally optimal vessel arrangements using the Simulated AnneaLing Vascular Optimization (SALVO) algorithm, which we have previously used to reproduce features of cardiac and cerebral vasculatures. Fundamental biophysical understanding of complex vascular structure has applications to modelling of cardiovascular diseases, and for improved representations of vasculatures in large artificial tissues. In order to progress in-silico methods for growing arterial networks, we need to understand the stability of computational arterial growth algorithms to complexity, variations in physiological parameters such as tissue demand, and underlying assumptions regarding the value of junction exponents. We determine the globally optimal structure of two-dimensional arterial trees; analysing sensitivity of tree morphology and optimal bifurcation exponent to physiological parameters. We find that, for physiologically relevant simulation parameters, arterial structure is stable, whereas optimal junction exponents vary. We conclude that the full complexity of arterial trees is essential for determining the fundamental properties of vasculatures. These results are important for establishing that optimisation-based arterial growth algorithms are stable against uncertainties in physiological parameters, while identifying that optimal bifurcation exponents (a key parameter for many arterial growth algorithms) are sensitive to complexity and the boundary conditions dictated by organs.'
author:
- Jonathan Keelan
- 'James P. Hague'
bibliography:
- 'references.bib'
title: The role of vascular complexity on optimal junction exponents
---
Introduction
============
[[Vascular systems are highly complex, multiscale systems, with dominant physics that changes with length scale.]{}]{} Within a typical organ, vascular trees connect major arteries of $\sim10$mm diameter to huge numbers of tiny arterioles with width of $\sim 10-100\mu$m. The physics of large vessels is often dominated by pulsatile flows and turbulence, while small vessels are microfluidic [@Nakamura2014]. Evolution must account for this complex and multiscale physics when optimizing arterial networks, since efficient vasculatures are crucial for supplying oxygen to tissues.
[[Computational techniques and analytic expressions for describing these complex and multiscale networks have applications in physiology, tissue engineering, and medical diagnosis.]{}]{} Beyond the desire to understand the fundamental biological properties of vascular networks, deviations from optimal flow conditions could be a sign of underlying disease, and vascularization is a key issue limiting the growth of large engineered tissues.
[[The challenge is that the complex and multiscale structures of vascular networks are difficult to reproduce [*in-silico*]{}.]{}]{} Within organs, there can be hundreds of thousands of arterioles dependent on every major artery. The number of possible combinations of vessels associated with these connections is enormous. Since it is not possible to deterministically search all these combinations for all but the smallest trees, stochastic methods are needed. We previously introduced the SALVO algorithm to find the globally optimal structure of arteries using simulated annealing to overcome these problems [@keelan2016; @keelan2019].
[[Vascular networks are primarily constructed from bifurcations [@bifexp], which can be characterized by defining a bifurcation exponent (also known as radius exponent and junction exponent), $\gamma$.]{}]{} The radii of the two output vessels, $r_{\rm out,A}$ and $r_{\rm out,B}$, are related to the radius of the input vessel, $r_{\rm in}$, via, $$r_{\rm in}^{\gamma} = r_{\rm out,A}^{\gamma}+r_{\rm out,B}^{\gamma}.
\label{eqn:murraylaw}$$
[[Murray carried out a single-vessel analysis, which when combined with conservation of flow, shows that when flow in vessels is approximately laminar (Poiseuille flow) the optimal junction exponent, $\gamma_{\rm opt}=3$ [@Murray1926a].]{}]{} In Murray’s analysis, there are two competing contributions to the metabolic demand of vessels: the power dissipated during Poiseuille flow, and the metabolic cost of maintaining a volume of blood. The former is minimized for wide vessels and the latter for narrow vessels, so the actual radius is a compromise.
[[In living systems, the bifurcation exponent is often measured to deviate from three [@Nakamura2014], which is not fully understood, although several factors are known to modify $\gamma_{\rm opt}$ in single-vessel analyses.]{}]{} Inclusion of pulsatile flow, elastic wall vessels, and turbulence contribute to reduction of the optimal junction exponent to $\gamma_{\rm opt}=2.33$ [@Nakamura2014]. A key assumption leading to allometric scaling laws is that cross-sectional area is conserved, i.e. $\gamma_{\rm opt}=2$ [@west1997general].
[[Values of $\gamma$, measured in many vascular networks, are larger than expected from single-segment analyses.]{}]{} In some organs, $\gamma$ is found to be slightly greater than three [@Nakamura2014; @keelan2019]. To our knowledge, no explanation of this effect is currently available, since corrections to flow in single artery analyses to include turbulence, pulsatile flow, and elastic wall vessels, lead to $\gamma_{\rm opt}<3$. This suggests a role for complexity in vascular network analysis.
[[We propose that, in order to fully understand the optimal branching exponents in vascular trees, it is essential to take into account the complexity of the entire arterial network in an organ, and the boundary conditions imposed by the organism.]{}]{} A single vessel is part of a much larger arterial tree for an organ, that is in turn part of an organism, and the role of this additional complexity is poorly understood. The metabolic demand of the organ determines the blood flow to the organ. The radius of the primary artery supplying that organ is determined by a compromise between the whole organism and the organ. These two properties define boundary conditions for arterial growth algorithms.
[[In this paper we carry out a theoretical and numerical analysis of the optimal bifurcation exponent for large and complex arterial trees with physiologically measured boundary conditions.]{}]{} The work goes beyond previous analyses [@Murray1926a; @Zamir1979; @Wischgoll2009; @Zamir2001; @Horsfield1989; @Nakamura2014], by optimizing entire trees, rather than a single arterial bifurcation. The constraints on flow and radius of root vessels in real organs are also taken into account. Further to this analysis, we use the SALVO algorithm [@keelan2016; @keelan2019] to determine the numerically exact globally optimal bifurcation exponent.
Methods
=======
Power cost
----------
The arterial tree is divided into straight segments and bifurcations, with Poiseuille flow assumed within segments. The power cost for a single arterial tree segment experiencing Poiseuille flow is, $$W_{j} = m_b \pi r_{j}^2 l_{j} + \frac{8 \mu f_{j}^{2} l_{j}}{\pi r_{j}^{4}}
\label{eq:singlesegment}$$ where $j$ denotes a segment, $r_{j}$ the segment radius, $l_{j}$ its length, $f_{j}$ its volumetric flow, $m_b$ the metabolic power cost of blood, and $\mu$ the dynamic viscosity of blood. The power cost associated with bifurcations is neglected.
The total cost, ${\mathcal{W}}$, of an arterial tree is the sum of these individual segment costs, $${\mathcal{W}}= \sum_{j \in \{\rm segments\} } W_{j}.\label{eq:costfunc}$$
Murray’s law
------------
Murray’s law is derived by optimizing total cost in a single segment (Eq. \[eq:singlesegment\]). By differentiating with respect to $r_{j}$, $$\frac{\partial W_{j}}{\partial r_{j}}=2m_{b}\pi f_{j}^{2}l_{j}-\frac{32\mu f_{j}^2l_{j}}{\pi r_{j}^{5}}.$$ When $\partial W_{j}/\partial r_{j}=0$, the optimal $r_{j}$ can be found. This leads to a relation for flow in terms of $r_{j}$ $$f_{j}=\frac{m_{b}^{1/2}\pi}{4\mu^{1/2}}r_{j}^{3}$$
In the following analysis, we will assume that $l=l_{\rm root}r^{\alpha}/r_{\rm root}^{\alpha}$, where $l_{\rm root}$ and $r_{\rm root}$ are the length and radius of the root segment respectively, and $\alpha$ is the length–radius exponent. This slightly modifies the preceding argument, so that, $$f_{j}=\frac{m_{b}^{1/2}\pi}{2(2\mu)^{1/2}}r_{j}^{3}\sqrt{\frac{2+\alpha}{4-\alpha}} = \frac{f_{\rm root}}{r_{\rm root}^{3}}{r_{j}^{3}},$$ where $f_{\rm root}$ is the flow through the root segment.
SALVO
-----
The power of the numerical SALVO algorithm is that the generated trees are globally optimized and therefore represent the lowest possible power cost, allowing the effects of evolutionary optimization to be investigated. While the globally optimized solution represents an idealized evolutionary endpoint, insight into the compromise associated with optimizing the competing costs of the different metabolic requirements associated with maintaining a complicated vasculature will be gained by this analysis.
The (SALVO) algorithm developed in earlier papers [@keelan2016; @keelan2019] can be used to generate arterial trees in the 2D plane. In this section, an outline of this algorithm is given. The algorithm is similar to the approach for growing cardiac and cerebral vasculature[@keelan2016; @keelan2019], with some differences relating to the use of fixed nodes to supply tissue.
Equation \[eq:costfunc\] is the cost function at the core of the SALVO algorithm. In this paper we study an idealized two-dimensional (2D) piece of ‘tissue’. Leaf nodes are fixed in place, and there is no supply penalty. Similarly, there is no penalty for tissue penetration of large vessels, since all vessels lie within the 2D tissue. The root node of the tree is fixed to the corner of a square region of side $a$. In contrast to earlier use of the algorithm, a Poisson disc process is used to place leaf nodes [@bridson2007a]. The whole 2D region was accessible by nodes, with only metabolic- and flow-related penalties consistent with Eq. \[eq:singlesegment\].
On each iteration, modifications to the binary tree are attempted by either (1) moving a node or (2) changing the tree structure by swapping node connections. These updates are sufficient to ensure ergodicity. Updates are summarized in Fig. \[fig:updatessummary\] and Table \[table:sweepparameters\]. The root node is never updated. In this version of the algorithm, leaf nodes are never moved. We use simulated annealing to optimize the cost function [@siman].
Acceptance of the updates is determined according to the probability, $$P_{\theta,\theta+1} = {\rm min}\left\{\exp(\frac{-\Delta {\mathcal{W}}^{(\theta,\theta+1})}{T}),1\right\}\label{annealing}$$ where $\Delta {\mathcal{W}}^{(\theta,\theta+1)}={\mathcal{W}}^{(\theta+1)}-{\mathcal{W}}^{(\theta)}$ is the change in cost associated with modifying the tree from configuration $(\theta)$ to configuration $(\theta+1)$, respectively. $T$ is the annealing temperature, which is slowly reduced using the common exponential schedule, $T_{\theta + 1} = \epsilon T_{\theta}$ where $\theta$ is the iteration number, $\epsilon = \exp( \ln T_{0} - \ln T_{\Theta})/\Theta$, $\Theta$ the total number of iterations, and $T_{0}$ ($T_{\Theta}$) are the initial (final) temperatures.
![Two types of update, that move node coordinates and swap the parent segments of nodes respectively, are required for ergodicity. The figure shows a summary of these updates. In panel (a) node a is moved. In panel (b) parents of two nodes (nodes b and d) are swapped. The parent of node b is node a, and the parent of node d is node c. After the swap, the parent of node b is node c and the parent of node d is node a.[]{data-label="fig:updatessummary"}](figure2.pdf){width="60mm"}
SALVO was implemented in C++ making full use of the 2011 standard library (g++ version 7.4.0 compiled with the -O3 flag). Generated trees were analyzed using Python 3.7. A Threadripper 2990WX processor was used for the calculations, with calculations for different $\gamma$, $\Omega$ and tree sizes sent to different threads by a Python script. The optimization of a 500 node tree takes approximately 30 minutes, and a 5000 node tree takes approximately 11 hours on a single thread for $10^8$ updates.
Analytical results
==================
Formalism and simplifications
-----------------------------
Arteries can be grouped together, so that each group comprises arteries with identical properties (e.g length, diameter, flow). In a real arterial system, this would not be true, but it would still be possible to group arteries with similar lengths, radii, and flows together.
Without loss of generality, the total power can be rewritten as, $${\mathcal{W}}= \sum_{j \in \{r,l,q\}} {\mathcal{N}}(r_j,l_j,f_j) \left( m_b \pi r_{j}^{2} l_j + \frac{8 \mu f_{j}^{2} l_j}{\pi r_{j}^{4}} \right), \label{eq:costequation}$$ where ${\mathcal{N}}(r_j,l_j,f_j)$ is the number of arterial segments with identical radii, lengths and flows.
Under the restriction that the flow in all leaf nodes is identical and equal to $f_{\rm leaf}$, the flow in each segment is, $$f_{n} = n f_{\rm leaf}$$ where $n$ is an integer and represents the total number of leaf nodes downstream of the segment.
Comparing Eq. \[eqn:murraylaw\] with flow conservation, a radius–flow relation is identified: $$f_{n} = f_{\rm leaf} (r_{n}/r_{\rm leaf})^{\gamma}
\label{eqn:radiusflow}$$ thus, $$r_{n}=r_{\rm leaf}\left(f_{n}/f_{\rm leaf}\right)^{1/\gamma} = r_{\rm leaf}n^{1/\gamma}.
\label{eqn:flowradius}$$
Experimental data suggest that the length of an arterial segment is proportional to a power of the radius, $$l_{n} = l_{\rm leaf} \left(r_{n}/r_{\rm leaf}\right)^{\alpha} = l_{\rm leaf} \left(f_{n}/f_{\rm leaf}\right)^{\alpha/\gamma},
\label{eqn:lengthscaling}$$ where the value of the exponent $\alpha$ is typically close to 1.0 [@Kamiya2007; @Nakamura2014].
By substituting Eqs. \[eqn:lengthscaling\], the power required to maintain blood flow through a segment depends only on the flow $f_{n}$, $$\begin{aligned}
W_{n} = W(f_{n}) & = & m_{b}\pi r_{\rm leaf}^{2}l_{\rm leaf} \left(f_{n}/f_{\rm leaf}\right)^{(2+\alpha)/\gamma}\nonumber\\
& & +\frac{8\mu l_{\rm leaf}}{\pi r_{\rm leaf}^{4}} f_{n}^{2} \left(f_{n}/f_{\rm leaf}\right)^{(\alpha-4)/\gamma}.\end{aligned}$$
Thus, the dimensionless *metabolic ratio*, defined as $\Omega = m_{b}\pi^2 r_{\rm leaf}^{6}/8\mu f_{\rm leaf}^2$, along with $N$, controls location in parameter space. $$\begin{aligned}
W_{n} & = & C \left(\Omega n^{(2+\alpha)/\gamma}+n^{2+(\alpha-4)/\gamma}\right)\\
& = & C n^{1+(\alpha-1)/\gamma}\left(\Omega n^{3/\gamma-1}+n^{1-3/\gamma}\right)
\label{eqn:seesymmetryclearly}\end{aligned}$$ $C=\frac{8 \mu f_{\rm leaf}^2 l_{\rm leaf}}{\pi r_{\rm leaf}^4}$. Both $C$ and $\Omega$ are defined in terms of the leaf node.
A similar ratio for the root node, $\Omega_{\rm root}=m_{b}\pi^2 r_{\rm root}^{6}/8\mu f_{\rm root}^2$ can be defined for convenient contact with experiment. The values $r_{\rm root}$ and $f_{\rm root}$ are often known from experiment, e.g. Doppler ultrasound, and $N$ can be estimated. This ratio can be related to $\Omega$ via $\Omega_{\rm root}=N^{6/\gamma-2} \Omega$.
The total power requirement is, $${\mathcal{W}}= \sum_{n} {\mathcal{N}}_n W_{n}.
\label{eqn:totalpowergeneral}$$ ${\mathcal{N}}_{n}$ is the number of segments with flow $n f_{\rm leaf}$, and simplifies the function ${\mathcal{N}}(r_i,l_i,f_i)$. For any tree structure, $N$ is always the number of leaf nodes, so ${\mathcal{N}}_{1}=N$. There is always a single root node with total flow $N f_{\rm leaf}$, so ${\mathcal{N}}_{N} = 1$. The remaining ${\mathcal{N}}_n$ are dependent on the structure of the tree. At each bifurcation, flow conservation requires that $n_{\rm in}=n_{\rm out,1} + n_{\rm out,2}$, so $n$ is an integer.
Total power is linear in length scale, so the structure of the power landscape (including the location of any minima with respect to $\gamma$) is independent of $a$. It is the global minimum with respect to $\gamma$ that sets the structure of the tree, and when locating the minimum, $\partial{\mathcal{W}}/\partial\gamma=0$, so the factors of $l_{\rm leaf}$ simply cancel, thus making the solution independent of $a$. Changes in $r_{\rm leaf}$ can be absorbed into the ratio $m_b/\mu$ and thus are similar to changing the metabolic requirements of the organ [@keelan2016].
There are two special cases: fully symmetric and fully asymmetric. In the first case, identified as a fully symmetric tree, the flow is split evenly at each bifurcation. For the case which we shall identify as fully asymmetric, a single leaf node emerges at each bifurcation and the rest of the flow passes down the other bifurcation. We will explore these special cases in the following two sections.
Fully symmetric vascular tree
-----------------------------
In a fully symmetric tree, all of the segments with flow $n$ exist at the same bifurcation layer. Each layer, $m$, has $2^{m}$ segments, where $m$ is the number of bifurcations upstream of that layer ($m = 0$ at the root segment). Within a layer, all segments have the same flow, and thus the same radius and length. The tree has a total of $M$ layers.
The total power cost can be determined by substituting the definitions ${\mathcal{N}}_{n} =2^{m}$ if $n=2^{M-m}$ otherwise ${\mathcal{N}}_{n}=0$, into Eq. \[eqn:totalpowergeneral\], $${\mathcal{W}}= C\sum_{m = 0}^{M} 2^{m} \left( \Omega 2^{(M-m)(2+\alpha)/\gamma}+2^{(M-m)(2+(\alpha-4)/\gamma)} \right). \label{eq:costreducedconstgam}$$
Thus, by summing the geometric series, the total power cost for a fully symmetric tree is, $${\mathcal{W}}= 2^{M} C\left(\Omega\frac{2^{-M(1-(2+\alpha)/\gamma)}-1}{1-2^{1-(2+\alpha)/\gamma}}+ \frac{2^{M(1+(\alpha-4)/\gamma)}-1}{1-2^{-(1+(\alpha-4)/\gamma)}}\right)
\label{eqn:symmetriccost}$$
Fully asymmetric tree
---------------------
The total power cost of the fully asymmetric tree may be calculated by noting that each discrete flow is represented once for all $n$, so ${\mathcal{N}}_{n} = 1$, except there are $N$ leaf nodes so ${\mathcal{N}}_{1}=N$.
Substitution into Eq. \[eqn:totalpowergeneral\] gives, $${\mathcal{W}}= C\left(\sum_{n=1}^{N} (\Omega n^{(2+\alpha)/\gamma}+n^{2+(\alpha-4)/\gamma})
+ (\Omega+1)(N-1)\right)$$
So the total power cost for an asymmetric tree is $$\begin{aligned}
{\mathcal{W}}& = & C\left(\Omega H_{N}^{(-(2+\alpha)/\gamma)}+H_{N}^{(-(2+(\alpha-4)/\gamma))}\right) \nonumber\\
& & \hspace{20mm} + C(\Omega+1)(N-1),
\label{eqn:asummetriccost}\end{aligned}$$ where $H_{n}^{(r)}$ is the generalized harmonic function, $\sum_{k=1}^{n} 1/k^{r}$.
Optimal bifurcation exponent
----------------------------
The optimal value of $\gamma$ is obtained by solving $\partial {\mathcal{W}}/\partial \gamma=0$ for Eqs. \[eqn:symmetriccost\] and \[eqn:asummetriccost\]. We numerically solve $\partial {\mathcal{W}}/\partial \gamma=0$ using Mathematica’s contour plot routines (Mathematica v8.0.4.0, Wolfram). Computations are much faster for the symmetric than the asymmetric trees.
![(a) Deviations from Murray’s law ($\gamma_{\rm opt}=3$) depend strongly on changes in the metabolic ratio, but are insensitive to the structure of the tree. The figure shows a comparison of $\gamma_{\rm opt}$ vs $\Omega$ for fully symmetric, asymmetric and numerical trees. (b) Optimal bifurcation exponent is insensitive to changes in $\alpha$.[]{data-label="fig:symmetricgamma"}](gammavsc2overc1.pdf "fig:"){width="80mm"} ![(a) Deviations from Murray’s law ($\gamma_{\rm opt}=3$) depend strongly on changes in the metabolic ratio, but are insensitive to the structure of the tree. The figure shows a comparison of $\gamma_{\rm opt}$ vs $\Omega$ for fully symmetric, asymmetric and numerical trees. (b) Optimal bifurcation exponent is insensitive to changes in $\alpha$.[]{data-label="fig:symmetricgamma"}](varyalpha.pdf "fig:"){width="80mm"}
\[fig:gammaalpha\]
The optimal bifurcation exponent is strongly dependent on the metabolic ratio, $\Omega$, which can change due to physiological boundary conditions on flow and radius at the input vessels. These constraints may be due to limits in the size of the largest vessel and changing flow demands of tissue. Figure \[fig:symmetricgamma\](a) shows the optimal value of $\gamma$. When $\Omega=1$ and $\alpha=1$ the result of Murray’s law ($\gamma_{\rm opt}=3$) is recovered.
$\gamma_{\rm opt}$ is qualitatively unchanged by the structure of the tree. Results for asymmetric and symmetric trees with $N=2.047\times 10^{3}$ follow essentially the same functional forms. The optimal bifurcation exponent for the asymmetric tree is closer to $\gamma=3$ than the symmetric tree. Also shown in Fig. \[fig:symmetricgamma\](a) are numerical values from SALVO, which will be discussed later.
The optimal bifurcation exponent is modified up or down from $\gamma=3$ by changes in the length exponent, $\alpha$ (Fig. \[fig:gammaalpha\](b)). This structural effect potentially has implications for the value of $\gamma_{\rm opt}$ in organs, since $\alpha$ can vary with organ type, with estimates ranging from $0.89-1.15$. In practice, changes in $\gamma_{\rm opt}$ for this variation in $\alpha$ are far less than the error for measurements of $\gamma$ and changes in $\alpha$ can essentially be neglected.
Deviations from Murray’s law are larger for smaller trees and strongly dependent on changes in the metabolic ratio. The larger the tree, the closer to Murray’s law $\gamma_{\rm opt}$ becomes. Figure \[fig:asymmetricgammavsN\] shows variation of $\gamma_{\rm opt}$ with $N$ for fully symmetric trees. For vascular tree sizes of between $10^{3}$ and $10^{6}$ segments, which are typical in organs, $\gamma_{\rm opt}$ ranges between 2 and 4.
![Deviations from Murray’s law are largest for small trees and strongly dependent on changes in the metabolic ratio. The figure shows $\gamma_{\rm opt}$ vs $N$ for a fully asymmetric tree.[]{data-label="fig:asymmetricgammavsN"}](plotgammavsN.pdf){width="80mm"}
Numerical results
=================
The generation of globally optimal trees using a numerical algorithm helps to test analytic expressions, and provides additional morphological measures that can be used to understand arterial networks. In this section, we use SALVO to investigate the role of vascular complexity and physiological boundary conditions on the properties of globally optimal trees. Several properties of the numerically generated trees are investigated. We determine the sensitivity of globally optimal trees to $\Omega$ and $\gamma$. Through examination of ${\mathcal{W}}_{\rm tot}$, we compute $\gamma_{\rm opt}$ for complex trees. For each value of $\gamma$ and $\Omega$ investigated, arterial trees with up to 5000 nodes were generated. Table \[table:sweepparameters\] summarizes the parameters used for the numerical calculations.
Name symbol range
-------------------------- --------------------- --------------------------------
bifurcation exponent $\gamma$ 1.0-5.0
metabolic ratio $\Omega$ 0.1-10
number of leaf nodes $N$ 100-5000
blood viscosity $\mu$ $3.6\times 10^{-3}$ Pa s
tissue size $a$ 1cm
SA steps $\Theta$ $10^{8}$ ($10^{9}$ for checks)
SA initial ‘temperature’ $T_{0}$ 1 Js$^{-1}$
SA final ‘temperature’ $T_{\Theta}$ $10^{-12}$ Js$^{-1}$
short move distance $d_{\mathrm{move}}$ $0.05$mm
long move distance $d_{\mathrm{move}}$ $0.5$mm
short move node weight 0.3
long move update weight 0.2
swap update weight 0.5
: Simulation parameters and their ranges.[]{data-label="table:sweepparameters"}
Three sectors of the parameter space have qualitatively different tree structures (Fig. \[fig:varyntrees\]): (1) for $\gamma\lesssim 2, \Omega>1$ and $\gamma\gtrsim 4, \Omega<1$, long and narrow leaf nodes originate from the vicinity of the root node; (2) for $\gamma\lesssim 2, \Omega<1$ and $\gamma\gtrsim 4, \Omega>1$ asymmetric and tortuous branches dominate; (3) for $2\lesssim \gamma \lesssim 4$ trees have a branching structure similar to the kinds of vasculature seen in living tissue. In the figure, the vessel widths are normalized to the root radius to improve visibility.
Trees with $\Omega\gg 1$, $\gamma<2$ and $\Omega\ll 1$, $\gamma>4$ are very similar, which is not a coincidence, and can be explained by examining the structure of Eq. \[eqn:seesymmetryclearly\]. When $\alpha\approx 1$, the power in a segment is $W_{n} = C n(\Omega n^{3/\gamma-1}+n^{1-3/\gamma})$. For $\gamma>3$, the exponents (which involve $3/\gamma-1$) have opposite sign to those for $\gamma<3$. So after the substitutions $\Omega=1/\Omega'$, $\gamma=3\gamma'/(2\gamma'-3), C'=\Omega C$, $W_{n} = C' n(\Omega' n^{3/\gamma'-1}+n^{1-3/\gamma'})$, and the sum has an equivalent structure. The substitution is determined by identifying where $1-3/\gamma=3/\gamma'-1$. Since the prefactor $C'$ scales the entire sum, then the minima of ${\mathcal{W}}$ and thus the results for $\gamma,\Omega$ and $\gamma',\Omega'$ are identical. This symmetry is only approximate if $\alpha\neq 1$.
For $\gamma\lesssim 2$ and small $\Omega$ (and $\gamma\gtrsim 4$ and large $\Omega$), the tree structure is highly asymmetric, with long trunks snaking through leaf node sites (see top left panels in Fig. \[fig:varyntrees\]). This is due to the domination of the $n^{1-3/\gamma}$ term due to Poiseuille flow for low $\gamma$, and the $n^{1-3/\gamma'}$ metabolic cost term for large $\gamma$. Thus terms with large $n$ (i.e. thick trunks) are favored.
For $\gamma\lesssim 2$ and large $\Omega$ (and $\gamma\gtrsim 4$ and small $\Omega$), long leaf segments connect root and leaf nodes (see top right panels in Fig. \[fig:varyntrees\]). This is due to the domination of the $n^{3/\gamma-1}$ term due to metabolic maintenance of blood for low $\gamma$, and the $n^{3/\gamma'-1}$ Poiseuille term for large $\gamma$. The term related to metabolic maintenance of blood (with $n^{3/\gamma-1}$) dominates. Thus, terms with small $n$ (i.e. leaf nodes) are favored.
![The structure of the globally optimal vasculature varies with $\gamma$ and $\Omega$. Trees have size $N=100$. Radii are normalized by the root radius for easier visualization.[]{data-label="fig:varyntrees"}](treegrid.pdf){width="120mm"}
For the biologically relevant regime, $2<\gamma<4$, trees have a symmetric structure. No single term in ${\mathcal{W}}$ dominates. There is surprisingly little variation between the tree structures in this region.
To quantify the effect of varying $\gamma$ and $\Omega$ on the network structure, we have examined average segment length, path length and radius asymmetry. The radius of an arterial segment is given by $r_j$, and the length by $l_j$. Average length is defined as $l=\sum l_j/N$. The average summed path length from root to leaf node is $L=\langle \sum_{\mathrm{path}} l_j \rangle $. Radius asymmetry is measured using $\langle r_{c>}/(r_{c<}+r_{c>})\rangle$ (where $r_{c>} \geq r_{c<}$). The sensitivities of these quantities to variations in $\gamma$ and $\Omega$ are presented in Figure \[fig:bexp\_sana\].
In the typical range of biological tissue ($2<\gamma<4$), the dominant factor controlling morphological properties is $\gamma$. All morphological properties are insensitive to variation in $\Omega$. Average segment length is short and path length is long in this region, consistent with the branching structures seen for intermediate $\gamma$ in Fig. \[fig:varyntrees\]. Bifurcation symmetry is in the range $0.58-0.62$, so bifurcations are moderately symmetric. Although $\Omega$ leads to minor changes in tree morphology in this regime, we note it can affect $\gamma_{\rm opt}$ and thus the tree morphology via $\gamma$ as a secondary effect.
In the regions $\gamma<2$ and $\gamma>4$, $\Omega$ is responsible for huge variations in the tree morphology, and $\gamma$ can also produce large variations in the various morphological and structural properties of the tree. Path length drops outside this region to approximately $a/\sqrt{2}$ consistent with a large number of straight paths from the root node to leaf nodes. For $\gamma<2, \Omega\ll 1$, the asymmetry increases dramatically. For all other regions of the parameter space, the asymmetry drops.
Morphological measurements are essentially insensitive to changes in $N$, consistent with additional segments adding more detail, but not qualitatively changing the tree structure. Panels on the left of Fig. \[fig:bexp\_sana\] show results for $N=2163$ and panels to the right for $N=3968$.
![(a) A well defined global minimum in total power cost means that optimal bifurcation exponent $\gamma_{\rm opt}$ can be determined without ambiguity. The figure shows total power cost as a function of bifurcation exponent $\gamma$ for several values of $\Omega$. (b) The relationship of $\gamma_{\rm opt}$ to $N$ and $\Omega$, numerically determined using SALVO, is qualitatively similar to the relationship determined from analytic expressions. The figure shows $\gamma_{\rm opt}$ vs $N$ for several $\Omega$.[]{data-label="fig:opcurves_r"}](c1c2_costs.pdf "fig:"){width="80mm"} ![(a) A well defined global minimum in total power cost means that optimal bifurcation exponent $\gamma_{\rm opt}$ can be determined without ambiguity. The figure shows total power cost as a function of bifurcation exponent $\gamma$ for several values of $\Omega$. (b) The relationship of $\gamma_{\rm opt}$ to $N$ and $\Omega$, numerically determined using SALVO, is qualitatively similar to the relationship determined from analytic expressions. The figure shows $\gamma_{\rm opt}$ vs $N$ for several $\Omega$.[]{data-label="fig:opcurves_r"}](c1c2.pdf "fig:"){width="80mm"}
\[fig:optgamc1c2\]
Optimal bifurcation exponent $\gamma_{\rm opt}$ can be determined without ambiguity from the minimum in ${\mathcal{W}}$. Figure \[fig:opcurves\_r\](a) shows how the total power cost varies with $\gamma$. There is a clearly defined global minimum for all values of $\Omega$ shown. $\gamma_{\rm opt}$ can be found by fitting a quadratic form to the bottom of the minimum.
The variation of $\gamma_{\rm opt}$ with $\Omega$ and $N$, numerically determined using SALVO, is qualitatively similar to the results from analytic expressions. Numerical values of $\gamma_{\rm opt}$ for various values of $\Omega$ vs $N$ are shown in Fig. \[fig:optgamc1c2\](b), and compare favorably to Fig. \[fig:asymmetricgammavsN\]. Several numerical values are compared with the analytic results in Fig. \[fig:symmetricgamma\](a), also showing good agreement for both symmetric and asymmetric trees.
A power-law relationship, $l=Ar^{\alpha}$, is found for the median segment length in terms of segment radius calculated using SALVO (Fig. \[fig:lvsrscatter\](a)). The variation of the value of $l/r_{\rm root}$ with $r/r_{\rm root}$ is analyzed using Python 3. The expression $l=Ar^{\alpha}$ is fitted to the median value. Figure \[fig:lvsrscatter\](a) shows the fit. Regions shaded light blue show the range between the 25th and 75th percentiles in the length histograms. For segments selected from trees with $N>2000$, $2.75<\gamma<3.25$, $\Omega=0.9$, the fit has exponent $\alpha=0.887\pm 0.088$, consistent with experimental values [@Nakamura2014]. As in experiment, there is a strong scatter on the length.
![(a) A power law relationship is found for the median segment length in terms of segment radius calculated using SALVO. The figure shows median values of $l/r_{\rm root}$ vs $r/r_{\rm root}$, a power law fit (dashed line), and the 25th and 75th percentiles (light blue shading). To calculate the length–radius relation, segments are binned from trees with $N>2000$, $2.75<\gamma<3.25$, $\Omega=0.9$. (b) The length–radius exponent, $\alpha$, is close to one for trees grown with $2.5<\gamma<3.5$. To calculate the length–radius relation, segments are binned from trees with $N>2000$ and specific $\gamma$ and $\Omega$ values.[]{data-label="fig:lvsrscatter"}](l_vs_r_wplaw_loglog.pdf "fig:"){height="50mm"} ![(a) A power law relationship is found for the median segment length in terms of segment radius calculated using SALVO. The figure shows median values of $l/r_{\rm root}$ vs $r/r_{\rm root}$, a power law fit (dashed line), and the 25th and 75th percentiles (light blue shading). To calculate the length–radius relation, segments are binned from trees with $N>2000$, $2.75<\gamma<3.25$, $\Omega=0.9$. (b) The length–radius exponent, $\alpha$, is close to one for trees grown with $2.5<\gamma<3.5$. To calculate the length–radius relation, segments are binned from trees with $N>2000$ and specific $\gamma$ and $\Omega$ values.[]{data-label="fig:lvsrscatter"}](alphadata_error.pdf "fig:"){height="50mm"}
\[fig:alphagrid\]
The length–radius exponent, $\alpha$, is consistent with experimental values for trees grown with realistic $2.5<\gamma<3.5$, but can become effectively negative when long leaf segments dominate outside this region (Fig. \[fig:alphagrid\](b)). To calculate the length–radius relation, segments are binned from trees with $N>2000$ and specific $\gamma$ and $\Omega$ values. Where exponents are negative, the relation only poorly follows a power law, and errors on $\alpha$ are large. The relation is well followed within the region $2.5<\gamma<3.5$, and this leads to smaller error bars. Overall, errors on $\alpha$ are relatively large, and could be reduced by making calculations for additional $N$.
Discussion and conclusions
==========================
[[In this paper we determined analytic expressions, and carried out numerical calculations, for the properties and structures of globally optimal vascular trees, with the aim of understanding how overall complexity and physiological boundary conditions contribute to the optimal junction exponent and other structural properties of arterial trees.]{}]{} Analytic expressions were derived for the special cases of maximally symmetric and asymmetric arterial trees. The parameter space of the arterial trees was more fully explored by making numerical calculations with SALVO, enabling globally optimal vasculatures to be found for arbitrary tree morphology. Tree structures, morphological properties, sensitivity to dimensionless parameters and optimal bifurcation (junction) exponent are calculated.
[[Our analytic expressions are consistent with numerical calculations, and predict that $\gamma_{\rm opt}$ is insensitive to tree symmetry, so we propose that the analytic expressions derived here are applicable to a wide range of vasculatures.]{}]{} Analytic expressions can be used for much larger trees, and would, therefore, be useful for predicting the properties of vasculatures within a range of organs where the number of vessel segments and overall complexity exceed the capabilities of current computers. We expect that it will be possible to extend the analytic expressions to include pulsatile flow and turbulence, and will investigate this possibility in future studies.
[[We predict that tree complexity is a significant contributor to the bifurcation exponents in living organisms.]{}]{} The deviations we find from complexity are of similar size to those predicted by including turbulence and pulsatile flow in previous analyses. These deviations are particularly significant if physiological boundary conditions lead to $\Omega\neq 1$. This may occur since all organs, with their dramatically varying demands, are connected to the same major vasculature. We expect that large variations of $\gamma_{\rm opt}$ with increasing complexity will also occur if a more detailed analysis including pulsatile flow and turbulence is carried out.
[[We predict that arterial tree complexity can lead to optimal bifurcation exponent, $\gamma_{\rm opt} > 3$, a situation which can be found in experiment, and is of interest since inclusion of turbulence and pulsatile flow in single artery analyses leads to $\gamma_{\rm opt}<3$.]{}]{} Large values of $\gamma$ are measured in e.g. the brain vasculature ($\gamma = 3.2$) [@keelan2019], retina ($\gamma=3.1$ [@habib2006], $\gamma=3.9\pm 0.12$ [@aldiri2010]) and other mammalian vasculatures where $\gamma$ can range as high as 4 [@Nakamura2014]. Such large $\gamma$ are not predicted by single segment analyses including effects related to pulsatile flow, elastic vessel walls and turbulence ($\gamma=2.3$), [@Nakamura2014]. Complexity and boundary conditions provide an additional contribution that can account for larger values of $\gamma_{\rm opt}$.
[[We predict that tree structures within the physiological regime are only sensitive to $\gamma$; outside the physiological regime structures are also highly sensitive to $\Omega$; and for all regimes tree structures are insensitive to $N$.]{}]{} Changes in $N$ do not qualitatively change the morphology of the tree, but add more detail. Outside the regime $2<\gamma<4$, structure can change dramatically with $\Omega$.
[[Accurate values of $\gamma_{\rm opt}$ are particularly relevant to computational techniques used for growing very large arterial trees *in-silico*, such as constrained constructive optimization (CCO).]{}]{} Such algorithms rely upon a fixed bifurcation exponent to set the radii in the generated trees [@Schreiner1993; @Schreiner2006; @Karch2000]. Similarly, allometric scaling arguments require knowledge of $\gamma$ [@west1997general], and variations of $\gamma_{\rm opt}$ could modify such approaches. $\gamma_{\rm opt}$ is quite hard to measure experimentally, leading to values with large uncertainties, and we consider the calculation of such values to be a useful application of our technique.
[[For values of $\gamma$ consistent with living systems, we find power law exponents in our computational trees that are consistent with the value $\alpha\sim 1$ obtained experimentally.]{}]{} Experimental values range from $0.85<\alpha<1.21$ [@Nakamura2014]. We find a similar range of values in our numerical calculations, and with improved description of the flow, the accuracy of the predictions could be improved. Values of $\alpha$ are also useful as input to other calculations.
[[Future work to include additional physics, such as pulsatile flow, turbulence and vessel elasticity, would lead to a computational model with enhanced predictive power.]{}]{} These improvements to the treatment of flow through vessels could be incorporated into both the analytic expressions derived in this paper, and into the cost function of SALVO without having to change the core algorithm. Once analytical expressions are modified to include this additional physics, we suggest that parameters such as $m_{b}$ could be determined from empirical results.
[[The significant structural changes visible at $\gamma\sim 2$ and $\gamma\sim 4$ would also be interesting areas for further study, since the rapid changes in the tree morphology are reminiscent of a phase transition.]{}]{} These changes are on the edge of the physiologically relevant regime. Confirmation of a phase transition would require the identification of the order parameter and the signatures of critical behavior.
[[Finally, we hypothesize that evolutionary compromises may favor closer adherence to the predictions of single segment analyses in organs with large flow demands to the detriment of less flow-hungry organs.]{}]{} Additional studies could be carried out to test this hypothesis. Overall, the computational and analytical approaches introduced here lead to a range of predictions regarding the structures of vascular trees, that provide interesting links to experimental and theoretical approaches.
Contributions {#contributions .unnumbered}
=============
JPH carried out the analytical calculations and led the study. JK carried out the numerical calculations. Both authors contributed to writing of the manuscript and analysis of the data.
Data availability {#data-availability .unnumbered}
=================
The datasets generated and analyzed during the current study are available in the ORDO repository:\
doi.org/10.21954/ou.rd.12220490 (note, these will be added at proof stage, data available on request).
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors have no competing interests. JK would like to acknowledge EPSRC grant no. EP/P505046/1.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- Carles Altimiras
- Olivier Parlavecchio
- Philippe Joyez
- Denis Vion
- Patrice Roche
- Daniel Esteve
- Fabien Portier
date: 'September 20, 2013\'
title: 'Supplemental Material for the article “Dynamical Coulomb Blockade of Shot Noise”'
---
[[^1]]{}
Calculation of the current noise
================================
The circuit we consider consists of a pure tunnel element connected in series with an arbitrary linear electromagnetic environment (See Fig. 1 of the main text). In this picture, the capacitance of the real tunnel junction has been incorporated into the impedance of the electromagnetic environment [[@ingoldnazarov1992DCB]]{}. The Hamiltonian of the circuit is: $$H=H_{0} +H_{\text{T}} .$$ with $$H_{0} = \sum_{\ell} \varepsilon_{\ell} c_{\ell}^{\dagger} c_{\ell} +
\sum_{r} \varepsilon_{r} c_{r}^{\dagger} c_{r} +H_{\text{{\ensuremath{\operatorname{env}}}}} .$$ Here, the indexes $\ell$ and $r$ span all quasiparticle states in the left and right electrodes, the $c_{\ell ,r}^{\dagger}$ (resp. $c_{\ell ,r}$) denote the fermionic quasiparticle creation (resp. destruction) operators, $H_{\text{{\ensuremath{\operatorname{env}}}}}$ is the Hamiltonian of the electromagnetic environment, and $H_{\text{T}} =T+T^{\dagger}$ is the tunneling Hamiltonian with $T$ and $T^{\dagger}$ implementing the transfer of one electron across the barrier from left to right and from right to left, respectiveley. This operator can be decomposed as $T=e^{i \varphi} \Theta$ where the $e^{i \varphi}$ operator acts only on the electromagnetic environment by translating the charge transfered through the impedance by $e$, the charge of the tunneling electron, while $\Theta = \sum_{\ell ,r} \tau_{\ell r} c_{\ell}^{\dagger} c_{r}$ acts only on the quasiparticles in the electrodes. In this writing $\tau_{\ell r}$ is the tunnel coupling amplitude of states $\ell$ and $r$, and $\varphi$ denotes the phase difference operator across the tunnel element, acting on the electromagnetic environment, and related to the voltage drop $V$ across the tunnel element by $V= \frac{\hbar}{e} \frac{\partial \varphi}{\partial t}$. The current operator through the tunnel element is given by $I=-
\frac{e}{\hbar} \frac{\partial H}{\partial \varphi} =-i \frac{e}{\hbar} (
T-T^{\dagger} )$.
The non-symmetrized noise current density $S_{{\mathrm{I}}} ( \nu )$ is the Fourier transform of the current-current correlator: $$S_{{\mathrm{I}}} ( \nu ) =2 \int_{- \infty}^{\infty} \langle I(t)I(0) \rangle
e^{-i2 \pi \nu t} dt \label{eq:SII}$$ In this convention positive (resp. negative) frequencies correspond to energy being emitted (resp. absorbed) by the quasiparticles to (resp. from) the electromagnetic modes, and the current-current correlator function reads $$\begin{aligned}
\langle I(t)I(0) \rangle & = & \frac{e^{2}}{\hbar^{2}} \{ \langle
T(t)T^{\dagger} (0) \rangle + \langle T^{\dagger} (t)T(0) \rangle \} .\end{aligned}$$ We evaluate the averages by taking separate thermal equilibrium averages over the unperturbed quasiparticles and environmental degrees of freedom: $$\begin{split}
\langle I(t)I(0) \rangle = \frac{e^{2}}{\hbar^{2}} \left\{ \langle e^{i \varphi
(t)} e^{-i \varphi (0)} \rangle \langle \Theta (t) \Theta^{\dagger} (0)
\rangle \right. \\ \left. + \langle e^{-i \varphi (t)} e^{i \varphi (0)} \rangle \langle
\Theta^{\dagger} (t) \Theta (0) \rangle \right\} . \label{eq:curcor}
\end{split}$$ Unless at zero bias voltage where Eq. \[eq:curcor\] is always exact, this amounts to assuming that there exist unspecified relaxation mechanisms fast enough to restore thermal equilibrium both in the electrodes and in the impedance between tunneling events. The validity of this latter assumption is highly dependent on the details of the system and should be checked on a case by case basis.
Within this assumption, the occupation probability of the quasiparticle energy levels in the electrodes is given by the Fermi function $f ( \epsilon )$ at the inverse temperature $\beta$. We further assume constant densities of states $\rho_{l} , \rho_{r}$ in the electrodes and we replace the $|
\tau_{\ell r} |^{2}$ by their average value $| \tau |^{2}$ over all $l,r$ states. The average over quasiparticles degrees of freedom thus takes the form $$\begin{aligned}
\langle \Theta (t) \Theta^{\dagger} (0) \rangle & = & \left\langle
\sum_{\ell ,r} | \tau_{\ell r} |^{2} c_{\ell}^{\dagger} (t)c_{\ell} (0)c_{r}
(t)c_{r}^{\dagger} (0) \right\rangle\\
& = & | \tau |^{2} \rho_{l} \rho_{r} \int d \varepsilon d \varepsilon' f
( \varepsilon ) ( 1-f( \varepsilon + \varepsilon' ) ) e^{-i \varepsilon' t/
\hbar}\\
& = & \frac{\hbar G_{T}}{2 \pi e^{2}} \int d \varepsilon' \gamma (
\varepsilon' ) e^{-i \varepsilon' t/ \hbar}\\
& = & \frac{\hbar G_{T}}{2 \pi e^{2}} \tilde{\gamma} (t) ,\end{aligned}$$ where $G_{T} =4 \pi^{2} G_{K} | \tau |^{2} \rho_{l} \rho_{r}$ denotes the tunnel conductance of the junction, which we assume much smaller than $G_{K}
=e^{2} /h$, allowing to treat $H_{T}$ at the lowest order in perturbation theory; $\tilde{\gamma} (t)$ denotes the inverse Fourier transform of $$\begin{array}{rcl}
\gamma ( \epsilon ) & = & \int d \varepsilon' f ( \varepsilon' ) ( 1-f(
\varepsilon' + \varepsilon ) )\\
& = & \frac{\epsilon}{1-e^{- \beta \epsilon}}\\
& = & \frac{\epsilon}{2} \left( 1+ \coth \frac{\beta \epsilon}{2}
\right)\\
& = & n_{B} ( | \epsilon | ) + \epsilon \hspace{0.25em} \theta (
\epsilon ) ,
\end{array}$$ where $n_{B}$ is the Bose distribution function. The function $\gamma (
\epsilon )$ is proportional to the number of possible electron-hole excitations in the Fermi seas of the electrodes with an energy difference of $\epsilon$ between the final and initial states. The same function is encountered in other contexts, such as in the noise of a resistor, for emission and absorption by bosonic degrees of freedom (see below). Even though $\gamma ( \epsilon )$ is not bounded, its inverse Fourier transform can nevertheless be expressed using distribution functions, meant to be integrated with proper functions [[@PhysRevB.56.1848; @odintsov_effect_1989-1]]{}: $$\tilde{\gamma} (t) =i \pi \hbar^{2} \frac{d}{dt} \delta (t) -
\frac{\pi^{2}}{\beta^{2}} \sinh^{-2} \frac{\pi t}{\hbar \beta} .
\label{eq:gammatilde}$$ The second quasiparticle term $\langle \Theta^{\dagger} (t) \Theta (0)
\rangle$ can be checked to give the same result, i.e. $\langle
\Theta^{\dagger} (t) \Theta (0) \rangle = \langle \Theta (t) \Theta^{\dagger}
(0) \rangle$.
To evaluate the average over the electromagnetic degrees of freedom, we decompose the phase difference across the tunnel element $\varphi (t) =
\varphi_{0} (t) + \tilde{\varphi} (t)$ into the deterministic part $\varphi_{0} (t) =eVt/ \hbar$ corresponding to the dc voltage $V$ across the junction, and a fluctuating random phase $\tilde{\varphi} (t)$ caused by the fluctuations in the electromagnetic environment. For a linear electromagnetic environment at equilibrium, phase fluctuations are Gaussian. In this case, the averages we need to evaluate in Supplemental Material Eq. \[eq:curcor\] can be expressed as $\langle \exp \pm i \tilde{\varphi} (t) \exp \mp i
\tilde{\varphi} (0) \rangle = \exp J (t)$, where $J (t) = \langle (
\tilde{\varphi} (t)- \tilde{\varphi} (0) ) \tilde{\varphi} (0) \rangle$ is the phase-phase correlation function [[@ingoldnazarov1992DCB]]{}. The phase being proportional to the time derivative of the voltage, the phase-phase correlation function is related to the voltage noise on impedance seen by the junction, which can be obtained by the quantum fluctuation-dissipation theorem [[@ingoldnazarov1992DCB]]{}: $$J (t) =h R^{^{-1}}_{K} \int_{- \infty}^{+ \infty} \frac{S_{\text{V \text{}}}
( \nu )}{(h \nu )^{2}} (e^{-i2 \pi \nu t} -1) d \nu , \label{eq:j(t)}$$ with $$S_{\text{V}} ( \nu ) =2 \text{{\ensuremath{\operatorname{Re}}}} Z ( \nu ) \gamma (h \nu )$$ being the (equilibrium) non-symmetrized voltage noise across the environment impedance [[@RevModPhys.82.1155]]{}. Collecting the above terms in Supplemental Material Eq. \[eq:curcor\], we obtain the final expression for the current-current correlator: $$\langle I(t)I(0) \rangle = \frac{G_{\text{T}}}{2 \pi \hbar} 2 \cos
\frac{eVt}{\hbar} e^{J (t)} \tilde{\gamma} (t) .$$ Inserting this result in the Fourier transform (\[eq:SII\]), the current noise can be expressed as $$S_{{\mathrm{I}}} ( \nu ) =2G_{\text{T}} [ \gamma \ast P(-h \nu +eV)+ \gamma \ast
P(-h \nu -eV) ] ,$$ where \* denotes convolution, and $P (E) = \frac{1}{2 \pi \hbar} \int_{-
\infty}^{\infty} e^{J (t) +iEt/ \hbar} \text{{{\em {\ensuremath{\operatorname{dt}}}\/}} {{\em \/}}}$ is the Fourier transform of $\exp [J(t)]$. This function $P (E)$ is interpreted as the probability for the environment to absorb the algebraical energy $E$ during a tunnel event, and the convolution product: $$\gamma \ast P (E) = \int d \varepsilon' \gamma ( \varepsilon' ) P (E-
\epsilon' )$$ simply accounts for all the possible ways to split an energy $E$ between the quasiparticles degrees of freedom and the environment modes. For the case of an environment of vanishing impedance, $P (E) \rightarrow \delta (E)$, and we recover a well known result $$S_{\text{I}} ( \nu ) \rightarrow S_{\text{I} } ( \nu ) =2 G_{\text{T}} [
\gamma (-h \nu +eV)+ \gamma (-h \nu -eV) ] .$$
Sample Fabrication
==================
The 300 nm thick gold ground plane of the resonator and thermalization pad were obtained by optical lithography, followed by evaporation and lift-off. SQUIDs where fabricated following the process described in Ref. [@PopReproducibleJunctionsArXiv2012]: the SQUIDs (see the top inset) are obtained by double angle deposition of ($20/40\,\mathrm{nm}$) thin aluminum electrodes, with a $20'$ oxydation of the first electrode at $400\,\mathrm{mBar}$ of a ($85\%\,\mathrm{O}_2/15\%\,\mathrm{Ar}$) mixture. Before the evaporation, the substrate was cleaned by rinsing in ethanol and Reactive Ion Etching in an oxygen plasma [@Plasma]. The normal junction was obtained using the same technique, with 30/60 nm thick copper electrodes and an aluminum oxyde tunnel barrier (5 nm thick aluminum oxidized for 15 minutes at a 800 mBar (85%O$_{2}$, 15%Ar) mixture).
Details on the Josephson transmission line
==========================================
For sample 1 (resp. 2), our resonator consists in a $360\,\mathrm{\mu m}$ (resp. $91\,\mathrm{\mu m}$) long Josephson metamaterial line containing 72 (resp. 38) lithographically identical and evenly spaced SQUIDs with a $5\,\mathrm{\mu m}$ (resp. $2.4\,\mathrm{\mu m}$) period. The SQUIDs tunnel barriers of sample 1 (resp. sample 2) have an area of $0.5\,\mathrm{\mu m^2}$ (resp. $0.5\,\mathrm{\mu m^2}$) each resulting in a room temperature tunnel resistance $R_\mathrm{N}=720\,\Omega$ (resp. $R_\mathrm{N}=1880 \,\Omega$). To assess that the SQUIDs in the array are identical, we have performed reproducibility tests, yielding constant values of $R_\mathrm{N}$ (within a few $\%$) over millimetric distances. Assuming a superconducting gap $\Delta=180\,\mathrm{\mu
eV}$ and a 17% increase of the tunnel resistance between room temperature and base temperature [@GloosGTTunnelAPL], one obtains a zero flux critical current for the SQUIDs $I_C= 671$nA for sample 1 and $I_C= $268 nA, corresponding to $L_J(\phi=0)=0.49$ nH for sample 1 and $L_J(\phi=0)=1.25$ nH for sample 2. This corresponds to an effective lineic inductance $\mathcal{L}\simeq\mathrm{100\,\mu H.m^{-1}}$ at zero magnetic flux and frequency much lower than the Josephson plasma frequency of the junctions $\nu_\textrm{P}$[@JosephsonRMP1964]. Assuming a capacitance for the junctions of the order of 80 fF/$\mu$m$^2$ yields $\nu_\textrm{P}\simeq$ 25 GHz. Note that our simple fabrication mask produces 10 times bigger Josephson junction in between adjacent SQUIDs, resulting in an additional $\sim \mathcal{L}\simeq\mathrm{10\,\mu H.m^{-1}}$ lineic inductance for sample 1 and $\sim \mathcal{L}\simeq\mathrm{25\,\mu H.m^{-1}}$ for sample 2. The $\sim \mathcal {L}\simeq\mathrm{1\,\mu H.m^{-1}}$ electromagnetic inductance associated to our geometry is negligible. With the designed lineic capacitance $\mathcal{C}=75 \,\mathrm{pF.m^{-1}}$, the length of the resonator sets the first resonance at $\nu_0\simeq 8\,\mathrm{GHz}$ for sample 1 and $\nu_0\simeq 12\,\mathrm{GHz}$ for sample 2. The 12 fF shunting capacitance of the themalization pad reduces these frequencies to $\nu_0 \simeq 6\,\mathrm{GHz}$ in both cases.
Details on setup and calibration
================================
We describe the calibration of the low frequency circuitry for voltage bias and current measurement as well the microwave components used to define the environment of the junction and to measure the emitted radiation.
Low frequency circuit
---------------------
In addition to the components depicted in Fig. 1 of the main text, the low frequency circuit includes a copper powder filter anchored on the mixing chamber, as well as a distributed RC filter made with a resistive wire (50 cm of IsaOhm 304 $\Omega$ m$^{-1}$) winded around a copper rod, and glued with silver epoxy on a copper plate in good thermal contact with the mixing chamber. Both are inserted between the 13 M$\Omega$ bias resistor and the bias T and are represented by the 170$ \Omega$/450 pF RC filter on the biasing line in Supplemental Material Fig. 1. The distributed RC filter has two benefits on the effective electron temperature of our experiment: it provides a high frequency filtering that reduces the polarisation noise as well as thermalisation of the electrons. The copper powder filter is meant to absorb parasitic microwave noise. The line allowing to measure the low frequency response of the junction is filtered by a multipole RC low pass filter, made with a succession of 2 k$\Omega$ Nickel-Chromium resistances and 1 nF capacitances to ground. The NiCr resistances were checked in an independent cool-down to change by less than 1%, which allows to calibrate the 13 M$\Omega$ resitor in-situ, with a precision better than 1%, which in turn allows us to determine the dc voltage $V$ applied to the tunnel junction. The validity of this callibrattion is confirmed by the quality of the comparison between the observed steps in $\partial S_{I} ( \nu ) / \partial V$ and our predictions.
Microwave circuit and calibration
---------------------------------
The microwave chain comprises a bias T, two 4-8 GHz cryogenic circulators anchored at the mixing chamber, as well as a 4-8 GHz bandpass filter and a 12 GHz low pass gaussian absorptive filter (see Supplemental Material Fig. 2). These elements are anchored on the mixing chamber and are meant to protect the sample from the back-action noise of the amplifier. The quantitative determination of the detection impedance relies on the detection of the power emitted by the shot noise of the tunnel junction in the high bias regime. We bias the junction at $\sim 1 {\ensuremath{\operatorname{mV}}}$, where DCB corrections are negligible, so that $S_{I} =2eI$ at frequencies $| \nu | \ll eV/h \simeq 0.5
{\ensuremath{\operatorname{THz}}}$. In order to separate this noise from the noise floor of the cryogenic amplifier, we then apply small variations of the bias voltage and measure the corresponding changes in the measured microwave power with a lock-in amplifier. The conversion of $S_{I}$ into emitted microwave power depends on the environment impedance $Z ( \nu )$ seen by the tunneling resistance $R_{T}$. First, only a fraction $R_{T}^{2} / |R_{T} +Z( \nu )|^{2}$ of the current noise is absorbed by the environment. The current noise in the environment has then to be multiplied by ${\ensuremath{\operatorname{Re}}} [ Z( \nu ) ]$ to obtain the microwave power emitted by the electronic shot noise: $$S_{P} ( \nu ) =2eV \frac{{\ensuremath{\operatorname{Re}}} [ Z( \nu ) ] G_{{\mathrm{T}}}^{-1}}{|Z( \nu
)+G_{{\mathrm{T}}}^{-1} |^{2}} \simeq 2eV \frac{{\ensuremath{\operatorname{Re}}} [ Z( \nu ) ]
G_{{\mathrm{T}}}^{-1}}{[ {\ensuremath{\operatorname{Re}}} [ Z( \nu ) ] +G_{{\mathrm{T}}}^{-1} ]^{2}} .
\label{eq:shotnoise}$$ The last approximation, ${\ensuremath{\operatorname{Im}}} [ Z( \nu ) ] \ll G_{{\mathrm{T}}}^{-1}$ is satisfied with a precision better than 2%. Finally, what is actually detected at room temperature is the amplified microwave power: $$S^{{\ensuremath{\operatorname{RT}}}}_{P} ( \nu ) =2eV G ( \nu ) \frac{{\ensuremath{\operatorname{Re}}} [ Z( \nu ) ]
G_{{\mathrm{T}}}^{-1}}{[ {\ensuremath{\operatorname{Re}}} [ Z( \nu ) ] +G_{{\mathrm{T}}}^{-1} ]^{2}} .
\label{eq:shotnoiseamplified}$$ Supplemental Material Eq. \[eq:shotnoiseamplified\] shows that the extracted ${\ensuremath{\operatorname{Re}}} [ Z( \nu ) ]$ depends on the gain of the microwave chain $G ( \nu
)$, which has to be determined in-situ and independently. To do so, we inserted a 20 dB directional coupler between the sample and the bias Tee, and injected through an independently calibrated injection line, comprising 70 dB attenuation distributed between 4.2 K and the mixing chamber temperature (see Supplemental Material Fig. 2). Both the attenuators and the directional coupler were calibrated at 4.2 K. The resonator is tuned at a resonance well below 4 GHz, so that the entire microwave power is reflected by the sample, allowing to calibrate the gain of the detection chain, albeit doubling the insertion losses of the bias Tee and of the 10 cm microwave cable connecting it to the sample. However, this parasitic contribution can be substracted thanks to independent calibrations. Due to lack of space, we then had to remove one of the circulators for this experiment, which increased the electronic temperature from 16 mK to 25 mK, which explains that we calibrated the detection impedance in an independant run of the experiment.
Extracting the current noise
============================
We discuss here the possible consequences of the fact that the detection impedance is not negligible compared to the tunneling resistance. More specifically, we show that due to the variations of the tunneling resistance with bias voltage, measuring $\partial S_{P} ( \nu ) / \partial V$ is not rigourously equivalent to measuring $\partial S_{I} / \partial V$. However, the error introduced by this approximation can be shown to be negligible.
Due to the non linearity of the tunnel transfer, the power emitted by the junction biased at bias $V {\ensuremath{\operatorname{reads}}}$ $$S_{P} ( \nu ) = {\ensuremath{\operatorname{Re}}} [ Z( \nu ) ] \left| \frac{Z_{\text{T}} ( \nu ,V
)}{Z( \nu )+Z_{\text{T}} ( \nu ,V )^{}} \right|^{2} S_{I} ( V, \nu )
\label{eq:sp} .$$ Here $Z_{\text{T}} ( \nu ,V )$ is the differential impedance of the junction, biased at voltage $V$, at the measurement frequency $\nu$. Supplemental Material Eq. \[eq:sp\] is valid as long as the ac current going through the junction as a consequence of the shot noise is small enough for the response of the junction $Z_{\text{T}} ( \nu ,V )$ to remain in the linear regime. In that case, the modulation of the output voltage of the quadractic detector that we measure is proportional to $$\begin{split}
\frac{\partial S_{P} ( \nu )}{\partial V} = {\ensuremath{\operatorname{Re}}} [ Z( \nu ) ]
\left[ \left| \frac{Z_{\text{T}} ( \nu ,V )}{Z( \nu )+Z_{\text{T}} ( \nu ,V
)^{}} \right|^{2} \frac{\partial S_{I} ( V, \nu )}{\partial V} \right.\\ \left. +S_{I} ( V,
\nu ) \frac{\partial}{\partial V} \left| \frac{Z_{\text{T}} ( \nu ,V )}{Z(
\nu )+Z_{\text{T}} ( \nu ,V )^{}} \right|^{2} \right] \label{eq:dspdv} .
\end{split}$$
We can deduce the expected variations of $Z_{\text{T}} ( \nu ,V )$ with bias voltage from the dc transport properties of the junction via $$Z^{-1}_{\text{T}} ( \nu ,V ) = \frac{e}{2 h \nu} [ I ( V+h \nu /e ) -I (
V-h \nu /e ) ] \label{eq:invzt} .$$ Inserting Eq. \[eq:invzt\] in Eq. \[eq:dspdv\] shows that the associated corrections are negligeable, so that detecting $\frac{\partial S_{P} ( \nu
)}{\partial V}$ gives direct access to $\frac{\partial S_{I} ( V, \nu
)}{\partial V}$ within a precision better than 1%.
Note on noise symetrization
===========================
Let us add a note about the symetrization of the correlator being measured in our high frequency noise measurements. On one hand the microwave amplifier probes its input voltage [@RevModPhys.82.1155], which can be written as
$$V_{\mathrm{in}}(t)=-\sqrt{\frac{\hbar Z_0}{4\pi}}\int_B \omega^{1/2}(i a_{\mathrm{in}} (\omega) e^{-i \omega (t - x/c)} + \mathrm{h.c.}) d\omega,$$
where $a_{\mathrm{in}}(\omega)$ stands for the destruction operator of a input (right moving) photon at frequency $\omega$, $c$ is the velocity of electromagnetic waves in the transmission line connecting the sample to the amplifier, of characteristic impedance $Z_0 = 50 \Omega$, and $B$ the measurement bandwidth. The output voltage reads:
$$V_{\mathrm{out}}(t)=-\sqrt{\frac{\hbar Z_0}{4\pi}}\int_B \omega^{1/2}(i a_{\mathrm{out}} (\omega) e^{-i \omega (t - x/c)} + \mathrm{h.c.}) d\omega,$$
with
$$a_{\mathrm{out}}=\sqrt{G} a_{\mathrm{in}} + \sqrt{G-1} f^\dagger$$
with $G$ the gain of the amplifier and $f^\dagger$ representing the amplifier’s noise [@RevModPhys.82.1155]. Thus the final power measurement $\propto <V_{\mathrm{out}}^2>$ contains, on top of the amplifier’s noise, a term proportional to
$$\begin{aligned}
\left\langle V^2_{\mathrm{in}}\right\rangle&=&\frac{\hbar Z_0}{4\pi}\int_B \omega \left\langle a_{\mathrm{in}}(\omega) a^\dagger_{\mathrm{in}}(\omega) + a^\dagger_{\mathrm{in}}(\omega) a_{\mathrm{in}}(\omega) \right\rangle d\omega \\
&=&\frac{\hbar Z_0}{2\pi}\int_B \omega \left[ \left\langle a^\dagger_{\mathrm{in}}(\omega) a_{\mathrm{in}}(\omega))\right\rangle + \frac{1}{2} \right] d\omega, \end{aligned}$$
thus containing a power representing the zero point motion of the line, associated to a power $\hbar \omega /2$ per unit bandwidth. In other words, the amplifier gives access to the sum of the absorption and emission noise of the input line, unlike quantum detectors [@PhysRevLett.105.166801; @PhysRevB.85.085435] which allow detecting them separately. However, this does not imply that we measure the *electronic current noise of the sample* symetrized with respect to frequency: In our experiment, we measure the excess output noise power associated to the dc biasing of our sample. This results in a excess population of the input field $<a^\dagger_{\mathrm{in}}(\omega) a_{\mathrm{in}}(\omega)>$, which is itself proportional to the photons leakage rate out of our resonator. Thanks to the circulators anchored at the mixing chamber temperature $T \ll h \nu_0 /k_\mathrm{B}$, the latter is kept very close to its ground state (the maximum average number of photons in the resonator is below 5% for the experiments reported in the paper), and can thus only absorb energy from the electronic current fluctuations (in other words $P(h\nu) \ll P(-h\nu)$). This is why our signal is proportional to the emission noise power $ S_{I}(\nu ,V)$ and not the noise power symetrized with respect to frequency $S^{\text{{\ensuremath{\operatorname{sym}}}}}_{{\mathrm{I}}}(\nu ,V) = [S_{I} (- \nu ,V)+S_{I} (\nu ,V) ] /2$, which also contains the electronic zero point motion.
[1]{} G.-L. Ingold and Y. V. Nazarov, “Charge tunneling rates in ultrasmall junctions,” in [[*[Single Charge Tunneling]{}*]{}]{} (H. Graber and M. H. Devoret, eds.), Plenum Press (New York and London), 1992.
P. Joyez and D. Esteve, “Single-electron tunneling at high temperature,” [[*[Phys. Rev. B]{}*]{}]{}, vol. 56, pp. 1848–1853, Jul 1997.
A. A. Odintsov, “Effect of dissipation on dynamic characteristics of small tunnel junctions in terms of the polaron model,” [[*[Soviet Journal of Low Temperature Physics]{}*]{}]{}, vol. 15, pp. 263–266, May 1989.
See A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” [[*[Rev. Mod. Phys.]{}*]{}]{}, vol. 82, pp. 1155–1208, Apr 2010 and references therein.
I. M. Pop, T. Fournier, T. Crozes, F. Lecocq, I. Matei, B. Pannetier, O. Buisson, and W. Guichard, “Fabrication of stable and reproducible submicron tunnel junctions,” J. Vac. Sci. Technol. B vol. 30, p 010607, Jan 2012.
Following Ref. [@PopReproducibleJunctionsArXiv2012], we performed Reactive Ion Etching under an oxygen pressure of 0.3 mbar and 10W RF power, during 15 seconds, in a Plassys MG -200- S RIE equipment.
K.Gloos, R. S. Poikolainen and J. P. Pekola, App. Phys. Lett. **77**,2915 (2000).
B. D. Josephson, Rev. Mod. Phys **36**, 216 (1964).
J. Basset, H. Bouchiat, and R. Deblock, Phys. Rev. Lett. **105**, 166801 (2010).
J. Basset, H. Bouchiat, and R. Deblock, Phys. Rev. B **85**, 085435 (2012).
[^1]: *Note:* Presently at NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The abundances of metals in the intergalactic medium (IGM) can be used to constrain the amount of star formation at high redshift and the spectral shape of the ionizing background radiation. For both purposes it is essential to measure the abundances in regions of low density, away from local sources of metals and ionizing photons. Here we report the first detection of in the low-density IGM at high redshift. We perform a pixel-by-pixel search for absorption in eight high quality quasar spectra spanning the redshift range $z=2.0$-4.5. At $2 \la z\la 3$, we clearly detect in the form of a positive correlation between the [Ly$\alpha$]{} optical depth and the optical depth in the corresponding pixel, down to $\tau_{\rm HI} \sim 10^{-1}$. This is an order of magnitude lower in $\tau_{\rm HI}$ than the best measurements can probe and constitutes the first clear detection of metals in *underdense* gas. The non-detection of at $z > 3$ is consistent with the enhanced photoionization from a hardening of the UV background below $z\sim 3$ but could also be caused by the high level of contamination from Ly series lines.'
author:
- 'Joop Schaye, Michael Rauch, Wallace L. W. Sargent, and Tae-Sun Kim'
title: 'The detection of oxygen in the low-density intergalactic medium'
---
Introduction
============
The abundance of metals in the low-density intergalactic medium (IGM) at high redshift is of interest for at least two reasons. Firstly, the metallicity can be used to distinguish between different enrichment mechanisms and to constrain the amount of star formation at high redshift. Secondly, the relative abundances of ions with different ionization potentials can be used to derive the spectrum of the integrated UV background from stars and quasars. For both purposes it is essential to measure the metal abundances in gas of low density, away from the influence of local sources of ionizing radiation and locally produced metals.
Resonant [Ly$\alpha$]{} absorption by neutral hydrogen along the line of sight to distant quasars results in a forest of absorption lines bluewards of the quasar’s [Ly$\alpha$]{} emission line. With the advent of the High-Resolution Echelle Spectrograph (HIRES) on the Keck telescope, it became clear that many of the high column density [Ly$\alpha$]{} lines ($N_{\rm HI} \ga 10^{14.5
}\,{\rm cm}^{-2}$) show associated absorption (e.g., Cowie et al. 1995). Although increasing the signal-to-noise ratio (S/N) reveals systems of progressively lower column density, lines corresponding to low column density [Ly$\alpha$]{} lines are generally far too weak to detect (e.g., Ellison et al. 2000).
The success of cosmological simulations in reproducing the observations of the [Ly$\alpha$]{} forest has convincingly shown that the low column density forest ($N_{\rm HI} \la 10^{14.5}\,{\rm cm}^{-2}$) at high redshift ($z \ga 2$) arises in a smoothly fluctuating IGM, with individual lines corresponding to local density maxima of moderate overdensity ($\rho/\bar{\rho} \la 10$). The interplay between photoionization heating and adiabatic cooling that is due to the universal expansion results in a tight temperature-density relation, which is well approximated by a power-law, $T=T_0(\rho/\bar{\rho})^{\gamma-1}$ (Hui & Gnedin 1997). The [Ly$\alpha$]{} optical depth is proportional to the neutral hydrogen density, which in photoionization equilibrium is proportional to $\rho^2 T^{-0.76}$. Hence the optical depth depends on the underlying baryon overdensity, $\tau_{\rm HI} \propto
(\rho/\bar{\rho})^{2.76-0.76\gamma}$. The constant of proportionality is an increasing function of redshift, at $z\sim 3$, an optical depth of one corresponds to slightly overdense gas.
Cowie & Songaila (1998) realized that the tight correlation between optical depth and gas overdensity makes a pixel-by-pixel analysis of metal abundances useful. By measuring the median / optical depth ratio as a function of $\tau_{\rm HI}$, they were able to show that the IGM is enriched with a roughly constant / ratio down to $\tau_{\rm HI}\sim 1$ at $z\sim 3$. Ellison et al. (2000) showed explicitly that directly detected lines account for only a small fraction of the total amount of metals in the forest, as inferred from a pixel analysis.
Although other metal lines have been detected, ($\lambda\lambda$1548, 1551) has so far proven the most sensitive since it is the strongest line redwards of [Ly$\alpha$]{} ($\lambda$1216), where there is little contamination from other absorption lines. However, simulations show that at high redshift, photoionized ($\lambda\lambda$1032, 1038) should be a much better probe of the metallicity of the IGM in regions close to the mean density (Chaffee et al. 1986; Rauch, Haehnelt, & Steinmetz 1997; Hellsten et al. 1998). In practice it has proven very difficult to detect at high redshift because it lies deep in the [Ly$\alpha$]{} and [Ly$\beta$]{} ($\lambda 1026$) forest. Since Lu & Savage (1993) established the presence of in intervening absorbers by stacking Lyman limit systems that show absorption, has been detected at high redshift in several Lyman limit systems (Vogel & Reimers 1995, Kirkman & Tytler 1997; 1999). Davé et al. (1998) (see also Cowie & Songaila 1998) carried out a thorough search for but found only evidence for associated with absorption in high density gas.
At low redshift ($z \la 1$), absorbers have been shown to be common (e.g., Bergeron et al. 1994), so common in fact that they may harbor a large fraction of the baryons (Burles & Tytler 1996; Tripp, Savage & Jenkins 2000). These low redshift systems probably correspond to hot, shock heated gas in the potential wells of (groups of) galaxies, which may well be collisionally ionized.
We have carried out a pixel-by-pixel search for in the spectra of eight quasars, spanning the redshift range $z\sim 2.0$–4.5. We clearly detect at $2\la z \la 3$, down to very low optical depths, $\tau_{\rm HI} \sim 10^{-1}$, which is an order of magnitude lower than the best measurements can probe.
Observations and sample definition
==================================
We analyzed spectra of the eight quasars listed in column 6 of Table 1. The spectra of Q1101, J2233 and Q1122 were taken during the Commissioning I and Science Verification observations of the UV-Visual Echelle Spectrograph (UVES) and have been released by ESO for public use. They were reduced with the ESO-maintained MIDAS ECHELLE package (see Kim, D’Odorico, & Cristiani 2000, in preparation) for details on the data reduction). The other spectra were obtained with the HIRES spectrograph (Vogt et al. 1994) on the Keck telescope. The reduction procedures for these quasars can be found in Barlow & Sargent (1997) and Sargent, Barlow & Rauch (2000, in preparation). The HIRES and UVES spectra have a nominal velocity resolution of 6.6 and 6.7 ${{\rm km}\,{\rm s}^{-1}}$ (FWHM) and a pixel size of 0.04 and 0.05 Å respectively. To avoid confusion with the [Ly$\beta$]{} forest, we only used the [Ly$\alpha$]{} region between the quasar’s [Ly$\alpha$]{} and [Ly$\beta$]{}emission lines. In addition, spectral regions close to the quasar (typically $5000~{{\rm km}\,{\rm s}^{-1}}$) were excluded to avoid proximity effects. Regions in the [Ly$\alpha$]{} forest thought to be contaminated by metal lines were excluded.
The mean absorption increases rapidly bluewards of the quasar’s [Ly$\beta$]{}emission line becaue of the growing number of absorption lines other than [Ly$\alpha$]{}. For quasars at $z \ga 3$ the increase in the absorption is dominated by higher order Ly lines, corresponding to [Ly$\alpha$]{} lines which fall in between the quasar’s [Ly$\beta$]{} and [Ly$\alpha$]{} emission lines (i.e. in the [Ly$\alpha$]{} forest). At lower redshift the hydrogen lines have smaller column densities and metal lines contribute significantly to the mean absorption. The increase in the mean absorption towards the blue, raises the threshold for detecting absorption in the [Ly$\alpha$]{}forest considerably. Moreover, the S/N in the region generally decreases towards the blue, raising the detection threshold further. We therefore analyzed the red and blue halves, i.e. the high and low redshift halves, of each [Ly$\alpha$]{} forest spectrum separately (for those quasars for which our spectral coverage extends far enough into the blue). This turned out to be crucial, as we were unable to detect in any of the low redshift half samples. All the high redshift half samples are listed in Table 1.
[lccrrcc]{}\
\
Sample & $z_{\min}$ & $z_{\max}$ & (S/N)$_{\rm OVI}$ & (S/N)$_{\rm
HI}$ & QSO & $z_{\rm em}$\
1101 & 1.97 & 2.10 & 21 & 78 & Q1101$-$264 & 2.14\
2233 & 2.07 & 2.18 & 12 & 30 & J2233$-$606 & 2.24\
1122 & 2.03 & 2.34 & 27 & 66 & Q1122$-$165 & 2.40\
1442 & 2.51 & 2.63 & 8 & 61 & Q1442+293 & 2.67\
1107 & 2.71 & 2.95 & 10 & 49 & Q1107+485 & 3.00\
1425 & 2.92 & 3.14 & 42 & 113 & Q1425+604 & 3.20\
1422 & 3.22 & 3.53 & 46 & 105 & Q1422+231 & 3.62\
2237 & 4.15 & 4.43 & 31 & 32 & Q2237$-$061 & 4.55\
\[tbl:samples\]
Method {#sec:method}
======
We search for using a variant of the pixel technique introduced by Cowie & Songaila (1998). We measure the optical depth in each [Ly$\alpha$]{} forest pixel and in the corresponding pixel. The -pixel pairs are binned according to their optical depth and the median and optical depths are computed for each bin. This technique enables us to probe the abundance down to much lower densities than is possible by fitting lines. Its main advantage over, for example, the stacking method is its robustness. Being a median, it is relatively insensitive to (non-Gaussian) noise, contamination from non- lines, problems that severely compromise the stacking method, which is a mean (see Ellison et al. 2000 for a critical assessment of both the stacking technique and the pixel technique for the case of ). Its main limitation is that it measures an *apparent* optical depth, $\tau_{\rm OVI, app}$, which is in general not the same as the true optical depth because of contamination.
The effect of contaminating lines is reduced by defining $\tau_{\rm
OVI, app} \equiv \min(\tau_{\rm OVI,\,1032},2\tau_{\rm OVI,\,1038})$, where we used the fact that $(f\lambda)_{1032} =
2(f\lambda)_{1038}$. Taking the minimum doublet component does of course not guarantee that $\tau_{\rm OVI, app} = \tau_{\rm OVI}$. For some pixels both components may be contaminated, which would result in an overestimate of the optical depth. However, if the contamination is negligible, then noise will cause us to underestimate $\tau_{\rm OVI}$. For , which falls redwards of the [Ly$\alpha$]{} forest, noise is the limiting factor and Ellison et al. (2000) therefore only used the weaker component if a $3\sigma$ detection was predicted, based on the signal in the stronger component. For however, contamination is a much greater problem and we therefore take the the minimum if the expected signal in the weaker component is greater than 0.5$\sigma$, i.e. if $\exp(-0.5\tau_{\rm OVI,\,1032}) <
1-0.5\sigma_{\rm OVI,\,1038}$, where $\sigma_{\rm OVI,\,1038}$ is the rms amplitude of the noise at the position of ,1038 (as given by the normalized noise array).
When a pixel is close to saturation \[$\exp(-\tau_{\rm HI}) <
0.5\sigma$\], we use up to 10 higher order Ly lines to determine the optical depth: $\tau_{{\rm Ly}\alpha}\equiv \min(\tau_{{\rm
Ly}n}f_{{\rm Ly}\alpha}\lambda_{{\rm Ly}\alpha}/f_{{\rm
Ly}n}\lambda_{{\rm Ly}n})$. We limit the effect of noise features by requiring $0.5\sigma_n < \exp(-\tau_{{\rm Ly}n}) < 1-0.5\sigma_n$ ($\sigma_n$ is the rms noise amplitude at the position of Ly$n$). The number of higher order lines available is different for each quasar spectrum and varies also within a single spectrum. However, since the number of pixels with a given $\tau_{\rm HI}$ decreases rapidly for $\tau_{\rm HI} \ga 1$, the misbinning of due to missing higher orders affects only the highest $\tau_{\rm HI}$ bins.
We use logarithmic $\tau_{\rm HI}$ bins of size 0.35 dex (0.5 dex for the two lowest redshift quasars) and estimate the error in the medians by bootstrap resampling[^1].
Results
=======
\[fig:uphalf\]
Figure 1 shows the results from a pixel analysis of the samples listed in Table 1. The apparent optical depth is clearly correlated with $\tau_{\rm HI}$ at $z\la 3$ down to $\tau_{\rm HI} \sim 10^{-1}$, indicating that is detected in the [Ly$\alpha$]{} forest. At $z\sim
3.2$–3.5 we detect only at very high optical depths, $\tau_{\rm HI} > 10$, while at $z > 4$ $\tau_{\rm OVI}$ and $\tau_{\rm
HI}$ appear uncorrelated. The transition seems to occur at $z\sim
3.0$–3.5. If contamination and noise, which are independent of $\tau_{\rm HI}$, dominate over absorption from , then $\tau_{\rm
OVI, app}$ will flatten off at the detection limit. Ellison et al.(2000) analyzed a very high quality spectrum of Q1422+231 ((S/N)$_{\rm
CIV} \sim 200$) and found that $\tau_{\rm CIV}$ flattens off below $\tau_{\rm HI} \sim 2$. We find that the correlation between $\tau_{\rm OVI, app}$ and $\tau_{\rm HI}$ continues down to much lower $\tau_{\rm HI}$, indicating that oxygen has been detected in gas of very low density contrast ($\rho/\bar{\rho} < 1$).
The horizontal, dashed lines indicate the median $\tau_{\rm OVI, app}$ corresponding to a random pixel, i.e. the median $\tau_{\rm OVI,
app}$ for the set of all pixels. This reference level includes contributions from , from contaminating non- lines and from noise. For $z \ga 3$, the contribution of to this level is negligible and the level is effectively the detection limit. For $z
\la 3$ on the other hand, accounts for a significant part of the total absorption. Hence the detection limit is lower than the reference level and $\tau_{\rm OVI, app}$ drops below the dashed line at low $\tau_{\rm HI}$.
We stress that because of noise and especially contamination, the measured, apparent optical depths could differ considerably from the true values. We tried to estimate the effects of noise and contamination, which vary systematically across the spectrum, by analyzing synthetic spectra generated by drawing Voigt profiles at random from observed line lists until the mean absorption in the [Ly$\alpha$]{} forest equals the observed value. We then added the corresponding absorption lines using a fixed / column density ratio and a fixed $b$-parameter ratio. Finally, higher order Ly lines and noise were added using the line list and the noise array respectively. We found that the simulations agree reasonably well with the observations for $N($$)/N($$) \sim
10^{-2}$–$10^{-1}$, although the results are sensitive to the assumed ratio of $b$-parameters. Although encouraging, these Monte Carlo simulations are clearly too simplistic to derive the intrinsic / ratios. The effects of noise and contamination can only be modeled accurately by calibrating the method against synthetic spectra extracted from hydrodynamic simulations. This should be done separately for each observed spectrum, using synthetic spectra that resemble the observed spectrum in detail. We leave such a quantitative analysis for a future paper but note that the effects that prevent us from deriving the actual abundance, will reduce any intrinsic correlation between $\tau_{\rm HI}$ and $\tau_{\rm OVI, app}$, making a detection of such a correlation a robust result.
For a fixed abundance, the value of the reference level (Fig. 1, dashed lines) increases with decreasing S/N and with increasing redshift. The latter effect is more important and is due to the increase in the number and strength of contaminating Ly series lines with redshift, which is a direct consequence of the expansion of the universe. As the universe expands, the column densities of lines corresponding to absorbers of a fixed overdensity decrease (the evolution of the ionizing background and the growth of structure are less important). This results in an increase of the amount of contamination, and thus the detection threshold, with redshift. Note that this effect is opposite to the increase in contamination towards the blue (i.e. low redshift) end of a single quasar spectrum, which is a consequence of the growing number of interlopers (see section 2). Like the column density, the optical depth corresponding to a fixed overdensity also increases with redshift. This complicates the comparison of the $\tau_{\rm HI}$ level below which the correlation with flattens off in samples of different redshifts. While an optical depth of one corresponds to about the mean density at $z\sim 3$, it corresponds to an overdensity of a few at $z\sim 2$ and to slightly underdense gas at $z\sim 4$.
For the quasars Q1107, Q1422 and Q2233 our spectral coverage is sufficient to analyze the blue half of the forest. We find no correlation between $\tau_{\rm OVI}$ and $\tau_{\rm HI}$ in any of these samples. As discussed above, this is a consequence of the large number of contaminating absorption lines from intervening absorbers and/or the low S/N of the data and does not imply that there is no . In principle, even more could be uncovered by looking at a smaller part of the spectrum. The contamination in the upper redshift quarter for example, will be lower than in the upper half. However, decreasing the number of pixels further would bring the redshift path into the regime of large-scale structures and the samples would therefore no longer be representative. This may already be the case for the two lowest redshift samples, for which the number of pixels showing absorption is much smaller than for the higher redshift samples. Nevertheless, we have tried smaller samples and found that the correlation strengthens somewhat for 1422 but not for 2237.
Figure 2 demonstrates that the that we detect at low $\tau_{\rm
HI}$ is not associated with detectable absorption. The solid line indicates the apparent absorption in 1442. The dashed line shows the result obtained after all pixels that have nonzero absorption at the $1\,\sigma$ level \[$\exp(-\tau_{\rm CIV}) <
1-1\sigma$\] have been excluded. The dashed curve only falls below the solid curve for $\tau_{\rm HI} \ga 2$, indicating that some of the stronger systems have been excluded. The fact that the two curves are almost indistinguishable for $\tau_{\rm HI} \la 2$ shows that is detectable in absorbers for which absorption is too weak to detect and is thus direct observational proof of the prediction that is a more sensitive probe of the metallicity in low-density gas than .
\[fig:civ\]
Discussion
==========
We have searched for absorption in the spectra of eight quasars spanning the redshift range $z\sim 2.0$–4.5 and reported the first detection of in the low-density IGM at high redshift. We analyzed the spectra pixel by pixel and detected a positive correlation between the optical depths in and in the corresponding pixels down to $\tau_{\rm HI}\sim 10^{-1}$ at $2
\la z\la 3$. This is an order of magnitude lower in $\tau_{\rm HI}$ than the best measurements can probe and constitutes the first firm detection of metals in *underdense* gas. We showed that the signal does not come from systems for which is detectable, and this confirms the prediction that is a much better tracer of metals in low-density gas than .
Although the very strong absorbers may well be collisionally ionized (e.g., Kirkman & Tytler 1999), the volume filling factor of this hot gas is much smaller than that of photoionized gas (e.g., Cen & Ostriker 1999). The ubiquitous detected here, arises in gas that is too dilute for collisional ionization to be effective. Photons of at least 8.4Ryd are needed to create and hence we would not expect to detect photoionized if the reionization of , for which photons of at least 4Ryd are required, is not yet complete. Measurements of the [Ly$\alpha$]{} opacity (Heap et al. 2000 and references therein) and the temperature of the IGM (Schaye et al.2000, Ricotti, Gnedin & Shull 2000) suggest that the reionization of helium may have been completed at $z\sim 3$. Although our non-detection of at $z > 3$ is consistent with enhanced photoionization from a hardening of the ionizing background below $z\sim 3$, it could also be caused by the increased level of contamination from Ly series lines.
Our detection of down to $\tau_{\rm HI}\sim 10^{-1}$ shows that the IGM is enriched down to much lower overdensities than have been probed up till now. A comparison with simulations could clarify how noise and contamination from Ly series lines affect the apparent optical depth, what fraction of the low-density gas is enriched and what the scatter in the / ratio is, thereby providing strong constraints on theoretical models for the enrichment of the IGM.
We are grateful to S. Ellison, M. Pettini and G. Efstathiou for discussions and a careful reading of the manuscript. JS thanks the Isaac Newton Trust and PPARC for support. WLWS acknowledges support from the NSF under grant AST-9900733.
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[^1]: The spectrum is divided into chunks of 2Å and 250 spectra of length equal to the original spectrum are generated by picking chunks at random (with replacement, each chunk can be picked more than once). The $1\sigma$ error in the median is then the square root of the variance in the median over the bootstrap realizations.
|
{
"pile_set_name": "ArXiv"
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**IMPOSSIBILITY OF OBTAINING SPLIT LINKS**
**FROM SPLIT LINKS VIA TWISTINGS**
MAKOTO OZAWA [^1]
*Department of Mathematics, School of Education,*
*Waseda University, Nishiwaseda 1-6-1, Shinjuku-ku,*
*Tokyo 169-8050, Japan*
*[email protected]*
Introduction
============
Let $L$ be a link in $S^3$ and $C$ a trivial knot in $S^3$ missing $N(L)$. Then we can get a new link $L^*$ in $S^3$ as the image of $L$ after doing $1/n$-Dehn surgery along $C$. We say that $L^*$ is obtained from $L$ by an [*$n$-twisting*]{} along $C$. In this paper, we consider the following problem.
[**Problem.**]{} Is it possible that both $L$ and $L^*$ are splittable?
For this problem, it is reasonable to make the following definition. An $n$-twisting is said to be [*non-trivial*]{} if $n\ne 0$ and the link $L\cup C$ is non-splittable. Then our result is stated as follows.
It is impossible to obtain a split link from a split link by a non-trivial twisting.
Next, we consider when a trivial link is obtained from a trivial link by an $n$-twisting. For a trivial knot, this problem has been solved as follows.
$($[@M], [@KMS]$)$ Suppose that a trivial knot $K^*$ is obtained from a trivial knot $K$ by an $n$-twisting along $C$. Then one of the following conclusions holds.
\(1) The link $K\cup C$ is a trivial link.
\(2) The link $K\cup C$ is a Hopf link.
\(3) The link $K\cup C$ is a torus link of type $(4,-2)$ or $(4,2)$, and $n=1$ or $-1$ respectively.
By Theorems 1 and 2, we obtain the next corollary.
Suppose that a trivial link $L^*=K_1^*\cup \ldots \cup K_l^*$ is obtained from a trivial link $L=K_1\cup \ldots \cup K_l$ by an $n$-twisting along $C$. Then one of the following conclusions holds.
\(1) The link $L\cup C$ is a trivial link.
\(2) The link $L\cup C$ is a split union of a Hopf link $K_i\cup C$ and a trivial link $L-K_i$ for some $i\in \{1, \ldots , l\}$
\(3) The link $L\cup C$ is a split union of a torus link $K_i\cup C$ of type $(4,-2)$ or $(4,2)$ and a trivial link $L-K_i$ for some $i\in \{1,\ldots ,l\}$, and $n=1$ or $-1$ respectively.
Preliminaries
=============
In this section, we prepare some lemmas for Theorem 1. All manifolds are assumed to be compact and orientable, and any srufaces in a 3-manifold are assumed to be properly embedded and in general position.
Let $M$ be a 3-manifold, and $F_1$ and $F_2$ two surfaces in $M$. Let $\hat{F_1}$, $\hat{F_2}$ be the closed surfaces obtained by capping off $\partial F_1$, $\partial F_2$ with disks. Then, for $\alpha \in \{ 1,2\}$, one defines a graph $G_{\alpha}$ in $\hat{F_{\alpha}}$, where the edges of $G_{\alpha}$ correspond to the arc components of $F_1\cap F_2$, and the vertices to the components of $\partial F_{\alpha}$. Recall that a [*1-sided face*]{} in a graph is a disk face with exactry one edge in its boundary.
Recall that if $M$ is a 3-mainfold with torus boundary and $\gamma$ is a slope on $\partial M$, then $M(\gamma )$ denotes the closed maifold obtained by attaching a solid torus $J$ to $M$ so that the boundary of a meridian disk of $J$ has slope $\gamma$ on $\partial M$. Recall also that if $\gamma _1,\gamma _2$ are two slopes on $\partial M$, then $\Delta (\gamma _1,\gamma _2)$ denotes the minimal geometric intersection number of $\gamma _1$ and $\gamma _2$.
The following lemma will be needed for Theorem 1.
Let $M$ be a 3-manifold with torus boundary and let $F_1,F_2$ be planar surfaces in $M$ with boundary slopes $\gamma _1,\gamma _2$. Suppose that the graphs $G_1,G_2$ contain no 1-sided faces, and that $\Delta (\gamma _1,\gamma_2)\ge 1$. Then either the first homology groups $H_1(M(\gamma _1))$ or $H_1(M(\gamma_2))$ has a torsion.
[*Proof.*]{} If $\Delta (\gamma _1,\gamma _2)\ge 2$, then Lemma 1 follows [@GL1 lemma 2.2]. Otherwise, by [@GL2 Proposition 2.0.1], $G_1$ contains a Scharlemann cycle or $G_2$ represents all $\{ 1,\ldots ,|\partial P_1|\}$-types. In the formar case, $M(\gamma _2)$ has a lens space as a connected summand. In the latter case, by [@P Theorem], $H_1(M(\gamma _1))$ has a torsion. This completes the proof of Lemma 2. 0.8pt
Proof of Theorem 1
==================
Suppose that a split link $L^*$ is obtained from a split link $L$ by a non-trivial $n$-twisting along $C$. Let $S$ and $S^*$ be the splitting spheres for $L$ and $L^*$ respectively. Put $M=S^3-intN(C)$. We may assume that $C$ intersects $S$ and $S^*$ transversely in the 3-spheres $M(1/0)$ and $M(1/n)$ respectively, and assume that $|C\cap S|$ and $|C\cap S^*|$ are minimal among all 2-spheres isotopic to $S$ and $S^*$ respectively. Then, since $L\cup C$ and $L^*\cup C$ are non-splittable, $|C\cap S|$ and $|C\cap S^*|$ are not equal to zero. Put $P_1=S-intN(C)$ and $P_2=S^*-intN(C)$. Then by the minimality of $|C\cap S|$ and $|C\cap S^*|$ and by the irreducibility of $M-L$, $P_1$ and $P_2$ satisfy the hypothesis of Lemma 2. Hence $H_1(M(1/0))$ or $H_1(M(1/n))$ has a torsion, this is impossible. 0.8pt
[Acknowledgement]{}
The author would like to thank Prof. Chuichiro Hayashi for his helpful comments.
[000]{} C. McA. Gordon and J. Luecke, [*Only integral Dehn surgeries can yield reducible manifolds,* ]{} Math. Proc. Camb. Phil. Soc. [**102**]{} (1987) 94-101.
C. McA. Gordon and J. Lueke, [*Knots are determined by their complements,* ]{} J. Amer. Math. Soc. [**2**]{} (1989) 371-415.
M. Kouno, K. Motegi and T. Shibuya, [*Twisting and knot types,* ]{} J. Math. Soc. Japan [**44**]{} (1992) 199-216.
Y. Mathieu, [*Unknotting, knotting by twists on disks and Property (P) for knots in $S^3$,* ]{} Knots 90 (ed Kawauchi, A), Proc. 1990 Osaka Conf. on Knot Theory and Related Topics, de Gruyter, (1992) 93-102.
W. Parry, [*All types implies torsion,* ]{} Proc. Amer. Math. Soc. [**110**]{} (1990) 871-875.
[^1]: The author was supported in part by Fellowship of the Japan Society for the Promotion of Science for Japanese Junior Scientists.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this work we study the problem of tax evasion on a fully-connected population. For this purpose, we consider that the agents may be in three different states, namely honest tax payers, tax evaders and undecided, that are individuals in an intermediate class among honests and evaders. Every individual can change his/her state following a kinetic exchange opinion dynamics, where the agents interact by pairs with competitive negative (with probability $q$) and positive (with probability $1-q$) couplings, representing agreement/disagreement between pairs of agents. In addition, we consider the punishment rules of the Zaklan econophysics model, for which there is a probability $p_{a}$ of an audit each agent is subject to in every period and a length of time $k$ detected tax evaders remain honest. Our results suggest that below the critical point $q_{c}=1/4$ of the opinion dynamics the compliance is high, and the punishment rules have a small effect in the population. On the other hand, for $q>q_{c}$ the tax evasion can be considerably reduced by the enforcement mechanism. We also discuss the impact of the presence of the undecided agents in the evolution of the system.'
address: |
Instituto de Física, Universidade Federal Fluminense\
Av. Litorânea s/n, 24210-340 Niterói - RJ, Brazil
author:
- Nuno Crokidakis
title: 'A three-state kinetic agent-based model to analyze tax evasion dynamics'
---
[^1]
Keywords: Dynamics of social systems, Collective phenomena, Computer simulations
Introduction
============
In the recent years, the statistical physics techniques have been successfully applied in the description of socioeconomic phenomena. Among the studied problems we can cite opinion dynamics, language evolution, biological aging, dynamics of stock markets, earthquakes and many others [@galam_book; @sen_book; @econ_book; @pmco_book]. These interdisciplinary topics are usually treated by means of computer simulations of agent-based models, which allow us to understand the emergence of collective phenomena in those systems.
A challenging interdisciplinary subject is tax evasion dynamics, which is an interesting practical topic to be studied because tax evasion remains to be a major predicament facing governments [@bloom; @prinz; @andreoni]. Models of tax evasion were firstly studied by economists [@gachter; @frey; @follmer; @slemrod; @davis], and more recently physicists became also interested in the subject [@zaklan; @lima1; @lima2; @lima3; @llacer; @seibold] (for recent reviews, see [@bloom; @prinz]). Experimental evidence provided by Gachter suggests that tax payers tend to condition their decision regarding whether to pay taxes or not on the tax evasion decision of the members of their group [@gachter]. In addition, Frey and Torgler also provide empirical evidence on the relevance of conditional cooperation for tax morale [@frey]. Based on these ideas, Zaklan *et al.* recently proposed a model that has been attracted attention [@zaklan]. In the so-called Zaklan model, the dynamics of tax payers and tax evaders is analyzed by means of the two-dimensional Ising model at a given temperature $T$. In this model, each agent $i$ may be in one of two possible states, namely $s_{i}=+1$ (honest) or $s_{i}=-1$ (cheater or tax evader). A transition $s_{i} \to -s_{i}$ (or a spin flip) is controlled by the “social temperature” $T$ and also depends on the nearest neighbors’ states of the agent (or spin) at site $i$. Thus, for low temperatures few spin flips occur and for high temperatures many spin flips occur. In other words, tax evaders have the greatest influence to turn honest citizens into tax evaders if they constitute a majority in the respective neighborhood. In addition, some punishment rules are applied: there is a probability $p_{a}$ of an audit each agent is subject to in every period and a length of time $k$ detected tax evaders remain honest [@zaklan]. In another work, the dynamics of the model was also controlled by another two-state model, namely the majority-vote model with noise [@maj_vot], where the noise $q$ plays the role of the temperature. In this case, similar results were found [@lima3], suggesting that the results of the Zaklan model are robust.
In this work we study the tax evasion dynamics by means of a three-state agent-based model. The agents interact by pairs considering kinetic exchanges of their states, in a way that the pairwise couplings may be positive of negative. In addition, we apply the punishment rules of the Zaklan econophysics model. Our results suggest that above the critical point of the opinion dynamics the tax evasion can be considerably reduced by the enforcement rules. On the other hand, below the critical point the compliance is high, and the punishment rules have a small impact on the evasion.
This work is organized as follows. In Section 2 we present the microscopic rules that define the model, and the numerical results are discussed in Section 3. Finally, our conclusions are presented in Section 4.
Model
=====
Our model is based on a kinetic exchange opinion model [@biswas]. A population of $N$ agents is defined on a fully-connected graph, i.e., each agent can interact with all others. In opposition to what occurs in the Zaklan model [@zaklan], for which the dynamics is governed by the Ising model (i.e., a two-state model), in our model each individual $i$ ($i=1,2,...,N$) carries one of three possible states or attitudes at a given time step $t$, represented by $s_{i}(t)=+1,-1$ or $0$. The dynamic rules are defined following the opinion model of Ref. [@biswas]. Each social interaction occurs between two given agents $i$ and $j$, and we considered that $j$ will influence $i$. First, this pair of agents $(i,j)$ is randomly chosen. Then, the state of the agent $i$ in the next time step $t+1$ will be updated according to $$\label{eq1}
s_{i}(t+1) = {\rm sgn}\left[ s_{i}(t) + \mu_{ij}\,s_{j}(t) \right]\,,$$ where the sign function is defined such that ${\rm sgn}(0)=0$ and the interaction strenghts $\{\mu_{ij}\}$ are quenched random variables given by the discrete bimodal probability distribution $$\label{eq2}
F(\mu_{ij}) = q\,\delta(\mu_{ij}+1) + (1-q)\,\delta(\mu_{ij}-1) ~.$$
Notice that we considered that each agent can in principle interact with all other agents, i.e. there is no specific underlying topology for the structure of the interaction network. So the model can be viewed as an infinite dimension (or “mean field”) Zaklan model. This is an almost realistic situation thanks to the modern social and communication networks.
First, let us elaborate upon the nature of the above-mentioned three states. The state $s_{i}=+1$ represents a honest tax payer, i.e., an individual 100$\%$ convinced of his/her honesty, who does not consider evasion. He/she is either habitually compliant or he/she is a recent evader who has become honest as a result of enforcement efforts or social norms. On the other hand, the state $s_{i}=-1$ represents a cheater, i.e, an individual who is an evading tax payer. Whether a tax payer continues to evade depends on both enforcement and the effect of social interactions.
Those two classes correspond to the $\pm 1$ states of the standard Zaklan model [@zaklan]. In addition, we have considered a third state, $s_{i}=0$, which can be interpreted as an *undecided* individual. However, notice that the above rules of the opinion dynamics \[Eqs. (\[eq1\]) and (\[eq2\])\] impose that for an agent to shift from state $s=+1$ to $s=-1$ or vice-versa it must to pass by the intermediate undecided state $s=0$ [@biswas]. Thus, an agent that is currently at the state $s=0$ was a honest tax payer ($s=+1$) or a tax evader ($s=-1$) before. In the first case, the individual is a honest tax payer and, due to social interactions, he/she becomes a tax payer who is dissatisfied with the tax system, perhaps as a result of seeing others evading without being punished. He/she is not actively evading, but he/she might if the perceived benefits of doing so exceed the perceived costs. For this group, evasion is an option. On the other hand, the second possibility is that the agent is a tax evader and, due to social interactions, he/she stops temporarily the evasion because he/she wondered whether it is worth to evade. This agent is fickle and is not 100$\%$ convinced of his/her honesty, and thus he/she can become a honest tax payer ($s=+1$) or he/she can come back to the tax evader state ($s=-1$), depending on the next interactions with his/her social contacts. The above discussion will become more clear in the following, when we will discuss the interpretation of the competitive interactions $\mu_{ij}$.
The pairwise couplings $\mu_{ij}$ in Eq. (\[eq2\]) may be either negative (with probability $q$) or positive (with probability $1-q$), such that $q$ represents the fraction of negative couplings [@biswas; @celia]. The above process given by Eqs. (\[eq1\]) and (\[eq2\]) is repeated $N$ times, which defines one time step in the dynamics. In addition to such basic dynamics of the model, after the $N$ interactions we have considered a policy makers’ tax enforcement mechanism consisting of two components, a probability $p_{a}$ of an audit each person is subject to in every period and a length of time $k$ detected tax evaders remain honest, as considered in the Zaklan model [@zaklan]. In other words, after the application of the above-mentioned kinetic exchange opinion dynamics, we have considered that each tax evader will be caught by an audit with probability $p_{a}$. In this case, the individual must remain honest for a given number $k$ of time steps. As mentioned before, the above rules of the opinion dynamics impose that for an agent to shift from state $s_{i}=+1$ to $s_{i}=-1$ or vice-versa it must to pass by the intermediate undecided state $s_{i}=0$. However, this kind of hierarchy is partially broken when we apply the enforcement rules. Indeed, if a tax evader (state $s_{i}=-1$) is caught by an audit, he/she changes directly to the honest state $s_{i}=+1$ and remains in this state at least during the next $k$ time steps.
Following ref. [@biswas], the dynamics defined by the kinetic exchange model can be interpreted as follows. If $\mu_{ij}$ is positive (with probability $1-q$), there is a kind of agreement between the agents $i$ and $j$. In this case, if $s_{i}(t)=0$ the agent $i$ is undecided and does not know what is the best choice (be honest or be evader), and thus he/she follows the decision of $j$, i.e., the state of $i$ is updated to $s_{i}(t+1)=s_{j}(t)$. If $s_{i}(t)=s_{j}(t)$ nothing occurs, and if $s_{i}(t)=-s_{j}(t)$ the agent $i$ will become undecided and change to state $s_{i}(t+1)=0$ in the next time step $t+1$, and he/she can become evader or a honest tax payer in a next interaction. On the other hand, if $\mu_{ij}$ is negative (with probability $q$), there is a kind of disagreement or mutual disliking between the agents $i$ and $j$. In this case, if $s_{i}(t)=s_{j}(t)$ the agent $i$ will become undecided and change to $s_{i}(t+1)=0$ and and if $s_{i}(t)=-s_{j}(t)$ the agent $i$ keeps his/her decision. Finally, if $s_{i}(t)=0$, the agent $i$ does not know what is the best choice (be honest or be evader) and due to the mentioned disliking his state is updated to $s_{i}(t+1)=-s_{j}(t)$.
![Magnetization per spin $m$ of the kinetic exchange opinion model of Ref. [@biswas] as a function of the fraction $q$ of negative interactions (i.e., where no punishment rules were considered). The system undergoes an order-disorder phase transition at $q_{c}=1/4$, with a paramagnetic disordered phase defined by the coexistence of the three states $s=+1, -1$ and $0$ with equal fractions ($1/3$ for each one). The population size is $N=10^{4}$, the squares are numerical results averaged over $100$ independent simulations and the dashed line is just a guide to the eyes.[]{data-label="fig1"}](figure1.eps){width="50.00000%"}
As discussed in [@biswas], the standard opinion dynamics defined by Eqs. (\[eq1\]) and (\[eq2\]), i.e., where no punishment rules were considered, undergoes a nonequilibrium phase transition at a critical fraction $q_{c}=1/4$. For $q<q_{c}$ we have a symmetry of the extreme opinions $s=\pm 1$, i.e., one of the extreme opinions $+1$ or $-1$ dominates the system, with consensus states occurring only for $q=0$, i.e., in the absence of negative interactions. On the other hand, for $q\geq q_{c}$ the system is in a disordered “paramagnetic” phase characterized by the coexistence of the three opinions, with the fraction of each opinion being $1/3$. This picture can be clearly seen in Fig. \[fig1\], where we exhibit the order parameter of the system, i.e., $$\label{eq3}
m = \left\langle \frac{1}{N}\left|\sum_{i=1}^{N} s_{i}\right|\right\rangle ~,$$ where $\langle\, ...\, \rangle$ denotes a disorder or configurational average taken at steady states. The Eq. (\[eq3\]) defines the “magnetization per spin” of the system, and the behavior of $m$ as a function of the fraction $q$ of negative interactions for a population of size $N=10^{4}$ agents is shown in Fig. \[fig1\].
Thus, the next step is to apply the enforcement rules of the Zaklan model [@zaklan] to the model of Ref. [@biswas]. The numerical results will be discussed in the next section.
Numerical Results
=================
We applied the enforcement rules of the Zaklan model [@zaklan] to the opinion dynamics model of Ref. [@biswas]. As usually occurs in Zaklan-like models [@zaklan; @lima1; @lima2; @lima3], we have considered that initially all agents are honest, i.e., we have $s_{i}(t=0)=+1$ for all individuals $i$. In this case, we broke the above-mentioned symmetry of the stationary states of the ordered phase: for $q<q_{c}$ the majority of agents will be in the honest state $s_{i}=+1$. All the following results are for a population of size $N=10^{4}$.
Following the previous studies of the Zaklan model [@zaklan; @lima1; @lima2; @lima3], one can start analyzing the time evolution of the tax evasion, i.e., the fraction of tax evaders ($s=-1$) in the population. In Fig. \[fig2\] we exhibit the tax evasion as a function of time for two distinct values of $q>q_{c}$, namely $q=0.8$ \[(a) and (b)\] and $q=0.5$ \[(c) and (d)\]. In this case, as $q>q_{c}$, the kinetic exchange dynamics defined by Eqs. (\[eq1\]) and (\[eq2\]) leads the system to a disordered state with an equal fraction of each state. In other words, considering only the opinion dynamics, the stationary fraction of evaders should be $1/3\approx 33\%$. Thus, one can see from Fig. \[fig2\] that if the audits are efficient ($p_{a}=90\%$) the tax evasion can be considerably reduced to $\approx 10\%$ for $k=10$ and for $\approx 3\%$ for $k=50$. In these cases, we observe similar fluctuations of the tax evasion as the ones reported in Zaklan models defined in regular lattice and networks [@zaklan; @lima1; @lima2; @lima3]. For the cases where we consider a realistic value of the efficient audits ($p_{a}=5\%$) the punishment is effective only if the penalty duration is high ($k=50$). In this case, the tax evasion can be reduced for values around $20\%$. Notice that when we decrease the value of $q$ the fraction of tax evaders decreases, as one can see in Fig. \[fig1\]. It can be understood as follows. As the opinion dynamics “coexists” in the system with the punishment rules, the system does not achieve in our model the steady states with equal fractions of the three opinions $+1,-1$ and $0$ for $q>q_{c}$. For high values of $q$, there are many negative couplings $\mu_{ij}$ in the population, which allows many transitions $s_{i}=+1 \to s_{i}=0$ and then $s_{i}=0 \to s_{i}=-1$. So, it is expected that for high $q$ the fraction of opinions $-1$ is greater than in the cases of lower values of $q$ (of course, we are talking about the disordered phase of the kinetic exchange opinions dynamics).
![(Color online) Time evolution of the tax evasion for different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (left side) and $p_{a}=0.05$ (right side). The results are for $q=0.8$ \[(a) and (b)\] and $q=0.5$ \[(c) and (d)\]. Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig2"}](figure2a.eps "fig:"){width="48.00000%"} ![(Color online) Time evolution of the tax evasion for different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (left side) and $p_{a}=0.05$ (right side). The results are for $q=0.8$ \[(a) and (b)\] and $q=0.5$ \[(c) and (d)\]. Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig2"}](figure2b.eps "fig:"){width="47.00000%"}\
![(Color online) Time evolution of the tax evasion for different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (left side) and $p_{a}=0.05$ (right side). The results are for $q=0.8$ \[(a) and (b)\] and $q=0.5$ \[(c) and (d)\]. Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig2"}](figure2c.eps "fig:"){width="48.00000%"} ![(Color online) Time evolution of the tax evasion for different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (left side) and $p_{a}=0.05$ (right side). The results are for $q=0.8$ \[(a) and (b)\] and $q=0.5$ \[(c) and (d)\]. Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig2"}](figure2d.eps "fig:"){width="47.00000%"}
As discussed above, when we decrease the value of the parameter $q$, the fraction of tax evaders decreases (for $q>q_{c}$). In this case, the compliance increases when we apply the same punishments for a population with lower values of $q$. Furthermore, one can conclude that for a population with a large fraction of negative interactions, i.e., with a high disagreement among the individuals, the compliance is low in the system, and thus it is necessary a strong enforcement by the public policies in order to control the tax evasion, which is a realistic feature of the model.
![(Color online) Time evolution of the tax evasion for $q=0.3$, different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (a) and $p_{a}=0.05$ (b). Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig3"}](figure3a.eps "fig:"){width="48.00000%"} ![(Color online) Time evolution of the tax evasion for $q=0.3$, different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (a) and $p_{a}=0.05$ (b). Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig3"}](figure3b.eps "fig:"){width="48.00000%"}
![(Color online) Time evolution of the tax evasion for $q=0.1$, different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (a) and $p_{a}=0.05$ (b). Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig4"}](figure4a.eps "fig:"){width="48.00000%"} ![(Color online) Time evolution of the tax evasion for $q=0.1$, different values of the number $k$ of periods that a detected tax evader must remain honest for and two distinct audit probabilities, namely $p_{a}=0.9$ (a) and $p_{a}=0.05$ (b). Each curve is a single realization of the dynamics for a population of size $N=10^{4}$.[]{data-label="fig4"}](figure4b.eps "fig:"){width="48.00000%"}
In Fig. \[fig3\] we exhibit results for $q=0.3$, i.e., another value of $q>q_{c}$, but now the system is near the critical point $q_{c}=1/4$. In this case, one can see that for a high audit probability like $p_{a}=90\%$ the tax evasion can be extremely reduced in a short run \[see Fig. \[fig3\] (a)\], even if only a small period like $k=10$ each individual is compelled to remain honest. Notice that even for $p_{a}=5\%$ the application of severe punishments as $k=50$ can lead the evasion to low levels like $10\%$, as one can see in Fig. \[fig3\] (b).
It is shown in Fig. \[fig4\] the time series of the tax evasion for $q=0.1$. In this case, as $q<q_{c}$, the kinetic exchange dynamics defined by Eqs. (\[eq1\]) and (\[eq2\]) leads the system to an ordered state with the majority of agents in the honest ($s=+1$) state, with a small fraction of tax evaders and undecided. For $N=10^{4}$ (see Fig. \[fig1\]), the fraction of $s=-1$ individuals is $\approx 1.5\%$. Thus, as one can see in Fig. \[fig4\], the consideration of the punishment rules togheter with the basic dynamics has a small effect on the tax evasion in the system, since the tendency of the agents is to be in state $s=+1$ and thus there are few tax evaders to be caught by the audits. For $p_{a}=5\%$ the stationary fraction of evaders is similar for small or large $k$, and for $p_{a}=90\%$ the tax evasion decreases to $\approx 1\%$, with a small difference between the cases $k=10$ and $k=50$. Thus, for a population with a small fraction of negative interactions, i.e., with low disagreement (or high agreement) among the individuals, the compliance is high in the system, and it is not necessary a strong control by the public policies.
![(Color online) Average stationary tax evasion as a function of the audit probability $p_{a}$. In the panels (a), (b) and (c) the results are for $q=0.1$, $q=0.3$ and $q=0.8$, respectively, and typical values of $k$. In the last panel (d) it is shown the results for $k=10$ and different fractions of negative interactions $q$. Each point is averaged over $100$ independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig5"}](figure5a.eps "fig:"){width="48.00000%"} ![(Color online) Average stationary tax evasion as a function of the audit probability $p_{a}$. In the panels (a), (b) and (c) the results are for $q=0.1$, $q=0.3$ and $q=0.8$, respectively, and typical values of $k$. In the last panel (d) it is shown the results for $k=10$ and different fractions of negative interactions $q$. Each point is averaged over $100$ independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig5"}](figure5b.eps "fig:"){width="47.00000%"}\
![(Color online) Average stationary tax evasion as a function of the audit probability $p_{a}$. In the panels (a), (b) and (c) the results are for $q=0.1$, $q=0.3$ and $q=0.8$, respectively, and typical values of $k$. In the last panel (d) it is shown the results for $k=10$ and different fractions of negative interactions $q$. Each point is averaged over $100$ independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig5"}](figure5c.eps "fig:"){width="47.00000%"} ![(Color online) Average stationary tax evasion as a function of the audit probability $p_{a}$. In the panels (a), (b) and (c) the results are for $q=0.1$, $q=0.3$ and $q=0.8$, respectively, and typical values of $k$. In the last panel (d) it is shown the results for $k=10$ and different fractions of negative interactions $q$. Each point is averaged over $100$ independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig5"}](figure5d.eps "fig:"){width="47.00000%"}
To better analyze the results obtained from the simulations, we exhibit in Fig. \[fig5\] the average tax evasion in the stationary states, i.e., in the long-time limit, as a function of the audit probability $p_{a}$. In Figs. \[fig5\] (a), (b) and (c) we present the results for $q=0.1$, $q=0.3$ and $q=0.8$, respectively, for typical values of $k$. In these panels one can clearly see the above-discussed effects. Indeed, for $q<q_{c}$ the tax evasion is low, but the punishment can effectively reduce such evasion if we consider large values of $k$ as $k=100$. On the other hand, the long-time tax evasion for the cases $q>q_{c}$ can be considerably decreased by the application of public policies. In other words, the tax evasion decreases for $\approx 33\%$ in the absence of punishment ($p_{a}=0$) until very small percentages like $\approx 1\%$. One can also see in these cases ($q>q_{c}$) that for large $k$ and $p_{a} > 50\%$ the compliance does not change considerably, suggesting that it is more important to monitor individuals caught in audits for a very long time than to perform extremely efficient audits. Similar conclusions can be obtained if one analyze the effect of changing $q$ for a fixed value of $k$, as shown in Fig. \[fig5\] (d).
![(Color online) Average stationary fractions of evaders (squares), undecided (circles) and honests (triangles, in the insets) as functions of $p_{a}$ for $k=10$ and typical values of $q$, namely $q=0.1$ (a), $q=0.3$ (b), $q=0.5$ (c) and $q=0.8$ (d). Each point is averaged over 100 independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig6"}](figure6a.eps "fig:"){width="48.00000%"} ![(Color online) Average stationary fractions of evaders (squares), undecided (circles) and honests (triangles, in the insets) as functions of $p_{a}$ for $k=10$ and typical values of $q$, namely $q=0.1$ (a), $q=0.3$ (b), $q=0.5$ (c) and $q=0.8$ (d). Each point is averaged over 100 independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig6"}](figure6b.eps "fig:"){width="47.00000%"}\
![(Color online) Average stationary fractions of evaders (squares), undecided (circles) and honests (triangles, in the insets) as functions of $p_{a}$ for $k=10$ and typical values of $q$, namely $q=0.1$ (a), $q=0.3$ (b), $q=0.5$ (c) and $q=0.8$ (d). Each point is averaged over 100 independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig6"}](figure6c.eps "fig:"){width="47.00000%"} ![(Color online) Average stationary fractions of evaders (squares), undecided (circles) and honests (triangles, in the insets) as functions of $p_{a}$ for $k=10$ and typical values of $q$, namely $q=0.1$ (a), $q=0.3$ (b), $q=0.5$ (c) and $q=0.8$ (d). Each point is averaged over 100 independent simulations for population size $N=10^{4}$, and the dashed lines are just guides to the eye.[]{data-label="fig6"}](figure6d.eps "fig:"){width="47.00000%"}
After the analysis of the tax evaders’ behavior, a question may arise: what is the behavior of the other two classes ($s=+1$ and $s=0$)? In particular, remembering that we are considering a 3-state model instead of an usual 2-state one, what is the impact of the presence of the third class, the undecided individuals, in the evolution of the system? In Fig. \[fig6\] we exhibit the stationary fractions of the three classes of agents as functions of the audit probability $p_{a}$ for $k=10$ and typical values of $q$. One can see that, in general, the stationary fraction of undecided individuals is greater than the stationary fraction of tax evaders, and the former is always $> 10\%$ of the population. The evaders in the long-time limit are majority in comparison with undecided agents only for large densities of negative interactions $q$, like $q=0.8$, and $p_{a}<0.7$ \[see Fig. \[fig6\] (d)\]. Even in this case, the honests are the majority in the population. For increasing values of $p_{a}$, the fraction of honest agents grows slowly for $q<q_{c}$ and fastly for $q>q_{c}$. One can also see from Fig. \[fig6\] that when we rise the fraction of negative interactions $q$, it is more difficult to control the tax evasion in the population, as discussed above. Indeed, for $q=0.3$ we have a density of $\approx 10\%$ of tax evaders for $p_{a}=0.2$, a value that is only reached for $q=0.5$ and $q=0.8$ at $p_{a}\approx 0.7$ and $p_{a}\to 1.0$, respectively. This is due to the high disagreement among the individuals in the population.
Summarizing, one can see that the presence of undecided agents in the population (state $s=0$) naturally reduces the number of evaders, and the implementation of public policies for punishment can be effective in controlling the tax evasion in the population, either due to fear of the undecided to being caught in an audit (if he/she become an evader), as by the monitoring of the tax evaders. This kind of behavior was not observed in the previous studies on the Zaklan model [@zaklan; @lima1; @lima2; @lima3; @seibold].
Final remarks
=============
In this work, we have studied the dynamics of tax evasion on a fully-connected population. Different from the previous studies on the Zaklan econophysics model, where the agents can be in two distinct states ($s=\pm 1$), the dynamics of interactions among the agents in our model follows a three-state ($s=+1, -1, 0$) kinetic exchange opinion model [@biswas], where individuals interact by pairs with competitive negative (with probability $q$) and positive (with probability $1-q$) couplings. Furthermore, we have considered the enforcement rules of the Zaklan model, where each agent is caught by an audit with probability $p_{a}$ and he/she is punished and remains honest during the following $k$ periods of time (or time steps).
Below the critical point $q_{c}=1/4$ of the opinion model, the dynamics leads the population to a state where the majority of the agents are honests ($s=+1$). In this case, the punishment rules do not affect considerably the system. On the other hand, for $q>q_{c}$ the kinetic exchange opinion model conducts the system to a disordered state where the three opinions coexist with equal fractions ($1/3$ for each one). In this case, we verified that the application of the enforcement mechanism can considerably reduce the tax evasion in the population. This reduction increases for decreasing values of $q$, which means that it is more difficult to control the compliance in populations with more disagreement among the individuals, i.e., with a large value of $q$.
We have also verified that the fraction of undecided or undecided individuals, i.e., agents with state $s=0$, affects the evolution of the tax evasion. These agents survive in the population in the long-time limit, which favors the reduction of the tax evasion. This fact together with the control given by the public policies may lead to low levels of evasion.
Regarding the critical behavior of the model, that is clearly observed in the absence of the enforcement rules (i.e., for $p_{a}=k=0$), one can easily see from our numerical results that the application of the punishment rules induces a decrease of the number of tax evaders ($s=-1$) in the system, and consequently an increase of the number of honest agents (state $s=+1$), which makes the order parameter defined in Eq. (\[eq3\]) always greater than zero, i.e., we have no disordered phase in the presence of the enforcement rules (i.e., for $p_{a}>0$ and $k>0$).
As a future work, it can be interesting to analyze how different initial fractions of undecided individuals affect the tax evasion dynamics. In addition, the effects of agents convictions in the model can also be a realistic feature to be addressed. This kind of heterogeneity can affect considerably the evolution of opinion models [@jstat_pmco; @celia; @meu; @conference; @xiong], and will certainly affect the dynamics of tax evasion considered here. Finally, the presence of inflexible agents [@galam] may also be important for better understanding of collective decision-making phenomena as occur in the dynamics of tax evasion.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author acknowledges financial support from Proppi - Universidade Federal Fluminense, Brazil, through the FOPESQ project.
[40]{}
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[^1]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have performed for the first time the simultaneous measurement of the two-body and three-body photodisintegration cross-sections of $^{4}$He in the energy range from 21.8 to 29.8 MeV using monoenergetic pulsed photons and a 4$\pi$ time projection chamber containing $^{4}$He gas as an active target in an event-by-event mode. The photon beam was produced via the Compton backscattering of laser photons with high-energy electrons. The $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross sections were found to increase monotonically with energy up to 29.8 MeV, in contrast to the result of a recent theoretical calculation based on the Lorentz integral transform method which predicted a pronounced peak at around 26$-$27 MeV. The energy dependence of the obtained $^{4}$He($\gamma$,n)$^{3}$He cross section up to 26.5 MeV is marginally consistent with a Faddeev-type calculation predicting a flat pattern of the excitation function. The cross-section ratio of $^{4}$He($\gamma$,p)$^{3}$H to $^{4}$He($\gamma$,n)$^{3}$He is found to be consistent with the expected value for charge symmetry of the strong interaction within the experimental uncertainty in the measured energy range. The present results for the total and two-body cross-sections of the photodisintegration of $^{4}$He are compared to previous experimental data and recent theoretical calculations.'
author:
- 'T. Shima'
- 'S. Naito'
- 'Y. Nagai'
- 'T. Baba'
- 'K. Tamura'
- 'T. Takahashi'
- 'T. Kii'
- 'H. Ohgaki'
- 'H. Toyokawa'
title: 'Simultaneous measurement of the photodisintegration of $^{4}$He in the giant dipole resonance region'
---
\[sec:level1\]Introduction
==========================
The unique feature of $^{4}$He as the lightest self-conjugate nucleus with the simplest closed-shell structure prompts both experimentalists and theorists to study its photodisintegration reaction in the giant dipole resonance (GDR) region. Since the reaction proceeds mainly by an electric dipole (E1) transition in the GDR region, the photodisintegration study provides a wealth of fundamental information on nucleon-nucleon (NN) interactions, meson exchange currents [@Unkelbach] as well as the possibility to study the charge symmetry of the strong interaction [@Barker]. The photodisintegration study also gives important insight on the rapid neutron capture process (r-process) nucleosynthesis induced by neutrino-driven wind from a nascent neutron star [@Woosley], since the neutrino transitions are the direct analogs of the giant electric dipole resonance observed in the photodisintegration [@Woosley; @Meyer].
A considerable amount of theoretical work on the photodisintegration of $^{4}$He has been carried out in the GDR region. Above 50 MeV, the two-body $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross sections as well as the total cross section are well described by a plane-wave approximation, in which final state interactions (FSI) are known to play a minor role [@Sofianos]. Below 30 MeV, however, these cross sections are sensitive to FSI, meson exchange currents as well as to the choice of NN interaction [@Sofianos; @Wachter]. Recently two different methods, one based on the Lorentz integral transform (LIT) [@Efros; @Quaglioni] and another based on Faddeev-type Alt-Grassberger-Sandhas (AGS) integral equations [@Ellerkmann], have been developed to accurately describe the low-energy dynamics of the $^{4}$He photodisintegration. Here it should be mentioned that although these models are quite different from each other, the calculated photodisintegration cross sections of $^{3}$H and $^{3}$He provided by these models agree with each other with high precision for the same NN interaction and three-nucleon forces (3NF) [@Golak]. However, the values of the photodisintegration cross section of $^{4}$He calculated by the same models differ significantly from each other. According to the calculation performed with the LIT method, both the total and two-body cross sections show a pronounced GDR peak at around 26$-$27 MeV, and the total cross section fully satisfies both the E1 sum rule and the inverse-energy-weighted E1 sum rule [@Efros; @Quaglioni]. On the other hand, the calculation based on the AGS method, carried out for the $^{4}$He($\gamma$,n)$^{3}$He cross section, shows a flat pattern below the three-body threshold energy of 26.1 MeV, and the calculated cross section at 26.1 MeV is only about 60% of the value derived by the LIT method [@Ellerkmann].
Experimentally the two-body, three-body, and total photodisintegration cross sections of $^{4}$He have been measured in the energy range from 20 to 215 MeV using quasi-monoenergetic photon beams and/or bremsstrahlung photon beams. Concerning the two-body $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He reactions, their inverse, the nucleon capture reactions, were used to derive the photodisintegration cross sections. Previous data for the two-body and total cross sections are shown in Figs. 1(a), 1(b), and 1(c), respectively. It is quite interesting to note that above 35$-$40 MeV most of the previous $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He data agree with each other within their respective data sets [@Ellerkmann]. However, there appear to be discrepancies especially in the peak region of 25$-$26 MeV, where the data show either a pronounced GDR peak or a fairly flat excitation function as shown in Figs. 1(a) and 1(b). The experimental methods and their results in the previous measurements are briefly described below to obtain some hints of the origin of the large discrepancies mentioned above. Here, it would be quite interesting to note the discrepancies related to different photon probes. The $^{4}$He($\gamma$,p)$^{3}$H cross section, $\sigma(\gamma,p)$, was measured by detecting the protons by means of a NE213 liquid scintillator [@Bernabei88] and/or a Si(Li) detector array [@Hoorebeke]. Note that the latest result by Hoorebeke [*et al.*]{} using 34 MeV end-point bremsstrahlung photons [@Hoorebeke] is larger than the data by Bernabei [*et al.*]{} [@Bernabei88] using a monochromatic photon beam by about 40% at around 30 MeV. The difference of these two data sets, however, becomes smaller with increasing the $\gamma$-ray energy, and they agree with each other at 33 MeV within the experimental uncertainty. The $^{3}$H(p,$\gamma$)$^{4}$He reaction cross section was measured using a tritium target absorbed into various metals by detecting a $\gamma$-ray by means of a NaI(Tl) detector \[13-19\]. Note that the latest result by Hahn [*et al.*]{} [@Hahn] is about 20% larger than that by Feldman [*et al.*]{} [@Feldman]. In summary, the $^{4}$He($\gamma$,p)$^{3}$H cross section derived from both the photodisintegration and the inverse reaction shows a large discrepancy between different data sets, and the difference is quite large (about 50%) at $E_{\gamma}=$ 25 MeV.
On the other hand, the $^{4}$He($\gamma$,n)$^{3}$He cross section, $\sigma(\gamma,n)$, was measured by detecting the neutrons with BF$_{3}$ neutron detectors and using bremsstrahlung photons [@Irish; @Malcom] and/or monoenergetic photons [@Berman]. The results obtained using bremsstrahlung photons are larger by about 30$\sim$100% than the result obtained using monoenergetic photons in the region between 25 and 30 MeV. Similarly to the case noted above, the difference between these data sets with different photon beams becomes smaller with increasing $\gamma$-ray energies, and they agree with each other at 35 MeV within an experimental uncertainty. The $^{3}$He(n,$\gamma$)$^{4}$He reaction cross section was measured by detecting $\gamma$-rays with a NaI(Tl) and/or a BGO detector [@Ward; @Komar], and their measured cross sections in the $\gamma$-ray energy range from 22 to 33 MeV agree with the $^{4}$He($\gamma$,n)$^{3}$He data by Berman [*et al.*]{} within the experimental uncertainty [@Berman].
Simultaneous measurements of the cross-sections for all reaction channels were performed by detecting charged fragments from the photodisintegration by means of cloud chambers using bremsstrahlung photon beams in the energy range from 21.5 to 215 MeV [@Gorbunov68; @Gorbunov58], from 20.5 to 150 MeV [@Arkatov74; @Arkatov70], and from 24 to 46 MeV [@Balestra77; @Balestra79], respectively. The results obtained with these measurements are 30$\sim$70% larger than the cross sections obtained with monoenergetic photon beams or tagged photon beams.
The elastic photon scattering of $^{4}$He was performed in the energy range from 23 to 34 MeV to derive indirectly the total photodisintegration cross section of $^{4}$He [@Wells]. The results by Gorbunov [*et al.*]{} [@Gorbunov58] agree with those by Arkatov [*et al.*]{} [@Arkatov74] and also by Wells [*et al.*]{} [@Wells] within the experimental uncertainty (see Fig. 1(c)).
{width=".6\linewidth"}
The electromagnetic property of the photodisintegration cross-section of $^{4}$He in the giant resonance region has been discussed in terms of the electric dipole (E1) radiation [@Ellerkmann]. Experimentally below 26.6 MeV the E1 dominance with a small M1 contribution of less than 2% has been shown by measuring angular distributions of cross-sections and/or analyzing powers for the inverse $^{3}$H(p,$\gamma$)$^{4}$He reaction [@McBroom; @Calarco; @Wagenaar]. Theoretically an E2 contribution to the total two-body cross-section is estimated to be small, about 6%, even at $E_{\gamma} =$ 60 MeV [@Ellerkmann].
The cross-section ratio of $^{4}$He($\gamma$,p)$^{3}$H to $^{4}$He($\gamma$,n)$^{3}$He, $R_{\gamma} =
\sigma(\gamma,p)/\sigma(\gamma,n)$, in the GDR region has been used to test the validity of the charge symmetry of the strong interaction. When charge symmetry is valid, the ratio is about unity for pure E1 excitations [@Barker]. $R_{\gamma}$ has been obtained experimentally with values ranging from 1.1 to 1.7 by separate measurements of $\sigma(\gamma,p)$ and $\sigma(\gamma,n)$ in the GDR region [@Calarco]. From simultaneous measurements of the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He reactions using cloud chambers and bremsstrahlung photon beams, $R_{\gamma}$ was obtained as 1.0$\sim$1.5 in the energy range from 23 to 44 MeV [@Gorbunov68; @Arkatov74; @Balestra77]. Recently $R_{\gamma}$ of 1.1 was obtained by a simultaneous ratio measurement of the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He reactions in the energy range from 25 to 60 MeV [@Florizone]. The measurement was performed by detecting a charged fragment emitted at 90$^{\circ}$ with respect to an incident tagged photon beam direction by means of windowless $\Delta$E-E telescopes. Here an angular distribution effect of a fragment was corrected for using theory.
In summary, although considerable experimental efforts have been made in determining $R_{\gamma}$, there remains a large discrepancy between separate measurements and simultaneous ratio measurements for the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He channels. Hence, one can hardly discuss the validity of the charge symmetry of the strong nuclear force using existing data. Hence it is highly required to accurately measure these cross sections with use of a new method in the GDR region, in particular between 22 and 32 MeV [@Bernabei88; @Hoorebeke; @Wells; @Vinokurov].
In designing a new experiment, it would be worthwhile to reconsider what we learned from previous data. Firstly, we notice that both the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross sections measured with bremsstrahlung photons are much larger than those measured with monoenergetic photons in the energy range from 22 to 30 MeV, and they agree with each other above $\sim$35 MeV. Theoretically the two-body as well as the total cross sections are well described by a plane-wave approximation and they agree with previous data above 50 MeV [@Sofianos]. Secondly, most experiments were performed separately for the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He channels via the photodisintegration reactions and/or the inverse nucleon capture reactions. Thirdly, the simultaneous two-body and three-body cross section measurements were performed using a cloud chamber, which did not allow us to take data in an event-by-event mode with a pulsed photon beam, which is necessary to reject background. One may conclude that the large discrepancies between different data sets could be due to background inherent to incident photon beams and/or due to an uncertainty of the normalization of the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross sections.
In the present study we have carried out the simultaneous measurement of the two-body and three-body $^{4}$He photodisintegration cross sections in the energy region between 21.8 and 29.8 MeV using a monoenergetic pulsed laser Compton backscattering photon beam by means of a newly developed 4$\pi$ time projection chamber containing $^{4}$He gas as an active target.
\[sec:level2\]Experiment
========================
\[sec:level2a\]Experimental method
----------------------------------
The experiment was carried out using a pulsed Laser Compton backscattering (LCS) photon beam at the National Institute of Advanced Industrial Science and Technology (AIST). The charged fragments from the photodisintegration of $^{4}$He were detected by means of a time projection chamber (TPC). A schematic view of an experimental setup is shown in Fig. 2.
![\[fig:Fig2\] Experimental setup for measurement of the photodisintegration of $^{4}$He at AIST.](Fig2){width=".9\linewidth"}
A quasi-monoenergetic pulsed LCS photon beam was produced via the Compton backscattering of the photons from a Nd:YLF laser in third harmonics ($\lambda=$ 351 nm) with electrons circulating in the 800 MeV storage ring TERAS at the AIST [@Ohgaki]. An LCS photon beam is well known to be an excellent probe to measure a photodisintegration cross section of a nucleus with little background associated with primary photon beam and with small uncertainty in determination of the LCS photon flux using a $\gamma$-ray detector. Even with this kind of setup, there are several difficulties inherent to the measurement of the photodisintegration cross section of $^{4}$He, among which the cross section is small (about $\sim$1 mb), the photon beam flux is low, the target density of $^{4}$He is low, and the energies of the fragments from the photodisintegration of $^{4}$He in the GDR region are quite low, typically less than a few MeV. Hence, it has been crucial to develop a new detector, which enabled us to make a simultaneous measurement of the two-body and three-body photodisintegration cross sections of $^{4}$He by detecting such a low energy fragment with an efficiency of 100% with a large solid angle of 4$\pi$, and with a large signal-to-noise ratio.
In the present study we constructed a TPC which meets the mentioned requirements.
\[sec:level2b\]Laser Compton backscattering (LCS) photon beam
-------------------------------------------------------------
A pulsed LCS $\gamma$-ray with the maximum energies $E_{max}=$ 22.3, 25, 28 and 32 MeV was used in the present experiment, obtained by changing the electron energy of the TERAS. The pulse width of the electron beam was 6 ns with a repetition rate of 100 MHz, while that of the laser photon beam was 150 ns with a repetition rate of 1 kHz. Pulsed laser photons scattered by electrons were collimated using a lead block with a hole of 2 mm in diameter and 200 mm in length to obtain quasi-monoenergetic LCS $\gamma$-rays. The absolute value of $E_{max}$ was determined with accuracy better than 1% from the wavelength of the laser light and the kinetic energy of the electron beam. The electron beam energy has been calibrated by measuring the LCS $\gamma$-ray energy generated with Nd:YAG laser photons in fundamental mode ($\lambda=$ 1064 nm) [@Ohgaki]. The half-width of the $\gamma$-ray energy distribution was 2.5 MeV at $E_{max}=$ 32 MeV, and the obtained $\gamma$-ray intensity was about 10$^{4}$ photons/s. The TPC was placed 3 m downstream of the lead collimator.
\[sec:level2c\]Time projection chamber (TPC)
--------------------------------------------
A 4$\pi$ time projection chamber containing $^{4}$He gas as an active target was constructed to detect the charged fragments from the photodisintegration of $^{4}$He with an efficiency of 100%. The TPC was contained in a vessel with a size of 244 mm in inner diameter and 400 mm in length. A mixed gas of 80% natural He and 20% CH$_{4}$ with a total pressure of 1000 Torr was filled in the vessel as a target for the photon-induced reactions and an operational gas of the TPC.
The TPC consisted of a drift region with a uniform electric field with an area of 60$\times$60 mm$^{2}$ and a length of 250 mm, and a multi-wire proportional counter (MWPC) region as shown in Fig. 3. The MWPC consisted of one anode plane and two cathode planes, which were set with a gap of 2 mm. Each plane had 30 wires with a spacing of 2 mm. In order to obtain two-dimensional track information of a charged fragment, cathode wires in front of and behind the anode plane were stretched along x- and y-axes, respectively. Here the x- and y-directions were defined to be parallel to and perpendicular to the anode wires, respectively.
![\[fig:Fig3\] Schematic drawing of the structure of the TPC.](Fig3){width=".9\linewidth"}
The TPC operates as follows. Electrons were produced by the interaction of a charged fragment with the mixed gas along the fragment path in the drift region. The electrons were drifted along the uniform electric field toward the MWPC region, where they were multiplied via an avalanche process. The avalanche signal was picked up with both the anode- and cathode-wires. The cathode signals were used to measure the track of a charged fragment on an x-y plane, since the directions of these cathode wires were perpendicular to each other. A z-position of a charged fragment was determined by measuring the drift time of the electrons with use of a time to digital converter as described below. An anode signal was used to determine the amount of energy loss of a fragment in the drift region of the TPC. Both, track and energy loss signals of a charged fragment were used to clearly identify a reaction channel. It should be noted that since a light charged fragment did not stop in the drift region, we observed various energy loss signals depending on a charged fragment type and on incident LCS $\gamma$-ray energy. An external magnetic field has not been used in the present TPC configuration.
The performance of the TPC was studied using the $^{241}$Am $\alpha$-ray source and a Si detector. The energy resolution of the TPC was measured as being 7.5% (FWHM) per anode wire. Since the energy measured by an anode wire depends on the emission angle of a fragment with respect to the anode wire direction, we collimated an $\alpha$-ray and determined its emission angle by using a coincidence signal between the TPC anode signal and the signal from the Si detector. A drift velocity of ionized electrons was measured as a function of the z-position using the same measuring system. A typical value of the drift velocity was 7.00$\pm$0.14 mm/$\mu$s. The time resolution was obtained as being 32 ns (1$\sigma$), which corresponded to the position resolution along the z-direction of 0.22 mm (1$\sigma$). Detailed description of the TPC will be published elsewhere [@Kii].
\[sec:level2d\]Electronics and data acquisition
-----------------------------------------------
A schematic diagram of data acquisition system is shown in Fig. 4. A linear signal from the preamplifier was used as a stop signal for a time to digital converter (TDC) after discriminating the electronic noise by a comparator. A common start signal for the TDC is obtained from the output of a pulsed laser clock. Both times, the leading edge and the trailing edge of an input signal are recorded on the TDC not only to determine the drift time of electrons but also to unambiguously identify the reaction channel. To measure the amount of energy loss of a charged fragment by integrating the current of a signal, we recorded its pulse shape using a flash ADC (FADC) and constructed a charge-integrated spectrum of a fragment in the off-line analysis.
![\[fig:Fig4\] Block diagram of the data acquisition system. LASER-CLK; laser clock pulse, ANODE-R(L); sum of the linear signals from the anode wires in right- (left-) hand side with respect to the LCS photon beam axis, CATHODE-i; linear signal from the i-th cathode wire, AMP; preamplifier, DSC; discriminator, DLY; delay circuit, and STR; pulse stretcher.](Fig4){width=".9\linewidth"}
Concerning the data acquisition, a trigger signal for the TPC is obtained from the clock pulse of the laser system. A logic signal from a cathode wire is sent to a discriminator to reject the noise signal and then sent to the TDC to measure a drift time of ionized electrons. An anode signal is used to generate a TPC-hit signal. When a 70 $\mu$s delayed signal of the laser clock pulse and the TPC-hit signal are in coincidence within a gate width of 100 $\mu$s, the data are acquired. The width is set longer than the maximum drift time (36 $\mu$s) of electrons in the TPC drift region in order to measure not only the photodisintegration event of $^{4}$He but also background events. Data from CAMAC modules are acquired by a personal computer and recorded on a hard disk drive in an event-by-event mode. The TPC count rate during the experiment was several tens of counts per second, and thus the dead time of the data taking system was a few % (monitored during the measurement). A PC-based pulse-height analyzer was used for monitoring the LCS $\gamma$-ray intensities with a BGO detector as described below.
\[sec:level3\]ANALYSIS AND RESULTS
==================================
\[sec:level3a\]Event identification
-----------------------------------
All pulse-height spectra taken by the FADC were analyzed to classify the observed events into photodisintegration events of $^{4}$He and $^{12}$C, and background events. We observed $^{12}$C events, since we used CH$_{4}$ gas. The total event rate in the present experiment was of several tens of counts per second, and the event rate from the photodisintegration of $^{4}$He and $^{12}$C was of several tens of counts per hour, less than one thousandth of the background events. A detailed description of electron background, natural background and photodisintegration events is given here below.
### \[sec:level3a1\]Background events
[*(1) Electron events.*]{} Most events taken by the FADC were due to background. The dominant background was originating from the interaction of LCS $\gamma$-rays with atomic electrons of $^{4}$He and CH$_{4}$ used for the TPC. Electron events were identified by their small pulse height. Note that electron energy loss rate in the TPC was small, of the order of 0.1 keV/mm, since electron energy was in the range from a few MeV to several tens of MeV. Therefore most electron events could be discriminated by a discriminator. A typical spectrum taken by the FADC is shown in Fig. 5. Here a dotted line indicates a threshold level, which was set to further remove electron background during the off-line analysis. Typical tracks of electron events, which were detected with one anode trigger signal and whose energies were above the threshold level, are shown in Fig. 6. Note that we could see several tracks for one anode trigger signal. In addition, observed tracks were not straight, seldom crossed the LCS $\gamma$-ray axis, and their track width was quite thin. These features allowed us to unambiguously identify electron events.\
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![\[fig:Fig5\] Pulse-height spectrum for all the acquired events. The huge component below $\sim$10 ch is mainly due to scattered electrons.](Fig5){width=".9\linewidth"}
![\[fig:Fig6\] Example of a track of a scattered electron. The $\gamma$-ray beam is coming from the left-hand side. The dots indicate the envelopes of the electron clouds ionized by the scattered electrons. The dashed line denotes the incident $\gamma$-ray beam axis. The box is the drift region of the TPC (side view).](Fig6){width=".9\linewidth"}
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[*(2) Natural background events.*]{} The natural background events are not correlated with the pulsed LCS $\gamma$-rays, and therefore the track did not cross the LCS $\gamma$-ray axis as shown in Fig. 7. Hence the natural background events could be clearly discriminated from the photodisintegration events of $^{4}$He and $^{12}$C. Since the track width of natural background is wider than that of electrons, the background might be due to an $\alpha$-particle from a natural radioactivity such as Rn contained in the TPC chamber or in the mixed gas of natural He and CH$_{4}$.
![\[fig:Fig7\] Example of a track of a natural background event.](Fig7){width=".9\linewidth"}
### \[sec:level3a2\]Photodisintegration events of $^{4}$He and $^{12}$C
Both electron and natural background events were identified as described above. Consequently, the background free (BF) events, which contained the photodisintegration events of $^{4}$He and $^{12}$C, were obtained from all the events recorded on the FADC. We checked the path length, the track width and the pulse height of each BF event to finally identify a reaction channel for the photodisintegration of $^{4}$He and $^{12}$C.
The calculated path length of the various fragments from the photodisintegration of $^{4}$He and $^{12}$C in the present experiment are listed in Table I. The path length of a light fragment such as p, $^{3}$H and $^{3}$He is much longer than that of a heavy fragment such as $^{11}$B and $^{11}$C. Hence, the photodisintegration of $^{4}$He can be separated from that of $^{12}$C by referring to the path length of a charged fragment.
------------------------------ --------- ---------- ------ ------ ------ ------
Reaction $Q$ Fragment
channel \[MeV\] 22.3 25 28 32
$^{4}$He($\gamma$,p)$^{3}$H -19.81 p 130 580 1310 2690
$^{3}$H 14.5 45 91 174
$^{4}$He($\gamma$,n)$^{3}$He -20.58 $^{3}$He 6 17 58 135
$^{4}$He($\gamma$,pn)$^{2}$H -26.07 p $-$ $-$ 84 590
$^{2}$H $-$ $-$ 34 214
$^{12}$C($\gamma$,p)$^{11}$B -15.96 p 1080 2230 3770 6440
$^{11}$B 4.3 6 7 9
$^{12}$C($\gamma$,n)$^{11}$C -18.72 $^{11}$C 3 4.3 5.5 7.4
------------------------------ --------- ---------- ------ ------ ------ ------
: \[tab:table1\] Maximum ranges of the fragments from the photodisintegrations of $^{4}$He and $^{12}$C (unit:mm).
The track width of a charged fragment was obtained by converting both times of the leading edge and the trailing edge of a cathode signal into the z-coordinate of the fragment track. Note that as the pulse height of a cathode signal becomes higher, the time difference between these two edges is larger, and thus the track width becomes wider. Hence, the track width of a charged fragment provides energy loss information of a fragment in the TPC. Since the energy loss rate of a fragment depends on the fragment type (p, $^{2}$H, $^{3}$H, $^{3}$He, $^{4}$He, $^{11}$B, and $^{11}$C), the track width of a fragment was used to identify its photodisintegration reaction channel together with the path length, the charge-integrated pulse height taken by the FADC and the reaction kinematics.
The measured pulse height spectrum of a fragment was compared to the spectrum calculated by a Monte-Carlo method. The Monte-Carlo calculation simulated the kinematics of the photodisintegration events, the migration of drift electrons, and the pulse shapes of the signals from the anode and cathode wires. The calculation has been performed as follows. Firstly, an incident intrinsic LCS photon spectrum of given energy was generated which reproduced a measured energy spectrum with a NaI(Tl) detector. Then, a reaction point was randomly chosen in the region irradiated by the LCS photon beam in the TPC. The track of a charged fragment emitted by a photodisintegration reaction of $^{4}$He and/or $^{12}$C was calculated by considering the LCS photon energy and the Q-value of the reaction. In order to calculate the emission angle of a charged fragment we assumed an E1 angular distribution and an isotropic distribution for the two-body and three-body channels of the photodisintegration of $^{4}$He, respectively. Note that the E1 dominance of the two-body channel was experimentally shown as mentioned above [@McBroom; @Calarco; @Wagenaar], and the isotropic fragment distribution was also observed for the $\gamma$-ray energy range from 28 to 60 MeV within the experimental uncertainty [@Gorbunov58; @Arkatov70; @Balestra79]. The energy deposited by a charged fragment was calculated as a function of the distance from a reaction point using the energy loss formula given by Ziegler [*et al.*]{} [@Ziegler], and was converted to the number of ionized electrons using the ionization energy of electrons in the TPC gas. The drift time of ionized electrons was calculated using the local drift velocity, which has been obtained as a function of the electric field strength in the TPC as described above. Using the drift time thus calculated, the shaping time of an amplifier, and the threshold level of a discriminator, we obtained the simulated data of FADC and TDC for each wire. The event data thus obtained were recorded and analyzed with the same procedure as for the data of the real measurements.\
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[*(1) Two-body channel of $^{4}$He photodisintegration.*]{} This channel is characterized by the fact that two fragments p (n) and $^{3}$H ($^{3}$He) are emitted in the opposite direction with respect to the LCS $\gamma$-ray beam direction, with equal momentum in the center-of-mass system. This channel can be separated from the two-body channel of $^{12}$C disintegration by the completely different path lengths of the charged fragments as mentioned above.\
(i) $^{4}$He($\gamma$,p)$^{3}$H channel\
Both the proton and the triton were detected by the TPC. An event, which meets the reaction kinematics conditions mentioned above, is selected as a candidate event of the $^{4}$He($\gamma$,p)$^{3}$H event. Since the energy loss of $^{3}$H is a few times larger than that of a proton, the track width of $^{3}$H is wider than that of a proton. A typical track of an event observed at $E_{max} =$ 28 MeV consistent with the above-mentioned feature is shown in Fig. 8(a). The sum spectrum of the measured pulse height of p and of $^{3}$H is in good agreement with that of a Monte-Carlo simulation as shown in Fig. 8(b). This event can be unambiguously assigned as a $^{4}$He($\gamma$,p)$^{3}$H event.\
(ii) $^{4}$He($\gamma$,n)$^{3}$He channel\
The TPC was not sensitive to neutrons, and therefore only the $^{3}$He, which crossed the LCS $\gamma$-ray axis, was detected for this reaction channel. A typical track of the event observed at $E_{max} =$ 28 MeV is shown in Fig. 9(a). The track of $^{3}$He is shown to extend to the opposite side across the central axis of the TPC. This is due to the finite size of the LCS photon beam and the diffusion of secondary electrons during the migration to the MWPC. The pulse height spectrum of $^{3}$He agrees nicely with a simulated one as shown in Fig. 9(b). Note that the track length of $^{3}$He is much longer compared to that of $^{11}$C as shown in Fig. 10 (also Fig. 12(a)), and therefore the $^{4}$He($\gamma$,n)$^{3}$He events can be clearly separated from the $^{12}$C($\gamma$,n)$^{11}$C events.\
(iii) $^{4}$He($\gamma$,d)$^{2}$H channel\
We did not observe any candidate of the $^{4}$He($\gamma$,d)$^{2}$H reaction. Note that the $^{4}$He($\gamma$,d)$^{2}$H cross section was measured to be about 3.2 $\mu$b at the peak of $E_{\gamma} =$ 29 MeV [@Weller], and it is therefore much smaller compared to the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross sections (a few mb at the corresponding $\gamma$-ray energy).\
![\[fig:Fig8\] (a) Example of the $^{4}$He($\gamma$,p)$^{3}$H event. (b) Total pulse height spectrum of the $^{4}$He($\gamma$,p)$^{3}$H reaction: open circles; observed, solid curve; fitting spectrum calculated with a Monte-Carlo simulation.](Fig8){width=".9\linewidth"}
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![\[fig:Fig9\] (a) Example of the $^{4}$He($\gamma$,n)$^{3}$He event. (b) Total pulse height spectrum of the $^{4}$He($\gamma$,n)$^{3}$He reaction.](Fig9){width=".9\linewidth"}
![\[fig:Fig10\] Distributions of the track length of charged fragments from the $^{4}$He($\gamma$,n)$^{3}$He and $^{12}$C($\gamma$,n)$^{11}$C reactions observed for $E_{max} =$ 28 MeV. The open circles are the experimental data. The solid curve and the dashed curve are the results of Monte-Carlo simulations for $^{4}$He($\gamma$,n)$^{3}$He and $^{12}$C($\gamma$,n)$^{11}$C, respectively.](Fig10){width=".9\linewidth"}
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[*(2) Two-body channel in the $^{12}$C photodisintegration.*]{} This channel is characterized by the fact that two fragments p (n) and $^{11}$B ($^{11}$C) are emitted in the opposite direction with respect to the LCS $\gamma$-ray beam axis with equal momentum in the center-of-mass system. The path length of $^{11}$B and $^{11}$C, however, are much shorter than that of $^{3}$H and $^{3}$He, and therefore this two-body channel can be clearly separated from that of $^{4}$He.\
(i) $^{12}$C($\gamma$,p)$^{11}$B channel\
Both the proton and $^{11}$B are detected by the TPC. A typical track of an event which meets the above-mentioned condition, observed at $E_{max} =$ 28 MeV, is shown in Fig. 11(a). The path length of the proton is much longer than that of $^{11}$B, and the track width of the proton is much narrower than that of $^{11}$B. The sum spectrum of the measured pulse height of $^{11}$B and p is in good agreement with the Monte-Carlo simulation as shown in Fig. 11(b).\
(ii) $^{12}$C($\gamma$,n)$^{11}$C channel\
Only the track of $^{11}$C, which crossed the LCS $\gamma$-ray beam axis, was observed for this reaction channel. A typical track of a $^{12}$C($\gamma$,n)$^{11}$C event observed at $E_{max} =$ 28 MeV is shown in Fig. 12(a). The path length of $^{11}$C is much shorter than that of $^{3}$He as shown in Fig. 10, and therefore we could unambiguously discriminate the $^{12}$C($\gamma$,n)$^{11}$C events from those of the $^{4}$He($\gamma$,n)$^{3}$He reaction. The pulse height spectrum of $^{11}$C also agrees nicely with the simulated one as shown in Fig. 12(b).\
![\[fig:Fig11\] (a) Example of the $^{12}$C($\gamma$,p)$^{11}$B event. (b) Total pulse height spectrum of the $^{12}$C($\gamma$,p)$^{11}$B reaction.](Fig11){width=".9\linewidth"}
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![\[fig:Fig12\] (a) Example of the $^{12}$C($\gamma$,n)$^{11}$C event. (b) Total pulse height spectrum of the $^{12}$C($\gamma$,n)$^{11}$C reaction.](Fig12){width=".9\linewidth"}
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[*(3) Three-body channels.*]{}\
(i) $^{4}$He($\gamma$,pn)$^{2}$H channel\
The Q-value of the $^{4}$He($\gamma$,pn)$^{2}$H reaction is -26.1 MeV, and therefore the reaction events could only be observed at $E_{max} =$ 28 and 32 MeV. This reaction event can be discriminated from that of the $^{4}$He($\gamma$,p)$^{3}$H reaction, because the tracks of the proton and deuteron are randomly oriented with respect to one another in the center-of-mass system and with respect to the LCS $\gamma$-ray beam axis, and the path length of the proton from the $^{4}$He($\gamma$,pn)$^{2}$H reaction is much shorter than that of the $^{4}$He($\gamma$,p)$^{3}$H reaction. A typical track of p and $^{2}$H observed at $E_{max} =$ 28 MeV is shown in Fig. 13. In this case, the track width of the proton is not constant and depends on the proton energy.
![\[fig:Fig13\] Example of the $^{4}$He($\gamma$,pn)$^{2}$H three-body event.](Fig13){width=".9\linewidth"}
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(ii) $^{12}$C($\gamma$,2$\alpha$)$^{4}$He channel\
This reaction event can be easily identified by three tracks of the particles as shown in Fig. 14, which was observed at $E_{max} =$ 28 MeV.
![\[fig:Fig14\] Example of the $^{12}$C($\gamma$,2$\alpha$)$^{4}$He event.](Fig14){width=".9\linewidth"}
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[*(4) Photodisintegration reaction of $^{4}$He and/or $^{12}$C without LCS photon beams.*]{} We checked for possible photodisintegration events of $^{4}$He and/or $^{12}$C caused by bremsstrahlung photons from the TERAS, but not LCS photons. Since such an event would occur continuously, the data corresponding to the drift time of between 40 and 64 $\mu$s were analyzed. We did not find any event which could be identified as any of the reaction channels of the photodisintegrations of $^{4}$He and/or $^{12}$C.\
\[sec:level3b\]Cross sections of the photodisintegrations of $^{4}$He ($\sigma_{i}(E_{\gamma})$)
------------------------------------------------------------------------------------------------
The incident LCS $\gamma$-ray has a finite energy spread, and the TPC efficiency depends on the incident $\gamma$-ray energy as described below. Hence, a partial cross section $\sigma_{i}(E_{\gamma})$ corresponding to the two-body and/or the three-body photodisintegration of $^{4}$He at a $\gamma$-ray energy $E_{\gamma}$ is given as follows: $$Y_{i}=N_{t} \cdot L \cdot \Phi \times
\frac{\int_{0}^{E_{max}} \varepsilon_{i}(E_{\gamma}) \cdot
\sigma_{i}(E_{\gamma}) \cdot \phi(E_{\gamma}) dE_{\gamma}}
{\int_{0}^{E_{max}} \phi(E_{\gamma}) dE_{\gamma}}
\label{eq:one}.$$
Here $Y_{i}$, $N_{t}$ and $L$ stand for the yield of a reaction channel $i$, the number density of the target nuclei, and the effective length of the TPC, respectively. $E_{max}$ denotes the maximum energy of the incident LCS $\gamma$-ray. The parameter $\varepsilon_{i}(E_{\gamma})$ is the detection efficiency of the TPC for a fragment emitted by the photodisintegration process at the $\gamma$-ray energy $E_{\gamma}$. The parameter $\phi(E_{\gamma})$ denotes the intensity of the incident LCS $\gamma$-ray at the energy $E_{\gamma}$. $\Phi$ is the incident LCS $\gamma$-ray flux, and is equal to the energy-integrated value of $\phi(E_{\gamma})$. The average cross section $<\sigma_{i}>$ and the weighted-mean reaction energy $<E_{\gamma}>_{i}$ are defined as
$$\begin{aligned}
<\sigma_{i}>=\frac{\int_{0}^{E_{max}} \varepsilon_{i}(E_{\gamma})
\cdot \sigma_{i}(E_{\gamma}) \cdot \phi(E_{\gamma}) dE_{\gamma}}
{\int_{0}^{E_{max}} \varepsilon_{i}(E_{\gamma}) \cdot
\phi(E_{\gamma}) dE_{\gamma}}\nonumber \\
= \frac{Y_{i}}
{N_{t} \cdot L \cdot \int_{0}^{E_{max}} \varepsilon_{i}(E_{\gamma})
\cdot \phi(E_{\gamma}) dE_{\gamma}},
\label{eq:two}\end{aligned}$$
$$<E_{\gamma}>_{i}=\frac{\int_{0}^{E_{max}} E_{\gamma} \cdot
\varepsilon_{i}(E_{\gamma}) \cdot \sigma_{i}(E_{\gamma})
\cdot \phi(E_{\gamma}) dE_{\gamma}}
{\int_{0}^{E_{max}} \varepsilon_{i}(E_{\gamma}) \cdot
\sigma_{i}(E_{\gamma}) \cdot \phi(E_{\gamma}) dE_{\gamma}}
\label{eq:three}.$$
The parameters $\varepsilon_{i}$, $N_{t}$, $L$, $\Phi$ and $\phi$ were determined as discussed in the following subsections.
### \[sec:level3b1\]Effective length ($L$) and detection efficiency ($\varepsilon_{i}(E_{\gamma})$) of the TPC
Any charged fragments produced by the photodisintegration of $^{4}$He and/or $^{12}$C in the TPC produces electrons by interacting with atomic electrons in the He and CH$_{4}$ mixed gas in the TPC. Since the signal of an electron is picked up by the anode and the cathode wires, the efficiency $\varepsilon_{i}(E_{\gamma})$ of the TPC is expected to be as high as 100% along the TPC geometrical drift length of 250 mm. However, since the electric field strength applied in the drift region is not uniform at both edges of the drift region, the efficiency is not constant in the whole length of the drift region. Hence, we measured a pulse height spectrum of $\alpha$-particles from the decay of $^{241}$Am to determine an effective length $L$, in which a pulse height was constant to provide a constant efficiency. The length $L$ is defined as 225 mm in the region between $z =$ 12.5 mm and 237.5 mm. Fig. 15 shows the position distribution of the photodisintegration event of $^{4}$He along the z-direction. It is clearly seen that the distribution is uniform within the effective length within the experimental uncertainty.
![\[fig:Fig15\] Position distribution of the $^{4}$He photodisintegration yield along the z-direction.](Fig15){width=".9\linewidth"}
One might expect a 100% efficiency $\varepsilon_{i}(E_{\gamma})$ within the effective length. However since the energy of a fragment from the photodisintegration of $^{4}$He and/or $^{12}$C is low and discriminators were used to reject electric noise of both the anode and cathode signals and electron background, it is necessary to investigate a possible change of the efficiency due to threshold levels of the discriminators of the cathode and anode signals.
It should be mentioned that the anode signal was obtained by summing signals from several anode wires on either right-hand side or left-hand side with respect to the LCS $\gamma$-ray beam axis as shown in the block diagram of the data acquisition system in Fig. 4. Since the average energy deposit by a charged fragment from the photodisintegration is above 500 keV, an average pulse height of the summed anode signal is above 500 keV. A threshold level of the anode signal was set at about 5 keV by referring to the $\alpha$-ray pulse height spectrum of $^{241}$Am not to decrease the efficiency.
On the other hand, a cathode signal was obtained from each cathode wire as shown in Fig. 4. A threshold level of the cathode signal was set at about 0.8 keV, and the effect of the discriminator on the efficiency was studied by making a pulse peak height spectrum of its signal. The spectrum was obtained by taking the maximum peak of all signals of a cathode wire taken by FADC. A typical spectrum for a proton and a $^{3}$H from the $^{4}$He($\gamma$,p)$^{3}$H channel is respectively shown in Fig. 16, where a solid curve indicates a peak height spectrum calculated by a Monte-Carlo method, and the dotted line is the threshold level set in the present measurement. The measured peak height spectrum is in good agreement with the simulated one both for the proton as well as for the $^{3}$H. Since the pulse peak height of the proton is well above the threshold level, the discriminator for the cathode signal does not decrease the TPC efficiency. Using the Monte-Carlo simulation, the efficiency $\varepsilon_{i}(E_{\gamma})$ is obtained as being 0.97(5)$\sim$1.00(1) depending on the reaction energy. Here the bracket indicates the uncertainty of $\varepsilon_{i}(E_{\gamma})$, which was obtained by fitting a measured pulse-height spectrum with the simulated one.
![\[fig:Fig16\] Peak pulse-height spectra of proton (open circles) and $^{3}$H (diagonal crosses) from the $^{4}$He($\gamma$,p)$^{3}$H reaction at $E_{max} =$ 28 MeV. Solid curves are the spectra calculated by a Monte-Carlo method.](Fig16){width=".9\linewidth"}
### \[sec:level3b2\]Target number density ($N_{t}$)
The target number density $N_{t}$ was determined from measured pressure $P$, temperature $T$ and chemical purity (99.999%) of the $^{4}$He gas in the TPC. The uncertainty in the determination of $N_{t}$ was evaluated to be 0.18% due to the uncertainty in the determination of $P$ and $T$.
### \[sec:level3b3\]Incident LCS $\gamma$-ray flux ($\Phi$)
The incident LCS $\gamma$-rays were measured using a BGO detector with a diameter of 50.8 mm and a length of 152.4 mm. A typical measured $\gamma$-ray spectrum is shown in Fig. 17, in which we see multiple peaks due to pile-up effects.
![\[fig:Fig17\] Typical $\gamma$-ray pulse height spectrum for $E_{max} =$ 28 MeV. The solid curve and the dashed curve represent the measured one and the Monte-Carlo simulation assuming an average photon multiplicity $M=$ 5.2, respectively.](Fig17){width=".9\linewidth"}
The laser photon beam was pulsed with a pulse width of 150 ns and a repetition rate of 1 kHz, while the electron beam was also pulsed with a width of 6 ns and a repetition rate of 100 MHz. Laser photons, therefore, collide several times with electron bunches circulating in the TERAS within one long laser pulse width, and LCS $\gamma$-rays with the same energy distribution were produced within a time interval of 150 ns. Note that this time interval was too short for the BGO system to decompose the multiple LCS $\gamma$-rays into an individual LCS $\gamma$-ray produced by one electron pulse. Consequently, the multiple $\gamma$-ray peaks were produced as pile-ups in the LCS $\gamma$-ray spectrum (see Fig. 17).
The photodisintegration yield of $^{4}$He is proportional to an averaged number $M$ of multiple LCS $\gamma$-rays per laser pulse. The number $M$ was obtained by comparing a measured BGO spectrum to a calculated one obtained from a Monte-Carlo simulation [@Toyokawa]. The calculated spectrum was obtained with the following assumptions. The LCS $\gamma$-ray yield is proportional to the number of electrons (electron currents) times the number of laser photons. The probability density for generating LCS $\gamma$-rays per laser pulse is so small that the LCS $\gamma$-ray yield can be assumed to follow a Poisson distribution. The electron beam in the TERAS can be assumed to be a continuous beam because its repetition rate is much higher than that of the laser photon beam. The observed multiple peaks of the LCS $\gamma$-ray spectrum are assumed to be the sum of the pulse height spectra of each LCS $\gamma$-ray. This assumption is reasonable since the BGO responds to each $\gamma$-ray independently. The response function of the BGO detector to the LCS $\gamma$-ray was obtained by measuring the $\gamma$-ray spectrum with low flux, which was free from multiple peaks. Finally, the pulse shape of the BGO detector for multiple LCS $\gamma$-rays was obtained using both the time distribution of the LCS $\gamma$-ray measured by a plastic scintillation counter and a shaping time of 1 $\mu$s of an amplifier used for the BGO detector system. Based on these assumptions, a response function of the BGO detector with an averaged number $M$ of multiple LCS $\gamma$-rays was calculated by a Monte-Carlo method, and the number $M$ was obtained by fitting a measured spectrum with the multiple peaks with the calculated response function. A typical measured spectrum is in good agreement with the calculated one with $M =$ 5.2 as shown in Fig. 17.
Using the number $M$ thus determined the LCS $\gamma$-ray total flux $\Phi$ is obtained as follows:
$$\Phi = M \times f \times T_{L}
\label{eq:four}.$$
Here $f$ is a frequency of the laser pulse, and $T_{L}$ is a live time of the measurement. A $\gamma$-ray flux thus obtained has an uncertainty of about 2%, which consists of statistics of the LCS $\gamma$-ray yield, an uncertainty of the response function of the BGO detector, and an uncertainty in the least-square fitting of the LCS $\gamma$-ray spectrum with multiple peaks measured with the BGO detector.
It is interesting to see a relation between the electron current in the TERAS and the average number M, which is shown in Fig. 18. Electron currents are shown after electrons were injected into the TERAS. While an electron beam current gradually decreases due to the collision of electrons with the residual gas containing in the ring, the average number $M$ remains almost constant.
![\[fig:Fig18\] Time dependences of the electron beam current $I_{B}$ (dashed curve), the average photon number $M$ (solid curve) and the photon production efficiency $M/I_{B}$ (dash-dotted curve) for an electron beam current of 1 mA.](Fig18){width=".9\linewidth"}
### \[sec:level3b4\] Energy spectrum of incident LCS $\gamma$-ray ($\phi(E_{\gamma})$)
To determine the photodisintegration cross section of $^{4}$He at a certain energy corresponding to an incident LCS $\gamma$-ray, it is necessary to measure the intrinsic energy spectrum $\phi(E_{\gamma})$ of incident LCS $\gamma$-rays. Note that the LCS $\gamma$-ray has a finite energy spread due to the finite widths of the lead collimator and of the emittance of electron beams of the TERAS. Hence the LCS $\gamma$-ray spectrum was measured using an anti-Compton NaI(Tl) spectrometer, which consisted of a central NaI(Tl) detector with a diameter of 76.2 mm and a length of 152.4 mm, and an annular one with an outer diameter of 254 mm and a length of 280 mm. A typical spectrum measured at $E_{max} =$ 28 MeV is shown in Fig. 19. Using a response function of the NaI(Tl) detector calculated with the GEANT4 simulation code [@GEANT4], the intrinsic energy spectrum of the LCS $\gamma$-ray was obtained as shown in Fig. 19. An energy spread of LCS $\gamma$-rays was determined as 6% (FWHM) at $E_{max} =$ 28 MeV.
![\[fig:Fig19\] NaI pulse height spectra obtained for $E_{max} =$ 28 MeV. The thick curve and the thin curve are the measured one and the calculated one, respectively. The dashed curve is an intrinsic energy distribution of the LCS $\gamma$-ray required to reproduce the measured spectrum.](Fig19){width=".9\linewidth"}
### \[sec:level3b5\] Photodisintegration of deuteron
The values of $Y_{i}$, $\varepsilon_{i}$, $N_{t}$, $L$, $\phi$, and $\Phi$ were accurately determined as described above, and therefore the photodisintegration cross section of $^{4}$He is determined accurately using the formula of Eq. 1. It is, however, worthwhile to measure the photodisintegration cross section of deuteron to learn about any possible systematic uncertainty of the present experimental method. Note that the cross section has been well studied both experimentally and theoretically in the wide energy range from 10 to 75 MeV [@Dgpn; @Arenhovel]. The present measurement was performed using CD$_{4}$ gas instead of CH$_{4}$ gas as the quenching gas of the TPC at $E_{max} =$ 22.3 MeV. The data were analyzed as extensively described above. The cross section turns out to be 0.56$\pm$0.04(stat)$\pm$0.03(syst) mb, which agrees nicely with both the previous data [@Dgpn] and the theoretical value of 0.55 mb [@Arenhovel] as shown in Fig. 20. The weighted-mean reaction energy was determined as being 21.0 MeV using Eq. 3 and the known energy dependence of the cross section [@Dgpn; @Arenhovel]. Hence, the validity of the present experimental method including its analysis was confirmed with a quite small systematic uncertainty within the statistical uncertainty.\
![\[fig:Fig20\] Cross section of the photodisintegration of deuteron. The open circle denotes the present result, while other symbols indicate the previous data [@Dgpn]: filled circles; Skopik [*et al.*]{}, diagonal crosses; Ahrens [*et al.*]{}, open squares; Bernabei [*et al.*]{}, open diamonds; Michel [*et al.*]{}, open upward triangles; Bosman [*et al.*]{}, open downward triangles; Dupont [*et al.*]{} The solid curve is the theoretical cross section calculated by means of the momentum-space approach [@Arenhovel].](Fig20){width=".9\linewidth"}
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![\[fig:Fig21\] Differential cross section of the $^{4}$He($\gamma$,p)$^{3}$H reaction at $<E_{\gamma}>=$ 29.8 MeV. The open circles denote the measured one in the present work. The solid curve is the fitted one with the least-square method.](Fig21){width=".9\linewidth"}
\
### \[sec:level3b6\] Angular distribution of proton from the $^{4}$He($\gamma$,p)$^{3}$H reaction
In order to determine the electromagnetic property of the photodisintegration at 29.8 MeV we analyzed the angular distribution of charged fragments from the $^{4}$He($\gamma$,p)$^{3}$H reaction at $E_{max} =$ 32 MeV. Note that the two-body total photodisintegration cross section below 26.6 MeV is dominated by E1 radiation as mentioned above [@Ellerkmann]. We analyzed data taken at $E_{max} =$ 32 MeV by performing a least-square fit to the data using the following formula [@Vinokurov]:
$$\frac{d\sigma}{d\Omega} = A_{0} \cdot
(sin^{2}\theta_{c.m.}+\beta \cdot sin^{2}\theta_{c.m.} \cdot
cos\theta_{c.m.} +\gamma \cdot sin^{2}\theta_{c.m.} \cdot
cos^{2}\theta_{c.m.} + \delta + \epsilon \cdot cos\theta_{c.m.})
\label{eq:five}.$$
Here, $\theta_{c.m.}$ is the angle formed by the proton trajectory from the $^{4}$He($\gamma$,p)$^{3}$H reaction with respect to the incident LCS $\gamma$-ray beam in the center of mass system. $A_{0}$ is determined by the E1 absorption contribution. $\beta$ is due to the interference of E1 and E2 electric multipoles, $\gamma$ is the ratio of E2 to E1 absorption probabilities, $\delta$ is the ratio of M1 to E1 absorption, and $\epsilon$ is an isotropic term, which is known experimentally to be approximately zero. Consequently, these parameters are determined as $A_{0}=$ 0.16$\pm$0.02 mb/sr, $\beta=$ 0.17$\pm$0.13, $\gamma=$ 0$\pm$0.14, and $\delta=$ 0.02$\pm$0.01. The results indicate the dominance of an electric dipole process in the photodisintegration at around 30 MeV, and the M1 strength is about 2% of the E1 strength, and the E2 strength is negligible compared to the E1 strength. The present result shown in Fig. 21 is in good agreement with previous data below 26.2 MeV [@Wagenaar] and with a theoretical calculation [@Ellerkmann].
### \[sec:level3b7\] Cross sections of the two-body and three-body photodisintegration of $^{4}$He
In order to calculate average cross sections $<\sigma_{i}>$ of the photodisintegration of $^{4}$He and the weighted-mean reaction energies $<E_{\gamma}>_{i}$ using Eqs. 2 and 3, we first determined the cross sections $<\sigma_{i}(E_{\gamma})>$ using the measured yields of the photodisintegration of $^{4}$He and Eq. 1 as discussed below. Since we made the measurements at four maximum $\gamma$-ray energies $E_{max}$, we can set up four simultaneous equations as follows:
$$\begin{aligned}
Y_{i}^{(k)}=N_{t} \cdot L \cdot \Phi \times
\frac{\int_{0}^{E_{max}^{(k)}} \varepsilon_{i}(E_{\gamma})
\cdot \sigma_{i}(E_{\gamma}) \cdot \phi^{(k)}(E_{\gamma})
dE_{\gamma}}
{\int_{0}^{E_{max}^{(k)}} \phi^{(k)}(E_{\gamma}) dE_{\gamma}}
\nonumber \\
(k = 0\sim3) \label{eq:six}.\hspace{6mm}\end{aligned}$$
Here $k=$ 0, 1, 2 and 3 stand for the measurements at $E_{max}=$ 22.3, 25, 28 and 32 MeV, respectively. $Y_{i}^{(k)}$ is the measured yield of a reaction channel $i$ of the photodisintegration of $^{4}$He in a measurement $k$. Since all the resonance states below 30 MeV are known to be quite broad [@Tilley], we assume that $\sigma_{i}(E_{\gamma})$ is a smooth function of $E_{\gamma}$ and practically expressed by a power series of the relative momentum $p$ of the particles in the exit channel as follows:
$$\sigma_{i}(E_{\gamma}) = \sum_{j=1}^{3} a_{j}p^{j} \ , \
p=\left( \frac{\mu (E_{\gamma}-E_{i}^{th})}{2} \right)^{1/2}
\label{eq:seven}.$$
Here $\mu$ and $E_{i}^{th}$ are the reduced mass of the emitted particles and the threshold energy in a reaction channel $i$, respectively. The coefficients $a_{j}$ were determined by solving Eq. 6 and Eq. 7 simultaneously. Substituting $\sigma_{i}(E_{\gamma})$ thus obtained for Eqs. 2 and 3, the average cross sections $<\sigma>_{i}$ and the weighted-mean reaction energies $<E_{\gamma}>_{i}$ were obtained. The results for the ($\gamma$,p), ($\gamma$,n), and total photodisintegration cross sections are given in Table II and shown in Figs. 22(a), 22(b), and 22(c), in which the solid curves represent the most probable functions obtained with the mentioned procedure in the present energy range up to 32 MeV. The systematic uncertainties associated with $<\sigma>_{i}$ were calculated from the uncertainties in $\varepsilon_{i}$, $N_{t}$, $L$ and $\Phi$. Due to the similar excitation functions for the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He reactions, $<E_{\gamma}>$ obtained for both channels agreed with each other within 100 keV as expected. Hence, we have used the same values of $<E_{\gamma}>$ for the ($\gamma$,p) and ($\gamma$,n) reaction channels.
$<E_{\gamma}>$ \[MeV\] $<\sigma(\gamma,p)>$ \[mb\] $<\sigma(\gamma,n)>$ \[mb\] $<\sigma(\gamma,pn)>$ \[mb\] $<\sigma_{total}>$ \[mb\]
------------------------ ----------------------------- ----------------------------- ------------------------------ ---------------------------
21.8 0.19$\pm$0.02$\pm$0.01 0.10$\pm$0.02$\pm$0.006 $-$ 0.29$\pm$0.03$\pm$0.02
24.3 0.71$\pm$0.05$\pm$0.03 0.63$\pm$0.05$\pm$0.03 $-$ 1.34$\pm$0.07$\pm$0.06
26.5 0.89$\pm$0.06$\pm$0.02 0.80$\pm$0.06$\pm$0.02 $-$ 1.69$\pm$0.09$\pm$0.04
29.8 1.39$\pm$0.08$\pm$0.03 1.35$\pm$0.08$\pm$0.03 0.04$\pm$0.01$\pm$0.001 2.78$\pm$0.11$\pm$0.06
{width=".6\linewidth"}
\[sec:level4\]Discussion
========================
\[sec:level4a\]Ratio of the $^{4}$He($\gamma$,p)$^{3}$H cross section to the $^{4}$He($\gamma$,n)$^{3}$He cross section
-----------------------------------------------------------------------------------------------------------------------
The cross section ratio $R_{\gamma} \equiv \sigma(\gamma,p)/\sigma(\gamma,n)$ has been determined accurately with an experimental uncertainty of about 10% and with small systematic uncertainties in the energy range from 21.8 to 29.8 MeV. The ratio is consistent with calculated values without charge symmetry breaking of the strong interaction [@Unkelbach; @Quaglioni; @Halderson; @Londergan] within the experimental uncertainty as shown in Fig. 23. There, the previous data taken simultaneously for the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He reactions are shown for a comparison. Note that the large ratio, approximately equal to 2.0 at 21.8 MeV, is due to the difference of the Q-values between n-$^{3}$He and p-$^{3}$H channels. The ratio at 26.5 and 29.8 MeV agrees with the latest result obtained by detecting emitted particles from these reactions simultaneously at 90$^{\circ}$ with respect to the incident beam direction. Our result is also consistent with simultaneous measurements on the $^{4}$He(e,e’X) reaction in the excitation energy region between 22 and 36 MeV [@Spahn] and the $^{4}$He(p,p’X) reaction [@Raue].
![\[fig:Fig23\] Ratio $R_{\gamma}$ of the $^{4}$He($\gamma$,p)$^{3}$H cross section to the $^{4}$He($\gamma$,n)$^{3}$He cross section: open circles; present result, open triangles; Gorbunov [@Gorbunov68], gray diamonds; Balestra [*et al.*]{} [@Balestra77], diagonal crosses; Florizone [*et al.*]{} [@Florizone]. Short-dashed curve and solid curve are the calculations of the recoil-corrected continuum shell model with and without extra CSB effect, respectively [@Halderson]. The long-dashed curve, the dotted curve, and the dash-dotted curve are the calculations without extra CSB based on the LIT method [@Quaglioni], the coupled-channel continuum shell model [@Londergan], and the resonating group model [@Unkelbach], respectively.](Fig23){width=".9\linewidth"}
\[sec:level4b\]The partial and total cross-sections of the photodisintegration of $^{4}$He
------------------------------------------------------------------------------------------
The present ($\gamma$,p), ($\gamma$,n), and total photodisintegration cross sections of $^{4}$He shown in Figs. 22(a), 22(b), and 22(c) differ significantly from previous data (see below for details).
### \[sec:level4b1\]$^{4}$He($\gamma$,p)$^{3}$H
The $^{4}$He($\gamma$,p)$^{3}$H cross section increases monotonically with energy up to 29.8 MeV. The cross section below 26.5 MeV does show agreements with none of the previous data while at 29.8 MeV it agrees nicely with some of the previous data [@Gorbunov68; @Arkatov74; @Hoorebeke] and marginally agrees with that by Bernabei [*et al.*]{} [@Bernabei88]. The value at 28.6 MeV in Ref. [@Bernabei88] is in good agreement with an interpolated value of the present data between at 26.5 and at 29.8 MeV. Note that the present cross section and excitation function significantly differ from the theoretical calculation of the LIT method which predicts a pronounced peak at around 26$-$27 MeV as shown in Fig. 22(a).
### \[sec:level4b2\]$^{4}$He($\gamma$,n)$^{3}$He
The $^{4}$He($\gamma$,n)$^{3}$He cross section shows similar energy dependence as that of the $^{4}$He($\gamma$,p)$^{3}$H as shown in Fig. 22(b), and the value up to 26 MeV marginally agree with the data of Berman [*et al.*]{} [@Berman] within the experimental uncertainty which includes a systematic error of 15%. The cross section at 29.8 MeV is larger than the previous data [@Berman; @Ward] by about 30% but agrees with the data of Gorbunov [@Gorbunov68]. The cross section follows the shape of the theoretical calculation based on the AGS method up to 26 MeV, although the experimental value is smaller by about 20% in comparison to the calculation. The experimental result does not agree with the calculation based on the LIT method which predicts a pronounced peak at around 26$-$27 MeV.
![\[fig:Fig24\] $^{4}$He($\gamma$,pn)$^{2}$H reaction cross section: open circles; present result, open upward triangles; Gorbunov [*et al.*]{} [@Gorbunov58], open downward triangles; Arkatov [*et al.*]{} [@Arkatov70], gray diamonds; Balestra [*et al.*]{} [@Balestra79].](Fig24){width=".9\linewidth"}
### \[sec:level4b3\]$^{4}$He($\gamma$,pn)$^{2}$H
The small value of the $^{4}$He($\gamma$,pn)$^{2}$H cross section at 29.8 MeV, 0.04$\pm$0.01 mb, agrees with previous data as shown in Fig. 24 [@Gorbunov58; @Arkatov70; @Balestra79]. The theoretical calculations on the three-body $^{4}$He($\gamma$,pn)$^{2}$H cross section are not available.\
\
### \[sec:level4b4\]Total cross section
The total cross section increases monotonically with energy up to 29.8 MeV as shown in Fig. 22(c). The cross section below 26.5 MeV is significantly smaller than previous data [@Gorbunov68; @Arkatov74] and a theoretical calculation based on the LIT method. The cross section at 29.8 MeV agrees with the previous data and with the calculation. Here, it is worthwhile to mention that the total photo-absorption cross-section is inferred from the elastic photon scattering data of $^{4}$He [@Wells] together with previous data of the shape of the photodisintegration cross section, which claim the GDR peak in the region of 25$-$26 MeV. The cross-section inferred turns out to be 2.86$\pm$0.12 mb at around 26 MeV [@Wells], which differs significantly from the present value of 1.69$\pm$0.09(stat)$\pm$0.04(syst) mb at 26.5 MeV. The origin of the discrepancy is not clear, but it could be due to the shape difference between the presently obtained cross section and the previous one. Naturally, it would be interesting to estimate the total photo-absorption cross-section using the shape of the total cross section derived in the present study and the photon scattering cross-section data [@Wells]. Note that the shape can be obtained by combining the present results up to 29.8 MeV with the previous data above around 33$-$35 MeV, where the previous data agree with each other, as shown in Fig. 22(c).
### \[sec:level4b5\]E1 sum rule
Since the present cross section is found to be smaller than many previous data and considering that the E1 transition dominates, it is important to investigate the energy distribution of the transition strength. It is well known that the integrated cross section $\sigma_{0}$ for E1 photoabsorption and the inverse-energy-weighted sum $\sigma_{B}$ can be related to the properties of the ground state of a nucleus through the following sum rules [@Levinger; @Foldy]:
$$\begin{aligned}
\sigma_{0}= \int_{0}^{E_{\pi}} \sigma_{E1}(E_{\gamma}) dE_{\gamma}
= \sigma_{TRK}(1+\kappa) = \frac{2\pi^{2}e^{2}\hbar}{mc}
\frac{NZ}{A}(1+\kappa) \ ,
\\
\sigma_{B}= \int_{0}^{E_{\pi}}
\frac{\sigma_{E1}(E_{\gamma})}{E_{\gamma}} dE_{\gamma}
= \frac{4\pi^{2}}{3}\frac{e^{2}}{\hbar c}\frac{NZ}{A-1}
\cdot (<r_{\alpha}^{2}> - <r_{p}^{2}>)
\label{eq:8-9}.\end{aligned}$$
Here $\sigma_{E1}(E_{\gamma})$ is the total cross section for E1 photoabsorption as a function of $E_{\gamma}$. $N$, $Z$, and $A$ are the numbers of the neutrons, protons, and nucleons, respectively. $m$ and $\kappa$ are the nucleon mass and the correction factor for the contribution of the exchange forces, respectively. $E_{\pi}$, $e$, $\hbar$, and $c$ stand for the pion threshold energy, electron charge, Planck’s constant, and speed of light, respectively. $\sigma_{TRK}$ stands for the Thomas-Reiche-Kuhn (TRK) sum rule. $<r_{\alpha}^{2}>$ and $<r_{p}^{2}>$ are the mean-square charge radii of $^{4}$He and the proton, respectively. The integrations in Eqs. 8 and 9 have been performed as follows: below 31 MeV we assumed $\sigma_{E1}(E_{\gamma})$ is given as the sum of the present $\sigma(\gamma,p)$ and $\sigma(\gamma,n)$, because the cross section is found to be dominated by the E1 photoabsorption in the two-body channels. Above 31 MeV we employed previous data of Refs. [@Gorbunov68; @Gorbunov58] and [@Arkatov74; @Arkatov80], which are in an overall agreement with each other as well as with recent theoretical calculations. The $\sigma_{0}$ and $\sigma_{B}$ values are listed in Table III. Here it should be noted that the data taken from Refs. [@Gorbunov68; @Gorbunov58] and from Refs. [@Arkatov74; @Arkatov80] correspond to the cross sections for the total photoabsorption and the E1 photoabsorption, respectively. Therefore the data set of the present result and the cross section from Refs. [@Gorbunov68; @Gorbunov58] provides upper limits on $\sigma_{0}$ and $\sigma_{B}$. The contributions of higher multipoles have been estimated to be of a few percent [@Gorbunov58; @Arkatov80]. Consequently, the present value of $\sigma_{0}$ is marginally lower than the value expected from the other light nuclei [@Ahrens] and from theoretically predicted values [@Efros; @Heinze]. As for $\sigma_{B}$, the present value is significantly smaller than the calculated value of 2.62$\pm$0.02 mb obtained from Eq. 8 using the known experimental values of $<r_{\alpha}^{2}>^{1/2} =$ 1.673$\pm$0.001 fm [@Borie] and $<r_{p}^{2}>^{1/2} =$ 0.870$\pm$0.008 fm [@PDG2004].
$E_{\gamma}$ \[MeV\] Data set $\sigma_{0}$ \[MeV$\cdot$mb\] $\sigma_{B}$ \[mb\]
---------------------- --------------------------------- ------------------------------- ---------------------
19.8$-$31 Present 18.1$\pm$2.1 0.67$\pm$0.07
19.8$-$135 Present $+$ Refs. \[25,52\][^1] 96$\pm$7 2.24$\pm$0.17
Present $+$ Refs. \[26,53\] 80.4$\pm$2.3 1.92$\pm$0.12
Sum rule (see text.) 100$\sim$128 2.62$\pm$0.02
### \[sec:level4b6\]Present results and previous data
In the present simultaneous measurement of the $^{4}$He($\gamma$,p)$^{3}$H and $^{4}$He($\gamma$,n)$^{3}$He cross-sections using the 4$\pi$ TPC we could get the cross-section ratio $R_{\gamma}= \sigma(\gamma,p)/\sigma(\gamma,n)$ with smaller systematic uncertainties, which is consistent with the results obtained by other reactions as well as with recent simultaneous measurement [@Florizone]. However, there are discrepancies between the present two-body and total cross sections and previous ones as described before. Although it is difficult to find a unique reason of the discrepancies since a special care has been taken for the normalization of each of the cross section measured here, it might be instructive to look at general trends recognized in the previous data and to compare them to the present results.
Firstly we discuss the latest $^{4}$He($\gamma$,p)$^{3}$H data using bremsstrahlung photons [@Hoorebeke], which are larger than the data [@Bernabei88] by about 40% at around 30 MeV and are in good agreement with the data [@Bernabei88] at 33 MeV within the experimental uncertainty. Note that the data [@Bernabei88] were obtained using a monochromatic photon beam. This fact may indicate that a possible origin of the above mentioned discrepancy is due to background events inherent to measurement with bremsstrahlung photons. The former group used Si(Li) detectors to detect protons from the $^{4}$He($\gamma$,p)$^{3}$H reaction. They carefully considered possible systematic errors such as the $^{4}$He gas purity, the gas pressure, the efficiency of the Si(Li) detectors, the energy losses in the gas target, the incident flux calibration, the bremsstrahlung shape, the effects due to the background subtraction and others. They had to subtract the background component in the Si(Li) detector assuming an exponential fit to the low-energy photon data. They claimed that the validity of their background subtraction method has been supported by the test measurement of the $^{16}$O($\gamma$,p)$^{15}$N experiment. However, the cross section of the $^{16}$O($\gamma$,p)$^{15}$N reaction is about 10 times larger than that of $^{4}$He, and therefore the ambiguity due to the background subtraction might not have been relevant in the test experiment. The contribution of background due to bremsstrahlung photons is expected to decrease with increasing the photon energy, since the energy of an emitted proton becomes higher. Consequently the energy dependence of the difference between the two data sets mentioned above can be explained in this way.
Secondly, we discuss the result of the $^{4}$He($\gamma$,n)$^{3}$He reaction obtained by Berman [*et al.*]{} [@Berman], which was carried out using annihilation photon beams and BF$_{3}$ tubes embedded in a paraffin matrix as a neutron counter. They carefully made various corrections due to the background from bremsstrahlung photons, the neutron detector efficiency, and others, and they concluded that their data points at 25.3, 26.3 and 28.3 MeV should have systematic uncertainties as large as 15%. If we take the systematic error in addition to statistical error, our data marginally agree with the data by Berman [*et al.*]{} It should be stressed that we used a quasi-monoenergetic pulsed Laser Compton backscattering (LCS) photon beam, which is free from background inherent to bremsstrahlung photon beams, and we detected $^{3}$He unambiguously by the nearly 4$\pi$ TPC containing $^{4}$He gas as an active target. Hence, we could determine the detection efficiency of $^{3}$He with high accuracy.
\[sec:level5\]Conclusion
========================
In the present work, we have carried out for the first time the direct simultaneous measurement of the two-body and three-body photodisintegration cross sections of $^{4}$He in the energy range from 21.8 to 29.8 MeV using a quasi-monoenergetic pulsed real photon beam by detecting a charged fragment with a nearly 4$\pi$ time projection chamber in an event-by-event mode. The validity of the present new experimental method, including its data analysis, has been accurately confirmed by measuring the photodisintegration cross section of deuteron. By accurately determining the ratio of the $^{4}$He($\gamma$,p)$^{3}$H to $^{4}$He($\gamma$,n)$^{3}$He cross sections, we have solved for the first time the long-standing problem of the large discrepancy in this ratio obtained in separate measurements and simultaneous ones. The $^{4}$He($\gamma$,p)$^{3}$H, $^{4}$He($\gamma$,n)$^{3}$He and total cross sections do not agree with the recent calculations based on the Lorentz integral transform method. The $^{4}$He($\gamma$,n)$^{3}$He cross section follows the shape of the calculation based on the AGS method up to 26.5 MeV, but it is smaller by about 20% with respect to the calculated values. We conclude that further theoretical work in the GDR energy region is necessary to elucidate the GDR property of $^{4}$He. Concerning the photonuclear reactions of three-nucleon systems, it has been known that 3NF reduces the peak cross section by about 10$-$20% [@Skibinski]. Since $^{4}$He is tightly bounded compared to the three-nucleon systems, one might expect significant 3NF effects in the photodisintegration of $^{4}$He. The present result would affect significantly the production yields of r-nuclei by the neutrino-induced r-process nucleosynthesis, since the neutral current neutrino spallation cross sections are quite sensitive to the peak energy of the GDR and the cross section in the GDR energy region.
We would like to thank Profs. H. Kamada and T. Kajino for discussions, and Dr. A. Mengoni for comments and careful reading of the manuscript. The present work was supported in part by Grant-in-Aid for Specially Promoted Research of the Japan Ministry of Education, Science, Sports and Culture and in part by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (JSPS).
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[^1]: Upper limits for E1 contribution.
|
{
"pile_set_name": "ArXiv"
}
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---
author:
- |
\
Department of Physics, University of Colorado, Boulder, CO 80309, USA\
E-mail:
- |
Oliver Witzel\
Department of Physics, University of Colorado, Boulder, CO 80309, USA\
E-mail:
bibliography:
- '../General/BSM.bib'
title: 'Continuous $\beta$ function for the SU(3) gauge systems with two and twelve fundamental flavors'
---
Introduction
============
Real-space renormalization group (RG) flows map out the phase structure of lattice models including properties of infrared and ultraviolet fixed points (IRFP, UVFP). Recently an interpretation of gradient flow (GF) transformations as a continuous real-space RG transformation has been proposed [@Carosso:2018bmz]. This opens a new way to determine the RG $\beta$ function and anomalous dimensions of strongly coupled gauge-fermion systems both within and outside the conformal window. In Reference [@Hasenfratz:2019hpg] we developed the steps to determine the continuous RG $\beta$ function of the GF coupling illustrating the method for the QCD-like SU(3) model with 2 fundamental flavors. The new approach has several advantages compared to commonly performed step-scaling calculations. It is equally applicable for confining, conformal, or infrared free systems. In this work we extend the calculation of Ref. [@Hasenfratz:2019hpg] to 12 fundamental flavors.
GF is a continuous transformation that can be used to define real-space RG blocked quantities. The coarse graining of the RG transformation can be incorporated when calculating expectation values [@Carosso:2018bmz]. Relating the dimensionless GF time $t/a^2$ to the RG scale change as $b \propto \sqrt{t/a^2}$, the GF transformation describes a continuous real-space RG transformation. In particular, expectation values of local operators, like the energy density, are identical with or without coarse graining and describe at flow time $t/a^2$ physical quantities at energy scale $\mu \propto 1/\sqrt{t}$. We sketch typical RG flows on the chiral $m=0$ critical surface in an asymptotically free gauge-fermion system in the left panel of [Fig. \[fig:RG-flow\]]{}. $g_1$ refers to the relevant gauge coupling, while $g_2$ indicates all other irrelevant couplings. The Gaussian FP (GFP) and the renormalized trajectory (RT) emerging from it describe the cut-off independent continuum limit. The RT is a 1-dimensional line. A dimensionless local operator with non-vanishing expectation value can therefore be used to define a running coupling along the RT. The simplest such quantity in gauge-fermion systems is ${\ensuremath{\left\langle t^2 E(t) \right\rangle} }$, the energy density multiplied by $t^2\propto b^4$. This is the quantity proposed in Ref. [@Luscher:2010iy] to define the gradient flow renormalized coupling ${\ensuremath{g_{\rm GF}^2} }$. Numerical simulations are performed with an action characterized by a set of bare couplings. If this action is in the vicinity of the GFP or its RT, the typical RG flow approaches the RT and follows it as the energy scale is decreased from the cut-off towards the infrared as indicated by the blue lines. RG flows starting at different bare couplings approach the RT differently. However, once the irrelevant couplings have died out, they all follow the same 1-dimensional renormalized trajectory and describe the same continuum physics. At large flow time, irrelevant terms in the lattice definition of $E(t)$ die out as well and ${\ensuremath{g_{\rm GF}^2} }$ approaches a continuum renormalized running coupling. Its derivative is the RG $\beta$ function $$\beta(g_{GF}^2) = \mu^2 \frac{d g_{GF}^2}{d \mu^2} = - t \frac{d g_{GF}^2}{d t}.
\label{eq:contbfn}$$
![Sketch of possible phase diagrams and RG flows in the multi-dimensional action parameter space. Left panel: QCD-like gauge-fermion system. $g_1$ refers to the relevant gauge coupling, while $g_2$ indicates all other irrelevant couplings. The blue lines represent RG flow trajectories. Right panel: Infrared conformal system with 4-fermion interaction. $G$ denotes the 4-fermion coupling and only the relevant $g_1$ coupling of the gauge-fermion system is shown. The black solid line denotes a 2nd order phase transition separating chirally symmetric and broken phases. The black dashed line indicates that the phase transition might turn 1st order but it remains phase separating. The RT emerging from the GFP and from a UVFP on the phase separating surface end at an IRFP. RG flows in the weak coupling side are similar to those on the left panel but can become complicated on the strong coupling side of the IRFP. []{data-label="fig:RG-flow"}](plots/RG_flow.pdf "fig:"){width="0.475\columnwidth"} ![Sketch of possible phase diagrams and RG flows in the multi-dimensional action parameter space. Left panel: QCD-like gauge-fermion system. $g_1$ refers to the relevant gauge coupling, while $g_2$ indicates all other irrelevant couplings. The blue lines represent RG flow trajectories. Right panel: Infrared conformal system with 4-fermion interaction. $G$ denotes the 4-fermion coupling and only the relevant $g_1$ coupling of the gauge-fermion system is shown. The black solid line denotes a 2nd order phase transition separating chirally symmetric and broken phases. The black dashed line indicates that the phase transition might turn 1st order but it remains phase separating. The RT emerging from the GFP and from a UVFP on the phase separating surface end at an IRFP. RG flows in the weak coupling side are similar to those on the left panel but can become complicated on the strong coupling side of the IRFP. []{data-label="fig:RG-flow"}](plots/phase_diagram_conformal2-crop.pdf "fig:"){width="0.495\columnwidth"}
The RT of chirally broken systems continues to $g_1 \to \infty$, whereas conformal systems have an IRFP on the RT that stops the flows from either direction. It was suggested that the IRFP of conformal systems is accompanied by a UVFP [@Miransky:1998dh; @Kaplan:2009kr; @Gorbenko:2018ncu]. The latter requires the emergence of a new relevant operator, possibly a 4-fermion interaction. In the right panel of [Fig. \[fig:RG-flow\]]{}, we sketch a possible phase diagram for this case[^1]. We show only the relevant $g_1$ coupling of the gauge-fermion system and include $G$, the coupling of the 4-fermion Nambu-Jona-Lasinio (NJL) interaction [@NJL1; @NJL2]. The NJL model has a second order phase transition separating chirally symmetric and broken phases at $g_1$=0. Lattice studies indicate that for $g_1>0$ at least initially the phase transition remains continuous [@Catterall:2013koa; @Rantaharju:2016jxy; @Rantaharju:2017eej]. At some point it might turn first order but the phase transition must persist as it separates distinct phases. A UVFP could sit on the continuous region of this phase separating surface and be connected to the IRFP as indicated. Since the phase transition of the un-gauged NJL model does not depend on the number of fermions, it is conceivable that the phase separating surface remains far from the $G=0$ gauge fermion system. This might imply a sharp turn of the RT making the numerical study of the “backward flow” particularly challenging.\
The above discussion and the definition of the $\beta$ function in Eq. (\[eq:contbfn\]) is valid in infinite volume. In our approach we extrapolate $L/a \to \infty$ at fixed $t/a^2$ which also sets the renormalization scheme $c=\sqrt{8t}/L = 0$. The continuum limit of the $\beta$ function is obtained at fixed ${\ensuremath{g_{\rm GF}^2} }$ while taking $t/a^2\to\infty$. In QCD-like systems this automatically forces the bare gauge coupling towards zero, the critical surface of the GFP. In a conformal system the bare gauge coupling is tuned to zero in the weak coupling regime, whereas in the strong coupling “backward flow” regime it should be tuned to the phase separating critical surface. Specifically, once the GF coupling and its derivative are determined, the continuous $\beta$ function calculation proceeds in two steps:
- Infinite volume extrapolation at every GF time: the leading order corrections at small GF values are $(a/L)^4$. We restrict the flow time such that the finite volume corrections are well described by the leading behavior.
- Infinite flow time extrapolation at every $g^2_{GF}$: this step removes irrelevant operator contributions (cut-off effects) and plays the role of the $a/L \to 0$ continuum limit extrapolation of the step-scaling function approach.
Step A) is new in the continuous $\beta$ function approach but is compensated by other advantages. Most importantly the flow time in the continuous $\beta$ function calculation is independent of the volume and can be kept small. This significantly reduces statistical errors.
Numerical details
=================
Our lattice studies of both 2 and 12-flavor SU(3) systems are based on gauge field configurations generated with tree-level improved Symanzik gauge action and chirally symmetric Möbius domain wall (DW) fermions with stout smeared gauge links. We generate configurations using `Grid` [@Boyle:2015tjk; @GRID] with $a m=0$ bare mass and chose the DW 5th dimension large enough to ensure that the residual mass is $a m_\text{res}<10^{-5}$. Details of the $N_f=2$ configurations are discussed in Ref. [@Hasenfratz:2019hpg]. The $N_f=12$ analysis uses configurations generated for the step-scaling study published in Refs. [@Hasenfratz:2017qyr; @Hasenfratz:2019dpr]. We have implemented three different flows, Wilson (W), Symanzik (S), and Zeuthen (Z), in `Qlua` [@Pochinsky:2008zz; @qlua] and three different operators Wilson plaquette (W), clover (C), and Symanzik (S) to estimate the energy density [@Ramos:2014kka; @Ramos:2015baa]. Our data analysis is performed using the $\Gamma$-method [@Wolff:2003sm] which is designed to estimate and account for autocorrelations.
The GF coupling is defined as $$\begin{aligned}
\label{eq:pert_g2}
{\ensuremath{g_{\rm GF}^2} }(t; L,g^2_0) = \frac{128\pi^2}{3(N^2 - 1)} \frac{1}{1+\delta} {\ensuremath{\left\langle t^2 E(t) \right\rangle} }\,.\end{aligned}$$ The normalization ensures that ${\ensuremath{g_{\rm GF}^2} }$ matches the ${\ensuremath{\overline{\textrm{MS}} } }$ coupling at tree level, and the term $1/(1+\delta)$ corrects for the gauge zero modes due to periodic gauge boundary conditions [@Fodor:2012td].[^2]
![ Top panels show the finite volume $\beta$ function at flow times $t/a^2=2.2$ and 4.2 ($a^2/t=0.455$ and $0.238$) for our three volumes for $N_f=2$. Dashed lines show a polynomial interpolation of the data points. Bottom panels present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times.[]{data-label="fig:Linf_extrap_Nf2"}](plots/{ContBetaFn_beta_vs_gSq_t2.200_Nf2_ZS}.pdf "fig:"){width="0.494\columnwidth"} ![ Top panels show the finite volume $\beta$ function at flow times $t/a^2=2.2$ and 4.2 ($a^2/t=0.455$ and $0.238$) for our three volumes for $N_f=2$. Dashed lines show a polynomial interpolation of the data points. Bottom panels present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times.[]{data-label="fig:Linf_extrap_Nf2"}](plots/{ContBetaFn_beta_vs_gSq_t4.200_Nf2_ZS}.pdf "fig:"){width="0.494\columnwidth"}\
![ Top panels show the finite volume $\beta$ function at flow times $t/a^2=2.2$ and 4.2 ($a^2/t=0.455$ and $0.238$) for our three volumes for $N_f=2$. Dashed lines show a polynomial interpolation of the data points. Bottom panels present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times.[]{data-label="fig:Linf_extrap_Nf2"}](plots/{ContBetaFn_beta_vs_Linv_t2.200_Nf2_ZS}.pdf "fig:"){width="0.494\columnwidth"} ![ Top panels show the finite volume $\beta$ function at flow times $t/a^2=2.2$ and 4.2 ($a^2/t=0.455$ and $0.238$) for our three volumes for $N_f=2$. Dashed lines show a polynomial interpolation of the data points. Bottom panels present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times.[]{data-label="fig:Linf_extrap_Nf2"}](plots/{ContBetaFn_beta_vs_Linv_t4.200_Nf2_ZS}.pdf "fig:"){width="0.494\columnwidth"}
$N_f=2$
-------
We generate $16^3\times64$, $24^3\times64$ and $32^3\times64$ gauge field ensembles at 10 bare gauge coupling values ($\beta\equiv 6/g_0^2$ = 8.50, 7.00, 6.20, 6.00, 5.60, 5.20, 5.00, 4.90, 4.80, 4.70) using periodic boundary conditions in space, antiperiodic in time for the fermions. All ensembles are in the chirally symmetric regime, i.e. above the finite temperature phase transition. This choice allows us to run the simulations with $am=0$ and covers the coupling range ${\ensuremath{g_{\rm GF}^2} }\lesssim 7.0$.
#### A) Infinite volume extrapolation:
The $L/a\to\infty$ limit has to be taken at fixed $t/a^2$ and $g^2_{GF}$. The finite volume effects depend on $t/L^2$ and at leading order are proportional to $t^2/L^4$. We restrict the GF time in our analysis such that the leading order contribution describes the data well. First we determine ${\ensuremath{g_{\rm GF}^2} }(t)$ and its derivative on every ensemble, then interpolate $\beta({\ensuremath{g_{\rm GF}^2} }(t);L)$ for each lattice volume with a 4th order polynomial. This predicts the finite volume $\beta$ function as the function of the renormalized coupling. The top panels of Fig. \[fig:Linf\_extrap\_Nf2\] show both the lattice data and the interpolations at $t/a^2=2.2$ and $4.2$ ($a^2/t = 0.455$ and $0.238$) for the ZS combination. Using the predicted $\beta({\ensuremath{g_{\rm GF}^2} })$ values we extrapolate in $(a/L)^4$ to the infinite volume limit. The lower panels of Fig. \[fig:Linf\_extrap\_Nf2\] show this for several ${\ensuremath{g_{\rm GF}^2} }$ at the same GF time as the top panels. We find that finite volume effects are negligible at small flow time and remain small even at $t/a^2=4.2$. As a consistency check we compare extrapolations using all three volumes to extrapolations using the two largest volumes only. While the errors of the infinite volume predictions change, the values are consistent. Other flow and operator combinations show similar volume dependence.
![Representative $a^2/t \to 0$ continuum limit extrapolation for $N_f=2$ at ${\ensuremath{g_{\rm GF}^2} }=3.2$ (left panel) and ${\ensuremath{g_{\rm GF}^2} }=4.8$ (right panel). We show results for two infinite volume extrapolations and three operators, fitting filled symbols in the range $2.00 \le t/a^2 \le 3.64$ ($0.500\ge a^2/t\ge 0.274$). The (uncorrelated) fits are independent but predict consistent $a^2/t=0$ continuum values.[]{data-label="fig:cont_extrap_Nf2"}](plots/{ContBetaFn_Nf2_Zflow_continuum_gSq3.2}.pdf "fig:"){width="0.494\columnwidth"} ![Representative $a^2/t \to 0$ continuum limit extrapolation for $N_f=2$ at ${\ensuremath{g_{\rm GF}^2} }=3.2$ (left panel) and ${\ensuremath{g_{\rm GF}^2} }=4.8$ (right panel). We show results for two infinite volume extrapolations and three operators, fitting filled symbols in the range $2.00 \le t/a^2 \le 3.64$ ($0.500\ge a^2/t\ge 0.274$). The (uncorrelated) fits are independent but predict consistent $a^2/t=0$ continuum values.[]{data-label="fig:cont_extrap_Nf2"}](plots/{ContBetaFn_Nf2_Zflow_continuum_gSq4.8}.pdf "fig:"){width="0.494\columnwidth"}
#### B) Infinite flow time extrapolation:
The continuum limit of the continuous $\beta$ function at fixed ${\ensuremath{g_{\rm GF}^2} }$ is predicted in the $t/a^2 \to \infty$ limit. The range of $t/a^2$ values in the extrapolation has to be chosen with some care. The minimum flow time must be large enough for the RG flow to reach the vicinity of the RT where all but at most one irrelevant operators are suppressed. The maximum flow time is restricted by the requirement that the finite volume dependence follows the leading order $a^4/L^4$ dependence. Any change of the continuum limit prediction due to varying the minimal or maximal flow time values can be incorporated as systematical uncertainty.
The functional form of the flow time dependence of $\beta({\ensuremath{g_{\rm GF}^2} })$ is expected to be $(t/a^2)^{\alpha/2}$ where $\alpha < 0$ is the scaling dimension of the least irrelevant operator. Around the GFP $\alpha=-2$ and we find that our data is well described by a linear $a^2/t$ dependence for $t/a^2 \gtrsim 2.0$. We show two examples of continuum extrapolation at ${\ensuremath{g_{\rm GF}^2} }=3.2$ and 4.8 in Fig. \[fig:cont\_extrap\_Nf2\]. In both cases we fit the data (filled symbols) in the range $2.00 \le t/a^2 \le 3.64$ ($0.500 \ge a^2/t \ge 0.274$). While the flow time $t$ is a continuous variable, in practice we evaluate ${\ensuremath{g_{\rm GF}^2} }(t)$ with finite step-size $\varepsilon=0.04$ and choose to dilute the data in $\Delta t=0.12$ intervals.We perform uncorrelated fits though correlations in $t$ could easily be accounted for in a bootstrap or jackknife analysis. Once sufficiently large flow times are reached, the lower flow times in the fit range impact only the size of the uncertainties in the continuum limit and the largest value of the coupling $g^2_{GF}$ which can be reached on a given data set. In Figure \[fig:cont\_extrap\_Nf2\] we compare continuum limit extrapolations obtained using Zeuthen flow with Wilson plaquette (ZW), Symanzik (ZS), and clover (ZC) operators. We consider two different infinite volume extrapolations, using all three volumes or only the largest two. For illustration we show additional data at larger flow time using open symbols. The excellent agreement of the different extrapolations at the $a^2/t = 0$ limit is a strong consistency check of the GF time range and the infinite volume extrapolation.
It is worth to point out that the Zeuthen flow, Wilson plaquette operator combination (green symbols in Fig. \[fig:cont\_extrap\_Nf2\]) show very little cut-off dependence and the data are nearly constant in $a^2/t$. This is true for both ${\ensuremath{g_{\rm GF}^2} }$ shown in Fig. \[fig:cont\_extrap\_Nf2\] as well as other values we have investigated. The ZS combination also has relatively small cut-off effects, but it is growing steadily as ${\ensuremath{g_{\rm GF}^2} }$ increases.
#### The continuous $\beta$ function:
The Wilsonian RG description suggests that lattice simulations at a single bare coupling can predict, up to controllable cut-off corrections, a finite section of the RG $\beta$ function. In practice, the finite lattice volume limits the range where the infinite volume $\beta$ function is well approximated. In the left panel of Figure \[fig:beta-direct\_Nf2\] the colored data points show the predictions for the RG $\beta$ function from raw ZS lattice data without infinite volume or continuum extrapolation. They trace out a single curve with overlapping predictions from different bare gauge couplings. The result of the full ZS analysis is shown by the gray band in Fig. \[fig:beta-direct\_Nf2\] which is in close agreement with the raw data. The continuum limit predicted by different flow/operation combinations are consistent as shown in the right panel of Figure \[fig:beta-direct\_Nf2\]. In the $N_f=2$ system the coupling predicted by Zeuthen flow and Symanzik operator shows only small cut-off effects in the range of ${\ensuremath{g_{\rm GF}^2} }\lesssim 6.0$ as is already evident from the continuum extrapolations shown in Fig. \[fig:cont\_extrap\_Nf2\]. The raw ZW lattice data show even smaller cut-off effects and completely overlap with the gray band of the full analysis. The continuous $\beta$ function approach predicts the running of the renormalized coupling in a transparent way where cut-off and finite volume effects are clearly identifiable.
![ Left panel: Continuous RG $\beta$ function of 2-flavor QCD in the GF scheme. The grey band is the result of our full analysis with statistical uncertainties only. The colored data points show the lattice predictions using the ZS combination for $32^3\times64$ (’$+$’) and $24^3\times64$ (’$\times$’) ensembles in a wide range of bare couplings without any extrapolation or interpolation. Only flow times $t/a^2 \in(2.0,3.64)$ are shown. The dashed and dash-dotted lines are the perturbative 1- and 2-loop $\beta(g^2)$ functions. Right panel: Continuum limit of the continuous GF $\beta$ function predicted by nine different flow/operator combinations with statistical errors only. The different combinations are barely distinguishable and appear to be close to the 1-loop perturbative curve. []{data-label="fig:beta-direct_Nf2"}](plots/{ContBetaFn_Nf2_ZS_continuum}.pdf "fig:"){width="0.494\columnwidth"} ![ Left panel: Continuous RG $\beta$ function of 2-flavor QCD in the GF scheme. The grey band is the result of our full analysis with statistical uncertainties only. The colored data points show the lattice predictions using the ZS combination for $32^3\times64$ (’$+$’) and $24^3\times64$ (’$\times$’) ensembles in a wide range of bare couplings without any extrapolation or interpolation. Only flow times $t/a^2 \in(2.0,3.64)$ are shown. The dashed and dash-dotted lines are the perturbative 1- and 2-loop $\beta(g^2)$ functions. Right panel: Continuum limit of the continuous GF $\beta$ function predicted by nine different flow/operator combinations with statistical errors only. The different combinations are barely distinguishable and appear to be close to the 1-loop perturbative curve. []{data-label="fig:beta-direct_Nf2"}](plots/{ContBetaFn_Nf2_continuum}.pdf "fig:"){width="0.494\columnwidth"}
$N_f=12$
--------
The analyses of the continuous $\beta$ function with 12 fundamental flavors closely follows the steps discussed above. The 12 flavor system has received a lot of attention but the predictions of its $\beta$ function are inconsistent[@Appelquist:2007hu; @Hasenfratz:2011xn; @Fodor:2011tu; @Aoki:2012eq; @Cheng:2013xha; @Cheng:2014jba; @daSilva:2015vna; @Fodor:2016zil; @Hasenfratz:2016dou; @Fodor:2017gtj; @Hasenfratz:2017qyr; @Hasenfratz:2019dpr]. The $\beta$ function is very small, so even small systematical errors can have a significant effect. At the same time, the RG structure can change suddenly when the conformal IRFP emerges as the sketch in Fig. \[fig:RG-flow\] implies. The continuous $\beta$ function analysis is a new and partially independent approach that can shed light on the origin of the controversies.
We use the ensembles generated for the step scaling study published in Refs. [@Hasenfratz:2017qyr; @Hasenfratz:2019dpr]. These $(L/a)^4$ configurations with $\beta\equiv 6/g_0^2$ = 4.15, 4.17, 4.20, 4.25, 4.30, 4.40, 4.50, 4.60, 4.70, 4.80, 5.00, 5.20, 5.50, 6.00, 6.50, and 7.00 have antiperiodic fermion boundary conditions in all four directions. In the analysis we only include the $L/a=20$, 24, 28 and 32 sets, though in the infinite volume extrapolation we also show data from $L/a=16$ configurations. From the preliminary analysis we show in the following only results using Zeuthen flow.
#### A) Infinite volume extrapolation:
![The panels in the first row show the finite volume $\beta$ function at flow times $t/a^2=2.20$ and 3.40 ($a^2/t=0.455$ and $0.294$) for our five $N_f=12$ volumes. Dashed lines show a polynomial interpolation of the data points. The panels at the bottom present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times. To resolve the small variations at weak coupling, the infinite volume extrapolations are shown in two panels. The upper ones have very small range allowing us to resolve a downward slope which however has only a tiny effect on the absolute value.[]{data-label="fig:Linf_extrap_Nf12"}](plots/{ContBetaFn_beta_vs_gSq_t2.200_Nf12_ZS}.pdf "fig:"){width="0.494\columnwidth"} ![The panels in the first row show the finite volume $\beta$ function at flow times $t/a^2=2.20$ and 3.40 ($a^2/t=0.455$ and $0.294$) for our five $N_f=12$ volumes. Dashed lines show a polynomial interpolation of the data points. The panels at the bottom present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times. To resolve the small variations at weak coupling, the infinite volume extrapolations are shown in two panels. The upper ones have very small range allowing us to resolve a downward slope which however has only a tiny effect on the absolute value.[]{data-label="fig:Linf_extrap_Nf12"}](plots/{ContBetaFn_beta_vs_gSq_t3.400_Nf12_ZS}.pdf "fig:"){width="0.494\columnwidth"}\
![The panels in the first row show the finite volume $\beta$ function at flow times $t/a^2=2.20$ and 3.40 ($a^2/t=0.455$ and $0.294$) for our five $N_f=12$ volumes. Dashed lines show a polynomial interpolation of the data points. The panels at the bottom present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times. To resolve the small variations at weak coupling, the infinite volume extrapolations are shown in two panels. The upper ones have very small range allowing us to resolve a downward slope which however has only a tiny effect on the absolute value.[]{data-label="fig:Linf_extrap_Nf12"}](plots/{ContBetaFn_beta_vs_Linv_t2.200_Nf12_ZS}.pdf "fig:"){width="0.494\columnwidth"} ![The panels in the first row show the finite volume $\beta$ function at flow times $t/a^2=2.20$ and 3.40 ($a^2/t=0.455$ and $0.294$) for our five $N_f=12$ volumes. Dashed lines show a polynomial interpolation of the data points. The panels at the bottom present the infinite volume extrapolation at several ${\ensuremath{g_{\rm GF}^2} }$ values for the same flow times. To resolve the small variations at weak coupling, the infinite volume extrapolations are shown in two panels. The upper ones have very small range allowing us to resolve a downward slope which however has only a tiny effect on the absolute value.[]{data-label="fig:Linf_extrap_Nf12"}](plots/{ContBetaFn_beta_vs_Linv_t3.400_Nf12_ZS}.pdf "fig:"){width="0.494\columnwidth"}
Figure \[fig:Linf\_extrap\_Nf12\] shows details of the infinite volume extrapolation. Since the $\beta$ function is small, finite volume effects are more noticeable than in the $N_f=2$ study. Taking advantage of additional volumes, we consider the infinite volume limit restricted to $L/a\ge 24$ or $L/a\ge 20$. $L/a=16$ data are shown in Fig. \[fig:Linf\_extrap\_Nf12\] only for further illustration of finite volume effects. Since the slowly varying $\beta$ function is small, we show the infinite volume extrapolations using two panels in Fig. \[fig:Linf\_extrap\_Nf12\]. The upper panel has an extremely small range in $\beta$ to show different extrapolations at weak coupling ($g_{GF}^2<4.0$), while the lower panel presents the stronger coupling range with a larger scale. Although the downward slope in the upper panels is resolved, the absolute variation is tiny.
#### B) Infinite flow time extrapolation:
![Representative $a^2/t \to 0$ continuum limit extrapolation for $N_f=12$ at ${\ensuremath{g_{\rm GF}^2} }=3.5$ (left panel) and ${\ensuremath{g_{\rm GF}^2} }=5.5$ (right panel). We show results for two infinite volume extrapolations and three operators, fitting filled symbols in the range $2.20 \le t/a^2 \le 3.44$ ($0.455\ge a^2/t\ge 0.299$). The (uncorrelated) fits are independent but predict consistent $a^2/t=0$ continuum values.[]{data-label="fig:cont_extrap_Nf12"}](plots/{ContBetaFn_Nf12_Zflow_continuum_gSq3.5}.pdf "fig:"){width="0.494\columnwidth"} ![Representative $a^2/t \to 0$ continuum limit extrapolation for $N_f=12$ at ${\ensuremath{g_{\rm GF}^2} }=3.5$ (left panel) and ${\ensuremath{g_{\rm GF}^2} }=5.5$ (right panel). We show results for two infinite volume extrapolations and three operators, fitting filled symbols in the range $2.20 \le t/a^2 \le 3.44$ ($0.455\ge a^2/t\ge 0.299$). The (uncorrelated) fits are independent but predict consistent $a^2/t=0$ continuum values.[]{data-label="fig:cont_extrap_Nf12"}](plots/{ContBetaFn_Nf12_Zflow_continuum_gSq5.5}.pdf "fig:"){width="0.494\columnwidth"}
Figure \[fig:cont\_extrap\_Nf12\] shows two examples of the continuum $a^2/t$ extrapolations. While the scaling exponent of the leading irrelevant operator is expected to change as we move away from the GFP, we cannot resolve such an effect. Fits using the form $(t/a^2)^{\alpha/2}$ predict $\alpha \approx -2.0$ with large uncertainties such that the fit is consistent with $\alpha=-2.0$. Simultaneous fits to two or three of the W, S, Z operators could provide sufficient information to resolve the scaling exponent. We will report on such an analysis in the future. Using a linear fit in the range $t \in (2.20,3.4)$, we predict consistent continuum limit values from W, S and C operators.
Similar to the $N_f=2$ case, the Wilson operator shows the smallest cutoff effects. The cyan/green data points are flat at both ${\ensuremath{g_{\rm GF}^2} }$ values. The clover operator has significantly larger cut-off corrections than the Wilson or Symanzik operators, rendering the continuum limit prediction from the ZC combination to be the least reliable of the three.
![ Continuous RG $\beta$ function of 12-flavor SU(3) model in the GF scheme using Zeuthen flow. The left panel shows the continuum limit obtained from the ZW combination using volumes with $L/a\ge 24$ overlayed with the raw ZW data (colored data points) from the $L/a=32$ ensembles. The colors indicate the 16 bare coupling values $\beta\equiv 6/g_0^2$ labeled by the colorbar on the right. In the panel on the right, we compare different continuum limit predictions using Zeuthen flow and the W, S, C operators as well as extrapolations based on $L/a\ge 20$ or 24. In addition to our nonperturbative results, we show predictions based on perturbation theory at 2-, 3-, 4-, and 5-loop order.[]{data-label="fig:beta-direct_Nf12"}](plots/{ContBetaFn_Nf12_ZW_continuum}.pdf "fig:"){width="0.494\columnwidth"} ![ Continuous RG $\beta$ function of 12-flavor SU(3) model in the GF scheme using Zeuthen flow. The left panel shows the continuum limit obtained from the ZW combination using volumes with $L/a\ge 24$ overlayed with the raw ZW data (colored data points) from the $L/a=32$ ensembles. The colors indicate the 16 bare coupling values $\beta\equiv 6/g_0^2$ labeled by the colorbar on the right. In the panel on the right, we compare different continuum limit predictions using Zeuthen flow and the W, S, C operators as well as extrapolations based on $L/a\ge 20$ or 24. In addition to our nonperturbative results, we show predictions based on perturbation theory at 2-, 3-, 4-, and 5-loop order.[]{data-label="fig:beta-direct_Nf12"}](plots/{ContBetaFn_Nf12_Zflow_continuum}.pdf "fig:"){width="0.494\columnwidth"}
#### The continuous $\beta$ function:
Figure \[fig:beta-direct\_Nf12\] shows our predicted RG $\beta$ function based on Zeuthen flow. In the left panel we include the raw data from the ZW flow/operator combination at GF time $t^2/a\in(2.20,3.4)$. The $\beta$ function of the $N_f=12$ system is small, thus the renormalized ${\ensuremath{g_{\rm GF}^2} }$ gauge coupling changes very slowly with the flow time. As a result the raw data in Fig. \[fig:beta-direct\_Nf12\] explore a very small range in $g_{GF}^2$ at any given value of the bare coupling $\beta\equiv 6/g^2_0$. There is a more pronounced fluctuation in $\beta({\ensuremath{g_{\rm GF}^2} })$ visible mostly due to the very small scale of the plots. As is predicted in Fig. \[fig:cont\_extrap\_Nf12\], the ZW data show only small cut-off effects resulting in raw data which are very close to the continuum limit predictions. The ZS combination has larger cutoff corrections, approaches the continuum predictions from above but exhibits also a somewhat larger reach in $g_{GF}^2$. While the raw data of the 12-flavor system does not offer the same description of the continuous $\beta$ function as we have seen in the 2-flavor case, it still offers intuition on how the continuum limit is approached.
It is worth mentioning that the $\beta$ function predicted by the raw ZW flow increases with flow time up to $6/g^2_0\le4.25$ but changes direction, i.e. decreases towards more negative values for $6/g^2<4.25$. This qualitative change could indicate that the influence of a possible nearby UVFP is getting strong on the RG flows.
In the right panel of Fig. \[fig:beta-direct\_Nf12\], we compare continuum predictions obtained using Zeuthen flow and the three different operators S, W, C. In general we observe very good agreement between the three flow/operator combinations shown, although ZC exhibits larger discretization errors as well as a shorter reach in $g_{GF}^2$ than ZS or ZW. In addition we check for finite volume effects by restricting the infinite volume extrapolations to volumes with $L/a \ge 20$ or $L/a\ge 24$. Again we observe that our results are consistent and we cannot resolve finite volume effects.
Discussion
==========
We presented a method based on a real-space RG transformation with continuous scale change to determine the continuous RG $\beta$ function. The validity of the approach relies on the nonperturbative Wilsonian RG transformations and is equally valid in the vicinity of the perturbative Gaussian FP, strongly coupled conformal IRFP or possible emerging UVFP both in conformal or infrared free systems.
First we outlined the steps of determining the continuous $\beta$ function and validated the method in 2-flavor QCD. Subsequently we followed the same steps to analyze existing GF data for the 12 flavor system. Results based on Zeuthen flow and Wilson plaquette, Symanzik and clover operators are consistent and predict an IRFP around ${\ensuremath{g_{\rm GF}^2} }=6$. The value of the FP is scheme dependent. The continuous $\beta$ function corresponds to $c=0$ renormalization scheme, the value of the FP is however similar to our determination in the $c=0.250$ gradient flow step-scaling scheme [@Hasenfratz:2019dpr]. An advantage of the continuous RG transfromation is the possibility to resolve the scaling dimension of the irrelevant operators around a non-perturbative fixed point. In the $N_f=2$ system we found the scaling dimension $\alpha=-2.0$, consistent with the expectations around the GFP. At present we are not able to resolve any difference in the strong coupling regime. In the $N_f=12$ flavor system our preliminary analysis predicts $\alpha=-2.0$ with large errors which we expect to reduce in the future with a more sophisticated analysis. A similar method to determine the continuous $\beta$ function from lattice data generated at finite mass in chirally broken systems is discussed in Ref. [@Fodor:2017die].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are very grateful to Peter Boyle, Guido Cossu, Anontin Portelli, and Azusa Yamaguchi who develop the `Grid` software library providing the basis of this work and who assisted us in installing and running `Grid` on different architectures and computing centers. A.H. and O.W. acknowledge support by DOE grant DE-SC0010005. We thank Slava Rychkov for correspondence and fruitful discussion that led to the phase diagram in the right panel of Fig. \[fig:RG-flow\]. We also thank Francesco Sannino for discussion on the 4-fermion system presented in Ref. [@Rantaharju:2016jxy; @Rantaharju:2017eej]. We are indebted to Daniel Nogradi for extending his original calculation of the finite volume correction factors on symmetric volumes to asymmetric lattices and sharing the result prior to publication. We thank Alberto Ramos for many enlightening discussions during the “37th International Symposium on Lattice Field Theory”, Wuhan, China, and Rainer Sommer and Stefan Sint for helpful comments. We benefited from many discussions with Thomas DeGrand, Ethan Neil, David Schaich, and Benjamin Svetitsky. A.H. would like to acknowledge the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 39083149) for enabling us to complete a portion of this work. O.W. partial support by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”. Computations for this work were carried out in part on facilities of the USQCD Collaboration, which are funded by the Office of Science of the U.S. Department of Energy and the RMACC Summit supercomputer [@UCsummit], which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1548562 [@xsede] through allocation TG-PHY180005 on the XSEDE resource `stampede2`. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. We thank Fermilab, Jefferson Lab, NERSC, the University of Colorado Boulder, TACC, the NSF, and the U.S. DOE for providing the facilities essential for the completion of this work.
[^1]: The first version of this phase diagram emerged in a discussion between Slava Rychkov and A. H. during the TASI Summer School in June 2019.
[^2]: $\delta$ depends on the flow time and the aspect ratio of the lattice volume. We thank D. Nogradi for sharing with us his results on the latter prior publication.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report on the passive measurement of time-dependent Green’s functions [in the optical frequency domain]{} with low-coherence interferometry. Inspired by previous studies in acoustics and seismology, we show how the correlations of a broadband and incoherent wave-field can directly yield the Green’s functions between scatterers of a complex medium. Both the ballistic and multiple scattering components of the Green’s function are retrieved. This approach opens important perspectives for optical imaging and characterization in complex scattering media.'
author:
- Amaury Badon
- Geoffroy Lerosey
- 'Albert C. Boccara'
- Mathias Fink
- Alexandre Aubry
title: 'Retrieving time-dependent Green’s functions in optics with low-coherence interferometry'
---
Waves propagating in complex media can experience complicated trajectories through scattering off objects or reflection and refraction at interfaces. All these events are nonetheless captured by the Green’s function formalism. In random multiple scattering and reverberating media, the temporal Green’s function thus provides a unique signature of the propagation of waves between the source and observation points. This property has been put to profit to focus waves in acoustics and electromagnetism, where temporal Green’s function are easily accessible, through the concept of time reversal [@derode0; @lerosey]. Similarly, temporal Green’s functions have been proved to allow imaging of complex media, either using statistical approaches or through numerical reconstruction [@borcea; @aubry2009random; @larose2; @krishnan]. In optics, there have been recently exciting proposals to measure Green’s functions of complex media [@popoff2010measuring; @choi; @popoff2; @choi2], and use them for imaging or focusing purposes, mostly owing to the development of wave front shaping techniques [@vellekoop2007focusing; @Mosk_review]. Incidentally, previous studies in acoustics have proposed a simpler and elegant route towards a passive measurement of temporal Green’s functions without the use of any source [@weaver; @derode]. The cross-correlation (or mutual coherence function) of an incoherent wave-field measured at two points A and B can yield the time dependent Green’s function between these two points. Provided that the ambient field is equipartitioned in energy, the time derivative of the correlation function at two positions is actually proportional to the difference between the anticausal and causal Green’s functions. This property is a signature of the universal fluctuation dissipation theorem [@rytov; @agarwal; @bart] and has been derived following different approches [@weaver; @bart; @wapenaar; @snieder; @wapenaar2; @larose]. An elegant physical picture is provided by an analogy with time reversal [@derode; @derode2; @supp]. In the frequency domain, this result manifests itself as the link between the spatial correlation of the wave-field and the local density of states [@Joulain; @caze]. This fundamental quantity is actually proportional to the imaginary part of the monochromatic *self*-Green’s function.
Previously and independently developed in helioseismology [@duvall], the Green’s function estimation from diffuse noise cross correlations has received a considerable attention in seismology in the 2000s [@campillo]. The cross-correlation of seismic noise recorded by two stations over months has allowed to retrieve the Green’s functions between these observation points as if one was replaced by a virtual coherent source. By passively measuring the elastic Green’s functions between a network of seismic stations, an imaging of the Earth’s crust has been obtained with unprecedented high resolution [@shapiro]. More recently, thermal radiation noise has also been taken advantage of to measure passively electromagnetic Green’s functions in the microwave [frequency domain]{} [@davy].
The aim of this paper is to demonstrate the passive measurement of time-dependent Green’s functions in optics. For this proof-of-concept, the first sample under study consists in dispersed microbeads that are used as passive sensors. This scattering sample is isotropically illuminated by an incoherent halogen light source. The correlation of the scattered wave-field is measured by means of a Michelson interferometer and recorded on a CCD camera. In this Letter, we first show that the cross-correlation of the wave-field coming from two scatterers $A$ and $B$ converges towards the ballistic Green’s function $g_{AB}(t)$ between them. In a second example, we show that the multiple scattering components of the Green’s function can also be retrieved. This is illustrated by measuring the autocorrelation of the field coming from the scatterer $A$. The resulting *self*-Green’s function $g_{AA}(t)$ is shown to exhibit the time-resolved multiple scattering echoes between the scatterers $A$ and $B$. At last, we show that our approach can also be extended to a strongly scattering medium made of ZnO [nanoparticles]{}. The correlation of the scattered wave-field directly yields the time-dependent Green’s functions between each pixel of the CCD camera. With a moderate integration time, the resulting Green’s functions are shown to emerge from noise for times of flight at least twenty five times larger than the transport mean free time. Hence, this simple and powerful approach directly yields a wealth of information about the scattering medium. It opens important perspectives for imaging and characterization in strongly scattering media.
![(a) Experimental set up. A broadband incoherent light source isotropically illuminates a scattering sample (here consisting of two beads). The spatio-temporal correlation of the scattered wave field is extracted by means of a Michelson interferometer and recorded by a CCD camera. HL : halogen lamp. MO : microscope objective. BS : beam splitter. M : mirror. PZT : piezoelectric actuator. (b) Intensity image recorded by the CCD camera displaying the two beads and their superimposed image at the center. (c,d,e) Field measured by the CCD camera *via* phase-shifting interferometry at different optical path difference $\delta$: (c) $\delta=0$, (d) $\delta=L$, (e) $\delta>L$. Each field has been normalized by its maximum [range]{}.[]{data-label="fig1"}](Fig1){width="8.5cm"}
The experimental set-up is displayed in Fig.\[fig1\]. An incoherent broadband light source (650-850 nm) isotropically illuminates a scattering sample [in a *dark-field* configuration]{}. . The scattered wave-field is collected by the microscope objective and sent to a Michelson interferometer. The beams coming from the two interference arms are then recombined and focused by a lens. The intensity recorded by the CCD camera at the focal plane can be expressed as, $$S(\mathbf{r},\mathbf{r}+\Delta \mathbf{r},t) = \int_{0}^T || \mathbf{E}(\mathbf{r},t+\tau)+\mathbf{E}(\mathbf{r}+\Delta \mathbf{r},\tau) ||^2 \mathrm{d}\tau
\label{intensity}$$ with $\tau$ the absolute time, $\mathbf{r}$ the position vector on the CCD screen, $\mathbf{E}(\mathbf{r},\tau)$ the scattered electric field associated to the first interference arm and $T$ the integration time of the CCD camera. The tilt of mirror $M_2$ allows a displacement $\Delta \mathbf{r}$ of the associated wave-field on the CCD camera. The motorized translation of mirror $M_1$ induces a time delay $t = \delta/c$ between the two interferometer arms, with $\delta$ the optical path difference (OPD) and $c$ the light celerity. The interference term is extracted from the intensity pattern (Eq.\[intensity\]) by phase shifting interferometry (“four phases method” [@popoff2010measuring]) using a piezoelectric actuator placed on mirror $M_1$. It directly yields the mutual coherence function $C$ of the scattered wave-field $\mathbf{E}$: $$C(\mathbf{r},\mathbf{r}+\Delta \mathbf{r},t) = \int_{0}^T \mathbf{E}(\mathbf{r},t+\tau) \cdot \mathbf{E}(\mathbf{r}+\Delta \mathbf{r},\tau) \mathrm{d} \tau
\label{cor}$$ If the incident light is spatially and temporally incoherent, the time derivative of the correlation function $C(\mathbf{r_A},\mathbf{r_B},t)$ between two points A and B should converge towards the difference between the causal and anti-causal Green’s function, such that $$\partial_t C(\mathbf{r_A},\mathbf{r_B},t) \underset{T\rightarrow \infty}{\sim} g_{AB}(t) - g_{AB}(-t) .
\label{green}$$
The aim of this Letter is to prove experimentally this result and measure time-dependent Green’s functions with the basic experimental setup displayed in Fig.\[fig1\]. As a proof-of-concept, we first study a sample made of 3$\mu$m-diameter magnetite beads (Fe$_3$O$_4$) randomly embedded in a transparent polymer matrix (poly-L-lysine) on a microscope slide. The correlation function between two scatterers can be measured by tilting mirror $M_2$ such that the images of the two beads are superimposed on the same CCD camera pixel \[see Fig.\[fig1\](b)\]. Figs.\[fig1\](c,d,e) display the interference pattern recorded by the CCD camera at different OPD for an isolated couple of beads, $A$ and $B$, separated by a distance $L=$7 $\mu$m. At $\delta=0$, the straight fringes observed on the CCD camera result from a residual coherence of the incident wave-field \[Fig.\[fig1\](c)\]. At $\delta=L$, a strong interference signal is observed in the area where the images of the two beads overlap \[Fig.\[fig1\](d)\]. As we will see, it corresponds to the direct echo between $A$ and $B$. For $\delta>L$, the field measured by the CCD camera corresponds to noise that results from the interference between uncorrelated random wave-fields \[Fig.\[fig1\](e)\].
![Passive measurement of the Green’s function $g_{AB}(t)$ between the two beads $A$ and $B$. (a,b,c) Cross-correlation signals versus time delay / OPD for different integration times: $T=$10 ms (a), $T=$40 ms (b) and $T=$750 ms (c). (d) Interferometric signal obtained for the isolated bead $D$ ($T=750$ ms). (e,f,g) Sketch of the scattering events accounting for the different pulses emerging from the signals in (c) and (d). []{data-label="fig2"}](Fig2){width="8.5cm"}
Fig.\[fig2\] displays the time-dependence of the cross-correlation function between beads $A$ and $B$ for different integration times. This interferometric signal contains a random contribution that should vanish with average and a deterministic contribution due to the stationary interferences between the two beads. The latter one should directly lead to the Green’s function $g_{AB}(t)$ as stated by Eq.\[green\]. For $T=10$ ms, noise predominates and no coherent signal can be clearly detected \[Fig.\[fig2\](a)\]. For $T=40$ ms, three time-resolved echoes start to emerge from noise but the signal-to-noise ratio is still perfectible \[Fig.\[fig2\](b)\]. At last, for $T=750$ ms, noise is sufficiently averaged out to obtain the stationary interference signal with a good precision \[Fig.\[fig2\](c)\]. As a reference, Fig.\[fig2\](d) displays the cross-correlation function obtained between an isolated bead $D$ and the background wave-field for the same integration time and tilt of $M_2$. Each interferometric signal in Figs.\[fig2\](c,d) exhibits an echo around $\delta=0$ that corresponds to the straight fringes displayed by Fig.\[fig1\](c). Although the two beads $A$ and $B$ are separated by a distance $L>l_c$, the incident wave fields, $E_0$ and $E'_0$, seen by each of them remain slightly correlated . This gives rise to a stationary interference between the single scattering paths $E_1$ and $E'_1$ around $\delta=0$ \[Fig.\[fig2\](f)\]. Unlike the signal associated to the isolated bead $D$ \[Fig.\[fig2\](d)\], the cross-correlation between beads $A$ and $B$ clearly exhibits two echoes around $\delta$ = $\pm L$ \[Fig.\[fig2\](c)\]. It corresponds to the strong interference signal previously highlighted in Fig.\[fig1\](d). These two echoes are the expected causal and anticausal parts of the ballistic Green’s function between $A$ and $B$. They originate from the interference between the single scattering path $E_1$ and the double scattering path $E_2$ depicted in Figs.\[fig2\](e,g). Normally, the causal and anticausal parts of the Green’s function should be of same amplitude due to reciprocity.
![Passive measurement of the *self*-Green’s function $g_{AA}(t)$ associated to the bead $A$. (a)-(b) Autocorrelation signal versus time delay / OPD for the isolated bead $D$ (a) and the bead $A$ placed at the vicinity of bead $B$ (b). (c)-(f) Sketch of the scattering events accounting for the different pulses emerging from the autocorrelation signals in (a) and (b).[]{data-label="fig4"}](Fig3){width="8.5cm"}
This first experiment has demonstrated the ability of measuring a direct echo between two scatterers with low-coherence interferometry. However, one can go beyond and also measure the multiple scattering components of the Green’s function. As a demonstration, we show the measurement of the autocorrelation function associated to the bead $A$. This is performed by simply canceling the tilt of mirror $M_2$ such that $\Delta \mathbf{r}$ = 0 in Eq.\[cor\]. The autocorrelation function $C(\mathbf{r_A},\mathbf{r_A},t)$ should lead to a measurement of the *self*-Green’s function $g_{AA}(t)$ \[Eq.\[green\]\]. In this case, the bead $A$ virtually acts both as the source and the receiver while the bead $B$ only acts as a passive scatterer. Our aim is to detect the presence of bead $B$ in $g_{AA}(t)$ through the multiple scattering echoes that take place between the two beads. Fig.\[fig4\] compares the autocorrelation signal obtained for the bead $A$ \[Fig.\[fig4\](b)\] and for the isolated bead $D$ \[Fig.\[fig4\](a)\]. In the latter case, the autocorrelation function only exhibits a single echo around $\delta=0$ due to the interference of the scattered field $E_1$ with itself \[Fig.\[fig4\](c)\]. For bead $A$, six supplementary echoes are visible at $\delta$ = $\pm L$, $\delta$ = $\pm2L$ and $\delta$ = $\pm3L$. Each of them can be associated to a stationary interference between multiple scattering paths depicted in Figs.\[fig4\](d,e,f). The signal at $\delta=\pm L$ results from the interference between the single scattering path $E_1$ and the double scattering path $E'_2$ \[Fig.\[fig4\](d)\]. It corresponds to the direct echo between the two beads. It would not emerge if the incident wave fields seen by the two beads, $E_0$ and $E'_0$, were totally uncorrelated. However, , a residual coherence subsists for the incident wave-field at such a distance $L$. The echo at $\delta=\pm 2L$ results from the interference between the single scattering path $E_1$ and the triple scattering path $E_3$ \[Fig.\[fig4\](e)\]. As these two paths involve the same first scattering event, it would exist even for a perfectly incoherent incident wave-field. This signal corresponds to a way and return echo between the two beads. At last, the echo at $\delta=\pm 3L$ results from the interference between the single scattering path $E_1$ and the quadruple scattering path $E'_4$ \[Fig.\[fig4\](f)\]. As for the echo at $\delta=L$, it is only visible because of the residual coherence of the incident wave-field. Hence, the relevant echoes here are the ones associated to a roundtrip scattering event between the beads as they would exist even for a perfectly incoherent wave-field. They correspond to the causal and anti-causal parts of the Green’s function $g_{AA}(t)$ associated to bead $A$ and yield an information about its local environment, *i.e* the presence of bead $B$ at a distance $L$. Hence this experiment demonstrates the ability of measuring passively the multiple scattering contribution of a temporal Green’s function from an incoherent wave-field. In this example, the single, double and triple scattering events are nicely retrieved. Note that, in principle, it would be possible to measure higher order scattering events. To that aim, the integration time should be increased in order to lower the noise level but it would be at the cost of a longer measurement.
![Passive measurement of the *self*-Green’s function in a multiple scattering sample made of ZnO nanoparticles. The autocorrelation signal obtained for one pixel of the CCD camera is shown as a function of the time delay / OPD. The CCD image of the sample as well as the location of the selected pixel are displayed in the bottom left inset. The intensity of the autocorrelation signal averaged over 24 neighbour pixels versus time delay / OPD is shown in the top right inset in a log-log scale. The averaged intensity (continuous blue line) is fitted with the expected power law $t^{-5/2}$ (dashed red line).[]{data-label="fig5"}](Fig4){width="8.5cm"}
Now that the ability of measuring time-dependent Green’s functions between individual scatterers has been demonstrated, the case of a strongly scattering medium is now investigated. The sample under study is [a layer of slighly pressed ZnO]{} nanoparticles (Sigma Aldrich 544906) on a microscope slide. The sample thickness and the transport mean free path $l^*$ are of the order of 50 $\mu$m and 5 $\mu$m, respectively. The measurement of the autocorrelation signal for one pixel of the CCD camera is shown in Fig.\[fig5\]. It directly yields the *self*-Green’s function $g_{AA}(t)$ for a virtual sensor $A$ placed at the surface of the sample. Its characteristic size is given by the coherence length $l_c$ of the wave-field. The integration time has been fixed to 1250 ms to get rid of noise in the interference signal. The Green’s function shown in Fig.\[fig5\] is characteristic of a strongly scattering sample with a long tail that results from the numerous multiple scattering events that take place within the scattering medium. Here, we have access to a satisfying estimation of the Green’s function over 130 $\mu$m in terms of OPD (or 430 fs in terms of time delay), which corresponds to scattering paths of $26 l^*$. In the multiple scattering regime, a probabilistic approach is generally adopted to extract information from the measured Green’s functions. One can for instance study the intensity of the *self*-Green’s function averaged over several pixels of the CCD camera \[see inset of Fig.\[fig5\]\]. This leads to an estimation of the *return probability*, a key quantity in multiple scattering theory [@skipetrov] that describes the probability for a wave to come back close to its starting point. For a source placed at the surface of a scattering medium, the return probability is supposed to decrease as $t^{-5/2}$ in the diffusive regime [@dogariu; @aubry_2014]. As shown in the top right inset of Fig.\[fig5\], such a power law decay is recovered in our measurements for an OPD $\delta>70$ $\mu$m, *i.e* when the diffusive regime is reached. This observation demonstrates that the measured Green’s functions follow the temporal behavior predicted by diffusion theory, thus confirming the validity of our approach. In summary, this study demonstrates for the first time the optical measurement of time-dependent Green’s functions with low-coherence interferometry. As a proof-of-concept, we have first been able to retrieve the time-resolved ballistic and multiple scattering contributions of the Green’s function between individual scatterers. This approach has also been successfully applied to a strongly scattering medium. A whole set of time-dependent Green’s functions can be measured between each point of the surface of a scattering sample. The experimental access to this Green’s matrix is potentially important in many applications of wave physics in complex media whether it be for imaging [@shahjahan], characterization [@aubry2], focusing [@popoff2010measuring; @choi; @popoff2; @choi2] or communication [@derode_2003; @popoff2010image] purposes.
The authors are grateful for funding provided by LABEX WIFI (Laboratory of Excellence within the French Program Investments for the Future, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL\*). A.B. acknowledges financial support from the French “Direction G�n�rale de l’Armement”(DGA). A. A. and G. L. would like to acknowledge funding from High Council for Scientific and Technological Cooperation between France and Israel under reference P2R Israel N$^o$ 29704SC.
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[**Local and Global Dynamic Bifurcations\
of Nonlinear Evolution Equations**]{}
Desheng Li
Zhi-Qiang Wang [^1]
We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form $u_t+A u=f_\lam(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $\lam\in\R$ is the bifurcation parameter. Suppose the equation has a trivial solution branch $\{(0,\lam):\,\,\lam\in\R\}$. Denote $\Phi_\lam$ the local semiflow generated by the initial value problem of the equation. It is shown that if the crossing number $n$ at a bifurcation value $\lam=\lam_0$ is nonzero and moreover, $S_0=\{0\}$ is an isolated invariant set of $\Phi_{\lam_0}$, then either there is a one-sided neighborhood $I_1$ of $\lam_0$ such that $\Phi_\lam$ bifurcates a topological sphere $\mathbb{S}^{n-1}$ for each $\lam\in I_1\setminus\{\lam_0\}$, or there is a two-sided neighborhood $I_2$ of $\lam_0$ such that the system $\Phi_\lam$ bifurcates from the trivial solution an isolated nonempty compact invariant set $K_\lam$ with $0\not\in K_\lam$ for each $\lam\in I_2\setminus\{\lam_0\}$. We also prove that the bifurcating invariant set has nontrivial Conley index. Building upon this fact we establish a global dynamical bifurcation theorem. Roughly speaking, we prove that for any given neighborhood $\Omega$ of the bifurcation point $(0,\lambda_0)$, the connected bifurcation branch $\Gamma$ from $(0,\lambda_0)$ either meets the boundary $\pa\Omega$ of $\Omega$, or meets another bifurcation point $(0,\lambda_1)$. This result extends the well-known Rabinowitz’s Global Bifurcation Theorem to the setting of dynamic bifurcations of evolution equations without requiring the crossing number to be odd.
As an illustration example, we consider the well-known Cahn-Hilliard equation. Some global features on dynamical bifurcations of the equation are discussed.
Evolution equation, invariant-set bifurcation, global dynamical bifurcation.
34C23, 34K18, 35B32, 37G99.
Introduction
============
Dynamic bifurcation concerns the changes in the qualitative or topological structures of limiting motions such as equilibria, periodic solutions, homoclinic orbits, heteroclinic orbits and invariant tori etc. for nonlinear evolution equations as some relevant parameters in the equations vary. Historically, the subject can be traced back in the very earlier work of Poincar$\acute{\mb e}$ [@Poin] around 1892. It is now a fundamental tool to study nonlinear problems in mathematical physics and mechanics [@Chow; @Kie; @Mars], and enables us to understand how and when a system organizes new states and patterns near the original “trivial” one when the control parameters cross some critical values.
A relatively simpler case for dynamic bifurcation is that of the bifurcations from equilibria. Generally speaking, there are two typical such bifurcations in the classical bifurcation theory. One is the bifurcation from equilibria to equilibria (static bifurcation), and the other is from equilibria to periodic solutions (Hopf bifurcation). The former usually requires a “crossing odd-multiplicity” condition, namely, the linearized equation of a system has an odd number of eigenvalues (counting with multiplicity) crossing the imaginary axis when the control parameter crosses a critical value (the Krasnosel’skii’s Bifurcation Theorem). We also know that in such a case the bifurcation has some global features, which fact is addressed by the well-known Rabinowitz’s Global Bifurcation Theorem. Situations become very complicated if one drops the “crossing odd-multiplicity” condition mentioned above. If the system under consideration is a gradient one, then by a classical bifurcation theorem on potential operator equations due to Krasnosel’skii (see [@Kie Chap.II, Sect.7] or [@Kras]), one can still have local bifurcation results. Whereas the global bifurcation remains an open problem. To deal with general systems without the “crossing odd-multiplicity” condition, Ma and Wang [@MW4] proved some new local and global static bifurcation theorems by using higher-order nondegeneracy conditions on singularities of the nonlinearities. The Hopf bifurcation theory has a long history and, to some extent, forms the central part of the classical dynamic bifurcation theory. It focuses on the case when a pair of conjugate eigenvalues of the linearized equation cross the imaginary axis, and was fully developed in the 20-th century. There has been a vast body of literature on how to determine Hopf bifurcation for nonlinear systems arising from applications. One can also find some nice results concerning global results in [@AY; @Wu], etc.
This present work is mainly concerned with the general case of the bifurcations from equilibria in terms of invariant-set bifurcation, where the number of eigenvalues of the linearized equation crossing the imaginary axis might be even and greater than two. A particular but very important case in this line is the theory of attractor bifurcation, which was first introduced by Ma and Wang in 2003 [@MW0] and was further developed by the authors into a dynamic transition theory [@MW5]. Roughly speaking, it states that if the trivial equilibrium solution $\theta$ of a system changes from an attractor to a repeller on the local center manifold when the bifurcation parameter $\lam$ crosses a critical value $\lam_0$, then the system bifurcates a compact invariant set $K$ which is an attractor of the system on the center manifold. It is also known that $K$ has the shape of an $n$-dimensional sphere, where $n$ denotes the [*crossing number*]{} at $\lam=\lam_0$ (the number of eigenvalues of the linearized equation crossing the imaginary axis); see [@san3 Theorem 1] or [@MW1 Theorem 6.1]. Note that a fundamental assumption of this theory is that the trivial equilibrium $\theta$ is an attractor (repeller) of the system on the center manifold at $\lam=\lam_0$. Hence it is no longer applicable when $S_0=\{\theta\}$ is only an isolated invariant set when $\lam=\lam_0$. Fortunately in such a case, we know that dynamic bifurcation still occurs as long as there are eigenvalues crossing the imaginary axis. This has already been addressed in the literature; see e.g. Rybakowski [@Ryba] (pp.101-102) and Ward [@Ward1].
An abstract global dynamic bifurcation theorem was also proved in Ward [@Ward1] in terms of semiflows on complete metric spaces. Let $\Phi_\lam$ be a family of dynamical systems on a complete metric space $X$, where $\lam\in\R$. Suppose that $\theta$ is an equilibrium solution for each $\Phi_\lam$. Let $[a,b]$ be a compact interval which contains exactly one bifurcation value $\lam_0\in [a,b]$. The Ward’s global bifurcation theorem states that if $h(\Phi_a,\{\theta\})\ne h(\Phi_b,\{\theta\})$, a continua $\Gamma\subset X\X \R$ of bounded solutions bifurcates from $(\theta,\lam_0)$, where $h(\Phi_\lam,\{\theta\})$ denotes the Conley index of $\{\theta\}$ with respect to $\Phi_\lam$. Moreover, either $\Gamma$ is unbounded in $X\X [a,b]$, or it intersects $X\X\{a,b\}$. Note that, due to the requirement on the uniqueness of bifurcation values in $[a,b]$, the theorem mentioned above may fail to work when a $\lam$-interval contains multiple bifurcation values. This is somewhat different from the situation of the Rabinowitz’s Global Bifurcation Theorem.
In this paper we consider the abstract evolution equation \[e:1.1\] u\_t+A u=f\_(u) on a Banach space $X$, where $A$ is a sectorial operator on $X$ with [*compact resolvent*]{}, $f_\lam(u)$ is a locally Lipschitz continuous mapping from $X^\alpha\X \R$ to $X$ for some $0\leq\alpha<1$, and $\lam\in\R$ is the bifurcation parameter. Our main goal is to establish new local and global dynamic bifurcation results. Suppose that \[e:1.2\] f\_(0)0,. Thus $u=0$ is always a trivial solution of (\[e:1.1\]) for all $\lam$. It is also assumed that $f_\lam(u)$ is differentiable in $u$ with $Df_\lam(u)$ being continuous in $(u,\lam)$.
First, as one of our main purposes here, we give some more precise and general results on local dynamic bifurcations in terms of invariant sets. In particular, we show that if the [ crossing number]{} $n$ at a bifurcation value $\lam=\lam_0$ is nonzero and moreover, $S_0=\{0\}$ is an isolated invariant set of the system, then either there is a one-sided neighborhood $I_1$ of $\lam_0$ such that the system bifurcates an $(n-1)$-dimensional topological sphere $\mathbb{S}^{n-1}$ for each $\lam\in I_1\setminus\{\lam_0\}$, or there is a two-sided neighborhood $I_2$ of $\lam_0$ such that the system bifurcates from the trivial solution an isolated nonempty compact invariant set $K_\lam$ with $0\not\in K_\lam$ for each $\lam\in I_2\setminus\{\lam_0\}$. Then we prove that the invariant set $K_\lam$ from bifurcation has nontrivial Conley index. This result plays a key role in establishing our global dynamic bifurcation theorem. However, it may be of independent interest in its own right.
Finally, as our main goal in this present work, we establish a global dynamic bifurcation theorem, extending the Rabinowitz’s Global Bifurcation Theorem on operator equations to dynamical systems without assuming the “crossing odd-multiplicity” condition and the uniqueness of bifurcation values in parameter intervals. Roughly speaking, given a neighborhood $\Omega\subset X^\alpha\X\R$ of the bifurcation point $(0,\lam_0)$, we prove that the connected bifurcation branch $\Gamma$ from $(0,\lam_0)$ either meets the boundary $\pa\Omega$ of $\Omega$, or meets another bifurcation point $(0,\lam_1)$. As an example, we consider the homogeneous Neumann boundary value problem of the Cahn-Hilliard equation $$u_t+\Delta\(\kappa\De u-f(u)\)=0$$ on a bounded domain $\Omega\subset R^d$ ($d\leq 3$) with sufficiently smooth boundary, where $$f(u)=a_1u+a_2u^2+a_3u^3,\hs \,a_3>0.$$ The local attractor bifurcation and phase transition of the problem have been extensively studied in Ma and Wang [@MW1; @MW1b; @MW2]. Other results relates to bifurcation of the problem can be found in [@BDW; @Mis2], etc. Here by applying the theoretical results obtained above, we give some more precise local dynamic bifurcation results and demonstrate global features of the bifurcations.
This paper is organized as follows. In Section 2 we make some preliminaries, and in Section 3 we present results on local invariant manifolds of the equation (\[e:1.1\]) and give a slightly modified version of a reduction theorem for Conley index in [@Ryba]. In Section 4 we prove some local dynamic bifurcation results. Section 5 is concerned with the nontriviality of the Conley indices of bifurcating invariant sets. Section 6 is devoted to the global dynamic bifurcation theorem. Section 7 consists of an example mentioned above.
Preliminaries
=============
This section is concerned with some preliminaries.
Basic topological notions and facts
-----------------------------------
Let $X$ be a complete metric space with metric $d(\cdot,\cdot)$. For convenience we will always identify a singleton $\{x\}$ with the point $x$ for any $x\in X$.
Let $A$ and $B$ be nonempty subsets of $X$. The [*distance*]{} $d(A,B)$ between $A$ and $B$ is defined as $$d(A,B)=\inf\{d(x,y):\,\,\,x\in A,\,\,y\in B\},$$ and the [*Hausdorff semi-distance*]{} and [*Hausdorff distance*]{} of $A$ and $B$ are defined, respectively, as $$d_{\mbox{\tiny H}}(A,B)=\sup_{x \in A}d(x,B),\hs
\delta_{\mbox{\tiny H}}(A,B) = \max\left\{d_{\mbox{\tiny H}}(A,B),
d_{\mbox{\tiny H}}(B,A)\right\}.$$ We also assign $d_{\tiny\mb{H}}(\emp,B)=0$.
The closure, interior and boundary of $A$ are denoted, respectively, by $\ol A$, int$\,A$ and $\pa A$. A subset $U$ of $X$ is called a [*neighborhood*]{} of $A$, if $\ol A\subset \mbox{int}\,U$. The [*$\ve$-neighborhood*]{} ${\mb{B}}(A,\ve)$ of $A$ is defined to be the set $\{y\in
X:\,d(y,A)<\ve\}.$ Let $A_\lam$ ($\lam\in\Lam$) be a family of nonempty subsets of $X$, where $\Lam$ is a metric space. We say that $A_\lam$ is [*upper semicontinuous*]{} in $\lam$ at $\lam_0\in\Lam$, this means $$d_{\tiny\mb{H}}(A_\lam,A_{\lam_0})\ra 0\hs \mb{as }\,\lam\ra\lam_0.$$
\[l:2.2\][@Rab] Let $X$ be a compact metric space, and let $A$ and $B$ be two disjoint closed subsets of $X$. Then either there exists a subcontinuum $C$ of $X$ such that $$\ba{ll}
A\cap C\ne \emp\ne B\cap C,\ea$$ or $X=X_A\cup X_B$, where $X_A$ and $X_B$ are disjoint compact subsets of $X$ containing $A$ and $B$, respectively.
\[l:2.3\] ([@CV], pp. 41) Let $X$ be a compact metric space. Denote ${{\mathscr K}}(X)$ the family of compact subsets of $X$ which is equipped with the Hausdorff metric $\de_{\mbox{\tiny H}}(\.,\.)$. Then ${{\mathscr K}}(X)$ is a compact metric space.
Criteria on homotopy equivalence
--------------------------------
We denote “$\simeq$” and “$\cong$” the [*homotopy equivalence*]{} and [*homeomorphism*]{}, respectively, between topological spaces.
Let $X$ be a topological space, and $A\subset X$ be closed. The following result can be found in many text books on general topology. -1cm If $A$ is a strong deformation retract of $X$, then $X\simeq A.$
Let $i_A:A\ra X$ be the inclusion. Denote $$\ba{ll}
M_{i_A}=(X\X\{0\})\cup (A\X I),\hs C_{i_A}=M_{i_A}/(A\X\{1\}).\ea$$ $M_{i_A}$ and $C_{i_A}$ are called the [*mapping cylinder*]{} and [*mapping cone*]{} of $i_A$, respectively.
The pair $(X,A)$ is said to have [*homotopy extension property*]{} (H.E.P in short), if for any space $Y$, any mapping $f:M_{i_A}\ra Y$ can be extended to a mapping $F:X\X I\ra Y$. \[l2.5\] ([@Hat], pp.14) $(X,A)$ has the H.E.P. iff $M_{i_A}$ is a retract of $X\X I$.
\[l2.4\]([@Hat], Theorem 0.17) Suppose $(X,A)$ has the H.E.P. If $A$ is contractible, then $
X/A\simeq X.
$
As a consequence of Lemma \[l2.4\], we have
\[c2.5\] Suppose $(X,A)$ has the H.E.P. Let $B$ be a closed subset of $A$. If $B$ is a strong deformation retract of $A$, then $
X/A\simeq X/B.
$
[**Proof.**]{} We observe that $
X/A\cong(X/B)/\widetilde{A},
$ where $\widetilde{A}=\pi_B(A)$, and $\pi_B:X\ra X/B$ is the projection. In the following we verify that $$(X/B)/\widetilde{A}\simeq X/B,$$ thus completing the proof of what we desired.
Since $(X,A)$ has the H.E.P., $M_{i_A}$ is a retract of $X\X I$. Let $f:X\X I\ra M_{i_A}$ be a retraction, $$f(x,t)=(\phi(x,t),\xi(x,t)),\Hs (x,t)\in X\X I,$$ where $\phi(x,t)\in X$, and $\xi(x,t)\in I$. Define $$h:X\X I\ra M_{i_{\widetilde{A}}}=((X/B)\X\{0\})\cup (\widetilde{A}\X I)$$ as $
h(x,t)=\(\pi_B\circ\phi(x,t),\,\xi(x,t)\)$ for $ (x,t)\in X\X I$. Let $$Q(x,t)=(\pi(x), t),\Hs (x,t)\in X\X I.$$ Then $Q:X\X I\ra (X/B)\X I$ is a quotient mapping. Observing that $$h(x,t)=\(\pi_B\circ\phi(x,t),\xi(x,t)\)= \(\pi_B(x),t\),\Hs (x,t)\in M_{i_A},$$ one finds that $h$ remains constant on $B\X \{t\}$ for each $t\in I$. Consequently $h\equiv const.$ on $Q^{-1}(y,t)$ for each $(y,t)\in (X/B)\X I$. Thus by the basic knowledge in the theory of general topology (see e.g. [@Munk], Chap. 2, Theorem 11.1), there is a mapping $g:(X/B)\X I\ra M_{i_{\widetilde{A}}}$ such that $h=g\circ Q$. It is trivial to verify that $g$ is a retraction from $(X/B)\X I$ to $M_{i_{\widetilde{A}}}$. Thus the pair $(X/B,\widetilde{A})$ has the H.E.P.
Since $B$ is a strong deformation retract of $A$, the singleton $\{[B]\}$ is a strong deformation retract of $\widetilde{A}$, that is, $\widetilde{A}$ is contractible. Lemma \[l2.4\] then asserts that $(X/B)/\widetilde{A}\simeq X/B.$ $\Box$
Wedge/smash product of pointed spaces
-------------------------------------
Let $(X,x_0)$ and $(Y,y_0)$ be two pointed spaces. The [*wedge product*]{} $(X,x_0)\vee (Y,y_0)$ and [*smash product*]{} $(X,x_0)\wedge (Y,y_0)$ are defined, respectively, as follows: $$(X,x_0)\vee (Y,y_0)=\({{\mathcal W}},\,(x_0,y_0)\),\hs
(X,x_0)\wedge (Y,y_0)=\((X\X Y)/{{\mathcal W}},\,[{{\mathcal W}}]\),$$ where ${{\mathcal W}}=X\X\{y_0\}\cup \{x_0\}\X Y$.
We denote $[(X,x_0)]$ the [*homotopy type*]{} of a pointed space $(X,x_0)$. Since the operations “$\vee$” and “$\wedge$” preserve homotopy equivalence relations, they can be naturally extended to the homotopy types of pointed spaces. Specifically, $$[(X,x_0)]\vee [(Y,y_0)]=\left[(X,x_0)\vee(Y,y_0)\right],$$ $$[(X,x_0)]\wedge[(Y,y_0)]=\left[(X,x_0)\wedge(Y,y_0)\,\right].$$
Denote $\ol0$ and $\Sigma^0$ the homotopy types of the pointed spaces $(\{p\},p)$ and $(\{p,q\},q)$, respectively, where $p$ and $q$ are two distinct points. Let $\Sigma^m$ be the homotopy type of pointed $m$-dimensional sphere. One easily verifies that $$[(X,x_0)]\vee \ol0=[(X,x_0)],\hs $$ and $$\Sigma^m \wedge\Sigma^n=\Sigma^{m+n},\Hs \A\,m,n\geq 0.$$
Local semiflows and basic dynamical concepts
--------------------------------------------
In this subsection we briefly recall some dynamical concepts and facts that will be used throughout the paper.
Let $X$ be a complete metric space.
A [*local semiflow*]{} $\Phi$ on $X$ is a continuous map from an open subset ${{\mathcal D}}_\Phi$ of $\R^+\X X$ to $X$ satisfying that (i) $\A\,x\in X$, $\E\,T_x\in(0,\8]$ such that $$(t,x)\in {{\mathcal D}}_\Phi \Longleftrightarrow t\in[0,T_x)\,;$$ and (ii) $\Phi(0,\.)=id_X$, furthermore, $$\Phi(s+t,x)=\Phi\(t,\Phi(s,x)\), \Hs \A\,x\in X,\, \,s,t\geq0$$ as long as $(s+t,x)\in{{\mathcal D}}_\Phi$. The number $T_x$ in the above definition is called the [*escape time*]{} of $\Phi(t,x)$. Let $\Phi$ be a given local semiflow on $X$. For notational simplicity, we will rewrite $\Phi(t,x)$ as $\Phi(t)x$.
A [*trajectory*]{} on an interval $J$ is a continuous mapping $\gamma: J\ra X$ such that $$\gamma(t)=\Phi(t-s)\gamma(s),\Hs \A \,t,\, s\in J,\,\,\,t\geq s.$$ If $J=\R$, then we simply call $\gamma$ a [*complete trajectory*]{}. The [*$\omega$-limit set*]{} $\omega(\gamma)$ and [*$\alpha$-limit set*]{} $\alpha(\gamma)$ of a [ complete trajectory]{} $\gamma$ are defined, respectively, as $$\omega(\gamma)=\{y:\,\,\,\mb{$\E$ $x_n\in A$ and $t_n\ra\8$ such that $\gamma(t_n)\ra y$}\},$$ $$\a(\gamma)=\{y:\,\,\,\mb{$\E$ $x_n\in A$ and $t_n\ra-\8$ such that $\gamma(t_n)\ra y$}\}.$$
Let $S\subset X$. $S$ is said to be [*positively invariant*]{} (resp. [*invariant*]{}), if $\Phi(t)S\subset S$ (resp. $\Phi(t)S=S$) for all $t\geq0$. A compact invariant set ${{\mathcal A}}$ is called an [*attractor*]{}, if it attracts a neighborhood $U$ of itself, namely, $$\lim_{t\ra\8}d_H\(\Phi(t)U,{{\mathcal A}}\)=0.$$ The [*attraction basin*]{} of an attractor ${{\mathcal A}}$, denoted by ${{\mathscr U}}({{\mathcal A}})$, is defined as $${{\mathscr U}}({{\mathcal A}})=\{x:\,\,\lim_{t\ra\8}d(\Phi(t)x,\,{{\mathcal A}})=0\}.$$ \[r2.7\]By definition one easily verifies that the [attraction basin]{} ${{\mathscr U}}({{\mathcal A}})$ of an attractor ${{\mathcal A}}$ is open. Furthermore, for any trajectory $\gamma:J\ra X$ of $\Phi$ (where $J$ is an interval), it holds that $$\mb{ either
$\gamma(J)\subset {{\mathscr U}}({{\mathcal A}})$,\hs or\, $\gamma(J)\cap {{\mathscr U}}({{\mathcal A}})=\emp$}.$$
Conley index
------------
In this subsection we recall briefly some basic notions and results in the Conley index theory. The interested reader is referred to [@Conley; @Mis] and [@Ryba] for details.
Let $\Phi$ be a given local semiflow on $X$, and let $M$ be a subset of $X$. We say that [*$\Phi$ does not explode*]{} in $M$, if $T_x=\8$ whenever $\Phi([0,T_x))x\subset M.$ $M$ is said to be [*admissible*]{} (see [@Ryba], pp. 13), if for any sequences $x_n\in M$ and $t_n\ra \8$ with $\Phi([0,t_n])x_n\subset M$ for all $n$, the sequence $\Phi(t_n)x_n$ has a convergent subsequence. $M$ is said to be [*strongly admissible*]{}, if it is admissible and moreover, $\Phi$ does not explode in $M$.
\[d2.10\] $\Phi$ is said to be asymptotically compact on $X$, if each bounded subset $B$ of $X$ is [ strongly admissible]{}.
From now on we always assume that ([**AC**]{}) [*$\Phi$ is asymptotically compact on $X$.*]{} This requirement is fulfilled by a large number of examples from applications.
A compact invariant set $S$ of $\Phi$ is said to be [*isolated*]{}, if there exists a bounded closed neighborhood $N$ of $S$ such that $S$ is the maximal invariant set in $N$. Consequently $N$ is called an [*isolating neighborhood*]{} of $S$.
Let there be given an isolated compact invariant set $S$. A pair of bounded closed subsets $(N,E)$ is called an [*index pair*]{} of $S$, if (i) $N\sm E$ is an isolating neighborhood of $S$; (ii) $E$ is $N$-invariant, namely, for any $x\in E$ and $t\geq 0$, $$\Phi([0,t])x\subset
N\Longrightarrow\Phi([0,t])x\subset E;$$ (iii) $E$ is an exit set of $N$. That is, for any $x\in N$, if $\Phi(t_1)x\not\in N$ for some $t_1>0$, then there exists $0\leq t_0\leq t_1$ such that $\Phi(t_0)x\in E.$
Index pairs in the terminology of [@Ryba] need not be bounded. However, the bounded ones are sufficient for our purposes here.
(homotopy index) Let $(N,E)$ be an index pair of $S$. Then the [ homotopy Conley index]{} of $S$ is defined to be the homotopy type $[(N/E,[E])]$ of the pointed space $(N/E,[E])$, denoted by $h(\Phi,S)$.
Denote $H_*$ and $H^*$ the singular homology and cohomology theories with coefficient group $\mathbb Z$, respectively. Applying $H_*$ and $H^*$ to $h(\Phi,S)$ one obtains the [*homology*]{} and [*cohomology Conley index*]{} $CH_*(\Phi,S)$ and $CH^*(\Phi,S)$, respectively.
An important property of the Conley index is its continuation property. Here we state a result in this line for the reader’s convenience, which is actually a particular case of [@Ryba], Chap.1, Theorem 12.2.
Let $\Phi_\lam$ be a family of semiflows with parameter $\lam\in \Lam$, where $\Lam$ is a connected compact metric space. Assume $\Phi_\lam(t)x$ is continuous in $(t,x,\lam)$. Denote $\~\Phi$ the [*skew-product flow*]{} of the family $\Phi_\lam$ on $X\X \Lam$ defined as follows: \[e2.3\] \~(t)(x,)=$\Phi_\lam(t)x,\lam$,(x,)X. \[t:2.14\] Suppose $\~\Phi$ satisfies the assumption (AC) on $X\X\Lam$. Let ${{\mathcal S}}$ be a compact isolated invariant set of $\~\Phi$. Then $h(\Phi_\lam,S_\lam)$ is constant for $\lam\in\Lam$, where $S_\lam=\{x:\,\,(x,\lam)\in {{\mathcal S}}\}$ is the $\lam$-section of ${{\mathcal S}}$.
[**Proof.**]{} Take a bounded closed isolating neighborhood ${{\mathcal U}}$ of ${{\mathcal S}}$ in $X\X\Lam$. Then the $\lam$-section ${{\mathcal U}}_\lam$ of ${{\mathcal U}}$ is an isolating neighborhood of $S_\lam$. Since ${{\mathcal S}}$ is compact in $X\X\Lam$, one easily verifies that $S_\lam$ is upper semicontinuous in $\lam$, namely, $d_{\mbox{\tiny H}}(S_{\lam'},S_\lam)\ra0$ as $\lam'\ra\lam$. Consequently for each fixed $\lam\in\Lam$, ${{\mathcal U}}_\lam$ is also an isolating neighborhood of $S_{\lam'}$ for $\lam'$ near $\lam$. Now the conclusion directly follows from [@Ryba], Chap.1, Theorem 12.2. $\Box$
Finally, let us also recall the concept of an isolating block.
Let $B\subset X$ be a bounded closed set and $x\in \pa B$ be a boundary point. $x$ is called a [*strict egress*]{} (resp. [*strict ingress*]{}, [*bounce-off*]{}) point of $B$, if for every trajectory $\gamma:[-\tau,s]\ra X$ with $\gamma(0)=x$, where $\tau\geq0$, $s>0$, the following two properties hold.
1. There exists $0<\ve<s$ such that $$\gamma(t)\not\in B \,\,\,(\mb{resp. }\, \gamma(t)\in \mb{int}B,\,\,\,\mb{resp. }\,\gamma(t)\not\in B),\Hs \A\,t\in (0,\ve);$$
2. If $\tau>0$, then there exists $0<\de<\tau$ such that $$\gamma(t)\in \mb{int}B \,\,\,(\mb{resp. }\, \gamma(t)\not\in B,\,\,\,\mb{resp. }\,\gamma(t)\not\in B),\Hs \A\,t\in (-\de, t).$$
Denote $B^e$ (resp. $B^i$, $B^b$) the set of all strict egress (resp. strict ingress, bounce-off) points of the closed set $B$, and set $B^-=B^e\cup B^b$.
A closed set $B\subset X$ is called an [*isolating block*]{} if $B^-$ is closed and $
\pa B=B^i\cup B^-.$ It is well known that if $B$ is a bounded isolating block, then $(B,B^-)$ is an index pair of the maximal compact invariant set $S$ (possibly empty) in $B$.
For convenience, if $B$ is an isolating block, we call $B^-$ the [*boundary exit set*]{}.
Local Invariant Manifolds
=========================
In this section we present some fundamental results on local invariant manifolds of (\[e:1.1\]). We also state a slightly modified version of a reduction property of the Conley index given in [@Ryba].
It is well known that under the hypotheses in Section 1, the initial value problem of (\[e:1.1\]) is well-posed in $X^\a$. That is, for each $u_0\in X^\a$ the problem has a unique solution $u(t)$ in $X^\a$ with $u(0)=u_0$ on some maximal existence interval $[0,T)$; see e.g. [@Henry], Theorem 3.3.3.
Denote $\Phi_\lam$ the local semiflow generated by the problem on $X^\a$. For convenience in statement, given $Z\subset \mathbb{C}$ and $\a\in\R$, we will write $$\mb{Re}(Z)<\a\,(\,>\a),$$ which means that $
\mb{Re}(\mu)< \a\,(\,>\a) $ for all $\mu\in Z.$
Let $L_{\lam}=A-Df_\lam(0)$. Suppose there exist a neighborhood $J_0=[\lam_0-\eta,\,\lam_0+\eta]$ of $\lam_0\in\R$ and $\de>0$ such that the following hypotheses are fulfilled.
1. The spectral $\sig(L_\lam)$ has a decomposition $\ba{ll}\sig(L_\lam)=\Cup_{1\leq i\leq 3}\,\sig_\lam^i\ea $ with \[e:2.4\] (\_\^1)< -\_1<-\_2(\_\^2) < \_3<\_4< (\_\^3) for $\lam\in J_0$, where $\a_i$ ($1\leq i\leq 4$) are positive constants independent of $\lam$.
2. For each $\lam\in J_0$, $X$ has a decomposition $X=X^1_\lam\oplus X^2_\lam\oplus X^3_\lam\,$ corresponding to the spectral decomposition in (H1), where $X^i_\lam$ $(i=1,2,3)$ are $L_{\lam}$-invariant subspaces of $X$. Moreover, $$\mb{dim}\,\,(X^1_\lam),\, \mb{dim}\,(X^2_\lam)<\8.$$
3. There is a family of invertible bounded linear operators $T=T_\lam$ on $X$ depending continuously on $\lam$ such that when $\lam\in J_0$, we have \[e:2.a4\] T X\^i\_=X\^i\_[\_0]{}:=X\^i,i=1,2,3.
Instead of (H3), a more natural hypothesis is to assume that
1. the projection operators $P_\lam^{i}:X\ra X^i_\lam$ ($i=1,2$) are continuous in $\lam$. Indeed, when (H3)$'$ is fulfilled, it can be shown that there is a family of invertible bounded linear operators $T=T_\lam$ on $X$ such that holds true; see [@LW2], Appendix A for details.
We rewrite $E=X^\a$ and set $$\ba{ll}
E^i=E\cap X^i,\hs E^{ij}=E\cap \(X^i\oplus X^j\),\ea$$ where $i,j=1,2,3$ ($i\ne j$). Then $$E=E^2\oplus E^{13}=E^3\oplus E^{12}.$$ Since $\mb{dim}\,\,(X^1_\lam),\, \mb{dim}\,(X^2_\lam)<\8$, we have $E^1=X^1,$ $E^2=X^2.$ \[l:2.1\] Assume (H1)-(H3) are fulfilled. Then
1. there exist an open convex neighborhood $W$ of $0$ in $E^2$ and a mapping $\xi=\xi_{\lam}(w)$ from $W\X J_0$ to $E^{13}$ which is continuous in $(w,\lam)$ and differentiable in $w$, such that for each $\lam\in J_0$, \[e:2.5\] [[M]{}]{}\^2\_:=T\^[-1]{}M\^2\_,M\^2\_:={w+\_(w):wW}, is a local invariant manifold of the system (\[e:1.1\]); and
2. there exist an open convex neighborhood $V$ of $0$ in $E^{12}$ and a mapping $\zeta=\zeta_{\lam}(v)$ from $V\X J_0$ to $E^3$ which is continuous in $(v,\lam)$ and differentiable in $v$, such that for each $\lam\in J_0$, \[e:2.5b\] [[M]{}]{}\^[12]{}\_:=T\^[-1]{}M\^[12]{}\_,M\^[12]{}\_:={v+\_(v):vV}, is a local invariant manifold of the system (\[e:1.1\]).
[**Proof.**]{} The above results are just slight modifications of the existing ones in the literature; see e.g. [@Ryba], Chap. II, Theorem 2.1. Here we give a sketch of the proof for the reader’s convenience.
Let $B_\lam=T L_\lam T^{-1}$, and define $$g_\lam(v)=T \(f_\lam(T^{-1} v)-Df_\lam(0)(T^{-1} v)\),\Hs v\in E.$$ Setting $u=T^{-1} v$, the system (\[e:1.1\]) can be transformed into an equivalent one: \[e:2.6\] v\_t+B\_v=g\_(v).It is trivial to check that $||Dg_\lam(v)||\ra 0$ as $||v||_\a\ra 0$ uniformly with respect to $\lam\in J_0$. Further by the Mean-value Theorem one easily verifies that for any $\ve>0$, there exists a neighborhood $U$ of $0$ in $E$ such that \[e:2.7\] ||g\_(u)-g\_(v)||||u-v||\_,u,vU,J\_0.
We observe that $$B_\lam-\mu I=T L_\lam T^{-1}-\mu I=T (L_\lam-\mu I)T^{-1},$$ where $I=id_X$ is the identity mapping on $X$, from which it can be easily seen that $\mu\in \mathbb{C}$ is a regular value of $B_\lam$ if and only if it a regular value of $L_\lam$. Hence one concludes that $$\sig(B_\lam)=\sig(L_\lam).$$ Since $X^i_\lam$ ($i=1,2,3$) are $L_{\lam}$-invariant, it follows by (\[e:2.a4\]) that $X^i$ are $B_\lam$-invariant for all $\lam\in J_0$. Now using some standard argument in the geometric theory of PDEs (see Henry [@Henry], Sect. 6 and Hale [@Hale], Appendix) and the uniform contraction principle, it can be shown that there exist an open convex neighborhood $W$ of $0$ in $E^2$ and a mapping $\xi=\xi_{\lam}(w)$ from $W\X J_0$ to $E^{13}$ which is continuous in $(w,\lam)$ and differentiable in $w$, such that for each $\lam\in J_0$, \[e:3c\] M\^2\_:={w+\_(w):wW} is a local invariant manifold of the system (\[e:2.6\]). Consequently ${{\mathcal M}}^2_\lam=T^{-1}M^2_\lam$ is a local invariant manifold of (\[e:1.1\]).
The proof of the part (2) follows a fully analogous argument. $\Box$
Let ${{\mathcal M}}^2_\lam$ and ${{\mathcal M}}_\lam^{12}$ be the local invariant manifolds given in Lemma \[l:2.1\], and $\Phi_\lam^2$ and $\Phi_\lam^{12}$ be the restrictions of $\Phi_\lam$ on ${{\mathcal M}}_\lam^2$ and ${{\mathcal M}}_\lam^{12}$, respectively, where $\Phi_\lam$ is the local semiflow generated by .
The following result is a parameterized version of [@Ryba], Chap. II, Theorem 3.1, and can be proved in the same manner as in [@Ryba]. We omit the details. \[l:3.2\] Assume (H1)-(H3). Then there exist a neighborhood $U$ of $0$ in $E$ and a number $\ve>0$ such that for every $\lam\in[\lam_0-\ve,\lam_0+\ve]$,
2. $K\subset U$ is a compact invariant set of $\Phi_\lam$ iff it is a compact invariant set of $\Phi_\lam^2$ (resp. $\Phi_\lam^{12}$) on ${{\mathcal M}}_\lam^2$ (resp. ${{\mathcal M}}_\lam^{12}$); and
3. $K\subset U$ is an isolated invariant set of $\Phi_\lam$ iff it is an isolated invariant set of $\Phi_\lam^2$ (resp. $\Phi_\lam^{12}$) on ${{\mathcal M}}_\lam^2$ (resp. ${{\mathcal M}}_\lam^{12}$); furthermore, $$h\(\Phi_\lam,K \)= h\(\Phi_\lam^{12},K \)=\Sigma^m\wedge h\(\Phi_\lam^2,K \),$$ where $m=\mb{dim}\,(X^1)$ is the dimension of $X^1$. -1cm
Local dynamic bifurcation
=========================
In this section we state and prove some local dynamic bifurcation results concerning (\[e:1.1\]) in terms of invariant sets, so we always assume $$n:=\mb{dim}\,(X^2)\geq 1.$$
In what follows, by a [*$k$-dimensional topological sphere*]{} we mean the boundary $\pa D$ of any contractible open subset $D$ of a $(k+1)$-dimensional manifold ${{\mathcal M}}$ without boundary. We will use the notation $\mathbb{S}^k$ to denote any $k$-dimensional topological sphere.
$\mu\in\R$ is called a (dynamic) [bifurcation value]{} of , if for any neighborhood $U$ of $0$ and $\ve>0$, there exists $\lam\in (\mu-\ve,\mu+\ve)$ such that $\Phi_{\lam}$ has a compact invariant set $K_\lam\subset U$ with $K_\lam\sm\{0\}\ne\emp$.
If $\mu$ is a bifurcation value, then we call $(0,\mu)$ a (dynamic) [bifurcation point]{}.
We are basically interested in the bifurcation phenomena of the system (\[e:1.1\]) near a bifurcation value $\lam=\lam_0$. So in addition to (H1)-(H3), we will also assume
4. there exists $\ve_0>0$ such that for $\lam\in[\lam_0-\ve_0,\,\lam_0+\ve_0]$, $$\mb{Re}\,(\sig_\lam^2)<0\,\,(\mb{if }\lam<\lam_0),\hs \mb{Re}\,(\sig_\lam^2)>0\,\, (\mb{if }\lam>\lam_0).$$
Let ${{\mathcal M}}^2_\lam$ and ${{\mathcal M}}_\lam^{12}$ be the local invariant manifolds given in Lemma \[l:2.1\], and $\Phi_\lam^2$ and $\Phi_\lam^{12}$ be the restrictions of $\Phi_\lam$ on ${{\mathcal M}}_\lam^2$ and ${{\mathcal M}}_\lam^{12}$, respectively.
[**Convention.**]{} [*For simplicity in statement, from now on we set $\lam_0=0$.*]{}
Attractor/repeller bifurcation
------------------------------
In this subsection we give an attractor/repeller-bifurcation theorem, which slightly generalizes some fundamental results in Ma and Wang [@MW1 Theorem 6.1] and [@MW3 Theorem 4.3]. For the reader’s convenience, we also present a self-contained proof for the theorem.
\[t:2.2\] Assume (H1)-(H4) are fulfilled (with $\lam_0=0$).
Suppose $0$ is an attractor (resp. repeller) of $\Phi_0^2$. Then there exists a closed neighborhood $U$ of $0$ in $E$ and a number $\ve>0$ such that for each $\lam\in[-\ve,0)$ (resp. $ (0,\ve]$), the system $\Phi_\lam$ bifurcates from $0$ a maximal compact invariant set $K_\lam\ne\emp$ in $U\sm\{0\}$ which contains an invariant topological sphere $\mathbb{S}^{n-1}$. Furthermore, \[e4.5b\]\_[0]{}d\_H$K_\lam,S_0$=0.
[**Proof.**]{} [Case 1)]{} $0$ is an attractor of $\Phi_0^2$. We first consider the equivalent system (\[e:2.6\]) for $\lam\in J_0$. When (\[e:2.6\]) is restricted on the local center manifold $M^2_\lam$ defined by (\[e:3c\]), it reduces to an ODE system on an open neighborhood $W$ (independent of $\lam$) of $0$ in $E^2$ : \[e:2.9\] w\_t=-B\_\^2 w+P\^2g\_(w+\_(w)):=F\_(w),where $B_\lam^2=P^2 B_\lam$, and $P^2$ is the projection from $E=X^\a$ to $E^2$. Applying Lemma \[l:3.2\] to (\[e:2.6\]) one deduces that there exist a neighborhood ${{\mathcal U}}$ of $0$ in $E$ and $\ve_0>0$ such that for $\lam\in[-\ve_0,\ve_0]$, $S$ is an isolated invariant set of (\[e:2.6\]) in ${{\mathcal U}}$ iff it is an isolated invariant set of the system restricted on the manifold $M^2_\lam$.
Denote $\phi_\lam$ the local semiflow on $W$ generated by (\[e:2.9\]). Since $0$ is an attractor of $\Phi_0^2$, we find that $S_0:=\{0\}$ is an attractor of $\phi_0$. Let $\Omega={{\mathscr U}}(S_0)$ be the attraction basin of $S_0$ in $W$ with respect to $\phi_0$. Then by converse Lyapunov theorem on attractors (see e.g. [@Li1 Theorems 3.1 and 3.2]), one can find a function $V\in C^\8(\Omega)$ with $V(0)=0$ and $\lim_{x\ra\pa\Omega}V(x)=+\8$ such that \[e:2.10\] V(x). F\_[0]{}(x)-v(x),x.where $v\in C(\Omega)$ and $v(x)>0$ for $x\ne0$. Let $$N=V_a:=\{x\in\Omega:\,\,V(x)\leq a\}.$$ Then $N$ is a compact neighborhood of $0$ in $E^2$. Pick two numbers $a, \rho>0$ sufficiently small so that \[e4.4\]:=N\_[E\^[13]{}]{}$\xi_0(N),\rho$[[U]{}]{},where $\xi_0$ is the mapping determining the local center manifold $M_0^2$ given in Lemma \[l:2.1\], and ${\mb{B}}_{E^{13}}\(\xi_0(N),\rho\)$ denotes the $\rho$-neighborhood of $\xi_0(N)$ in $E^{13}$.
By (\[e:2.10\]) we have \[V1\] V(x). F\_[0]{}(x)-,xN, where $\mu=\min_{x\in \pa N}v(x)>0$, and $\pa N$ is the boundary of $N$ in $E^2$. Further by the continuity of $F_\lam$ in $\lam$, there exists $0<\ve_1\leq \ve_0$ such that \[e:3.b\] V(x). F\_(x)-,xN for $\lam\in[-\ve_1,\ve_1]$, which implies that $N$ is a positively invariant set of $\phi_\lam$.
It can be assumed that $\ve_1$ is sufficiently small so that \[e:3.c\] \_(N)\_[E\^[13]{}]{}$\xi_0(N),\rho$,.
Now assume $\lam\in[-\ve_1,0)$. Consider the inverse flow $\phi_\lam^-$ of $\phi_\lam$ generated by the system \[e:2.11\] w\_t=-F\_(w):=B\_\^2 w-P\^2g\_(w+\_(w)).By (H4) we find that $\mb{Re\,}(\sig(B_\lam^2))<0,$ which implies that $S_0$ is an attractor of $\phi_\lam^-$. Let $G_\lam={{\mathscr U}}(S_0)$ be the attraction basin of $S_0$ in $W$ with respect to $\phi_\lam^{-}$. We infer from (\[e:3.b\]) that each $x\in \pa N$ is a strict ingress point of $\phi_\lam$, and hence is a strict egress point of $\phi_\lam^-$. Thus one necessarily has $G_\lam\subset N$. Therefore the boundary $\pa G_\lam$ of $G_\lam$ in $E^2$ is contained in $N$; see Fig. \[fg3-1\].
![Attractor-bifurcation[]{data-label="fg3-1"}](fig3-1.pdf "fig:"){width="8.5cm"}-0.3cm
We prove that $\pa G_\lam$ is an invariant set of $\phi_\lam^-$. For this purpose, it suffices to show that for each $x_0\in \pa G_\lam$, there is a complete trajectory $w(t)$ of $\phi_\lam^-$ (i.e., a solution of ) with $w(0)=x_0$ such that $w(t)\in \pa G_\lam$ for all $t\in\R$.
Note that always has a unique solution $w(t)$ defined on a maximal existence interval $J$ such that $w(0)=x_0$. Since $x_0\not\in G_\lam$, by Remark \[r2.7\] we deduce that $w(t)\not\in G_\lam$ for all $t\in J$. We claim that $w(t)\in \pa G_\lam$ for $t\in J$, and consequently one also has $J=\R$, thus completing the proof of the invariance of $\pa G_\lam$. We argue by contradiction and suppose the claim was false. Then there would exist $t_0\in J$ such that $w(t_0)\not\in \ol{G}_\lam$. Hence $d\(w(t_0),\ol{G}_\lam\)>0$. Take a sequence $x_k\in G_\lam$ such that $x_k\ra x_0$. Let $w_k(t)$ by the solution of with $w_k(0)=x_k$. Then by continuity properties on ODEs, we know that $t_0$ belongs to the maximal existence interval $J_k$ of $w_k(t)$ if $k$ is sufficiently large; furthermore, $w_k(t_0)\not\in \ol{G}_\lam$. But by Remark \[r2.7\], this leads to a contradiction since $w_k(0)=x_k\in G_\lam$.
Denote $A_\lam$ the maximal compact invariant set of $\phi_\lam$ in $N\sm G_\lam$. Clearly $\pa G_\lam\subset A_\lam$. It is trivial to check that $A_\lam$ is the maximal compact invariant set of $\phi_\lam$ in $N\sm S_0$. Since $N$ is an isolating neighborhood of $S_0$ with respect to $\phi_0$, by a simple argument via contradiction it can be shown that \[e4.5a\]\_[0]{}d\_H$A_\lam,S_0$=0. We claim that $\pa G_\lam$ is an $(n-1)$-dimensional topological sphere. Indeed, define $$H(s,x)=\left\{\ba{ll}\phi_\lam^{-}\(\frac{s}{1-s}\)x,\Hs\, &s\in[0,1),\,\,x\in G_\lam;\\[1ex]
0,& s=1,\,\,x\in G_\lam.\ea\right.$$ Then $H$ is a strong deformation retraction shrinking $G_\lam$ to the point $0$. This shows that $G_\lam$ is contractible and proves our claim.
Now we define $$\widetilde{K}_\lam=\{w+\xi_\lam(w):\,\,w\in A_\lam\},\hs \widetilde{\mathbb{S}}=\{w+\xi_\lam(w):\,\,w\in\pa G_\lam\},$$ where $\xi_\lam$ is the mapping in given by Lemma \[l:2.1\]. By (\[e:3.c\]) and (\[e4.4\]) we find that $\widetilde{K}_\lam\subset \widetilde{U}\subset {{\mathcal U}}$. $\widetilde{K}_\lam$ is the maximal compact invariant set of the system (\[e:2.6\]) in $\widetilde{U}\setminus \{0\}$. It follows by that $\lim_{\lam\ra 0}d_H(\widetilde{K}_\lam,S_0)=0$. Finally, let $U_\lam=T^{-1}\widetilde{U},$ where $T=T_\lam$ is the linear operator in (H3). Then one can find a closed neighborhood $U$ of $0$ in $E$ and a number $0<\ve\leq\ve_1$ such that $U\subset U_\lam$ for all $\lam\in[-\ve,0)$. Set $ K_\lam=T^{-1}\widetilde{K}_\lam$. Then $\lim_{\lam\ra 0}d_H({K}_\lam,S_0)=0$. Thus we may assume that $\ve$ is chosen sufficiently small so that $K_\lam\subset \mb{int}\,U$ for all $\lam\in[-\ve,0)$. It is easy to see that $U$ and $K_\lam$ fulfill all the requirements of the theorem.
The equilibrium $0$ is a repeller of $\Phi_0^2$.
This case can be treated by replacing (\[e:2.9\]) and (\[e:2.11\]) with each other and repeating the above argument. We omit the details. $\Box$
Invariant-set bifurcation
-------------------------
Now we state and prove a general local invariant-set bifurcation theorem.
\[t:2.7\] Assume that (H1)-(H4) are fulfilled. Suppose $S_0=\{0\}$ is an isolated invariant set of $\Phi_0$. Then one of the following assertions holds.
1. $S_0$ is an attractor (resp. repeller) of $\Phi_0^2$. In such a case, the system undergoes an attractor-bifurcation (resp. repeller-bifurcation) in Theorem \[t:2.2\].
2. There exist a closed neighborhood $U$ of $0$ in $E$ and a two-sided neighborhood $I_2$ of $\lam_0$ such that $\Phi_\lam$ has a nonempty maximal compact invariant set $K_\lam$ in $U\sm S_0$ for each $\lam\in I_2\sm\{\lam_0\}$.
Furthermore, in both cases the bifurcating invariant set $K_\lam$ is upper semicontinuous in ${\lam}$ with \[e3.18a\]\_[0]{}d\_$K_\lam,\,0$=0.
[**Proof.**]{} Let us first verify the bifurcation results in (1) and (2). For this purpose, it suffices to assume $S_0$ is neither an attractor nor a repeller of $\Phi_0^2$ and prove that the second assertion (2) holds true.
Let us start with the local semiflow $\phi_\lam$ generated by the bifurcation equation (\[e:2.9\]) on $W$. Since $S_0=\{0\}$ is an isolated invariant set of $\Phi_0$, by Lemma \[l:3.2\] it is isolated for $\Phi_0^2$. Because $\Phi_\lam^2$ and $\phi_\lam$ are conjugate, one concludes that $S_0$ is an isolated invariant set of $\phi_0$.
Note that (H4) implies $$\mb{Re\,}(\sig(B_\lam^2))<0\,\,\,(\lam<0),\hs \mb{Re\,}(\sig(B_\lam^2))>0\,\,\,(\lam>0),$$ where $B_\lam^2$ is the linear operator in . Hence $S_0$ is a repeller of $\phi_\lam$ when $\lam<0$, and an attractor when $\lam>0$. By Lemma \[l:3.2\] we also have for some $\ve_1>0$ that \[e:4.5\] h$\phi_\lam,S_0$=\^[n]{}().
Pick a closed neighborhood $W_0$ of $S_0$ in $E^2$ such that it is an isolating neighborhood of $S_0$ with respect to $\phi_0$. Then by a simple argument via contradiction, we deduce that $W_0$ is also an isolating neighborhood of the maximal compact invariant set $S_\lam$ of $\phi_\lam$ in $W_0$ provided $\lam$ is sufficiently small; furthermore, \[e3.18\] \_[0]{}d\_$S_\lam,\,S_0$=0.
Fix a positive number $\ve<\ve_1$ such that $W_0$ is an isolating neighborhood of $S_\lam$ for all $\lam\in [-\ve,\ve]$. Then Theorem \[t:2.14\] asserts that \[e:2.13\] h(\_,S\_)const.,.
In what follows we show that \[e:4.a1\]h$\phi_0,S_0$\^[0]{}.
Since $S_0$ is an isolated invariant set of $\phi_0$, by [@CE], Theorem 1.5, one can find a connected isolating block $B$ of $S_0$ (with respect to $\phi_0$) with smooth boundary $\pa B$. We claim that $B^-\ne \emp$, where $B^-$ is the boundary exit set of $B$ with respect to the flow $\phi_0$. Indeed, if $B^-= \emp$ then $B$ is positively invariant under the system $\phi_0$. Because $S_0$ is the maximal compact invariant set of $\phi_0$ in $B$, one easily deduces that it is an attractor of $\phi_0$, which contradicts the assumption that $S_0$ is not an attractor of $\Phi_0^2$ (recall that $\Phi_\lam^2$ and $\phi_\lam$ are conjugate).
Denote $H_*$ the singular homology theories with coefficient group $\mathbb Z$. Then $h\(\phi_0,S_0\)=[(B/B^-,[B^-])]$. Therefore $$H_0(h\(\phi_0,S_0\))=H_0((B/B^-,[B^-]))=H_0(B,B^-).$$ As $B$ is path-connected and $B^-\ne\emp$, by the basic knowledge in the theory of algebraic topology we find that $H_0(B,B^-)=0$. Consequently $
H_0(h\(\phi_0,S_0\))=0$. On the other hand, recalling that $\Sigma^{0}$ is the homotopy type of any pointed space $(\{p,q\},q)$ consisting of exactly two distinct points $p$ and $q$, one has $$H_0(\Sigma^{0})=H_0((\{p,q\},q))=\mathbb{Z}.$$ Hence we see that holds true.
Now assume $\lam\in(0,\ve]$. Combining (\[e:4.5\]) and (\[e:2.13\]) it yields $$h(\phi_\lam,S_\lam)=h(\phi_0,S_0)\ne h\(\phi_\lam,S_0\),$$ which implies that $S_\lam\sm S_0\ne\emp.$ Recall that $S_0$ is an attractor of $\phi_\lam$. Let $$R_\lam=\{x\in S_\lam:\,\,\omega(x)\cap S_0=\emp\}.$$ Then $R_\lam$ is a nonempty compact invariant set of $\phi_\lam$ with $(R_\lam,S_0)$ being a repeller-attractor pair of $S_\lam$; see [@Ryba], pp.141. Because $S_\lam$ is maximal in $W_0$, it can be easily seen that $R_\lam$ is precisely the maximal compact invariant set in $W_0\sm S_0$.
Consider the inverse flow $\phi_\lam^-$ of $\phi_\lam$ on $W$. Then we have \[e:4.5b\] h$\phi_\lam^-,S_0$=\^[0]{}(). Since $S_0$ is a repeller of $\phi_\lam$ for $\lam\in[-\ve,0)$, it is an attractor of $\phi_\lam^-$. Repeating the same argument above with $\phi_\lam$ replaced by $\phi_\lam^-$, one immediately deduces that $\phi_\lam$ has a nonempty maximal compact invariant set $R_\lam$ in $W_0\sm S_0$ for $\lam\in[-\ve,0)$.
We show that $R_\lam$ is upper semicontinuous in $\lam$. We only consider the case where $\lam\in(0,\ve]$. The argument for the case where $\lam\in[-\ve,0)$ can be performed in the same manner by considering the inverse flow $\phi_\lam^-$, and we omit the details.
Let ${{\mathscr U}}_\lam={{\mathscr U}}(S_0)$ be the attraction basin of $S_0$ in $W$ with respect to $\phi_\lam$. For each fixed $\lam> 0$, pick a number $r>0$ such that $\ol{\mb{B}}_r\subset {{\mathscr U}}_\lam$, where (and below) ${\mb{B}}_r$ denotes the ball in $E^2$ centered at $0$ with radius $r$. Then by the stability property of attraction basins (see e.g. Li [@Li0 Theorem 2.9]), there exists $\rho>0$ such that $\ol{\mb{B}}_{r/2}\subset {{\mathscr U}}_{\lam'}$ provided $|\lam'-\lam|\leq\rho$. This implies that $$\ba{ll}R_{\lam'}\cap \ol{\mb{B}}_{r/2}=\emp\ea$$ for all $\lam'\in(0,\ve]$ with $|\lam'-\lam|\leq\rho$. We check that $
\lim_{\lam'\ra\lam}d_{\mbox{\tiny H}}\(R_{\lam'},\,R_\lam\)=0,
$ thus proving what we desired.
Suppose the contrary. There would exist $\lam_k\ra\lam$ and $\de_0>0$ such that $$d_{\mbox{\tiny H}}\(R_{\lam_k},\,R_\lam\)\geq \de_0,\Hs\A\,k\geq 1.$$ We may assume $|\lam_k-\lam|\leq\rho$, hence $R_{\lam_k}\subset W_0\sm{\mb{B}}_{r/2}$ for all $k$. Thanks to Lemma \[l:2.3\], it can be assumed that $R_{\lam_k}$ converges to a nonempty compact subset $R'_\lam$ of $W_0\sm{\mb{B}}_{r/2}$ in the sense of Hausdorff distance $\de_{\mbox{\tiny H}}(\.,\.)$. Then $d_{\mbox{\tiny H}}\(R'_\lam,\,R_\lam\)\geq \de_0$. On the other hand, one trivially verifies that $R'_\lam$ is an invariant set of $\phi_\lam$. Thus $R_\lam\cup R'_\lam$ is a compact invariant set of $\phi_\lam$ in $W_0\sm S_0$. This contradicts the maximality of $R_\lam$ in $W_0\sm S_0$.
We are now ready to complete the proof of the theorem. Let $U$ be the neighborhood of $0$ given in Lemma \[l:3.2\]. We may restrict $U$ sufficiently small in advance so that $P^2T_\lam U\subset W_0$ for all $\lam\in[-\ve,\ve]$, where $P^2:E\ra E^2$ is the projection, and $T_\lam$ is the operator in (H3). Let $K_\lam=T_\lam^{-1}\widetilde{R}_\lam$, where $$\widetilde{R}_\lam=\{w+\xi_\lam(w):\,\,w\in R_\lam\}.$$ Then $K_\lam$ is upper semicontinuous in $\lam$ and is a compact invariant set of $\Phi_\lam$. By we have $\lim_{\lam\ra0}d_{\mbox{\tiny H}}\(R_\lam,\,S_0\)=0$. It follows that $\lim_{\lam\ra0}d_{\mbox{\tiny H}}(K_\lam,\,S_0)=0$. Thus one can assume $\ve$ is chosen sufficiently small so that $K_\lam\subset U$ for $\lam\in[-\ve,\ve]$. We claim that $K_\lam$ is the maximal compact invariant set of $\Phi_\lam$ in $U\sm S_0$, which completes the proof of the theorem. Indeed, if this was false, then $\Phi_\lam$ would have another compact invariant set $K'_\lam\subset U\sm S_0$ such that $K_\lam \varsubsetneq K'_\lam$. It follows that $$R_\lam=P^2T_\lam K_\lam \varsubsetneq P^2T_\lam K'_\lam:=R'_\lam.$$ By the invariance of $K'_\lam$ it is easy to deduce that $R'_\lam$ is a compact invariant set of $\phi_\lam$ in $W_0\sm S_0$. However, this contradicts the maximality of $R_\lam$ in $W_0\sm S_0$. $\Box$
Some remarks on static bifurcation
----------------------------------
It is worth noticing that Theorem \[t:2.7\] may also give us information on the static bifurcation of the system in some cases. For example, if the stationary problem \[3.27\] Au=f\_(u), uE:=X\^ has a variational structure, then (\[e:1.1\]) is a gradient-like system, and each nonempty compact invariant set $K$ of $\Phi_\lam$ contains at least one equilibrium point, which is precisely a solution of (\[3.27\]). On the other hand, it is also easy to see that if $K$ consists of at least two distinct points, then it contains at least two distinct equilibrium points of $\Phi_\lam$. Thus under the hypotheses of Theorem \[t:2.7\], one immediately concludes that either there is a one-sided neighborhood $I_1$ of $\lam_0$ such that (\[3.27\]) bifurcates two distinct nontrivial solutions for each $\lam\in I_1\sm\{\lam_0\}$, or there is a two-sided neighborhood $I_2$ of $\lam_0$ such that (\[3.27\]) bifurcates at least one nontrivial solution for each $\lam\in I_2\sm\{\lam_0\}$.
We refer the interested reader to [@CW; @Rab2; @RSW] and [@SW], etc. for more detailed bifurcation results on such operator equations.
As another example, we consider the particular but important case where $$n=\mb{dim}(X^2)=1.$$ We first claim that each compact invariant set $C_\lam$ of $\Phi_\lam$ close to $0$ contains at least one equilibrium point which is a solution of (\[3.27\]). Indeed, each such invariant set $C_\lam$ is contained in the local invariant manifold ${{\mathcal M}}_\lam^2$. Because ${{\mathcal M}}_\lam^2$ is a $C^1$ curve, every connected component $\ell$ of $C_\lam$ is a segment of ${{\mathcal M}}_\lam^2$. Since (\[e:1.1\]) reduces to a one-dimensional ODE on ${{\mathcal M}}_\lam^2$ (hence backward uniqueness holds on ${{\mathcal M}}_\lam^2$), by invariance of $\ell$ it is trivial to deduce that the end points of $\ell$ are equilibria of $\Phi_\lam$.
Using the above basic fact, we can also easily verify that $0$ is an isolated solution of (\[3.27\]) at $\lam_0$ if and only if $S_0=\{0\}$ is an isolated invariant set of $\Phi_{\lam_0}$. Thanks to Theorem \[t:2.7\], one immediately obtains the following bifurcation result, which generalizes Henry [@Henry], Theorem 6.3.2.
\[t:3.6\] Assume (H1)-(H4) are fulfilled with $\mb{dim}\,(X^2)=1$. Then one of the following alternatives occurs.
1. There is a sequence $u_k$ of nontrivial solutions of (\[3.27\]) at $\lam=\lam_0$ such that $u_k\ra 0$ as $k\ra\8$.
2. There is a one-sided neighborhood $I_1$ of $\lam_0$ such that (\[3.27\]) bifurcates at least two nontrivial solutions for each $\lam\in I_1\sm\{\lam_0\}$.
3. There is a two-sided neighborhood $I_2$ of $\lam_0$ such that (\[3.27\]) bifurcates at least one nontrivial solution for each $\lam\in I_2\sm\{\lam_0\}$.
When $\mb{dim}\,(X^2)=1$ we can also use the classical Crandall-Rabinowitz Theorem (see [@Kie], Theorem I.5.1) to derive more explicit static bifurcation results under some additional assumptions such as the [*transversality condition*]{}. (Some nice bifurcation results when the transversality condition mentioned above is violated can be found in [@Shi] etc.) Other general bifurcation theorems such as the Krasnosel’skii Bifurcation Theorem (see [@Kie], Theorem II.3.2) also apply to deal with this special case.
Whether the bifurcating invariant set $K_\lam$ contains equilibrium solutions is an interesting problem. In the case of attractor-bifurcation this problem has already been addressed by Ma and Wang [@MW1] (pp. 155, Theorem 6.1), where one can find an index formula on equilibrium solutions. For the general case treated here, results in this line will be reported in our forthcoming paper entitled “Equilibrium index of invariant sets and global static bifurcation for nonlinear evolution equations”.
Nontriviality of the Conley Indices of the Bifurcating Invariant Sets
=====================================================================
Our main goal in this section is to show that the bifurcating invariant set $K_\lam$ in Theorem \[t:2.7\] has nontrivial Conley index. This result will play a crucial role in establishing our global dynamic bifurcation theorem. However, it may also be of independent interest in its own right.
Let $m=\mb{dim}\,(X^1)$, $n=\mb{dim}\,(X^2)$ ($n\geq1$), and let $K_\lam$ be the bifurcating invariant set of $\Phi_\lam$ in Theorem \[t:2.7\]. \[t:4.1\]Suppose (H1)-(H4) are fulfilled (with $\lam_0=0$), and that $S_0=\{0\}$ is an isolated invariant set of $\Phi_0$. Then there exists $\ve>0$ such that
1. if $h(\Phi_0,S_0)\ne \Sigma^{m+n}$, then \[3.28\]h(\_, K\_)0,.
[**Proof.**]{} Let $U$ be the neighborhood of $0$ given in Theorems \[t:2.2\] and \[t:2.7\]. Since $S_0$ is an isolated invariant set of $\Phi_0$, we can pick an $\ve>0$ sufficiently small such that $U$ is an isolating neighborhood of the maximal compact invariant set $S_\lam$ of $\Phi_\lam$ for all $\lam\in[-\ve,\ve]$. We may also assume that $U$ and $\ve$ are chosen sufficiently small so that Lemma \[l:3.2\] applies.
\(1) Assume $h(\Phi_0,S_0)\ne \Sigma^{m+n}.$ Let $\lam\in[-\ve,0)$. Then by (H1) and (H4), $$h(\Phi_\lam,S_0)=\Sigma^{m+n}\ne h(\Phi_0,S_0).$$ and the system bifurcates in $U\sm S_0$ a maximal compact invariant set $K_\lam$. By Lemma \[l:3.2\] one has $$h(\Phi_\lam, K_\lam)=h(\Phi_\lam^{12}, K_\lam).$$ Therefore to prove (\[3.28\]) we need to check that $h(\Phi_\lam^{12}, K_\lam)\ne \ol 0$.
Choose an isolating block $N=N_\lam$ of $S_\lam$ in ${{\mathcal M}}_\lam^{12}$. Since $S_0$ is a repeller of $\Phi_\lam^{12}$ on ${{\mathcal M}}_\lam^{12}$ (by (H4)), one can find an isolating block $N_0$ of $S_0$ in ${{\mathcal M}}_\lam^{12}$ (depending upon $\lam$) with $K_\lam\cap N_0=\emp$ such that $N_0^-=\pa N_0$, where $\pa N_0$ is the boundary of $N_0$ in ${{\mathcal M}}_\lam^{12}$. Then $M=N\sm\mb{int} N_0$ is an isolating block of $K_\lam$; see Fig.5.1.
-0.8cm
{width="4.3cm"} {width="4.3cm"}
Figure 5.1: $\lam<0$ Figure 5.2: $\lam>0$
As $h(\Phi_\lam,S_0)=\Sigma^{m+n}$, one finds that \[e5.a1\] S\^[m+n]{}N\_0/N\_0= N/M(N/N\^-)/M, where $\~M=\pi_{N^-}(M)$, and $\pi_{N^-}:N\ra N/N^-$ is the projection. Now let us argue by contradiction and suppose that $h(\Phi_\lam^{12}, K_\lam)=\ol0$. Noticing that $(M/N^-,[N^-])\cong (\~M,[N^-])$ (here we have used the same notation $[N^-]$ to denote both the base points in $M/N^-$ and $N/N^-$), we deduce that $$[(\~M,[N^-])]=\left[(M/N^-,[N^-])\right]=h(\Phi_\lam^{12}, K_\lam)=\ol0,$$where “$\left[\,\.\,\right]$" denotes homotopy type. This implies that $\~M$ is contractible.
By a standard argument one can easily show that $\pa N_0$ is a strong deformation retract of $N_0\sm S_0$. Consequently $M$ is a strong deformation retract of $N\sm S_0$. It then follows that $\~M$ is a strong deformation retract of $(N\sm S_0)/N^-$. Hence by [@Ryba] Chap.I, Pro.3.6, we deduce that the pair $(N/N^-,\~M)$ has the homotopy extension property. Further by Lemma \[l2.4\] and it holds that \[e5.3\] N/N\^-(N/N\^-)/MS\^[m+n]{}.
On the other hand, by the continuation property of the index we have$$h(\Phi_\lam,S_\lam)= h(\Phi_0,S_0)\ne \Sigma^{m+n},\Hs \lam\in[-\ve,0).$$ Since $h(\Phi_\lam,S_\lam)=h(\Phi_\lam^{12},S_\lam)$, one finds that $$h(\Phi_\lam^{12},S_\lam)=\left[(N/N^-,[N^-])\right]\ne \Sigma^{m+n}.$$ This implies that $
N/N^-\not\simeq S^{m+n},
$ which contradicts (\[e5.3\]).
\(2) Now consider the case where $h(\Phi_0,S_0)\ne\Sigma^{m}$. Let $\lam\in(0,\ve]$. Then by (H1) and (H4), we have $h(\Phi_\lam,S_0)=\Sigma^{m}$. Hence \[e4.0\]h(\_,S\_0)h(\_0,S\_0),so the system bifurcates in $U\sm S_0$ a maximal compact invariant set $K_\lam\ne\emp$.
Let $S_\lam$ be the maximal compact invariant set of $\Phi_\lam$ in $U$. Then $S_\lam\subset {{\mathcal M}}_\lam^2$. By Lemma \[l:3.2\] we have \[e4.10\]h(\_,S\_)=\^mh(\_\^2,S\_).On the other hand, \[4.5\]h(\_,S\_)= h(\_0,S\_0)\^[m]{}.Thus by (\[e4.10\]) and (\[4.5\]) one concludes that \[4.6\] h(\_\^2,S\_)\^0.
As $\Phi_\lam^2$ and the semiflow $\phi_\lam$ generated by the ODE system (\[e:2.9\]) on $W$ are conjugate, in the following argument we [identify]{} $\Phi_\lam^2$ with $\phi_\lam$, regardless of the conjugacy between them. By [@CE], Theorem 1.5, one can find a connected isolating block $N$ of $S_0$ (with respect to $\phi_0$) with smooth boundary $\pa N$. Further by [@CE], Theorem 1.6, it can be assumed that $\ve$ is sufficiently small so that $N$ is an isolating block of $S_{\lam}$ (with respect to $\phi_\lam$) for all $\lam\in(0,\ve]$ with $$B_\lam^-\equiv B^-_0:=N^-,$$ where $B_\lam^-$ denotes the boundary exit set of $N$ with respect to $\phi_\lam$. We claim that \[4.7\] N\^-.Indeed, if this was false, $S_0$ would be an attractor of $\phi_0$ in $N$ that attracts $N$. As $S_0$ is a singleton, it follows that $N$ is contractible. Consequently $$h(\phi_\lam,S_\lam)=h(\phi_0,S_0)=\left[(N/N^-,[N^-])\right]=[(N,\emp)]=\Sigma^0$$ for $\lam\in(0,\ve]$, which contradicts (\[4.6\]).
Because $S_0$ is an attractor of $\phi_\lam$ in $W$ for $\lam\in(0,\ve]$ (by (H4)), using appropriate smooth Lyapunov function of $S_0$ one can find an arbitrarily small isolating block $N_0$ of $S_0$ (depending upon $\lam$) with smooth boundary $\pa N_0$ such that $N_0^-=\emp$, where $N_0^-$ is the boundary exit set of $N_0$ with respect to $\phi_\lam$. Note that $M:=N\sm\mb{int} N_0$ is then an isolating block of $K_\lam$ (with respect to $\phi_\lam$) with $M^-=N^-\cup \pa N_0;$ see Fig. 5.2. We show that \[4.8b\]CH\_\*(\_, K\_)=H\_\*$h(\phi_\lam, K_\lam)$0,where $CH_*(\phi_\lam, K_\lam)$ is the homology Conley index of $K_\lam$ with respect to $\phi_\lam$.
First, we infer from [@Ryba] that the inclusion $M^-\subset M$ has the homotopy extension property. This implies that $M^-$ is a strong deformation retract of one of its neighborhoods in $M$. As $N^-$ and $\pa N_0$ are disjointed compact subsets of $M$, each of them is a strong deformation retract of a neighborhood of itself in $M$. We collapse $N^-$ and $\pa N_0$ to two distinct points $z$ and $w$ (see Fig.5.3), respectively, and denote $\~M$ the corresponding quotient space. Let $\~M_0=\{z,w\}$. Then \[4.11\] h(\_, K\_)= \[(M/M\^-,\[M\^-\])\]=\[(M/M\_0,\[M\_0\])\].
Consider the mapping cone $C_f$ as depicted in Fig.5.3, where $f:\~M_0\ra\~M$ is the inclusion. Let $$C\~M_0=(\~M_0\X I)/(\~M_0\X\{1\}).$$ Then $C\~M_0$ is homeomorphic to $I=[0,1]$. Hence one can think of $C_f$ as the space obtained by identifying the end points $0$ and $1$ of $I$ with $z$ and $w$, respectively, in the disjoint union of $\~M$ and $I$. We observe that $\~M_0$ is a strong deformation retract of an appropriate neighborhood in $\~M$. Consequently $C\~M_0$ is a strong deformation retract of an appropriate neighborhood in $C_f$. Noticing that $C_f$ is metrizable, by [@Ryba] Chap.I, Pro.3.6, we deduce that the inclusion $C\~M_0\subset C_f$ has the homotopy extension property. Since $C\~M_0$ is contractible, by the basic knowledge on homotopy equivalence (see e.g. [@Hat], Pro. 0.17), we have \[4.12\] M/M\_0= C\_f/CM\_0C\_f. -0.8cm
{width="11.5cm"}
Figure 5.3: $M/M^-\simeq\~M/\~M_0\simeq C_f/C\~M_0\simeq C_f$
Because $M$ is a domain in $E^2$ with smooth boundary, we deduce that $M$ is path-connected. It then follows that $\~M$ is path-connected as well. Consequently $C_f$ is a path-connected space. Let $\gamma_1$ be a path in $\~M\X\{0\}$ from $(w,0)$ to $(z,0)$ (see Fig.5.3), and $\gamma_2$ be a path in $C_f$ from $(z,0)$ to $(w,0)$ along $C\~M_0$. Define a closed path $\gamma$ in $C_f$ from $(z,0)$ to $(z,0)$ to be the product $\gamma_1*\gamma_2$ of $\gamma_1$ and $\gamma_2$. Then by a simple continuity argument it can be easily shown that $\gamma$ is not homotopic to any constant path. Thus the fundamental group $\pi_1(C_f)\ne 0$. Further by some basic knowledge in the theory of algebraic topology we know that $H_1(C_f)\ne 0$. In view of (\[4.11\]) and (\[4.12\]) one immediately concludes that $H_1\(h(\phi_\lam, K_\lam)\)\ne 0$. This finishes the proof of (\[4.8b\]).
Now we verify that $h(\Phi_\lam,K_\lam)\ne\ol0$. By Lemma \[l:3.2\] it suffice to check that $h\(\Phi_\lam^{12},K_\lam\)\ne\ol0$. Suppose the contrary. Then we would have $CH_*\(\Phi_\lam^{12},K_\lam\)=0$. Invoking the Poincar$\acute{\mb{e}}$-Lefschetz duality theory on homology Conley index (see McCord [@McC], Theorem 2.1), it then holds that $CH^*\((\Phi_\lam^{12})^-,K_\lam\)=0$, where $(\Phi_\lam^{12})^-$ denotes the inverse flow of $\Phi_\lam^{12}$. On the other hand, for $(\Phi_\lam^{12})^-$ we have $$h\((\Phi_\lam^{12})^-,K_\lam\)=h\((\Phi_\lam^2)^-,K_\lam\)=h\(\phi_\lam^-,K_\lam\).$$ (Recall that we identify $\Phi_\lam^2$ with $\phi_\lam$, regardless of the conjugacy between them.) Hence $CH^*\(\phi_\lam^-,K_\lam\)=0$. Again by the Poincar$\acute{\mb{e}}$-Lefschetz duality theory we find that $CH_*\(\phi_\lam,K_\lam\)=0$, which contradicts (\[4.8b\]). $\Box$
Global Dynamic Bifurcation
==========================
In this section we establish a global dynamic bifurcation result.
Existence of a local bifurcation branch
---------------------------------------
We first prove an existence result for local bifurcation branch.
Set ${{\mathcal E}}=E\X\R$, where $E=X^\a$. ${{\mathcal E}}$ is equipped with the metric $\rho$ defined as $$\rho\((u,\lam),\,(v,\lam')\)=||u-v||_\a+|\lam-\lam'|, \Hs \A\, (u,\lam),\, (v,\lam')\in{{\mathcal E}}.$$ Let ${{\mathcal Z}}\subset {{\mathcal E}}$. For any $\lam\in\R$, denote ${{\mathcal Z}}_\lam$ the [*$\lam$-section*]{} of ${{\mathcal Z}}$, $${{\mathcal Z}}_\lam=\{u:\,\,(u,\lam)\in {{\mathcal Z}}\}.$$
Let $\~\Phi$ be the [*skew-product flow*]{} of the family $\Phi_\lam$ ($\lam\in \R$) on ${{\mathcal E}}$, \[e6.1\] \~(t)(u,)=$\Phi_\lam(t)u,\,\lam$,(u,)[[E]{}]{}. By the basic theory on abstract evolution equations (see e.g.[@Henry], Chap. 3 or [@Ryba], Chap. 1, Theorem 4.4 ), one can easily verify that $\~\Phi$ is [*asymptotically compact*]{}, i.e., $\~\Phi$ satisfies the hypothesis (AC) in Section 2. For each $\lam\in\R$, denote $\stackrel{\circ}{{{\mathscr K}}}_\lam$ the [*family of nonempty compact invariant sets $K$ of $\Phi_\lam$ with $0\not\in K$*]{}. Given ${{\mathcal U}}\subset {{\mathcal E}}$, define $$\ba{ll}
{{\mathscr C}}({{\mathcal U}})=\ol{\Cup\{K\X\{\lam\}\subset {{\mathcal U}}:\,\, K\in \stackrel{\circ}{{{\mathscr K}}}_\lam,\,\,\lam\in\R\}}.\ea$$
(Bifurcation branch) Let $(0,\lam_0)$ be a bifurcation point, and ${{\mathcal U}}\subset {{\mathcal E}}$ be a closed neighborhood of $(0,\lam_0)$. Then the [bifurcation branch]{} in ${{\mathcal U}}$ from $(0,\lam_0)$, denoted by $\Gamma_{{\mathcal U}}(0,\lam_0)$, is defined to be the connected component of ${{\mathscr C}}({{\mathcal U}})$ which contains $(0,\lam_0)$.
Now we prove the following interesting result which ensures the existence of local bifurcation branch.
\[t:4.3\]Suppose the hypotheses (H1)-(H4) in Theorem \[t:2.7\] are fulfilled with $\lam_0=0$, and that $S_0=\{0\}$ is an isolated invariant set of $\Phi_0$. Then there exists $\ve>0$ such that $$\Gamma\cap (U\X\{\pm\ve\})\ne\emp,$$ where $\Gamma=\Gamma_{{\mathcal U}}(0,0)$, and ${{\mathcal U}}=U\X[-\ve,\ve]$.
[**Proof.**]{} Let $U$ be the neighborhood of $0$ given in Theorem \[t:2.7\], and let $S_\lam$ be the maximal compact invariant set of $\Phi_\lam$ in $U$. Choose an $\ve>0$ such that the assertions in Theorem \[t:4.1\] hold. Let $K_\lam$ be the maximal compact invariant set of $\Phi_\lam$ in $U\sm S_0$. Since $\lim_{\lam\ra0}d_{\mbox{\tiny H}}(K_\lam,0)=0$, we may also assume $\ve$ is sufficiently small so that there exists $r>0$ such that \[e:5.3\](K\_,r)U,.We show that $\ve$ fulfills the requirement of the theorem.
For definiteness, by Theorem \[t:4.1\] it can be assumed that \[e6.2\] h(\_, K\_)0 for $\lam\in(0,\ve]$. We check that $$\Gamma\cap (U\X\{\ve\})\ne\emp,$$ thus completing the proof of the theorem.
We first prove that for any $0<\mu<\ve$, ${{\mathscr C}}({{\mathcal U}}_\mu)$ has a connected component ${{\mathcal Z}}$ such that \[e:4.10\] [[Z]{}]{}$U\X\{\mu\}$[[Z]{}]{}$U\X\{\ve\}$,where ${{\mathcal U}}_\mu=U\X[\mu,\ve]$. For this purpose, let us first verify that $$\ba{ll}{{\mathscr C}}({{\mathcal U}}_\mu)={\Cup_{\mu\leq\lam\leq\ve}\,K_\lam\X\{\lam\}}:={{\mathcal K}}.\ea$$ Indeed, we infer from the maximality of $K_\lam$ in $U\sm S_0$ that ${{\mathscr C}}({{\mathcal U}}_\mu)=\ol{{{\mathcal K}}}.$ On the other hand, it is clear that ${{\mathcal K}}$ is invariant under the skew-product flow $\~\Phi$. Hence by asymptotic compactness of $\~\Phi$ we deduce that ${{\mathcal K}}$ is pre-compact. Further by upper semicontinuity of $K_\lam$ in $\lam$ one can easily verify that ${{\mathcal K}}$ is closed. Thus ${{\mathcal K}}$ is compact. Consequently ${{\mathscr C}}({{\mathcal U}}_\mu)=\ol{{\mathcal K}}={{\mathcal K}}.$
The compactness of ${{\mathcal K}}$ also implies \[e:5.5\] d(0,K\_)2,, where $\eta>0$ is a positive number independent of $\lam$.
In what follows we argue by contradiction and suppose that (\[e:4.10\]) fails to be true. Then for any connected component ${{\mathcal Z}}$ of ${{\mathscr C}}({{\mathcal U}}_\mu)$ one has $$\ba{ll}\mb{either}\,\,\,{{\mathcal Z}}\cap \(U\X\{\mu\}\)=\emp,\hs \mb{or }\,{{\mathcal Z}}\cap\( U\X\{\ve\}\)=\emp.\ea$$ If there are only a finite number of components, then each component ${{\mathcal Z}}$ is isolated in ${{\mathcal U}}$. Because the $\lam$-section ${{\mathcal Z}}_\lam$ of ${{\mathcal Z}}$ is empty when $\lam$ is close to either $\mu$ or $\ve$, by the continuation property of Conley index we see that $h(\Phi_\lam,{{\mathcal Z}}_\lam)\equiv \ol0.$ Consequently the “sum” of these indices equals $\ol0$. This contradicts and justifies (\[e:4.10\]), as the union of ${{{\mathcal Z}}_\lam}'s$ is precisely $K_\lam$. However, in general there is also the possibility that ${{\mathscr C}}({{\mathcal U}}_\mu)$ may contain infinitely many components. We will employ the Separation Lemma given in Section 2 to overcome this difficulty.
Set ${{\mathcal O}}_\mu={{\mathcal U}}_\mu\sm\({\mb{B}}(0,\eta)\X[\mu,\ve]\)$. Then clearly ${{\mathscr C}}({{\mathcal O}}_\mu)={{\mathscr C}}({{\mathcal U}}_\mu)$. Denote ${{\mathscr F}}$ the family of connected components of ${{\mathscr C}}({{\mathcal O}}_\mu)$. By (\[e:5.3\]) and (\[e:5.5\]) we see that ${{\mathcal O}}_\mu$ is a neighborhood of ${{\mathcal Z}}$ in the space $${{\mathcal H}}=E\X[\mu,\ve]$$ for each ${{\mathcal Z}}\in {{\mathscr F}}$. This allows us to pick for each ${{\mathcal Z}}\in {{\mathscr F}}$ a closed neighborhood $\Omega_{{\mathcal Z}}$ in ${{\mathcal H}}$ with $\Omega_{{\mathcal Z}}\subset {{\mathcal O}}_\mu$ such that if ${{\mathcal Z}}\cap \(U\X\{\sig\}\)=\emp$ (where $\sig=\mu$ or $\ve$), then \[Om\]\_[[Z]{}]{}$U\X\{\sig\}$=;see Fig.6.1. -0.3cm
![Separating neighborhoods of ${{\mathcal Z}}$ in ${{\mathcal H}}$[]{data-label="fg4-1"}](fig6-1.pdf){width="6cm"}
For any ${{\mathcal O}}\subset {{\mathcal H}}$, denote $\pa_{{\mathcal H}}{{\mathcal O}}$ the boundary of ${{\mathcal O}}$ in ${{\mathcal H}}$. Given ${{\mathcal Z}}\in{{\mathscr F}}$, set $$\ba{ll}
\mathfrak{B}=\Cup\{{{\mathcal F}}\in {{\mathscr F}}:\,\,{{\mathcal F}}\cap \,\pa_{{\mathcal H}}\Omega_{{\mathcal Z}}\ne\emp\}, \hs \mathfrak{D}=\Cup\{{{\mathcal F}}\in{{\mathscr F}}:\,\,{{\mathcal F}}\cap\, \Omega_{{\mathcal Z}}\ne\emp\},\ea$$ We claim that both $\mathfrak{B}$ and $\mathfrak{D}$ are closed. Indeed, if $b\in \ol{\mathfrak{B}}$, then there exists a sequence $b_k\in \mathfrak{B}$ such that $b_k\ra b$. We may assume that $b_k\in {{\mathcal F}}_k$ for some ${{\mathcal F}}_k\in {{\mathscr F}}$ with ${{\mathcal F}}_k\cap \pa_{{\mathcal H}}\Omega_{{\mathcal Z}}\ne\emp$. By Lemma \[l:2.3\] we deduce that there exists a subsequence of ${{\mathcal F}}_k$, still denoted by ${{\mathcal F}}_k$, such that $$\lim_{k\ra\8}\de_{\mbox{\tiny H}}({{\mathcal F}}_k,{{\mathcal F}}_0)=0.$$ One trivially checks that ${{\mathcal F}}_0$ is connected and contained in ${{\mathscr C}}({{\mathcal O}}_\mu)$; moreover, ${{\mathcal F}}_0\cap \pa_{{\mathcal H}}\Omega_{{\mathcal Z}}\ne\emp$. Since $b\in{{\mathcal F}}_0$, we conclude that $b\in \mathfrak{B}$. Hence $\mathfrak{B}$ is closed. Likewise it can be shown that $\mathfrak{D}$ is closed.
Note that ${{\mathcal Z}}\cap\mathfrak{B}=\emp$. Since ${{\mathcal Z}}$ does not intersect any other connected component of $\mathfrak{D}$, by Lemma \[l:2.2\] there exist two disjoint closed subsets ${{\mathcal K}}_1$ and ${{\mathcal K}}_2$ of $\mathfrak{D}$ such that $\mathfrak{D}={{\mathcal K}}_1\cup{{\mathcal K}}_2$, and $${{\mathcal Z}}\subset {{\mathcal K}}_1,\hs \mathfrak{B}\subset {{\mathcal K}}_2\,.$$ It is clear that ${{\mathcal K}}_1$ is contained in the interior of $\Omega_{{\mathcal Z}}$ relative to ${{\mathcal H}}$.
Take a positive number $\de_{{\mathcal Z}}$ with $$\de_{{\mathcal Z}}<\frac{1}{8}\min\(d({{\mathcal K}}_1,{{\mathcal K}}_2),\,d({{\mathcal K}}_1,\pa_{{\mathcal H}}\Omega_{{\mathcal Z}})\).$$ Let ${{\mathcal V}}_{{\mathcal Z}}={\mb{B}}_{{{\mathcal H}}}({{\mathcal K}}_1,4\de_{{\mathcal Z}})$ be the $4\de_{{\mathcal Z}}$-neighborhood of ${{\mathcal K}}_1$ in ${{\mathcal H}}$. Then ${{\mathcal V}}_{{\mathcal Z}}\subset \Omega_{{\mathcal Z}}$, and \[e:5.6\]\_[[[H]{}]{}]{}(\_[[H]{}]{}[[V]{}]{}\_[[Z]{}]{},2\_[[Z]{}]{})[[C]{}]{}([[O]{}]{}\_)=.By the compactness of ${{\mathscr C}}({{\mathcal O}}_\mu)$ there exist a finite number of ${{\mathcal Z}}\in{{\mathscr F}}$, say, ${{{\mathcal Z}}_1},\cdots,{{\mathcal Z}}_l$, such that ${{\mathscr C}}({{\mathcal O}}_\mu)\subset\Cup_{1\leq k\leq l}{{\mathcal V}}_{{{\mathcal Z}}_k}\,.$ Set $$\ba{ll}
{{\mathcal W}}_k={{\mathcal V}}_{{{\mathcal Z}}_k}\sm\(\ol{{\mathcal V}}_{{{\mathcal Z}}_1}\cup\cdots\cup \ol{{\mathcal V}}_{{{\mathcal Z}}_{k-1}}\),\Hs k=1,2,\cdots,l.\ea$$ Then ${{{\mathcal W}}_k}'s$ are disjoint open subsets of ${{\mathcal H}}$. One can easily check that \[e6.7\]\_[[H]{}]{}[[W]{}]{}\_k\_[1ik]{}\_[[H]{}]{}[[V]{}]{}\_[[[Z]{}]{}\_i]{}.Thus we deduce that ${{\mathscr C}}({{\mathcal O}}_\mu)\subset\Cup_{1\leq k\leq l}{{\mathcal W}}_k\,.$ Let ${{\mathcal S}}_k={{\mathscr C}}({{\mathcal O}}_\mu)\cap{{\mathcal W}}_k$. We claim that \[e:5.7\] d${{\mathcal S}}_k,\pa_{{\mathcal H}}{{\mathcal W}}_k$>0. Indeed, if $w\in {{\mathcal S}}_k$ then by (\[e:5.6\]) we have $$d(w,\pa_{{\mathcal H}}{{\mathcal V}}_{{{\mathcal Z}}_i})\geq 2\de_{{{\mathcal Z}}_i}\geq 2\min_{1\leq j\leq l}\de_{{{\mathcal Z}}_j}:=\de_0>0,\Hs 1\leq i\leq l,$$ and the conclusion follows from (\[e6.7\]). It follows by (\[e:5.7\]) that ${{\mathcal S}}_k={{\mathscr C}}({{\mathcal O}}_\mu)\cap\ol{{\mathcal W}}_k$. Hence ${{\mathcal S}}_k$ is compact. It can be easily seen that ${{\mathcal S}}_k$ is the maximal compact invariant set of $\~\Phi$ in $\ol{{\mathcal W}}_k$. Since $\ol{{\mathcal W}}_k$ is a neighborhood of ${{\mathcal S}}_k$ in ${{\mathcal H}}$, by Theorem \[t:2.14\] we have \[e:4.12\] h(\_,[[S]{}]{}\_[k,]{}), , where ${{\mathcal S}}_{k,\lam}$ is the $\lam$-section of ${{\mathcal S}}_k$. On the other hand, by (\[Om\]) we have either ${{\mathcal S}}_{k,\mu}=\emp$, or ${{\mathcal S}}_{k,\ve}=\emp$. Hence by (\[e:4.12\]) it holds that \[e:4.12b\] h(\_,[[S]{}]{}\_[k,]{})0, ,
Now by (\[e:4.12b\]) we conclude that $$h(\Phi_\lam,K_{\lam})=h(\Phi_\lam,{{\mathcal S}}_{1,\lam})\vee \cdots \vee h(\Phi_\lam,{{\mathcal S}}_{l,\lam})
=\ol0.$$ This contradicts (\[e6.2\]) and completes the proof of (\[e:4.10\]). We are now ready to complete the proof of the theorem. Take a sequence of positive numbers $\mu_k\ra 0$. For each $\mu_k$, pick a connected component ${{\mathcal Z}}_k$ of ${{\mathscr C}}({{\mathcal O}}_{\mu_k})$ such that $$\ba{ll}{{\mathcal Z}}_k\cap \(U\X\{\mu_k\}\)\ne\emp\ne {{\mathcal Z}}_k\cap \(U\X\{\ve\}\).\ea$$ By Lemma \[l:2.3\] we may assume that $$\lim_{k\ra\8}\de_{\mbox{\tiny H}}({{\mathcal Z}}_k,{{\mathcal Z}}_0)=0.$$ Then ${{\mathcal Z}}_0$ is a continuum in ${{\mathscr C}}({{\mathcal U}})$ with $(0,0)\in {{\mathcal Z}}_0$ and ${{\mathcal Z}}_0\cap (U\X\{\ve\})\ne\emp$. $\Box$
Global bifurcation
------------------
For the sake of convenience in statement, we make a convection that $\8\in \pa \Omega$ if $\Omega$ is an unbounded subset of ${{\mathcal E}}$.
The main result in this section is the following theorem. \[gbt\](Global dynamic bifurcation) Assume that the hypotheses in Theorem \[t:2.7\] are fulfilled. Let $\Omega\subset {{\mathcal E}}$ be a closed neighborhood of the bifurcation point $(0,0)$. Suppose that $S_0=\{0\}$ is an isolated invariant set of $\Phi_0$.
Let $\Gamma=\Gamma_\Omega(0,0)$.Then one of the following cases occurs.
1. $\Gamma\Cap \pa\Omega\ne\emp$; see Fig. 6.3.
2. $0\in\ol{ \Gamma_0\sm\{0\}}$, where $\Gamma_0$ is the $0$-section of $\Gamma$; see Fig. 6.4.
3. There exists $\lam_1\ne0$ such that $(0,\lam_1)\in\Gamma$; see Fig. 6.5.
{width="4.3cm"} {width="4.3cm"}
Figure 6.3: $\Gamma\Cap \pa\Omega\ne\emp$ Figure 6.4: $0\in\ol{ \Gamma_0\sm\{0\}}$
{width="4.3cm"} {width="4.3cm"}
Figure 6.5: $(0,\lam_1)\in\Gamma$ Figure 6.6: This case never occurs
[**Proof.**]{} We argue by contradiction and suppose that none of the cases (1)-(3) occurs. Then $\Gamma$ is a bounded closed subset of ${{\mathcal E}}$ contained in the interior of $\Omega$ as depicted in Fig.6.6. It is easy to see that $\Gamma$ is invariant under the skew-product flow $\~\Phi$. Hence by asymptotic compactness of $\~\Phi$ we deduce that $\Gamma$ is compact.
Since $0\not\in\ol{ \Gamma_0\sm S_0}$, we can write $\Gamma_0$ as $\Gamma_0=S_0\cup A_0$, where $A_0$ is a compact invariant set of $\Phi_0$ with $A_0\cap S_0=\emp$. We only consider the case where $A_0\ne\emp$. The argument for the case where $A_0=\emp$ is a slight modification of that of the former one.
Let $U\subset E$ and $\ve>0$ be as in Theorem \[t:4.3\]. Then the system $\Phi_\lam$ bifurcates, say, for each $0<\lam\leq \ve$, a nonempty maximal compact invariant set $K_\lam$ in $U\sm S_0$ with \[e5.11\]\_[0]{}d\_(K\_,S\_0)=0and \[e:5.10\]h(\_,K\_)0,(0,\].Pick a closed neighborhood $V$ of $0$ with $V\subset U$ and $$d\(A_0,\,V\):=\sig_0>0.$$ By (\[e5.11\]) we can further restrict $\ve$ sufficiently small so that for some $r_0>0$, $${\mb{B}}(K_\lam,r_0)\subset V,\Hs \A\, \lam\in(0,\ve].$$
By the compactness of $\Gamma$ it is easy to verify that the $\lam$-section $\Gamma_\lam$ of $\Gamma$ is upper semicontinuous in $\lam$. Let $${{\mathcal Z}}=\Gamma\cap(V\X[0,\ve]).$$ Then $d_{\mbox{\tiny H}}({{\mathcal Z}}_\lam,S_0)\ra0$ as $\lam\ra 0$. As $A_0\cap S_0=\emp$, it also holds that $$\lim_{\lam\ra0}d_{\mbox{\tiny H}}(A_\lam,A_0)\ra 0,$$ where $A_\lam=\Gamma_\lam\sm{{\mathcal Z}}_\lam$ ($\lam\in[0,\ve]$). Thus there exist $\eta_0>0$ and $0<\ve'\leq\ve$ such that \[e:5.9\]([[Z]{}]{}\_,\_0)V,(A\_,\_0)V=for all $\lam\in[0,\ve']$. Note that both ${{\mathcal Z}}_\lam$ and $A_\lam$ are compact invariant sets of $\Phi_\lam$.
Let $M_0=\Cup_{\lam\geq\ve'}\Gamma_\lam$. It can be easily shown that $M_0$ is a compact subset of $E$. Clearly $0\not\in M_0$, hence \[e6.8\]d(0,M\_0):=\_0>0.Fix a number $0<r<\frac{1}{3}\min\(\eta_0,\de_0\)$. Utilizing the Separation Lemma, by a similar argument as in the proof of Theorem \[t:4.3\] we can find a closed neighborhood ${{\mathcal O}}$ of $\Gamma$ with ${{\mathcal O}}\subset {\mb{B}}_{{\mathcal E}}(\Gamma,r)$ such that \[5.10\] [[C]{}]{}()[[O]{}]{}=.Here ${\mb{B}}_{{\mathcal E}}(\Gamma,r)$ denotes the $r$-neighborhood of $\Gamma$ in ${{\mathcal E}}$. By the choice of $r$ it can be easily seen that if $\lam\in(0,\ve']$ then $${{\mathcal O}}_\lam\subset \ol{\mb{B}}({{\mathcal Z}}_\lam,\eta_0)\cup\ol{\mb{B}}(A_\lam,\eta_0);$$ see Fig.6.7. Set $$\ba{ll}
G_\lam={{\mathcal O}}_\lam\cap \ol{\mb{B}}({{\mathcal Z}}_\lam,\eta_0),\hs H_\lam={{\mathcal O}}_\lam\cap
\ol{\mb{B}}(A_\lam,\eta_0).\ea$$ By (\[e:5.9\]) we have \[e:4.17\][[O]{}]{}\_=G\_H\_,G\_H\_=for $\lam\in(0,\ve']$.
We claim that there exists $\sig>0$ such that $${\mb{B}}_{\sig}\subset G_\lam$$ for all $\lam$ sufficiently small, where (and below) ${\mb{B}}_R$ denotes the ball in $E$ centered at $0$ with radius $R$. Suppose the contrary. There would exist sequences $\lam_k\ra0$ and $x_k\in \pa G_{\lam_k}$ such that $x_k\ra 0$. Noticing that $(x_k,\lam_k)\in\pa{{\mathcal O}}$, one concludes that $(0,0)\in \pa{{\mathcal O}}$, a contradiction!
By (\[e5.11\]) one can find a number $0<\mu\leq{\ve'}/{2}$ such that \[e:4.15\] K\_\_G\_,(0, 2\].Using the upper semicontinuity of $K_\lam$ in $\lam$ (see Theorems [\[t:2.7\]]{}) one can easily show that $F=\Cup_{\mu\leq\lam\leq\ve'}K_\lam$ is closed in $E$. Because ${{\mathcal F}}=\Cup_{\mu\leq\lam\leq\ve'}K_\lam\X\{\lam\}$ is invariant under the system $\~\Phi$, by asymptotic compactness of $\~\Phi$ we deduce that ${{\mathcal F}}$ is pre-compact in ${{\mathcal E}}$. It then follows that $F$ if compact in $E$. Hence $$d(0,F):=d_0>0.$$
Take a $\Lam>0$ such that ${{\mathcal O}}\subset
E\X(-\Lam,\Lam)$. Let $\rho$ be a positive number with $\rho<\rho_0:=\frac{1}{2}\min(d_0,\de_0)$, where $\de_0$ is the number given in . Set $$\ba{ll}
{{\mathcal V}}={{\mathcal O}}\cap {{\mathcal H}},\hs
{{\mathcal W}}={{\mathcal V}}\sm\({\mb{B}}_\rho\X[\mu,\Lam]\),\ea$$where ${{\mathcal H}}=E\X[\mu,\Lam]$; see Fig.6.8. Clearly ${{\mathcal V}}$ is closed in ${{\mathcal H}}$. Since ${\mb{B}}_\rho\X[\mu,\Lam]$ is open in ${{\mathcal H}}$, we see that ${{\mathcal W}}$ is closed in ${{\mathcal H}}$ as well. We claim that \[5.16\][[C]{}]{}([[W]{}]{})=[[C]{}]{}([[V]{}]{}):=[[C]{}]{},’$. Then by the choice of
$$ we find that $[[O]{}]{}\_\_=$; see Fig.\,\,6.8. It follows that
$[[V]{}]{}\_=[[W]{}]{}\_$. This finishes the proof of what we desired. Hence (\ref{5.16}) holds true.
We show that $[[V]{}]{}$ is a neighborhood of $[[C]{}]{}$ in $[[H]{}]{}:=E$. Suppose the contrary. Then $[[C]{}]{}\_[[[H]{}]{}]{}[[V]{}]{}$, where $\_[[H]{}]{}[[V]{}]{}$ denotes the boundary of $[[V]{}]{}$ relative to $[[H]{}]{}$. Noticing that $$\ba{ll}\pa_{{{\mathcal H}}}{{\mathcal V}}=\pa_{{{\mathcal H}}}({{\mathcal O}}\cap {{\mathcal H}})\subset\pa{{\mathcal O}}\cap {{\mathcal H}},\ea$$ we have $$\ba{ll}{{\mathscr C}}\cap \pa {{\mathcal O}}={{\mathscr C}}\cap \(\pa{{\mathcal O}}\cap {{\mathcal H}}\)\supset {{\mathscr C}}\cap \pa_{{{\mathcal H}}}{{\mathcal V}}\ne\emp.\ea$$ This contradicts (\ref{5.10}).
By (\ref{5.16}) we can fix a $>0$ sufficiently small so that $[[W]{}]{}=[[V]{}]{}${\mb{B}}_\rho\X[\mu,\Lam]$$ is a neighborhood of $[[C]{}]{}$ in $[[H]{}]{}$. By the definitions of $[[C]{}]{}=[[C]{}]{}([[W]{}]{})$ and the skew-product flow one can easily see that $[[C]{}]{}$ is the maximal compact invariant set of $\~$ in $[[W]{}]{}$. Hence $[[W]{}]{}$ is an isolating neighborhood of $[[C]{}]{}$ in $[[H]{}]{}$.
It then follows by Theorem \ref{t:2.14} that
\be\label{e:4.16} h(\Phi_\lam,{{\mathscr C}}_\lam)\equiv h(\Phi_{\Lam},{{\mathscr C}}_{\Lam})= h(\Phi_{\Lam},\emp)=\ol0,\Hs \lam\in[\mu, \Lam].\ee
On the other hand, if $2$ then by (\ref{e:4.15}) and the choice of $$, we find that $\_:=G\_\_$ is a
neighborhood of $K\_$. Since $K\_$ is the maximal compact invariant of $\_$ in $VS\_0$ (and hence in $\_$), we infer from
(\ref{e:4.17}) that $[[C]{}]{}\_K\_$ is necessarily contained in $ H\_$ (note that $[[W]{}]{}\_=G\_H\_$). Thus
$$
h(\Phi_\lam,{{\mathscr C}}_\lam)=h(\Phi_\lam,K_\lam)\vee
h(\Phi_\lam,{{\mathscr C}}_\lam\sm K_\lam).
$$
(\ref{e:4.16}) then implies that $$h(\Phi_\lam,K_\lam)=\ol0.$$ This contradicts (\ref{e:5.10}), which completes the proof of the theorem. \,$$
$[[E]{}]{}\_$ the set of stationary solutions of the equation (\ref{e:1.1}).
$\_$) is called {\bf gradient-like}, if there exists a continuous function $V:E$ such that \begin{enumerate}\item[(1)] $V$\Phi_\lam(t)u$$ is nonincreasing in $t$, and \item[(2)] if $V$\Phi_\lam(t)u$=V(u)$ for some $t>0$, then $u[[E]{}]{}\_$.\end{enumerate}
$V$ as above will be referred to as a {\bf Lyapunov function} of the system.
$$. Then by the basic knowledge on gradient-like systems
$S$ of the system necessarily contains a stationary solution $u[[E]{}]{}\_$.
$[[E]{}]{}$ be a closed domain with $(0,0)$. Let $=()$ be the dynamic bifurcation branch of the trivial solution
$(0,0)$ in $$. Then one of the following cases occur: \begin{enumerate} \item[(1)] The trivial solution $0$ is not
$[[E]{}]{}\_0$;
$$;
$\_10$ such that $$\lim_{\lam\ra\lam_1}d\(0,\,\Gamma_\lam\)=0.$$ \end{enumerate} \et
$0$ is isolated in $[[E]{}]{}\_0$ and the system is gradient-like, one easily verifies that $S\_0={0}$ is an isolated
$\_0$. Further by Theorem \ref{gbt} it necessarily holds that $0$. We show that this leads to a contradiction.
$K\_k\_0$ of $\_0$ with $0K\_k$ such that
$\_[k]{}d(0,K\_k)=0.$ For each $k$ we pick a $u\_kK\_k$ such that $\_[k]{}d(0,u\_k)=0.$ Let $\_k$ be a complete trajectory in $K\_k$ through $u\_k$.
$(\_k),(\_k)\_0[[E]{}]{}\_0$.
$V$ be a Lyapunov function of the system. Then there exists a subsequence
$$M_k=\{z\in K_k:\,\,V(z)=\min_{u\in K_k}V(u)\}.$$ It is trivial to check that $M\_k[[E]{}]{}\_0$. Note that $\_[k]{}d(0,M\_k)=0.$ This contradicts to the assumption that $S\_0$ is an isolated invariant set of the system.
\section{An Example}
In this section we give an example to illustrate our theoretical results by considering the well-known Cahn-Hilliard equation describing the spinodal decomposition.
The nondimensional form of the equation reads (see \cite{MW2})
\be\label{e:6.2}\left\{\ba{lll}
u_t+\Delta^2 u+\lam\De u=\De(b_2 u^2+b_3 u^3),\Hs &(x,t)\in \Omega\times R^+,\\[1ex]
\frac{\pa u}{\pa \nu}=\frac{\pa (\De u)}{\pa \nu}=0,\Hs &(x,t)\in \partial\Omega\times R^+,\\[1ex]
m(u)=0,\ea\right.
\ee
where $R\^d$ ($d3$) is a bounded domain with smooth boundary
$$, $b\_3>0$, and $$
m(u)=\frac{1}{|\Omega|}\int_\Omega u\,dx.
$$
The local attractor bifurcation and phase transition of the system have been extensively studied in Ma and Wang \cite{MW2}. Other results relates to bifurcation of the problem can be found in \cite{BDW,Mis2}, etc.
Here by applying the theoretical results obtained above, we try to provide some new results about the dynamic bifurcation of the system and demonstrate global features of the bifurcations.
\subsection{Mathematical setting of the system}
Denote by $(,)$ and $||$ the usual inner product and norm of $L\^2()$, respectively.
For mathematical setting, we introduce the Hilbert space $H$ as follows:
$$
H=\{u\in L^2(\Omega):\,\,\,m(u)=0\}.$$ Let $A\_0=-$ be the Laplacian in $H$ associated with the homogeneous boundary condition$$
\frac{\pa u}{\pa \nu}=0,\Hs x\in\pa\Omega.$$
${w\_k}\_[k=1]{}\^$ the family of eigenvectors of $A\_0$ corresponding to the eigenvalues $$0<\mu_1\leq\mu_2\leq\cdots\leq \mu_n\ra+\8,$$ which forms a canonical basis of $H$.
Set $A=A\_0\^2$. Then $A$ is a positive-definite self-adjoint operator in $H$ (and hence is a sectorial operator) with compact resolvent, and
$$\ba{ll}
D(A)=\left\{u\in H^4(\Omega)\cap H\mid\,\,\, \frac{\pa u}{\pa \nu}=\frac{\pa (\De u)}{\pa \nu}=0\mb{ on }\pa\Omega\right\}.\ea$$
The spectral $(A\_0)$ of $A\_0$ consists of a countably infinitely many eigenvalues: $$0<\mu_1<\mu_2<\cdots< \mu_k\ra+\8.$$
$j>0$, denote $w\_[j1]{},w\_[j2]{},,w\_[jm\_j]{}$ the eigenvectors of $A\_0$ corresponding to $\_j$.
Let $V:=D(A\_0)=D(A\^[1/2]{})$.
Denote $||||$ the norm in $V$. $$((u,v))= (Au,Av), \Hs u,v\in V.$$
Define
$$
g_\lam(u)=\De(b_2 u^2+b_3 u^3), \Hs u\in V.$$
Then $g\_: VH$ is locally Lipschitz, and the system (\ref{e:6.2}) can be reformulated in an abstract form:
\be\label{e:6.3}
u_t+ L_\lam u=g_\lam(u),\ee
where $L\_=A\_0\^2-A\_0$. We infer from Henry \cite{Henry}, Chap. 3 that for each $u\_0V$, (\ref{e:6.3}) has a unique global strong solution $u(t)$ in $V$ with $u(0)=u\_0$.
It is worth noticing that the problem has a natural Lyapunov function $J(u)$,
$$
J(u)=\frac{1}{2}|\nab u|^2+\int_\Omega F_\lam(u)\,dx,
\hs \mb{
where }\,
F_\lam(s)=-\frac{\lam}{2}s^2+\frac{b_2}{3}s^3+\frac{b_3}{4}s^4.
$$
\subsection{Bifurcation from the trivial solution}
It is obvious that each eigenvector $w$ of $A\_0$ corresponding to $\_k$ is also an eigenvector of $L\_$ corresponding to the eigenvalue $$\b_k(\lam):=\mu_k^2-\lam\mu_k=\mu_k(\mu_k-\lam). $$ Because $H$ has a canonical basis consisting of eigenvectors of $A\_0$, we deduce that
$ \_k()$ $(k=1,2,$) are precisely all the eigenvalues of $L\_$.
Let $\_$ be the semiflow generated by the system. We have
\bt\label{t:6.2} Assume $b\_20$. Suppose $A\_0$ has an eigenvector $w$ corresponding to $\_j$ such that $\_w\^3dx0$, and that
$0$ is an isolated equilibrium of $\_[\_j]{}$.
Then
there exist a closed neighborhood $U$ of $0$ in $V$ and a two-sided neighborhood $I\_2$ of $\_j$ such that $\_$ has a nonempty maximal compact invariant set $K\_$ in $U{0}$ for each $I\_2{\_j}$.
$U$ of $0$ in $E$ and a one-sided neighborhood $I\_1$ of $\_j$ such that $\_$ has an invariant topological sphere $\^[n-1]{}$ in $U{0}$ for each $I\_1{\_j}$, where $n=(ker(A\_0-\_j))$.
Consequently for $I\_2{\_j}$, $\_$ has at least one nontrivial equilibrium.
\et
\noindent{\bf Proof.} Since the system is a gradient-like one, by assumption it is easy to check that $S\_0={0}$ is an isolated invariant set of $\_[\_j]{}$. In what follows we check that $S\_0$ is neither an attractor nor a repeller of the restriction $\_[\_j]{}\^c$ of $\_[\_j]{}$ on $[[M]{}]{}\^c$, and hence the conclusion of theorem immediately follows from Theorem \ref{t:2.7}.
Denote $E\_j$ the space spanned by the eigenvectors of $A\_0$ corresponding to $\_j$. Then $H=E\_jE\_j\^$. Let $V\_j\^=VE\_j\^$. Then $V=E\_j V\_j\^$.
We infer from \cite{Ryba}, Chap. II, Theorem 2.1 that there is a small neighborhood $W$ of $0$ in $E\_j$ and a $C\^1$ mapping $: WV\_j\^$ with
$$
\xi(v)=0(||v||^2)\hs(\mb{as }||v||\ra 0)
$$
such that $[[M]{}]{}\^c={v+(v):vW}$ is a local center manifold of $\_[\_j]{}$.
For $u=v+(v)$, where $vW$, simple computations show that
$$
J(u)=\frac{1}{2}\int_\Omega |\xi'(v)\nab v|^2\,dx+\frac{b_2}{3}\int_\Omega v^3\,dx+o(||v||^3).
$$
Here we have used the facts that $(v),(v)E\_j\^$. Setting $v=w$, where $w$ is the eigenvector of $A\_0$ given in the theorem, then
since $’(v)=0(||v||)$, we have \be\label{e7.16}
J(\tau w)=\tau^3\,\frac{b_2}{3}\int_\Omega w^3\,dx+o(|\tau|^3)\hs\mb{as }\tau\ra0.
\ee
As $ \_w\^3dx0$, by \eqref{e7.16} it is clear that $0$ is neither a local maximum nor minimum point of $J$, which completes the proof of what we desired. \,$$
\Vs
The following result demonstrate some global features of the dynamic bifurcation of the system.
\bt\label{t:6.3} Suppose $0$ is an isolated equilibrium of $\_[\_j]{}$\,. Let $$ be the bifurcation branch in $V$ from the bifurcation point $(0,\_j)$. Set
$$
\Lam_0=\inf\{\lam:\,\,\Gamma_\lam\ne\emp\},\hs \Lam_1=\sup\{\lam:\,\,\Gamma_\lam\ne\emp\}\,,
$$
where $\_={u:(u,)}$ is the $$-section of $$.
Then $-0, (\^+)={0},$$$$J((\^-))<0, (\^-)={0}.$$
It is worth noticing that both $\a(\sig^+)$ and $\omega(\sig^-)$ in (3) consist of nontrivial equilibrium points. Therefore when (3) occurs, $\Phi_{\lam_1}$ has at least two distinct nontrivial equilibria. When $\Gamma_{\lam_1}$ contains only a finite number of equilibria, each of the two limit sets $\a(\sig^+)$ and $\omega(\sig^-)$ consists of exactly one equilibrium. Consequently $\sig^\pm$ become heteroclinic orbits. [**Proof of Theorem \[t:6.3\].**]{} It can be easily shown that if $\lam<0$ is large enough, then the trivial solution $0$ is the global attractor of $\Phi_\lam$. Hence we necessarily have $\Lam_0>-\8$. The existence of local bifurcation branch also implies $\Lam_0<\Lam_1$.
Assume $\Lam_1<+\8$ (otherwise (1) holds true, and thus we are done). Then $I=[\Lam_0,\Lam_1]$ is a compact interval. Therefore we infer from the proof for the existence of a global attractor of the system in Temam [@Tem] (see also [@LZ] etc.) that the system is dissipative uniformly with respect to $\lam\in I$. Specifically, there is a bounded set $B\subset V$ such that \[e7.0\][[A]{}]{}\_B,I,where ${{\mathcal A}}_\lam$ is the global attractor of $\Phi_\lam$. Thus the bifurcation branch $\Gamma$ is bounded. Hence by Theorem \[gbt\] we conclude that either (2) holds, or there is a $\lam_1\ne \mu_j$ such that $(0,\lam_1)\in \Gamma$. To complete the proof of the theorem, there remains to check the alternatives in (3).
So we assume that $(0,\lam_1)\in \Gamma$ for some $\lam_1\ne \mu_j$. Suppose the first case (i) in (3) does not occur. Then $0$ is an isolated equilibrium of $\Phi_{\lam_1}$. Fix a $\de_1>0$ such that $\Phi_{\lam_1}$ has no equilibria other than the trivial one in the $\de_1$-neighborhood ${\mb{B}}_{\de_1}$ of $0$ in $V$. By the definition of bifurcation branch we deduce that there exists a sequence $\nu_k\ra \lam_1$ such that for each $k$, $\Phi_{\nu_k}$ has a nonempty compact invariant set $M_k\subset \Gamma_{\nu_k}$ with $0\not\in M_k$ such that \[e7.1\] \_[k]{}d(0,M\_k)=0. For convenience, denote ${{\mathscr E}}(\Phi_\lam,M)$ the set of equilibria of $\Phi_\lam$ in $M\subset V$. Let $${{\mathscr E}}_k:={{\mathscr E}}\(\Phi_{\nu_k},M_k\).$$ Then ${{\mathscr E}}_k$ is a nonempty compact subset of $M_k$. As we have assumed that (i) does not occur, it can be easily seen that there exists $0<\de<\de_1$ such that \[e:6.10\] \_[k]{}d(0,[[E]{}]{}\_k)4>0.
By , for each $k$ we can pick a $u_k\in M_k$ such that the sequence $u_k\ra0$ as $k\ra\8$. It can be assumed that \[e7.2\]||u\_k||<for all $k$ (hence $d(u_k,{{\mathscr E}}_k)>3\de $). Let $\gamma_k$ be a complete trajectory of $\Phi_{\nu_k}$ contained in $M_k$ with $\gamma_k(0)=u_k$. We have \[e:6.11\] \_[t0]{}J(\_k(t))=J(\_k(0))=J(u\_k)0,k. Set $$t_k=\min\{s<0:\,\,\max_{t\in[s,0]}||\gamma_k(t)-u_k||\leq 2\de\}.$$ Noticing that $\a(\gamma_k)\subset {{\mathscr E}}_k$, we deduce by (\[e:6.10\]) and that $t_k>-\8$, and hence $||\gamma_k(t_k)-u_k||=2\de.$ Thereby \[e7.13\]||\_k(t\_k)||3,k1.
Define a sequence of complete trajectories $\sig_k$ as \[e:6.12\] \_k(t)=\_k(t\_k+t),t. Since all these trajectories are contained in the bounded set $B$ in , by very standard argument it can be shown that $\sig_k$ has a subsequence (still denoted by $\sig_k$) converging uniformly on any compact interval to a complete trajectory $\sig^+$. It is trivial to check that $\sig^+$ is contained in $\Gamma_{\lam_1}$. Observing that $\sig^+(0)=\lim_{k\ra\8}\gam_k(t_k)$, by we deduce that \[e7.15\]||\^+(0)||3.Because $$J(\sig_k(0))\geq J(\sig_k(-t_k))=J(\gam_k(0))=J(u_k)\ra 0$$ as $k\ra\8$, we also have $J(\sig^+(0))\geq 0$.
On the other hand, as $\Phi_{\lam_1}$ has no equilibrium in ${\mb{B}}_{\de_1}\sm \{0\}$, by we see that $\sig^+(0)$ is not an equilibrium of $\Phi_{\lam_1}$. Hence there is a small open interval $I_\ve=(-\ve,\ve)$ such that $J(\sig^+(t))$ is strictly decreasing in $t$ on $I_\ve$. Consequently $$J(\a(\sig^+))\equiv \mb{const.}>J(\sig^+(0))\geq 0.$$
In what follows we show that $\omega(\sig^+)=\{0\}$. If $t_k$ has a bounded subsequence (still denoted by $t_k$) with $t_k\ra -\tau\leq 0$, then $$\sig^+(\tau)=\lim_{k\ra\8}\sig_k(-t_k)=\lim_{k\ra\8}\gam_k(0)=\lim_{k\ra\8}u_k=0.$$ Hence $\sig^+(t)\equiv 0$ for $t\geq\tau$, which contradicts . Thus we know that $t_k\ra-\8$. Since $
||\gam_k(t)-u_k||\leq 2\de$ for $ t\in[t_k,0],
$ we have $$||\gam_k(t)||\leq ||u_k||+ 2\de\leq 3\de,\Hs t\in[t_k,0].$$ Thereby $$||\sig_k(t)||\leq 3\de,\Hs t\in[0,-t_k],$$ from which it follows that $||\sig^+(t)||\leq 3\de$ for all $t\geq 0$. As $0$ is the unique equilibrium of $\Phi_{\lam_1}$ in ${\mb{B}}_{\de_1}$ and $3\de<\de_1$, we immediately conclude that $\omega(\sig^+)=\{0\}$.
Likewise, we can prove that there is a complete trajectory $\sig^-$ in $\Gamma_{\lam_1}$ such that $$J(\omega(\sig^-))\equiv \mb{const.}<0, \hs \a(\sig^-)=\{0\}.$$ The proof of the theorem is finished. $\Box$
We have assumed in Theorems \[t:6.2\] and \[t:6.3\] that the trivial solution $0$ of the system is an isolated equilibrium of $\Phi_{\mu_j}$. In general it seems to be difficult to verify this condition due to the degeneracy. However, in some particular but important cases one can really do so. For instance, if $b_2=0$ then it can be shown that the equilibrium $0$ is isolated with respect to $\Phi_{\mu_j}$ (see the proof of Theorem 9.4 in Ma and Wang [@MW1b]).
0.2in
[**Acknowledgements:**]{} D. Li is supported by NSFC-10771159, 11071185 and Z.-Q. Wang is partially supported by NSFC-11271201. The authors would like to express their gratitude to the referees for their valuable comments and suggestions which helped them greatly improve the quality of the paper.
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[^1]: Corresponding author
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this article we introduce [PINGSoft]{}, a set of IDL routines designed to visualise and manipulate, in an interactive and friendly way, Integral Field Spectroscopic data. The package is optimised for large databases and a fast visualisation rendering. Here we describe its major characteristics and requirements, providing examples and describing its capabilities. The [PINGSoft]{} package is freely available at: <http://www.ast.cam.ac.uk/research/pings>'
address:
- 'Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK.'
- 'Departamento de Astrofísica Molecular e Infrarroja, IEM-CSIC, C/Serrano 121, 28006, Madrid, Spain.'
author:
- 'F. F. Rosales-Ortega'
bibliography:
- 'elsarticle.bib'
title: |
[PINGSoft]{}: an IDL visualisation and manipulation tool\
for Integral Field Spectroscopic data
---
techniques: spectroscopic ; methods: data analysis ; integral field spectroscopy
Introduction
============
The data reduction, manipulation and visualisation of Integral Field Spectroscopy (IFS) observations possess an intrinsic complexity given the nature of the data and the vast amount of information recorded even in a single observation. In spite of the obvious advantages of this technique in tackling known scientific problems, in opening up new lines of research, and the increasing number of instruments available to astronomers, 2-dimensional (2D) spectroscopy is a technique that is relatively little used.
Nowadays only few groups around the world are capable of reducing and analysing, in a systematic and homogeneous way, the huge volume of data generated by IFS observations, and these groups tend to be involved with a particular instrument, implying that most 2D data reduction, visualization and analysis packages are orientated towards and limited to a single instrument, so that experience with one instrument does not necessarily guarantee the ability to work on another.
The data processing of IFS observations requires several steps before any science can be extracted, and some of them are particular for specific science cases. They include the (complex) data reduction, mapping, source extraction, continuum fitting, emission/absorption line fitting, deconvolution, cross-correlation, etc. Most of these operations are not independent tasks, and visual/spatial checking of the cube at every step is essential for a correct data treatment and eventual scientific interpretation. On this regard, exploring, visualising and manipulating IFS reduced data by standard means still represents a challenge in the 2D spectroscopy community, fact that perhaps has discouraged part of the community into getting involved to this field.
There have been important efforts towards a standardisation of IFS visualisation and manipulation tools, one is represented by the Euro3D Research Training Network [@Walsh:2002p3819], who developed the E3D visualization tool [@Sanchez:2004p2632], a package which allows the user to view an 2D image at any wavelength slice, to explore the spectra at any spaxel, and to perform simple analyses, being capable of handling several data formats. Its main limitation resides in the multiple library dependencies during installation, specially important for non-native linux platforms. [QFitsView]{} by Thomas Ott, is a generic FITS viewer program capable of handling IFS data and performing basic analysis operations on practically any OS flavour. However on most cases, the final reduced cubes need to be reformatted in order to use this tool.
Other major astronomical packages include visualisation and manipulation tools for IFS data as part of their releases, examples are the [GAIA]{} and [DATACUBE]{} packages, as part of the Starlink project[^1], the [wavextract]{} IRAF[^2] task (by Jeremy Walsh) and some expected 3D tools in the upgrade of the Groningen Image Processing System, [GIPSY]{} [@vanderHulst:1992p3831]. The recently launched [p3d]{} software for IFS data reduction [@Sandin:2010p3798] also integrates some level of visualisation, but it is restricted to the reduction pipeline[^3]. All these tools allow a quick-look and basic manipulation of IFS observations, but depend to a certain extent on their major software packages, and in some cases, on compilation libraries and data formats. Furthermore, they are restricted to a graphical user interface (GUI), without a command-line based option, which might be a restriction while handling large IFS databases.
In order to overcome some of these limitations, the [PINGSoft]{} package was envisaged, i.e. a set of tools especially designed to visualise and manipulate, in an interactive and friendly way, IFS data regardless of the original instrument and spaxel size/shape, able to run on practically any computer platform and with minimal library requirements. In this article, the [PINGSoft]{} package is presented, including a general description of the program, installation requirements, examples of its performance with real data and a summary of all routines included within the distribution, in the hope that the community will find it useful, and in a way to contribute with the proliferation of IFS-based astronomical research.
The [PINGSoft]{} package
========================
The [PINGS software]{}, or [PINGSoft]{}, was originally developed during the PPAK IFS Nearby Galaxies Survey, or [PINGS]{} project [@RosalesOrtega:2010p3794], which used the PMAS [@Roth:2005p2463] Integral Field Unit (IFU) in the PPAK mode [@Verheijen:2004p2481; @Kelz:2006p3341; @Kelz:2006p338] at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, Spain. Given the large size of the 2D spectroscopic mosaics observed by PINGS, these routines were conceived to handle a large amount of data, therefore their implementation for equal or smaller fields of view and/or with similar instrumental setups is straightforward.
[PINGSoft]{} is written in the IDL programming language[^4] and consists of a set of individual routines called as command lines with a specific syntax in an IDL running terminal. The main drawback of the IDL environment is its commercial character[^5]. However, nowadays practically any astronomical centre around the world has a running institutional license of IDL available for its community, making possible to install very easily any IDL-based software, without compilation/library dependencies and/or platform issues.
The [PINGSoft]{} routines should be seen as the starting point to visualise and manipulate the (usually) large data formats produced by any IFU observation. More sophisticated visualisation and analysis would require tailored-built codes for the specific instrument and scientific case. However, given that the [PINGSoft]{} codes are not in the form of pre-compiled binaries and the source is completely open, the user can extract and modify any of the routines for more specific and personalised tasks, which is another advantage compared to other pre-compiled visualisation software.
In terms of data formats, practically any IFS data can be adapted to work with [PINGSoft]{}, regardless of the original data format (e.g. 3D cubes, RSS, FITS tables, etc.), and the size/shape of the spaxel. Given that these routines were developed for the PPAK instrument, they are perfectly suited for they immediate implementation to the [CALIFA]{} data[^6].
The public version of [PINGSoft]{} includes the basic tools to visualise spatially and spectrally the IFS data, to extract regions of interest by hand or within a given geometric aperture, to integrate the spectra within a given region, to read, edit and write IFS FITS files, and to perform simple analyses to the IFS data. Additionally, some miscellaneous codes useful for generic tasks performed in astronomy and spectroscopy are also included. For more sophisticated visualisation/analyses the updated versions of the [E3D]{} and [R3D]{} software [@Sanchez:2004p2632; @Sanchez:2006p331] are recommended, which include several routines for a detailed semi-automated spectroscopic analysis (SSP continuum fitting, emission line fitting, etc.).
Installation requirements
=========================
All the installation requirements and instructions are explained in detailed in the [PINGSoft]{} documentation found at the project webpage. Here we just summarise the main requirements and installations steps to give the reader an idea of the [*complexity*]{} in the installation of [PINGSoft]{}.
In order to run properly, [PINGSoft]{} should be installed in a UNIX, Linux, Mac or Windows computer via a terminal window running any IDL version greater than 6.0. The user should download the TAR file containing the [PINGSoft]{} library (i.e. [pingsoft.tar.gz]{}) from the project webpage, then extract it into a folder of his preference. This will create a directory named [pingsoft/]{}, with all the codes of the distribution and additional subdirectories. In order to work properly, [PINGSoft]{} requires the entire content of both the [NASA IDL Astronomy User’s Library]{} and the set of routines created by David W. Fanning, known as the [Coyote Library]{}. As these are very common IDL routines, they are probably already installed as part of the IDL library of the user’s institute. Otherwise, a personal copy can be downloaded from <http://idlastro.gsfc.nasa.gov> and <http://www.dfanning.com/documents/programs.html> respectively; both libraries must be extracted and installed in the same way as [PINGSoft]{}. Finally, the [pingsoft/]{} directory (as well as any other new library) and a [PINGSOFT\_PATH]{} variable should be defined in the user’s system, e.g. at the startup script that controls the shell of the OS system.
This is everything that the user needs to install the package, to check if the installation was done correctly, open a [**new**]{} terminal and type:
% echo $PINGSOFT_PATH
which should print something like: [/path\_to\_your\_IDL\_directory/pingsoft]{}\
Then, in an IDL running terminal:
The RSS data format
===================
[PINGSoft]{} works with the Row Stacked Spectra or RSS format [@Sanchez:2004p2632; @Sanchez:2006p331], plus a corresponding position table in [ASCII]{} format. A RSS file consists in a 2D FITS image in which the $X$-axis corresponds to the dispersion axis, and the other one corresponds to a given spatial ordering of the spectra determined by the position table, i.e. the $N$-row in the $Y$-axis corresponds to the spectrum at the position $(X_N,\,Y_N) \equiv
(\Delta {\rm RA}_N,\,\Delta {\rm Dec}_N)$ from a $(0,\,0)$ reference point (in arcseconds), which is the $N$ entry of the [ASCII]{} file position table. The number of rows on the RSS FITS file is equal to the total number of spectra of the IFS data, i.e. the spectra are [*stacked one on top of each other*]{} in the $Y$-axis. This is the standard output format for reduced PMAS/PPAK data, and for the PINGS and CALIFA projects.
For instruments in which the standard output is a 3D spectral cube, individual spectra can be extracted at any spatial position and stored in a RSS file. The spatial location of the spectra can be then recorded in a [ASCII]{} file to create the position table. This technique has been tested successfully with data obtained with the VIMOS instrument [@LeFevre:1998p3053]. The [ cube2rss.pro]{} routine included in [PINGSoft]{} can easily convert a 3D cube into a RSS file.\
The format of the input position table should be the following:
where the first line determines the size and shape of the spaxel (see below), and the following entries correspond to: [ID$_{\#}$ X$_{\rm offset}$ Y$_{\rm offset}$ flag]{}, where [ID$_{\#}$]{} is the spectrum number identification (integer value) and [flag]{} might be any numerical value (used sometimes for internal quality control). The size and shape of the spaxels is determined by the [*first two entries*]{} of the first line of the position table, the first character should be either a [C]{} or [S]{}, which corresponds to a [**c**]{}ircular (e.g. fibre, PPAK) or [**s**]{}quare (e.g. PMAS, VIMOS) spaxel, and the second entry should be a floating number corresponding to the radius or side length respectively, e.g.:\
C 1.34 ('C'ircular spaxel, with a radius 1.34 arcsec, e.g. PPAK)
S 0.67 ('S'quare spaxel, with sides of length 0.67 arcsec, e.g. VIMOS)
the [\$PINGSOFT\_PATH/pos\_tables]{} directory contains more position table examples.\
In addition to the format restrictions mentioned above, the RSS file should be wavelength calibrated and the spectral information should be included in the FITS header, namely the [CRPIX1]{}, [CRVAL1]{} and [CDELT1]{} values. If the IFS data to analyse is in the RSS format with corresponding position tables, then [PINGSoft]{} should work smoothly and the visualisation/manipulation should be straightforward (e.g. [CALIFA]{} data).
[PINGSoft]{} by examples
========================
All the [PINGSoft]{} routines are called via command lines in a terminal running IDL. The syntax of any program can be obtained by entering the name of the procedure without any other parameter. The routines are also documented within the [.pro]{} files, where an explanation, syntax and examples of the routines are included as comments at the beginning of the file.
The potential of the software can be illustrated by introducing the prototype procedure from which the rest of the main routines were based. This code is called [view\_rss.pro]{}, which displays interactively a visualisation of any RSS file, including its spatial and spectral information. The [PINGSoft]{} distribution includes some RSS example files in the directory [\$PINGSOFT\_PATH/examples/]{}, they correspond to different versions of a PINGS dithered observation of the central part of , covering from 6000 to 6650 Å; the main files are the following:
- [pings.n4625\_331.fits]{}: single PPAK exposure, corresponding to the central hexagon of the instrument; the RSS file contains 331 spectra.
- [pings.n4625\_382.fits]{}: same as before, but including the sky and calibration fibres of the instrument, for a total of 382 spectra.
- [pings.n4625\_dither.fits]{}: mosaic of the dithered observation including the three exposures (without the sky/calibration fibres); the RSS file contains 993 spectra.
- [pings.n4625\_pos\#.fits]{}: individual pointings of the mosaic described above, where \# goes from 1 to 3; each RSS file contains 331 spectra.
These files will be used in the following sections in order to introduce the main routines of [PINGSoft]{}. All the example commands used in this article can be found in the file [PINGSoft\_examples.pro]{}.
view\_rss {#view_rss .unnumbered}
---------
This routine provides a 2D interactive visualisation of the spaxels and spectra of a RSS file. Generally the program requires the name (and path) of the RSS file and its position table (as IDL strings). However, there are a number of special cases in which the program identifies the format of the RSS file (by the number of spectra and header information), and therefore the user does not need to include the entry for the position table, these cases are:
1. PMAS single pointing, all three resolutions ($16\times16$ spaxels).
2. PPAK single pointing (331 spectra).
3. PPAK 3 pointings dither mosaic (993 spectra).
4. Full PPAK pointing, including the sky and calibration fibres (382 spectra).
5. VIMOS single pointing, all resolutions in both configurations$^1$ ($40\times40$ and $80\times80$ spaxels).
6. VIMOS HR dithered ‘super-cube’ (square 4 pointing dither pattern, $44\times44$ spaxels, $29.7''\times29.7''$).
For example, to visualise a single PPAK pointing of (central hexagon, 331 fibres), type:
which produces the same visualisation as:
i.e. including explicitly the name of the position table.\
The program displays two windows (see ), on the right a visualisation of the spatial distribution of spaxels. The color-scale corresponds to a pseudo-narrow band image of a certain width centered at a given wavelength[^7]. The spatial units are assumed to be arcseconds in a standard North (up) East (left) configuration. On the left, the window shows the spectrum of the spaxel corresponding to the position of the mouse, the wavelength range is extracted from the information on the RSS FITS header, the fibre ID shown on the top of the window corresponds to the position of the spectrum in the RSS file (in the IDL format, i.e. starting at zero).\
![Screen shots of the visualisation windows generated by the [ view\_rss]{} command, on the right the spatial distribution of spaxels, on the left the spectral window showing the spectrum of a particular region of . \[fig:view\_rss\] ](view_left "fig:"){height="5cm"} ![Screen shots of the visualisation windows generated by the [ view\_rss]{} command, on the right the spatial distribution of spaxels, on the left the spectral window showing the spectrum of a particular region of . \[fig:view\_rss\] ](view_right "fig:"){height="5cm"}
Additional information is shown on the IDL terminal where the program was called, a left-click prints the spaxel information including the fibre ID, the offset from the reference spaxel (in arcsec), and if the WCS information is included in the FITS header, it shows the coordinates of the spaxel in sexagesimal and degree units[^8], e.g.\
IDL> view_rss, 'pings.n4625_331.fits'
RSS spectra viewer
==================
Move mouse over the mosaic to plot the spectra
Options:
LEFT-click: spaxel information
MIDDLE-click: selects and stores spaxels by subsequent left-clicks
RIGHT-click: QUIT
Spaxel ID RA offset Dec offset RA Dec RA deg Dec deg
269 17.4200 -12.0779 12h 41m 54.6s 41d 16m 10.8s 190.47765 41.269655
164 -3.48400 0.00000 12h 41m 52.8s 41d 16m 22.8s 190.46992 41.273010
267 -8.71893 -21.1184 12h 41m 52.3s 41d 16m 1.7s 190.46799 41.267144
39 10.4520 12.0779 12h 41m 54.0s 41d 16m 34.9s 190.47507 41.276365
A middle-button click prompts for a [PREFIX]{} used to generate a new series of files, all the subsequent left-clicks over the interactive right window will store the position and index of the selected spaxels which are outlined in red. When the program is terminated (by a right-click on the right window) the following files are created:
and the spectral window shows the integrated spectrum of the selected spaxels.\
The left panel of shows an example of some selected spaxels from the previous visualisation. The extracted RSS can be visualised again using [view\_rss]{}, e.g. if the chosen prefix was [test1]{}, then the selected spaxels can be displayed by:
IDL> view_rss, 'test1_rss.fits', 'test1_pt.txt', /draw
where [test1\_rss.fits]{} is the new created RSS file and [test1\_pt.txt]{} is the corresponding position table file, both shown in the right panel of .\
![Visualisation examples of spaxel extraction using the middle-click option of the [view\_rss]{} command. On the left, a single frame of showing the selected spaxels outlined in red. On the right, visualisation of the selected spaxels by using the [view\_rss]{} command on the generated RSS and position table files. []{data-label="fig:middle"}](middle_1 "fig:"){height="7cm"} ![Visualisation examples of spaxel extraction using the middle-click option of the [view\_rss]{} command. On the left, a single frame of showing the selected spaxels outlined in red. On the right, visualisation of the selected spaxels by using the [view\_rss]{} command on the generated RSS and position table files. []{data-label="fig:middle"}](middle_2 "fig:"){height="7cm"}
Several options are available to better visualise the data, including the central wavelength, width and intensity scaling of the pseudo narrow-band image, different fonts and colour tables, flux intensity and spectral ranges, etc. If the command name of any [PINGSoft]{} routine is entered without any keywords, the user will get online help with the correct syntax and available options, e.g.\
IDL> view_rss
CALLING SEQUENCE:
view_rss, 'RSS.fits' [, 'pos_table.txt', MIN_FLUX=min_flux, MAX_FLUX=max_flux, LMIN=lmin, LMAX=lmax, $
BAND=band, WIDTH=width, CT=ct, FONT=font, /DRAW, /LINEAR, /GAMMA, /LOG, /ASINH, /PS]
'RSS.fits': String of the wavelength calibrated RSS FITS file.
'pos_table.txt': String of the position table of the RSS file in ASCII format (in NE configuration).
(compulsory if not included in the default instruments/setups)
MIN/MAX_FLUX: Minimum/maximum flux in the spectral window to be plot,
if not set these are floating value.
LMIN/LMAX: Defines wavelength range on the spectral window,
if not set values are taken from the RSS header.
BAND: Central wavelength of the narrow band used to display the data,
defaults: BAND=6563 (i.e. Halpha) if within the spectral range,
else BAND=mean(lambda).
WIDTH: Width of the band to display the data, if not set the default is 100 Angstroms.
CT: IDL Color Table used to display the data (default ct=1, BLUE/WHITE).
FONT: Vector-drawn IDL font to be used, default: 3 (Simplex Roman)
/DRAW: Draws the contour of the spaxels.
/LINEAR: Displays the range of intensities using a linear min/max scaling.
/GAMMA: Displays the range of intensities using a power-law (gamma) scaling.
/LOG: Displays the range of intensities using a logarithmic scaling.
/ASINH: Displays the range of intensities using an inverse hyperbolic sine function scaling.
/PS: Writes a Postscript file of the spaxels visualisation.
![Example of visualisation windows generated by the [view\_rss]{} command using the [/PS]{} option. The left panel corresponds to the full bundle of the PPAK instrument, including the sky and calibration fibres. The right panel corresponds to the dithered mosaic of the nuclear part of . See the text for more details. []{data-label="fig:examples"}](example_1 "fig:"){height="7cm"} ![Example of visualisation windows generated by the [view\_rss]{} command using the [/PS]{} option. The left panel corresponds to the full bundle of the PPAK instrument, including the sky and calibration fibres. The right panel corresponds to the dithered mosaic of the nuclear part of . See the text for more details. []{data-label="fig:examples"}](example_2 "fig:"){height="7cm"}
If the [/PS]{} keyword is set, the program does not display a visualisation on the screen, instead writes a Postscript file of the distribution of spaxels (right window) with the selected display options (see ).\
![Examples of different visualisations of generated by the [view\_rss]{} command using different intensity scaling functions. Top-left panel: a linear scaling, [ /LINEAR]{}; top-right: a power-law function, [/GAMMA]{}; bottom-left: a logarithmic scaling, [/LOG]{}, bottom-right: an inverse hyperbolic sine function, [/ASINH]{}. See the text for more details. []{data-label="fig:scaling"}](scale_1 "fig:"){height="7cm"} ![Examples of different visualisations of generated by the [view\_rss]{} command using different intensity scaling functions. Top-left panel: a linear scaling, [ /LINEAR]{}; top-right: a power-law function, [/GAMMA]{}; bottom-left: a logarithmic scaling, [/LOG]{}, bottom-right: an inverse hyperbolic sine function, [/ASINH]{}. See the text for more details. []{data-label="fig:scaling"}](scale_2 "fig:"){height="7cm"} ![Examples of different visualisations of generated by the [view\_rss]{} command using different intensity scaling functions. Top-left panel: a linear scaling, [ /LINEAR]{}; top-right: a power-law function, [/GAMMA]{}; bottom-left: a logarithmic scaling, [/LOG]{}, bottom-right: an inverse hyperbolic sine function, [/ASINH]{}. See the text for more details. []{data-label="fig:scaling"}](scale_3 "fig:"){height="7cm"} ![Examples of different visualisations of generated by the [view\_rss]{} command using different intensity scaling functions. Top-left panel: a linear scaling, [ /LINEAR]{}; top-right: a power-law function, [/GAMMA]{}; bottom-left: a logarithmic scaling, [/LOG]{}, bottom-right: an inverse hyperbolic sine function, [/ASINH]{}. See the text for more details. []{data-label="fig:scaling"}](scale_4 "fig:"){height="7cm"}
[Intensity scaling]{}
The default colour-scale of the visualisation is obtained by sampling the range of intensities within the chosen narrow band into an [*ad hoc*]{} colour dynamic range of 255 values (i.e. equal to the number of values in a given IDL colour table). However, very different ranges of intensities are expected depending of the object, spectral range and signal-to-noise of the observations. Given that the main purpose of this routine is to visualise easily the IFS data, several intensity scaling functions are available to the user in order to improve the contrast and the identification of spectral features: a full linear (min/max) sampling, a power-law (gamma) function, a logarithmic transformation, and an inverse hyperbolic sine scaling.
For the [/GAMMA]{} transformation, the default value is $\gamma = 0.7$; for the [/LOG]{} scaling, the exponent default value is 2; for the [/ASINH]{} function, the default $\beta$ value is 10. For a full explanation of the scaling functions and these parameters see the documentation of the individual routines. shows the visualisation of the dithered mosaic of using the different intensity scalings mentioned above.\
![ Examples of spectra extraction using a slit aperture (top-left) and a circular aperture (top-right) on the dithered mosaic of , using the [extract\_slit]{} and [extract\_aperture]{} respectively. The bottom panels show the integrated spectra extracted within each aperture. \[fig:extract\] ](n4625_slit "fig:"){height="7cm"} ![ Examples of spectra extraction using a slit aperture (top-left) and a circular aperture (top-right) on the dithered mosaic of , using the [extract\_slit]{} and [extract\_aperture]{} respectively. The bottom panels show the integrated spectra extracted within each aperture. \[fig:extract\] ](n4625_aper "fig:"){height="7cm"} ![ Examples of spectra extraction using a slit aperture (top-left) and a circular aperture (top-right) on the dithered mosaic of , using the [extract\_slit]{} and [extract\_aperture]{} respectively. The bottom panels show the integrated spectra extracted within each aperture. \[fig:extract\] ](spec_right "fig:"){height="6cm"} ![ Examples of spectra extraction using a slit aperture (top-left) and a circular aperture (top-right) on the dithered mosaic of , using the [extract\_slit]{} and [extract\_aperture]{} respectively. The bottom panels show the integrated spectra extracted within each aperture. \[fig:extract\] ](spec_left "fig:"){height="6cm"}
[PINGSoft]{} list of routines {#sec:list}
=============================
The main visualisation code in [PINGSoft]{} is [view\_rss]{}, there is an additional routine for visualising specific sections of a RSS file where the ID of the spaxels are known (see below). The rest of the routines in the [PINGSoft]{} can be classified into three categories: a) Spectra extraction and integration; b) RSS and FITS manipulation and c) Miscellaneous codes. In this section we include a description of all the available [PINGSoft]{} routines, it is beyond the scope of this article to explain every single code and give examples in each case. Detailed descriptions can be found in the documentation.\
[**Visualisation**]{}
[view\_rss]{}: Provides a 2D interactive visualisation of the spaxels and spectra of a RSS file.
[view\_spaxel]{}: Displays the spectra of previously selected spaxels.
![ Example of an average radial extraction using both the [extract\_radial]{} and [shift\_ptable]{} routines on a VIMOS field of the local LIRG , after @Arribas:2008p3550. Top left-panel: extraction performed at the centre of the field. Top right-panel: extraction performed with a shifted reference point, centered on the nucleus of the South-East galaxy. The bottom panels show the integrated spectra within each ring, with radius increasing from the bottom to top in each case. \[fig:radial\] ](radial_1 "fig:"){height="7cm"} ![ Example of an average radial extraction using both the [extract\_radial]{} and [shift\_ptable]{} routines on a VIMOS field of the local LIRG , after @Arribas:2008p3550. Top left-panel: extraction performed at the centre of the field. Top right-panel: extraction performed with a shifted reference point, centered on the nucleus of the South-East galaxy. The bottom panels show the integrated spectra within each ring, with radius increasing from the bottom to top in each case. \[fig:radial\] ](radial_2 "fig:"){height="7cm"} ![ Example of an average radial extraction using both the [extract\_radial]{} and [shift\_ptable]{} routines on a VIMOS field of the local LIRG , after @Arribas:2008p3550. Top left-panel: extraction performed at the centre of the field. Top right-panel: extraction performed with a shifted reference point, centered on the nucleus of the South-East galaxy. The bottom panels show the integrated spectra within each ring, with radius increasing from the bottom to top in each case. \[fig:radial\] ](spec_1 "fig:"){height="6cm"} ![ Example of an average radial extraction using both the [extract\_radial]{} and [shift\_ptable]{} routines on a VIMOS field of the local LIRG , after @Arribas:2008p3550. Top left-panel: extraction performed at the centre of the field. Top right-panel: extraction performed with a shifted reference point, centered on the nucleus of the South-East galaxy. The bottom panels show the integrated spectra within each ring, with radius increasing from the bottom to top in each case. \[fig:radial\] ](spec_2 "fig:"){height="6cm"}
[**Spectra extraction and integration**]{}
[extract\_index]{}: Extracts new a RSS file and generates a new position table based on an index vector.
[extract\_region]{}: Extracts the spectra of regions selected by hand.
[extract\_slit]{}: Extracts the spectra within a rectangular aperture, resembling a long-slit observation (see left-panel of ).
[extract\_aperture]{}: Extracts the spectra within a circular aperture (see right-panel of ).
[extract\_radial]{}: Extracts radial average spectra within consecutive rings from a reference point (see ).
[integrate\_rss]{}: Integrates the spectra contained in a RSS file into a single spectrum.
[**RSS and FITS manipulation**]{}
[read\_rss]{}: Reads a RSS FITS file and stores the data into an IDL vector.
[merge\_rss]{}: Merges a list of RSS files into a single RSS file.
[show\_hdr]{}: Shows on screen the header of a FITS file, which can be written to an ASCII file.
[write\_hdr]{}: Adds or updates an entry in the header of a FITS file, using the [fxaddpar.pro]{} utility.
[copy\_hdr]{}: Copies the header of one FITS file to another.
[**Miscellaneous codes**]{}
[cube2rss]{}: Converts a 3D FITS cube with dimensions $X$, $Y$, $\lambda$ to a RSS FITS file + an [*ad hoc*]{} position table in ASCII format.
[write\_wcs]{}: Adds or updates the WCS (World Coordinate Systems) entries in a FITS header.
[get\_new\_pt]{}: Generates a new position table based on an index of selected spaxels.
[shift\_ptable]{}: Shifts the reference point or applies an offset to a given position table (an application of this routine is shown in ).
[merge\_ptable]{}: Concatenates a list of position table files into a single one for mosaicking purposes.
[offset2radec]{}: Transforms small angle offsets in arcsec from a reference point to equatorial coordinates.
[radec2offset]{}: Transforms equatorial coordinates to small angle offsets from a given reference point.
Summary
=======
The [PINGSoft]{} package presented in this article is a set of IDL routines developed for the [PINGS]{} project with a special emphasis on visualisation and manipulation of IFS-based data. One of its major advantages with respect other IFS visualisation tools reside in its portability to practically any OS platform with a running version of the IDL data language, a common software in most astronomical research institutes nowadays. The code is completely transparent to the user, allowing to create tailored-based routines depending on each scientific case. The package is run via command lines, which allows more flexibility during repetitive tasks (especially when dealing with large databases), and more precision in some cases (e.g. when defining extraction position/apertures). On the other hand, the spatial and spectral visualisations are fully interactive and optimised for large data files (e.g. mosaics), making the visualisation rendering faster and less prone to errors than other GUI-based visualisation tools.
[PINGSoft]{} includes routines to extract regions of interest by hand or within a given geometric aperture, to integrate the spectra within a given region, to convert 3D cubes to the RSS format, to read, edit and write RSS FITS files, and some other miscellaneous codes especially useful in astronomy and spectroscopy. [PINGSoft]{} is far from being perfect or complete, the main intention is to help a broad audience to be more familiar with IFS data, but bugs, errors and inconsistencies (especially with instruments not tested so far) are expected. For comments, suggestions and bug reports please contact the [author](mailto:[email protected]). The [PINGSoft]{} package is freely available at:\
<http://www.ast.cam.ac.uk/research/pings>\
under the [Software]{} section. If you find this package useful on your research, please acknowledge the use of [PINGSoft]{} by citing the corresponding reference in your publications. [PINGSoft]{} is licensed under GPLv3[^9].
[*Acknowledgements.*]{} I would like to acknowledge the Mexican National Council for Science and Technology (CONACYT) and the Direcci[ó]{}n General de Relaciones Internacionales (SEP) for the financial support during the period in which PINGSoft was developed.
[^1]: See: <http://starlink.jach.hawaii.edu/starlink>
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^3]: More information on IFS software can found at the [IFS wiki]{} [@Westmoquette:2009p3557 <http://ifs.wikidot.com/>], which is a dedicated webpage with excellent information on nearly every area of the IFS technique, especially for the novice user.
[^4]: IDL, the Interactive Data Language, is a computing environment for data analysis, data visualization, and software application development, available from ITT Visual Information Solutions (<http://www.ittvis.com/ProductServices/IDL.aspx>).
[^5]: Up to date, [ PINGSoft]{} cannot be run natively on the GNU Data Language, or [GDL]{}, i.e. the free IDL compatible incremental compiler, since the built-in IDL [POLYFILL]{} procedure is still not available in the last GDL release, which is one of the main procedure on which [PINGSoft]{} relies. Once GDL includes this function, the [*commercial character*]{} of IDL preventing the use of [PINGSoft]{} will not be an issue.
[^6]: Calar Alto Legacy Integral Field spectroscopy Area survey, Sánchez et al. (in preparation) see:\
<http://www.caha.es/sanchez/legacy/oa/>
[^7]: Default width: 100 Å. Default wavelength: H$\alpha$, $\lambda$6563 if within the spectral range, otherwise is equal to the mean of the wavelength range.
[^8]: Set the width of the terminal big enough so that you can see correctly this information.
[^9]: The GNU General Public License, found at: <http://www.gnu.org/licenses/gpl.html>
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Metal microbridges with a high impurity content and shorter than the energy relaxation length are considered. Their conductance is calculated with allowance made for the Coulomb electron-electron interaction. It is shown that nonequilibrium electrons in the microbridges gives rise to a nonlinear current-voltage characteristic.'
address: ' Institute of Radioengineering and Electronics, Russian Academy of Sciences, Mokhovaya ulica 11, 103907 Moscow, Russia'
author:
- 'K. E. Nagaev'
title: |
Nonlinear conductivity of diffusive normal-metal contacts\
Physics Letters A [**189**]{}, 134 (1994)$^*$
---
[2]{}
It is common knowledge that tunnel junctions between disordered metals show nonlinear conductance at low temperatures (the so-called zero-bias anomaly), which results from the electron-electron interaction.[@1] On the other hand, short microbridges possess meny properties of tunnel junctions. Therefore, it is interesting to investigate their nonlinear properties at low temperatures.
Nonlinear conductivity of weakly disordered metals at low temperatures was investigated both experimentally[@2; @3] and theoretically[@4] in a number of papers. In these papers, the nonlinear behavior resulted from the heating of the electron gas by the current and from the temperature-dependent correction to the conductivity arising either from weak localization or electron-electron interaction effects. The resulting current-voltage characteristics were influenced, however, by several parameters describing the heat transfer from the electron gas in a massive sample, e.g., by the electron-phonon coupling constant and the acoustic transparency of the sample boundary. Therefore it was difficult to compare the theoretical results with theoretical predictions. For metal microbridges, the situation is different because the heat transfer is determined by the diffusion of hot electrons into the banks and no additional parameters are required.
In this paper we investigate the influence of Coulomb exchange electron-electron interactions on the current-voltage characteristics of disordered-metal microbridges shorter than the energy relaxation length. The characteristic energy scale in this case is the largest of the quantities $eV$ and $kT$, wher $V$ is the bias voltage and $T$ is the temperature. Therefore the electron distribution function is essentially nonequilibrium at low temperatures $kT \ll eV$, and one may expect a nonlinear current-voltage dependence.
Consider a metal microbridge connecting two massive banks with the $x$ axis directed along the contact. The microbridge length $L$ is assumed to be larger than the electron elastic mean free path and the microbridge width. Yet, $L$ is assumed to be shorter than the energy relaxation length.
In studying the current-voltage characteristics of the contact under nonequilibrium conditions, the Keldysh diagrammatic technique[@5] may be conveniently used. In this technique, three types of Green functions are used, $$G^R(1,2)
=
i\theta(t_2 - t_1)
{\langle}\psi(1)\psi^{+}(2)
+
\psi^{+}(2)\psi(1)
{\rangle},$$ $$G^A(1,2)
=
-i\theta(t_1 - t_2)
{\langle}\psi(1)\psi^{+}(2)
+
\psi^{+}(2)\psi(1)
{\rangle},$$ $$G^K(1,2)
=
-i
{\langle}\psi(1)\psi^{+}(2)
-
\psi^{+}(2)\psi(1)
{\rangle}.$$ In the absence of the electron-electron interaction, the impurity-averaged retarded and advanced Green functions are given by the well-known expressions $$G^{R(A)}({\varepsilon},{{\bf r}}, br')
=
\int\frac{d^3p}{ (2\pi)^3 }
\frac{
\exp[
i{{\bf p}}({{\bf r}}- {{\bf r}}')
]
}{
{\varepsilon}- p^2/2m \pm i/2\tau
},
\label{1}$$ where $\tau$ is the elastic relaxation time. The Green function $G^K$ obeys the integral equation $$G^K({\varepsilon}, {{\bf r}}, {{\bf r}}')
=
\frac{1}{2\pi N\tau}
\int d^3 r_1\,
G^R({\varepsilon}, {{\bf r}}, {{\bf r}}_1)$$ $$\times
G^K({\varepsilon}, {{\bf r}}_1, {{\bf r}}_1)
G^A({\varepsilon}, {{\bf r}}_1, {{\bf r}}'),
\label{2}$$ where $N$ is the electron density of states at the Fermi level. By setting ${{\bf r}}= {{\bf r}}'$ in (\[2\]), one may obtain a diffusion equation for the Green function $G^K({\varepsilon}, {{\bf r}}, {{\bf r}}')$ (see Ref. [@6]) $$D\nabla^2
G^K({\varepsilon}, {{\bf r}}, {{\bf r}})
=
0,
\qquad
D
=
\frac{1}{3}
v_F^2\tau.
\label{3}$$ In the case of massive banks, the electron distribution at the ends of the microbridge is equilibrium and therefore the boundary conditions for Eq. (\[3\]) are $$\begin{aligned}
\left.
G^K({\varepsilon}, {{\bf r}}, {{\bf r}})
\right| _{x = -L/2}
=
-2\pi N
[
1 - 2n({\varepsilon}- eV/2)
],
\nonumber\\
\left.
G^K({\varepsilon}, {{\bf r}}, {{\bf r}})
\right| _{x = L/2}
=
-2\pi N
[
1 - 2n({\varepsilon}+ eV/2)
],
\label{4}\end{aligned}$$ where $n({\varepsilon})$ is the Fermi distribution function and $V$ is the voltage drop across the contact. The current flowing through the contact may be expressed in terms of the Green function $G^K$ via the formula $$j_x
=
-\frac{e}{2m}
\int d{\varepsilon}\left(
\frac{\partial}{\partial x}
-
\frac{\partial}{\partial x'}
\right)
\left.
G^K({\varepsilon}, {{\bf r}}, {{\bf r}}')
\right| _{{{\bf r}}= {{\bf r}}'}.
\label{5}$$
7.5cm
The corrections to the current from the electron-electron interaction are represented by the three diagrams shown in Fig. 1 and their complex conjugates. In these diagrams, solid lines denote the advanced and retarded electron Green functions, the black circles denote the Green functions $G^K({{\bf r}}, {{\bf r}})$. The single dashed lines represent the impurity-potential correlator $(2\pi N\tau)^{-1}\delta({{\bf r}}- {{\bf r}}')$. The shaded rectangles denote the impurity-averaged two-particle Green functions $$P(\omega, {{\bf r}}, {{\bf r}}')
=
{\langle}G^A({\varepsilon}+ \omega, {{\bf r}}, {{\bf r}}')
G^R({\varepsilon}, {{\bf r}}', {{\bf r}})
{\rangle}_{imp},
\label{6}$$ which obey the equation $$(
i\omega + D\nabla^2
)
P(\omega, {{\bf r}}, {{\bf r}}')
=
-2\pi N\delta({{\bf r}}- {{\bf r}}'),$$ $$\left.
P(\omega, {{\bf r}}, {{\bf r}}')
\right| _{x = \pm L/2}
=
0.
\label{7}$$ The shaded semicircles denote the impurity-renormalized electron vertex $(2\pi N\tau)^{-1}P(\omega, {{\bf r}}, {{\bf r}}')$. The electron-electron interaction is represented by the wavy lines. Taking into account the Debye screening, one obtains the following equation for the retarded potential of the electron-electron interaction, $$D\nabla^2
V^R(\omega, {{\bf r}}, {{\bf r}}')
=
N^{-1}
(
-i\omega + D\nabla^2
)
\delta({{\bf r}}- {{\bf r}}'),$$ $$\left.
V^R(\omega, {{\bf r}}, {{\bf r}}')
\right| _{x = \pm L/2}
=
0.
\label{8}$$ The advanced potential $V^A$ is the complex conjugate of $V^R$.
These diagrams result in a contribution to the current density in the form $${{\bf j}}_1({{\bf r}})
=
2(2\pi)^{-5} eDN^{-3}$$ $$\times {\rm Im}\,
\int d{\varepsilon}\int d\omega
\int d^3r_1 \int d^3r_2\,
G^K({\varepsilon}, {{\bf r}}, {{\bf r}})$$ $$\times P(-\omega, {{\bf r}}, {{\bf r}}_1)
G^K({\varepsilon}- \omega, {{\bf r}}_1, {{\bf r}}_1)
V^R(\omega, {{\bf r}}_1, {{\bf r}}_2)$$ $$\times\nabla_{{{\bf r}}}
P(-\omega, {{\bf r}}_2, {{\bf r}}).
\label{9}$$ To take into account the current-conservation law, this current density should be averaged over the contact length.
Substitute the solutions of (\[3\]), (\[7\]), and (\[8\]) into (\[9\]). In the low-temperature limit $kT \ll eV$, the Fermi
distribution function is step-like and the integration over ${\varepsilon}$ may be easily performed. The integration with respect to $\omega$ in (\[9\]) may be conveniently replaced by the integration with respect to a dimensionless quantity $y$, which is related to $\omega$ by $y^2 =
(L^2/2D)\omega$. Therefore, the correction to the current flowing through the contact may be presented in the form $$I_1
=
-\frac{16}{\pi}
\frac{eD}{L^2}
\left(
s^2
\int\limits_s^{\infty} dy
\frac{ Q(y) }{ y^3 }
+
\int\limits_0^s dy
\frac{ Q(y) }{ y }
\right),
\label{10}$$ where $s = L(eV/2\hbar D)^{1/2}$ and $$Q(y)
=
\{
(1/4)y
[
\sinh(2y) - \sin(2y)
]$$ $$-
y^{-1}
[
\sinh(2y) + \sin(2y)
]$$ $$+
2y^{-1}
[
\sinh(y)\cos(y) + \sin(y)\cosh(y)
]
\}$$ $$\times
[
\cosh(2y) - \cos(2y)
]^{-1}.
\label{11}$$ The corresponding differential resistivity versus voltage curve is shown in Fig. 2. At high voltages $eV \gg \hbar D/L^2$, $I_1 \approx -(8/\pi)
(DeV/2\hbar L^2)^{1/2}$. Therefore, the voltage dependence of resistivity in this voltage range has the shape characteristic of the temperature dependence of resistivity in one-dimensional conductors.[@1] This is quite natural, because the width of the drop in the electron distribution function is now determined by the applied voltage and not by the temperature. Therefore, the characteristic momentum transfer in (\[9\]) is of the order of $(eV/\hbar D)^{1/2}$. In the opposite case of small voltages $eV \ll \hbar D/L^2$, the momentum transfer is limited by $\hbar/L$. Therefore, the conductivity tends to a constant value, i.e., $I_1 = -0.17(e^2/\hbar)V$. Hence, increasing voltage gives rise to a cross-over from a zero-dimensional case to a quasi one-dimensional case.
Note that the effect of the electric field is not reduced to a simple heating of the electron gas to some effective temperature. Instead, the electron distribution function consists of two subsequent steps positioned at ${\varepsilon}_F \pm eV/2$ (see the inset in Fig. 2) and therefore has an essentially non-Fermian shape. In principle, the nonequilibrium electron distribution may also affect the phase-breaking time and therefore the weak localization contribution to the conductance.[@4] However, weak localization corrections may be suppressed by a sufficiently strong magnetic field, and the universal conduction fluctuations[@6; @7] may be eleminated by averaging the results over a number of samples.
The author is grateful to M.E. Gershenson, G.A. Ovsyannikov, and A.V. Zaitsev for discussing the results. This work was partially supported by the Soros International Science Foundation and the Russian Foundation for Fundamental Research.
[99]{} Very recently, the nonlinear Aronov - Altshuler correction to the conductance of diffusive mesoscopic contacts was experimentally studied by H.B. Weber et al., Phys. Rev. B [**63**]{}, 165426 (2001). Though not cited there, these theoretical results are relevant to this problem. B.L. Altshuler and A.G. Aronov, in: [*Electron-electron interactions in disordered systems,*]{} eds. A.L. Efros and M .Pollak, North-Holland, Amsterdam, 1985, p.1. G.J. Dolan and D.D. Osheroff, Phys. Rev. Lett. [**43**]{}, 721 (1979). G. Bergmann, W. Wei, Y. Zou, and R.M. Mueller, Phys. Rev. B [**41**]{}, 7386 (1990). P.W. Anderson, E. Abrahams, and T.V. Ramakrishnan, Phys. Rev. Lett. [**43**]{}, 718 (1979). L.V. Keldysh, Zh. Eksp. Teor. Fiz. [**47**]{}, 1515 (1964) \[JETP [**20**]{}, 1018 (1964)\]. A.I. Larkin and D.I. Khmelnitskii, Zh. Eksp. Teor. Fiz. [**91**]{}, 1815 (1986)\[JETP [**64**]{}, 1075 (1986)\]. U. Murek, R. Schafer, and W. Langheinrich, Phys. Rev. Lett. [**70**]{}, 841 (1993).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present preliminary improved measurements of the branching fractions of the color-suppressed decays $\Bzerobar \to \Dzero \hzero$ where $\hzero$ represents the three light neutral mesons $\pizero$, $\eta$ and $\omega$. The measurements are based on a data sample of 140 $ \mathrm{fb}^{-1}\,$ collected at the $ \Upsilon(4S) $ with the Belle detector at the KEKB energy-asymmetric $e^+e^-$ collider, corresponding to seven times the luminosity of the previous Belle measurements.'
title: |
\
Improved Measurements of Color-Suppressed Decays , [$\Dzero \eta$ ]{} and [$\Dzero \omega$ ]{}
---
Introduction
============
The weak decays $\Bzerobar \ra \Dstze \hzero$ [@CC], where $\hzero$ represents a light neutral meson, are expected to proceed predominantly through internal spectator diagrams, as illustrated in Fig. \[dpi-feynman\]. The color matching requirement between the quarks from the virtual $\Wmi$ and the other quark pair results in these decays being “color-suppressed” relative to decays such as $\Bzerobar \ra \Dstpl \hmi$, which proceed through external spectator diagrams.
Previous measurements of $\Bzerobar$ decays into $\Dstze \pizero$, $\Dzero \eta$, $\Dzero \omega$, and $\Dzero \rho^0$ by the Belle collaboration [@ref:Belle; @ref:Belle2], and of $\Bzerobar$ into $\Dstze \pizero$ by the CLEO collaboration [@ref:CLEO], and of $\Bzerobar$ decays into $\Dstze \pizero$, $\Dstze \eta$, $\Dstze \omega$, $\Dzero \eta'$ by the BaBar collaboration [@ref:Babar] indicate color suppressed branching fractions in the approximate range $(2$–$4)\times 10^{-4}$. This is substantially in excess of theoretical expectations from “naive" factorization models [@ref:Beneke; @ref:NeuSte; @ref:NeuPet; @ref:Chua; @ref:Rosner; @ref:Deandrea; @ref:ChRos] in the range $(0.3$–$1.7) \times 10^{-4}$.
Several approaches to achieving a better theoretical description [@ref:NeuPet; @ref:Chua; @ref:SCET; @ref:pQCD] have been developed. They extend upon the factorization approach with consideration of final state interactions and consequent simultaneous treatment of isospin amplitudes of color-suppressed and color-allowed decays. The possibility that similar effects could have dramatic implications on the measurement potential of direct $CP$ violation asymmetries in charmless decays, together with some degree of discrepancy between the prior Belle [@ref:Belle] and BaBar [@ref:Babar] measurements provide strong motivation for more precise measurements of the color-suppressed decays.
![Tree level internal (left) and external (right) spectator diagrams for $\overline{B}\to D\pi$ decays.[]{data-label="dpi-feynman"}](figs/fig1b "fig:"){width="2in"} ![Tree level internal (left) and external (right) spectator diagrams for $\overline{B}\to D\pi$ decays.[]{data-label="dpi-feynman"}](figs/fig1a "fig:"){width="2in"}
In this paper we report improved branching fraction measurements of $\Bzerobar$ decays into $\Dzero \pizero$, $\Dzero \eta$, $\Dzero \omega$. The measurements are based on a $140~{\rm fb}^{-1}$ data sample, which contains 152 million $B\overline{B}$ pairs, collected with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ (3.5 on 8 GeV) collider [@KEKB] operating at the $\Upsilon(4S)$ resonance. This corresponds to seven times the luminosity of the previous Belle measurements [@ref:Belle] and almost twice that of the earlier BaBar measurements [@ref:Babar].
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a three-layer silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel threshold Čerenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect $K_L^0$ mesons and to identify muons (KLM). The detector is described in detail elsewhere [@Belle].
Event Selection
===============
Color-suppressed $\Bzerobar$ meson decays are reconstructed from candidate $\Dzero$ mesons that are combined with light neutral meson candidates $\hzero$. The $\Dzero$ mesons are reconstructed in three decay modes: $\Kpi$, $\Ktwopi$, and $\Kthreepi$ while the light neutral mesons $\hzero$ are reconstructed in the decay modes: $\pizero \to \gamgam$, $\eta \to \gamgam$, $\eta \to \threepi$ and $\omega \to \threepi$. The invariant masses at each stage of the decay chains are required to be consistent within $2.5-3 \sigma$ mass resolution or natural width windows around the nominal masses of the assumed particle types. Vertex and mass constrained fits are performed for decays with charged products such as the three $\Dzero$ decays and $\eta \to \threepi$; mass constrained fits are performed on the $\pizero \to \gamgam$ and $\eta \to \gamgam$ candidates; and vertex constrained fits are performed on $\omega \to \threepi$ candidates due to the large natural width of the $\omega$ meson. These kinematic fits result in improved energy and momenta of the candidate mesons.
Charged tracks are required to have impact parameters within $\pm 5 \cm$ of the interaction point along the positron beam axis and within $1 \cm$ in the transverse plane. Each track is identified as a kaon or pion according to a likelihood ratio derived from the responses of the TOF and ACC systems and energy loss measurements from the CDC. The likelihood ratio is required to exceed 0.6 for kaon candidates. This requirement is $88\%$ efficient for kaons with a misidentification rate for pions of $8.5\%$.
The photon pairs that constitute $\pizero$ candidates are required to have energies greater than 50 MeV and an invariant mass within a $\pm 3\sigma$ ($\sigma = 5.4 \MeVcsq$) mass window around the nominal $\pizero$ mass. Candidate $\eta$ mesons that decay to $\gamgam$ are required to have photon energies $E_{\gamma}$ greater than $100 \MeV$. In addition the energy asymmetry $\frac{|E_{\gamma_1}-E_{\gamma_2}|}{E_{\gamma_1}+E_{\gamma_2}}$, is required to be less than 0.9. The $\eta$ candidates are required to have invariant masses within $2.5\sigma$ mass windows of the nominal mass, where $\sigma = 10.6\MeVcsq$ for the $\eta \to \gamgam$ mode and $3.4 \MeVcsq$ for the $\eta \to \threepi$ mode. If the photons that comprise the $\eta \to \gamgam$ candidate are found to contribute to any $\pizero \to \gamgam$ the candidate is excluded. The $\pizero$ decay products of the $\eta \to \threepi$ and $\omega \to \threepi$ candidates are required to have CM momentum greater than $200$ and $500 \MeV$, respectively. The $\omega$ candidates are required to have invariant masses within $\pm3 \Gamma$ of the nominal mass value, where $\Gamma$ is the natural width of $8.9 \MeVcsq$.
Invariant masses of the $\Dzero$ candidates are required to be within $\pm 2\sigma$ of the nominal mass where $\sigma$ is $8, 12$ and $5 \MeVcsq$ for the $\Kpi$, $\Ktwopi$, and $\Kthreepi$ modes respectively. The CM momentum of the $\pizero$ in the $\Ktwopi$ mode is required to be greater than $400 \MeV$.
reconstruction
================
The $\Bzerobar$ candidates are reconstructed from combinations of $\Dzero$ and $\hzero$ using the improved energy and momenta resulting from the vertex and mass constrained fits.
Two kinematic variables are used to distinguish signal candidates from backgrounds: the beam-energy constrained mass $\Mb = \sqrt{ (\Eb)^2 - |\sum \vec{p}_{i}^{*}|^2 )}$ and energy difference $\De = \sum E_{i}^{*} - \Eb $ where $\Eb$ is the CM energy, and $E_{i}^{*} $ , $\vec{p}_{i}^{*} $ are the CM energy and momenta, respectively, which are summed over the $\Dzero$ and $\hzero$ meson decay candidates.
The resolution of $\Mb$ is approximately $3 \MeVcsq$ for all modes, dominated by the beam energy spread, whereas the $\De$ resolution varies substantially among modes depending particularly on the number of $\pizero$ in the final state. Candidates within the broad region $|\De| < 0.25 \GeV $ and $ 5.2 \GeVcsq < \Mb < 5.3 \GeVcsq$ are selected for further consideration. Where more than one candidate is found in a single event the one with the smaller $\sum \chi_{i}^{2} / N_{i} $ is chosen, where $\chi_{i}^{2}$ and the number of degrees of freedom $N_{i}$ are obtained from the the kinematic fits to the $\Dzero$ and $\hzero$.
A common $\Mb$ signal region of $ 5.27 \GeVcsq < \Mb < 5.29 \GeVcsq$ is used for all final states. The signal region definitions in $\De$ are mode dependent with $|\De| < 0.05 \GeV $ for $\omega \to \threepi$ and $\eta \to \threepi$ modes , and $|\De| < 0.08 \GeV $ for $\pizero \to \gamgam$ and $\eta \to \gamgam$ modes. The event yields and efficiencies presented in the following sections correspond to these signal regions.
Continuum Suppression
=====================
At energies close to the $\Upsilon(4S)$ resonance the production cross section of $\epem \to \qqbar$ $( q = u,d,s,c )$ is approximately three times that of $\BB$ production, making continuum background suppression essential in all modes. The jet-like nature of the continuum events allows event shape variables to discriminate between them and the more spherical $\BB$ events.
The discrimination power of seven event shape variables is combined into a single Fisher discriminant [@fw] whose variables include the angle between the thrust axis of the candidate and the thrust axis of the rest of the event ($\cos{\theta_{T}}$), the sphericity variable, and five modified Fox-Wolfram moments [@fw].
Monte Carlo event samples of continuum $\qqbar$ events and signal events for each of the final states considered are used to construct probability density functions (PDFs) for the Fisher discriminant [@fw] and $\cos{\theta_{B}}$, where $\theta_{B}$ is the angle between the flight direction and the beam direction in the $\Upsilon(4S)$ rest frame. The products of the PDFs for these two variables give signal and continuum likelihoods ${\cal L}_{s}$ and ${\cal L}_{\qqbar}$ for each candidate, allowing a selection to be applied to the likelihood ratio ${\cal L} = {\cal L}_{s} / ({\cal L}_{s} + {\cal L}_{\qqbar} )$.
Monte Carlo studies of the signal significance $N_{s}/\sqrt{N_{s}+N_{b}}$, where $N_s$ and $N_b$ are signal and background yields (using signal branching fractions from previous measurements), as a function of a cut on the likelihood ratio ${\cal L}$ indicate a rather smooth behavior. Although the optimum significance is generally in the range 0.6-0.7, a looser cut of ${\cal L} > 0.5 $ is applied for all modes in order to reduce systematic uncertainties.
For the $\Bzerobar \to \Dzero\omega$ mode the polarized nature of the $\omega$ allows additional discrimination against backgrounds to be achieved with an additional requirement of $|\cos{\theta_{hel}}|>0.3$, where the helicity angle $\theta_{hel}$ is defined as the angle between the flight direction in the $\omega$ rest frame and the vector perpendicular to the $\omega$ decay plane in the $\omega$ rest frame.
Backgrounds from other decays
=============================
Significant background contributions arise both from color favored decays and from contributions from other color suppressed decays (crossfeed) $\Bzerobar \to \Dstarzero \hzero $. Some backgrounds have the same final state as the signal while others mimic signal due to missing or extra particles.
Generic Monte Carlo samples of $\BB$ and continuum $\qqbar$ are used to study the background contributions in the $\Mb$ and $\De$ distributions. The $\BB$ event sample excludes the color suppressed modes under investigation and associated $\Dstarzero$ $\hzero$ modes. These signal modes for each of the decay chains considered and the corresponding $\Dstarzero$ $\hzero$ decays are generated and reconstructed separately. They are used to estimate the crossfeed contributions to the other modes using the branching fractions measured here and by the BaBar collaboration [@ref:Babar]. A combined generic Monte Carlo sample weighted according to the effective production cross sections and selection efficiencies of $\qqbar$ and $\BB$ is also used.
Figures \[fig-pi0\], \[fig-eta\] and \[fig-omega\] show the $\Mb$ and $\De$ distributions after application of all selection requirements and with the $\De$ signal requirement applied for the $\Mb$ distributions (a) and the signal requirement $ 5.27 \GeVcsq < \Mb < 5.29 \GeVcsq$ applied for the $\De$ distributions (b). The $\De$ signal requirements used are $|\De| < 0.08 \GeV $ for $\Dzero\pizero$ mode , $ -0.08 \GeV < \De < 0.05 \GeV $ for $\Dzero\eta$ , and $|\De| < 0.05 \GeV $ for $\Dzero\omega$. These regions are indicated by vertical dashed lines on the figures.
The dominant crossfeed contributions to the $\Dzero \hzero$ decays are found to arise from the corresponding $\Dstarzero \hzero$ decays. These contributions peak at same $\Mb$ as the signal but are shifted to the lower side in $\De$. As can be seen from the figures, the crossfeed contribution is substantial in the region $ -0.25 \GeV < \De < -0.10 \GeV$ but quite small in the signal region. Within the signal region the fraction of crossfeed is less than $10\%$ of the observed yield in all cases. For the $\Bzerobar \to \Dzero\pizero$ mode the color allowed $\Bmi \to \Dstze \rhomi $ modes are found to be the dominant backgrounds. Non-reconstructed soft $\pizero$ from $\Dstarzero \to \Dzero \pizero $, photons from $\Dstarzero \to \Dzero \gamma $ and $\pimi$ from $ \rhomi \to \pimi \pizero $ produce the same final state as the signal. However the missing particles cause a shift in $\De$ with a broad peak centered at approximately $\De = -0.2 \GeV$. In order to reduce contributions from this background, events that contain candidates reconstructed as $\Bmi \to \Dstze \rhomi $ within the signal region $ 5.27 \GeV < \Mb < 5.29 \GeV$ and $|\De| < 0.1 \GeV $ are rejected. This requirement reduces the color allowed contribution in the region $-0.25 \GeV < \De < -0.10 \GeV$ by about $60 \%$ it does little to reduce contributions in the signal region, but remains useful to facilitate background modelling. The $\Mb$ distribution of these backgrounds is found to contribute at and slightly below the the $\Mb$ signal region.
Data modelling and Signal Extraction
====================================
Independent unbinned extended maximum likelihood fits to the $\Mb$ and $\De$ distributions are performed to obtain the signal yields. The yields from the $\Mb$ fits are used to extract the branching fractions, while the yields from the $\De$ fits are used to cross-check the results. In most cases the shapes of the signal and background component distributions in $\Mb$ and $\De$ are obtained from fits to MC samples. The signal models used are the same for all modes, with the $\Mb$ signals modelled with a Gaussian function and the $\De$ signals with an empirical formula that accounts for the asymmetric calorimeter energy response, known as the Crystal ball line shape [@cbline], added to a Gaussian function of the same mean. The signal models are represented in Figures \[fig-pi0\]- \[fig-omega\] by the dotted lines. Fits of the $\De$ distributions to signal Monte Carlo for each final state are used to obtain the signal shape parameters; all the $\Mb$ signal shape parameters are allowed to float in fits to data.
The crossfeed contributions in $\Mb$ and $\De$ are studied using a combination of signal Monte Carlo samples from all other color suppressed modes, weighted according to the branching fractions obtained here or from the Babar measurements [@ref:Babar]. Smoothed histograms obtained from this combined sample are used as estimates of the crossfeed contributions. In the $\Mb$ case the $\De$ signal region requirement results in very small crossfeed contributions, which are fixed at the Monte Carlo expectation. For $\De$ there are considerable contributions in the region $ -0.25 \GeV < \De < -0.10 \GeV$; the normalization of this component is allowed to float in the fit.
Continuum like backgrounds in the $\Mb$ fits are modelled by an empirical threshold function known as the ARGUS function [@argus]. The small peaking background contributions are modelled by a Gaussian of mean and width and normalization obtained by a fit to the $B\bar{B}$ background Monte Carlo $\Mb$ distribution, using an ARGUS function plus a Gaussian. A systematic uncertainty of 50% is assigned to the determination of this small background distribution. This treatment allows the vast majority of the background to be simply modelled with the ARGUS shape leaving a small but less well known peaking background component that represents the deviation from the ARGUS shape.
In fits to data $\Mb$ distributions the ARGUS background function parameters are fixed to the values obtained from fits to combined Monte Carlo $B\bar{B}$ and continuum background samples. The signal parameters are free, as are the normalizations of signal and background. The small peaking background and crossfeed contributions are fixed at their expected values.
The $\De$ background distributions in the $\Bzerobar \to \Dzero\eta$ and $\Bzerobar \to \Dzero\omega$ are modelled using smoothed histograms obtained from a combined continuum and generic $\BB$ Monte Carlo sample.
For the $\Bzerobar \to \Dzero\pizero$ mode, the shapes of the $\De$ distribution arising from $\BB$ and continuum background are very different, necessitating separate modelling. The continuum shape is modelled with a first order polynomial with slope obtained from fits to the continuum Monte Carlo sample. The shape of the $\BB$ background is modelled with a Gaussian function plus a second order polynomial, with parameters determined from a fit to the generic $\BB$ Monte Carlo sample. In fits to data the large peak in the region $ -0.25 \GeV < \De < -0.10 \GeV $ that arises principally from the color allowed $\Bmi \to \Dstze \rhomi $ decays is found to be broader than the Monte Carlo expectation, thus all parameters of this color allowed Gaussian are allowed to float in the fit. The normalizations of the contributions from the remainder of the $\BB$ background, the continuum and the signal are also floated in the fit, with the small crossfeed fixed as discussed above.
The results of the $\Mb$ and $\De$ fits for the combined modes are presented in Figures \[fig-pi0\], \[fig-eta\] and \[fig-omega\].
Branching Fraction results
==========================
Results of $\Mb$ and $\De$ fits are consistent; the agreement is typically within 50% of the statistical uncertainty. The results from the $\Mb$ fits are found to have a slightly smaller total uncertainty in most cases and are used for the final result. Yields are obtained both from the individual subdecay mode samples and from samples with the three $\Dzero$ subdecay samples combined. The yields from the one dimensional $\Mb$ fits are shown in Table \[yield-mb-sub\]. Both peaking backgrounds and crossfeed contributions in the signal region can be seen to contribute substantially less than the extent of the statistical uncertainty on the signal yield.
The yields obtained are interpreted as branching fractions using the number of analyzed $\BB$ events, the product of subdecay fractions from PDG [@pdg] corresponding to the decay of $\Dzero \hzero$ into the observed final states and the total selection efficiency. The efficiency for each mode is first obtained from signal Monte Carlo samples and then corrected by comparing data and MC predictions for other processes. For the $\pizero$ reconstruction efficiency the correction is obtained from comparisons of $\eta \to \pizero \pizero \pizero $ to $\eta \to \gamgam$ and to $\eta \to \threepi$, for data and Monte Carlo. The MC efficiency, correction factor and corrected efficiency are presented in Table \[eff-mb-sub\]. The corrections are obtained from the product of correction factors relevant to the final state of each submode. The branching fraction results for the individual submodes are shown in Table \[brr-mb-sub\]. The combined submode systematic uncertainties and branching fraction results are shown in Tables \[tot-mb\] and \[brr-mb\], respectively.
Systematic Uncertainties
========================
Systematic uncertainties of the combined modes, estimated for the results based on the $\Mb$ fits are summarized in Table \[tot-mb\]. Uncertainties on the efficiency correction factors relevant to each final state are listed in the Tables together with other uncertainties. For the $\Mb$ fit the uncertainty from the peaking background, which is fixed at the MC expectation in the fit, is obtained by propagating a 50% uncertainty on the normalization of this contribution. The crossfeed uncertainty is estimated as 25% of the contribution from this source in the signal regions. This accounts for uncertainties on the branching fractions of the crossfeed contributions and also differences observed between the floated crossfeed contributions in $\De$ fits and the MC expectation. Uncertainties arising from the background and signal modelling used are estimated from the changes in the yields as a result of $\pm 1 \sigma$ variations on the model parameters. The total uncertainty is obtained regarding uncertainties from different sources as uncorrelated.
Conclusion
==========
Improved measurements of the branching fractions of the color-suppressed decays $\Bzerobar \to \Dzero \pizero$, $ \Dzero \eta $ and $ \Dzero \omega $ are presented. The results are consistent with the previous Belle measurements. The total uncertainty of the new results is two to three times smaller than the previous results, mostly due to the seven times larger data sample. However comparing the results with those of BaBar [@ref:Babar] and CLEO [@ref:CLEO] indicates an approximately 2$\sigma$ difference, with all three branching fractions measured here lower than the previous measurements.
All the branching fraction results are similar, in the range 1.8-2.4 $\times 10^{-4}$. The large values disfavour theoretical predictions based on naive factorization descriptions and indicate the need for models including final state interaction effects to satisfactorily describe the observations.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the KEKB group for the excellent operation of the accelerator, the KEK Cryogenics group for the efficient operation of the solenoid, and the KEK computer group and the National Institute of Informatics for valuable computing and Super-SINET network support. We acknowledge support from the Ministry of Education, Culture, Sports, Science, and Technology of Japan and the Japan Society for the Promotion of Science; the Australian Research Council and the Australian Department of Education, Science and Training; the National Science Foundation of China under contract No. 10175071; the Department of Science and Technology of India; the BK21 program of the Ministry of Education of Korea and the CHEP SRC program of the Korea Science and Engineering Foundation; the Polish State Committee for Scientific Research under contract No. 2P03B 01324; the Ministry of Science and Technology of the Russian Federation; the Ministry of Education, Science and Sport of the Republic of Slovenia; the National Science Council and the Ministry of Education of Taiwan; and the U.S.Department of Energy.
[99]{}
Throughout this paper, the inclusion of the charge conjugate mode decay is implied unless otherwise stated.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce the concepts of growth and spectral bound for strongly continuous semigroups acting on Fréchet spaces and show that the Banach space inequality $s(A)\leqslant\omega_0(T)$ extends to the new setting. Via a concrete example of an even uniformly continuous semigroup we illustrate that for Fréchet spaces effects with respect to these bounds may happen that cannot occur on a Banach space.'
author:
- 'Sven-Ake Wegner'
title: |
The growth bound for strongly continuous\
semigroups on Fréchet spaces
---
[^1] [^2] [^3]
Introduction {#SEC-1}
============
Not all results on strongly continuous semigroups carry over from the world of Banach spaces to those of Fréchet spaces. For example, there exist semigroups with super exponential growth or uniformly continuous semigroups whose exponential series representations are divergent. From another perspective, exactly these phenomena illustrate that in the Fréchet case interesting effects may happen to take place where the picture is clear if the underlying space is Banach. We refer to Albanese, Bonet, Ricker [@ABR13] and Frerick, Jordá, Kalmes, Wengenroth [@FEJKW] for very recent results of this nature.\
In this article we first show that the concept of growth bound can be transferred to Fréchet spaces in two different ways, namely in a purely topological sense or with respect to a fixed fundamental system of seminorms. Then we consider classical and recent approaches for non Banach spectral theories to define the spectral bound. We show that for all these spectral theories and for both definitions of the growth bound the classical inequality between spectral and growth bound extends to Fréchet spaces. Finally we consider an example of an even uniformly continuous semigroup and show that depending on the notion of growth resp. spectral bound which we choose, their equality may fail. This effect cannot occur on a Banach space.\
For the theory of Fréchet spaces we refer to Jarchow [@Jarchow] and Köthe [@KoetheII]. For the theory of semigroups we refer to Engel, Nagel [@EN], Albanese, Bonet, Ricker [@ABR13], Yosida [@Yosida] and K[ō]{}mura [@Komura Section 1].
Growth Bound {#SEC-2}
============
For the whole article let $X$ be a Fréchet space, i.e., a complete metrizable locally convex space. We denote by ${\operatorname{cs}\nolimits}(X)$ the set of all continuous seminorms on $X$, by $\mathcal{B}$ the collection of all bounded subsets of $X$ and by $L(X)$ the space of all linear and continuous maps from $X$ into itself. We write $L_b(X)$, if $L(X)$ is endowed with the topology of uniform convergence on the bounded subsets of $X$ given by the seminorms $q_B(S)=\sup_{x\in B}q(Sx)$ for $S\in L(X)$, $B\in\mathcal{B}$ and $q\in{\operatorname{cs}\nolimits}(X)$. By [@Jarchow Prop. 11.2.7] a subset of $L(X)$ is equicontinuous if and only if it is bounded in $L_b(X)$ and by [@KoetheII §39, 6(5)], $L_b(X)$ is a complete locally convex space.\
Also for the whole article let ${(T(t))_{t\geqslant0}}$ be a $C_0$-semigroup on $X$. That is, $T(t)\in L(X)$ for $t\geqslant0$, $T(0)={\operatorname{id}\nolimits}_X$, $T(t+s)=T(t)T(s)$ for $t$, $s\geqslant0$ and $\lim_{t\rightarrow t_0}T(t)x=T(t_0)x$ for $x\in X$ and $t_0\geqslant0$. We say that ${(T(t))_{t\geqslant0}}$ is exponentially equicontinuous of order $\omega\in\mathbb{R}$, if $$\forall\:q\in\Gamma\;\exists\:p\in\Gamma,\,M\geqslant1\;\forall\:t\geqslant0,\,x\in X\colon q(T(t)x)\leqslant M e^{\omega t} p(x)$$ holds for some or, equivalently, for every fundamental system of seminorms $\Gamma\subseteq{\operatorname{cs}\nolimits}(X)$. The $C_0$-semigroup is said to be exponentially equicontinuous, if the above condition holds for some $\omega$ and equicontinuous, if the latter holds with $\omega=0$. Note that ${(T(t))_{t\geqslant0}}$ is exponentially equicontinuous of order $\omega$ if and only if $\{e^{-\omega t}T(t)\}_{t\geqslant0}$ is equicontinuous or, equivalently, bounded in $L_b(X)$.
Let ${(T(t))_{t\geqslant0}}$ be an exponentially equicontinuous $C_0$-semigroup. We denote by $\omega_0(T)$ the infimum over all $\omega\in\mathbb{R}$ for which $\{e^{-\omega t}T(t)\}_{t\geqslant0}$ is equicontinuous and call this number the *growth bound* of ${(T(t))_{t\geqslant0}}$.
The main ingredient for the following Lemma \[LEM-0\] is the recalibration trick of Moore [@Moore Thm. 4]. According to Joseph [@J Dfn. 3.1] we call $T\in L(X)$ universally bounded with respect to a fundamental system $\Gamma$ if there exists $N\geqslant0$ such that $q(Tx)\leqslant Nq(x)$ holds for all $q\in\Gamma$ and $x\in X$.
\[LEM-0\] Let ${(T(t))_{t\geqslant0}}$ be a $C_0$-semigroup and $\omega\in\mathbb{R}$. The following are equivalent.
The semigroup ${(T(t))_{t\geqslant0}}$ is exponentially equicontinuous of order $\omega$.
There exists a fundamental system $\Gamma$ and a constant $M\geqslant1$ such that $$\forall\:q\in\Gamma,\,t\geqslant0,\,x\in X \colon q(T(t)x)\leqslant Me^{\omega t}q(x)$$ holds.
There exists a fundamental system $\Gamma$ such that every operator in the semigroup is universally bounded with respect to $\Gamma$ and there exists a constant $M\geqslant1$ such that $$\|T(t)\|_{\Gamma}=\sup_{q\in\Gamma}\sup_{q(x)\leqslant1}q(T(t)x)\leqslant M e^{\omega t}$$ holds for all $t\geqslant0$.
(i)$\Rightarrow$(ii) Let $\Gamma'$ be some fundamental system and let $\omega$ be such that $\{e^{-\omega t}T(t)\}_{t\geqslant0}$ is equicontinuous. Define $\Gamma=\{q\:;\:q'\in\Gamma'\}$ via $q(x)=\sup_{t\geqslant0}q'(e^{-\omega t}T(t)x)$ for $x\in X$. By [@ABR13 Rem. 2.2(i)], $\Gamma$ is a fundamental system for $X$ satisfying the desired property with $M=1$. The converse is trivial.\
(iii)$\Rightarrow$(ii) Let $\Gamma$ be as in (iii). For $t\geqslant0$ there exists $N$ such that $q(T(t)x)\leqslant Nq(x)$ holds for all $q\in\Gamma$ and $x\in X$. We note that therefore, $q(x)=0$ always implies $q(T(t)x)=0$. Now we fix $q\in\Gamma$ and select $M\geqslant1$ according to (iii). Let $t\geqslant0$ and $x\in X$ be given. If $q(x)=0$ it follows $q(T(t)x)=0$ by the first sentence of this paragraph. Otherwise, we can select $\lambda>0$ such that $q(\lambda x)=1$. Then, $\lambda q(T(t)x)=q(T(t)(\lambda x))\leqslant M e^{\omega t}= M e^{\omega t}q(\lambda x)=\lambda M e^{\omega t}q(x)$ holds and whence $q(T(t)x)\leqslant M e^{\omega t}q(x)$ is valid. Again, the converse is trivial.
Let us mention that ${(T(t))_{t\geqslant0}}$ can be exponentially equicontinuous but for some $\Gamma$ the condition in Lemma \[LEM-0\](ii) can fail for every $\omega$. Consider $X=\mathbb{C}^2$, $T(t)x=(x_1+tx_2, x_2)$ for $x=(x_1,x_2)$, $t\geqslant0$. Then $\omega_0(T)=0$, cf. [@EN Ex. I.5.7(i)]. Let $\omega>0$ and $\Gamma=\{p_1,p_2\}$, $p_1(x)=|x_1|$, $p_2(x)=\max_{i=1,2}|x_i|$. Let $M\geqslant1$, select $t>0$ and $x=(0,1)$. Then, $p_1(T(t)x)=t>0$ and $p_1(x)=0$ hold. It follows $\|T(t)\|_{\Gamma}=\infty$ for $t>0$.\
From Lemma \[LEM-0\] we get $$\omega_0(T)=\inf\{\omega\in\mathbb{R}\:;\:\exists\:\Gamma,\,M\geqslant1\;\forall\:q\in\Gamma,\,t\geqslant0,\,x\in X\colon q(T(t)x)\leqslant Me^{\omega t}q(x)\}$$ where we may also restrict ourselves to $M=1$ in view of the proof of Lemma \[LEM-0\]. If one allows $M\geqslant1$, in the Banach space analogue of the above formula [@EN Dfn. I.5.6] it is not necessary to take the infimum over all norms which induce the topology of the underlying space. For Fréchet spaces and fundamental systems this is not the case.
Let ${(T(t))_{t\geqslant0}}$ be an exponentially equicontinuous $C_0$-semigroup. We define $$\omega_{0,\Gamma}(T)=\inf\{\omega\in\mathbb{R}\:;\:\exists\:M\geqslant1\;\forall\:q\in\Gamma,\,t\geqslant0,\,x\in X\colon q(T(t)x)\leqslant Me^{\omega t}q(x)\}$$ for a fixed fundamental system $\Gamma$. By definition, $\omega_0(T)$ is the infimum over all the $\omega_{0,\Gamma}(T)$.
Using the fact that the proof of (ii)$\Leftrightarrow$(iii) in Lemma \[LEM-0\] worked without a recalibration, variants of Banach space growth bound formulas [@EN Section IV.2] for $\omega_{0,\Gamma}(T)$ can be established by following the lines of their classical proofs.
\[PROP-1\] Let ${(T(t))_{t\geqslant0}}$ be an exponentially equicontinuous $C_0$-semigroup and $\Gamma$ be a fundamental system which satisfies the equivalent conditions in Lemma \[LEM-0\](ii) and Lemma \[LEM-0\](iii). Then, $$\omega_{0,\Gamma}(T)=\inf_{t>0}{\textstyle\frac{1}{t}}\log\|T(t)\|_{\Gamma} = \lim_{t\rightarrow\infty}{\textstyle\frac{1}{t}}\log\|T(t)\|_{\Gamma}={\textstyle\frac{1}{t_0}}\log(\lim_{n\rightarrow\infty}\|T(t_0)^n\|_{\Gamma}^{1/n})$$ holds for arbitrary $t_0>0$.
We start with the first and the second equality and abbreviate $\omega_{0,\Gamma}=\omega_{0,\Gamma}(T)$. Define $\xi\colon[0,\infty)\rightarrow\mathbb{R}$, $\xi(t)=\log\|T(t)\|_{\Gamma}$, which is bounded on compact intervals. Fix $s$, $t\geqslant0$ and $q\in\Gamma$. Put $C=\sup_{q(y)\leqslant1}q(T(t)y)<\infty$. If $q(x)\not=0$ we select $\lambda>0$ such that $q(\lambda x)=1$ and compute $q(T(t)x)=\lambda^{-1}q(T(t)(\lambda x))\leqslant\lambda^{-1}C q(\lambda x)=C q(x)$. If $q(x)=0$, it follows $q(T(t)x)=0$, cf. the proof of Lemma \[LEM-0\]. Therefore the estimate $q(T(t)x)\leqslant Cq(x)$ is true for all $x\in X$. We thus get that $q(T(t)T(s)x)\leqslant \sup_{q(y)\leqslant1}q(T(t)y)\cdot q(T(s)x)$ holds for every $x\in X$ and obtain finally the estimate $\sup_{q(x)\leqslant1}q(T(s)T(t)x)\leqslant\sup_{q(x)\leqslant1}q(T(s)x)\cdot\sup_{q(x)\leqslant1}q(T(t)x)$. From the latter it follows that $\xi$ is subadditive. From [@EN Lem. IV.2.3] we obtain that $\omega'=\inf_{t\geqslant0}{\
textstyle \frac{1}{t}}\log\|T(t)\|_{\Gamma}=\lim_{t\rightarrow\infty}{\textstyle\frac{1}{t}}\log\|T(t)\|_{\Gamma}$ exists in $\mathbb{R}$. We deduce that $e^{
\omega't}\leqslant\|T(t)\|_{\Gamma}$ is valid for all $t\geqslant0$. Hence, $\omega'\leqslant\omega_{0,\Gamma}$ must hold. Let $\omega>\omega'$. Then there exists $t_0>0$ such that $\frac{1}{t}\log\|T(t)\|_{\Gamma}\leqslant\omega$, hence $\|T(t)\|_{\Gamma}\leqslant e^{\omega t}$, holds for all $t\geqslant t_0$. Since $\sup_{0\leqslant{}t\leqslant{}t_0}\|T(t)\|_{\Gamma}<\infty$, we find $M\geqslant1$ such that $\|T(t)\|_{\Gamma}\leqslant Me^{\omega t}$ holds for every $t\geqslant0$ and thus $\omega_{0,\Gamma}\leqslant\omega$ must hold. Since $\omega>\omega'$ was arbitrary, $\omega_{0,\Gamma}\leqslant\omega'$ follows. To finish the proof, we compute $\omega_{0,\Gamma}=\lim_{n\rightarrow\infty}({\textstyle\frac{1}{nt_0}}\log\|T(nt_0)\|_{\Gamma})={\textstyle\frac{1}{t_0}}\lim_{n\rightarrow\infty}(\log\|T(t_0)^n\|_{\Gamma}^{1/n})$.
If $X$ is a Banach space, $\omega_0(T)$ coincides with the classical growth bound; in contrast $\omega_{0,\Gamma}(T)$ must not even be finite. On the other hand a reasonable choice for $\Gamma$, e.g., $\Gamma=\{\|\cdot\|\}$ for any norm $\|\cdot\|$ inducing the topology of $X$, yields $\omega_0(T)=\omega_{0,\Gamma}(T)$ whenever $X$ is Banach. For Fréchet spaces, the equality $\omega_0(T)=\omega_{0,\Gamma}(T)$ can fail even for nice fundamental systems $\Gamma$, cf. Example \[EX-0\].
Spectral Bound {#SEC-3}
==============
The space $L_b(X)$ is an associative algebra with identity in which left resp. right multiplication with a fixed element defines a continuous map. Using [@Allan Prop. 2.6] it follows that $L_b(X)$ is a pseudo complete locally convex algebra in the sense of Allan [@Allan Terminology 1.1 and Dfn. 2.5]. According to [@Allan Dfn. 2.1] we say that $B\in L_b(X)$ is bounded if there exists $\mu\in\mathbb{C}\backslash\{0\}$ such that $\{(\mu{}B)^n\}_{n\in{\mathbb{N}_0}}\subseteq L_b(X)$ is bounded and denote the set of bounded elements by $L_b(X)_0$. We use the convention $\mathbb{N}=\{1,2,\dots\}$ and $\mathbb{N}_0=\{0,1,2,\dots\}$.
Let $D(A)\subseteq X$ be a linear subspace and $A\colon D(A)\rightarrow X$ be linear. We put $\delta_A=\{\infty\}$, if $A\in L_b(X)_0$, and $\delta_A=\varnothing$ otherwise. Then the resolvent set of $A$ is defined by $$\rho(A) = \big\{ \lambda\in\mathbb{C}\:;\:\lambda-A\colon D(A)\rightarrow X \text{ is bijective and }R(\lambda,A)=(\lambda-A)^{-1}\in L_b(X)_0 \big\} \cup \delta_A.$$ The spectrum is defined via $\sigma(A)=\overline{\mathbb{C}}\,\backslash\,\rho(A)$ and the spectral bound by ${\operatorname{s}\nolimits}(A)=\sup\{{\operatorname{Re}\nolimits}\lambda\:;\:\lambda\in\sigma(A)\cap\mathbb{C}\}$.
For $A\in L(X)$ the above definition of resolvent and spectrum coincides with the those given by Allan in [@Allan Dfn. 3.1 and Dfn. 3.2]. We need the following characterization of the complex part of the resolvent.
\[LEM-2\] Let $A\colon D(A)\rightarrow X$ be a linear operator and $\mathcal{U}\subseteq\mathbb{C}$. Assume that $R(\lambda,A)$ exists for every $\lambda\in\mathcal{U}$. Then, $\mathcal{U}\subseteq\rho(A)$ holds if and only if for every $\lambda\in\mathcal{U}$ there exists a fundamental system $\Gamma$ and a constant $M_{\lambda}>0$ such that $p(R(\lambda,A)x)\leqslant{}M_{\lambda}\,p(x)$ is valid for all $p\in\Gamma$ and $x\in X$.
$\Rightarrow$ Let $\Gamma$ be some fundamental system. For $\lambda\in\mathcal{U}\subseteq\rho(A)$ we select $\mu\in\mathbb{C}\backslash\{0\}$ such that $\{[\mu{}R(\lambda,A)]^n\}_{n\in{\mathbb{N}_0}}$ is equicontinuous. We define $\Gamma'=\{p'\:;\:p\in\Gamma\}$ via $p'(x)=\sup_{n\in{\mathbb{N}_0}}p([\mu{}R(\lambda,A)]^nx)$ for $x\in X$. For every $p\in\Gamma$ we find $q\in\Gamma$ and $M\geqslant0$ such that $p([\mu{}R(\lambda,A)]^nx)\leqslant{}M q(x)$ holds for all $n$ and $x$ and thus $p'\leqslant Mq$ is valid. On the other hand for every $p\in\Gamma$ we have $p\leqslant{}p'$. It follows that $\Gamma'$ is a fundamental system for $X$. We put $M_{\lambda}=1/\mu$ and compute $p'(R(\lambda,A)x)=\sup_{n\in{\mathbb{N}_0}}p([\mu{}R(\lambda,A)]^{n}R(\lambda,A)x)=\mu^{-1}\sup_{n\in{\mathbb{N}_0}}p([\mu{}R(\lambda,A)]^{n+1}x)\leqslant M_{\lambda}\,p'(x)$ for arbitrary $p'$ and $x$.\
$\Leftarrow$ For $\lambda\in\mathcal{U}$ we select $\Gamma$ and $M_{\lambda}$ as in the condition and put $\mu=M_{\lambda}^{-1}$. For $p\in\Gamma$ and $n\in{\mathbb{N}_0}$ we compute $p([\mu{}R(\lambda,A)]^nx)\leqslant{}M_{\lambda}^{-n}p(R(\lambda,A)^nx)\leqslant{}M_{\lambda}^{-n}M_{\lambda}^n\,p(x)=p(x)$, i.e., $R(\lambda,A)\in L_b(X)_0$.
Recently, Albanese, Bonet, Ricker [@ABR13 Section 3] defined resolvent and spectrum for non-continuous operators on locally convex spaces in a different way; in their notation $\lambda\in\mathbb{C}$ is in the resolvent if and only if $R(\lambda,A)$ exists in $L(X)$ and thus their spectrum is a subset of $\sigma(A)\cap\mathbb{C}$. In [@ABR13 Equation (3.7)], they studied the condition of Lemma \[LEM-2\] as an additional property of points in the resolvent.\
Arikan, Runov, Zahariuta [@ARZ] introduced the ultraspectrum for operators in $A\in L_b(X)$. In their notation, $\lambda\in\mathbb{C}$ is a strictly regular point of $A$ if $R(\lambda,A)$ exists in $L(X)$ and is tamable [@ARZ p. 29], i.e., there exists a fundamental system of seminorms $\Gamma$ for $X$ such that for all $q\in\Gamma$ there exists $C\geqslant0$ such that $q(R(\lambda,A)x)\leqslant{}C q(x)$ holds for all $x\in X$. The point $\lambda=\infty$ is strictly regular if $A$ itself is tamable. The ultraspectrum is the complement of the strictly regular points in $\overline{\mathbb{C}}$. See [@ARZ Dfn. 10 and Thm. 14] for details. By Lemma \[LEM-2\] it follows that the ultraspectrum is contained in $\sigma(A)$; for $\lambda=\infty$ an argument similar to those in the proof of Lemma \[LEM-2\] can be used.\
If $A$ is the generator of the $C_0$-semigroup ${(T(t))_{t\geqslant0}}$, i.e., $$Ax=\lim_{t\searrow0}{\textstyle\frac{T(t)x-x}{t}} \;\text{ for }\; x\in D(A)=\{x\in X\:;\:\lim_{t\searrow0}{\textstyle\frac{T(t)x-x}{t}}\text{ exists}\},$$ the spectral theory of Allan yields the following relation between the spectral bound of the generator and the semigroup itself if $A$ is a so-called power bounded operator. According to Allan, we put ${\operatorname{r}\nolimits}(B)=\sup\{|\lambda|\:;\:\lambda\in\sigma(B)\}$ for $B\in L(X)$ and $|\infty|=+\infty$.
\[PROP-2\] Let ${(T(t))_{t\geqslant0}}$ be an exponentially equicontinuous $C_0$-semigroup. Let $(A,D(A))$ be its generator and assume that $\{A^n\}_{n\in{\mathbb{N}_0}}\subseteq L(X)$ is equicontinuous. Then $s(A)=\log{\operatorname{r}\nolimits}(T(1))$ holds.
By our assumption we have $A\in L_b(X)_0$ and thus $\infty\not\in\sigma(A)$. By Allan’s spectral mapping theorem, see [@Allan Prop. 6.11] and [@Allan p. 414] for the definition of those class of functions for which Allan’s functional calculus is designed, we have $\sigma(e^{A})=e^{\sigma(A)}$. By [@Yosida Section IX.6] we have $T(1)x={\ensuremath{\mathop{\textstyle\sum}_{n=0}^{\infty}}}{\textstyle\frac{1}{n!}}A^nx=e^{A}x$ for every $x\in X$. We compute $e^{{\operatorname{s}\nolimits}(A)}=\sup\{e^{{\operatorname{Re}\nolimits}\lambda}\:;\:\lambda\in\sigma(A)\}=\sup\{|\lambda|\:;\:\lambda\in\sigma(e^{A})\}={\operatorname{r}\nolimits}(T(1))$ and obtain ${\operatorname{s}\nolimits}(A)=\log{\operatorname{r}\nolimits}(T(1))$.
If $X$ is Banach, the above formula is one of the ingredients to prove that $s(A)=\omega_0(T)$ holds for every uniformly continuous semigroup ${(T(t))_{t\geqslant0}}$, cf. the comments at the end of Section \[SEC-5\].
Inequality {#SEC-4}
==========
We have the following general inequality between spectral bound and growth bound, cf. [@EN Prop. II.2.2].
\[THM\] Let ${(T(t))_{t\geqslant0}}$ be an exponentially equicontinuous $C_0$-semigroup with generator $(A,D(A))$. Then we have $-\infty\leqslant{\operatorname{s}\nolimits}(A)\leqslant\omega_0(T)<+\infty$.
By rescaling we may assume w.l.o.g. that $\omega_0=\omega_0(T)=0$. Then, $(e^{-\omega{}t}T(t))_{t\geqslant0}\subseteq L_b(X)$ is equicontinuous for every $\omega>0$. With [@ABR13 Lem. 4.4] it follows that $R(\lambda,A)$ exists in $L(X)$ for every $\lambda\in\mathbb{C}_+=\{\lambda\in\mathbb{C}\:;\:{\operatorname{Re}\nolimits}\lambda>0\}$. We claim that for every $\lambda\in\mathbb{C}_+$ there exists $\mu\in\mathbb{C}\backslash\{0\}$ such that $\{[\mu{}R(\lambda,A)]^n\}_{n\in{\mathbb{N}_0}}\subseteq L(X)$ is equicontinuous.\
We fix $\omega>0$ such that ${(S(t))_{t\geqslant0}}$, $S(t)=e^{-\omega{}t}T(t)$, $t\geqslant0$, is equicontinuous. The generator of ${(S(t))_{t\geqslant0}}$ is $(B,D(B))=(A-\omega,D(A))$. By [@ABR13 last paragraph of Rem. 3.5(iv)] we have $$\forall\:\lambda'\in\mathbb{C}_+\;\exists\:M_{\lambda'}>0\;\forall\:p\in\Gamma,\,x\in X\colon p(R(\lambda',B)x)\leqslant{}M_{\lambda'}p(x).$$ From [@ABR13 proof of Lem. 4.4] we get that $R(\lambda,A)=R(\lambda-\omega,B)$ holds for every $\lambda\in\mathbb{C}$ with ${\operatorname{Re}\nolimits}\lambda>\omega$.\
Let now $\lambda\in\mathbb{C}_+$ be given. Select $0<\omega<{\operatorname{Re}\nolimits}\lambda$ and put $\lambda'=\lambda-\omega\in\mathbb{C}_+$. We select $M_{\lambda'}>0$ as above and put $\mu=1/M_{\lambda'}$. For given $p\in\Gamma$, $n\in{\mathbb{N}_0}$ and $x\in X$ we have by iteration $p(R(\lambda',B)^nx)\leqslant M_{\lambda'}^np(x)$ and thus $p([\mu{}R(\lambda,A)]^nx)=p([M_{\lambda'}^{-1}R(\lambda-\omega,B)]^nx)=M_{\lambda'}^{-n}p([R(\lambda',B)]^nx)\leqslant p(x)$ which establishes the claim. Consequently, we have $\mathbb{C}_+\subseteq\rho(A)$, i.e., every $\lambda\in\sigma(A)\cap\mathbb{C}$ satisfies ${\operatorname{Re}\nolimits}\lambda\leqslant0$. Therefore, ${\operatorname{s}\nolimits}(A)\leqslant0$ follows.
The inequality in Theorem \[THM\] remains valid if we replace our notion of spectrum by those of [@ABR13]. If $A$ belongs to $L(X)$ and we replace $\sigma(A)$ by the ultraspectrum [@ARZ] the estimate also applies.
Example {#SEC-5}
=======
The following example was studied by Albanese, Bonet, Ricker in [@ABR13 Rem. 3.5(v)]. Below, we correct some flaws and extend their investigation.
\[EX-0\] Let $X=\mathbb{C}^{\mathbb{N}}$ be the space of all complex sequences endowed with the topology of coordinate wise convergence given by $\Gamma=\{p_n\:;\:n\in\mathbb{N}\}$, $p_n(x)=\max_{j=1,\dots,n}|x_j|$ for $x=(x_1,x_2,\dots)$. The operator $A\colon X\rightarrow X$, $Ax=(0,x_1,x_2,\dots)$, denotes the right shift on $X$. We have $\{A^n\}_{n\in{\mathbb{N}_0}}\subseteq L(X)$ and the latter is an equicontinuous set. The strongly continuous semigroup ${(T(t))_{t\geqslant0}}$ generated by $A\colon X\rightarrow X$ is thus given by the power series $$T(t)x={\ensuremath{\mathop{\textstyle\sum}_{k=0}^{\infty}}}{\textstyle\frac{1}{k!}}(tA)^kx=(x_1,x_2+tx_1,x_3+tx_2+{\textstyle\frac{t^2}{2!}}x_1,x_4+tx_3+{\textstyle\frac{t^2}{2!}}x_2+{\textstyle\frac{t^3}{3!}}x_1,\cdots)$$ for $x\in X$ and $t\geqslant0$, see [@Yosida p. 245] and [@ABR13 Rem. 3.5(v)]. We observe that for given $\omega>0$ and $n\in\mathbb{N}$ there exists $M\geqslant 1$ such that $p_n(T(t)x)\leqslant Me^{\omega t}p_n(x)$ holds for all $t\geqslant0$ and $x\in X$. Whence, $\omega_0(T)$ must be less or equal to zero. On the other hand we compute $p_n(T(t)x)=1+t+\cdots+t^{n-1}/(n-1)!$ for $x=(1,1,\dots)\in X$. This shows that in the previous condition for $0<\omega<1$ the constant $M\geqslant1$ cannot be independent of $n\in\mathbb{N}$. Consequently, $\omega_{0,\Gamma}(T)=1$.\
For $\lambda=0$ the map $\lambda-A$ is not surjective, for $\lambda\not=0$ it can be checked that $R(\lambda,A)$ exists in $L(X)$ and is given by the formula $$R(\lambda,A)x=({\textstyle\frac{1}{\lambda}}x_1,{\textstyle\frac{1}{\lambda}}x_2+{\textstyle\frac{1}{\lambda^2}}x_1,{\textstyle\frac{1}{\lambda}}x_3+{\textstyle\frac{1}{\lambda^2}}x_2+{\textstyle\frac{1}{\lambda^3}}x_1,\dots)$$ for $x\in X$. We estimate $$\begin{aligned}
p_n(R(\lambda,A)x)
& \leqslant & \max\{{\textstyle\frac{1}{|\lambda|}}|x_1|,{\textstyle\frac{1}{|\lambda|}}|x_2|+{\textstyle\frac{1}{|\lambda|^2}}|x_1|,\dots,{\textstyle\frac{1}{|\lambda|}}|x_n|+\cdots+{\textstyle\frac{1}{|\lambda|^n}}|x_1|\}\\
&\leqslant& \max\{{\textstyle\frac{1}{|\lambda|}},{\textstyle\frac{1}{|\lambda|}}+{\textstyle\frac{1}{|\lambda|^2}},\dots,{\textstyle\frac{1}{|\lambda|}}+\cdots+{\textstyle\frac{1}{|\lambda|^n}}\}\,\max\{|x_1|,\dots,|x_n|\}\\
& \leqslant & {\ensuremath{\mathop{\textstyle\sum}_{j=1}^{n}}}({\textstyle\frac{1}{|\lambda|}})^{j}\,p_n(x)\;=\;{\textstyle\frac{1-1/|\lambda|^n}{|\lambda|-1}}\,p_n(x)\;\leqslant\; {\textstyle\frac{1}{|\lambda|-1}}\,p_n(x)\end{aligned}$$ for $|\lambda|>1$, $x\in X$ and $n\in\mathbb{N}$. Using Lemma \[LEM-2\] with $\mathcal{U}=\{z\in\mathbb{C}\:;\:|z|>1\}$ and $M_\lambda=(|\lambda|-1)^{-1}$ for $\lambda\in\mathcal{U}$ it follows that all $\lambda\in\mathbb{C}$ with $|\lambda|>1$ belong to $\rho(A)$. For $\mathcal{U}'=\{z\in\mathbb{C}\:;\:0<|z|\leqslant1\}$ the condition $$\forall\:\lambda\in\mathcal{U}'\;\exists\:M_{\lambda}>0\;\forall\:p\in\Gamma,\,x\in X\colon p(R(\lambda,A)x)\leqslant M_{\lambda}p(x)$$ is not satisfied. Let $|\lambda|\leqslant1$ with $\lambda\not=1$. For $x=(1,1,\dots)$ we get $$p_n(R(\lambda,A)x)=\max\{|{\textstyle\frac{1}{\lambda}}|,|{\textstyle\frac{1}{\lambda}}+{\textstyle\frac{1}{\lambda^2}}|,\dots,|{\textstyle\frac{1}{\lambda}}+\cdots+{\textstyle\frac{1}{\lambda^n}}|\}=\max_{1\leqslant{}k\leqslant{}n}\big|{\textstyle\frac{1-1/\lambda^k}{\lambda-1}}\big|\geqslant \big|{\textstyle\frac{1-1/\lambda^n}{\lambda-1}}\big|\stackrel{n\rightarrow\infty}{\longrightarrow}\infty.$$ For $\lambda=1$ and $x$ as above we get $p_n(R(\lambda,A)x)=n$. Therefore, for no $\lambda\in\mathbb{C}$ with $0<|\lambda|\leqslant1$ there can exist $M_{\lambda}$ such that $p_n(R(\lambda,A)x)\leqslant M_{\lambda}\,p_n(x)$ holds for every $x\in X$ and $n\in\mathbb{N}$. Nevertheless, $\mathcal{U}'\subseteq\rho(A)$ holds. In particular, the aforementioned inequality becomes true if we switch to another system of seminorms, see Lemma \[LEM-2\]. We fix $x\in X$ and $\lambda\in\mathcal{U}'$. By induction we show that $$[n]\;\;\;\;\;\;\forall\:k\in\mathbb{N}\;\colon\;(R(\lambda,A)^nx)_k ={\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}\lambda^{-(n+j-1)}\,{\textstyle\binom{n-2+j}{j-1}}\,x_{k-j+1}$$ holds for every $n\in\mathbb{N}$. We observe that we established the statement for $n=1$ already above. Moreover, we have the recursion formula $(R(\lambda,A)x)_k=\lambda^{-1}x_k+\lambda^{-1}(R(\lambda,A)x)_{k-1}$ for arbitrary $k$ if we put $(R(\lambda,A)x)_{0}=0$. Now we show $[n]\Rightarrow[n+1]$. In order to establish $[n+1]$ we proceed by induction over $k$. Using the recursion formula and then $[n]$ we get $(R(\lambda,A)^{n+1}x)_1=\lambda^{-1}(R(\lambda,A)^nx)_1=\lambda^{-(n+1)}x_1$ which coincides with the right hand side of the equation in $[n+1]$ for $k=1$. For $k\geqslant2$ we use the recursion formula to get $(R(\lambda,A)^{n+1}x)_k=\lambda^{-1}(R(\lambda,A)^nx)_k+\lambda^{-1} (R(\lambda,A)^{n+1}x)_{k-1}$. For the first summand we use the induction hypothesis of the induction over $n$ and for the second those of the induction over $k$. Then we obtain $$\begin{aligned}
(R(\lambda,A)^{n+1}x)_k & = & \lambda^{-1}{\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}\lambda^{-(n+j-1)}\,{\textstyle\binom{n-2+j}{j-1}}\,x_{k-j+1}+\lambda^{-1}{\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k-1}}}\lambda^{-(n+j+1)}\,{\textstyle\binom{n-1+j}{j-1}}\,x_{k+j}\\
& = & \lambda^{-(n+1)}\,x_{k}+{\ensuremath{\mathop{\textstyle\sum}_{j=2}^{k}}}\lambda^{-(n+j)}\,\big[{\textstyle\binom{n-2+j}{j-1}}+{\textstyle\binom{n-2+j}{j-2}}\big]\,x_{k-j+1}\\
& = & {\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}\lambda^{-(n+j)}\,{\textstyle\binom{n-1+j}{j-1}}\,x_{k-j+1}\end{aligned}$$ which is the right hand side of the equation in $[n+1]$ for $k$. This finishes both inductions. We claim $$\forall\:\lambda\in\mathcal{U}'\;\exists\:\mu\in\mathbb{C}\backslash\{0\}\;\forall\:m\in\mathbb{N}\;\exists\:K\geqslant0\;\forall\:n\in{\mathbb{N}_0},\,x\in X \colon p_{m}([\mu{}R(\lambda,A)]^nx)\leqslant K p_{m}(x)$$ which implies that $R(\lambda,A)\in L_b(X)_0$ holds for every $\lambda\in\mathcal{U}'$. For given $\lambda\in\mathcal{U}'$ select $\mu$ such that $|\mu|/|\lambda|\leqslant1/2$ holds. Let $m$ be given. Put $K=|\lambda|^{-(m-1)}e^{m}\sup_{n\in\mathbb{N}}2^{-n}(n+1)^{m}$. Let $n$ and $x$ be given. Since $K\geqslant1$ we are done if $n=0$. Otherwise we estimate $$\begin{aligned}
p_{m}([\mu{}R(\lambda,A)]^nx) & \leqslant & |\mu|^{n} \max_{1\leqslant{}k\leqslant{}m}\,{\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}|\lambda|^{-(n+j-1)}{\textstyle\binom{n-2+j}{j-1}}|x_{k-j+1}|\\
&\leqslant & \big({\textstyle\frac{|\mu|}{|\lambda|}}\big)^{n}\max_{1\leqslant{}k\leqslant{}m}\,{\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}|\lambda|^{-(j-1)}{\textstyle\binom{n-2+j}{j-1}}\,p_k(x)\\
&\leqslant & 2^{-n}\,|\lambda|^{-(m-1)}\,\big(\max_{1\leqslant{}k\leqslant{}m}\,{\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}{\textstyle\binom{n-2+j}{j-1}}\big)\,p_{m}(x)\;\leqslant\;K\,p_{m}(x)\end{aligned}$$ where the last inequality holds since $${\ensuremath{\mathop{\textstyle\sum}_{j=1}^{k}}}{\textstyle\binom{n-2+j}{j-1}} \leqslant {\ensuremath{\mathop{\textstyle\sum}_{j=0}^{k}}}{\textstyle\binom{n-1+j}{j}}={\textstyle\binom{n+k}{k}}\leqslant \big({\textstyle\frac{(n+k)e}{k}}\big)^{k}\leqslant e^k ({\textstyle\frac{n}{k}}+1)^k\leqslant{}e^{m}(n+1)^{m}$$ is true whenever $1\leqslant{}k\leqslant{}m$. Finally, we get $\sigma(A)=\{0\}$ and thus ${\operatorname{s}\nolimits}(A)=0$.
If $X$ is a Banach space then ${\operatorname{s}\nolimits}(A)=\omega_0(T)$ holds whenever ${(T(t))_{t\geqslant0}}$ is uniformly continuous. The proof for this equality employs Hadamard’s formula for the spectral radius ${\operatorname{r}\nolimits}(\cdot)$ and the Banach space versions of Proposition \[PROP-1\] and Proposition \[PROP-2\]. Indeed, Allan [@Allan Thm. 3.12 and Prop. 2.18] proved a Hadamard type formula for the spectral radius within his theory. But neither the formula ${\operatorname{r}\nolimits}(T(t_0))=\lim_{n\rightarrow\infty}\|T(t_0)^n\|_{\Gamma}^{1/n}$ nor the equality $\omega_{0,\Gamma}(T)=\frac{1}{t_0}\log{\operatorname{r}\nolimits}(T(t_0))$ extend in general to the Fréchet case—not even for nice fundamental systems $\Gamma$. Already for $t_0=1$ the above and the Propositions \[PROP-1\] and \[PROP-2\] would imply that ${\operatorname{s}\nolimits}(A)=\omega_{0,\Gamma}(T)$ holds in Example \[EX-0\]. On the other hand, it seems to be open if $s(A)=\omega_0(T)$ holds in general or can fail for uniformly continuous semigroups on Fréchet spaces when $s(A)$ is defined as in Section \[SEC-3\], i.e., with respect to the spectral theory of [@Allan], or with respect to the ultraspectrum [@ARZ].
[Acknowledgements. ]{}The author’s stay at the Sobolov Institute of Mathematics at Novosibirsk was supported by the German Academic Exchange Service (DAAD). The author likes to express his gratitude to the referee for pointing out a mistake in the original version of this paper and for giving several valuable comments and references.
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[^1]: 2010 *Mathematics Subject Classification*: Primary 47D06; Secondary 46A04, 34G10.
[^2]: *Key words and phrases*: semigroup, growth bound, spectral bound, power bounded operator, Fréchet space.
[^3]: $^{\text{a}}$Sobolev Institute of Mathematics, Pr. Akad. Koptyuga 4, 630090, Novosibirsk, Russia, Phone:+7(8)383/363-4648,Fax:+7(8)383/333-2598, eMail: [email protected].
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{
"pile_set_name": "ArXiv"
}
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abstract: 'Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded, are given. A “Cuntz-Krieger uniqueness theorem” for relative Cuntz-Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz-Pimsner rings are introduced.'
address:
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Department of Mathematical Sciences\
NTNU\
NO-7491 Trondheim\
Norway
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Department of Mathematical Sciences\
NTNU\
NO-7491 Trondheim\
Norway
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Departamento de Matemáticas, Facultad de Ciencias\
Universidad de Cádiz, Campus de Puerto Real\
11510 Puerto Real (Cádiz)\
Spain.
author:
- Toke Meier Carlsen
- Eduard Ortega
- Enrique Pardo
title: 'Simple Cuntz-Pimsner rings'
---
Introduction
============
In [@CaOr2011] the two first named authors introduced the notion of a relative Cuntz-Pimsner ring ${\mathcal{O}_{(P,Q,\psi)}(J)}$ as an algebraic analogue of (relative) Cuntz-Pimsner $C^*$-algebras (see for example [@MuSo1998], [@Pi1997], [@Ka2004a] and [@Ka2007]), and showed that for instance Leavitt path algebras (see for example [@AbAr2005], [@AbAr2008] and [@To2007]), crossed products of a ring by a single automorphism (also called a skew group ring, see for example [@Mo1980] and [@Pa1989]) and fractional skew monoid rings of a single corner isomorphism (see [@ArGoGo2004]) can be constructed as relative Cuntz-Pimsner rings. They also gave a complete description of the graded ideals of an arbitrary relative Cuntz-Pimsner ring $\mathcal{O}_{(P,Q,\psi)}(J)$. The purpose of this paper is to study the non-graded ideals of such a relative Cuntz-Pimsner ring ${\mathcal{O}_{(P,Q,\psi)}(J)}$. Although we do not reach a complete description of all (graded or non-graded) ideals of ${\mathcal{O}_{(P,Q,\psi)}(J)}$, we do find necessary and sufficient conditions for when every non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ contains a non-zero graded ideal (Theorem \[thm:L\]), when ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple (Theorem \[thm:simple\]), and when every ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is graded (Theorem \[thm:K\]). We also give a “Cuntz-Krieger uniqueness theorem” for ${\mathcal{O}_{(P,Q,\psi)}(J)}$ (Theorem \[thm:CK\]) and introduce condition (L) (Definition \[def:L\]) and condition (K) (Definition \[def:K\]) for relative Cuntz-Pimsner rings. These results and definitions are generalizations of similar results and definitions about Leavitt path algebras given in [@To2007], and analogues of similar results and definitions given in the $C^*$-algebraic setting for graph $C^*$-algebras (see for example [@Ra2005]), ultragraph $C^*$-algebras (see [@To2003d]), topological graph $C^*$-algebras (see [@Ka2006b]), and (relative) Cuntz-Krieger algebras of finitely aligned higher rank graphs (see for example [@Si2006]).
It is worth pointing out that analogues in the $C^*$-algebraic setting of these results do not exist in the generality of this paper. It does not seem unreasonable to believe that it should be possible to obtain such, but a different approach than the one used in this paper seems to be needed.
Contents {#contents .unnumbered}
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Section \[sec:preliminaries\] contains some preliminary results and the pivotal Proposition \[prop:main\]. In Section \[sec:condition-L\] condition (L) is introduced (Definition \[def:L\]), and sufficient and necessary conditions for when every non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ contains a non-zero graded ideal are given (Theorem \[thm:L\]). Section \[sec:cuntz-krieg-uniq\] contains the Cuntz-Krieger uniqueness theorem (Theorem \[thm:CK\]). In Section \[sec:simplicity-ock\] sufficient and necessary for when ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple are given (Theorem \[thm:simple\]), and in Section \[sec:condition-k\] condition (K) is introduced (Definition \[def:K\]), and sufficient and necessary conditions for when every ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is graded are given (Theorem \[thm:K\]). In Section \[sec:toeplitz-rings\] the case when $J=0$ and ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is the Toeplitz ring ${\mathcal{T}_{(P,Q,\psi)}}$ of $(P,Q,\psi)$ is considered. Finally, in Section \[sec:leavitt\] and Section \[sec:cross\] we illustrate the results obtained in the paper by applying them to Leavitt path algebras (Section \[sec:leavitt\]), and to crossed products of a ring by a single automorphism and fractional skew monoid rings of a single corner isomorphism (Section \[sec:cross\]), and thereby obtain characterizations of when these algebras are simple. The characterization of when a Leavitt path algebra is simple is well-know (see [@To2007 Theorem 6.18]), whereas the characterizations of when a crossed product of a ring by an automorphism and a fractional skew monoid ring by a corner isomorphism are simple, to the knowledge of the authors, are new.
Notation and conventions {#notation-and-conventions .unnumbered}
------------------------
In this paper every ideal will be a two-sided ideal. The set of integers will be denoted by ${\mathbb{Z}}$, the set of positive integers will be denoted by ${\mathbb{N}}$ and the set of non-negative integers will be denoted by ${{\mathbb{N}_{0}}}$.
We will use the same notation as in [@CaOr2011] with the addition that $R$ will always denote a fixed ring, $(P,Q,\psi)$ will be a fixed $R$-system satisfying condition (FS) and $J$ will be a fixed faithful and $\psi$-compatible ideal in $R$. To ease notation we will let $\sigma$, $S$, $T$ and $\pi$ denote $\iota_R^J$, $\iota_P^J$, $\iota_Q^J$ and $\pi^J$, respectively. We will repeatedly use that since $(P,Q,\psi)$ satisfies condition (FS), the $R$-system $(P^{\otimes n},Q^{\otimes n},\psi_n)$ will for each $n\in{\mathbb{N}}$ also satisfy condition (FS) (see [@CaOr2011 Lemma 3.8]).
Preliminaries {#sec:preliminaries}
=============
This section contains some preliminary results leading to Proposition \[prop:main\], which is pivotal for the rest of the paper.
\[lemma:1\] If $n\in{\mathbb{N}}$, $x_{-n}\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(-n)}}\setminus\{0\}$ and $x_n\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(n)}}\setminus\{0\}$, then there is a $p\in P^{\otimes n}$ and a $q\in Q^{\otimes n}$ such that $x_{-n}T^n(q)\ne 0$ and $S^n(p)x_n\ne 0$.
Write $x_n$ as $\sum_{i=1}^kT^n(q_i)y_i$ where $q_i\in Q^{\otimes n}$ and $y_i\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ for $i=1,2,\dots, k$. It follows from condition (FS) that there is a $\theta\in\mathcal{F}_{P^{\otimes n}}(Q^{\otimes n})$ such that $\theta q_i=q_i$ for each $i=1,2,\dots,k$. It follows that $S^n(p)x_n$ cannot be 0 for all $p\in P^{\otimes n}$. That $x_{-n}T^n(q)\ne 0$ for some $q\in Q^{\otimes n}$ can be proved in a similar way.
For an ideal $I$ in $R$, let $\psi^{-1}(I)$ be the ideal $$\bigl\{x\in R\mid \psi(px\otimes q)\in I\text{ for all }q\in Q\text{ and all }p\in P\bigr\},$$ and let $I^{[\infty]}$ be the ideal $$\bigcap_{k=1}^\infty I^{[k]}$$ where $I^{[k]}$ is defined recursively by $I^{[1]}=I$ and $I^{[k]}=\psi^{-1}\bigl(I^{[k-1]}\bigr)\cap I$ for $k> 1$.
Recall that if $I$ is an ideal in $R$, then $QI=\operatorname{span}\{qx\mid q\in Q,\ x\in I\}$ (see [@CaOr2011 Definition 7.1]).
\[lemma:2\] Let $x\in R$. Then $x\in\psi^{-1}(I)$ if and only if $xq\in QI$ for all $q\in Q$.
Assume first that $x\in\psi^{-1}(I)$ and that $q\in Q$. Then it follows from condition (FS) that there are $q_1,\dots,q_m\in Q$ and $p_1,\dots,p_m\in P$ such that $xq=\sum_{i=1}^mq_i\psi(p_i\otimes xq)$. Since each $\psi(p_i\otimes xq)\in I$, it follows that $xq\in QI$.
Assume then that $x\in R$ and $xq\in QI$ for all $q\in Q$, and let $q\in Q$ and $p\in P$. Then there are $q_1,\dots q_m\in Q$ and $x_1,\dots,x_m\in I$ such that $xq=\sum_{i=1}^mq_ix_i$, from which it follows that $\psi(px\otimes q)=\psi(p\otimes xq)=\sum_{i=1}^m\psi(p\otimes q_i)x_i\in I$. Thus $x\in \psi^{-1}(I)$.
Let us now specialise to the case where $I=J$.
\[lemma:3\] Let $k\in{\mathbb{N}}$ and $x\in R$. Then $x\in J^{[k]}$ if and only if $\sigma(x)\in\operatorname{span}\{T^k(q)S^k(p)\mid q\in Q^{\otimes k}, p\in P^{\otimes k}\}$.
We will prove the lemma by induction over $k$. For $k=1$ the lemma follows from [@CaOr2011 Proposition 3.28].
Assume now that $k>1$ and that $x\in J^{[k-1]}$ if and only if $\sigma(x)\in\operatorname{span}\{T^{k-1}(q)S^{k-1}(p)\mid q\in Q^{\otimes k-1}, p\in P^{\otimes k-1}\}$. We will then prove that $x\in J^{[k]}$ if and only if $\sigma(x)\in\operatorname{span}\{T^k(q)S^k(p)\mid q\in Q^{\otimes k},\ p\in P^{\otimes k}\}$ for all $x\in R$. If $x\in J^{[k]}=\psi^{-1}(J^{[k-1]})\cap J$, then it follows from [@CaOr2011 Proposition 3.28] that there are $q_1,\dots,q_m\in Q$ and $p_1,\dots,p_m\in P$ such that $\sigma(x)=\sum_{i=1}^mT(q_i)S(p_i)$. It follows from condition (FS) that there are $q'_1,\dots,q'_n\in Q$ and $p'_1,\dots,p'_n\in P$ such that $\sum_{j=1}^n\theta_{p'_j,q'_j}p_i=p_i$ for each $i$, from which it follows that $$\sigma(x)=\sum_{i=1}^mT(q_i)S(p_i)=\sum_{i=1}^m\sum_{j=1}^nT(q_i)S(p_i)T(q'_j)S(p'_j) =\sum_{j=1}^nT(xq'_j)S(p'_j).$$ It follows from Lemma \[lemma:2\] that there for each $j$ are $q_{j,1}\dots,q_{j,m_j}\in Q$ and $x_{j,1},\dots,x_{j,m_j}\in J^{[k-1]}$ such that $xq_j=\sum_{l=1}^{m_j}q_{j,l}x_{j,l}$, and it then follows from the induction hypothesis that $$\begin{split}
\sigma(x)&=\sum_{j=1}^nT(xq'_j)S(p'_j)\\
&=\sum_{j=1}^n\sum_{l=1}^{m_j}T(q_{j,l})\sigma(x_{j,l})S(p'_j)\in \operatorname{span}\{T^k(q)S^k(p)\mid q\in Q^{\otimes k}, p\in P^{\otimes k}\}.
\end{split}$$
Conversely, if $\sigma(x)=\sum_{i=1}^mT^k(q_i)S^k(p_i)$, then $\iota_R(x)-\sum_{i=1}^m\iota_Q^k(q_i)\iota_P^k(p_i)\in\mathcal{T}(J)$ (cf. [@CaOr2011 Definition 3.15 and 3.16]), so it follows from [@CaOr2011 Lemma 3.21] that $x\in J$. If $p\in P$ and $q\in Q$, then $$\begin{gathered}
\sigma\bigl(\psi(px\otimes q)\bigr)=S(p) \sum_{i=1}^mT^k(q_i)S^k(p_i) T(q)\\\in \operatorname{span}\{T^{k-1}(q')S^{k-1}(p')\mid q'\in Q^{\otimes k-1}, p'\in P^{\otimes k-1}\},
\end{gathered}$$ which together with the induction hypothesis implies that $\psi(px\otimes q)\in J^{[k-1]}$, and thus that $x\in \psi^{-1}(J^{[k-1]})\cap J=J^{[k]}$.
A subring $A$ of $\mathcal{O}_{(P,Q,\psi)}(J)$ has the *ideal intersection property* if the implication $K\cap A=\{0\}{\Rightarrow}K=\{0\}$ holds for every ideal $K$ in ${\mathcal{O}_{(P,Q,\psi)}(J)}$.
We of course have that ${\mathcal{O}_{(P,Q,\psi)}(J)}$ itself has the ideal intersection property. We will in this paper study when $\sigma(R)$ and ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ have the ideal intersection property. We begin with ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$.
Let $n\in{\mathbb{N}}$. Recall from [@CaOr2011 Section 2] that there for each $p\in P$ exists a unique $R$-bimodule homomorphism $S_p:Q^{\otimes n+1}\to Q^{\otimes n}$ characterised by $S_p(q\otimes q_n)=\psi(p\otimes q)q_n$ for $q\in Q$ and $q_n\in Q^{\otimes n}$. Similarly, there exists for each $q_n\in Q^{\otimes n}$ an $R$-bimodule homomorphism $T_{q_n}:Q\to Q^{\otimes n+1}$ given by $T_{q_n}(q)=q_n\otimes q$ for $q\in Q$. Notice that $T^n(S_pT_{q_n}(q))=S(p)T^n(q_n)T(q)$ for $p\in P$, $q_n\in Q^{\otimes n}$ and $q\in Q$.
\[prop:main\] The following 3 conditions are equivalent:
1. \[item:1\] The subring ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ does not have the ideal intersection property.
2. \[item:2\] There is a non-zero graded ideal $\bigoplus_{k\in{\mathbb{Z}}}H^{(k)}$ in ${\mathcal{O}_{(P,Q,\psi)}(J)}$, an $n\in{\mathbb{N}}$ and a family $(\phi_k)_{k\in{\mathbb{Z}}}$ of injective ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$-bimodule homomorphisms $\phi_k:H^{(k)}\to{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k+n)}}$ such that $x\phi_k(y)=\phi_{k+j}(xy)$ and $\phi_k(y)x=\phi_{k+j}(yx)$ for $k,j\in{\mathbb{Z}}$, $x\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(j)}}$ and $y\in H^{(k)}$.
3. \[item:3\] There is a non-zero $\psi$-invariant ideal $I_0$ of $R$, an $n\in{\mathbb{N}}$ and an injective $R$-bimodule homomorphism $\eta:I_0\to Q^{\otimes n}$ such that $S_pT_{\eta(x)}(q)=\eta(\psi(px\otimes q))$ for $p\in P$, $x\in I_0$ and $q\in Q$, and such that $I_0\subseteq J^{[\infty]}$.
$\eqref{item:1}{\Rightarrow}\eqref{item:2}$: Let $K$ be a non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ such that $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$. Let $N$ be the set of $n\in{\mathbb{N}}_0$ for which there are $x_i\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(i)}}$, $i=0,1,\dots,n$ with $x_0\ne 0$ such that $\sum_{i=0}^nx_i\in K$. Let $\sum_{i=j}^kx_i\in K$ where $j\le k\in{\mathbb{Z}}$, $x_i\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(i)}}$ for $i=j,j+1,\dots,k$ and $x_j\ne 0$. If $j\ne 0$, then it follows from Lemma \[lemma:1\] that there is a $y_{-j}\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(-j)}}$ such that either $y_{-j}x_j$ or $x_jy_{-j}$ is non-zero. It follows that $N\ne\emptyset$. Since $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$, it follows that $0\notin N$. Let $n=\min N$. Then $n\in{\mathbb{N}}$.
For each $k\in{\mathbb{Z}}$ let $$H^{(k)}:=\left\{x_k\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k)}}\Bigm| \exists x_{k+i}\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k+i)}},\ i=1,2,\dots,n:\sum_{i=0}^nx_{k+i}\in K\right\}.$$ If $x\in H^{(k)}$ and $y\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(j)}}$, then $xy,yx\in H^{(k+j)}$. It follows that $\bigoplus_{k\in{\mathbb{Z}}}H^{(k)}$ is a graded ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$, and since $H^{(0)}\ne\{0\}$, it must be the case that $\bigoplus_{k\in{\mathbb{Z}}}H^{(k)}$ is non-zero.
Let $k\in{\mathbb{Z}}$ and let $x_k\in H^{(k)}$. It follows from Lemma \[lemma:1\] and the minimality of $n$ that there is a unique $x_{k+n}\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k+n)}}$ satisfying that there exist $x_{k+i}\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k+i)}}$, $i=1,2,\dots,n-1$ such that $\sum_{i=0}^nx_{k+i}\in K$. It also follows from Lemma \[lemma:1\] and the minimality of $n$ that $x_{k+n}\ne 0$ if $x_k\ne 0$. Thus there is an injective map $\phi_k:H^{(k)}\to{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(k+n)}}$ sending $x_k$ to $x_{k+n}$. It is easy to check that $\phi_k$ is a ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$-bimodule homomorphism, and that $x\phi_k(y)=\phi_{k+j}(xy)$ and $\phi_k(y)x=\phi_{k+j}(yx)$ when $k,j\in{\mathbb{Z}}$, $x\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(j)}}$ and $y\in H^{(k)}$.
$\eqref{item:2}{\Rightarrow}\eqref{item:3}$: We will first prove that $H^{(0)}\cap\sigma(R)\ne\{0\}$, so assume, for contradiction, that $H^{(0)}\cap\sigma(R)=\{0\}$. Then it follows from [@CaOr2011 Lemma 3.21 and Theorem 7.27] that $$\begin{gathered}
H^{(0)}=\operatorname{span}\bigl(\{T^n(q)(\sigma(x)-\pi(\Delta(x)))S^n(p)\mid n\in{\mathbb{N}},\ q\in Q^{\otimes n},\ x\in J',\ p\in P^{\otimes n}\}\\\cup\{\sigma(x)-\pi(\Delta(x))\mid x\in J'\}\bigr)
\end{gathered}$$ for some faithful $\psi$-compatible ideal $J'$ of $R$ which contains $J$. We claim that $H^{(0)}$ must contain a non-zero element of the form $\sigma(x)-\pi(\Delta(x))$, $x\in J'$. To see that this is the case, let $y$ be a non-zero element of $H^{(0)}$ and write it as $$\sigma(x_0)-\pi(\Delta(x_0))+\sum_{i=1}^kT^{n_i}(q_i)\bigl(\sigma(x_i)-\pi(\Delta(x_i))\bigr)S^{n_i}(p_i)$$ where $k\in{\mathbb{N}}$, $x_0,x_1,\dots,x_k\in J'$ and $n_i\in{\mathbb{N}}$, $q_i\in Q^{\otimes n_i}$, $p_i\in P^{\otimes n_i}$ for each $i\in\{1,2,\dots,k\}$, and assume that $\sum_{i\in M} T^{n_i}(q_i)\bigl(\sigma(x_i)-\pi(\Delta(x_i))\bigr)S^{n_i}(p_i)\ne 0$ where $M$ is the set of those $i$’s for which $n_i$ is maximal among $\{n_1,n_2,\dots,n_k\}$. Let $n$ be the maximal value of $n_i$. It follows from condition (FS) that there are $q\in Q^{\otimes n}$ and $p\in P^{\otimes n}$ such that if we let $x=\sum_{i\in M}\psi_n(p\otimes q_i)x_i\psi_n(p_i\otimes q)$, then $$\sigma(x)-\pi(\Delta(x))=S^n(p) \sum_{i\in M} T^{n_i}(q_i)\bigl(\sigma(x_i)-\pi(\Delta(x_i))\bigr)S^{n_i}(p_i)T^n(q)\ne 0.$$ Since $(\sigma(x_0)-\pi(\Delta(x_0)))T^n(q)=0$ and $(\sigma(x_i)-\pi(\Delta(x_i)))S^{n_i}(p_i)T^n(q)=0$ for each $i\notin M$, it follows that $$\begin{gathered}
\sigma(x)-\pi(\Delta(x))\\=S^n(p)\biggl( \sigma(x_0)-\pi(\Delta(x_0))+\sum_{i=1}^kT^{n_i}(q_i)\bigl(\sigma(x_i)-\pi(\Delta(x_i))\bigr)S^{n_i}(p_i)\biggr)T^n(q)\in H^{(0)}.
\end{gathered}$$ Thus $H^{(0)}$ contains a non-zero element of the form $\sigma(x)-\pi(\Delta(x))$, $x\in J'$. If follows from condition (FS) that there is a $p'\in P^{\otimes n}$ such that $$S^n(p')\phi_0\bigl(\sigma(x)-\pi(\Delta(x)\bigr)\ne 0,$$ but since $S^n(p'')(\sigma(x)-\pi(\Delta(x)))=0$ for all $p''\in P^{\otimes n}$, it follows that $$S^n(p')\phi_0\bigl(\sigma(x)-\pi(\Delta(x)\bigr)=\phi_{-n}\bigl(S^n(p')(\sigma(x)-\pi(\Delta(x))\bigr)=0,$$ and we have reached a contradiction. Thus it must be the case that $H^{(0)}\cap\sigma(R)\ne\{0\}$. Let $I=\{x\in R\mid \sigma(x)\in H^{(0)}\}$. Then $I$ is a non-zero $\psi$-invariant ideal of $R$. For each $m\in{{\mathbb{N}_{0}}}$ let $$A_m=\operatorname{span}\bigl\{T^{n+k}(q)S^k(p)\mid k\in\{0,1,\dots,m\},\ q\in Q^{\otimes n+k},\ p\in P^{\otimes k}\bigr\} \subseteq {{\mathcal{O}_{(P,Q,\psi)}(J)}^{(n)}}$$ and $$I_m=\{x\in I\mid \phi_0(\sigma(x))\in A_m\}.$$ Then $I_0\subseteq I_1\subseteq I_2\subseteq\dots$ and each $I_m$ is a $\psi$-invariant two-sided ideal in $R$. In fact, $x\in I_{m+1}$, implies that $\psi(px\otimes q)\in I_m$ for all $p\in P$ and $q\in Q$. Since $I$ is nonzero, there exists an $x\ne 0$ and an $m\in{{\mathbb{N}_{0}}}$ such that $x\in I_m$. Choose $k\in{\mathbb{N}}$ such that $kn\ge m$. Then $$\phi_{(k-1)n}\circ\phi_{(k-2)n}\circ\dots\circ\phi_n\circ\phi_0(\sigma(x))\in{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(nk)}}\setminus\{0\}$$ so it follows from Lemma \[lemma:1\] that there is a $p\in P^{\otimes nk}$ such that $$\phi_{-n}\circ\phi_{-2n}\circ\dots\circ\phi_{-(k-1)n}\circ\phi_{-kn}(S^{nk}(px))= S^{nk}(p) \phi_{(k-1)n}\circ\phi_{(k-2)n}\circ\dots\circ\phi_n\circ\phi_0(\sigma(x))\ne 0,$$ from which it follows that $px\ne 0$. It follows from condition (FS) that there is a $q\in Q^{\otimes kn}$ such that $\psi_{kn}(px\otimes q)\ne 0$. We have that $\psi_{kn}(px\otimes q)\in I_0$, so $I_0\ne \{0\}$.
Since $\phi_0(\sigma(x))\in T^n(Q^{\otimes n})$ for every $x\in I_0$, and $T^n:Q^{\otimes n}\to{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(n)}}$ is injective, we can define $\eta:I_0\to Q^{\otimes n}$ by, for $x\in I_0$, letting $\eta(x)$ be the unique element of $Q^{\otimes n}$ such that $T^n(\eta(x))=\phi_0(\sigma(x))$. It is straightforward to check that $\eta$ is an injective $R$-bimodule homomorphism, and if $p\in P$, $x\in I_0$ and $q\in Q$, then $$\begin{split}
T^n\Bigl(\eta\bigl(\psi(px\otimes q)\bigr)\Bigr)&=\phi_0\Bigl(\sigma\bigl(\psi(px\otimes q)\bigr)\Bigr) =S(p)\phi_0\bigl(\sigma(x)\bigr)T(q)\\
&=S(p)T^n\bigl(\eta(x)\bigr)T(q)=T^n\bigl(S_pT_{\eta(x)}(q)\bigr),
\end{split}$$ from which it follows that $\eta(\psi(px\otimes q))=S_pT_{\eta(x)}(q)$.
If $x\in I_0$ then it follows from condition (FS) that there are $q_i\in Q^{(n)}$, $p_i\in P^{(n)}$, $i=1,2,\dots,m$ such that $\sum_{i=0}^m\theta_{q_i,p_i}\eta(x)=\eta(x)$. We then have that $$T^n(\eta(x))=T^n\left(\sum_{i=0}^m\theta_{q_i,p_i}\eta(x)\right) =\sum_{i=0}^mT^n(q_i)S^n(p_i)\phi_0(\sigma(x)) =\phi_0\left(\sum_{i=0}^mT^n(q_i)S^n(p_ix)\right)$$ from which it follow that $\sigma(x)=\sum_{i=0}^m T^n(q_i)S^n(p_ix)$. It now follows from Lemma \[lemma:3\] that $x\in J^{[n]}\subseteq J$. Since $I_0$ is $\psi$-invariant, it follows that $x\in J^{[\infty]}$.
$\eqref{item:3}{\Rightarrow}\eqref{item:1}$: Let $K$ be the ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ generated by $\{\sigma(x)-T^n(\eta(x))\mid x\in I_0\}$. Clearly, $K$ is non-zero, so we just have to prove that $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$. Using condition (FS) and the properties of $\eta$, one can show that if $p\in P$, $x\in I_0$ and $q\in Q$, then $$S(p)\bigl(\sigma(x)-T^n(\eta(x))\bigr)\in\operatorname{span}\bigl\{\bigl(\sigma(x')-T^n(\eta(x'))\bigr)S(p')\bigm| x'\in I_0,\ p'\in P\bigr\}$$ and $$\bigl(\sigma(x)-T^n(\eta(x))\bigr)T(q)\in\operatorname{span}\bigl\{T(q')(\sigma(x')-T^n(\eta(x')))\bigm| q'\in Q,\ x'\in I_0\bigr\}.$$ It follows that $$\begin{split}
K=\operatorname{span}\Bigl(&\bigl\{T^k(q)\bigl(\sigma(x)-T^n(\eta(x))\bigr)\bigm| k\in{\mathbb{N}},\ q\in Q^{\otimes k},\ x\in I_0\bigr\}\\
&\cup \bigl\{T^k(q)\bigl(\sigma(x)-T^n(\eta(x))\bigr)S^l(p)\bigm| k,l\in{\mathbb{N}},\ q\in Q^{\otimes k},\ x\in I_0,\ p\in P^{\otimes l}\bigr\}\\
&\cup\bigl\{\sigma(x)-T^n(\eta(x))\bigm| x\in I_0\bigr\}\\
&\cup \bigl\{T^k(q)\bigl(\sigma(x)-T^n(\eta(x))\bigr)\bigm| l\in{\mathbb{N}},\ x\in I_0,\ p\in P^{\otimes l}\bigr\}\Bigr),
\end{split}$$ so to show that $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$, it sufficies to show the following 3 things:
1. \[item:4\] if $l\in{\mathbb{N}}$, $A$ is a finite subset of $\{(k,q,x,p)\mid k\in{\mathbb{N}},\ q\in Q^{\otimes l+k},\ x\in I_0,\ p\in P^{\otimes k}\}$ and $B$ is a finite subset of $\{(q,x)\mid q\in Q^{\otimes l},\ x\in I_0\}$, then $$\sum_{(k,q,x,p)\in A}T^{l+k}(q)\sigma(x)S^k(p) + \sum_{(q,x)\in B}T^l(q)\sigma(x)=0$$ if and only if $$\sum_{(k,q,x,p)\in A}T^{l+k}(q)T^n(\eta(x))S^k(p) + \sum_{(q,x)\in B}T^l(q)T^n(\eta(x))=0,$$
2. \[item:12\] if $A$ is a finite subset of $\{(k,q,x,p)\mid k\in{\mathbb{N}},\ q\in Q^{\otimes k},\ x\in I_0,\ p\in P^{\otimes k}\}$ and $x_0\in I_0$, then $$\sum_{(k,q,x,p)\in A}T^{k}(q)\sigma(x)S^k(p) + \sigma(x_0)=0$$ if and only if $$\sum_{(k,q,x,p)\in A}T^{k}(q)T^n(\eta(x))S^k(p) + T^n(\eta(x_0))=0,$$
3. \[item:13\] if $l\in{\mathbb{N}}$, $A$ is a finite subset of $\{(k,q,x,p)\mid k\in{\mathbb{N}},\ q\in Q^{\otimes k},\ x\in I_0,\ p\in P^{\otimes l+k}\}$ and $B$ is a finite subset of $\{(x,p)\mid x\in I_0,\ p\in P^{\otimes l+k}\}$, then $$\sum_{(k,q,x,p)\in A}T^{k}(q)\sigma(x)S^{l+k}(p) + \sum_{(q,x)\in B}\sigma(x)S^{l+k}(p)=0$$ if and only if $$\sum_{(k,q,x,p)\in A}T^{k}(q)T^n(\eta(x))S^{l+k}(p) + \sum_{(x,p)\in B}T^n(\eta(x))S^l(p)=0.$$
We will just prove . The other two claims can be proved in a similar way.
To prove , notice first that if $x\in I_0$ and $k\in{\mathbb{N}}$, then, since $I_0\subseteq J^{[\infty]}\subseteq J^{[k]}$, it follows from Lemma \[lemma:3\] that there are $q_1,\dots,q_m\in Q^{\otimes k}$ and $p_1,\dots,p_m\in P^{\otimes k}$ such that $\sigma(x)=\sum_{i=1}^mT^k(q_i)S^k(p_i)$. It follows from condition (FS) that there are $q'_1,\dots,q'_r,q''_1,\dots,q''_s\in Q^{\otimes k}$ and $p'_1,\dots,p'_r,p''_1,\dots,p''_s\in P^{\otimes k}$ such that $$\begin{split}
\sigma(x)&=\sum_{i=1}^mT^k(q_i)S^k(p_i) =\sum_{j=1}^r\sum_{l=1}^s \sum_{i=1}^m T^k(q'_j)S^k(p'_j)T^k(q_i)S^k(p_i)T^k(q''_l)S^k(p''_l)\\
&=\sum_{j=1}^r\sum_{l=1}^sT^k(q'_j)S^k(p'_j)\sigma(x)T^k(q''_l)S^k(p''_l) =\sum_{j=1}^r\sum_{l=1}^sT^k(q'_j)\sigma(\psi_k(p'_jx\otimes q''_l))S^k(p''_l).
\end{split}$$ Since $I_0$ is $\psi$-invariant, it follows that each $\psi_k(p'_jx\otimes q''_l)\in I_0$ and thus that $$T^n(\eta(x))=\sum_{j=1}^r\sum_{l=1}^sT^k(q'_j)T^n(\eta(\psi_k(p'_jx\otimes q''_l)))S^k(p''_l).$$ Thus it sufficies to show that if $k,l\in{\mathbb{N}}$ and $C$ is a finite subset of $\{(q,x,p)\mid q\in Q^{l+k},\ x\in I_0,\ p\in P^{\otimes k}\}$, then it is the case that $\sum_{(q,x,p)\in C}T^{l+k}(q)\sigma(x)S^k(p)=0$ if and only if $\sum_{(q,x,p)\in C}T^{l+k}(q)T^n(\eta(x))S^k(p)=0$, and that can be done using condition (FS) and the properties of $\eta$.
Condition (L) {#sec:condition-L}
=============
In this section condition (L) is introduced (Definition \[def:L\]) and sufficient and necessary conditions for when every non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ contains a non-zero graded ideal (Theorem \[thm:L\]) are given.
\[def:L\] We say that a $\psi$-invariant ideal $I$ in $R$ is an *$\psi$-invariant cycle* if there exist $n\in{\mathbb{N}}$ and an injective $R$-bimodule homomorphism $\eta:I\to Q^{\otimes n}$ such that $S_pT_{\eta(x)}(q)=\eta(\psi(px\otimes q))$ for $p\in P$, $x\in I$ and $q\in Q$, and we say that $J$ satisfies *condition (L)* with respect to the $R$-system $(P,Q,\psi)$ if there are no non-zero $\psi$-invariant cycles $I$ in $R$ such that $I\subseteq J^{[\infty]}$.
We will often, when it is clear from the context which $R$-system $(P,Q,\psi)$ we are working with, simply call a $\psi$-invariant cycle for an invariant cycle, and say that $J$ satisfies condition (L) instead of saying that it satisfies condition (L) with respect to $(P,Q,\psi)$.
Recall that if $(S',T',\sigma''B)$ is a covariant representation of $(P,Q,\psi)$, then $J_{(S',T',\sigma',B)}$ is defined to be the ideal $\{x\in R\mid \sigma'(x)\in\pi_{T',S'}(\mathcal{F}_P(Q)\}$ (see [@CaOr2011 Definition 3.23]).
\[thm:L\] The following 4 conditions are equivalent:
1. \[item:14\] The ideal $J$ satisfies condition (L).
2. \[item:5\] The subring ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property.
3. \[item:6\] Every non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ contains a non-zero graded ideal.
4. \[item:15\] If $(S',T',\sigma''B)$ is an injective covariant representation of $(P,Q,\psi)$ and $J=J_{(S',T',\sigma',B)}$, then the ring homomorphism $\eta^J_{(S',T',\sigma',B)}:{\mathcal{O}_{(P,Q,\psi)}(J)}\to B$ from [@CaOr2011 Theorem 3.29 (ii)] is injective.
$\eqref{item:14}\Leftrightarrow\eqref{item:5}$ follows from Proposition \[prop:main\].
$\eqref{item:5}\Rightarrow\eqref{item:6}$: Let $K$ be a non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$. Then $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}\ne \{0\}$ by assumption, and it follows from [@CaOr2011 Lemma 3.35] that the ideal $H$ generated by $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ is graded. Since $H$ is obviously contained in $K$, this proves .
$\eqref{item:6}\Rightarrow\eqref{item:5}$: Let $K$ be a non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$. By assumption there is a non-zero graded ideal $H$ such that $H\subseteq K$. It follows from [@CaOr2011 Lemma 3.35] that $H\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}\ne\{0\}$, so also $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}\ne\{0\}$, which proves that ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property.
$\eqref{item:5}\Rightarrow\eqref{item:15}$: Let $H$ be the ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ generated by $\ker\eta^J_{(S',T',\sigma',B)}\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$, and let $\wp:{\mathcal{O}_{(P,Q,\psi)}(J)}\to{\mathcal{O}_{(P,Q,\psi)}(J)}/H$ be the quotient map. Then $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)$ is a surjective covariant representation of $(P,Q,\psi)$. It follows from [@CaOr2011 Lemma 3.35] that $H$ is graded, from which it follows that the representation $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)$ is graded (see [@CaOr2011 Definition 3.20]). Since $H\subseteq \ker\eta^J_{(S',T',\sigma',B)}$, it follows that there is a ring homomorphism $\phi:{\mathcal{O}_{(P,Q,\psi)}(J)}/H\to B$ such that $\phi\circ\wp=\eta^J_{(S',T',\sigma',B)}$ and $\phi\circ\wp\circ S=S'$, $\phi\circ\wp\circ T=T'$ and $\phi\circ\wp\circ\sigma=\sigma'$. Since $(S',T',\sigma',B)$ is an injective representation, it follows that also $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)$ is injective. It follows from [@CaOr2011 Remark 3.13] that $$J\subseteq J_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)} \subseteq J_{(S',T',\sigma',B)}=J.$$ Thus $J_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)}=J$, and it follows from [@CaOr2011 Theorem 3.29] that $\wp$ is an isomorphism, and thus that $\ker\eta^J_{(S',T',\sigma',B)}\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$. It follows by assumption that $\ker\eta^J_{(S',T',\sigma',B)}=\{0\}$, and thus that $\eta^J_{(S',T',\sigma',B)}$ is injective.
$\eqref{item:15}\Rightarrow\eqref{item:5}$: Let $K$ be an ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ such that $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$, and let $\wp:{\mathcal{O}_{(P,Q,\psi)}(J)}\to{\mathcal{O}_{(P,Q,\psi)}(J)}/K$ be the quotient map. Then $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)$ is a surjective covariant representation of $(P,Q,\psi)$. Since $\sigma(R)$ and $\pi_{T,S}(\mathcal{F}_P(Q))$ are subsets of ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ and $K\cap{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}=\{0\}$, it follows from [@CaOr2011 Proposition 3.28] that $$J_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)} = J_{(S,T,\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)})}=J.$$ Thus $\wp=\eta_{(\wp\circ S,\wp\circ T\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)}$ is injective by assumption, and $K=\{0\}$ which proves that ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property.
The Cuntz-Krieger uniqueness theorem {#sec:cuntz-krieg-uniq}
====================================
In this Section the *Cuntz-Krieger uniqueness property* is defined (Definition \[def:CK\]), and the *Cuntz-Krieger uniqueness result* is proved (Theorem \[thm:CK\]).
\[def:CK\] We say that the ideal $J$ has the *Cuntz-Krieger uniqueness property* with respect to the $R$-system $(P,Q,\psi)$ if the following holds:
If $(S_1,T_1,\sigma_1,B_1)$ and $(S_2,T_2,\sigma_2,B_2)$ are two injective covariant representations of $(P,Q,\psi)$ and they are both Cuntz-Pimsner invariant relative to $J$, then there is a ring isomorphism $\phi$ between $\mathcal{R}\langle S_1,T_1,\sigma_1\rangle$ and $\mathcal{R}\langle S_2,T_2,\sigma_2\rangle$ such that $\phi\circ\sigma_1=\sigma_2$, $\phi\circ S_1=S_2$ and $\phi\circ T_1=T_2$.
We will often, when it is clear from the context which $R$-system $(P,Q,\psi)$ we are working with, simply say that $J$ has the Cuntz-Krieger uniqueness property instead of saying that it has the Cuntz-Krieger uniqueness property with respect to $(P,Q,\psi)$.
Recall from [@CaOr2011 Definition 4.6] that $J$ is said to be a *maximal* $\psi$-compatible ideal if $J=J'$ for any faithful $\psi$-compatible ideal $J'$ in $R$ satisfying $J\subseteq J'$.
\[thm:CK\] The following 5 conditions are equivalent:
1. \[item:9\] The ideal $J$ has the Cuntz-Krieger uniqueness property.
2. \[item:10\] If $(S',T',\sigma',B)$ is an injective covariant representation of $(P,Q,\psi)$ which is Cuntz-Pimsner invariant relative to $J$, then the ring homomorphism $\eta_{(S',T',\sigma',B)}^J:{\mathcal{O}_{(P,Q,\psi)}(J)}\to B$ from [@CaOr2011 Theorem 3.18] is injective.
3. \[item:7\] The subring $\sigma(R)$ has the ideal intersection property.
4. \[item:8\] The subring ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property, and $J$ is a maximal $\psi$-compatible ideal.
5. \[item:11\] The ideal $J$ satisfies condition (L) and is a maximal $\psi$-compatible ideal.
$\eqref{item:9}\Rightarrow\eqref{item:10}$: The ring homomorphism $\eta_{(S',T',\sigma',B)}^J:{\mathcal{O}_{(P,Q,\psi)}(J)}\to B$ is the unique ring homomorphism from ${\mathcal{O}_{(P,Q,\psi)}(J)}$ to $B$ such that $\eta_{(S',T',\sigma',B)}^J\circ\sigma=\sigma'$, $\eta_{(S',T',\sigma',B)}^J\circ S=S'$ and $\eta_{(S',T',\sigma',B)}^J\circ T=T'$, so it follows by assumption that $\eta_{(S',T',\sigma',B)}^J$ is injective.
$\eqref{item:10}\Rightarrow\eqref{item:9}$: If $(S_1,T_1,\sigma_1,B_1)$ and $(S_2,T_2,\sigma_2,B_2)$ are two injective covariant representations of $(P,Q,\psi)$ and there are both Cuntz-Pimsner invariant relative to $J$, then $\phi=\eta_{(S_2,T_2,\sigma_2,B_2)}^J\circ (\eta_{(S_1,T_1,\sigma_1,B_1)}^J)^{-1}$ is a ring isomorphism between $\mathcal{R}\langle S_1,T_1,\sigma_1\rangle$ and $\mathcal{R}\langle S_2,T_2,\sigma_2\rangle$ such that $\phi\circ\sigma_1=\sigma_2$, $\phi\circ S_1=S_2$ and $\phi\circ T_1=T_2$.
$\eqref{item:10}\Rightarrow\eqref{item:7}$: Let $K$ be an ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ such that $K\cap\sigma(R)=\{0\}$, and let $\wp:{\mathcal{O}_{(P,Q,\psi)}(J)}\to{\mathcal{O}_{(P,Q,\psi)}(J)}/K$ be the quotient map. Then $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)$ is an injective and surjective covariant representation of $(P,Q,\psi)$ which is Cuntz-Pimsner invariant relative to $J$. It follows by assumption that $\wp=\eta_{(\wp\circ S,\wp\circ T\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)}$ is injective. Thus $K=\{0\}$, which proves that $\sigma(R)$ has the ideal intersection property.
$\eqref{item:7}\Rightarrow\eqref{item:8}$: Since $\sigma(R)\subseteq{{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$, it follows that ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property if $\sigma(R)$ has. If $J$ is not a maximal $\psi$-invariant ideal, then there exists a $\psi$-compatible ideal $J'$ such that $J\subsetneq J'$. It follows from [@CaOr2011 Remark 4.1] that $\rho_J(\mathcal{T}(J'))$ then would be a non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ with a zero intersection with $\sigma(R)$, which would mean that $\sigma(R)$ does not have the ideal intersection property. Thus it must be the case that $J$ is a maximal $\psi$-invariant ideal.
$\eqref{item:8}\Rightarrow\eqref{item:10}$: Since $J$ is a maximal $\psi$-compatible ideal by assumption, it follows that $J_{(S',T',\sigma',B)}=J$. Thus it follows from Theorem \[thm:L\] that $\eta^J_{(S',T',\sigma',B)}$ is injective.
$\eqref{item:8}\Leftrightarrow\eqref{item:11}$ follows from Theorem \[thm:L\].
Simplicity of ${\mathcal{O}_{(P,Q,\psi)}(J)}$ {#sec:simplicity-ock}
=============================================
In this section sufficient and necessary conditions for when ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple are given (Theorem \[thm:simple\]).
We say that $J$ is a *super maximal* $\psi$-compatible ideal if the only $T$-pairs $(I,J')$ of $(P.Q,\psi)$ which satisfies that $J\subseteq J'$, are $(0,J)$ and $(R,R)$.
Since $(0,J')$ is a $T$-pair of $(P.Q,\psi)$ for any any faithful $\psi$-compatible ideal $J'$ in $R$, it follows that if $J$ is a super maximal $\psi$-compatible ideal, then it is also a maximal $\psi$-compatible ideal.
\[remark:supermax\] It follows from [@CaOr2011 Theorem 7.27] that $J$ is a super maximal $\psi$-compatible ideal if and only if the only graded ideals in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ are $\{0\}$ and ${\mathcal{O}_{(P,Q,\psi)}(J)}$.
\[thm:simple\] The following 5 conditions are equivalent:
1. \[item:16\] The ring ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple.
2. \[item:17\] The subring $\sigma(R)$ has the ideal intersection property and $J$ is a super maximal $\psi$-compatible ideal.
3. \[item:18\] The subring ${{\mathcal{O}_{(P,Q,\psi)}(J)}^{(0)}}$ has the ideal intersection property and $J$ is a super maximal $\psi$-compatible ideal.
4. \[item:19\] The ideal $J$ satisfies condition (L) and is a super maximal $\psi$-compatible ideal.
5. \[item:20\] If $(S',T',\sigma',B)$ is a non-zero covariant representation of $(P,Q,\psi)$ which is Cuntz-Pimsner invariant relative to $J$, then the ring homomorphism $$\eta_{(S',T',\sigma',B)}^J:{\mathcal{O}_{(P,Q,\psi)}(J)}\to B$$ from [@CaOr2011 Theorem 3.18] is injective.
$\eqref{item:16}\Rightarrow\eqref{item:17}$: If ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple, then clearly $\sigma(R)$ has the ideal intersection property. If $(I,J')$ is a $T$-pair of $(P,Q,\psi)$ different from $(0,J)$, then it follows from [@CaOr2011 Theorem 7.27] that $H^J_{(I,J')}$ is a non-zero ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$. If ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple, then that would imply that $H^J_{(I,J')}={\mathcal{O}_{(P,Q,\psi)}(J)}$ and thus $(I,J')=(R,R)$ from which it follows that $J$ is a super maximal $\psi$-compatible ideal.
$\eqref{item:17}\Leftrightarrow\eqref{item:18}$ and $\eqref{item:18}\Leftrightarrow\eqref{item:19}$ follow from Theorem \[thm:CK\] and the fact that $J$ is a maximal $\psi$-compatible ideal if it is a super maximal $\psi$-compatible ideal.
$\eqref{item:17}\Rightarrow\eqref{item:20}$: It follows from [@CaOr2011 Proposition 7.8] that $(I_{(S',T',\sigma',B)},J_{(S',T',\sigma',B)})$ is a $T$-pair. Since $(S',T',\sigma',B)$ is Cuntz-Pimsner invariant relative to $J$, it follows from [@CaOr2011 Remark 3.25] that $J\subseteq J_{(S',T',\sigma',B)}$, and since $(S',T',\sigma',B)$ is non-zero, it follows from [@CaOr2011 Theorem 7.11] that $(I_{(S',T',\sigma',B)},J_{(S',T',\sigma',B)})\ne (R,R)$. Thus $(I_{(S',T',\sigma',B)},J_{(S',T',\sigma',B)})=(0,J)$ which implies that $(S',T',\sigma',B)$ is an injective representation. It then follows from Theorem \[thm:CK\] that $\eta_{(S',T',\sigma',B)}^J$ is injective.
$\eqref{item:20}\Rightarrow\eqref{item:16}$: Let $K$ be a proper ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$, and let $\wp:{\mathcal{O}_{(P,Q,\psi)}(J)}\to{\mathcal{O}_{(P,Q,\psi)}(J)}/K$ be the quotient map. Then $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)$ is a surjective covariant representation of $(P,Q,\psi)$ which is Cuntz-Pimsner invariant relative to $J$. It follows by assumption that $\wp=\eta_{(\wp\circ S,\wp\circ T\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/K)}$ is injective. Thus $K=\{0\}$ which proves that ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is simple.
Condition (K) {#sec:condition-k}
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In this section condition (K) is introduced (Definition \[def:K\]), and sufficient and necessary conditions for when every ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is graded are given (Theorem \[thm:K\]).
Recall from [@CaOr2011 Section 7] that if $I$ is a $\psi$-invariant ideal in $R$, then $R_I=R/I$, $Q_I=Q/QI$ and ${}_IP=P/IP$, and $\wp_I$ denote the corresponding quotient map. Recall also that there is an $R_I$-bimodule homomorphism $\psi_I:{}_IP\otimes Q_I\to R_I$ given by $\psi_I(\wp_I(p)\otimes\wp_I(q))=\wp_I(\psi(p\otimes q))$. The triple $({}_IP,Q_I,\psi_I)$ is then an $R_I$-system satisfying condition (FS) (see [@CaOr2011 Lemma 7.4]). When $(I,J')$ is a $T$-pair, then $J'_I$ denote the faithful $\psi_I$-compatible ideal $\wp_I(J')$ in $R_I$.
\[def:K\] We say that the ideal $J$ satisfies *condition (K)* with respect to the $R$-system $(P,Q,\psi)$ if $J'_I$ satisfies condition (L) with respect to the $R_I$-system $({}_IP,Q_I,\psi_I)$ whenever $(I,J')$ is a $T$-pair of $(P,Q,\psi)$ such that $J\subseteq J'$.
We will often, when it is clear from the context which $R$-system $(P,Q,\psi)$ we are working with, simply say that $J$ satisfies condition (K) instead of saying that it satisfies condition (K) with respect to $(P,Q,\psi)$.
\[thm:K\] The following 3 conditions are equivalent:
1. \[item:21\] Every ideal of ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is graded.
2. \[item:22\] The ideal $J$ satisfies condition (K).
3. \[item:23\] If $(S',T',\sigma',B)$ is a covariant representation of $(P,Q,\psi)$ which is Cuntz-Pimsner invariant relative to $J$, and $(I,J')=\omega_{(S',T',\sigma',B)}$, then the ring homomorphism $\eta^{(I,J')}_{(S',T',\sigma',B)}:\mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I)\to B$ from [@CaOr2011 Theorem 7.11 (ii)] is injective.
$\eqref{item:21}\Rightarrow\eqref{item:22}$: Let $\omega=(I,J')$ be a $T$-pair of $(P,Q,\psi)$ such that $J\subseteq J'$ and let $H$ be a non-zero ideal in $\mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I)$. Recall from [@CaOr2011 Page 36] that there is a covariant representation $(\iota^\omega_P, \iota^\omega_Q, \iota^\omega_R,\mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I))$ such that $\iota^\omega_P=\iota^{J'_I}_{{}_IP}\circ\wp_I$, $\iota^\omega_Q=\iota^{J'_I}_{Q_I}\circ\wp_I$ and $\iota^\omega_R=\iota^{J'_I}_{R_I}\circ\wp_I$. It follows from [@CaOr2011 Remark 3.25 and Theorem 3.29] that there is a surjective graded ring homomorphism $\phi:{\mathcal{O}_{(P,Q,\psi)}(J)}\to \mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I)$ which intertwines the two representations $(S,T,\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)})$ and $(\iota^\omega_P, \iota^\omega_Q, \iota^\omega_R,\mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I))$. We then have that $\phi^{-1}(H)$ is an ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$. Thus $\phi^{-1}(H)$ is graded by assumption. It follows that also $H$ is graded. It therefore follows from Theorem \[thm:L\] that $J'_I$ satisfies condition (L) with respect to the $R_I$-system $({}_IP,Q_I,\psi_I)$. This proves that $J$ satisfies condition (K).
$\eqref{item:22}\Rightarrow\eqref{item:23}$: It follows from [@CaOr2011 Lemma 7.10] that there is an injective covariant representation $(S_I,T_I,\sigma_I,B)$ of $({}_IP,Q_I,\psi_I)$ such that $S_I=S'\circ\wp_I$, $T_I=T'\circ\wp_I$ and $\sigma_I=\sigma'\circ\wp_I$. Since $\pi_{T_I,S_I}(\mathcal{F}_{{}_IP}(Q_I))=\pi_{T',S'}(\mathcal{F}_P(Q))$, it follows that $J_{(S_I,T_I,\sigma_I,B)}=\wp_I(J_{(S',T',\sigma',B)})=\wp_I(J')=J'_I$. It therefore follows from Theorem \[thm:L\] that $\eta^{(I,J')}_{(S',T',\sigma',B)}=\eta^{J'}_{(S_I,T_I,\sigma_I,B)}$ is injective.
$\eqref{item:23}\Rightarrow\eqref{item:21}$: Let $H$ be an ideal in ${\mathcal{O}_{(P,Q,\psi)}(J)}$ and let $\wp:{\mathcal{O}_{(P,Q,\psi)}(J)}\to{\mathcal{O}_{(P,Q,\psi)}(J)}/H$ be the quotient map. Then $(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)$ is a covariant representation which is Cuntz-Pimsner invariant relative to $J$. Let $(I,J')=\omega_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)}$. Then $\eta^{(I,J')}_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)}$ is injective by assumption. Since $\oplus_{n\in{\mathbb{Z}}}\mathcal{O}^{(n)}_{({}_IP,Q_I,\psi_I)}(J'_I)$ is a ${\mathbb{Z}}$-grading of $\mathcal{O}_{({}_IP,Q_I,\psi_I)}(J'_I)$, it follows that $$\oplus_{n\in{\mathbb{Z}}}\wp(\mathcal{O}^{(n)}_{(P,Q,\psi)}(J)) =\oplus_{n\in{\mathbb{Z}}} \eta^{(I,J')}_{(\wp\circ S,\wp\circ T,\wp\circ\sigma,{\mathcal{O}_{(P,Q,\psi)}(J)}/H)} (\mathcal{O}^{(n)}_{({}_IP,Q_I,\psi_I)}(J'_I))$$ is a ${\mathbb{Z}}$-grading of ${\mathcal{O}_{(P,Q,\psi)}(J)}/H$. Thus $H$ is graded.
\[remark\] It follows from the above theorem that if $J$ satisfies condition (K), then [@CaOr2011 Theorem 7.27] gives a bijective correspondence between the set of all ideals of ${\mathcal{O}_{(P,Q,\psi)}(J)}$ and the set of $T$-pairs $(I,J')$ of $(P,Q,\psi)$ satisfying $J\subseteq J'$.
Toeplitz rings {#sec:toeplitz-rings}
==============
When $J=\{0\}$, then ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is the Toeplitz ring ${\mathcal{T}_{(P,Q,\psi)}}$ and $J$ automatically satisfies condition (L). Thus the following 3 corollaries follow from Theorem \[thm:L\], Theorem \[thm:CK\] and Theorem \[thm:simple\], respectively.
If $(S',T',\sigma',B)$ is an injective covariant representation of $(P,Q,\psi)$, then the ring homomorphism $\eta_{(S',T',\sigma',B)}:{\mathcal{T}_{(P,Q,\psi)}}\to B$ from [@CaOr2011 Theorem 1.7] is injective if and only if $J_{(S',T',\sigma',B)}=\{0\}$.
\[cor:Tuniq\] Assume that there are no non-zero faithful $\psi$-compatible ideals of $R$. If $(S_1,T_1,\sigma_1,B_1)$ and $(S_2,T_2,\sigma_2,B_2)$ are two injective covariant representations of $(P,Q,\psi)$, then there is a ring isomorphism $\phi$ between $\mathcal{R}\langle S_1,T_1,\sigma_1\rangle$ and $\mathcal{R}\langle S_2,T_2,\sigma_2\rangle$ such that $\phi\circ\sigma_1=\sigma_2$, $\phi\circ S_1=S_2$ and $\phi\circ T_1=T_2$.
The Toeplitz ring ${\mathcal{T}_{(P,Q,\psi)}}$ is simple if and only if $(0,0)$ and $(R,R)$ are the only $T$-pairs of $(P,Q,\psi)$.
Leavitt path algebras {#sec:leavitt}
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We will in this section show how we can recover from the results obtained in this paper Theorem 6.8, Corollary 6.10, Theorem 6.16, Corollary 6.17 and Theorem 6.18 of [@To2007] and obtain an algebraic analogue of [@FoRa1999 Theorem 4.1].
Let $(E^0,E^1,r,s)$ be a directed graph (ie. $E^0$ and $E^1$ are sets and $r$ and $s$ are maps from $E^1$ to $E^0$) and let $F$ be a field. When $n$ is a positive integer, then we let $E^n$ be the set $\{(e_1,e_2,\dots,e_n)\in E^1\times E^1\times\dots\times E^1\mid r(e_i)=s(e_{i+1})\text{ for }i=1,2,\dots,n-1\}$. For $\alpha=(e_1,e_2,\dots,e_n)\in E^n$ we define $s(\alpha)$ to be $s(e_1)$ and $r(\alpha)$ to be $r(e_n)$. For each $v\in E^0$ we let $vE^n$ denote the set $\{\alpha\in E^n\mid s(\alpha)=v\}$ and we let $E^nv$ denote the set $\{\alpha\in E^n\mid r(\alpha)=v\}$. A *closed path* is an $\alpha\in E^n$ such that $r(\alpha)=s(\alpha)$. The element $s(\alpha)$ is called the *base* of $\alpha$. A closed path $\alpha=(e_1,e_2,\dots,e_n)$ is said to be *simple* if $s(e_i)\ne s(e_1)$ for each $i=2,3,\dots,n$, and to have an *exit* if $|s(e_i)E^1|>1$ for some $i\in\{1,2,\dots,n\}$.
Following [@CaOr2011 Example 5.8] we define $R$ be the ring $\oplus_{v\in E^0}R_v$ where each $R_v$ is a copy of $F$; we let $Q$ be $R$-bimodule $\oplus_{e\in E^1}Q_e$ where each $Q_e$ is a copy of $F$ and the left and the right multiplication are defined by $$\begin{aligned}
\left(\sum_{e\in E^1} q_e\textbf{1}_{e}\right)\cdot \left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right) &= \sum_{e\in E^1} q_e r_{r(e)}\textbf{1}_{e} \\
\left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right)\cdot\left(\sum_{e\in E^1} q_e\textbf{1}_{e}\right) &= \sum_{e\in E^1} r_{s(e)} q_e\textbf{1}_{e} \end{aligned}$$ where $\textbf{1}_v$ denotes the unit of $R_v$, $\textbf{1}_e$ denotes the unit of $Q_e$, and $\{r_v\}_{v\in E^0}$ and $\{q_e\}_{e\in E^1}$ are families of elements from $F$ with only a finite number of non-zero elements; we let $P$ be the $R$-bimodule $\oplus_{e\in E^1} P_e$ where each $P_e$ is a copy of $F$ and the left and the right multiplication are defined by $$\begin{aligned}
\left(\sum_{e\in E^1} p_e\textbf{1}_{\overline{e}}\right)\cdot \left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right) &= \sum_{e\in E^1} p_e r_{s(e)}\textbf{1}_{\overline{e}} \\
\left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right)\cdot\left(\sum_{e\in E^1} p_e\textbf{1}_{\overline{e}}\right) &= \sum_{e\in E^1} r_{r(e)} p_e\textbf{1}_{\overline{e}} \end{aligned}$$ where $\textbf{1}_{\overline{e}}$ denotes the unit of $P_e$, and $\{r_v\}_{v\in E^0}$ and $\{p_e\}_{e\in E^1}$ are families of elements from $F$ with only a finite number of non-zero elements; and we define $\psi:P\otimes_R Q \to R$ to be the $R$-bimodule homomorphism given by $$\left(\sum_{e\in E^1} p_e\textbf{1}_{\overline{e}}\right)\otimes \left(\sum_{e\in E^1} q_e\textbf{1}_{e}\right)\mapsto \sum_{v\in E^0} \left(\sum_{e\in E^1v} p_{e} q_e \right)\textbf{1}_{v},$$ then $(P,Q,\psi)$ is an $R$-system. Recall also that if we let $J$ be the ideal $\text{span}_F\{\textbf{1}_v\mid v\in E^0,\ 0<|vE^1|<\infty \}\subseteq R$, then $J$ is a maximal faithful $\psi$-compatible ideal and ${\mathcal{O}_{(P,Q,\psi)}(J)}$ is isomorphic to the Leavitt path algebra of $(E^0,E^1)$ (see for example [@AbAr2005; @AbAr2008] and [@To2007]). It is straightforward to check that $J^{[n]}=\text{span}_F\{\textbf{1}_v\mid v\in E^0,\ 0<|vE^n|<\infty \}$ for each $n\in{\mathbb{N}}$ from which it follows that $J^{[\infty]}=\text{span}_F\{\textbf{1}_v\mid v\in E^0,\ 0<|vE^n|<\infty \text{ for all }n\in{\mathbb{N}}\}$.
Suppose that $I$ is a non-zero $\psi$-invariant cycle and let $\eta:I\to Q^{\otimes n}$ be an injective $R$-bimodule homomorphism satisfying $S_pT_{\eta(x)}(q)=\eta(\psi(px\otimes q))$ for $p\in P$, $x\in I$ and $q\in Q$. We will prove that it follows that $(E^0,E^1,r,s)$ has a closed path without an exit. We can, and will, identify $Q^{\otimes n}$ with the $R$-bimodule $\oplus_{\alpha\in E^n}Q_\alpha$ where each $Q_\alpha$ is a copy of $F$ and the left and the right multiplication are defined by $$\begin{aligned}
\left(\sum_{\alpha\in E^n} q_\alpha\textbf{1}_{\alpha}\right)\cdot \left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right) &= \sum_{\alpha\in E^n} q_\alpha r_{r(\alpha)}\textbf{1}_{\alpha} \\
\left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right)\cdot\left(\sum_{\alpha\in E^n} q_\alpha\textbf{1}_{\alpha}\right) &= \sum_{\alpha\in E^n} r_{s(\alpha)} q_\alpha\textbf{1}_{\alpha} \end{aligned}$$ where $\textbf{1}_\alpha$ denote the unit of $Q_\alpha$, and $\{r_v\}_{v\in E^0}$ and $\{q_\alpha\}_{\alpha\in E^n}$ are families of elements of $F$ with only a finite number of non-zero elements. Likewise, we identify $P^{\otimes n}$ with the $R$-bimodule $\oplus_{\alpha\in E^n}P_\alpha$ where each $P_\alpha$ is a copy of $F$ and the left and the right multiplication are defined by $$\begin{aligned}
\left(\sum_{\alpha\in E^n} p_\alpha\textbf{1}_{\overline{\alpha}}\right)\cdot \left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right) &= \sum_{\alpha\in E^n} p_\alpha r_{r(\alpha)}\textbf{1}_{\overline{\alpha}} \\
\left(\sum_{v\in E^0} r_v\textbf{1}_{v}\right)\cdot\left(\sum_{\alpha\in E^n} p_\alpha\textbf{1}_{\overline{\alpha}}\right) &= \sum_{\alpha\in E^n} r_{s(\alpha)} p_\alpha\textbf{1}_{\overline{\alpha}} \end{aligned}$$ where $\textbf{1}_{\overline{\alpha}}$ denote the unit of $P_\alpha$, and $\{r_v\}_{v\in E^0}$ and $\{p_\alpha\}_{\alpha\in E^n}$ are families of elements of $F$ with only a finite number of non-zero elements. We then have that $\psi_n:P^{\otimes n}\otimes Q^{\otimes n}\to R$ is given by $$\left(\sum_{\alpha\in E^n} p_\alpha\textbf{1}_{\overline{\alpha}}\right)\otimes \left(\sum_{\alpha\in E^n} q_\alpha\textbf{1}_{\alpha}\right)\mapsto \sum_{v\in E^0} \left(\sum_{\alpha\in E^nv} p_{\alpha} q_\alpha \right)\textbf{1}_{v}.$$
Let $H$ be the set $\{v\in E^0\mid \textbf{1}_v\in I\}$. It follows from the $\psi$-invariance of $I$ that $H$ is hereditary (that is, whenever $e\in E^1$ with $s(e)\in H$, then $r(e)\in H$). Let $v\in H$. Then $\eta(\textbf{1}_v)=\sum_{\alpha\in K}f_\alpha\textbf{1}_\alpha$ for some non-empty finite subset $K\subseteq E^n$ and non-zero elements $f_\alpha\in F,\ \alpha\in K$. Since $\textbf{1}_v\eta(\textbf{1}_v)\textbf{1}_v= \eta(\textbf{1}_v\textbf{1}_v\textbf{1}_v)=\eta(\textbf{1}_v)$, it follows that $r(\alpha)=s(\alpha)=v$ for each $\alpha\in K$. Let $\alpha\in E^n$ with $r(\alpha)=s(\alpha)=v$. Since $$\psi_n\bigl(\textbf{1}_{\overline{\alpha}}\otimes\eta(\textbf{1}_v)\bigr)\textbf{1}_\alpha= \eta\bigl(\psi_n(\textbf{1}_{\overline{\alpha}}\textbf{1}_v\otimes\textbf{1}_\alpha)\bigr)= \eta(\textbf{1}_v),$$ it follows that $K\subseteq\{\alpha\}$. Hence it must be the case that there is exactly one $\alpha_v\in E^n$ with $r(\alpha)=s(\alpha)=v$, and that $K$ consists of this element. Thus there is for each $v\in H$ a unique $\alpha_v\in E^n$ with $r(\alpha)=s(\alpha)=v$ and $\eta(\textbf{1}_v)=f_{\alpha_v}\textbf{1}_{\alpha_v}$ for some $f_{\alpha_v}\in F\setminus\{0\}$.
Let $v\in H$, let $\alpha_v=(e_1,e_2,\dots,e_n)$ and assume that there is an $e'\in E^1\setminus\{e_1\}$ with $s(e)=v$. Then $$\eta(\textbf{1}_{r(e)})=\eta\bigl(\psi(\textbf{1}_{\overline{e}}\textbf{1}_v\otimes\textbf{1}_e)\bigr)= S_{\textbf{1}_{\overline{e}}}T_{\eta(\textbf{1}_v)}\textbf{1}_e= f_{\alpha_v}S_{\textbf{1}_{\overline{e}}}T_{\textbf{1}_{\alpha_v}}\textbf{1}_e=0$$ which contradicts the fact that $\eta$ is injective. Thus, for each $v\in H$ it is the case that $vE^1=\{e_1\}$ where $e_1$ is the initial part of $\alpha_v$. It follows that every $v\in H$ is the base of a closed path which has no exit. In particular, $(E^0,E^1,r,s)$ has a closed path which has no exit.
On the other hand, it is straightforward to check that if $\alpha_v=(e_1,e_2,\dots,e_n)$ is a closed path without an exit, then $H=\{s(e_i)\mid i\in\{1,2,\dots,n\}$ is a hereditary subset of $E^0$, $I=\text{span}_F\{\textbf{1}_v\mid v\in H\}$ is contained in $J^{[\infty]}$ and is a $\psi$-invariant ideal in $R$, and the $F$-linear map $\eta:I\to Q^{\otimes n}$ given by $\textbf{1}_{s(e_i)}\mapsto\textbf{1}_{(e_i,e_{i+1},\dots,e_n,e_1,e_2,\dots,e_{i-1})}$ for $i\in\{1,2,\dots,n\}$ is an injective $R$-bimodule homomorphism $\eta:I\to Q^{\otimes n}$ satisfying $S_pT_{\eta(x)}(q)=\eta(\psi(px\otimes q))$ for $p\in P$, $x\in I$ and $q\in Q$. Thus $J$ satisfies condition (L) if and only every closed path in $(E^0,E^1,r,s)$ has an exit (cf. [@To2007 Definition 6.3]). We therefore recover [@To2007 Theorem 6.8 and Corollary 6.10] from Theorem \[thm:CK\]. By combining [@To2007 Theorem 5.7 and Proposition 6.12] and [@CaOr2011 Example 7.31] with the above characterization of when $J$ satisfies condition (L), one sees that $J$ satisfies condition (K) if and only if every $v\in E^0$ is either the base of no closed path or the base of at least two simple closed paths (cf. [@To2007 Definition 6.11]). We therefore recover [@To2007 Theorem 6.16 and Corollary 6.17] from Theorem \[thm:K\] and Remark \[remark\]. Finally, it follows from [@To2007 Theorem 5.7] (cf. [@CaOr2011 Example 7.31]) and Remark \[remark:supermax\] that $J$ is super maximal if and only if the only saturated hereditary subsets of $E^0$ are $\emptyset$ and $E^0$, thus we recover [@To2007 Theorem 6.18] from Theorem \[thm:simple\] and the above characterization of when $J$ satisfies condition (L).
We will end this subsection by using Corollary \[cor:Tuniq\] to give a uniqueness theorem for the Toeplitz ring $\mathcal{T}_{(P,Q,\psi)}=\mathcal{O}_{(P,Q,\psi)}(0)$.
\[def:TCK\] Let $E=(E^0,E^1,r,s)$ be a directed graph, let $F$ be a field and $B$ an $F$-algebra. A *Toeplitz-Cuntz-Krieger $E$-family* in $B$ consists of a family $\{p_v\mid v\in E^0\}$ of pairwise orthogonal idempotents in $B$ together with a family $\{x_e,y_e\mid e\in E^1\}$ of elements in $B$ satisfying the following relations
1. $p_{s(e)}x_{e}=x_{e}=x_{e}p_{r(e)}$ for $e\in E^1$,
2. $p_{r(e)}y_{e}=y_{e}=y_{e}p_{s(e)}$ for $e\in E^1$,
3. $y_{e}x_{f}=\delta_{e,f}p_{r(e)}$ for $e,f\in E^1$,
where $\delta_{e,f}$ denotes the Kronecker’s delta function.
\[thm:TCK\] Let $E=(E^0,E^1,r,s)$ be a directed graph and let $F$ be a field. Let $R$ and $(P,Q,\psi)$ be as defined above and let $(S,T,\sigma,\mathcal{T}_{(P,Q,\psi)})$ be the Toeplitz representation of $(P,Q,\psi)$. Then $\{\sigma(\textbf{1}_v)\mid v\in E^0\}$ together with $\{T(\textbf{1}_e), S(\textbf{1}_{\overline{e}})\mid e\in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family. If $B$ is an $F$-algebra and $\{p_v\mid v\in E^0\}$ together with $\{x_e,y_e\mid e\in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family, then there exists a unique $F$-algebra homomorphism $\eta:\mathcal{T}_{(P,Q,\psi)}\to B$ satisfying $\eta(\sigma(\textbf{1}_v))=p_v$ for $v\in E^0$, and $\eta(T(\textbf{1}_e))=x_e$ and $\eta(S(\textbf{1}_{\overline{e}}))=y_e$ for $e\in E^1$. The homomorphism $\eta$ is injective if and only if $p_v\ne 0$ for each $v\in E^0$ and $p_v\ne\sum_{e\in vE^1}x_ey_e$ for $v\in E^0$ with $0<\abs{vE^1}<\infty$.
That $\mathcal{T}_{(P,Q,\psi)}$ is an $F$-algebra and that $\{\sigma(\textbf{1}_v)\mid v\in E^0\}\cup \{T(\textbf{1}_e), S(\textbf{1}_{\overline{e}})\mid e\in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family is proved in [@CaOr2011 Example 1.10]. It is also proved in [@CaOr2011 Example 1.10] that if $B$ is an $F$-algebra and $\{p_v\mid v\in E^0\}$ together with $\{x_e,y_e\mid e\in E^1\}$ is a Toeplitz-Cuntz-Krieger $E$-family, then there is a covariant representation $(S',T',\sigma',B)$ of $(P,Q,\psi)$ such that $S'(\lambda\textbf{1}_{\overline{e}})=\lambda y_e$ and $T'(\lambda \textbf{1}_e)=\lambda x_e$ for $e\in E^1$ and $\lambda\in F$, and $\sigma'(\lambda\textbf{1}_v)=\lambda p_v$ for $v\in E^0$ and $\lambda\in F$. It then follows from [@CaOr2011 Theorem 1.7] that there is a ring homomorphism $\eta:\mathcal{T}_{(P,Q,\psi)}\to B$ such that $\eta(\sigma(\lambda\textbf{1}_v))=\sigma'(\lambda\textbf{1}_v)=\lambda p_v$ for $v\in E^0$ and $\lambda\in F$, and $\eta(T(\lambda\textbf{1}_e))=T'(\lambda\textbf{1}_e)=\lambda x_e$ and $\eta(S(\lambda\textbf{1}_{\overline{e}}))=S'(\lambda\textbf{1}_{\overline{e}})=\lambda y_e$ for $e\in E^1$ and $\lambda\in F$. It follows that $\eta$ is a $F$-algebra homomorphism and that $\eta(\sigma(\textbf{1}_v))=p_v$ for $v\in E^0$, and $\eta(T(\textbf{1}_e))=x_e$ and $\eta(S(\textbf{1}_{\overline{e}}))=y_e$ for $e\in E^1$. Since $\mathcal{T}_{(P,Q,\psi)}$ is generated, as an $F$-algebra, by $\{\sigma(\textbf{1}_v)\mid v\in E^0\}\cup\{T(\textbf{1}_e), S(\textbf{1}_{\overline{e}})\mid e\in E^1\}$, there cannot be any other $F$-algebra homomorphism from $\mathcal{T}_{(P,Q,\psi)}$ to $B$ which for every $v\in E^0$ maps $\sigma(\textbf{1}_v)$ to $p_v$ and for any $e\in E^1$ maps $T(\textbf{1}_e)$ to $x_e$ and $S(\textbf{1}_{\overline{e}})$ to $y_e$.
The map $\sigma$ is injective by [@CaOr2011 Theorem 1.7]. It follows that if $\eta$ is injective, then $p_v\ne 0$ for each $v\in E^0$. Assume that $p_v\ne 0$ for each $v\in E^0$. Since $R=\oplus_{v\in E^0}R_v$ where each $R_v$ is a copy of $F$, it follows that $\sigma'$ is injective. Thus it follows from Corollary \[cor:Tuniq\] that $\eta$ is injective if and only if $J_{(S',T',\sigma',B)}=0$. It follows from [@CaOr2011 Lemma 3.24] that $$J_{(S',T',\sigma',B)}=\bigl\{r\in\Delta^{-1}(\mathcal{F}_P(Q))\mid \sigma'(r)=\pi_{T',S'}(\Delta(r))\bigr\}.$$ It is proved in [@CaOr2011 Example 5.8] that $$\Delta^{-1}(\mathcal{F}_P(Q))=\operatorname{span}_F\{\textbf{1}_v\mid 0<\abs{vE^1}<\infty\},$$ and is straightforward to check that $\Delta(\textbf{1}_v)=\sum_{e\in vE^1}\theta_{\textbf{1}_e,\textbf{1}\overline{e}}$ if $\textbf{1}_v\in \Delta^{-1}(\mathcal{F}_P(Q))$. It follows that $$J_{(S',T',\sigma',B)}=\operatorname{span}_F\Bigl\{\textbf{1}_v\bigm| 0<\abs{vE^1}<\infty,\ p_v=\sum_{e\in vE^1}x_ey_e\Bigr\}.$$ Thus $\eta$ is injective if and only if $p_v\ne 0$ for each $v\in E^0$ and $p_v\ne\sum_{e\in vE^1}x_ey_e$ for $v\in E^0$ with $0<\abs{vE^1}<\infty$.
Theorem \[thm:TCK\] is the algebraic analogue of [@FoRa1999 Theorem 4.1].
Crossed products of a ring by an automorphism and fractional skew monoid rings of a corner isomorphism {#sec:cross}
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We will in this section use Theorem \[thm:simple\] to give a characterization of when the fractional skew monoid ring of a ring isomorphism is simple (Corollary \[cor:fractional\]), and when the crossed product of a ring by an automorphism is simple (Corollary \[cor:auto\]).
A ring $R$ has *local units* if given any finite set $F\subseteq R$ there exists an idempotent $e\in R$ such that $er=re=r$ for every $r\in F$, in other words, the set of all idempotents of $R$, $\text{Idem}(R)$, is a directed system (with order $e\leq f$ if and only if $ef=fe=e$) and $R=\bigcup_{e\in \text{Idem}(R)} eRe$.
Let $R$ be a ring with local units and let $\alpha:R\to R$ be an injective ring homomorphism such that $\alpha(R)R\alpha(R)\subseteq\alpha(R)$ (notice this is equivalent to $\alpha(R)R\alpha(R)=\alpha(R)$ since $R$ has local units). Recall from [@CaOr2011 Example 5.6] that if $P$ is the $R$-bimodule which is equal to $\operatorname{span}\{r_1\alpha(r_2)\mid r_1,r_2,\in R\}$ as a set, has the additive structure it inherits from $R$, and has the left and right actions given by $r\cdot p=rp$ and $p\cdot r=p\alpha(r)$ for $r\in R$ and $p\in P$; $Q$ is the $R$-bimodule which is equal to $\operatorname{span}\{\alpha(r_1)r_2\mid r_1,r_2\in R\}$ as a set, has the additive structure it inherits from $R$, and has the left and right given by $r\cdot q=\alpha(r)q$ and $q\cdot r=qr$ for $r\in R$ and $q\in Q$; and $\psi:P\otimes Q\to R$ is the $R$-bimodule homomorphism given by $p\otimes q\mapsto pq$, then $(P,Q,\psi)$ is an $R$-system. Recall also that $R$ is a uniquely maximal, faithful, $\psi$-compatible ideal and that if $\alpha$ is an automorphism, then $\mathcal{O}_{(P,Q,\psi)}(R)$ is isomorphic to the crossed product $R\times_\alpha{\mathbb{Z}}$ of $R$ by $\alpha$. If $R$ is unital, and we let $e=\alpha(1)$ (where 1 denotes the unit of $R$), then $e$ is an idempotent and $\alpha(R)=\alpha(R)R\alpha(R)=eRe$. It follows from [@CaOr2011 Example 5.7] that we in this case have that $\mathcal{O}_{(P,Q,\psi)}(R)$ is isomorphic to the fractional skew monoid ring $R[t_+,t_-;\alpha]$ that Ara, González-Barroso, Goodearl and Pardo have constructed in [@ArGoGo2004]. We will use these facts together with Theorem \[thm:simple\] to give a characterization of when the crossed product $R\times_\alpha{\mathbb{Z}}$ is simple and when the fractional skew monoid ring $R[t_+,t_-;\alpha]$ is simple, but first we introduce some notions and results that we will use for this.
Unless otherwise stated, $\alpha$ will just be assumed to be an injective ring homomorphism such that $\alpha(R)R\alpha(R)\subseteq\alpha(R)$. We let $(P,Q,\psi)$ be the $R$-system defined above. Using that $R$ has local units, it is not difficult to see that for $n\in{\mathbb{N}}$, the $R$-bimodule $P^{\otimes n}$ is isomorphic to the $R$-bimodule which is equal to $\operatorname{span}\{r_1\alpha^n(r_2)\mid r_1,r_2,\in R\}$ as a set, has the additive structure it inherits from $R$, and has the left and right actions given by $r\cdot p=rp$ and $p\cdot r=p\alpha^n(r)$, respectively. Likewise, $Q^{\otimes n}$ is isomorphic to the $R$-bimodule which is equal to $\operatorname{span}\{\alpha^n(r_1)r_2\mid r_1,r_2\in R\}$ as a set, has the additive structure it inherits from $R$ and has the left and right given by $r\cdot q=\alpha^n(r)q$ and $q\cdot r=qr$, respectively. We will simply identify $P^{\otimes n}$ and $Q^{\otimes n}$ with these two $R$-bimodules. We will use a $\cdot$ to indicate the left and right actions of $R$ on $P^{\otimes n}$ and $Q^{\otimes n}$ to distinguish these actions from the ordinary multiplication in $R$. It is straightforward to check that if $q\in Q$, $q_n\in Q^{\otimes n}$ and $p\in P$, then $S_pT_{q_n}(q)=\alpha^n(p)\alpha(q_n)q$. Let $(S,T,\sigma,\mathcal{O}_{(P,Q,\psi)}(R))$ denote the Cuntz-Pimsner representation of $(P,Q,\psi)$ relative to $R$. Then $S^n(p_n)\sigma(r) = S^n(p_n\alpha^n(r))$, $\sigma(r)S^n(p_n) = S^n(rp_n)$, $S^n(p_n)S^{n'}(p'_{n'})=S^{n+n'}(p_n\alpha^n(p'_{n'}))$, $T^n(q_n)\sigma(r)=T^n(q_nr)$, $\sigma(r)T^n(q_n) = T^n(\alpha^n(r)q_n)$, $T^n(q_n)T^{n'}(q'_{n'})=T^{n+n'}(\alpha^{n'}(q_n)q'_{n'})$, $S^n(p)T^n(q)=\sigma(pq)$ and $T^n(q)S^n(p)=\sigma(\alpha^{-n}(q_np_n))$ for $n,n'\in{\mathbb{N}}$, $p_n\in P^n$, $r\in R$, $n'\in P^{\otimes n'}$, $q_n\in Q^{\otimes n}$ and $q_{n'}\in Q^{\otimes n'}$ where $p_n$, $p_{n'}$, $q_n$ and $q_{n'}$ are considered as elements of $R$ and the multiplication of $R$ is used. It follows that $\mathcal{O}_{(P,Q,\psi)}(R)^{(0)}=\sigma(R)$, and that $\mathcal{O}_{(P,Q,\psi)}(R)^{(n)}=T^n(Q^{\otimes n})$ and $\mathcal{O}_{(P,Q,\psi)}(R)^{(-n)}=S^n(P^{\otimes n})$ for $n\in{\mathbb{N}}$. We say that an ideal $I$ of $R$ is strongly $\alpha$-invariant if $\alpha(I)\subseteq I$ and $\alpha(R)I\alpha(R)\subseteq \alpha(I)$ (this is equivalent to $\alpha(R)I\alpha(R)= \alpha(I)$ since $R$ has local units).
Let $R$ be a ring with local units, $\alpha:R\to R$ an injective ring homomorphism satisfying $\alpha(R)R\alpha(R)\subseteq\alpha(R)$, and let $(P,Q,\psi)$ be the $R$-system defined above. Then there is a bijective correspondence between graded ideals of $\mathcal{O}_{(P,Q,\psi)}(R)$ and strongly $\alpha$-invariant ideals of $R$.
For each strongly $\alpha$-invariant ideal $I$ in $R$, let $H_I$ be the ideal in $\mathcal{O}_{(P,Q,\psi)}(R)$ generated by $\sigma(I)$; and let for each graded ideal $H$ in $\mathcal{O}_{(P,Q,\psi)}(R)$, $I_H=\{x\in R\mid \sigma(x)\in H\}$. We will show that $H_I$ is a graded ideal in $\mathcal{O}_{(P,Q,\psi)}(R)$, that $I_H$ is a strongly $\alpha$-invariant ideal in $R$, and that $I_{H_I}=I$ and $H_{I_H}=H$ for all strongly $\alpha$-invariant ideals $I$ in $R$ and all graded ideals $H$ in $\mathcal{O}_{(P,Q,\psi)}(R)$. This will establish the bijective correspondence between the graded ideals of $\mathcal{O}_{(P,Q,\psi)}(R)$ and the strongly $\alpha$-invariant ideals of $R$.
Let $I$ be a strongly $\alpha$-invariant ideal in $R$. It is not difficult to check that if we let $H^{(0)}=\sigma(I)$ and for each $n\in{\mathbb{N}}$ let $H^{(n)}=\operatorname{span}\{T^n(\alpha^n(r)x)\mid r\in R,\ x\in I\}$ and $H^{(-n)}=\operatorname{span}\{S^n(x\alpha^n(r))\mid x\in I,\ r\in R\}$, then $\oplus_{n\in{\mathbb{Z}}}H^{(n)}$ is an ideal in $\mathcal{O}_{(P,Q,\psi)}(R)$. Since $\oplus_{n\in{\mathbb{Z}}}H^{(n)}$ contains $\sigma(I)$ and itself must be contained in any ideal which contains $\sigma(I)$, it must be the case that $H_I=\oplus_{n\in{\mathbb{Z}}}H^{(n)}$. It follows that $H_I$ is graded and that $I_{H_I}=I$.
Let $H$ be a graded ideal in $\mathcal{O}_{(P,Q,\psi)}(R)$. It is clear that $I_H$ is an ideal in $R$. Assume that $x\in I_H$. Choose idempotens $e_1,e_2\in R$ such that $e_1\alpha(x)e_1=\alpha(x)$ and $e_2xe_2=x$. Then $$\sigma(\alpha(x))=S(e_1\alpha(e_2))\sigma(x)T(\alpha(e_2)e_1)\in H,$$ so $\alpha(x)\in I_H$. Assume then that $r_1,r_2\in R$. Choose idempotents $f_1,f_2\in R$ such that $f\alpha(r_1)f_1=\alpha(r_1)$ and $f_2\alpha(r_2)=\alpha(r_2)$. Then $$\sigma(\alpha^{-1}(\alpha(r_1)x\alpha(r_2)))= T(\alpha(r_1)f_1)\sigma(x)S(f_2\alpha(r_2))\in H,$$ so $\alpha(r_1)x\alpha(r_2)\in \alpha(I_H)$. This shows that $I_H$ is a strongly $\alpha$-invariant ideal in $R$. Since $\mathcal{O}_{(P,Q,\psi)}(R)^{(0)}=\sigma(R)$, it follows from [@CaOr2011 Lemma 3.35] that $H$ is generated by $\sigma(I_H)$. Thus $H=H_{I_H}$.
By combining the above result with Remark \[remark:supermax\] we get the following characterization of when $R$ is a super maximal $\psi$-compatible ideal.
\[cor:supermax\] Let $R$ be a ring with local units, $\alpha:R\to R$ an injective ring homomorphism satisfying $\alpha(R)R\alpha(R)\subseteq\alpha(R)$, and let $(P,Q,\psi)$ be the $R$-system defined above. Then the following three conditions are equivalent:
1. The ring $R$ is a super maximal $\psi$-compatible ideal.
2. The only graded ideals in $\mathcal{O}_{(P,Q,\psi)}(R)$ are $\{0\}$ and $\mathcal{O}_{(P,Q,\psi)}(R)$.
3. The only strongly $\alpha$-invariant ideals in $R$ are $\{0\}$ and $R$.
We next introduce the *multiplier ring* of $R$ (see for example [@ArPe2000]). A double centralizer on $R$ is a pair $(f,g)$ where $f:R\to R$ is a right $R$-module homomorphism and $g:R\to R$ is a left $R$-module homomorphism satisfying $r_1f(r_2)=g(r_1)r_2$ for all $r_1,r_2\in R$. The *multiplier ring* of $R$ is the ring $\mathcal{M}(R)$ of all double centralizers on $R$ with addition defined by $(f_1,g_1)+(f_2,g_2)=(f_1+f_2,g_1+g_2)$ and product defined by $(f_1,g_1)(f_2,g_2)=(f_1\circ f_2,g_2\circ g_1)$. Notice that $(\operatorname{Id}_R,\operatorname{Id}_R)$ is a unit of $\mathcal{M}(R)$. There is a ring homomorphism $\iota:R\to\mathcal{M}(R)$ given by $\iota(r)=(f_r,g_r)$ where $f_r(s)=rs$ and $g_r(s)=sr$ for $r,s\in R$. Since $R$ has local units, $\iota$ is injective. We will therefore simple regard $R$ as a subring of $\mathcal{M}(R)$. We then have that if $u=(f,g)\in\mathcal{M}(R)$ and $r\in R$, then $ur=f(r)$ and $ru=g(r)$. It follows that $R$ is an ideal in $\mathcal{M}(R)$. Notice that $R=\mathcal{M}(R)$ if and only if $R$ is unital.
\[def:inner\] Let $n\in{\mathbb{N}}$ and let $R$ be a ring with local units. A ring homomorphism $\alpha:R\to R$ is said to be *inner with periodicity $n$* if there exist $u,v\in\mathcal{M}(R)$ such that $vu=1$ (where $1$ denotes the unit of $\mathcal{M}(R)$), and $\alpha^n(r)=urv$ and $\alpha(ur)=u\alpha(r)$ for all $r\in R$. If $\alpha$ is not inner of any periodicity, then it is said to be *outer*.
\[remark:inner\] Notice that if $\alpha$ is an automorphism and $u,v$ are as above, then $v$ is the inverse of $u$.
In [@ArPe2000] the authors introduce a topology on $\mathcal{M}(R)$ in the following way. A net $(x_\lambda)_{\lambda\in \Lambda}$ of elements of $\mathcal{M}(R)$ converges *strictly* to an a element $x\in \mathcal{M}(R)$ if there for every $r\in R$ exists $\lambda_0\in \Lambda$ such that $(x_\lambda-x)r=r(x_\lambda-x)=0$ for $\lambda\geq \lambda_0$. Since $R$ has local units, a net in $\mathcal{M}(R)$ can at most converges strictly to one element. Such an element will, if it exists, be called the *strict limit* of the net. A net $(x_\lambda)_{\lambda\in \Lambda}$ is *Cauchy* if there for every $r\in R$ exists $\lambda_0\in \Lambda$ such that $r(x_\lambda-x_\mu)=(x_\lambda-x_\mu)r=0$ for $\lambda,\mu\geq \lambda_0$. It is shown in [@ArPe2000 Proposition 1.6] that if $R$ has local units, then every Cauchy net in $\mathcal{M}(R)$ converges strictly, and that every element of $\mathcal{M}(R)$ is the strict limit of a net of elements of $R$.
A net $(r_\lambda)_{\lambda\in \Lambda}$ of elements of $R$ that converges to the unit of $\mathcal{M}(R)$ is called an *approximate unit* for $R$. Notice that in case $R$ has local units we can construct an approximate unit $(e_\lambda)_{\lambda\in \Lambda}$ consisting of idempotents simple by letting $\Lambda$ be the directed set of finite subsets of $R$ ordered by inclusion, and then for every $\lambda\in\Lambda$ choosing an idempotent $e_\lambda$ such that $e_\lambda r=re_\lambda=r$ for every $r\in \lambda$.
\[def:strict\] Let $R$ be a ring with local units. A ring homomorphism $\alpha:R\to R$ is said to be *strict* if there exists an approximate unit $(e_\lambda)_{\lambda\in\Lambda}$ for $R$ consisting of idempotents such that $(\alpha(e_\lambda))_{\lambda\in\Lambda}$ converges strictly.
\[remark:strict\] Notice that if $\alpha$ is an automorphism, then it is strict (since $(\alpha(e_\lambda))_{\lambda\in\Lambda}$ converges strictly to the unit in that case). Notice also that if $R$ is unital, then every ring homomorphism $\alpha:R\to R$ is automatically strict (because the net consisting of just $1$ is an approximate unit in that case).
\[prop:inner\] Let $R$ be a ring with local units, $\alpha:R\to R$ an injective ring homomorphism satisfying $\alpha(R)R\alpha(R)\subseteq\alpha(R)$, and let $(P,Q,\psi)$ be the $R$-system defined above. Consider the following three conditions:
1. \[item:221\] There exists an $n\in{\mathbb{N}}$ such that the homomorphism $\alpha$ is inner with periodicity $n$.
2. \[item:222\] The ring $R$ is a $\psi$-invariant cycle.
3. \[item:223\] The ring $R$ does not satisfy condition (L) with respect to $(P,Q,\psi)$.
Then implies , and implies . If in addition $R$ is a super maximal $\psi$-compatible ideal, and $\alpha^n$ is strict for every $n\in{\mathbb{N}}$, then implies and the three conditions are equivalent.
$\eqref{item:221}{\Rightarrow}\eqref{item:222}$: Let $u$ and $v$ be elements in $\mathcal{M}(R)$ such that $vu=1$, and $urv=\alpha^n(r)$ and $\alpha(ux)=u\alpha(x)$ for all $r\in R$. Define $\eta:R\to R$ by $\eta(r)=ur$. Let $r\in R$. Choose $e\in R$ such that $er=r$. Then we have that $\eta(r)=ur=uer=uevur=\alpha^n(e)ur$. This shows that $\eta(R)\subseteq Q^{\otimes n}$. It is clear that $\eta$ is additive and injective. Let $r_1,r_2,\in R$. Then $\eta(r_1r_2)=ur_1r_2=\eta(r_1)r_2$ and $\eta(r_1r_2)=ur_1r_2=\alpha^n(r_1)ur_2=\alpha^n(r_1)\eta(r_2)$, which shows that $\eta$ is an $R$-bimodule homomorphism from $R$ to $Q^{\otimes n}$. Let $p\in P$, $r\in R$ and $q\in Q$. Then we have that $$\begin{split}
\eta\left(\psi(p\cdot r\otimes q)\right)
&=\eta\left(p\alpha(r)q\right)=up\alpha(r)q=\alpha^n(p)u\alpha(r)q\\
&=\alpha^n(p)\alpha(ur)q=\alpha^n(p)\alpha(\eta(r))q=S_pT_{\eta(r)}(q).
\end{split}$$ Thus $R$ is a $\psi$-invariant cycle.
$\eqref{item:222}{\Rightarrow}\eqref{item:223}$: It is easy to see that $\psi^{-1}(R)=R$ from which it follows that $R^{[\infty]}=R$. Thus, if $R$ is a $\psi$-invariant cycle, then $R$ does not satisfy condition (L) with respect to $(P,Q,\psi)$.
$\eqref{item:223}{\Rightarrow}\eqref{item:221}$: Assume that $R$ does not satisfy condition (L) with respect to $(P,Q,\psi)$. It then follows from Proposition \[prop:main\] that there is a non-zero graded ideal $\bigoplus_{k\in{\mathbb{Z}}}H^{(k)}$ in $\mathcal{O}_{(P,Q,\psi)}(R)$, an $n\in{\mathbb{N}}$ and a family $(\phi_k)_{k\in{\mathbb{Z}}}$ of injective $\mathcal{O}_{(P,Q,\psi)}(R)^{(0)}$-bimodule homomorphisms $\phi_k:H^{(k)}\to\mathcal{O}_{(P,Q,\psi)}(R)^{(k+n)}$ such that $x\phi_k(y)=\phi_{k+j}(xy)$ and $\phi_k(y)x=\phi_{k+j}(yx)$ for $k,j\in{\mathbb{Z}}$, $x\in\mathcal{O}_{(P,Q,\psi)}(R)^{(j)}$ and $y\in H^{(k)}$. Notice that also $\bigoplus_{k\in{\mathbb{Z}}}\phi_{k-n}(H^{(k-n)})$ is a non-zero graded ideal in $\mathcal{O}_{(P,Q,\psi)}(R)$. If $R$ is a super maximal $\psi$-compatible ideal, then it follows from Corollary \[cor:supermax\] that $\bigoplus_{k\in{\mathbb{Z}}}H^{(k)}=\bigoplus_{k\in{\mathbb{Z}}}\phi_{k-n}(H^{(k-n)})=\mathcal{O}_{(P,Q,\psi)}(R)$ from which it follows that $H^{(0)}=\phi_{-n}(H^{(-n)})=\mathcal{O}_{(P,Q,\psi)}(R)^{(0)}=\sigma(R)$, $\phi_0(H^{(0)})=\mathcal{O}_{(P,Q,\psi)}(R)^{(n)}=T^n(Q^{\otimes n})$ and $H^{(-n)}=\mathcal{O}_{(P,Q,\psi)}(R)^{(-n)}=S^n(P^{\otimes n})$. Suppose in addition that $\alpha^n$ is strict, and let $(e_\lambda)_{\lambda\in\Lambda}$ be an approximate unit for $R$ consisting of idempotents such that $(\alpha(e_\lambda))_{\lambda\in\Lambda}$ converges strictly. Since $T^n$ and $\phi_{-n}$ are injective, and $Q^{\otimes n}$ and $P^{\otimes n}$ are subsets of $R$, there exists for each $\lambda\in\Lambda$ a unique $u_\lambda\in R$ such that $T^n(u_\lambda)=\phi_0(\sigma(e_\lambda))$ and a unique $v_\lambda\in R$ such that $\phi_{-n}(S^n(v_\lambda))=\sigma(e_\lambda)$. Notice that $$T^n(u_\lambda)=\phi_0(\sigma(e_\lambda))
=\phi_0(\sigma(e_\lambda e_\lambda))
=\sigma(e_\lambda)\phi_0(\sigma(e_\lambda))
=\sigma(e_\lambda)T^n(u_\lambda)=T^n(\alpha^n(e_\lambda)u_\lambda).$$ It follows that $\alpha^n(e_\lambda)u_\lambda=u_\lambda$. If $\lambda,\lambda_1\in\Lambda$ and $e_{\lambda_1}e_\lambda=e_{\lambda_1}$, then $$\begin{split}
T^n(\alpha^n(e_{\lambda_1})u_\lambda)
&=\sigma(e_{\lambda_1})T^n(u_\lambda)
=\sigma(e_{\lambda_1})\phi_0(\sigma(e_\lambda))\\
&=\phi_0(\sigma(e_{\lambda_1}e_\lambda))
=\phi_0(\sigma(e_{\lambda_1}))
=T^n(u_{\lambda_1}),
\end{split}$$ from which it follows that $\alpha^n(e_{\lambda_1})u_\lambda=u_{\lambda_1}$. Let $r\in R$. Choose $\lambda_1,\lambda_2,\lambda_3\in\Lambda$ such that $r\alpha^n(e_\lambda)=r\alpha^n(e_{\lambda_1})$ for $\lambda\ge\lambda_1$, $e_{\lambda_1}e_\lambda=e_{\lambda_1}$ for $\lambda\ge\lambda_2$, and $e_\lambda r=r$ for $\lambda\ge \lambda_3$. If $\lambda\ge \lambda_1,\lambda_2,\lambda_3$, then $$ru_\lambda=r\alpha^n(e_\lambda)u_\lambda=r\alpha^n(e_{\lambda_1})u_\lambda
=ru_{\lambda_1},$$ and $$T^n(u_\lambda r)
=T^n(u_\lambda)\sigma(r)=\phi_0(\sigma(e_\lambda))\sigma(r)
=\phi_0(\sigma(e_\lambda r))=\phi_0(\sigma(r)).$$ This shows that $(u_\lambda)_{\lambda\in\Lambda}$ is Cauchy and hence converges strictly to an element $u\in\mathcal{M}(R)$. One can by a similar method show that $(v_\lambda)_{\lambda\in\Lambda}$ converges strictly to an element $v\in\mathcal{M}(R)$.
Let $\lambda\in\Lambda$. Then $$\begin{split}
\sigma(v_\lambda u_\lambda)&=S^n(v_\lambda)T^n(u_\lambda)
=S^n(v_\lambda)\phi_0(\sigma(e_\lambda))
=\phi_{-n}(S^n(v_\lambda)\sigma(e_\lambda))\\
&=\phi_{-n}(S^n(v_\lambda))\sigma(e_\lambda)
=\sigma(e_\lambda)\sigma(e_\lambda)=\sigma(e_\lambda),
\end{split}$$ from which it follow that $v_\lambda u_\lambda=e_\lambda$. Thus $vu=1$.
Let $r\in R$. Choose $\lambda_0\in\Lambda$ such that $re_\lambda=e_\lambda r=r$ for $\lambda\ge\lambda_0$. If $\lambda\ge\lambda_0$, then $$T^n(\alpha^n(r)u_\lambda)=\sigma(r)\phi_0(\sigma(e_\lambda))
=\phi_0(\sigma(re_\lambda))=\phi_0(\sigma(e_\lambda r))
=\phi_0(\sigma(e_\lambda))\sigma(r)=T^n(u_\lambda r).$$ It follows that $\alpha^n(r)u=ur$ and thus that $urv=\alpha^n(r)$.
Let $r\in R$. Choose $\lambda_0\in\Lambda$ such that $e_\lambda r=r$ and $e_\lambda\alpha(r)=\alpha(r)$ for $\lambda\ge\lambda_0$. If $\lambda\ge\lambda_0$ then $$\begin{split}
T^n(\alpha(u_\lambda r))
&=T^n\bigl(\alpha^{n+1}(e_\lambda)\alpha(u_\lambda)\alpha(r)\bigr)
=S(\alpha(e_\lambda))T^n(u_\lambda) T(\alpha(r))\\
&=S(\alpha(e_\lambda))\phi_0(\sigma(e_\lambda))T(\alpha(r))
= \phi_0\bigl(S(\alpha(e_\lambda))\bigr) \sigma(e_\lambda)T(\alpha(r))\\
&=\phi_0\bigl(\sigma(\alpha(e_\lambda e_\lambda r))\bigr)
=\phi_0\bigl(\sigma(\alpha(r))\bigr)
=\phi_0\bigl(\sigma(e_\lambda\alpha(r))\bigr)
=\phi_0(\sigma(e_\lambda))\sigma(\alpha(r))\\
&=T^n(u_\lambda)\sigma(\alpha(r))=T^n(u_\lambda\alpha(r)),
\end{split}$$ from which it follows that $\alpha(u_\lambda r)=u_\lambda\alpha(r)$. Thus $\alpha(ur)=u\alpha(r)$.
Hence $\alpha$ is inner with periodicity $n$ in this case.
By combining Theorem \[thm:simple\] and Corollary \[cor:supermax\] with Remark \[remark:strict\], Proposition \[prop:inner\], and the fact that $\mathcal{O}_{(P,Q,\psi)}(R)$ is isomorphic to the crossed product $R\times_\alpha{\mathbb{Z}}$ of $R$ by $\alpha$ when $\alpha$ is an automorphism, and to the fractional skew monoid ring $R[t_+,t_-;\alpha]$ when $R$ is unital and $\alpha$ is an injective homomorphism such that $\alpha(R)=eRe$ for some idempotent $e\in R$, we get the following two corollaries.
\[cor:fractional\] Let $R$ be a unital ring and let $\alpha:R\to R$ be an injective ring homomorphism such that $\alpha(R)=eRe$ for some idempotent $e\in R$. Then the following two statements are equivalent:
1. The fractional skew monoid ring $R[t_+,t_-;\alpha]$ is simple.
2. The homomorphism $\alpha$ is outer and the only strongly $\alpha$-invariant ideals in $R$ are $\{0\}$ and $R$.
\[cor:auto\] Let $R$ be a ring with local units and let $\alpha:R\to R$ be a ring automorphism. Then the following two statements are equivalent:
1. The crossed product $R\times_\alpha{\mathbb{Z}}$ is simple.
2. The automorphism $\alpha$ is outer and the only strongly $\alpha$-invariant ideals in $R$ are $\{0\}$ and $R$.
We end by noticing that when $\alpha$ is an automorphism, the condition of $\alpha$ being outer is equivalent with the seemingly stronger, and perhaps more familiar, condition that $\alpha$ is *strongly outer*.
Let $n\in{\mathbb{N}}$ and let $R$ be a ring with local units and $\alpha:R\to R$ a ring automorphism. If there exists an invertible element $u\in\mathcal{M}(R)$ such that $\alpha^n(r)=uru^{-1}$ for all $r\in R$, then $\alpha$ is said to be *weakly inner with periodicity $n$*. If $\alpha$ is not weakly inner of any periodicity, then it is said to be *strongly outer*.
Let $R$ be a ring with local units and let $\alpha:R\to R$ a ring automorphism. Then $\alpha$ is outer if and only if it is strongly outer.
It follows from Remark \[remark:inner\] that if $\alpha$ is strongly outer, then it is also outer.
Suppose that $\alpha$ is not strongly outer. Then there exist $n\in{\mathbb{N}}$ and an invertible element $u\in\mathcal{M}(R)$ such that $\alpha^n(r)=uru^{-1}$ for all $r\in R$. If $x=(f,g)\in\mathcal{M}(R)$ where $(f,g)$ is a double centralizer, then we let $\hat{\alpha}(x)$ denote the double centralizer $(\alpha\circ f\circ\alpha^{-1},\alpha\circ g\circ\alpha^{-1})$. It is easy to check that $x\mapsto\hat{\alpha}(x)$ defines an automorphism $\hat{\alpha}$ of $\mathcal{M}(R)$ and that $\hat{\alpha}^n(x)=uxu^{-1}$ for all $x\in\mathcal{M}(R)$. In particular $\hat{\alpha}^n(u)=uuu^{-1}=u$ and $\hat{\alpha}^n(u^{-1})=uu^{-1}u^{-1}=u^{-1}$. Let $$u'=u\hat{\alpha}(u)\dots\hat{\alpha}^{n-1}(u)\text{ and } v'=\hat{\alpha}^{n-1}(u^{-1})\dots\hat{\alpha}(u^{-1})u^{-1}.$$ Then $v'u'=1$. If $r\in R$, then $$\begin{split}
\alpha(u'r)&=\hat{\alpha}(u')\alpha(r)=\hat{\alpha}
\bigl(u\hat{\alpha}(u)\dots\hat{\alpha}^{n-1}(u)\bigr)\alpha(r)
=\hat{\alpha}(u)\hat{\alpha}^2(u)\dots\hat{\alpha}^n(u)\alpha(r)\\
&=\hat{\alpha}^{n+1}(u)\hat{\alpha}^{n+2}(u)\dots\hat{\alpha}^{2n}(u)\alpha(r)
=\hat{\alpha}^n
\bigl(\hat{\alpha}(u)\hat{\alpha}^2(u)\dots\hat{\alpha}^n(u)\bigr)\alpha(r)\\
&=u\hat{\alpha}(u)\hat{\alpha}^2(u)\dots\hat{\alpha}^n(u)u^{-1}\alpha(r)
=u\hat{\alpha}(u)\hat{\alpha}^2(u)\dots\hat{\alpha}^{n-1}(u)uu^{-1}\alpha(r)
=u'\alpha(r)
\end{split}$$ and $$\begin{split}
u'rv'&=u\hat{\alpha}(u)\dots\hat{\alpha}^{n-1}(u)r
\hat{\alpha}^{n-1}(u^{-1})\dots\hat{\alpha}(u^{-1})u^{-1}\\
&=\hat{\alpha}^n\bigl(\hat{\alpha}(u)\dots\hat{\alpha}^{n-1}(u)r
\hat{\alpha}^{n-1}(u^{-1})\dots\hat{\alpha}(u^{-1})\bigr)\\
&=\hat{\alpha}(u)\dots\hat{\alpha}^{n-1}(u)\alpha^n(r)
\hat{\alpha}^{n-1}(u^{-1})\dots\hat{\alpha}(u^{-1})\\
&=\hat{\alpha}(u)\dots\hat{\alpha}^{n-2}(u)\alpha^{n+1}(r)
\hat{\alpha}^{n-2}(u^{-1})\dots\hat{\alpha}(u^{-1})\\
&\phantom{=\hat{\alpha}(u)\alpha^{(n-1)n}(r)}\vdots\\
&=\hat{\alpha}(u)\alpha^{(n-1)n}(r)\hat{\alpha}(u^{-1})=\alpha^{n^2}(r).
\end{split}$$ Thus $\alpha$ is inner with periodicity $n^2$ and is therefore not outer.
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of this work was done during visits of the third author to the Institut for Matematik og Datalogi, Syddansk Universitet and to the Institut for Matematiske Fag, Københavns Universitet (Denmark). The third author thanks both host centers for their kind hospitality.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'X-ray observations of some short gamma-ray bursts indicate that a long-lived neutron star can form as a remnant of a binary neutron star merger. We develop a gravitational-wave detection pipeline for a long-lived binary neutron star merger remnant guided by these counterpart electromagnetic observations. We determine the distance out to which a gravitational-wave signal can be detected with Advanced LIGO at design sensitivity and the Einstein Telescope using this method, guided by X-ray data from GRB140903A as an example. Such gravitational waves can in principle be detected out to $\sim$ 20 Mpc for Advanced LIGO and $\sim$ 450 Mpc for the Einstein Telescope assuming a fiducial ellipticity of $10^{-2}$. However, in practice we can rule out such high values of the ellipticity as the total energy emitted in gravitational waves would be greater than the total rotational energy budget of the system. We show how these observations can be used to place upper limits on the ellipticity using these energy considerations. For GRB140903A, the upper limit on the ellipticity is $10^{-3}$, which lowers the detectable distance to $\sim$ 2 Mpc and $\sim$ 45 Mpc for Advanced LIGO and the Einstein Telescope, respectively.'
author:
- Nikhil Sarin
- 'Paul D. Lasky'
- Letizia Sammut
- Greg Ashton
bibliography:
- 'ref.bib'
title: 'X-ray guided gravitational-wave search for binary neutron star merger remnants\'
---
\[sec:intro\]Introduction
=========================
The era of gravitational-wave multi-messenger astrophysics has begun. On 17th August 2017, the Advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) [@aligo] and Advanced Virgo [@Virgo] made the first gravitational-wave observation of a binary neutron star merger, known as GW170817 [@GW170817]. This event was also detected 1.74 seconds later as a short gamma-ray burst (SGRB) by the Fermi and Integral telescopes [@GW170817A_GRB], confirming that binary neutron star mergers can be the progenitors of SGRBs. There are competing hypotheses for the fate of the post-merger remnant. Some analyses of the electromagnetic observations support a hypermassive neutron star that collapsed to form a black hole in $\lesssim 1$s [@Metzger2018; @Pooley2017; @Margalit2017]. Others support the formation of a stable, rapidly spinning, long-lived magnetar [@Yu2018].
In either case, a short- or long-lived post-merger remnant emits gravitational waves. The detection of such gravitational waves will have significant implications for the understanding of neutron-star physics including the nuclear equation of state. A search for short and intermediate duration gravitational-wave signals from a post-merger remnant of GW170817 did not return a significant result [@postmerger2017]. This lack of detection was expected given theoretical models [@DallOsso2014; @Lasky2016a; @Doneva2015] and current aLIGO sensitivity. However, the proximity of GW170817, in conjunction with planned upgrades to aLIGO and Virgo sensitivity [@observing_aligo] and improved algorithms, suggests, that we may be able to detect post-merger gravitational waves from GW170817-like remnants in the future.
In general, the merger of two neutron stars could result in four different outcomes, which depend on the mass and spin of the remnant and the equation of state - a stable neutron star, a supramassive neutron star, a hyper massive neutron star or the direct collapse to a black hole. A supramassive neutron star is initially supported against gravitational collapse by rigid-body rotation but will collapse to form a black hole on timescales of $10 \textrm{s} - 10^4 \textrm{s}$ [@Ravi2014]. A hypermassive neutron star is supported against gravitational collapse through differential rotation but collapses to a black hole in $\leq 1$s (see @Baiotti2017 for a recent review).
In this paper, we focus on the scenario where a neutron star merger produces a supramassive or stable neutron star remnant. This rapidly spinning star spins down through a combination of electromagnetic and gravitational-wave radiation. The latter is likely produced by the non-zero stellar ellipticity in conjunction with the spin-flip instability [@Cutler2002; @Lasky2016a], unstable r-modes [@ANDERSSON2001; @Owen1998] or the secular Chandrasekhar-Friedmann-Schutz bar-mode instability [@Lai1995; @Shapiro1998; @Coyne2016; @Shibata2000; @Corsi2009; @Doneva2015].
The extended X-ray emission of many SGRBs has been observed by satellites such as *Swift* and Chandra, and used to determine parameters of the neutron star remnant [e.g., @Rowlinson2013; @Lu2015; @LaskyLeris2017]. @Rowlinson2010[@Rowlinson2010; @Rowlinson2013] showed that models of magnetic dipole radiation from spinning down millisecond magnetars [@Zhang2001; @Dai1998] agree with X-ray afterglow observations of several SGRBs. GRB170817A had an extended emission of a different structure [e.g., @JJRuan2018; @Troja2017].
In this paper, we present a method to search for gravitational waves from a long-lived post-merger neutron star remnant. In Sec. \[sec:waveform\] we derive a model for the gravitational waves emitted from a rapidly spinning down millisecond magnetar while also describing the parameters and the parameter space. In Sec. \[sec:GRBafterglow\] we discuss how we can utilize observations of X-ray afterglows from SGRBs to constrain parameters and run a targeted gravitational-wave search. We continue in Sec. \[sec:pipeline\] with a discussion of the detection statistics for our pipeline and conclude in Sec. \[sec:conclusion\] with a brief discussion on the extensions that will improve the analysis and physical theory.
Gravitational waveform from millisecond magnetars {#sec:waveform}
=================================================
A long-lived post-merger remnant spins down due to electromagnetic and gravitational-wave radiation. We start with the general torque equation. $$\label{Ch2: Eq. Torque}
\dot{\Omega} = -k\Omega^{n},$$ where $\Omega$ and $\dot{\Omega}$ are the star’s angular frequency and its time derivative, respectively, $k$ is a constant of proportionality, and $n$ is the braking index. The gravitational-wave frequency is a function of the star’s spin frequency. Throughout this work, we assume the gravitational waves are emitted at twice the star’s spin frequency, which is true for an orthogonal rotator. The following equations are therefore not valid for gravitational waves from $r$-mode emission; we discuss generalizations of our model in Sec. \[sec:conclusion\].
The braking index is related to the emission mechanism; $n = 3$ implies that the neutron star is spun down only through a dipole magnetic field in vacuum [@1983Shapiro], while $n = 5$ implies that the neutron star is spun down through gravitational-wave radiation [@Yue2006; @Bonazzola1996]. A braking index of $n = 7$ is conventionally associated with spin down through unstable $r$ modes [e.g. @Owen1998], although the true value can be less for different saturation mechanisms [@Alford2014a; @Alford2014b]. Inference of the braking index for two millisecond magnetars born in SGRBs give $n = 2.9 \pm 0.1$ and $2.6 \pm 0.1$ for GRB130603B and GRB140903A, respectively [@LaskyLeris2017].
Integrating Eq. (\[Ch2: Eq. Torque\]) and solving for the gravitational-wave frequency gives the gravitational-wave frequency evolution $$\label{Ch2: Eq.fgw(t)}
\fgw(t) = \fgwo\left(1+ \frac{t}{\tau}\right)^{\frac{1}{1 - n}},$$ where $$\tau = \frac{\left(\fgwo\pi\right)^{1 - n}}{-k(1-n)},$$ is the spin-down timescale and $\fgwo$ is the gravitational-wave frequency at $ t = 0$.
The dimensionless gravitational-wave strain amplitude for a non-axisymmetric, rotating body obeying Eq. (\[Ch2: Eq. Torque\]) is given by $$\label{Ch2: Eq.strainamp(t)}
h_0(t) = \frac{4\pi^2 G I_{zz}}{c^4} \frac{\epsilon}{d} \fgwo^2\left(1+ \frac{t}{\tau}\right)^{\frac{2}{1 - n}}.$$ Here, $I_{zz}$ is the principle moment of inertia, $\epsilon$ is the ellipticity of the rotating body, $d$ is the distance to the source, $G$ is the gravitational constant, and $c$ is the speed of light. The gravitational-wave strain at a detector $h(t)$ is a combination of the $h_+$ and $h_{\times}$ polarisations, $$\label{Ch2: Eq. ht}
h(t) = h_{0}(t)\left[F_{+}\frac{1+\cos^{2}(\iota)}{2}\cos\Phi(t) + F_{\times}\cos(\iota)\sin\Phi(t)\right],$$ where, $\iota$ is the inclination angle, and $$\label{Ch2: Eq. phi(t)}
\Phi(t) = \Phi_{0} + 2\pi\int_{0}^{t}dt'\fgw(t'),$$ is the phase, with $\Phi_0 = \Phi(t = 0)$. In Eq. (\[Ch2: Eq. ht\]), $F_+$ and $F_{\times}$ are the antenna pattern functions [@Jaranowski1998] for each of the polarisations. In reality, $F_+$ and $F_{\times}$ are functions of time. In this work, we have ignored this complication and assumed constant $F_+$ and $F_{\times}$ which we determine using the sky location of GRB140903A. This does not significantly affect our quantitative results, although it will need to be included when the full pipeline is developed to search for gravitational waves.
Substituting the gravitational-wave frequency evolution from Eq. (\[Ch2: Eq.fgw(t)\]) into Eq. (\[Ch2: Eq. phi(t)\]) gives $$\label{Ch2: Eq. Phi}
\Phi(t) = \Phi_0 + 2\pi\tau \fgwo\left(\frac{1 - n}{2 - n}\right)\left[\left(1 + \frac{t}{\tau}\right)^{\frac{2-n}{1 - n}} - 1\right].$$ The full waveform model for a rapidly rotating neutron star spinning down due to gravitational wave radiation with an arbitrary braking index consists of Eq. (\[Ch2: Eq.strainamp(t)\]), (\[Ch2: Eq. ht\]), and (\[Ch2: Eq. Phi\]). We refer to this waveform model as the magnetar waveform model, which is parameterized by the initial gravitational-wave frequency $f_{gw,0}$, the spin-down timescale $\tau$, braking index $n$, inclination $\iota$, initial phase $\Phi_0$ and scaling parameters $I_{zz}$, $\epsilon$, $d$.
In the following, we develop an algorithm for a matched-filter search for gravitational waves using the magnetar waveform model. We construct a template bank by choosing physical parameters for $\fgwo$, $\tau$, $n$, $\iota$, and $\Phi_0$ from a prior. We quantify in Sec. \[sec:pipeline\] that a template bank constructed from physically motivated but unconstrained priors is computationally expensive for detecting gravitational waves, but these priors can be further constrained using X-ray afterglow observations which reduce the computational cost of searches and increase the sensitivity. The scaling parameters do not require priors as they only affect the amplitude of the gravitational wave which is normalised in a matched-filter search. Throughout this work, we assume a fiducial moment of inertia, $I_{zz} = 10^{45}$ $\text{g cm}^{2}$, an optimal orientation $\iota = 0$, and a constant ellipticity $\epsilon$. We note that the strain scales linearly with the moment of inertia, which may be a factor of a few larger than our fiducial value. In principle, we can choose to model the ellipticity as a function of time. However, over the long timescales considered here, the ellipticity is not expected to evolve significantly; the internal magnetic field that likely causes the stellar deformation gets wound up on the Alfvén timescale, which for these systems is $\ll 1$s [e.g., @Shapiro2000]. Although it is possible to have an evolution of the ellipticity through other mechanisms such as stellar cooling, the effect is similar to the angle between the star’s principal moment of inertia and its rotation axis evolving due to, for example, the spin-flip instability (see Sec. \[sec:conclusion\]). We leave this generalization for future work.
Gravitational-wave energy budget {#subsec:Egwt}
--------------------------------
We also consider the energy budget of the gravitational wave emission to determine allowed regions of the parameter space. The total power emitted in gravitational waves is $$\dot{E}_{\textrm{gw}}(t) = -\frac{32G}{5c^5}I_{zz}^{2}\epsilon^2\Omega^{6}(t).
\label{Eq. power}$$ We substitute our gravitational-wave frequency evolution Eq. (\[Ch2: Eq.fgw(t)\]) for the evolution of the star’s angular frequency and integrate to determine the energy emitted in gravitational waves for a constant braking index $$\label{Eq. Egw}
E_{\textrm{gw}}(t) = -\frac{32\pi^6G}{5c^5}I_{zz}^2\fgwo^6\epsilon^2\tau\frac{n - 1}{n - 7}\left[\left(1 + \frac{t}{\tau}\right)^{\frac{7 - n}{1 - n}} - 1 \right].$$ This energy evolution is different to a standard continuous-wave signal as the strain evolves as a function of time. The total energy emitted in gravitational waves must be less than the initial rotational energy, $E_{\textrm{rot}}$ of the system $$\label{eq. energy_budget}
\lvert E_{\textrm{gw}}(t)\rvert < E_{\textrm{rot}},$$ where $$E_{\textrm{rot}} = \frac{1}{2}I_{zz}\fgwo^2\pi^2.$$ We can use this condition to check if a given parameter space is physical.
![The energy budget of a post-merger remnant inferred from GRB140903A with ellipticity $\epsilon = 10^{-2}$ (solid curves) and $10^{-3}$ (dashed curves) with the red shaded region indicating the 2$\sigma$ confidence interval. The grey shaded region above the solid black horizontal line is nonphysical as discussed in Sec. \[subsec:Egwt\].[]{data-label="fig:Energy Budget"}](Plots/Energy_Budget.pdf){width="50.00000%"}
Figure \[fig:Energy Budget\] illustrates, for a post-merger remnant inferred from GRB140903A with a fiducial $I_{zz} = 10^{45}$ $\text{g cm}^{2}$, an ellipticity $\epsilon = 10^{-2}$ violates the energy-budget constraint. Based on these energy considerations the upper limit on ellipticity for GRB140903A is $\epsilon \approx 10^{-3}$. In reality, the moment of inertia for a long-lived post-merger remnant is likely higher than the fiducial value we use here, however all our limits can be scaled appropriately for different values of $I_{zz}$. In particular, the moment of inertia is inversely proportional to the inferred upper limit on ellipticity, because the rotational energy grows linearly with $I_{zz}$, but the gravitational-wave energy grows quadratically. Our fiducial moment of inertia therefore provides a conservative limit on the ellipticity.
Optimal matched filter statistic {#subsec:rho_opt}
--------------------------------
The matched-filter signal-to-noise ratio $\rho$ is given by [@Cutler1994] $$\rho = \frac{\langle h|u\rangle}{\sqrt{\langle u|u\rangle}},
\label{eq:snr}$$ where $h = s + n$ is the combination of signal $s$ and noise $n$, $u$ is the template, and $\langle a|b\rangle$ denotes the noise-weighted inner product [@Cutler1994], defined by $$\label{Ch3: Eq. inner_product}
\langle a|b\rangle = 4 \Re \int_{0}^{\infty} \frac{\tilde{a}^{\ast}(f)\tilde{b}(f)}{S_h(f)}df.$$ Here $\tilde{a}$ denotes the Fourier transform of $a$, $\tilde{a}^\star$ its complex conjugate, and $S_{h}(f)$ is the noise power spectral density. The optimal matched-filter signal-to-noise ratio $\rho_{\textrm{opt}}$ is achieved when the template matches the data precisely: $$\label{eq:snr_exp}
\rho_{\textrm{opt}} = \sqrt{\langle h|h\rangle}.$$ In this analysis, the threshold signal-to-noise ratio required to make a detection is $\rho_{\textrm{threshold}} = 4.4$, which is derived in Sec. \[sec:pipeline\]. In Fig. \[Fig. rho\_opt\] we show the region of parameter space where we could detect a signal from a post-merger remnant at the same distance as GW170817 ($40$ Mpc). We assume $I_{zz} = 10^{45}$ $\text{g cm}^{2}$, $\epsilon = 0.01$ (top panel) and $\epsilon = 0.001$ (bottom panel), $n = 2.71$ and $\fgwo = 2050$ Hz. We use these values of $\fgwo$ and $n$ as they are the maximum likelihood parameters from GRB140903A using the method detailed in Sec. \[sec:GRBafterglow\].
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The left-hand side of Fig. \[Fig. rho\_opt\] shows it is theoretically possible for gravitational waves from such an object to be observable by aLIGO operating at design sensitivity [@observing_aligo] if $\tau \gtrsim 4 \times 10^4$ s and $t_{\textrm{obs}} \gtrsim 4 \times10^4$ s. The right-hand side shows that the Einstein Telescope (ET), a proposed third generation detector [@ET], can detect such a signal if $\tau \gtrsim 10^2$ s and $t_{\textrm{obs}} \gtrsim 10^2$ s for $\epsilon = 10^{-2}$. We note that GRB140903A has $\tau = 17207 \pm 1880$ s. However, as shown in Sec. \[subsec:Egwt\] this large ellipticity is nonphysical for GRB140903A-like post-merger remnant in all of the parameter space required to detect a signal with aLIGO. A physically realistic ellipticity $\epsilon = 10^{-3}$ rules out any prospect of detection with aLIGO for a GRB140903A-like post-merger signal at $40$ Mpc and requires $\tau \gtrsim 10^4$ s and $t_{\textrm{obs}} \gtrsim 10^4$ s for detecting the same signal with ET.
The optimal matched filter signal-to-noise ratio (Eq. \[eq:snr\_exp\]) can also be used to estimate the distance out to which we can detect a signal.
![The optimal matched-filter signal-to-noise ratio $\rho_{\textrm{opt}}$ as a function of distance for a millisecond magnetar inferred from GRB140903A for aLIGO (top panel) and ET (bottom panel) for two different ellipticities; $\epsilon = 10^{-2}$ (solid curves) and $\epsilon = 10^{-3}$ (dashed curves). The red shaded region indicates the $2\sigma$ confidence interval from the posteriors shown in Fig. \[fig:corner\]. A threshold $\rho_{\textrm{opt}} = 4.4$ is indicated by a black horizontal dotted line. Any value above this threshold is detectable by aLIGO at design sensitivity. All curves are constructed using an observation time of $5\times10^4$ s.[]{data-label="fig. kappa GRB140903A"}](Plots/rho_vs_D.pdf){width="50.00000%"}
Figure \[fig. kappa GRB140903A\] shows that with aLIGO at design sensitivity the furthest distance we can detect a signal with maximum likelihood parameters inferred from GRB140903A is $40$ and $4$ Mpc for $\epsilon = 10^{-2}$ and $10^{-3}$ respectively, while with ET the distances are $900$ Mpc and $90$ Mpc respectively. As we showed in Sec. \[subsec:Egwt\], for the parameters inferred from GRB140903A only an ellipticity $\epsilon \leq 10^{-3}$ is physical, post-merger remnants with longer spin-down timescale, $\tau$, can be detected to larger distances assuming that $\epsilon \sim 10^{-3}$ is physical for those parameters.
The optimal matched filter is the maximum signal-to-noise ratio one can achieve in a matched filter search. In practice, this limit is unobtainable with current computational resources. As shown by Fig. \[Fig. rho\_opt\] and Fig. \[fig. kappa GRB140903A\], to achieve $\rho_{\textrm{opt}} \geq 4.4$ and make a detection of gravitational waves, we need to observe a signal for at least $ \sim 10^4$ seconds with aLIGO at design sensitivity. At large observation times, the volume of parameter space imposed by uniform priors becomes unfeasible for a realistic gravitational-wave search (see Sec. \[sec:pipeline\]). In the following section, we demonstrate how to constrain the priors, and hence the search parameter space, using X-ray observations of SGRBs
\[sec:GRBafterglow\]X-ray afterglow
===================================
Short gamma-ray bursts are often followed by X-ray emission lasting up to many tens of thousands of seconds [@Rowlinson2010; @Rowlinson2013; @LaskyLeris2017; @Lu2015]. Such an X-ray afterglow was not observed for GRB170817A. In Fig. \[fig:lightcurve\] we show the X-ray afterglow of GRB140903A with data from the Neil Gehrels *Swift* and Chandra satellites [@Troja2016].
![$\gamma$ and X-ray lightcurves for GRB140903A. Black points are data from *Swift* and Chandra satellites. The blue curve shows the maximum likelihood model described in Sec. \[sec:GRBafterglow\]. The dark red band is the superposition of 800 models randomly drawn from the posterior distribution (shown in Fig. \[fig:corner\]). The dashed black curve is the model for the luminosity from the nascent neutron star (Eq. \[eq:luminosity\]).[]{data-label="fig:lightcurve"}](Plots/GRB140903A_chains_unfixed.pdf){width="50.00000%"}
@Rowlinson2013 modelled the X-ray afterglows of several SGRBs with two components. Firstly, an initial power-law decay, $$\label{Eq. Powerlaw}
L(t) = At^{-r},$$ where $L$ is the luminosity, $A$ is the power-law amplitude, and $r$ is the power-law exponent. Here, the decay exponent can be fixed to $r = \Gamma_{\gamma} + 1$, where $\Gamma_{\gamma}$ is the photon index of the prompt emission, or allowed to vary. The second component is a luminosity law to model the energy injection from a millisecond magnetar that is spinning down through magnetic dipole radiation ($n = 3$) [@Zhang2001; @Dai1998]. @LaskyLeris2017 extended this model to include other forms of radiation causing spin-down, which is derived by utilising the general torque equation (Eq. \[Ch2: Eq. Torque\]). The luminosity of the second component therefore comes directly from the nascent neutron star, and can be expressed as $$\label{eq:luminosity}
L(t) = L_{0}\left(1+ \frac{t}{\tau}\right)^{\frac{1 + n}{1 - n}},$$ where, $L_0$ is the initial luminosity at the onset of the plateau phase and is related to the initial gravitational-wave frequency $\fgwo$ by $$L_0 = \frac{\fgwo^2\pi^{2}I_{zz}\eta}{2\tau},$$ where $\eta$ encodes the efficiency of converting spin-down energy to X-rays. Our numerical model involves fitting Eq. (\[Eq. Powerlaw\]) and (\[eq:luminosity\]) to the X-ray observations from *Swift* and Chandra. However, instead of fitting $L_0$ we fit our initial gravitational-wave frequency $\fgwo$. We use a Markov Chain Monte Carlo algorithm [@Foreman-Mackey2013] to fit the X-ray afterglow of SGRBs with our model using uniform priors for $f_{\text{gw}}$, $n$, $\tau$, $A$, and $r$ between \[$\log_{10}(-1)$, $\log_{10}(5)$\], \[$\log_{10}(2)$, $\log_{10}(6)$\], \[$0$, $6$\], \[$\log_{10}(-10)$, $\log_{10}(5)$\], and \[$-2$, $5$\] respectively. Fits we have made to GRB140903A are shown in Fig. \[fig:lightcurve\]. We determine the posterior distribution on our parameters $\fgwo$, $\tau$, and $n$ which are shown in Fig. \[fig:corner\]. In the following section, we discuss how these posteriors can be used as priors for a targeted search for the post-merger remnant associated with an SGRB.
![Posterior distribution for $\fgwo$, $n$, and $\tau$ for GRB140903A. These posteriors are used as priors to build a GRB specific template bank. Shown are one-,two-, and three-sigma confidence levels. This figure is generated using the ChainConsumer software package [@Hinton2016].[]{data-label="fig:corner"}](Plots/Corner.pdf){width="50.00000%"}
\[sec:pipeline\]Gravitational-wave search pipeline
==================================================
Here we describe a pipeline to search for gravitational waves from a spinning down millisecond magnetar. The algorithm can be summarised as follows:
1. Generate posterior distributions on the three waveform parameters $\fgwo$, $n$ and $\tau$ using the X-ray afterglow observations of a specific SGRB as described in Sec. \[sec:GRBafterglow\].
2. These posterior distributions, along with uniform priors on $\Phi_0$ and $\cos\iota \in [0,1]$, serve as priors for our waveform model. Template waveforms are generated from points in these priors.
3. Templates are used to calculate the matched filter signal-to-noise ratio using LIGO data at the time of the SGRB.
The same pipeline can also be adopted with unconstrained uniform priors in step 1, in the case where no X-ray data is available. However, the number of templates required for a matched-filter search becomes computationally unfeasible. We quantify this throughout this section.
We calculate the fitting factor $FF$ [@Apostolatos1995], also commonly referred to as the overlap [e.g., @Cornish2012]. The fitting factor is the penalty in signal-to-noise ratio one suffers due to comparing templates that do not precisely match the signal: $FF = \rho/\rho_{\textrm{opt}}$. We want to minimize this penalty while maximizing the signal-to-noise ratio.
To calculate the $FF$ we randomly draw one value of each parameter from our priors and construct a model waveform using the waveform model described in Sec \[sec:waveform\]. We assume this is our true template, $h_{\textrm{T}}$. We determine the optimal matched filter signal-to-noise ratio for this template using Eq. (\[eq:snr\_exp\]), We randomly draw from our priors excluding our ‘true template’ and create a random template, $h_{\textit{i}}$, where $\textit{i}$ labels the $\textrm{i}^{\textrm{th}}$ drawn sample. We compute the matched filter signal-to-noise ratio (Eq. \[eq:snr\]), $\rho_{\textit{i}}$. We calculate $\rho_{i}$ for N random templates. In the limit of infinite templates, $\textrm{max}(\rho_{\textrm{i}}) \to \rho_{\textrm{opt}}$.
The maximum fitting factor is defined as $$\label{Ch3: Eq. FF}
FF = \frac{\text{max}(\rho_{\textrm{i}})}{\rho_{\textrm{opt}}},$$ where $\text{max}(\rho_{\textrm{i}})$ is the maximum matched-filter signal-to-noise ratio from a population of N templates. In the limit of an infinite number of templates, $FF \to 1$, assuming our signal parameters are within our template parameter space. Creating a large number of templates is computationally expensive. We therefore want to minimise the number of templates we need. Additionally, we want to maximise our signal-to-noise ratio by creating templates for a longer duration.
[cc!]{} {width="90.00000%"}
In Fig. \[fig: FF\] we show the scaling of $FF$ with the number of templates in the template bank for different $t_{\textrm{obs}}$ and two different priors: an unconstrained uniform prior (left panel) where the priors on $f_{\text{gw,0}},$ $n$ and $\tau$ are $[500, 3000]$ Hz, $[2.5, 5]$ and $[350, 35000]$ s, respectively, and the constrained posterior priors from using X-ray afterglow observations (right panel). The error bars indicate one sigma confidence levels, generated by repeating the analysis with 1000 different noise realizations.
Figure \[fig: FF\] shows that for $10^5$ templates, $FF = 0.62$ for $t_{\textrm{obs}} = 10$ s with uniform priors. A fitting factor $FF = 0.62$ implies that we lose $38 \%$ of the optimal matched-filter signal-to-noise ratio when running a matched-filter search. This recovery percentage is even worse for longer observation times, with $t_{\textrm{obs}} = 100$ s having $FF = 0.12$ for uniform priors with $10^5$ templates, indicating we lose $88 \%$ of the optimal matched-filter signal-to-noise ratio. Although $FF$ scales up for an increasing number of templates, the amount of templates required to construct a search that could detect potential signals is unfeasible computationally for uniform priors. Furthermore, as shown in Sec. \[sec:waveform\], real astrophysical signals likely require $t_{\textrm{obs}} > 1000$ s, and $FF$ at these $t_{\textrm{obs}}$ is significantly worse. Fortunately, $FF$ is comparatively better for constrained priors (right panel). For example, for $t_{\textrm{obs}} = 100$ seconds with $10^5$ templates, $FF = 0.72$ with constrained priors as opposed to $0.12$ with uniform priors. In a real search we will likely require $t_{\textrm{obs}} > 10^3$ seconds and $10^6$ templates. We have not calculated the $FF$ for these parameters as it is computationally expensive and requires an optimization step in the template generation to avoid using the high sampling frequencies throughout that are required at the beginning of the waveform. Furthermore, for aLIGO, detectable astrophysical signals require large $\tau$ values which are ruled out by the energy budget constraint; see Sec. \[subsec:Egwt\]. In addition, constructing searches with observation times significantly larger than $\tau$ gives worse results as one no longer accumulates significant signal-to-noise for $t \gg \tau$. Noting the scaling observed in $FF$, we expect $FF \approx 0.4$ for $t_{\textrm{obs}} = 10^4$ seconds with $10^6$ templates, an acceptable loss considering the gains from a longer signal duration.
We calibrate our pipeline by injecting signals into Gaussian noise coloured to match that of the expected strain sensitivity. This calibration is parameter dependent, so in a real search, we will need to do this for each SGRB. We use the posteriors from GRB140903A to create a fake signal. In Sec. \[sec:waveform\] we used the optimal matched filter signal-to-noise ratio (Eq. \[eq:snr\_exp\]) to determine an optimistic estimate for the distance out to which we can detect a signal (shown in Fig. \[fig. kappa GRB140903A\]). These distances are optimistic, and as we quantified with $FF$, we suffer a loss in signal-to-noise due to having imperfect templates.
We define a horizon distance as the distance to which a detector with a given sensitivity can observe events with a given significance in a real matched-filter search. We start with the matched filter signal-to-noise ratio $\rho$ (Eq. \[eq:snr\]) We determine a signal-to-noise ratio threshold $\rho_{\textrm{threshold}}$, which is the minimum signal-to-noise ratio to claim a detection with aLIGO at design sensitivity with a single detector. To determine this threshold, we calculate $\rho$ using Eq. (\[eq:snr\]) with noise-only realisations ($s = 0$) and for *N* templates. We take the maximum $\rho$ from *N* templates and do this for multiple realisations of noise retaining the maximum $\rho$ each time. We determine the 99.7 percentile of our probability distribution on $\rho$ with no signal, which indicates that 99.7 % of the time noise can mimic a signal (a false alarm). Any detection needs $\rho > \rho_{\text{threshold}}$ to be significant. For our pipeline, the $3\sigma$ $\rho_{\text{threshold}}$ is $4.4$ with $10^{4}$ templates and $1000$ realisations of noise, however the choice of this false-alarm rate is arbitrary.
We also establish a false dismissal probability, which quantifies when a real signal present in the data cannot be disassociated from the noise. As a result, it fails to be identified. To determine a horizon distance, we find the distance where our false dismissal probability is less than $10$ %, which is done by repeating the procedure for determining $\rho_{\textrm{threshold}}$, but injecting signals at fixed distances. We then determine at what distance less than $10$ % signals have $\rho < \rho_{\textrm{threshold}}$.
Prior to this point, we have only considered a single detector; the signal-to-noise ratio grows approximately in quadrature for a network of $N$ similar detectors and therefore having an aLIGO-Virgo triple detector network will increase the horizon distance accordingly. In the future, with a network of 3G detectors such as ET and Cosmic Explorer, a similar increase in signal-to-noise ratio can be expected. Other factors such as sky localization and time-varying $F_+$ and $F_\times$ will also affect the horizon distance. Considering these factors, in a real search we can expect our horizon distance for a GRB140903A inferred post-merger signal to be half the optimal matched-filter distance indicated by Fig. \[fig. kappa GRB140903A\] as $\sim 2$ and $\sim 45$ Mpc for $\epsilon = 10^{-3}$ for aLIGO and ET respectively.
Conclusion {#sec:conclusion}
==========
We have developed an algorithm to search for gravitational waves from a long-lived post-merger remnant of a binary neutron star merger. In Sec. \[sec:waveform\], we derive a waveform model for gravitational waves emitted from a spinning down millisecond magnetar. We detail and analyze a matched filter detection pipeline using this waveform model. We find that using X-ray observations from SGRB afterglows results in a significant decrease in parameter space resulting in a much improved and targeted search for a post-merger remnant. These X-ray guided priors can also be applied in other post-merger search pipelines. Our analysis indicates for an ellipticity $\epsilon = 10^{-2}$ our pipeline can, in principle detect gravitational waves with aLIGO at design sensitivity out to $\sim 20$ Mpc for a fiducial moment of inertia $10^{45}$ g cm$^{2}$. If one ignores the energy-budget constraint, this fiducial value implies a conservative limit on the gravitational-wave strain and therefore horizon distance. In reality, the moment of inertia of the remnant may be a factor few larger than this fiducial value; as the strain scales linearly with the moment of inertia, this implies the horizon distance may also be a factor of a few larger. However, when including the energy-budget constraint, the horizon distance implied by a higher moment of inertia is lower due to the inverse relationship between the moment of inertia and the ellipticity.
It is the energy-budget constraint that ultimately sets the distance to which these post-merger remnants can be detected. A large region of the parameter space is implausible, which lowers the horizon distance to $\sim 2$ Mpc for GRB140903A-like post-merger signals. The Einstein Telescope can detect a similar signal out to $\sim 45$ Mpc. Post-merger signals with longer spin-down timescale $\tau$ will be detectable out to larger distances.
We are also investigating a more realistic model. The waveform model introduced here is simplified as the model assumes the neutron star is an orthogonal rotator. In this state, the principal eigenvector of the moment of inertia tensor is orthogonal to the star’s rotation axis making the star an optimal emitter of gravitational waves. The neutron star is possibly driven to this orientation through the spin-flip instability [@Cutler2002; @Mestel1981; @1976Jones], but the timescales involved are uncertain [@DallOsso2014; @DallOsso2011; @Lasky2016a]. As the system is driven to orthogonalization, it emits gravitational waves which we can include in our waveform model. We also have not accounted for time-varying $F_+$ and $F_\times$ terms.
Another extension is to constrain our parameter space further by including information obtained through parameter estimation on the binary neutron star inspiral gravitational-wave signal. Specifically, we can constrain the inclination of the source which should increase the pipeline sensitivity. The X-ray afterglow observations also suggest an evolution of the braking index with time with the system evolving from gravitational-wave dominated spin-down to magnetic dipole. This evolution of the braking index is something we can include in our model.
Acknowledgments
===============
We are grateful to David Keitel, Hou-Jun Lü, and the anonymous referee for comments on the manuscript as well as the LIGO post-merger group for insightful discussions. NS is supported through an Australian Postgraduate Award. PDL is supported through Australian Research Council Future Fellowship FT160100112 and ARC Discovery Project DP180103155.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Formalism based on equilibrium statistical thermodynamics is applied to communication networks of decision making individuals. It is shown that in statistical ensembles for choice models, properly defined disutility can play the same role as energy in statistical mechanics. We demonstrate additivity and extensivity of disutility and build three types of equilibrium statistical ensembles: the canonical, the grand canonical and the super-canonical. Using Boltzmann-like probability measure one reproduce the logit choice model. We also propose using $q$-distributions for temperature evolution of moments of stochastic variables. The formalism is applied to three network topologies of different degrees of symmetry, for which in many cases analytic results are obtained and numerical simulations are performed for all of them. Possible applications of the model to airline networks and its usefulness for practical support of economic decisions is pointed out.'
author:
- Arkadiusz Majka
- |
and\
Wojciech Wiślicki
title: Statistical thermodynamics for choice models on graphs
---
\[sec:level1\]Introduction
==========================
Models used in demand analysis of transportation and communication problems usually assume that demand represents the result of decisions of each individual in the population. These decisions consist of choices made among finite sets of possibilities. As an example of a sequence of choices, in the context of airline transportation demand, consider decision to be taken by potential air passenger. If the passenger wills to fly from the origin port $O$ to destination port $D$, he has to choose among offers of carriers, accounting for many factors relevant for decision, as e.g. scheduled departure and arrival times and how they relate to his needs, flight durations, flight fares, numbers of stops and changes on the route $(OD)$, availability of seats in first class, probability of delays, declared and expected quality of service and many other aspects, too numerous to itemize. In order to quantify such process of decision making, one incorporates discrete choice models in hope of better understanding and predicting behaviour of such complex system as transportation network and thus obtaining hints for marketing and revenue management decisions.
One of the key concepts used in decision theory, dating back to the XIX-th century economy by Jeremy Bentham and other representatives of so called utilitarians economy, is the concept of [*utility*]{}, which had been further evolving since that time and found its more rigorous utterance in the classical work of von Neumann and Morgenstern [@neumann1] and in neoclassical economic theory (cf. e.g. refs [@hennings1], [@bierlaire1]). Utility function ascribes real numbers to alternatives and so defines the preference order on the choice set and puts the choice making process on more rational footing.
An approach to communication based on discrete choice models has long history and abundant literature (cf. e.g. refs [@benakiva1], [@louviere1]). Utility function in neoclassical economic theory is defined axiomatically. In order to use it in specific applications of choice models, one has to postulate a lot from outside of the theory and to find an effective way to either derive it from any fundamental or general theory or to parametrize it and estimate from data. Derivation of the utility function from first principles represents major theoretical problem because of lack of fundamental theory of human behaviour. Normally, one has to rely on partially justified models using power series approximations for utility functions, often restricted to linear or quadratic terms (cf. ref. [@dobson1]), and finding empirically relevant variables and estimators for parameters, if any data are available at all. Apart from those problems, numerous conceptual difficulties arise when applying utility functions to quantify the behaviour of the choice makers and to predict them. In particular, classical Bentham’s approach of maximizing the overall utility, integrated over individuals, when taking market decisions, is often a subject of serious objections [@straffin1].
Being aware of a long-lasting debate around the meaning and the role of utility functions in economy, social sciences and, more generally, in game theory, in this paper we try to incorporate it into economic games on somewhat different way. First, extending from the concept of utility, we find an analogy between the utility function, as it is used in choice models, and the energy function for physical systems. After specifying the system, consisting of an airline network and a set of passengers, and its state space, we discuss assumptions normally taken in theories of systems in thermal equilibrium and build quasistatic probability measures on the state space for equilibrium ensembles. This allows us to derive complete classical thermodynamics for the communication network. We consider the canonical and the grand canonical statistical ensembles for systems with fixed and variable numbers of clients.
As a possible extension of the model, we propose to consider also the network topology and the number of network vertices as a multidimensional random variable and formulate generalized canonical ensemble where the existence of any connection between sites may fluctuate and is represented by a binomial random variable. In the frameworks of the canonical and the grand canonical formalisms, we find some analytical results for thermodynamics of particular network topologies and perform numerical simulations. We study thermodynamical potentials, response functions and correlations between extensive variables in the system. The case of random topology is discussed in general terms only. It needs more study of a couple of specific and self-contained problems and this, altogether with numerical results, is relegated to future work.
We also develop another approach, based on so called $q$-distributions or [*escort*]{} distributions, which allows us to monitor fluctuations and correlations of interesting characteristics of the network, their temperature evolution and phase relations in the network.
Building our formalism we keep in mind its specific application to networks of airline traffic and often use its terminology, although we believe that this framework and most of its features are general enough to be used for other network-related problems, as various kinds of telecommunication, communications in social groups, traffic routing, energy networks, etc. On the other hand, the model we build does not account for many features of real airline markets, either because of lack of knowledge about utility functions or simplifications of the model at this stage.
\[sec:level2\]Assumptions and fundamentals of the model
=======================================================
\[ssec:level1\]Axiomatics of utility
------------------------------------
In choice models considered here we assume that decision makers are individuals and that these individuals are independent of each other. Such models are usually called [*disaggregate*]{}. Assumption on independence of decisions between individuals is rather poorly justified and often just not true in reality. But for many practical applications this assumption is not restrictive and depends on further details of the model, in particular on the utility function. In many cases, also in this paper, one may redefine the notion of a decision maker and consider a group of individuals as the decision maker, without loss of generality, and only mutual independence of such defined decision makers is relevant. This assumption can be further criticized because the ways decisions are taken by groups, even for very restricted class of consumer decisions considered here, certainly depend on the size and internal communication structure of groups and are qualitatively different than in case of individuals. Thus the utility function for aggregated decision makers may differ from that of individuals. In order to give full account of these diffences in general case, which we do not pretend to do in this paper, one has to consider many additional effects, as internal negotiations inside groups, their dynamics, possible splits, compactness of groups etc. However, at this stage of model development and for the case of limited number of variables relevant for utility, and for reasonably homogeneous groups, we feel it is fair to assume that utilities for all decision makers are independent and identically distributed stochastic variables. In other words, we assume high degree of [*decision coherence*]{} of groups, in the sense that decisions of aggregated individuals do not differ significantly from that of real individuals.
Analysis of the choice requires knowledge of the complete set of options disponsible for the choice maker. The set of options, or alternatives, $a,b,c,\ldots$ is called [*the choice set*]{} ${\cal C}=\{a,b,c,\ldots\}$. The complete set of all possible options available to any individual is called [*the universal choice set*]{} and a subset of the universal choice set considered by a particular individual is called [*the reduced choice set*]{}. If ${\cal C}$ is the universal choice set then all possible reduced choice sets correspond to events and form what is called [*$\sigma$-algebra over ${\cal C}$*]{} in the probability calculus. Hereon we assume that results of a choice performed by any decision maker are unique for given specification of the choice set. This means that results of a choice are those elements of $\sigma$-algebra which correspond to elementary events of the probability theory. Finally, we consider only [*discrete choice models*]{}, i.e. those for which their choice sets are finite and all options can be explicitly enumerated.
Each alternative of the choice set can be characterized by a set of [*attributes*]{}. Generally, attributes are random variables in the sense used in probability theory, i.e. they are funcions defined on reduced choice sets with values in other sets. We consider both directly observable attributes and functions of attributes.
In order to construct decision rules, the decision maker has to be able to compare alternatives. In neocalassical economy [@bierlaire1] two alternatives $a$ and $b$ are comparable using [*the preference-indifference*]{} operator $\succeq$ which orders ${\cal C}$ linearly, i.e.
1. The $\succeq$ is reflexive: $$\begin{aligned}
\forall_{a\in{\cal C}} \quad a\succeq a, \label{eq1} \end{aligned}$$
2. The $\succeq$ is transitive: $$\begin{aligned}
\forall_{a,b,c\in{\cal C}} \quad (a\succeq b \; \land \; b\succeq c) \Rightarrow (a\succeq c), \\ \nonumber \label{eq2} \end{aligned}$$
3. Any two alternatives are comparable: $$\begin{aligned}
\forall_{a,b\in{\cal C}} \quad a\succeq b \; \lor \; b\succeq a. \label{eq3} \end{aligned}$$
Since the choice set ${\cal C}$ is finite, the existence of the most preferred alternative $a^{\ast}$ is guaranteed $$\begin{aligned}
\exists_{a^{\ast}}\; \forall_{a\in{\cal C}} \quad a^{\ast}\succeq a.
\label{eq4}\end{aligned}$$ Because of the properties (\[eq1\]-\[eq3\]) there exists finite random variable $U:{\cal C}\longrightarrow \mathbb{R}^1$, refered to as [*the utility function*]{}, such that $$\begin{aligned}
\forall_{a,b\in {\cal C}} \quad a\succeq b \; & \Leftrightarrow & \; U(a)\geq U(b).
\label{eq5}\end{aligned}$$ From eq. (\[eq4\]) it follows that the most prefered alternative $a^{\ast}$ has the largest utility $U$ $$\begin{aligned}
\forall_{a\in{\cal C}} \quad U(a^{\ast})\ge U(a)
\label{eq6}\end{aligned}$$ or the smallest [*disutility*]{} $\bar{U}=-U$. Hence, important property of $\bar{U}$
Disutility function is bounded from bottom.
\[ssec:level1\]Specification of the system
------------------------------------------
Our system ${\cal S}$ consists of a complex communication network ${\cal G}$, represented by a set of directed graphs, and set of decision makers. Nodes of graphs in ${\cal G}$ represent airports and its directed edges correspond to air connections between them. Each ordered pair $(OD)$ of the origin $O$ and destination $D$ ports and set of all directed paths, called [*routes*]{}, connecting $O$ and $D$, excluding loops, constitute the $k$-th [*market*]{} and is represented by the graph ${\cal G}_k$ ($k=1,\ldots,M$). The whole network ${\cal G}$ is the sum of market graphs $$\begin{aligned}
{\cal G}={\cal G}_1\cup\ldots\cup{\cal G}_M.
\label{eq6_1}\end{aligned}$$ Using graph-theoretic terminology, markets are represented by sets of Hamilton paths, i.e. no one site on the route is visited more than once. So defined markets can overlap. The system is further specified by defining all features relevant for its state as sets of potential passengers (decision makers) for each market, resources of aircrafts, flight schedules, flight fares, etc. [*The state*]{} of the system is defined using results of choices performed by all decision makers, which means that the state-space consists of all possible distributions of all decision makers for all markets of the network.
We assume quasistatic approximation, meaning that any time variation of the system as a whole is slow compared to the time of individual decision. Moreover, specifying the state of the system, we accumulate decisions taken by individuals in time interval at least an order of magnitude shorter than the system time scale, but not necessarily short compared to individual decision time. In airline practice, one day is normally a good time bin for accumulation of decisions.
In present model we do not embed the network in any metric space, such that only the topology, or connectivity scheme, is relevant and not the geometry. Accounting for geometry may appear necessary along with sofistication of our model for utility.
For each market $k$ we define the total [*market disutility*]{} as a sum of disutilities of all passengers from this market $\bar{U}_{i_k}$ where $i_k$ stands for index of the $i$-th passenger in the $k$-th market $(i_k=1,\ldots,I_k)$ $$\begin{aligned}
\bar{U}_k=\sum_{i_k=1}^{I_k}\bar{U}_{i_k}.
\label{eq7}\end{aligned}$$ Overall [*network disutility*]{} $\bar U$ is a sum of market disutilities over all $M$ markets $$\begin{aligned}
\bar U=\sum_{k=1}^M \bar U_k.
\label{eq8}\end{aligned}$$
States of the system are defined in discretized time, which means that for definitions of functions of states we use all individual decisions taken in given time interval, e.g. one day.
\[ssec:level1\]Additivity and extensivity of the utility
--------------------------------------------------------
In order to find the probability of the state of the network using utility function and incorporating classical Boltzmann-Gibbs arguments, the utility has to be [*an additive stochastic variable*]{}. Disutility $\bar{U}$ of the system is called additive with respect to any two subsystems $A$ and $B$ if their disutilities add up to $\bar{U}$, i.e. $\bar{U}=\bar{U}_{A}+\bar{U}_{B}$ (cf. e.g. ref. [@touchette1] for more detailed discussion). Therefore, additivity requires the notion of the subsystem to be clarified first. In our case, the subsystems cannot be defined by dividing the connectivity net only, because the smallest meaningful entities in the theory are the markets, represented by graphs ${\cal G}_k$ $(k=1,\ldots,M)$. We decompose the system ${\cal S}$ into the sum of markets $$\begin{aligned}
{\cal S}={\cal S}_1\oplus\ldots\oplus{\cal S}_M,
\label{eq8_1}\end{aligned}$$ where $\oplus$ means the set sum of corresponding graphs [*and*]{} individuals being ascribed to corresponding markets. Fig. \[fig:fg0\] illustrates decomposition of an example system into markets.
![\[fig:fg0\] Decomposition of the network into markets](./1.eps){width="80mm" height="55mm"}
Each market $(O,D)$ can be further split into routes $[O,C_1,\ldots,C_L,D]$ in the same way. For our example from Fig. \[fig:fg0\] decomposition of markets into routes is the following: $$\begin{aligned}
(1,2) & = & [1,2] \nonumber \\
(1,3) & = & [1,2,3]\oplus [1,2,4,3] \nonumber \\
(1,4) & = & [1,2,4]\oplus [1,2,3,4] \nonumber \\
(2,3) & = & [2,3] \oplus [2,4,3] \nonumber \\
(2,4) & = & [2,4] \oplus [2,3.4] \nonumber \\
(3,4) & = & [3,4] \oplus [3,2,4]
\label{eq8_2}\end{aligned}$$ and reversed $(O,D)\rightarrow(D,O)$ pairs.
For subsystems defined in this way, from the definition of network disutility (\[eq7\],\[eq8\]), it follows by construction
Network disutility is an additive random function of state.
In addition, also [*the extensivity*]{} of disutility, in the sense used in ref. [@touchette1], can be easily justified. If $n$ is the number of decision makers in the system, then thermodynamic limit of disutility exists $$\begin{aligned}
\lim_{n\rightarrow\infty}\frac{\bar{U}}{n}<\infty
\label{eq8_3}\end{aligned}$$ for $\bar{U}$ proportional to $n^{\alpha}$ ($\alpha\le 1$). By definition, this is our case with $\alpha=1$, hence
Network disutility is an extensive random function of state.
From properties 1, 2 and 3 it follows, that network disutility function has all the same properties relevant for the construction of probability measure as the total energy function of finite physical system with no long-range interactions. Therefore it can play the same role in construction of statistical ensemble of systems as hamiltonians do, viz. it can be used for definition of the exponential probability measure on the state-space. For doing this we note that because of assumption of disaggregate choice model and additivity of disutilities, for any two subgroups of passengers, either belonging to different markets or from the same market, disutilities of subgroups are independent of each other. Hence, we find
For any independent subgroups of decision makers, disutilities of subgroups are mutually independent random variables.
Validity of Boltzmann and Gibbs arguments, as applied to $\bar{U}$, follows from properties 1-4. Hence, the probability of the state of the network for given value $u$ of the network disutility $\bar U$ is equal to $$\begin{aligned}
{\cal P}(\bar U=u)\sim \exp(-\beta u),\;\;\;\;\;\;\;\;\;\;\beta>0.
\label{eq9}\end{aligned}$$ An important consequence of this is that one can consider statistical ensemble of networks and statistical thermodynamics for them with the probability density (\[eq9\]), and there is no need for non-extensive statistics with power-law stochastic measure. At first sight, this seems counter intuitive because of apparent long-range correlations in the communication network which usually entails usage of non-extensive statistics (cf. e.g. ref. [@tsallis1]). This is understandable when realizing that only sites, and not markets, are really correlated in this model. Markets, representing elementary objects in the system, are uncorrelated since decision makers decide on routes only, as mentioned above. The choice of the market is not a subject of the game considered here. Correlations between markets appear when additional constraints are taken into account, as e.g. finite transmittance of nodes.
In the following we restrict our model to equilibrium thermodynamics, or require existence of stationary states, which on turn requires assumption on existence of the first moment of disutility function $$\begin{aligned}
\langle\bar U\rangle<\infty.
\label{eq10}\end{aligned}$$ Otherwise, the equilibrium temperature $T_{eq}=1/\beta_{eq}$ would not exist. We are not particularly concerned here neither in studying processes leading to equilibrium nor in all sufficient conditions for existence of equilibria in general. Therefore we do not start our discussion of ensembles from defining transition probabilities and deriving probability densities from detailed balance.
In our approach we do not incorporate random intensive variables. It has been recently shown [@beck1] that in the most general case of all variables of the system being stochastic, including intensive ones, one arrives to the class of ensembles with even more general proability densities than Levy and Tsallis power-law functions. Our assumption purposefully limits generality of statistical ensembles considered here, since our original intent was to consider intensive variables of the system as simple steering parameters.
\[sec:level3\]Equilibrium statistical ensembles
===============================================
Assuming existence of stationary regime for time evolution of complex network and using utility functions discussed above, it is straightforward to find probability measure on the state-space of the network. As known from statistical thermodynamics, the specific form of the probability density is determined by the nature of constraints on the system, or the set of variables of merit which are allowed to be stochastic.
\[ssec:level1\]The microcanonical ensemble
------------------------------------------
In the simplest case of disutilities being the same for all states of the system or, for non-discrete case, localized in very narrow interval $[\bar{U_0},\bar{U_0}+\delta\bar{U}]$, $(\delta\bar{U}/\bar{U_0}\ll 1)$, one normally assumes the probabilities for all states being the same and calls it [*the principle of a priori equal probabilities*]{}. Ensemble of systems so defined is called [*the microcanonical ensemble*]{} and corresponds to the extremly tight, and rather unrealistic in our case, constraints imposed on the system, and on the set of decision makers in particular. This means that the overall disutility is insensitive to the distribution of decision makers which could be interpreted as either lack of decision makers’ sensitivity to the conditions of the network or as an extremly poor offer of the carrier, giving no choice to its customers. This is rather trivial case and we do not discuss it furtheron.
\[ssec:level2\]The canonical ensemble
-------------------------------------
As the first non-trivial case consider [*the canonical ensemble*]{} of systems where the number $N$ of decision makers is fixed and the structure of connectivity, or the topology of the connections network, does not vary in time. One defines [*the partition function*]{} or [*the statistical sum*]{} of the system $$\begin{aligned}
Z(\beta) & = & \prod_{k=1}^M Z_k(\beta) \nonumber \\
& = & \prod_{k=1}^M \sum_{j_k=1}^{J_k} e^{-\beta\bar{U}_{j_k}}.
\label{eq11}\end{aligned}$$ The $Z_k(\beta)$ stands for the partition function of the $k$-th market and index $j_k$ runs over all $J_k$ distributions of passengers over routes belonging to the $k$-th market. Note that each market itself is treated here as a canonical ensemble, which means that also numbers of clients $N_k$ ($k=1,\ldots,M$) belonging to each market is fixed. This means that potential passengers from given origin node are at least decided as for their destination node, although they may not [*a priori*]{} know which route to fly there.
Due to additivity of disutility for each market (\[eq7\]) the $Z_k$ can be rewritten in terms of explicit sums over $R_k$ routes for the $k$-th market $$\begin{aligned}
Z_k(\beta) & = & Z(\beta,N_k) \nonumber \\
& = & \sum_{j_1,\ldots,j_{N_k}=1}^{R_k} e^{-\beta\sum_{l=1}^{N_k}\bar{U}_{l,j_l}} \nonumber \\
& = & \prod_{l=1}^{N_k}\sum_{j_l=1}^{R_k} e^{-\beta\bar{U}_{l,j_l}},
\label{eq12}\end{aligned}$$ where $j_1,\ldots,j_{N_k}$ are passenger’s indices running over $R_k$ routes and $l$ runs over passengers. Assuming further the [*non-subjectivity*]{} of the utility, i.e. that utility for given route does not depend on the decision maker, $\bar{U}_{l,j_l}=\bar{U}_j$ ($l=1,\ldots,N_k$), the partition function for the $k$-th market can be written as $$\begin{aligned}
Z_k(\beta) & = & \Big(\sum_{j=1}^{R_k}e^{-\beta\bar{U}_j}\Big)^{N_k} \nonumber \\
& = & Z_k^1(\beta)^{N_k},
\label{eq13}\end{aligned}$$ where $Z_k^1(\beta)$ stands for the partition function for the $k$-th market with one passenger on it.
The state of the system in the canonical ensemble is defined by specifying disutility for the complete set of distributions of passengers over all markets, i.e. $\Big\{\big\{\bar{U}_{j_k}\big\}_{j_k=1}^{J_k}\Big\}_{k=1}^M$. Probabilities of states are $$\begin{aligned}
{\cal P}_{j_k}(\beta)=\frac{e^{-\beta\bar{U}_{j_k}}}{Z(\beta)}.
\label{eq14}\end{aligned}$$ Noteworthly, the choice probability given by formula (\[eq14\]) is the same as obtained in the framework of [*the multinomial logit choice models*]{}, usually using strong assumptions on the type of distributions, extreme value or Gumbel, for the stochastic components of the utility (cf. e.g. ref. [@train1]).
Mean disutility can be calculated using (\[eq14\]), or by diffentiation over the Lagrange multiplier $\beta$: $$\begin{aligned}
\langle\bar{U}\rangle & = & \sum_{k=1}^M \sum_{j_k=1}^{J_k} {\cal P}_{j_k}(\beta)\bar{U}_{j_k} \nonumber \\
& = & -\frac{\partial}{\partial\beta}\ln Z(\beta)
\label{eq15}\end{aligned}$$ and if its value $\langle\bar{U}\rangle_{meas}$ is also known from measurements, their equality determines the equilibrium temperature $\beta_{eq}=1/T_{eq}$. The entropy and disutility fluctuations are given by formulae $$\begin{aligned}
S(\beta) & = & -\sum_{k=1}^M \sum_{j_k=1}^{J_k} {\cal P}_{j_k}(\beta) \ln {\cal P}_{j_k}(\beta) \nonumber \\
& = & -\beta^2 \frac{\partial}{\partial\beta}\frac{1}{\beta}\ln Z(\beta) \nonumber \\
& = & \beta\langle\bar{U}\rangle+\ln Z(\beta)
\label{eq16}\end{aligned}$$ and $$\begin{aligned}
\mbox{{\it Var}}\,(\bar{U}) & = & \frac{\partial^2}{\partial\beta^2}\ln Z(\beta) \nonumber \\
& = & -\frac{\partial}{\partial \beta}\langle\bar{U}\rangle.
\label{eq17}\end{aligned}$$
\[ssec:level3\]The grand canonical ensemble
-------------------------------------------
In [*the grand canonical*]{} case, both disutility $\bar{U}$ and the number of decision makers $N$ represent random variables, but the connectivity of the network remains fixed. The grand partition function or grand canonical sum is given by $$\begin{aligned}
\Xi(\beta,\vec{\mu})=\sum_{N_1,\ldots,N_M}e^{\beta\sum_{k=1}^M\mu_k N_k}Z_{N_k}(\beta),
\label{eq18}\end{aligned}$$ where $Z_{N_k}(\beta)=\prod_{k=1}^M Z_k(\beta)$ and $\vec{\mu}=(\mu_1,\ldots,\mu_M)$ is the vector of chemical potentials for markets, corresponding to the vector of passengers numbers $\vec{N}=(N_1,\ldots,N_M)$. Here $Z_k$ is the canonical partition function for the $k$-th market (\[eq13\]) but with $N_k$ being stochastic variable. Following intuitive meaning of the chemical potential for physico-chemical systems, [*a priori*]{} one could think about different chemical potentials for different markets, as individual passengers from different markets may contribute unequally to the overall disutility of the system. Here we assume, however, [*the hypothesis of chemical uniformity*]{} of the system which states that chemical potentials of all markets are equal: $\mu:=\mu_1=\ldots =\mu_M$. This hypothesis does not exclude multiphase systems from our considerations, so that our communication network corresponds to the case of a one-component, though not necessarily one-phase, chemical system. In further sofistication of this formalism one should categorize passengers regarding professions, wealth, aims of travels etc, and consider different chemical potentials for different categories. Those characteristics are in many cases correlated with markets but, obviously, the scopes of those features are not in one-to-one relations to the markets. Formally, $\mu$ is the Lagrange multiplier corresponding to the constraint $$\begin{aligned}
\langle N\rangle =\sum_{k=1}^M\langle N_k\rangle .
\label{eq19}\end{aligned}$$
Following the same reasoning as in the canonical case, the grand partition function can be rewritten $$\begin{aligned}
\Xi(\beta,\mu) & = & \sum_{N_1,\ldots,N_M}\prod_{k=1}^M e^{\beta\mu N_k}Z(\beta,N_k) \nonumber \\
& = & \prod_{k=1}^M \frac{Z(\beta,N_k+1)e^{N_k}-e^{-\beta\mu}}{Z_k(\beta,1)-e^{-\beta\mu}} \nonumber \\
& = & \prod_{k=1}^M \frac{[e^{\beta\mu}Z_k^1(\beta)]^{N_k+1}-1}{e^{\beta\mu}Z_k^1(\beta)-1}.
\label{eq20}\end{aligned}$$
The state of the system in the grand canonical ensemble is defined by specifying disutility for the complete set of distributions of passengers over all markets and for all numbers of passengers, i.e. $\Big\{\big\{\{\bar{U}_{j_k}(N_k)\}_{j_k(N_k)=1}^{J_k(N_k)}\big\}_{N_k}\Big\}_{k=1}^M$. Probabilities of states are $$\begin{aligned}
{\cal P}_{j_k(N_k)}(\beta,\mu)=\frac{e^{\beta(\mu N_k-\bar{U}_{j_k(N_k)})}}{\Xi(\beta,\mu)}.
\label{eq21}\end{aligned}$$ This reminds again the choice probability from the multinomial logit choice model, where $\beta\mu N_k$ plays the role of [*the alternative specific constant*]{} (cf. e.g. refs [@louviere1; @train1]). Here, however, this constant is specific for the whole class of routes for the $k$-th market with $N_k$ individuals on it and for the particular state $j_k(N_k)$.
First moments of $\bar{U}$ and $N$ and their correlations can be found by differentiation of $\Xi$ over Lagrange multipliers $\beta$ and $\mu$ as follows: $$\begin{aligned}
\langle\bar{U}\rangle & = & \sum_{k=1}^M\sum_{N_k}\sum_{j_k(N_k)=1}^{J_k(N_k)} {\cal P}_{j_k(N_k)}(\beta,\mu) \bar{U}_{j_k(N_k)} \nonumber \\
& = & -\Big(\frac{\partial}{\partial\beta}\ln \Xi(\beta,\mu)\Big)_{\mu\beta} \nonumber \\
\langle N\rangle & = & \sum_{k=1}^M\sum_{N_k}\sum_{j_k(N_k)=1}^{J_k(N_k)} {\cal P}_{j_k(N_k)}(\beta,\mu) N_k \nonumber \\
& = & \frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\ln \Xi(\beta,\mu)\Big)_{\beta} \nonumber \\
\mbox{{\it Var}}\,(\bar{U}) & = & \Big(\frac{\partial^2}{\partial\beta^2}\ln \Xi(\beta,\mu)\Big)_{\mu\beta} \nonumber \\
& = & -\Big(\frac{\partial}{\partial\beta}\langle\bar{U}\rangle\Big)_{\mu\beta} \nonumber \\
\mbox{{\it Var}}\,(N) & = & \frac{1}{\beta^2}\Big(\frac{\partial^2}{\partial\mu^2}\ln \Xi(\beta,\mu)\Big)_{\beta} \nonumber \\
& = & \frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\langle N\rangle\Big)_{\beta} \nonumber \\
\mbox{{\it cov}}(\bar{U},N) & = & -\frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\big(\frac{\partial}{\partial\beta}\ln\Xi(\beta,\mu)\big)_{\mu\beta}\Big)_{\beta} \nonumber \\
& = & \frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\langle\bar{U}\rangle\Big)_{\beta}
\label{eq22}\end{aligned}$$ and the entropy is given by $$\begin{aligned}
S(\beta,\mu) & = & -\mu\Big(\frac{\partial}{\partial\mu}\ln\Xi(\beta,\mu)\Big)_{\beta}-\beta^2\Big(\frac{\partial}{\partial\beta}\frac{1}{\beta}\ln\Xi(\beta,\mu)\Big)_{\beta\mu} \nonumber \\
& = & -\beta\mu\langle N\rangle + \beta\langle\bar{U}\rangle +\ln\Xi(\beta,\mu).
\label{eq23}\end{aligned}$$ Similarily to the canonical statistics, provided the first moments $\langle\bar{U}\rangle_{meas}$ and $\langle N\rangle_{meas}$ are known from measurements, the temperature $1/\beta$ and the chemical potential $\mu$ can be determined from eqns (\[eq22\]).
\[ssec:level4\]An outreach: possible extensions towards stochastic network topology – the super-canonical ensemble
------------------------------------------------------------------------------------------------------------------
For the network ensembles considered so far the network topology was fixed. In real communication networks it is not so and the connection graph has to be considered as a stochastic object. This can be realized by representing each pair of sites as a binary random variable and making the number of vertices random, and defining [*the super-canonical ensemble*]{} with new, multidimensional extensive random variable $\vec{L}=(L_1,\ldots,L_M)$, where $L_k$ represents the total path length for the $k$-th market $(OD)$ $$\begin{aligned}
L(OD)=\sum_{C_{\sigma(1)},\ldots,C_{\sigma(S)}}L[O,C_{\sigma_1},\ldots,C_{\sigma_S},D],
\label{eq23_1}\end{aligned}$$ where $\sigma(1),\ldots,\sigma(S)$ are sequencies of permutations of sites corresponding to existing routes. The $L_k$ is an analogy of the volume and we call it [*the volume of the $k$-th market*]{}, represented by the graph ${\cal G}_k$. For example, for the network in Fig. \[fig:fg0\], the volumes of markets are $$\begin{aligned}
L(1,2) & = & L[1,2]=1 \nonumber \\
L(1,3) & = & L[1,2,3]+L[1,2,4,3]=2+3=5 \nonumber \\
L(1,4) & = & L[1,2,4]+L[1,2,3,4]=2+3=5 \nonumber \\
L(2,3) & = & L[2,3]+L[2,4,3]=1+2=3 \nonumber \\
L(2,4) & = & L[2,4]+L[2,3,4]=1+2=3 \nonumber \\
L(3,4) & = & L[3,4]+L[3,2,4]=1+2=3
\label{eq23_2}\end{aligned}$$ such that the volume of the whole market, including reversed $(O,D)\rightarrow(D,O)$ pairs, amounts to 40.
Then we define the super partition function $$\begin{aligned}
Y(\beta,\mu,p) & \nonumber \\
= \sum_{L_1,\ldots,L_M}\sum_{N_1,\ldots,N_M}\prod_{k=1}^M e^{-\beta(pL_k-\mu N_k)}Z_k(\beta) &
\label{eq23a}\end{aligned}$$ where $p$ stands for the intensive variable, coupled to extensive variables $L_k$ and being direct analog to the pressure, and $Z_k(\beta)$ is the partition function given by eq. (\[eq13\]). Formally, $p$ is the Lagrange multiplier corresponding to the constraint $$\begin{aligned}
\langle L\rangle =\sum_{k=1}^M \langle L_k\rangle
\label{eq23b}\end{aligned}$$ where $L=L_1+\ldots +L_M$ will be called [*the total volume of the network*]{}.
The state of the system in the super-canonical ensemble is defined by specifying disutility for complete set of distributions of passengers over all markets and for all possible numbers of passengers, and for all possible random routes, i.e. $\Big\{\big\{\{\bar{U}_{j_k}(N_k,L_k)\}_{j_k(N_k,L_k)=1}^{J_k(N_k,L_k)}\big\}_{N_k,L_k}\Big\}_{k=1}^M$. Probabilities of states are $$\begin{aligned}
{\cal P}_{j_k(N_k,L_k)}(\beta,\mu,p)=\frac{e^{\beta(\mu N_k-pL_k-\bar{U}_{j_k(N_k,L_k)})}}{Y(\beta,\mu,p)}.
\label{eq23_3}\end{aligned}$$ The analogy with multinomial logit choice model is again close, with the logit alternative specific constant being equal to $\beta(\mu N_k-pL_k)$.
For completness, we list the formulae for first moments and correlations of extensive variables, including the cummulant $\langle\langle \bar{U}NL\rangle\rangle$ of $\bar{U}$, $N$ and $L$ accounting for true triple correlations between them, irreducible to pairwise corrlations: $$\begin{aligned}
\langle \bar{U}\rangle & = & -\Big(\frac{\partial}{\partial\beta}\ln Y(\beta,\mu,p)\Big)_{\beta\mu,\beta p} \nonumber \\
\langle N\rangle & = & \frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\ln Y(\beta,\mu,p)\Big)_{\beta,p} \nonumber \\
\langle L\rangle & = & -\frac{1}{\beta}\Big(\frac{\partial}{\partial p}\ln Y(\beta,\mu,p)\Big)_{\beta,\mu} \nonumber \\
\mbox{{\it Var}}\,(\bar{U}) & = & \Big(\frac{\partial^2}{\partial\beta^2}\ln Y(\beta,\mu,p)\Big)_{\beta\mu,\beta p} \nonumber \\
\mbox{{\it Var}}\,(N) & = & \frac{1}{\beta^2}\Big(\frac{\partial^2}{\partial\mu^2}\ln Y(\beta,\mu,p)\Big)_{\beta,p} \nonumber \\
\mbox{{\it Var}}\,(N) & = & \frac{1}{\beta^2}\Big(\frac{\partial^2}{\partial p^2}\ln Y(\beta,\mu,p)\Big)_{\beta,\mu} \nonumber \\
\mbox{{\it cov}}\,(\bar{U},N) & = & -\frac{1}{\beta}\Big(\frac{\partial}{\partial\mu}\big(\frac{\partial}{\partial\beta}\ln Y(\beta,\mu,p)\big)_{\beta\mu,\beta p}\Big)_{\beta,p} \nonumber \\
\mbox{{\it cov}}\,(\bar{U},L) & = & \frac{1}{\beta}\Big(\frac{\partial}{\partial p}\big(\frac{\partial}{\partial\beta}\ln Y(\beta,\mu,p)\big)_{\beta\mu,\beta p}\Big)_{\beta,\mu} \nonumber \\
\mbox{{\it cov}}\,(N,L) & = & -\frac{1}{\beta^2}\Big(\frac{\partial}{\partial p}\big(\frac{\partial}{\partial\mu}\ln Y(\beta,\mu,p)\big)_{\beta,p}\Big)_{\beta,\mu} \nonumber \end{aligned}$$ and $$\begin{aligned}
\langle\langle\bar{U}NL\rangle\rangle & \nonumber \\
=\langle\bar{U}NL\rangle-\langle\bar{U}N\rangle\langle L\rangle-\langle\bar{U}L\rangle\langle N\rangle & \nonumber \\
-\langle NL\rangle\langle\bar{U}\rangle+\langle\bar{U}\rangle\langle N\rangle\langle L\rangle & \nonumber \\
=\frac{1}{\beta^3}\bigg(\frac{\partial}{\partial p}\Big(\frac{\partial}{\partial\mu}\big(\frac{\partial}{\partial\beta}\ln Y(\beta,\mu,p)\big)_{\beta\mu,\beta p}\Big)_{\beta,p}\bigg)_{\beta,\mu} &
\label{eq23_4p}\end{aligned}$$ and for the entropy $$\begin{aligned}
S(\beta,\mu,p) & \nonumber \\
=\beta\langle\bar{U}\rangle-\beta\mu\langle N\rangle+\beta p\langle L\rangle + \ln Y(\beta,\mu,p). &
\label{eq23_4}\end{aligned}$$
We note that randomization of the network topology includes a couple of specific issues and its complete treatment needs further clarification. The set of extensive variables $\vec{L}$ is related to the network itself and not so much to the choice model, as it does not [*a priori*]{} reflect any preferences of decision makers. Even statistical ensembles of bare graphs, being less complex objects with no individuals making their choice among graph’s properties, are already complicated enough to be described with a few extensive variables and distinguishing interesting cases of the microcanonical, the canonical and the grand canonical, depending on the contruction procedures. The concept of bare random networks, unrelated to decision theory in the sense used here, is a subject of interest since pioneering works of Erdös and Rényi [@erdos1]. Various procedures of randomization of graphs with certain constraints on their characteristics, defining Hamilton paths and building probability measures for their ensembles are proposed e.g. in refs [@dorogovtsev1] and [@burda1] and summarized in refs [@dorogovtsev2] and [@barabasi1].
\[ssec:level4\][Escort moments]{}
---------------------------------
There is a useful approach to study of temperature dependence of thermodynamic functions, incorporating escort distribution [@beck2], which for the probability measure $\mu$ is defined as $$\begin{aligned}
P_{q_i}=\frac{p_i^q}{\sum_{i=1}^K p_i^q}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (i=1,\ldots,K)
\label{eq23aa}\end{aligned}$$ where $p_i=\int_{\Delta_i}\,d\mu$, $\Delta_i$ is the $i$-th phase space cell from the set of non-overlaping cells $\{\Delta_i\}_{i=1}^K$ covering completly the whole phase space, and $q\in\mathbb{R}^1$. From sum over cells one excludes those where $p_i=0$.
Using $q$-parametrized escort distributions, instead of ordinary probability density, one can efficiently probe those regions of the phase space where the probability measure is most concentrated (large positive $q$) or most rarified (large negative $q$). It was shown [@wislicki1; @majka1] that using Rényi entropies [@renyi2] $$\begin{aligned}
I_q=\frac{1}{1-q}\ln\sum_{i=1}^K p_i^q
\label{eq23ad}\end{aligned}$$ instead of ordinary Kolmogorov-Shannon entropies, and $q$-derivatives [@jackson] $$\begin{aligned}
\frac{d_q F(\beta)}{d_q \beta}=\frac{F(q\beta)-F(\beta)}{\beta(q-1)}
\label{eq23ae}\end{aligned}$$ instead of ordinary derivatives over inverse temperature, one effectively evolves the partition functions and all thermodynamic functions. For example, for the partition functions $Z$ and $\Xi$ one gets formulae $$\begin{aligned}
I_q=\frac{1}{1-q}[\ln Z(q\beta)-q\ln Z(\beta)] \nonumber \\
I_q=\frac{1}{1-q}[\ln \Xi(q\beta,q\mu)-q\ln \Xi(\beta,\mu)]
\label{eq23af}\end{aligned}$$ which control evolution of thermodynamic potentials and response functions with temperature and chemical potential.
Generally, for any stochastic function of temperature $A(\beta)$, the temperature scaling is equivalent to modification of the averaging prescription $$\begin{aligned}
\langle A(q\beta)\rangle=\langle A(\beta)\rangle_q
\label{eq23ab}\end{aligned}$$ and by iterating this procedure $$\begin{aligned}
\langle A(q_1q_2\ldots q_n\beta)\rangle=\langle\langle\ldots \langle A(\beta)\rangle_{q_n}\ldots\rangle_{q_2}\rangle_{q_1}
\label{eq23ac}\end{aligned}$$ where $\langle ..\rangle_q$ stands for average over the escort distribution (\[eq23aa\]). This is understandable because any temperature dependence of macroscopic observables, being themselves mean values or functions of them, stays in the probability measure. Modification of the measure $p_i^q$ in case of Boltzmann-like $p_i=P(\bar{U}=\bar{u}_i)\sim \exp(-\beta\bar{u}_i)$ is just temperature rescaling. For $q\rightarrow 1$ one recovers classical thermodynamics. In order to follow the temperature evolution of ordinary mean value or variance of $A$ without measuring it, one has to calculate corresponding $q$-moment for given temperature and vary $q$. Obviously, the same applies to any moment of the random variable $A(\beta)$, if it exists.
The concept of escort moments is also useful for monitoring of desired regions of the state space, e.g. those where experimental information is known most precisely or where the data exhibit the most advantageous signal-to-noise ratio and systematic bias is small (cf. ref. [@majka1] for detailed discussion).
\[sec:level4\]Applications
==========================
\[ssec:level4\][The model: analytical results and numerical simulations]{}
--------------------------------------------------------------------------
The formalism can be directly applied for study of relations between first moments of disutilities and market occupancies and their correlations and their dependencies on intensive variables. For any communication network, direct numerical simulations and evaluation of partition functions and probabilities on the state space can be performed. In many cases, however, this may be very demanding computationally because of factorial dependence of numbers of configurations on the graph size. For particular network topologies analytical shortcuts can be found and also some qualitative features of the networks can be inferred by pure reasoning.
We consider three network topologies (cf. Fig. \[fig:fg1\]):
1. [*The maximum connectivity*]{} network, defined by a complete graph, where any two sites are directly connected by one-segment path (Fig. \[fig:fg1\]a),
2. [*The hub-and-spoke*]{} network, where every site is only connected to the central hub (Fig. \[fig:fg1\]b),
3. [*The spider-web*]{}, representing an intermediate case between the previous ones, where the hub-and-spoke structure is complemented by connections between spokes forming the ring-like structure (Fig. \[fig:fg1\]c).
![\[fig:fg1\] The maximum connectivity graph (a), the hub-and-spoke (b) and the spider-web (c).](./2.eps){width="80mm" height="60mm"}
For exploratory study of basic properties of the model we assume the passengers disutility being proportional to the number of segments of the $j$-th route, $n_j$ $(n_j=0,1,\ldots)$, disregarding any additional dependencies, i.e. $\bar{U}=v_0+n_j v$, where $v$ is a positive constant and $v_0$ corresponds to disutility of making no trip and we assume $v_0=0$ without loss of genarality. This form corresponds to one of the basic features of the consumers disutility, which is his large reluctance to choose a route with large number of stops and changes. Such utility function allows us to find some analytical results and, in addition, it exhibits clear physical analogy to the one-dimensional quantum harmonic oscillator, where the energy of its $n$-th state depends linearly on $n$, $E_n=\hbar\omega(n+\frac{1}{2})$ $(n=0,1,\ldots)$. The no-trip disutility $v_0$ corresponds to vacuum energy $\hbar\omega/2$. The hierarchy of states of the harmonic oscillator corresponds to the market consisting of non-forking routes of increasing lengths, as illustrated in Fig. \[fig:fg2\].
![\[fig:fg2\] The hierarchy of routes of lengths $n=0,1,\ldots$, corresponding to the energy spectrum of one-dimensional quantum harmonic osillator. Crossed $n=0$ path (tadpole) corresponds to exclusion of the no-trip alternative from our model.](./3.eps){width="60mm" height="40mm"}
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![\[fig:fg3a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the maximum connectivity graph with 5 vertices. In the lower right panel, the values of $\bar{U}=3.062$ corresponding to $\beta>0$ extend to the left and those for $\beta<0$ - to the right from the $\beta=0$ point of $\langle\bar{U}\rangle=3.062$, indicated as vertical line. Values of entropies are divided by 1000. Curves correspond to analytical results and points are from numerical simulation.](./4a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the maximum connectivity graph with 5 vertices. In the lower right panel, the values of $\bar{U}=3.062$ corresponding to $\beta>0$ extend to the left and those for $\beta<0$ - to the right from the $\beta=0$ point of $\langle\bar{U}\rangle=3.062$, indicated as vertical line. Values of entropies are divided by 1000. Curves correspond to analytical results and points are from numerical simulation.](./4b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg3a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the maximum connectivity graph with 5 vertices. In the lower right panel, the values of $\bar{U}=3.062$ corresponding to $\beta>0$ extend to the left and those for $\beta<0$ - to the right from the $\beta=0$ point of $\langle\bar{U}\rangle=3.062$, indicated as vertical line. Values of entropies are divided by 1000. Curves correspond to analytical results and points are from numerical simulation.](./4c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the maximum connectivity graph with 5 vertices. In the lower right panel, the values of $\bar{U}=3.062$ corresponding to $\beta>0$ extend to the left and those for $\beta<0$ - to the right from the $\beta=0$ point of $\langle\bar{U}\rangle=3.062$, indicated as vertical line. Values of entropies are divided by 1000. Curves correspond to analytical results and points are from numerical simulation.](./4d.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$(\,\bar{U})$ [*vs.*]{} $U$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the maximum connectivity graph with 5 vertices. The values of $\langle\bar{U}\rangle=3.062$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./5f.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg3c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$, [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the maximum connectivity graph with 5 vertices. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=3.062$ corresponds to $\beta=0$.](./6f.eps "fig:"){width="40mm" height="40mm"}
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This analogy is more than purely academic observation since the harmonic oscillator is one of the best understood non-trivial systems in physics and thus the market of Fig. \[fig:fg2\] is an ideal testground for any concept developed within our choice model for networks. Our definition of the system leaves some freedom for including or not the no-trip alternative to the set of routes available to the choice maker, provided one does not consider the no-trip event as a double visit in given site, which would contradict the Hamilton property of the path. The model is perhaps more realistic excluding the no-trip choice ($n_j=0$) which means that the initial set of decision makers is restricted only to those determined to go. Otherwise, potential passengers are indistinguishable from those who do not enter the game at all and for which the choice set ${\cal C}$ is undetermined. Closed unit-edge loops, or [*tadpoles*]{}, can be easily avoided when the network topology is defined and non-random but it is known that special care in ascribing probabilities to the states is needed in case of random topologies [@dorogovtsev2].
The statistical sum is trivially calculable fo both cases $$\begin{aligned}
Z(\beta) & = & \left\{\begin{array}{ll}
(1-e^{-\beta v})^{-1} & \;\;\;(\mbox{no-trip included}) \\
(e^{\beta v}-1)^{-1} & \;\;\;(\mbox{no-trip excluded})
\end{array}\right .
\label{eq23bb}\end{aligned}$$ The first moments of disutility in the low- and high-temperature (resp. cool and hot) limits are $$\begin{aligned}
\langle \bar{U}\rangle & = & \frac{v}{e^{\beta v}-1} \nonumber \\
& \sim & \left\{\begin{array}{ll}
0, & \beta v\gg 1 \;\;\;(\mbox{cool}) \\
1/\beta, & \beta v\ll 1 \;\;\;(\mbox{hot})
\end{array}\right .
\label{eq23c}\end{aligned}$$ and $$\begin{aligned}
\langle \bar{U}\rangle & = & \frac{v}{1-e^{-\beta v}} \nonumber \\
& \sim & \left\{\begin{array}{ll}
v, & \beta v\gg 1 \;\;\;(\mbox{cool}) \\
1/\beta, & \beta v\ll 1 \;\;\;(\mbox{hot})
\end{array}\right .
\label{eq23d}\end{aligned}$$ for [*the no-trip included*]{} and [*the no-trip excluded*]{} model, respectively. Asymptotic behaviour is the same for both cases.
The variance of disutility, being proportional to the derivative of $\langle U\rangle$, is the same for both models: $$\begin{aligned}
\mbox{\it Var}\,(\bar{U}) & = & \frac{v^2 e^{-\beta v}}{(1-e^{-\beta v})^2} \nonumber \\
& \sim & \left\{\begin{array}{ll}
0, & \beta v\gg 1 \;\;\;(\mbox{cool}) \\
1/\beta^2, & \beta v\ll 1 \;\;\;(\mbox{hot})
\end{array}\right .
\label{eq23e}\end{aligned}$$
Our calculations and simulations are performed in the framework of grand canonical ensemble and we incorporate the notation $$\begin{aligned}
X_k=e^{\beta\mu}Z_k^1(\beta),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (k=1,\ldots,M)
\label{eq23f}\end{aligned}$$ Then the grand partition function (\[eq20\]) is given by $$\begin{aligned}
\Xi(\beta,\mu)=\prod_{k=1}^M\frac{1}{1-X_k}
\label{eq23ff}\end{aligned}$$ and formulae (\[eq22\]) can be rewritten in more specific and simpler form: $$\begin{aligned}
\langle\bar{U}_k\rangle & = & -\langle N_k\rangle \frac{\partial}{\partial\beta}\ln Z_k^1(\beta) \nonumber \\
& = & \langle N_k\rangle\langle\bar{U}_k\rangle_{\upharpoonleft}
\label{eq23g}\end{aligned}$$ where $\langle\bar{U}_k\rangle_{\upharpoonleft}$ stands for mean disutility of the one-passenger market $k$, where $$\begin{aligned}
\langle N_k\rangle = \frac{X_k}{1-X_k}
\label{eq23h}\end{aligned}$$ From eqn (\[eq23h\]) it follows that $X_k=\langle N_k\rangle/(\langle N_k\rangle+1)<1$ and the grand partition function (\[eq23ff\]) is always non-singular.
Second moments and correlations are equal to $$\begin{aligned}
\mbox{{\it Var}}\,(\bar{U}) & = & \sum_{k=1}^M \mbox{{\it Var}}\,(\bar{U}_k) \nonumber \\
& = & \sum_{k=1}^M \big(\langle N_k\rangle\mbox{{\it Var}}\,(\bar{U}_k)_{\upharpoonleft}+\langle\bar{U}_k\rangle_{\upharpoonleft}^2\mbox{{\it Var}}\,(N_k)\big) \nonumber \\
\mbox{{\it Var}}\,(N) & = & \sum_{k=1}^M \mbox{{\it Var}}\,(N_k) \nonumber \\
& = & \sum_{k=1}^M \big(\langle N_k\rangle +\langle N_k\rangle^2\big) \nonumber \\
\mbox{{\it cov}}\,(\bar{U},N) & = & \sum_{k=1}^M \mbox{{\it cov}}\,(\bar{U}_k,N_k) \nonumber \\
& = & \sum_{k=1}^M \langle\bar{U}_k\rangle_{\upharpoonleft}\mbox{{\it Var}}\,(N_k)
\label{eq23i}\end{aligned}$$ where $\mbox{{\it Var}}\,(\bar{U}_k)_{\upharpoonleft}$ is the variance of disutility for the one-passenger market $k$.
The canonical case is recovered for non-random $\langle N_k\rangle = N_k$ for all $k$, when $\mbox{{\it Var}}\,(N_k)=0$, as expected.
Worthwhile to note, for non-random market distility $\mbox{{\it Var}}\,(\bar{U}_k)=0$, moments of $\bar{U}$ and $N$ are the same as for the system of ideal bosons (cf. ref. [@majka1]).
\[ssec:level4\][Case study 1: the maximum connectivity network]{}
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![\[fig:fg4a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./7a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./7b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg4a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./7c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./7d.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$ [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $U$ (lower right) for the hub-and-spoke graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=1.630$ corresponding to $\beta=0$ are indicated by vertical lines in figures with $\langle\bar{U}\rangle$ on the abscissae.](./8f.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg4c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the hub-and-spoke graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=1.630$ corresponds to $\beta=0$.](./9f.eps "fig:"){width="40mm" height="40mm"}
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If the network graph consists of $\Gamma$ nodes, the total number of markets is $M=\Gamma^2-\Gamma$. Using combinatorics and denoting by $v$ disutility for one segment of any route, same everywhere, one finds the partition function for the $k$-th market $$\begin{aligned}
Z_k(\beta) & = & \Big(\sum_{l=1}^{\Gamma-1}\frac{(\Gamma-2)!}{(\Gamma-1-l)!}e^{-\beta vl}\Big)^{N_k}
\label{eq24}\end{aligned}$$ and $Z(\beta)$ is given by eqn. (\[eq11\]). The combinatorial factor accounts for the degeneration of the state of given disutility and is equal to the number of routes of the same length. For fixed $N=\sum_{k=1}^M N_k$ $$\begin{aligned}
\langle \bar{U}\rangle & = & Nv\Big(\Gamma-1-\frac{\sum_{l=0}^{\Gamma-2}le^{\beta vl}/l!}{\sum_{l=0}^{\Gamma-2}e^{\beta vl}/l!}\Big) \nonumber \\
& \stackrel{\Gamma\gg 1}{\longrightarrow} & Nv(\Gamma-e^{\beta v})
\label{eq25}\end{aligned}$$ and $$\begin{aligned}
\mbox{\it Var}\,(\bar{U}) & \stackrel{\Gamma\gg 1}{\longrightarrow} & N v^2e^{\beta v} \nonumber \\
& = & v(Nv\Gamma-\langle \bar{U}\rangle)
\label{eq26}\end{aligned}$$ where we used equilibrium temperature $\beta_{eq}=\frac{1}{v}\ln(\Gamma-\frac{\langle\bar{U}\rangle}{Nv})$ determined from eqn. (\[eq25\]).
In general case we failed to find analytic formulae for moments of $N$ and correlations between $N$ and $\bar{U}$. The chemical potential $\mu$ can be determined, provided $\langle N\rangle$ is known.
It is instructive to discuss analytic solutions for the simplest non-trivial case of one market on the $\Gamma=3$ graph and assuming $v=1$ in order to elliminate trivial factors. Using eqns (\[eq23g\]) and (\[eq23h\]), one easily finds equations for equilibrium $\beta$ and $\mu$ $$\begin{aligned}
\beta_{eq}=\ln\frac{2\langle N\rangle -\langle\bar{U}\rangle}{\langle\bar{U}\rangle-\langle N\rangle}
\label{eq27}\end{aligned}$$ and $$\begin{aligned}
\nu_{eq} & = & (\beta\mu)_{eq} \nonumber \\
& = & 2\ln\frac{2\langle N\rangle -\langle\bar{U}\rangle}{1+\langle N\rangle}-\ln\frac{\langle\bar{U}\rangle-\langle N\rangle}{1+\langle N\rangle}
\label{eq28}\end{aligned}$$ Using eqns (\[eq27\]),(\[eq28\]) and (\[eq23i\]), we obtain $$\begin{aligned}
\mbox{{\it Var}}\,(\bar{U}) & = & -2\langle N\rangle+3\langle\bar{U}\rangle+\langle\bar{U}\rangle^2 \nonumber \\
& = & (-\langle\bar{U}\rangle^2/\langle N\rangle+3\langle\bar{U}\rangle-2\langle N\rangle) \nonumber \\
& + & (\langle\bar{U}\rangle^2/\langle N\rangle+\langle\bar{U}\rangle^2)
\label{eq29}\end{aligned}$$ where the first term corresponds to the canonical ensemble, with fixed number of individuals $\langle N\rangle$, and the second term comes from randomization of the number of individuals in the grand canonical ensemble. Since in our model $\langle\bar{U}\rangle$ is proportional to $\langle N\rangle$, the asymptotic behaviour of both terms can be investigated expanding $\langle\bar{U}\rangle$ around $\langle N\rangle$. Asymptotically, we find that the first term vanishes and the second term behaves as $\langle N\rangle^2$ for $\langle N\rangle\gg 1$.
In addition, we calculate for this case $$\begin{aligned}
\mbox{{\it Var}}\,(N)=\langle N\rangle(\langle N\rangle+1)
\label{eq30}\end{aligned}$$ and $$\begin{aligned}
\mbox{{\it cov}}\,(\bar{U},N)=\langle\bar{U}\rangle(\langle N\rangle+1)
\label{eq31}\end{aligned}$$ and we see that both [*Var*]{}$\,(N)$ and covariance of $\langle\bar{U}\rangle$ and $\langle N\rangle$ depend quadratically on $\langle N\rangle$ in the limit of $\langle N\rangle\rightarrow\infty$. For completness, the entropy in this case is equal to $$\begin{aligned}
S & = & -\ln\frac{1}{1+\langle N\rangle}+(-2\langle N\rangle+\langle\bar{U}\rangle)\ln\frac{2\langle N\rangle-\langle\bar{U}\rangle}{1+\langle N\rangle} \nonumber \\
& + & (\langle N\rangle-\langle\bar{U}\rangle)\ln\frac{-\langle N\rangle+\langle\bar{U}\rangle}{1+\langle N\rangle}.
\label{eq31a}\end{aligned}$$
We also investigated analytically higher $\Gamma$ and found qualitatively the same bahaviour as for $\Gamma=3$.
We performed numerical simulation for the maximum conectivity network with $\Gamma=5$ and $\langle N\rangle=1000$ and displayed our results in Figs \[fig:fg3a\] and \[fig:fg3b\]. Both $\beta$ and $\mu$ were treated formally as Lagrange multipliers and thus we allowed for both positive and negative values of those parameters. The behaviour of $\langle\bar{U}\rangle$, [*Var*]{}$\,(\bar{U})$ and $S$ with temperature is intuitively appealing for $\beta>0$.
The dependence $S(\beta)$ exhibits the same tendency for systems of bosons with inversed populations alowed, i.e. the entropy increases with temperature for both positive and nagative temperatures. Moreover, the entropy $S$ exists in the limits of $\beta\rightarrow\pm\infty$, where $S\stackrel{\beta\rightarrow\infty}{\longrightarrow}{\it const.}$ and $S\stackrel{\beta\rightarrow -\infty}{\longrightarrow}\mbox{\it const.}\,'$. For the canonical ensemble ${\it const.}=0$ and for random $N$, in general ${\it const.}\ne 0$.
Dependence of $\nu$ on $\beta$ exhibits steeper slope for $\beta<0$ which is related to less steep dependence of $\langle\bar{U}\rangle(\beta)$ or $\beta<0$, as compared to $\beta>0$.
It is interesting to note that [*Var*]{}$\,(N)$ is insensitive to neither $\beta$ nor $\langle\bar{U}\rangle$ which reflects the maximal symmetry of the maximum connectivity network. Whatever the temperature is, relative populations for all routes are the same and the width of distribution of $N$ does not depend on temperature.
Results of simulations for $q$-moments are displayed in Figs \[fig:fg3c\]. One observes the same dependence of the $q$-moments on $q$ and explicit $\beta$-dependence of the ordinary moments in Figs \[fig:fg3b\]. It clearly confirms interpretation of $q$ as a temperature scaling parameter, as mentioned in chapt. III.E. Consistently, the values of ordinary moments for $\beta=0$ are the same as $q$-moments for $q=0$.
Fig. \[fig:fg3b\] (bottom right) presents the ratio of passengers in a given market as a function of utility, where each point correspons to one overall mean disutility, or temperature. Clearly, there is only one type of markets because of the maximal symmetry of this network. Vertical line at $\langle\bar{U}\rangle=3.062$ divides the domain of $\langle\bar{U}\rangle$ into subregions corresponding to $\beta>0$ (to the left) and $\beta<0$ (to the right).
\[ssec:level4\][Case study 2: the hub-and-spoke network]{}
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![\[fig:fg5a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./10a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./10b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg5a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./10c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5a\] Mean disutility $\langle\bar{U}\rangle$ [*vs.*]{} inverse temperature $\beta$ (upper left), the $\nu=\beta\mu$ [*vs.*]{} $\beta$ (upper right), the entropy $S$ [*vs.*]{} $\beta$ (lower left) and $S$ [*vs.*]{} $\langle\bar{U}\rangle$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./10d.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5b\] The variance of disuility [*Var*]{}$\,(\bar{U})$ [*vs.*]{} inverse temperature $\beta$ (upper left), the variance of the number of individuals [*Var*]{}$\,(N)$ [*vs.*]{} $\beta$ (upper right), [*Var*]{}$\,(\bar{U})$ [*vs.*]{} $\bar{U}$ (middle left), [*Var*]{}$\,(N)$ [*vs.*]{} $\bar{U}$ (middle right), the covariance of $\bar{U}$ and $N$, [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\beta$ (lower left) and [*cov*]{}$\,(\bar{U},N)$ [*vs.*]{} $\bar{U}$ (lower right) for the spider-web graph with 1 hub and 4 spokes. The values of $\langle\bar{U}\rangle=2.954$ corresponding to $\beta=0$ are indicated by vertical lines for figures with $\bar{U}$ on the abcissae.](./11f.eps "fig:"){width="40mm" height="40mm"}
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![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12a.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12b.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12c.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12d.eps "fig:"){width="40mm" height="40mm"}
![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12e.eps "fig:"){width="40mm" height="40mm"} ![\[fig:fg5c\] The $q$-mean disutility $\langle\bar{U}\rangle_q$ [*vs.*]{} $q$ (upper left) and the $q$-mean number of individuals $\langle N\rangle_q$ [*vs.*]{} $q$ (upper right), the $q$-variance of disutility [*Var*]{}$\,(\bar{U})_q$ (middle left), the $q$-variance of the number of individuals [*Var*]{}$\,(N)_q$ (middle right) and the $q$-covariance of $\bar{U}$ and $N$ [*cov*]{}$_q\,(\bar{U},N)$ (bottom left) for the spider-web graph with 1 hub and 4 spokes. The ratio of passengers in a given type of market as a function of mean disutility is displayed in the lower right panel. The vertical line at $\langle\bar{U}\rangle=2.954$ corresponds to $\beta=0$.](./12f.eps "fig:"){width="40mm" height="40mm"}
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For this topology, the one-passenger disutility in the $k$-th market is equal to $$\begin{aligned}
\bar{U}_{k_i} & = & \left\{\begin{array}{lc}
2v, & \mbox{$O$ and $D$ spokes $\;\;\;\;$ (case (i))} \\
v, & \mbox{otherwise $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (case (ii))}
\end{array}\right .
\label{eq32}\end{aligned}$$ Denoting by $M_{1(2)}$ the number of markets for the case (i) and (ii), which for the total numbers of spokes $\Gamma-1$ are equal to $\Gamma^2-3\Gamma+2$ and $\Gamma(\Gamma-1)$, respectively, and using $(\ref{eq13})$, one gets for the partition function $$\begin{aligned}
Z(\beta)=\prod_{k=1}^{M_1}e^{-2\beta vN_k}\prod_{k=M_1}^{M_1+M_2}e^{-\beta vN_k}
\label{eq33}\end{aligned}$$ and, for the canonical enesemble, the mean disutility is equal to $$\begin{aligned}
\langle\bar{U}\rangle=(2N_O+N_H)v
\label{eq34}\end{aligned}$$ where $N_O=\sum_{k=1}^{M_1}N_k$ and $N_H=\sum_{k=M_1+1}^{M_1+M_2}N_k$ are total numbers of individuals for cases (i) and (ii), respectively.
For large $N_k$s, the grand partition function is equal to $$\begin{aligned}
\Xi(\beta,\mu)=\frac{1}{(1-e^{\beta(\mu-2v)})^{M_1}}\cdot\frac{1}{(1-e^{\beta(\mu-v)})^{M_2}}
\label{eq35}\end{aligned}$$ and the first moments of $\bar{U}$ and $N$ are equal to $$\begin{aligned}
\langle\bar{U}\rangle & = & \frac{2vM_1}{e^{-\beta(\mu-2v)}-1}+\frac{vM_2}{e^{-\beta(\mu-v)}-1} \nonumber \\
\langle N\rangle & = & \frac{M_1}{e^{-\beta(\mu-2v)}-1}+\frac{M_2}{e^{-\beta(\mu-v)}-1}
\label{eq36}\end{aligned}$$ Following the same procedure as for the maximum connectivity network and assuming $v=1$, one finds the variances: $$\begin{aligned}
{\it Var}\,(\bar{U}) & = & \langle N\rangle\big(-2+\frac{5}{6}\langle N\rangle\big) \nonumber \\
& + & \langle\bar{U}\rangle\big(3-\frac{7}{6}\langle N\rangle+\frac{11}{24}{\langle\bar{U}\rangle}\big) \nonumber \\
{\it Var}\,(N) & = & \langle N\rangle\big(1-\frac{7}{12}\langle N\rangle\big) \nonumber \\
& - & \langle\bar{U}\rangle\big(\frac{2}{3}\langle N\rangle-\frac{5}{24}\langle\bar{U}\rangle\big)
\label{eq37}\end{aligned}$$ the covariance $$\begin{aligned}
{\it cov}\,(\bar{U},N)=\frac{2}{3}\langle N\rangle^2+\langle\bar{U}\rangle\big(1-\frac{5}{6}\langle N\rangle+\frac{7}{24}\langle\bar{U}\rangle\big)
\label{eq38}\end{aligned}$$ and the entropy $$\begin{aligned}
S & = & \langle\bar{U}\rangle\ln\frac{(-12+\langle N\rangle-\langle\bar{U}\rangle)(2\langle N\rangle-\langle\bar{U}\rangle)}{(\langle N\rangle-\langle\bar{U}\rangle)(8+2\langle N\rangle-\langle\bar{U}\rangle)} \nonumber \\
& - & \langle N\rangle\ln\frac{(-12+\langle N\rangle-\langle\bar{U}\rangle)(2\langle N\rangle-\langle\bar{U}\rangle)^2}{(\langle N\rangle-\langle\bar{U}\rangle)(8+2\langle N\rangle-\langle\bar{U}\rangle)^2} \nonumber \\
& - &8\ln\frac{8}{8+2\langle N\rangle-\langle\bar{U}\rangle} \nonumber \\
& - &12\ln\frac{12}{12-\langle N\rangle+\langle\bar{U}\rangle}.
\label{eq39}\end{aligned}$$
As for the previous case, we performed numerical simulation for the hub-and-spoke network with $\Gamma=5$ (one hub and four spokes) and $\langle N\rangle=1000$ and show the results, together with analytic curves, in Figs \[fig:fg4a\] and \[fig:fg4b\]. Quantitatively, the $\langle\bar{U}\rangle$, $\nu$ and $S$ with $\beta$ exhibit the same type of behaviour, including asymptotics, as observed for the maximal graph. Dependence of $\langle\bar{U}\rangle$ on $\beta$ for the hub-and-spoke for small $\beta$ is steeper than for the maximum connectivity graph. The maximum of entropy at $\beta=0$ is sharper for the hub-and-spoke than for the maximum connectivity graph.
Contrary to the maximum connectivity graph, the variance of $N$ does depend on temperature (Fig. \[fig:fg4b\] upper and middle right) and decreases with temperature for $\beta>0$. The same tendency is observed for variance of $\bar{U}$ (Fig. \[fig:fg4b\] upper and middle left). This behaviour reflects lower symmetry of the hub-and-spoke network, as compared to the maximum connectivity, which means that not all markets are identical. There are two types of markets, spoke-spoke and spoke-hub (cases (i) and (ii) in eqn (\[eq32\])) and relative occupancy of the second one increases with temperature, making the market probability distribution narrower, eventually reaching its minimum for $\beta\rightarrow 0$ ($T\rightarrow\infty$). The mirror-like behaviour is observed for negative $\beta$ but the plateau values for [*Var*]{}$\,(\bar{U})$, [*Var*]{}$\,(N)$ and [*cov*]{}$\,(\bar{U},N)$ are different than for positive $\beta$.
Simulations of $q$-moments are displayed in Figs \[fig:fg4c\]. Again, simulating the $q$-dependence one reproduce all features of explicit $\beta$-dependences of moments and correlations observed in previous figures.
Fig. \[fig:fg4c\] (bottom right) presents the ratio of passengers in a given market as a function of utility, where each point corresponds to one mean overall disutility. Asterisks correspond to the market with disutility $v$, i.e. the spoke-hub market, and triangles to the spoke-spoke market with disutility $2v$. The region of $\beta>0$ extends to the left of the vertical line at $\langle\bar{U}\rangle=1.630$ and $\beta<0$ - to the right.
\[ssec:level4\][Case study 3: the spider-web network]{}
-------------------------------------------------------
The spider-web network represents an intermediate case between the maximum connectivity and the hub-and-spoke toplogies, discussed above. But concerning symmetry, the spider topology is less symmetric than previous two, in a sense that there are three different types of markets in this network. We found analytic formulae also for this case but they are very lenghty and we do not print them here. All the results shown in Figs \[fig:fg5a\], \[fig:fg5b\] and \[fig:fg5c\] come from analytic formulae and from direct simulations with the same conditions as for previous cases. Many qualitative characteristics of the expected values and $q$-expected values of $\bar{U}$ and $N$, and also of the entropy $S$, are the same as for more symmetric topologies, but inspection of Figs \[fig:fg5b\] and \[fig:fg5c\] reveals also some interesting structures in $\beta$-dependence of the ($q$-)variances and ($q$-)covariances, deserving more attention. Contrary to more symmetric topologies, one observes two minima: one at $\beta=0$ or $q=0$, and another one for $\beta>0$ or $q>0$, and two maxima at non-zero values. Qualitatively, such behaviour can be understood by following temperature dependence of probabilities, as illustrated in Fig. \[fig:fg6\], where probability distributions for binary and ternary random variables are shown as functions of $\beta$.
![\[fig:fg6\] Various $\beta$-dependent probability distributions leading to single or multiple extrema for second moments of disutilities and numbers of individuals, located at different values of $\beta$. Upper panels present two types of probability distributions for binary random variable: the maximum variance at $\beta=0$ (upper left) and the maximum variance at $\beta>0$ (upper right). Lower panels show two types of ternary random variables: one with maxima at $\beta=0$ and at $\beta>0$ (lower left) and with three local maxima, two of them for $\beta>0$ (lower right).](./13.eps){width="80mm" height="80mm"}
For the binary case (cf. Figs \[fig:fg6\] upper), the widest distribution corresponds to equal probabilities $p_1=p_2$ which can happen either for $\beta=0$ or $\beta>0$. More complex structure of maxima for variance can easily arise for multinomial probability density function where probablilities intersect pairwise in more than one regions, or points of intersections clusterize. An example of ternary random variable is presented in lower Figs \[fig:fg6\], where distributions for multiple maxima are illustratively drawn.
In order to find equilibrium values $\beta_{eq}$ and $\nu_{eq}$, one has to solve equations for $\langle\bar{U}\rangle$ and $\langle N\rangle$. We observe that degree of those equations depends on the symmetry of the network and is equal to the number of topologically different markets for given networks: for the maximum connectivity all markets are the same, for hub-and-spoke there are two types of markets and for the spider-web there are three types of markets. We further observe that the number of local minima is less by one than the degree of those equations. We strongly suspect that this is true for higher numbers of different markets. At the moment we do not have any proof for the general case and leave this statement as a hypothesis.
Fig. \[fig:fg5c\] (bottom right) presents the ratio of passengers in a given market as a function of utility, where each point corresponds to one mean overall disutility, or one temperature. Triangles represent market where $O$ is located on the rim and $D$ in the hub, such that the set of possible routes for this market can be represented by the set of disutilities $\{v,2v,2v,3v,3v,4v,4v\}$. Asterisks correspond to the market with $O$ and $D$ on the rim and non-adjacent, such that the set of routes is $\{2v,2v,2v,3v,3v,3v,4v,4v\}$. Finally, diamonds are for the market of adjacent non-hub endpoints, where the routes are $\{v,2v,3v,3v,3v,4v,4v,4v\}$. The region of $\beta>0$ extends to the left of the vertical line at $\langle\bar{U}\rangle=2.954$ and $\beta<0$ - to the right. It is interesting to observe that for certain values of $\langle\bar{U}\rangle$, where the hub-rim market occupancy increases with temperature, two other sorts of markets tend to decrease, reaching local or global maxima. Such non-trivial behaviour demonstrates potential power of the model developed here, for estimation of market occupancies.
\[sec:level5\] Conclusions and final remarks
============================================
Summarizing, we proposed a formalism, based on the equilibrium statistical thermodynamics, describing complex communication systems, consisting of sets of communicating nodes and individuals able to make decisions on how they travel. In classical approach to the problem of choice making, based on game theory, the key concept is the utility function which quantifies choice probabilities. Utility function was defined axiomatically, according to neoclassical economic theory. We defined our system as a sum of markets, each market represented by a directed graph with two communicating endpoints and set of Hamilton paths connecting them. Our networks were not embeded in metric spaces and thus the model deals with topologies and not with geometry. The state of the system is defined by ascribing each individual to the market and specifying its choice of the comunication route. Disutility of the whole system, ascribed to each state of it, is defined as a sum of individual disutilities, integrated over all markets. We have shown that such disutility is an additive and extensive stochastic function and therefore the probability measure on the state space is of Boltzmann type, thus reproducing the multinomial logit choice model, known from decision theory, using quite general and fundamental reasoning. Further, we considered statistical ensembles: the microcanonical, canonical, grand canonical and super canonical. The first three are commonly known in statistical mechanics and are rather straightforward to construct and use, whereas the super canonical ensemble incorporates random network topology and needs much further and more detailed specifications. We did not randomize intensive variables and kept them as nonrandom variables or control parameters. Generalizations in this direction and using random intensive fields of temperature, pressure etc. seem quite natural and are certainly ahead of us in further developments. Incorporation of geometry, i.e. using metric characteristics of connections like distances, shapes etc., are quite straightforward but not equally relevant in all applications. In particular, for airline networks, topology plays principal role, as long as intercontinental connections are not concerned.
Following usual procedures, known from classical equilibrium therodynamics, we find the entropy and first moments of extensive random variables of the model and their correlations. We also propose using escort distributions for monitoring of temperature evolution of moments of stochastic variables.
We applied our model to networks with specified topologies and we obtained analytical results for two topologies: the maximum connectivity and the hub-and-spoke, using rather simplistic disutility function, proportional to the number of route segments. For this disutility we found physical analogy, as the hierarchy of routes of lengths increasing in steps of one unit can be directly mapped onto the one-dimensional quantum harmonic oscillator and exhibits close analogy to systems of bosons. For the spider-web topology the formulae can be also found but are very complex.
Any network can be in principle simulated numerically. We performed such simulations, in the framework of the grand canonical ensemble for all topologies, in order to crosscheck our analytical findings. Our simulations, however, were done so far for rather small networks consisting of five nodes and average number of thousand decision makers. Evaluation of statistical sums requires summing over all possible distributions of decision makers over all Hamiltonian paths on the graph and this is in general very demanding computationally. Since the world air traffic integrated over one day incorporates millions of passengers, its realistic simulations, even with tight constraints not accounted so far, will require more efficient software tools.
We see considerable economic potential of our thermodynamic approach to the process of decision making in communicating population. After complementing the model with constraints making it more suitable for real life applications in air transportation, as bunching of passengers in finite-size planes, realistic flight schedules, availability of air space and time slots, finite node transmittance, accounting for delays and other perturbations, and finally, using realistic disutility function, and many other factors, it can be used for monitoring and forecasting the state of the network. Predictions for market occupancies and fluctuations of numbers of individuals depending on passenger’s disutility and all types of correlations between disutilities and occupancies are useful for management of resources.
We see three principal directions of development of our model:
- Network description in the framework of super-canonical ensemble, i.e. using random topology, and incorporation of geometric characteristics of the network by embeding the graph in metric space,
- Monitoring of collective phenomena in the communication network using entropies, response functions and an approach based on the phase space uniformity, developed in ref. [@wislicki1],
- Transforming our choice model into the full game, with airline operators and carriers being also active players in such game. This needs finding a coupling mechanism between utility functions of all parties, being e.g. typical predator-prey coupling of the Lotka-Volterra type, as a good starting point for further developments.
We acknowledge support granted to this work by Boeing Company in the framework of joint Boeing–ICM UW [*Capstone*]{} project.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We measured the optical absorptance of superconducting nanowire single photon detectors. We found that 200-nm-pitch, 50%-fill-factor devices had an average absorptance of 21% for normally-incident front-illumination of 1.55-$\upmu$m-wavelength light polarized parallel to the nanowires, and only 10% for perpendicularly-polarized light. We also measured devices with lower fill-factors and narrower wires that were five times more sensitive to parallel-polarized photons than perpendicular-polarized photons. We developed a numerical model that predicts the absorptance of our structures. We also used our measurements, coupled with measurements of device detection efficiencies, to determine the probability of photon detection after an absorption event. We found that, remarkably, absorbed parallel-polarized photons were more likely to result in detection events than perpendicular-polarized photons, and we present a hypothesis that qualitatively explains this result. Finally, we also determined the enhancement of device detection efficiency and absorptance due to the inclusion of an integrated optical cavity over a range of wavelengths (700-1700 nm) on a number of devices, and found good agreement with our numerical model.'
address: |
$^1$Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA\
$^2$Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02420, USA\
$^3$Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
author:
- 'Vikas Anant,$^1$ Andrew J. Kerman,$^2$ Eric A. Dauler,$^{1,2}$ Joel K. W. Yang,$^1$ Kristine M. Rosfjord,$^3$ and Karl K. Berggren$^1$'
title: 'Optical Properties of Superconducting Nanowire Single-Photon Detectors'
---
Introduction
============
The short reset-time, low jitter, and broad wavelength response of superconducting nanowire single photon detectors (SNSPDs) makes them attractive candidates to replace other single photon detectors, including avalanche photodiodes [@apd], in applications such as free-space optical communications [@eric], quantum cryptography networks [@quantum-crypto; @takesue], and quantum computation [@knill]. To fully take advantage of the boost in speed afforded by these properties, SNSPDs with high system detection efficiencies ($SDE$) are needed. But SNSPDs typically exhibit $SDE$s of only 0.2-10% for 1.55-$\upmu$m-wavelength single-photons [@verevkin; @korneev; @hadfield]. This limitation is currently believed to be due to poor coupling efficiency ($\eta_\text{c}$); however, this may not be the entire story. To understand where the losses come from, we need to examine the factors that contribute to the $SDE$.
The $SDE$ is a product of two lumped quantities, the coupling efficiency ($\eta_\text{c}$) and the device detection efficiency ($DDE$). The quantity $\eta_\text{c}$ encapsulates all loss mechanisms encountered between the photon source and SNSPD, and is defined as the ratio of the number of photons that reach the active area to the number of photons emitted by the photon source. It is relatively straightforward to realize $\eta_\text{c}>0.9$ through careful optical design that minimizes these losses. On the other hand, maximizing $DDE$ is not as simple.
The $DDE$ is the probability that a photon incident on the active area results in a voltage pulse. $DDE$ depends on two quantities, the absorptance $A$, and the probability of resistive state formation due to an absorption event $P_\text{R}$ through $$DDE=P_\text{R}\, A.
\label{eq.DE}$$ While $P_\text{R}$ depends on the microscopic physics of the nanowires, $A$ depends only on the optical properties of our fabricated structures and the incident field. $A$ needs to be maximized to make efficient detectors; however, there are many ways an incident photon can remain unabsorbed. For example, the photon can pass through open gaps between the nanowires or be transmitted through the subwavelength thickness of NbN. In addition, a low value for $P_\text{R}$ will further suppress $DDE$ (and therefore also $SDE$).
In this paper, we experimentally demonstrate that the optical polarization and changes in the geometry (pitch and fill-factor) of SNSPDs impact their absorptance and efficiency. For example, for devices with 200 nm pitch and 50% fill factor, we found that 21% of light incident from the front was absorbed for parallel polarization, while only 10% was absorbed for perpendicular polarization. The absorptance was reduced to 14% and 6% for parallel and perpendicular polarizations, respectively, for devices with the same pitch but a 25% fill-factor. We describe both a numerical model that predicts the absorptance of our structures, and experiments that directly measure absorptance. We found that the experimental data matches the predictions of our model to within the uncertainties in our knowledge of physical and optical parameters. We also measured the $DDE$ for the same devices and compared it to the measured absorptance. We discovered, remarkably, that $P_\text{R}$ is different for photons polarized perpendicular ($\perp$) to the nanowires ($P_{\text{R},\perp}$) than for photons polarized parallel to the nanowires ($P_{\text{R},||}$). We propose a model that qualitatively explains our observation of $P_{\text{R},||} \neq P_{\text{R},\perp}$.
Finally, we also made measurements to confirm that our numerical model and modeling parameters are accurate over a range of wavelengths. We measured the factor of enhancement of $DDE$ due to the addition of an integrated optical cavity for single-photon wavelengths of 700-1700 nm, and found good agreement with our numerical model. As a result, we have shown that optical modelling can be used to design future SNSPDs in new regimes of operation.
This paper is organized as follows. In Section \[sec.model\], we will describe a numerical model that predicts the absorptance of our structures. In Section \[sec.expt\], we will describe the setup and experiments conducted to measure absorptance on our devices. In Section \[sec.wavelength\], we will describe the experiments conducted to measure the wavelength dependence of the enhancement due to the addition of an optical cavity. We will not describe fabrication details or $DDE$ measurement details as these were discussed in [@joel] and [@rosfjord], respectively.
Optical Model {#sec.model}
=============
We modeled the absorption process of photons by an SNSPD as a plane wave interacting with an infinite grating. These two approximations are justified because in our experiments, the photon was launched at normal incidence and had plane-wave phase-fronts when it reached the detectors, and because our beam spot was smaller than the test grating which will be described in Section \[sec.expt\].
While analytic methods, such as form birefringence theory, have been used in the past to analyze grating structures such as wire-grid polarizers [@wgp-yu; @wgp-yeh], they break down for subwavelength grating thicknesses, in particular for electric-field polarized perpendicular to the wires in the grating (i.e. in the direction of grating periodicity)[^1] [@lalanne]. Recently, a numerical technique (finite-element analysis) was used to approach this problem [@majedi]; but because of either a difference in material parameters or simulation conditions, their findings were not consistent with our experimental or theoretical results.
![Infinite wire grid schematized in (a) is further reduced via symmetry to the unit cell shown in (b) in numerical modeling of the absorptance, $A$. (The schematic is not drawn to scale.) Plots of the calculated $A$ as a function of fill-factor, $f$ and pitch $p$ are shown in (c) and (d) for parallel ($||$) and perpendicular ($\perp$) electric field polarization. Inset in (d) shows how the calculated cross-sectional time-averaged electric field magnitude, $|\vec{E}_{\text{cc}^\prime}|$ varies with position $x$ for $||$ and $\perp$ polarizations across a $f=\frac{1}{2}$, $p=200\,\text{nm}$ structure. The NbN film thickness ($t_\text{NbN}$) was 4 nm and incident electric field magnitude $|\vec{E}_\text{0}|=1$. The NbN region extends from $x=-50\,\text{nm}$ to $x=+50\,\text{nm}$.[]{data-label="fig.snspd-geometry"}](schematic-and-A-vs-pitch-ff-combined)
We used a different finite-element analysis software (Comsol Multiphysics v3.2b, EM module) than the one used in [@majedi] to calculate the absorptance of the geometry shown in Fig. \[fig.snspd-geometry\](a). We took advantage of symmetry by using the ‘In-plane Hybrid-Mode Waves’ implementation and defined the unit cell[^2] shown in Fig. \[fig.snspd-geometry\](b) with the desired NbN thickness ($t_\text{NbN}$), pitch ($p$), wire width ($w$), fill-factor ($f=w/p$), and a 1.55-$\upmu$m-wavelength. The thickness of NbN$_x$O$_y$ was held constant at because it was observed by transmission electron microscopy (TEM) to be the same thickness for different NbN thicknesses.[^3] The sapphire wafer thickness was not included because an anti-reflection coating was added to the sapphire-air interface in experiments. We applied periodic boundary conditions on the left (XY) and right ($\text{X}^\prime\text{Y}^\prime$) edges of the unit cell, scattering boundary conditions on the bottom (XX$^\prime$) and top (YY$^\prime$) edges. We applied either an out-of-plane ($E_{||}$), or in-plane ($E_\perp$) electric field on the top edge (front-illumination), which matches the experimental setup described in Sec. \[sec.expt\]. The material parameters were defined in terms of complex refractive indices collected either from measurements[^4] or the literature. We used and We then applied a Lagrange (quadratic) mesh constrained to 0.25 nm in NbN and $\text{NbN}_x\text{O}_y$, and made to be denser at edge XY than $\text{X}^\prime\text{Y}^\prime$ to maintain accuracy when using periodic boundary conditions. Moving to a denser mesh did not improve accuracy. We used a direct linear solver (UMFPACK) to solve for the spatial distribution of electric ($\vec{E}$) field, $\vec{E}(x,y)$.
The absorptance $A$ is proportional to the integral of the time-averaged electric field intensity in the NbN film. Because the electric field does not vary significantly over the thickness of the NbN, we can express $A$ in terms of the cross-sectional electric field $|\vec{E}_{\text{cc}^\prime}(x)|$ in NbN $\big($as pictured in Fig. \[fig.snspd-geometry\](b)$\big)$ through $$A = \int_{-p/2}^{p/2}\,\int_0^{t_\text{NbN}} Q(x,y)\, \text{d}x\, \text{d}y\, \bigg/ \int_{-p/2}^{p/2}\, I_\circ\, \text{d}x$$ where $Q(x,y)$ is the time-average resistive dissipation in the nanowire, and $I_\circ$ is the time-average incident Poynting power density given by the following equation $$I_\circ = \frac{1}{2}\,(\epsilon_\circ/\mu_\circ)^{1/2}\, |\vec{E}_\circ|^2$$ where $|\vec{E}_\circ|$ is the time-averaged incident electric field magnitude, $\epsilon_\circ$ is the permittivity of air, and $\mu_\circ$ is the permeability of air. For our wires which have typical $t_\text{NbN}$ of 4-6 nm, $Q(x,y)\approx Q(x)$ and is given by Ohm’s law to be $$Q(x) = \frac{1}{2}\, \omega\, \text{Im}[\epsilon] |\vec{E_{\text{cc}^\prime}}(x)|^2.$$
Figure \[fig.snspd-geometry\](c) shows the calculated dependence of $A$ on $f$ for constant values of $p$ and electric field polarizations that were either parallel ($||$) or perpendicular ($\perp$) to the periodicity of the nanowires, while Fig. \[fig.snspd-geometry\](d) shows $A$ as a function of $p$. Both figures show that $A_{||}$ is invariant with $p$ when $f$ is kept constant, and that $A$ can be maximized by narrowing the gaps between nanowires. We understand that $A_{||}$ remains constant with varying $p$ because the electric field intensity is continuous across the air-NbN boundaries as a result of tangential $\vec{E}_{\text{cc}^\prime}^{||}$ continuity, as shown in Fig. \[fig.snspd-geometry\](d) inset. Only the fraction of the electric field magnitude $|\vec{E}_{\text{cc}^\prime}^{||}|$ in NbN compared to air (i.e. $f$) that changes $A_{||}$. On the other hand, decreasing $p$ while keeping $f$ constant decreases $A_\perp$. This effect is observed because there are more air-NbN interfaces per unit area for a grating with a smaller pitch than one with a larger pitch, and the boundary conditions dictate a lower $|\vec{E}_{\text{cc}^\prime}^{\perp}|$ in the NbN at each air-NbN interface.
The plots also illustrate an important limitation that $A$ poses to the photon detection process: the maximum $A$ (and therefore also $DDE$) can not exceed 30% with the configuration shown. There are two ways that $A$ can be increased that have already been demonstrated: (1) by using back-illumination (i.e. through sapphire edge $\text{XX}^\prime$) instead of front-illumination (i.e. through air edge $\text{YY}^\prime$), and thereby reducing the index mismatch with NbN, $A$ can be increased up to 45%; (2) by fabricating an optical cavity designed to intensify the field in the NbN nanowires [@rosfjord]. An understanding of the impact of SNSPD geometry and optical polarization on $A$ and $P_\text{R}$ will be helpful in finding other ways to improve $A$.
We will now describe how we determined the optical absorptance of the devices and how the results compared to the model. We will also compare our measurements of the absorptance to the $DDE$ measurements of the same devices.
Absorptance measurements {#sec.expt}
========================
Figure \[fig.snspd-testing-schematic\] shows a schematic of the optical apparatus we used to measure absorptance. Free-space optics were attached to a three-axis, computer-controlled stage (motorized actuator: Newport LTA-HS, motion controller: Newport ESP300). The stage could be affixed to a cryostat specially designed to allow free-space optical coupling to a cold sample, or be used without a cryostat for room-temperature measurements. We mounted a single-mode optical fiber (Corning SMF-28) to the stage that was illuminated by a 1.55-$\upmu$m-wavelength diode laser (Thorlabs S1FC1550) and polarized using a state of polarization (SOP) locker (Thorlabs PL100S). The SOP locker was preferable to a manual polarizer because it provided mechanical isolation to our setup when we varied polarization and permitted automation of the data acquisition process. The polarized light was collimated and split equally into two arms. In one arm, a Ge-based sensor (Thorlabs S122A) connected to a power meter (Thorlabs PM100) was used to measure the incident light intensity, mainly for troubleshooting purposes. In the other arm, a long-working-distance microscope objective (Mitutoyo M Plan APO NIR 20X) was used to focus the light onto a device with a spot-size of $4\, \upmu\text{m}$ full-width half-max (FWHM). Light reflected from the device was collected and measured using another power meter. The transmitted light was measured by a third power meter. We controlled the power meters, SOP locker, and motion controller using the Instrument Control Toolbox (ver. 2.4) for Matlab (ver. 7.2/R2006a) and custom software.
![Schematic of the optical setup used to measure the optical absorptance. A state of polarization (SOP) locker was used to set the input polarization of the incident 1.55-$\upmu$m-wavelength light. Light was focused to a 4 $\upmu$m (FWHM) spot onto a device and the incident, reflected, and transmitted power were measured. The absorptance was calculated from these measurements and earlier calibrations discussed in the main text.[]{data-label="fig.snspd-testing-schematic"}](optical-setup6)
It was necessary to fabricate devices which were specially designed to facilitate both absorptance and $DDE$ measurements. A typical device is pictured in Figure \[fig.sems\] where the active area ($3\, \upmu\text{m} \times 3\, \upmu\text{m}$) of the SNSPD is centrally located within a large grating structure ($30\, \upmu\text{m} \times 10\,\upmu\text{m}$). The small active area facilitated $DDE$ measurements because it enabled better uniformity in electrical response across the device. The larger grating structure, which has the same $p$ and $f$ as the smaller active area, facilitated measurements of $A$ because it closely matched our simulation geometry.[^5] Thus, five groups of devices with identically sized large gratings and small active areas were fabricated with different $(f,\, p\, \text{[nm]})$ combinations of $(\frac{1}{2}, 200)$, $(\frac{1}{3}, 300)$, $(\frac{1}{4}, 400)$, $(\frac{1}{3}, 150)$, and $(\frac{1}{4}, 200)$ on the same chip.
![Scanning electron micrograph of a 50% fill-factor, 200-nm-pitch SNSPD is shown in (a). The central, photon-detecting region of the SNSPD structure outlined in (a) is magnified in (b). The extended parallel grating structure outside the active area was necessary for proximity-effect correction in the fabrication process, but was also useful in expanding the optically testable region of the device.[]{data-label="fig.sems"}](sem_v3)
The procedure for collecting absorptance data was as follows. We first measured the reflected and transmitted power from gold and sapphire in the vicinity of the device. These measurements were repeated over polarizations spaced by 10$^\circ$ on the Poincaré sphere. We calibrated the measured reflected and transmitted power to the theoretical reflectance and transmittance $(R,\, T)$ of gold[^6] $(0.98,\, 0)$ and sapphire $(0.07,\, 0.93)$ calculated from their refractive index values at $\lambda=1.55\,\upmu\text{m}$. We then measured the reflected and transmitted power with the laser centered in the active area of the SNSPD, and calculated $R$, $T$, and $A=1-R-T$. We then repeated our measurements on 10-13 identical devices in each of the five groups of devices.
Figure \[fig.A-vs-device-type\] shows our measurements of $A_{||}$ and $A_\perp$ at room temperature. Room temperature measurements did not differ significantly from measurements of $A$ made at 6 K. We attempted to fit the model by using $w$ and $t_\text{NbN}$ as free parameters, and assuming $t_\text{NbN}$ was between and . Our fit required wire widths that were systematically 10-15% larger than the nominal $w$ used in our electron-beam-lithography pattern. There are many uncertainties in our model parameters that can contribute to this. One cause is that we may have inadvertently exposed the resist adjacent to the intended structure causing $w$ to be larger than desired. We conducted scanning electron microscopy of our nanowires and found that $w$ predicted by the model were within the uncertainty caused by increased secondary electron emission from the edges of the nanowires. However, there are other factors that can contribute to a fitting error. One factor is that the NbN refractive index we used in our simulations may not be accurate because the refractive index was measured on a thicker film (12 nm) than the film we used to fabricated our devices on (4-6 nm). In addition, the films were from different growth batches so may have had slight differences in optical properties. Another factor is that our simple model of infinite gratings may not be sufficient, i.e. a 3D model may be needed. Considering these uncertainties, the qualitative and quantitative fit of our measurements with the model was good.
![Statistical plot of the measured parallel ($||$) and perpendicular ($\perp$) absorptance for devices with different pitch $p$ and fill-factor $f$. Each data symbol represents measurements of 10-13 devices for the polarization that yielded the maximum ($||$) or minimum ($\perp$) absorptance. The arrows indicate the calculated absorptance for structures with fitted wire widths (from left to right) $w$=104 nm, 108 nm, 114 nm, 58 nm, and 55 nm, NbN thickness , and nominal values of pitch $p$.[]{data-label="fig.A-vs-device-type"}](absorptance-vs-device-type_v4_labels_with_sems)
Comparison of device detection efficiency to absorptance
--------------------------------------------------------
A comparison of device detection efficiency to absorptance can answer one important question about our device: is every absorbed photon actually detected? To answer this question, we measured both $DDE$ (using the measurement apparatus described in [@rosfjord] but with front-illumination instead of back-illumination) and $A$ for the same devices. The two quantities are plotted against each other in Fig. \[fig.snspd-testing-DE-vs-A\] where, for reference, we have also plotted lines with constant slope $P_\text{R}$. It should be noted that the value for $DDE$ is measured with a relative accuracy of $\pm 25\%$ due to uncertainties in fiber output power calibration. Some features of this plot were as expected; for example, all of the devices had higher absorptance than device detection efficiency ($P_\text{R}<1$). But there were other features that were unexpected. We will now discuss these other features and see how they yield an unexpected new insight into the microscopic physics of SNSPDs.
![Plot of device detection efficiency as a function of absorptance for the same devices. Dotted lines have a constant slope given by $P_\text{R}$, the probability of resistive state formation.[]{data-label="fig.snspd-testing-DE-vs-A"}](DE_vs_A_legend_v3)
There are two remarkable features in the data plotted in Fig. \[fig.snspd-testing-DE-vs-A\]. First, $P_\text{R}$ had a peak value of 0.85$\genfrac{}{}{0pt}{3}{+0.15}{-0.21}$ for parallel polarization. The natural question one can ask is what happens to absorbed photons that do not lead to detection events? One possible explanation is that the devices were in some way not biased at their “true” critical currents, i.e. that they were constricted in some way so that the absorption of photons does not cause a voltage pulse. Another explanation is that there is some other absorbing medium that was not accounted for. We aim to address this issue in future experiments.
A second remarkable feature of the data in Fig. \[fig.snspd-testing-DE-vs-A\] is that $P_\text{R}$ for parallel polarization ($0.55\genfrac{}{}{0pt}{3}{+0.14}{-0.14} < P_{\text{R},||} < 0.85\genfrac{}{}{0pt}{3}{+0.15}{-0.21}$) is different than $P_\text{R}$ for perpendicular polarization ($ 0.35\genfrac{}{}{0pt}{3}{+0.09}{-0.09} < P_{\text{R},\perp} < 0.75\genfrac{}{}{0pt}{3}{+0.19}{-0.19}$). While we do not yet completely understand the origin of the disparity between $P_{\text{R},||}$ and $P_{\text{R},\perp}$, we have developed a model that can form a starting point for further analysis.
The main assumption of this model is a position-dependent (but polarization-independent) probability *density* of resistive state formation, $\psi(x)$, where $x$ is the distance across the nanowire. The second element of this model is the time-averaged resistive dissipation $Q(x)$ which we know to be both position-dependent and polarization-dependent. These two elements can be combined to give the device detection efficiency and absorptance for parallel and perpendicular polarization: $$\label{eq.DE-psi}
DDE = \int_{-p/2}^{p/2}\int_0^{t_\text{NbN}} \psi(x) \frac{Q(x)}{I_\circ}\, \text{d}x\, \text{d}y.$$ We can use this equation, Eq. , and a calculated $Q(x)$ to find a $\psi(x)$ that gives a different $P_\text{R}$ for parallel and perpendicular polarizations. Using our data, we found that $\psi(x)$ needs to be larger at the edges of the nanowire than in the center to explain $P_{\text{R},||}>P_{\text{R},\perp}$, however the five geometries we investigated did not provide enough resolution to fit $\psi(x)$ to a specific shape at this point.
Verification of the model through measurement of the cavity enhancement factor {#sec.wavelength}
==============================================================================
In the previous section, we verified that our model predicted the absorptance for our structures to within experimental uncertainties. But this verification was done only at a single wavelength of 1550 nm, mainly because all of our optical components, i.e., the laser, fiber, lenses, SOP locker, microscope objective, beam splitter, collimator, and photodiodes were optimized for that wavelength. In order to verify our modeling parameters for a range of wavelengths, we measured the cavity enhancement factor $\mathcal{E}$ for SNSPDs over a range of wavelengths. The cavity enhancement factor $\mathcal{E}$ is the ratio of the absorptance of an SNSPD with an integrated cavity to the absorptance without a cavity and depends on the refractive indices of the materials and the geometry. $\mathcal{E}$ is also the factor by which the intensity of the electric field in NbN, and therefore $Q(x)$, is increased due to the presence of a cavity. In view of Eq. , and because $\psi(x)$ should not depend on the intensity, we can determine $\mathcal{E}$ by measuring $DDE$ for an SNSPD with a cavity and the $DDE$ of an SNSPD without a cavity and taking the ratio of these two quantities for each wavelength.
We generated radiation spanning the range from 600-1700 nm using a supercontinuum source (Toptica photonics). This source consisted of a power-amplified, modelocked fiber laser with 100 fs pulses at 1550 nm, which could either be coupled into a highly-nonlinear step-index fiber (generating a frequency comb from $\sim$600-1050 nm) or first doubled using a periodically-poled Lithium-niobate crystal and then coupled into a photonic crystal fiber (Blazed photonics) (generating a frequency comb from $\sim$ 1150-1700 nm). We used a Pellin-Broca prism to select a wavelength band out of one or the other of these outputs (with a FWHM $\Delta \lambda/\lambda \sim 50$) which we then coupled into an optical fiber (SMF-28, single mode from $\sim$1200-1600 nm). The wavelength of the light output from the fiber could then be tuned by rotating the prism.
We used this tunable output to measure the enhancement $\mathcal{E}$. The most obvious way to do this would simply be to make calibrated $DDE$ measurements at all wavelengths before and after adding the cavity. However, we chose not to attempt this, since we had no way to accurately calibrate optical power levels at or near the single-photon level over such a wide wavelength range. To avoid the need for this calibration, we instead measured only detection efficiency ratios for pairs of detectors side-by-side. To do this, we fabricated a chip with 225 pairs of 3$\times$3.3 $\upmu$m detectors, where the two in each pair could be read out separately, and were spaced by only 100 $\upmu$m. We measured critical currents and room-temperature resistances of all 450 detectors, and identified $\sim$ 50 pairs for which these values were within 5% of each other. We then measured (calibrated) $DDE$s at 1550 nm for this subset of detectors, and further narrowed the experimental sample to the pairs having $DDE$s within 5% of each other. Then, we added optical cavities to one detector of each pair, and re-measured at 1550 nm. Although we commonly observe that $DDE$ for a given device can vary between cooldowns (particularly if some additional processing has occurred), these variations are nearly always strongly correlated with a variation of the critical current (a reduced $DDE$ correlates with a reduced critical current). So, we selected from our pairs of detectors only those where the critical currents remained the same to within 5% after the addition of cavities, and where the detector which did not have a cavity added had the same $DDE$ as previously to within 5%. Lastly, we further reduced our experimental sample by selecting only the subset of devices which were relatively unconstricted [@jamie]. In the present context, this corresponded to those devices with $DDE$ at 1550 nm $>$17% for front illumination. Our final sample consisted of five detector pairs. For these ten detectors, we measured count rates at each of a sequence of wavelengths spanning the entire accessible range. By measuring both detectors in a pair in succession without adjusting the source in any way, we ensured that the optical power incident on the two detectors was identical, and therefore that their count rates could be compared directly to give $\mathcal{E}$. This precaution was important since the optical power obtainable in a given wavelength band was not repeatable, nor was the coupling into the SMF-28 optical fiber, which was not single-mode over a fraction of our wavelength range. Given the latter issue, we restricted ourselves to the wavelength range 700-1700 nm where the fiber output could be fairly well polarized (extinction ratio $>$ 20).
Figure \[fig.snspd-testing-DE-vs-wavelength\] shows the enhancement measured in this manner, and a calculation of $\mathcal{E}$ as a function of wavelength. We can see that the data has good experimental agreement with the calculation. Inset Fig. \[fig.snspd-testing-DE-vs-wavelength\](a) shows the geometry that was used in the calculation while Fig. \[fig.snspd-testing-DE-vs-wavelength\](b) shows a plot of the real and imaginary parts of refractive index for NbN, NbN$_x$O$_y$, and HSQ[^7].
![Plot of enhancement factor, $\mathcal{E}$ as a function of wavelength $\lambda$. The dotted line shows the calculation that was carried out on the unit cell shown in inset (a) using values for the refractive indices shown in inset (b).[]{data-label="fig.snspd-testing-DE-vs-wavelength"}](DE_ratio_vs_wavelength_with_insets_v2)
Conclusion
==========
In this paper, we reported the first measurements of the absorptance of SNSPDs and showed how $A$ changed with the fill-factor and pitch of SNSPDs and optical polarization. We confirmed that our numerical model made accurate predictions for a range of geometry, polarization, and wavelength. We found that 200-nm-pitch, 50% fill-factor devices had an average absorptance of 21% for light polarized parallel to the nanowires, and only 10% for perpendicularly-polarized light. This disparity in polarization-sensitive absorptivity was even more evident in lower fill-factor and narrow wire-width devices, where we measured that parallel-polarized photons were more than 5 times as likely to be absorbed over perpendicularly polarized photons. We also found that potentially, some absorbed photons do not result in detection events, and that this quantity is smaller for photons with an orthogonal polarization. These results present new challenges that need to be understood and overcome if higher efficiency devices will be possible.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Prof. Hank Smith for the use of his facilities and equipment, James Daley and Mark Mondol for technical assistance, and Ayman Abouraddy, Scott Hamilton, Rich Molnar, Prof. Steven Johnson, and Prof. Terry Orlando for helpful discussions. The authors would also like to thank Jeffrey Stern for useful discussions about using effective index theory for optical modeling and for pointing out the formula for $n_\text{eff}$ for parallel electric-field polarization. This work made use of MIT’s shared scanning-electron-beam-lithography facility in the Research Laboratory of Electronics (SEBL at RLE).
This work was sponsored in part by the United States Air Force under Air Force Contract \#FA8721-05-C-0002 and IARPA. Opinions, interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the United States Government.
[^1]: For an electric field polarized parallel to the wires in a subwavelength grating, an accurate result for the $A_{||}$ can be obtained with the Fresnel equations where an effective index $n_\text{eff} = ((1-f)\,n_\text{air}^2 + f\,n_\text{NbN}^2)^{1/2}$ is used for the thin film consisting of NbN subwavelength gratings ($n_\text{air}=1$, $n_\text{NbN}=5.23-i\,5.82$, and $f$ is the fill-factor). A simple effective index model that only depends on $f$ will not work for perpendicular polarization since $A_\perp$ depends on both $f$ and $p$.
[^2]: While our fabrication process [@joel; @rosfjord] leaves behind 10-40 nm of residual resist on top of the nanowires, we found that including the resist in our geometry did not affect our results.
[^3]: TEM imaging services were provided by Materials Analytical Services, Inc.
[^4]: Measurements of the refractive indices of NbN and $\text{NbN}_x\text{O}_y$ made at room temperature by J. A. Woolam, Inc. using spectroscopic ellipsometry on a 12-nm-thick b film deposited on a sapphire wafer.
[^5]: 98% of the laser intensity in a 4 $\upmu$m FWHM laser spot was incident on the large grating structure.
[^6]: The patterned gold film was approximately 100 nm thick. We used bulk values for the complex refractive index of gold found in [@palik]. For a thin film thicker than 30 nm, the bulk refractive index can be used in this wavelength range [@reale].
[^7]: Measurements of the refractive indices of NbN and $\text{NbN}_x\text{O}_y$ were made at room temperature by J. A. Woolam, Inc. using spectroscopic ellipsometry on a 12-nm-thick b film deposited on a sapphire wafer. Measurements of refractive index of HSQ were also performed by J. A. Woolam, Inc. using spectroscopic ellipsometry. $n$ and $k$ for gold were taken from Ref. [@palik]
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[**EFFECT OF FINITE NUCLEON MASS ON PRIMORDIAL NUCLEOSYNTHESIS**]{}\
Geza Gyuk$^{1}$ and Michael S. Turner$^{1,2,3}$\
$^1$[*Department of Physics,\
The University of Chicago, Chicago, IL 60637-1433*]{}\
$^2$[*NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, IL 60510-0500*]{}\
$^3$[*Department of Astronomy & Astrophysics,\
Enrico Fermi Institute, The University of Chicago, Chicago, IL 60637-1433*]{}\
**ABSTRACT**
We have modified the standard code for primordial nucleosynthesis to include the effect of finite nucleon mass on the weak-interaction rates as calculated by Seckel [@seckel]. We find a small, systematic increase in the $^4$He yield, $\Delta Y \simeq 0.0057\,Y$, which is insensitive to the value of the baryon-to-photon ratio $\eta$ and slightly larger than Seckel’s estimate. The fractional changes in the abundances of D, $^3$He, and $^7$Li range from 0.08% to 3% for $10^{-11}\le \eta \le 10^{-8}$.
Introduction
============
Primordial nucleosynthesis is one of the cornerstones of the hot big-bang cosmology. The agreement between its predictions for the abundances of D, $^3$He, $^4$He and $^7$Li and their inferred primordial abundances provides its earliest, and perhaps most, stringent test. Further, big-bang nucleosynthesis has been used to provide the best determination of the baryon density [@ytsso; @walker] and to test particle-physics theories, e.g., the stringent limit to the number of light neutrino species [@nulimit].
The scrutiny of primordial nucleosynthesis, both on the theoretical side and on the observational side, has been constant: Reaction rates have been updated and the effect of their uncertainties quantified [@rates], finite-temperature corrections have been taken into account [@dicusetal], the effect of inhomogeneities in the baryon density explored [@matthews], and the slight effect of the heating of neutrinos by $e^\pm$ annihilations has been computed [@dt]; the primordial abundance of $^7$Li has been put on a firm basis [@lithium], the production and destruction of D and $^3$He have been studied carefully [@dhe3], and astrophysicists now argue about the third significant figure in the primordial $^4$He abundance [@he4].
A measure of the progress in this endeavour is provided by the shrinking of the “concordance region” of parameter space. The predicted and measured primordial abundances agree provided: the baryon-to-photon ratio lies in the narrow interval $3 \times 10^{-10}\la \eta \la
4\times 10^{-10}$ and the equivalent number of light neutrino species $N_\nu \la 3.3$ [@walker]. The shrinking of the concordance interval motivates the study of smaller and smaller effects.
Seckel [@seckel] has recently calculated the corrections to the weak-interaction rates that arise from taking account of the finite nucleon mass (in the standard code these rates are computed in the infinite-nucleon-mass limit). The corrections involve terms of order $m_e/m_N$, $T/m_N$, and $Q/m_N$, which are all of the order of 0.1%. Here $m_e$ is the electron mass, $m_N$ is the nucleon mass, $T\sim {\cal O}(\MeV )$ is the temperature during the epoch of nucleosynthesis, and $Q=m_n-m_p=1.293\MeV$ is the neutron-proton mass difference. The weak interaction rates govern the neutron-to-proton ratio and thereby are crucial to the outcome of nucleosynthesis; e.g., the mass fraction of $^4$He produced is roughly twice the neutron fraction at the time nucleosynthesis commences ($T\sim 0.07\MeV$).
The net effect of the finite-nucleon-mass corrections is to decrease the weak rates by about 1% around the time of nucleosynthesis. Based upon a simple code that follows the neutron fraction Seckel estimated that the cumulative effect of all the corrections increase the mass fraction of $^4$He synthesized by $\Delta Y \approx
0.0012$. Because the third significant figure of the primordial $^4$He abundance is now very relevant, we decided to incorporate the finite-nucleon-mass corrections to the weak-interaction rates into the standard nucleosynthesis code [@kawano]. In the next Section we describe the modifications we made; we finish with a discussion of our results for the change in the yield of $^4$He, which is slightly larger than Seckel’s estimates, and for the changes in the yields of the other light elements.
Modifications to the Standard Code
==================================
Role of weak interactions
-------------------------
The weak interactions that interconvert neutrons and protons, $n\leftrightarrow p + e +\nu$, $n+e\leftrightarrow
p+\nu$, and $n+\nu \leftrightarrow p+e$, play a crucial role as they govern the neutron fraction, and the neutron fraction ultimately determines the amount of nucleosynthesis that takes place. (Here and throughout we use $e$ to indicate electron or positron, and $\nu$ to indicate electron neutrino or antineutrino; the appropriate particle or antiparticle designation follows from charge and lepton number conservation.)
The weak-interaction rate per nucleon is very roughly $\Gamma_{n\rightarrow p}\sim \Gamma_{p\rightarrow n}
\sim G_F^2T^5$, while the expansion rate of the Universe $H\sim
T^2/\mpl$; here $G_F = 1.1664\times 10^{-5}\GeV^{-2}$ is the Fermi constant and $\mpl =1.22\times 10^{19}\GeV$ is the Planck mass. At temperatures greater than about $0.8\MeV$, $\Gamma$ is greater than $H$, and the neutron-to-proton ratio tracks its equilibrium value $$\left({n\over p}\right)_{\rm EQ} =
(m_n/m_p)^{3/2}e^{-Q/T}.$$ Two comments are in order. First, in the infinite-nucleon-mass limit the prefactor is unity; taking account of this factor tends to increase the equilibrium ratio by about 0.2%, suggesting that the neutron abundance and final $^4$He abundance should be correspondingly larger. Second, at a temperature of about $1\MeV$, the neutrino and photon temperatures begin to deviate as neutrinos decouple from the electromagnetic plasma ($e^\pm$, $\gamma$) and $e^\pm$ annihilations begin to heat the photons relative to the neutrinos as $e^\pm$ pairs transfer their entropy to the photons. When this happens the equilibrium value of the neutron-to-proton ratio is no longer given by such a simple formula.
When the temperature of the Universe drops below about $0.8\MeV$ the weak-interaction rate are no longer greater than the expansion rate and the neutron-to-proton ratio ceases to track its equilibrium value, and is said to “freeze out.” Until nucleosynthesis begins in earnest ($T\sim 0.07\MeV$) the neutron-to-proton ratio decreases slowly due to weak interactions (especially neutron decay). For $\eta\sim
3\times 10^{-10}$ it decreases from about 1/6 at freeze out to about 1/7 when nucleosynthesis begins, finally resulting in a mass fraction of $^4$He equal to about 25%. For a detailed discussion of the physics of primordial nucleosynthesis see Refs. [@Weinberg; @KT].
In the infinite-nucleon-mass limit the rates (per particle) for the six reactions that interconvert neutrons and protons are given by [@Weinberg] $$\begin{aligned}
\label{eq:weak}
\lambda (n+\nu \rightarrow p+e) & = &{\cal C}\,\int_Q^\infty\,
{p\,E\,(E-Q)^2\,dE \over
[e^{(E-Q)/T_\nu} + 1]\,[e^{-E/T} +1]} ;\nonumber\\
\lambda (n+ e \rightarrow p+\nu ) & = & {\cal C}\,\int_{m_e}^\infty\,
{p\,E\,(E+Q)^2\,dE \over
[e^{-(E+Q)/T_\nu} + 1]\,[e^{E/T} +1]} ;\nonumber\\
\lambda (n \rightarrow p+e+\nu ) & = & {\cal C}\,\int_{m_e}^Q\,
{p\,E\,(E-Q)^2\,dE \over
[e^{(E-Q)/T_\nu} + 1]\,[e^{-E/T} +1]} ;\nonumber\\
\lambda (p+e\rightarrow n+\nu ) & = & {\cal C}\,\int_Q^\infty\,
{p\,E\,(E-Q)^2\,dE \over
[e^{(Q-E)/T_\nu} + 1]\,[e^{E/T} +1]} ;\nonumber\\
\lambda (p+\nu \rightarrow n+e) & = & {\cal C}\,\int_{m_e}^\infty\,
{p\,E\,(E+Q)^2\,dE \over
[e^{(E+Q)/T_\nu} + 1]\,[e^{-E/T} +1]} ;\nonumber\\
\lambda (p+e+\nu \rightarrow n) & = & {\cal C}\,\int_{m_e}^Q\,
{p\,E\,(E-Q)^2\,dE \over
[e^{(Q-E)/T_\nu} + 1]\,[e^{E/T} +1]} ;\end{aligned}$$ where $T$ denotes the photon temperature, $T_\nu$ the neutrino temperature, and the common factor $${\cal C} = {G_F^2 \cos^2\theta_C (1+3c_A^2)\over
2\pi^3}\,f_{\rm EM}.$$ Here $\theta_C$ is the Cabibbo angle ($\cos\theta_C =0.975$), $c_A = 1.257$ is the ratio of the axial vector to vector coupling of the nucleon, and $f_{\rm EM} \sim 1.08$ quantifies the radiative and Coulomb corrections, which for our purposes here are not important (for more details see Ref. [@dicusetal]).
Finite-mass corrections
-----------------------
Seckel has calculated the corrections to the weak rates due to finite nucleon mass [@seckel]. He has grouped them into three categories; the corrections due to: (i) thermal motion of the target nucleon; (ii) the recoil energy of the outgoing nucleon; and (iii) weak magnetism. The net effect of these corrections is to reduce the weak rates by about 1% around the time the neutron-to-proton ratio freezes out. This in turn causes freeze out to occur slightly earlier, at a higher value of the neutron-to-proton ratio, resulting in an increase in $^4$He production.
The corrections to the rates for neutron decay and inverse decay are the simplest, just the factor that accounts for time dilation, $$\lambda (n\leftrightarrow p+e+\nu ) \longrightarrow
\left( 1 -{3\over 2}{T\over m_N}\right) \lambda (n\leftrightarrow p+e +\nu ).$$ Since the neutron lifetime is used to normalize all the weak rates, the recoil effect is automatically taken into account.
The corrections to the weak interactions that involve $2\leftrightarrow 2$ scatterings are organized in a way such that the integrands in the previous expressions, cf. Eqs. (\[eq:weak\]), are simply multiplied by a correction factor,[^1] $$1-\delta_n + \gt_{\rm wm} + \gt_{\rm rec} + \gt_{\rm th}
+\gt_{\rm etc},$$ where $\delta_n = -0.00201$ “uncorrects” the neutron lifetime for recoil effects, which are taken into account in the term $\gt_{\rm rec}$, $\gt_{\rm wm}$ is the weak magnetism correction, $\gt_{\rm th}$ is the thermal correction, and $\gt_{\rm etc}$ is the sum of three smaller (by a factor of 100) and less important terms, which we have included in our computations, but which have negligible effect.
The weak magnetism, recoil, and thermal corrections are given by [@seckel] $$\begin{aligned}
\gt_{\rm wm} & = & {2c_Af_2\over 1+3c_A^2}\,
{E_3k_1^2+E_1k_3^2\over m_NE_1E_3}; \\
\gt_{\rm rec} & = & {\gt_{\rm wm}\over f_2}
-{2E_1k_3^2 +E_3(k_1^2+k_3^2) \over 2m_Nk_3^2}
{m_1^2-m_3^2-Q^2 \over 2(1+3c_A^2)m_NE_3} \nonumber\\
& \ & +{c_A^2\over 1+3c_A^2}\,{6E_1^2E_3 -6E_1E_3^2
-3E_1k_1^2-4E_3k_1^2-E_1k_3^2 \over 2m_NE_1E_3} ;\\
\gt_{\rm th} & = & \left({T\over m_N}\right)\,
\left[ {3E_1^2 +2k_1^2\over 2E_1E_3} +{3k_1^2E_1 +3E_1^2E_3
+2k_1^2E_3 \over 2E_1k_3^2} -{k_1^2E_3^2\over 2k_3^4}\right];\end{aligned}$$ subscript 1 (3) refers to the incoming (outgoing) lepton, subscript 2 (4) refers to the incoming (outgoing) nucleon, $E_i$ is the energy of the $i$th particle, $k_i$ is its momentum, and $f_2=\pm 3.62$ where $+$ applies to reactions with leptons and $-$ to reactions with antileptons. For the three smaller terms embodied in $\gt_{\rm etc}$ we refer the reader to Ref. [@seckel].
Three small points; the limits of integration are those in the infinite-nucleon-mass limit, cf. Eqs. (\[eq:weak\]), with one exception (see below). In Eqs. (\[eq:weak\]) the integrals are performed over the energy of the electron; the neutrino energy is just that of the electron plus or minus $Q$. The integral of the final term in $\gt_{\rm th}$, which is proportional to $k_3^{-4}$, diverges for the reaction $p+\nu \rightarrow
n+e$ for the infinite-nucleon-mass limits because $k_3\rightarrow 0$ at threshold. For this term, and only this term, the lower limit of integration involves the finite-nucleon-mass kinematics: the minimum electron momentum is: $(k_3)_{\rm min} = (m_e/m_N)^{1/2}(Q+m_e)$ [@seckel2]. Because essentially all of the integral accumulates near $(k_3)_{\rm min}$ this term can be integrated analytically: $$\left({T\over m_N}\right)\int {p\,E\,(E+Q)^2\,dE\over [e^{(E+Q)/T_\nu}+1]
[e^{-E/T}+1]} {k_1^2E_3^2\over 2k_3^4}
= {(T/m_N)(Q+m_e)^3m_e^{3/2}m_N^{1/2}\over 2
[e^{(Q+m_e)/T_\nu}+1][e^{-m_e/T}+1]} .$$
Finally, we mention that Seckel [@seckel] has derived linear fits to the perturbed weak-interaction scattering rates: $$\begin{aligned}
\lambda (n\rightarrow p) & \rightarrow & \left[ 1 -\delta_n -0.00185
-0.01032\left({T\over m_N}\right)\right]\lambda (n\rightarrow p);\\
\lambda (p\rightarrow n) & \rightarrow & \left[ 1-\delta_n +0.00136
-0.01067\left({T\over m_N}\right)\right]\lambda (p\rightarrow n);\end{aligned}$$ where the first linear correction applies to the $2\leftrightarrow 2$ reactions that convert neutrons to protons, $n+\nu\rightarrow p+e$ and $n+e\rightarrow p+\nu$, and is valid for $2\MeV \ga T \ga 0.3 \MeV$, and the second applies to the $2\leftrightarrow 2$ reactions that convert protons to neutrons, $p+\nu\rightarrow n+e$ and $p+e\rightarrow n+\nu$, and is valid for $2\MeV\ga T \ga 0.7\MeV$.
Results and Conclusions
=======================
We have modified the integrands in the subrountines that compute the weak-interaction rates in the standard nucleosynthesis code [@kawano] as outlined above. For comparison, we have also modified the standard code just using Seckel’s linear fit, which is much easier to implement since the usual rate is multiplied by a factor outside the integral. Our results, which were obtained by taking three massless neutrino species and a mean neutron lifetime of $889\sec$, are shown in Figs. 1 and 2.
In Fig. 1 the change in the mass fraction of $^4$He and 0.0057 times the $^4$He yield are shown. While over the range $\eta = 10^{-11}$ to $10^{-8}$ $\Delta Y$ varies from about 0.0004 to 0.0015, $\Delta Y$ is remarkably close to $0.0057\,Y$. We also computed the change in $^4$He yield using Seckel’s linear fit; for the values of $\eta$ above, $\Delta Y/Y$ was slightly lower, by about 0.05%. Neglecting the subdominant terms ($\gt_{\rm etc}$) decreases $\Delta Y/Y$ from about 0.57% to about 0.53%.
In Fig. 2 we show the fractional changes in the abundances of D, $^3$He, and $^7$Li (relative to H). These changes, over the same interval in baryon-to-photon ratio, range from 0.08% to almost 3%. Since the inferred primeval abundances of these elements are no where near as well known as that of $^4$He, these changes are of little relevance at present.
1.5cm We thank David Seckel for many helpful conversations. This work was supported in part by the DOE (at Chicago and Fermilab), by the NASA through NAGW-2381 (at Fermilab), and GG’s NSF predoctoral fellowship.
2 cm
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[**FIGURE CAPTIONS**]{}
[**Figure 1:**]{} The change in the yield of $^4$He as a function of the baryon-to-photon ratio $\eta$, and, for comparison, 0.0057 times the $^4$He mass fraction.
[**Figure 2:**]{} The fractional changes in the yields of D (solid curve), $^3$He (broken curve), and $^7$Li (dotted curve) as a function of the baryon-to-photon ratio $\eta$.
[^1]: Our notation differs very slightly from that of Ref. [@seckel]; $\gt_i = \gamma_i/\gamma_0$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study a superlinear and subcritical Kirchhoff type equation which is variational and depends upon a real parameter $\lambda$. The nonlocal term forces some of the fiber maps associated with the energy functional to have two critical points. This suggest multiplicity of solutions and indeed we show the existence of a local minimum and a mountain pass type solution. We characterize the first parameter $\lambda_0^*$ for which the local minimum has non-negative energy when $\lambda\ge \lambda_0^*$. Moreover we characterize the extremal parameter $\lambda^*$ for which if $\lambda>\lambda^*$, then the only solution to the Kirchhoff equation is the zero function. In fact, $\lambda^*$ can be characterized in terms of the best constant of Sobolev embeddings. We also study the asymptotic behavior of the solutions when $\lambda\downarrow 0$.'
address: ' Instituto de Matemática e Estatística. Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil'
author:
- Kaye Silva
bibliography:
- 'Ref.bib'
title: 'The bifurcation diagram of an elliptic Kirchhoff-type equation with respect to the stiffness of the material '
---
Introduction
============
In this work we study the following Kirchhoff type equation $$\label{p}
\left\{
\begin{aligned}
-\left(a+\lambda\int |\nabla u|^2\right)\Delta u&= |u|^{\gamma-2}u &&\mbox{in}\ \ \Omega, \\
u&=0 &&\mbox{on}\ \ \partial\Omega,
\end{aligned}
\right.$$ where $a>0$, $\lambda>0$ is a parameter, $\Delta$ is the Laplacian operator and $\Omega\subset \mathbb{R}^3$ is a bounded regular domain. Let $H_0^1(\Omega)$ denote the standard Sobolev space and $\Phi_\lambda:H_0^1(\Omega)\to \mathbb{R}$ the energy functional associated with (\[p\]), that is $$\label{energyfunctional}
\Phi_\lambda(u)=\frac{a}{2}\int |\nabla u|^2+\frac{\lambda}{4}\left(\int |\nabla u|^2\right)^2-\frac{1}{\gamma}\int |u|^\gamma.$$ We observe that $\Phi_\lambda$ is a $C^1$ functional. By definition a solution to equation (\[p\]) is a critical point of $\Phi_\lambda$. Our main result is the following
\[T\] Suppose $\gamma\in(2,4)$. Then there exist parameters $0<\lambda_0^*<\lambda^*$ and $\varepsilon>0$ such that:
1. For each $\lambda\in(0,\lambda^*]$ problem has a positive solution $u_\lambda$ which is a global minimizer for $\Phi_\lambda$ when $\lambda\in(0,\lambda_0^*]$, while $u_\lambda$ is a local minimizer for $\Phi_\lambda$ when $\lambda\in(\lambda_0^*,\lambda^*)$. Moreover $\Phi_{\lambda}''(u_{\lambda})(u_{\lambda},u_{\lambda})>0$ for $\lambda\in(0,\lambda^*)$ and $\Phi_{\lambda^*}''(u_{\lambda^*})(u_{\lambda^*},u_{\lambda^*})=0$.
2. For each $\lambda\in(0,\lambda_0^*+\varepsilon)$ problem has a positive solution $w_\lambda$ which is a mountain pass critical point for $\Phi_\lambda$.
3. If $\lambda\in(0,\lambda_0^*)$ then $\Phi_\lambda(u_\lambda)<0$ while $\Phi_{\lambda_0^*}(u_{\lambda_0^*})=0$ and if $\lambda\in (\lambda_0^*,\lambda^*]$ then $\Phi_\lambda(u_\lambda)>0$.
4. $\Phi_\lambda(w_\lambda)>0$ and $\Phi_\lambda(w_\lambda)>\Phi_\lambda(u_\lambda)$ for each $\lambda\in(0,\lambda_0^*+\varepsilon)$.
5. If $\lambda>\lambda^*$ then the only solution $u\in H_0^1(\Omega)$ to the problem (\[p\]) is the zero function $u=0$.
Kirchhoff type equations have been extensively studied in the literature. It was proposed by Kirchhoff in [@Kir] as an model to study some physical problems related to elastic string vibrations and since then it has been studied by many author, see for example the works of Lions [@Lions], Alves et al. [@clafra], Wu et al. [@wuch], Zhang and Perera [@perzha] and the references therein. Physically speaking if one wants to study string or membrane vibrations, one is led to the equation , where $u$ represents the displacement of the membrane, $|u|^{p-2}u$ is an external force, $a$ and $\lambda$ are related to some intrinsic properties of the membrane. In particular, $\lambda$ is related to the Young modulus of the material and it measures its stiffness.
Our main interest here is to analyze equation with respect to the parameter $\lambda$ (stiffness) and provide a description of the bifurcation diagram. To this end, we will use the fibering method of Pohozaev [@poh] to analyse how the Nehari set (see Nehari [@neh; @neh1]) change with respect to the parameter $\lambda$ and then apply this analysis to study bifurcation properties of the problem (\[p\]) (see Chen et al. [@wuch] and Zhang et al. [@zhangnieto]). In fact, the extremal parameter $\lambda^*$ (see Il’yasov [@ilyasENMM]) which appears in the Theorem \[T\] can be characterized variationally by $$\lambda^*=C_{a,\gamma}\sup\left\{\left(\displaystyle\frac{\left(\int |u|^\gamma\right)^{\frac{1}{\gamma}}}{\left(\int |\nabla u|^2\right)^{\frac{1}{2}}}\right)^{\frac{2\gamma}{\gamma-2}}: u\in H_0^1(\Omega)\setminus\{0\}\right\},$$ where $C_{a,\gamma}$ is some positive constant. One can easily see from the last expression that $\lambda^*=C_{a,\gamma}S_\gamma^{\frac{2\gamma}{2-\gamma}}$, where $S_\gamma$ is best Sobolev constant for the embedding $H_0^1(\Omega)\hookrightarrow L^\gamma(\Omega)$.
In this work the extremal parameter $\lambda^*$ has the important role that if $\lambda>\lambda^*$ then the Nehari set is empty while if $\lambda\in(0,\lambda^*)$ then the Nehari set is not empty. Another interesting paramenter is $\lambda_0^*<\lambda^*$ which is characterized by the property that if $\lambda\in (0,\lambda_0^*)$, then $\inf_{u\in H_0^1(\Omega)}\Phi_\lambda(u)<0$ while if $\lambda\ge \lambda_0^*$ the infimum is zero. When $\lambda\in (0,\lambda_0^*]$ one can easily provide a Mountain Pass Geometry and a global minimizer for the functional $\Phi_\lambda$. Although here we characterize $\lambda_0^*$ variationally, one can see that the parameter $a^*$ defined in Theorem 1.3 (ii) of Sun and Wu [@sunwu] serves to the same purpose as $\lambda_0^*$ and hence our result for $\lambda\in(0,\lambda_0^*)$ is not new, however, when $\lambda>\lambda_0^*$ we could not find this result in the literature and in this case we need to provide some finer estimates on the Nehari sets in order to solve some technical issues to obtain again a Mountain Pass Geometry and a local minimizer for the functional $\Phi_\lambda$.
The hypothesis $\gamma\in(2,4)$ has the fundamental role that it forces the problem to be superlinear, subcritical and it allows the existence of fiber maps with two critical points. The existence of these kinds of fiber maps implies multiplicity of solutions (at least two solutions) and once for $\lambda>\lambda^*$ there is no solution at all, the parameter $\lambda^*$ is a bifurcation point where these solutions collapses. We refer the reader to the recently works of Siciliano and Silva [@gaeka], Il’yasov and Silva [@YaKa], Silva and Macedo [@KaAb], where the extremal parameters of some indefinite nonlinear elliptic problems were analyzed.
Concerning the asymptotic behavior of the solutions when $\lambda\downarrow 0$ we prove the following
\[T2\]There holds
1. $\Phi_\lambda(u_\lambda)\to -\infty$ and $\|u_\lambda\|\to\infty$ as $\lambda\downarrow 0$.
2. $w_\lambda \to w_0$ in $H_0^1(\Omega)$ where $w_0\in H_0^1(\Omega)$ is a mountain pass critical point associated to the equation $-a\Delta w=|w|^{p-2}w$.
This work is organized as follows: In Section \[S2\] we provide some definitions and prove technical results which will be used in the next sections. In Section \[S3\] we show the existence of local minimizers for the functional $\Phi_\lambda$. In Section \[S4\] we prove the existence of a mountain pass critical point for the functional $\Phi_\lambda$. In Section \[S5\] we prove Theorem \[T\]. In Section \[S6\] we prove Theorem \[T2\]. In Section \[S7\] we provide a picture detailing the bifurcation diagram with respect to the energy and make some conjectures and in the Appendix we prove some auxiliary results.
Technical Results {#S2}
=================
We denote by $\|u\|$ the standard Sobolev norm on $H_0^1(\Omega)$ and $\|u\|_\gamma$ the $L^\gamma(\Omega)$ norm. It follows from (\[energyfunctional\]) that $$\Phi_\lambda(u)=\frac{a}{2}\|u\|^2+\frac{\lambda}{4}\|u\|^4-\frac{1}{\gamma}\|u\|_\gamma^\gamma,\ \forall\ u\in H_0^1(\Omega).$$ For each $\lambda>0$ consider the Nehari set $$\mathcal{N}_\lambda=\{u\in H_0^1(\Omega)\setminus\{0\}:\ \Phi_\lambda'(u)u=0 \}.$$ To study the Nehari set we will make use of the fiber maps: for each $\lambda>0$ and $u\in H_0^1(\Omega)\setminus\{0\}$ define $\psi_{\lambda,u}:(0,\infty)\to \mathbb{R}$ by $$\psi_{\lambda,u}(t)=\Phi_\lambda(tu).$$ It follows that $$\mathcal{N}_\lambda=\{u\in H_0^1(\Omega)\setminus\{0\}:\ \psi'_{\lambda,u}(1)=0 \}.$$ We divide the Nehari set into three disjoint sets as follows: $$\mathcal{N}_\lambda=\mathcal{N}_\lambda^+\cup \mathcal{N}_\lambda^0\cup \mathcal{N}_\lambda^-,$$ where $$\mathcal{N}_\lambda^+=\{u\in H_0^1(\Omega)\setminus\{0\}:\ \psi'_{\lambda,u}(1)=0,\ \psi''_{\lambda,u}(1)>0 \},$$ $$\mathcal{N}_\lambda^0=\{u\in H_0^1(\Omega)\setminus\{0\}:\ \psi'_{\lambda,u}(1)=0,\ \psi''_{\lambda,u}(1)=0 \},$$ and $$\mathcal{N}_\lambda^-=\{u\in H_0^1(\Omega)\setminus\{0\}:\ \psi'_{\lambda,u}(1)=0,\ \psi''_{\lambda,u}(1)<0 \}.$$ By using the Implicit Function Theorem one can prove the following
\[Neharimanifold\] If $\mathcal{N}_\lambda^+,\mathcal{N}_\lambda^-$ are non empty then $\mathcal{N}_\lambda^+,\mathcal{N}_\lambda^-$ are $C^1$ manifolds of codimension $1$ in $H_0^1(\Omega)$. Moreover if $u\in\mathcal{N}_\lambda^+\cup\mathcal{N}_\lambda^-$ is a critical point of $\left(\Phi_\lambda\right)_{|\mathcal{N}_\lambda^+\cup\mathcal{N}_\lambda^-}$, then $u$ is a critical point of $\Phi_\lambda$.
In order to understand the Nehari set $\mathcal{N}_\lambda$ we study the fiber maps $\psi_{\lambda,u}$. Simple Analysis arguments show that
\[fibering\]For each $\lambda>0$ and $u\in H_0^1(\Omega)\setminus\{0\}$, there are only three possibilities for the graph of $ \psi_{\lambda,u}$
1. The function $\psi_{\lambda,u}$ has only two critical points, to wit, $0<t_\lambda^-(u)<t_\lambda^+(u)$. Moreover, $t_\lambda^-(u)$ is a local maximum with $\psi''_{\lambda,u}(t_\lambda^-(u))<0$ and $t_\lambda^+(u)$ is a local minimum with $\psi''_{\lambda,u}(t_\lambda^+(u))>0$;
2. The function $\psi_{\lambda,u}$ has only one critical point when $t>0$ at the value $t_\lambda(u)$. Moreover, $\psi''_{\lambda,u}(t_\lambda(u))=0$ and $\psi_{\lambda,u}$ is increasing;
3. The function $\psi_{\lambda,u}$ is increasing and has no critical points.
It follows from Proposition \[fibering\] that $\mathcal{N}_\lambda^+,\mathcal{N}_\lambda^-$ are non empty if and only if the item $I)$ is satisfied. Therefore, it remains to show whether $I)$ is satisfied or not. For this purpose we study for what values of $\lambda$ there holds $\mathcal{N}_\lambda^0\neq \emptyset$. Note that $tu\in \mathcal{N}_\lambda^0$ for $t>0$ and $u\in H_0^1(\Omega)\setminus\{0\}$ if and only if $$\left\{
\begin{aligned}
\psi'_{\lambda,u}(t) &= 0, \\
\psi''_{\lambda,u}(t) &= 0,
\end{aligned}
\right.$$ or equivalently $$\label{extremal}
\left\{
\begin{aligned}
a\|u\|^2+\lambda \|u\|^4t^{2}-\|u\|_\gamma^\gamma t^{\gamma-2} &= 0, \\
a\|u\|^2+3\lambda \|u\|^4t^{2}-(\gamma-1)\|u\|_\gamma^\gamma t^{\gamma-2} &= 0.
\end{aligned}
\right.$$ We solve the system (\[extremal\]) with respect to the variable $(t,\lambda)$ to obtain for each $u\in H_0^1(\Omega)\setminus \{0\}$ a unique pair $(t(u),\lambda(u))$ such that $$\label{tu}
t(u)=\left(\frac{2a}{4-\gamma}\frac{\|u\|^2}{\|u\|_\gamma^\gamma}\right)^{\frac{1}{\gamma-2}},$$ $$\label{extrefunction}
\lambda(u)=C_{a,\gamma}\left(\frac{\|u\|_\gamma}{\|u\|}\right)^{\frac{2\gamma}{\gamma-2}},$$ where $$C_{a,\gamma}=a\left(\frac{\gamma-2}{4-\gamma}\right)\left(\frac{4-\gamma}{2a}\right)^{\frac{2}{\gamma-2}}.$$ We define the extremal parameter (see Il’yasov [@ilyasENMM]) $$\label{extremalpara}
\lambda^*=\sup_{u\in H_0^1(\Omega)\setminus \{0\}}\lambda(u).$$ We also consider another parameter which is defined as a solution of the system $$\left\{
\begin{aligned}
\psi_{\lambda,u}(t) &= 0, \\
\psi'_{\lambda,u}(t) &= 0,
\end{aligned}
\right.$$ or equivalently $$\label{zeroenergy}
\left\{
\begin{aligned}
\frac{a}{2}\|u\|^2+\frac{\lambda}{4} t^{2}\|u\|^4-\frac{1}{\gamma}t^{\gamma-2}\|u\|_\gamma^\gamma &= 0, \\
a\|u\|^2+\lambda t^2\|u\|^4-t^{\gamma-2}\|u\|_\gamma^\gamma &= 0.
\end{aligned}
\right.$$ Similar to the system (\[extremal\]) we can solve the system with respect to the variable $(t,\lambda)$ to find a unique pair $(t_0(u),\lambda_0(u))$. Moreover, one can easily see that $$\lambda_0(u)=C_{0,a,\gamma}\lambda(u),\ \forall\ u\in H_0^1(\Omega)\setminus \{0\},$$ where $$C_{0,a,\gamma}=2\left(\frac{2}{\gamma}\right)^{\frac{2}{\gamma-2}}.$$ Observe that $C_{0,a,\gamma}<1$. We define $$\label{zeroenergypara}
\lambda_0^*=\sup_{u\in H_0^1(\Omega)\setminus \{0\}}\lambda_0(u).$$ The functions $\lambda(u)$ and $\lambda_0(u)$ has the following geometrical interpretation
\[fiberingvariation\] For each $u\in H_0^1(\Omega)\setminus\{0\}$ there holds
1. $\lambda(u)$ is the unique parameter $\lambda>0$ for which the fiber map $\psi_{\lambda,u}$ has a critical point with second derivative zero at $t(u)$. Moreover, if $0<\lambda<\lambda(u)$, then $\psi_{\lambda,u}$ satisfies I) of Proposition \[fibering\] while if $\lambda>\lambda(u)$, then $\psi_{\lambda,u}$ satisfies III) of Proposition \[fibering\].
2. $\lambda_0(u)$ is the unique parameter $\lambda>0$ for which the fiber map $\psi_{\lambda,u}$ has a critical point with zero energy at $t_0(u)$. Moreover, if $0<\lambda<\lambda_0(u)$, then $\inf_{t>0}\psi_{\lambda,u}(t)<0$ while if $\lambda>\lambda(u)$, then $\inf_{t>0}\psi_{\lambda,u}(t)=0$.
$i)$ The uniqueness of $\lambda(u)$ comes from equation . Assume that $\lambda\in(0,\lambda(u))$, then $\psi_{\lambda,u}$ must satisfies $I)$ or $III)$ of Proposition \[fibering\]. We claim that it must satisfies $I)$. Indeed, suppose on the contrary that it satisfies $III)$. Once $$\psi'_{\lambda(u),u}(t)>\psi'_{\lambda,u}(t)>0, \forall\ t>0,$$ we reach a contradiction since $\psi'_{\lambda(u),u}(t(u))=0$ where $t(u)$ is given by , therefore $\psi_{\lambda,u}$ must satisfies $I)$. Now suppose that $\lambda>\lambda(u)$, then $$\psi'_{\lambda,u}(t)>\psi'_{\lambda(u),u}(t)\ge0, \forall\ t>0,$$ and hence $\psi_{\lambda,u}$ must satisfies $III)$.
$ii)$ The uniqueness of $\lambda_0(u)$ comes from equation . If $0<\lambda<\lambda_0(u)$ then from the definition we have $$\psi_{\lambda,u}(t_0(u))<\psi_{\lambda_0(u),u}(t_0(u))=0,$$ which implies that $\inf_{t>0}\psi_{\lambda,u}(t)<0$. If $\lambda>\lambda_0(u)$ then $$\psi_{\lambda,u}(t))>\psi_{\lambda_0(u),u}(t)\ge0, \forall t>0,$$ and therefore $\inf_{t>0}\psi_{\lambda,u}(t)=\psi_{\lambda,u}(0)=0$.
Now we turn our attention to the parameters $\lambda^*$ and $\lambda_0^*$.
\[extremalparameter\] There holds $\lambda_0^*<\lambda^*<\infty$. Moreover, there exists $ u\in H_0^1(\Omega)\setminus \{0\}$ such that $\lambda(u)=\lambda^*$ and $\lambda_0(u)=\lambda_0^*$.
Indeed, from the Sobolev embedding it follows that $\lambda_0,\lambda^*<\infty$. Now observe that $\lambda(u)$ is $0$-homogeneous, that is $\lambda(tu)=\lambda(u)$ for each $t>0$. It follows that there exists a sequence $u_n\in H_0^1(\Omega)\setminus \{0\}$ such that $\|u_n\|=1$ and $\lambda(u_n)\to \lambda^*$ as $n\to \infty$. We can assume that $u_n \rightharpoonup u$ in $H_0^1(\Omega)$ and $u_n\to u$ in $L^\gamma(\Omega)$. Moreover, from (\[extrefunction\]) it follows that $u\neq 0$. We conclude that $$\lambda\left(\frac{u}{\|u\|}\right)=\lambda(u)\ge C_{a,\gamma}\left(\frac{\lim_{n\to \infty}\|u_n\|_\gamma}{\liminf_{n\to \infty}\|u_n\|}\right)^{\frac{2\gamma}{\gamma-2}}\ge \limsup_{n\to \infty}\lambda(u_n)=\lambda^*,$$ and hence $u_n\to u$ in $H_0^1(\Omega)$ and $u$ satisfies $\lambda(u)=\lambda^*$. Once $\lambda_0(u)$ is a mulitple of $\lambda(u)$ it follows also that $\lambda_0(u)=\lambda_0^*$ and from $C_{0,a,\gamma}<1$, we conclude that $\lambda_0^*<\lambda^*$.
As a consequence of Proposition \[extremalparameter\] we have the following
\[Neharimanifoldsandzeroenergy\] There holds
1. For each $\lambda\in (0,\lambda^*)$ we have that $\mathcal{N}_\lambda^+$ and $\mathcal{N}_\lambda^-$ are non empty. Moreover, if $\lambda>\lambda^*$ then $\mathcal{N}_\lambda=\emptyset$.
2. For each $\lambda\in (0,\lambda_0^*)$ there exists $u\in H_0^1(\Omega)\setminus\{0\}$ such that $\Phi_\lambda(u)<0$. Moreover, if $\lambda\ge\lambda_0^*$ then $\inf_{t>0}\psi_{\lambda,u}(t)=0$ for each $u\in H_0^1(\Omega)\setminus\{0\}$.
$i)$ From Proposition \[extremalparameter\], there exists $u\in H_0^1(\Omega)\setminus \{0\}$ such that $\lambda(u)=\lambda^*$. It follows from Proposition \[fiberingvariation\] that for each $\lambda\in (0,\lambda^*)$ the fiber map $\psi_{\lambda,u}$ satisfies $I)$ of Proposition \[fibering\] and hence $t_\lambda^-(u)u\in \mathcal{N}_\lambda^-$ and $t_\lambda^+(u)u\in \mathcal{N}_\lambda^+$. Now suppose that $\lambda>\lambda^*$, then it follows that $\lambda>\lambda^*\ge \lambda(u)$ for each $u\in H_0^1(\Omega)\setminus \{0\}$, which implies from Proposition \[fiberingvariation\] that $\psi_{\lambda,u}$ satisfies $III)$ of Proposition \[fibering\] and hence $\mathcal{N}_\lambda=\emptyset$.
$ii)$ From Proposition \[extremalparameter\], there exists $u\in H_0^1(\Omega)\setminus \{0\}$ such that $\lambda(u)=\lambda_0^*$. It follows from Proposition \[fiberingvariation\] that for each $\lambda\in (0,\lambda_0^*)$, there exists $t>0$ such that $\Phi_\lambda(tu)<0$. Now assume that $\lambda\ge \lambda_0^*$. Therefore $\lambda>\lambda_0^*\ge \lambda_0(u)$ for each $u\in H_0^1(\Omega)\setminus \{0\}$, which implies from Proposition \[fiberingvariation\] that $\inf_{t>0}\psi_{\lambda,u}(t)=0$.
From Proposition \[Neharimanifoldsandzeroenergy\] we obtain the following nonexistence result.
\[nonexistence\] For each $\lambda>\lambda^*$ the functional $\Phi_\lambda$ does not have critical points other than $u=0$.
Indeed, observe that for each $\lambda>\lambda^*$ there holds $\mathcal{N}_\lambda=\emptyset$.
Now we turn our attention to some estimates which will prove to be useful on the next section. We start with:
\[distanceorigin\] Suppose that $\lambda\in (0,\lambda^*]$, then there exists $r_\lambda>0$ such that $\|u\|\ge r_\lambda$ for each $u\in \mathcal{N}_\lambda$.
The existence of $r_\lambda$ is straightforward from $$a\|u\|^2+\lambda\|u\|^4-C\|u\|^\gamma\le a\|u\|^2+\lambda\|u\|^4-\|u\|_\gamma^\gamma=0,\ \forall\ u\in \mathcal{N}_\lambda,$$ where $C>0$ comes from the Sobolev embedding.
\[n0\] For each $\lambda\in(0,\lambda^*]$, there holds $$\Phi_\lambda(u)=\frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda},\ \forall\ u\in \mathcal{N}_\lambda^0.$$
In fact, if $u\in \mathcal{N}_\lambda^0$, then $$\label{E1}
\left\{
\begin{aligned}
a\|u\|^2+\lambda \|u\|^4-\|u\|_\gamma^\gamma &= 0, \\
2a\|u\|^2+4\lambda \|u\|^4-\gamma\|u\|_\gamma^\gamma &= 0.
\end{aligned}
\right.$$ It follows from (\[E1\]) that $$\label{E2}
\|u\|^2=\frac{\gamma-2}{4-\gamma}\frac{a}{\lambda}.$$ Moreover, from (\[E1\]) we also have that $$\label{E3}
\Phi_\lambda(u)=\frac{\gamma-2}{2\gamma}a\|u\|^2-\frac{4-\gamma}{4\gamma}\lambda\|u\|^4, \forall\ u\in \mathcal{N}_\lambda^0.$$ We combine (\[E2\]) with (\[E3\]) to prove the proposition.
We conclude this Section with some variational properties related to the functional $\Phi_\lambda$.
\[variatonal\] For each $\lambda\in (0,\lambda^*)$ there holds
1. The functional $\Phi_\lambda$ is weakly lower semi-continuous and coercive.
2. Suppose that $u_n$ is a Palais-Smale sequence at the level $c\in\mathbb{R}$, that is $\Phi_\lambda(u_n)\to c$ and $\Phi_\lambda'(u_n)\to 0$ as $n\to \infty$, then $u_n$ converge strongly to some $u$.
3. There exist $C_\lambda>0$ and $\rho_\lambda>0$ satisfying $$\Phi_\lambda(u)\ge C_\lambda,\ \forall\ u\in H_0^1(\Omega),\ \|u\|=\rho_\lambda,$$ and $$\lim_{C_\lambda\to 0}\rho_\lambda=0.$$
$i)$ is obvious. To prove $ii)$, observe from $i)$ that $u_n$ is bounded and therefore we can assume that $u_n\rightharpoonup u$ in $H_0^1(\Omega)$ and $u_n\to u$ in $L^\gamma(\Omega)$. From the limit $\Phi_\lambda'(u_n)\to 0$ as $n\to \infty$ we infer that $$\limsup_{n\to \infty}[-(a+\lambda\|u_n\|^2)\Delta u_n(u_n-u)]=\limsup_{n\to \infty}|u_n|^{\gamma-2}u_n(u_n-u)=0,$$ which easily implies that $u_n \to u$ in $H_0^1(\Omega)$.
$iii)$ It follows from the inequality $$\Phi_\lambda(u)\ge \frac{a}{2}\|u\|^2+\frac{\lambda}{4}\|u\|^4-\frac{C}{\gamma}\|u\|^\gamma,\ \forall\ H_0^1(\Omega),$$ where the constant $C$ is positive.
Local Minimizers for $\Phi_\lambda$
===================================
$\label{S3}$ In this section we prove the following
\[GMl0\] For each $\lambda\in(0,\lambda^*)$ the functional $\Phi_\lambda$ has a local minimizer $u_\lambda\in H_0^1(\Omega)\setminus\{0\}$. Moreover, if $\lambda\in(0,\lambda_0^*)$ then $\Phi_\lambda(u_\lambda)<0$ while $\Phi_{\lambda_0^*}(u_{\lambda_0^*})=0$ and if $\lambda\in(\lambda_0^*,\lambda^*)$ then $\Phi_\lambda(u_\lambda)>0$.
In fact if $\lambda\in(0,\lambda_0^*]$ then the local minimizer given by the Lemma \[GMBl0\] is a global minimizer.
We divide the proof of Proposition \[GMl0\] in some Lemmas.
\[GMBl0\] For each $\lambda\in(0,\lambda_0^*)$ the functional $\Phi_\lambda$ has a global minimizer $u_\lambda$ with negative energy.
It is a consequence of Lemma \[variatonal\] and Proposition \[Neharimanifoldsandzeroenergy\].
\[GMB00\] The functional $\Phi_{\lambda_0^*}$ has a global minimizer $u_{\lambda_0^*}\neq 0$ with zero energy.
Suppose that $\lambda_n\uparrow\lambda_0^*$ as $n\to \infty$ and for each $n$ choose $u_n\equiv u_{\lambda_n}$, where $u_{\lambda_n}$ is given by Lemma \[GMBl0\]. From the inequality $\Phi_{\lambda_n}(u_n)<0$ for each $n$ and Lemma \[variatonal\] we obtain that $u_n$ is bounded. Therefore we can assume that $u_n\rightharpoonup u$ in $H_0^1(\Omega)$ and $u_n \to u$ in $L^\gamma(\Omega)$. From Lemma \[variatonal\] we have that $$\Phi_{\lambda_0^*}(u)\leq\liminf_{n\to \infty}\Phi_{\lambda_n}(u_n)\le 0.$$ From Proposition \[Neharimanifoldsandzeroenergy\] we conclude that $\Phi_{\lambda_0^*}(u)=0$ and hence $ \Phi_{\lambda_0^*}(u)=\lim_{n\to \infty}\Phi_{\lambda_n}(u_n)$. Therefore $u_n\to u$ in $H_0^1(\Omega)$ and from Proposition \[distanceorigin\] we obtain that $u\neq 0$. If $u_{\lambda_0^*}\equiv u$ the proof is complete.
\[Rm1\] Observe that $\lambda_0^*(u_{\lambda_0^*})=\lambda_0^*$ and hence $\lambda^*(u_{\lambda_0^*})=\lambda^*$.
In order to show the existence of local minimizers when $\lambda>\lambda_0^*$ we need the following definition: for $\lambda\in(0,\lambda^*)$ define $$\label{minneh}
\hat{\Phi}_\lambda=\inf\{\Phi_\lambda(u):\ u\in\mathcal{N}_\lambda^+\cup\mathcal{N}_\lambda^0\}.$$
From the definitions, Proposition \[fibering\] and Proposition \[Neharimanifoldsandzeroenergy\] we conclude that $$\hat{\Phi}_\lambda=\inf_{u\in H_0^1(\Omega)}\Phi_\lambda(u), \forall \lambda\in(0,\lambda_0^*].$$
\[nearzeroenergy\] For each $\lambda\in (\lambda_0^*,\lambda^*)$ there holds $$\hat{\Phi}_\lambda<\frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda}.$$
Indeed, first observe from Remark \[Rm1\] that $t_\lambda^+(u_{\lambda_0^*})$ is defined for each $\lambda\in (\lambda_0^*,\lambda^*)$. From Proposition \[decre\] in the Appendix we know that $t_\lambda^-(u_{\lambda_0^*})<t_{\lambda_0^*}(u_{\lambda_0^*})<t_\lambda^+(u_{\lambda_0^*})$ for each $\lambda\in (\lambda_0^*,\lambda^*)$ and therefore $$\begin{aligned}
\label{ttt}
\hat{\Phi}_\lambda&\le& \Phi_\lambda(t_\lambda^+(u_{\lambda_0^*})u_{\lambda_0^*}) \nonumber\\
&<& \Phi_\lambda(t_{\lambda^*}(u_{\lambda_0^*})u_{\lambda_0^*}) \nonumber\\
&<& \Phi_{\lambda^*}(t_{\lambda^*}(u_{\lambda_0^*})u_{\lambda_0^*}) \nonumber\\
&=& \frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda^*},\ \forall\ \lambda\in(\lambda_0^*,\lambda^*),
\end{aligned}$$ where the equality comes from Proposition \[n0\]. We combine with $\lambda<\lambda^*$ to complete the proof.
\[ekeland\] For each $\lambda\in(\lambda_0^*,\lambda^*)$ there exists $u_\lambda\in \mathcal{N}_\lambda^+$ such that $\Phi_\lambda(u_\lambda)=\hat{\Phi}_\lambda$.
Indeed, suppose that $u_n\in \mathcal{N}_\lambda^+\cup \mathcal{N}_\lambda^0$ satisfies $\Phi_{\lambda}(u_n)\to \hat{\Phi}_\lambda$. From Lemma \[variatonal\] we have that $u_n$ is bounded and therefore we can assume that $u_n \rightharpoonup u$ in $H_0^1(\Omega)$ and $u_n\to u$ in $L^\gamma(\Omega)$. From $a\|u_n\|^2+\lambda\|u_n\|^{4}-\|u_n\|_\gamma^\gamma=0$ for all $n$ and Proposition \[distanceorigin\] we conclude that $u\neq 0$. We claim that $u_n\to u$ in $H_0^1(\Omega)$. In fact, suppose on the contrary that this is false. It follows that $$\psi'_{\lambda,u}(1)=a\|u\|^2+\lambda\|u\|^4-\|u\|_\gamma^\gamma< \liminf_{n\to \infty}(a\|u_n\|^2+\lambda\|u_n\|^{4}-\|u_n\|_\gamma^\gamma)=0,$$ and hence we conclude that the fiber map $\psi_{\lambda,u}$ satisfies $I)$ of Proposition \[fibering\] and $t_\lambda^-(u)<1<t_\lambda^+(u)$. It follows that $$\Phi_\lambda(t_\lambda^+(u)u)< \Phi_\lambda(u)\le \liminf_{n\to \infty}\Phi_\lambda(u_n)=\hat{\Phi}_\lambda,$$ which is a contradiction since $t_\lambda^+(u)u\in \mathcal{N}_\lambda^+$. We conclude that $u_n\to u$ in $H_0^1(\Omega)$ and hence $\Phi_\lambda(u)=\hat{\Phi}_\lambda$. From Propositions \[n0\] and \[nearzeroenergy\] we obtain that $u\in \mathcal{N}_\lambda^+$.
The Lemmas \[GMBl0\] and \[GMB00\] guarantee the existence of a global minimizer $u_\lambda$ for the functional $\Phi_\lambda$ satisfying: if $\lambda\in(0,\lambda_0^*)$ then $\Phi_\lambda(u_\lambda)<0$ while $\Phi_{\lambda_0^*}(u_{\lambda_0^*})=0$. For $\lambda\in(\lambda_0^*,\lambda^*)$ we use Lemma \[ekeland\] in order to obtain a local minimizer for the functional $\Phi_\lambda$. It remains to show that $\Phi_\lambda(u_\lambda)>0$ for $\lambda\in(\lambda_0^*,\lambda^*)$, however, once $\hat{\Phi}_{\lambda_0^*}=0$ this is a consequence of Proposition \[contidecre\].
Mountain Pass Solution for $\Phi_\lambda$
=========================================
$\label{S4}$ In this Section we show the exsitence of a mountain pass type solution to equation . In order to formulate our result we need to introduce some notation. For each $\lambda\in(0,\lambda^*)$ define $$\label{MPE}
c_\lambda=\inf_{\varphi\in \Gamma_\lambda}\max_{t\in [0,1]}\Phi_\lambda(\varphi(t)),$$ where $\Gamma_\lambda=\{\varphi\in C([0,1],H_0^1(\Omega)): \varphi(0)=0,\ \varphi(1)=\bar{u}_\lambda\}$ with $\bar{u}_\lambda=u_{\lambda_0^*}$ if $\lambda\in (0,\lambda_0^*]$ and $\bar{u}_\lambda=u_\lambda$ for $\lambda\in (\lambda_0^*,\lambda^*)$.
\[MPS\] There exists $\varepsilon>0$ such that for each $\lambda\in(0,\lambda_0^*+\varepsilon)$ one can find $w_\lambda\in H_0^1(\Omega)$ satisfying $\Phi_\lambda(w_\lambda)=c_\lambda$ and $\Phi'_\lambda(w_\lambda)=0$. Moreover $c_\lambda>0$ and $c_\lambda>\hat{\Phi}_\lambda$.
To prove Proposotion \[MPS\] we need some auxiliary results.
\[nearzeroenergyE\] Given $\delta>0$, there exists $\varepsilon_\delta>0$ such that $$0< \hat{\Phi}_\lambda\le \delta,\ \forall\lambda\in(\lambda_0^*,\lambda_0^*+\varepsilon_\delta).$$
The inequality $\hat{\Phi}_\lambda> 0$ follows from Proposition \[GMl0\]. Let $u_{\lambda_0^*}$ be given as in Proposition \[GMB00\]. Observe that if $\lambda\downarrow \lambda_0^*$, then $\Phi_\lambda(u_{\lambda_0^*})\to \Phi_{\lambda_0^*}(u_{\lambda_0^*})=0$. Moreover, since from Remark \[Rm1\] the fiber map $\psi_{\lambda_0^*,u_{\lambda_0^*}}$ satisfies $I)$ of Proposition \[fibering\], we have from Proposition \[fiberingvariation\] that $\lambda_0^*<\lambda(u_{\lambda_0^*})$. It follows that there exists $\varepsilon_1>0$ such that $\lambda_0^*+\varepsilon_1<\lambda(u_{\lambda_0^*})$. From Propositions \[fibering\] and \[fiberingvariation\], for each $\lambda\in (\lambda_0^*,\lambda_0^*+\varepsilon_1)$, there exists $t_\lambda^+(u_{\lambda_0^*})>0$ such that $t_\lambda^+(u_{\lambda_0^*})u_{\lambda_0^*}\in \mathcal{N}_\lambda^+$. Note that $t_\lambda^+(u_{\lambda_0^*})\to 1$ as $\lambda\downarrow \lambda_0^*$ and therefore $$\hat{\Phi}_\lambda\le \Phi_\lambda(t_\lambda^+(u_{\lambda_0^*})u_{\lambda_0^*})\to \Phi_{\lambda_0^*}(u_{\lambda_0^*})=0,\ \lambda\downarrow \lambda_0^*.$$ If $\varepsilon_{2,\delta}>0$ is choosen in such a way that $\Phi_\lambda(t_\lambda^+(u_{\lambda_0^*})u_{\lambda_0^*})<\delta$ for each $\lambda\in (\lambda_0^*,\lambda_0^*+\varepsilon_{2,\delta})$, then we set $\varepsilon_\delta=\min\{\varepsilon_1,\varepsilon_{2,\delta}\}$ and the proof is complete.
\[DEFI\] For $\lambda\in(0,\lambda^*)$ denote $$\label{D1}
M_\lambda=\min\left\{C_\lambda,\frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda}\right\},$$ where $C_\lambda$ is given by Lemma \[variatonal\] and $\frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda}$ is given by Proposition \[n0\]. We assume that $\rho_\lambda<r_\lambda$ where both numbers are given by Lemma \[variatonal\] and Proposition \[distanceorigin\] respectively. Choose $0<\delta<M_\lambda$ and from Proposition \[nearzeroenergy\] we take the corresponding $\varepsilon_\delta$.
Now we are in position to prove Proposition \[MPS\]
The proof will be done once we show that the functional $\Phi_\lambda$ has a mountain pass geometry (remember that $u_\lambda$ is a local minimizer for $\Phi_\lambda$), however, one can see from Definition \[DEFI\] that $$\label{MPGI}
\inf_{\|u\|=\rho_\lambda}\Phi_\lambda(u)\ge M_\lambda>\max\{\Phi_\lambda(0),\Phi_\lambda(\bar{u}_\lambda)\},$$ which is the desired mountain pass geometry. It follows that $c_\lambda\ge M_\lambda>\Phi_\lambda(\bar{u}_\lambda)$ and $\Phi_\lambda(\bar{u}_\lambda)\ge \hat{\Phi}_\lambda$ if $\lambda\in (0,\lambda_0^*]$ and $\Phi_\lambda(\bar{u}_\lambda)= \hat{\Phi}_\lambda$ otherwise.
We infer that there exists a Palais-Smale sequence for the functional $\Phi_\lambda$ at the level $c_\lambda$, that is, there exists $w_n\in H_0^1(\Omega)$ such that $\Phi_\lambda(w_n)\to c_\lambda$ and $\Phi_\lambda(w_n)\to 0$. From Lemma \[variatonal\] we have that $w_n\to w$ in $H_0^1(\Omega)$ and hence $\Phi_\lambda(w_\lambda)=c_\lambda$ and $\Phi'_\lambda(w_\lambda)=0$.
Proof of Theorem \[T\]
======================
$\label{S5}$ In this Section we prove our main result
The existence of $u_\lambda$ and $w_\lambda$ are given by Propositions \[GMl0\] and \[MPS\]. Observe that $u_\lambda$ being a global minimizer for $\Phi_\lambda$ when $\lambda\in(0,\lambda_0^*]$ it is obviously a critical point for $\Phi_\lambda$ and hence a solution to (\[p\]). If $\lambda\in(\lambda_0^*,\lambda^*)$ we saw in Lemma \[ekeland\] that $u_\lambda\in \mathcal{N}_\lambda^+$ and hence from Lemma \[Neharimanifold\] it is a critical point for the functional $\Phi_\lambda$. The case $\lambda=\lambda^*$ goes as following. Choose a sequence $\lambda\uparrow \lambda^*$ and a corresponding sequence $u_n\equiv u_{\lambda_n}$ such that $\Phi_{\lambda_n}(w_n)
=\hat{\Phi}_{\lambda_n}$ and $\Phi'_{\lambda_n}(u_n)=0$ for each $n\in \mathbb{N}$. Observe from the proof of Proposition \[nearzeroenergy\] that $$\hat{\Phi}_{\lambda_n}< \frac{(\gamma-2)^2}{4\gamma(4-\gamma)}\frac{a^2}{\lambda^*},\ \forall\ n\in\mathbb{N},$$ and therefore from Lemma \[variatonal\] we conclude that $u_n\to u$ in $H_0^1(\Omega)$. From Proposition \[contidecre\] we obtain that $$\Phi_{\lambda^*}(u)=\lim_{n\to \infty}\Phi_{\lambda_n}(w_n)=\lim_{n\to \infty}\hat{\Phi}_{\lambda_n}>0,$$ and hence $u\neq 0$. By passing the limit it follows that $\Phi'_{\lambda^*}(u)=0$. Moreover from the definition of $\lambda^*$ we also obtain that $\Phi''_{\lambda^*}(u)(u,u)$=0. If we set $u_{\lambda^*}\equiv u$ the proof of Theorem \[T\] items $1)$, $2)$ and $3)$ is complete.
The item $4)$ is a consequence of Proposition \[MPS\]. Item $5)$ is proved by using the fact that every critical point of $\Phi_\lambda$ lies in $\mathcal{N}_\lambda$ and Proposition \[Neharimanifoldsandzeroenergy\]. To conclude we observe that standard arguments using the fact that $\Phi_\lambda(u)=\Phi_\lambda(|u|)$ provide positive solutions.
Asymptotic Behavior of $u_\lambda$ and $w_\lambda$ as $\lambda\downarrow 0$
===========================================================================
$\label{S6}$ Define $\Phi_0:H_0^1(\Omega)\to \mathbb{R}$ by $$\label{lambda=0}
\Phi_0(u)=\frac{a}{2}\|u\|^2-\frac{1}{\gamma}\|u\|_\gamma^\gamma,$$ and observe that $\Phi_0(u_{\lambda_0^*})<\Phi_{\lambda_0^*}(u_{\lambda_0^*})=0$, where $u_{\lambda_0^*}$ is given by Theorem \[T\]. Define $$\label{MP00}
c_0=\inf_{\varphi\in \Gamma}\max_{t\in [0,1]}\Phi_0(\varphi(t)),$$ where $\Gamma=\{\varphi\in C([0,1]:H_0^1(\Omega)):\varphi(0)=0,\ \varphi(1)=u_{\lambda_0^*} \}$. Standard arguments provide a function $w_0\in H_0^1(\Omega)$ such that $\Phi_0(w_0)=c_0>0$ and $\Phi'_0(w_0)=0$. For $\lambda\in (0,\lambda^*_0)$, let us assume that $u_\lambda,w_\lambda$ are given by Theorem \[T\]. In this section we prove the following
\[asym\] There holds
1. $\Phi_\lambda(u_\lambda)\to -\infty$ and $\|u_\lambda\|\to\infty$ as $\lambda\downarrow 0$.
2. $w_\lambda \to w_0$ in $H_0^1(\Omega)$ where $w_0\in H_0^1(\Omega)$ satisfies $\Phi_0(w_0)=c_0$ and $\Phi'_0(w_0)=0$.
$i)$ Indeed, choose any $u\in H_0^1(\Omega)$ and suppose without loss of generality that $\lambda\in(0,\lambda(u))$. It follows from Proposition \[fibering\] that $\psi_{\lambda,u}(t)\ge \psi_{\lambda,u}(t_\lambda^+(u))\ge \inf_{u\in H_0^1(\Omega)}\Phi_\lambda(u)=\hat{\Phi}_\lambda$. Now observe that for fixed $t>0$ there holds $$\label{AA1}
\psi_{\lambda,u}(t)\to \frac{a}{2}\|u\|^2 t^2-\frac{1}{\gamma}\|u\|_\gamma^\gamma t^\gamma,\ \mbox{as}\ \lambda\downarrow 0.$$ Once $$\lim_{t\to \infty}\left(\frac{a}{2}\|u\|^2 t^2-\frac{1}{\gamma}\|u\|_\gamma^\gamma t^\gamma\right)=-\infty,$$ it follows from that given $M<0$ there exists $t>0$ and $\delta>0$ such that if $\lambda\in(0,\delta)$, then $\psi_{\lambda,u}(t)<M$ and hence $\hat{\Phi}_\lambda<M$, which proves that $\Phi_\lambda(u_\lambda)\to -\infty$ as $\lambda\downarrow 0$. One can easily infer from the last convergence that $\|u_\lambda\|\to\infty$ as $\lambda\downarrow 0$.
To prove the item $ii)$ of Proposition \[asym\] we need to establish some results.
\[las1\] The function $[0,\lambda_0^*)\ni\lambda\mapsto c_\lambda=\Phi_\lambda(w_\lambda)$ is non-decreasing. Moreover $c_\lambda\to c_0$ as $\lambda\downarrow 0$.
First observe that $\Gamma_\lambda=\Gamma$ for each $\lambda\in (0,\lambda_0^*]$. Suppose that $0\le\lambda<\lambda'<\lambda_0^*$ and fix any $\varphi\in \Gamma$. It follows that $\max_{t\in [0,1]}\Phi_\lambda(\varphi(t))<\max_{t\in [0,1]}\Phi_{\lambda'}(\varphi(t))$ and by taking the infimum in both sides we conclude that $c_\lambda\le c_{\lambda'}$.
Once $c_\lambda$ is non-decreasing, we can assume that $c_\lambda\to c\ge c_0$ as $\lambda \downarrow 0$. Suppose on the contrary that $c>c_0$. Given $\delta>0$ such that $c_0+\varepsilon<c$ choose $\varphi\in \Gamma$ such that $c_0\le \max_{t\in [0,1]}\Phi_0(\varphi(t))<c_0+\delta$. If $\lambda$ is sufficiently close to $0$, then $c_0\le \max_{t\in [0,1]}\Phi_0(\varphi(t))< \max_{t\in [0,1]}\Phi_\lambda(\varphi(t))<c_0+\delta$ and consequently $c_0\le c_\lambda<c_0+\delta<c<c_\lambda$ which is clearly a contradiction and therefore $c_\lambda\to c_0$ as $\lambda\downarrow 0$.
Now we may finish the proof of Proposition \[asym\]:
Indeed, suppose that $\lambda_n\downarrow 0$ and for each $n\in \mathbb{N}$ choose $w_n\equiv w_{\lambda_n}$ such that $\Phi_{\lambda_n}(w_n)=c_{\lambda_n}$ and $\Phi'_{\lambda_n}(w_n)=0$. We claim that $\lambda_n\|w_n\|^4\to 0$ as $n\to \infty$. In fact, for each $n$ we can find a path $\varphi_n\in \Gamma_{\lambda_n}=\Gamma$ and a function $v_n$ such that $\Phi_{\lambda_n}(v_n)=\max_{t\in[0,1]}\Phi_{\lambda_n}(\varphi(t))$ and $$\label{tt1}
0<\Phi_{\lambda_n}(v_n)-c_{\lambda_n}\to 0,\ \|v_n-w_n\|\to 0,\ \|v_n-w_n\|_\gamma\to 0,\ \mbox{as}\ n\to \infty.$$ Now observe from the definition of $c_0$, Lemma \[las1\] and (\[tt1\]) that $$\label{tt2}
0<\lim_{n\to \infty}\Phi_0(v_n)-c_0\le \lim_{n\to \infty}\Phi_{\lambda_n}(v_n)-c_0=\lim_{n\to \infty}(\Phi_{\lambda_n}(v_n)-c_{\lambda_n})=0.$$ It follows from and that $$\frac{a}{2}\|v_n\|^2-\frac{1}{p}\|v_n\|_\gamma^\gamma\to 0\ \mbox{and}\ \frac{a}{2}\|v_n\|^2+\frac{\lambda_n}{4}\|v_n\|^4-\frac{1}{p}\|v_n\|_\gamma^\gamma\to 0,\ \mbox{as}\ n\to \infty,$$ which implies that $\lambda_n\|v_n\|^4\to 0$ as $n\to \infty$. From we conclude that $$|\lambda_n\|w_n\|^4-\lambda_n\|v_n\|^4|\to 0,\ \mbox{as}\ n\to \infty,$$ and hence $\lambda_n\|w_n\|^4\to 0$ as $n\to\infty$ as we desired. Now note from the equations $\Phi_{\lambda_n}(w_n)=c_{\lambda_n}$ and $\Phi'_{\lambda_n}(w_n)=0$, $n\in\mathbb{N}$ that $$\label{pt1}
\left\{
\begin{aligned}
\frac{a}{2}\|w_n\|^2+\frac{\lambda_n}{4} \|w_n\|^4-\frac{1}{\gamma}\|w_n\|_\gamma^\gamma &= c_{\lambda_n}, \\
a\|w_n\|^2+\lambda_n \|w_n\|^4-\|w_n\|_\gamma^\gamma &= 0,
\end{aligned}
\right.$$ which combined with the limit $\lambda_n\|w_n\|^4\to 0$ as $n\to\infty$ and the Lemma \[las1\] implies that $$\left\{
\begin{aligned}
\frac{a}{2}\lambda_n\|w_n\|^2-\frac{\lambda_n}{\gamma}\|w_n\|_\gamma^\gamma &= o(1), \\
a\lambda_n\|w_n\|^2-\lambda_n\|w_n\|_\gamma^\gamma &= 0.
\end{aligned}
\right.$$ We multiply the first equation by $-\gamma$ and sum with the second equation to obtain that $$\left(-\frac{\gamma}{2}+1\right)a\lambda_n\|w_n\|^2=o(1),$$ which implies that $\lambda_n\|w_n\|^2\to 0$ as $n\to \infty$. Now we claim that $\|w_n\|$ is bounded. In fact, suppose on the contrary that up to a subsequence $\|w_n\|\to \infty$ as $n\to \infty$. From we obtain that $$\left\{
\begin{aligned}
\frac{a}{2}+\frac{\lambda_n}{4} \|w_n\|^2-\frac{1}{\gamma}\frac{\|w_n\|_\gamma^\gamma}{\|w_n\|^2} &= o(1), \\
a+\lambda_n \|w_n\|^2-\frac{\|w_n\|_\gamma^\gamma}{\|w_n\|^2} &= 0.
\end{aligned}
\right.$$ Once $\lambda_n\|w_n\|^2\to 0$ as $n\to \infty$ we conclude that $\gamma=2$ which is a contradiction. Since $\|w_n\|$ is bounded we obtain that $\Phi_0(w_n)\to c_0$ and $\Phi'_0(w_n)\to 0$ as $n\to \infty$ and hence $w_n\to w_0$ as $n\to \infty$, where $w_0$ satisfies $\Phi_0(w_0)=c_0$ and $\Phi'_0(w_0)=0$.
It is a consequence of Proposition \[asym\].
Some Conclusions and Remarks
============================
$\label{S7}$ If we plot the energy of the two solutions as a function of $\lambda$ we obtain the following picture:
(-1,0) – (5,0) node\[below\] [$\lambda$]{}; in (0pt,2pt) – (0pt,-2pt) node\[below\] [$\x$]{}; (0,-2) – (0,2) node\[left\] [$\mbox{Energy}$]{}; in (2pt,0pt) – (-2pt,0pt) node\[left\] [$\y$]{}; at (0,0) [$0$]{}; (0,-2) .. controls (0,-1.5) and (0,0) .. (4,1.5); (0,.5) .. controls (0,0.5) and (0,0.6) .. (3,1.3); (3,1.3) .. controls (3,1.3) and (4,1.5) .. (4,1.5); (1.16,-.1) node\[below\][$\lambda_0^*$]{} – (1.16,0.1); (3.1,-.1) node\[below\][$\lambda_0^*+\varepsilon$]{} – (3.1,0.1); (4,-.1) node\[below\][$\lambda^*$]{} – (4,0.1); (-.1,-2) node\[left\][$-\infty$]{} – (.1,-2); at (1,1.5) [ [$\Phi_\lambda(w_\lambda)$]{}]{}; at (3,0.5) [ [$\Phi_\lambda(u_\lambda)$]{}]{};
Observe from Proposition \[contidecre\] that the energy of the local minimum depending on $\lambda$ is continuous and increasing (red plot) and although we could not prove it, we believe that the same holds true for the energy of the mountain pass solution (blue plot). We also believe that $\lambda^*$ is a bifurcation turning point, that is, the two types of solutions must coincide at $\lambda^*$ as Figure 1 suggests.
\[contidecre\] The function $(0,\lambda^*)\ni \lambda\mapsto\hat{\Phi}_\lambda$ is continuous and increasing.
First we prove that $(0,\lambda^*)\ni \lambda\mapsto\hat{\Phi}_\lambda$ is decreasing. Indeed, suppose that $\lambda<\lambda'$. From Lemmas \[GMBl0\], \[GMB00\] and \[ekeland\], there exists $u_{\lambda'}$ such that $\hat{\Phi}_{\lambda'}=\Phi_{\lambda'}(u_{\lambda'})$. Since the fiber map $\psi_{\lambda',u_{\lambda'}}$ obvioulsy satisfies I) of Proposition \[fibering\] it follows from Proposition \[fiberingvariation\] that $\psi_{\lambda,u_{\lambda'}}$ also satisfies I) of Proposition \[fibering\] and then $$\hat{\Phi}_{\lambda}\le \Phi_{\lambda}(t_{\lambda}^+(u_{\lambda'})u_{\lambda'})<\Phi_{\lambda}(t_{\lambda'}^+(u_{\lambda'})u_{\lambda'})=\Phi_{\lambda'}(u_{\lambda'})=\hat{\Phi}_{\lambda'}.$$ Now we prove that $(0,\lambda^*)\ni \lambda\mapsto\hat{\Phi}_\lambda$ is continuous. In fact, suppose that $\lambda_n\uparrow \lambda\in (0,\lambda^*)$ and choose $u_n\equiv u_{\lambda_n}$ such that $\hat{\Phi}_{\lambda_n}= \Phi_{\lambda_n}(u_n)$ for all $n$. Similar to the proof of Lemma \[ekeland\] we may assume that $u_n\to u\in\mathcal{N}_\lambda^+$. We claim that $\hat{\Phi}_{\lambda_n}\to \hat{\Phi}_\lambda$ as $n\to \infty$. Indeed, once $(0,\lambda^*)\ni \lambda\mapsto\hat{\Phi}_\lambda$ is increasing, we can assume that $\hat{\Phi}_{\lambda_n}< \hat{\Phi}_\lambda$ for each $n$ and $\hat{\Phi}_{\lambda_n}\to \Phi_\lambda(u) \le\hat{\Phi}_\lambda$ as $n\to \infty$, wich implies that $\Phi_\lambda(u) =\hat{\Phi}_\lambda$.
Now suppose that $\lambda_n\downarrow \lambda\in (0,\lambda^*)$. Once $(0,\lambda^*)\ni \lambda\mapsto\hat{\Phi}_\lambda$ is increasing, we can assume that $\hat{\Phi}_{\lambda_n}> \hat{\Phi}_\lambda$ for each $n$ and $\lim_{n\to \infty}\hat{\Phi}_{\lambda_n}\ge\hat{\Phi}_\lambda$. Choose $u_\lambda$ such that $\hat{\Phi}_\lambda=\Phi_\lambda(u_\lambda)$ and observe that $\hat{\Phi}_\lambda\le \lim_{n\to \infty}\hat{\Phi}_{\lambda_n}\le\lim_{n\to \infty}\Phi_{\lambda_n}(t_{\lambda_n}^+(u_\lambda)u_\lambda)=\hat{\Phi}_\lambda$.
For the next proposition we assume that $u_{\lambda_0^*}$ is given as in Lemma \[GMB00\] and $t(u_{\lambda_0^*})$ is defined in . Observe from Remark \[Rm1\] that $t_\lambda^+(u_{\lambda_0^*})$ is well defined for each $\lambda\in (0,\lambda^*)$.
\[decre\] There holds
1. The function $(0,\lambda^*)\ni \lambda\mapsto t_\lambda^+(u_{\lambda_0^*})$ is decreasing and continuous.
2. The function $(0,\lambda^*)\ni \lambda\mapsto t_\lambda^-(u_{\lambda_0^*})$ is increasing and continuous.
Moreover $$\lim_{\lambda\uparrow \lambda^*}t_\lambda^+(u_{\lambda_0^*})=\lim_{\lambda\uparrow \lambda^*}t_\lambda^-(u_{\lambda_0^*})=t(u_{\lambda_0^*}).$$
Indeed, let $t_\lambda\equiv t_\lambda^+(u_{\lambda_0^*})$ and note that $t_\lambda$ satisfies $\psi'_\lambda(t_\lambda)=0$ for each $\lambda\in (0,\lambda^*)$. By implicit differentiation and the fact that $\psi''_\lambda(t_\lambda)>0$, we conclude that $(0,\lambda^*)\ni \lambda\mapsto t_\lambda^+(u_{\lambda_0^*})$ is decreasing and continuous, which proves $i)$ The proof of $ii)$ is similar and the limits $$\lim_{\lambda\uparrow \lambda^*}t_\lambda^+(u_{\lambda_0^*})=\lim_{\lambda\uparrow \lambda^*}t_\lambda^-(u_{\lambda_0^*})=t(u_{\lambda_0^*}),$$ are straightforward from the definitions.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Since the discovery of high-$T_c$ and other such systems based on FeAs layers, several proposals have been made for the superconducting order parameter $\Delta_\k$, on both phenomenological and microscopic grounds. Here we discuss how the symmetry of $\Delta_\k$ in the bulk can be determined, assuming that single crystals will soon be available. We suggest that a measurement of the dependence of the low temperature specific heat on the angle of a magnetic field in the FeAs plane is the simplest such method, and calculate representative specific heat vs. field angle oscillations for the various candidate states, using a phenomenological band structure fitted to the DFT Fermi surface.'
author:
- 'S. Graser'
- 'G.R. Boyd'
- Chao Cao
- 'Hai-Ping Cheng'
- 'P. J. Hirschfeld'
- 'D. J. Scalapino'
title: 'Determining gap nodal structures in Fe-based superconductors: angle-dependence of the low temperature specific heat in an applied magnetic field '
---
The recent discovery of superconductivity with onset temperature of 26K in [@LOFA_JACS] was followed rapidly by the development of materials with $T_c$ up to $\sim$50K[@NLWang1; @HHWen1; @HHWen2; @Jin; @HHWen3; @NLWang2; @NLWang3; @HHWen4; @ChenXH], which possess a similar structure but where La has been replaced by Sm, Pr, Nd or Ce. Common to all such materials is an electronically layered structure, where according to electronic structure theories a rare-earth oxide layer where F substitutions for O dope an FeAs layer. The iron atoms are arranged in a simple square lattice, separated by arsenic atoms above and below this plane. The FeAs complex provides the bands at the Fermi level, and the Fermi surface consists of sheets around the $\Gamma$ point and the $M$ point of the Brillouin zone[@LDA0; @Xuetal; @Mazin:2008; @Cao:2008; @LDA4; @Haule:2008].
Beyond this general initial consensus on the commonalities of the different materials, electronic structure calculations differ on the details of the ground state and the band structure near the Fermi level. Both paramagnetic and antiferromagnetic (sublattice and linear SDW) ground states have been reported, with some authors claiming that the system is close to a Mott transition and also possibly to a ferromagnetic state. Crude support for the proximity of competing magnetic states is provided by the known helimagnetism in the layered iron monoarsenide system.
Within weeks of the discovery of the systems, theoretical analyses of various possibilities for the mechanism of superconductivity and the symmetry of the superconducting order parameter have appeared[@Mazin:2008; @Xuetal; @Kuroki:2008; @XDai; @Hanetal]. Eliashberg style calculations based on density functional theory (DFT) determination of electron-phonon coupling constants[@Phonon] suggest that conventional electron-phonon interactions are not sufficient to generate the observed transition temperatures. Thus several authors have discussed electronic pairing mechanisms of the spin fluctuation type[@Mazin:2008; @Xuetal; @Kuroki:2008; @XDai; @Hanetal], but disagree about the symmetry of the ground state, apparently because of the details of the electronic structure used as an input to the calculation. Given the past history of theoretical approaches to unconventional superconductors, it may be some time before a consensus on the correct microscopic approach is forged.
In the intervening period, it would clearly be useful to have some information on the symmetry of the order parameter to guide such theoretical discussions. Evidence for nodes in the order parameter has already been provided by point contact tunnelling[@Shan], which has reported a zero bias state in a series of relatively high-transparency junctions, and specific heat measurements in a magnetic field $H$[@Muetal], which indicate a $C_V/T\sim
\sqrt{H}$ term similar to that predicted by Volovik for a $d$-wave (or, more generally, nodal) superconductor. Because the current experiments have been performed on powdered samples, however, the distribution on the Fermi surface of order parameter nodes, which could provide some information on the symmetry of the pair state is not yet determined. In addition, the point contact measurements probe only the superconducting state at the surface, whereas ideally one would prefer to extract information on the bulk superconducting state. When single crystals are produced, it will be possible to perform what is possibly the simplest bulk probe of the distribution of gap nodes, a measurement of the specific heat of a sample in the presence of a field in the FeAs plane as a function of its angle relative to the crystal axes. The superflow field in the vortex state of the type-II superconductor is known to “Doppler shift" the energies of quasiparticles, changing their local occupation and giving rise to a residual density of states[@volovik] which depends on the angle the field makes with the nodes[@volovik2; @VHCN]. This method of “nodal mapping" was proposed in the context of the cuprates, where the experiment is difficult due to the large phonon background, but has found more fruitful application in lower-$T_c$ materials[@Matsudareview].
In this paper we calculate the specific heat oscillations with magnetic field angle to be expected in the presence of a variety of candidate superconducting pair states. Rather than tie ourselves to any particular microscopic electronic structure calculation, we use a phenomenological two-band model [@Raghu] which captures the essential qualitative features of the bands near the Fermi surface. We find that various extended-$s$ like states can be distinguished from, e.g. $d$-wave or $p$-wave like states by the positions of their nodes. There are also cases, however, where nodes lie in positions on the Fermi surface where $\k_n$ and the Fermi velocity ${\bf v}_F$ are not parallel. In this case the minimum of the specific heat does not correspond precisely to the nodal position and the structure of the set of minima must be examined in detail.
[*Effective band structure.*]{} The crystal structure of LaOFeAs consists of alternating layers of FeAs and LaO. Density functional theory (DFT) calculations show that the energy bands crossing the Fermi level can be assigned to the Fe $3d$ and the As $4p$-orbitals [@LDA0; @Xuetal; @Mazin:2008; @Haule:2008; @Cao:2008; @LDA4; @Kuroki:2008]. Thus, to describe superconducting properties we can consider the LaO layers mainly as spacing layers and possible charge reservoirs. The FeAs layers can be further subdivided into a square Fe lattice with an Fe-Fe spacing $d_{Fe-Fe}=2.82$ Å and an As square lattice displaced by a vector $(1/2,1/2)$ in the $x$-$y$-plane to a position in the center of the Fe squares. Additionally, the As atoms are displaced alternately above or below the Fe plane leading to a pyramidal Fe-As configuration. Due to the alternating sign of the As $z$-displacements the primitive unit cell of LaOFeAs contains two Fe and two As atoms. The axes of the corresponding Brillouin zone (BZ) are aligned in the next nearest neighbor Fe-Fe direction and the BZ has a size of $2 \pi/a \times 2\pi/a$. However, due to the high degeneracy of the two As positions we will treat an effective model consisting of a smaller unit cell having only one Fe and one As atom. This leads to a larger effective BZ that has axes that are aligned to the nearest neighbor Fe-Fe direction. In this case the [*real*]{} BZ occupies a diamond shaped region within the [*effective*]{} BZ.
![(Color online) The two hybridized bands of our model. The white region depicts the narrow energy window around the Fermi energy (dashed line) where the model reproduces the semiquantitative aspects of the LDA band structure. The momenta refer to points in the 2D effective large Brillouin zone (see Fig. \[FS\_fit\]). []{data-label="BS"}](BandStructure_Chao.eps){width=".8\columnwidth"}
Since the thermodynamic properties of the superconducting state are governed by low lying quasiparticle excitations, we restrict our considerations in the following to the region where the energy bands cross the Fermi level, forming the different Fermi surface sheets in the BZ. To simplify the rather complicated band structure, we use a two band model that takes only the iron $d_{xz}$ and $d_{yz}$ orbitals into account [@Raghu]. Here the basic symmetry of the hopping parameters is determined from the direct overlap of the Fe $d$ orbitals as well as from the hopping mediated by the As $p$ orbitals. The model neglects contributions from other orbitals, e.g. hybridization due to the other Fe $d$ orbitals, and the hopping parameters are adjusted to give the generic form of the Fermi surface sheets determined by bandstructure calculations.
![(Color online) The different FS sheets in the large effective BZ calculated within our two-band model. The two hybridized bands result in two sets of Fermi surface sheets, centered around the $\Gamma$ point (blue) and the $M$ point (red) of the real BZ (dashed black line). Backfolding of the large into the small BZ produces the dashed sheets in the small zone. The black crosses show the FS of the paramagnetic ground state determined by DFT [@Cao:2008].[]{data-label="FS_fit"}](FS_Chao_Fit.eps){width=".8\columnwidth"}
Calculating the hopping from the direct overlap of the Fe $d_{xz}$ and $d_{yz}$ orbitals as well as the effective hopping in second order perturbation theory on the path Fe-As-Fe, taking the As $p_x$, $p_y$ and $p_z$ into account, leads to a tight binding Hamiltonian with nearest and next-nearest neighbor hopping between the same orbitals and a next-nearest neighbor exchange hopping between the two bands. Due to the choice of the orbitals there are different nearest neighbor hopping values $t_1$ and $t_2$ for hopping in the $x$ and $y$ directions in one band which are interchanged in the other band. The intraband next nearest neighbor hopping $t_3$ is the same for both bands and both directions, while the interband hopping $t_4$ has a different sign for the $(1,1)$ compared to the $(1,-1)$ direction. After the usual Fourier transformation we can write the intraband energies in momentum space as $$\begin{aligned}
\epsilon_{11} &=& -2 t_1 \cos k_x -2 t_2 \cos k_y - 4 t_3 \cos k_x \cos k_y \\
\epsilon_{22} &=& -2 t_2 \cos k_x -2 t_1 \cos k_y - 4 t_3 \cos
k_x \cos k_y\end{aligned}$$ and the interband exchange energy is $$\epsilon_{12} = \epsilon_{21} = - 4 t_4 \sin k_x \sin k_y$$ Taking the hybridization of the two bands into account one finds for the two bands $$\begin{aligned}
\epsilon^{\alpha} &=& \frac{1}{2} \left(
\epsilon_{11}+\epsilon_{22} -
\sqrt{(\epsilon_{11}-\epsilon_{22})^2
+4 \epsilon_{12}^2} \right)\\
\epsilon^{\beta} &=& \frac{1}{2} \left(
\epsilon_{11}+\epsilon_{22} +
\sqrt{(\epsilon_{11}-\epsilon_{22})^2
+4 \epsilon_{12}^2} \right)\end{aligned}$$ The hopping parameters $t_i$ and the chemical potential $E_F$ can be used to fit the FS of the paramagnetic ground state found within the DFT calculations [@Cao:2008]. We find a reasonable agreement of the FS for the following values $t_1 = -1.2$, $t_2 = 1.35$, $t_3 =
-0.8$, $t_4 = -0.8$ and $E_F=2$. These values lead to the bandstructure shown in Fig. \[BS\].
![(Color online) Candidate order parameter states considered in this work. a) $d_{x^2-y^2}$ state; b) $d_{xy}$ state; c) extended-$s$ state from Ref. [@Mazin:2008] ; d) generalized extended-$s$; e) $p_x$ state ($\Delta_\k\propto \sin
k_x$); f) $p_x+ip_y$ state ($\Delta_\k=\sin k_x +i \sin k_y$) [@Xuetal]. The dashed orange line denotes the small Brillouin zone and the green line denotes locus of gap nodes. []{data-label="Gaps"}](Gaps.eps){width=".8\columnwidth"}
[*Order parameters.*]{} Because of the small coherence length of ${\cal O}$(30 Å), these systems are strongly type-II and it may be appropriate to think of the range of the pairing interaction as being very short, of order the lattice spacing. Generally speaking, order parameters involving pairing on nearest neighbor sites are also those which have been proposed for these systems. We therefore consider as representative candidates the states listed in Fig. \[Gaps\], proposed by various authors, beginning with nearest-neighbor a) $d_{x^2-y^2}$ state ($\Delta_\k\propto\cos k_x
-\cos k_y$)[@Hanetal] ; b) $d_{xy}$ state $(\Delta_\k \propto
\sin k_x \sin k_y)$; and c) extended $s$-wave state ($\Delta_\k
\propto \cos k_x +\cos k_y$) [@Mazin:2008], and e) a $p_x$-wave state. The extended $s$-wave state shown in c) changes sign on the Fermi surface of the model system, as seen in the Figure. On the other hand, its nodes are located at the 45$^\circ$ directions relative to the sheet center at the M point, which is not generic for a state with $s$ ($A_{1g}$) symmetry. We therefore show in Fig. \[Gaps\]d) the result of adding to this state a higher order $s$-harmonic $\Delta_\k \propto (1-\gamma)(\cos k_x +\cos k_y) +
\gamma ( \cos 2k_x +\cos 2k_y)$ with $\gamma=0.05$. For example, the RPA spin fluctuation calculations of Kuroki et al. appear to lead to a more general extended $s$-wave state. Similarly, it can be seen in Fig. \[Gaps\]d) that the points where the nodes cross the Fermi surface sheets are away from the 45$^\circ$ directions (relative to the center of the sheet on the zone face). Finally, we show in Fig. \[Gaps\]f) the nodeless $p_x+ip_y$ state proposed by Xu et al. [@Xuetal].
[*Specific heat.*]{} To get a qualitative understanding of the specific heat oscillations as a function of the rotation angle of an in-plane magnetic field at low temperature it is sufficient to study the spectrum of low energy excitations. To calculate the spectrum in the vortex state we want to follow a semiclassical approach, that neglects the core states and considers only the shift of the quasiparticle energies of the extended nodal states in the presence of a magnetic field. Following [@VHCN] we approximate the vortex lattice using a circular unit cell with radius $R$ and winding angle $\beta$. Then the Doppler shifted quasiparticle energy is $$\delta E^{(i)} = m {\bf v}_F^{(i)} {\bf v}_s = \frac{E_H}{\rho}
\left(\hat{v}_{F,y}^{(i)} \cos \alpha - \hat{v}_{F,x}^{(i)}
\sin \alpha \right) \sin \beta$$ Here ${\bf v}_F^{(i)}$ denotes the Fermi velocity on band $i$, ${\bf v}_s$ is the gauge invariant expression of the quasiparticle flow field around the vortex core and $\alpha$ is the angle between the magnetic field and the $x$-axis of our coordinate system. The dimensionless radial variable $\rho = r/R$ and $\hat{v}_{F,x/y}^{(i)}$ are the components of the Fermi velocity calculated from $\nabla \epsilon_k^{(i)}$, normalized by a Fermi surface averaged value of $v_F^{(i)}$. The energy scale $E_H$ associated with the Doppler shift is $$E_H^{(i)} = \frac{a}{2} \tilde{v}_F^{(i)} \sqrt{\pi H/\Phi_0}$$ where $a$ is a geometric constant characteristic of the vortex lattice, $\Phi_0$ is the flux quantum, and $\tilde{v}_F^{(i)}$ is an averaged Fermi velocity on band $i$ determined from DFT calculations. This procedure of normalizing the Fermi velocities prevents us from overestimating the differences in the energy gradients at the Fermi level of the simplified two-band model. Using the Doppler shifted energy in a BCS-like density of states we can calculate the low energy spectrum as $$N_0^{(i)} = \mathrm{Re} \left\langle \left\langle \frac{
|\delta E^{(i)}|}
{\sqrt{\left(\delta E^{(i)} \right)^2 - \left(\Delta_k /E_H^{(i)} \right)^2 \rho^2}}
\right\rangle_{H} \right\rangle_{FS}$$ where the angular brackets denote an average over the vortex cell ($H$) and over the Fermi surface ($FS$), respectively. The integral over the vortex cell can be done analytically leading to $$N_0^{(i)} (\alpha)= \left\langle \mathrm{min} \left[ 1,
(E_H^{(i)}/\Delta_k)^2 \left(\hat{v}_{F,y}^{(i)} \cos \alpha -
\hat{v}_{F,x}^{(i)} \sin \alpha \right)^2 \right]
\right\rangle_{FS}$$ The last average is to be performed over the different Fermi surface sheets in the unfolded Brillouin zone leading to an oscillation of the low energy spectrum as a function of the angle of the applied magnetic field. These oscillations can be directly determined by low temperature thermodynamic measurements, like specific heat or the thermal conductivity.
![(Color online) Residual density of states $N(\omega=0,{\bf H})$ vs. $\alpha$, the angle $\bf H$ makes with the $x$-axis for the different superconducting states shown in Fig. \[Gaps\]. $N(\omega=0,{\bf H})$ is proportional to the linear specific heat coefficient at low temperature. Note all curves have been offset by a constant amount for clarity.[]{data-label="N0alpha"}](N0alpha.eps){width=".8\columnwidth"}
In Fig. \[N0alpha\], we show the residual angle-dependent density of states, or linear specific heat coefficient as a function of field angle $\theta$. For the most part, one expects fairly straightforward generalizations of the results for a circular Fermi surface[@VHCN], as seen for the $p_x$ and $d$-symmetry states: minima in the specific heat at low temperatures $T\ll E_H^{(i)}$ lie at the expected nodal positions. The nodeless $p_x+ip_y$ state produces no Volovik effect, is therefore not plotted in Fig. \[N0alpha\] and is apparently not a candidate for the Fe-based materials. In the extended-$s$ cases, some interesting points arise. It is seen from Fig. \[Gaps\]c) that in the simple extended-$s$ case, the nodes are located along the 45$^\circ$ directions. Nevertheless the minima in Fig. \[N0alpha\] are slightly displaced symmetrically with respect to these nodes; this is due to the fact that the M sheets are elliptical, with the consequence that the Fermi velocities are not parallel to nodal $\k_n$ measured from the sheet center. When there are higher harmonics, such as in the generalized extended $s$-wave case, the nodes themselves actually are displaced from the 45$^\circ$ directions. Thus a measurement of this kind can identify an extended $s$ state by the displacements of the minima, but a direct correspondence with the nodal positions requires a precise knowledge of the underlying Fermi surface.
[*Conclusions.*]{} In this paper, we have proposed that the measurement of specific heat oscillations as a function of the magnetic field angle in the FeAs plane of the new iron-based superconductors could be the simplest and most straightforward bulk determination of gap symmetry, once single crystals or highly oriented powders are available. To simplify the calculation, we used an effective two-band model with parameters chosen to reproduce the DFT Fermi surface. We then calculated the low-temperature linear term in the specific heat to be expected as a function of field angle for a variety of candidate states. The elliptical Fermi surface pockets near the M points introduce some interesting complications in the problem relative to the usual picture of $C_V({\bf H})$ oscillations over the field angle.
This work is supported by DOE DE-FG02-02ER45995, NSF/DMR/ITR-0218957 (HPC and CC), and DOE DE-FG02-05ER46236 (PJH). SG gratefully acknowledges support by the Deutsche Forschungsgemeinschaft. DJS acknowledges support from the Center for Nanophase Material Science, ORNL.
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|
---
abstract: 'We present a hybrid approach to groundwater transport modeling, “CTRW-on-a-streamline”, that allows continuous-time random walk (CTRW) particle tracking on large-scale, explicitly-delineated heterogeneous groundwater velocity fields. The combination of a non-Fickian transport model (in this case, the CTRW) with general heterogeneous velocity fields represents an advance of the current state of the art, in which non-Fickian transport models *or* heterogeneous velocity fields are employed, but generally not both. We present a general method for doing this particle tracking that fully separates the model parameters characterizing macroscopic flow, subscale advective heterogeneity, and mobile-immobile mass transfer, such that each can be directly specified a priori from available data. The method is formalized and connections to classic CTRW and subordination approaches are made. Numerical corroboration is presented.'
author:
- 'Scott K. Hansen[^1] and Brian Berkowitz[^2]'
bibliography:
- 'auroralit.bib'
- 'auroralit-rev.bib'
title: 'Modeling non-Fickian solute transport due to mass transfer and physical heterogeneity on arbitrary groundwater velocity fields'
---
Introduction
============
Motivation
----------
Classical Eulerian numerical models that employ a discretized advection-dispersion equation (ADE) are often not sufficient to model groundwater solute transport with needed realism because they fail to capture important physical heterogeneity and mass transfer at scales beneath their discretization scale. Additionally, their flow models may be oversmoothed even above their discretization scale due to a lack of constraining information. Non-Fickian transport models are a generalization of the classical transport models that can capture a wider range of physics, and have demonstrated success at capturing realistic chemical physics that cannot be modeled by classic approaches by introducing additional model flexibility. In the case of the popular continuous-time random walk (CTRW) approach, the additional flexibility comes in the free selection of a waiting time probability distribution, $\psi(t)$, that better describes the dispersion process.
However, when capturing the effects of groundwater velocity heterogeneity, there has commonly been an “all or nothing” choice. Either model all the heterogeneity explicitly and then use an ADE-based transport model, or model all heterogeneity implicitly by considering its effects, combined with those of any other sources of non-Fickian behavior, in selection of the non-Fickian model parameters (e.g., $\psi(t)$). A consequence is that it is difficult to incorporate partial knowledge about large-scale flow into non-Fickian solute transport models, and it is generally difficult to predict correct model parameters a priori, as these depend in non-straightforward ways on the interaction of multiple physical and chemical sources of heterogeneity. This limitation may have reduced the penetration of advanced transport models from academic research into hydrogeologic practice. But whatever the underlying reasons, practitioners have typically favored explicitly heterogeneous models incorporating large-scale flow information coupled with ADE-based transport, whereas researchers have often chosen conversely: employing homogeneous, often quasi-1D, flow alongside more exotic non-Fickian transport models. Ideally, a hybrid approach to modeling would be available (a) allowing usage of explicitly-delineated, spatially-heterogeneous groundwater velocity fields, with non-Fickian transport modeling used *only* to capture small-scale physics, and (b) whose non-Fickian model parameters have a straightforward relationship to the physics of the natural system they model. Developing such a hybrid approach is the goal of this work.
Sources of heterogeneity beneath flow model support scale {#sec: hetero below support}
---------------------------------------------------------
Before it developing a hybrid approach in which small-scale flow heterogeneity and mass transfer are captured with a stochastic transport model, it is naturally important to understand these processes and their impact on transport, and why a stochastic approach is valuable.
Consider an ideal scenario in which the groundwater flow field is known down to the pore scale in the plume region and computational power is unlimited, with the only stochastic process operative being molecular diffusion. In this situation, solute transport could in theory be captured exactly by a deterministic model, and perhaps by a discretized advection-dispersion equation. However, in realistic scenarios only the volume average groundwater velocity over some larger scale may be known with any precision, and the local-scale velocity fluctuations are random in an epistemic sense: unknown except potentially for their statistics. The first effort to deal with this problem was the so-called macrodispersion theory, which attempts encapsulate the effects of all the unknown velocity fluctuations into an additional Fickian dispersive term, justified by the Central Limit Theorem. While this approach is appropriate for large distances from the solute source and smaller heterogeneity [@Hansen2018], frequently we are interested in understanding solute breakthrough in systems for which these restrictions are not appropriate. In such cases, non-Fickian behavior is generally observed and a more general approach such as usage of the continuous time random walk (CTRW)—where the travel time is captured by a travel time pdf—is appropriate.
The groundwater velocity field is determined by solution of the groundwater flow equation on a (generally) heterogeneous hydraulic conductivity ($K$) field, and it is reasonable that its heterogeneity statistics will determine flow heterogeneity. On the assumption of lognormality of the $K$ field, (typically unrealistically) small log-$K$ variance, and multi-Gaussian correlation structure with light-tailed (exponentially decaying or finitely supported) semivariogram, the macrodispersion theory [@Rubin2003] provides analytical expressions for the solute transport behavior. However, when these assumptions are violated, we must rely on numerical studies linking heterogeneity and breakthrough curve statistics. There are not many of these in the literature, but to our knowledge, those that exist point strongly in two directions. For mild-to-moderate hydraulic conductivity heterogeneity and multi-Gaussian correlation structure, numerical studies have indicated a lognormal distribution for $\psi(t)$, gradually acquiring power law behavior in the tails as heterogeneity increases [@Gotovac2009; @Hansen2018]. We stress that sub-Darcy-scale heterogeneity and trapping may still cause power law tailing even under mild or nonexistent $K$-field heterogeneity, as numerous experiments have shown (see discussion in [@Berkowitz2006]). For lognormally distributed $K$ and heavy-tailed (i.e., power law) semivariograms, studies have pointed in different directions. [@Moslehi2017] analyzed breakthrough in conductivity fields with a single, moderate, heterogeneity ($\sigma^2_{\ln K}$ = 4) and different strengths of power law semivariogram, finding their breakthrough curves generally well described by lognormal distributions, except for some heavy tailing. [@Bolster2012] considered mildly heterogeneous, but power law correlated, *velocity* fields, showing analytically, under the validity conditions of their perturbation expansion, that breakthrough curves show power law tailing with the same exponent as the semivariogram. [@Zhang2013] performed a large-scale simulation study on highly heterogeneous realizations generated with depositional simulators (i.e., non-multivariate-Gaussian realizations), finding generally power-law behavior across realizations. They further found that where the islands of the lowest-$K$ material had a power law size distribution (loosely analogous to the correlation length in a semivariogram), that this determined the tail exponent. However, the relationship between exponents that they uncovered differed from that of [@Bolster2012]. [@Edery2014] performed a numerical particle tracking study on moderate-to-large heterogeneity ($\sigma^2_{\ln K}$ = 3 to 7) multivariate Gaussian fields, finding their breakthrough curves generally well described by a truncated power law distribution; we also speculate, based on analysis for [@Hansen2018], that the lower-heterogeneity breakthrough curves would also be well described by lognormal distributions, again deviating towards power law behavior at late time. Finally, [@Tyukhova2016] considered an idealized problem of transport in a homogeneous medium with embedded spheres, each homogeneous and each with random (uniform) conductivity. They found that for truncated power law conductivity distribution, breakthrough curves were truncated power law, and that for (truncated) lognormal conductivity distributions, breakthrough curves were lognormal.
In addition to the heterogeneous advection just discussed, a second category of transport process may also cause sub-support-scale heterogeneity: mobile-immobile mass transfer (MIMT), or broadly speaking, trapping processes. This category naturally includes adsorption, in which solute is truly immobilized, but also includes processes in which diffusion roughly orthogonal to the local advection direction moves solute into low-permeability regions in which advection is essentially inoperative. It is known that such physics may be captured by CTRW [@Berkowitz2006; @Berkowitz2016]. However, they are also frequently treated by a variety of specialized modeling approaches, including retardation factors, explicit two-domain models [e.g., @Neretnieks1980], dual porosity approaches [@Gerke1993], kinetic sorption models [see, e.g., @Fetter1999], multi-rate mass transfer (MRMT) [@Haggerty1995], and fractal-MRMT / memory function approaches [@Schumer2003]. We note that heterogeneous advection has also been modeled with MRMT (the so-called MRMT-1 of [@Dentz2003]), but for our purposes we treat it separately.
Numerical modeling of transport with particle tracking
------------------------------------------------------
A common approach to numerical modeling of transport is to write the governing partial differential or integro-differential equation and then solve it analytically or semi-analytically using transform methods. However, this approach is generally only tenable in systems with spatially-uniform parameters. For systems with spatially non-uniform parameters, solution of the governing equation must be accomplished by spatial discretization. This introduces the problems of numerical dispersion and (in reactive models) numerical mixing. Where the governing equation contains a temporal integral, as in non-Fickian transport models, numerical instability becomes a concern, too.
Particle tracking approaches avoid these difficulties, and have been found to work well under very general conditions, as long as enough particles are used. A classic particle tracking algorithm typically uses a constant temporal user-specified step size, $\Delta_t$. The simplest possible particle update conditions apply in the case of pure advection (i.e., streamline tracing) iteration of each particle’s position would be performed according to the following equations:
$$\begin{aligned}
\bm{x}_{n+1} &=& \bm{x}_n + \bm{v}(\bm{x}_n) \Delta_t \label{eq: pure adv}\\
t_{n+1} &=& t_n + \Delta_t \label{eq: time update}
\end{aligned}$$
To model Fickian dispersion one modifies to read
$$\bm{x}_{n+1} = \bm{x}_n + \bm{v}(\bm{x}_n) \Delta_t + \bm{\Delta}_{\bm{x},n}^D, \label{eq: with disp}$$
where in the case of isotropic diffusion
$$\Delta_{\bm{x},n}^D \equiv \bm{\eta},
\label{eq: fickian disp}$$
where $\bm{\eta}$ is a random 3-vector, each of whose components $\eta_i \sim N(0,2 D \Delta_t)$, and $D$ is the Fick’s Law constant. For CTRW particle tracking, either or may be used, but the time step selection in is modified so as to be selected at random:
$$\Delta_t \sim \psi(\Delta_t),$$
where $\psi$ is a probability distribution function with strictly non-negative support.
Below, we will see how to modify these equations to capture the sources of heterogeneity discussed in Section \[sec: hetero below support\] while also incorporating coarse-scale flow information. Our strategy is to perform a CTRW that makes essentially fixed-length steps down the streamlines of the coarse-scale flow field. Such an approach is sometimes referred to as a time-domain random walk (TDRW), a special case of CTRW originally developed in 1D based on ADE transport solutions [@Banton1997; @Reimus2002], and subsequently developed to model MIMT in discrete fractures by [@Delay2001]. This approach was formalized using CTRW transition-time distribution theory by [@Cvetkovic2002] and explicit solutions for rock matrix residence times were presented by [@Painter2008]. Subsequently, the TDRW approach was employed in a large number of particle tracking studies on discrete fracture networks. For transport in heterogeneous porous media, [@Hansen2014] introduced a cognate CTRW approach, capturing transport as a series of transitions among parallel planes orthogonal to mean flow (leading to a quasi-1D upscaled transport model). Subsequently, another quasi-1D approach that approximates flow in heterogeneous media with an effective medium and also incorporates MIMT was discussed by [@Cvetkovic2016]. TDRW solutions based on classical (ADE) ideas and that include transverse dispersion were also presented by [@Bodin2015].
Some other works whose concepts we build upon include [@Cortis2004], who introduced a zoned approach to CTRW modeling of subsurface transport, with different CTRW parameters manually defined for different regions. The idea that a travel time distribution for advection with random motion can be expressed as the product of an average velocity and a distribution representing heterogeneity, is latent in the [@Kreft1978] solution for advective-dispersive breakthrough (see the factorization we present in ). This idea has been employed in the context of scaling CTRW transition time distributions to a local advection velocity by [@Srinivasan2010] and by [@Kang2014]. The concept of mapping an advective travel time to a random total time including multiple immobilization events was presented by [@Benson2009] for exponentially-distributed mobile and immobile times. A similar approach has been adopted for arbitrary immobile time distributions by [@Hansen2016] and by [@Russian2016]. Finally, the CTRW-on-a-streamline approach can be seen as extending the ideas of [@Cirpka2003] to general, potentially non-Fickian, transport: instead of using an additional dispersion term to capture small-scale flow field variability excluded from a smoothed deterministic model, we propose to use a CTRW transition time distribution.
Developing from the ideas in these works, we present a complete treatment capturing local-scale heterogeneous advection, MIMT, and transverse dispersion using fully 3D streamlines and presenting explicit, physics-based formulae for the relevant CTRW transition-time distributions.
Development of the CTRW-on-a-streamline approach
================================================
Theory underlying the approach {#sec: theory}
------------------------------
For pure streamline tracing, it is possible to choose a fixed *distance*, $d$, that is traversed in equation by adjusting the time step with reference to the local velocity: $\Delta_{t_O,n} = \frac{d}{\| \textbf{v}(\textbf{x}_n) \|}$. We add the subscript $n$ on each time step, $\Delta t_n$, because the time taken with each advective step is now variable and the variable $t_O$ stands for “operational time”, whose significance will be explained below. This *fixed distance increment, variable time* approach to large-scale advection is advantageous versus the common *fixed time increment, variable distance* approach because it sets us up to use a CTRW transition time distribution to capture small-scale advective heterogeneity and mass transfer. To wit: we first *define* a random walk transition to occur whenever another increment $d$ downgradient is traversed. We then seek to define the conditional pdf, $\psi(\Delta_{t_C}|\Delta_{t_O})$, for the true or *clock* time, $\Delta_{t_C}$, taken to traverse that increment when small-scale physical heterogeneity and mass transfer are considered, conditioned on the notional or *operational* time that would be taken to traverse the same segment by pure advection on the large-scale streamlines.
Each segment of length $d$ of a large-scale streamline may be conceptualized as representing a flux-weighted ensemble, or “bundle”, of small-scale, adjacent stream tubes of length $d$ with varying local harmonic mean hydraulic conductivity. Each small-scale stream tube has a different advection time, $\Delta_{t_A}$, and because the small-scale stream tubes in an ensemble are epistemically interchangeable (we have no knowledge of their exact position or of the particular small-scale tube to which a particle being tracked along a large-scale streamline belongs), it is sensible to define a conditional probability distribution, $\phi(\Delta_{t_A}|\Delta_{t_O})$.
The nature of $\phi(\Delta_{t_A}|\Delta_{t_O})$ can be characterized by noting that the small-scale stream tubes in an ensemble share approximately the same gradient: justified by the fact that the global head field is an integral quantity and is smooth relative to the underlying hydraulic conductivity ($K$) field. Because $K$ is everywhere independent of the boundary conditions on head that determine the magnitude of the large-scale effective $\bm{v}$, and the head drop is shared by small-scale stream tubes in an ensemble, it follows that the Darcy velocities in each of the tubes in the ensemble always maintain a same constant of proportionality to $\|\bm{v}\|$, regardless of system boundary conditions. And because $d$ is fixed, the same is true of travel time, $\Delta_{t_A}$. Thus a simple scaling law applies: $\phi(\Delta_{t_A}|\Delta_{t_O})\Delta_{t_O}=\phi(\Delta_{t_A}/\Delta_{t_O}|1)=:f(r)$, where $f$ is some unknown pdf, solely determined by the statistics of small-scale heterogeneity, which contains all relevant information for mapping from operational time to true advection time, and $r=\Delta_{t_A}/\Delta_{t_O}$. Another way of looking at this is that the average of the ratio $\Delta_{t_A}$ to $\Delta_{t_O}$ is independent of the actual magnitude of $\Delta_{t_O}$ (and hence the magnitude of the groundwater velocity). The distribution $f(r)$ is determined by flow heterogeneity alone.
This informal argument is supported by a number of additional lines of evidence. We note that this is true for all advective-dispersive systems (see equation ). The formal analysis of [@Tyukhova2016] considers transport in a regularly-spaced grid of equal-sized circular inclusions with random (low) conductivity in a uniform higher-conductivity matrix and shows mathematically that the velocity in any each inclusion varies in proportion with the system average flux. [@Comolli2016] show how under pure advection, the pdf for fixed distance travel time can be written in terms of the Lagrangian velocity pdf, and that under ergodic conditions the Eulerian velocity pdf determines the Lagrangian. Because the Eulerian pdf naturally scales linearly with global hydraulic gradient, the ratio of $\Delta_{t_A}$ to $\Delta_{t_O}$ must be independent of average velocity under ergodic conditions.
Once advective heterogeneity is specified by $f(r)$, any MIMT, whether due to diffusion into secondary porosity or to chemical adsorption, can be captured by defining two additional probability distributions. The first distribution is exponential, with a rate constant, $\lambda$, representing the probability of immobilization of mobile solute per unit time traveled. The second, not necessarily exponential, distribution, $g(t)$, represents the time for the length of a single particle immobilization event. The exponential form of the mobile-time distribution is justified by the assumption of randomly located sites that are dense relative to length scale $d$. We note that the $\lambda$ and $g$ approach captures even multiple types of immobilization sites with different capture and release distributions (i.e. multi-rate mass transfer), as the time until the first event for multiple simultaneous exponentially-distributed processes is itself exponential, and we can create an immobilization-site-prevalence-weighted average of the immobilization time pdf’s for the various site types.
To quantify the travel time increase due to MIMT, we can develop a conditional pdf, $\zeta(\Delta_{t_C}|\Delta_{t_A})$. We note that the total (clock) time is the advection time, $\Delta_{t_A}$ plus the delay due to $n$ independent immobilization events each of whose lengths is drawn from $g(t)$. The probability mass function for $i$ immobilization events, $w(i)$, is Poisson distributed with parameter $\lambda \Delta_{t_A}$, leading [@Billingsley1986 p. 262] to probability mass function
$$w(i) = e^{-\lambda \Delta_{t_A}}\frac{(\lambda \Delta_{t_A})^i}{i!},$$
Our conditional pdf, $\zeta$, is thus a weighted average of $i$-fold autoconvolutions of $g$:
$$\zeta(\Delta_{t_C}|\Delta_{t_A}) = \sum_{i=0} w(i)\times g^{(i*)}(\Delta_{t_C}-\Delta_{t_A}).
\label{eq: zeta}$$
Where sites are sparse, an exponential distribution whose rate constant represents probability of immobilization of mobile solute per streamline *distance* traveled may instead be more appropriate [@Margolin2003]. On this conception, we must understand $\lambda$ as capture probability per unit *distance*, and $\zeta$ becomes no longer dependent on $\Delta_{t_A}$:
$$w(i) = e^{-\lambda d}\frac{(\lambda d)^i}{i!},$$
and we have
$$\zeta(\Delta_{t_C}|\Delta_{t_O},d) = \sum_{i=0} w(i)\times g^{(i*)}(\Delta_{t_C}-\Delta_{t_O}).
\label{eq: zeta dist}$$
we will not discuss this second conception further, but it is important to stress its compatibility with our mathematics.
Although it is most convenient computationally to work with $f$, $g$, and $\lambda$, we can easily relate the above analysis to the CTRW transition distribution, $\psi$, by constructing the conditional distribution
$$\psi(\Delta_{t_C}|\Delta_{t_O}) = \int_0^{\Delta_{t_C}} \zeta(\Delta_{t_C}|\Delta_{t_A})\ \phi(\Delta_{t_A}|\Delta_{t_O})\ d\Delta_{t_A}.$$
This can be converted into a classic (unconditional) CTRW distribution by marginalizing
$$\psi(\Delta_{t_C}) = \int_0^{\Delta_{t_C}}\psi(\Delta_{t_C}|\Delta_{t_O})\ h_d(\Delta_{t_O}),$$
where $h_d$ is an unknown but determinate pdf for the operational time for all transitions of length $d$ anywhere in the domain. We see here two reasons why it is advantageous to employ the CTRW-on-a-streamline approach when large-scale information is available. First, the move $\psi(\Delta_{t_C}|\Delta_{t_O})$ to $\psi(\Delta_{t_C})$ involves a loss of information: under a classic approach, the same spatially-averaged $\psi(\Delta_{t_C})$ is used at all locations, inducing a loss of fidelity. Second, it is actually *harder* to compute this distribution because it depends on an additional a priori unknown pdf that must be constrained by statistical information.
Finally, we note that our analysis has so far detailed no mechanism for transfer of solute laterally between streamlines. This small-scale transverse dispersion is a Fickian process may be incorporated via the $\bm{\Delta}_{\bm{x},n}^D$ term in . Unlike the isotropic diffusion case , here we are only looking to model dispersion in the plane transverse to the streamline. First, we need to compute two orthonormal vectors which are also orthogonal to $\bm{v}$. To do so, an arbitrary vector $\textbf{a}$, not collinear with $\textbf{v}(\textbf{x})$, is specified. Then, orthonormal vectors are computed $$\begin{aligned}
\bm{n}_1 &=& \frac{\bm{a} \times \bm{v}(\bm{x})}{\|\bm{a} \times \bm{v}(\bm{x}) \|_2} \label{eq: n1}\\
\bm{n}_2 &=& \frac{\bm{n}_1 \times \bm{v}(\bm{x})}{\|\bm{n}_1 \times \bm{v}(\bm{x}) \|_2} \label{eq: n2}
\end{aligned}$$ For compactness, we define the matrix $\bm{N}(\bm{v})\equiv[\bm{n}_1\ \bm{n}_2]$ (i.e., the matrix whose columns are the streamline-orthogonal unit vectors $\bm{n}_1$ and $\bm{n}_2$), then we define $\bm{\eta}$ to be a 2-vector, each of whose components, $\eta_i \sim \mathcal{N}(0,2 \alpha_t d)$, where $\alpha_t$ represents the transverse dispersivity. Finally, the perturbation between streamlines is computed by matrix-vector multiplication:
$$\bm{\Delta}_{\bm{x},n}^D = \bm{N}(\bm{v})\bm{\eta}.$$
Formal specification
--------------------
It is straightforward to distill the above analysis into a set of formal equations that define the particle position and time updating rules for the CTRW-on-a-streamline approach. The top level equations are: $$\begin{aligned}
\bm{x}_{n+1} &=& \bm{x}_n + \bm{v}(\bm{x}_n) \Delta_{t_O,n} + \bm{N}(\bm{v}(\bm{x}_n))\bm{\eta}, \label{eq: space update formal}\\
t_{C,n+1} &=& t_{C,n} + \Delta_{t_C,n}, \label{eq: time update formal}
\end{aligned}$$ where $$\begin{aligned}
\Delta_{t_O,n} &=& \frac{d}{\| \bm{v}(\bm{x}_n) \|}, \label{eq: operational from velocity} \\
\eta_i &\sim& \mathcal{N}(0,2 \alpha_t d),\label{eq: dispersion formal}\\
\Delta_{t_C,n} &\sim& \psi(\Delta_{t_C,n}|\Delta_{t_O,n}).\label{eq: clock update formal}
\end{aligned}$$ Placed in this form, it is clear that equations (\[eq: space update formal\]-\[eq: clock update formal\]) represent an evolution of the classic CTRW equations, with the conditioning of $\psi$ on $\Delta_{t_O,n}$ representing the major novelty (the restriction of transverse dispersion to the plane orthogonal to the streamline is a minor novelty). It is also clear from the examination of equations (\[eq: space update formal\]-\[eq: time update formal\]) that CTRW-on-a-streamline can be seen as a form of subordination technique [@Benson2009; @Hansen2016]: particle location is updated according to a notional operational time, whereas particle time is updated in accordance with a true clock time.
Numerical implementation
========================
Pseudocode
----------
As mentioned, it is not optimal to compute $\Delta_{t_C,n}$ by parameterizing the distribution $\psi(\Delta_{t_C,n}|\Delta_{t_O,n})$ and then drawing random variables from it. Rather, it is easier to build up $\Delta_{t_C,n}$ by first computing $\Delta_{t_A,n}$ by drawing from the pdf $f(\cdot)$, defining the advection time per unit operational time, and then computing the additional immobile time by making draws from $\mathrm{Poisson}(\cdot;\lambda \Delta_{t_A,n})$ and $g(\cdot)$. The following algorithm simulates a particle’s transport in accordance with equations (\[eq: space update formal\]-\[eq: clock update formal\]), by means of these intermediate distributions:
1. For step $n$ = 0, initialize the particle’s position in space and time by constructing the tuple $(\bm{x}_0,t_C)$ and decide on streamline tracing increment $d$.
2. Referring to the explicit large-scale discrete velocity field, determine $\bm{v}(\bm{x}_n)$.
3. Compute the operational time $\Delta_{t_O,n}$ by using .
4. Compute actual advection time $\Delta_{t_A,n}$ by drawing randomly from the advection-time-per-unit-operational-time pdf $f$, and multiplying that result by $\Delta_{t_O,n}$.
5. Compute the number of immobilization events, $i$, in advection time $\Delta_{t_A,n}$ by drawing randomly from the distribution $\mathrm{Poisson}(\cdot;\lambda \Delta_{t_A,n})$.
6. Draw $i$ random numbers independently from the $g(\cdot)$, the pdf for the length of a single sojourn in the immobile state, and sum them to yield the total time immobile time while traveling $d$ units down the streamline.
7. Compute $\Delta_{t_C,n}$ by adding total immobile time and $\Delta_{t_A,n}$.
8. Determine transverse dispersion by computing $\bm{N}(\bm{v}(\bm{x}_n))$ with (\[eq: n1\]-\[eq: n2\]), drawing the random vector, $\bm{\eta}$, according to , and multiplying.
9. Update the particle’s position, $\bm{x}_{n+1}$, according to .
10. Update the particle’s clock time, $t_{C,n+1}$, according to .
11. Record any events of interest (breakthroughs or particle location snapshots).
12. If $t_{C,n+1}$ is less than the end time for the simulation, increment the step index, $n$, by one and loop back to step 2.
Note that because particles do not interact and have independent clocks, (a) this algorithm can be performed in parallel for multiple particles, and (b) particles can be initialized at different clock times. A schematic diagram summarizing the essential procedural steps is given in Figure \[fig: schematic\].
![Schematic diagram outlining the essential conceptual steps of the CTRW-on-a-streamline approach.[]{data-label="fig: schematic"}](FIG_schematic.pdf)
Of course, to completely specify the algorithm, we need to specify $f(\cdot)$, $\lambda$, and $g(\cdot)$. How this is done depends on the underlying physics to be modeled; considerations are discussed immediately below.
Specification of advective heterogeneity
----------------------------------------
Advective heterogeneity refers to varying local pore water velocities experienced by particles as they are advected along a stream tube and experience transverse diffusion. For our purposes, it also incorporates physics that are sometimes considered using MRMT techniques (i.e., the “MRMT-1” class outlined in [@Dentz2003]), namely advection into discrete low-velocity features. The CTRW-on-a-streamline approach captures advective heterogeneity by directly encoding the spectrum of times taken to travel $d$ in the small-scale stream tubes associated with a large-scale streamline in the distribution $\phi(\Delta_{t_A}|\Delta_{t_O})$, or equivalently in $f(\cdot)$. For predictive modeling purposes, we would like to express the parameters defining the distribution $f(r)$ in terms of well-known parameters describing the subsurface heterogeneity, as in the case of MIMT. For the case of homogeneous local-scale Fickian transport, this is easily accomplished analytically: an inverse Gaussian distribution applies. However, heterogeneity at all scales complicates the picture.
Although, as outlined in the introduction, there is no general theory relating ensemble breakthrough parameters to conductivity field statistics, there are three classes of pdf supported by the literature for capturing advective heterogeneity pdf $f(r)$, each suitable for different problems. These classes are: (i) inverse Gaussian, which derives from Fickian analysis, (ii) lognormal, applicable to mildly or moderately heterogeneous systems, and (iii) power law / truncated power law, perhaps the most widely applicable, capturing the strong heterogeneity often encountered at field scale. We discuss how to specify each for use with the CTRW-on-a-streamline approach in turn.
### Fickian dispersion: inverse Gaussian behavior
For longitudinal, local-scale Fickian dispersion, the travel time pdf for a transition of length $d$, is simply the well-known Inverse Gaussian distribution, as calculated by, e.g., [@Kreft1978]. We write it in factored form as
$$\phi(\Delta_{t_A}) = \frac{1}{\Delta_{t_A}} \left[\frac{d}{\sqrt{4\pi\alpha_l\|v\|\Delta_{t_A}}}\exp\left\{ \frac{-(d - \|v\|\Delta_{t_A})^2}{4\alpha_l\|v\|\Delta_{t_A}} \right\}\right],
\label{eq: KZ inv gauss}$$
where, $\alpha_l$ is the Fickian longitudinal dispersivity. Note that the quantity in square brackets is dimensionless. Further noting that $r = \Delta_{t_A}/\Delta_{t_O} = \|v\|\Delta_{t_A}/d$, we may rephrase in terms of $r$:
$$f(r)=\frac{1}{r\sqrt{4\pi A r}}\exp\left\{-\frac{(r-1)^2}{4Ar}\right\},
\label{eq: f ig}$$
where we also define the dimensionless variable $A\equiv\alpha_l/d$.
In Section \[sec: theory\], we present a number of strands of evidence supporting the contention that the distribution of the ratio $\Delta_{t_A}/\Delta_{t_0}$ is independent of the large-scale groundwater velocity, and is rather a proxy for small-scale heterogeneity. The analysis deriving , which does not depend explicitly on either $\Delta_{t_A}$ or $\Delta_{t_O}$ from , shows that this is true for all systems described by the advection-dispersion equation.
### Lognormal behavior
The lognormal travel time distribution is defined by the variance of log arrival time distributions, $\sigma_{\ln t}^2$, rather than $\alpha_l$, but is otherwise similar. For moderate levels of heterogeneity, $\sigma_{\ln t}^2$ can be predicted from $\sigma_{\ln K}^2$ and correlation length by numerical study [@Gotovac2009; @Beaudoin2013; @Hansen2018]. Provided that $\Delta_{t_O}$ is the geometric mean of the travel times in its corresponding stream tube bundle, we may write exactly:
$$f(r)=\frac{1}{r\sqrt{2\pi\sigma_{\ln t}^2}} \exp\left\{-\frac{(\ln r+\frac{1}{2}\sigma_{\ln t}^2)^2}{2\sigma_{\ln t}^2}\right\}.
\label{eq: f ln}$$
The assumption that large-scale streamline velocity *is* the geometric mean of the corresponding small-scale stream tube velocities restates the assumption that the effective hydraulic conductivity for a region is its geometric mean conductivity [@Tsang1994] (discussed at length in chapter 5 of [@Rubin2003]), combined with our general assumption that the head field being smooth relative to the conductivity field.
### Power law behavior
It is straightforward to write a transition time pdf for a pure power law distribution (i.e. Lomax distribution):
$$f(r)=\frac{\alpha\lambda}{(r+\lambda)^{1+\alpha}}.
\label{eq: f pl}$$
General theory relating the parameters $\alpha$ and $\lambda$ to aquifer heterogeneity statistics does not appear to exist in the literature, unlike for Fickian and lognormal breakthrough, although relations have been developed [e.g. @Edery2014; @Nissan2019].
Specification of MIMT {#sec: MIMT}
---------------------
The formulation of MIMT is that first employed by [@Margolin2003], corresponding to an Eulerian transport equation of the form
$$\frac{\partial c_\mathrm{m}}{\partial t} + \lambda c_\mathrm{m} - \int_{0}^{t} g(t-\tau) \lambda c_\mathrm{m}(\tau) d\tau = L\left\{c_\mathrm{m}\right\},
\label{eq: our MIMT}$$
where $L$ is a linear transport operator, and we use $c_\mathrm{m}$ to refer explicitly to the mobile concentration. This differs superficially from many of the approaches to MIMT in the literature, so it is necessary to outline how this relates to them, and how $\lambda$ and $g(t)$, introduced in Section \[sec: theory\], may be determined explicitly from the parameters defining other MIMT formulations seen in the literature. We consider three such cases in the next subsections.
### Connection to MRMT memory function
MRMT is a formulation that obviously generalizes single-rate mass transfer, but also dual-porosity models, explicit models of diffusion into secondary structures such as slabs, spheres, and cylinders, and slow kinetic sorption. The general MRMT transport equation has the form
$$\frac{\partial c_\mathrm{m}}{\partial t} + \frac{\partial c_\mathrm{im}}{\partial t} = L\left\{c\right\},
\label{eq: normal MIMT}$$
and needs to be closed by means of a relationship between $c_\mathrm{im}$ and $c_\mathrm{m}$. This relationship is most commonly expressed as a convolution,
$$\frac{\partial c_\mathrm{im}}{\partial t} = G*\frac{\partial c_\mathrm{m}}{\partial t},
\label{eq: standard im-m relation}$$
where $G(\cdot)$ is a “memory function”. This is the formulation used, for example, in the papers of [@Carrera1998] and [@Haggerty2000] (although a typo in the first reference’s equation 13 causes it to be missing the time derivative on its LHS), and also in the fractal MIM formulation of [@Schumer2003]. Similarly to [@Haggerty2000], we may integrate by parts to yield
$$\frac{\partial c_\mathrm{im}}{\partial t} = G(0)c_\mathrm{m}(t) - \cancelto{0}{G(t)c_\mathrm{m}(0)} + \frac{\partial G}{\partial t}*c_\mathrm{m}.
\label{eq: standard im-m final}$$
The canceled term arises from the fact that is derived for an initially zero-concentration domain. (See equation 5 of [@Schumer2003] for the equivalent version of with a general initial condition.) An apparently different formulation relating the immobile concentration to the mobile concentration is used by [@Dentz2003]:
$$c_\mathrm{im} = \int_0^t G(t-\tau)c_\mathrm{m}(\tau)d\tau.
\label{eq: dentz im-m relation}$$
However, when differentiated with respect to $t$, it yields
$$\frac{\partial c_\mathrm{im}}{\partial t} = G(0)c_\mathrm{m}(\tau) + \frac{\partial G}{\partial t}*c_\mathrm{m},
\label{eq: dentz im-m final}$$
which is consistent with . From inspection of either or , it is obvious that the definitions $$\begin{aligned}
\lambda &\equiv& G(0),
\label{eq: MRMT-CTRW equivalence lambda}\\
g(t) &\equiv& \frac{\partial G}{\partial t},
\label{eq: MRMT-CTRW equivalence g}
\end{aligned}$$ make equivalent to combined with .
The careful reader may note that [@Dentz2003] *also* showed how a CTRW memory function could be derived that is equivalent to any MRMT model, and that the corresponding CTRW transition time pdf is different from our $\zeta$, when defined according to and (\[eq: MRMT-CTRW equivalence lambda\]-\[eq: MRMT-CTRW equivalence g\]). The reason for this is that the earlier authors used a different definition of “immobile” from that introduced by [@Margolin2003] and used in this paper. work by upscaling a microscopic (discrete site) formulation of the CTRW and consequently *define* immobile particles to be those that are, at any instant, not completing transitions. However, in this paper, mesoscopic transitions are defined by periodic arrivals at “milestones” separated by distance $d$ on a streamline. Thus, we must consider mobile particles that are at a given moment undergoing advection but are not presently arriving at a milestone and completing a transition. In our conception, the particles that are immobilized due to sorption or diffusion into secondary porosity are a subset of the particles not presently completing transitions. This difference in conceptions accounts for the difference in equations.
### Connection to first-order mass transfer coefficient
First-order mass transfer is described by the governing equation
$$\frac{\partial c_\mathrm{m}}{\partial t} + \mu (c_\mathrm{im}-c_\mathrm{m}) = L\left\{c_\mathrm{m}\right\}.
\label{eq: first-order MIMT}$$
While this is plainly a special case of single rate mass transfer with equal rate constants for both mobilization and immobilization, and this is itself a subset of MRMT, there is no memory function in this formulation, and so separate analysis is required to derive $\lambda$ and $g(t)$. We note that if we define $g(t)=\mu \exp(-\mu t)$, then the probability of a particle remaining immobile for at least $T$ is $\int_T^\infty \mu \exp(-\mu t) dt = \exp(-\mu T)$. The immobile concentration at time $t$ is the integral of the mobile-immobile flux at each previous time, $\tau$, weighted by the probability of remaining immobile for at least $T=t-\tau$. Thus, it follows that $$\begin{aligned}
c_\mathrm{im} &=& \int_{0}^{t} e^{-\mu(t-\tau)} [-\lambda c_\mathrm{m}(\tau)] d\tau,\\
-\mu c_\mathrm{im} &=& \int_{0}^{t} g(t-\tau) \lambda c_\mathrm{m}(\tau) d\tau.
\end{aligned}$$ It then follows from direct inspection that and are equivalent, as long as the following identification is made: $$\begin{aligned}
\lambda &\equiv& \mu,
\label{eq: FO-CTRW equivalence lambda}\\
g(t) &\equiv& \mu e^{-\mu t}.
\label{eq: FO-CTRW equivalence g}
\end{aligned}$$
### Connection to retardation factor
The final case we consider is a transport equation where MIMT is encoded by multiplication of the time derivative by a retardation factor, R:
$$R\frac{\partial c_\mathrm{m}}{\partial t} = L\left\{c_\mathrm{m}\right\}.
\label{eq: retardation MIMT}$$
We note that this model is an inherently defective simplification of first-order mass transfer. The defect lies in the fact that MIMT always causes dispersion which is erroneously disregarded by use of the retardation factor alone. The so-called “local equilibrium assumption” actually amounting to the assumption that dispersion due to MIMT is small relative to other sources of dispersion [@Hansen2018a]. Thus, our approach is to begin with a single-rate mass transfer model, and define its parameters so that its effective velocity corresponds to that of the retardation model, and its dispersion due to mass transfer is “small” in some sense. This can not be said to have less realism than , and can generate arbitrarily close behavior. We begin from a slightly generalized form of , with distinct mobilization and immobilization rates:
$$\frac{\partial c_\mathrm{m}}{\partial t} + \mu c_\mathrm{im} - \lambda c_\mathrm{m} = L\left\{c_\mathrm{m}\right\}.
\label{eq: first-order MIMT relaxed}$$
It is easy to show [e.g., @Hansen2018a] that an equivalent $R$ for any $\lambda$ and $\mu$ is defined
$$R = 1 + \frac{\lambda}{\mu}.$$
It has been similarly been shown by [@Michalak2000] and [@Uffink2012] that the effective late-time Fickian dispersion coefficient, $D_\mathrm{eff}$, corresponding to the MIMT in is
$$D_\mathrm{eff} = \frac{\lambda\mu v^2}{(\lambda+\mu)^3},$$
where $v$ is the magnitude of the streamline advection velocity. Thus, for to match , we must select $\lambda$ and $\mu$ in to match $R$, and so that $D_\mathrm{eff}$ is sufficiently small. This is attainable because $R$ fixes only the ratio of $\lambda$ and $\mu$, and their respective magnitudes can be scaled by any factor, $k$, to scale $D_\mathrm{eff}$ by the factor $k^{-1}$. We can bring and into arbitrary close alignment by choosing sufficiently large $\mu$ and making the following identification: $$\begin{aligned}
\lambda &\equiv& (R-1)\mu,
\label{eq: retardation-CTRW equivalence lambda}\\
g(t) &\equiv& \mu e^{-\mu t}.
\label{eq: retardation-CTRW equivalence g}
\end{aligned}$$
Numerical corroboration
=======================
While the arguments connecting common MIMT models to the $\lambda$ and $g(t)$ used for CTRW-on-a-streamline modeling in Section \[sec: MIMT\] are mathematically exact and do not need further support, the arguments supporting use of the single pdf, $f(r)$, defined by one of the equations (\[eq: f ig\]-\[eq: f pl\]) to capture the effects of small scale heterogeneous advection are more qualitative. As a result it is useful to demonstrate their successful use numerically. We shall do so, simultaneously demonstrating the entire CTRW-on-a-streamline procedure.
Our approach is to generate a single “true” multi-Gaussian log hydraulic conductivity field and discretize it on a structured grid, generating a matrix of log hydraulic conductivity values. By convolution smoothing of this master field, a derivative field with less small-scale detail but sharing the same large-scale features may be created. By solving the steady-state groundwater flow equation on both conductivity fields with the same boundary conditions, discretized velocity fields may be created for each, and CTRW-on-a-streamline particle tracking performed on each. We seek to show that we recover concentration profiles and breakthrough curves that closely match those computed using the true velocity field by using the smoothed velocity field as well as an appropriate $f(r)$.
In more detail: a multivariate 20 m square master hydraulic conductivity field was generated using the GSTools package [@Muller2019] with an 0.1 m discretization length and geometric mean conductivity of $10^{-2}\ \mathrm{ms^{-1}}$. The (base 10) log hydrualic conductivity featured a multivariate Gaussian correlation structure with $\sigma^2_{\log K}$ = 0.5 and an exponential semivariogram with correlation length 0.5 m. From the 200 by 200 matrix representing the master conductivity field, the true conductivity field was derived by convolving against a uniform kernel with 0.8 m square support (represented by an 8 by 8 matrix). The smoothed conductivity field was derived by convolving against a uniform kernel with 1.6 m square support (represented by a 16 by 16 matrix). Both of the derived $\log K$ fields are shown in Figure \[fig: k and h comparison\]. Both matrix convolutions, and all of the numerical analysis was performed in Python using the Numpy/Scipy ecosystem [@Oliphant2007].
For each of the true and smoothed conductivity fields, a head field was derived by numerically solving the steady-state governing equation
$$\nabla \cdot \left( K(x,y) \nabla h(x,y) \right) = 0,$$
subject to boundary conditions $$\begin{aligned}
h(x, 0) &=& 0, \\
h(x, 20) &=& 1, \\
\frac{\partial h}{\partial x}(0, y) &=& 0, \\
\frac{\partial h}{\partial x}(20, y) &=& 0,
\end{aligned}$$ where $h(x,y)\ [\mathrm{m}]$ represents the hydraulic head at coordinate $(x,y)$. Solution was computed using second-order accurate finite difference techniques, employing the Scipy sparse matrix solver tools. The resulting head distributions are shown in Figure \[fig: k and h comparison\], and the relative smoothness of the $h$ field relative to the $\log K$ field, as posited when developing the CTRW-on-a-streamline approach, is plainly apparent.
From each head field, a velocity field, $\bm{v}(x,y)$ was determined by solving
$$\bm{v}(x,y) = \frac{K(x,y) \nabla h(x,y)}{\theta},$$
where $\theta=0.25$ was taken to be the spatially-uniform porosity, and cell center velocities were stored for each cell. These velocities are displayed as a quiver plot, again in Figure \[fig: k and h comparison\].
Finally, three different particle tracking simulations were performed, each by flux-weighted injection of 10000 particles along the top ($y=20$) boundary between $x=1$ and $x=19$. This was implemented by assigning weights to each cell adjacent to the relevant portion of the $y=20$ boundary according to the magnitude of the $y$-component of $\bm{v}$ at their respective cell centers. Subsequently, a weight-proportional portion of the 10000 particles was assigned to each of these border cells and these particles were initially positioned uniformly randomly within said cell. For each of the three simulations, particle tracking was performed using the CTRW-on-a-streamline approach and two events were recorded: (a) the location of all particles when $t = 7\times 10^3$ s, and (b) the time of breakthrough of each particle at $y=0$. The three simulations were:
1. Classical physics on the highly resolved field: $f(t)=\delta(t)$, $\alpha_t = 0.01$ m, with no MIMT.
2. Classical physics on the smoothed field: $f(t)=\delta(t)$, $\alpha_t = 0.01$ m, with no MIMT.
3. Additional CTRW physics on the smoothed field: $f(t)$ inverse Gaussian with $\alpha_l=0.152$ m, $\alpha_t = 0.01$ m, with no MIMT.
We see from qualitative comparison of the plume snapshots in Figure \[fig: plume comparison\] and breakthrough curves in Figure \[fig: btc comparison\] that the additional CTRW physics (in the form of $f(t)$) do capture much of the transport information that was lost when moving from the highly resolved to the smoothed field. This qualitative assessment is backed by comparison of the 2-norms (also known as Euclidian or Frobenius norms) of the difference between each of the two discretized concentration fields computed using the smoothed velocity field versus the true concentration field. The 2-norm of the error of the concentration field computed without additional CTRW physics is 1.59 times as large as that of the field computed with the additional physics. A similar picture emerges when the $\infty$-norms (largest absolute divergence) are compared. The $\infty$-norm of the no-additional-physics case is 1.68 times as large.
[> m[0.5cm]{} > m[8cm]{} > m[8cm]{}]{} &**True fields** & **Smoothed fields**\
& &\
& &
[> m[0.5cm]{} > m[8cm]{} > m[8cm]{}]{} &**On true velocity field** & **On smoothed velocity field**\
& &\
& &
![Breakthrough curves at $y=0$ for each of the three simulations.[]{data-label="fig: btc comparison"}](FIG_btc.pdf)
Summary and conclusion
======================
We have presented a hybrid approach to modeling solute transport on velocity fields that are explicitly characterized at a coarse scale, but which contain small-scale physical heterogeneity and/or mass transfer leading to non-Fickian transport. The approach, which we call “CTRW-on-a-streamline” is a particle tracking technique that represents an advance of the current state of the art in which, for technical reasons, non-Fickian transport models *or* explicitly-delineated heterogeneous velocity fields are employed, but generally not both. Our approach separates the model parameters characterizing macroscopic flow, subscale advective heterogeneity, and mobile-immobile mass transfer, such that each can be directly specified a priori from available data, where relevant theory exists.
We have presented theory justifying our approach, formalized it in a set of governing equations that illustrate its connection to both classic CTRW and to subordination approaches, and operationalized it: listing an exact algorithm that one can implement. Finally, we presented a numerical demonstration of that implementation that supports the qualitative arguments for capturing small-scale heterogeneous advection validation with the single pdf, $f(r)$.
Acknowlegments {#acknowlegments .unnumbered}
==============
This paper does not concern a data set. Source code for the simulations is archived at <https://doi.org/10.5281/zenodo.3558792>. SKH holds the Helen Unger Career Development Chair in Desert Hydrogeology.
[^1]: Zuckerberg for Water Research, Ben-Gurion University of the Negev, Israel
[^2]: Department of Earth and Planetary Science, Weizmann Institute of Science, Israel
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study gradient estimates and convergence to the equilibrium for solutions of the parabolic equation which is associated to degenerate hypoelliptic diffusion operators. The method relies on a generalized Bakry-Émery type criterion that applies to this kind of operators. Our approach includes as a special case the kinetic Fokker-Planck equation and allows, in that case, to recover hypocoercive estimates obtained by Villani.'
author:
- Fabrice Baudoin
date: |
Department of Mathematics, Purdue University\
West Lafayette, IN, USA
title: 'Bakry-Emery meet Villani'
---
Introduction
============
We study gradient bounds and convergence to equilibrium for the semigroup generated by a diffusion operator of the form $L=\sum_{i=1}^n X_i^2 +Y$, where $X_1,\cdots,X_n,Y$ are vector fields. Though our results are more general, we are particularly interested in the case where $L$ is hypoelliptic and $\sum_{i=1}^n X_i^2$ is not, that is the hypoellipticity comes from the first order operator $Y$. The problem of convergence to an equilibrium in this type of situation has attracted a lot of interest in the literature, because evolution equations involving a degenerate dissipative operator and a conservative operator naturally arise in many fields of applied mathematics: We refer to Villani’s memoir [@Villani1] and to the references therein for a discussion about this.
There have been different approaches to tackle this problem. A functional analytic approach, based on previous ideas by Kohn and Hörmander, uses pseudo-differential calculus and delicate spectral localization tools to prove exponential convergence to equilibrium with explicit bounds on the rate. For this approach, we refer to Eckmann and Hairer [@EH], Hérau and Nier [@HN1], and Heffer and Nier [@HN2].
Villani in his memoir [@Villani1] introduces the concept of hypocoercivity and derives very general sufficient conditions ensuring the convergence to an equilibrium. The main strategy is to work in a suitable Hilbert space associated to the equation and to find in this Hilbert space a nice norm which is equivalent to the original one, but with respect to which convergence to equilibrium is easy to obtain; We refer to Section 4.1 in [@Villani1] for a more precise description. L. Wu in [@Wu] and Bakry, Cattiaux and Guillin in [@BCG] use the powerful method of Lyapunov functions to prove the exponential convergence to equilibrium.
All these approaches have in common to use global methods to prove the convergence to equilibrium in the sense that the functional inequalities that are used are written in an integrated form with respect to the invariant measure. Our approach parallels the Bakry-Emery approach to hypercontractivity [@BE] and is only based on local computations: We just compute derivatives. To perform these computations the existence of an invariant measure is not even required. Let us describe this approach and what is the main idea of the present work.
We can associate to $L=\sum_{i=1}^n X_i^2 +Y$ its *carré du champ* operator $$\Gamma(f,g)=\frac{1}{2}\left(L(fg)- fLg-gLf\right)$$ and its iteration $$\Gamma_2(f,g)=\frac{1}{2}\left(L\Gamma(f,g)-\Gamma(f,Lg)-\Gamma(g,Lf)\right).$$
If the operator $L$ admits a symmetric measure $\mu$ and if for every $f$, $\Gamma_2(f,f)\ge \rho \Gamma(f,f)$ for some positive constant $\rho$, then it is known the semigroup $P_t$ generated by $L$ will converge exponentially fast to an equilibrium (see [@BE]). However in a number of interesting situations including the ones described before, it is impossible to bound from below $\Gamma_2$ by $\Gamma$ alone. Actually, the Bakry-Emery criterion $\Gamma_2 \ge \rho \Gamma$ requires some form of ellipticity of the operator $L$ and fails to hold for strictly subelliptic operators. In the recent few years, there have been several works, extending the Bakry-Emery approach to subelliptic operators. We mention in particular Baudoin and Bonnefont [@BB], Baudoin and Garofalo [@BG], and Baudoin and Wang [@BW] where a generalized curvature dimension inequality is shown to be satisfied for a large class of geometrically relevant subelliptic operators. In particular, under suitable conditions, explicit rates of convergence to equilibrium are obtained for the semigroup. The hypoelliptic situations treated in these works are quite different from the ones we have in mind here, because in [@BB], [@BG] or [@BW] the operator $\sum_{i=1}^n X_i^2$ is hypoelliptic and $Y$ is in the linear span of $X_1,\cdots,X_n$. Here, we really are interested in situations where $\sum_{i=1}^n X_i^2$ is fully degenerate.
Since it is impossible to bound from below $\Gamma_2$ by $\Gamma$ alone, our main idea will be to introduce a *vertical* first order bilinear form $\Gamma^Z$ and to compute the curvature of $L$ in this new vertical direction: $$\Gamma^Z_2(f,g)=\frac{1}{2}\left(L\Gamma^Z(f,g)-\Gamma^Z(f,Lg)-\Gamma^Z(g,Lf)\right).$$ As it turns out, in the degenerate situations mentioned above, it is possible to find, under some conditions, such a form $\Gamma^Z$ with the property that $$\begin{aligned}
\label{CD2}
\Gamma_2(f,f)+\Gamma^Z_2(f,f) \ge -K \Gamma(f,f)+ \rho \Gamma^Z(f,f),\end{aligned}$$ where $K \in \mathbb{R}$ and where $\rho >0$. The important point here is that $\rho$ is positive and will hence induce a convergence to equilibrium in the missing vertical direction. The bound is local and implies several interesting pointwise bounds for the semigroup $P_t$. Let us observe that these bounds can not be obtained by Villani’s method where inequalities are always given in an integrated form with respect to an invariant measure which is not even assumed to exist here. In the case when there is an invariant probability measure, that satisfies the Poincaré inequality with respect to the new gradient $\Gamma+\Gamma^Z$, then the inequality $\eqref{CD2}$ implies exponential convergence to equilibrium with an explicit rate, and we recover then an important Villani’s result.
The paper is organized as follows. In Section 2, we study in details the case of the kinetic Fokker-Planck equation. The study of this important equation was the motivation of the present work. In that case, we recover the convergence results by Villani but also prove pointwise gradient bounds that do not seem to have been known before. It is a very special case of equation that we can treat by our methods, however we think that the main ideas are already there so that, for pedagogical reasons, we start with the study of this equation. Section 3 gives more general results. We study in details the structure of the $\Gamma_2$ associated to degenerate hypoelliptic operators and give sufficient conditions to obtain lower bounds on $\Gamma_2+\Gamma^Z_2$. As a consequence, we get sufficient conditions for gradient bounds and convergence to the equilibrium.
The kinetic Fokker-Planck equation
==================================
In this section we study the kinetic Fokker-Planck equation which is an important example of equation to which our methods apply and which is the one that motivated our study.
Let $V:\mathbb{R}^n \to \mathbb{R}$ be a smooth function. The kinetic Fokker-Planck equation with confinement potential $V$ is the parabolic partial differential equation: $$\label{FP1}
\frac{\partial h}{\partial t}=\Delta_v h - v \cdot \nabla_v h+\nabla V \cdot \nabla_v h -v\cdot \nabla_x h , \quad (x,v) \in \mathbb{R}^{2n}.$$ It is the Kolmogorov-Fokker-Planck equation associated to the stochastic differential system $$\begin{cases}
dx_t =v_t dt \\
dv_t=-v_t dt -\nabla V (x_t) dt +dB_t,
\end{cases}$$ where $(B_t)_{ t\ge 0}$ is a Brownian motion on $\mathbb{R}^n$. The operator $$L=\Delta_v - v \cdot \nabla_v +\nabla V \cdot \nabla_v -v\cdot \nabla_x$$ is not elliptic but it can be written in Hörmander’s form $$L=\sum_{i=1}^n X_i^2 +X_0+Y,$$ where $X_i=\frac{\partial}{\partial v_i}$, $X_0=- v \cdot \nabla_v$ and $Y=\nabla V \cdot \nabla_v -v\cdot \nabla_x $. The vectors $$(X_1,\cdots,X_n, [Y,X_1],\cdots,[Y,X_n])$$ form a basis of $\mathbb{R}^{2n}$ at each point. This implies from Hörmander’s theorem that $L$ is hypoelliptic. The operator $L$ admits for invariant measure the measure $$d\mu=e^{-V(x)-\frac{\| v \|^2}{2}} dxdv.$$ It is readily checked that $L$ is not symmetric with respect to $\mu$ but that the adjoint $L^*$ in $L^2(\mu)$ is given by $$L^*=\sum_{i=1}^n X_i^2 +X_0-Y.$$ The operator $L$ is the generator of a strongly continuous sub-Markov semigroup $(P_t)_{t \ge 0}$. If we assume that the Hessian $\nabla^2 V$ is bounded, then $P_t$ is Markovian (that is $P_t 1=1$) and for any bounded Borel function $f:\mathbb{R}^{2n} \to \mathbb{R}$, $(t,x),v \to P_t f(x,v)$ is the unique solution of the Cauchy problem $$\begin{cases}
\frac{\partial h}{\partial t}=Lh \\
h(0,x,v)=f(x,v).
\end{cases}$$
One of the main results of Villani (see also Hellfer and Nier [@HN1] for related results) concerning the convergence to equilibrium of $P_t$ is the following theorem:
Define $H^1(\mu)=\{ f \in L^2(\mu), \| \nabla f \| \in L^2(\mu)\}$. Assume that there is a constant $c>0$ such that $\| \nabla^2 V \| \le c( 1+ \| \nabla V \|)$ and that the normalized invariant measure $d\mu=\frac{1}{Z}e^{-V(x)-\frac{\| v \|^2}{2}} dxdv$ is a probability measure that satisfies the classical Poincaré inequality $$\int_{\mathbb{R}^{2n}} \| \nabla f \|^2 d\mu \ge \kappa \left[ \int_{\mathbb{R}^{2n}} f^2 d\mu -\left( \int_{\mathbb{R}^{2n}} f d\mu\right)^2 \right].$$ Then, there exist constants $C>0$ and $\lambda >0$ such that for every $f \in H^1(\mu)$, with $\int_{\mathbb{R}^{2n}} f d\mu=0$, $$\int_{\mathbb{R}^{2n}} (P_t f)^2 d\mu + \int_{\mathbb{R}^{2n}} \| \nabla P_t f \|^2 d\mu \le C e^{-\lambda t} \left( \int_{\mathbb{R}^{2n}} f^2 d\mu + \int_{\mathbb{R}^{2n}} \| \nabla f \|^2 d\mu\right)$$
It is worth observing that since $\mu$ is a product, it satisfies the Poincaré inequality as soon as the marginal measure $d\mu_x= e^{-V(x)} dx$ satisfies the Poincaré inequality on $\mathbb{R}^n$.
In this section, we give a new proof of this result under the further assumption that the Hessian $\nabla^2 V$ is bounded. This is a stronger assumption than in Villani’s result. However our method gives pointwise gradient estimates that can not be obtained by Villani’s method. It also provides a better control of the constants $C$ and $\lambda$. Finally, as we shall see in the next section, a very small variation of our method will almost immediately give an entropic convergence of $P_t$, under the assumption that $\mu$ satisfies the log-Sobolev inequality. This latter result is also obtained by Villani under the assumption that $\nabla^2 V$ is bounded (see Theorem 35 in [@Villani1]). So, from now on, and in all this section we assume that $\nabla^2 V$ is bounded.
$\Gamma_2$ calculus for the kinetic Fokker-Planck equation
----------------------------------------------------------
Following Bakry and Émery [@BE] we associate to $L=\Delta_v - v \cdot \nabla_v +\nabla V \cdot \nabla_v -v\cdot \nabla_x$ the operator, $$\Gamma(f,g)=\frac{1}{2}\left(L(fg)- fLg-gLf\right)=\nabla_v f \cdot \nabla_v g=\sum_{i=1}^n \frac{\partial f}{\partial v_i}\frac{\partial g}{\partial v_i}$$ and its iteration $$\Gamma_2(f,g)=\frac{1}{2}\left(L\Gamma(f,g)-\Gamma(f,Lg)-\Gamma(g,Lf)\right).$$ For simplicity of notations, we will denote $\Gamma(f):=\Gamma(f,f)$ and $\Gamma_2(f):=\Gamma_2(f,f)$. A straightforward computation shows that:
For $f \in C^\infty(\mathbb{R}^{2n})$, $$\begin{aligned}
\Gamma_2(f)& =\| \nabla_v^2 f \|^2 +\Gamma(f)+ \nabla_x f\cdot \nabla_v f.
\end{aligned}$$
The term $\nabla_x f\cdot \nabla_v f$ makes impossible to bound from below $\Gamma_2$ by $\Gamma$ alone. As a consequence the Bakry-Émery curvature of $L$ is $-\infty$. The idea is to introduce a carefully chosen vertical gradient and to compute the corresponding curvature of $L$ in this vertical direction. For $i=1,\cdots,n$, we denote $$Z_i= 2\frac{\partial }{\partial x_i }+\frac{\partial }{\partial v_i }.$$ We define then $$\Gamma^Z(f,g)=\sum_{i=1}^n Z_i fZ_ig$$ and $$\Gamma^Z_2(f,g)=\frac{1}{2}\left(L\Gamma^Z(f,g)-\Gamma^Z(f,Lg)-\Gamma^Z(g,Lf)\right).$$
For $f \in C^\infty(\mathbb{R}^{2n})$, $$\begin{aligned}
\Gamma_2^Z(f)& =\| \nabla_v Z f\|^2 +\frac{1}{2}\Gamma^Z(f) +\frac{1}{2}\nabla_v f\cdot Z f-2\nabla^2V(\nabla_v f,Zf) \\
&=\sum_{i,j=1}^n \left( \frac{\partial}{\partial v_i} Z_j f \right)^2+\frac{1}{2}\Gamma^Z(f)+\frac{1}{2}\sum_{i=1}^n\frac{\partial f}{\partial v_i}Z_if-2\sum_{i,j=1}^n \frac{\partial^2V}{\partial x_i \partial x_j}\frac{\partial f}{\partial v_i} Z_jf.
\end{aligned}$$
Let us write $$L=\sum_{i=1}^n X_i^2 +X_0+Y,$$ where $X_i=\frac{\partial}{\partial v_i}$, $X_0=- v \cdot \nabla_v$ and $Y=\nabla V \cdot \nabla_v -v\cdot \nabla_x $. We have $$\begin{aligned}
\Gamma^Z_2(f,g)&=\frac{1}{2}\left(L\Gamma^Z(f)-2\Gamma^Z(f,Lf)\right) \\
&=\frac{1}{2}\left(L\left(\sum_{i=1}^n (Z_if)^2\right)-2\sum_{i=1}^n Z_if Z_iLf\right)\\
&=\sum_{i=1}^n \Gamma(Z_i f) +\sum_{i=1}^n Z_if [L,Z_i]f \\
&=\sum_{i=1}^n \Gamma(Z_i f) +\sum_{i=1}^n Z_if [X_0,Z_i]f +\sum_{i=1}^n Z_if [Y,Z_i]f.\end{aligned}$$ We now compute, $$[X_0,Z_i]=X_i$$ and $$[Y,Z_i]=\frac{1}{2}Z_i-\frac{1}{2}X_i-2 \sum_{j=1}^n \frac{\partial^2V}{\partial x_i \partial x_j}\frac{\partial }{\partial v_j} .$$ The result follows then easily.
A consequence of the previous computations is the following lower bound for $\Gamma_2+\Gamma_2^Z$.
\[lowerbound\] For every $0 < \eta <\frac{1}{2}$, there exists $K(\eta) \ge -\frac{1}{2}$ such that for every $f \in C^\infty(\mathbb{R}^{2n})$, $$\Gamma_2(f)+\Gamma_2^Z(f) \ge -K(\eta) \Gamma(f)+\eta \Gamma^Z(f).$$
As a consequence of the previous lemmas, we have $$\Gamma_2(f) \ge \frac{1}{2} \Gamma(f)+ \frac{1}{2}\nabla_v f\cdot Z f$$ and $$\Gamma_2^Z(f) \ge \frac{1}{2}\Gamma^Z(f) +\frac{1}{2}\nabla_v f\cdot Z f-2\nabla^2V(\nabla_v f,Zf).$$ We deduce $$\Gamma_2(f) + \Gamma_2^Z(f) \ge \frac{1}{2} \Gamma(f)+\frac{1}{2}\Gamma^Z(f) +\nabla_v f\cdot Z f-2\nabla^2V(\nabla_v f,Zf).$$ We now pick $0<\eta <\frac{1}{2}$ and $K \in \mathbb{R}$ and write $$\begin{aligned}
& \frac{1}{2} \Gamma(f)+\frac{1}{2}\Gamma^Z(f) +\nabla_v f\cdot Z f-2\nabla^2V(\nabla_v f,Zf) \\
=&\eta \Gamma^Z(f) -K\Gamma(f) +\left( \frac{1}{2}-\eta\right) \Gamma^Z(f)+\left(\frac{1}{2}+K\right)\Gamma(f) +(\mathbf{Id}-2\nabla^2V)(\nabla_v f,Zf).
\end{aligned}$$ The bilinear form $\left( \frac{1}{2}-\eta\right) \Gamma^Z(f)+\left(\frac{1}{2}+K\right)\Gamma(f) +(\mathbf{Id}-2\nabla^2V)(\nabla_v f,Zf)$ can now be made non negative as soon as, in the sense of symmetric matrices, $$4 \left( \frac{1}{2}-\eta\right) \left(\frac{1}{2}+K\right) \ge ( \mathbf{Id}-2\nabla^2V )^2.$$ The claim follows then from the fact that we assume that $\nabla^2V$ is bounded on $\mathbb{R}^n$
- The keypoint of the previous proposition is the positivity of $\eta$ which will imply the coercivity of $P_t$ in the vertical direction. The sign of $K(\eta)$ is not that relevant in the sense that the coercivity of $P_t$ in the horizontal direction can be obtained as a consequence of a Poincaré inequality for the invariant measure.
- If there are constants $0<a<b<1$ such that for every $x \in \mathbb{R}^n$, $a \le \nabla^2 V \le b$, then the previous proof shows that we can chose $K(\eta)$ to be negative.
Gradient bounds
---------------
We prove now some global bounds for the gradient of the semigroup $(P_t)_{t \ge 0}$. Related pointwise gradient bounds in this kinetic model were also obtained by Guillin and Wang [@FYW], but our bounds have the advantage to be global. Let us also observe that such bounds can not be obtained by Villani’s method.
The previous computations have shown that for $f \in C^\infty(\mathbb{R}^{2n})$, $$\Gamma_2(f)+\Gamma_2^Z(f) \ge \lambda(\eta)( \Gamma(f)+ \Gamma^Z(f)).$$ where $\lambda(\eta)= \min (-K(\eta),\eta)$. With this lower bound in hands we can use the methods introduced by Baudoin-Bonnefont [@BB] and F.Y. Wang [@FYW2] to obtain the following results.
\[GB\] If $f$ is a bounded Lipschitz function on $\mathbb{R}^{2n}$, then for every $t \ge 0$, $P_tf$ is a bounded and Lipschitz function. More precisely, with the notations of Lemma \[lowerbound\], for every $(v,x) \in \mathbb{R}^{2n}$, $$\Gamma(P_tf) (x,v)+\Gamma^Z(P_tf)(x,v) \le e^{-2\lambda(\eta)t} P_t ( \Gamma(f) +\Gamma^Z(f) )(x,v),$$ where $\lambda(\eta)= \min (-K(\eta),\eta)$.
The heuristic argument is the following: We fix $(v,x) \in \mathbb{R}^{2n}$, $t >0$ and consider the functional $$\Psi(s)=P_s (\Gamma(P_{t-s} f) +\Gamma^Z(P_{t-s} f) )(x,v).$$ Differentiating $\Psi$ , leads to $$\Psi'(s)=2P_s (\Gamma_2(P_{t-s} f) +\Gamma_2^Z(P_{t-s} f) )(x,v)\ge 2 \lambda(\eta) P_s (\Gamma(P_{t-s} f) +\Gamma^Z(P_{t-s} f) )(x,v)=2\lambda(\eta) \Psi(s).$$ Therefore, we obtain $\Psi(t) \ge e^{\lambda(\eta) t} \Psi(0)$, which is the claimed inequality. In order to rigorously justify this argument, we observe that since $\nabla V$ is Lipschitz, the function $W(x,v)=1+\| x \|^2 +\| v\|^2$ is a Lyapunov function such that, for some constant $C>0$, $LW \le C W$ and $\| \nabla W \| \le C W$. We can then use Proposition \[lowerbound\] and argue like [@BB] Proposition 2.2, or [@FYW2], Lemma 2.1.
A direction computation shows that, since the vector fields $X_i$ and $Z_j$ commute, we do have the following intertwining of the quadratic forms $\Gamma$, $\Gamma^Z$: For every $f \in C^\infty(\mathbb{R}^{2n})$, $$\Gamma(f ,\Gamma^Z(f))=\Gamma^Z(f,\Gamma(f)).$$ This intertwining leads to the following entropic type pointwise bound:
\[EB\] Let $f \in C^\infty(\mathbb{R}^{2n})$ be a positive function such that $\sqrt{f}$ is bounded and Lipschitz, then for $t \ge 0$, $\sqrt{P_t f}$ is bounded and Lipschitz. More precisely, with the notations of Lemma \[lowerbound\], for every $(v,x) \in \mathbb{R}^{2n}$, $$P_t f (x,v)\Gamma(\ln P_tf) (x,v)+P_t f (x,v)\Gamma^Z(\ln P_tf)(x,v) \le e^{-2\lambda(\eta)t} P_t (f \Gamma(\ln f) +f\Gamma^Z(\ln f) )(x,v),$$ where $\lambda(\eta)= \min (-K(\eta),\eta)$.
Again, we first give the heuristic argument. We consider the functional $$\Psi(s)=P_s ( P_{t-s} f \Gamma(\ln P_{t-s} f) +P_{t-s} f \Gamma^Z(\ln P_{t-s} f) )(x,v),$$ which by differentiation gives $$\Psi'(s)=2P_s (\ln P_{t-s} f \Gamma_2(\ln P_{t-s} f) +\ln P_{t-s} f \Gamma_2^Z(\ln P_{t-s} f) )(x,v)\ge 2 \lambda(\eta) \Psi(s).$$ Let us observe that the computation of the derivative crucially relies on the fact that $ \Gamma(P_tf ,\Gamma^Z(\ln P_t f))=\Gamma^Z(P_t f,\Gamma(\ln P_t f))$. The rigorous justification can be given as above by using the Lyapunov function $W$ and Proposition 2.1 in [@BB].
Convergence in $H^1$
--------------------
We can now, in particular, recover Villani’s convergence result.
\[conve\] Assume that the normalized invariant measure $d\mu=\frac{1}{Z}e^{-V(x)-\frac{\| v \|^2}{2}} dxdv$ is a probability measure that satisfies the Poincaré inequality $$\int_{\mathbb{R}^{2n}}( \Gamma(f) +\Gamma^Z(f)) d\mu \ge \kappa \left[ \int_{\mathbb{R}^{2n}} f^2 d\mu -\left( \int_{\mathbb{R}^{2n}} f d\mu\right)^2 \right].$$ With the notations of Lemma \[lowerbound\]:
- If $K(\eta)+\eta >0$, then for every $f \in H^1(\mu)$, with $\int_{\mathbb{R}^{2n}} f d\mu=0$, $$\begin{aligned}
& (\eta +K(\eta))\int_{\mathbb{R}^{2n}} (P_t f)^2d\mu +\int_{\mathbb{R}^{2n}} (\Gamma(P_tf) +\Gamma^Z(P_tf) )d\mu \\
\le & e^{-\lambda t}\left( (\eta +K(\eta)) \int_{\mathbb{R}^{2n}} f^2d\mu + \int_{\mathbb{R}^{2n}}(\Gamma(f) +\Gamma^Z(f)) d\mu \right) ,\end{aligned}$$ where $\lambda=\frac{2\eta \kappa}{\kappa+\eta +K(\eta)}$.
- If $K(\eta)+\eta \le 0$, then for every $f \in H^1(\mu)$, with $\int_{\mathbb{R}^{2n}} f d\mu=0$,
$$\begin{cases}
\int_{\mathbb{R}^{2n}} \Gamma(P_tf) +\Gamma^Z(P_tf) )d\mu \le e^{-2\eta t} \int_{\mathbb{R}^{2n}} ( \Gamma(f) +\Gamma^Z(f)) d\mu \\
\int_{\mathbb{R}^{2n}} (P_tf)^2 d\mu \le \frac{1}{\kappa} e^{-2\eta t} \int_{\mathbb{R}^{2n}} ( \Gamma(f) +\Gamma^Z(f)) d\mu .
\end{cases}$$
The two norms $(\eta +K(\eta)) \int_{\mathbb{R}^{2n}} f^2d\mu + \int_{\mathbb{R}^{2n}}(\Gamma(f) +\Gamma^Z(f)) d\mu$ and $\int_{\mathbb{R}^{2n}} \| \nabla f \|^2 d\mu$ are equivalent on $H^1(\mu)$, so the previous result implies Villani’s theorem.
By a density argument we can and will prove these inequalities when $f$ is smooth, bounded and Lipschitz.
Let us first assume that $K(\eta)+\eta >0$. We fix $t>0$ and consider the functional $$\Psi(s) =(K(\eta)+\eta)P_s( (P_{t-s} f)^2 ) + P_s( \Gamma(P_{t-s}f) +\Gamma^Z(P_{t-s}f) ).$$ By repeating the arguments of the previous section, we get the differential inequality $$\Psi(s)-\Psi(0) \ge 2 \eta \int_0^s P_u(\Gamma(P_{t-u}f) +\Gamma^Z(P_{t-u}f) )du.$$ Denote now $\varepsilon=\frac{\eta +K(\eta)}{\kappa +\eta +K(\eta)}$. We have $$\varepsilon\int_{\mathbb{R}^{2n}} \Gamma(P_{t-u}f) +\Gamma^Z(P_{t-u}f) d\mu \ge \varepsilon \kappa \int_{\mathbb{R}^{2n}} (P_{t-u}f)^2 d\mu .$$ Therefore, denoting $\Theta(s)=\int_{\mathbb{R}^{2n}} \Psi(s)d\mu$, we obtain $$\begin{aligned}
\Theta(s)-\Theta(0)& \ge 2 \eta(1-\varepsilon)\int_0^s \int_{\mathbb{R}^{2n}} \Gamma(P_{t-u}f) +\Gamma^Z(P_{t-u}f) d\mu du+2\varepsilon \kappa \int_0^s \int_{\mathbb{R}^{2n}} (P_{t-u}f)^2 d\mu du \\
&\ge \lambda \int_0^s \Theta (u) du.\end{aligned}$$ We conclude with Gronwall’s differential inequality.
If $K(\eta)+\eta \le 0$, then the argument is identical by considering then the functional $$\Psi(s) =P_s (\Gamma(P_{t-s}f) +\Gamma^Z(P_{t-s}f) ).$$
Entropic convergence to the equilibrium
---------------------------------------
We now prove the entropic convergence of $P_t$ to the equilibrium if the invariant measure satisfies log-Sobolev inequality. The result is obtained in a very similar way as the convergence in $H^1(\mu)$.
\[entropy\] Assume that the normalized invariant measure $d\mu=\frac{1}{Z}e^{-V(x)-\frac{\| v \|^2}{2}} dxdv$ is a probability that satisfies the log-Sobolev inequality $$\int_{\mathbb{R}^{2n}}(f \Gamma(\ln f) +f\Gamma^Z(\ln f)) d\mu \ge \kappa \left[ \int_{\mathbb{R}^{2n}} f \ln f d\mu -\left( \int_{\mathbb{R}^{2n}} f d\mu\right)\ln \left( \int_{\mathbb{R}^{2n}} f d\mu\right) \right].$$ With the notations of Lemma \[lowerbound\]:
- If $K(\eta)+\eta >0$, then for every positive and bounded $f \in C^\infty(\mathbb{R}^{2n})$, such that $\sqrt{f}$ is Lipschitz and $\int_{\mathbb{R}^{2n}} f d\mu=1$, $$\begin{aligned}
& 2(\eta +K(\eta))\int_{\mathbb{R}^{2n}} P_t f \ln P_t f d\mu +\int_{\mathbb{R}^{2n}} (P_tf \Gamma(\ln P_tf) +P_tf \Gamma^Z(\ln P_tf) )d\mu \\
\le & e^{-\lambda t}\left( 2 (\eta +K(\eta)) \int_{\mathbb{R}^{2n}} f \ln f d\mu + \int_{\mathbb{R}^{2n}}(f\Gamma(\ln f) +f\Gamma^Z(\ln f )) d\mu \right) ,\end{aligned}$$ where $\lambda=\frac{2\eta \kappa}{\kappa+2(\eta +K(\eta))}$.
- If $K(\eta)+\eta \le 0$, then for every positive and bounded $f \in C^\infty(\mathbb{R}^{2n})$, such that $\sqrt{f}$ is Lipschitz and $\int_{\mathbb{R}^{2n}} f d\mu=1$,
$$\begin{cases}
\int_{\mathbb{R}^{2n}} P_t f \Gamma(\ln P_tf) +P_t f \Gamma^Z(\ln P_tf) )d\mu \le e^{-2\eta t} \int_{\mathbb{R}^{2n}} (f \Gamma(\ln f) +f \Gamma^Z(\ln f)) d\mu \\
\int_{\mathbb{R}^{2n}} P_tf \ln P_t f d\mu \le \frac{1}{\kappa} e^{-2\eta t} \int_{\mathbb{R}^{2n}} ( f\Gamma(\ln f) +f\Gamma^Z(\ln f)) d\mu .
\end{cases}$$
The proof is identical to the proof of Theorem \[conve\] by considering now the functional $$\Psi(s) =2 \max(K(\eta)+\eta, 0)P_s ( P_{t-s} f \ln P_{t-s} f) +P_s ( P_{t-s} f \Gamma(\ln P_{t-s} f) +P_{t-s} f \Gamma^Z(\ln P_{t-s} f) ).$$
$\Gamma_2$ calculus for degenerate diffusion operators
======================================================
Our goal is now to considerably extend the scope of the previous section to handle much more general situations. As it appeared, the crucial ingredient is $\Gamma_2$ calculus. This section is devoted to the analysis of $\Gamma_2$ for a large class of degenerate diffusion operators.
Let $X_1,\cdots,X_n$ be smooth vector fields on $\mathbb{R}^{d}$. We consider on $\mathbb{R}^{d} $ a diffusion operator $L$ that can be written as $$L=L_0 +Y,$$ where $L_0=\sum_{i=1}^n X_i^2 $ and $Y$ is a smooth vector field on $\mathbb{R}^{d}$. For the range of applications we have in mind (like the kinetic models previously studied), $L_0$ can be thought as a diffusion operator which is elliptic on an integral submanifold of $\mathbb{R}^{d}$ and $Y$ can be thought as a vector field such that, at every point, $\mathbf{span} \{ X_1,\cdots, X_n, [Y,X_1],\cdots, [Y,X_n]\}=\mathbb{R}^{d}$. For instance in the kinetic Fokker-Planck model, $X_i=\frac{\partial }{\partial v_i}$ and $Y=- v \cdot \nabla_v +\nabla V \cdot \nabla_v -v\cdot \nabla_x$. However, our framework covers much more general situations.
Bochner’s type identities
-------------------------
In order to simplify notations in the subsequent analysis, we introduce the following concept which parallels the notion of relative boundedness of operators introduced by Villani [@Villani1]: If $(T_i)_{i\in I}, (U_j)_{j \in J}$ are two families of vector fields on $\mathbb{R}^{d}$ , we shall say that the family $(T_i)_{i\in I}$ is bounded relatively to the family $(U_j)_{j \in J}$, if there exist smooth and bounded functions $a_i^j$ on $\mathbb{R}^{d}$ such that $$T_i =\sum_{j \in J} a_i^j U_j.$$ In that case, we shall denote $$(T_i)_{i\in I} \prec (U_j)_{j \in J}.$$ In the coefficients $a_i^j$ are moreover such that $\Gamma(a_i^j)$ is always a bounded function, then we shall say that that the family $(T_i)_{i\in I}$ is bounded relatively to the family $(U_j)_{j \in J}$ with $\Gamma$-Lipschitz coefficients.
Our first objective is to prove Bochner’s type identities for the operator $L$. For that purpose, we introduce, as before, the following differential bilinear forms: The carré du champ and its iteration $$2\Gamma(f,g)=L(fg)-fLg-gLf=\sum_{i=1}^n X_i f X_i g,$$ $$2\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(f,Lg).$$ We also assume given a family of smooth vector fields $Z_1,\cdots, Z_m$ such that: $$\{ [X_i, X_j] \}_{1 \le i,j \le n} ,\{ [X_i, Z_k] \}_{1 \le i \le n, 1 \le k \le m} , \{ [Y, X_i] \}_{1 \le i \le n} , \{ [Y,Z_k]\}_{1 \le k \le m} \prec \{ X_i ,Z_k \}_{1 \le i \le n, 1 \le k \le m},$$ and $\{ [X_i, X_j] \}_{1 \le i,j \le n} ,\{ [X_i, Z_k] \}_{1 \le i \le n, 1 \le k \le m} $ have $\Gamma$-Lipschitz coefficients.
We now denote, as before $$\Gamma^Z(f,g)=\sum_{i=1}^n Z_i fZ_ig$$ and $$\Gamma^Z_2(f,g)=\frac{1}{2}\left(L\Gamma^Z(f,g)-\Gamma^Z(f,Lg)-\Gamma^Z(g,Lf)\right).$$ Finally, we introduce the functions $\omega_{ij}^k, \gamma_{ij}^k,\alpha_{ij}^k, \beta_{ij}^k, \sigma_j^k,\lambda_j^k $ such that $$\begin{aligned}
\label{commu1}
[X_i,X_j]=\sum_{k=1}^n \omega_{ij}^k X_k +\sum_{k=1}^m \gamma_{ij}^k Z_k,
\end{aligned}$$ $$[X_i,Z_j]=\sum_{k=1}^n \alpha_{ij}^k X_k +\sum_{k=1}^m \beta_{ij}^k Z_k,$$ $$[Y,Z_i]=\sum_{j=1}^n \sigma_i^j X_j +\sum_{j=1}^m \lambda_i^j Z_j.$$
With these preliminaries in hands, we now give the first central result of this section, which can be thought as a horizontal Bochner’s type identity for $L$:
For $f \in C^\infty(\mathbb{R}^{d})$, $$\Gamma_2 (f)= \| \nabla^2_X f \|^2 + \sum_{{\ell},j = 1}^n\bigg(\sum_{k=1}^m\gamma^{k}_{{\ell}j} Z_{k} f\bigg)^2-2 \sum_{i,j=1}^n \sum_{k=1}^m \gamma_{ij}^{k} (X_j Z_{k} f) (X_i f)+\sum_{i=1}^n X_i f T_i f,$$ where $$\| \nabla^2_X f \|^2= \sum_{{\ell}=1}^n \left( X^2_{\ell}f-\sum_{i=1}^n
\omega_{i{\ell}}^{\ell}X_i f \right)^2 + 2 \sum_{1 \le {\ell}<j \le n}
\left( \frac {X_j X_{\ell}+ X_{\ell}X_j}{2}f -\sum_{i=1}^n \frac{\omega_{ij}^{\ell}+\omega_{i{\ell}}^j}{2} X_i f \right)^2$$ and the $T_i$’s are vector fields such that $ T_i \prec \{ X_i ,Z_k \}_{1 \le i \le n, 1 \le k \le m}$.
Let us preliminarily observe that $$X_i X_j f = f_{,ij} + \frac 12 [X_i,X_j]f,$$ where we have let $$f_{,ij} = \frac 12 (X_i X_j + X_j X_i) f.$$ Using , we obtain $$\label{nonsym}
X_iX_jf = f_{,ij} + \frac{1}{2} \sum_{{\ell}=1}^n \omega^{\ell}_{ij} X_{\ell}f + \frac{1}{2} \sum_{k=1}^m \gamma^{k}_{ij} Z_{k} f.$$ Starting from the definition of $\Gamma_2(f)$, we have $$\begin{aligned}
\Gamma_2(f) = \sum_{i,j=1}^n (X_jX_i f)^2 - 2
\sum_{i,j=1}^n X_if [X_i,X_j]X_j f + \sum_{i,j=1}^n X_i f
[[X_i,X_j],X_j]f + \sum_{i=1}^n X_i f [Y,X_i]f.\end{aligned}$$ From we have $$\begin{aligned}
\sum_{i,j=1}^n (X_jX_i f)^2 & = \sum_{i,j=1}^n f_{,ij}^2 + \frac{1}{2} \sum_{1\le
i<j\le n}\left(\sum_{{\ell}=1}^n \omega^{\ell}_{ij} X_{\ell}f\right)^2 +
\frac{1}{2} \sum_{1\le i<j\le n}\left(\sum_{k=1}^m
\gamma^{k}_{ij} Z_{k} f\right)^2 \\
& + \sum_{1\le i<j\le n}\sum_{{\ell}=1}^n \sum_{k=1}^m
\omega^{\ell}_{ij} \gamma^{k}_{ij} Z_{k}f X_{\ell}f
\notag\end{aligned}$$ and therefore, $$\begin{aligned}
\Gamma_2(f) & = \sum_{i,j=1}^n f_{,ij}^2- 2
\sum_{i,j=1}^n X_if [X_i,X_j]X_j f + \sum_{i,j=1}^n X_i f
[[X_i,X_j],X_j]f
\\
& + \sum_{i=1}^n X_i f [Y,X_i]f + \frac{1}{2} \sum_{1\le
i<j\le d}\left(\sum_{{\ell}=1}^n \omega^{\ell}_{ij} X_{\ell}f\right)^2 +
\frac{1}{2} \sum_{1\le i<j\le n}\left(\sum_{k=1}^m
\gamma^{k}_{ij} Z_{k} f\right)^2
\notag\\
& + \sum_{1\le i<j\le n}\sum_{{\ell}=1}^n \sum_{k=1}^m
\omega^{\ell}_{ij} \gamma^{k}_{ij} Z_{k}f X_{\ell}f.
\notag\end{aligned}$$ We now compute $$\begin{aligned}
& \sum_{i,j=1}^n f_{,ij}^2 - 2 \sum_{i,j=1}^n X_i f [X_i,X_j]X_j f
\\
& = \sum_{\ell=1}^n \left(f_{,\ell\ell}^2 - 2 \left(\sum_{i=1}^n
\omega^{\ell}_{i{\ell}} X_i f\right) f_{,{\ell}{\ell}}\right)
\\
& + 2 \sum_{1 \le {\ell}<j \le n}\left( f_{,j\ell}^2 - 2 \sum_{1\le
{\ell}<j\le n} \left(\sum_{i=1}^n \frac{\omega_{ij}^{\ell}+
\omega_{i{\ell}}^j}{2}
X_i f\right) f_{,{\ell}j} \right)
\\
& - \sum_{i,j=1}^n \sum_{{\ell}, k=1}^n \omega_{ij}^{\ell}\omega^k_{{\ell}j} X_k f X_i f - \sum_{i,j=1}^n \sum_{{\ell}=1}^n \sum_{k=1}^m
\omega_{ij}^{\ell}\gamma^{k}_{{\ell}j} Z_{k} f\ X_i f
\\
& - 2 \sum_{i,j=1}^n \sum_{k=1}^m \gamma_{ij}^{k} Z_{k}X_j f\
X_i f.\end{aligned}$$ Completing the squares in the latter expression we find $$\begin{aligned}
& \sum_{i,j=1}^n f_{,ij}^2 - 2 \sum_{i,j=1}^n X_i f [X_i,X_j]X_j f
\\
& = \sum_{{\ell}=1}^n \left( f_{,\ell\ell} -\sum_{i=1}^n
\omega_{i{\ell}}^{\ell}X_i f \right)^2 + 2 \sum_{1 \le {\ell}<j \le n}
\left( f_{,j\ell} -\sum_{i=1}^n \frac{\omega_{ij}^{\ell}+\omega_{i{\ell}}^j}{2} X_i f \right)^2
\notag\\
& - \sum_{{\ell}=1}^n \left(\sum_{i=1}^n \omega_{i{\ell}}^{\ell}X_i f
\right)^2 - 2 \sum_{1 \le {\ell}<j \le n} \left(\sum_{i=1}^n
\frac{\omega_{ij}^{\ell}+\omega_{i{\ell}}^j}{2} X_i f \right)^2
\notag\\
& - \sum_{i,j,k,{\ell}=1}^n \omega_{ij}^{\ell}\omega^k_{{\ell}j} X_k f
X_i f - \sum_{i,j=1}^n \sum_{{\ell}=1}^n \sum_{k=1}^m
\omega_{ij}^{\ell}\gamma^{k}_{{\ell}j} Z_{k} f\ X_i f
\notag\\
& - 2 \sum_{i,j=1}^n \sum_{k=1}^m \gamma_{ij}^{k} X_j Z_{k}f\
X_i f - 2 \sum_{i,j=1}^n \sum_{k=1}^m \gamma_{ij}^{k}
[Z_{k},X_j] f\ X_i f. \notag\end{aligned}$$ The conclusion then easily follows by putting the $\Gamma_2$ pieces together.
The second central result of the section is the following vertical Bochner’s formula:
For $f \in C^\infty(\mathbb{R}^{d})$, $$\begin{aligned}
\Gamma^Z_2 (f) = & \| \nabla^2_{X,Z} f \|^2 +\sum_{j,k=1}^m \left(\lambda_{j}^k + \sum_{ i,{\ell}=1}^n\alpha_{ik}^l \gamma_{li}^j +\sum_{i=1}^n X_i \beta_{ik}^j +\sum_{ i,{\ell}=1}^m \beta_{ik}^{\ell}(\beta_{i {\ell}}^j - \beta_{ij}^{\ell}) \right)Z_j f Z_k f \\
&+2 \sum_{k=1}^m \sum_{i,j=1}^n \alpha_{ik}^j Z_k f X_i X_j f+\sum_{i=1}^n X_i f U_i f\end{aligned}$$ where $$\| \nabla^2_{X,Z} f \|^2= \sum_{i=1}^n \sum_{j=1}^m \left(X_iZ_j f +\sum_{k=1}^m \beta_{ik}^j Z_k f\right)^2.$$ and the $U_i$’s are vector fields such that $ U_i \prec \{ X_i ,Z_k \}_{1 \le i \le n, 1 \le k \le m}$.
We start from the definition $$\Gamma^Z_2(f)=\frac{1}{2}\left(L\Gamma^Z(f)-2\Gamma^Z(f,Lf)\right),$$ which by similar computations as before easily leads to $$\begin{aligned}
\Gamma^Z_2(f) &=\sum_{i=1}^n \sum_{k=1}^m (X_i Z_k)^2 + 2\sum_{i=1}^n \sum_{k=1}^m Z_kf X_i [X_i,Z_k]f +\sum_{i=1}^n\sum_{k=1}^m (Z_kf) [[X_i,Z_k],X_i]f +\sum_{k=1}^m Z_k f [Y,Z_k]f \\
&=\sum_{i=1}^n \sum_{k=1}^m (X_i Z_k)^2 + 2\sum_{i=1}^n \sum_{k=1}^m Z_kf X_i \left(\sum_{j=1}^n \alpha_{ik}^jX_jf +\sum_{{\ell}=1}^m\beta_{ik}^{\ell}Z_{\ell}f \right) \\
& +\sum_{i=1}^n\sum_{k=1}^m (Z_kf) [[X_i,Z_k],X_i]f +\sum_{k=1}^m Z_k f [Y,Z_k]f \\
&=\sum_{i=1}^n \sum_{k=1}^m (X_i Z_k)^2 + 2 \sum_{i=1}^n\sum_{k,{\ell}=1}^m \beta_{ik}^{\ell}Z_kf X_iZ_{\ell}f+ 2\sum_{k=1}^m\sum_{i,j=1}^n \alpha_{ik}^j Z_kf X_iX_jf\\
& +2 \sum_{i=1 }^n\sum_{j,k=1}^m X_i\beta_{ik}^j Z_k f Z_j f+2\sum_{i,j=1}^n \sum_{k=1}^m X_i \alpha_{ik}^j X_j f Z_kf \\
& +\sum_{i=1}^n\sum_{k=1}^m (Z_kf) [[X_i,Z_k],X_i]f +\sum_{k=1}^m Z_k f [Y,Z_k]f \\
\end{aligned}$$ We then complete the squares $$\begin{aligned}
& \sum_{i=1}^n \sum_{k=1}^m (X_i Z_k f)^2 +2\sum_{i=1}^n \sum_{j,k=1}^m \beta_{ik}^j Z_k f X_iZ_jf \\
=& \sum_{i=1}^n \sum_{j=1}^m (X_i Z_j f+\sum_{k=1}^m \beta_{ik}^j Z_k f)^2 - \sum_{i=1}^n \sum_{j=1}^m \left( \sum_{k=1}^m \beta_{ik}^j Z_k f\right)^2\end{aligned}$$ and compute that $$\begin{aligned}
\sum_{k=1}^m \sum_{i=1}^n [[X_i,Z_k],X_i]f Z_k f &=\sum_{k=1}^m \sum_{i=1}^n \left[\sum_{j=1}^n \alpha_{ik}^j X_j +\sum_{{\ell}=1}^m \beta_{ik}^{\ell}Z_l ,X_i\right]f Z_k f.\end{aligned}$$ We then have $$\begin{aligned}
& \left[\sum_{j=1}^n \alpha_{ik}^j X_j +\sum_{{\ell}=1}^m \beta_{ik}^{\ell}Z_l ,X_i\right] \\
=& \sum_{j=1}^n \alpha_{ik}^j [X_j , X_i]+\sum_{{\ell}=1}^m \beta_{ik}^{\ell}[Z_{\ell},X_i]- \sum_{j=1}^n(X_i \alpha_{ik}^j) X_j-\sum_{{\ell}=1}^m(X_i \beta_{ik}^{\ell}) Z_{\ell},\end{aligned}$$ and the result follows by putting the $\Gamma_2^Z$ pieces together.
Combining the two Bochner’s identities leads to the following result:
\[CD\] Assume that there exists a constant $\rho \in \mathbb{R}$ such that for every $f \in C^\infty(\mathbb{R}^{d})$, $$\begin{aligned}
\sum_{j,k=1}^m \left( \lambda_j^k+\sum_{ i,{\ell}=1}^n(\alpha_{ik}^l+\gamma_{li}^k) \gamma_{li}^j -\alpha_{ik}^{\ell}\alpha_{ij}^{\ell}+\sum_{i=1}^n X_i \beta_{ik}^j +\sum_{ i,{\ell}=1}^m \beta_{ik}^{\ell}(\beta_{i {\ell}}^j - \beta_{ij}^{\ell}) \right)Z_j f Z_k f \ge \rho \Gamma^Z(f).\end{aligned}$$ Then, for every $\eta < \rho$, there exists $K(\eta) \in \mathbb{R}$ such that for every $f \in C^\infty(\mathbb{R}^{d})$, $$\Gamma_2(f)+\Gamma^Z_2 (f) \ge -K(\eta) \Gamma(f)+\eta \Gamma^Z(f).$$
Adding the formulas for $\Gamma_2$ and $\Gamma_2^Z$ and completing the squares in the sums $$\| \nabla^2_{X,Z} f \|^2-2 \sum_{i,j=1}^n \sum_{k=1}^m \gamma_{ij}^{k} (X_j Z_{k} f) (X_i f)$$ and $$\| \nabla^2_X f \|^2+2 \sum_{k=1}^m \sum_{i,j=1}^n \alpha_{ik}^j Z_k f X_i X_j f$$ leads to the inequality $$\Gamma_2(f)+\Gamma^Z_2 (f) \ge \rho \Gamma^Z(f) +\sum_{i=1}^n X_i f V_i f,$$ where the $V_i$’s are vector fields such that $ V_i \prec \{ X_i ,Z_k \}_{1 \le i \le n, 1 \le k \le m}$. If we now pick $\eta <\rho$ it is clear that for $K(\eta)$ big enough the bilinear form $$(\rho-\eta) \Gamma^Z(f) +\sum_{i=1}^n X_i f V_i f+K(\eta)\Gamma(f)$$ can be made positive. As a consequence, we have $$\Gamma_2(f)+\Gamma^Z_2 (f) \ge -K(\eta) \Gamma(f)+\eta \Gamma^Z(f).$$
In order to use the previous results in concrete situations, we of course need to identify the vector fields $Z_i$’s and try to chose them in such a way that $\rho >0$ .
The following result is an explicit example of choice for the $Z_i$’s under boundedness conditions on the vector fields $[X_i,X_j]$, $ [X_i, [Y,X_j]]$, and $ [Y, [Y,X_i]] $. It is comparable to Theorem 24 in Villani [@Villani1] and applies to kinetic type Fokker-Planck operators.
\[example\] Assume that the family of vector fields $\{ [X_i, X_j] \}_{1 \le i,j \le n}$ is bounded relatively with respect to the family $\{ X_i\}_{1 \le i \le n}$ with $\Gamma$-Lipschitz coefficients and that the two families $\{ [X_i, [Y,X_j]] \}_{1 \le i \le n, 1 \le j \le n} $,and $ \{ [Y, [Y,X_i]] \}_{1 \le i \le n}$ are bounded relatively to
$\{ X_i ,[Y,X_k] \}_{1 \le i \le n, 1 \le k \le n}$ with $\Gamma$-Lipschitz coefficients, then there exist vector fields $Z_1,\cdots,Z_m$ and a constant $\rho >0$ such that for every $\eta < \rho$, there exists $K(\eta) \in \mathbb{R}$ such that for every $f \in C^\infty(\mathbb{R}^{d})$ $$\Gamma^Z_2 (f) \ge -K(\eta) \Gamma(f)+\eta \Gamma^Z(f).$$
We can write $$[X_i,X_j]=\sum_{k=1}^n \omega_{ij}^k X_k$$ $$[X_i, [Y,X_j]]=\sum_{k=1}^n \alpha_{ij}^ k X_k +\sum_{k=1}^n \beta_{ij}^k [Y,X_k],$$ for some bounded and $\Gamma$-Lipschitz functions $\omega_{ij}^k,\alpha_{ij}^ k,\beta_{ij}^k$. Consider then $$Z_i=X_i +\varepsilon [Y,X_i],$$ where $\varepsilon >0$ is to be chosen later. We have $$\begin{aligned}
[X_i,Z_j]& =[X_i,X_j]+\varepsilon [X_i,[Y,X_j]] \\
&=\sum_{k=1}^n \omega_{ij}^k X_k+\varepsilon\sum_{k=1}^n \alpha_{ij}^ k X_k +\varepsilon\sum_{k=1}^n \beta_{ij}^k [Y,X_k] \\
&=\sum_{k=1}^n( \omega_{ij}^k +\varepsilon \alpha_{ij}^ k)X_k+\varepsilon \sum_{k=1}^n\beta_{ij}^k(Z_k-X_k) \\
&=\sum_{k=1}^n( \omega_{ij}^k +\varepsilon \alpha_{ij}^ k-\varepsilon \beta_{ij}^k)X_k+\varepsilon \sum_{k=1}^n\beta_{ij}^k Z_k\end{aligned}$$ and $$\begin{aligned}
[Y,Z_i] & =[Y,X_i]+\varepsilon[Y, [Y,X_i]] \\
&=\frac{1}{\varepsilon}(Z_i-X_i)+\varepsilon[Y, [Y,X_i]].\end{aligned}$$ We can then chose $\varepsilon$ small enough in such a way that the assumptions of Proposition \[CD\] are satisfied with $\rho>0$.
Gradient estimates and convergence to the equilibrium
-----------------------------------------------------
Collecting the previous $\Gamma_2$ estimates and reasoning as for the kinetic Fokker-Planck model we can now generalize the results of Section 2. As in the previous section, we consider $$L=L_0 +Y,$$ and we now assume that there exist smooth vector fields $Z_1,\cdots, Z_m$ and constants $\rho_1\ge 0,\rho_2 >0$ such that for every $f \in C^\infty(\mathbb{R}^d)$, $$\Gamma_2(f)+\Gamma_2^Z(f) \ge -\rho_1 \Gamma(f) +\rho_2\Gamma^Z(f).$$ This is for instance the case if the assumptions of Proposition \[example\] are fulfilled.
Borrowing an hypothesis introduced by F.Y. Wang [@FYW2], we also assume that there exists a function $W$ with compact level sets such that $W \ge 1$, $\Gamma(W)+\Gamma^Z(W) \le C W^2$, $LW \le C W$ for some constant $C>0$. In that case, $L$ is the generator of a Markov semigroup $(P_t)_{t \ge 0}$ and the following result is a consequence of [@BB] Proposition 2.2, or [@FYW2], Lemma 2.1.
If $f$ is a bounded Lipschitz function on $\mathbb{R}^{d}$, then for every $t \ge 0$, $P_tf$ is a bounded and Lipschitz function. More precisely, for every $x \in \mathbb{R}^{d}$, $$\Gamma(P_tf) (x)+\Gamma^Z(P_tf)(x) \le e^{-2 \lambda t} P_t ( \Gamma(f) +\Gamma^Z(f) )(x),$$ where $\lambda= \min (-\rho_1,\rho_2)$.
The proof is identical to the proof of Lemma \[GB\].
We also have the following result:
Assume that for every $f \in C^\infty(\mathbb{R}^{2n})$, $$\Gamma(f ,\Gamma^Z(f))=\Gamma^Z(f,\Gamma(f)).$$
Let $f \in C^\infty(\mathbb{R}^{d})$ be a positive function such that $\sqrt{f}$ is bounded and Lipschitz, then for $t \ge 0$, $\sqrt{P_t f}$ is bounded and Lipschitz. More precisely, for every $x \in \mathbb{R}^{d}$ $$P_t f (x)\Gamma(\ln P_tf) (x)+P_t f (x)\Gamma^Z(\ln P_tf)(x) \le e^{-2\lambda t} P_t (f \Gamma(\ln f) +f\Gamma^Z(\ln f) )(x),$$ where $\lambda(\eta)= \min (-\rho_1,\rho_2)$.
The proof is identical to the proof of Lemma \[EB\].
The following theorems are obtained in the very same way as Theorems \[conve\] and \[entropy\].
Assume that the operator $L$ admits an invariant probability measure $\mu$ that satisfies the Poincaré inequality $$\int_{\mathbb{R}^{d}}( \Gamma(f) +\Gamma^Z(f)) d\mu \ge \kappa \left[ \int_{\mathbb{R}^{2n}} f^2 d\mu -\left( \int_{\mathbb{R}^{2n}} f d\mu\right)^2 \right].$$ Then, for every bounded function $f$ such that $\Gamma(f)+\Gamma^Z(f)$ is bounded and $\int_{\mathbb{R}^{d}} f d\mu=0$, $$\begin{aligned}
& (\rho_1+\rho_2)\int_{\mathbb{R}^{d}} (P_t f)^2d\mu +\int_{\mathbb{R}^{d}} (\Gamma(P_tf) +\Gamma^Z(P_tf) )d\mu \\
\le & e^{-\lambda t}\left( (\rho_1+\rho_2) \int_{\mathbb{R}^{d}} f^2d\mu + \int_{\mathbb{R}^{d}}(\Gamma(f) +\Gamma^Z(f)) d\mu \right) ,\end{aligned}$$ where $\lambda=\frac{2\rho_2 \kappa}{\kappa+\rho_1+\rho_2}$.
Assume that for every $f \in C^\infty(\mathbb{R}^{d})$, $$\Gamma(f ,\Gamma^Z(f))=\Gamma^Z(f,\Gamma(f)).$$ Assume also that the operator $L$ admits an invariant probability measure $\mu$ that satisfies the log-Sobolev inequality $$\int_{\mathbb{R}^{d}}(f \Gamma(\ln f) +f\Gamma^Z(\ln f)) d\mu \ge \kappa \left[ \int_{\mathbb{R}^{d}} f \ln f d\mu -\left( \int_{\mathbb{R}^{d}} f d\mu\right)\ln \left( \int_{\mathbb{R}^{d}} f d\mu\right) \right].$$ Then for every positive and bounded $f \in C^\infty(\mathbb{R}^{d})$, such that $\Gamma(\sqrt{f})+\Gamma^Z(\sqrt{f})$ is bounded and $\int_{\mathbb{R}^{d}} f d\mu=1$, $$\begin{aligned}
& 2(\rho_1+\rho_2)\int_{\mathbb{R}^{d}} P_t f \ln P_t f d\mu +\int_{\mathbb{R}^{d}} (P_tf \Gamma(\ln P_tf) +P_tf \Gamma^Z(\ln P_tf) )d\mu \\
\le & e^{-\lambda t}\left( 2 (\rho_1+\rho_2) \int_{\mathbb{R}^{d}} f \ln f d\mu + \int_{\mathbb{R}^{d}}(f\Gamma(\ln f) +f\Gamma^Z(\ln f )) d\mu \right) ,\end{aligned}$$ where $\lambda=\frac{2\rho_2 \kappa}{\kappa+2(\rho_1 +\rho_2)}$.
[10]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the use of generative adversarial networks (GANs) in speech dereverberation for robust speech recognition. GANs have been recently studied for speech enhancement to remove additive noises, but there still lacks of a work to examine their ability in speech dereverberation and the advantages of using GANs have not been fully established. In this paper, we provide deep investigations in the use of GAN-based dereverberation front-end in ASR. First, we study the effectiveness of different dereverberation networks (the generator in GAN) and find that LSTM leads to a significant improvement as compared with feed-forward DNN and CNN in our dataset. Second, further adding residual connections in the deep LSTMs can boost the performance as well. Finally, we find that, for the success of GAN, it is important to update the generator and the discriminator using the same mini-batch data during training. Moreover, using reverberant spectrogram as a condition to discriminator, as suggested in previous studies, may degrade the performance. In summary, our GAN-based dereverberation front-end achieves 14%$\sim$19% relative CER reduction as compared to the baseline DNN dereverberation network when tested on a strong multi-condition training acoustic model.'
address: |
$^1$Shaanxi Provincial Key Laboratory of Speech and Image Information Processing,\
School of Computer Science, Northwestern Polytechnical University, Xi’an, China\
$^2$Xiaomi, Beijing, China
bibliography:
- 'interspeech2018-GAN.bib'
title: Investigating Generative Adversarial Networks based Speech Dereverberation for Robust Speech Recognition
---
[0.80]{}
**Index Terms**: Speech dereverberation, robust speech recognition, generative adversarial nets, residual networks
Introduction {#section:introduction}
============
The performance of automatic speech recognition (ASR) has been boosted tremendously in the last several years and state-of-the-art systems can even reach the performance of professional human transcribers in some conditions [@xiong2017microsoft; @kurata2017language]. However, room reverberation often seriously degrades the ASR performance, especially in far-field speech recognition where the talker is away from the microphone [@kinoshita2016summary; @wu2017reverberation]. Therefore, more attention has been paid recently in the research community to address this issue.
In theory, reverberant speech can be regarded as a room impulse response (RIR) convolving with the clean speech in the time domain [@neely1979invertibility]. A straightforward approach is called *speech dereverberation*, i.e., remove the reverberation from the contaminated speech. In this track, microphone array and multi-channel signal processing are very helpful [@delcroix2007dereverberation; @kumatani2012microphone], but single-channel speech reverberation is still desirable in many real applications in which using multiple microphones may be impractical. Single-microphone speech dereverberation has been intensively studied in the signal processing community and a variety of approaches have been proposed [@neely1979invertibility; @wu2006two; @kinoshita2009suppression; @mosayyebpour2013single; @mohammadiha2016speech]. Another approach to deal with reverberation (and noise) in speech recognition is multi-condition training (MCT), in which speech contaminated with reverberation, either simulated or real-recorded, is added in the acoustic model training set, letting the model learn the reverberation effects automatically. Although the above approaches are reasonably effective, it is still far away from claiming success in the fight against reverberation in speech recognition.
Recently, due to their strong regression learning abilities, deep neutral networks (DNNs) have been used in speech enhancement [@xu2014experimental] and later in speech dereverberation [@kinoshita2016summary; @wu2017reverberation; @kinoshita2017derev]. The deep structure can be naturally regarded as a dereverberation filter that can learn the essential relationship between the reverberant speech and its counterpart, i.e., the clean speech, through a set of multi-condition data. Various deep structures, e.g., feed-forward [@han2015learning], recurrent [@weninger2014deep] and convolutional [@park2016fully], have been explored in the field. Either direct spectral mapping [@wu2017reverberation; @han2015learning] or masking [@williamson2017time] can be considered in the dereverberation network. In the typical spectral mapping approach [@xu2014experimental], the multi-condition data set used in the network training usually consists of pairs of reverberant and clean speech represented by log-power spectra (LPS). Note that in speech recognition, the output of the dereverberation network can be features like FBanks or MFCCs, which do not need to be inverted back to waveforms.
[0.3]{} {width="70.00000%"} \[fig:dnn\]
[0.3]{} {width="100.00000%"} \[fig:rced\]
[0.3]{} {width="40.00000%"} \[fig:lstm\]
All the above DNN-based speech dereverberation approaches aim to minimize the mean square error (MSE) between the outputs of network and the ground truth. Hence, there is an underlying hypothesis that the enhanced speech has the minimal value in the MSE loss with the referenced clean speech. However, the MSE objective function has very strong implicit assumptions, e.g., independence of temporal or spatial, equal importance of all signal samples, and inaccurate to describe the degree of signal fidelity [@wang2009mean]. To remedy this deficiency, generative adversarial networks (GANs) [@goodfellow2014generative], which consist of a generator network ($G$) and a discriminator network ($D$), learned through a min-max adversarial game, might be a good choice. Specifically, Pascual *et al.* have recently demonstrated the promising performance of GAN in speech enhancement [@pascual2017segan] in the presence of additive noise. In the SEGAN approach [@pascual2017segan], the generator $G$ tries to learn the distribution of the clean data and generate enhanced samples from noisy speech to fool the discriminator $D$; while $D$ aims to discriminate between the clean and enhanced samples (generated from $G$), which captures the essential difference between them. While SEGAN works on the waveform level, which targets to improve the perceptual speech quality, Donahue *et al.* [@donahue2017exploring] have explored GAN-based speech enhancement for robust speech recognition. Specifically, in [@donahue2017exploring], GAN works on the log-Mel filter-bank spectra instead of waveforms. The results have shown that GAN enhancement improves the performance of a clean-trained ASR system on noisy speech but it falls short of the performance achieved by conventional MCT. By appending the GAN-enhanced features to the noisy inputs and retraining, a 7% WER improvement relative to the MTR system was achieved.
While the major goal of the above GAN approaches is to remove additive noises, in this paper, we investigate the use of GANs in the mapping-based speech dereverberation for robust speech recognition. Although the same framework can be borrowed from these previous studies, we provide a series of deep investigations in the use of dereverberation front-end in ASR. First, we study the effectiveness of different dereverberation networks (used later as the GAN generator) and find that LSTM dereverberating network can achieve superior speech recognition performance as compared with feed-forward DNN and CNN. Second, further adding residual connections in the deep LSTMs can continuously boost the performance. Finally, we find that it is important to update the generator $G$ and the discriminator $D$ using the same mini-batch data during training for the success of GAN. Moreover, we discover that, using reverberant spectrogram as a condition to $D$, as suggested in [@donahue2017exploring; @michelsanti2017conditional], may degrade the performance of $G$. In summary, using the dereverberation GAN can achieve 14%$\sim$19% relative character error rate (CER) reduction as compared with the DNN dereverberation baseline when tested on a strong multi-condition training acoustic model.
Mapping based speech dereverberation {#section:mapping_methods}
====================================
Speech dereverberation can be achieved by a typical mapping approach [@xu2014experimental], in which a regression DNN (shown in Fig. \[fig:dnn\]) is trained by pairs of reverberant and clean LPS and a linear activation function at the output of DNN is adopted instead of a nonlinear one. Moreover, the target LPS feature is usually normalized globally over all training utterances into zero mean and unit variance (CMVN). In the dereverberation stage, the LPS features of input speech are fed into the well-trained regression DNN to generate the corresponding enhanced LPS features. Finally, the dereverberated waveform is reconstructed from the predicted spectral magnitude and the reverberant speech phase with an overlap-add algorithm.
Besides LPS, the input and output of the dereverberation DNN can be other speech features, e.g., MFCC and FBank. The speech features do not need to be inverted back to waveforms, when used for robust ASR. In [@han2015deep], results show that the mapping from LPS to MFCC can achieve lower word error rate than the mapping from MFCC to MFCC in a speech recognition task under additive noise conditions. This also indicates that the transformation for different feature domains and nonlinear dereverberation function can be learned by the neural network simultaneously. Furthermore, as shown in Fig. \[fig:rced\] and \[fig:lstm\], CNN and LSTM can be used as enhancers as well. We expect that these more powerful network structures can bring further improvements in the speech dereverberation task. We will elaborate the network configurations and evaluate the performances of different networks later in Section \[section:exp\_mapping\].
Dereverberation GAN {#section:gan}
===================
GAN
---
![GAN based speech dereverberation framework.[]{data-label="fig:gan"}](gan.pdf){width="0.6\linewidth"}
Generative adversarial networks (GANs) [@goodfellow2014generative] are generative models implemented by two neural networks competing with each other in a two-player min-max game. Specifically, the generator network $G$ tries to learn a distribution $P_{g}(x)$ over data $x$ and a prior input noise variables $p_z(z)$. The aim is to match the true data distribution $P_{data}(x)$ to fool the discriminator $D$. The discriminator network $D$ serves as a binary classifier which aims to determine the probability that a given sample comes from the real dataset rather than $G$. Because of the weak guidance, the vanilla generative model cannot generate desirable samples. Hence the conditional GAN (CGAN) [@mirza2014conditional] was proposed to steer the generation by considering extra information $x_c$ with the following objective function:
$$\label{eq:cgan}
\begin{split}
\min_G\max_D V(G,D) = \mathbb{E}_{x\sim p_{data}(x,x_c)}[logD(x,x_c)] \\[-0.1cm]
+ \mathbb{E}_{x_c\sim p_{data}(x_c),z\sim p_z(z)}[log(1-D(G(z,x_c),x_c))]\,.
\end{split}
\vspace{-0.5cm}$$
In order to stabilize training and increase the quality of the generated samples in $G$, least-squares GAN (LSGAN) [@mao2016least] was further proposed and the objective function changes to
$$\label{eq:lsgan_d}
\begin{split}
%\min_{D}V_{\text{LSGAN}}(D) = \frac{1}{2}\mathbb{E}_{x\sim p_{data}(x,x_c)}[(D(x,x_c)-1)^2] \\[-0.2cm]
% + \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c),z\sim p_z(z)}[D(G(z, x_c))^2]\,,
\min_{D}V(D) = \frac{1}{2}\mathbb{E}_{x\sim p_{data}(x,x_c)}[(D(x,x_c)-1)^2] \\[-0.2cm]
+ \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c),z\sim p_z(z)}[D(G(z, x_c), x_c)^2]\,,
\end{split}$$
$$\label{eq:lsgan_g}
%\min_{G}V_{\text{LSGAN}}(G) = \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c),z\sim p_z(z)}[(D(G(z,x_c))-1)^2]\,.
\min_{G}V(G) = \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c),z\sim p_z(z)}[(D(G(z,x_c), x_c)-1)^2]\,.$$
Speech dereverberation with GAN
-------------------------------
It is straightforward to use GAN in speech dereverberation and Fig. \[fig:gan\] illustrates such a kind of architecture. It consists of a $G$ and a $D$, where $G$, serving as the mapper in conventional methods, tries to learn a transformation from reverberant speech to clean speech and $D$ tries to determine whether the input samples come from $G(x_c)$ or real-data $x$. Similar to [@han2015deep], $G$ aims to learning a mapping from the LPS feature input to the MFCC feature output which can be directly used in ASR. In some works [@donahue2017exploring; @michelsanti2017conditional], the latent code $z$ is excluded from the generator $G$ to learn a direct mapping instead of a diversified translation in the original image-to-image translation task [@isola2016image]. We borrowed this idea, but we remove the reverberant spectrogram as a condition to $D$. As we will report in Section \[section:exp\_gan\], the added reverberant spectrogram as a condition to $D$ not only increases the parameter size of $D$, but also degrades the performance of $G$. Therefore, we learn a generator distribution $P_g(x)$ over the conditional data $P_{data}(x_c)$ with the following proposed objective function:
$$\label{eq:lsgan_d_new}
\begin{split}
\min_{D}V(D) &= \frac{1}{2}\mathbb{E}_{x\sim p_{data}(x)}[(D(x)-1)^2] \\[-0.1cm]
&+ \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c)}[D(G(x_c))^2]\,,
\end{split}$$
$$\label{eq:lsgan_g_new}
\min_{G}V(G) = \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c)}[(D(G(x_c))-1)^2]\,.$$
To further improve the ability of the adversarial component, previous CGAN approaches have indicated that it is beneficial to mix the GAN objective function with some numerical loss functions [@mirza2014conditional]. We follow this approach in the dereverberation GAN approach and the MSE loss is controlled by a new hyper-parameter $\lambda$. Finally Eq. (\[eq:lsgan\_g\_new\]) becomes
$$\label{eq:lsgan_mse}
\begin{split}
\min_{G}V(G) &= \frac{1}{2}\mathbb{E}_{x_c\sim p_{data}(x_c)}[(D(G(x_c))-1)^2] \\[-0.1cm]
&+ \frac{1}{2} \lambda \mathcal{L}_{\text{MSE}}(G(x_c), x)\,.
\end{split}$$
In practice, the generator $G$ can be a feed-forward network, a convolutional network or a LSTM RNN network, as we described in Section \[section:mapping\_methods\]. Note that the discriminator $D$ is only used in the training and discarded in the dereverberation stage. In our approach, a 2-layer LSTM without residual connection is set to be the architecture of $D$.
Experiments and results {#section:experiment}
=======================
\[1.0\]
[\*[6]{}[c]{}]{} & **Clean** & **Real** & **Simu**\
& $7.86$ & $23.85$ & $20.24$\
[MCT]{} & $7.81$ & $16.02$ & $13.99$\
Datasets
--------
In the experiments, we used a Mandarin corpus as our source of clean speech data, which consists of 103,000 utterances (about 100hrs). The RIRs were from [@ko2017study], including real-recorded RIRs and simulated RIRs for small, medium and large rooms. We randomly selected 97,000 utterances for network training and 3000 utterances for validation, and convolved with the RIRs (both real-recorded and simulated) to obtain the reverberant utterances. The rest 3000 utterances were used for testing and convolved with the real RIR and the simulated RIRs for small, medium and large rooms. Finally we obtained a testing set named ‘Real’ that contains 3000 reverberant speech utterances convolved with real RIRs and another testing set named ‘Simu’ that contains 9000 reverberant speech utterances convolved with simulated RIRs (3000 for small/medium/large). To test the generalization ability of our approach, we ensured the RIRs used for training and testing were totally different. All waveforms were sampled at 16kHz. We used Kaldi [@povey2011kaldi] to generate the reverberant speech by convolving the clean signal with the corresponding RIR. As for feature extraction, the frame length was set to 25ms with a frame shift of 10ms.
ASR back-end
------------
Our speech dereverberation front-end was used for speech recognition experiments. We used Kaldi to train our back-end ASR system with the similar acoustic model architecture and features in [@peddinti2015time]. The original training dataset consists of 1600hrs Mandarin speech data. We used speed-perturbation and volume-perturbation techniques [@ko2015audio] to do data augmentation. Hence the clean model were trained using 4800hrs of speech data (1600 $\times$ 3). We also trained an acoustic model (AM) using multi-condition training (MCT) strategy. The training data for the MCT model is 6400hrs (1600 $\times$ 4), including the above 4800hrs of clean data and 1600hrs of reverberant data generated by convolving the clean data with the RIRs in [@ko2017study] as the dereverberation front-end.
The time delay neural network (TDNN) acoustic model (AM) had 6 layers, and each layer had 850 rectified linear units (ReLUs) with batch renormalization (BRN) [@ioffe2017batch]. The input contexts of TDNN AM were set to \[$-$2,2\]-{$-$1,2}-{$-$3,3}-{$-$7,2}-{$-$3,3}-{0} and the output softmax layer had 5795 units. The notation \[$-$2,2\] means we splice together frames $t-2$ through $t+2$ at the input layer and the notation {$-$1,2} means we splice together the input at the current frame minus 1 and the current frame plus 2. The input of the AM was 40-dimensional MFCC. All the speech dereverberation front-ends were tested on both Clean and MCR AMs. A trigram language model (LM), which was trained on about 2TB scripts with more than 100,000 words in the vocabulary, was used for decoding in the experiments. We also used entropy-based parameter pruning [@stolcke2000entropy] and the threshold was set to be $10^{-8}$.
The baseline results of Clean and MCT model are shown in Table \[tab:am\]. We can see a significant increase in CER when speech is contaminated with reverberations. In extending the training data of acoustic model by adding reverberant speech, the MCT AM can greatly reduce CER.
Mapping-based speech dereverberation {#section:exp_mapping}
------------------------------------
We first investigated the speech dereverberation performances of different networks and input features in the mapping-based approach. Later we will select the best network as the generator in the GAN-based speech dereverberation. Specifically, we tested three different dereverberation networks, i.e., feed-forward DNN, redundant convolutional encoder decoder (RCED) and LSTM. As shown in Fig. \[fig:base\_architecture\], the DNN has 4 hidden layers and each of which contains 1024 ReLU neurons. The structure of the RCED is similar with [@park2016fully] except the last layer. We changed the last filter CNN layer to a fully connected output layer as shown in Fig. \[fig:rced\], because our input and target features were not in the same dimension. The input feature contains a context window of 11 frames ($t\pm 5$) for the DNN and the RCED. The number of filters and filter width of RCED model were set to 12-16-20-24-32-24-20-16-12 and 13-11-9-7-7-7-9-11-13, respectively. The learning rate was set to 0.001 with a mini-batch size of 256. Moreover, BRN was also used for DNN and RCED training. Instead of using vanilla LSTM, we adopted an LSTM with recurrent projection layer (LSTMP) [@sak2014long], which means we do not need to add an extra layer to do residual add like sDNN2 in [@tu2017speech] to avoid dimension mismatch. The LSTM has 4 LSTMP layers followed by a linear output layer. Each LSTMP layer has 760 memory cells and 257 projection units and the input to the LSTM is a single acoustic frame. The learning rate was set to 0.0003 and the model was trained with 8 full-length utterances parallel processing.
\[1.0\]
[\*[7]{}[c]{}]{} & & &\
& & **Real** & **Simu** & **Real** & **Simu**\
& DNN & $17.86$ & $16.63$ & $16.31$ & $14.72$\
& RCED & $18.28$ & $16.73$ & $16.76$ & $15.09$\
& LSTM & $15.38$ & $13.37$ & $14.21$ & $12.46$\
& DNN & $16.62$ & $15.33$ & $15.35$ & $14.03$\
& RCED & $15.55$ & $14.15$ & $14.15$ & $13.09$\
& LSTM & $15.04$ & $13.16$ & $13.97$ & $12.20$\
\[1.0\]
-------------- -------------- -------------- -------------- -------------- -- --
**Real** **Simu** **Real** **Simu**
2-layer LSTM $\bm{15.41}$ $\bm{13.50}$ $\bm{14.25}$ $\bm{12.55}$
*+ Res-I* $16.18$ $14.41$ $14.99$ $13.06$
*+ Res-L* $16.13$ $13.74$ $14.61$ $12.65$
4-layer LSTM $15.04$ $13.16$ $13.97$ $\bm{12.20}$
*+ Res-I* $15.81$ $13.48$ $14.60$ $12.47$
*+ Res-L* $\bm{14.99}$ $\bm{13.13}$ $\bm{13.90}$ $12.22$
8-layer LSTM
*+ Res-I* $15.53$ $13.55$ $14.48$ $12.49$
*+ Res-L* $\bm{14.67}$ $\bm{12.75}$ $\bm{13.62}$ $\bm{12.04}$
-------------- -------------- -------------- -------------- -------------- -- --
: CER (%) comparisons for different layers and residual connection architectures.[]{data-label="tab:resnet"}
\[1.0\]
------------------- ---------------------- ---------------------- ---------------------- ---------------------- --
**Real** **Simu** **Real** **Simu**
SEGAN $32.98~(-98.44)$ $37.14~(-142.27)$ $30.18~(-96.61)$ $32.37~(-130.72)$
DNN $16.62~(0.00)$ $15.33~(0.00)$ $15.35~(0.00)$ $14.03~(0.00)$
LSTM $15.04~(9.51)$ $13.16~(14.16)$ $13.97~(8.99)$ $12.20~(13.04)$
*+ Res* $14.99~(9.81)$ $13.13~(14.35)$ $13.90~(9.45)$ $12.22~(12.90)$
*+ GAN* $\bm{14.07~(15.34)}$ $12.02~(21.59)$ $13.15~(14.33)$ $11.42~(18.60)$
*+ GAN+Res* $14.10~(15.16)$ $\bm{11.96~(21.98)}$ $\bm{13.14~(14.40)}$ $\bm{11.40~(18.75)}$
*+ GAN+Res (DB)* $15.72~(5.42)$ $13.95~(9.00)$ $14.60~(4.89)$ $12.83~(8.55)$
*+ GAN+Res+CD* $14.27~(14.14)$ $12.19~(20.48)$ $13.38~(12.83)$ $11.43~(18.53)$
------------------- ---------------------- ---------------------- ---------------------- ---------------------- --
All the models explored here were optimized with the Adam [@kingma2014adam] method and initialized with the Xavier [@glorot2010understanding] algorithm. We also used exponential decay to decrease the learning rate which was similar with Kaldi nnet3[^1] and the terminated learning rate was 5 orders of magnitude smaller than the initial learning rate.
In Table \[tab:baseline\], we list all experimental results on both Clean and MCT AMs. Firstly, we observe consistent improvement on all dereverberation networks by replacing MFCC with LPS features as the network input. Here the LPS feature is 257 dimension and the MFCC feature is 40 dimensions. Note that the output of all the dereverberation networks is 40-dimension MFCC which is fed into the ASR system. This conclusion is consistent with that in [@han2015deep], where LPS performs better than MFCC when used as the input of a denoising network.
When we compare Table \[tab:baseline\] with Table \[tab:am\], we can find that the mapping-based dereverberation works quite well. When tested on the Clean AM, all the dereverberation networks are effective with significant CER reduction; when tested on the MCT AM, the dereverberation networks with the LPS input are still effective with apparent CER reduction. Comparing different model structures, we discover that LSTM achieves the best performance. For instance, the LPS-LSTM dereverberation network reduces the CER from 23.85% (real-reverberation added) to 15.04% for the Clean AM and reduces the CER from 16.02% (real-reverberation added) to 13.97% for the MCT AM. We believe that the superior peformance is because of the LSTM’s ability to model long-term contextual information that is essential is the speech dereverberation task. We also find RCED-CNN is not good when MFCC is used as the input. We will use LSTM as our network in the rest of the experiments.
Adding ResNet
-------------
Table \[tab:resnet\] shows the results of different residual connection architectures. The layer-wise residual connection (Res-L) structure can be seen in Fig. \[fig:lstm\]; while the input residual connection (Res-I) structure is similar with Res-L and more details can be found in [@chen2017improving]. As we expected, it’s not necessary to add residual connections to shallow networks. Performances degrade when residual connections are used in a 2-layer LSTM. Res-L always performs better than Res-I. This is reasonable because Res-L tries to learn the residue of the high-level abstract feature while Res-I just learns the residue of the input feature. When the LSTM is as deep as 4 layers, Res-L starts to work and the lowest CERs are achieved when the LSTM has 8 layers. As training a 8-layer LSTM is time-consuming, we perform the GAN experiments with a 4-layer LSTM generator in the following.
Speech dereverberation with GAN {#section:exp_gan}
-------------------------------
We finally investigated the ability of GAN in mapping-based speech dereverberation. We also reproduced the SEGAN approach [@pascual2017segan] with the open-source codes[^2] as a comparison. As shown in Table \[tab:res\_gan\], SEGAN degrades the ASR performance within our expectation, which is consistent with the reported results in [@donahue2017exploring]. We believe this is because SEGAN aims to improve the perception of noisy speech and time-domain enhancement may be not appropriate for reverberant speech recognition.
In the proposed GAN-based methods, the architecture of $G$ is consistent with that in Fig. \[fig:lstm\] with 4 hidden LSTMP layers. The architecture of $D$ is similar with $G$ but contains only 2 LSTMP layers and the cell number and the projection dimension are set to 256 and 40, respectively. The hyper-parameter $\lambda$ in Eq. (\[eq:lsgan\_mse\]) was set to 200 and the learning rate of $G$ and $D$ were set to $0.00008$ and $0.0003$, respectively. In each iteration, we updated the parameters of $G$ twice and the parameters of $D$ once. To stabilizing GAN training, we also add instance Gaussian noise to the MFCC input of $D$[^3]. In Table \[tab:res\_gan\], we demonstrate that using GAN (in LSTM+GAN) is not only viable but also outperforms the LSTMs.
At the early stage of our experiments, we updated the parameters of $G$ and $D$ using different mini-batch data like the ways they do in image tasks. In other words, the parameters of $D$ were updated using one mini-batch data and then the parameters of $G$ were updated using a new mini-batch data. We found that this training strategy—LSTM+GAN+Res (DB) in the second-to-last row of Table \[tab:res\_gan\]—was quite unstable in our experiments and we always achieved results worse than the non-adversarial training (e.g., LSTM+Res) as shown in Table \[tab:res\_gan\]. Instead, when we tried to update the parameters of $G$ and $D$ using the same mini-batch data, we achieved consistently better results (LSTM+GAN+Res in Table \[tab:res\_gan\]). We believe that this strategy is essential in making our GAN approach performing well. Adding residual connections works for most cases. LSTM+GAN+Res lowered the MCT AM CER from 15.35% down to 13.14% with 14.4% relative CER reduction for the Real set and lowered the MCT AM CER from 14.03% down to 11.40% with 18.75% relative CER reduction for the Simu set. Finally, we also find the performance of LSTM+GAN+Res+CD is worse than LSTM+GAN+Res. This means that adding the reverberant spectrogram as a condition to D is useless to the dereverberation performance.
Summary {#section:conclusion}
=======
In this paper, we provide a deep investigation of GAN in mapping-based speech dereverberation for robust speech recognition. In the selection of the generator network, we find that LSTM achieves superior performance, while adding residual connections (ResNets) in deep LSTMs can further boost the performance. In the use of GAN, we find that it is essential to update the generator and the discriminator using the same mini-batch data during model training; and using reverberant spectrogram as a condition to the discriminator may degrade the performance. With the above findings, we are able to achieve 14%$\sim$22% relative CER reduction in ASR as compared with a DNN baseline, while the SEGAN baseline even does not work on the ASR task. In the future, we plan to further explore the use of GAN in more adverse conditions (both reverberant and noisy) and try to combine the framework with joint-training strategy to further improve the ASR performance.
Acknowledgements {#section:acknowledgment}
================
The authors would like to thank Shan Yang from Northwestern Polytechnical University, Dr. Bo Li from Google and Dr. Bo Wu from Xidian University for their helpful comments and suggestions on this work. The research work is supported by the National Key Research and Development Program of China (Grant No.2017YFB1002102) and the National Natural Science Foundation of China (Grant No.61571363).
[^1]: egs/wsj/s5/steps/libs/nnet3/train/common.py(get\_learning\_rate)
[^2]: https://github.com/santi-pdp/segan
[^3]: http://www.inference.vc/instance-noise-a-trick-for-stabilising-gan-training/
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We give a geometric description of variational principles in mechanics, with special attention to constrained systems. For the general case of nonholonomic constraints, a unified variational approach is given, and the equations of motion of both vakonomic and nonholonomic frameworks are obtained. We study specifically the existence of infinitesimal variations in both cases. When the constraints are integrable, both formalisms are compared and it is proved that they coincide. As examples, we give geometric formulations of the equations of motion for the case of optimal control and for vakonomic and nonholonomic mechanics with constraints linear in the velocities.'
---
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‘@=12 =cmssi10 scaled 1
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=
=
[**SOME GEOMETRIC ASPECTS OF VARIATIONAL CALCULUS IN CONSTRAINED SYSTEMS**]{}\
Xavier Gràcia\
[*Departament de Matemàtica Aplicada i Telemàtica\
Universitat Politècnica de Catalunya\
Campus Nord UPC, edifici C3, C. Jordi Girona 1, 08034 Barcelona, Spain\
[[email protected]]{}*]{}\
Jesús Mar[í]{}n-Solano\
[*Departament de Matemàtica Econòmica, Financera i Actuarial\
Universitat de Barcelona\
Av. Diagonal 690, 08034 Barcelona, Spain\
[[email protected]]{}*]{}\
Miguel-C. Muñoz-Lecanda\
[*Departament de Matemàtica Aplicada i Telemàtica\
Universitat Politècnica de Catalunya\
Campus Nord UPC, edifici C3, C. Jordi Girona 1, 08034 Barcelona, Spain\
[[email protected]]{}*]{}
[*PACS: 02.40.Vh, 02.30.Wd, 45.20.Jj*]{} [*MSC: 70F25, 70H30, 58F05, 49S05*]{}
Introduction
============
To describe the motion of mechanical systems there is a variety of mathematical models which are based on different principles. Most of the physical models are obtained using an appropriate variational principle in a certain evolution space. But variational principles are not only important in physics, but also in many branches of engineering [@Ze] or economics [@In], where one is interested in optimizing a given functional, possibly subject to some restrictions. In fact, constraints are ubiquitous in many mechanical systems and much more different situations.
In this paper we are going to study Lagrangian systems, i.e., dynamical systems in which the equations of motion are obtained by finding the critical paths of a functional $$\int_{t_1}^{t_2} L \,\dif t\; ,$$ where $L$ is a function defined on the tangent bundle $\Tan Q$ of a given differentiable manifold $Q$, the configuration space. We will not consider arbitrary variations, but only variations satisfying certain conditions. These conditions arise from some given constraints on the dynamics of the system. We will analize the case when the constraints are defined by a certain submanifold $C$ of the tangent bundle $\Tan Q$. Such constraints are usually called nonholonomic.
There are two different approaches when dealing with constraints. The first one is based on the idea of understanding the constraints as constraint forces. This point of view, that seems very natural in a physical context, gives rise to the classical d’Alembert-Lagrange principle. Mechanics of Lagrangian systems with nonholonomic constraints based on this principle is called nonholonomic mechanics [@Ar][@Ga][@Ve]. But there is another different point of view, that seems more natural when one is interested in optimizing a given functional defined as above when there are constraints. For example, if we wish to change the state of a given system minimizing a cost functional (a typical problem in engineering or economics), it is not natural to understand the constraints as forces acting on the system. In this case one is interested in minimizing the functional considering only the variations allowed by the constraints. Mechanics of Lagrangian systems with nonholonomic constraints based on this idea is often called vakonomic mechanics (mechanics of [**v**]{}ariational [**a**]{}xiomatic [**k**]{}ind [@Ar]). For example, optimal control theory is a typical example of vakonomic mechanics. It is interesting to notice that both mechanics do not coincide in general, but they agree when the constraints are integrable. Several references on these topics are [@Ar], [@Bli], [@Els] and [@Le]. See also [@LMM; @SC].
The paper is organized as follows. In section 2 we give a generalized notion of a variational problem. Section 3 is devoted to study vakonomic mechanics from a geometric point of view, with special attention to the existence of admissible variations in order to obtain the equation of motion. In section 4 we do the same for nonholonomic mechanics. Both mechanics are understood as generalized variational problems. In section 5 it is proved that, when the constraints are integrable, vakonomic and nonholonomic mechanics coincide. In section 6, as an example of vakonomic mechanics, a geometric formulation of optimal control theory is studied. Finally, in section 7, we give a geometric formulation of vakonomic mechanics when the constraints are defined by a distribution (linear constraints in the velocities).
Basic knowledge of differential geometric structures is assumed. The presentation is almost self-contained but the interested reader may consult the bibliography for more specific topics as the vertical lift, the fibre derivative and the canonical involution [@Ab], or the Euler-Lagrange operator [@Ca].
Variational problems
====================
Elements of a variational problem
---------------------------------
First, we are going to define what we mean by a variational problem. A variational problem consists of the data $(Q,L,C,{\cal C},{\cal W})$ where:
- $Q$ is a $n$-dimensional differentiable manifold, the [*configuration space*]{}.
- $L$ is the [*lagrangian function*]{} defined on the tangent bundle, $L \colon \Tan Q\longrightarrow\R$.
- $C$ is the [*constraint submanifold*]{}, and it is a submanifold of $\Tan Q$.
- $\cal{C}$ is the family of [*admissible paths*]{}. Given two points $q_1, q_2 \in Q$ and a compact interval $I = [t_1,t_2]$, we will say that a path $\gamma
\colon I \to Q$ of class $C^2$ is admissible if:
$\gamma(t_1) = q_1$, $\gamma(t_2) = q_2$ and
$\dot{\gamma}(t)\in C$, for all $t\in I$.
- ${\cal W}$ are the [*admissible variation fields*]{} (or infinitesimal variations). For a given admissible path $\gamma$, ${\cal W}_\gamma$ consists on a certain set of $C^1$ vector fields along $\gamma$.
Notice that we do not consider variations of a path $\gamma$, but variation fields along $\gamma$. Now we are ready to define the variational problem associated to $(Q,L,C,\cal{C},\cal{W})$. The [*action*]{} of $L$ along a path $\gamma$ is the functional $S \colon {\cal C}\longrightarrow\R$ given by the integral \[accion\] S\[\] = \_I L((t)) t .
A variational problem consists in finding the critical admissible paths of the functional $S$, in a sense that will be precised later.
Variations and variation fields
-------------------------------
Let $\gamma \colon I\longrightarrow Q$, $\gamma(t_1)=q_1$, $\gamma(t_2)=q_2$, be an admissible path. A [*variation*]{} of $\gamma$ is a ${\rm C}^2$ function $\Gamma \colon (-\delta,\delta)\times I\longrightarrow Q$ such that:
1. $\Gamma_\eps = \Gamma(\eps,\cdot)$ is a one-parameter family of paths defined on $I$ with fixed end-points, $\Gamma(\eps,t_i)=q_i$, $\forall\eps\in(-\delta,\delta)$, $i=1,2$, and
2. $\Gamma(0,t) = \gamma(t)$, $\forall t\in I$ (if there is no variation, $\eps=0$, we obtain the original path $\gamma$).
Given a function $\Gamma(\eps,t)$ of two real variables, we will denote its derivatives with respect to $\eps$ and $t$ as $\Gamma'$ and $\dot\Gamma$, respectively. It is clear that $\Gamma'$ and $\dot\Gamma$ are vector fields along $\Gamma$.
Now, we are in conditions to define a variation field of $\Gamma$.
Given a variation $\Gamma(\eps,t)$ of an admissible path $\gamma$, the [*variation field*]{} of $\Gamma$ is the vector field ${\bf
w}$ along $\gamma$ defined by $$\bfw(t)= \Tan_{(0,t)}\Gamma \cdot \left. \deriv{}{\eps} \right|_{(0,t)} =
\Gamma'(0,t)\; .$$
Notice that $$\bfw(t_1) = 0 , \quad \bfw(t_2) = 0\; ,$$ since the $\Gamma_\eps$ have fixed end-points.
Therefore, given a family of variations of $\gamma$, we can associate to them a family of variation vector fields along $\gamma$. We will say that a variation $\Gamma$ of $\gamma$ is [*admissible*]{} if its associated variation vector field along $\gamma$, $\bfw(t)=\Gamma'(0,t)$, is an admissible variation field of $\gamma$, i.e., $\bfw\in{\cal W}_\gamma$.
We finish this description about variations and variation fields with a useful lemma, whose proof is straighforward in local coordinates.
\[lambda-w\] For any $\bfw$ vector field along $\gamma$ and any function $\lambda\colon I\to\R$, $$\invol \circ (\lambda \bfw)^{\textstyle.} =
(\Dif \lambda)\, \vl(\dot\gamma,\bfw) + \lambda \,\invol \circ \dot\bfw\; .$$ $\Box$
Notice that, if $\bfw$ is a vector field along $\gamma$, then $\dot\bfw$ is a vector field along $\bfw$ and $\invol \circ \dot\bfw$ is a vector field along $\dot\gamma$. The function $\lambda \colon I \to \R$ denotes a function of time, and it is clear that $\lambda \bfw$ is another vector field along $\gamma$. $\Dif$ is the usual derivative with respect to the time. The map $\vl$ denotes the vertical lift $\vl\colon\Tan Q\times_Q\Tan
Q\to\Tan(\Tan Q)$. Its local expression is $\vl (q,v,u)=(q,v;0,u)$. Finally, $\invol \colon \Tan(\Tan Q) \to \Tan(\Tan Q)$ denotes the canonical involution, which is an isomorphism between the two vector bundle structures of $\Tan(\Tan Q)$. Its local expression is $\invol(q,v;u,a) = (q,u;v,a)$.
Critical admissible paths
-------------------------
An admissible path $\gamma$ is said to be [*critical*]{} if, for each admissible variation $\Gamma_\eps$, the first variation of $S[\Gamma_\eps]$ is zero; i.e., $$\left. \deriv{}{\eps} S[\Gamma_\eps] \right|_{\eps=0} = 0$$ for each admissible variation $\Gamma_\eps$ of $\gamma$.
The main purpose of this paper consists in discussing the criticity conditions for different variational problems and describing the solutions. First of all, we are going to describe the criticity condition for a general problem.
It is clear that, if $g \colon Q \to \R$ is a function, then, for any function $\Gamma(\eps,t)$ ($\Gamma \colon
U\subset\R^2\longrightarrow Q$) of two real variables, $$\derpar{}{\eps} g(\Gamma(\eps,t)) =
\langle \dif g(\Gamma(\eps,t)) , \Gamma'(\eps,t) \rangle\; ,$$ and similarly for $\derpar{}{t}$.
Now, suppose that $\Gamma(\eps,t)$ is a variation of a path $\gamma$. Let us consider $\dot\Gamma \colon (-\delta,\delta) \times I \to \Tan Q$. Derivation of $\dot\Gamma$ with respect to $\eps$ and $t$ yields $(\dot\Gamma)'$ and $(\dot\Gamma)^{\textstyle.}$, which are now vector fields along $\dot\Gamma$. Taking $\eps=0$ yields two vector fields along $\dot\gamma$, which are $\invol \circ \dot{\bfw}$ and $\ddot\gamma$.
Then, if $f \colon \Tan Q \to \R$ is a function, we have . |\_[=0]{} f((,t)) = f((t)) , ((t)) . \[epsder\] (Remember that $\dot\bfw$ is a vector field along $\bfw$, and $\invol \circ \dot\bfw$ is a vector field along $\dot\gamma$, so the contraction makes sense.)
Now, we can characterize the criticity condition in a more manageable way.
\[PEL\] Given a variational problem $(Q,L,C,\cal{C},\cal{W})$, an admissible path $\gamma$ is critical if and only if $$\int_I
\left\langle \dif L(\dot\gamma(t)) , \invol(\dot\bfw(t)) \right\rangle
\,\dif t\ = 0\; ,$$ for each admissible vector field $\bfw\in{\cal W}_\gamma$.
[**Proof:**]{} Using (\[epsder\]) in (\[accion\]), we obtain $$\left. \deriv{}{\eps} S[\Gamma_\eps] \right|_{\eps=0}
=
\int_I
\left\langle \dif L(\dot\gamma(t)) , \invol(\dot\bfw(t)) \right\rangle
\,\dif t\; ,$$ and the result follows. $\Box$
Observe that this condition does not depend on the full variation $\Gamma(\eps ,t)$, but only on its variation field (see also [@Calo]). Therefore, in our study of variational calculus, we will shift our attention to infinitesimal variations rather to finite variations.
The Euler-Lagrange operator
---------------------------
To obtain a more manageable condition of criticity, it is convenient to perform an integration by parts. First, let us define the Euler-Lagrange operator of $L$.
\[EL\] The [*Euler-Lagrange operator*]{} associated with a function $L \colon \Tan Q\longrightarrow\R$ is a mapping $\EL_L
\colon \Tan^2Q \to \Tan^*Q$ defined by the relation $$\langle \EL_L \circ \ddot\gamma , \bfw \rangle
=
\langle \dif L \circ \dot\gamma , \invol \circ \dot\bfw \rangle
-
\Dif \langle \FD L \circ \dot\gamma , \bfw \rangle\; ,$$ for any path $\gamma\colon I\to Q$ and vector field $\bfw$ along $\gamma$.
Here, the map $\FD L \colon \Tan Q \to \Tan^*Q$ is the fibre derivative of $L$. Recall that, given a vector bundle $E \to B$, if $f \colon E \to \R$ is a function, then the derivatives of the restrictions of $f$ to the fibres define the fibre derivative of $f$, which is a map $\FD f \colon E \to E^*$. Its local expression is $\FD f(b;a) = (b; \derpar fa)$.
It is easy to check (in coordinates) that the Euler-Lagrange operator is well-defined by this relation.
Therefore, the Euler-Lagrange operator is a one-form along the projection $\Tan^2Q \to Q$, and also an affine bundle map along $\Tan Q \to Q$. The expression in local coordinates of $\EL_L$ is the usual one, $$\EL_L =
\left( \derpar Lq - \deriv{}{t}\left( \derpar Lv \right) \right) \dif q\; ,$$ where $\deriv{}{t}$ is the total time-derivative operator. The Euler-Lagrange operator can be extended in the same way to a time dependent Lagrangian.
Using the definition of the Euler-Lagrange operator, we can characterize the criticity condition in the usual form.
\[ELgen\] Given a variational problem $(Q,L,C,\cal{C},\cal{W})$, an admissible path $\gamma$ is critical if and only if $$\int_I \left\langle \EL_L(\ddot\gamma(t)) , \bfw(t) \right\rangle \,\dif t
=0\; ,$$ for each admissible variation field $\bfw\in{\cal W}_\gamma$.
[**Proof:**]{} From proposition \[PEL\] and the definition \[EL\] of the Euler-Lagrange operator we obtain that $$\left. \deriv{}{\eps} S[\Gamma_\eps] \right|_{\eps=0}
=
\int_I \left\langle \EL_L(\ddot\gamma(t)), \bfw(t) \right\rangle \,\dif t +
\bigg[\langle \FD L(\dot\gamma(t)) , \bfw(t) \rangle \bigg]_{t_1}^{t_2}\; .$$ The result follows observing that, since $\bfw$ is a variation field, the last term vanishes ($\bfw(t_1)=0$, $\bfw(t_2)=0$). $\Box$
For the case when there are no constraints, $C=\Tan Q$ and ${\cal W}_\gamma$ is the set of all the vector fields along $\gamma$, we obtain the well-known [*Euler-Lagrange equation*]{}.
Given a unconstrained variational problem, a path $\gamma\in{\cal C}$ is critical if and only if $$\EL_L \circ \ddot\gamma = 0\; .$$
In this paper we will be interested in variational problems when $C\subseteq
\Tan Q$, $C\neq \Tan Q$. Hence, given a set of admissible paths, it is necessary to select a set of admissible variation fields (or infinitesimal variations) along the admissible paths. We will consider two different approaches to this problem. The first one is vakononomic mechanics, which can be considered as a strictly variational approach. The second one is nonholonomic mechanics, which is variational in our generalized sense, but not in the classical one. Nonholonomic mechanics is the usual way to describe the dynamics of a mechanical system with constraints. In the next two sections we will describe the dynamical equations obtained in each case. It is interesting to remark that both approaches are equivalent when the constraints are integrable (holonomic constraints).
We finish this section with a useful property of the Euler-Lagrange operator that will be used in many calculations in the following. The proof is straighforward in local coordinates.
\[mu-f\] For any $f \colon \Tan Q \to \R$ and $\mu \colon I \to \R$ (a function of time), $$\EL_{\mu f} = \mu\, \EL_f - (\Dif \mu) \FD f \circ \tau^2_{\,1}\; ,$$ where $\tau^2_{\,1} \colon \Tan^2 Q \to \Tan Q$ is the canonical projection. $\Box$
Vakonomic mechanics
===================
Roughly speaking, vakonomic mechanics is the result of variational calculus when the variations are restricted by some constraints on the positions and also the velocities.
Our initial setting is therefore a submanifold $C \subset \Tan Q$ of codimension $m < n$; let us denote by $j$ the inclusion of $C$ in $\Tan Q$. A [*constraint*]{} is any function $\phi$ vanishing on $C$. Locally $C$ is defined by the vanishing of some constraints $\phi^i \colon \Tan Q \to \R$ ($i=1,\dots ,m$) whose differentials $\dif \phi^i$ are linearly independent at each point of $C$.
We will assume that [*the projection of $C$ to $Q$, $\tau_Q \circ j \colon C \longrightarrow Q$, is a submersion*]{}. It can be easily proved that this statement is equivalent to say that the constraints $\phi^i$ can be chosen such that their fibre derivatives $\FD
\phi^i$ are linearly independent at every point of $C$. In local coordinates, this means that $\derpar{\phi^i}{v^k}$ has maximal rank. That is, the constraints restrict the velocities, not the positions.
With the assumptions above, the image $(\tau_Q\circ j)(C) \subset Q$ is open, so we may assume that the projection $C \to Q$ is a [*surjective*]{} submersion. Then there exists the vertical subbundle $\Ver(C) \subset
\Tan(C)$, which has rank $n-m$ (the dimension of the fibres of the submersion). Indeed, at each $v_q \in C$ we have $\Ver_{v_q}(C) =
\Tan_{v_q}(C) \cap \Ver_{v_q}(\Tan Q)$.
To obtain the equations of motion of vakonomic mechanics, we need first to describe which are the admissible variations.
The variations of vakonomic mechanics
-------------------------------------
We remember that an admissible path is a mapping $\gamma \colon I \to Q$ such that $\dot\gamma$ takes its values in the submanifold $C \subset \Tan
Q$. Due to our assumptions on $C$, there exist vector fields locally defined on $Q$ taking values in $C$. Their integral curves have their derivatives in
$C$, so there are many admissible paths.
Let $\Gamma$ be a variation of an admissible path $\gamma$. The variation $\Gamma$ is called a [*strongly admissible variation*]{} of $\gamma$ if every path $\Gamma_\eps$ is admissible.
If $\Gamma$ is a strongly admissible variation, then $\phi(\dot\Gamma(\eps,t)) = 0$, for any constraint $\phi$. Taking the derivative with respect to $\eps$ at $\eps=0$ and using (\[epsder\]) we have $$\langle \dif \phi \circ \dot\gamma , \invol \circ \dot\bfw \rangle = 0$$ for every constraint $\phi$. This can also be expressed as $$\invol(\dot\bfw(t)) \in \Tan_{\dot\gamma(t)}(C)$$ for each $t \in I$.
\[varvak\] A variation field $\bfw$ of an admissible path $\gamma$ is called an [*admissible variation field*]{} for a given vakonomic problem if $$\invol(\dot\bfw(t)) \in \Tan_{\dot\gamma(t)}(C)\; ,$$ that is, $\langle \dif \phi \circ \dot\gamma , \invol \circ \dot\bfw \rangle
= 0$.
From the definition of the Euler-Lagrage operator (definition \[EL\]), we obtain that $\bfw$ is a variation field if and only if $$\langle \EL_\phi \circ \ddot\gamma , \bfw \rangle
=
- \Dif \langle \FD \phi \circ \dot\gamma , \bfw \rangle\$$ for every constraint $\phi$.
It is important to remark that an admissible path may not have any nontrivial strongly admissible variation, and so an admissible variation field may not arise from a strongly admissible variation. One may say that the variations having admissible variation fields are the variations that preserve the constraints up to first order in $\eps$. These variations may be called [*weakly admissible variations*]{}.
Next, we are going to give a more detailed description of admissible variation fields. Among all the vector fields $\bfw$ along $\gamma$, we consider a particular submodule. Take the subbundle $$\LD_\gamma^C \subset \gamma^* \Tan(Q)=I\times_\gamma \Tan (Q)$$ whose sections are the vector fields $\bfw$ along $\gamma$ of class $C^1$ whose vertical lifts $\vl(\dot\gamma,\bfw)$ are tangent to $C$. Using this subbundle we can express the admissible variation fields in a more manageable way. First, notice that, since $I$ is an interval, both $\LD_\gamma^C$ and $\gamma^* \Tan (Q)$ are trivializable. Therefore there exists a global frame for $\gamma^* \Tan (Q)$, $(\bfw_k)$ ($k=1,\dots ,n$). Since $\LD_\gamma^C$ is a subbundle of rank $n-m$, we can assume that the last $n-m$ of the $\bfw_k$ span this subbundle.
Any vector field along $\gamma$ can be thus uniquely written $\bfw =
\sum_{k=1}^{n} \lambda^k \bfw_k$, where $\lambda^k$ are functions of time. Then $\bfw$ is a variation field if and only if the coefficients $\lambda^k$ vanish at the end-points of $I$. Moreover, according to definition \[varvak\], $\bfw$ is an admissible variation field if it is a variation field and $\langle \dif \phi^i \circ \dot\gamma, \invol \circ \dot\bfw
\rangle = 0$, for $i=1,\dots ,m$. Taking into account lemma \[lambda-w\], this condition can be written \[des\] \_[k=1]{}\^[n]{} \^i , \_k \^k + \_[k=1]{}\^[n]{} \^i , \_k \^k = 0 . Notice that, since the fibre derivatives $\FD \phi^i$ are linearly independent at each point, the matrix with entries $\langle \FD \phi^i \circ \dot\gamma, \bfw_k
\rangle$ has maximal rank, $m$. By the special choice of the $\bfw_k$, the last $n-m$ of them vanish under the $\FD \phi^i$, and hence the square matrix $A = (\langle \FD \phi^i \circ \dot\gamma, \bfw_j
\rangle)_{i,j=1,\dots ,m}$ is invertible. So, writing the equation as $$\label{humm}
\sum_{j=1}^{m} A^i_{\,j} \, \Dif \lambda^j +
\sum_{j=1}^{m} B^i_{\,j} \, \lambda^j + \sum_{l=m+1}^{n} C^i_{\,l} \,
\lambda^l
= 0\; ,$$ we can isolate the $\Dif \lambda^j$ ($j=1,\dots ,m$) linearly in terms of all the $\lambda^k$ ($k=1,\dots, n$). This determines uniquely $\lambda^j$, $j=1,\dots,m$ as functions of $\lambda^l$, $l=m+1,\dots,n$, due to the initial condition $\lambda^j(t_1)=0$. However, not any $\lambda^l$, $l=m+1,\dots,n$ vanishing on $t_1,t_2$ are admissible. Notice that the solutions $\lambda^j$, $j=1,\dots, m$ must vanish also in $t_2$. In fact, the existence of solutions of (\[humm\]) satisfying $\lambda^k(t_1)=\lambda^k(t_2)=0$, $k=1,\dots n$, is not guaranteed in principle. If we write (\[humm\]) as $$A \, \dot\lambda_{(1)} = -
B \, \lambda_{(1)} -
C \, \lambda_{(2)}\; ,$$ the solution satisfying $\lambda_{(1)}(t_1)=0$ is $$\lambda_{(1)}(t)=-\nu(t)\int^t_{t_1} [\nu(s)]^{-1}\, A^{-1}(s)\,
C(s)\,\lambda_{(2)}(s)\,\dif s\; ,$$ where $\nu(t)$ is the fundamental matrix of the homogeneous system $A \,
\dot\lambda_{(1)} = -
B \, \lambda_{(1)}$ satisfying the initial condition $\nu^j_{\,i}(t_2)=\delta^j_{\,i}$. If $\lambda_{(1)}(t_2)=0$, then necessarily $$\label{uultima}
\int^{t_2}_{t_1} [\nu(s)]^{-1}\, A^{-1}(s)\, C(s)\,\lambda_{(2)}(s)\,\dif s
= 0\; .$$ If the system is homogeneous ($C=0$) we obtain the trivial solution $\lambda^j(t)=0$, $j=1,\dots,m$, remaining $\lambda^l$, $l=m+1,\dots,n$ as arbitrary functions satisfying the boundary conditions $\lambda^l(t_1)=\lambda^l(t_2)=0$. As for the general case, in the following section we will show that admissible variations always exist in vakonomic mechanics.
Condition (\[des\]) is very useful to study variation fields in vakonomic mechanics, as we show in the following example.
[**Example**]{} Let $Q=\R^2$ be the configuration space, with coordinates $(x,y)$, and consider a Lagrangian function and a constraint both depending only on the velocities, i.e., $L=L(\dot{x},\dot{y})$ and $\phi = \phi(\dot{x},\dot{y})$.
From our assumptions on the constraints we can write locally $\phi =
\dot{y}-f(\dot{x})\equiv 0$. In this case, using theorem \[importante\], it is a simple calculus to show that the equations of motion of the vakonomic problem are $\ddot{x}=0$, that is $x(t)=a t+b$ and $y(t)=f(a)
t+c$. The parameters $a$, $b$ and $c$ are obtained from the boundary conditions $x(t_1)$, $y(t_1)$, $x(t_2)$ and $y(t_2)$. If $\gamma(t)=(x(t),y(t))$ is a straight line satisfying the boundary conditions, using (\[des\]), the reader can check that there exist admissible variation fields, and they are vector fields along $\gamma(t)$ of the form $\lambda (t) \bfw$, where $\lambda(t_1)=\lambda(t_2)=0$ and $\bfw(x,y)=(x,y;1,f'(a))$.
For example, if $\phi(\dot{x},\dot{y})\equiv \dot{y}-\sqrt{1+\dot{x}^2}=0$ and $x(t)$ is a linear function of time, there are not strongly admissible variations (see [@Ar]). But there exist admissible variation fields, so there are weakly admissible variations. In fact, the weakly admissible variations $\Gamma(\eps,t)=(x(\eps,t),y(\eps,t))$ have the form $x(\eps,t)=x(t)+\lambda(t)\eps + o(\eps)$, $y(\eps,t)=y(t)+(a/\sqrt{1+a^2})\lambda (t)\eps+o(\eps)$, where $\lambda(t_1)=\lambda(t_2)=0$.
The equations of motion of vakonomic mechanics
----------------------------------------------
As we have shown, a critical path of the action with constraints is an admissible path $\gamma$ such that $\ds \int_I \left\langle
\EL_L(\ddot\gamma(t)) , \bfw(t) \right\rangle \dif t$ vanishes for each [*admissible*]{} variation field $\bfw$ (theorem \[ELgen\]). To obtain the corresponding Euler-Lagrange equation, we first establish the following proposition.
\[importante0\] Given a variational problem $(Q,L,C,{\cal C},\cal{W})$, where $\cal{W}$ are the variation fields satisfying definition \[varvak\], let $\gamma$ be an admissible path. Then, for any family of functions $\mu_i(t)$, $i=1,\dots,m$, the first-order variations of the $\int_{\dot\gamma}L\,\dif
t$ and $\int_{\dot\gamma} (L + \sum_{i=1}^{m} \mu_i \phi^i) \,\dif t$ with respect to an admissible variation field $\bfw \in\cal{W}$ coincide.
[**Proof:**]{} In principle, since the variations may not be strongly admissible, it is not clear that the variations of both actions yield the same result. However, using theorem \[ELgen\] and definition \[EL\], the difference of the first-order variations of the actions is $$\int_I
\left\langle \EL_{\sum_{i=1}^{m} \mu_i \phi^i} \circ \ddot\gamma, \bfw
\right\rangle \dif t
=
\sum_{i=1}^{m} \int_I \mu_i
\left\langle \dif \phi^i \circ \dot\gamma, \invol \circ \dot\bfw
\right\rangle\dif t
-
\sum_{i=1}^{m} \int_I
\Dif \left\langle \mu_i (\FD \phi^i \circ \dot\gamma), \bfw
\right\rangle\dif t\; ,$$ and both terms vanish whenever $\bfw$ is an admissible variation field. Therefore, the variations of the two actions coincide when $\bfw$ is an admissible variation field. $\Box$
\[importante\] Given a variational problem $(Q,L,C,{\cal C},\cal{W})$, where $\cal{W}$ are the variation fields satisfying definition \[varvak\], let $\gamma$ be an admissible path. Then $\gamma$ is critical if and only if there exist functions $\mu_j \colon I\to\R$, $j=1,\dots m$, such that $$\label{VAK}
\EL_{L + \sum_{i=1}^{m} \mu_i \phi^i} \circ \ddot\gamma = 0 \; .$$ This is the equation of motion of vakonomic mechanics.
[**Proof:**]{} If equation (\[VAK\]) holds then, for each admissible variation field $\bfw$, $\int_I\langle \EL_{L + \sum_{i=1}^{m} \mu_i \phi^i}
\circ \ddot\gamma , \bfw \rangle\,\dif t = 0$, which, according to Proposition (\[importante0\]), is equivalent to $\int_I\langle \EL_{L}
\circ \ddot\gamma , \bfw \rangle\,\dif t = 0$. This shows that $\gamma$ is a critical path. So it remains to prove the converse: that equation (\[VAK\]) is a necessary condition for the criticity of an admissible path $\gamma$. First, notice that the $\mu_i$ can be chosen such that \_[L + \_[i=1]{}\^[m]{} \_i \^i]{} , \_j = 0 \[mu-part\] for $j=1,\dots ,m$. Indeed, by lemma \[mu-f\] this equation can be written as $$\langle \EL_L \circ \ddot\gamma , \bfw_j \rangle +
\sum_{i=1}^{m}
\mu_i \,\langle \EL_{\phi^i} \circ \ddot\gamma , \bfw_j \rangle -
\sum_{i=1}^{m}
\Dif \mu_i \,\langle \FD \phi^i \circ \dot\gamma , \bfw_j \rangle
= 0$$ for each $j=1,\dots ,m$. From definition \[EL\] and the choice of $\bfw_j$ we have $$\langle \EL_{\phi^i} \circ \ddot\gamma , \bfw_j \rangle =
\langle \dif \phi^i \circ \dot\gamma , \invol \circ \dot\bfw_j \rangle
-
\Dif \langle \FD \phi^i \circ \dot\gamma , \bfw_j \rangle = \langle \dif
\phi^i \circ \dot\gamma , \invol \circ \dot\bfw_j \rangle \; ,$$ for $j=1,\dots,m$. That is, we have $$\label{hummm}
(\Dif \mu_i) \,A^i_{\,j} - \mu_i \,B^i_{\,j} - D_j = 0\; ,$$ where $A$ and $B$ are the matrices we have used previously (\[humm\]). So again we have a linear differential equation that determines the functions $\mu_i$ (up to initial conditions). From now on we assume that $A$ is the identity matrix; this can be easily done through a linear change of the basis $(\bfw_i)$.
If we apply the variational principle for the modified Lagrangian $L +
\sum_{i=1}^{m} \mu_i \phi^i$, we have $$\sum_{k=1}^{n}
\int_{I} \langle
\EL_{L + \sum_{i=1}^{m} \mu_i \phi^i} \circ \ddot\gamma, \bfw_k
\rangle \lambda^k\,\dif t = 0$$ for each set of functions $\lambda^k$ yielding an admissible variation field.
If we choose the functions $\mu_i$ satisfying (\[mu-part\]), then the sum is only from $m+1$ to $n$: $$\label{otramas}
\sum_{j=m+1}^{n}
\int_{I} \langle
\EL_{L + \sum_{i=1}^{m} \mu_i \phi^i} \circ \ddot\gamma, \bfw_j
\rangle \lambda^j\,\dif t = 0\;.$$ This must hold for any choice of the functions $\lambda^{m+1}, \ldots
,\lambda^{n}$ giving an admissible variation field. However, as we have shown in the preceding section, the functions $\lambda^{m+1},\ldots,\lambda^{n}$ are not arbitrary in general, due to the final conditions $\lambda^1(t_2)=\cdots=\lambda^m(t_2)=0$. Let $(\bar{\mu}_i)_{1\leq i\leq m}$ be the particular solution of (\[hummm\]) satisfying $\bar{\mu}_i(t_2)=0$, and let $(\bar{\nu}^j_{\,i})$ be the transpose of the fundamental matrix of the homogeneous system of (\[hummm\]) satisfying the initial condition $\bar{\nu}^j_{\,i}(t_2)=\delta^j_{\,i}$. Notice that $\bar{\nu}=(\nu)^{-1}$. (In general, if $\nu$ is a fundamental matrix of $\dot{x}=A\cdot x$, then $(\nu^t)^{-1}$ is a fundamental matrix of $\dot{x}=-A^t\cdot x$). Then the general solution of (\[hummm\]) is $\mu_i=\bar{\mu}_i+\sum_{j=1}^{m}\rho_j\bar{\nu}^j_{\,i}$, where $\rho_j$ are arbitrary constants. Suppose for a while that there exist admissible variation fields $\bfw = \sum_{k=1}^n \lambda^k\bfw_k$ with $$\label{fiinal}
\lambda^l=\langle \EL_{L + \sum_{i=1}^{m} (\bar{\mu}_i
+\sum_{j=1}^{m}\rho_j\bar{\nu}^j_{\,i})\phi^i} \circ \ddot\gamma, \bfw_l
\rangle\; ,$$ for $l=m+1\dots,n$. Then (\[otramas\]) is a vanishing sum of integrals of squares, which, combined with (\[mu-part\]), yields the equation of motion (\[VAK\]). It remains to prove that the choice of such $\lambda^l$, $l=m+1,\ldots,n$, gives an admissible variation field. From (\[uultima\]) and $\bar{\nu}=(\nu)^{-1}$, the variations defined by (\[fiinal\]) are admissible if and only if $$\label{ajj}
\int^{t_2}_{t_1} \bar{\nu}(s)\, C(s)\,\, [\langle \EL_{L + \sum_{i=1}^{m}
(\bar{\mu}_i +\sum_{j=1}^{m}\rho_j\bar{\nu}^j_i)\phi^i} \circ \ddot\gamma,
\bfw_{(2)}
\rangle]^t\dif s = 0\; ,$$ where $\bfw_{(2)}$ denotes the last $n-m$ vector fields. Now, from lemma \[mu-f\] and definition \[EL\], we have $$\label{ujj}
\langle \EL_{\sum_{j=1}^m\bar{\nu}^i_{\,j}\phi^j}\circ\ddot{\gamma}, \bfw_l
\rangle =
\sum_{j=1}^m\bar{\nu}^i_{\,j}\langle\EL_{\phi^j}\circ\ddot{\gamma},\bfw_l
\rangle -
\sum_{j=1}^m(\hbox{D}\bar{\nu}^i_{\,j})\langle\FD\phi^j\circ\dot{\gamma},\bf
w_l\rangle =
\sum_{j=1}^m\bar{\nu}^i_{\,j}\langle\dif\phi^j\circ\gamma,\invol\circ\bfw_l\
rangle\; .$$ Using that $C^i_{\,l}=\langle\dif\phi^i\circ\dot{\gamma},\invol\circ{\bf\dot{w}}_l
\rangle$ and combining (\[ajj\]) and (\[ujj\]), we obtain the linear system for the $\rho_j$ $$\sum_{h=1}^m\rho_h\int_{t_1}^{t_2}\sum_{l=m+1}^n
\langle\EL_{\sum_{j=1}^m\bar{\nu}^i_{\,j}\phi^j}\circ\ddot{\gamma}, \bfw_l
\rangle\,\,
\langle \EL_{\sum_{k=1}^m\bar{\nu}^h_{\,k}\phi^k}\circ\ddot{\gamma}, \bfw_l
\rangle =$$ $$\label{sistemita}
= - \int_{t_1}^{t_2}\sum_{l=m+1}^n
\langle\EL_{\sum_{j=1}^m\bar{\nu}^i_{\,j}\phi^j}\circ\ddot{\gamma}, \bfw_l
\rangle\,\,
\langle \EL_{L+\sum_{k=1}^m\bar{\mu}_k\phi^k}\circ\ddot{\gamma}, \bfw_l
\rangle; .$$ If this system has any solution, then we can find values for $\rho_j$, $j=1,\dots,m$, such that the functions $\lambda^l$ defined in (\[fiinal\]) give rise to admissible variations. Now, we prove that this system has always solution. Consider the pre-Hilbert space of continuous vector-valued functions ${\bf f}:[t_1,t_2]\to\R^{n-m}$ with the usual scalar product $\langle {\bf f},{\bf g}\rangle
=\sum_{l=m+1}^n\int_{t_1}^{t_2} f_l\cdot g_l$. Let $V$ be the finite–dimensional subspace spanned by the $m$ vectors $${\bf e}_i =
\left(\langle\EL_{\sum_{j=1}^m\bar{\nu}^i_{\,j}\phi^j}\circ\ddot{\gamma},
\bfw_{m+1} \rangle, \ldots,
\langle\EL_{\sum_{j=1}^m\bar{\nu}^i_{\,j}\phi^j}\circ\ddot{\gamma}, \bfw_n
\rangle\right)\; ,$$ $i=1,\ldots,m$. Then we can write the system (\[sistemita\]) as $$\sum_{h=1}^m\rho_h\langle{\bf e}_i,{\bf e}_h\rangle = \langle{\bf e}_i,{\bf
v}\rangle\; ,$$ where ${\bf v}=-(\langle
\EL_{L+\sum_{k=1}^m\bar{\mu}_k\phi^k}\circ\ddot{\gamma}, \bfw_{m+1}
\rangle,\ldots,\langle
\EL_{L+\sum_{k=1}^m\bar{\mu}_k\phi^k}\circ\ddot{\gamma}, \bfw_{n} \rangle)$.
The solutions of this system are any constants $\rho_h$ such that $\sum_{h=1}^m \rho_h\cdot{\bf e}_h$ is the orthogonal projection of ${\bf
v}$ onto $V$. This is well defined, since $V$ is finite–dimensional. (Notice that the $\rho_h$ may not be unique, since the ${\bf e}_h$ are not necessarily independent). This completes the proof. $\Box$
[**Remark:**]{} Notice that, using lemma \[mu-f\], the equation of motion may also be written as \_L = \_[i=1]{}\^[m]{} ( (\_i) \^i - \_i \_[\^i]{} ) . \[VAK’\] In the proofs of the equation of motion of vakonomic mechanics that one can find in the literature, it is usually assumed that the functions $\lambda^l$, $l=m+1,\ldots,n$, giving the admissible variations are free. Then, the equation of motion is obtained as a straight consequence of (\[otramas\]). However, in general, these functions are not absolutely free.
The variations and equations of motion of nonholonomic mechanics
================================================================
In this section we are going to show that nonholonomic mechanics may be understood as a variational problem.
Our initial setting is also the submanifold $C \subset \Tan Q$, which may be locally defined by the vanishing of the constraints $\phi^i$. An [*admissible path*]{} is still a path $\gamma \colon I \to Q$ such that $\dot\gamma$ takes its values in $C$. Let us define which are the admissible variation fields in nonholonomic mechanics.
\[varnohol\] A variation field $\bfw$ of an admissible path $\gamma$ is called an admissible variation field (in nonholonomic mechanics) if it is a section of the subbundle $\LD_\gamma^C \subset \gamma^* \Tan Q$. That is, $$\vl(\dot\gamma(t),\bfw(t)) \in \Tan_{\dot\gamma(t)}(C)\; .$$
Using the constraints, equivalent statements are $$\langle \dif \phi \circ \dot\gamma , \vl(\dot\gamma,\bfw) \rangle = 0\; ,$$ or $$\label{varnohol1}
\langle \FD \phi \circ \dot\gamma , \bfw \rangle = 0\; ,$$ for each constraint $\phi$.
Notice the key difference with respect to vakonomic mechanics: now the admissibility is a $C^1(I)$-linear condition on $\bfw$. This linearity makes things easier. Next, we obtain the equation of motion of nonholonomic mechanics.
Given a variational problem $(Q,L,C,{\cal C},\cal{W})$, where ${\cal W}$ are the admissible variation fields satisfying definition \[varnohol\], an admissible path $\gamma$ is critical if and only if there exist functions $\mu_j$, $j=1,\dots m$, such that $$\label{nonhol}
\EL_L \circ \ddot\gamma =
\sum_{i=1}^{m} \mu_i \, \FD \phi^i \circ \dot\gamma .$$
[**Proof:**]{} A [*critical path*]{} for nonholonomic mechanics is an admissible path $\gamma$ such that the first-order variation of the action, $\ds
\int_I \left\langle \EL_L(\ddot\gamma(t)) , \bfw(t) \right\rangle \dif t ,
$ vanishes for each admissible variation field $\bfw$. By equation (\[varnohol1\]), this means that $\EL_L(\ddot\gamma(t))$ is a linear combination of the $\FD \phi^i \circ \dot\gamma$. Thus, the result follows. $\Box$
[**Remark:**]{} It is obvious that there always exist admissible variation fields in nonholonomic mechanics. For example, if we calculate the admissible variation fields of the example in section 3, we will find that, in this case, they coincide with the admissible variation fields of vakonomic mechanics. But this will not be true in general if the constraints are not integrable.
The case of integrable constraints
==================================
Let us consider the problem of holonomic constraints in the usual sense.
Given a differentiable manifold $Q$ and a Lagrangian function $L \colon \Tan Q\longrightarrow\R$, a [*holonomic problem*]{} is a variational problem where
- The constraint submanifold is given by a submanifold $P\subset Q$, thus $C=\Tan P$.
- Admissible paths are paths $\gamma \colon I\longrightarrow P\subset
Q$.
- Variation fields $\bfw$ along $\gamma$ are admissible if they are tangent to $P$.
Notice that, from any admissible variation field along an admissible path $\gamma$, one may construct a variation $\Gamma$ contained in $P$. Therefore, it is clear that the problem with holonomic constraints is equivalent to the unconstrained variational problem defined on $P$ by taking the restriction of the Lagrangian $L$ to $\Tan P\subset \Tan Q$.
Now, consider the cases of both vakonomic and nonholonomic mechanics when the constraints are defined by an integrable subbundle $C\subset \Tan Q$. In this situation, we have the following equivalence.
If the constraint submanifold $C$ is an integrable subbundle of $\Tan Q$, then both vakonomic and nonholonomic mechanics coincide, and they are equivalent to a holonomic constrained problem on each integral submanifold of $C$.
[**Proof:**]{} First of all, notice that, in both cases, a path $\gamma \colon
I\longrightarrow Q$ is admissible (i.e., $\dot{\gamma}$ is in $C$) if and only if it is contained in an integral submanifold $P\subset Q$ of $C$. Recall that $\Tan_q(P)=C_q$, for each point in $P$.
Let $\bfw$ be a variation field of $\gamma$. We know that $\bfw$ is admissible in the nonholonomic framework if $\vl(\dot\gamma(t),\bfw(t)) \in
\Tan_{\dot\gamma(t)}(C)$ (see definition \[varnohol\]). Since the vertical lift restricts naturally to subbundles, this is equivalent to say that $\vl(\dot\gamma(t),\bfw(t)) \in \hbox{V}_{\dot{\gamma}(t)} C\subset
\hbox{V}_{\dot\gamma(t)}(\Tan Q)$, that is, $\bfw (t)\in C_{\gamma(t)}$ or, what is the same, $\bfw (t)\in \Tan_{\gamma(t)} P$. And this last condition says that $\bfw$ is admissible for the holonomic problem. Therefore, we have proved that a nonholonomic problem with integrable constraints is equivalent to a holonomic problem on each integral submanifold of $C$.
Now, we show the equivalence with the vakonomic problem. Since $C$ is an integrable subbundle of $\Tan Q$, it is known that the integral submanifolds of $C$ can be locally described as $\psi=\hbox{constant}$, for some independent functions $\psi$ on $Q$. This implies that the constraint submanifold $C$ can be locally described by $\phi=\hbox{\~{d}}\psi=0$. Here, $\hbox{\~{d}}\psi \colon \Tan Q\longrightarrow\R$ is the differential d$\psi$ of $\psi \colon Q\longrightarrow\R$ considered as a function on the tangent bundle.
Let $\bfw$ be a variation field of $\gamma$, in the sense of vakonomic mechanics. Then, for any $\phi=\hbox{\~{d}}\psi$, we have $$0=\langle \hbox{d}\phi\circ\dot{\gamma},\invol\circ{\bf \dot{w}}\rangle =
\hbox{\~{d}\~{d}}\psi\circ\invol\circ{\bf\dot{w}}\; .$$ Using the property $\hbox{\~{d}\~{d}}\psi =
\hbox{\~{d}\~{d}}\psi\circ\invol$, we obtain $$0=\hbox{\~{d}\~{d}}\psi\circ{\bf\dot{w}} =
\langle\hbox{d\~{d}}\psi\circ{\bf w},{\bf\dot{w}}\rangle =
\langle\hbox{d}\phi\circ{\bf w},{\bf\dot{w}}\rangle = \Dif (\phi\circ\bfw) =
\Dif\langle\hbox{d}\psi\circ\gamma,\bfw\rangle\; .$$ Thus, $\bfw$ is an admissible variation field in vakonomic mechanics if and only if $\langle\hbox{d}\psi\circ\gamma,\bfw\rangle$ is constant. Since $\bfw (t_1)=0$, this constant is zero, $\langle\hbox{d}\psi\circ\gamma,\bfw\rangle=0$, which means that $\bfw$ is tangent to $P$. Therefore, $\bfw$ is admissible for the vakonomic problem if and only if it is admissible for the holonomic problem on $P$. $\Box$
Optimal control and vakonomic mechanics
=======================================
A problem of optimal control may be given by the following data: a configuration space $B$ describing the state variables, a fibre bundle $\pi\colon M\longrightarrow B$ whose fibres describe the control variables, a vector field $Y$ along the projection of the bundle, $Y\colon
M\longrightarrow\Tan B$, and a “Lagrangian function" $L\colon
M\longrightarrow\R$. For a path $\gamma\colon I\to M$ where $\pi\circ\gamma$ (not $\gamma$!) has fixed end-points, the problem is to find an extremum of the action $$\int_\gamma L(\gamma(t))\,\dif t$$ when $\gamma$ satisfies the differential equation $$\label{difcon}
(\pi\circ\gamma)^{\textstyle.} = Y\circ\gamma$$ that rules the evolution of the state variables.
It is easy to show that this is indeed a vakonomic problem on the manifold $M$, in which the Lagrangian $L$ is very singular, since it does not depend on the velocities. The constraint submanifold $C\subset \Tan M$, given by the differential equation above, is $$C=\{ w_u\in \Tan M\mid\Tan\pi (w_u)=Y(u)\}\; .$$ In this way, a path $\gamma$ is admissible if and only if it is a solution of the differential equation (\[difcon\]) or, equivalently, if it takes values in the affine subbundle $C$ of $\Tan M$. On the other hand, from the special characteristics of optimal control problems, we can relax the boundary conditions as we have done. Observe that theorems \[ELgen\] and \[importante\] remain true since $L$ does not depend on the velocities and the structure of the constraints (\[difcon\]) (they do not depend on the derivatives on the control variables). In coordinates, if $x^i$, $i=1,\dots
m$, are local coordinates in $B$ and $(x^i,u^\alpha)$, $i=1,\dots m$, $\alpha=1,\dots,n-m$, are local coordinates in $M$, the action is given by $$\int_{t_1}^{t_2} L(x^i,u^\alpha)\,\dif t$$ and the constraints are given by a set of first order differential equations $$\dot{x}^i=Y^i(x,u)~,~~i=1,\dots,m,$$ with boundary conditions $x^i_1=x^i(t_1)$ to $x^i_2=x^i(t_2)$ (there are no boundary conditions on the control variables). Notice that for this vakonomic problem the constraints are very particular: they express the velocities of the state variables in terms of the state and control variables.
Let us identify which variation fields $\bfw$ are admissible. Notice that frames for the bundles $\LD_\gamma^C \subset \gamma^* \Tan(M)$ are provided with $(\bfw_\alpha)$ and $(\bfw_i,\bfw_\alpha)$, where for instance $$\bfw_\alpha = \derpar{}{u^\alpha} \circ \gamma , \quad
\bfw_i = \derpar{}{x^i} \circ \gamma ,$$ if we have coordinates $(x^i,u^\alpha)$ on $M$. Writing $\bfw = \lambda^i \bfw_i + \lambda^\alpha \bfw_\alpha$, the differential equation (\[des\]) turns out to be $$\Dif \lambda^i - \derpar{y^i}{x^j} \lambda^j = \derpar{y^i}{u^\alpha}
\lambda^\alpha .$$
Optimal control theory admits several geometric formulations and expressions of the equation of motion. First, we can give a [*lagrangian description*]{} on the configuration manifold $\Tan^*B \times_B M$. We only need to define a lagrangian on its tangent space: $${\cal L}(x,p,u,\dot{x},\dot p,\dot u) = L(x,u) + \langle p, \dot x - Y(x,u)
\rangle$$ where we write $(x;p,u)$ for the variables of $\Tan^*B \times_B M$—recall that $\Tan(\Tan^*B \times_B M) = \Tan(\Tan^*B) \times_{\Tan B} \Tan M$; its elements are pairs of tangent vectors $(\dot{x},\dot p,\dot u)$ projecting to the same tangent vector $\dot x$. For a path $\eta$ on $\Tan^*B \times_B
M$ the Euler-Lagrange equation $\EL_{\cal L} \circ \ddot\eta$ is readily seen to be equivalent to the vakonomic equation (\[VAK\]).
Let us recall that, given a $l$-dimensional differentiable manifold $Q$ with local coordinates $(q^A)$, there is a canonical tensor field on $\Tan Q$, the vertical endomorphism $S$ which is a rank-$l$ (1,1) tensor field on $\Tan Q$ such that $\Ker S = \hbox{Im}S$ and whose Nijenhuis tensor $N_S$ vanishes. In natural coordinates $(q^A,\dot{q}^A)$, the local expression of $S$ is given by $S=\dif q^A\otimes\frac{\partial}{\partial\dot{q}^A}$. Also, we have the Liouville vector field $\Delta$ (the infinitesimal generator of the dilations along the fibres on $\Tan Q$), whose local expression is $\Delta=\dot{q}^A\partial
/\partial\dot{q}^A$. A vector field $X$ in $\Tan Q$ is called a second order differential equation (SODE) if $S(X)=\Delta$. Now, if ${\cal L} \colon \Tan
Q\longrightarrow\R$ is a Lagrangian function, we can construct the Cartan 1-form associated with ${\cal L}$, given by $\theta_{\cal L}=S^*\circ\dif
{\cal L}$, the Cartan 2-form $\omega_{\cal L}=-\dif\theta_{\cal L}$ and the energy function $E_{\cal L}=\Delta ({\cal L})-{\cal L}$. Then the paths $\eta$ solution of the Euler-Lagrange equations are the integral curves of a second order differential equation $X$ in $\Tan Q$ satisfying the dynamical equation $i_X\omega_{\cal L}=\dif E_{\cal L}$.
Taking $Q=\Tan^*B\times_B M$ and ${\cal L}= L(x,u) + \langle p, \dot x -
Y(x,u)\rangle$, we obtain a geometrical expression of the equation of motion of optimal control theory, $$i_X\omega_{\cal L}=\dif E_{\cal L}\; ,$$ where $E_{\cal L}=\langle p, Y(x,u)\rangle - L(x,u)$ and $\omega_{\cal
L}=-\dif\theta_{\cal L} = -\dif(p_i \dif x^i) = \dif x^i\wedge\dif p_i$.
Associated to this lagrangian description we can consider the hamiltonian formalism of ${\cal L}$. This means to consider the manifold $\Tan^*(\Tan^*B
\times_B M)$ with its canonical symplectic structure, the Legendre’s transformation of ${\cal L}$, $\FD{\cal L} \colon \Tan(\Tan^*B \times_B M)
\to \Tan^*(\Tan^*B \times_B M)$, and to push forward through it the energy function to a hamiltonian function ${\cal H} \colon \Tan^*(\Tan^*B \times_B
M) \to \R$.
However, the most interesting geometric description of optimal control theory is a [*presymplectic description*]{} which can be constructed on the manifold $\Tan^*B \times_B M$. Here we consider the 2-form $\omega$ obtained by pull-back through $\Tan^*B \times_B M \to \Tan^*B$ of the canonical 2-form of the last manifold. In local coordinates, $\omega=\dif q \wedge \dif p$. Taking the hamiltonian function defined by $$H(x,u,p) = \langle p,Y(x,u) \rangle - L(x,u)\; ,$$ if $\eta$ is a path on $\Tan^*B \times_B M$, the presymplectic equation $$i_{\dot\eta} \omega = \dif H \circ \eta$$ is equivalent to the equation of motion of vakonomic mechanics, in the sense that there is a natural bijection between both sets of solutions. To show this is enough to write the local expressions.
In optimal control theory the hamiltonian function is usually written as $H(x,u,p) = \langle p,Y(x,u) \rangle - \mu_0 L(x,u)$, where $\mu_0=0,1$. When $\mu_0=0$ we recover the so called abnormal solutions (see [@LiuS]). However, in the vakonomic approach, there are not abnormal solutions. The key issue is that we work with admissible variation fields, not admissible variation curves.
Constraints defined by a distribution
=====================================
In this section we present a geometric framework for constrained systems when the constraint submanifold $C$ is a distribution (or vector subbundle) of the tangent manifold $\Tan Q$. In local coordinates, this means that the constraints are linear functions on the velocities. The subbundle $C\subset\Tan Q$ can be described in terms of its annihilator, $C^0\subset\Tan^* Q$. If this is locally described in terms of 1-forms, $\alpha^i=\alpha^i_a(q)\dif q^a$ ($i=1,\dots,m$, where $m$ is the codimension of $C$ and $(q^a)$ are local coordinates of $Q$), then $C$ is locally described in terms of the constraints $$\phi^i(v_q) = \langle \alpha^i(q),v_q \rangle = 0 , \quad i=1,\dots, m\; .
%\phi^i(q,\dot{q})=\alpha^i_a(q)\,\dot{q}^a=0\;, \quad i=1,\dots, m\; .$$
Let us consider the vector bundle $\Tan Q\oplus C^0$, in which we will set up the dynamics.
On the one hand, given the Lagrangian function $L$ on the tangent bundle $\Tan Q$, let $\theta_L=S^*\circ\dif L$ be the Lagrange 1-form on $\Tan Q$. Its pull-back along the projection $\pi_1\colon\Tan Q\oplus C^0\longrightarrow\Tan Q$ yields the 1-form $$\theta_1=\pi_1^*\theta_L$$ on $\Tan Q\oplus C^0$. Also, using the Liouville vector field $\Delta$ on $\Tan Q$, the energy function associated with $L$ in $\Tan Q\oplus C^0$ is $$E=\pi_1^*(\Delta (L)-L) .$$
On the other hand, let $\theta_Q$ be the canonical 1-form defined on the cotangent bundle $\Tan^*Q$. If $j_0\colon C^0\longrightarrow \Tan^*Q$ denotes the canonical inclusion and $\pi_2 \colon\Tan Q\oplus C^0\longrightarrow C^0$ is the projection onto the second factor, then we can take the pull-back of these mappings to construct a 1-form $\theta_2$ on $\Tan Q\oplus C^0$ as $$\theta_2=(j_0\circ\pi_2)^*\theta_Q .$$
Using the 1-forms $\theta_1$ and $\theta_2$, we have a presymplectic form $$\Omega=-\dif(\theta_1+\theta_2) .$$ By means of the energy function $E$, we obtain a presymplectic dynamics on the extended phase space $\Tan Q\oplus C^0$ which is equivalent to vakonomic mechanics:
\[vaklineal\] Let $L \colon \Tan Q \to \R$ be a Lagrangian, and $C \subset \Tan Q$ a vector subbundle. For a path $\xi$ in the manifold $\Tan Q \oplus C^0$, consider the differential equation i\_ = E . \[vaksum\] This equation is equivalent to the equation of motion of vakonomic mechanics in the following sense:
- If $\xi = (\xi_1,\xi_2)$ is a solution of (\[vaksum\]) and $\xi_1$ is the lift of a path in $Q$, $\xi_1 = \dot\gamma$, then $\gamma$ is an admissible path ($\dot\gamma$ is in $C$) and is a solution of the equation of motion of vakonomic mechanics (\[VAK\]).
- Conversely, given an admissible path $\gamma$ which is a solution of (\[VAK\]), together with the multipliers $\mu^i$, then the path $\xi(t) = (\dot\gamma(t), \sum \mu_i(t) \dif \phi^i(\dot\gamma(t)))$ is a solution of equation (\[vaksum\]).
If the Lagrangian is regular then equation (\[vaksum\]) already implies that $\xi_1$ is the lift of a path in $Q$.
[**Proof:**]{} It is enough to check the equivalence in local coordinates. We take $(q^a,v^a,\lambda_i)$, $a=1,\dots,n$, $i=1,\dots,m$, as local coordinates in $\Tan Q \oplus C^0$ (we represent an element of $C^0_q$ as $\sum \lambda_i \alpha^i(q)$). Then we have $$\begin{aligned}
\theta_1 &=& \frac{\partial L}{\partial v^a} \dif q^a ,
\\
\theta_2 &=& \lambda_i \alpha^i_a(q) \dif q^a ,
\\
\Omega &=&
\left( \frac{\partial^2 L}{\partial v^a\partial q^b} +
\lambda_i \frac{\partial \alpha^i_a}{\partial q^b} \right)
\dif q^a \wedge \dif q^b
+
\frac{\partial^2 L}{\partial v^a \partial v^b}
\dif q^a \wedge \dif v^b
+
\alpha^i_a \dif q^a \wedge \dif \lambda_i .\end{aligned}$$ Since $E = v^a \left(\partial L/\partial v^a\right) - L$, we also have $$\dif E =
\left( v^b \frac{\partial^2 L}{\partial q^a\partial v^b} -
\frac{\partial L}{\partial q^a} \right) \dif q^a +
v^b \frac{\partial^2 L}{\partial v^a \partial v^b} \dif v^a .$$
Now let us consider the path $\xi(t) = (q^a(t),v^a(t),\lambda^i(t))$, with velocity $\dot\xi = (q,v,\lambda;\dot q,\dot v,\dot \lambda)$. Then $$\begin{aligned}
i_{\dot\xi}\Omega & = &
\left(
\dot q^b \frac{\partial^2 L}{\partial q^a \partial v^b}
+ \dot q^b \frac{\partial \alpha^i_b}{\partial q^a}\lambda_i
- \dot q^b \frac{\partial^2 L}{\partial v^a \partial q^b}
- \dot q^b \frac{\partial \alpha^i_a}{\partial q^b}\lambda_i
- \dot v^b \frac{\partial^2 L}{\partial v^a \partial v^b}
- \dot \lambda_i \alpha^i_a
\right) \dif q^a
\nonumber \\
&& \hbox{}
+ \dot q^b \frac{\partial^2 L}{\partial v^a \partial v^b} \dif v^a
+ \dot q^a \alpha^i_a \dif \lambda_i .
\nonumber\end{aligned}$$
Therefore, equation $i_{\dot\xi}\Omega = \dif E$ is equivalent to the three equations $$\label{un}
(\dot q^b-v^b) \frac{\partial^2 L}{\partial q^a \partial v^b}
+ \dot q^b \frac{\partial \alpha^i_b}{\partial q^a} \lambda_i
- \dot q^b \frac{\partial^2 L}{\partial v^a \partial q^b}
- \dot q^b \frac{\partial \alpha^i_a}{\partial q^b} \lambda_i
- \dot v^b \frac{\partial^2 L}{\partial v^a \partial v^b}
- \dot \lambda_i \alpha^i_a
= - \frac{\partial L}{\partial q^i} ,$$ $$\label{dos}
\dot q^b \frac{\partial^2 L}{\partial v^a \partial v^b}
=
v^b \frac{\partial^2 L}{\partial v^a \partial v^b} ,$$ $$\label{tres}
\dot q^a \alpha^i_a = 0 .$$
The fact that $\xi_1$ is the lift of a path $\gamma$ in $Q$ means in coordinates that $v(t) = \dot q(t)$, so equation (\[dos\]) is an identity. Notice also that if the Lagrangian is regular then the Hessian matrix $\left(\frac{\partial^2 L}{\partial v^a \partial v^b}\right)$ is invertible, therefore in this case equation (\[dos\]) implies that $v(t) = \dot q(t)$, that is to say, $\xi_1$ is the lift of a path in $Q$. Then, in equation (\[tres\]) we obtain the constraints $\phi^i(q,\dot q) = \alpha^i_a(q) \dot{q}^a = 0$, that is, $\gamma$ is an admissible path. Finally, we can write equation (\[un\]) as $$\ddot q^b \frac{\partial^2 L}{\partial v^a \partial v^b} +
\dot q^b \frac{\partial^2 L}{\partial v^a \partial q^b} +
\dot q^b \frac{\partial \alpha^i_a}{\partial q^b} \lambda_i +
\dot \lambda_i \alpha^i_a
=
\frac{\partial L}{\partial q^a} +
\dot q^b \frac{\partial \alpha^i_b}{\partial q^a} \lambda_i .$$ But these are the vakonomic equations (\[VAK\]) of the extended Lagrangian ${\cal L} = L + \mu_i \alpha^i_a v^a$, using the natural identification between the functions $\mu_i$ and the coordinates $\lambda_i$ of the cotangent vectors. $\Box$
[**Remark:**]{} In a similar way, the vakonomic dynamics can be also defined on the manifold $C\oplus C^0$. Since $C\oplus C^0$ is a vector subbundle of $\Tan Q\oplus C^0$, we can pull-back the $2$-form $\Omega$ and the energy function $E$ to define a $2$-form $\tilde{\Omega}$ and a new function $\tilde{E}$ in $C\oplus C^0$. The reader can check that, then the equation of motion of vakonomic mechanics (\[VAK\]) is equivalent to find the paths $\xi = (\xi_1,\xi_2)$ in $C\oplus C^0$, where $\xi_1$ is the lift of a path in $Q$, such that $$\label{vakCC}
i_{\dot\xi} \tilde{\Omega} = \dif\tilde{E} \circ \xi .$$ Moreover, if the lagrangian is regular, then $\tilde\Omega$ is a symplectic form. Notice also that this equation, as well as equation (\[vaksum\]), can also be expressed in terms of vector fields. For instance, when $\tilde\Omega$ is symplectic, the solutions of equation (\[vakCC\]) are the integral curves of the vector field $\tilde X$ such that $$i_{\tilde X} \tilde{\Omega} = \dif\tilde{E} .$$
In the case of nonholonomic mechanics, a similar result can be proved, in the same way as for theorem \[vaklineal\]. Let us denote $\Omega_1 = -\dif \theta_1$. Then we have:
\[nhlineal\] Let $L \colon \Tan Q \to \R$ be a Lagrangian, and $C \subset \Tan Q$ a vector subbundle. For a path $\xi$ in the manifold $\Tan Q \oplus C^0$, consider the differential equation i\_ \_1 = E + \_2 . \[nhsum\] This equation is equivalent to the equation of motion of nonholonomic mechanics in the following sense:
- If $\xi = (\xi_1,\xi_2)$ is a solution of (\[nhsum\]) where $\xi_1$ is the lift of a path $\gamma$ in $Q$, $\xi_1 = \dot\gamma$, then $\gamma$ is a solution of the equation of motion of nonholonomic mechanics (\[nonhol\]).
- Conversely, given a path $\gamma$ which is a solution of (\[nonhol\]), together with the multipliers $\mu^i$, then the path $\xi(t) = (\dot\gamma(t), \sum \mu_i(t) \dif \phi^i(\dot\gamma(t)))$ is a solution of equation (\[nhsum\]).
If the Lagrangian is regular then equation (\[nhsum\]) already implies that $\xi_1$ is the lift of a path in $Q$. $\Box$
Conclusions
===========
In this paper we have presented variational calculus (in one dimension) in a geometric framework, aiming to study dynamical systems with non-holonomic constraints ([*i.e.*]{}, constraints depending on the positions and the velocities). We have shown that a generalised formulation of variational calculus, in which the admissible paths and the admissible infinitesimal variations are not necessarily related, makes room for the study of dynamical systems subject to non-holonomic constraints from different points of view. This generalized variational calculus encompasses the often-called vakonomic mechanics (which is a strict variational problem with constraints) and the non-holonomic mechanics (based on d’Alembert’s principle).
In the case of vakonomic mechanics, we have provided a geometric procedure to obtain the equation of motion, choosing an appropriate set of admissible infinitesimal variations proving that they always exist.
In the case of non-holonomic mechanics, it is far more simple than in vakonomic mechanics to choose an appropriate set of admissible infinitesimal variations, and the corresponding equation of motion is readily obtained.
Our formulation also provides a neat equivalence between both vakonomic and non-holonomic mechanics when the constraints are integrable (also called holonomic).
We have also found the geometry lying on some particular cases of vakonomic mechanics, namely the case of optimal control and the case where the constraint submanifold is a vector subbundle of the tangent bundle.
All the paper is written for the case of time-independent lagrangian and constraints, but the reader may check that the time-dependent case may be dealt with by adjunction of the time variable in a not too involved way.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank N. Román-Roy for useful discussions. X.G. and M.C.M.L. acknowledge partial financial support from CICYT TAP 97–0969–C03–01 and PB98–0920. J.M.S. acknowledge partial financial support from CICYT projects PB98–0821 and PB98–0920.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Unsupervised neural machine translation (UNMT) has recently achieved remarkable results for several language pairs. However, it can only translate between a single language pair and cannot produce translation results for multiple language pairs at the same time. That is, research on multilingual UNMT has been limited. In this paper, we empirically introduce a simple method to translate between thirteen languages using a single encoder and a single decoder, making use of multilingual data to improve UNMT for all language pairs. On the basis of the empirical findings, we propose two knowledge distillation methods to further enhance multilingual UNMT performance. Our experiments on a dataset with English translated to and from twelve other languages (including three language families and six language branches) show remarkable results, surpassing strong unsupervised individual baselines while achieving promising performance between non-English language pairs in zero-shot translation scenarios and alleviating poor performance in low-resource language pairs.'
author:
- |
Haipeng Sun[$^1$]{}[^1], Rui Wang[$^2$]{}, Kehai Chen[$^2$]{},\
**Masao Utiyama[$^2$]{}, Eiichiro Sumita[$^2$]{}, and Tiejun Zhao[$^1$]{}**\
$^1$Harbin Institute of Technology, Harbin, China\
$^2$National Institute of Information and Communications Technology (NICT), Kyoto, Japan\
`[email protected]`, `[email protected]`\
`{wangrui, khchen, mutiyama, eiichiro.sumita}@nict.go.jp`\
\
bibliography:
- 'arxiv.bib'
title: |
Knowledge Distillation for\
Multilingual Unsupervised Neural Machine Translation
---
Introduction
============
Recently, neural machine translation (NMT) has been adapted to the unsupervised scenario in which NMT is trained without any bilingual data. Unsupervised NMT (UNMT) [@DBLP:journals/corr/abs-1710-11041; @lample2017unsupervised] requires only monolingual corpora. UNMT achieves remarkable results by using a combination of diverse mechanisms [@lample2018phrase] such as an initialization with bilingual word embeddings, denoising auto-encoder [@DBLP:journals/jmlr/VincentLLBM10], back-translation [@P16-1009], and shared latent representation. More recently, @DBLP:journals/corr/abs-1901-07291 achieves better UNMT performance by introducing the pretrained language model. However, conventional UNMT can only translate between a single language pair and cannot produce translation results for multiple language pairs at the same time [@nlp-2020].
Multilingual UNMT (MUNMT) translating multiple languages at the same time can save substantial training time and resources. Moreover, the performance of MUNMT in similar languages can promote each other. Research on MUNMT has been limited and there are only a few pioneer studies. For example, @ijcai2019-739 and @sen-etal-2019-multilingual proposed a multilingual scheme that jointly trains multiple languages with multiple decoders. However, the performance of their MUNMT is much worse than our re-implemented individual baselines (shown in Tables \[Tab:all-en-non\] and \[Tab:all-non-en\]) and the scale of their study is modest (i.e., 4-5 languages).
In this paper, we empirically introduce an unified framework to translate among thirteen languages (including three language families and six language branches) using a single encoder and single decoder, making use of multilingual data to improve UNMT for all languages. On the basis of these empirical findings, we propose two knowledge distillation methods, i.e., self-knowledge distillation and language branch knowledge distillation, to further enhance MUNMT performance. Our experiments on a dataset with English translated to and from twelve other languages show remarkable results, surpassing strong unsupervised individual baselines.This paper primarily makes the following contributions:
- We propose a unified MUNMT framework to translate between thirteen languages using a single encoder and single decoder. This paper is the first step of multilingual UNMT training on a large scale of European languages.
- We propose two knowledge distillation methods for MUNMT and our proposed knowledge distillation methods consider linguistic knowledge in the specific translation task.
- Our proposed MUNMT system achieves state-of-the-art performance on the thirteen languages. It also achieves promising performance in zero-shot translation scenarios and alleviates poor performance in low-resource language pairs.
The remainder of the paper is organized as follows. In Section \[sec:second\], the background of UNMT is briefly described. MUNMT is proposed in Section \[sec:third\]. In Section \[sec:fourth\], we propose methods to train MUNMT with knowledge distillation. Sections \[sec:fifth\] describes experiments and evaluates the performance of our proposed methods and Section \[sec:sixth\] analyzes the results. Some related work is discussed in Section \[sec:seventh\]. We conclude the paper in Section \[sec:eighth\].
Background of UNMT {#sec:second}
==================
UNMT can be decomposed into four components: cross-lingual language model pretraining, denoising auto-encoder, back-translation, and shared latent representations. For UNMT, two monolingual corpora $X^1=\{X_i^1\}$ and $X^2=\{X_i^2\}$ in two languages $L_1$ and $L_2$ are given. $|X^1|$ and $|X^2|$ are the number of sentences in monolingual corpora $\{X_i^1\}$ and $\{X_i^2\}$ respectively.
Cross-lingual Language Model Pretraining
----------------------------------------
A cross-lingual masked language model, which can encode two monolingual sentences into a shared latent space, is first trained. The pretrained cross-lingual encoder is then used to initialize the whole UNMT model [@DBLP:journals/corr/abs-1901-07291]. Compared with previous bilingual embedding pretraining [@DBLP:journals/corr/abs-1710-11041; @lample2017unsupervised; @P18-1005; @lample2018phrase; @sun-etal-2019-unsupervised], this pretraining can provide much more cross-lingual information, causing the UNMT model to achieve better performance and faster convergence.
Denoising Auto-encoder
----------------------
Noise obtained by randomly performing local substitutions and word reorderings [@DBLP:journals/jmlr/VincentLLBM10; @DBLP:conf/naacl/HillCK16; @DBLP:conf/nips/HeXQWYLM16], is added to the input sentences to improve model learning ability and regularization. Consequently, the input data are continuously modified and are different at each epoch. The denoising auto-encoder model objective function can be minimized by encoding a noisy sentence and reconstructing it with the decoder in the same language:
[$$\begin{aligned}
\mathcal{L}_{D}
&=\sum_{i=1}^{|X^1|} -log P_{L_1 \to L_1}(X_i^1|C(X_i^1)) \\&+ \sum_{i=1}^{|X^2|} -log P_{L_2 \to L_2}(X_i^2|C(X_i^2)),
\end{aligned}$$]{}where $\{C(X_i^1)\}$ and $\{C(X_i^2)\}$ are noisy sentences. $P_{L_1 \to L_1}$ and $P_{L_2 \to L_2}$ denote the reconstruction probability in language $L_1$ and $L_2$, respectively.
Back-translation
----------------
Back-translation [@P16-1009] plays a key role in achieving unsupervised translation that relies only on monolingual corpora in each language. The pseudo-parallel sentence pairs $\{(M^2(X_i^1),X_i^1)\}$ and $\{(M^1(X_i^2),X_i^2)\}$ produced by the model in the previous iteration are used to train the new translation model. Therefore, the back-translation objective function can be optimized by minimizing:
[$$\begin{aligned}
\mathcal{L}_{B} &= \sum_{i=1}^{|X^1|} -log P_{L_2 \to L_1}(X_i^1|M^2(X_i^1))\\&+\sum_{i=1}^{|X^2|} -log P_{L_1 \to L_2}(X_i^2|M^1(X_i^2)),
\end{aligned}$$]{}where $P_{L_1 \to L_2}$ and $P_{L_2 \to L_1}$ denote the translation probability across the two languages.
Sharing Latent Representations
------------------------------
Encoders and decoders are (partially) shared between $L_1$ and $L_2$. Therefore, $L_1$ and $L_2$ must use the same vocabulary. The entire training of UNMT needs to consider back-translation between the two languages and their respective denoising processes. In summary, the entire UNMT model can be optimized by minimizing:
[$$\begin{aligned}
\mathcal{L}_{all} = \mathcal{L}_{D} + \mathcal{L}_{B}.
\end{aligned}$$]{}
Multilingual UNMT (MUNMT) {#sec:third}
=========================
![\[fig:MUNMT\] [MUNMT architecture. We take $L_1\leftrightarrow L_j$ time-step as an example. The grey symbols indicate that the corresponding data are not used or generated during this time-step.]{}](architecture3){width="0.8\linewidth"}
Multilingual Pretraining
------------------------
Motivated by @DBLP:journals/corr/abs-1901-07291, we construct a multilingual masked language model, using a single encoder. For each language, the language model is trained by encoding the masked input and reverting it with this encoder. This pretrained multilingual language model is used to initialize the full set of parameters of MUNMT.
Multilingual UNMT Training
--------------------------
We have established a MUNMT model on $N$ languages with a single encoder and single decoder. We denote a sentence in language $L_j$ as $X_i^j$. For example, $L_1$ indicates English. $|X^j|$ is the number of sentences in the corpus $X^j=\{X_i^j\}$.
As Figure \[fig:MUNMT\] shows, the entire training process of the MUNMT model is performed through the denoising and back-translation mechanisms, between English and non-English language pairs, by minimizing:
[$$\begin{aligned}
\mathcal{L}_{MUNMT} = \mathcal{L}_{MD} + \mathcal{L}_{MB},
\end{aligned}$$]{}where $\mathcal{L}_{MD}$ denotes the denoising function and $\mathcal{L}_{MB}$ denotes the back-translation function.
In the denoising training, noise (in the form of random token deletion and swapping) is introduced into the input sentences for any language $L_j$. The denoising auto-encoder, which encodes a noisy version and reconstructs it with the decoder in the same language, is optimized by minimizing:
[$$\begin{aligned}
\mathcal{L}_{MD} &= \sum_{j=1}^{N} \sum_{i=1}^{|X^j|} -log P_{L_j \to L_j}(X_i^j|C(X_i^j)) ,
\end{aligned}$$]{}where $\{C(X_i^j)\}$ is a set of noisy sentences for language $L_j$. $P_{L_j \to L_j}$ denotes the reconstruction probability in $L_j$.
In this paper, we primarily focus on the translation from English to other languages or from other languages to English. This is because most test dataset contains English. In the process of back-translation training, we only conduct back-translation from language $L_1$ (English) to other languages and back-translation from other languages to language $L_1$. For any non-English language $L_j$, the pseudo-parallel sentence pairs $\{(M^j(X_i^1),X_i^1)\}$ and $\{(M^1(X_i^j),X_i^j)\}$ are obtained by the previous model in the $L_1 \to L_j$ and $L_j \to L_1$ direction, respectively. Therefore, the back-translation objective function can be optimized on these pseudo-parallel sentence pairs by minimizing:
[$$\begin{aligned}
\mathcal{L}_{MB} &=\sum_{j=2}^{N} \sum_{i=1}^{|X^1|} -log P_{L_j \to L_1}(X_i^1|M^j(X_i^1))\\&+\sum_{j=2}^{N}\sum_{i=1}^{|X^j|} -log P_{L_1 \to L_j}(X_i^j|M^1(X_i^j)),
\end{aligned}$$]{}where $P_{L_1 \to L_j}$ and $P_{L_j \to L_1}$ denote the translation probabilities, in each direction, between any non-English language and English.
Knowledge Distillation for MUNMT {#sec:fourth}
================================
\
Monolingual training data $X^1,X^2,\cdots,X^N$;\
The pretrained model $\theta_0$; Number of steps $K$\
Sample batch $\{X_i^j\}$ from $X^j$\
Compute denoising loss $\mathcal{L}_{MD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MD}$) Sample batch $\{X_i^1\}$from $X^1$\
Compute back-translation loss $\mathcal{L}_{MB}$ Randomly select another language $L_z$ and compute distillation loss $\mathcal{L}_{SKD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MB}+\mathcal{L}_{SKD}$) Sample batch$\{X_i^j\}$ from $X^j$\
Compute back-translation loss $\mathcal{L}_{MB}$ Randomly select another language $L_z$ and compute distillation loss $\mathcal{L}_{SKD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MB}+\mathcal{L}_{SKD}$)
To further enhance the performance of our proposed MUNMT described in Section \[sec:third\], we propose two knowledge distillation methods: self-knowledge distillation (Algorithm \[alg:SKD\]) and language branch knowledge distillation (Algorithm \[alg:LBKD\]). Figure \[fig:architecture\] illustrates the architecture of MUNMT and the proposed knowledge distillation methods.
Generally, during UNMT training, an objective function $\mathcal{L}_{KD}$ is added, to enhance the generalization ability of the MUNMT model. The general MUNMT objective function can be reformulated as follows:
[$$\begin{aligned}
\mathcal{L}_{MUNMT} = \mathcal{L}_{MD} + \mathcal{L}_{MB'},\\
\mathcal{L}_{MB'} = (1-\alpha)\mathcal{L}_{MB} + \alpha T^2\mathcal{L}_{KD},
\end{aligned}$$]{}where $\alpha$ is a hyper-parameter that adjusts the weight of the two loss functions during back-translation. $T$ denotes the temperature used on the softmax layer. If the temperature is higher, the probability distribution obtained would be softer [@DBLP:journals/corr/HintonVD15].
Self-knowledge Distillation
---------------------------
On the basis of the existing architecture of MUNMT, we introduce self-knowledge distillation [@hahn2019self] (SKD) during back-translation, to enhance the generalization ability of the MUNMT model, as shown in Figure \[fig:SKD\]. Unlike @hahn2019self’s method, using two soft target probabilities that are based on the word embedding space, we make full use of multilingual information via self-knowledge distillation.
{width="0.9\linewidth"}
During back-translation, only language $L_j$ sentences $M^j(X_i^1)$ are generated before training the MUNMT model in the $L_j \to L_1$ direction. However, other languages, which have substantial multilingual information, are not used during this training. Motivated by this, we propose to introduce another language $L_z$ (randomly chosen but distinct from $L_1$ and $L_j$) during this training. We argue that the translation from the source sentences through different paths, $L_1 \to L_j \to L_1$ and $L_1 \to L_z \to L_1$, should be similar. The MUNMT model matches not only the ground-truth output of language $L_j$ sentences $M^j(X_i^1)$, but also the soft probability output of language $L_z$ sentences $M^z(X_i^1)$. The opposite direction is similar. Therefore, this MUNMT model is optimized by minimizing the objective function:
[$$\begin{aligned}
&\mathcal{L}_{MB'} = (1-\alpha)\mathcal{L}_{MB} + \alpha T^2\mathcal{L}_{SKD},\\
\mathcal{L}_{SKD} &= \sum_{j=2}^{N}\sum_{i=1}^{|X^1|} KL(X^1(M^j(X_i^1)),X^1(M^z(X_i^1))) \\
&+\sum_{j=2}^{N}\sum_{i=1}^{|X^j|} KL(X^j(M^1(X_i^j)),X^j(M^z(X_i^j))),
\end{aligned}$$ ]{}where $KL(\cdot)$ denotes the KL divergence. It is computed over full output distributions to keep these two probability distributions similar. For any language $L_j$, $X^1(M^j(X_i^1))$ and $X^1(M^z(X_i^1))$ denote the softened $L_1$ sentence probability distribution after encoding $M^j(X_i^1)$ and $M^z(X_i^1)$, respectively. $M^j(X_i^1)$ and $M^z(X_i^1)$ were generated by the previous model in the $L_1 \to L_j$ and $L_1 \to L_z$ directions, respectively. $X^j(M^1(X_i^j))$ and $X^j(M^z(X_i^j))$ denote the softened $L_j$ sentence probability distribution after encoding $M^1(X_i^j)$ and $M^z(X_i^j)$, respectively. $M^1(X_i^j)$ and $M^z(X_i^j)$ were generated by the previous model in the $L_j \to L_1$ and $L_j \to L_z$ directions, respectively. Note that zero-shot translation was used to translate language $L_j$ to language $L_z$. The direction $L_j \to L_z$ was not trained during MUNMT training.
Language Branch Knowledge Distillation
--------------------------------------
\
Monolingual training data $X^1,X^2,\cdots,X^N$;\
LBUNMT models $\theta_1^{LB},\theta_2^{LB},\cdots,\theta_M^{LB}$;\
The pretrained model $\theta_0$; Number of steps $K$\
Sample batch $\{X_i^j\}$ from $X^j$\
Compute denoising loss $\mathcal{L}_{MD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MD}$) Sample batch $\{X_i^1\}$from $X^1$\
Compute back-translation loss $\mathcal{L}_{MB}$ Select LBUNMT language $L_1$ belongs and compute distillation loss $\mathcal{L}_{LBKD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MB}+\mathcal{L}_{LBKD}$) Sample batch$\{X_i^j\}$ from $X^j$\
Compute back-translation loss $\mathcal{L}_{MB}$ Select LBUNMT language $L_j$ belongs and compute distillation loss $\mathcal{L}_{LBKD}$ Update $\theta\leftarrow$optimizer($\mathcal{L}_{MB}+\mathcal{L}_{LBKD}$)
We consider thirteen languages: Czech (Cs), German (De), English (En), Spanish (Es), Estonian (Et), Finnish (Fi), French (Fr), Hungarian (Hu), Lithuanian (Lt), Latvian (Lv), Italian (It), Romanian (Ro), and Turkish (Tr), which belong to three language families including several language branches [@lewis2009ethnologue] as shown in Figure \[fig:language\_family\].
As shown in Figure \[fig:LFKD\], we propose knowledge distillation within a language branch (LBKD), to improve MUNMT performance through the existing teacher models. To the best of our knowledge, this is the first proposal that aims to distill knowledge within a language branch. As the number of languages increases, the cost of training time and resources to train an individual model on any two languages increases rapidly. An alternative knowledge distillation method within a language branch can avoid this prohibitive computational cost. Because languages in the same language branch are similar, we first train small multilingual models across all languages in the same language branch (LBUNMT) before training MUNMT. The LBUNMT model trained in the same language branch performed better than the single model because similar languages have a positive interaction during the training process as shown in Tables \[Tab:all-en-non\] and \[Tab:all-non-en\]. Therefore, the distilled information of LBUNMT is used to guide the MUNMT model during back-translation. The MUNMT model matches both the ground-truth output and the soft probability output of LBUNMT. Therefore, this MUNMT model is optimized by minimizing the objective function:
[$$\begin{aligned}
&\mathcal{L}_{MB'} = (1-\alpha)\mathcal{L}_{MB} + \alpha T^2\mathcal{L}_{LBKD},\\
\mathcal{L}_{LBKD} &=\sum_{j=2}^{N}\sum_{i=1}^{|X^1|} KL(X^1(M^j(X_i^1)),{LB}^1(M^j(X_i^1))) \\
&+\sum_{j=2}^{N}\sum_{i=1}^{|X^j|} KL(X^j(M^1(X_i^j)),{LB}^j(M^1(X_i^j))),
\end{aligned}$$]{}where $X^1(M^j(X_i^1))$ and ${LB}^1(M^j(X_i^1))$ denote the softened $L_1$ sentence probability distribution of the MUNMT and LBUNMT models, respectively, after encoding $M^j(X_i^1)$ generated by the previous MUNMT model in the $L_1 \to L_j$ direction. $X^j(M^1(X_i^j))$ and ${LB}^j(M^1(X_i^j))$ denote the softened $L_j$ sentence probability distribution of the MUNMT and LBUNMT models, respectively, after encoding $M^1(X_i^j)$ generated by the previous MUNMT model in the $L_j \to L_1$ direction.
Experiments {#sec:fifth}
===========
Datasets
--------
To establish an MUNMT system, we considered 13 languages from WMT monolingual news crawl datasets: Cs, De, En, Es, Et, Fi, Fr, Hu, It, Lt, Lv, Ro, and Tr. For preprocessing, we used the `Moses` tokenizer [@koehn-etal-2007-moses]. For cleaning, we only applied the `Moses` script `clean-corpus-n.perl` to remove lines in the monolingual data containing more than 50 words. We then used a shared vocabulary for all languages, with 80,000 sub-word tokens based on BPE [@sennrich2015neural]. The statistics of the data are presented in Table \[Tab:Statistics\]. For Cs,De,En, we randomly extracted 50M monolingual news crawl data after cleaning; For other languages, we used all news crawl data after cleaning as shown in Table \[Tab:Statistics\].
\[Tab:all-en-non\]
We report the results for WMT newstest2013 for Cs-En, De-En, Es-En, and Fr-En. We can evaluate the translation performance between pairs of non-English languages because newstest2013 includes these five languages parallel to each other. For other language pairs, we chose the newest WMT newstest set. That is, we reported the results on WMT newstest2019 for Fi-En and Lt-En; WMT newstest2018 for Et-En and Tr-En; WMT newstest2017 for Lv-En; WMT newstest2016 for Ro-En; and WMT newstest2009 for Hu-En and It-En. Note that the versions of newstest2019 on Fi/Lt$\rightarrow$ En and En $\rightarrow$ Fi / Lt are different. We chose the corresponding newstest2019 for each direction.
Language Model and UNMT Settings
--------------------------------
We used a transformer-based `XLM` toolkit to train a multilingual masked language model and followed the settings used in @DBLP:journals/corr/abs-1901-07291: six layers were used for the encoder. The dimension of hidden layers was set to 1024. The Adam optimizer [@kingma2014adam] was used to optimize the model parameters. The initial learning rate was 0.0001, `\beta_1 = 0.9`, and `\beta_2 = 0.98`.
We used the same toolkit and followed the settings of UNMT used in [@DBLP:journals/corr/abs-1901-07291]: six layers were used for the encoder and decoder. The batch size was set to 2000 tokens. The other parameters were the same as those used for training language model. For our proposed knowledge distillation method, $\alpha$ was set to 0.1 and $T$ was set to 2 (the parameters are empirically selected by small-scale experiments and most of the settings achieved good results). The cross-lingual language model was used to pretrain the encoder and decoder of the whole UNMT model. All monolingual data, described in Table \[Tab:Statistics\], were used in the pretraining and MUNMT training phase. The parameters of the multilingual and single models were the same.
For evaluation, we used the case-sensitive BLEU scores computed by the `Moses` script `multi-bleu.perl`. We executed a single model (two languages) for 60,000 iterations, a small multilingual model (three to five languages) for 30,000 iterations, and a large multilingual model (13 languages) for 15,000 iterations. Eight V100 GPUs were used to train all UNMT models. The single model was trained for approximately two days; the multilingual model (13 languages) costs approximately six days since 13 languages participated in the training.
Main Results
------------
Tables \[Tab:all-en-non\] and \[Tab:all-non-en\] present the detailed BLEU scores of all systems on the English and non-English language pairs, in each direction[^2]. Our observations are as follows:
1\) Our proposed LBUNMT model trained in the same language branch performed better than the single model (SM) because similar languages have a positive interaction during the training process. Moreover, SM performed very poorly on low-resource language pairs such as En-Lt and En-Lv in the Baltic language branch.
2\) Our proposed MUNMT model trained in all languages significantly outperformed the previous work [@sen-etal-2019-multilingual; @ijcai2019-739] by 4$\sim$12 BLEU scores. Moreover, the MUNMT model could alleviate the poor performance achieved with low-resource language pairs, such as En-Lt and En-Lv. However, the performance of MUNMT is slightly worse than SM in some language pairs.
3\) Our proposed knowledge distillation methods outperformed the original MUNMT model by approximately 1 BLEU score. Moreover, our proposed MUNMT with knowledge distillation performed better than SM in all language pairs with fewer training iterations. Regarding our two proposed methods, LBKD achieved better performance since it could obtain much more knowledge distilled from LBUNMT model.
4\) There is a gap between the performance of our proposed MUNMT model and that of the supervised NMT systems. To bridge this gap, relying solely on monolingual training data, is worthy of being studied in the future.
Discussion {#sec:sixth}
==========
Zero-shot Translation Analysis
------------------------------
We also studied the zero-shot translation accuracy of the MUNMT model. Although MUNMT could be trained on all translation directions (ordered language pairs), it would require an extremely long training time. Our proposed MUNMT model was trained in 24 translation directions (all English and non-English language pairs, in each direction), whereas 156 translation directions exist. As the number of languages increases, the number of translation directions increases quadratically. Therefore, zero-shot translation accuracy is important to the MUNMT model.
Table \[Tab:non-English\] shows the performance of translation between non-English language pairs in the zero-shot translation scenario. Note that @ijcai2019-739 (2019) shows the results of direct translation between the two languages, not the result of zero-shot translation. Compared with previous works, our MUNMT model outperformed the previous systems in almost all translation directions, particularly the direct translation results reported in @ijcai2019-739. Compared with the original MUNMT model, our proposed knowledge distillation methods further improved the performance of zero-shot translation. Regarding our two proposed methods, SKD significantly outperformed LBKD by approximately 3 BLEU scores since the third language was introduced during SKD translation training for two language pairs, achieving much more cross-lingual knowledge.
Further Training (Fine-tuning) Analysis
---------------------------------------
To better assess the effectiveness of our proposed MUNMT model, we further trained the MUNMT and LBKD model individually on each language pair for 15,000 iterations. As shown in Tables \[Tab:ft-en-non\] and \[Tab:ft-non-en\], after further training, the model outperformed the original single model on each language pair by approximately 4 BLEU scores. Actually, the number of iterations of the whole process (including training the MUNMT model) is half that of the original single model. This demonstrates that our proposed MUNMT model is a robust system and contains substantial cross-lingual information that could improve translation performance.
Related Work {#sec:seventh}
============
Multilingual NMT has attracted much attention in the machine translation community. @dong-etal-2015-multi first extended NMT from the translation of a single language pair to multiple language pairs, using a shared encoder and multiple decoders and multiple attention mechanisms, for each language. @DBLP:journals/corr/LuongLSVK15 translated multiple source languages to multiple target languages using a combination of multiple encoders and multiple decoders. @firat-etal-2016-multi used a shared attention mechanism but multiple encoders and decoders for each language. @DBLP:journals/corr/HaNW16 and @DBLP:journals/tacl/JohnsonSLKWCTVW17 proposed a simpler method to use one encoder and one decoder to translate between multiple languages. Recently, many methods [@lakew-etal-2018-comparison; @platanios-etal-2018-contextual; @sachan-neubig-2018-parameter; @blackwood-etal-2018-multilingual; @DBLP:conf/wmt/LuKLBZS18; @wang19iclr; @aharoni-etal-2019-massively; @wang-etal-2019-compact; @wang-neubig-2019-target] have been proposed to boost multilingual NMT performance. In particular, Tan et al. proposed a knowledge distillation method [@DBLP:conf/iclr/TanRHQZL19] and a language clustering method [@tan-etal-2019-multilingual] to improve the performance of multilingual NMT. @ren-etal-2018-triangular propose a triangular architecture to tackle the problem of low-resource pairs translation by introducing another rich language.
To further tackle the problem of low-resource pairs translation, UNMT [@DBLP:journals/corr/abs-1710-11041; @lample2017unsupervised] has been proposed, using a combination of diverse mechanisms such as initialization with bilingual word embeddings, denoising auto-encoder [@DBLP:journals/jmlr/VincentLLBM10], back-translation [@P16-1009], and shared latent representation. @lample2018phrase concatenated two bilingual corpora as one monolingual corpus, and used monolingual embedding pretraining in the initialization step, to achieve remarkable results with some similar language pairs. @DBLP:journals/corr/abs-1901-07291 achieved better UNMT performance by introducing a pretrained language model. @sun-etal-2019-unsupervised [@9043536] proposed to train UNMT with cross-lingual language representation agreement, to further improve UNMT performance. Moreover, an unsupervised translation task that evaluated in the WMT19 news translation task [@barrault-etal-2019-findings] attracted many researchers to participate [@marie-etal-2019-nicts; @li-etal-2019-niutrans].
For Multilingual UNMT, @ijcai2019-739 exploited multiple auxiliary languages for jointly boosting UNMT models via the Polygon-Net framework. @sen-etal-2019-multilingual proposed an MUNMT scheme that jointly trains multiple languages with a shared encoder and multiple decoders. In contrast with their use of multiple decoders, we have constructed a simpler MUNMT model with one encoder and one decoder. Further, we have extended the four or five languages used in their work to thirteen languages, for training our MUNMT model.
Conclusion and Future Work {#sec:eighth}
==========================
In this paper, we have introduced a unified framework, using a single encoder and decoder, for MUNMT training on a large scale of European languages. To further enhance MUNMT performance, we have proposed two knowledge distillation methods. Our extensive experiments and analysis demonstrate the effectiveness of our proposed methods. In the future, we intend to extend the work to include language types such as Asian languages. We will also introduce other effective methods to improve zero-shot translation quality.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the anonymous reviewers and the area chair for their insightful comments and suggestions. The corresponding authors are Rui Wang and Tiejun Zhao. Rui Wang was partially supported by JSPS grant-in-aid for early-career scientists (19K20354): “Unsupervised Neural Machine Translation in Universal Scenarios" and NICT tenure-track researcher startup fund “Toward Intelligent Machine Translation". Tiejun Zhao was partially supported by National Key Research and Development Program of China via grant 2017YFB1002102. Masao Utiyama was partially supported by JSPS KAKENHI Grant Number 19H05660.
[^1]: Haipeng Sun was an internship research fellow at NICT when conducting this work.
[^2]: The translation quality of pretrained model was not presented in the Tables \[Tab:all-en-non\] and \[Tab:all-non-en\]. The result was poor because the pretrained model (cross-lingual language model) was trained within an encoder. The encoder and decoder of UNMT was initialized with the same parameters of pretrained language model (just an encoder).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We calculate the power spectrum of electric field-driven ion transport through cylindrical nanometer-scale pores using both linearized mean-field theory and Langevin dynamics simulations. With the atom-sized cutoff radius as the only fitting parameter, the linearized mean-field theory accurately captures the dependence of the simulated power spectral density on the pore radius and the applied electric field. Remarkably, the linearized mean-field theory predicts a plateau in the power spectral density at low frequency $\omega$, which is confirmed by the Langevin dynamics simulations at low ion concentration. At high ion concentration, however, the power spectral density follows a power law that is reminiscent of the $1/\omega^{\alpha}$ dependence found experimentally at low frequency. Based on simulations with and without ion-ion interactions, we attribute the low-frequency power law dependence to ion-ion correlations. Finally, we show that the surface charge density has no effect on the frequency dependence of the power spectrum.'
author:
- Mira Zorkot
- Ramin Golestanian
- Douwe Jan Bonthuis
title: 'The Power Spectrum of Ionic Nanopore Currents: The Role of Ion Correlations'
---
Introduction
============
Measuring the ionic current passing through a nanometer-scale membrane pore has emerged over the past couple of decades as a versatile technique to study nanometer-scale transport processes [@2007_Dekker]. In particular, nanopores are being used to study the properties of translocating biological molecules; to count them, measure their size and translocation velocity, or even – in the case of DNA – determine their sequence [@1996_Kasianowicz; @2000_Deamer; @2000_Meller]. The sensitivity of such measurements is limited by the noise level, which typically increases with decreasing frequency $\omega = 2 \pi f$ [@1988_Weissman_RevModPhys]. A high signal-to-noise ratio, as well as a consistent noise spectrum across individual devices, are among the requirements for a nanopore system to be of practical use. Apart from limiting the measurement accuracy, however, the noise level can sometimes be used to study the microscopic properties of a nanofluidic system, such as the adsorption of molecules on the walls of a nanometer-scale cavity [@2011_Singh-Lemay]. At low frequency, the power spectrum $S\left(\omega\right)$ of the ionic current through an electrolyte-filled nanopore typically follows a power law $S\left(\omega\right) \propto 1/\omega^{\alpha}$ with $\alpha \approx 1$, which is referred to as pink noise, or flicker noise, or $1/f$ noise [@2015_Heerema; @2008_Smeets_PNAS]. At high frequency, $S\left(\omega\right)$ is dominated by thermal or capacitive noise [@2008_Smeets_PNAS; @2007_Tabard-Cossa].
Ever since the discovery of the near-universal appearance of pink noise in electrical systems, its microscopic origins have been heavily debated. Measurements in protein channels indicate that the pink noise in biological nanopores is related to small conformational fluctuations of the open nanopore [@2000_Bezrukov_PRL; @1997_Wohnsland]. These results are supported by experiments on nanopores in membranes with varying flexibility, showing pink noise in pores in flexible membranes, and a quadratic noise spectrum ($\alpha = 2$) in pores in solid membranes [@2002_Siwy-Fulinski_PRL]. Other explanations of pink noise include surface charge fluctuations [@2009_Hoogerheide] and the rectifying nature of conical pores [@2009_Powell-Siwy_PRL]. Alternatively, the noise is thought to originate not in the properties of the nanopore, but instead in the nonequilibrium dynamics of charge carriers under confinement. The first of these possible explanation stems from Hooge, whose measurements on homogeneous metal samples showed that the amplitude of the pink noise is proportional to the concentration of mobile charge carriers [@1970_Hooge]. Further evidence is provided by the observation that stationary confining walls affect the velocity autocorrelation function of colloidal particles at long time scales [@1997_Hagen]. Fractional Brownian motion, which can be used to describe pink noise, also governs subdiffusive motion of molecules under confinement [@2010_Jeon-Metzler]. Based on careful analysis of noise measurements in solid-state nanopores filled with ionic liquids, Tasserit *et al.* reached the conclusion that pink noise is caused by a cooperative effect in the ionic motion [@2010_Tasserit]. In computer simulations, pink noise has only been observed in nanochannels when the single-file motion is enforced artificially [@2008_Fulinski].
Identifying the source of noise in nanopores is essential for efforts to optimize experimental setups, and to design nanopore systems for use in large-scale technological applications. Moreover, a comprehensive understanding of the power spectrum of nanopore ion currents would constitute the basis of a new probe of the pore’s microscopic properties, allowing researchers to extract a wealth of information from a part of the measured signal that has traditionally been discarded. However, a systematic theoretical investigation of the noise spectrum in nanopores filled with an aqueous electrolyte has been lacking so far.
In this paper, we present the first comprehensive theoretical analysis of the nonequilibrium noise spectrum of an ion current through a rigid nanopore using implicit water. We derive an analytical expression for the noise spectrum in the mean-field regime, providing a tool to analyze and interpret experimental results in both the high and the low frequency limits. We compare our analytical expressions with the results of Langevin dynamics simulations, showing excellent agreement at high frequency. Although the low frequency results follow our theoretical linearized mean-field prediction at low ion concentration, at high ion concentration the simulation results deviate from the linearized mean-field prediction, and show a nonzero $\alpha$, indicative of pink noise. Our results suggest that ion correlations are a source of pink noise in nanopores.
Linearized Mean Field Theory
============================
We consider a system consisting of a cylindrical nanopore of length $L$ and radius $R$ connecting two reservoirs. Because the ion current through the system is ultimately determined by the flux through the pore, we base our model on the ion flux density ${\boldsymbol{J}}^{\pm}\left({\boldsymbol{x}},t\right)$ inside the nanopore, with ${\boldsymbol{x}}$ denoting the position in three dimensions and $t$ denoting the time. We model the electrolyte filling the nanopore as ions of valency $\pm 1$ in implicit water. Note that the model implies that nonlinear terms originating in the coupling of the ionic motion to the fluid velocity are ignored. The ion concentrations $C^{\pm}\left({\boldsymbol{x}},t\right)$ are governed by the continuity equation, $$\label{eqn:fick}
\frac{\partial C^{\pm}}{\partial t} + \nabla \cdot {\boldsymbol{J}}^{\pm} = 0.$$ The corresponding flux densities ${\boldsymbol{J}}^{\pm}\left({\boldsymbol{x}},t\right)$ are given by the Nernst-Planck equation, $$\label{eqn:nernst-planck}
{\boldsymbol{J}}^{\pm} = -D^{\pm} \nabla C^{\pm} \mp D^{\pm}C^{\pm} \frac{e {\boldsymbol{E}}}{k_{B}T} + {\boldsymbol{\eta}},$$ where ${\boldsymbol{E}}\left({\boldsymbol{x}},t\right)$ is the applied electric field, $e$ denotes the ion charge, and ${\boldsymbol{\eta}}\left({\boldsymbol{x}},t\right)$ denotes the thermal noise that accounts for fluctuations in the environment, most importantly the effect of collisions between water molecules and ions. From here on, we restrict ourselves to the case where the ion diffusion coefficients are equal, $D^{+} = D^{-} = D$, which is accurate for many common salts, such as KCl and KNO$_3$. Because of the cylindrical geometry we have only two independent coordinates, one parallel to the length of the pore ($\parallel$) an one perpendicular to the pore wall ($\perp$). We are interested in the current in parallel direction in response to an electric field that is constant inside the pore and nonzero only in parallel direction ${\boldsymbol{E}}\left({\boldsymbol{x}},t\right) = E_{\parallel}$. We determine the Fourier transforms (denoted by $\widetilde{...}$) of Eqs. \[eqn:fick\] and \[eqn:nernst-planck\] as a function of the wave vectors $q_{\parallel}$ and $q_{\perp}$ and the frequency $\omega$ [@supp], and calculate the power spectral density $S\left(\omega\right)$ from the Fourier-transformed current density $\widetilde{J}_{\parallel}^{+}\left(q_{\parallel},q_{\perp},\omega\right) - \widetilde{J}_{\parallel}^{-}\left(q_{\parallel},q_{\perp},\omega\right)$. In terms of the Cartesian coordinates (Fig. \[fig:setup\]a), the perpendicular wave vector is defined by $q_{\perp}^2 = q_{\perp1}^2 + q_{\perp2}^2$. Multiplying by the Fourier-transformed lateral surface area $\widetilde{A}(q_\perp)$ of the pore, and integrating over all wave vectors inside the nanopore, we arrive at the power spectral density $S\left(\omega\right)$ [@supp], $$\label{eqn:psd}
S = 2 \int^{\varLambda^{-1}}_{R^{-1}} \frac{\textrm{d}^2 q_{\perp}}{(2\pi)^2}
\int^{\varLambda^{-1}}_{L^{-1}} \frac{\textrm{d} q_{\parallel}}{2\pi} \, \widetilde{A}^2 \, \langle |\widetilde{J}_{\parallel}^{+}-\widetilde{J}_{\parallel}^{-}|^2\rangle$$ with $R$ and $L$ being the radius and the length of the pore, respectively, and $\Lambda$ being the cut-off length, which is of the order of the particle size. The factor $2$ stems from the integral over negative wave vectors $q_{\parallel}$. The ensemble average is denoted by $\langle...\rangle$ and the squared area function $\widetilde{A}(q_\perp)^2$ is given by [@supp] $$\label{eqn:area}
\widetilde{A}^2 = \left(2\pi\right)^2 \left[\frac{R^2}{q_{\perp}^2} - \frac{2R\sin{R q_{\perp}}}{q_{\perp}^{3}} - \frac{2 \cos{R q_{\perp}}}{q_{\perp}^{4}} + \frac{2}{q_{\perp}^{4}} \right] .$$ The average modulus square Fourier-transformed current density $\langle|\widetilde{J}_{\parallel}^{+}\left(q_{\parallel},q_{\perp},\omega\right)-\widetilde{J}_{\parallel}^{-}\left(q_{\parallel},q_{\perp},\omega\right)|^2\rangle$ is calculated from Eqs. \[eqn:fick\] and \[eqn:nernst-planck\] as [@supp] $$\label{eqn:current}
\begin{split}
\langle| & \widetilde{J}_{\parallel}^{+} - \widetilde{J}_{\parallel}^{-}|^2\rangle = \\
& \frac{8 D C_{0}\Big[\frac{eE_{\parallel}}{kT}\Big]^2 \Big[\frac{\omega^2}{D^2} + q_{\perp}^{4}\Big] \Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]}
{\Big(\Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]^2 \!\! + \Big[\frac{ eE_{\parallel} }{kT}\Big]^2 \! q_{\parallel}^2 \! - \frac{\omega^2}{D^2} \! \Big)^2
\!\! + 4 \frac{\omega^2}{D^2} \Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]^2},
\end{split}$$ with $C_0$ being the bulk ion density. Together with Eqs. \[eqn:area\] and \[eqn:current\], Eq. \[eqn:psd\] is solved numerically to get the linearized mean-field prediction.
Remarkably, Eq. \[eqn:current\] – and therefore $S\left(\omega\right)$ – is independent of the frequency $\omega$ in the limit $\omega \to 0$, in clear contradiction to the low-frequency $S\left(\omega \to 0\right) \propto 1/\omega^{\alpha}$ behavior observed experimentally. It is interesting to note that the same calculation in one dimension leads to $S\left(\omega \to 0\right) \propto \omega^2$, which is incompatibly not only with experimental results, but also with effectively one-dimensional simulations [@2008_Fulinski]. At vanishing electric field $E_{\parallel}$, the noise spectrum becomes independent of $\omega$.
![ Snapshots from our simulations with (a) the membrane with the nanopore, (b) the top view of the membrane with the radius $R$ indicated, and (c) the side view of the simulation box with the membrane thickness $L$ indicated. The red and blue particles depict the charged ions, the yellow particles depict the membrane, and the white particles depict the pore wall that might be charged or uncharged. In (a) and (b) the ions are not shown. []{data-label="fig:setup"}](setup_v02){width="0.5\columnwidth"}
Simulations
===========
We perform Langevin dynamics simulations of ions passing a membrane pore under influence of an applied potential difference using the molecular simulation package Espresso [@2006_Limbach]. The Langevin equation for particle $i$ reads $$\label{eqn:langevin}
m_{i} \frac{\partial {\boldsymbol{u}}_{i}}{\partial t} = -\sum_{j\neq i} \nabla V_{ij} + {\boldsymbol{F}}_{i} - {\gamma} {\boldsymbol{u}}_{i} + {\boldsymbol{\xi}}_{i},$$ with ${\boldsymbol{u}}_{i}\left(t\right)$ being the particle’s velocity in units Å/$\tau$, $\gamma = 1$ $k_{B} T \tau$/Å$^2$ being the friction coefficient, with $\tau$ being the time scale and $k_{B}T$ being the thermal energy. The random force ${\boldsymbol{\xi}}_{i}\left(t\right)$ satisfies $\langle \xi_{i}\left(t\right)\xi_{i}\left(t'\right)\rangle = 6 k_{B} T \gamma \delta\left(t-t'\right)$. Because the only dynamical particles in our simulation are the ions, the particles are assumed to be of equal mass $m_{i} = 1$ $k_{B}T\tau^2/$Å$^2$, effectively incorporating the mass into the time scale $\tau$. The index $j$ runs over all other particles of the system and the potential $V_{ij}\left(r_{ij}\right)$ (in units of $k_{B}T$) comprises the Coulomb interactions and the Lennard-Jones interactions, truncated and shifted to form a Weeks-Chandler-Andersen potential, $$\label{eqn:potential}
V_{ij}= l_{B}\frac{q_{i}q_{j}}{r_{ij}} + 4 \epsilon_{ij} \left[ \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right] + V_{\textrm{shift}},$$ with $r_{ij}$ being the distance between particles $i$ and $j$ and $q_i$ being the charge of particle $i$ in units of the elementary charge $e$. The Bjerrum length $l_{B} = e^2/\left(4\pi\varepsilon\varepsilon_0 k_B T\right)$, with $\varepsilon_0$ being the permittivity of vacuum, is set to $l_{B} = 7$ Å, modeling electrostatic interactions in water with a dielectric constant of $\varepsilon = 80$. The long-ranged electrostatic interactions are treated using P$^3$M (particle-particle-particle mesh) Ewald summation with an automatically determined real-space cutoff [@2006_Limbach]. The Lennard-Jones potential is truncated and shifted to zero at $r_{ij} = 2^{1/6}\sigma_{ij}$, using $V_{\textrm{shift}} = \epsilon_{ij}/4$. We use the Lennard-Jones parameters $\epsilon_{ij} = 0.5$ $k_{B}T$ for all particles, $\sigma_{ij} = 3$ Å for ion-ion interactions, and $\sigma_{ij} = 9$ Å for ion-membrane interactions. The applied potential difference across the membrane is modeled by the external force ${\boldsymbol{F}}_{i}$, where we use the approximation that the electric field exerts a force in $x_{\parallel}$-direction inside the pore only, $$\label{eq:external-force}
{\boldsymbol{F}}_{i} = \Bigg\{ {\begin{array}{l l}
q_{i} {\boldsymbol{E}} & \quad \text{if $0<x_{\parallel}<L$ }\\
0 & \quad \text{otherwise},
\end{array}}$$ with ${\boldsymbol{E}}\left({\boldsymbol{x}},t\right) = E_{\parallel}$. The electric field is varied between $E_{\parallel} = 0.3$ $k_{B}T/\left(e\text{\AA}\right)$ and $E_{\parallel} = 1.6$ $k_{B}T/\left(e\text{\AA}\right)$, which at $L=48$ Å corresponds to a potential difference between $0.4$ V and $2$ V.
The time scale $\tau$ is calibrated by calculating the conductivity in simulations of a bulk system with an applied homogeneous electric field, and comparing it to the conductivity of a solution of potassium chloride, leading to $\tau = 5$ ps [@supp].
We construct a membrane of thickness $L=48$ Å consisting of membrane particles on a cubic lattice with a lattice constant of $6$ Å, connected on both sides to a reservoir of width $W = 96$ Å and height $H = 100$ Å. Permeating the membrane, we construct a cylindrical pore of varying radii $R = \{6,13,19,25,28\}$ Å. The reservoirs and the pore are filled with positive and negative ions at three different concentrations, $C_0 = \{3\cdot 10^{-6},3\cdot 10^{-5},3\cdot 10^{-4}\}$ Å$^{-3}$, corresponding to $C_0 = \{0.005, 0.05, 0.5\}$ mol/l. The membrane is verified to be impermeable to ions. The membrane particles are frozen. We use a simulation time step of $0.06 \tau$ and perform simulations of $3 \cdot 10^7$ time steps (Figs. \[fig:concentrations\]-\[fig:electric-field\]). For Fig. \[fig:interactions\] we use $1 \cdot 10^8$ time steps (with interactions) and $3 \cdot 10^8$ time steps (without interactions). We discard $5000$ time steps for equilibration and calculate the current every $100$ time steps. We use periodic boundary conditions in all directions. We verify that the effect of the periodic boundary conditions, which affects fluctuations in general [@2014_Villamaina-Trizac], is negligible in our case [@supp].
We calculate the current $I_{\parallel} = \int (J^{+}_{\parallel}-J^{-}_{\parallel})\textrm{d}A$, equal to the integral of the current density over the lateral area $A$ of the pore, from the integrated velocity in parallel direction $$\label{eqn:flux}
I_{\parallel} = \frac{1}{L} \sum_{i}
\Bigg\{ {\begin{array}{l l}
q_i\left({\boldsymbol{u}}_{i}\right)_{\parallel} & \quad \text{if $0 < x_{\parallel} < L$ }\\
0 & \quad \text{otherwise},
\end{array}}$$ where the index $i$ runs over all positive and negative ions.
From the current we calculate the power spectrum using the Welch method with a Hamming window, an overlap of 0.5, and 100 windows [@1967_Welch].
![ The power spectral density $S$ of the ionic current in units of the inverse time scale $\tau$, as a function of the frequency $\omega$ on a log-log scale, for ion concentrations (a) $C_0 = 3\cdot 10^{-6}$ Å$^{-3}$, (b) $C_0 = 3\cdot 10^{-5}$ Å$^{-3}$, and (c) $C_0 = 3\cdot 10^{-4}$ Å$^{-3}$. Solid curves denote $S$ from simulations for 5 different radii, and dashed lines denote fits with Eq. \[eqn:psd\]. The fitting parameter $\Lambda = 7$ Å for $R = 28$, $25$, and $19$ Å, $\Lambda = 6$ Å for $R=13$ Å, and $\Lambda=2.6$ Å for $R=6$ Åfor all concentrations. The applied electric field is $E_{\parallel}=0.8$ Å$^{-1}$. []{data-label="fig:concentrations"}](concentrations_v04.pdf){width="0.5\columnwidth"}
Results and Discussion
======================
**Pore radius & ion concentration.** – In Fig. \[fig:concentrations\] we show the power spectral density $S(\omega)$, calculated from the Langevin dynamics simulations, as a function of the frequency $\omega$ (solid lines). The curves have been fitted with the linearized mean-field expression of Eq. \[eqn:psd\] (dashed lines), using the high wave vector cutoff $\Lambda$ as the only fitting parameter. Eq. \[eqn:psd\] fits the curves with remarkable accuracy for all pore radii. All curves show a transition around $\omega = 10^{-2}$ $\tau^{-1}$, and a power law decrease $S\propto 1/\omega^{b}$ at high frequency. At even higher frequency, $S$ is dominated by white noise (not shown). At low ion concentrations ($C_0 = 10^{-6}$ Å$^{-3}$ and $C_0 = 10^{-5}$ Å$^{-3}$, Figs. \[fig:concentrations\]a-b), $S$ exhibits a plateau at low frequency, as predicted by the linearized mean-field theory. At higher ion concentration ($C_0 = 10^{-4}$ Å$^{-3}$, Fig. \[fig:concentrations\]c), the linearized mean-field theory still captures the high frequency simulation results. At low frequency, however, we do not find a plateau, but $S$ continues to increase with decreasing frequency instead, following a power law $S\propto 1/\omega^{a}$. This behavior is reminiscent of the $1/\omega^{\alpha}$ dependence of $S$ found experimentally at low frequency.
![ The power spectra of the ion current through a pore of $R = 25$ Å at a concentration of $C_0 = 3 \cdot 10^{-4}$ Å$^{-3}$, at different applied electric field strengths inside the pore (0.3, 0.8, 1.2, and 1.6 $k_BT/\left(e\text{\AA}\right)$). With increasing electric field strength, the transition frequency shifts upward, and the background white noise decreases. []{data-label="fig:electric-field"}](electric-field_v02.pdf){width="0.5\columnwidth"}
**Electric field.** – We perform simulations of the pore of $R = 25$ Å at a concentration of $C_0 = 3\cdot 10^{-4}$ Å$^{-3}$, varying the electric field from $0.3$ to $1.6$ $k_B T/(e\text{\AA})$ (Fig. \[fig:electric-field\]). Again, the linearized mean-field theory (dashed lines) fits the transition in the frequency domain very well for all electric fields, without further adjustable parameters. The transition frequency shifts to higher frequencies for higher electric fields. As predicted, the dependence of $S\left(\omega\right)$ on $\omega$ vanishes for vanishing electric field.
![ (a) The power $a$ of the fits $S\left(\omega\right) \propto \omega^{-a}$ ($\omega < 10^{-2}$ $\tau^{-1}$) as a function of concentration $C_0$. The dashed line denotes the linearized mean-field prediction of Eqs. \[eqn:psd\]–\[eqn:current\]. (b) The power $b$ of the fits $S\left(\omega\right) \propto \omega^{-b}$ ($\omega > 10^{-2}$ $\tau^{-1}$) as a function of the radius $R$. The electric field is set to $E_{\parallel} = 0.8$ $k_B T/(e\text{\AA})$. []{data-label="fig:powers"}](powers_v02){width="0.5\columnwidth"}
**Limiting behavior.** – We fit the low frequency regime ($\omega < 10^{-2}\tau^{-1}$) and the high frequency regime ($\omega > 10^{-2} \tau^{-1}$) with power laws with exponents $a$ and $b$, respectively (see Fig. \[fig:concentrations\]c). Apart from the case $R = 6$ Å, the exponent $a$ shows a strong dependence on the ion concentration $C_0$ (Fig. \[fig:powers\]a), increasing from $a < 0.05$ for $C_0 = 3\cdot 10^{-6}$ to $a > 0.15$ for $C_0 = 3\cdot 10^{-4}$. For radii $R > 6$ Å, $a$ is largely independent of $R$, suggesting that the power law behavior is an intrinsic property of nonequilibrium ion transport, rather than a consequence of the confinement. In contrast, the exponent $b$ depends on the radius, but is independent of the ion concentration, as predicted by Eq. \[eqn:psd\] (Fig. \[fig:powers\]b).
![ The power spectrum with the ion-ion interaction potential $V_{ion-ion}\left(r_{ij}\right)$ given by Eq. \[eqn:potential\] (with interactions, red line) and with $V_{ion-ion}\left(r_{ij}\right) = 0$ (without interactions, blue line). The spectrum is calculated in a $R = 25$ Å channel with $E_{\parallel} = 0.8$ $k_{B}T/(e\text{\AA})$. []{data-label="fig:interactions"}](interactions_v02){width="0.5\columnwidth"}
**Ion correlations.** – The strong dependence of $a$ on $C_0$, which is not predicted by mean-field theory, suggests a strong effect of ion correlations. To investigate this low-frequency power law dependence of the power spectrum in more detail, we perform simulations with $V_{ion-ion} = 0$ for the ion-ion interactions at the high concentration $C_0 = 4 \cdot 10^{-4}$, eliminating ion correlations. Ion-membrane interactions are left unchanged. We perform extra long simulations in a pore with $R = 25$ Å. In Fig. \[fig:interactions\], we show the power spectrum obtained with $V_{ion-ion}\left(r_{ij}\right) = 0$, and compare it with the the power spectrum obtained with full interactions ($V_{ion-ion}\left(r_{ij}\right)$ given by Eq. \[eqn:potential\]). Clearly, with $V_{ion-ion}\left(r_{ij}\right) = 0$ the power spectrum saturates at low frequency, in agreement with the linearized mean-field prediction. Therefore, the departure from linearized mean-field theory at high concentrations shown in Fig. \[fig:powers\]a is due to the ion correlations caused by the ion-ion interaction term $V_{ion-ion}\left(r_{ij}\right)$.
![The power spectral density of the ionic current through charged and uncharged pores of radii 25 Å and 13 Å at a concentration of $3\cdot 10^{-4}$ Å$^{-3}$ and an electric field of $0.8$ $k_BT/(e\text{\AA})$.[]{data-label="fig:charge"}](charge_v01.pdf){width="0.5\columnwidth"}
**Surface charge.** – Typically, both synthetic and biological aqueous pores have a finite surface charge density. The interactions between the charged surface and the ions has been suggested as the source of the $1/\omega$ power law at low frequency [@2009_Powell-Siwy_PRL; @2009_Hoogerheide]. We investigate the effect of surface charges by allocating negative charges on the surface of the pores, balanced by extra positive ions in solution. The charges are located at the membrane particles located in a cylindrical shell between $R < r < R + 6$ Å, with the total charge equal to $Q = -19e$ for $R=13$ Å and $Q = -26e$ for $R=25$ Å. The corresponding power spectra are shown in Fig. \[fig:charge\]. No differences are observed in the shape, the transition frequency, or the power laws at low and high frequency, in agreement with Ref. [@2010_Tasserit]. In the charged pores, the magnitude of the power spectral density is $20\%$ higher than in the uncharged pores, which we attribute to the increased ion density due to ions screening the surface charge. The increase in the power spectral density corresponds indeed to the increased number of ions in the pore, which we determine independently.
Conclusions
===========
We calculate the power spectrum of electric field driven ion transport through cylindrical nanometer scale pores using both linearized mean-field theory and Langevin dynamics simulations. We derive that the linearized mean-field theory predicts a plateau in the power spectrum at low frequency, which is not found in experiments. Furthermore, the linearized mean-field theory predicts a decreasing power law at high frequency, where the power depends on the applied electric field. At low ion concentrations (0.005 mol/l - 0.05 mol/l), the Langevin dynamics simulations confirm the mean-field predictions with high accuracy, including the electric field dependence of the power law decrease at high frequency and the plateau at low frequency. Our linearized mean-field expression fits the simulation data using only one adjustable parameter, the cut-off small length scale $\Lambda$, varying between 6 Å and 7 Å for all curves, apart from $R=6$ Å, which gives $\Lambda = 3$ Å. At high ion concentration (0.5 mol/l), the simulated power spectrum is still accurately described by our linearized mean-field theory at high frequency, but at low frequency the simulation curves do not exhibit a plateau. Instead, the power spectrum increases with decreasing frequency with a power law $S(\omega) \propto 1/\omega^a$, with $0.14 < a < 0.18$. We attribute this deviation from the mean-field prediction to ion-ion correlations that are not present in the linearized mean-field theory. Finally, we study the effect of a finite surface charge density on the inside of the nanopore. Contrary to reports in literature [@2009_Powell-Siwy_PRL; @2009_Hoogerheide] we find no significant dependence of the power spectrum on the surface charge density.
Appendix
========
**Derivation of the governing equations.** – To derive $\tilde{J}^+_{\parallel}\left({\boldsymbol{q}},\omega\right) - \tilde{J}^-_{\parallel}\left({\boldsymbol{q}},\omega\right)$, we switch to index notation where $\alpha$, $\beta$, and $\gamma$ correspond to the three components of our coordinate system. After applying a standard Fourier transform to Eqs. \[eqn:fick\] and \[eqn:nernst-planck\], $$\label{eq:er8}
\widetilde{J}^{z}_{\alpha} = \sum^{3}_{\beta=1} \left[ -\frac{iD}{\omega} \, q_{\alpha} q_{\beta} \widetilde{J}^z_{\beta} - \frac{D\,zeE_{\alpha}}{\omega\, k_BT} \, q_{\beta} \widetilde{J}^z_{\beta} - \widetilde{\eta}_{\alpha} \right],$$ with $\widetilde{...}$ denoting the Fourier transform, ${\boldsymbol{q}}$ being the wave vector, $\omega$ being the frequency, and $z = \pm 1$ denoting the ion charge. Rewriting Eq. \[eq:er8\] leads to: $$\label{eq:er10}
\begin{split}
- \widetilde{\eta}_{\alpha} &=
\sum^{3}_{\beta=1} \widetilde{J}^z_{\beta} \left[\delta_{\alpha \beta} + \frac{iD}{\omega}\,q_{\alpha} q_{\beta} + \frac{D\,zeE_{\alpha}}{\omega \, k_BT} \, q_{\beta} \right] \\
&= \sum^{3}_{\beta=1} \widetilde{J}^z_{\beta} M_{\alpha\beta},
\end{split}$$ where $M_{\alpha\beta}$ denotes the matrix $$\label{eq:er11}
M_{\alpha\beta}=\delta_{\alpha \beta} + \frac{iD}{\omega} \, q_{\alpha}q_{\beta} + \frac{D\,zeE_{\alpha}}{\omega \, k_BT} \, q_{\beta}.$$ Combining Eqs. \[eq:er10\] and \[eq:er11\] and solving for $\widetilde{J}^z_{\gamma}$, we find $$\label{eq:er13}
\begin{split}
\widetilde{J}^z_{\gamma} &= \frac{1}{\rm{det}(M)}\left[\sum^{3}_{\beta=1} \left[-\frac{iD}{\omega}q_{\gamma}{q_{\beta}\widetilde{\eta}_{\beta}} - \frac{D\,zeE_{\gamma}}{\omega\,k_BT} \, q_{\beta} \widetilde{\eta}_{\beta} \right] \right. \\
&\qquad + \left. \widetilde{\eta}_{\gamma}\sum^{3}_{\beta=1}{\left[\frac{1}{3} + \frac{iD}{\omega}q_{\beta}{q_{\beta}}
+ \frac{D\,ze{E}_{\beta}}{\omega \, k_BT} \, q_{\beta}\right]} \right].
\end{split}$$ Because of our cylindrical symmetry, the flux has two unique components: Parallel ($\parallel$) and perpendicular($\perp$) (Fig. 1a). The electric-field is nonzero only in parallel direction ${\boldsymbol{E}} = (0,0,E_{\parallel})$. Because we are interested in the longitudinal flow of ions through the pore, we concentrate on the flux in the parallel direction: $$\label{eq:er14}
\begin{split}
\widetilde{J}^z_{\parallel}({\boldsymbol{q}},\omega) &=\frac{ \frac{iD}{\omega} \big[{q_{\perp 1}q_{\parallel}\widetilde{\eta}_{\perp 1}} + {q_{\perp 2}q_{\parallel} \widetilde{\eta}_{\perp 2}} + {q_{\perp 1}^{2} \widetilde{\eta}_{\parallel}} + {q_{\perp 2}^{2} \widetilde{\eta}_{\parallel}}\big] }{1 + \frac{iD}{\omega} \big[q_{\parallel}^2 + q_{\perp 1}^2 + q_{\perp 2}^2 \big] + \frac{D\,zeE_{\parallel}}{\omega\,k_BT} q_{\parallel}}\\
&\qquad + \frac{ -\frac{D\,zeE_{\parallel}}{\omega\,k_BT}\big[{q_{\perp 1} \widetilde{\eta}_{\perp 1}} + {q_{\perp 2} \widetilde{\eta}_{\perp 2}}\big] + \widetilde{\eta}_{\parallel}}{1 + \frac{iD}{\omega}\big[q_{\parallel}^2 + q_{\perp 1}^2 + q_{\perp 2}^2\big] + \frac{D\,ze{E}_{\parallel}}{\omega\,k_BT}q_{\parallel}},
\end{split}$$ with $q_{\perp 1}$ and $q_{\perp 2}$ being the two perpendicular wave vectors. The power spectrum of the noise is proportional to the bulk concentration $C_0$, $$\label{eq:er17}
\begin{split}
\langle \eta_{\gamma}\left({\boldsymbol{x}},t\right)\eta_{\beta}({\boldsymbol{x}}',t')\rangle &= 2DC_{0}\delta_{\gamma\beta}\delta({\boldsymbol{x}}-{\boldsymbol{x}}')\delta(t-t') \\
\langle \widetilde{\eta}_{\gamma}({\boldsymbol{q}},\omega)\widetilde{\eta}_{\beta}({\boldsymbol{q}}',\omega')\rangle &= 2DC_{0}\delta_{\gamma\beta}(2\pi)^{4}\delta({\boldsymbol{q}}+{\boldsymbol{q}}') \delta(\omega+\omega').
\end{split}$$ Using short-hand notation, we derive from Eqs. \[eq:er14\]-\[eq:er17\] $$\label{eq:er18}
\begin{split}
\langle| & \widetilde{J}_{\parallel}^{+} - \widetilde{J}_{\parallel}^{-}|^2\rangle \equiv \int\!\!\int\!\!\int \frac{\textrm{d}q_{\perp1}'\textrm{d}q_{\perp2}'\textrm{d}q_{\parallel}'}{(2\pi)^3}\int \frac{\textrm{d}\omega'}{2\pi} \Big[\\
\langle\big[ &\widetilde{J}^{+}_{\parallel}({\boldsymbol{q}},\omega)-\widetilde{J}^{-}_{\parallel}({\boldsymbol{q}},\omega)\big]
\big[\widetilde{J}^{+}_{\parallel}({\boldsymbol{q}}',\omega')-\widetilde{J}^{-}_{\parallel}({\boldsymbol{q}}',\omega')\big]\rangle \Big] = \\
& \frac{8 D C_{0} \Big[\frac{eE_{\parallel}}{kT}\Big]^2 \Big[\frac{\omega^2}{D^2} + q_{\perp}^{4}\Big] \Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]}
{\Big(\Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]^2 \!\! + \Big[\frac{ eE_{\parallel} }{kT}\Big]^2 \! q_{\parallel}^2 \! - \frac{\omega^2}{D^2} \! \Big)^2
\!\! + 4 \frac{\omega^2}{D^2} \Big[q_{\perp}^{2} + q_{\parallel}^{2}\Big]^2}.
\end{split}$$
The power spectral density $S\left(\omega\right)$ of the current $I_{\parallel}\left(t\right)$ defined on the domain $0 < t <T$ is given by the limit of $T \to \infty$ of $$S\left(\omega\right) = \frac{1}{T} \langle | \tilde{I}_{\parallel}\left(\omega\right) |^2 \rangle = \frac{1}{T} \langle \tilde{I}_{\parallel} \left(\omega\right) \tilde{I}_{\parallel}\left(-\omega\right) \rangle,$$ which can be written as $$S\left(\omega\right) = \frac{1}{T} \int_T \!\textrm{d}t \int_T \!\textrm{d}t' \, e^{-i\omega\left(t-t'\right)} \, \langle I_{\parallel}\left(t\right) I_{\parallel}\left(t'\right) \rangle.$$ We write $I_{\parallel}\left(t\right)$ as the integral of the current density $J^+_{\parallel}\left({\boldsymbol{x}},t\right)-J^-_{\parallel}\left({\boldsymbol{x}},t\right)$ at a given position in the direction of $x_{\parallel}$ over the lateral surface area $A$ of the pore, $$\label{eqn:S_of_J}
\begin{split}
& S\left(\omega\right) = \frac{1}{T} \int_T \!\!\textrm{d}t \! \int_T \!\!\textrm{d}t'
\, e^{-i\omega\left(t-t'\right)} \\
& \! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1} \textrm{d}x_{\perp2} \! \int \!\!\textrm{d}x_{\parallel}
\! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1}' \textrm{d}x_{\perp2}'\! \int \!\!\textrm{d}x_{\parallel}' \delta(x_{\parallel})\delta(x_{\parallel}')\\
&\langle
\big[J^+_{\parallel}\left({\boldsymbol{x}},t\right) - J^-_{\parallel}\left({\boldsymbol{x}},t\right)\big]
\big[J^+_{\parallel}\left({\boldsymbol{x}}',t'\right) - J^-_{\parallel}\left({\boldsymbol{x}}',t'\right)\big]
\rangle.
\end{split}$$ Expressing the delta functions and the current density in Eq. \[eqn:S\_of\_J\] in terms of their Fourier transforms, we arrive at $$\begin{split}
& S\left(\omega\right) = \frac{1}{T} \int_T \!\!\textrm{d}t \! \int_T \!\!\textrm{d}t'
\, e^{-i\omega\left(t-t'\right)} \\
& \! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1} \textrm{d}x_{\perp2} \! \int \!\!\textrm{d}x_{\parallel}
\! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1}' \textrm{d}x_{\perp2}'\! \int \!\!\textrm{d}x_{\parallel}'\\
& \! \int \! \frac{\textrm{d}q_{\perp1} }{2\pi} \! \int \! \frac{\textrm{d}q_{\perp2} }{2\pi} \! \int \! \frac{\textrm{d}q_{\parallel} }{2\pi}
\, e^{-i(q_{\perp1}x_{\perp1}+q_{\perp2}x_{\perp2}+q_{\parallel}x_{\parallel})} \\
& \! \int \! \frac{\textrm{d}q_{\perp1}' }{2\pi} \! \int \! \frac{\textrm{d}q_{\perp2}' }{2\pi} \! \int \! \frac{\textrm{d}q_{\parallel}' }{2\pi}
\, e^{-i(q_{\perp1}'x_{\perp1}'+q_{\perp2}'x_{\perp2}'+q_{\parallel}'x_{\parallel}')} \\
& \! \int \! \frac{\textrm{d}\omega'}{2\pi} \!\int\!\frac{\textrm{d}\omega''}{2\pi} \, e^{i\omega' t} e^{i\omega'' t'}
\! \int\frac{\textrm{d}q_{\parallel}''}{2\pi} e^{iq_{\parallel}'' x_{\parallel}} \!\! \int\frac{\textrm{d}q_{\parallel}'''}{2\pi} e^{iq_{\parallel}''' x_{\parallel}'} \\
&\langle
\big[\widetilde{J}^+_{\parallel}\left({\boldsymbol{q}},\omega'\right) - \widetilde{J}^-_{\parallel}\left({\boldsymbol{q}},\omega'\right)\big]
\big[\widetilde{J}^+_{\parallel}\left({\boldsymbol{q}}',\omega''\right) - \widetilde{J}^-_{\parallel}\left({\boldsymbol{q}}',\omega''\right)\big]
\rangle.
\end{split}$$ Performing the integrals over $\omega''$, ${\boldsymbol{q}}'$, $q_{\parallel}''$, $q_{\parallel}'''$, $x_{\parallel}$, and $x_{\parallel}'$, and using the short-hand notation of Eq. \[eq:er18\] leads to $$\begin{split}
& S\left(\omega\right) = \frac{1}{T} \int_T \!\!\textrm{d}t \! \int_T \!\!\textrm{d}t'
\, e^{-i\omega\left(t-t'\right)} \\
& \! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1} \textrm{d}x_{\perp2}
\! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1}' \textrm{d}x_{\perp2}'\\
& \! \int \! \frac{\textrm{d}q_{\perp1} }{2\pi} \! \int \! \frac{\textrm{d}q_{\perp2} }{2\pi} \! \int \! \frac{\textrm{d}q_{\parallel} }{2\pi}
e^{-i(q_{\perp1}(x_{\perp1}-x_{\perp1}')+q_{\perp2}(x_{\perp2}-x_{\perp2}'))} \\
& \! \int \! \frac{\textrm{d}\omega'}{2\pi} e^{i\omega' (t-t')}
\langle |\big[\tilde{J}^+_{\parallel}\left( {\boldsymbol{q}}, \omega'\right) - \tilde{J}^-_{\parallel}\left( {\boldsymbol{q}}, \omega'\right)\big] |^2 \rangle.
\end{split}$$ We rearrange the exponential functions and perform the integrals over $t$, $t'$ and $\omega'$, yielding $$\label{eqn:psd-derivation}
\begin{split}
S\left(\omega\right) = \int \! \frac{\textrm{d}q_{\perp1} }{2\pi} \! & \int \! \frac{\textrm{d}q_{\perp2} }{2\pi} \! \int \! \frac{\textrm{d}q_{\parallel} }{2\pi} \widetilde{A}^2(q_{\perp}) \\
& \langle |\big[\tilde{J}^+_{\parallel}\left( {\boldsymbol{q}}, \omega\right) - \tilde{J}^-_{\parallel}\left( {\boldsymbol{q}}, \omega\right)\big] |^2 \rangle,
\end{split}$$ with the squared Fourier-transformed area function being given by $$\label{eqn:area-derivation}
\begin{split}
\widetilde{A}^2(q_{\perp}) & =
\! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1} \textrm{d}x_{\perp2}
\! \int\!\!\!\int_A \!\!\textrm{d}x_{\perp1}' \textrm{d}x_{\perp2}' \\
& \quad\quad\quad\quad e^{-i(q_{\perp1}(x_{\perp1}-x_{\perp1}')+q_{\perp2}(x_{\perp2}-x_{\perp2}'))} \\
& = \int_{0}^{2\pi}\!\!\!\textrm{d}\theta \! \int_{0}^{2\pi}\!\!\! \textrm{d}\theta' \! \int_{0}^{R}\!\!\! \textrm{d}x_{\perp} \! \int_{0}^{R} \!\!\! \textrm{d}x_{\perp}' \, x_{\perp} \, x_{\perp}' \, e^{-iq_{\perp}\left(x_{\perp}-x_{\perp}'\right)},
\end{split}$$ with $x_{\perp} = \sqrt{x_{\perp1}^2 + x_{\perp2}^2}$ and $\theta = \arctan(x_{\perp2}/x_{\perp1})$ being the cylindrical coordinates and $q_{\perp} = \sqrt{q_{\perp1}^2+q_{\perp2}^2}$. Combining Eqs. \[eq:er18\] and \[eqn:psd-derivation\], and performing the integrals in Eq. \[eqn:area-derivation\], we arrive at Eqs. \[eqn:psd\]-\[eqn:current\].
![ The electrical conductivity of a bulk system of ions ($100\times100\times100$ Å) as a function of the ionic concentration $C_0$. Inset: The current as a function of the electric field. []{data-label="fig:conductivity"}](conductivity_v02){width="0.5\columnwidth"}
**Time scale.** – We calibrate the time scale $\tau$ from the conductivity of a bulk system of $100^3$ Å$^3$, calculated from a linear fit of the current as a function of the electric field $E$ (Fig. \[fig:conductivity\]). Experimentally, the conductivity is given by $$\label{eqn:conductivity}
\frac{1}{C_0}\frac{\textrm{d}I}{\textrm{d}E} = A\left(D^+ + D^-\right)\frac{e^2}{k_BT},$$ with $A$ the lateral surface area and $D^{+} = D^{-} = 0.2$ Å$^2$/ps being the diffusion coefficient of K$^+$ and Cl$^-$. We calculate the time scale $\tau$ from a linear fit to the curve in Fig. \[fig:conductivity\] up to $C_0 = 3\cdot 10^{-4}$ and comparison with Eq. \[eqn:conductivity\], giving $\tau = 5$ ps.
![ The power spectra of pores with radius $R = 13$ Åand length $L = 48$ Å, each connected to two reservoirs of size $W\times W\times H$. The total simulation box is $W \times W \times L+2H$, with periodic boundary conditions in all directions. The simulations are performed with $V_{\textit{ion-ion}} = 0$. []{data-label="fig:size-dependence"}](size-dependence_v02){width="0.5\columnwidth"}
**Finite-size effects.** – To check the effect of the simulation box size, we vary the reservoir size between $50$ Å and $288$ Å in $x_{\parallel}$-direction and between $96$ Å and $288$ Å in $x_{\perp}$-direction. The power spectra are independent of the reservoir size (Fig. \[fig:size-dependence\]), from which we conclude that the periodic boundary conditions do not affect the ion fluctuations.
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Bradley J. Kavanagh'
bibliography:
- 'Directional.bib'
title: Discretising the velocity distribution for directional dark matter experiments
---
Introduction
============
The measurement of the directionality of nuclear recoils in dedicated low-background detectors has long been proposed as a method to detect particle dark matter (DM) [@Spergel:1988]. The motion of the Sun through the Galactic DM halo generates a so-called ‘wind’ of DM particles, which would appear to originate from the direction of the constellation of Cygnus. The resulting recoil spectrum will be peaked in the opposing direction (or for high mass DM, in a ring around this direction [@Bozorgnia:2011]). Such a directional signal would be a smoking gun for DM against the typically isotropic distribution of residual backgrounds [@Copi:1999; @Copi:2002; @Morgan:2005; @Billard:2010c; @Green:2010b]. Directional detection may also provide a way to circumvent the neutrino floor [@Grothaus:2014] and to probe the local structure of the DM velocity distribution [@Billard:2013; @Ohare:2014].
A number of directional experiments are currently in the prototype stage. The predominant technology is the low-pressure time-projection chamber (TPC), in which nuclear recoils leave an $\mathcal{O}$(1 mm) track of ionised electrons, which can be imaged by drifting the electrons (or electron-transport gas) to an anode grid. This technology is currently utilised by the DRIFT [@Daw:2012; @Battat:2014], MIMAC [@Riffard:2013; @Santos:2013], DMTPC [@Monroe:2012; @Battat:2013], NEWAGE [@Miuchi:2010; @Miuchi:2012] and D3 [@Vahsen:2012] collaborations and the problem of reconstructing the recoil direction using this technology has been much studied [@Burgos:2007; @Billard:2012]. More recent proposals for directional detectors include nuclear emulsions [@Naka:2012], electronic scattering in crystals [@Essig:2012b], DNA-based methods [@Drukier:2012] and columnar recombination in Xenon targets [@Nygren:2013nda; @Mohlabeng:2015efa].
Directional experiments such as these may be the only possibility for probing the full 3-dimensional velocity distribution of DM, $f(\mathbf{v})$, which is *a priori* unknown. The standard analysis of direct detection experiments assumes the so-called Standard Halo Model (SHM), a simplified model for the DM halo as an isotropic, isothermal sphere of particles [@Green:2012]. However, such a simple description is unlikely to be an accurate description of the Milky Way halo [@Evans:2000; @Widrow:2000; @Lisanti:2010; @Bhattacharjee:2012; @Fornasa:2013]. In addition, $N$-body simulations raise the possibility of more complicated distributions, including debris flows [@Lisanti:2013; @Kuhlen:2012], tidal streams [@Freese:2004; @Freese:2005] or a dark disk [@Read:2009; @Read:2010; @Pillepich:2014]. The impact of these astrophysical uncertainties on directional signals has previously been studied [@Bozorgnia:2011; @Billard:2013; @Ohare:2014]. While non-standard astrophysics should not severely affect the ‘smoking gun’ directional signature of DM recoils [@Billard:2013], it may still pose a potential problem for future data. In particular, trying to reconstruct the DM parameters (such as mass $m_\chi$ and interaction cross section $\sigma$) from a potential signal requires some assumptions to be made about the form of $f(\mathbf{v})$. As with non-directional detection techniques, poor assumptions about the astrophysics of DM can lead to biased reconstructions of these parameters [@Peter:2011; @Fairbairn:2012; @Kavanagh:2012; @Kavanagh:2013a; @Kavanagh:2014].
Astrophysics-independent approaches to *non-directional* experiments have received significant attention in the past few years, including so-called ‘halo-independent’ methods [@Fox:2011b; @Fox:2011c; @Frandsen:2012; @Gondolo:2012; @DelNobile:2013a; @Bozorgnia:2013; @DelNobile:2013b; @Feldstein:2014a; @Feldstein:2014b; @Fox:2014; @Cherry:2014] and attempts at parametrising the 1-dimensional speed distribution [@Strigari:2009; @Peter:2011; @Kavanagh:2012; @Kavanagh:2013a]. There have also been several attempts to construct a suitable parametrisation for the 3-dimensional velocity distribution, relevant for directional experiments [@Billard:2010b; @Lee:2012; @Alves:2012; @Lee:2014]. This is significantly more difficult than in the non-directional case, owing to the presence of the angular components of $f(\mathbf{v})$. A single 1-dimensional function is therefore no longer sufficient to describe the full distribution function and a suitable angular basis must be found. Several bases have been suggested (e.g. Refs. [@Alves:2012; @Lee:2014]) but all typically suffer from the same problem, which has not previously been addressed in the literature: they allow $f(\mathbf{v})$ to take negative - and therefore unphysical - values, which can lead to spurious results. We discuss this problem in more detail in Sec. \[sec:problems\].
In this work, we present a framework which allows the velocity distribution to be parametrised in a model-independent way, while guaranteeing that it is everywhere positive. This consists of an angular discretisation of the velocity distribution, decomposing it into a series of 1-dimensional functions, each of which is constant over a given bin in the angular coordinates. This is motivated in the first instance by its ability to describe the simplest directional signal, a forward-backward asymmetry in the scattering rate. However, we extend the discretisation to an arbitrary number of bins $N$ in the angular variables, such that in the limit of large $N$, the full 3-dimensional distribution can be recovered. For arbitrary $N$, we also demonstrate how to calculate the Radon transform (the function which encodes the angular dependence of the scattering rate) and therefore how to compute the full scattering rate.
Here, we focus on the angular discretisation itself, and do not attempt a full reconstruction of the DM velocity distribution and particle physics parameters (as in e.g. [@Peter:2011; @Kavanagh:2012]). Instead, we investigate the magnitude of the error introduced by this angular discretisation for two different benchmark velocity distributions. To do this, we compare the event rate within each angular bin obtained using the full and discrete velocity distributions. Qualitatively, we discuss and compare the energy dependence of the event rate within each bin. Quantitatively, we can calculate the number of signal events in each angular bin for the full and discrete distributions, and compare the result. This allows us to evaluate whether (and in which scenarios) fitting to data using such an angular discretisation gives accurate results.
In order to perform this analysis, we assume that the velocity distribution is known exactly. In the analysis of real experiments, a parametrisation of $f(\mathbf{v})$ within each bin must be chosen and fit to the data. We leave the choice of a suitable parametrisation of the remaining 1-dimensional functions of $v$ to future work, meaning that the results presented here represent a lower limit on the error induced using the discretised velocity distribution. However, we lay out a framework which can be used in the analysis of future data to extract coarse-grained information about the DM velocity distribution in a way which was not previously possible.
In Sec. \[sec:formalism\], we present the directional detection formalism, including calculation of the scattering cross section and the Radon transform, and a brief discussion of the associated astrophysical uncertainties. This is followed in Sec. \[sec:problems\] by a discussion of previous attempts to overcome these uncertainties. In Sec. \[sec:discretisation\], we present the discretised velocity distribution which is the focus of this work. In Sec. \[sec:compare\], we investigate how this discretised approximation compares to the exact result, both at the level of the Radon transform and the event rate. Finally, we discuss the significance of these results and plans for future work in Sec. \[sec:discussion\]. The procedure for calculating the Radon transform from the discretised distribution is presented in Appendix \[app:Radon\], while we relegate the full derivation of this formula to Appendix \[app:RadonDeriv\].
Directional formalism {#sec:formalism}
=====================
The directional rate per unit detector mass for recoils of energy $E_R$ can be written as [@Gondolo:2002]
$$\label{eq:Rate}
\frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}\Omega_q} = \frac{\rho_0}{4\pi \mu_{\chi p}^2 m_\chi} \sigma^p \mathcal{C}_N F^2(E_R) \hat{f}(v_\textrm{min}, \hat{\mathbf{q}}) \,,$$
where $\hat{\mathbf{q}} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\phi)$ is a unit vector pointing in the direction of the recoil. We have written the rate $R$ in terms of the DM-proton cross section at zero-momentum transfer $\sigma_p$, which may be spin-independent or spin-dependent; the form factor $F^2(E_R)$, which describes the loss of coherence due to the finite size of the nucleus; and an enhancement factor $\mathcal{C}_N$, which describes the enhancement of the rate for a nucleus $N$ relative to the DM-proton rate. The local DM density is written as $\rho_0$, the DM mass as $m_\chi$ and the reduced DM-proton mass as $\mu_{\chi p} = m_\chi m_p/(m_\chi + m_p)$.
For the gaseous targets used in low-pressure TPCs, relatively light nuclei are typically used. For example, the DRIFT experiment [@Daw:2012; @Battat:2014] uses $\mathrm{CF}_4$ as the target gas, with spin-dependent (SD) scattering from $^{19}\mathrm{F}$ nuclei expected to be the dominant interaction with DM particles. For SD interactions, the enhancement factor can be written as [@Cerdeno:2010; @Cannoni:2013]
$$\mathcal{C}_N^{SD} = \frac{4}{3}\frac{J+1}{J} \left|\langle S_p \rangle + a_n/a_p \langle S_n \rangle\right|^2\,,$$
where $J$ is the total nuclear spin, $\langle S_{p,n} \rangle$ is the expectation value of the proton and neutron spin in the nuclear ground state with maximal magnetic quantum number, and $a_n/a_p$ is the ratio of the DM-neutron and DM-proton couplings. The ratio $a_n/a_p$ depends on the specific model of DM under consideration, although simplifying assumptions are often used, including pure-nucleon couplings ($a_p = 0$ or $a_n = 0$) [@Tovey:2000] or values motivated by specific models (e.g. $a_n/a_p = \pm 1$) [@Vasquez:2012].
The SD form factor can be written in terms of the spin structure functions of the nucleus $S_{ij}(q)$, which depend on the momentum transfer $q = \sqrt{2 m_N E_R}$, for a nucleus of mass $m_N$:
$$F^2_{SD}(q) = S(q)/S(0)\,,$$
with $$S(q) = a_0^2S_{00}(q) + a_0 a_1 S_{01}(q) + a_1^2 S_{11}(q)\,.$$ The isoscalar and isovector couplings are related to the proton and neutron couplings by $a_0 = a_p + a_n$ and $a_1 = a_p - a_n$ [@Cannoni:2013]. The spin structure functions (and the proton and neutron spin matrix elements) can be calculated using nuclear shell models (see e.g. Refs. [@Ellis:1988; @Engel:1989; @Iachello:1991; @Ressel:1993; @Dimitrov:1995; @Menendez:2012]). Here, we focus on Fluorine targets, assuming $a_p = a_n$, with spin structure functions and matrix elements taken from Ref. [@Divari:2013]. The spin structure functions lead to a roughly exponential suppression of the recoil spectrum as a function of recoil energy.
The DM velocity distribution enters into the scattering rate through the Radon transform $$\hat{f}({v_\mathrm{min}}, {\hat{\mathbf{q}}}) = \int_{\mathbb{R}^3} f(\mathbf{v}) \delta \left(\mathbf{v} \cdot {\hat{\mathbf{q}}}- {v_\mathrm{min}}\right) \mathrm{d}^3 \mathbf{v}\,,$$ where ${v_\mathrm{min}}$ is the minimum DM speed required to excite a nuclear recoil of energy $E_R$: $$\label{eq:vmin}
v_\mathrm{min} = v_\mathrm{min}(E_R) = \sqrt{\frac{m_N E_R}{2 \mu_{\chi N}^2}}\\.$$ Geometrically, the Radon transform corresponds to an integral over $f(\textbf{v})$ on a plane perpendicular to $\hat{\textbf{q}}$, which has a perpendicular distance $v_\mathrm{min}$ from the origin. Physically, this corresponds to integrating over all DM velocities which satisfy the kinematic constraints for exciting a nuclear recoil of energy $E_R$ in direction $\hat{\mathbf{q}}$.
The typical assumption for the form of the DM velocity distribution is the Standard Halo Model (SHM). For a spherically symmetric, isothermal DM halo, with density profile $\rho(r) \propto r^{-2}$, the resulting velocity distribution has a Maxwell-Boltzmann form in the reference-frame of the Galaxy, [@Gondolo:2002] $$f(\mathbf{v}) = \frac{1}{(2\pi \sigma_v^2)^{3/2}} \exp\left(-\frac{\mathbf{v}^2}{2\sigma_v^2}\right)\,,$$ with velocity dispersion $\sigma_v$. The corresponding Radon transform also takes the form of a Gaussian, $$\label{eq:analRadon}
\hat{f}({v_\mathrm{min}}, \hat{\mathbf{q}}) = \frac{1}{(2\pi \sigma_v^2)^{1/2}} \exp\left(-\frac{{v_\mathrm{min}}^2}{2\sigma_v^2}\right)\,.$$ Using the properties of the Radon transform we can obtain the corresponding result in the Earth frame by performing a Galilean transformation $\mathbf{v} \rightarrow \mathbf{v} + \mathbf{v}_e$, with $\mathbf{v}_\mathrm{e}$ the velocity of the Earth with respect to the rest frame of the halo, in which case
$$\hat{f}({v_\mathrm{min}}, \hat{\mathbf{q}}) \rightarrow \hat{f}({v_\mathrm{min}}+ \mathbf{v}_\mathrm{e}\cdot \hat{\mathbf{q}}, \hat{\mathbf{q}})\,.$$
The SHM velocity distribution in the laboratory frame is then $$\label{eq:SHM_Radon}
\hat{f}({v_\mathrm{min}}, \hat{\mathbf{q}}) = \frac{1}{(2\pi \sigma_v^2)^{1/2}} \exp\left(-\frac{({v_\mathrm{min}}+ \mathbf{v}_\mathrm{e}\cdot\hat{\mathbf{q}}) ^2}{2\sigma_v^2}\right)\,.$$ Within the SHM, one typically assumes values of $v_\mathrm{e} \approx 220 {\textrm{ km s}^{-1}}$ and $\sigma_v = v_\mathrm{e}/\sqrt{2} \approx 156 {\textrm{ km s}^{-1}}$. However, there are still uncertainties on these values of at least $10\%$ [@Feast:1997; @Schonrich:2012; @Bovy:2012a]. Introducing a cut off in the distribution at the Galactic escape speed $v_\mathrm{esc}$ introduces further uncertainty into the model [@Rave:2014].
Despite its widespread use, the SHM is unlikely to be an accurate representation of the DM halo. Observations and N-body simulations indicate that the halo should deviate from a $1/r^2$ profile and may not be spherically symmetric. As a result alternative models have been proposed. Speed distributions associated with triaxial halos [@Evans:2000] or with more realistic density profiles [@Widrow:2000] have been suggested, as well as analytic parametrisations which should provide more realistic behaviour at low and high speeds [@Lisanti:2010]. Self-consistent distribution functions reconstructed from the potential of the Milky Way have also been obtained [@Bhattacharjee:2012; @Fornasa:2013].
It is also possible to extract the speed distribution from N-body simulations. Such distribution functions tend to peak at lower speeds than the SHM and have a more populated high speed tail [@Vogelsberger:2009; @Kuhlen:2010; @Mao:2012]. There are also indications that DM substructure may be significant, causing ‘bumps’ in the speed distribution, or that DM which has not completely phase-mixed - so-called ‘debris flows’ - may have a contribution [@Kuhlen:2012].
Another result obtained from simulations is the possibility of a dark disk. When baryons are included in simulations of galaxy formation, this can result in DM subhalos being preferentially dragged into the disk plane where they are tidally stripped [@Read:2009; @Read:2010]. The result is a dark disk which corotates with the stellar disk, though with a smaller velocity dispersion, typically $\sigma_v^{DD} \sim 50 {\textrm{ km s}^{-1}}$. Such a dark disk could comprise a large fraction (up to 50%) of the total DM density, though more recent results [@Pillepich:2014] suggest a smaller dark disk contribution (around 10%), depending on the merger history of the Milky Way.
Finally, direct detection experiments probe the DM halo on sub-milliparsec scales and there is therefore the possibility that DM sub-structures could dominate the local distribution. Analyses of N-body simulations suggest that no individual subhalos should dominate the local density [@Helmi:2002; @Vogelsberger:2007]. However, local structures could still be significant, especially streams of DM from the tidal disruption of Milky Way satellites, such as Sagittarius [@Freese:2004; @Freese:2005; @Savage:2006].
Such a wide range of possibilities for the form of $f(\mathbf{v})$ leads to substantial uncertainties on the predicted event rate for directional detectors. This in turn can lead to potential bias in the reconstruction of other DM parameters, if these uncertainties are not properly accounted for.
Problems with parametrising the velocity distribution {#sec:problems}
=====================================================
With promising developments in directional detector technology, it is interesting to ask what information about the velocity distribution we could, in principle, extract from a directional signal. Early attempts to address this question involved extending the SHM to allow anisotropic velocity dispersions and fitting these new parameters to mock data [@Billard:2010b]. The possibility of including additional structures, such as streams and dark disks, has also been considered [@Lee:2012]. However, such methods are likely to be accurate only if the underlying velocity distribution is well-described by the chosen model.
Alves, Hendri and Wacker [@Alves:2012] investigated the more general possibility of describing $f(\textbf{v})$ in terms of a series of special functions of integrals of motion (energy and angular momentum). These can then be fit to data, with around 1000 events required to distinguish between the SHM and a Via Lactea II distribution function [@Kuhlen:2008]. However, the special, separable form of the velocity distribution requires that the dark matter halo is in equilibrium. More troubling is that the resulting velocity distribution is not guaranteed to be everywhere positive and therefore not all combinations of parameters correspond to physical distribution functions.
A more general parametrisation for the velocity distribution was recently proposed by Lee [@Lee:2014]. In this approach, the velocity distribution is decomposed into products of Fourier-Bessel functions and spherical harmonics. This is completely general and does not require assumptions about the halo being in equilibrium. Lee also gives an analytic expression for the Radon transform of the Fourier-Bessel basis, making this approach computationally efficient. However, this approach can also produce negative-valued, non-physical distribution functions, as in the case of the method of Alves et al..
In fact, any decomposition in terms of spherical harmonics leads to this problem, because the spherical harmonic basis functions can have negative values. It is unclear how this issue will affect parameter reconstruction. It may lead to parameter estimates (e.g. for $m_\chi$ or $\sigma_p$) which are unphysical, as they require an unphysical distribution function to fit the data well. Without some criteria which determines which coefficients of the spherical harmonics lead to strictly positive distribution functions, it may not be possible to reject such parameter points. We may attempt to numerically test each parametrised distribution function for negative values but for a real function of three parameters $f(\textbf{v}) = f(v_x, v_y, v_z)$ this would require a very large number of evaluations, which may not be computationally feasible (and does not absolutely guarantee positive-definiteness). In addition, physical distributions may occupy only a small fraction of the total space of parameters making parameter sampling and reconstruction difficult.
Even if positive-definiteness could be ensured, it is not clear how to interpret the parameters reconstructed in this way. Spherical harmonic approximations of typical velocity distributions such as the SHM (obtained by integrating out the coefficients of the basis functions) tend to produce distributions which do contain negative values. This is especially true when only a small number of basis functions is used. However, the ‘true’ spherical harmonic coefficients obtained in this way cannot be obtained by fitting to data (because we would reject those which lead to negative values). This may again lead to a bias in the reconstructed DM parameters, because the ‘true’ values do not lie in the allowed parameter space. Such potential problems have not previously been noted in the literature.
In order to fit to data, then, it is necessary to decompose $f(\textbf{v})$ into a series of angular components $A^i$: $$f(\textbf{v}) = f(v, \cos\theta', \phi')= f^1(v) A^1(\theta',\phi') + f^2(v) A^2(\theta',\phi') +f^3(v) A^3(\theta',\phi') +...\,.$$ We then truncate the series at some order, leaving only a finite number of 1-dimensional functions $f^i(v)$ which are unknown. This reduces the problem of attempting to fit a function of the 3-dimensional variable $\textbf{v}$ to the problem of parametrising a series of 1-dimensional functions, which is much more tractable. Of course, we should be careful that this truncation preserves enough angular information to still provide a good approximation to $f(\textbf{v})$. However, as more data becomes available, we can add more terms to the series to capture more angular features in the distribution.
As we have discussed, the spherical harmonic basis may not be an appropriate choice for this decomposition. In the next section, I will present an alternative decomposition which can guarantee that the velocity distribution is everywhere positive and therefore represents a promising and general method for extracting information from directional experiments.
A discretised velocity distribution {#sec:discretisation}
===================================
In order to ensure that the velocity distribution is everywhere positive, we propose that the velocity distribution be discretised into $N$ angular components:
$$\label{eq:discretisedf}
f(\textbf{v}) = f(v, \cos\theta', \phi') =
\begin{cases}
f^1(v) & \textrm{ for } \theta' \in \left[ 0, \pi/N\right]\,, \\
f^2(v) & \textrm{ for } \theta' \in \left[ \pi/N, 2\pi/N\right]\,, \\
& \vdots\\
f^k(v) & \textrm{ for } \theta' \in \left[ (k-1)\pi/N, k\pi/N\right]\,, \\
& \vdots\\
f^N(v) & \textrm{ for } \theta' \in \left[ (N-1)\pi/N, \pi\right]\,. \\
\end{cases}$$
Over each bin in $\theta'$, $f(\textbf{v})$ has no angular dependence and depends only on a single function of the DM speed. The resulting velocity distribution will be everywhere positive, as long as a suitable parametrisation for the $f^k(v)$ is chosen which is itself everywhere positive.
We consider for simplicity only a discretisation in $\cos\theta'$. We note that in this work we will only be considering the azimuthally-averaged event rate (i.e. integrated over the recoil angle $\phi$). In this case, we emphasise that no assumptions about the $\phi'$-dependence of $f(\mathbf{v})$ are required, as the integral over $\phi'$ can be performed exactly, and the azimuthally-averaged rate depends only on the azimuthally-averaged velocity distribution (see Appendix \[app:RadonDeriv\]). We can therefore take the velocity distribution to be independent of $\phi'$ without loss of generality. However, this analysis could be extended to consider bins in the $\phi$ angle, with an additional discretisation in $\phi'$ if required.
The motivation for this discretised description is that the simplest signal (beyond an isotropic $N=1$ signal) which can be observed with a directional detector is an asymmetry between the event rates in, say, the forward and backward directions. Shortly after the confirmation of a dark matter signal at a directional detector, the number of events may still be quite small (for example, the roughly 10 events required to distinguish from an isotropic background [@Morgan:2005], or 30 events required to confirm the peak recoil direction [@Green:2010b]). In this small statistics scenario, constraining a large number of free functions is not feasible. However, if we discretise $f(\textbf{v})$ into $N=2$ angular components, it may be possible to extract some meaningful directional information with only a small number of events. With larger numbers of events, $N$ can be increased to allow more directional information to be extracted.
We show in Fig. \[fig:Discrete\] some examples of this discretised velocity distribution. We show the SHM velocity distribution (top left), as well as the $N=2$ (middle left) and $N=3$ (bottom left) discretised approximations, in all cases integrated over $\phi'$. These approximations are obtained by averaging the full velocity distribution over each bin in $\theta'$:
$$\label{eq:averagef}
f^{k}(v) = \frac{1}{\cos((k-1)\pi/N) - \cos(k\pi/N)}\int_{\cos(k\pi/N)}^{\cos((k-1)\pi/N)} f(\mathbf{v}) \, \mathrm{d}\cos\theta'\,.$$
For comparison, the results for a stream distribution are also shown in the right column. We describe these two distribution functions in more detail at the start of Sec. \[sec:compare\].
![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](SHMpolar.pdf "fig:"){width="49.00000%"} ![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](Streampolar.pdf "fig:"){width="49.00000%"}
![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](SHMpolarN2.pdf "fig:"){width="49.00000%"} ![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](StreampolarN2.pdf "fig:"){width="49.00000%"}
![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](SHMpolarN3.pdf "fig:"){width="49.00000%"} ![The full velocity distribution (top) as well as $N=2$ (middle) and $N=3$ (bottom) discretised approximations for two examples: the SHM with $v_\mathrm{e} = 220 {\textrm{ km s}^{-1}}$, $\sigma_v = 156 {\textrm{ km s}^{-1}}$ (left column) and a stream with $v_\mathrm{e} = 500 {\textrm{ km s}^{-1}}$, $\sigma_v = 20 {\textrm{ km s}^{-1}}$ (right column). In each case, we have integrated over the $\phi'$ direction and only show $f(v, \cos\theta')$. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The same colour scale is used in each plot. In the case of the full stream distribution (top right), large values have been truncated for easier comparison with the other plots. []{data-label="fig:Discrete"}](StreampolarN3.pdf "fig:"){width="49.00000%"}
Upon discretisation, the distribution which is initially focused in one direction (towards $\theta' = 0^\circ$) now becomes constant over $\theta'$ in each bin. For the SHM (left column), there is a predominantly forward-going component ($\cos\theta' > 0$), as well as a smaller backwards component ($\cos\theta' < 0$). For the stream (right column), which has a much narrower dispersion and higher peak speed, the $N=2$ discretised distribution is almost entirely in the forward direction. However, we note that due to the discretisation, there is now a sizeable population of particles with transverse velocities ($\theta' = 90^\circ$), which was not the case in the full distribution. In the $N=3$ discretisation (bottom row), the velocity distributions become more focused and begin to recover more of the directionality of the full distributions. In the limit of large $N$, the full distribution can be recovered.
In order to determine the corresponding event spectrum, we must calculate the Radon transform from this discretised $f(\mathbf{v})$. We introduce the integrated Radon transform (IRT), integrated over the same angular bins as for $f(\mathbf{v})$:
$$\label{eq:discreteRadon}
\hat{f}^j(v_\textrm{min}) = \int_{\phi = 0}^{2\pi} \int_{\cos(j\pi/N)}^{\cos((j-1)\pi/N)} \hat{f}(v_\textrm{min}, \hat{\textbf{q}})\, \mathrm{d}\cos\theta\mathrm{d}\phi\,,$$
where $\hat{\mathbf{q}} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\phi)$.[^1] In principle, we can define the IRT over a set of bins different from those used to define the discretised distribution function. However, using the same bins in both cases typically simplifies the calculation of the IRT. A notable exception would be to consider fewer bins for the Radon transform than for the distribution function (in order reduce possible discretisation error). In this scenario, calculation of the IRT would be even simpler.
The IRT arises from the calculation of the directional recoil rate, integrated over a given angular bin, which could then be compared with the number of events observed in that bin. We consider the IRT for two reasons. First, it allows us to perform all of the relevant angular integrals in the calculation of the Radon transform, eliminating the need for computationally-intensive numerical integrals over the angular variables. Secondly, the loss of angular information in discretising the velocity distribution means that the full Radon transform of this discretised distribution is unlikely to provide a good fit to the data on an event-by-event basis. Instead, considering the IRT (and correspondingly binned data) should reduce the error which is introduced by using a discretised approximation to the velocity distribution. This in turn allows us to parametrise the $v$-dependence of each angular bin and mitigate uncertainties in the velocity distribution.
The full details of the derivation of $\hat{f}^j(v_\textrm{min})$ for arbitrary $N$ are included in Appendix. \[app:RadonDeriv\]. We include this derivation as a pedagogical tool in the interests of anyone who wishes to modify or extend the approach presented here (for example, by considering different definitions for the bins in $\cos\theta'$ or $\cos\theta$). However, for the reader interested in simply calculating $\hat{f}^j(v_\textrm{min})$, given a set of $f^k(v)$, the algorithm is given in Appendix \[app:Radon\]. We note that all of the angular integrals involved can be performed analytically, meaning that computing the discretised Radon transform reduces to performing a series of 1-dimensional integrals over $v$ (one integral for each $\hat{f}^j$), which can be performed numerically. A python code which implements the full calculation of Appendix \[app:Radon\] is available from the author.
We note that in the case $N=1$, the IRT simply reduces to an integral over all angles, exactly recovering the velocity integral which appears in non-directional experiments. Also, in the $N=2$ case, the IRT takes the following simple form: $$\begin{aligned}
\label{eq:directional:N2result}
\hat{f}^1({v_\mathrm{min}}) &= 4\pi\int_{{v_\mathrm{min}}}^{\infty} v \left\{ \pi f^1(v) + {\tan^{-1}}\left(\frac{\sqrt{1-\beta^2}}{\beta}\right)\left[f^2(v) - f^1(v)\right] \right\} \, \mathrm{d}v \,,\\
\hat{f}^2({v_\mathrm{min}}) &= 4\pi\int_{{v_\mathrm{min}}}^{\infty} v \left\{ \pi f^2(v) + {\tan^{-1}}\left(\frac{\sqrt{1-\beta^2}}{\beta}\right)\left[f^1(v) - f^2(v)\right] \right\} \, \mathrm{d}v\,,\end{aligned}$$ where $\beta = {v_\mathrm{min}}/v$. We have also checked using Monte Carlo calculations that our calculations give the correct forms for the IRT in the cases $N=2$ and $N=3$. In order to do this, we randomly sample particles from the discretised distributions. For each particle, we simulate a random scattering event and store the values of ${v_\mathrm{min}}$ and $\hat{\mathbf{q}}$. We then bin these events in angle and compare with the calculated distribution of $\hat{f}^j({v_\mathrm{min}})$. The results are in perfect agreement with the results expected from Appendix \[app:Radon\].
Comparison with exact results {#sec:compare}
=============================
We now wish to compare these approximate IRTs with the IRTs obtained from the full (non-discretised) velocity distribution. To do this, we select a benchmark velocity distribution (such as the SHM) and calculate the $f^{k}(v)$ of Eq. \[eq:discretisedf\] by averaging over $\cos\theta'$ in each bin, as in Eq. \[eq:averagef\] and Fig. \[fig:Discrete\]. We then insert these into the algorithm presented in Appendix \[app:Radon\] to obtain the corresponding IRTs. We refer to these as the *approximate* IRTs. For comparison, we use the full Radon transform of Eq. \[eq:analRadon\] to obtain the *exact* IRTs by integrating over $\cos\theta$. We fix the peak of the underlying speed distribution to be aligned in the forward direction $\theta' = 0^\circ$. However, we discuss the consequences for ‘misaligned’ distributions in Sec. \[sec:discussion\].
We discuss qualitatively the differences in shape between the exact and approximate IRTs, as any such differences could potentially lead to biases in the reconstruction of particle physics parameters and the velocity distribution. However, directional direct detection experiments do not directly measure the Radon Transform, but rather the differential recoil rate $\mathrm{d}R/\mathrm{d}E_R\mathrm{d}\Omega_q$ of Eq. \[eq:Rate\]. Analogously to the previous section, we define the directionally-integrated recoil spectrum:
$$\label{eq:discreteRate}
\frac{\mathrm{d}R^j}{\mathrm{d}E_R} = \int_{\phi = 0}^{2\pi} \int_{\cos(j\pi/N)}^{\cos((j-1)\pi/N)} \frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}\Omega_q} \, \mathrm{d}\cos\theta \, \mathrm{d}\phi\,.$$
The integrated recoil spectrum is then related to the IRT by $$\label{eq:discreteRate2}
\frac{\mathrm{d}R^j}{\mathrm{d}E_R} \propto \hat{f}^j({v_\mathrm{min}}(E_R)) F^2(E_R)\,,$$ where ${v_\mathrm{min}}(E_R)$ is defined in Eq. \[eq:vmin\]. The form factor $F^2(E_R)$ leads to a roughly exponential suppression of the rate (relative to the IRT) with increasing $E_R$. Because of this simple proportionality relationship between the IRT and the rate, we do not discuss the shape of the rates in detail.
Instead, we perform a more quantitative comparison between the full and discretised event rates. Assuming a particular idealised detector, we compare the expected number of events $N_j$ obtained within each bin using the exact and approximate IRTs presented in the previous section: [^2] $$\label{eq:Nevents}
N_j = \int_{E_\mathrm{min}}^{E_\mathrm{max}} \frac{\mathrm{d}R^j}{\mathrm{d}E_R} \, \mathrm{d}E_R \,.$$ In converting from ${v_\mathrm{min}}$ to recoil energy $E_R$, it is necessary to fix the values of the DM and nuclear masses. We therefore take as an example a Fluorine target and DM mass of $m_\chi = 50 \,\,\mathrm{ GeV}$. Direct detection experiments do not probe down to arbitrarily low energies and typically an energy threshold is set below which either the signal may be dominated by unrejected backgrounds or the direction of the recoil cannot be reconstructed. We consider a typical threshold energy of 20 keV [@Daw:2011].
We fix the normalisation of the event rate by requiring a total of 50 signal events across all bins. As previously noted, the anisotropy of a DM signal can be confirmed with around 10 signal events [@Morgan:2005], while the median recoil direction can be confirmed to match that expected due to the Earth’s motion with of order 30 events [@Green:2010b]. It is reasonable then to expect that the approach presented here might be employed once around 50 signal events are observed. In addition, we include a background of 1 event, distributed isotropically. We discuss the impact of a larger number of signal events in Sec. \[sec:discussion\].
The error between the exact and approximate results can then be compared with the typical Poissonian uncertainty on the expected number of events in each bin $\Delta N_j \sim \sqrt{N_j}$. If the discrepancy between the exact and approximate results is smaller than this Poisson error in the number of events, we would expect the two methods to be in agreement within statistical uncertainties. This means that the data should be approximately as well-fit by the discretised velocity distribution as by the full distribution. This in turn indicates that the discretised velocity distribution can be used, along with a suitable parametrisation, to fit and extract information about the underlying $f(\mathbf{v})$.
We consider two example distributions, which have already been illustrated in Fig. \[fig:Discrete\]. The first is the canonical Standard Halo Model (SHM), with parameters $v_{e} = 220 {\textrm{ km s}^{-1}}$ and $\sigma_v = 156 {\textrm{ km s}^{-1}}$. We use this not only because it is so often studied in the literature, but also because it has a relatively smooth, simple structure and is not too strongly peaked. If the method presented here is to be useful, it should give accurate results for such a simple underlying distribution. For simplicity we do not truncate the SHM at the Galactic escape speed. The inclusion of such a cut-off would reduce the high-speed DM population, reducing the directionality of the signal and therefore improving the results presented here. In the reconstruction of real data, an escape speed cut-off could be included easily in the chosen parametrisation for $f^k(v)$.
As a comparison, we also consider a stream distribution, modeled using Eq. \[eq:SHM\_Radon\], but with parameters $v_{e} = 500 {\textrm{ km s}^{-1}}$ and $\sigma_v = 20 {\textrm{ km s}^{-1}}$. This leads to a sharply peaked, strongly directional distribution function, which allows us to compare the SHM with a more extreme case. We note in particular that these parameters lead to a sharper distribution (and therefore a more directional rate) than values typically assumed, for example, for the Sagittarius stream [@Freese:2004; @Freese:2005; @Savage:2006]. This stream should therefore be considered a ‘worst-case’ scenario which is difficult to approximate accurately.
N=2 discretisation
------------------
Figure \[fig:Compare-N=2\] shows the comparison between the exact and approximate IRTs for the $N=2$ discretisation. The exact IRT is shown as a solid line, while the approximate IRT obtained from the discretisation is shown as a dashed line. We show the results for both the SHM (blue) and stream distributions (red). Values of ${v_\mathrm{min}}$ in the shaded region lie below the assumed 20 keV energy threshold for a Fluorine target and 50 GeV DM particle. In the case of the SHM, the shape of the Radon transform is well reconstructed in both the forward (left panel) and backward (right panel) directions, even with the crude $N=2$ discretisation. For the forward IRT ($j = 1$), the ratio between the exact and approximate results is less than $20\%$, while for the backward ($j=2$) case, the error is typically larger ($50-100\%$), though the absolute value is smaller.
For the stream distribution, the differences between the exact and approximate results are more significant. The results using the discretised velocity distribution underestimate the forward IRT at low ${v_\mathrm{min}}$, while overestimating the backward IRT for all values of ${v_\mathrm{min}}$. The reason for this can be seen by examining the velocity distributions illustrated in Fig. \[fig:Discrete\]. The full stream distribution (top right) is strongly focused in the forward ($\theta' = 0^\circ$) direction, meaning that particles must scatter through almost $90^\circ$ to produce a recoil in the backward direction. This is kinematically allowed only for a small fraction of the population. Due to the angular averaging, however, the $N=2$ velocity distribution (middle right) has a significant population of DM particles with velocities at right angles ($\theta' = 90^\circ$) to the forward direction. Thus, the discretised velocity distribution has a greater chance of producing scatters in the backward direction. Overall, the discretised distribution is less focused in the forward direction than the full distribution, resulting in a reduced asymmetry between the forward and backward scattering rates.
![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$, defined in Eq. \[eq:discreteRadon\], for $N=2$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The shaded grey region lies below the energy threshold of 20 keV for a DM mass of 50 GeV.[]{data-label="fig:Compare-N=2"}](Compare_N=2_1.pdf "fig:"){width="32.00000%"} ![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$, defined in Eq. \[eq:discreteRadon\], for $N=2$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. The shaded grey region lies below the energy threshold of 20 keV for a DM mass of 50 GeV.[]{data-label="fig:Compare-N=2"}](Compare_N=2_2.pdf "fig:"){width="32.00000%"}
Figure \[fig:Rate-N=2\] shows the comparison between the number of events $N_j$ in each of the angular bins, defined in Eq. \[eq:Nevents\], for the case of $N=2$ discretisation. The exact and approximate calculations are shown as solid and hatched bars respectively, while the SHM and stream distributions are shown separately in the left and right panels. In addition, the ‘exact’ event numbers in each bin are assigned an error bar given by the Poissonian standard deviation $\sqrt{N_j}$.
In the case of the SHM (left), the number of events in the forward direction ($j=1$) is roughly in agreement (within Poisson uncertainties) when calculated using the exact and approximate IRTs. However, in the backward direction ($j=2$), the approximate calculation overestimates the number of events by a factor of 3. As shown in the left hand panel of Fig. \[fig:Compare-N=2\], in the forward direction, the greatest discrepancy between the IRTs occurs at low ${v_\mathrm{min}}$. When we consider events only above the energy threshold (i.e. above the shaded grey region), this discrepancy is therefore minimal. In the right hand panel of Fig. \[fig:Compare-N=2\], however, the fractional discrepancy grows as a function of ${v_\mathrm{min}}$, and is greatest above the energy threshold of the experiment. This is due to the enhancement in DM particles travelling in the backward direction $\theta' = 180^\circ$ when using the discretised velocity distribution (see Fig. \[fig:Discrete\]) and leads to the large discrepancy in the number of backward-going events.
This problem is even greater for the stream distribution (right), for which neither the forward nor backward event numbers agree within the statistical errors. In the backwards direction, the exact calculation predicts much less than 1 event, while the approximate calculation overestimates this by a factor of $\sim 30$. For such a highly directional distribution, such a small number of bins is clearly not suitable.
We note briefly that in the case of $N=2$ discretisation, the approximate calculation of event numbers gives results which are almost indistinguishable for the SHM and stream distributions. This is because the discretised forms of the SHM and stream distributions differ most substantially at low ${v_\mathrm{min}}$. This can be seen in the middle row of Fig. \[fig:Discrete\]; eliminating those particles which are below the threshold speed of $\sim 300 {\textrm{ km s}^{-1}}$ gives two similar distributions. Given that there is poor agreement between the number of expected events for the $N=2$ discretisation, and that the discretised SHM and stream distributions are largely indistinguishable, it is clear that such a coarse angular approximation is not suitable for fitting to data.
![Comparison of the number of events in each angular bin (Eq. \[eq:Nevents\]) for N=2 bins, for a Fluorine target with 20 keV threshold and $m_\chi = 50 \,\, \mathrm{GeV}$. Event numbers are calculated using the exact (solid bars) and approximate (hatched bars) IRTs. The SHM distribution is shown in the left panel, while the stream distribution is shown on the right. A total of 50 signal events are assumed (as well as 1 isotropically distributed background event). The error bars on the ‘exact’ number of events are the Poissonian errors $\Delta N_j \sim \sqrt{N_j}$.[]{data-label="fig:Rate-N=2"}](Numbers_SHM_N=2.pdf "fig:"){width="45.00000%"} ![Comparison of the number of events in each angular bin (Eq. \[eq:Nevents\]) for N=2 bins, for a Fluorine target with 20 keV threshold and $m_\chi = 50 \,\, \mathrm{GeV}$. Event numbers are calculated using the exact (solid bars) and approximate (hatched bars) IRTs. The SHM distribution is shown in the left panel, while the stream distribution is shown on the right. A total of 50 signal events are assumed (as well as 1 isotropically distributed background event). The error bars on the ‘exact’ number of events are the Poissonian errors $\Delta N_j \sim \sqrt{N_j}$.[]{data-label="fig:Rate-N=2"}](Numbers_stream_N=2.pdf "fig:"){width="45.00000%"}
N=3 discretisation
------------------
Figure \[fig:Compare-N=3\] compares the approximate and exact IRTs for the $N=3$ discretisation illustrated on the bottom row of Fig. \[fig:Discrete\]. In the case of the SHM, the agreement between the two functions is improved compared to the $N=2$ case. The slope of the event rate is degenerate with the DM mass $m_\chi$, so any error in the shape of the IRTs could translate to a bias in the reconstructed value of $m_\chi$. In the forward direction (left panel), the fractional error is reduced to at most $10\%$ and the shape of the IRT appears to be agree closely in all three directional bins. This suggests that any such bias should be minimal.
![As Fig. \[fig:Compare-N=2\], for N = 3 angular bins. In the right panel, the exact Radon transform for the stream is indistinguishable from zero.[]{data-label="fig:Compare-N=3"}](Compare_N=3_1.pdf "fig:"){width="32.00000%"} ![As Fig. \[fig:Compare-N=2\], for N = 3 angular bins. In the right panel, the exact Radon transform for the stream is indistinguishable from zero.[]{data-label="fig:Compare-N=3"}](Compare_N=3_2.pdf "fig:"){width="32.00000%"} ![As Fig. \[fig:Compare-N=2\], for N = 3 angular bins. In the right panel, the exact Radon transform for the stream is indistinguishable from zero.[]{data-label="fig:Compare-N=3"}](Compare_N=3_3.pdf "fig:"){width="32.00000%"}
However, there is still a significant difference between the shapes of the exact and approximate IRTs for the stream distribution. For example, in the forward direction (left panel), the exact calculation predicts scattering events occurring only with a narrow range of ${v_\mathrm{min}}$, from $250 {\textrm{ km s}^{-1}}$ to $500 {\textrm{ km s}^{-1}}$. This is because almost all particles are moving in the forward direction with speed $v = 500 {\textrm{ km s}^{-1}}$. The kinematics of the scattering requires that $\mathbf{v}\cdot\hat{\mathbf{q}} = {v_\mathrm{min}}$, meaning that scattering in the forward direction ($\theta = 0^\circ$) is only allowed for ${v_\mathrm{min}}\approx 500 {\textrm{ km s}^{-1}}$. Scattering away from the forward direction, but still included in the first angular bin ($\theta < 60^\circ$), is kinematically allowed for ${v_\mathrm{min}}> \cos(60^\circ) \times 500 {\textrm{ km s}^{-1}}\approx 250 {\textrm{ km s}^{-1}}$. For the discretised velocity distribution, there is a population of DM particles initially directed away from the forward direction ($\theta \neq 0^\circ$) which can scatter through larger angles and still produce scattering events within the first angular bin. These large-angle scattering events mean that values of ${v_\mathrm{min}}$ down to zero are now kinematically allowed.
Considering the $j=3$ angular bin (right panel), we note that the exact stream distribution predicts no scattering events in the backward direction. This is because the particles would have to scatter through an angle much larger than $90^\circ$ to produce events in the $j=3$ bin, which is not kinematically allowed. However, for the discretised distribution (bottom right panel of Fig. \[fig:Discrete\]), it is still kinematically possible for particles on the edge of the forward bin ($\theta' \approx 60^\circ$) to scatter into the backward direction ($\theta > 120^\circ$). Thus, the approximate calculation still predicts a small but non-zero IRT for the $j=3$ case.
![As Fig. \[fig:Rate-N=2\], for N = 3 angular bins.[]{data-label="fig:Rate-N=3"}](Numbers_SHM_N=3.pdf "fig:"){width="45.00000%"} ![As Fig. \[fig:Rate-N=2\], for N = 3 angular bins.[]{data-label="fig:Rate-N=3"}](Numbers_stream_N=3.pdf "fig:"){width="45.00000%"}
Figure \[fig:Rate-N=3\] shows the event numbers expected in each of the $N = 3$ angular bins calculated using the exact and approximate IRTs. For the SHM (left panel), in the forward direction ($j=1$), the approximate calculation underestimates the rate by around 13%, while in the transverse direction ($j=2$), the approximate calculation overestimates the rate by around 30%. Finally, in the backwards direction ($j=3$) the rate is overestimated by a factor of 2. However, these discrepancies are roughly the same magnitude as the typical Poisson error in each bin for a sample of 50 events. In contrast to the $N=2$ case then, the $N=3$ case can provide a reasonable approximation to the event rate for the SHM.
For the stream distribution (right panel), there are significant discrepancies, which remain larger than the typical statistical error. In particular, there is a substantial leakage of events from the forward direction ($j=1$) into the transverse direction ($j=2$) in the approximate calculation. This can be seen in the middle panel of Fig. \[fig:Compare-N=3\], in which the exact IRT is zero above the experimental threshold, while the approximate IRT is substantially non-zero. This in turn arises due to the ‘smearing’ of the distribution away from the forward direction, as previously discussed. In contrast, in the backward direction ($j=3$), the approximate calculation now agrees with the exact calculation, giving zero events. As shown in the right panel of Fig. \[fig:Rate-N=3\], the discrepancy between the exact and approximate IRTs arises only at low ${v_\mathrm{min}}$ (below the energy threshold of the experiment), where scattering into the backwards bin is kinematically allowed.
### The folded rate
The $N=3$ case is important for the scenario where head-tail discrimination is not possible when reconstructing recoil tracks. Head-tail discrimination of tracks has previously been demonstrated [@Burgos:2008], but may not be possible with 100% accuracy in near-future detectors [@Billard:2012]. Within the formalism considered here, detectors without sense discrimination cannot distinguish between a recoil with direction $\cos\theta$ and another with direction $-\cos\theta$. They therefore probe the so-called ‘folded’ recoil spectrum:
$$\label{eq:folded}
\frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}|\cos\theta|} = \frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}\cos\theta} + \frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}(-\cos\theta)}\,.$$
For the $N=2$ case, the folded spectrum is entirely isotropic, as the forward and backward IRTs differ only in the sign of $\cos\theta$. However, in the $N=3$ case, the transverse IRT, given by
$$\label{eq:transverse}
\hat{f}^T({v_\mathrm{min}}) = \hat{f}^2({v_\mathrm{min}}) = \int_{\phi = 0}^{2\pi} \int_{-1/2}^{1/2} \hat{f}({v_\mathrm{min}}, \cos\theta) \,\mathrm{d}\cos\theta \, \mathrm{d}\phi\,,$$
is invariant under $\hat{f}({v_\mathrm{min}}, \cos\theta) \rightarrow \hat{f}({v_\mathrm{min}}, -\cos\theta)$. That is, the transverse event rate ‘folds’ back onto itself. Thus, even without sense discrimination, directional experiments will still be sensitive to this transverse scattering rate. By comparison, if the forward and backward directions cannot be distinguished, the remaining two IRTs (the left and right panels in Fig. \[fig:Compare-N=3\]) are folded together, to obtain the longitudinal Radon transform $$\label{eq:longitudinal}
\hat{f}^L({v_\mathrm{min}}) = \int_{\phi = 0}^{2\pi}\int_{-1}^{-1/2} \hat{f}({v_\mathrm{min}}, \cos\theta, \phi) \,\mathrm{d}\cos\theta \, \mathrm{d}\phi + \int_{\phi = 0}^{2\pi}\int_{1/2}^{1} \hat{f}({v_\mathrm{min}}, \cos\theta) \,\mathrm{d}\cos\theta \, \mathrm{d}\phi\,.$$ Analogously, we can define the longitudinal and transverse event spectra, [$\textrm{d}R^{L,T}/\textrm{d}E_R$]{}, and event numbers $N^{L,T}$, derived from the corresponding Radon transforms.
Figure \[fig:Rate-N=3\_folded\] shows the event numbers for this folded rate in the longitudinal and transverse bins. For the SHM (left panel), there is a slight improvement in the agreement between the exact and approximate results compared with Fig. \[fig:Rate-N=3\]. The backward bin, which was previously overestimated by a factor of 2, has been removed and merged with the forward bin to form the longitudinal bin. Because the approximate result previously underestimated the true number of events in the forward bin, this combination of the bins leads to marginally improved agreement, though the effect is small in this example.
In the case of the stream (right panel), the number of events in each bin appears almost indistinguishable from those of Fig. \[fig:Rate-N=3\]. This is because the leakage of the events into the backwards direction when using the approximate distribution is minimal and therefore the effect of folding is also minimal. However, we have demonstrated that this method can still be employed when sense discrimination of recoils is not possible and, in general, it should lead to a reduction in the associated discretisation error.
![As Fig. \[fig:Rate-N=2\], for the folded event rate with $N=3$ underlying bins, producing one longitudinal and one transverse directional bin (see Eq. \[eq:folded\] and associated text).[]{data-label="fig:Rate-N=3_folded"}](Numbers_SHM_N=3_folded.pdf "fig:"){width="45.00000%"} ![As Fig. \[fig:Rate-N=2\], for the folded event rate with $N=3$ underlying bins, producing one longitudinal and one transverse directional bin (see Eq. \[eq:folded\] and associated text).[]{data-label="fig:Rate-N=3_folded"}](Numbers_stream_N=3_folded.pdf "fig:"){width="45.00000%"}
N=5 discretisation
------------------
We now consider a more finely discretised velocity distribution, namely $N=5$. The comparison of the exact and approximate IRTs is shown in Fig. \[fig:Compare-N=5\]. The results for the SHM, showing typical discrepancies at the 5% level, indicate that the approximate Radon transform does in fact tend to the true Radon transform in the limit of large $N$. The results for the stream again show much poorer agreement. We expect that the agreement will not improve significantly until the angular size of each bin is close to the angular extent of the underlying distribution function. For the stream, we can see by eye in Fig. \[fig:Discrete\] that most of the distribution is distributed over an angle $\theta' \lesssim 10^\circ$, meaning that roughly $N=18$ bins are required to prevent the smearing of the distribution function visible for $N=2$ and $N=3$. In spite of this, some structures (such as the peak in the forward direction of the upper left panel) are better reconstructed than in the $N=3$ case, indicating that additional information is still gained by adding more bins. In principle, there is no obstacle to increasing the number of bins up to (and beyond) $N=18$, as the formalism of Appendix \[app:Radon\] is applicable for arbitrary $N$. However, the appropriate number of bins would typically be dictated by the amount of data available, with a large amount of data required to justify fitting the number of parameters associated with $N=18$ bins.
We note that for the bin focused in the backward direction, $j=5$ (bottom right), the exact and approximate IRTs for the stream are indistinguishable and very close to zero. As in the $N=3$ case, scattering in the backward direction is not possible for the full stream distribution because of the strong directionality. For the discretised distribution, most of the distribution is focused in the forward-directed bin ($k=1$ in Eq. \[eq:averagef\]). In contrast to the $N=3$ result, however, this is sufficiently directional that scattering into the backward direction (into the $j=5$ bin) is no longer kinematically allowed, resulting in significantly closer agreement in the backward direction.
![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$ defined in Eq. \[eq:discreteRadon\], for $N=5$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. In the lower left panel, the exact IRT for the stream is indistinguishable from zero, while in the lower right panel, both the approximate and exact IRTs are indistinguishable from zero.[]{data-label="fig:Compare-N=5"}](Compare_N=5_1.pdf "fig:"){width="32.00000%"} ![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$ defined in Eq. \[eq:discreteRadon\], for $N=5$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. In the lower left panel, the exact IRT for the stream is indistinguishable from zero, while in the lower right panel, both the approximate and exact IRTs are indistinguishable from zero.[]{data-label="fig:Compare-N=5"}](Compare_N=5_2.pdf "fig:"){width="32.00000%"} ![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$ defined in Eq. \[eq:discreteRadon\], for $N=5$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. In the lower left panel, the exact IRT for the stream is indistinguishable from zero, while in the lower right panel, both the approximate and exact IRTs are indistinguishable from zero.[]{data-label="fig:Compare-N=5"}](Compare_N=5_3.pdf "fig:"){width="32.00000%"} ![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$ defined in Eq. \[eq:discreteRadon\], for $N=5$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. In the lower left panel, the exact IRT for the stream is indistinguishable from zero, while in the lower right panel, both the approximate and exact IRTs are indistinguishable from zero.[]{data-label="fig:Compare-N=5"}](Compare_N=5_4.pdf "fig:"){width="32.00000%"} ![Exact (solid) and approximate (dashed) integrated Radon transforms, $\hat{f}^j$ defined in Eq. \[eq:discreteRadon\], for $N=5$. Results are shown for the SHM (blue) and stream (red) distribution functions. The approximate Radon transforms are obtained by discretising the full velocity distribution into $N$ angular bins. The vector $\textbf{v}_\textrm{e}$ is aligned along $\theta' = 0$. In the lower left panel, the exact IRT for the stream is indistinguishable from zero, while in the lower right panel, both the approximate and exact IRTs are indistinguishable from zero.[]{data-label="fig:Compare-N=5"}](Compare_N=5_5.pdf "fig:"){width="32.00000%"}
In fig. \[fig:Rate-N=5\], we show the event numbers in each bin for the $N=5$ discretisation. For the SHM (left panel), there is now close agreement between the exact and approximate results, with less than 10% discrepancy for the forward bins $j=1,2$ and an error between 20% and 40% in the transverse and backward bins $j=3,4,5$. For all bins, the agreement between the exact and approximate results is significantly smaller than the Poisson uncertainty for the example of 50 signal events. This closely matches the expectation from Fig. \[fig:Compare-N=5\], which shows close agreement between the exact and approximate IRTs.
For the stream (right panel), there is an improved fit between the exact and approximate results, with bins $j=1,2,4,5$ showing agreement within the statistical uncertainty. Leakage of events between the two forward bins leads to an slight overestimation of the $j=2$ rate. However, the biggest problem remains leakage of events into the transverse bin $j=3$ which gives a significant overestimation of the number of events when using the approximate IRT. Once again, this is seen easily in the upper right panel of Fig. \[fig:Rate-N=5\], where the approximate IRT is non-zero above the energy threshold of the experiment.
We note that while there remains a notable discretisation error for the stream distribution, it is now possible to distinguish the results for the SHM and stream distribution when using the approximate IRTs, in contrast to $N=2$ discretisation. For example, in the $j=3$ bin, the approximate calculation predicts around $\sim8$ events for the SHM, compared to $\sim 2$ events for the stream. This difference is larger than the error induced by the discretisation (a discrepancy of $\sim 2$ events for both the SHM and stream). This indicates that while the $N=5$ discretisation may not be able to accurately reproduce the event rate for a stream distribution, it may still be useful for distinguishing between different distributions. For example, if the $N=5$ discretisation were used to fit to data and the results indicated a smaller number of events than expected in the $j=3$ bin, this could point to deviations from the SHM.
![As Fig. \[fig:Rate-N=2\], for N=5 angular bins.[]{data-label="fig:Rate-N=5"}](Numbers_SHM_N=5.pdf "fig:"){width="45.00000%"} ![As Fig. \[fig:Rate-N=2\], for N=5 angular bins.[]{data-label="fig:Rate-N=5"}](Numbers_stream_N=5.pdf "fig:"){width="45.00000%"}
Forward-backward asymmetry
--------------------------
Finally, we calculate the forward-backward asymmetry in the number of events, $A_\mathrm{FB}$, and compare the exact and approximate results. In terms of the number of forward and backward events between the threshold and maximum energies $E_\mathrm{th}$ and $E_\mathrm{max}$,
$$\begin{aligned}
N_\mathrm{F} &= \int_{E_{\mathrm{th}}}^{E_\mathrm{max}} \int_{\phi = 0}^{2\pi} \int_{0}^{1} \frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}\Omega_q} \, \mathrm{d}\cos\theta\, \mathrm{d}\phi \, \mathrm{d}E_R \\
N_\mathrm{B} &= \int_{E_{\mathrm{th}}}^{E_\mathrm{max}} \int_{\phi = 0}^{2\pi} \int_{-1}^{0} \frac{\mathrm{d}R}{\mathrm{d}E_R\mathrm{d}\Omega_q} \, \mathrm{d}\cos\theta\, \mathrm{d}\phi \, \mathrm{d}E_R\,,\end{aligned}$$
the asymmetry is given by
$$\label{eq:AFB}
A_\mathrm{FB} = \frac{N_\mathrm{F} - N_\mathrm{B}}{N_\mathrm{F} + N_\mathrm{B}}\,.$$
By comparing the values of $A_\mathrm{FB}$ obtained from the full and discretised distributions, we can obtain a measure of how well the discretised distribution captures the directionality of the signal, as well as showing how this improves with increasing $N$. In addition, this forward-backward asymmetry would potentially be the first directional signal measured with directional detectors and so determining how well it can be recovered is of substantial importance. Using the current formalism, we can consider only even values of $N$, for which each angular bin lies entirely in either the forward or backward direction. In principle, the calculation can also be extended to include odd values of $N$ using the framework of Appendix \[app:RadonDeriv\].
Figure \[fig:RateComparison\] shows the forward-backward asymmetry obtained from the discretised velocity distribution for $N=2,4,6,8,10$ (filled squares), compared to the true forward-backward asymmetry, obtained from the full velocity distribution (solid lines). In the case of the stream (red), the event rate is strongly asymmetric ($A_{\mathrm{FB}} \approx 1$) as the velocity distribution is highly focused in the forward direction. By comparison, the SHM (blue) has a smaller asymmetry ($A_{\mathrm{FB}} \approx 0.9$) due to its wider velocity dispersion. Dotted lines show the $1\sigma$ statistical uncertainties on the measured value of $A_\mathrm{FB}$ assuming 50 signal events.
![Forward-backward asymmetry of the event rate $A_{\mathrm{FB}}$, defined in Eq. \[eq:AFB\], using the full velocity distribution (solid lines) and the discretised velocity distribution (filled squares). Results are shown for the SHM and stream distributions in blue and red respectively. Dotted lines show the $1\sigma$ statistical uncertainties on the measured value of $A_\mathrm{FB}$ assuming 50 signal events.[]{data-label="fig:RateComparison"}](RateComparison.pdf){width="50.00000%"}
For both the SHM and stream, the asymmetry is significantly underestimated when using only the simple forward-backward ($N=2$) discretisation. This matches the analysis of the previous sections, which demonstrated that the discretised velocity distribution tends to lead to a leakage of events from the forward to the backward direction. However, for both distributions, the asymmetry obtained from the discretised distribution rapidly approaches the true value with increasing $N$. For the stream distribution, the approximation converges more rapidly than for the SHM. This is because of the strong directionality of the stream velocity distribution, so only relatively few bins are required to capture a simple forward-backward asymmetry. Even with only 4 angular bins, the error in $A_\mathrm{FB}$ for both velocity distributions is less than 10% and lies within the $1\sigma$ statistical uncertainty expected from a sample of 50 signal events.
Discussion {#sec:discussion}
==========
We have demonstrated that for smooth SHM-like distributions, the discretised velocity distribution allows us to obtain a good approximation to the integrated Radon transform (IRT) with relatively few angular bins. Though the $N=2$ discretisation is too simple an approximation, for 3 or more bins the discretised distribution leads to a number of events in each bin which matches the true number within statistical uncertainties (assuming 50 signal events). This means that we should be able to parametrise the speed distributions in each angular bin $f^k(v)$ and account for astrophysical uncertainties without introducing a large error in the number of observed events. For comparison, we have also considered a more extreme and highly-directional stream distribution. The disagreement between the true recoil spectrum and that obtained from the discretised velocity distribution is significantly larger in this case. Though increasing the number of bins $N$ improves the agreement, a rather large number of bins ($N \sim 18$) may be needed to accurately describe the stream distribution.
The results we have presented here have assumed an idealised future directional detector. However, application of these results is not necessarily as straightforward in a realistic experiment. Conversion from ${v_\mathrm{min}}$ to $E_R$ requires knowledge of the DM mass $m_\chi$ and attempting to fit both the velocity distribution and mass simultaneously can lead to spurious results [@Peter:2011; @Kavanagh:2012]. Furthermore, for comparison with experimental data, we must take into account the fact that experiments have finite angular resolution, typically in the range 20$\,^{\circ}$-80$\,^{\circ}$, with higher resolution at high energy [@Billard:2012]. Finally, throughout this work, we have assumed that the basis for discretising the velocity distribution is aligned with the peak of the underlying velocity distribution (i.e. $\mathbf{v}_e$ aligned along $\theta' = 0$). However, this is not known *a priori* and a reliable method of selecting the orientation of the angular basis is required.
In spite of these open issues, the work presented here is general and conservative. We have demonstrated that the stream distribution is more poorly fit than the SHM distribution with few angular bins. However, the stream is an extreme example, as discussed in Sec. \[sec:compare\], and it is unlikely the discretisation error will be any greater than observed in that case. Other possibilities for the velocity distribution would tend to be less directional than the stream example, leading to better agreement between the exact and approximate rates. For a dark disk, the low value of $v_e$ means that the velocity distribution appears almost isotropic in the lab frame, which will only reduce the angular discretisation error. A misalignment between $\mathbf{v}_e$ and $\theta' = 0^\circ$ would also lead to a distribution more isotropic in $\theta'$. Similarly, including finite angular resolution will smear the velocity distribution (in a style similar to the discretisation), reducing the directional asymmetry and therefore the discretisation error. We have also demonstrated that this method is applicable in detectors where sense recognition is not possible.
For a realistic analysis, all of these issues should be taken into account. Based on mock (or future) data, the optimal direction for $\theta' = 0^\circ$ should be selected (potentially based on the observed median recoil direction [@Morgan:2005; @Green:2010b]). An appropriate number of bins $N$ should then be chosen, which may be influenced by the angular resolution of the experiment and the number of observed events. In some scenarios, such as for strongly peaked signals, deviations from the smooth SHM may be obvious from the data. In this case, a definition of the angular bins different from that used here (including differently-sized bins) may be optimal to reduce the discretisation error. Using the results of Appendix \[app:RadonDeriv\], such a scenario can be straightforwardly accommodated, although we leave the issue of choosing the number and size of the bins to future studies.
In order to make more quantitative statements, we have focused on a specific scenario, in which 50 signal events (and 1 background event) are observed. This has allowed us to compare the typical Poisson uncertainties on the number of observed events with the error induced by using the discretised velocity distribution. Though a signal consisting of 50 events is somewhat arbitrary, it is representative of the lower limit on the number of events required before such an analysis would be reasonable. As the number of signal events increases, the typical statistical uncertainty decreases. Eventually, this uncertainty becomes smaller than the error between the approximate and exact results. Increasing the number of angular bins would then reduce the discretisation error and increase the statistical uncertainty on the number of events in each bin, reducing any bias which may be induced by the discretisation. This therefore provides a natural scheme for deciding how many angular bins are required, based on the number of signal events observed.
Once the number of bins $N$ has been fixed and a suitable parametrisation for the radial functions $f^k(v)$ has been chosen, the full parameter space (of both DM particle physics parameters and astrophysics parameters) should be fit to the data. We note in particular that by summing the radial functions $f^k(v)$, the 1-dimensional speed distribution is recovered, meaning that non-directional experiments can easily be included in this framework. As emphasised in this section, the fitting process will be highly dependent on the underlying DM and experimental parameters, as well as on the chosen parametrisation for $f^k(v)$. We have therefore focused here on the discretisation of $f(\mathbf{v})$ and left more involved investigations for future work.
Conclusions {#sec:conclusions}
===========
In this work, we have presented an angular basis which can be used to parametrise the DM velocity distribution. This involves discretising the velocity distribution into $N$ angular bins: $$f(\textbf{v}) = f(v, \cos\theta', \phi') =
\begin{cases}
f^1(v) & \textrm{ for } \theta' \in \left[ 0, \pi/N\right]\,, \\
f^2(v) & \textrm{ for } \theta' \in \left[ \pi/N, 2\pi/N\right]\,, \\
& \vdots\\
f^k(v) & \textrm{ for } \theta' \in \left[ (k-1)\pi/N, k\pi/N\right]\,, \\
& \vdots\\
f^N(v) & \textrm{ for } \theta' \in \left[ (N-1)\pi/N, \pi\right]\,. \\
\end{cases}$$ In Appendices \[app:RadonDeriv\] and \[app:Radon\], we have provided recipes for calculating the corresponding directionally-integrated Radon transforms (IRTs), which appear in the directional event rate. Alternative methods for parametrising the DM velocity distribution (such as Refs. [@Alves:2012; @Lee:2014], and in particular those based on spherical harmonics) may lead to negative values $f(\mathbf{v})$, leading to unphysical distribution functions. It is not clear what affect this might have on parameter reconstruction or how this problem can be mitigated. The advantage of the basis presented here is that it guarantees that the resulting distribution function is everywhere positive and therefore physical.
We have investigated the possible size of discretisation errors in the directional event rate by comparing the IRT in each bin obtained from the discretised and full velocity distributions. The simplest possible discretisation ($N=2$) is unsuitable for fitting to data, as it substantially underestimates the forward scattering rate, while underestimating the backwards scattering rate. However, starting from $N=3$, the standardly-assumed SHM can be well fit by the discretised velocity distribution. We have shown that once the DM nature of the signal is confirmed (with around 50 events), the $N=3$ discretisation allows us to calculate the number of events in each angular bin and obtain good agreement with the true event numbers (within statistical uncertainties). Increasing the number of bins to $N=5$ improves this agreement, giving discretisation errors in the range 10-50% depending which angular bin is considered.
For comparison, we also consider an extreme, highly directional stream distribution. In this case, the discretisation error is much larger and exceeds the statistical uncertainties in the $N=5$ case. An estimated $N=18$ angular bins would be required to accurately map the full velocity distribution. However, increasing $N$ does reduce the discrepancy between the exact and approximate results. In addition, with as few as $N=5$ bins, it is possible to distinguish between the SHM and stream distribution. This suggests that even though the discretised distribution gives a poor approximation to the stream, it may be useful in distinguishing different distributions, even for relatively few bins.
Finally, we have considered the forward-backward asymmetry $A_{\mathrm{FB}}$ in the event rate, calculated using both the full velocity distribution and the discretised velocity distribution. The latter calculation rapidly converges to the correct value with increasing $N$. For both the SHM and stream distributions, the error in $A_\mathrm{FB}$ is below $10\%$ for $N \geq 4$ bins, within the statistical error expected for 50 signal events. We have also demonstrated that the method presented here can still be used when head-tail discrimination of the recoil tracks is not possible, in which case $N \geq 3$ bins are required.
In this work, we have only considered discretising the angular component of the DM velocity distribution, leaving the $N$ radial functions $f^{k}(v)$ fixed to their ‘true’ values. In future work, it will be necessary to combine the discretisation presented here with a parametrisation of these radial functions (such as the parametrisations of Refs. [@Peter:2011; @Kavanagh:2012; @Kavanagh:2014]), in order to reconstruct the velocity distribution (and other DM parameters) from mock data. In addition, several questions are still to be addressed, including how to decide on the optimal ‘forward’ direction for the discretisation, and how realistic angular resolution would impact the results presented here. However, these additional considerations are only expected to improve the agreement between the true and approximate event rates. This work represents an initial step towards a parametrisation of the full DM velocity distribution, which would allow future data from directional experiments to be analysed without astrophysical uncertainties and which would potentially allow the velocity distribution itself to be probed and reconstructed.
The author thanks Anne M. Green, Julien Billard and Samuel K. Lee for helpful discussions on the subject of directional detection. The author also thanks Anne M. Green and Ciaran A. J. O’Hare for critical comments on this manuscript. The author is supported by the European Research Council (ERC) under the EU Seventh Framework Programme (FP7/2007- 2013)/Erc Starting Grant (agreement n. 278234 – ’NewDark’ project).
Calculating the Radon transform {#app:RadonDeriv}
===============================
In this appendix, we derive the Radon transform corresponding to the discretised velocity distribution. The final result can be found in Appendix \[app:Radon\]. The Radon transform of $f(\textbf{v})$ is defined by:
$$\hat{f}({v_\mathrm{min}}, {\hat{\mathbf{q}}}) = \int_{\mathbb{R}^3} f(\mathbf{v}) \, \delta\left(\mathbf{v}\cdot{\hat{\mathbf{q}}}- {v_\mathrm{min}}\right) \, \mathrm{d}^3\mathbf{v}\,.$$
We write the coordinates of the velocity and recoil momentum as
$$\begin{aligned}
\begin{split}
\mathbf{v} &= v\left(\sin\theta'\cos\phi', \sin\theta'\sin\phi', \cos\theta'\right) \\
{\hat{\mathbf{q}}}&= \left(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta\right) \,,
\end{split}\end{aligned}$$
and consider here only the azimuthally integrated Radon transform. We write this as $\hat{f}({v_\mathrm{min}}, \cos\theta)$, which is given by
$$\begin{aligned}
\begin{split}
\hat{f}({v_\mathrm{min}}, \cos\theta) &= \int_{0}^{2\pi} \hat{f}({v_\mathrm{min}}, \cos\theta,\phi) \, \mathrm{d}\phi \\
&= \int_{0}^{2\pi} \left( \int_{\mathbb{R}^3} f(\mathbf{v}) \, \delta\left(\mathbf{v}\cdot{\hat{\mathbf{q}}}- {v_\mathrm{min}}\right) \, \mathrm{d}^3\mathbf{v}\right) \, \mathrm{d}\phi \\
& = \int_{\mathbb{R}^3} f(\mathbf{v}) \left(\int_{0}^{2\pi} \, \delta\left(\mathbf{v}\cdot{\hat{\mathbf{q}}}- {v_\mathrm{min}}\right) \, \mathrm{d}\phi \right) \, \mathrm{d}^3\mathbf{v}\\
&\equiv \int_{\mathbb{R}^3} f(\mathbf{v}) \, D\left({v_\mathrm{min}}, \cos\theta, \mathbf{v} \right) \, \mathrm{d}^3\mathbf{v}\,.
\end{split}\end{aligned}$$
We expand the $\delta-$function explicitly in terms of the angular coordinates:
$$\delta\left(\mathbf{v}\cdot{\hat{\mathbf{q}}}- {v_\mathrm{min}}\right) = \frac{1}{v}\delta\left(\sin\theta\sin\theta'\cos(\phi - \phi') + \cos\theta\cos\theta' - {v_\mathrm{min}}/v \right) \equiv \frac{1}{v}\delta\left(g(\phi)\right)\,.$$
We rewrite the argument of the $\delta-$function as a function $\phi$: $$\label{eq:deltadecomp}
\delta\left( g(\phi) \right) = \sum_{i} \frac{\delta(\phi - \phi_i)}{\left| g'(\phi_i) \right|}\,.$$ Here, we sum over those values of $\phi_i$ satisfying $g(\phi_i) = 0$:
$$\label{eq:gamma}
\cos(\phi_i - \phi') = \frac{\beta - \cos\theta\cos\theta'}{\sin\theta\sin\theta'} \equiv \alpha\,,$$
where we have also defined $\beta = {v_\mathrm{min}}/v$. The solutions for $\phi \in [0, 2\pi]$ are:
$$\begin{aligned}
\phi_1 &= \phi' + {\cos^{-1}}\alpha, & \textrm{ for } & \phi' \in \left[0, 2\pi - {\cos^{-1}}\alpha\right] \notag\\
\phi_2 &= \phi' + 2\pi - {\cos^{-1}}\alpha, & \textrm{ for } & \phi' \in \left[0, {\cos^{-1}}\alpha\right] \notag\\
\phi_3 &= \phi' + {\cos^{-1}}\alpha -2\pi, & \textrm{ for } & \phi' \in \left[2\pi - {\cos^{-1}}\alpha, 2\pi\right] \notag\\
\phi_4 &= \phi' -{\cos^{-1}}\alpha, & \textrm{ for } & \phi' \in \left[{\cos^{-1}}\alpha, 2\pi\right] \,.\end{aligned}$$
We note that these solutions exist only for $\beta \in \left[0,1\right]$ (or equivalently $v > {v_\mathrm{min}}$) and for $\alpha \in [-1,1]$, otherwise Eq. \[eq:gamma\] cannot be satisfied. If these constraints are satisfied, there exist exactly 2 solutions for a given value of $\phi'$ and therefore 2 $\delta$-functions in Eq. \[eq:deltadecomp\].
For the derivative of $g(\phi)$ we obtain $$\begin{aligned}
g'(\phi) = -\sin\theta\sin\theta'\sin(\phi-\phi')\,.\end{aligned}$$ Substituting the values of $\phi_{1,2,3,4}$, we see that $$\begin{aligned}
|g'(\phi_{1,2,3,4})| = \sqrt{\left(\sin\theta\sin\theta'\right)^2 - \left(\beta - \cos\theta\cos\theta'\right)^2}\,.\end{aligned}$$ Each of the two $\delta$-functions therefore contributes the same amount to the integral, regardless of the value of $\phi'$. We can now perform the integral over $\phi$:
$$\begin{aligned}
\begin{split}
D\left({v_\mathrm{min}}, \cos\theta, \mathbf{v} \right) &= \frac{1}{v} \int_{0}^{2\pi} \, \delta\left(g(\phi)\right) \, \mathrm{d}\phi \\
&= \frac{2 C(\alpha)}{v\sqrt{\left(\sin\theta\sin\theta'\right)^2 - \left(\beta - \cos\theta\cos\theta'\right)^2}}\Theta(v - {v_\mathrm{min}}) \\
&\equiv \frac{2}{v}C(\alpha) I(\beta, \cos\theta, \cos\theta') \Theta(v - {v_\mathrm{min}})
\end{split}\end{aligned}$$
where $C(\alpha) = 1$ for $\alpha \in [-1,1]$ and vanishes otherwise. We note that $D$ is independent of the azimuthal angle $\phi'$. The constraint $\alpha \in [-1,1]$ is satisfied for $\cos\theta' \in [x_-, x_+]$, where
$$x_\pm= \beta \cos\theta \pm \sqrt{1-\beta^2}\sin\theta \,.$$
We therefore obtain $$\hat{f}({v_\mathrm{min}}, \cos\theta) = 2 \int_{x_-}^{x_+} \int_{{v_\mathrm{min}}}^{\infty} f(v, \cos\theta') I(\beta, \cos\theta, \cos\theta')\, v \,\mathrm{d}v\, \mathrm{d}\cos\theta'\,,$$ where we have performed the $\phi'$ integral over the velocity distribution, and defined $$f(v, \cos\theta') = \int_{0}^{2\pi} f(v, \cos\theta', \phi') \, \mathrm{d}\phi'\,,$$ because $I$ does not depend on $\phi'$. We note from this that the azimuthally-integrated Radon transform is unaffected by the exact $\phi'$-dependence of $f(\mathbf{v})$. Instead, the azimuthally-integrated Radon transform depends only on the azimuthally-integrated velocity distribution. For the framework considered here, then, we can assume that $f(\mathbf{v})$ is independent of $\phi'$ without loss of generality.
We will consider a velocity distribution which is discretised into $N$ angular pieces: $$f(\textbf{v}) = f(v, \cos\theta', \phi') =
\begin{cases}
f^1(v) & \textrm{ for } \theta' \in \left[ 0, \pi/N\right]\,, \\
f^2(v) & \textrm{ for } \theta' \in \left[ \pi/N, 2\pi/N\right]\,, \\
& \vdots\\
f^k(v) & \textrm{ for } \theta' \in \left[ (k-1)\pi/N, k\pi/N\right]\,, \\
& \vdots\\
f^N(v) & \textrm{ for } \theta' \in \left[ (N-1)\pi/N, \pi\right]\,. \\
\end{cases}$$ We would then ultimately like to calculate the directionally-integrated Radon transform $\hat{f}^j$, integrated over the same bins in $\cos\theta$ as for the velocity distribution,
$$\begin{aligned}
\label{eq:fj1}
\begin{split}
\hat{f}^j({v_\mathrm{min}}) &= \int_{\cos(j\pi/N)}^{\cos((j-1)\pi/N)} \hat{f}({v_\mathrm{min}}, \cos\theta) \, \mathrm{d}\cos\theta \\
&= 2 \int_{a_{j}}^{a_{j-1}} \int_{{v_\mathrm{min}}}^{\infty} \left( \int_{x_-}^{x_+} f(v,\cos\theta') I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta' \right) \, v \,\mathrm{d}v \,\mathrm{d}\cos\theta \,,
\end{split}\end{aligned}$$
Here, we have introduced the notation $a_k = \cos(k\pi/N)$ and defined $f(v, \cos\theta') = 2\pi f(v, \cos\theta', \phi')$. We divide the integration range for $\cos\theta'$ into different regions:
$$\begin{aligned}
\begin{split}
\cos\theta' &\in [x_-, a_{k_--1}] \\
\cos\theta' &\in [a_{k_--1}, a_{k_--2}]\\
\vdots&\\
\cos\theta' &\in [a_{k_+ + 1}, a_{k_+}]\\
\cos\theta' &\in [a_{k_+}, x_+]\,,
\end{split}\end{aligned}$$
where $k_\pm$ are defined such that:
$$x_\pm \in [a_{k_\pm}, a_{k_\pm - 1}]\,.$$
These regions are chosen such that $f(v, \cos\theta')$ is independent of $\cos\theta'$ within each region. We note that if $k_+ = k_-$, there is only one region: $\cos\theta' \in [x_-, x_+]$. Using these definitions we can therefore rewrite the term in brackets in Eq. \[eq:fj1\] as
$$\begin{aligned}
\begin{split}
\label{eq:stepone}
&\int_{x_-}^{x_+} f(v,\cos\theta') I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta' \\
&=2\pi f^{k_-}(v) \int_{x_-}^{a_{k_--1}} I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta' \\
&+\sum_{k=k_- - 1}^{k_+ +1} 2 \pi f^{k}(v) \int_{a_k}^{a_{k-1}} I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta' \\
&+ 2\pi f^{k_+}(v) \int_{a_{k_+}}^{x_+} I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta'\,.
\end{split}\end{aligned}$$
We can now explicitly perform the integral over $\cos\theta'$,
$$\begin{aligned}
\begin{split}
&\int_{a_k}^{a_{k-1}} I(\beta, \cos\theta, \cos\theta') \,\mathrm{d}\cos\theta' \\
&= \int_{a_{k}}^{a_{k-1}} \frac{\mathrm{d}\cos\theta'}{\sqrt{(\sin\theta\sin\theta')^2 - (\beta - \cos\theta\cos\theta')^2}} \\
&= \left[ {\sin^{-1}}\left( \frac{\cos\theta' - \beta\cos\theta}{\sqrt{1-\beta^2}\sin\theta} \right) \right]_{a_k}^{a_{k-1}} \\
&= F(a_{k-1}, \cos\theta) - F(a_k,\cos\theta)\,,
\end{split}\end{aligned}$$
where we have defined $$F(y, \cos\theta) = {\sin^{-1}}\left( \frac{y - \beta\cos\theta}{\sqrt{1-\beta^2}\sin\theta} \right) \,.$$ We note that $$F(x_\pm, \cos\theta) = \pm \frac{\pi}{2}\,,$$ for all $\cos\theta$. We also note that $F(y, \cos\theta)$ can be integrated analytically:
$$\begin{aligned}
\label{eq:F}
\begin{split}
&\int^x F(y, \cos\theta) \, \cos\theta\\
&= x {\sin^{-1}}\left(\frac{y-\beta x}{\sqrt{1-x^2}\sqrt{1-\beta^2}}\right)\\
&\, + y {\tan^{-1}}\left(\frac{x-y\beta}{\sqrt{t}}\right) \\
&\, + \frac{1}{2} {\tan^{-1}}\left(\frac{1 - y^2 - \beta^2 - x + y\beta(1+x)}{(y-\beta) \sqrt{t}}\right) \\
&\, - \frac{1}{2} {\tan^{-1}}\left(\frac{1 - y^2 - \beta^2 + x - y\beta(1-x)}{(y+\beta) \sqrt{t}}\right)\\
&\equiv J(x,y) + C
\end{split}\end{aligned}$$
where it is understood that $J(x,y)$ is a function of $\beta$ and where $$t = (1-x^2)(1-\beta^2) - (y - \beta x)^2\,.$$
It is now possible to also complete the integration over $\cos\theta$. Assuming that $k_\pm$ do not change over a range $\cos\theta \in [b_i, b_{i+1}]$, we can combine Eqs. \[eq:stepone\] and \[eq:F\] to obtain
$$\begin{aligned}
\label{eq:steptwo}
\begin{split}
&\int_{b_i}^{b_{i+1}} \left(\int_{x_-}^{x_+} f(v, \cos\theta') I(\beta, \cos\theta, \cos\theta') \, \mathrm{d}\cos\theta' \right)\, \mathrm{d}\cos\theta \\
&= 2\pi f^{k_-}(v) \left( \frac{\pi}{2}(b_{i+1} - b_i) + J(b_{i+1}, a_{k_- - 1}) - J(b_{i}, a_{k_-- 1}) \right) \\
&+\sum_{k=k_- - 1}^{k_+ +1} 2 \pi f^{k}(v) \left( J(b_{i+1},a_{k-1}) - J(b_{i},a_{k-1}) - J(b_{i+1},a_{k}) + J(b_{i},a_{k})\right) \\
&+ 2\pi f^{k_+}(v) \left( \frac{\pi}{2}(b_{i+1} - b_i) - J(b_{i+1}, a_{k_+}) - J(b_{i}, a_{k_+}) \right)\,.
\end{split}\end{aligned}$$
However, we must take into account the fact that the values of $x_\pm$ and therefore of $k_\pm$ depend on $\beta$ and $\cos\theta$. We now discuss how to divide up the integration regions of $v$ and $\cos\theta$ such that we can apply Eq. \[eq:steptwo\].
As an illustration, we show in Fig. \[fig:integlimits1\] the values of $x_+ (x_-)$ as solid (dashed) lines as a function of $\cos\theta$, for a fixed value of $\beta$. In evaluating Eq. \[eq:fj1\], we wish to integrate over the region enclosed by the solid and dashed lines, for the relevant range of $\cos\theta$. As an example, we show with horizontal and vertical dotted lines the edges of the discretised regions in $\cos\theta$ and $\cos\theta'$ for $N=3$. That is, the dotted lines show the values $\cos(n \pi/N)$ for $n = 0, 1, 2, 3$. If we wish to evaluate $\hat{f}^1({v_\mathrm{min}})$, we need to integrate over the shaded region in Fig. \[fig:integlimits1\].
In Fig. \[fig:integlimits2\], we show $x_\pm$ for different values of $\beta$. We give two examples, for two different ranges of $\beta$: $\beta < 1/2$ (black) and $\beta > \sqrt{3}/2$ (blue). The values of $k_\pm$ (and therefore the relevant $f^k(v)$) will clearly depend on the value of $\beta$, so we must be careful to account for this properly.
![Integration limits for Eq. \[eq:fj1\] as a function of $\cos\theta$, for a fixed value of $\beta$. The limits $x_+$ and $x_-$ are shown as solid and dashed lines respectively. The dotted vertical and horizontal lines mark the values $\cos(n \pi/N)$ for $n = 0, 1, 2, 3$ and $N=3$. If we wish to calculate $\hat{f}^1({v_\mathrm{min}})$, we must perform the angular integral over the shaded region.[]{data-label="fig:integlimits1"}](Integlimits1.pdf){width="50.00000%"}
![As Fig. \[fig:integlimits1\], but for two different values of $\beta$. The black lines show $x_\pm$ for a value of $\beta$ in the range $\beta < 1/2$ while blue lines show $x_\pm$ for a value in the range $\beta > \sqrt{3}/2$.[]{data-label="fig:integlimits2"}](Integlimits2.pdf){width="50.00000%"}
Calculation of the values of $k_\pm$ and the corresponding bin edges in $\cos\theta$ ($\left\{b_i\right\}$) is involved, though not technically difficult. We therefore sketch the procedure here. We begin with the definition of $x_+$:
$$\label{eq:xplusdef}
x_+ = x_+(\cos\theta) = \beta \cos\theta + \sqrt{1-\beta^2}\sin\theta\,.$$
By straight-forward differentiation, we observe that this function has a maximum at $\cos\theta = \beta$. For concreteness, we consider $\beta$ in the range,
$$\beta \in \left[\cos(n\pi/N), \cos((n-1) \pi/N) \right] \qquad \textrm{ for } n = 1, 2, ..., N/2 \,.$$
We must consider three separate cases, depending on $j$ (i.e. which $\cos\theta$ bin we are considering):
(i) $j \leq (n-1)$
In this case, $\cos\theta > \beta$, meaning that $\partial x_+ / \partial \cos\theta < 0$, so $x_+$ is monotonically decreasing in $\cos\theta$.
(ii) $j \geq (n+1)$
In this case, $\cos\theta < \beta$, meaning that $\partial x_+ / \partial \cos\theta > 0$, so $x_+$ is monotonically increasing in $\cos\theta$.
(iii) $j = n$
In this case, the maximum of $x_+$ lies in the $j^\mathrm{th}$ bin.
For each value of $j$, we can now determine the maximum and minimum value of $x_+$ within that $\cos\theta$ bin by using the definition in Eq. \[eq:xplusdef\] . This allows us to determine the value of $k_+$ (i.e. the $\cos\theta'$ bin into which $x_+$ falls). In fact, it can be shown that for each value of $j$ (apart from $j=n$), $x_+$ crosses one of the $\cos\theta'$ bin edges exactly once, when $\cos\theta = \cos\theta_+^j$, satisfying
$$x_+(\cos\theta_+^j) = \cos(l \pi/N)\,,$$
for some $l = 1,2,...,N$. This means that for a given $j$, there are two distinct regions in $\cos\theta$ ($\cos\theta < \cos\theta_+^j$ and $\cos\theta > \cos\theta_+^j$) each with a different value of $k_+$.
We can perform a similar analysis for $x_-$ to obtain the values of $k_-$ for a given $n$ and $j$, as well as the values of $\cos\theta_-^j$, where the curve $x_-$ crosses from one bin to the next. However, we can only apply Eq. \[eq:steptwo\] if both $k_+$ and $k_-$ do not change over the range $\cos\theta \in [b_i, b_{i+1}]$. It is clear then that we must subdivide each $\cos\theta$ bin into 3 smaller bins. The edges of these sub-bins will be $\cos\theta_+^j$ and $\cos\theta_-^j$, at which the value of either $k_+$ or $k_-$ changes.
Finally, we need to determine which of these crossings $\cos\theta_\pm^j$ occurs first. The crossings occur at the same value of $\cos\theta$ if $\cos\theta_+^j$ = $\cos\theta_-^j$, which occurs for $\beta = \cos\left((n-1/2)\pi/N\right)$. For each value of $n$, we therefore need to distinguish two regimes, depending on which of the crossings $\cos\theta_\pm^j$ is larger. It is helpful then to further subdivide the range of $\beta$, writing $$\beta \in \left[\cos(m\pi/(2N)), \cos((m-1) \pi/(2N)) \right] \qquad \textrm{ for } m = 1, 2, ..., N \,,$$ from which we obtain $$n = \begin{cases}
(m+1)/2 & \textrm{ for } m \textrm{ odd}\\
m/2 & \textrm{ for } m \textrm{ even}\,.
\end{cases}$$ We can now determine which of $\cos\theta_\pm^j$ is greater, depending on whether $m$ is even or odd. For a given value of $m$ then, the values of the sub-bin edges in $\cos\theta$ ($\left\{b_i\right\}$) are now well determined, along with the values of $k_\pm$ within each sub-bin. This allows us to apply Eq. \[eq:steptwo\] to each of these sub-bins.
Finally, we can perform the integral over $v$ described in Eq. \[eq:fj1\]. However, we note that we should sub-divide the range of integration, depending on the value of $m$. That is, the integral over $v$ becomes,
$$\int_{{v_\mathrm{min}}}^{\infty} \, \mathrm{d}v \rightarrow \sum_{m = 1}^N \int_{{v_\mathrm{min}}/a_{(m-1)/2}}^{{v_\mathrm{min}}/a_{m/2}} \, \mathrm{d} v\,,$$
where we remind the reader that $a_k = \cos(k\pi/N)$. For each term in this sum, the value of $m$ does not change within the range of integration, so we can apply the results which have been obtained so far. We give the full expression for the IRT, as well as the values of $k_\pm$ and $\cos\theta_\pm^j$ (calculated as described here) in Appendix \[app:Radon\].
The directionally-integrated Radon transform (IRT) {#app:Radon}
==================================================
The discretised velocity distribution, with $N$ angular bins, is defined as
$$f(\textbf{v}) = f(v, \cos\theta', \phi') =
\begin{cases}
f^1(v) & \textrm{ for } \theta' \in \left[ 0, \pi/N\right]\,, \\
f^2(v) & \textrm{ for } \theta' \in \left[ \pi/N, 2\pi/N\right]\,, \\
& \vdots\\
f^k(v) & \textrm{ for } \theta' \in \left[ (k-1)\pi/N, k\pi/N\right]\,, \\
& \vdots\\
f^N(v) & \textrm{ for } \theta' \in \left[ (N-1)\pi/N, \pi\right]\,. \\
\end{cases}$$
The integrated Radon transform (IRT), integrated over the same angular bins, is defined as
$$\hat{f}^j(v_\textrm{min}) = \int_{\phi = 0}^{2\pi} \int_{a_{j}}^{a_{j-1}} \hat{f}(v_\textrm{min}, \hat{\textbf{q}})\, \mathrm{d}\cos\theta \,\mathrm{d}\phi\,,$$
where have defined the shorthand $a_k = \cos(k\pi/N)$. The IRT is given in terms of the distribution functions $f^k(v)$ as
$$\begin{aligned}
\begin{split}
\hat{f}^j({v_\mathrm{min}}) &= 4\pi\sum_{m = 1}^N \int_{{v_\mathrm{min}}/a_{(m-1)/2}}^{{v_\mathrm{min}}/a_{m/2}} v \, \textrm{d}v\sum_{i = 0,1,2} \Big[\frac{\pi}{2} (f^{k_+^{i}}(v) + f^{k_-^{i}}(v)) \left(b_{i+1} - b_i\right) \\
& + (1-\delta_{k_+^{i} k_-^{i}}) f^{k_-^{i}}(v)\left( J(b_{i+1}, a_{k_-^{i} - 1}) - J(b_{i}, a_{k_-^{i} - 1}) \right) \\
& -(1-\delta_{k_+^{i} k_-^{i}}) f^{k_+^{i}}(v)\left( J(b_{i+1}, a_{k_+^{i}}) - J(b_{i}, a_{k_+^{i}}) \right) \\
& + \sum_{k = k_+^{i} + 1}^{k_-^{i}- 1} f^{k}(v)\left(J(b_{i+1},a_{k-1}) - J(b_{i},a_{k-1}) \right. \\
&\left. \qquad \qquad \qquad- J(b_{i+1},a_{k}) + J(b_{i},a_{k})\right) \big]\, .
\end{split}\end{aligned}$$
The function $J(x,y)$ depends implicitly on the speed $v$ and is defined below. For each value of $j$, we integrate over N ranges in $v$, labeled by the index $m$. We also sum over three ‘bins’, labeled by the index $i = 0,1,2$, which are distinguished by their values of the integers $k_\pm^{i}$. We note that the values of $k_\pm^{i}$ will depend on the values of $j$ and $m$ under consideration, as well as the index $i$. We write $\mathbf{k_{\pm}} = (k_\pm^{0}, k_\pm^{1}, k_\pm^{2})$ and give the values below.
For odd $m$, we define $n = (m+1)/2$ and bin edges
$$\begin{aligned}
\begin{split}
b_0 &= \cos\left(j \pi/N\right)\\
b_1 &= \cos\left(\gamma + (j + n - 1)\pi/N\right) \\
b_2 &= \cos\left(\gamma + (j -n)\pi/N\right) \\
b_3 &= \cos\left((j-1) \pi/N\right)\,,
\end{split}\end{aligned}$$
with $\gamma = {\cos^{-1}}\left({v_\mathrm{min}}/v\right)$. The corresponding $\mathbf{k}_\pm$ values are:
$$\mathbf{k}_+ =
\begin{cases}
\left(n - j, n - j, n - j + 1 \right) & \textrm{ for } j \leq (n-1) \\
\left(1,1,1\right) & \textrm{ for } j = n \\
\left(j-n+1, j-n+1, j-n \right) & \textrm{ for } j \geq (n+1)\,, \\
\end{cases}$$
and $$\mathbf{k}_- =
\begin{cases}
\left(n+j, n+j -1, n+j-1 \right) & \textrm{ for } j \leq (N-n) \\
\left(N,N,N\right) & \textrm{ for } j = (N+1-n)\\
\left(2N - n -j +1, 2N - n - j +2, 2N-n-j + 2 \right) & \textrm{ for } j \geq (N+2-n)\,. \\
\end{cases}$$
For even $m$, we write $n= m/2$ and bin edges $$\begin{aligned}
\begin{split}
b_0 &= \cos\left(j \pi/N\right)\\
b_1 &= \cos\left(\gamma + (j - n)\pi/N\right) \\
b_2 &= \cos\left(\gamma + (j+n -1)\pi/N\right) \\
b_3 &= \cos\left((j-1) \pi/N\right)\,.
\end{split}\end{aligned}$$ The corresponding $\mathbf{k}_\pm$ values are: $$\mathbf{k}_+ =
\begin{cases}
\left(n - j, n - j+1, n - j + 1 \right) & \textrm{ for } j \leq (n-1) \\
\left(1,1,1\right) & \textrm{ for } j = n \\
\left(j-n+1, j-n, j-n \right) & \textrm{ for } j \geq (n+1)\,, \\
\end{cases}$$ and $$\mathbf{k}_- =
\begin{cases}
\left(n+j, n+j, n+j-1 \right) & \textrm{ for } j \leq (N-n) \\
\left(N,N,N\right) & \textrm{ for } j = (N+1-n)\\
\left(2N - n -j +1, 2N - n - j +1, 2N-n-j + 2 \right) & \textrm{ for } j \geq (N+2-n)\,. \\
\end{cases}$$
Finally, the angular function $J$ is given by:
$$\begin{aligned}
\begin{split}
J(x, y) &= x {\sin^{-1}}\left(\frac{y-\beta x}{\sqrt{1-x^2}\sqrt{1-\beta^2}}\right)\\
&\, + y {\tan^{-1}}\left(\frac{x-y\beta}{\sqrt{t}}\right) \\
&\, + \frac{1}{2} {\tan^{-1}}\left(\frac{1 - y^2 - \beta^2 - x + y\beta(1+x)}{(y-\beta) \sqrt{t}}\right) \\
&\, - \frac{1}{2} {\tan^{-1}}\left(\frac{1 - y^2 - \beta^2 + x - y\beta(1-x)}{(y+\beta) \sqrt{t}}\right)\,,
\end{split}\end{aligned}$$
where we define $\beta = {v_\mathrm{min}}/v$ and $t = (1-x^2)(1-\beta^2) - (y - \beta x)^2$. We note that the angular integrals in the Radon transform lead to this analytic form for arbitrary $N$, meaning that only 1-dimensional integrals over $v$ need to be performed. A python implementation of this algorithm is available on request from the author.
[^1]: The integration over $\phi$ essentially reduces the experiment to what is referred to as 1-dimensional directional detection, in which only information about $\theta$ is available. However, it has been shown that this does not significantly reduce the discovery power of the detector when compared with 3-dimensional detection [@Billard:2015].
[^2]: Because the $N=1$ case (full average over all angles) is exact, the discretised velocity distribution will always recover the correct total number of signal events. We have checked this explicitly up to $N=10$ by summing the number of events in each angular bin.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In excited molecules, the interaction between the covalent Rydberg and ion-pair channels forms a unique class of excited Rydberg states, in which the infinite manifold of vibrational levels are the equivalent of atomic Rydberg states with a heavy electron mass. Production of the ion pair states usually requires excitation through one or several interacting Rydberg states; these interacting channels are pathways for loss of flux, diminishing the rate of ion pair production. Here, we develop an analytical, asymptotic charge transfer model for the interaction between ultracold Rydberg molecular states, and employ this method to demonstrate the utility of off-resonant field control over the ion pair formation, with near unity efficiency.'
author:
- Samuel Markson
- 'H. R. Sadeghpour'
bibliography:
- 'ionbib.bib'
title: 'Charge Transfer in Ultracold Rydberg-Ground State Atomic Collisions'
---
Introduction
============
Ultracold atomic systems allow for precise control via laser and static fields and are employed for simulating strongly-correlated many-body condensed matter and optical systems, for performing chemistry in the quantum regime, and for executing quantum computing protocols [@bloch_many-body_2008; @regal_creation_2003; @carr_cold_2009]. Excitations in ultracold atomic traps have ushered in the Rydberg blockade regime [@lukin_dipole_2001; @saffman_quantum_2010] (with promise for quantum information processing and quantum bit operations), ultracold plasmas [@castro_role_2009] (with application in recombination, ion crystal order and heating), and ultra long range Rydberg molecules [@greene_creation_2000; @bendkowsky_observation_2009] (for studies of few-body molecular systems, symmetry breaking and coherent control [@rittenhouse_ultracold_2010]).
Another class of molecular states, the ion pair states, form channels when the covalent Rydberg channels couple to the long-range ion-pair potential. These atomic cation-anion pair states share several properties with ionic molecules; an infinite spectrum of vibrational levels which follow a Rydberg progression with a heavy electron mass, and large permanent electric dipole moments. They are also long-range states and have typically negligible Franck-Condon (FC) overlap with the usual short-range molecular levels. These heavy Rydberg states (HRS) have been experimentally observed in several molecular species, relying on excitation from bound molecular levels [@vieitez_observation_2008; @vieitez_spectroscopic_2009; @mollet_dissociation_2010; @ekey_spectroscopic_2011].
The prevailing issue with indirect excitation of bound molecules, such as in H$_2$ and Cl$_2$ [@vieitez_observation_2008; @vieitez_spectroscopic_2009; @mollet_dissociation_2010; @ekey_spectroscopic_2011], is that it is not [*a priori*]{} possible to identify a set of long-lived intermediate heavy Rydberg states to which ion the pair states couple. In a recent work [@kirrander_approach_2013], it was proposed to directly pump long-lived HRS from ultracold Feshbach molecular resonances, just below the avoided crossings between the covalent potential energy curves and the ion-pair channel. The predominant excitation to HRS occurs near the avoided crossings, because the non-adiabatic mixing allows for favorable electronic transitions to the HRS. When the nuclear HRS wave function peaks at the classical turning point, the internuclear FC overlap increases.
In this work, we develop an analytical, but asymptotic model for one-electron transfer, merging single-center potentials for the Rb atom, and demonstrate efficient field control over the rate of ion pair formation. Adiabatic potential energy curves are calculated along with the radial non-adiabatic coupling and dipole transition matrix elements. We compare our adiabatic potential energies with the Born-Oppenheimer (BO) potential energy curves from Ref. [@park_theoretical_2001]. We find that with modest off-resonant external fields, we can alter the avoided crossing beween the covalent HRS and ion pair channels, and modify the behavior of the nuclear wavefunction at the classical turning points; hence, control the FC overlaps and rate of ion pair formation.
This excitation occurs at larger internuclear separations, where the overlap and transition dipole moments between the ground and ion pair states are significant. The method proposed in [@kirrander_approach_2013] requires adiabatic rapid passage or multiphoton transitions to enhance this excitation, while in the present work, only off-resonant fields must be employed to increase the efficiency of ion pair production.
![Scheme from [@kirrander_approach_2013] relied on excitation to the Rb(5s)+Rb(6p)/Rb$^+$+Rb$^-$ (ion pair) or Rb(5p)+Rb(5p)/ion pair crossings (excitation 2 and 3). Here, we focus on the Rb(5s)+Rb(7p)/ion pair crossing (excitation 1). Subsequent to excitation, both schemes from [@kirrander_approach_2013] require multiphoton excitation or chirped pulse rapid adiabatic passagee. Excitation 1, directly to the Rb(5s)+Rb(7p)/ion pair crossing, can more easily be controlled with dc electric fields. Additionally, the Franck-Condon factors for this excitation are favorable, as demonstrated by the $\nu=124$ Feshbach molecule vibrational wavefunction. The dashed curve is the ion pair potential.](arrowexcite.pdf){width="45.00000%"}
\[sec:level2\]Methodology and Results
=====================================
The present charge transfer model relies on one active electron participating in the process of ion pair excitation. For the charge transfer to occur, the Rb valence electron is ionized from one center, separated from the other neutral center by a distance $R$. The ionization process is calculated in the one-electron effective potential model of Ref. [@marinescu_dispersion_1994]. The interaction of the electron with the neutral Rb atom is modeled by the short-range scattering of the low-energy electron from the ground-state Rb atom. This is done with a symmetric inverse hyperbolic cosine (Eckart) potential.
The two-center Hamiltonian matrix elements are constructed by expanding in a basis of atomic Rb Rydberg orbitals and the single bound orbital in the Eckart potential. The resulting generalized eigenvalue equation is solved for the adiabatic potential energy curves, radial coupling and electronic dipole matrix elements. Our control scheme hinges on modifying the avoided crossing gap between the 5s+7p and ion pair state by applying an off-resonant field, thereby tuning the Landau-Zener crossing probabilities between the ion pair and covalent states. The two-center potential for the Rb$_2$ Rydberg-excited molecule is the sum of two single-center potentials, $$V_{\text{model}} (r_a, r_b|R) = V_1 (r_a|R) + V_2 (r_b|R)
\label{eq:potential}$$ where $V_1$ represents the potential felt by the valence electron due to the Rb core, and the pseudopotential $V_2$ that of the binding to the neutral rubidium atom; $r_a$ and $r_b$ represent the distances between the electron and the cation/neutral atom, respectively.
We require that the potential in Eq. \[eq:potential\] correctly reproduce the asymptotic dissociation energies for all the relevant Rydberg and ion-pair states [@NIST_table]. We further enforce that the known energy-independent $e^{-}\text{-Rb}$ scattering length and the Rb electron affinity are reproduced, i.e. $a_{sc} =-16.1$ a.u. [@bahrim_3se_2001] and EA=-0.01786 a.u. [@NIST_table].
The valence-electron potential, $V_1$ [@marinescu_dispersion_1994] correctly reproduces the observed atomic energies [@NIST_table], $$V_1(r_a)= - \frac{Z_l (r_a)}{r_a} - \frac{\alpha_c}{2 r_a^4} (1- e^{-(r_a/r_c)^6})
\label{eq:leftpotential}$$ where $$Z_l(r_a)=1+(z-1) e^{-\alpha_1 r_a} -r (a_3 + a_4 r_a) e^{-a_2 r_a}$$ with $z$ being the nuclear charge. The terms of the potential account for screening of the nuclear charge due to core electrons and effect of core polarizability.
For the neutral rubidium electron affinity, we use the radial Eckart potential [@eckart_penetration_1930]. $$V_2(r_b)=V_0 \cosh^{-2} {(\frac{r_b}{r_0})}
\label{eq:rightpotential}$$ where $V_0$ and $r_0$ are parameters related in a set of coupled equations to the electron affinity and $e^{-}\text{-Rb}$ scattering length; see appendix A.
The Hamiltonian matrix is constructed from a suitable basis for each center and may be conveniently represented in prolate spheroidal coordinates $$r=\frac{R}{2} (\xi + \kappa \eta),\ \cos{\theta} = \frac{1+\kappa \xi \eta}{\xi + \kappa \eta}$$ where $\kappa = +1(-1)$ represents the radial coordinate of the left(right) centers, $\xi \in [1,\infty),\ \eta \in [-1,1]$.
We solve the generalized eigenvalue problem: $$\mathbf{H} \vec{\Psi} = E(R) \mathbf{S} \vec{\Psi}\label{eq:scheq}$$ where $\mathbf{H}$ and $\mathbf{S}$ are the Hamiltonian and overlap matrices in the truncated basis: $$\mathbf{H}_{jj'}=\bra{\phi_j^{(a)}} H_0^{(a)}+V_2 \ket{\phi_j^{(a)}}
\tag{4a}$$ $$\mathbf{H}_{kk'}=\bra{\phi_k^{(b)}} H_0^{(b)}+V_1 \ket{\phi_k^{(b)}}
\tag{4b}$$ $$\mathbf{H}_{jk}=\bra{\phi_j^{(a)}} H_0^{(b)}+V_1 \ket{\phi_k^{(b)}} = \bra{\phi_k^{(b)}} H_0^{(a)} + V_2 \ket{\phi_j^{(a)}}
\tag{4c}$$ $$\mathbf{S}_{jj'}=\braket{\phi_j^{(a)} | \phi_j'^{(a)}} = \delta_{j j'}
\tag{4d}$$ $$\mathbf{S}_{kk'}=\braket{\phi_k^{(b)} | \phi_k'^{(b)}} = \delta_{k k'}
\tag{4e}$$ $$\mathbf{S}_{jk}=\braket{\phi_j^{(a)} | \phi_k^{(b)}}
\tag{4f}$$
The full wave function is comprised of $\{\phi_j^{(a)}(\frac{R}{2} (\xi + \eta)\}$ centered on $V_1$, and $\{\phi_k^{(b)}(\frac{R}{2} (\xi - \eta)\}$ centered on $V_2$, i. e. $\Psi_i= \sum_j^{n_a} c_{ij} \phi_j^{(a)} + \sum_k^{n_b} c_{i k} \phi_k^{(b)}$. The truncated basis set contains atomic orbitals, $\{\phi_j^{(a)}\} = \{(5\text{-}9)s,(5\text{-}11)p,(4\text{-}6)d\}$, and the short-range wave function for scattering the of electrons from $V_2$, $\{\phi_k^{(b)}\} = \{\cosh^{-2 \lambda}\frac{r_b}{r_0} \sqrt{z}\ {}_2 F_1(0,-2 \lambda,\frac{3}{2},z)\}$, with $z=-\sinh^2{(\frac{r_b}{r_0})},\ \mbox{and}\ \lambda=0.0321816$; see Appendix \[sec:eckappendix\]. ${}_2 F_1(a,b;c;z)$ is the hypergeometric function [@bateman_manuscript_project_tables_1954].
One major feature of this asymptotic approach is that the permanent and transition dipole, and non-adiabatic radial coupling matrix elements can now be calculated from the eigenstates of Eq. \[eq:scheq\]. Full details are available in Appendix \[sec:eckappendix\]. The adiabatic potential energy curves, $E(R)$, are shown in Fig. \[fig:potential\]. By construction, these adiabatic potentials correlate to the asymptotic dissociation energies for the Rydberg and ion pair states and have avoided crossings between covalent and ion pair channels. The Rb$_2$ BO potential energies in the region of Rb(5s)+Rb(6p) dissociation energy are superposed on the adiabatic potentials for comparison. The non-adiabatic coupling matrix element between the ion pair channel with the molecular Rydberg curve dissociating to Rb(5s)+Rb(7s) is shown in the inset of Fig. \[fig:potential\].
![Nuclear potential energy curves, overlaid with calculations from other publications. Calculated curves are well-behaved in the large R limit. Inset shows avoided crossing at $R_c=116.2$, along with non-adiabatic coupling matrix element between Rb(5s)+Rb(7p)/ion pair channels. Rb(5s)+Rb(6p), Rb(5p)+Rb(5p), and ion pair curves taken from Rb(5s)+Rb(6p), Rb(5p)+Rb(5p), Rb(5s)+Rb(7p), and ion pair curves taken from [@park_theoretical_2001] [@bellos_upper_2013]. Ion pair curve is fitted to $-\frac{1}{R}-\frac{\alpha_{\text{Rb}_{-}}+\alpha_{\text{Rb}_{+}}}{2 R^4}$, where $\alpha_{\text{Rb}_{-}}=526.0$ a.u [@fabrikant_polarizability], $\alpha_{\text{Rb}_{+}}=9.11$ a.u [@clark_polarizability] are the polarizabilities of the anion and cation respectively. []{data-label="fig:potential"}](truepecs.pdf){width="50.00000%"}
Field Control Covalent/Ion Pair Channels
========================================
The avoided crossing in the interaction of ion pair channel (Rb$^+$+Rb$^-$) and Rydberg channel (Rb(5s)+Rb(7p)) is nearly diabatic, i. e. the covalently populated vibrational states will predominantly dissociate to neutral atoms with little possibility of forming HRS and ion pair states. This is reflected in the narrowness of the avoided crossing and the strength of the radial non-adiabatic matrix element. The two channels have non-zero dipole transition moments, so in an external field, they will mix and modify the transition probability for populating ion pair states.
Near the avoided crossing, the electronic wave function becomes hybridized in the field, as in the first order of perturbation theory, $\ket{\psi} = \ket{\psi_0} + \sum_{k \neq 0} \frac{\bra{ \psi_k} - \mathbf{F} \cdot \mathbf{z} \ket{\psi_0}}{E_0-E_k} \ket{\psi_k}$, where $\ket{\psi_k}$ are the dipole-allowed states which couple to the initial state, $\ket{\psi_0}$. This hybridization is demonstrated in Fig. \[fig:hybrid\] for the two channels of interest. When the field is off, the wave function amplitude in the covalent Rydberg channel peaks near the avoided crossing. With the field on, the two amplitudes become comparable.
![Hybridization of the wavefunctions at the avoided crossing by the field. The upper panel show the cross section of the field-free electronic wavefunction along the internuclear axis—states associated with the ion pair and Rb(5s)+Rb(7p) channels have very little overlap. The lower panel is field-dressed ($\mathbf{F}=1300$ V/cm, showing significant overlap of the wavefunctions associated with the two channels. The blue line is the pseudopotential along the internuclear axis for $R=116.2$ a.u.[]{data-label="fig:hybrid"}](./hybridizedwavefunctions.pdf){width="50.00000%"}
The probability that ion-pair states survive the single-pass traversal through the avoided crossing region in Fig. \[fig:potential\] is given approximately by the Landau-Zener-Stückelberg formula: $$P_{\text{ad}}=1-\exp{[-2 \pi \frac{a^2}{|v \frac{\partial (E_2-E_1)}{\partial R} |}]}
\label{eq:lzform}$$ where $v = \sqrt{\frac{2(E- V(R_c))}{m}}$ is the velocity of the wavepacket at crossing, $R=R_c$, $a$ is the off-diagonal element coupling the two states at $R_c$ (half the crossing size in the fully diagonalized picture), and $\frac{\partial (E_2(R)-E_1(R))}{\partial R}|_{R=R_c}$ is the relative slope of the intersecting curves at $R=R_c$.
![How the ion-pair survival probability may be controlled with modest off-resonant electric fields: Perturbation theory becomes unreliable for **F** $>1700$ V/cm, and classical ionization occurs for **F** $>5$ MV/cm. The vertical width of contour regions gives a measure of the survival probability above a desired threshold for a laser-pulse-excited vibrational wavepacket. For example, at $\mathbf{F} \simeq 1000 $V/cm., more than 2 $\sigma_{\text{std}}$ will survive if the laser linewidth can be constrained to $\sim150$ MHz. []{data-label="fig:lzc"}](./LZContours.png){width="50.00000%"}
Fig. \[fig:lzc\] illustrates how the ion=pair probability can be controlled with modest electric fields. Each curve represents a contour of cross-sectional survival probability for a vibrational wavepacket excited to a certain energy above the crossing threshold at a given field strength. For example, at field strengths of $1000 $V/cm., more than 2 $\sigma_{\text{std}}=95.45\%$ will survive if the laser linewidth can be constrained to $\sim150$ MHz.
As the loss probability of a given vibrational level goes at weak field strength as $e^{-\mathbf{F}^4}$, even relatively weak fields have a profound effect on ion pair state survival.
Conclusion
==========
We have shown that application of weak off-resonant electric fields provide a simple yet potent means of drastically increasing the survivability of the ion pair states in Rb dimers. To this end, we have introduced a simple and intuitive analytical pseudopotential which reproduces the correct asymptotic behavior of the dimer. This model provides an easy and computationally straightforward way for us to calculate dipole and non-adiabatic coupling matrix elements for excited states of alkali dimers.
This work was supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular, and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory; S. Markson was additionally supported through a graduate research fellowship through the National Science Foundation.
The Eckart potential {#sec:eckappendix}
====================
Bound states
------------
Taking the Schrödinger equation with $V(r)$ from \[eq:rightpotential\], we make the substitutions [@gol?dman_problems_2006] $$\psi = (\cosh \frac{r}{r_0})^{-2 \lambda} u,\ \text{where}\ \lambda=\frac{1}{4} (\sqrt{\frac{8 \mu V_0 {r_0}^2}{ \hbar^{2}} +1} -1)$$ The Schrödinger equation then becomes $$\frac{d^2 u}{dr^2} - \frac{4 \lambda}{r_0} \tanh{(\frac{r}{r_0})} \frac{du}{dr} + \frac{4}{{r_0}^2} (\lambda^2-\chi^2) u = 0
\label{eq:transformedeckarteq}$$ with $$\chi = \sqrt{-\frac{\mu E {r_0}^2}{2 \hbar^2}}$$ Letting $z=-\sinh^2 (\frac{r}{r_0})$ leads us to the hypergeometric equation $$z(1-z) \frac{d^2 u}{dz^2} + (\gamma-(\alpha + \beta +1) z ) \frac{du}{dz} - \alpha \beta u = 0
\label{eq:hypergeo}$$ where $\gamma=\frac{1}{2},\ \alpha= \chi - \lambda,\ \beta=-\chi -\lambda$.
In spherical coordinates, only odd hypergeometric solutions are valid: $$u=\sqrt{z}\ F (-\lambda + \chi + \frac{1}{2},-\lambda -\chi+\frac{1}{2};\frac{3}{2};z)
\label{eq:hgl}$$ By enforcing asymptotic boundary conditions for the bound states ($u\rightarrow 0$ as $ r \rightarrow \infty$), we have that $$\lambda - \chi = n+\frac{1}{2},\ n=0,1,2,\cdots$$ giving the set of bound states $$\psi_n=(\cosh \frac{r}{r_0})^{-2 \lambda} u_n
\label{eq:boundstates}$$ where $$u_n=N \sinh( \frac{r}{r_0}) F(-n,-2\lambda +n+1;\frac{3}{2};-\sinh^2( \frac{r}{r_0}))$$ (N being a normalizing constant), with corresponding energies $$E_n = -\frac{2 \hbar^2}{\mu {r_0}^2} (\frac{1}{4} \sqrt{\frac{8 \mu V_0 {r_0}^2}{ \hbar^{2}} +1} -n -\frac{3}{4})^2
\label{eq:eckarteigenenergies}$$
Scattering States
-----------------
With $E=\frac{\hbar^2 k^2}{2 \mu}$, $\chi^2 = -\frac{\mu E {r_0}^2}{2 \hbar^2} \rightarrow i k r_0 = 2 \chi$ [@gol?dman_problems_2006], and
$$\begin{gathered}
u(r) = N \sinh{( \frac{r}{r_0})} F(-\lambda + \frac{i k r_0}{2}+\frac{1}{2}, \\ -\lambda-\frac{i k r_0}{2}+\frac{1}{2}, \frac{3}{2},-\sinh^2 \frac{r}{r_0})\end{gathered}$$
fulfilling the boundary conditions $$\lim_{r\rightarrow 0}u(r)=0,\ \lim_{r\rightarrow \infty} u(r) \simeq \sin ( k r + \delta_0)$$ where $\delta_0$ is the usual phase shift.
We note that $F(\alpha,\beta; \gamma; 0)=1,\ \forall \alpha,\beta,\gamma$, and that $$\lim_{r \to \infty} { \sinh^2} (\frac{r}{r_0}) \approx { \frac{1}{2}} e^{\frac{2 r}{r_0}}$$
Using the asymptotic behavior of the Gaussian hypergeometric functions (see [@bateman_manuscript_project_tables_1954], p. 108, equation 2). $$u(r) = A e^{i k r} + B e^{-i k r}$$ where $$A= 2^{-i k r_0} \frac{\Gamma(i k r_0)}{\Gamma(\frac{1-2 \lambda+i k r_0}{2}) \Gamma(1+\frac{2 \lambda+ i k r_0}{2})}, \ B=A^\dagger$$ where we neglect a real factor common to $A$ and $B$ which will not affect the scattering phase shift. The phase shift itself has the form: $$\delta_0 = \frac{1}{2 i} \ln (-\frac{A}{A^\dagger})
\label{eq:phaseshift}$$ In the limit $ka \ll 1$, the Gamma functions may be expanded in a Taylor series,
$$\Gamma(i k r_0) \simeq \frac{1}{i k r_0} \Gamma(1+ i k r_0) \simeq \frac{1 + i k r_0 \psi(1) } {i k r_0} \Gamma(1)$$ $$\Gamma(\frac{1-2 \lambda}{2} + \frac{i k r_0}{2}) \simeq \Gamma (\frac{1-2 \lambda}{2}) (1+ \frac{i k r_0}{2} \psi^{(0)} (\frac{1-r_0}{2}) )$$ $$\Gamma(1+\frac{2 \lambda}{2} + \frac{i k r_0}{2}) \simeq \Gamma (1+\lambda) (1+ \frac{i k r_0}{2} \psi^{(0)} (1+\lambda))$$ where $\psi^{(0)}$ is the $m^{th}$-order polygamma function, i.e. $\psi^{(0)}(x) = \frac{d}{dx} \ln \Gamma(x) $. The final expression for $\delta_0$ now becomes: $$\delta_0= k r_0 [-\ln(2) + \psi^{(0)}(1) -\frac{1}{2} \psi^{(0)} (\frac{1-2 \lambda}{2}) - \frac{1}{2} \psi^{(0)} (1+ \lambda)]
\label{eq:finaldelta}$$ which is related to the s-wave scattering length by the usual formula, $$\lim_{k\rightarrow 0} k \cot{ \delta_0} = -\frac{1}{a_{sc}}$$ The parameters for the pseudopotential $V_2(r)$ ( \[eq:rightpotential\]) are hence found by solving the system of equations:
$$a_s = r_0 (-ln(2) + \psi(1) -\frac{1}{2} \psi(\frac{1-2 \lambda}{2}) -\frac{1}{2} \psi(1+\lambda))
\label{eq:condfirst}$$
and $$EA=-\frac{\hbar^2}{2 \mu {r_0}^2} (\frac{1}{2} \sqrt{\frac{8 \mu V_0 {r_0}^2}{\hbar^2}+1} -\frac{3}{2})^2
\label{eq:condsecond}$$
There are multiple solutions to Eqs. (A8 - A9). However, since it is known that Rb${}^{-}$ has only one bound state, we impose the additional restriction: $$N_{\text{bound}} = \lfloor \frac{1}{4} \sqrt{\frac{8 \mu V_0 r_0^2}{\hbar^2} +1} + \frac{1}{4}\rfloor =1$$ The solutions to Eqs. (A8-A9) yield $r_0=9.004786$ a.u. and $V_0=0.061675$ a.u.
Dipole and Non-Adiabatic Coupling Matrix Elements
=================================================
Dipole elements take the usual form: $$\vec{d}_{mn}=\bra{\psi_m} r \cos{\theta} \ket{\psi_n}$$ Non-adiabatic coupling matrix elements are computed via the finite difference formula: $$A_{mn}=\frac{1}{2 \Delta R} (\gamma_{mn}(R|R+\Delta R) - \gamma_{mn}(R|R-\Delta R))$$ where $$\gamma_{mn}(R|R\pm \Delta R)=\bra{\psi_m(r|R)}\ket{\psi_n(r|R\pm \Delta R)}$$
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Zhan Shi[^1]\
The University of Texas at Austin\
`[email protected]`\
Kevin Swersky, Daniel Tarlow, Parthasarathy Ranganathan, Milad Hashemi\
Google Research\
`{kswersky, dtarlow, parthas, miladh}@google.edu`\
bibliography:
- 'dyn\_gnn.bib'
title: |
Learning Execution through\
Neural Code Fusion
---
[^1]: Work completed during an internship at Google.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain character formulas are invertible, yielding expressions for the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group.'
address:
- |
Department of Mathematics\
Bar-Ilan University\
52900 Ramat-Gan\
Israel
- |
Department of Mathematics\
Bar-Ilan University\
52900 Ramat-Gan\
Israel
author:
- 'Ron M. Adin'
- Yuval Roichman
date: 'Sep. 11, ’13'
title: 'Matrices, Characters and Descents'
---
[^1]
Introduction {#section:intro}
============
Many character formulas involve the descent set of a permutation or of a standard Young tableau. We propose here a general setting for such formulas, involving a new family of asymmetric matrices of Walsh-Hadamard type. These matrices turn out to have fascinating properties, some of which are studied here using a transformation based on Möbius inversion. These include the evaluation of determinants, entries of transformed matrices and their inverses, and eigenvalues.
The inverse matrices lead to formulas expressing the cardinalities of sets of combinatorial objects with prescribed descent sets in terms of character values of the symmetric group. Examples of such objects include permutations of fixed length, involutions, standard Young tableaux, and more. It also follows that certain statements in permutation statistics have equivalent formulations in character theory. For example, the fundamental equi-distribution Theorem of Foata and Schützenberger, independently proved by Garsia and Gessel, is equivalent to a theorem of Lusztig and Stanley in invariant theory.
The organization of this paper is as follows: Section \[section:preliminaries\] contains the necessary definitions and background material, ending with a statement of the central motivating question. Section \[section:matrices\] introduces the main tool – a family (actually, two “coupled” families) of square matrices – and states some of their properties. Section \[section:AM\_BM\] contains a proof of the invertibility of these matrices, using a transformation corresponding to Möbius inversion. Properties of the transformed matrices are described in Section \[section:matrix\_entries\]. The application to character formulas, involving the concept of fine sets, is described in Section \[section:rep\_theory\_fine\].
Preliminaries and notation {#section:preliminaries}
==========================
Intervals, compositions, partitions and runs {#subs.prelim_compositions}
--------------------------------------------
\
For positive integers $m, n$ denote $$[m,n]:= \begin{cases}
\{m, m+1, \ldots, n\}, & \text{if $m \le n$;}\\
\emptyset, & \text{otherwise.}
\end{cases}$$ Denote also $[n] := [1,n] = \{1, \ldots, n\}$.
A [*composition*]{} of a positive integer $n$ is a vector $\mu=(\mu_1, \ldots, \mu_t)$ of positive integers such that $\mu_1+\cdots+\mu_t=n$. A [*partition*]{} of $n$ is a composition with weakly decreasing entries $\mu_1 \ge \ldots \ge \mu_t > 0$. The [*underlying partition*]{} of a composition is obtained by reordering the entries in weakly decreasing order.
For each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ define the set of its partial sums $$S(\mu) := \{ \mu_1, \mu_1+\mu_2, \ldots, \mu_1 + \ldots + \mu_t = n\} \subseteq [n],$$ as well as its complement $$I(\mu) := [n] \setminus S(\mu) \subseteq [n-1].$$ For example, for the composition $\mu=(3,4,2,5)$ of $14$: $S(\mu) = \{3,7,9,14\}$ and $I(\mu) = \{1,2,4,5,6,8,10,11,12,13\}$.
The correspondence $\mu \longleftrightarrow I(\mu)$ is a bijection between the set of all compositions of $n$ and the power set (set of all subsets) of $[n-1]$. The [*runs*]{} (maximal consecutive intervals) in $I(\mu)$ correspond to some of the components of $\mu$ – those satisfying $\mu_k > 1$. The length of the run corresponding to $\mu_k$ is $\mu_k-1$.
Permutations, Young tableaux and descent sets
---------------------------------------------
\
Let $S_n$ be the symmetric group on the letters $1,\dots,n$. For $1 \le i \le n-1$ denote $s_i := (i,i+1)$, a simple reflection (adjacent transposition) in $S_n$. For a composition $\mu = (\mu_1, \dots, \mu_t)$ of $n$ let $$s_\mu := (1, 2, \ldots, \mu_1)(\mu_1+1, \mu_1+2, \ldots, \mu_1+\mu_2) \cdots \in S_n,$$ a product of $t$ cycles of lengths $\mu_1, \mu_2, \ldots, \mu_t$ consisting of consecutive letters. The permutation $s_\mu$ may be obtained from the product $s_1 s_2 \cdots s_{n-1}$ of all simple reflections (in the usual order) by deleting the factors $s_{\mu_1+\ldots+\mu_k}$ for all $1 \le k < t$; equivalently, $$s_\mu = \prod_{i \in I(\mu)} s_i.$$
The [*descent set*]{} of a permutation $\pi\in S_n$ is ${\operatorname{Des}}(\pi):=\{i : \ \pi(i)>\pi(i+1)\}$.
The [*descent set*]{} of a standard Young tableaux $T$ is the set ${\operatorname{Des}}(T):=\{1 \le i \le n-1 : i+1 \hbox{ lies southwest of } i\}$.
$\mu$-unimodality {#subs.prelim_unimodality}
-----------------
\
A sequence $(a_1, \ldots, a_n)$ of distinct positive integers is [*unimodal*]{} if there exists $1 \le m\le n$ such that $a_1 > a_2 > \ldots > a_m < a_{m+1} < \ldots < a_n$. (This definiton differs slightly from the commonly used one, where all inequalities are reversed.)
Let $\mu=(\mu_1,\dots,\mu_t)$ be a composition of $n$. A sequence of $n$ positive integers is $\mu$-[*unimodal*]{} if the first $\mu_1$ integers form a unimodal sequence, the next $\mu_2$ integers form a unimodal sequence, and so on. A permutation $\pi \in S_n$ is $\mu$-[*unimodal*]{} if the sequence $(\pi(1), \ldots, \pi(n))$ is $\mu$-unimodal. For example, $\pi=936871254$ is $(4,3,2)$-unimodal, but not $(5,4)$-unimodal.
Let $U_\mu$ be the set of all $\mu$-unimodal permutations in $S_n$.
A family of character formulas {#section:character formulas}
------------------------------
\
Let $\lambda$ and $\mu$ be partitions of $n$. Let $\chi^\lambda$ be the $S_n$-character of the irreducible representation $S^\lambda$, and let $\chi^\lambda_\mu$ be its value on a conjugacy class of cycle type $\mu$. The following formula for the irreducible characters is a special case of [@Ro2 Theorem 4]. For a direct combinatorial proof see [@Ra2].
\[t.c1\][@Ro2 Theorem 4]$$\chi^\lambda_\mu =
\sum\limits_{\pi\in {\mathcal C}\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $\mathcal C$ is any Knuth class of RSK-shape $\lambda$.
Let $\chi^{(k)}$ be the $S_n$-character defined by the symmetric group action on the $k$-th homogeneous component of the coinvariant algebra. Then
\[t.c2\][@APR1 Theorem 5.1]$$\chi^{(k)}_\mu =
\sum\limits_{\pi\in L(k)\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $L(k)$ is the set of all permutations of length $k$ in $S_n$.
A complex representation of a group or an algebra $A$ is called a [*Gelfand model*]{} for $A$ if it is equivalent to the multiplicity free direct sum of all the irreducible $A$-representations. Let $\chi^G$ be the character of the Gelfand model of $S_n$ (or of its group algebra).
\[t.c3\][@APR2 Theorem 1.2.3] $$\chi^G_\mu =
\sum\limits_{\pi\in I_n\cap U_\mu}
(-1)^{|{\operatorname{Des}}(\pi)\cap I(\mu)|},$$ where $I_n:=\{\sigma\in S_n : \sigma^2=id\}$ is the set of all involutions in $S_n$.
More character formulas of this type are described in Subsection \[sec:fine\_sets\].
In this paper we propose a general setting for all of these results. In particular, we provide an answer to the following question.
\[q.invertible\] Are these character formulas invertible?
Two families of matrices {#section:matrices}
========================
It is well known that [*partitions*]{} of $n$ are the natural indices for the characters and conjugacy classes of $S_n$. It turns out that a major step towards an answer to Question \[q.invertible\] is to use, instead, [*compositions*]{} of $n$ (or, equivalently, subsets of $[n-1]$), in spite of the apparent redundancy. A surprising structure arises, in the form of a certain matrix $A_{n-1}$. In fact, it is convenient to define two “coupled” families of matrices, $(A_n)$ and $(B_n)$; for each nonnegative integer $n$, $A_n$ and $B_n$ are square matrices of order $2^n$, with entries $0$, $\pm 1$, which may be viewed as asymmetric variants of Walsh-Hadamard matrices. These matrices and some of their properties will be presented in this section.
We shall give two equivalent definitions for these matrices. The explicit definition is closer in spirit to the subsequent applications, but the recursive definition is very simple to describe and easy to use, and will therefore be presented first.
A recursive definition
----------------------
\
Recall the well known [*Walsh-Hadamard (Sylvester)*]{} matrices, defined by the recursion $$H_n = \left(\begin{array}{cc}
H_{n-1} & H_{n-1} \\
H_{n-1} & -H_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $H_0 = (1)$.
\[d.AB\_recursion\] Define, recursively, $$A_n = \left(\begin{array}{cc}
A_{n-1} & A_{n-1} \\
A_{n-1} & -B_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $A_0 = (1)$, and $$B_n = \left(\begin{array}{cc}
A_{n-1} & A_{n-1} \\
0 & -B_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $B_0 = (1)$.
Each of the matrices $A_n$ and $B_n$ may be obtained from the corresponding Walsh-Hadamard matrix $H_n$, all the entries of which are $\pm 1$, by replacing some of the entries by $0$.
$$A_1 = \left(\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right)
\qquad B_1 = \left(\begin{array}{cc}
1 & 1 \\
0 & -1
\end{array}\right)$$
$$A_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 \\
1 & -1 & 0 & 1
\end{array}\right)
\qquad B_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1 \\
0 & 0 & -1 & -1 \\
0 & 0 & 0 & 1
\end{array}\right)$$
An explicit definition
----------------------
\
It will be convenient to index the rows and columns of $A_n$ and $B_n$ by subsets of the set $\{1,\ldots,n\}$.
\[def.Pn\] Let $P_n$ be the power set (set of all subsets) of $[n]:=\{1,\ldots,n\}$. Endow $P_n$ with the anti-lexicographic linear order: for $I, J \in P_n$, $I \ne J$, let $m$ be the largest element in the symmetric difference $I \triangle J := (I \cup J) \setminus (I \cap J)$, and define: $I < J \iff m \in J$.
The linear order on $P_3$ is $$\emptyset < \{1\} < \{2\} < \{1, 2\} < \{3\} < \{1, 3\} < \{2, 3\} < \{1, 2, 3\}.$$
\[def.intervals\] For $I \in P_n$ let $I_1, \ldots, I_t$ be the sequence of [*runs*]{} (maximal consecutive intervals) in $I$, namely: $I$ is the disjoint union of the $I_k$ ($1\le k \le t$), and each $I_k$ is a nonempty set of the form $\{ m_k+1, m_k+2, \ldots, m_k+\ell_k \}$ with $\ell_k\ge 1$ $(\forall k)$ and $0\le m_1 < m_1+\ell_1 < m_2 < m_2+\ell_2 < \ldots < m_t < m_t+\ell_t \le n$. In particular, $|I| = \ell_1 + \ldots + \ell_t$.
For $I = \{1,2,4,5,6, 8, 10\}\in P_{10}$: $I_1 = \{1,2\}$, $I_2 =
\{4,5,6\}$, $I_3 = \{8\}$, $I_4 = \{10\}$.
Order $P_n$ as in Definition \[def.Pn\]. The entries of the Walsh-Hadamard matrix $H_n = (h_{I,J})_{I, J \in P_n}$ are explicitly given by the formula $$h_{I,J} := (-1)^{|I \cap J|} \qquad(\forall I, J \in P_n).$$
A [*prefix*]{} of an interval $I =\{m+1, \ldots, m + \ell\}$ is an interval of the form $\{m +1, \ldots, m + p\}$, for $0 \le p \le \ell$.
\[t.AB\_explicit\][(Explicit definition)]{} Order $P_n$ as in Definition \[def.Pn\], and let $I_1, \ldots, I_t$ be the runs of $I\in P_n$. Then:
- $A_n = (a_{I,J})_{I, J \in P_n}$, where $$a_{I,J} =
\begin{cases}
(-1)^{|I \cap J|}, & \text{if $I_k \cap J$ is a prefix of $I_k$ for each $k$;}\\
0, & \text{otherwise.}
\end{cases}$$
- $B_n = (b_{I,J})_{I, J \in P_n}$, where: $$b_{I,J} =
\begin{cases}
(-1)^{|I \cap J|}, & \text{if $I_k \cap J$ is a prefix of $I_k$ for each $k$, and}\\
& n \not\in I \setminus J;\\
0, & \text{otherwise.}
\end{cases}$$
It will be convenient here to define $A_n$ and $B_n$ explicitly as in the lemma, and then show that they satisfy the recursions in Definition \[d.AB\_recursion\].
We shall start with $A_n$. Clearly $A_0 = (1)$.
For $I, J \in P_n$ ($n \ge 1$) denote $I' := I \setminus \{n\}$ and $J' := J \setminus \{n\}$.
The “upper left” quarter of $A_n$ corresponds to $I, J \in P_n$ such that $n \not\in I$ and $n \not\in J$. In this case, clearly $a_{I,J}$ in $A_n$ is the same as $a_{I',J'}$ in $A_{n-1}$.
Similarly when $n \not\in I$ and $n \in J$, and also when $n \in I$ and $n \not\in J$: $|I \cap J| = |I' \cap J'|$, and $I_k \cap J$ is a prefix of $I_k$ for all $k$ if and only if $I'_k \cap J$ is a prefix of $I'_k$ for all $k$.
The “lower right” quarter of $A_n$ corresponds to $I, J \in P_n$ such that $n \in I \cap J$. If $n-1 \not\in I$ then $I_k \cap J$ is a prefix of $I_k$ for all $k$ if and only if $I'_k \cap J$ is a prefix of $I'_k$ for all $k$. Also $|I \cap J| = |I' \cap J'| + 1$, so that $a_{I,J}$ in $A_n$ is equal to $-a_{I',J'}$ in $A_{n-1}$ and also to $-b_{I',J'}$ in $B_{n-1}$ (since $n-1 \not\in I'$ so $n-1 \not\in I' \setminus J'$). If $n-1 \in I \cap J$ then, again, $a_{I,J}$ in $A_n$ is equal to $-a_{I',J'}$ in $A_{n-1}$ and also to $-b_{I',J'}$ in $B_{n-1}$ (since $n-1 \in J'$ so $n-1 \not\in I' \setminus J'$). Finally, if $n-1 \in I$ but $n-1 \not\in J$ then, for the last run $I_t$ of $I$, $I_t \cap J$ is not a prefix of $I_t$, and thus $a_{I,J} = 0$ in $A_n$ as well as $-b_{I',J'} = 0$ in $B_{n-1}$ (since $n-1 \in I' \setminus J'$).
We have proved the recursion for $A_n$. The entries of $B_n$ are equal to the corresponding entries of $A_n$, except for those in the quarter corresponding to $(I,J)$ with $n \in I$ and $n \not\in J$, which are all zeros (since $n \in I \setminus J$). This proves the recursion for $B_n$ as well.
Determinants
------------
\
It turns out that the invertibility of $A_n$ is the key factor in an answer to Question \[q.invertible\].
\[t.An-determinant\] $A_n$ and $B_n$ are invertible for all $n \ge 0$. In fact, $$\det(A_n) = (n+1) \cdot \prod_{k=1}^{n} k^{2^{n-1-k} (n+4-k)} \qquad(n \ge 2)$$ while $\det(A_0) = 1$ and $\det(A_1) = -2$, and $$\det(B_n) = \prod_{k=1}^{n} k^{2^{n-1-k} (n+2-k)} \qquad(n \ge 2)$$ while $\det(B_0) = 1$ and $\det(B_1) = -1$.
A proof of Theorem \[t.An-determinant\] will be given in the next section. For comparison, $$\det(H_n) = 2^{2^{n - 1} n} \qquad(n \ge 2)$$ with $\det(H_0) = 1$ and $\det(H_1) = -2$.
Eigenvalues
-----------
\
Consider the matrix $$A_2 = \left(\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 \\
1 & -1 & 0 & 1
\end{array}\right).$$ As an asymmetric matrix, it might conceivably have non-real eigenvalues. Surprisingly, computation shows that its characteristic polynomial is $$(x^2 - 3)(x^2 - 4),$$ and thus all its eigenvalues are (up to sign) square roots of positive integers!
This is not a coincidence. The following combinatorial description of the eigenvalues of $A_n$ and $B_n$, which was stated as a conjecture in an earlier version of this paper, has recently been proved by Gil Alon.
\[t.GilAlon\] [(G. Alon [@Alon])]{}
- The roots of the characteristic polynomial of $A_n$ are in $2:1$ correspondence with the compositions of $n$: each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ corresponds to a pair of eigenvalues $\pm \sqrt{\pi_\mu}$ of $A_n$, where $$\pi_\mu := \prod_{i=1}^{t} (\mu_i+1).$$
- The roots of the characteristic polynomial of $B_n$ are in $2:1$ correspondence with the compositions of $n$: each composition $\mu = (\mu_1, \ldots, \mu_t)$ of $n$ corresponds to a pair of eigenvalues $\pm \sqrt{\pi'_\mu}$ of $B_n$, where $$\pi'_\mu := \prod_{i=1}^{t-1} (\mu_i+1).$$
Surprising connections between the eigenvalues and the diagonal elements of $A_n^2$, as well as the column sums of some related matrices, appear in Theorem \[t.col\_sums\] below.
Möbius inversion {#section:AM_BM}
================
In this section we prove Theorem \[t.An-determinant\]. Our approach is to study certain matrices, with more transparent structure, obtained from $A_n$ and $B_n$ by a transformation corresponding to (poset theoretic) Möbius inversion.
Auxiliary definitions
---------------------
\
Let us define certain auxiliary families of matrices.
\[d.ZM\_recursion\] Define, recursively, $$Z_n = \left(\begin{array}{cc}
Z_{n-1} & Z_{n-1} \\
0 & Z_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $Z_0 = (1)$, as well as $$M_n = \left(\begin{array}{cc}
M_{n-1} & -M_{n-1} \\
0 & M_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $M_0 = (1)$.
$Z_n$ is the [*zeta matrix*]{} of the poset $P_n$ with respect to [*set inclusion*]{} (not with respect to its linear extension, described in Definition \[def.Pn\]). Thus $Z_n = (z_{I,J})_{I,J \in P_n}$ is a square matrix, with entries satisfying $$z_{I,J} =
\begin{cases}
1, & \text{if $I \subseteq J$;}\\
0, & \text{otherwise.}
\end{cases}$$
$M_n = Z_n^{-1}$ is the corresponding [*Möbius matrix*]{}, expressing the Möbius function (see [@Rota]) of the poset $P_n$. Thus $M_n = (m_{I,J})_{I,J \in P_n}$ has entries satisfying $$m_{I,J} =
\begin{cases}
(-1)^{|J \setminus I|}, & \text{if $I \subseteq J$;}\\
0, & \text{otherwise.}
\end{cases}$$
Denote $AM_n := A_n M_n$, $BM_n := B_n M_n$ and $HM_n := H_n M_n$.
It follows from Definitions \[d.AB\_recursion\] and \[d.ZM\_recursion\] that $$\label{eq.AMn}
AM_n = \left(\begin{array}{cc}
AM_{n-1} & 0 \\
AM_{n-1} & -(AM_{n-1} + BM_{n-1})
\end{array}\right)
\qquad(n \ge 1)$$ with $AM_0 = (1)$ and $$\label{eq.BMn}
BM_n = \left(\begin{array}{cc}
AM_{n-1} & 0 \\
0 & -BM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $BM_0 = (1)$, as well as $$\label{eq.HMn}
HM_n = \left(\begin{array}{cc}
HM_{n-1} & 0 \\
HM_{n-1} & -2HM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ with $HM_0 = (1)$.
The block triangular form of $AM_n$ and block diagonal form of $BM_n$ facilitate a recursive computation of the determinants of $A_n$ and $B_n$.
A proof of Theorem \[t.An-determinant\]
---------------------------------------
\
By recursion (\[eq.BMn\]), $$\det(BM_n) = \det(AM_{n-1}) \det(-BM_{n-1}) \qquad(n \ge 1).$$ Now $M_n$ is an upper triangular matrix with $1$-s on its diagonal, so that $$\det(M_n) = 1.$$ We conclude that $$\label{eq.Bn}
\det(B_n) = \delta_{n-1} \det(A_{n-1}) \det(B_{n-1}) \qquad(n\ge 1),$$ where $$\delta_n = (-1)^{2^{n}} =
\begin{cases}
-1, & \text{if $n = 0$;} \\
1, & \text{if $n \ge 1$.}
\end{cases}$$ Similarly, for any scalar $t$, $$AM_n + tBM_n = \left(\begin{array}{cc}
(t+1)AM_{n-1} & 0 \\ AM_{n-1} & -AM_{n-1} - (t+1)BM_{n-1}
\end{array}\right)
\qquad(n \ge 1)$$ and a similar argument yields $$\det(A_n + tB_n) = \delta_{n-1} \det((t+1)A_{n-1})
\det(A_{n-1} + (t+1)B_{n-1}) \qquad(n\ge 1).$$
It follows that $$\begin{aligned}
\det(A_n)
&= \delta_{n-1} \det(A_{n-1}) \det(A_{n-1} + B_{n-1}) \\
&= \delta_{n-1} \det(A_{n-1}) \delta_{n-2} \det(2 A_{n-2}) \det(A_{n-2} + 2B_{n-2}) \\
&= \ldots \\
& = \left( \prod_{k=1}^{n} \delta_{n-k} \det(k A_{n-k}) \right) \cdot \det(A_0 + n B_0) = \\
& = -(n+1) \cdot \prod_{k=1}^{n} k^{2^{n-k}} \cdot \prod_{k=1}^{n} \det(A_{n-k}) \qquad(n \ge 1).\end{aligned}$$ Since $A_0 = (1)$ it follows that $\det(A_n) \ne 0$ for any nonnegative integer $n$, and therefore $$\det(A_n) / \det(A_{n-1}) =
\frac{-(n+1)}{-n} \cdot n \cdot \prod_{k=1}^{n-1} k^{2^{n-1-k}} \cdot \det(A_{n-1}) \qquad(n\ge 2).$$ The solution to this recursion, with initial value $\det(A_1) = -2$, is $$\det(A_n) = (n+1) \cdot \prod_{k=1}^{n} k^{2^{n-1-k} (n+4-k)} \qquad(n \ge 2).$$ Recursion (\[eq.Bn\]) above, with initial value $\det(B_1) = -1$, now yields $$\det(B_n) = \prod_{k=1}^{n} k^{2^{n-1-k} (n+2-k)} \qquad(n \ge 2).$$ For comparison, $$\det(H_n) = 2^{2^{n-1}} \det(H_{n-1})^2 \qquad(n\ge 2)$$ with initial value $\det(H_1) = -2$, so that $$\det(H_n) = 2^{2^{n - 1} n} \qquad(n \ge 2).$$
We can also write $$\det(A_n) = \prod_{k=1}^{n+1} k^{a_{n+1-k}}\qquad(n \ge 2),$$ where the sequence $(a_0, a_1, \ldots) = (1, 2, 5, 12, 28, 64, \ldots)$ coincides with [@Sloane sequence A045623].
Properties of the transformed matrices {#section:matrix_entries}
======================================
In this section we describe some additional properties of the transformed matrices $AM_n$ and $BM_n$, namely: Explicit expressions for their entries, for the entries of their inverses, and for their row sums and column sums. The latter are surprisingly related to the eigenvalues of $A_n$ and $B_n$ described in Theorem \[t.GilAlon\] above. The proofs of most of the results in this section follow from recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]), and are therefore indicated only when additional ingredients are present.
Matrix entries
--------------
\
We shall now compute explicitly the entries of $AM_n$ and $BM_n$, starting with $HM_n$ as a “baby case”.
$$HM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -2 & 4 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -2 & 4 & 0 & 0 \\
1 & 0 & -2 & 0 & -2 & 0 & 4 & 0 \\
1 & -2 & -2 & 4 & -2 & 4 & 4 & -8
\end{array}\right)$$
This generalizes to an explicit description of the entries of $HM_n$, which follows easily from recursion (\[eq.HMn\]).
\[t.HMentries\]$$(HM_n)_{I,J} \ne 0 \iff J \subseteq I$$ and $$(HM_n)_{I,J} \ne 0 \,{\Longrightarrow}\, (HM_n)_{I,J} = (-2)^{|J|}.$$
The corresponding results for $AM_n$ and $BM_n$ are much more subtle (and interesting). Their proofs follow, in general, from recursions (\[eq.AMn\]) and (\[eq.BMn\]).
\[t.LT\] For every $n\ge 0$, the matrices $AM_n$ and $BM_n$ are lower triangular.
$(AM_n) \cdot Z_n$ is an $LU$ factorization of $A_n$; similarly for $B_n$.
$$AM_1 = \left(\begin{array}{cc}
1 & 0 \\
1 & -2
\end{array}\right)
\qquad BM_1 = \left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right)$$
$$AM_2 = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 \\
1 & 0 & -2 & 0 \\
1 & -2 & -1 & 3
\end{array}\right)
\qquad BM_2 = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)$$
$$AM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -1 & 3 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & -2 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -2 & 4 & 0 & 0 \\
1 & 0 & -2 & 0 & -1 & 0 & 3 & 0 \\
1 & -2 & -1 & 3 & -1 & 2 & 1 & -4
\end{array}\right)$$ $$BM_3 = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & -2 & 0 & 0 & 0 & 0 & 0 \\
1 & -2 & -1 & 3 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -1
\end{array}\right)$$
Apparently, $AM_n$ has the same zero pattern and the same sign pattern as $HM_n$. $BM_n$ also has the same sign pattern, but has more zero entries. The absolute values of entries in both families are more intricate than in $HM_n$.
\[t.AB\_entries\] [(Entries of $AM_n$ and $BM_n$)]{}
- [**Zero pattern:**]{} $$(AM_n)_{I,J} \ne 0 \iff J \subseteq I$$ and $$(BM_n)_{I,J} \ne 0 \iff J \subseteq I \text{\ and\ } {\operatorname{maxout}}(J) =
{\operatorname{maxout}}(I),$$ where $${\operatorname{maxout}}(I) := \max\{0 \le i \le n \,|\, i \not\in I\} \qquad(\forall
I \in P_n).$$
- [**Signs:**]{} $$(AM_n)_{I,J} \ne 0 {\Longrightarrow}{\operatorname{sign}}((AM_n)_{I,J}) = (-1)^{|J|}$$ and $$(BM_n)_{I,J} \ne 0 {\Longrightarrow}{\operatorname{sign}}((BM_n)_{I,J}) = (-1)^{|J|}.$$
- [**Absolute values:**]{} For $I, J \in P_n$, let $J_1,
\ldots, J_t$ be the runs (maximal consecutive intervals) in $J$. For $J_k = \{m_k + 1, \ldots, m_k + \ell_k\}$ $(1 \le k \le t)$, let $$c_k(I) =
\begin{cases}
0, & \text{if $m_k \in I$;} \\
1, & \text{otherwise.}
\end{cases}$$ Then $$(AM_n)_{I,J} \ne 0 {\Longrightarrow}|(AM_n)_{I,J}| = \prod_{k=1}^{t} (|J_k| +
1)^{c_k(I)}$$ and $$(BM_n)_{I,J} \ne 0 {\Longrightarrow}|(BM_n)_{I,J}| = \prod_{k=1}^{t'} (|J_k|
+ 1)^{c_k(I)},$$ where $$t' =
\begin{cases}
t-1, & \text{if $n \in I$ (equivalently, $n \in J$);} \\
t, & \text{otherwise.}
\end{cases}$$
It is clear from recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]) that all the entries in column $J$ of $AM_n$ (or $BM_n$) have sign $(-1)^{|J|}$ or are zero, exactly as in $HM_n$.
Comparison of the two recursions shows that wherever $AM_n$ has a zero entry so does $BM_n$, but not conversely. The zero pattern of $AM_n + BM_n$ is therefore the same as that of $AM_n$, and thus recursions (\[eq.AMn\]) and (\[eq.HMn\]) imply that $AM_n$ and $HM_n$ have the same zero pattern. The zero pattern of $BM_n$ now follows from recursion (\[eq.BMn\]).
Finally, the explicit formulas for the absolute values of entries are relevant, of course, only when $J \subseteq I$. They are a little difficult to come up with, but easy to confirm by recursion.
Let $I, J \in P_n$ satisfy $J \subseteq I$. Then:
- $$|(AM_n)_{I,J}| \le |(HM_n)_{I,J}| = 2^{|J|},$$ with equality if and only if $|J_k| = 1$ for each $k$ for which $m_k \not\in I$.
- $$|(BM_n)_{I,J}| \le |(AM_n)_{I,J}|,$$ with equality if and only if either $n \not\in I$ or $m_t \in I$.
$|J_k| + 1 \le 2^{|J_k|}$, with equality if and only if $|J_k| =
1$.
An alternative description of the entries may be given in terms of compositions, using the correspondence $\mu \longleftrightarrow I(\mu)$ described in Subsection \[subs.prelim\_compositions\] above.
\[t.AB\_entries\_comp\] [(Entries of $AM_n$ and $BM_n$, composition version)]{}\
Let ${\lambda}$ and $\mu$ be compositions of $n+1$. Write $(AM_n)_{{\lambda}, \mu}$ instead of $(AM_n)_{I({\lambda}), I(\mu)}$, and similarly for $BM_n$.
- [**Zero pattern:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \iff \text{\rm $\mu$ is a refinement of ${\lambda}$}$$ and $$\begin{aligned}
(BM_n)_{{\lambda},\mu} \ne 0 &\iff& \text{\rm $\mu$ is a refinement of ${\lambda}$ and}\\
& & \text{\rm the last component of ${\lambda}$ is unrefined in $\mu$}.\end{aligned}$$
- [**Signs:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, {\operatorname{sign}}((AM_n)_{{\lambda},\mu}) = (-1)^{n+1-\ell(\mu)}$$ and $$(BM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, {\operatorname{sign}}((BM_n)_{{\lambda},\mu}) = (-1)^{n+1-\ell(\mu)},$$ where $\ell(\mu)$ is the number of components of $\mu$.
- [**Absolute values:**]{} $$(AM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, |(AM_n)_{{\lambda}, \mu}| = \prod_i \mu_{\rm init}({\lambda}_i)$$ and $$(BM_n)_{{\lambda},\mu} \ne 0 \,{\Longrightarrow}\, |(BM_n)_{{\lambda}, \mu}| = \prod_i ' \mu_{\rm init}({\lambda}_i),$$ where $\mu_{\rm init}({\lambda}_i)$ is the first component in the subdivision (in $\mu$) of the component ${\lambda}_i$ of ${\lambda}$, and $\prod_i '$ is a product over all values of $i$ except the last one.
Diagonal entries
----------------
\
The following corollary of Theorem \[t.AB\_entries\](3) and Theorem \[t.AB\_entries\_comp\](3) is stated, simultaneously, in terms of a composition $\mu$ of $n+1$ and the corresponding subset $J = I(\mu)$ of $[n]$. Different indices ($i$ and $k$) are used for the components of $\mu$ and the runs of $J$, since runs correspond only to the components statisfying $\mu_i > 1$.
\[t.AB\_diag\] [(Diagonal and last row of $AM_n$)]{}
- The diagonal entries of $AM_n$ are $$|(AM_n)_{J,J}| = \prod_i \mu_i = \prod_k (|J_k| + 1)$$ and the entries in its last row are $$|(AM_n)_{[n],J}| = \mu_1 =
\begin{cases}
|J_1| + 1, & \text{if $1 \in J$;}\\
1, & \text{otherwise.}
\end{cases}$$
- Each nonzero entry $(AM_n)_{I,J}$ divides the corresponding diagonal entry $(AM_n)_{J,J}$ and is divisible by the corresponding last row entry $(AM_n)_{[n],J}$.
For any finite set $J$ of positive integers, let ${\operatorname{ord}}(J) :=\sum_{j \in J} 2^{j-1}$. This is the ordinal number of $J$ in the anti-lexicographic order on $P_n$ (see Definition \[def.Pn\]), where the enumeration starts with $0$ for the empty set. The number ${\operatorname{ord}}(J)$ is independent of $n$, as long as $J \in P_n$. Define $$a_{{\operatorname{ord}}(J)} := |(AM_n)_{J,J}|, $$ and consider the resulting sequence $(a_m)_{m\ge 0}$ of absolute values of diagonal entries of the “limit matrix” $AM_{\infty} = \lim_{n \to \infty} AM_n$.
The sequence $(a_m)$ satisfies the recursion $$a_0 = 0,\ \ \ a_{2m} = a_m,\ \ \ a_{4m+1} = 2a_{2m},\ \ \ a_{4m+3}
= 2a_{2m+1} - a_m \quad(\forall m \ge 0).$$
Consider the formula in Corollary \[t.AB\_diag\](1) expressing a diagonal entry of $AM_n$ in terms of the corresponding run lengths. We shall not distinguish a set $J$ from its ordinal number ${\operatorname{ord}}(J)$.
(The set corresponding to) $2m$ has the same runs as $m$, shifted forward by $1$, so that $a_{2m} = a_m$. $4m+1$ has the same runs as $2m$, shifted forward by $1$, plus a singleton run $\{1\}$, so that $a_{4m+1} = 2a_{2m}$.
If $m$ is even then $a_{4m+3} = 3a_m$ and $a_{2m+1} = 2a_m$, so that $a_{4m+3} = 2a_{2m+1} - a_m$. If $m$ is odd, let $\ell$ be the length of the first run in $m$. The corresponding runs in $2m+1$ and in $4m+3$ have lengths $\ell+1$ and $\ell+2$, respectively, so that again $a_{4m+3} = 2a_{2m+1} - a_m$.
The sequence $(a_m)$ coincides with [@Sloane sequence A106737]. Thus, in particular, $$a_m = \sum_{k=0}^m \left[ {m+k \choose m-k} {m \choose k} \mod 2 \right],$$ where the expression in the square brackets is interpreted as either $0$ or $1$ and summed as an ordinary integer.
Row sums and column sums
------------------------
\
The following two results, regarding row and column sums of $AM_n$ and $BM_n$, are stated, for simplicity, almost entirely in the language of compositions.
\[t.row\_sums\] [(Row sums of $AM_n$, $BM_n$)]{}\
Let ${\lambda}$ be a composition of $n+1$, and let $I = I({\lambda})$ be the corresponding subset of $[n]$.
- The sum of all entries in row $I$ of $AM_n$ (or $BM_n$, or $HM_n$) is $(-1)^{|I|}$.
- The sum of absolute values of all entries in row $I$ of $AM_n$ is $$\prod_i (2^{{\lambda}_i} - 1).
$$ The sum of absolute values of all entries in row $I$ of $BM_n$ is $$\prod_i ' (2^{{\lambda}_i} - 1),
$$ where $\prod_i '$ is a product over all values of $i$ except the last. In $HM_n$ the corresponding sum is $3^{|I|}$.
\[t.col\_sums\] [(Column sums of $AM_n$, $BM_n$ and diagonal entries of $A_n^2$, $B_n^2$)]{}\
Let $\mu$ be a composition of $n+1$, and let $J = I(\mu)$ be the corresponding subset of $[n]$. Let $\mu^*$ be the composition of $n$ obtained from $\mu$ by reducing its first component by $1$, without changing the other components: $\mu_1^* = \mu_1 - 1$, $\mu_i^* = \mu_i$ $(\forall i > 1)$.
- The sum of absolute values (also: absolute value of the sum) of all the entries in column $J$ of $AM_n$ is equal to the diagonal entry $(A_n^2)_{J,J}$, which in turn is equal to $$\prod_i (\mu_i^* + 1).
$$
- The sum of absolute values (also: absolute value of the sum) of all the entries in column $J$ of $BM_n$ is equal to the diagonal entry $(B_n^2)_{J,J}$, which in turn is equal to $$\prod_i ' (\mu_i^* + 1),
$$ where $\prod_i '$ is a product over all values of $i$ except the last.
- For comparison, the sum of absolute values of all the entries in column $J$ of $HM_n$ is equal to the diagonal entry $(H_n^2)_{J,J}$, which in turn is equal to the constant $2^n$.
The recursions for $A_n^2$ and $B_n^2$ are $$A_n^2 = \left(\begin{array}{cc}
2 A_{n-1}^2 & A_{n-1}(A_{n-1} - B_{n-1}) \\
(A_{n-1} - B_{n-1})A_{n-1} & A_{n-1}^2 + B_{n-1}^2
\end{array}\right)
\qquad(n \ge 1)$$ and $$B_n^2 = \left(\begin{array}{cc}
A_{n-1}^2 & A_{n-1}(A_{n-1} - B_{n-1}) \\
0 & B_{n-1}^2
\end{array}\right)
\qquad(n \ge 1),$$ with $A_0^2 = B_0^2 = (1)$. Denoting $\alpha_n(J) := (A_n^2)_{J,J}$, $\beta_n(J) := (B_n^2)_{J,J}$ and $J' := J \setminus \{n\}$, it follows that $$\alpha_n(J) =
\begin{cases}
2 \alpha_{n-1}(J'), & \text{if $n \not\in J$;} \\
\alpha_{n-1}(J') + \beta_{n-1}(J'), & \text{otherwise}
\end{cases}$$ and $$\beta_n(J) =
\begin{cases}
\alpha_{n-1}(J'), & \text{if $n \not\in J$;} \\
\beta_{n-1}(J'), & \text{otherwise,}
\end{cases}$$ with $\alpha_0(\emptyset) = \beta_0(\emptyset) = 1$.
A short look at recursions (\[eq.AMn\]) and (\[eq.BMn\]) shows that the above recursions also hold if $\alpha_n(J)$ and $\beta_n(J)$ denote the sum of absolute values of all the entries in column $J$ of $AM_n$ and of $BM_n$, respectively. The same recursions also hold if $\alpha_n(J)$ and $\beta_n(J)$ stand for the explicit product formulas in the theorem, since if $J = I(\mu)$ and $J' = J \setminus \{n\} = I(\mu')$ then $n \not\in J$ means that $\mu$ is obtained from $\mu'$ by appending a new component of size $1$, while $n \in J$ means that $\mu$ is obtained from $\mu'$ by increasing the last component by $1$.
Comparing Theorem \[t.col\_sums\] with Theorem \[t.GilAlon\] gives a surprising conclusion.
The multiset of eigenvalues, counted by algebraic multiplicity, of $A_n^2$ (or $B_n^2$) is equal to the multiset of diagonal entries of this matrix.
This is remarkable since, apparently, for $n \ge 3$ the matrices $A_n^2$ are not even diagonalizable!
Inverse matrix entries
----------------------
\
We would like to have explicit expressions for the entries of $A_n^{-1}$, for use in Section \[section:rep\_theory\_fine\]. This turns out to be difficult to do directly, and we shall compute, as an intermediate step, the entries of $AM_n^{-1}$. Note that $A_n^{-1} = M_n \cdot AM_n^{-1}$.
$$A_3^{-1} = \left(\begin{array}{cccccccc}
1/24 & 1/24 & 1/12 & 1/12 & 1/8 & 1/8 & 1/4 & 1/4 \\
1/8 & -1/24 & 1/12 & -1/12 & 5/24 & -1/8 & 1/12 & -1/4 \\
5/24 & 5/24 & -1/12 & -1/12 & 1/8 & 1/8 & -1/4 & -1/4 \\
1/8 & -5/24 & -1/12 & 1/12 & 1/24 & -1/8 & -1/12 & 1/4 \\
1/8 & 1/8 & 1/4 & 1/4 & -1/8 & -1/8 & -1/4 & -1/4 \\
5/24 & -1/8 & 1/12 & -1/4 & -5/24 & 1/8 & -1/12 & 1/4 \\
1/8 & 1/8 & -1/4 & -1/4 & -1/8 & -1/8 & 1/4 & 1/4 \\
1/24 & -1/8 & -1/12 & 1/4 & -1/24 & 1/8 & 1/12 & -1/4
\end{array}\right)$$ $$AM_3^{-1} = \left(\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 \\
1/2 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 \\
1/6 & -1/3 & -1/6 & 1/3 & 0 & 0 & 0 & 0 \\
1/2 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 \\
1/4 & -1/4 & 0 & 0 & -1/4 & 1/4 & 0 & 0 \\
1/6 & 0 & -1/3 & 0 & -1/6 & 0 & 1/3 & 0 \\
1/24 & -1/8 & -1/12 & 1/4 & -1/24 & 1/8 & 1/12 & -1/4
\end{array}\right)$$
We shall attempt an inductive computation of $AM_n^{-1}$. Recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]) yield corresponding recursions for the inverse matrices: $$AM_n^{-1} = \left(\begin{array}{cc}
AM_{n-1}^{-1} & 0 \\
(AM_{n-1} + BM_{n-1})^{-1} & -(AM_{n-1} + BM_{n-1})^{-1}
\end{array}\right)
\qquad(n \ge 1)
$$ and $$BM_n^{-1} = \left(\begin{array}{cc}
AM_{n-1}^{-1} & 0 \\
0 & - BM_{n-1}^{-1}
\end{array}\right)
\qquad(n \ge 1),
$$ with $AM_0^{-1} = BM_0^{-1} = (1)$; however, the recursion for $AM_n^{-1}$ involves the inverse of a new matrix, $AM_{n-1} + BM_{n-1}$, which in turn involves the inverse of $AM_{n-2} + 2 BM_{n-2}$, and so forth. We are thus led to consider a more general situation.
\[def.Mnx\] For any real number $x$ let $$M_n(x) := x AM_n + (1-x) BM_n.$$
In particular, $M_n(0) = BM_n$ and $M_n(1) = AM_n$.
For each $n \ge 0$ and $x > 0$, $$M_n^{-1}(x)_{I,J} \ne 0 \iff J \subseteq I$$ and, for $J \subseteq I$, $$M_n^{-1}(x)_{I,J} = (-1)^{|J|} \prod_{i \in I} \frac{d_{I, J, x}(i)}{e_{I, J, x}(i)},$$ where $I_1, \ldots, I_t$ are the runs of $I$ and, for $i \in I_k$:
- If $n \not\in I_k$ then $$d_{I, J, x}(i) :=
\begin{cases}
\max(I_k) - i + 1, & \text{if $i \in J$;}\\
1, & \text{otherwise}
\end{cases}$$ and $$e_{I, J, x}(i) := \max(I_k) - i + 2.$$
- If $n \in I_k$ (and thus necessarily $k = t$) then $$d_{I, J, x}(i) :=
\begin{cases}
(\max(I_k) - i) \cdot x + 1, & \text{if $i \in J$;}\\
x, & \text{otherwise}
\end{cases}$$ and $$e_{I, J, x}(i) := (\max(I_k) - i + 1) \cdot x + 1.$$
Let $x > 0$. By Definition \[def.Mnx\] and recursion formulas (\[eq.AMn\]) and (\[eq.BMn\]), $$\begin{aligned}
M_n(x)
&= \left(\begin{array}{cc}
AM_{n-1} & 0 \\
x AM_{n-1} & -(x AM_{n-1} + BM_{n-1})
\end{array}\right) \\
&= \left(\begin{array}{cc}
M_{n-1}(1) & 0 \\
x M_{n-1}(1) & -(1+x) M_{n-1}\left(\frac{x}{1+x}\right)
\end{array}\right)
\qquad(n \ge 1)\end{aligned}$$ with $M_0(x) = (1)$. Invertibility of $M_{n-1}(x)$ for all $x > 0$ clearly implies the invertibility of $M_{n}(x)$ for all $x > 0$. The inverse satisfies $$\label{e.Mnx_inverse_rec}
M_n^{-1}(x) = \left(\begin{array}{cc}
M_{n-1}^{-1}(1) & 0 \\
\frac{x}{1+x} M_{n-1}^{-1}\left(\frac{x}{1+x}\right) & \frac{-1}{1+x} M_{n-1}^{-1}\left(\frac{x}{1+x}\right)
\end{array}\right)
\qquad(n \ge 1)$$ with $M_0^{-1}(x) = (1)$, for all $x > 0$.
Recursion (\[e.Mnx\_inverse\_rec\]) shows that, indeed, for $x > 0$: $M_n^{-1}(x)_{I,J} \ne 0 \iff J \subseteq I$, and that the sign of this entry is $(-1)^{|J|}$.
Regarding the absolute value of this entry, assume by induction that the prescribed formula holds for $M_{n-1}^{-1}(x)$, $\forall x > 0$.
If $n \not\in I$ then also $n \not\in J$, and clearly $M_{n}^{-1}(x)_{I,J} = M_{n-1}^{-1}(1)_{I,J}$ satisfies the required formula.
If $n \in I$, let $I' := I \setminus \{n\}$, $J' := J \setminus \{n\}$ and $x' := \frac{x}{1+x}$. The assumed formula for $M_{n-1}^{-1}(x')_{I',J'}$ and the claimed formula for $M_{n}^{-1}(x)_{I,J}$ have exactly the same factors for all $i \not\in I_t$, so we need only consider $i \in I_t$.
If $|I_t| = 1$ (i.e., $n-1 \not\in I$) then there is nothing else in $M_{n-1}^{-1}(x')_{I',J'}$, but according to (\[e.Mnx\_inverse\_rec\]) there is an extra factor $\frac{1}{1+x}$ or $\frac{x}{1+x}$ in $M_{n}^{-1}(x)_{I,J}$ (depending on whether or not $n \in J$), and this is exactly the missing $d_{I, J, x}(n)/e_{I, J, x}(n)$.
Finally, assume that $|I_t| > 1$. Again, the extra factor $\frac{1}{1+x}$ or $\frac{x}{1+x}$ is exactly $d_{I, J, x}(n)/e_{I, J, x}(n)$. The other factors in $M_{n-1}^{-1}(x')_{I',J'}$, corresponding to $i \in I'_t$, are (if $i \in J$) $$\frac{d_{I', J', x'}(i)}{e_{I', J', x'}(i)} =
\frac{(n-1-i)x'+1}{(n-i)x'+1} =
\frac{(n-1-i)x+1+x}{(n-i)x+1+x}=
\frac{d_{I, J, x}(i)}{e_{I, J, x}(i)}$$ or (if $i \not\in J$) $$\frac{d_{I', J', x'}(i)}{e_{I', J', x'}(i)} =
\frac{x'}{(n-i)x'+1} =
\frac{x}{(n-i)x+1+x}=
\frac{d_{I, J, x}(i)}{e_{I, J, x}(i)},$$ exactly as claimed for $M_{n}^{-1}(x)_{I,J}$.
We are especially interested, of course, in the special case $x = 1$.
[(Entries of $AM_n^{-1}$)]{}\[t.AM\_inverse\]\
For each $n \ge 0$ $$(AM_n^{-1})_{I,J} \ne 0 \iff J\subseteq I$$ and, for $J\subseteq I$, $$(AM_n^{-1})_{I,J} = (-1)^{|J|} \prod_{i \in I} \frac{d_{I, J}(i)}{e_{I, J}(i)},$$ where $I_1, \ldots, I_t$ are the runs of $I$ and, for $i \in I_k$: $$d_{I, J}(i) :=
\begin{cases}
\max(I_k) - i + 1, & \text{if $i \in J$;}\\
1, & \text{otherwise}
\end{cases}$$ and $$e_{I, J}(i) := \max(I_k) - i + 2.$$ Equivalently, for $J\subseteq I$, $$(AM_n^{-1})_{I,J} = (-1)^{|J|} \prod_{k=1}^{t} \frac{1}{(|I_k|+1)!} \prod_{i \in I_k \cap J} (\max(I_k) - i + 1).
$$
Note that the denominator $\prod_{k=1}^{t} (|I_k| + 1)!$ is the cardinality of the parabolic subgroup $\langle I \rangle$ of $S_{n+1}$ generated by the simple reflections $\{s_i\,:\, i \in I\}$.
\[t.rows\_of\_inverse\]
- Each nonzero entry of $AM_n^{-1}$ is the inverse of an integer.
- In each row of $AM_n^{-1}$, the sum of absolute values of all the entries is $1$.
- In each row $I$ of $AM_n^{-1}$, the first entry $$(AM_n^{-1})_{I, \emptyset} = \prod_{k=1}^{t} \frac{1}{(|I_k|+1)!}$$ divides all the other nonzero entries and the diagonal entry $$(AM_n^{-1})_{I, I} = (-1)^{|I|} \prod_{k=1}^{t} \frac{1}{|I_k|+1}$$ is divisible by all the other nonzero entries, where a rational number $r$ is said to [*divide*]{} a rational number $s$ if the quotient $s/r$ is an integer.
Fine sets {#section:rep_theory_fine}
=========
The concept {#sec:concept}
-----------
\
A general setting for character formulas is introduced in this section. It will serve as a framework for the answer to Question \[q.invertible\].
Recall from Subsection \[subs.prelim\_compositions\] the definition of $I(\mu)$ for a composition $\mu$.
\[def.unimodal\_set\] Let $\mu=(\mu_1, \ldots, \mu_t)$ be a composition of $n$. A subset $J \subseteq [n-1]$ is [*$\mu$-unimodal*]{} if each run of $J \cap I(\mu)$ is a prefix of the corresponding run of $I(\mu)$; in other words, if $J \cap I(\mu)$ is a disjoint union of intervals of the form $\left[ \sum_{i=1}^{k-1}\mu_i+1, \sum_{i=1}^{k-1}\mu_i + \ell_k \right]$, where $0 \le \ell_k \le \mu_k-1$ for every $1 \le k \le t$.
A permutation $\pi \in S_n$ is $\mu$-unimodal according to the definition in Subsection \[subs.prelim\_unimodality\] if and only if its descent set ${\operatorname{Des}}(\pi)$ is $\mu$-unimodal according to Definition \[def.unimodal\_set\].
\[defn-fine\] Let ${{\mathcal{B}}}$ be a set of combinatorial objects, and let ${\operatorname{Des}}: {{\mathcal{B}}}\to P_{n-1}$ be a map which associates with each element $b\in {{\mathcal{B}}}$ a subset ${\operatorname{Des}}(b) \subseteq [n-1]$. Denote by ${{\mathcal{B}}}^\mu$ the set of elements in ${{\mathcal{B}}}$ whose “descent set” ${\operatorname{Des}}(b)$ is $\mu$-unimodal. Let $\rho$ be a complex $S_n$-representation. Then ${{\mathcal{B}}}$ is called a [*fine set*]{} for $\rho$ if, for each composition $\mu$ of $n$, the character of $\rho$ at a conjugacy class of cycle type $\mu$ satisfies $$\label{defn-fine1}
\chi^\rho_\mu=\sum\limits_{b\in {{\mathcal{B}}}^\mu} (-1)^{|{\operatorname{Des}}(b)\cap I(\mu)|}.$$
It follows from Theorems \[t.c1\], \[t.c2\] and \[t.c3\] that
\[fine-examples\]
- Any Knuth class of RSK-shape $\lambda$ is a fine set for the Specht module $S^\lambda$.
- The set of permutations of a fixed Coxeter length $k$ in $S_n$ is a fine set for the $k$-th homogeneous component of the coinvariant algebra of $S_n$.
- The set of involutions in $S_n$ is a fine set for the Gelfand model of $S_n$.
More examples of fine sets are given in Subsection \[sec:fine\_sets\].
Distribution of descent sets
----------------------------
\
\[sec:descent\_sets\] We are now ready to state our main application.
\[t.main\] If ${{\mathcal{B}}}$ is a fine set for an $S_n$-representation $\rho$ then the character values of $\rho$ determine the distribution of descent sets over ${{\mathcal{B}}}$. In particular, for every $I\subseteq [n-1]$, the number of elements in ${{\mathcal{B}}}$ whose descent set contains $I$ satisfies $$|\{b\in B : {\operatorname{Des}}(b) \supseteq I\}| =
\frac{1}{|\langle I \rangle|}
\sum\limits_{J\subseteq I} (-1)^{|J|}\chi^\rho(c_J)
\prod_{k=1}^{t} \prod_{i \in I_k \cap J} (\max(I_k) - i + 1),$$ where $I_1, \ldots, I_t$ are the runs in $I$, $|\langle I \rangle|$ is the cardinality of the parabolic subgroup of $S_n$ generated by $\{s_i : i\in I\}$, and $c_I$ is any Coxeter element in this subgroup.
The mapping $\mu \mapsto I(\mu)$ (see Subsection \[subs.prelim\_compositions\]) is a bijection between the set of all compositions of $n$ and the set $P_{n-1}$ of all subsets of $[n-1]$. For a subset $J=\{j_1,\dots,j_k\}\subseteq [n-1]$ with $j_1<j_2<\cdots<j_k$ let $c_J$ be the product $s_{j_1}s_{j_2}\cdots s_{j_k} \in S_n$. This is a Coxeter element in the parabolic subgroup generated by $\{s_i : i\in J\}$, and its cycle type is (the partition corresponding to) the composition $\mu$, where $J = I(\mu)$. Let $x^\rho$ be the vector with entries $\chi^\rho(c_J)$, where the subsets $J \in P_{n-1}$ are ordered anti-lexicographically as in Definition \[def.Pn\].
Similarly, let $v^{{\mathcal{B}}}=(v_J^{{\mathcal{B}}})_{J \in P_{n-1}}$ be the vector with entries $$v^{{\mathcal{B}}}_J:=|\{b\in {{\mathcal{B}}}: {\operatorname{Des}}(b)=J\}| \qquad(\forall J \in P_{n-1}).$$
By Definition \[defn-fine\] and Lemma \[t.AB\_explicit\]($i$), ${{\mathcal{B}}}$ is a fine set for $\rho$ if and only if $$\label{e.xAv}
x^\rho = A_{n-1} v^{{\mathcal{B}}},$$ where $x^\rho$ and $v^{{\mathcal{B}}}$ are written as column vectors. By Theorem \[t.An-determinant\], $A_{n-1}$ is an invertible matrix, which proves that $x^\rho$ uniquely determines $v^{{\mathcal{B}}}$.
The explicit formula follows from Corollary \[t.AM\_inverse\], as soon as equation (\[e.xAv\]) is written in the form $$Z_{n-1} v^{{\mathcal{B}}}= AM_{n-1}^{-1} x^\rho.$$
The Inclusion-Exclusion Principle (namely, multiplication by $M_{n-1}$) gives an equivalent form of the explicit formula.
Let $B$ be a fine set for an $S_n$-representation $\rho$. For every $I\subseteq [n-1]$, the number of elements in $B$ with descent set exactly $D$ satisfies $$|\{b\in B : {\operatorname{Des}}(b) = D\}| =
\sum_J \chi^\rho(c_J) \sum_{I: D \cup J \subseteq I} (-1)^{|I \setminus D|} (AM_{n-1}^{-1})_{I,J}
$$ where $(AM_{n-1}^{-1})_{I,J}$ is as in Corollary \[t.AM\_inverse\] and the notation is as in Theorem \[t.main\].
Permutation statistics versus character theory {#sec:versus}
----------------------------------------------
\
By Theorem \[t.main\], certain statements in permutation statistics have equivalent statements in character theory. In particular,
\[ps-rt\] Given two symmetric group modules with fine sets, the isomorphism of these modules is equivalent to equi-distribution of the descent set on their fine sets.
Combining Theorem \[t.main\] with Definition \[defn-fine\].
Here is a distinguished example. Recall the major index of a permutation $\pi$, $${\operatorname{maj}}(\pi):=\sum\limits_{i\in {\operatorname{Des}}(\pi)} i.$$ For a subset $I\subseteq [n-1]$ denote ${{\mathbf x}}^I:=\prod_{i\in I} x_i$. The following is a fundamental theorem in permutation statistics.
\[FS-thm\] [(Foata-Schützenberger)]{} [@FS79] $$\sum_{\pi\in S_n} {{\mathbf x}}^{{\operatorname{Des}}(\pi)}q^{\ell(\pi)} =
\sum_{\pi\in S_n} {{\mathbf x}}^{{\operatorname{Des}}(\pi)}q^{{\operatorname{maj}}(\pi^{-1})}.$$
See also [@GG].
For $0\le k\le {n\choose 2}$ denote by $R_k$ the $k$-th homogeneous component of the coinvariant algebra of the symmetric group $S_n$. The following is a classical theorem in invariant theory.
\[St-thm\] [(Lusztig-Stanley)]{} [@St79 Prop. 4.11] For a partition $\lambda$ denote by $m_{k,\lambda}$ the number of standard Young tableaux of shape $\lambda$ with major index $k$. Then $$R_k \cong \bigoplus_{\lambda\vdash n} m_{k,\lambda} S^\lambda,$$ where the sum is over all partitions of $n$ and $S^\lambda$ denotes the irreducible $S_n$-module indexed by $\lambda$.
It follows from Corollary \[ps-rt\] that
The Foata-Schützenberger Theorem is equivalent to the Lusztig-Stanley Theorem.
First, notice that the set of permutations $B_k=\{\pi\in S_n:\
{\operatorname{maj}}(\pi^{-1})=k\}$ is a disjoint union of Knuth classes, where for each partition $\lambda\vdash n$, there are exactly $m_{k,\lambda}$ Knuth classes of RSK-shape $\lambda$ in this disjoint union. Combining this fact with Proposition \[fine-examples\](1) implies that $B_k$ is a fine set for the representation $\rho_k:=
\bigoplus_{\lambda\vdash n} m_{k,\lambda} S^\lambda$.
On the other hand, by Proposition \[fine-examples\](2), the set of permutations $L_k=\{\pi\in S_n:\ \ell(\pi)=k\}$ is a fine set for $R_k$.
Combining these facts with Corollary \[ps-rt\], $\rho_k\cong
R_k$ if and only if the distributions of the descent set over $B_k$ and $L_k$ are equal.
A combinatorial proof of the Lusztig-Stanley Theorem as an application of Foata-Schützenberger’s Theorem appears in [@Ro-Schubert]. The opposite implication is new.
More examples of fine sets {#sec:fine_sets}
--------------------------
\
The following criterion for fine sets is useful.
\[t.condition-fine\] Let $\rho$ be an $S_n$-representation, let $\{C_b : b\in {{\mathcal{B}}}\}$ be a basis for the representation space, and let ${\operatorname{Des}}: {{\mathcal{B}}}\to P_{n-1}$ be a map. If for every $1 \le i
\le n-1$ and $b, v \in {{\mathcal{B}}}$ there are suitable coefficients $a_i(b,v)$ such that $$s_i (C_b)=
\begin{cases}
-C_b, &\hbox{\rm if } i\in {\operatorname{Des}}(b);\\
C_b+\sum\limits_{v \in {{\mathcal{B}}}\text{\rm\ s.t.\ } i\in{\operatorname{Des}}(v)}
a_i(b,v) C_v, &\hbox{\rm otherwise}
\end{cases}$$ then ${{\mathcal{B}}}$ is a fine set for $\rho$.
The proof is a natural extension of the proof of [@Ro-Schubert Theorem 1] and is omitted.
Two well known bases which satisfy the assumptions of Proposition \[t.condition-fine\] are the Kazhdan-Lusztig basis for the group algebra [@KL (2.3.b), (2.3.d)] and the Schubert polynomial basis for the coinvariant algebra [@BGG Theorem 3.14(iii)][@APR11]. Since $S_n$ embeds naturally in classical Weyl groups of rank $n$, it follows that Kazhdan-Lusztig cells, as well as subsets of elements of fixed Coxeter length in these groups, are fine sets for the $S_n$-action.
Another useful criterion is the following. For a partition $\nu\vdash n$ let ${{\rm {SYT}}}(\nu)$ be the set of standard Young tableaux of shape $\nu$.
\[criterion1\] A subset ${{\mathcal{B}}}\subseteq S_n$ is a fine set if and only if for every partition $\nu\vdash n$ there exist a nonnegative integer $m(\nu,{{\mathcal{B}}})$ such that $$\sum\limits_{\pi\in {{\mathcal{B}}}} {\bf x}^{{\operatorname{Des}}(\pi)}=
\sum\limits_{\nu\vdash n} m(\nu,{{\mathcal{B}}}) \sum\limits_{T\in SYT(\nu)}
{\bf x}^{{\operatorname{Des}}(T)}.$$
Follows from Theorem \[t.main\].
It follows that a subset ${{\mathcal{B}}}\subseteq S_n$ is a fine set if and only if the sum of quasi-symmetric functions $\sum\limits_{\pi \in
{{\mathcal{B}}}} F_{{\operatorname{Des}}(\pi)}$ is Schur positive. By [@GR Thm. 2.1] (as reformulated in [@R13 Thm. 2.2]), conjugacy classes in the symmetric group satisfy this criterion. It follows that subsets of the symmetric group, which are closed under conjugation, are fine sets.
Another example which satisfies this criterion has been given recently. A permutation $\pi\in S_n$ is called an [*arc permutation*]{} if for every $1\le k\le n$ the set $\{\pi(1),\dots,\pi(k)\}$ forms an interval in the cyclic group ${\mathbb{Z}}_n$ (where $n$ is identified with 0). By [@ER Theorem 5], the subset of arc permutations in $S_n$ satisfies the criterion of Proposition \[criterion1\]; thus, the subset of arc permutations is a fine set in $S_n$.
We conclude with a list of the known fine subsets of the symmetric group.
The following subsets of $S_n$ are fine sets:
- Subsets closed under Knuth relations. In particular, Knuth classes, inverse descent classes and $321$-avoiding permutations.
- Subsets closed under conjugation.
- Permutations of fixed Coxeter length.
- The set of arc permutations.
The first two examples appear in [@GR Thm. 5.5]. It is a challenging problem to find a characterization of fine subsets in $S_n$.
[99]{}
R. M. Adin, A. Postnikov, and Y. Roichman, [*Hecke algebra actions on polynomial rings*]{}, J. Algebra [**233**]{} (2000), 594–613.
R. M. Adin, A. Postnikov, and Y. Roichman, [*On characters of Weyl groups*]{}, Discrete Math. [**226**]{} (2001), 355–358.
R. M. Adin, A. Postnikov, and Y. Roichman, [*Combinatorial Gelfand models*]{}, J. Algebra [**320**]{} (2008), 1311–1325.
G. Alon, [*Eigenvalues of the Adin-Roichman matrices*]{}, preprint, 2013. [arXiv:1301.5760]{}.
I. N. Bernstein, I. M. Gelfand, S. I. Gelfand, [*Schubert cells and cohomology of the spaces $G/P$*]{}, Usp. Mat. Nauk. [**28**]{} (1973), 3–26.
S. Elizalde and Y. Roichman, [*Arc permutations*]{}, J.Algebraic Combin., to appear. D. Foata and M. P. Schützenberger, [*Major index and inversion number of permutations*]{}, Math. Nachr. [**83**]{} (1978), 143–159.
A. M. Garsia and I. Gessel, [*Permutation statistics and partitions*]{}, Adv. Math. [**31**]{} (1979), 288–305.
I. M. Gessel and C. Reutenauer, [*Counting permutations with given cycle structure and descent set*]{}, J. Combin. Theory Ser. A 64 (1993), 189�-215.
D. Kazhdan and G. Lusztig, [*Representations of Coxeter groups and Hecke algebras*]{}, Invent. Math. [**53**]{} (1979), 165–184.
A. Ram, [*An elementary proof of Roichman’s rule for irreducible characters of Iwahori-Hecke algebras of type A*]{}, in: Mathematical essays in honor of Gian-Carlo Rota, Progr. Math., 161, Birkhäuser, Boston, 1998, 335–342.
Y. Roichman, [*A recursive rule for Kazhdan-Lusztig characters*]{}, Adv. in Math. [**129**]{} (1997), 24–45.
Y. Roichman, [*Schubert polynomials, Kazhdan-Lusztig basis and characters*]{}, Formal Power Series and Algebraic Combinatorics (Vienna, 1997). Discrete Math. [**217**]{} (2000), 353–365.
Y. Roichman, [*A note on the number of $k$-roots in $S_n$*]{}, preprint, 2013.
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[^1]: Both authors were partially supported by Internal Research Grants from the Office of the Rector, Bar-Ilan University
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: '[Using an equivariant version of Connes’ Thom Isomorphism,w]{}e prove that equivariant $K$-theory is invariant under strict deformation quantization for a compact Lie group action.'
author:
- 'Xiang Tang and Yi-Jun Yao'
title: 'K-theory of Equivariant Quantization'
---
Introduction {#sec:intro}
============
Let $\alpha$ be a strongly continuous action of ${{\mathbb R}}^n$ on a $C^*$-algebra $A$, and $J$ be a skew-symmetric matrix on ${{\mathbb R}}^n$. Rieffel [@rieffel:quantization] constructed a strict deformation quantization $A_J$ of $A$ via oscillatory integrals $$\label{eq:star}
a\times_J b:=\int_{{{\mathbb R}}^n\times {{\mathbb R}}^n}
\alpha_{Ju}(a)\alpha_{v}(b)e^{2\pi i u\cdot v} du dv,$$ for $u,v\in {{\mathbb R}}^n$, and $a,b\in A^\infty$ (the smooth subalgebra of $A$ for $\alpha$). Such a construction gives rise to many interesting examples of noncommutative manifolds, e.g. quantum tori, $\theta$-deformation of $S^4$, etc. In [@rieffel:k-quant], Rieffel proved that the $K$-theory of $A_J$ is equal to the $K$-theory of the original algebra $A$, [by using Connes’ Thom isomorphism of $K$-theory]{}.
In this paper, we are interested in examples that the algebra $A$ is also equipped with a strongly continuous action $\beta$ by a compact group $G$. When the two actions commute, [the results in [@rieffel:k-quant]]{} naturally generalize to the equivariant setting. An easy observation is that, as the $G$-action $\beta$ commutes with the ${{\mathbb R}}^n$-action $\alpha$, naturally $\alpha$ can be lifted to a strongly continuous action $\tilde{\alpha}$ on the crossed product algebra $A\rtimes_\beta G$. Rieffel’s construction (\[eq:star\]) applies to the ${{\mathbb R}}^n$-action $\tilde{\alpha}$ on $A\rtimes_\beta G$, and defines a quantization algebra $(A\rtimes_\beta
G)_J$. By the commutativity between $\alpha$ and $\beta$, we easily check that $\beta$ lifts to a strongly continuous action $\tilde{\beta}$ on $A_J$, and $A_J\rtimes_{\tilde{\beta}}G$ is isomorphic to $(A\rtimes_\beta G)_J$. Now by [the results]{} on the $K$-theory of strict deformation quantization [@rieffel:k-quant], we conclude that $$K_\bullet(A\rtimes_\beta G)=K_\bullet((A\rtimes_\beta
G)_J)=K_\bullet(A_J\rtimes_{\tilde{\beta}}G).$$
In this paper, we generalize the above discussion of equivariant quantization to a situation where the actions $\alpha$ and $\beta$ do not commute. Define $GL(J)$ to be the group of invertible matrices $g$ such that $g^t J g =J$, and $SL_n({{\mathbb R}},
J):=SL_n({{\mathbb R}})\bigcap GL(J)$. We remark that when $J$ is the standard skew-symmetric matrix on ${{\mathbb R}}^{2n}$, $GL(J)$ is the linear symplectic group. Let $\rho: G\rightarrow SL_n({{\mathbb R}}, J)$ be a group homomorphism such that $$\label{eq:action}
\beta_g\alpha_x=\alpha_{\rho_g(x)}\beta_g,\ \ \ \text{for any}\ g\in
G, x\in {{\mathbb R}}^n.$$ When $\rho$ is a trivial group homomorphism, the actions $\alpha$ and $\beta$ commute.
A natural example of such a system appears as follows.
\[ex:z-2\] Let $G={{\mathbb Z}}_2={{\mathbb Z}}/2{{\mathbb Z}}$ act on ${{\mathbb R}}^{2n}$ by reflection with respect to the origin. Let ${{\mathbb Z}}^{2n}$ be the integer lattice in ${{\mathbb R}}^{2n}$. The $2n$-torus ${\mathbb
T}^{2n}={{\mathbb R}}^{2n}/{{\mathbb Z}}^{2n}$ inherits an action of ${{\mathbb Z}}_2$ from the ${{\mathbb Z}}_2$ action on ${{\mathbb R}}^{2n}$. The group ${{\mathbb R}}^{2n}$ acts on ${{\mathbb R}}^{2n}$ by translation and descends to act on $\mathbb {T}^{2n}$. Let $A$ be the $C^*$-algebra of continuous functions on ${\mathbb T}^{2n}$, and $J$ be the standard symplectic matrix on ${{\mathbb R}}^{2n}$. The action $\alpha$ (and $\beta$) of ${{\mathbb R}}^{2n}$ (and ${{\mathbb Z}}_2$) on $A$ is the dual action of the corresponding actions on ${\mathbb T}^{2n}$. We easily check that Eq. (\[eq:action\]) holds in this case with $\rho$ being the natural inclusion ${{\mathbb Z}}_2\hookrightarrow
SL_{2n}({{\mathbb R}}, J)$.
Different from the case where the actions $\alpha$ and $\beta$ commute, for a nontrivial $\rho:G\rightarrow SL_n({{\mathbb R}}, J)$, the ${{\mathbb R}}^n$-action $\alpha$ on $A$ does not lift naturally to an action on $A\rtimes_\beta G$. Therefore, we cannot apply Rieffel’s deformation construction to the algebra $A\rtimes_\beta G$. Nevertheless, a simple calculation shows that $$\beta_g(a\times_J b)=\beta_g(a)\times_J \beta_g(b),\ \
\beta_g(a^*)=\beta_g(a)^*,$$ which shows that the $G$-action $\beta$ is still well-defined on $A_J$. Accordingly, we can consider the crossed product algebra $A_J\rtimes_\beta G$. Applying this construction to Ex. \[ex:z-2\], we obtain $A_J\rtimes_\beta {{\mathbb Z}}_2$, which is well studied in literature, e.g. [@ech-luck-phil:k-qtorus], [@farsi:qorbfld], [@ku:k-theory], and [@walters:k-theory].
In this paper, we prove the following theorem about the $K$-theory groups of $A_J\rtimes_\beta G$.
\[thm:main-k\]If the actions $\alpha$, $\beta$ and the group homomorphism $\rho$ satisfy (\[eq:action\]), then $$K_\bullet(A_J\rtimes_\beta G)\cong K_{\bullet}(A\rtimes_\beta G),\ \
\ \bullet=0,1.$$
The proof of this theorem will be presented in the next section. As applications of our theorem, we recover some results of [@ech-luck-phil:k-qtorus] on the computation of the $K$-groups of ${\mathbb Z}_i$-quantum tori for $i=2,3,4,6$, and we [also apply]{} these results to the $\theta$-deformation [@connes-landi:theta] of $S^4$.\
[**Acknowledgments:**]{} We would like to thank Professors S. Echterhoff and N. Higson for explaining the relationship between the equivariant Thom isomorphism theorem (Theorem \[thm:equ-thom\]) and the Connes-Kasparov conjecture. We also want to thank Professor H. Li for interesting discussions and comments which greatly helped us to improve the readability of the paper. [We thank Professor H. Oyono-Oyono for helping us to remove a separability assumption in a previous version.]{} And we are grateful to an anonymous referee for pointing out one mistake and various places to improve accuracy in a previous version of this paper. Tang’s research is partially supported by NSF grant 0900985. Tang would like to thank the School of Mathematical Sciences of Fudan University and the Max-Planck Institute for their warm hospitality of his visits. Yao’s research is partially supported by NSF grant 0903985 and NSFC grant 10901039 [and 11231002]{}.
Proof of the main theorem
=========================
Our proof of Theorem \[thm:main-k\] is an equivariant generalization of Rieffel’s proof in [@rieffel:k-quant]. We first prove the theorem under the assumption that $A$ is separable. Following [@rieffel:k-quant], we will decompose our proof into 3 steps. In our proof, we need an equivariant generalization of Connes’ Thom isomorphism Theorem [@connes:thom]. We cannot find a proof[^1] of such a theorem in literature and therefore have decided to provide it below.
[**Step I.**]{} Following [the notations in]{} [@rieffel:k-quant], we let ${{\mathcal B}}^A$ be the space of smooth $A$-valued functions on ${{\mathbb R}}^n$ whose derivatives together with themselves are bounded on ${{\mathbb R}}^n$. Let ${{\mathcal S}}^A$ be the space of $A$-valued Schwartz functions on ${{\mathbb R}}^n$. The integral $$\langle f, g\rangle_A:=\int f(x)^*g(x)dx$$ defines an $A$-valued inner product on ${{\mathcal S}}^A$. Rieffel generalized the definition to ${{\mathcal B}}^A$ by using oscillatory integrals. Namely, given $J$, we define a product on ${{\mathcal B}}^A$ by $$(F\times_J G)(x):=\int F(x+Ju)G(x+v)e^{2\pi i u\cdot v}dudv,\qquad F,G\in
{{\mathcal B}}^A.$$ Furthermore, ${{\mathcal B}}^A$ acts on ${{\mathcal S}}^A$ by $$(L^J_F f)(x):=\int F(x+Ju)f(x+v)e^{2\pi i u\cdot v}dudv,\qquad F\in
{{\mathcal B}}^A,\ f\in {{\mathcal S}}^A.$$ The above two integrals are both oscillatory ones. Via the $A$-valued inner product on ${{\mathcal S}}^A$, we can equip ${{\mathcal B}}^A$ with the operator norm $\|\ \|_J$, and obtain a pre-$C^*$-algebra $({{\mathcal B}}^A_J,\times_J, \|\ \|_J)$. Denote the corresponding $C^*$-algebra by $\overline{{{\mathcal B}}}^A_J$. Meanwhile, ${{\mathcal S}}^A$ viewed as a $*$-ideal of ${{\mathcal B}}^A_J$ (cf. Rieffel, [@rieffel:quantization]), denoted by ${{\mathcal S}}^A_J$, can be completed into $\overline{{{\mathcal S}}}^A_J$.
With an action $\alpha$ of ${{\mathbb R}}^n$ on $A$, Rieffel [@rieffel:k-quant Prop. 1.1] introduced a strongly continuous ${{\mathbb R}}^n$-action $\nu$ on $\overline{{{\mathcal B}}}^A_J$ and also on $\overline{{{\mathcal S}}}^A_J$ by $$(\nu_t(F))(x):=\alpha_t(F(x-t)).$$ The fixed point subalgebra of this action $\nu$ is identified [@rieffel:k-quant Prop. 2.14] with the $C^*$-subalgebra of $\overline{{{\mathcal B}}}^A_J$ generated by elements $$\tilde{a}(x):=\alpha_x(a),\qquad a\in A^\infty,$$ which is exactly $A_J$.
[In [@rieffel:k-quant Thm. 3.2], it is ]{}proved that $A_J$ is strongly Morita equivalent to $\overline{{{\mathcal S}}}^A_J\rtimes_\nu
{{\mathbb R}}^n$. We will generalize this theorem to the equivariant setting with the $G$-action $\beta$. We introduce the $G$-action $\overline{\beta}$ on ${{{\mathcal B}}}^A_J$ by $$\overline{\beta}_g(F)(x):=\beta_g(F(\rho_{g^{-1}}(x))).$$ The exactly same arguments as [in]{} [@rieffel:k-quant Prop. 1.1] prove that the $G$-action $\overline{\beta}$ is strongly continuous on ${{\mathcal S}}^A$, therefore so is it on $\overline{{{\mathcal S}}}^A_J$.
\[prop:fixed-pt\]The crossed product algebras $A_J\rtimes_\beta
G$ and $(\overline{{{\mathcal S}}}^A_J\rtimes_\nu
{{\mathbb R}}^n)\rtimes_{\bar{\beta}}G$ are strongly Morita equivalent.
We will apply Combes’ theorem [@combes:crossed Sec. 6] on equivariant Morita equivalence [after proving]{} that the $G$-actions $\beta$ and $\bar{\beta}$ are Morita equivalent, which [will imply]{} the Morita equivalence we seek.
According to [@combes:crossed], two $G$-actions $\beta^1,\beta^2$ on $A$ and $B$ are Morita equivalent if there is a strong Morita equivalence bimodule $X$ between $A$ and $B$ such that there is a $G$-action $\beta$ on $X$ satisfying $$\begin{split}
\beta_g(a\xi)=\beta^1_g(a)\beta_g(\xi),\ \ \qquad\qquad
\qquad&\beta_g(\xi
b)=\beta_g(\xi)\beta^2_g(b),\\
_A\langle \beta_g(\xi_1),
\beta_g(\xi_2) \rangle=\beta^1_g(_A \langle \xi_1,
\xi_2\rangle),\qquad &\langle \beta_g(\xi_1),
\beta_g(\xi_2)\rangle_B=\beta^2_g(\langle \xi_1, \xi_2\rangle_B).
\end{split}$$ for $\xi, \xi_1, \xi_2\in X$.
Rieffel [@rieffel:k-quant] constructed a Morita equivalence bimodule between $A_J$ and $\overline{{{\mathcal S}}}^A_J\rtimes _\nu {{\mathbb R}}^n$. We recall it now. Let $C_\infty({{\mathbb R}}^n, A)$ be the $C^*$-algebra of $A$-valued functions on ${{\mathbb R}}^n$ that vanish at infinity. Let $\tau$ be the ${{\mathbb R}}^n$-action on ${{\mathcal B}}^A_J$ by translation, $(\tau_tF)(x)=F(t+x)$, and $\mu$ be the action of ${{\mathbb R}}^n$ on $C_\infty({{\mathbb R}}^n,A)$ by $$\mu_s(f)(x)=e^{2\pi i s\cdot x}f(x).$$ Define an action $\alpha$ of ${{\mathbb R}}^n$ on $\overline{{{\mathcal S}}}^A_J$ by $$\alpha_t(F)(x)=\alpha_t(F(x)).$$ Both $\mu$ and $\tau$ act on $C_\infty({{\mathbb R}}^n,A)$ and their combination gives an action of the Heisenberg group $H$ of dimension $2n+1$ on $C_\infty({{\mathbb R}}^n, A)$. This Heisenberg group action commutes with $\alpha$ and defines an $H\times {{\mathbb R}}^n$-action $\sigma$ on $C_\infty({{\mathbb R}}^n,
A)$. Define $X_0$ to be the subspace of $C_\infty({{\mathbb R}}^n, A)$ of $\sigma$-smooth vectors. Rieffel [@rieffel:k-quant Prop. 2.2] proved that $X_0$ is a $*$-subalgebra of ${{\mathcal S}}^A_J$ for any $J$, and a suitable completion $\overline{X}_0$ of $X_0$ serves as a strong Morita equivalence bimodule, which we refer to [@rieffel:k-quant] for details.
Define a right $A_J$-module structure on $\overline{X}_0$ by identifying $A_J$ with the subspace of $\nu$-invariant vectors in $\overline{{{\mathcal B}}}^A_J$, i.e. $$f\cdot a:=f\times_J \tilde{a},\qquad \text{for}\ a\in A^\infty$$ where $\tilde{a}\in C^\infty({{\mathbb R}}^n, A)$ is defined by $\tilde{a}(x)=\alpha_x(a)$. The algebra $\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n$ acts on $\overline{X}_0$ by $$\psi(f):=\int \psi(t)\times_J\nu_{t}(f)dt,\qquad \psi\in
\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n,\ f\in \overline{X}_0.$$ We define an $\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n$-valued inner product on $\overline{X}_0$ by $$_{\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n}\langle
f,g\rangle(x):=f\times_J \nu_x(g^*),\qquad x\in {{\mathbb R}}^n,\ f,g\in
\overline{X}_0,$$ and an $A_J$-valued inner product on $\overline{X}_0$ by $$\langle f,g\rangle_{A_J}:=\left(\int \alpha_t(f^*\times_J g(-t))dt\right),\qquad f,g\in
\overline{X}_0.$$ [We also know from [@rieffel:k-quant] that]{} $\left(\overline{X}_0,\,\,
_{\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n}\langle\ ,\ \rangle,
\langle\ ,\ \rangle_{A_J}\right)$ is a strong Morita equivalence bimodule between $\overline{{{\mathcal S}}}^A_J\rtimes_\nu {{\mathbb R}}^n$ and $A_J$.
We easily check the following identities between the actions $$\overline{\beta}_g\alpha_t=\alpha_{\rho_g(t)}\overline{\beta}_g,\qquad
\overline{\beta}_g\tau_t=\tau_{\rho_g(t)}\overline{\beta}_g, \qquad
\overline{\beta}_g\mu_{t}=\mu_{(\rho_g^T)^{-1}(t)}\overline{\beta}_g,\qquad
g\in G,\ t\in {{\mathbb R}}^n.$$ where $\rho_g^{T}$ is the transpose of $\rho_g$. These identities show that the $G$-action $\overline{\beta}$ on $C_\infty({{\mathbb R}}^n,
A)$ preserves the subspace $X_0$ of $\sigma$-smooth vectors. Using the property that $\beta$ and $\overline{\beta}$ act strongly continuously on $\overline{{{\mathcal S}}}^A_J$, we can easily check that $\beta$ and $\overline{\beta}$ are Morita equivalent $G$-actions in the sense of Combes [@combes:crossed]. Therefore, $A_J\rtimes_\beta G$ is strongly Morita equivalent to $(\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n)\rtimes_{\overline{\beta}}
G$.
As $A$ is separable, $A$ has a countable approximate identity. This implies that [@rieffel:k-quant Cor. 3.3] $A_J$ (and $\overline{{{\mathcal S}}}^A_J$) has a countable approximate identity. Accordingly, $A_J\rtimes_\beta G$ (and $(\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n)\rtimes_{\overline{\beta}}
G$) also has a countable approximate identity, and therefore has strictly positive elements. This together with the above Morita equivalence result shows that $A_J\rtimes_\beta G$ and $(\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n)\rtimes_{\overline{\beta}}
G$ are stably isomorphic. As stably isomorphic $C^*$-algebras have isomorphic $K$-groups, we conclude that $$K_\bullet(A_J\rtimes_\beta G)\cong
K_{\bullet}((\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n)\rtimes_{\beta}G).$$ [**Step II.**]{} [As we know, one powerful tool in dealing with the $K$-theory of $C^*$-algebras is Connes’ Thom isomorphism, which remains to this day one of the few ways to prove isomorphism results of $K$-groups for crossed products.]{} Let ${{\mathbb C}}_n$ be the complex Clifford algebra associated with ${{\mathbb R}}^n$. We first observe that the semidirect product group ${{\mathbb R}}^n\rtimes_{\rho} G $ is amenable, hence by Kasparov[^2] [@kasp:nov §6, Thm. 2.] we know that for a separable $G$-$C^*$-algebra $B$, there exists an isomorphism from $KK^i({{\mathbb C}}, B\rtimes ({{\mathbb R}}^n\rtimes G))$ to $KK^i({{\mathbb C}}, ((B\otimes {{\mathbb C}}_n)\rtimes G)$. [In other words, we need to use the following equivariant Thom isomorphism Theorem, which is a generalization of Connes’ Thom isomorphism Theorem [@connes:thom]. This is a key ingredient of the whole approach.]{}
\[thm:equ-thom\]Let ${{\mathbb R}}^n$ and $G$ act strongly continuously on a separable $C^*$-algebra $B$ with the actions denoted by $\alpha$ and $\beta$. Let $\rho:G\rightarrow
GL(n, {{\mathbb R}})$. If the actions $\alpha$ and $\beta$ satisfy Equation (\[eq:action\]), then $$K_\bullet((B\rtimes_\alpha
{{\mathbb R}}^n)\rtimes_{\beta} G)\cong
K^G_\bullet\big(B\rtimes_\alpha
{{\mathbb R}}^n\big)\cong K^G_{\bullet}(B\otimes {{\mathbb C}}_n)\cong
K_{\bullet}((B\otimes {{\mathbb C}}_n)\rtimes_{\beta} G),$$ where ${{\mathbb C}}_n$ is the complex Clifford algebra associated with ${{\mathbb R}}^n$.
Taking $B=\overline{{{\mathcal S}}}^A_J$ in the above theorem which is separable (as $A$ is separable), we conclude that $K_\bullet((\overline{{{\mathcal S}}}^A_J\otimes {{\mathbb C}}_n)\rtimes_{\bar{\beta}} G)$ is isomorphic to $K_{\bullet}((\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}{{\mathbb R}}^n)\rtimes_{\overline{\beta}}G)$.
[**Step III.**]{} Rieffel proved [@rieffel:quantization Prop. 5.2] that there is an isomorphism $$\label{eq:iso}
\overline{{{\mathcal S}}}^A_J\cong A\otimes {\mathcal K}\otimes C_\infty(V_0),$$ where ${\mathcal K}$ is the algebra of compact operators on an infinite dimensional separable Hilbert space ${{\mathcal H}}$, and $V_0$ is the kernel of $J$ in ${{\mathbb R}}^n$. Let $U$ be the orthogonal complement of $V_0$ in ${{\mathbb R}}^n$. It is easy to check that $U$ is a $J$-invariant subspace, and both $U$ and $V_0$ are $G$-invariant subspaces. As $G$ is compact, there is a $G$-invariant complex structure on $U$ compatible with $J|_U$ (viewed a symplectic form on $U$). Without loss of generality, we will just assume that $G$ preserves the standard Euclidean structure on $U$. The key observation in the proof of [@rieffel:quantization Prop. 5.2] is that when $A$ is the trivial $C^*$-algebra ${{\mathbb C}}$ and $J$ invertible, $\overline{{{\mathcal S}}}^{{\mathbb C}}_J$ is naturally identified as the space of compact operators, still denoted by $\mathcal K$, on the subspace ${{\mathcal H}}$ of $L^2(U)$ generated by elements $$g(\bar{z})e^{-\frac{\|z\|^2}{2}},$$ where $g$ is an anti-holomorphic function. As ${{\mathcal H}}$ is a $G$-invariant subspace, we can conclude that Rieffel’s isomorphism (\[eq:iso\]) is $G$-equivariant (note that $G$ acts on ${\mathcal {K}}$ by conjugation). By Combes’ result on G-equivariant Morita equivalence, $(A\otimes {\mathcal K}\otimes
C_\infty(V_0)\otimes{{\mathbb C}}_n)\rtimes_{\bar{\beta}} G$ is strongly Morita equivalent to $(A\otimes C_{\infty}(V_0)\otimes
{{\mathbb C}}_n)\rtimes_{\bar{\beta}} G$.
Now we look at the decomposition of ${{\mathbb R}}^n$ as $V_0\oplus U$. The Clifford algebra ${{\mathbb C}}_n$ associated with ${{\mathbb R}}^n$ is $G$-equivariantly isomorphic to ${{\mathbb C}}_{V_0}\otimes {{\mathbb C}}_{U}$, where ${{\mathbb C}}_{V_0}$ and ${{\mathbb C}}_U$ are the complex Clifford algebras associated with $V_0$ and $U$, respectively. Notice that $J$ restricts to define a symplectic form on $U$, and that the action of $G$ preserves both the restricted $J$ and the metric on $U$. Therefore the $G$-action on $U$ is $spin^c$. Hence, the algebra $(A\otimes C_{\infty}(V_0)\otimes
{{\mathbb C}}_n)\rtimes_{\bar{\beta}} G$ is $KK$-equivalent to $(A\otimes
C_{\infty}(V_0)\otimes {{\mathbb C}}_{V_0})\rtimes_{\bar{\beta}} G$. Again by the $G$-equivariant Thom isomorphism Thm. \[thm:equ-thom\] for the trivial $V_0$ action on $A$, we conclude that $$\begin{split}
K_{\bullet}((\overline{{{\mathcal S}}}^A_J\otimes{{\mathbb C}}_n)\rtimes_{\bar{\beta}}
G)&=K_{\bullet}((A\otimes {\mathcal K}\otimes
C_\infty(V_0)\otimes{{\mathbb C}}_n)\rtimes_{\bar{\beta}}
G)\\
&=K_\bullet(\big(A\otimes C_\infty(V_0)\otimes
{{\mathbb C}}_{V_0}\big)\rtimes_{\bar{\beta}} G)=K_\bullet(A\rtimes_\beta
G).
\end{split}$$
Summarizing Step I-III, we have the following equality, $$K_\bullet(A_J\rtimes_\beta G)\stackrel{\text{Step
I}}{===}K_\bullet((\overline{{{\mathcal S}}}^A_J\rtimes_{\nu}
{{\mathbb R}}^n)\rtimes_{\bar{\beta}} G)\stackrel{\text{Step
II}}{===}K_{\bullet}((\overline{{{\mathcal S}}}^A_J\otimes{{\mathbb C}}_n)\rtimes_{\bar{\beta}}
G)\stackrel{\text{Step III}}{===}K_\bullet(A\rtimes_\beta G).$$ [This completes the proof of Theorem \[thm:main-k\] under the assumption that $A$ is separable. For a general $C^*$-algebra $A$, we can write $A$ as an inductive limit of a net $A^I$ of separable ${{\mathbb R}}^n\rtimes_\rho G$-algebras. Then $A_J$ is an inductive limit of the net $A^I_J$ of separable $G$-algebras. As $K$-groups commutes with inductive limit, we conclude that $$K_\bullet(A_J\rtimes_\beta G)=\lim\limits_I K_\bullet(A_J^I\rtimes_\beta G)=\lim\limits_I K_\bullet(A^I\rtimes_\beta G)=K_\bullet(A\rtimes_\beta G).$$ This completes the proof of Theorem \[thm:main-k\] for general $C^*$-algebras.]{}
Equivariant Thom isomorphism {#sec:thom}
----------------------------
In [@connes:thom], Connes proved a fundamental theorem in noncommutative geometry which is a generalization of the classical Thom isomorphism in topology. Let ${{\mathbb R}}^n$ act strongly continuously on a separable $C^*$-algebra $A$. Connes’ theorem states that $$K_\bullet(A\rtimes {{\mathbb R}}^n)\cong K_{\bullet+n}(A).$$
\[thm:equ-thom\]Let ${{\mathbb R}}^n$ and $G$ act strongly continuously on a separable $C^*$-algebra $A$ with the actions denoted by $\alpha$ and $\beta$. Let $\rho:G\rightarrow
GL(n, {{\mathbb R}})$. If the actions $\alpha$ and $\beta$ satisfy Equation (\[eq:action\]), then $$K_\bullet(((A\otimes{{\mathbb C}}_n)\rtimes_\alpha
{{\mathbb R}}^n)\rtimes_{\beta} G)\cong
K^G_\bullet\big((A\otimes{{\mathbb C}}_n)\rtimes_\alpha
{{\mathbb R}}^n\big)\cong K^G_{\bullet}(A)\cong
K_{\bullet}(A\rtimes_{\beta} G),$$ where ${{\mathbb C}}_n$ is the complex Clifford algebra associated with ${{\mathbb C}}^n$.
The idea is to construct an invertible element $\varphi^G_\alpha$ in $KK_{n}^G(A, (A\otimes {{\mathbb C}}_n)\rtimes_\alpha {{\mathbb R}}^n)$ as a $G$-equivariant version of Fack-Skandalis’ proof [@fack-skandalis:thom] of Connes’ Thom isomorphism [@connes:thom].
As $G$ is compact, there is always a $G$-invariant metric on ${{\mathbb R}}^n$. Without loss of generality, we will assume in what follows that $G$ preserves the standard metric on ${{\mathbb R}}^n$. When the algebra $A$ is the complex numbers ${{\mathbb C}}$, Kasparov [@kasp:k-theory] defined two elements $$x_n\in KK^G({{\mathbb C}}, C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n),\qquad y_n\in
KK^G(C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n, {{\mathbb C}})$$ using the Hodge-de Rham operator $d+d^*$ on ${{\mathbb R}}^n$, where we equip ${{\mathbb R}}^n$ with the standard metric and $d^*$ is the adjoint of $d$. The key point is that using the symbol calculus introduced by Connes [@connes:ncg], we can twist the Hodge-de Rham operator by a $C^*$-algebra $A$, and therefore leads to generalizations of $x_n$ and $y_n$.
Let $\Delta=dd^*+d^*d$ be the Laplace-Beltrami operator, and $\Sigma$ be the principal symbol of $(d+d^*)(1+\Delta)^{-1/2}$, then $$\Sigma(x, \xi)=\frac{c(\xi)}{(1+\|\xi\|^2)^{\frac{1}{2}}},\qquad
x\in {{\mathbb R}}^n, \xi\in {{{\mathbb R}}^n} ^*,$$ where $c(\xi)$ is the Clifford multiplication of $\xi$ on $\wedge^\bullet {{{\mathbb C}}^n}^*$. A crucial property here is that $\Sigma(x,\xi)$ is independent of $x$ and only a function of $\xi$, which allows us to apply Connes’ pseudodifferential calculus [@connes:ncg]. Furthermore, we also observe that $(d+d^*)(1+\Delta)^{-1/2}$ is $G$-invariant, as $G$ is assumed to preserve the metric.
For a $C^*$-dynamical system $(A, {{\mathbb R}}^n, \alpha)$, let $t\mapsto
V_t$ be the canonical representation of ${{\mathbb R}}^n$ in $M(A\rtimes_\alpha {{\mathbb R}}^n)$, the multiplier algebra, with $V_{x}aV_{x}^*=\alpha_{x}(a)$ ($a\in A$).
Consider $B=A\otimes {{\mathbb C}}_n$ with an extended action $\alpha$ by ${{\mathbb R}}^n$, which acts trivially on the component ${{\mathbb C}}_n$. The smooth subalgebra $B^\infty$ of $\alpha$ is identified with $A^\infty\otimes {{\mathbb C}}_n$. As $\Sigma\in S^m({{\mathbb R}}^n, B^\infty)$ is a symbol of order $0$, $$\widehat{\Sigma}(x)=\int_{{{{\mathbb R}}^n}^*}\Sigma(\xi)e^{-i\xi(x)}d\xi$$ is a well-defined distribution on ${{\mathbb R}}^n$ with values in $B^\infty$. Following [@connes:ncg Prop. 8], we define $D_\alpha\in M(B^\infty)$ by $$D_\alpha:=\int \widehat{\Sigma}(x)V_x dx.$$
As $(d+d^*)(1+\Delta)^{-1/2}$ defines an element in $KK^G(B,
{{\mathbb C}})$, it is straightforward to check that $(B^\infty, id,
D_\alpha)$ defines an element $t^G_\alpha$ in $KK^G(A,
B\rtimes_\alpha {{\mathbb R}}^n)$. A direct computation via Fourier transform shows the following properties of $t^G_\alpha$: When $A$ is the trivial $C^*$-algebra of complex numbers, $t^G_\alpha$ is computed to be equal to $x_n\in KK^G({{\mathbb C}},
C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n)$ following Kasparov. Let $F_n=c(\xi)(1+\|\xi\|^2)^{-1/2}$ be a multiplier on $C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n$, and $x_n=(C_0({{\mathbb R}}^n)\otimes
{{\mathbb C}}_n, id, F_n)$. When $A$ is $C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n$ equipped with the standard ${{\mathbb R}}^n$-translation action $\hat{\alpha}$, $t^G_{\hat{\alpha}}$ is identified with the element $y_n\in
KK^G(C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n, {{\mathbb C}})$ via the $G$-equivariant Morita equivalence between $C_0({{\mathbb R}}^n)\rtimes_{\hat{\alpha}}{{\mathbb R}}^n$ and ${{\mathbb C}}$. For $\mathbb H = L^2({{\mathbb R}}^n, \wedge^\bullet {{\mathbb C}}^n)$, the algebra $C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n$ acts on $\mathbb H$ by Clifford multiplication, and $D_{\hat{\alpha}}$ is identified to be the operator $(d+d^*)(1+\Delta)^{-1/2}$ on $\mathbb H$. Now let $\hat{\alpha}$ be the dual action of ${{\mathbb R}}^n$ on $A\rtimes_\alpha {{\mathbb R}}^n$ and also on $B\rtimes_\alpha {{\mathbb R}}^n$, and $t^G_{\hat{\alpha}}$ be the element defined by $D_{\hat{\alpha}}$ in $KK^G(B, (B\otimes
{{\mathbb C}}_n)\rtimes_{\hat{\alpha}}{{\mathbb R}}^n)$. Via the $G$-equivariant Morita equivalence (the Takai duality) $B\otimes {{\mathbb C}}_n=A\otimes
{{\mathbb C}}_n\otimes {{\mathbb C}}_n\cong A$ and the $G$-equivariant Takai duality, we want to prove that the Kasparov products $$\label{eq:tensor}
t^G_\alpha \otimes_{B\rtimes_\alpha {{\mathbb R}}^n}
t^G_{\hat{\alpha}},\qquad t^G_{\hat{\alpha}}\otimes_{A} t^G_\alpha$$ are the element $1$ in $KK^G(A, A)$ and $KK^G(B\rtimes_{\hat{\alpha}} {{\mathbb R}}^n,
B\rtimes_{\hat{\alpha}}{{\mathbb R}}^n)$, respectively.
The proof of Equation (\[eq:tensor\]) is exactly identical to that one of [@fack-skandalis:thom Thm. 2]. The main step is to define a deformation $\alpha^s$ of the action $\alpha$ for $s\in
[0,1]$. Define $\alpha^s:{{\mathbb R}}^n\times A\to A$ by $$\alpha^s_x(a):=\alpha_{sx}(a),\qquad a\in A, x\in {{\mathbb R}}^n.$$ Then the same formula defines $t^G_{\alpha^s}\in KK^{G}(A, (A\otimes
{{\mathbb C}}_n)\rtimes_{\alpha}{{\mathbb R}}^n)$. Furthermore, $t^G_{\alpha^s}\otimes t^G_{\hat{\alpha}^s}$ defines a homotopy in $KK^G(A,A)$ for $s\in [0,1]$. This implies that $$t^G_\alpha \otimes_{B\rtimes_{\alpha} {{\mathbb R}}^n}
t^G_{\hat{\alpha}}=t^G_{\alpha^0}\otimes_{B\rtimes_{\alpha^0}{{\mathbb R}}^n}
t^G_{\hat{\alpha}^0}.$$ The product $t^G_{\alpha^0}\otimes_{B\rtimes_{\alpha^0}{{\mathbb R}}^n}
t^G_{\hat{\alpha}^0}$ is reduced to the computation [@kasp:k-theory] of $$x_n\otimes_{C_0({{\mathbb R}}^n)\otimes {{\mathbb C}}_n}y_n$$ as the action of ${{\mathbb R}}^n$ on $A$ is trivial. We conclude that $t^G_{\alpha^0}\otimes_{B\rtimes_{\alpha^0}{{\mathbb R}}^n}
t^G_{\hat{\alpha}^0}$ is equal to $1$ in $KK^G(A, A)$ by [@kasp:k-theory]. And similar computation shows that $t^G_{\hat{\alpha}}\otimes_{A} t^G_\alpha$ is 1 in $KK^G(B, B)$.
The Kasparov tensor product with the element $t^G_{\alpha}$ defines an isomorphism between $K^G_\bullet(A)$ and $K^G_{\bullet+n}(A\rtimes_\alpha {{\mathbb R}}^n)$.
Our Theorem \[thm:equ-thom\] is a generalization of Phillips’ theorem [@phi:c-star Thm. 6.3.2], where the $G$-action $\beta$ is assumed to commute with the ${{\mathbb R}}^n$-action $\alpha$.
Examples
========
In this section, we discuss some applications of Theorem \[thm:main-k\].
Noncommutative toroidal orbifolds
---------------------------------
We identify a 2-torus ${\mathbb T}^2$ by ${{\mathbb R}}^2/{{\mathbb Z}}^2$. ${{\mathbb R}}^2$ acts on itself by translation and induces an action $\alpha$ on ${\mathbb T}^2$. For $\theta\in{{\mathbb R}}$, we consider the symplectic form $J=\theta dx_1\wedge dx_2$ on ${{\mathbb R}}^2$. The group $SL_2({{\mathbb Z}})$ acts on ${{\mathbb R}}^2$ preserving the lattice ${{\mathbb Z}}^2$ and therefore also acts on ${\mathbb T}^2$, which is denoted by $\beta$. Inside $SL_2({{\mathbb Z}})$, there are cyclic subgroups generated by $$\begin{split}
\sigma_2=\left(\begin{array}{cc}-1&0\\
0&-1\end{array}\right),\qquad\qquad &\sigma_3=\left(\begin{array}{cc}-1&-1\\
1&0\end{array}\right)\\
\sigma_4=\left(\begin{array}{cc}0&-1\\
1&0\end{array}\right),\qquad\qquad\ \ &\sigma_6=\left(\begin{array}{cc}0&-1\\
1&1\end{array}\right).
\end{split}$$ The element $\sigma_i$ generates a cyclic subgroup ${{\mathbb Z}}_i$ of $SL_2({{\mathbb Z}})$ of order $i=2,3,4,6$. In this example, the group $SL_2({{\mathbb R}}, J)$ is identical to the group $SL_2({{\mathbb R}})$. Define $\rho:{{\mathbb Z}}_i\to SL_2({{\mathbb R}})$ to be the inclusion. And it is straightforward to check the actions $\beta$ of ${{\mathbb Z}}_i$ on ${\mathbb T}^2$, $\rho$ of ${{\mathbb Z}}_i$ on ${{\mathbb R}}^2$, and $\alpha$ of ${{\mathbb R}}^2$ on ${\mathbb T}^2$ satisfy Eq. (\[eq:action\]). As is explained in Sec. \[sec:intro\], the group ${{\mathbb Z}}_i$ naturally acts on Rieffel’s deformation $A_J$, which is the quantum torus $A_\theta$. Theorem \[thm:main-k\] states that $$K_\bullet(A_J\rtimes {{\mathbb Z}}_i)=K_\bullet(A\rtimes {{\mathbb Z}}_i).$$ We recover with a completely different proof the result of [@ech-luck-phil:k-qtorus Cor. 2.2]. We have brought the question of computation of $K$-groups of these noncommutative orbifolds to a purely topological setting, and we refer to [@ech-luck-phil:k-qtorus] and references therein for the explicit computation of the $K$-groups of the undeformed algebras $A\rtimes {{\mathbb Z}}_i$, $i=2,3,4,6$. For example, when $i=2$, the $K$-groups of $A\rtimes{{\mathbb Z}}_2$ are $$K_\bullet(A\rtimes {{\mathbb Z}}_2)\cong\left\{\begin{array}{ll}{{\mathbb Z}}^6,&\bullet=0,\\ 0,&\bullet=1.\end{array}\right.$$
Theta deformation
-----------------
Consider a 4-sphere $S^4$ centered at $(0,0,0,0,0)$ in ${{\mathbb R}}^5$ with radius $1$. In coordinates, it is the set $$\left\{(x_1, \cdots, x_5)|
x_1^2+x_2^2+x_3^2+x_4^2+x_5^2={1}\right\}.$$ Defines ${\mathbb T}^2$-action on $S^4$ by, for $0\leq t_1, t_2 < 2\pi$, $$\big((t_1,t_2),(x_1, \cdots, x_5)\big)\longrightarrow
(x_1, \cdots, x_5)\left(\begin{array}{ccccc}\cos(t_1)&\sin(t_1)&0&0&0\\
-\sin(t_1)&\cos(t_1)&0&0&0\\0&0&\cos(t_2)&\sin(t_2)&0\\
0&0&-\sin(t_2)&\cos(t_2)&0\\0&0&0&0&1
\end{array}\right).$$ The same formula as above also defines an ${{\mathbb R}}^2$-action $\alpha$ on $S^4$. The action $\beta$ of ${{\mathbb Z}}_2$ on $S^4$ is by reflection $$(\sigma_2, (x_1, \cdots, x_5))\longrightarrow (x_1, -x_2, x_3, -x_4,
x_5).$$ The group ${{\mathbb Z}}_2$ also acts on ${{\mathbb R}}^2$ by reflection $$\rho: \sigma_2\longrightarrow \left(\begin{array}{cc}-1&0\\
0&-1
\end{array}\right).$$ On ${{\mathbb R}}^2$, for $\theta\in {{\mathbb R}}$, consider the same symplectic form $J=\theta
dx_1\wedge dx_2$. It is easy to check that the actions $\alpha,
\beta, \rho$ satisfy Eq. (\[eq:action\]). Consider the algebra $C(S^4)$ of continuous functions on $S^4$. Rieffel’s construction defines a deformation $C(S^4_\theta)$ of $C(S^4)$ by $J$ and the action $\alpha$, which is the $\theta$-deformation [@connes-landi:theta] introduced by Connes and Landi. As is explained in Sec. \[sec:intro\], ${{\mathbb Z}}_2$ acts strongly continuously on $C(S^4_\theta)$. Theorem \[thm:main-k\] states that $$K_\bullet(C(S^4)\rtimes {{\mathbb Z}}_2)=K_\bullet(C(S^4_\theta)\rtimes
{{\mathbb Z}}_2).$$ The $K$-theory of $C(S^4)\rtimes {{\mathbb Z}}_2$ can be computed [@phi:k-theory] topologically as the Grothendieck group of the monoid of all isomorphism classes of ${{\mathbb Z}}_2$-equivariant vector bundles on $S^4$.
Notice that the quotient $S^4/{{\mathbb Z}}_2$ is an orbifold homeomorphic to $S^4$. As an orbifold, $S^4/{{\mathbb Z}}_2$ [@mo-pr:covering] has a good covering $\{U_i\}$ such that each $U_i$ and any none empty finite intersection $U_{i_1}\cap \cdots\cap
U_{i_k}$ is a quotient of a finite group action on ${{\mathbb R}}^4$. Such a good covering allows to compute the topological ${{\mathbb Z}}_2$-equivariant $K$-theory of $S^4$ by the Čech cohomology on $S^4/{{\mathbb Z}}_2$ of the sheaf ${\mathcal
K}^\bullet_{{{\mathbb Z}}_2}$ introduced by Segal [@segal:equivariant]. The restriction of ${\mathcal
K}^\bullet_{{{\mathbb Z}}_2}$ to an open chart $U$ of $S^4/{{\mathbb Z}}_2$ is defined to be the ${{\mathbb Z}}_2$-equivariant $K$-theory of $\pi^{-1}(U)$ with $\pi$ the canonical projection $S^4\to
S^4/{{\mathbb Z}}_2$. Locally, when $U$ is sufficiently small, we can compute $K^\bullet_{{{\mathbb Z}}_2}(U)$ to be $K^\bullet(\pi^{-1}(U)^{\sigma_2})\oplus K^\bullet(U)$, where $\pi^{-1}(U)^{\sigma_2}$ is the $\sigma_2$-fixed point submanifold. When $\bullet=0$, it is equal to ${{\mathbb Z}}|_{\pi^{-1}(U)^{\sigma_2}}\oplus {{\mathbb Z}}_{U}$, and when $\bullet=1$, it is zero. Gluing this local computation by the Mayer-Vietoris sequence, we conclude that $$K_0(C(S^4_\theta)\rtimes {{\mathbb Z}}_2)={{\mathbb Z}}^4,\qquad
K_1(C(S^4_\theta)\rtimes {{\mathbb Z}}_2)=0.$$
We observe that in the above example, the group ${{\mathbb Z}}_2$ is not essential. Our computations generalize to $K_\bullet(C^\infty(S^4_\theta)\rtimes {{\mathbb Z}}_i)$, for $i=3,4,6$.
[99]{} Combes, F., Crossed products and Morita equivalence, [*Proc. London Math. Soc.*]{} (3) 49 (1984), no. 2, 289–306.
Connes, A., $C^{\ast} $-algébres et géométrie différentielle. (French) [*C. R. Acad. Sci. Paris Sér.*]{} A-B 290 (1980), no. 13, A599–A604.
Connes, A., An analogue of the Thom isomorphism for crossed products of a $C^{\ast} $-algebra by an action of ${\mathbb R}$., [*Adv. in Math*]{}, 39 (1981), no. 1, 31–55.
Connes, A., Landi, G., Noncommutative manifolds, the instanton algebra and isospectral deformations, [*Comm. Math. Phys.*]{} 221 (2001), no. 1, 141–159.
Echterhoff, S., Lück, W., Phillips, N. C., and Walters, S., The structure of crossed products of irrational rotation algebras by finite subgroups of ${\rm SL}_2(\mathbb Z)$, [*J. Reine Angew. Math.*]{} 639 (2010), 173–221.
Farsi, C., Watling, N., Symmetrized non-commutative tori, [*Math. Ann.*]{} 296 (1993), 739-741.
719.
Kasparov, G., K-theory, group $C^*$-algebras, and higher signatures (conspectus). [*Novikov conjectures, index theorems and rigidity*]{}, Vol. 1 (Oberwolfach, 1993), 101-146, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, (1995).
Kumjian, A., On the K-theory of the symmetrized noncommutative torus, [*C.R. Math. Rep. Acad. Sci. Canada*]{} 12, 87-89 (1990).
Berlin, 1987.
Phillips, N. C., [*Equivariant $K$-theory for proper actions*]{}, Pitman Research Notes in Mathematics Series, 178, Longman Scientific $\&$ Technical, Harlow; copublished in the United States with John Wiley $\&$ Sons, Inc., New York, (1989).
Rieffel, M., Deformation quantization for actions of ${\mathbb
R}^d$, [*Mem. Amer. Math. Soc.*]{} 106 (1993), no. 506.
Rieffel, M., $K$-groups of $C^*$-algebras deformed by actions of ${\mathbb R}^d$, [*J. Funct. Anal.*]{} 116 (1993), no. 1, 199–214.
Moerdijk, I., Pronk, D. A. Simplicial cohomology of orbifolds, [*Indag. Math.*]{} 10 (1999), no. 2, 269–293.
Segal, G., Equivariant $K$-theory, [*Inst. Hautes Études Sci. Publ. Math.*]{} no. 34 (1968) 129–151.
Walters, S., Projective modules over the non-commutative sphere, [*J. London Math. Soc.*]{} (2) 51 (1995), no. 3, 589-602.
[Xiang Tang]{}, Department of Mathematics, Washington University, St. Louis, MO, 63130, U.S.A., Email: [email protected].
[Yi-Jun Yao]{}, School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R.China., Email: [email protected].
[^1]: As Prof. Echterhoff pointed out, this result is basically a corollary of the validity Connes-Kasparov Conjecture for the group ${{\mathbb R}}^n\rtimes
G$. The proof that we give here is relatively pedestrian using Connes’ pseudodifferential calculus [@connes:ncg].
[^2]: In [@kasp:nov §6, Thm. 2.], the connectivity of the group $G$ is assumed. But this assumption can be easily dropped using the same idea of the proof.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Stochastic gradient algorithms have been the main focus of large-scale learning problems and they led to important successes in machine learning. The convergence of SGD depends on the careful choice of learning rate and the amount of the noise in stochastic estimates of the gradients. In this paper, we propose a new adaptive learning rate algorithm, which utilizes curvature information for automatically tuning the learning rates. The information about the element-wise curvature of the loss function is estimated from the local statistics of the stochastic first order gradients. We further propose a new variance reduction technique to speed up the convergence. In our preliminary experiments with deep neural networks, we obtained better performance compared to the popular stochastic gradient algorithms.'
author:
- |
Caglar Gulcehre\
Université de Montréal\
`[email protected]` Marcin Moczulski\
University of Oxford\
`[email protected]` Yoshua Bengio\
Université de Montréal\
`[email protected]`
bibliography:
- 'myref.bib'
title: 'ADASECANT: Robust Adaptive Secant Method for Stochastic Gradient'
---
Introduction
============
In this paper we develop a stochastic gradient algorithm that reduces the burden of extensive hyper-parameter search for the optimization algorithm. The proposed algorithm exploits a low variance estimator of curvature of the cost function and uses it to obtain an automatically tuned adaptive learning rate for each parameter.
In the deep learning and numerical optimization, several papers suggest using a diagonal approximation of the Hessian (second derivative matrix of the cost function with respect to parameters), in order to estimate optimal learning rates for stochastic gradient descent over high dimensional parameter spaces [@becker1988improving; @schaul2012no; @lecun1993automatic]. A fundamental advantage of using such approximation is that inverting it is a trivial and cheap operation. However generally, for neural networks, the inverse of the diagonal Hessian is usually a bad approximation of the diagonal of the inverse of Hessian. Examples options of obtaining a diagonal approximation of Hessian are the Gauss-Newton matrix [@lecun2012efficient] or by finite differences [@schaul2013adaptive]. Such estimations may however be sensitive to noise due to the stochastic gradient. [@schaul2012no] suggested a reliable way to estimate the local curvature in the stochastic setting by keeping track of the variance and average of the gradients.
In this paper, we followed a different approach: instead of using a diagonal estimate of Hessian, we proposed to estimate curvature along the direction of the gradient and we apply a new variance reduction technique to compute it reliably. By using root mean square statistics, the variance of gradients are reduced adaptively with a simple transformation. We keep track of the estimation of curvature using a technique similar to that proposed by [@schaul2012no], which uses the variability of the expected loss. Standard adaptive learning rate algorithms only scale the gradients, but regular Newton-like second order methods, can perform more complicate transformations, e.g. rotating the gradient vector. Newton and quasi-newton methods can also be invariant to affine transformations in the parameter space. Our proposed **Adasecant** algorithm is basically a stochastic rank-$1$ quasi-Newton method. But in comparison with other adaptive learning algorithms, instead of just scaling the gradient of each parameter, Adasecant can also perform an affine transformation on them.
Directional Secant Approximation {#sec:dir_sec_approx}
================================
Directional Newton is a method proposed for solving equations with multiple variables [@levin2002directional]. The advantage of directional Newton method proposed in [@levin2002directional], compared to Newton’s method is that, it does not require a matrix inversion and still maintains a quadratic rate of convergence.
In this paper, we developed a second-order directional Newton method for nonlinear optimization. Step-size $\vt^k$ of update $\Delta^k$ for step $k$ can be written as if it was a diagonal matrix: $$\label{eqn:dir_newton}
\Delta^k = \vt^k \odot \nabla_{\TT} \f(\TT^k) = \diag(\vt^k) \nabla_{\TT} \f(\TT^k) = - \diag(\vd^k)(\diag(\H \vd^k))^{-1} \nabla_{\TT} \f(\TT^k)$$ where $\TT^k$ is the parameter vector at update $k$, $\f$ is the objective function and $\vd^k$ is a unit vector of direction that the optimization algorithm should follow. Denoting by $\vh_i=\nabla_{\TT}\frac{\partial \f(\TT^k)}{\partial \theta_i}$ the $i^{th}$ row of the Hessian matrix $\H$ and by $\nabla_{\TT_i} f(\TT^k)$ the $i^{th}$ element of the gradient vector at update $k$, a reformulation of Equation \[eqn:dir\_newton\] for each diagonal element of the step-size $\diag(\vt^k)$ is: $$\Delta_i^k =
- t_i^k \nabla_{\TT_i }\f(\TT^k) =
- d_i^k \frac{\nabla_{\TT_i }\f(\TT^k)}{\vh^k_i\vd^k}
\text{, so effectively } t_i^k=\frac{d_i^k}{\vh^k_i\vd^k}$$ We can approximate the per-parameter learning rate $t_{i}^k$ following [@an2005directional]: $$\label{eqn:step_size_eq}
t_i^k = \frac{d_i^k}{\vh_i^k \vd^k} \approx \frac{t_i^{k} d_i^k}{\nabla_{\TT_i} \f(\TT^k + \vt^{k} \vd^k) - \nabla_{\TT_i} f(\TT^k)}$$ Please note that alternatively one might use the R-op to compute the Hessian-vector product for the denominator in Equation \[eqn:step\_size\_eq\] [@schraudolph2002fast].
To choose a good direction $\vd^k$ in the stochastic setting, we use a block-normalized gradient vector for each weight matrix $\mW^i_k$ and bias vector $\vb^i_k$ where $\TT=\left\{\mW^i_k, \vb^i_k\right\}_{i=1\cdots k}$ at each layer $i$ and update $k$, i.e. $\vd_{\mW^i_k}^k = \frac{\nabla_{\mW^i_k} \f(\TT)}{||\nabla_{\mW^i_k} \f(\TT)||_2}$ and $\vd_k = \left[ \vd_{\mW^0_k}^k \vd_{\vb^0_k}^k \cdots \vd_{\vb^l_k}^k\right]$ for a neural network with $l$ layers. Block normalization of the gradient adds an additional noise, but in practice we did not observe any negative impact of it. We conjecture that this is due to the angle between the stochastic gradient and the block-normalized gradient still being less than $90$ degrees. The update step is defined as $\Delta_{i}^k = t_i^k d_i^k$. The per-parameter learning rate $t_i^k$ can be estimated with the finite difference approximation,
$$t_i^k = \frac{\Delta_{i}^k}{\nabla_{\TT_i} \f(\TT^k + \Delta^k) - \nabla_{\TT_i} \f(\TT^k)},$$
since, in the vicinity of the quadratic local minima, $$\begin{aligned}
\nabla_{\TT} \f(\TT^k + \Delta^k) - \nabla_{\TT} \f(\TT^k) \approx \H^{k} \Delta^k\end{aligned}$$ We can therefore recover $\vt^k$ as $$\vt^k = \diag(\Delta^k)(\diag(\H^{k} \Delta^k))^{-1}.$$ The directional secant method basically scales the gradient of each parameter with the curvature along the direction of the gradient vector and it is numerically stable.
Variance Reduction for Robust Stochastic Gradient Descent {#sec:var_reduction}
=========================================================
Variance reduction techniques for stochastic gradient estimators have been well-studied in the machine learning literature. Both [@wang2013variance] and [@johnson2013accelerating] proposed new ways of dealing with this problem. In this paper, we proposed a new variance reduction technique for stochastic gradient descent that relies only on basic statistics related to the gradient. Let $\g_i$ refer to the $i^{th}$ element of the gradient vector $\vg$ with respect to the parameters $\TT$ and $\E[\cdot]$ be an expectation taken over minibatches and different trajectories of parameters.
We propose to apply the following transformation to reduce the variance of the stochastic gradients: $$\label{eqn:tilde_gamma_i_def}
\tilde{g_i}=\frac{g_i + \gamma_i\E[g_i]}{1+\gamma_i}$$ where $\gamma_i$ is strictly a positive real number. Let us note that: $$\begin{aligned}
\E[\tilde{g_i}] = \E[g_i]
\text{ and }
\var(\tilde{g_i}) = \frac{1}{(1+\gamma_i)^2}\var(g_i)\end{aligned}$$ So variance is reduced by a factor of $(1+\gamma_i)^2$ compared to $\var(g_i)$. In practice we do not have access to $\E[g_i]$, therefore a biased estimator $\overline{g_i}$ based on past values of $g_i$ will be used instead. We can rewrite the $\tilde{g_i}$ as: $$\tilde{g_i} = \frac{1}{1+\gamma_i}g_i + (1 - \frac{1}{1+\gamma_i}) \E[g_i]$$ After substitution $\beta_i = \frac{1}{1+\gamma_i}$, we will have: $$\tilde{g_i} = \beta_i g_i + (1 - \beta_i) \E[g_i]$$ By adapting $\gamma_i$ or $\beta_i$, it is possible to control the influence of high variance, unbiased $g_i$ and low variance, biased $\overline{g_i}$ on $\tilde{g_i}$. Denoting by $\vg^{\prime}$ the stochastic gradient obtained on the next minibatch, the $\gamma_i$ that well balances those two influences is the one that keeps the $\tilde{g_i}$ as close as possible to the true gradient $\E[g_i^{\prime}]$ with $g_i^{\prime}$ being the only sample of $\E[g_i^{\prime}]$ available: $$\label{eqn:exp_gamma_cri}
\argmin_{\beta_i} \E[||\tilde{g_i} - g_i^{\prime}||^2_2]$$ It can be shown that this a convex problem in $\beta_i$ with a closed-form solution (details in appendix) and we can obtain the $\gamma_i$ from it: $$\label{eqn:gamma_i_formula}
\gamma_i = \frac{\E[(g_i - g_i^{\prime})(g_i - \E[g_i])]}{\E[(g_i-\E[g_i])(g^{_i\prime}-\E[g_i]))]}$$ As a result, in order to estimate $\gamma$ for each dimension, we keep track of a estimation of $\frac{\E[(g_i - g_i^{\prime})(g_i - \E[g_i])]}{\E[(g_i - \E[g_i])(g_i^{\prime}-\E[g_i]))]}$ during training. The necessary and sufficient condition here, for the variance reduction is to keep $\gamma$ positive, to achieve a positive estimate of $\gamma$ we used the root mean square statistics for the expectations.
Adaptive Step-size in Stochastic Case
=====================================
In the stochastic gradient case, the step-size of the directional secant can be computed by using an expectation over the minibatches: $$\label{eqn:adapt_stepsize}
\E_k[t_{i}] = \E_k[\frac{\Delta_i^k}{\nabla_{\TT_i} \f(\TT^k + \Delta^k) - \nabla_{\TT_i} \f(\TT^k)}]$$ The $E_k[\cdot]$ that is used to compute the secant update, is taken over the minibatches at the past values of the parameters.
Computing the expectation in Eq \[eqn:adapt\_stepsize\] was numerically unstable in stochastic setting. We decided to use a more stable second order Taylor approximation of Equation \[eqn:adapt\_stepsize\] around $(\sqrt{\E_k[(\alpha_i^k)^2]}, \sqrt{\E_k[(\Delta_i^k)^2]})$, with $\alpha_i^k = \nabla_{\TT_i} \f(\TT^k + \Delta^k) - \nabla_{\TT_i} \f(\TT^k)$. Assuming $\sqrt{\E_k[(\alpha_i^k)^2]} \approx \E_k[\alpha_i^k]$ and $\sqrt{\E_k[(\Delta_i^k)^2]} \approx \E_k[\Delta_i^k]$ we obtain always non-negative approximation of $\E_k[t_{i}]$: $$\begin{aligned}
\label{eqn:step_size_stoc_der1}
& \E_k[t_{i}] \approx \frac{\sqrt{\E_k[(\Delta_i^k)^2]}}{\sqrt{\E_k[(\alpha_i^k)^2]}} -
\frac{\cov(\alpha_i^k,\Delta_i^k)}{\E_k[(\alpha_i^k)^2]}
%& \E_k[t_{i}] \approx \frac{2\sqrt{\E_k[(\Delta_i^k)^2]}}{\sqrt{\E_k[(\alpha_i^k)^2]}} -
%\frac{\E_k[\alpha_i^k\Delta_i^k]}{\E_k[(\alpha_i^k)^2]} \label{eqn:step_size_stoc_der2}\end{aligned}$$ In our experiments, we used a simpler approximation, which in practice worked as well as formulations in Eq \[eqn:step\_size\_stoc\_der1\]: $$\label{eqn:step_size_stoc2}
\E_k[t_{i}] \approx \frac{\sqrt{\E_k[(\Delta_i^k)^2]}}{\sqrt{\E_k[(\alpha_i^k)^2]}} -
\frac{\E_k[\alpha_i^k\Delta_i^k]}{\E_k[(\alpha_i^k)^2]}$$
Algorithmic Details
===================
Approximate Variability
-----------------------
To compute the moving averages as also adopted by [@schaul2012no], we used an algorithm to dynamically decide the time constant based on the step size being taken. As a result algorithm that we used will give bigger weights to the updates that have large step-size and smaller weights to the updates that have smaller step-size.
By assuming that $\bar{\Delta}_i[k]\approx\E[\Delta_i]_k$, the moving average update rule for $\bar{\Delta}_i[k]$ can be written as, $$\begin{aligned}
\label{eqn:approx_var}
& \bar{\Delta}_i^2[k] = (1~-~\tau_i^{-1}[k])\bar{\Delta}_i^2[k-1] + \tau_i^{-1}[k](t_i^k \tilde{\vg}_i^k), \text{ and }
\bar{\Delta}_i[k] = \sqrt{\bar{\Delta}_i^2[k]}\end{aligned}$$ This rule for each update assigns a different weight to each element of the gradient vector . At each iteration a scalar multiplication with $\tau_i^{-1}$ is performed and $\tau_i$ is adapted using the following equation: $$\label{eqn:approx_var_update}
\tau_i[k] = (1~-~\frac{\E[\Delta_i]_{k-1}^2}{\E[(\Delta_i)^2]_{k-1}})\tau_i[k-1] + 1$$
Outlier Gradient Detection
--------------------------
Our algorithm is very similar to [@schaul2013adaptive], but instead of incrementing $\tau_i[t+1]$ when an outlier is detected, the time-constant is reset to $2.2$. Note that when $\tau_i[t+1] \approx 2$, this assigns approximately the same amount of weight to the current and the average of previous observations. This mechanism made learning more stable, because without it outlier gradients saturate $\tau_i$ to a large value.
Variance Reduction
------------------
The correction parameters $\gamma_i$ (Eq \[eqn:gamma\_i\_formula\]) allows for a fine-grained variance reduction for each parameter independently. The noise in the stochastic gradient methods can have advantages both in terms of generalization and optimization. It introduces an exploration and exploitation trade-off, which can be controlled by upper bounding the values of $\gamma_i$ with a value $\rho_i$, so that thresholded $\gamma_i^{\prime} = \min(\rho_i, \gamma_i)$. We block-wise normalized the gradients of each weight matrix and bias vectors in $\vg$ to compute the $\tilde{\vg}$ as described in Section \[sec:dir\_sec\_approx\]. That makes Adasecant scale-invariant, thus more robust to the scale of the inputs and the number of the layers of the network. We observed empirically that it was easier to train very deep neural networks with block normalized gradient descent.
Improving Convergence
=====================
Classical convergence results for SGD are based on the conditions: $$\sum_i (\eta^{(i)})^2 < \infty \text{ and } \sum_i \eta^{(i)} = \infty$$ such that the learning rate $\eta^{(i)}$ should decrease [@robbins1951stochastic]. Due to the noise in the estimation of adaptive step-sizes for Adasecant, the convergence would not be guaranteed. To ensure it, we developed a new variant of Adagrad [@duchi2011adaptive] with thresholding, such that each scaling factor is lower bounded by $1$. Assuming $a_i^k$ is the accumulated norm of all past gradients for $i^{th}$ parameter at update $k$, it is thresholded from below ensuring that the algorithm will converge: $$a_i^k = \sqrt{\sum_{j=0}^k (g_i^j)^2} \text{ and } \rho_i^k = \text{maximum}(1, a_i^k) \text{ giving } \Delta_i^k = \frac{1}{\rho_i}\eta_i^k\tilde{\vg}_i^k$$ In the initial stages of training, accumulated norm of the per-parameter gradients can be less than $1$. If the accumulated per-parameter norm of a gradient is less than $1$, Adagrad will augment the learning-rate determined by Adasecant for that update, i.e. $\frac{\eta_i^k}{\rho_i^k} > \eta_i^k$ where $\eta_i^k=\E_k[t_i^k]$ is the per-parameter learning rate determined by Adasecant. This behavior tends to create unstabilities during the training with Adasecant. Our modification of the Adagrad algorithm is to ensure that, it will reduce the learning rate determined by the Adasecant algorithm at each update, i.e. $\frac{\eta_i^k}{\rho_i^k} \le
\eta_i^k$ and the learning rate will be bounded. At the beginning of the training, parameters of a neural network can get $0$-valued gradients, e.g. in the existence of dropout and ReLU units. However this phenomena can cause the per-parameter learning rate scaled by Adagrad to be unbounded.
In Algorithm \[alg:adasecant\], we provide a simple pseudo-code of the Adasecant algorithm.
Experiments
===========
We have performed our experiments on MNIST with Maxout Networks [@goodfellow2013maxout] comparing Adasecant with popular stochastic gradient learning algorithms: Adagrad, Rmsprop [@graves2013generating], Adadelta [@zeiler2012adadelta] and SGD+momentum (with linearly decaying learning rate). Results are summarized in Figure \[fig:depthexp\] at the appendinx and we showed that Adasecant converges as fast or faster than other techniques, including the use of hand-tuned global learning rate and momentum for SGD, RMSprop, and Adagrad.
Conclusion
==========
We described a new stochastic gradient algorithm with adaptive learning rates that is fairly insensitive to the tuning of the hyper-parameters and doesn’t require tuning of learning rates. Furthermore, the variance reduction technique we proposed improves the convergence when the stochastic gradients have high variance. According to preliminary experiments presented, we were able to obtain a better training performance compared to other popular, well-tuned stochastic gradient algorithms. As a future work, we should do a more comprehensive analysis, which will help us to better understand the algorithm both analytically and empirically.
### Acknowledgments {#acknowledgments .unnumbered}
We thank the developers of Theano [@Bastien-2012] and Pylearn2 [@pylearn2_arxiv_2013] and the computational resources provided by Compute Canada and Calcul Québec. This work has been partially supported by NSERC, CIFAR, and Canada Research Chairs, Project TIN2013-41751, grant 2014-SGR-221. We would like to thank Tom Schaul for the valuable discussions. We also thank Kyunghyun Cho and Orhan Firat for proof-reading and giving feedbacks on the paper.
Appendix
========
Derivation of Eq \[eqn:exp\_gamma\_cri\]
----------------------------------------
$$\begin{aligned}
\label{eqn:exp_gamma_opt2}
\frac{\partial \E[(\frac{g_i+\mathbf{\gamma_i}\E[g_i]}{1+\mathbf{\gamma_i}} - g_i^{\prime})^2]}{\partial \gamma_i}&=0 \\
\E[(\frac{g_i + \mathbf{\gamma_i}\E[g_i] }{1+\mathbf{\gamma_i}} - g_i^{\prime}) \frac{\partial (\frac{(g_i+\mathbf{\gamma_i}\E[g_i])}{1+\mathbf{\gamma_i}} - g_i^{\prime})}{\partial \gamma_i}]&=0 \\
\E[(\frac{g_i + \mathbf{\gamma_i} \E[g_i]}{1 + \mathbf{\gamma_i}} - g_i^{\prime})
(\frac{(g_i+\mathbf{\gamma_i}\E[g_i])-(1+\gamma_i)(\E[g_i])}{(1 + \mathbf{\gamma_i})^2})]&=0 \\
\E[(g_i + \mathbf{\gamma_i} \E[g_i] - (1+\gamma_i)g_i{\prime})(g_i - \E[g_i])]&=0 \\
\E[(g_i - g^{_i\prime} +\mathbf{\gamma_i}(\E[g_i]-g_i^{\prime}))(g_i - \E[g_i])]&=0 \\
\E[(g_i - g_i^{\prime})(g_i - \E[g_i])] = \gamma_i \E[(g_i-\E[g_i])(g_i^{\prime}-\E[g_i]))]&=0 \\\end{aligned}$$
$$\begin{aligned}
\label{eqn:exp_gamma_opt2}
\frac{\partial \E[(\beta_i g_i + (1 - \beta_i) \E[g_i] - g_i^{\prime})^2]}{\partial \beta_i}&=0 \\
\E[(\beta_i g_i + (1 - \beta_i) \E[g_i] - g_i^{\prime})\frac{\partial (\beta_i g_i + (1 - \beta_i) \E[g_i] - g_i^{\prime})}{\partial \beta_i}]&=0 \\
\E[(\beta_i g_i + (1 - \beta_i) \E[g_i] - g_i^{\prime})(g_i - \E[g_i])]&=0 \\
\E[(\beta_i g_i (g_i - \E[g_i]) + (1 - \beta_i) \E[g_i] (g_i - \E[g_i]) - \E[g_i](g_i - \E[g_i])]&=0 \\
\beta_i = \frac{\E[(g_i - \E[g_i])(g_i^{\prime} - \E[g_i])]}{\E[(g_i - \E[g_i])(g_i - \E[g_i])]}
= \frac{\E[(g_i - \E[g_i])(g_i^{\prime} - \E[g_i])]}{\var(g_i)} \end{aligned}$$
Algorithm pseudo-code
---------------------
Algorithm \[alg:adasecant\] contains the pseudo-code of the Adasecant algorithm.
\[alg:adasecant\]
Further Experimental Details
----------------------------
In our experiments with Adasecant algorithm, adaptive momentum term $\gamma_i^k$ was clipped at $1.8$. In $2$-layer Maxout network experiments for SGD-momentum experiments, we used the best hyper-parameters reported by [@goodfellow2013maxout], for Rmsprop and Adagrad, we crossvalidated learning rate for $15$ different learning rates sampled uniformly from the log-space. We crossvalidated $30$ different pairs of momentum and learning rate for SGD+momentum, for Rmsprop and Adagrad, we crossvalidated $15$ different learning rates sampled them from log-space uniformly for deep maxout experiments. In Figure \[fig:adasecant\_clock\], we analyzed the effect of using different minibatch sizes for Adasecant and compared its convergence with Adadelta in wall-clock time. For minibatch size $100$ Adasecant was able to reach the almost same training negative log-likelihood as Adadelta after the same amount of time, but its convergence took much longer. With minibatches of size $500$ Adasecant was able to converge faster in wallclock time to a better local minima.
\[fig:adasecant\_clock\] ![In this plot, we compared adasecant trained by using minibatch size of $100$ and $500$ with adadelta using minibatches of size $100$. We performed these experiments on MNIST with 2-layer maxout MLP using dropout.[]{data-label="fig:adasecant_clock"}](adasecant_vs_adadelta_mnist_wallclock.pdf "fig:"){width="10cm"}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We investigate the rotational evolution of solar-like stars with a focus on the internal angular momentum transport processes. The double zone model, in which the star’s radiative core and convective envelope are assumed to rotate as solid bodies, is used to test simple relationships between the core-envelope coupling timescale, $\tau_c$, and rotational properties, like the envelope angular velocity or the differential rotation at the core-envelope interface. The trial relationships are tested by fitting the model parameters to available observations via a Monte Carlo Markov Chain method. The synthetic distributions are tested for compatibility with their observational counterparts by means of the standard Kolmogorov-Smirnov (KS) test.
A power-law dependence of $\tau_c$ on the inner differential rotation leads to a more satisfactory agreement with observations than a two-value prescription for $\tau_c$, which would imply a dichotomy between the initially slow ($P_\mathrm{rot} \gtrsim 3$ d) and fast ($P_\mathrm{rot} \lesssim 3$ d) rotators. However, we find it impossible to reconcile the high fraction of fast rotators in $\alpha$ Per with the rotation period distributions in stellar systems at earlier and later evolutionary stages. This could be explained by local environmental effects (e.g. early removal of circumstellar discs due to UV radiation and winds from nearby high-mass stars) or by observational biases.
The low KS probability that the synthetic and observed distributions are not incompatible, found in some cases, may be due to over-simplified assumptions of the double zone model, but the large relative uncertainties in the age determination of very young clusters and associations are expected to play a relevant role. Other possible limitations and uncertainties are discussed.
title: 'Modelling the rotational evolution of solar-like stars: the rotational coupling timescale'
---
\[firstpage\]
Methods: numerical – stars: rotation – stars: interiors – stars: late-type – stars: pre-main sequence
Introduction
============
Rotation is an important parameter for solar-like stars. It is an essential ingredient of the dynamo process [e.g. @Subramanian_Brandenburg:2005], which in turn powers magnetic activity, it can influence the surface abundance of light elements through rotationally-induced mixing [@Zahn:1993], and it is related to the formation and evolution of planetary systems [see, e.g., @Chang_ea:2010; @Lanza:2010].
Rotation period measurements in open clusters and young associations provide a valuable source of information on angular momentum evolution of late-type stars. In recent years, photometric measurements of the stellar rotation period, based on the rotational modulation of the light curve induced by photospheric starspots, have replaced the measurements based on the $v\, \sin i$ as derived from spectral line profiles. This has considerably improved both the number and the precision of available rotation periods. Such observational constraint, however, has not been exploited in its full potential yet. In fact, most of the analyses carried out so far have considered only limiting cases such as the subsets of the faster and slower rotators at given ages.
During the pre-main sequence (PMS) phase, solar-mass stars undergo a global contraction. However, for the first few Myr, the interaction with a circumstellar disc drains angular momentum from the star, thus delaying its spin up for the (variable) duration of the disc lifetime. This process is not understood in detail yet (see, e.g., @Cameron_Campbell:1993 [@Shu_ea:1994; @Matt_ea:2010]) and is usually modelled by means of very simplified assumptions, e.g., the disc-locking hypothesis [@Koenigl:1991]. Once on the main sequence (MS), the rotational evolution is driven by the loss of angular momentum through a magnetised stellar wind [@Weber_Davis:1967; @Kawaler:1988; @Chaboyer:1995a; @Chaboyer:1995b].
Surface period measurements are also an indirect probe of the internal rotation profile of these stars. Phenomenological modelling of MS angular momentum evolution [the so-called double zone model, DZM in the following: @MacGregor_Brenner:1991; @Keppens_ea:1995; @Allain:1998; @Bouvier:2008b see also Sect. \[sec:dzmodel\]], assuming that the radiative core and the convective envelope rotate rigidly at all ages, has successfully reproduced the observations provided a certain amount of differential rotation between the core and the envelope is allowed. This is parameterised through a coupling timescale $\tau_{\rm c}$, which determines the rate of angular momentum exchange between the two regions. From the comparison of synthetic rotational tracks and the upper and lower percentiles of the observed period distributions, @Bouvier:2008b concluded that $\tau_{\rm c}$ is of the order of $10$ Myr for the stars that begun their evolution on the ZAMS as fast rotators (i.e., with an initial rotation period of $\sim$ 1 d), while it is remarkably longer ($\tau_{\rm c} \sim 100$ Myr) for initially slow rotators (i.e., with initial periods $\sim$ 10 d). A possible connection between this dichotomy and the presence or absence of a planetary system orbiting the star is further discussed in @Eggenberger_ea:2010. An understanding of the physical processes that give rise to such a dichotomy and eventually establish a nearly solid-body rotation within the age of the Sun [as observed through helioseismology, see e.g. @Thompson_ea:2003] is still lacking.
Our aim is twofold: on the one hand, we try to extract the whole statistical information from the observed rotation period distributions. In recent years, rotation period measurements have improved significantly both in number and in precision. In spite of the age spread within cluster members, uncertainties on cluster age estimates and other systematic effects, good sample completeness and homogeneity has already been achieved. We exploit this information comparing the observed period distributions with synthetic ones, constructed by evolving the Orion Nebula Cluster (hereafter ONC) distribution by means of the DZM. The modelling of the period distributions based on the DZM requires some prescription on the dependence of $\tau_c$ on the angular velocity, to comply with the findings of @Bouvier:2008b mentioned above. On the basis of intuitive physical arguments, we formulate some hypotheses on the scaling of $\tau_c$. The set of parameters in the DZM that produces the best fit of the synthetic distributions to the observed ones is determined by means of an iterative procedure based on a Markov Chain Monte Carlo (MCMC) sampling. This suffices to the purpose of comparing the different hypotheses on $\tau_c$ and to test the overall capability of the DZM to satisfy the latest observational constraints.
A direct comparison of theoretical modelling of stellar rotation with observed period distributions has already been attempted in the past by, e.g., @Tinker_ea:2002. They evolved the ONC period distribution to the age of the Pleiades and the Hyades by means of the Yale Rotating Evolution Code [@Guenther_ea:1992], mainly aiming at determining the parameters in the wind braking law. The present study is novel in many respects, because we rely upon a simpler modelling, focussing on the coupling timescale. Moreover, a much larger sample of observed period distributions is used.
Rotational period data {#sec:obs}
======================
We use the rotation period distributions extracted from photometric surveys of stellar open clusters and young associations, as available in the literature (see Table \[tab:data\] for the references).
The rotational data for the associations TW Hydrae, $\beta$ Pictoris, Tucana–Horologium, Columba, Carina (age $\lesssim 30$ Myr), AB Doradus (age $\sim 120$ Myr) are taken from [@Messina_ea:2010a] and for the cluster M 11 (age $\sim 230$ Myr) from [@Messina_ea:2010b] and are used for the first time for this kind of investigation. Remarkably, these young associations fill an age gap previously present between Tau-Aur ($\sim 6$ Myr old) and $\alpha$ Per ($\sim 70$ Myr) in the mass range $0.9 \leq M_*/M_\odot \leq 1.1$.
The ages reported here are the most recent literature estimates. In particular, the association AB Dor is considered almost coeval with the Pleiades, according to the analysis of @Messina_ea:2010b, who revised a previously proposed age of $70$ Myr. A remarkably extensive rotational dataset for the Pleiades have been produced by [@Hartman_ea:2010].
The rotational evolution significantly depends on stellar mass, owing to different durations of the PMS phase, convective envelope depths, etc. As we are primarily interested in the rotational evolution of solar analogues, a subsample of each period distribution in the mass range $0.9 \leq M_*/M_\odot \leq 1.1$ was selected. For very young clusters (i.e. age $\leq 10$ Myr), which are affected by a substantial reddening, the effective temperature of the stars was determined on the basis of their spectral type as given by @Rebull_ea:2004. For MS objects, the dereddened $(B-V)$ and $(V-I)$ colours were employed for the same purpose. In both cases, the determination of the mass from $T_\mathrm{eff}$ or the colour index was performed by direct comparison with the theoretical isochrones of [@Baraffe_ea:1998].
Object name Age \[Myr\] \# stars Reference
------------------------------------------- ------------- ---------- ----------------------
ONC 2 57 \[reb\]
NGC 2264 4 52 \[reb\]
NGC 2362 5 43 \[ir08\]
[**Tau-Aur**]{} 6 25 \[reb\]
[**TW Hya**]{}, [**$\beta$ Pic**]{} 10 12 \[smass\]
[**Tuc-Hor**]{}, [**Col**]{}, [**Car**]{} 30 46 \[smass\]
$\alpha$ Per 70 43 \[smcomp\]
[**AB Dor**]{}, Pleiades 120 109 \[smass\], \[har10\]
M 50 130 62 \[ir09\]
M 35 150 88 \[mei\]
M 34 220 17 \[ir06\]
M 11 230 15 \[sm10\]
M 37 550 89 \[h09\]
: Names, estimated ages and number of stars in the mass range $0.9 \leq M_*/M_\odot \leq 1.1$ for the stellar systems used in this work. Stellar associations are in bold.[]{data-label="tab:data"}
(a) \[reb\]: compilation, see [@Rebull_ea:2004] and references therein;
(b) : \[ir08\] [@Irwin_ea:2008];
(c) : \[smass\] [@Messina_ea:2010a];
(d) : \[smcomp\] compilation, see [@Messina_ea:2001], [@Messina_ea:2003] and references therein;
(e) : \[har10\] [@Hartman_ea:2010];
(f) : \[ir09\] [@Irwin_ea:2009];
(g) : \[mei\] [@Meibom_ea:2009];
(h) : \[ir06\] [@Irwin_ea:2006];
(i) : \[sm10\] [@Messina_ea:2010b];
(j) : \[h09\] [@Hartman_ea:2009].
![Observed period distribution for the $57$ stars of ONC in the selected mass range.[]{data-label="fig:onc"}](figure1.eps){width="45.00000%"}
The ONC period distribution, providing the initial conditions for our rotational modelling, is shown in Figure \[fig:onc\]. In the restricted mass range which is of interest to us, the distribution is unimodal, with a fairly pronounced peak of fast rotators (i.e., with $P_\mathrm{rot}\leq 4$ days).
Numerical procedure {#sec:method}
===================
We construct a series of synthetic rotation period distributions at the ages reported in Table \[tab:data\]. Each period in the ONC solar analogue subsample is evolved forwards in time with the DZM, calculated for a $1~M_\odot$ star. The synthetic distributions are tested for compatibility with their observational counterparts by means of the standard Kolmogorov-Smirnov (KS) test [see, e.g., sect. 14.3 of @NR:1992]. The input to our numerical procedure is an initial guess for the parameters appearing in the equations of the DZM. The parameter space is then sampled by means of the MCMC method, which performs random jumps in each parameter, guided by a probabilistic acceptance rule (see Sect. \[sec:mcmc\]). The best-fitting set of model parameters is obtained in output.
Parameters in the DZM {#sec:dzmodel}
---------------------
Our simple model for the rotational evolution of a solar-like star assumes that the radiative core and the convective envelope both rotate as solid bodies [@MacGregor_Brenner:1991; @Keppens_ea:1995; @Allain:1998]. The envelope is expected to be strongly coupled by the very efficient angular momentum redistribution associated with turbulent convection. On the other hand, a large scale magnetic field, even as weak as $\lesssim 1$ G, can maintain a condition of rigid rotation in most of the core (see, e.g. @Spada_ea:2010).
At any given time $t$, the angular momentum of the core, $J_c$, and of the envelope, $J_e$, are: $$\begin{aligned}
J_c(t) = I_c(t)\ \Omega_c(t) \ \ \ ; \ \ \ J_e(t) = I_e(t)\ \Omega_e(t) ,\end{aligned}$$ where $I_c$, $I_e$ and $\Omega_c$, $\Omega_e$ are the moments of inertia and the angular velocities of the core and the envelope, respectively. Our evolution begins at the ONC age, i.e. $t_0 \sim 2$ Myr. Stars are fully convective until $t_0$, thus justifying a solid-body rotation as our initial condition, i.e. $\Omega_e(t_0) = \Omega_c(t_0)$.
The evolution of the angular momenta is governed by two coupled differential equations: $$\begin{aligned}
\nonumber
\frac{dJ_c}{dt} &=& -\frac{\Delta J}{\tau_c} + \left( \frac{2}{3}~R_c^2~\frac{dM_c}{dt} \right) \Omega_e ; \\
\label{eq:dzm}
\\
\nonumber
\frac{dJ_e}{dt} &=& +\frac{\Delta J}{\tau_c} - \left( \frac{2}{3}~R_c^2~\frac{dM_c}{dt} \right) \Omega_e - \left. \frac{dJ_e}{dt} \right|_\mathrm{wind} ,\end{aligned}$$ where $R_c$ and $M_c$ are the radius and mass of the radiative core. The stellar structure quantities and their time derivatives are taken from PMS and MS evolutionary models by @Baraffe_ea:1998. In equations , the core and the envelope exchange an amount of angular momentum $\Delta J \equiv \dfrac{I_e\ J_c - I_c\ J_e}{I_e+I_c}$ at a rate determined by the coupling timescale $\tau_c$. During the PMS, the growth of the core at the expenses of the envelope produces an angular momentum transfer, accounted for by the second term in the right-hand side of both equations . Because of the magnetised wind, the envelope loses angular momentum at a rate $ \left. \dfrac{dJ_e}{dt} \right|_\mathrm{wind}$, for which the following parametric formula is used [@Kawaler:1988]: $$\label{eq:wind}
\left. \frac{dJ_e}{dt} \right|_\mathrm{wind} = K_w \left( \frac{R_*/R_\odot}{M_*/M_\odot} \right)^{1/2} \mathrm{min}(\Omega_e^3,\Omega_\mathrm{sat}^2\ \Omega_e) ,$$ where $K_w$ and $\Omega_\mathrm{sat}$ determine the braking intensity and saturation threshold, respectively, while $R_*/R_\odot$ and $M_*/M_\odot$ are the stellar radius and mass in solar units. Finally, the angular momentum loss due to the interaction with the disc during early PMS evolution is taken into account with the disc-locking hypothesis [@Koenigl:1991], i.e.: $$\frac{d \Omega_e}{dt} = 0 \ \ \mathrm{while}\ \ \ t \leq \tau_\mathrm{disc} .$$ In other words, the net effect of the interaction with the disc is that of keeping the surface angular velocity of the star constant for the whole disc lifetime.
The solutions of equations depend on the parameters $K_w$, $\Omega_\mathrm{sat}$, $\tau_c$ and $\tau_\mathrm{disc}$. In applying the DZM to a whole distribution of rotators, the wind law parameters $K_w$ and $\Omega_\mathrm{sat}$ can be assumed to be the same for all the stars; for $\tau_c$ and $\tau_\mathrm{disc}$, on the contrary, such an assumption is rather crude. It has already been established that it is impossible to reconcile the lower and upper envelopes of the rotation period distributions with the same value of $\tau_c$ [e.g. @Bouvier:2008b]. This is dealt with by simple assumptions on the dependence of $\tau_c$ on stellar rotation, as discussed below (see Sect. \[sec:tauc\]). A realistic treatment of the star-disc interaction is a complex task. Even under the disc-locking assumption, the value of $\tau_\mathrm{disc}$ for each star in the sample should be extracted from a distribution.
Although information on the distribution of disc lifetime, as well as on its dependence on the presence of high mass stars in the neighbourhood, is becoming increasingly available [e.g. @Hernandez_ea:2008; @Kennedy_Kenyon:2009], we refrain from exploiting it in full details so to keep our model as simple as possible.
Hypotheses on the coupling timescale {#sec:tauc}
------------------------------------
We consider the following prescriptions on the dependence of $\tau_c$ on stellar rotation:
- A two-valued function of the form:
$$\label{eq:2val}
\tau_c = \left\{
\begin{array}{cc}
10 \ \mathrm{Myr} & \mathrm{if} \ \ \Omega_e(t_0) \geq \Omega_\mathrm{crit} \\
\tau_0 & \mathrm{otherwise}
\end{array}
\right. .$$
This is the simplest implementation of the early empirical finding (e.g. @Krishnamurthi_ea:1997, see also @Bouvier:2008b) that fast rotators are well described by a nearly solid-body inner profile, while a sizeable decoupling is necessary for slow rotators. The value of $10$ Myr used for initially fast rotators is so short as to ensure nearly instantaneous coupling at all ages [see also @Denissenkov_ea:2010]. The timescale for slow rotators, $\tau_0$, and the initial angular velocity threshold, $\Omega_\mathrm{crit}$, are treated as free parameters to be determined by means of the MCMC procedure.
- A power law dependence on the instantaneous amount of differential rotation: $$\label{eq:powl}
\tau_c(t) = \tau_0 ~ \left[ \frac{\Delta \Omega_\odot}{ \Delta \Omega(t)} \right]^{\alpha} \ ,$$ where $\Delta \Omega(t) = \Omega_c(t) - \Omega_e(t)$ and $\Delta \Omega_\odot$ is the inner differential rotation of the present Sun, assumed as a reference value (we used $\Delta \Omega_\odot = 0.2 \ \Omega_\odot$).
Note that this choice produces a time-dependent $\tau_c$. This is an attempt to account for the effect of an enhanced angular momentum transport due to turbulent viscosity. The viscosity enhancement could be the result of either a rotational mixing [e.g. @Zahn:1993] or of hydromagnetic instabilities [@Spruit:1999; @Spruit:2002] powered by the available kinetic energy stored in the differential rotation. The form of equation , with $\alpha>0$, is chosen to produce a shorter coupling timescale (which mimics a greater effective viscosity, ensuring a stronger coupling) when a greater differential rotation is present. $\tau_0$ and $\alpha$ are regarded as free parameters.
Other simple prescriptions, e.g. a dependence of $\tau_{c}$ on the surface angular velocity $\Omega_e(t)$, failed to give suitable results and are not discussed in the following.
The MCMC method {#sec:mcmc}
---------------
Our application of the MCMC method [see, e.g. @Tegmark_ea:2004] constructs a sequence of values for each parameter of the DZM, approaching the best fitting region of the parameter space quite rapidly (burn-in phase, usually a few hundreds of steps), and then sampling this restricted neighbourhood by performing a random walk within it.
The routine is initialised providing a starting guess for the parameters. At the beginning of each step, the current set of parameters is used to evolve forwards in time the ONC period distribution to each of the ages in Table \[tab:data\] by means of the DZM. A KS test is then performed comparing the synthetic period distributions with their observational counterparts. Calling $\bar P_{\mathrm{KS},i}$ the probability of the synthetic and observed distributions to be significantly different for a certain cluster $i$ and viewing the individual KS tests as independent from each other, we define the quantity $Q$ by adding these probabilities together: $$Q = \sum_{i=1}^N \bar{P}_{\mathrm{KS},i} \ ,$$ with $N$ being the number of rotation period distributions used. Two additional constraints come from the rotational properties of the present Sun, namely: $$\label{eq:sun_con}
\Omega_e(t_\odot) = \Omega_\odot \ \ \ ; \ \ \ \frac{\Omega_c(t_\odot)-\Omega_e(t_\odot)}{\Omega_e(t_\odot)} \leq 0.2 ,$$ which are taken into account by suitable additive terms in $Q$. The MCMC procedure is used to iterate the model parameters towards the minimum of $Q$, which corresponds to the best fit.
After an evaluation of $Q$, the following step begins with a tentative random jump in each parameter. The new set of parameters just generated is retained in the chain or rejected following the stochastic Metropolis-Hastings rule [@Metropolis_ea:1953; @Hastings:1970]: $$\begin{aligned}
\Delta Q < 0 &:& \ \mathrm{accept} ; \\
\Delta Q > 0 &:& \ \mathrm{accept } \ \mathrm{with} \ \mathrm{probability} \propto e^{-\Delta Q} ,\end{aligned}$$ where $\Delta Q$ is the difference between the current value of $Q$ and its value at the last accepted step. This rule ensures a quick descent towards the global minimum, with a non-zero probability of escaping from possible local minima encountered during the path.
[c|c]{} two-valued $\tau_c$ & power law $\tau_c$\
(cf. equation \[eq:2val\]) & (cf. equation \[eq:powl\])\
$\left< K_w \right>=(5.40 \pm 0.054) \cdot 10^{47}$ & $ \left<K_w \right>=(5.97 \pm 0.13) \cdot 10^{47}$\
\
$\left< \tau_0 \right>= 128 \pm 3.84$ Myr & $\left< \tau_0 \right>= 57.7 \pm 5.24$ Myr\
\
$\left< \Omega_\mathrm{crit} \right>= 3.89 \pm 0.14 \ \Omega_\odot$ & $ \left<\alpha \right>=0.076 \pm 0.02$\
\[tab:results\]
Results {#sec:results}
=======
The procedure described above was applied to determine the best values of $K_w$, $\tau_0$ and $\Omega_\mathrm{crit}$, in equation , or $\alpha$, in the case of equation , respectively. The parameters $\Omega_\mathrm{sat}$ and $\tau_\mathrm{disc}$ were held fixed. We used the following fiducial values: $$\begin{aligned}
\Omega_\mathrm{sat} = 5.5 \, \Omega_\odot \ \ \ ; \ \ \ \tau_\mathrm{disc} = 5.8 \, \mathrm{Myr}.\end{aligned}$$ These choices are motivated by independent observational estimates, namely the angular velocity saturation threshold for the X-ray emission of late-type stars [see figure $3$ of @Pizzolato_ea:2003] for $\Omega_\mathrm{sat}$, and the age at which about $10\, \%$ of stars still possess a circumstellar disc [see figure $1$ of @Mamajek:2009] for $\tau_\mathrm{disc}$. This is also in good quantitative agreement with earlier works [e.g. @Bouvier:2008b; @Irwin_Bouvier:2009].
The best-fitting values of $K_w$, $\tau_0$ and $\Omega_\mathrm{crit}$ were calculated from the portion of the MCMC chain that effectively sampled the region of the parameter space around the minimum (in both cases, we had $\gtrsim 10^4$ steps). For each parameter, we used the mean value and the standard deviation as estimates of the best fitting value and its uncertainty, respectively. The results are summarised in Table \[tab:results\]. Note that the value of $K_w$ found, of the order of $10^{47}$ g cm$^2$ s, is in good agreement with previous works (see, e.g. @Kawaler:1988, @Allain:1998).
To allow the comparison of the two different prescription for $\tau_c$, we show in Figure \[fig:tauc\_plaw\] the evolution of $\tau_c$ in the power-law case for two initial values of $\Omega_e$ representative of slow and fast rotators. Remarkably, the best-fitting parameters produce a $\tau_c \gtrsim 40$ Myr for fast rotators, significantly longer than $10$ Myr, which was used in the two-valued case to ensure nearly solid-body rotation at all ages, as suggested by previous works.
![Values of $\tau_c$ determined by the power-law prescription with the best-fitting $\tau_0$ and $\alpha$ (see Table \[tab:results\]) for two initial velocities representative of slow (dotted line) and fast (dashed line) rotators, respectively.[]{data-label="fig:tauc_plaw"}](figure2.eps){width="45.00000%"}
A comparison of the quality of the fitting in the two cases, based on the individual KS tests, is presented in Figures \[fig:pks\_2val\] and \[fig:pks\_plaw\], which show the synthetic and observed cumulative distributions (CDF) for the various clusters. The KS compatibility probability, $P_{\mathrm{KS},i}$, i.e., the probability of the two distributions for cluster $i$ of being not incompatible, is reported in each panel along with the number of stars in the respective observed distribution (note that $P_{\mathrm{KS},i} = 1 - \bar P_{\mathrm{KS},i}$). The smaller the value of $P_{\mathrm{KS},i}$, the worse the fitting between the calculated and the observed distributions for cluster $i$.
The photometric periods on which our analysis is based are Fourier-derived from the rotational light modulation induced by starspots. The amplitude of such a modulation falls below the minimum value currently detectable from the ground ($\simeq 0.01$ mag) for MS stars with $P_\mathrm{rot} \gtrsim 10$ d (see also Sect. \[sec:disc\]). As a consequence, the photometric rotation period of slow rotators is hardly derived and the observed distributions may suffer a lack of completeness beyond this value of the rotation period. To account for this effect, periods in the synthetic distribution greater than the maximum of the corresponding observed distribution are not considered when calculating $ \bar P_{\mathrm{KS},i}$ and $P_{\mathrm{KS},i}$.
A good agreement ($P_{\mathrm{KS},i} \gtrsim 0.37$) is achieved for the first $\sim 10$ Myr in both Figures. The first two panels refer to clusters younger than the assumed $\tau_\mathrm{disc}$; as a consequence, the corresponding synthetic distributions show no evolution with respect to ONC, by virtue of the disc-locking constraint. For ages greater than $\tau_\mathrm{disc}$ but lower than $\sim 30$ Myr, the spin-up dominates due to the stellar contraction. At $6$ and $10$ Myr, therefore, the agreement with observations mainly rely on $\tau_\mathrm{disc}$ and on the stellar structure and evolution model, while it is quite insensitive to $\tau_c$ and $K_w$. For ages older than approximately 30 Myr, the agreement is strongly dependent on $K_w$ and on the model assumed for $\tau_c$. For the two-value $\tau_c$ model, the only satisfactory agreement is found for M 34 (220 Myr). For the power-law $\tau_c$ model, a satisfactory agreement is found for the Pleiades (120 Myr) and M 37 (550 Myr). We note, however, that the CDFs derived from the observations do not follow an evolution as regular as expected from the models. The $\alpha$ Persei CDF, in particular, displays a sharp rise at short rotation periods due to an excess of fast and ultra-fast rotators. Such a sharp rise is not seen in any other CDFs, except, perhaps, in M 34 and M 11, although the number of periods available is too low to draw any definitive conclusions. The two-value $\tau_c$ model can account for this sharp rise, but then fails to reproduce the smooth CDF shape of AB Dor/Pleiades, M 50, etc. The power-law $\tau_c$ model, on the other hand, cannot reproduce any sharp rise in the CDF and therefore fails to reproduce the $\alpha$ Persei, M 34 and M 11 CDFs.
Without any angular momentum transfer from the core to the envelope, the maximum angular velocity is expected at about the Tuc-Hor / Col / Car age, i.e., $\approx 30$ Myr (see also Figures \[fig:ev\_2val\] and \[fig:ev\_plaw\]). However, this is not observed because the evolution from the Tuc-Hor / Col / Car age (30 Myr) to $\alpha$ Persei (70 Myr) appears consistent with a continuation of the spin-up after the ZAMS. The spin-up of fast and ultra-fast rotators from the Tuc-Hor / Col / Car age (30 Myr) to the $\alpha$ Persei (70 Myr) age seems even faster than for slower rotators. In the DZM framework, such a behaviour cannot be reproduced unless, for the fast and ultra-fast rotators, a remarkable amount of angular momentum is transferred from the core to the envelope within a very short timescale ($\sim 30-40$ Myr). In this case, however, very little angular momentum would be left in the core for the later evolution.
After the peak in the angular velocity, a generally monotonic spin-down is expected. The fast rotator excess in M 34 (220 Myr) and M 11 (230 Myr), however, hampers the possibility of obtaining a satisfactory fit at all ages. In fact, the percentage of observed fast rotators ($P_{\rm rot}\sim$ 1 - 2 d) tends to decrease from $\alpha$ Per to M 35, but then increases abruptly in M 34 and M 11. Such a distribution evolution is clearly inconsistent with the DZM alone.
Despite such limitations, which will be discussed in some more details in Sect.\[sec:disc\], the power-law $\tau_c$ model produces a more satisfactory fit to the data, particularly since it reproduces the shape of most of the observed CDFs also giving a high $P_{\mathrm{KS},i}$ value for AB Dor / Pleaides and M 37. It requires, however, an alternative explanation for the excess of fast and ultra-fast rotators in $\alpha$ Per, M 34, and M 11.
{width="80.00000%"}
{width="80.00000%"}
Two rotational tracks, with initial conditions representative of slow ($\Omega_e(t_0)=\Omega_c(t_0)= 3~\Omega_\odot$) and fast rotators ($\Omega_e(t_0)=\Omega_c(t_0)= 25~\Omega_\odot$), as calculated with the DZM and prescriptions or for $\tau_c$, are compared with the observed distributions and the solar-age angular velocity in Figures \[fig:ev\_2val\] and \[fig:ev\_plaw\], respectively. Assigning a constant value to $\tau_c$ for the whole rotational evolution results in slow and fast tracks that intersect with each other at ages later than about $1$ Gyr (see Figure \[fig:ev\_2val\]). This is another consequence of the high percentage of fast rotators in $\alpha$ Per, which imposes a fast angular momentum transfer from the core to the envelope just after the ZAMS. This diminishes dramatically the core angular momentum reservoir available for later stages, producing a faster spin-down for fast rotators. On the other hand, the power-law prescription for $\tau_c$ leads to a remarkably greater internal differential rotation for the fast rotators and a somewhat lower internal differential rotation for the slow rotators than the two-value prescription.
The solar constraint (equation \[eq:sun\_con\]) is reasonably matched by the power-law prescription. The large differential rotation developed by fast rotators is due to the relatively large value of $\tau_c$, as previously noted. In contrast, the poor matching found for the slow rotators in the two-value case may seem to be in contradiction with previous results (e.g. @Bouvier:2008b). Note, however, that in the present work an attempt is made to fit the rotation period distributions, not just their percentiles. A value of $\tau_c \sim 130$ Myr, which is the result of our best-fitting procedure (and as such is the outcome of taking into account all the period distributions plus the solar constraint), turns out to be too large to recover the almost rigid rotation at the age of the Sun (cf. also Figure \[fig:tauc\_plaw\] for the power-law case). The slightly lower value of $K_w$ with respect to the power-law case (see Table \[tab:results\]), moreover, explains the excess of rotation at $t_\odot$. Indeed, this disagreement is partly relieved if the $\alpha$ Per distribution, which is likely prone to observational biases (see Sect. \[sec:disc\]), is excluded from the fitting procedure. A run performed for comparison purposes without the $\alpha$ Per constraint in the two-value case gives KS probabilities above $20 \, \%$ for the intermediate age clusters (i.e. M 50 to M 11) and different values of the best fitting parameters, namely $K_w \sim 6.3 \cdot 10^{47}$ g cm$^2$ s and $\tau_c \sim 80$ Myr. A similar test for the power-law case results in negligible differences both in KS probabilities and in best-fit parameters, consistently with a moderate sensitivity to the $\alpha$ Per constraint.
In conclusion, the power-law prescription is more suited to fit the shape of the rotational distributions, while the two-value models better reproduces the basic features of the two extreme populations (i.e. the slow and fast rotators). The two prescriptions also show different sensitivity to peculiar characteristics (either real or due to biases) of the distributions. Clearly, none of the two models is able to completely describe rotational evolution.
{width="80.00000%"}
{width="80.00000%"}
Discussion {#sec:disc}
==========
We modelled the evolution of rotation period distributions from the PMS to the solar age, using the available observations to determine the best-fitting parameters of a modified version of the DZM. We aimed at including in the analysis the whole statistical information contained in the observed distributions as available so far; at the same time, we tested the possibility of including in the DZM some kind of dependence of the coupling timescale on the rotational state of the star. A numerical procedure was implemented to evolve forwards in time the solar analogues’ subsample of the ONC period distribution, used as our initial condition. The agreement of the synthetic distributions with the observed ones was evaluated by means of the KS test. This procedure was used to select the values of model parameters ensuring the best fitting of the observational constraints.
A prescription for the dependence of $\tau_c$ on the rotation of the star is required to fit the evolution of the whole period distributions. It is already known, in fact, that the slow and fast percentiles of the period distributions require significantly different values of $\tau_c$ [e.g. @Bouvier:2008b]. In formulating these prescriptions (see equations \[eq:2val\] and \[eq:powl\]) we relied on basic physical intuition, keeping the number of free parameters to a minimum. Equation is the simplest way to mimic the dichotomy in $\tau_c$ between slow and fast rotators as suggested by earlier works. It implies that different $\tau_c$ values are set at early stages by some mechanism that quickly distinguishes between two separate rotation regimes, but is poorly sensitive to internal differential rotation. A different magnetic field configuration within the stellar interior might account for such a behaviour (see @Spruit:1999; @Spada_ea:2010). Equation , on the other hand, assumes that the available content of rotational kinetic energy, as measured in the DZM by the amount of differential rotation, can trigger angular momentum transport processes with an efficiency parameterised by the exponent $\alpha$. Other formulations, with a power law dependence of $\tau_c$ on the surface angular velocity $\Omega_e(t)$, with or without saturation mechanisms like that in equation , proved to be completely unsuccessful.
A major limitation of this work comes from the uncertainty in the ages of the clusters [e.g. @Mayne_Naylor:2008 and references therein]. This is particularly relevant for the very young clusters and for the stellar associations, which are also more likely to be prone to remarkable internal age scattering.
During the PMS, the fitting is dependent essentially on $\tau_\mathrm{disc}$ and on the radius contraction as predicted by the evolutionary model. Considering the uncertainties in age and the crude approximation on the role of the circumstellar disc, the fitting can be considered quite satisfactory in this part of the evolution. After the ZAMS, when the rotational evolution becomes very sensitive to angular momentum transfer from the core to the envelope and to wind-braking, the observed CDFs do not show a regular evolution as expected from the DZM assumptions. The $\alpha$ Per CDF, in particular, shows a marked (ultra) fast-rotators excess, which is difficult to reconcile with the CDF of younger systems, either assuming a two-value or a power-law prescription for $\tau_c$ in the DZM. This distribution feature is not seen in any other cluster or association, except perhaps M 34 and M 11. The consequences on the model fitting have been discussed in Sect.\[sec:results\].
Obviously, the reasons for such a discrepancy could be: a) the DZM is inadequate or contains oversimplified assumptions; b) the stellar rotation period distributions in open clusters and young associations may be affected by characteristics of the (parent) cluster as a whole, particularly those leading to the evaporation of circumstellar discs by UV radiation and wind from neighbour high-mass stars [e.g. @Hernandez_ea:2008; @Guarcello_ea:2010]; c) the observed period distributions are incomplete and/or biased.
Indeed, our model does not include differences in the rotational history of clusters’ stars that may arise from the presence of high mass stars, multiple or prolonged star formation events, the compactness and richness of the clusters, and the stars’ dispersal into the field. Therefore, despite the DZM may contain the essential physics to describe the stellar angular momentum evolution, a satisfactory modelling may require taking into account the peculiarities of the cluster to which the stars belong. In the case of $\alpha$ Per, for instance, high mass stars might have influenced the rotational history of a fraction of low mass stars by evaporating their circumstellar discs at an early stage. Without an efficient disc-locking, these stars may have reached the ZAMS (first 30 Myr) with a very high rotation and have had no time to spin-down significantly in the subsequent $\sim 40$ Myr.
Other possible sources of uncertainties in the period distribution include clusters contamination by field stars, the presence of close binaries whose high rotation rate is maintained by the tidal synchronisation between the components, and observational biases. Concerning these latter, photometric monitoring with a limited temporal extension (generally of $10$–$15$ d as in the cases of M 11, M 34, M 35, M 50, NGC 2362 and of numerous members of $\alpha$ Per) prevents us from detecting the slow rotation periods, making the derived distributions incomplete. Moreover, for MS stars with $P_\mathrm{rot} \gtrsim 10$ d the amplitude of the optical light curve is very small ($< 0.001 - 0.01$ mag), so their rotational modulation cannot be measured by ground-based photometry [see, e.g., @Radick_ea:1998; @Messina_ea:2003]. The absence of rotators with periods longer than $\sim 10$ d in the observed distributions shown in Figures \[fig:pks\_2val\] and \[fig:pks\_plaw\] can therefore be due to such limitations, although this is unlikely to hamper the analysis to a significant extent as the percentage of rotators with $P_\mathrm{rot} \gtrsim 10$ is predicted to be rather low at young ages. On the other hand, a poor sampling from ground-based observations (for example, $1$–$2$ d in the case of young associations considered in our analysis) makes it hard to detect the ultra-fast rotators, making the derived period distributions incomplete also on the fast side. Despite high-precision CCD photometry and multi-site monitoring campaigns may overcome most of these difficulties, the data available to date are still rather inhomogeneous and far from being complete. Most stars, especially in MS, do not show a permanent periodic modulation, and therefore deriving a rotation period is sometimes a matter of chance, unless stars are observed for several seasons [see, e.g., @Parihar_ea:2009]. Often a beat period is detected and prolonged monitoring is required to remove aliasing effects. The effect of large spotted areas localised on opposite stellar hemispheres, producing a rotational modulation with half the true period, should also be taken into account [see, e.g., @Cameron_ea:2009].
The dispersal of clusters’ stars into the field may add other selection effects. In order to assemble a complete sample one should, in principle, collect data for all stars in each cluster as well as data for all cluster’s stars that have been dispersed into the field, a task which is far from being accomplished. Regarding $\alpha$ Per, which is particularly important to constrain the rotational evolution just after the ZAMS, an alternative explanation for the presence of a high percentage of fast and ultra-fast rotators is therefore that the dataset available may miss information about a significant fraction of slower rotators, either because of observational biases or because of an intrinsic incompleteness of the sample due to stars’ dispersal into the field. We note in passing that the fraction of ultra-fast rotators (i.e., $P_\mathrm{rot} \lesssim 2$ d) in the CDF of AB Dor / Pleiades was reduced from $\sim0.2$ to the current $\sim0.12$ by the data recently acquired by [@Hartman_ea:2010]. The same reasoning may also apply to $\alpha$ Per, M50, M35, M 34 and M 11.
Some indication of possible observational biases in the alpha Per periods can be derived from a comparison with $v \sin i$ measurements. Using the @Gaige:1993 inversion procedures for a random orientation of rotational axes and $v \sin i$ data from [@Stauffer_ea:1985; @Stauffer_ea:1989], we have derived the cumulative distribution of equatorial velocities for alpha Per stars in our mass range. Such procedures have been applied, for instance, by @Queloz_ea:1998 in an analysis of rotational velocities in the Pleiades. We have then compared such a distribution with the cumulative distribution of equatorial velocities derived from the period measurements assuming that the radius of all stars in our mass range is approximately $1\, R_\odot$ (Fig. \[fig:aperbias\]).
![Cumulative distributions for the equatorial rotational velocities derived from the $v \sin i$ data (thin line) and from the rotation period data (thick line) in $\alpha$ Per.[]{data-label="fig:aperbias"}](figure7.eps){width="45.00000%"}
For the slowest rotators, [@Stauffer_ea:1985; @Stauffer_ea:1989] are able to derive only an upper limit of $v \sin i$, which for the stars in our mass range is $\sim 10$ km/s. This implies that the cumulative distribution of recovered equatorial velocities is rather uncertain at the lower end of the velocity range. Numerical experiments with several assumptions on the $v \sin i$ distribution below $10$ km/s have shown, however, that above $v_{\rm eq} \sim 15$ km/s the derived cumulative distribution function is unaffected by any assumption made on the $v \sin i$ distribution below $10$ km/s. On the other hand, the expected differences in stellar radii in our mass range have negligible effects on the cumulative distribution of equatorial velocities derived from the period measurements. Therefore, a meaningful comparison can be done between the fractions of stars with rotational velocities lower than, say, $30$ km/s in both distributions, which are approximately $80\, \%$ according to the $v \sin i$ measurements and $50\, \%$ according to the rotational period measurements. Furthermore, from the [@Chandra:1950] relationships (the overline denotes average) $$\overline{v \sin i} = \frac{\pi}{4} \bar{ v}$$ and $$\sigma^2= \overline{(v-\bar v)^2} = \frac{3}{2} \left[ \overline{(v \sin i)^2} - \frac{16}{\pi^2} \left(\overline{v \sin i} \right)^2 \right]$$ we derive $\bar v = 35.5$ km/s, and $\sigma^2 =2800$ (km/s)$^2$, while the rotational velocities inferred from the period measurements have $\bar v = 63.9$ km/s and $\sigma^2=3700$ (km/s)$^2$. Finally, the KS test applied to these cumulative distribution functions gives a probability of only $0.008$ that they are drawn from the same distribution. We conclude that there is a convincing evidence of a bias in the measured period distribution in $\alpha$ Per towards an excess of fast rotators. Having carried out such a test on the problematic period distribution of $\alpha$ Per, a similar comparison for other clusters in our sample would be desirable but it is outside the scope of this paper and is deferred to future work.
The DZM with a power-law prescription for $\tau_c$ is found quite effective in describing the rotational evolution from AB Dor / Pleiades to M 37. Between these two, however, it predicts more rotators with $P_\mathrm{rot} \gtrsim 5$ d than observed in M 50 and M 35. For M 34 (which seems to have also a higher fraction of fast rotators than predicted) and M 11 the number of available periods seems too low to draw any definitive conclusion. Assuming that the DZM contains the essential physics to describe the evolution from AB Dor / Pleiades to M 37 and that the age estimate is not too poor, one may reach the conclusion that a significant fraction of rotators with $P_\mathrm{rot} \gtrsim 5$ d are missing in M 50 and M 35. This conclusion is also supported by a crude comparison between the slower rotator tail in the observed CDFs, from which M 50 and M 35 would appear rotationally older than M 37. At ages between the ZAMS and AB Dor /Pleiades, the DZM with a power-law prescription for $\tau_c$ is unable to describe the high percentage of fast and ultra-fast rotators in $\alpha$ Per, as discussed above, and is not able to fit the Tuc-Hor/Col/Car, despite the shape of the observed and fitted CDFs’ are not too different. For these latter associations, however, it should be noted that, besides the issue of completeness discussed above, the modelling is particular sensitive to age uncertainties in this evolution phase and to the $\alpha$ Per constraint at 70 Myr.
The fit using the two-value prescription for $\tau_c$ is more affected by the high percentage of (ultra) fast rotators in $\alpha$ Per than the power-law prescription. This prevents from obtaining a satisfactory fit for older systems, where such a high percentage of fast and ultra-fast rotators is not present.
Only for M 34, which also seems to show a high percentage of fast rotators, $P_{\rm KS,i}$ is close to unity. Also for Tuc-Hor/Col/Car the two-value prescription for $\tau_c$ produces a synthetic CDF which is slightly higher than the power-law prescription for the faster rotators partly because of the constraint imposed by $\alpha$ Per.
Conclusions
===========
We studied the rotational evolution of solar-mass stars from PMS to the solar age, using the most complete sample of observations on rotation period distributions available to date, filling the gap between $\sim 6$ and $70$ Myr with recently acquired data and implementing a novel prescription on the internal rotational coupling in the framework of the DZM.
Although the completeness and accuracy of observational data have considerably improved in recent years, possible selection effects (e.g. scarce sensitivity to rotational modulation with $P_\mathrm{rot} \gtrsim 10$ d, limits for detection of fast or slow rotation periods due to insufficient time sampling or extension of the observations), as well as the uncertainties in clusters’ ages, are still severe limitations for the use of the period distributions to study stellar rotational evolution. Nevertheless, we investigated the dependence of the core-envelope coupling timescale $\tau_c$ on stellar rotation.
After the ZAMS, a power-law dependence of $\tau_c$ on the stellar rotational velocity produces CDFs whose shapes are more similar to observations than the two-value prescription and gives an excellent fit for AB Dor / Pleiades and M37. While M34 and M11 have too few period measurements to draw firm conclusions, the discrepancies in $\alpha$ Per, M50, and M35 could be due either to observational biases or to environmental effects, such as those causing early destruction of circumstellar discs.
Comparing the results obtained with the two-value prescription for $\tau_c$ with those obtained with the power-law, it appears that the former better reproduce the extrema while the latter the shape of the observed period distributions. This conclusion may be a useful guide to formulate more sophisticated models of rotational evolution in the future.
Part of the discrepancy between the models and the observations here may also arise from the assumption of a single disk lifetime ($\tau_{\rm disc}=5.8$ Myr). Assuming instead a distribution of disk lifetimes would yield more fast rotators at a given age and a broader rotation period distribution at the ZAMS, but including this feature in the model is beyond the scope of this paper.
To put our conclusions on a firm ground, however, more observations of clusters and associations in the age range $30 - 100$ Myr are required, together with an improvement of the model, e.g., by including correlations with the global properties of each cluster.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank an anonymous referee for her/his valuable comments. FS thanks Professor J. Bouvier for interesting discussion. This research has made use of the ADS-CDS databases, operated at the CDS, Strasbourg, France.
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[**New classes of bi-axially symmetric solutions\
to four-dimensional Vasiliev higher spin gravity**]{}\
Per Sundell[^1] and Yihao Yin[^2]\
*Departamento de Ciencias Físicas, Universidad Andres Bello\
Republica 220, Santiago de Chile*\
#### Abstract
We present new infinite-dimensional spaces of bi-axially symmetric asymptotically anti-de Sitter solutions to four-dimensional Vasiliev higher spin gravity, obtained by modifications of the Ansatz used in arXiv:1107.1217, which gave rise to a Type-D solution space. The current Ansatz is based on internal semigroup algebras (without identity) generated by exponentials formed out of the bi-axial symmetry generators. After having switched on the vacuum gauge function, the resulting generalized Weyl tensor is given by a sum of generalized Petrov type-D tensors that are Kerr-like or 2-brane-like in the asymptotic AdS$_4$ region, and the twistor space connection is smooth in twistor space over finite regions of spacetime. We provide evidence for that the linearized twistor space connection can be brought to Vasiliev gauge.
Introduction
============
Vasiliev’s equations [@Vasiliev:1990en] (for a recent review, see [@Didenko:2014dwa]) provide a fully nonlinear description of higher spin gauge fields in four dimensions coupled to gravity and matter fields. The basic feature of Vasiliev’s theory is that the full field configurations are captured by master fields that live on an extension of spacetime by a noncommutative twistor space. The equations admit an exact solution given by the direct product of anti-de Sitter spacetime and an undeformed twistor space. In a specific gauge, certain linearized perturbations of the noncommutative twistor space structure give rise to Fronsdal fields. This suggests a holographic relationship to three-dimensional conformal field theories [@WittenSeminar2001NovJHSchwarz60Birthday; @Sezgin:2002rt; @Klebanov:2002ja]; see also [@Sundborg:1999ue; @Leigh:2003gk; @Koch:2010cy]. In [@Giombi:2009wh; @Giombi:2010vg] this relation was examined under the assumption that the Gubser-Klebanov-Polyakov-Witten (GKPW) prescription [@Gubser:1998bc; @Witten:1998qj] for on-shell computations of Witten diagrams can be applied to classical field configurations obtained from Vasiliev’s equations.
However, the Fronsdal fields embedded into Vasiliev’s master fields have non-local interactions [@Sezgin:2002ru; @Kristiansson:2003xx; @Boulanger:2015ova] [^3] that belong to a functional class widely separated [@Taronna:2016ats] from that of the quasi-local Fronsdal theory [@Bekaert:2015tva], which is built by applying the canonical Noether approach to Fronsdal fields in anti-de Sitter spacetime. The GKPW prescription applies to the quasi-local theory by construction, as its action has self-adjoint kinetic terms, and the resulting holographic correlation functions indeed correspond to free three-dimensional conformal field theories.[^4] Recent work [@Vasiliev:2016xui] shows that there exists an explicit field redefinition that maps Vasiliev’s theory to a quasi-local theory on-shell, obtained by carefully fine-tuning the perturbative expansion on the Vasiliev side, though it remains to be seen whether it coincides with that of [@Bekaert:2015tva]. Moreover, as later shown in [@Taronna:2016xrm] the required field redefinition is large, and hence it is unclear to what extent the method can be used to actually compute any holographic correlation functions. Thus, to our best understanding, the issue of whether holographic amplitudes can be extracted by applying the GKPW prescription to the Fronsdal fields embedded into Vasiliev’s master fields remains an open problem.
An alternative approach, pursued in [@Boulanger:2015kfa], is to seek a weaker relation between the two theories, namely at the level of two distinct effective actions, derived in their own rights following different principles, and then evaluated subject to suitable dual boundary conditions. To this end, one starts from Hamilton’s principle applied to a covariant Hamiltonian action formulated using Weyl order on a noncommutative manifold whose boundary is given by the direct product of spacetime and twistor space [@Boulanger:2011dd; @Boulanger:2015kfa]; the Weyl order is required for the noncommutative version of the Stokes’ theorem to hold and for the imposition of boundary conditions. The resulting variational principle yields Vasiliev’s equations in Weyl order, that can be mapped back to Vasiliev’s normal order for special classes of initial data in twistor space following the perturbative scheme set up in [@Iazeolla:2011cb; @Iazeolla:ToAppear]. The resulting form of the higher spin amplitudes [@Colombo:2010fu; @Colombo:2012jx; @Didenko:2012tv] is closely related to first-quantized topological open string amplitudes [@Engquist:2005yt], but nonetheless reproduce exactly the same correlation functions as the Witten diagrams computed in the quasi-local theory. We would like to stress the fact that the Hamiltonian form of the action implies that the dependence of the classical Vasiliev master fields on classical sources are of a different type than for fields obeying equations of motion following from an action with self-adjoint kinetic terms. Indeed, instead of applying the GKPW prescription, the higher spin amplitudes are obtained from functionals given by topological boundary terms added to the Hamiltonian action [@Sezgin:2011hq; @Colombo:2012jx; @Boulanger:2015kfa], whose on-shell values are given by higher spin invariants, as we shall comment on further below.
In this paper, we shall construct new perturbatively defined solution spaces to Vasiliev’s equations in Weyl order, by taking into account classes of functions that resemble closely those used in [@Iazeolla:2011cb]. We shall then demonstrate explicitly that they can be mapped to Vasiliev’s normal order, at least at the linearized level, thus providing further evidence in favour of the covariant Hamiltonian approach outlined above.
To this end, we recall that at the linearized level, the fluctuations in the master fields that are asymptotic to anti-de Sitter spacetime form various representation spaces of the anti-de Sitter isometry algebra, including lowest-weight spaces as well as spaces associated to linearized solitons [@Iazeolla:2008ix] and generalized Petrov type-D solutions [@Didenko:2009td; @Iazeolla:2008ix; @Iazeolla:2011cb]. Nonlinear completions of various Type-D solution spaces were constructed in [@Didenko:2009td; @Iazeolla:2008ix; @Iazeolla:2011cb]; for a review, see [@Iazeolla:2012nf]. Of direct relevance for the work in this paper is the subspace that contains the the black-hole-like solutions,[^5] including spherically symmetric solutions. In these solutions, each individual Fronsdal field has a point-like source at the origin, showing up as a divergence in its Weyl tensor. However, upon packing all curvatures into a master zero-form, one obtains the symbol of a quantum-mechanical operator that approaches a delta function distribution at the origin [@Iazeolla:2011cb], which defines a smooth state as seen via classical observables given by zero-form charges [@Sezgin:2005pv; @Colombo:2012jx; @Didenko:2012tv]. In this sense, the black-hole-like Type-D solutions to Vasiliev’s theory are source free at the origin.[^6] Furthermore, it is possible to dress these solutions with lowest-weight space modes [@Iazeolla:ToAppear] at the fully nonlinear level; in doing so, the latter modes induce Type-D modes already at the second order of classical perturbation theory.[^7] Clearly, the full extent of the moduli space of the theory yet remains to be determined. In this paper, we shall present a new infinite-dimensional class of bi-axially symmetric exact solutions that are asymptotic to anti-de Sitter spacetime and singularity free at the level of zero-form charges. We shall furthermore propose a super-selection mechanism based on requiring that the solutions can be brought to Vasiliev gauge (where the asymptotic linearized fluctuations are in terms of Fronsdal fields). Our construction method follows closely the one devised in [@Iazeolla:2011cb] using gauge functions and separation of twistor space variables, which is in effect equivalent to starting from an Ansatz in Weyl order. The key difference is that we shall expand the master fields over a new set of elements in the associative fiber algebra, thus adding a branch to the existing moduli space. In a generic gauge, the expansion coefficients are functions on the base manifold. However, in the holomorphic gauge of [@Iazeolla:2011cb] the Weyl zero-form is a constant while the twistor space one-form is given by a universal set of functions, related to Wigner’s deformed oscillators, originally derived within the context of three-dimensional matter coupled higher spin gravity [@Prokushkin:1998bq]. The resulting solution space is then mapped to Vasiliev gauge in which the spacetime one-form consists of nonlinear Fronsdal tensors (after a suitable field redefinition in order to reinstate manifest Lorentz covariance). This map is achieved by means of two consecutive (large) gauge transformations: First, one uses a vacuum gauge function in SO(2,3)/SO(1,3).[^8] Provided that the resulting twistor space connection is smooth at the origin of the base of the twistor space, Vasiliev gauge can be reached by means of a second perturbatively defined gauge transformation. As we shall see, the real-analyticity requirement constrains the initial data in the Weyl zero-form already at the linearized level.[^9]
More specifically, the new sector of the fiber algebra is isomorphic to the group algebra $\Comp[\mathbb Z\times \mathbb Z]$ where $\mathbb Z\times \mathbb Z$ is generated by two elements in Sp$(4;\Comp)$ given by exponentials of a pair of Cartan generators of sp$(4;\Real)$. These correspond to linear symmetries of the two-dimensional harmonic oscillator, and generate the Killing symmetries of the solutions (including higher spin symmetries). As we shall see, the aforementioned super-selection rule amounts to restricting the master fields to a subalgebra of the group algebra not containing the unity.
The paper is organized as follows: In Section \[Sec Vasiliev eqs\] we review parts of Vasiliev’s bosonic higher spin gravity model that we shall use in constructing and interpreting the exact solutions. Solution spaces based on (semi)group algebras are constructed in Section \[Sec solve eqs\] using the aforementioned method; the singular nature of the contribution from the identity is pointed out in Section \[Sec singular id\]. In Section \[Sec Weyl tensor\], we show that the Weyl tensor is given by a sum of Petrov type-D tensors that are Kerr-like or 2-brane-like in the asymptotic AdS$_4$ region, and we compute higher spin curvature invariants. In Section \[Sec analyticity\], we show in special cases that the twistor space one-form is real-analytic in twistor space over finite regions of spacetime, and that its linearized part can be brought to Vasiliev gauge. We conclude in Section \[Sec Conclusions\].
Bosonic Vasiliev model \[Sec Vasiliev eqs\]
===========================================
In this section, we describe the non-minimal bosonic higher spin gravity model of Vasiliev type [@Vasiliev:1990en],[^10] for which we shall present exact solutions in the next section. The model is characterized by the fact that it admits a linearization consisting of real Fronsdal fields in four-dimensional anti-de Sitter spacetime of spins $s=0,1,2,\dots$ with each spin occurring once; for further details, we refer to [@Sezgin:2002ru; @Iazeolla:2011cb] and the review [@Didenko:2014dwa].
We first provide the formal definition in terms of master fields on the direct product of a commuting space and a noncommutative twistor space. We then spell out the component form of the equations, including their reformulation in terms of deformed oscillators. Finally, we remark on choices of bases for the internal algebra, and the Lorentz covariant weak field expansion scheme leading to Fronsdal fields, stressing the role of Vasiliev gauge and smoothness in twistor space.
Master field equations
----------------------
Vasiliev’s original formulation of higher spin gravity is given in terms of two master fields $\Phi $ and $A$ of degrees $0$ and $1$, respectively, and two closed and twisted-central elements $I$ and $\overline I$ of degree $2$, all of which are elements of a differential graded associative algebra $\Omega({\cal M})$ of forms on a non-commutative manifold ${\cal M}$, valued in an internal associative algebra ${\cal A}$. Letting $\star$ denote the associative product of $\Omega({\cal M})\otimes {\cal A}$, which is assumed to be compatible with $d$, the fully nonlinear master field equations read $$\begin{aligned}
F+ {\cal B}\star\Phi \star I- \overline {\cal B}\star\Phi \star \overline I &=&0 \ ,\label{originalEqA}\\
D\Phi &=& 0\ , \label{originalEqAPhi}\end{aligned}$$ where $$F:=dA+A\star A \ ,\qquad D\Phi:= d\Phi +A\star \Phi -\Phi \star \pi \left( A\right) \ ,$$ and $\pi$ denotes an automorphism of the differential graded associative algebra. The two-forms are characterised by the subsidiary constraints $$\begin{aligned}
dI &=&0\text{ ,} \qquad I\star f \ =\ \pi (f)\star I \label{Jf=pifJ}\ ,\end{aligned}$$ for any $f\in \Omega({\cal M})\otimes {\cal A}$, *idem* $\overline I$. Finally, the star functions $${\cal B}:=\sum_{n=0}^\infty b_n (\Phi\star \pi(\Phi))^{\star n}\ ,\qquad
\overline {\cal B}:=\sum_{n=0}^\infty \bar b_n (\Phi\star \pi(\Phi))^{\star n}\ ,$$ where $b_n, \bar b_n\in \Comp$. It follows that $\Phi\star \pi(\Phi)$ and hence ${\cal B}$ is covariantly constant, *viz.* $$d{\cal B}+A\star {\cal B}-{\cal B}\star A=0\ ,$$ *idem* $\overline {\cal B}$. As $\Phi \star I$ and $\Phi \star \overline I$ are covariantly constant as well, it follows that the constraint on $F$ is compatible with its Bianchi identity. The integrability of the constraint on $D\Phi$, on the other hand, requires $F\star \Phi-\Phi\star \pi(F)$ to vanish, which is indeed a consequence of the constraint on $F$. The resulting Cartan integrability, *i.e.* consistency with $d^2\equiv 0$, holds for any dimension of ${\cal M}$ and any star functions ${\cal B}$ and $\overline {\cal B}$, which are hence not fixed uniquely by the requirement of higher spin symmetry alone.
In the context of higher spin gravity, it is usually assumed that =[X]{}\_4 \_4 ,\[directproduct\]where ${\cal X}_4$ is a four-dimensional real commuting manifold, with coordinates $x^{\mu }$, and ${\cal Z}_4$ is a four-dimensional real non-commutative symplectic manifold, with canonical coordinates $Z^{\underline{\alpha }}$. The compatibility between the star product and the differential amounts to the Leibniz’ rule $$d(f\star g)=df\star g+ (-1)^{{\rm deg}(f)} f\star dg\ .$$ The differential star product algebra is assumed to be trivial in strictly positive degrees, in the sense that $d\Xi^M:=(dx^\mu,dz^\a,d\bar z^{\dot{\alpha}})$ are taken to be graded anti-commuting elements obeying $$d\Xi^M \star f= d\Xi^M \wedge f\ ,\qquad
f\star d\Xi^M = f\wedge d\Xi^M \ ,$$ which are consistent with associativity. The algebra $\Omega({\cal M})\otimes {\cal A}$ is also assumed to be equipped with an anti-linear anti-automorphism $\dagger$, for which we use the convention $$\left( f_{1}\star f_{2}\right) ^{\dag }=(-1)^{{\rm deg}(f_1){\rm deg}(f_2)}
f_{2}^{\dag }\star f_{1}^{\dag }\ ,\qquad (df)^\dagger= d(f^\dagger)
\text{ .}$$ In case of the basic bosonic models, without internal Yang-Mills symmetries, the internal algebra ${\cal A}$ consists of classes of functions on yet one more four-dimensional real non-commutative symplectic manifold, that we shall denote by ${\cal Y}_4$, with canonical coordinates $Y^{\underline{\alpha}}$. We shall refer to ${\cal Y}_4\times{\cal Z}_4$ as the full twistor space, and ${\cal Y}_4$ and ${\cal Z}_4$, respectively, as the internal and external twistor spaces.[^11] The Sp(4;$\Real$) quartets are split into SL(2;$\mathbb{C}$) doublets, *viz.*[^12] $$Y^{\underline{\alpha}}=( y^{\alpha },\bar{y}^{\dot{\alpha}})\ ,\qquad
Z^{\underline{\alpha }}= (z^{\alpha },\bar{z}^{\dot{\alpha}})\ ,$$ obeying $$\bar{y}^{\dot{\alpha}}=\left( y^{\alpha }\right) ^{\dag }\text{ , }\bar{z}^{
\dot{\alpha}}=-\left( z^{\alpha }\right) ^{\dag }\text{ ,} \label{realityyz}$$ The automorphism $\pi $ and its hermitian conjugate $\bar{\pi}$ are defined by $$\begin{aligned}
\pi \left( x^{\mu };y^{\alpha },\bar{y}^{\dot{\alpha}};z^{\alpha },\bar{z}^{\dot{\alpha}}\right) &=&\left( x^{\mu };-y^{\alpha },\bar{y}^{\dot{\alpha}};-z^{\alpha },\bar{z}^{\dot{\alpha}}\right) \text{ ,} \\
\bar{\pi}\left( x^{\mu };y^{\alpha },\bar{y}^{\dot{\alpha}};z^{\alpha },\bar{z}^{\dot{\alpha}}\right) &=&\left( x^{\mu };y^{\alpha },-\bar{y}^{\dot{\alpha}};z^{\alpha },-\bar{z}^{\dot{\alpha}}\right) \text{ ,}\end{aligned}$$ and $\pi \circ d=d\circ \pi$ *idem* $\bar \pi$. Imposing $$\begin{aligned}
\Phi ^{\dag } &=&\pi \left( \Phi \right) \text{ ,} \qquad
A^{\dag } \ =\ -A\text{ ,} \qquad I^\dag \ =\ \overline I\ , \label{realityA}\end{aligned}$$ and $${\cal B}^\dag=\overline{\cal B}\ ,$$ that is, $(b_n)^\dag=\bar b_n$, and $$\begin{aligned}
\pi \bar{\pi}\left( \Phi \right) &=&\Phi \text{ ,} \qquad \pi \bar{\pi}\left( A\right) \ =\
A\text{ ,}\qquad \pi \bar{\pi}\left( I \right) \ =\ I\ ,\qquad \pi \bar{\pi}\left(\overline
I \right) \ =\ \overline I\label{bosonA}\end{aligned}$$ yields a model with a perturbative expansion around four-dimensional anti-de Sitter spacetime in terms of Fronsdal fields of all integer spins. The equations given so far provide a formal definition of the basic bosonic model.
Star product, twisted central element and traces {#starstraces}
------------------------------------------------
In what follows, we shall use Vasiliev’s original realization of the $\star$-product given by $$\begin{aligned}
&&f_{1}\left( y,\bar{y},z,\bar{z}\right) \star f_{2}\left( y,\bar{y},z,\bar{z}\right) \notag \\
&=&\!\!\!\!\!\int \frac{d^{2}ud^{2}\bar{u}d^{2}vd^{2}\bar{v}}{\left( 2\pi \right) ^{4}}e^{i v^{\alpha }u_{\alpha }+i\bar{v}^{\dot{\alpha}}\bar{u}_{\dot{\alpha}} }\ f_{1}\left( y+u,\bar{y}+\bar{u};z+u,\bar{z}-\bar{u}\right)
f_{2}\left( y+v,\bar{y}+\bar{v};z-v,\bar{z}+\bar{v}\right) \text{ .} \notag
\\
&& \label{star prod def}\end{aligned}$$ We shall encounter $\star $-product compositions leading to Gaussian integrals involving indefinite bilinear forms. To define these we use the fact that the auxiliary integration is a formal representation of the original Moyal-like contraction formula, which means that the integration must be performed by means of analytical continuations of the eigenvalues of the bilinear forms.
#### Symbol calculus.
The star product rule implies that $$\left[ f_{1}\left( y,\bar{y}\right) ,f_{2}\left( z,\bar{z}\right) \right]
_{\star }=0\text{ ,}$$ that is, the variables $Y^{\underline{\alpha}}$ and $Z^{\underline{\alpha }}$ are mutually commuting. Moreover, from $$y_{\alpha }\star y_{\beta }=y_{\alpha }y_{\beta }+i\varepsilon _{\alpha
\beta }\text{ , \ }y_{\alpha }\star z_{\beta }=y_{\alpha }z_{\beta
}-i\varepsilon _{\alpha \beta }\text{ , \ }z_{\alpha }\star y_{\beta
}=z_{\alpha }y_{\beta }+i\varepsilon _{\alpha \beta }\text{ , \ }z_{\alpha
}\star z_{\beta }=z_{\alpha }z_{\beta }-i\varepsilon _{\alpha \beta }\text{ ,} \label{yz comm}$$ it follows that $$a_{\alpha }^{\pm }:=\frac{1}{2}\left( y_{\alpha }\pm z_{\alpha }\right) \text{
,}$$ obey $$\left[ a_{\alpha }^{-},a_{\beta }^{+}\right] _{\star }=
\left[ a_{\alpha }^{+},a_{\beta }^{-}\right] _{\star }=
i\varepsilon _{\alpha \beta }\text{ , }
\left[ a_{\alpha }^{+},a_{\beta }^{+}\right] _{\star }=
\left[ a_{\alpha }^{-},a_{\beta }^{-}\right] _{\star }=0\text{ .}$$ Letting ${\cal O}_{\rm Weyl}$ and ${\cal O}_{\rm Normal}$ denote the Wigner maps that send a classical function $f$ to the operator with symbol $f$ in the Weyl and normal order, respectively, where an operator is said to be in normal order if all ${\cal O}_{\rm Normal}(a^+_\a)$ stand to the left of all ${\cal O}_{\rm Normal}(a^-_\a)$. As a result, one has \_[Normal]{}(f\_[1]{}( y,z) f\_[2]{}( y,z))= [O]{}\_[Normal]{}(f\_[1]{}( y,z)) [O]{}\_[Normal]{}(f\_[2]{}( y,z)) One also has \_[Weyl]{}(f(y))=[O]{}\_[Normal]{}(f(y)) ,\_[Weyl]{}(f(z))=[O]{}\_[Normal]{}(f(z)) ,resulting in that $$\begin{aligned}
{\cal O}_{\rm Weyl}(f_{1}\left( y\right) \star f_{2}\left( y\right)) &=&
{\cal O}_{\rm Weyl}(f_{1}\left( y\right) ) {\cal O}_{\rm Weyl}( f_{2}\left( y\right)) \text{ ,} \\
{\cal O}_{\rm Weyl}(f_{1}\left( z\right) \star f_{2}\left( z\right)) &=&
{\cal O}_{\rm Weyl}(f_{1}\left( z\right) ) {\cal O}_{\rm Weyl}(f_{2}\left( z\right))\text{ ,}\end{aligned}$$ and also \_[Normal]{}( f\_[1]{}( y) f\_[2]{}( z)) =\_[Weyl]{}( f\_[1]{}( y) f\_[2]{}( z)) = [O]{}\_[Weyl]{}( f\_[1]{}( y) ) [O]{}\_[Weyl]{}( f\_[2]{}( z)) \[f1f2 Weyl\]
#### Twisted central element.
The condition can be solved by $$I=j_z\star \kappa_y\ ,\qquad j_z=\frac{i}{4}dz^\alpha \wedge dz^\b \varepsilon _{\alpha \beta }\kappa_z
\text{ , \qquad }\kappa _{y}=2\pi \delta
^{2}\left( y\right) \text{ , \qquad }\kappa _{z}=2\pi \delta ^{2}\left( z\right)
\text{ ,}\label{factorized}$$ where $\kappa_y$ is an inner Klein operator obeying $$\kappa_y \star f(y)\star \kappa_y= f(-y) \text{ , \qquad }\kappa _{y}\star
\kappa _{y}\ =\ 1\ ,$$ *idem* $\kappa_z$. Thus, one may write I=dz\^dz\^\_ ,:=\_[y]{}\_[z]{}=(i y\^z\_) ,\[kappa\]where thus f(y,z)=f(z,y) ,f(y,z)=f(-z,-y) ,f(y,z)=(f(y,z)) ,=1 .By hermitian conjugation one obtains $$\overline {I}=-I ^{\dag }=\frac{i}{4}d\bar z^{\dot\alpha} \wedge d\bar z^{\dot\beta}
\varepsilon _{\dot{\alpha}\dot{\beta}}\bar{\kappa}\ .$$ The two-forms $j_z$ and $\overline j_{\bar z}$ can be extended to globally defined forms on a non-commutative space ${\cal Z}_4$ having the topology of a direct product of two complexified two-spheres [@Iazeolla:2011cb; @Boulanger:2015kfa], with nontrivial flux \_[[Z]{}\_4]{} j\_zj\_[|z]{}=- . In this topology, it is furthermore assumed that $\Phi$ belongs to a section that is bounded at infinity, while the twistor-space one-form is a connection whose curvature two-form falls off at infinity.
We note that the form of $I$ given in Eq. is useful in deriving the perturbative expansion in terms of Fronsdal fields in Vasiliev gauge, while the factorized form in Eq. is useful in finding exact solutions.
#### Trace operations.
The detailed form of the symbol of an operator depends on the basis with respect to which it is defined. Its trace, on the other hand, is basis independent, and in addition gauge invariant. The star product algebra admits two natural trace operations. The basic operation is given by the integral over phase space using the symplectic measure, *viz.* f := \_[[Z]{}\_4\_4]{} j\_yj\_[|y]{} \_y |\_[|y]{} f ,f([Z]{}\_4) ,where $j_y$ is given by replacing $z^\alpha$ by $y^\alpha$ in $j_z$ defined in Eq. . An alternative trace operation, of relevance to higher spin gauge theory, can be defined if ${\cal A}$ admits the decomposition =\_[n,|n=0,1]{} [A]{}\_[n,|n]{}(\_y)\^n (|\_[|y]{})\^[|n]{} , \[calAnbarn\]where ${\cal A}_{n,\bar n}$ consist of operators whose symbols in Weyl order are regular at the origin of ${\cal Y}_4$. One may then define the trace operation ’ f := \_[ [Y]{}\_4]{} j\_yj\_[|y]{} f\_[1,|1]{}=- f\_[1,|1]{}|\_[y=0=|y]{} , using the decomposition , with the convention that \_y|\_[|y]{}f=f’ f =18 f|\_[y=0=|y]{} .One may view ${\rm Tr}'$ as a regularized version of ${\rm Tr}$ in the sense that if $f$ admits a decomposition of the form then $$\begin{aligned}
{\rm Tr} f &=& \sum_{n,\bar n=0,1} {\rm Tr} f_{n,\bar n}\star
(\kappa_y)^n \star (\bar\kappa_{\bar y})^{\bar n}\\
&=& {\rm Tr}' f+ {\rm Tr} (f_{0,\bar 0} +f_{1,\bar 0}\star
\kappa_y + f_{0,\bar 1} \star \bar\kappa_{\bar y})\ ,\end{aligned}$$ that is, ’ f=[Tr]{} f - [Tr]{} (f\_[0,|0]{} +f\_[1,|0]{}\_y + f\_[0,|1]{} |\_[|y]{}) . Indeed, in several applications it turns out that ${\rm Tr} f $ is ill-defined while ${\rm Tr}' f $ is well-defined, as for example in the case that $f$ is a polynomial on ${\cal Y}_4$.
Equations in components and deformed oscillators
------------------------------------------------
We decompose the master one-form into locally defined components as follows: $$A=U_{\mu }dx^{\mu }+V_{\alpha }dz^{a}+V_{\dot{\alpha}}d\bar{z}^{\dot{\alpha}}\ ,$$ The reality condition (\[realityA\]) and the bosonic projection (\[bosonA\]) imply $$\begin{aligned}
U_{\mu }^{\dag } &=&-U_{\mu }\text{ ,} \qquad
V_{\alpha }^{\dag } \ =\ \bar{V}_{\dot{\alpha}}\text{ ,}\\
\pi \bar{\pi}\left( U_{\mu }\right) &=&U_{\mu }\text{ ,} \qquad
\pi \bar{\pi}\left( V_{\alpha }\right) \ =\ -V_{\alpha }\text{ .}\end{aligned}$$ Decomposing master equations into components using inner derivatives $\imath_{\partial_\mu}$, $\imath_{\partial_\a}$ and $\imath_{\partial_{\dot\a}}$, where $\partial_\a\equiv \partial/\partial z^\a$ *idem* $\partial_{\dot\a}$, one has $$\begin{aligned}
\partial _{\lbrack \mu }U_{\nu ]}+U_{[\mu }\star U_{\nu ]} &=&0\text{ ,}
\label{eqxU} \\
\partial _{\mu }\Phi +U_{\mu }\star \Phi -\Phi \star \pi \left( U_{\mu
}\right) &=&0\text{ ,} \label{eqxPhi}\end{aligned}$$ the mixed components $$\partial _{\mu }V_{\alpha }-\partial _{\alpha }U_{\mu }+\left[ U_{\mu
},V_{\alpha }\right] _{\star }=0\text{ , \qquad }\partial _{\mu }\bar{V}_{\dot{
\alpha}}-\partial _{\dot{\alpha}}U_{\mu }+\left[ U_{\mu },\bar{V}_{\dot{
\alpha}}\right] _{\star }=0\text{ ,} \label{eqxZ}$$ which are related by hermitian conjugation, and $$\begin{aligned}
\partial _{\lbrack \alpha }V_{\beta ]}+V_{[\alpha }\star V_{\beta ]}+\frac{i}{4}\varepsilon _{\alpha \beta }{\cal B}\star \Phi \star \kappa &=&0 \ ,\qquad
\partial_{\lbrack \dot{\alpha}}\bar{V}_{\dot{\beta}]}+\bar{V}_{[\dot{\alpha}}\star
\bar{V}_{\dot{\beta}]}+\frac{i}{4}\varepsilon _{\dot{\alpha}\dot{\beta}}
\overline {\cal B}\star\Phi
\star \bar{\kappa}\ =\ 0\ ,\qquad \label{eqZ-VJ} \\
\partial _{\alpha }\Phi +V_{\alpha }\star \Phi -\Phi \star \bar{\pi}\left(
V_{\alpha }\right) &=&0 \label{eqZ-PhiV} \ ,\qquad
\partial _{\dot{\alpha}}\Phi +\bar{V}_{
\dot{\alpha}}\star \Phi -\Phi \star \pi \left( \bar{V}_{\dot{\alpha}}\right)
\ =\ 0\text{ ,}\\
\partial _{\alpha }\bar{V}_{\dot{\alpha}}-\partial _{\dot{\alpha}}V_{\alpha
}+\left[ V_{\alpha },\bar{V}_{\dot{\alpha}}\right] _{\star } &=&0\text{ ,}
\label{eqZ-V}\end{aligned}$$ where the two equations in Eq. are related by hermitian conjugation *idem* Eq. . The twistor space equations – can be rewritten by introducing Vasiliev’s deformed oscillators[@Vasiliev:1990en] $$S_{\alpha }=z_{\alpha }-2iV_{\alpha }\text{ , \ }\bar{S}_{\dot{\alpha}}=\bar{z}_{\dot{\alpha}}-2i\bar{V}_{\dot{\alpha}}\text{ ,}$$ for which the reality condition and the bosonic projection take the form: $$\begin{aligned}
\left( S_{\alpha }\right) ^{\dag } &=&-\bar{S}_{\dot{\alpha}}\text{ ,} \\
\pi \bar{\pi}\left( S_{\alpha }\right) &=&-S_{\alpha }\text{ .}\end{aligned}$$ In terms of the new fields, the aforementioned equations read $$\begin{aligned}
\left[ S_{\alpha },S_{\beta }\right] _{\star } &=&-2i\varepsilon _{\alpha
\beta }\left( 1-{\cal B}\star \Phi \star \kappa \right) \text{ \ and h.c.\ ,} \\
S_{\alpha }\star \Phi +\Phi \star \pi \left( S_{\alpha }\right) &=&0\text{ \
and h.c.\ ,} \\
\left[ S_{\alpha },\bar{S}_{\beta }\right] _{\star } &=&0\text{ ,}\end{aligned}$$ as can be seen using $$\begin{aligned}
\left[ z_{\alpha },f\right] _{\star } &=&-2i\partial _{\alpha }f\text{ , \ }\left[ \bar{z}_{\dot{\alpha}},f\right] _{\star }=-2i\partial _{\dot{\alpha}}f\text{ ,} \\
\left[ z_{\alpha },z_{\beta }\right] _{\star } &=&-2i\varepsilon _{\alpha
\beta }\text{ , \ }\left[ \bar{z}_{\dot{\alpha}},\bar{z}_{\dot{\beta}}\right]
_{\star }=-2i\varepsilon _{\dot{\alpha}\dot{\beta}}\text{ ,} \\
\left[ z_{\alpha },\bar{z}_{\dot{\alpha}}\right] _{\star } &=&0\text{ .}\end{aligned}$$ As we shall see below, the deformed oscillators are useful in defining the field redefinition to Lorentz covariant basis. They also provide a useful basis for finding exact solutions as they convert the differential equations on ${\cal Z}_4$ into algebraic equations that can be solved using Laplace transformation methods [@Prokushkin:1998bq]; for related details, see [@Iazeolla:2011cb].
Lorentz covariance, Fronsdal fields and Weyl tensors {#LorentzFronsdal}
----------------------------------------------------
To arrive at a perturbative formulation in terms of Fronsdal fields on ${\cal X}_4$, one first solves Eqs. (\[eqxZ\])–(\[eqZ-V\]) subject to an initial datum for $\Phi$ and $U_\mu$ at $Z^{\underline\a}=0$ in a perturbative expansion in the zero-form initial data in Vasiliev gauge[^13] $$z^\alpha V_\alpha=0\ .\label{vasgauge}$$ In this gauge, initial data for the zero-form given by generic smooth symbols on ${\cal Y}_4$ yields twistor space configurations that are smooth functions on ${\cal Y}_4\times {\cal Z}_4$. Letting $\omega_\mu^{\alpha\beta}$ denote the canonical Lorentz connection, one can show that [@Vasiliev:1999ba] $\Phi$, $V_\a$ and[^14] $$W_\mu:=U_\mu- \frac1{4i} \left( \omega_\mu^{\alpha\beta} {M}_{\alpha\beta}
+\bar{\omega}_\mu^{\dot{\alpha}\dot{\beta}} M_{\dot{\alpha}\dot{\beta}}\right)\ ,
\label{fieldredef}$$ where $$M_{\alpha\beta}:=y_\alpha y_\beta -z_\alpha z_\beta+S_{\alpha}
\star S_{\beta}\ ,$$ have Taylor expansions in $(Y^{\underline\a},Z^{\underline\a})$ around $Y^{\underline\a}=Z^{\underline\a}=0$ in terms of Lorentz tensors. The redefinition induces a shift symmetry that can be used to set the coefficient of $y_\alpha y_\beta$ in $W_\mu$ to zero, such that $$W_\mu|_{Z=0}=e_\mu+W'_\mu\ ,\qquad e_\mu=\frac1{2i}
e^{\alpha\dot\alpha}_\mu y_\alpha \bar y_{\dot \alpha}\ ,$$ where $W'_\mu$ consists of a spin-one field and a tower of higher spin gauge fields with $s=3,4,\dots$. Proceeding by assuming that $e^{\alpha\dot\alpha}_\mu$ defines a vierbein, and taking $\Phi|_{Z=0}$ and $W'_\mu$ to be weak fields in which the couplings in Eqs. (\[eqxU\])–(\[eqxPhi\]) can be expanded perturbatively, one can show that the resulting algebraically independent fields are given by the Lorentz scalar $$\varphi:=\Phi|_{Y=Z=0}\ ,\label{Vasilievgauge}$$ the metric $$g_{\mu\nu}:=e_\mu^a e_{\nu,a}\ ,$$ and the tower of doubly traceless tensor gauge fields $$\varphi_{a_1\dots a_s}:=(e^{-1})_{(a_1}{}^\mu
W'_{\mu,a_2\dots a_s)}\ ,\qquad s=1,3,4,\dots\ ,\label{Fronsdal}$$ where $W'_{\mu,a_1\dots a_n}$ is the coefficient in $W'_\mu$ of $(\sigma^{a_1})_{\a\dot\a}y^{\a}\bar y^{\dot{\a}}\cdots
(\sigma^{a_n})_{\a\dot\a}y^{\a}\bar y^{\dot{\a}}$. These fields obey equations of motion on the Lorentzian manifold $({\cal X}_4,g_{\mu\nu})$ with second-order kinetic terms, critical masses and dynamical metric.[^15]
The virtue of Vasiliev gauge is that the metric and the gauge fields are identical to the Fronsdal tensors that can be obtained at the linearized level by integrating the generalized Weyl tensor C\_[\_1…\_[2s]{}]{}=.( )|\_[Y=Z=0]{} ,s=1,2,3,… ,using the generalized Poincare lemma (for example, see [@DuboisViolette:1999rd; @DuboisViolette:2001jk; @Bekaert:2002dt]). In other words, an asymptotic observer who sources the bulk using a linearized spin-$s$ Fronsdal field will activate the corresponding component field given above, whose boundary value can thus be identified with a dual conformal field theory source coupled to a conserved spin-$s$ current.
The higher order couplings depend on the choice of gauge as well as the initial data for $\Phi$ and $W_\mu$; as proposed by Vasiliev [@Vasiliev:2016xui], these initial data can be fine-tuned at higher orders in order to obtain quasi-local equations of motion in the gauge .
An alternative approach, which we shall follow here, is to restrict the initial data for the zero-form to specific classes of functions on ${\cal Y}_4$, corresponding to associative subalgebras of ${\cal A}$ leading to well-defined field configurations obeying physical boundary conditions on ${\cal M}$.
Internal star product algebras and solution spaces
--------------------------------------------------
A parameterised set $(\Phi(\nu,G),U(\nu,G),V(\nu,G),
\bar V(\nu,L))_{}$, where $\nu$ belongs to a parameter space and $G$ is a gauge function, obeying the master field equations form an admissible solution space if they generate a free differential algebra together with $I$ and $\overline I$ (for each fixed value of $\nu$). To construct such spaces we use associative star product algebras[^16] \_[S]{}=\_ T\_ł ,that are closed under the actions of $\pi$, $\bar\pi$, $\dagger$ and star multiplication by $\kappa_y$ and $\bar\kappa_{\bar y}$, and whose basis elements $T_\l$, labeled by $\lambda$ in a discrete set ${\cal S}$, have finite traces. We say that ${\cal A}_{{\cal S}}$ is contained in ${\cal A}_{{\cal S}'}$ if there exists a monomorphism $\rho:{\cal A}_{{\cal S}'}\to
{\cal A}_{{\cal S}}$ such that ${\rm Tr}'\circ \rho={\rm Tr}'$ *i.e.* if the elements in ${\cal A}_{{\cal S}}$ can be expanded in terms of the elements in ${\cal A}_{{\cal S}'}$ in a way compatible with the trace operation.
Expanding the master fields over ${\cal A}_{\cal S}$ yields a set of modes on ${\cal X}_4$ and ${\cal Z}_4$ that forms a free differential algebra together with $j_z$ and its hermitian conjugate. Using Cartan integration methods, the modes can be expressed locally in terms of zero-form integration constants, which define the $\nu$ parameters, and gauge functions. These data can then be adapted to boundary conditions, which may require a change of basis from ${\cal A}_{{\cal S}}$ to a basis ${\cal A}_{{\cal S}'}$ containing ${\cal A}_{{\cal S}}$; for example, in asymptotically anti-de Sitter spacetimes, it makes sense to impose boundary conditions in a Lorentz covariant basis adapted to a dual conformal field theory. We shall say that a subalgebra ${\cal A}_{\cal S}$ yields a higher spin gravity solution space if the resulting Lorentz covariant master fields in Vasiliev gauge have symbols defined in normal order that can be expanded over finite regions of ${\cal X}_4$ in terms of the set of monomials on ${\cal Y}_4\times {\cal Z}_4$ that vanish at the origin of ${\cal Y}_4\times {\cal Z}_4$, *i.e.* they are real-analytic at this point.
The resulting moduli spaces can be coordinatized by higher spin invariant functionals, playing the role of classical higher spin observables [@Colombo:2010fu; @Sezgin:2011hq; @Vasiliev:2015mka; @Vasiliev:2016sgg]. By choosing a structure group [@Sezgin:2011hq] and fixing a topology for the base manifold, one may extend the locally defined solutions to globally defined higher spin geometries supporting various types of topologically nontrivial observables. Working locally on ${\cal X}_4$, the accessible observables are on-shell closed zero-forms on ${\cal X}_4$ given by combined integrals over ${\cal Z}_4$ and traces over ${\cal Y}_4$ of adjoint constructs built from $(\Phi,V_\a,\bar V_{\dot\a};I,\overline I;\kappa,\bar\kappa)$, referred to as zero-form charges. Evaluated on solutions that are asymptotical to anti-de Sitter spacetime, these observables have been shown to have a physical interpretation as generating functionals for correlation functions of holographically dual conformal field theories.
We remark that various subalgebras of ${\cal A}$ can be obtained from different quantum mechanical systems in four-dimensional phase space. It is an interesting problem to examine which of these are admissible in the above sense, and to furthermore distinguish between these systems using higher spin invariant observables.
New class of biaxially symmetric solutions\[Sec solve eqs\]
===========================================================
In this section, we construct a new class of exact solutions to Vasiliev’s equations on a direct product manifold of the form using a gauge function and expansions in terms of exponentials of two Cartan generators of ${\rm sp}(4;\Comp)$, which leads to biaxial symmetry.
Gauge function {#Sec Gauge function}
--------------
From Eq. (\[eqxU\]) and the fact that ${\cal X}_4$ is commuting, it follows that $U_{\mu}$ can be expressed in terms of a gauge function $G$ defined locally on ${\cal X}_4\times {\cal Z}_4$ . Thus, setting[^17] $$\begin{aligned}
U^{(G)}_{\mu } &=&G^{-1}\star \partial _{\mu }G\text{ ,} \label{AnsatzUG} \\
\Phi^{(G)} &=&G^{-1}\star \Phi ^{\prime }\star \pi {}\left( G\right)
\label{AnsatzPhiG} \\
V^{(G)}_{\alpha } &=&G^{-1}\star \partial _{\alpha }G+G^{-1}\star V_{\alpha
}^{\prime }\star G\text{ , }\qquad\bar{V}^{(G)}_{\dot{\alpha}}=G^{-1}\star \partial _{\dot{\alpha}}G+G^{-1}\star \bar{V}_{\alpha }^{\prime }\star G \text{ ,}
\label{AnsatzVG}\end{aligned}$$ Eqs. (\[eqxPhi\]) and (\[eqxZ\]) reduce to $$\partial_\mu \Phi ^{\prime }=0\ ,\qquad \partial_\mu V_{\alpha
}^{\prime }=0\ ,\qquad \partial_\mu \bar{V}_{\alpha }^{\prime }=0\ ,$$ *i.e.* the primed fields are constant on ${\cal X}_4$, and Eqs. (\[eqZ-VJ\])-(\[eqZ-V\]) take the form $$\begin{aligned}
\partial _{\lbrack \alpha }V_{\beta ]}^{\prime }+V_{[\alpha }^{\prime }\star
V_{\beta ]}^{\prime }+\frac{i}{4}\varepsilon _{\alpha \beta }{\cal B}'\star \Phi ^{\prime
}\star \kappa &=&0\text{ \ and h.c.\ ,} \label{X-indep-VJ} \\
\partial _{\alpha }\Phi ^{\prime }+V_{\alpha }^{\prime }\star \Phi ^{\prime
}-\Phi ^{\prime }\star \bar{\pi}\left( V_{\alpha }^{\prime }\right) &=&0\text{ \ and h.c.\ ,} \label{X-indep-PhiV} \\
\partial _{\alpha }\bar{V}_{\dot{\alpha}}^{\prime }-\partial _{\dot{\alpha}}V_{\alpha }^{\prime }+\left[ V_{\alpha }^{\prime },\bar{V}_{\dot{\alpha}}^{\prime }\right] _{\star } &=&0\text{ ,} \label{X-indep-V}\end{aligned}$$ where ${\cal B}':=\sum_{n=0}^\infty b_n (\Phi'\star \pi(\Phi'))^{\star n}$.
In order to obtain solutions that are asymptotic to AdS$_4$, we choose[^18] $$G=L\star H\ ,$$ where $L$, which we shall refer to as the vacuum gauge function, is a locally defined map from ${\cal X}_4$ to SO(2,3)/SO(1,3) that is constant on ${\cal Z}_4$, *i.e.* $$\partial _{\alpha}L=\partial _{\dot{\alpha}}L=0\ ,$$ and $H$ is determined by imposing the Vasiliev gauge condition , *viz.* $$z^\a V^{(G)}_{\alpha } =0\ ,\qquad
\bar z^{\dot\alpha} \bar{V}^{(G)}_{\dot{\alpha}}=0\ ,$$ in a perturbative expansion $$H=1+\sum_{n=1}^\infty H^{(n)}\ ,$$ where the superscript $(n)$ denotes an $n$-linear function of $\Phi'$. Thus, the master fields in Vasiliev gauge are given by perturbative corrections of $$\begin{aligned}
U^{(L)}_{\mu } &=&L^{-1}\star \partial _{\mu }L\text{ ,} \label{AnsatzU} \\
\Phi^{(L)} &=&L^{-1}\star \Phi ^{\prime }\star \pi {}\left( L\right)
\label{AnsatzPhi} \\
V^{(L)}_{\alpha } &=&L^{-1}\star \partial _{\alpha }L+L^{-1}\star V_{\alpha
}^{\prime }\star L\text{ , }\qquad\bar{V}^{(L)}_{\dot{\alpha}}=L^{-1}\star \partial _{\dot{\alpha}}L+L^{-1}\star \bar{V}_{\alpha }^{\prime }\star L \text{ ,}
\label{AnsatzV}\end{aligned}$$ where the Maurer-Cartan form $U^{(L)}_{\mu}$ consists of the frame field and Lorentz connection on the anti-de Sitter background spacetime, for which we shall use the explicit form in stereographic coordinates given in Appendix \[Sec coordinates\]. As for $H$, its existence requires that $V^{(L)}_{\underline{\alpha}}$ admits a power series expansion on ${\cal Z}_4$ around $Z^{\underline{\alpha}}=0$, to be examined in more detail in Section \[Sec analyticity\].
Thus, the dependence on ${\cal X}_4$ arises via the gauge function, leaving ${\cal X}_4$-independent equations (\[X-indep-VJ\])–(\[X-indep-V\]), to which we turn next.
Exact solutions in holomorphic gauge from abelian group algebras {#Sec general ansatz}
----------------------------------------------------------------
One class of solution spaces arise from star product algebras \_=\_[n,|n=0,1]{}\_ (T\_[ł]{} \_y\^[ n]{}|\_[|y]{}\^[ |[n]{}]{}) ,\[calALambda\]where $\vec\l=(\l_1,\dots,\l_N)$ belongs to an $N$-dimensional lattice $\Lambda$ and T\_[ł]{}T\_[ł’]{}=T\_[ł+ł’]{} ,\_=0 ,( T\_[ł]{})\^= T\_[c(ł)]{} ,(T\_[ł]{})=T\_[(ł)]{} , \[basis\]for $c,\pi:\Lambda\to \Lambda$. The second relation, which is equivalent to the bosonic projection $\pi \bar{\pi}\left( T_{\vec\l}\right) =T_{\vec\l}$, makes it possible to decompose under $$\Pi _{\sigma }:=\frac{1}{2}\left( 1+\sigma \kappa _{y}\star \bar{\kappa}_{
\bar{y}}\right) =\frac{1}{2}\left( 1+\sigma \kappa _{y}\bar{\kappa}_{\bar{y}
}\right) \text{ ,} \label{Pidef}$$ by expanding $$\begin{aligned}
\Phi ^{\prime } &=&\sum_{\sigma; \vec\l}T_{\vec\l}
\star \Pi _{\sigma }\star ( \nu _{\sigma; \vec\l} \kappa _{y}+ \check
\nu _{\sigma; \vec\l})\text{ ,} \qquad \ \nu _{\sigma;\vec\l}\ , \ \check\nu _{\sigma;\vec\l}\in \mathbb{C}\ , \label{Phiprime}\\
V_{\alpha }^{\prime } &=&\sum_{\sigma;\vec\l} T_{\vec\l}\star \Pi
_{\sigma }\star \left( a_{\sigma;\vec\l;\a}+ \check a_{\sigma;\vec\l;\a}\star \kappa _{y}\right)\text{ ,}
\label{Vprime}\end{aligned}$$ where $ a_{\sigma;\vec\l;\a}$ and $\check a_{\sigma;\vec\l;\a}$ are holomorphic functions on ${\cal Z}_4$ and are constant over ${\cal Y}_4$, which may be viewed as a gauge choice (for given zero-form initial data). Expanding[^19] ’’\_y=\_[; ł]{}T\_[ł]{} \_( \_[; ł]{} + \_[; ł]{} \_[y]{}) \_[;ł]{} , \_[;ł]{} ,\[defmu\]and introducing $$\mathring{\mu}_{\sigma }( \vec\zeta) :=\sum_{\vec\l}\mu
_{\sigma;\vec\l}(\vec\zeta)^{\vec\l}\ ,\qquad
\mathring{\check\mu}_{\sigma }( \vec\zeta) :=\sum_{\vec\l}\check\mu
_{\sigma;\vec\l}(\vec\zeta)^{\vec\l}\ ,\ee
\be
\mathring{a}_{\sigma }(\vec\zeta):= \sum_{\vec\l} dz^\a {a}_{\sigma;\vec\l;\alpha }
(\vec\zeta)^{\vec\l}\ ,
\qquad
\mathring{\check a}_{\sigma }(\vec\zeta):= \sum_{\vec\l} dz^\a {\check a}_{\sigma;\vec\l;\alpha }
(\vec\zeta)^{\vec\l}\ ,$$ where $\vec\zeta:=(\zeta_1,\dots,\zeta_N)\in \Comp^N$ and $(\vec\zeta)^{\vec\l}:=(\zeta_1)^{\l_1}\cdots(\zeta_N)^{\l_N}$, the remaining equations on ${\cal Z}_4$ take the form d\_+\_\_+\_\_+ j\_z \_=0 , \[3.23\]d\_+\_\_+\_\_+ j\_z \_=0 ,where the element $\gamma$ obeys ()\^[ł]{}=()\^[(ł)]{} ,\_=0 .Defining \_\^=\_\_ , \_\^= \_\_ ,\[defamu\]we obtain two decoupled systems of the form d\_\^+\_\^\_\^+ j\_z \_\^=0 ,\[3.27\]that can be solved using the method of [@Iazeolla:2011cb] (see also [@Iazeolla:2012nf]), drawn from the original method devised in [@Prokushkin:1998bq]. Omitting discrete moduli which arise via projector algebras on ${\cal Z}_4$, two particular solutions that we label by $\varsigma =\pm 1$, are given by $$\left( \mathring{a}_{\sigma;\varsigma }^\pm \right) _{\alpha }=2iz_{\alpha
}\int_{-1}^{1}\frac{d\tau }{\left( \tau
+1\right) ^{2}} j_{\sigma }\left( \varsigma \mathring{\mu}
_{\sigma}^\pm \ ;\ \tau \right)\text{exp}\left[ \varsigma c\left( \tau \right) U^{\beta \gamma
}z_{\beta }z_{\gamma }\right] \text{ ,} \label{a-circle soln}$$ where j\_(\_\^;) :=- \_[1]{}F\_[1]{} c( ) := i and $$U^{\beta \gamma }:=\left( u^{+}\right) ^{(\beta }\left( u^{-}\right)
^{\gamma )}\text{,}$$ where $u^{+}$ and $u^{-}$ are a set of spinor basis vectors obeying $$\left( u^{+}\right) ^{\alpha }\left( u^{-}\right) _{\alpha }=1\text{ , \ }\left( u^{+}\right) ^{\alpha }\left( u^{+}\right) _{\alpha }=\left(
u^{-}\right) ^{\alpha }\left( u^{-}\right) _{\alpha }=0\text{ .}$$ Using (\[eps explicit\]), we can choose $$\left( u^{+}\right) ^{\alpha }=\left[
\begin{array}{c}
0 \\
1\end{array}\right] \text{ , \ }\left( u^{-}\right) ^{\alpha }=\left[
\begin{array}{c}
1 \\
0\end{array}\right] \text{ .}$$ The original twistor space connection can thus be obtained by expanding the confluent hypergeometric function in a power series, followed by identifying powers of $\vec\zeta$ and $\gamma$, though in what follows we shall mainly work directly with the generating functions.
Twistor space connection in Weyl order in holomorphic gauge {#Singular Integrand}
-----------------------------------------------------------
The twistor space connection $V'_\a$ is given in the holomorphic gauge by . From Eq. , it follows that $$\begin{aligned}
&&{\cal O}_{\rm Normal}\left(\sum_{\sigma;\vec\l} T_{\vec\l}\star \Pi
_{\sigma }\star \left( a_{\sigma;\vec\l;\a}+ \check a_{\sigma;\vec\l;\a}\star \kappa _{y}\right)\right)
\notag \\
&=& {\cal O}_{\rm Weyl}\left(\sum_{\sigma;\vec\l} \left( (T_{\vec\l}\star \Pi
_{\sigma }) a_{\sigma;\vec\l;\a} + (T_{\vec\l}\star \Pi
_{\sigma }\star \kappa _{y}) \check a_{\sigma;\vec\l;\a}\right)\right)\ ,\end{aligned}$$ that is, the symbol in Weyl order of $V'_\a$ is given by the argument of the Wigner map on the right-hand side. This quantity contains singular distributions on ${\cal Y}_4$, which we shall examine in more detail later, and on ${\cal Z}_4$, which we shall examine in what follows. To this end, we observe that the integrand in (\[a-circle soln\]) has potential divergences at $\tau=0$, where log($\tau^2$) goes to infinity, and at $\tau=-1$, where denominators vanish.
As for the potential divergence at $\tau=0$, it does not lead to any non-real-analyticity in ${\cal Z}_4$ to any finite order in perturbation theory as follows from the fact that[^20] $$\left\vert {}_{1}F_{1}
\left[ \frac{1}{2};2;\frac{\varsigma \mathring{\mu}^\pm_\sigma}{2}\log
\left( \tau ^{2}\right) \right] \right\vert \leq \left\vert {}\tau
^{\varsigma \mathring {\mu}^\pm_\sigma}\right\vert\ ,\qquad
{\rm Re} (\varsigma \mathring{\mu}^\pm_\sigma) <0\ ,$$ for $\tau\in [-1,1]$, while the same quantity is bounded for $\tau\in [-1,1]$ if ${\rm Re} (\varsigma \mathring{\mu}^\pm_\sigma) \geqslant 0$. Thus, at $\tau=0$ there is no singularity as long as $\mathring{\mu}^\pm_\sigma$ lies inside the unit disc; indeed, for $\mathring{\mu}^\pm_\sigma$ sufficiently close to zero, the power series expansion of the confluent hypergeometric function yields a basis of functions of $\tau$ that can be used to convert the integral equation, obtained by inserting Eq. (\[a-circle soln\]) into the deformed oscillator equation, into an algebraic equation for symbols (for details, see [@Prokushkin:1998bq; @Iazeolla:2011cb]). Thus, in order for (\[a-circle soln\]) to provide a solution, there has to exist an annulus of convergence in the $\vec \zeta$-space for the Laurent series defining $\mathring{\mu}^\pm_\sigma$ where its modulus is less than one, which can be achieved by tuning the overall strength of the $\nu$- and $b_n$-parameters. In other words, the contribution to (\[a-circle soln\]) from the region around $\tau=0$ is real-analytic on ${\cal Z}_4$ to any finite order in perturbation theory.
Turning to the divergence at $\tau=-1$, it induces a simple pole $\mathring a^\pm_\a|_{\rm pole}$ in $\mathring a^\pm_\a$ at $z^\a=0$, which can be extracted using the formula \_[-1]{}\^1 e\^[p ]{}= , p>0 ,and analytical continuation of $U^{\beta \gamma }z_{\beta
}z_{\gamma }$. It follows that $$\begin{aligned}
(\mathring a^\pm_{\sigma;\varsigma})_\a|_{\rm pole}&=&\left.2iz_{\alpha }\int_{-1}^{1}
\frac{d\tau}{\left( \tau +1\right) ^{2}} j_{\sigma }\left( \varsigma \mathring{\mu}^\pm
_{\sigma };\tau \right)\text{exp}\left[
\frac{\varsigma i(\tau-1)}{\tau +1}U^{\beta \gamma }z_{\beta }z_{\gamma }\right]
\right|_{\rm pole}
\notag \\
&=&\left.-iz_{\alpha }\frac{\varsigma \mathring{\mu}^\pm_{\sigma }}{2}\int_{-1}^{1}
\frac{d\tau}{\left( \tau +1\right) ^{2}}\text{ exp}\left[
\frac{\varsigma i(\tau-1)}{\tau +1}U^{\beta \gamma }z_{\beta }z_{\gamma }\right]
\right|_{\rm pole}
\notag \\
&=&- \frac{ \mathring{\mu}^\pm_{\sigma }z_{\alpha }}{4U^{\beta \gamma }z_{\beta }z_{\gamma }}
\text{ \ \ for \ \ Re}\left( 2\varsigma iU^{\beta \gamma }z_{\beta
}z_{\gamma }\right) >0\text{ .}\end{aligned}$$ Indeed, taking the exterior derivative of the right-hand side one obtains a delta function on the holomorphic slice of ${\cal Z}_4$ that cancels the linear source term in Eq. . As for the higher order corrections to $\mathring a_\a$ in the $\nu$-expansion, they are finite but not analytic at $z^\a=0$, given by combinations of positive powers and logarithms of $z^\a$.
As we shall see in Section \[Sec analyticity\], the nature of the twistor space connection as a distribution on ${\cal Y}_4\times {\cal Z}_4$, changes drastically once the vacuum gauge function is switched on and the connection is given in normal order.
Singularities in $L$-gauge from $T_{\vec 0}$ {#Sec singular id}
--------------------------------------------
We note that if the unity $T_{\vec 0}$ of the star product algebra ${\cal A}_{\Lambda}$ in is represented by the constant symbol on ${\cal Y}_4$, then its contributions to both $V^{(L)}_\alpha$ and $\Phi^{(L)}$ that are not real-analytic at the origin of ${\cal Y}_4\times {\cal Z}_4$ for generic points in ${\cal X}_4$. More precisely, the singular contributions to $V^{(L)}_\alpha$ are given by $\Pi_\sigma \star a_{\sigma;\vec 0;\alpha}$, where $a_{\sigma;\vec 0;\alpha}$ is given by $\vec\zeta$-independent contribution to ; and those to $\Phi^{(L)}$ are given by $\nu_{\sigma;\vec 0} \Pi_\sigma\star \kappa_y$. They are hence singular at $Z^{\underline\alpha}=0$ and $Y^{\underline\alpha}=0$, respectively. Thus, in order for a star product algebra to give rise to proper higher spin gravity configurations, it cannot contain the constant symbol on ${\cal Y}_4$; in the case of a group algebra this can be achieved by a truncation to a proper semigroup algebra (without the unity), as we shall analyse in more detail in Sections \[Sec Weyl tensor\] and \[Sec analyticity\].
In the remainder of this section, however, we shall proceed with the construction of solution spaces in the holomorphic gauge without truncating the underlying group algebras.
Abelian group algebra from Cartan subalgebra of ${\rm sp}(4;\Real)$
-------------------------------------------------------------------
In what follows, we shall give an explicit example of a solution space of the type introduced above in the case when the lattice is two-dimensional, *i.e.* $\vec{\l}=(m,\tilde m)$ with $m,\tilde m\in \mathbb Z$. The underlying group algebra $\Comp[\mathbb{Z}\times \mathbb Z]$ is realized as \_[E,J]{}:= \_[=]{} [A]{}\_[E,J;]{} ,\_[E,J;]{}:= \_[m,m]{} ( T\_[m,]{}\_) ,in terms of group elements $$T_{m,\tilde{m}}:=e_{\star }^{-4m\theta E}\star e_{\star }^{-4\tilde{m}\tilde{
\theta} J}\ , \label{Tdef}$$ generated by the anti-de Sitter energy and spin operators[^21] $$\begin{aligned}
E &=&\frac{1}{8}E_{\underline{\alpha \beta }}Y^{\underline{\alpha }}\star Y^{\underline{\beta }}=\frac{1}{8}E_{\underline{\alpha \beta }}Y^{\underline{\alpha }}Y^{\underline{\beta }}\text{ \ ,} \\
J &=&\frac{1}{8}J_{\underline{\alpha \beta }}Y^{\underline{\alpha }}\star Y^{\underline{\beta }}=\frac{1}{8}J_{\underline{\alpha \beta }}Y^{\underline{\alpha }}Y^{\underline{\beta }}\text{ \ ,}\end{aligned}$$ respectively, using ${\rm sp}(4;\Real)$ valued matrices obeying[^22] (E\^2)\_\^=(J\^2)\_\^=-\_\^ \[EJProp1\] (EJ)\_\^=(JE)\_\^ (EJ)\_+(JE)\_=0 (EJ)\_\^=0 , \[EJProp2\] from which it follows that ( \_\^+aE\_\^J\_\^) =( 1-a\^[2]{}) \^[2]{} \[EJProp4\]As for the parameters, we take $$\theta\in \mathbb{R}\cup i\mathbb{R}\ ,\qquad
i\mathbb{Z} \theta \cap \left( \frac{\pi }{2}+\mathbb{Z\pi }\right)
=\emptyset \text{ ,}$$ *idem* $\tilde{\theta}$. The basis elements obey , viz. $$T_{m,\tilde{m}}\star T_{n,\tilde{n}}=T_{m+n,\tilde{m}+\tilde{n}}\text{ ,}\qquad
\left[ T_{m,\tilde{m}},\Pi _{\sigma }\right] _{\star }=0\text{ ,}
\qquad \pi(T_{m,\tilde{m}})=T_{-m,\tilde{m}}\ .$$ To compute the symbol of $T_{m,\tilde m}$ in Weyl order, we first use (\[star and ordinary exponent\]) with $N=4$ to compute $$e_{\star }^{-4m\theta E}=\mathbf{S}^{2}\,
e^{-4\mathbf{T} E} \text{ \ , \ \ }e_{\star }^{-4\tilde{m}\tilde{
\theta}{J}}=
\widetilde{\mathbf{S}}^{2} \,e^{-4\widetilde{\mathbf{T}} J}\text{ ,}
\label{star and ordinary exponent E J}$$ where $$\mathbf{S}:= \text{sech}\left( m\theta \right) \text{ , \quad }\mathbf{T}
:= \text{tanh}\left( m\theta \right) \text{ , \quad }\widetilde{\mathbf{S}}
:= \text{sech}\left( \tilde{m}\tilde{\theta}\right) \text{ , \quad }
\widetilde{\mathbf{T}}:= \text{tanh}\left( \tilde{m}\tilde{\theta}
\right) \text{ . \ }$$ In what follows, we make the convention that all boldfaced quantities depend on $m\theta$ and $\tilde{m}\tilde{\theta}$. The symbol of $T_{m,\tilde{m}}$ is thus given by $$\begin{aligned}
T_{m,\tilde{m}} &=&\left[ \mathbf{S}^{2}e^{-4\mathbf{T}E}\right] \star \left[
\widetilde{\mathbf{S}}^{2}e^{-4\widetilde{\mathbf{T}}J}\right]
\notag \\
&=&\left( \mathbf{S}\widetilde{\mathbf{S}}\right) ^{2}\int \frac{d^{4}Ud^{4}V}{\left( 2\pi \right) ^{4}}\text{ exp}\left\{ i\left( V^{\underline{\alpha }}-Y^{\underline{\alpha }}\right) \left( U_{\underline{\alpha }}-Y_{\underline{\alpha }}\right) \right\} \notag \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{exp}\left\{ -\frac{1}{2}\left[ \mathbf{T}E_{\underline{\alpha \beta }}U^{\underline{\alpha }}U^{\underline{\beta }}+\widetilde{\mathbf{T}}J_{\underline{\alpha \beta }}V^{\underline{\alpha }}V^{\underline{\beta }}\right] \right\} \text{ .}\end{aligned}$$ By performing the Gaussian integrals, we obtain $$T_{m,\tilde{m}}=\mathbf{A}\text{exp}\left\{ -\frac{1}{2} \mathbf{K}_{\underline{\alpha \beta }} Y^{\underline{\alpha }}Y^{\underline{\beta }}\right\} \text{ ,}\qquad
\mathbf{K}_{\underline{\alpha \beta }}:=\mathbf{B}E_{\underline{
\alpha \beta }}+\mathbf{C}J_{\underline{\alpha \beta }}\text{ ,}
\label{Tmm EJ ABC}$$ where $$\mathbf{A}:= \frac{\left( \mathbf{S}\widetilde{\mathbf{S}}\right) ^{2}}{1-\left( \mathbf{T}\widetilde{\mathbf{T}}\right) ^{2}}\text{ , \quad }\mathbf{B}:=\frac{\mathbf{T}\left( 1-\widetilde{\mathbf{T}}^{2}\right) }{1-\left(
\mathbf{T}\widetilde{\mathbf{T}}\right) ^{2}}\text{ , \quad }\mathbf{C}:=
\frac{\widetilde{\mathbf{T}}\left( 1-\mathbf{T}^{2}\right) }{1-\left(
\mathbf{T}\widetilde{\mathbf{T}}\right) ^{2}}\text{ .} \label{ABC notation}$$ We also need the symbol of $T_{m,\tilde{m}}\star \kappa_{y}\bar{\kappa}_{\bar{y}}$, which is given by $$\begin{aligned}
T_{m,\tilde{m}}\star \kappa _{y}\bar{\kappa}_{\bar{y}} \notag
&=&\left( 2\pi \right) ^{2}\mathbf{A}\text{exp}\left\{ -\frac{1}{2}\mathbf{K}_{\underline{\alpha \beta }}Y^{\underline{\alpha }}Y^{\underline{\beta }}\right\} \star \delta ^{4}\left( Y\right) \notag \\
&=&\mathbf{A}\int \frac{d^{4}Ud^{4}V}{\left( 2\pi \right) ^{2}}e^{iV^{\underline{\alpha }}U_{\underline{\alpha }}}e^{-\frac{1}{2}\mathbf{K}_{\underline{\alpha \beta }}\left( Y^{\underline{\alpha }}+U^{\underline{\alpha }}\right) \left( Y^{\underline{\beta }}+U^{\underline{\beta }}\right)
}\delta ^{4}\left( Y+V\right) \notag \\
&=&\mathbf{A}\int \frac{d^{4}Ud^{4}V}{\left( 2\pi \right) ^{2}}e^{-iY^{\underline{\alpha }}U_{\underline{\alpha }}}e^{-\frac{1}{2}\mathbf{K}_{\underline{\alpha \beta }}U^{\underline{\alpha }}U^{\underline{\beta }}}
\notag \\
&=&\frac{\mathbf{A}}{\sqrt{\text{det}\left( \mathbf{K}\right) }}\text{exp}\left\{ -\frac{1}{2}\left( \mathbf{K}^{-1}\right) ^{\underline{\alpha \beta }}Y_{\underline{\alpha }}Y_{\underline{\beta }}\right\} \text{ ,}
\label{Tkappa}\end{aligned}$$ where $$\mathbf{K}_{\underline{\alpha \beta }}\left( \mathbf{K}^{-1}\right) ^{\underline{\beta \gamma }}:=\delta _{\underline\alpha }{}^{\underline\gamma }\text{ .}$$
New exact biaxially symmetric solutions in holomorphic gauge {#Sec biaxial sym}
------------------------------------------------------------
The above construction of ${\cal A}_{E,J}$ thus allows us to solve equations (\[X-indep-VJ\])–(\[X-indep-V\]) using the Ansatz –. In order to keep matters simple, we shall assume that $\check \nu = \check \a = 0$, and work with the following reduced version: $$\begin{aligned}
\Phi ^{\prime } &=&\sum_{\sigma ;m,\tilde{m}}\nu _{\sigma ;m,\tilde{m}}T_{m,\tilde{m}}\star \Pi _{\sigma }\star \kappa _{y}\text{ ,} \label{ansatzF_YZ}
\\
V_{\alpha }^{\prime } &=&\sum_{\sigma ;m,\tilde{m}}T_{m,\tilde{m}}\star \Pi
_{\sigma }\star \left( a_{\sigma ;m,\tilde{m}}(z)\right) _{\alpha }\text{ ,}
\label{ansatzV_YZ}\\
\bar{V}_{\dot{\alpha}}^{\prime }&=&(V_{\alpha }^{\prime })^\dag=\sum_{\sigma ;m,\tilde{m}}T_{m,\tilde{m}}^{\dag }\star \Pi _{\sigma }\star \left( \bar{a}_{\sigma ;m,\tilde{m}
}(\bar z)\right) _{\dot{\alpha}}\text{ , }
\label{ansatzVbar_YZ}\end{aligned}$$ where thus $\nu _{\sigma;m,\tilde{m}}\in \mathbb{C}$ and we recall that the twistor space connection is (anti-)holomorphic, as indicated above. From $$\begin{tabular}{c|cc}
& $\theta \in \mathbb{R}$ & $\theta \in i\mathbb{R}$ \\ \hline
$\tilde{\theta}\in \mathbb{R}$ & $T_{m,\tilde{m}}^{\dag }=T_{m,\tilde{m}}$ &
$T_{m,\tilde{m}}^{\dag }=T_{-m,\tilde{m}}$ \\
$\tilde{\theta}\in i\mathbb{R}$ & $T_{m,\tilde{m}}^{\dag }=T_{m,-\tilde{m}}$
& $T_{m,\tilde{m}}^{\dag }=T_{-m,-\tilde{m}}$
\end{tabular}$$ it follows that the reality condition $(\Phi')^\dag=\pi(\Phi')$ implies that $$\begin{tabular}{c|cc}
& $\theta \in \mathbb{R}$ & $\theta \in i\mathbb{R}$ \\ \hline
$\tilde{\theta}\in \mathbb{R}$ & $\nu _{\sigma ;m,\tilde{m}}^{\ast }=\sigma
\nu _{\sigma ;m,\tilde{m}}$ & $\nu _{\sigma ;m,\tilde{m}}^{\ast }=\sigma \nu
_{\sigma ;-m,\tilde{m}}$ \\
$\tilde{\theta}\in i\mathbb{R}$ & $\nu _{\sigma ;m,\tilde{m}}^{\ast }=\sigma
\nu _{\sigma ;m,-\tilde{m}}$ & $\nu _{\sigma ;m,\tilde{m}}^{\ast }=\sigma
\nu _{\sigma ;-m,-\tilde{m}}$
\end{tabular}\text{\ \ .}
\label{realcond-nu-sigma}$$ We note that the Ansatz – identically obeys (\[X-indep-PhiV\]) and (\[X-indep-V\]) since \_=0 ,while (\[X-indep-VJ\]) reduces to $$\partial _{\lbrack \alpha }\left( a_{\sigma ;m,\tilde{m}}\right) _{\beta
]}+\sum_{n,\tilde{n}}\left( a_{\sigma ;n,\tilde{n}}\right) _{[\alpha }\star
\left( a_{\sigma ;m-n,\tilde{m}-\tilde{n}}\right) _{\beta ]}+\frac{i}{4}\varepsilon _{\alpha \beta }\mu _{\sigma ;m,\tilde{m}}\kappa _{z}=0\text{ \
and h.c.\ ,} \label{z-dependent eq}$$ where $\mu _{\sigma ;m,\tilde{m}}$ are defined as in . Finally, multiplying (\[z-dependent eq\]) with $\zeta ^{m}\tilde{\zeta} ^{\tilde{m}}$, where $\zeta,\tilde{\zeta}\in \mathbb{C}$, and summing over $m$ and $\tilde{m}$, yields the equivalent equation $$\partial _{\lbrack \alpha }\left( \mathring{a}_{\sigma }\right) _{\beta
]}+\left( \mathring{a}_{\sigma }\right) _{[\alpha }\star \left( \mathring{a}
_{\sigma }\right) _{\beta ]}+\frac{i}{4}\varepsilon _{\alpha \beta }
\mathring{\mu}_{\sigma }\kappa _{z}=0\text{ \ and h.c.\ ,}
\label{z-dependent eq simp}$$ where the generating functions $$\left( \mathring{a}_{\sigma }\right) _{\alpha }\left( \zeta ,\tilde{\zeta}
\right) := \sum_{m,\tilde{m}}\left( {a}_{\sigma ,m,\tilde{m}
}\right) _{\alpha }\zeta ^{m}\tilde\zeta ^{\tilde{m}}\text{\ \ , \ \ }\mathring{\mu
}_{\sigma }\left( \zeta ,\tilde{\zeta}\right) := \sum_{m,\tilde{m}}\mu
_{\sigma ;m,\tilde{m}}\zeta ^{m}\tilde\zeta ^{\tilde{m}}\text{ ,}$$ for which we shall use the particular solutions in with $\check \nu = \check a = 0$.
By definition, the symmetries of the solution are generated by generalized Killing gauge parameters $\epsilon^{(G)}$ leaving $(\Phi^{(G)},U_\mu^{(G)},V_\alpha^{(G)})$ invariant. Locally, the space of such parameters is given by \^[(G)]{}= G\^[-1]{}’ G ,’=’(E,J) ,where the parameters are arbitrary star polynomials in $E$ and $J$; and globally, a Killing parameter belongs to an adjoint section obeying suitable boundary conditions, and we shall assume that $\epsilon^{(G)}$ is real-analytic on ${\cal Y}_4\times {\cal Z}_4$ and falls off at infinity of ${\cal X}_4$, such that they leave the background spacetime invariant. This implies that the solutions have time-translational and rotational symmetries generated by $E$ and $J$, respectively. Furthermore, if the Ansatz is expanded over only $T_{m,0}$ or $T_{0,\tilde m}$, respectively, then the symmetry is further enhanced to the enveloping algebras of ${\rm so}(2)_E \oplus {\rm so}(3)$ or ${\rm so}(1,2)\oplus {\rm so}(2)_J$, where ${\rm so}(3)$ is the subalgebras of ${\rm sp}(4;\Real)$ commuting to $E$ *idem* ${\rm so}(1,2)$ and $J$. Acting on the solutions with the full higher spin algebra leads to an orbit that forms a higher spin representation space. The trace operation ${\rm Tr}'$ equips this space with an indefinite sesqui-linear form, as we shall comment on below in the context of higher spin invariant functionals.
Weyl zero-form and Weyl tensors {#Sec Weyl tensor}
===============================
In this section we compute the Weyl zero-form, Weyl tensors and higher spin invariants formed out of them.
The Weyl zero-form in $L$-gauge
-------------------------------
The Weyl tensors in the $L$-gauge are contained in the zero-form master field. From (\[AnsatzPhi\]) and (\[ansatzF\_YZ\]) it follows that $$\begin{aligned}
\Phi^{(L)}&=& \frac{1}{2}\sum_{\sigma ,m,\tilde{m}}\nu _{\sigma ,m,\tilde{m}}L^{-1}\star T_{m,\tilde{m}}\star \left( \kappa _{y}+\sigma \bar{\kappa}_{\bar{y}}\right) \star \pi \left( L\right) \text{ ,}\notag\\
&=&\frac{1}{2}\sum_{\sigma ,m,\tilde{m}}\nu _{\sigma ,m,\tilde{m}}\left( L^{-1}\star T_{m,\tilde{m}}\star L\right) \star \left( \kappa
_{y}+\sigma \bar{\kappa}_{\bar{y}}\right) \notag \\
&=&\sum_{m,\tilde{m}}\left( \nu _{1,m,\tilde{m}}T_{m,\tilde{m}}^L\star \kappa _{y}+\nu _{2,m,\tilde{m}}T_{m,\tilde{m}}^L\star \bar{\kappa}_{\bar{y}}\right) \text{ ,} \label{Phi two terms}\end{aligned}$$ where T\_[m,]{}\^L:=L\^[-1]{}T\_[m,]{}L ,and the parameters $$\nu _{1,m,\tilde{m}} := \frac{1}{2}\left( \nu _{+,m,\tilde{m}}+\nu _{-,m,\tilde{m}}\right) \text{ \ , \ }\nu _{2,m,\tilde{m}} := \frac{1}{2}\left(
\nu _{+,m,\tilde{m}}-\nu _{-,m,\tilde{m}}\right) \text{ ,}\label{nurelation}$$ obey the reality conditions $$\begin{tabular}{c|cc}
& $\theta \in \mathbb{R}$ & $\theta \in i\mathbb{R}$ \\ \hline
$\tilde{\theta}\in \mathbb{R}$ & $\nu _{1,m,\tilde{m}}^{\ast }=\nu _{2,m,\tilde{m}}$ & $\nu _{1,m,\tilde{m}}^{\ast }=\nu _{2,-m,\tilde{m}}$ \\
$\tilde{\theta}\in i\mathbb{R}$ & $\nu _{1,m,\tilde{m}}^{\ast }=\nu _{2,m,-\tilde{m}}$ & $\nu _{1,m,\tilde{m}}^{\ast }=\nu _{2,-m,-\tilde{m}}$\end{tabular}$$ To compute $ T_{m,\tilde{m}}^L$ we use the lemma $$L^{-1}\star f\left( Y_{\underline{\alpha }}\right) \star L=f\left( L_{\underline{\alpha }}{}^{\underline{\beta }}Y_{\underline{\beta }}\right)
\text{ ,}$$ where $L_{\underline{\alpha }}{}^{\underline{\beta }}$ is a matrix that depends on the spacetime coordinates (see Appendix \[Sec coordinates\] for an explicit expression). It follows from (\[Tmm EJ ABC\]) that $$T_{m,\tilde{m}}^L=\mathbf{A}\text{exp}\left\{ \left( -\frac{1}{2}\right) \mathbf{K}_{\underline{\alpha \beta }}^{L} Y^{\underline{\alpha }}Y^{\underline{\beta }}\right\} \text{ ,} \label{LTmL EJ ABC}$$ where $$\mathbf{K}_{\underline{\alpha \beta }}^{L}:=
\mathbf{B}E_{\underline{\alpha \beta }}^{L}+\mathbf{C}J_{
\underline{\alpha \beta }}^{L}\ ,\qquad
E^{L}_{\underline{\alpha \beta }}:= E_{\underline{\gamma
\delta }}L^{\underline{\gamma }}{}_{\underline{\alpha }}L^{\underline{\delta
}}{}_{\underline{\beta }}\text{ , \qquad}J^{L}_{\underline{\alpha
\beta }}:=J_{\underline{\gamma \delta }}L^{\underline{\gamma }}{}_{
\underline{\alpha }}L^{\underline{\delta }}{}_{\underline{\beta }}\text{ .}$$ Under $Y^{\underline{\alpha }}=\left\{ y^{\alpha },
\bar{y}^{\dot{\alpha}}\right\} $, the above matrices decompose into $$E_{\underline{\alpha \beta }}^{L}=:\left(
\begin{array}{cc}
(\kappa_E^{L})_{\alpha \beta } & (v_E^{L})_{\alpha \dot{\beta}} \\
(\bar{v}_E^{L})_{\dot{\alpha}\beta } & (\bar{\kappa}_E^{L})_{\dot{\alpha}\dot{\beta}}
\end{array}
\right) \text{ ,}$$ *idem* $J_{\underline{\alpha \beta }}^{L}$, whose components obey[^23] $$v_{\alpha \dot{\beta}}^{L}=\bar{v}_{\dot{\beta}\alpha }^{L}\text{ , \qquad }
\left( v^{L}\right) _{\alpha \dot{\beta}}\left(\bar v^{L}\right) ^{\dot{\beta}
\gamma }=\left( v^{L}\right) ^{2}\delta _{\alpha }{}^{\gamma }\text{ , \qquad }
\left( \bar{v}^{L}\right) _{\dot{\alpha}\beta }\left( v^{L}\right)
^{\beta \dot{\gamma}}=\left( v^{L}\right) ^{2}\delta _{\dot \alpha }{}^{\dot \gamma }
\text{ ,}
\label{v-prop}$$ where $\left( v^{L}\right)^{2}:= \frac{1}{2}\left( v^{L}\right) _{\alpha \dot{\beta}}\left( v^{L}\right) ^{\alpha \dot{\beta}}$, and $$\left(\kappa^L\right)^2:= \frac{1}{2}\left(\kappa^L\right)_{\alpha \beta }\left(\kappa^L\right)^{\alpha
\beta }=\text{det}\left(\kappa^L\right)
\ \ \ \ , \ \ \ \ \ \ \
\left(\kappa^L\right)_{\alpha\beta }\left(\kappa^L\right)^{\beta \gamma }=\left(\kappa^L\right)^{2}\delta _{\alpha }{}^{\gamma } \ \ ,$$ *idem* $\bar\kappa^L$, which are derived from general properties of any 2$\times$2 symmetric matrix. Furthermore, from it follows that $$\left( \kappa ^{L}\right) _{\alpha \beta }\left( v^{L}\right) ^{\beta \dot{\gamma}}+\left( v^{L}\right) _{\alpha \dot{\beta}}\left( \bar{\kappa}^{L}\right) ^{\dot{\beta}\dot{\gamma}}=0\text{ , \qquad }\left( \kappa
^{L}\right) ^{2}+\left( v^{L}\right) ^{2}=\left( \bar{\kappa}^{L}\right)
^{2}+\left( v^{L}\right) ^{2}=1\text{ ,}$$ which in its turn implies $$\left( \bar{\kappa}^{L}\right) ^{2}\left( \kappa ^{L}\right) _{\alpha \beta
}-\left( \bar{\kappa}^{L}\right) ^{\dot{\alpha}\dot{\beta}}\left(
v^{L}\right) _{\alpha \dot{\alpha}}\left( v^{L}\right) _{\beta \dot{\beta}}=\left( \kappa ^{L}\right) _{\alpha \beta }\text{ ,} \label{kk-kvv}$$ which will be useful later when we determine the Petrov type.
Returning to (\[LTmL EJ ABC\]), we thus have $$\mathbf{K}_{\underline{\alpha \beta }}^{L} = \left(
\begin{array}{cc}
\mathbf{F}_{\alpha \beta } & \mathbf{G}_{\alpha \dot{\beta}} \\
\mathbf{G}_{\dot{\alpha}\beta } & \mathbf{H}_{\dot{\alpha}\dot{\beta}}\end{array}\right) :=\left(
\begin{array}{cc}
\mathbf{B}\left( \kappa _{E}^{L}\right) _{\alpha \beta }+\mathbf{C}\left(
\kappa _{J}^{L}\right) _{\alpha \beta } & \mathbf{B}\left( v_{E}^{L}\right)
_{\alpha \dot{\beta}}+\mathbf{C}\left( v_{J}^{L}\right) _{\alpha \dot{\beta}}
\\
\mathbf{B}\left( \bar{v}_{E}^{L}\right) _{\dot{\alpha}\beta }+\mathbf{C}
\left( \bar{v}_{J}^{L}\right) _{\dot{\alpha}\beta } & \mathbf{B}\left( \bar{
\kappa}_{E}^{L}\right) _{\dot{\alpha}\dot{\beta}}+\mathbf{C}\left( \bar{
\kappa}_{J}^{L}\right) _{\dot{\alpha}\dot{\beta}}
\end{array}
\right) \text{ ,}$$ where $\mathbf{G}_{\dot{\alpha}\beta }=\mathbf{G}_{\beta\dot{\alpha}}$, and correspondingly $$T_{m,\tilde{m}}^L=\mathbf{A}\text{exp}\left\{ \left( -\frac{1}{2}\right) \left[ y^{\alpha }\mathbf{F}_{\alpha \beta }y^{\beta }+\bar{y}^{\dot{\alpha}}\mathbf{H}_{\dot{\alpha}\dot{\beta}}\bar{y}^{\dot{\beta}}+2y^{\alpha }\mathbf{G}_{\alpha \dot{\beta}}\bar{y}^{\dot{\beta}}\right]
\right\} \text{ .}$$ Finally, for $(m,\tilde m)\neq (0,0)$, by performing Gaussian integrals we obtain[^24] $$\begin{aligned}
&&T_{m,\tilde{m}}^L\star \kappa _{y} \notag \\
&=&\!\!\!\!\frac{\mathbf{A}}{\sqrt{\mathbf{F}^{2}}}\text{exp}\left\{ \frac{1}{2\mathbf{F}^{2}}\left[ \left( \mathbf{F}^{\alpha \beta }\mathbf{G}_{\alpha
\dot{\alpha}}\mathbf{G}_{\beta \dot{\beta}}-\mathbf{F}^{2}\mathbf{H}_{\dot{\alpha}\dot{\beta}}\right) \bar{y}^{\dot{\alpha}}\bar{y}^{\dot{\beta}}-\mathbf{F}_{\alpha \beta }y^{\alpha }y^{\beta }+2i\mathbf{F}_{\alpha
}{}^{\beta }\mathbf{G}_{\beta \dot{\beta}}y^{\alpha }\bar{y}^{\dot{\beta}}\right] \right\} \text{ ,} \notag \\
&&\end{aligned}$$ and $$\begin{aligned}
&&T_{m,\tilde{m}}^L\star \bar{\kappa}_{\bar{y}} \notag \\
&=&\!\!\!\!\frac{\mathbf{A}}{\sqrt{\mathbf{H}^{2}}}\text{exp}\left\{ \frac{1}{2\mathbf{H}^{2}}\left[ \left( \mathbf{H}^{\alpha \beta }\mathbf{G}_{\alpha
\dot{\alpha}}\mathbf{G}_{\beta \dot{\beta}}-\mathbf{H}^{2}\mathbf{F}_{\dot{\alpha}\dot{\beta}}\right) y^{\alpha }y^{\beta }-\mathbf{H}_{\dot{\alpha}\dot{\beta}}\bar{y}^{\dot{\alpha}}\bar{y}^{\dot{\beta}}+2i\mathbf{H}_{\dot{\alpha}}{}^{\dot{\beta}}\mathbf{G}_{\beta \dot{\beta}}\bar{y}^{\dot{\alpha}}y^{\beta }\right] \right\} \text{ ,} \notag \\
&&\end{aligned}$$ while $T_{0,\tilde{0}}^L\star \kappa_{y}= \kappa_{y}$ and $T_{0,\tilde{0}}^L\star \bar\kappa_{\bar y}= \bar \kappa_{\bar y}$.
Substituting the above formulae into (\[Phi two terms\]), we obtain $$\begin{aligned}
&\Phi^{(L)} \notag \\
=& \ \nu_{1,0,0}\kappa_{y}\ +\ \nu_{2,0,0} \bar \kappa_{\bar y}\ +
\notag\\
&\sum_{(m,\tilde{m})\neq (0,0)}\mathbf{A}\left( \frac{\nu _{1,m,\tilde{m}}}{\sqrt{\mathbf{F}^{2}}}\text{exp}\left\{ \frac{1}{2\mathbf{F}^{2}}\left[ \left(
\mathbf{F}^{\alpha \beta }\mathbf{G}_{\alpha \dot{\alpha}}\mathbf{G}_{\beta
\dot{\beta}}-\mathbf{F}^{2}\mathbf{H}_{\dot{\alpha}\dot{\beta}}\right) \bar{y}^{\dot{\alpha}}\bar{y}^{\dot{\beta}}-\mathbf{F}_{\alpha \beta }y^{\alpha
}y^{\beta }+2i\mathbf{F}_{\alpha }{}^{\beta }\mathbf{G}_{\beta \dot{\beta}}y^{\alpha }\bar{y}^{\dot{\beta}}\right] \right\} \right. \notag \\
& \ \ \ \left. +\frac{\nu _{2,m,\tilde{m}}}{\sqrt{\mathbf{H}^{2}}}\text{exp}\left\{ \frac{1}{2\mathbf{H}^{2}}\left[ \left( \mathbf{H}^{\dot{\alpha}\dot{\beta}}\mathbf{G}_{\alpha \dot{\alpha}}\mathbf{G}_{\beta \dot{\beta}}-\mathbf{H}^{2}\mathbf{F}_{\alpha \beta }\right) y^{\alpha }y^{\beta
}-\mathbf{H}_{\dot{\alpha}\dot{\beta}}\bar{y}^{\dot{\alpha}}\bar{y}^{\dot{\beta}}+2i\mathbf{H}_{\dot{\alpha}}{}^{\dot{\beta}}\mathbf{G}_{\beta \dot{\beta}}\bar{y}^{\dot{\alpha}}y^{\beta }\right] \right\} \right) \text{ .}
\notag \\
&\end{aligned}$$ The expression $\mathbf{H}^{2}\mathbf{F}_{\alpha\beta}-
\mathbf{H}^{\dot{\alpha}\dot{\beta}}\mathbf{G}_{\alpha\dot{\alpha}}
\mathbf{G}_{\beta\dot{\beta}}$ can be factorized as $$\mathbf{H}^{2}\mathbf{F}_{\alpha\beta}-
\mathbf{H}^{\dot{\alpha}\dot{\beta}}\mathbf{G}_{\alpha\dot{\alpha}}
\mathbf{G}_{\beta\dot{\beta}}\ =\ \left(\mathbf{B}^2-\mathbf{C}^2\right)\mathbf{\breve F}_{\a\b} \ ,$$ where $\mathbf{\breve F}_{\a\b}$ satisfies \^2 = \^2 . Then, assuming that \_[1,0,0]{}=0=\_[2,0,0]{} ,the resulting generalized spin-$s$ Weyl tensor in the $L$-gauge reads $$\begin{aligned}
&&C_{\alpha _{1}\cdots \alpha _{2s}} \notag \\
&:=&\left[ \frac{\partial }{\partial y^{\alpha _{1}}}\cdots \frac{\partial }{\partial y^{\alpha _{2s}}}\Phi^{(L)} \right] _{Y=0} \notag \\
&=&\frac{\left( 2s\right) !}{s!}\sum_{(m,\tilde{m})\neq (0,0)}\mathbf{A}\left\{ \frac{\nu _{1,m,\tilde{m}}}{\sqrt{\mathbf{F}^{2}}}\left( \frac{-1}{2\mathbf{F}^{2}}\right) ^{s}\mathbf{F}_{(\alpha _{1}\alpha _{2}}\cdots \mathbf{F}_{\alpha
_{2s-1}\alpha _{2s})}\right. \notag \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+\left(\mathbf{B}^2-\mathbf{C}^2\right)^{s} \frac{\nu _{2,m,\tilde{m}}}{\sqrt{\mathbf{\mathbf{\breve{F}}}^{2}}}\left( \frac{-1}{2\mathbf{\breve{F}}^{2}}\right) ^{s}
\mathbf{\breve F}_{(\alpha _{1}\alpha _{2}}\cdots \mathbf{\breve F}_{\alpha
_{2s-1}\alpha _{2s})}
\right\} \text{ ,}
\label{Weyl tensor}\end{aligned}$$ where there are two separate generalized Petrov type-D tensors summed for each $(m, \tilde{m})$ for positive $s$.[^25]
Petrov types of the Weyl tensors
--------------------------------
In what follows, we analyze in a few special cases whether the Weyl tensor as the sum is of Petrov type D.
#### The case $\protect\theta\tilde{\protect\theta}=0$.
If $\theta\neq 0$ and $\tilde{\theta}=0$, then $\widetilde{\mathbf{S}}=1$, $\widetilde{\mathbf{T}}=0$ and $$\mathbf{A}=\mathbf{S}^{2}\text{ , \ }\mathbf{B}=\mathbf{T}\text{ , \ }\mathbf{C}=0\text{ .}$$ It follows that $$\left(
\begin{array}{cc}
\mathbf{F}_{\alpha \beta } & \mathbf{G}_{\alpha \dot{\beta}} \\
\mathbf{G}_{\dot{\alpha}\beta } & \mathbf{H}_{\dot{\alpha}\dot{\beta}}\end{array}\right) =\mathbf{T}\left(
\begin{array}{cc}
\left( \kappa _{E}^{L}\right) _{\alpha \beta } & \left( v_{E}^{L}\right)
_{\alpha \dot{\beta}} \\
\left( \bar{v}_{E}^{L}\right) _{\dot{\alpha}\beta } & \left( \bar{\kappa}_{E}^{L}\right) _{\dot{\alpha}\dot{\beta}}\end{array}\right) \text{ .}$$ Furthermore, using (\[kk-kvv\]) we obtain $$\mathbf{\breve F}_{\alpha\beta} =
\mathbf{T} \left( \kappa
_{E}^{L}\right) _{\alpha\beta} \text{ .}$$ The resulting spin-$s$ Weyl tensor reads[^26] $$\begin{aligned}
&&C_{\alpha _{1}\cdots \alpha _{2s}}|_{\tilde{\theta}=0} \\
&=&\frac{\left( 2s\right) !}{s!}\sum_{m\neq0}\frac{\mathbf{S}^{2}}{\sqrt{\mathbf{T}^{2}\left( \kappa _{E}^{L}\right) ^{2}}}\frac{1}{\left[ -2\left( \kappa
_{E}^{L}\right) ^{2}\right] ^{s}}\left( \nu _{1,m}\mathbf{T}^{-s}+\nu _{2,m}\mathbf{T}^{s}\right) \left( \kappa _{E}^{L}\right) _{(\alpha _{1}\alpha
_{2}}\cdots \left( \kappa _{E}^{L}\right) _{\alpha _{2s-1}\alpha _{2s})}\text{ ,}\end{aligned}$$$$$$where $\nu _{1,m}=\sum_{\tilde{m}}\nu _{1,m,\tilde{m}}$ and $\nu
_{2,m}=\sum_{\tilde{m}}\nu _{2,m,\tilde{m}}$, which we assume to be finite and vanishing for $m=0$.
If instead $\theta =0$ and $\tilde{\theta}\neq 0$, then $\mathbf{S}=1$, $\mathbf{T}=0$, and $$\mathbf{A}=\widetilde{\mathbf{S}}^{2}\text{ , \ }\mathbf{B}=0\text{ , \ }\mathbf{C}=\widetilde{\mathbf{T}}\text{ .}$$ Then we have $$\left(
\begin{array}{cc}
\mathbf{F}_{\alpha \beta } & \mathbf{G}_{\alpha \dot{\beta}} \\
\mathbf{G}_{\dot{\alpha}\beta }^{T} & \mathbf{H}_{\dot{\alpha}\dot{\beta}}\end{array}\right) =\widetilde{\mathbf{T}}\left(
\begin{array}{cc}
\left( \kappa _{J}^{L}\right) _{\alpha \beta } & \left( v_{J}^{L}\right)
_{\alpha \dot{\beta}} \\
\left( \bar{v}_{J}^{L}\right) _{\dot{\alpha}\beta } & \left( \bar{\kappa}_{J}^{L}\right) _{\dot{\alpha}\dot{\beta}}\end{array}\right) \text{ ,}$$ and hence using (\[kk-kvv\]) it follows that $$\mathbf{\breve F}_{\alpha\beta} =
-\widetilde{\mathbf{T}} \left( \kappa
_{J}^{L}\right) _{\alpha\beta} \text{ ,}$$ and the spin-$s$ Weyl tensor becomes$$\begin{aligned}
&&C_{\alpha _{1}\cdots \alpha _{2s}}|_{\theta =0} \\
&=&\frac{\left( 2s\right) !}{s!}\sum_{\tilde{m}\neq 0}\frac{\widetilde{\mathbf{S}}^{2}}{\sqrt{\widetilde{\mathbf{T}}^{2}\left( \kappa _{J}^{L}\right) ^{2}}}\frac{1}{\left[ -2\left( \kappa _{J}^{L}\right) ^{2}\right] ^{s}}\left( \nu
_{1,\tilde{m}}\widetilde{\mathbf{T}}^{-s}+\nu _{2,\tilde{m}}\widetilde{\mathbf{T}}^{s}\right) \left( \kappa _{J}^{L}\right) _{(\alpha _{1}\alpha
_{2}}\cdots \left( \kappa _{J}^{L}\right) _{\alpha _{2s-1}\alpha _{2s})}\text{ ,}\end{aligned}$$$$$$where $\nu _{1,\tilde{m}}=\sum_{m}\nu _{1,m,\tilde{m}}$ and $\nu _{2,\tilde{m}}=\sum_{m}\nu _{2,m,\tilde{m}}$, which we assume to be finite and vanishing for $\tilde m=0$.
Thus, to summarize, if the Weyl zero-form depends on either $E$ or $J$, but not both, in the holomorphic gauge, then Weyl tensors in $L$-gauge become proportional to direct products of $\kappa^L$’s, which means they are of generalized Petrov type D.
#### The case $\protect\theta \tilde{\protect\theta}\neq 0$.
If both $\theta $ and $\tilde{\theta}$ are non-zero, *i.e.* if both $E$ and$\ J$ are present in the Weyl zero-form in the holomorphic gauge, then we can simplify the analysis by substituting the explicit expressions provided in Appendices [Sec gamma]{} and \[Sec coordinates\] into the spin-$s$ Weyl tensor (\[Weyl tensor\]) in $L$-gauge.
If $\mathbf{B}^{2}-\mathbf{C}^{2}=0$ *i.e.* $m\theta =\pm \tilde{m}\tilde{\theta}$, then the second set of terms in (\[Weyl tensor\]) vanishes, and in the first set of terms $\mathbf{F}_{\alpha \beta }=\mathbf{B}\left[\left( \kappa _{E}^{L}\right) _{\alpha \beta }\pm \left(\kappa _{J}^{L}\right) _{\alpha \beta }\right]$. This means that if $\theta/\tilde{\theta}$ is a rational number, and if furthermore we turn on only the terms with $m\theta =\pm \tilde{m}\tilde{\theta}$, then the Weyl tensors become proportional to direct products of $\left(\kappa _{E}^{L} \pm \kappa _{J}^{L} \right)$’s, *i.e.* they are of generalized Petrov type D.[^27] However, for generic values of $\theta/\tilde{\theta}$, (\[Weyl tensor\]) is not of type D,[^28] though it is a sum of type-D tensors.
Asymptotic behaviour of the Weyl tensors
----------------------------------------
By using the gamma matrix realization in Appendix \[Sec gamma\] and the global coordinates in Appendix \[Sec coordinates\], we can investigate the asymptotic behaviour of the Weyl tensors. When $r\rightarrow \infty$, we have $$\mathbf{F}^2|_{r\rightarrow\infty}\ =\ \mathbf{\breve F}^2|_{r\rightarrow\infty}
\ =\ \lambda^2 r^2\left[-\mathbf{B}^2+\mathbf{C}^2 {\rm sin}^2 (\vartheta)\right] \ ,$$ Then the terms in (\[Weyl tensor\]) of spin-$s$ Weyl tensor at large radius, by a simple power counting, scale as $$\left\{\lambda^2 r^2\left[-\mathbf{B}^2+\mathbf{C}^2 {\rm sin}^2 (\vartheta)\right]\right\}^{-\frac{1}{2}(s+1)} \ ,$$ and hence each term is either Kerr-like (when $\mathbf{B}^2\neq \mathbf{C}^2$) or 2-brane-like (when $\mathbf{B}^2 = \mathbf{C}^2$) in the asymptotic region. The Weyl tensor as the sum of these terms falls off as $\frac{1}{r^{s+1}}$, which is the regular boundary condition of asymptotically AdS$_{4}$ solutions.
Zero-form charges {#Sec 0-form charges}
-----------------
Although the separate spin-$s$ Weyl tensors blow up at the origin of spacetime, the limit of the full Weyl zero-form remains well-defined as the symbol of an operator. From this operator, it is possible to obtain higher spin gauge invariant quantities given by $$\mathcal{I}_{2p}:=\int_{\mathcal{Z}_{4}} \text{Tr}^{\prime }\left\{
I\star \bar{I}\star \left[ \Phi \star \pi \left( \Phi \right) \right]
^{\star p}\right\} \text{ ,}\label{calI2p}$$ which are referred to as zero-form charges [@Sezgin:2005pv] and that are related to higher spin amplitudes [@Colombo:2010fu; @Colombo:2012jx; @Didenko:2012tv]. On our exact solutions, *i.e.* by substituting (\[AnsatzPhi\]) and (\[ansatzF\_YZ\]), these charges are given by $$\left. \mathcal{I}_{2p}\right\vert _{\text{on-solution}}:=\frac{1}{32}\sum
_{\substack{ \sigma , \\ m_{1},m_{2},\cdots ,m_{2p}, \\ \tilde{m}_{1},\tilde{m}_{2},\cdots ,\tilde{m}_{2p}}}\text{ }\mathbf{A}_{{\textstyle\sum\nolimits}_{j=1}^{2p}
\left(-1\right) ^{j+1}m_{j},{\textstyle\sum\nolimits}_{j=1}^{2p}\tilde{m}_{j}}{\textstyle\prod\nolimits}_{j=1}^{2p}\nu _{\sigma ,m_{j},\tilde{m}_{j}}\text{ ,}$$where$$\mathbf{A}_{m,\tilde{m}}:=\frac{\left[ \text{sech}(
m\theta ) \ \text{sech}( \tilde{m}\tilde{\theta}) \right] ^{2}}{1-\left[ \text{tanh}( m\theta ) \
\text{tanh}( \tilde{m} \tilde{\theta}) \right] ^{2}}\text{
.}$$The simplest case is $p=1$:$$\left. \mathcal{I}_{2}\right\vert _{\text{on-solution}}=\frac{1}{32}\sum
_{\substack{ \sigma ,m,\tilde{m},n,\tilde{n}}}\text{ }\mathbf{A}_{m-n,\tilde{m}+\tilde{n}}\nu _{\sigma ,m,\tilde{m}}\nu _{\sigma ,n,\tilde{n}}\text{ .}$$ In [@Boulanger:2015kfa], this zero-form charge has been proposed to be one of the contributions to the effective action for higher spin gravity in asymptotically anti-de Sitter spacetimes. As noted at the end of Section \[Sec biaxial sym\], the resulting contribution to the free energy functional is not positive definite.
Twistor space connection {#Sec analyticity}
========================
In this section, we first compute the twistor space connection $V^{(L)}_{\underline{\alpha}}$, and show in special cases that it admits a regular power series expansion on ${\cal Z}_4$ around $Z^{\underline{\alpha}}=0$ over finite regions of spacetime provided that the group algebra $\Comp[\mathbb Z\times\mathbb Z]$ is truncated down to a non-unital subalgebra. We then demonstrate the existence of the linearized gauge function $H^{(1)}$ taking the linearized twistor space connection to Vasiliev gauge in a special case.
Generating function for twistor space connection in $L$-gauge \[Sec one-form analyticity\]
------------------------------------------------------------------------------------------
In order to facilitate the analysis, we write $$\begin{aligned}
V_{\alpha }^{(L)} &=&L^{-1}\star V_{\alpha }^{\prime }\star L \notag \\
&=&\sum_{\sigma ,m,\tilde{m}}L^{-1}\star T_{m,\tilde{m}}\star \Pi _{\sigma
}\star L\star \left( a_{\sigma ,m,\tilde{m}}\right) _{\alpha }\nonumber\\
&=&\sum_{\sigma ,m,\tilde{m}}
\oint_{0}\frac{d\zeta }{2\pi i\zeta ^{m+1}}\oint_{0}\frac{d\tilde{\zeta}}{2\pi i\tilde{\zeta}^{\tilde m+1}}
\frac{1}{2}\left(\mathring V_{0;\sigma,m,\tilde m}^{(L)}
+\sigma \mathring V_{1;\sigma,m,\tilde m}^{(L)}\right)_\a
\ ,\label{VL}\end{aligned}$$ in terms of the generating functions ($n=0,1$) $$\left(\mathring V_{n;\sigma,m,\tilde m}^{(L)}\right)_\a :=
2i\frac{\partial}{\partial \rho^\a}
\int_{-1}^1\frac{d\tau j_\s(\tau )
}{\left( \tau +1\right) ^{2}}
T^L_{m,\tilde{m}} \star (\kappa _{y}\bar{\kappa}_{\bar{y}})^n \star
\left\{ \text{exp}\left[\varsigma c\left( \tau \right) U^{\beta \gamma }z_{\beta
}z_{\gamma }+\rho ^{\beta }z_{\beta }\right] \right\}
_{\rho=0}
\text{ ,} \qquad \label{analy2}$$ where $\rho^\a$ is an auxiliary commuting spinor, and we denote $j_\s(\tau ) \equiv j_{\sigma }\left( \varsigma \mathring{\mu}_{\sigma };\tau \right)$. Thus, if these two integrals are finite for bounded $\mathring{\mu}_{\sigma }$ and finite $\rho^\a$, then $V_{\alpha }$ is real-analytic in ${\cal Y}_4\times {\cal Z}_4$.
Singular twistor space connection in $L$-gauge from $T_{0,0}$ \[Sec truncation\]
--------------------------------------------------------------------------------
From the discussion in Section \[Singular Integrand\] and the fact that $T^L_{0,0}=1$, it follows that contains a term given by $(a_{\sigma,0,0})_\a$, which is not real-analytic in ${\cal Z}_4$. Thus, real-analyticity of $V_\a^{(L)}$ in ${\cal Z}_4$ requires (a\_[,0,0]{})\_=0 .This can be achieved by a consistent truncation of the Ansatz (\[ansatzF\_YZ\])–(\[ansatzVbar\_YZ\]) by taking ${\cal A}_{E,J}$ to be a semigroup without identity.[^29]
If $\theta \tilde{\theta} \neq 0$ this can be achieved by taking \_[,m,]{}=(a\_[,m,]{})\_=0In other words, in the original Ansatz we sum over $m,\tilde m\in\mathbb{Z}$, but due to the requirement of real-analyticity, we instead sum over only positive $m$ and/or positive $\tilde{m}$. Furthermore, as can be seen from the table (\[realcond-nu-sigma\]), for compatibility with the reality condition, along with the truncation we must set $\theta $ and/or $\tilde{\theta}$ to be real. If $\theta= 0$ (or $\tilde{\theta}=0$) then we need to restrict $\tilde{m}\in \mathbb{Z}^{+}$, $\tilde{\theta} \in \mathbb{R}$ (or $m\in \mathbb{Z}^{+}$, $\theta \in \mathbb{R}$).
$\theta $ and $\tilde{\theta}$ cannot be both zero.
To summarize, in the following table, we give notations to the consistent truncations, and “$\times $” means that the situation either includes the unity or is inconsistent with the reality condition.
------------------------------------------------- ----------------------------------- ----------------------------------------- ------------------------- ------------------------------------------ --------------------- ----------------------
$\theta \in \mathbb{R}\backslash \{0\}$ $\theta \in i\mathbb{R}\backslash \{0\}$ $\ \ \
\theta =0\ \ \ $
$m\in \mathbb{Z}$ $m\in \mathbb{Z}^{+}$ $m\in \mathbb{Z}$ $m\in
\mathbb{Z}^{+}$
$\tilde{\theta}\in \mathbb{R}\backslash \{0\}$ $\tilde{m}\in \mathbb{Z}$ $\times $ $\mathcal{A}_{+,\pm }$ $\times $ $\times $ $\times $
$\tilde{\theta}\in \mathbb{R}\backslash \{0\}$ $\tilde{m}\in \mathbb{Z}^{+} $\mathcal{A}_{\pm ,+}$ $\mathcal{A}_{+,+}$ $\mathcal{A}_{\pm i,+}$ $\times $ $\mathcal{A}_{0,+}$
$
$\tilde{\theta}\in i\mathbb{R}\backslash \{0\}$ $\tilde{m}\in \mathbb{Z}$ $\times $ $\mathcal{A}_{+,\pm i}$ $\times $ $\times $ $\times $
$\tilde{\theta}\in i\mathbb{R}\backslash \{0\}$ $\tilde{m}\in \mathbb{Z} ^{+}$ $\times $ $\times $ $\times $ $\times $ $\times $
$\tilde{\theta}=0$ $\times $ $\mathcal{A}_{+,0}$ $\times $ $ \times $ $\times $
------------------------------------------------- ----------------------------------- ----------------------------------------- ------------------------- ------------------------------------------ --------------------- ----------------------
\[TableTruncation\]
Regularity of twistor space connection in $L$-gauge for non-unital ${\cal A}_{E,J}$
-----------------------------------------------------------------------------------
Under the assumption that ${\cal A}_{E,J}$ does not contain the unity, we proceed by investigating (\[analy2\]). From (\[Tkappa\]) it follows that $$T_{m,\tilde{m}}\star \kappa _{y}\bar{\kappa}_{\bar{y}}=\frac{\mathbf{A}}{
\sqrt{\text{det}\left( \mathbf{K}\right) }}\text{exp}\left\{ -\frac{1}{2}\left( \mathbf{K}^{-1}\right) ^{\underline{\alpha \beta }}Y_{\underline{
\alpha }}Y_{\underline{\beta }}\right\} \text{ ,}$$ where ( \^[-1]{}) \^ = ( ) =( \^[2]{}- \^[2]{}) \^[2]{} \[explicit detK\] Thus the case of $n=1$ is equivalent to the case of $n=0$ by replacing $\mathbf{B}$ and $\mathbf{C}$ with $\frac{\mathbf{B}}{\mathbf{B}^{2}-\mathbf{C}^{2}}$ and $\frac{-\mathbf{C }}{\mathbf{B}^{2}-\mathbf{C}^{2}}$, respectively, and multiplying an overall factor $\frac{1}{\sqrt{\left(\mathbf{B}^{2}-\mathbf{C}^{2}\right)^2}}$, which is possible provided that $\mathbf{B}
^{2}\neq \mathbf{C}^{2}$ *i.e.* if $m\theta \neq \pm \tilde{m}\tilde{\theta}$ for all allowed values of $m$ and $\tilde{m}$ (in the non-unital case). This can be achieved by a suitable choice of $\theta $ and $\tilde{\theta}$.
For $n=0$ we performing the $\star $-products between $T_{m,\tilde{m}}^L$ and the $Z$-dependent exponential in (\[analy2\]), which yields
&(V\_[0;m,m]{}\^[(L)]{})\_\
=& 2i\_[-1]{}\^1 { -iz\^y\_- |[y]{}\^\_|[y]{}\^+( iz\_+\_|[y]{}\^) ( iz\_+\_|[y]{}\^) }\
&{ { } } \_[=0]{}\
=& 2i\_[-1]{}\^1\
& { -iz\^y\_-|[y]{}\^[ ]{}\_|[y]{}\^+( iz\_+\_[ ]{}|[y]{}\^) ( iz\_+\_|[y]{}\^) .\
& . +} \[analy1convert\]
where $$\mathbf{M}_{\alpha \beta }(\tau)\equiv \frac{\mathbf{F}_{\alpha \beta }}{
\mathbf{F}^{2}}-2\varsigma c\left( \tau \right) U_{\alpha \beta } \text{ ,}$$ The integrand has potential divergencies at $\tau=0$, $\tau=-1$ and any value for $\tau$ where $\mathbf{F}^{2}$ or $\mathbf{M}^{2}(\tau)$ vanishes. As analysed in Section \[Singular Integrand\], the potential divergencies in $j_{\sigma }\left( \tau \right) $ at $\tau=0$ do not spoil the convergence of the integral provided that the $\nu$- and $b_n$-parameters are sufficiently small. Furthermore, since $\mathbf{M}_{\alpha \beta }(\tau)\sim \left( \tau +1\right)
^{-1}$ as $(\tau+1)\rightarrow 0$, it follows that both the prefactor and the exponent are bounded at $\tau=-1$.
To facilitate the investigation of $\mathbf{F}^{2}$ and $\mathbf{M}^{2}(\tau)$, which are thus functions of $m\theta$, $\tilde{m}\tilde{\theta}$, ${\cal X}_4$ and $\tau$, we use the gamma matrix realization in Appendix \[Sec gamma\] and the coordinates for $L$ in Appendix \[Sec coordinates\]. We have not succeeded in a complete analysis, but we have been able to cover a few important special cases as follows:
#### The case $\mathcal{A}_{+,0}$.
In this case, we have $\theta \in \mathbb{R}\backslash \{0\}$, $\tilde{\protect\theta}=0$, $m\in \mathbb{Z}^{+}$, and hence $\mathbf{A}={\rm sech}^2 m\theta$, $\mathbf{B}={\rm \tanh}\, m\theta$, $\mathbf{C}=0$. Using the explicit matrices and spherical coordinates defined in the appendices, we obtain $$\begin{aligned}
\mathbf{F}^{2} &=&-\mathbf{B}^{2}\lambda ^{2}r^{2}\text{ ,} \\
\mathbf{M}^{2}(\tau)&=&-\frac{\left[\varsigma c\left( \tau \right) \mathbf{B}\lambda
r+e^{i\vartheta }\right] \left[ \varsigma c\left( \tau \right) \mathbf{B}\lambda
r+e^{-i\vartheta }\right] }{\mathbf{B}^{2}\lambda ^{2}r^{2}}\text{ .}\end{aligned}$$ From $m\theta \neq 0$ it follows that $\mathbf{B}\neq 0$, and hence $\mathbf{F}^{2}$ does not vanish except at $r=0$. Moreover, since $c( \tau )$ is purely imaginary, the quantity $\varsigma c\left( \tau \right) \mathbf{B}\lambda r$ is purely imaginary as well. Thus $\mathbf{M}^{2}(\tau)$ vanishes iff = ,{ - , - } . \[equatorialsing\] Thus, in this case the twistor space connection is real-analytic everywhere away from the equatorial plane in the spherical coordinates.[^30]
#### The case $\tilde{\protect\theta}\neq 0$.
When $\tilde{\theta}\neq 0$, we resort to case-by-case investigation. We will only show two examples below.
For example, if we consider the region of small $r$, *i.e.* a small spatial sphere around the origin point, we have $$\begin{aligned}
\mathbf{F}^{2} &=&\mathbf{C}^{2}-2i\mathbf{BC}\lambda r\ \text{cos}\left(
\vartheta \right) +O\left( r^{2}\right) \text{ ,} \\
\mathbf{M}^{2}(\tau) &=&-\mathbf{C}^{-2}\left[\varsigma c\left( \tau \right) \mathbf{C}-i\right] ^{2}-2\mathbf{BC}^{-3}\left[\varsigma c\left( \tau \right) \mathbf{C}-i\right]
\lambda r\ \text{cos}\left( \vartheta \right) +O\left( r^{2}\right) \text{ .}\end{aligned}$$ A valid choice of the parameters is $\theta \in \mathbb{R}\backslash \{0\}$, $\tilde{\theta}\in i\mathbb{R}\backslash \{0\}$ and $m\in \mathbb{Z}^{+}$, *i.e.* the truncation $\mathcal{A}_{+,\pm i}$. With this choice, we have $\mathbf{B}\in \mathbb{R}\backslash \{0\}$, $\mathbf{C}\in i\mathbb{R}$. Then for $\mathbf{C}\neq 0$ *i.e.* $\tilde{m}\neq
0$, both first leading terms of $\mathbf{F}^{2} $ and $\mathbf{M}^{2}(\tau) $ are non-zero. For $\tilde{m}=0$, the discussion is the same as the above $\tilde\theta=0$ case. To summarize, the one-form field in this case is real-analytic in the small sphere except on the equatorial plane.
For another example, we consider the region of small $\vartheta $, *i.e.* a narrow cone around the axis of symmetry $\vartheta =0$, we have $$\begin{aligned}
\mathbf{F}^{2} &=&\left( \mathbf{C}-i\mathbf{B}\lambda r\right) ^{2}+O\left(
\vartheta^{2}\right) \text{ ,} \\
\mathbf{M}^2(\tau)&=&\left[ \frac{1}{\mathbf{C}-i\mathbf{B}\lambda r}-i\varsigma c\left(
\tau \right) \right] ^{2}+O\left( \vartheta ^{2}\right) \text{ .}\end{aligned}$$ A valid choice of the parameters is $\theta \in \mathbb{R}\backslash \{0\}$, $
\tilde{\theta}\in \mathbb{R}\backslash \{0\}$ and $m\in \mathbb{Z}^{+}$ *i.e.* the truncation $\mathcal{A}_{+,\pm }$. With this choice, we have $\mathbf{B}\in \mathbb{R}\backslash \{0\}$, $
\mathbf{C}\in \mathbb{R}$. Then for $r\neq 0$ we have $\mathbf{C}-i\mathbf{B}\lambda r\notin \mathbb{R}$, and thus, with $i\varsigma c\left( \tau \right)\in
\mathbb{R}$, both first leading terms of $\mathbf{F}^{2} $ and $\mathbf{M}^2(\tau)$ are non-zero. The one-form field in this case is real-analytic in the narrow cone around the axis $\vartheta =0$ excluding the origin point.
Linearized twistor space connection in Vasiliev gauge
-----------------------------------------------------
Finally, let us check in a special case that it is indeed possible to bring the linearized twistor space connection to Vasiliev gauge by means of a linearized gauge transformation, as described in Section \[Sec Gauge function\], *viz.* $$V_{\alpha }^{(G)(1)}=V_{\alpha }^{(L)(1)}+\partial _{\alpha }H^{(1)}\text{ ,}$$ where $H^{(1)}$ is formally given by $$H^{(1)}=H^{(1)}\vert_{Z=0}-\frac{1}{z^{\beta }\partial _{\beta }}\left( z^{\alpha }V_{\alpha
}^{(L)(1)}\right) \text{ .}$$ Note that, as explained around Eq. (\[equatorialsing\]), in the case $\tilde{\theta}=0$ with the truncation $\mathcal{A}_{+,0}$, the regularity of the twistor space connection at $\vartheta =\frac{\pi }{2}$ has not yet been verified in the $L$-gauge. However, we expect that this problem would not exist in Vasiliev gauge.
To perform the check, we set $\vartheta =\frac{\pi }{2}$, $t=\phi=0$, $y^{\a}=\bar{y}^{\dot\a}=0$ and $\lambda =1$. The resulting expression of the generating function for the $L$-gauge twistor space connection reads $$\left. \left( \mathring{V}_{n=0;\sigma ,m}^{(L)}\right) _{\alpha
}\right\vert _{\lambda =1;\ \vartheta =\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0}=-2i \mathbf{A}\int_{-1}^{1}\frac{d\tau j_{\sigma }\left( \tau \right) }{\left( \tau
+1\right) ^{2}} P_{\alpha \beta }z^{\beta }\text{exp}\left\{ Q_{\alpha \beta }z^{\alpha }z^{\beta }\right\} \text{ ,}$$where$$P_{\alpha \beta }=-\left[ 1+\mathbf{B}^{2}c^{2}\left( \tau \right) r^{2}\right] ^{-\frac{3}{2}}\left(
\begin{array}{cc}
\mathbf{B}\varsigma c\left( \tau \right) r & -1 \\
1 & \mathbf{B}\varsigma c\left( \tau \right) r\end{array}\right) \text{ ,}$$$$Q_{\alpha \beta }=-\frac{1}{2}\varsigma c\left( \tau \right) \left[ 1+\mathbf{B}^{2}c^{2}\left( \tau \right) r^{2}\right] ^{-1}\left(
\begin{array}{cc}
-\mathbf{B}\varsigma c\left( \tau \right) r & 1 \\
1 & \mathbf{B}\varsigma c\left( \tau \right) r\end{array}\right) \text{ .}$$ Going to Vasiliev gauge, we obtain $$\begin{aligned}
&&\left. \left( \mathring{V}_{n=0;\sigma ,m}^{(G)(1)}\right) _{\alpha
}\right\vert _{\lambda =1;\ \vartheta =\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0}\notag
\\
&=&-\frac{i\varsigma b_{0}\mathring{\nu}_{\sigma }}{2}\mathbf{A}z_{\alpha
}\int_{-1}^{1}\frac{d\tau }{\left( \tau +1\right) ^{2}}\left\{ Pe^{Q(z)}+\frac{e^{Q(z)}-1-Q(z)e^{Q(z)}}{Q^2(z)}z^{\alpha }S_{\alpha
}{}^{\beta }Q_{\beta \gamma }z^{\gamma }\right\} \text{,}
\label{VG10PQ}\end{aligned}$$where$$Q(z):=Q_{\alpha \beta }z^{\alpha }z^{\beta }\text{ ,}$$ and we have decomposed $$P_{\alpha \beta }=:P\varepsilon _{\alpha \beta }+S_{\alpha \beta }\text{ ,}\quad S_{[\alpha\beta]}=0\ ,$$*i.e.*$$P=\left[ 1+\mathbf{B}^{2}c^{2}\left( \tau \right) r^{2}\right] ^{-\frac{3}{2}}\text{ , \quad }S_{\alpha \beta }=-\left[ 1+\mathbf{B}^{2}c^{2}\left( \tau
\right) r^{2}\right] ^{-\frac{3}{2}}\mathbf{B}\varsigma c\left( \tau \right) r\left(
\begin{array}{cc}
1 & 0 \\
0 & 1\end{array}\right) \text{ .}$$The integrand of (\[VG10PQ\]) can be converted into a total derivative of $\tau $:$$\begin{aligned}
&&\left. \left( \mathring{V}_{n=0;\sigma ,m}^{(G)(1)}\right) _{\alpha
}\right\vert _{\lambda =1;\ \vartheta =\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0} \notag \\
&=&-\frac{\varsigma b_{0}\mathring{\nu}_{\sigma }}{2}\mathbf{A}z_{\alpha
}\int_{-1}^{1}d\tau \frac{\partial }{\partial \tau } \left\{
\frac{(e^{Q(z)}-1)(Q(z)P-z^{\alpha }S_{\alpha
}{}^{\beta }Q_{\beta \gamma }z^{\gamma })}{\frac{\partial {Q^2(z)}}{\partial c}}\right\}\ ,\end{aligned}$$ where, more explicitly, $$\begin{aligned}
&&\frac{(e^{Q(z)}-1)(Q(z)P-z^{\alpha }S_{\alpha
}{}^{\beta }Q_{\beta \gamma }z^{\gamma })}{\frac{\partial {Q^2(z)}}{\partial c}}
\notag
\\
&=&
-\frac{\sqrt{1+
\mathbf{B}^{2}c^{2}\left( \tau \right) r^{2}}\left[ 1-\text{exp}\left( \frac{ \mathbf{B}c\left( \tau \right) ^{2}r\left( z^{1}+z^{2}\right) \left(
z^{1}-z^{2}\right) -2\varsigma c\left( \tau \right) z^{1}z^{2}}{2\left[ 1+\mathbf{B} ^{2}c^{2}\left( \tau \right) r^{2}\right] }\right) \right] }{\mathbf{B} \varsigma c\left( \tau \right) r\left( z^{1}+z^{2}\right) \left( z^{1}-z^{2}\right)
-2z^{1}z^{2}}\ .\end{aligned}$$ Thus, assigning the singularity in the interior of the integration domain its principal value, and using separate analytical continuations above and below the singularity, one finds that it does not contribute, and hence . ( \_[n=0;,m]{}\^[(G)(1)]{}) \_\_[=1; =, t==0; y=|[y]{}=0]{} =z\_ \[VG10\] We note that the limit $z^\alpha\rightarrow 0$ must be taken after the integration over $\tau$ has been performed. This yields a well-defined limit, such that the twistor space connection is indeed real-analytic at $z^\a=0$. If one instead takes the limit $z^\alpha\rightarrow 0$ under the integral, one ends up with a divergent integral; this divergence cannot, however, be interpreted as any pole or other singularity at $z^\a=0$. Thus, the prescription that we use is the unique one leading to a sensible result.[^31] We note, however, that in the holomorphic gauge the corresponding operations commute, and, correspondingly, the twistor space connection is non-real-analytic at $z^\a=0$ in this gauge; see Section \[Singular Integrand\].
Similarly, we can also calculate for $n=1$: $$\left. \left( \mathring{V}_{n=1;\sigma ,m}^{(G)(1)}\right) _{\alpha
}\right\vert _{\lambda =1;\ \vartheta =\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0}=\frac{ b_{0}\mathring{\nu}_{\sigma }}{2}z_{\alpha }\mathbf{A}\frac{1-\text{exp}\left[ \frac{\mathbf{B}}{2r}\left( z^{1}+z^{2}\right)
\left( z^{1}-z^{2}\right) \right] }{\mathbf{B}^{2}\left( z^{1}+z^{2}\right)
\left( z^{1}-z^{2}\right) }\text{ .} \label{VG11}$$ Finally, using the analog of Eq. (\[VL\]) in Vasiliev gauge, *i.e.* replacing the label $(L)$ with $(G)(1)$ and substituting (\[VG10\]) and (\[VG11\]), we obtain $$\begin{aligned}
&&\left. \mathring{V}_{\alpha }^{(G)(1)}\right\vert _{\lambda =1;\ \vartheta
=\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0} \notag \\
&=&\frac{ b_{0}}{4}z_{\alpha }\sum_{\sigma ,m,\tilde{m}}\mathbf{A}\nu _{\sigma ,m,\tilde{m}}\left[ \frac{1-\text{exp}\left[ \frac{1}{2\mathbf{B}r}\left( z^{1}+z^{2}\right) \left( z^{1}-z^{2}\right) \right] }{\left(
z^{1}+z^{2}\right) \left( z^{1}-z^{2}\right) }\right.\notag
\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left.+\sigma \frac{1-\text{exp}\left[ \frac{\mathbf{B}}{2r}\left( z^{1}+z^{2}\right) \left(
z^{1}-z^{2}\right) \right] }{\mathbf{B}^{2}\left( z^{1}+z^{2}\right) \left(
z^{1}-z^{2}\right) }\right] \text{ .}
\label{VG1sum}\end{aligned}$$
Indeed, starting in Vasiliev gauge, one can integrate the equations of motion for the linearized twistor space connection directly without factorizing the inner Klein operator $\kappa$, with the result [@Vasiliev:1990en] $$V_{\alpha }^{(G)(1)}=-\frac{ib_0}{2}z_{\alpha }\int_{0}^{1}d\tau \ \tau e^{iy^{\alpha
}z_{\alpha }\tau }\left( \left. \Phi ^{(G)(1)}\right\vert _{y\rightarrow -z\tau
}\right) \text{ .} \label{Vas integral}$$ We note that, unlike the solution for the twistor space connection obtained starting in the holomorphic gauge, which refers to a splitting of $z^\alpha$ into $z^\pm$ as in Section \[Sec general ansatz\], the above expression does not refer to any auxiliary spinor frame in $Z$ space. From $\Phi^{(G)(1)}=\Phi^{(L)(1)}=\Phi^{(L)}$ it follows that implies that $$\begin{aligned}
&&\left. \mathring{V}_{\alpha }^{(G)(1)}\right\vert _{\lambda =1;\ \vartheta
=\frac{\pi }{2},\ t=\phi =0;\ y=\bar{y}=0} \notag \\
&=&\frac{b_0}{2}z_{\alpha }\sum_{m,\tilde{m}}\mathbf{A}\left[ \nu _{1,m,\tilde{m}}\frac{1-\text{exp}\left[ \frac{1}{2\mathbf{B}r}\left( z^{1}+z^{2}\right) \left(
z^{1}-z^{2}\right) \right] }{\left( z^{1}+z^{2}\right) \left(
z^{1}-z^{2}\right) }\right.\notag
\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\left. +\nu _{2,m,\tilde{m}}\frac{1-\text{exp}\left[ \frac{\mathbf{B}}{2r}\left( z^{1}+z^{2}\right) \left( z^{1}-z^{2}\right) \right] }{\mathbf{B}^{2}\left( z^{1}+z^{2}\right) \left( z^{1}-z^{2}\right) }\right]
\text{ ,}\end{aligned}$$ which one can readily identify with upon using .
Conclusion {#Sec Conclusions}
==========
In this paper, we have given a new class of bi-axially symmetric solutions to Vasiliev’s bosonic higher spin gravity model using an Ansatz based on gauge functions and separation of the dependence on the coordinates in twistor space.
This facilitates the construction of perturbatively exact solutions in a holomorphic gauge. In this gauge, the spacetime connection vanishes, the Weyl zero-form is constant, *i.e.* it depends only on the fiber coordinates, while the twistor space connection depends on the twistor space via a universal holomorphic function on $Z$-space with singularties at $z^\alpha=0$ that we have exhibited in the Weyl order in Section \[Singular Integrand\], and on the fiber coordinates via the zero-form integration constants. We have then expanded the dependence on the fiber coordinates in terms of the basis of a group algebra generated by the exponents of $\theta E$ and $\tilde\theta J$, where $E$ and $J$ are the generators the time-translational and rotational symmetries of the solutions.
We have then switched on the spacetime dependence using a vacuum gauge function $L$. In the resulting gauge, which we refer to as $L$-gauge, the spacetime connetion describes an anti-de Sitter spacetime. The terms containing the unity of the internal algebra need to be removed, in order for the Weyl zero-form in $L$-gauge to be real-analytic on twistor space. The resulting generalized spin-$s$ Weyl tensor, which thus obeys the Bargmann-Wigner equation, is given by a sum of generalized Petrov type-D tensors that are asymptotically Kerr-like or 2-brane-like. For special values of the parameters, including the symmetry enhanced cases, the spin-$s$ Weyl tensor is of generalized Petrov type D.
We have also shown that the twistor space connection in $L$-gauge, provided that the group algebra is truncated to a non-unital semigroup algebra as summarized in the table (\[TableTruncation\]), is real-analytic in finite spacetime regions for a number of choices of parameters. In particular, in the spherically symmetric case, it is real-analytic everywhere away from the equatorial plane. The Ansatz introduces a fixed frame in $Z$-space that breaks the manifest spherical symmetry upon going to the normal order in master fields with a $Z$-dependence. At this plane, singularities may appear in auxiliary integrals, whose treatment requires analytical continuations in twistor space. We have not spelled out the nature of the resulting contributions to the twistor space connection in $L$-gauge in this work.
Finally, we have examined the problem of transforming the master fields from the $L$-gauge to Vasiliev gauge at the linearized level. It is trivial in the case of the Weyl zero-form. As for the linearized twistor space connection, we have argued that the transformation exists at spacetime points where the connection is real-analytic in twistor space in $L$-gauge. Among the remaining cases, we have focused on the potential divergence at the equatorial plane in the spherically symmetric case, which should be removed by the transformation, as the twistor space connection in Vasiliev’s gauge does not refer to any fixed frame in $Z$-space. Indeed we have verified that this is the case at the origin of the fiber space (*i.e.* at $Y^{\underline \a}=0$), for general $z^\alpha$, and consequently we have found agreement with the expression for the twistor space connection in Vasiliev gauge obtained by direct integration.
Thus, more briefly, we have found families of exact bi-axially symmetric solutions in the holomorphic and $L$-gauges, and we have verified that they can be brought to Vasiliev gauge at the linearized level in a special case, leaving the more general case as well as higher order perturbation for future study.
We end our conclusions by commenting on future directions. We have left a number of technical details unattended, that we would like to examine more carefully. Besides the issues related to real-analyticity of the linearized master fields in Vasiliev gauge, there is the intriguing degenerate case $\mathbf B=\mathbf C$. Moreover, by taking limits for $\theta$ and $\nu$-parameters it is possible to make contact with the solutions found in [@Iazeolla:2011cb], and more general Kerr-like extensions thereof by expanding the fiber subalgebra using a combination of group algebra elements and endomorphisms in Fock spaces.
More generally, we recall that the importance of Vasiliev’s gauge at linearized level is that, when combined with normal order, the linearized spacetine connection $W_\mu^{(G)(1)}$ has a $Y$-expansion at $Z=0$ in terms of unfolded Fronsdal tensors and the initial data $H^{(1)}\vert_{Z=0}$ modulo gauge transformations.[^32] Exact solutions, however, are easier to find in Weyl order using the gauge function method. As far as we can see from the results here and elsewhere, we expect there to be an agreement at the linearized level between the holomorphic and Vasiliev gauges for a fairly large class of linearized zero-form initial data $\Phi^{\prime(1)}(Y)$, and it would be desirable to establish this correspondence more precisely, *e.g.* by expanding $\Phi^{\prime(1)}(Y)$ in terms of twistor space plane waves.
Turing to higher order perturbations, the next step is to compute the first subleading corrections to all master fields in Vasiliev gauge, and examine whether real-analyticity in twistor space for generic spacetime points constrains the initial data $\Phi^{\prime(n)}(Y)$ for the zero-form and $H^{(n)}\vert_{Z=0}$ for the gauge function, for $n=1,2$. This may lead to modified asymptotic boundary conditions in AdS$_4$ and corresponding corrections to the zero-form charges. In particular, as proposed in [@Boulanger:2015kfa], the zero-form charge ${\cal I}_2$ is a contribution to the free energy functional. The corresponding sesqui-linear form is not definite on the representation space of the underlying higher spin symmetry algebra containing the initial data of our solutions. There are additional contributions to the free energy, however, that may lead to an interesting phase diagram.
The above analysis can also be performed for the closely related Kerr-like solutions outlined above. More generally, one may consider relaxing the Vasiliev gauge as well as the smoothness conditions in twistor space, which may lead to more general noncommutive geometries with interesting properties.
#### Acknowledgements
We are thankful to V. Didenko and C. Iazeolla for useful discussions and correspondence. We have also benefited from conversations with C. Arias, R. Aros, R. Bonezzi, N. Boulanger, K. Crysostomos, K. Morand, R. Olea, E. Sezgin, E. Skvortsov, M. Taronna, A. Torres Gomez, M. Valenzuela and M. Vasiliev. The work of P. S. is supported by Fondecyt Regular grant N$^{\rm o}$ 1140296, Conicyt grant DPI 20140115 and UNAB internal grant DI-1382-16/R. The work of Y. Y. is supported by the Chilean Fondecyt Postdoc Project N$^{\rm o}$ 3150692.
The $\star $-exponent
=====================
Let $Y^\alpha$, $\alpha=1,\dots, N$, be oscillator variables obeying \_= 2iC\^ ,where $N$ is even and $C^{\alpha\beta}$ is invertible. Denote$$w=\frac{1}{4}K_{\alpha \beta }Y^{\alpha }\star Y^{\beta }\text{ ,}
\label{def_w}$$where $K_{\alpha\beta}$ is a constant matrix obeying $$K_{\alpha \beta }=K_{\beta \alpha }\text{ ,\qquad }
K_{\alpha \beta }K^{\beta \gamma }=\delta _{\alpha }{}^{\gamma }\text{ ,}$$ where indices are raised and lowered using the conventions $Y^\alpha=C^{\alpha\beta}Y_\beta$, $Y_\beta=Y^\beta C_{\beta\alpha}$, and $
C^{\alpha \beta }C_{\alpha \gamma }=\delta _{\gamma }{}^{\beta }$. The $\star $-exponent is defined by the Taylor series of exponential function with $\star $-products replacing ordinary products. In what follows we will compute the symbol in Weyl order of the $\star $-exponent$$g\equiv e_{\star }^{-2tw}\text{ .} \label{def_g}$$ From (\[def\_g\]) we can derive$$w\star g=-\frac{1}{2}\frac{\partial g}{\partial t}\text{ ,} \label{wg_dg}$$and to proceed we will compute the symbol of $w\star g$. To do so we use the identity $$Y_{\alpha }\star f\left( Y\right) =Y_{\alpha }f\left( Y\right) +i\frac{\partial }{\partial Y^{\alpha }}f\left( Y\right) \text{ .}$$ Thus $$Y^{\alpha }\star Y^{\beta }=Y^{\alpha }Y^{\beta }+iC^{\alpha \beta }\text{ .}$$Hence (\[def\_w\]) can also be written as$$w=\frac{1}{4}K_{\alpha \beta }Y^{\alpha }Y^{\beta }\text{ .}$$ Using this, we can show that $$w\star g=wg-\frac{N}{8}\frac{\partial g}{\partial w}-\frac{1}{4}w\frac{\partial ^{2}g}{\partial w^{2}}\text{ .} \label{w_star_g}$$
$$\begin{aligned}
w\star g &=&\frac{1}{4}K^{\alpha \beta }Y_{\alpha }\star Y_{\beta }\star g \\
&=&\frac{1}{4}K^{\alpha \beta }Y_{\alpha }\star \left( Y_{\beta }g+i\frac{\partial g}{\partial Y^{\beta }}\right) \\
&=&\frac{1}{4}K^{\alpha \beta }\left( Y_{\alpha }Y_{\beta }g+iY_{\alpha }\frac{\partial g}{\partial Y^{\beta }}+i\frac{\partial \left( Y_{\beta
}g\right) }{\partial Y^{\alpha }}-\frac{\partial ^{2}g}{\partial Y^{\alpha
}\partial Y^{\beta }}\right) \\
&=&wg+\frac{i}{2}K^{\alpha \beta }Y_{\alpha }\frac{\partial g}{\partial
Y^{\beta }}-\frac{1}{4}K^{\alpha \beta }\frac{\partial ^{2}g}{\partial
Y^{\alpha }\partial Y^{\beta }}\text{ .}\end{aligned}$$
The last two terms can be further converted:$$\begin{aligned}
\frac{i}{2}K^{\alpha \beta }Y_{\alpha }\frac{\partial g}{\partial Y^{\beta }}
&=&\frac{i}{2}K^{\alpha \beta }Y_{\alpha }\frac{\partial g}{\partial w}\frac{\partial w}{\partial Y^{\beta }} \\
&=&\frac{i}{2}K^{\alpha \beta }Y_{\alpha }\left( \frac{1}{2}K_{\beta \gamma
}Y^{\gamma }\right) \frac{\partial g}{\partial w} \\
&=&\frac{i}{4}Y_{\alpha }Y^{\alpha }\frac{\partial g}{\partial w} \\
&=&0\text{ ,}\end{aligned}$$$$\begin{aligned}
-\frac{1}{4}K^{\alpha \beta }\frac{\partial ^{2}g}{\partial Y^{\alpha
}\partial Y^{\beta }} &=&-\frac{1}{4}K^{\alpha \beta }\frac{\partial }{\partial Y^{\alpha }}\left( \frac{\partial g}{\partial w}\frac{\partial w}{\partial Y^{\beta }}\right) \\
&=&-\frac{1}{4}K^{\alpha \beta }\frac{\partial }{\partial Y^{\alpha }}\left(
\frac{1}{2}K_{\beta \gamma }Y^{\gamma }\frac{\partial g}{\partial w}\right)
\\
&=&-\frac{1}{8}\frac{\partial }{\partial Y^{\alpha }}\left( Y^{\alpha }\frac{\partial g}{\partial w}\right) \\
&=&-\frac{1}{8}\delta _{\alpha }{}^{\alpha }\frac{\partial g}{\partial w}-\frac{1}{8}Y^{\alpha }\frac{\partial ^{2}g}{\partial w^{2}}\frac{\partial w}{\partial Y^{\alpha }} \\
&=&-\frac{N}{8}\frac{\partial g}{\partial w}-\frac{1}{8}Y^{\alpha }\left(
\frac{1}{2}K_{\alpha \beta }Y^{\beta }\right) \frac{\partial ^{2}g}{\partial
w^{2}} \\
&=&-\frac{N}{8}\frac{\partial g}{\partial w}-\frac{1}{4}w\frac{\partial ^{2}g}{\partial w^{2}}\text{ .}\end{aligned}$$Thus (\[w\_star\_g\]) is proven.
By substituting (\[w\_star\_g\]), (\[wg\_dg\]) can be converted to$$wg-\frac{N}{8}\frac{\partial g}{\partial w}-\frac{1}{4}w\frac{\partial ^{2}g}{\partial w^{2}}=-\frac{1}{2}\frac{\partial g}{\partial t}\text{ .}
\label{wg_no_star_dg}$$This differential equation can be solved by substituting the Ansatz$$g=a\left( t\right) e^{b\left( t\right) w}\text{ ,} \label{Ansatz_g}$$ which gives$$a\left( t\right) we^{b\left( t\right) w}-\frac{N}{8}a\left( t\right) b\left(
t\right) e^{b\left( t\right) w}-\frac{1}{4}a\left( t\right) b^{2}\left(
t\right) we^{b\left( t\right) w}=-\frac{1}{2}a^{\prime }\left( t\right)
e^{b\left( t\right) w}-\frac{1}{2}a\left( t\right) b^{\prime }\left(
t\right) we^{b\left( t\right) w}\text{ ,} \label{dg_ab}$$and this equation requires the following set of ordinary differential equations for $a\left( t\right) $ and $b\left( t\right) $ to be satisfied:$$\begin{aligned}
-\frac{1}{2}a^{\prime }\left( t\right) &=&-\frac{N}{8}a\left( t\right)
b\left( t\right) \text{ ,} \\
-\frac{1}{2}a\left( t\right) b^{\prime }\left( t\right) &=&a\left( t\right)
\left( 1-\frac{1}{4}b^{2}\left( t\right) \right) \text{ .}\end{aligned}$$The general solution is given by $$\begin{aligned}
a\left( t\right) &=&C_{2}\left[ \text{sech}\left( t+C_{1}\right) \right] ^{\frac{N}{2}}\text{ ,} \label{a(t)withConst} \\
b\left( t\right) &=&-2\text{tanh}\left( t+C_{1}\right) \text{ ,}
\label{b(t)withConst}\end{aligned}$$where $C_{1}$ and $C_{2}$ are constants. These are determined by requiring that (\[Ansatz\_g\]) and (\[def\_g\]) stand for the same solution of (\[wg\_no\_star\_dg\]). It is obvious that $$\left. g\right\vert _{t=0}=e_{\star }^{0w}=1\text{ ,}$$and hence$$\left. \left( wg-\frac{N}{8}\frac{\partial g}{\partial w}-\frac{1}{4}w\frac{\partial ^{2}g}{\partial w^{2}}\right) \right\vert _{t=0}=\left. w\star
g\right\vert _{t=0}=w\text{ .}$$Consequently we have$$\begin{aligned}
a\left( 0\right) e^{b\left( 0\right) w} &=&1\text{ ,} \\
a\left( 0\right) we^{b\left( 0\right) w}-\frac{N}{8}a\left( 0\right) b\left(
0\right) e^{b\left( 0\right) w}-\frac{1}{4}a\left( 0\right) b^{2}\left(
0\right) we^{b\left( 0\right) w} &=&w\text{ .}\end{aligned}$$Therefore,$$a\left( 0\right) =1\text{ and }b\left( 0\right) =0\text{\ .}$$By substituting them into (\[a(t)withConst\]) and (\[b(t)withConst\]) we can determine that$$C_{1}=0\text{ and }C_{2}=1\text{ .}$$Then we derive that$$\begin{aligned}
a\left( t\right) &=&\left[ \text{sech}\left( t\right) \right] ^{\frac{N}{2}}\text{ ,} \\
b\left( t\right) &=&-2\text{tanh}\left( t\right) \text{ .}\end{aligned}$$In this way we conclude $$e_{\star }^{-2tw}=g=\left[ \text{sech}\left( t\right) \right] ^{\frac{N}{2}}e^{-2\text{tanh}\left( t\right) w}\text{ .}
\label{star and ordinary exponent}$$
Van der Waerden symbols and gamma matrices \[Sec gamma\]
========================================================
To simplify some of the calculations in this paper, one can use a set of explicit matrix expressions of Pauli matrices and gamma matrices, which for example is given in this appendix.
Pauli matrices
--------------
We define the $\sigma $-matrices with two lower spinor indices$$\left( \sigma _{0}\right) _{\alpha \dot{\alpha}}=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1\end{array}\right) \text{\ , \ }\left( \sigma _{1}\right) _{\alpha \dot{\alpha}}=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \text{\ , \ }\left( \sigma _{2}\right) _{\alpha \dot{\alpha}}=\left(
\begin{array}{cc}
0 & -i \\
i & 0\end{array}\right) \text{\ , \ }\left( \sigma _{3}\right) _{\alpha \dot{\alpha}}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1\end{array}\right) \text{ ,}$$where $\sigma _{0}$ is the identity matrix and $\sigma _{1,2,3}$ are the usual Pauli matrices. We also define their complex conjugate:$$\left( \bar{\sigma}_{0}\right) _{\dot{\alpha}\alpha }=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1\end{array}\right) \text{\ , \ }\left( \bar{\sigma}_{1}\right) _{\dot{\alpha}\alpha
}=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \text{\ , \ }\left( \bar{\sigma}_{2}\right) _{\dot{\alpha}\alpha
}=\left(
\begin{array}{cc}
0 & i \\
-i & 0\end{array}\right) \text{\ , \ }\left( \bar{\sigma}_{3}\right) _{\dot{\alpha}\alpha
}=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1\end{array}\right) \text{ .}$$Then obviously we have$$\left( \sigma _{a}\right) _{\alpha \dot{\alpha}}=\left( \bar{\sigma}_{a}\right) _{\dot{\alpha}\alpha }\text{ .}$$Furthermore, we use$$\varepsilon ^{\alpha \beta }=\varepsilon _{\alpha \beta }=\varepsilon ^{\dot{\alpha}\dot{\beta}}=\varepsilon _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
0 & 1 \\
-1 & 0\end{array}\right) \label{eps explicit}$$to raise or lower indices (by NW-SE rules).
We also define$$\begin{gathered}
\left( \sigma _{ab}\right) _{\alpha \beta }=-\left( \sigma _{ba}\right)
_{\alpha \beta }=\left( \sigma _{\lbrack a}\right) _{\alpha }^{\ \ \dot{\gamma}}\left( \bar{\sigma}_{b]}\right) _{\dot{\gamma}\beta }\text{ ,} \\
\left( \bar{\sigma}_{ab}\right) _{\dot{\alpha}\dot{\beta}}=-\left( \bar{\sigma}_{ba}\right) _{\dot{\alpha}\dot{\beta}}=\left( \bar{\sigma}_{[a}\right) _{\dot{\alpha}}^{\ \ \gamma }\left( \sigma _{b]}\right)
_{\gamma \dot{\beta}}\text{ .}\end{gathered}$$To write them explicitly: $$\begin{gathered}
\left( \sigma _{01}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
-1 & 0 \\
0 & 1\end{array}\right) \text{ \ , }\left( \sigma _{02}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
i & 0 \\
0 & i\end{array}\right) \text{ \ , }\left( \sigma _{03}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \text{ \ ,} \notag \\
\left( \sigma _{12}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
0 & i \\
i & 0\end{array}\right) \text{ \ , }\left( \sigma _{13}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1\end{array}\right) \text{ \ , }\left( \sigma _{23}\right) _{\alpha \beta }=\left(
\begin{array}{cc}
-i & 0 \\
0 & i\end{array}\right) \text{ \ ,} \notag \\
\left( \bar{\sigma}_{01}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
-1 & 0 \\
0 & 1\end{array}\right) \text{ \ , }\left( \bar{\sigma}_{02}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
-i & 0 \\
0 & -i\end{array}\right) \text{ \ , }\left( \bar{\sigma}_{03}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \text{ \ ,} \notag \\
\left( \bar{\sigma}_{12}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
0 & -i \\
-i & 0\end{array}\right) \text{ \ , }\left( \bar{\sigma}_{13}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
1 & 0 \\
0 & 1\end{array}\right) \text{ \ , }\left( \bar{\sigma}_{23}\right) _{\dot{\alpha}\dot{\beta}}=\left(
\begin{array}{cc}
i & 0 \\
0 & -i\end{array}\right) \text{ \ .}\end{gathered}$$As shown above, the pair of spinor indices are symmetric.
Gamma matrices
--------------
We construct the explicit expressions of gamma matrices in the following way:$$\left( \Gamma _{a}\right) _{\underline{\alpha }}^{\ \ \underline{\beta }}=\left(
\begin{array}{cc}
0 & \left( \sigma _{a}\right) _{\alpha }^{\ \ \dot{\beta}} \\
\left( \bar{\sigma}_{a}\right) _{\dot{\alpha}}^{\ \ \beta } & 0\end{array}\right) \text{ ,}$$whose explicit expressions are$$\begin{gathered}
\left( \Gamma _{0}\right) _{\underline{\alpha }}^{\ \ \underline{\beta }}=\left(
\begin{array}{cccc}
0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0\end{array}\right) \text{ , }\left( \Gamma _{1}\right) _{\underline{\alpha }}^{\ \
\underline{\beta }}=\left(
\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & -1 \\
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0\end{array}\right) \text{ ,} \notag \\[5pt]
\text{ }\left( \Gamma _{2}\right) _{\underline{\alpha }}^{\ \ \underline{\beta }}=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0\end{array}\right) \text{ , \ }\left( \Gamma _{3}\right) _{\underline{\alpha }}^{\ \
\underline{\beta }}=\left(
\begin{array}{cccc}
0 & 0 & 0 & -1 \\
0 & 0 & -1 & 0 \\
0 & -1 & 0 & 0 \\
-1 & 0 & 0 & 0\end{array}\right) \text{ ,}\end{gathered}$$One can check the property that$$\left( \Gamma _{(a}\right) _{\underline{\alpha }}^{\ \ \underline{\gamma }}\left( \Gamma _{b)}\right) _{\underline{\gamma }}^{\ \ \underline{\beta }}=\eta _{ab}\delta _{\underline{\alpha }}{}^{\underline{\beta }}\text{ ,}$$where $\eta _{ab}=$ diag$\left\{ -1,1,1,1\right\} $.
We further use$$C^{\underline{\alpha \beta }}=\left(
\begin{array}{cc}
\varepsilon ^{\alpha \beta } & 0 \\
0 & \varepsilon ^{\dot{\alpha}\dot{\beta}}\end{array}\right) \text{ and }C_{\underline{\alpha \beta }}=\left(
\begin{array}{cc}
\varepsilon _{\alpha \beta } & 0 \\
0 & \varepsilon _{\dot{\alpha}\dot{\beta}}\end{array}\right) \text{ ,}$$to raise or lower the spinor indices of gamma matrices (by NW-SE rules). For example, by lowering the second spinor index, we get $$\left( \Gamma _{a}\right) _{\underline{\alpha \beta }}=\left( \Gamma
_{a}\right) _{\underline{\alpha }}^{\ \ \underline{\gamma }}C_{\underline{\gamma \beta }}=\left(
\begin{array}{cc}
0 & \left( \sigma _{a}\right) _{\alpha \dot{\beta}} \\
\left( \bar{\sigma}_{a}\right) _{\dot{\alpha}\beta } & 0\end{array}\right) \text{ .}$$One can check that in this way of construction, the pair spinor indices are symmetric, *i.e.* $\left( \Gamma _{a}\right) _{\underline{\alpha \beta }}=\left( \Gamma _{a}\right) _{\underline{\beta \alpha }}$.
We also define$$\left( \Gamma _{ab}\right) _{\underline{\alpha }}^{\ \ \underline{\beta }}=\left( \Gamma _{\lbrack a}\right) _{\underline{\alpha }}^{\ \ \underline{\gamma }}\left( \Gamma _{b]}\right) _{\underline{\gamma }}^{\ \ \underline{\beta }}\text{ .}$$One can easily check that$$\left( \Gamma _{ab}\right) _{\underline{\alpha \beta }}=\left(
\begin{array}{cc}
\left( \sigma _{ab}\right) _{\alpha \beta } & 0 \\
0 & \left( \bar{\sigma}_{ab}\right) _{\dot{\alpha}\dot{\beta}}\end{array}\right) \text{ .}$$In this way of construction, $\left( \Gamma _{ab}\right) _{\underline{\alpha
\beta }}=\left( \Gamma _{ab}\right) _{\underline{\beta \alpha }}$.
Now we define$$\begin{aligned}
E_{\underline{\alpha \beta }} &=&-\left( \Gamma _{0}\right) _{\underline{\alpha \beta }}=\left(
\begin{array}{cccc}
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
-1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0\end{array}\right) \text{ ,} \label{explicit E} \\[5pt]
J_{\underline{\alpha \beta }} &=&-\left( \Gamma _{12}\right) _{\underline{\alpha \beta }}=\left(
\begin{array}{cccc}
0 & -i & 0 & 0 \\
-i & 0 & 0 & 0 \\
0 & 0 & 0 & i \\
0 & 0 & i & 0\end{array}\right) \text{ .} \label{explicit J}\end{aligned}$$One can for instance check the properties (\[EJProp1\])-(\[EJProp4\]) using the above explicit matrix expressions.
Spacetime gauge function\[Sec coordinates\]
===========================================
The four-dimensional anti-de Sitter spacetime, AdS$_{4}$, of inverse radius $\lambda$, is the hyperbola $$X^{A}X^{B}\eta _{AB}=-\lambda ^{-2}\,\ \text{,}$$ in the five-dimensional space with coordinates $X^A$, $A=\left\{ 0,1,2,3,0^{\prime }\right\} $ and flat metric $\eta _{AB}=$diag$\left\{ -1,1,1,1,-1\right\} $. A set of global coordinates $$\left( t,r,\vartheta ,\phi \right)\ ,\qquad
0\leqslant \lambda t< 2\pi\ ,\quad
r\geqslant 0\ ,\quad 0\leqslant \vartheta \leqslant \pi\ ,\quad
0\leqslant \phi<2\pi\ ,$$ can be introduced by taking $$\begin{aligned}
X^{0} &=&-\sqrt{\lambda ^{-2}+r^{2}}\ \text{sin} \lambda t \text{
\ \ , \ \ \ }X^{0^{\prime }}=-\sqrt{\lambda ^{-2}+r^{2}}\ \text{cos}
\lambda t \text{ \ \ ,} \notag \\
X^{1} &=&r\ \text{sin}\vartheta \ \text{cos}\phi \text{ \ \ , \ \ \ }X^{2}=r\ \text{sin}\vartheta \ \text{sin}\phi \text{ \ \ , \ \ \ }X^{3}=r\
\text{cos}\vartheta \text{ \ \ .}\end{aligned}$$ The resulting induced metric is $$ds^{2}=-\left( 1+\lambda ^{2}r^{2}\right) dt^{2}+\left( 1+\lambda
^{2}r^{2}\right) ^{-1}dr^{2}+r^{2}\left( d\vartheta ^{2}+\text{sin}^{2}\vartheta \ d\phi ^{2}\right) \text{ .}$$ The stereographic coordinates $$x^{\mu }\equiv \delta _{a}^{\mu }x^{a}=\frac{X^{a}}{1+|X^{0'}|}\ ,$$ where $a=\left\{ 0,1,2,3\right\} $, $\eta_{ab} =\ $diag$\left\{ -,+,+,+\right\}$ and $x^{2}:=x^{a}x^{b}\eta_{ab}$, maps the two halves $X^{0'}>0$ and $X^{0'}<0$ of AdS$_4$ to the region $-1<\lambda^2 x^2<1$ of $\Real^{3,1}$. From the inverse relation given by $$X^{a}=\frac{2x^{a}}{1-\lambda ^{2}x^{2}}\ ,\quad X^{0'}=\pm \lambda^{-1}
\frac{1+\lambda^{2}x^2}{1-\lambda^{2}x^2}\ ,$$ it follows that $X^{0'}\rightarrow -X^{0'}$ corresponds to $x^a\rightarrow -(\lambda^2 x^2)^{-1}x^a$. Thus, the extension of the stereographic coordinates $x^a$ to the entire $\Real^{3,1}$ provides a global coordinate of AdS$_4$; the boundary of AdS$_4$ is mapped to the hyperbola $\lambda^2 x^2=1$ in $\Real^{3,1}$.
The gauge function $$L\left( x;y,\bar{y}\right) =\frac{2h}{1+h}\text{exp}\left( \frac{i\lambda }{1+h}x^{\alpha \dot{\alpha}}y_{\alpha }\bar{y}_{\dot{\alpha}}\right)\ ,
\qquad x_{\alpha \dot{\alpha}}:=x^{a}\left( \sigma
_{a}\right) _{\alpha \dot{\alpha}}\ ,\quad h:=\sqrt{1-\lambda
^{2}x^{2}}\text{ ,}$$ which is defined in the region $\lambda^2 x^2<1$, leads to $$U_{\mu }=L^{-1}\star \partial _{\mu }L=-\frac{i}{2}e_{\mu }^{\alpha \dot{\alpha}}y_{\alpha }\bar{y}_{\dot{\alpha}}-\frac{i}{4}\left( \omega _{\mu
}^{\alpha \beta }y_{\alpha }y_{\beta }+\bar{\omega}_{\mu }^{\dot{\alpha}\dot{\beta}}\bar{y}_{\dot{\alpha}}\bar{y}_{\dot{\beta}}\right) \text{ ,}$$ where $$\begin{aligned}
e_{\mu }^{\alpha \dot{\alpha}} &=&-\frac{\lambda \delta _{\mu }^{a}\left(
\sigma _{a}\right) ^{\alpha \dot{\alpha}}}{h^{2}}\text{\ ,} \\
\omega _{\mu }^{\alpha \beta } &=&-\frac{\lambda ^{2}\delta _{\mu
}^{a}x^{b}\left( \sigma _{ab}\right) ^{\alpha \beta }}{h^{2}}\text{\ , \ }\bar{\omega}_{\mu }^{\dot{\alpha}\dot{\beta}}=-\frac{\lambda ^{2}\delta
_{\mu }^{a}x^{b}\left( \bar{\sigma}_{ab}\right) ^{\dot{\alpha}\dot{\beta}}}{h^{2}}\text{ ,}\end{aligned}$$are the vierbein and Lorentz connection of AdS$_{4}$ in stereographic coordinates, with flat indices converted to spinor ones using van der Waerden symbols. One also has $$L^{-1}\star Y_{\underline{\alpha }}\star L=L_{\underline{\alpha }}{}^{\underline{\beta }}Y_{\underline{\beta }}\text{ ,}$$with the matrix $$L_{\underline{\alpha }}{}^{\underline{\beta }}=h^{-1}\left[
\begin{array}{cc}
\delta _{\alpha }{}^{\beta } & \lambda x_{\alpha }{}^{\dot{\beta}} \\
\lambda x_{\dot{\alpha}}{}^{\beta } & \delta _{\dot{\alpha}}{}^{\dot{\beta}}\end{array}\right] \text{ .}$$ As an Sp(4;$\mathbb{R}$) group element, $L(x;y,\bar y)$ corresponds to the transvection in AdS$_{4}$ that sends all the information of the classical solution encoded at the origin of the stereographic coordinate system to the point $x^\mu$.
Determination of Petrov type of spin-2 Weyl tensor\[Sec Petrov type\]
=====================================================================
In this appendix, we briefly explain how to check (only for spin-2) the Petrov type of a Weyl tensor by using the eigenvalue method. For more details on this topic one can check [@Stephani:2003tm].
The restricted Lorentz group SO$^{+}$(3,1,$\mathbb{R}$) is isomorphic to SO(3,$\mathbb{C}$), and a Weyl tensor can be converted into its equivalent form with SO(3,$\mathbb{C}$) indices. We can convert the Weyl tensor $C_{\alpha \beta \gamma \delta }$ with four symmetric SL(2;$\mathbb{C}$) indices into an equivalent tensor $Q_{IJ}$ with two symmetric and traceless SO(3,$\mathbb{C}$) indices, simply by using the Pauli matrices:$$Q_{IJ}=\left( \sigma _{I}\right) ^{\alpha \beta }\left( \sigma _{J}\right)
^{\gamma \delta }C_{\alpha \beta \gamma \delta }\text{ ,}$$where $\left( \sigma _{I}\right) ^{\alpha \beta }=\varepsilon ^{\alpha
\alpha ^{\prime }}\left( \sigma _{I}\right) _{\alpha ^{\prime }}{}^{\beta }$ and we can explicitly choose$$\left( \sigma _{1}\right) _{\alpha }{}^{\beta }=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0\end{array}\right) \text{\ , \ }\left( \sigma _{2}\right) _{\alpha }{}^{\beta }=\left(
\begin{array}{cc}
0 & -i \\
i & 0\end{array}\right) \text{\ , \ }\left( \sigma _{3}\right) _{\alpha }{}^{\beta }=\left(
\begin{array}{cc}
1 & 0 \\
0 & -1\end{array}\right) \text{ .}$$The indices $I,J=1,2,3$ should be raised or lowered by the Kronecker delta, so whether they are upper or lower indices does not make a difference.
If we treat $Q$ as a 3$\times $3 matrix, then observing its eigenvalues and eigenvectors is sufficient for determining its Petrov type. Below we list all Petrov types and their corresponding $Q$-matrix criteria:$$\begin{aligned}
&&\begin{tabular}{cc}
Petrov types & $Q$-matrix criteria \\
I & $\left[ Q-\lambda _{1}I\right] \left[ Q-\lambda _{2}I\right] \left[
Q-\lambda _{3}I\right] =0$ \\
D & $\left[ Q-\left( -\frac{1}{2}\lambda \right) I\right] \left[ Q-\lambda I\right] =0$ \\
II & $\left[ Q-\left( -\frac{1}{2}\lambda \right) I\right] ^{2}\left[
Q-\lambda I\right] =0$ \\
N & $Q^{2}=0$ \\
III & $Q^{3}=0$ \\
O & $Q=0$\end{tabular}
\notag \\
&&\end{aligned}$$In the list, $\lambda _{1,2,3}$, $\lambda $ and $\left( -\frac{1}{2}\lambda
\right) $ are eigenvalues of $Q$, $\lambda _{1}+\lambda _{2}+\lambda _{3}=0$ and $I$ is the identity matrix. In particular, being Petrov type D means the matrix $Q$ has three independent eigenvectors while two of them correspond to equal eigenvalues.
Using the explicit matrices and coordinates provided in Appendices \[Sec gamma\] and \[Sec coordinates\], for spin $s=2$ we can evaluate the Weyl tensor (\[Weyl tensor\]) at a given spacetime point with a chosen set of parameters, and then we can evaluate the corresponding $Q$ matrix to check its Petrov type. We have found that in general the $Q$ matrix has three distinct eigenvalues (type I) and thus is not of type D, unless we choose some special parameters or consider only some special spacetime locations.
[^1]: [email protected]
[^2]: [email protected]
[^3]: For a review, see [@Bekaert:2010hw].
[^4]: The functional class encountered in the quasi-local Fronsdal theory in [@Bekaert:2015tva] (within the AdS/CFT context) has not yet been identified completely; for a discussion, see [@Taronna:2016ats], [@Bekaert:2016ezc] and Section 7 of [@Sleight:2016hyl]. At the cubic order, the separation between the functional classes of this theory and the Vasiliev theory has been spelled out in [@Sleight:2016dba].
[^5]: This subspace is related to the massless spectrum by means of a $\mathbf Z_2$-operation [@Iazeolla:2011cb], reminiscent of a U-duality transformation [@Bossard:2015foa].
[^6]: It remains to be examined whether additional topological two-forms describing Dirac strings need to be activated in the dynamical two-form [@Vasiliev:2015mka; @Boulanger:2015kfa].
[^7]: This phenomena resembles some of the scattering processes in U-duality covariant field theory [@Bossard:2015foa].
[^8]: Whether a more general vacuum gauge function can introduce additional classical moduli remains an open problem.
[^9]: An optional criterion is that the fiber algebra is a unitarizable representation of the higher spin algebra and hence the anti-de Sitter isometry algebra; we expect this property to arise at higher orders of classical perturbation theory by requiring positivity of a suitable free energy functional.
[^10]: For recent reformulations containing the original Vasiliev system as consistent truncations, see [@Vasiliev:2015mka; @Boulanger:2015kfa].
[^11]: Taking the master fields to be smooth functions of ${\cal Y}_4$ yields an anti-de Sitter analog of the Penrose-Newman transformation; to our best understanding, the precise relation between ${\cal Y}_4\times{\cal Z}_4$ and the original (commuting) twistor space of Penrose remains to be spelled out in detail.
[^12]: The doublet indices are raised and lowered using $f^{\alpha }=\varepsilon ^{\alpha \beta }f_{\beta }$ and $f_{\beta}=f^{\alpha }\varepsilon _{\alpha \beta }$ *idem* $f^{\dot{\alpha} }$.
[^13]: At the linearized level, this gauge yields the canonical basis for unfolded linearized Fronsdal fields [@Vasiliev:1990en]; for further details, see [@Sezgin:2002ru] and the review [@Didenko:2014dwa]. Beyond the linearized approximation, it has been used in amplitude computations [@Giombi:2009wh; @Giombi:2010vg; @Colombo:2010fu; @Colombo:2012jx] and related recent works[@Vasiliev:2015mka; @Vasiliev:2016sgg]. Most exact solutions found so far, however, have been given in other gauges argued to be equivalent to Vasiliev gauge; for example, see [@Sezgin:2005pv; @Didenko:2008va; @Iazeolla:2011cb].
[^14]: The resulting manifestly Lorentz covariant form of the master field equations can be found in [@Iazeolla:2011cb; @Colombo:2010fu].
[^15]: Whether the resulting system admit any consistent truncation to a pure higher-derivative gravity theory remains an open problem.
[^16]: The multiplication table of ${\cal A}_{\cal S}$ may involve fusion rules [@Iazeolla:2011cb; @Boulanger:2013naa], which stipulate which pairs of basis elements that have nontrivial star products and which basis elements that are to be used to expand the result.
[^17]: We denote the star product inverse of $G$ by $G^{-1}$, that is, $G\star G^{-1}=1$.
[^18]: The reality condition and bosonic projection of a gauge function $G$ takes the form $G^{\dag }=G^{-1}$ and $\pi \bar{\pi}\left( G\right)= G$.
[^19]: We use a convention such that if ${\cal B}'=b_0$ then $\mu _{\sigma; \vec\l}=b_0 \nu _{\sigma; \vec\l}$ and $\check
\mu _{\sigma; \vec\l}=b_0 \check
\nu _{\sigma; \vec\l}$.
[^20]: The confluent hypergeometric function ${}_1F_1(a;b;x):=\sum_{n=0}^\infty \frac{(a)_n x^n }{(b)_n n!}$ obeys $0<{}_1F_1(a;b;x)< e^x$ for $b>a>0$ and $x>0$. Its asymptotic form for large $|x|$ is given by ${}_1F_1(a;b;x)\sim
\frac{\Gamma(b)}{ \Gamma(a)} x^{a-b} e^x +
\frac{\Gamma(b)}{ \Gamma(b-a)} (-x)^{-a}$.
[^21]: Inequivalent exact solution spaces can be obtained by replacing $E$ and $J$ by Cartan subalgebra generators in ${\rm sp}(4;\Comp)$, which we leave for future work.
[^22]: We have suppressed the dummy indices, which are contracted using the north-west to south-east convention.
[^23]: Eqs. – hold, if we label all components with either “$E$” or “$J$” (not a mixture of both).
[^24]: We note the useful relations $\mathbf{F}^{2}:= \frac{1}{2}\mathbf{F}_{\alpha \beta }(\mathbf{F})^{\alpha
\beta }=\text{det}\left( \mathbf{F}\right)$ and $
\mathbf{F}_{\alpha
\beta }\mathbf{F}^{\beta \gamma }=\mathbf{F}^{2}\delta _{\alpha }{}^{\gamma }$, *idem* $\mathbf{H}$.
[^25]: A generalized spin-$s$ Petrov type-D tensor is defined as a symmetric rank-$2s$ tensor with spinor indices that can be decomposed into the products of two spinors, each of which has the power $s$ [@Iazeolla:2011cb].
[^26]: One can show that $\left(\kappa_E^{L}\right)^2=-\lambda^2 r^2$. The analytical continuation involves the choice of sign in front of the square roots. These must be correlated to analogous choices in the expression for the twistor space connection.
[^27]: Note, however, the matrix $\mathbf{K}_{\underline{\alpha \beta }}\equiv \left[ \mathbf{B}E_{\underline{\alpha \beta }}+\mathbf{C}J_{\underline{ \alpha \beta }}\right] $ in this special case has determinant $\left(
\mathbf{B}^{2}-\mathbf{C}^{2}\right) ^{2}=0$, which has consequences for the twistor space connection; see Eq. .
[^28]: See Appendix \[Sec Petrov type\] for details on spin-2.
[^29]: Removing the unity from the presentation of ${\cal A}$ also removes a singularity from $\Phi^{(L)}$.
[^30]: On the equatorial plane, for a certain value of $\tau$ between the integration limits, zero-denominators appear in the integrand of both on the exponent and in the factor in the front. We leave the consequence of this for future work.
[^31]: This suggests that in more general perturbatively defined solutions to Vasiliev’s equations obtained by repeated homotopy integration [@Vasiliev:1990en; @Sezgin:2002ru] (see also [@Didenko:2014dwa]), the resulting auxiliary integrals should be performed prior to taking the limit $z^\alpha\rightarrow 0$; whether this prescription is actually unique and correct, remains to be investigated.
[^32]: If $H^{(1)}\vert_{Z=0}$ and the gauge parameters belong to the same class of functions then $H^{(1)}\vert_{Z=0}$ describes pure gauge degrees of freedom.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In the Cosmological Slingshot Scenario, our Universe is a $D3$-brane that extends in the $4d$ noncompact directions of a warped Calabi-Yau compactification of IIB Supergravity. Early time cosmology corresponds to a period in which the brane moves inside a warped throat where a non-vanishing angular momentum ensures that the trajectory of the brane has a turning point. The corresponding induced metric on the $D3$-brane experiences a cosmological evolution with a bounce. In this framework, the homogeneity, flatness, and isotropy problems of standard cosmology might be avoided. The power spectrum of primordial perturbations of the brane embedding can be found and it is shown to be in agreement to WMAP data.'
author:
- Cristiano Germani
- Nicolás Grandi
- Alex Kehagias
title: 'The Cosmological Slingshot Scenario: a Stringy Proposal for the Early Time Cosmology[^1]'
---
[ address=[SISSA and INFN - via Beirut 4, 34014 Trieste, Italy - [email protected]\
]{} ]{}
[ address=[IFLP, CCT La Plata, CONICET - Dto. de Física, UNLP\
Casilla de Correos 67, CP1900 La Plata, Argentina - [email protected]\
]{} ]{}
[ address=[Department of Physics, National Technical University of Athens\
GR-15773, Zografou, Athens, Greece - [email protected]]{} ]{}
Introduction
============
The Cosmological Slingshot Scenario [@Germani:2006pf; @Germani:2007ub; @Germani:2007uc] is a proposal for the cosmic early-time evolution in the String Theory context. According to that, our Universe is a $D3$-brane moving in a String Theory background of the form ${\cal M}^4\times K^6$. ${\cal M}^4$ is a warped Minkowskian space-time and $K^6$ is a compact Calabi-Yau (CY) space. The latter includes a throat sourced by a stack of a large number ($N$) of other $D3$-branes. The Slingshot is characterized by a non-trivial orbital motion of the Universe in the compact space around the stack of $D3$-branes. If back-reaction can be neglected (probe brane approximation), a brane observer measures a $4d$ metric induced in terms of the brane embedding, that defines a cosmological brane evolution commonly called Mirage Cosmology [@Kehagias:1999vr].
The early-time evolution ([*i.e.*]{} well before nucleosynthesis) corresponds to the motion of the $D3$-brane deep into the throat (Slingshot era) moving towards the hat of the compact space. Since $N$ is taken to be large, close to the stack the probe brane approximation can be used. The late-time cosmology starts when the $D3$-brane reaches the hat of the CY, the probe brane approximation breaks down and local gravity [*à la*]{} Randall-Sundrum [@Randall:1999vf; @Shiromizu:1999wj] dominates the cosmological evolution. Under this approximation, the Slingshot brane observer experiences a non-singular bouncing cosmology. Additionally, as we shall show later on, the Standard Cosmology problems ([*i.e.*]{} homogeneity, isotropy and flatness) might be avoided in the brane induced cosmology.
The Cosmological Slingshot Scenario {#sKS}
===================================
Setup
-----
To make the discussion concrete, we will consider a probe $D3$-brane moving in a throat of a Calabi-Yau (CY) compact manifold, whose metric and Ramond-Ramond $5$-form can be written as $$\begin{aligned}
ds^2=h^{-\frac{1}{2}}ds^2_{\mbox{\tiny ${\cal M}_4$}} +
h^{\frac{1}{2}}\left(dr^2+r^2 ds^2_5)\right)\ ,\ \ \ \ \ \ \ C_{0123}=1-\frac1h\,,
\label{metrik}\end{aligned}$$ where $h$ is a function of $r$ only, and $ds^2_5$ is the base manifold characterizing the remaining part of the transverse space.
The dynamics of a probe brane is governed by the Dirac-Born-Infeld action with the Wess-Zumino coupling $$\begin{aligned}
S_{DBI}+S_{WZ}= -T_3\int \sqrt{-g_i}\,d^4\xi - T_3\int C_{(4)}
\, . \label{actionn}\end{aligned}$$ We assume that all other fields on the brane are switched off and matter is created later. The sign of the Wess-Zumino term has been chosen to represent a $D3$-brane and $T_3=1/{(2\pi)^3g_sl_s^4}$ is the tension of the probe. The probe brane is extended along the ${\cal M}_4$ directions, so that it looks like a point particle moving in the transverse space. In the static gauge the resulting induced metric is $$ds_i^2=h^{-1/2}\left[-\left(1-h(r'^2+r^2\Omega_5'^2){\phantom{z\!\!}}\right)dt^2+ d\vec x\cdot d\vec x\right]\ ,
\label{induced}$$where a prime $(')$ denotes a derivative with respect to $t$ and we have assumed that the only non-vanishing transverse momenta are in the $r$, $\Omega_5$ directions, $\Omega_5$ representing an angle in the transverse space. Replacing this induced metric into the brane action (\[actionn\]) we get $$\begin{aligned}
S&=&
-T_3V_3
\int dt \,\left[\frac1h\sqrt{1-h\left(
r'^2+
r^2\Omega_5'^2
\right)}+1-\frac1h\right]\ , \label{dbikt}\end{aligned}$$ where $V_{3}$ is the un-warped volume of the directions parallel to the probe. The resulting equations of motion have first integrals provided by $$U = \frac1h\left[\frac1{ \sqrt{1-h(r'^2+r^2\Omega_5'^2)} }-1\right]
\,,\ \ \ \ \ \ \ \
J= \frac {r^2}{\sqrt{1-h(r'^2+r^2\Omega_5'^2)}}\,\Omega_5'\ ,
\label{enn}$$ that can be inverted to get $$r'^2=\frac1{(1+hU)^2}\left(2U-\frac{J^2}{r^2}+hU^2\right)\, .
\label{pin}$$ The motion will take place at the values of $r$ that make this expression positive. Moreover wherever the expression in parenthesis vanishes, the trajectory will have a turning point.
To interpret the induced metric (\[induced\]) as the cosmology experienced by an observer on the brane, we need to define the cosmic time according to $$d\tau=h^{-1/4}\sqrt{1-h(r'^2+r^2\Omega_5'^2){\phantom{z\!\!}}}dt\,,$$ in terms of which the metric (\[induced\]) is identified as a Friedman-Robertson-Walker metric with scale factor $$\label{scale}
a=h^{-1/4}(r) \ .$$ When $r(t)$ has a turning point, it is easy to see that the same happens to $a(\tau)$, generating a nonsingular bouncing cosmology. A Friedmann equation can be written for such cosmology by changing to cosmic time in Eq.(\[pin\]) and dividing the result by $a^2$ $$H^2 =\frac1{16} \,a^6h_r^2(a) \left[2U-\frac{J^2}{r^2(a)}+\frac{U^2}{a^4}\right] \ ,
\label{hub}$$ where $H=\dot a/a$ is the (mirage) Hubble constant (here $\dot a=\partial_\tau a$).
The model is completed by smoothly pasting this Mirage era to a local gravity driven late evolution when the brane reaches the top of the CY and gravity becomes localized [*á la*]{} Randall-Sundrum [@Randall:1999vf; @Shiromizu:1999wj]. There, the standard late time evolution of the observed Universe is supposed to be well reproduced by the brane dynamics. This assumption involves a transition from a mirage dominated era with a moving brane without any matter, into a local gravity dominated era with an static brane and matter fields excited on it. This transition has to be understood as an analogous of the reheating process in standard inflationary models. It entails a dynamical mechanism under which the kinetic energy of the brane is passed to matter fields. The description of this dynamics as well as the robustness of our predictions for physical observables is an open point that is left for future research.
A concrete example of the above proposed situation is given by a probe moving in the $AdS_5\times S_5$ background. In it, the metric takes the form (\[metrik\]) with $$h_{AdS}=\frac{L^4}{r^4}\, , \ \ \ \ \ \ \ \ L^4=4\pi l_s^4 N g_s\, ,$$ where $l_s$ is the string length and $g_s$ is the string coupling. The supergravity approximation is valid as long as the curvature radius of the solution is large compared to the string length $l_s$. String perturbation theory on the other hand requires $g_s\ll 1$.
A close look to Eq.(\[pin\]) shows that, in the present case, the second factor is a quadratic function of the variable $r^{-2}$, that will have a root as long as its discriminant is positive $J^4-8U^3L^4>0$. For values of $r$ larger that that of the root the function is positive. Then whenever this inequality is satisfied, the probe orbit has an inner turning point.
The resulting scale factor (\[scale\]) reads $$a_{AdS}=\frac rL\,.
\label{Ads}$$ In this background, the induced Friedmann equation (\[hub\]) becomes $$H_{AdS}^2=\frac1{L^2}\left[\frac{2U}{a^4_{AdS}}-\frac{(J/L)^2}{a^6_{AdS}}+\frac{U^2}{a^8_{AdS}}\right]\ ,
\label{hub2}$$where $H_{AdS}=\dot a_{AdS}/a_{AdS}$ is the (mirage) Hubble constant. Since when $J^4-8U^3L^4>0$ the orbit has a turning point, in that case the corresponding cosmology has a bounce.
Another example is that of a probe brane motion in a Klebanov-Strassler (KS) throat [@Klebanov:2000hb]. In the region far from the tip of the throat, KS geometry can be well approximated by the Klebanov-Tseytlin (KT) metric [@Klebanov:2000nc] that takes the form (\[enn\]). The warp factor reads $$h_{KS}=\frac{L^4}{r^4}\ln(\frac r{r_s})\, , \ \ \ \ \ \ \ \ L^2=(9/\sqrt8)\,
l_s^2 M g_s\, ,$$where $r_s$ is proportional to the radius of the blown up sphere at the tip of the cone. To trust the KT approximation we need to ensure that the probe brane will never reach $r\simeq r_s$. This will be the case iff $r_s$ lies inside the forbidden region $r'^2<0$. Going back to equation (\[pin\]) and evaluating it at $r_s$, we see that this is true when $2r_sU-J^2<0$, the probe motion having a turning point at some value of $r$ larger than $r_s$.
The resulting scale factor is $$a_{KS}=\frac{r}{L} \,\ln^{-1/4}(\frac r{r_s})\, ,
\label{ver}$$ and, under the assumption $2r_sU-J^2<0$, it corresponds to a bouncing cosmology.
An important ingredient in our argument is that the scale factor for a brane moving in a KS throat (\[ver\]) can be rewritten as a conformal re-scaling of the corresponding scale factor for a brane moving in $AdS_5$. Indeed, when written in conformal time, the induced metric on the brane reads $ds^2_i= a^{-2}ds^2_{\mbox{\tiny ${\cal M}_4$}} $ and we can write $$a_{ KS} = \Omega(a_{AdS})\,a_{AdS}\,, \ \ \ \ \ \ \ \ \mbox{with}\ \Omega(a_{AdS}) = \log^{\!\!-1\!/4}\!(a_{AdS}L/r_s)
\label{ads}\ .$$It should be kept in mind that our approximations are valid whenever $a_{AdS}\gg r_s/L$. Under such a re-scaling, the Hubble constant changes as $$H_{KS} = \left(1+a_{AdS}\frac{d\ln\Omega}{da_{AdS}}\right)H_{AdS}
\label{hubblee}\ .$$
The Problems of Standard non-Inflationary Cosmology
---------------------------------------------------
We are now ready to study how standard cosmological problems are solved in the Slingshot scenario.
.2cm \[intro\] [**[A. Homogeneity.]{}**]{} As explained above, in both the $AdS$ case and the KS throat the probe brane experiences a bounce in the String frame. This immediately ensures that homogeneity problem is solved. To check this explicitly, we write the co-moving horizon $$\Delta\eta = \int_{\eta_i}^{\eta_0}d\eta\, ,$$where $\eta$ is conformal time and $\eta_i$ is its smallest value. To solve homogeneity problem it is required that $\Delta\eta > H_0^{-1}$. Since we have $\eta_i\to-\infty$ due to the absence of a cosmological singularity, this condition is trivially satisfied.
.2cm [**[B. Isotropy.]{}**]{} In the $AdS$ case, mirage matter contributes to Friedmann equation (\[hub2\]) with a term $\rho\sim a^{-8}_{AdS}$. This term dominates over the shear $\rho_{shear}\sim a^{-6}_{AdS}$ at early times, avoiding the chaotic behavior [@Erickson:2003zm]. To check whether this is true in the KS case, we should verify that the corresponding mirage contribution dominates over the shear. The form of this contribution can be read from (\[hubblee\]), and we can write the quotient $$\sqrt{\frac{\rho_{shear}}{\rho}}\propto\left(1+a_{AdS}\frac{d\ln\Omega}{da_{AdS}}\right)^{-1}\frac {a_{AdS}}{\Omega^3}\ .
\label{shear}$$ The proportionality constant in (\[shear\]) parameterizes the anisotropic perturbations in the pre-bounce era. It is simple to check that (\[shear\]) is an increasing function of $a_{AdS}$ in the region $a \gtrsim e^{((2\sqrt{2}-1)/4)} \,r_s/L\simeq1.57 \,r_s/L$. As we assumed that the Slingshot brane never approaches the tip of the KS throat, this condition is automatically satisfied. Therefore, ${\rho_{shear}}/{\rho}$ decreases very rapidly close to the bouncing point in the pre-bounce era, solving the isotropy problem.
.2cm [**[C. Flatness.]{}**]{} The curvature contribution to the Hubble equation[^2] can be disregarded if the quantity $|\Omega_{\mbox{\tiny Total}}-1|\,=\,{1}/{a^2H^2}$ passes through a minimum where it satisfies the phenomenological constraint $$|\Omega_{\mbox{\tiny Total}}-1|_{min}<10^{-8}
\, .\label{cons}$$
For the $AdS$ case, the above quantity evaluated at its minimum reads $$|\Omega_{\mbox{\tiny Total}}-1|_{min}=
\frac{(J^2+\sqrt{J^4-6L^4U^3})^3}{4L^2U^2(J^4-4L^4U^3+J^2\sqrt{J^4-6L^4U^3})}
\simeq
\left(\frac{J}{2LU}\right)^2\!\!+\!{\cal O}\!\left(\!\frac{L^4U^3}{J^4}\!\right)
<10^{-8}.$$ This condition is not a fine tuning in parameter space, but just a restriction to a two dimensional region. In this sense flatness problem might be alleviated in the Slingshot scenario.
For the KS case we have, after conformal re-scaling $$|\Omega_{\mbox{\tiny Total}}-1|=\frac {f^2}{a_{AdS}^2H_{AdS}^2}\ ,
\ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \
f=\frac{4\ln(a_{AdS} L/r_s)}{4\ln (a_{AdS} L/r_s)-1}\ .
\label{f1}$$ The KT approximation is valid for $r_{min}\gg r_s$; to fix ideas we will use $r_{min}>10^2\ r_s$. In this region we have $f={\cal O}(1)$ and decreasing in $a_{AdS}$. Consequently, the flatness problem in the KS space might, in good approximation, be alleviated by the same choice of parameters used in the $AdS$ case.
Primordial Perturbations {#sPert}
========================
The Hollands-Wald Mechanism
---------------------------
In inflationary scenarios the primordial perturbations are produced by quantum fluctuations of the inflaton field and are codified into its two point correlation function in the vacuum state. However, these fluctuations are over-damped by the expansion of the Universe at super-horizon scales. At these scales then, the quantum state becomes characterized by a large occupation number and the system collapses into a classical state. This classical state represents a random spectrum of perturbations whose variance is given by the quantum correlations evaluated at the quantum-to-classical transition point [@Liddle:2000cg].
Let us now turn our attention into the mechanism proposed by Hollands and Wald in [@Hollands:2002yb]. A perturbation of wavelength $\lambda$ smaller than a typical quantum scale, say $l_c$, is in its quantum vacuum. In an expanding background, the wavelength of a perturbation grows in time ($\lambda\propto a$) and whenever $\lambda\sim l_c$, or in other words, as soon as the perturbation becomes macroscopic, wavelengths bigger than the horizon scale collapse into a classical random state. In the proposal of [@Hollands:2002yb], the relevant fluctuations are so continuously “created” at “super-horizon” scales. Thus, a coherent spectrum of classical perturbations is produced with variance given by matching the classical correlations with the quantum correlations at the quantum-to-classical transition point.
It has been suggested [@Li:1996rp] that a space-time uncertainty relation $\Delta X\Delta T\gtrsim l_s^2$ should be realized in String Theory, $\Delta X$ and $\Delta T$ representing the uncertainties in measuring space and time distances. Since the smallest length that can be probed in String Theory is the $11$-dimensional Planck length $\Delta X>l_{P^{11}}\sim g_s^{\mbox{\tiny$1/3$}}l_s$, we obtain that the smallest measurable time is $\Delta T\gtrsim g_s^{\mbox{\tiny$-1/3$}}l_s$. The period of a wave propagating in a $D$-brane is $2\pi\omega^{-1}\sim \lambda$ and cannot be smaller than that $\Delta T$, implying $$\lambda>l_s g_s^{-1/3}\ .\label{second}$$ We therefore have a minimal wavelength for a perturbation on the brane as in the Hollands-Wald mechanism. This strongly suggests to use such mechanism to study the cosmological perturbations in the Slingshot model[^3]$^,$[^4].
Technically, the mechanism explained above introduces a vacuum state in which the perturbation is destroyed (coming from the pre-bounce era) and then created again (after the bounce) at the time $\eta_*$ in which the proper wavelength of the corresponding quantum mode reaches the value $$a(\eta_*)/k\equiv a_*/k = l_c\, . \label{lc}$$ We start by perturbing the embedding of the probe brane by writing $r = r(\eta)+\delta r(\eta,\vec x)$ and $\Omega_5 = \Omega_5(\eta)+\delta \Omega(\eta,\vec x)$, where $r(\eta),\Omega_5(\eta)$ are the solutions of the equations of motion obtained from action (\[actionn\]), written as functions of the conformal time $\eta$. In the non-relativistic approximation $hU\ll1$ we have $\eta\equiv t$ and Eqs.(\[enn\]) are integrated to $$r(\eta)=\sqrt{2U\,\eta^2+\frac{J^2}{2 U}}\, ,
~~~~~\Omega_5(\eta)=\arctan\left(\frac{2U}{J}\,\eta\right)\, .\label{8}$$Note that we have a turning point at $r_{min}=J/\sqrt{2U}$.
In what follows, we will use as our variable the Bardeen potential $\delta \Phi_k = \delta r_k/r$ [@Durrer:1993db], in terms of which the action (\[actionn\]) can be expanded to quadratic order in $\delta$’s and their derivatives, getting (in Fourier space) $$\!S\!=\!T_3\sum_k\!\int
d\eta \left( \phantom{\frac12}\!\!\! \! \! \! \! \! \frac{r^2}2\! \left(\! {\delta
\Phi'}_k ^2 + \delta {\Omega}'^2_k -k^2( \delta\Phi_k^2
+\delta\Omega_k^2) \right)+ J\,\delta {\Omega}_k'\delta \Phi_k
-J\,\delta {\Omega}_k\delta \Phi'_k \right) . \label{dos}$$
Power Spectrum and Spectral Index
---------------------------------
The canonical quantization procedure applied to the action (\[dos\]) provide the normalized operators $$\begin{aligned}
&&\delta\hat\Phi_k=u_1 \hat a_1+u_2 \hat a_2+c.c.\,, \ \ \ \ \ \ \ \ \ \ \delta\hat \Omega_k=v_1 \hat a_1+v_2\hat a_2+c.c.\ ,
\label{quant}\end{aligned}$$ where $a_i,a_i^\dag$ are standard annihilation and creation operators, and $$\begin{aligned}
&&\!\!\!\!\!
u_1=\sqrt{\frac{U}{k T_3}}\, \frac{\eta}{r^2}\,e^{-ik\eta}\, ,
~~~~~~~~~\ \ \ \ \ \ \ \ u_2=\sqrt{\frac{1}{U k T_3}}\, \frac{J}{2r^2}\,e^{-ik\eta}\,,\\
&&\!\!\!\!\!
v_1=u_2=\sqrt{\frac{1}{U k T_3}}\, \frac{J}{2r^2}\,e^{-ik\eta}\,,
~~\ \ v_2=-u_1=-\sqrt{\frac{U}{k T_3}}\, \frac{\eta}{r^2}\,e^{-ik\eta}\ .
\label{quantII}\end{aligned}$$ We are interested in the correlation of the Bardeen potential $\delta \hat \Phi$ at the time of creation $\eta_*$. Using the above formulas to define $r_*=r(\eta_*)$, it is straightforward to check that $$\langle\delta\hat\Phi_k\delta\hat\Phi_{k'}\rangle=\delta_{k,k'}\frac{1}{2kT_3r_*^2}\ .$$ We will consider the transition point of the quantum to the classical description in the region in which $k\ll J/r^2$. In this limit, we can discard the $k^2$ term in the action (\[dos\]) and write its classical solutions as $$\begin{aligned}
\label{solutions}
\delta\Phi_k=\frac{C_k}{2J}+A_k\sin\left(2\theta+\phi_k\right)\ ,\ \ \ \ \ \
\delta\Omega=-\frac{D_k}{2J}+A_k\cos\left(2\theta+\phi_k\right)\ ,\end{aligned}$$ where $\theta=\Omega_5(\eta)-\Omega_5(\eta_*)$ and $\phi_k,C_k,D_k,A_k$ are constants of integration, that can be written as $$\begin{aligned}
C_k=r^2\delta\Omega_k'+2J\delta\Phi_k\ ,\ \ \ \ \ \ \ \,
D_k=r^2\delta\Phi_k'-2J\delta\Omega_k\, ,
\nonumber\\
A_k=\frac{r^2}{2J}\left[\delta\Phi_k'\cos\left(2\theta+\phi_k\right)-\delta\Omega_k'\sin\left(2\theta+\phi_k\right)\right]\ .\end{aligned}$$ We now consider initial conditions arising from the matching of the classical to the quantum system at the time $\eta=\eta_*$. Therefore $C_k,D_k,A_k$ will be taken as Gaussian stochastic variables with correlations $\langle...\rangle_c$ matching the quantum correlators $\langle...\rangle$ at $\eta=\eta_*$.
Using the quantum solutions described above at the matching point $\eta=\eta_*$ after a lengthly but straightforward calculation we have $$\begin{aligned}
&&\langle C_{k} C_{k'}\rangle_{c}=\delta_{k,k'}\frac{k^2r_*^2+2U}{2kT}\ ,\ \ \ \ \ \ \
\langle A_{k} A_{k'}\rangle_{c}=\delta_{k,k'}\frac{k^2r_*^2+2U}{8J^2kT}\ ,
\nonumber\\&&
\langle A_{k} C_{k'}\rangle_{c}=\delta_{k,k'}\frac{\sin\phi}{4kTJr_*^2}\{2J^2-2Ur_*^2-k^2r_*^4
-4JU\eta_*\cot\phi_k\}\ .\end{aligned}$$ The matching of $$\langle \delta\Phi_k\delta\Phi_{k'}\rangle_{c}=\langle \delta\hat\Phi_k\delta\hat\Phi_{k'}\rangle\ ,$$requires $\phi_k=\pi/2$; this is the selection of positive frequencies.
In general correlators depend on time through $\theta$. However in the region $k\ll J/r^2$, the oscillation rapidly stabilizes in time when $2U\eta_{\mbox{\tiny asymp.}}>2\pi J$. We will consider this to happen well before the nucleosynthesis. At this time then $$\delta\Phi_k=\frac{C_k}{2J}-A_k\cos(2\Omega_5(\eta_*))=\frac{C_k}{2J}+A_k\left(1-\frac{2r_{min}^2}{r_*^2}\right)\ .$$Using the initial conditions found above we then get in the limit $k\ll J/r_*^2<J/r_{min}^2$ $$\langle \delta\Phi_k\delta\Phi_{k'}\rangle_{c}\Big|_{\eta>\eta_{\mbox{\tiny asymp.}}}\simeq \frac{\delta_{k,k'}}{2kTr_*^2}\left[1-
\left(\frac{r_{min}}{r_*}\right)^2\right]\ ,$$so the power spectrum of temperature fluctuations is $$P(k)\simeq\frac{1}{2k\,T_3\,r_*^2}\left[1-
\left(\frac{r_{min}}{r_*}\right)^2\right]\ . \label{power}$$A consistency condition for the production of the perturbation is that $r_{min}<r_*$. So we see that in the limit $r_{min}\ll r_*$ we obtain the power spectrum introduced in [@Germani:2006pf]. .2cm Since we assumed that a perturbation is created when its physical wavelength reaches a fixed value $l_c$, we have from Eq.(\[lc\]), $k l_c= a_*$. In the $AdS$ metric Eq.(\[Ads\]) implies $kl_cL=r_*$, resulting in the power spectrum $$P(k)\simeq\frac{1}{2\,T_3(l_cL)^2 \,k^3}\left[1-\frac{r_{min}^2}{(l_c
L)^2\,k^2}
\right]\, , \label{powerAds}$$for which the scalar spectral index $n_s-1=d\ln (k^3P(k))/d\ln k$ reads $$n_s\simeq1+\frac{2}{\frac{(l_cL)^2}{r_{min}^2}k^2-1}\, \label{indexAds}$$and we see that the flat spectrum found in [@Germani:2006pf] is blue-shifted by the subsequent time evolution. .2cm In the Klebanov-Tseytlin (KT) metric on the other hand, the condition $kl_c=a_*$ is solved by $r_* = r_s e^{{-\,W_{-1}(-\zeta)}/4}$ where $\zeta=4(r_s/Ll_ck)^4\leq
e^{-1}$ and $W_{-1}(x)$ is the negative branch of Lambert’s $W$-function. Then the power spectrum (\[power\]) is explicitly written as $$P(k)=\frac{1}{2T_3\, k \, r_s^2} \,e^{{\frac12
W_{-1}(-\zeta)}}\!\left(1\!-\!\left(\frac{r_{min}}{r_s}\right)^2e^{\frac12
\,
W_{-1}(-\zeta)}\right)\,,$$whereas the scalar spectral index turns out to be $$\begin{aligned}
n_s &=& 1 + \frac{2}{1+W_{-1}(-\zeta)}\!\left(1-\frac{W_{-1}(-\zeta)}{1\!-\!({r_s}/{r_{min}})^2e^{-\frac12W_{-1}(-\zeta)}}\right)
\nonumber\\
&\simeq& 1 +
\frac{2}{\ln(\zeta)}-\frac{2\sqrt\zeta}{\sqrt\zeta-({r_s}/{r_{min}})^2}\,
,
\label{hi}\end{aligned}$$where the expansion of the Lambert W function for small argument $W(-\zeta)\simeq \ln(\zeta)+\cdots$ was used in the second line. Since in this limit $\ln(\zeta)<0$, the first correction on $n_s$ is negative. On the other hand, the second correction is red or blue according to the sign of its denominator. It will be negative whenever $$\sqrt\zeta>{r_s^2}/{r_{min}^2}\, ,
\label{hola}$$from which we immediately see that long wavelengths are red-shifted.
If the last term is instead positive, then $\sqrt\zeta<{r_s^2}/{r_{min}^2}$ and the overall sign of the correction has to be evaluated taking into account the joint contribution of both terms in (\[hi\]). After some manipulations we find that the correction is red whenever $${\sqrt\zeta}\left(1-2\,{\log\!\sqrt\zeta}\right)<({r_s}/{r_{min}})^2\, ,$$from which we conclude that short wavelengths are also red-shifted, and there is an intermediate range of wavelengths that is blue-shifted.
Back-Reaction and Effective 4d Theory
=====================================
The 4D effective theory for warped compactifications of IIB supergravity with (static) D-branes has been derived by a perturbative approach in [@GM], and by a gradient expansion method in [@KK2]. Using these results, in [@Germani:2007ub] the following effective 4d Lagrangian describing Slingshot cosmology has been found $$\! \!\! \! \! \! \!\! \! \! \! \!\!\!\!\!\!\!\!\! S_{brane}=\frac{L^2}2\int d^4x \sqrt{-g}\left[\left(\frac{1}{\kappa^2r^2}-
\frac{T_3}{6 N}\right)R+\frac{6}{\kappa^2}\frac{(\nabla r)^2}{r^4}-\frac{T_3}{N}(\nabla\Omega_5)^2\right]\ .
\label{eff2}$$ The resulting equations of motion, specialized to a Friedmann-Robertson-Walker background with scale parameter $a(\eta)$, result into the following set of equations $$\begin{aligned}
&&\frac{r''}{r}+2\frac{r'}{r}\left(\frac{a'}{a}-\frac{r'}{r}\right)+{\Omega'^2_5}=0\ ,
\ \ \ \ \ \ \frac{d~}{d\eta}\!\!\left(a^2\Omega'_5\right)=0\,,
\nonumber\\
&&\frac{T_3\kappa^2}{N}\left(\frac{a'^2}{a^2}+\Omega_5'^2\right)=
\,\frac{6}{r^2}\!\left(\frac{a'}{a}-\frac{r'}{r}\right)^2\!.
%\nonumber \\\end{aligned}$$ As can be checked by direct substitution, an exact solution of the full system is $$\begin{aligned}
a=\frac1L\sqrt{\frac{J^2}{2U}+{2U}\eta^2}\ ,\ \ \ \ \ \ \Omega_5'=\frac{J}{L^2}\,\frac1{a^2}\,, \ \ \ \ \
r=\frac{a\,L}{1+\kappa\sqrt{\frac{UT_3}{3N}}\,\eta}\ .\end{aligned}$$ We obtained the same scale factor evolution as in the mirage approximation in [@Germani:2007ub], but now considering local gravity back-reactions. The only difference from the mirage approximation is on the $r$ evolution, due to the denominator $1+ \kappa\sqrt{UT_3/3N}\eta$. Since $\kappa\sqrt{UT_3/3N}$ is supposed to be small, the mirage approximation breaks down at very late or very early times, [*i.e.*]{} when the brane leaves the throat, as expected. When this denominator is taken into account, the system is no longer time symmetric and there is a field singularity at the time in which the denominator vanishes. However, since the extra-dimensional space is compact $r<r_{max}$, where $r_{max}$ defines the cut-off of the compact space, this singularity is just fictitious, and the effective action cannot be trusted when $1+ \kappa\sqrt{UT_3/3N}\eta\simeq 0$. There a description [*á la*]{} Randall-Sundrum [@Shiromizu:1999wj] must be used.
Summary
=======
The Cosmological Slingshot scenario provides an example of a bouncing cosmology in which the dynamics of the bounce is under control. It provides alternative solutions to the problems of standard cosmology, and the resulting perturbations spectrum is in agreement with WMAP data. In the way it has been presented here it still has to be checked that all the constraints, that appeared during our calculations due to our approximations and to phenomenological inputs, are mutually compatible. Even if in [@Germani:2006pf] some of these cross checks have been performed successfully, a complete cross-checking is still needed, and it may be the case that all the problems of standard cosmology cannot be solved at the same time. For example, to make our solution of flatness problem compatible with perturbation spectrum, a very strong lower bound on the conserved quantity $U$ is needed. The model is currently under research.
Acknowledgements {#acknowledgements .unnumbered}
================
C.G. wishes to thank Cliff Burgess and Toni Riotto for useful discussions. This work is partially supported by the European Research Training Network MRTN-CT-2004-005104 and the PEVE-NTUA-2007/10079 programme. N.E.G. wants to thank Department of Particle Physics of Santiago de Compostela University for hospitality during part of this work.
[10]{}
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S. Hollands and R. M. Wald, “An alternative to inflation,” Gen. Rel. Grav. [**34**]{}, 2043 (2002) \[arXiv:gr-qc/0205058\].\
S. Hollands and R. M. Wald, “Comment on inflation and alternative cosmology,” arXiv:hep-th/0210001. M. Li and T. Yoneya, “D-particle dynamics and the space-time uncertainty relation,” Phys. Rev. Lett. [**78**]{}, 1219 (1997) \[arXiv:hep-th/9611072\]. R. Brandenberger, H. Firouzjahi and O. Saremi, “Cosmological Perturbations on a Bouncing Brane,” arXiv:0707.4181 \[hep-th\]. L. Kofman, A. Linde and V. F. Mukhanov, “Inflationary theory and alternative cosmology,” JHEP [**0210**]{}, 057 (2002) \[arXiv:hep-th/0206088\]. R. Durrer, “Gauge Invariant Cosmological Perturbation Theory: A General Study And Its Application To The Texture Scenario Of Structure Formation,” Fund. Cosmic Phys. [**15**]{}, 209 (1994) \[arXiv:astro-ph/9311041\]. S. B. Giddings and A. Maharana, “Dynamics of warped compactifications and the shape of the warped landscape,” Phys. Rev. D [**73**]{}, 126003 (2006) \[arXiv:hep-th/0507158\]. K. Koyama, K. Koyama and F. Arroja, “On the 4D effective theory in warped compactifications with fluxes and branes,” Phys. Lett. B [**641**]{}, 81 (2006) \[arXiv:hep-th/0607145\].
[^1]: Based on [@Germani:2006pf] and [@Germani:2007ub].
[^2]: The details of how to include a curvature term in the mirage Hubble equation can be found in [@Germani:2006pf].
[^3]: To be precise, the Hollands-Wald mechnism can be used only for perturbations with wavenumbers $k\geq k_{min}$, where $k_{min}=a_{min}l_c^{-1}$. Perturbations with $k<k_{min}$ never enter in the quantum region. These perturbations are therefore normalized in the past infinity and their associated spectrum is generically blue \[13\]. However, as the size of these perturbations can be taken to be much larger than the Hubble horizon today \[3\], they can be safely excluded from current CMBR observations.
[^4]: In the original proposal of [@Hollands:2002yb] the perturbation was produced by the same radiation which sets the CMB. However, as pointed out by [@Kofman:2002cj], the perturbation coming out from the horizon today, was necessarily born when the energy density of radiation was much bigger than the Planck energy, which makes the mechanism unreliable. In the Slingshot instead, perturbations are created by brane fluctuations in a regime in which the supergravity approximation is still valid, and no extra quantum effect takes place. Moreover, super-horizon causality is not required, since in the Slingshot perturbations are overdamped at sub-horizon scales $k<J/r^2$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Different structural features of human language change at different rates and thus exhibit different temporal stabilities. Existing methods of linguistic stability estimation depend upon the prior genealogical classification of the world’s languages into language families; these methods result in unreliable stability estimates for features which are sensitive to horizontal transfer between families and whenever data are aggregated from families of divergent time depths. To overcome these problems, we describe a method of stability estimation without family classifications, based on mathematical modelling and the analysis of contemporary geospatial distributions of linguistic features. Regressing the estimates produced by our model against those of a genealogical method, we report broad agreement but also important differences. In particular, we show that our approach is not liable to some of the false positives and false negatives incurred by the genealogical method. Our results suggest that the historical evolution of a linguistic feature leaves a footprint in its global geospatial distribution, and that rates of evolution can be recovered from these distributions by treating language dynamics as a spatially extended stochastic process.
**Keywords:** linguistic typology; stability estimation; complex systems
author:
- 'Henri Kauhanen[^1]'
- Deepthi Gopal
- Tobias Galla
- 'Ricardo Bermúdez-Otero'
bibliography:
- 'tau.bib'
title: |
Geospatial distributions reflect rates of evolution\
of features of language
---
Language change and linguistic stability
========================================
Languages differ from each other in respect of a finite number of structural features. These features determine how individual words are formed, how words are combined into phrases and sentences, and which sounds and sound sequences are available in a language. For example, some languages place the verb before the object (e.g. English *Mary loves John*), while others place the object before the verb (e.g. Turkish *Mary John’u seviyor*). Similarly, some languages employ a marker of definiteness (e.g. English *the* in *the car*) whereas others have no such device (e.g. Finnish $\varnothing$ *auto*). The set of all linguistic features defines the variation space of human language, with each individual language occupying a specific point in this space.
Languages are not immutable entities, however, but rather change over time through complex processes of cultural evolution. For instance, a number of languages have undergone change from object–verb to verb–object order [@Hro2000; @PinTay2006; @Zar2011], and several languages have independently innovated a definite marker [@DeMulCar2011]. Languages thus sometimes adopt features they formerly lacked, or lose features they formerly possessed. These processes of change may be either vertical (from ancestor to descendant, e.g. from Old English to Modern English) or horizontal (between contemporary geographically close languages in extensive contact, e.g. Norman French and Middle English), loosely paralleling the distinction between inheritance and horizontal gene transfer in biological evolution [@OchLawGro2000].
One of the most important findings of modern linguistics is that different linguistic features evolve at widely disparate rates. This fact calls for an explanation: it is, for example, not predicted by replicator-neutral models of cultural evolution, according to which cultural change is largely driven by random processes akin to genetic drift and neutral evolution in biology [@BenEtal2004; @BenEtal2007]. Yet substantial evidence exists that certain linguistic features are more stable—harder to lose and harder to innovate—than others [@DedCys2013]. Although the causes of these differences remain poorly understood, efforts have been made to develop techniques for estimating the relative stabilities of individual features. Current methods of stability estimation, however, depend upon assumptions of genealogy (linguistic relatedness) that incur serious problems, resulting in unreliable stability estimates, as we argue in detail below.
In this paper, we put forward the proposition that significant information about the rate of evolution of a linguistic feature is encoded in its global geospatial distribution. Building on early, qualitative work in linguistic typology [@Gre1978], we suggest that features which are present in geographically scattered samples of languages are unstable (rapidly changing), while features which cluster together in geographical space are stable (slow to change). Thus, the contemporary geospatial distribution of a linguistic feature carries a signal about its past. Based on this idea, we develop a technique of linguistic stability estimation from geospatial distributions alone, using methods from statistical physics and without recourse to assumptions of genealogical language relatedness. Comparing the predictions of our technique against those of a genealogical method, we report broad agreement but also important differences: specifically, we show that our method is not liable to some of the false positives and false negatives incurred by the latter. We thus conclude that genealogical methods not only incorrectly predict the stability of certain problematic features, but may also be unnecessary—a model that relies solely on directly observable geospatial information fares no worse. More generally, our results demonstrate that significant information about the evolution of a linguistic feature is retained in its current geospatial distribution, and that it is possible to tap into this signal by treating language dynamics as a spatially extended stochastic process.
Problems of genealogical stability estimation
=============================================
The majority of existing linguistic stability estimation techniques rely on the genealogical grouping of the world’s languages into language families such as Indo-European, Uralic and Austronesian [@Mas2004; @Par2008; @Ded2011; @DedCys2013; @GreEtal2017]. These families are established using standard comparative reconstruction techniques [@Bow2014]. Although implementation details vary, all genealogical stability estimation methods work on the basic assumption that stable linguistic features ought to be conserved within language families, unstable features exhibiting within-family variation instead. For instance, the basic order of verb and object is verb–object in all existing Romance languages [@Led2010]; verb–object order may thus be considered stable within the Romance family. On the other hand, the expression of the subject exhibits considerable variation among the Romance languages: many of these languages allow null subjects (e.g. Spanish $\varnothing$ *voy* ‘I go’), but this option has been partially or fully lost in French, Rhaeto-Romance, Provençal, Northern Italo-Romance and Brazilian Portuguese, which require a pronoun or pronoun-like element (e.g. French *je vais* ‘I go’). The possibility of dropping pronominal subjects is then a relatively unstable feature within the Romance languages. Moreover, aggregating data from several language families suggests that this conclusion holds universally: the basic order of verb and object tends to be a stable, and the expression of subjects an unstable, feature among the world’s languages [@DedCys2013].
In general, the genealogical classification procedures used in modern linguistics are highly reliable: the family trees they yield are rarely in dispute, except in respect of the fine structure of otherwise uncontroversial families, or the very distant putative kinship relationships lying beyond the reach of the standard comparative method of linguistic reconstruction [@Bow2014]. There is, in other words, little uncertainty as to which languages ought to belong to which language family. The high reliability of linguistic genealogies, however, does not by itself render them an appropriate tool for estimating the stability of linguistic features. We here outline two limitations shared by all genealogical methods: the problem of time depths, and the problem of horizontal transfer.
Firstly, no agreement exists on how to estimate the time depths (or, ages) of otherwise undisputed language families [@McMMcM2005; @PerLew2015; @Bow2017]. We illustrate this problem with the Uralic and Germanic family trees in Fig. \[fig:tree\]. In each tree, each branching point corresponds to at least one linguistic innovation (‘mutation’), whereby the daughter languages diverge from their ancestor in the specification of at least one linguistic feature. The degree of variation among the surviving members of a family must then increase with the age of the family (unless later changes precisely undo the effects of earlier ones, but this is rare). In our example trees, this is illustrated by the fact that while all surviving members of the Germanic family employ a definite marker, only about half of the surviving Uralic languages do so, reflecting the greater time depth of the latter family. The problem for genealogical stability estimation techniques arises from the fact that no agreed methods exist for establishing the ages of individual families; no way exists for controlling that the families employed in the estimation are of similar time depths. Some studies [@DedCys2013] have attempted to overcome this problem by making comparisons across the so-called genera identified in the World Atlas of Language Structures (WALS) [@WALS]. This genalogical classification into genera is intended to yield groupings of comparable time depths, and indeed WALS treats Germanic as a single genus, whereas Uralic (like Indo-European) is seen as comprising several genera. This expedient, however, mitigates but does not solve the problem: the WALS editors themselves describe WALS genera as ‘highly tentative’, ‘based on meagre initial impressions’ and consisting of no more than ‘educated guesses’ [@Dry1989; @WALS].
![Some languages of the Uralic and Germanic families with respect to the presence of a definite marker (such as English *the*).[]{data-label="fig:tree"}](problems_new){width="\textwidth"}
Secondly, genealogical stability estimation techniques necessarily miss out the effect of horizontal transfer—the borrowing of a feature from one family to another—on a feature’s stability. For example, Old Written Estonian was in extensive contact with German and borrowed a definite marker from it [@Met2011]; this relationship is visualized as the purple arrow in Fig. \[fig:tree\]. The problem for genealogical stability estimation methods arises from the fact that some linguistic features (e.g. inflectional markers) are more resistant to horizontal transfer than others [@GarArkAmi2014], while some (e.g. case systems) are highly vulnerable to simplification in contact situations involving large numbers of second-language learners [@BenWin2013]. Combining genealogical and areal groupings [@Par2008] is not a solution, however, as no agreed methods exist for delimiting linguistic areas or for estimating the time depths of areal relationships.
Stability estimation from geospatial distributions
==================================================
The above criticisms motivate the search for a stability measure that reflects the relatedness of languages without presorting them into predefined groupings and can take horizontal transfer effects into account. Here, we propose such a technique by modelling language dynamics as a stochastic process on a spatial substrate; this model can be studied in computer simulations and mathematically. While the model dynamics continue indefinitely, the statistical properties of the distribution of features over physical space becomes stationary after the simulation has been run for a sufficiently long time. Using techniques from statistical physics, this stationary state can be characterized mathematically. In what follows, we show how this analytical solution can be utilized to estimate the tendency of individual linguistic features to change based solely on their contemporary geospatial distributions, measured from the WALS atlas [@WALS].
![Markov chain dynamics of a single linguistic feature. F: feature present; $-$F: feature absent.[]{data-label="fig:markov"}](markovchain)
Following an early but underexplored proposal of Greenberg’s [@Gre1978], we treat the evolution of binary linguistic features as a memoryless stochastic process. The dynamics of each feature are then given by a Markov chain with two parameters, $p_I$ and $p_E$ (Fig. \[fig:markov\]). The former parameter gives the probability of a language adopting the feature in question; we call it the feature’s *ingress rate*. The latter parameter, in turn, gives the probability of a language losing the feature; we will refer to it as the feature’s *egress rate*. We assume language communities to be distributed on a spatial substrate which, for reasons of mathematical tractability, we take to be a square lattice, i.e. language communities reside at regularly spaced positions over physical space. Each community is assumed to be subject to an ingress–egress dynamics as described by the Markov chain model of Fig. \[fig:markov\]. To account for horizontal effects, we assume the existence of an interaction process between spatially contiguous language communities. This interaction process is based on the so-called voter model [@CliSud1973; @Kra1992; @Lig1997; @CasForLor2009; @FerEtal2014] and operates as follows. In each interaction event, a ‘target’ community is chosen, and the presence or absence of a feature in a randomly selected neighbouring community is copied into the target community. All in all, the model is iterated as follows:
1. Initialize the lattice in a random state (for each feature $F$ and community $C$, $F$ is present in $C$ with probability $0.5$).
2. Pick a random community $C$ and a random feature $F$.
3. Execute one of the following steps:
1. with probability $q$: pick a random lattice neighbour $C'$ of $C$, and set the value for feature $F$ in $C$ to that in $C'$; or
2. with probability $1 - q$: if $F$ is absent from $C$, acquire $F$ with probability $p_I$ (ingress); if $F$ is present in $C$, lose $F$ with probability $p_E$ (egress).
4. Go to 2.
Inevitably, this model idealizes away from many of the complexities of real world language dynamics. What matters for present purposes is that the model should be able to capture two key elements contributing to language change. First, the evolution of a linguistic community over time is subject to both vertical and horizontal effects: the vertical effects arise mainly from the transmission of linguistic knowledge across generations through language acquisition; the horizontal effects reflect not only contact between speakers of different languages, but also contact between speakers of varieties of the same language [@Lab2007]. Secondly, both vertical and horizontal effects can result in the faithful transmission of a feature or in a mutation. For example, contact between speakers of different languages can result in the simple transfer of a feature (borrowing), but it can also result in mutation, as when the interaction between two languages with different inflectional systems leads to the emergence of a simplified system that is different from both its predecessors [@BenWin2013]. In some instances of mutation, horizontal and vertical effects interact in highly intricate ways, e.g. during the formation of creole languages [@DeGra2009]. Our model reflects this state of affairs: faithful transmission occurs both vertically, with probability $(1 - q)(1 - p_I - p_E)$, and horizontally, with probability $q$. In turn, the ingress–egress dynamics (probabilities $p_I$ and $p_E$) covers processes of mutation and is agnostic about their causes (vertical or horizontal).
![In the model, linguistic features evolve on a square lattice with periodic boundary conditions. The figure shows three *ad hoc* illustrations (red: feature present; blue: feature absent). In each case, the feature frequency is $\rho = 0.5$ (half of the sites are blue, the other half are red). However, the isogloss density $\sigma$, defined as the proportion of disagreeing lattice interfaces (yellow dots), depends on the spatial distribution of the feature. It is low when a feature is present throughout extended domains, and larger when a feature is scattered.[]{data-label="fig:lattice"}](lattice)
The statistical properties of the stationary distribution of features at long times depends on the model parameters $p_I$, $p_E$ and $q$ (for further details see the SI). For the purposes of stability estimation, we are interested in two quantities in particular (illustrated in Fig. \[fig:lattice\]): the frequency, $\rho$, with which a particular feature is present across the lattice, and the feature’s associated *isogloss density*. This latter quantity indicates the probability of finding a dialect boundary (an isogloss) between two neighbouring communities such that the feature is found on one side of the boundary but not on the other. We define this as the proportion of pairs of adjacent lattice cells that differ in the feature value, and denote it by $\sigma$; similar quantities are sometimes also found as the ‘density of reactive interfaces’ in literature on interacting particle systems [@Kra1992; @KraRedBen2010]. The frequency of a feature in the stationary state is given by $$\label{eq:A:steady-rho}
\rho = \frac{p_I}{p_I + p_E}.$$ This can be demonstrated mathematically (see SI), but is also clear intuitively; the higher the ingress rate $p_I$ of a feature is in relation to its egress rate $p_E$, the more prevalent the feature will be. Obtaining the stationary isogloss density is more intricate mathematically (see again the SI). We find $$\label{eq:A:steady-sigma}
\sigma = 2H(\tau) \rho (1- \rho)$$ with $$\label{eq:A:Htau}
H(\tau) = \frac{\pi (1+\tau)}{2K\left(\frac{1}{1+\tau}\right)} - \tau$$ and $$\label{eq:A:tau}
\tau = \frac{(1-q)(p_I + p_E)}{q}.$$ The function $K(\cdot)$ denotes the complete elliptic integral of the first kind (see SI for full technical details). Thus, from Eq. (\[eq:A:steady-sigma\]), the stationary-state isogloss density $\sigma$ is found to be a parabolic function of the feature’s overall frequency $\rho$. The height of this parabola is controlled by the parameter $\tau$ (Fig. \[fig:mainfig\]A); this parameter gives the relative rate of ingress–egress events (i.e., mutations) over faithful transmission (Eq. \[eq:A:tau\]). For example, a value of $\tau=10^{-5}$ would indicate that faithful copying events between neighbouring communities are $10^5$ times more frequent on average than mutations. This suggests that $\tau$ can be interpreted as a *temperature*, measuring the amount of noise in the dynamics: lower values of $\tau$ signify a stable feature, higher values indicating instability.
![(A) Statistical properties of the model. At long times the state of the lattice is characterized by the two quantities feature frequency $\rho$ and isogloss density $\sigma$. We show computer simulations (markers) and analytical solution (curves) for different values of $\tau$ (determined by the combination of $p_I$, $p_E$ and $q$). Simulation snapshots are shown for two different values of $\tau$. (B) Empirical measurements of feature frequency $\rho$ and isogloss density $\sigma$ for 35 linguistic features, identified by their WALS feature IDs (see Methods for details).[]{data-label="fig:mainfig"}](mainfig){width="\textwidth"}
We next describe how the above mathematical solution may be used to obtain estimates of the stability of linguistic features using contemporary geospatial data. Our particular aim is to estimate the temperature parameter $\tau$ for individual features, using geospatial information contained in the WALS database [@WALS], focusing on 35 binary features (see Methods). For each feature, the frequency $\rho$ is given by the proportion of languages possessing that feature in the WALS language sample. The isogloss density $\sigma$ is calculated as follows: for each language, we first establish its nearest geographical neighbour in the relevant language sample. The empirical isogloss density is then given by the proportion of nearest-neighbour language pairs differing in their values for the feature in question. Our data are summarized in Fig. \[fig:mainfig\]B, which supplies $\rho$ and $\sigma$ for each of the 35 features; a lower value of isogloss density $\sigma$ signals a geographically clustered feature, whilst a higher value implies a feature with a scattered geographical distribution. Fig. \[fig:maps\] illustrates this difference with two features, definite marker (WALS feature 37A) and order of object and verb (WALS feature 83A).
![Empirical geospatial distributions of two linguistic features on the hemisphere from 30 W to 150 E (red: feature present, blue: feature absent): (A) definite marker (WALS feature 37A), (B) object–verb order (WALS feature 83A). Shown are both individual empirical data points (languages, as given by WALS coordinates) and a spatial interpolation (inverse distance weighting) on these points. Map projection: Albers equal-area.[]{data-label="fig:maps"}](interp_13A.png "fig:")\
![Empirical geospatial distributions of two linguistic features on the hemisphere from 30 W to 150 E (red: feature present, blue: feature absent): (A) definite marker (WALS feature 37A), (B) object–verb order (WALS feature 83A). Shown are both individual empirical data points (languages, as given by WALS coordinates) and a spatial interpolation (inverse distance weighting) on these points. Map projection: Albers equal-area.[]{data-label="fig:maps"}](interp_38A.png "fig:")
For a given feature frequency $\rho$, the isogloss density $\sigma$ is fixed by the value of $H(\tau)$ (Eq. \[eq:A:steady-sigma\]); this quantity itself is an increasing function of $\tau$ (Eq. \[eq:A:Htau\]). Since each of our 35 empirical features lies on a unique parabola in the space spanned by $\rho$ and $\sigma$ (Fig. \[fig:mainfig\]), estimating its temperature is now a matter of inverting the function $H(\tau)$. For each feature, we measure $\rho$ and $\sigma$ as described above. From Eq. (\[eq:A:steady-sigma\]), we then obtain the value of $H(\tau)$ and invert this to recover $\tau$. Although the elliptic integral in Eq. (\[eq:A:Htau\]) cannot be expressed in terms of elementary functions and $H(\tau)$ thus cannot be inverted analytically, the inversion can be performed numerically. Using this procedure we obtain an estimate of $\tau$ for any feature for which empirical measurements of frequency $\rho$ and isogloss density $\sigma$ exist. Table \[tab:tau-estimates\] supplies these estimates for the least stable and most stable features in our dataset (for a full listing of $\tau$ estimates for the entire dataset, see Table S2 in the SI).
[ ]{}
Comparison with a genealogical method
=====================================
The technique proposed by Dediu [@Ded2011] represents the state of the art in genealogy-based stability estimation. Using a Bayesian phylogenetic algorithm, this method produces a posterior distribution of rates of evolution for each linguistic feature within a predefined genealogical grouping. Dediu tests two phylogenetic algorithms and draws data from two sources—WALS and Ethnologue [@Ethnologue]—to control for implementation effects. His stability estimates are then expressed as the additive inverse of the first component (PC1) of a principal component analysis on the stability ranks predicted by each combination of phylogenetic algorithm and dataset (i.e. the higher the PC1 value, the less stable the feature).
In Fig. \[fig:comparison\]A, we consider the 24 features which are both in our list of 35 features and in Dediu’s list. Regressing our estimates for $\tau$ against Dediu’s PC1 (red regression line), we find a moderate correlation between the estimates predicted by the two methods (Pearson’s $r = 0.53$, $p = 0.008$). A number of features, however, are clearly outliers of the regression. To detect these outliers more objectively, we pruned the regression recursively by removing those data points that contribute the greatest error in terms of sum of squared residuals; Fig. \[fig:comparison\]B gives the reduction of error at each step of this pruning. The reduction profile prompts us to classify as outliers the first five data points, corresponding to the following WALS features: 11A (front rounded vowels), 107A (passive construction), 8A (lateral consonants), 44A (gender in independent personal pronouns) and 57A (possessive affixes). Regressing the pruned dataset (Fig. \[fig:comparison\]A, black regression line), we find a high correlation between our $\tau$ estimates and Dediu’s PC1 (Pearson’s $r = 0.90$, $p = 1.613 \times 10^{-7}$).
![Regression of our temperature ($\tau$) estimates against Dediu’s PC1. (A) Red line: regression with all 24 data points (Pearson’s $r = 0.53$, $p = 0.008$). Black line: regression with five outliers (red crosses) removed (Pearson’s $r = 0.90$, $p = 1.613 \times 10^{-7}$). (B) Outliers were detected by pruning the dataset recursively for those data points that contribute most to the regression error, quantified as the sum of squared residuals. This identified features 11A, 107A, 8A, 44A and 57A as outliers (see text).[]{data-label="fig:comparison"}](dediu_regression "fig:")![Regression of our temperature ($\tau$) estimates against Dediu’s PC1. (A) Red line: regression with all 24 data points (Pearson’s $r = 0.53$, $p = 0.008$). Black line: regression with five outliers (red crosses) removed (Pearson’s $r = 0.90$, $p = 1.613 \times 10^{-7}$). (B) Outliers were detected by pruning the dataset recursively for those data points that contribute most to the regression error, quantified as the sum of squared residuals. This identified features 11A, 107A, 8A, 44A and 57A as outliers (see text).[]{data-label="fig:comparison"}](dediu_outliers "fig:")
We suggest that, rather than representing different views on stability, these outliers are false positives and negatives of the genealogical method. We illustrate this with the case of (the presence or absence of) front rounded vowels (WALS feature 11A), i.e. the vowels /y/ (e.g. Finnish *kyy*), // (German *schön*), // (French *bœuf*) and // (Danish *grøn*). This is one of the most stable features in the genealogical analysis[^2] but one of the least stable features in ours (Table \[tab:tau-estimates\]); we argue that evidence from both language change and language acquisition supports our position. On the one hand, front rounded vowels are frequently innovated: historical fronting of the back rounded vowel /u/ to \[y\] (with or without subsequent phonemicization to /y/) has been documented in a number of languages, including but not limited to Armenian, Attic-Ionic Greek, French, Frisian, Lithuanian, Old Scandinavian, Oscan, Parachi, Umbrian, West Syriac, Yiddish, Zuberoan Basque, and numerous dialects of English [@Oha1981; @LabEtal1972; @Dre1974; @Las1988; @Lab1994; @Egu2017; @Sam2017]; additionally, front rounded vowels can arise through the influence of /i/ or /j/ on a neighbouring back rounded vowel [@Oha1981; @IveSal2003]. On the other hand, front rounded vowels are difficult to acquire in situations of language contact: there is experimental evidence that second-language learners whose first language lacks these vowels perceive them as more similar to back vowels than front vowels [@StrLevLaw2009]. This perceptual assimilation is mirrored in speech production: productions of /y/ by second-language learners are far less advanced in phonetic space than native speakers’ productions, and are indeed often perceived as /u/ by the latter [@Roc1995]. The fact that front rounded vowels are readily innovated points to a high ingress rate, while frequent acquisition failure by second-language learners in situations of language contact points to a high egress rate. These facts are inconsistent with the high stability predicted by the genealogical method, but consistent with our approach, in which high ingress and high egress make for high temperature (Eq. \[eq:A:tau\]) and thus low stability.
Although space limitations preclude a full linguistic analysis, we point out that similar arguments can be made for the remaining outliers. For instance, all Uralic languages employ possessive affixes (e.g. Finnish *auto-ni* ‘my car’, *auto-si* ‘your car’, etc.), and the appearance of this system of possession can be dated back to Pre-Proto-Uralic by standard reconstructive techniques [@Jan1982]. Possessive affixes are also old in the unrelated Turkic language family [@Erd2004]. There is, then, reason to believe that WALS feature 57A on possessive affixes is a false negative of the genealogical method, which classifies possessive affixes as one of the least stable features (Fig. \[fig:comparison\]).
Discussion
==========
The challenge of linguistic stability estimation arises, essentially, from having to work with a poor signal. Although evolutionary and anthropological evidence suggests that human language in its modern form has existed for at least 100,000 years [@Bic2007; @Tal2012], the historical evolution of languages is (necessarily) poorly documented. Such documentation only captures a few thousand years for languages with the best coverage, cannot in principle go beyond the introduction of the first writing systems, and does not exist at all for the majority of the world’s languages. The rest of the historical evolution of language must be reconstructed based on available data. In this paper, we have argued that stability estimation methods relying on even the most accurate reconstructive methods have their shortcomings: there is no guarantee that the genealogical classifications assumed in such estimation reflect equivalent time depths between different language families, and the methods do not control for horizontal transfer between languages belonging to different families. We have introduced a stability estimation procedure that does without genealogies and infers stability estimates from contemporary geospatial distributions of linguistic features alone. The method is relatively easy to implement: all that is needed are measures of feature frequency and isogloss density for a large enough sample of languages, and inversion of Eq. (\[eq:A:Htau\]).
We have offered some evidence in support of our method, in the sense that this method is not liable to some of the false positives and false negatives incurred by genealogical methods. Turning now to its limitations, we note that our approach currently only applies to binary features, i.e. features which are either present in or absent from a language community. Most genealogical methods do not suffer from this limitation: Dediu’s [@Ded2011] procedure, in particular, can be applied to polyvalent as well as binary features. Interestingly, however, Dediu finds a correlation between estimates generated for polyvalent and binary (or binarized) features. This suggests that the resolution at which the values of a linguistic variable are recorded may be a minor issue: after all, any polyvalent classification can be reduced to a hierarchy of binary oppositions. Another limitation of our technique, shared by all existing methods, is that it treats the evolution of individual features independently of the rest. Feature interactions are known to exist, however—for example, a language which places objects before verbs is far more likely to also place adverbs before verbs, rather than after them [@Gre1963; @Dry2003]. It is in principle possible to generalize our method to cater for polyvalent features and feature interactions, by extending the lattice model in the direction of the Axelrod model of cultural dissemination [@Axe1997]. It is at the moment unclear, however, whether the behaviour of such a generalized model can be solved analytically so that stability estimates may be derived in the way we have described.
While these remain tasks for future research, the present study enables us to conclude that, though hitherto underexploited, geospatial distributions provide one of the best sources of evidence on the rates of evolution of features of human language.
Methods {#methods .unnumbered}
=======
Data preparation {#data-preparation .unnumbered}
----------------
The WALS Online database [@WALS] was downloaded on 18 July 2017 and used as the empirical basis for measures of feature frequency $\rho$ and isogloss density $\sigma$. Since WALS employs a polyvalent coding for most features, a manual pass was made, retaining only those features that are binary or binarizable on uncontroversial linguistic grounds. Features with fewer than 300 languages in their language sample were discarded to ensure statistically robust results. This procedure resulted in 35 binary features (see Table S1 in the SI for a listing, together with full information about our binarization scheme). Nearest neighbours of languages were determined by the great-circle distance, calculated from WALS coordinate data using the haversine formula with the assumption of a perfectly spherical earth.
Analysis {#analysis .unnumbered}
--------
To eliminate any possible effect the differing language sample sizes of different WALS features might have on our statistics, a fixed number of 300 languages was considered for each feature, with languages sampled uniformly at random from the feature’s language sample. This procedure was repeated 10,000 times for each feature to generate the bootstrap averages reported in Fig. \[fig:mainfig\]B. For comparison with the genealogical method, Table S1 in the SI to Ref. [@Ded2011] was consulted and only those features were selected for comparison for which our binarization schemes agreed; the PC1 values for the intersecting features were then gathered from Table S4. Correlation strengths were measured using the Pearson correlation coefficient; significance was tested with a two-tailed $t$-test. The analytical solution of the lattice model (Eqs. \[eq:A:steady-rho\]–\[eq:A:Htau\]) appears in the SI.
Data availability {#data-availability .unnumbered}
-----------------
WALS is freely available at http://wals.info; the binarization scheme used to prepare the data appears in the SI.
Code availability {#code-availability .unnumbered}
-----------------
Computer code for data analysis, stability estimation and numerical simulations may be obtained from the corresponding author.
Author contributions {#author-contributions .unnumbered}
====================
Designed the study: HK, DG, TG and RB-O. Analysed the data: HK, DG, TG and RB-O. Solved the mathematical model: HK, DG and TG. Wrote the paper: HK, DG, TG and RB-O. Wrote visualization routines: HK and DG. Wrote the data analysis and simulation code: HK.
Author information {#author-information .unnumbered}
==================
The authors declare no conflict of interest and no competing financial interests. Correspondence and requests for materials should be addressed to [email protected].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Dan Dediu, Danna Gifford and George Walkden for comments and discussions. HK acknowledges financial support from Emil Aaltonen Foundation and The Ella and Georg Ehrnrooth Foundation.
Supplementary information
Features consulted
==================
Table \[tab:features\] provides a listing of the 35 WALS [@WALS] features consulted in this study, together with our scheme for feature binarization. Each WALS feature is a variable of either nominal or ordinal level, whose possible values are recorded in the WALS database using integer labels. The meanings of these labels are explained at length in Section \[sec:wals-details\], below; Table \[tab:features\] indicates which values of each variable were folded into the ‘feature absent’ category and which values to the ‘feature present’ category in our binarization.
description WALS abs. pres. $N$
----- ------------------------------------------------------- ------ ------ ------- ------
1. adpositions 48A 1 2–4 378
2. definite marker 37A 4–5 1–3 620
3. *hand* and *arm* identical 129A 2 1 617
4. *hand* and *finger(s)* identical 130A 2 1 593
5. front rounded vowels 11A 1 2–4 562
6. gender in independent personal pronouns 44A 6 1–5 378
7. glottalized consonants 7A 1 2–8 567
8. grammatical evidentials 77A 1 2–3 418
9. indefinite marker 38A 4–5 1–3 534
10. inflectional morphology 26A 1 2–6 969
11. inflectional optative 73A 2 1 319
12. lateral consonants 8A 1 2–5 567
13. morphological second-person imperative 70A 5 1–4 547
14. order of adjective and noun is AdjN 87A 2 1 1366
15. order of degree word and adjective is DegAdj 91A 2 1 481
16. order of genitive and noun is GenN 86A 2 1 1249
17. order of numeral and noun is NumN 89A 2 1 1153
18. order of object and verb is OV 83A 2 1 1519
19. order of subject and verb is SV 82A 2 1 1497
20. ordinal numerals 53A 1 2–8 321
21. passive construction 107A 2 1 373
22. plural 33A 9 1–8 1066
23. possessive affixes 57A 4 1–3 902
24. postverbal negative morpheme 143F 4 1–3 1324
25. preverbal negative morpheme 143E 4 1–3 1324
26. productive reduplication 27A 3 1–2 368
27. question particle 92A 6 1–5 884
28. shared encoding of nominal and locational predication 119A 1 2 386
29. tense-aspect inflection 69A 5 1–4 1131
30. tone 13A 1 2–3 527
31. uvular consonants 6A 1 2–4 567
32. velar nasal 9A 3 1–2 469
33. verbal person marking 100A 1 2–6 380
34. voicing contrast 4A 1 2–4 567
35. zero copula for predicate nominals 120A 1 2 386
: The 35 binary features considered in this study. The ‘WALS’ column gives the WALS feature mined; values indicated in the ‘abs.’ column were folded into our ‘binary feature absent’ value, whilst values indicated in the ‘pres.’ column were folded into our ‘binary feature present’ value (see Section \[sec:wals-details\], below). The final column gives the size of the WALS language sample (number of languages) for each feature.[]{data-label="tab:features"}
Temperature estimates
=====================
Table \[tab:results\] gives the temperature ($\tau$) estimates found by our method for the 35 features.
feature $\rho$ $\sigma$ $\tau$
----- ------------------------------------------------------- ------------ ------------ ------------
1. order of object and verb is OV 0.50352113 0.12087912 0.00001910
2. order of numeral and noun is NumN 0.44097222 0.12454212 0.00003306
3. order of genitive and noun is GenN 0.59420290 0.12204724 0.00003339
4. order of subject and verb is SV 0.86071429 0.07722008 0.00049325
5. order of adjective and noun is AdjN 0.29856115 0.14843750 0.00120923
6. tone 0.41666667 0.17333333 0.00128340
7. velar nasal 0.50000000 0.18333333 0.00183200
8. order of degree word and adjective is DegAdj 0.54135338 0.21212121 0.00569870
9. uvular consonants 0.17000000 0.13000000 0.00979939
10. ordinal numerals 0.89666667 0.08666667 0.01181972
11. shared encoding of nominal and locational predication 0.30333333 0.21000000 0.01732702
12. glottalized consonants 0.27666667 0.21000000 0.02605203
13. inflectional optative 0.15000000 0.14333333 0.04120610
14. inflectional morphology 0.85333333 0.14000000 0.04215630
15. possessive affixes 0.71333333 0.23666667 0.04784846
16. passive construction 0.43333333 0.29333333 0.05949459
17. tense-aspect inflection 0.86666667 0.13666667 0.05994843
18. *hand* and *arm* identical 0.37000000 0.28000000 0.06211254
19. productive reduplication 0.85000000 0.15333333 0.06274509
20. voicing contrast 0.68000000 0.26333333 0.06769913
21. adpositions 0.83333333 0.17000000 0.06908503
22. postverbal negative morpheme 0.46333333 0.30333333 0.06961203
23. *hand* and *finger(s)* identical 0.12000000 0.13000000 0.07032095
24. morphological second-person imperative 0.77666667 0.21666667 0.08655531
25. preverbal negative morpheme 0.70666667 0.27333333 0.11292895
26. plural 0.91000000 0.11000000 0.12817720
27. zero copula for predicate nominals 0.45333333 0.33333333 0.13213325
28. grammatical evidentials 0.56666667 0.33333333 0.14401769
29. front rounded vowels 0.06666667 0.08666667 0.19468340
30. lateral consonants 0.83333333 0.20000000 0.21489842
31. indefinite marker 0.44666667 0.35666667 0.22952822
32. gender in independent personal pronouns 0.33000000 0.32333333 0.24577576
33. question particle 0.59666667 0.36666667 0.34336306
34. definite marker 0.60666667 0.36333333 0.35396058
35. verbal person marking 0.78000000 0.27666667 0.50590667
: Feature frequency $\rho$, isogloss density $\sigma$ and estimated temperature $\tau$ for the 35 features considered in this study (to eight significant decimals), ordered by increasing $\tau$ (from stable to unstable).[]{data-label="tab:results"}
Analytical solution of lattice model {#app:solution}
====================================
We will treat the model as a spin system on a two-dimensional regular square lattice with $N = L\times L$ sites and periodic boundary conditions (for comparable approaches to the voter model without an ingress–egress dynamics, see Refs. [@Kra1992; @KraRedBen2010]). Our model is conceptually similar to a voter model with noise, which has been treated with similar methods in Ref. [@Oli2003]. We write $s({\mathbf{x}}) \in \{-1,1\}$ for the spin at lattice site ${\mathbf{x}} = (x_1, x_2)$; $S({\mathbf{x}}) = \langle s({\mathbf{x}})\rangle$ for the average spin of ${\mathbf{x}}$ (over realizations of the stochastic process); and $m = \sum_{{\mathbf{x}}} S({\mathbf{x}})/N$ for the mean magnetization over the entire lattice. The feature frequency $\rho$, or fraction of up-spins in the system, is related to $m$ by the identity $\rho = (m+1)/2$. We further write $S({\mathbf{x}}, {\mathbf{y}}) = \langle s({\mathbf{x}})s({\mathbf{y}})\rangle$ for the pair correlation of $s({\mathbf{x}})$ and $s({\mathbf{y}})$. In summations, ${\mathbf{x}}'$ is understood to index the set of von Neumann neighbours of site ${\mathbf{x}}$, i.e. the set $$\{(x_1 - 1, x_2), (x_1 + 1, x_2), (x_1, x_2 - 1), (x_1, x_2 + 1)\}.$$
Spin flip probability
---------------------
Central to our analytical derivations is the spin flip probability, i.e. the probability with which the spin at site ${\mathbf{x}}$ changes its state from $-1$ to $+1$ or vice versa, if it is selected for potential update. In our model this is of the form $$w({\mathbf{x}}) = (1-q) A({\mathbf{x}}) + q B({\mathbf{x}}),$$ where $A({\mathbf{x}})$ is the contribution of the ingress–egress process and $B({\mathbf{x}})$ the contribution of the spatial (voter) process. These are $$A({\mathbf{x}}) = \frac{1 - s({\mathbf{x}})}{2} p_I + \frac{1 + s({\mathbf{x}})}{2} p_E
= \frac{1}{2} \left[p_I + p_E + (p_E - p_I)s({\mathbf{x}})\right]$$ and $$B({\mathbf{x}}) = \frac{1}{4}\sum_{{\mathbf{x}}'} \frac{1 - s({\mathbf{x}})s({\mathbf{x}}')}{2}
= \frac{1}{2} \left[1 - \frac{1}{4} \sum_{{\mathbf{x}}'} s({\mathbf{x}})s({\mathbf{x}}') \right],$$ where it is important to remember that the summation over ${\mathbf{x}}'$ runs over the four nearest neighbours of ${\mathbf{x}}$. Hence we have $$\label{eq:spin-flip-probability}
w({\mathbf{x}}) = \frac{1-q}{2} \left[p_I + p_E + (p_E - p_I)s({\mathbf{x}})\right]
+ \frac{q}{2} \left[1 - \frac{1}{4}\sum_{{\mathbf{x}}'} s({\mathbf{x}})s({\mathbf{x}}') \right].$$
Stationary-state feature frequency $\rho$
-----------------------------------------
The spin at ${\mathbf{x}}$ changes by the amount $-2s({\mathbf{x}})$ with probability $\frac{1}{N} w({\mathbf{x}})$, the prefactor $1/N$ representing the probability of ${\mathbf{x}}$ being picked for update. Consequently, the mean spin $S({\mathbf{x}}) = \langle s({\mathbf{x}})\rangle$ evolves as $$S({\mathbf{x}}, t + \Delta t) - S({\mathbf{x}}, t) = \frac{1}{N} \langle -2w({\mathbf{x}})s({\mathbf{x}}) \rangle,$$ where $\Delta t$ is the time step associated with each attempted spin flip. Bearing in mind that $s({\mathbf{x}})s({\mathbf{x}}) = 1$ and plugging Eq. in, this implies $$\frac{S({\mathbf{x}}, t + \Delta t) - S({\mathbf{x}},t)}{1/N}
= (1-q) \left[p_I - p_E - (p_I + p_E)S({\mathbf{x}}) \right]
+ q\left[ -S({\mathbf{x}}) + \frac{1}{4} \sum_{{\mathbf{x}}'} S({\mathbf{x}}') \right].$$ Taking the sum over all sites ${\mathbf{x}}$, one has $$\begin{split}
\frac{m(t + \Delta t) - m(t)}{1/N}
&= (1-q)(p_I - p_E) - (1-q)(p_I + p_E) \frac{1}{N} \sum_{{\mathbf{x}}} S({\mathbf{x}}) - \\
&\quad\quad - \frac{q}{N} \sum_{{\mathbf{x}}} S({\mathbf{x}}) + \frac{q}{4N} \sum_{{\mathbf{x}}}\sum_{{\mathbf{x}}'} S({\mathbf{x}}').
\end{split}$$ Now $\sum_{{\mathbf{x}}}\sum_{{\mathbf{x}}'} S({\mathbf{x}}') = 4\sum_{{\mathbf{x}}} S({\mathbf{x}})$, since the LHS is the sum of the four von Neumann neighbours of all lattice sites, so that each site, having four neighbours, gets counted four times. Using $m = \sum_{{\mathbf{x}}} S({\mathbf{x}})/N$, we then have $$\frac{m(t + \Delta t) - m(t)}{1/N} = (1-q)\left[p_I - p_E - (p_I + p_E)m\right].$$ With the standard choice $\Delta t = 1/N$, and taking the limit $N \to \infty$ (i.e. the continuous-time limit $\Delta t \to 0$), we thus find $$\frac{dm}{dt} = (1-q)\left[p_I - p_E -(p_I + p_E)m\right].$$ Hence the mean magnetization in the stationary state ($dm/dt = 0$) is $$m = \frac{p_I - p_E}{p_I + p_E}.$$ From this, using $\rho = (m+1)/2$, we find $$\label{eq:steady-rho}
\rho = \frac{p_I}{p_I + p_E}$$ for the fraction of up-spins in the stationary state.
Pair correlation function
-------------------------
To compute the pair correlation $S({\mathbf{x}}, {\mathbf{y}}) = \langle s({\mathbf{x}})s({\mathbf{y}}) \rangle$, we note that $s({\mathbf{x}})s({\mathbf{y}})$ changes by the amount $-2s({\mathbf{x}})({\mathbf{y}})$ if either ${\mathbf{x}}$ or ${\mathbf{y}}$ flips spin. Assuming ${\mathbf{x}} \neq {\mathbf{y}}$, and working directly in the continuous-time limit, we have $$\frac{dS({\mathbf{x}}, {\mathbf{y}})}{dt} = \langle -2\left[w({\mathbf{x}}) + w({\mathbf{y}})\right]s({\mathbf{x}})s({\mathbf{y}})\rangle.$$ After some algebra we find $$\begin{split}
\frac{d S({\mathbf{x}}, {\mathbf{y}})}{dt} &= (1-q) \left[ (p_I - p_E) [S({\mathbf{x}}) + S({\mathbf{y}})] - 2(p_I + p_E)S({\mathbf{x}}, {\mathbf{y}}) \right] + \\
&\quad\quad + q \left[ - 2S({\mathbf{x}}, {\mathbf{y}}) + \frac{1}{4} \sum_{{\mathbf{x}}'} S({\mathbf{x}}', {\mathbf{y}}) + \frac{1}{4} \sum_{{\mathbf{y}}'} S({\mathbf{x}}, {\mathbf{y}}') \right],
\end{split}$$ where the summation over ${\mathbf{y}}'$ is over the four nearest neighbours of ${\mathbf{y}}$.\
We now assume translation invariance and write $C({\mathbf{r}}) = C({\mathbf{x}} - {\mathbf{y}}) = S({\mathbf{x}}, {\mathbf{y}})$. Then, for ${\mathbf{r}} \neq {\mathbf{0}}$, $$\begin{split}
\frac{dC({\mathbf{r}})}{dt} &= (1-q) \left[ (p_I - p_E) [S({\mathbf{x}}) + S({\mathbf{y}})] - 2(p_I + p_E) C({\mathbf{r}}) \right] + \\
&\quad\quad + q \left[ -2C({\mathbf{r}}) + \frac{1}{4} \sum_{{\mathbf{x}}'} C({\mathbf{x}}' - {\mathbf{y}}) + \frac{1}{4}\sum_{{\mathbf{y}}'} C({\mathbf{x}} - {\mathbf{y}}') \right].
\end{split}$$ Due to translation invariance the two summations on the RHS coincide, and we have (always restricting ${\mathbf{r}} \neq {\mathbf{0}}$) $$\label{eq:pair-correlation-function}
\frac{dC({\mathbf{r}})}{dt} = (1-q) \left[ (p_I - p_E) [S({\mathbf{x}}) + S({\mathbf{y}})] - 2(p_I + p_E)C({\mathbf{r}}) \right] + 2q\Delta C({\mathbf{r}}),$$ where $\Delta$ is the lattice Laplace operator $$\Delta f({\mathbf{x}}) = -f({\mathbf{x}}) + \frac{1}{4}\sum_{{\mathbf{x}}'} f({\mathbf{x}}').$$ We note that we always have the boundary condition $C({\mathbf{0}}, t) = 1$ for all $t$ (self-correlation is $1$ at all times, as $s({\mathbf{x}})^2=1$).
Stationary-state isogloss density $\sigma$
------------------------------------------
If $C({\mathbf{e}}_1 ,t)$ were known, where ${\mathbf{e}}_1$ is the unit vector $(1,0)$, the isogloss density could be obtained via the identity $$\sigma (t) = \frac{1 - C({\mathbf{e}}_1 ,t)}{2}.$$ Thus, knowing the limiting ($t \to \infty$) value of $C({\mathbf{e}}_1)$ would imply the stationary-state isogloss density $\sigma$.
At the stationary state, $S({\mathbf{x}}) + S({\mathbf{y}}) = 2m$ and $\frac{dC({\mathbf{r}})}{dt} = 0$. Assuming ${\mathbf{r}} \neq {\mathbf{0}}$, Eq. then implies $$0 = (1-q) \left[ 2(p_I - p_E)m - 2(p_I + p_E)C({\mathbf{r}}) \right] + 2q \Delta C({\mathbf{r}}),$$ in other words, $$\label{eq:before-alpha}
\begin{split}
0 &= (1-q) \left[ (p_I - p_E) m - (p_I + p_E) C({\mathbf{r}}) \right] + q\Delta C({\mathbf{r}}) \\
&= (1-q) (p_I + p_E) \left[ \frac{p_I - p_E}{p_I + p_E}m - C({\mathbf{r}}) \right] + q\Delta C({\mathbf{r}}) \\
&= (1-q)(p_I + p_E)\left[ m^2 - C({\mathbf{r}}) \right] + q \Delta C({\mathbf{r}}).
\end{split}$$ In the special case $q = 0$ (i.e., no spatial interaction between neighbouring sites), this implies $C({\mathbf{r}}) = m^2$ for all ${\mathbf{r}} \neq {\mathbf{0}}$. Thus for any two sites ${\mathbf{x}} \neq {\mathbf{y}}$, $\langle s({\mathbf{x}})s({\mathbf{y}})\rangle = m^2$, indicating that spins at different sites are fully uncorrelated. This is what one would expect, as all sites operate independently when $q=0$.
In the special case $q = 1$ (no ingress–egress dynamics within the sites), on the other hand, we obtain $\Delta C({\mathbf{r}}) = 0$ for ${\mathbf{r}} \neq 0$. We also have $C({\mathbf{0}}) = 0$. This implies $C({\mathbf{r}}) = 1$ everywhere, so either all spins are up or all are down. This is the only possible stationary state when the only dynamics is through nearest-neighbour interactions.
Now suppose $0 < q < 1$. Dividing Eq. by $q$ we obtain $$\label{eq:to-be-solved}
\begin{split}
0 &= \frac{1-q}{q}(p_I + p_E) \left[m^2 - C({\mathbf{r}})\right] + \Delta C({\mathbf{r}}) \\
&= \frac{(q-1)(p_I + p_E)}{q} \left[C({\mathbf{r}}) - m^2\right] + \Delta C({\mathbf{r}}) \\
&= \alpha C({\mathbf{r}}) - \alpha m^2 + \Delta C({\mathbf{r}}),
\end{split}$$ where we write $$\label{eq:alpha}
\alpha = \frac{(q-1)(p_I + p_E)}{q}.$$ Now let $$\label{eq:defc}
c({\mathbf{r}}) = C({\mathbf{r}}) - m^2.$$ We note that $\Delta C({\mathbf{r}})=\Delta c({\mathbf{r}})$. To solve Eq. , it then suffices to solve (for ${\mathbf{r}} \neq {\mathbf{0}}$) $$\label{eq:little-c}
\Delta c({\mathbf{r}}) + \alpha c({\mathbf{r}}) = 0$$ subject to the condition $$\label{eq:little-c-condition}
c({\mathbf{0}}) = 1 - m^2.$$
We now first focus on the equation $$\label{eq:G}
\Delta G({\mathbf{r}}) + \alpha G({\mathbf{r}}) = -\delta_{{\mathbf{r}},{\mathbf{0}}},$$ for all ${\mathbf{r}}$ (including ${\mathbf{r}} = {\mathbf{0}}$), and where $\delta_{{\mathbf{x}},{\mathbf{y}}}$ is the Kronecker delta.
Let $G_{\alpha}({\mathbf{r}})$ be a solution of Eq. (\[eq:G\]). Then $$\label{eq:little-c-G}
c({\mathbf{r}}) = (1-m^2) \frac{G_{\alpha}({\mathbf{r}})}{G_{\alpha}({\mathbf{0}})}$$ is a solution of Eqs. and . This can be seen as follows: first, from Eq. , $$c({\mathbf{0}}) = (1-m^2)\frac{G_{\alpha}({\mathbf{0}})}{G_{\alpha}({\mathbf{0}})} = 1-m^2,$$ so the condition in Eq. is met. Second, we need to show that Eqs. and imply $\Delta c({\mathbf{r}}) + \alpha c({\mathbf{r}}) = 0$ for ${\mathbf{r}} \neq {\mathbf{0}}$. For ${\mathbf{r}} \neq {\mathbf{0}}$ we have $$\label{eq:G-with-r-nonzero}
\Delta G({\mathbf{r}}) + \alpha G({\mathbf{r}}) = 0$$ from Eq. . The quantity $c({\mathbf{r}})$ in Eq. is proportional to $G_{\alpha}({\mathbf{r}})$ with a proportionality constant $(1-m^2)/G_{\alpha}({\mathbf{0}})$ which is independent of ${\mathbf{r}}$. Given that $G_{\alpha}({\mathbf{r}})$ fulfills Eq. for ${\mathbf{r}} \neq {\mathbf{0}}$ it is then clear that $c({\mathbf{r}})$ fulfills $\Delta c({\mathbf{r}}) + \alpha c({\mathbf{r}}) = 0$ for ${\mathbf{r}} \neq {\mathbf{0}}$, i.e. Eq. .\
So we are left with the task of finding a solution of $$\label{eq:big-GG}
\Delta G({\mathbf{r}}) + \alpha G({\mathbf{r}}) = -\delta_{{\mathbf{r}},{\mathbf{0}}}.$$ Let us write $G({\mathbf{r}})$ in Fourier representation: $$\label{eq:fourier}
G({\mathbf{r}})=\frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi}dk_1 dk_2~ e^{i{\mathbf{k}} \cdot {\mathbf{r}}} \widehat G({\mathbf{k}}),$$ where ${\mathbf{k}}=(k_1,k_2)$ and $\widehat G({\mathbf{k}})$ is the Fourier transform of $G({\mathbf{r}})$, i.e. $$\widehat G({\mathbf{k}}) = \sum_{{\mathbf{r}}} e^{-i {\mathbf{k}} \cdot {\mathbf{r}}} G({\mathbf{r}}) = \sum_x \sum_y e^{-i x k_1 -i y k_2} G(x,y).$$ Applying this to both sides of Eq. leads to $$\label{eq:help}
\sum_{x,y} e^{-i x k_1 -i y k_2} \left[\Delta G({\mathbf{r}}) + \alpha G({\mathbf{r}})\right]= - 1.$$ Next, notice $$\begin{split}
\sum_{x,y} e^{-i x k_1 -i y k_2} \Delta G(x,y ) &= \frac{1}{4}\sum_{x,y} e^{-i x k_1 -i y k_2} \left[G(x+1,y)+G(x-1,y)\right. + \\
&\quad\quad \left. +G(x,y+1)+G(x,y-1)-4G(x,y)\right] \\
&= \frac{1}{4}\sum_{x,y} \left[e^{-i (x-1) k_1 -i y k_2} +e^{-i (x+1) k_1 -i y k_2} \right. + \\
&\quad\quad \left. +e^{-i x k_1 -i (y-1) k_2} + e^{-i x k_1 -i (y+1) k_2} -4 e^{-i x k_1 -i y k_2}\right ] G(x,y) \\
&= \frac{1}{4}\sum_{x,y} \left[\underbrace{e^{i k_1 } +e^{-i k_1 }}_{=2\cos k_1} +\underbrace{e^{i k_2} + e^{ -i k_2}}_{=2\cos k_2} -4 \right ] e^{-i x k_1 -i y k_2} G(x,y) \\
&= \frac{1}{2}\left[ \cos k_1 + \cos k_2 -2 \right] \sum_{x,y} e^{-i x k_1 -i y k_2} G(x,y) \\
&= \frac{1}{2}\left[ \cos k_1 + \cos k_2 -2 \right] \widehat G({\mathbf{k}}).
\end{split}$$ Using this in Eq. gives $$\left[\frac{1}{2}\left( \cos k_1 + \cos k_2 -2 \right )+\alpha\right] \widehat G({\mathbf{k}})
=-1,$$ in other words $$\widehat G({\mathbf{k}})=\frac{1}{ 1-\alpha-\frac{1}{2}\left[\cos k_1 + \cos k_2\right ] }.$$ Using Eq. we then have $$G_\alpha ({\mathbf{r}}) = \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \frac{e^{i{\mathbf{k}} \cdot {\mathbf{r}}} dk_1 dk_2}{1 - \alpha - \frac{1}{2}(\cos k_1 + \cos k_2)}.$$ Hence $$G_\alpha ({\mathbf{0}}) = \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \frac{dk_1dk_2}{1 - \alpha - \frac{1}{2}(\cos k_1 + \cos k_2)}.$$ The integrand is symmetric with respect to $k_1 \leftrightarrow -k_1$ and $k_2 \leftrightarrow -k_2$ respectively, due to the identity $\cos(k)=\cos(-k)$. Hence $$\begin{split}
G_\alpha ({\mathbf{0}})
&= \frac{1}{\pi^2} \int_0^{\pi} \int_0^{\pi} \frac{dk_1 dk_2}{1 - \alpha - \frac{1}{2}(\cos k_1 + \cos k_2)} \\
&= \frac{1}{1-\alpha} \frac{1}{\pi^2} \int_0^{\pi} \int_0^{\pi} \frac{dk_1 dk_2}{1 - \frac{1}{2(1-\alpha)}(\cos k_1 + \cos k_2)}.
\end{split}$$ This expression is related to the complete elliptic integral of the first kind, see for example Ref. [@Mor1971]. We use the following notation: $$K(z)=\frac{1}{2\pi}\int_0^\pi du \int_0^\pi dv \frac{1}{1-\frac{z}{2}(\cos u + \cos v)}.$$ Hence we have $$\label{eq:gk}
G_\alpha({\mathbf{0}}) = \frac{2}{\pi(1-\alpha)} K\left(\frac{1}{1-\alpha}\right) = \frac{2K_\alpha}{\pi (1-\alpha)},$$ where we abbreviate $$K_\alpha = K\left( \frac{1}{1-\alpha} \right)$$ for convenience.
Next, we write the Laplacian $\Delta G_{\alpha} ({\mathbf{0}})$ in full: $$\Delta G_{\alpha} ({\mathbf{0}}) = -G_{\alpha}({\mathbf{0}}) + \frac{1}{4} [G_{\alpha}(1,0) + G_{\alpha}(-1,0) + G_{\alpha}(0,1) + G_{\alpha}(0,-1)].$$ Assuming isotropy, each term in the square brackets is equal to $G_{\alpha}({\mathbf{e}}_1) = G_{\alpha} (1,0)$. Hence Eq. (\[eq:big-GG\]), evaluated at ${\mathbf{r}}={\mathbf{0}}$, takes the form $$G_{\alpha} ({\mathbf{e}}_1) + (\alpha - 1)G_{\alpha}({\mathbf{0}}) = -1,$$ in other words $$G_{\alpha}({\mathbf{0}}) - G_{\alpha}({\mathbf{e}}_1) = 1 + \alpha G_{\alpha} ({\mathbf{0}}).$$ From this we find $$1 - \frac{G_{\alpha}({\mathbf{e}}_1)}{G_{\alpha}({\mathbf{0}})} = \left[\alpha + \frac{1}{G_{\alpha}({\mathbf{0}})} \right].$$ Now using Eqs. (\[eq:defc\]) and (\[eq:little-c-G\]), $$C({\mathbf{r}}) = m^2 + c({\mathbf{r}}) = m^2 + (1-m^2) \frac{G_{\alpha}({\mathbf{r}})}{G_{\alpha}({\mathbf{0}})}.$$ The stationary-state isogloss density $\sigma$ is then $$\begin{split}
\sigma &= \frac{1}{2} [1 - C({\mathbf{e}}_1)] \\
&= \frac{1}{2} \left[1 - m^2 - (1-m^2) \frac{G_{\alpha}({\mathbf{e}}_1)}{G_{\alpha} ({\mathbf{0}})} \right] \\
&= \frac{1}{2} (1-m^2) \left[1 - \frac{G_{\alpha}({\mathbf{e}}_1)}{G_{\alpha}({\mathbf{0}})} \right] \\
&= \frac{1}{2} (1-m^2) \left[\alpha + \frac{1}{G_{\alpha}({\mathbf{0}})}\right] \\
&= \frac{1}{2} (1-m^2) \left[\alpha + \frac{\pi(1-\alpha)}{2K_{\alpha}}\right],
\end{split}$$ where we have used Eq. (\[eq:gk\]). On the other hand, $$1-m^2 = 1 - \frac{(p_I - p_E)^2}{(p_I + p_E)^2}
= 4\left(\frac{p_I}{p_I + p_E}\right)\left(\frac{p_E}{p_I + p_E}\right)
= 4\rho (1-\rho),$$ so that $$\sigma = 2\rho (1-\rho) \left[\alpha + \frac{\pi (1-\alpha)}{2K_{\alpha}} \right].$$ Recalling that $$\alpha = -\tau = - \frac{(1-q)(p_I + p_E)}{q},$$ see Eq. (\[eq:alpha\]), and defining $\tau=-\alpha$, i.e. $K_{\alpha} = K_{-\tau}$, yields our final equation for the stationary-state isogloss density: $$\label{eq:parabola}
\sigma = 2 H(\tau) \rho (1-\rho),$$ where $$\label{eq:Htau}
H(\tau) = \frac{\pi (1 + \tau)}{2K\left(\frac{1}{1+\tau}\right)} - \tau.$$ The expression in Eq. (\[eq:parabola\]) is a downward-opening parabola in $\rho$ whose height is fixed by $H(\tau)$, where $$\label{eq:tau}
\tau = \frac{(1-q)(p_I + p_E)}{q}.$$
Sanity check: the limits $\tau \to 0$ and $\tau \to \infty$
-----------------------------------------------------------
The complete elliptic integral $K(z)$ has the following known properties [@Fer2016]: $$\lim_{z\to 0} K(z) = \frac{\pi}{2}
\quad\textnormal{and}\quad
\lim_{z\to 1} K(z) = \infty.$$
Now consider the limit $\tau \to 0$. This limit is relevant when either $q\to 1$, or both $p_E\to 0$ and $p_I\to 0$, see Eq. (\[eq:tau\]). These are situations in which the spatial (voter) process dominates the ingress–egress dynamics. In this limit $1/(1+\tau) \to 1$, so that $K_{-\tau} = K(1/(1+\tau)) \to \infty$. Noting that Eq. can be written in the form $$\label{eq:Htau-different-form}
H(\tau) = \tau \left[ \frac{\pi}{2K_{-\tau}} - 1\right] + \frac{\pi}{2K_{-\tau}},$$ we then find that $H(\tau) \to 0$. Thus, in the limit where the spatial (voter) process completely overtakes the ingress–egress process, the stationary-state isogloss density is zero, indicating that all sites agree in their spin.
Next consider $\tau \to \infty$. This limit occurs when $p_I+p_E>0$ and $q\to 0$, i.e. the ingress–egress process dominates. Then $1/(1+\tau) \to 0$, so that $K_{-\tau} \to \pi/2$. From Eq. , we then find that $H(\tau) \to 1$ in this case. Thus, in the limit where the ingress–egress process completely overtakes the spatial process, the stationary-state isogloss density is given by the parabola $\sigma = 2\rho (1-\rho)$, indicating complete independence of the individual spins.
WALS feature levels {#sec:wals-details}
===================
The following list gives the values of the WALS features mined; the italicized part after each value gives its value in our binarization (*present* for ‘feature present’, *absent* for ‘feature absent’ and *N/A* if the WALS value was excluded from the binarization as irrelevant). Languages attesting irrelevant values were excluded from the corresponding feature language sample for the purposes of calculating our statistics.
1. adpositions {#adpositions .unnumbered}
--------------
- WALS feature mined: ‘Person Marking on Adpositions’ (48A)
- Values:
1. No adpositions (*absent*)
2. No person marking (*present*)
3. Pronouns only (*present*)
4. Pronouns and nouns (*present*)
2. definite marker {#definite-marker .unnumbered}
------------------
- WALS feature mined: ‘Definite Articles’ (37A)
- Values:
1. Definite word distinct from demonstrative (*present*)
2. Demonstrative word used as definite article (*present*)
3. Definite affix (*present*)
4. No definite, but indefinite article (*absent*)
5. No definite or indefinite article (*absent*)
3. *hand* and *arm* identical {#hand-and-arm-identical .unnumbered}
-----------------------------
- WALS feature mined: ‘Hand and Arm’ (129A)
- Values:
1. Identical (*present*)
2. Different (*absent*)
4. *hand* and *finger(s)* identical {#hand-and-fingers-identical .unnumbered}
-----------------------------------
- WALS feature mined: ‘Finger and Hand’ (130A)
- Values:
1. Identical (*present*)
2. Different (*absent*)
5. front rounded vowels {#front-rounded-vowels .unnumbered}
-----------------------
- WALS feature mined: ‘Front Rounded Vowels’ (11A)
- Values:
1. None (*absent*)
2. High and mid (*present*)
3. High only (*present*)
4. Mid only (*present*)
6. gender in independent personal pronouns {#gender-in-independent-personal-pronouns .unnumbered}
------------------------------------------
- WALS feature mined: ‘Gender Distinctions in Independent Personal Pronouns’ (44A)
- Values:
1. In 3rd person + 1st and/or 2nd person (*present*)
2. 3rd person only, but also non-singular (*present*)
3. 3rd person singular only (*present*)
4. 1st or 2nd person but not 3rd (*present*)
5. 3rd person non-singular only (*present*)
6. No gender distinctions (*absent*)
7. glottalized consonants {#glottalized-consonants .unnumbered}
-------------------------
- WALS feature mined: ‘Glottalized Consonants’ (7A)
- Values:
1. No glottalized consonants (*absent*)
2. Ejectives only (*present*)
3. Implosives only (*present*)
4. Glottalized resonants only (*present*)
5. Ejectives and implosives (*present*)
6. Ejectives and glottalized resonants (*present*)
7. Implosives and glottalized resonants (*present*)
8. Ejectives, implosives, and glottalized resonants (*present*)
8. grammatical evidentials {#grammatical-evidentials .unnumbered}
--------------------------
- WALS feature mined: ‘Semantic Distinctions of Evidentiality’ (77A)
- Values:
1. No grammatical evidentials (*absent*)
2. Indirect only (*present*)
3. Direct and indirect (*present*)
9. indefinite marker {#indefinite-marker .unnumbered}
--------------------
- WALS feature mined: ‘Indefinite Articles’ (38A)
- Values:
1. Indefinite word distinct from ‘one’ (*present*)
2. Indefinite word same as ‘one’ (*present*)
3. Indefinite affix (*present*)
4. No indefinite, but definite article (*absent*)
5. No definite or indefinite article (*absent*)
10. inflectional morphology {#inflectional-morphology .unnumbered}
---------------------------
- WALS feature mined: ‘Prefixing vs. Suffixing in Inflectional Morphology’ (26A)
- Values:
1. Little affixation (*absent*)
2. Strongly suffixing (*present*)
3. Weakly suffixing (*present*)
4. Equal prefixing and suffixing (*present*)
5. Weakly prefixing (*present*)
6. Strong prefixing (*present*)
11. inflectional optative {#inflectional-optative .unnumbered}
-------------------------
- WALS feature mined: ‘The Optative’ (73A)
- Values:
1. Inflectional optative present (*present*)
2. Inflectional optative absent (*absent*)
12. lateral consonants {#lateral-consonants .unnumbered}
----------------------
- WALS feature mined: ‘Lateral Consonants’ (8A)
- Values:
1. No laterals (*absent*)
2. /l/, no obstruent laterals (*present*)
3. Laterals, but no /l/, no obstruent laterals (*present*)
4. /l/ and lateral obstruent (*present*)
5. No /l/, but lateral obstruents (*present*)
13. morphological second-person imperative {#morphological-second-person-imperative .unnumbered}
------------------------------------------
- WALS feature mined: ‘The Morphological Imperative’ (70A)
- Values:
1. Second singular and second plural (*present*)
2. Second singular (*present*)
3. Second plural (*present*)
4. Second person number-neutral (*present*)
5. No second-person imperatives (*absent*)
14. order of adjective and noun is AdjN {#order-of-adjective-and-noun-is-adjn .unnumbered}
---------------------------------------
- WALS feature mined: ‘Order of Adjective and Noun’ (87A)
- Values:
1. Adjective-Noun (*present*)
2. Noun-Adjective (*absent*)
3. No dominant order (*N/A*)
4. Only internally-headed relative clauses (*N/A*)
15. order of degree word and adjective is DegAdj {#order-of-degree-word-and-adjective-is-degadj .unnumbered}
------------------------------------------------
- WALS feature mined: ‘Order of Degree Word and Adjective’ (91A)
- Values:
1. Degree word-Adjective (*present*)
2. Adjective-Degree word (*absent*)
3. No dominant order (*N/A*)
16. order of genitive and noun is GenN {#order-of-genitive-and-noun-is-genn .unnumbered}
--------------------------------------
- WALS feature mined: ‘Order of Genitive and Noun’ (86A)
- Values:
1. Genitive-Noun (*present*)
2. Noun-Genitive (*absent*)
3. No dominant order (*N/A*)
17. order of numeral and noun is NumN {#order-of-numeral-and-noun-is-numn .unnumbered}
-------------------------------------
- WALS feature mined: ‘Order of Numeral and Noun’ (89A)
- Values:
1. Numeral-Noun (*present*)
2. Noun-Numeral (*absent*)
3. No dominant order (*N/A*)
4. Numeral only modifies verb (*N/A*)
18. order of object and verb is OV {#order-of-object-and-verb-is-ov .unnumbered}
----------------------------------
- WALS feature mined: ‘Order of Object and Verb’ (83A)
- Values:
1. OV (*present*)
2. VO (*absent*)
3. No dominant order (*N/A*)
19. order of subject and verb is SV {#order-of-subject-and-verb-is-sv .unnumbered}
-----------------------------------
- WALS feature mined: ‘Order of Subject and Verb’ (82A)
- Values:
1. SV (*present*)
2. VS (*absent*)
3. No dominant order (*N/A*)
20. ordinal numerals {#ordinal-numerals .unnumbered}
--------------------
- WALS feature mined: ‘Ordinal Numerals’ (53A)
- Values:
1. None (*absent*)
2. One, two, three (*present*)
3. First, two, three (*present*)
4. One-th, two-th, three-th (*present*)
5. First/one-th, two-th, three-th (*present*)
6. First, two-th, three-th (*present*)
7. First, second, three-th (*present*)
8. Various (*present*)
21. passive construction {#passive-construction .unnumbered}
------------------------
- WALS feature mined: ‘Passive Constructions’ (107A)
- Values:
1. Present (*present*)
2. Absent (*absent*)
22. plural {#plural .unnumbered}
----------
- WALS feature mined: ‘Coding of Nominal Plurality’ (33A)
- Values:
1. Plural prefix (*present*)
2. Plural suffix (*present*)
3. Plural stem change (*present*)
4. Plural tone (*present*)
5. Plural complete reduplication (*present*)
6. Mixed morphological plural (*present*)
7. Plural word (*present*)
8. Plural clitic (*present*)
9. No plural (*absent*)
23. possessive affixes {#possessive-affixes .unnumbered}
----------------------
- WALS feature mined: ‘Position of Pronominal Possessive Affixes’ (57A)
- Values:
1. Possessive prefixes (*present*)
2. Possessive suffixes (*present*)
3. Prefixes and suffixes (*present*)
4. No possessive affixes (*absent*)
24. postverbal negative morpheme {#postverbal-negative-morpheme .unnumbered}
--------------------------------
- WALS feature mined: ‘Postverbal Negative Morphemes’ (143F)
- Values:
1. VNeg (*present*)
2. (*present*)
3. VNeg&\[V-Neg\] (*present*)
4. None (*absent*)
25. preverbal negative morpheme {#preverbal-negative-morpheme .unnumbered}
-------------------------------
- WALS feature mined: ‘Preverbal Negative Morphemes’ (143E)
- Values:
1. NegV (*present*)
2. (*present*)
3. NegV&\[Neg-V\] (*present*)
4. None (*absent*)
26. productive reduplication {#productive-reduplication .unnumbered}
----------------------------
- WALS feature mined: ‘Reduplication’ (27A)
- Values:
1. Productive full and partial reduplication (*present*)
2. Full reduplication only (*present*)
3. No productive reduplication (*absent*)
27. question particle {#question-particle .unnumbered}
---------------------
- WALS feature mined: ‘Position of Polar Question Particles’ (92A)
- Values:
1. Initial (*present*)
2. Final (*present*)
3. Second position (*present*)
4. Other position (*present*)
5. In either of two positions (*present*)
6. No question particle (*absent*)
28. shared encoding of nominal and locational predication {#shared-encoding-of-nominal-and-locational-predication .unnumbered}
---------------------------------------------------------
- WALS feature mined: ‘Nominal and Locational Predication’ (119A)
- Values:
1. Different (*absent*)
2. Identical (*present*)
29. tense-aspect inflection {#tense-aspect-inflection .unnumbered}
---------------------------
- WALS feature mined: ‘Position of Tense-Aspect Affixes’ (69A)
- Values:
1. Tense-aspect prefixes (*present*)
2. Tense-aspect suffixes (*present*)
3. Tense-aspect tone (*present*)
4. Mixed type (*present*)
5. No tense-aspect inflection (*absent*)
30. tone {#tone .unnumbered}
--------
- WALS feature mined: ‘Tone’ (13A)
- Values:
1. No tones (*absent*)
2. Simple tone system (*present*)
3. Complex tone system (*present*)
31. uvular consonants {#uvular-consonants .unnumbered}
---------------------
- WALS feature mined: ‘Uvular Consonants’ (6A)
- Values:
1. None (*absent*)
2. Uvular stops only (*present*)
3. Uvular continuants only (*present*)
4. Uvular stops and continuants (*present*)
32. velar nasal {#velar-nasal .unnumbered}
---------------
- WALS feature mined: ‘The Velar Nasal’ (9A)
- Values:
1. Initial velar nasal (*present*)
2. No initial velar nasal (*present*)
3. No velar nasal (*absent*)
33. verbal person marking {#verbal-person-marking .unnumbered}
-------------------------
- WALS feature mined: ‘Alignment of Verbal Person Marking’ (100A)
- Values:
1. Neutral (*absent*)
2. Accusative (*present*)
3. Ergative (*present*)
4. Active (*present*)
5. Hierarchical (*present*)
6. Split (*present*)
34. voicing contrast {#voicing-contrast .unnumbered}
--------------------
- WALS feature mined: ‘Voicing in Plosives and Fricatives’ (4A)
- Values:
1. No voicing contrast (*absent*)
2. In plosives alone (*present*)
3. In fricatives alone (*present*)
4. In both plosives and fricatives (*present*)
35. zero copula for predicate nominals {#zero-copula-for-predicate-nominals .unnumbered}
--------------------------------------
- WALS feature mined: ‘Zero Copula for Predicate Nominals’ (120A)
- Values:
1. Impossible (*absent*)
2. Possible (*present*)
[^1]: Corresponding author: [email protected]
[^2]: Front rounded vowels are the fourth most stable feature (out of 86) in Dediu’s study (Ref. [@Ded2011], Table S4) and the second most stable (out of 62) in Dediu and Cysouw’s metastudy (Ref. [@DedCys2013], Table 7).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Deep Learning (DL) is one of the hottest trends in machine learning as DL approaches produced results superior to the state-of-the-art in problematic areas such as image processing and natural language processing (NLP). To foster the growth of DL, several open source frameworks appeared providing implementations of the most common DL algorithms. These frameworks vary in the algorithms they support and in the quality of their implementations. The purpose of this work is to provide a qualitative and quantitative comparison among three of the most popular and most comprehensive DL frameworks (namely Google’s TensorFlow, University of Montreal’s Theano and Microsoft’s CNTK). The ultimate goal of this work is to help end users make an informed decision about the best DL framework that suits their needs and resources. To ensure that our study is as comprehensive as possible, we conduct several experiments using multiple benchmark datasets from different fields (image processing, NLP, etc.) and measure the performance of the frameworks’ implementations of different DL algorithms. For most of our experiments, we find out that CNTK’s implementations are superior to the other ones under consideration.'
author:
- |
Ghadeer Al-Bdour\
Faculty of Computer and Information Technology\
Jordan University of Science and Technology\
Irbid, Jordan\
`[email protected]`\
Raffi Al-Qurran\
Faculty of Computer and Information Technology\
Jordan University of Science and Technology\
Irbid, Jordan\
`[email protected]`\
Mahmoud Al-Ayyoub\
Faculty of Computer and Information Technology\
Jordan University of Science and Technology\
Irbid, Jordan\
`[email protected]`\
Ali Shatnawi\
Faculty of Computer and Information Technology\
Jordan University of Science and Technology\
Irbid, Jordan\
`[email protected]`\
title: A Detailed Comparative Study of Open Source Deep Learning Frameworks
---
Introduction {#sec:intro}
============
Deep Learning (DL) is the hottest trend in Machine Learning (ML). The idea of DL is to train a multi-layer Artificial Neural Network (ANN) on a dataset in order to allow it to handle real world tasks. Although the theoretical concepts behind DL are not new, DL has enjoyed a surge of interest over the past decade due to many factors including its successful application to several problems (many of which have great commercial potentials) and the improved affordability of the required computing infrastructure.
DL approaches have significantly outperformed state-of-the-art approaches in many classical problems of many fields such as image processing, computer vision, speech processing, natural language processing (NLP), etc. Moreover, the scientific community (from both the academia and the industry) has quickly and massively adopted DL. Open source implementations of successful DL algorithms quickly appeared on code sharing websites, and were subsequently used by many researchers in different fields.
Several DL frameworks exist such as TensorFlow, Theano, CNTK, Caffe, Torch, Neon, pylearn, etc. Each one of these frameworks has different features and performance characteristics. Further, each framework utilizes different techniques to optimize its implementation of DL algorithms. Although the same algorithm is implemented in different frameworks, the performance of the different implementations can vary greatly. A researcher/implementer willing to employ such an algorithm in his research or application faces a difficult choice, since the number of different implementations is high and the effort invested by the research community in scientifically comparing these implementations is limited.
In this work, we aim at providing qualitative and quantitative comparison between popular open source DL frameworks. To be more specific, we focus on three very popular DL frameworks, namely Theano (from LISA Lab, University of Montreal), TensorFlow (from Google), and CNTK (from Microsoft). These frameworks support multi-core CPUs as well as multiple GPUs. All of them import cuDNN, which is a DL library from NVIDIA that supports highly tuned implementations for standard routines such as forward and backward convolution, normalization, pooling, and activation layers. We compare these frameworks by training different neural network (NN) architectures on five different standard benchmark datasets for various tasks in image processing, computer vision and NLP. In an earlier version of this work [@shatnawi2018comparative], we only considered two datasets. Despite their importance, comparative studies like ours that focus on performance issues are rare. A comparative study of the frameworks is important in order to enable people who are interested in applying DL in their research and/or applications to make informed decisions about which of the existing frameworks suits their needs. The rest of this paper is organized as follows. The following section discusses the existing efforts on comparing DL frameworks highlighting what distinguishes our work. Sections \[sec:related\] presents our survey of the literature, while Section \[sec:nn\] explains in details what NN are. Section \[sec:frameworks\] discusses the frameworks, the way they were used to train the datasets and a brief comparison between them. The methodology we follow is discussed in Section \[sec:method\]. Experimental results and the discussion are detailed in Sections \[sec:method\] and \[sec:disc\]. The work is concluded with final thoughts presented in Section \[sec:conc\].
Literature Survey {#sec:related}
=================
Only few researchers did a comparative study between state-of-the-art DL frameworks running on different hardware platforms (CPU and GPU) to highlight the advantages and limitations for each framework when applied on different deep NN architectures, and to enable developers to optimize the running performance of DL frameworks.
One of the first and most prominent examples of comparative studies between DL frameworks was carried out by Bahrampour et al. [@bahrampour2015comparative]. The authors compared five DL frameworks: Caffe, Neon, TensorFlow, Theano, and Torch, in terms of speed (gradient computation time and forward time), hardware utilization and extensibility (ability to support different types of DL architectures) after applying various convolutional algorithms on the aforementioned frameworks. They conducted their experiments on a single machine for both CPU (multithreaded) and GPU (NVIDIA Titan X) environments. The comparison between frameworks was carried out by training convolutional and stacked autoencoder (AE) networks on the MNIST and ImageNet datasets. They also trained long short-term memory (LSTM) networks [@hochreiter1997long] on the IMDB dataset [@maas2011learning].[^1] The authors reported several observations/findings. In terms of extensibility, they found that Theano and Torch were the best as they can support various DL architectures and libraries. Also, they found that TensorFlow was a very flexible framework especially when used in different parts of the computational graph. Finally, emphasizing ease of use, they noticed that Caffe was the easiest. In terms of performance, they noticed that Torch was the best for training and testing their DL architectures on a CPU platform. Theano came in second while Neon gave the worst performance. On a GPU platform, for convolutional and fully connected networks, they found that Torch was the best followed by Theano. Also, they noticed that Theano was the fastest on small networks and Torch was the fastest on large networks followed by Neon. For recurrent networks (LSTM), they found that Theano’s results were the best in terms of performance. TensorFlow on single GPU was the worst compared to other studied frameworks.
Shi et al. [@shi2016benchmarking] did a comparative study between several DL frameworks including Caffe, MXNet, CNTK, TensorFlow, and Torch. They considered three types of NN including; fully connected NN (FCN), convolutional NN (CNN) and recurrent NN (RNN). Moreover, they used different hardware environments including two CPU platforms and three GPU platforms. They considered the running time and the convergence rate as the metrics to evaluate the selected frameworks. They used synthetic datasets to measure running time performance and real-world datasets to measure the convergence rate in their experiments. The results were as follows. In synthetic datasets, they evaluated the performance of FCN using a large NN (FCN-s). They used AlexNet and ResNet-50 on ImageNet dataset. For real-world datasets, they applied MNIST dataset using a small FCN (FCN-R). Also, they applied CIFAR10 dataset using AlexNet-R and ResNet-56. For RNN, they chose two LSTM layers for testing. After experimentation, they found that all tested frameworks achieved significant speedup using GPU over CPU. For CPU platform, they found that TensorFlow was the best compared to other tools. On a single GPU, Caffe, CNTK and Torch performed better than MXNet and TensorFlow on FCN implementations. For small CNN, Caffe and CNTK achieved good performances. For RNN (LSTM), CNTK was the fastest as it was five to ten times better than the other tools. Finally, on multi-GPU platforms, all implementations had higher throughput and convergence rate.
Goldsborough [@goldsborough2016tour] showed the timeline of ML software libraries for DL. He focused on TensorFlow’s results and its basic properties including computational paradigms, its distributed model and programming interface. He compared TensorFlow to other DL frameworks including Theano, Torch and Caffe, qualitatively and quantitatively. In qualitative terms, he compared aforementioned frameworks using several categories including frontends, programming model style, how gradients are computed, and distributing the execution of computational graph. Table \[table:peter\] shows a summary of this comparison. In quantitative terms, he reviewed works (such as [@bahrampour2015comparative; @chintalaconvnet; @al2016theano]), which contain comparisons between TensorFlow and other DL frameworks. From LeNet benchmark in [@samuel1959some], he noted that TensorFlow was ranked second after Torch in forward and backward measures, but, in terms of performance, TensorFlow came at the last rank compared to tested frameworks. From the results on [@hinton2006reducing]’s benchmark for convolutional network models, he noted that TensorFlow came at the second place behind Torch in forward and backward propagation time. Finally, in [@martinez2013learning], the benchmarks were CNN models including AlexNet architecture, and an LSTM network operating on the Penn TreeBank dataset [@marcus1993building]. He noted that TensorFlow was the best framework for small model followed by Theano then Torch. For large models, TensorFlow came at the second rank after Theano, and Torch came at the last place.
Library Frontends Style Gradients Distributed Execution
------------ ------------ ------------- ----------- -----------------------
TensorFlow Python,C++ Declarative Symbolic
Theano Python Declarative Symbolic X
Torch LuaJIT Imperative Explicit X
Caffe Protobuf Imperative Explicit X
: Goldsborough’s qualitative comparison of DL frameworks[]{data-label="table:peter"}
Chintala [@chintalaconvnet] applied different ImageNet benchmarks for a variety convolutional network types including AlexNet, GoogleNet, Overfeat, and OxfordNet using different open source DL frameworks such as Caffe, Theano, Torch, TensorFlow, Chainer, etc. He conducted his experiments on NVIDIA Titan X GPU and two new packages for Fast Fourier Transform (FFT) computation [@vasilache2014fast]. The first one is based on the NVIDIA library (cuFFT) and another on a Facebook FFT (fbfft). Moreover, he used native version of each DL frameworks. After experimentation, he found that using fbfft resulted in a speedup over cuFFT for all applied CNNs. Also, he found that the fastest framework for CNN was Torch followed by Tensorflow.
A Theano development team of Al-Rfou et al. [@al2016theano] discussed the Theano framework, its features, how to use it, and showed recent improvements on it. They did a performance comparison between Theano and other frameworks including Torch and TensorFlow on three types of ML models including CNN, RNN and sequence-to-sequence mapping RNN. Finally, they showed the computation speed using multiple GPUs. The results were as follows. On a single GPU platform, for CNN, they found the processing time for four different convolutional models (AlexNet [@krizhevsky2014one], OverFeat [@sermanet2013overfeat], VGG [@simonyan2014very], and GoogLeNet [@szegedy2015going]) on ImageNet dataset. They reported results for each framework per minibatch for forward and backward pass. They found that Theano was slower than Torch and TensorFlow. However, in overall performance-wise, they were close to each other. For RNN, they used LSTM on Penn TreeBank dataset [@marcus1993building] and reported the results on small, medium and large LSTM models. They found that Theano was in the second place after TensorFlow for the small model, but Theano was the fastest for the medium and large models. They also showed that Torch was slower than Theano on all tested models. Finally, for sequence-to-sequence model [@yao2015describing], the input was video frames and the output was the English sentence describing the input. The input video frame was preprocessed by a GoogLeNet (pre-trained for classification on ImageNet). They compared Theano to TensorFlow and excluded Torch because there was no available implementation in Torch. They reported the processing time per minibatch using three different batch sizes (32, 64 and 128). They found that Theano was the fastest on smaller batches. However, on large ones, TensorFlow was the superior one. They repeated the previous RNN model (LSTM) on multi-GPU platforms (2-GPUs and 4-GPUs) using platoon. Their measured processing speed when synchronizing after each batch was found to provide speedups between 1.6X and 1.7X for 2-GPUs and 3.2X for 4-GPUs. When synchronizing after every 100 batches, they found a 2X speedup for 2-GPUs and 3.9X-4X speedup for 4-GPUs.
Kovalev et al. [@kovalev2016deep] presented a comparative study between five DL frameworks: Theano, Torch, Caffe, TensorFlow, and DeepLearning4J, in terms of training and prediction speed and classification accuracy. They used MNIST dataset of handwritten digits for testing five FCN frameworks. Their computation experiments were applied only on CPU; they reported out the results for two kinds of scaling factors applied on FCN networks including changing the network’s *depth* (number of internal layers) and changing the network’s *width* (number of neurons). Also, they tested the NN with two different activation functions, Tanh and ReLU. In Tanh nonlinearity function, they found that the training time is approximately 30 seconds when they changed the number of layers from one to four in all frameworks except for DeepLearning4J. For DeepLearning4J, the training time started from 140 seconds for one layer and grew up to 210 seconds when they used four layers. For the prediction time, they found that Theano, Torch, Caffe, and TensorFlow consumed less than 0.4 second. However, for DeepLearning4J, it started from 0.75 second when using one layer and increased up to 1.1 second for four layers. In terms of classification accuracy, they found that Theano, DeepLearning4J, and Caffe achieved high accuracy starting from 94% for one layer and going up to 98% for four layers. For Torch and TensorFlow, the accuracy dropped with increasing the number of network layers. With ReLU nonlinearity function, the training time was much lower compared with the case of Tanh function. As for the prediction time, they observed that the use of ReLU gave results that were similar to Tanh results. In terms of accuracy, they found that Torch’s accuracy grew up while increasing the number of layers, whereas other frameworks had the same behavior as with Tanh. Finally, they changed the number of neurons in the hidden layers of networks for ReLU function only, and reported the speed and accuracy values. They found that DeepLearning4J framework was the slowest in training and prediction times, and time consumed increased with the increase in the number of neurons. The final classification accuracy when changing the internal layer sizes from 64 to 1024 neurons remained around 97% for Theano, Caffe, TensorFlow, and DeepLearning4J. However, in the case of Torch, the classification accuracy grew with the growing layer size (started from 70% and reached 98%).
Bastien et al.[@bastien2012theano] suggested a new feature to be added to the main features of Theano in order to improve its performance in different benchmarks. They conducted a comparative study (in terms of features and performance) between Theano and Torch7 on NN benchmarks, and between Theano and RNNLM on RNN benchmarks. In their comparison, they used three learning architectures: *logistic regression*, *NN with one hidden layer* (500 units) and *DNN with three hidden layers* (1000 hidden units each). They found that when applying NN on CPU using one hidden layer models without using mini-batches, Theano’s results overcome Torch7. However, on the logistic regression benchmark, Torch7 thrived because, at each call, it decreased the amount of performed computation. On the GPU, with batch size equal to one, Torch7 overcame Theano. When using mini-batches, Theano was faster than (or has an equivalent speed to) Torch7 on all three learning architectures under consideration. When they applied RNN on Theano and RNNLM with batch size of one, they found that RNNLM was faster than Theano on smaller models. However, for bigger sizes, Theano was faster. Ding et al. [@ding2014theano] introduced an implementation of Theano-based AlexNet on ImageNet dataset and applied it on multiple GPUs in order to accelerate the training process. They compared their results of Theano to Caffe library which runs on single GPU in terms of training time. On a single GPU with batch size equal to 256, Caffe showed to be faster than Theano. However, on 2-GPUs with batch size equal to 128, they found that Theano was faster than Caffe on 1-GPU.
In previous studies, the comparison goal focused only on processing time. None of those comparative studies dealt with CPU and GPU utilization or memory consumption. This work covered these metrics to find which of the considered frameworks achieve the best performance. Finally and most importantly, the comparisons involved more datasets from more fields compared with previous studies.
Neural Networks {#sec:nn}
===============
We start this section with a glimpse of history related to neural networks.
Single Layer Neural Network
---------------------------
A single layer NN is a network that consists of a single hidden layer between the input layer and the output layer. The hidden layer has many units called neurons. See Figure \[fig:fig8\].
![Single layer Neural Network []{data-label="fig:fig8"}](fig8){width="50.00000%"}
### Artificial Neurons
A neuron is one of the most important parts of human brain, where a bunch of neurons build what is called a neural network. This network with its tiny components processes and transmits signals in a form of electrical and chemical pulses. These signals are passed through the neurons via synapses. Same theory applies to artificial neurons where these units when combined with each other perform as a single impenetrable network which takes input signals and performs complex calculations to produce output. This process is depicted in Equation \[eq:an\], where $w_i$ represents connection weights and $x_i$ represents the input values. The left side of the equation $Y$ represents the outputs.
$$Y = f (\sum_{i=0}^{n} w_ix_i)
\label{eq:an}$$
### The Perceptron
The perceptron algorithm was devised in 1957 by Rosenblatt [@rosenblatt1957perceptron] for image recognition tasks. A perceptron is the basic form of a NN as it does not contain any hidden layer and the input signals are fed directly to output neurons using a series of weights as shown in Figure \[fig:fig10\]. The perceptron contains only an input layer and one output layer that consists of one or more artificial neurons.
![Rosenblatt’s perceptron[]{data-label="fig:fig10"}](fig10){width="60.00000%"}
A perceptron is considered a binary classifier as it has only two possible results 1 or 0, which is determined by computing a single output from multiple input values. This is done by evaluating a linear combination of input values and their weights and then putting the output through some nonlinear activation function like the Heaviside step function (as Threshold Function) [@debnath2014integral], as shown in Figure \[fig:fig11\]. The output neuron in the output layer connects to all inputs to produce one output value.
In spite of the successes of the complex architectures of DL, the perceptron is still used because of the simplicity of its architecture and its “light” computation requirement allowing its efficient use with very large datasets. The main drawback of the perceptron is that it only learns linearly-separable functions. In order to solve this problem, a multilayer perceptron (also known as Deep Neural Network (DNN)) was suggested by Ivakhnenko et al. [@ivakhnenko1966cybernetic] to get more powerful learning mechanisms.
![Perceptron’s activation function[]{data-label="fig:fig11"}](fig11){width="50.00000%"}
### Activation Function
An activation function is a nonlinear function that is used in different types of NN. It takes weighted input data and transforms them into a nonlinear output by performing some mathematical operations on them such as matrix multiplication between inputs and weights. Regarding DL implementations, nonlinear activation functions create complex features with every layer. Implementations with a linear activation function would behave like a single-layer network (no matter how many hidden layers) because summing these layers would give just another linear function. This is the reason why nonlinear activation functions are used more widely at DL networks. However, it is possible that some NN may contain neurons with linear activation function in the output layer. These neurons require a nonlinear activation function in previous parts of the network. There are several types of activation functions including the sigmoid function shown in Figure \[fig:fig3\].
![Sigmoid function []{data-label="fig:fig3"}](fig3){width="50.00000%"}
The sigmoid function has the following mathematical form. $$\sigma(x)=\frac{1}{1+e^{-x}}$$ It takes a real-value number and converts it into values in the range \[0, 1\] (large negative numbers convert to 0 and large positive numbers convert to 1). Its main drawbacks are the vanishing gradient problem and the fact that its output is not zero-centered.
Another activation function called *Tanh* is shown in Figure \[fig:fig4\]. $$Tanh(x)=2\sigma (2x)-1$$ It is a nonlinear activation function that takes a real-value number and converts it into the range \[-1, 1\]. It causes the vanishing gradient like sigmoid, but its output is zero-centered which enables the Tanh nonlinearity to be used more widely than the sigmoid function.
![Tanh function []{data-label="fig:fig4"}](fig4){width="50.00000%"}
The *Rectified Linear Unit (ReLU)* activation function (Figure \[fig:fig5\]) has formula as in the following equation. $$f(x)=\max(0,x)$$ It was found to be more accelerated than the Tanh and sigmoid functions due to its linear and non-saturating form. Moreover, it can be implemented in a less expensive way compared with Tanh and sigmoid. Unfortunately, ReLU unit may “die” during training where it outputs zero value for any given input. This happens when the input to its units are negative or after a large negative bias value is inputted for its weights (gradients will be zero). ReLUs cannot recover from this problem because they will not modify the weights (block backpropagation).
![ReLU function []{data-label="fig:fig5"}](fig5){width="50.00000%"}
The *Maxout* [@goodfellow2013maxout] is a nonlinear activation function that applies dot product between the weights and data, and its output is the maximum of a set of inputs. The Maxout neuron uses the following formula. $$\max(w_{1}^{T}x+b_1,w_{2}^{T}x+b2)$$
It is to be noted that the ReLU function is a special case of the Maxout function. The Maxout neuron has all the benefits of a ReLU unit and does not have its drawbacks (dying unit), which makes Maxout one of the most common activation functions used in DL networks.
Another form of activation functions is called the Logistic Regression. It is a regression model developed by Cox in 1958 [@cox1958regression] that estimates the relationship between statistical input variables to make prediction of an output variable. It uses the logistic sigmoid function to generate a prediction as to which of multiple classes the input data belongs. Logistic regression is used for different areas like medical and social sciences where it is used for analytical purposes and interpretation of results from experiments. It is used for very large datasets because of its simplicity and speed. The final layer at a DL algorithm can be constructed using a logistic regression, where the network has multiple feature learning layers that pass features into a logistic regression layer to classify inputs.
Logistic regression can be binomial, ordinal or multinomial. In the binomial type, the observed output for a dependent variable has two possibilities, either zero or one. In multinomial logistic regression, the output can have more than two types (e.g., “Disease A” vs “Disease B” vs “Disease C”) which are not ordered. In ordinal type, the dependent variables must be ordered.
In linear regression algorithms (using least square), Gradient Descent (GD) is an optimization algorithm that is used to find the values of parameters in a way that minimizes the cost function and the least square error. If a huge dataset is trained using GD, calculating the parameters will be expensive and take long time. If there are millions of sample points, every iteration, GD must go through these points to calculate the parameters. To solve these problems, a variation of GD called Stochastic GD (SGD) is used. SGD is an optimization method used to train models including support vector machines (SVM), logistic regression, graphical models, etc. To calculate the parameters in SGD, a sample of training set or one training value is used instead of using the entire sample in every iteration. This method is much faster and less costly than GD.
Backpropagation of errors is a learning method that is used to train NN. It is used along with an optimization algorithm such as GD. The modern version of backpropagation was proposed by Linnainmaa [@linnainmaa1970representation], where he published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions. Backpropagation repeatedly performs two steps for training the network; propagation and weights update. At the first step, the input vector is forward propagated layer-by-layer until it reaches the output layer and produces the output of this vector. For each of the neurons in the output layer, an error value is calculated using a loss function, which compares the output of the network to the desired output. It then calculates the gradient of the loss function with respect to the weights. Backpropagation sends these error values backwards starting from the output layer until it reaches the first layer. At the second step, this gradient value is fed to an optimization method (e.g., GD) to update the weights in order to minimize the loss function. Backpropagation is considered as a supervised learning method, because it requires the knowledge of the desired output to calculate the loss function gradient, but some unsupervised networks such as autoencoders (AE) can use it.
The rapid improvement in DL methods makes the training of any DL network a complex and time consuming process. To address these issues, many software tools (frameworks) have appeared to develop these methods in an easy and efficient manner.
Multilayer Neural Networks
--------------------------
A multilayer NN is a network consisting of more than one hidden layer between the input layer and the output layer, as shown in Figure \[fig:fig9\].
![Multilayer Neural Network[]{data-label="fig:fig9"}](fig9){width="50.00000%"}
One popular example is the Multilayer Perceptron (MLP) network, which consists of an input layer, one or more hidden layers of computation neurons, and an output layer. MLP can learn linear and nonlinear functions in contrast to the single layer perceptron that only supports linear functions. MLP has a large number of features. Also, it uses backpropagation technique for training the network. Each node in its hidden layers is a neuron that applies a nonlinear activation function. The input values are passed from the input nodes to the first hidden layer, which applies some calculations to them using the activation function. The resulting signals are then passed as input signals to the next hidden layer. This procedure is repeated until the signals reach the output layer.
Deep Learning (DL)
------------------
The wide adoption of Deep Neural Networks (DNN) gave rise to new field of *Deep Learning* (DL). The DNN is simply a NN with more than one hidden layer of nonlinear processing units which extract the features by passing input data from a layer to another until a desirable output is produced. One of the most used algorithms in DL is the Backpropagation algorithm, which is used to train the network by updating the weights of the connections between sequential hidden layers. This process is repeated many times until the output matches the desired output.
There are several types of DL architectures such as DNN, Convolutional DNN (CDNN), Deep Belief Networks (DBN) and Recurrent NN (RNN). Approaches based on these architectures have been achieving significant performance improvements when applied to several tasks in speech recognition, computer vision, natural language processing (NLP), etc.
More than half a century ago, Ivakhnenko et al. [@ivakhnenko1966cybernetic] introduced deep MLP (Figure \[fig:fig1\]), where thin but deep models (three hidden layers) with polynomial activation functions were used. The authors used statistical methods to select the best features in each layer, and forwarded these features to the next layer until the output layer is reached. Finally, they used layer-by-layer backpropagation algorithm to train the network. A deeper network with eight layers was introduced in [@ivakhnenko1971polynomial] which was trained using the Group Method of Data Handling (GMDH) algorithm. In 1980, a network with multiple convolutional and pooling layers was introduced in [@fukushima1982neocognitron], where it was trained using a reinforcement learning. The challenge for this model was the training of the multiple layers. At that time, backpropagation of errors was an inefficient and incomplete form to train such deep models. LeCun in 1989 gave the first efficient and practical application of backpropagation at Bell Labs [@lecun1989backpropagation]. He applied backpropagation to a deep convolutional network in order to classify the handwritten digits of the MNIST dataset. This approach achieved good results. Unfortunately, it consumed a lot of time which made it useless for many years later. In 1993, recurrent neural networks (RNNs) were introduced to solve the time consumption problem. RNNs learned by unsupervised learning, which was implemented and used with very deep learning tasks (more than 1,000 subsequent layers) [@hochreiter1997long]. After that, a DL method called the long short-term memory (LSTM) for RNN was proposed by Hochreiter and Schmidhuber in 1997 [@hochreiter1997long], where it was used in the deep learning tasks that require memories of events (like speech). Also, it avoided the vanishing gradient problem at which no learning signals reached to early layers in the network during training the deep network.
![The architecture of the first known deep network by Ivakhnenko[]{data-label="fig:fig1"}](fig1){width="50.00000%"}
A big shift in the field of DL occurred when more people started to use the graphics processing units (GPUs) in the training process. This increased the computational speed allowing NN to produce better results by using more training data. However, training using huge amounts of data brought back to light the vanishing gradient problem. To solve this problem, the model was learned in a layer-by-layer fashion using unsupervised learning. This requires the features of early layers to be initialized with suitable features beforehand (pre-trained). However, in supervised learning, the features at early layers need to be adjusted during learning process. One solution, which is the pre-training solution, was initially developed for RNN in 1992 [@schmidhuber1992learning] and for feedforward networks in 2006 [@hinton2006reducing]. A second solution for the vanishing gradient problem in RNN was the LSTM [@hochreiter1997long].
In the year 2011, the rapid increase in speed of GPUs reached its glory which led many researchers such as Ciresan et al. to train deep networks without using pre-training techniques and started to introduce deep learning networks that were constructed from convolutional layers, max-pooling layers, and several fully connected layers followed by a final classification layer [@ciresan2011flexible; @martinez2013learning]. Moreover, Krizhevsky et al. used a similar architecture with rectified linear activation functions and dropout function [@krizhevsky2012imagenet]. Since then, the research of DL field using GPUs has accelerated rapidly.
Convolutional Neural Networks (CNN)
-----------------------------------
Convolution, which is widely used in DL networks, is a mathematical process used to mix input data function, $g$, with convolutional kernel (filter), $f$, in order to produce a transformed feature map (a modified version of the original input) as shown in the following equation. $$(f\times g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau
=\int_{-\infty}^{\infty}f(t-\tau)g(\tau)d\tau$$
Convolution has different applications in many fields such as probability, statistics, computer vision, imaging, etc. In probability theory, convolution is similar to cross-correlation, while in statistics, it updates the weights over normalization of input vector. Figure \[fig:fig2\] shows a Convolutional Neural Network (CNN or ConvNet), depicted in Figure \[fig:fig2\], is a network with multiple layers of convolutions that apply a nonlinear activation function like ReLU or Tanh (mostly used for image processing). CNN layers filter input data to produce useful feature map information. These layers have parameters that are updated repeatedly to produce the desired output. In feedforward NN with fully connected layers, each input neuron is connected to all neurons of the next layer. However, with CNN, after calculation of the output by applying convolutions on the input data, each node only connects itself with the closest neighboring neurons (local connectivity). Convolutional layers sizes shrink as they become deeper.
![Convolutional Neural Network []{data-label="fig:fig2"}](fig2){width="50.00000%"}
The architecture of CNN primarily has three types of layers including; *convolutional layers*, *pooling layers*, and *fully connected layers*. The convolutional layers are the main block of a CNN that do most of the computational operations. The pooling layers are applied after the convolutional layers, where these layers partition the input data into non-overlapping sets (windows), and reduce each set to a single value (subsampling) by applying a max operation (outputs the maximum value in each set) to the result of each filter. The pooling layers have benefits of reducing the spatial size of data, reducing the number of parameters, reducing computations, and controlling overfitting. Finally, after all generated features are combined, they are used to find the final classification via fully connected layers, whose neurons are fully connected to all activations in the previous layer.
Many CNN architectures exist such as:
- LeNet: Developed by LeCun et al. in 1990’s [@lecun1998gradient], LeNet was used to read zip codes, recognize characters, etc.
- AlexNet: Developed by Krizhevsky et al. [@krizhevsky2012imagenet], AlexNet was used in computer vision tasks. It has a similar architecture to LeNet, but with deeper, bigger, and more stacked convolutional layers on top of each other.
- GoogleNet: Introduced by Szegedy et al. from Google [@szegedy2015going], GoogleNet had a reduced number of parameters in the network (it had 4M parameters, compared to AlexNet’s 60M).
- VGGNet: Introduced by Simonyan and Zisserman [@simonyan2014very], the goal of VGGNet’s architecture was to prove that a good performance depends on the depth of the network.
- ResNet (Residual Network): Introduced by He et al. [@he2016deep], ResNet’s architecture does not have fully connected layers at the end of the network. Also, it uses batch normalization.
Recurrent Neural Networks (RNN)
-------------------------------
RNN is a type of ANN, first introduced by Elman in 1990 [@elman1990finding]. After that, Elman and others began to develop the concept of RNN. In 1993, a modified RNN model was developed to solve very deep learning tasks which required more than 100 subsequent layers. RNN connections between neurons form a directed cycle (fed data from previous layer and from themselves) as shown in Figure \[fig:fig6\]. This makes it available to be used for sequential information. Because RNN is built in a way that fits sequential information, it is used in many tasks in NLP, speech recognition, image capturing, language translation, etc.
![Recurrent Neural Network []{data-label="fig:fig6"}](fig6){width="40.00000%"}
Simple ANN and CNN models assumed that all inputs are independent of each other. However, in RNN, the output is dependent on the previous computations. Thus, a RNN has an internal memory to save previous computations. For instance, RNN is popular in NLP tasks where it predicts next word depending on previous words in any given sequence using the internal memory in neurons of the hidden layers.
Long short-term memory (LSTM) is a special type of RNN (shown in Figure \[fig:fig7\]) proposed by Hochreiter and Schmidhuber in 1997 [@hochreiter1997long] in order to solve the vanishing gradient problem. It uses LSTM neurons (memory cells with three gates: input, output and forget) instead of simple neurons in the hidden layers. In addition, LSTMs are designed in a way to avoid the long-term dependency problem. While simple RNN can “remember” previous information for short time periods, LSTM can remember them for much longer time periods. However, if the information is not used, it will be lost.
![LSTM Neural network []{data-label="fig:fig7"}](fig7){width="40.00000%"}
An LSTM cell is smarter than a simple neurons as it has a memory to store previous sequences. Each cell contains gates that manage its state and output. Each gate within a unit uses the sigmoid activation function to decide whether it is triggered or not, which makes it conditional to change state or add information.
These three gates within a memory cell are **forget gate** to decide what information to discard from each LSTM cell, **input gate** to decide the update of memory state depending on input values and **output gate** to decide what to output based on the input and the memory of the LSTM cell.
Deep Learning Frameworks {#sec:frameworks}
========================
The frameworks considered in this comparative study are: CNTK, TensorFlow and Theano. We also use Keras on top of these frameworks as will be discussed later. All of these frameworks provide flexible APIs and configuration options for performance optimization. Software versions of the frameworks[^2] are shown in Table \[table:comparative\] and their properties are shown in Table \[table:properties\].
Framework Major Version Github Commit ID
------------ --------------- ------------------
CNTK 2.0 7436a00
TensorFlow 1.2.0 49961e5
Theano 0.10.0.dev1 8a1af5b
Keras 2.0.5 78f26df
: Frameworks used for this comparative study[]{data-label="table:comparative"}
Property CNTK TensorFlow Theano Keras
-------------------- ------ ------------ ------------------------------ --------
Core C++ C++ Python Python
CPU
Multi-Threaded CPU Eigen Blas, conv2D, Limited OpenMP
GPU
Multi-GPU X (experimental version)
NVIDIA cuDNN
: Properties of the considered frameworks[]{data-label="table:properties"}
CNTK
----
Microsoft Cognitive Toolkit (CNTK) is an Open source DL framework developed by Microsoft Research [@yu2014introduction] for training and testing many types of NN across multiple GPUs or servers. CNTK supports different DL architectures like Feedforward, Convolutional, Recurrent, LSTM, and Sequence-to-Sequence NN.
A Computational Network learns any function by converting it to a directed graph where each leaf node consists of an input value or a learning parameter, whereas, each other node represents a matrix operation applied on its children. In this case, CNTK has an advantage as it can automatically find the derive gradients for all the computations which are required to learn the parameters. In CNTK, users specify their networks using a configuration file that contains information about the network type, where to find input data, and the way to optimize parameters [@yu2015computational].
CNTK interface supports different APIs of several languages such as Python, C++ and C\# across both GPU (CUDA) or CPU platforms. According to its developers [@cntkmicro], CNTK was written in C++ in an efficient way, where it removes duplicated computations in forward and backward passes, uses minimal memory and reduces memory reallocation by reusing them. The framework’s installation is discussed in [@ghadeer_thesis].
Theano
------
Theano[^3] is an open source Python library developed at MILA lab at University of Montreal as a compiler for mathematical expressions that let users and developers optimize and evaluate their expressions using a NumPy’s syntax (a Python library that supports a large and multi-dimensional arrays) [@bergstra2010theano; @al2016theano]. Theano starts performing computations automatically by optimizing the selection of computations, translates them into other machine learning languages such as C++ or CUDA (for GPU) and then compiles them into Python modules in an efficient way on CPUs or GPUs.
Theano is being actively developed since 2008, which makes it more popular on a research and ecosystem platform than many DL libraries. Several software packages have been developed to build on top of Theano, with a higher-level user interface which aims to make Theano easier to express and train different architectures of deep learning models, such as Pylearn2, Lasagne, and Keras. The framework’s installation is discussed in [@ghadeer_thesis].
TensorFlow
----------
TensorFlow is an open source framework developed by Google Brain Team [@abadi2016tensorflow], that uses a single data flow graph, expressing all numerical computations, to achieve excellent performance. TensorFlow constructs large computation graphs where each node represents a mathematical operation, while the edges represent the communication between nodes. This data flow graph executes the communication between sub-computations explicitly, which makes it possible to execute independent computations in parallel or to use multiple devices to execute partition computations [@abadi2016tensorflow2]. The framework’s installation is discussed in [@ghadeer_thesis].
Programmers of TensorFlow define large computation graphs from basic operators, then distribute the execution of these graphs across a heterogeneous distributed system (can deploy computation to one or more CPUs or GPUs on a different hardware platforms such as desktops, servers, or even mobile devices). The flexible architecture of TensorFlow allows developers and users to experiment and train a wide variety of deep neural network models, and it is used for deploying machine learning systems into production for different fields including speech recognition, NLP, computer vision, robotics, and computational drug discovery. TensorFlow uses different APIs of several languages such as Python, C++, and Java for constructing and executing a graph (Python API is the most complete and the easiest to use) [@TensorFlowwebsite]. The framework’s installation is discussed in [@ghadeer_thesis].
Keras
-----
Keras is an open source DL library developed in python. It runs on top of CNTK, Theano or TensorFlow frameworks. Keras was founded by Google engineer Chollet [@chollet2015keras] in 2015 as a part of the research project ONEIROS (Open-ended Neuro-Electronic Intelligent Robot Operating System). Keras is designed in a way that allows fast expression with deep neural networks and easy and fast prototyping (modularity and extensibility) [@chollet2017keras]. The framework’s installation is discussed in [@ghadeer_thesis].
Methods {#sec:method}
=======
The goal of this experimental study is to compare the aforementioned frameworks (Theano, TensorFlow and CNTK) by using them to train CNN and RNN models on standard benchmark datasets of classical problems in image processing (MNIST, CIFAR-10, and Self-driving Car) and NLP (Penn TreeBank and IMDB). We then evaluate each framework’s performance through the following metrics:
- Running time
- Memory consumption
- CPU and GPU utilization
- Number of epochs
We aim at comparing the aforementioned frameworks using a GPU-equipped laptop that runs Windows 10 operating system, and has the following specifications:
- Intel Core i7-6700HQ CPU @ 2.60GHz (4 cores)
- 16 GB RAM
- 64-bit operating system, x64-based processor
- NVIDIA GEFORCE GTX 960m graphics card (Laptop), with PCI Express 3.0 bus support, equipped with 4 GB GDDR5 memory and 640 CUDA cores.
It is worth mentioning that our goal is to compare the resources consumed by each framework to reach a certain accuracy level for each problem. So, we experimented with different epoch counts in order to make sure the accuracy for all frameworks are close to each other.
CNN is applied for different classical problems including: MNIST, CIFAR-10, IMDB and Self-Driving Car datasets, where each dataset has a different network architecture. When applying CNN datasets on both Theano and Tensorflow, Keras is used for coding all of these datasets, while, in CNTK, Keras is used only when applying Self-Driving car and IMDB datasets. Thus, MNIST and CIFAR-10 are implemented without Keras.
For the MNIST and CIFAR-10 datasets, two convolutional layers with ReLU activation function are used after the input layer. The activation function is used to reduce the training time and to prevent vanishing gradients. After each CNN layer, a max-pooling layer is added in order to down-sample the input and to reduce overfitting. In the max-pooling layer, the stride value must be specified with which the filter is slid. When the stride is one, the filter (window) is moved one pixel at a time. When the stride is two, the filter moved two pixels at a time. This will produce smaller output volumes spatially. After each max-pooling layer, the dropout method is used in order to reduce overfitting by forcing the model to learn many independent representations of the same data through randomly disabling neurons in the learning phase. The sequential architecture (layers part only) used on the MNIST and CIFAR-10 datasets are shown in [@ghadeer_thesis].
Another example of applying CNN is on the Self-Driving Car dataset, where the network has the same components as the ones used with the MNIST and CIFAR-10 datasets, but with deeper model that consists of five convolutional layers with Exponential Linear Unit (ELU) activation function. The sequential architecture of this CNN is shown in [@ghadeer_thesis]. The convolutional layers are used for feature engineering. The fully connected layer is used for predicting the steering angle (final output). The dropout avoids overfitting and, finally, the ELU activation function is used to solve the problem of the vanishing gradient.
The final example of applying CNN is on the IMDB dataset. The movie reviews in this dataset are composed of sequences of words of different lengths. These words are encoded by mapping movie reviews to sequences of word embeddings where words are mapped to vectors of real numbers; the network architecture consists of an embedding layer followed by a 1D convolution layer which is used for temporal data followed by a global max-pooling operation. These sequences are padded to have the same size as the largest sequence because they have different lengths. The sequential architecture used on IMDB dataset is shown in [@ghadeer_thesis].
The other neural network type we consider is RNN with LSTM. One of the most popular uses of LSTM is for text analysis tasks such as the ones associated with the Penn TreeBank (PTB) dataset. Word-level prediction experiments on PTB was adopted, which consists of 929k training words, 73k validation words, and 82k test words. It has 10k words in its vocabulary. We trained models of two sizes (small LSTM and medium LSTM) and we uses the same architecture presented in [@zaremba2014recurrent].
To evaluate language models of the PTB implementation, a special metric called a perplexity is used, where better prediction accuracy is achieved when perplexity value is as low as possible. Perplexity is the inverse of probability definition. This means that minimizing perplexity value is the same as maximizing probability. The goal of applying PTB dataset is to match a probabilistic form which assigns probabilities to sentences. This process is done by predicting next words in a text given a history of previously located words. LSTM cell presents the core of the model which processes one word at a time and computes probabilities of the possible values for the next word in the sentence. A vector of zeros is used for the memory state of the network to get initialized and updated after reading each word.
In the small LSTM model, two hidden layers (with 200 LSTM units per layer) are used with Tanh activation function. Its weights are initialized to 0.1. We train it for four epochs with a learning rate of one (number of epochs trained with initial learning rate), and then the learning rate is decreased by a factor of two after each epoch (the decay of the learning rate for each epoch after four epochs), for a total of 13 training epochs. The size of each batch is 20, then the network is unrolled for 20 steps. The sequential architecture used for PTB dataset is shown in [@ghadeer_thesis].
Benchmark Datasets
------------------
In this subsection, we discuss the datasets used in our experiments.
### MNIST
The MNIST (Mixed National Institute of Standards and Technology) dataset (shown in Figure \[fig:mnist\]) is a computer vision database for handwritten digits. It is widely used for training and testing in the field of machine learning [@lecun1998gradient; @lecun2009mnist]. MNIST has a training set of 60,000 images and a testing set of 10,000 images. It is a subset of a larger set available from NIST. Each image is $28\times~28$ pixels which can be represented as a big array of numbers. We can flatten this array into a vector of $28\times 28= 784$ numbers. Each image in MNIST has a corresponding label, a number between 0 and 9 representing the digit appearing in the image. See Figure \[fig:mnist\].
![MNIST digits []{data-label="fig:mnist22"}](mnist22){width="40.00000%"}
Our goal is to construct a CNN to classify MNIST images. The training will be carried out on both CPU and GPU environments using different frameworks including the aforementioned ones. We then evaluate the performance of each framework.
### CIFAR-10
The CIFAR-10 dataset is one of the 80 million images datasets, collected by Krizhevsky et al. [@krizhevsky2012imagenet; @krizhevsky2009learning]. It consists of 60,000 $32\times32$ color images evenly distributed over ten classes: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, and truck. There are 50,000 training images and 10,000 test images. Figure \[fig:cifar10\] shows the classes in the dataset, as well as 10 random images from each class. The classes are completely mutually exclusive. I.e., there is no overlap between them. For instance, the “Automobile” class includes sedans, SUVs, etc. On the other hand, the “Truck” class includes only big trucks. To avoid overlap, neither one of these two classes includes pickup trucks.
![CIFAR-10 dataset classes []{data-label="fig:cifar10"}](cifar10){width="50.00000%"}
### Penn TreeBank
In 1993, Marcus et al. [@marcus1993building] wrote a paper on constructing a large annotated corpus of English called the Penn TreeBank (PTB). They reviewed their experience with constructing one large annotated corpus that consists of over 4.5 million words of American English. The project was divided into phases. For the first three-year phase, the corpus was annotated for part-of-speech (POS) tag information (See Figure \[fig:tagset.png\]). Moreover, half of the corpus was annotated for skeletal syntactic structure.
![PTB POS tagset []{data-label="fig:tagset.png"}](tagset){width="60.00000%"}
The dataset is large and diverse. It includes the Brown Corpus (retagged) and the Wall Street Journal Corpus, as well as Department of Energy abstracts, Dow Jones Newswire stories, Department of Agriculture bulletins, Library of America texts, MUC-3 messages, IBM Manual sentences, WBUR radio transcripts, and ATIS sentences.
### IMDB
IMDB dataset [@maas2011learning; @maas-EtAl:2011:ACL-HLT2011] is another example of applying CNN, which is an online dataset of information regarding films, TV programs and video games. It consists of 25,000 reviews labeled by the sentiment (positive/negative) of each review. The reviews have been preprocessed and encoded as integers in a form of a sequence of word indexes (See Figure \[fig:imdb.png\]). Words are indexed by overall frequency in the dataset, so that the index $i$ encodes the $i$th most frequent word in the data in order to allow operations of quick filtering.
![Positive/Negative movie reviews sentiment for IMDB []{data-label="fig:imdb.png"}](imdb){width="70.00000%"}
### Self-Driving Car
This dataset uses a Udacity’s Self-Driving Car simulator as a testbed for training an autonomous car. This work started in the 1980s with Carnegie Mellon University’s Navlab and ALV projects [@wallace1985first]. The training phase starts with activating the simulator which is an executable application. A user initiates the service of collecting the data for training followed by collecting the data as images and saving them locally on the computer. So, the framework can take these images and train them. The training is done via distinguishing the image’s edges which are taken by the three cameras laying on the front of the car in the simulator. After the training phase is done, the testing phase begins by taking the file generated whenever the performance in the epoch is better than the previous best. Finally,the last generated file is executed in order to make the car drive autonomously to observe the testing phase results. See figure \[fig:self.jpg\].
![Training Self-Driving Car dataset via Udacity’s simulator[]{data-label="fig:self.jpg"}](self){width="70.00000%"}
Results {#sec:res}
=======
After implementing the experiments of each dataset on each framework the target was to find which of these frameworks had the best performance. Relating to table \[table:proc\], regarding image recognition datasets (MNIST and CIFAR-10), one can observe the superiority of CNTK over Tensorflow and Theano in terms of GPU and CPU multithreading, but in CIFAR-10 using 8,16 and 32 threads in CPU Tensorflow was faster than CNTK. On the other hand, Theano revealed to be more time consuming than other frameworks. Transitioning to sentiment analysis dataset (IMDB), CPU multithreading was not performed because CNTK is written in python in which multithreading is not supported. Without CPU multithreading (CPU uses the default number of existing physical cores which are equal one thread per core), the superiority of TensorFlow is revealed in both CPU and GPU environments. In text analysis dataset (Penn TreeBank), as mentioned in IMDB dataset (CNTK does not support python multithreading), the default number of threads (8) is taken in consideration while comparing the results. This shows the superiority of Tensorflow over CNTK and Theano, as well as the case in GPU. Moving forward to video analysis dataset (Self-Driving Car), using 8 CPU threads only for the same reason mentioned in IMDB and Penn TreeBank datasets, the superiority of TensorFlow is revealed in both CPU and GPU environments, while CNTK showed to be more time consuming than the other two frameworks.
Dataset Environment Threads CNTK TensorFlow Theano
--------- ------------- --------- --------------- --------------- --------------- -- -- --
CPU 1 847 5130 3560
CPU 2 630 3180 2500
CPU 4 574 2070 2260
CPU 8 560 1740 2060
CPU 16 567 1920 2050
CPU 32 588 2010 2050
GPU - 66.67 328.93 377.86
CPU 1 20196 25905 26700
CPU 2 14520 16610 18700
CPU 4 13662 11550 17250
CPU 8 11484 9955 15800
CPU 16 11550 10340 15850
CPU 32 11649 10835 15750
GPU - 926 2166.4 2386.1
CPU 1 - 1244 538
CPU 2 - 642 412
CPU 4 - 390 380
CPU 8 486 290 368
CPU 16 - 249 368
CPU 32 - 302 384
GPU - 73.1 62.4 220.41
CPU 1 - \~ 33.3 hours \~ 50 hours
CPU 2 - \~ 19.8 hours \~ 44.2 hours
CPU 4 - \~ 15 hours \~ 42.6 hours
CPU 8 \~ 47.6 hours \~ 14.1 hours \~ 43.5 hours
CPU 16 - \~ 16.4 hours \~ 43.5 hours
CPU 32 - \~ 16.4 hours \~ 43.5 hours
GPU - 8.7 hours 6 hours 6.8 hours
CPU 1 - 40560 27066
CPU 2 - 26819 23244
CPU 4 - 18733 21541
CPU 8 4290 16407 21450
CPU 16 - 16848 21476
CPU 32 - 18369 21541
GPU - 2106 1342.28 1630
: Processing time for each dataset (measured in seconds)[]{data-label="table:proc"}
Discussion {#sec:disc}
==========
At this section, a visual representation of tables that appeared in the results section is presented. Figures (\[fig:mnist\] \[fig:gpu\]) include CPU multithreading and GPU processing time, as well as figures (\[fig:cntk\], \[fig:tf\], \[fig:theano\]) represent memory utilization for both CPU and GPU for each framework. The processing times clearly show the advantage of GPU over CPU for training deep convolutional and recurrent neural networks. The advantage of fast GPU would be more significant when training complex models with larger data as in the Self-Driving Car dataset. From the CPU results, the best performance occurred when the number of threads is equal to the number of physical CPU cores, where each thread possesses a single core. In our work we used a laptop with 8 cores, so in each dataset the best performance in terms of processing time was achieved while using 8 threads as shown in figures (\[fig:mnist\] \[fig:gpu\]). The metrics measurement of each framework was conducted to explain the failure of one of the selected frameworks.
We noticed poor performance of Theano at most datasets comparing to CNTK and TensorFlow. This could be explained because of low CPU utilization comparing to aforementioned frameworks, meanwhile CNTK and TensorFlow use all available resources (high CPU utilization). CNTK outperformed both TensorFlow and Theano while training MNIST and CIFAR-10 datasets. This achievement is highly likely due to the use of BrainScript [@brainsc1] format which is a custom network description language that makes CNTK more flexible for neural networks customization. On the other hand, TensorFlow uses Eigen [@eigen1], which is a C++ template library (BLAS library) for linear algebra including matrices, vectors, numerical solvers, and related algorithms. It is used to make TensorFlow perform better than CNTK and Theano in RNN.
Comparing our work to previous work such as S.Bahrampour’s et al. work presented at [@bahrampour2015comparative] and S.Shi et al. work presented at [@shi2016benchmarking], we reveal the following findings. S.bahrampour et.al based their comparative study on three main aspects including speed, hardware utilization, and extensibility. Besides they used three neural network types such as CNNs, AE, and Recurrent LSTM to train MNIST, ImageNet, and IMDB datasets on Caffe, Neon, Tensorflow, Theano and Torch frameworks. They used the following hardware specs; Intel Xeon CPU E5-1650 v2 @3.5GHz (with multi-threading), NVIDIA GeForce GTX Titan X/PCI/SSE2, 32GB DDR3 Memory, and a SSD drive to come up with the following results; while training on CPU, Torch performed the best followed by Theano, but Neon had the worst performance, also Theano and Torch are the best in extensibility term, as well as TensorFlow and Theano were very flexible and Caffe was the easiest to find the performance. Regarding training datasets on GPU, and for larger convolutional and fully connected networks, Torch was the best followed by Neon. For smaller networks Theano was the best. For LSTM, Theano results was the best in performance, while TensorFlow performance is not competitive compared to other studied frameworks.
On the other hand, S.Shi et al. based their comparative study on two comparative terms including processing time and convergence rate. The neural networks used are fully connected NN, CNNs and RNNs to train ImageNet, MNIST, and CIFAR10 datasets on Caffe, CNTK, MXNet, TensorFlow, and Torch frameworks. They used the following hardware specs; Two types of multi-threaded CPU: desktop CPU (intel i7-3820) (8 threads) and server-grade CPU (intel xeon E5-2630) (32 threads), as well as three types of NVIDIA GPU (GTX980, GTX1080, Tesla K80), and Two Tesla K80 cards used to evaluate the multi-GPU performance. The results of TensorFlow were the best while using CPU. While using single GPU; on FCNs, Caffe, CNTK and Torch performed better than MXNet and TensorFlow. As for small CNN; Caffe and CNTK achieved a good performance, and for RNN (LSTM), CNTK was the fastest (5-10x faster than other frameworks). Using Multi-GPU implementation, all frameworks had higher throughput and accelerated the convergence speed.
Our comparative study was based on measuring performance for CNTK, TensorFlow, and Theano using two types of networks CNNs and RNNs to train MNIST, CIFAR-10, Penn TreeBank, IMDB, and Self-Driving Car datasets on CNTK, TensorFlow, and Theano. The results were as follows: regarding image recognition datasets (MNIST and CIFAR-10) one can observe the superiority of CNTK over Tensorflow and Theano in terms of GPU and CPU multithreading, but in CIFAR-10 using 8,16 and 32 threads in CPU Tensorflow was faster than CNTK. On the other hand, Theano revealed to be more time consuming than be aforementioned frameworks. Transitioning to sentiment analysis dataset (IMDB), CPU multithrading was not performed because of CNTK is written in python in which the multithreading is not supported yet, but while not using CPU multithreading (CPU uses the default number of existing physical cores which are equal one thread per core) the superiority of TensorFlow reveals in both CPU and GPU environments. In text analysis dataset (Penn TreeBank), as mentioned in IMDB dataset CNTK does not support python multithreading, so the default number of threads (8) are taken in consideration while comparing the results which shows the superiority of Tensorflow over CNTK and Theano, as well as the case in GPU. Moving forward to video analysis dataset (Self-Driving Car), using 8 CPU threads only for the same reason mentioned in IMDB and Penn TreeBank datasets, the superiority of TensorFlow reveals in both CPU and GPU environments while CNTK showed to be more time consuming than the other two frameworks.
![CPU processing time for MNIST dataset[]{data-label="fig:mnist"}](mnist){width="70.00000%"}
![CPU processing time for CIFAR-10 dataset[]{data-label="fig:cifar"}](cifar){width="70.00000%"}
![CPU processing time for IMDB dataset []{data-label="fig:imdb2"}](imdb2){width="70.00000%"}
![CPU processing time for Self-Driving Car dataset []{data-label="fig:car"}](car "fig:"){width="70.00000%"}\
![CPU processing time for Penn TreeBnk dataset []{data-label="fig:penn"}](penn){width="70.00000%"}
![GPU processing time for each dataset []{data-label="fig:gpu"}](gpu){width="70.00000%"}
![Memory utilization for CNTK framework[]{data-label="fig:cntk"}](cntk){width="70.00000%"}
![Memory utilization for Tensorflow framework[]{data-label="fig:tf"}](tf){width="70.00000%"}
![Memory utilization for Theano framework[]{data-label="fig:theano"}](theano){width="70.00000%"}
Conclusions and Future Work {#sec:conc}
===========================
Conclusion
----------
In this paper, we have provided a qualitative and quantitative comparison between three of the most popular and most comprehensive DL frameworks (namely Microsoft’s CNTK, Google’s TensorFlow and University of Montreal’s Theano). The main goal of this work was to help end users make an informed decision about the best DL framework that suits their needs and resources. To ensure that our study is as comprehensive as possible, we have used multiple benchmark datasets namely MNIST, CIFAR-10, Self-Driving Car, and IMDB which were trained via multilayer CNN network architecture and Penn TreeBank dataset which was trained via RNN architecture. We have run our experiments on a laptop with windows 10 operating system. We have measured performance and utilization of CPU multithreading , GPU and memory.\
After the evaluation of the datasets, we have observed the following:
- In all implementations, the processing time clearly shows the advantage of GPU over CPU for training networks.
- For GPU utilization metric in MNIST and CIFAR-10 datasets, TensorFlow had the lowest utilization, followed by Theano and CNTK. For CPU utilization metric, Theano had the lowest utilization followed by TensorFlow and CNTK. For Memory utilization while using CPU and GPU, the results were close to each other.
- For GPU utilization metric in IMDB dataset, the results were close to each other. For CPU utilization metric, Theano had the lowest utilization followed by TensorFlow and CNTK. For Memory utilization while using CPU and GPU, TensorFlow exhibited the worst memory consumption.
- For GPU utilization metric in Self-Driving Car dataset, the results were close to each other. For CPU utilization metric, Theano had the lowest utilization followed by TensorFlow and CNTK. For Memory utilization while using CPU and GPU, Theano revealed the best memory consumption.
- For GPU utilization metric in Penn TreeBank dataset, the results were close to each other. For CPU utilization metric, Theano had the lowest utilization followed by TensorFlow and CNTK. For Memory utilization while using CPU and GPU, CNTK revealed the best memory consumption.
- In MNIST and CIFAR-10 datasets one can observe the superiority of CNTK over Tensorflow and Theano in terms of GPU and CPU multithreading, but in CIFAR-10 using 8, 16 and 32 threads in CPU Tensorflow was faster than CNTK. On the other hand Theano revealed to be more time consuming than be aforementioned frameworks.
- In sentiment analysis dataset (IMDB), CPU multithrading was not performed because of CNTK is written in python in which the multithreading is not supported yet, but while not using CPU multithreading (CPU uses the default number of existing physical cores which are equal one thread per core) the superiority of TensorFlow reveals in both CPU and GPU environments.
- In text analysis dataset (Penn TreeBank), as mentioned in IMDB dataset CNTK does not support python multithreading, so the default number of threads (8) are taken in consideration while comparing the results which shows the superiority of Tensorflow over CNTK and Theano, as well as the case in GPU.
- Moving forward to video analysis dataset Self-Driving Car, using 8 CPU threads only for the same reason mentioned in IMDB and Penn TreeBank datasets, the superiority of TensorFlow reveals in both CPU and GPU environments while CNTK showed to be more time consuming than the other two frameworks.
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[^1]: More details will be given later about these neural network architectures and these datasets.
[^2]: This work was conducted in the summer of 2017. The versions we consider were the latest ones. Since then, these frameworks have been updated.
[^3]: Theano is no longer supported but it was so when this paper has been written.\
https://groups.google.com/forum/\#!topic/theano-users/7Poq8BZutbY
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'In [@Bogom] it was proposed, that the nearest-neighbor distribution $P(s)$ of the spectrum of the Bohr-Mottelson model is similar to the semi-Poisson distribution. We show however, that $P(s)$ of this model differs considerably in many aspects from semi-Poisson. In addition we give an asymptotic formula for $P(s)$ as $s \to 0$, which gives $P''(0) = \pi\sqrt{3}/2$ for the slope at $s=0$. This is different not only from the GOE case but also from the semi-Poisson prediction that leads to $P''(0) = 4$'
---
**Comment on “Models of Intermediate Spectral Statistics”**
**T. Gorin$^{1}$, M. Müller$^{2}$ and P. Seba$^{3,4}$**
**
$^1$ Centro de Ciencias Físicas, UNAM, Campus Morelos, Cuernavaca, Mexico, 62251, A.P. 48-3\
$^2$ Facultad de Ciencias, UAEM, C.P. 62210, Cuernavaca, Morelos, Mexico\
$^3$ Institute of Physics, Czech Academy of Science, Cukrovarnicka 10, Prague, Czech Republic\
$^4$ Pedagogical University, Department of Physics, Hradec Kralove, Czech Republic\
PACS: 03.65, 05.45
The motivation for this comment stems from the impression, that the article “models of intermediate spectral statistics” can easily be misinterpreted in two ways: One might be led to believe that (i) the semi-Poisson distribution is universal, and (ii) the universality class of “intermediate statistics” is as well defined and established as for example the Poisson ensemble or the GOE. In this comment, we will argue, that both statements are wrong.
The purpose of [@Bogom] is to present models which could constitute a “third” universality class of systems which show so called “intermediate statistics”, previously introduced by Shklovskii in [@Shklovskii]. The Poissonian and the Gaussian ensemble (for definiteness, consider orthogonal ensembles only) are considered as the first two universality classes in this list.
As in the Poissonian and in the GOE case, where the respective members have common and unique statistical properties, one would expect the same to hold for the models with intermediate statistics. In [@Bogom] the authors concentrate on the distribution of nearest neighbor spacings. In the Poissonian case it is given by $P(s)= \exp(-s)$, in the GOE case it is close to the well known Wigner surmise $P(s)\approx (\pi/2) \exp(-\pi /4
s^2)$, whereas in the case of the “intermediate statistics” the candidate proposed in [@Bogom] is the semi-Poisson distribution $P(s)= 4s\; \exp(-2s)$.
In what follows we will show, that the level spacing distribution in the case of the Bohr-Mottelson model, that represents one of the candidate systems for the intermediate statistics mentioned in [@Bogom] is in fact very different from the proposed semi-Poisson distribution. This discrepancy can be found in fact already on a figure published in [@Montambaux] but without discussing the problem. However, [@Montambaux] gives an overview over the statistical properties of different variants of the Bohr-Mottelson model.\
In [@Bogom] the following matrix model (originally introduced by Bohr and Mottelson) is presented as a possible candidate showing statistical properties similar to the semi-Poisson distribution: $$H_{mn} = e_n \delta_{mn} + t_m t_n \; .
\label{model}$$ $H$ is a $N\times N$-matrix, $e_n$ are mutually independent random variables uniformly distributed over a finite interval, and the $t_n$ are chosen with equal absolute value squared $t_n^2 = r$. The authors of [@Bogom] sketch a procedure for calculating analytically the 2-point correlation function. For small distances it should agree with $P(s)$, so that one can derive the slope $P'(0)= \pi\sqrt{3}/2$ of the spacing distribution at $s=0$. This slope is different from the GOE case, where it equals to $\pi^2 /6$ as well as from the slope of the semi Poisson distribution that equals to $4$. The difference to the GOE case is remarked, but a similar remark on the difference to the semi-Poisson is avoided. An unprejudiced reader might believe, that the correlation properties of the matrix model are similar to semi-Poisson, even though that the spectral statistics of this model differ remarkably.\
In Figure 1 we present the numerical result for $P(s)$ obtained for an ensemble of 1000 matrices of dimension 750. For the statistical analysis we used only one third of the states in the center of the spectral region. The numbers $e_n$ are uniformly distributed over an interval $[-1,1]$ and the elements of the vector $\vec t$ are chosen as $t_i=\sqrt{\alpha/(\pi\rho N)}$, where $\rho$ is the level density in the center of the spectrum, $N$ is the dimension of the matrix and $\alpha=10$ is the coupling constant. (We checked, that larger coupling does not change the numerical results). Figure 1.a demonstrates the qualitative differences in the behavior of $P(s)$ between the random matrix model and the semi-Poisson distribution. The slope at $s=0$ is smaller, the maximum of $P(s)$ is slightly shifted to the right, and for values $s>1$ it shows significant deviations well above of the statistical error. Hence the spacing distribution of the Bohr-Mottelson model is not close to semi-Poison.
![(a) $P(s)$ for the model (\[model\]) compared with the the semi-Poisson distribution, (b) the same as in (a) for short distances. In addition the theoretical result (\[3point\]) is drawn as a dashed line and the GOE result as a dashed dotted curve.](Fig1)
Figure 1.b. shows a magnification of the interval $0 \le s \le 1/2$ using the same data as in figure 1.a. In this figure we additionally plotted the asymptotic result (\[model\]) for the present model as a dashed line. The basic idea of the approximation is the following: In order to get a short distance between two neighbored levels in the spectrum of $H$, three eigenvalues of $H_0$ have to come close together. Then the levels which are farther away can be neglected. Therefore we can restrict the sum $$K(E) = \sum_{i=1}^N \frac{t_i^2}{E-e_i}$$ whose roots define the eigenvalues of $H$, to those terms with the three consecutive eigenvalues. Resolving for the two roots, calculating their distance, and averaging over the levels $e_i$ leads to the following formula $$P(s)= \frac{9 s}{4} \intop_0^{\pi/2} d\phi \frac{{\rm exp} \left[
-\frac{3s}{2} (\cos\phi + \sin\phi)/\sqrt{1+\sin(2\phi)/2} \right ]}
{1 + \sin(2\phi)/2} \; .
\label{3point}$$ The dashed curve in Figure 1.b. is obtained from a numerical integration of (\[3point\]). At short distances, this approximation describes the numerical data much better than the semi-Poisson. A Taylor expansion of the integrand of (\[3point\]) gives $P'(0)= \pi\sqrt{3}/2$ for the slope at $s=0$, the same result as found in [@Bogom].
A detailed numerical investigation of several statistical properties of the type of models can be found also in [@dittes; @gorin; @Montambaux].\
To conclude, if one wants to insist on the introduction of a “third” universal ensemble, one should possibly use a criterium similar to the following (cited from [@Bogom]: “…the main features are (i) the existence of level repulsion (as in random matrix ensembles), and (ii) slow (approximately exponential) fall-off …”. Even though this definition is quite “spongy”, it seems to be the only way to make sure, that the systems discussed fit into this class.
<span style="font-variant:small-caps;">Acknowledgements</span> M. Müller acknowledges financial support from the CONNACyT (No.32101-E).
[1]{}
E.B. Bogomolny, U. Gerland and C. Schmit, Phys. Rev. E [**59**]{} (1999) R1315
F.-M. Dittes, I. Rotter and T.H. Seligman, Phys. Lett A [**158**]{} (1991) 14
T. Gorin, F.-M. Dittes, M. Müller, I. Rotter and T.H. Seligman, Phys. Rev. E [**56**]{} (1997) 2481
M. Pascaud and G. Montambaux, Ann. Phys. [**7**]{} (1998) 406
B. I. Shklovskii, Phys. Rev. B [**47**]{} (1993) 11487
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This paper presents a neural network-based end-to-end clustering framework. We design a novel strategy to utilize the contrastive criteria for pushing data-forming clusters directly from raw data, in addition to learning a feature embedding suitable for such clustering. The network is trained with weak labels, specifically partial pairwise relationships between data instances. The cluster assignments and their probabilities are then obtained at the output layer by feed-forwarding the data. The framework has the interesting characteristic that no cluster centers need to be explicitly specified, thus the resulting cluster distribution is purely data-driven and no distance metrics need to be predefined. The experiments show that the proposed approach beats the conventional two-stage method (feature embedding with k-means) by a significant margin. It also compares favorably to the performance of the standard cross entropy loss for classification. Robustness analysis also shows that the method is largely insensitive to the number of clusters. Specifically, we show that the number of dominant clusters is close to the true number of clusters even when a large $k$ is used for clustering.'
author:
- |
Yen-Chang Hsu\
School of Electrical and Computer Engineering\
Georgia Institute of Technology\
Atlanta, GA 30332, USA\
`[email protected]`\
Zsolt Kira\
Georgia Tech Research Institute\
Atlanta, GA 30318, USA\
`[email protected]`\
bibliography:
- 'iclr2016\_NNclustering.bib'
title: 'Neural network-based clustering using pairwise constraints'
---
Introduction
============
Performing end-to-end training and testing using deep neural networks to solve various tasks has become a dominant approach across many fields due to its performance, efficiency, and simplicity. Success across a diverse set of tasks has been achieved in this manner, including classification of pixel-level information into high level categories [@Alex2012imagenet], pixel-level labeling for image segmentation [@long2015segmentation; @zheng2015conditional], robot arm control [@levine2015end], speech recognition [@graves2014towards], playing Atari games [@mnih2015human] and Go [@clark2015training; @David2016go]. All of the above techniques largely avoid sophisticated pipeline implementations and human-in-the-loop tuning by adopting the concept of training the networks to learn the target problem directly.
Clustering, a classical machine learning problem, has not yet been fully explored in a similar manner. Although there are some two-stage approaches that have tried to learn the feature embedding specifically for clustering, they still require using other clustering algorithms such as k-means to determine the actual clusters at the second step. Specifically, the first stage of previous works usually assume how the data is distributed in the projected space using human-chosen criteria such as self-reconstruction, local relationship preservation, sparsity [@tian2014learning; @huang2014deep; @shao2015deep; @wang2015learning; @chen2015deep; @song2013auto], fitting predefined distributions [@xie2015unsupervised], or strengthening of neighborhood relationships [@rippel2015metric] to learn the feature embedding. Furthermore, all of these techniques then use a metric, such as Euclidean or cosine distance, in the second stage. This further introduces human-induced bias via strong assumptions and the chosen metric may not necessarily be appropriate for the embedded space. In other words, there has not been a method to solve the two sub-problems (learning a feature space and performing clustering within that feature space) jointly in an end-to-end manner.
In this work, we propose a framework which minimizes such assumptions by training a network that can directly assign the clusters at the output layer. We specifically use weak labels, in the form of pairwise constraints or similar/dis-similar pairs, to learn the feature space as well as output a clustering. It is worth emphasizing that such weak labels could be obtained automatically (in an unsupervised manner) based on spatial or temporal relationships, or using a neighborhood assumption in the feature space similar to the above works. One could also get the weak labels from the ground-truth obtained from crowd-sourcing. In many cases, it may be an easier task for a human to provide pair-wise relationships rather than direct assignment of class labels (e.g. when dealing with attribute learning).
In order to adopt the raw data and weak labels for end-to-end clustering, we present the novel concept of constructing the cost function in a manner that incorporates contrastive KL divergence to minimize the statistical distance between predicted cluster probabilities for similar pairs, while maximizing the distance for dissimilar pairs. In the latter sections, we will show that the framework is extremely easy to realize by rearranging existing functional blocks of deep neural networks, so it has large flexibility to adopt new layer types, network architectures, or optimization strategies for the purpose of clustering.
One significant property of the proposed end-to-end clustering is that there are no cluster centers explicitly represented. This largely differs from all of the works mentioned above. Without the centers, no explicit distance metrics need to be involved for deciding the cluster assignment. The learning of the cluster assignments is purely data-driven and is implicitly handled by the parameters and the non-linear operations of the network. Of course the outputs of the last hidden layer could be regarded as the learned features, however it is not necessary to interpret it using predefined metrics such as Euclidean or cosine distance. The networks will find the best way to utilize the embedded feature space during the same training process in order to perform clustering. The experimental sections will demonstrate this property, in addition to strong robustness when the number of output clusters is varied. In such cases, the network tends to output a clustering that only utilizes the same number of nodes as there are clusters intrinsically in the data.
Furthermore, since the proposed framework can learn the cluster assignments using the proposed contrastive loss, it opens up the possibility of directly comparing its accuracy to the standard cross-entropy loss when full labels are available. This is achieved by developing an implementation that can efficiently utilize such dense pairwise information. This implementation strategy and experiments are also presented in sections \[impl\] and \[cvc\], showing favorable results compared to the standard classification approach. Source code in Torch is provided on-line.
Related Works
-------------
A common strategy to utilize pairwise relationship with neural networks is the Siamese architecture [@bromley93]. The concept had been widely applied to various computer vision topics, such as similarity metric learning [@chopra2005learning], dimensionality reduction [@hadsell2006dimensionality], semi-supervised embedding [@weston2008deep], and some applications to image data, such as in learning to match patches [@han2015matchnet; @Zagoruyko_2015_CVPR] and feature points [@simo2015discriminative]. The work of @mobahi2009deep uses the coherence nature of video as a way to collect the pairwise relationship and learn its features with a Siamese architecture. The similar idea of leveraging temporal data is also presented in the report of @goroshin2015unsupervised. In addition, the triplet networks, which could be regarded as an extension of Siamese, gained significant success in the application of learning fine-grained image similarity [@wang2014learning] and face recognition [@schroff2015facenet]. Despite the wide applicability of the Siamese architecture, there is no report exploring them from the clustering perspective. Furthermore, while some works try to maximize the information in a training batch by carefully sampling the pair [@han2015matchnet] or by formulating it as a triplet [@wang2014learning], there is no work showing how to use dense pairwise information directly and efficiently.
Our proposed implementation strategy can efficiently utilize any amount of pairwise constraints from a dataset to train a neural network to perform clustering. When the full set of constraints is given, it can compare to the vanilla networks trained using supervised classification. If only partial pairwise constraints are available, the problem is similar to semi-supervised clustering. There is a long list of previous work related to the problem. For example, COP-Kmeans [@wagstaff01] forced the clusters to comply with constraints and @rangapuram12 added terms in spectral clustering to penalize the violation of constraints. The more closely related works perform metric learning [@bilenko04] or feature re-weighting [@minkowski12] during the clustering process. The recent approaches TVClust and RDP-means [@khashabi2015clustering] address the problem with probabilistic models. None of these approaches, however, jointly learn the feature space in addition to clustering.
In the next sections we will explain how to inject the concept of clusters into a neural network formulations. The experiments on two image datasets will be presented in the third section, demonstrating the efficacy of the approach.
The End-to-End Clustering Networks {#gen_inst}
==================================
Consider the vanilla multilayer perceptron (MLP) used for classification tasks: Each output node is associated with predefined labels and the optimization minimizes a cost function, such as cross entropy, that compares the output labels (or the distribution over the labels) provided by the network for a set of instances and the corresponding ground truth labels. We start from this model and remove the hard association between labels and network outputs. The idea is to only use pairwise information and define the output nodes in a manner such that they can represent a clustering of the data. In other words, which node will correspond to which cluster (or object class) is dynamically decided during the training process. To achieve this, we formulate an approach that only needs to modify the cost criterion above the softmax layer of any neural network which was designed for a classification task. We therefore present a new pairwise cost function for clustering that can take the place of, or be combined with, the traditional supervised classification loss functions. This flexibility allows the network to use both types of information, depending on which is available.
Pairwise KL-Divergence
----------------------
{width="0.8\linewidth"}
While the output of the traditional softmax layer represents the probability that a sample belongs to the class labels (or clusters in our problem), the outputs of the whole softmax layer could be viewed as the distribution of possible clusters given a sample (Figure \[fig:concept\_diagram\]a). If the data only contains a single concept, such as in the case of hand-written digits, then the distributions between the softmax output for a similar pair should be similar. Conversely, the distribution over the class labels should be dissimilar if the pair belongs to different clusters (Figure \[fig:concept\_diagram\]b). The similarity between distributions could be evaluated by statistical distance such as Kullback-Leibler (KL) divergence. Traditionally this can be used to measure the distance between the output distribution and ground truth distribution. In our case, however, it can instead be used to measure the distance between the two output distributions given a pair of instances. Given a pair of distributions $\bm{P}$ and $\bm{Q}$, obtained by feeding data $x_p$ and $x_q$ into network $f$, we will fix $\bm{P}$ first and calculate the divergence of $\bm{Q}$ from $\bm{P}$. Assume the network has $k$ output nodes, then the total divergence will be the sum over $k$ nodes. To turn the divergence into a cost, we define that if $\bm{P}$ and $\bm{Q}$ come from a similar pair, the cost will be plain KL-divergence; otherwise, it will be the hinge loss (still using divergence). The indicator functions $I_s$ in equation \[eq2\] will be equal to one when $(x_p,x_q)$ is a similar pair, while $I_{ds}$ works in reverse manner. In other words:
$$\bm{P}=f(x_p), \bm{Q}=f(x_q),$$ $$KL(\bm{P} \parallel \bm{Q})=\sum_{i=1}^k{P_i log(\frac{P_i}{Q_i})},$$
$$\label{eq2}
loss(\bm{P} \parallel \bm{Q})=I_s(x_p,x_q)KL(\bm{P} \parallel \bm{Q})+I_{ds}(x_p,x_q)\max{(0,margin-KL(\bm{P} \parallel \bm{Q}))}.$$
Since the cost should be calculated from fixing both $P$ or $Q$ (i.e. symmetric), the total cost $L$ of the pair $x_p$, $x_q$ is the sum of both directions: $$\label{eq3}
L(\bm{P},\bm{Q})=loss(\bm{P} \parallel \bm{Q})+loss(\bm{Q} \parallel \bm{P}).$$
To calculate the derivative of cost $L$, it is worth to note that the $P$ in the first term of equation \[eq3\] (and $Q$ in the second term) is regarded as constant instead of variable. Thus, the derivative could be formulated as: $$\label{eq4}
\begin{split}
\frac{\partial}{\partial Q_i}L(\bm{P},\bm{Q})&=\frac{\partial}{\partial Q_i}loss(\bm{P} \parallel \bm{Q}), \\
&=\left\{\begin{array}{ccc}-\frac{P_i}{Q_i} & \mbox{if} & I_s(x_p,x_q)=1, \\
\frac{P_i}{Q_i} & \mbox{elseif} & KL(\bm{P} \parallel \bm{Q}))<margin, \\
0 & \mbox{otherwise}. & \end{array}\right.
\end{split}$$ $$\label{eq5}
\begin{split}
\frac{\partial}{\partial P_i}L(\bm{P},\bm{Q})&=\frac{\partial}{\partial P_i}loss(\bm{Q} \parallel \bm{P}), \\
&=\left\{\begin{array}{ccc}-\frac{Q_i}{P_i} & \mbox{if} & I_s(x_p,x_q)=1, \\
\frac{Q_i}{P_i} & \mbox{elseif} & KL(\bm{Q} \parallel \bm{P}))<margin, \\
0 & \mbox{otherwise}. & \end{array}\right.
\end{split}$$
With the defined derivatives of cost, the standard back-propagation algorithm can be applied without change.
Efficient implementation to Utilize Pairwise Constraints {#impl}
--------------------------------------------------------
Equation \[eq2\] is in the form of contrastive loss that is suitable to be trained with Siamese networks [@hadsell2006dimensionality]. However, when the amount of pairwise constraints increases, it is not efficient to enumerate all pairs of data and feed them into Siamese networks. Specifically, if there is a mini-batch that has pairwise constraints between any two samples, the number of pairs that have to be fed into the networks will be $n(n-1)/2$ where $n$ is mini-batch size. However, a redundancy occurs when a sample has more than one constraint associated with it. In such cases the sample will be fed-forward multiple times. However, feed-forward once for each sample is sufficient for calculating the pairwise cost in a mini-batch. Figure \[fig:architecture\_diagram\]c demonstrates an example for the described situation. The data with index 1 and 3 are fed-forward twice in vanilla Siamese networks to enumerate the three pairwise relationships: (1,2),(1,3), and (3,4). To avoid the redundancy of computation, we apply a strategy of enumerating the pairwise relationships only in the cost layer, instead of instantiating the Siamese architecture. This strategy simplified the implementation of neural networks which utilize pairwise relationship. Our proposed architecture is shown in the Figure \[fig:architecture\_diagram\]b. The pairwise constraints only need to be presented to the cost layer in the format of tuples $T:(i,j,relationship)$ where $i$ and $j$ are the index of sample inside the mini-batch and $relationship$ indicates similar/dissimilar pair. Each input data is therefore only fed-forward once in a mini-batch and its full/partial pairwise relationships are enumerated as tuples.
![The comparison between (a) classification networks, (b) our proposed networks, and (c) Siamese networks. The parts that differ across architectures are shown with distinct colors. In (a) and (b), the numbers in the data represent the index of the input data in a mini-batch.[]{data-label="fig:architecture_diagram"}](figure1.pdf){width="1\linewidth"}
Concretely, the gradients for the back-propagation in a mini-batch are calculated as: $$\frac{\partial}{\partial f(x_i)}\hat{L}=\sum_{\forall{j};(i,j)\in{T}}\frac{\partial}{\partial f(x_i)}L(f(x_i),f(x_j)).$$
One could see our proposed architecture (Figure \[fig:architecture\_diagram\]b) is highly similar to the standard classification networks (Figure \[fig:architecture\_diagram\]a). As a result of this design, ideas in the above two sections could be easily implemented as a cost criterion in the torch [*nn*]{} module. Then a network could be switched to either classification mode or clustering mode by simply changing the cost criterion. We therefore implemented our approach in Torch, and have released the source on-line [^1].
It is also worth mentioning that the presented implementation trick is not specifically for the designed cost function. Any contrastive loss could benefit from the approach. Since there is no openly available implementation to address this aspect, we include it in our released demo source.
Experiments
===========
We evaluate the proposed approach on the MNIST [@lecun98] and CIFAR-10 [@krizhevsky09] datasets. The two datasets are both normalized to zero mean and unit variance. The convolutional neural networks architecture used in these experiments is similar to LeNet [@lecun98]. The network has 20 and 50 5x5 filters for its two convolution layers with batch normalization [@ioffe2015batch] and 2x2 max-pooling layers. We use the same number of filters for both MNIST and CIFAR-10 experiments. The two subsequent fully connected layers have 500 and 10 nodes. Both convolutional and the first fully connected layers are followed by rectified linear units. The only hyper-parameter in our cost function is the $margin$ in equation \[eq2\]. The margin was chosen by cross-validation on the training set. There is no significant difference when the margin was set to 1 or 2. However, it has a higher chance of converging to a lower training error when the margin is 2, thus we set it to the latter value across the experiments. To minimize the cost function, we applied mini-batch stochastic gradient descent.
Clustering with partial constraints
-----------------------------------
We performed three sets of experiments to evaluate our approach. The first experiment seeks to demonstrate how the approach works with partial constraints. In this case, we use a clustering metric to demonstrate how good the resulting clustering is. The constraints are uniformly sampled from the full set, i.e, ${\#full\textendash{constraints}=n(n-1)/2}$, where $n$ is the size of training set. The pairwise relationship is converted from the class label. If a pair has the same class label, then it is a similar pair, otherwise it is dissimilar. We did not address the fact that the amount of dissimilar pairs usually dominates the pairwise relationship (which is more realistic in many application domains), especially when the number of classes is large. In our experiments for this section, the ratio between the number of similar and dissimilar pairs is roughly 1:9.
We evaluate the resulting clusters with the purity measure and normalized mutual information (NMI) [@strehl2003cluster]. The index of cluster for each sample is obtained by feed-forwarding the training/testing data into the trained networks. Note that we collect the clustering results of training data after the training error has converged, i.e, feed the training data one more time to collect the outputs after the training phase. We picked the networks which have the lowest training loss among five random restarts while the set of constraints are kept the same.
![The results of clustering with partial pairwise constraints. The \#constraint axis is the number of sampled pairwise relationship in the training data. The clustering and training is simultaneously applied on the training data (red line). The testing data (for blue, green, black lines) is used to validate if the feature space learned during clustering has generalizability. NNclustering is our proposed method. The *NNclustering feature + kmeans* uses the outputs at the last hidden layer (500-D) as the input for k-means. The baseline networks (black line) were trained with hinge loss of Euclidean distance. The evaluation metric in the first column is purity, while the second column shows the NMI score.[]{data-label="fig:cluster_result"}](figure2.pdf){width="0.9\linewidth"}
Figure \[fig:cluster\_result\] shows that on MNIST the clustering could still achieve high accuracy when constraints are extremely sparse. With merely 1200 constraints, which were randomly sampled from the pairwise relationship of full (60000 samples) training set, it achieves >0.9 purity and >0.8 NMI scores. Note that the training samples without any constraint associated to it has no contribution to the training. Thus, the scheme is not the same as the semi-supervised clustering framework in previous works [@wagstaff01; @bilenko04; @minkowski12] where their unlabeled data contribute to calculating the centers of the clusters. The lack of explicit cluster centers provides the flexibility to learn more complex non-linear representations, so the proposed algorithm could still predict the cluster of unseen data without knowing the cluster centers. In the experiments with MNIST, we could see the performance of testing data has no degradation. It is mainly because the networks could learn the clustering with so few constraints such that most of the training data have no constraints and act like unseen data. Note that although no directly comparable results for MNIST have been reported for our specific problem formulation, results for the closest problem setting can be seen in Figure 4(b) of @li2009constrained which achieves similar results for a subset of the classes and a much smaller number of constraints (only about 3k). Hence, our results are competitive with theirs but our approach is scalable enough to allow the use of many more constraints and continues to improve while their approach seems to plateau.
![The visualization of clustering. The figure was created by using the outputs of the softmax layer as the input for t-SNE [@van2008visualizing]. Only testing data are shown. The networks used in the first row are trained with 300, 1200, and 12000 pairs of constraints in MNIST training set. The second row is trained with 50k, 200k, and 800k constraints in CIFAR-10.[]{data-label="fig:vis_clustering"}](figure3.jpg){width="0.8\linewidth"}
To demonstrate the advantage of performing joint clustering and feature learning, we also applied the k-means algorithm with the features learned at the last hidden layer, which has 500 dimensions. The k-means algorithm used Euclidean or cosine distance and was deployed with 50 random restarts on the testing set. We report the clustering results of k=10 which has the lowest sum of point-to-centroid distances among 50 restarts. Since the dimensionality is relatively high, the performance of using Euclidean and cosine distance showed minor difference. The results in Figure \[fig:cluster\_result\] show that the jointly trained last layer utilize the outputs of last hidden layer much better than k-means.
To construct the baseline approach, we use the common strategy of training a Siamese networks with standard hinge loss embedding criteria in *torch nn* package, then perform k-means on the networks’ outputs. The baseline networks have the same architecture except the softmax layer and the loss function. Figure \[fig:cluster\_result\] shows that the proposed clustering framework beats the baseline with a significant margin when the number of constraints is few in the easy dataset (purity is ${\sim}5\%$ better in MNIST) or when the dataset is harder (purity is $15{\sim}50\%$ better in CIFAR-10).
The experiments with CIFAR-10 provides some idea of how the approach works on a more difficult dataset. The required constraints to achieve reasonable clustering is much higher. Eight constraints/sample (400,000 total constraints) is required to reach a 0.8 purity score with the same network. The performance on unseen data is also degraded because the networks is over-fitting the constraints. The degradation could possibly be mitigated by adding some regularization terms such as dropout. While any general regularization strategy could be applied in the proposed scheme, we do not address it in this work. Nevertheless, the clustering on the training set is still effective with sparse constraints, e.g., it is able to reach a purity of ${\sim}1$ with only 16 constraints/sample on CIFAR-10. The visualization in Figure \[fig:vis\_clustering\] provides more intuition about the clustering results trained with different numbers of constraints.
Robustness of Clustering
------------------------
### Adding Noise
Noisy constraints are likely to occur when the pairwise relationships are generated in an automatic/unsupervised way. We simulated this scenario by flipping the sampled pairwise relationship. Since the ratio of similar pair and dissimilar pair is 1:9, adding 10% noise will introduce equal amount of false-similar pair as the amount of true-similar pair. The clustering performance in Figure \[fig:robustness\] (left) shows the reasonable tolerance against noise. We would like to point out that when noise is less than 10%, the performance degradation is reduced when the number of constraints increased. This means that the proposed method could achieve higher performance by adding more pairwise information while keeping the ratio of noise the same. Real applications would benefit from this property since adding more weakly labeled data is cheap and the noise level of automatically generated constraints are usually the same.
### Changing Number of Clusters
![The robustness evaluation of the proposed clustering method. Left figure is the result of adding noisy constraints into MNIST, while the right figure simulates the case when the number of clusters is unknown.[]{data-label="fig:robustness"}](figureAdd2.pdf){width="1\linewidth"}
![The contingency tables of resulting clusters. It only shows k=30 (same experiment in the right part of figure \[fig:robustness\]) for the ease of visualization. NNclustering produces similar result even when k=100. The numbers in the table show the amount of samples been assigned to the cluster, while the blank rows indicate empty clusters. Higher numbers in fewer positions is the preferred result for clustering.[]{data-label="fig:confusion"}](confusion30_mat.pdf){width="0.6\linewidth"}
Another common scenario is that the number of target clusters is unknown. Since the purity metric is not sensitive to the number of clusters, NMI is more appropriate in this evaluation. We performed the experiment with 12,000 constraints for training, which include ${\approx}$1200 similar pairs in MNIST. The testing results in the right of figure \[fig:robustness\] show that the proposed method is almost not affected by increasing the number of clusters. Even in the condition of 100 clusters (by setting 100 output nodes in our networks), the performance only decreases by a very small amount. In fact, in figure \[fig:confusion\] most of the data were assigned to $\approx$10 major clusters and left other clusters being empty. In contrast, the kmeans-based approach (hinge loss embedding + kmeans) is susceptible to the number of clusters and usually divide a class into many small clusters.
Clustering VS Classification {#cvc}
----------------------------
The final set of experiments compares the accuracy of our approach with a pure classification task in order to get an upper bound of performance (since full labels can be used to create a full set of constraints) and see whether our approach can leverage pairwise constraints to achieve similar results. To make the results of clustering (contrastive loss) comparable to classification (cross-entropy loss), the label of each cluster is obtained from the training set. Specifically, we make the number of output nodes to be the same as the true number of classes, thus we could assign each output node with a distinct label using the optimal assignment. The results in Table \[table:cls\_vs\_clu\] show our cost function achieved slightly higher or comparable accuracy in most of the experiment settings. The exception is MNIST with 6 samples/class. The reason is that the proposed cost function creates more local minimum. If the training data is too few, then the training will be more likely to be trapped in certain local minimum. Note that we also applied a random restart strategy (randomly initializing the parameters of the network) to find a better clustering result based on the training set, which is a common strategy used in typical clustering procedures. We ran 5 randomly initialized networks to perform clustering and chose the network that had the highest training accuracy and then used the resulting network to predict the clusters on the testing set.
We also performed the experiments using the same architecture applied to a harder dataset, i.e, CIFAR-10. We did not pursue optimal performance on the dataset, but instead used it to compare the performance difference of learning between the classification and clustering networks. The results show that they are fully comparable. Since CIFAR-10 is a much more difficult dataset compared to MNIST, the overall drop of accuracy on CIFAR-10 is reasonable. Even in the extreme case when the number of training samples is small, the proposed architecture and cost function proved effective.
[|P[5cm]{}|P[2cm]{}|P[2cm]{}|]{} Training approach & Classification & Clustering\
Training data: & &\
MNIST 6 sample/class & **82.4%** & 79.4%\
MNIST 60 sample/class & 94.7% & **95.1%**\
MNIST 600 sample/class & 98.3% & **98.8%**\
MNIST full ($\approx$6000 sample/class) & 99.4% & **99.6%**\
Training data: & &\
CIFAR-10 5 sample/class & 21.3% & **22.0%**\
CIFAR-10 50 sample/class & 34.6% & **37.0%**\
CIFAR-10 500 sample/class & **55.0%** & 53.2%\
CIFAR-10 full (5000 sample/class) & **73.7%** & 73.4%\
\[table:cls\_vs\_clu\]
Conclusion and Future Works {#headings}
===========================
We introduce a novel framework and construct a cost function for training neural networks to both learn the underlying features while, at the same time, clustering the data in the resulting feature space. The approach supports both supervised training with full pairwise constraints or semi-supervised with only partial constraints. We show strong results compared to traditional K-means clustering, even when it is applied to a feature space learned by a Siamese network. Our robustness analysis not only shows good tolerance to noise, but also demonstrates the significant advantage of our method when the number of clusters is unknown. We also demonstrate that, using only pairwise constraints, we can achieve equal or slightly better results than when explicit labels are available and a classification criterion is used. In addition, our approach is both easy to implement for existing classification networks (since the modifications are in the cost layer) and can be efficiently implemented.
In future work, we plan to deploy the approach using deeper network architectures on datasets that have a larger number of classes and instances. We hope that this work inspires additional investigation into feature learning via clustering, which has been relatively less explored. Given the abundance of available data and recent emphasis on semi or unsupervised learning as a result, we believe this area holds promise for analyzing and understanding data in a manner that is flexible to the available amount and type of labeling.
### Acknowledgments {#acknowledgments .unnumbered}
This work was supported by the National Science Foundation and National Robotics Initiative (grant \# IIS-1426998).
[^1]: http://github.com/yenchanghsu/NNclustering
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I. Jack and D. R. T. Jones
*DAMTP, University of Liverpool, Liverpool L69 3BX, U.K.*
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We show that a particular “universal” form for the soft-breaking couplings in a softly broken $N=1$ supersymmetric gauge theory is renormalisation-group invariant through two loops, provided we impose one simple condition on the dimensionless couplings. The universal form for the trilinear couplings and mass terms is identical to that found in popular derivations of the soft-breaking terms from strings or supergravity.
If we take the , generalise to two Higgs doublets, supersymmetrise, impose R-parity, and add all possible soft breaking terms then we have the . The resulting theory has an alarming number of arbitrary parameters; far more than the . It is customary to assume that the plethora of possible independent soft terms undergo a form of unification, at the same scale where the gauge couplings meet. At this scale it is supposed that the soft terms consist simply of a common scalar mass, a common gaugino mass, and $\phi^3$ and $\phi^2$ interactions proportional to the analogous terms in the superpotential; the constants of proportionality being denoted $A$ and $B$ respectively. This simplification can be motivated to some extent by appeal to $N=1$ supergravity, and in particular to the idea that the breaking occurs in a hidden sector and is communicated to the observable sector via gravitational interactions (for a review, see \[\]). It also arises in superstring phenomenology\[\] \[\].
In this note we attempt to motivate a simple form for the soft breakings in a different way. We explore the consequences of imposing that the soft breakings in the theory at the unification scale be form invariant under renormalisation. In other words we require that the theory be renormalisable, in the usual sense that counter-terms generated by shifting parameters and fields in the Lagrangian suffice to remove the divergences encountered in perturbation theory. In general, of course, imposing strict renormalisability requires us to write down all interactions permitted by the symmetries. We will find, however, that a particular universal form for the soft-breaking couplings (one which is compatible with the desired pattern of breaking described above) is renormalisation-group (RG) invariant at least through two loops provided we impose one simple condition on the dimensionless coupling sector of the theory. Theories with this property would have the attractive feature that the universal form of the soft breaking terms (which is presumably generated by supersymmetry breaking of the underlying supergravity or superstring theory at or near the Planck scale) would be exactly preserved down to the gauge unification scale. The Lagrangian $L_{\rm SUSY} (W)$ is defined by the superpotential $L_{\rm SUSY}$ is the Lagrangian for the $N=1$ supersymmetric gauge theory, containing the gauge multiplet $\{A_{\m},\l\}$ ($\l$ being the gaugino) and a chiral superfield $\Ph_i$ with component fields $\{\ph_i,\psi_i\}$ transforming as a (in general reducible) representation $R$ of the gauge group $\cal G$. We assume that there are no gauge-singlet fields and that $\cal G$ is simple. (The generalisation to a semi-simple group is trivial.) The soft breaking is incorporated in $L_{\rm SB}$, given by (Here and elsewhere, quantities with superscripts are complex conjugates of those with subscripts; thus $\ph^i\equiv(\phi_i)^*$.) Aside from the terms included in $L_{SB}$ in Eq. , one might in general have $\ph^2\ph^*$-type couplings, $\psi\psi$ mass terms or $\l\psi$-mixing terms (as long as they satisfy a constraint that quadratic divergences are not produced). However, the soft-breaking terms we have included are those which would be engendered by an underlying supergravity theory and which are therefore considered most frequently in the literature.
The non-renormalisation theorem tells us that the superpotential $W$ undergoes no infinite renormalisation so that we have, for instance where $\g$ is the anomalous dimension for $\Ph$. The one-loop results for the gauge coupling $\b$-function $\b_g$ and for $\g$ are given by where $$\eqalignno{
\lf Q&=T(R)-3C(G),\q\hbox{and}\q &\Aac a\cr
\lf P^i{}_j&={1\over2}Y^{ikl}Y_{jkl}-2g^2C(R)^i{}_j. &\Aac b\cr}$$ Here The one-loop $\b$-functions for the soft-breaking couplings are given by $$\eqalignno{
\lf\b_h^{(1)ijk}&=U^{ijk}+U^{kij}+U^{jki},&\Ac a\cr
\lf[\b_{m^2}^{(1)}]^i{}_j&=W^i{}_j+2g^2(R_A)^i{}_j\tr[R_Am^2],&\Ac b\cr
\lf\b_b^{(1)ij}&=V^{ij}+V^{ji},&\Ac c\cr
\lf\b_M^{(1)}&=2g^2QM,&\Ac d\cr}$$ where $$\eqalignno{
U^{ijk}&=h^{ijl}P^k{}_l+Y^{ijl}X^k{}_l,&\Aaa a\cr
V^{ij}&=b^{il}P^k{}_l+{1\over2}Y^{ijl}Y_{lmn}b^{mn}
+\m^{il}X^j{}_l,&\Aaa b\cr
W^j{}_i&={1\over2}Y_{ipq}Y^{pqn}(m^2)^j{}_n+{1\over2}Y^{jpq}Y_{pqn}(m^2)^n{}_i
+2Y_{ipq}Y^{jpr}(m^2)^q{}_r\cr &\quad
+h_{ipq}h^{jpq}-8g^2MM^* C(R)^j{}_i,&\Aaa c\cr}$$ with Our assumption that the group $\cal G$ is semi-simple implies that the $\tr[R_Am^2 ]$ term in Eq. is zero, while the absence of gauge singlets means that (for instance in Eq. ) we have We then claim that the conditions $$\eqalignno{h^{ijk}&=-MY^{ijk},&\Aj a\cr
(m^2)^i{}_j&={1\over3}(1-{1\over{\lf}}{2\over3}g^2Q)MM^*\d^i{}_j,&\Aj b\cr
b^{ij}&=-{2\over3}M\m^{ij}&\Aj c\cr}$$ are RG invariant through at least two loops, provided we impose the condition (The idea of seeking relations amongst dimensionless couplings which are preserved by renormalisation has been explored in the coupling constant reduction programme of Zimmermann [*et al.*]{}\[\].) We first demonstrate the RG invariance of the conditions Eq. . The invariance of Eq. requires The strategy we adopt to verify equations such as Eq. is to simplify the $\b$-functions and anomalous dimensions as follows: firstly we use Eq. to replace $P^i{}_j$ by $Q$. We also use Eqs. to replace $h^{ijk}$, $m^2$ and $b$ wherever they occur. Having done this, we find that any occurrences of $Y_{ikl}Y^{jkl}$, $C(R)$, $C(G)$ or $T(R)$ can be written in terms of $P$ and $Q$ according to Eq. . We can now use Eq. again if necessary to replace $P$ by $Q$. For instance, we find, applying our strategy of imposing the condition Eq. in Eq. , and using Eqs. , , Henceforth we shall simply assume that this procedure is followed where possible. For instance, from Eqs. , , we find which, using Eqs. , ensures that Eq. is satisfied at one loop. The RG invariance of Eq. requires that At one loop we readily find, from Eqs. , , which, with Eqs. implies Eq. at one loop. (The additional, two-loop term in Eq. will be required later.) Finally, for the RG invariance of Eq. we need From Eqs. , , we obtain which, using Eqs. , leads immediately to Eq. at one loop. Finally, it behoves us to check that the condition Eq. is itself RG invariant. This amounts to the condition which is easily verified at one loop using Eqs. . The fact that the conditions Eq. , are preserved by renormalisation at one loop seems to us remarkable enough; however, they are actually preserved even at the two-loop level as well. The two-loop $\b$-functions for the dimensionless couplings were calculated in Ref. \[\]; they can be written in the form $$\eqalignno{ \llf\b_g^{(2)}&=2g^5C(G)Q-2g^3r^{-1}C(R)^i{}_jP^j{}_i
&\Au a\cr
\llf\g^{(2)i}{}_j&=[-Y_{jmn}Y^{mpi}-2g^2C(R)^p{}_j\d^i{}_n]P^n{}_p+
2g^4C(R)^i{}_jQ,&\Au b\cr}$$ where $Q$ and $P^i{}_j$ are given by Eq. , and $r=\d_{AA}$.
The calculation of the two-loop $\b$-functions for the soft breaking couplings raises interesting issues concerning the use of dimensional reduction in non- theories \[\].
The results are as follows\[\]–\[\]: $$\eqalignno{
\llf\b_h^{(2)ijk}&=-\Bigl[h^{ijl}Y_{lmn}Y^{mpk}+2Y^{ijl}Y_{lmn}
h^{mpk}-4g^2MY^{ijp}C(R)^k{}_n\Bigr]P^n{}_p\cr&\quad-2g^2U^{ijl}C(R)^k{}_l
+g^4(2h^{ijl}-8MY^{ijl})C(R)^k{}_lQ-Y^{ijl}Y_{lmn}Y^{pmk}X^n{}_p\cr
&\quad+(k\leftrightarrow i)+(k\leftrightarrow j),&\Av a\cr
\llf[\b_{m^2}^{(2)}]^j{}_i
&=\biggl(-\Bigl[(m^2)_i{}^lY_{lmn}Y^{mpj}
+{1\over2}Y_{ilm}Y^{jpm}(m^2)^l{}_n+{1\over2}Y_{inm}Y^{jlm}(m^2)^p{}_l
\cr&\quad+Y_{iln}Y^{jrp}(m^2)^l{}_r+h_{iln}h^{jlp}\cr&\quad+
4g^2MM^*C(R)^j{}_n\d^p{}_i+2g^2(R_A)^j{}_i(R_Am^2)^p{}_n\Bigr]P^n{}_p\cr
&\quad+\bigl[2g^2M^*C(R)^p{}_i\d^j{}_n-h_{iln}Y^{jlp}\bigr]X^n{}_p
-{1\over2}\bigl[Y_{iln}Y^{jlp}+2g^2C(R)^p{}_i\d^j{}_n\bigr]W^n{}_p\cr
&\quad +12g^4MM^* C(R)^j{}_iQ+4g^4SC(R)^j{}_i\biggr)+{\rm h.c.},&\Av b\cr
\llf\b_b^{(2)ij}&=\Bigl[-b^{il}Y_{lmn}Y^{mpj}
-2\m^{il}Y_{lmn}h^{mpj}-Y^{ijl}Y_{lmn}b^{mp}
\cr&\quad+4g^2MC(R)^i{}_k\m^{kp}\d^j{}_n\Bigr]P^n_p
-\bigl[\m^{il}Y_{lmn}Y^{mpj}+{1\over2}Y^{ijl}Y_{lmn}\m^{mp}\bigr]X^n{}_p
\cr&\quad-2g^2C(R)^i{}_kV^{kj}+g^2C(R)^i{}_kY^{kjl}Y_{lmn}b^{mn}\cr &\quad
+2g^4(b^{ik}-4M\m^{ik})C(R)^j{}_kQ+(i\leftrightarrow j),&\Av c\cr
\llf\b_M^{(2)}&=g^2\Bigl(8g^2C(G)QM-4r^{-1}C(R)^i{}_jP^j{}_iM
+2r^{-1}X^i{}_jC(R)^j{}_i\Bigr),&\Av d\cr}$$ where The expressions given in Eq. (and in particular Eq. ) correspond to the use of a particular subtraction scheme whereby the mass of the $\e$-scalars decouples from the evolution of the other parameters. For a discussion, see refs. , .
At two loops we find, applying the usual procedure to Eqs. , , $$\eqalignno{
\llf\g^{(2)i}{}_j&=-{2\over9}g^4Q^2\d^i{}_j,&\Aw a\cr
\llf\b^{(2)}_g&=-{2\over3}g^5Q^2,&\Aw b\cr
\llf\b^{(2)}_M&=-{8\over3}g^4Q^2M.&\Aw c}$$ Now we can go on to check the RG invariance of Eqs. to two-loop order. Using Eqs. , in Eq. , we find Inserting Eqs. , and into Eq. , we immediately verify the two-loop RG-invariance of Eq. . Now using Eqs. , in Eq. , we obtain Hence, from Eqs. , , , we obtain Using Eqs. , , , , in Eq. , we see that Eq. is RG invariant throughtwo loops. Using Eqs. , , , in Eq. , we find On substituting Eqs. and into Eq. , we see that Eq. is RG invariant at two loops. Finally, using Eqs. , we verify Eq. at two loops, ensuring the RG invariance of Eq. at this level. Thus we have demonstrated the RG invariance of Eqs. and through two loops.
We turn now to the possibility of constructing realistic models satisfying our constraints. The main impact on low-energy physics, is that from Eq. we have (in the usual notation) a universal scalar mass $m_0$ and universal $A$ and $B$ parameters related (to lowest order in $g^2$) to the gaugino mass $M$ as follows: $$\eqalignno{
m_0 &= {1\over{\sqrt{3}}}M,&\Baa a\cr
A &= -M,&\Baa b\cr
B &= -{2\over3}M.&\Baa c\cr}$$ Evidently it will be interesting to explore the region of the usual parameter space consistent with Eq. ; current experimental constraints will probably not rule out the scenario [*per se*]{}, but the various super-partner masses will be more tightly correlated than in the usual approach.
It follows from our results that if $P^i{}_j=Q=0$, (guaranteeing that the dimensionless coupling $\b$-functions are zero to two loops) then soft-breaking couplings related by Eq. will also have vanishing $\b$-functions, leading to the possibility of finite softly-broken theories. This has already been pointed out at the one-loop level in Ref. \[\] and at the two-loop level in Ref. . In Ref. it was remarked that Eqs. are consistent with the pattern of soft-breaking terms which emerges from breaking in the hidden sector of an underlying supergravity theory with a “minimal” Kähler potential. Even more interestingly, Eqs. are identical to relations which arise in effective supergravity theories motivated by superstring theory, where breaking is assumed to occur purely via a vacuum expectation value for the dilaton. More general scenarios involving vacuum expectation values for other moduli fields are also possible. To be more specific, we follow Ref. in concentrating on the modulus $T$ whose classical value gives the size of the manifold, and parametrising the ratio of the auxiliary fields $F^S$ and $F^T$ for the dilaton $S$ and for $T$ by an angle $\th$–so that $\th$ characterises the extent to which supersymmetry-breaking is dominated by $S$ or $T$. We also simplify still further by assuming a vanishing cosmological constant and by ignoring string loop corrections, and also the phases of $F^S$ and $F^T$. In this more general case, the gaugino mass is related to the gravitino mass $m_{3\over2}$ by $M=\sqrt3m_{3\over2}\sin\th$, and the soft-breaking parameters $m_0$ and $A$ are still given by Eqs. , while $B$ is either given by or\[\] depending on whether the $\m$ term is generated by an explicit $\m$-term in the supergravity superpotential, or by a special term in the Kähler potential. In the first case the value $\th={4\pi\over3}$ reproduces our constraint Eq. ; however, in the second case there is no value of $\th$ which is consistent with this constraint.
In addition to Eq. , we also need to to impose Eq. as a condition on the theory at the unification scale. It is not clear at present how such a constraint would naturally emerge from string theory. The special case $P^i{}_j=Q=0$, corresponding, as we have remarked, to two-loop finite theories, was tabulated in Ref. \[\]. They found a fair number of possibilities, including a few of phenomenological interest: in particular a simple $SU_5$ model \[\] \[\].
We can anticipate, therefore, that it will be possible to construct unified models satisfying Eq. . The obvious try is a simple generalisation of the finite $SU_5$ model first analysed in ref. . The superpotential is where $i,j: 1\ldots x$ and $\a,\b : 1\ldots y$ so that we have $x$ generations, $y$ sets of Higgs multiplets $(H + \Hbar)$ and a single adjoint $(24)$. It is straightforward to write down Eq. for this model; tracing on all indices we obtain the relations:
$$\eqalignno{
|A|^2 + {8\over5}|C|^2 &= g^2 y\left({8\over5}+{1\over9}Q\right),&\jrmod a\cr
|B|^2 + {6\over5}|C|^2 &= g^2 y\left({6\over5}+{1\over{12}} Q\right),
&\jrmod b\cr
|B|^2 &= g^2 x\left({6\over5}+{1\over{12}} Q\right),&\jrmod c\cr
3|A|^2 + 2|B|^2 &= g^2 x\left({36\over5}+{1\over{3}} Q\right),&\jrmod d\cr
{189\over5}D^2 + C^2 &= g^2 \left( 10 +{1\over{3}} Q\right).&\jrmod e\cr}$$ where here $Q = 2x + y - 10$. It is easy to show, however, that Eqs. do not have a solution unless $Q=0$, which corresponds to the finite case. This outcome is not generic, however, and it is easy to modify the theory so as to produce candidate theories that do satisfy Eq. , for example by including one or more sets of $10 + \10bar$ multiplets. It remains to be seen, however, whether there exists a compelling unified theory with universal soft breaking terms. Meanwhile, if we conjecture that such a theory leads to the same low energy physics as the , we can at least explore the consequences of Eq. for the super-particle spectrum. We will report on these calculations elsewhere.
We thank Luis Ibáñez for communications, and in particular for drawing Ref to our attention. IJ thanks Carlos Muñoz and Dennis Silverman for useful conversations, and also thanks PPARC for financial support.
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[*Introduction.—*]{} Flux pinning by correlated disorder in superconductors has recently been the subject of intense investigations because of its ability to immobilize vortices, reduce dissipation effects, and create high critical currents [@reviews]. In particular, the introduction of correlated disorder by heavy ion irradiation can create columnar defects in which vortices are pinned along their entire length [@Civale]. Such systems are predicted to form a Bose glass at low temperatures when the vortices localize at randomly-placed columnar pins [@Nelson; @Tauber]. The behavior of this state depends on whether the magnetic flux density $B$ is below, equal to, or greater than $B_{\phi}$, the matching field at which the number of flux lines equals the number of columnar pins [@Shapiro; @Leo; @Rosenbaum]. In addition, an important role is played by the critical state [@Bean; @Kim], in which a gradient in the flux profile arises as individual vortices enter the sample and become pinned at defects. This gradient determines $ J_{c}(B)$ [@reviews; @Bean; @Kim]. Detailed information on the vortex dynamics in the critical state, including [*local*]{} and averaged physical quantities, is required to explain the effects induced by the addition of columnar defects. Furthermore, understanding the effect of the spatial distribution of the pinning sites on the Bose glass can facilitate creating samples with optimal pinning to enhance $J_c$, which is of technological importance. To investigate the [*flux-gradient-driven*]{} dynamics of vortices in superconductors with columnar defects, we perform simulations of individual vortices [*and*]{} antivortices interacting with strong columnar pinning sites, as an external field $H(t)$ is swept through a complete loop. We monitor distinct changes in relevant physical quantities (e.g., $B$, $M$, $J_{c}$).
[*Simulation.—*]{} We simulate an infinite slab with a magnetic field $ {\bf H} = H {\bf z} $ applied parallel to the surface so that there are no demagnetization effects. We consider rigid vortices and straight columnar pins of uniform strength (all $\parallel {\bf z}$); thus we only need to model a transverse 2D slice ($x$–$y$ plane) from the 3D slab. Our results are for a $36 \lambda \times 36 \lambda $ system with periodic boundary conditions, where $ \lambda $ is the penetration depth. Our superconducting system has a pinned region from $ x = 6\lambda $ to $ x = 30\lambda $; thus $1/3$ of the system is unpinned and $2/3$ has randomly-placed non-overlapping parabolic traps with radius $\xi_p = 0.15 \lambda$. The flux lines evolve according to a $T=0$ molecular dynamics algorithm. Thus, thermal effects are neglected and we consider a situation deep in the Bose glass regime, where the Mott insulator phase should be observable [@Nelson; @Tauber; @Leo]. Note that pinning due to columnar disorder is much less sensitive to thermal effects than pinning due to point defects. For instance, for $B < B_{\phi}$ Täuber [*et al.*]{}[@Tauber], working with Bi$_{2}$Sr$_{2}$CaCu$_{2}$0$_8$, find that thermal effects on pinning become relevant only for $ T_1 \approx 0.9 T_c \approx 78 $ K $ \approx T_{irrev}$. Therefore, in this case all temperatures $ T < T_1 $ may be considered low.
We simulate an increasing external field $H(t)$ as described in [@Richardson; @Reichhardt] where we quasi-statically add flux lines to the unpinned region; these flux lines attain a uniform density $n$, so that an external applied field $H$ may be defined as $H=n\Phi_{0}$. As the external field increases, the flux lines—by their own mutual repulsion—are forced into the pinned region where their motion is impeded by the defects. We correctly model the vortex-vortex force by a modified Bessel function $ K_{1}(r/\lambda) $, which falls off exponentially so that a cutoff at $ 6\lambda $ can be safely imposed. The overdamped equation of motion is: ${\bf f}_{i} = {\bf f}_{i}^{vv} + {\bf f}_{i}^{vp} = \eta{\bf v}_{i}$, where the total force ${\bf f}_{i}$ on vortex $i$ (due to other vortices $ {\bf f}_{i}^{vv} $, and pinning sites $ {\bf f}_{i}^{vp} $) is $
{ \bf f}_{i} = \ \sum_{j=1}^{N_{v}}\, f_{0} \,
K_{1} ( |{\bf r}_{i} - {\bf r}_{j}| / \lambda )
\, {\bf {\hat r}}_{ij}
+ \sum_{k=1}^{N_{p}}
\frac{f_{p}}{\xi_{p}}
\ |{\bf r}_{i} - {\bf r}_{k}^{(p)}|\ \
\Theta ( ( \xi_{p} - |{\bf r}_{i} - {\bf r}_{k}^{(p)}| ) / \lambda )
\ {\bf {\hat r}}_{ik} \
$. Here $\Theta $ is the Heaviside step function, $ {\bf r}_{i} $ ($ {\bf v}_{i}) $ is the location (velocity) of the $i$th vortex, $ {\bf r}_{k}^{(p)} $ is the location of the $k$th pinning site, $ \xi_{p}$ is the radius of the pinning well, $ f_{p}$ is the maximum pinning force, and we take $\eta=1$. We measure lengths in units of the penetration depth $\lambda$, forces in terms of $ f_{0} = \Phi_{0}^{2}/8\pi^{2}\lambda^{3}$ ($=2\epsilon_0/\lambda$), energies in units of $\epsilon_0=(\Phi_0/4\pi\lambda)^2$ ($=f_0\lambda/2$), and magnetic fields in units of $\Phi_0/\lambda^2$. The sign of the interaction between vortices is determined by $ f_{v} $; we take $ f_{v} = +f_{0} $ for repulsive vortex-vortex interactions and $ f_{v} = -f_{0} $ for attractive vortex-antivortex interactions. A vortex and antivortex annihilate and are removed from the system if they come within $ 0.3\lambda $ of one another. For columnar pinning the condensation energy is $ \epsilon_{0} = (\Phi_{0}/4\pi\lambda)^{2} $, which can be related to the maximum pinning force $f_p$ to give $ f_p/f_0 = \lambda/\xi_{p} = \lambda/0.15\lambda \approx 7 $ . However the measurable quantities of the system do not change for pinning forces greater than $ 2.0f_{0}$. We systematically vary one parameter such as the density of pins, $ n_{p}$, pinning strength $f_{p}$, or the spatial distribution of pinning, while keeping the other parameters fixed. Previous simulations of flux-gradient-driven vortices [@Richardson; @Reichhardt] considered situations where $B < B_{\phi}$ and $f_{p}/f_{0} < 1$ and found a $J_{c}(H)$ dependence as $1/H$. This corresponds to the experimentally common situation described by the Kim model[@Kim]. However, when we consider very strong columnar pins ($f_p/f_0 > 2.0$) we find a very different behavior.
[*Flux Profiles.—*]{} From the location of the vortices, we compute their local density $B(x,y,H(t))$ and profile $B(x,H(t))$ inside the sample. Figure 1 shows a typical example of $B(x)$ for a hysteresis loop. When $ |B| < B_{\phi}$, the gradient of the flux profile (i.e., the current $J_c$) is very large. Above $ B_{\phi} $ we see an abrupt change in the flux profile to a much shallower slope. As the external field is ramped still higher, the slope of $B$ decreases gradually. These results are very similar to those observed experimentally [@Rosenbaum; @Majer] for samples with columnar pins as the local field is increased beyond $B_{\phi}$ and quite different from the ones seen in other types of samples (e.g., [@Richardson; @Reichhardt]). Such results have been interpreted in terms of a change in the pinning force: when $B < B_{\phi}$ the vortices are pinned at individual defects, and when $B > B_{\phi}$ many vortices are pinned at interstitial sites due to the repulsion from the defect-pinned vortices.
To give a more microscopic idea of the behavior in this system, in Fig. 2(a–c) we present snapshots of the positions of the pins and the vortices in an $ 8\lambda\times8\lambda $ region of the sample (while the external field is gradually increased). The vortices are entering from the left of the figures and thus there is a flux gradient in the three frames. In Fig. 2(a), most of the vortices are pinned at defects; only two (located near other defect-pinned flux lines) are at interstitial sites. Since most of the vortices at these low fields are trapped by defects, the effective pinning force is very high and hence, as seen in Fig. 1, the flux gradient for low fields is quite steep. In Fig. 2(b), near $ B_{\phi} $, a greater fraction of interstitially pinned vortices is present, producing a much lower flux gradient. The fact that unoccupied pins are typically located very close to occupied pins indicates that many unoccupied pins are being “screened" by the vortex repulsion from the occupied pins. Also, because of the flux gradient, there are more interstitially pinned vortices near the left of the figure than the right. Thus, the presence of a flux gradient limits the number of accessible pinning sites, when these sites are randomly distributed. Finally in Fig. 2(c), above $ B_{\phi} $, the flux gradient is small and hence the effective pinning force is low. Here a large number of vortices are weakly trapped at interstitial sites and the majority of the defects are now occupied by vortices. In the lower left corner a small domain of triangularly-arranged vortices can be seen—indicating that the vortex-vortex repulsion is starting to dominate over the vortex-pin attraction.
[*Magnetization loops and Critical Current.—*]{} From the microscopic vortex dynamics of individual vortices described above, we can compute macroscopic measurable quantities. In particular, we quantify the vortex behavior described above, as the field is brought through $ B_{\phi} $, by measuring $ M(H) $ and $ J_{c}(B) $ for samples with different pinning parameters (see Fig. (3)). For instance, the effect of changing the matching field $ B_{\phi} $ is examined in Fig. 3(a,b) by fixing $ f_{p} $ at $ 2.5f_{0}$ and changing the pin density $n_p$. For all the cases we considered, the width of the magnetization curve is much broader when $ | B | < B_{\phi} $, falling off very rapidly when $ B > B_{\phi} $. Fig. 3 shows the enhancement of $J_{c}(B)$, for $B < B_{\phi}$, as the field is swept. We obtain $ J_{c}(B) $ directly from the flux density profiles using Maxwell’s equation $ dB/dx = \mu_{0}J $. For a fixed field $H$, we average the slope of $B$ over the entire sample. In all cases the enhancement of $ J_{c}(B) $ is restricted to $ | B | < B_{\phi} $. This is very similar to the results seen in [@Majer] at low $ T $, where $ J_{c}(B) $ was also obtained directly from the flux gradient. It can also be seen from Fig. 3 that $J_{c}$ has a definite dependence on $B$, which is different from the step-like two-current model suggested in [@Rosenbaum] (indicated by a dashed line in Fig. 3(b)).
We next examine the effect of varying the pinning force from $ f_{p} = 0.3f_{0} $ to $f_{p} = 5.0f_{0}$ (see Fig. 3(c,d)). There is little change in $M(H)$ and $J_c(B)$ as the pinning force is increased from $ f_{p} = 2.5f_{0}$ to $ f_{p} = 5.0f_{0}$. This reflects the fact that once vortices are trapped strongly enough, so that they cannot be unpinned, any further increase in the pinning strength does not significantly affect $M(H)$ and $J_c(B)$. As the pinning force is decreased in Fig. 3(c), the width of the magnetization loop narrows. A crossover at $B_{\phi}$ is no longer observed when the pinning force from the individual defects becomes on the order of the interstitial pinning force (when $f_p \sim f_0/2$). This behavior is emphasized in Fig. 3(d) for $ J_{c}(B) $. At the lowest pinning strengths, only a small increase in $ J_{c}(B) $ can be seen when $|B|<B_{\phi}/3$, and not when $|B|<B_{\phi}$ as in the strongly pinned ($f_p \gtrsim 2.0 f_0$) cases. These results are very similar to those seen in [@Majer] when the temperature was raised, which reduces the effective pinning strength.
The theory of a Mott insulator phase in the Bose glass [@Nelson] leads one to expect an enhancement in $ J_{c}(B) $ at $ B_{\phi} $. At this field the vortices become strongly localized at the pinning sites, due to the repulsion of the other vortices, and produce a Meissner-like phase which prevents increased flux penetration over a range of field. However, we do [*not*]{} observe any particular enhancement of $ J_{c} $ at $ B = B_{\phi} $. This is due to the combined effects of [*both*]{} the critical state, which prevents a uniform field in the sample, and the randomness in the spatial distribution of defects, which prevents all the pins from being occupied due to “screening" effects. We point out that near $B_{\phi}$ there can still be a decrease in the magnetization relaxation rate since it depends on thermal activation and tunneling.
We can significantly reduce screening effects (due to closely-spaced defects) by changing the spatial arrangement of the pinning sites, thus enhancing $J_{c}$. Figure 3(e,f) presents $M(H)$ and $J_c(B)$ for three systems with pinning sites which are (1) randomly located, (2) placed in a triangular lattice (see, e.g., [@Baert]), and (3) randomly displaced, up to $0.25 \lambda$ from a triangular lattice. All three systems have $ B_{\phi} = 1/\lambda^{2}$ and $ f_{p} = 2.5f_{0} $. For fields less then $ B_{\phi} $, the width of $M(H)$ and $J_{c}(B)$ for the triangular pinning is nearly twice that for the random case. The asymmetry in $ M(H) $ and $ J_{c}(B) $ is much more pronounced when pins are placed in a triangular lattice. For this case, there is also a very sharp drop in $ M(H) $ just past $ B_{\phi} $, very similar to that observed in [@Baert], since here the distinction between vortex pinning at defects and interstitial pinning is much better defined. This result is very important since it indicates that substantial increases in the critical current for fields less then $ B_{\phi} $ can be achieved by evenly spacing the pins. In this case, $M(H)$ and $ J_{c}(B) $ are enhanced because all the pins are accessible to the vortices, as opposed to the random case where it is difficult for vortices to reach closely-spaced defects. Moreover, in the system with triangular pinning, trapped vortices are better localized at the defects due to the interactions with other pinned vortices (since an additional energy minima is present when the vortices are in a favorable lattice configuration). This indicates that the Mott insulator phase is more accessible in a triangular array of pinning sites than in a Bose glass with randommly-located defects. In the case where the pinning sites are randomly displaced (up to $0.25\lambda$) from the triangular sites, there is still a clear enhancement of $ J_{c}(B) $; however, it is much smaller than in the ordered triangular pinning array.
[*Vortex Plastic Flow.—*]{} To investigate the dynamics of the flux-gradient-driven vortices near $B_{\phi}$ and higher fields, we present in Fig. 4(a,b) the trajectories of the vortices for a system with (a) strong pinning, $ f_{p} = 2.5f_{0} $, and with (b) weaker pinning, $ f_{p} = 0.3f_{0} $, both with $B_{\phi} = 1.0\Phi_{0}/\lambda^{2}$. For both panels the external field $H$ is increased from $ 0.9\Phi_{0}/\lambda^{2} $ to $ 1.4\Phi_{0}/\lambda^{2} $. In (a) the moving vortices follow specific winding channels indicative of [ *plastic flow*]{} of interstitial vortices around domains of flux lines which are strongly pinned at defects. These “vortex rivers" often develop well defined tight [*bottlenecks*]{} as well as much broader “vortex streets" with several lanes. At the top left side of Fig. 4(a), two bottlenecks can be seen. Each of these “interstitial vortex paths" is located between flux lines that are strongly pinned at defects, indicating that the interstitial vortices are flowing through the energy minima created by the strongly pinned flux lines. These interstitial-vortex-channels form for both the entrance and exit of vortices and are a general feature of all simulations with randomly placed strong pinning sites. It is interesting to note that although a large number of vortices are clearly immobile at the defects, $J_{c}$ is quite low for this region of $B > B_{\phi}$ indicating that the channels are acting as [*weak links*]{}. In Fig. 3(b) with $ f_{p} = 0.3f_{0} $ we see a much different behavior with only a few channels. Further, it can be observed from the lines that cross through pinning sites that vortices can become unpinned since the pinning force is on the order of the interstitial pinning. Thus vortex motion consists of both interstitial trajectories and also pin-to-pin motion.
[*Conclusions.—*]{} Our simulations of flux-gradient-driven rigid vortices interacting with columnar pinning sites are consistent with recent experiments [@Rosenbaum] which indicate that a sharp change in the magnetic flux gradient occurs above $B_{\phi}$. Our results quantitatively predict how $ J_{c} $ varies with field and pinning parameters (e.g., strength and location). Moreover, we can monitor the spatio-temporal dynamics of vortices as the pinning mechanism evolves from strong columnar pinning (for $B<B_{\phi}$) to weaker interstitial pinning (for $B>B_{\phi}$). In particular, we quantify the enhancement of $J_{c}$, when $B < B_{\phi}$, by varying the spatial arrangements of the pinning sites. When the pinning strength of the columnar defects becomes on the order of the interstitial pinning strength, this sharp transition in $J_{c}$ is greatly reduced. We compute the spatio-temporal dynamics of vortices and show that for strongly pinned samples the vortex transport is dominated by the plastic flow of interstitially-pinned vortices around regions of flux lines strongly pinned at defects. In the weaker pinning samples, vortices can jump from defect to defect site. Finally, $ M $ and $ J_{c} $ can be considerably enhanced by placing the defects on a regular triangular lattice, for a given pin density, so that all pins are equally accessible, thus preventing the screening of closely-spaced defects occurring in a random distribution.
This work was supported in part by the NSF under grant No. DMR-92-22541. CR and CJO acknowledge support from Rackham graduate fellowships. We also acknowledge the UM Center for Parallel Computing for computer time, and J. Siegel for a critical reading of this manuscript.
The literature on vortices is vast and we do not attempt a review here. For recent reviews, and an extensive list of references, see, e.g., G. Blatter [*et al.*]{}, Rev. Mod. Phys. [**66**]{}, 1125 (1994); E.H. Brandt, preprint.
See, e.g., L. Civale [*et al.*]{}, Phys. Rev. Lett. [**67**]{}, 648 (1991); W. Gerhäuser [*et al.*]{}, [*ibid*]{} [**68**]{}, 879 (1992); R.C. Budhani [*et al.*]{}, [*ibid*]{} [**69**]{}, 3816 (1992); M. Konczykowski [*et al.*]{}, Phys. Rev. B [**44**]{}, 7167 (1991).
D.R. Nelson and V.M. Vinokur, Phys. Rev. B [**48**]{}, 13060 (1993).
U.C. Täuber [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 5132 (1995); H. Dai [*et al.*]{}, Science [**265**]{}, 1552 (1994); U.C. Täuber and D.R. Nelson, preprint.
I.B. Khalfin, B.Ya. Shapiro, Physica [**C207**]{}, 359 (1993).
L. Radzihovsky, Phys. Rev. Lett. [**74**]{}, 4919, 4923 (1995).
K.M. Beauchamp [*et al.*]{}, Phys. Rev. B [**52**]{}, 13025 (1995).
C.P. Bean, Rev. Mod. Phys. [**36**]{}, 31 (1964).
Y.B. Kim, [*et al.*]{} Rev. Mod. Phys. [**36**]{}, 43 (1964).
R.A. Richardson, O. Pla, and F. Nori, Phys. Rev. Lett. [**72**]{}, 1268 (1994).
C. Reichhardt, C.J. Olson, J. Groth, S. Field, and F. Nori, Phys. Rev. B [**52**]{}, 10411 (1995).
K.M. Beauchamp [*et al.*]{}, Phys. Rev. Lett. [**75**]{}, 3942 (1995).
M. Konczykowski [*et al.*]{}, Physica [**C235-240**]{}, 2965 (1994).
M. Baert [*et al.*]{}, Phys. Rev. Lett. [**74**]{}, 3269 (1995).
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In spite of enormous theoretical and experimental progresses in quantum uncertainty relations, the experimental investigation of most current, and universal formalism of uncertainty relations, namely majorization uncertainty relations (MURs), has not been implemented yet. A significant problem is that previous studies on the classification of MURs only focus on their mathematical expressions, while the physical difference between various forms remains unknown. First, we use a guessing game formalism to study the MURs, which helps us disclosing their physical nature, and distinguishing the essential differences of physical features between diverse forms of MURs. Second, we tighter the bounds of MURs in terms of flatness processes, or equivalently, in terms of majorization lattice. Third, to benchmark our theoretical results, we experimentally verify MURs in the photonic systems.'
author:
- Yuan Yuan
- Yunlong Xiao
- Zhibo Hou
- 'Shao-Ming Fei'
- Gilad Gour
- 'Guo-Yong Xiang'
- 'Chuan-Feng Li'
- 'Guang-Can Guo'
title: 'Strong Majorization Uncertainty Relations: Theory and Experiment'
---
***Introduction.—***In the quantum world, measurements allow us to gain information from a system, and the action of measurements on quantum systems is fully embraced in the areas of quantum optics, quantum information theories, and quantum communication tasks. It is therefore of great practical interest to study the limitations and precisions of quantum measurements. In taking the measurements on board, however, it appears that quantum mechanics imposes strict limitation on our ability to specify the precise outcomes from incompatible measurements simultaneously, which is known as “[*Heisenberg Uncertainty Principle*]{}” [@Heisenberg1927].
In the context of the uncertainty principle, both variance [@Kennard1927; @Weyl1927; @Robertson1929; @Schrodinger1930; @Huang2012; @Maccone2014; @Xiao2016W; @Xiao2016M; @Xiao2016S; @Guise2018; @Qiao2018T; @Xiao2017I] and entropies [@Deutsch1983; @Partovi1983; @Kraus1987; @Maassen1988; @Ivanovic1992; @Sanchez1993; @Ballester2007; @Wu2009; @Berta2010; @Li2011; @Prevedel2011; @Huang2011; @Tomamichel2011; @Coles2012; @Coles2014; @Kaniewski2014; @Furrer2014; @Li2015; @Berta2016; @Xiao2016St; @Xiao2016QM; @Xiao2016U; @review; @Xiao2018H; @Xiao2018Q; @Chen2018; @Coles2019; @Li2019; @Wang2019E; @Xiao2019CIP; @XiaoPhD] are by no reason the most adequate to use. The attempt to find all suitable uncertainty measures has triggered the interest of the scientific community in the quest for a better understanding and exploitation of the precisions of quantum measurements. As previously shown in [@PRL; @Narasimhachar2016], any eligible candidate of uncertainty measures should be: ([1]{}) non-negative; ([2]{}) a function only of the probability vector associated with the measurement outcomes; ([3]{}) invariant under permutations; ([4]{}) nondecreasing under a random relabeling. According to these restrict conditions, a qualified uncertainty measure should be a non-negative Schur-concave function, and the [*majorization uncertainty relations*]{} (MURs) arise from the fact that all Schur-concave functions can, in general, preserve the partial order induced by majorization [@Hardy1929; @Partovi2011; @Majorization]. Based on the mathematical expressions, the notions of MURs are classified into two categories; that are direct-product MUR (DPMUR) [@PRL; @JPA] and direct-sum MUR (DSMUR) [@PRA; @M7]. In the original work of [@PRA], the essential differences of mathematical features between DPMUR and DSMUR (i.e. tensor and direct-sum) are compared and analyzed. However, it is fair to say, that our understanding of the physical essences of MURs is still very limited.
In this work, our first contribution, which also reflects the original intention of this work, is to characterize the essential differences of physical features between DPMUR and DSMUR theoretically. More precisely, we show that the difference between these MURs are more than its mathematical expressions, what really matters is the joint uncertainty they represent. DPMUR is identified as a type of [*spatially-separated joint uncertainty*]{}, and meanwhile DSMUR is recognized as a type of [*temporally-separated joint uncertainty*]{}. Despite previous developments on MURs, there is still a gap between their optimal bounds and the ones constructed in [@PRL; @JPA; @PRA; @M7]. Our second contribution is to fill this gap by applying a technique, called flatness process [@Cicalese2002], which is also known as concave envelope in Mathematics.
Besides theoretical advancements, the experimentally implementations of quantum uncertainty relations are also already of great interest, as they are a pioneering demonstration of the limitations on quantum measurements, and may also inspire breakthrough in modern quantum technologies. So far, the uncertainty relations based on variance and entropies have been successfully realized in various physical systems, including neutronic systems [@neutron1; @neutron2; @neutron3], photonic systems [@photon1; @photon2; @photon3; @photon4; @photon5; @photon6; @photon7], nitrogen-vacancy (NV) centres [@NV], nuclear magnetic resonance (NMR) [@NMR], and so forth. However, an experimental demonstration of the uncertainty relations given by majorization has never been shown. To boost the experimentally study of the uncertainty relations, it is highly desirable to know how to investigate MURs in a physical system. The third contribution of this work is that we implement the MURs by measuring a qudit state encoded with the path and polarization degree of the freedom of a photon system for the first time.
at (-2.7,-0.4) [(a) DPMUR.]{};
(-5,3) rectangle (-0.75,0); (-4.8,2.5) rectangle (-3.8,1.8) node\[pos=0.5,white\] [$\Gamma_\rho$]{}; (-3,2.5) rectangle (-2,1.8);
(-2.5,1.95) – (-2.35,2.4); (-2.15,1.95) arc (10:170:0.35);
(-4.8,1.2) rectangle (-3.8,0.5) node\[pos=0.5,white\] [$\Gamma_\rho$]{}; (-2.5,0.65) – (-2.35,1.1);
(-2.15,0.65) arc (10:170:0.35); (-3,1.2) rectangle (-2,0.5);
at (-2.5,2.7) [$M$]{}; at (-2.5,0.3) [$N$]{};
(-3.8,2.15) – (-3,2.15); (-3.8,0.8) – (-3,0.8);
(-2,2.15) – (-1.15,2.15) – (-1.15,1.75); (-2,0.8) – (-1.15,0.8) – (-1.15,1.2); at (-1.6,2.4) [$a$]{}; at (-1.6,0.55) [$b$]{}; at (-1.15,1.475) [$(a,b)$]{};
at (-3.4,2.4) [$\mathcal H$]{}; at (-3.4,0.55) [$\mathcal H$]{};
(-0.2,3) rectangle (3.7,0); (0,2.5) rectangle (1,1.8) node\[pos=0.5,white\] [$\Gamma_\rho$]{}; (1.8,2.5) rectangle (2.8,1.8);
(2.3,1.95) – (2.45,2.4); (2.65,1.95) arc (10:170:0.35);
(1.8,1.2) rectangle (2.8,0.5);
(1,2.15) – (1.8,2.15) node\[pos=0.5,shift=[(0,0.22)]{}\] [$\mathcal H$]{}; at (2.3,2.7) [$M/N$]{}; at (2.3,0.85) [$R$]{};
(2.3,1.2) – (2.3,1.8); (2.8,2.15) – (3.3,2.15); at (3.48,2.15) [${\mathbf{p}}$]{}; at (2.1,1.4) [$0$]{}; at (2.5,1.4) [$1$]{}; at (2,-0.4) [(b) DSMUR.]{};
***Direct-Product.—***The construction of DPMUR proposed in [@PRL; @JPA] is best formulated as a game, shown in Fig. (\[comp\]a), between an experimentalist (Alice) and a referee (Bob) trying to guess the measurement outcomes. More explicitly, the game considered here is as follows: two black boxes $\mathrm{\Gamma}_{\rho}$ are located in different positions, each of them provides a quantum state $\rho$ to Alice and She implements her measurements $M$ and $N$ to the input state separately in each round. Alice knows the measurement outcome from experiments, but she does not know the actual state given to her. By repeating the same procedure a sufficient number of times, Alice derive distinct pairs of measurement outcomes, and the goal of Bob is to guess $k$ distinct pairs of them correctly.
Mathematically, $\mathrm{\Gamma}_{\rho}$ is a preparation channel, generating quantum state $\rho$ on a Hilbert space $\mathcal{H} \cong \mathbb{C}^{d}$ [@Kraus1983]. The outcome $a$ of the positive operator valued measure (POVM) $M = \{M_{a}\}$ occurs with probability $p_{a} := {\mbox{Tr}}(M_{a}~\rho)$ ($a=1, \ldots, n$). Similarly, we implement the measurement $N$, and denote the corresponding probability distribution by $q_{b} := {\mbox{Tr}}(N_{b}~\rho)$ ($b=1, \ldots, m$). We collect the numbers $p_{a}$ and $q_{b}$ into two probability vectors ${\mathbf{p}}$ and ${\mathbf{q}}$, respectively.
In the present scheme, the joint uncertainty between ${\mathbf{p}}$ and ${\mathbf{q}}$ is captured by the maximal probability of Bob in winning the game. For example, when Alice receives outcome $(a, b)$ from measurements, Bob will have a maximal probability $\max_{\rho}p_{a}q_{b}$ to win. In general, if Alice receive $k$ distinct pairs of outcomes, then the quantum mechanics gives Bob $R_{k}$ chance to win, with $$\begin{aligned}
R_{k} := \max\limits_{I_{k}}\max\limits_{\rho}\sum\limits_{(a,b)\in I_{k}}
p_{a}q_{b},\notag\end{aligned}$$ where $I_{k} \subset [n] \times [m]$ is a subset of $k$ distinct pair of indices. Here $[n] = \{ 1, \ldots, n\}$ is the set of natural numbers ranging from $1$ to $n$, and $k \in [mn]$. Clearly, such guessing game can be reformulated as the following $mn$ inequalities $$\begin{aligned}
\sum\limits_{(a,b)\in I_{k}}
p_{a}q_{b} \leqslant R_{k}. \quad\forall k\in[mn] \notag\end{aligned}$$ A concise approach of expressing the inequalities mentioned above is to use the majorization “$\prec$” [@Majorization]; A probability vector ${\mathbf{x}}\in \mathbb{R}^n$ is majorizied by ${\mathbf{y}}\in \mathbb{R}^n$, i.e. ${\mathbf{x}}\prec {\mathbf{y}}$, if and only if $\sum_{j=1}^{k} x^{\downarrow}_{j} \leqslant \sum_{j=1}^{k} y^{\downarrow}_{j}$ for all $1\leqslant k \leqslant n-1$. Here the down-arrow indicates that the components of the vectors are arranged in a non-increasing order. Now we can abbreviate the guessing game into one inequality $$\begin{aligned}
\label{dp}
{\mathbf{p}}\otimes{\mathbf{q}}\prec \bm{r},\end{aligned}$$ with $\bm{r} :=(R_{1}, R_{2}-R_{1}, \ldots, R_{mn}-R_{mn-1})$. Consequently, the essence of DPMUR is captured by our framework of guessing game, which demonstrates a [*spatially-separated joint uncertainty*]{}. Note that $R_{k}$ can be in general difficult to calculate explicitly, as they involve an optimization problem. However, the authors of [@PRL] provide us a calculate-friendly bound $\bm{t}$, satisfying ${\mathbf{p}}\otimes{\mathbf{q}}\prec \bm{r} \prec \bm{t}$.
Physically, MURs are very general; they encompass the most well-known entropic functions used in quantum information theory, but they are not restricted to these functions. Mathematically, majorization lattice forms a [*complete lattice*]{}; the optimal bounds for MURs exist. To obtain the optimal bounds, it suffices to perform a standard process (flatness process) $\mathcal{F}$. Hence, the implementation of the process $\mathcal{F}$ on ${\mathbf{p}}\otimes{\mathbf{q}}\prec \bm{r} \prec \bm{t}$ could lead to a new relation $$\begin{aligned}
\label{fdp}
{\mathbf{p}}\otimes{\mathbf{q}}\prec \mathcal{F} (\bm{r}) \prec \bm{r} \prec \mathcal{F} (\bm{t})
\prec \bm{t},\end{aligned}$$ where $\bm{r}$ and $\bm{t}$ are the bounds given in [@PRL]. Because of the mathematical properties of flatness process (concave envelope), the vector $\mathcal{F} (r)$ is optimal. However, a major drawback of $\mathcal{F} (\bm{r})$ is that the calculation of $\mathcal{F} (\bm{r})$ is even harder than $\bm{r}$. With the help of flatness process, we also obtain an effectively computable bound $\mathcal{F} (\bm{t})$, which is tighter than the original $\bm{t}$. We defer the construction of $\bm{t}$, and the rigorous definition of flatness process to the Supplementary Material [@SM].
{width="16cm" height="7.6cm"}
***Direct-Sum.—***Our protocol of DSMUR combines guessing game with a binary random number generator $R$, shown in Fig. (\[comp\]b); in each round, the measurement is determined by $R$. More specifically, $R$ outputs number $0$ with probability $\lambda$, and $1$ with probability $1-\lambda$. After receiving $0$, Alice performs $M$, otherwise she implements $N$. Again the goal of Bob is to guess the measurement outcome of Alice. The maximal probability for Bob to guess $k$ outcomes correctly is given by $$\begin{aligned}
\label{sk}
S_{k} := \max\limits_{|I|+|J|=k}\max\limits_{\rho}\sum\limits_{\substack{a\in I \subset [n] \\b\in J \subset [m] }}
\left(\lambda p_{a} + (1-\lambda) q_{b}\right) \notag\end{aligned}$$ where $|\bigcdot|$ denotes the cardinality of $\bigcdot$. There exists an efficient way of computing the success probability $S_{k}$. Let us define an operator $G_{c}$ as $$G_{c}(\lambda) := \left\{
\begin{aligned}
&\lambda M_{c} ~ & 1 \leqslant c \leqslant n, \\
&(1-\lambda) N_{c-n} ~ & n+1 \leqslant c \leqslant n+m. \notag
\end{aligned}
\right.$$ Then the quantity $S_{k}$ becomes $$\begin{aligned}
S_{k}(\lambda) = \max\limits_{|I|=k}
\lambda_{1}\left( \sum\limits_{c\in I \subset [n+m]}G_{c}(\lambda) \right),\notag\end{aligned}$$ where $\lambda_{1}(\bigcdot)$ denotes the maximum eigenvalue of the argument. Now we can conclude our guessing game within one inequality by using majorization; that is $$\begin{aligned}
\label{ds}
\lambda{\mathbf{p}}\oplus(1-\lambda){\mathbf{q}}\prec \bm{s}(\lambda),\end{aligned}$$ with $\bm{s}(\lambda):=(S_{1}(\lambda), S_{2}(\lambda)-S_{1}(\lambda), \ldots, S_{m+n}(\lambda)-S_{m+n-1}(\lambda))$. In the framework of DSMUR, classical uncertainty of the random number generator is injected into the guessing game, and as a consequence $\lambda{\mathbf{p}}\oplus(1-\lambda){\mathbf{q}}$ is a hybrid type of uncertainty, mingling both classical and quantum uncertainties. Quite remarkably, the measurements, monitored by $R$, can be implemented in the same position but cannot performed simultaneously, and hence $\lambda{\mathbf{p}}\oplus(1-\lambda){\mathbf{q}}$ reveals a [*temporally-separated joint uncertainty*]{}. It should be stressed here that the original DSMUR [@PRA; @M7] is a special case of our notion by first taking $\lambda=1/2$, and then timing the scalar 2, i.e. ${\mathbf{p}}\oplus{\mathbf{q}}\prec2\bm{s}(1/2)$.
Let us now consider the DSMUR after flatness process$$\begin{aligned}
\label{fds}
\lambda{\mathbf{p}}\oplus(1-\lambda){\mathbf{q}}\prec \mathcal{F} (\bm{s}(\lambda)) \prec \bm{s}(\lambda).\end{aligned}$$ Unlike the case of DPMUR, the vector $\mathcal{F} (\bm{s}(\lambda))$ is optimal and can be calculate explicitly. Moreover, for DSMUR with uniform distribution, i.e. $\lambda=1/2$, one can easily show that ${\mathbf{p}}\oplus{\mathbf{q}}\prec 2 \mathcal{F} (\bm{s}(1/2)) \prec 2\bm{s}(1/2)$. Note that, the flatness process cannot be applied to ${\mathbf{p}}\oplus{\mathbf{q}}\prec2\bm{s}(1/2)$ directly [@Li2019; @Wang2019E], since the results presented in [@Cicalese2002] are only designed for probabilities. To accommodate this, a more general lemma is proved in our Supplementary Material [@SM].
***Experimental setup.—***The experimental setup used for verifications of DPMUR and DSMUR is shown in Fig. \[fig:setup\]. It consists of single-photon source (see Supplementary Material for details), state preparation, and measurement modules.
In the state preparation module, we prepare a family of $4$-dimensional states with parameters $\theta$ and $\phi$, $|\psi_{\theta,\phi}\rangle=\cos\theta\sin\phi|0\rangle+\cos\theta\cos\phi|1\rangle+\sin\theta|2\rangle+0|3\rangle$, which is encoded by four modes of a single photon. States $|0\rangle$ and $|1\rangle$ are encoded by different polarizations of the photon in the lower mode, and $|2\rangle$ and $|3\rangle$ are encoded by polarization of the photon in the upper mode. The beam displacer (BD) causes the vertical polarized photons to be transmitted directly, and the horizontal polarized photons to undergo a $4$ mm lateral displacement. When the photon passes through a half-wave plate (H$_{1}$) with a certain setting angle, it is splited by BD1 into two parallel spatial modes – upper and lower modes. Therefore the photon is prepared in the desired state $|\psi_{\theta,\phi}\rangle$, with parameters $\theta$ and $\phi$ are controlled by the plates H$_{1}$ and H$_{2}$, respectively.
In the measurement module, we consider a setting with a pair of measurements
$$\label{2measurement}
\begin{aligned}
&A=\left\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\right\}
\\&B=\begin{aligned}&\{(|0\rangle-i|1\rangle-i|2\rangle+|3\rangle)/2, (|0\rangle-i|1\rangle+i|2\rangle-|3\rangle)/2, \\& (|0\rangle+i|1\rangle-i|2\rangle-|3\rangle)/2,(|0\rangle+i|1\rangle+i|2\rangle+|3\rangle)/2\}\end{aligned}
\end{aligned}$$
and another one with multi-measurements $$\label{3measurement}
\begin{aligned}
&C_{1}=\left\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\right\}
\\&C_{2}=\left\{|0\rangle, \frac{|2\rangle+|3\rangle}{\sqrt{2}}, \frac{|1\rangle+|2\rangle-|3\rangle}{\sqrt{3}}, \frac{2|1\rangle-|2\rangle+|3\rangle}{\sqrt{6}}\right\}
\\&C_{3}=\left\{\frac{|2\rangle+|3\rangle}{\sqrt{2}}, |1\rangle, \frac{|0\rangle+|2\rangle-|3\rangle}{\sqrt{3}}, \frac{2|0\rangle-|2\rangle+|3\rangle}{\sqrt{6}}\right\}.
\end{aligned}$$
at (0,0) [![(color online) Experimental investigation of DPMUR and DSMUR with two measurements. Lorenz curves in (a) and (b) show the experimental datum for DPMUR and DSMUR with states $|\psi_{\pi/4,\phi}\rangle$, and the Lorenz curves in (c) and (d) exhibit the joint uncertainties of DPMURs and DSMURs with states $|\psi_{\theta,\pi/4}\rangle$. Blue curves represent the previous bounds $\bm{t}$ ($\bm{s}(1/2)$), and our improved bounds $\mathcal{F} (\bm{t})$ ($ \mathcal{F} ( \bm{s}(1/2) )$)) are highlighted in red. The dotted lines marked with different colours indicate joint uncertainties with different parameters.[]{data-label="DSandDP_twoM"}](Figure3.pdf "fig:"){width="8.8cm" height="8cm"}]{};
at (-1,4.35) [$\phi = 0^{\circ}$]{}; at (-0.93,4.02) [$\phi = 10^{\circ}$]{}; at (-0.93,3.69) [$\phi = 20^{\circ}$]{}; at (-0.93,3.36) [$\phi = 30^{\circ}$]{}; at (-0.93,3.03) [$\phi = 40^{\circ}$]{}; at (-0.93,2.70) [$\phi = 50^{\circ}$]{}; at (-0.93,2.37) [$\phi = 60^{\circ}$]{}; at (-0.93,2.04) [$\phi = 70^{\circ}$]{}; at (-0.93,1.71) [$\phi = 80^{\circ}$]{}; at (-0.93,1.38) [$\phi = 90^{\circ}$]{}; at (-0.93,1.05) [$\bm{t}$]{}; at (-0.93,0.72) [$\mathcal{F} ( \bm{t} )$]{};
at (-1,-1) [$\theta = 0^{\circ}$]{}; at (-0.93,-1.33) [$\theta = 10^{\circ}$]{}; at (-0.93,-1.66) [$\theta = 20^{\circ}$]{}; at (-0.93,-1.99) [$\theta = 30^{\circ}$]{}; at (-0.93,-2.32) [$\theta = 40^{\circ}$]{}; at (-0.93,-2.65) [$\theta = 50^{\circ}$]{}; at (-0.93,-2.98) [$\theta = 60^{\circ}$]{}; at (-0.93,-3.31) [$\theta = 70^{\circ}$]{}; at (-0.93,-3.64) [$\theta = 80^{\circ}$]{}; at (-0.93,-3.97) [$\theta = 90^{\circ}$]{}; at (-0.93,-4.30) [$\bm{t}$]{}; at (-0.93,-4.63) [$\mathcal{F} ( \bm{t} )$]{};
at (4.9,4.35) [$\phi = 0^{\circ}$]{}; at (4.97,4.02) [$\phi = 10^{\circ}$]{}; at (4.97,3.69) [$\phi = 20^{\circ}$]{}; at (4.97,3.36) [$\phi = 30^{\circ}$]{}; at (4.97,3.03) [$\phi = 40^{\circ}$]{}; at (4.97,2.70) [$\phi = 50^{\circ}$]{}; at (4.97,2.37) [$\phi = 60^{\circ}$]{}; at (4.97,2.04) [$\phi = 70^{\circ}$]{}; at (4.97,1.71) [$\phi = 80^{\circ}$]{}; at (4.97,1.38) [$\phi = 90^{\circ}$]{}; at (4.97,1.05) [$\bm{s}(1/2)$]{}; at (5.05,0.72) [$\mathcal{F} ( \bm{s}(1/2) )$]{};
at (4.9,-1) [$\theta = 0^{\circ}$]{}; at (4.97,-1.33) [$\theta = 10^{\circ}$]{}; at (4.97,-1.66) [$\theta = 20^{\circ}$]{}; at (4.97,-1.99) [$\theta = 30^{\circ}$]{}; at (4.97,-2.32) [$\theta = 40^{\circ}$]{}; at (4.97,-2.65) [$\theta = 50^{\circ}$]{}; at (4.97,-2.98) [$\theta = 60^{\circ}$]{}; at (4.97,-3.31) [$\theta = 70^{\circ}$]{}; at (4.97,-3.64) [$\theta = 80^{\circ}$]{}; at (4.97,-3.97) [$\theta = 90^{\circ}$]{}; at (4.97,-4.30) [$\bm{s}(1/2)$]{}; at (5.05,-4.63) [$\mathcal{F} ( \bm{s}(1/2) )$]{};
In Fig. (\[fig:setup\]), device (a) is used to realize measurements $A$ and $C_{1}$. In the presence of quarter-wave plates with an angle of $45^{\circ}$, device (b) is used to realize measurement $B$, and the setting angles of H$_{3}$–H$_{6}$ are $45^{\circ}$, $0^{\circ}$, $22.5^{\circ}$, and $22.5^{\circ}$. On the other hand, in the absence of quarter-wave plates, device (b) is exploited to implement measurement $C_{2} ( C_{3} )$ when the setting angles of H$_{3}$–H$_{6}$ are $22.5^{\circ}$, $0^{\circ}(45^{\circ})$, $27.4^{\circ}$, and $0^{\circ}$.
at (0,0) [![(color online) Experimental investigation of DPMURs and DSMURs with three measurements. The plots in (a) and (b) show the joint uncertainties of the quantum state $|\psi_{\pi,\phi}\rangle$ varied with $\phi$ under measurements $C_{1}$, $C_{2}$, $C_{3}$, and our bound $\mathcal{F}(\bm{t}^{\prime})$, $\mathcal{F}(\bm{s}^{\prime}(1/3))$ by means of the Lorenz curves. The plots in (c) and (d) show the Shannon entropic uncertainty relations, with measurements $C_{1}$, $C_{2}$, $C_{3}$, of states $|\psi_{\pi,\phi}\rangle$ and $|\psi_{\theta,\pi/2}\rangle$ respectively. Here the curves marked with magenta, green, and blue stand for the uncertainty associated with measurements $C_{1}$, $C_{2}$ and $C_{3}$; that are $H(C_{1})$, $H(C_{2})$ and $H(C_{3})$, and the curve dyed red represents their joint uncertainties $\sum_{i} H ( C_{i} )$. The dotted line ($H ( \mathcal{F}(\bm{t}^{\prime}) )=0.7651$) and solid line ($H ( 3\mathcal{F}(\bm{s}^{\prime}(1/3)))=0.7979$) are the bounds of DPMUR and DSMUR. Error bars emphasize the standard deviation of our experimental datum.[]{data-label="DPDS_shannon"}](Figure4.pdf "fig:"){width="8.8cm" height="8.4cm"}]{};
at (-1.6,2.73) [$\phi = 0^{\circ}$]{}; at (-1.53,2.40) [$\phi = 10^{\circ}$]{}; at (-1.53,2.07) [$\phi = 20^{\circ}$]{}; at (-1.53,1.74) [$\phi = 30^{\circ}$]{}; at (-1.53,1.41) [$\phi = 40^{\circ}$]{}; at (-1.53,1.08) [$\mathcal{F}(\bm{t}^{\prime})$]{};
at (4.26,2.65) [$\phi = 0^{\circ}$]{}; at (4.33,2.32) [$\phi = 10^{\circ}$]{}; at (4.33,1.99) [$\phi = 20^{\circ}$]{}; at (4.33,1.66) [$\phi = 30^{\circ}$]{}; at (4.33,1.33) [$\phi = 40^{\circ}$]{}; at (4.63,1.00) [$\mathcal{F}(\bm{s}^{\prime}(1/3))$]{};
at (-4.8,-0.1) [$H(C_{1})$]{}; at (-3.6,-0.1) [$H(C_{2})$]{}; at (-2.4,-0.1) [$H(C_{3})$]{}; at (-1.0,-0.1) [$\sum_{i} H(C_{i})$]{}; at (0.7,-0.06) [$H ( \mathcal{F}(\bm{t}^{\prime}) )$]{}; at (2.7,-0.06) [$H ( 3\mathcal{F}(\bm{s}^{\prime}(1/3)))$]{};
at (-3.3,-5.38) [$\phi$]{}; at (2.45,-5.38) [$\theta$]{};
***Experimental results.—***The experimental datum induced by performing measurements (\[2measurement\], \[3measurement\]) on $|\psi_{\theta,\phi}\rangle$ are acquired, and the target of verifying the MURs is fulfilled. In order to unfold the MURs intuitively and geometrically, we employ the technique of [*Lorentz curve*]{} [@Majorization]; for an non-negative vector ${\mathbf{x}}=(x_{i})_{i=1}^{n}$ with non-increasing order, the corresponding Lorenz curve $\mathcal{L}({\mathbf{x}})$ is defined as the linear interpolation of the points $\{(k,\sum_{i=1}^{k}x_{i})_{k=0}^{n}\}$ with the convention $(0, 0)$ for $k=0$. Based on Lorenz curves, we have $\mathcal{L}({\mathbf{x}})$ lays everywhere below $\mathcal{L}({\mathbf{y}})$ if and only if ${\mathbf{x}}\prec {\mathbf{y}}$.
For measurements $A$ and $B$, the bound $\bm{t}$ for DPMUR ${\mathbf{p}}\otimes{\mathbf{q}}$, introduced in [@PRL; @JPA], is given by $(0.5625, 0.1661, 0.2714)$, and the bound $2\bm{s}(1/2)$ for DSMUR ${\mathbf{p}}\oplus{\mathbf{q}}$, introduced in [@PRA], is given by $(0.5, 0.2071, 0.2929)$. To further improve previous results on MURs, we apply the flatness process $\mathcal{F}$ to the bounds $\bm{t}$, $\bm{s}(1/2)$, and acquire $\mathcal{F} (\bm{t})=(0.5625, 0.21875, 0.21875)$, $\mathcal{F} (\bm{s}(1/2)) = (0.5, 0.25, 0.25)$. In Fig. (\[DSandDP\_twoM\]), the dotted lines are obtained by transforming the experimental datum into Lorenz curves. The experimental plots depicted in Fig. (\[DSandDP\_twoM\]) confirm the betterments of our bounds by showing that all experimental datum-induced Lorenz curves lay below our bounds $\mathcal{F} (\bm{t})$ ($\mathcal{F} (\bm{s}(1/2))$), and our bounds are under the previous ones $\bm{t}$ ($\bm{s}(1/2)$).
For measurements $C_{1}$, $C_{2}$ and $C_{3}$, the bound $\mathcal{F}(\bm{t}^{\prime})$ for DPMUR is given by $(0.7773, 0.2227)$, and the bound $\mathcal{F}(\bm{s}^{\prime}(1/3))$ for DSMUR is given by $(1, 1, 0.7583, 0.2417)/3$. In Fig. \[DPDS\_shannon\] (a) and (b) we see that the joint uncertainties associated with different parameters $\phi$ of the states $|\psi_{\theta,\phi}\rangle$ are marjorized by our bounds $\mathcal{F}(\bm{t}^{\prime})$ and $\mathcal{F}(\bm{s}^{\prime}(1/3))$. Entropies are important tools in both information theory and quantum information theory, and they are closely related to the majorization. From the properties of majorization, it follows the entropic uncertainty relations $\sum_{i} H ( C_{i} ) \geqslant H ( \mathcal{F}(\bm{t}^{\prime}) )$ and $\sum_{i} H ( C_{i} ) \geqslant H ( 3\mathcal{F}(\bm{s}^{\prime}(1/3))) $ with $H$ stands for the [*Shannon entropy*]{}. All of this can be seen in Fig. \[DPDS\_shannon\] (c) and (d).
***Conclusions.***–Our guessing game formalism of MURs enable us to classify DPMUR and DSMUR into spatially-separated and temporally-separated joint uncertainties accordingly, which differs from previous developments and, more important, exhibit the essential differences of physical features between DPMUR and DSMUR theoretically. We also implemented an optical experiment that demonstrates the MURs. In order to present the experimental data efficiently, a novel technique, called Lorenz curve, has been employed. Furthermore, it is advantageous to apply the techniques of flatness process to tighter the bounds of MURs, and its efficiency is confirmed by our experiment. The existence of MURs provides tremendous flexibility in formulating uncertainty relations, and greatly enhance our understanding of quantum mechanics. Therefore, the new formalism, and tighter bounds, as well as the corresponding experimental investigation presented in this work would deeper our knowledge of the quantum world.
[**Acknowledgements:**]{} This work is supported by the National Natural Science Foundation of China (Grants No. 11574291 and No. 11774334), China Postdoctoral Science Foundation (Grant No. 2016M602012 and No. 2018T110618), National Key Research and Development Program of China (Grants No. 2016YFA0301700 and No.2017YFA0304100), and Anhui Initiative in Quantum Information Technologies. Y. Xiao and G. Gour acknowledge NSERC support. S.-M. Fei acknowledges financial support from the National Natural Science Foundation of China under Grant No. 11675113 and Beijing Municipal Commission of Education (KZ201810028042).
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See Supplemental Material for full discussions and mathematical details of the completeness of majorization lattice, the flatness process, bounds for both DPMUR and DSMUR, bounds for multi-measurements MURs, and mathematical comparisons between DPMUR and DSMUR, as well as Refs [@Rapat1991; @Bosyk2019; @Shannon1948]. Besides the well-known Shannon entropy, other super additive functions have also been discussed in our Supplemental Material.
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This supplemental material contains a more detailed analysis and extensions of the results presented in the main text. We may reiterate some of the definitions and concepts in the main text to make the supplemental material more explicit and self-contained.
Majorization Lattice
====================
Before proceeding, it is worth introducing the basic concepts of lattice.
A partial order is a binary relation “$\prec$” over a set $\mathcal{L}$ satisfying reflexivity, antisymmetry, and transitivity. That is, for all $x$, $y$, and $z$ in $\mathcal{L}$, we have
1. Reflexivity: $x \prec x$,
2. Antisymmetry: If $x \prec y$ and $y \prec x$, then $x=y$,
3. Transitivity: If $x \prec y$ and $y \prec z$, then $x \prec z$.
Note that without the antisymmetry, “$\prec$” is just a preorder. Let us now define the set of all $n$-dimensional probability vectors as $$\begin{aligned}
\mathcal{P}^{n} = \left\{{\mathbf{p}}=\left(p_{1}, \ldots, p_{n}\right)~|~
p_{j}\in[0,1], \sum\limits_{j=1}^{n}p_{j}=1, p_{j}\geqslant p_{j+1}
\right\},\end{aligned}$$ with components in non-increasing order. Accordingly, majorization is a partial order over $\mathcal{P}^{n}$, i.e. $\langle\mathcal{P}^{n}, \prec\rangle$ is a poset.
A poset $\langle \mathcal{L}, \prec \rangle$ is called a join-semilattice, if for any two elements $x$ and $y$ of $\mathcal{L}$, it has a unique least upper bound (lub,supremum) $x \lor y$ satisfying
1. $x \lor y \in \mathcal{L}$,
2. $x \prec x \lor y$ and $y \prec x \lor y$.
On the other hand, $\langle \mathcal{L}, \prec \rangle$ is called a meet-semilattice, if for any two elements $x$ and $y$ of $\mathcal{L}$, it has a unique greatest lower bound (glb,infimum) $x \land y$ satisfying
1. $x \land y \in \mathcal{L}$,
2. $x \land y \prec x$ and $x \land y \prec y$.
$\langle \mathcal{L}, \prec \rangle$ is called a lattice if it is both a join-semilattice and a meet-semilattice, and denote it as a quadruple $\langle \mathcal{L}, \prec, \land, \lor \rangle$.
A lattice $\langle \mathcal{L}, \prec, \land, \lor \rangle$ is called complete, if for any subset $\mathcal{S}\subset\mathcal{L}$, it has a greatest element, denoted by $\top$ and a least element, denoted by $\bot$ which satisfy
1. $x \prec \top$, for all $x\in\mathcal{S}$ and $x \prec y$ for all $x\in\mathcal{S}$ $\Rightarrow \top \prec y$,
2. $\bot \prec x$, for all $x\in\mathcal{S}$ and $y \prec x$ for all $x\in\mathcal{S}$ $\Rightarrow y \prec \bot$.
By embedding the majorization “$\prec$”, the quadruple $\langle \mathcal{P}^{n}, \prec, \land, \lor \rangle$ forms a complete lattice. We remark that the result of completeness follows directly from the work presented in [@Rapat1991SM], and the algorithm in finding the greatest element and the least element of a subset $\mathcal{S}$ (also known as flatness process) was first introduced in [@Cicalese2002SM]. As we are trying to connect the structure of majorization lattice with MURs, here we are only interested in the construction of the greatest element of $\mathcal{S} \subset \mathcal{P}^{n}$. Roughly speaking, there are two steps in finding it; that are
- Step 1: Finding the largest partial sums; for each ${\mathbf{x}}= (x_{1}, \ldots, x_{n}) \in \mathcal{S}$, we need to evaluate the following quantities $$\begin{aligned}
Y_{k} := \max_{{\mathbf{x}}\in \mathcal{S}}\sum_{i=1}^{k} x_{i},\end{aligned}$$ and collect these numbers into a vector ${\mathbf{y}}:= (Y_{1}, Y_{2} - Y_{1}, \ldots, Y_{n} - Y_{n-1}) := (y_{1}, y_{2}, \ldots, y_{n})$. Clearly, we have ${\mathbf{x}}\prec {\mathbf{y}}$ for all ${\mathbf{x}}\in \mathcal{S}$. From now on, we denote the vector ${\mathbf{y}}$ as $\vee \mathcal{S}$; that is ${\mathbf{y}}:= \vee \mathcal{S}$, and ${\mathbf{x}}\prec \vee \mathcal{S}$ for all ${\mathbf{x}}\in \mathcal{S}$. For the set with finite elements, such as $\mathcal{S} = \{ {\mathbf{x}}_{1}, \ldots, {\mathbf{x}}_{k}\}$, we can also use ${\mathbf{x}}_{1}\vee \ldots \vee {\mathbf{x}}_{k}$ to stand for $\vee \mathcal{S}$.
- Step 2: Flatness process; let $j$ be the smallest integer in $\left\{2, \ldots, n\right\}$ such that $y_{j}>y_{j-1}$, and $i$ be the greatest integer in $\left\{1, \ldots, j-1\right\}$ such that $y_{i-1} \geqslant (\sum_{k=i}^{j} y_{k})/(j-i+1):=a$. Define $$\begin{aligned}
\label{eq; mj bound t}
\mathcal{F} ( {\mathbf{y}}) := \left(F_{1}, \ldots, F_{n}\right) \quad \text{with} \quad F_{k} =
\begin{cases}
a & \text{for}\quad k = i, \ldots, j \\
y_{k} & \text{otherwise.} \\
\end{cases}\end{aligned}$$ Here we also use the notation $\mathcal{F} ( \vee \mathcal{S} )$ to denote $\mathcal{F} ( {\mathbf{y}})$.
Rigorously speaking, to obtain the optimal bounds from a set $\mathcal{S}$, which contains infinite number of elements, by only applying Steps 1 and 2 is far from enough [@Li2019; @Wang2019E]. Actually, we should first guarantee the target set $\mathcal{S}$ is a subset of some complete lattice, which ensures the existence of the optimal bounds.
A key lemma in proving the optimality is the following lemma, which was first proved in [@Cicalese2002SM]
\[lem1\] Let ${\mathbf{x}}$, ${\mathbf{y}}\in \mathcal{P}^{n}$, there exists a unique optimal upper bound $\mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}) \in \mathcal{P}^{n}$, satisfying
- ${\mathbf{x}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}})$, and ${\mathbf{y}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}})$;
- for any ${\mathbf{z}}\in \mathcal{P}^{n}$ satisfying ${\mathbf{x}}\prec {\mathbf{z}}$ and ${\mathbf{y}}\prec {\mathbf{z}}$, it follows $\mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}) \prec {\mathbf{z}}$.
The domain of Lemma \[lem1\] is the set $\mathcal{P}^{n}$. However, its generalization is also correct. Let us now consider the following set $$\begin{aligned}
\mathcal{P}^{n}_{c} = \left\{{\mathbf{p}}=\left(p_{1}, \ldots, p_{n}\right)~|~
p_{j} \geqslant 0, \sum\limits_{j=1}^{n}p_{j}=c, p_{j}\geqslant p_{j+1}
\right\},\end{aligned}$$ with a constant $c$. Then we have
\[lem2\] Let ${\mathbf{x}}$, ${\mathbf{y}}\in \mathcal{P}^{n}_{c}$, there exists a unique optimal upper bound $\mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}) \in \mathcal{P}^{n}_{c}$, satisfying
- ${\mathbf{x}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}})$, and ${\mathbf{y}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}})$;
- for any ${\mathbf{z}}\in \mathcal{P}^{n}_{c}$ satisfying ${\mathbf{x}}\prec {\mathbf{z}}$ and ${\mathbf{y}}\prec {\mathbf{z}}$, it follows $\mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}) \prec {\mathbf{z}}$.
By dividing the constant $c$, we obtain $$\begin{aligned}
\frac{1}{c}{\mathbf{x}}\in \mathcal{P}^{n}, \quad \frac{1}{c}{\mathbf{y}}\in \mathcal{P}^{n},\end{aligned}$$ which implies $$\begin{aligned}
\frac{1}{c}{\mathbf{x}}\prec \mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}), \quad \frac{1}{c}{\mathbf{y}}\prec \mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}).\end{aligned}$$ Then for the positive vector ${\mathbf{x}}$ and ${\mathbf{y}}$ we have that $$\begin{aligned}
{\mathbf{x}}\prec c \, \mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}), \quad
{\mathbf{y}}\prec c \, \mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}).\end{aligned}$$ Now due to the fact that $\mathcal{F}$ is scalar-multiplication-preserving, we get $c \, \mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}) = \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}})$, and hence $$\begin{aligned}
{\mathbf{x}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}), \quad {\mathbf{y}}\prec \mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}).\end{aligned}$$ For any ${\mathbf{z}}\in \mathcal{P}^{n}_{c}$ satisfying ${\mathbf{x}}\prec {\mathbf{z}}$ and ${\mathbf{y}}\prec {\mathbf{z}}$, we have $$\begin{aligned}
\mathcal{F} ( \frac{1}{c}{\mathbf{x}}\vee \frac{1}{c}{\mathbf{y}}) \prec \frac{1}{c} {\mathbf{z}},\end{aligned}$$ which immediately yields $\mathcal{F} ({\mathbf{x}}\vee {\mathbf{y}}) \prec {\mathbf{z}}$ and completes the proof.
As a corollary of our Lemma \[lem2\], previous statement remains valid when the domain has been replaced by the set $S = \{{\mathbf{p}}=\left(p_{1}, \ldots, p_{n}\right)~|~
p_{j}\in[0,1], \sum_{j=1}^{n}p_{j}=c, p_{j}\geqslant p_{j+1}
\}$, and this proves the key lemma used in [@Li2019]. Here we only show the proof of two elements, but actually it works for any countable elements [@Bosyk2019SM].
Bounds for DPMUR
================
Now we are in the position to construct the optimal for DPMUR. Note that the set of spatially-separated joint uncertainty ${\mathbf{p}}\otimes{\mathbf{q}}$ forms a subset of $\mathcal{P}^{n}$, i.e. here $\mathcal{S} = \{ {\mathbf{p}}\otimes{\mathbf{q}}\} \subset \mathcal{P}^{n}$. From Step 1, we have $Y_{k} = R_{k}$ with $R_{k}$ defined in the main text. For the collection of quantities $R_{k}$, we apply the flatness process and obtain $\mathcal{F} (\bm{r})$. Therefore, we have ${\mathbf{p}}\otimes{\mathbf{q}}\prec \mathcal{F} (\bm{r})$ for all probability vector ${\mathbf{p}}$ and ${\mathbf{q}}$, and $\mathcal{F} (\bm{r})$ is the largest element for $\mathcal{S} = \{ {\mathbf{p}}\otimes{\mathbf{q}}\}$, and hence optimal.
However, the vector $\bm{r}$ can be in general difficult to calculate explicitly, as they involve a complicated optimization problem. Fortunately, we still have the following relaxing method, $$\begin{aligned}
R_{k} := \max\limits_{I_{k}}\max\limits_{\rho}\sum\limits_{(a,b)\in I_{k}}
{\mathbf{p}}_{a}(\rho)~{\mathbf{q}}_{b}(\rho)
\leqslant \max\limits_{I_{k_{1}}, I_{k_{2}}}
\max\limits_{\rho} \left( \sum\limits_{a\in I_{k_{1}}} {\mathbf{p}}_{a}(\rho) \right) \left( \sum\limits_{b\in I_{k_{2}}} {\mathbf{p}}_{b}(\rho) \right)
\leqslant \max\limits_{I_{k_{1}}, I_{k_{2}}} \max\limits_{\rho}
\left( \frac{\sum_{a\in I_{k_{1}}} {\mathbf{p}}_{a}(\rho) + \sum_{b\in I_{k_{2}}} {\mathbf{p}}_{b}(\rho) }{2}\right)^{2},\end{aligned}$$ with $$\begin{aligned}
\max\limits_{\rho}
\left( \frac{\sum_{a\in I_{k_{1}}} {\mathbf{p}}_{a}(\rho) + \sum_{b\in I_{k_{2}}} {\mathbf{p}}_{b}(\rho) }{2}\right)^{2}
= \left( \frac{ \lambda_{1}( \sum_{a\in I_{k_{1}}} M_{a} + \sum_{b\in I_{k_{2}}} N_{b} ) }{2} \right)^{2},\end{aligned}$$ and their indices $k_{1}$ and $k_{2}$ satisfying $k_{1} + k_{2} = k+1$. Let us define $T_{k}$ as $$\begin{aligned}
T_{k} &:= \max\limits_{I_{k_{1}}, I_{k_{2}}} \left( \frac{ \lambda_{1}( \sum_{a\in I_{k_{1}}} M_{a} + \sum_{b\in I_{k_{2}}} N_{b} ) }{2} \right)^{2},\notag\\
t_{k} &:= T_{k} - T_{k-1}, \notag\\
\bm{t} &:= (t_{1}, \ldots, t_{mn}).\end{aligned}$$ Note that here the vector $\bm{t}$ can be computed explicitly, satisfying the following inequalities $$\begin{aligned}
\label{UUR}
{\mathbf{p}}\otimes{\mathbf{q}}\prec \bm{r} \prec \bm{t}.\end{aligned}$$ Eq. (\[UUR\]) is the main result of [@PRL], which is also the implementation of Step 1 presented in the previous section. In order to obtain a better bound, the flatness process $\mathcal{F}$ is needed; that is $$\begin{aligned}
\label{improvedUUR}
{\mathbf{p}}\otimes{\mathbf{q}}\prec \mathcal{F} ( \bm{r} ) \prec \bm{r} \prec \mathcal{F} ( \bm{t} ) \prec \bm{t}.\end{aligned}$$ The proof of (\[improvedUUR\]) follows [@Cicalese2002SM] straightforwardly.
Bounds for DSMUR
================
In the cases of (weighted) DSMUR, we have $\mathcal{S} = \{ \lambda{\mathbf{p}}\oplus (1-\lambda){\mathbf{q}}\} \subset \mathcal{P}^{n}$. From Step 1, we have $$\begin{aligned}
S_{k} &:= \max\limits_{|I|+|J|=k}\max\limits_{\rho}\sum\limits_{\substack{a\in I\\b\in J}} \left(\lambda{\mathbf{p}}_{a}(\rho)+(1-\lambda){\mathbf{q}}_{b}(\rho)\right)
= \max\limits_{|I|+|J|=k} \max\limits_{\rho} {\mbox{Tr}}\left[\rho \left( \sum\limits_{\substack{a\in I\\b\in J}} (\lambda M_{a} + (1-\lambda) N_{b} ) \right) \right]
= \max\limits_{|I|+|J|=k} \lambda_{1} \left( \sum\limits_{\substack{a\in I\\b\in J}} (\lambda M_{a} + (1-\lambda) N_{b} ) \right) \notag\\
&= \max\limits_{|I|=k}\sum\limits_{c\in I}
\lambda_{1}(G_{c}(\lambda)).\end{aligned}$$ Unlike the cases of DPMUR, here the quantities $S_{k}$ can be computed explicitly. Based on these notations, we construct $\bm{s}(\lambda)$ as $(S_{1}(\lambda), S_{2}(\lambda)-S_{1}(\lambda), \ldots, S_{m+n}(\lambda)-S_{m+n-1}(\lambda))$, which meets the following relation $$\begin{aligned}
\lambda {\mathbf{p}}\oplus (1-\lambda) {\mathbf{q}}\prec \bm{s}(\lambda).\end{aligned}$$ Applying the flatness process, we immediately obtain $$\begin{aligned}
\label{FDSMUR}
\lambda {\mathbf{p}}\oplus (1-\lambda) {\mathbf{q}}\prec \mathcal{F} ( \bm{s}(\lambda) ) \prec \bm{s}(\lambda).\end{aligned}$$ Again, the optimality of $\mathcal{F} ( \bm{s}(\lambda) )$ follows from the completeness of $\mathcal{P}^{n}$ and the flatness process $\mathcal{F}$ directly, not just because of the flatness process [@Li2019; @Wang2019E]. For random number generator $R$ with uniform distribution, i.e. $\lambda = 1/2$, (\[FDSMUR\]) implies that $$\begin{aligned}
\frac{1}{2} {\mathbf{p}}\oplus \frac{1}{2} {\mathbf{q}}\prec \mathcal{F} ( \bm{s}(1/2) ) \prec \bm{s}(1/2),\end{aligned}$$ and hence we have $$\begin{aligned}
{\mathbf{p}}\oplus {\mathbf{q}}\prec 2 \mathcal{F} ( \bm{s}(1/2) ) \prec 2 \bm{s}(1/2).\end{aligned}$$ Note that, the flatness process cannot be applied to the DSMUR ${\mathbf{p}}\oplus{\mathbf{q}}\prec2\bm{s}(1/2)$ directly [@Li2019; @Wang2019E], since the results presented in [@Cicalese2002SM] are only designed for the vector belongs to $\mathcal{P}^{n}$. Otherwise, an appropriate modification of the proof, i.e. our Lemma \[lem2\], is needed. This is another reason, from mathematical viewpoints, why our forms of DSMUR are valuable.
Bounds for Multi-measurements MURs
==================================
Uncertainty relation is not the patent of two measurements, so what to make of this? We checked in with a multi-measurements MURs to meake more sense of the ruling. First, we consider DPMUR with multi-measurements. Assume we have a set of POVMs $\{ M_{x} \}_{x=1}^{n}$ with $M_{x} = \{ M_{a|x} \}_{a=1}^{d}$, and the denote outcome probability distribution as $p(a(x)|x) := {\mbox{Tr}}[\rho M_{a|x}]$. By collecting these numbers into the probability vectors, we have $p_{x} := (p(a(x)|x))_{a}$, and their spatially-separated joint uncertainty becomes $\bigotimes_{x} p_{x}$. In order to obtain a computing-friendly bound, we apply the Geometric-Arithmetic mean inequality, i.e. $$\begin{aligned}
\max\limits_{I_{k}}\max\limits_{\rho}\sum\limits_{(a(x))_{x}\in I_{k}}
\prod\limits_{x} p(a(x)|x)
\leqslant \max\limits_{\sum_{x} I_{x} = k}\max\limits_{\rho}
\prod\limits_{x}
\sum\limits_{a(x) \in I_{x}} p(a(x)|x)
\leqslant \max\limits_{\sum_{x} I_{x} = k}\max\limits_{\rho}
\left( \frac{\sum_{x} \sum_{a(x) \in I_{x}} p(a(x)|x)}{n}\right)^{n}
= \max\limits_{\sum_{x} I_{x} = k}
\left( \frac{ \lambda_{1}(\sum_{x} \sum_{a(x) \in I_{x}} M_{a|x})}{n}\right)^{n}.\end{aligned}$$ Similarly, define $$\begin{aligned}
T_{k}^{\prime} &:= \max\limits_{\sum_{x} I_{x} = k}
\left( \frac{ \lambda_{1}(\sum_{x} \sum_{a(x) \in I_{x}} M_{a|x})}{n}\right)^{n},\notag\\
t_{k}^{\prime} &:= T_{k}^{\prime} - T_{k-1}^{\prime}, \notag\\
\bm{t}^{\prime} &:= (t_{1}^{\prime}, \ldots, t_{d^{n}}^{\prime}),\end{aligned}$$ which satisfying the following multi-measurements DPMUR $$\begin{aligned}
\bigotimes_{x} p_{x} \prec \bm{t}^{\prime}.\end{aligned}$$ Moreover, by apply the flatness process $\mathcal{F}$ again, we obtain a tighter bound $\mathcal{F} (\bm{t}^{\prime})$; that is $$\begin{aligned}
\bigotimes_{x} p_{x} \prec \mathcal{F} (\bm{t}^{\prime}) \prec \bm{t}^{\prime}.\end{aligned}$$ Remark that the bound $\mathcal{F} (\bm{t}^{\prime})$ outperforms the one constructed in [@PRL], i.e $\bm{t}$. On the other hand, for probability vectors $p_{x}$, we can also consider their joint uncertainties in temporally-separated forms, i.e. $\bigoplus_{x} c_{x} p_{x}$ with $\bm{c} := (c_{x})_{x}$ a probability vector. To find its bound, consider the following equation $$\begin{aligned}
\max\limits_{\sum_{x} I_{x} = k}\max\limits_{\rho}
\sum\limits_{x}
\sum\limits_{a(x) \in I_{x}} c_{x} p(a(x)|x)
= \max\limits_{\sum_{x} I_{x} = k}
\lambda_{1} ( \sum\limits_{x}
\sum\limits_{a(x) \in I_{x}} c_{x} M_{a(x)|x} ) := S_{k}^{\prime},\end{aligned}$$ and define $\bm{s}^{\prime}$ as $(S_{1}^{\prime}, S_{2}^{\prime}-S_{1}^{\prime}, \ldots, S_{nd}^{\prime}-S_{nd-1}^{\prime})$. Based on these notations, we have the following multi-measurements DSMUR $$\begin{aligned}
\bigoplus_{x} c_{x} p_{x} \prec \bm{s}^{\prime}.\end{aligned}$$ The optimal bounds of $\bigoplus_{x} c_{x} p_{x}$ is obtained as $$\begin{aligned}
\bigoplus_{x} c_{x} p_{x} \prec \mathcal{F}(\bm{s}^{\prime}) \prec \bm{s}^{\prime},\end{aligned}$$ by performing the flatness process $\mathcal{F}$. Therefore the construction of the optimal bound for $\bigoplus_{x} p_{x}$ is also straightforward.
Mathematical Comparisons between DPMUR and DSMUR
================================================
With the majorization relation for vectors, we now present DPMUR and DSMUR as $$\begin{aligned}
{\mathbf{p}}\otimes{\mathbf{q}}& \prec \bm{t} := {\mathbf{x}}, \\
{\mathbf{p}}\oplus{\mathbf{q}}& \prec 2\bm{s}(1/2) := {\mathbf{y}},\end{aligned}$$ where $\rho$ runs over all quantum states in Hilbert space $\mathcal{H}$ with ${\mathbf{x}}$, ${\mathbf{y}}$ standing for the state-independent bound of DPMUR and DSMUR respectively. Let us take any nonnegative Schur-concave function $\mathcal{U}$ to quantify the uncertainties and apply it to DPMUR and DSMUR, which leads to $$\begin{aligned}
\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}}) &\geqslant \mathcal{U}( {\mathbf{x}}),\\
\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}}) &\geqslant \mathcal{U}( {\mathbf{y}}).\end{aligned}$$ The universality of MURs comes from the diversity of uncertainty measures $\mathcal{U}$ and DPMUR, DSMURs stand for different kind of uncertainties.
We next move to describe the additivity of uncertainty measures, and call a measure $\mathcal{U}$ [*direct-product additive*]{} if $\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}})=\mathcal{U}({\mathbf{p}}) + \mathcal{U}({\mathbf{q}})$. Instead of direct-product between probability distribution vectors, one can also consider direct-sum and define [*direct-sum additive*]{} for $\mathcal{U}$ whenever it satisfies $\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}})=\mathcal{U}({\mathbf{p}}) + \mathcal{U}({\mathbf{q}})$. Note that the joint uncertainty ${\mathbf{p}}\oplus{\mathbf{q}}$ considered here is unnormalized and comparison between DPMUR and normalized DSMUR is detailed later. Once an uncertainty measure $\mathcal{U}$ is evolved to both direct-product additive and direct-sum additive, then we call it [*super additive*]{} for uncertainties. It is worth to mention that $\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}})=\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}})$ whenever the uncertainty measure is super additive. Consequently, the bound ${\mathbf{y}}$ for DSMUR performs better than ${\mathbf{x}}$ in the case of super additive, $$\begin{aligned}
\label{eqs}
\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}})=\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}}) \geqslant \mathcal{U}( {\mathbf{y}}) \geqslant \mathcal{U}( {\mathbf{x}}),\end{aligned}$$ since ${\mathbf{y}}\prec \left\{1\right\} \oplus {\mathbf{x}}$ [@PRA]. We remark that the well known Shannon entropy is super additive and only by applying super additive functions, like Shannon entropy, DPMUR and DSMUR are comparable. It should also be clear that DPMUR and DSMUR have been employed to describe different type of uncertainties. For an uncertainty measure $\mathcal{U}$, in general, it can be checked that $\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}}) \neq \mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}})$ and hence it is meaningless to state that DSMUR performs better than DPMUR and vice versa.
One of the main goals in the study of uncertainty relations is the quantification of the joint uncertainty of incompatible observables. DPMUR and DSMUR provide us two different methods to quantify joint uncertainty between incompatible observables. Relations between DPMUR and DSMUR are of fundamental importance both for the theoretical characterization of joint uncertainties, as well as the experimental implementation. Quite uncannily, we find that for some eligible uncertainty measure $\mathcal{U}$, DPMUR and DSMUR are given by $$\begin{aligned}
\label{eq1}
\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}}) \geqslant \mathcal{U}( {\mathbf{y}}) > \mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}}) \geqslant \mathcal{U}( {\mathbf{x}}),\end{aligned}$$ for some quantum state $\rho$.
Let us now construct such uncertainty measure $\mathcal{U}$. First define the summation function $\mathcal{S}$ as $\mathcal{S}(\mathbf{u}):=\sum_{l}u_{l}=\left\lVert \mathbf{u} \right\rVert_{1}$ with $\mathbf{u}=(u_{1}, u_{2}, \ldots, u_{d})$. Another important function $\mathcal{M}$ is defined as $\mathcal{M}(\mathbf{u}):=\max_{l}u_{l}=2^{-H_{\text{min}}(\mathbf{u})}$. And hence it is easy to check that $\mathcal{U}:=\mathcal{S}-\mathcal{M}$ is a nonnegative Schur-concave functions; take two vectors satisfying $x \prec y$, and based on the definition of $\mathcal{U}$ we have $\mathcal{U}(x)=\sum^{d}_{j=2}x_{j}^{\downarrow}\geqslant \sum^{d}_{j=2}y_{j}^{\downarrow}=\mathcal{U}(y)$. Specifically this function, which combines $\mathcal{S}$ and $\mathcal{M}$ together, is a qualified uncertainty measure and satisfies Eq. (\[eq1\]) for some quantum states and measurements. Moreover, specific examples are given in the following experimental demonstration.
In principle, DPMUR and DSMUR do not have to be comparable and their joint uncertainty can be quantified by their bound. However, we can compare their differences by checking which bound approximates their joint uncertainty better since joint uncertainties are often classified by their bounds. Take any nonnegative Schur-concave function $\mathcal{U}$, which leads to two nonnegative quantities $\xi_{DS}:=\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}}) - \mathcal{U}( {\mathbf{y}})$ and $\xi_{DP}:=\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}}) - \mathcal{U}( {\mathbf{x}})$. To determine whether the bound ${\mathbf{x}}$ approximates DPMUR better than ${\mathbf{y}}$ approximates DSMUR, we simply compare the numerical value of $\xi_{DS}$ and $\xi_{DP}$. And how such bounds contribute to the joint uncertainties are depicted in our experiment.
The above discussion on DSMUR is based on its unnormalized form ${\mathbf{p}}\oplus {\mathbf{q}}$, since it was first given in [@PRA] with the form ${\mathbf{p}}\oplus{\mathbf{q}}\prec {\mathbf{y}}$ for probability distributions ${\mathbf{p}}$ and ${\mathbf{q}}$. However, unlike ${\mathbf{p}}\otimes{\mathbf{q}}$ constructed in DPMUR [@PRL], ${\mathbf{p}}\oplus{\mathbf{q}}$ is not even a probability distribution. In order to derive a normalized DSMUR, we simply take the weight $1/2$ $$\begin{aligned}
\frac{1}{2}{\mathbf{p}}\oplus\frac{1}{2}{\mathbf{q}}& \prec \frac{1}{2}{\mathbf{y}}.\end{aligned}$$ And now we compare the normalized DSMUR $\frac{1}{2}{\mathbf{p}}\oplus\frac{1}{2}{\mathbf{q}}\prec \frac{1}{2}{\mathbf{y}}$ with DPMUR ${\mathbf{p}}\otimes{\mathbf{q}}\prec {\mathbf{x}}$; by taking the quantum states shown in the main text $$\begin{aligned}
|\psi_{\theta,\phi}\rangle&=\cos\theta\sin\phi|0\rangle+\cos\theta\cos\phi|1\rangle+\sin\theta|2\rangle
\\&=(\cos\theta\sin\phi,\cos\theta\cos\phi,\sin\theta,0)^{\top},
\end{aligned}$$ and measurements $A$, $B$ with the following eigenvectors $$\begin{aligned}
&A=\left\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\right\}\\
&B=\left\{\frac{|0\rangle-i|1\rangle-i|2\rangle+|3\rangle}{2}, \frac{|0\rangle-i|1\rangle+i|2\rangle-|3\rangle}{2}, \frac{|0\rangle+i|1\rangle-i|2\rangle-|3\rangle}{2},\frac{|0\rangle+i|1\rangle+i|2\rangle+|3\rangle}{2}\right\}.
\end{aligned}$$ We depicted the pictures of $H\left({\mathbf{p}}\otimes{\mathbf{q}}\right)$, $H\left(\frac{1}{2}{\mathbf{p}}\oplus\frac{1}{2}{\mathbf{q}}\right)$, $H\left({\mathbf{x}}\right)$, and $H\left(\frac{1}{2}{\mathbf{y}}\right)$ in Fig. \[shannontwo\].
{width="16cm" height="6.4cm"}
Super Additivity
================
To be comparable for DPMUR and DSMUR, we should choose an uncertainty measure $\mathcal{U}$ that are both Schur-concave and super additive. Clearly Shannon entropy is a qualified candidate. The question, thus, naturally arises: is there another function satisfies the following properties:
Property 1
: $\mathcal{U}$ should be continuous in ${\mathbf{p}}$ and ${\mathbf{q}}$.
Property 2
: $\mathcal{U}$ should be a Schur-concave function.
Property 3
: $\mathcal{U}$ should be super additive, i.e. $$\begin{aligned}
\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}})&=\mathcal{U}({\mathbf{p}}) + \mathcal{U}({\mathbf{q}}),\\
\mathcal{U}({\mathbf{p}}\oplus{\mathbf{q}})&=\mathcal{U}({\mathbf{p}}) + \mathcal{U}({\mathbf{q}}).
\end{aligned}$$
Or will these properties lead to a unique function (up to a scalar)? Since we can take ${\mathbf{q}}$ as $\left(1, 0, \ldots, 0\right)$, and then $\mathcal{U}({\mathbf{p}}\otimes{\mathbf{q}})= \mathcal{U}({\mathbf{p}})$ which is continuous in the $p_{i}$ while ${\mathbf{p}}=\left(p_{i}\right)_{i}$. Moreover, due to the Schur-concavity, $\mathcal{U}$ is a monotonic increasing function of $d$ when taking $p_{i}=\frac{1}{d}$. In addition, if $\mathcal{U}$ complies with the composition law for compound experiments, then there is only one possible expression for $\mathcal{U}$, i.e. Shannon entropy (up to a scalar). Namely, if there is a measure, say $\mathcal{U}({\mathbf{p}})=\mathcal{U}\left(p_{1}, p_{2}, \ldots, p_{d}\right)$ which is required to meet the following three properties:
Property 4
: $\mathcal{U}$ should be continuous in ${\mathbf{p}}$.
Property 5
: If all the $p_{i}$ are equal, $p_{i}=\frac{1}{d}$, then $\mathcal{U}$ should be a monotonic increasing function of $d$. With equally $d$ likely events there is more choice, or uncertainty, when there are more possible events.
Property 6 (Composition Law)
: If a choice be broken down into two successive choices, the original $\mathcal{U}$ should be the weighted sum of the individual values of $\mathcal{U}$.
Then the only $\mathcal{U}$ satisfying the three above assumptions is of the form [@Shannon1948]: $$\begin{aligned}
\mathcal{U}({\mathbf{p}})=k\cdot\left(-\sum\limits_{i=1}^{d}p_{i}\log p_{i}\right),\end{aligned}$$ where $k$ is a positive constant. Whenever a function $\mathcal{U}$ satisfies Property 1 and Property 2, it will meet Property 4 and Property 5 automatically. However, super additivity differs with the Composition Law, and this leads to function satisfied Property 1, 2, and 3 other than Shannon entropy.
For example, consider the composition between logarithmic function and elementary symmetric function: $$\begin{aligned}
\mathcal{V}({\mathbf{p}}):=\log\left(\prod\limits_{i=1}^{d}p_{i}\right).\end{aligned}$$ Here $\mathcal{V}$ satisfies Properties 1, 2, and the DPMUR is read as $$\begin{aligned}
\mathcal{V}({\mathbf{p}}\otimes{\mathbf{q}})&=\log\left(\prod\limits_{i, j}p_{i}q_{j}\right)\notag\\
&=\log\left(\prod\limits_{i}p_{i}\cdot\prod\limits_{j}q_{j}\right)\notag\\
&=\log\left(\prod\limits_{i}p_{i}\right)+\log\left(\prod\limits_{j}q_{j}\right)\notag\\
&=\mathcal{V}({\mathbf{p}})+\mathcal{V}({\mathbf{q}}),\end{aligned}$$ where the probability distributions ${\mathbf{p}}$ and ${\mathbf{q}}$ are defined as $\left(p_{i}\right)_{i}$ and $\left(q_{j}\right)_{j}$. On the other hand, DSMUR is written as $$\begin{aligned}
\mathcal{V}({\mathbf{p}}\oplus{\mathbf{q}})&=\log\left(\prod\limits_{i, j}p_{i}q_{j}\right)=\mathcal{V}({\mathbf{p}}\otimes{\mathbf{q}}),\end{aligned}$$ hence, $\mathcal{V}$ meets Property 3. To summarize, we derive a function $\mathcal{V}$, which is valid for Properties 1, 2, and 3. However $\mathcal{V}$ is not a good uncertainty measure, since $\mathcal{V}({\mathbf{y}})$ and $\mathcal{V}({\mathbf{x}})$ are not well defined (due to the occurrence of $\log0$). Whether there exists another function that obeys Properties 1, 2, and 3 remains an open question, and one may conjecture that $\mathcal{V}$, Shannon entropy $H$ and the convex combinations of $\mathcal{V}$ and $H$ are the only suitable candidates.
[99]{}
R. B. Rapat, Majorization and singular values. [slowromancap3@]{}, [*Linear Algebra Its Appl.*]{} **145**, 59 (1991).
F. Cicalese and U. Vaccaro, Supermodularity and Subadditivity Properties of the Entropy on the Majorization Lattice, [*IEEE Trans. Inf. Theory*]{} **48**, 933 (2002).
J.-L. Li and C.-F. Qiao, Quantum Uncertainty Relation: The Optimal Uncertainty Relation, [*Ann. Phys. (Berlin)*]{} **10**, 531 (2019).
H. Wang, J.-L. Li, S. Wang, Q.-C. Song, and C.-F. Qiao, Experimental investigation of the uncertainty relations with coherent light, [*Quantum Inf. Proc.*]{} **19**, 38 (2019).
G. M. Bosyk, G. Bellomo, F. Holik, H. Freytes, and G. Sergioli, Optimal common resource in majorization-based resource theories, [*New J. Phys.*]{} **21** 083028 (2019).
S. Friedland, V. Gheorghiu, and G. Gour, Universal Uncertainty Relations, [*Phys. Rev. Lett.*]{} [**111**]{}, 230401 (2013).
. Rudnicki, Z. Pucha[ł]{}a, and K. Życzkowski, Strong majorization entropic uncertainty relations, [*Phys. Rev. A*]{} [**89**]{}, 052115 (2014).
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|
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"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study spacetime singularities in a general five-dimensional braneworld with curved branes satisfying four-dimensional maximal symmetry. The bulk is supported by an analog of perfect fluid with the time replaced by the extra coordinate. We show that contrary to the existence of finite distance singularities from the brane location in any solution with flat (Minkowski) branes, in the case of curved branes there are singularity-free solutions for a range of equations of state compatible with the null energy condition.'
author:
- |
[<span style="font-variant:small-caps;">Ignatios Antoniadis$^{1,2}$[^1], Spiros Cotsakis[^2]$^{3}$[^3]</span>]{}, [<span style="font-variant:small-caps;">Ifigeneia Klaoudatou$^{3}$[^4],</span>]{}\
$^1$LPTHE, UMR CNRS 7589, Sorbonne Universités, UPMC Paris 6,\
4 place Jussieu, T13-14, 75005 Paris, France\
$^2$ Albert Einstein Center for Fundamental Physics, ITP,\
University of Bern, Sidlerstrasse 5 CH-3012 Bern, Switzerland\
$^{3}$Department of Mathematics, American University of the Middle East\
P. O. Box 220 Dasman, 15453, Kuwait
---
Introduction
============
In previous work we studied the singularity structure of a braneworld model consisting of a flat $3$-brane embedded in a five-dimensional bulk space filled with an analogue of a perfect fluid (the fifth coordinate $Y$ playing the role of time). The perfect fluid satisfied a linear equation of state with a constant parameter $\gamma$, $P=\gamma\rho$, where $P$ is the ‘pressure’ and $\rho$ is the ‘density’. In [@ack3] we showed that for a flat brane there exist singularities that appear within finite distance $Y_{0}$ from the position of the brane supposedly located at the origin, for all values of $\gamma$. A way to avoid such singularities is to exploit the natural $Z_2$ symmetry introduced by the existence of the brane by cutting the bulk space and considering a slice of it which is free from finite-distance singularities. Although this matching mechanism is possible for all values of $\gamma$, the requirement for localised gravity on the brane restricts $\gamma$ in the interval $(-2,-1)$. On the other hand, further requirements for physical conditions, such as energy conditions, restrict $\gamma$ in values greater than $-1$. Therefore within the framework of our flat brane model it is not possible to satisfy at the same time the positive energy conditions and the condition for localised gravity on the brane.
A question that naturally arises is whether any of the above conclusions about the existence of singularities are sensitive to the geometry of the brane so that singularities are absent when we consider a curved brane. It was proposed in [@ack1], [@ack2] that the singularity present in the flat brane model moves to infinite distance when the brane becomes curved, such as de Sitter (dS) or anti-de Sitter (AdS) in the maximally symmetric case, which was in accordance with previous claims made in [@gubser]. In this paper, we show that this is indeed possible. In particular, we show that for curved branes there exist ranges of $\gamma$ for which finite-distance singularities are avoided; these are: $\gamma>-1/2$ (for positively curved brane) and $-1<\gamma<-1/2$ (for negatively curved brane). For each type of brane geometry and values of $\gamma$ outside these regions, we find that finite-distance singularities continue to exist. A way to be removed is by using the cutting and matching procedure mentioned above to construct a slice of non-singular bulk space, when that is possible. Moreover, imposing the null energy condition (guaranteeing the absence of ghosts in the bulk) excludes de Sitter (dS) branes and one is left only with the second region of $\gamma$ for Anti-de Sitter (AdS) branes. However, we further show that this region in AdS branes is incompatible with having also localised gravity on the brane. The situation is not improved when allowed non-singular solutions obtained by the cutting and matching procedure. The plan of this paper is the following: In Sections $2$ and $3$, we present the model and give the exact solutions respecting 4d maximal symmetry, as well as the complete list of all asymptotic behaviours for all ranges of the parameters in our model. In Section $4$, we analyse in detail the two non-singular solutions found in Section $3$. In Section $5$, we derive the null energy condition and investigate its consequences. In Section $6$ we construct non-singular orbifold-like solutions, called matching in the following, obtained by the cutting and matching procedure for those cases that allow it, and study the null energy condition. In Section $7$, we examine whether the previously derived solutions also satisfy the condition for localisation of gravity on the brane. Section $8$ contains a summary of our results and some concluding remarks. Finally, in Appendix A we derive the solutions found for two special values of the parameter $\gamma$ that cannot be incorporated in the solutions given in Section $3$.
The setup for a curved brane model
==================================
Our braneworld model consists of a 3-brane embedded in a five-dimensional bulk space $\mathcal{M}\times\mathbb{R}$ filled with an analogue of a perfect fluid with equation of state $P=\gamma \rho$, where the ‘pressure’ $P$ and the ‘density’ $\rho$ are functions only of the fifth dimension, $Y$. The bulk metric is of the form \[warpmetric\] g\_[5]{}=a\^[2]{}(Y)g\_[4]{}+dY\^[2]{}, where $g_{4}$ is the four-dimensional de Sitter or anti de Sitter metric, [*i.e.*]{}, \[branemetrics\] g\_[4]{}=-dt\^[2]{}+f\^[2]{}\_[k]{}g\_[3]{}, with \[g\_3\] g\_[3]{}=dr\^[2]{}+h\^[2]{}\_[k]{}g\_[2]{}, and \[g\_2\] g\_[2]{}=d\^[2]{}+\^[2]{}d\^[2]{}, where $f_{k}=\cosh (H t)/H$ or $\cos (H t)/H $ ($H^{-1}$ is the de Sitter (or AdS) curvature radius) and $ h_{k}=\sin r$ or $\sinh r $, respectively.
The metric (\[warpmetric\]) is a warped product on $\mathbb{R}\times_{a}\mathcal{M}$ the warping factor being positive $a(Y)>0$, it may be considered as a generalization of the standard Riemannian cone metric $dS^{2}=dY^{2}+Y^{2}g_{4}$ on $\mathbb{R}\times_{a}\mathcal{M}$ [@peterson], [@o'neill]. The bulk fluid has an energy-momentum tensor of the form \[T old\] T\_[AB]{}=(+P)u\_[A]{}u\_[B]{}-Pg\_[AB]{}, where $A,B=1,2,3,4,5$ and $u_{A}=(0,0,0,0,1)$, with the 5th coordinate corresponding to $Y$. The five-dimensional Einstein equations, G\_[AB]{}=\^[2]{}\_[5]{}T\_[AB]{}, can be written in the following form: \[syst2i\] &=&-2 A,\
\[syst2iii\] &=& +, where $A=\kappa_{5}^{2}/4$, $k=\pm 1$, and the prime $(\,')$ denotes differentiation with respect to $Y$. The equation of conservation, \_[B]{}T\^[AB]{}=0, becomes \[syst2ii\] ’+4(1+)=0. Integration of the continuity equation (\[syst2ii\]) gives the following relation between the density and the warp factor, \[rho to a\] = c\_[1]{} a\^[-4(+1)]{}, where $c_{1}$ is an arbitrary integration constant. Substitution of Eq. (\[rho to a\]) in Eq. (\[syst2iii\]), gives a’\^2=A c\_[1]{}a\^[-2(2+1)]{}+kH\^2, and after setting $C=2/3Ac_{1}$ (note the the sign of $C$ is the same with the sign of $\rho$), we have the Friedman constraint in the form, \[integration eq\] a’\^2=Ca\^[-2(2+1)]{}+kH\^2. The left hand side of this equation restricts the signs of $C$ and $k$, as well as the range of $a$. As a result, the case $C<0$ and $k<0$ becomes automatically impossible.
On the other hand, the case $C<0$, $k>0$ is possible only for, $$\label{bound ds}
\textrm{dS braneworld:}\quad
0<a^{-2(2\gamma+1)}<-\dfrac{kH^2}{C},$$ while, the case $C>0$, $k<0$ is possible only for $$\label{bound ads}
\textrm{AdS braneworld:}\quad a^{-2(2\gamma+1)}>-\dfrac{kH^2}{C}>0.$$ It is straightforward to see that these two cases offer the possibility for avoidance of singularities. More generally, our solutions below are characterised by these three constants, namely, the curvature constant $k$, the fluid constant $\gamma$, and the constant $C$, where as mentioned above the sign of $C$ controls that of the density $\rho$. The case $C<0$, $k>0$ of a dS brane with $\gamma>-1/2$, implies that $a^{2(2\gamma+1)}>-C/(kH^{2})>0$, so that the warp factor, $a$, is bounded away from zero, excluding therefore collapse singularities from happening. The only way that this case may introduce a finite-distance singularity is to have a warp factor that becomes divergent within a finite distance (big-rip singularity). However, it will follow from our analysis in the next Section that this behaviour is also excluded and therefore this case does indeed lead to the avoidance of finite-distance singularities. On the other hand, the case $C>0$, $k<0$ of an AdS brane with $\gamma<-1/2$, implies that the warp factor takes only values greater than $-kH^2/C>0$, thus excluding the existence of collapse singularities, as well. As we will show later on, this latter case requires a further restriction on $\gamma$, $-1<\gamma<-1/2$, in order to avoid a finite-distance big-rip singularity.
Analysis of the Friedman equation
=================================
Eq. (\[integration eq\]) can be integrated out to give a solution represented by the Gaussian hypergeometric function $_{2}F_{1}$, for all possible cases defined by the signs of $C$ and $k$ and the range of $\gamma$ [^5]. These solutions along with the asymptotic behaviours they introduce are presented in what follows. The values $\gamma=-1$ and $\gamma=-1/2$ are special values for our system of equations (\[syst2i\]), (\[syst2iii\]) and (\[syst2ii\]) in the sense that they reduce it to a significantly simpler form which leads to solutions that cannot be incorporated into the solutions found below with the use of the Gaussian hypergeometric function. The solutions for these values of $\gamma$ are studied separately in Appendix A, however, in order to have a complete view of the asymptotics for all possible values of $\gamma$ immediately, we present the behaviors of the solutions for $\gamma=1,-1/2$ in the Subsections 3.1, 3.2 and 3.3 that follow.
dS branes with positive density
-------------------------------
We begin our study with the case of a dS braneworld with positive density. This corresponds to $C>0$, $k>0$, and depending on the range of $\gamma$ we have the following sub-cases:
- For $\gamma<-1/2$, the solution is, \[sol type Ia\] (Y-Y\_[0]{})=\_[2]{}F\_[1]{}(,-, ,-a\^[-2(2+1)]{}).
- For $\gamma>-1/2$, we have, \[sol type Ib\] (Y-Y\_[0]{})=a\^[2(+1)]{} \_[2]{}F\_[1]{}(, ,, -a\^[2(2+1)]{}).
The possible asymptotic behaviours follow from those of the hypergeometric function $_{2}F_{1}$:
1. $\gamma\geq-1/2$ we find that $a\rightarrow 0$, as $Y\rightarrow Y_{0}$. This is a *collapse* type singularity and it appears within a finite distance, at $Y_{0}$.
2. $\gamma\geq-1/2$ we have that $a\rightarrow \infty$, as $Y\rightarrow\infty$, which describes the behaviour of the warp factor at infinite distance.
3. $\gamma\leq-1/2$ the behaviour here is $a\rightarrow 0$, as $Y\rightarrow Y_{0}$, so that we have a collapse singularity at $Y_{0}$.
4. $-1\leq\gamma\leq-1/2$ we get $a\rightarrow\infty$, as $Y\rightarrow\infty$, which is as before the behaviour of the warp factor at infinite distance.
5. $\gamma<-1$, in this case $a\rightarrow\infty$, as $Y\rightarrow Y_{0}\pm \delta$, where the constant $\delta$ is given by: =C\^ where $\Gamma$ is the Gamma function. This is a *big rip* singularity and it appears within finite distance, at $Y_{0}\pm \delta$.
For $\gamma<-1$ there are two types of finite-distance singularities: a collapse singularity located at $Y_{0}$, *and* a big-rip singularity located at $Y_{0}\pm \delta$. As we will show later, the coexistence of these two types of singularity not only does not lead to non-singular spacetimes, but it also impedes the construction of any non-singular matching solution for $\gamma<-1$.
On the other hand, for $\gamma\geq -1$, the braneworld suffers only from a finite-distance singularity of the collapse type, which allows for the construction of a matching non-singular solution.
dS branes with negative density
-------------------------------
We now consider a dS braneworld with negative density, corresponding to $C<0$, $k>0$, and let $\gamma$ varying as follows:
- $\gamma<-1/2$, the solution is given by (\[sol type Ia\]).
- $\gamma>-1/2$, the solution is, \[sol type IIb\] (Y-Y\_[0]{})&=&\
& &\_[2]{}F\_[1]{}(,,, 1+a\^[2(2+1)]{}).
The solutions in this case satisfy the bounds (\[bound ds\]) and we arrive at the following asymptotic behaviours:
1. $\gamma\leq-1/2$ here $a\rightarrow 0$ as $Y\rightarrow Y_{0}$, and this is a collapse singularity appearing at $Y_{0}$. There is no other singularity since $a$ is bounded from above and never diverges.
2. $\gamma>-1/2$ we see that $a\rightarrow\infty$ as $Y\rightarrow\infty$, which means that this case is free from finite-distance singularities, since $a$ is bounded from below and never vanishes.
AdS branes with positive density
--------------------------------
The last possible case is that of an open universe with positive density support on the bulk which translates to considering $C>0$ and $k<0$ (AdS braneworld). Taking into account the possible ranges of $\gamma$ we have the following outcomes:
- $\gamma>-1/2$ or, $\gamma<-1$ the solution is given by Eq. (\[sol type Ib\]).
- $-1<\gamma<-1/2$ the solution is, \[sol type IIIb\] (Y-Y\_[0]{})&=&-\
&& \_[2]{}F\_[1]{} (,, , 1+a\^[-2(2+1)]{}).
As we mentioned earlier, this case is subject to the bound (\[bound ads\]) The possible asymptotic behaviours for this case are then as follows:
1. $\gamma\geq-1/2$ we find $a\rightarrow 0$, as $Y\rightarrow Y_{0}$, which implies a collapse singularity at $Y_{0}$; the warp factor is bounded from above.
2. $-1\leq\gamma<-1/2$ we find $a\rightarrow\infty$ as $Y\rightarrow\infty$, so that this region of $\gamma$ is free form finite distance singularities, since $a$ is again bounded from below and never vanishes.
3. $\gamma<-1$ we have $a\rightarrow\infty$, as $Y\rightarrow Y_{0}$. This is a big-rip singularity located at $Y_{0}$. There is no collapse singularity since $a$ is bounded from below.
Non-singular solutions
======================
We saw in the previous Section that there are solutions free from finite-distance singularities in the following cases:
- dS brane with negative density and $\gamma> -1/2$
- AdS brane with positive density and $-1\leq\gamma<-1/2$.
In this Section we analyse the complete character of these two non-singular solutions. The first non-singular solution is given by Eq. (\[sol type IIb\]) for $\gamma>-1/2$. The two branches of this solution may be matched in the following way. Let $\hat{Y}$ denote the value of $Y$ for which $a^{-2(2\gamma+1)}$ is equal to $-kH^{2}/C$, that is \[a hat\] (a())\^[-2(2+1)]{}=-. Then we see from Eq. (\[sol type IIb\]) that the value of the Gaussian hypergeometric function at $\hat{Y}$ is equal to one, while from the left-hand side of the same equation we find that $\hat{Y}=Y_{0}$. We note that $Y_{0}$ is a regular point of the solution. Putting Eq. (\[a hat\]) in Eq. (\[integration eq\]) we find that a’(Y\_[0]{})=0, so that $Y_{0}$ is a critical point. We now check the second derivative of $a$, $a''$. For $C<0$ it follows from Eq. (\[rho to a\]) and the fact that $c_{1}=3C/(2A)$ that also $\rho<0$. Further assuming $\gamma>-1/2$, we find from Eq (\[syst2i\]) that a”(Y\_[0]{})>0. Thus, for this choice of parameters we see that at $Y_{0}$ the warp factor takes its minimum value and then it starts to increase, avoiding in this way collapse singularities.
To study the behaviour of the solution at infinity i.e. as $a\rightarrow\infty$, we expand the hypergeometric function as follows [@wang], & &\_[2]{}F\_[1]{}(,,, 1+a\^[2(2+1)]{})= \_[1]{}(-1-a\^[2(2+1)]{})\^[-1/2]{}\
&&\_[2]{}F\_[1]{}(,0,, (1+a\^[2(2+1)]{})\^[-1]{}) +\_[2]{}(-1-a\^[2(2+1)]{})\^[-]{}\
&& \_[2]{}F\_[1]{}(, -,, ([1+a\^[2(2+1)]{}]{})\^[-1]{}), where \_[1]{}= and \_[2]{}=. Substituting in the solution (\[sol type IIb\]) we find, \[asympt 1\] (Y-Y\_[0]{})\~a, and so we see from (\[asympt 1\]) that $a\rightarrow \infty$ is only possible for $Y\rightarrow\pm\infty$. The behaviour of $a$ is shown in Fig. \[non singular solution\].
![**[]{data-label="non singular solution"}](nonsing.eps "fig:")\
For an AdS braneworld on the other hand, the solution (\[sol type IIIb\]) is non-singular for $-1<\gamma<-1/2$. We note that because of Eq. (\[bound ads\]), the warp factor $a$ for this range of $\gamma$ cannot approach zero and therefore all collapse singularities are excluded from happening, however, the possibility of the warp factor becoming *divergent* within finite distance is not a priori prohibited. If such behaviour is encountered then we end up with an even stronger type of singularity, a big rip.
Let us suppose that the warp factor does become divergent, $a\rightarrow \infty$, but restrict $\gamma$ in the interval $(-1,-1/2)$. The hypergeometric function appearing in Eq. (\[sol type IIIb\]) has the argument $1+C a^{-2(2\gamma+1)}/(kH^{2})$ which is diverging so to study its behaviour at infinity, we first expand it in the following way, & &\_[2]{}F\_[1]{}( ,,, 1+a\^[-2(2+1)]{})=\
& &\_[3]{}(-1-a\^[-2(2+1)]{})\^[-]{}\
& &\_[2]{}F\_[1]{}(, , , (1+a\^[-2(2+1)]{})\^[-1]{})+\
& & +\_[4]{}(-1-a\^[-2(2+1)]{})\^[-1/2]{} \_[2]{}F\_[1]{}(,0,, (1+a\^[-2(2+1)]{})\^[-1]{}), where $\Gamma_{3}$ and $\Gamma_{4}$ are the constants, \_[3]{}=, and \_[4]{}=. Substituting the above expression of the hypergeometric function in the solution (\[sol type IIIb\]), we deduce the asymptotic behaviour, \[asyminfty\] (Y-Y\_[0]{})\~a\^[2(+1)]{}, so that for $a\rightarrow\infty$, we get $Y\rightarrow\pm\infty$. Therefore the divergence of the warp factor is only possible at infinite distance which means that finite-distance big rip singularities are excluded. The behaviour of $a$ is similar to the one shown in Fig. \[non singular solution\].
The null energy condition
=========================
In this Section we study the null energy condition for our type of matter (\[T old\]) and then examine for which ranges of $\gamma$ it holds true.
We note that our metric (\[warpmetric\]) and our fluid are static with respect to the time coordinate $t$. We may reinterpret our fluid analogue as a real anisotropic fluid having the following energy momentum tensor: \[T new\] T\_[AB]{}= (\^[0]{}+ p\^[0]{})u\_[A]{}\^[0]{}u\_[B]{}\^[0]{} +p\^[0]{}g\_\_[A]{}\^\_[B]{}\^+ p\_[Y]{}g\_[55]{}\_[A]{}\^[5]{}\_[B]{}\^[5]{}, where $u_{A}^{0}=(a(Y),0,0,0,0)$, $A,B=1,2,3,4,5$ and $\alpha,\beta=1,2,3,4$. When we combine (\[T old\]) with (\[T new\]), we get the following set of relations, \[p y to rho\] p\_[Y]{}&=&\
\[rho new\] \^[0]{}&=&p\
\[p new\] p\^[0]{}&=&-p. The last two relations imply that \[p new to rho new\] p\^[0]{}=-\^[0]{}, which means that this type of matter satisfies a cosmological constant-like equation of state. Imposing further $p=\gamma\rho$, and using (\[p y to rho\]), leads to, \[py to p\] p\_[Y]{}=. Substituting (\[p new\]), (\[p new to rho new\]) and (\[py to p\]) in (\[T new\]), we find that T\_[AB]{}= -p g\_\_[A]{}\^\_[B]{}\^+ g\_[55]{}\_[A]{}\^[5]{}\_[B]{}\^[5]{}. We are now ready to form the null energy condition for our type of matter. According to the null energy condition, every future-directed null vector $k^{A}$ should satisfy [@poisson] T\_[AB]{}k\^[A]{}k\^[B]{}0. This condition implies that the energy density should be non-negative. Here we find that it translates to p+0, or, in terms of $\rho$ (+1)0, which leads to two possible cases, namely, 0 -1,0 -1. With the use of (\[rho to a\]), in which $c_{1}=3/(2A)C$, these two conditions may be written equivalently with respect to $C$ instead of $\rho$ as, \[nec\_c\_1\] C0 -1, and \[nec\_c\_2\] C0 -1.
The conditions (\[nec\_c\_1\]) and (\[nec\_c\_2\]) show that the requirement of satisfying the null energy condition leads to restrictions on both the range of $\gamma$ and the sign of the constant $C$. We conclude that the only range for $C$ and $\gamma$ that is compatible with a non-singular solution, and at the same time also satisfies the null energy condition, is $C>0$ and $-1<\gamma<-1/2$ combined with $k<0$, that is AdS braneworld with positive density and $\gamma\in(-1,-1/2)$. In particular, dS non-singular braneworlds are incompatible with the null energy condition holding in the bulk.
Matching solutions
==================
In this Section we will examine those solutions from the Sections 3.1-3.3, that allow a jump in the derivative of the warp factor, $a'$, across the brane, and also satisfy the null energy condition. These are the cases $Ia)$ and $Ib)$.
For the case $Ia)$, we have $\gamma<-1/2$, and for the null energy condition we should further restrict to $-1<\gamma<-1/2$. Setting $c_{2}=\mp Y_{0}$, and choosing the $+$ sign of $Y$ for $Y>0$ and the $-$ sign for $Y<0$, the solution (\[sol type Ia\]) can be written in the form \[matching sol 1\] |Y|+c\_[2]{}\^=\_[2]{}F\_[1]{}(,-, ,-a\^[-2(2+1)]{}). For $c_{2}^{\pm}>0$, we see that collapse singularities are excluded. Since we have restricted $\gamma$ to take values greater than $-1$, big-rip singularities are also excluded. Assuming a continuous warp factor at the position of the brane $Y=0$, we get from (\[matching sol 1\]) a condition for $c_{2}$ which reads, \[match\_1\] c\_[2]{}\^[+]{}=c\_[2]{}\^[-]{}, where $c_{2}^{\pm}$ denote the values of $c_{2}$ at $Y=0^{\pm}$. Note that $c_{2}^{\pm}$ are both positive from (\[matching sol 1\]) which is compatible with our choice of sign for $c_{2}$. Further imposing continuity of $\rho$ at $Y=0$, we find from (\[rho to a\]) and $c_{1}=3/(2A)C$ that \[match\_2\] C\^[+]{}=C\^[-]{}. Next, we take into account the jump of the derivative of the warp factor across the brane. For our type of geometry this junction condition reads \[match\_3\] a’(0\^[+]{})-a’(0\^[-]{})=-f((0))a(0), where $f(\rho)$ is the tension of the brane. For our solution the above condition translates to \[match\_4\] f((0))=-6, from which we note that the brane tension is negative.
\[matching two solution\] \
In Fig. $2$, we depict the two branches of the solution and the way they may be matched together to give the non-singular solution described above. Similarly, we may match the two branches of solution of case $Ib)$ for $\gamma>-1/2$, and find, \[matching sol 2\] |Y|+c\_[2]{}\^=a\^[2(+1)]{} \_[2]{}F\_[1]{}(, ,, -a\^[2(2+1)]{}). It follows that conditions (\[match\_1\])-(\[match\_4\]) are also true in this case. The two branches of the solution and the way they may be matched together to give the non-singular solution are shown in Fig. $3$.
\[matching one solution\] \
We can also have a matching solution by cutting the regular solutions $IIb)$, or, $IIIb)$ at a point different from the minimum (cutting the regular solutions at the minimum would lead to a vanishing brane tension). For example by taking $Y_{0}\neq 0$ and putting the brane at $Y=0$ we can derive the corresponding junction conditions which are again given by (\[match\_1\])-(\[match\_3\]). The brane tension, however, now reads f((0))=-12, and we note that it is again negative.
The rest of the cases $II$ and $III$ that satisfy the null energy condition for $\gamma<-1$ ($IIa$) and $\gamma>-1$ ($IIIa$), respectively, are not suitable for constructing non-singular matching solutions since they exhibit either two collapse (case $IIIa$), or two big rip singularities (case $IIIa)$) that restrict $Y$ to take values only between the interval with endpoints the two finite singularities. That means that the resulting matching solutions cannot be extended to the whole real line of $Y$.
Localisation of gravity
=======================
Another question we would like to answer is whether the non-singular solutions we have found for the warp factor $a$, satisfying the null energy condition, lead to a finite four-dimensional Planck mass, thus localising 4d gravity on the brane for some range of the parameter $\gamma$. The value of the four-dimensional Planck mass, $M_{p}^{2}=8\pi/\kappa$, is determined by the following integral [@forste], =\_[-Y\_[c]{}]{}\^[Y\_[c]{}]{}a\^[2]{}(Y)dY. For our first matching solution, Eq. (\[matching sol 1\]), the behaviour of $a^{2}$ at large $|Y|$ is \[asq linear\] a\^[2]{}\~(|Y|+c\_[2]{})\^[2]{}, and the above integral becomes, \_[-Y\_[c]{}]{}\^[Y\_[c]{}]{}(|Y|+c\_[2]{})\^[2]{}dY =(-(-Y+c\_[2]{})\^[3]{}|\_[-Y\_[c]{}]{}\^[0]{}+(Y+c\_[2]{})\^[3]{}|\_[0]{}\^[Y\_[c]{}]{}). In the limit $Y_{c}\rightarrow\infty$, the Planck mass becomes infinite.
The same behavior is valid also for the regular solution $IIb)$ found for a dS brane with negative density and $\gamma>-1/2$. Placing the brane at $Y=0$ and using the line of thinking of Section $6$ we can bring the left hand side of solution (\[sol type IIb\]) into a form involving the absolute value of $Y$. Then Eq. (\[asympt 1\]) can take the form of (\[asq linear\]) which lead to an infinite Planck mass.
For our second matching solution (\[matching sol 2\]), the behaviour of $a^{2}$ is \[asq gamma\] a\^[2]{}\~(|Y|+c\_[2]{})\^. Integration of $a^{2}$ gives an expression with $Y$ raised to the exponent, which is positive for this case since here $\gamma>-1/2$. Therefore we see that the Planck mass is infinite also in this case.
As before the same behavior is valid for the other regular solution $IIIb)$ of an AdS brane with positive density and $-1<\gamma<-1/2$. For this solution we could also place the brane at $Y=0$ and by using its asymptotic behavior given by Eq. (\[asyminfty\]) we see that it could take the form of (\[asq gamma\]) and therefore lead to an infinite Planck mass since $\gamma$ does not lie in $(-2,-1)$.
Conclusions
===========
In this paper, we have analysed braneworld singularities in the presence of dS or AdS branes and found one non-singular solution for a dS brane with negative (bulk) density and another one for an AdS brane with positive (bulk) density, for particular ranges of the parameter space, that we constructed explicitly. As we showed in [@ack3] this was impossible for Minkowski branes. In the case of AdS branes the null energy condition is also satisfied.
Comparing and contrasting the results of the asymptotic behaviour of the solutions found in this paper to those of our previous work [@ack2] which was implemented with a different method of asymptotic analysis we extract the following conclusions: The case of a dS brane with positive density, described in Section 3.1, was asymptotically constructed by the two balances simultaneously $_{\gamma}\mathcal{B}_{1}$ and $_{\gamma}\mathcal{B}_{2}$. The first balance described the behavior of $a$ around a finite collapse singularity, while the second balance described the behavior of $a$ at infinity. In particular, the balance $_{\gamma}\mathcal{B}_{2}$ gave the behavior, $a\rightarrow\infty$ as $Y\rightarrow\infty$, however, this case is characterised as singular because of the finite-distance singularity of collapse type introduced by the balance $_{\gamma}\mathcal{B}_{1}$.
On the other hand, the case of an AdS brane with positive density, described in Section 3.2, was depicted by the balance $_{\gamma}\mathcal{B}_{2}$ which is non-singular for $\gamma>-1/2$. The balance $_{\gamma}\mathcal{B}_{1}$ which leads to finite-distance singularities for flat and positively curved branes is not valid in this case since it assumes only positive density, whereas, this case is characterised by a negative density. Lastly, the third case, described in Section 3.3, can be described by the balance $_{\gamma}\mathcal{B}_{1}$ which allows for a non-singular solution for $-1<\gamma<-1/2$.
It is possible that non-singular solutions that satisfy the null energy condition in the bulk and at the same time localize gravity in the braneworld exist for models of interacting matter as in [@ack4], or, for homogeneous but anisotropic (eg. Bianchi I, V, or VIII, IX) braneworlds. Exploring such models will help us decide about the stability of our non-singular solutions discovered here with respect to more general (anisotropic) perturbation. We leave this to a future publication.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank an anonymous referee of [@ack3] for suggesting and giving a first analysis on the problem of curved braneworlds studied in this paper. I.K. is grateful to LPTHE for making her visit there possible and for financial support.
Appendix: Solutions for curved branes and special fluids
========================================================
In this Appendix we analyze the behavior of the system of equations (\[syst2i\])-(\[syst2iii\]) and (\[syst2ii\]) for special values of the parameter $\gamma$ of the fluid. These are values of $\gamma$ that simplify the dynamical system significantly and lead to solutions that cannot be incorporated to the solutions found in Section $3$. These special values are $\gamma=-1/2$ and $\gamma=-1$.
Consider first $\gamma=-1/2$. Eqs. (\[syst2i\])-(\[syst2iii\]) and (\[syst2ii\]) for $\gamma=-1/2$ become \[syst g=-1/2 i\] a”&=&0\
\[syst g=-1/2 ii\] &=&+\
\[syst g=-1/2 iii\] ’+2&=&0. Eq. (\[syst g=-1/2 i\]) gives directly \[a for g=-1/2\] a=c\_[1]{}Y+c\_[2]{}, where $c_{1}$ and $c_{2}$ are arbitrary constants. Inputting (\[a for g=-1/2\]) in Eq. (\[syst g=-1/2 iii\]) we find \[rho for g=-1/2\] =, where $c_{3}$ is an arbitrary constant. We substitute Eqs. (\[a for g=-1/2\]) and (\[rho for g=-1/2\]) in Eq. (\[syst g=-1/2 ii\]) to derive the relation between the three arbitrary constants which reads c\_[3]{}=([c\_[1]{}]{}\^[2]{}-kH\^[2]{}). The linear solution (\[a for g=-1/2\]) shows that for $\gamma=-1/2$, $a\rightarrow 0$ at a finite-distance Y\_[0]{}=-, and also $a\rightarrow\infty$ as $Y\rightarrow \infty$. This case therefore suffers from a finite-distance collapse singularity.
For $\gamma=-1$, on the other hand, the dynamical system given by Eqs. (\[syst2i\])-(\[syst2iii\]) and (\[syst2ii\]) takes the form \[syst g=-1 i\] &=&\
\[syst g=-1 ii\] &=&+\
\[syst g=-1 iii\] ’&=&0. Eq. (\[syst g=-1 iii\]) implies that \[rho g=-1 c\_3<0\] =c\_[3]{}, with $c_{3}$ an arbitrary constant. By substitution of this in Eq. (\[syst g=-1 i\]) we get the following second order differential equation with constant coefficients a”-\_[5]{}\^[2]{}a=0 which has the characteristic equation \^[2]{}-\_[5]{}\^[2]{}=0. For $c_{3}>0$ the above equation has two distinct real roots \[char-eq\] =\_[5]{} and so the general solution has the form \[gen sol g=-1\] a=c\_[1]{}e\^[[\_[5]{}]{}Y]{}+c\_[2]{}e\^[[-\_[5]{}]{}Y]{}, c\_[3]{}>0, where $c_{1}$ and $c_{2}$ are arbitrary constants. Substituting (\[gen sol g=-1\]) in Eq. (\[syst g=-1 ii\]) we find the relation connecting the three arbitrary constants which reads c\_[3]{}=-. Since we have taken $c_{3}>0$ we need to have the following restrictions on the signs of $c_{1}$, $c_{2}$ and $k$ c\_[1]{}c\_[2]{}<0 k>0, c\_[1]{}c\_[2]{}>0 k<0. For $c_{1}c_{2}<0$ there is a finite-distance singularity at Y\_[0]{}=(-). We also see from the solution (\[gen sol g=-1\]) that $a$ becomes infinite only at infinite $Y$. For $c_{1}c_{2}>0$ and $k<0$, however, we see that the solution is free from finite-distance singularities.
For $c_{3}<0$, on the other hand, we have a different behavior. Taking $c_{3}<0$ translates first from Eq. (\[rho g=-1 c\_3<0\]) to having negative density and then from Eq. (\[syst g=-1 ii\]) to allowing only for a dS brane. For $c_{3}<0$ the characteristic equation has imaginary roots and the general solution (\[gen sol g=-1\]) becomes complex. However, we can still obtain a real general solution from the complex one by imposing real initial conditions. The real general solution obtained in this way is given by a=c\_[1]{}( \_[5]{}Y)+ c\_[2]{}( \_[5]{}Y), where $C_{3}=-c_{3}>0$. This solution has an infinite number of finite-distance singularities.
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Enveloping branes and braneworld singularities*, Eur. Phys. J. C74 (2014) 3192, \[arXiv:hep-th/1406.0611v2\].
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Braneworld cosmological singularities*, Proceedings of MG11 meeting on General Relativity, vol. 3, pp. 2054-2056, \[arXiv:gr-qc/0701033\].
I. Antoniadis, S. Cotsakis, I. Klaoudatou, *Brane singularities and their avoidance*, Class. Quant. Grav. 27 (2010) 235018 \[arXiv:gr-qc/1010.6175\].
S. S. Gubser, *Curvature singularities: The good, the bad, and the naked*, Adv. Theor. Math. Phys. 4 (2000) 679, \[arXiv:hep-th/0002160\].
P. Peterson, *Riemannian geometry*, Springer 2006.
B. O’ Neill, *Semi-Riemannian geometry with applications to relativity*, Academic Press 1983.
Z. X. Wang, D. R. Guo, *Special Functions*, World Scientific, 1989.
E. Poisson, A Relativist’s Toolkit, Cambridge University Press, 2004.
S. Forste, H. P. Nilles and I. Zavala, *Nontrival Cosmological Constant in Brane Worlds with Unorthodox Lagrangians*, JCAP [**1107**]{} (2011) 007, \[arXiv:hep-th/1104.2570\].
Antoniadis, I., Cotsakis, S. and Klaoudatou, I., *Brane singularities with mixtures in the bulk*, Fortschr. Phys. 61 (2013) 20-49.
[^1]: `[email protected]`
[^2]: On leave from the University of the Aegean, 83200 Samos, Greece.
[^3]: `[email protected]`
[^4]: `[email protected]`
[^5]: The classification of all the possible cases studied in Sections 3.1-3.3 is defined through the restriction of obtaining a valid integral representation of the Gaussian hypergeometric function from Eq. (\[integration eq\]), [@wang].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The hollow inside of single-wall carbon nanotubes (SWCNT) provides a unique degree of freedom to investigate chemical reactions inside this confined environment and to study the tube properties. It is reviewed herein, how encapsulating fullerenes, magnetic fullerenes, $^{13}$C isotope enriched fullerenes and organic solvents inside SWCNTs enables to yield unprecedented insight into their electronic, optical, and interfacial properties and to study their growth. Encapsulated C$_{60}$ fullerenes are transformed to inner tubes by a high temperature annealing. The unique, low defect concentration of inner tubes makes them ideal to study the effect of diameter dependent treatments such as opening and closing of the tubes. The growth of inner tubes is achieved from $^{13}$C enriched encapsulated organic solvents, which shows that fullerenes do not have a distinguished role and it opens new perspectives to explore the in-the-tube chemistry. Encapsulation of magnetic fullerenes, such as N@C$_{60}$ and C$_{59}$N is demonstrated using ESR. Growth of inner tubes from $^{13}$C enriched fullerenes provides a unique isotope engineered heteronuclear system, where the outer tubes contain natural carbon and the inner walls are controllably $^{13}$C isotope enriched. The material enables to identify the vibrational modes of inner tubes which otherwise strongly overlap with the outer tube modes. The $^{13}$C NMR signal of the material is specific for the small diameter SWCNTs. Temperature and field dependent $^{13}$C $T_1$ studies show a uniform metallic-like electronic state for all inner tubes and a low energy, 3 meV gap is observed that is assigned to a long sought Peierls transition.'
author:
- |
Ferenc Simon , Rudolf Pfeiffer, and Hans Kuzmany\
*Institut für Materialphysik, Universität Wien*\
*Strudlhofgasse 4, A-1090 Wien, Austria*\
`[email protected]`\
bibliography:
- 'CACReview.bib'
title: 'Recent advances in the internal functionalization of carbon nanotubes: synthesis, optical, and magnetic resonance studies'
---
Introduction
============
The era of nanotechnology received an enormous boost with the discovery of carbon nanotubes (CNTs) by Sumio Iijima in 1991 [@IijimaNAT1991]. Before 1991 nano- and nanotechnology usually meant small clusters of atoms or molecules. The originally discovered CNTs contain several coaxial carbon shells and are called multi-wall CNTs (MWCNTs). Soon thereafter single-wall CNTs (SWCNTs), i.e. a carbon nanotube consisting of a single carbon shell were discovered [@IijimaNAT1993; @BethuneNAT1993]. The principal interest in CNTs comes from the fact that they contain carbon only and all carbon are locally sp$^{2}$ bound, like in graphite, which provides unique mechanical and transport properties. This, combined with their huge, $>$ 1000, aspect ratio (the diameters being 1-20 nm and their lengths over a few micron or even exceeding cms) gives them an enormous application potential. The not exhaustive list of applications includes field-emission displays (epxloiting their sharp tips) [@Obraztsov], cathode emitters for small sized x-ray tubes for medical applications [@ZhouAPL2002], reinforcing elements for CNT-metal composites, tips for scanning probe microscopy [@HafnerNAT], high current transmitting wires, cables for a future space elevator, elements of nano-transistors [@BachtoldSCI2001], and elements for quantum information processing [@HarneitPSS].
Carbon nanotubes can be represented as rolled up graphene sheets, i.e. single layers of graphite. Depending on the number of coaxial carbon nanotubes, they are usually classified into multi-wall carbon nanotubes (MWCNTs) and single-wall carbon nanotubes (SWCNTs). Some general considerations have been clarified in the past 14 years of nanomaterial research related to these structures. MWCNTs are more homogeneous in their physical properties as the large number of coaxial tubes smears out individual tube properties. This makes them suitable candidates for applications where their nanometer size and the conducting properties can be exploited such as e.g. nanometer sized wires. In contrast, SWCNT materials are grown as an ensemble of weakly interacting tubes with different diameters. The physical properties of similar diameter SWCNTs can change dramatically as the electronic structure is very sensitive on the rolling-up direction, the so-called chiral vector [@HamadaPRL1992; @DresselhausTubes]. The chiral vector is characterized by the $(n,m)$ vector components which denote the direction along which a graphene sheet is rolled up to form a nanotube. Depending on the chiral vector, SWCNTs can be metallic or semiconducting [@DresselhausTubes]. This provides a richer range of physical phenomena as compared to the MWCNTs, however significantly limits the range of applications. To date, neither the directed growth nor the controlled selection of SWCNTs with a well defined chiral vector has been performed successfully. Thus, their broad applicability is still awaiting. Correspondingly, current research is focused on the post-synthesis separation of SWCNTs with a narrow range of chiralities [ChattopadhyayJACS,KrupkeSCI,RinzlerNL,StranoSCI]{} or on methods which yield information that are specific to SWCNTs with different chiralities. Examples for the latter are the observation of chirality selective band-gap fluorescence in semiconducting SWCNTs [@Bachilo:Science298:2361:(2002)] and chirality assigned resonant Raman scattering [@FantiniPRL2004; @TelgPRL2004].
Clearly, several fundamental questions need to be answered before all the benefits of these novel nanostructures can be fully exploited. Recent theoretical and experimental efforts focused on the understanding of the electronic and optical properties of single-wall carbon nanotubes. It has been long thought that the one-dimensional structure of SWCNTs renders their electronic properties inherently one-dimensional [@HamadaPRL1992; @DresselhausTubes]. This was suggested to result in a range of exotic correlated phenomena such as the Tomonaga-Luttinger (TLL) state [@EggerPRL1997], the Peierls transition [@BohnenPeierlsPRL2004; @ConnetablePeierlsPRL2005], ballistic transport [@DekkerNAT1997], and bound excitons [@KanePRL2003; @LouiePRL2004; @AvourisPRL2004; @AvourisPRL2005]. The presence of the TLL state is now firmly established [@BockrathNAT; @KatauraNAT2003; @PichlerPRL2004], there is evidence for the ballistic transport properties [@DekkerNAT1997] and there is growing experimental evidence for the presence of excitonic effects [@HeinzSCI2005; @MaultzschPRB2005]. The Peierls transition, however remains still to be seen.
![HR-TEM image of C$_{60}$@SWCNT peapods.[]{data-label="CACReview_PeapodHRTEM"}](./CACReview_PeapodHRTEM.eps){width="0.9\linewidth"}
An appealing tool to study the SWCNT properties originates from the discovery of fullerenes encapsulated inside SWCNTs by Smith, Monthioux, and Luzzi [@SmithNAT]. This peapod structure is particularly interesting as it combines two fundamental forms of carbon: fullerenes and carbon nanotubes. A high-resolution transmission electron microscopy (HR-TEM) image of a peapod is shown in Fig. \[CACReview\_PeapodHRTEM\]. It was also shown that macroscopic filling with the fullerenes can be achieved [@LuzziCPL1999; @KatauraSM2001]. This, in principle, opens the way to encapsulate magnetic fullerenes which would enable the study of the tube electronic properties using electron spin resonance as it is discussed in this review. Another interesting follow-up of the peapod structure discovery is that the encapsulated fullerenes can be fused into a smaller diameter inner tube [@LuzziCPL2000; @BandowCPL2001] thus producing a double-wall carbon nanotube (DWCNT). DWCNTs were first observed to form under intensive electron radiation [@LuzziCPL1999] in a high resolution transmission electron microscope from C$_{60}$ peapods. Following the synthesis of C$_{60}$ peapods in macroscopic amounts [@KatauraSM2001], bulk quantities of the DWCNT material are available using a high temperature annealing method [@BandowCPL2001]. Alternatively, DWCNTs can be produced with usual synthesis methods such as arc-discharge [@HutchisonCAR2001] or CVD [@ChengCPL2002] under special conditions. According to the number of shells, DWCNTs are between SWCNTs and MWCNTs. Thus, one expects that DWCNTs may provide a material where improved mechanical stability as compared to SWCNTs coexists with the rich variety of electronic properties of SWCNTs. There are, of course, a number of yet unanswered questions e.g. if the outer tube properties are unaffected by the presence of the inner tube or if the commensurability of the tube structures plays a role. These questions should be answered before the successful application of these materials.
![Schematic structure of an isotope engineered DWCNT with (14,6) outer and (6,4) inner tubes. $^{12}$C and $^{13}$C are shown in black and blue, respectively. The inner tube is 89 % $^{13}$C enriched and the outer contains natural carbon (1.1 % $^{13}$C abundance), which are randomly distributed for both shells.[]{data-label="CACReview_DWCNTwirediagram"}](./CACReview_DWCNTwirediagram.eps){width="0.9\linewidth"}
The inner tubes grown inside SWCNTs from peapods turned out to be a particularly interesting system as they are remarkably defect free which results in very long phonon life-times, i.e. very narrow vibrational modes [@PfeifferPRL2003]. In addition, their smaller diameters results in a larger energy spread, i.e. larger spectral splitting, for diameter dependent phonon modes such as e.g. the radial breathing mode (RBM). These two effects make the inner tubes very suitable to study diameter dependent physics of the small diameter tubes with precision. Here, we review how to employ the inner tubes as $probes$ of the outer tube properties. The additional benefit of the inner tube growth from fullerenes is that the starting carbon source can be tailored at wish, e.g. when $^{13}$C isotope enriched fullerenes are encapsulated inside the SWCNT host tubes, $^{13}$C isotope enriched inner tubes are grown. In Fig. \[CACReview\_DWCNTwirediagram\] we show the schematics of such a DWCNT.
Here, we review the efforts to study the SWCNTs properties through encapsulation using Raman and magnetic resonance spectroscopy. The reviewed phenomena include the precise characterization of diameter distribution of SWCNTs, the study of reversible hole engineering on the SWCNTs, study of the inner tube growth mechanism with the help of $^{13}$C isotope labeling, the study of local density of states on the tubes using nuclear magnetic resonance (NMR) on the $^{13}$C isotope enriched inner tubes, and the electron spin resonance (ESR) studies of the SWCNTs using encapsulated magnetic fullerenes. This review is organized as follows. First, we present the general properties of DWCNTs using Raman, discuss the electronic and vibrational properties of the inner tubes, which are the probes in the subsequent studies. Second, we present the use of the inner tubes to probe the host outer tube diameter distribution and to study the opening and closing of holes on the outer tubes. Third, we present a study on the inner tube growth mechanism using isotope enriched carbon. Fourth, we discuss the efforts related to studying the SWCNT properties by encapsulating magnetic fullerenes using ESR. Fifth, we discuss the NMR results on the isotope enriched inner tubes and in particular we present the observation of a low energy spin-gap in the density of states of SWCNTs.
Experimental methods and sample preparation
===========================================
Sample preparation
------------------
**The starting SWCNT samples**
SWCNTs from different sources and prepared by different methods were used. Commercial arc-discharge grown SWCNTs with 50 % weight purity (Nanocarblab, Moscow, Russia) and laser ablation prepared SWCNTs with 10 % weight purity (Tubes@Rice, Houston, USA) were used. The latter material was purified through repeated steps of air oxidation and washing in HCl. Some laser ablation prepared and purified samples were obtained from H. Kataura. The purified samples are usually well opened to enable fullerene encapsulation. If not, annealing in air at 450 $^{\circ }$C for 0.5 hour makes them sufficiently open. The HiPco samples used as reference were purchased from CNI (Carbon Nanotechnologies Inc., Houston, USA). Most samples were used in the form of a buckypaper, which is prepared by filtering a suspension of SWCNTs. We found that commercially available SWCNTs already meet a required standard in respect of purity and quality. In addition, for the amount of experimental work described here, reproducible samples i.e. a large amount of SWCNTs from similar quality, were required. Commercial samples meet this requirement, which compensates for their slightly inferior quality compared to laboratory prepared ones.
**Synthesis of peapods**
Encapsulation of fullerenes at low temperatures inside SWCNTs (solvent method) was performed by sonicating the fullerene and opened SWCNT suspensions together in organic solvents following Refs. [@YudasakaCPL; @SimonCPL2004; @Monthioux2004; @BriggsJMC]. For fullerene encapsulation at high temperatures (the vapor method), the SWCNTs and the fullerenes were sealed under vacuum in a quartz ampoule and annealed at 650 $^{\circ }$C for 2 hours [@KatauraSM2001]. Fullerenes enter the inside of the SWCNTs at this temperature due to their high vapor pressure that is maintained in the sealed environment. Non-encapsulated fullerenes were removed by dynamic vacuum annealing at the same temperature for 1 hour. High purity fullerenes were obtained from a commercial source (Hoechst AG, Frankfurt, Germany). The filling of SWCNTs with the fullerenes was characterized by observing the peapod structure in high-resolution transmission electron microscopy (HR-TEM), by x-ray studies of the one-dimensional array of fullerenes inside the SWCNTs and by the detection of the fullerene modes from the cages encapsulated inside the SWCNTs using Raman spectroscopy [@KatauraSM2001; @PichlerPRL2001].
**Synthesis of DWCNTs**
DWCNTs were prepared by two routes: from fullerene peapods and using chemical vapor deposition (CVD) growth technique [@SimonCPL2005]. The peapods were transformed to DWCNTs by a dynamic vacuum treatment at 1250 $^{\circ }$C for 2 hours following Ref. [@BandowCPL2001]. Again, the DWCNT transformation was followed by HR-TEM and by the observation of the DWCNT structure factors using x-ray studies. In addition, new Raman modes emerge after the 1250 $^{\circ}$C heat treatment particularly in a frequency range that is clearly upshifted from the outer tube RBMs. For the CVD DWCNT growth [@SimonCPL2005], the catalyst was a modified version of the Fe/Mo/MgO system developed by Liu *et al.* [@LiuCPL2004] for SWCNT synthesis.
Both kinds of DWCNTs have advantageous and disadvantageous properties. For peapod template grown DWCNTs, the inner tube is known to fill only up to $\sim$ 70 % of the outer tube length [@AbePRB2003]. This is the consequence of insufficient carbon in the fullerenes: the C$_{60}$ peapods have 60 carbon atoms per 1 nm (the lattice constant of the peapod) whereas the (9,0) inner tube with $d=0.708$, which is representative of the most abundant 7 nm diameter inner tube contains 36 carbon atoms per the $c_{0}=0.424$ nm lattice constant [@ZolyomiPRB2004]. In contrast, CVD grown inner tubes fill up to the total length of the outer tubes, however such samples have usually a less well defined tube diameter distribution due to the inevitable growth of small diameter SWCNTs and large diameter DWCNTs [@EndoNAT]. Peapod template grown DWCNTs can be grown with relatively narrow diameter distribution due to the available narrow diameter distribution of the SWCNT host tubes. This also allows for a good control over the DWCNT diameter as described in Ref. [@SimonPRB2005] and is discussed below.
**Synthesis of isotope engineered DWCNTs**
Commercial $^{13}$C isotope enriched fullerenes (MER Corp., Tucson, USA) were used to prepare fullerene peapods C$_{60}$,C$_{70}$@SWCNT with enriched fullerenes. Two supplier specified grades of $^{13}$C enriched fullerene mixtures were used: 25 and 89 %, whose values were slightly refined based on the Raman spectroscopy. The 25 % grade was nominally C$_{60}$, and the 89 % grade was nominally C$_{70}$ with C$_{60}$/C$_{70}$/higher fullerene compositions of 75:20:5 and 12:88:$<$ 1, respectively. The above detailed standard routes were performed for the peapod and the DWCNT productions.
Experimental methods
--------------------
**Raman spectroscopy**
Raman spectra were measured with a Dilor xy triple spectrometer using various lines of an Ar/Kr laser, a He/Ne laser and a tunable Ti:sapphire and Rhodamin dye-laser in the 1.54-2.54 eV (805-488 nm) energy range. Tunable lasers allow to record the so-called Raman map [@FantiniPRL2004; @TelgPRL2004] i.e. to detect the SWCNT resonance energies through the Raman resonance enhancement [@KuzmanyBook], which ultimately allows the chiral index assignment. The spectra can be recorded in normal (NR) and high resolution (HR) mode, respectively ($\Delta\bar{\nu}_{\text{NR}}=1.3 {\text{ cm}^{-1}}$ for blue and $\Delta\bar{\nu}_{\text{HR}}=0.4 {\text{ cm}^{-1}}$ in the red). The samples in the form of bucky-paper are kept in dynamic vacuum and on a copper tip attached to a cryostat, which allows temperature variation in the 20-600 K temperature range. Raman spectroscopy was used to characterize the diameter distribution of the SWCNTs, to determine the peapod concentrations, and to monitor the DWCNT transformation of the peapod samples.
**Electron spin resonance**
The peapod and the reference SWCNT materials were mixed with the ESR silent high purity SnO$_{2}$ in a mortar to separate the pieces of the conducting bucky-papers. The samples were sealed under dynamic vacuum. A typical microwave power of 10 $\mu$W and 0.01 mT magnetic field modulation at ambient temperature were used for the measurements in a Bruker Elexsys X-band spectrometer.
**Nuclear magnetic resonance**
Nuclear magnetic resonance (NMR) is usually an excellent technique for probing the electronic properties at the Fermi level of metallic systems. The examples include conducting polymers, fullerenes, and high temperature superconductors. However the 1.1% natural abundance of $^{13}$C with nuclear spin $I$=1/2 limits the sensitivity of such experiments. As a result, meaningful NMR experiments have to be performed on $^{13}$C isotope enriched samples. NMR data were taken with the samples sealed in quartz tubes filled with a low pressure of high purity Helium gas [@SimonPRL2005]. We probed the low frequency spin dynamics (or low energy spin excitations, equivalently) of the inner-tubes using the spin lattice relaxation time, $T_{1}$, defined as the characteristic time it takes the $^{13}$C nuclear magnetization to recover after saturation. The signal intensity after saturation, $M(t)$, was deduced by integrating the fast Fourier transform of half the spin-echo for different delay times, $t$.
Results and discussion
======================
Inner tubes in DWCNTs as local probes
-------------------------------------
### Electronic and vibrational properties of DWCNTs
![Transformation of fullerene peapods to DWCNTs as followed with Raman spectroscopy at 496.5 nm laser excitation and 90 K. The SWCNT Raman spectra (lower curve) is shown as reference. The fullerene related peapod modes (dots) in the middle curve disappear upon the heat treatment. Note the sharp RBMs appearing in the 250-450 cm$^{-1}$ for the DWCNT sample.[]{data-label="CACreview_DWCNTtransformation"}](./CACreview_DWCNTtransformation.eps){width="1\linewidth"}
Encapsulating fullerenes and transforming them into inner tubes by the high temperature annealing process [@BandowCPL2001] provides a unique opportunity to study the properties of the host outer tubes. In Fig. \[CACreview\_DWCNTtransformation\] we show the evolution of the SWCNT Raman spectrum upon C$_{60}$ fullerene encapsulation and the DWCNT transformation after Ref. [@PfeifferPRL2003]. The series of sharp modes in the peapod spectrum, which are related to the encapsulated fullerenes [@PichlerPRL2001], disappear upon the heat treatment and a series of sharp modes appear in the 250-450 cm$^{-1}$ spectral range. The presence of inner tubes after this protocol have been independently confirmed by HR-TEM [@LuzziCPL2000]. The small diameter tubes with $d \sim$ 0.7 nm would have RBM modes in the $\sim$ 250-450 cm$^{-1}$ spectral range, which clarifies the identification of these modes. The identification of the inner tube RBMs is possible due to the strong $d$ dependence of this Raman mode [@Kuerti:PhysRevB58:R8869:(1998)]. Assignment of less diameter dependent modes such as the G mode [@DresselhausTubes] to inner and outer tubes are more difficult although a number of small intensity new modes are observed for the DWCNT sample in Fig. \[CACreview\_DWCNTtransformation\]. It is shown in Section \[isotope\_labeled\] that unambiguous assignment can be given with the help of selective isotope enrichment of the inner walls.
![Raman spectra of the RBMs in DWCNT and HiPco (SWCNT) samples at 594 nm laser excitation and 90 K in the high resolution spectrometer mode.[]{data-label="CACReview_DWCNTHR"}](./CACReview_DWCNTHR.eps){width="0.9\linewidth"}
A variety of additional information can be gained about the inner tube properties when their RBMs are studied using the additive mode, i.e. high-resolution of the Raman spectrometer. In Fig. \[CACReview\_DWCNTHR\], we show the inner tube RBMs at 90 K with high-resolution in comparison with an SWCNT sample with similar tube diameter prepared by the HiPco process. Three striking observations are apparent in the comparison of the two spectra: i) there are a larger number of inner tube RBMs than geometrically allowed and they appear to cluster around the corresponding modes in the SWCNT sample, ii) the inner tube RBMs are on average an order of magnitude narrower than the SWCNT RBMs in the HiPco sample [@PfeifferPRL2003] and iii) the Raman intensity of the inner tubes is large in view of the $\sim$ 3 times less number of carbon atoms on them [@PfeifferPRB2004]. Points ii) and iii) are explained by the long phonon and quasi-particle life-times of inner tubes which are discussed further below.
![Raman map of DWCNTs. Circles and squares are the $E_{22}^{\text{s}}$ and $E_{11}^{\text{m}}$ peaks as measured in a HiPco sample [@FantiniPRL2004], respectively. The family numbers and the chiral indexes for the (6,5) and (6,4) tubes are indicated. Dashed lines join chiralities in the same family. Laser excitation was not available in the missing area. Reprinted figure with permission from Ref. [@PfeifferPRB2005b], R. Pfeiffer *et al.* Phys. Rev. B **72**, 161404 (2005). Copyright (2005) by the American Physical Society.[]{data-label="CACReview_DWCNTRamanMapFull"}](./CACReview_DWCNTRamanMapFull.eps){width="\linewidth"}
Observation i), i.e. the clustering behavior of the observed inner tube RBMs around SWCNT RBMs, is further evidenced in energy dispersive Raman measurements. In Fig. \[CACReview\_DWCNTRamanMapFull\], we show the Raman map for the DWCNTs from Ref. [@PfeifferPRB2005b]. The advantage of studying Raman maps is that the optical transition energies are also contained in addition to the Raman shifts. These two quantities uniquely identify the chirality of a nanotube [@DresselhausTubes; @DresselhausTubesNew; @KatauraSM1999]. The analogous Raman map for HiPco SWCNTs were measured by Fantini *et al.* [@FantiniPRL2004] and Telg *et al.* [@TelgPRL2004]. Their results are also shown in Fig. \[CACReview\_DWCNTRamanMapFull\]) with squares and circles for metallic and semiconducting tubes, respectively. It turns out that family patterns with $2n+m=$ const can be identified for which the tube resonance energies and Raman shifts are closely grouped together [@Bachilo:Science298:2361:(2002)]. The comparison of the HiPco results and the DWCNT Raman map confirms the above statement, i.e. that a number of inner tube modes are observed for the DWCNT where only a few (or one) SWCNT chirality is present. This is best seen for the (6,5) and (6,4) chiralities which are well resolved from other modes.
![Raman map comparison of the (6,5) and (6,4) tube RBM ranges for DWCNT and SWCNT (HiPco) samples. Ellipsoids indicate the corresponding tube modes. Note the progressive transition energy downshift for the split components of the inner tubes and the 30 meV transition energy difference between the two kinds of samples, which are discussed in the text. []{data-label="CACReview_DWCNTHIPCORamanMaps"}](./CACReview_DWCNTHIPCORamanMaps.eps){width="\linewidth"}
In Fig. \[CACReview\_DWCNTHIPCORamanMaps\], we show the Raman maps for the two samples near the energy and Raman shift regions for the (6,5) and (6,4) tube modes [@Bachilo:Science298:2361:(2002); @FantiniPRL2004; @TelgPRL2004]. The comparison of the Raman maps of the two kinds of samples shows that the corresponding tube modes are split into up to 15 components for the inner tube RBMs. This is explained by the inner-outer tube interaction in the DWCNT samples: an inner tube with a particular chirality can be grown in outer tubes with different diameters (chiralities). The varying inner-outer tube spacing can give rise to a different Raman shift for the split components. The large number of split components is a surprising result as it is expected that an inner tube with a given diameter is grown in maximum 1-2 outer tubes where its growth is energetically preferred.
To further prove the origin of the splitting and to quantify this effect, model calculations on the inner-outer tube interactions were performed [@PfeifferPRB2005b; @PfeifferEPJB2004] following the continuum model of Popov and Henrard [@Popov:PhysRevB65:235415:(2002)]. These calculations showed that the interaction of inner and outer tubes can gives rise to a shift in the inner tube RBM frequency up to 30 cm$^{-1}$.
### Phonon and quasi-particle life-times in DWCNTs
![High resolution Raman spectra taken at 676 nm laser excitation and 90 K on the CVD- and PEA-DWCNT and an SWCNT reference (HiPco) sample. The deconvoluted spectrum is also shown for the CVD-DWCNT sample. The narrow line-widths indicate the long RBM phonon life-times of the inner tubes in both DWCNT materials. Reprinted figure with permission from Ref. [@SimonCPL2005], F. Simon *et al.* Chem. Phys. Lett. **413**, 506 (2005). Copyright (2005) by Elsevier.[]{data-label="CACReview_CVDDWCNT"}](./CACReview_CVDDWCNT.eps){width="0.9\linewidth"}
Now, we turn to discussion of the observed very narrow line-widths of the RBMs. This is the most important property of the inner tube RBMs, which will be exploited throughout in this work. Intrinsic line-widths can be determined by deconvoluting the experimental spectra with a Voigtian fit, whose Gaussian component describes the spectrometer resolution and the Lorentzian gives the intrinsic line-width. The Lorentzian component for some inner tube RBMs is as small as 0.4 cm$^{-1}$ [@PfeifferPRL2003], which is an order or magnitude smaller than the values obtained for isolated individual tubes in a normal SWCNT sample [@Jorio:PhysRevLett86:1118:(2001)]. The narrow line-widths, i.e. long phonon life-times of the inner tube RBMs was originally associated to the perfectness of the inner tubes grown from the peapod templates [@PfeifferPRL2003]. It was found, however, that inner tubes in chemical vapor deposition (CVD) grown DWCNTs have similarly small line-widths [@SimonCPL2005]. In Fig. \[CACReview\_CVDDWCNT\], the high resolution spectra for the inner tube RBMs in CVD and peapod template grown DWCNTs is shown. This suggests, that the tube environment plays an important role in the magnitude of the observable RBM line-width.
The tube-tube interactions have been shown to give rise to up to $\approx$ 30 cm$^{-1}$ extra shift to the RBMs [@PfeifferPRB2005b]. The principal difference between SWCNTs and inner tubes in DWCNTs (irrespective whether these are CVD or peapod template grown) is the different surrounding of a small diameter SWCNT with a given chirality: for the SWCNT sample, each tube is surrounded by the ensemble of other SWCNTs. For a close packed hexagonal bundle structure [@Thess:Science273:483:(1996)], this involves 6 nearest neighbors with random chiralities. This causes an inhomogeneous broadening of the RBMs. However, the nearest-neighbor of an inner tube with a given chirality is an outer tube also with a well defined chirality. A given inner tube can be grown in several outer tubes with different diameters, however the chiralities of an inner-outer tube pair is always well defined, therefore the nearest neighbor interaction acting on an inner tube is also well-defined.
![Raman resonance profile for the (6,4) tubes in the SWCNT (CoMoCat) and DWCNT samples, $\blacksquare$: 80 K, $\bigcirc$: 300 K, $\blacktriangle$: 600 K. Solid curves show fits with the RRS theory. Dashed curve is a simulation for the 80 K SWCNT data with $\Gamma=10$ meV. Arrows indicate the incoming and outgoing resonance energies. Note the much narrower widths for the DWCNT sample. Reprinted figure with permission from Ref. [@SimonPRB2006], F. Simon *et al.* Phys. Rev. B **74**, 121411(R) (2006). Copyright (2006) by the American Physical Society. []{data-label="CACReview_DWCNTCOMOCAT_ERG_PROFILE"}](./CACReview_DWCNTCOMOCAT_ERG_PROFILE.eps){width="1.0\hsize"}
In addition to the long phonon life-times of inner tubes, the life-time of optical excitations, i.e. the life-time of the quasi-particle associated with the Raman scattering is unexpectedly long. To demonstrate this, we compare the resonant Raman scattering data for an inner tube and a SWCNT with the same chirality following Ref. [@SimonPRB2006]. In Fig. \[CACReview\_DWCNTCOMOCAT\_ERG\_PROFILE\] we show the energy profile of the resonant Raman scattering at some selected temperatures for two 6,4 tube modes: one is an inner tube in a DWCNT sample, the other is a SWCNT in a CoMoCat sample. Such energy profiles are obtained by taking an energy (vertical) cross section of a Raman map such as shown in Fig. \[CACReview\_DWCNTHIPCORamanMaps\]. The Raman intensities for a given excitation energy were obtained by fitting the spectra with Voigtian curves for the tube modes, whose Gaussian component accounts for the spectrometer resolution and whose Lorentzian for the intrinsic line-width. For the DWCNT sample, the strongest (6,4) inner tube component at 347 cm$^{-1}$ and for the SWCNT CoMoCat sample the (6,4) tube mode at 337 cm$^{-1}$ is shown. The temperature dependent resonant Raman data can be fitted with the conventional resonance Raman theory for Stokes Raman modes [@KuzmanyBook; @KuzmanyEPJB]:
$$\begin{aligned}
I(E_{\text{l}}) =M_{\text{eff}}^{4}\left\vert
\frac{\left(E_{\text{l}}-E_{\text{ph}}\right)^4\left(n_\text{BE}(E_{\text{ph}})+1\right)}{\left(
E_{\text{l}}-E_{\text{ii}}-i\Gamma \right) \left(
E_{\text{l}}-E_{\text{ph}}-E_{\text{ii}}-i\Gamma \right)
}\right\vert ^{2} \label{CACReview_Resonance_Raman}\end{aligned}$$
Here, the electronic density of states of SWCNTs is assumed to be a Dirac function and the effective matrix element, $M_{\text{eff}}$, describing the electron-phonon interactions is taken to be independent of temperature and energy. $E_{\text{l}}$, $E_{22}$ and $E_{\text{ph}}$ are the exciting laser, the optical transition and the phonon energies, respectively. $n_\text{BE}(E_{\text{ph}})=(\exp(
E_{\text{ph}}/\text{k}_{\text{B}}T)-1)^{-1}$ is the Bose-Einstein function and accounts for the thermal population of the vibrational state [@KuzmanyBook] and $n_\text{BE}(E_{\text{ph}})+1$ changes a factor $\sim$ 2 between 80 and 600 K. The temperature dependence of $E_{\text{ph}}$ is $\sim$ 1 % for the studied temperature range [@AjayanPRB2002] thus it can be neglected. The first and second terms in the denominator of Eq. \[CACReview\_Resonance\_Raman\] describe the incoming and outgoing resonances, respectively and are indicated on a simulated curve by arrows in Fig. \[CACReview\_DWCNTCOMOCAT\_ERG\_PROFILE\]. These are separated by $E_{\text{ph}}$. This means the apparent width of the resonance Raman data does not represent $\Gamma$.
Clearly, the resonance width is always smaller for the DWCNT than for the SWCNT sample. In other words, the life-time of the optically excited quasi-particle is longer lived for the DWCNT. The quasi-particle life-time is an important parameter for the application of carbon nanotubes in optoelectronic devices [@AvourisPRL2004; @AvourisPRL2005]. As a result, DWCNTs appear to be superior in this respect than their one-walled counterparts.
### Probing the SWCNT diameter distribution through inner tube growth
As discussed above, the Raman spectra of inner tubes have several advantages compared to that of the outer tubes: i) their RBMs have about a factor 2 times larger splitting due to the smaller diameters, ii) the line-widths are about 10 times narrower. The larger spectral splitting and narrower line-widths of the inner tube RBMs enable to characterize the inner tube diameter distribution with a spectral resolution that is about $20$ times larger as compared to the analysis on the outer tubes. To prove that studying the inner tubes can be exploited for the study of outer ones, here we show that there is a one-to-one correspondence between the inner and outer tube diameter distributions following Ref. [@SimonPRB2005].
![As measured Raman spectra of the inner nanotube RBMs for four DWCNT samples (lower curves in each quarter) at 647 nm laser excitation. The upper spectra (shown in red) are “smart-scaled” from the lower left spectrum. The Gaussian diameter distribution is shown for the DWCNT-L sample. Reprinted figure with permission from Ref. [@SimonPRB2005], F. Simon *et al.* Phys. Rev. B **71**, 165439 (2005). Copyright (2005) by the American Physical Society.[]{data-label="CACReview_FourSamples"}](./CACReview_FourSamples.eps){width="0.85\hsize"}
In Fig. \[CACReview\_FourSamples\], we compare the inner tube RBM Raman spectra for four different DWCNT materials based on SWCNTs with different diameters and produced with different methods. The SWCNTs were two arc-discharge grown SWCNTs (SWCNT-N1 and N2) and two laser ablation grown tubes (SWCNT-R and SWCNT-L). The diameter distributions of the SWCNT materials were determined from Raman spectroscopy [@KuzmanyEPJB] giving $d_{\text{N}1}=$ 1.50 nm, $\sigma _{\text{N1}}$ = 0.10 nm, $d_{\text{N}2}=$ 1.45 nm, $\sigma
_{\text{N}1}$ = 0.10 nm, $d_{\text{R}}=$ 1.35 nm, $\sigma
_{\text{R}}$ = 0.09 nm, and $d_{\text{L}}=$ 1.39 nm, $\sigma
_{\text{L}}$ = 0.09 nm for the mean diameters and the variances of the distributions, respectively.
The spectra shown are excited with a 647 nm laser that is representative for excitations with other laser energies. The RBMs of all the observable inner tubes, including the split components [@PfeifferPRL2003], can be found at the same position in all DWCNT samples within the $\pm $0.5 cm$^{-1}$ experimental precision of the measurement for the whole laser energy range studied. This proves that vibrational modes of DWCNT samples are robust against the starting material.
As the four samples have different diameter distributions, the overall Raman patterns look different. However, scaling the patterns with the ratio of the distribution functions (“smart-scaling”) allows to generate the overall pattern for all systems, starting from e.g. DWCNT-L in the bottom-left corner of Fig. \[CACReview\_FourSamples\]. It was assumed that the inner tube diameter distributions follow a Gaussian function with a mean diameter 0.72 nm smaller than those of the outer tubes following Ref. [@AbePRB2003] and with the same variance as the outer tubes. The empirical constants from Ref. [@KrambergerPRB2003] were used for the RBM mode Raman shift versus inner tube diameter expression. The corresponding Gaussian diameter distribution of inner tubes is shown for the DWCNT-L sample in Fig. \[CACReview\_FourSamples\]. A good agreement between the experimental and simulated patterns for the DWCNT-R sample is observed. A somewhat less accurate agreement is observed for the DWCNT-N1, N2 samples, which may be related to the different growth method: arc discharge for the latter, as compared to laser ablation for the R and L samples. The observed agreement has important consequences for the understanding of the inner tube properties. As a result of the photoselectivity of the Raman experiment, it proves that the electronic structure of the inner tubes is identical in the different starting SWCNT materials.
The scaling of the inner tube Raman spectra with the outer tube distribution shows that the inner tube abundance follows that of the outer ones. This agrees with the findings of x-ray diffractomery on DWCNTs [@AbePRB2003] and is natural consequence of the growth of inner tubes inside the outer tube hosts.
### Studying the reversible hole engineering using DWCNTs
Soon after the discovery of the peapods [@SmithNAT], it was recognized [@KatauraSM2001; @SmithCPL2000] that opening the SWCNTs by oxidation in air or by treating in acids is a prerequisite for good filling. Good filling means a macroscopic filling where the peapods are observable not only by local microscopic means such as HR-TEM but also by spectroscopy such as Raman scattering. On the other hand, a heat treatment around 1000 $^{\circ }$C was known to close the openings which results in a low or no fullerene filling. It was also shown that the geometrically possible maximum of filling can be achieved when purified SWCNTs were subject to a 450 $^{\circ
}$C heat treatment in flowing oxygen [@LiuPRB2002]. However, these studies have concerned the overall fullerene filling, with no knowledge on the precise dependence on the thermal treatment or tube diameter specificity.
The high diameter and chirality sensitivity of Raman spectroscopy for the inner tubes allows to study the behavior of tube openings when subject to different treatments. More precisely, openings which allow fullerenes to enter the tubes can be studied. This is achieved by studying the resulting inner tube RBM pattern when the outer tube host was subject to some closing or opening treatments prior to the fullerene encapsulation [@HasiJNN]. Annealing of as purchased or opened tubes was performed at various temperatures between 800 $^{\circ }$C and 1200 $^{\circ }$C in a sealed and evacuated quartz tube at a rest gas pressure of 10$^{-6}$ mbar. Opening of the tubes was performed by exposure to air at various temperatures between 350 $^{\circ }$C and 500 $^{\circ }$C.
![Raman spectra in the spectral range of the inner shell tube RBM for nanotubes after special pre-treatment. Bottom: after filling the tubes with C$_{60}$ and standard transformation; Center: after annealing the tubes at 1000 $^{\circ }$C, filling with C$_{60}$ and standard transformation; Top: after re-opening the annealed samples, filling with C$_{60}$, and standard transformation. All spectra recorded at 90 K and $\lambda=647$ nm. Insert: the RBM of the outer tubes before (a) and after (b) annealing at 1000 $^{\circ }$C. Reprinted figure with permission from Ref. [@HasiJNN], F. Hasi *et al.* J. Nanosci. Nanotechn. **5**, 1785 (2005). Copyright (2005) by the American Scientific Publishers.[]{data-label="CACReview_CloseOpen"}](./CACReview_CloseOpen.eps){width="0.8\hsize"}
Figure \[CACReview\_CloseOpen\] shows the Raman response of tubes after the standardized DWCNT transformation conditions but different pre-treatment. Only the spectral range of the inner tube is shown in the main part of the figure. The spectrum at the center was recorded under identical conditions but the SWCNT was pre-annealed before the standardized filling and standardized transformation. Almost no response from inner shell tubes is observed for this material, which means no fullerenes had entered the tubes: the tubes were very efficiently closed by the annealing process. The insert in Fig. \[CACReview\_CloseOpen\] depicts the RBM response from the outer tubes before and after annealing. The two spectra are almost identical, which proves that no outer tube coalescence had occurred at the temperature applied. The spectrum at the top in Fig. \[CACReview\_CloseOpen\] was recorded after reopening the annealed tubes at 500 $^{\circ }$C on air and standard filling and transformation. The spectra derived from the pristine and from the reopened tubes are identical in all details. This means no dramatic damages by cutting a large number of holes into the sidewalls have happened. Consequently, the sidewalls of the tubes remain highly untouched by the opening process. Thus, it is suggested that fullerenes enter the tubes through holes at the tube ends.
Growth mechanism of inner tubes studied by isotope labeling {#isotope_labeled}
-----------------------------------------------------------
The growth of inner tubes from fullerenes raises the question, whether the fullerene geometry plays an important role in the inner tube growth or it acts as a carbon source only. Theoretical results suggest the earlier possibility [@SmalleyPRL2002; @TomanekPRB]. In addition, it needs clarification whether carbon exchange occurs between the two tube walls. Here, we review $^{13}$C isotope labeled studies aimed at answering these two open questions. $^{13}$C is a naturally occurring isotope of carbon with 1.1 % abundance. In general, isotope substitution provides an important degree of freedom to study the effect of change in phonon energies while leaving the electronic properties unaffected. This has helped to unravel phenomena such as e.g. the phonon-mediated superconductivity [@BCS].
First, we discuss the inner tube growth from isotope labeled fullerenes [@SimonPRL2005], and second we present the growth of inner tubes from isotope labeled organic solvents [@SimonCPL2006].
Commercial $^{13}$C isotope enriched fullerenes with two different enrichment grades were used to grow isotope enriched inner tubes. Fullerene encapsulation [@KatauraSM2001] and inner tube growth was performed with the conventional methods [@BandowCPL2001]. This results in a compelling isotope engineered system: double-wall carbon nanotubes with $^{13}$C isotope enriched inner walls and outer walls containing natural carbon [@SimonPRL2005].
![Raman spectra of DWCNTs with natural carbon and $^{13}$C enriched inner tubes at 676 nm laser excitation and 90 K. The inner tube RBM (a) and D and G mode spectral ranges (b) are shown. Arrows and filled circles indicate the D (left) and G (right) modes corresponding to the inner and outer tubes, respectively. Reprinted figure with permission from Ref. [@SimonPRL2005], F. Simon *et al.* Phys. Rev. Lett. **95**, 017401 (2005). Copyright (2005) by the American Physical Society.[]{data-label="CACReview_13DWCNT_RBM"}](./CACReview_13DWCNT_RBM.eps){width="0.9\hsize"}
In Fig. \[CACReview\_13DWCNT\_RBM\]a, we show the inner tube RBM range Raman spectra for a natural DWCNT and two DWCNTs with differently enriched inner walls, 25 % and 89 %. These two latter samples are denoted as $^{13}$C$_{25}$- and $^{13}$C$_{89}$-DWCNT, respectively. The inner wall enrichment is taken from the nominal enrichment of the fullerenes used for the peapod production, whose value is slightly refined based on the Raman data. An overall downshift of the inner tube RBMs is observed for the $^{13}$C enriched materials accompanied by a broadening of the lines. The downshift is clear evidence for the effective $^{13}$C enrichment of inner tubes. The magnitude of the enrichment and the origin of the broadening are discussed below.
The RBM lines are well separated for inner and outer tubes due to the $\nu _{\text{RBM}}\propto 1/d$ relation and a mean inner tube diameter of $d \sim $ 0.7 nm [@AbePRB2003; @SimonPRB2005]. However, other vibrational modes such as the defect induced D and the tangential G modes strongly overlap for inner and outer tubes. Arrows in Fig. \[CACReview\_13DWCNT\_RBM\]b indicate a gradually downshifting component of the observed D and G modes. These components are assigned to the D and G modes of the inner tubes. The sharper appearance of the inner tube G mode, as compared to the response from the outer tubes, is related to the excitation of semiconducting inner tubes and metallic outer tubes [@PfeifferPRL2003; @SimonPRB2005].
The shifts for the RBM, D and G modes can be analyzed for the two grades of enrichment. The average value of the relative shift for these modes was found to be $\left( \nu _{0}-\nu \right) /\nu
_{0}=0.0109(3)$ and $0.0322(3)$ for the $^{\text{13}}$C$_{0.25}$- and $^{\text{13}}$C$_{0.89}$-DWCNT samples, respectively. Here, $\nu
_{0}$ and $\nu $ are the Raman shifts of the same inner tube mode in the natural carbon and enriched materials, respectively. In the simplest continuum model, the shift originates from the increased mass of the inner tube walls. This gives $\left( \nu _{0}-\nu
\right) /\nu _{0}=1-\sqrt{\frac{12+c_{0}}{12+c}}$, where $c$ is the concentration of the $^{13}$C enrichment on the inner tube, and $c_{0}=0.011$ is the natural abundance of $^{13}$C in carbon. The resulting values of $c$ are $0.277(7)$ and $0.824(8)$ for the 25 and 89 % samples, respectively.
![G’ spectral range of DWCNTs with natural carbon and $^{13}$C enriched inner walls with 515 nm laser excitation. Note the unchanged position of the outer tube G’ mode indicated by a vertical line.[]{data-label="CACReview_13DWCNT_Dst"}](./CACReview_13DWCNT_Dst.eps){width="0.8\hsize"}
The growth of isotope labeled inner tubes allows to address whether carbon exchange between the two walls occurs during the inner tube growth. In Fig. \[CACReview\_13DWCNT\_Dst\], we show the G’ spectral range for DWCNTs with natural carbon and $^{13}$C enriched inner walls with 515 nm laser excitation. The G’ mode of DWCNTs is discussed in detail in Ref. [@PfeifferEPJB2004]: the upper G’ mode component corresponds to the outer tubes and the lower to the inner tubes. The outer tube G’ components are unaffected by the $^{13}$C enrichment within the 1 cm$^{-1}$ experimental accuracy. This gives an upper limit to the extra $^{13}$C in the outer wall of 1.4 %. This proves that there is no sizeable carbon exchange between the two walls as this would result in a measurable $^{13}$C content on the outer wall, too.
The narrow RBMs of inner tubes and the freedom to control their isotope enrichment allows to precisely compare the isotope related phonon energy changes in the experiment and in *ab-initio* calculations. This was performed by J. Kürti and V. Zólyomi in Ref. [@SimonPRL2005]. The validity of the above simple continuum model for the RBM frequencies was verified by performing first principles calculations on the $(n,m)=(5,5)$ tube as an example. In the calculation, the Hessian matrix was determined by DFT using the Vienna Ab Initio Simulation Package [@KressePRB1999]. Then, a large number of random $^{13}$C distributions were generated and the RBM vibrational frequencies were determined from the diagonalization of the dynamical matrix for each individual distribution. It turns out that the calculation can account for the above mentioned broadening of the RBM lines due to the random distribution of the $^{12}$C and $^{13}$C nuclei [@SimonPRL2005].
The known characteristics of isotope labeled inner tubes allow to study the possibility of inner tube growth from non-fullerene carbon sources [@SimonCPL2006]. For this purpose, we chose organic solvents containing aromatic rings, such as toluene and benzene. These are known to wet the carbon nanotubes and are appropriate solvents for fullerenes. As described in the following, the organic solvents indeed contribute to the inner tube growth, however only in the presence of C$_{60}$ “stopper" molecules [@SimonCPL2006]. In the absence of co-encapsulated fullerenes the solvents alone give no inner tube.
The fullerene+organic solvents encapsulation was performed by dissolving typically 150 $\mu $g fullerenes in 100 $\mu $l solvents and then sonicating with 1 mg SWCNT in an Eppendorf tube for 1 h. The weight uptake of the SWCNT is $\sim $15 % [@SimonPRL2005] that is shared between the solvent and the fullerenes. The peapod material was separated from the solvent by centrifuging and it was then greased on a sapphire substrate. The solvent prepared peapods were treated in dynamic vacuum at 1250 $^{\circ}$C for 2 hours for the inner tube growth. The inner tube growth efficiency was found independent of the speed of warming.
![a) The G’ mode of toluene+C$_{60}$ peapod based DWCNTs with varying $^{13}$C enrichment at 515 nm laser excitation. From top to bottom: 74 %, 54 %, 26.5 % and natural $^{13}$C content. b) The G’ mode of the inner tubes after subtracting the experimental SWCNT spectrum. A small residual peak is observed around 2710 cm$^{-1}$ (denoted by an asterisk) due to the imperfect subtraction. Arrows indicate the spectral weight shifted toward lower frequencies. Reprinted figure with permission from Ref. [@SimonCPL2006], F. Simon and H. Kuzmany, Chem. Phys. Lett. **425**, 85 (2006). Copyright (2006) by Elsevier.[]{data-label="CACReview_varying13C"}](./CACReview_varying13C.eps){width="0.9\hsize"}
The growth of inner tubes from the solvents can be best proven by the use of C$_{60}$ containing natural carbon and a solvent mixture consisting of $^{13}$C enriched and natural carbon containing solvents with varying concentrations. Toluene was a mixture of ring $^{13}$C labeled ($^{13}$C$_{6}$H$_{6}$-$^{\text{NAT}}$CH$_{3}$) and natural toluene ($^{\text{NAT}}$C$_{7}$H$_{8}$). Benzene was a mixture of $^{13}$C enriched and natural benzene. The labeled site was $>$ 99 % $^{13}$C labeled for both types of molecules. The $^{13}$C content, $x$, of the solvent mixtures was calculated from the concentration of the two types of solvents and by taking into account the presence of the naturally enriched methyl-group for the toluene. In Fig. [CACReview\_varying13C]{}a, we show the G’ modes of DWCNTs with varying $^{13}$C labeled content in toluene+C$_{60}$ based samples and in Fig. \[CACReview\_varying13C\]b, we show the same spectra after subtracting the outer SWCNT component. A shoulder appears for larger values of $x$ on the low frequency side of the inner tube mode, whereas the outer tube mode is unchanged. Similar behavior was observed for the benzene+C$_{60}$ based peapod samples (spectra not shown) although with a somewhat smaller spectral intensity of the shoulder. The appearance of this low frequency shoulder is evidence for the presence of a sizeable $^{13}$C content in the inner tubes. This proves that the solvent indeed contributes to the inner tube formation as it is the only sizeable source of $^{13}$C in the current samples. The appearance of the low frequency shoulder rather than the shift of the full mode indicates an inhomogeneous $^{13}$C enrichment. A possible explanation is that smaller diameter nanotubes might be higher $^{13}$C enriched as they retain the solvent better than larger tubes.
![$^{13}$C content of inner tubes based on the first moment analysis as explained in the text as a function of $^{13}$C enrichment of benzene and toluene. Lines are linear fits to the data and are explained in the text. Reprinted figure with permission from Ref. [@SimonCPL2006], F. Simon and H. Kuzmany, Chem. Phys. Lett. **425**, 85 (2006). Copyright (2006) by Elsevier.[]{data-label="CACReview_13Cscaling"}](./CACReview_13Cscaling.eps){width="0.9\hsize"}
To quantify the $^{13}$C enrichment of the inner tubes, the downshifted spectral weight of the inner tube G’ mode was determined from the subtracted spectra in Fig. \[CACReview\_13Cscaling\]b. The subtraction does not give a flat background above 2685 cm$^{-1}$, however it is the same for all samples and has a small spectral weight, thus it does not affect the current analysis. The line-shapes strongly deviate from an ideal Lorentzian profile. Therefore the line positions cannot be determined by fitting, whereas the first moments are well defined quantities. The effective $^{13}$C enrichment of the inner tubes, $c$, is calculated from $\left( \nu _{0}-\nu \right) /\nu
_{0}=1-\sqrt{\frac{12+c_{0}}{12+c}}$, where $\nu _{0}$ and $\nu $ are the first moments of the inner tube G’ mode in the natural carbon and enriched materials, respectively, and $c_{0}=0.011$ is the natural abundance of $^{13} $C in carbon. The validity of this “text-book formula" is discussed above and it was verified by *ab-initio* calculations for enriched inner tubes in Ref. [@SimonPRL2005]. In Fig. \[CACReview\_13Cscaling\], we show the effective $^{13}$C content in the inner tubes as a function of the $^{13}$C content in the starting solvents. The scaling of the $^{13}$C content of the inner tubes with that in the starting solvents proves that the source of the $^{13}$C is indeed the solvents. The highest value of the relative shift for the toluene based material, $\left( \nu _{0}-\nu \right) /\nu _{0}=0.0041(2)$, corresponds to about 11 cm$^{-1}$ shift in the first moment of the inner tube mode. The shift in the radial breathing mode range (around 300 cm$^{-1}$) [@DresselhausTubesNew] would be only 1 cm$^{-1}$. This underlines why the high energy G’ mode is convenient for the observation of the moderate $^{13}$C enrichment of the inner tubes. When fit with a linear curve with $c_{0}+A\ast x$, the slope, $A$ directly measures the carbon fraction in the inner tubes that originates from the solvents.
The synthesis of inner tubes from organic solvent proves that any form of carbon that is encapsulated inside SWCNTs contributes to the growth of inner tubes. As mentioned above, inner tubes are not formed in the absence of fullerenes but whether the fullerene is C$_{60}$ or C$_{70}$ does not play a role. It suggests that fullerenes act only as a stopper to prevent the solvent from evaporating before the synthesis of the inner tube takes place. It also clarifies that the geometry of fullerenes do not play a distinguished role in the inner tube synthesis as it was originally suggested [@SmalleyPRL2002; @TomanekPRB]. It also proves that inner tube growth can be achieved irrespective of the carbon source, which opens a new prospective to explore the in-the-tube chemistry with other organic materials.
ESR studies on encapsulated magnetic fullerenes
-----------------------------------------------
Observation of the intrinsic ESR signal of pristine SWCNTs remains elusive [@NemesPRB2000; @SalvetatPRB2005]. Now, it is generally believed that intrinsic ESR of the tubes can not be observed as conduction electrons on metallic tubes are relaxed by defects too fast to be observable. In addition, one always observes a number of ESR active species in a sample, such as graphitic carbon or magnetic catalyst particles, which prevent a meaningful analysis of the signal. In contrast, local probe studies could still allow an ESR study of tubes, provided the local spin probe can be selectively attached to the tubes. This goal can be achieved by using magnetic fullerenes, such as e.g. N@C$_{60}$ or C$_{59}$N, since fullerenes are known to be selectively encapsulated inside SWCNTs [@SmithNAT] and can be washed from the outside using organic solvents [@KatauraSM2001]. As the properties and handling of the two magnetic fullerenes are quite different, the synthesis of the corresponding peapods and the results are discussed separately.
N@C$_{60}$ is an air stable fullerene [@WeidingerPRL] but decays rapidly above $\sim$ 200 $^{\circ}$C [@WaiblingerPRB] which prevents the use of the conventional vapor method of peapod preparation which requires temperatures above 400 $^{\circ}$C. To overcome this limitation and to allow in general the synthesis of temperature sensitive peapod materials, low temperature peapod synthesis (solvent method) was developed independently by four groups [@YudasakaCPL; @SimonCPL2004; @Monthioux2004; @BriggsJMC]. These methods share the common idea of mixing the opened SWCNTs with C$_{60}$ in a solvent with low fullerene solubility such as methanol [@YudasakaCPL] or n-pentane [@SimonCPL2004]. The encapsulation is efficient as it is energetically preferred for C$_{60}$ to enter the tubes rather than staying in the solution. After the solvent filling, excess fullerenes can be removed by sonication in toluene, which is a good fullerene solvent as it was found that fullerenes enter the tube irreversibly. HR-TEM has shown an abundant filling with the fullerenes [@YudasakaCPL; @Monthioux2004] and a more macroscopic characterization using Raman spectroscopy has proven that peapods prepared by the solvent method are equivalent to the vapor prepared peapods [@SimonCPL2004].
![X-band electron spin resonance spectrum of the a.) pristine SWCNT, b.) crystalline N@C$_{60}$:C$_{60}$, c.) (N@C$_{60}$:C$_{60}$)@SWCNT and d.) the triplet component of the (N@C$_{60}$:C$_{60}$)@SWCNT ESR spectrum at ambient temperature. Reprinted figure with permission from Ref. [@SimonCPL2004], F. Simon *et al.* Chem. Phys. Lett. **383**, 362 (2004). Copyright (2004) by Elsevier.[]{data-label="CACReview_lowTfillesr"}](./CACReview_lowTfillesr.eps){width="0.9\hsize"}
The low temperature synthesis allows to encapsulate the N@C$_{60}$ fullerene. The N@C$_{60}$:C$_{60}$ endohedral fullerene: fullerene solid solution can be produced in a N$_{2}$ arc-discharge tube following Ref. [@PietzakCPL] with a typical yield of a few 10 ppm [@JanossyKirch2000]. In Fig. \[CACReview\_lowTfillesr\]., the ESR spectra of the starting SWCNT, (N@C$_{60}$:C$_{60}$)@SWCNT, and N@C$_{60}$:C$_{60}$ are shown. The ESR spectrum of the pristine SWCNT for the magnetic field range shown is dominated by a signal that is assigned to some residual carbonaceous material, probably graphite. Fig. \[CACReview\_lowTfillesr\]c. shows, that after the solvent encapsulation of N@C$_{60}$:C$_{60}$ in the NCL-SWCNT, a hyperfine N triplet ESR is observed, similar to that in pristine N@C$_{60}$:C$_{60}$, superimposed on the broad signal present in the pristine nanotube material. Fig. \[CACReview\_lowTfillesr\]d. shows the triplet component of this signal after subtracting the signal observed in pristine SWCNT. The hyperfine triplet in N@C$_{60}$:C$_{60}$ is the result of the overlap of the $^{4}$S$_{3/2}$ state of the three 2p electrons of the N atom and the $^{14}$N nucleus, with nuclear spin, $I=1$. The isotropic hyperfine coupling of N@C$_{60}$:C$_{60}$ is unusually high as a result of the strongly compressed N atomic 2p$^{3}$ orbitals in the C$_{60}$ cage thus it unambiguously identifies this material [@WeidingerPRL]. The hyperfine coupling constant observed for the triplet structure in the encapsulated material, $A_{\text{iso}}=0.57\pm 0.01 $ mT, agrees within experimental precision with that observed in N@C$_{60}$:C$_{60}$ [@WeidingerPRL], which proves that the encapsulated material is (N@C$_{60}$:C$_{60}$)@SWCNT. The ESR line-width for the encapsulated material, $\Delta H_{pp}$ = 0.07 mT, is significantly larger than the resolution limited $\Delta H_{pp}$ =0.01 mT in the pristine N@C$_{60}$:C$_{60}$ material, the lines being Lorentzian. The most probable cause for the broadening is static magnetic fields from residual magnetic impurities in the SWCNT [@TangNMRSCI]. The ESR signal intensity is proportional to the number of N spins, and this allows the quantitative comparison of N concentrations in (N@C$_{60}$:C$_{60}$)@SWCNT and N@C$_{60}$:C$_{60}$. It was found that the number of observed N@C$_{60}$ spins is consistent with the number expected from a good filling efficiency [@SimonCPL2004].
As seen from the ESR results on encapsulated N@C$_{60}$, relatively limited information can be deduced about the tubes themselves. This stems from the fact that the N spins are well shielded in N@C$_{60}$ [@DinsePCCP] and are thus are relatively insensitive to the SWCNT properties. In contrast, N@C$_{60}$@SWCNT peapods might find another application as building elements of a quantum computer as proposed by Harneit *et al.* [@HarneitPSS]. A better candidate for magnetic fullerene encapsulation is the C$_{59}$N monomer radical as here the unpaired electron is on the cage and is a sensitive probe of the environment. This material can be chemically prepared [@WudlSCI], however it forms a non-magnetic dimer crystal (C$_{59}$N)$_{2}$. It appears as a spinless monomer in an adduct form [@WudlReview] or attached to surface dangling bonds [@PrassidesPRL]. The magnetic C$_{59}$N monomer radical is air sensitive but it can be stabilized as a radical when it is dilutely mixed in C$_{60}$[FulopCPL]{}. As a result, a different strategy has to be followed to encapsulate C$_{59}$N inside SWCNT, which is discussed in the following along with preliminary ESR results [@SimonCAR2006].
To obtain C$_{59}$N peapods, air stable C$_{59}$N derivatives, (C$_{59}$N-der in the following) were prepared chemically by A. Hirsch and F. Hauke following standard synthesis routes [@WudlReview; @HirschC59NReview]. The C$_{59}$N-der was 4-Hydroxy-3,5-dimethyl-phenyl-hydroazafullerene. The C$_{59}$N derivatives were encapsulated either pure or mixed with C$_{60}$ as C$_{59}$N-der:C$_{60}$ with 1:9 concentrations using a modified version of the low temperature encapsulation method. In brief, the mixture of the dissolved fullerenes and SWCNTs were sonicated in toluene and filtered. It is expected that the C$_{59}$N monomer radical can be obtained after a heat treatment in dynamic vacuum, which is discussed below.
Raman spectroscopy was performed to characterize the SWCNT filling with the C$_{59}$N-der [@SimonCAR2006]. The major Raman modes of the pristine C$_{59}$N-der are similar to those of the (C$_{59}$N)$_{2}$ dimer [KuzmanyPRB1999]{}. The strongest mode is observed at 1459.2 cm$^{-1}$ which is derived from the C$_{60}$ $A_{\text{g}}$(2) mode and is downshifted to 1457 cm$^{-1}$ after the encapsulation procedure. The 2.2 cm$^{-1}$ downshift proves the encapsulation of the molecule inside the SWCNT. When encapsulated inside SWCNTs, the corresponding A$_{\text{g}}$(2) mode of C$_{60}$ downshifts with 3 cm$^{-1}$ , which is assigned to the softening of the C$_{60}$ $A_{\text{g}}$(2) vibrational mode due to the interaction between the ball and the SWCNT wall [@PichlerPRL2001].
![Raman spectra of the encapsulated C$_{59}$N-der:C$_{60}$ mixture at the 488 nm laser excitation. The spectra for the C$_{59}$N-der and C$_{60}$ peapods is shown together with their weighted sum as explained in the text. A and B mark the components coming nominally from the superposing two phases. The asterisk marks a mode that is present in the pristine SWCNT material. Note the different scale for the C$_{59}$N-der peapod material. Reprinted figure with permission from Ref. [@SimonCAR2006], F. Simon *et al.* Carbon, **44**, 1958 (2006). Copyright (2006) by Elsevier.[]{data-label="CACReview_C59Ndermixedpeapodspectra"}](./CACReview_C59Ndermixedpeapodspectra.eps){width="0.9\hsize"}
The integrated intensity of the observed A$_{\text{g}}$(2) derived mode of the C$_{59}$N is approximately 5 times larger than that of a C$_{60}$ peapod prepared identically when normalized by the SWCNT G mode intensity. This, however, can not be used to measure the encapsulation efficiency as Raman intensities depend on the strength of the Raman resonance enhancement and the Raman scattering matrix elements [@KuzmanyBook]. For C$_{60}$ peapods the Raman signal was calibrated with independent and carbon number sensitive measurements: EELS studies gave the total number of C$_{60}$ related and non-C$_{60}$ related carbons [@LiuPRB2002] and the mass of encapsulated C$_{60}$s was determined from NMR studies using $^{13}$C enriched fullerenes [@SimonPRL2005; @SingerPRL2005]. In the current case, neither methods can be employed and we determined the filling efficiency for the azafullerene by encapsulating a mixture of the azafullerene and C$_{60}$. In Fig. \[CACReview\_C59Ndermixedpeapodspectra\], the Raman spectra of the encapsulated C$_{59}$N-der:C$_{60}$ mixture with weight ratios of 1:9 in the starting solvent is shown. The Raman spectrum of the encapsulated mixture was simulated with a weighted sum of the separately recorded spectra for encapsulated C$_{59}$N-der and C$_{60}$. The best agreement between the simulated and the experimental spectra is for a C$_{59}$N-der content of 0.12(2). This value is close to the expected value of 0.1 and it proves that the azafullerene enters the tubes with the same efficiency as C$_{60}$.
![ESR spectra of crystalline C$_{59}$N:C$_{60}$ (a) and (C$_{59}$N:C$_{60}$)@SWCNT obtained by annealing the (C$_{59}$N-der:C$_{60}$)@SWCNT. Solid curves show the deconvolution of the different ESR components for the encapsulated material. Reprinted figure with permission from Ref. [@SimonPRL2005], F. Simon *et al.* Phys. Rev. Lett. **97**, 136801 (2006). Copyright (2006) by the American Physical Society.[]{data-label="CACReview_C59N_ESR"}](./CACReview_C59N_ESR.eps){width="0.9\hsize"}
Fig. \[CACReview\_C59N\_ESR\] shows the room temperature ESR spectra of C$_{59}$N:C$_{60}$@SWCNT after 600 $^{\circ }$C vacuum annealing from Ref. [@SimonPRL2006]. The spectra of C$_{59}$N:C$_{60}$, a C$_{59}$N monomer embedded in C$_{60}$ [@FulopCPL], is also shown for comparison. This latter spectrum was previously assigned to the superposition of rotating C$_{59}$N monomers and bound C$_{59}$N-C$_{60}$ heterodimers [@RockenbauerPRL2005]. The large spin density at the $^{14}$N nucleus of the rotating C$_{59}$N molecule results in an ESR triplet signal and the C$_{59}$N-C$_{60}$ heterodimer has a singlet signal (arrow in Fig. \[CACReview\_C59N\_ESR\]) as the spin density resides on the C$_{60}$ molecule. $^{14}$N triplet structures are observed in the peapod samples with identical hyperfine coupling as in the crystalline sample and are thus identified as the ESR signals of rotating C$_{59}$N monomer radicals encapsulated inside SWCNTs. The additional component (arrow in Fig. \[CACReview\_C59N\_ESR\]) observed for sample B, which contains co-encapsulated C$_{60}$, is identified as C$_{59}$N-C$_{60}$ heterodimers encapsulated inside SWCNTs since this signal has the same $g$-factor as in the crystalline material. This singlet line is absent in sample A which does not contain C$_{60}$. For both peapod samples a broader line with HWHM of $\Delta H \sim 0.6$ mT is also observed. The broader component appears also on heat treatment of reference samples without encapsulated C$_{59}$N-der and is identified as a side-product. Annealing at 600 $^{\circ }$C is optimal: lower temperatures result in smaller C$_{59}$N signals and higher temperatures increase the broad impurity signal without increasing the C$_{59}$N intensity.
The observation of the ESR signal of C$_{59}$N related spins proves that after the 600 $^{\circ }$C heat treatment, a sizeable amount of rotating C$_{59}$N monomer radicals are present in the sample. This is not surprising in the view of the ability to form C$_{59}$N monomers from C$_{59}$N at similar temperatures [@SimonJCP], however the current process is not reversible and the remnants of the side-groups are probably removed by the dynamic pumping.
NMR studies on isotope engineered heteronuclear nanotubes
---------------------------------------------------------
The growth of the “isotope engineered" nanotubes, i.e. DWCNTs with highly enriched inner wall allows to study the electronic properties of small diameter carbon nanotubes with an unprecedented specificity using NMR. For normal SWCNTs, either grown from natural or $^{13}$C enriched carbon, the NMR signal originates from all kinds of carbon like amorphous or graphitic carbon.
![NMR spectra normalized by the total sample mass, taken with respect to the tetramethylsilane (TMS) shift. (a) Static spectrum for non-enriched SWCNT enlarged by 15. Smooth solid line is a chemical shift anisotropy powder pattern simulation with parameters published in the literature [TangNMRSCI]{}. (b) Static and (c) MAS spectra of $^{13}$C$_{0.89}$-DWCNT, respectively. Asterisks show the sidebands at the 8 kHz spinning frequency. Reprinted figure with permission from Ref. [@SimonPRL2005], F. Simon *et al.* Phys. Rev. Lett. **95**, 017401 (2005). Copyright (2005) by the American Physical Society.[]{data-label="CACReview_Static_NMR"}](./CACReview_Static_NMR.eps){width="0.9\hsize"}
NMR allows to determine the macroscopic amount of enriched tubes as it is sensitive to the number of $^{13}$C nuclei in the sample. In Fig. \[CACReview\_Static\_NMR\], we show the static and magic angle spinning spectra of $^{13}$C enriched DWCNTs, and the static spectrum for the SWCNT material. The mass fraction which belongs to the highly enriched phase can be calculated from the integrated signal intensity by comparing it to the signal intensity of the 89 % $^{13}$C enriched fullerene material. It was found that the mass fraction of the highly enriched phase relative to the total sample mass is 13(4) % which agrees with the expected value of 15 %. The latter is obtained from the SWCNT purity (50 %), $\sim $70 % volume filling for peapod samples [@LiuPRB2002], and the mass ratio of encapsulated fullerenes to the mass of the SWCNTs. This suggests that the NMR signal comes nominally from the inner tubes, and other carbon phases such as amorphous or graphitic carbon are non $^{13}$C enriched.
The typical chemical shift anisotropy (CSA) powder pattern was observed for the SWCNT sample in agreement with previous reports [TangNMRSCI,GozeBacCAR2002]{}. However, the static DWCNT spectrum cannot be explained with a simple CSA powder pattern even if the spectrum is dominated by the inner tube signal. The complicated structure of the spectrum suggests that the chemical shift tensor parameters are highly distributed for the inner tubes. It is the result of the higher curvature of inner tubes as compared to the outer ones: the variance of the diameter distribution is the same for the inner and outer tubes [@SimonDWCNTReview] but the corresponding bonding angles show a larger variation [@KuertiNJP2003]. In addition, the residual line-width in the MAS experiment, which is a measure of the sample inhomogeneity, is 60(3) ppm, i.e. about twice as large as the $\sim $35 ppm found previously for SWCNT samples [@TangNMRSCI; @GozeBacCAR2002]. The isotropic line position, determined from the MAS measurement, is 111$(2)$ ppm. This value is significantly smaller than the isotropic shift of the SWCNT samples of 125 ppm [@TangNMRSCI; @GozeBacCAR2002]. However, recent theoretical *ab-initio* calculations by F. Mauri and co-workers have successfully explained this anomalous isotropic chemical shift [@MauriPRB2006]. It was found that diamagnetic demagnetizing currents on the outer walls cause the diamagnetic shift of the inner tube NMR signal.
In addition to the line position, dynamics of the nuclear relaxation is a sensitive probe of the local electronic properties [@SlichterBook]. The electronic properties of the nanotubes was probed using the spin lattice relaxation time, $T_{1}$, defined as the characteristic time it takes the $^{13}$C nuclear magnetization to recover after saturation [@SingerPRL2005]. The signal intensity after saturation, $S(t)$, was deduced by integrating the fast Fourier transform of half the spin-echo for different delay times $t$. The data were taken with excitation pulse lengths $\pi/2
=$ 3.0 $\mu$s and short pulse separation times of $\tau=$ 15 $\mu$s [@SlichterBook]. The value of $T_{1}$ was obtained by fitting the $ t$ dependence of $S(t)$ to the form $ S(t) \ = \ S_{a} - S_{b}
\cdot M(t)$, where $S_{a} \simeq S_{b}$ ($>0$) are arbitrary signal amplitudes, and $$M(t) = \exp\left[-\left(t/T_{1}^{e}\right)^{\beta}\right],
\label{CACReview_stretched}$$ is the reduced magnetization recovery of the $^{13}$C nuclear spins. It was found that $M(t)$ does not follow the single exponential form with $\beta=1$, but instead fits well to a stretched exponential form with $\beta \simeq 0.65(5)$, implying a distribution in the relaxation times $T_{1}$. For a broad range of experimental conditions, the upper 90 % of the $M(t)$ data is consistent with constant $\beta \simeq 0.65(5)$, implying a field and temperature independent underlying distribution in $T_{1}$. The collapse of the data with constant $\beta = 0.65(5)$ is a remarkable experimental observation and it implies that each inner-tube in the powder sample has a different value of $T_{1}$, yet *all* the $T_{1}$ components and therefore all the inner-tubes follow the same $T$ and $H$ dependence within experimental uncertainty. This finding is in contrast to earlier reports in SWCNTs where $M(t)$ fits well to a bi-exponential distribution, 1/3 of which had a short $T_{1}$ value characteristic of fast relaxation from metallic tubes, and the remaining 2/3 had long $T_{1}$ corresponding to the semiconducting tubes [@TangNMRSCI; @GozeBacCAR2002; @ShimodaPRL2002; @KleinhammesPRB2003], as expected from a macroscopic sample of SWCNTs with random chiralities.
![Temperature dependence of spin-lattice relaxation rate divided by temperature, $ 1/T_{1}^{e}T$, in units of $(10^{3}\times$ s$^{-1}$ K $^{-1}$) for 3.6 T ($\bigcirc$) and 9.3 T ($\blacksquare$). Grey curves are best fits to Eq. (\[CACReview\_gapT1\]) with $2\Delta = 46.8 (40.2)$ K for $H= 3.6
(9.3)$ Tesla, respectively. Inset shows the suggested DOS with a small energy, $2\Delta$ secondary gap at the Fermi level of metallic inner tubes, which is displayed not to scale. Note the van Hove singularities (vHs) at $\pm\Delta$. Reprinted figure with permission from Ref. [@SingerPRL2005], P. M. Singer *et al.* Phys. Rev. Lett. **95**, 236403 (2005). Copyright (2005) by the American Physical Society.[]{data-label="CACReview_T1T_DOS"}](./CACReview_T1T_DOS.eps){width="1\hsize"}
The bulk average $T_{1}^{e}$ defined in Eq. (\[CACReview\_stretched\]) can be considered and its uniform $T$ and $H$ dependence can be followed. The $M(t)$ data can be fitted with the constant exponent $\beta=0.65(5)$, which reduces unnecessary experimental scattering in $T_{1}^{e}$. In Fig. \[CACReview\_T1T\_DOS\], we show the temperature dependence of $1/T_{1}^{e}T$ for two different values of the external magnetic field, $H$. The data can be separated into two temperature regimes; the high temperature regime $\gtrsim$ 150 K, and the low $T$ regime $\lesssim$ 150 K. At high temperatures, $1/T_{1}^{e}T$ is independent of $T$ which indicates a metallic state [@SlichterBook] for all of the inner tubes. A strong magnetic field dependence for $T_{1}$ was also observed, which was explained by a 1D spin diffusion mechanism for $T_{1}$ [@SingerPRL2005].
The experimentally observed uniform metallicity of inner tubes is a surprising observation. This was suggested to be caused by the shifting of the inner tube Fermi levels due to charge transfer between the two tube walls. Indeed, using *ab-initio* calculations Okada and Oshiyama have found that DWCNTs made of non-metallic zig-zag inner-outer tubes, such as the (7,0)@(16,0) DWCNT, are metallic [@OkadaPRL2003]. The direction of the charge transfer goes against the Faraday effect as inner tubes are electron and outer tubes are hole doped. Although, calculations are difficult if not impossible for an arbitrary inner-outer tube pair, this result confirms that two non-metallic tubes when producing an inner tube can render the electronic structure metallic.
The origin of the unusual $T$ dependence of 1/$T_{1}^{e}T$ in the low temperature regime ($\lesssim 150$ K) is peculiar. Some explanations can be ruled out as its origin. Firstly, one can rule out the possibility of an activation type mechanism where $T_{1}$ is dominated by fluctuating hyperfine fields with a characteristic time scale $\tau $ which increases with decreasing $T$ (i.e. glassy slowing). This would result in a peaked relaxation with a strongly field dependent peak value [@SlichterBook], which is clearly not the case. Furthermore at low $T$, 1/$T_{1}^{e}T$ drops below its high temperature value, which rules out the possibility of a $T $ independent component in 1/$T_{1}^{e}T$ plus an activated component on top. Secondly, the possibility of a simple $1/T $ Curie-like $T$ dependence in 1/$T_{1}^{e}T$ as a result of paramagnetic centers in the sample can be ruled out. This can be inferred from the pronounced gap in 1/$ T_{1}^{e}T$, together with the fact that no loss of $^{13}$C NMR signal intensity in the entire temperature range of the experiment was observed.
The simplest possible explanation for the experimental data is a non-interacting electron model of a 1D semiconductor with a small secondary gap (SG). The 1/$ T_{1}^{e}T$ data can be fitted using this model with only one parameter, the homogeneous SG, 2$\Delta $. The normalized form of the gapped 1D density-of-states $n(E)$ $$n(E)\ =\
\begin{cases}
\frac{E}{\sqrt{E^{2}-\Delta ^{2}}} & \text{for }|E|>\Delta \cr0 &
\text{ otherwise}\cr
\end{cases}
\label{CACReview_newDOS}$$ here, $E$ is taken with respect to the Fermi energy). The total DOS of an inner tube is shown schematically in the insert of Fig. \[CACReview\_T1T\_DOS\]. Eq. (\[CACReview\_newDOS\]) is used to calculate 1/$T_{1}^{e}T$ [@Moriya] as such $$\frac{1}{T_{1}^{e}T}\ =\ \alpha (\omega )\int_{-\infty }^{\infty
}n(E)n(E+\omega )\left( -\frac{\delta f}{\delta E}\right) dE,
\label{CACReview_gapT1}$$ where $E$ and $\omega$ are in temperature units for clarity, $f$ is the Fermi function $f = [\exp(E/T) + 1 ]^{-1}$, and the amplitude factor $\alpha(\omega)$ is the high temperature value for $1/T_{1}^{e}T$. The results of the best fit of the data to Eq. (\[CACReview\_gapT1\]) are presented in Fig. \[CACReview\_T1T\_DOS\], where $2\Delta = 43(3)$ K ($\equiv$ 3.7 meV) is $H$ independent within experimental scattering between 9.3 and 3.6 Tesla.
The origin of the experimentally observed gap still remains to be clarified. Tight binding calculations predict that applied magnetic fields can induce SG’s of similar magnitude for metallic SWCNT [@LuPRL1995]. However, such a scenario can be excluded due to the absence of any field dependence of the gap. The NMR data would be more consistent with a curvature induced SG for metallic tubes [@HamadaPRL1992; @KanePRL1997; @MintmirePRL1998; @ZolyomiPRB2004], however for the typical inner-tubes the predicted values, $\sim 100$ meV, are over an order of magnitude larger than our experimental data. Other scenarios, such as quantization of levels due to finite short lengths of the nanotubes could be considered as well, however, in all these cases a behavior independent of tube size and chirality is certainly not expected.
This suggests that electron-electron interactions may play an important role for the metallic inner tubes. It has been predicted that electron-electron correlations and a Tomonaga-Luttinger (TLL) state leads to an increase in 1/$T_{1}T$ with decreasing $T$ [@Yoshioka], which is a direct consequence of the 1D electronic state. The correlated 1D nature may also lead to a Peierls instability [@DresselhausTubes] with the opening of a small collective gap $2 \Delta$ and a sharp drop in 1/$T_{1}T$ below $\Delta \sim 20$ K. Therefore, the presence of both a TLL state and a Peierls instability could possibly account for the data.
Summarizing the NMR studies, it was shown that $T_{1}$ has a similar $T$ and $H$ dependence for all the inner-tubes, with no indication of a metallic/semiconducting separation due to chirality distribution. At high temperatures, ($T\gtrsim 150$ K) 1/$T_{1}^{e}T$ of the inner tubes exhibit a metallic 1D spin diffusion state. Below $\sim $150 K, 1/$T_{1}^{e}T$ increases dramatically with decreasing $T$, and a gap in the spin excitation spectrum is found below $ \Delta \simeq $ 20 K, which is suggested to be caused by a Peierls instability [@DresselhausTubes; @DresselhausTubesNew].
Summary
=======
In summary, we reviewed how in-the-tube functionalization of SWCNTs can be used to study various properties of the tubes themselves. Inner tubes grown from encapsulated fullerenes were shown to be an excellent probe of diameter dependent reactions on the outer tubes. Inner tubes grown from isotope labeled fullerenes and organic solvents allowed to understand the role of the different carbon phases in the growth of the inner tubes. In addition, isotope labeled inner tubes were shown to yield an unparalleled precision to study the density of states near the Fermi level using NMR. It was reviewed how magnetic fullerenes can be encapsulated inside SWCNTs yielding linear spin chains with sizeable spin concentrations and also to allow ESR studies of the tube properties.
Acknowledgements
================
This work was supported by the Austrian Science Funds (FWF) project Nr. 17345, by the Hungarian State Grants No. TS049881, F61733 and NK60984, by the EU projects BIN2-2001-00580 and MERG-CT-2005-022103, by the Zoltán Magyary and Bolyai postdoctoral fellowships. J. Bernardi and Ch. Schaman are acknowledged for the HR-TEM contribution and for preparing some of the figures, respectively.
Present address: Budapest University of Technology and Economics, Institute of Physics and Solids in Magnetic Fields Research Group of the Hungarian Academy of Sciences, H-1521, Budapest P.O.Box 91, Hungary
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present an in-depth analysis of the geometrical percolation behavior in the continuum of random assemblies of hard oblate ellipsoids of revolution. Simulations where carried out by considering a broad range of aspect-ratios, from spheres up to aspect-ratio 100 plate-like objects, and with various limiting two particle interaction distances, from 0.05 times the major axis up to 4.0 times the major axis. We confirm the widely reported trend of a consistent lowering of the hard particle critical volume fraction with the increase of the aspect-ratio. Moreover, assimilating the limiting interaction distance to a shell of constant thickness surrounding the ellipsoids, we propose a simple relation based on the total excluded volume of these objects which allows to estimate the critical concentration from a quantity which is quasi-invariant over a large spectrum of limiting interaction distances. Excluded volume and volume quantities are derived explicitly.'
author:
- Gianluca Ambrosetti
- Niklaus Johner
- Claudio Grimaldi
- Andrea Danani
- Peter Ryser
title: Percolative properties of hard oblate ellipsoids of revolution with a soft shell
---
Introduction
============
A central problem in materials science is the precise evaluation of the percolation threshold of random particle dispersions embedded in a continuous medium. This occurs typically in composite materials and is of importance for the prediction of relevant properties such as the electrical conduction in insulator-conducting composites. Practical examples include carbonaceous fillers like carbon fibers, graphite, carbon black, carbon nanotubes and fullerenes, but also metallic and ceramic ones, while matrices can be for instance polymeric, metallic or ceramic. Despite that the most studied particle form is the sphere, see e.g. [@Heyes2006; @Lee1998; @Wang1993; @Balberg1987; @Shante1971; @Scher1970], a broad range of fillers in real composites have forms which deviate consistently from the sphere. Previous investigations have considered different particle shapes like e.g. sticks [@Neda1999; @Bug1986; @Bug1985; @Balberg1984a; @Balberg1984b],wavy sticks [@Berhan2007b], plates [@Celzard1996; @Charlaix1986; @Charlaix1984] or ellipsoids [@Yi2004; @Garboczi1995; @Sevick1988; @Skal1974], but always in the fully penetrable case, where the particles are allowed to freely overlap. Only in few cases were hard sticks with a soft shell considered [@Schilling2007; @Berhan2007a; @Ogale1993; @Balberg1987], and a recent paper [@Akagawa2007] contemplated, as in the present study, the case of hard ellipsoids of revolution, but in the prolate domain.
The widespread use of composites containing fibrous fillers has made the stick, or other elongated objects, the favorite non-spherical shape in many studies. Nevertheless, some other fillers, notably graphite, have shapes which are better assimilable to flattened ellipsoids or platelets, and over a broad range of aspect-ratios i.e. longer dimension to shorter dimension ratios. The exploration of the relatively uncharted terrain of the percolative properties of oblate objects as a function of their aspect-ratio is then the aim of the present study.
In this paper we consider the special case of oblate *ellipsoids of revolution*, usually called (oblate) *spheroids*, which are ellipsoids with two equal (major) axes and may be obtained by rotation of a 2D ellipse around its minor axis. The reasons for this choice are twofold: first, spheroids are characterized by a smaller number of parameters (7 against 9 of the general ellipsoid); second, experimental measurement techniques of the filler particle size distributions are generally able to extract only major and minor dimensions, making it difficult to quantitatively define a size distribution for the third axis.
Our model is defined by a dispersion of impenetrable spheroids of identical dimensions with isotropic distribution of the symmetry axis orientation. Given any two spheroids, a connectivity criterion is introduced by allowing an upper cutoff distance beyond which the two spheroids are considered disconnected. More precisely, each spheroid is coated with a penetrable shell of constant thickness, and two particles are connected if their shells overlap. In a system of conducting spheroids dispersed in an insulating continuum host, the shell thickness can be physically interpreted as a typical tunneling length between the particles, governing the electrical connectivity of the composite.
To carry out our investigation we exploit a simulation algorithm, described in the following section, that allows to determine the percolation behavior of a random distribution of impenetrable spheroids as a function of their volume fraction, aspect-ratio, shell thickness and simulation cell size.
The Simulation Algorithm
========================
Even if the ellipsoid is a sufficiently simple geometrical form, much less simple is the construction of an algorithm which involves random distributions of them, since the computation of relevant functions like the particle inter-distance or the overlap criterion is far from trivial, as opposed to the sphere. Now, if we want to build an algorithm to carry out the proposed simulations, we first need a routine that generates a distribution of ellipsoids and that calculates the inter-distance between them. In order to do that we require two functions, an ellipsoid overlap criterion and the distance between two ellipsoids, the first being needed of course only if it can be computed in a time consistently shorter than the second. None of these functions allows simple closed form solutions, but some evaluation techniques are nevertheless available [@Donev2005; @Yi2004; @Garboczi1995; @Perram1985; @Vieillard1972]. We have chosen the approach proposed by Rimon and Boyd (R&B) [@Rimon1997; @Rimon1992] which was used for an obstacle collision detection procedure for robots, where short computational times are essential. The R&B technique allows two key benefits: 1) A quick estimation procedure of the distance between two ellipsoids that uses standard computation routines and that can be made sufficiently precise. 2) An overlap criterion between two ellipsoids as an intermediate result to the inter-distance computation that can be calculated in about half the time needed for the complete calculation. The computation is based on a formula for the distance of a point from an ellipsoid which reduces the problem to the calculation of the minimal eigenvalue of an auxiliary matrix constructed from the geometrical data.
We are now going to briefly outline how the distribution algorithm is constructed. First, a spheroid distribution is created inside a cubic cell of volume $L^3$ by random sequential addition: for every new particle, random placing is attempted and accepted as valid only if there is no overlap with any neighboring particle. To speed up the search for neighbors the main cubic cell is subdivided in discrete binning cells of size comparable to the major dimension of the spheroids. Moreover, to avoid unnecessary computations of the overlap function, cases are discarded which unavoidably lead to overlap or that can anyway not lead to overlap via simple geometrical rejection criteria. Periodic boundary conditions are imposed on the main cell. Second, the inter-particle distance is computed. An interaction inter-distance is chosen so that spheroids separated by a distance greater than it are considered non-interacting. Again, the same neighbor search and rejection criteria are used and, if it is the case, the distance computation is performed. To do this in an efficient way, a first R&B calculation is executed and if the resulting first distance estimate is clearly beyond the interaction range even when considering the worst possible error the calculation is stopped. Otherwise, the computation is continued by performing the R&B calculation with inverted spheroids, comparing with the first calculation and retaining the shorter of the two and finally reiterating part of the R&B procedure to obtain a further correction. This procedure leads to a distance estimate that has an average error of about +1 % on a wide range of distance to major spheroid dimension ratios (from $\textrm{10}^{\textrm{-4}}$ to 10), as obtained by comparing the R&B results with a more accurate but much slower distance evaluation procedure. Figure \[fig:SIM1\] shows one of such distributions as it appears by loading the algorithm output file in a viewer.
Once the desired distribution has been created and the neighboring particles inter-distances computed, the distribution algorithm output data are fed into the part of a program which isolates the connected cluster using a modified version of the Hoshen-Kopelman algorithm [@Johner2008; @Al-Futaisi2003; @Hoshen1976]. Finally, it is verified if the connected cluster spans two specific opposite sides of the simulation cell.
Simulation Results
==================
To explore the percolative properties of hard oblate spheroids surrounded with a penetrable shell of constant thickness, we considered spheroids with an aspect-ratio, i.e., spheroid major axis $a$ to minor (simmetry) axis $b$ ratio $a/b$ between $1$ (spheres) and $100$. The shell thickness $d$ to spheroid major axis ratio $d/a$ was chosen to variate between $0.05$ and $4.0$.
To extrapolate the percolation threshold from the simulation algorithm we followed finite-size scaling arguments as described in Ref. [@Rintoul1997], and briefly outlined below. For a given size $L$ of the cube, we obtained the spanning probability as a function of the spheroids volume fractions by recording the number of times a percolating cluster appeared over a given number of realizations. The resulting spanning probabilities were then plotted against the volume fraction and fitted with the sigmoidal function $$\label{eq:sigmoid}
f=\frac{1}{2}\bigg[1+\tanh{\bigg(\frac{\phi-\phi^{\textrm{eff}}_{c}}{\Delta}\bigg)}\bigg]\quad,$$ where $\phi^{\textrm{eff}}_{c}$ is the percolation threshold for a given value of $L$ and corresponds to the hard particle volume fraction at which the spanning probability is equal to $\frac{1}{2}$, while $\Delta$ represents the width of the percolation transition. Both $\phi^{\textrm{eff}}_{c}$ and $\Delta$ depend on the size $L$ of the system and, by following the scaling arguments of [@Rintoul1997], allow to deduct the percolation threshold $\phi_{c}$ for the infinite system through the following scaling relations: $$\begin{aligned}
\label{eq:scaling1}
&\Delta(L)\propto L^{-\frac{1}{\nu}},\\
\label{eq:scaling2} &\phi^{\textrm{eff}}_{c}(L)-\phi_{c}\propto
L^{-\frac{1}{\nu}},\end{aligned}$$ where $\nu$ is the correlation length exponent. By repeating the simulation procedure for different cell sizes it is possible, via the percolation transition widths $\Delta$ and the inversion of Eq. , to extract $\nu$ and consequently, from Eq. , the percolation threshold $\phi_{c}$ for $L=\infty$. We choose to simulate ten different cell sizes, $L=
10$, $13$, $15$, $17$, $20$, $23$, $25$, $27$, $30$ and $35$ times the major spheroid dimension, i.e., twice the major axis $a$. For thick shells ($d/a\geq 1.0$) the cell sizes were increased further. The spheroid number was in the order of thousands for the smallest cells up to about $70'000$ for the largest. The number of realizations per volume fraction step varied from $50$ for the smallest shell thickness up to $400$ for the thicker ones. Higher realization numbers did not show appreciable improvements. In all cases, the correlation length exponent $\nu$ had a value around 0.9, in good agreement with previous results on spheres [@Johner2008; @Lee1998; @Rintoul1997]. However, sometimes the fluctuations of the $\phi^{\textrm{eff}}_{c}$ where too large and a simple average of the results provided a more significative result that the one obtained from the finite size analysis.
In Fig. \[fig:SIM2\] we report the obtained spanning probability as a function of $\phi$ for $a/b=1$ and $a/b=2$ and for selected values of the cell size $L$. The shell thickness $d$ to major axis ratio was set equal to $d/a=0.1111$. From the figure it is clear that increasing the aspect-ratio from $a/b=1$ (spheres) to $a/b=2$ leads to a lowering of the percolating volume fraction. This trend is confirmed in Fig. \[fig:SIM02\], where the critical hard particle volume fraction $\phi_{c}$ is plotted as a function of $a/b$ and for several values of the penetrable shell thickness. For the thinnest shells we find that $\phi_c$ can be reduced by about one order of magnitude in going from $a/b=1$ up to $a/b=100$. This result is fully consistent with the frequently reported trend that assemblies of oblate objects with high aspect-ratios entail a lower percolation threshold. For example, several studies of graphite-polymer composites reported a consistent lowering of the electrical conductivity percolation threshold when very high aspect-ratio graphite nanosheets [@Lu2006; @Chen2003; @Celzard1996] or graphene flakes [@Stankovich2006] were used.
Besides $\phi_c$, another quantity characterizing the percolation threshold is the reduced critical density $\eta_c$ defined as $$\eta_{c}= \rho_{c}V_{d}=\phi_{c}\frac{V_{d}}{V},$$ where $\rho_c$ is the number density at percolation and $V_{d}$ is the total object volume, comprising the volume of the hard spheroid, $V$, plus that of the penetrable shell. $V_d$ is explicitly calculated in the appendix, see Eq. (\[eq:Vd\]). The behavior of $\eta_c$, plotted in Fig. \[fig:SIM3\] as a function of the penetrable shell thickness $d/a$ and for several aspect-ratios, accounts for the dependence of the percolation threshold on the geometry of the total object (hard-core plus penetrable shell). Indeed, for $d/a=4$ the reduced critical density is almost independent of the aspect-ratio $a/b$ while, for thinner penetrable shells, $\eta_c$ acquires a stronger dependence on $a/b$. This is due to the fact that, for large $d/a$ values, the form of the total object does not deviate much from that of a sphere, so that $\eta_c\simeq 0.34$ as for fully penetrable spheres. On the contrary, for smaller values of $d/a$, the geometry of the total object is more similar to that of an oblate ellipsoid, with a consequently stronger dependence of $\eta_c$ on the aspect-ratio.
\
Quasi-invariants at the Percolation Threshold {#sec:general SEE}
=============================================
In continuum percolation, an important quantity providing information on the local topology of the percolating cluster is the average number $B_c$ of objects connected to a given particle. For fully penetrable objects, and since in this case there is no spatial correlation, $B_c$ is simply given by [@Balberg1984a]: $$\label{eq:Bcspheres}
B_c=\rho_{c}V_{ex},$$ where $V_{ex}$ is the excluded volume defined by the volume around an object where the center of another object cannot penetrate if overlap is to be avoided. For penetrable spheres each of volume $V$, the excluded volume is $V_{ex}=8V$ and, by using $\rho_c=\eta_c/V$ with $\eta_c\simeq 0.34$, the resulting connectivity number is $B_c\simeq 2.74$, which agrees well with the evaluation of $B_c$ from a direct enumeration of connections in assemblies of penetrable spheres at percolation [@Balberg1987; @Heyes2006]. Indeed, for fully penetrable spheres, for which the sphere centers are distributed randomly, Eq. simply states that $B_c$ is equivalent to the average number of centers found within an excluded volume, irrespectively of the spatial configuration of the percolating objects. However, for semi-penetrable spheres, the presence of hard-core introduces a spatial correlation (see below), so that $B_c$ is expected to deviate from the uncorrelated case of Eq. . In particular, $B_c$ is found to decrease as the hard-core portion of the sphere increases, reaching $B_c\simeq 1.5$ for very thin penetrable shells [@Balberg1987; @Heyes2006], as a result of the repulsion of the impenetrable hard-cores.
Let us now consider the case of assemblies of oblate ellipsoids. In Fig. \[fig:SEE3\] we plot the computed values of $B_c$ as a function of the penetrable shell thickness $d/a$ and for selected values of the aspect-ratio $a/b$. For $a/b=1$ we recover the results for the spheres: $B_c\simeq 2.7$ for large values of $d/a$ while $B_c\simeq 1.5$ for $d/a=0.0526$. For $a/b>1$ and thick penetrable shells, $B_c$ remains close to the spherical case also for larger aspect-ratios because, as said before, for large $d/a$ values the entire object (hard-core plus penetrable shell) is basically a semi-penetrable sphere with a small hard-core spheroid. However, by decreasing $d/a$, we find that $B_c$ continues to remain very close to the $a/b=1$ case also for the thinnest penetrable shells, irrespectively of the aspect-ratio. This is well illustrated by the inset of Fig. \[fig:SEE3\] where the calculated $B_c$ for $d/a=0.0526$ does not show appreciable variations over a two-order of magnitude change of $a/b$. This result is rather interesting in view of the fact that a quasi-invariance of $B_c$ with respect to the aspect-ratio in oblate spheroids is in striking contrast to what is found in prolate objects such as the spherocylinders studied in Refs.[@Balberg1987; @Schilling2007]. For example, for spherocylinders made of hard cylinders on length $H$ and diameter $D$ capped by hemispheres and with penetrable shells of thickness $0.1D$, $B_c$ is found to decreases from $B_c=1.61$ for $H/D=4$ down to $B_c=1.29$ at $H/D=25$ [@Schilling2007], consistently deviating therefore from $B_c\simeq 1.76$ obtained for spheres of diameter $D$ and the same penetrable shell thickness [@Heyes2006]. Different behaviors of quasi-impenetrable oblate and prolate objects noted here are also found in the fully penetrable case. Indeed $B_c$ of prolate objects decreases as the aspect-ratio is increased and is expected to approach unity in the extreme prolate limit as a consequence of the vanishing critical density [@Bug1985], while $B_c$ of oblate objects remains close to $B_c\simeq 3$ all the way from the moderate- to extreme-oblate regimes [@Garboczi1995].
Now, we can write a general relation between the average connection number $B_c$ at percolation and the critical number density $\rho_c$. If we consider hard spheroids with penetrable shell and with an isotropic distribution of orientations, then $B_c$ reduces to: $$\label{eq:Bgeneral}
B_c=\rho_c \int_{0}^{2\pi} \mathrm{d}\theta \int_{0}^{\pi}\mathrm{d}\varphi \,\Phi(\theta,\varphi)\int_{V_{exd}(\theta,\varphi)}\!\!\!\!\mathrm{d}^{3}\mathbf{r}\, g(\mathbf{r},\theta,\varphi),$$ where $\theta$ and $\varphi$ are the angles between the major axes of two spheroids separated by $\mathbf{r}$ and $g(\mathbf{r},\theta, \varphi)$ is the radial distribution function: given a particle centered in the origin, $\rho_c\,g(\mathbf{r},\theta, \varphi)$ represents the mean particle number density at position $\mathbf{r}$ with an orientation $\theta, \varphi$. The integration in $\mathbf{r}$ is performed over the total excluded volume $V_{exd}(\theta,\varphi)$ (hard-core plus penetrable shell) centered at the origin and having orientation $\theta,\varphi$.
We observe that all the information about the presence of a hard core inside the particles is included in the radial distribution function, which will be zero in the volume occupied by the hard core of the particle centered at the origin. However, $g(\mathbf{r},\theta,\varphi)$ is a rather complex function and even for the case of spheres there are only approximate theoretical expressions [@Trokhymchuk2005]. Also the construction of a fitted expression to simulation data may result to be excessively complicated when the respective orientation of the particles has to be taken into account.
The lowest order approximation which we may then consider, and which is exact in the case of fully penetrable particles, is the one where $g(\mathbf{r},\theta,\varphi)=1$. This is equivalent to neglect all contributions of the radial distribution function which spur from the presence of the hard core. The resulting connectivity number, which in this approximation we denote by $\bar{B}_c$, is then given by: $$\label{Bbarra}
\bar{B}_c=\rho_c \int_{0}^{2\pi} \mathrm{d}\theta \int_{0}^{\pi}\mathrm{d}\varphi \,\Phi(\theta,\varphi)
\int_{V_{exd}(\theta,\varphi)}\!\!\!\!\mathrm{d}\mathbf{r}=\rho_c\langle V_{exd}\rangle,$$ where $\langle V_{exd}\rangle$ is the orientation averaged total excluded volume. Given the averaged excluded volume of spheroids surrounded with a shell of constant thickness $\langle
V_{exd}\rangle$ (\[eq:<Vexd>isotr.\]), together with the hard spheroid excluded volume expression (\[eq:<Vex>isotr.\]) or (\[eq:<Vex>I.O.W..\]), we can calculate $\bar{B}_c$ from the percolation threshold results obtained from the simulations: $$\label{Bbarra2}
\bar{B}_{c}=\rho_{c}\langle V_{exd}\rangle=\phi_{c}\frac{\langle V_{exd}\rangle}{V}\quad,$$ where we have used the hard core volume fraction $\phi_c$. The full details of the calculation of the excluded volume quantities can be found in the appendix \[sec:excluded volume quantities\]. The resulting values of $\bar{B}_c$ are plotted in Fig. \[fig:SEE4\] as a function of the penetrable shell thickness and for several aspect-ratios. Comparing Fig. \[fig:SEE4\] with Fig. \[fig:SEE3\] we note that for $d/a>1$, i.e., for thick penetrable shells, $\bar{B}_c$ does not deviate much from $B_c$, indicating that the effect of the hard-core is, in this regime, rather weak. On the contrary, for thinner shells, $\bar{B}_c$ increasingly deviates from $B_c$ because the correlation driven by the hard-core is stronger.
Despite that $\bar{B}_c$ overestimates the number of connected particles, its behavior is nevertheless rather intriguing. Indeed the dependence of $\bar{B}_c$ on the penetrable shell thickness $d/a$ appears to be universal with respect to the aspect-ratio, for all $d/a$ values larger than $d/a>0.1$. Furthermore, in this region of $d/a$, $\bar{B}_c$ has a rather weak dependence on the shell thickness, not deviating much from $\bar{B}_c\simeq 2.8$. This must be contrasted to $B_c$ which, from $d/a=4$ down to $d/a=0.1$, decreases from about $2.8$ to only $1.6$. The quasi-invariance of $\bar{B}_c$ has therefore some practical advantages since, by using Eq. , the percolation threshold $\phi_c$ can be estimated from $\langle V_{exd}\rangle$ and $\bar{B}_c\simeq 2.8$, in a wide interval of $d/a$ and aspect-ratio values.
Conclusions
===========
The geometrical percolation threshold in the continuum of random distributions of oblate hard ellipsoids of revolution surrounded with a soft shell of constant thickness has been investigated. Simulation results spanning a broad range of aspect-ratios and shell thickness values have been reported. It is found that larger aspect-ratios entail lower percolation thresholds, in agreement with the behavior observed experimentally in insulator-conductor composites where the conducting phase is constituted by oblate objects, such as graphite nanosheets. Furthermore, the number $B_c$ of connected object at percolation is a quasi-invariant with respect to the aspect-ratio, in contrast with what has been previously reported for prolate objects. Finally, we have derived an additional quasi-invariant based on the excluded volume concept which allows to infer the system percolation threshold.
Acknowledgments
===============
This study was supported by the Swiss Commission for Technological Innovation (CTI) through project GraPoly, (CTI grant 8597.2), a joint collaboration led by TIMCAL Graphite & Carbon SA. Simulations were performed at the ICIMSI facilities with the help of Eric Jaminet. Data analysis was carried out with the help of Ermanno Oberrauch. Figures \[fig:EV2\] and \[fig:EV3\] are due to Raffaele Ponti. Comments by I. Balberg were greatly appreciated.
Evaluation of Excluded Volume Quantities {#sec:excluded volume quantities}
========================================
We report in the following the derivation of the excluded volume of two oblate spheroids, the excluded volume of two oblate spheroids surrounded with a shell of constant thickness and their angular averages. We follow a route due to the pioneering work of Isihara [@Isihara1951] which is somehow more laborious than the one used by the same author [@Isihara1950] and the authors of [@Ogston1975] to derive the widely used Isihara-Ogston-Winzor spheroid excluded volume formula. The advantage is that it is possible to obtain, albeit in a series expansion form, the excluded quantities with their full angle dependence. The average on the spheroid angle distribution function is performed in a second time and can be easily extended to non-isotropic cases. Let us consider the case of two identical spheroids of major axis $a$ and minor axis $b$ in contact as illustrated in Fig. \[fig:EV2\].
We then have that the geometrical quantities $H$ and $K$, which represent the distances from the spheroid centers to the tangent plane to the two spheroids in the contact point, may be written as $$\label{eq:H3}H(\alpha)=a\sqrt{1-\epsilon^2\cos^{2}{\alpha}},$$ $$\label{eq:K1}
K(\alpha')=a\sqrt{1-\epsilon^2\cos^{2}{\alpha'}},$$ where $\epsilon$ represents the *eccentricity* (for oblate spheroids) $$\label{eq:eccentricity}
\epsilon\equiv\sqrt{1-\frac{b^{2}}{a^{2}}}.$$ Furthermore, we have $$\begin{aligned}
\label{eq:cosalphaprime}
\cos^{2}{\alpha'}&=[\sin{\varphi}\sin{\alpha}(\cos{\theta}cos{\beta}
+\sin{\theta}\sin{\beta})+\cos{\varphi}\cos{\alpha}]^{2}\nonumber \\
&=[\sin{\varphi}\sin{\alpha}\cos{(\beta-\theta)}
+\cos{\varphi}\cos{\alpha}]^{2},\end{aligned}$$ where $\theta$ and $\varphi$ are the angles which define the rotation that transforms the symmetry axis vector $\mathbf{b}$ of spheroid A in the one of B, $\mathbf{b}'$.
We can then write the excluded volume of two identical spheroids, or more generally two identical ovaloids, as [@Isihara1951; @Isihara1950]: $$\label{eq:Vex1b}
V_{ex}=2V+\int K(H,H)\textrm{d}\omega=2V+\int_{0}^{2\pi}\textrm{d}\beta\int_{0}^{\pi}\textrm{d}\alpha\sin{\alpha}K(H,H),$$ where $\textrm{d}\omega$ is the infinitesimal surface element of the unit sphere centered in the origin which, by using the reference frame choice of fig. \[fig:EV2\], takes the form $$\label{eq:domega}
\textrm{d}\omega=\sin{\alpha}\textrm{d}\alpha\textrm{d}\beta.$$ Furthermore, in Eq. (\[eq:Vex1b\]) we have introduced the differential operator on the unit sphere which for two equal scalar quantities $F$ takes the form $$\label{eq:diffop}
(F,F)\equiv2\Bigg\{\Bigg(\frac{\partial^{2}F}{\partial\alpha^{2}}+F\Bigg)\Bigg(\frac{1}{\sin^{2}{\alpha}}\frac{\partial^{2}F}{\partial\beta^{2}}+
\frac{\cos{\alpha}}{\sin{\alpha}}\frac{\partial F}{\partial\alpha}+F\Bigg)-\Bigg[\frac{\partial}{\partial\alpha}\Bigg(\frac{1}{\sin{\alpha}}\frac{\partial F}{\partial\beta}\Bigg)\Bigg]^{2}\Bigg\},$$ while $V$ is the volume of the spheroid.
With the explicit form of $H$, Eq. (\[eq:H3\]), $K$, Eq. (\[eq:K1\]), and relation (\[eq:cosalphaprime\]) we can write for the excluded volume (\[eq:Vex1b\]) in the case of the two spheroids the integral form: $$\begin{aligned}
\label{eq:Vex2}
V_{ex}(\theta,\varphi)&=2V+2a^{3}(1-\epsilon^{2})\int_{0}^{2\pi}
\textrm{d}\beta\int_{0}^{\pi}\textrm{d}\alpha\sin{\alpha}\frac{ \sqrt{1-\epsilon^{2}\cos^{2}{\alpha'}}}{(1-\epsilon^{2}\cos^{2}{\alpha})^{2}}\nonumber \\
&=2V+2a^{3}(1-\epsilon^{2})\underbrace{\int_{0}^{2\pi}\textrm{d}
\beta\int_{0}^{\pi}\textrm{d}\alpha\sin{\alpha}\frac{ \sqrt{1-\epsilon^{2}(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}}}
{(1-\epsilon^{2}\cos^{2}{\alpha})^{2}}}_{I},\end{aligned}$$ where we have used the fact that $$\label{eq:notheta}
\int_{0}^{2\pi}\mathrm{d}\beta\sqrt{1-\epsilon^{2}[\sin{\varphi}\sin{\alpha}\cos{(\beta-\theta)}
+\cos{\varphi}\cos{\alpha}]^{2}} =\int_{0}^{2\pi}\mathrm{d}\beta\sqrt{1-\epsilon^{2}
(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}},$$ because of the $2\pi$ periodicity of the integrand and which means that $V_{ex}$ is $\theta$-independent.
We now may expand the $1-\epsilon^{2}(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}$ square root: $$\begin{aligned}
\label{eq:sqrt exp}
\sqrt{1-\epsilon^{2}(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}}
&=1-\frac{1}{2\sqrt{\pi}}\sum_{k=1}^{\infty}\Gamma(k-\frac{1}{2})\frac{\epsilon^{2k}}{k!}
(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2k}\nonumber \\
&=1-\frac{1}{2\sqrt{\pi}}\sum_{k=1}^{\infty}\Gamma(k-\frac{1}{2})\frac{\epsilon^{2k}}{k!}
\sum_{i=0}^{k}\binom{k}{i}(\sin{\varphi}\sin{\alpha}\cos{\beta})^{2i}
(\cos{\varphi}\cos{\alpha})^{2k-2i}.\end{aligned}$$ Substituting this in integral $I$ of (\[eq:Vex2\]) and integrating in $\beta$ in the first resulting term we obtain: $$\begin{aligned}
\label{eq:int}
I=& 2\pi\int_{0}^{\pi}\textrm{d}\alpha\frac{\sin{\alpha}}{(1-\epsilon^{2}\cos^{2}{\alpha})^{2}}\nonumber \\
&-\frac{1}{2\sqrt{\pi}}\sum_{k=1}^{\infty}\Gamma(k-\frac{1}{2})\frac{\epsilon^{2k}}{k!}
\sum_{i=0}^{k}\binom{k}{i}\sin^{2i}{\varphi}\cos^{2k-2i}{\varphi}
\int_{0}^{2\pi}\textrm{d}\beta\cos^{2i}{\beta}\int_{0}^{\pi}\textrm{d}\alpha\frac{ \sin^{2i+1}{\alpha}\cos^{2k-2i}{\alpha}}{(1-\epsilon^{2}\cos^{2}{\alpha})^{2}}.\end{aligned}$$ The integration follows then with the aid of formulas 2.153 (3.), 3.682, 3.681 (1.) of [@Gradshteyn2000] obtaining with (\[eq:Vex2\]) the expression for the *excluded volume of two identical oblate spheroids*: $$\begin{aligned}
\label{eq:VexFinal}
V_{ex}(\varphi)=&2V+2a^{3}(1-\epsilon^{2})\biggl[4\pi F(\scriptstyle 2 \,,\,\frac{1}{2}\,,\,\frac{3}{2}\,,\,\epsilon^{2}\textstyle)\nonumber \\
&-\sqrt{\pi}\sum_{k=1}^{\infty}\Gamma( k-\frac{1}{2})\epsilon^{2k}
\sum_{i=0}^{k}\frac{\sin^{2i}{\varphi}\cos^{2k-2i}{\varphi}}{2^{i}(k-i)!(i!)^{2}}
B(\scriptstyle i+1\,,\,\frac{2k-2i+1}{2}\textstyle)
F(\scriptstyle 2\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)\biggr].\end{aligned}$$
\
Let us now consider the situation depicted in fig. \[fig:EV3\] which represent two (identical) spheroids surrounded with a shell of constant thickness $d$. We are again interested in evaluating the excluded volume of these objects, which, because of the constant shell offset, will not be anymore ellipsoids. Nevertheless, we see that in this case we can again construct geometrical quantities like $H$ and $K$ of the two spheroids of Fig. \[fig:EV2\] and that these, which we will call $H'$ and $K'$ are parallel to the old $H$ and $K$ respectively. Then it follows: $$\begin{aligned}
\label{eq:Hprime}
H'(\alpha)&=H(\alpha)+d \\
K'(\alpha')&=K(\alpha')+d,\end{aligned}$$ and $H$ and $K$ will be given by (\[eq:H3\]) and (\[eq:K1\]). Now, also in this case expression (\[eq:Vex1b\]) holds true and, observing that the volume of an ovaloid may be written as [@Isihara1950; @Isihara1951] $$\label{eq:Vovaloid}V=\frac{1}{6}\int G(G,G)\mathrm{d}\omega,$$ where $G$ is a geometric quantity constructed like $H, K, H', K'$, we are able to write for the excluded volume of the two spheroids with shell: $$\begin{aligned}
\label{eq:Vexd}
V_{exd}=&2V'+\int K'(H',H')\mathrm{d}\omega\nonumber \\
=&V_{ex}+\frac{4d}{3}\underbrace{\int (H,H)\mathrm{d}\omega}_{I_{1}}+
2d\underbrace{\int\bigg(\frac{H}{3}+\frac{4d}{3}\bigg)
\bigg(\frac{\partial^{2}H}{\partial\alpha^{2}}+\frac{\cos{\alpha}}{\sin{\alpha}}\frac{\partial H}{\partial\alpha}+2H+d\bigg)\mathrm{d}\omega}_{I_{2}}\nonumber \\
&+2d\underbrace{\int K\bigg(\frac{\partial^{2}H}{\partial\alpha^{2}}
+\frac{\cos{\alpha}}{\sin{\alpha}}\frac{\partial H}{\partial\alpha}
+2H+d\bigg)\mathrm{d}\omega}_{I_{3}},\end{aligned}$$ and $V_{ex}$ is the excluded volume of the two spheroids (\[eq:VexFinal\]).
Integrals $I_{1}$ and $I_{2}$ are straightforward and may be solved with the aid of formulas 3.682, 2.583 (3.), 2.584 (3.) and 2.584 (39.) of [@Gradshteyn2000]: $$I_{1}=2a^{2}(1-\epsilon^{2})\int_{0}^{2\pi}\textrm{d}\beta\int_{0}^{\pi}\textrm{d}\alpha\frac{ \sin{\alpha}}{(1-\epsilon^{2}\cos^{2}{\alpha})^{2}}=8\pi a^{2}(1-\epsilon^{2})
F(\scriptstyle 2\,,\,\frac{1}{2}\,,\,\frac{3}{2}\,,\,\epsilon^{2}\textstyle),$$
$$\begin{aligned}
\label{eq:I2b}
I_{2}&=\int_{0}^{2\pi}\textrm{d}\beta\int_{0}^{\pi}\textrm{d}
\alpha\sin{\alpha}\bigg(\frac{a\sqrt{1-\epsilon^{2}\cos^{2}{\alpha}}}{3}+\frac{4d}{3}\bigg)
\bigg[\frac{a}{\sqrt{1-\epsilon^{2}\cos^{2}{\alpha}}}+\frac{a(1-\epsilon^{2})}{(1-\epsilon^{2}\cos^{2}
{\alpha})^{\frac{3}{2}}}+d\bigg]\nonumber \\
&=\frac{4\pi}{3}(a^{2}+4d^{2})+6\pi ad\bigg(\sqrt{1-\epsilon^{2}}+\frac{\arcsin{\epsilon}}{\epsilon}\bigg)
+\frac{4\pi}{3}a^{2}(1-\epsilon^{2})\frac{\textrm{arctanh}\,\epsilon}{\epsilon}.\end{aligned}$$
Regarding $I_{3}$ we have, using Eq. (\[eq:K1\]), Eq. (\[eq:cosalphaprime\]) and Eq. (\[eq:notheta\]): $$\begin{aligned}
I_{3}=&a\int_{0}^{2\pi}\!\!\!\textrm{d}\beta\int_{0}^{\pi}\!\!\textrm{d}\alpha
\sin{\alpha}\sqrt{1-\epsilon^{2}(\sin{\varphi}\sin{\alpha}
\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}}\nonumber \\
&\times\bigg[\frac{a}{\sqrt{1-\epsilon^{2}\cos^{2}{\alpha}}}+
\frac{a(1-\epsilon^{2})}{(1-\epsilon^{2}\cos^{2}
{\alpha})^{\frac{3}{2}}}+d\bigg],\end{aligned}$$ and we can again expand the $1-\epsilon^{2}(\sin{\varphi}\sin{\alpha}\cos{\beta}+\cos{\varphi}\cos{\alpha})^{2}$ square root obtaining $$\begin{aligned}
I_{3}=&a\int_{0}^{2\pi}\!\!\!\textrm{d}\beta\int_{0}^{\pi}\!\!\textrm{d}\alpha
\sin{\alpha}\bigg[\frac{a}{\sqrt{1-\epsilon^{2}\cos^{2}{\alpha}}}+
\frac{a(1-\epsilon^{2})}{(1-\epsilon^{2}\cos^{2}
{\alpha})^{\frac{3}{2}}}+d\bigg]\nonumber \\
&-\frac{a}{2\sqrt{\pi}}\sum_{k=1}^{\infty}\Gamma(k-\frac{1}{2})\frac{\epsilon^{2k}}{k!}
\sum_{i=0}^{k}\binom{k}{i}\sin^{2i}{\varphi}\cos^{2k-2i}{\varphi}
\int_{0}^{2\pi}\!\!\!\textrm{d}\beta\cos^{2i}{\beta}\nonumber \\
&\times\Bigg[a\!\int_{0}^{\pi}\!\!\mathrm{d}\alpha\frac{\sin^{2i+1}{\alpha}
\cos^{2k-2i}{\alpha}}{\sqrt{1-\epsilon^{2}\cos^{2}{\alpha}}}+
a(1-\epsilon^{2})\!\int_{0}^{\pi}\!\!\mathrm{d}\alpha
\frac{\sin^{2i+1}{\alpha}\cos^{2k-2i}{\alpha}}{(1-\epsilon^{2}\cos^{2}
{\alpha})^{\frac{3}{2}}}+d\!\int_{0}^{\pi}\!\!\mathrm{d}
\alpha\sin^{2i+1}{\alpha}\cos^{2k-2i}{\alpha}\Bigg].\end{aligned}$$ These integrals may be solved again with the use of the formulas 2.153 (3.), 3.682, 3.681 (1.), 2.583 (3.), 2.584 (39.) and 3.621 (5.) of [@Gradshteyn2000], yielding $$\begin{aligned}
\label{eq:I3b}
I_{3}=&4\pi a\bigg(\frac{a\arcsin{\epsilon}}{\epsilon}+a\sqrt{1-\epsilon^{2}}+2d\bigg)
-a\sqrt{\pi}\sum_{k=1}^{\infty}\Gamma(k-\frac{1}{2})\epsilon^{2k}
\sum_{i=0}^{k}\frac{\sin^{2i}{\varphi}\cos^{2k-2i}{\varphi}}{2^{i}(k-i)!(i!)^{2}}\nonumber \\
&\times B(\scriptstyle i+1\,,\,\frac{2k-2i+1}{2}\textstyle)
\bigg[a F(\scriptstyle \frac{1}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+a(1-\epsilon^{2})
F(\scriptstyle \frac{3}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+d \bigg].\end{aligned}$$ We can then combine all these results together with property [@Maple] $$F(\scriptstyle 2\,,\,\frac{1}{2}\,,\,\frac{3}{2}\,,\,\epsilon^{2}\textstyle)=
\frac{1}{2}\bigg(\frac{1}{1-\epsilon^{2}}+\frac{\textrm{arctanh}\,\epsilon}{\epsilon}\bigg)$$ and Eq. (\[eq:Vexd\]) to write the *excluded volume of two oblate spheroids surrounded with a shell of constant thickness* $V_{exd}$: $$\begin{aligned}
\label{eq:VexdFinal}
V_{exd}=&V_{ex}+\frac{8\pi}{3} d(3a^{2}+4d^{2}+3ad)+4\pi ad(2a+3d)\bigg(\sqrt{1-\epsilon^{2}}+\frac{\arcsin{\epsilon}}{\epsilon}\bigg)\nonumber \\
&+8\pi a^{2}d(1-\epsilon^{2})\frac{\textrm{arctanh}\,\epsilon}{\epsilon}
-2ad\sqrt{\pi}\sum_{k=1}^{\infty}\Gamma{(k-\frac{1}{2})}\epsilon^{2k}
\sum_{i=0}^{k}\frac{\sin^{2i}{\varphi}\cos^{2k-2i}{\varphi}}{2^{i}(k-i)!(i!)^{2}}\nonumber \\
&\times B(\scriptstyle i+1\,,\,\frac{2k-2i+1}{2}\textstyle)
\bigg[a F(\scriptstyle \frac{1}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+a(1-\epsilon^{2})
F(\scriptstyle \frac{3}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+d \bigg].\end{aligned}$$ We note that the above procedure allowed to obtain an expression for $V_{exd}$ with an angular dependence only upon $\varphi$. However, the orientation of the surface enclosing this volume will be dependent also on $\theta$, which is why it is needed e.g. in (\[eq:Bgeneral\]).
The above results can also be easily used to compute the *total volume of the spheroid with the shell* starting from Eq. (\[eq:Vovaloid\]) with Eq. (\[eq:Hprime\]): $$V_{d}=V+\frac{d}{6}\int (H,H)\mathrm{d}\omega+
\frac{d}{3}\int(H+d)\bigg(\frac{\partial^{2}H}{\partial\alpha^{2}}+\frac{\cos{\alpha}}{\sin{\alpha}}\frac{\partial H}{\partial\alpha}+2H+d\bigg)\mathrm{d}\omega,$$ which is very similar to the first part of Eq. (\[eq:Vexd\]) and can be integrated alike, obtaining $$\label{eq:Vd}
V_{d}=V+\frac{2\pi d}{3}\bigg[3a^{2}(1-\epsilon^{2})\frac{\textrm{arctanh}\,\epsilon}{\epsilon}+3ad\bigg(\sqrt{1-\epsilon^{2}}+
\frac{\arcsin{\epsilon}}{\epsilon}\bigg)+3a^{2}+2d^{2}\bigg].$$
We now want to calculate the averaged excluded volume starting from the angle distribution functions which arise in the spheroid distributions of the simulation algorithm. For axially symmetric objects the angle distribution function $\Phi(\varphi)$ is dependent only on the angle between the symmetry axes $\varphi$. In the case of an isotropic (or Poissonian) angle distribution, where any orientation is equally probable, it is easy to find $$\label{eq:PhiIsotr}
\Phi_{isotr.}(\varphi)=\frac{\sin{\varphi}}{4\pi}.$$ To verify that this situation occurs unbiasedly in the simulations we used a modified version of the spheroid distribution creation algorithm where, after the distribution was realized, for every spheroid it was searched for neighbors which lied within a certain radius from its center and the angles between their symmetry axis were recorded. We then fitted the function to the simulated angle distribution results and, although we may expect that this distribution function will deviate from the purely isotropic case when highly packed assemblies are realized due to local orientation, we obtained no deviation for all binning radiuses and all volume fractions considered in the present research. The averaged excluded volume of the two spheroids will then be $$\langle V_{ex}\rangle_{isotr.}=\int_{0}^{2\pi}\mathrm{d}\theta\int_{0}^{\pi}\mathrm{d}\varphi\,
\Phi_{isotr.}(\varphi)V_{ex}(\varphi)=
\frac{1}{2}\int_{0}^{\pi}\mathrm{d}\varphi\,\sin{\varphi}V_{ex}(\varphi).$$ This easily leads with (\[eq:VexFinal\]) and 3.621 (5.) of [@Gradshteyn2000] to the *averaged excluded volume of two oblate spheroids*: $$\begin{aligned}
\label{eq:<Vex>isotr.}
\langle V_{ex}\rangle=&2V+8\pi a^{3}(1-\epsilon^{2}) F(\scriptstyle 2 \,,\,\frac{1}{2}\,,\,\frac{3}{2}\,,\,\epsilon^{2}\textstyle)\nonumber \\
&-\sqrt{\pi}a^{3}(1-\epsilon^{2})\sum_{k=1}^{\infty}\Gamma( k-\frac{1}{2})\epsilon^{2k}
\sum_{i=0}^{k}\frac{[B(\scriptstyle i+1\,,\,\frac{2k-2i+1}{2}\textstyle)]^{2}}{2^{i}(k-i)!(i!)^{2}}
F(\scriptstyle 2\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle),\end{aligned}$$ and with Eq. (\[eq:VexdFinal\]) and the same formula of [@Gradshteyn2000] to the *averaged excluded volume of two oblate spheroids surrounded with a shell of constant thickness*: $$\begin{aligned}
\label{eq:<Vexd>isotr.}
\langle V_{exd}\rangle=&\langle V_{ex}\rangle+\frac{8\pi}{3} d(3a^{2}+4d^{2}+3ad)+4\pi ad(2a+3d)\bigg(\sqrt{1-\epsilon^{2}}+\frac{\arcsin{\epsilon}}{\epsilon}\bigg)\nonumber \\
&+8\pi a^{2}d(1-\epsilon^{2})\frac{\textrm{arctanh}\,\epsilon}{\epsilon}
-ad\sqrt{\pi}\sum_{k=1}^{\infty}\Gamma{(k-\frac{1}{2})}\epsilon^{2k}
\sum_{i=0}^{k}\frac{[B(\scriptstyle i+1\,,\,\frac{2k-2i+1}{2}\textstyle)]^{2}}{2^{i}(k-i)!(i!)^{2}}\nonumber \\
&\times \bigg[a F(\scriptstyle \frac{1}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+a(1-\epsilon^{2})
F(\scriptstyle \frac{3}{2}\,,\,\frac{2k-2i+1}{2}\,,\,\frac{2k+3}{2}\,,\,\epsilon^{2}\textstyle)+d \bigg].\end{aligned}$$ The quantities involved in Eq. (\[eq:<Vex>isotr.\]) and Eq. (\[eq:<Vexd>isotr.\]) can then be easily evaluated with a mathematical software like Maple [@Maple] .
The averaged excluded volume of the hard spheroids (\[eq:<Vex>isotr.\]) is of course equivalent to the Isihara-Ogston-Winzor expression [@Ogston1975; @Isihara1950]: $$\label{eq:<Vex>I.O.W..}
\langle V_{ex}\rangle_{I.O.W.}=\frac{4}{3}\pi a^{2}b\bigg\{2+\frac{3}{2}\bigg[1+\frac{\arcsin{\epsilon}}{\epsilon\sqrt{1-\epsilon^{2}}}\bigg]
\bigg[1+\frac{(1-\epsilon^{2})}{2\epsilon}\ln{\bigg(\frac{1+\epsilon}{1-\epsilon}\bigg)}\bigg]\bigg\}.$$
These expressions have then been successfully verified through simulation by generating a great number of randomly placed spheroids couples with fixed reciprocal orientation and seeing how many times their shells overlapped. The ratio of overlaps to the total trial number will then be equal to the ratio of the excluded volume to the volume of the simulation cell. Convergence tests on the series were also performed.
It is then interesting to observe that the behavior ratio between $\langle V_{exd}\rangle$ and the spheroid volume $V$ is roughly linearly dependent upon the spheroid aspect-ratio and deviates slightly from it only close to the sphere case. The same holds true for the averaged excluded volume $\langle V_{ex}\rangle$, showing that interpreting the influence of the spheroid aspect-ratio as an excluded volume effect is a consistent approach.
[41]{}
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Our proposed deeply-supervised nets (DSN) method simultaneously minimizes classification error while making the learning process of hidden layers direct and transparent. We make an attempt to boost the classification performance by studying a new formulation in deep networks. Three aspects in convolutional neural networks (CNN) style architectures are being looked at: (1) transparency of the intermediate layers to the overall classification; (2) discriminativeness and robustness of learned features, especially in the early layers; (3) effectiveness in training due to the presence of the exploding and vanishing gradients. We introduce “companion objective” to the individual hidden layers, in addition to the overall objective at the output layer (a different strategy to layer-wise pre-training). We extend techniques from stochastic gradient methods to analyze our algorithm. The advantage of our method is evident and our experimental result on benchmark datasets shows significant performance gain over existing methods (e.g. all state-of-the-art results on MNIST, CIFAR-10, CIFAR-100, and SVHN).'
author:
- |
Chen-Yu Lee [^1]\
Dept. of EECS, UCSD\
`[email protected]`\
Saining Xie $^{*}$\
Dept. of CSE and CogSci, UCSD\
`[email protected]`\
Patrick Gallagher\
Dept. of CogSci, UCSD\
`[email protected]`\
Zhengyou Zhang\
Microsoft Research\
`[email protected] `\
Zhuowen Tu [^2]\
Dept. of CogSci, UCSD\
`[email protected]`\
title: 'Deeply-Supervised Nets'
---
Introduction
============
Much attention has been given to a resurgence of neural networks, deep learning (DL) in particular, which can be of unsupervised [@Hinton06], supervised [@CNN], or a hybrid form [@Lee09]. Significant performance gain has been observed, especially in the presence of large amount of training data, when deep learning techniques are used for image classification [@dropout; @Le10] and speech recognition [@Dahl12]. On the one hand, hierarchical and recursive networks [@Elman91; @Hinton06; @CNN] have demonstrated great promise in automatically learning thousands or even millions of features for pattern recognition; on the other hand concerns about deep learning have been raised and many fundamental questions remain open.
Some potential problems with the current DL frameworks include: reduced transparency and discriminativeness of the features learned at hidden layers [@Zeiler13]; training difficulty due to exploding and vanishing gradients [@Glorot10; @Pascanu14]; lack of a thorough mathematical understanding about the algorithmic behavior, despite of some attempts made on the theoretical side [@Eigen14]; dependence on the availability of large amount of training data [@dropout]; complexity of manual tuning during training [@imagenet]. Nevertheless, DL is capable of automatically learning and fusing rich hierarchical features in an integrated framework. Recent activities in open-sourcing and experience sharing [@dropout; @DeCAF; @theano] have also greatly helped the adopting and advancing of DL in the machine learning community and beyond. Several techniques, such as dropout [@dropout], dropconnect [@dropcon], pre-training [@Dahl12], and data augmentation [@mulCOL], have been proposed to enhance the performance of DL from various angles, in addition to a variety of engineering tricks used to fine-tune feature scale, step size, and convergence rate. Features learned automatically by the CNN algorithm [@CNN] are intuitive [@Zeiler13]. Some portion of features, especially for those in the early layers, also demonstrate certain degree of opacity [@Zeiler13]. This finding is also consistent with an observation that different initializations of the feature learning at the early layers make negligible difference to the final classification [@Dahl12]. In addition, the presence of vanishing gradients also makes the DL training slow and ineffective [@Glorot10]. In this paper, we address the feature learning problem in DL by presenting a new algorithm, deeply-supervised nets (DSN), which enforces direct and early supervision for both the hidden layers and the output layer. We introduce [*companion objective*]{} to the individual hidden layers, which is used as an additional constraint (or a new regularization) to the learning process. Our new formulation significantly enhances the performance of existing supervised DL methods. We also make an attempt to provide justification for our formulation using stochastic gradient techniques. We show an improvement of the convergence rate of the proposed method over standard ones, assuming local strong convexity of the optimization function (a very loose assumption but pointing to a promising direction).
Several existing approaches are particularly worth mentioning and comparing with. In [@pretrain], layer-wise supervised pre-training is performed. Our proposed method does not perform pre-training and it emphasizes the importance of minimizing the output classification error while reducing the prediction error of each individual layer. This is important as the backpropagation is performed altogether in an integrated framework. In [@Snoek12], label information is used for unsupervised learning. Semi-supervised learning is carried in deep learning [@Weston12]. In [@Tang13], an SVM classifier is used for the output layer, instead of the standard softmax function in the CNN [@CNN]. Our framework (DSN), with the choice of using SVM, softmax or other classifiers, emphasizes the direct supervision of each intermediate layer. In the experiments, we show consistent improvement of DSN-SVM and DSN-Softmax over CNN-SVM and CNN-Softmax respectively. We observe all state-of-the-art results on MNIST, CIFAR-10, CIFAR-100, and SVHN. It is also worth mentioning that our formulation is inclusive to various techniques proposed recently such as averaging [@mulCOL], dropconnect [@dropcon], and Maxout [@maxout]. We expect to see more classification error reduction with careful engineering for DSN.
Deeply-Supervised Nets
======================
In this section, we give the main formulation of the proposed deeply-supervised nets (DSN). We focus on building our infrastructure around supervised CNN style frameworks [@CNN; @DeCAF; @theano] by introducing classifier, e.g. SVM model [@vapnik], to each layer. An early attempt to combine SVM with DL was made in [@Tang13], which however has a different motivation with ours and only studies the output layer with some preliminary experimental results.
Motivation
----------
{width="0.8\linewidth"}
----------------------- ----------------
\(a) DSN illustration \(b) functions
----------------------- ----------------
We are motivated by the following simple observation: in general, a discriminative classifier trained on highly discriminative features will display better performance than a discriminative classifier trained on less discriminative features. If the features in question are the hidden layer feature maps of a deep network, this observation means that the performance of a discriminative classifier trained using these hidden layer feature maps can serve as a proxy for the quality/discriminativeness of those hidden layer feature maps, and further to the quality of the upper layer feature maps. By making appropriate use of this feature quality feedback at each hidden layer of the network, we are able to directly influence the hidden layer weight/filter update process to favor highly discriminative feature maps. This is a source of supervision that acts deep within the network at each layer; when our proxy for feature quality is good, we expect to much more rapidly approach the region of good features than would be the case if we had to rely on the gradual backpropagation from the output layer alone. We also expect to alleviate the common problem of having gradients that “explode” or “vanish”. One concern with a direct pursuit of feature discriminativeness at all hidden layers is that this might interfere with the overall network performance, since it is ultimately the feature maps at the output layer which are used for the final classification; our experimental results indicate that this is not the case.
Our basic network architecture will be similar to the standard one used in the CNN framework. Our additional deep feedback is brought in by associating a companion local output with each hidden layer. We may think of this companion local output as analogous to the final output that a truncated network would have produced. Backpropagation of error now proceeds as usual, with the crucial difference that we now backpropagate not only from the final layer but also simultaneously from our local companion output. The empirical result suggests the following main properties of the companion objective: [(1) it acts as a kind of feature regularization (although an unusual one), which leads to significant reduction to the testing error but not necessarily to the train error; (2) it results in faster convergence, especially in presence of small training data (see Figure (\[fig:mnist\]) for an illustration on a running example).]{}
Formulation
-----------
We focus on the supervised learning case and let $S=\{({{\bf X}}_i, y_i),i=1..N\}$ be our set of input training data where sample ${{\bf X}}_i \in R^n$ denotes the raw input data and $y_i \in \{1,..,K\}$ is the corresponding groundtruth label for sample $X_i$. We drop $i$ for notational simplicity, since each sample is considered independently. The goal of deep nets, specifically convolutional neural networks (CNN) [@CNN], is to learn layers of filters and weights for the minimization of classification error at the output layer. Here, we absorb the bias term into the weight parameters and do not differentiate weights from filters and denote a recursive function for each layer $m=1..M$ as: $$\begin{aligned}
& & {{\bf Z}}^{(m)} = f({{\bf Q}}^{(m)}), \;\; and \quad {{\bf Z}}^{(0)}\equiv {{\bf X}},\label{eq:Z} \\
& & {{\bf Q}}^{(m)} = {{\bf \mathsf{W}}}^{(m)} * {{\bf Z}}^{(m-1)}. \label{eq:Q}\end{aligned}$$ $M$ denotes the total number of layers; ${{\bf \mathsf{W}}}^{(m)}, m=1..M$ are the filters/weights to be learned; ${{\bf Z}}^{(m-1)}$ is the feature map produced at layer $m-1$; ${{\bf Q}}^{(m)}$ refers to the convolved/filtered responses on the previous feature map; $f()$ is a pooling function on ${{\bf Q}}$; Combining all layers of weights gives $${{\bf \mathsf{W}}}= ({{\bf \mathsf{W}}}^{(1)},...,{{\bf \mathsf{W}}}^{(M)}).$$ Now we introduce a set of classifiers, e.g. SVM (other classifiers like Softmax can be applied and we will show results using both SVM and Softmax in the experiments), one for each hidden layer, $${{\bf w}}= ({{\bf w}}^{(1)},...,{{\bf w}}^{(M-1)}),$$ in addition to the ${{\bf \mathsf{W}}}$ in the standard CNN framework. We denote the ${{\bf w}^{(out)}}$ as the SVM weights for the output layer. Thus, we build our overall combined objective function as: $${\lVert {{\bf w}^{(out)}}\rVert^2 + {\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}^{(out)}}) + \sum_{m=1}^{M-1} \alpha_m [\lVert {{\bf w}}^{(m)} \rVert^2 + {\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)}) - \gamma]_+,}
\label{eq:total2}$$ where $$\label{eq:layer}
{{\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}}^{(out)}) = \sum_{y_k \ne y} [1- <{{\bf w}}^{(out)}, \phi({{\bf Z}}^{(M)}, y) - \phi({{\bf Z}}^{(M)}, y_k)>]_{+}^2 }$$ and $$\label{eq:closs}
{{\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)}) = \sum_{y_k \ne y} [1- <{{\bf w}}^{(m)}, \phi({{\bf Z}}^{(m)}, y) - \phi({{\bf Z}}^{(m)}, y_k)>]_{+}^2 }$$
We name ${\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}}^{(M)})$ as the [*overall loss*]{} (output layer) and ${\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)})$ as the [*companion loss*]{} (hidden layers), which are both squared hinge losses of the prediction errors. The above formulation can be understood intuitively: in addition to learning convolution kernels and weights, ${{\bf \mathsf{W}}}^{\star}$, as in the standard CNN model [@CNN], enforcing a constraint at each hidden layer for directly making a good label prediction gives a strong push for having discriminative and sensible features at each individual layer. In eqn. (\[eq:total2\]), $\lVert {{\bf w}}^{(out)} \rVert^2$ and ${\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}}^{(out)})$ are respectively the margin and squared hinge loss of the SVM classifier (L2SVM [^3]) at the output layer (we omit the balance term $C$ in front of the hinge for notational simplicity); in eqn. (\[eq:layer\]), $\lVert {{\bf w}}^{(m)} \rVert^2$ and ${\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)})$ are respectively the margin and squared hinge loss of the SVM classifier at each hidden layer. Note that for each ${\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)})$, the ${{\bf w}}^{(m)}$ directly depends on ${{\bf Z}}^{(m)}$, which is dependent on ${{\bf \mathsf{W}}}^{1},..,{{\bf \mathsf{W}}}^{m}$ up to the $m$th layer. ${\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}}^{(out)})$ depends on ${{\bf w}}^{(out)}$, which is decided by the entire ${{\bf \mathsf{W}}}$. The second term in eqn. (\[eq:total2\]) often goes to zero during the course of training; this way, the overall goal of producing good classification of the output layer is not altered and the companion objective just acts as a proxy or regularization. This is achieved by having $\gamma$ as a threshold (a hyper parameter) in the second term of eqn. (\[eq:total2\]) with a hinge loss: once the overall value of the hidden layer reaches or is below $\gamma$, it vanishes and no longer plays role in the learning process. $\alpha_m$ balances the importance of the error in the output objective and the companion objective. In addition, we could use a simple decay function as $\alpha_m \times 0.1 \times (1-t/N) \rightarrow \alpha_m$ to enforce the second term to vanish after certain number of iterations, where $t$ is the epoch step and $N$ is the total number of epochs (wheather or not to have the decay on $\alpha_m$ might vary in different experiments although the differences may not be very big).
To summarize, we describe this optimization problem as follows: we want to learn filters/weights ${{\bf \mathsf{W}}}$ for the entire network such that an SVM classifier ${{\bf w}}^{(out)}$ trained on the output feature maps (that depend on those filters/features) will display good performance. We seek this output performance while also requiring some “satisfactory” level of performance on the part of the hidden layer classifiers. We are saying: restrict attention to the parts of feature space that, when considered at the internal layers, lead to highly discriminative hidden layer feature maps (as measured via our proxy of hidden-layer classifier performance). The main difference between eqn. (\[eq:total2\]) and previous attempts in layer-wise supervised training is that we perform the optimization altogether with a robust measure (or regularization) of the hidden layer. For example, greedy layer-wise pretraining was performed as either initialization or fine-tuning which results in some overfitting [@pretrain]. The state-of-the-art benchmark results demonstrate the particular advantage of our formulation. As shown in Figure \[fig:mnist\](c), indeed both CNN and DSN reach training error near zero but DSN demonstrates a clear advantage of having a better generalization capability.
To train the DSN model using SGD, the gradients of the objective function w.r.t the parameters in the model are: $$\label{eq:grad}
\begin{split}
& {\scriptstyle \frac{\partial F}{\partial {{\bf w}}^{(out)}} = 2{{\bf w}}^{(out)} - 2 \sum_{y_k \ne y} [\phi({{\bf Z}}^{(M)}, y) - \phi({{\bf Z}}^{(M)}, y_k)][1- <{{\bf w}}^{(out)}, \phi({{\bf Z}}^{(M)}, y) - \phi({{\bf Z}}^{(M)}, y_k)>]_{+}} \\
& {\scriptstyle \frac{\partial F}{\partial {{\bf w}}^{(m)}}} = \begin{cases} {\scriptstyle \alpha_m \left\{ 2{{\bf w}}^{(m)} - 2 \sum_{y_k \ne y} [\phi({{\bf Z}}^{(m)}, y) - \phi({{\bf Z}}^{(m)}, y_k)][1- <{{\bf w}}^{(m)}, \phi({{\bf Z}}^{(m)}, y) - \phi({{\bf Z}}^{(m)}, y_k)>]_{+} \right\} },
\quad \text{otherwise} \\
0, \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \text{if} \;\; {\scriptstyle \lVert {{\bf w}}^{(m)} \rVert^2+{\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)}) \le \gamma}
\end{cases}
\end{split}$$ The gradient w.r.t ${{\bf \mathsf{W}}}$ just follows the conventional CNN based model plus the gradient that directly comes from the hidden layer supervision.
Next, we provide more discussions to and try to understand intuitively about our formulation, eqn. (\[eq:total2\]). For ease of reference, we write this objective function as $$F({{\bf \mathsf{W}}}) \equiv {{\mathcal{P}}}({{\bf \mathsf{W}}}) + {{\mathcal{Q}}}({{\bf \mathsf{W}}}),$$ where ${{\mathcal{P}}}({{\bf \mathsf{W}}})\equiv \lVert {{\bf w}^{(out)}}\rVert^2 +{\mathcal{L}}({{\bf \mathsf{W}}}, {{\bf w}^{(out)}})$ and ${{\mathcal{Q}}}({{\bf \mathsf{W}}}) \equiv \sum_{m=1}^{M-1} \alpha_m [\lVert {{\bf w}}^{(m)} \rVert^2+{\ell}({{\bf \mathsf{W}}}, {{\bf w}}^{(m)})-\gamma]_+$.
Stochastic Gradient Descent View
--------------------------------
We focus on the convergence advantage of DSN, instead of the regularization to the generalization aspect. In addition to the present problem in CNN where learned features are not always intuitive and discriminative [@Zeiler13], the difficulty of training deep neural networks has been discussed [@Glorot10; @Pascanu14]. As we can observe from eqn. (\[eq:Z\]) and (\[eq:Q\]), the change of the bottom layer weights get propagated through layers of functions, leading to exploding or vanishing gradients [@Pascanu14]. Various techniques and parameter tuning tricks have been proposed to better train deep neural networks, such as pre-training and dropout [@dropout]. Here we provide a somewhat loose analysis to our proposed formulation, in a hope to understand its advantage in effectiveness.
The objective function in deep neural networks is highly non-convex. Here we make the following assumptions/observations: (1) the objective/energy function of DL observes a large “flat” area around the “optimal” solution where any result has a similar performance; locally we still assume a convex (or even $\lambda$-strongly convex) function whose optimization is often performed with stochastic gradient descent algorithm [@Bottou98].
The definition of $\lambda$-strongly convex is standard: A function $F({{\bf \mathsf{W}}})$ is $\lambda$-strongly convex if $\forall, {{\bf \mathsf{W}}}, {{\bf \mathsf{W}}}' \in \mathcal{W}$ and any subgradient ${{\bf g}}$ at ${{\bf \mathsf{W}}}$, $$F({{\bf \mathsf{W}}}') \ge F({{\bf \mathsf{W}}}) + <{{\bf g}}, {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}> + \frac{\lambda}{2} \lVert {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}\rVert^2,
\label{eq:overall}$$ and the update rule in Stochastic Gradient Descent (SGD) at step $t$ is ${{\bf \mathsf{W}}}_{t+1} = \Pi_{\mathcal{W}}({{\bf \mathsf{W}}}_{t} - \eta_t {{\bf \hat{g}}})$, where $\eta_t=\Theta(1/t)$ refers to the step rate and $\Pi_{\mathcal{W}}$ helps to project onto the space of $\mathcal{W}$. Let ${{\bf \mathsf{W}}}^{*}$ be the optimum solution, upper bounds for ${{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T-{{\bf \mathsf{W}}}^{*} \rVert^2]$ and ${{\mathbb{E}}}[(F({{\bf \mathsf{W}}}_T)-F({{\bf \mathsf{W}}}^{*})^2]$ in [@Rakhlin12] for the strongly convex function, and ${{\mathbb{E}}}[(F({{\bf \mathsf{W}}}_T)-F({{\bf \mathsf{W}}}^{*})^2]$ for convex function in [@Shamir13]. Here we make an attempt to understand the convergence of eqn. (\[eq:overall\]) w.r.t. ${{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T-{{\bf \mathsf{W}}}^{*} \rVert^2]$, due to the presence of large area of flat function shown in Figure (\[fig:architecture\].b). In [@Loh13], a convergence rate is given for the M-estimators with locally convex function with compositional loss and regularization terms. Both terms in eqn. (\[eq:overall\]) here refer to the same class label prediction error, a reason for calling the second term as [*companion objective*]{}. Our motivation is two-fold: [**(1)**]{} encourage the features learned at each layer to be directly discriminative for class label prediction, while keeping the ultimate goal of minimizing class label prediction at the output layer; [**(2)**]{} alleviate the exploding and vanishing gradients problem as each layer now has a direct supervision from the ground truth labels. One might raise a concern that learning highly discriminative intermediate stage filters may not necessarily lead to the best prediction at the output layer. An illustration can been seen in Figure (\[fig:architecture\].b). Next, we give a loose theoretical analysis to our framework, which is also validated by comprehensive experimental studies with overwhelming advantages over the existing methods.
We name $\mathcal{S}_{\gamma}(F)=\{{{\bf \mathsf{W}}}: F({{\bf \mathsf{W}}}) \le \gamma \}$ as the $\gamma$-feasible set for a function $F({{\bf \mathsf{W}}}) \equiv {{\mathcal{P}}}({{\bf \mathsf{W}}}) + {{\mathcal{Q}}}({{\bf \mathsf{W}}})$.
First we show that a feasible solution for ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$ leads to a feasible one to ${{\mathcal{P}}}({{\bf \mathsf{W}}})$. That is: [\[lm:1\] $\forall m,m'=1..M-1, and \; m' > m$ if $\; \lVert {{\bf w}}^{(m)} \rVert^2+ {\ell}((\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)}), {{\bf w}}^{(m)}) \le \gamma$ then there exists $(\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)},..,\hat{{{\bf \mathsf{W}}}}^{(m')})$ such that $\; \lVert {{\bf w}}^{(m')} \rVert^2 + {\ell}((\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)}..,\hat{{{\bf \mathsf{W}}}}^{(m')}), {{\bf w}}^{(m')}) \le \gamma$. [^4] ]{}
As we can see from an illustration of our network architecture shown in fig. (\[fig:architecture\].a), for $\forall \;(\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)})$ such that $\; {\ell}((\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)}), {{\bf w}}^{(m)}) \le \gamma$. Then there is a trivial solution for the network for every layer $j>m$ up to $m'$, we let $\hat{{{\bf \mathsf{W}}}}^{(j)} = \mathbf{I}$ and ${{\bf w}}^{(j)}={{\bf w}}^{(m)}$, meaning that the filters will be identity matrices. This results in $\; {\ell}((\hat{{{\bf \mathsf{W}}}}^{(1)},..,\hat{{{\bf \mathsf{W}}}}^{(m)}..,\hat{{{\bf \mathsf{W}}}}^{(m')}), {{\bf w}}^{(m')}) \le \gamma$. $\square$
Lemma \[lm:1\] shows that a good solution for ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$ is also a good one for ${{\mathcal{P}}}({{\bf \mathsf{W}}})$, but it may not be the case the other way around. That is: a ${{\bf \mathsf{W}}}$ that makes ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ small may not necessarily produce discriminative features for the hidden layers to have a small ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$. However, ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$ can be viewed as a regularization term. Since ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ observes a very flat area near even zero on the training data and it is ultimately the test error that we really care about, we thus only focus on the ${{\bf \mathsf{W}}}$, ${{\bf \mathsf{W}}}^{\star}$, which makes both ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$ and ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ small. Therefore, it is not unreasonable to assume that $F({{\bf \mathsf{W}}})\equiv {{\mathcal{P}}}({{\bf \mathsf{W}}}) + {{\mathcal{Q}}}({{\bf \mathsf{W}}})$ and ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ share the same optimal ${{\bf \mathsf{W}}}^{\star}$.
Let ${{\mathcal{P}}}({{\bf \mathsf{W}}}))$ and ${{\mathcal{P}}}({{\bf \mathsf{W}}}))$ be strongly convex around ${{\bf \mathsf{W}}}^{\star}$, $\lVert{{\bf \mathsf{W}}}'- {{\bf \mathsf{W}}}^{\star}\rVert^2 \le D$ and $\lVert{{\bf \mathsf{W}}}- {{\bf \mathsf{W}}}^{\star}\rVert^2 \le D$, with ${{\mathcal{P}}}({{\bf \mathsf{W}}}') \ge {{\mathcal{P}}}({{\bf \mathsf{W}}}) + <{{\bf gp}}, {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}> + \frac{\lambda_1}{2} \lVert {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}\rVert^2$ and ${{\mathcal{Q}}}({{\bf \mathsf{W}}}') \ge {{\mathcal{Q}}}({{\bf \mathsf{W}}}) + <{{\bf gq}}, {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}> + \frac{\lambda_1}{2} \lVert {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}\rVert^2$, where ${{\bf gp}}$ and ${{\bf gq}}$ are the subgradients for ${{\mathcal{P}}}$ and ${{\mathcal{Q}}}$ at ${{\bf \mathsf{W}}}$ respectively. It can be directly seen that $F({{\bf \mathsf{W}}})$ is also strongly convex and for subgradient ${{\bf gf}}$ of $F({{\bf \mathsf{W}}})$ at ${{\bf \mathsf{W}}}$, ${{\bf gf}}= {{\bf gp}}+ {{\bf gq}}$.
[\[lm:2\] Suppose ${{\mathbb{E}}}[\lVert \hat{{{\bf gp}}}_t \rVert^2] \le G^2$ and ${{\mathbb{E}}}[\lVert \hat{{{\bf gq}}}_t \rVert^2] \le G^2$, and we use the update rule of ${{\bf \mathsf{W}}}_{t+1} = \Pi_{\mathcal{W}}({{\bf \mathsf{W}}}_{t} - \eta_t (\hat{{{\bf gp}}}_t+\hat{{{\bf gq}}}_t))$ where ${{\mathbb{E}}}[\hat{{{\bf gp}}}_t]={{\bf gp}}_t$ and ${{\mathbb{E}}}[\hat{{{\bf gq}}}_t]={{\bf gq}}_t$. If we use $\eta_t=1/(\lambda_1+\lambda_2)t$, then at time stamp $T$ $${{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T - {{\bf \mathsf{W}}}^{\star} \rVert^2] \le \frac{12G^2}{(\lambda_1+\lambda_2)^2 T}$$ ]{}
Since $F({{\bf \mathsf{W}}}) = {{\mathcal{P}}}({{\bf \mathsf{W}}}) + {{\mathcal{Q}}}({{\bf \mathsf{W}}})$, it can be directly seen that $$F({{\bf \mathsf{W}}}') \ge F({{\bf \mathsf{W}}}) + <{{\bf gp}}+{{\bf gq}}, {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}> + \frac{\lambda_1+\lambda_2}{2} \lVert {{\bf \mathsf{W}}}'-{{\bf \mathsf{W}}}\rVert^2.$$ Based on lemma 1 in [@Rakhlin12], this upper bound directly holds. $\square$
[\[lm:3\] Following the assumptions in lemma \[lm:2\], but now we assume $\eta_t=1/t$ since $\lambda_1$ and $\lambda_2$ are not always readily available, then started from $\lVert {{\bf \mathsf{W}}}_1-{{\bf \mathsf{W}}}^{\star} \rVert^2 \le D$ the convergence rate is bounded by $${{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T - {{\bf \mathsf{W}}^{\star}}\rVert^2] \le e^{-2\lambda (\ln T + 0.578)} D + (T-1) e^{-2\lambda \ln(T-1)} G^2$$ ]{}
Let $\lambda = \lambda_1 + \lambda_2$, we have $$F({{\bf \mathsf{W}}^{\star}})- F({{\bf \mathsf{W}}}_t) \ge \;\;<{{\bf gf}}_t, {{\bf \mathsf{W}}^{\star}}- {{\bf \mathsf{W}}}_t> + \frac{\lambda}{2} \lVert {{\bf \mathsf{W}}^{\star}}- {{\bf \mathsf{W}}}_t \rVert^2, \;\;and$$
$$F({{\bf \mathsf{W}}^{\star}}) - F({{\bf \mathsf{W}}}_t) \ge \frac{\lambda}{2} \lVert {{\bf \mathsf{W}}}_t- {{\bf \mathsf{W}}^{\star}}\rVert^2.$$ Thus, $$<{{\bf gf}}_t, {{\bf \mathsf{W}}}_t - {{\bf \mathsf{W}}^{\star}}> \;\; \ge \lambda \lVert {{\bf \mathsf{W}}}_t- {{\bf \mathsf{W}}^{\star}}\rVert^2$$ Therefore, with $\eta_t=1/t$, $$\begin{aligned}
{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_{t+1} - {{\bf \mathsf{W}}^{\star}}\rVert^2] &=& {{\mathbb{E}}}[\lVert \Pi_{\mathcal{W}}({{\bf \mathsf{W}}}_{t} - \eta_t \hat{{{\bf gf}}}_t) - {{\bf \mathsf{W}}^{\star}}\rVert^2] \nonumber \\
&\le& {{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_{t} - \eta_t \hat{{{\bf gf}}}_t - {{\bf \mathsf{W}}^{\star}}\rVert^2] \nonumber \\
&=& {{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_{t} - {{\bf \mathsf{W}}^{\star}}\rVert^2] - 2 \eta_t {{\mathbb{E}}}[<{{\bf gf}}_t, {{\bf \mathsf{W}}}_t-{{\bf \mathsf{W}}^{\star}}>] + \eta_t {{\mathbb{E}}}[\lVert \hat{{{\bf gf}}}_t \rVert^2] \nonumber \\
&\le& (1-2\lambda/t) {{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_{t} - {{\bf \mathsf{W}}^{\star}}\rVert^2] + G^2/t^2\end{aligned}$$ With $2\lambda/t$ being small, we have $1-2\lambda/t \approx e^{-2\lambda/t}.$ $$\begin{aligned}
{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T - {{\bf \mathsf{W}}^{\star}}\rVert^2] &\le& e^{-2\lambda(\frac{1}{1}+\frac{1}{2}+,..,\frac{1}{T})} D + \sum_{t=1}^{T-1} \frac{G^2}{t^2} e^{-2\lambda(\frac{1}{t}+\frac{1}{t+1}+,..,\frac{1}{T-1})} \nonumber \\
&=& e^{-2\lambda (\ln T + 0.578)} D + G^2 \sum_{t=1}^{T-1} e^{-2 \ln(t)- 2 \lambda(\ln(T-1)-2\lambda \ln(t)} \nonumber \\
&\le& e^{-2\lambda (\ln T + 0.578)} D + (T-1) e^{-2\lambda \ln(T-1)} G^2 \nonumber \quad\quad\quad\quad\quad\quad\quad\quad\quad \square\end{aligned}$$
[Let ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ be $\lambda_1$-strongly convex and ${{\mathcal{Q}}}({{\bf \mathsf{W}}})$ be $\lambda_2$-strongly convex near optimal ${{\bf \mathsf{W}}^{\star}}$ and denote ${{\bf \mathsf{W}}}_T^{(F)}$ and ${{\bf \mathsf{W}}}_T^{({{\mathcal{P}}})}$ as the solution after $T$ iterations when applying SGD on $F({{\bf \mathsf{W}}})$ and ${{\mathcal{P}}}({{\bf \mathsf{W}}})$ respectively. Then our deeply supervised framework in eqn. (\[eq:total2\]) improves the the speed over using top layer only by $\frac{{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T^{({{\mathcal{P}}})}- {{\bf \mathsf{W}}^{\star}}\rVert^2]}{{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T^{(F)} - {{\bf \mathsf{W}}^{\star}}\rVert^2]} = \Theta (1+\frac{\lambda_2^2}{\lambda_1^2}), \; when \; \eta_t=1/\lambda t, \quad and,$ $\frac{{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T^{({{\mathcal{P}}})}- {{\bf \mathsf{W}}^{\star}}\rVert^2]}{{{\mathbb{E}}}[\lVert {{\bf \mathsf{W}}}_T^{(F)} - {{\bf \mathsf{W}}^{\star}}\rVert^2]} = \Theta (e^{\ln(T) \lambda_2}), \; when \; \eta_t=1/t.$ ]{}
Lemma \[lm:1\] shows the compatibility of the companion objective of ${{\mathcal{Q}}}$ w.r.t the output objective ${{\mathcal{P}}}$. The first equation can be directly derived from lemma \[lm:2\] and the second equation can be seen from lemma \[lm:3\]. In general $\lambda_2 \gg \lambda_1$ which leads to a great improvement in convergence speed and the constraints in each hidden layer also helps to learning filters which are directly discriminative. $\square$
Experiments
===========
We evaluate the proposed DSN method on four standard benchmark datasets: MNIST, CIFAR-10, CIFAR-100 and SVHN. We follow a common training protocol used by Krizhevsky et al. [@imagenet] in all experiments. We use SGD solver with mini-batch size of $128$ at a fixed constant momentum value of $0.9$. Initial value for learning rate and weight decay factor is determined based on the validation set. For a fair comparison and clear illustration of the effectiveness of DSN, we match the complexity of our model with that in network architectures used in [@NIN] and [@maxout] to have a comparable number of parameters. We also incorporate two dropout layers with dropout rate at $0.5$. Companion objective at the convolutional layers is imposed to backpropagate the classification error guidance to the underlying convolutional layers. Learning rates are annealed during training by a factor of $20$ according to an epoch schedule determined on the validation set. The proposed DSN framework is not difficult to train and there are no particular engineering tricks adopted. Our system is built on top of widely used Caffe infrastructure [@caffe]. For the network architecture setup, we adopted the mlpconv layer and global averaged pooling scheme introduced in [@NIN]. DSN can be equipped with different types of loss functions, such as Softmax and SVM. We show performance boost of DSN-SVM and DSN-Softmax over CNN-SVM and CNN-Softmax respectively (see Figure (\[fig:mnist\].a)). The performance gain is more evident in presence of small training data (see Figure (\[fig:mnist\].b)); this might partially ease the burden of requiring large training data for DL. Overall, we observe state-of-the-art classification error in all four datasets (without data augmentation), $0.39\%$ for MINIST, $9.78\%$ for CIFAR-10, $34.57\%$ for CIFAR-100, and $1.92\%$ for SVHN ($8.22\%$ for CIFAR-10 with data augmentation). All results are achieved without using averaging [@mulCOL], which is not exclusive to our method. Figure (\[fig:visualization\]) gives an illustration of some learned features.
MNIST
-----
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Classification error on MNIST test. (a) shows test error of competing methods; (b) shows test error w.r.t. the training sample size. (c) training and testing error comparison.[]{data-label="fig:mnist"}](Fig2a.pdf "fig:"){width="0.3\linewidth" height="0.3\linewidth"} ![Classification error on MNIST test. (a) shows test error of competing methods; (b) shows test error w.r.t. the training sample size. (c) training and testing error comparison.[]{data-label="fig:mnist"}](Fig2b.pdf "fig:"){width="0.3\linewidth"} ![Classification error on MNIST test. (a) shows test error of competing methods; (b) shows test error w.r.t. the training sample size. (c) training and testing error comparison.[]{data-label="fig:mnist"}](Fig2c.pdf "fig:"){width="0.3\linewidth"}
(a) (b) (c)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
We first validate the effectiveness of the proposed DSN on the MNIST handwritten digits classification task [@Lecun98], a widely and extensively adopted benchmark in machine learning. MNIST dataset consists of images of 10 different classes (0 to 9) of size $28 \times 28$ with 60,000 training samples and 10,000 test samples. Figure \[fig:mnist\](a) and (b) show results from four methods, namely: (1) conventional CNN with softmax loss (CNN-Softmax), (2) the proposed DSN with softmax loss (DSN-Softmax), (3) CNN with max-margin objective (CNN-SVM) , and (4) the proposed DSN with max-margin objective (DSN-SVM). DSN-Softmax and DSN-SVM outperform both their competing CNN algorithms (DSN-SVM shows classification error of $0.39\%$ under a single model without data whitening and augmentation). Figure \[fig:mnist\](b) shows classification error of the competing methods when trained w.r.t. varying sizes of training samples ($26\%$ gain of DSN-SVM over CNN-Softmax at $500$ samples. Figure \[fig:mnist\](c) shows a comparison of generalization error between CNN and DSN.
----------------------------- ----------
CNN [@objREC] 0.53
Stochastic Pooling [@SPool] 0.47
Network in Network [@NIN] 0.47
Maxout Networks[@maxout] 0.45
**DSN (ours)** **0.39**
----------------------------- ----------
: MNIST classification result (without using data augmentation).
CIFAR-10 and CIFAR-100
----------------------
-------------------------------------------------------------------------------------------------------------------------------------------------
![Visualization of the feature maps learned in the convolutional layer.[]{data-label="fig:visualization"}](visualization4 "fig:"){width="90mm"}
\(a) by DSN (b) by CNN
-------------------------------------------------------------------------------------------------------------------------------------------------
CIFAR-10 dataset consists of $32 \times 32$ color images. A total number of 60,000 images are split into 50,000 training and 10,000 testing images. The dataset is preprocessed by global contrast normalization. To compare our results with the previous state-of-the-art, in this case, we also augmented the dataset by zero padding 4 pixels on each side, then do corner cropping and random flipping on the fly during training. No model averaging is done at the test phase and we only crop the center of a test sample. Table (\[cifar100-table\]) shows our result. Our DSN model achieved an error rates of $9.78\%$ without data augmentation and $8.22\%$ with data agumentation (the best known result to our knowledge).
DSN also provides added robustness to hyperparameter choice, in that the early layers are guided with direct classification loss, leading to a faster convergence rate and relieved burden on heavy hyperparameter tuning. We also compared the gradients in DSN and those in CNN, observing $4.55$ times greater gradient variance of DSN over CNN in the first convolutional layer. This is consistent with an observation in [@maxout], and the assumptions and motivations we make in this work. To see what the features have been learned in DSN vs. CNN, we select one example image from each of the ten categories of CIFAR-10 dataset, run one forward pass, and show the feature maps learned from the first (bottom) convolutional layer in Figure (\[fig:visualization\]). Only the top 30% activations are shown in each of the feature maps. Feature maps learned by DSN show to be more intuitive than those by CNN.
+:---------------------------------:+:---------------------------------:+
| CIFAR-10 classification error | CIFAR-100 classification error |
+-----------------------------------+-----------------------------------+
| [ll]{} &\ | ----------------------------- - |
| \ | ---------- |
| Stochastic Pooling [@SPool] | Stochastic Pooling [@SPool] 4 |
| &15.13\ | 2.51 |
| Maxout Networks [@maxout] &11.68\ | Maxout Networks [@maxout] 3 |
| Network in Network [@NIN] &10.41\ | 8.57 |
| **DSN (ours)** & **9.78**\ | Tree based Priors [@tree] 3 |
| \ | 6.85 |
| Maxout Networks [@maxout] &9.38\ | Network in Network [@NIN] 3 |
| DropConnect [@dropcon] &9.32\ | 5.68 |
| Network in Network [@NIN]&8.81\ | **DSN (ours)** * |
| **DSN (ours)** &**8.22**\ | *34.57** |
| | ----------------------------- - |
| | ---------- |
+-----------------------------------+-----------------------------------+
CIFAR-100 dataset is similar to CIFAR-10 dataset, except that it has 100 classes. The number of images for each class is then $500$ instead of $5,000$ as in CIFAR-10, which makes the classification task more challenging. We use the same network settings as in CIFAR-10. Table (\[cifar100-table\]) shows previous best results and $34.57\%$ is reported by DSN. The performance boost consistently shown on both CIFAR-10 and CIFAR-100 again demonstrates the advantage of the DSN method.
Street View House Numbers
-------------------------
----------------------------- ----------
Stochastic Pooling [@SPool] 2.80
Maxout Networks [@maxout] 2.47
Network in Network [@NIN] 2.35
Dropconnect [@dropcon] 1.94
**DSN (ours)** **1.92**
----------------------------- ----------
: SVHN classification error.[]{data-label="svhn-table"}
Street View House Numbers (SVHN) dataset consists of $73,257$ digits for training, $26,032$ digits for testing, and $53,1131$ extra training samples on $32 \times 32$ color images. We followed the previous works for data preparation, namely: we select 400 samples per class from the training set and 200 samples per class from the extra set. The remaining 598,388 images are used for training. We followed [@maxout] to preprocess the dataset by Local Contrast Normalization (LCN). We do not do data augmentation in training and use only a single model in testing. Table \[svhn-table\] shows recent comparable results. Note that Dropconnect [@dropcon] uses data augmentation and multiple model voting.
Conclusions
===========
In this paper, we have presented a new formulation, deeply-supervised nets (DSN), attempting to make a more transparent learning process for deep learning. Evident performance enhancement over existing approaches has been obtained. A stochastic gradient view also sheds light to the understanding of our formulation.
Acknowledgments
===============
This work is supported by NSF award IIS-1216528 (IIS-1360566) and NSF award IIS-0844566 (IIS-1360568). We thank Min Lin, Naiyan Wang, Baoyuan Wang, Jingdong Wang, Liwei Wang, and David Wipf for help discussions. We are greatful for the generous donation of the GPUs by NVIDIA.
[^1]: equal contribution
[^2]: Corresponding author. Patent disclosure, UCSD Docket No. SD2014-313, filed on May 22, 2014.
[^3]: It makes negligible difference between L1SVM and L2SVM.
[^4]: Note that we drop the ${{\bf \mathsf{W}}}^{(j)}, j>m$ since the filters above layer $m$ do not participate in the computation for the objective function of this layer.
|
{
"pile_set_name": "ArXiv"
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